The 3d O(4) model as an effective approach to the QCD phase diagram

Authors

  • Edgar López-Contreras Insituto de Ciencias Nucleares, UNAM
  • José Antonio García-Hernández Instituto de Ciencias Nucleares, UNAM
  • Elías Natanael Polanco-Euán Instituto de Ciencias Nucleares, UNAM
  • Wolfgang Bietenholz Instituto de Ciencias Nucleares, UNAM

DOI:

https://doi.org/10.31349/SuplRevMexFis.3.020727

Keywords:

QCD phase diagram, non-linear sigma model, lattice simulations

Abstract

The QCD phase diagram is one of the most prominent outstanding puzzles within the Standard Model. Various experiments, which aim at its exploration beyond small baryon density, are operating or in preparation. From the theoretical side, this is an issue of non-perturbative QCD, and therefore of lattice simulations. However, a finite baryon density entails a technical problem (known as the “sign problem”), which has not been overcome so far. Here we present a study of an effective theory, the O(4) non-linear sigma model. It performs spontaneous symmetry breaking with the same Lie group structure as 2-flavor QCD in the chiral limit, which strongly suggests that they belong to the same universality class. Since we are interested in high temperature, we further assume dimensional reduction to the 3d O(4) model, which implies topological sectors. As pointed out by Skyrme, Wilczek and others, its topological charge takes the role of the baryon number. Hence the baryon chemical potential µB appears as an imaginary vacuum angle, which can be included in the lattice simulation without any sign problem. We present numerical results for the critical line in the chiral limit, and for the crossover in the presence of light quark masses. Their shapes are compatible with other predictions, but up to the value of about µB ≈ 300 MeV we do not find the notorious Critical Endpoint (CEP).

References

H.-T. Ding et al. (HotQCD Collaboration), Chiral phase transition temperature in (2+1)-Flavor QCD, Phys. Rev. Lett. 123 (2019) 062002 https://doi.org/10.1103/PhysRevLett.123.062002.

A.Yu. Kotov, M.P. Lombardo and A. Trunin, QCD transition at the physical point, and its scaling window from twisted mass Wilson fermions, arXiv:2105.09842 [hep-lat].

S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, C. Ratti, K. K. Szabo, Is there still any Tc mystery in lattice QCD? Results with physical masses in the continuum limit III, JHEP 1009 (2010) 073 https://doi.org/10.1007/JHEP09(2010)073. T. Bhattacharya et al. (HotQCD Collaboration), QCD Phase Transition with Chiral Quarks and Physical Quark Masses, Phys. Rev. Lett. 113 (2014) 082001 https://doi.org/10.1103/PhysRevLett.113.082001. A. Bazavov et al. (HotQCD Collaboration), Chiral crossover in QCD at zero and non-zero chemical potentials, Phys. Lett. B 795 (2019) 15 https://doi.org/10.1016/j.physletb.2019.05.013.

D. Blaschke et al., Searching for a QCD mixed phase at the Nuclotron-based Ion Collider fAcility (NICA White Paper) 2014 http://mpd.jinr.ru/wp-content/uploads/2016/04/WhitePaper_10.01.pdf. V. Abgaryan et al. (MPD Collaboration), Status and initial physics performance studies of the MPD-NICA experiment, in preparation.

P. de Forcrand, Simulating QCD at finite density, PoS LAT2009 (2009) 010 https://doi.org/10.22323/1.091.0010. L. Levkova, QCD at nonzero temperature and density, PoS LATTICE2011 (2011) 011 https://doi.org/10.22323/1.139.0011. C. Ratti, Lattice QCD and heavy ion collisions: a review of recent progress, Rept. Prog. Phys. 81 (2018) 084301 https://doi.org/10.1088/1361-6633/aabb97.

