Polarimeter vector field#
Amplitude model#
The helicity amplitude for the \(\Lambda_c \to p K \pi\) decay reads as a sum of three partial wave series, incorporating \(\Lambda^{**}\) resonances, \(\Delta^{**}\) resonances, and \(K^{**}\) resonances. The particles are ordered as \(\Lambda_c(\mathbf{0}) \to p(\mathbf{1}) \pi(\mathbf{2}) K(\mathbf{3})\).
(1)#\[\begin{split}
\begin{align}
\mathcal{A}_{\nu,\lambda}(m_{K\pi},m_{pK}) &=
\sum_{\nu',\lambda'} \big[\\
&\qquad d_{\nu,\nu'}^{1/2}(\zeta_{1(1)}^0) \mathcal{A}_{\nu',\lambda'}^{K} d_{\lambda',\lambda}^{1/2}(\zeta_{1(1)}^1)
&\mathbf{subsystem\,1\;}(\to 23)\\
&\qquad + d_{\nu,\nu'}^{1/2}(\zeta_{2(1)}^0) \mathcal{A}_{\nu',\lambda'}^{\Lambda} d_{\lambda',\lambda}^{1/2}(\zeta_{2(1)}^1)
&\mathbf{subsystem\,2\;}(\to 31)\\
&\qquad + d_{\nu,\nu'}^{1/2}(\zeta_{3(1)}^0) \mathcal{A}_{\nu',\lambda'}^{\Delta} d_{\lambda',\lambda}^{1/2}(\zeta_{3(1)}^1)\big]\,.
&\mathbf{subsystem\,3\;}(\to 12)\\
\end{align}
\end{split}\]
where \(\zeta^{i}_{j(k)}\) is the Wigner rotation for particle \(i,k\) and chain \(j\). The number in brackets indicates the overall definition of the helicity states \(|1/2,\nu\rangle\) and \(|1/2,\lambda\rangle\) for \(\Lambda_c\) and proton, respectively.
We use the particle-2 convention for the helicity couplings, which leads to the phase factor in the transitions \(\Lambda_c\to K^{**}p\), and \(\Lambda\to K p\):
\[\begin{split}
\begin{align}
\mathcal{A}^{K}_{\nu,\lambda} &= \sum_{j,\tau} \delta_{\nu,\tau - \lambda}\mathcal{H}^{\Lambda_c \to K^{**} p}_{\tau,\lambda} (-)^{1/2 - \lambda} \,d^{j}_{\lambda,0} (\theta_{23}) \, \mathcal{H}^{K^{**} \to \pi K}_{0,0}\\
%
\mathcal{A}^{\Lambda}_{\nu,\lambda} &= \sum_{j,\tau}
\delta_{\nu,\tau}
\mathcal{H}^{\Lambda_c \to \Lambda^{**} \pi}_{\tau,0} d^{j}_{\tau,-\lambda} (\theta_{31}) \mathcal{H}^{\Lambda^{**} \to K p}_{0,\lambda} (-)^{1/2-\lambda} \\
%
\mathcal{A}^{\Delta}_{\nu,\lambda} &= \sum_{j,\tau}
\delta_{\nu,\tau}
\mathcal{H}^{\Lambda_c \to \Delta^{**} K}_{\tau,0} d^{j}_{\tau,\lambda}(\theta_{12}) \mathcal{H}^{\Delta^{**} \to p\pi}_{\lambda,0} \,.
