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: The impact of the phase transition on Quark-Gluon Plasma with an extremely strong magnetic field in holographic QCD.


APS



caoxm@jnu.edu.cn
 Department of Physics and Siyuan Laboratory, Jinan University, Guangzhou 510632, China

tliuhui@jnu.edu.cn
 Department of Physics and Siyuan Laboratory, Jinan University, Guangzhou 510632, China

§ ABSTRACT 
We investigate the phase transition within an extremely strong magnetic background field, employing a holographic Quantum Chromodynamics (QCD) model with a focus on entropy and pressure properties. At relatively modest magnetic field strengths, our study discerns a crossover transition between the normal phase and the Quark-Gluon Plasma (QGP) phase as the temperature rises. In contrast, under the influence of an extremely strong magnetic field, a first-order phase transition is observed. A critical point is identified at (eB_c, T_c) ≈ (2.8623  GeV^2, 0.1191  GeV), which corresponds to a second-order phase transition. This phase structure is found to be in qualitative agreement with lattice simulation predictions reported in [Phys. Rev. D 105, 034511 (2022)].
Furthermore, we explore the impact of the magnetic field on the jet quenching parameter across various phases. At zero magnetic field (eB = 0 ), the normalized jet quenching parameter q̂ / T^3 exhibits a monotonic increase with temperature. However, in the presence of a magnetic background field, the normalized jet quenching parameters not only display directional anisotropy but also experience a universal enhancement, particularly in the vicinity of the critical temperature region. This observation suggests that the jet quenching parameter could potentially act as an indicator of phase transitions.


The impact of the phase transition on  Quark-Gluon Plasma with an extremely strong magnetic field in holographic QCD
    Hui Liu
    August 5, 2024
====================================================================================================================






§ INTRODUCTION

The investigation of the quantum chromodynamics (QCD) properties under a magnetic field is one of the most important fields in high energy physics <cit.>. The background magnetic field could be generated and widely exist in non-central heavy ion collisions <cit.>, in the early cosmology <cit.>, and in magnetar <cit.>. Many important and interesting phenomena are found in this area, such as the (inverse) magnetic catalyses <cit.> and the chiral magnetic effects <cit.>.

The chiral and confining properties of QCD are directly influenced by the presence of a magnetic field. An extremely strong magnetic field can even change the order of the phase transition, which could be important for our understanding of the early universe <cit.>. One of the most powerful methods for studying QCD in a finite magnetic background field is the lattice simulation.  Lattice simulations <cit.>, which employed standard staggered fermions, demonstrated a rise of transition temperature in chiral condensates with Polyakov loops. However, these findings are in contrast with the results of lattice simulations  <cit.> that incorporated improved staggered fermions.
 The discrepancy is primarily attributed to lattice cutoff effects <cit.>. Moreover, lattice simulations have demonstrated that a first-order transition exists when eB ≳ 11 m_π^2 with N_f=3 <cit.>. Extrapolation of the lattice results predicts a critical point beyond eB=3.25   GeV^2 with N_f=1+1+1 <cit.>. In more recent investigations, a lattice simulation for N_f = 2 + 1 <cit.>, which employs stout-improved staggered fermions and try to control the cut-off effect, has indicated that the critical temperature decreases as the magnetic field strengthens. Despite the absence of technical issues such as the sign problem at finite chemical density, lattice simulations remain constrained by limitations in computational power. They observed a crossover transition at eB=4  GeV^2 and a first order phase transition at eB=9  GeV^2, indicating the presence of a critical end point between these values. 

The phase structure of QCD under magnetic fields has also been extensively investigated using low energy effective theories and models, such as the bag model <cit.>, chiral perturbation theory (χPT) <cit.>, the Nambu-Jona- Lasinio (NJL) model <cit.> and the quark-meson (QM) model <cit.>. However, a unified consensus on the phase diagram at finite magnetic fields and temperatures has not yet been reached across these various studies. The phase diagram in an extremely strong magnetic field remains an open topic. Therefore, it is necessary to conduct further research in this domain with a variety of numerical and theoretical methods.

In addition to the traditional methods, holographic duality is another powerful approach for studying strongly coupled systems <cit.>. Given that the quark-gluon-plasma (QGP)  exists as a strongly coupled state <cit.>, holographic duality has become an important tool in the QCD investigations <cit.>. Regarding investigations involving magnetic backgrounds, various studies have employed different  constructive models, including bottom-up holographic models  in the Veneziano limit <cit.>, magnetized Einstein-Maxwell model <cit.> and Magnetized Einstein-Maxwell  model <cit.>. An abundance of research has been dedicated to studying the Quantum Chromodynamics (QCD) phase diagram within the finite temperature and chemical potential plane <cit.>. However, in this study, we employ the magnetized Einstein-Maxwell-Dilaton (EMD) model to explore the phase transition in the context of an extended magnetic field region, especially the extremely strong magnetic field. In the previous investigations, they utilized the magnetized EMD model <cit.> and have yielded some findings. They have predominantly concentrated on crossover transitions and have not yet provided an exhaustive examination of the domain characterized by extremely strong magnetic fields. Building upon this foundation, we have developed numerical code to extend the parameter space  of the magnetic field and temperature.
Then, we study the extremely strong magnetic field region and observe first-order phase transitions. 

Taking into account the first-order transitions, we discuss their influence on the properties of QGP.  Among various QGP signals in heavy-ion collisions, jet quenching is a crucial one <cit.>. The holographic calculation of the jet quenching parameter q̂ has been previously provided  <cit.> and extensively tested in some holographic models <cit.>. In this work, we focus on the behaviors of anisotropic jet quenching parameters under both crossover and first order phase transitions.
 


This paper is organized as follows. In section II, we introduce the construction of EMD model and  how it realizes duality to QCD. In section III, we numerically obtain the mapping between the input parameters (ϕ,ℬ) and the physical parameters (T,eB). We then study the entropy variation with temperature at fixed magnetic field, determine the phase boundary of the first-order transition and identify the critical end point in T-eB plane. In section IV, we investigate the effect of the first-order transition to the jet quenching parameters. Section V is a brief conclusion.



§ THE EINSTEIN-MAXWELL-DILATION MODEL

In this section, we provide a brief introduction to the EMD model with a finite magnetic field B≠ 0 and zero chemical potential μ_B=0, as constructed in Refs. <cit.>. This bottom-up approach aims to mimic the QCD system by fitting lattice results from the holographic perspective, a method originally proposed by Gubser and Nellore in Ref. <cit.>.


In the EMD model, a scalar field ϕ, i.e., a dilaton field, and an abelian gauge field A_μ are included.  The five dimension action is

    S=1/2 κ^2∫ d^5 x√(-g)[R-1/2(∂_μϕ)^2-V(ϕ)-f(ϕ)/4F_μν^2],

with

    V(ϕ)   =   -v_0  cosh(v'_0 ϕ)+ v_2 ϕ^2+v_4 ϕ^4+v_6 ϕ^6,
    κ^2   =   8π G_5,
    
        f(ϕ)   =   c_3  sech(c_0+c_1 ϕ+c_2ϕ^2),

and the field strength tensor F_μν=∂_μ A_ν-∂_ν A_μ. The dilaton potential V(ϕ)  and gravitational constant κ^2≡ 8π G_5 is determined by fitting the lattice data of equation of state at B=0 in Ref. <cit.> [For the fitting of dilaton potential, the Breitenlohner-Freedman bound should be satisfied <cit.>.]. The coupling between the gauge field and dilaton f(ϕ) is determined by fitting the lattice results of the magnetic susceptibility at B=0 in Ref. <cit.>. The values of these parameters are summarised in Table <ref>. For the asymptotically AdS_5 geometries, the ultraviolet asymptotically expansion should satisfy V(ϕ→ 0)≈ -12/L^2+m^2ϕ^2/2 with L the asymptotic AdS_5 radius, in which the mass m and the scaling dimension Δ of the dual operator in the gauge theory satisfy the relation

    m^2 L^2=-νΔ=(4-Δ)Δ.

Here,  the radius is always set to be unity, i.e., L=1. Thus, one can derive Δ= 2.73294.




In this EMD model, the magnetic field ℬ is introduced through the abelian gauge field at the boundary and set to be a constant in the z direction. Therefore, one can take the metric ansatz as

    ds^2   =   e^2a(r)[-h(r)dt^2+dz^2]+e^2c(r)(dx^2+dy^2)
          +e^2b(r)dr^2/h(r),
    ϕ   =   ϕ(r),
    
        A   =   A_μ dx^μ=ℬ xdy⇒ F=dA= ℬdx∧ dy,

with five dimension coordinates (t,x,y,z,r).  As a result of the magnetic field, the rotation symmetry is broken from SO(3) to SO(2). Along the holographic direction, the boundary of the asymptotically AdS_5 is at r→∞ and the horizon of the black hole at r_H, which defined by h(r_H)=0. Under this metric in (<ref>), the equations of motion (EOMs) can be derived as,



    ϕ”+ (2 a'+2 c'-b'+h'/h )ϕ'-e^2b/h(∂ V(ϕ)/∂ϕ +ℬ^2 e^-4c/2∂ f(ϕ)/∂ϕ)   =   0,
    
        a”+(14/3c'-b'+4h'/3h)a'+8/3a'^2+2/3c'^2+2h'/3hc'+2e^2b/3hV(ϕ)-1/6ϕ'^2   =   0,
    
        c”-(10/3a'+b'+h'/3h )c'+2/3c'^2-4/3a'^2-2h'/3ha'-e^2b/3hV(ϕ)+1/3ϕ'^2   =   0
    
        h”+(2a'+2c'-b')h'   =   0,

with a constraint equation,

    a'^2+c'^2-1/4ϕ'^2+(a'/2+c')h'/h+4a'c'+e^2b/2h(V(ϕ)+ℬ^2 e^-4c/2f(ϕ))   =   0,



where the prime denotes the derivative with respect to the fifth dimension r.

To investigate the properties of the EMD model, one should solve the EOMs in Eqs. (<ref>) and extract the thermaldynamic quantities. The first step involves obtaining the near-boundary and near-horizon asymptotic expansions for the bulk field ϕ(r) and a(r), c(r), h(r). We adopt the domain-wall gauge b(r)=0. Consequently, the ultraviolet asymptotic expansions can be derived as follows:

    
    a(r)   =   α(r)+⋯,
    
            c(r)   =   α(r)+c_0^ far-a_0^ far+⋯,
    
            h(r)   =   h_0^  far+h_4^  far e^-4α(r)+⋯,
    ϕ(r)   =   ϕ_Ae^-να(r)+ϕ_B e^-Δα(r)+⋯,

with 

    α(r)   =   a_0^ 
     far+r/√(h_0^ 
     far).


and ⋯ denote higher-order terms. The integration constants a_0^ far, c_0^ far, h_0^ far and ϕ_A are related to the thermodynamic quantities. Near the horizon r_H=0, the asymptotic expansions of the bulk fields are given by:

    Y(r)=∑_n=0^∞Y_n (r-r_H)^n,

where Y=a,c,h,ϕ. In the expansion coefficients, due to the freedom in rescaling properties, one can set r_H=a_0=c_0=0 and h_1=1. One can also assume h_0=0. After these settings and assumptions, only ϕ_0 and ℬ remain undetermined among the seven parameters in the near horizon expansions.  Once ϕ_0 and ℬ are given, the values X(ϵ) [Due to the singularity at horizon, a small deviation ϵ is used in the numerical strategy. We choose ϵ=10^-15 in this work.] and their first-order derivative values X'(ϵ) of the near horizon expansion are determined. One can numerically solve the EOMs in Eqs (<ref>)  by fitting the UV asymptotic expansions in Eq. (<ref>) with these numerical solutions to obtain the corresponding integration constants a_0^ far,c_0^ far, h_0^ far and ϕ_A. From our numerical tests, we find that choosing the maximum radial coordinate r_max=10 is sufficient for the near-boundary asymptotic expansions. Thus, the numerical solutions are all given in the interval r∈ [ϵ, r_max].

As previously shown, we can obtain the numerical results with ϕ_A≠ 0, which can be understood as renormalization group flows induced  by a slight deformation of a relevant operator. However, to accurately mimic QCD, ϕ_A must be maintained constant.  To accomplish this, it is necessary to perform a coordinate transformation from the numerical coordinates to the standard coordinates. In the standard coordinates, the metric and the UV asymptotic expansions are

    ds̃^2   =   e^2ã(r̃)[-h̃(r̃)dt̃^2+dz̃^2]+e^2c̃(r̃)(dx̃^2+dỹ^2)+dr̃^2/h̃(r̃),
    ϕ̃   =   ϕ̃(r̃),
    Ã   =   Ã_μ dx̃^μ=B̂x̃dỹ⇒F̃=dÃ=B̂dx̃∧ dỹ,

and 


    ã(r̃)   =   r̃+⋯,
    c̃(r̃)   =   r̃+⋯,
    h̃(r̃)   =   1+⋯,
    ϕ̃(r̃)   =   e^-νr̃+⋯.


with the boundary at r̃→∞ and the horizon at r̃=r̃_H. The numerical coordinates and the standard coordinates are equivalent and satisfy

    ds̃^2=ds^2,    ϕ̃(r̃)=ϕ(r),    B̂dx̃∧ dỹ=ℬ dx∧ dy.

Using the relation in Eq. (<ref>) and comparing the UV asymptotic expansions in numerical and standard coordinates in Eqs. (<ref>) and  (<ref>), one can derive


    r̃=r/√(h_0^ far)+a_0^ far-ln(ϕ_A^1/ν),
       t̃=ϕ_A^1/ν√(h_0^ far)t,
       x̃=ϕ_A^1/ν e^c_0^ far-a_0^ farx,
       ỹ=ϕ_A^1/ν e^c_0^ far-a_0^ farx,
       z̃=ϕ_A^1/νz,
    


    
    ã(r̃)=a(r)-ln (ϕ_A^1/ν),
       c̃(r̃)=c(r)-(c_0^ far-a_)^ far)-ln (ϕ_A^1/ν),
       h̃(r̃)=h(r)/h_0^ far,
       ϕ̃(r̃)=ϕ(r),


and

    B̂=e^2(a_0^ far-c_0^ far)/ϕ_A^2/νℬ.




Therefore, one can solve the EOMs in the  numerical coordinates and obtain the numerical solutions  of the QCD quantities in the standard coordinates through the relations in Eqs. (<ref>) and (<ref>). In the upper panel of Fig. <ref>, the curve represents the upper bound of the the input parameters (ϕ_0,ℬ).
From the relation in Eqs. (<ref>) and asymptotic expansions at horizon in Eqs. (<ref>), one can obtain the QCD temperature as

    T̂=.√(-g̃_t̃t̃^'g̃^'r̃r̃)/4π |_r̃=r̃_H=e^ã(r̃_H)/4π|h̃'(r̃_H)|=1/4πϕ_A^1/ν√(h_0^ far),




To mimic to the real QCD, the temperature corresponding to the minimum of the sound speed c_s^2 in the holographic results is compared to the lattice results. Based on this comparison, a scaling factor is determined,

    λ=T^ lattice_ min c_s^2/T̂_ min c_s^2≈ 1058.83  MeV.

Therefore, the QCD quantities X can be obtained by scaling X̂ with λ, following the relation X=X̂λ^p [MeV^p], where p is determined by dimensional analysis. For example T=T̂λ [ MeV] and B=B̂λ^2 [ MeV^2].
In the lower panel of Fig. <ref>, we show the numerical results for (T, eB), where each curve corresponds to a fixed ϕ_0 and a series of ℬ. Normally, small ϕ_0 maps to high T, large ϕ_0 maps to low T and small ℬ maps to small eB. As ℬ approaches its upper bound, a slight variation in ℬ induces a large change of eB. Noticing that the fixed ϕ_0 curves in  lower panel of Fig. <ref> are convex for large ϕ_0 ( i.e., low T) but concave for small ϕ_0 ( i.e., high T), these curves may exhibit overlaps at the moderate temperatures in the range of T=100∼ 150 MeV.







§ THE PHASE TRANSITION WITH FINITE MAGNETIC FIELD

In this section, we investigate the QCD properties in a magnetic background field. The strategy is as follows: first, we  draw a vertical line to intersect with the curves in the lower panel of Fig. <ref>, and then numerically map from the intersection points back to determine the corresponding input parameters (ϕ_0,ℬ). For example, as shown in Fig. <ref>, we plot three curves of the selected input parameters (ϕ_0,ℬ), which are corresponding to the magnetic background fields set at eB=0.1, 1 and 10 GeV^2, respectively. Finally, after determining several sets of the input parameters for fixed magnetic fields, we solve the EOMs in Eq. (<ref>) and obtain the corresponding values of the QCD quantities.




 §.§ The entropy and pressure

For the mapping between (ϕ_0,ℬ) and (T,B) shown in the lower panel of Fig. <ref>, there is a special region where a pair of (T,B) corresponds to different pairs of (ϕ_0,ℬ) [A similar result of the multiple correspondence, has been observed in Ref. <cit.>, in which the authors studied the critical point at finite temperature and chemical potential.]. Specifically, as eB increases from small to large, this special region begins at a particular point. This behavior could be a signal of the first-order phase transition. To confirm this conjecture, we  study the behaviors of the entropy density s and pressure P in this region in the following part.

Entropy is an extensive quantity, which can be extracted through the Bekenstein-Hawking relation as S=A/4G_N with A the area of the horizon. Thus, the entropy density can be obtained by dividing the volume V,

    s=S/V=2π/κ^2e^ã(r̃_H)+2c̃(r̃_H)=2π e^2(a_0^ far-c_0^ far)/κ^2ϕ_A^3/ν.


