Paper Title: System size dependence of energy loss and correlations of heavy mesons at LHC energies.




SUBATECH, Nantes University, IMT Atlantique, IN2P3/CNRS, 

4 rue Alfred Kastler, 44307 Nantes cedex 3, France



§ ABSTRACT  
We study the system size dependence of heavy quark (HQ) observables at a center of mass energy of √(s_ NN)=5.02 TeV  to explore whether it can provide further constraints on the physical processes which are involved: energy loss of HQs in the quark gluon plasma (QGP), hadronization and hadronic rescattering. We use the EPOS4HQ approach to study p-p and 0-10% central O-O, Ar-Ar, Kr-Kr and Pb-Pb reactions and investigate in detail the momentum change of heavy quarks from creation until their detection as part of a hadron as well as the enhancement of heavy baryon production at low p_T.  We investigate furthermore the origin and the system size dependence of the azimuthal correlations between the heavy quark Q and the heavy antiquark Q̅ and how one can bypass the problem that correlations are washed out due to the combinatorial background. We conclude that a systematic study of the system size dependence of the momentum loss allows to separate the momentum loss due to the passage through the QGP from the  momentum change due to hadronization and the QQ̅ correlations allow to gain inside into the different pQCD processes in which the heavy quarks are created. 


12.38Mh

System size dependence of energy loss and correlations of heavy mesons at LHC energies 
    Jiaxing Zhao, Joerg Aichelin, Pol Bernard Gossiaux, and Klaus Werner
    August 5, 2024
=======================================================================================






§ INTRODUCTION




Theory predicts that at high temperature and/or high density a plasma of   quarks and gluons (QGP) is formed in which partons are not confined anymore. One of the main goals of heavy-ion reactions at ultra-relativistc energies is to study this new form of deconfined matter. In the last years ample evidence has been gathered that this state exists. The observation of strangeness enhancement <cit.>, quark scaling of the elliptic flow <cit.>, jet quenching <cit.>, and quarkonium suppression <cit.> points in this direction.  It is the task of future experiments and the challenge for theory to study the properties of the QGP, not only at the moment when it hadronizes into mesons and baryons but also before, during the expansion of the QGP when the temperature is considerably higher than the hadronization temperature of about T_c ≈ 160  MeV.
Because the multiplicity of mesons, which contain only light valence quarks, is well described by a statistical model with T=T_c and the spectra of light hadrons are perturbed by hadronic rescattering, the efforts concentrated in the last year on the study of penetrating probes, which include jets and heavy flavour hadrons.
Heavy flavour quarks have the advantage that they are created only in the initial hard collisions between projectile and target nucleons because their mass m_Q is large compared to QGP temperatures and that, due to m_Q >Λ_QCD, their production can be described by perturbative QCD (pQCD). 

In the last years many theoretical approaches have been advanced to describe the passage of heavy quarks through the medium, the subsequent hadronization and hadronic rescattering. They include PHSD <cit.>, in which the heavy quark physics is embedded in an approach, which described the light hadrons as well. It is based on the dynamical quasi particle model (DQPM), which respects the equation of state of strongly interacting matter. The Catania model <cit.> is also based on
a DQPM approach and describes the expanding medium by a Boltzmann equation. The LBT model <cit.> solves as well a Boltzmann equation for the heavy quarks including elastic and inelastic collisions but the medium is described by viscous hydrodynamics. Other models like TAMU<cit.>, Duke<cit.>, and Torino <cit.> replace the Boltzmann equation by a Langevin equation to describe the dynamics of heavy quarks and model the expanding QGP by ideal or viscous hydrodynamics, fitted to different key observables of light hadrons. 

Recently we advanced the EPOS4HQ model, which is based on the EPOS4 approach, <cit.>, to describe the physics of heavy quarks.  EPOS4 has been very successful in describing a multitude of light hadron observables from RHIC to LHC energies <cit.>. 
The agreement between EPOS4 and the experimental observables indicates that the time evolution of the QGP, including hadronization and hadronic rescattering, is well reproduced.  EPOS4 creates heavy quarks as well but they do not interact with the QGP. EPOS4HQ embeds the heavy quark physics in the EPOS4 approach. There the heavy quarks,  created in the EPOS4 interaction points,  interact with the QGP partons by elastic and inelastic collisions, described by pQCD elastic and inelastic cross sections. Heavy quarks, which have passed a QGP,  hadronize either by coalescence or by fragmentation. The relative weight of both processes depends on the transverse momentum, p_T, of the heavy quark.  A detailed description of EPOS4HQ, including the scattering with the plasma constituents and hadronization,  can be found in <cit.>. There is also shown that EPOS4HQ describes all the presently available data quite well.
 
In <cit.> the EPOS4HQ approach has been applied to pp collisions and it has been demonstrated that it  describes equally well the heavy hadron production in pp collision, assuming that in  pp collisions  as well as, in heavy-ion collisions the same criterion for the formation of a QGP applies, see Ref. <cit.>. 

The agreement of the EPOS4HQ approach with p-p as well as with Pb-Pb data motivates to study systematically the system size dependence of several key observables. We start out with a short overview over the heavy quark production processes (section <ref>) and of the QGP in EPOS4 (section <ref>). The momentum difference between the c and c̅-quark at creation and between the finally observed D-mesons as well as the momentum loss of HQ in a QGP is shown in section <ref>. This is followed by a study of  the system size dependence of the heavy baryon enhancement (section <ref>) as well as of the change of the azimuthal correlation between a D and a D̅ meson, created in the same elementary collision, due to the QGP (section <ref>). Finally, in section <ref>, we present our conclusions.

The calculations are based on version EPOS4.0.1.s9.





§ HEAVY QUARK PRODUCTION AND EVOLUTION IN EPOS4




In  EPOS4HQ  the heavy quarks are produced together with light quarks and gluons. Energy and momentum is conserved at each vertex. The interaction vertices are given by EPOS4. Therefore the kinetic variables as well as the initial heavy quark coordinates are identical in EPOS4 and EPOS4HQ.  The difference is, as said, that in EPOS4HQ we follow the heavy quarks through the QGP where they may interact with the QGP partons and that heavy quarks, which pass a QGP, may hadronize not only by fragmentation but also by coalescence. 




In EPOS4 QQ̅ pairs can be produced in different processes, which are shown in  Fig. <ref>. Before the partons from projectile and target interact in a hard process they develop a space like DGLAP cascade (SLC), displayed in the left panel of Fig. <ref>. In the DGLAP process gluons can be emitted, which can split into a heavy quark-antiquark pair. This is a next to leading order process. In the hard process,  Fig. <ref>, middle, collisions of gluons or light quark-antiquark pairs can create a QQ̅ pair as well. This is the leading order (Born) process. The gluons, created in the hard process, are time like (TLC) and may therefore, in a next to leading order process, disintegrate into a QQ̅ pair. This process is shown in the right panel of    Fig. <ref>.  As we will see later,  the correlations between the Q and the Q̅  are rather different in the three processes.

In EPOS4HQ heavy quarks interact with the partons of the QGP, formed by the light core partons and gluons. We include  both, elastic <cit.> and radiative <cit.> collisions.  To calculate the energy  exchange of the heavy quark with the medium we first calculate the interaction rate for elastic and inelastic collisions. If an interaction takes place we select randomly the momentum of the scattering partner, a light quark or a gluon, from their corresponding thermal distribution. The thermal distribution is determined by the local temperature and the local mean parton velocity, at the position where the heavy quark is localized. The parton momentum is then boosted into the computational frame, the projectile/target center of mass system in which the heavy quark momentum is defined. The scattering cross sections of the heavy quark with gluons and light quarks are calculated by pQCD matrix elements with a running coupling constant. The kinematic variables of the final state particles are drawn randomly from the cross section integrated over energy and momentum conserving delta functions.

When the QGP has expanded to the critical value of the local temperature or energy density  (T_ FO=167  MeV or ϵ_ FO=0.57  GeV/fm^3), the system hadronizes and the heavy quarks, which are localized in the QGP, are converted into a heavy flavor (HF) hadron. In the EPOS4HQ framework, there are two ways in which the  heavy quark can hadronize, either by fragmentation or by coalescence. The momentum distribution of a HF hadron, which is produced via the coalescence process, can be calculated via,

    dN d^3 P   =   ∑ g_H∫∏_i=1^kd^3p_i (2π)^3E_i f_i( p_i)
       ×   W_H( p_1,.., p_i) δ^(3)( P-∑_i=1^k p_i),
   
where g_H is the degeneracy factor of color and spin. k equals 2(3) for mesons (baryons). P and p_i are the momenta of heavy flavor hadrons and the quark momenta, respectively. The delta function conserves the momentum.  
f_1( p_1)=(2π)^3δ^(3)( p_c- p_1) is the normalized momentum space distribution of the heavy quark and  f_i( p_i) for i>1 is the momentum space distribution of the light quarks at the hadronization point.  W_H( p_1,.., p_i) is the momentum distribution of the quarks bound in a heavy hadron H in momentum space, corresponding to the Wigner density distribution after integration over coordinate space. It can be constructed from the hadron wave function. The hadron wave function can be approximated by a three-dimensional harmonic oscillator state with the same root mean square radius. For the ground state of charmed mesons, the Wigner density in the center-of-mass (CM) frame can be expressed as,

    W(p_r)   =   (2√(π)σ)^3e^-σ^2p_r^2,
 
where p_r =|E_2 p_1-E_1 p_2|/( E_1+E_2) is the relative momentum between the  two constituent quarks in their CM frame. It is normalized, ∫ W(p_r)d^3 p_r/(2π)^3=1. 
E_1 ( p_1) and E_2( p_2) are the energies and momenta of the quark or antiquark in the CM frame, respectively. The width σ in the Wigner density is controlled by the root-mean-radius. 
If heavy quarks hadronize by fragmentation we apply  the HQET-based fragmentation function <cit.>.
The probability for coalescence, P_coal, and of fragmentation, 1-P_coal, depends on the momentum  of the heavy quark. It is displayed in <cit.>.
Fragmentation and coalescence act differently: Whereas the momentum of the D-meson formed by coalescence is usually larger than that of the c-quark due to the momentum of the light quark, which  is picked up, the momentum of the D-meson formed by fragmentation is smaller. 








§ HOT QCD MEDIUM IN EPOS4



EPOS4 and EPOS4HQ use event by event hydrodynamics. This means that in every simulation the interaction points and the 
parton productions are different. The parton energy distribution is the initial condition for the hydrodynamic evolution
of QGP. 
Typical temperature profiles of a single event in the plane perpendicular to the beam direction for z=0 at the time τ_0  (τ_0=0.4, 1.024, 1.078, 1.167, 1.457 fm/c for p-p, 0-10% O-O, Ar-Ar, Kr-Kr, and Pb-Pb collisions)  are shown in Fig. <ref> for central O-O, Ar-Ar, Kr-Kr, and Pb-Pb events.  x is the direction of the impact parameter.  We observe strong fluctuations of the temperature whose scale is given in GeV.  Large areas are above the critical temperature of 0.167 GeV above which a QGP exists. So almost all heavy quarks,  in O-O as well as in Pb-Pb collisions, pass a QGP.  The hydrodynamic evolution of the QGP stops when the critical energy density of ϵ_ FO = 0.57 GeV/fm^3 is reached. Then the system hadronizes, as described in <cit.>.




§ HEAVY QUARK MOMENTUM LOSS









On the way to the detector heavy quarks, which are later converted into heavy mesons, change their momentum by three processes: a) by interaction with the QGP partons, b) by hadronization, when the heavy quark becomes a heavy hadron and c) by hadronic rescattering. 

In Fig. <ref>  we display the transverse momentum distribution of heavy quarks and heavy mesons created at midrapidity (|y|<0.5) in minimum bias (MB) for p-p and central collisions for different symmetric heavy-ion collisions. We select here only those heavy quarks which hadronize to D^0-mesons. Therefore the norm of the spectra for c-quarks and D-mesons
is identical. The black line shows the distribution of the charm quarks when they are created in the pQCD processes, the blue line that at the moment before they hadronize into D^0-mesons. The red and green lines show the transverse momentum spectrum of the D-mesons immediately after hadronization and after hadronic rescattering, described by UrQMD, respectively. 

We see, first of all, that the red and green lines are almost identical, so the hadronic rescattering does not modify the spectra considerably. We observe furthermore that the difference between the black and the blue lines increases with the size of the colliding system, as we expect, because the spatial extension of the QGP increases with the system size and we expect a larger momentum loss if system gets larger. It is remarkable that already in p-p collisions the black and blue line differ, indicating the formation of a QGP, as we have pointed out recently <cit.>. Hadronization changes the spectra in a non negligible way. For c-quarks with a  p_T > 5 GeV hadronization by fragmentation is the dominant process and so we expect that the p_T of the D-mesons is smaller that that of the c-quark before hadronization. This shift is seen in the spectra. At low p_T coalescence dominates and therefore we expect a momentum gain in the hadronization process, as also seen in the spectra.

To elucidate the physics from a different point of view we display in Fig. <ref> ratios of the p_T spectra for different projectile-target combinations. Top left we display the ratio of the c-quark spectra just before hadronization and at creation. Top right shows the ratio of the D-meson p_T distribution after hadronization and the p_T distribution of the charm quark before hadronization, hence the change of the spectrum due to hadronization. The bottom figure shows the ratio of the D-meson spectra after and before UrQMD and consequently the influence of the hadronic rescattering. 

For p_T>5 GeV the ratio before hadronization/at creation depends strongly on the system size. For Pb-Pb the spectrum before hadronization is suppressed by more than a factor of  5 but even for a system as small as O-O the suppression  is close to a factor of 2, so clearly visible.
At low p_T the heavy quark rescattering leads to a partial thermalization of the heavy quarks with the plasma partons. This enhances the hadronization p_T spectra around the thermal equilibrium value of p_T. The ratio of the p_T spectrum of the D-meson immediately after hadronization and of the c-quark before hadronization, displayed on the right hand side, shows that the D-meson spectrum is suppressed as compared to that of the c-quarks. For p_T > 5 GeV this suppression of the order of 2 is almost independent of the size of the colliding system.  
This confirms that hadronization is a local process, which happens when the expanding QGP has reached the critical temperature and which is therefore almost independent of the system size.  This shift is a superposition  between the momentum increase due to hadronization by coalescence and the momentum loss due to hadronization by fragmentation. Fig. <ref>  bottom, shows the ratio of the p_T spectrum after and before hadronic rescattering, which is close to one for 
p_T > 5 GeV.




In Fig. <ref> we compare  integrated p_T spectra (for p_T > 7 GeV)  as a function of A^1/3 to quantify more their system size dependence. On the left hand side we present the ratio before hadronization/at creation, on the right hand side before UrQMD/at creation. As shown, the final spectrum equals the spectrum before UrQMD. 
We see again from the left figure that larger systems shift the high momentum particles more to lower p_T values than smaller systems. In Pb-Pb 90% of the heavy quarks with an initial  p_T > 7 GeV are shifted to momenta below 7 GeV. In p-p collisions this is only the case for 20%.  For smaller systems the fraction of c-quarks shifted to lower p_T is roughly proportional to A^1/3, and so proportional to the average length, which the heavy quarks have to travel in the QGP.  If we compare the ratio after UrQMD/at creation on the left hand side we find a curve which is characteristic for the relative fraction of  the momentum loss 
due to collisions in the QGP and due to hadronization.  This curve can be normalized 
to the pp values and compared to the experimental finding.  Therefore, a systematic study of the system size dependence offers the possibility to separate experimentally the momentum loss of heavy quarks in the QGP (which increases with the system size) and that due to hadronization (which is system size independent and dominates at small systems) just by comparing the integrated  p_T spectra of D mesons. This would also allow to calibrate better the transport models which do not agree on the relative fraction of both momentum losses <cit.>. 



The microscopic description of the time evolution of heavy quarks in EPOS4HQ allows to follow the heavy quarks individually until their hadronization. We can therefore calculate the momentum transfer, which a heavy quark suffers while it is moving in the QGP, as well as the momentum difference between the c-quark before hadronization and the D-meson after hadronization.



For √(s_ NN) = 200 GeV Au-Au collisions, this momentum loss has already been calculated using another hydrodynamical model <cit.>. There a momentum shift of approx 7 GeV was observed for an initial c-quark momentum of 20 GeV. 

In Fig. <ref> we present the results for Δ p_T (top row), as well as those
for Δ p_T/p_T (bottom row). On the left hand side we see, for large p_T, an increase of the momentum loss in the QGP  with p_T. Because for all systems  Δ p_T/p_T is almost constant at high p_T, the momentum change of a heavy quark in a Pb-Pb collisions with p_T= 20 GeV is almost a factor of four larger than for a quark at 5 GeV. This is a consequence of the energy dependence of the cross sections and the kinematic of the reaction. Although for all systems the momentum shift as a function of p_T is almost linear, the slope is different for the different systems due to different lengths, which the heavy quarks travel in the QGP. At low p_T, where the heavy quarks are slow and stay for a long time in contact with the QGP particles, we see that the c-quarks are in the process to come to an equilibrium with the QGP, as already observed in <cit.>. Therefore the heavy quarks may gain momentum.   

Also the momentum shift due to hadronization, displayed at the right hand side, depends on p_T. Here we observe roughly a factor of two between p_T= 10 GeV and p_T= 20 GeV: The fragmentation function scales with the momentum fraction of the D-meson with respect to the momentum of the c-quark. Hence the momentum shift of a D-meson produced by a p_T = 10 GeV c-quark should be roughly half of that produced by a p_T = 20 GeV c-quark, what is indeed observed. The momentum shift due to hadronization is system size independent due to the local character of hadronization. At low p_T the momentum change is positive due to the coalescence process.

To explore the relative importance of the physical processes to which the finally observed D-mesons (here are all charmed hadrons before UrQMD) are sensitive to, we divide both ratios, the ratio of the momentum shift of the c-quark due to interactions with the QGP and the momentum shift caused by hadronization. The result is displayed in Fig. <ref>. 
Let us start with the high p_T part. Since both, momentum shift in the plasma and momentum shift due to hadronization, increase with p_T, the ratio is rather flat. For light projectile-target combination the momentum shift due to hadronization is larger than that due to interactions with the QGP. The final D-meson p_T spectrum is there more sensitive to the fragmentation process (and its correct description in the medium) then to the energy loss of the c-quark in the QGP and is therefore little informative about the c-quark interactions with the QGP. For larger systems this ratio may decrease  by a factor of three and therefore the final spectra become more sensitive to the c-quark interactions with the QGP.  






§ MULTIPLICITY OF HEAVY BARYONS



 



The enhanced production of heavy baryons at low p_T as compared to the value expected
from e^+e^- fragmentation functions was one of the surprising results in p-p collisions. Such an enhancement has also been observed in Pb-Pb and there it is attributed the the hadronization of heavy quarks at the end of the expansion phase of the QGP, where heavy quarks may form heavy baryons by coalescence. A smooth system size dependence of this enhancement may therefore verify the conjecture that also in very small systems like p-p a QGP is formed, where heavy baryons can be produced by coalescence. This conjecture is based on the successful description of different observables in p-p collision in the EPOS4HQ approach <cit.>.

The system size dependence of the enhancement of heavy baryons at low p_T is shown in Fig. 
<ref>, which displays from left to right the ratio of the p_T spectra of
Λ_c/D^0, Ξ_c/D^0 and Ω_c/D^0 for p-p, O-O, Ar-Ar, Kr-Kr and Pb-Pb collisions.  We observe
that this enhancement is rather similar for all systems even if a detailed analysis shows that the maximum of the enhancement shifts from p_T≈ 3 GeV for p-p to p_T≈4 GeV for Pb-Pb.  Also the enhancement at the maximum increases by 40% from p-p to Pb-Pb.  If this smooth system size dependence, and explicitly the small difference between p-p and O-O, is verified by experiments it is hard to imagine that a p-p collision proceeds differently than a heavy ion collision.
 






§ CORRELATIONS



Another source of information, not only on the interaction of c-quarks with the QGP but also on the c-quark production mechanism itself, are correlations among heavy hadrons. They have been measured in p-p collisions at forward rapidity by the LHCb collaboration <cit.> and been successfully described by EPOS4HQ <cit.>. 







As discussed in section <ref>, in collisions among hadrons a heavy flavour quark-antiquark (QQ̅) pair may be produced in three different ways, shown in Fig. <ref>. The correlation between Q and Q̅ is rather different for the three different pQCD creation processes.




 §.§ Correlations in rapidity

The rapidity correlation between Q and Q̅ for the three production processes as well as for all three processes together is displayed in Fig. <ref>.

We see that the correlations for flavour excitation and gluon splitting are centered around y_1=y_2.  For flavor excitation the correlations are rather weak and therefore the Q and Q̅, created at the same vertex, may have rather different rapidities. This means that 
in experiments a large rapidity interval is necessary to register both heavy quarks.
For gluon splitting the distribution is considerably narrower because there is no hard interaction which may cause a strong rapidity shift. For the flavour creating process, however, the distribution has a maximum at a finite rapidity difference between the heavy quarks. Parton distribution functions increase strongly with decreasing x = p_parton/p_proton values. Therefore collisions with different x values for projectile and target are more probable than symmetric collisions. This is also true for collisions with a sufficient √(s)
value to produce a QQ̅ pair.

In Fig. <ref> we display the distributions of the total rapidity, y_1+y_2, of the QQ̅ pairs. They are normalized to one. All peak at y_total=0  but the distribution caused by the flavor creation is much broader.




 §.§ Correlation in transverse momentum

The correlation between the absolute value of the transverse momentum of the Q and Q̅ of those pairs, where both quarks have a rapidity |y|<1, is displayed for central Pb-Pb collisions in Fig. <ref>, as well separated for the three different processes. We display the initial  distribution (top) and the distribution before hadronization, after the passage through the QGP (bottom).  

Due to momentum conservation we expect that initially quarks, coming from the flavour creation process, have very similar transverse momenta and this is indeed seen in this LO process where the cc̅ pair is emitted nearly back-to-back.
For flavour excitation, where one of the heavy quarks gets a kick due to the hard interaction, we expect a transverse momentum difference between the heavy quarks, what is also observed.  










Gluon splitting shows a rather weak correlation, which peaks at p_T,1≈ p_T,2.  The dominating process for the creation of a Q Q̅ pair is flavour excitation.


If we compare the top and the bottom figure we see that the correlations are considerably weakened during the travel of the heavy quarks through the QGP, due to their elastic and inelastic collisions.  In addition, the transverse momentum is lowered. The  p_T,1≈ p_T,2 correlation of the pairs created by flavor creation has disappeared and for all creation processes most of the pairs have quite different  p_T,1 and p_T,2 values.  If we investigate smaller systems or more peripheral reactions we expect correlations intermediate between the top and the bottom figure.

This figure shows as well that, if we demand that the p_T of both heavy quarks is larger than 4 GeV, we select pairs which are created by flavour excitation and not by flavour creation. 

To quantify the influence of the QGP on the relative momentum between the heavy quarks
| Δ p_T| = |p_T1-p_T2|  we display in Fig. <ref> this quantity for Pb-Pb collisions at √(s_ NN)=5.02 TeV for the different production mechanisms, on the left 
hand side at production, on the right hand side before hadronization.  Here both quarks have to have a p_T > 3 GeV and the distributions are normalized to one. We see that initially QQ̅ pairs created by  flavour creation have a small momentum difference whereas those produced in flavor excitation have on the average a considerably larger Δ p_T. Those from gluon splitting are in between. After traversing the QGP all distributions are rather similar and even that for flavor excitation increases up to Δ p_T→ 0 (see left hand side).  Therefore a Δ p_T
cut cannot serve as possibility to separate the three production processes in central Pb-Pb collisions.




 §.§ Azimuthal correlations











Next we study the azimuthal correlations between the transverse momenta of the heavy quarks for the different production mechanisms of the QQ̅ pairs. These are  displayed in Fig. <ref> for pairs in which both quarks have |y|<1. On the left hand side we display the distribution of the difference of the azimuthal angle, Δϕ, between the Q and Q̅ momenta, separated for the three different production processes, for HQ which both have p_T> 3 GeV.  In the middle column we display this correlation of the HQ after hadronization and in the right column we display the sum of all production processes for
three different p_T cuts for both HQ. From top to bottom we display these correlations
for p-p and central Ar-Ar, Kr-Kr and Pb-Pb collisions.

In the top row, p-p collisions,  we see that each of the production processes creates a different correlation, which is preserved during the hadronization process.   QQ̅s created by flavour creation show an enhancement for back to back production, as expected by the two body kinematics, which is at the origin of this type of production. Gluon splitting, on the contrary, produces an enhancement around Δϕ = 0 because the direction of the momentum of both quarks is close to that of the emitting gluon. The correlation between the transverse momenta of those pairs, which are produced by flavour excitation, is weak due to the kick, which one of the heavy quarks obtains  in the hard process. 
Summing up the different correlations the correlation function looks smoother. For p_T<1 GeV there are practically no strong azimuthal correlations, which show up when we demand p_T>3 GeV. There we see a strong enhancement at Δϕ =0 and  Δϕ = π coming from flavour creation and gluon splitting.  We can therefore conclude that the experimentally observed forward-backward enhancement of the correlation function is due to the three different production processes <cit.>. 



When we increase the system size, displayed in the lower rows of Fig. <ref> for Ar-Ar, Kr-Kr  and Pb-Pb, we observe that the initial correlations are rather similar. The passage through the QGP changes the correlations in a different way. The correlation due to gluon splitting, where both HQ have a similar p_T, passes the same region of the QGP and remains rather unchanged whereas that due to flavor creation is strongly weakened. There are two reasons.  One is the collective radial velocity of the QGP, which is communicated by collisions to the heavy quarks. If heavy quarks are emitted back to back, the one with a large momentum component anti-parallel to the radial collective velocity gets decelerated due to collisions and can inverse the direction of its momentum, what is not the case for the other quark which has a large momentum component parallel to the collective velocity.  The other reason is the path length difference in the QGP  of  back to back emitted particles . Both processes weaken the azimuthal correlations between the heavy quarks and depend on the size of the collective expansion velocity (which increases with the system size) as well as on the size of the QGP.  As a consequence, the correlation function for heavy systems at  Δϕ =0 is enhanced and the peak at Δϕ =π disappears.  A similar evolution of the correlation function, especially for flavour creation, has been observed in PHSD calculations for Au-Au collisions at √(s_ NN) = 200 GeV, using initial cc̅ correlations provided by a tuned Pythia <cit.> calculation,  as well as in other transport approaches <cit.>.




Because it is in discussion to extend the rapidity range of the ALICE detector at LHC, in which heavy quarks can be detected, we display in Fig. <ref> the results if both HQ can be observed in a rapidity interval  of |y|<4. We see that the correlations 
are only slightly modified. The flavor excitation process remains rather flat and the enhancement around Δϕ≈ 0 is reduced for the gluon splitting process.  In the right column we see that for this larger rapidity interval the forward-backward enhancement, seen in p-p, is recovered for Pb-Pb collisions if the p_T of both HQ is larger than 6 GeV. 












Up to now we have investigated the ideal situation that one knows at which vertex the QQ̅ is created. In experiments the vertex cannot be identified and therefore the experimentally observed correlation function is that created by all possible DD̅ pairs. Their number increases with the system size and the DD̅ pairs from different vertices are uncorrelated. The correlation function, obtained by including pairs from different vertices, means the combinatorial background, is shown in Fig. <ref> for different system sizes. We see that pairs coming from different vertices, whose number increases strongly as compared to those coming from the same vertex,  wash out rather fast the correlation function. For O-O we see for p_T > 3 GeV for both heavy quarks still a structure of the correlation function but very reduced as compared to p-p. For Ar-Ar the structure of the correlation function has disappeared. This is also the case if one widens the rapidity interval as can be seen from the top row of Fig. <ref>, where we display the same correlation function. The combinatorial background destroys also here the structure of the correlation function of pairs coming from the same vertex.

The middle row show the correlation function of two D-mesons.  In heavy-ion collisions the two D-mesons come in their large majority from different pomerons and therefore from  different production vertices and should not show a structure in the correlation function. However, if both D mesons contain heavy quarks from the same pomeron  they are correlatedin EPOS4  <cit.>. This is the case for p-p collisions. 

To quantify these correlations we display in Table <ref> the fraction of DD̅ mesons which come from the different processes in the intervals Δϕ>5π/6 and Δϕ<π/6, for p-p collisions with √(s_ NN)=5.02 TeV. We display this percentage for different cuts in the transverse momentum of the D and D̅. We see that without any cut flavour excitation is dominant in both intervals but cuts in the transverse momentum allows to select the pQCD process. For Δϕ>5π/6 this is the flavour creation and for Δϕ<π/6 gluon splitting. So with with a combination of a high p_T cut and a cut in the opening angle one can exclusively study these pQCD processes individually. 

Can this combinatorial background be removed? for experiments event mixing is a possibility, which is however too costly for transport approaches. Having the complete information about all Q and Q̅ quarks available a simpler method is to subtract from the DD̅ correlation the DD correlation, for which the number of possible pairs is almost identical. In the bottom row of Fig. <ref> we display the difference between the DD̅ and DD correlation and indeed we recover a structure, which resembles the structure of the original DD̅ correlation function of DD̅ pairs, which are coming from the same vertex, shown in Fig. <ref>. It remains to be seen how this recovery potential depends on the acceptance of the detectors.

In heavy ion collisions the structure of the correlation function depends only slightly on the centrality of the collision. This is shown in Fig. <ref>, where we display for three centrality intervals [0-10%], [30-50%] and [60-80%]  the DD̅ correlation function for three different p_T cuts. In the [30-50%] centrality bin a slight enhancement at Δϕ≈π appears, which becomes even more visible in peripheral collisions, [60-80%] centrality, where also a small enhancement   at Δϕ≈ 0 appears. The structure remains, however less pronounced than in the combinatorial background corrected central collisions (see Fig. <ref>.)






§ CONCLUSION



We presented a study about the system size dependence of several heavy hadron observables, which have been identified as being sensitive to the interaction between heavy quarks and the QGP, created in heavy ion reactions. These observables have the potential to elucidate further the properties of the expanding QGP, the key objective of the experimental study of ultrarelativistic heavy ion collisions. For this investigation we used the recently advanced EPOS4HQ approach, which has been able to describe the heavy meson single particle spectra in p-p as well as in heavy ion collisions and reproduced in addition quite nicely the presently available data on heavy meson correlations. 

We observe that indeed the interaction of the heavy quarks with the QGP hadrons changes the heavy quark momentum distribution or, the other way round, that the heavy quark spectra carry information about the constituents of the QGP and the interaction strength.  The momentum change depends strongly on the system size. The other source
of the momentum difference between the initial c-quark and the finally observed hadron
is hadronization, which is a process which depends only mildly on the system size. A systematic study of the momentum change (or the change of the form of the p_T spectrum) may separate the two sources.

The heavy baryon enhancement at low p_T is a smooth function of the system size and continuous from p-p to Pb-Pb. If verified experimentally, this strengthen the conjecture that this enhancement is a consequence of the production of a QGP, even in a system as small as p-p. 

Azimuthal correlations between the Q and the Q̅ at creation are only weakly modified by shadowing effect. The form of the correlation function is determined by the three different creation mechanisms of QQ̅ pairs.  Depending on the creation mechanism the momentum difference between the pair particles is rather differently modified when the heavy quarks pass the QGP.  Also hadronization modifies the different QQ̅ correlation functions.  If one sums over all different production mechanisms the correlation function is much smoother than that for the individual processes. 

In p-p, a cut in p_T and in the opening angle allows to separate experimentally gluon splitting and flavour creation from the other QQ̅ production mechanisms. This is a promising perspective to study these pQCD processes separately.  

The heavy hadrons coming from the same vertex are still correlated, even in Pb-Pb collisions.  Experimentally the vertex, at which the QQ̅ pair is produced, cannot be identified.  With increasing system size this leads to a the strong increase of the  multiplicity of heavy quark pairs from different vertices (as compared to those from the same vertex).  This combinatorial background  makes the correlation function featureless and insensitive to the interaction of the heavy quark with QGP partons.

The correlation function can be recovered if one subtracts from the correlation function for Q Q̅ pairs that for  QQ pairs. If this is experimentally possible the system size dependence of the correlation function
is another observable which sheds light on the interaction of heavy quarks with the QGP.    






Acknowledgement: 
This work is supported by the European Union's Horizon 2020 research and innovation program under grant agreement No 824093 (STRONG-2020).


apsrev4-1.bst





 Paper Title: Bayesian analysis of (3+1)D relativistic nuclear dynamics with the RHIC beam energy scan data.




 hm0746@wayne.edu
Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA
 Hendrik.Roch@wayne.edu
Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA
 chunshen@wayne.edu
Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA


§ ABSTRACT 
This work presents a Bayesian inference study for relativistic heavy-ion collisions in the Beam Energy Scan program at the Relativistic Heavy-Ion Collider.
The theoretical model simulates event-by-event (3+1)D collision dynamics using hydrodynamics and hadronic transport theory.
We analyze the model's 20-dimensional posterior distributions obtained using three model emulators with different accuracy and demonstrate the essential role of training an accurate model emulator in the Bayesian analysis.
Our analysis provides robust constraints on the Quark-Gluon Plasma's transport properties and various aspects of (3+1)D relativistic nuclear dynamics.
By running full model simulations with 100 parameter sets sampled from the posterior distribution, we make predictions for p_ T-differential observables and estimate their systematic theory uncertainty.
A sensitivity analysis is performed to elucidate how individual experimental observables respond to different model parameters, providing useful physics insights into the phenomenological model for heavy-ion collisions. 

Bayesian analysis of (3+1)D relativistic nuclear dynamics with the RHIC beam energy scan data
    Chun Shen
    August 5, 2024
=============================================================================================





§ INTRODUCTION


Relativistic heavy-ion collision programs at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) have been a key tool in studying the properties of the Quark Gluon Plasma (QGP), a state of matter exhibiting quarks and gluon degrees of freedom <cit.>.
These experiments allow high-energy nuclear physicists to quantitatively characterize the strongly interacting nuclear matter under extreme temperatures and densities, thereby exploring the phase structure of Quantum Chromodynamics (QCD).
Especially with the systems studied in the RHIC Beam Energy Scan (BES) program, where the center-of-mass energy is varied, it is possible to study the QGP over a wide range of temperatures and densities <cit.>.
With precision measurements from this experimental program, it is exciting to investigate the transition between the QGP and hadronic matter, search for a critical point and the associated first-order phase boundary in the QCD phase diagram, and study the emergent properties of the produced state of matter (see reviews <cit.>).

Modeling heavy-ion collisions event-by-event is a complex and computationally demanding task <cit.>.
Because the relevant length scales change dynamically as the collision systems evolve, it requires matching different types of physical models in a multi-stage approach.
Fluid dynamics serves as an effective long-wavelength description of the collective behavior of the strongly coupled QGP produced in heavy-ion collisions.
Relativistic viscous hydrodynamics can well describe the time evolution of the produced bulk medium, efficiently transforming the initial-state spatial inhomogeneities to anisotropies in the final-state hadrons momentum distributions. This theoretical description allows us to access information about the initial conditions of heavy-ion collisions and transport properties of the QGP from the experimental measurements.

It is challenging to compute the QGP transport coefficients from first principles <cit.>. There have been several attempts using lattice QCD techniques to compute the plasma's shear viscosity for a pure gluon system <cit.>.
On the other hand, extensive phenomenological studies were able to show that the hadronic observables measured from heavy-ion collisions are sensitive to the viscosities in the QGP <cit.>.
There have been many studies using hydrodynamics in heavy-ion collision simulations to estimate the specific shear viscosity η/s for the QGP <cit.>.
Nowadays, there is more effort put into the direction of extracting the QGP-specific shear and bulk viscosities, together with their uncertainties from varying other aspects of the theoretical model, based on large-scale model-to-data comparisons with Bayesian inference analysis in a high dimensional model parameter space <cit.>.

The RHIC BES program generates a wealth of data for heavy-ion collisions at different collision energies. These measurements enable phenomenological extraction of the QGP transport properties at finite baryon density via the global Bayesian inference analysis.
For collisions with energies around 𝒪(10) GeV, full (3+1)D simulations are required to perform precision comparisons <cit.>. 

This work expands upon Ref. <cit.> and performs a systematic Bayesian inference analysis of RHIC BES measurements using full (3+1)D event-by-event simulations. In addition to the constraints on the QGP-specific viscosity, we will report and analyze the posterior distribution for all model parameters in Sec. <ref>.
We improve the constraints on the model parameters by applying a more accurate Gaussian Process (GP) model emulator <cit.> than the one used in <cit.>. Sec. <ref> will further provide a selection of model predictions by performing numerical simulations with the parameter sets drawn from the posterior distribution. The relative variations in the model predictions from different parameter sets provide the theoretical uncertainty from the Bayesian calibration. We will perform a detailed sensitivity analysis between the model parameters and experimental observables in Sec. <ref>.



§ THE HYBRID THEORETICAL FRAMEWORK AND MODEL PARAMETERS



The theoretical simulations for Au+Au collisions at √(s_ NN)=7.7, 19.6, and 200 GeV used for this Bayesian analysis were performed using the  framework <cit.>.
To simulate the full (3+1)D dynamical evolution, a  model is coupled with relativistic viscous hydrodynamics () <cit.> and hadronic transport () <cit.>.
The  model considers the finite thickness of nuclei along the beam direction as they pass through each other, which becomes important as the center of mass collision energy decreases to 𝒪(10) GeV <cit.>. The transverse collision geometry is controlled by the shape of nuclei and their relative impact parameter. Each nucleon inside the colliding nuclei contains four hotspots related to three constituent quarks and one soft gluon. These hotspots lose energy and longitudinal momentum by being decelerated by strings between them during nucleon-nucleon collisions <cit.>.
The averaged rapidity loss is parametrized as a function of the incoming rapidity y_ init in the collision pair rest frame <cit.>:

    ⟨ y_ loss⟩ = 
            y_ loss,2y_ init/2,  0<y_ init≤ 2
    
            y_ loss,2 + (y_ loss,4 - y_ loss,2)y_ init-2/2,  2<y_ init<4
    
            y_ loss,4 + (y_ loss,6 - y_ loss,4)y_ init-4/2,  y_ init≥ 4

where y_ loss, n is the average amount of rapidity loss for y_ init=n.
The event-by-event fluctuations of the rapidity loss are modeled by the parameter σ_y_ loss, which controls the variance of the fluctuations <cit.>.
The parameter α_ shadowing takes into account coherence effects when the incoming nucleons go through multiple collisions.
With α_ shadowing=0, all binary collisions will produce strings while α_ shadowing≠ 0 sets a finite probability to shadow string production from secondary collisions between nucleon pairs.
The parameter α_rem controls the amount of energy-momentum lost by the collision remnants <cit.>.
These wounded partons are connected via strings and decelerated in the longitudinal direction until τ_ hydro=0.5 fm/c in the collision rest frame.
Then, they are fed as source terms for the system's energy-momentum tensor and net baryon current,

    ∂_μ T^μν   = J^ν, 
    ∂_μ J^μ_B    = ρ_B.

The definitions of the source terms were explained in detail in Ref. <cit.>.
Here, we introduce a longitudinal tilted string profile by parameterizing the transverse profile as,

    f(x⃗_⊥, η_s) = 1/2π (σ_x^string)^2 
               exp[ - [x - x_c(η_s)]^2 + [y - y_c(η_s)]^2/2 (σ_x^string)^2],

where the position of the string center (x_c, y_c) depends on the space-time rapidity,

    x_c (η_s)    = x^P + x^T/2 + α_string tiltx^P - x^T/η_s^P - η_s^T(η_s - η_s^P + η_s^T/2), 
    
        y_c (η_s)    = y^P + y^T/2 + α_string tilty^P - y^T/η_s^P - η_s^T(η_s - η_s^P + η_s^T/2).

Here, (x^P/T, y^P/T, η_s^P/T) are the space-time coordinates for the projectile (P) and target (T) hotspots after the string deceleration.

Motivated by the baryon junction model <cit.>, we allow for the incoming baryon numbers to be located at the string ends or fluctuate along the string with a parameter λ_B <cit.>,

    P(y^B_P/T) = (1 - λ_B) y_P/T + λ_B e^y^B_P/T - (y_P + y_T)/2/4 sinh[(y_P - y_T)/4].


The pre-equilibrium evolution of the system is modeled via a blast-wave-like transverse flow profile for the individual strings developed in the time ≤τ_ hydro with a transverse flow rapidity <cit.>

    η_⊥(𝐱_⊥) = α_ preFlow|𝐱̃_⊥|,

where 𝐱̃_⊥=(x-x_c, y-y_c), with x_c and y_c being the string's center in the transverse plane in Eqs. (<ref>) and (<ref>).

The hydrodynamic equation of motion, Eqs. (<ref>) and (<ref>), are evolved with the lattice-QCD-based  equation of state at finite densities, which assumes strangeness neutrality and n_Q=0.4n_B for Au+Au collisions <cit.>.
The system's viscous stress tensors are evolved with the Denicol-Niemi-Molnar-Rischke (DNMR) theory <cit.>.
The μ_B dependence of QGP-specific shear viscosity is parametrized by 

    η̃(μ_B)=η_0 + (η_2 - η_0)μ_B/0.2,  0<μ_B≤ 0.2 GeV
    η_2 + (η_4 - η_2)(μ_B - 0.2)/0.2,  0.2<μ_B<0.4 GeV,
    η_4, μ_B≥ 0.4 GeV

where η̃≡η T/(e+P) and η_0, η_2 and η_4 are the values of the specific shear viscosity at μ_B=0,0.2,0.4 GeV respectively. We did not introduce an explicit temperature dependence in the parameterization to keep the number of model parameters manageable. A previous analysis showed that a temperature-independent η/s is compatible with experimental data at high energy collisions <cit.>.

We parameterize the specific bulk viscosity by

    ζ̃(T,μ_B)=ζ_ maxexp[-[T - T_ζ(μ_B)]^2/2σ^2_ζ,-],  T<T_ζ(μ_B) 
    ζ_ maxexp[-[T - T_ζ(μ_B)]^2/2σ^2_ζ,+],  T≥ T_ζ(μ_B)
        ,

where ζ̃≡ζ T/(e+P) with the bulk pressure peak temperature T_ζ(μ_B)≡ T_ζ,0-0.15/1 GeVμ_B^2.
This parameterization follows the constant energy density curve e=e(T_ζ,0,μ_B=0) of the  equation of state closely and ensures that the parameterization peaks close to the phase crossover at μ_B>0 <cit.>. 

We choose to normalize the QGP shear and bulk viscosity by T/(e + P) instead of the local entropy density s to better control the variations of the viscous relaxation times in numerical simulations. Take the shear relaxation time, for example,

    τ_π≡ C_ηη/(e + P)   = C_ηη̃1/T
       = C_ηη/s1/T1/1 + μ_B n_B/Ts.

One can see that the shear relaxation time τ_π in Eq. (<ref>) remains well behaved when we vary η̃ within the prior range. If we parameterize η/s as in Eq. (<ref>), then τ_π in Eq. (<ref>) could potentially reduce to very small values and violate causality conditions <cit.> when the factor μ_B n_B/(Ts) becomes large during the dynamical evolution.

All fluid cells dropping below an energy density e_ sw are particlized using a Cooper-Frye freeze-out procedure including δ f corrections for multiple conserved charge currents (B, Q, S) using the Grad moment method <cit.>.
The hadronic rescatterings and decays in the afterburner phase are modeled using  <cit.>.

All 20 model parameters, together with their prior ranges, are listed in Table <ref>.






§ MODEL EMULATION AND BAYESIAN INFERENCE SETUP


For the exploration of the 20-dimensional parameter space shown in Table <ref> with Bayesian inference, we need efficient Gaussian Process (GP) emulators to replace the computationally expensive high-fidelity model calculations with the  framework.
In this work, we will make use of the PCSK emulator implemented in the  package <cit.> and a standard GP emulator from the Scikit-learn Python package <cit.>.
For the details of the implementation of these emulators, we refer to our previous comparison of these GP emulators in Ref. <cit.>.

For an efficient exploration of the parameters in Table <ref>, we generate model simulations at 1,000 different parameter sets in 20 dimensions using the Maximum Projection Latin Hypercube Design (LHD). At every design point, the model simulations consist of 1,000 minimum bias Au+Au collisions at 200 GeV and 2,000 minimum bias events each at 19.6 GeV and 7.7 GeV.
Additionally, we make use of 95 additional full model simulations, where the parameters were sampled from the posterior distribution.
This additional dataset is denoted as High Probability Posterior (HPP) points.
In our standard setup, we train the GP emulator with the LHD + HPP datasets combined, which improves the emulator predictions in the parameter region with higher probability.

The fast prediction of model outputs at different points in the parameter space using GP emulators allows us to make use of Bayes' rule:

    𝒫(θ |  y_ exp) = 𝒫( y_ exp | θ) 𝒫(θ)/𝒫( y_ exp).

𝒫(θ |  y_ exp) represents the posterior probability density function, 𝒫(θ) is the prior probability density function, 𝒫( y_ exp) stands for the total evidence, and 𝒫( y_ exp | θ) is the likelihood function.
In this work, we choose uniform prior distributions for all the model parameters listed in Table <ref>.
The likelihood function is defined as,

    𝒫( y_ exp | θ) = 1/√(|2πΣ|)
         ×exp[-1/2( y_ sim(θ)- y_ exp)^𝖳Σ^-1( y_ sim(θ)- y_ exp)],

where Σ≡Σ_model + Σ_exp is the covariance matrix. The GP emulator provides the model covariance matrix Σ_model. Lacking the knowledge of the full covariance matrix for the experimental measurements, we assume a diagonal form for Σ_exp using the square of the statistical errors for its elements. 

To tackle the computation of the posterior distribution numerically, we employ Markov Chain Monte Carlo (MCMC) techniques <cit.>.
In Ref. <cit.>, we have shown that the affine invariant MCMC implemented in the  Python package <cit.> and the Parallel Tempering Langevin Monte Carlo (PTLMC) implemented in  lead to consistent results.
The parallel tempering methods generally work better when there are multiple modes in the posterior distributions.
This was the case for the closure test performed there, where the uncertainty of the test point was sufficiently small.
For real experimental data with a larger uncertainty, we will therefore use a method that can deal with multi-modal posterior distributions.
Since the previously tested PTLMC method does not compute the evidence 𝒫( y_ exp) (this factor is essential for computing the Bayes factor, which we use to compare different training processes in our analysis), we will employ another efficient MCMC sampler routine implemented in the  package <cit.>.
This sampler implements Preconditioned Monte Carlo (PMC), where first normalizing flow is applied to decorrelate the parameters and the distributions, and then adaptive Sequential Monte Carlo (SMC) is applied.

The Bayesian inference analysis in this paper is performed using the STAR and PHOBOS measurements for Au+Au collisions at the RHIC BES program. The details are listed in Table <ref>.



In Ref. <cit.>, we systematically studied the performance of different model emulation setups on these experimental measurements and quantified their impacts on the closure test.



§ POSTERIOR ANALYSIS



In this section, we perform Bayesian analyses on the RHIC measurements using three different GP models: PCSK and Scikit GP trained with LHD + HPP simulations, and PCSK trained only with the LHD dataset.
We will first evaluate and compare the general aspects of the posterior distributions from the three GP models and then discuss the physics insights from the constraints on model parameters.



In Fig. <ref>, we show the posterior marginal distributions for the 20 model parameters using the PCSK emulator (blue), the Scikit GP emulator (green), and the PCSK emulator trained only with the 1,000 LHD points (magenta). The former two GP emulators were trained with the 1,095 LHD + HPP training points.
We showed that the PCSK emulator performed much better than the Scikit GP in the closure test in Ref. <cit.>. However, when applying the Bayesian inference analysis to the real experimental measurements, the two emulators result in similar posterior distributions for the model parameters. 
For most of the parameters, we find that the PCSK emulator still gives slightly more peaked distributions than those from the Scikit GP.

Comparing the blue and magenta posterior distributions, we find that including the additional 95 HPP points in training the GP emulator results in slightly stronger constraints on the model parameters. 

The amount of information gained in our Bayesian inference analysis can be quantified using the Kullback–Leibler (KL) divergence, which is defined as,

    D_KL(p || q) = ∫ dθ  p(θ) ln(p(θ)/q(θ)).

Here, we compute the KL divergence with p(θ) as the posterior distribution 𝒫(θ |  y_ exp) in Eq. (<ref>) and q(θ) as the uniform prior distribution 𝒫(θ). 
Using Eq. (<ref>), we can compute the KL divergence from the sampled posterior chains as

    D_KL(p || q) = 1/N_samples∑_iln(𝒫( y_ exp | θ_i)/𝒫( y_ exp)),

where the index i runs through the N_samples parameter samples in the posterior chains.



Table <ref> reports the KL divergence for the three posterior distributions presented in Fig. <ref>. The ordering of the D_KL is consistent with our observation in Fig. <ref> discussed above. The more accurate model emulator, PCSK, gives stronger constraints on model parameters than those from the Scikit GP. Including 95 high probability posterior (HPP) points in training the GP emulator further tightens the model constraints.

Another metric to compare the posterior distributions is the Bayes factor. 
The Bayes factor is defined as the ratio of posterior probabilities from two models, A and B, given the experimental data 𝐲_ exp,

    ℬ_A/B≡𝒫(A |𝐲_ exp)/𝒫(B |𝐲_ exp) = 𝒫(𝐲_ exp| A)𝒫(A)/𝒫(𝐲_ exp| B)𝒫(B).

Here, we compare the three GP emulator models shown in Fig. <ref>. Because the three GP models are emulating the same theoretical model, we assume the prior model probability is the same for all the models, namely 𝒫(A) = 𝒫(B) in Eq. (<ref>). Thus, the Bayes factor reduces to the ratio of marginalized likelihoods, also known as Bayesian evidence:

    ℬ_A/B = 𝒫(𝐲_ exp| A)/𝒫(𝐲_ exp| B).

The Bayesian evidence for a model A on the given experimental data 𝐲_ exp, 𝒫(𝐲_ exp| A), can be computed by integrating the likelihood function over the model parameter set across the entire parameter space:

    𝒫(𝐲_ exp| A) = ∫dθ_A 𝒫(𝐲_ exp|θ_A, A)𝒫(θ_A).

We compute the Bayesian evidence for the three models using the  package. The Bayes factors are reported in Table <ref>.

