Available Resonances Model
1. "default", "BWR"
(Particle
)
\[R(m) = \frac{1}{m_0^2 - m^2 - i m_0 \Gamma(m)}\]Argand diagram
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2. "x"
(ParticleX
)
simple particle model for mass, (used in expr)
\[R(m) = m\](
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3. "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 ( \(m_0 = 0.1 < 0.1 + 0.1 = m_1 + m_2\)).(
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4. "BWR_below"
(ParticleBWRBelowThreshold
)
\[R(m) = \frac{1}{m_0^2 - m^2 - i m_0 \Gamma(m)}\]
5. "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.\[R(m) = \frac{1}{m_0^2 - m^2 - i m_0 \Gamma_0 \frac{q}{m} q^{2l} B_L'^2(q, 1/d, d)}\](
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6. "BWR_normal"
(ParticleBWR_normal
)
\[R(m) = \frac{\sqrt{m_0 \Gamma(m)}}{m_0^2 - m^2 - i m_0 \Gamma(m)}\]
7. "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\[R(m) = \frac{1 + D \Gamma_0 / m_0}{(m_0^2 -m^2) + f(m) - i m_0 \Gamma(m)}\]\[f(m) = \Gamma_0 \frac{m_0 ^2 }{q_0^3} \left[q^2 [h(m)-h(m_0)] + (m_0^2 - m^2) q_0^2 \frac{d h}{d m}|_{m0} \right]\]\[h(m) = \frac{2}{\pi} \frac{q}{m} \ln \left(\frac{m+2q}{2m_{\pi}} \right)\]\[\frac{d h}{d m}|_{m0} = h(m_0) [(8q_0^2)^{-1} - (2m_0^2)^{-1}] + (2\pi m_0^2)^{-1}\]\[D = \frac{f(0)}{\Gamma_0 m_0} = \frac{3}{\pi}\frac{m_\pi^2}{q_0^2} \ln \left(\frac{m_0 + 2q_0}{2 m_\pi }\right) + \frac{m_0}{2\pi q_0} - \frac{m_\pi^2 m_0}{\pi q_0^3}\]
8. "BW"
(ParticleBW
)
\[R(m) = \frac{1}{m_0^2 - m^2 - i m_0 \Gamma_0}\](
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9. "LASS"
(ParticleLass
)
\[R(m) = \frac{m}{q cot \delta_B - i q} + e^{2i \delta_B}\frac{m_0 \Gamma_0 \frac{m_0}{q_0}} {(m_0^2 - m^2) - i m_0\Gamma_0 \frac{q}{m}\frac{m_0}{q_0}}\]\[cot \delta_B = \frac{1}{a q} + \frac{1}{2} r q\]\[e^{2i\delta_B} = \cos 2 \delta_B + i \sin 2\delta_B = \frac{cot^2\delta_B -1 }{cot^2 \delta_B +1} + i \frac{2 cot \delta_B }{cot^2 \delta_B +1 }\]
10. "one"
(ParticleOne
)
\[R(m) = 1\](
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11. "exp"
(ParticleExp
)
\[R(m) = e^{-|a| m}\]
12. "exp_com"
(ParticleExpCom
)
13. "poly"
(ParticlePoly
)
\[R(m) = \sum c_i (m-m_0)^{n-i}\]lineshape when \(c_0=1, c_1=c_2=0\)
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14. "Flatte"
(ParticleFlatte
)
Flatte like formula
\[R(m) = \frac{1}{m_0^2 - m^2 + i m_0 (\sum_{i} g_i \frac{q_i}{m})}\]\[\begin{split}q_i = \begin{cases} \frac{\sqrt{(m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2)}}{2m} & (m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2) >= 0 \\ \frac{i\sqrt{|(m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2)|}}{2m} & (m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2) < 0 \\ \end{cases}\end{split}\](
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,Required input arguments
mass_list: [[m11, m12], [m21, m22]]
for \(m_{i,1}, m_{i,2}\).
15. "FlatteC"
(ParticleFlatteC
)
Flatte like formula
\[R(m) = \frac{1}{m_0^2 - m^2 - i m_0 (\sum_{i} g_i \frac{q_i}{m})}\]\[\begin{split}q_i = \begin{cases} \frac{\sqrt{(m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2)}}{2m} & (m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2) >= 0 \\ \frac{i\sqrt{|(m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2)|}}{2m} & (m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2) < 0 \\ \end{cases}\end{split}\]Required input arguments
mass_list: [[m11, m12], [m21, m22]]
for \(m_{i,1}, m_{i,2}\).(
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16. "FlatteGen"
(ParticleFlateGen
)
More General Flatte like formula
\[R(m) = \frac{1}{m_0^2 - m^2 - i m_0 [\sum_{i} g_i \frac{q_i}{m} \times \frac{m_0}{|q_{i0}|} \times \frac{|q_i|^{2l_i}}{|q_{i0}|^{2l_i}} B_{l_i}'^2(|q_i|,|q_{i0}|,d)]}\]\[\begin{split}q_i = \begin{cases} \frac{\sqrt{(m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2)}}{2m} & (m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2) >= 0 \\ \frac{i\sqrt{|(m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2)|}}{2m} & (m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2) < 0 \\ \end{cases}\end{split}\]Required input arguments
mass_list: [[m11, m12], [m21, m22]]
for \(m_{i,1}, m_{i,2}\). And addition argumentsl_list: [l1, l2]
for \(l_i\)
has_bprime=False
to remove \(B_{l_i}'^2(|q_i|,|q_{i0}|,d)\).
