QCD topological susceptibility from the nonlocal chiral quark model

Abstract

We investigate the QCD topological susceptibility $\chi_t$ by using the nonlocal chiral quark model (NL$\chi$QM). This model is based on the liquid instanton QCD-vacuum configuration in which $\mathrm{SU}(3)$ flavor symmetry is explicitly broken by the current quark mass $(m_{u,d},m_s)\approx(5,135)$ MeV. To compute $\chi_t$, the local topological charge density operator $Q_t(x)$ is derived from the effective partition function of NL$\chi$QM. We take into account the contributions from the leading-order (LO) ones $\sim\mathcal{O}(N_c)$ in the $1/N_c$ expansion. We also verify that the analytical expression of $\chi_t$ in NL$\chi$QM satisfy the Witten-Veneziano (WV) and the Leutwyler-Smilga (LS) formulae. Once the average instanton size and inter-instanton distance are fixed with $\bar{\rho}=1/3$ fm and $\bar{R}=1$ fm, respectively, all the associated model parameters are all determined self-consistently within the model, including the $\eta$ and $\eta'$ weak decay constants. We obtain the results such as $F_{\eta}=96.77$ MeV and $F_{\eta'}=102.53$ MeV for instance. Numerically we observe that $\chi_{t}=(165.57\,\mathrm{MeV})^4$ in our full calculation. This value is comparable with its empirical one $\chi_t=(175\pm5\,\mathrm{MeV})^4$. We also find that our $\chi^\mathrm{WV}_t=\chi^\mathrm{QL}_t=(194.30,\mathrm{MeV})^4$ in the quenched limit and $\chi^\mathrm{LS}_t=(162.54\,\mathrm{MeV})^4$ in the chiral limit. Consequently, we conclude that $\chi_{t}<\chi^\mathrm{QL}_t$. Our result also implies that the $(10\sim20)\,\%$ decrease with the dynamical quark contributions.