Abstract
I discuss the possible effects of a finite density of localised near-zero Dirac modes in the chiral limit of gauge theories with Nf degenerate fermions. I focus in particular on the fate of the massless quasi-particle excitations predicted by the finite-temperature version of Goldstone’s theorem, for which I provide an alternative and generalised proof based on a Euclidean SU(Nf )A Ward-Takahashi identity. I show that localised near-zero modes can lead to a divergent pseudoscalar-pseudoscalar correlator that modifies this identity in the chiral limit. As a consequence, massless quasi-particle excitations can disappear from the spectrum of the theory in spite of a non-zero chiral condensate. Three different scenarios are possible, depending on the detailed behaviour in the chiral limit of the ratio of the mobility edge and the fermion mass, which I prove to be a renormalisation-group invariant quantity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Wuppertal-Budapest collaboration, Is there still any Tc mystery in lattice QCD? Results with physical masses in the continuum limit III, JHEP 09 (2010) 073 [arXiv:1005.3508] [INSPIRE].
A. Bazavov et al., Polyakov loop in 2 + 1 flavor QCD from low to high temperatures, Phys. Rev. D 93 (2016) 114502 [arXiv:1603.06637] [INSPIRE].
G. Boyd et al., Thermodynamics of SU(3) lattice gauge theory, Nucl. Phys. B 469 (1996) 419 [hep-lat/9602007] [INSPIRE].
P.H. Damgaard, U.M. Heller, A. Krasnitz and T. Madsen, A Quark-anti-quark condensate in three-dimensional QCD, Phys. Lett. B 440 (1998) 129 [hep-lat/9803012] [INSPIRE].
F. Karsch and M. Lütgemeier, Deconfinement and chiral symmetry restoration in an SU(3) gauge theory with adjoint fermions, Nucl. Phys. B 550 (1999) 449 [hep-lat/9812023] [INSPIRE].
J. Engels, S. Holtmann and T. Schulze, Scaling and Goldstone effects in a QCD with two flavors of adjoint quarks, Nucl. Phys. B 724 (2005) 357 [hep-lat/0505008] [INSPIRE].
F. Karsch, E. Laermann and C. Schmidt, The Chiral critical point in three-flavor QCD, Phys. Lett. B 520 (2001) 41 [hep-lat/0107020] [INSPIRE].
P. de Forcrand and O. Philipsen, The QCD phase diagram for three degenerate flavors and small baryon density, Nucl. Phys. B 673 (2003) 170 [hep-lat/0307020] [INSPIRE].
P. de Forcrand and O. Philipsen, The Chiral critical point of Nf = 3 QCD at finite density to the order (μ/T)4, JHEP 11 (2008) 012 [arXiv:0808.1096] [INSPIRE].
G. Bergner, C. López and S. Piemonte, Study of center and chiral symmetry realization in thermal \( \mathcal{N} \) = 1 super Yang-Mills theory using the gradient flow, Phys. Rev. D 100 (2019) 074501 [arXiv:1902.08469] [INSPIRE].
M. Göckeler, P.E.L. Rakow, A. Schäfer, W. Soldner and T. Wettig, Calorons and localization of quark eigenvectors in lattice QCD, Phys. Rev. Lett. 87 (2001) 042001 [hep-lat/0103031] [INSPIRE].
C. Gattringer, M. Göckeler, P.E.L. Rakow, S. Schaefer and A. Schaefer, A Comprehensive picture of topological excitations in finite temperature lattice QCD, Nucl. Phys. B 618 (2001) 205 [hep-lat/0105023] [INSPIRE].
A.M. García-García and J.C. Osborn, Chiral phase transition and anderson localization in the instanton liquid model for QCD, Nucl. Phys. A 770 (2006) 141 [hep-lat/0512025] [INSPIRE].
A.M. García-García and J.C. Osborn, Chiral phase transition in lattice QCD as a metal-insulator transition, Phys. Rev. D 75 (2007) 034503 [hep-lat/0611019] [INSPIRE].
R.V. Gavai, S. Gupta and R. Lacaze, Eigenvalues and Eigenvectors of the Staggered Dirac Operator at Finite Temperature, Phys. Rev. D 77 (2008) 114506 [arXiv:0803.0182] [INSPIRE].
T.G. Kovács, Absence of correlations in the QCD Dirac spectrum at high temperature, Phys. Rev. Lett. 104 (2010) 031601 [arXiv:0906.5373] [INSPIRE].