C. Laflamme et al., CI P(N − 1) quantum field theories with alkaline-earth atoms in optical lattices, Annals Phys. 370 (2016) 117 https://doi.org/10.1016/j.aop.2016.03.012; Proposal for the Quantum Simulation of the CI P(2) Model on Optical Lattices, PoS LATTICE2015 (2016) 311 https://doi.org/10.22323/1.251.0311.

Z.F. Cui, J.L. Zhang and H.S. Zong, Proper time regularization and the QCD chiral phase transition, Sci. Rep. 7 (2017) 45937 https://doi.org/10.1038/srep45937. J. Braun, M. Leonhardt and M. Pospiech, Fierz-complete NJL model study III: Emergence from quark-gluon dynamics, Phys. Rev. D 101 (2020) 036004 https://doi.org/10.1103/PhysRevD.101.036004.

G.A. Contrera, A.G. Grunfeld and D. Blaschke, Supporting the search for the CEP location with nonlocal PNJL models constrained by lattice QCD, Eur. Phys. J. A 52 (2016) 231 https://doi.org/10.1140/epja/i2016-16231-x. K. Xu, Z. Li and M. Huang, QCD critical end point from a realistic PNJL model, EPJ Web Conf. 192 (2018) 00019 https://doi.org/10.1051/epjconf/201819200019. M. Motta, W.M. Alberico, A. Beraudo, P. Costa and R. Stiele, Exploration of the phase diagram and the thermodynamic properties of QCD at finite temperature and chemical potential with the PNJL effective model, J. Phys. Conf. Ser. 1667 (2020) 012029 https://doi.org/10.1088/1742-6596/1667/1/012029. Y.P. Zhao, S.Y. Zuo and C.M. Li, QCD chiral phase transition and critical exponents within the nonextensive Polyakov-Nambu-Jona-Lasinio model, Chin. Phys. C 45 (2021) 073105 https://doi.org/10.1088/1674-1137/abf8a2. A. Sarkar, P. Deb and R. Bose, Study of finite volume number density fluctuation at RHIC energies in PNJL model for the search of QCD Critical Point, arXiv:2202.00034 [hep-ph].

A. Ayala, S. Hernandez-Ortiz and L.A. Hernandez, QCD phase diagram from chiral symmetry restoration: analytic approach at high and low temperature using the linear sigma model with quarks, Rev. Mex. Fıs. 64 (2018) 302 https://doi.org/10.31349/RevMexFis.64.302. A. Ayala, S. Hernandez-Ortiz, L.A. Hernandez, V. Knapp-Perez and R. Zamora, Fluctuating temperature and baryon chemical potential in heavy-ion collisions and the position of the critical end point in the effective QCD phase diagram, Phys. Rev. D 101 (2020) 074023 https://doi.org/10.1103/PhysRevD.101.074023. A. Ayala, B. Almeida Zamora, J.J. Cobos-Martınez, S. Hernandez-Ortiz, L.A. Hernandez, A. Raya and M.E. Tejeda-Yeomans, Collision energy dependence of the critical end point from baryon number fluctuations in the Linear Sigma Model with quarks, Eur. Phys. J. A 58 (2022) 87 https://doi.org/10.1140/epja/s10050-022-00732-8.

O. DeWolfe, S.S. Gubser and C. Rosen, Dynamic critical phenomena at a holographic critical point, Phys. Rev. D 84 (2011) 126014 https://doi.org/10.1103/PhysRevD.84.126014. R. Critelli, J. Noronha, J. Noronha-Hostler, I. Portillo, C. Ratti and R. Rougemont, Critical point in the phase diagram of primordial quark-gluon matter from black hole physics, Phys. Rev. D 96 (2017) 096026 https://doi.org/10.1103/PhysRevD.96.096026. Z. Li, K. Xu and M. Huang, The entanglement properties of holographic QCD model with a critical end point, Chin. Phys. C 45 (2021) 013116 https://doi.org/10.1088/1674-1137/abc539.

