# Aspects on Effective Theories and the QCD Transition

## Abstract

**:**

## 1. Introduction and Motivation: Remarks on the QCD Phase Diagram

## 2. Effective Theories below the Transition

## 3. The Role of Thermal Resonances

#### 3.1. Spectral Properties of Hadrons in the Thermal Bath

#### 3.2. The Thermal ${f}_{0}\left(500\right)$ and Chiral Symmetry Restoration

## 4. The Nature of the Transition: Partners and Patterns

#### 4.1. Ward Identities, $O\left(4\right)$ vs. $U{\left(1\right)}_{A}$ Restoration and Screening Masses

#### 4.2. $U\left(3\right)$ ChPT Analysis of the Scalar and Pseudoscalar Nonet

#### 4.3. The Topological Susceptibility

## 5. Conclusions and Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Adams, J.; et al. [STAR Collaboration] Experimental and theoretical challenges in the search for the quark gluon plasma: The STAR Collaboration’s critical assessment of the evidence from RHIC collisions. Nucl. Phys. A
**2005**, 757, 102–183. [Google Scholar] [CrossRef] [Green Version] - Adcox, K., et al. [PHENIX Collaboration] Formation of dense partonic matter in relativistic nucleus-nucleus collisions at RHIC: Experimental evaluation by the PHENIX collaboration. Nucl. Phys. A
**2005**, 757, 184–283. [Google Scholar] [CrossRef] [Green Version] - Ratti, C. Lattice QCD and heavy ion collisions: A review of recent progress. Rept. Prog. Phys.
**2018**, 81, 084301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bazavov, A.; Karsch, F.; Mukherjee, S.; Petreczky, P. Hot-dense Lattice QCD: USQCD whitepaper 2018. Eur. Phys. J. A
**2019**, 55, 194. [Google Scholar] [CrossRef] - Alford, M.G.; Schmitt, A.; Rajagopal, K.; Schafer, T. Color superconductivity in dense quark matter. Rev. Mod. Phys.
**2008**, 80, 1455–1515. [Google Scholar] [CrossRef] [Green Version] - Bazavov, A.; Brambilla, N.; Ding, H.-T.; Petreczky, P.; Schadler, H.-P.; Vairo, A.; Weber, J.H. Polyakov loop in 2+1 flavor QCD from low to high temperatures. Phys. Rev. D
**2016**, 93, 114502. [Google Scholar] [CrossRef] [Green Version] - Smilga, A.V.; Verbaarschot, J.J.M. Scalar susceptibility in QCD and the multiflavor Schwinger model. Phys. Rev. D
**1996**, 54, 1087. [Google Scholar] [CrossRef] [Green Version] - Pisarski, R.D.; Wilczek, F. Remarks on the Chiral Phase Transition in Chromodynamics. Phys. Rev. D
**1984**, 29, 338–341. [Google Scholar] [CrossRef] - Aoki, Y.; Borsanyi, S.; Durr, S.; Fodor, Z.; Katz, S.D.; Krieg, S.; Szabo, K.K. The QCD transition temperature: Results with physical masses in the continuum limit II. JHEP
**2009**, 0906, 088. [Google Scholar] [CrossRef] [Green Version] - Borsanyi, S.; Fodor, Z.; Hoelbling, C.; Katzc, S.D.; Krieg, S.; Ratti, C.; Szabo, K.K. Is there still any T
_{c}mystery in lattice QCD? Results with physical masses in the continuum limit III. JHEP**2010**, 1009, 073. [Google Scholar] [CrossRef] [Green Version] - Bazavov, A.; Bhattacharya, T.; Cheng, M.; DeTar, C.; Ding, H.-T.; Gottlieb, S.; Gupta, R.; Hegde, P.; Heller, U.M.; Karsch, F.; et al. The chiral and deconfinement aspects of the QCD transition. Phys. Rev. D
**2012**, 85, 054503. [Google Scholar] [CrossRef] [Green Version] - Bazavov, A.; Ding, H.-T.; Hegde, P.; Kaczmarek, O.; Karsch, F.; Karthik, N.; Laermann, E.; Lahiri, A.; Larsen, R.; Li, S.-T.; et al. Chiral crossover in QCD at zero and non-zero chemical potentials. Phys. Lett. B
**2019**, 795, 15–21. [Google Scholar] [CrossRef] - Ding, H.T.; Hegde, P.; Kaczmarek, O.; Karsch, F.; Lahiri, A.; Li, S.-T.; Mukherjee, S.; Ohno, H.; Petreczky, P.; Schmidt, C.; et al. Chiral Phase Transition Temperature in (2+1)-Flavor QCD. Phys. Rev. Lett.
**2019**, 123, 062002. [Google Scholar] [CrossRef] [PubMed] [Green Version] - D’Elia, M.; Lombardo, M.P. Finite density QCD via imaginary chemical potential. Phys. Rev. D
**2003**, 67, 014505. [Google Scholar] [CrossRef] [Green Version] - De Forcrand, P.; Philipsen, O. The Chiral critical line of N(f) = 2+1 QCD at zero and non-zero baryon density. JHEP
**2007**, 0701, 077. [Google Scholar] [CrossRef] [Green Version] - Fodor, Z.; Katz, S.D. Critical point of QCD at finite T and mu, lattice results for physical quark masses. JHEP
**2004**, 0404, 050. [Google Scholar] [CrossRef] [Green Version] - Aarts, G. Can stochastic quantization evade the sign problem? The relativistic Bose gas at finite chemical potential. Phys. Rev. Lett.