F. Bruckmann, T.G. Kovács and S. Schierenberg, Anderson localization through Polyakov loops: lattice evidence and Random matrix model, Phys. Rev. D 84 (2011) 034505 [arXiv:1105.5336] [INSPIRE].
T.G. Kovács and F. Pittler, Poisson to Random Matrix Transition in the QCD Dirac Spectrum, Phys. Rev. D 86 (2012) 114515 [arXiv:1208.3475] [INSPIRE].
G. Cossu and S. Hashimoto, Anderson Localization in high temperature QCD: background configuration properties and Dirac eigenmodes, JHEP 06 (2016) 056 [arXiv:1604.00768] [INSPIRE].
L. Holicki, E.-M. Ilgenfritz and L. von Smekal, The Anderson transition in QCD with Nf = 2 + 1 + 1 twisted mass quarks: overlap analysis, PoS LATTICE2018 (2018) 180 [arXiv:1810.01130] [INSPIRE].
M. Giordano and T.G. Kovács, Localization of Dirac Fermions in Finite-Temperature Gauge Theory, Universe 7 (2021) 194 [arXiv:2104.14388] [INSPIRE].
T.G. Kovács and R.Á. Vig, Localization transition in SU(3) gauge theory, Phys. Rev. D 97 (2018) 014502 [arXiv:1706.03562] [INSPIRE].
R.Á. Vig and T.G. Kovács, Localization with overlap fermions, Phys. Rev. D 101 (2020) 094511 [arXiv:2001.06872] [INSPIRE].
M. Giordano, Localisation in 2 + 1 dimensional SU(3) pure gauge theory at finite temperature, JHEP 05 (2019) 204 [arXiv:1903.04983] [INSPIRE].
M. Giordano, S.D. Katz, T.G. Kovács and F. Pittler, Deconfinement, chiral transition and localisation in a QCD-like model, JHEP 02 (2017) 055 [arXiv:1611.03284] [INSPIRE].
M. Giordano, T.G. Kovács and F. Pittler, Localization and chiral properties near the ordering transition of an Anderson-like toy model for QCD, Phys. Rev. D 95 (2017) 074503 [arXiv:1612.05059] [INSPIRE].
T.G. Kovács and F. Pittler, Anderson Localization in Quark-Gluon Plasma, Phys. Rev. Lett. 105 (2010) 192001 [arXiv:1006.1205] [INSPIRE].
F. Bruckmann and J. Wellnhofer, Anderson localization in sigma models, EPJ Web Conf. 175 (2018) 07005 [arXiv:1710.05662] [INSPIRE].
C. Bonati, M. Cardinali, M. D’Elia, M. Giordano and F. Mazziotti, Reconfinement, localization and thermal monopoles in SU(3) trace-deformed Yang-Mills theory, Phys. Rev. D 103 (2021) 034506 [arXiv:2012.13246] [INSPIRE].
G. Baranka and M. Giordano, Localization of Dirac modes in finite-temperature Z2 gauge theory on the lattice, Phys. Rev. D 104 (2021) 054513 [arXiv:2104.03779] [INSPIRE].
M. Cardinali, M. D’Elia, F. Garosi and M. Giordano, Localization properties of Dirac modes at the Roberge-Weiss phase transition, Phys. Rev. D 105 (2022) 014506 [arXiv:2110.10029] [INSPIRE].
M. Giordano, T.G. Kovács and F. Pittler, An Ising-Anderson model of localisation in high-temperature QCD, JHEP 04 (2015) 112 [arXiv:1502.02532] [INSPIRE].
M. Giordano, T.G. Kovács and F. Pittler, An Anderson-like model of the QCD chiral transition, JHEP 06 (2016) 007 [arXiv:1603.09548] [INSPIRE].
T. Banks and A. Casher, Chiral Symmetry Breaking in Confining Theories, Nucl. Phys. B 169 (1980) 103 [INSPIRE].
D. Diakonov, Chiral symmetry breaking by instantons, Proc. Int. Sch. Phys. Fermi 130 (1996) 397 [hep-ph/9602375] [INSPIRE].
R.Á. Vig and T.G. Kovács, Ideal topological gas in the high temperature phase of SU(3) gauge theory, Phys. Rev. D 103 (2021) 114510 [arXiv:2101.01498] [INSPIRE].
P.W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109 (1958) 1492 [INSPIRE].