P. Kovacs and G. Wolf, Phase diagram and isentropic curves from the vector meson extended Polyakov quark meson model, Acta Phys. Polon. Supp. 10 (2017) 1107 https://doi.org/10.5506/APhysPolBSupp.10.1107.

C. Shi, Y.-L. Du, S.-S. Xu, X.-J. Liu and H.-S. Zong, Continuum study of the QCD phase diagram through an OPE-modified gluon propagator, Phys. Rev. D 93 (2016) 036006 https://doi.org/10.1103/PhysRevD.93.036006. B.L. Li, Z.F. Cui, B.W. Zhou, S. An, L.P. Zhang and H.S. Zong, Finite volume effects on the chiral phase transition from Dyson-Schwinger equations of QCD, Nucl. Phys. B 938 (2019) 298 https://doi.org/10.1016/j.nuclphysb.2018.11.015. Y.P. Zhao, R.R. Zhang, H. Zhang and H.S. Zong, Chiral phase transition from the Dyson-Schwinger equations in a finite spherical volume, Chin. Phys. C 43 (2019) 063101 https://doi.org/10.1088/1674-1137/43/6/063101. C. Shi, X.- T. He, W.-B. Jia, Q.-W. Wang, S.-S. Xu and H.-S. Zong, Chiral transition and the chiral charge density of the hot and dense QCD matter, JHEP 06 (2020) 122 https://doi.org/10.1007/JHEP06(2020)122. F. Gao and J.M. Pawlowski, QCD phase structure from functional methods Phys. Rev. D 102 (2020) 034027 https://doi.org/10.1103/PhysRevD.102.034027.

Z.Q. Wu, J.L. Ping and H.S. Zong, QCD phase diagram at finite isospin and baryon chemical potentials with the self-consistent mean field approximation. Chin. Phys. C 45 (2021) 064102 https://doi.org/10.1088/1674-1137/abefc3.

E.S. Fraga, L.F. Palhares and P. Sorensen, Finite-size scaling as a tool in the search for the QCD critical point in heavy ion data, Phys. Rev. C 84 (2011) 011903 https://doi.org/10.1103/PhysRevC.84.011903. N.G. Antoniou, F.K. Diakonos, X.N. Maintas and C.E. Tsagkarakis, Locating the QCD critical endpoint through finite-size scaling, Phys. Rev. D 97 (2018) 034015 https://doi.org/10.1103/PhysRevD.97.034015.

R. D. Pisarski and F. Wilczek, Remarks on the chiral phase transition in chromodynamics, Phys. Rev. D 29 (1984) 338 https://doi.org/10.1103/PhysRevD. 29.338. F. Wilczek, Application of the renormalization group to a second order QCD phase transition, Int. J. Mod. Phys. A 7 (1992) 3911 https://doi.org/10.1142/S0217751X92001757 [Erratum: Int. J. Mod. Phys. A 7 (1992) 6951 https://doi.org/10.1142/S0217751X92003665]. K. Rajagopal and F. Wilczek, Static and dynamic critical phenomena at a second order QCD phase transition, Nucl. Phys. B 399 (1993) 395 https://doi.org/10.1016/0550-3213(93)90502-G. A. Yu. Kotov, M. P. Lombardo and A. Trunin, Gliding Down the QCD Transition Line, from Nf = 2 till the Onset of Conformality, Symmetry 13 (2021) 1833 https://doi.org/10.3390/sym13101833.

W. Bietenholz, On the Isomorphic Description of Chiral Symmetry Breaking by Non-Unitary Lie Groups, Int. J. Mod. Phys. A 25 (2010) 1699 https://doi.org/10.1142/S0217751X10048123.