**2009**, 102, 131601. [Google Scholar] [CrossRef] - Adamczyk, L.; et al. [STAR Collaboration] Bulk Properties of the Medium Produced in Relativistic Heavy-Ion Collisions from the Beam Energy Scan Program. Phys. Rev. C
**2017**, 96, 044904. [Google Scholar] [CrossRef] [Green Version] - Andronic, A.; Braun-Munzinger, P.; Redlich, K.; Stachel, J. Decoding the phase structure of QCD via particle production at high energy. Nature
**2018**, 561, 321–330. [Google Scholar] [CrossRef] [Green Version] - Bazavov, A.; Ding, H.-T.; Hegde, P.; Kaczmarek, O.; Karsch, F.; Laermann, E.; Maezawa, Y.; Mukherjee, S.; Ohno, H.; Petreczky, P.; et al. Additional Strange Hadrons from QCD Thermodynamics and Strangeness Freezeout in Heavy Ion Collisions. Phys. Rev. Lett.
**2014**, 113, 072001. [Google Scholar] [CrossRef] [Green Version] - Luo, X.; Xu, N. Search for the QCD Critical Point with Fluctuations of Conserved Quantities in Relativistic Heavy-Ion Collisions at RHIC: An Overview. Nucl. Sci. Tech.
**2017**, 28, 112. [Google Scholar] [CrossRef] [Green Version] - Tanabashi, M.; et al. [Particle Data Group] Review of Particle Physics. Phys. Rev. D
**2018**, 98, 030001. [Google Scholar] [CrossRef] [Green Version] - Hagedorn, R. Hadronic matter near the boiling point. Nuovo Cim. A
**1968**, 56, 1027–1057. [Google Scholar] [CrossRef] [Green Version] - Karsch, F.; Redlich, K.; Tawfik, A. Thermodynamics at nonzero baryon number density: A Comparison of lattice and hadron resonance gas model calculations. Phys. Lett. B
**2003**, 571, 67–74. [Google Scholar] [CrossRef] [Green Version] - Karsch, F.; Redlich, K.; Tawfik, A. Hadron resonance mass spectrum and lattice QCD thermodynamics. Eur. Phys. J. C
**2003**, 29, 549–556. [Google Scholar] [CrossRef] - Tawfik, A.; Toublan, D. Quark-antiquark condensates in the hadronic phase. Phys. Lett. B
**2005**, 623, 48–54. [Google Scholar] [CrossRef] [Green Version] - Leupold, S. Four-quark condensates and chiral symmetry restoration in a resonance gas model. J. Phys. G
**2006**, 32, 2199–2218. [Google Scholar] [CrossRef] [Green Version] - Huovinen, P.; Petreczky, P. QCD Equation of State and Hadron Resonance Gas. Nucl. Phys. A
**2010**, 837, 26–53. [Google Scholar] [CrossRef] [Green Version] - Megias, E.; Arriola, E.R.; Salcedo, L.L. The Polyakov loop and the hadron resonance gas model. Phys. Rev. Lett.
**2012**, 109, 151601. [Google Scholar] [CrossRef] [Green Version] - Jankowski, J.; Blaschke, D.; Spalinski, M. Chiral condensate in hadronic matter. Phys. Rev. D
**2013**, 87, 105018. [Google Scholar] [CrossRef] [Green Version] - Andronic, A.; Braun-Munzinger, P.; Stachel, J. Hadron production in central nucleus-nucleus collisions at chemical freeze-out. Nucl. Phys. A
**2006**, 772, 167–199. [Google Scholar] [CrossRef] [Green Version] - Andronic, A.; Braun-Munzinger, P.; Stachel, J.; Winn, M. Interacting hadron resonance gas meets lattice QCD. Phys. Lett. B
**2012**, 718, 80–85. [Google Scholar] [CrossRef] - Huovinen, P.; Petreczky, P. Hadron Resonance Gas with Repulsive Interactions and Fluctuations of Conserved Charges. Phys. Lett. B
**2018**, 777, 125–130. [Google Scholar] [CrossRef] - Weinberg, S. Phenomenological Lagrangians. Physica A
**1979**, 96, 327–340. [Google Scholar] [CrossRef] - Kapusta, J.I.; Gale, C. Finite Temperature Field Theory. Principles and Applications; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Gasser, J.; Leutwyler, H. Chiral Perturbation Theory to One Loop. Ann. Phys.
**1984**, 158, 142–210. [Google Scholar] [CrossRef] [Green Version] - Gasser, J.; Leutwyler, H. Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark. Nucl. Phys. B
**1985**, 250, 465–516. [Google Scholar] [CrossRef] [Green Version] - Ecker, G.; Gasser, J.; Pich, A.; de Rafael, E. The Role of Resonances in Chiral Perturbation Theory. Nucl. Phys. B
**1989**, 321, 311–342. [Google Scholar] [CrossRef] [Green Version] - Ecker, G.; Gasser, J.; Leutwyler, H.; Pich, A.; de Rafael, E. Chiral Lagrangians for Massive Spin 1 Fields. Phys. Lett. B
**1989**, 223, 425–432. [Google Scholar] [CrossRef] - Meissner, U.G. Recent developments in chiral perturbation theory. Rept. Prog. Phys.
**1993**, 56, 903–996. [Google Scholar] [CrossRef] [Green Version] - Bernard, V.; Kaiser, N.; Meissner, U.G. Chiral dynamics in nucleons and nuclei. Int. J. Mod. Phys. E
**1995**, 4, 193–346. [Google Scholar] [CrossRef] [Green Version] - Ecker, G. Chiral perturbation theory. Prog. Part. Nucl. Phys.
**1995**, 35, 1–80. [Google Scholar] [CrossRef] [Green Version] - Pich, A. Chiral perturbation theory. Rept. Prog. Phys.
