# Multiple dispersive bounds. I) The z-expansion

## TL;DR

This paper enhances the $z$-expansion method for hadronic form factors by introducing unitarity filters and multiple dispersive bounds, improving model-independent analysis of physical processes.

## Contribution

It introduces a unitarity filter and multiple dispersive bounds into the $z$-expansion framework, connecting it with the dispersion matrix method for more robust phenomenological applications.

## Key findings

- Unitarity filter constrains form factor data more effectively.
- Multiple dispersive bounds improve the analysis of hadronic form factors.
- Framework applicable to various physical processes like meson form factors and semileptonic decays.

## Abstract

We propose the implementation of two ingredients in the phenomenological applications of the unitary approach based on the $z$-expansion of hadronic form factors, commonly referred to as the Boyd-Grinstein-Lebed (BGL) $z$-expansion [1-4]. The first ingredient is the explicit addition of a unitarity filter applied to a given set of input data for the hadronic form factors. This further constraint is not usually taken into account in the phenomenological applications of the BGL $z$-expansion. We show that it follows from the equivalence between the BGL approach and the Dispersion Matrix (DM) method [5]}, which also describes hadronic form factors in a completely model-independent and non-perturbative way. The second ingredient is represented by the introduction of suitable kernel functions in the evaluation of unitarity bounds, leading to the application of multiple dispersive bounds to hadronic form factors, whenever data and/or (non-)perturbative techniques allow to do so. This idea may be useful for the investigation of many physical processes, from the analysis of the electromagnetic form factors of mesons and baryons to the study of weak semileptonic decays of hadrons. An explicit numerical application will be presented in the companion paper [6], where the effects of sub-threshold branch-cuts are analyzed.