T. H. R. Skyrme, A Nonlinear field theory, Proc. Roy. Soc. Lond. A 260 (1961) 127 https://doi.org/10.1098/rspa.1961.0018; A unified field theory of mesons and baryons, Nucl. Phys. 31 (1962) 556 https://doi.org/10.1016/0029-5582(62)90775-7. G. S. Adkins, C. R. Nappi and E. Witten, Static properties of nucleons in the Skyrme model, Nucl. Phys. B 228 (1983) 552 https://doi.org/10.1016/0550-3213(83)90559-X. I. Zahed and G.E. Brown, The Skyrme Model, Phys. Rept. 142 (1986) 1 https://doi.org/10.1016/0370-1573(86)90142-0.

B. Berg and M. Lüscher, Definition and Statistical Distributions of a Topological Number in the Lattice O(3) Sigma Model, Nucl. Phys. B 190 (1981) 412 https://doi.org/10.1016/0550-3213(81)90568-X.

J. Murakami, Volume formulas for a spherical tetrahedron, Proc. Amer. Math. Soc. 140 (2012) 3289 https://doi.org/10.1090/S0002-9939-2012-11182-7.

M. A. Nava Blanco, W. Bietenholz and A. Fernandez Tellez, Conjecture about the 2-Flavour QCD Phase Diagram, J. Phys. Conf. Ser. 912 (2017) 012048 https://doi.org/10.1088/1742-6596/912/1/012048. M. A. Nava Blanco, Estudio del diagrama de fase de QCD con dos sabores usando el modelo 3d O(4), M.Sc. thesis, Benemerita Universidad Autonoma de Puebla, 2019.

U. Wolff, Collective Monte Carlo Updating for Spin Systems, Phys. Rev. Lett. 62 (1989) 361 https://doi.org/10.1103/PhysRevLett.62.361.

C.M. Fortuin and P.W. Kasteleyn, On the random-cluster model: I. Introduction and relation to other models, Physica 57 (1972) 536 https://doi.org/10.1016/0031-8914(72)90045-6.

E. Lopez-Contreras, Phase Diagram of the 3d O(4) model as a conjecture for Quantum Chromodynamics in the chiral limit, B.Sc. thesis, Universidad Nacional Autonoma de Mexico, 2021.

K. Kanaya and S. Kaya, Critical exponents of a threedimensional O(4) spin model, Phys. Rev. D 51 (1995) 2404 https://doi.org/10.1103/PhysRevD.51.2404.

M. Oevers, Kritisches Verhalten in O(4)-symmetrischen Spinmodellen und 2-flavour QCD, Diploma thesis, Universität Bielefeld, 1996.

J. Engels, L. Fromme and M. Seniuch, Correlation lengths and scaling functions in the three-dimensional O(4) model, Nucl. Phys. B 675 (2003) 533 https://doi.org/10.1016/j.nuclphysb.2003.09.060.

K. Binder and D.W. Heermann, Monte Carlo Simulation in Statistical Physics, Springer, 1997.

J.-S. Wang, Clusters in the three-dimensional Ising model with a magnetic field, Physica A 161 (1989) 249 https://doi.org/10.1016/0378-4371(89)90468-8.

J.A. Garcıa-Hernandez, Conjecture about the Quantum Chromodynamics Phase Diagram with two Light Quark Flavors, B.Sc. thesis, Universidad Nacional Autonoma de Mexico, 2020.

J. A. Garcıa-Hernandez, E. Lopez-Contreras, E. N. Polanco-Euan and W. Bietenholz, Conjecture about the QCD phase diagram, PoS LAT2021 (2021) 595. https://doi.org/10.48550/arXiv.2111.01954.

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Published

2022-06-30

How to Cite

1.
López-Contreras E, García-Hernández JA, Polanco-Euán EN, Bietenholz W. The 3d O(4) model as an effective approach to the QCD phase diagram. Supl. Rev. Mex. Fis. [Internet]. 2022 Jun. 30 [cited 2022 Oct. 4];3(2):020727 1-13. Available from: https://rmf.smf.mx/ojs/index.php/rmf-s/article/view/6173