**1995**, 58, 563–610. [Google Scholar] [CrossRef] [Green Version] - Witten, E. Current Algebra Theorems for the U(1) Goldstone Boson. Nucl. Phys. B
**1979**, 156, 269–283. [Google Scholar] [CrossRef] - Vecchia, P.D.; Veneziano, G. Chiral Dynamics in the Large n Limit. Nucl. Phys. B
**1980**, 171, 253–271. [Google Scholar] [CrossRef] [Green Version] - Herrera-Siklody, P.; Latorre, J.I.; Pascual, P.; Taron, J. Chiral effective Lagrangian in the large N(c) limit: The Nonet case. Nucl. Phys. B
**1997**, 497, 345–386. [Google Scholar] [CrossRef] [Green Version] - Kaiser, R.; Leutwyler, H. Large N(c) in chiral perturbation theory. Eur. Phys. J. C
**2000**, 17, 623–649. [Google Scholar] [CrossRef] [Green Version] - Aoki, S.; Aoki, Y.; Becirevic, D.; Blum, T.; Colangelo, G.; Collins, S.; Morte, M.D.; Dimopoulos, P.; Durr, S.; Fukaya, H.; et al. FLAG Review 2019. Eur. Phys. J. C
**2020**, 80, 113. [Google Scholar] [CrossRef] [Green Version] - Gerber, P.; Leutwyler, H. Hadrons Below the Chiral Phase Transition. Nucl. Phys. B
**1989**, 321, 387–429. [Google Scholar] [CrossRef] - Leutwyler, H. QCD: Low Temperature Expansion and Finite Size Effects. Nucl. Phys. Proc. Suppl.
**1988**, 4, 248–258. [Google Scholar] [CrossRef] - Fernandez-Fraile, D.; Gómez Nicola, A. Bulk viscosity and the conformal anomaly in the pion gas. Phys. Rev. Lett.
**2009**, 102, 121601. [Google Scholar] [CrossRef] [Green Version] - Fernandez-Fraile, D.; Nicola, A.G. Transport coefficients and resonances for a meson gas in Chiral Perturbation Theory. Eur. Phys. J. C
**2009**, 62, 37. [Google Scholar] [CrossRef] - Fernandez-Fraile, D.; Nicola, A.G. The Electrical conductivity of a pion gas. Phys. Rev. D
**2006**, 73, 045025. [Google Scholar] [CrossRef] [Green Version] - Dobado, A.; Llanes-Estrada, F.J.; Torres-Rincon, J.M. Eta/s and phase transitions. Phys. Rev. D
**2009**, 79, 014002. [Google Scholar] [CrossRef] - Dobado, A.; Llanes-Estrada, F.J.; Torres-Rincon, J.M. Bulk viscosity of low-temperature strongly interacting matter. Phys. Lett. B
**2011**, 702, 43. [Google Scholar] [CrossRef] [Green Version] - Dobado, A.; Llanes-Estrada, F.J.; Torres-Rincon, J.M. Bulk viscosity and energy-momentum correlations in high energy hadron collisions. Eur. Phys. J. C
**2012**, 72, 1873. [Google Scholar] [CrossRef] [Green Version] - Das, S.K.; Ghosh, S.; Sarkar, S.; Alam, J.E. Drag and diffusion coefficients of B mesons in hot hadronic matter. Phys. Rev. D
**2012**, 85, 074017. [Google Scholar] [CrossRef] [Green Version] - Tolos, L.; Torres-Rincon, J.M.; Das, S.K. Transport coefficients of heavy baryons. Phys. Rev. D
**2016**, 94, 034018. [Google Scholar] [CrossRef] [Green Version] - Abhishek, A.; Mishra, H.; Ghosh, S. Transport coefficients in the Polyakov quark meson coupling model: A relaxation time approximation. Phys. Rev. D
**2018**, 97, 014005. [Google Scholar] [CrossRef] [Green Version] - Dashen, R.; Ma, S.K.; Bernstein, H.J. S Matrix formulation of statistical mechanics. Phys. Rev.
**1969**, 187, 345. [Google Scholar] [CrossRef] - Venugopalan, R.; Prakash, M. Thermal properties of interacting hadrons. Nucl. Phys. A
**1992**, 546, 718. [Google Scholar] [CrossRef] - Dobado, A.; Pelaez, J.R. Chiral symmetry and the pion gas virial expansion. Phys. Rev. D
**1999**, 59, 034004. [Google Scholar] [CrossRef] [Green Version] - Pelaez, J.R. The SU(2) and SU(3) chiral phase transitions within chiral perturbation theory. Phys. Rev. D
**2002**, 66, 096007. [Google Scholar] [CrossRef] [Green Version] - Martin, R.G.; Pelaez, J.R. Chiral condensate thermal evolution at finite baryon chemical potential within Chiral Perturbation Theory. Phys. Rev. D
**2006**, 74, 096003. [Google Scholar] [CrossRef] [Green Version] - Gómez Nicola, A.; Pelaez, J.R.; de Elvira, J.R. Scalar susceptibilities and four-quark condensates in the meson gas within Chiral Perturbation Theory. Phys. Rev. D
**2013**, 87, 016001. [Google Scholar] [CrossRef] [Green Version] - Broniowski, W.; Giacosa, F.; Begun, V. Cancellation of the σ meson in thermal models. Phys. Rev. C
**2015**, 92, 034905. [Google Scholar] [CrossRef] [Green Version] - Ferreres-Solé, S.; Gómez Nicola, A.; Vioque-Rodríguez, A. Role of the thermal f
_{0}(500) in chiral symmetry restoration. Phys. Rev. D**2019**, 99, 036018. [Google Scholar] [CrossRef] [Green Version] - Albaladejo, M.; Oller, J. On the size of the sigma meson and its nature. Phys. Rev. D
**2012**, 86, 034003. [Google Scholar] [CrossRef] [Green Version] - Pelaez, J.R. From controversy to precision on the sigma meson: A review on the status of the non-ordinary f
_{0}(500) resonance. Phys. Rept.**2016**, 658, 1. [Google Scholar] [CrossRef] [Green Version] - Gell-Mann, M.; Levy, M. The axial vector current in beta decay. Nuovo Cim.