\end{align}
\end{split}\]
The helicity couplings in the particle-2 convention obey simple properties with respect to the parity transformation:
\[
\begin{align}
\mathcal{H}^{A\to BC}_{-\lambda,-\lambda'} = P_A P_B P_C (-)^{j_A-j_B-j_C} \mathcal{H}^{A\to BC}_{\lambda,\lambda'}
\end{align}
\]
It reduced amount of the couplings in the strong decay of isobars. Moreover the magnitude of the couplings cannot be determined separately, therefore, it is set to 1:
\[\begin{split}
\begin{align}
\mathcal{H}^{\Lambda^{**} \to K p}_{0,1/2} &= 1\,, &
\mathcal{H}^{\Delta^{**} \to p\pi}_{1/2,0} &= 1\,,&
\mathcal{H}^{K^{**} \to \pi K}_{0,0} &= 1, \\
\mathcal{H}^{\Lambda^{**} \to K p}_{0,-1/2} &= -P_\Lambda (-)^{j-1/2}\,, &
\mathcal{H}^{\Delta^{**} \to p\pi}_{-1/2,0} &= -P_\Delta (-)^{j-1/2}\,, &&
\end{align}
\end{split}\]
The helicity couplings for the \(\Lambda_c^+\) decay are fit parameters. There are four of them for the \(K^{**}\) chain, and two for both the \(\Delta^{**}\) and \(\Lambda^{**}\) chains.
Resonances and LS-scheme#
Resonance choices and their \(LS\)-couplings are defined as follows:
resonance_names = {
1: ["K*(892)0"],
2: ["Lambda(1520)", "Lambda(1670)"],
3: ["Delta(1232)++"],
}
| resonance | \(j^P\) | \(m\) (MeV) | \(\Gamma_0\) (MeV) | \(l_R\) | \(l_{\Lambda_c}^\mathrm{min}\) |
|---|---|---|---|---|---|
| \(K^{*}(892)^{0} \to \pi^{+} K^{-}\) | \(1^-\) | 895 | 47 | 1 | 0 |
| \(\Lambda(1520) \to p K^{-}\) | \(3/2^-\) | 1519 | 16 | 2 | 1 |
| \(\Lambda(1670) \to p K^{-}\) | \(1/2^-\) | 1674 | 30 | 0 | 0 |
| \(\Delta(1232)^{++} \to p \pi^{+}\) | \(3/2^+\) | 1232 | 117 | 1 | 1 |
Aligned amplitude#
\[\displaystyle \sum_{\lambda^{\prime}=-1/2}^{1/2} \sum_{\nu^{\prime}=-1/2}^{1/2}{A^{K}_{\nu^{\prime}, \lambda^{\prime}} d^{\frac{1}{2}}_{\lambda^{\prime},\lambda}\left(\zeta^1_{1(1)}\right) d^{\frac{1}{2}}_{\nu,\nu^{\prime}}\left(\zeta^0_{1(1)}\right) + A^{\Delta}_{\nu^{\prime}, \lambda^{\prime}} d^{\frac{1}{2}}_{\lambda^{\prime},\lambda}\left(\zeta^1_{3(1)}\right) d^{\frac{1}{2}}_{\nu,\nu^{\prime}}\left(\zeta^0_{3(1)}\right) + A^{\Lambda}_{\nu^{\prime}, \lambda^{\prime}} d^{\frac{1}{2}}_{\lambda^{\prime},\lambda}\left(\zeta^1_{2(1)}\right) d^{\frac{1}{2}}_{\nu,\nu^{\prime}}\left(\zeta^0_{2(1)}\right)}\]
Dynamics#
The lineshape function is factored out of the \(\Lambda_c\) helicity coupling:
\[
\begin{align}
\mathcal{H}^{\Lambda_c \to R x}_{\lambda,\lambda'} = \hat{\mathcal{H}}^{\Lambda_c \to R x}_{\lambda,\lambda'}\,\mathcal{R}(s)\,.
\end{align}
\]
The relativistic Breit-Wigner parametrization reads:
\[
\begin{align}
\mathcal{R}(s) =
\left(\frac{q}{q_0}\right)^{l_{\Lambda_c}^\text{min}} \frac{F_{l_{\Lambda_c}^\text{min}}(q R)}{F_{l_{\Lambda_c}^\text{min}}(q_0 R)}\,
\frac{1}{m^2-s-im\Gamma(s)}
\left(\frac{p}{p_0}\right)^{l_R} \frac{F_{l_R}(pR)}{F_{l_R}(p_0R)},
\end{align}
\]
with energy-dependent width given by
\[
\begin{align}
\Gamma(s) = \Gamma_0 \left(\frac{p}{p_0}\right)^{2l_R+1} \frac{m}{\sqrt{s}} \, \frac{F_{l_R}^2(pR)}{F_{l_R}^2(p_0R)}\,,
\end{align}
\]
The form-factor \(F\) is the Blatt-Weisskopf factor with the length factor \(R=5\,\)GeV\(^{-1}\):
\[
\begin{align}
F_0(pR) &= 1\,,&
F_1(pR) &= \sqrt{\frac{1}{1+(pR)^2}}\,,&
F_2(pR) &= \sqrt{\frac{1}{9+3(pR)^2+(pR)^4}}\,.