Following the Jeffreys' Scale of the Bayes factor in Ref. <cit.>, we find strong evidence in favor of the analysis with the PCSK emulator than the one with the Scikit GP. This result emphasizes the importance of getting an accurate model emulator in Bayesian studies.
We also compare the two model setups with different training data sets using the PCSK emulator. In this case, we find a Bayes factor with a magnitude between 1 and 2.5, which suggests that the PCSK trained with only the LHD points is favored but only with weak evidence. 
Naively, one would expect the GP emulator with more training data to be more favored. The reversed hierarchy here is because the PCSK emulator trained with the LHD + HPP dataset gave a smaller model covariance matrix Σ_model, which leads to an overall lower likelihood in Eq. (<ref>) compared to the results from PCSK trained with LHD only. 


Considering the KL divergence and Bayes factor results, we will perform further analysis on the posterior distribution from the PCSK emulator (LHD + HPP) from now on.







Now, we discuss the physics implications of the constraints on model parameters from the posterior distribution. 


    
  * The parameter B_G is the width of the nucleon to sample its hot spots <cit.>. The posterior range B_G = 15.99^+7.14_-9.66 GeV^-2 translates to the nucleon mass width between 0.5 and 0.95 fm at 90% confidence level.
    
  * The parameter α_shadowing controls the probability of producing strings from secondary binary collisions in the  initial state model <cit.>. Our Bayesian analysis prefers α_shadowing∼ 0.15, indicating that only 15% of the binary collisions are shadowed for soft string production. The numbers of produced strings scale closer to the number of binary collisions than the number of participants.
    
  * The rapidity loss fluctuation parameter σ_y_loss∼ 0.3 in our analysis, which is smaller than the value σ_y_loss = 0.6 used in Ref. <cit.>. The latter was calibrated using the multiplicity distribution in p+p collisions, which has a stronger sensitivity to this model parameter than the 10% centrality binned particle yields in Au+Au collisions <cit.>. The particle multiplicity fluctuations in Au+Au collisions include the number of participant fluctuations, further diluting the sensitivity to the rapidity loss fluctuation parameter.
    
  * The parameter for the nucleons' remnant rapidity loss α_rem = 0.54 ± 0.15 is constrained by the shape of the pseudo-rapidity distribution dN^ch/ dη. Its value is consistent with the previous simulations <cit.>.
    
  * The parameter λ_B controls the amount of initial-state baryon charge transport to the string junction <cit.>. Its constrained value λ_B ∼ 0.15 is consistent with previous analyses <cit.>.
    
  * Our Bayesian analysis prefers α_preflow∼ 0, suggesting small pre-equilibrium flow in the transverse plane for the string source terms before depositing into hydrodynamics.
    
  * The posterior distribution for the hotspot transverse size σ^string_x peaks at the lower bound of the prior, suggesting a small hotspot size is preferred. 
    
  * The parameter σ^string_η controls the string profile's fall-off along the longitudinal direction. Its marginal distribution has a double-peaked structure, indicating this parameter may have a √(s_ NN) dependence.



To quantify the constraints on the functional dependence for the initial-state rapidity loss and the QGP-specific viscosities, we display the 90% confidence level of the prior distributions (gray) with the 68% and 90% confidence levels of the posterior distributions (green) for these parameters in Fig. <ref>.

The average rapidity loss in the initial state is, compared to the prior distribution (gray), tightly constrained by the particle yield measurements at multiple collision energies. Despite using different types of GP emulators and MCMC algorithms, our results are consistent with those in Ref. <cit.>.
We also obtain strong constraints on the specific shear and bulk viscosities of QGP as functions of μ_B and T, respectively.
The values of QGP-specific shear viscosity show a significant increase with μ_B within [0, 0.2] GeV. This constraint primarily comes from the pseudo-rapidity dependence of elliptic flow v_2(η) measurements <cit.>.
The values of η̃ at larger μ_B remain around 0.3 closer to the upper bound of the prior for the parameter η_4.
For the QGP's specific bulk viscosity, we find a peaked distribution around a temperature of ≈ 200 MeV with a maximum value of 0.12. The values of ζ̃ fall off to zero quickly when departing from the maximum value.




§ MODEL SIMULATIONS WITH POSTERIOR DISTRIBUTIONS


After using the GP emulators to obtain the posterior distributions of model parameters displayed in Fig. <ref>, we will verify these constraints with full model simulations and further make model predictions. We randomly choose 100 samples {θ^post} from the posterior distributions generated with the PCSK emulator (LHD + HPP) and then perform event-by-event simulations with the  framework.
At each posterior model parameter set θ^post_i, we simulate 1,000 minimum bias Au+Au collisions at √(s_ NN)=200 GeV, 2,000 events at √(s_ NN)=19.6 GeV, and 3,000 collisions at √(s_ NN)=7.7 GeV.
With these simulations, we compute the averages of the observables from the 100 posterior parameter sets and denote their standard deviations as the remaining systematic uncertainties after the Bayesian analysis.
Here, we assume the statistical errors of observables at individual posterior parameter sets are much smaller compared to the standard deviations of the 100 posterior parameter sets.
In Fig. <ref>, we show the comparison to the experimental observables (see Tab. <ref>) used in the Bayesian inference.





In general, the full model calculations with parameters sampled from the posterior distribution can provide a good description of the 544 data points in the Bayesian inference. The agreement is generally better for central collisions and near the mid-rapidity region. This behavior is related to the fact that the GP emulator has smaller emulation errors for those observables than the observables in peripheral centralities and forward rapidity regions <cit.>. 

Looking into details of the mid-rapidity observables at √(s_ NN)=200 GeV (right columns of Fig. <ref>), our model slightly overpredicted the particle yields of K^+ mesons. The centrality dependencies of proton and anti-proton yields are somewhat flatter than the STAR measurements. All these observables are sensitive to the switching energy density from hydrodynamics to hadronic transport, the current model discrepancy could be reduced by allowing the parameter e_sw to be centrality dependent.
The identified particles' mean p_ T and mass hierarchy reflect the amount of radial flow developed in the hydrodynamic phase. Our model provides a good description of the STAR measurements for pions and kaons, while slightly underestimating the proton and anti-proton mean p_ T but still within the experimental uncertainties. The averaged transverse momenta of protons is more sensitive to the scattering cross sections and baryon-anti-baryon annihilation processes in the hadronic transport model <cit.>.

For the mid-rapidity observables at √(s_ NN)=19.6 GeV, the description of kaon's particle yields is slightly improved compared to that at 200 GeV. At this collision energy, the averaged net baryon chemical potential can reach up to 150-200 MeV at mid-rapidity <cit.>. Our model gives a ∼ 50 MeV larger mean p_ T for anti-protons than those for protons at finite baryon density. The current STAR measurement can not distinguish this difference. In our model, the mean p_ T difference between protons and anti-protons primarily comes from the  hadronic scattering phase as anti-protons emit relatively earlier than protons from the hydrodynamic hypersurface and experience more hadronic interactions <cit.>. This phenomenon is more pronounced at √(s_ NN)=7.7 GeV, at which the anti-proton mean p_ T is about 100 MeV larger than that of protons in our calculations. At √(s_ NN)=7.7 GeV, our model underestimates the mean p_ T of pions and kaons, suggesting the systems do not develop enough radial flow in the hydrodynamic phase. Introducing an explicit μ_B dependence on the QGP-specific bulk viscosity would help to reduce the tension here. Because the hydrodynamic phase at √(s_ NN)=7.7 GeV is limited, we observe that the triangular flow coefficients v_3^ ch{2} start to deviate from the experimental data in peripheral collisions, similar to the finding in Ref. <cit.>.

Lastly, we look at the pseudorapidity-dependent observables. Our model gives a good description of the PHOBOS charged hadron dN^ch/ dη measurements across collision centrality at 200 and 19.6 GeV.
Our model gives a flatter shape for v_2^ ch(η) compared to the PHOBOS measurements. This is because our parameterization for the QGP shear viscosity is temperature-independent. Introducing a η/s(T) will improve the model-to-data comparison <cit.>.


The full model simulations allow us to go beyond the calibrated experimental measurements and make model predictions with controlled systematic uncertainties. In this work, we select p_ T-differential observables to demonstrate the predictive power of our model. We will report a systematic extrapolation of our model to different collision systems such as (p, d, ^3He)+Au, O+O, Ru+Ru, Zr+Zr collisions, and Au+Au collisions at other collision energies in the RHIC BES program in future work.

Since our model is calibrated using only the p_ T-integrated observables in Table <ref>, the model-to-data comparison for p_ T-differential observables can effectively test the consistency of our Bayesian analysis.

Figure <ref> shows our model predictions for the p_ T-differential spectra for pions and protons and charged hadron v_2(p_ T) in Au+Au collisions at 7.7, 19.6, and 200 GeV compared with the STAR measurements. We find a good description of the identified particle p_ T spectra at 200 and 19.6 GeV, while the p_ T spectra are slightly steeper than the STAR data at 7.7 GeV. These results are consistent with our description of the identified particle mean p_ T in Fig. <ref>. Our results show that the first two moments of the particle p_ T spectra, namely the particle yield and mean p_ T, are sufficient to produce reasonable p_ T spectra in the Bayesian calibration <cit.>.

For the charged hadron p_ T-differential elliptic flow, our model prediction can reproduce the experimental data fairly well up to 2 GeV. We notice that the systematic uncertainty in v_2(p_ T) grows with p_ T in the calculations. The effects from the out-of-equilibrium δ f corrections in the Cooper-Frye particlization become important at high p_ T. The v_2(p_ T) at 200 GeV shows a strong suppression in p_ T > 2 GeV from the δ f based on the Grad's moment method.




§ HIGH LIKELIHOOD PARAMETERS



Although our approach in Sec. <ref> is a preferred way to make model predictions with controlled systematic errors, it is computationally demanding to perform event-by-event simulations at 𝒪(100) parameter sets sampled from the posterior distribution, especially for observables which require high statistics. 







The Bayesian analysis usually reports the parameter set θ^MAP at the MAP (maximum a posteriori probability) estimate using the GP emulator. In our case, we find the MAP parameter set in the entire prior space gives a reverse ordering for some rapidity loss parameters, namely y_ loss,4>y_ loss,6. In the Bayesian analysis above, we did not impose any constraints on the prior distribution of the rapidity loss parameters. As shown in Fig. <ref>, the experimental measurements at 200 GeV give strong constraints on the initial-state rapidity loss near its beam rapidity, y_ beam≡arccosh(√(s_ NN)/(2m_ N))=5.36. Therefore, the values of y_ loss,4 and y_ loss,6 are anti-correlated (see Fig. <ref> below), and both orderings are allowed in the posterior distribution.
Although the MAP parameter set with y_ loss,4>y_ loss,6 is sufficient to perform simulations for the RHIC BES program, we expect it to fail when extrapolating to the higher LHC energies. Therefore, we choose to report the highest likelihood parameter set with the monotonic constraint y_ loss,2≤ y_ loss,4≤ y_ loss,6 in Table <ref>. We compute the Bayesian evidence without and with the monotonic constraint and obtain a Bayes factor of ln (ℬ_w.o. constr./w. constr.) = 0.85 ± 0.02, which falls in the “inconclusive” category based on the Jeffreys' Scale <cit.> to distinguish the two setups.




Figure <ref> compares the posterior distribution for the averaged rapidity loss ⟨ y_ loss⟩ as a function of y_ init with and without the monotonic constraints, y_ loss,2≤ y_ loss,4≤ y_ loss,6. The 90% posterior band shrinks almost by half for y_ init∈ [3, 6]. Because the values for y_ loss,2 and y_ loss,4 are also anti-correlated in the posterior distribution, the ⟨ y_ loss⟩ at small y_ init also receive some minor effects from the monotonic constraint.


Figure <ref> shows the marginalized posterior distributions for individual rapidity loss parameters on the diagonal and the correlations between the different parameters in the off-diagonal corners.

The diagonal panels show that imposing the monotonic constraint on the rapidity loss parameters leads to more peaked marginalized posterior distributions because the prior space is smaller with the constraint.
While the peak position of y_ loss,2 is barely affected by the constraint, the peak position of y_ loss,4 is shifted to a slightly smaller value and the position of y_ loss,6 to a slightly larger value.
The marginal distribution for the rapidity loss fluctuations σ_y_ loss is almost unaffected. We have further checked that the marginalized posterior distributions of the other 16 parameters are not affected by imposing the monotonic constraint.

The upper right off-diagonal panels show the correlation between the rapidity loss parameters without the monotonic constraint.
We find strong anti-correlations between y_ loss, 2 and y_ loss, 4 and between y_ loss, 4 and y_ loss, 6.
Such anti-correlation is a consequence of the piece-wise parameterization in Eq. (<ref>) when the experimental measurements at 200 and 19.6 GeV set constraints on the average rapidity loss near their beam rapidity 5.36 and 3.04, respectively.
Looking at the constrained case in the lower left part of the off-diagonal panels, we also observe a sharp cut in the y_ loss,6 vs. y_ loss,4 distribution, which is a direct consequence of the y_ loss,4≤ y_ loss,6 constraint in the prior.





§ SENSITIVITY ANALYSIS



The fast model emulator lets us quantitatively study how individual model parameters affect different experimental observables. 

Starting with a global sensitivity analysis, we compute the first-order Sobol' indices <cit.> for individual experimental observables over the entire prior region.
Similar analyses were performed for heavy-ion collisions with (2+1)D simulations assuming longitudinal boost invariance  <cit.>.
The first-order Sobol' indices quantify the importance of each parameter for a given observable 𝕆(θ) by performing a decomposition of the variance of 𝕆(·) over the parameter space θ≡ (θ_1, …θ_m).
The first-order Sobol' index for the model parameter θ_i is defined as

    S_i^𝕆≡Var_θ_i(𝔼_θ_-i(𝕆(θ)|θ_i))/Var_θ(𝕆(θ)),   i=1,…,m.

Here, the index i runs over the dimension of the model parameter space (m = 20). We sample each model parameter θ_i independently from a uniform distribution over its corresponding prior range in Table <ref>. 
The expression 𝔼_θ_-i(𝕆(θ)|θ_i) performs an average of the observable 𝕆 over a parameter subspace θ_-i≡θ\θ_i by holding the i-th parameter θ_i at a fixed value.
Then Sobol' index S_i^𝕆 measures the fraction of the variance of the observable 𝕆 from the parameter θ_i relative to its total variance.



Figure <ref> shows the first-order Sobol' indices for identified particle yields dN/dy and ⟨ p_ T⟩ in the most central events and charged hadron v_2^ ch{2} at 0-5% and 20-30% centrality bins.
The particle yields of charged pions have strong sensitivities to the rapidity loss parameters y_loss, n. The measurements at 200 GeV are sensitive to the rapidity loss parameters at incoming rapidity y_init = 4 and y_init = 6, while the strongest sensitivity shifts to y_loss, 2 for the lower collision energy measurements. Comparing the Sobol' indices between pion and proton yields, we find that the proton yields show additional sensitivity to the baryon stopping parameter λ_B and the string's longitudinal profile parameter σ^string_η.

The averaged transverse momenta of pions and protons are mostly sensitive to the initial-state hotspot size in the transverse plane σ^string_x and the magnitude of pre-equilibrium flow α_preFlow.
Both model parameters have strong effects on the development of hydrodynamic radial flow. 
The hotspot's transverse size controls the magnitude of the early-stage pressure gradients.
A stronger pre-equilibrium flow leads to faster fireball expansion at the early time. 
We also find the pion's mean p_ T at 200 GeV shows some sensitivity to the specific bulk viscosity of the QGP. 
However, the magnitudes of their Sobol' indices are dwarfed compared to those model parameters in the initial-state and pre-equilibrium stages.
A similar situation is observed for mid-rapidity anisotropic flow. 
The charged hadron elliptic flow coefficients at central and semi-peripheral collisions are mostly sensitive to the hotspot's transverse size and pre-equilibrium flow over the entire prior range. 
The sensitivity to the specific shear viscosity of the QGP is very small. 
We checked that the charged hadron triangular flow shows very similar global sensitivity as the elliptic flow observables.



Figure <ref> further represents the first-order Sobol' indices for rapidity-dependent observables. 
We find that the charged hadron yields at forward rapidity show a stronger sensitivity to the specific bulk viscosity of the QGP and also the amount of rapidity loss in the beam remnant α_rem.
We find a similar global sensitivity for charged hadron elliptic flow at central and forward rapidity regions.

Although it is informative to quantify the global sensitivity of experimental observables to the model parameter over the entire prior parameter space shown in Figs. <ref> and <ref>, the responses could be different in different local regions of the parameter space. Therefore, we further compute the response coefficients between experimental observables and model parameters using the posterior distribution. We follow the approach in Ref. <cit.> and define the ensemble-averaged response coefficient matrix,

    ℛ_ai≡⟨∂ y_a/∂θ_i⟩_post = ∑_j ⟨δ y_a δθ_j ⟩_post⟨δθ_j δθ_i ⟩_post^-1.

Here, the index a runs over individual experimental observables in Table <ref>, and the index i runs over the model parameters in Table <ref>. The ensemble average ⟨·⟩_post is performed on the posterior distribution to extract the model's response to the experimental observables near the real measurements. To make the response coefficients unitless, we define the normalized experimental observables δ y_a = (y_a(θ_j) - y̅_a)/σ_y_a and model parameters δθ_i = (θ_i - θ̅_i)/σ_θ_i, where y̅_a(θ̅_i) and σ_y_a(σ_θ_i) are the mean and standard deviation of the observable y_a (parameter θ_i) over the posterior distribution, respectively. The right-hand side of Eq. (<ref>) should be interpreted as a matrix multiplication. 



Figure <ref> shows the response matrix between the experimental observables and model parameters over the posterior distribution. We observe that identified particle yields show strong sensitivity to the rapidity loss parameters, consistent with the results from the global analysis above. The signs of the response coefficients provide additional information compared to Sobol' indices. Interestingly, the mid-rapidity charged hadron yields at 19.6 GeV show positive correlations with the rapidity loss parameters while the forward dN^ch/ dη negatively correlates with these parameters. This pattern results from the energy-momentum conservation imposed in the  model. More particle production near mid-rapidity would require more energy loss during the initial state, which leaves less available energy at forward rapidity regions. Particle yields show significant anti-correlation with the parameter α_shadowing because a smaller value of α_shadowing produces more string sources from the binary collisions and increases mid-rapidity particle production.

Different from the global sensitivity analysis above, we find the observables' response coefficients to the hotspot transverse size σ^string_x and pre-equilibrium flow α_preFlow are small in the posterior distribution. This difference is because the marginalized posterior distributions for these two parameters are sharply peaked around the edge of their prior range, as shown in Fig. <ref>. The correlations in the local region are small and different from the global averages. 

We find that the observables' responses to the QGP transport properties in Fig. <ref> are more in line with our physics intuition. 

The identified particles' mean p_ T shows sizable negative correlations with the parameters related to the specific bulk viscosity of the QGP. This result aligns with our intuition that the bulk viscosity acts as a resistance to reduce the development of hydrodynamic radial flow.

The parameters related to the specific shear viscosity of the QGP have a significant negative correlation with the charged hadrons' anisotropic flow coefficients at 200 GeV. The pseudo-rapidity dependent v_2(η) measurements also show strong sensitivity to the QGP shear viscosity, especially the v_2 at forward rapidity shows some sensitivity to the η_2 parameter. Although the correlation is relatively weak, we find a negative correlation between charged hadron v_n at 19.6 GeV and the shear viscosity at μ_B = 200 MeV. And the η_4 parameter, which controls the QGP shear viscosity at μ_B = 400 MeV, is negatively correlated with the charged hadron v_n at 7.7 GeV.



§ CONCLUSION


This work presents a systematic Bayesian analysis of particle production and flow measurements at the RHIC BES program using an event-by-event (3+1)D hydrodynamics + hadronic transport framework. We obtain robust statistical constraints on the QGP transport properties and various aspects of relativistic nuclear dynamics.
This work expands upon Ref. <cit.> and provides the complete 20-dimensional model posterior distribution using an improved Gaussian Process model emulator. Utilizing the statistical tools developed in Ref. <cit.>, we systematically analyze the model's posterior distributions obtained from three model emulators with different accuracy. Using the statistical measures, namely the Kullback-Leibler divergence and Bayes factor, we demonstrate that the accuracy of the model emulator plays a crucial role in getting robust posterior distribution from the experimental measurements.
With an independent Bayesian analysis framework <cit.>, we verify that the posterior distributions of the initial-state rapidity loss and the specific shear and bulk viscosity of QGP obtained in this work are consistent with the results in Ref. <cit.>, demonstrating the reproducibility of our results.

We sample 100 parameter sets from the posterior distributions and perform large-scale event-by-event simulations at each parameter set using our (3+1)D hybrid model. These results verify that the posterior distribution obtained with the model emulator is reliable for the real model. We further make predictions for a set of p_ T-differential observables and estimate their systematical errors based on the variations among the 100 parameter sets. Good agreements with the experimental data are found. Based on this approach, we will perform more systematic model predictions in future work.
For community users who do not have sufficient computing resources to perform simulations at multiple posterior parameter sets, we provide a high-likelihood parameter set in Table <ref>.

Lastly, we report a sensitivity analysis of how individual experimental observables respond to different model parameters, enabled by the fast model emulator. We find commons and differences between the global sensitivity analysis over the entire prior region and the response matrix from the posterior distribution. Both sensitivity analyses provide useful physics insights into the phenomenological modeling of relativistic heavy-ion collisions. We will extend the sensitivity analysis to a more systematic experimental design study in future work.

We provide the following results from this work for community use:

    
  * Training datasets in Table <ref> <cit.>
    
  * Trained model emulator PCSK (LHD + HPP) <cit.>
    
  * Posterior parameter samples from the MCMC <cit.>
    
  * An updated Bayesian analysis package with the  integration <cit.>




We thank Björn Schenke and Wenbin Zhao for their involvement during the early stage of the project and useful discussions.
This work is supported in part by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under DOE Award No. DE-SC0021969 and DE-SC0024232. H. R. and C. S. were supported in part by the National Science Foundation (NSF) within the framework of the JETSCAPE collaboration (OAC-2004571).
C. S. acknowledges a DOE Office of Science Early Career Award.
S.-A. J. acknowledges support from the BAND Fellowship funded by the National Science Foundation CSSI program under award number OAC-2004601. Numerical simulations presented in this work were partly performed at the Wayne State Grid, and we gratefully acknowledge their support.
This research was done using resources provided by the Open Science Grid (OSG) <cit.>, which is supported by the National Science Foundation award #2030508 and #1836650.






 Paper Title: A simple high-accuracy method for solving bound-state equations with the Cornell potential in momentum space.





    et al.
            #1#2#1/#2
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            apsrev4-2
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             LIPCFTP
            stadler@uevora.ptDepartamento de Física, Universidade de Évora, 7000-671 Évora, PortugalLIP Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, PortugalDepartamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugalelmar.biernat@tecnico.ulisboa.pt
            CFTP, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 
            1049-001 Lisboa, PortugalDepartamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049-001 Lisboa, PortugalLIP Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugalv.valverde@campus.fct.unl.ptNOVA School of Science and Technology, 2829-516 Caparica, Portugal
            
             
            
            
             
             
            
             § ABSTRACT 
            We present a simple, high-accuracy method for solving bound-state integral equations in momentum space with singular kernels. For the case of the linear-confining potential, the associated Cauchy-principal-value singularity is removed by subtraction. Derivatives of the wave function that appear as a result of the subtraction technique are approximated by means of interpolating functions. The resulting non-singular integral equation is solved using the Nyström method. The results show excellent agreement with exactly known energy eigenvalues. By further increasing the number of integration points and the order of the Lagrange interpolation, results with extremely high accuracy can be achieved.     
             
            11.10.St, 14.40.Pq, 12.39.Pn, 03.65.Ge
            
            A simple high-accuracy method for solving bound-state equations with the Cornell potential in momentum space 
                Vasco Valverde
                August 5, 2024
            =============================================================================================================
            
            
            
            
            § INTRODUCTION
            
            The nonrelativistic Cornell potential model <cit.>, and  relativized” variations of it <cit.>, have been very successful in describing the masses and decay rates of heavy quarkonia. They describe the interaction between heavy quarks and antiquarks typically in terms of a combination of a long-range linear potential for confinement and a short-range color-Coulomb potential for one-gluon exchange. Generally formulated within Schrödinger theory, these models are not able to account for the dynamical covariant structure of light quarks. But for mesons with at least one light quark, a relativistic treatment is necessary, and in addition, for the description of the light mesons, in particular the pions, the implementation of dynamical mass generation and spontaneous chiral symmetry breaking (SχSB), is indispensable. 
            
            Nonperturbative approaches based on quantum field theory, such as the Dyson-Schwinger/Bethe-Salpeter (DSBS) formalism <cit.> and the Covariant Spectator Theory (CST) <cit.> are manifestly Poincaré covariant frameworks that are able incorporate SχSB in a consistent way. In CST, even in the presence a confining interaction with a combination of Lorentz scalar and pseudoscalar structures, one can still demonstrate consistency with chiral symmetry and its dynamical breaking, which goes against common expectations <cit.>. Allowing the possibility of scalar structures in the kernel is interesting, because other approaches suggest that the confining interaction is actually mostly scalar <cit.>. In addition, while the DSBS equations are usually formulated in Euclidean momentum space, the CST equations are solved directly in Minkowski space. Potential ambiguities that may arise for observables like form factors associated with the Wick rotation from Euclidean- to Minkowski-space momenta are not present in CST. Finally, the CST formalism allows for a covariant generalization of the Cornell potential model, i.e. the CST equations for mesons reduce in the nonrelativistic limit to the Cornell-potential Schrödinger equation. Successful attempts to employ a Poincaré invariant generalization of the Cornell potential also exist in the DSBS approach, however it is not entirely clear whether or not the corresponding covariant off-shell extension of the linear potential actually describes the confinement of quarks consistently <cit.>.  
            
            Ambiguities of this kind do not show up in the CST, as demonstrated by the proof that a CST-generalization of the linear potential leads to quark confinement <cit.>. A CST model for mesons can therefore be viewed as the most natural covariant extension of the Cornell model that incorporates both SχSB and confinement. Because of its fully-relativistic nature and its nonrelativistic Cornell limit, the CST model emerges as a particularly suited candidate for a unified description of all quark-antiquark mesons, from the pion to bottomonium.
            
            When solving integral equations with singular kernels, efficient numerical methods are crucial for performing global fits of many states to obtain a unified description. In the Cornell potential case, the complexity arises from the linear confining kernel in momentum space, leading to a Cauchy-principal-value singularity in the Schrödinger equation. Even though this singularity is integrable, the direct application of standard numerical solution techniques becomes intricate. One way to avoid this problem in the nonrelativistic case is to solve the Schrödinger equation for the linear potential directly in coordinate space <cit.>. However, if one includes the lowest-order relativistic corrections to the potential, which might already be important for charm quarks, non-local terms show up, apart from the well-known spin-spin, spin-orbit and tensor force contributions <cit.>. It is not obvious how to treat non-local terms in coordinate space and therefore they are usually neglected <cit.>. Even more drastic, a fully-relativistic treatment, as needed for the light quarks, is unfeasible in coordinate space and requires a momentum-space formulation. 
            
            Several works have addressed the linear-confining problem in momentum-space. They either use a non-singular screened linear potential <cit.>, or they start with a screened potential, perform a partial wave expansion and then take the  unscreened” limit <cit.>. The numerical integration of the principal-value singularity is then performed by expanding the wave function in a suitable set of basis functions and the applying the Sloan method, i.e. integrating by placing the Gaussian quadrature points symmetrically around the singularity. This procedure works in principle, but is impractical, for instance, in fitting procedures because the set of basis functions depends on the fitting parameters, i.e. the quark masses and the coupling. Alternatively, in Ref. <cit.>, the linear-confining problem with l=0 was solved using a quadrature method where the singularity of the integral is included in the weight functions. This method gives remarkable high-precision results, but is very involved in practice, because the weight functions must be adapted specifically for each integral appearing in the Schrödinger equation.  
            
            These difficulties are resolved by removing the principal-value singularity altogether by subtraction <cit.>. This occurs at the cost of introducing the derivatives of the wave function under the integral. Prior to the present work, we have dealt with this difficulty by applying the so-called Galerkin method, i.e. by expanding the wave function in basis functions whose derivatives can be easily calculated. For this method to work in practice, one needs to carefully choose appropriate basis functions that reproduce the wave function's correct asymptotic behavior determined by the potential. It is this inflexibility together with long computation times that makes this method often impractical.
            
            In view of these challenges our aim here was to develop a simpler, more flexible and faster numerical method. We show that the Nyström collocation method can be applied to the linear confining potential in momentum space after the principal-value singularity has been removed. This procedure provides extremely accurate results and can directly be extended to the fully-relativistic linear confining problem, because the singularity structure remains unchanged under covariant generalization. 
            
            In Section <ref> we show how the Cauchy-principal value singularity is removed by subtraction, leading to ordinary integrals that involve the first and second derivatives of the wave function. Section <ref> deals with the numerical evaluation of the integrals using the Nyström collocation method. The numerical results are presented and discussed in Section <ref>, and finally Section <ref> gives a summary of the work and conclusions. 
            
            
            
            
            
            
            
            § THE CORNELL POTENTIAL IN MOMENTUM SPACE
            
            The Cornell quark-antiquark potential <cit.> in coordinate space is composed of a Coulomb-like short-range interaction, originating from one-gluon-exchange, and a linearly rising confining potential, of the form
            
                Ṽ(r)= -α/r+σ r   ,
            
            where r=| r| is the distance between the two quarks, α is the strong coupling constant, including a color factor, and the constant σ characterizes the strength of the confining potential.
            
            The Fourier-transform of the Coulomb potential
            
                Ṽ_C( r)= -α/r
            
             is very well known,
            
                V_C( q) 
                 =∫ d^3r Ṽ_C( r) e^i q· r
                 =-4πα/ q^2  ,
            
            which is obtained by first transforming a screened version of it and then taking the unscreened limit. The momentum transfer q= k- p is the difference between the relative momenta in the initial and final state.
            
            In order to find the momentum-space version of the linear potential, 
            
                Ṽ_L( r)=σ r   ,
            
            one can apply the same general idea, but different screening methods can be used. In <cit.>, we found that starting from a particular kind of screening leads to the convenient representation
            
                V_L( q) = [ V_A( q) -(2π)^3δ^(3)( q)∫d^3 q'/(2π)^3 V_A( q') ]   ,
            
            where 
            
                V_A( q) = - 8πσ/ q^4  .
            
            The form of the “potential” (<ref>) is somewhat peculiar and should be interpreted as a distribution. It will always be multiplied with a wave function and integrated over the loop momentum k, which produces meaningful expressions.  
            
            This becomes clear when the momentum-space Cornell potential,
            
                V( q)=V_C( q)+V_L( q)   ,
            
             is now inserted into the Schrödinger equation for the momentum-space wave function ψ ( p ) of a two-body system with reduced mass m_R,
            
                p^2/2m_Rψ ( p )
                +  
                ∫d^3 k/(2π)^3 V_C ( p- k)
                ψ ( k )
                
                 
                +  
                P∫d^3 k/(2π)^3 V_A ( p- k)
                 [   ψ ( k )
                 -   ψ ( p )] =
                 E   ψ ( p )  .
             
            Both integrands are singular  at k= p, and it appears that the singularity in  V_A is even much stronger than the one in V_C. However, it is weakened by the wave-function subtraction in the numerator that arises from the Dirac-delta term in V_L of Eq. (<ref>). As a result, the integral over V_A in (<ref>) reduces to a Cauchy principal value integral, indicated by the symbol P∫, and is therefore well defined <cit.>.
            
            Next, we expand Eq. (<ref>) into partial waves. Introducing the unit vectors k̂ and p̂, such that p=p p̂  and k=k k̂, and abbreviating x=k̂·p̂, the potentials, 
            depending only on q^2=p^2+k^2-2pkx, can be written as functions of p, k, and x. Let's first look at the Coulomb potential. The angular dependence is 
            represented as a series of Legendre polynomials P_ℓ(x),
            
                V_C(p,k,x) =  ∑_ℓ4π/2ℓ+1 V_C,ℓ(p,k)P_ℓ(x)   .
            
            The expansion coefficients are determined through the angular integral
            
                V_C,ℓ(p,k)
                    =2π∫_-1^1 dx P_ℓ(x) V_C(p,k,x) 
                   =-8π^2 α∫_-1^1 dx P_ℓ(x)/p^2+k^2-2pkx
                    = -8π^2 α/pk Q_ℓ(y) 
                
                
                   ,
            
            where the Legendre functions of the second kind, Q_ℓ(y), depend on
            
                y=p^2+k^2/2pk  ,
            
            and are singular at y=1, which occurs when k=p. 
            The representation
            
                Q_ℓ(y) = P_ℓ(y)Q_0(y)-W_ℓ-1(y)   ,
            
            with
            
                W_ℓ-1(y) = ∑_m=1^ℓ1/m P_ℓ-m(y) P_m-1(y)   ,
            
            is very useful here.
            It shows that the singularities in the potential matrix elements for all angular momenta have their origin in Q_0(y), which can also be written
            
                Q_0(y) = 1/2ln| y+1/y-1| 
                = 1/2ln( p+k/p-k)^2   ,
            
            whereas W_ℓ-1(y) is not singular and contributes only for ℓ≥ 1. 
            
            The partial-wave decomposition of the linear potential V_L( q) of (<ref>) has been derived in <cit.>. Because of its nature as a distribution, it is easiest to write how it acts on the partial-wave components ψ_ℓ(p) of the wave function[Because of rotational symmetry, neither the partial-wave potentials nor wave functions depend on the eigenvalue of the z-component of the angular momentum, so it is suppressed throughout.]. We find
            
                ∫_0^∞d k k^2/(2π)^3
                V_L,ℓ(p,k)ψ_ℓ(k) 
                
                = 
                ∫_0^∞d k k^2/(2π)^3[
                V_A,ℓ(p,k)  ψ_ℓ(k) 
                  - 
                V_A,0(p,k)  ψ_ℓ(p)
                 ]   ,
            
            where the partial-wave elements of V_A are
            
                V_A,ℓ(p,k) 
                   =2π∫_-1^1 dx P_ℓ(x) V_A(p,k,x) 
                    =
                8π^2σ/(pk)^2[ 
                 P_ℓ(y) Q'_0(y)
                 +
                 P'_ℓ(y) Q_0(y) 
                 -W'_ℓ-1(y) ] 
                
                    = -8π^2σ/(pk)^2[ 
                (2pk / p^2-k^2)^2 P_ℓ(y)
                -
                P'_ℓ(y) Q_0(y)
                +
                W'_ℓ-1(y)
                ] .
            
            In addition to the singularity in Q_0(y), that is also present in the Coulomb potential (<ref>), the first term, due to the derivative Q'_0(y), is even stronger singular and  the source of all the difficulties in this calculation.
            
            We can now write the partial-wave Schrödinger equation for the Cornell potential as
            
            
                p^2/2m_Rψ_ℓ(p)
                 -α/π p ∫_0^∞  dk   k
                 [ P_ℓ(y)Q_0(y)-W_ℓ-1(y)]
                ψ_ℓ(k) 
                
                
                 -
                 σ/π p^2P∫_0^∞ dk 
                 {4p^2 k^2/(k^2-p^2)^2[ P_ℓ(y) ψ_ℓ(k)-ψ_ℓ(p) ] 
                 -
                Q_0(y)
                 P'_ℓ(y)ψ_ℓ(k)
                 +W'_ℓ-1(y)ψ_ℓ(k)
                 }
                 = E ψ_ℓ(p)   .
            
            Next we turn our attention to the problem of solving this equation numerically.
            
            
            
             §.§ The Nyström method
            
            The Schrödinger equation in momentum space (<ref>) is a homogeneous Fredholm integral equation of the form
            
                ∫_a^b dk  K(p,k) ϕ(k)=E ϕ(p)   ,
            
            where K(p,k) is its kernel. One convenient method to solve it numerically is the Nyström method <cit.>, which is based on the numerical integration of a function f(k) by means of a quadrature rule,
            
                ∫_a^b  f(k) dk ≈∑_j=1^N w_j f(k_j)   .
            
            The quadrature points k_j and weights w_j are determined by the choice of the quadrature method and the integration interval, which in our case extends from 0 to ∞. After applying (<ref>) to (<ref>), one obtains a closed set of N linear equations if for the external variable p we choose each of the points {k_j} of the quadrature rule,
            
                ∑_j=1^N w_j K(k_i,k_j) ϕ(k_j)=E ϕ(k_i)   .
            
            The continuous variables p and k have thereby been discretized to take on only values from the same set, and they are now essentially only distinguished by their indices. From here on we will refer to the set of quadrature points as {p_i} and use the notation p_i or p_j rather than k_i or k_j.
            
            If we introduce the notation ϕ_i = ϕ(p_i) and M_ij=w_j K(p_i,p_j), the discretized Schrödinger equation can be written as a linear eigenvalue problem for the N× N matrix M_ij, 
            
                ∑_j=1^N M_ijϕ_j=E ϕ_i   ,
            
            where the eigenvector for each eigenvalue E is a set of values of the corresponding eigenfunction at the quadrature points, ϕ(p_i). 
            
            
            
             §.§ Subtraction of logarithmic singularities from the kernel
            
            Several terms in the kernel of Eq. (<ref>) are singular when k=p (and y=1). Therefore, the diagonal elements M_ii in (<ref>) don't exist and the Nyström method cannot be applied. 
            
            However, by means of a trick due to Landé <cit.>, the logarithmic singularities in the Q_0(y) terms can be eliminated.
            Taking advantage of the well-known result <cit.>
                ∫_0^∞ dk Q_0(y)/k=π^2/2  ,
            
            and using P_ℓ(1)=1, we can write
            
                ∫_0^∞ dk     k P_ℓ(y) Q_0(y) ψ_ℓ(k)  
                    = ∫_0^∞ dk [ k P_ℓ(y) Q_0(y)ψ_ℓ(k)-p^2/kQ_0(y)ψ_ℓ(p) ]
                
                   +  ∫_0^∞ dk p^2/kQ_0(y) ψ_ℓ(p)
                   =
                p ∫_0^∞ dk [ k/p P_ℓ(y) Q_0(y)ψ_ℓ(k)-p/kQ_0(y)ψ_ℓ(p) ] 
                
                   + p^2 π^2/2ψ_ℓ(p)   .
            
            The integrand in square brackets vanishes in the limit k→ p and therefore does not contribute to the diagonal of the matrix M. 
            The complete Coulomb term in the Schrödinger equation (<ref>) can now be written 
            
                -α/π p ∫_0^∞  dk   k
                 [ P_ℓ(y)Q_0(y)-W_ℓ-1(y)]
                ψ_ℓ(k) 
                
                =  -α/π p ∫_0^∞  dk  [ k   Q_ℓ(y)ψ_ℓ(k)-p^2/kQ_0(y)ψ_ℓ(p) ] 
                
                -απ p/2ψ_ℓ(p)   .
            
            For the Nyström method, we need to know the limit of the integrand of the second line of (<ref>) as k → p, which comes only from the W_ℓ-1(y) term hidden in Q_ℓ(y),
            
                lim_k→ p[k    Q_ℓ(y)ψ_ℓ(k)-p^2/kQ_0(y)ψ_ℓ(p)] = -p W_ℓ-1(1) ψ(p)   ,
            
            and it is easy to see that 
            
                W_ℓ-1(1) =  ∑_m=1^ℓ1/m  .
            
            
            Using the Landé subtraction technique, the pure Coulomb problem can therefore be solved quite easily with the Nyström method.
             
            A similar singular term as in the Coulomb potential appears also in the kernel (<ref>) of the linear potential in all partial waves with ℓ≥ 1. We can treat it with a slight variation of the same idea,
            
                ∫_0^∞  dk  Q_0(y) P'_ℓ(y)ψ_ℓ(k) 
                    = ∫_0^∞  dk [ Q_0(y) P'_ℓ(y)ψ_ℓ(k) - p/k Q_0(y) P'_ℓ(1)ψ_ℓ(p) ] 
                    +∫_0^∞  dk p/k Q_0(y) P'_ℓ(1)ψ_ℓ(p)  
                    = ∫_0^∞  dk [ Q_0(y) P'_ℓ(y)ψ_ℓ(k) - p/k Q_0(y) P'_ℓ(1)ψ_ℓ(p) ] 
                    +p π^2/2 P'_ℓ(1)ψ_ℓ(p)   ,
            
            where 
            P'_ℓ(1)=ℓ (ℓ+1)/2.
            
            Again, the integrand vanishes when k→ p and can therefore be omitted from the diagonal elements of the matrix M when the Nyström method is used. There is no need to write P∫_0^∞ in the first line of (<ref>) because this is now a regular integral.
            
            
            
             §.§ Subtraction of the principal value singularities
            
            
            There is only one singular term remaining in the kernel of the linear potential in (<ref>), but this is the most problematic one. After pulling out a factor 2p^2 for convenience, we have to deal with
            
                P∫_0^∞ dk 
                 2 k^2/(k^2-p^2)^2[ P_ℓ(y) ψ_ℓ(k)-ψ_ℓ(p) ]   .
            
            
             There appears to be a double pole at k=p, but the numerator goes to zero like (k-p),  so it is actually only a single pole, and the principal value integral exists.
             
            To see this, we expand the factor in brackets in the numerator of (<ref>) in a Taylor series around k=p,
            
                P_ℓ(y) ψ_ℓ(k)-ψ_ℓ(p) = (k-p) ψ'_ℓ(p) 
                
                 + (k-p)^2/2![  P'_ℓ(1)/p^2ψ_ℓ(p) +ψ”_ℓ(p)
                ]
                +(k-p)^3/3!  R_ℓ(k)   ,
            
            where the function R_ℓ(k) is the remainder of the Taylor series  after the terms up to second order have been subtracted and (k-p)^3/3! has been factored out. The only important property of R_ℓ(k) in this context is that it is finite at k=p. 
            Equation (<ref>) confirms that one power of k-p in the denominator of (<ref>) is canceled and that the singularity is therefore only a single pole.
            
            Substituting (<ref>) into (<ref>) shows that the integrand in the vicinity of k=p behaves like
            
                2k^2/(k^2-p^2)^2[ P_ℓ(y) ψ_ℓ(k)-ψ_ℓ(p) ] 
                
                 =
                 2k^2/k+pψ'_ℓ(p)/k^2-p^2 
                 +  k^2/(k+p)^2[  P'_ℓ(1)/p^2ψ_ℓ(p) +ψ”_ℓ(p)
                ] 
                
                +k^2(k-p)/3(k+p)^2R_ℓ(k) 
                
                 =  pψ'_ℓ(p)/k^2-p^2 +2k+p/(k+p)^2ψ'_ℓ(p)
                
                 +  k^2/(k+p)^2[  P'_ℓ(1)/p^2ψ_ℓ(p) +ψ”_ℓ(p)
                ]
                +k^2(k-p)/3(k+p)^2R_ℓ(k)   .
            
            
            
            
            For the Nyström method to work, this singularity has to be removed. 
            In <cit.> we have shown that this can be done with another Landé subtraction. The reason why we have separated the term proportional to ψ'_ℓ(p) in(<ref>) into two parts is that, when the first one is subtracted, 
                P∫_0^∞ dk 
                 2 k^2/(k^2-p^2)^2[ P_ℓ(y) ψ_ℓ(k)-ψ_ℓ(p) ] 
                =
                 ∫_0^∞ dk {2 k^2/(k^2-p^2)^2[ P_ℓ(y) ψ_ℓ(k)-ψ_ℓ(p) ] -  pψ'_ℓ(p)/k^2-p^2}
                 +p ψ'_ℓ(p)P∫_0^∞dk/k^2-p^2  ,
            
            the well-known integral
            
                P∫_0^∞dk/k^2-p^2=0
            
            takes care of the principal value singularity. The final result is an ordinary integral, free of singularities,
            
            
                P∫_0^∞ dk 
                 2 k^2/(k^2-p^2)^2[ P_ℓ(y) ψ_ℓ(k)-ψ_ℓ(p) ] 
                 = 
                 ∫_0^∞ dk {2 k^2/(k^2-p^2)^2[ P_ℓ(y) ψ_ℓ(k)-ψ_ℓ(p) ] -  pψ'_ℓ(p)/k^2-p^2}  .
            
            The price to pay for this simplification is that the derivative of the wave function makes its entrance into the integrand, which does not seem compatible with the Nyström method. In <cit.> we therefore followed a different strategy to solve the—now singularity-free—integral equation for the Cornell potential, 
            
            
                (p^2/2m_R 
                -απ p/2 
                +σπ/2p) ψ_ℓ(p)
                 -α/π p ∫_0^∞  dk  [ k   Q_ℓ(y)ψ_ℓ(k)-p^2/kQ_0(y)ψ_ℓ(p) ] 
                
                 -
                 2σ/π∫_0^∞ dk 
                 {2 k^2/(k^2-p^2)^2[ P_ℓ(y) ψ_ℓ(k)-ψ_ℓ(p) ] 
                 -  pψ'_ℓ(p)/k^2-p^2
                 -
                Q_0(y)/2p^2[ 
                 P'_ℓ(y)ψ_ℓ(k) - p/k 
                 P'_ℓ(1)ψ_ℓ(p)
                 ]
                 +W'_ℓ-1(y)/2p^2ψ_ℓ(k)
                 }
                
                 = E ψ_ℓ(p)   ,
            
            namely to expand ψ_ℓ(p) into a set of basis functions whose derivatives can be easily calculated. In <cit.> we chose a basis of modified cubic B-spline functions. 
            
            In this method, sometimes referred to as the Galerkin method, one solves for the expansion coefficients instead of the function values at grid points. The method works well and gives accurate results, but the choice of basis functions requires special care. It is especially important that the basis is able to accurately reproduce the behavior of the solutions at very small and very large momenta. This asymptotic behavior needs to be determined before the equation is solved, which is not always easy to do. It is one of the advantages of the Nyström method that no advance knowledge of the properties of the solutions is needed.
            
            
            
             §.§ The Nyström method for the subtracted kernels
            
            The purpose of this work is to show how the Nyström method can be applied to solve the integral equation (<ref>). As already mentioned before, one of the requirements is to be able to calculate the kernel along the diagonal k=p. 
            That the kernel is not singular at k=p is a necessary, but not a sufficient condition. In the case of the subtracted Coulomb potential, (<ref>) provides these matrix elements. We need to do the same for the subtracted linear potential.
            
            From (<ref>) we can read off
            
                lim_k→ p{2 k^2/(k^2-p^2)^2[ P_ℓ(y) ψ_ℓ(k)-ψ_ℓ(p) ] -  pψ'_ℓ(p)/k^2-p^2}
                
                 = 3/4ψ'_ℓ(p) + 1/4[  P'_ℓ(1)/p^2ψ_ℓ(p) +ψ”_ℓ(p)
                ]   .
            
            The Q_0(y) term of the linear potential in  (<ref>) vanishes in this limit, so only the unproblematic W'_ℓ -1(1)ψ_ℓ(p)/2p^2 has to be added to (<ref>).
            
            At this point, the situation seems to be even worse than expected from looking at (<ref>), because we need to know the first and second derivatives of the unknown function ψ_ℓ(p) to construct the complete kernel matrix along the diagonal.
            
            The new idea of this work is to calculate the derivatives of the wave function at the grid points p_i by means of interpolating functions that depend linearly on the values of the wave function at the grid points  in the vicinity of p_i. The Schrödinger equation can thereby be cast into the form (<ref>) and solved by standard numerical methods to determine the eigenvalues and eigenvectors of square matrices.
            
            
             
             
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
                
            
            
            
             
            
            
            
            
            § THE DISCRETIZED FORM OF THE MOMENTUM-SPACE SCHRÖDINGER EQUATION
             
            As already outlined in Sec. <ref>, we discretize the integrations over k according to (<ref>), rename the grid points k_j → p_j, and choose the points p on the same grid of points {p_i}. Furthermore, we introduce the abbreviations ψ_i=ψ_ℓ(p_i) and y_ij=(p_i^2+p_j^2)/(2p_ip_j). With these conventions, the Schrödinger equation (<ref>) becomes
            
            
                (p_i^2/2m_R 
                -απ p_i/2 
                +σπ/2p_i) ψ_i
                 -α/π p_i ∑_j  i w_j  [ p_j   Q_ℓ(y_ij)ψ_j-p_i^2/p_jQ_0(y_ij)ψ_i ] 
                 + α/π w_i W_ℓ-1(1)ψ_i 
                
                 -
                 2σ/π∑_j i w_j 
                 {2 p_j^2/(p_j^2-p_i^2)^2[ P_ℓ(y_ij) ψ_j-ψ_i ] 
                 -  p_iψ'_i/p_j^2-p_i^2
                 -
                Q_0(y_ij)/2p_i^2[ 
                 P'_ℓ(y_ij)ψ_j - p_i/p_j 
                 P'_ℓ(1)ψ_i
                 ]
                 +W'_ℓ-1(y_ij)/2p_i^2ψ_j
                 }
                
                 + 2σ/π w_i
                 {3/4ψ'_i + 1/4[  P'_ℓ(1)/p_i^2ψ_i +ψ”_i ] + W'_ℓ-1(1)/2p_i^2ψ_i }
                 = E ψ_i   .
            
            Note that, for both potentials, the summations over j exclude the diagonal terms j=i, whose explicit expressions are written right after the corresponding summation.
            
            As already mentioned above, we approximate now the derivatives ψ'_ℓ(p_i) and ψ”_ℓ(p_i) through the derivatives of an interpolating function, which represents ψ_ℓ(p) as a linear combination of the wave function at some subset of the grid points,
            
                ψ_ℓ(p)=∑_j ψ_ℓ(p_j) L_j(p)   ,
            
            where the functions L_j(p) are often taken to be polynomials. For instance, for Lagrange interpolation using n points { p_k } near point p, they are
            
                L_j(p)=∏_k=1 
                 k ≠ j^n p-p_k/p_j-p_k  .
            
            With this we get the derivatives at the quadrature points as 
            
                ψ'_l(p_i)=∑_j  L'_j(p_i)ψ_l(p_j)    ,     ψ”_l(p_i)=∑_j L”_j(p_i) ψ(p_j)    .
            
            How many and exactly which indices j are summed over here depends on the order of the interpolating polynomials.
            It seems best to choose the interpolation points p_k to be the closest quadrature points to the left and right of p_i. For instance, for a 5-point Lagrange interpolation, the p_k should be
            p_i-2,p_i-1,p_i,p_i+1,p_i+2. Exceptions must be made for the points close to the ends, where not enough interpolation points to the left or right are available.
            