**1960**, 16, 705. [Google Scholar] [CrossRef] - Hatsuda, T.; Kunihiro, T. Fluctuation Effects in Hot Quark Matter: Precursors of Chiral Transition at Finite Temperature. Phys. Rev. Lett.
**1985**, 55, 158. [Google Scholar] [CrossRef] - Bernard, V.; Meissner, U.G.; Zahed, I. Properties of the Scalar σ Meson at Finite Density. Phys. Rev. Lett.
**1987**, 59, 966. [Google Scholar] [CrossRef] [PubMed] - Bochkarev, A.; Kapusta, J.I. Chiral symmetry at finite temperature: Linear versus nonlinear sigma models. Phys. Rev. D
**1996**, 54, 4066. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ayala, A.; Sahu, S. Pion propagation in the linear sigma model at finite temperature. Phys. Rev. D
**2000**, 62, 056007. [Google Scholar] [CrossRef] [Green Version] - Masjuan, P.; Sanz-Cillero, J.J.; Virto, J. Some Remarks on the Pade Unitarization of Low-Energy Amplitudes. Phys. Lett. B
**2008**, 668, 14. [Google Scholar] [CrossRef] [Green Version] - Truong, T.N. Chiral Perturbation Theory and Final State Theorem. Phys. Rev. Lett.
**1988**, 61, 2526. [Google Scholar] [CrossRef] [PubMed] - Dobado, A.; Herrero, M.J.; Truong, T.N. Unitarized Chiral Perturbation Theory for Elastic Pion-Pion Scattering. Phys. Lett. B
**1990**, 235, 134. [Google Scholar] [CrossRef] [Green Version] - Dobado, A.; Pelaez, J.R. The inverse amplitude method in Chiral Perturbation Theory. Phys. Rev. D
**1997**, 56, 3057. [Google Scholar] [CrossRef] [Green Version] - Oller, J.A.; Oset, E. Chiral symmetry amplitudes in the S wave isoscalar and isovector channels and the σ, f
_{0}(980), a_{0}(980) scalar mesons. Nucl. Phys. A**1997**, 620, 438, Erratum in**1999**, 652, 407. [Google Scholar] [CrossRef] [Green Version] - Oller, J.A.; Oset, E.; Pelaez, J.R. Meson meson interaction in a nonperturbative chiral approach. Phys. Rev. D
**1999**, 59, 074001, Erratum in**1999**, 60, 099906; Erratum in**2007**, 75, 099903. [Google Scholar] [CrossRef] [Green Version] - Gómez Nicola, A.; Pelaez, J.R. Meson meson scattering within one loop chiral perturbation theory and its unitarization. Phys. Rev. D
**2002**, 65, 054009. [Google Scholar] [CrossRef] [Green Version] - Pelaez, J.; Gómez Nicola, A. Light meson resonances from unitarized chiral perturbation theory. AIP Conf. Proc.
**2003**, 660, 102–115. [Google Scholar] - Gómez Nicola, A.; Pelaez, J.R.; Rios, G. The Inverse Amplitude Method and Adler Zeros. Phys. Rev. D
**2008**, 77, 056006. [Google Scholar] [CrossRef] [Green Version] - Albaladejo, M.; Oller, J.; Roca, L. Dynamical generation of pseudoscalar resonances. Phys. Rev. D
**2010**, 82, 094019. [Google Scholar] [CrossRef] [Green Version] - Dobado, A.; Pelaez, J. On the large N(f) limit of chiral perturbation theory. Phys. Lett. B
**1992**, 286, 136–146. [Google Scholar] [CrossRef] - Dobado, A.; Morales, J. Pion mass effects in the large N limit of chi(PT). Phys. Rev. D
**1995**, 52, 2878–2890. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cortés, S.; Gómez Nicola, A.; Morales, J. Large-N pion scattering at finite temperature: The f
_{0}(500) and chiral restoration. Phys. Rev. D**2016**, 93, 036001. [Google Scholar] [CrossRef] [Green Version] - Cortés, S.; Gómez Nicola, A.; Morales, J. Chiral Symmetry Restoration for the large-N pion gas. Phys. Rev. D
**2016**, 94, 116008. [Google Scholar] [CrossRef] [Green Version] - Gasser, J.; Leutwyler, H. Light Quarks at Low Temperatures. Phys. Lett. B
**1987**, 184, 83. [Google Scholar] [CrossRef] - Schenk, A. Pion propagation at finite temperature. Phys. Rev. D
**1993**, 47, 5138. [Google Scholar] [CrossRef] - Goity, J.L.; Leutwyler, H. On the Mean Free Path of Pions in Hot Matter. Phys. Lett. B
**1989**, 228, 517. [Google Scholar] [CrossRef] - Pisarski, R.D.; Tytgat, M. Propagation of cool pions. Phys. Rev. D
**1996**, 54, R2989. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Adamczyk, L.; et al. [STAR Collaboration] Measurements of Dielectron Production in Au+Au Collisions at $\sqrt{{}^{S}NN}$ = 200 GeV from the STAR Experiment. Phys. Rev. C
**2015**, 92, 024912. [Google Scholar] [CrossRef] [Green Version] - Adare, A.; Aidala, C.; Ajitanand, N.N.; Akiba, Y.; Akimoto, R.; Alexander, J.; Alfred, M.; Al-Ta’ani, H.; Angerami, A.; Aoki, K.; et al. Dielectron production in Au+Au collisions at $\sqrt{{}^{S}NN}$ = 200 GeV. Phys. Rev. C
**2016**, 93, 014904. [Google Scholar] [CrossRef] [Green Version] - Acharya, S.; et al. [ALICE Collaboration] Measurement of dielectron production in central Pb-Pb collisions at $\sqrt{{}^{S}NN}$ = 2.76 TeV. Phys. Rev. C
**2019**, 99, 024002. [Google Scholar] [CrossRef] [Green Version] - Adamczewski-Musch, J.; et al. [HADES Collaboration] Probing dense baryon-rich matter with virtual photons. Nat. Phys.
