Abstract
We determine the QCD equation of state at nonzero temperature in the presence of an isospin asymmetry between the light quark chemical potentials on the lattice. Our simulations employ Nf = 2 + 1 flavors of dynamical staggered quarks at physical masses, using three different lattice spacings. The main results, obtained at the individual lattice spacings, are based on a two-dimensional spline interpolation of the isospin density, from which all relevant quantities can be obtained analytically. In particular, we present results for the pressure, the interaction measure, the energy and entropy densities, as well as the speed of sound. Remarkably, the latter is found to exceed its ideal gas limit deep in the pion condensed phase, the first account of the violation of this limit in first principles QCD. Finally, we also compute the phase diagram in the temperature — isospin density plane for the first time. Even though the results are not continuum extrapolated and thus not final, the data for all observables will be useful for the benchmarking of effective theories and low-energy models of QCD and are provided in ancillary files for simple reuse.
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References
Y. Aoki et al., The Order of the quantum chromodynamics transition predicted by the standard model of particle physics, Nature 443 (2006) 675 [hep-lat/0611014] [INSPIRE].
T. Bhattacharya et al., QCD Phase Transition with Chiral Quarks and Physical Quark Masses, Phys. Rev. Lett. 113 (2014) 082001 [arXiv:1402.5175] [INSPIRE].
D. Teaney, J. Lauret and E.V. Shuryak, A Hydrodynamic Description of Heavy Ion Collisions at the SPS and RHIC, nucl-th/0110037 [INSPIRE].
P.F. Kolb and U.W. Heinz, Hydrodynamic description of ultrarelativistic heavy ion collisions, nucl-th/0305084 [INSPIRE].
D. Boyanovsky, H.J. de Vega and D.J. Schwarz, Phase transitions in the early and the present universe, Ann. Rev. Nucl. Part. Sci. 56 (2006) 441 [hep-ph/0602002] [INSPIRE].
J.M. Lattimer and M. Prakash, Neutron star structure and the equation of state, Astrophys. J. 550 (2001) 426 [astro-ph/0002232] [INSPIRE].
I. Tews, J. Carlson, S. Gandolfi and S. Reddy, Constraining the speed of sound inside neutron stars with chiral effective field theory interactions and observations, Astrophys. J. 860 (2018) 149 [arXiv:1801.01923] [INSPIRE].
E. Annala et al., Evidence for quark-matter cores in massive neutron stars, Nature Phys. 16 (2020) 907 [arXiv:1903.09121] [INSPIRE].
I.M. Oldengott and D.J. Schwarz, Improved constraints on lepton asymmetry from the cosmic microwave background, EPL 119 (2017) 29001 [arXiv:1706.01705] [INSPIRE].
M.M. Wygas, I.M. Oldengott, D. Bödeker and D.J. Schwarz, Cosmic QCD Epoch at Nonvanishing Lepton Asymmetry, Phys. Rev. Lett. 121 (2018) 201302 [arXiv:1807.10815] [INSPIRE].
M.M. Middeldorf-Wygas, I.M. Oldengott, D. Bödeker and D.J. Schwarz, Cosmic QCD transition for large lepton flavor asymmetries, Phys. Rev. D 105 (2022) 123533 [arXiv:2009.00036] [INSPIRE].
V. Vovchenko et al., Pion Condensation in the Early Universe at Nonvanishing Lepton Flavor Asymmetry and Its Gravitational Wave Signatures, Phys. Rev. Lett. 126 (2021) 012701 [arXiv:2009.02309] [INSPIRE].
D.T. Son and M.A. Stephanov, QCD at finite isospin density, Phys. Rev. Lett. 86 (2001) 592 [hep-ph/0005225] [INSPIRE].
J.B. Kogut and D.K. Sinclair, Quenched lattice QCD at finite isospin density and related theories, Phys. Rev. D 66 (2002) 014508 [hep-lat/0201017] [INSPIRE].
J.B. Kogut and D.K. Sinclair, Lattice QCD at finite isospin density at zero and finite temperature, Phys. Rev. D 66 (2002) 034505 [hep-lat/0202028] [INSPIRE].
J.B. Kogut and D.K. Sinclair, The Finite temperature transition for 2-flavor lattice QCD at finite isospin density, Phys. Rev. D 70 (2004) 094501 [hep-lat/0407027] [INSPIRE].
G. Endrödi, Magnetic structure of isospin-asymmetric QCD matter in neutron stars, Phys. Rev. D 90 (2014) 094501 [arXiv:1407.1216] [INSPIRE].