            
            
            Equation (<ref>) can now be written as an eigenvalue equation of the form M ψ = E ψ, where ψ is a column vector with elements ψ_i ≡ψ_ℓ(p_i), and M is a matrix (with elements M_ij). For the derivatives we write  
            
                ψ'_i = ∑_j D^(1)_ijψ_j   ,     ψ”_i = ∑_j D^(2)_ijψ_j   ,
            
            where the matrices D^(1)_ij=L'_j(p_i) and D^(2)_ij=L”_j(p_i) depend only on the grid points close to p_i. Other interpolation methods can of course be used as well, such as spline interpolation. This will lead to different matrices D^(1)_ij and D^(2)_ij, but the general form (<ref>) will stay the same.
            
            It is convenient to slightly reorganize (<ref>). The subtraction terms in the summations over j contain ψ_i and other factors that don't depend on the summation index and can be pulled out, leaving the following sums: 
            
                S_i^(1)=∑_j ≠ i  w_j 2p_j^2/(p_j^2-p_i^2)^2  ,     
                S_i^(2)=∑_j ≠ iw_j/p_j^2-p_i^2  ,     
                S_i^(3)=∑_j ≠ iw_j/p_j Q_0(y_ij)   .
            
            Then we get
            
                {p_i^2/2m_R 
                +α p_i/π( S_i^(3) -π^2/2 + w_i W_ℓ-1(1)/p_i)
                +2σ/π[
                 π^2/4p_i 
                -P'_ℓ(1)/2 p_i  S_i^(3) 
                + w_i/2p_i^2(   P'_ℓ(1)/2  + W'_ℓ-1(1) )
                +S_i^(1)]
                }ψ_i
                
                
                 -α/π p_i ∑_j  i w_j    
                  p_j   Q_ℓ(y_ij)ψ_j 
                
                 -
                 2σ/π∑_j i w_j 
                 [
                 2 p_j^2/(p_j^2-p_i^2)^2
                 P_ℓ(y_ij) 
                 -
                Q_0(y_ij)/2p_i^2
                 P'_ℓ(y_ij) 
                 +W'_ℓ-1(y_ij)/2p_i^2] ψ_j 
                 
                
                  + 2σ/π∑_j 
                 [
                 ( p_i S_i^(2)+3w_i/4) D_ij^(1)+
                 w_i/4  D_ij^(2)]
                 ψ_j
                 = E ψ_i   .
            
            Equation (<ref>) is organized in such a way that all diagonal kernel elements appear in the first line, the off-diagonal elements in the second line, and in the third line the contributions from the derivatives of the wave function, which are usually contained in a more or less narrow band along the diagonal. 
            
            The form of this equation is
            
                ∑_j=1^N M_ijψ_j = E ψ_i   ,
            
            and we can read off the elements of the matrix M from (<ref>):
            
                M_ij=
                δ_ij{p_i^2/2m_R 
                +α p_i/π( S_i^(3) -π^2/2 + w_i W_ℓ-1(1)/p_i)
                +2σ/π[
                 π^2/4p_i 
                -P'_ℓ(1)/2 p_i  S_i^(3) 
                + w_i/2p_i^2(   P'_ℓ(1)/2  + W'_ℓ-1(1) )
                +S_i^(1)]
                }
                
                +(1-δ_ij)w_j{
                 -α/π p_i    
                  p_j   Q_ℓ(y_ij) 
                
                 -
                 2σ/π[
                 2 p_j^2/(p_j^2-p_i^2)^2
                 P_ℓ(y_ij) 
                 -
                Q_0(y_ij)/2p_i^2
                 P'_ℓ(y_ij) 
                 +W'_ℓ-1(y_ij)/2p_i^2]  
                 }
                
                  + 2σ/π[
                 ( p_i S_i^(2)+3w_i/4) D_ij^(1)+
                 w_i/4  D_ij^(2)]
                   .
            
            
            § NUMERICAL RESULTS
            
            In this section we present the results of calculations we performed for the linear confining part of the Cornell potential with the Nyström method, by numerically determining the eigenvalues and eigenvectors of the matrix M of (<ref>), where we have set α=0. 
            
            The reason for this restriction is simply that the Coulomb problem has already been solved with the Nyström method in the past. We have of course also solved the pure Coulomb problem as a test case, but we do not display here any results because there are no new insights to be gained. Suffice it to say that we reproduce the exactly known energy levels and wave functions with excellent accuracy.
            
            For the quadrature rule (<ref>), we start
            with Gauss-Legendre quadrature points { x_i} and weights { w_x,i} for the interval x ∈ [-1,+1], such that
            
                ∫_-1^+1 g(x) dx ≈∑_i=1^N w_x,i  g(x_i)   ,
            
            and then map from the interval [-1,+1] to [0,∞) with the transformation 
            
                p(x)=p_0 1+x/1-x  ,
            
            such that 
            
                p_i = p_0 1+x_i/1-x_i   ,      w_i =  p_0 2 w_x,i/(1-x_i)^2  ,
            
            where p_0>0 is a scaling parameter.
            
            The Hamiltonian contains two parameters, namely the string tension σ of the linear potential and the reduced mass m_R of the 2-body system. Because the energy eigenvalues
            scale with (σ^2/2m_R)^1/3, it is sufficient to solve Eq. (<ref>) for σ = 2m_R = 1.
            
            The linear potential in S-waves is the ideal case to test our numerical methods, because its exact solutions in coordinate space are known in terms of the Airy functions Ai, with the corresponding energies given by
            
                E_n^ℓ=0=-z_n(σ^2/2m_R)^1/3 , with   Ai (z_n)=0 ,
            
            where z_n<0 is the n-th root of the Airy function Ai(z).
             
            First we demonstrate for the linear potential the numerical convergence of the energy eigenvalues to the exact values as the number of Gauss-Legendre points N increases. We calculated the lowest energy eigenvalues n=1 to n=10 using 5-point Lagrange interpolation to approximate the first and second derivatives of the wave function. Table <ref> shows that our numerical S-wave energies converge quickly to the exact eigenvalues. For N=1000 Gauss-Legendre points we find excellent agreement up to the 9th decimal place for the lowest eigenvalues, while for the highest eigenvalues we still reach very good agreement up to the 7th decimal place. 
            
            
            In Table <ref> the S-wave results for the energy eigenvalues n=1 to n=10, obtained with four different methods, are compared. For the Nyström methods, 1000 Gauss-Legendre integration points were used, while for the Galerkin method with expansion in a basis of 64 B-spline functions the numerical integration was performed using the  NIntegrate” routine in Mathematica. Excellent agreement with the exact results was achieved with 5-point and higher Lagrange interpolation, whereas the B-spline method with 64 basis functions still yields accurate results, but clearly less than what is achieved with the Nyström methods.    
            
            The higher-order Lagrange interpolation results turned out to be so precise that we wanted to see how high their accuracy can go when the number of integration points and the order of the interpolating polynomials is increased. In Tables <ref>, <ref>, and <ref>, we show the differences between the numerical results and the exact energies given by Eq. (<ref>) with higher precision. The exact energies are given up to 20 decimal places, together with the difference between the numerical and the exact results.
            
            In Tab. <ref> we see that, for N=200 integration points, the accuracy increases with the interpolation order up to about 11-point interpolation, then it drops again to levels similar to 3-point interpolation, except that the accuracy of the higher states remains somewhat higher. Similarly, Tab. <ref> for N=600 shows that higher-order interpolations improve the accuracy further until a drop that happens at a higher order. For N=1000 in  Tab. <ref>, a clear drop is no longer discernible, and an impressive accuracy is achieved even in the highly excited states.
            
            We can conclude that, as expected, increasing the number of integration points improves the accuracy of the results. So does increasing the order of the Lagrange interpolation polynomial. However, if the number of integration points is low one should not choose a high-order interpolation. The plausible explanation for this behaviour is that high-order interpolating poynomials tend to oscillate between the node points, such that the derivatives become unreliable. Increasing the number of nodes (here the integration points) reduces their relative distance and lowers this tendency, such that higher-order interpolation actually improves the results.
            
            It should be mentioned that the calculations for Tables <ref> to <ref> were performed with Matematica with 30-digit precision. In normal machine-precision (16 digits), the roundoff errors accumulate too much to allow such high accuracy. In applications to actual physical problems, like calculations of the meson spectra with the Cornell potential, a much lower accuracy is of course sufficient.
            
            For higher partial waves, with ℓ > 0, no exact solutions are available. However, given the extreme accuracy of our Nyström method, we list in, Tab. <ref>, the converged energies, up to 10 decimal places, for angular momenta up to ℓ=5, calculated with N=1000 and 15-point Lagrange interpolation.
            Partial waves like P-, D-, and F-waves with ℓ=1, 2, 3, respectively, are of practical importance as they contribute to the wave functions of scalar, vector, axial-vector, and tensor mesons. 
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            § SUMMARY AND CONCLUSIONS
            
            In this work, we propose an accurate and simple method for solving the  Schrödinger equation with a linear confining potential in momentum space. The linear potential gives rise to a Cauchy principal-value singularity and a logarithmic singularity which complicate the direct application of standard numerical solution techniques, such as the Nyström collocation method. In a preceding paper <cit.>, we have showed how both, the principal-value and logarithmic singularity can be subtracted making the Schrödinger equation  singularity free. However, as a consequence of the removal of the principal singularity, the derivatives of the – a priori – unknown wave function appear in the equation. One way to deal with this difficulty is the so-called Galerkin method, i.e. the expansion of the wave function in a basis of appropriate functions whose derivatives are known or can be calculated. Then the problem becomes a generalized eigenvalue problem for the expansion coefficients that can be solved and the energy eigenvalues and eigenfunctions are obtained. In Ref. <cit.> we followed this approach and solved the singularty-free Schrödinger equation by means of an expansion in a basis of cubic B-splines. We found that the solutions converge quickly with increasing number of basis functions, especially, for the lowest energy eigenvalues. For the higher energy eigenstates, however, a rather large number of basis functions was needed to find good agreement with the exactly known values. A large number of basis functions made the numerical computations also time consuming and is therefore not very practicable for performing global fits in the future that involve many states of the meson spectrum. But the main drawback of the expansion in B-splines is, that the appropriate basis functions must be chosen for each problem individually. For instance, if the basis functions do not have the same asymptotic behavior as the wave function determined by the potential in the Schrödinger equation, then this can lead to problems with convergence. These issues can be resolved by modifying the basis functions accordingly in order to reproduce the correct behavior of the wave function. But because the basis functions also depend on the fitting parameters of the problem, this makes the method is often impractical and sometimes unreliable.  
            
            Our objective here was to develop a simpler, more reliable and yet sufficiently general numerical method that can in principle be applied to any type of realistic bound-state problem. In this work, we demonstrated that the Nyström collocation method can be used to solve the Schrödinger equation with the linear confining potential after the principal singularity has been subtracted accordingly. The first and second derivatives of the wave function, that enter the equation, can be re-expressed in terms of the wave function itself by means of an expansion in interpolating polynomials. The problem then reduces to an ordinary eigenvalue problem that can be easily solved.
            
            
            
            We tested different interpolation methods and already with a 5-point Lagrange interpolation with 600 Gauss-Legendre integration points very good agreement with the exactly known S-wave values was achieved. With 1000 we reached excellent agreement up to the 9th decimal place for the lowest energy eigenvalues and for the highest energy eigenvalues agreement up to the 7th decimal place. Even though, the computations were performed in  Mathematica programming language, the computing times were reasonably short (orders of several seconds) such that this method is suitable for near-future global fits of the meson spectrum. For comparison, for the cubic B-spline method with 64 basis functions, the computing time was by a factor of 1000 longer and at the same time the results were significantly less accurate.
            In addition, we found that by further increasing the number of integration points and the order of the Lagrange interpolation, extreme accuracy can be achieved, which requires to go beyond normal machine precision in the numerical computations. With 1000 integration points and a 13-point Lagrange interpolation the differences between the numerical results obtained from a calculation in  Mathematica with 30-digit precision and the exact values are of the order 10^-15 or smaller. For higher partial waves, for which there exist no exact values, we list the converged energies, up to 10 decimal places, for angular momenta up to ℓ=5, calculated with N=1000 and 15-point Lagrange interpolation. Partial waves like P-, D-, and F-waves are of practical importance as they contribute to the wave functions of scalar, vector, axial-vector, and tensor mesons. 
            
            
            
            
            
            This work was supported by FCT under project No. CERN/FIS-PAR/0023/2021. E.B. was also supported by FCT and IST-ID under contract No. IST-ID/148/2018.  
            
            
            § APPROXIMATE DERIVATIVES WITH LAGRANGE INTERPOLATION
            
            In this appendix, we list the explicit expressions for the matrices that determine the approximate first and second derivatives of the wave function at the grid points.
            
            
             §.§ Construction of the Lagrange polynomial
            
            
            We start with a set of N points {p_j}, where  j=1,…,N, and the respective values of a function f(p) at these points, { f_j }, where we abreviate f_j ≡ f(p_j). 
            The Lagrange interpolating polynomial is the unique polynomial L(p)  of order N-1 that passes exactly through these given points (p_j,f_j), i.e.,
            
                L(p_j)=f(p_j)  ,    j=1,…,N   .
            
            It can be written in the form
            
                L(p)=∑_j=1^N f_j L_j(p)   ,
            
            with
            
                L_j(p) = ∏_k=1 
                 k j^Np-p_k/p_j-p_k
            
            To find the first derivative of the interpolating polynomial,
            
                L'(p)=∑_j=1^N f_j L'_j(p)   ,
            
             we only need to find L'_j(p). Applying the rule for the derivative of a product, we get
            
                L'_j(p)= ∑_m =1 
                m j^N 1/p_j-p_m∏_k=1
                k j 
                 k  m^N p-p_k/p_j-p_k  .
            
            Similarly, we get the second derivative through
            
                L”(p)=∑_j=1^N f_j L”_j(p)   ,
            
            and
            
                L”_j(p)= ∑_m =1 
                m j^N 1/p_j-p_m∑_l =1 
                l j 
                l m^N 1/p_j-p_l∏_k=1
                k j 
                 k  l 
                 k  m ^N p-p_k/p_j-p_k  .
            
            
             §.§ Derivatives at the nodes
            
            An important application of the Lagrange polynomials is the numerical approximation of the derivatives of the unknown function f(p) that is only known at a list of points. The derivatives at arbitrary points can be calculated from the expressions (<ref>)-(<ref>) and (<ref>)-(<ref>), respectively. If the derivatives are needed only at the nodes themselves, this can be somewhat simplified.
            
            First, note that for any specific node p_i, which is one of the set  {p_j}, we get
            
                L_j(p_i) = ∏_k=1 
                 k j^Np_i-p_k/p_j-p_k=δ_ij  ,
            
            because if i  j, i will coincide with one of the k's and therefore the product is zero because it must include a factor p_i-p_i, whereas each factor is 1 if i=j. Inserting in (<ref>), this also implies
            
                L(p_i)=∑_j=1^N f_j L_j(p_i)=∑_j=1^N f_j δ_ij =f_i   ,
            
            as it must.
            
            Next, we calculate (<ref>) for p=p_i. 
            
                L'(p_i)=∑_j=1^N f_j L'_j(p_i)   ,
            
                L'_j(p_i)= ∑_m =1 
                m j^N 1/p_j-p_m∏_k=1
                k j 
                 k  m ^N p_i-p_k/p_j-p_k  ,
            
                L”(p_i)=∑_j=1^N f_j L”_j(p_i)   ,
            
                L”_j(p_i)= ∑_m =1 
                m j^N 1/p_j-p_m∑_l =1 
                l j 
                l m^N 1/p_j-p_l∏_k=1
                k j 
                 k  l 
                 k  m ^N p_i-p_k/p_j-p_k  .
            
            Note that m=i and l=i are not excluded from the sums, and that k=i gives 0 in the product.
            
            
            
             §.§ Interpolation on a subset of points
            
            Usually we have a table of n>N points and first need to select a subset of points on which to perform the N-point Lagrange interpolation. The result of the interpolated function and its derivatives will depend on the selected subset, and clearly different reasonable choices are possible.
            
            We adopted an algorithm for the selection of the subset of points for the case that we want to calculate the derivatives at one of the n given points p_i. We assume that the closest neighbouring points should selected, i.e., we need to determine the offset s, such that the interpolation will use all N points between j=s+1 and j=s+N:
            
                L(p_i)     =∑_j=s+1^s+N f_j L_j(p_i)=f_i   , 
                
                L'(p_i)     =∑_j=s+1^s+N f_j L'_j(p_i)   , 
                
                L'_j(p_i)     = ∑_m =s+1 
                m j^s+N1/p_j-p_m∏_k=s+1
                k j 
                 k  m ^s+Np_i-p_k/p_j-p_k  , 
                
                L”(p_i)     =∑_j=s+1^s+N f_j L”_j(p_i)   , 
                
                L”_j(p_i)     = ∑_m =s+1 
                m j^s+N1/p_j-p_m∑_l =s+1 
                l j 
                l m^s+N1/p_j-p_l∏_k=s+1
                k j 
                 k  l 
                 k  m ^s+Np_i-p_k/p_j-p_k  .
            
            With our map (<ref>), the points are distributed such that the distance between them increases as one moves to larger values. In this situation, it makes sense to take an equal number of points to the left and to the right of i if N is odd, but one more point to the left if N is even.
            
            Our algorithm for determining s is:
            
              * For odd N
              (i)If i< N+1/2:s=0
              (ii)If i>n-N-1/2:s=n-N
              (iii)Else:s=i-N+1/2
            
              * For even N
              (i)If i< N+2/2:s=0
              (ii)If i>n-N-2/2:s=n-N
              (iii)Else:s=i-N+2/2.


 Paper Title: Fully strange tetraquark resonant states as the cousins of $X(6900)$.













yaoma@pku.edu.cn
School of Physics and Center of High Energy Physics, Peking University, Beijing 100871, China

wlwu@pku.edu.cn
School of Physics, Peking University, Beijing 100871, China

lu.meng@rub.de
Institut für Theoretische Physik II, Ruhr-Universität Bochum,  D-44780 Bochum, Germany 

chenyanke@stu.pku.edu.cn
School of Physics, Peking University, Beijing 100871, China

zhusl@pku.edu.cn
School of Physics and Center of High Energy Physics, Peking University, Beijing 100871, China


§ ABSTRACT 

We conduct systematic calculations of the S-wave fully strange systems with “normal" (J^P C=0^++, 1^+-, 2^++) and “exotic" (J^P C=0^+-, 1^++, 2^+-) C-parities, which are the 
strange analogue of the fully charmed tetraquark state X(6900).
Within a constituent quark potential model, we employ the Gaussian expansion method to solve the four-body Schrödinger equation and the complex scaling method to identify resonant states. We obtain a series of resonant states and zero-width states in the mass range of 2.7 to 3.3 GeV, with their widths ranging from less than 1 MeV to about 50 MeV. Their rms radii strongly indicate that they are compact tetraquark states. Among these states, the T_4s,2^++(2714) may be the most likely one to be observed experimentally. We urge the experimental exploration of the 2^++ s s s̅s̅ state around 2.7 GeV in the ϕϕ channel. Since the lowest S-wave s s s̅s̅ state is around 2.7 GeV, the compact P-wave s s s̅s̅ states are expected to be heavier. Hence, ϕ(2170) and X(2370) are unlikely to be compact tetraquark states.





Fully strange tetraquark resonant states as the cousins of X(6900)
    Shi-Lin Zhu 0000-0002-4055-6906
    August 5, 2024
==================================================================






§ INTRODUCTION
 

Since 2003, the interest in multiquark states has been reignited by the experimental discovery of a series of candidates (for recent reviews, see Refs. <cit.>). These discoveries have significantly enriched the hadron spectrum. The study of their properties and structures enhances our understanding of the non-perturbative nature of the strong interaction.

Inspired by their many strangeness decays, there are several promising candidates for the fully strange tetraquark states. 


 
  * The state ϕ(2170) (was Y(2175)) was first reported in 2006 by the BaBar Collaboration in the process of e^+ e^-→ϕ(1020) f_0(980) <cit.>. Later it was confirmed by the BES and BESIII Collaborations in the J / ψ→ηϕ f_0(980) process <cit.>, and the Belle Collaboration in the e^+ e^-→ϕπ^+π^-and e^+ e^-→ϕ f_0(980) processes <cit.>. 
 
 
  * The state X(2370) was first observed in 2010 by the BESIII Collaboration in the π^+π^-η^' invariant-mass spectrum <cit.>. Later it was confirmed in 2019 by the BESIII Collaboration in the K K̅η^' invariant-mass distribution <cit.>, which suggests that the X(2370) likely contains four valence s or s̅ quarks.

 
  * The pseudoscalar state X(2500) was first observed in 2016 by the BES Collaboration in the process J / ψ→γϕϕ <cit.>, together with the scalar state f_0(2100), and three tensor states, the f_2(2010), f_2(2300) and f_2(2340).

 
  * The X(2239) was seen in 2018 by the BESIII Collaboration in the cross section of e^+ e^-→ K^+ K^- <cit.>.
 
 
  * An evidence for a structure X(2100) in the ϕη^' mass spectrum of the J / ψ→ϕηη^' decay was reported in 2019 by the BESIII Collaboration <cit.>.

 
  * Another structure in the ϕ f_0 cross section around 2.4 GeV, labelled as X(2436) <cit.>, was reported in 2007 by the BaBar Collaboration <cit.>. It is also seen by the BES Collaboration <cit.> and Belle Collaboration <cit.>, and evidenced by the BESIII Collaboration <cit.>. However, the latest BESIII  results indicate that its statistical significance is no more than 2σ <cit.>.


Among the aforementioned possible T_s s s̅s̅ candidates, all the established ones have negative parity. The existence of the positive parity states remains uncertain. Additionally, the LHCb Collaboration recently discovered a fully charmed tetraquark candidate X(6900) <cit.>, which was later confirmed by the CMS and ATLAS <cit.>. 
The QCD Lagrangian leads to the same quark gluon interaction if one omits the quark mass difference. The discovery of X(6900) implies the possible existence of the strange analogue T_s s s̅s̅. 

Inspired by these experimental observations, various theoretical investigations have been conducted on the fully strange tetraquark systems. Regarding the ϕ(2170) state with J^P C=1^–, some authors used the QCD sum rule <cit.>, quark model <cit.> and diquark-antidiquark scheme <cit.> to interpret it as an s s s̅s̅ tetraquark state, although other interpretations were not  excluded <cit.>. Similarly, the X(2370) state was also explained as an s s s̅s̅ tetraquark state with 0^-+ <cit.>. The pseudoscalar state X(2500) was interpreted as a P-wave s s s̅s̅ tetraquark state of J^P C=0^-+ <cit.>. In Ref. <cit.>, the X(2500), X(2239), f_2(2300) were assigned as 0^-+, 1^–, 2^++ s s s̅s̅ tetraquark states, respectively. Moreover, the structure X(2100) was interpreted as a 1^+- s s s̅s̅ state <cit.>. In Ref. <cit.>, the resonant states X(2100) and X(2239) were treated as the axial-vector and vector tetraquarks with the quark content s s s̅s̅, respectively. The structure X(2436) was explained as a P-wave s s s̅s̅ tetraquark state of J^P C=1^– <cit.>. The s s s̅s̅ tetraquark states with other quantum numbers were also expolred  <cit.>.

So far, there is no experimental conclusion regarding the existence of positive parity s s s̅s̅ states, and theoretical investigations on these states are also limited. Although some theoretical calculations on the fully strange tetraquark system in quark models have been conducted <cit.>, they are relatively sparse and less refined compared to those on tetraquark systems containing heavy quarks. The studies in Refs. <cit.> focus exclusively on the diquark-antidiquark picture, obtaining only the states above the dimeson thresholds. These states would become scattering states or resonant states once the dimeson configuration is incorporated. Only the resonant states have physical significance and should be identified using nontrivial approaches. In Refs. <cit.>, the s s s̅s̅ tetraquark resonant states were investigated via the real scaling method. This method detects resonant states by avoiding the cross pattern of energy levels across a parameter controlling the spatial extension of the wave function, which is usually quite rough. 

To address these issues, we will conduct systematic calculations of the positive parity S-wave fully strange system. We will employ the Gaussian expansion method (GEM) <cit.> to solve the four-body Schrödinger equation within a nonrelativistic constituent quark potential model. Additionally, we will adopt the complex scaling method (CSM) <cit.> to identify genuine resonant states from meson-meson scattering states. In our recent works, we have incorporated both the dimeson and diquark-antidiquark configurations to carry out benchmark calculations for tetraquark bound states <cit.> and resonant states <cit.>. We will continue to include both configurations for the fully strange tetraquark system. In this study, we will investigate both the “normal" C-parity states (J^PC=0^++, 1^+-, 2^++) and the “exotic" C-parity states (J^PC=0^+-, 1^++, 2^+-), with the “exotic” C-parity systems referring to those that cannot be composed of two S-wave ground quarkonia.

This paper is arranged as follows. 
In Sec. <ref>, we introduce the theoretical framework, including the Hamiltonian in the constituent quark potential, the construction of the wave function, the complex scaling method, and the approach to analyzing the spatial structures.
In Sec. <ref>, we present the numerical results for the properties of the fully strange resonant states.
Finally, we give a brief summary and discussion in Sec. <ref>.




§ THEORETICAL FRAMEWORK
 



 §.§ Hamiltonian
 

The nonrelativistic Hamiltonian of a four-quark system reads

    H=∑_i^4(m_i+p_i^2/2m_i)+∑_i<j=1^4 V_i j ,

where m_i and p_i are the mass and momentum of quark i.

In this study, we will employ the AL1 constituent quark potential model <cit.> containing the one-gluon-exchange (OGE) interaction and a linear confinement term.
The AL1 potential reads 

    V_i j=-3/16   λ_i^c ·λ_j^c(-κ/r_ij+λ r_i j-Λ
       +8πκ'/3m_im_jexp(-r_ij^2/r_0^2)/π^3/2r_0^3s_i·s_j),






where r_0=A(2m_im_j/m_i+m_j)^-B. λ^c represents the SU(3) color Gell-Mann matrix and s_i is the spin operator of quark i. The parameters of the potential have been determined by fitting the meson spectra in Ref. <cit.> and are listed in Table <ref>. The calculated meson masses in AL1 model are shown in Table <ref>. 





















 §.§ Wave function construction
 

The wave function bases of the sss̅s̅ tetraquark states can be expressed as the direct product of spatial wave function ϕ, spin wave function χ_s, and color wave function χ_c.

    ψ=𝒜(ϕ⊗χ_s⊗χ_c),

where 𝒜 is the antisymmetrization operator, representing the exchange of identical quarks. For the sss̅s̅ system, 𝒜=(1-P_12)(1-P_34), where P_i j permutes the ith and jth (anti)quarks. 


For the spatial wave function, the Gaussian expansion method (GEM) <cit.> is employed. Namely, the spatial wave function is expanded using the following basis:

    ϕ_n l m(r)=√(2^l+5 / 2/Γ(l+3/2) r_n^3)(r/r_n)^l e^-r^2/r_n^2 Y_l m(r̂),

where the r_n is taken in geometric progression, r_n=r_0 a^n-1. Y_l m is the spherical harmonics.

For a four-body system, one can work on the Jacobi coordinates to exclude the center-of-mass motion. In principle, one could construct a complete basis functions using any type of Jacobi coordinates. However, to get rid of the complexity from the angular momentum, we use only S-wave bases constructed in different types of Jacobi coordinates to partially compensate for higher partial waves (see Ref. <cit.> for details). In this work, we include two types of Jacobi coordinates, the diquark-antidiquark structure and the dimeson structures, as shown in Fig. <ref>.


In our calculation, we choose n_max=12 bases for each coodinate, and the settings of the basis parameters are 

    {[ r_0=0.4 fm, r_n_max=2.0 fm              s-s  or s̅-s̅; r_0=0.4 fm, r_n_max=2.0 fm               (s s)-(s̅s̅); r_0=0.4 fm, r_n_max=1.3 fm                       s-s̅; r_0=0.5 fm, r_n_max=4.5 fm              (s s̅)-(s s̅) ],.

where we use relatively large r_n_max for the coordinate between two s s̅ clusters to better describe the meson-meson scattering states and possible molecular states.

For the spin wave function, the bases for total spin S=0,1,2 are

    S=0:{[ [(s_1 s_2)_0(s̅_3 s̅_4)_0]_0; [(s_1 s_2)_1(s̅_3 s̅_4)_1]_0 ],. 
        S=1:{[ [(s_1 s_2)_0(s̅_3 s̅_4)_1]_1; [(s_1 s_2)_1(s̅_3 s̅_4)_0]_1; [(s_1 s_2)_1(s̅_3 s̅_4)_1]_1 ],. 
        S=2:[ [(s_1 s_2)_1(s̅_3 s̅_4)_1]_2 ].


Two optional color wave function bases sets are

    color-I: {[ [(s_1 s_2)_3̅(s̅_3 s̅_4)_3]_1;  [(s_1 s_2)_6(s̅_3 s̅_4)_6]_1 ],.


    color-II: {[ [(s_1 s̅_3)_1(s_2 s̅_4)_1]_1; [(s_1 s̅_3)_8(s_2 s̅_4)_8]_1 ]..

Since both sets are orthogonal and complete, using either set will yield exactly the same results <cit.>.

Besides the antisymmetrization operation in Eq. (<ref>), since the s s s̅s̅ system carries definite C-parity, it is necessary to add (or subtract) the C-transformation of each term in Eq. (<ref>) to ensure that the overall wave function has the desired positive (or negative) C-parity. The specific C-transformation is

    [(q_1 q_2^')^s_1_c̅(q̅_3 q̅_4^')_c^s_2]^S_1_cϕ(r_1, r_2, r_3, r_4)
       .(-1)^S-s_1-s_2[(q_1 q_2^'))_c̅^s_2(q̅_3 q̅_4^')_c^s_1]^S_1_cϕ(r_3,r_4,r_1,r_2)

for the bases with color-I set, and

    [(q_1 q̅_3)_c^s_1(q_2^'q̅_4^')_c^s_2]^S_1_cϕ(r_1, r_2, r_3, r_4)
       (-1)^s_1+s_2[(q_1 q̅_3)_c^s_1(q_2^'q̅_4^')_c^s_2]^S_1_cϕ(r_3,r_4,r_1,r_2)

for the bases with color-II set. The particle q and q' can be different quarks, although they both represent the same s quark in the s s s̅s̅ system.




 §.§ Complex scaling method
 

The complex scaling method (CSM) is a direct approach to obtain the energies and the decay widths of resonant states in a many-body system by performing an analytical continuation of the Schrödinger equation <cit.>. This is achieved by carrying out a complex rotation on the coordinate r and momentum p, given by

    U(θ) r=r e^i θ,    U(θ) p=p e^-i θ.

Under the rotation, the Hamiltonian in Eq. (<ref>) becomes

    H(θ)=∑_i=1^4(m_i+p_i^2 e^-2 i θ/2 m_i)+∑_i<j=1^4 V_i j(r_i j e^i θ).

Meanwhile, for the resonant states with pole positions within the range of the rotated angle, their wave functions become normalizable by integration, thereby solvable through localized Gaussian bases in the same way as bound states. As a result, solving the complex-scaled Schrödinger equation will simultaneously yield the eigenenergies of bound states and resonant states within the rotated angle.

A typical pattern of the solved eigenenergies in the complex energy plane is: The bound states lie on the negative real axis of the energy plane. The continuum states align along beams originating from thresholds with Arg(E)=-2 θ. The resonant states with mass M_R and width Γ_R are located at E_R=M_R-i Γ_R / 2, and only those within |Arg(E_R)|<2 θ can be solved. The positions of the bound and resonant states remain unchanged with the variation of the rotation angle.  One can find more details in Refs. <cit.>.



 §.§ Spatial structure


The root-mean-square (rms) radius is a good physical quantity for reflecting the spatial structure of the tetraquark states, and a widely used criterion for distinguishing between compact and molecular tetraquark states.

The conventional definition of the rms radius under CSM is 

    r_i j^rms,C≡Re[√((Ψ(θ)|r_i j^2 e^2 i θ| Ψ(θ))/(Ψ(θ) |Ψ(θ)))],

where the Ψ(θ) is the obtained complex wave function of the tetraquark state. The round bra-ket represents the so-called c-product <cit.> defined as

    (ϕ_n |ϕ_m) ≡∫ϕ_n(r) ϕ_m(r) d^3r,

without taking complex conjugate of the bra-state. This procedure ensures the function inside the integral is analytic, thereby the expectation value of the physical quantity remains stable as the rotation angle changes. The rms radius calculated from the c-product is generally not real; however, its real part can still reflect the internal quark clustering behavior if the resonant state is not too broad, as discussed in Ref. <cit.>.

Nevertheless, for tetraquark states containing identical quarks, this definition may fail to identify the molecular nature of their spatial structure due to the antisymmetrization of identical quarks, as discussed in Ref. <cit.>. To eliminate this ambiguity, we have proposed a new approach to calculate the rms radii of the fully heavy Q Q Q̅Q̅ system in Ref. <cit.>, which can also be applied to the s s s̅s̅ system. In this new approach, the total wave function is non-orthogonally decomposed into

    Ψ(θ) =     ∑_s_1 ≤ s_2[[(q_1 q̅_3^')_1_c^s_1(q_2 q̅_4^')_1_c^s_2]_1_c^S ϕ(r_1, r_2, r_3, r_4;θ).
        - [(q_2 q̅_3^')_1_c^s_1(q_1 q̅_4^')_1_c^s_2]_1_c^S ϕ(r_2, r_1, r_3, r_4;θ) 
        - [(q_1 q̅_4^')_1_c^s_1(q_2 q̅_3^')_1_c^s_2]_1_c^S ϕ(r_1, r_2, r_4, r_3;θ)
        + .[(q_2 q̅_4^')_1_c^s_1(q_1 q̅_3^')_1_c^s_2]_1_c^S ϕ(r_2, r_1, r_4, r_3;θ)]
    
    =    𝒜[ ∑_s_1 ≤ s_2[(q_1 q̅_3^')_1_c^s_1(q_2 q̅_4^')_1_c^s_2]_1_c^S ϕ(r_1, r_2, r_3, r_4;θ)]
    ≡   𝒜Ψ_13,24(θ),

where s_1, s_2 sum over all spin configurations with total spin S. The particle q and q' can be different quarks, although they both represent s quark in the s s s̅s̅ system. We use the decomposed wave function Ψ_13,24(θ) to define the new rms radius

    r_ij^rms,M≡Re[√((Ψ_13,24(θ)|r_i j^2 e^2 i θ| Ψ_13,24(θ))/(Ψ_13,24(θ) |Ψ_13,24(θ)))] .

This newly defined rms radius can better describe the spatial structure of the molecular-type tetraquark states containing identical quarks. We will provide the results for both definitions of the rms radius in our calculation.




§ NUMERICAL RESULTS
 



 §.§ States with “normal" C-parity


The “normal” C-parity s s s̅s̅ systems refer to those that can be composed of two S-wave ground quarkonia, namely, the  systems with J^PC=0^++,1^+-,2^++. The complex eigenenergies yielded are shown in the upper panel of Fig. <ref>. We choose different complex scaling angles θ=9^∘,12^∘,15^∘ to distinguish bound states and resonant states, which remain stationary as the angle changes, from the continuum states. The markers that align along beams originating from meson-meson thresholds with Arg(E) = -2 θ are the continuum state eigenenergies. The results show that no bound state is found below any of the thresholds, while a series of resonant states are obtained and marked out by black circles. Their complex energies, proportions of different color configurations, and two kinds of rms radii are summarized in Table <ref>. For convenience, we label the tetraquark resonant states obtained in our calculations as T_4s, J^PC(M), where M is the mass of the state.










The masses of the resonant states are in the region (2.7,3.3) GeV, and their widths range from a few MeV to several tens of MeV. For the resonant states above the 1S2S di-meson threshold, our calculation actually underestimates their widths because the decay width of the 2S meson has not been accounted for. The rms radii of all the obtained states approximately match the sizes of the 2S mesons. Furthermore, the newly defined r_ij^rms,M does not exhibit any molecular characteristics, indicating that all these states are compact tetraquark states.



In tetraquark systems, there are abundant dimeson thresholds. Each of these thresholds represents a branch cut connecting two different Riemann sheets. In our calculations, we must choose appropriate rotation angle to make the pole of interest emerge. Meanwhile, due to numerical reasons, the pole close to the continuum solutions will be affected by them, as illustrated in Refs. <cit.>. Therefore, to reduce uncertainties, the rotation angle should be chosen such that the continuum states are not too close to the resonant state. In practices, we adjust the angle to make each pole and its rms radii stable. 






Comparing the fully strange systems with ccc̅c̅ using the AL1 model in our previous work <cit.>, we observe a similar overall pattern. However, the fully strange systems exhibit an additional quasi-bound state with 2^++ below the ϕϕ(2S) threshold. This difference may stem from the mass-dependent color-magnetic term in the potential.

For the 0^++ states, the five resonant states can all decay into η'η', ϕϕ, η'η'(2S), ϕϕ(2S) final states, although there may be some effects if the mixing between the I=0 η(nn̅) and η'(ss̅) is taken into account. The T_4s,0^++(2917) and T_4s,0^++(3133) have an additional decay channel into η'η'(3S). The higher T_4s,0^++(3175) and T_4s,0^++(3248) have two additional channel into η'η'(3S) and η'(2S)η'(2S). For the 1^+- states, the five resonant states can all decay into η'ϕ, η'ϕ(2S) and η'(2S)ϕ. The  T_4s,1^+-(3142) has an additional decay channel into η'ϕ(3S). And the higher T_4s,1^+-(3228) and T_4s,1^+-(3237) have two extra η'ϕ(3S) and η'(3S)ϕ decay channels. For the 2^++ states, the lowest T_4s,2^++(2714) below the ϕϕ(2S) threshold have only one decay channel ϕϕ. Other two 2^++ resonant states T_4s,2^++(2993) and T_4s,2^++(3164) have an additional decay channel into ϕϕ(2S). The T_4s,2^++(3266) has an extra decay channel to ϕϕ(3S).

Among these states, the state T_4s,2^++(2714) is the most likely one to be observed experimentally. Its mass is the lowest, and its decay channel ϕϕ predominantly decays into K^+K^-K^+K^-, which are all charged particles that can be detected directly by detectors, leading to high detection efficiency and resolution.




 §.§ States with “exotic" C-parity


The “exotic” C-parity systems refer to those that cannot be composed of two S-wave ground quarkonia, namely, the system with J^PC=0^+-,1^++,2^+-. The complex eigenenergies yielded are shown in the lower panel of Fig. <ref>. We obtain a series of resonant states and zero-width states, whose energies, proportions of different color configurations, and two kinds of rms radii are summarized in Table <ref>. The rms radii approximately match the sizes of the 2S mesons. The newly defined r_ij^rms,M does not exhibit any molecular characteristics, indicating that all these states are compact tetraquark states. 

For the J^PC=0^+- and 2^+- systems, there is no S-wave di-quarkonium threshold. So the calculated states appear as bound states on the complex energy plane. However, there may actually be higher-wave thresholds below these states, which are not included in the calculation. If the higher-wave mesons are included, the properties of these states may change and their widths may no longer be zero.

For the J^PC=1^++ system, the masses of the resonant states range in (2.7,3.3) GeV, and their widths are small, with the largest being only a few MeV. Generally, one would expect the widths of these resonant states to arise from their decays into lower-energy meson-meson channels. However, their narrow widths suggest that such decays are suppressed. The reason for the suppression has been discussed in detail in Ref. <cit.>. Simply put, when the rms radii between different i,j quarks are roughly equal, the transition operator of the color-electric term becomes approximately proportional to the identity operator. Therefore, the color mixing transition is suppressed. In the s s s̅s̅ system, however, the contribution from the color-magnetic term is not as significantly suppressed by the quark mass as it is in the heavy quark sector. Therefore, the transition can still occur via spin mixing, which, although limited, can lead to slightly larger widths of these resonant states. All three J^PC=1^++ resonant states can decay via ϕϕ(2S), though this process is suppressed.








§ SUMMARY AND DISCUSSION
 

We calculate the mass spectrum of the S-wave fully strange tetraquark systems with “normal" C-parity J^PC=0^++,1^+-,2^++ and “exoitic" C-parity J^PC=0^+-,1^++,2^+- using the AL1 quark potential model. We employ the GEM to solve the four-body Schrödinger equation, and the CSM to identify genuine resonant states from meson-meson scattering states.

We obtain a series of resonant and zero-width states in the mass region (2.7,3,3) GeV, with widths ranging from less than 1 MeV to around 50 MeV. Apart from the state T_4s,2^++(2714), all the others lie above the M(1 S) M^'(2 S) dimeson thresholds.

We calculate two kinds of rms radius for all states, the conventional rms radius r_ij^rms,C and the newly defined rms radius r_ij^rms,M based on the decomposed wave function. For all states, both kinds of rms radii approximately match the sizes of the 2S mesons, and the newly defined r_ij^rms,M does not exhibit any molecular characteristics, indicating that all these states are compact tetraquark states. 

Among these states, the state T_4s,2^++(2714) is the most likely to be observed experimentally due to its lowest mass and its decay into ϕϕ, which predominantly produces directly detectable charged K^+K^- pairs, ensuring high detection efficiency and resolution.

Recalling the experimental candidates for s s s̅s̅ tetraquark states, their masses are all in range of 2.1 to 2.5 GeV, whereas the lowest S-wave resonant state in our calculations is around 2.7 GeV. This suggests the states like ϕ(2170) and X(2370) are unlikely to be compact s s s̅s̅ tetraquark states. Since the lowest S-wave state is already as high as 2.7 GeV, it's unlikely that a P-wave compact state would be lower than the S-wave ones. However, they could still be P-wave molecular-type resonant states. In this work, a state T_4s,2^++(2714) is found near the threshold of ground ϕ combined with a radially excited ϕ(2S), although it does not exhibit a molecular spatial structure. Introducing P-wave thresholds might yield molecular-type resonant states near the threshold of an S-wave meson combined with a P-wave meson, which could be lower in mass. This possibility awaits careful investigation in the future.

We emphasize that the fully charmed and fully strange tetraquark resonant states share many common characteristics. Their internal structures and underlying dynamics are nearly the same. All of them are compact states. The discovery of X(6900) implies the existence of its fully strange cousins, which can be searched for in the ϕϕ or ϕϕ(2S) channels. While X(6900) can only be produced at LHC now, the fully strange tetraquark resonances may be observed at BESIII, Belle II and LHC. The future discovery of these states shall shed light on both the charmed and strange compact tetraquark states.     












We are grateful to Zi-Yang Lin, Jun-Zhang Wang and Liang-Zhen Wen for the helpful discussions. This project was supported by the National Natural Science Foundation of China (11975033 and 12070131001). This project was also funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, Project ID 196253076-TRR 110). The computational resources were supported by High-performance Computing Platform of Peking University.










 Paper Title: Weak rates in strongly coupled cold quark matter.

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             Weak rates in strongly coupled cold quark matter
                Carlos Hoyos^1[hoyoscarlos@uniovi.es], Andrea Olzi^2[andrea.olzi@unifi.it],
            David Rodriguez-Fernandez^3[david.rfernandez@upm.es]
                August 5, 2024
            ====================================================================================================================================
            
            
            
            0.75[][s]^1Department of Physics and Instituto de Ciencias and Tecnologías Espaciales de Asturias (ICTEA)[][l] Universidad de Oviedo, c/ Leopoldo Calvo Sotelo 18, ES-33007 Oviedo, Spain[][s]^2INFN, Sezione di Firenze and Dipartimento di Fisica e Astronomia, Università di Firenze,[][l] Via G. Sansone 1,I-50019 Sesto Fiorentino (Firenze), Italy.[][s]^3Departamento de matemática aplicada a las TIC, Universidad Politécnica de Madrid[][l] Nikola Tesla, s/n, ES-28031 Madrid, Spain
            
            § ABSTRACT 
            The rates of flavor-changing weak processes are crucial in determining the conditions of beta equilibrium in neutron stars and mergers, influencing the damping of oscillations, the stability of rotating pulsars, and the emission of gravitational waves. We derive a formula for these rates at nonzero temperature, to leading order in the Fermi coupling and exact in the QCD coupling. Utilizing a simple phenomenological holographic model dual to QCD, we study massless unpaired quark matter at high densities. We numerically compute the rate for small deviations from beta equilibrium and derive an analytic approximation for small temperatures. Our findings reveal that, compared to the perturbative result, the rate is suppressed by logarithmic factors of the temperature.
            
            
            
            
            
            
            
            
            § INTRODUCTION
            
            
            
            Weak interactions play a crucial role in determining the properties of neutron stars and their binary mergers. The equation of state, which dictates the mass and radius of the star, is constrained by the condition of beta equilibrium. This condition establishes relationships among the relative amounts of different quark (or baryon) and lepton species. In mergers, the beta equilibrium condition may be modified due to the temperature dependence of the rates <cit.>. When a perturbation in the density drives the matter of the star out of beta equilibrium, weak processes work to restore it and dampen the perturbation. This results in an effective bulk viscosity that is significantly enhanced when the characteristic time of the perturbation resonates with the timescale determined by the rate of weak processes and the density in the star <cit.> (see <cit.> for a review of transport in neutron stars).
            
            Oscillations of a rotating neutron star produce gravitational waves that cause the star to lose angular momentum and spin down, a process that is more pronounced for rapidly rotating stars. Despite this, fast-rotating pulsars with periods on the order of milliseconds are observed within a limited range of temperatures. The effective bulk viscosity induced by beta equilibration is likely necessary to explain the existence and location of this stability window <cit.>. This viscosity also affects the continuous emission of gravitational waves by a quiescent star <cit.> and could impact the gravitational wave signal produced during the inspiral <cit.> and post-merger evolution of a neutron star collision <cit.>.
            
            Although most matter in a neutron star and the remnant of a merger is expected to be primarily composed of hadrons, there are indirect hints of quark matter in the core of heavy neutron stars <cit.>. Simulations of binary mergers suggest that pockets of quark matter may be produced just after the collision <cit.>. Therefore, it is interesting to evaluate the bulk viscosity of quark matter induced by weak processes. At the densities involved, quantum chromodynamics (QCD) cannot be treated perturbatively, and lattice QCD suffers from the sign problem, necessitating other non-perturbative approaches. Holographic models can circumvent these issues and have been applied to the study of dense matter in neutron stars (see <cit.> for comprehensive reviews). Specifically, the bulk viscosity induced by weak processes was computed using holographic models in <cit.>[The holographic estimate of the QCD contribution to the bulk viscosity was computed previously and found to be much smaller than the expected weak contribution <cit.>.]. However, the bulk viscosity formula involves weak reaction rates that have been computed only perturbatively, so it is not a completely self-contained approach. Furthermore, the existing formulas for weak rates are obtained through tree-level scattering matrix elements of quarks and leptons <cit.>. Since the observables obtained from gauge/gravity duality are correlators of gauge-invariant operators, there is no straightforward way to obtain the rates by extrapolating the tree-level formulas. Unfortunately, a main feature of the bulk viscosity, namely its temperature dependence, is highly sensitive to the behavior of the rate. Thus, the present estimation of the bulk viscosity is hindered by significant uncertainty in this respect.
            
            As a step towards improving the bulk viscosity estimates, we will provide a non-perturbative formula for the rates based on correlators of gauge-invariant operators. We will then use it to obtain the temperature dependence of the rates in a simple holographic model of unpaired quark matter. The model, a phenomenological holographic dual description of QCD <cit.>, has been used extensively, with recent related works including the description of nuclear matter in neutron stars and mergers <cit.> and the calculation of neutrino emissivities from flavor current correlators <cit.>. Our goal is not to obtain completely realistic values for the rates, for which more refined models would be needed, but to ascertain possible qualitative changes in the temperature dependence at strong coupling.
            
            Holographic models, despite their utility, come with inherent limitations. They are typically constructed as approximations that capture certain features of QCD, but they do not encompass all the complexities of the theory. One significant limitation is the lack of a first-principles derivation from QCD, leading to uncertainties in their applicability to specific phenomena. Furthermore, these models often rely on large-N_c (number of colors) approximations, which may not accurately reflect the behavior of QCD at the finite N_c relevant to real-world scenarios. The models also face challenges in accurately representing the dynamics at low temperatures and densities where confinement and chiral symmetry breaking are essential. Therefore, while holographic models provide valuable insights and a non-perturbative framework, their predictions should be interpreted with caution and validated against other approaches and experimental data.
            
            Keeping the limitations above in mind, for direct application to the description of neutron stars and their binary mergers, the rate should be computed in the model used to describe matter in the star. The most employed holographic models so far have been those inspired by top-down brane intersections, namely the D3-D7 <cit.> and Witten-Sakai-Sugimoto models <cit.>, and a bottom-up model designed to fit lattice QCD data and other known observables, the V-QCD model <cit.>. The D3-D7 model is the simplest but is limited to describing quark matter in the star <cit.> and must be combined with other nuclear matter models to construct hybrid equations of state for the matter inside the neutron star. The V-QCD model can describe confined matter but is used primarily at large densities, requiring a hybrid approach to describe matter at lower densities outside the core of the star <cit.>. Hybrid equations of state have also been constructed with the Sakai-Sugimoto model <cit.>, but there are proposals to describe nuclear matter in neutron stars at all densities purely holographically <cit.>. The Sakai-Sugimoto model has also been used to describe more exotic quark stars <cit.>. Other proposals to describe dense matter in neutron stars include Einstein-Maxwell-Dilaton theories and other holographic phenomenological models <cit.>. Computing the weak rate and bulk viscosity to estimate the location of the stability window of rotating pulsars would serve as an additional constraint on these models, as well as on any other phenomenological model for which such a calculation is possible.
            
            The outline of the paper is as follows. In Section <ref>, we introduce the weak reaction rates for beta equilibrium of quark matter and derive a non-perturbative formula in terms of correlators of flavor currents. Next, in Section <ref>, we use a simple holographic model to compute flavor current correlators in a high-density state and study their dependence on frequency and spatial momentum. Finally, in Section <ref>, we combine the results of the previous sections to produce analytical and numerical estimates of the weak rates at low temperatures. We compare these to the perturbative results and discuss potential improvements in Section <ref>. Technical details regarding the calculation of correlators in the holographic model are provided in the appendices.
            