**2019**, 15, 1040–1045. [Google Scholar] - Rapp, R.; Wambach, J. Chiral symmetry restoration and dileptons in relativistic heavy ion collisions. Adv. Nucl. Phys.
**2000**, 25, 1. [Google Scholar] - Jung, C.; Rennecke, F.; Tripolt, R.A.; von Smekal, L.; Wambach, J. In-Medium Spectral Functions of Vector- and Axial-Vector Mesons from the Functional Renormalization Group. Phys. Rev. D
**2017**, 95, 036020. [Google Scholar] [CrossRef] [Green Version] - Rapp, R.; van Hees, H. Thermal Dileptons as Fireball Thermometer and Chronometer. Phys. Lett. B
**2016**, 753, 586. [Google Scholar] [CrossRef] [Green Version] - Turbide, S.; Rapp, R.; Gale, C. Hadronic production of thermal photons. Phys. Rev. C
**2004**, 69, 014903. [Google Scholar] [CrossRef] [Green Version] - Paquet, J.F.; Shen, C.; Denicol, G.S.; Luzum, M.; Schenke, B.; Jeon, S.; Gale, C. Production of photons in relativistic heavy-ion collisions. Phys. Rev. C
**2016**, 93, 044906. [Google Scholar] [CrossRef] - Gómez Nicola, A.; Llanes-Estrada, F.J.; Pelaez, J. Finite temperature pion scattering to one loop in chiral perturbation theory. Phys. Lett. B
**2002**, 550, 55–64. [Google Scholar] [CrossRef] [Green Version] - Dobado, A.; Gómez Nicola, A.; Llanes-Estrada, F.J.; Pelaez, J. Thermal rho and sigma mesons from chiral symmetry and unitarity. Phys. Rev. C
**2002**, 66, 055201. [Google Scholar] [CrossRef] [Green Version] - Fernandez-Fraile, D.; Gómez Nicola, A.; Herruzo, E. Pion scattering poles and chiral symmetry restoration. Phys. Rev. D
**2007**, 76, 085020. [Google Scholar] [CrossRef] [Green Version] - Cabrera, D.; Fernandez-Fraile, D.; Gómez Nicola, A. Chiral Symmetry and light resonances in hot and dense matter. Eur. Phys. J. C
**2009**, 61, 879–892. [Google Scholar] [CrossRef] - Gómez Nicola, A.; de Elvira, J.R.; Andres, R.T. Chiral Symmetry Restoration and Scalar-Pseudoscalar partners in QCD. Phys. Rev. D
**2013**, 88, 076007. [Google Scholar] [CrossRef] [Green Version] - Weldon, H. Simple Rules for Discontinuities in Finite Temperature Field Theory. Phys. Rev. D
**1983**, 28, 2007. [Google Scholar] [CrossRef] - Ghosh, S.; Sarkar, S.; Mallik, S. Analytic structure of rho meson propagator at finite temperature. Eur. Phys. J. C
**2010**, 70, 251–262. [Google Scholar] [CrossRef] - Gómez Nicola, A.; Llanes-Estrada, F.J.; Pelaez, J. Finite temperature pion vector form-factors in chiral perturbation theory. Phys. Lett. B
**2005**, 606, 351–360. [Google Scholar] [CrossRef] [Green Version] - Song, C.; Koch, V. Pion electromagnetic form-factor at finite temperature. Phys. Rev. C
**1996**, 54, 3218–3231. [Google Scholar] [CrossRef] [Green Version] - Gao, R.; Guo, Z.; Pang, J. Thermal behaviors of light scalar resonances at low temperatures. Phys. Rev. D
**2019**, 100, 114028. [Google Scholar] [CrossRef] [Green Version] - Hanhart, C.; Pelaez, J.R.; Rios, G. Quark mass dependence of the rho and sigma from dispersion relations and Chiral Perturbation Theory. Phys. Rev. Lett.
**2008**, 100, 152001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pelissetto, A.; Vicari, E. Relevance of the axial anomaly at the finite-temperature chiral transition in QCD. Phys. Rev. D
**2013**, 88, 105018. [Google Scholar] [CrossRef] [Green Version] - Eser, J.; Grahl, M.; Rischke, D.H. Functional Renormalization Group Study of the Chiral Phase Transition Including Vector and Axial-vector Mesons. Phys. Rev. D
**2015**, 92, 096008. [Google Scholar] [CrossRef] [Green Version] - Fejos, G. Functional dependence of axial anomaly via mesonic fluctuations in the three flavor linear sigma model. Phys. Rev. D
**2015**, 92, 036011. [Google Scholar] [CrossRef] [Green Version] - Gross, D.J.; Pisarski, R.D.; Yaffe, L.G. QCD and Instantons at Finite Temperature. Rev. Mod. Phys.
**1981**, 53, 43. [Google Scholar] [CrossRef] - Mitter, M.; Schaefer, B.J. Fluctuations and the axial anomaly with three quark flavors. Phys. Rev. D
**2014**, 89, 054027. [Google Scholar] [CrossRef] [Green Version] - Kapusta, J.I.; Kharzeev, D.; McLerran, L.D. The Return of the prodigal Goldstone boson. Phys. Rev. D
**1996**, 53, 5028. [Google Scholar] [CrossRef] [Green Version] - Csorgo, T.; Vertesi, R.; Sziklai, J. Indirect observation of an in-medium η’ mass reduction in √$\sqrt{{}^{S}NN}$
= 200 GeV Au+Au collisions. Phys. Rev. Lett.