In  the upper panel of Fig. <ref>, we show the numerical results of the temperature dependence of the normalized entropy density s/T^3 for four different fixed magnetic fields, eB=0, 1, 2.8326, and 4 GeV^2.  In the absence of a magnetic background field, the normalized entropy density is small at low temperatures and exhibits a monotonic increase as the temperature rises.  At high temperatures, it behaves according to the relation s∼ T^3. This may indicate that the QCD system undergoes a crossover from the low-temperature normal phase to the high-temperature quark-gluon-plasma (QGP) phase. 
With a finite magnetic background field,  the normalized entropy density in both the low and high-temperature limits consistently approaches the values observed in the absence of a magnetic field. In the intermediate temperature region, there is a noticeable enhancement of the normalized entropy density, which becomes more pronounced with an increase in the magnetic field. When eB≳ 2.8326  GeV^2, as shown in the inset in the upper panel of Fig. <ref>, the normalized entropy density as a function of temperature exists multiple value region, signaling the onset of the first-order phase transition. When eB≤ 2.8326  GeV^2, we the specific heat C=T∂ s/∂ T with μ_B=0 shown in the lower panel of Fig. <ref> diverges at a certain temperature T_c≈ 0.1191   GeV. Below the critical magnetic field eB_c=2.8326  GeV^2, the susceptibility of entropy density with respect to the temperature is smooth and continuous, indicating a crossover transition. Therefore, the point (eB_c, T_c)=(2.8326  GeV^2, 0.1191   GeV) is a critical endpoint, corresponding to a second-order phase transition, where the first-order transition changes to a crossover.



To determine the phase boundary of the first-order transition, we study the behavior of the pressure P. At a fixed magnetic background field, the pressure can be approximately determined as

    P(T)≈∫_T_ low^T s(T)  dT,

where we select a low temperature T_ low as the lower boundary rather than absolute zero due to the impracticality of achieving zero temperature within this model. This selection introduces only a constant difference and does not affect the results of our analysis. For convenience, we choose T_ low=0.053  GeV in our calculations. In Fig. <ref>, we show the temperature dependence of the entropy density (S-shape) and pressure (Swallowtail shape) in the upper panels and lower panels, respectively. The corresponding turning points of the entropy and pressure are labeled with red cross symbols. The first-order phase transition temperature and the unstable region can be  determined  by employing the Maxwell construction, where the areas separated by the isotherm in the thermodynamic curves are equal.  In the lower left panel, the five-pointed star symbol determines the transition temperature T_t=0.114  GeV,  at which the low-temperature branch of the entropy density curve directly jumps to the high-temperature branch through the red vertical line of left upper panel, with the remaining part representing the unstable or nonphysical states. For any other magnetic background field eB>2.8326  GeV^2, we can also determine the transition temperature using this method. Finally, we show the phase diagram in the magnetic field-temperature plane in Fig. <ref>,  where the five-points star symbol represents the critical endpoint (CEP) and the black line represents the first-order boundary.  As the results demonstrate, the temperature at which the first-order phase transition occurs diminishes with an increase of the magnetic field.



§ JET QUENCHING PARAMETER

From the study of the phase transition in the temperature and magnetic field plane in the frame of holographic QCD, we find there are crossover, first-order and second-order phase transitions. It would be interesting to investigate the jet quenching parameter within these different phase transition backgrounds. This investigation of jet quenching parameter is essential for gaining insights into the QGP from both theoretical and experimental perspectives, as well as for understanding the phase transitions within the framework of QCD.  Although laboratory-generated magnetic fields may not reach extremely high values, the investigation for the extremely magnetic field region remains meaningful. On one hand, relevant phenomena can be compared to the high density case. On the other hand, extremely strong magnetic fields might have existed in the early universe making the study of first-order transition very important.









Within the framework of holographic QCD, the calculation of the jet quenching parameter was first obtained by Hong Liu, Krishna Rajagopal, and Urs Achim Wiedemann in Ref. <cit.>. Since then, many studies have been conducted under various conditions and using different holographic models. However, the jet quenching parameter in extremely strong magnetic fields remains unexplored. Our calculations suggest that the phase transition in this condition could be of the first-order,  thereby rendering the discussion of jet quenching parameters in this scenario both compelling and significant.

Due to the magnetic background field, the spatial SO(3) symmetry is broken down to SO(2). In the anisotropic medium, there are three different types of jet quenching parameters. First, the light parton moves along the direction of the magnetic field, with transverse momentum broadening perpendicular to the magnetic field direction. Second, the light parton moves perpendicular to the magnetic field, with transverse momentum broadening in the magnetic field direction. Third, the light parton moves perpendicular to the magnetic field direction, with transverse momentum broadening also perpendicular to the magnetic field direction.
Following the steps in Refs. <cit.>, one can derive the jet quenching parameters q̂ as [In the numerical process, the integration interval is chosen to be [ϵ,r_max].]


    q̂_p(k)=√(λ_t)/π(∫_r̃_H^∞ dr̃1/g̃_kk√(g̃_rr/g̃_tt+g̃_pp))^-1,



where p is the direction of the light parton moving and k is the direction of the transverse momentum broadening. λ_t is the 't Hooft coupling. Substituting the relation Eq. (<ref>) into the expression of the jet quenching parameter Eq. (<ref>), one can obtain the expressions for these normalized jet quenching parameters.


    
    q̂_||(⊥)/T^3= 64π^2 √(λ_t) h_0^ far/∫_r_H^r_maxd re^-2c(r)-a(r)+2(c_0^ far-a_0^ far)/√(h(r)[h_0^ far-h(r)])
    q̂_⊥(||)/T^3=32π^2 √(λ_t) h_0^ far/∫_r_H^r_maxd re^-2a(r)/√(h(r)[h_0^ fare^2c(r)-2(c_0^ far-a_0^ far)-h(r)e^2a(r)])
    q̂_⊥(⊥)/T^3=32π^2 √(λ_t) h_0^ far/∫_r_H^r_maxd re^-2c(r)+2(c_0^ far-a_0^ far)/√(h(r)[h_0^ fare^2c(r)-2(c_0^ far-a_0^ far)-h(r)e^2a(r)])




In Fig. <ref>, we show the results of q̂ without a magnetic field.
As shown by the red curve, we have the normalized jet quenching parameter q̂/T^3 as a function of the temperature. In our calculations, we choose the 't Hooft coupling λ_t=1/2.  We have found that the holographic outcomes are in agreement with the Monte Carlo simulations, which were conducted using an initial quark jet with an energy of  E=10 GeV, as well as the results from the deep inelastic scattering (DIS)  <cit.>. The holographic curve lies within the region of the Monte Carlo and the DIS results. The significant difference between the low and high temperatures indicates a crossover from the normal phase to the QGP phase.


In Fig. <ref>, we show the normalized jet quenching parameter q̂_||(⊥) as a function of temperature for different fixed magnetic background fields. Comparing the sub-figures, one can see that q̂_||(⊥) is generally enhanced by the magnetic field, favoring  the transverse momentum broadening. This enhancement is especially significant in the phase transition temperature region and tends to match the values of the zero magnetic field case in both the low and high-temperature limits. Furthermore, with relatively small magnetic fields, q̂_||(⊥)/T^3 remains a monotonic function of temperature.  However,  in the presence of an extremely strong magnetic field, for instance, eB=4 and 10  GeV^2, q̂_||(⊥)/T^3, there exists a region around the phase transition temperature where q̂_||(⊥)/T^3 takes on multiple values. This results in three different values of q̂_||(⊥)/T^3 for a single temperature, as indicated by the red dashed lines in Fig. <ref>, which is the typical characteristic of the first-order transition. At the critical value of the magnetic field, i.e., eB=2.8326  GeV^2, the first-order derivative of jet quenching parameter with respect to the temperature ∂q̂_||(⊥)/∂ T diverges at T_c.

For the other two types of jet quenching parameters, q̂_⊥(||) and q̂_⊥(⊥), the numerical results are shown in Fig. <ref>. The overall trends of these jet quenching parameters are almost the same as q̂_||(⊥) shown in Fig. <ref>. Specifically, q̂_⊥(||)+q̂_⊥(⊥) is larger than q̂_||(⊥); q̂_⊥(⊥) is larger than q̂_⊥(||) under the same temperature and magnetic field conditions [This is consistent with the results in a narrower region of magnetic field  <cit.>].





§ CONCLUSIONS

In this paper, we investigate the impact of a magnetic background field on QCD properties using the 
 EMD model. Our findings reveal that at extremely high magnetic fields, the QCD phase transition is of first-order. As the magnetic field decreases, the first-order phase transition boundary terminates at a critical endpoint, located at (eB,T_C)= (2.8326  GeV^2, 0.1191   GeV), before which the transition turns into a crossover.  The entropy density and pressure behavior confirmed this phase transition boundary, with typical S-shape and Swallowtail shape curves. The first-order transition temperature decreases with the increasing of magnetic field. These are qualitatively consistent with lattice results in Ref. <cit.>. Additionally, both the entropy and pressure are significantly enhanced by the magnetic fields. 

Furthermore, we also discuss the variation of the jet-quenching parameter during the phase transition process in the presence of a magnetic field. For comparison, in the absence of a magnetic field, our holographic results are primarily consistent with the Monte Carlo and the DIS experiment  <cit.>, provided that λ_t=1/2. In the presence of a magnetic field, it highlights the anisotropic nature of QGP  in heavy ion collisions. The magnetic field reduces the spatial SO(3) symmetry to SO(2), leading to distinct jet quenching parameters that vary with the direction of the parton’s motion and the orientation of the transverse momentum broadening. Under the phase transition, the jet quenching parameters are greatly affected by the magnetic field. The normalized jet quenching parameters are enhanced around the phase transition temperature, suggesting that magnetic fields promote the transverse momentum broadening as well as the jet energy loss in the QGP. The parameters converge to the zero magnetic field cases at both low and high-temperature extremes. Moreover, as a consequence of the first-order phase transition, they can exhibit multiple values which indicates that jet quenching could serve as a signal to probe the phase diagram of QCD phase transition.



§ ACKNOWLEDGEMENT

We thank Danning Li for his beneficial discussion. This work is supported by the Guangdong Basic and Applied Basic Research Foundation under Grant No. 2024A1515012931, the National Natural Science Foundation of China (NSFC) under Grant Nos. 12305142, 12275108  and the Science and Technology Planning Project of Guangzhou, China under Grant No. 2024A04J3243.





 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: Prospects for Cosmological Research with the FAST Array: 21-cm Intensity Mapping Survey Observation Strategies.


	
	
	
 
	
	
	Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China
	
	Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China
	
	Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China
	
	Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China
	
	
	
	
	
	Corresponding author.
zhangxin@mail.neu.edu.cn
	
	
	Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China
	Key Laboratory of Data Analytics and Optimization for Smart Industry (Ministry of Education), Northeastern University, Shenyang 110819, China
	National Frontiers Science Center for Industrial Intelligence and Systems Optimization, Northeastern University, Shenyang 110819, China
	
	
		
	
	
	
	
	
	§ ABSTRACT 
	Precise cosmological measurements are essential for understanding the evolution of the universe and the nature of dark energy. The Five-hundred-meter Aperture Spherical Telescope (FAST), the most sensitive single-dish radio telescope, has the potential to provide the precise cosmological measurements through neutral hydrogen 21-cm intensity mapping sky survey. This paper primarily explores the potential of technological upgrades for FAST in cosmology. The most crucial upgrade begins with equipping FAST with a wide-band receiver (0 < z < 2.5). This upgrade can enable FAST to achieve higher precision in cosmological parameter estimation than the Square Kilometre Array Phase-1 Mid frequency. On this basis, expanding to a FAST array (FASTA) consisting of six identical FAST would offer significant improvements in precision compared to FAST. Additionally, compared with the current results from the data combination of cosmic microwave background, baryon acoustic oscillations (optical galaxy surveys), and type Ia supernovae, FASTA can provide comparable constraints. Specifically, for the dark-energy equation-of-state parameters, FASTA can achieve σ(w_0) = 0.09 and σ(w_a) = 0.33.

		
	
	
	
	Prospects for Cosmological Research with the FAST Array: 21-cm Intensity Mapping Survey Observation Strategies
    Xin Zhang
    
==============================================================================================================



§ INTRODUCTION

 
    Dark energy is a component of the universe that causes the late-time cosmic acceleration. Although it is known to comprise 68% of the total energy density of the universe, the nature of dark energy remains unknown. The Λ cold dark matter (ΛCDM) model, the standard model of cosmology, treats dark energy as a constant energy density filling space homogeneously. However, the latest baryon acoustic oscillation (BAO) observation from the Dark Energy Spectroscopic Instrument (DESI) suggests that dark energy may potentially be a dynamical energy density evolving with time <cit.>. This finding indicates the possibility of dark-energy dynamical evolution, but more observations and advanced instruments are needed to fully confirm this and to understand the evolutionary dynamics of dark energy.

BAO from the large-scale structure (LSS) is a late-time cosmological probe that can be used as a standard ruler to study the expansion history of the universe <cit.>. Currently, BAO studies primarily rely on optical galaxy surveys <cit.>. While effective, these surveys are limited in terms of efficiency and coverage.  To overcome these limitations and achieve more precise cosmological measurements, advancements in radio astronomy have developed a promising method known as 21 cm intensity mapping (21 cm IM). After reionization, neutral hydrogen (H  I) mainly exists within galaxies. This technique measures the intensity of the characteristic 21 cm emission line from H  I to study the LSS <cit.>. This method does not require resolving individual galaxies, thus it has high survey efficiency. The characteristics of the 21 cm signal allow us to probe deeper into the universe and directly obtain redshift information. Recent forecasts and observational studies have confirmed the potential of 21 cm IM in cosmology <cit.>.





	
	To achieve more precise cosmological measurements, advanced radio telescopes are utilized in 21 cm IM. These telescopes include those currently in use, such as the Five-hundred-meter Aperture Spherical Telescope (FAST) <cit.>, the MeerKAT <cit.>, the Tianlai <cit.>, the Canadian Hydrogen Intensity Mapping Experiment (CHIME) <cit.>, and those planned for future use, such as the Square Kilometre Array (SKA) <cit.>, the Baryon Acoustic Oscillations from Integrated Neutral Gas Observations (BINGO) <cit.>, and the Hydrogen Intensity and Real-time Analysis eXperiment (HIRAX) <cit.>.

    
    
    
    Among these, FAST is particularly significant due to its large aperture and sensitivity. The current FAST has achieved substantial success in many areas, particularly in fast radio burst and gravitational wave research <cit.>. However, to harness its potential for 21 cm IM and to extend its capability for cosmological studies, several upgrades are planned for FAST  <cit.>. 
    
    This paper aims to explore how potential future upgrades to FAST could promote cosmological research by using 21 cm IM. For the expansion of FAST itself, equipping it with a wide-band receiver aims to extend its 21 cm signal detection capability, enabling broader redshift coverage <cit.>. This will allow us to observe a larger volume of the universe, aiding in constructing a more precise three-dimensional map of the LSS. Moreover, a phased array feed (PAF) is planned to enhance its survey efficiency <cit.>.
    Additionally, the expansion of the construction around FAST is also considered. The plan involves building 40-meter diameter spherical radio telescopes (40 m dishes) in two phases around FAST: the first phase involves constructing six 40 m dishes, and the second phase will increase this number to twenty. 
	The most significant upgrade is the FAST Array (FASTA), which plans to add five more identical 500-meter diameter telescopes to join the current FAST <cit.>. This network will significantly improve sensitivity and survey efficiency, enhancing the ability to explore LSS and providing higher precision in cosmological parameter estimation.
	
    In this paper, we present a forecast for estimating cosmological parameters using simulated 21 cm IM data and discuss how the various extensions of FAST will improve the precision of these constraints.
    
    In Section <ref>, we introduce the 21 cm IM simulation and the methods used to constrain cosmological parameters. In Section <ref>, we discuss the performance of various configurations for 21 cm IM and the resulting cosmological constraints. In Section <ref>, we conclude with perspectives for future research.
	
	In our analysis, we adopt the Planck best-fit ΛCDM model as fiducial cosmology  <cit.>. The parameters are H_0 = 67.3 km s^-1 Mpc^-1, Ω_m = 0.317, Ω_b = 0.0495, Ω_k = 0, σ_8 = 0.812, and n_s = 0.965.
	
	



§ 21 CM INTENSITY MAPPING

   

 §.§ Signal

   21 cm IM measures the intensity of 21 cm signals and converts it into brightness temperature.  The mean temperature of H  I can be expressed as <cit.>
   
    T_b = 3/32 π𝒽 c^3 A_10/k_B m_pν_21^2(1+z)^2/H(z)Ω_H   I(z) ρ_c, 0 ,

   where 𝒽 is the Planck constant, c is the speed of light, A_10 is the Einstein coefficient for spontaneous emission <cit.>, k_B is the Boltzmann constant, m_p is the proton mass, ν_21 is the H  I line frequency, H(z) is the Hubble parameter, Ω_H  I(z) is the fractional density of H  I, and ρ_c, 0 is the current critical density of the universe.
   
    The signal covariance is expressed in terms of the power spectrum
    <cit.>
		
    C^S(q, y) = T_b^2(z_i)/r^2 r_ν (b_H   I + fμ^2)^2 exp(-k^2μ^2σ_NL^2)  P(k, z),

	where q = k_⊥ r and y = k_ r_ν. r denotes the comoving distance to z, and r_ν corresponds to the comoving distance for the frequency interval. b_H   I represents the bias factor of H  I <cit.>. f is the linear growth rate of the structure. μ = k_ / k is the cosine of the angle between the wavevector k and the line of sight. σ_NL=7 Mpc represents the non-linear dispersion scale <cit.>. P(k, z) is the matter power spectrum at wavenumber k and z, which can be calculated by CAMB <cit.>.
	
