Available Resonances Model

"default", "BWR" (Particle)

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

(Source code, png, hires.png, pdf)
"x" (ParticleX)

simple particle model for mass, (used in expr)

\[R(m) = m\]

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

"BWR2" (ParticleBWR2)

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

The difference of BWR, BWR2 is the behavior when mass is below the threshold ( \(m_0 = 0.1 < 0.1 + 0.1 = m_1 + m_2\)).

(Source code, png, hires.png, pdf)
"BWR_below" (ParticleBWRBelowThreshold)

\[R(m) = \frac{1}{m_0^2 - m^2 - i m_0 \Gamma(m)}\]
"BWR_coupling" (ParticleBWRCoupling)

Force \(q_0=1/d\) to avoid below theshold condition for BWR model, and remove other constant parts, then the \(\Gamma_0\) is coupling parameters.

\[R(m) = \frac{1}{m_0^2 - m^2 - i m_0 \Gamma_0 \frac{q}{m} q^{2l} B_L'^2(q, 1/d, d)}\]

(Source code, png, hires.png, pdf)
"BWR_normal" (ParticleBWR_normal)

\[R(m) = \frac{\sqrt{m_0 \Gamma(m)}}{m_0^2 - m^2 - i m_0 \Gamma(m)}\]
"GS_rho" (ParticleGS)

Gounaris G.J., Sakurai J.J., Phys. Rev. Lett., 21 (1968), pp. 244-247

c_daug2Mass: mass for daughter particle 2 (\(\pi^{+}\)) 0.13957039

c_daug3Mass: mass for daughter particle 3 (\(\pi^{0}\)) 0.1349768

\[R(m) = \frac{1 + D \Gamma_0 / m_0}{(m_0^2 -m^2) + f(m) - i m_0 \Gamma(m)}\]

\[f(m) = \Gamma_0 \frac{m_0 ^2 }{q_0^3} \left[q^2 [h(m)-h(m_0)] + (m_0^2 - m^2) q_0^2 \frac{d h}{d m}|_{m0} \right]\]

\[h(m) = \frac{2}{\pi} \frac{q}{m} \ln \left(\frac{m+q}{2m_{\pi}} \right)\]

\[\frac{d h}{d m}|_{m0} = h(m_0) [(8q_0^2)^{-1} - (2m_0^2)^{-1}] + (2\pi m_0^2)^{-1}\]

\[D = \frac{f(0)}{\Gamma_0 m_0} = \frac{3}{\pi}\frac{m_\pi^2}{q_0^2} \ln \left(\frac{m_0 + 2q_0}{2 m_\pi }\right) + \frac{m_0}{2\pi q_0} - \frac{m_\pi^2 m_0}{\pi q_0^3}\]
"BW" (ParticleBW)

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

(Source code, png, hires.png, pdf)
"LASS" (ParticleLass)

\[R(m) = \frac{m}{q cot \delta_B - i q} + e^{2i \delta_B}\frac{m_0 \Gamma_0 \frac{m_0}{q_0}} {(m_0^2 - m^2) - i m_0\Gamma_0 \frac{q}{m}\frac{m_0}{q_0}}\]

\[cot \delta_B = \frac{1}{a q} + \frac{1}{2} r q\]

\[e^{2i\delta_B} = \cos 2 \delta_B + i \sin 2\delta_B = \frac{cot^2\delta_B -1 }{cot^2 \delta_B +1} + i \frac{2 cot \delta_B }{cot^2 \delta_B +1 }\]
"one" (ParticleOne)

\[R(m) = 1\]

(Source code, png, hires.png, pdf)
"exp" (ParticleExp)

\[R(m) = e^{-|a| m}\]
"exp_com" (ParticleExp)

\[R(m) = e^{-(a+ib) m^2}\]

lineshape when \(a=1.0, b=10.\)

(Source code, png, hires.png, pdf)
"KMatrixSingleChannel" (KmatrixSingleChannelParticle)

K matrix model for single channel multi pole.

\[K = \sum_{i} \frac{m_i \Gamma_i(m)}{m_i^2 - m^2 }\]

\[P = \sum_{i} \frac{\beta_i m_0 \Gamma_0 }{ m_i^2 - m^2}\]

the barrier factor is included in gls

\[R(m) = (1-iK)^{-1} P\]
"KMatrixSplitLS" (KmatrixSplitLSParticle)

K matrix model for single channel multi pole and the same channel with different (l, s) coupling.

\[K_{a,b} = \sum_{i} \frac{m_i \sqrt{\Gamma_{a,i}(m)\Gamma_{b,i}(m)}}{m_i^2 - m^2 }\]

\[P_{b} = \sum_{i} \frac{\beta_i m_0 \Gamma_{b,i0} }{ m_i^2 - m^2}\]

the barrier factor is included in gls

\[R(m) = (1-iK)^{-1} P\]
"BWR_LS" (ParticleBWRLS)

Breit Wigner with split ls running width

\[R_i (m) = \frac{g_i}{m_0^2 - m^2 - im_0 \Gamma_0 \frac{\rho}{\rho_0} (\sum_{i} g_i^2)}\]

, \(\rho = 2q/m\), the partial width factor is

\[g_i = \gamma_i \frac{q^l}{q_0^l} B_{l_i}'(q,q_0,d)\]

and keep normalize as

\[\sum_{i} \gamma_i^2 = 1.\]

The normalize is done by (\(\cos \theta_0, \sin\theta_0 \cos \theta_1, \cdots, \prod_i \sin\theta_i\))
"interp" (Interp)