A pion decay constant in the multi-flavor Schwinger model

The pion decay constant $F_{\pi }$ plays an important role in QCD and in Chiral Perturbation Theory. It is hardly known, however, that a corresponding constant exists in the Schwinger model with $N_{\mathrm{f}}\ge 2$ degenerate fermion flavors. In this case, the pion'' does not decay and $F_π$is dimensionless. Still, $F_π$ can be defined by 2d analogies to the Gell-Mann--Oakes--Renner relation, the Witten--Veneziano formula and the residual pion’’ mass in the $\delta$-regime. With suitable assumptions, and by inserting simulation data, these QCD-inspired relations are all compatible with $F_{\pi }\simeq 1\mathrm{/}\sqrt{2\pi }$ at zero fermion mass, as we observe for $N_{\mathrm{f}}=2,\dots ,6$. We conclude that this is a meaningful constant in the multi-flavor Schwinger model.