An analogue to the pion decay constant in the multi-flavor Schwinger model

We study the Schwinger model with $N_{\rm f} \geq 2$ degenerate fermion flavors, by means of lattice simulations. We use dynamical Wilson fermions for $N_{\rm f} = 2$, and re-weighted quenched configurations for overlap-hypercube fermions with $N_{\rm f} \leq 6$. In this framework, we explore an analogue of the QCD pion decay constant $F_{\pi}$, which is dimensionless in $d=2$, and which has hardly been considered in the literature. We determine $F_{\pi}$ by three independent methods, with numerical and analytical ingredients. First, we consider the 2-dimensional version of the Gell-Mann--Oakes--Renner relation, where we insert both theoretical and numerical values for the quantities involved. Next we refer to the $\delta$-regime, {\it i.e.\ a small spatial volume, where we assume formulae from Chiral Perturbation Theory to apply even in the absence of Nambu-Goldstone bosons. We further postulate an effective relation between $N_{\rm f}$ and the number of relevant, light bosons, which we denote as "pions". Thus $F_{\pi}$ is obtained from the residual "pion" mass in the chiral limit, which is a finite-size effect. Finally, we address to the 2-dimensional Witten--Veneziano formula: it yields a value for $F_{\eta}$, which we identify with $F_{\pi}$, as in large-$N_{\rm c}$ QCD. All three approaches consistently lead to $F_{\pi} \simeq 1/\sqrt{2 \pi}$ at fermion mass $m=0$, which implies that this quantity is meaningful.