Abstract
We analyze the contribution of the η′ (958) meson in the first two non-trivial moments of the QCD topological charge distribution, namely, the topological susceptibility and the fourth-order cumulant of the vacuum energy density. We perform our study within U(3) Chiral Perturbation Theory up to next-to-next-to-leading order in the combined chiral and large-Nc expansion. We also describe the temperature dependence of these two quantities and compare them with previous analyses in the literature. In particular, we discuss the validity of the thermal scaling of the topological susceptibility with the quark condensate, which is intimately connected with a Ward Identity relating both quantities. We also consider isospin breaking corrections from the vacuum misalignment at leading order in the U(3) framework.
References
E. Witten, Current algebra theorems for the U(1) Goldstone boson, Nucl. Phys. B 156 (1979) 269 [INSPIRE].
G. Veneziano, U(1) without instantons, Nucl. Phys. B 159 (1979) 213 [INSPIRE].
P. Di Vecchia and G. Veneziano, Chiral dynamics in the large N limit, Nucl. Phys. B 171 (1980) 253 [INSPIRE].
J. Gasser and H. Leutwyler, Chiral perturbation theory: expansions in the mass of the strange quark, Nucl. Phys. B 250 (1985) 465 [INSPIRE].
H. Leutwyler and A.V. Smilga, Spectrum of Dirac operator and role of winding number in QCD, Phys. Rev. D 46 (1992) 5607 [INSPIRE].
JLQCD, TWQCD collaboration, Topological susceptibility in two-flavor lattice QCD with exact chiral symmetry, Phys. Lett. B 665 (2008) 294 [arXiv:0710.1130] [INSPIRE].
F. Bernardoni et al., Probing the chiral regime of Nf = 2 QCD with mixed actions, Phys. Rev. D 83 (2011) 054503 [arXiv:1008.1870] [INSPIRE].
RBC, UKQCD collaboration, 2 + 1 flavor domain wall QCD on a (2 fm)∗83 lattice: Light meson spectroscopy with L(s) = 16, Phys. Rev. D 76 (2007) 014504 [hep-lat/0701013] [INSPIRE].
TWQCD collaboration, Topological susceptibility in 2 + 1 flavors lattice QCD with domain-wall fermions, Phys. Lett. B 671 (2009) 135 [arXiv:0810.3406] [INSPIRE].
V. Bernard, S. Descotes-Genon and G. Toucas, Topological susceptibility on the lattice and the three-flavour quark condensate, JHEP 06 (2012) 051 [arXiv:1203.0508] [INSPIRE].
S. Aoki et al., Review of lattice results concerning low-energy particle physics, Eur. Phys. J. C 77 (2017) 112 [arXiv:1607.00299] [INSPIRE].
C. Bonati et al., Axion phenomenology and θ-dependence from Nf = 2 + 1 lattice QCD, JHEP 03 (2016) 155 [arXiv:1512.06746] [INSPIRE].
P. Petreczky, H.-P. Schadler and S. Sharma, The topological susceptibility in finite temperature QCD and axion cosmology, Phys. Lett. B 762 (2016) 498 [arXiv:1606.03145] [INSPIRE].
S. Borsányi et al., Calculation of the axion mass based on high-temperature lattice quantum chromodynamics, Nature 539 (2016) 69 [arXiv:1606.07494] [INSPIRE].
P. Dimopoulos et al., Topological susceptibility and η′ meson mass from Nf = 2 lattice QCD at the physical point, Phys. Rev. D 99 (2019) 034511 [arXiv:1812.08787] [INSPIRE].
F. Burger, E.-M. Ilgenfritz, M.P. Lombardo and A. Trunin, Chiral observables and topology in hot QCD with two families of quarks, Phys. Rev. D 98 (2018) 094501 [arXiv:1805.06001] [INSPIRE].
F.C. Hansen and H. Leutwyler, Charge correlations and topological susceptibility in QCD, Nucl. Phys. B 350 (1991) 201 [INSPIRE].
A. Gómez Nicola and J. Ruiz de Elvira, Pseudoscalar susceptibilities and quark condensates: chiral restoration and lattice screening masses, JHEP 03 (2016) 186 [arXiv:1602.01476] [INSPIRE].
V. Azcoiti, Topology in the SU(N F) chiral symmetry restored phase of unquenched QCD and axion cosmology, Phys. Rev. D 94 (2016) 094505 [arXiv:1609.01230] [INSPIRE].
A. Gómez Nicola and J. Ruiz De Elvira, Chiral and U (1)A restoration for the scalar and pseudoscalar meson nonets, Phys. Rev. D 98 (2018) 014020 [arXiv:1803.08517] [INSPIRE].
M.I. Buchoff et al., QCD chiral transition, U(1)A symmetry and the Dirac spectrum using domain wall fermions, Phys. Rev. D 89 (2014) 054514 [arXiv:1309.4149] [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].
