breit_wigner

This module provides functions to describe the lineshapes of the intermediate particles, namely generalized Breit-Wigner function. Users can also define new lineshape using the function wrapper regist_lineshape().

BW(m, m0, g0, *args)

Breit-Wigner function

\[BW(m) = \frac{1}{m_0^2 - m^2 - i m_0 \Gamma_0 }\]

BWR(m, m0, g0, q, q0, L, d)

Relativistic Breit-Wigner function (with running width). It’s also set as the default lineshape.

\[BW(m) = \frac{1}{m_0^2 - m^2 - i m_0 \Gamma(m)}\]

BWR2(m, m0, g0, q2, q02, L, d)

Relativistic Breit-Wigner function (with running width). Allow complex \(\Gamma\).

\[BW(m) = \frac{1}{m_0^2 - m^2 - i m_0 \Gamma(m)}\]

BWR_normal(m, m0, g0, q2, q02, L, d)

Relativistic Breit-Wigner function (with running width) with a normal factor.

\[BW(m) = \frac{\sqrt{m_0 \Gamma(m)}}{m_0^2 - m^2 - i m_0 \Gamma(m)}\]

Bprime(L, q, q0, d)

Blatt-Weisskopf barrier factors. E.g. the first three orders

\(L\)	\(B_L'(q,q_0,d)\)
0	1
1	\(\sqrt{\frac{(q_0d)^2+1}{(qd)^2+1}}\)
2	\(\sqrt{\frac{(q_0d)^4+3*(q_0d)^2+9}{(qd)^4+3*(qd)^2+9}}\)

\(d\) is 3.0 by default.

Bprime_num(L, q, d)

The numerator (as well as the denominator) inside the square root in the barrier factor

Bprime_polynomial(l, z)

It stores the Blatt-Weisskopf polynomial up to the fifth order (\(L=5\))

Parameters:

• l – The order

• z – The variable in the polynomial

Returns:

The calculated value

Bprime_q2(L, q2, q02, d)

Blatt-Weisskopf barrier factors.

GS(m, m0, g0, q, q0, L, d, c_daug2Mass=0.13957039, c_daug3Mass=0.1349768)

Gamma(m, gamma0, q, q0, L, m0, d)

Running width in the RBW

\[\Gamma(m) = \Gamma_0 \left(\frac{q}{q_0}\right)^{2L+1}\frac{m_0}{m} B_{L}'^2(q,q_0,d)\]

Gamma2(m, gamma0, q2, q02, L, m0, d)

Running width in the RBW

\[\Gamma(m) = \Gamma_0 \left(\frac{q}{q_0}\right)^{2L+1}\frac{m_0}{m} B_{L}'^2(q,q_0,d)\]

barrier_factor(l, q, q0, d=3.0, axis=0)

Barrier factor multiplied with \(q^L\), which is used as a combination in the amplitude expressions. The values are cached for \(L\) ranging from 0 to l.

barrier_factor2(l, q, q0, d=3.0, axis=-1)

???

chew_mandelstam(m, m1, m2)

Chew-Mandelstam function in PDG 2024 Eq 50.44 multiply \(16\pi\) factor.

\[\Sigma(m) = \frac{1}{\pi}\left[ \frac{2q}{m} \ln \left(\frac{ m_1^2 + m_2^2 - m^2 + 2mq }{ 2 m_1 m_2}\right) - (m_1^2 - m_2^2) (\frac{1}{m^2} - \frac{1}{(m_1+m_2)^2}) \ln \frac{m_1}{m_2} \right]\]

for \(m>(m_1+m_2)\)

\[Im\Sigma(m) = \frac{1}{i}\frac{1}{\pi} \frac{2q}{m} \ln (-1) = \frac{2q}{m}\]

\[Re\Sigma(m) = \frac{1}{\pi}\left[ \frac{2q}{m} \ln \left( \frac{ m^2 - m_1^2 - m_2^2 - 2mq }{ 2 m_1 m_2}\right) - (m_1^2 - m_2^2) (\frac{1}{m^2} - \frac{1}{(m_1+m_2)^2}) \ln \frac{m_1}{m_2} \right]\]

chew_mandelstam_l(m, m1, m2, l)

TODO. Function from J.Math.Phys. 25 (1984) 3540 , with some modifies to be same as chew_mandelstam function.

compare with chew_mandelstam function.

\(m_1=0.4, m_2=0.1\)

(Source code, png, hires.png, pdf)

\(m_1=m_2=0.4\)

(Source code, png, hires.png, pdf)

(Source code, png, hires.png, pdf)

complex_q(s, m1, m2)

dFun(s, daug2Mass, daug3Mass)

dh_dsFun(s, daug2Mass, daug3Mass)

fsFun(s, m2, gam, daug2Mass, daug3Mass)

get_bprime_coeff(l)

The coefficients of polynomial in Bprime function.

\[|\theta_{l}(jw)|^2 = \sum_{i=0}^{l} c_i w^{2 i}\]

hFun(s, daug2Mass, daug3Mass)

one(*args)

A uniform function

regist_lineshape(name=None)

It will be used as a wrapper to define various Breit-Wigner functions

Parameters:

name – String name of the BW function

Returns:

A function used in a wrapper

reverse_bessel_polynomials(n, x)

Reverse Bessel polynomials.

\[\theta_{n}(x) = \sum_{k=0}^{n} \frac{(n+k)!}{(n-k)!k!} \frac{x^{n-k}}{2^k}\]

to_complex(i)

twoBodyCMmom(m_0, m_1, m_2)

relative momentum for 0 -> 1 + 2