Available Resonances Model

  1. "default", "BWR" (Particle)

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

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

    _images/particle_model-1.png

  2. "x" (ParticleX)

simple particle model for mass, (used in expr)

\[R(m) = m\]

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

_images/particle_model-2.png

  1. "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)

    _images/particle_model-3.png

  2. "BWR_below" (ParticleBWRBelowThreshold)

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

  3. "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)

    _images/particle_model-4.png

  4. "BWR_normal" (ParticleBWR_normal)

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

  5. "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

    \[R(m) = \frac{1 + D \Gamma_0 / m_0}{(m_0^2 -m^2) + f(m) - i m_0 \Gamma(m)}\]
    \[f(m) = \Gamma_0 \frac{m_0 ^2 }{q_0^3} \left[q^2 [h(m)-h(m_0)] + (m_0^2 - m^2) q_0^2 \frac{d h}{d m}|_{m0} \right]\]
    \[h(m) = \frac{2}{\pi} \frac{q}{m} \ln \left(\frac{m+q}{2m_{\pi}} \right)\]
    \[\frac{d h}{d m}|_{m0} = h(m_0) [(8q_0^2)^{-1} - (2m_0^2)^{-1}] + (2\pi m_0^2)^{-1}\]
    \[D = \frac{f(0)}{\Gamma_0 m_0} = \frac{3}{\pi}\frac{m_\pi^2}{q_0^2} \ln \left(\frac{m_0 + 2q_0}{2 m_\pi }\right) + \frac{m_0}{2\pi q_0} - \frac{m_\pi^2 m_0}{\pi q_0^3}\]

  6. "BW" (ParticleBW)

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

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

    _images/particle_model-5.png

  7. "LASS" (ParticleLass)

    \[R(m) = \frac{m}{q cot \delta_B - i q} + e^{2i \delta_B}\frac{m_0 \Gamma_0 \frac{m_0}{q_0}} {(m_0^2 - m^2) - i m_0\Gamma_0 \frac{q}{m}\frac{m_0}{q_0}}\]
    \[cot \delta_B = \frac{1}{a q} + \frac{1}{2} r q\]
    \[e^{2i\delta_B} = \cos 2 \delta_B + i \sin 2\delta_B = \frac{cot^2\delta_B -1 }{cot^2 \delta_B +1} + i \frac{2 cot \delta_B }{cot^2 \delta_B +1 }\]

  8. "one" (ParticleOne)

    \[R(m) = 1\]

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

    _images/particle_model-6.png

  9. "exp" (ParticleExp)

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

  10. "exp_com" (ParticleExp)

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

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

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

    _images/particle_model-7.png

  11. "Flatte" (ParticleFlatte)

    Flatte like formula

\[R(m) = \frac{1}{m_0^2 - m^2 + m_0 (\sum_{i} g_i \frac{q_i}{m})}\]
\[\begin{split}q_i = \begin{cases} \frac{\sqrt{(m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2)}}{2m} & (m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2) >= 0 \\ \frac{i\sqrt{|(m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2)|}}{2m} & (m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2) < 0 \\ \end{cases}\end{split}\]

Required input arguments mass_list: [[m11, m12], [m21, m22]] for \(m_{i,1}, m_{i,2}\).

  1. "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

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

  2. "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

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

  3. "KmatrixSimple" (KmatrixSimple)

    simple Kmatrix formula.

    K-matrix

    \[K_{i,j} = \sum_{a} \frac{g_{i,a} g_{j,a}}{m_a^2 - m^2+i\epsilon}\]

    P-vector

    \[P_{i} = \sum_{a} \frac{\beta_{a} g_{i,a}}{m_a^2 - m^2 +i\epsilon} + f_{bkg,i}\]

    total amplitude .. math:

    R(m) = n (1 - K i \rho n^2)^{-1} P

    barrief factor .. math:

    n_{ii} = q_i^l B'_l(q_i, 1/d, d)

    phase space factor

    \[\rho_{ii} = q_i/m\]

    \(q_i\) is 0 when below threshold

  4. "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 (\(\cos \theta_0, \sin\theta_0 \cos \theta_1, \cdots, \prod_i \sin\theta_i\))

  5. "interp" (Interp)

linear interpolation for real number

  1. "interp_c" (Interp)

linear interpolation for complex number

  1. "spline_c" (Interp1DSpline)

Spline interpolation function for model independent resonance

  1. "interp1d3" (Interp1D3)

Piecewise third order interpolation

  1. "interp_lagrange" (Interp1DLang)

Lagrange interpolation

  1. "interp_hist" (InterpHist)

Interpolation for each bins as constant

  1. "hist_idx" (InterpHistIdx)

Interpolation for each bins as constant

  1. "spline_c_idx" (Interp1DSplineIdx)

Spline function in index way