We study the bulk viscosity of moderately hot and dense neutrinotransparent color superconducting quark matter arising from weakinteractiondriven direct URCA processes. The quark matter is modeled using the NambuJonaLasinio model improved to account for vector and t Hooft interactions as well as antisymmetric pairing among the redgreen up and down quarks. The unpaired excitations are the strange quarks and the blue up or down quarks. We compute the relaxation rates associated with inlineformulammlmath displayinlinemmlmrowmmlmidmmlmimmlmrowmmlmathinlineformula and inlineformulammlmath displayinlinemmlmismmlmimmlmathinlineformulaquark decay and electron capture processes on the inlineformulammlmath displayinlinemmlmiummlmimmlmathinlineformula quark for blue color. The resulting bulk viscosity for density oscillations in the 110 kHz range shows a resonant peak at inlineformulammlmath displayinlinemmlmrowmmlmiTmmlmimmlmommlmommlmn5mmlmnmmlmtext mmlmtextmmlmtext mmlmtextmmlmiMeVmmlmimmlmrowmmlmathinlineformula and the damping time may drop below 10 ms. This is short enough to affect the postmerger evolution and is very similar to the damping predicted in nuclear matter.
['Nuclear Theory', 'Astrophysics - High Energy Astrophysical Phenomena', 'High Energy Physics - Phenomenology']
Bulk viscosity of twocolor superconducting quark matter in neutron star mergers
{'Bulk viscosity of two-color superconducting quark matter in neutron star mergers': "Mark Alford, 1, ∗ Arus Harutyunyan, 2, 3, † Armen Sedrakian, 4, 5, ‡ and Stefanos Tsiopelas 4, §  \n1 Department of Physics, Washington University, St. Louis, Missouri 63130, USA 2 Byurakan Astrophysical Observatory, Byurakan 0213, Armenia 3 Department of Physics, Yerevan State University, Yerevan 0025, Armenia 4 Institute of Theoretical Physics, University of Wrocglyph[suppress]law, 50-204 Wrocglyph[suppress]law, Poland 5 Frankfurt Institute for Advanced Studies, D-60438 Frankfurt am Main, Germany (Dated: 22 August 2024)  \nWestudy the bulk viscosity of moderately hot and dense, neutrino-transparent color superconducting quark matter arising from weak-interaction-driven direct URCA processes. The quark matter is modeled using the Nambu-Jona-Lasinio model improved to account for vector and 't Hooft interactions as well as antisymmetric pairing among the red/green up and down quarks. The unpaired excitations are the strange quarks and the blue up or down quarks. We compute the relaxation rates associated with d and s -quark decay and electron capture processes on u quark for blue color. The resulting bulk viscosity for density oscillations in the 1-10 kHz range shows a resonant peak at T ∼ 5 MeV, and the damping time may drop below 10 ms. This is short enough to affect the postmerger evolution and is very similar to the damping predicted in nuclear matter.", 'I. INTRODUCTION': "The recent discovery of gravitational waves followed by electromagnetic counterparts originating from binary neutron star mergers, as observed by the LIGO-Virgo Collaboration [1], has spurred interest in the transport characteristics of hot and dense matter. Numerical simulations of binary neutron star mergers typically conducted within the framework of nondissipative hydrodynamics have predicted significant density oscillations and robust gravitational wave emissions in the initial tens of milliseconds following the merger [2-7]. These oscillations are expected to eventually dissipate due to various dissipative processes within the postmerger matter, ultimately impacting the gravitational wave signal. Among these processes, bulk viscous dissipation facilitated by weak interactions appears to be the most efficacious mechanism in damping these oscillations as demonstrated by the initial estimates [8], first-principle computations of rates and associated relaxations times [9-12] and dissipative numerical simulations [13-15].  \nIn the cold regime pertinent to mature compact stars, extensive studies on bulk viscosity have been conducted, largely building upon the seminal work of Ref. [16]. The case of cold quark matter in hybrid and strange stars was studied in Refs. [17-25] in the context of damping of r -mode oscillations of cold stars. In particular, twoflavor ('2SC') pairing was studied in Ref. [19] including finite temperatures within a model defined in terms of the 2SC gap and chemical potentials (without electrical neutrality) while the numerical results were focused on non-leptonic processes [35].  \n‡  \nElectronic address: sedrakian@fias.uni-frankfurt.de  \nThe study of bulk viscosity of quark matter in the context of binary neutron star (BNS) mergers is still in its early stages [26, 27] and has focused on hot, unpaired quark matter. In this work, we present the first computation of the bulk viscosity of a BNS postmerger object that incorporates color superconductivity at intermediate densities and nonzero temperatures. We also consider conditions specific to BNS mergers, such as charge and color neutrality, and the density and temperature dependence of gaps and chemical potentials, which are accounted for self-consistently within our model.  \nWe focus on the weak leptonic quark URCA processes within the phase structure of color superconducting matter relevant to compact stars. At intermediate densities, quark matter is expected to be in the two-color superconducting (2SC) phase, where red-green light quarks pair while light blue quarks, as well as s quarks, remain unpaired [28]. Our study, using the vector-interactionimproved Nambu-Jona-Lasinio (NJL) model [29], examines bulk viscosity primarily from the URCA processes involving these unpaired blue quarks.", 'II. BULK VISCOSITY OF udse MATTER': "We consider neutrino-transparent matter composed of u, d, s quarks and electrons in the core of the compact star. The simplest (semileptonic) β equilibration processes in this system are the d and s -quark decay and the electron capture processes of the direct URCA type,  \nd → u + e -+ ¯ ν e , (1)  \nu + e -→ d + ν e , (2)  \ns → u + e -+ ¯ ν e , (3)  \nu + e -→ s + ν e , (4)  \nwhere ν e and ¯ ν e are the electron neutrino and antineutrino, respectively. The processes (1)-(4) proceed only  \nin the direction from left to right because in neutrinotransparent matter neutrinos/antineutrinos can appear only in final states.  \nThe rates of the URCA processes (1)-(4) can be written in full analogy with the nucleonic counterparts, see for example, Eqs. (8)-(10) of [11] (for quark matter one simply needs to replace g A → 1 in the matrix element and also cos θ c → sin θ c for processes including s -quarks, where θ c = 13 . 04 o is the Cabibbo angle).  \nWe will show below that, as has previously been noted [19, 26, 27], the nonleptonic processes  \nu + d ↔ u + s (5)  \nproceed much faster than the URCA processes. Consequently, in the regime where URCA-process-driven bulk viscosity is at its maximum the non-leptonic equilibration effectively takes place immediately, and we can assume µ s = µ d at all times (see [19], Appendix A). In this regime, there is only a single equilibrating quantity µ ∆ ≡ µ d -µ u -µ e , which is the relevant measure of how much the system is driven out of β -equilibrium state by a cycle of compression and rarefaction.  \nThe bulk viscosity coefficient of neutrino-transparent udse matter arising from the URCA processes (1)-(4) for small-amplitude density oscillations with a frequency ω is given by  \nζ = C 2 A γ ω 2 + γ 2 , (6)  \nwhich has the classic resonant form depending on two quantities: the susceptibility prefactor C 2 /A , which depends only on the equation of state, and the relaxation rate γ = λ d A , which depends additionally on the microscopic interaction rates via the equilibration coefficient  \nλ ≡ d Γ dµ ∆ , (7)  \nwhere Γ is the total rate of the URCA process. Analogous quantities for nonleptonic processes are defined in a similar fashion, e.g., λ ' ≡ d Γ ' /dµ ' ∆ where Γ ' is the rate of nonleptonic process and µ ' ∆ is the deviation from chemical equilibrium due to nonleptonic processes. The susceptibilities are defined as  \nA = -1 n B ∂µ ∆ ∂x u ∣ ∣ ∣ ∣ n B , C = n B ∂µ ∆ ∂n B ∣ ∣ ∣ ∣ x u , (8)  \nwhere n B is the baryon density and x u is the fraction of blue up quarks. The derivatives in Eq. (8) and the relaxation rate γ should be computed in β -equilibrium. (In the regime of interest the density oscillations are supposed to be adiabatic, however, the difference between adiabatic and isothermal susceptibilities is negligible in this case [12, 33].)  \nTo describe the properties of 2SC quark matter we  \nadopt an NJL-type Lagrangian which is given by  \nL Q = ¯ ψ ( iγ µ ∂ µ -ˆ m ) ψ + G V ( ¯ ψiγ µ ψ ) 2 + G S 8 ∑ a =0 [( ¯ ψλ a ψ ) 2 +( ¯ ψiγ 5 λ a ψ ) 2 ] + G D ∑ γ,c [ ¯ ψ a α iγ 5 ϵ αβγ ϵ abc ( ψ C ) b β ][( ¯ ψ C ) r ρ iγ 5 ϵ ρσγ ϵ rsc ψ 8 σ ] -K { det f [ ¯ ψ (1 + γ 5 ) ψ ] + det f [ ¯ ψ (1 -γ 5 ) ψ ] } , (9)  \nwhere the quark spinor fields ψ a α carry color a = r, g, b and flavor ( α = u, d, s ) indices, the matrix of quark current masses is given by ˆ m = diag f ( m u , m d , m s ), λ a where a = 1 , ..., 8 are the Gell-Mann matrices in the color space, and λ 0 = (2 / 3) 1 f . The charge conjugate spinors are defined as ψ C = C ¯ ψ T and ¯ ψ C = ψ T C , where C = iγ 2 γ 0 is the charge conjugation matrix. The original NJL model, which contains the scalar coupling term with coupling G S has been extended here to include the 't Hooft interaction with coupling K and the vector interactions with coupling G V ; this model is referred to as the vector-NJL model. The pairing gap in the quasiparticle spectrum is ∆ c ∝ ( ¯ ψ C ) a α iγ 5 ϵ αβc ϵ abc ψ b β . The numerical values of the parameters of the Lagrangian are the same as in Ref. [29] and we fix the strength of the diquark interaction at G D /G S = 1 . 1, this ratio being somewhat larger than the value 0.75 suggested by the Fierz transformation. Note that the larger the gap, the stiffer the equation of state and the larger the maximum mass of a hybrid star.  \nWe assume 2SC pairing by adopting a diquarkantidiquark interaction [fourth term in Eq. (9)] that has color and flavor antisymmetric pairing of u and d quarks while s quarks remain unpaired.  \nThe 2SC pairing gap depends weakly on the density: it varies in the range 130 ≤ ∆ ≤ 150 MeV in the number density range 0 . 6 ≤ n B ≤ 1 . 14 fm -3 . It also depends weakly on the temperature in the range of interest ∆ ≫ T ∼ (1 -10) MeV. The masses of quarks in the same density interval vary in the range 10 ≤ m u,d ≤ 30 and 300 ≤ m s ≤ 400 MeV and are weakly dependent on temperature in the range 1 ≤ T ≤ 10 MeV. Further input characteristics of the unperturbed quark matter model forming the background equilibrium are given in Supplemental Material [30].  \nThe equilibration coefficients for the URCA processes (1)-(4) as well as for the nonleptonic processes (5) are plotted in Fig. 1 for fixed baryon density n B /n 0 = 7. The nonleptonic processes have a much higher rate than the URCA processes, as mentioned above, whereas the rates of the d -URCA and s -URCA processes are comparable in the temperature range 0 . 1 ≤ T ≤ 10 MeV. This implies that the bulk viscosity produced by the nonleptonic processes will, for typical BNS merger oscillation frequencies in the kilohertz range, be peaked at much lower temperatures which lie out of the temperature range of interest of BNS mergers. This argument justifies our assumption that the bulk viscosity at temperatures 1 ≤ T ≤ 10 MeV can be computed from URCA  \nFIG. 1: The total equilibration coefficients λ for processes (1) and (2) (solid line), for processes (3) and (4) (dashed line) and for the nonleptonic processes (dash-dotted line) as functions of the temperature for baryon density fixed at n B /n 0 = 7.  \n<!-- image -->  \nFIG. 2: The β -relaxation rate γ as a function of the temperature for two values of the baryon density. The horizontal lines show where γ = 2 πf for selected values of oscillation frequency f = 1 kHz (solid line) and f = 10 kHz (dashed line).  \n<!-- image -->  \nprocesses only, with the additional constraint µ d = µ s .  \nTo gain physical insight, we quote here approximate analytical low-temperature results for the equilibration coefficients λ d ≃ 0 . 2 G 2 F cos 2 θ c p 2 Fd T 5 , and λ s ≃ 0 . 