Available Resonances Model

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

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

Argand diagram

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

Pole position

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

2. "x" (ParticleX)

simple particle model for mass, (used in expr)

\[R(m) = m\]

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

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 ( \(m_0 = 0.1 < 0.1 + 0.1 = m_1 + m_2\)).

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

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.

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

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

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)

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

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

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)

\[R(m) = 1\]

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

11. "exp" (ParticleExp)

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

12. "exp_com" (ParticleExpCom)

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

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

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

13. "poly" (ParticlePoly)

\[R(m) = \sum c_i (m-m_0)^{n-i}\]

lineshape when \(c_0=1, c_1=c_2=0\)

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

14. "MLP" (ParticleMLP)

Multilayer Perceptron like model.

\[R(m) = \sum_{k} w_k activation(m-m_0+b_k)\]

lineshape when interp_N: 11, activation: relu, \(b_k=(k-5)/10\), \(w_k = exp(k i\pi/2)\)

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

15. "DI" (DispersionIntegralParticle)

Dispersion Integral model. In the model a linear interpolation is used to avoid integration every times in fitting. No paramters are allow in the integration.

\[f(s) = \frac{1}{m_0^2 - s - \sum_{i} g_i^2 [Re\Pi_i(s) -Re\Pi_i(m_0^2) + i Im \Pi_i(s)] }\]

where \(Im \Pi_i(s)=\rho_i(s)n_i^2(s)\), \(n_i(s)={q}^{l} {B_l'}(q,1/d, d)\).

The real parts of \(\Pi(s)\) is defined using the dispersion intergral

\[Re \Pi_i(s) = \frac{\color{red}(s-s_{0,i})}{\pi} P \int_{s_{th,i}}^{\infty} \frac{Im \Pi_i(s')}{(s' - s){\color{red} (s'-s_{0,i})}} \mathrm{d} s'\]

By default, \(s_{0,i}=0\), it can be change to other value though option s0: value. value=sth for \(s_{th,i}\).

Note

Small int_N will have bad precision.

The shape of \(\Pi(s)\) and comparing to Chew-Mandelstam function \(\Sigma(s)\)

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

The Argand plot

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

16. "DI_a0" (DispersionIntegralParticleA0)

“DI_a0” model is the model used in PRD78,074023(2008) . In the model a linear interpolation is used to avoid integration every times in fitting. No paramters are allowed in the integration, unless dyn_int=True.

\[f(s) = \frac{1}{m_0^2 - s - \sum_{i} [Re \Pi_i(s) - Re\Pi_i(m_0^2)] - i \sum_{i} \rho'_i(s) }\]

where \(\rho'_i(s) = g_i^2 \rho_i(s) F_i^2(s)\) is the phase space with barrier factor \(F_i^2(s)=\exp(-\alpha k_i^2)\).

The real parts of \(\Pi(s)\) is defined using the dispersion intergral

\[Re \Pi_i(s) = \frac{1}{\pi} P \int_{s_{th,i}}^{\infty} \frac{\rho'_i(s')}{s' - s} \mathrm{d} s' = \lim_{\epsilon \rightarrow 0} \left[ \int_{s_{th,i}}^{s-\epsilon} \frac{\rho'_i(s')}{s' - s} \mathrm{d} s' +\int_{s+\epsilon}^{\infty} \frac{\rho'_i(s')}{s' - s} \mathrm{d} s'\right]\]

The reprodution of the Fig1 in PRD78,074023(2008) .

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

The Argand plot

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

17. "Flatte" (ParticleFlatte)

Flatte like formula

\[R(m) = \frac{1}{m_0^2 - m^2 + i m_0 (\sum_{i} g_i \frac{q_i}{m})}\]

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

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

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

18. "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 \(m_{i,1}, m_{i,2}\).

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

19. "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 arguments l_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 \(m_0=0.7, m_{0,1}=m_{0,2}=0.1, m_{1,1}=m_{1,2}=0.3, g_0=0.3,g_1=0.2,l_0=0,l_1=1\).

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

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 \(q_{i0}=1\).

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

20. "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.

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

21. "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] and width_list: [width1, width2].

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

22. "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

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

23. "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

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

25. "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

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

26. "MultiBWR" (ParticleMultiBWR)

Combine Multi BWR into one particle

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

27. "MultiBW" (ParticleMultiBW)

Combine Multi BW into one particle

28. "linear_npy" (InterpLinearNpy)

Linear interpolation model from a npy file with array of [mi, re(ai), im(ai)]. Required file: path_of_file.npy, for the path of npy file.

The example is exp(5 I m).

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

29. "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 of txt file.

The example is exp(5 I m).

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

30. "interp" (Interp)

linear interpolation for real number

31. "interp_c" (Interp)

linear interpolation for complex number

32. "spline_c" (Interp1DSpline)

Spline interpolation function for model independent resonance

33. "interp1d3" (Interp1D3)

Piecewise third order interpolation

34. "interp_lagrange" (Interp1DLang)

Lagrange interpolation

35. "interp_hist" (InterpHist)

Interpolation for each bins as constant

36. "hist_idx" (InterpHistIdx)

Interpolation for each bins as constant

use

min_m: 0.19
max_m: 0.91
interp_N: 8
with_bound: True

for 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.

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

37. "spline_c_idx" (Interp1DSplineIdx)

Spline function in index way.

use

min_m: 0.19
max_m: 0.91
interp_N: 8
with_bound: True

for 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.

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

38. "sppchip" (InterpSPPCHIP)

Shape-Preserving Piecewise Cubic Hermite Interpolation Polynomial. It is monotonic in each interval.

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