Abstract
We analyze the contribution of the η′ (958) meson in the first two non-trivial moments of the QCD topological charge distribution, namely, the topological susceptibility and the fourth-order cumulant of the vacuum energy density. We perform our study within U(3) Chiral Perturbation Theory up to next-to-next-to-leading order in the combined chiral and large-Nc expansion. We also describe the temperature dependence of these two quantities and compare them with previous analyses in the literature. In particular, we discuss the validity of the thermal scaling of the topological susceptibility with the quark condensate, which is intimately connected with a Ward Identity relating both quantities. We also consider isospin breaking corrections from the vacuum misalignment at leading order in the U(3) framework.
Highlights
The topological susceptibility is meant to be connected with the η mass through the UA(1) anomaly
We discuss the validity of the thermal scaling of the topological susceptibility with the quark condensate, which is intimately connected with a Ward Identity relating both quantities
We have provided a full calculation of the topological susceptibility and the fourthorder cumulant up to next-to-nextleading order (NNLO) in U(3) Chiral Perturbation Theory
Summary
The above result is consistent with the large-Nc scaling analysis of the topological susceptibility provided in [35]. The diagram (b) in figure 1 comes from the third-order derivative induced vertex (3.4) and the LO contribution in the δ expansion to the single-derivative vertex, i.e., first term in (3.2) It involves a single η( ) propagator at vanishing momentum and terms proportional to F 2M02v4(0)/M02η( ), where η( ) stands for a η or η field, that contribute at O(δ2). Diagram (c) is coming from the product of two second-order derivative vertices at NLO in the δ expansion, involving two η( ) propagators It contributes only at O(δ4) and it will not enter in the NNLO. The topology shown in figure 1 (d) is produced from the NLO U(3) contribution of the second derivative vertex in (3.3) and two single-derivative vertices at LO in the δ expansion It involves two η( ) propagators and terms proportional to M04v2(2)/M04η( ) and M04v4(0)/M04η( ). Where, as explained, the first term in the r.h.s. is O 1/Nc2 and the rest of the displayed terms are O 1/Nc3
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