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 ${\eta }^{\prime }$ weak decay constants. We obtain the results such as $F_{\eta }=96.77$ MeV and $F_{{\eta }^{\prime }}=102.53$ 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 ${\chi }_t=(175\pm 5\mathrm{MeV}{)}^4$. We also find that our ${\chi }_t^{\mathrm{WV}}={\chi }_t^{\mathrm{QL}}=(194.30,\mathrm{MeV}{)}^4$ in the quenched limit and ${\chi }_t^{\mathrm{LS}}=(162.54\mathrm{MeV}{)}^4$ in the chiral limit. Consequently, we conclude that ${\chi }_t<{\chi }_t^{\mathrm{QL}}$. Our result also implies that the $(10\sim 20)\%$ decrease with the dynamical quark contributions.

10 pages, 1 figure

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