Determining the chiral condensate from the distribution of the winding number beyond topological susceptibility

Abstract

The first two non-trivial moments of the distribution of the topological charge (or gluonic winding number), i.e., the topological susceptibility and the fourth cumulant, can be computed in lattice QCD simulations and exploited to constrain the pattern of chiral symmetry breaking. We compute these two topological observables at next-to-leading order in three-flavour Chiral Perturbation Theory, and we discuss the role played by the η propagation in these expressions. For hierarchies of light-quark masses close to the physical situation, we show that the fourth cumulant has a much better sensitivity than the topological susceptibility to the three-flavour quark condensate, and thus constitutes a relevant tool to determine the pattern of chiral symmetry breaking in the limit of three massless flavours. We provide the complete formulae for the two topological observables in the isospin limit, and predict their values in the particular setting of the recent analysis of the RBC/UKQCD collaboration. We show that a combination of the topological susceptibility and the fourth cumulant is able to pin down the three flavour condensate in a particularly clean way in the case of three degenerate quarks.

First determination of f +(0)|V us | from a combined analysis of τ → Kπν τ decay and πK scattering with constraints from K ℓ3 decays

“…As for ∆ CT the error bars are large and the result is again compatible with χPT calculations and chiral extrapolation of the lattice data [68][69][70]91]. In the former case on gets∆ CT = 0.03 at NLO in the isospin limit [51], while at two loop order two low energy constants enter [92] leading to the following estimate −0.035 <∆ CT < 0.11 [5].…”

“…(4.11) one gets ∆ CT = (−1.29 ± 1.28) × 10 −2 , (4.13) whose central value is somewhat larger than expected from χPT calculations. However considering the large error bar the value of ∆ CT is compatible with the NLO χPT result in the isospin limit [51] higher order corrections), NNLO estimates with isospin breaking given in [39,90], and chiral extrapolations to lattice data [68][69][70]91]. Similarly for the form factor at the soft kaon analog point one gets…”

“…Usually F (3) is traded with F π , the difference being of higher orders and thus a 0 = 1.16 GeV −2 . However there are some indications from several studies that possibly significative differences of patterns exist between the N f = 2 and N f = 3 chiral limits [63,64]- [68][69][70]. Such differences can be interpreted as a paramagnetic suppression of chiral order parameters when the number of massless flavors in the theory increases, in relation with the role ofss vacuum pairs in chiral dynamics.…”

Review of lattice results concerning low-energy particle physics

“…[213]. The potential usefulness of higher moments of the topological charge distribution to determine LECs has been investigated in [214].…”

Constraints on low energy QCD parameters from $$\eta \rightarrow 3\pi $$ η → 3 π and $$\pi \pi $$ π π scattering

“…If we roughly estimated the theoretical uncertainty of these fits as the distance of the corresponding points in the (X, Z ) plane, then these fits might look quite compatible with our distribution. The apparent inconsistency with the result of [12] is intriguing. It uses resummed χ PT as well, paired with lattice data.…”