		
	

 §.§ Noise

	The noise covariance can be written as <cit.>
	
    C^N(q, y)=T_sys ^2/t_tot Δν N_b N_d U_bin B_⊥^-2 B_^-1,

	where T_sys includes the instrumental temperature T_inst and the sky background radiation T_sky≈ 60  K× (ν / 300 MHz)^-2.5.  t_tot represents the total integration time, N_b is the number of beams, N_d is the number of dishes in the array, and Δν is the total bandwidth. U_bin = S_areaΔν̃ represents the survey volume element at a given redshift, where S_area is the survey area and Δν̃ = Δν/ν_21 is the dimensionless bandwidth for a given redshift bin. B_ is the radial beam response
	
    B_(y)=exp(-(y δν / ν_21)^2/16 ln 2),

	where δν is the bandwidth of an individual frequency channel. B_⊥ is the transverse beam response in single dish mode
	
    B_⊥(q)=exp(-(q λ/D_dish)^2/16 ln 2),

	where D_dish is the diameter of the dish.

	

 §.§ Foreground

	The foreground is an important component of the signal received by the telescope. It primarily originates from the galaxy and extragalactic point sources. At this stage, the effectiveness of methods for subtracting foregrounds from observational data has been very limited. The mainstream methods mainly include Polynomial Fitting, Principal Component Analysis, Independent Component Analysis, and Artificial Intelligence <cit.>. We assume a method that can effectively subtract the foreground, and the residual foreground model can be written as <cit.>
	
    C^F(q, y)=ε_FG^2 ∑_X A_X(l_p/2 π q)^n_X(ν_p/ν_i)^m_X,

	where ε_FG is a factor that describes the efficiency of foreground subtraction. ε_FG = 1 represents no foreground subtraction, while ε_FG = 0 represents perfect foreground subtraction. We assume ε_FG=10^-6, as this level is effective for extracting cosmological signals. X is the different foreground component shown in Table <ref>. 




    






 §.§  Instrument parameters

    

    
    
In this paper, we consider several different configurations of FAST and Square Kilometre Array Phase 1 - Mid frequency (SKA1-Mid) for simulating the 21 cm IM. The specific instrument parameters are listed in Table <ref>.

The current capability of FAST to detect 21 cm signals covers the redshift range 0 < z < 0.35. A new wide-band receiver under test extends this range to 0 < z < 1.84 <cit.>. Previous galaxy surveys, however, have performed precise BAO measurements within z < 2.5, achieving 1%-2% accuracy <cit.>. This paper assumes the upgraded new wide-band receiver on FAST can detect 21 cm signals up to 0 < z < 2.5. 

Additionally, constructing 6×40 m dishes or 20×40 m dishes around FAST is considered. These dishes will use the same receiver and feed system as FAST, so their frequency bands and N_b will be identical. Given the outstanding performance of FAST, we also consider constructing five additional identical FAST telescopes to form FASTA, significantly improving survey efficiency. Furthermore, we consider equipping FAST with a PAF to increase the value of N_b to 100.

Regarding instrument temperature, in Ref. <cit.> the author measured the temperature of FAST near boresight to be 20 K in the L-band. We assume that the frequency band can be broadened while maintaining the instrument temperature at 20 K. For the PAF, we assume the temperature to be approximately 30 K due to the increase in the number of beams. The total observation time is assumed to be 10000 hours. The calculation for the survey area is as follows

	
    S_area = η∫_ϕ∫_θsinθdθdϕ ,

	where η represents the fraction of the initial survey area that remains usable for scientific observation after excluding the Milky Way. For FAST/FASTA, we considered η = 0.7 and the range of declination from -14^∘ to +66^∘, yielding an area of approximately 8100 deg^2.

For SKA1-Mid, we considered band I (0.35<z<3) planned for 21 cm IM, with the survey area of 20000 deg^2 and the total observation time of 10000 hours. The instrument temperature is as follows <cit.>

    T_inst=15  K+30  K(f/GHz-0.75)^2 .

	

 §.§ The Fisher matrix

	
	
	
	
    
	
	
	
	
	
	
	
	
	
	To achieve optimal observational results, different telescopes require different survey strategies, so the methods for calculating C^tot also vary. For a single dish or identical dishes observing the same area of the sky (such as FAST or FASTA), the total covariance matrix is expressed as

    C^tot = C^S + C^N + C^F.


For the combination of different telescopes, we consider two scenarios. The first scenario involves two telescopes in close geographic proximity observing the same area of the sky, such as combining 40 m dishes and FAST. This approach maximizes the reduction of thermal noise impact on observations. Since C^S and C^F are only related to the observed sky area, they remain fixed. Different telescopes have varying C^N due to their differing thermal noise integration effects. Therefore, the total covariance matrix can be expressed as

    C^tot = C^S + ((C_1^N)^-1 + (C_2^N)^-1)^-1 + C^F,

where C_1^N and C_2^N are the noise covariance matrices of the different dishes, with (C^N)^-1 denoting their inverse matrices. The second scenario involves two telescopes with significant differences in latitude, resulting in a very small common observable sky area. Therefore, they must observe different areas of the sky, which also maximizes the survey volume. Since both the observed sky areas and the telescopes are different, the total covariance matrix is given by

    C^tot = ((C_1^tot)^-1 + (C_2^tot)^-1)^-1,

where C_1^tot and C_2^tot are the total covariance matrices of the different dishes.

	After obtaining the total Fisher matrix C^tot, we can transform it into the Fisher matrix of observations {p_i} <cit.>
	
    F_i j = 1/2 U_bin∫d^2 q  d y/(2 π)^3[∂ln C^tot(q, y)/∂ p_i∂ln C^tot(q, y)/∂ p_j].

	In this paper, we mainly calculate the Fisher matrix for several key observables to constain the evolution of LSS, as well as the growth and geometry of the universe. These include the transverse distance indicator D_A(z), radial distance indicator H(z) and the parameter combination fσ_8(z) from redshift space distortions (RSDs), where f(z) is growth rate and σ_8 is the amplitude of matter density fluctuations on a scale of 8 h^-1 Mpc (with h the dimensionless Hubble constant).
    
   
    
    
	

 §.§ Cosmological parameter constraints

    We use the Fisher matrix to propagate errors from observational errors to the BAO parameters D_A(z), H(z), and fσ_8(z). Then, we use the errors of BAO parameters to form the likelihood function for cosmological models. Finally, we employ Markov Chain Monte Carlo (MCMC) analysis to constrain the cosmological parameters <cit.>.
 
 To compare with the mainstream observations, we calculated the constraints from the latest observations. The data used include: CMB – Planck 2018 high-ℓ TT, TE, EE, and low-ℓ TT, EE, and PR4 lensing data <cit.>; BAO – DESI data <cit.>; and supernova (SN) – Pantheon+ data <cit.>. The data combination of CMB, BAO, and SN in this paper is also abbreviated as CBS.

	The equation of state (EoS) of dark energy is given by w = p/ρ, where p is the pressure and ρ is the energy density. We employ different models to describe its dynamics. (i) ΛCDM: w = -1. (ii) wCDM: w(z) = w, where w is a constant EoS for dark energy. (iii) w_0w_aCDM: w varies with redshift according to the parameterization w(z) = w_0 + w_az / (1 + z), where w_0 is the present-day value of the equation of state parameter and w_a describes its evolutionary behavior.
	


§ RESULTS AND DISCUSSION

	In this section, we  first discuss the capabilities of various instruments and their combinations in 21 cm IM using the relative errors of BAO parameters {ε(D_A(z)), ε(H(z)), ε(fσ_8(z))}, where ε(ξ) = σ(ξ)/ξ. We then present the cosmological parameter constraints employing FAST, SKA1-Mid, FASTA, and CBS, followed by comparison and discussion of the results.
    
    
    
    
    
	
	

 §.§ Combination of FAST and SKA-Mid

 
 In this subsection, we discuss the capabilities of SKA1-Mid and FAST with a wide-band receiver, as well as their combination in 21 cm IM. Due to geographical factors, the overlapping observable sky area for FAST and SKA1-Mid is very limited, which necessitates a strategy where each telescope observes different sky regions.

From Figure <ref>, it is evident that FAST and SKA1-Mid have distinct advantages across different redshift ranges. SKA1-Mid achieves tight constraints within the range of 0.35 < z < 0.8 due to its high survey efficiency. In this range, the relative errors in BAO parameters are typically below 3%. On the other hand, FAST attains robust constraints within the range of 0.8 < z < 2.5 because of its high transverse resolution and sensitivity. In this range, the relative errors in BAO parameters are typically around or below 4%.


Their combined observations enable precise measurement of BAO parameters across the redshift range of 0 < z < 2.5, achieving constraints on D_A(z), H(z), and fσ_8(z) to below 2% for most of this range, and to as low as 1% at z ≈ 1.

Upgrading the existing receiver to detect 21 cm signals up to 0 < z < 2.5 will enable FAST to utilize its high-redshift advantage in 21 cm IM. Combined with SKA1-Mid, this complementary strategy will improve measurements of BAO parameters across a wide redshift range.

    

 §.§ Combination of FAST and 40 m dishes

	In this subsection, we discuss the extent to which incorporating 40 m dishes with FAST enhances the constraints on BAO parameters in 21 cm IM. An array of 40 m dishes, located in close geographical proximity to FAST, enhances the survey efficiency by enabling observations of the same sky regions.

Figure <ref> illustrates the fractional constraints achievable on D_A(z), H(z), and fσ_8(z) using 21 cm IM with an array of 40 m dishes, FAST, and their combined configurations. The 40 m dishes perform better at low redshifts because they have higher survey efficiency. FAST is more effective at higher redshifts due to its greater sensitivity and resolution. The combination of FAST and 40 m dishes significantly reduces observational errors, particularly at high redshifts. The relative errors in BAO parameters are typically below 4%, with errors reaching as low as 2% at z ≈ 1.

For the configurations of 6× 40 m dishes and 20× 40 m dishes combined with FAST, Figure <ref> shows that the larger array provides only marginal improvements. This result suggests that the combination of FAST with 6× 40 m dishes offers a more optimal configuration for the 21 cm IM survey.

     
    

 §.§ FASTA

	In this subsection, we present the cosmological parameters constraints by using simulations of 21 cm IM data with different instruments and current observational data from CBS. Figure <ref> shows the  parameter constraints under the 1σ and 2σ confidence intervals for the ΛCDM, wCDM, and w_0w_aCDM models. The specific values are listed in Table <ref>.

Compared to the results in Ref. <cit.> for FAST (0 < z < 0.35), a wide-band receiver proposed in this paper utilizes the resolution advantage of the FAST at high redshifts, significantly enhancing the constrains on cosmological parameters. The main specific parameter constraints are as follows: for the ΛCDM model, FAST achieves σ(Ω_m) = 0.0076 and σ(H_0) = 0.51  km s^-1 Mpc^-1. For the wCDM model, the constraint is σ(w) = 0.036. For the w_0w_aCDM model, the constraints are σ(w_0) = 0.14, and σ(w_a) = 0.57. Compared to SKA1-Mid, FAST also has higher sensitivity and lower receiver noise, which enables it to achieve better parameter constraints. For the dark-energy EoS parameters, there is an improvement of approximately 20% for w, 12.5% for w_0, and 21.9% for w_a.

	Further advances are observed with FASTA. From Figure <ref>, FASTA performs well in observing BAO parameters across the entire band, achieving an observational error as low as 1%. This precise observation of BAO parameters translates into tight constraints on cosmological parameters. 
 
In the ΛCDM model, FASTA provides best constraints, σ(Ω_m) = 3.6 × 10^-3 and σ(H_0) = 0.26  km s^-1 Mpc^-1, which surpasses other instruments and CBS.
In the wCDM model, FASTA also provides best constraints, σ(Ω_m) = 0.0037, σ(H_0) = 0.39  km s^-1 Mpc^-1, and σ(w) = 0.02. For the parameter w, FASTA represents an improvement of approximately 55.56% compared to SKA1-Mid, 44.44% compared to FAST, and 16.67% compared to CBS.
In the w_0w_aCDM model, FASTA achieves highly precise parameter constraints, with σ(Ω_m) = 0.012, σ(H_0) = 0.89  km s^-1 Mpc^-1, σ(w_0) = 0.09, and σ(w_a) = 0.33. For w_0 and w_a, FASTA represents an improvement of approximately 43.75% and 54.79% compared to SKA1-Mid, and 35.71% and 42.11% compared to FAST, while achieving constraints comparable to CBS using 21 cm IM  alone.

In summary, FASTA significantly improves cosmological parameter constraints across various models, surpassing other instruments and achieving results comparable to CBS using 21 cm IM  alone.



 
 
 
	
        
	

 §.§ Phased array feed

	In this subsection, we discuss the enhancements and challenges of upgrading the feed of FAST to PAF for 21 cm IM. PAF can achieve various beamforming modalities by adjusting the phase and amplitude of each element in the array, thereby generating multiple beams simultaneously. This capability covers a larger field of view (FOV) and enables rapidly beam steering and scanning <cit.>.  At the same time, different beamforming strategies significantly impact noise characteristics, and data calibration and processing.  We considered two beamforming strategies in terms of survey efficiency.
 
    In the first scenario, we consider a fixed FOV and maximize the N_b within this field. The size of a beam is given by θ = c/(ν D_dish). If there is no overlap between the beams, the maximum N_b can be written as
    
    N_b = FOV·(ν D/c)^2 .


    In this case, C^N follows the same pattern as described in Ref. <cit.>
    
    C^N(q, y) → C^N(q, y) ×1,    ν>ν_crit 
    (ν_crit  / ν)^2,    ν⩽ν_crit  .

	We assume ν_crit=1420 MHz. As shown in Figure <ref>, the errors increase significantly at low frequencies. This is because at low frequencies, the beam size increases, resulting in a decrease in N_b.
 
	Another scenario is a fixed N_b. We assume the case of 100 beams. As shown in Figure <ref>, the increase in the N_b significantly enhances survey efficiency, achieving best constraint result of 2% around z = 1. However, in practice, the FOV does not increase indefinitely as the beam size increases at lower frequencies. As the FOV certainly deviates from the boresight, the survey efficiency decreases. This requires actual measurements to PAF, but here we make an ideal estimate. The real situation is expected to lie between the fixed FOV and the fixed N_b scenarios.


	

§ CONCLUSION

	In this paper, we explore several potential upgrades to FAST using 21 cm IM to enhance cosmological research. Those upgrades include FAST with a wide-band receiver, the combination of 40 m dishes and FAST, FASTA, and FAST with a PAF. We simulated the full 21 cm IM power spectrum to constrain the BAO and RSD parameters D_A(z), H(z), and fσ_8(z), and used these results to evaluate the capabilities of these different configurations. Finally, we used the simulated BAO and RSD data to constrain the cosmological parameters of the ΛCDM, wCDM, and w_0w_aCDM models.

	Firstly, the wide-band receiver utilizes the capabilities of FAST in the redshift range 0.8 < z < 2.5, achieving better cosmological parameter constraints than SKA1-Mid. It also enables FAST to form a complementary observation with SKA1-Mid, which has strengths in the redshift range of 0 < z < 0.8, providing high-precision observations across a wider redshift range.
 
	Secondly, we discuss the combination of the 40 m dishes and FAST. Incorporating 40 m dishes with FAST significantly enhances survey efficiency, thereby reducing BAO parameter observational errors to typically below 2%. For the combination with FAST, 6×40 m dishes are a more balanced configuration compared to 20×40 m dishes for 21 cm IM.
	
	Next, we discuss the cosmological parameter constraints for FASTA. Through BAO parameter measurements approaching 1% accuracy, it shows excellent performance in constraining cosmological parameters. Specifically, in the w_0w_aCDM model, FASTA achieves highly precise parameter constraints, with improvements of 43.75% for w_0 and 54.79% for w_a over SKA1-Mid, and 35.71% for w_0 and 42.11% for w_a over FAST, achieving results comparable to CBS using 21 cm IM  alone.


	Lastly, we analyze FAST with a PAF. We consider two beamforming strategies, a fixed FOV and a fixed N_b. The real situation is likely to fall between these two cases. This provides a basis for understanding the advantages and limitations of different beamforming strategies for 21 cm IM. A balanced beamforming strategy will be beneficial for effectively utilizing PAF.
	
	In summary, these upgrades to FAST have enormous potential to improve the precision of cosmological measurements. Despite facing some technical challenges, they can greatly enhance our understanding of the evolution of the universe and the nature of dark energy.
	
	
	
	We thank Guang-Chen Sun, Ji-Guo Zhang,  Zi-Qiang Zhao, and Lu Feng for helpful discussions and suggestions. We are grateful for the support from the National SKA Program of China (grant Nos. 2022SKA0110200 and 2022SKA0110203), the National Natural Science Foundation of China (grant Nos. 11975072, 11875102, and 11835009), and the National 111 Project (Grant No. B16009).
		
	
	

	


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





    et al.
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            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: Captured molecules could make a Bose star visible.



v.flambaum@unsw.edu.au
igor.samsonov@unsw.edu.au
School of Physics, University of New South Wales, Sydney 2052, Australia

§ ABSTRACT 
A Bose star passing through cold molecular clouds may capture atoms, molecules and dust particles. The observational signature of such an event would be a relatively small amount of matter that is gravitationally bound. This binding may actually be provided by invisible dark matter forming the Bose star. 
We may expect a relative excess of heavier atoms, molecules, and solid dust compared to the content of giant cold molecular clouds since the velocity of heavy particles at a given temperature is lower and it may be small compared to the escape velocity, v_rms = √(3k_B T/m_gas)≪ v_esc=√(2GM/R). Finally, the velocity of this captured matter cloud may correlate with the expected velocity of free dark matter particles (e.g. expected  axion wind velocity relative to Earth).



Captured molecules could  make a Bose star visible
    I. B. Samsonov
    August 5, 2024
==================================================






§ INTRODUCTION

Bose stars  are assumed to be formed from cold dark matter fields (scalars, axions) which may form Bose condensate and solitons bound mainly by the gravitational  attraction <cit.>. 
Bose condensation temperature T_B ∝ n^2/3/m, so in thermal equilibrium axions (or scalars) form Bose condensate for sufficiently small mass m  and large  density n.  

