Generalization of Weinberg's Compositeness Relations

TL;DR

This paper extends Weinberg's compositeness relations by incorporating range corrections through a general form factor, providing more accurate analytic expressions and applying the formalism to the deuteron.

Contribution

It introduces a generalized analytic form factor for Weinberg's relations, including range effects, and derives an integral representation of compositeness with reduced uncertainty.

Findings

Derived an analytic form factor including range corrections.

Established an exact relation between bound state wave function and scattering phase.

Applied the formalism to analyze the deuteron.

Abstract

We generalize the time-honored Weinberg's compositeness relations by including the range corrections through considering a general form factor. In Weinberg's derivation, he considered the effective range expansion up to $\mathcal{O}(p^2)$ and made two additional approximations: neglecting the non-pole term in the Low equation; approximating the form factor by a constant. We lift the second approximation, and work out an analytic expression for the form factor. For a positive effective range, the form factor is of a single-pole form. An integral representation of the compositeness is obtained and is expected to have a smaller uncertainty than that derived from Weinberg's relations. We also establish an exact relation between the wave function of a bound state and the phase of the scattering amplitude neglecting the non-pole term. The deuteron is analyzed as an example, and the formalism can be applied to other cases where range corrections are important.