A. Gomez Nicola and J. Ruiz de Elvira, Patterns and partners for chiral symmetry restoration, Phys. Rev. D 97 (2018) 074016 [arXiv:1704.05036] [INSPIRE].
TWQCD collaboration, Topological susceptibility to the one-loop order in chiral perturbation theory, Phys. Rev. D 80 (2009) 034502 [arXiv:0903.2146] [INSPIRE].
G. Grilli di Cortona, E. Hardy, J. Pardo Vega and G. Villadoro, The QCD axion, precisely, JHEP 01 (2016) 034 [arXiv:1511.02867] [INSPIRE].
M. Gorghetto and G. Villadoro, Topological susceptibility and QCD axion mass: QED and NNLO corrections, JHEP 03 (2019) 033 [arXiv:1812.01008] [INSPIRE].
R.D. Peccei and H.R. Quinn, CP conservation in the presence of instantons, Phys. Rev. Lett. 38 (1977) 1440 [INSPIRE].
S. Weinberg, A new light boson?, Phys. Rev. Lett. 40 (1978) 223 [INSPIRE].
M. Spalinski, Chiral corrections to the axion mass, Z. Phys. C 41 (1988) 87 [INSPIRE].
V. Bernard, S. Descotes-Genon and G. Toucas, Determining the chiral condensate from the distribution of the winding number beyond topological susceptibility, JHEP 12 (2012) 080 [arXiv:1209.4367] [INSPIRE].
F.-K. Guo and U.-G. Meißner, Cumulants of the QCD topological charge distribution, Phys. Lett. B 749 (2015) 278 [arXiv:1506.05487] [INSPIRE].
E. Vicari and H. Panagopoulos, Theta dependence of SU(N) gauge theories in the presence of a topological term, Phys. Rept. 470 (2009) 93 [arXiv:0803.1593] [INSPIRE].
H. Panagopoulos and E. Vicari, The 4D SU(3) gauge theory with an imaginary θ term, JHEP 11 (2011) 119 [arXiv:1109.6815] [INSPIRE].
C. Bonati, M. D’Elia, P. Rossi and E. Vicari, θ dependence of 4D SU(N) gauge theories in the large-N limit, Phys. Rev. D 94 (2016) 085017 [arXiv:1607.06360] [INSPIRE].
T. Vonk, F.-K. Guo and U.-G. Meißner, Aspects of the QCD θ-vacuum, JHEP 06 (2019) 106 [Erratum ibid. 10 (2019) 028] [arXiv:1905.06141] [INSPIRE].
HotQCD collaboration, The chiral transition and U (1)A symmetry restoration from lattice QCD using Domain Wall Fermions, Phys. Rev. D 86 (2012) 094503 [arXiv:1205.3535] [INSPIRE].
C. Bonati, M. D’Elia, H. Panagopoulos and E. Vicari, Change of θ dependence in 4D SU(N) gauge theories across the deconfinement transition, Phys. Rev. Lett. 110 (2013) 252003 [arXiv:1301.7640] [INSPIRE].
P. Herrera-Siklody, J.I. Latorre, P. Pascual and J. Taron, Chiral effective Lagrangian in the large Nc limit: the nonet case, Nucl. Phys. B 497 (1997) 345 [hep-ph/9610549] [INSPIRE].
R. Kaiser and H. Leutwyler, Large Nc in chiral perturbation theory, Eur. Phys. J. C 17 (2000) 623 [hep-ph/0007101] [INSPIRE].
A. Gomez Nicola and R. Torres Andres, Isospin-breaking quark condensates in chiral perturbation theory, J. Phys. G 39 (2012) 015004 [arXiv:1009.2170] [INSPIRE].
R. Escribano, F.S. Ling and M.H.G. Tytgat, Large Nc, chiral approach to M (η′) at finite temperature, Phys. Rev. D 62 (2000) 056004 [hep-ph/0003052] [INSPIRE].
X.-W. Gu, C.-G. Duan and Z.-H. Guo, Updated study of the η- η′ mixing and the thermal properties of light pseudoscalar mesons at low temperatures, Phys. Rev. D 98 (2018) 034007 [arXiv:1803.07284] [INSPIRE].
M. Ishii, H. Kouno and M. Yahiro, Model prediction for temperature dependence of meson pole masses from lattice QCD results on meson screening masses, Phys. Rev. D 95 (2017) 114022 [arXiv:1609.04575] [INSPIRE].
A.Yu. Kotov, M.P. Lombardo and A.M. Trunin, Fate of the η′ in the quark gluon plasma, Phys. Lett. B 794 (2019) 83 [arXiv:1903.05633] [INSPIRE].
C. Rosenzweig, J. Schechter and C.G. Trahern, Is the effective Lagrangian for QCD a σ-model?, Phys. Rev. D 21 (1980) 3388 [INSPIRE].
Z.-H. Guo, J.A. Oller and J. Ruiz de Elvira, Chiral dynamics in U(3) unitary chiral perturbation theory, Phys. Lett. B 712 (2012) 407 [arXiv:1203.4381] [INSPIRE].