A simple estimate, similar to that for hydrogen atom,  makes relation between mass M≈ N m and size R  of Bose star, where N is the number of bosons and m is their mass. 
Estimating kinetic energy of the ground state using uncertainty relation for minimal momentum p  ∼ħ/r and estimating gravitational energy we obtain equation for the non-relativistic energy of Bose star: 

    E=a_1 Nħ^2/mr^2 - a_2 N(N-1)/2Gm^2/r,

 where a_1, a_2 are constants ∼ 1. Minimizing energy by varying  size r or using virial theorem (U=- 2E_k, E=-E_k) we obtain size of gravitationally bound  Bose star   

    R=4 a_1  ħ^2/a_2 (N-1)G m^3,

 binding energy

    E=N (N-1)^2  a_2^2 G^2 m^5/16 a_1ħ^2,

and number density 

    n  ∼N/V∼ 0.004 N (N-1)^3  G^3 m^9/ħ^6,

where the star volume is V ∼ 4 π R ^3/3.
We see very strong dependence on the boson mass m. We also see that the density n in   the ground state is proportional to N^4 so any, even small non-gravitational self interaction between bosons may become important  for large N.
We may add the self-interaction  contribution of the self-interaction term λϕ^4 to the Bose star energy 

    E=a_1 Nħ^2/mr^2 - a_2 N(N-1)/2Gm^2/r+a_3  λN(N-1)/2 m^2 r^3  ,

where a_1, a_2 a_3 are constants ∼ 1. Using dE/dr=0,  this expression  gives quadratic equation for r with two solutions. For the QCD axion, the selfinteraction constant λ=- m^2/(24 f_a^2) is expressed via the axion decay constant f_a (see e.g. <cit.>).  In the case of λ<0 the energy tends to minus infinity for r → 0, so the Bose star is metastable. Solution with larger r corresponds to a minimum whereas the smaller r solution is maximum. More accurate analytical calculations using variational approach will be given in the next section.

Distribution of ordinary matter is given by Boltzmann distribution in the gravitational field of Bose star. Velocity distribution is given by Maxwell distribution. We can  add molecular matter terms E_m to the energy E:

    E_m=   a_4 N_m 5/2 kT(r) - a_5 N_M N Gm_m m/r
       - a_6 N_m(N_m-1)/2Gm_m^2/r .

If the total mass of the molecular cloud bound to the Bose star, M_m=N_m m_m, is relatively small as compared with the mass of the Bose star, N_m m_m ≪ Nm, we may assume that in the leading approximation molecular distribution does not affect Bose star parameters. Then we may include gravitational effect of molecules using iterations. 

Time of formation of Bose stars very strongly depends on the Boson  mass m. For m >  10^-22 eV the formation time is smaller than the lifetime of the Universe 13.7 billion years (see e.g. <cit.>).

Total mass of the Bose star M is strongly model dependent and depends on time. There are estimates of a  typical scale for expected mass M. The QCD axion dark matter field with axion mass m ∼ 10^-5  eV makes  Bose star with mass  M ∼  10^-15 M_⊙  <cit.>. Fuzzy dark matter with particle mass m ∼ 10^-22  eV may compose a Bose star with mass M ∼  10^8 M_⊙  <cit.>. Bose star accumulating  certain critical mass becomes unstable, collapses  and decays to relativistic jets <cit.>.
 
Axions and scalars are neutral, so in zero approximation Bose star is transparent and is hardly observable by telescopes.  In principle, Bose stars may be observed using gravitational lensing of passing light and axion decays to photons. We discuss a different possibility: Bose star may  capture atoms, molecules and dust. Radiation and absorption of light by this matter could make Bose stars visible. 




§ VARIATIONAL APPROACH


The variational approach allows one to get an approximate analytic solution to the particle density which minimizes total energy of the system of interacting particles. We consider N bosonic DM particles with position vectors r⃗_i and momenta p⃗_i. Given that each particle has mass m, they can interact gravitationally. The corresponding non-relativistic Hamiltonian is

    H  = -ħ^2/2m∑_i=1^N ∇_k^2 - ∑_i<j^N G m^2/r_ij ,

where G is the gravitational constant and r_ij≡ |r⃗_i - r⃗_j|. 

In the ground state, all particles are in the same state with the wave function ψ(r⃗_i), and the full system is described by a product of these wave functions, 

    Ψ = ∏_i=1^N ψ(r⃗_i) .

Below we consider two approximate analytic solutions to the function ψ(r⃗).



 §.§ Exponential ansatz


The ground state wave function may be searched for within the ansatz

    ψ(r) = 1/√(π) R^-3/2e^-r/R ,

where R is a free parameter. The function ψ(r) is the 1s solution of the stationary Schrödinger equation with the potential V = -ħ^2/m Rr and energy E_0 = -ħ^2/2 mR^2.

Making use of the function (<ref>) we find the expectation value of the Hamiltonian (<ref>):

    E = ⟨Ψ|H|Ψ⟩ 
        =N/2[
        ħ^2/m R^2 - 5/8Gm^2/R (N-1)
        ].

This energy reaches its minimum value 

    E_min = -25/512G^2 m^5/ħ^2 N(N-1)^2

at

    R = 16/5ħ^2/Gm^3 (N-1) .


The energy of the solution (<ref>) agrees at 90% level with the corresponding numerical result in Ref. <cit.>.
The root-mean-squared (RMS) radius of this solution is

    r_rms = (⟨Ψ | r_i^2 | Ψ⟩)^1/2
        =√(3) R .


As an example, consider a Bose star with mass of Bennu asteroid, M=7× 10^10 kg composed of QCD axions with mass m=10^-5 eV. It consists of N=4× 10^51 particles. Its size is

    r_rms = 4× 10^10 .

Note that the diameter of the Bennu asteroid is 0.5 km.

The RMS speed:

    v_rms = 1/m(
        ⟨Ψ |p⃗_i^2| Ψ⟩)^1/2 =ħ/m R .

For the axion star considered above, it is v≈ 8× 10^-16c. Thus, the non-relativistic approximation is justified. However, for solutions with boson  mass m ∼ 10^-5 eV and total mass of order the Solar mass, a relativistic treatment is needed.





 §.§ Gaussian ansatz


The simple exponential ansatz (<ref>) for the wave function provides a rather low accuracy for the energy. A better accuracy may be achieved within the Gaussian ansatz <cit.>

    ψ(r) = ( 2/π R^2)^3/4
        e^-r^2/R^2 ,

where R is the parameter to be found. Making use of this wave function, we find the expectation value of the Hamiltonian (<ref>):

    E = ⟨ H ⟩ = 3Nħ^2/2m R^2 - N(N-1)Gm^2/√(π)R .

This energy reaches its minimum value 

    E_min = -N(N-1)^2/6πG^2 m^5/ħ^2

at

    R = 3√(π)ħ^2/(N-1)Gm^3 .


The energy (<ref>) agrees at 98% level with the numerical solution presented in Ref. <cit.>. Therefore, we will consider only the Gaussian ansatz in what follows.

The RMS radius calculated with the wave function (<ref>) is 

    r_rms = √(3) R/2 = 3√(3π)ħ^2/2(N-1)Gm^3 ,

and the RMS speed is

    v_rms
        =√(3)ħ/m R .


The non-relativistic approximation is applicable for  v_rms≲ 0.1 c. Making use of equations (<ref>) and (<ref>), this relation translates into the constraint on the particle number and mass,

    Nm^2<0.1 
        ħ c√(3π)/G≈ 1.5 × 10^-10 g^2 .






 §.§ Gravitational potential


The Bose star solution is given by the product of N identical wave functions (<ref>) each describing a particle with mass m. They create a gravitational potential V(r) obeying the Poisson equation

    ∇^2 V(r) = -4π G ρ(r) ,

where the mass density is

    ρ = N m ψ^†ψ
        =N m( 2/π R^2)^3/2
        e^-2r^2/R^2 .

The solution to Eq. (<ref>) reads

    V(r)    = - G N m erf (√(2) r/R)/r
       ≈ -GNm{[ 2√(2)/√(π)R
        -4√(2) r^2/3√(π)R^3                            (r≪ R) ,;                                 1/r                            (r≫ R) . ].


On the boundary r=r_rms, the gravitational potential is

    V(r_rms) = -2G Nm/√(3) Rerf( √(3/2))
        ≈ - 0.2 N^2 G^2 m^4/ħ^2 .

Thus, the escape velocity on the boundary is

    v_esc = √(2|V(r_rms)|)≈ 0.6 NGm^2/ħ .


Consider, for example, the QCD axion with mass m=10^-5 eV and the total mass comparable with the mass of the Earth, Nm = M_⊕. The size of this object is about r_rms≈ 0.4 m. The corresponding escape velocity at the boundary is v_esc≈ 0.14 c.

If the DM particle mass is m=10^-9 eV, the corresponding RMS radius and escape velocity are r_rms≈ 6.3 R_⊕, v_esc≈ 4.3 km/s.



 §.§ Adding λϕ^4 selfinteraction


The scalar field ϕ is related to the wave function (<ref>) as

    ϕ = 1/√(m)Ψ .

Making use of the Gaussian ansatz (<ref>), we find

    λ∫ϕ^4 d^3r = λ/π^3/2 m^2 R^3 .

In the contribution to the energy, this expression should be multiplied by N(N-1)/2 which accounts for the pairwise particle interaction. Thus, the total energy of this field configuration reads



    E    = E_kin + E_grav + E_λ ,
    
        E_kin   = 3Nħ^2/2m R^2,
    
        E_grav   = - N(N-1)Gm^2/√(π)R ,
    
        E_λ   =  N(N-1)/2λħ^3/π^3/2 c m^2 R^3 .


This expression reaches its extrema at

    R_± = 3√(π)ħ^2/2G m^3(N-1)( 
            1±√(1+2λ G m^2(N-1)^2/3π^2ħ c)).


The case λ<0 corresponds to unstable solution because the energy (<ref>) is unbounded from below in this case. However, there is a potential barrier separating energy minimum at R_+ in Eq. (<ref>) from the collapse area close to r=0, with the barrier maximum at R_-. In the case of small λ, we obtain minimum of energy

    E_min = -N(N-1)^2/6πG^2 m^5/ħ^2 + O(λ)

at

    R = 3√(π)ħ^2/G m^3(N-1)
        +λħ(N-1)/2π^3/2mc+O(λ^2) ,

and maximum at small R ∝λ,

    R = -λħ(N-1)/2π^3/2mc + O(λ^2) .

In the case of small λ this barrier will be sufficiently  high to make  the   decay time of the  metastable Bose star  exponentially  long, making this kind of Bose star potentially observable.  

For positive selfinteraction constant, λ>0, only “+” solution in Eq. (<ref>) is physical. If λ is sufficiently small, the leading contribution has the same form as in Eq. (<ref>).
For strong selfinteraction, 2λ G m^2(N-1)^2/3π^2ħ c≫1,
the parameter R is independent of N,

    R = ħ^3/2/m^2√(3λ/2π cG) .






§ IDEAL GAS DISTRIBUTION AROUND THE BOSE STAR


Let us consider an ideal gas with particles of mass m_gas in the gravitational potential V(r)<0 at temperature T. Assuming that the total mass of the gas is sufficiently small so that the gravitational potential is not changed significantly, we can write the density distribution in the gas cloud,

    ρ(r) = ρ_0 exp( m_gas V(r)/k_BT) .

The total mass of the gas cloud is

    M_gas = 4πρ_0 ∫_0^∞exp( m_gas V(r)/k_BT) r^2 dr .


The gas at every point should follow the Maxwell-Boltzmann distribution

    f(v) = 4π v^2 ( m_gas/2π k_BT)^3/2 e^-m_gas v^2/2 k_B T .

This distribution, however, should be truncated at v=v_esc. Note that v_rms = √(3k_B T/m_gas).

At the Earth's surface, the temperature is about T=300 K. Empirically, we find the condition for the gas molecules to be captured by the gravitational field of the Earth:

    v_rms≲ 0.1v_esc .

This corresponds to the escape probability suppressed by the  exponential factor in the distribution (<ref>) exp(-3/2v_esc^2/v_rms^2)≲exp(-150). For the Earth, this condition implies that the gas molecules with mass number A<6 escape from the gravitational attraction. This is confirmed by the fact that the fractions of hydrogen and helium gases in the Earth's atmosphere are very low. 

In the case of Bose stars, making use of Eq. (<ref>) the above condition may be cast in the form

    m_gas >830k_BTħ^2/N^2 G^2 m^4 ,

Representing the gas particle mass as m_gas = A m_p, with m_p the proton mass, we find the condition on the atomic number A of gas particles captured near the surface of the Bose star,

    A>   830k_BTħ^2/N^2 G^2 m^4 m_p
       = 1.35 · 10^-29(M_⊙/M)^2(eV/m)^2
        T/10 K .




The temperature in cold molecular clouds in our Galaxy is typically in the range from 10 to 20 K. Assuming T=10 K, we estimate the possibility of capture of gases in the fuzzy dark matter with m=10^-22 eV and total mass of the object Nm=10^7M_⊙:

    A>13 .

Thus, this object can accumulate only heavy atoms and molecules, as well as dust particles. 

In Fig. <ref>, we present plots of mass (and radius) of the Bose star as a function of scalar particle mass m which can capture molecules with atomic mass numbers A=1, 10, and 100.

If radius R of Bose star is small, there  will be too little matter inside the star and impossible to observe it. However, matter may be captured to distant orbits, but this will require bigger mass M  of the star, since the escape velocity will be smaller; for matter at distance r, the  required star mass is M(r)=M_0 R/r, where M_0 is the star mass required to capture matter inside. 

Cold molecular gas and dust may also be captured by large axion clusters.

If matter capture conditions  are  fulfilled inside Bose star for all A, there will be violation of such conditions outside Bose star at some distance r from the centre. Firstly, this condition will be violated for A=1, going further, for higher A.  Therefore, there will be separation of heavy and light elements outside the Bose star. This is important for Bose star of large mass and small radius.




§ SUMMARY


We applied the analytic Gaussian solution for Bose star density to study the distribution of interstellar gas particles within the gravitational field of a Bose star, assuming the gas does not significantly perturb the star's gravitational field. We examined the conditions under which interstellar gas particles with a given atomic number A could escape the gravitational attraction of a Bose star with a total mass M composed of scalar dark matter particles of mass m. Our findings indicate that, for a range of values of m and M, the Bose star can capture atoms, molecules, and dust particles.

A small cloud of interstellar gas cannot form a gravitationally bound state without an additional gravitational source. Thus, the observation of such a gravitationally bound gas cloud would suggest it is held by the gravitational field of a Bose star or a dark matter cluster. Due to differences in escape velocities between light and heavy molecules, molecular clouds captured by a Bose star may exhibit higher metallicity compared to typical giant molecular clouds. The discovery of small, gravitationally bound molecular clouds with high metallicity in the interstellar medium would provide compelling evidence supporting this dark matter model.




Acknowledgements.— 
We are grateful to John Webb for interest in this work and useful discussions. The work was supported by the Australian Research Council Grants No. DP230101058 and No. DP200100150.










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 Paper Title: Perturbative results of matrix elements of the axial current and their relation with the axial anomaly.







































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    [5]
        Physics Letters Bsort   compressmodel1a-num-names
        
        
        
        
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           DESY-24-117
            
           JLAB-THY-24-4130
        
        JCaddr]Ignacio Castellijorge.castelli@temple.eduAFaddr]Adam Freeseafreese@jlab.orgCLaddr]Cédric Lorcécedric.lorce@polytechnique.eduJCaddr]Andreas Metzmetza@temple.eduBPaddr1,BPaddr2]Barbara Pasquinibarbara.pasquini@unipv.itSRaddr]Simone Rodini
        
        simone.rodini@desy.de[JCaddr]Department of Physics, SERC, Temple University, Philadelphia, PA 19122, USA[AFaddr]Theory Center, Jefferson Lab, Newport News, Virginia 23606, USA[CLaddr]CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France[BPaddr1]Dipartimento di Fisica, Università degli Studi di Pavia, 27100 Pavia, Italy[BPaddr2]Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, 27100 Pavia, Italy[SRaddr]Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany§ ABSTRACT  
        In the Standard Model of particle physics, the axial current is not conserved,
        due both to fermion masses and to the axial anomaly.
        Using perturbative quantum chromodynamics, we calculate matrix elements of the local and non-local axial current for a gluon target, clarifying their connection with the axial anomaly.
        In so doing, we also reconsider classic results obtained in the context of the nucleon spin sum rule as well as recent results for off-forward kinematics.
        An important role is played by the infrared regulator, for which we put a special emphasis on the nonzero quark mass.
        We highlight cancellations that take place between contributions from the axial anomaly and the quark mass, and we elaborate on the relation of those cancellations with the conservation of angular momentum.
        
        Perturbative QCD; axial anomaly
        
        
        
        § INTRODUCTION
        
        Unlike the vector current, the axial current of spin-half fermions is not conserved.
        In quantum chromodynamics (QCD), the non-conservation of the flavor-singlet axial current, J_5^μ(x) = ∑_q q̅(x)  γ^μγ_5   q(x), is expressed via 
        
            ∂_μ J_5^μ(x) = ∑_q 2 i m_q  q̅(x)  γ_5   q(x) - α_s  N_f/4 π Tr ( F^μν(x) F_μν(x) )  ,
        
        where m_q is the quark mass, α_ s the strong coupling, N_f the number of quark flavors, and F_μν the dual field strength tensor defined through F_μν(x) = 1/2ε_μνρσ F^ρσ(x) (with ε^0123 = 1).
        The second term on the r.h.s. of Eq. (<ref>) represents the axial anomaly, which is generated through radiative corrections in quantum field theory <cit.>.
        When considering the divergence of the axial current, the focus is mostly concentrated on the axial anomaly while the fermion-mass term is often neglected.
        One motivation of the present work is to take the quark-mass term in Eq. (<ref>) into account, which can qualitatively change results as we discuss in more detail below.
        
