Aspects of the QCD θ-vacuum

  • Thomas VonkEmail author
  • Feng-Kun Guo
  • Ulf-G. Meißner
Open Access
Regular Article - Theoretical Physics


This paper addresses two aspects concerning the θ-vacuum of Quantum Chro-modynamics. First, large-Nc chiral perturbation theory is used to calculate the first two non-trivial cumulants of the distribution of the winding number, i.e. the topological susceptibility, χtop, and the fourth cumulant, c4, up to next-to-leading order. Their large-Nc scaling is discussed, and compared to lattice results. It is found that \( {\chi}_{\mathrm{top}} = \mathcal{O}\left({N}_c^0\right) \), as known before, and \( {c}_4 = \mathcal{O}\left({N}_c^{-3}\right) \), correcting the assumption of \( \mathcal{O}\left({N}_c^{-2}\right) \) in the literature. Second, we discuss the properties of QCD at θπ using chiral perturbation theory for the case of 2 + 1 light flavors, i.e. by taking the strange quark mass heavier than the degenerate up and down quark masses. It is shown that — in accordance with previous findings for Nf = 2 and Nf = 3 mass-degenerate flavors — in the region θπ two vacuum states coexist, which become degenerate at θ = π. The wall tension of the energy barrier between these degenerate vacua is determined as well as the decay rate of a false vacuum.