            
            
            
            
            
            § WEAK REACTION RATES
            
            
            
            In this section, we derive non-perturbative formulas in the coupling of the strong interactions for the weak reaction rates of quark matter. The formulas correspond to the leading contribution of the electroweak interactions, whose corrections could in principle be computed perturbatively. We also assume that the times and length scales involved are much larger than the scale set by the inverse of the W boson mass.
            
            For the densities expected in the interior of neutron stars and mergers, one should be able to restrict to three quark flavors (q=u,d,s) and two lepton flavors (ℓ=e,μ). The relevant weak processes, or reactions, are produced by a W-boson exchange or emission:
            
                u+d⇌ u+s,   u+ℓ^- →  d,s+ν_ℓ,    d,s → u+ℓ^- +ν̅_ℓ,  ℓ_1^- →ℓ_2^-+ν_ℓ_1+ν̅_ℓ_2 .
            
            Assuming neutrinos are not trapped, the beta-equilibrium conditions in quark matter implied by these reactions are
            
                μ_u+μ_ℓ=μ_d=μ_s,    μ_ℓ=μ_e=μ_μ .
            
            Even if quarks are confined to hadrons the same reactions will change their type among protons, neutrons, and hyperons. In this situation, the beta equilibrium conditions should be given in terms of baryon chemical potentials. Note that in order for any of the species of fermions to be present it is necessary that the chemical potential is larger than the mass of the corresponding particle.
            
            
            If a perturbation takes the chemical potentials out of their equilibrium values, the weak reactions will not be in balance, and the densities will change according to the weak rates Γ:[We have omitted the label on the neutrinos when there is just one lepton involved and put the same numerical sub-index as the corresponding lepton when there are two.]
                d n_u/dt   =   ∑_ℓ=e,μ( Γ_d→ uℓν̅-Γ_uℓ→ dν+ Γ_s→ u ℓν̅-Γ_uℓ→ sν) ,
                d n_d/dt   =   Γ_su→ ud-Γ_ud→ su+∑_ℓ=e,μ( Γ_uℓ→ dν-Γ_d→ uℓν̅) ,
                d n_s/dt   =   Γ_ud→ su-Γ_su→ ud+ ∑_ℓ=e,μ(Γ_uℓ→ sν-Γ_s→ u ℓν̅) ,
                d n_ℓ_1/dt   =   Γ_d→ uℓ_1ν̅-Γ_uℓ_1→ dν+Γ_s→ u ℓ_1ν̅-Γ_uℓ_1→ sν+Γ_ℓ_2→ℓ_1ν_2ν̅_1-Γ_ℓ_1→ℓ_2ν_1ν̅_2 .
            
            It is easy to confirm that the total baryon number and electric charge are conserved by these processes
            
                n_B=1/3(n_u+n_d+n_s),   n_Q=2/3n_u-1/3n_d-1/3n_s-n_e-n_μ,    dn_B/dt=dn_Q/dt=0 .
            
            
            Assuming the deviation of the chemical potentials from their beta-equilibrium values δμ_f^a is small compared to the baryon chemical potential μ_B=μ_u+μ_d+μ_s, we can expand the rates to linear order in the differences
            
                Γ_ud→ su-Γ_su→ ud   ≈   λ_ds (δμ_s-δμ_d),
                Γ_uℓ→ dν-Γ_d→ uℓν̅   ≈   λ_ud^ℓ (δμ_d-δμ_u-δμ_ℓ),
                Γ_uℓ→ sν-Γ_s→ uℓν̅   ≈   λ_us^ℓ (δμ_s-δμ_u-δμ_ℓ),
                Γ_e→μν_2ν̅_1-Γ_μ→ eν_1ν̅_2   ≈   λ_eμ (δμ_μ-δμ_e).
            
            
            In some cases, for instance when quark matter is close to being symmetric, the lepton chemical potential could be below the muon mass μ_ℓ<m_μ. When this happens, muons quickly decay and the only leptons present are electrons. The equilibration equations simplify somewhat
            
                d n_u/dt   =   Γ_d→ u eν̅-Γ_ue→ dν+ Γ_s→ u eν̅-Γ_u e→ sν ,
                d n_d/dt   =   Γ_su→ ud-Γ_ud→ su+Γ_u e→ dν-Γ_d→ u eν̅ ,
                d n_s/dt   =   Γ_ud→ su-Γ_su→ ud+ Γ_u e→ sν-Γ_s→ u eν̅ ,
                d n_e/dt   =   Γ_d→ ueν̅-Γ_ue→ dν+Γ_s→ u eν̅-Γ_ue→ sν .
            
            For small deviations from beta-equilibrium, the rates are expanded as (<ref>), but taking into account only the electron contribution. Moreover, the coefficient λ_eμ does not involve strong coupling physics, so we will ignore it in the following, as it can be computed using the standard methods. Our strategy to derive the other coefficients is to arrive at an expression like (<ref>) from Ward identities capturing the non-conservation of flavor currents due to the coupling to the weak sector. 
            
            
            
            
             §.§ Flavor Ward identities
            
            
            The theory of strong interactions, QCD, respects flavor symmetries so that there are no flavor-changing processes induced by them. Weak interactions on the other hand are not flavor symmetric, and processes such as those in (<ref>) are possible. At energy scales much below the W mass, the flavor-changing processes are captured by adding Fermi's interaction term to the effective action
            
                L_ Fermi=-2√(2) G_F (J_ch^μ)^† J_ch μ,
            
            where the J_ch^μ is the left-charged current
            
                J_ch^μ=ν_e Lγ^μ e_L+ν_μ Lγ^μμ_L+cosθ_C u_Lγ^μ d_L+sinθ_C u_Lγ^μ s_L,
            
            with G_F Fermi's constant and θ_C the Cabibbo angle giving the change of basis between quark mass and flavor eigenstates. In addition to Fermi's term, flavor symmetry can also be broken by flavor-preserving mass terms and electromagnetic and weak neutral interactions.
            
            In the absence of any flavor-breaking terms, there would be a SU(3)_L× SU(3)_R global symmetry for flavor rotations among quarks and a SU(4)_L× SU(4)_R flavor global symmetry for leptons. To each of these symmetries, we can associate a conserved current J^μ_f χ, where f=l,q for leptons or quarks and χ=L,R for left or right chirality. Since the symmetry is non-Abelian the current components transform under flavor rotations. Denoting by T^A_f χ the generators of the flavor symmetry, the transformation of the current is
            
                δ_θ_f χ J^μ_f χ=iθ_f χ^A[J^μ_f χ,T^A_f χ] .
            
            We will take the usual normalization for the generators tr(T_f χ^A T_f χ^B)=1/2δ^AB. 
            
            A vector flavor transformation corresponds to a transformation with equal angles for both chiralities θ_f L^A=θ_f R^A=θ_f^A. In the following vector currents and generators will be denoted as J^μ_f and T^A_f. If there are flavor-symmetry breaking terms in the action L_ br, the Ward identity associated with a vector transformation becomes
            
                2θ_f^A tr(∂_μ J^μ_f T^A_f)=-δ_θ^A_f L L_ br-δ_θ^A_f R L_ br|_θ_f L^A=θ_f R^A=θ_f^A.
            
            In the following, it will be convenient to introduce the notation for the currents
            
                (J_f χ^μ)^a_ b=f̅_χ bγ^μ f_χ^a ,     f^a=q^a,l^a;   q^a=u,d,s;  l^a=ν_e,e,ν_μ,μ.
            
            
            We are interested in the flavor-changing contributions introduced by Fermi's interaction. We can simplify the analysis by noticing that since Fermi's interaction is symmetric under a rotation between the two lepton generations (ν_e,e)↔ (ν_μ,μ), we can actually restrict lepton transformations to SU(2)× SU(2)⊂ SU(4), with each SU(2) rotating a charged lepton and its corresponding neutrino within each generation. We leave details of the calculation, which is straightforward but tedious, to Appendix <ref>. The Ward identities for the diagonal components of the vector flavor currents are
            
                ∂_μ(J_q^μ)^u_ u   =     i√(2)G_F η_μν∑_ℓ=e,μ[cosθ_C(J_q L^μ)^u_ d(J_l L^ν)^d_ u-(J_q L^μ)^d_ u(J_l L^ν)^ν_ℓ_ ℓ.
                
                        . +sinθ_C(J_q L^μ)^u_ s(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^s_ u(J_l L^ν)^ν_ℓ_ ℓ],
                
                ∂_μ(J_q^μ)^d_ d    =    i√(2) G_Fη_μν[cosθ_C sinθ_Cη_μν(J_q  L^μ)^u_ s (J_q L^ν)^d_ u-(J_q L^μ)^s_ u(J_q L^ν)^u_ d.
                
                        .-cosθ_C∑_ℓ=e,μ(J_q L^μ)^u_ d(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^d_ u(J_l L^ν)^ν_ℓ_ ℓ],
                
                ∂_μ(J_q^μ)^s_ s   =    i√(2) G_Fη_μν[-cosθ_C sinθ_Cη_μν(J_q  L^μ)^u_ s (J_q L^ν)^d_ u-(J_q L^μ)^s_ u(J_q L^ν)^u_ d.
                
                        .-sinθ_C∑_ℓ=e,μ(J_q L^μ)^u_ s(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^s_ u(J_l L^ν)^ν_ℓ_ ℓ],
                
                ∂_μ(J_l^μ)^ν_ℓ_ ν_ℓ   =    i√(2)G_F η_μν[-cosθ_C(J_q L^μ)^u_ d(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^d_ u(J_l L^ν)^ℓ_ ν_ℓ.
                
                        . -sinθ_C(J_q L^μ)^u_ s(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^s_ u(J_l L^ν)^ν_ℓ_ ℓ.
                
                        .-(J_l L^μ)^ν_ℓ'_ ℓ'(J_l L^ν)^ℓ_ ν_ℓ-(J_l L^μ)^ℓ'_ ν_ℓ'(J_l L^ν)^ℓ_ ν_ℓ],
                
                ∂_μ(J_l^μ)^ℓ_ ℓ   =    i√(2)G_F η_μν[cosθ_C(J_q L^μ)^u_ d(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^d_ u(J_l L^ν)^ν_ℓ_ ℓ.
                
                        . +sinθ_C(J_q L^μ)^u_ s(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^s_ u(J_l L^ν)^ν_ℓ_ ℓ.
                
                        .+(J_l L^μ)^ν_ℓ'_ ℓ'(J_l L^ν)^ℓ_ ν_ℓ-(J_l L^μ)^ℓ'_ ν_ℓ'(J_l L^ν)^ν_ℓ_ ℓ].
            
            Where (ℓ,ℓ')=(e,μ) or (μ,e). Note that other terms that break the non-Abelian flavor symmetry, but are diagonal in flavor, do not contribute to the Ward identities of the diagonal components. One can easily confirm that the expressions above are consistent with the conservation of baryon and lepton numbers as well as the electric charge:
            
                ∂_μ(J_q^μ)^u_ u+(J_q^μ)^d_ d+(J_q^μ)^s_ s=0,  ∂_μ(J_l^μ)^ν_ℓ_ ν_ℓ+(J_l^μ)^ℓ_ ℓ=0, 
                   ∂_μ2/3(J_q^μ)^u_ u-1/3(J_q^μ)^d_ d-1/3(J_q^μ)^s_ s-∑_ℓ=e,μ (J_l^μ)^ℓ_ ℓ=0.
            
            The time components of the diagonal components of the flavor vector currents equal the corresponding quark or lepton densities
            
                n_u=(J_q^0)^u_ u, n_d=(J_q^0)^d_ d, n_s=(J_q^0)^s_ s, n_ℓ=(J_l^0)^ℓ_ ℓ.
            
            Compared with (<ref>) for homogeneous configurations, the rates are determined by the expectation value of a non-diagonal operator quadratic in the flavor currents appearing on the right-hand side of the Ward identities. We can think of such a quadratic operator as obtained from a two-point correlator in the (properly regularized) limit of coincident points. In the absence of flavor-breaking terms, the most general form of the flavor current correlators would be
            
                (J_f  L^μ)^a_ b (J_f L^ν)^c_ d= G_1 δ^a_bδ^c_d+G_2 δ^a_d δ^c_b .
            
            If there are mass terms in the action of the form L_m=f̅_a M^a_f b f^b, or chemical potentials introduced through a background flavor gauge field A^a_ f b, there could also be terms where we replace some or all of the deltas in (<ref>) by M_f or A_f, or contractions like M^a_f c M^c_f b,  M^a_f c A^c_f b, etc. This potentially can produce many new terms, however, for diagonal mass matrices and chemical potentials, the non-diagonal current correlators appearing on the right-hand side of the Ward identities would still vanish, and similarly for contributions originating from interactions that are diagonal in flavor. Therefore, the non-diagonal current correlators contributing to the Ward identities must be proportional to the flavor-changing coupling G_F. We will derive in the following a closed expression for the O(G_F^2) contribution to the rates in thermal field theory.
            
            
            
            
            
             §.§ Leading contribution in the Fermi coupling
            
            
            We will assume the theory is at finite temperature and use the real-time formalism to compute the current correlators. In general, thermal correlation functions are computed by insertions in a path integral defined along a contour in complex time C. The two-point function of two operators O_1 and O_2 is
            
                T_C[ O_1(x_1) O_2(x_2)].
            
            The complex time path, or Schwinger-Keldysh path, extends along (t_i,t_f)∪ (t_f,t_f-iσ)∪ (t_f-iσ,t_i -iσ)∪ (t_i-iσ,t_i-iβ), with σ>0 and  t_i→-∞, t_f→+∞. We take the point x_1 to be at the first leg of the complex time path on the real axis. Then we have two possible contributions, if x_2 is on the same leg, then the thermal correlator coincides with the usual time-ordered Feynman correlator
            
                T_C[ O_1(x_1) O_2(x_2)]=T[ O_1(x_1) O_2(x_2)]=G^F_ O_1 O_2 (t_1-t_2,x_1-x_2).
            
            If, on the other hand, x_2 is on the third leg (t'=-σ), the correlator is the Wightman correlator
            
                T_C[ O_1(x_1) O_2(x_2)]= O_2(x_2) O_1(x_1)=G^<_ O_1 O_2 (t_1-t_2+iσ,x_1-x_2).
            
            With this in mind, let us find the effect of Fermi's interaction (<ref>) when we treat it perturbatively. The thermal expectation value of an operator O will be, to leading order in the Fermi coupling,
            
                O(x)_G_F=T_C[ O(x) e^i∫_C   L_ Fermi]_0≈ O(x)_0+i∫_C d^4 x'T_C[ O(x)   L_ Fermi(x')]_0.
            
            In the expression above X_0 indicates that the expectation value of X is computed in the absence of Fermi's interaction. Since x is in the first leg of the complex time contour and  x' runs over its full extent, we have contributions from both the first and third leg. 
            
                ∫_C d^4 x'T_C[ O(x)   L_ Fermi(x')]_0=∫ d^4x' [G_ O L_ Fermi^F(t-t',x-x')+G_ O L_ Fermi^<(t-t'+iσ,x-x')].
            
            In the evaluation of the right-hand side of the Ward identities (<ref>)O is quadratic in the flavor currents, as well as L_ Fermi. Therefore, the leading O(G_F^2) contribution is proportional to correlators of the form
            
                [(J_f^μ)^a_ b(x) (J_f^ν)^c_ d(x)][(J_f'^α)^a'_ b'(x') (J_f'^β)^c'_ d'(x')]_0≈
                   δ_ff'[(J_f^μ)^a_ b(x)(J_f^α)^a'_ b'(x')_0(J_f^ν)^c_ d(x) (J_f^β)^c'_ d'(x')_0+(J_f^μ)^a_ b(x) (J_f^β)^c'_ d'(x')_0(J_f^ν)^c_ d(x)(J_f^α)^a'_ b'(x')_0].
            
            The factorization above is an approximation. For holographic models dual to a large-N_c theory it will happen naturally as part of the large-N_c expansion. More generally, the non-factorizable contributions would originate from effective eight-fermion interactions, so it is perhaps not unreasonable to assume that they would introduce relatively small corrections even in the finite-N_c strongly coupled theory. 
            
            As we discussed before, in the absence of the Fermi interaction, non-vanishing current correlators should be diagonal in flavor, thus
            
                G^μα_f;ab,a'b'(x-x')≡(J_f^μ)^a_ b(x)(J_f^α)^a'_ b'(x')_0≠ 0,  ⇒  (a=b   a'=b')   (a=b'   a'=b).
            
            This limits the number of non-zero contributions to the four-current thermal correlators. We will now write the leading contributions in the Fermi coupling to the Ward identities (<ref>) in terms of the Fourier transforms of the Feynman and Wightmann correlators of the flavor currents
            
                G^K μα_f;ab,a'b'(x-x')=∫d^4 q/(2π)^4e^iq· (x-x') G^K μα_f;ab,a'b'(q),     K=F,<  .
            
            It will be convenient to define
            
                γ__f_1^a f_2^b→ f_1^cf_2^d=γ^F__f_1^af_2^b→ f_1^c f_2^d+γ^<__f_1^af_2^b→ f_1^c f_2^ d,
            
            with (K=F,<)
            
                γ_f_1^a f_2^b→ f_1^c f_2^d^K=η_μνη_αβ∫d^4 q/(2π)^4[G^K μα_f_1;ac,ca(q)G^K νβ_f_2;bd,db(-q)-G^K μα_f_1;ca,ac(q)G^K νβ_f_2;db,bd(-q)] .
            
            In terms of these quantities, the leading order contributions in the Fermi coupling to the flavor Ward identities are
            
                ∂_μ(J_q^μ)^u_ u   ≈    -4 G_F^2∑_ℓ=e,μ(cos^2θ_C γ_uℓ→ dν+sin^2θ_C γ_uℓ→ sν),
                ∂_μ(J_q^μ)^d_ d   ≈    4 G_F^2 (-sinθ_C^2cos^2θ_C γ_ud→ su+cos^2θ_C ∑_ℓ=e,μγ_uℓ→ dν),
                ∂_μ(J_q^μ)^s_ s   ≈    4 G_F^2 (sinθ_C^2cos^2θ_C γ_ud→ su+sin^2θ_C ∑_ℓ=e,μγ_uℓ→ sν),
                ∂_μ(J_l^μ)^ν_ℓ_ ν_ℓ   ≈    4 G_F^2 (cos^2θ_C γ_uℓ→ dν+sin^2θ_C γ_uℓ→ sν+γ_ℓν'→νℓ'),
                ∂_μ(J_l^μ)^ℓ_ ℓ   ≈    -4 G_F^2 (cos^2θ_C γ_uℓ→ dν+sin^2θ_C γ_uℓ→ sν+γ_ℓν'→νℓ').
            
            Where in the last two rows (ℓ,ℓ')=(e,μ) or (μ,ℓ) and ν,ν' are the neutrinos corresponding to ℓ,ℓ' respectively.
            
            
            
            
             §.§ Formulas for the rates
            
            
            A direct comparison of the equilibration equations  (<ref>) and the Ward identities (<ref>) results in the identification of the leading order contribution in the Fermi coupling to the rates as
            
                Γ_uℓ→ dν-Γ_d→ u ℓν̅   ≈    4 G_F^2cos^2θ_C γ_uℓ→ dν  ,
                Γ_u ℓ→ sν-Γ_s→ u ℓν̅   ≈     4 G_F^2sin^2θ_C γ_uℓ→ sν  ,
                Γ_ud → su-Γ_su → ud   ≈     4 G_F^2sinθ_C^2cos^2θ_C γ_ud→ su  ,
                Γ_ℓ_1→ℓ_2ν_1ν̅_2-Γ_ℓ_2→ℓ_1ν_2ν̅_1   ≈    4 G_F^2γ_ℓ_1ν_2→ν_1ℓ_2 .
            
            With the factors γ defined in (<ref>) and (<ref>) in terms of thermal correlation functions. We can further simplify the expressions by first noting that, if the system has parity and time reversal invariance, the Feynman correlator satisfies
            
                G_f;ab,ba^F μν(-q_0,-q)=G_f;ba,ab^F νμ(q_0,q).
            
            A straightforward calculation then shows that γ^F__f_1^af_2^b→ f_1^c f_2^d=0 in this case. The Wightmann correlator on the other hand is proportional to the spectral function, defined as the imaginary part of the retarded correlator:
            
                G^< μν_f;ab,ba(q_0,q)=n_f;ab(q_0)ρ_f;ab^μν(q_0,q),  ρ_f;ab^μν=-2 (G^R μν_f;ab,ba),
             
            where the factor multiplying the spectral function is the thermal distribution
            
                n_f;ab(q_0)=1/e^β(q_0+μ_f^a-μ_f^b)-1 ,
            
            with β=1/T the inverse of the temperature and μ_f^a the chemical potential for the fermion species f^a. We now introduce (<ref>) in (<ref>), and use that
            
                n_f;ab(-q_0)=-(1+n_f;ba(q_0)),     ρ_f;ab^μα(-q_0,-q)=-ρ_f;ba^αμ(q_0,q) ,
            
            where the last identity follows from time-reversal invariance and parity. After this, one arrives at
            
                γ__f_1^a f_2^b→ f_1^cf_2^d=γ^<__f_1^af_2^b→ f_1^c f_2^ d=η_μνη_αβ∫d^4 q/(2π)^4[(n_f_1;ac(q_0)-n_f_2;db(q_0))ρ^μα_f_1;ac(q_0,q)ρ^βν_f_2;db(q_0,q)] .
            
            This quantity will vanish when the chemical potentials satisfy the relations
            
                μ_f_1^a-μ_f_1^c=μ_f_2^d-μ_f_2^b.
            
            Then, demanding that each of the rates in (<ref>) vanish gives the conditions
            
                μ_u-μ_d    =    μ_ν_ℓ-μ_ℓ,
                μ_u-μ_s    =   μ_ν_ℓ-μ_ℓ,
                μ_u-μ_s    =   μ_u-μ_d,
                μ_ν_ℓ_1-μ_ℓ_1   =   μ_ν_ℓ_2-μ_ℓ_2.
            
            Setting μ_ν_ℓ=0, it is easy to check that this coincides with the beta equilibrium conditions in (<ref>). We want to expand (<ref>) to the lowest order around the equilibrium values, μ_f^a=μ_f^a^eq+δμ_f^a. The thermal distributions have a pole at small frequencies, that we need to deal with. Let us introduce an arbitrary frequency ω_0 such that δμ_f_1^a-δμ_f_1^c∼δμ_f_2^d-δμ_f_2^b≪ω_0 ≪ T. Then, there is a contribution that we can expand to linear order in the chemical potentials plus a correction from the poles
            
                γ__f_1^a f_2^b→ f_1^cf_2^d≈(δμ_f_2^d-δμ_f_2^b-(δμ_f_1^a-δμ_f_1^c))Λ_f_1^a f_2^b→ f_1^cf_2^d^0+δγ__f_1^a f_2^b→ f_1^cf_2^d ,
            
            where, after defining Δμ^eq=μ_f_1^a^eq- μ_f_1^c^eq=μ_f_2^d^eq- μ_f_2^v^eq, the coefficient Λ is
            
                Λ__f_1^a f_2^b→ f_1^cf_2^d^0=   η_μνη_αβ∫d^4 q/(2π)^4ρ^μα_f_1;ac(q_0,q)ρ^βν_f_2;db(q_0,q) 
                   ×[1/4Tsinh^2(q_0+Δμ^eq/2T)-T/(q_0+Δμ^eq)^2Θ(ω_0^2-(q^0+Δμ^eq)^2)].
            
            We remind the reader that in this formula the fermion species are f_1,f_2=q (quarks) or l (leptons) and the possible flavors are q^a=u,d,s, l^a=ν_e,e,ν_μ,μ. For q_0+Δμ^eq≳ T the function decays exponentially ∼ e^-(q_0+Δμ^eq)/T, thus most of the contributions in the frequency integral will come from an interval of width ∼ T around q^0+Δμ^eq. 
            
            The correction to the rate originating from the poles is approximately
            
                δγ__f_1^a f_2^b→ f_1^cf_2^d=   η_μνη_αβ∫_|q^0+Δμ^eq|<ω_0d^4 q/(2π)^4ρ^eq μα_f_1;ac(q_0,q)ρ^eq βν_f_2;db(q_0,q) 
                   ×[T/q^0+Δμ^eq+δμ_f_1^a-δμ_f_1^c -T/q^0+Δμ^eq+δμ_f_2^d-δμ_f_2^b] .
            
            We will use the principal value integral
            
                ∫_-ω_0-Δμ^eq^ω_0-Δμ^eqd q^0/2πT/q^0+Δμ^eq+δμ S(q^0) ≈T/2πS(-Δμ^eq)logω_0+δμ/ω_0-δμ≈T/πω_0S(-Δμ^eq) δμ.
            
            Thus, we can write
            
                γ__f_1^a f_2^b→ f_1^cf_2^d≈(δμ_f_2^d-δμ_f_2^b-(δμ_f_1^a-δμ_f_1^c))Λ_f_1^a f_2^b→ f_1^cf_2^d ,
            
            with
            
                Λ__f_1^a f_2^b→ f_1^cf_2^d≈Λ__f_1^a f_2^b→ f_1^cf_2^d^0+η_μνη_αβT/πω_0∫d^3 q/(2π)^3ρ^eq μα_f_1;ac(-Δμ^eq,q)ρ^eq βν_f_2;db(-Δμ^eq,q).
            
            If the spectral function vanishes at q^0=-Δμ^eq the last term in (<ref>) is zero and one can take the ω_0→ 0 limit, in which case the expression for the rate simplifies
            
                Λ__f_1^a f_2^b→ f_1^cf_2^d≈η_μνη_αβ∫d^4 q/(2π)^4ρ^eq μα_f_1;ac(-Δμ^eq,q)ρ^eq μα_f_2;db(-Δμ^eq,q)/4Tsinh^2(q_0+Δμ^eq/2T).
            
            
            Combining the expression (<ref>) with (<ref>) and (<ref>), we arrive at the following formulas for the coefficients of the rates
            
                λ_ds   ≈    4 G_F^2 sin^2θ_Ccos^2θ_C Λ_ud→ su,
                λ_ud^ℓ   ≈    4 G_F^2 cos^2θ_C Λ_uℓ→ dν,
                λ_us^ℓ   ≈    4 G_F^2 sin^2θ_C Λ_uℓ→ sν,
                λ_eμ   ≈    4 G_F^2 Λ_μν_e→ν_μ e.
            
            When leptonic contributions can be neglected, as it is usually the case in the evaluation of the bulk viscosity in neutron stars, the only relevant coefficient is λ_1=λ_ds, following the conventions in <cit.>. In the remaining of the paper, we will give an estimate of λ_1 at strong coupling using a holographic model.
            
            
            
            
            
            
            § HOLOGRAPHIC MODEL
            
            
            
            We will use a holographic model dual to a SU(N_c) gauge theory with N_f flavors to compute the rate. We use a bottom-up approach, where we construct a gravitational action with the necessary ingredients to capture the physics of interest, but that does not correspond to a string theory construction where both sides of the duality are known.
            
            The glue sector is given by an action consisting of just the Einstein-Hilbert action plus a negative cosmological constant
             
            S_c = 1/16π G_5∫ d^5 x  √(-g)[ R - 2Λ],  Λ=-6/L^2 . 
             
            In the absence of matter, the maximally symmetric solution is an AdS_5 space with radius L. Including the flavor sector will modify the geometry, but it will still asymptote to the same AdS_5, so we can apply the usual holographic dictionary between the fields in the gravity theory and sources and operators in the gauge theory dual.
            
            The flavor sector consists of U(N_f)_L× U(N_f)_R gauge fields, L_M and R_M, dual to the flavor currents, and a complex scalar field X in the bifundamental representation dual to a quark bilinear. The action for the flavor fields is the one commonly used for holographic QCD models, originating with <cit.>
            S_f = 1/16π G_5N_f/N_c∫ d^5 x √(-g)  Tr[- | D X|^2 + 3 | X|^2 -L^2/4( F_(R)^2 + F_(L)^2) ] .
             
            The field strength is defined as F_(A)MN = ∂_M A_N - ∂_N A_M - i[ A_M,A_N]. The covariant derivative of the scalar field is defined as follows 
             
            D_N X = ∂_N X - i L_N X + i X R_N ,      D_N X^† = ∂_N X^† + i X^† L_N - i  R_N X^† .
             
            The mass of the scalar is fixed in such a way that the field is dual to a scalar operator of scaling dimensions Δ=3, as corresponds to a quark bilinear.
            
            The action is fixed up to an overall normalization of the scalar and gauge fields. In <cit.> (see also <cit.>) the normalization was fixed by matching the large Euclidean momentum form of current correlators derived from the holographic model to the OPE of QCD. The gauge fields and scalars are rescaled to put the flavor sector of the action in the form  
            
            S_f =∫ d^5 x √(-g)  Tr[g_X^2(- | D X|^2 + 3 | X|^2) -1/4 g_5^2( F_(R)^2 + F_(L)^2) ] .
             
            Then, the matching to QCD in the UV fixes 
            
                1/g_5^2=N_c/12π^2,   g_X^2=N_c/4π^2 .
            
            
            The equations of motion for the metric are Einstein's equations with a cosmological constant and the energy-momentum tensor sourced by the gauge fields and the scalars
            
                R_MN-1/2g_MNR+ g_MNΛ=N_f/N_cT_MN.
            
            The equations of motion for the gauge and scalar fields read
            1/√(-g) D_M [√(-g) F^MN_(R)]    =     j^N_(R) ,
            1/√(-g) D_M [√(-g) F^MN_(L)]    =    j^N_(L) ,
            1/√(-g) D_M [√(-g)  D^M X  ]    =      -3 X .
            
            
            where the bulk currents appearing in the equations of the gauge fields are
            
            j^M_(R)    =    i [( D^M X)^† X - X^† D^M X] , 
            
            j^M_(L)    =    i [ ( D^M X) X^† - X (D^M X)^†] .
            
            We will look for solutions dual to a deconfined state of the gauge theory at nonzero temperature and baryon density, which are charged black brane geometries. In order to simplify the analysis we will turn off the scalar field, which amounts to having zero quark mass and condensate for all the flavors. In this case, imposing the condition of beta equilibrium fixes all quark chemical potentials to be equal, since deviations from symmetric quark matter due to nonzero quark masses are absent. Since we are ultimately interested in comparing with QCD results, we will fix the rank of the flavor fields N_f=3.
            
            
            
            
             §.§ Background solution
            
            
            The metric of the charged black brane takes the general form, in Poincaré coordinates,
            
                ds^2 = L^2z^2(-f(z)dt^2+dz^2f(z)+dx ^2),      f(z)=1-M z^4/z_h^4+Q^2 z^6/z_h^6 .
            
            The AdS_5 asymptotic boundary is at z=0, and the horizon is at the value where the black body factor vanishes f(z_h)=0. The values of M and Q depend on the background scalar and gauge fields. The leading term in the asymptotic expansion of the background scalar at the boundary determines the quark masses 
            
                X ∼([ m_u        ;     m_d    ;         m_s ])z+O(z^3) .
            
            We will set all quark masses to zero m_u=m_d=m_s=0, which allows to have a trivial solution of the scalar X^ bkg=0. The background gauge fields dual to the quark densities are
            
                (L_0^ bkg)^a_ b=(R_0^ bkg)^a_ b=μ_q/2(1-z^2/z_h^2)δ^a_ b .
            
            Where μ_q=μ_u=μ_d=μ_s is the quark chemical potential, which is the same for all flavors at beta equilibrium. The backreaction over the metric is limited to the gauge fields, so the geometry is the AdS Reissner-Nordstrom charged black brane (<ref>), with 
            
                M=1+Q^2 ,      Q^2=z_h^2μ_q^2/2 .
            
            The temperature in the dual field theory is equal to the Hawking temperature, given by
            
                T = 1π z_h(1-Q^22) .
            
            One can use these formulas to obtain the position of the horizon as a function of the temperature and quark chemical potential
            
                z_h=2/μ_q(√(1+(π T/μ)^2)-π T/μ) .
            
            Later we will often use
            
                T̅= z_h T ≈2 T/μ_q .
            
            
            
            
             §.§ Two-point functions of flavor currents
            
            
            
            The spectral function entering the formulas for the rates can be obtained from the imaginary part of the retarded correlator of the left flavor currents. Following the usual holographic dictionary, we can extract the correlator from the solutions to the linearized equations of fluctuations. We will impose the boundary conditions 
            
                lim_z→ 0 L_μ(x,z)=l_μ(x),  lim_z→ 0R_μ(x,z)=0,  lim_z→ 01/zX(x,z)=0.
            
            This turns off the sources for the right flavor currents and the quark bilinears and sets l_μ(x) as the source for the left flavor currents. The boundary asymptotic expansion of the Fourier transform of the left gauge fields is
            
                (L_μ)^a_ b(ω,k,z)∼( δ^ ν_μ+(c_μ^ ν)_ab z^2 logz/L+z^2 (g_μ^ ν)_ab+⋯)(l_ν)^a_ b(ω,k).
            
            Where, keeping the dependence on the quark masses,
            
                (c_μ^ ν)_ab = 12((k_α k^α+m_a^2+m_b^2)δ_μ^ν - k_μ k^ν).
            
            The two-point function of the left flavor current can be obtained from the one-point function identifying the coefficient of the source l_μ. The one-point function is extracted from the boundary value of the regularized canonical momentum
            
                J_q L^μ=-lim_z→ 0L^2/16π G_5N_f/N_c[√(-g) F_L^zμ+(logz/L+log(LΛ))(L D_α(√(-γ) F_L^αμ)-1/L√(-γ) j_(L)^α)],
            
            where γ_μν=g_μν is the induced metric on a radial slice and Λ is an energy scale of the theory. The additional terms are obtained from the variation of the counter-term action
            
                S_c.t=L/16π G_5N_f/N_c(logz/L+log(LΛ))∫ d^4 x√(-γ)[ L^2/4 F_L^μνF_L μν+(D_μ X)^† D^μ X] .
            
            
            Using the expansion (<ref>), the one-point function to linear order in the fluctuation is
            
                (J_q L^μ)^a_ b= L^3/8π G_5N_f/N_c G^μν_ab(l_ν)^a_ b,     G^μν_ab=-η^μα((g_α^ ν)_ab-(c_α^ ν)_ablogΛ L/e^1/2)   .
            
            Taking the variation with respect to the source gives the two-point function
            
                (J_q L^μ)^a_ b (J_q L^ν)^a'_ b'= L^3/8π G_5N_f/N_c G_ab^μνδ^a_ b'δ^a'_b  .
            
            In cases where the holographic dual is a known gauge theory L^3/8π G_5∼ N_c^2, so the overall factor appearing on the correlator is expected to be proportional to N_c N_f. In the following, we will denote the normalization constant simply as
            
                N=L^3/8π G_5N_f/N_c .
            
            The fit of the model to QCD at large Euclidean momentum (<ref>) and (<ref>) fixes
            
                N= N_c/6π^2 .
            
            The spectral function is obtained from the imaginary part of the retarded correlator, which can only receive contributions from the coefficient of the O(z^2) term in the boundary expansion (<ref>)
                ρ^μν_ab,ba=2 Nη^μα Im (g_α^ ν)_ab .
            
            In order to compute the correlators of flavor currents involved in the formulas for the weak reaction rates, we must consider linearized fluctuations of the gauge fields around the background solution. We can restrict to fluctuations that are off-diagonal in flavor, and take into account that the background is diagonal. When the quark masses are zero and the scalar profile is trivial, the energy-momentum tensor of the gauge fields is at least quadratic in the fluctuations. Therefore, metric fluctuations can be turned off. Furthermore, the fluctuations of the left gauge fields decouple from the right gauge field and the scalar, so it is enough to restrict the analysis to the left gauge fields.
            
            
            
            
             §.§ Linearized fluctuations
            
            
            We will work in the radial gauge L_z=R_z=0 and expand the fluctuations in Fourier modes
            
                Y(x,z)=∫dω/2π∫d^3 k/(2π)^3 e^-iω x^0+ik·x Y(ω,k,z) ,  
                Y=L_μ,R_μ,X,X^† .
            
            Our fluctuations correspond to the off-diagonal flavor components Y≡ Y^a_ b, but we will omit flavor indices unless there is some ambiguity.
            
            It will be convenient to rescale coordinates by the horizon position 
            
                z=z_h Z,    x^μ=z_h X^μ ,
            
            and work with the dimensionless frequency and  momentum  
            
                ω̅=z_h ω,  k̅=z_h k ,  (k̅=|k̅|) .
            
            
            We are interested in temperatures much smaller than the chemical potential T/μ_q ≪ 1. Since the integral over frequencies that determines the rates (<ref>) is concentrated on frequencies smaller or of the order of the temperature, we can use a small frequency approximation ω/μ≪ 1. We will use the approach of <cit.> to compute low-frequency correlators. When the temperature is zero, the background solution is an extremal black brane that develops an AdS_2 throat at the horizon. One can then divide the geometry into an inner and outer region relative to this throat. When the temperature is non-zero, but very small compared to the chemical potential, there is still a throat but the inner geometry is a non-extremal AdS_2 with a horizon. From (<ref>), we see that the near-extremal solutions are for Q^2≲ 2. Let us parametrize
            
                Q^2=2/Z_*^6,     Z_*-1=|ω̅| Z_*^2/12 ζ_0,     Z_*-Z=|ω̅| Z_*^2/12 ζ .
            
            The temperature in units of the horizon position is πT̅=1-Z_*^-6. This gives the approximate relation
            
                ζ_0≈|ω̅|/2πT̅=|ω|/2π T .
            
            Expanding the metric (<ref>) for |ω̅|→ 0 and Z_*≈ 1 (since T̅≪ 1), one gets
            
                ds^2≈L^2/12 ζ^2(-F(ζ)dτ̅^2+dζ^2/F(ζ))+L^2dX ^2 ,      F(ζ)=1-ζ^2/ζ_0^2,  τ̅=|ω̅| X^0=|ω| x^0  .
            
            This is AdS_2×ℝ^3 Schwarzschild geometry with an AdS_2 radius L^2__AdS_2=L^2/12 and a horizon at ζ=ζ_0. In the zero temperature limit Z_*→ 1, ζ_0→∞ the throat extends to values ζ→∞.
            
            The equations for linearized fluctuations can be solved in the inner region imposing an ingoing boundary condition at the horizon ζ→ζ_0. In the region towards the end of the AdS_2 throat, the solutions have an expansion
            
                L_μ∼ A_μ^-ζ^Δ_-+A_μ^+ ζ^Δ_+ ,    Δ_+>Δ_-.
             
            This matches with the behaviour of solutions in the outer region close to the horizon Z→ 1
                L_μ = C^-_μη_-(Z)+C^+_μη_+(Z),  η_±(Z)∼ (1-Z)^-Δ_± ,
            
            which allows the identification
            
                C^±_μ=A_μ^±(|ω̅|/12)^Δ_± .
            
            The ingoing condition fixes a relation between A_μ^+ and A_μ^- that translates into a relation between C_μ^+ and C_μ^-:
            
                C_μ^+=𝒢(ω,k) C_ν^- ,
            
            with 𝒢(ω,k) usually interpreted as an infrared (IR) Green's function associated with a 0+1 CFT dual to the AdS_2 gravitational theory. This IR function determines the Green's function of the full theory. In order to see this, we need to translate the above relation between coefficients to the outer normalizable and non-normalizable solutions
            
                L_μ = l_μϕ_NN(Z)+g_μϕ_N(Z),
            
            with expansions close to the boundary Z→ 0
                ϕ_NN(Z)∼ 1+k̅^2/2Z^2log Z+O(Z^4),  ϕ_N(Z)=Z^2+O(Z^4) .
            
            The two sets of outer solutions are related by the connection coefficients
            
                η_±(Z)=α_±ϕ_NN(Z)+β_±ϕ_N(Z) .
            
            Thus, the full solution is
            
                L_μ =C_μ^-( η_-(Z)+𝒢η_+(Z))=C_μ^-(α_- +α_+𝒢)ϕ_NN(Z)+C_μ^-(β_- +β_+𝒢)ϕ_N(Z) .
            
            Fixing the coefficient of the non-normalizable solution to be the source l_μ, we get that the coefficient of the normalizable term is
            
                g_μ=β_- +β_+𝒢/α_- +α_+𝒢 l_μ .
             
            More precisely, the connection coefficients and IR Green's function depend on whether the fluctuation is transverse or longitudinal. The calculation of the solutions and connection coefficients up to O(ω^2/μ_q^2) corrections is detailed in Appendix <ref>. It turns out that at leading order in ω/T the longitudinal components do not contribute.
            
            
            
            
             §.§ Spectral function
            
            
            Since only the transverse components of the left gauge fields are relevant at low frequencies, the spectral function determined by (<ref>) is (ρ^00≈ 0, ρ^0i≈ 0),
            
                ρ^ij_ab,ba≈ Nμ_q^2/2(δ^ij-k^i k^j/k^2)ρ_L(ω/2π T,k̅,T̅) ,  ρ_L(ω/2π T,k̅,T̅)=β_+α_- - β_-α_+(α_-+ α_+Re 𝒢_L)^2 + (α_+)^2(Im 𝒢_L)^2Im 𝒢_L .
            
            The IR Green's function of the transverse modes of the left gauge fields is
            
                𝒢_L(ω̅,k̅,T̅)≈Γ(-ν_k) Γ(1/2-i  ω/2π T+ν_k)/Γ(ν_k) Γ(1/2-i  ω/2π T-ν_k)(πT̅/12)^2ν_k.
            
            Up to corrections suppressed by factors of T/μ_q. At large spatial momentum the IR Greens' function approaches
            
                𝒢_L(ω̅,k̅,T̅)∼ -cos(πk̅/√(12)+iω/2T)/sin(πk̅/√(12))(πT̅/12)^k̅/√(3) .
            
            At low temperatures, there is a strong suppression due to the T̅-dependent factor. When T/μ_q→ 0 (T̅→ 0), there is a strong suppression of the correlator already at small momenta
            
                𝒢_L∼(πT̅/12)e^-k̅^2/12log12/πT̅ .
            
            The support of the Green's function is concentrated on values k̅≲ |log(T/μ_q)|^-1≪ 1. In this case, we can approximate the Green's function by
            
                𝒢_L(ω̅,k̅,T̅)≈(πT̅/12)e^-(2ν_k-1)log12/πT̅(iω/π T+ (2ν_k-1))+O((2ν_k-1)^2,iω̅(2ν_k-1)) .
            
            The connection coefficients are real and only depend on the momentum. We have computed them numerically and determined their asymptotic behaviour at large spatial momentum (c.f. Appendix <ref>)
            
                α_±∼∓ 6^∓k̅/√(12)k̅^-1/2 ,
                   β_±∼∓ 6^∓k̅/√(12) k̅^3/2 logk̅.
            
            Considering the large momentum asymptotic behaviour of the IR Green's function (<ref>) and connection coefficients (<ref>), we see that the spectral function vanishes exponentially fast at large spatial momentum. A special point is k̅=3 (ν_k=1), where 𝒢_L diverges and the spectral function has its first zero. For a numerical estimate, one can thus restrict the momentum integral to smaller values.
            
            When T/μ_q→ 0 we can approximate the connection coefficients by their k̅→ 0 values, that we can compute analytically
            
                α_- ≈ 1+O(ν_k-1/2),   β_-≈ -3/2(2ν_k-1)+O[(ν_k-1/2)^2],
                   α_+≈4/3(2ν_k-1)+O(1),  β_+≈ 4+O(ν_k-1/2).
            
            Thus β_+α_–β_-α_+≈ 6 and
            
                ρ_L(ω/2π T,k̅,T̅)≈6(2ν_k-1)^2(2ν_k-1)^2 + (4 A_k(T̅) ω/3π T)^2  A_k(T̅)ω/π T≈ 6A_k(T̅)ω/π T .
            
            Where we have defined
            
                A_k(T̅)=(πT̅/12)e^-(2ν_k-1)log12/πT̅ .
            
            
            Numerically, at low frequencies the IR Green's function is well approximated by
            
                𝒢_L(ω̅,k̅,T̅)≈   Γ(-ν_k) Γ(1/2+ν_k)/Γ(ν_k) Γ(1/2-ν_k)(πT̅/12)^2ν_k(1-i/2tan(πν_k)ω/T +
                   + π ^2 tan ^2(πν_k )-ψ ^(1)(1/2-ν_k )+ψ ^(1)(1/2+ν_k)8π^2ω^2/T^2)+O(ω^3/T^3),
            
            where ψ^(1) is a polygamma function. In order to see how good is the approximation we plot the real and the imaginary part of the full IR Green function and its approximated expression. We define a reduced IR Green function
            
                g_L(ω̅,k̅) = Γ(ν_k)/Γ(-ν_k) (12/πT̅)^2ν_k𝒢_L(ω̅,k̅,T̅),
            
            so that the plots will be independent of T̅. In Figure <ref> we plot the full imaginary part of g_L (dashed lines) with the approximated formula (solid lines) for some values of ν_k, as a function of ω/T ∈ (0,2). Then, we plot the full real part of g_L (dashed lines) with the approximated formula (solid lines), as a function of ω/T ∈ (0,2), keeping just the first term in the expansion, Figure <ref>, and then adding also the quadratic correction, Figure <ref>. The values of ν_k are 0.6,0.7,0.75,0.8,0.9 which correspond to colors from blue to light purple. We observe that as k becomes smaller, the real part becomes better approximated by the leading term.
            
            
            
            
            
            § ESTIMATE OF WEAK REACTION RATES AT STRONG COUPLING
            
            
            
            We will proceed to estimate the weak rate. Since all the quark chemical potentials are equal at beta equilibrium and the spectral function (<ref>) vanishes at zero frequency, we can use the simplified formula (<ref>) to compute the rate. Taking into account that only the transverse components contribute to leading order, and rescaling the variables in the momentum integral q^0=2 π T s, k=μ_q k̅/2, 
            
                Λ_ud→ us≈ N^2μ_q^7/2^7π^2∫_-∞^∞ ds ∫_0^∞ dk̅ k̅^2 ρ_L^2(s,k̅,T̅)/sinh^2(π s) .
            
            Introducing (<ref>), and changing variables to k̅^2=3(4ν^2-1),
            
                Λ_ud→ us≈ N^26^2μ_q^7/2^7π^22(πT̅/12)^2
                  ∫_0^∞ ds (2s)^2/sinh^2(π s)∫_1/2^∞ dν 12 ν√(3(4ν^2-1))  e^-2(2ν-1)log12/πT̅ .
            
                Λ_ud→ us≈ N^23√(3)μ_q^5 T^2/4∫_0^∞ ds s^2/sinh^2(π s)∫_1/2^∞ dν ν√((4ν^2-1))  e^-2(2ν-1)log12/πT̅ .
            
            We have
            
                ∫_0^∞ ds s^2/sinh^2(π s)=1/6π
            
            Changing variables ν=1/2+u,
            
                Λ_ud→ us≈ N^2√(3)μ_q^5 T^2/8π∫_0^∞ du  (1+2u) √(u(u+1))  e^-4ulog12/πT̅ .
            
            We use that the main contributions to the integral come from u≪ 1:
            
                ∫_0^∞ du  (1+2u) √(u(u+1))  e^-4ulog12/πT̅≈∫_0^∞ du  u^1/2  e^-4ulog12/πT̅=Γ(3/2)/(4log12/πT̅)^3/2=√(π)/2(4log12/πT̅)^3/2 .
            
            Then,
            
                Λ_ud→ us≈ N^2/128√(3/π)μ_q^5 T^2(log6μ_q/π T)^-3/2 .
            
            We have computed numerically the rate for values of the temperature T/μ_q=10^-1,10^-2,10^-3,10^-4, considering the full dependence of the connection coefficients, and several approximations for the IR Green's function, namely the leading order (<ref>), the second order approximation (<ref>), and the full result (<ref>). In the numerical integration over momentum, we cut the integral at ν_k=1, where the spectral function has its first zero. The exponential suppression already present at this value of the spatial momentum makes higher momentum contributions to be exceedingly small, so we believe this is a good approximation. We show our numerical results together with the analytic approximation at low temperatures (<ref>) in Figure <ref>. We see that indeed the analytic formula approximates well the numerical result at low temperatures, even up to T/μ_q ≲ 10^-2, but for T/μ_q∼ 10^-1 the deviations are already sizeable, although the analytic formula still serves as an estimate of the order of magnitude.
            
            
            
            
            
            
            § DISCUSSION AND OUTLOOK
            
            
            
            In this work, we have derived formulas (<ref>) and (<ref>) for the weak equilibration rate of quark flavor densities n_a (a=u,d,s) to leading order in the Fermi coupling and exact in the QCD coupling. Although we have focused on quark matter, the formulas are derived in an explicitly gauge-invariant way and rely solely on symmetry arguments. Thus, they should be valid for any state of QCD at nonzero temperature, including hadronic (confined) phases. In a confined phase, the natural degrees of freedom are baryons rather than quarks. Baryons are composed of three quarks and can be grouped according to their flavor composition. Therefore, the chemical potential for a baryon B^abc∼ q^a q^b q^c would be μ_B^abc = μ_a + μ_b + μ_c, with μ_a,b,c being the quark chemical potentials corresponding to each flavor. The associated baryon densities should satisfy the conditions
            
                n_a = 3 n_B^aaa + 2 ∑_b ≠ a n_B^aab + ∑_b,c ≠ a n_B^abc,
            
            where the order of the flavor labels is immaterial, and n_B^abc would, in principle, include all contributions from baryon resonances with the same flavor composition and with a mass below the value of the baryon chemical potential, assuming the decay channels to lower resonances become closed. With these identifications, one can combine the expressions in (<ref>) to obtain the rates for the evolution of the density of different baryon species, or rather for the evolution of some combinations of them.
            
            Returning to our analysis, we have obtained the leading order low-temperature behavior of the weak reaction rate in a large-density state of strongly coupled quark matter at beta equilibrium. For simplicity, we have only considered massless quarks, from which we were able to extract an analytic expression (<ref>) for the rate coefficient as defined in (<ref>) and (<ref>), resulting in
            
                λ_ds≈ G_F^2 sin^2 θ_C cos^2 θ_C  N^2/32√(3/π)μ_q^5 T^2 ( log6 μ_q/π T)^-3/2.
            