**2010**, 105, 182301. [Google Scholar] [CrossRef] [Green Version] - Kotov, A.Y.; Lombardo, M.P.; Trunin, A.M. Fate of the η? in the quark gluon plasma. Phys. Lett. B
**2019**, 794, 83. [Google Scholar] [CrossRef] - Shuryak, E.V. Which chiral symmetry is restored in hot QCD? Comments Nucl. Part. Phys.
**1994**, 21, 235. [Google Scholar] - Cohen, T.D. The High temperature phase of QCD and U(1)-A symmetry. Phys. Rev. D
**1996**, 54, R1867. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lee, S.H.; Hatsuda, T. U-a(1) symmetry restoration in QCD with N(f) flavors. Phys. Rev. D
**1996**, 54, R1871. [Google Scholar] [CrossRef] [Green Version] - Meggiolaro, E.; Morda, A. Remarks on the U(1) axial symmetry and the chiral transition in QCD at finite temperature. Phys. Rev. D
**2013**, 88, 096010. [Google Scholar] [CrossRef] [Green Version] - Heller, M.; Mitter, M. Pion and η-meson mass splitting at the two-flavor chiral crossover. Phys. Rev. D
**2016**, 94, 074002. [Google Scholar] [CrossRef] [Green Version] - Ishii, M.; Yonemura, K.; Takahashi, J.; Kouno, H.; Yahiro, M. Determination of U(1)
_{A}restoration from pion and a_{0}-meson screening masses: Toward the chiral regime. Phys. Rev. D**2016**, 93, 016002. [Google Scholar] [CrossRef] [Green Version] - Glozman, L. Chiralspin Symmetry and Its Implications for QCD. Universe
**2019**, 5, 38. [Google Scholar] [CrossRef] [Green Version] - Gómez Nicola, A.; de Elvira, J.R. Pseudoscalar susceptibilities and quark condensates: Chiral restoration and lattice screening masses. JHEP
**2016**, 1603, 186. [Google Scholar] [CrossRef] [Green Version] - Azcoiti, V. Topology in the SU(Nf) chiral symmetry restored phase of unquenched QCD and axion cosmology. Phys. Rev. D
**2016**, 94, 094505. [Google Scholar] [CrossRef] [Green Version] - Gómez Nicola, A.; de Elvira, J.R. Patterns and partners for chiral symmetry restoration. Phys. Rev. D
**2018**, 97, 074016. [Google Scholar] [CrossRef] [Green Version] - Gómez Nicola, A.; Elvira, J.R.D. Chiral and U(1)
_{A}restoration for the scalar and pseudoscalar meson nonets. Phys. Rev. D**2018**, 98, 014020. [Google Scholar] [CrossRef] [Green Version] - Buchoff, M.I.; Cheng, M.; Christ, N.H.; Ding, H.-T.; Jung, C.; Karsch, F.; Lin, Z.; Mawhinney, R.D.; Mukherjee, S.; Petreczky, P.; et al. QCD chiral transition, U(1)A symmetry and the Dirac spectrum using domain wall fermions. Phys. Rev. D
**2014**, 89, 054514. [Google Scholar] [CrossRef] [Green Version] - Aoki, S.; Fukaya, H.; Taniguchi, Y. Chiral symmetry restoration, eigenvalue density of Dirac operator and axial U(1) anomaly at finite temperature. Phys. Rev. D
**2012**, 86, 114512. [Google Scholar] [CrossRef] [Green Version] - Cossu, G.; Aoki, S.; Fukaya, H.; Hashimoto, S.; Kaneko, T.; Matsufuru, H.; Noaki, J.I. Finite temperature study of the axial U(1) symmetry on the lattice with overlap fermion formulation. Phys. Rev. D
**2013**, 87, 114514, Erratum in**2013**, 88, 019901. [Google Scholar] [CrossRef] [Green Version] - Tomiya, A.; Cossu, G.; Aoki, S.; Fukaya, H.; Hashimoto, S.; Kaneko, T.; Noaki, J. Evidence of effective axial U(1) symmetry restoration at high temperature QCD. Phys. Rev. D
**2017**, 96, 034509. [Google Scholar] [CrossRef] [Green Version] - Brandt, B.B.; Francis, A.; Meyer, H.B.; Philipsen, O.; Robaina, D.; Wittig, H. On the strength of the U(1)
_{A}anomaly at the chiral phase transition in N_{f}=2 QCD. JHEP**2016**, 1612, 158. [Google Scholar] [CrossRef] [Green Version] - Brandt, B.B.; Cè, M.; Francis, A.; Harris, T.; Meyer, H.B.; Wittig, H.; Philipsen, O. Testing the strength of the U
_{A}(1) anomaly at the chiral phase transition in two-flavour QCD. arXiv**2019**, arXiv:1904.02384. [Google Scholar] - Lee, J.W.; Lucini, B.; Piai, M. Symmetry restoration at high-temperature in two-color and two-flavor lattice gauge theories. JHEP
