A simple alternative to the Breit-Wigner distribution

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.