            A fit to the large momentum behavior of QCD correlators fixes the normalization constant to N = N_c / (6 π^2), Eq. (<ref>). We can compare this with the perturbative result <cit.> including the leading QCD correction <cit.>
                λ_ds≈ G_F^2 sin^2 θ_C cos^2 θ_C 64/4 π^2μ_q^5 T^2 (1 + 4 α_s/9 πlogκ α_s μ_q/T)^4,
            
            where κ≈  0.28(2/3π)^1/2≈ 0.13. The logarithmic enhancement of the rate at low temperatures originates from a non-Fermi-liquid behavior of quark matter due to long-range interactions (see also <cit.>). Although the logarithm should not be too large in order for the perturbative expansion to be valid, the fourth power with which it appears can produce a significant effect on the rate. Considering the strong coupling result, the holographic dual geometry has also been conjectured to describe a non-Fermi (“marginal”) liquid <cit.>. However, the logarithmic factor operates in the opposite way, suppressing the rate. Furthermore, compared to the perturbative result, the logarithm will be much larger, as there are no coupling-dependent factors in the argument suppressing the ratio μ_q/T. As shown in Figure <ref>, the analytic formula (<ref>) provides a good approximation for temperatures T/μ_q below 10^-2, but at higher temperatures, corrections to this formula start to become significant. However, their effect is to further suppress the rate.
            
            Several improvements could be applied to the model, with perhaps the most significant being the inclusion of non-zero quark masses. In the perturbative calculation of the rate, the mass enters the dispersion relation of the fermions in the scattering amplitude and removes the low-energy contributions from the massive quarks. In the holographic model, the scalar field dual to the quark bilinears would be non-zero, and its backreaction can change the near-horizon geometry, thus affecting the rate behavior. A generic expectation is that the AdS_2 throat in the near-horizon geometry would be replaced by a hyperscaling Lifshitz geometry (see <cit.> and references therein), which would probably result in the rate becoming proportional to a different power of the temperature depending on the dynamical exponent and the hyperscaling parameters.
            
            Another extension of the model would be to study phases of paired quark matter, as quark pairing can affect transport properties <cit.>. A paired state would have a quark condensate with a non-trivial flavor structure, which could be captured by turning on a profile of the scalar field reproducing the same structure.
            
            In conclusion, our study provides a framework for understanding weak interaction rates in strongly coupled quark matter and highlights the importance of non-perturbative methods in studying dense QCD phases. Future work should focus on refining these models, incorporating more realistic features such as non-zero quark masses and quark pairing, to enhance our understanding of neutron star interiors and their observational signatures.
            
            
            
            
            
            § ACKNOWLEDGMENTS
            
            
            We thank Niko Jokela for useful discussions. This work is partially supported by the AEI and the MCIU through the
            Spanish grant PID2021-123021NB-I00. A.O. thanks the University of Oviedo for the kind hospitality during the preparation of this work. 
            
            
            
            
            
            
            
            
            § NON-CONSERVATION EQUATIONS OF FLAVOR CURRENTS
            
            
            
            With three massless flavors of quarks q^i=(u,d,s) and one flavor of massless leptons l^a=(ν_e,e) (with only left-handed component for the neutrino), the flavor symmetry group is SU(3)_q_L× SU(3)_q_R× SU(2)_l_L× U(1)_e_R. Unequal mass terms for the quarks and the electron break explicitly the flavor group to U(1)^5, where the quarks and electron U(1)s are vector-like.
            
            The EW sector, in particular, W-boson exchange introduces a term in the effective action of the form (Fermi model)
            
                Δ L_EW=-2√(2) G_F (J_ch^μ)^† J_ch μ,
            
            where the J_ch^μ is the left-charged current
            
                J_ch^μ=ν_e Lγ^μ e_L+cosθ_C u_Lγ^μ d_L+sinθ_C u_Lγ^μ s_L,
            
            with θ_C the Cabbibo angle, which gives the change of basis between mass and flavor eigenstates. Let us introduce the notation
            
                (J_q^μ)^i_ j=q̅_j γ^μ q^i,  (J_l^μ)^a_ b=l̅_b γ^μ l^a .
            
            Then, in terms of current components,
            
                J_ch^μ=(J_l L^μ)^2_ 1+cosθ_C (J_q L^μ)^2_ 1+sinθ_C  (J_q L^μ)^3_ 1,
            
            Where J_l L^μ is the current for the SU(2)_L lepton symmetry and J_q L^μ is the current for the SU(3)_L flavor symmetry. The EW term becomes
            
                Δ L_EW=   -2√(2) G_F [ cos^2θ_C (J_q L^μ)^ 2_ 1 (J_q L μ)^1_ 2+sin^2 θ_C  (J_q L^μ)^ 1_ 3 (J_q L μ)^3_ 1.
                   +cosθ_C sinθ_C( (J_q L^μ)^ 2_ 1 (J_q L^μ)^ 1_ 3+(J_q L^μ)^3_ 1(J_q L^μ)^ 1_ 2)
                   +cosθ_C ((J_l L^μ)^2_ 1(J_q L μ)^1_ 2+(J_l L^μ)^1_ 2(J_q L μ)^2_ 1)
                   .+sinθ_C ((J_l L^μ)^2_ 1(J_q L μ)^ 1_ 3+(J_l L^μ)^ 1_ 2(J_q L μ)^3_ 1)] .
            
            Let us neglect the terms coupling quarks and leptons. We can rewrite the remaining terms in the following way
            
                Δ L_EW=   -2√(2) G_F [ cos^2θ_C  tr(P_11 J_q L^μ P_22 J_q L μ) +sin^2 θ_C  tr(P_11 J_q L^μ P_33 J_q L μ).
                   .+cosθ_C sinθ_C( tr(P_11 J_q L^μ P_32 J_q L μ)  + tr(P_11 J_q L^μ P_23 J_q L μ) )] .
            
            Where we have defined the projectors (P_kl)^i_ j=δ^i_kδ_j^l.
            
            By doing SU(3)_L× SU(3)_R rotations of equal angle we can derive Ward identities for the vector SU(3) flavor currents. The currents transform as
            
                δ_θ J_q^μ=iθ_A [T^A,J^μ_q]
            
            
            Defining J_q^μ=J_q L^μ+J_q R^μ and J_l^μ=J_l L^μ+J_l R^μ. The covariant derivative under a background flavor gauge potential A_μ^f is
            
                D_μ J_q^μ=∂_μ J_q^μ-i[A_μ^f,J_q^μ] .
            
            We will take the normalization for the gauge group generators to be tr(T^A T^B)=1/2δ^AB. Then, the Ward identities are
            
                2tr(D_μ J_q^μ T^A)=
                   2i√(2) G_F[ cos^2θ_C  (tr(P_11 [T^A, J_q L^μ] P_22 J_q L μ)+tr(P_11 J_q L^μ P_22 [T^A,J_q L μ]  )).
                    +sin^2 θ_C  ( tr(P_11 [T^A,J_q L^μ] P_33 J_q L μ)+tr(P_11J_q L^μ P_33 [T^A,J_q L μ]  ))
                   +cosθ_C sinθ_C( tr(P_11 [T^A,J_q L^μ] P_23 J_q L μ)+tr(P_11 J_q L^μ P_23 [T^A,J_q L μ]  ). 
                   .. + tr(P_11 [T^A,J_q L^μ] P_32 J_q L μ)+ tr(P_11 J_q L^μ P_32[T^A, J_q L μ]  ) )] .
            
            Combining the traces and using the cyclic property we have that
            
                D_μ J_q^μ =
                   i√(2) G_F( cos^2θ_C  [ J_q L μ , P_11 J_q L^μ P_22 +P_22 J_q L^μ P_11 ].
                    +sin^2 θ_C  [ J_q L μ , P_11 J_q L^μ P_33 +P_33 J_q L^μ P_11 ]
                   .+cosθ_C sinθ_C(  [ J_q L μ , P_11 J_q L^μ P_23 +P_23 J_q L^μ P_11 ]+  [ J_q L μ , P_11 J_q L^μ P_32 +P_32 J_q L^μ P_11 ])) .
            
            In general, there will also be contributions on the right-hand side depending on the quark masses, but they should vanish for diagonal components.
            
            We can do a similar derivation taking into account the leptons if we introduce projectors
            
                (P̃_ai)^b_ j=δ_a^bδ^i_j,  (P̃^T_ia)^j_ b=δ^a_bδ^j_i,
            
            Then
            
                P̃J^μ_qP̃^T∼ J_l^μ,  P̃^T J^μ_lP̃∼ J_q^μ.
            
            And introduce the SU(2)_L× SU(2)_R rotations of the leptons, such that the diagonal rotations transform the vector currents as
            
                δ_ϕ J_l^μ=iϕ_α[τ^α,J^μ_l].
            
            With similar normalization for the generators tr(τ^ατ^β)=1/2δ^αβ.
            
            
            
            
            
            
            § LINEARIZED EQUATIONS OF MOTION OF FLUCTUATIONS
            
            
            
            The transverse spatial components of the gauge fields k^i L_i=k^i R_i=0 decouple from the other fluctuations. Their equations are
             
            0    =   L_i”+ (f'/f-1/Z)L_i'+
            ( ω̅^2/f^2-k̅^2/f)L_i  ,
            
            
             0    =   R_i”+ (f'/f-1/Z)R_i'+
            ( ω̅^2/f^2-k̅^2/f)R_i  , 
            
            When the quark masses are nonzero, the time L_0, R_0 and longitudinal L_∥ =k^i/k L_i, R_∥=k^i/k R_i components of the gauge fields are coupled to the scalar fluctuations, but they decouple otherwise. The left gauge field equations are
            
                0   =   L_∥”+(f'/f-1/Z)L_∥'+ω̅^2/f^2L_∥+k̅ω̅/f^2L_0, 
                
                     0   =   L_0”-1/ZL_0'-k̅^2/fL_0-k̅ω̅/fL_∥ ,
                
                     0   =   ω̅L_0'+k̅ f L_∥' .
            
            The right gauge field equations are
            
                0   =   R_∥”+(f'/f-1/Z)R_∥'+ω̅^2/f^2R_∥+k̅ω̅/f^2R_0,
                
                     0   =   R_0”-1/ZR_0'-k̅^2/fR_0-k̅ω̅/fR_∥,
                
                     0   =   ω̅R_0'+k̅ f R_∥' .
            
            The scalar equations are     
            
                0    =   X”+(f'/f-3/Z)X'+(ω̅^2/f^2-k̅^2/f+3/Z^2 f)X  .
            
            We can simplify the equations in the longitudinal sector by introducing the electric fields
            
                E_L=k̅ L_0+ω̅L_∥,     E_R=k̅ R_0+ω̅R_∥ .
            
            The constraints (<ref>) and (<ref>) fix the time and longitudinal components of the gauge fields in terms of the electric fields and the scalars
            
                L_∥'=ω̅E_L'/ω̅^2-k̅^2 f ,     L_0'=- k̅ fE_L'/ω̅^2-k̅^2 f ,
                   R_∥'=ω̅E_R'/ω̅^2-k̅^2 f ,     R_0'=- k̅ fE_R'/ω̅^2-k̅^2 f ,
            
            The combinations ω̅f/Z(<ref>)+k̅/Z(<ref>), and  ω̅f/Z(<ref>)+k̅/Z(<ref>) produce the equations for the left and right electric fields respectively
            
                0   =   ∂_Z (f/ZE_L'/ω̅^2-k̅^2 f)+1/f ZE_L, 
                
                0   =   ∂_Z (f/ZE_R'/ω̅^2-k̅^2 f)+1/f ZE_R .
            
            
            
            
            § SOLUTIONS IN ADS REISSNER-NORSDTROM
            
            
            
            We want to solve (<ref>) and (<ref>) without the terms depending on the masses, in the charged black brane geometry (<ref>) with parameters (<ref>). We can solve the equations of motion in the near-extremal AdS_2 throat (<ref>) and in the asymptotically AdS_5 metric (<ref>). When |ω̅|→ 0 the two sets of solutions have an overlapping region ζ→ 0 in the throat and Z→ 1 in the outer region. To leading order in the small |ω̅| expansion, the equations of motion of the left gauge fields (<ref>) and (<ref>) in the inner region are
            
                0    =   ∂_ζ^2 L_i +F'F∂_ζ L_i +1/ζ^2 F^2(ζ^2 -k̅^2/12F)L_i ,
                
                 0    =   ∂_ζ^2 E_L + ( F'/F-2/ζk̅^2/121/ζ^2-k̅^2/12F)∂_ζ E_L +1/ζ^2 F^2(ζ^2-k̅^2/12F)E_L .
            
            Outside the AdS_2 throat, we can approximate the geometry by the extremal (zero temperature) Reissner-Norstrom solution, with f(Z)=1-3Z^4+2Z^6. The equations in the outer region to leading order in the small frequency expansion are
            
                0    =   ∂_Z(f(Z)Z∂_Z L_i ) -k̅^2ZL_i+O(ω̅^2) ,
                
                 0    =   ∂_Z(∂_Z E_LZ) -k̅^2Z f(Z)E_L +O(ω̅^2) .
            
            
            
             §.§ Inner solutions of the transverse components
            
            
            Let us first find the solutions to the equations in the inner region (<ref>). At T=0 (F=1), there is a simple solution to the equation of the transverse components
            
                L_i,I(ζ) =√(ζ) ( C_i^(1)Θ(ω) H_ν_k^(1)(ζ)+C_i^(2)Θ(-ω)  H_ν_k^(2)(ζ)) ,
            
            where the coefficients have been chosen so that the solution satisfies ingoing boundary conditions and we have defined
            
                ν_k=√(1/4+k̅^2/12) .
            
            These solutions should match with the solutions of the outer region for ζ→ 0. The leading terms in the expansion are
            
                L_i,I(ζ) ∼  A_i ζ^1/2-ν_k+B_i ζ^1/2+ν_k ,
            
            where
            
                A_i    =      -i2^ν_kΓ(ν_k)/π(C_i^(1)Θ(ω)-C_i^(2)Θ(-ω)) ,
                
                B_i    =   1/2^ν_kΓ(ν_k+1)(C_i^(1)Θ(ω)+C_i^(2)Θ(-ω)+i(πν_k)(C_i^(1)Θ(ω)-C_i^(2)Θ(-ω))  ) .
            
            In terms of the original coordinate Z, related to ζ through (<ref>), the expansion (<ref>) is
            
                L_i,I(ζ) ∼  A_i(|ω̅|/12)^1/2-ν_k[(1-Z)^-1/2+ν_k+𝒢_L(ω̅,k̅) (1-Z)^-1/2-ν_k) ,
            
            where the IR Green's function is 
            
                𝒢_L(ω̅,k̅)=π2^2ν_kΓ(ν_k)Γ(1+ν_k)(-(πν_k)+i sign(ω))(|ω̅|/12)^2ν_k.
            
            At non-zero temperatures, it is still possible to find analytic solutions
            
                L_i,I(ζ) = A_i φ_ν_k^-(ζ)+B_iφ_ν_k^+(ζ) ,
            
            where
            
                φ_ν_k^±(ζ)=(1-ζ^2/ζ_0^2)^-i/2sign(ω)ζ_0(ζ/ζ_0)^1/2±ν_k_2 F_1(1/4-i/2sign(ω)ζ_0±ν_k/2 , 3/4-i/2sign(ω)ζ_0±ν_k/2 ,  1±ν_k ; ζ^2/ζ_0^2) .
            
            The condition that the solution is ingoing fixes
            
                B_i=Γ(-ν_k) Γ(1/2-i  sign(ω)ζ_0+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-i  sign(ω) ζ_0-ν_k)  A_i .
            
            Using that Z_*≈ 1, we arrive at 
            
                L_i,I(ζ) ∼  A_i(|ω̅|/12 ζ_0)^1/2-ν_k[(1-Z)^-1/2+ν_k+𝒢_L(ω̅,k̅,ζ_0) (1-Z)^-1/2-ν_k) ,
            
            where now
            
                𝒢_L(ω̅,k̅,ζ_0)=Γ(-ν_k) Γ(1/2-i  sign(ω)ζ_0+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-i  sign(ω) ζ_0-ν_k)(|ω̅|/12 ζ_0)^2ν_k.
            
            In terms of temperature the IR Green's function is approximately
            
                𝒢_L(ω̅,k̅,T̅)≈Γ(-ν_k) Γ(1/2-i  ω/2π T+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-i  ω/2π T-ν_k)(πT̅/6)^2ν_k.
            
            For frequencies smaller than the temperature, we can expand the IR Green's function as
            
                𝒢_L(ω̅,k̅,T̅)≈Γ(-ν_k) Γ(1/2+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-ν_k)(πT̅/6)^2ν_k(1-i/2tan(πν_k)ω/T)+O(ω^2/T^2).
            
            The imaginary term produces the leading contribution to the spectral function. We can obtain these first terms in the ω/T by first extracting the ingoing factor from the solutions
            
                L_i,I(ζ)=(1-ζ^2/ζ_0^2)^-i/2sign(ω)ζ_0ψ_i(ζ),
            
            and then expanding the equations in powers of ζ_0 and solving order by order. Changing variables to u=ζ^2/ζ_0^2, the equations for the first two orders ψ_i=ψ_i^0+ζ_0ψ_i^1+⋯ are
            
                0   =   ∂_u(√(u)(1-u)∂_u ψ_i^0)+1-4ν_k^2/16 u^3/2ψ_i^0 ,
                
                 0   =   ∂_u(√(u)(1-u)∂_u ψ_i^1)+1-4ν_k^2/16 u^3/2ψ_i^1+i sign(ω)√(u)(∂_uψ_i^0+1/4 uψ_i^0 ) .
            
            The solutions to zeroth order are
            
                ψ_i^0(u)= A_i^0 φ_ν_k^0,-(u)+B_iφ_ν_k^0,+(u),
            
            with
            
                φ_ν_k^0,±(u)= u^1/4(1± 2ν_k)_2 F_1(1/4±ν_k/2 , 3/4±ν_k/2 ,  1±ν_k ; u ) .
            
            Imposing regularity of this solution when u→ 1 (ζ→ζ_0), i.e. removing logarithmic terms, fixes the relation between the coefficients
            
                B_i^0=Γ(-ν_k) Γ(1/2+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-ν_k)  A_i^0 .
            
            The next-order solution can be obtained using a Green's function. First we identify the subleading solution when u→ 0 (ζ→ 0), φ^<, and the ingoing solution when u→ 1, φ^>,
            
                φ^<(u)=φ_ν_k^0,+(u) ,  φ^>(ζ)=φ_ν_k^0,-(u)+Γ(-ν_k) Γ(1/2+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-ν_k)φ_ν_k^0,+(u) .
            
            With these and the Wronskian of the two subleading and ingoing functions we construct a Green's function that maintains the boundary conditions of the leading order solution 
            
                G(u,u')=-1/ν_k{[ φ^<(u)φ^>(u')             ,          u<u'; φ^>(u)φ^<(u')             ,          u>u' ].
            
            Then, taking into account that ψ_i^0=A_i^0φ^>(u), the subleading solution is
            
                ψ_i^1(u)=-i sign(ω)A_i^0∫_0^1 du' G(u,u')√(u')(∂_u'φ^>(u')+1/4 u'φ^>(u') ) .
            
            We can now expand the leading and subleading solutions for u→ 0. One can check that all contributions from ψ_i^1 are subleading compared to the leading term ∼ u^1/4(1-2 ν_k). Also, there is just one contribution to the term ∼ u^1/4(1+2 ν_k), from a total derivative 
            
                ∫_0^1 du' G(u,u')√(u')(∂_u'φ^>(u')+1/4 u'φ^>(u') )∼φ^<(u)lim_u→ 0∫_u^1 du' ∂_u'(√(u')(φ^>(u'))^2/2)=φ^<(u)  (φ^>(1))^2/2 .
            
            Taking this into account, the relevant terms in the expansion are
            
                ψ_i^0    ∼   ( u^1/4(1- 2ν_k)+Γ(-ν_k) Γ(1/2+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-ν_k)  u^1/4(1+2 ν_k))A_i^0  ,
                ψ_i^1    ∼    -i sign(ω)πtan(πν_k) Γ(-ν_k) Γ(1/2+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-ν_k)  u^1/4(1+2 ν_k)  A_i^0 .
            
            It is straightforward to check that this reproduces the first two terms in the ω/T expansion of the  IR Greens' function (<ref>).
            
            
            
            
            
             §.§ Inner solutions of the longitudinal components
            
            
            The equation for the electric field in (<ref>) does not have a simple analytic solution, but since we are interested in frequencies ω/T≲ 1, we can apply the same expansion as for the transverse components to find the leading contribution to the spectral function.
            
            We first extract the ingoing factor from the solutions
            
                E_L,I(ζ)=(1-ζ^2/ζ_0^2)^-i/2sign(ω)ζ_0ψ_i(ζ),
            
            and then expand the equations in powers of ζ_0 and solve them order by order. Changing variables to u=ζ^2/ζ_0^2, the equations for the first two orders ψ_i=ψ_i^0+ζ_0ψ_i^1+⋯ are
            
                0   =   ∂_u(u^3/2∂_u ψ_i^0)+1-4ν_k^2/16 u^1/2(1-u)ψ_i^0 ,
                
                 0   =   ∂_u(u^3/2∂_u ψ_i^1)+1-4ν_k^2/16 u^1/2(1-u)ψ_i^1+i sign(ω)u^3/2/1-u(∂_uψ_i^0+3-u/4 u(1-u)ψ_i^0 ) .
            
            The solutions to zeroth order are
            
                ψ_i^0(u)= A_i^0 φ_ν_k^0,-(u)+B_iφ_ν_k^0,+(u),
            
            with
            
                φ_ν_k^0,±(u)= u^-1/4(1∓ 2 ν_k)_2 F_1(-1/4±ν_k/2 , 1/4±ν_k/2 ,  1±ν_k ; u ) .
            
            Imposing regularity of this solution when u→ 1 (ζ→ζ_0), i.e. removing logarithmic terms, fixes the relation between the coefficients
            
                B_i^0=Γ(-ν_k) Γ(3/2+ν_k)/2^2ν_kΓ(ν_k) Γ(3/2-ν_k)  A_i^0 .
            
            It turns out that this makes the solution vanish at the horizon, something that will be relevant below.
            
            The next-order solution can be obtained in terms of a Green's function. First, we identify the subleading solution when u→ 0 (ζ→ 0), φ^<, and the ingoing solution when u→ 1, φ^>,
            
                φ^<(u)=φ_ν_k^0,+(u) ,  φ^>(ζ)=φ_ν_k^0,-(u)+Γ(-ν_k) Γ(3/2+ν_k)/2^2ν_kΓ(ν_k) Γ(3/2-ν_k)φ_ν_k^0,+(u) .
            
            With these and the Wronskian of the two subleading and ingoing functions we construct a Green's function that maintains the boundary conditions of the leading order solution 
            
                G(u,u')=-1/ν_k{[ φ^<(u)φ^>(u')             ,          u<u'; φ^>(u)φ^<(u')             ,          u>u' ].
            
            Then, taking into account that ψ_i^0=A_i^0φ^>(u), the subleading solution is
            
                ψ_i^1(u)=-i sign(ω)A_i^0∫_0^1 du' G(u,u')u'^3/2/1-u'(∂_u'φ^>(u')+3-u'/4 u'(1-u')φ^>(u') ) .
            
            We can now expand the leading and subleading solutions for u→ 0. One can check that all contributions from ψ_i^1 are subleading compared to the leading term ∼ u^-1/4(1+2 ν_k). Also, there is just one potential contribution to the term ∼ u^-1/4(1-2 ν_k), from a total derivative 
            
                ∫_0^1 du' G(u,u')u'^3/2/1-u'(∂_u'φ^>(u')+3-u'/4 u'(1-u')φ^>(u') )
                       ∼φ^<(u)lim_u→ 0∫_u^1 du' ∂_u'(u'^3/2/1-u'(φ^>(u'))^2/2)=φ^<(u)  lim_u'→ 1(φ^>(u'))^2/2(1-u')=0 .
            
            Taking this into account, the relevant terms in the expansion are
            
                ψ_i^0    ∼   ( u^-1/4(1+2 ν_k)+Γ(-ν_k) Γ(3/2+ν_k)/2^2ν_kΓ(ν_k) Γ(3/2-ν_k)  u^-1/4(1-2 ν_k))A_i^0  ,
                ψ_i^1    ∼    0 ×  A_i^0  u^-1/4(1-2 ν_k) .
            
            Thus, the leading ω/T correction actually vanishes for the longitudinal components. This implies that for the calculation of the spectral function, it is enough to compute the transverse contribution.
            
            
            
            
            
             §.§ Outer solutions of the longitudinal components
            
            
            Although we will not need these solutions to compute the leading contributions to the rates, it will be useful to study them as they can be found analytically.
            
            We are looking for solutions of the electric field in (<ref>). First we will change coordinates u=Z^2, so the equations takes the form of a Schroedinger equation
            
                E_L,O”(u)-k̅^24u(2u+1)(u-1)^2E_L,O=0 .
            
            We can transform the above equation into a hypergeometric equation by the following change of coordinate and field redefinition
            
                E_L,O(u) = (1+2u)^c_±(1-u)^b_±t_±(u),      c_± = 12±ν_k,     b_± = 1-c_± = 12∓ν_k,
                   x=3u1+2u,           u =x3-2x.
            
             Then, we get the following hypergeometric equation
                
                x(1-x)t_±”(x)+(± 2ν_k -1)x t_±'(x) -k^212 t_±(x)=0.
            
                We can compare the above equation to the standard form x(1-x)t_±”(x) + (c_±-(a_±+b_±+1)x)t_±'(x) -a_± b_± t(x) =0, to get
            
                c_± = 0,         a_± = {1/2∓ν_k,-1/2∓ν_k},     b_± = {-1/2∓ν_k,1/2∓ν_k}.
            
            A general solution can be written as a linear combination of a normalizable and a non-normalizable mode at the boundary. In terms of the original radial variable, they have the form
            
                E_O^(0) = α ϕ_NN(Z) + β ϕ_N(Z),
                   ϕ_NN(Z) = (1-Z^2)^1/2+ν_k(1+2Z^2)^1/2-ν_k[((12-2ν_k^2)H_-1/2+ν_k-ν_k^2(-1+log 3)+
                             + 112(1+3log 3))3Z^21 + 2Z^2 _2F_1(12+ν_k,32+ν_k,2,3Z^21 + 2Z^2)+
                             +1Γ(-1/2+ν_k)Γ(1/2+ν_k)G_2,2^2,2(
                [ 3Z^21+2Z^2 ]|[ 1/2-ν_k,32-ν_k;            0,1;                ])],
                   ϕ_N(Z) = (1-Z^2)^1/2+ν_k(1+2Z^2)^1/2-ν_k3Z^21+2Z^2 _2F_1(12+ν_k,32+ν_k,2,3Z^21 + 2Z^2),
            
            where H_-1/2+ν_k is the (-1/2+ν_k)^th harmonic number and G is the Meijer function.
            
            The non-normalizable and normalizable solutions have been chosen so that their near-boundary expansion is
            
                ϕ_NN(Z) = 1 + 32(-1+4ν_k^2)Z^2log Z + O(Z^4log Z),               ϕ_N(Z) = Z^2 + O(Z^4) .
            
            We want to determine the coefficients α and β for solutions with a selected behaviour close to the horizon Z→ 1. The expansion close to the horizon is 
            
                E_L,O(Z) ∼ A(1-Z)^1/2+ν_k + B(1-Z)^1/2-ν_k +...         Z→ 1,
            
            These match with the expansions of the φ_ν_k^0,± inner solutions (<ref>) in the ζ→ 0 limit.
            By requiring that A = 1 and B = 0 we then determine the value of α and β for the minus mode while, if we impose A = 0 and B = 1 we find the coefficients for the plus mode. They turn out to be
            
                α_∓ =6^±ν_k -1/2Γ (±ν_k +1)√(π) Γ(±ν_k +3/2),
                   β_- = -2^ν_k -5/2 3^ν_k -1/2ν_k  Γ (ν_k ) ((6-24 ν_k ^2) H_ν -1/2-12 ν_k ^2 (1+log (3))+1+log (27))√(π) Γ(ν_k +3/2),
                   β_+ = -πν_k  ( (2 πν_k ) ((6-24 ν_k ^2) H_ν_k -1/2-12 ν_k ^2 (log (3)-1)+1+log (27))+3 π(4 ν_k ^2-1)  ^2(πν_k ))2^-ν_k +3/2 3^ν_k +1/2Γ(1/2-ν_k ) Γ(3/2-ν_k ) Γ (2 ν_k +1) .
            
            
            Since in order to compute the rates we have to perform an integral over all momentum, it is interesting to ascertain the behaviour of the spectral function at large spatial momentum, which will be in part determined by the behaviour of the connection coefficients. From the analytic formulas above one can see that, for ν_k→∞, their asymptotic form is
            
                α_- ∼ 6^ν_k-1/2(πν_k)^-1/2,  α_+ ∼ -6^-ν_k-1/2(πν_k)^-1/2(πν_k) ,
                   β_-∼ 6^ν_k+1/2ν_k^3/2log(3^1/2e^-1/2e^γ_Eν_k),
                   β_+ ∼  6^-ν_k+1/2ν_k^3/2√(π)(1-(πν_k)/πlog(3^1/2e^-1/2e^γ_Eν_k)) .
            
            We can recover this behaviour of the connection coefficients by using a WKB approximation at large spatial momentum. The equation of motion (<ref>) is already in Schroedinger form with a potential
            
                U_E(u)=k̅^24u(2u+1)(u-1)^2 .
            
            We expand the solutions as
            
                E_L,O(u) = Cexp(k̅ S_0(u) + S_1(u) + O(1/k̅)).
            
            The equation for the leading term in the 1/k̅ expansion is
            
                S_0'(u)^2-18 u^4-12 u^3+4 u = 0,
            
            which has an analytical solution
            
                S_0(u) = ±1√(3)tanh ^-1( √(3u2u+1)) + c_0.
            
            Then, the subleading term is given by
            
                S_1(u) = log(√(2) √(1- u) √(u (1+2u))) + c_1.
            
            Then the solution up to this order reads
            
                E_L,O(u)≈ψ(u)= C_± √(1- u) √(u (1+2u))exp(±k̅√(3)tanh ^-1( √(3u1+2u))).
            
            The WKB solution near the boundary has the following expansion
            
                ψ(u)∼ C_± u^1/4e^± k√(u)          u→ 0.
            
            Note that the two terms in the solution directly correspond to the "plus" and "minus" modes at the horizon
            
                ψ(u) ∼C_±(11-u)^±k̅/√(12)         u→ 1.
            
            We have to fix the overall coefficient to properly normalize the WKB solutions in the original coordinate z which are
            
                E(z) = C_± √(1- z^2) √(z) (1+2z^2)^1/4exp(±k̅√(3)tanh ^-1( √(3z^21+2z^2))).
            
            Then, the overall coefficients are found to be
            
                C_± = 6^∓k̅/√(12).
            
            Now we have to match these results with the solutions close to the boundary and read the boundary coefficients. Close to the boundary, the potential is to leading order
            
                U_E(u) ∼k̅^24u.
             
            A solution for this approximated potential can be given in terms of modified Bessel functions
            
                ψ(u)≈ c_1 √(u) K_1(k̅ √(u))+ c_2√(u) I_1(k̅ √(u)).
            
            The expansion for large k̅ is
            
                √(u) K_1(k̅ √(u))∼√(π2k̅)u^1/4e^-k̅√(u),                √(u) I_1(k̅ √(u))∼u^1/4√(2πk̅)e^k̅√(u),
            
            which is consistent with the WKB solution. We fix the coefficient c_1,2 with the formulas for C_± so that the WKB solution for z→ 0 and the solution close to the boundary for large k̅ coincide. Finally, we can read off from the z→ 0 expansion the coefficients α_± and β_± which are respectively the coefficients of the non-normalizable and normalizable mode. In a compact form, they are
            
                α_±∼∓ 6^∓k̅/√(12)k̅^-1/2 ,
                   β_±∼∓ 6^∓k̅/√(12) k̅^3/2 logk̅.
            
            Comparing to (<ref>), and taking into account ν_k∼k̅/√(12) in this limit, the WKB approximation captures the leading k̅ dependence up to the oscillatory factors.
            
            
            
            
             §.§ Outer solutions of the transverse components
            
            
            We are looking for solutions of the transverse components in (<ref>). We do not have a close analytic solution in this case, so we must resort to numerics. The general solution is a linear combination of two modes η_L,∓(Z) which when Z→ 1 match with the inner solutions (<ref>) for ζ→ 0,
            
                η_L,±(Z) ∼(1-Z)^-1/2∓ν_k + ...          Z→ 1.
            
            There is another basis of solutions we can use to write a general solution, consisting of the normalizable and non-normalizable modes, with a boundary expansion at Z→ 0
                ϕ_NN(Z) = 1 + 32(-1+4ν_k^2)Z^2log Z + O(Z^4log Z),               ϕ_N(Z) = Z^2 + O(Z^4)
            
            The change of basis is determined by the connection coefficients α_±,β_±:
            
                η_L,±(Z) = α_±ϕ_NN(Z) +β_±ϕ_N(Z).
            
            The connection coefficients can be determined from the solutions by matching the η_L ± solutions and their first derivative with a linear combination of the normalizable and non-normalizable solutions at any intermediate point 0<Ẑ<1:
            
                [ α_±; β_± ]
                =
                1W(Z)[ -∂_Zϕ_N(Z)     ϕ_N(Z); ∂_Zϕ_NN(Z)   -ϕ_NN(Z) ][    η_L,±(Z); ∂_Zη_L,±(Z) ]|_Z = Ẑ ,
            
            where W(Z) = ϕ_N(Z)∂_Zϕ_NN(Z)-∂_Zϕ_N(Z)ϕ_NN(Z).  We will obtain the connection coefficients using this formula, by finding numerically the η_L ± solutions using shooting from the horizon, and the ϕ_N,ϕ_NN solutions using shooting from the boundary. We use Ẑ=0.2 as the matching point, but we have checked that changing this value through the interval of numerical integration does not affect the result within our numerical precision.
            
            For the shooting from the boundary, we introduce a cutoff at Z=10^-5 and use a series expansion to give initial values for the solution and its first derivative at the cutoff. The expansion is 
            
                ϕ_N(Z) = ∑_n = 1^N_a a_n Z^2n,
                   ϕ_NN(Z) = b_0 + ∑_n = 2^N_b b_n Z^2n +log(Z)∑_n = 1^N_c c_n Z^2n.
            
            Where we take N_a=8, N_b=6 and N_c=7. The first coefficients of the expansion given that a_1 = b_0 = 1 are
            
                a_2 = k̅^28,    a_3 = 1 + k̅^4192,    a_4 = -12 + 5k̅^224 + k̅^69216, ...
                   b_2 = -3k̅^464,    b_3 = k̅^26 - 7k̅^62304  ...      c_1 =k̅^22,    c_2 = k̅^416, …
            
            
            For the shooting from the horizon we introduce a cutoff at Z=1-10^-5. We extract the leading power in the horizon expansion from the solutions
            
                η_L,∓(Z) = (1-Z)^-1/2±ν_kp_∓(Z) .
            
            The equation of motion we solve is
            
                ∂_Z^2 p_∓(Z) +(1+2Z(-1 ±ν_k)Z(-1+Z) + f'(Z)f(Z))∂_Z p_∓(Z)+
                   +(±ν_k-1)[(2-5Z ± 2Zν_k)f(Z) + 2(1-Z)Z(6(-1+Z)(1± 2ν_k) -f'(Z))]4 (Z-1)^2 Z f(Z) p_∓(Z).
            
            To give initial values at the cutoff we use the series expansion 
            
                p_∓ = ∑_n=0^N_j j_∓,n(1-Z)^n,
            
            and shooting it from the horizon towards the boundary. We expand up to N_j=8 terms, with the first coefficients in the expansion, given that j_∓,0=1, are
            
                j_∓,1 = 8√(3)√(3+k̅^2)∓ 24 ± 7k̅^212(√(3)√(3+k̅^2)± 3),    j_∓,2 = k̅^2 (49 √(3)√(k̅^2+3)±534)-156 (√(3)√(k̅^2+3)∓3)864 (√(3)√(k̅^2+3)±6), …
            
            
            Although we do not have an analytic formula for the connection coefficients, we can obtain analytic expressions for their asymptotic behaviour at large spatial momentum, by applying the same WKB approximation we used for the coefficients of the longitudinal components. First, we should put the equation of motion for L_i in (<ref>) in Schroedinger form. This is achieved by the change of variables u=Z^2 followed by the redefinition
            
                L_i(u)= (1-u) (1+2 u)^-1/2ψ_L_i(u) .
            
            The equation of motion is now
            
                ψ_L_i”(u)-U_L_i(u)ψ_L_i(u) = 0 ,      U_L_i(u) = k^2/4 (1-u)^2 u (2 u+1)-3 (u+1)/(1-u) (2 u+1)^2 .
            
            At large k̅ the potential is to leading order the same as for the longitudinal modes (<ref>)
                U_L_i(u)∼k^2/4 (1-u)^2 u (2 u+1)=U_E(u) .
            
            Therefore, the large k̅ WKB analysis proceeds in the same way as for the longitudinal components, and the connection coefficients have the save asymptotic behaviour given in (<ref>).
            
            
            We have plotted the connection coefficients in Figures <ref>- <ref>. In Figures <ref> and <ref> we plot α_+ and α_-. The coefficient of the non-normalizable mode, α_+, is multiplied by the factor 2(1-ν_k)(2ν_k-1) in order to remove the divergences in k̅ =0, 3. We will examine the origin of this divergence in more detail below. 
            In Figures <ref> and <ref> are shown the plots of 2(1-ν_k)β_+ and (2ν_k-1)^-1β_-. 
            
            
            
            
              §.§.§ Low momentum behavior of connection coefficients
            
            
            When k̅=0 (ν_k=1/2) the solutions in the outer region are
            
                η_L,-^0(Z)=1,    η_L,+^0(Z)=2/1-Z^2+4/3log[1+2Z^2/1-Z^2]-1/2-4/3log3/2 .
            
            The expansions at the boundary and the horizon of the η_L,+^0 solution are
            
                η_L,+^0(Z)    ∼   3/2-4/3log3/2+6Z^2+⋯,
                η_L,+^0(Z)    ∼    (1-Z)^-1-4/3log(1-Z)+⋯ .
            
            Therefore, the connection coefficients of these solutions are
            
                α_-^0=1, β_-^0=0,  α_+^0=3/2-4/3log3/2, β_+^0=6.
            
            We can relate these to the k̅→ 0 limit of the connection coefficients of the non-zero momentum solutions. First, note that the coefficient j_-,n in (<ref>) go to zero in the k̅→ 0 limit, so η_L,-→η_L,-^0. On the other hand j_+,1 in (<ref>) is singular in this limit, but the singularity does not propagate to other coefficients in the expansion, one can already see that it is absent in j_+,2. In order to remove the singular term we can take the following combination of the solutions
            
                η̃_L,+(Z)=η_L,+(Z)-j_1,+η_L,-(Z) .
            
            Expanding this solution at the horizon and for k̅→ 0 one can check that it coincides with the expansion in (<ref>). We then conclude that  η̃_L,→η_L,+^0 in the k̅→ 0 limit. This determines the k̅→ 0 behaviour of the connection coefficients
            
                α_-∼α_-^0,     β_-∼β_-^0,    α_+∼α_+^0+j_+,1α_-^0,     β_+∼β_+^0+j_+,1β_-^0 .
            
            Since j_+,1→∞ and β_-→ 0, in order to completely determine β_+ we need to find the non-zero subleading contribution to β_-. We expand
            
                η_L,-(Z)= (1-Z)^-1/2(1-2ν_k)(1+(ν_k-1/2)η_-^1(Z)+O[(ν_k-1/2)^2]) .
            
            The solution for η_-^1(Z) can be found explicitly
            
                η_-^1(Z)=   1/9(log (1-Z) (9-12 log (Z))+9 log (1+Z)+π ^2+9-9 log (2)-6Li_2(-2).
                   .+6 log (Z) (2 log(1+2
                   Z^2/1+Z)+3 Z^2/1-Z^2) + 6Li_2(-Z^2)-6Li_2(Z^2)) .
            
            The boundary and horizon expansions are
            
                η_L,-^1(Z)    ∼   1/9(9 + π^2 - 9log(2)- 6Li_2(-2)) + 3 Z^2 (-1 +2 log Z)+⋯,
                η_L,-^1(Z)    ∼   log(1-Z)+ 11/6(1-Z)+⋯ .
            
            Therefore
            
                β_-∼ -3/2(2ν_k-1) .
            
            Using now that to leading order
            
                j_+,1∼4/3(2ν_k-1),
            
            We find that when k̅→ 0
                α_-∼ 1+O(ν_k-12),    β_-∼ -3/2(2ν_k-1)+O[(ν_k-1/2)^2] ,
                   α_+∼43(2ν_k-1)+O(1),    β_+∼ 4+O(ν_k-1/2) .
            utphys ] ]


 Paper Title: Considerations on measuring spin-spin correlation of hyeprons in heavy-ion experiments.






dyshen@fudan.edu.cn
Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China

Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China

Brookhaven National Laboratory, Upton, New York 11973


§ ABSTRACT 
     The significant global spin alignment observed for ϕ mesons in heavy-ion collisions has sparked intense discussions about its origin and implications. One explanation suggests that fluctuations in the strong force field may introduce strong spin correlations between strange (s) and anti-strange (s̅) quarks, leading to the global spin alignment of ϕ mesons. Extending this line of research, the theoretical community has proposed studying the spin correlation between Λ and Λ̅ hyperons. In this paper,  we construct experimental observables and make connections between them and the theoretical proposed quantities.
    

Considerations on measuring spin-spin correlation of hyeprons in heavy-ion experiments
    Aihong Tang
    July 2024
======================================================================================

The concept of global polarization in heavy-ion collisions was first proposed in Ref.<cit.>. In off-central heavy-ion collisions, quarks are predicted to be globally polarized along the direction of angular momentum due to spin-orbital coupling in strong interactions. This effect has recently been confirmed experimentally through measurements of Λ global polarization <cit.>. However, another related phenomenon, vector meson spin alignment, cannot be explained solely by vorticity or other  mechanisms, most of which are conventional <cit.>. The measured ρ_00, the 00-component of the spin density matrix (with 1/3 indicating no spin alignment), of ϕ mesons is significantly above 1/3 <cit.>, while predictions from the quark coalescence model, considering Λ data, suggest <cit.>  ρ_00 below 1/3 with a much smaller deviation from 1/3. This discrepancy between hyperon polarization and vector meson spin alignment implies richer spin physics beyond global quark polarization. It was later realized that spin correlation between quarks and anti-quarks plays a crucial role in determining vector meson spin <cit.>. For instance, short-range correlations between s and s̅ quarks due to strong-force fields successfully explain the observed ϕ meson spin alignment <cit.>.

The spin-spin correlation of spin 1/2 particles was usually studied in helicity-frame in e^+e^- collisions or p+p collisons <cit.>, and the correlation function is defined as 

    c_ij = f_↑↑ + f_↓↓ - f_↓↑ - f_↑↓/f_↑↑ + f_↓↓ + f_↓↑ + f_↑↓,

where f_m_im_j=⟨ m_im_j | ρ̂ | m_im_j ⟩ denotes the fraction of fermion pairs in the spin state of | m_im_j  ⟩ along spin quantization direction, with ρ̂ being the spin density matrix. In general, the quantization directions of particle i and j differ in helicity-frame. However, in heavy-ion collisions, we are interested in global spin-spin correlations, meaning the quantization directions of particle i and j are fixed. In this case, Eq. <ref> is modified as <cit.>:

    c'_ij = c_ij - P_i P_j,

where P_i and P_j are the polarization of particle i and j respectively. It is clear that the c'_ij represents ⟨σ̂_m_iσ̂_m_j⟩ - ⟨σ̂_m_i⟩⟨σ̂_m_j⟩, where σ̂_m is the Pauli operator along spin quantization direction. 

The left-hand side of Eq. <ref> represents the global spin correlations in heavy-ion collisions and can be calculated by measuring the c_ij and global polarization. In the following, we demonstrate how to measure c_ij using the example of Λ-Λ spin correlation. This method can be easily extended to other hyperons. 

We begin with the proton distribution from Λ weak decay,

    dN/dcosθ^*=1/2(1+α_Λ P_Λcosθ^*),

where θ^* is the angle, in Λ's rest frame, between proton momentum and Λ spin quantization direction. P_Λ defines Λ polarization and α_Λ is a decay constant. For Λ pairs in the spin state | ↑↑⟩ (both are fully polarized thus P_Λ=1), the distribution of proton-pairs can be written as

    dN_↑↑/dcosθ_i^*dcosθ_j^*=1/4 [1+α_Λ (cosθ_i^*+cosθ_j^*) + α_Λ^2cosθ_i^*cosθ_j^*  ],

where N_↑↑ denotes proton pairs from Λ pairs in | ↑↑⟩ state. The footnote i and j denote the first proton and the second proton, respectively. Similarly, for Λ pairs in the spin state of | ↓↓⟩, | ↑↓⟩, and | ↓↑⟩, the proton pairs are described by 

    dN_↓↓/dcosθ_i^*dcosθ_j^*   =   1/4 [1-α_Λ (cosθ_i^*+cosθ_j^*) + α_Λ^2cosθ_i^*cosθ_j^*  ], 
    dN_↑↓/dcosθ_i^*dcosθ_j^*   =   1/4 [1+α_Λ (cosθ_i^*-cosθ_j^*) - α_Λ^2cosθ_i^*cosθ_j^*  ], 
    dN_↓↑/dcosθ_i^*dcosθ_j^*   =   1/4 [1-α_Λ (cosθ_i^*-cosθ_j^*) - α_Λ^2cosθ_i^*cosθ_j^*  ].

In reality, Λ pairs are mixture of the above spin states, with the fractions given by f_m_im_j. Therefore, the general distribution of daughter proton pairs can be expressed as follows :

    dN/dcosθ_i^*dcosθ_j^*   =    f_↑↑dN_↑↑/dcosθ_i^*dcosθ_j^* + f_↓↓dN_↓↓/dcosθ_i^*dcosθ_j^* + f_↑↓dN_↑↓/dcosθ_i^*dcosθ_j^* + f_↓↑dN_↓↑/dcosθ_i^*dcosθ_j^*
       =   1/4 [ 1 + Aα_Λcosθ_i^* 
    + Bα_Λcosθ_j^* +Cα_Λ^2cosθ_i^*cosθ_j^*  ],

where 

    A=f_↑↑-f_↓↓+f_↑↓-f_↓↑/f_↑↑ + f_↓↓ + f_↓↑ + f_↑↓, 
    
    B=f_↑↑-f_↓↓-f_↑↓+f_↓↑/f_↑↑ + f_↓↓ + f_↓↑ + f_↑↓, 
    
    C=f_↑↑+f_↓↓-f_↑↓-f_↓↑/f_↑↑ + f_↓↓ + f_↓↑ + f_↑↓.

The parameter C is the c_ij defined in Eq. <ref>. Indeed, the parameter A could be reduced to (f_↑-f_↓)/(f_↑+f_↓) which represents the polarization of the first Λ by definition, where f_↑=f_↑↑+f_↑↓ (f_↓=f_↓↑+f_↓↓) is the fraction of it in the spin state | ↑⟩ (| ↓⟩). Similarly, the parameter B represents the polarization of the second Λ. It is reasonable to expect that parameter C reflects the spin correlation between the first and second Λ. 
Mathematically, the c_ij can be obtained by the integral of cosθ^*_icosθ^*_j, it is equivalent to calculate the average ⟨cosθ^*_icosθ^*_j ⟩ in the experiment. 

Based on Eq. <ref>, we establish the connection between experimental observable and the Λ spin correlation as used in the context of theoretical work: 

    c'_ΛΛ=9/α^2_Λ⟨cosθ^*_icosθ^*_j ⟩ - P_Λ^2.

In experiment, the spin quantization direction (orbital angular momentum direction) is estimated from the event plane, which has a finite resolution. To correct for event plane resolution, similar to Ref. <cit.>, we project Eq. <ref> onto transverse plane, resulting in :

    dN/dΔϕ_i^*dΔϕ_j^* = 1/16π(1+Aα_ΛsinΔϕ_i^* + Bα_ΛsinΔϕ_j^*+Cα_Λ^2π/4sinΔϕ_i^*sinΔϕ_j^*),

and c'_ΛΛ can be calculated by

    c'_ΛΛ=64/π^2α_Λ^2⟨sinΔϕ_i^*sinΔϕ_j^* ⟩ - P_Λ^2,

where Δϕ_i^* (Δϕ_j^*) is the azimuthual angle relative to the reaction plane for proton i (j) in first (second) Λ rest frame. Considering a finite event plane resolution, Eq. <ref> can be expressed as

    c'_ΛΛ=64/π^2α_Λ^2⟨sin(ϕ_i^*-Ψ_ EP)sin(ϕ_j^*-Ψ_ EP) ⟩/⟨cos^2(Ψ_ EP - Ψ_ RP) ⟩ - P_Λ^2,

where Ψ_ EP and Ψ_ RP denote the event plane and the reaction plane, respectively. Note that

    ⟨cos^2(Ψ_ EP - Ψ_ RP) ⟩ =   1 + ⟨cos(2(Ψ_ EP - Ψ_ RP)) ⟩/2,

and ⟨cos(2(Ψ_ EP - Ψ_ RP)) ⟩ can be determined from sub-event method in experiment <cit.>.

Building on the structure of balance function <cit.>, we propose a spin balance function to detect the difference of spin correlation between “same sign” pairs (Λ-Λ or Λ̅-Λ̅) and “opposite sign” pairs (Λ-Λ̅),

    B(c'_OS,c'_SS) = 1/2(c'_ΛΛ̅ - c'_ΛΛ/c'_ΛΛ̅ + c'_ΛΛ + c'_Λ̅Λ - c'_Λ̅Λ̅/c'_Λ̅Λ + c'_Λ̅Λ̅ ).

The advantage of the above equation is that the detector effect and flavor independent correlation will be canceled, only additional correlation between s and s̅ (in Λ-Λ̅ pairs) remains.

To summarize, in this paper, we investigated the correlation function used in a theoretical context, identified its connection to experimental observables, and outlined the procedure to measure it in experiments.  Additionally, we proposed a method for measuring  the  correlation using a spin balance function, which minimize the impact of detector and other trivial effects. This work will be instrumental in studying hyperon spin correlations and facilitating the understanding of spin-spin correlations and their implications for fluctuations in the strong force field. 




§ ACKNOWLEDGMENT

This work was inspired by the theoretical study <cit.> and benefited from discussions with X.-N. Wang, Z.-T. Liang, Q. Wang, X.-L. Sheng., and Z. Tu.  