P. de Forcrand, M.A. Stephanov and U. Wenger, On the phase diagram of QCD at finite isospin density, PoS LATTICE2007 (2007) 237 [arXiv:0711.0023] [INSPIRE].
P. Cea et al., The critical line of two-flavor QCD at finite isospin or baryon densities from imaginary chemical potentials, Phys. Rev. D 85 (2012) 094512 [arXiv:1202.5700] [INSPIRE].
W. Detmold, K. Orginos and Z. Shi, Lattice QCD at non-zero isospin chemical potential, Phys. Rev. D 86 (2012) 054507 [arXiv:1205.4224] [INSPIRE].
B.B. Brandt, G. Endrodi and S. Schmalzbauer, QCD phase diagram for nonzero isospin-asymmetry, Phys. Rev. D 97 (2018) 054514 [arXiv:1712.08190] [INSPIRE].
B.B. Brandt and G. Endrodi, Reliability of Taylor expansions in QCD, Phys. Rev. D 99 (2019) 014518 [arXiv:1810.11045] [INSPIRE].
B.B. Brandt et al., New class of compact stars: Pion stars, Phys. Rev. D 98 (2018) 094510 [arXiv:1802.06685] [INSPIRE].
A. Cherman, T.D. Cohen and A. Nellore, A Bound on the speed of sound from holography, Phys. Rev. D 80 (2009) 066003 [arXiv:0905.0903] [INSPIRE].
B. Brandt, F. Cuteri and G. Endrödi, Dataset for “Equation of state and speed of sound of isospin-asymmetric QCD on the lattice”, Bielefeld University (2023), https://pub.uni-bielefeld.de/record/2980217https://doi.org/10.4119/UNIBI/2980217].
B.B. Brandt, G. Endrodi and S. Schmalzbauer, QCD at finite isospin chemical potential, EPJ Web Conf. 175 (2018) 07020 [arXiv:1709.10487] [INSPIRE].
B.B. Brandt, G. Endrodi and S. Schmalzbauer, QCD at nonzero isospin asymmetry, PoS Confinement2018 (2018) 260 [arXiv:1811.06004] [INSPIRE].
B.B. Brandt, F. Cuteri and G. Endrodi, QCD thermodynamics at non-zero isospin asymmetry, PoS LATTICE2021 (2022) 132 [arXiv:2110.14750] [INSPIRE].
S. Borsanyi et al., The QCD equation of state with dynamical quarks, JHEP 11 (2010) 077 [arXiv:1007.2580] [INSPIRE].
S. Borsanyi et al., Full result for the QCD equation of state with 2+1 flavors, Phys. Lett. B 730 (2014) 99 [arXiv:1309.5258] [INSPIRE].
HotQCD collaboration, Equation of state in (2+1)-flavor QCD, Phys. Rev. D 90 (2014) 094503 [arXiv:1407.6387] [INSPIRE].
J. Engels et al., Nonperturbative thermodynamics of SU(N) gauge theories, Phys. Lett. B 252 (1990) 625 [INSPIRE].
T. Blum, L. Karkkainen, D. Toussaint and S.A. Gottlieb, The beta function and equation of state for QCD with two flavors of quarks, Phys. Rev. D 51 (1995) 5153 [hep-lat/9410014] [INSPIRE].
J. Engels et al., Thermodynamics of four flavor QCD with improved staggered fermions, Phys. Lett. B 396 (1997) 210 [hep-lat/9612018] [INSPIRE].
Y. Aoki, Z. Fodor, S.D. Katz and K.K. Szabo, The Equation of state in lattice QCD: With physical quark masses towards the continuum limit, JHEP 01 (2006) 089 [hep-lat/0510084] [INSPIRE].
C.R. Allton et al., The Equation of state for two flavor QCD at nonzero chemical potential, Phys. Rev. D 68 (2003) 014507 [hep-lat/0305007] [INSPIRE].
K. Iida and E. Itou, Velocity of sound beyond the high-density relativistic limit from lattice simulation of dense two-color QCD, PTEP 2022 (2022) 111B01 [arXiv:2207.01253] [INSPIRE].
E. Itou and K. Iida, Bump of sound velocity in dense 2-color QCD, PoS LATTICE2022 (2023) 151 [arXiv:2210.14385] [INSPIRE].
H. Akaike, Information Theory and an Extension of the Maximum Likelihood Principle, Springer Series in Statistics, Springer New York (1993), p. 610–624 [https://doi.org/10.1007/978-1-4612-0919-5_38].