03 G 2 F sin 2 θ c µ ∗ s m 2 s T 4 , where G F is the Fermi coupling constant and p Ff is the Fermi momentum of quark of flavor f = u, d, s . These formulas were derived under the  \nassumptions T ≪ µ i , m d , m s → 0 and ω 0 -ϕ 0 ≪ m s and are accurate to a few percent when compared to numerical results that do not assume these approximations. The rates of d -URCA processes include an extra power of T because the scattering matrix element contains the four-product of the momenta of the u -quark and electron, which are nearly collinear at chemical equilibrium. This near collinearity occurs because the light quarks are almost massless, with a small angle deviation allowed only by the thermal smearing of these momenta around the Fermi spheres. Consequently, the scattering matrix element adds an extra power of T to the process rates, in addition to the standard T 5 scaling for massive particles, which arises from the available scattering phase space. This difference in temperature scaling leads to the intersection of the λ d ( T ) and λ s ( T ) curves at a temperature in the MeV range, as seen in Fig. 1. Two more aspects are worth noticing: (i) λ s ( T ) is suppressed by a factor of sin 2 θ c = 0 . 05 compared to λ d ( T ) which has cos 2 θ c = 0 . 95 and has a smaller numerical prefactor arising from the phase-space integration. The remaining factors such as m s , p Fd , and µ ∗ s are all of the same order of magnitude. (ii) Because the interactions modify the chemical potentials of light quarks via the isoscalar ω -field, p Fd = p fu + p Fe in β -equilibrium for massless quarks, which in turn implies that the direct URCA channel is only thermally open. In contrast, s -URCA processes are widely open with p Fu + p Fe -p Fs ≥ 70 MeV as a result of large s -quark mass [31, 32].  \nThe temperature dependence of the total β -relaxation rate γ due to the URCA processes (1)-(4) is shown in Fig. 2. It follows a power-law scaling γ ∝ T 4 . 5 in the temperature range 1 ≤ T ≤ 10 MeV, and increases weakly with the density. We see that γ matches the constant angular frequency ω = 2 πf at a temperature around T ≃ 3-4 MeV for f = 1kHz and around T ≃ 5-6 MeV for f = 10kHz, the exact values of the temperature being dependent on the baryon density.  \nFigure 3 shows the temperature dependence of the bulk viscosity of 2SC quark matter for the density oscillation frequency fixed at a characteristic to BNS mergers value f = 1 kHz. The bulk viscosity has a maximum at the temperature where ω = γ . The point of the maximum shifts slowly to lower temperatures with increasing density. The maximum value and the overall magnitude of the bulk viscosity is a sensitive function of density through its dependence on the susceptibility factor C 2 /A , see Eq. (6). It decreases with the density, and, e.g., at n B /n 0 = 7 the maximum bulk viscosity is more than an order of magnitude smaller than at n B /n 0 = 4. For comparison, we show also the bulk viscosity of ude matter [ i.e. , when the processes (3) and (4) are not taken into account] with short dashed lines at n B /n 0 = 4. We see that neglecting the s -quark contribution underestimates the bulk viscosity at low temperatures and overestimates it at higher temperatures. However, the location of the maximum does not change significantly, because at these temperatures the d URCA and s URCA processes have  \nFIG. 3: The bulk viscosity of udse 2SC quark matter as a function of temperature for two values of baryon density. The frequency of density oscillations is fixed at f = 1 kHz. The bulk viscosity of ude matter is shown by short dashed lines at n B /n 0 = 4. In addition, the bulk viscosity of npe matter from Ref. [12] is shown for comparison by dotted (model DDME2) and dot-dot-dashed (model NL3) lines at density n B /n 0 = 4.  \n<!-- image -->  \nFIG. 4: The damping timescale of density oscillations for 2SC quark matter as a function of temperature for f = 1 kHz. The results for nucleonic matter from Ref. [12] are shown for comparison by dotted (model DDME2) and dot-dot-dashed (model NL3) lines at density n B /n 0 = 4.  \n<!-- image -->  \ncomparable rates, and the inclusion of s quarks increases the relaxation rate by a factor between 1 and 2, leading to a slight shift of the maximum to lower temperatures.  \nFor comparison, we show also the bulk viscosity of  \nFIG. 5: Same as Fig. 4 but for f = 10 kHz.  \n<!