 Paper Title: Improving relativistic energy density functionals with tensor couplings.



Improving relativistic energy density functionals with tensor couplings]Improving relativistic energy density functionals with tensor couplings













[1,2,3]Stefan Typelstypel@ikp.tu-darmstadt.de

4]Shalom Shlomos-shlomo@tamu.edu





*[1]Technische Universität Darmstadt, Fachbereich Physik, Institut für Kernphysik, Schlossgartenstraße 9, Darmstadt, 64289, 
    Germany

[2]GSI Helmholtzzentrum für Schwerionenforschung, Theorie,  Planckstraße 1, Darmstadt, 64291, 
    Germany

[3]Helmholtz Forschungsakademie Hessen für FAIR (HFHF), GSI Helmholtzzentrum für Schwerionenforschung, Campus Darmstadt,  Schlossgartenstraße 2, Darmstadt, 64289, 
    Germany

[4]Texas A&M University, Cyclotron Institute,
  
  College Station, 77843, Texas, USA





Energy density functionals (EDFs)
    have been used extensively
    with great success to calculate properties
    of nuclei and to predict the equation of state (EOS)
    of dense nuclear matter.
    Besides non-relativistic EDFs, mostly of the Skyrme or Gogny type,
    relativistic EDFs of different types are in widespread use. In these
    latter approaches, the effective in-medium interaction is described by an
    exchange of mesons between nucleons. In most cases, only minimal
    meson-nucleon couplings are considered. The effects of additional
    tensor couplings were rarely investigated. In this work, a new relativistic
    EDF with tensor couplings and density dependent minimal meson-nucleon
    couplings will be presented. The parameters of the model are
    determined using a carefully selected set of experimental data with
    realistic uncertainties that are determined self-consistently. Predictions
    for various nuclear observables, the nuclear matter
    equation of state, and properties of neutron stars
    are discussed.

  



















[
    [
    August, 2024
================





§ INTRODUCTION


For more than half a century, energy density functionals (EDFs) have been used with great success in the description of atomic nuclei across the nuclear chart and nuclear matter over a large range of densities and isospin asymmetries. Many improvements were introduced in these phenomenological models leading to a more and more precise prediction of nuclear observables but, nevertheless, the quest for further improvements remains. There are different strategies to develop more accurate EDFs. They either follow a systematic approach, e.g., using expansions in power of small quantities, or try to incorporate known effects in a more heuristic way.

EDFs exist in different versions. Most well known are non-relativistic approaches that employ convenient representations of effective in-medium interactions between nucleons, e.g., the Skyrme or Gogny type models
<cit.>.
Relativistic or covariant EDFs start from a field-theoretic approach where the interaction is described by meson-exchange taking medium dependent effects into account in different ways
<cit.>.
Either self-interactions of the mesons are introduced in so-called nonlinear (NL) models or a density dependence (DD) of the the meson-nucleon couplings is utilized. These extensions of the original Walecka model
<cit.> are essential to obtain reasonable properties of nuclei and nuclear matter. Applying certain approximations to the full many-body wave function of the system of interest, e.g., of Hartree, Hartree-Fock or Hartree-Fock-Bogoliubov type, a set of coupled equations is found which has
to be solved self-consistently.

In the present work, new parameterisations of a relativistic EDF will be presented that include tensor couplings between nucleons and mesons in addition to the standard minimal couplings. Such an extension has been considered before in early applications of relativistic EDFs, see, e.g., Ref. <cit.>,
but it has been explored scarcely in depth.
Including ρ-tensor, but no ω-tensor,
contributions in relativistic Hartree-Fock calculations showed that
the single-nucleon spectrum could be improved <cit.>.
The interest in relativistic EDFs with tensor couplings has been grown recently as it can affect the description of surface properties of nuclei and (indirectly) the equation of state of nuclear matter.
New parametrisations were developed in Ref.<cit.> for relativistic models with meson-nucleon couplings that depend on vector or scalar densities of the medium in various combinations,
but using a more limited number of nuclear observables in the parameter fit
as compared to the present work.
The effects of tensor couplings on different properties 
of nuclei and, using the resulting equation of state (EOS) of dense matter, on neutron stars were explored in Ref. <cit.>, however without a thorough determination of model parameters. Here, we shall embark on the development of a new relativistic EDF with minimal and tensor couplings using a careful selection of nuclear observables that are used in the fit of the model parameters. The medium dependence of the interaction is taken into account with a density dependence of the minimal couplings using the functional forms introduced in Ref. <cit.>.

The content of this paper is structured as follows:
The formalism of the relativistic energy density functional is presented
in Section <ref>. The main quantities are introduced and 
the notation is established, in particular regarding the contribution
with the tensor couplings. This section also provides the relevant equations
for the description of spherical nuclei and nuclear matter in two subsections.
In Section <ref> the approach to determine the model parameters
is outlined. The functional form of the density dependent couplings is
specified and the selection of the observables and nuclei in the fit is
presented. Details of the fitting strategy are also given. Results and their
discussion follow in Section <ref>, where properties of nuclei,
nuclear matter, and neutron stars are considered
in individual subsections. Finally, we close with a summary and outlook
in Section <ref>.
  


§ RELATIVISTIC ENERGY DENSITY FUNCTIONAL



In a relativistic theory of nuclei and nuclear matter, nucleons and mesons are assumed to be the relevant degrees of freedom. Protons and neutrons are represented as Dirac spinors Ψ_η (with η=π for protons and η=ν for neutrons) following Fermi statistics, whereas mesons are described by bosonic fields. The strong interaction between nucleons can be represented effectively by an exchange of mesons of different type. They often share their properties with those mesons of the same name and quantum numbers that are observed in experiments but are not necessarily identical. Actually, it is possible to describe the main features of the strong interaction by a rather small set of effective mesons that reduces the complexity of the calculations and the number of model parameters.

In relativistic EDFs, it is common practise to introduce an iso-scalar, Lorentz-scalar σ meson to model the attraction and an iso-scalar, Lorentz-vector ω meson to model the repulsion between nucleons. In order to take the isovector dependence of the interaction into account, an iso-vector, Lorentz-scalar δ meson and an iso-vector, Lorentz-vector ρ meson are also considered. The notation of these fields follows their properties, i.e., σ, ω^μ, δ⃗, and ρ⃗^μ with Lorentz indices μ and the vector symbol for the three isovector components. The pion as another important meson is not included as it does not appear in the final equations after applying the basic approximations of the approach. Besides the above mentioned mesons, the photon field, denoted by the vector A^μ, completes the number of degrees of freedom to include also the electromagnetic interaction in the model.

The derivation of an EDF in a relativistic approach follows several steps that are detailed below. First, a Lagrangian density is formulated containing the contributions for the free propagation of the fields and their interaction. From it, the coupled field equations are derived that have to be solved self-consistently. This can be achieved in certain approximations and expressions for the energy of the system and other quantities of interest can be obtained. They are functions of the particle fields and their densities, so that the EDF is a generalized form of a density functional. It depends on a rather small number of parameters that need to be determined by appropriate procedures.
In the formulas given below, we use a system of units
where ħ = c = 1. The value ħ = 197.3269804 MeV fm/c
is used for the conversion of units.



 §.§ Lagrangian density, field equations, and energy density



The development of the relativistic EDF starts from a
covariant Lagrangian density

    ℒ    =    ∑_η=π,νΨ_η( γ_μi𝒟^μ - σ_μλ𝒯^μλ
    - ℳ_η) Ψ_η
          
    + 1/2( 
     ∂^μσ∂_μσ - M_σ^2 σ^2
     + ∂^μδ⃗·∂_μδ⃗ 
     - M_δ^2 δ⃗·δ⃗
           
     - 1/2 G^μλ G_μλ
     + M_ω^2 ω^μω_μ
     - 1/2H⃗^μλ·H⃗_μλ
     + M_ρ^2 ρ⃗^μ·ρ⃗_μ
      - 1/2 F^μλ F_μλ)   ,

with several contributions. Here, the field tensors of the mesons and of the
massless electromagnetic field are given by the usual expressions;

    G^μλ  =  ∂^μω^λ
      - ∂^λω^μ  ,
          H⃗^μλ  =  ∂^μρ⃗^λ
      - ∂^λρ⃗^μ  ,
          
      F^μλ  =  ∂^μ A^λ - ∂^λ A^μ  .

The minimal coupling to the nucleons
is realized by the corresponding terms in the covariant derivative

    i𝒟^μ = i ∂^μ - Γ̃_ωω^μ
      - Γ̃_ρρ⃗^μ·τ⃗
      - Γ_γ  A^μ1+τ_3/2  ,

for the Lorentz-vector mesons and the photon, and in the mass operator

    ℳ_η =  M_η - Γ̃_σσ 
     - Γ̃_δδ⃗·τ⃗  ,

for the Lorentz-scalar mesons, respectively.
The term in Eq. (<ref>) with the quantity

    𝒯^μλ = Γ_Tω/2M_π G^μλ
      + Γ_Tρ/2M_πH⃗^μλ·τ⃗  ,

accounts for the additional tensor coupling of the nucleons to the mesons.
Tensor couplings to the electromagnetic field are not included.
The masses of the nucleons and the mesons are denoted
by M_η (η=π,ν) and M_j (j=σ,δ,ω,ρ),
respectively.
The isospin dependence is expressed with help of the
isopin matrices τ_k (k=1,2,3)
in analogy to the Pauli spin matrices σ_k.
The coupling factors Γ̃_j
in Eqs. (<ref>), and (<ref>)
are assumed to be functionals of the nucleon fields Ψ_η, and
Ψ_η= Ψ_η^†γ^0. 
In contrast,
the tensor couplings Γ_Tω, and Γ_Tρ,
in Eq. (<ref>) as well as
the electromagnetic coupling Γ_γ
in Eq. (<ref>) are simple constants.
The devision by the proton mass in Eq. (<ref>)
is introduced to have dimensionless
couplings Γ_Tω and Γ_Tρ.

The field equations of the nucleons, mesons, and the photon
can be derived in the usual way from the Euler-Lagrange equations.
However, they cannot be solved in the field-theoretical sense and
approximations are needed to arrive at tractable equations.
The fields of the mesons and the photons are not quantized but considered
as classical fields and will be denoted by the same symbols as in the
original Lagrangian density (<ref>) in the following. 
For the nucleons, the Hartree approximation is applied, i.e.,
the many-nucleon wave function Ψ of the system of interest
is written as a product of single-nucleon states Ψ_η k_ηm_η,

    Ψ = ∏_η=π,ν∏_k_η,m_η
      w_η k_ηm_ηΨ_η k_ηm_η  ,


with occupation factors w_η k_ηm_η.
Only values of w_η k_ηm_η=0 and 1 are allowed.
This condition reflects the fermionic nature of the nucleons.
The meaning of the quantum numbers k_η, and m_η, is explained
in Subsection <ref>.
Only single-nucleon states of positive-energy are considered here,
corresponding to the no-sea approximation
and only a finite number of states contributes in the
many-nucleon wave function.

For the dependence of the couplings Γ̃_j
in Eqs. (<ref>), and (<ref>), a so-called
vector density dependence is assumed, i.e., with the approximation
mentioned above, they become functions Γ_j of the vector density,

    ϱ_v = √(j_μj^μ)  ,

with the total current,

    j^μ = ∑_η=π,ν∑_k_η,m_η w_ik_ηm_η Ψ_ik_ηm_ηγ^μΨ_ik_ηm_η  ,

of the nucleons. The quantity ϱ_v is in fact a Lorentz scalar
but has to be distinguished from the actual scalar density

    ϱ_s = ∑_η=π,ν∑_k_η,m_ηw_ik_ηm_η Ψ_η k_ηm_ηΨ_η k_ηm_η  ,

that is used for couplings with a scalar density dependence,
c.f. Ref. <cit.>.

In the actual application to nuclei and nuclear matter, a specific frame
of reference is chosen and the covariance of the equations will be lost.
In particular, a frame is chosen where the system is at rest,
the couplings depend only on the zero-component of the current
ϱ_v = j^0, and j⃗=0.
Since only static solutions without a change of isospin are considered,
only the zero-component of the Lorentz-vector fields and
a single component of the isovector quantities have to be taken
into account, since all other components vanish. 
Assuming spherical symmetry, the field equation of the nucleons assumes
the form of a modified Dirac equation,

    Ĥ_ηΨ_η k_ηm_η(r⃗)
      = [ α⃗·p̂⃗̂
        + β( M_η - S_η)
        + V_η + i γ⃗·r⃗/r T_η]
      Ψ_η k_ηm_η(r⃗)
      = E_η k_ηΨ_η k_ηm_η(r⃗)   ,

with α⃗=γ^0γ⃗, and β=γ^0,
for single-nucleon states of energy E_η k_η.
The interaction with the
meson and photon fields is expressed through the scalar potentials

    S_η(r⃗)  =  Γ_σσ
     + g_ηΓ_δδ  ,

vector potentials,

    V_η(r⃗)  =  Γ_ωω_0
     + g_ηΓ_ρρ_0 
     + 1+g_η/2Γ_γ A_0 + V^(R)  ,

and tensor potentials,

    T_η(r⃗) = - Γ_Tω/M_πr⃗/r·∇⃗ω_0
      - g_ηΓ_Tρ/M_πr⃗/r·∇⃗ρ_0  ,

with factors g_π=1, and g_ν=-1.
The vector density dependence of the couplings generates the
rearrangement contribution

    V^(R)(r⃗) =
      d Γ_ω/d n^(v) n_ωω_0
     + d Γ_ρ/d n^(v)  n_ρρ_0
     -  d Γ_σ/d n^(v) n_σσ
     -  d Γ_δ/d n^(v) n_δδ  ,

in the vector potential (<ref>). It contains derivatives of the
couplings with respect
to the vector density n^(v)=ϱ_v, as well as meson fields and
source densities n_j (j=σ,δ,ω,ρ)
that also appear in the meson field equations;

    -Δσ + M_σ^2σ    =    Γ_σ n_σ  ,
      
    
      -Δδ + M_δ^2δ    =    Γ_δ n_δ  ,
      
    
      -Δω_0 + M_ω^2ω_0    =    Γ_ω n_ω
      + Γ_Tω/m_p∇⃗·^(t)_ω  ,
      
    
      -Δρ_0 + M_ρ^2ρ_0    =    Γ_ρ n_ρ
      + Γ_Tρ/m_p∇⃗·^(t)_ρ  ,

for the non-vanishing components of the fields.
They are given by

    n_σ = n^(s) = n_π^(s)+n_ν^(s)  ,    
      n_δ = n_π^(s)-n_ν^(s)  ,
      
           n_ω = n^(v) = n_π^(v)+n_ν^(v)}: ,    
      n_ρ = n_π^(v)-n_ν^(v)  ,

with the vector and scalar densities;

    n_η^(v)(r⃗) =
      ∑_k_η,m_η w_η k_ηm_η n_η k_ηm_η^(v)
      = ∑_k_η,m_η w_η k_ηm_ηΨ_η k_ηm_ηγ^0Ψ_η k_ηm_η  ,

and

    n_η^(s)(r⃗) =
      ∑_k_η,m_η w_η k_ηm_η n_η k_ηm_η^(s)
      = ∑_k_ηm_η w_η k_ηm_ηΨ_η k_ηm_ηΨ_η k_ηm_η  ,

of nucleons, respectively.
In Eqs. (<ref>), and (<ref>), also
the divergences of the tensor currents

    ^(t)_ω = _π^(t) - _ν^(t)  ,
           ^(t)_ρ = _π^(t) - _ν^(t)  ,

with,

    _η^(t)(r⃗) =
      ∑_k_η,m_η w_η k_ηm_η _η k_ηm_η^(t)
      = ∑_k_η,m_η w_η k_ηm_η Ψ_η k_ηm_η i α⃗Ψ_η k_ηm_η  ,

contribute to the source terms of the Lorentz-vector meson fields
in Eqs. (<ref>), and (<ref>).
Note that these tensor currents are not the vector part of the usual
  four-currents,
  
    j_η^μ(r⃗) =
        ∑_k_η,m_η w_η k_ηm_η Ψ_η k_ηm_ηγ^μΨ_η k_ηm_η  .

Finally, the field equation for the electromagnetic field is given by
the usual Poisson's equation

    -Δ A_0 =  Γ_γ n_γ  ,

with the source density n_γ=n_π^(v).

In the application to nuclear matter, there is no spatial dependence
of the source densities and meson fields. They become constants and there
are no contributions from the tensor currents in Eqs. (<ref>),
and (<ref>). The field equations are solved trivially
and the tensor potential (<ref>) in the Dirac equation
(<ref>) vanishes too.
Nucleon states are then described by plane waves.

The energy density of the system
can be obtained as a component of the energy-momentum tensor

    T^μλ = ∑_φ∂ℒ/∂ (∂_μφ)∂^λφ - g^μλℒ  ,

where the sum runs over all fields
φ=Ψ_η k_ηm_η,Ψ_η k_ηm_η,
σ,δ,ω_0,ρ_0,A_0,
and the metric tensor is g^μλ=(-1,1,1,1).
It can be expressed as a sum,

    ε(r⃗) = T^00 =
      ε_nucleon(r⃗)
      + ε_field(r⃗)   ,

with the contribution

    ε_nucleon(r⃗) =
      ∑_η=π,ν∑_k_η,m_η
      w_η k_ηm_η E_η k_η n^(v)_η k_ηm_η  ,

of the nucleons and that of the fields

    ε_field(r⃗)     =    1/2( 
     ∇⃗σ·∇⃗σ + M_σ^2σ^2
     + ∇⃗δ·∇⃗δ + M_δ^2δ^2. 
          .
      - ∇⃗ω_0·∇⃗ω_0
      - M_ω^2ω_0^2
      - ∇⃗ρ_0·∇⃗ρ_0 - M_ρ^2ρ_0^2
      - ∇⃗ A_0·∇⃗ A_0)
     - V^(R) n^(v)

with an explicit rearrangement term.




 §.§ Spherical nuclei



For describing nuclei, the coupled differential equations (<ref>)
for the single-nucleon wave functions, and
(<ref>) - (<ref>), have to be solved self-consistently.
In the case of spherical symmetry, it is convenient to represent the
single-nucleon wave functions as

    Ψ_η k_ηm_η(r⃗) = 1/r( [    F_η k_η(r) 𝒴_κ_ηm_η(r̂); i G_η k_η(r) 𝒴_-κ_ηm_η(r̂) ])   ,

with radial functions F_η k_η, and G_η k_η,
and tensor spherical harmonics,

    𝒴_κ m(r̂) = ∑_m_l m_s
      (l_κ  m_l  s   m_s | j_κ  m)
      Y_l_κ m_l(r̂) χ_sm_s  ,

in the upper and lower components of the Dirac spinor.
Each state is identified by the index η=π,ν, and two quantum numbers
n_η, and κ_η, that are collectively denoted as k_η, and
a total angular momentum projection m_η.
The first, n_η=0,1,2,…,
is the principal quantum number related to the number of
nodes of the function F_η k_η;
the second, κ_η=± 1, ± 2,…, defines
the total angular momentum j_κ=|κ|-1/2
and the orbital angular momentum
l_κ=j_κ-κ/(2|κ|) of a state.
Then the Dirac equation (<ref>) is equivalent to the
set of two coupled differential equations;

    ( E_η k_η - V_η - M_η + S_η)
      F_η k_η  
      +( d/dr + κ_k/r - T_η)
      G_η k_η  =  0   ,

and,

    ( d/dr -κ_k/r + T_η)
      F_η k_η
      - ( E_η k_η - V_η + M_η - S_η)
      G_η k_η  =  0   ,

for the radial functions that have to be solved with the boundary conditions
F_η k_η(0)=G_η k_η(0)=0, and
lim_r → 0F_η k_η(r)= lim_r → 0G_η k_η(r) = 0.

The representation (<ref>) of the single-nucleon wavefunctions leads
to the expressions

    n_η k_ηm_η^(s)(r)  =  1/4π r^2[ | F_η k_η(r)|^2
        - | G_η k_η(r)|^2]

and

    n_η k_ηm_η^(v)(r)  =  1/4π r^2[ | F_η k_η(r)|^2
       + | G_η k_η(r)|^2]

for the scalar and vector single-nucleon densities
in Eqs. (<ref>), and (<ref>), respectively,
if an average over the quantum number m_η is applied.
This procedure is used to obtain spherical source densities
in the field equations for the mesons and the 
photon in case that sub-shells of given j are not completely filled.
The vector densities are normalized as,

    ∫ d^3r   n_η k_ηm_η^(v)(r)  = 1   ,

but, there is no such condition for the scalar densities.
The single-nucleon tensor currents have only
a component in radial direction in a spherical system. Then one can define

    n_η k_ηm_η^(t)  =  1/4π r^2[ F_η k_η^∗(r) G_η k_η(r)
       + F_η k_η(r) G_η k_η^∗(r) ]    ,

so that,

    _η k_ηm_η^(t)
      = n_η k_ηm_η^(t)r⃗/r  ,
              ∇⃗·_η k_ηm_η^(t)
      = 1/r^2d/dr( r^2 n_η k_ηm_η^(t))   ,

in Eq. (<ref>) after applying the averaging procedure.
The occupation factors in the calculation of the densities are determined
such that,

    ∑_k_π,m_π w_π k_πm_π = Z   ,         ∑_k_νm_ν w_ν k_νm_ν = N   ,

with the proton and neutron numbers Z and N of the nucleus,
respectively, with mass number A=Z+N.

The energy of the nucleus is finally obtained as

    E_nucleus = E_sp + E_field - E_corr  ,

with the contributions of the single-particle states

    E_sp = ∑_η=π,ν∑_k_η,m_η w_η k_ηm_η E_η k_η  ,

and of the fields, 

    E_field    =    1/2∫ d^3r  {Γ_σ n_σσ
      + Γ_δ n_δδ
      - Γ_γ n_γ A_0
          
        - [ Γ_ω n_ω
        + Γ_Tω/M_p∇⃗·_Tω] ω_0
      - [ Γ_ρ n_ρ
        + Γ_Tρ/M_p∇⃗·_Tρ] ρ_0
       - V^(R) n^(v)}  ,

where the field equations were exploited after a partial integration
using Eq. (<ref>). Since the translational symmetry is broken,
a correction for the center-of-mass motion has to be applied. It is calculated
as the expectation value

    E_corr = ⟨Ψ | P⃗^2/2M(Z,N) | Ψ⟩  ,

of the total momentum

    P⃗= ∑_j=1,Zp⃗_π j + ∑_j=1,Np⃗_ν j  ,

in non-relativistic form
with the total mass M(Z,N)=ZM_π+NM_ν of the nucleus,
neglecting the binding energy.

For the numerical solution of the coupled differential equations
(<ref>), and (<ref>), the Lagrange-mesh technique
is utilized, see Ref. <cit.> and references therein.  
This approach combines the virtues of an expansion in terms of
basis functions and a discretization on a grid. At the same time
it is numerically efficient and precise.
The radial wave functions are represented with their values at grid points
r_i=h x_i^(α), where h is a scale parameter, and
x_i^(α) are the zeroes
i=1,…,N_zero of the Laguerre
polynomials L_N^(α)(x) with N_zero=25,
and α=2|κ|.
The scale parameter h is chosen such that the largest zero
of L_N^(0) is at a radius r=0.95R, with R=r_0 A^1/3+10 fm, and
r_0=1.2 fm depending
on the mass number A of the nucleus. The field equations of the mesons,
(<ref>) - (<ref>),
and the photon, (<ref>),
are solved by expanding the fields and sources in terms of
Riccati-Bessel functions. Special care has to be taken for the electromagnetic
potential A_0 to obtain the correct asymtotic behavior
for large radii. For more details on the numerical techniques,
see Ref. <cit.>. 




 §.§ Nuclear matter


The solution of the field equations
(<ref>), (<ref>) - (<ref>), and (<ref>)
simplifies considerably in homogeneous and isotropic nuclear matter.
The source densities and meson fields become constants. The electromagnetic
field has to be discarded and all contributions of the tensor terms vanish. 
Nucleons are described by plane waves of momentum k_η
following the dispersion relation

    E_η k_η = √(k_η^2+( M_η^∗)^2) + V_η  ,

where the Dirac effective mass

    M_η^∗ = M_η - S_η  ,

and constant scalar and vector potentials (<ref>), and (<ref>), appear.
At zero temperature, all relevant quantities can be calculated analytically.
The scalar and vector densities assume the form;

    n_η^(s) = γ_η∫_0^k_η^∗d^3k_η/(2π)^3 M_η^∗/√((k_η)^2
          +(M_η^∗)^2)
       =   γ_ηM_η^∗/4π^2[ k_η^∗μ_η^∗
        - ( M_η^∗)^2lnk_η^∗+μ_η^∗/M_η^∗]   ,

and,

    n_η^(v) = γ_η∫_0^k_η^∗d^3k_η/(2π)^3 =
      g_η/6π^2(k_η^∗)^3  ,

in the continuum approximation, with the spin degeneracy factor γ_η=2.
The Fermi momenta k_η^∗ are related to the
effective chemical potentials 

    μ_η^∗ = μ_η - V_η
      = √((k_η^∗)^2+( M_η^∗)^2)  ,

of the nucleons.

The energy density of cold nuclear matter can be written as,

    ε = ∑_η=π,ν1/4[ 3μ_η^∗n_η^(v)
        + M_η^∗n_η^(s)]
      + 1/2∑_j=σ,δ,ω,ρ C_j n_j^2  ,

with the coupling-to-mass ratios 

    C_j = Γ_j^2/M_j^2  .

The pressure

    P = ∑_η=π,ν1/4[ μ_η^∗n_η^(v)
        - M_η^∗n_η^(s)] 
      - 1/2∑_j=σ,δ( C_j + n^(v) C_j^') n_j^2
      + 1/2∑_j=ω,ρ( C_j + n^(v) C_j^') n_j^2  ,

also depends on the derivatives

    C_j^' = dC_j/dn^(v)  ,

and can be derived using the thermodynamic definition
P = d(ε/n^(v))/dn^(v), or from the energy-momentum tensor
(<ref>), as P = (1/3)∑_μ=1^3T^μμ.
It is easily shown that energy density and pressure
fulfill the Hugenberg-van-Hove relation
<cit.>

    ε + P = ∑_η=π,νμ_η n_η^(v)  ,

since the rearrangement terms in the pressure combine with those
in the chemical potentials μ_η appearing in Eq. (<ref>).

The energy density (<ref>), and pressure (<ref>), are functions
of two independent densities, the proton and neutron vector densities
n_π^(v), and n_ν^(v).
Alternatively, they can be regarded
as functions of the baryon density

    n_b = ϱ_v = n^(v) = n_ν^(v) + n_π^(v)  ,

and the neutron-proton asymmetry

    α = n_ν^(v) - n_π^(v)/n_b  ,

that lies inside the interval [-1,+1]. Then it is convenient to define
the energy per nucleon

    E/A(n_b,α) = ε/n_b - M_nuc  ,

substracting the average nucleon mass M_nuc=(M_ν+M_π)/2.
A second derivative,

    E_sym(n_b) =
      1/2. d^2/dα^2E/A|_α=0  ,

gives the symmetry energy.
It characterizes how the isospin dependence of the energy per nucleon
changes with the baryon density.




§ DETERMINATION OF MODEL PARAMETERS



The relativistic EDF depends on several parameters: the masses of the
particles and the coefficients that describe the strength of the
couplings and their density dependence.
The masses of the nucleons and the mesons, except that of the
σ meson are usually chosen to be identical or close to the experimental
values. In the present application of the relativistic EDF
the δ meson is not included in the model.
The remaining fixed mass parameters of the nucleons and Lorentz-vector mesons
are given in Table <ref> for the models in this work.
The mass of the σ meson is used as an independent parameter because
it is not well defined by experimental data and affects the surface
properties of nuclei.



The actual values of the model parameters are determined by fitting selected
observables of finite nuclei that are sensitive to different contributions
in the EDF and are obtained more-or-less directly from experiment.
More indirectly obtained quantities,
e.g., characteristic nuclear matter parameters, that can depend
strongly on the extraction method from actual experimental data,
or other constraints, e.g.,  from ab-initio calculations
of nuclear matter or from astrophysical observations, are not used
in this procedure. A comparison to these data will be performed after
the model parameters have been determined.



 §.§ Functional form of density dependent couplings


The meson-nucleon couplings Γ_j 
are assumed to be functions
of the vector density ϱ_v=n^(v). In this work, the functional
form of the dependence is chosen as introduced
originally in <cit.> with

    Γ_j(ϱ) = Γ_j^(0) f_j(x)   ,

where Γ_j^(0)=Γ_j(ϱ_ref)
is the coupling at a reference density
ϱ_ref and the functions f_j depend on the
parameter x=ϱ_v/ϱ_ref≥ 0.
For the isoscalar mesons j=σ,ω a rational form

    f_j(x) = a_j1+b_j(x+d_j)^2/1+c_j(x+d_j)^2  ,

with four coefficients a_j, b_j, c_j, and d_j is used.
To avoid a pole in f_j, the coefficients
c_j, and d_j have to be non-negative.
The constraints f_j(1)=1, and f_j^''(0)=0,
reduce the number of independent coefficients to two with
a_j=[1+c_j(1+d_j)^2]/[1+b_j(1+d_j)^2],
and c_j=1/(3d_j^2). The coupling function for the isovector mesons
j=ρ is chosen as an exponential,

    f_ρ(x) = exp[-a_ρ(x-1)]   ,

with only one coefficient a_ρ.
Obviouly, other functions can be and have been used
to describe the density dependence of the couplings, see, e.g., Ref.<cit.>.
Also a dependence on the scalar density n^(s) or other densities
can be explored, see, e.g., Ref. <cit.>.

In the model parameterisations of this work,
there are three independent parameters for the isoscalar mesons
σ and for ω
and two for the isovector meson ρ,
thus eight parameters in total that need to be determined
for the nuclear-matter calculations besides the mass of the σ meson.
In the actual fitting procedure, the saturation density of symmetric nuclear
matter n_sat is used as reference density ϱ_ref,
but the couplings Γ_j^(0)
at ϱ_ref and the coefficients
a_j, b_j, c_j, and d_j are not used directly as independent
quantities that are varied.
Instead, the following characteristic nuclear
matter parameters are selected as variable quantities:


  * the effective Dirac mass at saturation M_sat^∗,

  * the binding energy per nucleon at saturation
  B_sat, 

  * the incompressibility coefficient at saturation K,

  * the symmetry energy at saturation J,

  * the slope parameter of the symmetry energy at saturation L.

Together with the condition of vanishing pressure at n_sat,
this gives already six parameters. 
The remaining two quantities
that are used in the actual fit are the ratios

    r_10 = f_ω^'(1)/f_ω(1)  ,
              
      r_21 = f_ω^''(1)/f_ω^'(1)  ,

with the density dependent function f_ω(x) of the ω meson.
It is assumed that the coupling functions are decreasing
with the density, hence f_ω^'(1) has to be smaller than zero.
Then the functional form of f_ω leads to the constraint
-3 < r_21 < 0 
for the second ratio in Eq. (<ref>).
The actual procedure of the parameter conversion is given in detail
in Appendix <ref>.

In the description of finite nuclei, two additional tensor
couplings Γ_Tω and Γ_Tρ appear if the tensor
terms in the Lagrangian density (<ref>) are considered. These constants
are used directly as independent parameters in the fit.
Hence there are in total 11 (9) independent parameters in models with (without)
tensor couplings taking into account the mass of the σ meson
M_σ as another undetermined quantity.




 §.§ Observables and nuclei in the fit



The choice of the observables that are included in fitting the model
parameters will affect the final results. Only quantities should be
selected that are well determined by experiments. Different
strategies are followed by various authors.
In the present work, the following nuclear observables of a nucleus are used:


  * the binding energy B,

  * the charge radius r_c,

  * the diffraction radius r_d,

  * the surface thickness σ,

  * the root-mean-square radius r_n of a single valence
  neutron level above a shell closure,
  
  * the spin-orbit splitting Δ E_so
  for two levels of a nucleon η
  in states with the same quantum numbers n_η and l_η but different
  j_η=l_η± 1/2,

  * the constraint isoscalar monopole giant resonance energy E_mono.

They are calculated after the coupled equations for a nucleus (Z,N)
are solved and can be compared with experimental data.



  §.§.§ Values of observables from theory


The binding energy is derived as

    B(Z,N) = M(Z,N)-E_nucleus(Z,N)    ,

from the energy (<ref>).
The charge radius, the diffraction radius, and the surface thickness
are obtained from the charge form factor of a spherical nucleus. It depends
on a momentum q⃗
and is calculated as,

    F_Z,N(q) = [F_Z(q)F_π(q)
        +F_N(q)F_ν(q)] F_corr(q)   ,

with form factors of the proton and neutron distributions in the nucleus,
single-nucleon form factors 
and a center-of-mass correction.
The density distributions (<ref>) of neutron and protons are
used to calculate the form factors

    F_Z(q) = ∫ d^3r  ∑_k_π,m_π w_π k_πm_π n_π k_πm_π(r⃗)  exp( iq⃗·r⃗)   ,

and,

    F_N(q)= ∫ d^3r  ∑_k_ν,m_ν w_ν k_νm_ν n_ν k_νm_ν(r⃗)  exp( iq⃗·r⃗)   ,

that are spherically symmetric in momentum space due to the spherical
symmetry of the density distributions.
The single-nucleon form factors are parameterized
<cit.> as

    F_η(q) = ∑_k=1^4A_η k/1+B_η kq^2  ,

with coefficients A_η k, and B_η k,
that are given in Table <ref>.
The center-of-mass correction

    F_corr(q) =
      exp[3q^2/16E_corrM(Z,N)]   ,

depends on the center-of-mass correction (<ref>) of the energy.
The charge density of the nucleus is finally obtained from

    n_charge(r) = ∫d^3q/(2π)^3 
      F_Z,N(q)  exp(-i q⃗·r⃗)   .

It is used to calculate the root-mean-square charge radius

    r_c = [ 1/Z∫ d^3r   r^2  n_charge(r)
        ]^1/2  ,

of the nucleus. The form factor (<ref>)
at zero momentum has the values F_Z,N(0) = Z and decreases with q
until it reaches a first zero at q_0. This value is used to obtain
the diffraction radius

    r_d = 4.493409458/q_0  ,

where the numerical factor is the first zero of the Bessel function
j_1(x)=-j_0^'(x). Further increasing q leads to the first
minimum of F_Z,N(q) at q_m that is used to calculate
the surface thickness

    σ(Z,N) = 1/q_m{ 2 ln[ZF_box(q_m)/F_Z,N(q_m)]
      }^1/2  ,

with the form factor
F_box(q) = 3 j_1(qr_d)/(qr_d) 
of a box of radius r_d,
c.f. Ref. <cit.>, and
Eq. (18a) of Ref. <cit.> with corrections.



The root-mean-square radius of a single-neutron state is defined as

    r_ν = [  ∫ d^3r   r^2  n_ν k_νm_ν(r) ]^1/2  ,

similar as the charge radius (<ref>).
The spin-orbit splitting is given by the difference

    Δ E_so(η n_ηl_η) = E_η k_η^--E_η k_η^+  ,

of the single-nucleon energies of the levels with
identical principal quantum number n_η, and
k_η^± = (n_η,κ_η^±),
where
κ_η^±=(j_η^±-l_η)(2j_η^±+1)
for j_iη^±=l_η± 1/2.
The constraint isoscalar monopole giant resonance energy E_mono
is obtained from a constraint calculation of the nucleus as described
in Ref. <cit.>. A contribution

    V_λ = -λ r^2  ,

with parameter λ,
is added to the vector potential (<ref>) of the nucleons
and the coupled equations for the fields are solved self-consistently
for given λ.  Then the root-mean-square radius

    ⟨ r^2⟩_λ = ∫ d^3r   r^2 
      n^(v)_λ(r⃗)   ,

of the total vector density n^(v)_λ of the obtained
configuration is calculated.
The constraint isoscalar monopole giant resonance energy

    E_mono = √(m_1/m_-1)  ,

can be found from the ratio of moments 

    m_1 = 2 ħ^2/m_nuc⟨ r^2⟩_λ=0  ,

and,

    m_-1 = 1/2.
      d/dλ⟨ r^2⟩_λ|_λ=0  .
 
In the actual calculation, a five-point formula is used with
λ_n=n × 0.125 keV and n=-2,-1,0,1,2
to calculate the derivative.



  §.§.§ Selection of nuclei and experimental data


The description of nuclei with the relativistic EDF presented here can
only be applied to nuclei with spherical symmetry. Also pairing effects
are not taken into account. Thus a comparison of theory and experiment
can be reasonably performed only for magic or semi-magic nuclei. This limits
the amount of data that can be included in the fit of the model parameters.
However, this restriction is not too severe since the essential features
of strongly interacting systems are sufficiently well constrained by the
selection of the fitted data
in this approach. In order to fix the isospin dependence of the
effective interaction well, nuclei with large |N-Z| should be included,
preferably on both sides of the valley of stability.
The nuclei actually considered can be read off from Tables <ref>,
and <ref>.
There are 18 (semi)magic nuclei and 2 with
one valence neutron above a doubly magic core.
Not all data from the list in Subsection <ref>
are always available for all nuclei. 

Binding energies B(Z,N)
of nuclei are obtained from the Atomic Mass Evaluation 2020
<cit.> by multiplying the binding energy per nucleon given there
with the mass number A=Z+N. These data
are tabulated in Table <ref>.
Experimental data for the charge radius, diffraction radius, surface thicknesses
and neutron radii can be found in Table <ref>.
The charge radii and quantities related to the charge form factor
are extracted from
<cit.>
using the Fourier-Bessel parameterisation if available.
The experimental valence neutron radii are taken from
<cit.>.
Experimental spin-orbit splittings are given in Table <ref>.
They are derived from the nuclear level schemes collected
at the National Nucleat Data Center <cit.>.
Experimental constraint isoscalar monopole giant resonance energies
<cit.>
are listed in Table <ref> for selected nuclei.
The choice of the constraint isoscalar monopole giant resonance energy
as an observable in the fit greatly helps to obtain a reasonable
incompressibility coefficient
K of nuclear matter, a quantity that
is often set to a predefined value since it is not well constrained
by other quantities of nuclei.




 §.§ Objective function and uncertainties



In order to determine optimal parameter values of the relativistic EDF,
an appropriate fitting procedure has to be applied. This is usually done
by minimizing an objective function that quantifies the deviation of the
theoretical predictions for certain observables from experimental data.
A crucial question is the size of the uncertainties that are used for the
different observables, e.g., in a χ^2 minimization. In the
parameter fit of an EDF it is generally not reasonable to use the
experimental uncertainties because they are for most quantities
much smaller than the precision that can be achieved by an EDF. Instead,
often some 'reasonable' values are chosen that may, however, not reflect
the 'true' uncertainties that are expected for such an EDF.
Thus, in the present work a fitting scheme is employed that
also attempts to determine the uncertainties, c.f. Ref. <cit.>.

In the present approach to fit the model parameters,
N_obs=7 types of observables are considered
as specified in Section <ref> with
N_i^(obs) data per observable (i=1,…,N_obs).
Thus an objective function,

    χ^2({p_k})
      =  ∑_i=1^N_obsχ_i^2({p_k})   ,

is defined that depends on N_par parameters p_k
(k=1,…,N_par) with N_par=11 (9) for
models with (without) tensor couplings. The contributions
for each observable 𝒪_i to
(<ref>) are given by the standard expression

    χ_i^2({p_k})
      =  ∑_n=1^N_i^(obs)[ 𝒪_i^(model)(n,{p_k})
          - 𝒪_i^(exp)(n)/Δ𝒪_i]^2  ,

where the n=1,…,N_i^obs
data of an observable 𝒪_i depend on the parameters {p_k}.
The total number of data is given by

    N_data = ∑_i=1^N_obs N_i^(obs)  ,

and thus the number of degrees of freedom
is N_dof = N_data - N_par. 

The uncertainties for each observable 𝒪_i
are denoted by Δ𝒪_i. Since binding energies of stable nuclei
are much more precisely known than
those of exotic nuclei, their uncertainties vary over
a large range. Thus the uncertainty of the observable 𝒪_i = B(Z,N)
in the function (<ref>) is assumed to compose of two contributions as,

    Δ𝒪_i =
      [ (Δ𝒪_i^(exp))^2
        + (Δ𝒪_i^(fit))^2]^1/2  ,

with an experimental uncertainty Δ𝒪_i^(exp)
and a fit uncertainty Δ𝒪_i^(fit).
For all other observables, the experimental uncertainty is neglected in the
fits of this work. A reasonable estimate of the fit uncertainties will be
obtained if,

    χ^2({p_k})/N_dof = 1    ,

and,

    χ_i^2({p_k})/N_i^(obs) =
      χ^2({p_k})/N_data  ,

i.e., each observable contributes on the average equally
to the total χ^2.
Theses conditions allow to determine the fit errors
Δ𝒪_i^(fit) during the fit by a rescaling.
Thus the general fit procedure proceeds in two
steps: variation of the
model parameter to minimize (<ref>) until a certain convergence
is reached, and a rescaling of the fit uncertainties.
These two steps are executed repeatedly until the fit uncertainties
and model parameters have converged to the desired precision.
To find the optimal values of the model parameters
in their space of dimension N_par, a simplex method
<cit.> was employed as in Ref. <cit.>. It only relies
on values of the χ^2 function and does not need derivatives.
Once the minimum of the total χ^2 function
is found, the solution can be represented as a vector
p⃗^ min=(p_1^ min, … ,  p_N_ par^ min)
in the space of parameters.
A true (local) minimum of χ^2 is reached when
the matrix of second derivatives, 

    ℳ_ij = 1/2. ∂^2χ^2/∂ p_i∂ p_j|_p⃗_ min  ,

has only positive eigenvalues.
If this is not the case, the eigenvectors with negative eigenvalues
indicate directions for a further reduction of χ^2. This information
is used every now and then to accelerate the minimization. However, the
calculation of ℳ_ij is expensive as it needs
2N_par^2+1 function evaluations.
Finally, it is possible to determine the uncertainty

    Δ𝒪 =
      √((Δ𝒪)^2)  ,

of any observable 𝒪
and the correlation coefficient

    c_𝒪_1𝒪_2 =
      Δ𝒪_1Δ𝒪_2/√((Δ𝒪_1)^2 (Δ𝒪_2)^2)  ,

between two observables 𝒪_1 and 𝒪_2
from

    Δ𝒪_1Δ𝒪_2
      = ∑_ij∂𝒪_1/∂ p_i( ℳ^-1)_ij∂𝒪_2/∂ p_j  ,

that contains the inverse of the matrix ℳ and first derivatives
of the observables with respect to the parameters.




 §.§ Types of energy density functionals


The main aim of the present work is the development of a new parameterisation
of a relativistic EDF with tensor couplings, denoted by 'DDT' in the
following, in addition to the usual
minimal couplings of the mesons to the nucleons.
In order to compare the main differences of this EDF
to a model without tensor interactions,
the results of the DD2 model <cit.> will be shown,
that uses the same form of the relativistic EDF.

A particular issue is the effect of exchange contributions
that are neglected in the relativistic EDF, which is constructed 
in Hartree approximation from the Lagrangian density. This concerns
particularly the Coulomb interaction.
The source density of the electromagnetic field, that is
found by solving Eq. (<ref>), includes the charge distribution
of all protons so that the Coulomb field in the vector potential
(<ref>), which a proton feels in the mean-field created by all particles,
also contains a self-interaction contribution
that would not arise with the proper use
of an antisymmetrized many-body wave function.
The exchange term is sometimes added in form of the
Slater approximation <cit.>.
However, this is not done in this work. As an alternative,
a simple correction to
reduce the self-interaction term is to multiply the field A_0 in Eq.(<ref>) by the factor (Z-1)/Z in the calculation of nuclei,
as used in the DD2 parameterisation. Then the
vector potential V_π(r⃗) has the correct asymptotic form
for large radii
when a proton is separated from a nucleus. To study the effect of
this modification, a variation with this reduction factor,
DDTC, of the original DDT parameterisation is also developed.

A more detailed consideration of the contributions to the energy of a nucleus
reveals, however, that it is admissible to neglect the exchange part
in the Coulomb energy. In fact, there are two contributions that in total
compensate for the exchange term. On the one hand,
there is the Coulomb displacement energy
of analog states in mirror nuclei that, in the mean-field approximation, e.g.,
for the ^41Sc - ^41Ca pair is smaller by about 7%.
This is known as the Nolen-Schiffer anomaly <cit.>
which is solved by the taking into account the
charge symmetry breaking (CSB) contribution in the nucleon-nucleon
interaction, and
long-range correlation (LRC) effects <cit.>.
Ignoring the factor (Z-1)/Z
in the Coulomb direct term, which leads to including the
Coulomb self-interaction terms, and ignoring the Coulomb exchange term
corresponds to including the contributions of the CSB and LRC effects.
Thus only the
direct part of the Coulomb contribution to the energy should be included in
the calculation, see also subparagraph
IIA in Ref. <cit.>.



§ RESULTS AND DISCUSSION



After performing the fit of the parameters for the two
models with tensor couplings,
various quantities for nuclei, nuclear matter and neutron stars
can be calculated and compared to experimental data. Before 
they are presented, in the following subsections, a discussion
of the model parameters and of the obtained
density dependence of the coupling functions
is of interest since they are essential to obtaining
a good quality fit.

The parameters of the three relativistic EDF models of the
present work are specified in Table <ref>.
For the DDT and DDTC parameterisations,
uncertainties are given as determined with the method described
in Subsection <ref>. For the DD2 model, these are not available
because the parameter set does not correspond to a minimum of χ^2 with
the present set of observables and model uncertainties.
The most obvious differences are the large values of the tensor couplings
Γ_Tω, and Γ_Tρ, and the small σ
meson masses M_σ
as compared to the value of the DD2 model
without tensor couplings. This is
not surprising since Γ_Tω and M_σ
directly affect the description
of the surface properties of nuclei and thus there is a strong correlation
between them. The DD2 model is also characterized by a rather small
saturation density.
The isoscalar couplings Γ_σ^(0) and Γ_ω^(0)
at the reference density of the DDT and DDTC models are considerably smaller
than those of the DD2 model. On the other hand, the couplings
Γ_ρ^(0) and the coefficient a_ρ are larger.
Particularly striking are the large coefficients b_ω and
c_ω with a small d_ω for the DDT model.
This will lead to an unusual density dependence
of the coupling Γ_ω, see below.






The correlations coefficients of the fit parameters
in the DDT model are given in Figure <ref> where boxes in red color
indicate the largest absolute values
whereas grey boxes correspond to the smallest
moduli of the coefficients. As expected,
a first group with M_σ, M_nuc^∗,
and Γ_Tω shows strong correlations since they all affect
the description of the nuclear surface. In contrast, Γ_Tρ
is hardly correlated to any other parameter of the fit. The other two
parameters related to the isovector part of the interaction,
J, and L, however, show a strong correlation. Apart from this
dependence, J seems to be practically uncorrelated to all other
fit parametes. Another pair of parameters,
r_10, and r_21, determines the density dependence of the
isoscalar couplings. They are also strongly correlated. A somewhat lesser
strength of correlation is found for the incompressibility coefficient K,
and r_10 with the first group of parameters.





The coupling functions Γ_i(n_b) generally decrease
with increasing baryon density n_b, see Figure <ref>.
They approach a finite value in the high-density limit by construction, and
go to zero for Γ_ρ, in particular. The isoscalar couplings for the
models with tensor couplings (DDT, and DDTC)
are significantly smaller than those of the model without tensor couplings
(DD2). This can also be observed for the absolute values
of the couplings at the reference density in Table <ref>.
The strong rise of the ω meson coupling
in the DDT model towards very small densities is particularly noteworthy.
This behavior is a consequence of the large values of the parameters
b_ω and c_ω in combination with a small d_ω.
The isoscalar couplings, Γ_ω and Γ_σ,
of the DDTC parameterisation have a much larger uncertainty as compared to those
of the DDT model, but their shape is very similar to the corresponding
couplings of the DD2 model.
The density dependence of the
ρ meson couplings is very similar for all models with slightly
higher values for the models with tensor couplings.




 §.§ Nuclei



Before analyzing the predictions of individual properties
of finite nuclei, it is worthwhile to look at the fit uncertainties of the
observables, see Table <ref>,
and their contributions
to the χ^2 function that are collected in Table
<ref> for the three parameterisations. The uncertainties of the
energy quantities, i.e., the binding energy, the spin-orbit splitting,
and the constraint isocalar monopole giant resonance energy, are all
on a sub-MeV level, whereas the uncertainties of
the size quantities, i.e., radii and skin thickness,
are typically in the order of a few hundredths of a fm with the
error in the root-mean-square (rms)
valence neutron radius beeing the smallest. These results give
an impression on the actual quality a description of nuclei with
a relativistic EDF can achieve and how reasonable values have to be chosen
in other attempts to find parameters of such models from fits
to finite nuclei.






A comparison of the total χ^2, see Table <ref>,
and the contributions for the three models reveals interesting differences.
First of all, it is evident that the DDT model gives the best description
of the data with the lowest total χ^2. By construction, its value
is given by the number of degrees of freedom N_dof=39.
The other parameterisations, in particular the one without tensor couplings,
are considerably worse. In case of the DDTC parameterisation
this is mostly caused by the larger errors in the
binding energies and the spin-orbit splittings.
The values of χ^2 of the DD2 model is completely
dominated by the difference of the valence neutron radii from the experimental
values. Even neglecting their contribution to χ^2, the descripton
of experimental data with the DD2 model is almost always much worse than
that of the DDT and DDTC models. All observables
that are determined strongly by the surface description of a nucleus, viz.,
the radii, the skin thickness, and the spin-orbit splitting,
are on the average better described in the EDFs with tensor couplings.
Only the surface thickness σ is slightly better described
by the DD2 model.






The first quantity of major interest is the binding energy
B(Z,N) of a nucleus that is predicted by a given parameterisation
of a relativistic EDF.
The results of the binding energy calculations
of the three models of the present work
are listed in Table <ref>, for nuclei considered in the fit,
in comparison to experimental values from the Atomic Mass Evaluation
2020 <cit.>. In order to have a better impression
of the quality of the models, the differences between the experimental
and theoretical data are depicted in Figure <ref>. It is seen
that the models with tensor coupling show less fluctuations and deviate
on the average less strongly from the experimental data than the models without
tensor couplings. This is also reflected in the root-mean-square deviation
of the theoretical values from the experimental data in the bottom
line of Table <ref>. The trend of these numbers agrees with the
variation of the binding energy contribution to the total χ^2
shown in Table <ref>.