cut_phsp=True
to set \(q_i = 0\) when \((m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2) < 0\)The plot use parameters \(m_0=0.7, m_{0,1}=m_{0,2}=0.1, m_{1,1}=m_{1,2}=0.3, g_0=0.3,g_1=0.2,l_0=0,l_1=1\).
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no_m0=True
to set \(i m_0 => i\) in the width part.
no_q0=True
to remove \(\frac{m_0}{|q_{i0}|}\) and set others \(q_{i0}=1\).(
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17. "Flatte2"
(ParticleFlate2
)
General Flatte like formula.
\[R(m) = \frac{1}{m_0^2 - m^2 - i m_0 [\sum_{i} \color{red}{g_i^2}\color{black} \frac{q_i}{m} \times \frac{m_0}{|q_{i0}|} \times \frac{|q_i|^{2l_i}}{|q_{i0}|^{2l_i}} B_{l_i}'^2(|q_i|,|q_{i0}|,d)]}\]\[\begin{split}q_i = \begin{cases} \frac{\sqrt{(m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2)}}{2m} & (m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2) >= 0 \\ \frac{i\sqrt{|(m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2)|}}{2m} & (m^2-(m_{i,1}+m_{i,2})^2)(m^2-(m_{i,1}-m_{i,2})^2) < 0 \\ \end{cases}\end{split}\]It has the same options as
FlatteGen
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18. "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
\[R(m) = (1-iK)^{-1} P\]requird
mass_list: [pole1, pole2]
andwidth_list: [width1, width2]
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19. "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
\[R(m) = (1-iK)^{-1} P\]
20. "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
\[R(m) = n (1 - K i \rho n^2)^{-1} P\]barrief factor
\[n_{ii} = q_i^l B'_l(q_i, 1/d, d)\]phase space factor
\[\rho_{ii} = q_i/m\]\(q_i\) is 0 when below threshold
21. "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 (\(\cos \theta_0, \sin\theta_0 \cos \theta_1, \cdots, \prod_i \sin\theta_i\))
22. "BWR_LS2"
(ParticleBWRLS2
)
Breit Wigner with split ls running width, each one use their own l,
\[R_i (m) = \frac{1}{m_0^2 - m^2 - im_0 \Gamma_0 \frac{\rho}{\rho_0} (g_i^2)}\], \(\rho = 2q/m\), the partial width factor is
\[g_i = \gamma_i \frac{q^l}{q_0^l} B_{l_i}'(q,q_0,d)\]
23. "MultiBWR"
(ParticleMultiBWR
)
Combine Multi BWR into one particle
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24. "MultiBW"
(ParticleMultiBW
)
Combine Multi BW into one particle
25. "linear_npy"
(InterpLinearNpy
)
Linear interpolation model from a
npy
file with array of [mi, re(ai), im(ai)]. Requiredfile: path_of_file.npy
, for the path ofnpy
file.The example is
exp(5 I m)
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26. "linear_txt"
(InterpLinearTxt
)
- Linear interpolation model from a
txt
file with array of [mi, re(ai), im(ai)].Required
file: path_of_file.txt
, for the path oftxt
file.The example is
exp(5 I m)
.(
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27. "interp"
(Interp
)
linear interpolation for real number
28. "interp_c"
(Interp
)
linear interpolation for complex number
29. "spline_c"
(Interp1DSpline
)
Spline interpolation function for model independent resonance
30. "interp1d3"
(Interp1D3
)
Piecewise third order interpolation
31. "interp_lagrange"
(Interp1DLang
)
Lagrange interpolation
32. "interp_hist"
(InterpHist
)
Interpolation for each bins as constant
33. "hist_idx"
(InterpHistIdx
)
Interpolation for each bins as constant
use
min_m: 0.19 max_m: 0.91 interp_N: 8 with_bound: Truefor mass range [0.19, 0.91] and 7 bins
The first and last are fixed to zero unless set
with_bound: True
.This is an example of \(k\exp (i k)\) for point k.
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34. "spline_c_idx"
(Interp1DSplineIdx
)
Spline function in index way.
use
min_m: 0.19 max_m: 0.91 interp_N: 8 with_bound: Truefor mass range [0.19, 0.91] and 8 interpolation points
The first and last are fixed to zero unless set
with_bound: True
.This is an example of \(k\exp (i k)\) for point k.
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35. "sppchip"
(InterpSPPCHIP
)
Shape-Preserving Piecewise Cubic Hermite Interpolation Polynomial. It is monotonic in each interval.
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