Extracting the electro-magnetic pion form factor from QCD in a finite volume

We consider finite volume effects on the electromagnetic form factor of the pion. We compute the peudoscalar-vector-pseudoscalar correlator in the $\epsilon$ expansion of chiral perturbation theory up to the next-to-leading order and find a way to remove the dominant part, which comes from a contribution of the pion zero-mode. Inserting non-zero momentum to relevant operators (or taking a subtraction of the correlators at different time-slices), and taking an appropriate ratio of them, one can automatically cancel the zero-mode's contribution, which becomes non-perturbatively large $\sim \mathcal{O}(100 \%)$ in the $\epsilon$ regime. The remaining finite volume dependence, which comes from the non-zero momentum modes, is shown to be perturbatively small even in such an extremal case. Since the zero-mode's dominance is universal in any finite volume scaling, and we do not rely on any particular feature of the $\epsilon$ expansion, our method has a wide application to many other correlators of QCD.