Dynamics lineshapes

7.1. Dynamics lineshapes#

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import sympy as sp
from ampform.kinematics.phasespace import Kallen
from ampform_dpd.dynamics import (
    BlattWeisskopf,
    BreitWignerMinL,
    BuggBreitWigner,
    EnergyDependentWidth,
    FlattéSWave,
    P,
    Q,
)

from polarimetry.io import display_doit, display_latex

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z = sp.Symbol("z", positive=True)
L = sp.Symbol("L", integer=True, nonnegative=True)
display_doit(BlattWeisskopf(z, L))
\[\begin{split}\displaystyle \begin{array}{rcl} F_{L}\left(z\right) &=& \begin{cases} 1 & \text{for}\: L = 0 \\\frac{1}{\sqrt{z^{2} + 1}} & \text{for}\: L = 1 \\\frac{1}{\sqrt{z^{4} + 3 z^{2} + 9}} & \text{for}\: L = 2 \end{cases} \\ \end{array}\end{split}\]

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x, y, z = sp.symbols("x:z")
display_doit(Kallen(x, y, z))
\[\begin{split}\displaystyle \begin{array}{rcl} \lambda\left(x, y, z\right) &=& x^{2} - 2 x y - 2 x z + y^{2} - 2 y z + z^{2} \\ \end{array}\end{split}\]

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s, m0, mi, mj, mk = sp.symbols("s m0 m_i:k", nonnegative=True)
display_doit(P(s, mi, mj))
display_doit(Q(s, m0, mk))
\[\begin{split}\displaystyle \begin{array}{rcl} p_{_{m_{i},m_{j}}}\left(s\right) &=& \frac{\sqrt{\lambda\left(s, m_{i}^{2}, m_{j}^{2}\right)}}{2 \sqrt{s}} \\ \end{array}\end{split}\]
\[\begin{split}\displaystyle \begin{array}{rcl} q_{_{m_{0},m_{k}}}\left(s\right) &=& \frac{\sqrt{\lambda\left(s, m_{0}^{2}, m_{k}^{2}\right)}}{2 m_{0}} \\ \end{array}\end{split}\]

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R = sp.Symbol("R")
l_R = sp.Symbol("l_R", integer=True, positive=True)
m, Γ0, m1, m2 = sp.symbols("m Γ0 m1 m2", nonnegative=True)
display_doit(EnergyDependentWidth(s, m, Γ0, m1, m2, l_R, R))
\[\begin{split}\displaystyle \begin{array}{rcl} \Gamma_{l_{R}}\left(s\right) &=& Γ_{0} \frac{m}{\sqrt{s}} \frac{F_{l_{R}}\left(R p_{_{m_{1},m_{2}}}\left(s\right)\right)^{2}}{F_{l_{R}}\left(R p_{_{m_{1},m_{2}}}\left(m^{2}\right)\right)^{2}} \left(\frac{p_{_{m_{1},m_{2}}}\left(s\right)}{p_{_{m_{1},m_{2}}}\left(m^{2}\right)}\right)^{2 l_{R} + 1} \\ \end{array}\end{split}\]

7.1.1. Relativistic Breit-Wigner#

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m_top, m_spec = sp.symbols(R"m_\mathrm{top} m_\mathrm{spectator}")
R_dec, R_prod = sp.symbols(R"R_\mathrm{res} R_{\Lambda_c}")
l_Λc = sp.Symbol(R"l_{\Lambda_c}", integer=True, positive=True)
display_doit(BreitWignerMinL(s, m_top, m_spec, m, Γ0, m1, m2, l_R, l_Λc, R_dec, R_prod))
\[\begin{split}\displaystyle \begin{array}{rcl} \mathcal{R}^\mathrm{BW}_{l_{R},l_{\Lambda_c}}\left(s\right) &=& \frac{\frac{F_{l_{R}}\left(R_\mathrm{res} p_{_{m_{1},m_{2}}}\left(s\right)\right)}{F_{l_{R}}\left(R_\mathrm{res} p_{_{m_{1},m_{2}}}\left(m^{2}\right)\right)} \frac{F_{l_{\Lambda_c}}\left(R_{\Lambda_c} q_{_{m_\mathrm{top},m_\mathrm{spectator}}}\left(s\right)\right)}{F_{l_{\Lambda_c}}\left(R_{\Lambda_c} q_{_{m_\mathrm{top},m_\mathrm{spectator}}}\left(m^{2}\right)\right)} \left(\frac{p_{_{m_{1},m_{2}}}\left(s\right)}{p_{_{m_{1},m_{2}}}\left(m^{2}\right)}\right)^{l_{R}} \left(\frac{q_{_{m_\mathrm{top},m_\mathrm{spectator}}}\left(s\right)}{q_{_{m_\mathrm{top},m_\mathrm{spectator}}}\left(m^{2}\right)}\right)^{l_{\Lambda_c}}}{m^{2} - i m \Gamma_{l_{R}}\left(s\right) - s} \\ \end{array}\end{split}\]