**2017**, 1704, 036. [Google Scholar] [CrossRef] [Green Version] - Chandrasekharan, S.; Li, A. Anomaly and a QCD-like phase diagram with massive bosonic baryons. JHEP
**2010**, 1012, 021. [Google Scholar] [CrossRef] [Green Version] - Aarts, G.; Allton, C.; Hands, S.; Jäger, B.; Praki, C.; Skullerud, J.I. Nucleons and parity doubling across the deconfinement transition. Phys. Rev. D
**2015**, 92, 014503. [Google Scholar] [CrossRef] [Green Version] - Aarts, G.; Allton, C.; Boni, D.D.; Hands, S.; Jäger, B.; Praki, C.; Skullerud, J.I. Light baryons below and above the deconfinement transition: Medium effects and parity doubling. JHEP
**2017**, 1706, 034. [Google Scholar] [CrossRef] [Green Version] - Bochicchio, M.; Maiani, L.; Martinelli, G.; Rossi, G.C.; Testa, M. Chiral Symmetry on the Lattice with Wilson Fermions. Nucl. Phys. B
**1985**, 262, 331. [Google Scholar] [CrossRef] - Boucaud, P.; Leroy, J.P.; Yaouanc, A.L.; Micheli, J.; Pene, O.; Rodriguez-Quintero, J. Quark pseudoscalar vertex and quark mass function with clover fermions: Spontaneous symmetry breaking, OPE, symmetry restoration at small volume. Phys. Rev. D
**2010**, 81, 094504. [Google Scholar] [CrossRef] [Green Version] - Karsch, F.; Laermann, E. Thermodynamics and in medium hadron properties from lattice QCD. arXiv
**2003**, arXiv:hep-lat/0305025. [Google Scholar] - Ishii, M.; Kouno, H.; Yahiro, M. Model prediction for temperature dependence of meson pole masses from lattice QCD results on meson screening masses. Phys. Rev. D
**2017**, 95, 114022. [Google Scholar] [CrossRef] [Green Version] - Cheng, M.; Datta, S.; Francis, A.; van der Heide, J.; Jung, C.; Kaczmarek, O.; Karsch, F.; Laermann, E.; Mawhinney, R.D.; Miao, C.; et al. Meson screening masses from lattice QCD with two light and the strange quark. Eur. Phys. J. C
**2011**, 71, 1564. [Google Scholar] [CrossRef] - Cheng, M.; Datta, S.; Francis, A.; van der Heide, J.; Jung, C.; Kaczmarek, O.; Karsch, F.; Laermann, E.; Mawhinney, R.D.; Miao, C.; et al. The QCD equation of state with almost physical quark masses. Phys. Rev. D
**2008**, 77, 014511. [Google Scholar] [CrossRef] [Green Version] - Maezawa, Y.; Bazavov, A.; Karsch, F.; Petreczky, P.; Mukherjee, S. Meson screening masses at finite temperature with Highly Improved Staggered Quarks. arXiv
**2013**, arXiv:1312.4375. [Google Scholar] - Gómez Nicola, A.; Elvira, J.R.D.; Vioque-Rodríguez, A. The QCD topological charge and its thermal dependence: The role of the η
^{′}. JHEP**2019**, 11, 086. [Google Scholar] - Guo, X.K.; Guo, Z.H.; Oller, J.A.; Sanz-Cillero, J.J. Scrutinizing the η-η
^{′}mixing, masses and pseudoscalar decay constants in the framework of U(3) chiral effective field theory. JHEP**2015**, 1506, 175. [Google Scholar] [CrossRef] [Green Version] - Di Cortona, G.G.; Hardy, E.; Vega, J.P.; Villadoro, G. The QCD axion, precisely. JHEP
**2016**, 1601, 034. [Google Scholar] [CrossRef] - Bernard, V.; Descotes-Genon, S.; Toucas, G. Topological susceptibility on the lattice and the three-flavour quark condensate. JHEP
**2012**, 1206, 051. [Google Scholar] [CrossRef] [Green Version] - Aoki, S.; Aoki, Y.; Bernard, C.; Blum, T.; Colangelo, G.; Della Morte, M.; Durr, S.; El-Khadra, A.X.; Fukaya, H.; Horsley, R.; et al. Review of lattice results concerning low-energy particle physics. Eur. Phys. J. C
**2017**, 77, 112. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Burger, F.; Ilgenfritz, E.M.; Lombardo, M.P.; Trunin, A. Chiral observables and topology in hot QCD with two families of quarks. Phys. Rev. D
**2018**, 98, 094501. [Google Scholar] [CrossRef] [Green Version] - Leutwyler, H.; Smilga, A.V. Spectrum of Dirac operator and role of winding number in QCD. Phys. Rev. D
**1992**, 46, 5607. [Google Scholar] [CrossRef] - Veneziano, G. U(1) Without Instantons. Nucl. Phys. B
**1979**, 159, 213. [Google Scholar] [CrossRef] [Green Version] - Mao, Y.-Y.; Chiu, T.-W. Topological Susceptibility to the One-Loop Order in Chiral Perturbation Theory. Phys. Rev. D
**2009**, 80, 034502. [Google Scholar] [CrossRef] [Green Version] - Bonati, C.; D’Elia, M.; Mariti, M.; Martinelli, G.; Mesiti, M.; Negro, F.; Sanfilippo, F.; Villadoro, G. Axion phenomenology and θ-dependence from N
_{f}= 2+1 lattice QCD. JHEP**2016**, 1603, 155. [Google Scholar] [CrossRef] - Vonk, T.; Guo, F.K.; Meißner, U.G. Aspects of the QCD θ-vacuum. JHEP
**2019**, 1906, 106. [Google Scholar] [CrossRef] [Green Version] - Vicari, E.; Panagopoulos, H. Theta dependence of SU(N) gauge theories in the presence of a topological term. Phys. Rept.
**2009**, 470, 93. [Google Scholar] [CrossRef] [Green Version] - Borsanyi, S.; Fodor, Z.; Guenther, J.; Kampert, K.-H.; Katz, S.D.; Kawanai, T.; Kovacs, T.G.; Mages, S.W.; Pasztor, A.; Pittler, F.; et al. Calculation of the axion mass based on high-temperature lattice quantum chromodynamics. Nature
**2016**, 539, 69. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Schematic representation of the Quantum Chromodynamics (QCD) phase diagram in the plane of temperature and baryon chemical potential.

