I=2 Pion Scattering Length from Two-Pion Wave Functions

TL;DR

This study computes the I=2 pion scattering length using two-pion wave functions in lattice QCD, confirming the finite-volume method's validity and achieving more precise results than traditional correlator methods.

Contribution

It introduces a wave function-based approach to determine the pion scattering length, validating the finite-volume condition and reducing statistical errors in lattice QCD calculations.

Findings

Finite-volume condition is satisfied for L≥24 (3.92 fm).

Scattering length obtained with smaller statistical error than from correlators.

Method confirms validity of Lüscher's finite-volume approach in this context.

Abstract

We calculate the two-pion wave function in the ground state of the I=2 $S$-wave system and find the interaction range between two pions, which allows us to examine the validity of the necessary condition for the finite-volume method for the scattering length proposed by L\"uscher. We work in the quenched approximation employing a renormalization group improved gauge action for gluons and an improved Wilson action for quarks at $1/a=1.207(12) {\rm GeV}$ on $16^3 \times 80$, $20^3 \times 80$ and $24^3 \times 80$ lattices. We conclude that the necessary condition is satisfied within the statistical errors for the lattice sizes $L\ge 24$ ($3.92 {\rm fm}$) when the quark mass is in the range that corresponds to $m_\pi^2 = 0.273-0.736 {\rm GeV}^2$. We obtain the scattering length with a smaller statistical error from the wave function than from the two-pion time correlator.