A simple alternative to the Breit-Wigner distribution

TL;DR

This paper introduces the Sill distribution, a simple, normalized, and versatile alternative to the Breit-Wigner distribution for modeling resonance spectral functions, with applications to various mesons and baryons.

Contribution

It proposes the Sill distribution, a new relativistic energy distribution that addresses normalization, threshold effects, and generalization issues of traditional Breit-Wigner models.

Findings

The Sill distribution fits experimental data for multiple mesons and baryons.

It compares favorably with spectral functions from quantum field theory models.

The distribution's limitations are discussed using specific resonance examples.

Abstract

First, we discuss the conditions under which the non-relativistic and relativistic types of the Breit-Wigner energy distributions are obtained. Then, upon insisting on the correct normalization of the energy distribution, we introduce a Flatt\'{e}-like relativistic distribution -- denominated as Sill distribution -- that (i) contains left-threshold effects, (ii) is properly normalized for any decay width, (iii) can be obtained as an appropriate limit in which the decay width is a constant, (iv) is easily generalized to the multi-channel case (v) as well as to a convoluted form in case of a decay chain and -- last but not least -- (vi) is simple to deal with. We compare the Sill distribution to spectral functions derived within specific QFT models and show that it fairs well in concrete examples that involve a fit to experimental data for the $\rho$, $a_1(1260)$, and $K^*(982)$ mesons as well as the $\Delta(1232)$ baryon. We also present a study of the $f_2(1270)$ which has more than one possible decay channels. Finally, we discuss the limitations of the Sill distribution using the $a_0(980)$-$a_0(1450)$ and the $K_0^\ast(700)$-$K_0^\ast(1430)$ resonances as examples.