7.1.2. Bugg Breit-Wigner#

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mKπ, m0, Γ0, mK, mπ, γ = sp.symbols(R"m_{K\pi} m0 Gamma0 m_K m_pi gamma")
bugg = BuggBreitWigner(mKπ**2, m0, Γ0, mK, mπ, γ)
q = P(mKπ**2, mK, mπ)
s_A = sp.Symbol("s_A")
definitions = {
    s_A: mK**2 - mπ**2 / 2,
    q: q.evaluate(),
}
display_latex({bugg: bugg.evaluate().subs({v: k for k, v in definitions.items()})})
display_latex(definitions)
\[\begin{split}\displaystyle \begin{array}{rcl} \mathcal{R}^\mathrm{Bugg}\left(m_{K\pi}^{2}\right) &=& \frac{1}{- \frac{i \Gamma_{0} m_{0} \left(m_{K\pi}^{2} - s_{A}\right) e^{- \gamma m_{K\pi}^{2}}}{m_{0}^{2} - s_{A}} + m_{0}^{2} - m_{K\pi}^{2}} \\ \end{array}\end{split}\]
\[\begin{split}\displaystyle \begin{array}{rcl} s_{A} &=& m_{K}^{2} - \frac{m_{\pi}^{2}}{2} \\ p_{_{m_{K},m_{\pi}}}\left(m_{K\pi}^{2}\right) &=& \frac{\sqrt{\lambda\left(m_{K\pi}^{2}, m_{K}^{2}, m_{\pi}^{2}\right)}}{2 \sqrt{m_{K\pi}^{2}}} \\ \end{array}\end{split}\]

One of the models uses a Bugg Breit-Wigner with an exponential factor:

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q = Q(s, m0, m1)
alpha = sp.Symbol("alpha")
bugg * sp.exp(-alpha * q**2)
\[\displaystyle e^{- \alpha q_{_{m_{0},m_{1}}}\left(s\right)^{2}} \mathcal{R}^\mathrm{Bugg}\left(m_{K\pi}^{2}\right)\]

7.1.3. Flatté for S-waves#

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Γ1, Γ2, m1, m2, mπ, mΣ = sp.symbols("Gamma1 Gamma2 m1 m2 m_pi m_Sigma")
display_doit(FlattéSWave(s, m, (Γ1, Γ2), (m1, m2), (mπ, mΣ)))
\[\begin{split}\displaystyle \begin{array}{rcl} \mathcal{R}^\mathrm{Flatté}\left(s\right) &=& \frac{1}{m^{2} - i m \left(\frac{\Gamma_{1} m p_{_{m_{1},m_{2}}}\left(s\right)}{\sqrt{s} p_{_{m_{\pi},m_{\Sigma}}}\left(m^{2}\right)} + \frac{\Gamma_{2} m p_{_{m_{\pi},m_{\Sigma}}}\left(s\right)}{\sqrt{s} p_{_{m_{\pi},m_{\Sigma}}}\left(m^{2}\right)}\right) - s} \\ \end{array}\end{split}\]

where, in this analysis, we couple the \(\Lambda(1405)\) resonance to the channel \(\Lambda(1405) \to \Sigma^-\pi^+\).

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