Z.-H. Guo, J.A. Oller and J. Ruiz de Elvira, Chiral dynamics in form factors, spectral-function sum rules, meson-meson scattering and semi-local duality, Phys. Rev. D 86 (2012) 054006 [arXiv:1206.4163] [INSPIRE].
X.-K. Guo, Z.-H. Guo, J.A. Oller and J.J. Sanz-Cillero, Scrutinizing the η- η′ mixing, masses and pseudoscalar decay constants in the framework of U(3) chiral effective field theory, JHEP 06 (2015) 175 [arXiv:1503.02248] [INSPIRE].
A. Gomez Nicola, J.R. Pelaez and J. Ruiz de Elvira, Non-factorization of four-quark condensates at low energies within chiral perturbation theory, Phys. Rev. D 82 (2010) 074012 [arXiv:1005.4370] [INSPIRE].
A. Gomez Nicola, J.R. Pelaez and J. Ruiz de Elvira, Scalar susceptibilities and four-quark condensates in the meson gas within Chiral Perturbation Theory, Phys. Rev. D 87 (2013) 016001 [arXiv:1210.7977] [INSPIRE].
H. Leutwyler, Implications of η- η′ mixing for the decay η → 3π, Phys. Lett. B 374 (1996) 181 [hep-ph/9601236] [INSPIRE].
Flavour Lattice Averaging Group collaboration, FLAG review 2019, arXiv:1902.08191 [INSPIRE].
F. Karsch, K. Redlich and A. Tawfik, Hadron resonance mass spectrum and lattice QCD thermodynamics, Eur. Phys. J. C 29 (2003) 549 [hep-ph/0303108] [INSPIRE].
A. Tawfik and D. Toublan, Quark-antiquark condensates in the hadronic phase, Phys. Lett. B 623 (2005) 48 [hep-ph/0505152] [INSPIRE].
P. Huovinen and P. Petreczky, QCD equation of state and hadron resonance gas, Nucl. Phys. A 837 (2010) 26 [arXiv:0912.2541] [INSPIRE].
J. Jankowski, D. Blaschke and M. Spalinski, Chiral condensate in hadronic matter, Phys. Rev. D 87 (2013) 105018 [arXiv:1212.5521] [INSPIRE].
A. Gomez Nicola, J. Ruiz de Elvira and R. Torres Andres, Chiral symmetry restoration and scalar-pseudoscalar partners in QCD, Phys. Rev. D 88 (2013) 076007 [arXiv:1304.3356] [INSPIRE].
E.V. Shuryak, Which chiral symmetry is restored in hot QCD?, Comments Nucl. Part. Phys. 21 (1994) 235 [hep-ph/9310253] [INSPIRE].
T.D. Cohen, The high temperature phase of QCD and U(1)A symmetry, Phys. Rev. D 54 (1996) R1867 [hep-ph/9601216] [INSPIRE].
S.H. Lee and T. Hatsuda, U(1)A symmetry restoration in QCD with N (f) flavors, Phys. Rev. D 54 (1996) R1871 [hep-ph/9601373] [INSPIRE].
E. Meggiolaro and A. Morda, Remarks on the U (1) axial symmetry and the chiral transition in QCD at finite temperature, Phys. Rev. D 88 (2013) 096010 [arXiv:1309.4598] [INSPIRE].
S. Aoki, H. Fukaya and Y. Taniguchi, Chiral symmetry restoration, eigenvalue density of Dirac operator and axial U(1) anomaly at finite temperature, Phys. Rev. D 86 (2012) 114512 [arXiv:1209.2061] [INSPIRE].
G. Cossu et al., Finite temperature study of the axial U(1) symmetry on the lattice with overlap fermion formulation, Phys. Rev. D 87 (2013) 114514 [Erratum ibid. D 88 (2013) 019901] [arXiv:1304.6145] [INSPIRE].
V. Dick et al., Microscopic origin of UA (1) symmetry violation in the high temperature phase of QCD, Phys. Rev. D 91 (2015) 094504 [arXiv:1502.06190] [INSPIRE].
A. Tomiya et al., Evidence of effective axial U(1) symmetry restoration at high temperature QCD, Phys. Rev. D 96 (2017) 034509 [arXiv:1612.01908] [INSPIRE].
B.B. Brandt et al., On the strength of the U(1)A anomaly at the chiral phase transition in Nf = 2 QCD, JHEP 12 (2016) 158 [arXiv:1608.06882] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1907.11734
Electronic supplementary material
ESM 1
(PDF 1628 kb)
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Nicola, A.G., de Elvira, J.R. & Vioque-Rodríguez, A. The QCD topological charge and its thermal dependence: the role of the η′. J. High Energ. Phys. 2019, 86 (2019). https://doi.org/10.1007/JHEP11(2019)086
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2019)086
Keywords
- Chiral Lagrangians
- 1/N Expansion
- Anomalies in Field and String Theories
- Phase Diagram of QCD