\end{align}
\]
The break-up momenta is calculated for every decay chain separately. Using the notations \(0->R(\to ij) k\), one writes:
\[
\begin{align}
p &= \lambda^{1/2}(s,m_i^2,m_j^2)/(2\sqrt{s})\,, &
q &= \lambda^{1/2}(s,m_0^2,m_k^2)/(2m_0)\,.
\end{align}
\]
The momenta with subindex zero are computed for nominal mass of the resonance, \(s=m^2\). The three-argument Källén function reads:
(2)#\[
\begin{align}
\lambda(x,y,z) = x^2+y^2+z^2 - 2xy-2yz-2zx\,.
\end{align}
\]
Formulation with SymPy
\[\begin{split}\displaystyle \begin{align*}
F_{L}\left(z\right) = & \begin{cases} 1 & \text{for}\: L = 0 \\\frac{1}{\sqrt{z^{2} + 1}} & \text{for}\: L = 1 \\\frac{1}{\sqrt{z^{4} + 3 z^{2} + 9}} & \text{for}\: L = 2 \end{cases}
\end{align*}\end{split}\]
\[\displaystyle \begin{eqnarray}
\lambda\left(x, y, z\right) & = & x^{2} - 2 x y - 2 x z + y^{2} - 2 y z + z^{2}
\end{eqnarray}\]
\[\displaystyle \begin{eqnarray}
p_{s} & = & \frac{\sqrt{\lambda\left(s, m_{i}^{2}, m_{j}^{2}\right)}}{2 \sqrt{s}}
\end{eqnarray}\]
\[\displaystyle \begin{eqnarray}
q_{s} & = & \frac{\sqrt{\lambda\left(s, m_{0}^{2}, m_{k}^{2}\right)}}{2 m_{0}}
\end{eqnarray}\]
\[\displaystyle \begin{eqnarray}
\Gamma\left(s\right) & = & Γ_{0} \frac{m}{\sqrt{s}} \frac{F_{l_{R}}\left(R p_{s}\right)^{2}}{F_{l_{R}}\left(R p_{m^{2}}\right)^{2}} \left(\frac{p_{s}}{p_{m^{2}}}\right)^{2 l_{R} + 1}
\end{eqnarray}\]
\[\displaystyle \begin{eqnarray}
\mathcal{R}\left(s\right) & = & \frac{\frac{F_{l_{R}}\left(R p_{s}\right)}{F_{l_{R}}\left(R p_{m^{2}}\right)} \frac{F_{l_{\Lambda_c}}\left(R q_{s}\right)}{F_{l_{\Lambda_c}}\left(R q_{m^{2}}\right)} \left(\frac{p_{s}}{p_{m^{2}}}\right)^{l_{R}} \left(\frac{q_{s}}{q_{m^{2}}}\right)^{l_{\Lambda_c}}}{m^{2} - i m \Gamma\left(s\right) - s}
\end{eqnarray}\]
Decay chain amplitudes#
\[\displaystyle \sum_{\tau=-1}^{1}{\left(-1\right)^{\frac{1}{2} - \lambda} \delta_{\nu, - \lambda + \tau} \mathcal{H}^\mathrm{decay}_{K^{*}(892)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{K^{*}(892)^{0}, \tau, \lambda} \mathcal{R}\left(\sigma_{1}\right) d^{1}_{\tau,0}\left(\theta_{23}\right)}\]
\[\displaystyle \sum_{\tau=-3/2}^{3/2}{\left(-1\right)^{\frac{1}{2} - \lambda} \delta_{\nu \tau} \mathcal{H}^\mathrm{decay}_{\Lambda(1520), 0, \lambda} \mathcal{H}^\mathrm{production}_{\Lambda(1520), \tau, 0} \mathcal{R}\left(\sigma_{2}\right) d^{\frac{3}{2}}_{\tau,- \lambda}\left(\theta_{31}\right)} + \sum_{\tau=-1/2}^{1/2}{\left(-1\right)^{\frac{1}{2} - \lambda} \delta_{\nu \tau} \mathcal{H}^\mathrm{decay}_{\Lambda(1670), 0, \lambda} \mathcal{H}^\mathrm{production}_{\Lambda(1670), \tau, 0} \mathcal{R}\left(\sigma_{2}\right) d^{\frac{1}{2}}_{\tau,- \lambda}\left(\theta_{31}\right)}\]
