Quark mass dependence of the pseudoscalar hairpin vertex

Abstract

In a recent investigation of chiral behavior in quenched lattice QCD, the flavor-singlet pseudoscalar ``hairpin'' vertex associated with the ${\ensuremath{\eta}}^{\ensuremath{'}}$ mass was studied for pion masses ranging from $\ensuremath{\approx}275$ to 675 MeV. It has been observed by Sharpe that, with a consistent NLO calculation in quenched chiral perturbation theory, the quark-mass dependence of the hairpin vertex is uniquely predicted in terms of the single-pole coefficient ${\ensuremath{\alpha}}_{\ensuremath{\Phi}}$ and the Leutwyler parameter ${L}_{5}$. Since ${\ensuremath{\alpha}}_{\ensuremath{\Phi}}$ is found to be $\ensuremath{\approx}0$, the chiral slope of the double-pole residue determines a value for ${L}_{5}$, providing a consistency check between the chiral slope of the hairpin mass insertion and that of the pion decay constant. We investigate the consistency of these mass dependences in our Monte Carlo results at two values of lattice spacing. Within statistics, the slopes are found to be consistent with the $Q\ensuremath{\chi}PT$ prediction.