Generalization of Weinberg’s compositeness relations

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 nonpole term in the Low equation and 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 nonpole term. The deuteron is analyzed as an example, and the formalism can be applied to other cases where range corrections are important.