Pseudoscalar susceptibilities and quark condensates: chiral restoration and lattice screening masses

Abstract

Abstract:We derive the formal Ward identities relating pseudoscalar susceptibilities and quark condensates in three-flavor QCD, including consistently the η-η sector and the U A (1) anomaly. These identities are verified in the low-energy realization provided by ChPT, both in the standard SU(3) framework for the octet case and combining the use of the U(3) framework and the large-N c expansion of QCD to account properly for the nonet sector and anomalous contributions. The analysis is performed including finite temperature corrections as well as the calculation of U(3) quark condensates and all pseudoscalar susceptibilities, which together with the full set of Ward identities, are new results of this work. Finally, the Ward identities are used to derive scaling relations for pseudoscalar masses which explain the behavior with temperature of lattice screening masses near chiral symmetry restoration.

Patterns and partners for chiral symmetry restoration

“…Let us remark that in the light chiral limitm → 0 þ , neithermχ 5;disc nor χ 8A P in (19) vanishes at T ¼ 0. In fact, the latter vanish at T ¼ 0 only when the anomalous part of the η 0 mass goes to zero for fixedm [31] 2 Therefore, we expect χ 5;disc ∼ 1=m and χ top ∼m away from T c [the latter from (44)]. This is supported also by [39], where it is argued that χ top ∼mhqqi l in the chiral limit, as hqqi l is regular in that limit [2,6].…”

“…To test this assumption, we compare 1 The normalization factor (43) is chosen so that the definition of χ top coincides with [5]. Such factor comes from our normalization of AðxÞ and our definition of Euclidean gauge fields, which follows [31]. 2 There is a missing multiplying M 2 0 in the LO ChPT expression for χ 8A P in Eq.…”

“…Consequently, its connected part χ con S would also peak at Oð4Þ × Uð1Þ A restoration. Actually, in the physical massive case, χ con S ¼ χ δ S =2 grows with T below the transition [5,37,40] and, because χ π P decreases like hqqi l [31], their degeneracy would give rise to a maximum for χ δ S near U A ð1Þ restoration. A hint of that behavior for χ con S is seen at around T ∼ 190 MeV in the lattice [4].…”

“…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].…”

“…Applying (1) to a single bilinear O a ¼ iψγ 5 λ a ψ ≡ P a , one obtains WI relating quark condensates and pseudoscalar susceptibilities [31]. In particular, for our present purposes, let us take O P ¼ π b ; η l ; η s ; K b ; A in (1), which gives…”