Finite volume scaling of the electro-magnetic pion form factor in the $\epsilon$ regime

Abstract

We consider finite volume effects on the electro-magnetic pion form factor near the chiral limit, in the so-called $\epsilon$ regime. The pseudoscalar-vector-pseudoscalar three-point function is calculated in the $\epsilon$ expansion of chiral perturbation theory to the next-to-leading order. In the $\epsilon$ regime, finite volume effects are non-perturbatively large in general. However, we find a way to remove its dominant part, by inserting momenta to the correlators, and taking an appropriate ratio of them. The subleading contribution is, then, shown to be perturbatively small, and one can extract the form factor as in a similar way to that in the $p$ regime.