A simple alternative to the relativistic 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 Flatté-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 $$\rho $$ ρ , $$a_1(1260)$$ a 1 ( 1260 ) , and $$K^*(982)$$ K ∗ ( 982 ) mesons as well as the $$\varDelta (1232)$$ Δ ( 1232 ) baryon. We also present a study of the $$f_2(1270)$$ 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 ( 980 ) -$$a_0(1450)$$ a 0 ( 1450 ) and the $$K_0^*(700)$$ K 0 ∗ ( 700 ) -$$K_0^*(1430)$$ K 0 ∗ ( 1430 ) resonances as examples.

The phenomenology of the exotic hybrid nonet with $π_1(1600)$ and $η_1(1855)$

“…Using spectral distribution functions that inherently take into account the thresholds (e.g, Ref. [55]), one can study the contributions of these decay channels. This will be attempted in the future.…”

From well-known tensor mesons to yet unknown axial-tensor mesons

“…In order to describe them, we introduce the properly normalized "Sill" spectral function discussed in Ref. [50] for the ρ-meson as…”

“…The Sill spectral function is useful because the normalization is guaranteed also for broad states and because the contribution of virtual particles (the pion-pion loop for the ρ meson) has a vanishing real part, thus only the imaginary part describing the decay width is sufficient (in other words, no modification of the ρ(770)-mass part of the denominator appears), see details in Ref. [50].…”

Multichannel decay: alternative derivation of the $i$-th channel decay probability

“…An interesting model, discussed in Ref. [51], postulates Π i (s) = i Γi √ s − s th,i , for which Γ i (s) = Γi (s − s th,i )/sθ (s − s th,i ) (that reduces to a constant for large s). Despite its simplicity, it allows to fit quite well the spectral functions of various broad hadrons.…”