        
        
        It was noted early on that the axial anomaly could play an important role when trying to understand the spin structure of the nucleon in QCD <cit.>; see  Refs. <cit.> for reviews.
        Research in this area intensified after the discovery of the proton `spin crisis' by the European Muon Collaboration <cit.>, after which works by Altarelli and Ross (AR) <cit.> and by Carlitz, Collins and Mueller (CCM) <cit.> received considerable attention.
        These papers studied the photon-gluon fusion process, γ^∗ g → q q̅, in polarized inclusive deep-inelastic lepton-proton scattering (DIS), ℓ⃗p⃗→ℓ X.
        The cross section of the γ^∗ g → qq̅ process is represented at leading order by the imaginary part of the box diagram in Fig. <ref>(a).
        In the Bjorken limit Q^2 →∞ and x_ B = Q^2 / 2 P_N · q fixed, where P_N   (q) is the four-momentum of the nucleon (virtual photon) and Q^2 = - q^2 > 0, the relevant contribution of the box diagram is fully captured by the triangle diagram in Fig. <ref>(b) <cit.>.
        It is this triangle diagram from which the axial anomaly in QCD can be computed <cit.>. 
        Considering the full DIS process through O(α_ s), AR <cit.> and CCM <cit.> obtained 
        
            ΔΣ = ΔΣ  - α_ s  N_f/2 π Δ G  ,
        
        in terms of the measured quark-spin contribution ΔΣ, the `intrinsic' quark-spin contribution ΔΣ, and the amount of the proton spin due to the gluon spin Δ G.
        Equation (<ref>) was considered a potential explanation of the discrepancy between the small experimental result for ΔΣ and the (much) larger ΔΣ obtained in quark models.
        This development caused some excitement, especially since the Δ G term in Eq. (<ref>) could be attributed to the axial anomaly.
        
        For several reasons, however, concerns came up early.
        First, the pre-factor of Δ G in Eq. (<ref>) depends on the infrared (IR) regulator used for the calculation of the box/triangle diagram <cit.>.
        Second, the connection of the α_ s term in Eq. (<ref>) with the anomaly in Eq. (<ref>) was questioned <cit.>.
        Third, the way how one factorizes the box diagram into a perturbative and a non-perturbative part can modify Eq. (<ref>) <cit.>.
        (This is related to the first point about the IR regulator.)
        In this paper, we mainly concentrate on the first two points.
        
        Recently, Tarasov and Venugopalan <cit.>, as well as Bhattacharya, Hatta and Vogelsang <cit.> took a fresh look at this topic.
        Among other things, they pursued the idea of using a nonzero momentum transfer as an IR regulator of the box diagram with massless quarks <cit.>.
        They argued that, in contrast to the forward kinematics of the DIS process, off-forward kinematics would allow one to (fully) capture the physics of the anomaly. 
        They furthermore argued that, in the forward limit, the anomaly would give rise to a pole term in perturbation theory, potentially endangering QCD factorization.
        (Note, however, that in the latest of these papers it was shown that the anomaly contribution can very well be compatible with factorization <cit.>.)
        The use of off-forward kinematics suggests that deeply virtual Compton scattering (DVCS) off the proton <cit.>, ℓ p →ℓ p γ, is well suited for studying the axial anomaly.
        This result was implied by Refs. <cit.> and elaborated on in great detail in Refs. <cit.>.
        Both the real and imaginary part of the box diagram for the process γ^∗ g →γ g contribute to the DVCS scattering amplitude at O(α_ s) <cit.>.
        Here, we revisit pertinent perturbative calculations, and we confirm that for off-forward kinematics there is (also) a clear connection with the axial anomaly. 
        On the other hand, we show by exploiting physical polarization vectors for the gluons that the axial anomaly does not generate a pole in such calculations when taking the forward limit.
        In fact, for a nonzero quark mass there is even an exact cancellation between the anomaly and quark-mass contributions in that limit, a result which is actually required by the conservation of angular momentum.
        
        In Sec. <ref>, we calculate, to the lowest non-trivial order in perturbative QCD, the helicity-dependent parton distribution function (PDF) defined through the non-local axial current using a gluon target. 
        Integrating this PDF over the momentum fraction x of the quark provides the matrix element of the local axial current in the forward limit, which gives the pre-factor of Δ G in Eq. (<ref>) <cit.>.
        We discuss the dependence of the result on the IR regulator and, in particular, confirm the nonzero CCM result for off-shell gluons <cit.>.
        In Sec. <ref>, we consider the matrix element of the local axial current for arbitrary momentum transfer, and we discuss its relation with the matrix element of the anomaly and the quark-mass term in Eq. (<ref>).
        We show that for on-shell gluons, the axial current vanishes for forward kinematics.
        (If the calculation is not gauge invariant the result can be nonzero, which applies to the classic AR work <cit.>.)
        This result can be understood as a consequence of the cancellation between the anomaly and quark-mass terms.
        In Sec. <ref>, we consider the helicity-dependent generalized parton distributions (GPDs) <cit.> that parameterize the matrix element of the non-local axial current for off-forward kinematics. 
        One of the GPDs is unambiguously related to the axial anomaly, as already emphasized in Refs. <cit.>.
        However, we argue that the contribution of that GPD vanishes in the forward limit when using the finite quark mass as IR regulator.
        We conclude in Sec. <ref>.
        
        
        
        
        
        
        § PARTON DISTRIBUTION FUNCTION
        
        
        
        We consider the light-cone operator of the axial quark current, evaluated between gluon states, and we define the corresponding PDF 
        (denoted by g_1(x)) according to[For a generic four-vector a = (a^0, a^1, a^2, a^3), we define the light-cone components via a^+ = 1/√(2)(a^0 + a^3), a^- = 1/√(2)(a^0 - a^3), and a⃗_⊥ = (a^1, a^2).]
            Φ_λλ'^[γ^+ γ_5](x) = ∫dz^-/4π  e^i k · z ⟨ g (p, λ')   |  q̅(- z2)  γ^+ γ_5   W(- z2, z2)   q(z2)   |   g (p, λ) ⟩|_z^+ = 0, z⃗_⊥ = 0⃗_⊥ 
            = - i/p^+ ε^+    p  g_1(x)  ,
        
        with x = k^+ / p^+.
        Here λ  (λ') indicates the polarization state of the incoming (outgoing) gluon, W(- z2, z2) represents the Wilson line rendering the bilocal quark operator gauge invariant, and ε^+    p = ε^+  μ ν ρ _μ _ν  p_ρ.[To keep the notation simple, we do not show the dependence of the PDF on the renormalization scale, and we write μ instead of μ (λ) (and likewise for the outgoing gluon).  
        Furthermore, in what follows we mostly consider one quark flavor only, and we denote the quark mass by m.]
        It is worth emphasizing that, due to the forward kinematics, ε^+    p is the only structure that appears when parameterizing the light-cone correlator Φ_λλ'^[γ^+ γ_5](x).
        Evaluating that structure for different polarization states of the gluons provides
        
            g_1(x) =1/2(Φ_++^[γ^+ γ_5](x) -
            Φ_–^[γ^+ γ_5](x) )  ,
        
        where +   (-) means positive (negative) helicity. 
        This implies, in particular, that the matrix element in Eq. (<ref>) is nonzero only if the helicity of the incoming and outgoing gluon is the same.
        A gluon helicity flip is forbidden due to conservation of angular momentum. 
        (Note that the two quark fields of the light-cone operator in Eq. (<ref>) have the same helicity.  Even if their helicity was different, a gluon helicity-flip is not allowed for forward kinematics since it requires a change of two units of angular momentum.)
        We will come back to this point below.
        
        We compute g_1 to lowest non-trivial order in perturbative QCD (see Fig. <ref>) by keeping a finite quark mass and space-like off-shellness p^2 < 0 for the gluon.
        We use dimensional regularization (DR) to deal with ultraviolet (UV) divergences.
        To define the matrix γ_5 in DR, we employ the Larin scheme <cit.>, along with the replacement γ^μγ_5 →1/2 (γ^μγ_5 - γ_5 γ^μ); see, for instance, Ref. <cit.>.
        For the positive-x region, we find
        
            g_1(x; m, p^2) = 
            α_ s/4 π [ ( 1/ε - lnm^2 - p^2 x (1 - x)/μ̅^2)   (2x - 1) + p^2 x (1 - x)/m^2 - p^2 x (1 - x)] +  O(ε)      0 ≤ x ≤ 1  ,
        
        where μ̅^2 = 4π e^- γ_Eμ^2,
        with μ being the DR scale and γ_E the Euler constant.
        (The corresponding result in quantum electrodynamics is obtained by α_ s / 2 →α_ em, where 1/2 is the color factor of the QCD diagram.)
        The result for the negative-x region can be found by substituting x → -x in Eq. (<ref>).
        The UV divergence of g_1 is reflected by the 1/ε pole.
        Both m and p^2 serve as IR regulators.
        Notice that in order to obtain a IR-finite result just one of these two regulators is needed.
        (However, when using nonzero p^2 only, a logarithmic endpoint singularity emerges.)
        
        The lowest moment of g_1, which provides the local axial current for forward kinematics, is UV-finite and given by
        
            ∫_-1^1 dx   g_1(x; m, p^2) = α_ s/2π [ - 1 + ∫_0^1 dx  2 m^2 (1 - x)/m^2 - p^2 x (1 - x)]
            = α_ s/2 π [ - 1 + 2/√(η  (η + 4)) ln√(η + 4) + √(η)/√(η + 4) - √(η)]  .
        
        The expression after the first equal sign in Eq. (<ref>) matches Eq. (12) of the CCM paper <cit.>.
        The moment of g_1 depends on m and p^2 only through the ratio η = -p^2 / m^2 > 0.
        It is IR-finite but does depend on the numerical values of the IR regulators.
        For the limits η→ 0 (corresponding to p^2 → 0 and finite m, or finite p^2 and m →∞) and η→∞ (corresponding to finite p^2 and m → 0, or p^2 →∞ and finite m), we find
        
            ∫_-1^1 dx   g_1(x; m, p^2)    = α_ s/2 π [ - η/6 +  O(η^2 )  ]  η →  0→  0  ,         
            ∫_-1^1 dx   g_1(x; m, p^2)    = α_ s/2 π [ - 1 + 2/ηlnη +  O(1/η^2)  ]  η → ∞→  - α_ s/2 π ,
        
        respectively.
        Multiplying the result in Eq. (<ref>) by the number of quark flavors N_f provides the pre-factor of Δ G in Eq. (<ref>); see also Ref. <cit.>.
        We repeat that the dependence of the lowest moment of g_1 on the IR regulator gave rise to extensive discussions <cit.>, putting into question Eq. (<ref>) as a viable explanation of the proton spin crisis.
        Here we do not elaborate further on this point.
        In the next section, however, we make explicit the connection between the result in Eq. (<ref>) and the operators on the r.h.s. of Eq. (<ref>).
        
        
        
        
        
        
        § LOCAL AXIAL CURRENT
        
        We proceed to the discussion of the matrix element of the local axial current J_5^μ(x) for nonzero momentum transfer.
        We first consider the matrix element of the divergence of the current,
        
            ⟨ g (p', λ')   |  ∂_μ J_5^μ (0)  |   g (p, λ) ⟩
            = -2  ε^  P  Δ D(Δ^2) = - 2  ε^  P  Δ( D_a(Δ^2) + D_m(Δ^2) )  ,
        
        with P = (p + p')/2 and Δ = p' - p.
        The quantity D_a   (D_m) is the contribution of the anomaly (mass) term in Eq. (<ref>).
        (The factor -2 on the r.h.s. of Eq. (<ref>) has been introduced for later convenience.)
        Because of the structure ε^  P  Δ in Eq. (<ref>), off-forward kinematics is required to obtain a nonzero result.
        However, this does not imply that the moment of g_1 in Eq. (<ref>) is unrelated to the anomaly (nor to the quark mass term), as we will discuss below in this section.
        
        We evaluate the matrix element in Eq. (<ref>) for two cases: (i) arbitrary Δ^2, nonzero quark mass, on-shell gluons; (ii) zero Δ^2, nonzero quark mass, off-shell gluons (p^2 = p'^2 < 0).
        The anomaly contribution follows from a tree-level calculation, while the quark-mass term is obtained by evaluating the triangle diagram for the pseudo-scalar current.
        (This triangle diagram is UV finite.  Its evaluation is simpler than the direct calculation of the axial current where a UV regulator is needed at intermediate steps.)
        
        For the case (i), we find
        
            D_a(Δ^2; m, 0) = - α_ s/2 π ,
                
            D_m(Δ^2; m, 0) = α_ s/2 π 1/τ ln^2 √(τ + 4) + √(τ)/√(τ + 4) - √(τ) ,
        
        with τ = - Δ^2 / m^2.
        Note that the result for D_a does not depend on the momentum transfer or the IR regulator.
        In the limits τ→ 0 and τ→∞, we have
        
            D(Δ^2; m, 0)    = α_ s/2 π [ - τ/12 +  O(τ^2 )  ]  τ →  0→  0  ,         
            
            D(Δ^2; m, 0)    = α_ s/2 π [ - 1 + 1/τln^2 τ +  O(1/τ^2)  ]  τ → ∞→  - α_ s/2 π ,
        
        respectively.
        It is very interesting that for τ→ 0, there is an exact cancellation between D_a and D_m, a result that has been known for quite some time; see, e.g., Ref. <cit.>.
        (Generally, if the quark mass is much larger than any other scale, the matrix element in Eq. (<ref>) vanishes.)
        We emphasize that this cancellation exists here for any finite value of the quark mass.
        On the other hand, for τ→∞ (corresponding to Δ^2 finite and m → 0) the divergence of the axial current is exclusively determined by the anomaly, as expected.
        Given the qualitative difference between Eq. (<ref>) and Eq. (<ref>), and the fact that in nature quarks have a finite mass, it is prudent to keep the quark-mass term in Eq. (<ref>).
        In the next section on GPDs, we will add further discussion related to this point.
        The function (2 π / α_ s)  D(Δ^2; m, 0) is displayed in Fig. <ref>.
        It deviates from the value -1 due to the contribution of the mass term, which is non-negligible over a significant range of τ.
        
        
        
        For the case (ii), we obtain (see also Ref. <cit.>)
        
            D_a(0; m, p^2) = - α_ s/2 π ,
                
            D_m(0; m, p^2) = α_ s/2 π 2/√(η  (η + 4)) ln√(η + 4) + √(η)/√(η + 4) - √(η) .
        
        These expressions for D_a and D_m agree exactly with the two terms of the result in Eq. (<ref>), strongly suggesting that—despite the forward kinematics—the nonzero result of CCM for the lowest moment of g_1 <cit.> indeed has a robust connection with the anomaly and with the quark-mass term in Eq. (<ref>). 
        Below, we will make this connection explicit by considering the local axial current (rather than its divergence).
        Before that, the η→ 0 and η→∞ limits of D(0; m, p^2) are given by the corresponding expressions for ∫ dx   g_1 in Eqs. (<ref>), (<ref>).
        In particular, according to Eq. (<ref>), for η→ 0 there is again an exact cancellation between the anomaly and the quark-mass terms in Eq. (<ref>).
        
        Moving on to the matrix element of the local axial current itself, we first elaborate on its general structure (see also, e.g., Refs <cit.> and references therein),
        
            Γ_5^μ = ⟨ g(p', λ')   |   J_5^μ(0)    |   g(p, λ) ⟩ 
            = ∑_i = 1^3    G_i(Δ^2)   A_i^μ ,
        
        where the vectors A_i^μ, which are multiplied by the form factors G_i, are defined as
        
            A_1^μ = - 2i  ε^μ   P ,     
            A_2^μ = 2i/Δ^2 Δ^μ ε^  P  Δ ,     
            A_3^μ = 4i/Δ^2 ( · P  ε^μ   P  Δ + · P  ε^μ   P  Δ)
             .
        
        (For simplicity, we do not display the dependence of Γ_5^μ and the A_i^μ on λ and λ'.)
        The A_i^μ are symmetric under the exchange ↔, p ↔ - p'.
        In principle, in Eq. (<ref>) one could also include the vector
        
            A_4^μ = 2i/Δ^2 ( ·Δ ε^μ  P  Δ - ·Δ ε^μ   P  Δ)  ,
        
        but A_4^μ is not independent due to the relation
        
            A_4^μ = - A_1^μ + A_2^μ ,
        
        which follows from the Schouten identity
        
            g^αβ ε^μνρσ
            = g^μβ ε^ανρσ
            + g^νβ ε^μαρσ
            + g^ρβ ε^μνασ
            + g^σβ ε^μνρα .
        
        In order to proceed, we consider the Ward identity (gauge invariance) for the vector current to which the two gluons couple.
        Defining Γ_5^ μ = Γ_5^μ (→ p), the Ward identity related to the initial-state gluon reads Γ_5^ μ = 0. 
        This implies
        
            G_1 (Δ^2) = ( 1 - 4 p^2/Δ^2)   G_3 (Δ^2)  ,
        
        allowing us to write the general form of the axial current as
        
            Γ_5^μ = G_1(Δ^2)  ( A_1^μ + Δ^2/Δ^2 - 4 p^2  A_3^μ) + G_2(Δ^2)   A_2^μ .
        
