Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multiple dispersive bounds. I) The z-expansion

Silvano Simula, Ludovico Vittorio

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.

Multiple dispersive bounds. I) The z-expansion

Abstract

We propose the implementation of two ingredients in the phenomenological applications of the unitary approach based on the -expansion of hadronic form factors, commonly referred to as the Boyd-Grinstein-Lebed (BGL) -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 -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.

Paper Structure

This paper contains 10 sections, 69 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The BGL expansion
  3. The dispersive susceptibility as the upper bound $\chi^U$
  4. Unitary BGL expansion
  5. Multiple dispersive bounds
  6. Implementation of the multiple dispersive bounds in the BGL expansion
  7. Evaluation of the multiple dispersive bounds $\chi_p^U$
  8. Conclusions
  9. Parameters of the outer functions for semileptonic meson decays $M_1 \to M_2 \ell \nu_\ell$
  10. Additive form of the BGL expansion

Figures (1)

  • Figure 1: Left panel: the three Euclidean-time kernels $K_{SW}(\tau)$, $K_{IW}(\tau)$ and $K_{LW}(\tau)$ versus the time-distance $\tau$, given by Eqs. (\ref{['eq:kernels_tau_RBC']})-(\ref{['eq:parms_RBC']}) according to Ref. RBC:2018dos. Right panel: the three corresponding energy kernels $\widetilde{K}_{SW}(\omega)$, $\widetilde{K}_{IW}(\omega)$ and $\widetilde{K}_{LW}(\omega)$ with $\omega$ given by Eq. (\ref{['eq:omega']}), obtained using Eq. (\ref{['eq:kernel_LQCD_Q0=0']}) at $Q_0^2 = 0$ and $n = 2$.