Also interesting observations for individual sets of nuclei
can be made. In constrast to the DD2 parameterisation,
the binding energy of ^28O is smaller than that of ^24O
in the DDT and DDTC parameterisations, a result that is difficult
to achieve in most other relativistic EDFs.
Similar patterns can be seen in Figure <ref> when
comparing different parameterisations, e.g., for the heaviest nuclei.
There are also striking effects for
the set of Ni nuclei with different trends when comparing the models.
It is seen in Table <ref>
that the calculated binding energies of ^48Ni
obtained for the DDT model without the (Z-1)/Z factor
is significantly closer to the experimental value than that of the
DDTC and DD2 models with the (Z-1)/Z factor.



Theoretical values of the quantities related to the 
charge form factor are given in Table <ref> in comparison
to experimental data for the nuclei in the fit. The theoretical
uncertainties are generally in the order of 0.005 fm.
Differences between theory and experiment
are depicted in Figure <ref> with three panels
for the charge radius, the diffraction radius, and the surface
thickness. There is a good agreement with typical differences
between theory and experiment of
less than 0.02 fm, and a clear trend in the deviations
is not seen. Only the surface thickness of ^90Zr is overestimated
by more than 0.04 fm. The values of the valence neutron radii of ^17O
and ^41Ca can also be found in Table <ref>. The
deviations between theory and experiment
are slightly larger than those of the
quantities related to the charge form factor, but the agreement with
the experimental data is still satisfactory, given the large
theoretical uncertainties.



The differences between experimental and theoretical
values of the spin-orbit splittings can be explored by studying
Table <ref>. Considering the theoretical uncertainties of
somewhat less than 0.1 MeV, there is a fair agreement of these
data. The difference between the experimental
spin-orbit splittings of the ν 0 p and π 0p levels
in ^16O (-0.140 MeV) is not reproduced by the theoretical models
as their differences are positive, 0.182 MeV and
0.170 MeV for the DDT and DDTC models, respectively.
The spin-orbit splitting of the ν 0 p levels is
larger than that of the π 0 p levels in contrast
to the experiment, in particular for the DD2 parameterisation.
The spin-orbits splittings of the
ν 1 p levels in ^48Ca are underestimated in the theoretical
models in comparison to the experimental value.
Larger differences between theory and experiment
are also observed for the π 1d levels in
^132Sn, and ^208Pb.
In general, the results of the DD2 model show a larger
deviation from the experiment as compared to the models with tensor
couplings.




The energies of the isoscalar monopole giant
resonance are crucial, in the fit of the model parameters,
to obtain
a reasonable value of the incompressibility coefficient K of
nuclear matter. Experimental data and theoretical values are
given in Table <ref> with theoretical uncertainties
in the order of 0.2 MeV. In general, there is a good reproduction
of the experimental values by theory, by the models with and without
tensor couplings. Only the prediction for the
resonance energy of ^208Pb is on the lower side as compared to the
experiment.





In the determination of the model
parameters, several exotic nuclei on both sides of the valley of
stability were included in order to better constrain the isovector
part of the effective in-medium interaction. In contrast to
most earlier relativistic EDFs with density dependent couplings,
three independent isovector
parameters are considered in the parameterisations
that include tensor couplings. It is interesting to study, how this
affects the predictions of the neutron skin thickness of nuclei,
Δ r_skin = r_ν-r_π,
that is obtained by the difference
between the neutron and proton rms radii. The theoretical values
for the nuclei in the fit are collected in Table <ref>.
Nuclei with N=Z exhibit a small negative value of
Δ r_skin that becomes even more negative
for neutron deficient nuclei like ^34Ca and ^48Ni.
On the other hand, neutron-rich nuclei can develop a
substantial neutron skin. This is most clearly seen for
^24O, ^28O, ^78Ni, and ^132Sn.
The theoretical uncertainties, however, become much larger
for the most exotic nuclei as compared to stable nuclei.
The neutron skin thicknesses of more exotic nuclei are generally
larger for the DD2 models as compared to the DDT and DDTC models. 
Also stable nuclei such as ^48Ca and ^208Pb
show a sizeable neutron skin thickness. It can be determined
experimentally by studying the parity-violation asymmetry in
high-energy electron scattering experiments.
A comparison of the theoretical predictions in Table
<ref> with the experimental data of the
CREX experiment <cit.> with
Δ r_skin(^48) =
0.121± 0.025  ± 0.024 fm
and those of the combined PREX experiments <cit.>
with Δ r_skin(^208) = 0.283± 0.071 fm
shows that the theory can nicely reproduce the size of the
neutron skin in ^48Ca within the uncertainties
but it predicts a considerably smaller
neutron skin thickness in ^208Pb, an issue that
is not solved so far. The neutron skin thickness can
also be determined using hadronic probes using precisely
known proton radii from electron scattering.
A former analysis of 1 GeV proton scattering
on ^48Ca <cit.> gave a neutron
skin thickness of Δ r_skin=0.10 ± 0.03 fm only
slightly smaller but consistent within the errors with the CREX
result. A more recent study of neutron radii using proton scattering
<cit.> gave ranges of
0.056  < Δ r_skin(^48) < 0.102 fm
and
0.083  < Δ r_skin(^208) < 0.111 fm
that point to rather small neutron thicknesses of these
nuclei.

The inclusion of pairs of mirror nuclei
in the fit also allows to better constrain the mirror displacement
energies. The binding energy differences of three pairs of
mirror nuclei are given in Table <ref>. There is a good
agreement between experiment and theory with theoretical results
of the DDT model that are somewhat larger than the experimental values.
The differences between experimental and theoretical values
are 0.374,  0.883, and 1.132 MeV for the pairs with mass number
A=41, 34, and 48 for the DDT model, respectively, thus overestimating the
binding energy differences. In contrast, they are systematically smaller
with 0.072, -1.055, and -1.318 MeV for the DDTC model
and -0.237, -1.738, and -3.836 MeV for the DD2 model, respectively,
where the Z(Z-1) factor in the Coulomb field was introduced.
In addition, 
the neutron-deficient nuclei were not considered in the fit of the parameters
in the latter parameterisation.
Thus the parameter set DDT gives the best description on the average.
The theoretical uncertainties increase with the isospin difference of the
pair of nuclei.
  






 §.§ Nuclear matter




The energy per nucleon E/A, Eq.  (<ref>),
in homogeneneous nuclear matter is a
function of the baryon density n_b=n_ν+n_π and the
isospin asymmetry α = (n_ν-n_π)/n_b.
For α=0, the energy per nucleon of symmetric nuclear matter
E_0=E/A(n_b,0) is obtained. It is depicted in Figure
<ref> for the 
parameterisations DDT, DDTC, and DD2.
The minimum of E_0 for these
equations of state defines the saturation density
n_sat, see Table <ref>.
The region of negative slope, i.e., negative pressure, signals an
instability of homogeneous matter that is related to the
occurrence of the liquid-gas phase transition.
At high densities, the predictions for the
energy per nucleon become less certain as the widening
of the uncertainty bands of the DDT and,
significantly larger, DDTC models shows.
The energy per nucleon
of pure neutron matter, i.e., E/A for α=1,
is also shown in Figure <ref>.
It increases monotonically with the
baryon density.

The curves of the DDT and DD2
parameterisations in Fig. <ref> look very similar
at first sight. But there are differences in detail, in particular
at lower densities.
The curves of the parameterisation DDTC follow along the
lower limit of the
uncertainty bands of the DDT model, hence signaling a softer equation of state.
The neutron matter equation of state of the DDT model
shows a peculiar feature at the lowest densities. This can be seen
in the inset of Figure <ref>. The energy per nucleon does not
approach zero as smoothly as in the DD2 and DDTC models.
This is related to the
specific density dependence of the ω-meson coupling Γ_ω,
see panel (b) of Figure <ref>. The increase of the coupling
at very small densities causes an additional repulsion, which also applies
for symmetric nuclear matter at very low densities (not shown in the inset).



Assuming a quadratic dependence of E/A on
α for given baryon density n_b, the symmetry
energy (<ref>) can be obtained as a difference
E_sym(n_b)=E/A(n_b,1)-E/A(n_b,0). This quantity
is shown in Figure <ref> as function of n_b.
Particularly noteworthy is the change of the slope
above saturation density,
for the DDT and DDTC models,
with a widening of the uncertainty bands.
At densities below saturation, symmetry energy is
well constrained by experiments and ab-initio calculations
of pure neutron matter, see, e.g., Refs.<cit.> and references therein.
At densities above saturation,
the predictions for the equation of state of nuclear matter
are less precise as indicated by the increasing uncertainty
bands.

The E_sym curves of the DDT, DDTC, and DD2 model
cross at a density below n_sat.
It has been observed before that fitting model parameters of EDFs
to properties of nuclei effectively
determines the symmetry energy around this subsaturation density,
see, e.g., Ref. <cit.>. A comparison of the low-density
symmetry energy of the DDT, DDTC, and DD2 models with experimental data 
is depicted in the inset of Figure <ref>. These data come from
the determination of the dipole polarizability of ^208Pb (violet),
the analysis of EDFs (green, light blue), and isobaric analog states (orange),
see Ref. <cit.> and references therein.
The symmetry energy of the
parameterisations DDT and DDTC lie slightly above these points, corresponding
to the small slope parameter L, whereas the DD2 curve
matches the data somewhat better.


The energy per nucleon of symmetric matter
near saturation can be expanded as

    E_0(n_b) = -B_sat
      
      + 1/2 K x^2 + 1/6 Q x^3 + …  ,

with x=(n_b-n_sat)/(3n_sat)
without a term linear in x because
the pressure at saturation is
p(n_sat)=0.
The incompressibility coefficient

    K = 9 n_ sat^2.  d^2/d(n_b)^2 E_0|_n_sat
      = 9 .  dp/dn_b|_n_sat  ,

and skewness parameter

    Q = 27 n_ sat^3.  d^3/d(n_b)^3 E_0|_n_ sat  ,

are found as second and third derivatives of E_0.
Similarly, the symmetry energy can be written as

    E_sym(n_b) = J + L x  
      + 1/2 K_sym x^2 + …  ,

with the symmetry energy at saturation J=E_sym(n_sat),
the slope parameter

    L = 3 n_sat. dE_sym/dn_b|_n_sat  ,

and the symmetry incompressibility

    K_ sym = 9 n_ sat^2.  d^2 E_sym/d(n_b)^2|_n_sat  .

It is also useful to define the socalled volume part of the isospin
compressibility 

    K_τ,v = K_ sym-6L-QL/K  ,

that describes the change of the incompressibility with the variation
in α <cit.>.
These quantities, together with the effective Dirac mass at saturation
(M_nuc^∗)_sat=
M_nuc-C_σn_sat^(s),
are convenient to characterize the equation of state of nuclear matter
around saturation. They are given in Table <ref> for the
models DDT, DDTC, and DD2.
Most remarkable are the increased effective mass
and the reduced incompressibility coefficient
of the models with tensor couplings
(DDT and DDTC) in comparison to that without these couplings (DD2).
Most existing relativistic models show values of K that are rather high
when compared with non-relativistic EDFs, cf. Refs.<cit.>.
The tensor models are an exception and they
are also characterized by rather large saturation densities.
The higher-order coefficients Q and K_sym show considerable
differences and are not very well determined by the parameter fits as they have
large uncertainties.
The smaller values of K and Q of the DDTC model
indicate a softer equation of state as compared to the DDT model as seen already
in Figure <ref>.
The DDT and DDTC models also have symmetry energy slope
parameters that are smaller than the value of L of the DD2 model. This
difference correlates with the smaller neutron skin thicknesses for exotic
nuclei as discussed in Subsection <ref>.
  









 §.§ Neutron stars




The properties of a 
non-rotating, spherically symmetric neutron star, are obtained
by solving the first-order
Tolman-Oppenheimer-Volkoff equation
<cit.>

    dp/dr = -G  M(r)ε(r)/r^2[ 1 + P(r)/ε(r)]
      [ 1 + 4π r^3P(r)/M(r)]
      [ 1 - 2 G M(r)/r]^-1  ,

with the mass

    M(r) = 4π∫_0^r dr^'  (r^')^2 ε(r^')   ,

inside the volume of
radius r. The factors in square brackets are
the general relativistic corrections to the classical
expression for the change of the pressure with the radius.
Assuming a particular density
n_central
in the center of the star, the first order differential equation
(<ref>) can be solved by an outward integration until
the pressure P(r) becomes sufficiently small. This defines the
stellar radius R and the total mass M(R). The essential
ingredient in the calculation is the equation of state
of charge-neutral matter in β equilibrium. It is obtained
by adding the leptonic contributions (electrons and muons) to the
nuclear equation of state and determining the isospin asymmetry
α for given baryon density n_b, by requiring vanishing
lepton chemical potentials μ_e+μ_q, and
μ_μ+μ_q, with charge chemical potential
μ_q=μ_π-μ_ν.
At densities below nuclear saturation,
an appropriate crust equation of state has to be used as the
matter is no longer homogeneous. Here, the
crust description of the generalized relativistic density functional
with the DD2 parameterisation of the couplings <cit.>
is used where the masses of the nuclei in the crust are taken from
the Duflo-Zuker model DZ31 <cit.>,
if not available from the atomic mass
evaluation AME2020 <cit.>.


The mass-radius relations of neutron stars
for the different parameterisations
are depicted in Figure <ref>. Open circles on
the curves denote the properties of neutron stars
with central densities that are an integer multiple
of the saturation density n_sat.
The curves show the well-known inverse
S-shaped rise of the mass with increasing central density
and the corresponding variation of the radius. Ordinary neutron
stars with masses of 1.4 times the solar mass M_sol
have central densities that are typically between twice
and three times the saturation density. At certain central
densities n_central^(max), the most massive
neutron star configurations are obtained.
They have central densities of five to six times
the saturation density. Neutron stars with
higher central density are not stable any more and this last
branch of the curve will not be populated. The maximum masses
of the shown models are always above the largest
masses of observed neutron stars
<cit.>
since their equation of state
is sufficiently stiff. Masses and radii are avalaible from
NICER observations
<cit.>,
but these data have large uncertainties. All models are consistent
with these observations.
Characteristic properties of neutron stars
of the models in this work are collected in Table <ref>.
The properties of the maximum mass stars are very similar in all models
but the radii of the DD2 model are systematically larger that those
of the parameterisation DDT and DDTC. The latter model
predicts the lowest maximum mass, a higher central density at this mass,
and smaller radii as compared to the other models,
corresponding to the softer equation of state.






§ SUMMARY AND CONCLUSIONS



In this work, an extension of a relativistic EDF is
presented that includes tensor couplings between the Lorentz vector mesons,
ω and ρ, and nucleons in addition to the standard minimal
meson-nucleon couplings. The latter are assumed to depend on the baryon
density. The field equation are derived in the mean-field approximation
from a covariant Lagrangian density
and solved self-consistently for nuclei and nuclear matter.

The parameters of this phenomenological approach are found from a fit
to properties of 20 selected nuclei that comprise binding energies, charge and
diffraction radii, surface thicknesses, spin-orbit splittings,
valence neutron radii,
and constraint isoscalar monopole giant resonance energies.
The model uncertainties
of these quantities are determined self-consistently during
the fitting procedure.
The model predictions are compared to experimental data and given with
their theoretical uncertainties, which were, to our knowledge,
not specified in earlier parameterisations of relativistic EDFs.
Two new sets of parameters, DDT and DDTC, are developed that differ
in the treatment of the Coulomb interaction.

The inclusion of tensor couplings in the relativistic EDF
leads to a considerable improvement
in the description of nuclear properties, in particular their binding energies.
Quantities related to the nuclear charge form factor show a similar quality
as those of an earlier parameterisation (DD2) without tensor couplings.
The binding energy differences of mirror nuclei are also closer to the
experimental data than in the DD2 model, that did not consider exotic,
neutron deficient nuclei in the determination of the parameters.
Predictions for the neutron skin thickess of bound and unstable nuclei are made.

The equations of state for symmetric nuclear matter and pure neutron matter
are calculated with the new parameters. They look similar as those of the
DD2 model but show some specific differences. These are most
apparent by comparing characteristic nuclear matter parameters.
The new parameterisations
exhibit larger saturation densities and effective
nucleon masses, and smaller incompressibility coefficients and symmetry
energy slope parameters.

The description of charge-neutral nuclear matter in β equilibrium
with the extended EDFs allows to calculate properties of neutron stars and,
in particular, their mass-radius relation. The results are similar as
for the DD2 model but with slightly smaller radii. Due to the stiffness
of the EOS at high densities, the maximum masses 
are predicted to lie above the data of observed heavy neutron stars.
The obtained mass-radius relations are consistent with results from
NICER measurements that still have substantial uncertainties, in particular
for the radii.

The EDFs with the new parameterisations can be studied in the future for larger
sets of nuclei, including deformed ones, taking other effects
like pairing into account. The additional tensor couplings,
however, require some extra work to be considered in such deformed calculations.
The new models can also be used to prepare EOS tables
of stellar matter at finite temperature for larger ranges in isospin
asymmetry and density, in order to be used in simulations of core-collapse
supernovae and neutron-star mergers <cit.>.
For this, the calculations have to
be extended by including additional degrees of freedom, e.g., light clusters
at subsaturation densities <cit.>,
or hyperons at high densities, see, e.g., Ref. <cit.>.









§ ACKNOWLEDGMENTS


S. Typel acknowledges the support by the Cyclotron Institute,
Texas A&M University.
S. Shlomo is partially supported by the US Department of Energy under Grant no. DE-FG03-93ER-40773.


  
  

§ TRANSFORMATION BETWEEN PARAMETERS OF NUCLEAR MATTER AND
    OF COUPLING FUNCTIONS

  

  The parameters of the coupling functions (<ref>) can be derived
  from the characteristic nuclear matter parameters.
  To simplify this conversion, the average nucleon mass
  M_nuc=(M_π+M_ν)/2 is used instead of proton and
  neutron masses in the full calculation.
  However, this does not pose any problem.
  
  
  * The reference mass ϱ_ref is chosen as the
    baryon density at saturation n_sat and defines the
    Fermi momentum
    
    k_sat^∗ =
          ( 3π^2/2 n_sat)^1/3  .

  
  * The Dirac effective mass at saturation M_sat^∗
    gives the effective chemical potential
    
    μ_sat^∗ = √((k_sat^∗)^2
           + (M_sat^∗)^2)  ,

    and the scalar density at saturation
    
    n_sat^(s) = M_sat^∗/π^2[ k_sat^∗μ_sat^∗
              - (M_sat^∗)^2ln( k_sat^∗
                +μ_sat^∗/M_sat^∗) ]
              .

    This determines the scalar coupling at saturation
    
    C_σ =
          M_nuc-M_sat^∗/n_sat^(s)  ,

    and the auxiliary quantity
    
    f_sat = 3 (
          n_sat^(s)/M_sat^∗
           - n_sat/μ_sat^∗)   ,

    that will be used below.
  
  * The energy per nucleon at saturation (E/A)_sat
    gives the energy density at saturation
    
    ε_sat = [ (E/A)_sat
           + M_nuc] n_sat  ,

    and the vector coupling at saturation
    
    C_ω = 2/n_sat^2[ ε_sat
            - 1/4( 3 μ_sat^∗  n_sat
            + M_sat^∗ n_sat^(s))
            - 1/2 C_σ( n_sat^(s))^2]
             .

    Furthermore, the chemical potential
    
    μ_sat = ε_sat/n_sat
           =  (E/A)_sat + M_sat  ,

    and the rearrangement potential
    
    V^(R) = μ_sat - μ_sat^∗
           - C_ω n_sat  ,

    are found.
  
  * The ratio r_10 determines the derivatives
    
    C_ω^' = 2 r_10C_ω/n_sat  ,

    and
    
    C_σ^' = 1/( n_sat^(s))^2( C_ω^' n_sat^2
          - 2 V^(R))   ,

    at saturation.
    Then, the second and third auxiliary variables
    
    g_sat =
          M_sat^∗/μ_sat^∗
            - C_σ^'n_sat^(s)f_sat/1+
          C_σf_sat  ,

    and,
    
    h_sat = - C_σ^'n_sat^(s)
            + C_σM_sat^∗/μ_sat^∗/1
          + C_σf_sat  ,

    can be calculated.
  
  * The ratio
    r_21 is used to obtain the second derivative
    
    C_ω^'' = 2 r_10 (r_10+r_21)
          C_ω/n_sat^2  ,

    for the ω meson at saturation.
  
  * The incompressibility coefficient K gives the
    second derivative,
    
    C_σ^''    =    2/n_sat( n_sat^(s))^2[ 1/3( μ_sat^∗
            - M_sat^∗ g_sat)
            - K/9
            + ( C_ω + 2 C_ω^' n_sat
            + 1/2 C_ω^'' n_sat^2) n_sat. 
          .
            - ( C_σ + C_σ^' n_sat) n_sat^(s) g_sat
            - C_σ^'( n_sat^(s))^2]   ,

    for the σ meson.
  
  * The symmetry energy at saturation J is used to calculate
    the coupling of the ρ meson at saturation,
    
    Γ_ρ^(0) = √(C_ρ) M_ρ  ,

    where
    
    C_ρ = 2/n_sat[ J - (k_sat^∗)^2/6
              μ_sat^∗]   .

  
  * The slope parameter L of the symmetry energy at saturation
    gives the derivative
    
    C_ρ^' = 2/3n_sat^2{ L - 3J - (k_sat^∗)^2/3
            μ_sat^∗[ 1/2( M_sat^∗/μ_sat^∗)^2( 1 - 3 n_sat/M_sat^∗
            h_sat) - 1 ]
          }  ,

    for the ρ meson at saturation.
    Then the parameter
    
    a_ρ = -1/2C_ρ^'/C_ρ n_sat  ,

    in the function f_ρ(x) is found.
  
  * The values of C_j and their derivatives determine
    the couplings and first derivatives
    
    Γ_j^(0) = √(C_j) M_j  ,
                  Γ_j^' = 1/2Γ_j^(0)C_j^'/C_j  ,

    as well as the second derivatives
    
    Γ_j^'' = 1/2Γ_j^(0)[ C_j^''/C_j
            - 1/2( C_j^'/C_j)^2]   ,

    for j=σ and ω.
    Then also the derivatives 
    
    f_j^'(1) = Γ_j^'/Γ_j^(0)  ,
                  
          f_j^''(1) = Γ_j^''/Γ_j^(0)  ,

    of the functions f_j(x) are known.
  
  * Finally, the parameters of the rational functions f_j(x)
    for the isoscalar mesons can be found with help of the explicit
    expressions;
    
    f_j^'(x)     =     2 a_j(b_j-c_j)
          x+d_j/[1+c_j(x+d_j)^2]^2  ,
          
    
          f_j^''(x)     =     2 a_j(b_j-c_j)
          1-3c(x+d_j)^2/[1+c_j(x+d_j)^2]^3  ,

    for the first and second derivatives of the functions f_j(x). The ratio,
    
    r_21 = f_ω^''(1)/f(1)
           = d_j^2-(1+d_j)^2/[d_j^2+(1+d_j)^2/3](1+d_j)  ,

    is a function of d_j only because of the constraint
    1=3c_jd_j^2
    that follows from f_j^''(0)=0. Thus the value of d_j can
    be obtained by a simple numerical search or iteration.
    Then c_j=1/(3d_j^2) is known
    as well and b_j is calculated from
    
    b_j = c_j z + r_10/z-(1+d_j)^2 r_10  ,

    with the auxiliary quantity
    
    z = 2(1+d_j)/1+c_j(1+d_j)^2  .

    Finally, the normalization condition f_j(1)=1 gives
    
    a_j = 1+c_j(1+d_j)^2/1+b_j(1+d_j)^2  .

  
  




















 Paper Title: Determination of electron screening potential of 6 Li(p,α)3 He reaction using MultiLayer Perceptron based neural network.














    [ dipayanchattopadhyay90@gmail.com^1Department of Physics, The ICFAI University Tripura, Agartala 799210, India§ ABSTRACT 
          Background Understanding the nuclear reactions between light charged nuclei at sub-coulomb energy region holds significant importance in several astrophysical processes. Determination of the precise reaction cross-section within the astrophysically important Gamow range is difficult because of electron screening. Various polynomial fits, R-Matrix and Indirect Trojan horse method estimate much higher electron screening energies as compared to the adiabatic limit.
        
          Purpose Obtain the bare astrophysical S-factor of ^6Li(p,α)^3He using Multi-Layer Perceptron based Artificial Neural Network based analysis and extract the electron screening energies. 
        
          Methods Experimental S-factor of ^6Li(p,α)^3He, available in literature, are reanalyzed using the Multi-Layer Perceptron based Artificial Neural Network based algorithm to obtain the energy dependent astrophysical S-factor. Bare astrophysical S-factor is also calculated using the same Feed-forward Artificial Neural Network from the data range above 60 keV where the electron screening effect is expected to be negligible. Electron screening potential is then obtained by taking the ratio of total shielded S-factor with the bare S-factor.    
        
          Results and Conclusions The electron screening potential obtained from the Present work through the Artificial Neural network based algorithm is found to be 220 eV. The extracted electron screening potential through the present analysis indicates that the Artificial Neural Network might be an alternative tools for estimation the electron screening potential involving light nuclei.      
        25.70.Gh, 25.70.Jj, 25.60.Gc
                                      
        Determination of electron screening potential of ^6Li(p,α)^3He reaction using Multi Layer Perceptron based neural network
             D. Chattopadhyay^1
            August 5, 2024
        =========================================================================================================================
        
        
        
        § INTRODUCTION
        
        Understanding the nuclear reactions mechanism between light charged nuclei at ultra low energies relevant to astrophysical energies is emerging as a burning topic in recent times. Measurement of precise cross-section is essential to understand the stellar evolution as well as abundances of various nuclides observed in the universes <cit.>.  Low energy nuclear reaction process involve transfer and capture processes, with the former primarily influenced by the nuclear force and the latter predominantly driven by the electromagnetic interaction. Determination of reaction cross-section at ultra low energies (much below the Coulomb barrier) of astrophysical interest requires considerable effort. Direct determination of cross-section in many cases is found to be very difficult or even impossible owing to small cross-section because of Coulomb repulsion of the interacting particles <cit.>.  Extrapolation of direct cross-sections down to stellar energies requires extra theoretical effort. Different Indirect method has been evolved in recent times to overcome this problems <cit.>. 
        
        In this context, the low-energy cross sections governing reactions that produce or deplete lithium isotopes are pivotal. These cross sections are indispensable for addressing unresolved astrophysical questions, including the intricacies of Big Bang nucleosynthesis and the observed lithium depletion in celestial bodies like the Sun and other stars within our galaxy <cit.>. The precise estimation of bare astrophysical cross-sections from experimental measurement is essential as they serve as key inputs in the stellar evolutionary codes. However, as the Gamow energy region is far below the Coulomb barrier, the cross-sections are very small and also very difficult to predict due to presence of electrons inside the target. Electrons inside the target nucleus effectively screen the coulomb potential at ultra low energies and thus reducing the height of the coulomb barrier. As a result the cross-section is enhanced by a factor exp(U_eπηE) <cit.> and is given by 
        
            σ(E)=σ_b(E)exp(U_eπηE)
        
        
        where, U_e is the electron screening potential U_e(=Z_1Z_2e^2R_a where, R_a is atomic radius) arising from the interaction of the charged nuclei of the projectile with the electrons inside the target. In terms of astrophysical S-factor, the above expression can be written as  
        
        
            S(E)=S_b(E)exp(U_eπηE)
        
        
        Similarly, in stellar plasma, ionized atoms are also surrounded by a sea of electrons within the Debye-Huckle radius and there also screening effect is important which depends on the temperature and density of the plasma. This plasma screening factor is different from the lab screening factor. Hence, it is essential to determine bare cross-section as accurate as possible by eliminating the electron screening effect from laboratory experiment. From the bare cross-section, the actual cross-section in stellar plasma can be determined through
        
            σ_pl(E)=σ_b(E)f_pl
        
        
        where, f_pl(E) is the plasma screening enhancement factor and is related with the plasma screening factor by 
        
            f_pl(E)=exp(U_plπηE)
        
        
        U_pl(E) is the plasma screening potential and can be calculated within the framework of Debye-Huckel theory.  
        
        Several attempts by different authors have been taken to understand the electron screening factor. Engstler  et al. <cit.> have extracted screening potential for both the reactions ^6Li(p,α)^3He  using polynomial fit S(E)=a+bE+cE^2+dE^3. The sensitivity of the ratio of E to U_e has also been addressed by them. Extracted screening potential for both the reactions are found to be higher than the adiabatic limit (U_e=175 eV) <cit.>. Later Cruz et al. <cit.> have used lithium implanted targets and attempted to extract the screening potential for both ^6Li(p,α)^3He reaction. They have extracted potential as U_e=273±111 eV  for ^6Li(p,α)^3He. The values are lower than those obtained by Engstler  et al. <cit.> but still higher than the adiabatic limit. The uncertainty in the extracted electron screening potential is too large to make any feasible comment. In a systematic work by Barker <cit.> have also pointed out the higher value of lab screening potential with respect to the atomic adiabatic limit. 
        
        Indirect Trojan Horse Method has been taken by Lamia et al. <cit.> to obtain electron screening potential. They have also determined much higher potential than the predicted value of adiabatic limit. It is also interesting that the screening potential thus obtained for ^6Li(p,α)^3He (U_e=350±100 eV) is less than that for ^7Li(p,α)α (U_e=425±60 eV) which is also in contrary to the results obtained by Engstler et al. <cit.> . Phenomenological R-Matrix method has been applied by Angulo et al. <cit.> for ^6Li(p,α)^3He to obtain electron screening for both solid and gas target. Extracted electron screening potential is also found to higher for both cases.  
        
        Artificial neural networks (ANN) serve as a powerful mathematical tool for estimating various values in science and technology, including nuclear physics studies. This method is particularly important in providing accurate results for highly nonlinear relationships between dependent and independent data variables. Recently, ANN has found application in numerous areas of nuclear physics, such as constructing consistent physical formulas for detector counts in neutron exit channel selection <cit.>, determining one and two proton separation energies <cit.>, developing nuclear mass systematic <cit.>, determining ground state energies of nuclei <cit.>, identifying impact parameters in heavy-ion collisions <cit.>, determining beta-decay energies <cit.>, and estimating nuclear rms charge radius <cit.>.
        
        In this investigation, the Multi-Layer Perceptron based Feed-forward Artificial Neural Networks (ANN) <cit.> using scikit-learn is utilized to estimate the bare astrophysical S-factor, total S-factor and subsequently the electron screening potential for ^6Li(p,α)^3He reaction. The experimental datas were extracted from existing literature <cit.>. Through training the ANN with known experimental cross-section values for various combinations of beam, target, and bombarding energy, bare S-factor and total S-factor is successfully predicted. From the ratio of total S-factor with bare S-factor, the electron screening potential has been calculated. The details of Artificial Neural Network method will be discussed in next section followed by Calculations of Screening potentials and Conclusions.
         
        
        
        
        § ARTIFICIAL NEURAL NETWORK
        
        An artificial neural network (ANN) is a computational model that draws inspiration from the structure and functionality of biological neural networks found in the human brain. It serves as a crucial element in machine learning and artificial intelligence systems, aiming to emulate the information processing mechanism of the human brain. Artificial neural networks are constructed with neurons as their fundamental units. Each neuron receives inputs, undergoes computation, and generates an output. These neurons are arranged into layers, comprising an input layer, one or more hidden layers, and an output layer. In the case of a feed-forward neural network, information progresses unidirectionally from the input layer to the output layer without loops. The network includes multiple layers of nodes (neurons), featuring at least one hidden layer positioned between the input and output layers. Neurons establish connections through edges, each associated with a weight representing the strength of the connection. Additionally, a bias term is often employed to adjust the output of a neuron. Throughout the training phase, the weights and biases undergo adjustments to optimize the overall performance of the neural network. The activation function plays a crucial role in determining a neuron's output based on its input, introducing non-linearity and enabling the network to grasp intricate patterns. With a single output node, the network is well-suited for regression tasks, where the objective is to predict a continuous variable. During the feed-forward operation, input data progresses through the network layer by layer. Each neuron's output is computed by evaluating the weighted sum of its inputs and applying the activation function. Training a feed-forward neural network entails fine-tuning the weights and biases by considering the disparity between the predicted output and the actual target. This is typically accomplished through supervised learning using an optimization algorithm like gradient descent. The gradients of the loss concerning the weights and biases are computed, indicating how much the loss would change with adjustments to each weight and bias. The weights and biases are then modified in a direction that minimizes the loss, often employing optimization algorithms such as stochastic gradient descent or Adam. In the current analysis, a grid search will explore both `Adam' and `Stochastic Gradient Descent,' determining the most effective solver during the hyperparameter tuning process. These processes iterate for a specified number of training epochs (maxiter), progressively refining the model to enhance its predictive capabilities.
        
        The subsequent discussion explores the mathematical framework of a straightforward feedforward neural network, specifically a Multi-Layer Perceptron (MLP) based for regression. I will start by breaking down the network layer by layer, and then proceed to introduce the training process, which incorporates gradient descent and adam optimization. 
        
        
         §.§ Forward Pass:
        
        
         §.§ Input Layer:
        
        The neural network takes information about a single data point through a vector called X. If there are m features, X is like a vertical list with m rows and 1 column, often represented as:
        
            X    = [ x_1; x_2;   ⋮; x_m ]
           
        Here, each x_i represents a feature of the input data point. The information (x_1,x_2,...,x_m) passes to the nodes in the hidden layer. 
        
        
        
         §.§ Hidden Layer:
        
        For a single hidden layer consisting of n neurons, the output Z_hidden is calculated as:
        
            Z_hidden=(W_hidden· X+b_hidden)
         
        where,
        
        
        
        W_hidden is the weight matrix of size n × m.
        
        b_hidden is the bias vector of size n × 1.
        
        After computing Z_hidden, it is passed through an activation function σ:
        
            A=σ(Z_hidden)
        
        Where, A represents the output of the hidden layer. If the weighted sum of inputs to a neuron exceeds the threshold or activation layer, the neuron fires. In this analysis, the activation function for the hidden layers is being investigated with two choices: `ReLU' (Rectified Linear Unit) and `tanh' (Hyperbolic Tangent). The model will undergo training and evaluation using both activation functions to determine which one yields better performance for the provided data.
        
        So, the overall computation can be written as:
        
            A=σ(W_hidden· X+b_hidden)
        
        
        This represents the forward propagation through the first hidden layer of a neural network. The activation function σ introduces non-linearity to the model, allowing it to learn complex patterns in the data.
        
        In the present analysis four hidden layers are used in the neural network, the computation of the output at the first hidden layer Z_hidden1 with n_1 neurons in the layer and the subsequent hidden layers Z_hidden2 with n_2 neurons, Z_hidden3 with n_3 neurons and Z_hidden4 with n_4 neurons can be expressed as follows:
        
        For the first hidden layer:
        
            Z_hidden1=σ(W_hidden1· X+b_hidden1)
         
        Here:
        
            W_hidden1 is the weight matrix of the first hidden layer with dimensions n_1× m,
            
            b_hidden1 is the bias vector of the first hidden layer with dimensions n_1× 1,
        
            X is input vector with dimensions m× 1, 
            
            σ represents the activation function already mentioned above.
        
        For the second hidden layer:
        
            Z_hidden2=σ(W_hidden2· Z_hidden1+b_hidden2)
         
        
        Here:
        
            W_hidden2 is the weight matrix of the second hidden layer with dimensions n_2× n_1,
            
            b_hidden2 is the bias vector of the second hidden layer with dimensions n_2× 1.
        
            Z_hidden1 is output of the first hidden layer with dimensions n_1× 1.  
        
        For the third hidden layer:
        
            Z_hidden3=σ(W_hidden3· Z_hidden2+b_hidden3)
         
        
        Here:
        
            W_hidden3 is the weight matrix of the second hidden layer with dimensions n_3× n_2,
            
            b_hidden3 is the bias vector of the second hidden layer with dimensions n_3× 1.
        
            Z_hidden2 is output of the first hidden layer with dimensions n_2× 1.    
        
        For the fourth hidden layer:
        
            Z_hidden4=σ(W_hidden4· Z_hidden3+b_hidden4)
         
        
        Here:
        
            W_hidden4 is the weight matrix of the second hidden layer with dimensions n_4× n_3,
            
            b_hidden4 is the bias vector of the second hidden layer with dimensions n_4× 1.
        
            Z_hidden3 is output of the first hidden layer with dimensions n_3× 1.        
        
        This process enhances the neural network's ability to learn hierarchical features and intricate representations from the input data.
        
        
        
         §.§ Output Layer:
        
        For regression, there is typically no activation function in the output layer, and the output Y is given by:
        
            Y=W_output· Z_hidden4+b _output
         
        
        where, W_output is a weight matrix of size 1 × n_4
        
        Z_hidden4 is the output from the fourth hidden layer,
        
        b_output is the bias term for the output layer.
        
        
        
         §.§ Training with Stochastic Gradient Descent:
        
        
         §.§ Loss Function:
        
        
        The Mean Squared Error (MSE) is commonly used for regression:
        
            MSE=∑_i=1^N (Ŷ_̂î-Y_i)^2N
         
        where,
        N is the data points.
        Ŷ_̂î is the predicted output for data point i.
        Y_i is the actual output for data point i.
        Normalized Mean Squared Error(NMSE) is defined as:
        
            NMSE=MSES_max-S_min
         
        where, S_max and S_min is the maximum and minimum value of astrophysical S-factor.
        Furthermore, a regularization term (R) is incorporated into the loss function during training to counteract overfitting by imposing penalties on large weights in the model. In the case of L2 regularization, the regularization term is directly linked to the square of the weights:
        
            R_L2=α2∑_i=1^n W_i^2
         
        Here, α represents the strength of the regularization term. Higher values of α result in stronger regularization.
        
        
         §.§ Back Propagation:
        
        
        Backpropagation (short for “backward propagation of errors”) is a widely used supervised learning algorithm for training artificial neural networks. It is a gradient-based optimization algorithm that aims to minimize the error between the predicted output and the actual target output. The key steps in the backpropagation process for an artificial neural network (ANN) are as follows:
        
        (1) Compute the gradient of the regularization term with respect to the output layer weights: ∂ R∂ W_output.
        
        (2) Compute the gradient of the mean squared error with respect to the output layer weights: ∂ MSE∂ W_output.
        
        (3) Compute the gradient of the mean squared error with respect to the output layer biases: ∂ MSE∂ b_output.
        
        (4) Propagate the gradients through the network to compute the gradient with respect to the fourth hidden layer weights and biases: ∂ R∂ W_hidden4,∂ MSE∂ W_hidden4, ∂ MSE∂ b_hidden4 .
        
        (5) Propagate the gradients through the network to compute the gradient with respect to the third hidden layer weights and biases: ∂ R∂ W_hidden3,∂ MSE∂ W_hidden3, ∂ MSE∂ b_hidden3 .
        
        (6) Propagate the gradients through the network to compute the gradient with respect to the second hidden layer weights and biases: ∂ R∂ W_hidden2,∂ MSE∂ W_hidden2, ∂ MSE∂ b_hidden2 .
        
        (7) Propagate the gradients through the network to compute the gradient with respect to the first hidden layer weights and biases: ∂ R∂ W_hidden1,∂ MSE∂ W_hidden1, ∂ MSE∂ b_hidden1 .
        
        The process involves the backward propagation of gradients through each layer of the network. The chain rule is applied to calculate the gradients at each layer based on the gradients computed in subsequent layers. This iterative process helps update the weights and biases of the network, refining its parameters for improved performance over time.
        
        
        
        
         §.§ Update Weights and Biases:
        
        
        
        Update the weights and biases using the gradients and a learning rate η:
        
            W_output← W_output-η(α∂ R∂ W_output+∂ MSE∂ W_output)
        
            b_output← b_output-η∂ MSE∂ b_output
        
            W_hidden4← W_hidden4-η(α∂ R∂ W_hidden4+∂ MSE∂ W_hidden4)
        
            b_hidden4← b_hidden4-η∂ MSE∂ b_hidden4
        
            W_hidden3← W_hidden3-η(α∂ R∂ W_hidden3+∂ MSE∂ W_hidden3)
        
            b_hidden3← b_hidden3-η∂ MSE∂ b_hidden3
        
            W_hidden2← W_hidden2-η(α∂ R∂ W_hidden2+∂ MSE∂ W_hidden2)
        
            b_hidden2← b_hidden2-η∂ MSE∂ b_hidden2
        
            W_hidden1← W_hidden1-η(α∂ R∂ W_hidden1+∂ MSE∂ W_hidden1)
        
            b_hidden1← b_hidden1-η∂ MSE∂ b_hidden1
         
        where, R represents the regularization term added to the loss function during training. 
        
        This process is repeated for maximum iterations until convergence.
        The training with Stochastic Optimization using Adam(Adaptive Moment Estimation) has been described in  <cit.>. 
        
        This is a simplified overview, and the actual scikit-learn implementation may involve additional details. 
        
        
        § CALCULATIONS OF ELECTRON SCREENING POTENTIAL
        
        Multi Layer Perceptron based ANN algorithm has been used to find the S-factor from the available experimentally measured literature data. Those data sets are chosen in present calculations which are experimentally well established and have better accuracy. Reliable experimental data are crucial as it will contribution to the robustness of the present ANN model calculations, that can help in drawing meaningful conclusions. Table <ref> summarizes the different data sets used for the present analysis. The data sets are directly taken from  EXFOR database. Some datasets in the existing literature exhibit notable discrepancies with the majority of previous findings are excluded in the present analysis. The utilized data sets cover an energy range in which both electronic screening effects are expected to be present(below ≈ 60 keV) and in which they are not expected to be present(above ≈ 60 keV). The collecting dataset has then been splitted randomly into train and test part. The train part contains 70% and test part contains 30% of total data set. Model is trained using the train dataset. During the initial phase of the study, following numerous trials, the optimal number of hidden layer neurons was identified and employed in a consolidated four-in-one layer which demonstrated the most effective outcomes for the present problem. Present analysis employs RandomizedSearchCV with a comprehensive grid search for optimal hyperparameters. Hyperparameters play a crucial role in machine learning algorithms by controlling the learning process and shaping the final model's behavior. Optimized hyperparameters are used to obtain the best correlation coefficient. Higher correlation coefficient indicates the lower uncertainty and vice versa.  In the present calculations, one input layer, four hidden layers with size (128,64,32,16) and one output layer is used after obtaining this configuration from several trials for optimization. The number of hidden layer and hidden units are so chosen to get the minimum loss as much as possible. The maximum iteration is used as 5000. In the Adam algorithm, the initial learning rate is 0.001 and decay constants are 0.9 and 0.999. Once the model are trained, the same set of Hyperparameters are used to predict the performance of the model on test data-set.  In the second part, data's are above 60 keV are used to find the bare S-factor using the same ANN code with same hyperparameters. Once the bare S-factor is obtained the electron screening potential is then obtained by taking the ratio of the total S-factor with the bare S-factor. The total S-factor and bare S-factor obtained from the present ANN based model is shown in Fig. <ref>. Bare astrophysical S-factor at zero energy calculated from the present model is 3.36 Mev-barn and it is tabulated in Table <ref> and compared with the value obtained by the different methods in Literature. The values obtained from the present calculation is in excellent agreement with that obtained from the R-MATRIX calculation <cit.> and THM calculation <cit.> by taking into account the systematic uncertainty. The variation of bare astrophysical S-factor with that obtained from R-MATRIX and THM has been compared in Fig. <ref> and Fig. <ref> respectively. It has been observed from the comparison that the present calculations reasonably matches with the well established R-MATRIX and THM calculation. Electron screening potential obtained from the present calculation (U_e=220 eV) has been tabulated in Table <ref> and compared with the electron screening potential obtained from the other methods as shown in Fig. <ref>. The electron screening potential thus obtained from the present ANN based model nicely matches with the other methods within the quoted uncertainty. This results indicates that Multi-Layer Perceptron based Artificial Neural Network model can be used an alternative tools to find the the electron screening potential in light charged particle induced reactions of astrophysical interest.   
        
        
        
        § CONCLUSIONS
        
        In this study, the electron screening potential is obtained for ^6Li(p,α)^3He reactions by using the Multi-Layer Perceptron based Artificial Neural Network based algorithm for the first time. From the present analysis, the electron screening potential is obatined as 220 eV. The electron screening potential matches resonably well with other welll estrablished techniques like R-MATRIX calculation and THM method indicating the ANN method as an alternative powerful tool for extraction of such astrophysical quantity. The variation of the bare S-factor with R-MATRIX and THM calculation also supports the claim.   
        
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 Paper Title: Investigating the role of nuclear parameters on oscillation modes in hot Neutron Stars.



APS/123-QED





nilaksha.barman@iucaa.in
Inter-University Centre for Astronomy and Astrophysics

bikramp@iucaa.in
Inter-University Centre for Astronomy and Astrophysics

debarati@iucaa.in
Inter-University Centre for Astronomy and Astrophysics






             

§ ABSTRACT 
 Recent studies have revealed that certain nuclear parameters are more dominant than others in governing global neutron star properties, such as its structure or oscillation mode characteristics. Although neutron stars can in general assumed to be cold, in astrophysical scenarios such as newly born neutron stars or remnants of binary neutron star mergers, finite temperature effects play a non-negligible role. In this work, we perform a consistent and systematic investigation of the role of nuclear parameters and thermal effects on neutron star properties and fluid oscillation modes within a full general relativistic scheme. We impose constraints on the parameter space of the relativistic mean field model using state-of-the-art information from terrestrial experiments and multi-messenger astrophysical data. We find effective nucleon mass to be the most important nuclear parameter controlling astrophysical observables of hot neutron stars, similar to the cold beta equilibrated matter. However, we conclude that the interplay among saturation properties and astrophysical observables depends not only on the thermal configurations considered but also on the constraints imposed. We also investigated the role of nuclear saturation parameters on some universal relations for hot NSs which are important in gravitational wave asteroseismology. Our investigation confirmed that these relations are mostly insensitive to nuclear saturation properties and mainly affected by variation of charge fraction in the star. 



                              
Investigating the role of nuclear parameters on oscillation modes
 in hot Neutron Stars
    Debarati Chatterjee
    August 5, 2024
=======================================================================================







§ INTRODUCTION




Neutron stars (NSs) are the densest known compact objects in the observable universe. They provide us unique opportunity to study matter under extreme conditions and hence sometimes referred to as cosmic laboratories. They are formed at the end stage of stellar evolution of massive stars known as core-collapse supernovae or type-II supernovae. Hot newly born neutron stars undergo cooling via neutrino emission and mature into cold degenerate compact objects. Cold NSs are primarily composed of neutrons, protons, electrons and muons  (free neutrinos escape at low temperature < 1 MeV). However, in the high density environment of the NS core other exotic particle degrees of freedom such as hyperons, quarks, kaon condensates etc. might appear in addition to nucleons via weak interactions and  can be in equilibrium with the mixture <cit.>. 





The composition of the cold NS interior can be described by Equation of State (EoS) models that relate the pressure and the density <cit.>. The EoS connects the microscopic properties of the NS to its global macroscopic structure, such as its mass or its radius. However, the ultradense nuclear matter in the NS core requires solving the complicated many-body problem. There are different approaches e.g. microscopic models <cit.> which calculate the nucleon-nucleon interactions ab-initio, while phenomenological models <cit.> adjust the nucleonic parameters to satisfy observed nuclear saturation properties.
Chiral effective field theory (χEFT) <cit.> is a microscopic method which uses chiral expansion for a reliable description of the sub-saturation region of pure neutron matter. χEFT results are being used widely in phenomenological or parametric models as a robust method to constrain the parameter space of dense matter EoS at low densities e.g. <cit.>.





Being highly relativistic and compact, neutron stars can emit gravitational waves (GWs) via both binary neutron star (BNS) mergers as well as in isolated systems. Any non-axisymmetric perturbations, e.g. quasi-normal modes, deformations such as mountains or magnetic deformations, can lead to time varying mass quadrupole moment in isolated neutron stars and may manifest themselves as gravitational waves. The frequencies of some of the oscillation modes lie in the high frequency band of GWs (e.g. f-mode frequencies lie in 1-2 KHz) and provide interesting prospects of detection by current and future generation GW detectors.  The first detection of BNS (GW170817) via GW and electromagnetic (EM) signals opened up a new window in the multimessenger astronomy <cit.>. In this context previous works have shown that spin and eccentricity effects lead to excitation of f-modes during the inspiral phase of BNS mergers <cit.>. With LIGO O4 run and subsequent future GW observations many more BNS merger events are expected to be detected. 





Any realistic model of neutron stars should satisfy astrophysical constraints from EM and GW observations. The maximum predicted theoretical NS mass should be able to explain the highest observed NS mass of 2.08^+0.07_-0.07M_⊙ <cit.> or 2.078^+0.067_-0.066 M_⊙ <cit.>. EM observations from the recently launched NICER put constraints on masses and radii two pulsars namely PSR J0030-0451 <cit.> and PSR J0740-6620 <cit.>. Analysis of GW170817 revealed that the radius of a 1.4M_⊙ neutron star is 13.4±0.2 km <cit.> and the estimates for effective tidal deformability for the binary system is 300^+420_-230 <cit.>. These multi-messenger observations are being used to constrain the high and intermediate density region of nuclear EoSs <cit.>. 





Astrophysical events such as proto-neutron stars (PNSs) born out of supernovae (SN) and binary mergers exceed temperatures of tens of MeV and the cold matter approximations are no longer valid

 <cit.>.
It is important to take into account thermal effects as they play a significant role in supernova dynamics, oscillations in PNS as well as in determining stability of the post-merger remnants, which along with the composition  can change the threshold mass limit for prompt collapse to a black hole <cit.>. The EoS of a finite temperature system can be uniquely characterized by three quantities, temperature T, baryon number density n_b and the charge fraction of strongly interacting particles Y_Q.  In order to be applicable in numerical simulations, a wide range and finely spaced sets of parameters should be considered, along with different possible composition in the neutron star interior. The EoS of hot homogeneous matter spans 0-100 MeV in temperature, 0.01-0.60 in charge fraction and 0.5-6.0 times in nuclear saturation densities in BNS mergers and supernovae <cit.>. In the absence of EoS models spanning the required parameter ranges and composition, the so-called Γ-law (supplementing the cold EoS with ideal gas thermal contributions) has often been employed to incorporate finite temperature effects in simulations. However, this method of constructing EoS has been shown to be inappropriate particularly for matter consisting of non-nucleonic degrees of freedom <cit.>. It has been described in a recent work in quark-hadron phase transition <cit.> that Γ value should be different in hadronic (Γ∼ 1.75) and quark (Γ∼ 4/3) phases. The Γ-law also does not take into account the effect of charge fraction across all densities. Therefore it is crucial to incorporate composition and thermal EoS effects consistently into simulations.