Chiral Lagrangians Effective Field Theories 1/N Expansion 


Open Access

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  1. [1]
    A.A. Belavin et al., Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59 (1975) 85 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    T. Schäfer and E.V. Shuryak, Instantons in QCD, Rev. Mod. Phys. 70 (1998) 323 [hep-ph/9610451] [INSPIRE].
  3. [3]
    C.G. Callan Jr., R.F. Dashen and D.J. Gross, The structure of the gauge theory vacuum, Phys. Lett. B 63 (1976) 334 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    C.G. Callan Jr., R.F. Dashen and D.J. Gross, Toward a theory of the strong interactions, Phys. Rev. D 17 (1978) 2717 [INSPIRE].ADSGoogle Scholar
  5. [5]
    H. Leutwyler and A.V. Smilga, Spectrum of Dirac operator and role of winding number in QCD, Phys. Rev. D 46 (1992) 5607 [INSPIRE].ADSMathSciNetGoogle Scholar
  6. [6]
    E. Witten, Large N chiral dynamics, Annals Phys. 128 (1980) 363 [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    A.V. Smilga, QCD at θπ, Phys. Rev. D 59 (1999) 114021 [hep-ph/9805214] [INSPIRE].
  8. [8]
    C.A. Baker et al., An improved experimental limit on the electric dipole moment of the neutron, Phys. Rev. Lett. 97 (2006) 131801 [hep-ex/0602020] [INSPIRE].
  9. [9]
    J.M. Pendlebury et al., Revised experimental upper limit on the electric dipole moment of the neutron, Phys. Rev. D 92 (2015) 092003 [arXiv:1509.04411] [INSPIRE].ADSGoogle Scholar
  10. [10]
    V. Baluni, CP violating effects in QCD, Phys. Rev. D 19 (1979) 2227 [INSPIRE].ADSGoogle Scholar
  11. [11]
    F.K. Guo et al., The electric dipole moment of the neutron from 2 + 1 flavor lattice QCD, Phys. Rev. Lett. 115 (2015) 062001 [arXiv:1502.02295] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    J. Dragos et al., Confirming the existence of the strong CP problem in lattice QCD with the gradient flow, arXiv:1902.03254 [INSPIRE].
  13. [13]
    J.E. Kim, Light pseudoscalars, particle physics and cosmology, Phys. Rept. 150 (1987) 1 [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    J.E. Kim and G. Carosi, Axions and the strong CP problem, Rev. Mod. Phys. 82 (2010) 557 [arXiv:0807.3125] [INSPIRE].ADSGoogle Scholar
  15. [15]
    G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
  16. [16]
    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].ADSCrossRefGoogle Scholar
  17. [17]
    B. Lucini and M. Panero, SU(N) gauge theories at large N, Phys. Rept. 526 (2013) 93 [arXiv:1210.4997] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    F. Luciano and E. Meggiolaro, Study of the θ dependence of the vacuum energy density in chiral effective Lagrangian models at zero temperature, Phys. Rev. D 98 (2018) 074001 [arXiv:1806.00835] [INSPIRE].ADSGoogle Scholar
  19. [19]
    M.H.G. Tytgat, QCD at θπ reexamined: domain walls and spontaneous CP-violation, Phys. Rev. D 61 (2000) 114009 [hep-ph/9909532] [INSPIRE].
  20. [20]
    J. Gasser and H. Leutwyler, Chiral perturbation theory to one loop, Annals Phys. 158 (1984) 142 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    J. Gasser and H. Leutwyler, Chiral perturbation theory: expansions in the mass of the strange quark, Nucl. Phys. B 250 (1985) 465 [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    V. Bernard and U.-G. Meißner, Chiral perturbation theory, Ann. Rev. Nucl. Part. Sci. 57 (2007) 33 [hep-ph/0611231] [INSPIRE].
  23. [23]
    S. Scherer and M.R. Schindler, A primer for chiral perturbation theory, Lecture Notes Physics volume 830, Springer, Germany (2012).CrossRefGoogle Scholar
  24. [24]
    TWQCD collaboration, Topological susceptibility to the one-loop order in chiral perturbation theory, Phys. Rev. D 80 (2009) 034502 [arXiv:0903.2146] [INSPIRE].
  25. [25]
    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].ADSCrossRefGoogle Scholar
  26. [26]
    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].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    F.-K. Guo and U.-G. Meißner, Cumulants of the QCD topological charge distribution, Phys. Lett. B 749 (2015) 278 [arXiv:1506.05487] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    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].
  29. [29]
    P. Dimopoulos et al., Topological susceptibility and ηmeson mass from N f = 2 lattice QCD at the physical point, Phys. Rev. D 99 (2019) 034511 [arXiv:1812.08787].ADSGoogle Scholar
  30. [30]
    M. Cè, C. Consonni, G.P. Engel and L. Giusti, Non-Gaussianities in the topological charge distribution of the SU(3) Yang-Mills theory, Phys. Rev. D 92 (2015) 074502 [arXiv:1506.06052].ADSGoogle Scholar
  31. [31]
    M. Cè, M. García Vera, L. Giusti and S. Schaefer, The topological susceptibility in the large-N limit of SU(N ) Yang-Mills theory, Phys. Lett. B 762 (2016) 232 [arXiv:1607.05939].
  32. [32]
    R.J. Crewther, Chirality selection rules and the U(1) problem, Phys. Lett. 70B (1977) 349 [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    P. Di Vecchia and G. Veneziano, Chiral dynamics in the large N limit, Nucl. Phys. B 171 (1980) 253 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    C. Rosenzweig, J. Schechter and C.G. Trahern, Is the effective Lagrangian for QCD a σ-model?, Phys. Rev. D 21 (1980) 3388 [INSPIRE].ADSGoogle Scholar
  35. [35]
    R. Kaiser and H. Leutwyler, Large N c in chiral perturbation theory, Eur. Phys. J. C 17 (2000) 623 [hep-ph/0007101] [INSPIRE].
  36. [36]
    K. Kawarabayashi and N. Ohta, The problem of η in the large N limit: effective Lagrangian approach, Nucl. Phys. B 175 (1980) 477.ADSCrossRefGoogle Scholar
  37. [37]
    N. Ohta, Vacuum structure and chiral charge quantization in the large N limit, Prog. Theor. Phys. 66 (1981) 1408 [Erratum ibid. 67 (1982) 993].Google Scholar
  38. [38]
    G. Veneziano, U(1) without instantons, Nucl. Phys. B 159 (1979) 213 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    P. Herrera-Siklody, J.I. Latorre, P. Pascual and J. Taron, Chiral effective Lagrangian in the large N c limit: the nonet case, Nucl. Phys. B 497 (1997) 345 [hep-ph/9610549] [INSPIRE].
  40. [40]
    H. Leutwyler, Implications of ηηmixing for the decay η → 3π, Phys. Lett. B 374 (1996) 181 [hep-ph/9601236] [INSPIRE].
  41. [41]
    G.M. Shore, The U(1)A anomaly and QCD phenomenology, Lect. Notes Phys. 737 (2008) 235 [hep-ph/0701171] [INSPIRE].
  42. [42]
    Particle Data Group collaboration, Review of particle physics, Phys. Rev. D 98 (2018) 030001 [INSPIRE].
  43. [43]
    B. Lucini and M. Teper, SU(N) gauge theories in four-dimensions: exploring the approach to N = ∞, JHEP 06(2001) 050 [hep-lat/0103027] [INSPIRE].
  44. [44]
    L. Del Debbio, H. Panagopoulos and E. Vicari, θ dependence of SU(N) gauge theories, JHEP 08 (2002) 044 [hep-th/0204125] [INSPIRE].
  45. [45]
    N. Cundy, M. Teper and U. Wenger, Topology and chiral symmetry breaking in SU(N c) gauge theories, Phys. Rev. D 66 (2002) 094505 [hep-lat/0203030] [INSPIRE].
  46. [46]
    B. Lucini, M. Teper and U. Wenger, Topology of SU(N ) gauge theories at T = 0 and T = T(c), Nucl. Phys. B 715(2005) 461 [hep-lat/0401028] [INSPIRE].
  47. [47]
    M. D’Elia, Field theoretical approach to the study of theta dependence in Yang-Mills theories on the lattice, Nucl. Phys. B 661 (2003) 139 [hep-lat/0302007] [INSPIRE].
  48. [48]
    L. Giusti, S. Petrarca and B. Taglienti, Theta dependence of the vacuum energy in the SU(3) gauge theory from the lattice, Phys. Rev. D 76 (2007) 094510 [arXiv:0705.2352] [INSPIRE].ADSGoogle Scholar
  49. [49]
    R. Dashen, Some features of chiral symmetry breaking, Phys. Rev. D 3 (1971) 1879.ADSMathSciNetGoogle Scholar
  50. [50]
    D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, time reversal and temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    D. Gaiotto, Z. Komargodski and N. Seiberg, Time-reversal breaking in QCD 4 , walls and dualities in 2 + 1 dimensions, JHEP 01 (2018) 110 [arXiv:1708.06806] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    M. Creutz, Quark masses and chiral symmetry, Phys. Rev. D 52 (1995) 2951 [hep-th/9505112] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical PhysicsUniversität BonnBonnGermany
  2. 2.CAS Key Laboratory of Theoretical Physics, Institute of Theoretical PhysicsChinese Academy of SciencesBeijingChina
  3. 3.School of Physical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  4. 4.Institute for Advanced SimulationInstitut für Kernphysik and Jülich Center for Hadron PhysicsJülichGermany
  5. 5.Tbilisi State UniversityTbilisiGeorgia

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