# Model-independent constraints on hadronic form factors with above-threshold poles

## TL;DR

This paper explores how above-threshold resonances, which appear as poles on unphysical Riemann sheets, do not affect model-independent constraints on hadronic form factors and discusses how they can indirectly inform form factor behavior.

## Contribution

It introduces a method to incorporate above-threshold poles into model-independent bounds on hadronic form factors, expanding the understanding of their influence beyond below-threshold resonances.

## Key findings

- Above-threshold poles do not affect original model-independent constraints.
- Residues of above-threshold poles can provide indirect information on form factors.
- Constraints can be improved by including additional information from above-threshold poles.

## Abstract

Model-independent constraints on hadronic form factors, in particular those describing exclusive semileptonic decays, can be derived from the knowledge of field correlators calculated in perturbative QCD, using analyticity and unitarity. The location of poles corresponding to below-threshold resonances, i.e., stable states that cannot decay into a pair of hadrons from the crossed channel of the form factor, must be known a priori, and their effect, accounted for through the use of Blaschke factors, is to reduce the strength of the constraints in the semileptonic region. By contrast, above-threshold resonances appear as poles on unphysical Riemann sheets, and their presence does not affect the original model-independent constraints. We discuss the possibility that the above-threshold poles can provide indirect information on the form factors on the first Riemann sheet, either through information from their residues or by constraining the discontinuity function. The bounds on form factors can be improved by imposing, in an exact way, the additional information in the extremal problem. The semileptonic $K\to \pi\ell \nu$ and $D\to \pi\ell\nu$ decays are considered as illustrations.