**Figure 2.**Subtracted quark condensate and disconnected scalar susceptibility from the lattice work [11] plotted for different actions and lattice resolutions.

**Figure 3.**Particle density for pions, kaons and the rest of states in the Particle Data Group (PDG) with masses below 2 GeV.

**Figure 5.**One-loop Feynman diagrams contributing to two-particle scattering to order $\mathcal{O}\left({p}^{4}\right)$ in Chiral Perturbation Theory. The different diagrams correspond to (

**a**): s-channel, (

**b**): t-channel, (

**c**): u-channel, (

**d**): tadpoles from six-point vertices, (

**e**): tadpoles from external legs renormalization. All vertices come from the ${\mathcal{L}}_{2}$ lagrangian.

**Figure 7.**Thermal evolution of the real and imaginary parts of the thermal pole ${s}_{p}={({M}_{p}-i{\Gamma}_{p}/2)}^{2}$ for the $IJ=00$ and $IJ=11$ channels, from [103,106]. (

**Left**): real part ${M}_{p}$ and the scalar mass corresponding to the real part of the self-energy as explained in the text. (

**Right**): imaginary part ${\Gamma}_{p}$, compared to the thermal phase space in the vector channel.

**Figure 10.**Scalar susceptibility within the UChPT saturated approach fitting the A parameter to lattice data [67] with the central values of the LEC given in [112]. Fit 1 corresponds to fitting data up to $T\le {T}_{c}=155$ MeV while in fit 2 we include two more lattice points, up to $T=163$ MeV. The uncertainties in A and the bands correspond to the 95% confidence level of the fit. The lattice data and errors are from [9]. The ${\chi}^{2}$/dof values are 6.2 and 4.9 for fits 1 and 2 respectively.

**Figure 11.**HRG fits for the scalar susceptibilty [67]. Fit 1 corresponds to fitting data up to $T\le {T}_{c}=155$ MeV while in fit 2 we include two more lattice points, up to $T=163$ MeV. The uncertainties in B and the bands correspond to the 95% confidence level of the fit. The lattice data and errors are from [9]. The ${\chi}^{2}$/dof values are 1.3 and 10.3 for fits 1 and 2 respectively.

**Figure 12.**Comparison of lattice data for screening masses with those for the combinations of quark condensates predicted by the scaling law (48) and the WIs (38)–(41), from [131]. The lattice data are taken from [146] (masses) and [147] (condensates) with the same lattice action and resolution and ${T}_{0}=145$ MeV. the definition of the subtracted condensates ${\Delta}_{i}$ can be found in [128,131].

**Figure 13.**$U\left(3\right)$ ChPT results from [131] where the bands correspond to the LEC uncertainties given in [150]. (

**Left**): $I=0,1$ susceptibilities for the ${\pi}^{a}$, ${\delta}^{a}$, ${\sigma}_{l}$, ${\eta}_{l}$ states in (29), (30), where ${B}_{0}^{r}$ is the renormalized chiral ${B}_{0}$ parameter. (

**Right**): $I=1/2$ susceptibilities for K and $\kappa $ states.

**Figure 14.**$U\left(3\right)$ ChPT result for the chiral limit behaviour of different chiral and $U{\left(1\right)}_{A}$ restoration temperatures, from [131].

**Figure 15.**$U\left(3\right)$ ChPT results for the T-dependence of the topological susceptibility [149]. (

**Left**): result of the $U\left(3\right)$ calculation compared to the lattice results in [158,161], where the band corresponds to the LEC uncertainty. (

**Right**): comparison of the $U\left(3\right)$${\chi}_{top}$ temperature scaling with the quark condensate in different approximations.

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gómez Nicola, A.
Aspects on Effective Theories and the QCD Transition. *Symmetry* **2020**, *12*, 945.
https://doi.org/10.3390/sym12060945

**AMA Style**

Gómez Nicola A.
Aspects on Effective Theories and the QCD Transition. *Symmetry*. 2020; 12(6):945.
https://doi.org/10.3390/sym12060945

**Chicago/Turabian Style**

Gómez Nicola, Angel.
2020. "Aspects on Effective Theories and the QCD Transition" *Symmetry* 12, no. 6: 945.
https://doi.org/10.3390/sym12060945