\[\displaystyle \sum_{\tau=-3/2}^{3/2}{\delta_{\nu \tau} \mathcal{H}^\mathrm{decay}_{\Delta(1232)^{++}, \lambda, 0} \mathcal{H}^\mathrm{production}_{\Delta(1232)^{++}, \tau, 0} \mathcal{R}\left(\sigma_{3}\right) d^{\frac{3}{2}}_{\tau,\lambda}\left(\theta_{12}\right)}\]
Angle definitions#
Angles with repeated lower indices are trivial. The other angles are computed from the invariants and these are the angles with sign
\[\begin{split}
\begin{align}
\zeta_{1(1)}^{0} &= \hat{\theta}_{1(1)}^{0} = 0\,, & \zeta_{1(1)}^{1} &= 0\,,\\
\zeta_{2(1)}^0 &=\hat{\theta}_{2(1)} =-\hat{\theta}_{1(2)}\,,\\
\zeta_{3(1)}^0 &= \hat{\theta}_{3(1)}\,, &\zeta_{3(1)}^1 &= -\zeta_{1(3)}^1\,,
\end{align}
\end{split}\]
The expressions for the cosine of the positive (anticlockwise) angles, \(\theta_{12}, \theta_{23}, \theta_{13}\) and \(\hat\theta_{1(2)}, \hat\theta_{3(1)}, \zeta^1_{1(3)}\) can be expressed in terms of Mandelstam variables \(\sigma_1, \sigma_2, \sigma_3\) using [Mikhasenko et al., 2020], Appendix A:
\[\begin{split}\displaystyle \begin{array}{rcl}
\theta_{12} & = & \operatorname{acos}{\left(\frac{2 \sigma_{3} \left(- m_{K}^{2} - m_{p}^{2} + \sigma_{2}\right) - \left(- m_{K}^{2} + m_{\Lambda_c}^{2} - \sigma_{3}\right) \left(m_{p}^{2} - m_{\pi}^{2} + \sigma_{3}\right)}{\sqrt{\lambda\left(m_{\Lambda_c}^{2}, m_{K}^{2}, \sigma_{3}\right)} \sqrt{\lambda\left(\sigma_{3}, m_{p}^{2}, m_{\pi}^{2}\right)}} \right)} \\
\theta_{23} & = & \operatorname{acos}{\left(\frac{2 \sigma_{1} \left(- m_{p}^{2} - m_{\pi}^{2} + \sigma_{3}\right) - \left(- m_{K}^{2} + m_{\pi}^{2} + \sigma_{1}\right) \left(- m_{p}^{2} + m_{\Lambda_c}^{2} - \sigma_{1}\right)}{\sqrt{\lambda\left(m_{\Lambda_c}^{2}, m_{p}^{2}, \sigma_{1}\right)} \sqrt{\lambda\left(\sigma_{1}, m_{\pi}^{2}, m_{K}^{2}\right)}} \right)} \\
\theta_{31} & = & \operatorname{acos}{\left(\frac{2 \sigma_{2} \left(- m_{K}^{2} - m_{\pi}^{2} + \sigma_{1}\right) - \left(m_{K}^{2} - m_{p}^{2} + \sigma_{2}\right) \left(- m_{\pi}^{2} + m_{\Lambda_c}^{2} - \sigma_{2}\right)}{\sqrt{\lambda\left(m_{\Lambda_c}^{2}, m_{\pi}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{K}^{2}, m_{p}^{2}\right)}} \right)} \\