        (The Ward identity related to the final-state gluon does not provide any additional constraint.)
        At this point we use the Lorenz condition · p = · p' = 0, for which A_3^μ = A_4^μ.
        We can therefore eliminate A_3^μ utilizing Eq. (<ref>), leading to
        
            Γ_5^μ|_ real   = ( G_1(Δ^2; m, 0) + G_2(Δ^2; m, 0) )   A_2^μ = G(Δ^2; m, 0)   A_2^μ , 1/1
            Γ_5^μ|_ virtual   = - 4 p^2/Δ^2 - 4 p^2  G_1(Δ^2; m, p^2)   A_1^μ 
            +  ( G_2(Δ^2; m, p^2) + Δ^2/Δ^2 - 4 p^2  G_1(Δ^2; m, p^2) )   A_2^μ .
        
        This means that for on-shell (real) gluons, there is just one independent form factor, which multiplies the vector A_2^μ and which we denote by G = G_1 + G_2.
        On the other hand, for off-shell (virtual) gluons the parameterization of the matrix element of the local axial current contains two form factors.
        
        Finally, we consider the (anomalous) axial Ward identity, which provides the connection with the matrix element of the divergence of the axial current in Eq. (<ref>),
        
            i Δ_μ Γ_5^μ = ⟨ g (p', λ')   |  ∂_μ J_5^μ (0)  |   g (p, λ) ⟩ .
        
        For on-shell gluons, Eq. (<ref>) readily implies 
        
            G(Δ^2; m, 0) = D(Δ^2; m, 0)  ,
        
        with D(Δ^2; m, 0) given by the sum of the anomaly term and quark-mass term in Eq. (<ref>). 
        Therefore, the local axial current for on-shell gluons is completely fixed by the divergence of the current.
        We repeat that computing D(Δ^2; m, 0) is simpler than G(Δ^2; m, 0) as defined through Eq. (<ref>).
        However, we have performed both calculations and verified that our results satisfy Eq. (<ref>).
        Using Eq. (<ref>) for off-shell gluons and focusing on the limit Δ^2 → 0 leads to
        
            G_1(0; m, p^2) + G_2(0; m, p^2) = G_1(0; m, p^2)= D(0; m, p^2)  ,
        
        with D(0; m, p^2) given by the sum of the two terms in Eq. (<ref>). 
        We note that by explicit calculation we find G_2(0; m, p^2) = 0, giving rise to the first equal sign in Eq. (<ref>).
        Although for off-shell gluons and arbitrary Δ^2 only one linear combination of the two form factors is related to the divergence of the current, the axial Ward identity fully fixes the matrix element of the current in the forward limit.
        Equation (<ref>) allows us to write
        
            lim_Δ^2  →  0Γ_5^μ|_ virtual   = G_1(0; m, p^2)   A_1^μ .
        
        Comparing Eq. (<ref>) for μ=+ with Eq. (<ref>) integrated over x provides the relation
        
            ∫_-1^1 dx   g_1(x; m, p^2) = G_1(0; m, p^2)  .
        
        We have now achieved three things in connection with the classic result of the CCM paper <cit.> (see Eq. (<ref>)).
        First, we showed that this result is unambiguously related to the matrix element of the operators on the r.h.s. of Eq. (<ref>). 
        Second, by using Eqs. (<ref>) and (<ref>), we
        decomposed the result into the contributions from the anomaly and from the classical quark-mass source term, finding that the vanishing result in Eq. (<ref>) is caused by an exact cancellation of these two contributions.
        Third, with the help of Ward identities we derived
        Eqs. (<ref>) and (<ref>),
        allowing Eq. (<ref>) to be found through a simpler, UV-finite calculation
        (thus also avoiding the need to choose a γ_5 scheme).
        
        We have derived and explained the result in Eq. (<ref>) by starting from the general form of the axial current for off-shell gluons in Eq. (<ref>), then taking the limit Δ^2 → 0, followed by the limit p^2 → 0.
        However, it is instructive to repeat the analysis by first taking p^2 → 0 and afterwards Δ^2 → 0.
        This means we will start from Eq. (<ref>).
        We will also evaluate A_1^μ and A_2^μ using physical polarization vectors of the gluons.
        For this analysis, we choose the symmetric reference frame in which
        
            P = ( P^+,  Δ⃗_⊥^  2/8 (1 - ξ^2) P^+,  0⃗_⊥)  ,     Δ = ( -2 ξ P^+,  ξΔ⃗_⊥^  2/4 (1 - ξ^2) P^+,  Δ⃗_⊥)  ,
        
        but we emphasize that our general conclusions do not depend on this choice.
        The so-called skewness variable ξ defines the longitudinal momentum transfer to the gluons.
        In addition to the frame-independent relations P ·Δ = 0 and
        P^2 = - Δ^2 / 4, we have Δ^2 = - Δ⃗_⊥^  2 / (1 - ξ^2).
        We employ the polarization vectors specified in Ref. <cit.> using the light-cone gauge.
        In our notation, for the (on-shell) initial-state gluon they are given by
        
            _(1)^μ = - 1/√(N) ( 2 ξ P^μ + Δ^μ + Δ^2  n^μ/2 P · n)  ,     _(2)^μ = + 1/√(N) ε^μ  n   P  Δ/P · n ,
        
        where N = Δ⃗_⊥^ 2, and n is the light-cone four-vector for which n · a = a^+ given a generic four-vector a.
        The polarization vectors in Eq. (<ref>) satisfy
        _(i)·_(j) = - δ_ij for i,j∈{1,2},
        and
        _(i)· p = 0.
        The linear combinations _(±)= ∓ ( _(1)± i  _(2)) / √(2) describe states of definite (light-cone) helicity.
        The polarization vectors for the final-state gluon are obtained from Eq. (<ref>) through the replacement Δ→ - Δ (which implies ξ→ - ξ) and an overall sign change for _(1)^μ and _(2)^μ <cit.>.
        With the notation A_1   (ij)^μ = -2 i  ε^μ _(i) _(j) P etc. we find
        
            A_1   (11)^μ   = - 2i/(1 - ξ^2) P · n ε^μ  n   P  Δ ,      
            A_1   (12)^μ   = i/1 + ξ ( 2 P^μ - Δ^μ + Δ^2  n^μ/2 P · n)  , 
            
            
            A_1   (21)^μ   = - i/1 - ξ ( 2 P^μ + Δ^μ + Δ^2  n^μ/2 P · n)  ,     
            A_1   (22)^μ   = 0  ,
            
            
            A_2   (11)^μ   = 0  ,      
            A_2   (12)^μ   = - i  Δ^μ , 
            
            
            A_2   (21)^μ   = - i  Δ^μ ,     
            A_2   (22)^μ   = 0  .
         
        For the specific case μ = +, the only nonzero expressions are
        
            A_1   (12)^+ = - A_1   (21)^+ = 2 i   P^+  ,     
            A_2   (12)^+ = A_2   (21)^+ = 2 i  ξ P^+  .
        
        Using helicity states, we can write
        
            A_1   (12)^+ - A_1   (21)^+ = i (
            A_1   (++)^+ - A_1   (–)^+ )  ,     
            A_2   (12)^+ + A_2   (21)^+ = i(
            A_2   (+-)^+ - A_2   (-+)^+ )  .
        
        Since the structure in front of g_1 on the r.h.s. of Eq. (<ref>) is nothing but A_1^+ / 2p^+, the result in Eq. (<ref>) is equivalent to the first relation in (<ref>).
        While Eq. (<ref>) means that, for forward kinematics, the axial current can only generate a helicity-conserving transition, in Eq. (<ref>)A_2^μ appears which, according to the second relation in (<ref>), implies a helicity flip.
        Upon a closer inspection, we see that there is no contradiction though.
        Since the PDF g_1 is defined for Δ = 0 we first consider Γ_5^+ for this specific kinematical point. 
        The second relation in (<ref>) then leads to
        
            Γ_5^+ (ξ = 0, Δ⃗_⊥) |_ real =
            Γ_5^+ (ξ = 0, Δ⃗_⊥ = 0⃗_⊥) |_ real = 0  .
        
        (See also Ref. <cit.> for a related discussion.)
        We emphasize that this result, which is fully compatible with the conservation of angular momentum, is not based on the calculation of Feynman diagrams.
        Equation (<ref>) can be compatible with Eq. (<ref>) only if 
        
            ∫_-1^1 dx   g_1(x) |_ real = 0  ,
        
        which must hold for any IR regulator.
        Taking as an example a nonzero quark mass as IR regulator and the result in Eq. (<ref>), we see agreement with Eq. (<ref>).
        Put differently, the steps leading to Eq. (<ref>) can be considered an alternative derivation of the result in Eq. (<ref>), which was first obtained by CCM <cit.>.
        
        Overall, we find consistent results regardless of the order that the
        Δ^2 → 0 and p^2 → 0 limits are taken.
        However, both ways of taking these limits for the local axial current provide unique insights.
        We also emphasize that Eq. (<ref>) does not imply that there is no contribution from the anomaly;
        based on the discussion above, it rather means that for on-shell gluons and forward kinematics, the anomaly contribution is cancelled by the quark-mass term.
        Alternatively, one could say that the anomaly exactly cancels the (classical) quark-mass term—and
        that this cancellation is needed for consistency with 
        Eq. (<ref>), which is a consequence of angular momentum conservation.
        
        Before moving on, we point out that the conservation of angular momentum implies the (stronger) constraint Γ_5^+ (ξ, Δ⃗_⊥ = 0⃗_⊥) |_ real = 0.
        Since A_2^+ can be nonzero for ξ≠ 0, this constraint requires that the form factor G(0) vanishes, which it does for m ≠ 0; see Eqs. (<ref>) and (<ref>).
        Keeping the quark mass is therefore necessary to ensure the conservation of angular momentum.
        In the next section, we will see the same result in the context of the GPDs.
        
        Now we briefly comment on the work by AR <cit.>, in which the box diagram for polarized DIS was analyzed.
        The authors considered exact forward kinematics and on-shell gluons.
        However, they neglected certain quark-mass terms in the numerator, implying that their result is not gauge invariant, as was already pointed out in Ref. <cit.>. 
        Deriving the general structure of the axial current for on-shell gluons without using the gauge-invariance constraint leads to 
        
            Γ_5^μ  (ngi)|_ real
            = G_1^ (ngi)  A_1^μ + G_2^ (ngi)  A_2^μ ,
        
        that is, there are two independent form factors instead of just one as in Eq. (<ref>).
        When taking the forward limit of  Eq. (<ref>) we can get a nonzero result related to G_1^ (ngi), while G_2^ (ngi) drops out since A_2^μ vanishes.
        This finding is consistent with the nonzero result obtained by AR <cit.>.
        (In a closely related study we computed ∫ dx   g_1 by neglecting all quark-mass terms in the numerator, finding a nonzero result as well.)
        Since Eq. (<ref>) is in conflict with Eq. (<ref>), we consider the lack of gauge invariance a deficiency of the AR paper <cit.>,
        although this does not put into question the importance of this pioneering work.
        
        The last point we want to address in this section is the behavior of the vector A_2^μ when taking the forward limit.
        (Our discussion here also applies to A_3^μ and A_4^μ.)
        Because of Eqs. (<ref>) and (<ref>), this vector is associated with the axial anomaly (and the quark-mass term) appearing in the divergence of the axial current. 
        The factor 1/Δ^2 in the definition of A_2^μ gave rise to extensive discussions about an `anomaly pole' that would emerge for Δ^2 → 0; see, in particular, Refs. <cit.>.
        However, when evaluated for physical polarization vectors, according to (<ref>) one finds either zero or a finite result, depending on the polarization state of the gluons.
        
        
        
        
        
        
        § GENERALIZED PARTON DISTRIBUTIONS
        
        In this section we return to the light-cone operator of the non-local axial current, now evaluated between gluon states with different momenta.
        Here we exclusively consider the case of on-shell gluons.
        Using the Schouten identity in Eq. (<ref>), the Ward identity for both gluons, as well as · p = · p' = 0, we find
        
            F_λλ'^[γ^+ γ_5](x,Δ)    = 
            ∫dz^-/4π  e^i k · z ⟨ g (p', λ')   |  q̅(- z2)  γ^+ γ_5   W(- z2, z2)   q(z2)   |   g (p, λ) ⟩|_z^+ = 0, z⃗_⊥ = 0⃗_⊥
               = ( B_1 - B_2 + ξ B_3 + B_4 )   H_1(x, ξ, Δ^2) + B_2   H_2(x, ξ, Δ^2)  ,
        
        with the two GPDs H_1 and H_2.
        Furthermore, we have B_i = A_i^+ / (2P^+) for i = 1,2, and 
        
            B_3 = -2i/Δ^2   P^+ ( · P  ε^+    P  Δ - · P  ε^+    P  Δ) 
             ,     
            B_4 = -i/2(P^+)^2 ( + ε^+    P  Δ + + ε^+    P  Δ)  .
        
        To arrive at the decomposition in Eq. (<ref>) we followed to some extent the classification of deuteron GPDs presented in Ref. <cit.>.
        We agree with Refs. <cit.> that the matrix element in Eq. (<ref>) defines (just) two GPDs.
        The only difference compared to those papers is that the structure in front of H_1 in Eq. (<ref>) is gauge invariant.
        In analogy with Eq. (<ref>), we can address the GPDs through specific helicity combinations,
        
            H_1(x, ξ, Δ^2) =1/2(1-ξ^2)(F_++^[γ^+ γ_5](x, Δ) - F_–^[γ^+ γ_5](x, Δ) ) 
             ,     
            H_2(x, ξ, Δ^2) = - 1/2ξ(F_+-^[γ^+ γ_5](x, Δ) - F_-+^[γ^+ γ_5](x, Δ) )  .
        
        While H_1 (like g_1) is associated with helicity-conserving transitions, a helicity flip is needed in the case of H_2.
        However, for Δ⃗_⊥ = 0⃗_⊥, a gluon helicity flip is forbidden due to the conservation of angular momentum (see also the discussion after Eq. (<ref>)), and therefore the B_2   H_2 term in Eq. (<ref>) must drop out for that kinematics.
        Since B_2 (ξ≠ 0,Δ⃗_⊥=0⃗_⊥) ≠ 0, the GPD H_2 should vanish for a vanishing transverse momentum transfer.
        
        Before discussing our results for the GPDs, we list a few additional constraints the GPDs must satisfy.
        By definition, in the forward limit the GPD correlator in Eq. (<ref>) reduces to the one for the PDF in Eq. (<ref>), leading to
        
            H_1(x, 0, 0) = g_1(x)  .
        
        Equation (<ref>) assumes that the forward limit of the GPD correlator exists, which actually is not the case if a nonzero momentum transfer is taken as the sole IR regulator.
        For our calculation, this situation arises if one uses m = 0, as we discuss in a bit more detail below.
        Further constraints follow when integrating the GPD correlator in Eq. (<ref>) over x, which provides the local current. 
        Comparing the result with Eq. (<ref>) we find
        
            ∫_-1^1 dx   H_1(x, ξ, Δ^2) = 0 
             ,     ∫_-1^1 dx   H_2(x, ξ, Δ^2) = G(Δ^2)  .
        
        The relation for H_1 can be considered a generalization of Eq. (<ref>).
        Furthermore, using Eq. (<ref>) we see that the GPD H_2 is clearly related with the axial anomaly, confirming the corresponding statement in Refs. <cit.>.
        
        To find the two GPDs to lowest non-trivial order in perturbative QCD, we evaluate the same two Feynman diagrams that contribute to the PDF g_1, but now for off-forward kinematics.
        The diagram in Fig. <ref> contributes to the positive DGLAP region (ξ≤ x ≤ 1) and the ERBL region (-ξ≤ x ≤ξ), 
        whereas the second diagram contributes to the negative DGLAP region and the ERBL region.
        For convenience, we introduce the variable κ = τ  (1 - x)^2 / (1 - ξ^2).
        Our calculation provides 
        
            H_1(x, ξ, Δ^2; m)     = α_ s/4 π2x -1 - ξ^2/1 - ξ^2 [ 1/ε - lnm^2/μ̅^2] - 1 
            
            
            + 4 + (1 + ξ^2) κ - 2 x (κ + 2)/1 - ξ^21/√(κ (κ + 4))ln√(κ + 4) + √(κ)/√(κ + 4) - √(κ)   -ξ≤ x ≤ 1  ,  
            
            
            - (1 - ξ)(ξ +x)/2 ξ (1 + ξ) [ 1/ε - lnm^2/μ̅^2] - ξ + x/2 ξ
            
            - ξ^2 (2 - x) - x/2 ξ (1 - ξ^2)ln[ 1 + (1 - ξ) (ξ + x) (ξ^2 + ξ (1 - x) - x) κ/4 ξ^2 (1 - x)^2]
            
            
            + 4 + (1 + ξ^2) κ -2x (κ + 2)/2 (1 - ξ^2) 1/√(κ(κ + 4))lnh_+/h_-  +   (x → -x)
                -ξ≤ x ≤ξ ,
            
            
            H_2(x, ξ, Δ^2; m)     = α_ s/4 π2(1 - x)/1 - ξ^2 [ - 1 + 2/√(κ(κ + 4))ln√(κ + 4) + √(κ)/√(κ + 4) - √(κ) ] 
               -ξ≤ x ≤ 1  ,  
            
            2/1 + ξ [ - ξ + x/2 ξ + 1 - x/1 - ξ1/√(κ(κ + 4))lnh_+/h_-]
              +   (x → -x)
                -ξ≤ x ≤ξ .
        
        with the auxiliary functions
        
            h_± = 4 ξ (1-x) ± (1-ξ) (ξ + x)  √(κ) ( √(κ + 4)±√(κ) )  .
        
        Results for -1 ≤ x ≤ -ξ can be obtained from the formulas for the positive DGLAP region by substituting x → -x.
        The results for H_1 and H_2 are continuous at x = ±ξ (as they should be) and satisfy the constraints in Eqs. (<ref>) and (<ref>).
        (We verified H_1(x, 0, 0; m) = g_1(x; m, 0), since for our perturbative calculation Eq. (<ref>) is meaningful only for m ≠ 0.)
        Also note that only H_1 is UV-divergent.
        
