A simple alternative to the relativistic Breit–Wigner distribution
Francesco Giacosa, Anna Okopińska, Vanamali Shastry
Abstract
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 $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> , $$a_1(1260)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1260</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , and $$K^*(982)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>982</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> mesons as well as the $$\varDelta (1232)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1232</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> baryon. We also present a study of the $$f_2(1270)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1270</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> which has more than one possible decay channels. Finally, we discuss the limitations of the Sill distribution using the $$a_0(980)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>980</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> - $$a_0(1450)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1450</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and the $$K_0^*(700)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mn>0</mml:mn> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>700</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> - $$K_0^*(1430)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mn>0</mml:mn> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1430</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> resonances as examples.