B.B. Brandt and G. Endrodi, QCD phase diagram with isospin chemical potential, PoS LATTICE2016 (2016) 039 [arXiv:1611.06758] [INSPIRE].
S. Borsanyi et al., QCD equation of state at nonzero chemical potential: continuum results with physical quark masses at order mu2, JHEP 08 (2012) 053 [arXiv:1204.6710] [INSPIRE].
J.N. Guenther et al., The QCD equation of state at finite density from analytical continuation, Nucl. Phys. A 967 (2017) 720 [arXiv:1607.02493] [INSPIRE].
D. Bollweg et al., Equation of state and speed of sound of (2+1)-flavor QCD in strangeness-neutral matter at non-vanishing net baryon-number density, arXiv:2212.09043 [INSPIRE].
S. Borsanyi et al., Fluctuations of conserved charges at finite temperature from lattice QCD, JHEP 01 (2012) 138 [arXiv:1112.4416] [INSPIRE].
P. Adhikari, J.O. Andersen and M.A. Mojahed, Condensates and pressure of two-flavor chiral perturbation theory at nonzero isospin and temperature, Eur. Phys. J. C 81 (2021) 173 [arXiv:2010.13655] [INSPIRE].
R.R. Silbar and S. Reddy, Neutron stars for undergraduates, Am. J. Phys. 72 (2004) 892 [Erratum ibid. 73 (2005) 286] [nucl-th/0309041] [INSPIRE].
I. Sagert, M. Hempel, C. Greiner and J. Schaffner-Bielich, Compact stars for undergraduates, Eur. J. Phys. 27 (2006) 577 [astro-ph/0506417] [INSPIRE].
Y. Fujimoto, K. Fukushima, L.D. McLerran and M. Praszalowicz, Trace Anomaly as Signature of Conformality in Neutron Stars, Phys. Rev. Lett. 129 (2022) 252702 [arXiv:2207.06753] [INSPIRE].
B.B. Brandt, F. Cuteri and G. Endrödi, Equation of state and Taylor expansions at nonzero isospin chemical potential, PoS LATTICE2022 (2023) 144 [arXiv:2212.01431] [INSPIRE].
T. Kojo and D. Suenaga, Peaks of sound velocity in two color dense QCD: Quark saturation effects and semishort range correlations, Phys. Rev. D 105 (2022) 076001 [arXiv:2110.02100] [INSPIRE].
L. McLerran and S. Reddy, Quarkyonic Matter and Neutron Stars, Phys. Rev. Lett. 122 (2019) 122701 [arXiv:1811.12503] [INSPIRE].
K.S. Jeong, L. McLerran and S. Sen, Dynamically generated momentum space shell structure of quarkyonic matter via an excluded volume model, Phys. Rev. C 101 (2020) 035201 [arXiv:1908.04799] [INSPIRE].
N. Kovensky and A. Schmitt, Holographic quarkyonic matter, JHEP 09 (2020) 112 [arXiv:2006.13739] [INSPIRE].
T. Kojo, Stiffening of matter in quark-hadron continuity, Phys. Rev. D 104 (2021) 074005 [arXiv:2106.06687] [INSPIRE].
R. Somasundaram, I. Tews and J. Margueron, Investigating signatures of phase transitions in neutron-star cores, Phys. Rev. C 107 (2023) 025801 [arXiv:2112.08157] [INSPIRE].
E. Annala et al., Multimessenger Constraints for Ultradense Matter, Phys. Rev. X 12 (2022) 011058 [arXiv:2105.05132] [INSPIRE].
S. Altiparmak, C. Ecker and L. Rezzolla, On the Sound Speed in Neutron Stars, Astrophys. J. Lett. 939 (2022) L34 [arXiv:2203.14974] [INSPIRE].
C. Ecker and L. Rezzolla, A General, Scale-independent Description of the Sound Speed in Neutron Stars, Astrophys. J. Lett. 939 (2022) L35 [arXiv:2207.04417] [INSPIRE].
M. Marczenko, L. McLerran, K. Redlich and C. Sasaki, Reaching percolation and conformal limits in neutron stars, Phys. Rev. C 107 (2023) 025802 [arXiv:2207.13059] [INSPIRE].
B.B. Brandt, F. Cuteri, G. Endrődi and S. Schmalzbauer, The Dirac spectrum and the BEC-BCS crossover in QCD at nonzero isospin asymmetry, Particles 3 (2020) 80 [arXiv:1912.07451] [INSPIRE].