D. Thouless, Electrons in disordered systems and the theory of localization, Phys. Rep. 13 (1974) 93.
P.A. Lee and T.V. Ramakrishnan, Disordered electronic systems, Rev. Mod. Phys. 57 (1985) 287 [INSPIRE].
B. Kramer and A. MacKinnon, Localization: theory and experiment, Rep. Prog. Phys. 56 (1993) 1469.
F. Evers and A.D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80 (2008) 1355 [arXiv:0707.4378] [INSPIRE].
E. Abrahams, ed., 50 Years of Anderson Localization, World Scientific, Singapore (2010) [DOI].
R. Nandkishore and D.A. Huse, Many body localization and thermalization in quantum statistical mechanics, Annu. Rev. Condens. Matter Phys. 6 (2015) 15 [arXiv:1404.0686] [INSPIRE].
M. Giordano, T.G. Kovács and F. Pittler, Universality and the QCD Anderson Transition, Phys. Rev. Lett. 112 (2014) 102002 [arXiv:1312.1179] [INSPIRE].
S.M. Nishigaki, M. Giordano, T.G. Kovács and F. Pittler, Critical statistics at the mobility edge of QCD Dirac spectra, PoS LATTICE2013 (2014) 018 [arXiv:1312.3286] [INSPIRE].
M. Giordano, T.G. Kovács and F. Pittler, Anderson localization in QCD-like theories, Int. J. Mod. Phys. A 29 (2014) 1445005 [arXiv:1409.5210] [INSPIRE].
L. Ujfalusi, M. Giordano, F. Pittler, T.G. Kovács and I. Varga, Anderson transition and multifractals in the spectrum of the Dirac operator of Quantum Chromodynamics at high temperature, Phys. Rev. D 92 (2015) 094513 [arXiv:1507.02162] [INSPIRE].
J. Goldstone, A. Salam and S. Weinberg, Broken Symmetries, Phys. Rev. 127 (1962) 965 [INSPIRE].
R.V. Lange, Goldstone Theorem in Nonrelativistic Theories, Phys. Rev. Lett. 14 (1965) 3 [INSPIRE].
D. Kastler, D.W. Robinson and A. Swieca, Conserved currents and associated symmetries; Goldstone’s theorem, Commun. Math. Phys. 2 (1966) 108 [INSPIRE].
J.A. Swieca, Range of forces and broken symmetries in many-body systems, Commun. Math. Phys. 4 (1967) 1 [INSPIRE].
G. Morchio and F. Strocchi, Mathematical Structures for Long Range Dynamics and Symmetry Breaking, J. Math. Phys. 28 (1987) 622 [INSPIRE].
F. Strocchi, Symmetry Breaking, Lecture Notes in Physics 732, Springer, Berlin (2008) [DOI].
A.J. McKane and M. Stone, Localization as an alternative to Goldstone’s theorem, Annals Phys. 131 (1981) 36 [INSPIRE].
M. Golterman and Y. Shamir, Localization in lattice QCD, Phys. Rev. D 68 (2003) 074501 [hep-lat/0306002] [INSPIRE].
S. Aoki, New Phase Structure for Lattice QCD with Wilson Fermions, Phys. Rev. D 30 (1984) 2653 [INSPIRE].
M. Giordano, Localised Dirac eigenmodes, chiral symmetry breaking, and Goldstone’s theorem at finite temperature, J. Phys. A 54 (2021) 37LT01 [arXiv:2009.00486] [INSPIRE].
M. Giordano, Localised Dirac eigenmodes and Goldstone’s theorem at finite temperature, PoS LATTICE2021 (2022) 401 [arXiv:2110.12250] [INSPIRE].
V. Dick, F. Karsch, E. Laermann, S. Mukherjee and S. Sharma, Microscopic origin of UA(1) symmetry violation in the high temperature phase of QCD, Phys. Rev. D 91 (2015) 094504 [arXiv:1502.06190] [INSPIRE].
H.T. Ding, S.T. Li, S. Mukherjee, A. Tomiya, X.D. Wang and Y. Zhang, Correlated Dirac Eigenvalues and Axial Anomaly in Chiral Symmetric QCD, Phys. Rev. Lett. 126 (2021) 082001 [arXiv:2010.14836] [INSPIRE].
O. Kaczmarek, L. Mazur and S. Sharma, Eigenvalue spectra of QCD and the fate of UA(1) breaking towards the chiral limit, Phys. Rev. D 104 (2021) 094518 [arXiv:2102.06136] [INSPIRE].
