# Model-independent constraint on the pion scalar form factor and light quark masses

## TL;DR

This paper derives a model-independent constraint linking light quark masses to the pion scalar form factor's shape, using phase shift data and QCD inputs, and finds consistency with recent lattice results.

## Contribution

It introduces a novel, model-independent method to constrain light quark masses using the pion scalar form factor and phase shift data without assumptions on the form factor's zeros.

## Key findings

- The constraint is consistent with recent lattice QCD results.
- A strong correlation between the pion scalar radius and curvature is established.
- High curvature values are predicted for the pion scalar form factor.

## Abstract

We investigate the pion scalar form factor in the Meiman-Okubo framework, implementing the phase below the inelastic $K\bar K$ threshold, where it is known from the $\pi\pi$ scalar isoscalar phase shift $\delta_0^0$ by Watson theorem. State-of-the-art knowledge of the perturbative QCD expansion of the scalar correlator and the phase shift $\delta_0^0$ is used as input. No assumptions about the phase above the inelastic threshold or the possible zeros of the form factor in the complex plane are necessary. We obtain a model-independent constraint relating the sum of the light quark masses to the slope and the curvature of the pion scalar form factor at the origin. The recent lattice results for the light quark masses and the pion scalar radius are found to satisfy this constraint. We obtain also a strong correlation between the pion scalar radius and the curvature of the form factor, with rather high values predicted for the curvature.