-- image -->  \nneutrino-transparent nucleonic matter as was computed in Ref. [12] for models DDME2 (dotted line) and NL3 (dashed line) for n B /n 0 = 4. The main difference between these two models is that the low-temperature direct URCA channel is allowed for NL3 and is blocked for DDME2 at the density value considered. As a result, the bulk viscosities for the two models differ significantly at temperatures T ≤ 4 MeV, but are rather similar at T ≥ 4 MeV. At these temperatures, the bulk viscosity of 2SC quark matter is not very different from the bulk viscosity of nucleonic matter. In the low-temperature regime, 2SC quark matter's bulk viscosity has intermediate values compared to the two nucleonic models. The reason for this is the fact that the u -quark fraction is below, but very close to the direct URCA threshold for the reactions (1) and (2): the difference between the Fermi momenta of initial and final state particles p Fd -p Fu -p Fe is less than 1 MeV for densities 4 ≤ n B /n 0 ≤ 7. This implies that already at temperature T ≥ 1 MeV thermal smearing is sufficient to open up phase space for lightquark URCA processes.  \nSince the bulk viscosity comes from semileptonic weak interactions of the unpaired blue quarks, one would expect that calculations for unpaired quark matter would see a similar maximum in the bulk viscosity at MeV temperatures. This is visible in Ref. [22] (Fig. 3), and also in Ref. [27] (Fig. 4) though their maximum bulk viscosity is smaller because their strange quark is lighter.  \nNext, we estimate the bulk viscous damping timescales of density oscillations. The damping timescale is given by Refs. [8, 9, 33]  \nτ ζ = 1 9 Kn B ω 2 ζ , (10)  \nwhere the incompressibility is K = 9( ∂P/∂n B ) , and P is the pressure. The damping timescale exhibits a temper-  \ne dependence that is inverse to that of bulk viscosity, as K remains insensitive to the temperature within the range 1 ≤ T ≤ 10 MeV. As seen from Figs. 4 and 5 the time-scale τ ζ is shortest when the bulk viscosity is largest, at T ≃ 3-4 MeV for oscillations of frequency f = 1 kHz, and at T ≃ 5-6 MeV for f = 10 kHz. At this minimum, the bulk viscous damping timescales of 2SC quark matter are in the range from a few milliseconds to a few hundred milliseconds, which coincides with the typical short-term evolution timescales of the postmerger object. The lightshaded region in Figs. 4 and 5 corresponds τ ζ ≤ 1 s, whereas the heavy-shaded region in Fig. 5 corresponds to τ ζ ≤ 10 ms.  \nFor a typical oscillation frequency of f = 1 kHz, our results predict that bulk viscous damping would be marginally relevant over the short BNS postmerger evolution timescales of approximately 10 ms, but noticeable for longer-lived remnants in the temperature range of 2-8 MeV. For higher frequencies, such as f = 10 kHz, bulk viscous damping significantly affects the short-term evolution timescale of approximately 10 ms within the temperature range of 5-8 MeV at relatively low densities. It is also noteworthy that the damping timescales for 2SC quark matter are quite similar to those found for nucleonic matter.  \nThis implies that, according to our NJL model of quark matter, at temperatures in the 1-10 MeV range it will be difficult to distinguish 2SC quark matter from nuclear matter based on its bulk viscous dissipation properties.", 'III. CONCLUSIONS': "In this work, we studied the bulk viscosity of neutrinotransparent 2SC quark matter driven by direct URCA processes in the temperature range 1 ≤ T ≤ 10MeV and density range 4 n 0 ≤ n B ≤ 7 n 0 . This complements our recent study of relativistic npeµ matter [12]. 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The resulting damping timescales, down to tens of milliseconds, suggest that bulk viscosity may significantly contribute to energy dissipation in BNS mergers at these temperatures.  \nThe bulk viscosity and damping time for 2SC quark matter closely resemble those of nucleonic matter [12], making it challenging to distinguish these states via their bulk viscous behavior. However, at lower temperatures, quark matter shows a second resonant peak in bulk viscosity at T ∼ 0 . 05 MeV, absent in nuclear matter [19, 26, 27].  \nInterestingly, semileptonic rates for unpaired quark matter may be similar to those for 2SC matter, suggesting that unpaired quark matter at T ≈ 1 to 10 MeV may also be difficult to distinguish from nuclear matter. A recent study of unpaired quark matter [27] found a peak in bulk viscosity in this temperature range, though with longer damping times, likely due to different susceptibilities in their model. Future studies should address these model-dependent uncertainties in quark matter phenomenology under BNS merger conditions.", 'Acknowledgments': "M. A. is partly supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Award No. DE-FG02-05ER41375. A. H. and A. S. were supported by the Volkswagen Foundation (Hannover, Germany) Grant No. 96 839. A. S. and S. T. acknowledge the support of the Polish National Science Centre (NCN) Grant 2020/37/B/ST9/01937. A. S. acknowledges Grant No. SE 1836/5-3 from Deutsche Forschungsgemeinschaft (DFG). S. T. is supported by the IMPRS for Quantum Dynamics and Control at Max Planck Institute for the Physics of Complex Systems, Dresden, Germany.  \n- [5] A. Bauswein, N.-U. F. Bastian, D. B. Blaschke, K. Chatziioannou, J. A. Clark, T. Fischer et al., Identifying a First-Order Phase Transition in Neutron-Star Mergers through Gravitational Waves , Phys. Rev. Lett. 122 (2019) 061102, [ 1809.01116 ].\n- [6] D. 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[19] where the bulk viscosity is derived for coupled nonleptonic and leptonic processes, similar to our approach here.", 'Supplemental Material Bulk viscosity of two-color superconducting quark matter in neutron star mergers': 'Here, we show the particle fractions of unpaired particles and their chemical potentials in finite-temperature β -equilibrated 2SC quark matter [28] within the vector-interaction-improved NJL model described in detail in Ref. [29]. These are the key input characteristics of the unperturbed quark matter model needed for analyzing perturbations.  \nThe quark (thermodynamic) chemical potentials are given by  \nµ f,c = 1 3 µ B + µ Q Q f + µ 3 T c 3 + µ 8 T c 8 , (11)  \nwith µ B and µ Q being the baryon and charge chemical potential and  \nQ f = diag f ( 2 3 , -1 3 , -1 3 ) , (12)  \nT c 3 = 1 2 diag c (1 , -1 , 0) , T c 8 = 1 2 √ 3 diag c (1 , 1 , -2) . (13)  \nThe quark effective chemical potentials are modified by the vector fields µ ∗ = diag f ( µ u -ω 0 , µ d -ω 0 , µ s -ϕ 0 ).  \nFIG. 6: Particle fractions of unpaired particles in finite-temperature β -equilibrated 2SC quark matter as functions of the baryon density n B (in units of nuclear saturation density n 0 = 0 . 16 fm -3 ) at T = 1 MeV. They are insensitive to the value of the temperature within the range 1 ≤ T ≤ 10 MeV.  \n<!-- image -->  \nFigure 6 shows the finite-temperature composition of the 2SC phase under β -equilibrium, relevant for the computation of the bulk viscosity, where we show only the population of the blue (unpaired color) u and d quarks, as well as the population of s quarks of blue color. (The blue s -quark density differs slightly from red-green s -quark densities, with the difference in chemical potentials being on the order of 2%, approximately 10 MeV.) Note that the substantial s -quark population is present in matter which quickly equilibrates with the light flavor sector via the non-leptonic processes. It is also seen that s -quarks reduce the lepton fraction in matter, as already noticed long ago [31, 32].  \nFigure 7 displays the effective chemical potentials of the species in β -equilibrated 2SC matter. We show only the chemical potentials of blue quarks, which are relevant for the computation of the bulk viscosity. The temperature dependence of the chemical potentials is weak in the shown density range. Note that the fast equilibration via nonleptonic processes implies µ d = µ s and the difference in the effective chemical potentials of d and s quarks seen in Fig. 7 is seen in Fig. 7 is due to their coupling to ω and ϕ mesons.  \nFIG. 7: The same as in Fig. 6 but for effective chemical potentials of unpaired particles. They depend weakly on the temperature in the range 1 ≤ T ≤ 10 MeV .  \n<!-- image -->'}