H.-T. Ding, W.-P. Huang, M. Lin, S. Mukherjee, P. Petreczky and Y. Zhang, Correlated Dirac eigenvalues around the transition temperature on Nτ = 8 lattices, PoS LATTICE2021 (2022) 591 [arXiv:2112.00318] [INSPIRE].
H.-T. Ding, S.-T. Li, X.-D. Wang, Y. Zhang, A. Tomiya and S. Mukherjee, Correlated Dirac Eigenvalues and Axial Anomaly in Chiral Symmetric QCD, PoS LATTICE2021 (2022) 619 [arXiv:2112.00465] [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].
J.I. Kapusta and C. Gale, Finite-Temperature Field Theory, Cambridge University Press (2006) [DOI].
M. Laine and A. Vuorinen, Basics of Thermal Field Theory, Lecture Notes in Physics 925, Springer (2016) [DOI] [arXiv:1701.01554] [INSPIRE].
S. Fulling and S. Ruijsenaars, Temperature, periodicity and horizons, Phys. Rep. 152 (1987) 135.
J. Bros and D. Buchholz, Axiomatic analyticity properties and representations of particles in thermal quantum field theory, Ann. Inst. H. Poincare Phys. Theor. 64 (1996) 495 [hep-th/9606046] [INSPIRE].
G. Cuniberti, E. De Micheli and G.A. Viano, Reconstructing the thermal Green functions at real times from those at imaginary times, Commun. Math. Phys. 216 (2001) 59 [cond-mat/0109175] [INSPIRE].
H.B. Meyer, The Bulk Channel in Thermal Gauge Theories, JHEP 04 (2010) 099 [arXiv:1002.3343] [INSPIRE].
H.B. Meyer, Transport Properties of the Quark-Gluon Plasma: A Lattice QCD Perspective, Eur. Phys. J. A 47 (2011) 86 [arXiv:1104.3708] [INSPIRE].
R. Haag, N.M. Hugenholtz and M. Winnink, On the Equilibrium states in quantum statistical mechanics, Commun. Math. Phys. 5 (1967) 215 [INSPIRE].
R. Kubo, Statistical mechanical theory of irreversible processes. I. General theory and simple applications in magnetic and conduction problems, J. Phys. Soc. Jap. 12 (1957) 570 [INSPIRE].
P.C. Martin and J.S. Schwinger, Theory of many particle systems. I, Phys. Rev. 115 (1959) 1342 [INSPIRE].
C.W. Bernard, Feynman Rules for Gauge Theories at Finite Temperature, Phys. Rev. D 9 (1974) 3312 [INSPIRE].
V.N. Gribov, Quantization of Nonabelian Gauge Theories, Nucl. Phys. B 139 (1978) 1 [INSPIRE].
I.M. Singer, Some Remarks on the Gribov Ambiguity, Commun. Math. Phys. 60 (1978) 7 [INSPIRE].
A. Alexandru and I. Horváth, Unusual Features of QCD Low-Energy Modes in the Infrared Phase, Phys. Rev. Lett. 127 (2021) 052303 [arXiv:2103.05607] [INSPIRE].
A. Alexandru and I. Horváth, Anderson metal-to-critical transition in QCD, Phys. Lett. B 833 (2022) 137370 [arXiv:2110.04833] [INSPIRE].
I. Horváth and P. Markoš, Super-Universality in Anderson Localization, Phys. Rev. Lett. 129 (2022) 106601 [arXiv:2110.11266] [INSPIRE].
C. Vafa and E. Witten, Restrictions on Symmetry Breaking in Vector-Like Gauge Theories, Nucl. Phys. B 234 (1984) 173 [INSPIRE].
S.L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev. 177 (1969) 2426 [INSPIRE].
J.S. Bell and R. Jackiw, A PCAC puzzle: π0 → γγ in the σ model, Nuovo Cim. A 60 (1969) 47 [INSPIRE].
K. Fujikawa, Path Integral Measure for Gauge Invariant Fermion Theories, Phys. Rev. Lett. 42 (1979) 1195 [INSPIRE].
K. Fujikawa, Path Integral for Gauge Theories with Fermions, Phys. Rev. D 21 (1980) 2848 [INSPIRE].
J.C. Ward, An Identity in Quantum Electrodynamics, Phys. Rev. 78 (1950) 182 [INSPIRE].