        As a next step, we expand the results in Eqs. (<ref>) and (<ref>) for τ→ 0 (corresponding to Δ⃗_⊥→0⃗_⊥ and m finite) and for τ→∞ (corresponding to Δ⃗_⊥ finite and m → 0).
        In both cases we keep the full dependence on ξ.
        For τ→ 0, the Taylor expansions of the GPDs read
        
            H_1(x, ξ, Δ^2; m)     = α_ s/4 π2x - 1 - ξ^2/1 - ξ^2[ 1/ε - lnm^2/μ̅^2 -1 ] +  O( τ) 
               -ξ≤ x ≤ 1  ,  
            
            
            - 1 - ξ/1 + ξ[ 1/ε - lnm^2/μ̅^2 -1 ] +  O( τ)
                -ξ≤ x ≤ξ ,
            
            
            H_2(x, ξ, Δ^2; m)     = α_ s/4 π
            - (1 - x)^3/3 (1 - ξ^2)^2 τ +  O( τ^2) 
             τ →  0→  0 
               -ξ≤ x ≤ 1  ,  
            
            
            - (ξ + x)^2   (ξ^2 + 2 ξ (1 - x) -x)/12 ξ^3 (1 + ξ)^2 τ + (x → - x) +  O( τ^2)
             τ →  0→  0
                -ξ≤ x ≤ξ .
        
        We highlight that H_2 does vanish for τ→ 0, and we repeat that this result must hold, for any x and ξ, due to the conservation of angular momentum; see the discussion after Eq. (<ref>). 
        (Based on the relation for H_2 in Eq. (<ref>) and our result for the form factor G(Δ^2; m, 0), it was clear before explicitly computing H_2 that the integral of this GPD must vanish for τ→ 0.
        However, this does not imply that H_2 has to vanish for arbitrary x and ξ.)
        We proceed to the limit τ→∞ for which we find
        
            H_1(x, ξ, Δ^2; m)     = α_ s/4 π2x - 1 - ξ^2/1 - ξ^2 [ 1/ε - ln( - Δ^2/μ̅^2) - ln(1 - x)^2/1 - ξ^2] - 1 +  O( 1/τ) 
               -ξ≤ x ≤ 1  ,  
            
            
            - 1 - ξ/1 + ξ [ 1/ε - ln( - Δ^2/μ̅^2) ] 
            - 1/1 - ξ^2 [ 2 ξln (ξ^2 - x^2)      
             + (1 + ξ^2) ln(1 + ξ)^2/1 - x^2 + 2 x ln(1 - x) (x + ξ)/(1 + x) (ξ - x) - 4 ξln (2 ξ) ] - 1 +  O( 1/τ)
               -ξ≤ x ≤ξ ,
            
            
            H_2(x, ξ, Δ^2; m)     = α_ s/4 π
            - 2 (1 - x)/1 - ξ^2  +  O( lnτ/τ) 
             τ → ∞→  
            - 2 (1 - x)/1 - ξ^2   -ξ≤ x ≤ 1  ,  
            
            
            - 2/1 + ξ  +  O( lnτ/τ)
             τ → ∞→  
            - 2/1 + ξ   -ξ≤ x ≤ξ .
        
        These expanded results fully agree with those presented in Ref. <cit.>, where the quark mass was neglected right form the start of the calculation.
        Since m = 0 for the leading terms in Eq. (<ref>), a (logarithmic) singularity arises in H_1 for Δ⃗_⊥→0⃗_⊥.
        This is just a manifestation of the fact that a nonzero transverse momentum transfer acts as the IR regulator.
        The expressions for H_2 in Eq. (<ref>) for both the DGLAP and ERBL regions do not diverge for Δ⃗_⊥→0⃗_⊥.
        However, as we already pointed out above, using those results for Δ⃗_⊥ = 0⃗_⊥ would contradict the conservation of angular momentum.
        Using m = 0 means that H_2 is just given by the anomaly.
        (The x-integral of H_2 in Eq. (<ref>) is nothing but D_a in Eq. (<ref>).)
        But we have already seen above that the combination of the anomaly and the quark-mass term is needed to get consistent results for the forward limit of the local current.
        It is therefore not too surprising that an issue arises for the non-local axial current when neglecting the quark mass.
        
        Overall, we believe the most important result of this section was that—in perturbation theory—the (anomaly-related) helicity-flip contribution to the off-forward matrix element of the non-local axial current vanishes in the limit Δ⃗_⊥→0⃗_⊥ when taking the quark mass into account.
        We repeat that this result is required by the conservation of angular momentum. 
        This result is at odds with statements in the literature suggesting an anomaly-related pole in perturbation theory <cit.>.
        Finally, we argue that Eq. (<ref>) may be considered the non-local generalization of Eq. (<ref>), showing that the contribution due to the quark-mass term cancels the contribution from the axial anomaly (or vice versa).
        
        
        
        
        
        § CONCLUSIONS
        
        In this work, we presented several perturbative-QCD results related to the local and non-local axial current evaluated for gluon states.
        We confirmed the classic CCM result for the local current for forward kinematics for both on-shell and off-shell gluons <cit.>, found a simpler way of obtaining this result, and made clear the connection of this result with the matrix elements of the axial anomaly and the quark-mass term that appear in the divergence of the axial current.
        We emphasize that the axial anomaly does contribute to the axial current for forward kinematics for on-shell and off-shell gluons. 
        
        We agree with Refs. <cit.> that the anomaly does play an important role in off-forward kinematics as well.
        However, in the case of on-shell gluons, the anomaly contribution is associated with a gluon helicity flip which, for a vanishing transverse momentum transfer to the gluons, is forbidden by the conservation of angular momentum.
        Using physical polarization vectors for the gluons and a nonzero quark mass as the IR regulator, we found an exact cancellation between the anomaly term and the quark-mass term in this kinematic limit.
        We conclude that perturbative calculations related the axial current are guaranteed to be physically meaningful only if the quark mass is not neglected; see also, e.g., Ref. <cit.>.
        In this context, one should keep in mind that in nature quarks have finite masses. 
        
        Recent works had argued that—in perturbative calculations—the axial anomaly would give rise to a pole when approaching the forward limit <cit.>.
        It was also suggested that for a full physical process involving the proton, a non-perturbative mechanism related to the generation of the mass of the η' meson <cit.> could cancel that pole <cit.>.
        While we did not address a full physical process, we repeat that in perturbation theory the axial anomaly does not generate a pole.
        Regarding that discussion, we point out that the argument about the conservation of angular momentum can also directly be applied to the full box diagram in Fig. <ref>(a) for both the DIS and the DVCS processes.
        It is known that in DVCS there can be a gluon helicity flip for forward kinematics, but only if there is a corresponding helicity flip for the photons <cit.>.
        If the photon helicity is conserved, a gluon helicity flip for the box is forbidden for forward kinematics. 
        We therefore expect that a full calculation of the box diagram for this case—using a nonzero quark mass—will lead to the same vanishing result in the forward limit that we found in our GPD analysis.
        (In this context, see also Ref. <cit.> and references therein.)
        
        
        
        
        
        
        § ACKNOWLEDGEMENT
        
        We have used the FeynCalc <cit.> and JaxoDraw <cit.> software.
        We gratefully acknowledge a discussion with Markus Diehl.
        The work of I.C. was supported by the National Science Foundation under the Grant No. PHY-2110472, and by a Summer Research Grant from Temple University.
        The work of A.F. was supported by the U.S. Department of Energy contract No. DE-AC05-06OR23177, under which Jefferson Science Associates, LLC operates Jefferson Lab.
        The work of A.M. was supported by the National Science Foundation under the Grant No. PHY-2110472, and by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under the Quark-Gluon Tomography (QGT) Topical Collaboration with Award DE-SC0023646.
        The work of S.R. was supported by the German Science Foundation (DFG), grant number 409651613 (Research Unit FOR 2926), subproject 430915355.


 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: The Holographic QCD Running Coupling Constant from the Ricci Flow.


Einstein metrics on aligned homogeneous spaces with two factors
    Cynthia Will
    August 5, 2024
===============================================================



§ ABSTRACT 
Through a holographic model of QCD, we present a phenomenological approach to study the running of the strong coupling constant α_s in both non-perturbative and perturbative regimes. The renormalization of the metric tensor, driven by the Ricci Flow, and the breaking of conformal and chiral symmetries -thanks to introducing a double dilaton model and large-N_c corrections- allow us to relate the existence of an infrared fixed point in the coupling constant with a smooth matching to pQCD well above 2 GeV. The proposed dilaton model yields linear Regge trajectories and vector decay constants similar to their experimental counterparts.




§ INTRODUCTION


AdS/CFT conjecture has been used extensively in the search for a possible holographic model dual to QCD in the non-perturbative regime. First introduced as an equivalence between strongly coupled 𝒩=4 super Yang-Mills theory and supergravity <cit.>, it was shown later it was possible to break conformal symmetry to obtain approximate large-N_c phenomenological models of QCD at low energies. In particular Hard <cit.> and Soft Wall <cit.> models provide a way to compute masses, decay constants, and Regge trajectories of mesons of arbitrary spin. In both cases, their phenomenological scope is still limited <cit.>, yet some advances are promising <cit.>.

In this work, we are interested in obtaining the running coupling constant in the strongly coupled regime and extending it to the well-known perturbative one. We can get a QCD-like running from dual models like Einstein-dilaton gravity <cit.>, or by considering Light-Front QCD embedded in a AdS background <cit.>. In our case we will explore the QCD running coupling constant from a different procedure: we will relate it to the running of the metric tensor on the other side of the duality. Intuitively, an abrupt change of the running coupling constant at Λ_QCD could be related to an expansion of spacetime on the holographic side, which can be thought of as an exponential ansatz for the running of the metric tensor driven by the so-called Ricci Flow. The procedure above provides for a family of dilaton fields which beyond returning a running strong-coupling constant will, as a byproduct, return masses and decay constants of vector mesons.

This letter is organized as follows. In Section <ref>, we present some generalities of the Ricci Flow and its relation to the QCD running coupling constant. In Section <ref>, we propose a new holographic QCD model, and we explore its running coupling constant. A discussion of its prediction for masses and decay constants is presented in Section <ref>, including a comparison of previous results, and we finalize the letter with concluding remarks in Section <ref>.



§ RICCI FLOW


The objective is to establish a precise novel relation between the running of the metric tensor and the strong coupling constant α_s. In the process, we derive the well-known formula for the Ricci Flow, and we do it so that this derivation ultimately implies the relation as mentioned above between g_μν and QCD β-function.

Consider a generic AdS background of the form:

    ds^2=R^2/z^2(dz^2+η_μνdx^μdx^ν).

Here R is the AdS radius, which is related to 't Hooft's coupling λ by the formula:

    R^2/α'=√(λ),

where λ=g_YM^2N_c. We are assuming λ≫ 1, which is equivalent to supposing we are considering the large-N_c limit in the quantum field theory at the boundary of AdS. Our interest is in QCD so g_YM is the QCD strong coupling. In this way since α_s=g_YM^2/(4π), from Eq.(<ref>) we can write:

    α_s=R^4/4π N_c α'^2.

If the boundary theory has exact conformal symmetry then α_s is constant and the background remains a pure AdS spacetime. If we want to break conformal symmetry, we can consider a deviation from pure AdS imposing R=R(z). In this way, α_s will depend on the holographic coordinate. More precisely, if we return to Eq.(<ref>) and write it in the form

    ds^2=R^2/z^2dz^2+g_μνdx^μdx^ν,

then g_μν=(R^2/z^2)η_μν where R^2∼√(α_s), see Eq.(<ref>). We can now relate a running of α_s with a running of g_μν. In terms of μ= z^-1, the renormalization group equation for the metric tensor g_μν(μ)=μ^2R^2(μ)η_μν, with R^2(μ)= α'√(4π N_cα_s(μ^2))η_μν, will be:

    μ∂ g_μν/∂μ=( 2μ^2 R^2(μ)+2μ^3 R(μ)d R(μ)/dμ)η_μν.

Multiplying and dividing by R(μ) in the second term we can write:

    μ∂ g_μν/∂μ   =(2g_μν(μ)+2μ g_μν(μ)d/dμlog R(μ))
       =2g_μν(μ)(1+μ/4d/dμlog (4π N_cα'^2α_s(μ^2)))
       =2g_μν(μ)(1+μ/4d/dμlog (α_s(μ^2))).

In this way, we can identify the second term β_μν=g_μνμ^2/4d/dμlogα_s  as an homologous to a β-function of the metric g_μν, and relate naturally the running of α_s(μ^2) with the running of the metric tensor g_μν(μ). This second running is, at first order in α', β_μν=-2R_μν, where R_μν is the Ricci tensor of the metric g_μν <cit.>. This result is specific for the metric Eq.(<ref>), but motivates the introduction of the running of a general metric tensor as:

    ∂ g_μν/∂ t=2g_μν-2R_μν,

where t=logμ. This formula is known as the Ricci Flow. Introduced in the context of string theory <cit.>, and in mathematics <cit.>, Ricci Flow has been a method to compute the renormalization of non-linear sigma models via the above equation. Also, we can conceive the running of the metric tensor as the radial evolution equation in Hamilton-Jacobi formulation (see <cit.> for more details about this connection).

From the previous results, if α_s does not run at all, solving the Ricci Flow should return a metric tensor corresponding to AdS background: assume R_μν=0, then the solution for the flow equation

    ∂ g_μν/∂ t=2g_μν

is, in terms of the holographic coordinate z,

    g_μν(z)=1/z^2η_μν,

considering R=1. We obtain the expected AdS background:

    ds^2=1/z^2(dz^2+η_μνdx^μdx^ν).

For a non-trivial running of α_s, we have to break conformal symmetry in the geometric side of the duality by performing a deformation from the AdS metric. Consider the parametrization R_μν=Λ(t)g_μν for Λ(t) an unknown function. In this way we must solve the flow:

    ∂ g_μν/∂ t=2(1-Λ(t))g_μν.

As observed in the introduction, an abrupt change of α_s at low energies could be related to an expansion in AdS spacetime, cf. Eq.(<ref>), since growth in α_s implies an increase in AdS radius R. We assume an exponential varying function of the form Λ(t)=c_0 e^c_1 t, where c_0 has units of energy squared and c_1 is unitless. We softly break the conformal symmetry by allowing t dependence in Λ. The exponential parameterization will enable us to perform the desired abrupt change of α_s at low energies. Other models of conformal symmetry breaking are possible as, for example, we could model Λ(t) as a step function to represent Hard Wall models.

We stick to Eq.(<ref>) as it can be considered a convenient ansatz for solving the flow. Using the aforementioned exponential dependence, Eq.(<ref>), leads us to:

    g_μν(t)=η_μνexp(2 t -2c_0/c_1e^c_1 t+k),

where k is an integration constant. In conformal coordinates (t=-log z):

    g_μν(z)=η_μνexp(-2logz -2c_0/c_1z^-c_1+k).

We fix c_0=2/5  GeV^2 to reproduce the usual volume form, and c_1=-2 to obtain the correct z^-2 term. We obtain:

    g_μν(z)=e^2/5z^2+k/z^2η_μν.

The interpretation is clear: Λ=Λ(t) leads to the presence of a dilaton ϕ(z), the field in charge of conformal symmetry breaking in the holographic model <cit.> as deformation of the metric tensor. An exponential ansatz is enough to find a Soft Wall-like dilaton plus a constant. The integration constant k could be interpreted as an AdS radius R_0^2, giving a warp factor of the form R_0^2 e^-(2/5)ϕ(z)/z^2 if k=2logR_0, but for now, let us consider it part of the dilaton ϕ.



With the previous choice of c_0 and c_1, we obtain the following volume form:

    √(g)=e^-ϕ(z)/z^5.

The denominator gives the measure, and the numerator is the usual dilaton term in the action of holographic models <cit.>.

As a consistency check, we observe that if ϕ(z)=ϕ(t(z)) is the obtained dilaton, then dϕ(t)/dt=-2Λ(t). In this way, the function that induces a deviation from pure AdS spacetime, Λ(t), is related directly to the field that breaks conformal symmetry in the holographic model, ϕ(z).

As summary, in presence of a dilaton ϕ(z), we identify the AdS radius of Eq.(<ref>) as R^4(z)=exp(-(4/5)ϕ(z)). As a consequence, we have a z-dependent coupling constant:

    α_s(z)=1/4π N_c α'^2e^-4/5ϕ(z).

From this viewpoint, Hard Wall models are obtained after considering a step function for exp(-ϕ(z)). 

The above procedure allows us to define a general QCD running coupling constant for models with a dilaton. In other words, each holographic model will induce a particular running of α_s as, for example, the one obtained in Ref.<cit.> using an embedding of Light-Front QCD into a Soft Wall model. In the next section, we propose a dilaton model that will induce a QCD running coupling constant with an infrared fixed point and with a matching to pQCD. 



§ DOUBLE DILATON SOFT WALL MODEL AND ITS RUNNING


Our attempt is to explore what sort of dilaton field could return a running coupling constant resembling the one in QCD. A confining dilaton was used in <cit.>, which had the opposite sign in comparison with the original dilaton presented in <cit.>. We now introduce a model with two dilatons, one with a positive and one with a negative sign, all together in a single model. This choice of signs will be responsible for breaking the chiral symmetry straight away and will enable us to study properties of the running of α_s at the same time. Physically, this setup should correspond to a system with vector and axial-vector mesons. We see that this system naturally provides a running of the strong coupling constant with an infrared fixed point. A Soft Wall dilaton would have an ambiguity sign for this system: Regge trajectories are the same regardless of the sign of the dilaton. To fix the sign of the dilaton, one can relate to higher spin mesons as done in <cit.>. Alternatively, we consider that vector and axial-vector mesons see different geometries <cit.>.
In reference <cit.>, different warp factors for vector and axial-vector mesons were introduced in order to reproduce the well-known QCD OPE of the vector and axial-vector two-point correlation functions.
As such, each type of meson sees dilatons with opposite signs <cit.>.
In particular, Ref.<cit.> remarks on the importance of having opposite dilaton signs for axial-vector mesons. The dilaton sign choice impacts the IR part of the action. Consequently, it seems reasonable to think that sign choices for vector and axial-vector mesons have an impact both in the IR regime where the running of α_s is not yet known, and in the UV, where the connection of α_s with perturbative QCD running have not been fully accomplished. 
So we consider potentially different warped factors to describe dynamics. In this way, think of a background defined in AdS × AdS'. Ricci tensor factorizes in the product, so we can conceive each piece evolves with the equations:

    ∂ g_μν/∂ t=2g_μν-2R_μν, ∂ g'_αβ/∂ t'=2g'_αβ-2R'_αβ.