F. Cuteri, B.B. Brandt and G. Endrődi, Searching for the BCS phase at nonzero isospin asymmetry, PoS LATTICE2021 (2022) 232 [arXiv:2112.11113] [INSPIRE].
M. Leonhardt et al., Symmetric nuclear matter from the strong interaction, Phys. Rev. Lett. 125 (2020) 142502 [arXiv:1907.05814] [INSPIRE].
S. Gandolfi et al., Quantum Monte Carlo calculation of the equation of state of neutron matter, Phys. Rev. C 79 (2009) 054005 [arXiv:0903.2610] [INSPIRE].
I. Tews, T. Krüger, K. Hebeler and A. Schwenk, Neutron matter at next-to-next-to-next-to-leading order in chiral effective field theory, Phys. Rev. Lett. 110 (2013) 032504 [arXiv:1206.0025] [INSPIRE].
T. Gorda et al., Next-to-Next-to-Next-to-Leading Order Pressure of Cold Quark Matter: Leading Logarithm, Phys. Rev. Lett. 121 (2018) 202701 [arXiv:1807.04120] [INSPIRE].
K. Hebeler, J.M. Lattimer, C.J. Pethick and A. Schwenk, Equation of state and neutron star properties constrained by nuclear physics and observation, Astrophys. J. 773 (2013) 11 [arXiv:1303.4662] [INSPIRE].
A. Kurkela, E.S. Fraga, J. Schaffner-Bielich and A. Vuorinen, Constraining neutron star matter with Quantum Chromodynamics, Astrophys. J. 789 (2014) 127 [arXiv:1402.6618] [INSPIRE].
E. Annala, T. Gorda, A. Kurkela and A. Vuorinen, Gravitational-wave constraints on the neutron-star-matter Equation of State, Phys. Rev. Lett. 120 (2018) 172703 [arXiv:1711.02644] [INSPIRE].
E.R. Most, L.R. Weih, L. Rezzolla and J. Schaffner-Bielich, New constraints on radii and tidal deformabilities of neutron stars from GW170817, Phys. Rev. Lett. 120 (2018) 261103 [arXiv:1803.00549] [INSPIRE].
M.-Z. Han, J.-L. Jiang, S.-P. Tang and Y.-Z. Fan, Bayesian Nonparametric Inference of the Neutron Star Equation of State via a Neural Network, Astrophys. J. 919 (2021) 11 [arXiv:2103.05408] [INSPIRE].
M.-Z. Han, Y.-J. Huang, S.-P. Tang and Y.-Z. Fan, Plausible presence of new state in neutron stars with masses above 0.98MTOV, Sci. Bull. 68 (2023) 913 [arXiv:2207.13613] [INSPIRE].
H. Akaike, Information Theory and an Extension of the Maximum Likelihood Principle, in E. Parzen, K. Tanabe and G. Kitagawa eds., Selected Papers of Hirotugu Akaike, Springer Series in Statistics (1998) pp. 199–213 [INSPIRE].
G. Endrodi, Multidimensional spline integration of scattered data, Comput. Phys. Commun. 182 (2011) 1307 [arXiv:1010.2952] [INSPIRE].
Acknowledgments
The authors are grateful to Szabolcs Borsányi, Gergely Markó, Guy Moore and Aleksi Vuorinen for useful discussions and to Kálmán Szabó for providing the parameterization for the lattice QCD EoS at μI = 0. We also thank Prabal Adhikari, Jens Oluf Andersen and Martin Mojahed for discussions and for providing the chiral perturbation theory data from ref. [45]. The authors acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the CRC-TR 211 “Strong-interaction matter under extreme conditions” – project number 315477589 – TRR 211. F.C. acknowledges the support by the State of Hesse within the Research Cluster ELEMENTS (Project ID 500/10.006). The authors also gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUC-NG at Leibniz Supercomputing Centre (www.lrz.de). Parts of the computations in this work were performed on the GPU cluster at Bielefeld University and at Goethe-HLR at Goethe-University Frankfurt. We thank the computing staff of both institutions for their support.
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Brandt, B.B., Cuteri, F. & Endrődi, G. Equation of state and speed of sound of isospin-asymmetric QCD on the lattice. J. High Energ. Phys. 2023, 55 (2023). https://doi.org/10.1007/JHEP07(2023)055
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DOI: https://doi.org/10.1007/JHEP07(2023)055