Y. Takahashi, On the generalized Ward identity, Nuovo Cim. 6 (1957) 371 [INSPIRE].
B.B. Brandt, A. Francis, H.B. Meyer and D. Robaina, Chiral dynamics in the low-temperature phase of QCD, Phys. Rev. D 90 (2014) 054509 [arXiv:1406.5602] [INSPIRE].
H.J. Rothe, Lattice gauge theories: An Introduction, World Scientific, Singapore (1992) [DOI].
I. Montvay and G. Münster, Quantum Fields on a Lattice, Cambridge University Press (1994) [DOI].
C. Gattringer and C.B. Lang, Quantum chromodynamics on the lattice, Lecture Notes in Physics 788, Springer, Berlin (2010) [DOI].
H.B. Nielsen and M. Ninomiya, No Go Theorem for Regularizing Chiral Fermions, Phys. Lett. B 105 (1981) 219 [INSPIRE].
H.B. Nielsen and M. Ninomiya, Absence of Neutrinos on a Lattice. I. Proof by Homotopy Theory, Nucl. Phys. B 185 (1981) 20 [INSPIRE].
H.B. Nielsen and M. Ninomiya, Absence of Neutrinos on a Lattice. II. Intuitive Topological Proof, Nucl. Phys. B 193 (1981) 173 [INSPIRE].
P.H. Ginsparg and K.G. Wilson, A Remnant of Chiral Symmetry on the Lattice, Phys. Rev. D 25 (1982) 2649 [INSPIRE].
P. Hasenfratz and F. Niedermayer, Perfect lattice action for asymptotically free theories, Nucl. Phys. B 414 (1994) 785 [hep-lat/9308004] [INSPIRE].
T.A. DeGrand, A. Hasenfratz, P. Hasenfratz and F. Niedermayer, The Classically perfect fixed point action for SU(3) gauge theory, Nucl. Phys. B 454 (1995) 587 [hep-lat/9506030] [INSPIRE].
D.B. Kaplan, A Method for simulating chiral fermions on the lattice, Phys. Lett. B 288 (1992) 342 [hep-lat/9206013] [INSPIRE].
Y. Shamir, Chiral fermions from lattice boundaries, Nucl. Phys. B 406 (1993) 90 [hep-lat/9303005] [INSPIRE].
R. Narayanan and H. Neuberger, Chiral determinant as an overlap of two vacua, Nucl. Phys. B 412 (1994) 574 [hep-lat/9307006] [INSPIRE].
R. Narayanan and H. Neuberger, Chiral fermions on the lattice, Phys. Rev. Lett. 71 (1993) 3251 [hep-lat/9308011] [INSPIRE].
H. Neuberger, Exactly massless quarks on the lattice, Phys. Lett. B 417 (1998) 141 [hep-lat/9707022] [INSPIRE].
H. Neuberger, More about exactly massless quarks on the lattice, Phys. Lett. B 427 (1998) 353 [hep-lat/9801031] [INSPIRE].
M. Lüscher, Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation, Phys. Lett. B 428 (1998) 342 [hep-lat/9802011] [INSPIRE].
P. Hasenfratz, Lattice QCD without tuning, mixing and current renormalization, Nucl. Phys. B 525 (1998) 401 [hep-lat/9802007] [INSPIRE].
Y. Kikukawa and A. Yamada, Axial vector current of exact chiral symmetry on the lattice, Nucl. Phys. B 547 (1999) 413 [hep-lat/9808026] [INSPIRE].
P. Hasenfratz, S. Hauswirth, T. Jörg, F. Niedermayer and K. Holland, Testing the fixed point QCD action and the construction of chiral currents, Nucl. Phys. B 643 (2002) 280 [hep-lat/0205010] [INSPIRE].
R. Frezzotti and G.C. Rossi, Chirally improving Wilson fermions. I. O(a) improvement, JHEP 08 (2004) 007 [hep-lat/0306014] [INSPIRE].
F. Karsch, E. Laermann, P. Petreczky and S. Stickan, Infinite temperature limit of meson spectral functions calculated on the lattice, Phys. Rev. D 68 (2003) 014504 [hep-lat/0303017] [INSPIRE].
G. Aarts and J.M. Martínez Resco, Continuum and lattice meson spectral functions at nonzero momentum and high temperature, Nucl. Phys. B 726 (2005) 93 [hep-lat/0507004] [INSPIRE].
Y. Burnier et al., Thermal quarkonium physics in the pseudoscalar channel, JHEP 11 (2017) 206 [arXiv:1709.07612] [INSPIRE].