The evolution allows us to consider two different dilatons, allowing choosing a dilaton ϕ in AdS and its opposite -ϕ in AdS'. This setup enables us to work with a ten-dimensional model in AdS× AdS' with action:

    S=∫ d^5xd^5x'√(g_×)(-1/2g_5^2(F_L^2+F_R^2)).

x and x' are spacetime coordinates in AdS and AdS', respectively. g_× is the determinant of the metric tensor in the product AdS× AdS'. This is an action of a theory of gauge fields A_L and A_R dual to the QCD currents J_L and J_R in ten dimensions. Also, g_5 is a Yang-Mills coupling and F_L=F_L(x) and F_R=F_R(x') are field strength tensors defined in the usual way <cit.>.

By using the exponential ansatz for Λ(t) used in the previous section

    Λ(t)=2/5e^-2 t,

we can obtain dilatons ϕ(z)=-z^2-k and ϕ'(z)=+z'^2+k', being k and k' integration constants. From this point of view AdS and AdS' will have the following actions, respectively:

    S=∫ d^5xe^-z^2/z^5(-1/2g_5^2F_L^2),


    S'=∫ d^5x'e^+z'^2/z'^5(-1/2g_5^2F_R^2).



To obtain a five-dimensional model we consider the diagonal Δ of AdS× AdS', defined as the submanifold x=x'. The action on the diagonal is given by the sum of the above two actions evaluated at the diagonal:

    S^Δ=∫ d^5xe^-ϕ^Δ(z)/z^5(-1/2g_5^2(F_L^2+F_R^2)).

This choice not only gives a five-dimensional holographic model but also breaks chiral symmetry. For instance  AdS contains a U(N_f)_L theory and AdS' contains a U(N_f)_R theory, being N_f the number of flavours. The spacetime product AdS× AdS' contains a theory with gauge symmetry U(N_f)_L× U(N_f)_R, as can be seen from the action in Eq.(<ref>). By selecting the diagonal one obtains a U(N_f)_V theory, breaking chiral symmetry geometrically. This is the only choice in our setup that will give a reasonable holographic model of QCD. The diagonal dilaton obtained is given by:

    ϕ^Δ(z)=log(2cosh(λ^2(z^2+1)))  ,

where we have used the identity cosh(x)=(1/2)(e^+x+e^-x). The constant λ is introduced to render the argument of cosh unitless, and we have chosen k=1  GeV^-2 for simplicity.
To verify ∫_0^∞e^-ϕ^Δ(z)dz<∞ we consider a positive dilaton. From now on we will refer to this model as the Double Dilaton Soft Wall model (DDSW).

At this point, we shall comment about what would happen if other signs for the dilaton were chosen. If both signs are positive, we would obtain again a Soft Wall-like dilaton as the one considered in <cit.>; and if both signs are negative we would obtain a running coupling constant with no physical significance for QCD. In this way, our choice of opposite signs is the appropriate one if matching to pQCD is pursued.

In Ref.<cit.> the running of strong coupling constant at low energies was obtained with the embedding of Light-Front QCD into a Soft Wall model with a positive dilaton background, obtaining the so-called Holographic Light-Front QCD model. The properties of a low-energy running coupling constant and its β-function were studied since low energies are the natural landscape of application of holographic models. In our case, since no intrinsic scale is used in holographic models of QCD, the distinction between low and high energy is blurred, yet a matching with pQCD should be possible. Our objective is to find a running with an infrared fixed point and low energies and approach to pQCD result at high energies. A possible avenue would follow Ref.<cit.> and match both regimes using an analytical continuation of the gauge/gravity duality using a Holographic Light-Front QCD model. We will follow a different approach and explore such matching from the large-N_c expansion.

To do that, and at leading order in 1/N_c, let us use the expression deduced from the AdS/CFT dictionary formula seen in Eq.(<ref>).
As usual we identify μ∼ Q so we can define α_s(Q)=α_s(1/z). Using Eq.(<ref>) in Eq.(<ref>) with N_c=3, and recalling  experimental data is given in terms of α_s(Q), the DDSW model predicts:

    α_s(Q)/π=a/cosh^4/5(b(Q^2+1)).


Since the argument of the cosh function must be dimensionless we have considered a constant b different from the previous λ. The parameters a and b are determined after data fitting. We use data for the so-called effective strong coupling or effective charge α_g1(Q)/π collected in Ref.<cit.>. The effective strong coupling
constant is related to α_s from the perturbative series of an observable truncated to its first order in α_s <cit.>. Using first-order pQCD equations makes the effective coupling renormalization-scheme and gauge independent, and free of divergence at low Q^2. Data in Ref.<cit.> are obtained from JLab experiments Hall A/EG4 (from Q=0.143  GeV to 0.704  GeV), CLAS EG4/E977110 (from Q=0.187  GeV to 0.490  GeV) and EG1dvcs (from Q=0.775  GeV to 2.177  GeV), represented in Fig.<ref> as solid star, solid circle, and solid triangle, respectively. Notice the normalization at the fixed-point α_s(Q ≃ 0)/π≃ 1 in Fig.<ref>. As stated, these data extract the strong coupling constant in the effective charge approach, offering a non-perturbative renormalization-scale invariant treatment of α_s at low energies. From a fit to data with the model in Eq.(<ref>), we obtain a = 1.545 ± 0.047 b=1.150 ±  0.041  GeV^-2 with a reasonable χ^2/DOF = 1.45. As can be seen in Fig.<ref>, we obtain naturally an infrared fixed point. At Q≃ 1.5 GeV we observe a slight deviation from the experimental result and a tendency for α_s(Q)/π to go to zero quite fast.
 
Even though the fit is reasonably good, we do not find pQCD running coupling constant at high energies. This is expected since holographic models of QCD work only in the strongly coupled regime. At higher energies, the running should have a transition to the well-known perturbative one. Nonetheless, this deviation occurs at higher energies compared to, for example, the Light-Front QCD AdS model employed in <cit.>, which found a transition between the AdS and the pQCD running at around 1 GeV. The same authors improve it by joining their model with the fit parameterization to the effective strong coupling data from Ref.<cit.>.

Our running has been deduced from a holographic model based on AdS/CFT duality, so one should expect large-N_c corrections to this result. Since the large-N_c limit corresponds to the non-relativistic limit, corrections to that limit shall allow going beyond the non-relativistic limit. In practice, corrections to the large-N_c limit should allow matching to pQCD to higher scales, beyond 1 GeV. These large-N_c corrections are difficult to derive from first principles. We follow a more phenomenological approach. At finite N_c, quark loops are suppressed by 1/N_c, and non-planar (self-intersecting) diagrams are suppressed by 1/N_c^2 <cit.>. Any observable can be expanded in the form <cit.>:

    ∑_g=0^∞∑_L=0^∞N_c^2-2g-Lf_g(t^2).

Here t=α'/R^2, and χ=2-2g-L is the Euler characteristic that classifies Feynman diagrams in planar and non-planar diagrams by its embedding on closed surfaces <cit.>. g is the genus of the surface and L is the number of quark loops (number of boundaries).

For example, planar diagrams correspond to g=L=0 and can be embedded in the sphere (χ=2), while g>0 correspond to non-planar diagrams and are embedded in surfaces with holes (χ<2). This classification allows χ to run from 2 to -∞, including odd values because of the boundaries. The Ricci Flow is first order in α' so a factor f_g(t^2) is expected in the expansion.
This expansion can be understood as quantum corrections to the AdS/CFT dictionary on Eq.(<ref>)  as <cit.>:

    α_s(z)=R_0^4/4π N_cα'^2+R^4(z)/4π N_cα'^2+(R^4(z)/4π N_cα'^2)^2... ,

where R_0 is the usual AdS radius. We choose the above form to be able to perform a resummation.
The resummation of pQCD running coupling constant returns a Landau pole at low energies. Since Eq.(<ref>) relates α_s with AdS geometry, a Landau pole in α_s at IR would imply a divergent metric tensor at finite z>0 in the UV, which is absent from our formulation. Therefore, the observed finitude of our background metric tensor is fundamental to establishing that α_s at low energies is finite. Imposing a finite metric tensor in the UV returns an IR fixed point in the running of α_s. In practice, the first term of the above sum will avoid the Landau pole to find a smooth match with pQCD running. 

Eq.(<ref>) can be rewritten using then Eq.(<ref>) as:

    α_s(z)=∑_χe^-4/5χϕ(z)(1/4π N_cα'^2)^χ.



Now, observe that the term N_c^2-b can be resumed and be considered as a normalization constant c for N_c=3. With all these considerations, our final parametrization is the following partial sum:

    α_s(Q)/π=c∑_g=0^n-1(a/cosh^4/5(b(Q^2+1)))^g.

Here we have included the term N_c^-2g inside a^g. Thanks to the g=0 term, which allows a horizontal asymptote, we can match to perturbative QCD running at four loops with reference scale M_Z=91.18  GeV, α_s(M_z)=0.118. The parameters a,b,c must be determined either from data or from matching to pQCD or a combination of both. In the following, we opt for matching only the parameter c and let the data tell us about a and b. To determine c, we can search for a matching energy point Q_0 in which pQCD running coupling, coming from high energies, meets the model of Eq.(<ref>), coming from low energies, with the following iterative procedure. However, the matching energy point Q_0 is not known, which implies we must solve a system of two equations: 

    α_s^DDSW(Q_0)=α_s^pQCD(Q_0),


    dα_s^DDSW/dQ|_Q=Q_0=dα_s^pQCD/dQ|_Q=Q_0.

We solve the system recursively, starting with the initial value c=α_s(M_z)/π = 0.118/π≃ 0.0376, for then fit the data to obtain a and b. Solving the system we determine Q_0. So, if c is unfrozen, the previously determined Q_0 would return a new value for c which comes along with a new pair of fit parameters (a,b), and reiterate the process until the distance between two consecutive c parameters is equal to 10^-10. Convergence is very quick, giving concrete values of the c parameter and matching point Q_0, see Table <ref>.
The first non-constant version of the model in Eq.(<ref>) has 3 terms (n=2). A fit to data returns a=18.87 ± 0.60, b=(1.289 ± 0.041)  GeV^-2, c=0.082 and a χ^2/DOF = 0.99. This result is depicted in Fig.<ref> as a purple-dashed line (notice we plot α_s(Q)/π), with a matching with pQCD at 2.39 GeV. 

We can add more and more terms to the partial sum in Eq.(<ref>). Since it is a geometric series it can be fully resumed assuming the ratio is less than unity (suggested by the fitted coefficients), to obtain:

    α_s(Q)/π=c/1-a  sech^4/5(b(Q^2+1)).

In this case, and with the fitted coefficients shown in Table <ref>, the matching point is moved up to Q=3.79  GeV, see Fig.<ref> and Table <ref>. Interestingly, the resummation has a slower decrease at around 1-2 GeV, allowing a better fit to the data.

 In the end, requesting a matching at a given energy point implies the non-perturbative running is predicting perturbative β-function coefficients. Indeed, using the fitted values for a, b with fixed values for c and Q_0 in Eq.(<ref>) we can compute the β_0 coefficient using Eq.(<ref>) with the 1-loop pQCD running coupling constant:

    β_0=1 - a  sech^4/5(b (Q_0^2 + 1))-π c/α_s^pQCD(M_Z)/c log(Q_0^2/M_Z^2)

We obtain β_0 = 2.04 ± 0.01, to be compared to the theoretical pQCD result β_0=1.917 for N_f=5, so we obtain a departure below 10%. The higher-order β-function coefficients β_1, β_2 and β_3 can be obtained using a Monte Carlo method and the corresponding pQCD expression for α_s, but are more difficult to determine since they are suppressed by powers of α_s(M_z)/π, so they can be predicted with huge errors only.
We could also attempt to predict the leading-order term of the QCD β-function in the large-N_c expansion. Again, current experimental uncertainties mask the significant results.

We can also determine the non-perturbative β-function analytically following:

    β(Q)= Qd/d Qα_s(Q)/π.

For a finite n in Eq.(<ref>), β-function reads:

    β(Q)   =-8/5bc Q^2 tanh(b(Q^2+1))
       ·∑_g=1^n-2g(a/cosh^4/5(b(Q^2+1)))^g.



 
The resummed version reads:

    β(Q)=-8/5abcQ^2 sinh(b(Q^2+1)) sech^9/5(b(Q^2+1))/(1-a sech^4/5(b(Q^2+1)))^2.

The predicted β-function, depicted in Fig.<ref>, has all the desired properties: it is negative, approaches zero at high and low energies, and has a minimum that characterizes the transition between the two regimes.



§ MASSES AND DECAY CONSTANTS


The proposed dilaton Eq.(<ref>) can be used to predict the vector mesons Regge trajectory after solving their corresponding equation of motion <cit.>:

    ∂_z(e^-ϕ^Δ(z)/z∂_zψ_n(z))=-M_n^2e^-ϕ^Δ(z)/zψ_n(z).

Here M_n are interpreted as vector meson masses and the eigenfunctions ψ_n(z) are normalized according to:

    ∫_0^+∞dze^-ϕ^Δ(z)/z|ψ_n(z)|^2=1.

To solve the equation of motion Eq.(<ref>) the lightest mass must be fixed to be the lightest resonance in nature, the ρ-meson mass M_0=M_ρ=775  MeV. To do so,  we impose λ=0.4152 in Eq.(<ref>).






Regge trajectories for our DDSW model are linear as in the Soft Wall model, that is, M_n^2∼ n.  
We depict our predictions using the DDSW model for the ρ-meson family (red points in Fig.<ref>) together with the physical values from the PDG <cit.> (purple boxes) where errors are obtained via de half-width rule (i.e, M_n^2 ± M_n Γ_n, with Γ_n the total decay width of the particle <cit.>), and the prediction of Soft Wall model <cit.> (blue stars).
Comparison to experimental data demands merging all ρ states with total angular momentum J=1,  including 1S and 2S states, as we understand the holographic model cannot distinguish among them. In this respect, the Regge slope for the radial trajectory for the 7 known members of the ρ family would read 0.87(7)GeV^-2 as an average of the 1S and 2S trajectories from Ref.<cit.>. The DDSW model yields a slope 0.6962(18) GeV^-2.

The prediction of the ρ-meson decay constant using the DDSW model is also possible as is related to the residue at the pole M_0 on the equation of motion. It reads F_0^1/2=F_ρ^1/2=346  MeV which departs from the PDG value F_ρ^1/2=348  MeV by 0.57% only <cit.>. This is an improvement with respect to the Soft Wall (SW) result, which reads F_ρ^1/2=260  MeV <cit.>.
Other approaches such as the Semi-Hard Wall model (SHW, <cit.>) or the Tachyon Condensation model (TC <cit.>) have linear Regge trajectories only for large n. The departure for F_ρ^1/2 to the PDG one in these cases reaches 9.7% and 10%, respectively (see Table <ref>).




From the ρ decay constant predicted by our DDSW, we can use the relation

    F_ρ^2=3M_ρ^3/4πα^2Γ(ρ^0→ e^+e^-).

to obtain Γ(ρ^0→ e^+e^-)=6.87  keV. This gives a departure of 2.41% in comparison to the experimental value 7.04(6) keV <cit.>. For the usual Soft Wall model the value is 2.19  keV with a departure of 68.9%.






§ CONCLUSIONS


In this work, we have explored a new holographic model of QCD, named Double Dilaton Soft Wall model (DDSW), which from the Ricci flow allow us to obtain a parameterization for α_s able to match an infrared fixed point at low energies with pQCD at high energies. We used an AdS/CFT dictionary formula defined in Eq.(<ref>), to build the DDSW raising from a dilaton background defined in Eq.(<ref>) that breaks conformal symmetry geometrically thanks to including two dilaton fields of opposite signs (cf. <cit.>), each of them understood as vector and axial-vector mesons seeing different geometries <cit.>.
The matching with pQCD, which is found way above 2 GeV, is successful thanks to including in the holographic side large-N_c corrections. In the process, we use experimental data from Ref. <cit.> to fit the free leftover parameters of the α_s parameterization. 

As a by-product, we have studied the spectrum of vector mesons in the DDSW model by solving the equation of motion of the vector gauge field <cit.> with the dilaton mentioned above. This allowed us to determine masses as eigenvalues of the differential operator defined by the equation of motion, obtaining linear Regge trajectories. The predicted masses are heavier than in the usual Soft Wall model, thus closer to experimental results. Moreover, the squared root of the decay constant of the ρ-meson is improved compared to usual SW models as well, giving a departure of less than 1% when compared to the experimental result.

These results open the possibility of exploring other phenomenological results. In particular, our method could be applied to other dilaton backgrounds, obtaining strong coupling runnings at low energies and corresponding matching to pQCD.



§ ACKNOWLEDGEMENTS


The work of PM has been supported by the Ministerio de Ciencia e Innovación under grant PID2020-112965GB-I00 and by the Secretaria d’Universitats i Recerca del Departament d’Empresa i Coneixement de la Generalitat de Catalunya under grant 2021 SGR 00649. IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.

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 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
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            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 ] ]


