Quark pseudoscalar vertex and quark mass function with clover fermions: Spontaneous symmetry breaking, operator product expansion, symmetry restoration at small volume

Abstract

We study the quark mass function on hypercubic lattices in a large range of physical volumes and cutoffs. To avoid the very large Wilson term artefact, we exploit the relation between the quark mass function and the pseudoscalar vertex in the continuum. We extrapolate to the chiral limit. In function of the physical volume, we observe a striking discontinuity in the properties of chiral extrapolation around a physical volume L c ' 6 GeV À1 ¼ 1:2 fm. It is present in the quark mass function, which collapses to zero, as well as in the pion mass and the quark condensate as directly calculated from the pseudoscalar correlator. It is strongly reminiscent of the phenomenon of chiral symmetry restoration observed by Neuberger and Narayanan at N C ¼ 1 around the same physical length. In the case of spontaneous symmetry breaking, we confirm that the operator product expansion of the quark mass function, involving the quark condensate, is not operative at the available momenta, even taking into account the unusually large high order corrections to the Wilson coefficient calculated by Chetyrkin and Maier; the gap remains large, around a factor 2, even at the largest momenta available to us (p ' 6 GeV).

Patterns and partners for chiral symmetry restoration

“…Hence, up to renormalization ambiguities related to the fields and vertices involved [33,34], they should be respected by any model or lattice calculation. In fact, the one-point WI relating quark condensates and pseudoscalar susceptibilities have been verified recently in the hadronic sector through chiral perturbation theory (ChPT) [31], and the identities (11) and (13) have been tested in the lattice [5].…”

“…The four partners in (34) and (35) would become degenerate within a Oð4Þ × Uð1Þ A pattern, where χ 5;disc ∼ 0. In turn, in a full Uð3Þ restoring scenario, all members of the scalar/pseudoscalar nonets would become degenerate.…”

“…Nevertheless, the latter limit is meant to be reached at a much higher temperature than chiral restoration, as seen, for instance, in the degeneration of screening masses [35]. The four correlators (34) and (35) have been actively investigated in lattice and theoretical analyses to study partner degeneracy [5,[22][23][24][25]27,36,37].…”

“…However, the ls correlators should be exactly zero at Oð4Þ restoration, i.e., in the regime where the Oð4Þ chiral partners in (34) with a ¼ 1, 2, 3 can be written as a sum of P-odd (first) and P-even (second) contributions. Thus, it is possible to choose an angle α a in (36) such that the P-odd part vanishes.…”

Chiral and U(1)A restoration for the scalar and pseudoscalar meson nonets

“…Thus, in principle, they are subject to renormalization ambiguities related to the fields and vertices involved [63,64]. It is therefore important to provide a specific low-energy realization of WI and the observables entering them, such as the scalar and pseudoscalar susceptibilities that we have been analyzing in previous sections.…”