S. Nussinov and M.A. Lampert, QCD inequalities, Phys. Rep. 362 (2002) 193 [hep-ph/9911532] [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].
L. Del Debbio, L. Giusti, M. Lüscher, R. Petronzio and N. Tantalo, Stability of lattice QCD simulations and the thermodynamic limit, JHEP 02 (2006) 011 [hep-lat/0512021] [INSPIRE].
L. Giusti and M. Lüscher, Chiral symmetry breaking and the Banks-Casher relation in lattice QCD with Wilson quarks, JHEP 03 (2009) 013 [arXiv:0812.3638] [INSPIRE].
A. Bazavov et al., Meson screening masses in (2 + 1)-flavor QCD, Phys. Rev. D 100 (2019) 094510 [arXiv:1908.09552] [INSPIRE].
E.H. Lieb and M. Loss, Analysis: Second Edition, Graduate Studies in Mathematics 14, American Mathematical Society, Providence (2001).
T.G. Kovács, Localization at the quenched SU(3) phase transition, PoS LATTICE2021 (2022) 238 [arXiv:2112.05454] [INSPIRE].
R.D. Pisarski and F. Wilczek, Remarks on the Chiral Phase Transition in Chromodynamics, Phys. Rev. D 29 (1984) 338 [INSPIRE].
G. Fejős, Second-order chiral phase transition in three-flavor quantum chromodynamics?, Phys. Rev. D 105 (2022) L071506 [arXiv:2201.07909] [INSPIRE].
F. Cuteri, O. Philipsen and A. Sciarra, On the order of the QCD chiral phase transition for different numbers of quark flavours, JHEP 11 (2021) 141 [arXiv:2107.12739] [INSPIRE].
HotQCD collaboration, Chiral Phase Transition Temperature in (2 + 1)-Flavor QCD, Phys. Rev. Lett. 123 (2019) 062002 [arXiv:1903.04801] [INSPIRE].
L.Y. Glozman, Three regimes of QCD, Int. J. Mod. Phys. A 36 (2021) 2044031 [arXiv:1907.01820] [INSPIRE].
A. Alexandru and I. Horváth, Possible New Phase of Thermal QCD, Phys. Rev. D 100 (2019) 094507 [arXiv:1906.08047] [INSPIRE].
M. Cardinali, M. D’Elia and A. Pasqui, Thermal monopole condensation in QCD with physical quark masses, arXiv:2107.02745 [INSPIRE].
R.E. Norton and J.M. Cornwall, On the Formalism of Relativistic Many Body Theory, Annals Phys. 91 (1975) 106 [INSPIRE].
M.B. Kislinger and P.D. Morley, Collective Phenomena in Gauge Theories. II. Renormalization in Finite Temperature Field Theory, Phys. Rev. D 13 (1976) 2771 [INSPIRE].
R.D. Pisarski, Computing Finite Temperature Loops with Ease, Nucl. Phys. B 309 (1988) 476 [INSPIRE].
A. Vladikas, Three Topics in Renormalization and Improvement, in Les Houches Summer School: Session 93: Modern perspectives in lattice QCD: Quantum field theory and high performance computing, Les Houches France, August 3–28 2009 [Oxford Academic (2011), pp. 161–222, DOI] [arXiv:1103.1323] [INSPIRE].
E.C. Titchmarsh, The theory of functions, Oxford University Press, Oxford (1939).
C.W. Bernard and M.F.L. Golterman, Partially quenched gauge theories and an application to staggered fermions, Phys. Rev. D 49 (1994) 486 [hep-lat/9306005] [INSPIRE].
M. Lüscher, Topological effects in QCD and the problem of short distance singularities, Phys. Lett. B 593 (2004) 296 [hep-th/0404034] [INSPIRE].
K. Cichy, E. García-Ramos and K. Jansen, Short distance singularities and automatic O(a) improvement: the cases of the chiral condensate and the topological susceptibility, JHEP 04 (2015) 048 [arXiv:1412.0456] [INSPIRE].
J.J.M. Verbaarschot and T. Wettig, Random matrix theory and chiral symmetry in QCD, Ann. Rev. Nucl. Part. Sci. 50 (2000) 343 [hep-ph/0003017] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2206.11109
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
Giordano, M. Localisation of Dirac modes in gauge theories and Goldstone’s theorem at finite temperature. J. High Energ. Phys. 2022, 103 (2022). https://doi.org/10.1007/JHEP12(2022)103
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2022)103