Currently, there exist only a limited number of available finite temperature EoSs which are widely applied in simulations e.g. Lattimer Swesty (LS220) <cit.> and Shen et al. (STOS) <cit.>, both for nucleonic matter and based on single nucleus approximation (SNA). The more recent EoS model of Hempel and Schaffner-Bielich (HS) <cit.> is based on nuclear statistical equilibrium (NSE) which incorporates realistic nuclear masses and goes beyond single-nucleus approximation. The high density core can be modelled using the RMF formalism and is used in constructing non-linear (NL) RMF EoSs such as SFHo/SFHx <cit.>, HS(IUF) <cit.>, QMC-RMF <cit.>. The RMF EoSs based on linear density dependent (DD) models use the formalism of Typel and Wolter <cit.>. They include nucleonic EoSs such as HS(DD2) <cit.> as well as models with exotic matter such as hyperons and Δ-baryons e.g. BHB(DD2Lϕ) <cit.>, OMHN(DD2Y) <cit.>, R(DD2YΔ) <cit.>. Other finite temperature EoSs include hybrid Chiral Mean field (CMF) model <cit.>. 



The existing finite temperature EoS models, which consider a fixed parametrization, correspond to certain values of nuclear saturation properties but they do not span the entire range of uncertainty allowed by current terrestrial experiments. However recent studies have shown that nuclear saturation properties are correlated with astrophysical observables such as radius and tidal deformability in cold neutron stars <cit.>. These studies provide an interesting prospect to investigate the role of nuclear saturation effects on NS observable properties at finite temperature. There have been a few recent works in this direction. An EoS formalism within the covariant density functional approach with density dependent couplings employing the posteriors of a Bayesian inference (DDB*) formalism <cit.> has been extended to finite temperature by Raduta et al. <cit.>. In another approach, the effect of slope of symmetry energy L_sym and skewness Q_sat on finite temperature EoS has been studied within the covariant density functional scheme for fixed DD-RMF parametrizations in  <cit.>. 




Significant efforts have been dedicated for advancing our understanding of GW asteroseismology for PNSs, through the study of GWs emitted by PNS oscillation modes <cit.>. Both   simulations <cit.> and perturbative approaches <cit.> have been considered in studying the PNS oscillations.
In the simulation by Torres et al. <cit.>, it was shown that most of the GW energy is stored in the lowest-order eigenmodes, in particular ^2g_1 and ^2f. The same group in their previous study in Cowling approximation showed that higher order p_i-modes upto 5 nodes and g_1-modes appear in GW spectrogram <cit.>. PNS simulations by Sotani et al. <cit.> also indicate that g_1 and f-modes appear to dominate GW spectra in the earlier and later stages of PNS evolution respectively.
 Further analyses involve assessing the detectability of GWs from PNSs and inferring PNS properties from the GW data <cit.>.
 


Cold NSs are known to follow certain EoS independent universality such as C-Love, I̅-Λ-Q̅, f-Love <cit.> among others. Much effort has been dedicated into deriving universal relations (URs) that link the oscillation mode properties of PNSs with their physical characteristics, such as mass and radius, which are crucial for GW asteroseismology <cit.> of proto-neutron stars. Raduta et al. <cit.> obtained I-Λ-Q-C relations for hot NSs using a DD-RMF parameterization.  This was extended by Khadkikar et al. <cit.> to explore URs between static and Kepler rotation properties such as mass, radius and frequency etc. and   fit relations were provided for I-C and Q-C for different thermal configurations with various tabulated EoSs. Similar UR study for nucleonic and hybrid hot stars were performed by Largani et al. <cit.> where they provided fits for normalized moment of inertia with compactness at Keplerian limit for fixed values of entropy per baryon and lepton fraction. Laskos-Patkos et al. <cit.> studied deviations of hot neutron stars from cold universality for C-Love relations.





In a series of recent works <cit.>, we developed a formalism to constrain the EoS of cold dense NS matter using χEFT at low densities, heavy-ion data at intermediate densities and multi-messenger astrophysical data at high densities, within a non-linear relativistic mean field model (NL-RMF) framework. These works also investigated the correlation of nuclear empirical parameters with NS macroscopic properties (radius R, tidal deformability Λ) and found the effective nucleon mass m^* to be strongly correlated to these observables. Further, recent investigations within the same formalism also showed that m^* is the dominant parameter related to oscillation modes such as fundamental (f-) mode in NSs <cit.>. However, it is not clear whether such correlations will hold in finite temperature scenarios, such as oscillations in newly born neutron stars or binary neutron star mergers. A few interesting studies recently probed thermal effects on the NS EoS imposing constraints from χEFT and NS maximum mass within the NL-RMF formalism (see e.g. <cit.>). In another recent work <cit.>, thermal effects were studied by considering a few density-dependent (DD) RMF tabulated EoSs from CompOSE <cit.>. The effect of finite temperature on f- and first overtone pressure (p_1)-modes was studied within Cowling approximation for the chosen EoSs. However, the correlations among nuclear and NS properties were investigated only for cold NS matter employing the DDB* model. From these studies, the correlations of the nuclear saturation properties on NS global properties and general relativistic mode characteristics at finite temperature are not directly evident. 




The aim of this work is to perform a systematic and consistent investigation of the role of nuclear saturation parameters as well as thermal properties i.e., entropy per baryon, S/A and out-of-β equilibrium condition (charge/electron fraction, Y_Q) on the EoS of hot and dense matter. For this we extend our recently developed formalism of imposing constraints at different densities on the parameter space in the RMF scheme for cold matter EoS to incorporate thermal effects.  The finite temperature EoS is then used to derive macroscopic properties of hot neutron stars.  The EoS framework is further applied for the study of oscillation modes within a fully general relativistic treatment. With the posteriors, we search for possible correlations between nuclear saturation parameters as well as thermal properties with astrophysical observables. We also investigate the effect of nuclear saturation and thermal properties on the Universal relations. Note that we have not included the effects of neutrino trapping in our study. 






The paper is structured as follows: In section <ref> we provide a description of the formalism involved. In subsection <ref> the RMF model and its extension to include thermal effects have been given. We also derive the expressions of various thermodynamic quantities of interest. In the next subsection <ref> we determine macroscopic NS properties, while in the subsection <ref> the fully General Relativistic (GR) description of calculating the oscillation modes are given. In section <ref> we discuss results for thermal effects on EoSs consistent with χEFT, astrophysical and heavy ion collision constraints as shown in subsection <ref> and their global properties in subsection <ref>. 

We then discuss results for oscillation mode frequencies and damping timescales in <ref>. We also present the results of the investigation of correlations of the nuclear properties with observable NS properties.

In <ref> we show deviation of universality of oscillation modes at finite temperature compared to cold β equilibrated matter. Then we summarise our results in section <ref>. Throughout the paper we use the natural units (c=ħ=k_B=G=1).



§ FORMALISM






 §.§ Equation of State

The EoS of the NS core is modelled using a non-linear relativistic mean field (NL-RMF) formalism  <cit.>. The nucleon interactions occur via exchange of σ, ω and ρ mesons. σ is a scalar meson that facilitates long range attraction while ω is a vector meson for nucleon-nucleon repulsion. ρ vector meson is responsible for isospin asymmetry and to maintain charge neutrality. The non-linear terms are the scalar self-interaction term U_σ  given by U_σ = 1/3b m_N (g_σ Nσ)^3 + 1/4c (g_σ Nσ)^4, and Λ_ω which mimics the ω-ρ interaction.
The Lagrangian density for the homogenous nuclear matter is given below: 

    ℒ   = ∑_i ∈ Bψ_i (i γ_μ∂^μ - m_i + g_σ iσ - g_ω iγ^μω_μ - g_ρ iγ^μρ_μ^a τ^a )  ψ_i 
       + 1/2∂^μσ∂_μσ - 1/2 m_σ^2 σ^2 - 1/4ω_μνω^μν + 1/2 m_ω^2 ω^2 
       - 1/2ρ_μνρ^μν + 1/2 m_ρ^2 ρ^2 + Λ_ω (g_ρ N^2 ρ_μ.ρ^μ) (g_ω N^2 ω_μω^μ) 
       - U_σ + ψ_e (i γ_μ∂^μ - m_e) ψ_e 
       +∑_i = B,eψ_i(-q_i γ_μ A^μ)ψ_i -1/4F_μνF^μν ,

where, ψ_i are the Dirac spinor for nucleon(s). The isoscalar couplings (g_σ B,g_ω B, b, c) are fitted to reproduce isoscalar saturation properties (saturation density n_sat, energy per particle E_sat, compressibility K_sat, effective nucleon mass m^*) while isovector coupling constants (g_ρ B, Λ_ω) 
 are fitted to reproduce isovector saturation properties (symmetry energy J_sym and its slope L_sym). The leptonic species present are non-interacting electrons (e^-) obeying Fermi-Dirac (FD) statistics; muons (μ^-) are not considered as they are expected to appear after the star reaches β equilibrium. Photons (γ) are bosonic non-interacting degrees of freedom described by A_μ and F^μν = ∂^μ A^ν - ∂^ν A^μ. Under RMF approximation the meson mean field equations become,

    m_σ^2 σ̅   = ∑_i ∈ B g_σ i n_i^s - ∂ U/∂σ̅
    
        m_ω^2 ω̅_0    = ∑_i ∈ B g_ω i n_i - 2 Λ_ω g_ρ^2 g_ω^2 ρ̅_03^2 ω̅_0   
    
        m_ρ^2 ρ̅_03   = ∑_i ∈ B  g_ρ i t_3i n_i - 2 Λ_ω g_ρ^2 g_ω^2 ρ̅_03ω̅_0^2  ,

where σ̅, ω̅_̅0̅, ρ̅_̅0̅3̅ are the mean meson fields and t_3i is the third component of the isospin of the i-th baryonic species.




Including thermal effects, the number density and scalar density of the i-th baryonic species respectively are <cit.>:

    n_i    = 2J_i+1/2π^2∫ dk k^2 [f_FD(E_i(k) - μ_i^*/T). 
       . - f_FD(E_i(k) + μ_i^*/T) ]  , 
    
        n_i^s    = 2 J_i+1/2π^2∫ dk k^2 m_i^*/E_i(k)[f_FD(E_i(k) - μ_i^*/T) . 
        .+ f_FD(E_i(k) + μ_i^*/T) ]  .

In eqs. <ref> and <ref>, m_i^* = m_i - g_σ iσ̅ is the effective baryon mass and E_i(k) = √(k^2+m_i^*^2), the effective energy dispersion of the i-th baryonic species. J_i is the spin of the i-th baryonic species. f_FD(x)=1/(1+e^x) is the FD distribution function, T is the equilibrium temperature of the mixture and μ_i^* is the effective chemical potential of the i-th species given by,

    μ_i^* = μ_i - g_ω iω̅_0 - g_ρ i t_3iρ̅_03 .

where μ_i is the actual chemical potential of the corresponding species and can be written in terms of baryon (μ_B) and charge (μ_Q) chemical potentials. If B_i and Q_i are baryon and electric charges of the i-th species then, μ_i = B_i μ_B + Q_i μ_Q. Both particle and anti-particle species contributes to entropy, as do leptons/anti-leptons and radiation. The effective chemical potentials for anti-particles are negative of corresponding particle effective chemical potentials. Chemical potential for electrons is μ_e = μ_L-μ_Q, where μ_L is the lepton chemical potential. The total entropy density (s) is given by

    s    = -∑_i ∈ B2 J_i+1/2π^2∫ dk k^2 [f_FD(E_i(k) - μ_i^*/T). 
        .ln f_FD(E_i(k) - μ_i^*/T) + f_FD(E_i(k) - μ_i^*/T). 
        . lnf_FD(E_i(k) - μ_i^*/T)] 
       + anti-baryon contribution (μ^*_i→-μ^*_i) 
       + s_e^- + s_e^+
       + s_γ ,


where, f_FD(x) = 1 - f_FD(x) is the Fermi-Dirac distribution function for vacancy of state.
The electronic contribution to the entropy density (s_e^-) is found by substituting electron chemical potential (μ_e) in place of effective chemical potential (μ_i^*) in eq. <ref>, while the radiation (γ) contribution is s_γ = 4 π^2 T^3 / 45. The entropy per baryon is then given by S/A=s/n_B. Also radiation energy density, ϵ_γ = π^2 T^4 / 15. From the Lagrangian described in eq. <ref>, the energy density can be evaluated from the energy momentum tensor T^μν in the mean field approximation, and the pressure can be evaluated using the Gibbs-Duhem relation, as given below:





    ϵ   = <T^00> 
       = 1/2 m_σ^2 σ^2 + 1/2 m_ω^2 ω_0^2 + 1/2 m_ρ^2 ρ_03^2 + U_σ
       + 3Λ_ω (g_ρ N g_ω Nρ_03.ω_0)^2 
       + ∑_i2 J_i+1/2π^2∫ dk k^2 E_i(k)  [f_FD(E_i(k) - μ_i^*/T). 
       +. f_FD(E_i(k) + μ_i^*/T) ] 
       + ϵ_e^- + ϵ_e^+
       + ϵ_γ , 
    
        p    = ∑_i = B, eμ_i n_i + Ts - ϵ .

The electron contribution to the energy density is again found by replacing effective chemical potential (μ_i^*) with electron chemical potential (μ_e). For describing the low density part of the EoS the HS(DD2) model is used <cit.>, taken from CompOSE database <cit.>. For each thermal case, the crust and core are matched by keeping S/A and Y_Q same for crust [HS(DD2)] and core (NL-RMF) ensuring thermodynamic consistency.



The direct consequences of finite temperature are effects such as out-of-β equilibrium, neutrino trapping among others. These effects can manifest themselves in the hot EoS and consequently in the macroscopic properties. At postbounce evolution in PNSs, the thermal properties control the EoS until the neutron star becomes cold and degenerate. These thermal properties are entropy per baryon (S/A) and electric charge (out-of-β equilibrium) and/or lepton fractions (in case of neutrino trapping). Initially the star is in a low entropy state with high charge/lepton fractions, it then becomes a hot (high entropy) star with low charge/lepton fractions and eventually attains a deleptonized cold β equilibrium state <cit.>. It has been shown in previous works that this leads to deviations from universal relations (URs) usually observed in cold neutron stars <cit.>.



The core temperature profiles corresponding to different thermodynamic conditions for nucleonic matter N(S/A,Y_Q) i.e., different values of entropy per baryon S/A and charge fraction Y_Q, are shown in fig. <ref>. It can be seen that higher entropy leads to larger temperatures and higher proton fraction results in lower temperatures at high densities.




 §.§ Macroscopic and Tidal properties



We assume a perfect fluid configuration which is static and spherically symmetric. The space time geometry for such a configuration is given by Schwarzschild interior metric 

    ds^2 = -e^2Φ (r) dt^2 + e^2λ (r) dr^2 + r^2 (dθ^2 + sin^2θ dϕ^2) .

consisting of metric functions Φ (r) and λ (r) which depend on the underlying equation of state [p = p(ϵ)].
The macroscopic configuration of the hot neutron star is determined by the hydrostatic equilibrium condition and is obtained by solving the Tolman-Oppenheimer-Volkoff (TOV) equations:

    dm (r)/dr   = 4 π r^2 ϵ(r) , 
    dp (r)/dr   = -[p(r)+ϵ(r)] [m(r)+4π r^3 p(r)]/[r^2(1-2m(r)/r)] , 
    
        e^2 λ (r)   = 1/[1-2m(r)/r] , 
    d ϕ(r)/dr   = - 1/[ϵ (r) + p(r)]dp (r)/dr .

Here m(r) is the mass enclosed within a spherical shell of radius r and therefore if R is the radius of the star then m(r=R) = M, the NS mass. The tidal properties of the star are encoded in tidal love number k_2 <cit.> and it is related to dimensionless tidal deformability Λ through stellar compactness C = M/R.

    Λ = 2/3k_2/C^5 .


We display the NS mass-radius relations in fig. <ref> corresponding to the thermodynamic conditions mentioned in the previous section. The case N(T=0) stands for β equilibrated nucleonic matter.

It can be observed from fig. <ref> that the increase of charge fraction leads to a decrease in maximum mass supported by the hot neutron star. Both the increase in S/A and Y_Q results in higher radii for the same mass configurations in intermediate mass ranges. As an example we have also shown the isothermal configuration with T = 20 MeV and Y_Q=0.2. For this configuration low density part is taken again from HS(DD2) EoS with T = 20 MeV and Y_Q=0.2 from CompOSE <cit.> to ensure thermodynamic consistency. This configuration has the flattest mass-radius relations among all the cases.




 §.§ Oscillation modes




In general relativity, oscillation modes in neutron stars can be analyzed by considering perturbations in both the fluid and the background metric. These modes are categorized into polar and axial types. Here, we focus on polar nonradial oscillations, as they are strongly coupled to the fluid and are dominant for gravitational wave radiation. In GR, the modes are quasi-normal, characterized by complex eigenfrequencies (ω), where the real part relates to the mode's frequency (f) and the imaginary part corresponds to the gravitational damping time (τ). However, a simplified approach known as the relativistic Cowling approximation, which neglects metric perturbations, can solve for the oscillation modes, providing information solely on the oscillation frequency. While GR offers a comprehensive understanding of oscillation modes and GW damping times, it is more complex and computationally involved. Meanwhile, the simplified relativistic Cowling approximation can be used to draw many qualitative conclusions and study the oscillation of the PNS.  It has been shown that for the cold NS  f -mode, Cowling approximation can overestimate the mode frequency by ∼ 30% compared to the frequency obtained with complete linearized equations of general relativity. Similar conclusions have also been drawn for the PNS cite<cit.>. Both the relativistic Cowling and GR formalisms are employed in this study to solve for the NS oscillation modes and described separately in  <ref> respectively.


  §.§.§ Relativistic Cowling Approximation

Within the Cowling approximation, to find mode frequencies one has to solve the following differential equations  (Sotani, Vasquez, Sandoval,PRC): 

    d W(r)/dr   =   d ϵ/dp[ω^2r^2e^Λ (r)-2ϕ (r)V (r)+d Φ(r)/dr W (r)] 
       -    l(l+1)e^Λ (r)V (r) 
    d V(r)/dr   =    2dΦ (r)/dr V (r)-1/r^2e^Λ (r)W (r)

Where,

    d Φ(r)/dr=-1/ϵ(r)+p(r)dp/dr

The functions V (r) and W (r) along with frequency ω, characterize the Lagrange displacement vector  (ζ) associate to perturbed fluid,

    ζ^i= ( e^-Λ (r)W (r),-V (r)∂_θ,-V (r) sin^-2θ ∂ _ϕ)r^-2 Y_lm (θ,ϕ)

where Y_lm (θ,ϕ) is the  lm-spherical harmonic. Solution of eq. (<ref>) with the fixed background metric  (<ref>) near origin will behave as follows:



    W (r)=Ar^l+1,  V (r)=-A/l r^l

The vanishing  perturbed Lagrangian pressure at the surface will provide another constraint to be included while solving eq.(<ref>) which is given by,

    ω^2e^Λ (R)-2Φ (R)V (R)+1/R^2dΦ (r)/dr|_r=RW (R)=0

The differential eqs. (<ref>) are integrated along with the TOV <ref> from center to the surface of the star with different guesses of ω, among the solutions, those that satisfy the boundary condition given by Eq. (<ref>) are the eigenfrequencies of the star. In this work, the solutions were obtained using the Ridder's method.



In general relativity (GR), NS oscillation modes are analyzed by considering perturbations in both the fluid and the background metric. These modes are classified into polar and axial types depending on their transformation under parity <cit.>. Our focus here is on polar (which are strongly coupled to the fluid) and non-radial (dominant in the GW emission) oscillations. The theory of non-radial oscillations of non-rotating NSs, along with the mathematical formalism in the context of GR, was developed in the pioneering work of Thorne <cit.>. In GR, these modes are quasi-normal, characterized by complex eigenfrequencies (ω). The real part of ω corresponds to the mode frequency (ν), while the imaginary part is related to the gravitational damping time (τ).


A simplified approach known as the relativistic Cowling approximation, which neglects metric perturbations, is frequently used to solve for the oscillation modes, providing information solely on the oscillation frequency. While GR offers a comprehensive understanding of oscillation modes and GW damping times, it is more complex and computationally involved. Meanwhile, the simplified relativistic Cowling approximation has been used to draw many qualitative conclusions and study PNS oscillations. The accuracy of the oscillation mode solutions in the relativistic Cowling approximation of the PNS has been demonstrated in <cit.>. Similar to the cold NSs, the relativistic Cowling approximation can overestimate the f-mode frequency by ∼20 % compared to that obtained in the GR formalism. Further investigations involve searching for the dependence of the mode properties on the outer boundary conditions <cit.> and numerical dimensions <cit.>. Additionally,  in the work of <cit.>, the post-bounce oscillation modes of PNSs have also been studied in general relativity, including spacetime perturbations. 


Several approaches have been developed in the past few decades to solve for the oscillation modes within the linearized theory of GR, including the direct integration method <cit.>, WKB approximation, the CF method by Chandrasekhar and Ferrari <cit.> and the method of continued fraction <cit.>. The differential equations for the fluid and metric variables for solving the non-radial modes can be found in <cit.>. We adopt the direct integration method developed by  Lindblom and  Detweiler <cit.> to solve for the non-radial QNM modes. In short, the coupled perturbation equations for perturbed metric and fluid variables are integrated within the NS interior with appropriate boundary conditions. We define the PNS boundary as the radius obtained from the TOV solutions. Further, the fluid variables are set to zero outside the star, and the Zerilli wave equation <cit.> is integrated far away from the star. Further, a search is carried out for the complex mode frequency (ω=2πν+1/τ) corresponding to only the outgoing wave solution to the Zerilli equation at infinity. For finding the mode characteristics, we use the numerical methods developed in our previous work <cit.>. 

We display f- and p_1-mode frequencies of some specific thermal cases  in figs. <ref> and <ref> respectively calculated in general relativistic treatment. The p_1-mode frequencies are highest for the cold configuration and they are found to decrease with an increase in entropy per baryon and charge fraction. The isothermal configuration for T=20 MeV has the lowest p_1-mode frequencies. A crossing appears between N(1, 0.4) and N(2, 0.2) configurations for the p_1-mode at high mass. This behaviour is also seen in the full parameter space allowed by various constraints (discussed in subsection <ref>). On the other hand, f-mode frequencies for all cases have some overlapping regions at intermediate and high mass ranges.











 §.§ Constraining the model parameters




The parameter space of the EoS considered should be consistent with microscopic calculations, state-of-the-art astrophysical observations and terrestrial nuclear experiments. In previous works by some of the authors of this paper <cit.>, a Bayesian-like cut-off scheme described in this section has been used to restrict the parameter space of EoS of cold nucleonic and hyperonic NS matter by imposing the above constraints. In Shirke et al. (2023) <cit.>, this formalism was extended to cold hybrid stars with phase transition using constraints from perturbative QCD (p-QCD). In this work, we have applied a similar scheme to extend the study to finite temperature, as described below.

    
  * Chiral Effective Field Theory (χEFT): The χEFT takes into account nuclear many body interactions using order-by-order expansion in terms of contact interactions and long-range pion exchange interactions and low momentum expansion of nuclear forces related to QCD symmetries are applied. These are based on microscopic calculations from the works of Drischler et al. <cit.>. These constraints can been used in the low density range 0.07-0.20fm^-3 for cold Pure Neutron Matter (PNM). It should be noted that χEFT calculations have recently been extended to finite temperature asymmetric nuclear systems <cit.>. These have also been applied recently to impose constraints on finite temperature EoSs <cit.>. However as these calculations deal with isothermal (fixed temperature) systems, these results have not been implemented in this work as it focuses on isentropic (fixed entropy per baryon) cases.
    
  * Astrophysical Observations (Astro): These are appropriate for the high density part of  Asymmetric Nuclear Matter (ANM) and β equilibrated EoS. We impose astrophysical constraints in which the maximum mass supported by a cold neutron star has a lower limit of 2.01 M_⊙, compatible with the current data. GW170817 observation <cit.> puts an upper limit on effective tidal deformability (Λ̃) to be 720 <cit.> and radius of a 1.4 M_⊙ neutron star to be 13.5 km <cit.>.
    
  * Heavy Ion Collisions (HIC): These constraints have been derived from heavy-ion experiments at GSI in Germany, namely KaoS <cit.>, FOPI <cit.> and ASY-EOS <cit.>, which leads to constraints at intermediate densities in the range 1-3n_sat. The KaoS experiment can be used to put upper limits on nucleon potential (U_N) for ANM in the density range 2-3n_sat. The FOPI experiment can be used to constrain binding energy per nucleon (E/A) of Symmetric Nuclear Matter (SNM) in the density range 1.4-2n_sat, while ASY-EOS results put bounds on the Symmetry Energy of ANM in the supra-saturation density range 1.1-2n_sat.


We provide the uncertainty ranges in nuclear saturation parameters from state-of-the-art experimental data in Table <ref> As in <cit.>, a hard cut-off filter scheme has been used, in which nuclear saturation properties are varied within their allowed uncertainty range to generate a flat prior and only the results which are within the permitted uncertainty region of the above mentioned constraints are accepted. Imposing these constraints reduces the parameter space of nuclear saturation properties and in turn also of the couplings used in constructing the NL-RMF Lagrangian <ref>.


 
tableUncertainty ranges in nuclear saturation parameters used in this study: the 2nd row provides the lower limit and 3rd row gives upper limit <cit.>. 
 
n_sat     E_sat     K_sat     J_sym     L_sym     m^*/m 

(fm^-3)     (MeV)     (MeV)     (MeV)     (MeV)     

0.14     -16.2     200     28     40     0.55 
 
0.17     -15.8     300     34     70     0.75 
 








§ RESULTS



The composition as well as thermodynamical properties of the neutron star change with the evolving temperature during the PNS formation or neutron star merger. 
To investigate the effects of temperature and out-of-β equilibrium, in this work we consider two thermal configurations, N(S/A = 1, Y_Q = 0.4) and N(S/A = 2, Y_Q = 0.2) representing two different stages of evolution of a hot PNS. The earlier evolution state corresponds to a hot lepton/electron rich configuration while at later times the state corresponds to a hotter core with deleptonized configuration <cit.>. 
We also show comparison with cold β equilibrated state in our study.



 §.§ Equation of State



We investigate the effect of non-zero temperature and out-of-β equilibrium condition on the macroscopic NS properties. The results after imposing constraints from χEFT, Astro and HIC constraints are shown for the thermal case N(2, 0.2) in fig. <ref>. The bands of EoSs corresponds to different sets of nuclear saturation properties consistent with filters mentioned above. The imposition of Astro and HIC constraints eliminates certain stiffer EoSs from the parameter set as evident from the figure. 




The temperature profiles in the core of these stars using  the χEFT + Astro constraints are shown in fig. <ref>. For low density (n_B<0.1 fm^-3) there is no significant variation in the temperature profiles for the thermal cases. However, saturation properties do induce appreciable changes in temperature at intermediate densities. For N(1, 0.4) the variation is limited to about n_B<0.9 fm^-3 while that of N(2, 0.2) extends to the entire density range considered. fig. <ref> shows temperature profiles after inclusion of HIC filter and we see that temperature spreads across intermediate and high densities have reduced. However qualitative difference of spreads between N(1, 0.4) and N(2, 0.2) across densities still persists.




We observe that at higher densities the temperature profiles get flattened. This is more prominent in case of N(1, 0.4) case as there are higher nucleation of positrons at high densities due to higher charge fraction. The new anti-particle degrees of freedom contribute to the overall entropy and hence temperature do not increase rapidly with increasing density. check. Compared to which work?




 §.§ Mass-Radius Relations


The Mass-Radius relations for N(2, 0.2) are shown in fig. <ref>. The filter imposed by Astro constraints on stiffer EoSs translates to lower values of maximum masses and radii supported by the configurations. We will see later on in subsection <ref> that the maximum mass is strongly anti-correlated with effective nucleon mass (m^*/m) and therefore this parameter controls the stiffness of the EoS. It also strongly effects other macroscopic properties such as radii and tidal deformability. 


We also display these results for the thermal case N(1, 0.4) and their comparison with cold β equilibrated matter in fig. <ref>. One can see that the cold β equilibrated configuration is the most compact one and the highest maximum mass it supports  is almost identical to N(2, 0.2). The N(2, 0.2) configuration is more compact for higher masses while for N(1, 0.4) this is the case for low and intermediate mass configurations. After imposition of HIC constraints (fig. <ref>), the maximum masses supported by each thermal configuration further reduce. Most notably M_max>2.5M_⊙ configurations are eliminated for all three cases. We discuss in subsection <ref> how the imposition of these constraints affect masses, radii and tidal deformability.





 §.§ f-mode frequency and damping time


As discussed in subsection <ref>, after imposing the constraints mentioned above, the f-mode frequencies and damping times are calculated. The results for the case N(2, 0.2) are shown in fig. <ref> and fig. <ref>.


Next we show the combined f-mode spectrum for all the three cases with χEFT+Astro constraints in fig. <ref>. From the figure we can see that f-mode frequencies are lower for low mass (M ≲1.1 M_⊙) NSs for both the thermal cases in comparison with cold and β equilibrated stars. For higher masses all the three configurations have overlapping regions similar to their mass-radius relations (fig. <ref>). The effective nucleon mass again plays a vital role in determining f-mode frequencies and damping times as have discussed in subsection <ref>. n_sat also plays some role in f-mode oscillations of intermediate mass configurations (see subsection <ref>).


    For the case N(1, 0.4) the f-mode frequencies are highest for M ≳1.4 M_⊙ if the EoS is softer and M>1.7 M_⊙ if it is stiffer. All the N(2, 0.2) states have the lowest f-mode frequencies across the entire mass range. ?? DID NOT FOLLOW



Fig. <ref> shows damping timescales for the three cases. Cold neutron stars have the fastest damping for low mass (M≲1.1M_⊙) cases. N(2, 0.2) configuration have the slowest damping in low and intermediate mass regions. For higher masses we again have overlapping regions. 

    For N(1, 0.4), damping times are almost similar to that of N(T=0) for M≳1.7 M_⊙ in case of stiffer EoSs and for M≳1.4 M_⊙ in case of softer EoSs. All the N(2, 0.2) states show the slowest damping across the entire mass range. ?? DID NOT FOLLOW



The f-mode frequencies and damping times obtained after applying HIC constraints are shown in figs. <ref> and <ref>. These constraints eliminate certain regions of figs. <ref> and <ref>, however the qualitative behaviour does not change much. 
We see that the frequencies and damping times are inversely correlated for the thermal cases, a fact which is already known for cold NSs. The correlations of f-mode frequencies and damping times with nuclear saturation properties are discussed in <ref>.






 §.§ Universal relations




In this section, we discuss the effects of nuclear saturation properties on the Universal Relations (URs) of hot NSs. We focus on the two well-studied URs of cold neutron stars, i.e., Love-Compactness (Λ-C) and f-mode frequency-Love (f-Λ) relations. In addition to the two thermal cases considered earlier, we also show the variation of these URs with a wider range of thermal conditions.


  §.§.§ C-Love Relations
 
The EoS independent UR between dimensionless tidal deformability (Λ) and compactness (C) was studied in Maselli et al., where they provide the fitting coefficients in equation (4) of their paper <cit.>. We display ln(Λ)-C plots from EoSs generated using constraints from χEFT+Astro in fig. <ref>. We see that like cold β equilibrated stars (N(T=0)), the plots for two thermal cases also fall in their respective curves which are distinct from that of N(T=0) curve, indicating that these configurations do not follow the same URs. The deviations from the cold case are more for higher values of Λ. We also see that compactness corresponding to a particular Λ is highest for N(T=0) followed by N(2, 0.2) and N(1, 0.4) indicating the order in compactness in the three configurations. The C-ln(Λ) polynomial fitting can be given as - 


    C = ∑^2_i=0 a_i [ln(Λ)]^i .





 
tableFit relations for C-ln(Λ) UR for the two thermal configurations

 
    a_0     a_1     a_2  

N(1, 0.4)     3.556×10^-1     -4.209×10^-2     1.173×10^-3  
 
N(2, 0.2)     3.598×10^-1     -4.070×10^-2     1.182×10^-2  
 





To further investigate how the thermal effects break this universality, we separately vary entropy per baryon (S/A) and charge fractions (Y_Q) keeping nuclear saturation properties fixed. First we show the variation on changing S/A from 1-2 by fixing Y_Q=0.4, and in β equilibrium. Next we fix S/A=1 and vary Y_Q in the range 0.01-0.40. 



It is clear that Λ-C relation depends not only on charge fraction but also on whether the matter is in β equilibrium when we vary S/A. The spread in universality in fig. <ref> is much less than that of fig. <ref>. From the results in fig. <ref> one can obtain universal fits similar to the works of Maselli et al. <cit.>. We provide the universal fits for the two thermal configurations in Table <ref>. On the other hand, we see from fig. <ref> that Y_Q, which is a measure of out-of-β equilibrium, is mainly responsible for breaking this universality and there is no possibility to obtain a fit relation.




  §.§.§ f-Love Relations
 
It is known that for cold NSs f-mode frequencies scale with certain combinations of astrophysical observables, such as M, R and Λ. It was shown that f-mode frequency has an approximately linear relation with square root of average density (√(M/R^3)) while the normalized damping time (τ_f R^4/M^3) scales with stellar compactness <cit.>. These fit relations are found in the works of Andersson and Kokkotas <cit.>, Benhar and Ferrari <cit.>, Doneva et al. <cit.>, Pradhan and Chatterjee <cit.>. 


However, it turns out that these fits are composition dependent and have large deviations depending on the EoSs considered. Pradhan et al. <cit.> also provided other improved universal fits which correlate mass-scaled frequencies (M ω) (real and imaginary) with logarithm of dimensionless tidal deformability (ln(Λ)). This polynomial fitting can be given as - 

    Mω = ∑^5_i=0α_i[ln(Λ)]^i .


Here we examine this f-Love universality in the context of hot NSs. The f-Love relations after imposing constraints from χEFT and Astro are shown in figs. <ref> and <ref>. Unlike fig. <ref>, we observe negligible spreads for both thermal cases. Small deviations from the universality appear for NSs with Λ>e^8∼3000 or with masses close to or less than solar mass. After the post-bounce stage, a PNS becomes more and more compact as it evolves, consequently Λ goes from high to low and the star approaches a cold β equilibrated state. The f-Love fit coefficients for the two thermal configurations are given in Table <ref>.



These findings are reasonable, as at higher Λ the star can be assumed to be in the N(1, 0.4) representative state and this configuration shows the highest deviation from cold f-Love universality. The N(2, 0.2) state is representative of intermediate Λ and it has close resemblance with the cold β equilibrated configuration. The close resemblance these URs have with N(T=0) promotes us to study them across ranges of S/A and Y_Q. The conclusions are somewhat different to those for C-Love relations (see Appendix subsections <ref> and <ref>).









 
tableFit relations for f-Love UR for the two thermal configurations 
 
    α_0     α_1     α_2     α_3     α_4     α_5 

N(1, 0.4)     1.812×10^-1     -5.435×10^-3     -4.999×10^-3     7.106×10^-4     -3.798×10^-5     7.034×10^-7 
 
    +i2.237×10^-5     +i5.599×10^-5     -i1.863×10^-5     +i2.142×10^-6     -i1.004×10^-7     +i1.452×10^-9 

N(2, 0.2)     1.809×10^-1     -5.110×10^-3     -5.141×10^-3     7.405×10^-4     -4.114×10^-5     8.486×10^-7 
 
    +i1.411×10^-5     +i6.519×10^-5     -i2.239×10^-5     +i2.849×10^-6     -i1.619×10^-7     +i3.466×10^-9 






















    This part may be moved to the Appendix. We should keep Real(Mω_f) v/s ln(Λ) figs. <ref> and <ref> and the discussions. We can remove figs. <ref> and <ref> as they point to similar conclusions.








 §.§ Correlation studies



From the correlation studies in the case of cold β equilibrated matter, it was found that the most important nuclear saturation parameter which affects the NS observable properties is the nucleon effective mass m^* and to a certain extent the nuclear saturation density n_sat  <cit.>. In this section we will probe possible correlations of the nuclear parameters with NS observables and also among each for the thermal cases. First we study the effect of imposing the χEFT + Astro constraints on the correlations. Next, we also include the HIC constraints, and investigate how the correlations get altered. As the HIC results are known to be dependent on the transport models, we have treated them separately with caution (see <cit.> for a more detailed discussion). 


    For this analysis we use the Pearson’s linear correlation coefficient. For two random variables A and B, it is given by:

    R(A,B) = Cov(A,B)/S(A) × S(B)

where Cov(A,B) is the covariance of two random variables and S(A) and S(B) are standard deviations of A and B variables respectively.





  §.§.§ χEFT + Astro

The correlations among the nuclear saturation parameters and the NS observable properties on imposing χEFT + Astro constraints are shown for the two thermal cases in figs. <ref> and <ref>. We describe the findings in detail below. 


    
  * Correlations among nuclear saturation properties: 
    
        
  * Effective nucleon mass m^* shows moderate anti-correlation with n_sat.
        
  * Relatively strong correlation is observed between J_sym and L_sym. This is consistent with previous studies <cit.>.
        
  * E_sat and K_sat do not show correlations with other saturation properties.
    
    
    
  * Correlations of nuclear saturation properties with NS observables: 
    
        
  * For both thermal cases, n_sat shows moderate correlations/anti-correlations with properties (R, Λ, ν_f, τ_f) of canonical 1.4M_⊙ hot NSs.
        
  * On the other hand n_sat shows weak correlations/anti-correlations with properties (R, Λ, ν_f, τ_f) of canonical 2M_⊙ hot NSs and it does not have any effect on maximum mass supported by the corresponding  EoS.
        
  * Effective nucleon mass m^* has strong anti-correlation with M_max, it also shows relatively stronger correlations/anti-correlations with R_2M_⊙, Λ_2M_⊙, ν_f, 2M_⊙ and τ_f, 2M_⊙ for both cases.
       
        
        
  * However for 1.4M_⊙ hot NSs, m^*/m has weak correlations/anti-correlations with the observables for the case N(1, 0.4) but moderate correlations/anti-correlations for N(2, 0.2) observables. Previous study with cold NSs have also shown moderate correlations of  m^*/m with 1.4M_⊙ observables <cit.>. We can conclude that during the early phase of hot NS with low entropy and high lepton fraction, m^*/m have lesser effect on properties of intermediate mass configurations.
        
    

    
  * Correlations among NS observables:
    
        
  * Maximum mass is strongly correlated/anti-correlated with properties of canonical 2M_⊙ observables for both cases.
        
  * With observables of 1.4M_⊙, these correlations are moderate for the state N(1, 0.4) and strong for the case N(2, 0.2). 
       
        
  * Correlations among radii, Λ, f-mode frequencies and damping times are similar in both cases and they are similar to that of cold NSs <cit.>.
    




  §.§.§ χEFT + Astro + HIC

The correlations on imposing χEFT + Astro + HIC constraints are shown for the two thermal cases in figs. <ref> and <ref>. We describe the key results and differences from χEFT + Astro filters below.

    
  * Correlations among nuclear saturation properties: 
    
        
  * Correlation between J_sym and L_sym gets weaker.
        
  * Anti-correlation between m^*/m and n_sat becomes relatively stronger.
        
  * K_sat and L_sym now have moderate correlations with m^*/m.
    
    
    
  * Correlations of nuclear saturation properties with NS observables: 
    
        
  * m^*/m anti-correlation with maximum mass remains strong for both cases but they are relatively weaker than those of correlations without HIC constraints.
        
  * Masses, radii, f-mode frequencies and damping times of 1.4M_⊙ PNSs now have poor correlation with effective mass. 
        
  * Correlations/anti-correlations of effective mass with properties (R, Λ, ν_f, τ_f) of canonical 2M_⊙ PNSs become moderate for both cases.
        
  * Correlations/anti-correlations of n_sat with observables (R, Λ, ν_f, τ_f) of canonical 1.4M_⊙ PNSs become stronger for both cases. However, its correlation with canonical 2M_⊙ hot NS observables remain weak.
        
        
    

    
  * Correlations among NS observables:
    
       
  * Correlations/anti-correlations of M_max with observables of 1.4M_⊙  hot NSs become weaker for both cases.
      
      
    















§ DISCUSSIONS



Summary of the results: 
In this work, we develop a systematic and consistent formalism for nucleonic equations of state in neutron stars at finite temperature, by extending a cold matter EoS whose parameter space of allowed uncertainties of the model is restricted by demanding compatibility with current nuclear experimental data, multi-messenger astrophysical observations as well as heavy-ion physics. The developed framework was then applied to the calculation of polar f-modes and p_1-modes within a general relativistic scheme. The full ranges of allowed nuclear saturation properties for nucleonic matter used in our work allowed for the extensive study of correlations of the nuclear and thermal properties on macroscopic and oscillation properties of hot NSs.  We specifically chose a low entropy high charge fraction state and a high entropy deleptonized low charge fraction state and compared it with cold β equilibrated NSs. 




On including thermal contributions to the EoS, we found that increasing S/A leads to higher core temperatures while Y_Q has an opposite effect and deleptonized configurations are more compact. Increasing S/A results in larger radii and flatter mass-radius relations. These observations are consistent with previous studies <cit.>. The f- and p_1-mode frequencies calculated in a full general relativistic scheme are found to be the highest for the cold configuration and decrease with an increase in entropy per baryon and charge fraction.



On imposing χEFT, Astro and HIC constraints, we found that nuclear saturation parameters affect temperature profiles at intermediate and high densities. There are overlapping regions in the posteriors for M-R, f-mode frequency and damping timescales for high masses among N(T=0), N(1, 0.4), N(2, 0.2) states. 
From the correlation study of hot NSs, we found on imposing χEFT and Astro constraints that the most important saturation properties having effect on astrophysical observables, such as maximum mass, radii, tidal deformation as well as f-mode characteristics, are found to be m^*/m followed by n_sat.  
 However their correlations are qualitatively different depending on the thermal state after imposing χEFT + Astro constraints and N(2, 0.2) state seems to resemble closely to that of cold NS results <cit.>. After applying HIC constraints, we found that correlations of m^* with NS observables are significantly weaker. These findings are qualitatively similar to the results in cold β equilibrated matter case <cit.>.
 
 Hence, the interplay among saturation properties and astrophysical observables depends not only on the thermal configurations considered but also on the constraints imposed.
It should be noted that previous study of cold β equilibrated NSs by Pradhan et al. <cit.> found strong correlations/anti-correlations of effective nucleon mass with all astrophysical observables but did not include constraints from χEFT and HIC. 



Using the posteriors from the constraint filters, we further investigated the role of nuclear saturation parameters on the universal relations for hot NSs, in particular Love-compactness and f-Love relations. We concluded that, hot NS configurations at higher Λ values show significant deviations in C-Love and f-Love relations compared to the cold β equilibrated case, although these variations are more prominent in the former case. Further investigation confirmed that these URs are mostly insensitive to nuclear saturation properties and is mainly affected by charge fraction variation, which is a measure of out-of-β equilibrium condition.



Comparison with other works: 
There exist other EoS models in the RMF scheme that have been extended to finite temperature and tested against current constraints. In the extension of the FSU2H parametrization based on the RMF scheme to finite temperature <cit.>, nucleonic and hyperonic matter have been considered. The resulting FSU2H* model is compatible with astrophysical constraints (maximum mass of 2 M_⊙ and radii below 13 km) as well as saturation properties of nuclear matter and finite nuclei, and pressure constraint from heavy-ion collisions. 
Alford et al. <cit.> also constructed QMC-RMF EoS models for finite temperature NSs, informed by χEFT and astrophysical constraints. These works demonstrate the strong influence of thermal corrections on the EoS and thermodynamic properties. However, they only consider fixed RMF parametrizations and therefore the role of the nuclear parameters in governing NS observable properties is not evident from such studies, but this is possible with a systematic variation of nuclear parameters as done in our work. Further, as these calculations are performed for isothermal and not isentropic conditions, it is not easy to estimate their effects on astrophysical observables.



It is interesting to note that, several numerical simulations of CCSN, PNS as well as BNS mergers, employing different EoS models, have also reported the influence of the nucleon effective mass <cit.>. Using a selection of purely nucleonic EoS models from CompOSE (including Skyrme, covariant density functional models, CMF) in agreement with constraints from astrophysical observations and nuclear data, Raduta et al. <cit.>, reviewed thermal properties relevant for PNS and BNS mergers and concluded that they are strongly influenced by the effective nucleon mass. This was further corroborated by using a covariant density functional theory with density dependent couplings employing the posteriors of a Bayesian inference (DDB*), which encompasses a large number of EoS models <cit.>. 
These EoSs were also employed in
 Thapa et al. <cit.> to calculate the f- and p_1-mode frequencies within Cowling approximation. Although the thermal and composition effects on the oscillation modes were studied for the selection of EoSs from CompOSE, the correlations were calculated within the cold EoS approximation, and therefore did not provide any information about the thermal conditions.



A few recent works by Ghosh et al. <cit.> and Kumar et al. <cit.> studied oscillations within Cowling approximation including neutrino contributions for hot nucleonic and hybrid neutron stars. It should be noted that Burgio et al. <cit.> used a model where core and envelope entropy per baryon are different, and the former rises and then falls and the latter falls along with deleptonization during its evolution, to study g_1-, f- and p_1-modes using a BHF model extended to finite temperature. In our work, the entropy per baryon has been kept fixed throughout the star in our study. 



In the recent work by Tsiopelas et al. <cit.>, which employed a density dependent covariant density functional model, the effect of L_sym was found to be diminishing with increasing temperature. In our work using NL-RMF formalism we found that after applying χEFT + Astro filters, L_sym has no impact on macroscopic or oscillation properties for N(1, 0.4) configuration. On the other hand for N(2, 0.2) configuration, these correlations are qualitatively similar to that of cold NS results <cit.>.




Previous studies have indicated deviations of I̅-Q̅ relations in hot environment <cit.>. A few studies <cit.> showed that universality is restored if finite T configurations for different EoS with same S/A and electron fraction are considered. PNS simulations by Sotani et al. <cit.> explored the prospect of measurement PNS mass and radius using simultaneous measurements of f- (or p_1-) and w_1-mode frequencies by relating those to average mass density and stellar compactness respectively. Torres-Forné et al. <cit.> have demonstrated how g_1 and f-mode frequencies scale with surface gravity in their PNS simulations. In our article, we have shown that C-Love and f-Love URs are mostly not sensitive to nuclear saturation properties but they change depending on the thermal state. In particular charge fraction significantly affects these deviations and they occur at high Λ-values (or low compactness). A recent simulation <cit.> has also shown that due to thermal effects deviations from URs i.e., f-√(M/R^3), I̅-Λ-Q̅, f-Love occur during PNS evolution at earlier stages where the configuration has large tidal deformability.




Limitations and future work:
It is worth mentioning that, we define the modes following the classification by Cowling <cit.>. However, a recent investigation done in <cit.>, considering the nucleonic EoS models, demonstrates that in the initial phase (within  0.4 s) of the PNS evolution, the Cowling classification scheme does not hold but after 0.4 sec the classification of oscillation modes with different approaches are in very good agreement. We further note that neutrinos have not been included in this work. Neutrinos may not remain free streaming for the entire temperature regime considered in this study, however we restrict ourselves to this approximation for this work and defer calculations including neutrino trapping to our forthcoming studies. 




This study can be extended to include non-nucleonic degrees of freedom such as hyperons, delta baryons and quark matter in hot EoS that satisfy microscopic, experimental and astrophysical constraints. These new degrees of freedom are known to soften the EoSs and might also lead to decrease in core temperature due to entropy being contributed by these new species. It would also be interesting to include the effect of neutrinos as they are expected to be in equilibrium with the thermal/isentropic bath at high temperatures.







§ APPENDIX




 §.§ Effect of entropy per baryon on f-Love relations
 



We fix the nuclear saturation properties as in fig. <ref>. Figs. <ref> and <ref> show effect of changing S/A between 1-2 in Y_Q=0.4 and β equilibrated states.
From fig. <ref>, we see that Y_Q=0.4 case shows deviations from β equilibtrated hot NSs for higher Λ and they are overlapping otherwise.  This is in contrast to Λ-C URs shown in fig. <ref> where we did not get any overlapping region. The relation between Im (Mω) and ln(Λ) shown in fig. <ref> also establishes deviations at high Λ values. There are also qualitatively negligible deviations for both real and imaginary parts across the entropy per baryon range considered as compared to Λ-C URs.

These deviations shown between states with β equilibration and Y_Q=0.4 clearly shows that at higher Λ universality is broken by out-of-β equilibrium effects. 

The spreads however in figs. <ref> and <ref> are qualitatively much lower than that of Λ-C relations shown in fig. <ref> where deviations are observed across the entire range of dimensionless tidal deformability.












 §.§ Effect of charge fraction on f-Love relations
 
Here we fix entropy per baryon at S/A=1 and vary Y_Q from 0.01 - 0.40. We see in figs. <ref> and <ref> that this has no effect on both real and imaginary parts of f-Love relations for low Λ values and at higher values there are small spreads. These deviations appear for Λ>e^8∼3000. The conclusion is that charge fraction is responsible for deviations in the universality like C-Love relations albeit to a much lesser degree and only at higher Λ.













§ ACKNOWLEDGEMENTS


The authors are grateful to Suprovo Ghosh for providing the posterior data that was used in this work. D.C. would like to thank Adriana R. Raduta and Mikhail V. Beznogov for their warm hospitality at IFIN-HH, Bucharest which led to several interesting discussions related to this work. D.C. would also like to thank Prof. J. Schaffner-Bielich and his students at ITP Frankfurt for insightful discussions on finite temperature models of neutron stars. N.B., B.K.P. and D.C. acknowledge the usage of the IUCAA HPC computing facility for numerical calculations of oscillation modes. 











