Abstract
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.
Highlights
Another interesting way to determine non-perturbative features of the QCD vacuum consists in exploiting the connection between the fluctuation of the winding number, or topological charge, defined from the gluonic strength tensor G as
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 constitutes a relevant tool to determine the pattern of chiral symmetry breaking in the limit of three massless flavours
In order to avoid any confusion and make a direct link with phenomenological analyses performed in N = 2 χPT [1], one should always use the expressions eqs. (5.2) and (5.4) to deal with the two-flavour chiral expansions of the topological susceptibility and the fourth cumulant. Due to their connection with the UA(1) anomaly, topological observables describing the distribution of the topological charge are able to probe the structure of low-energy QCD, and in particular the pattern of chiral symmetry breaking
Summary
Let us first define these two quantities related to topological aspects of QCD [6, 8, 37]. If we work in a finite volume V with an Euclidean metric, we can define the vacuum energy density as ǫvac(M, θ) Both quantities can be interpreted as quantities describing the distribution of the topological charge Q χ=. C4 measures the kurtosis of the distribution, i.e., its more or less peaked nature (a Gaussian distribution has a vanishing kurtosis) Both quantities can be defined non-ambiguously in terms of Green functions of scalar and pseudoscalar densities at vanishing momentum, so that they are renormalisation-group invariant [38]. [39], the effective theory of QCD in a finite volume with periodic boundary conditions amounts to the same Lagrangian as in the infinite volume The properties of this partition function have been extensively studied in refs. The properties of this partition function have been extensively studied in refs. [6, 40], and in particular the distribution of the winding number Q according to the leading-order (LO) chiral Lagrangian
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