Quark mass dependence of on-shell and half off-shell scattering amplitudes from Bethe-Salpeter wave function inside the interaction range

Abstract

We evaluate scattering amplitudes at on-shell and half off-shell for I = 2 S-wave two-pion system using the Bethe-Salpeter wave function inside the interaction range in the quenched QCD. The scattering length and effective range are extracted from these scattering amplitudes. Quark mass dependence of them is investigated with the pion mass ranged in 0.52-0.86 GeV. We examine consistency between a result by the conventional finite volume method and our estimate, as well as the phenomenological value.

Two-pion scattering amplitude from Bethe-Salpeter wave function at the interaction boundary

“…It is found that the results of δ (k) obtained from the two methods, the finite volume method and the extended relation, completely agree with each other, and the half-off-shell amplitude can be calculated in a wide range of the momentum with reasonable statistical error. Furthermore, we confirm that similar results for the on-shell and half-off-shell amplitudes are obtained in smaller m π [6].…”

“…In this report, we discuss conditions for this t independence through a definition of the scattering amplitude on the lattice under some assumptions. The results in this report have already been presented in our paper [6].…”

“…It is noted that since the summation in Eq. (2.1) can be replaced by the one inside R because of (∆ + k 2 )ϕ (x; k) = 0 in x > R, H L (k; k) is expressed by ϕ (x; k) at the interaction boundary with a different form of surf(k; k) [6]. A similar relation to Eq.…”

“…(2.2). Note that it is straightforward to include π ′ π ′ scattering states in this discussion [6]. Using a relation of surf(k; q),…”

Renormalization of a Contact Interaction on a Lattice

“…While alternatives, including the potential method (Refs. [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]), the mapping onto harmonic oscillators (Ref. [28]) and the imposition of spherical walls (Refs.…”