\zeta^0_{1(1)} & = & 0 \\
\zeta^0_{2(1)} & = & - \operatorname{acos}{\left(\frac{- 2 m_{\Lambda_c}^{2} \left(- m_{p}^{2} - m_{\pi}^{2} + \sigma_{3}\right) + \left(m_{p}^{2} + m_{\Lambda_c}^{2} - \sigma_{1}\right) \left(m_{\pi}^{2} + m_{\Lambda_c}^{2} - \sigma_{2}\right)}{\sqrt{\lambda\left(m_{\Lambda_c}^{2}, m_{\pi}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(m_{\Lambda_c}^{2}, \sigma_{1}, m_{p}^{2}\right)}} \right)} \\
\zeta^0_{3(1)} & = & \operatorname{acos}{\left(\frac{- 2 m_{\Lambda_c}^{2} \left(- m_{K}^{2} - m_{p}^{2} + \sigma_{2}\right) + \left(m_{K}^{2} + m_{\Lambda_c}^{2} - \sigma_{3}\right) \left(m_{p}^{2} + m_{\Lambda_c}^{2} - \sigma_{1}\right)}{\sqrt{\lambda\left(m_{\Lambda_c}^{2}, m_{p}^{2}, \sigma_{1}\right)} \sqrt{\lambda\left(m_{\Lambda_c}^{2}, \sigma_{3}, m_{K}^{2}\right)}} \right)} \\
\zeta^1_{1(1)} & = & 0 \\
\zeta^1_{2(1)} & = & \operatorname{acos}{\left(\frac{2 m_{p}^{2} \left(- m_{K}^{2} - m_{\Lambda_c}^{2} + \sigma_{3}\right) + \left(- m_{K}^{2} - m_{p}^{2} + \sigma_{2}\right) \left(m_{p}^{2} + m_{\Lambda_c}^{2} - \sigma_{1}\right)}{\sqrt{\lambda\left(m_{\Lambda_c}^{2}, m_{p}^{2}, \sigma_{1}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{p}^{2}, m_{K}^{2}\right)}} \right)} \\
\zeta^1_{3(1)} & = & - \operatorname{acos}{\left(\frac{2 m_{p}^{2} \left(- m_{\pi}^{2} - m_{\Lambda_c}^{2} + \sigma_{2}\right) + \left(- m_{p}^{2} - m_{\pi}^{2} + \sigma_{3}\right) \left(m_{p}^{2} + m_{\Lambda_c}^{2} - \sigma_{1}\right)}{\sqrt{\lambda\left(m_{\Lambda_c}^{2}, m_{p}^{2}, \sigma_{1}\right)} \sqrt{\lambda\left(\sigma_{3}, m_{p}^{2}, m_{\pi}^{2}\right)}} \right)} \\
\end{array}\end{split}\]
where \(m_0\) is the mass of the initial state \(\Lambda_c\) and \(m_1, m_2, m_3\) are the masses of \(p, \pi, K\), respectively:
\[\begin{split}\displaystyle \begin{array}{rcl}
m_{\Lambda_c} & = & 2.28646 \\
m_{p} & = & 0.938272081 \\
m_{\pi} & = & 0.13957039 \\
m_{K} & = & 0.493677 \\
\end{array}\end{split}\]
Helicity coupling values#
\[\begin{split}\displaystyle \begin{array}{rcl}
\mathcal{H}^\mathrm{decay}_{K^{*}(892)^{0}, 0, 0} & = & 1 \\
\mathcal{H}^\mathrm{decay}_{\Lambda(1520), 0, \frac{1}{2}} & = & 1 \\
\mathcal{H}^\mathrm{decay}_{\Lambda(1520), 0, - \frac{1}{2}} & = & -1 \\
\mathcal{H}^\mathrm{decay}_{\Lambda(1670), 0, \frac{1}{2}} & = & 1 \\
\mathcal{H}^\mathrm{decay}_{\Lambda(1670), 0, - \frac{1}{2}} & = & 1 \\
\mathcal{H}^\mathrm{decay}_{\Delta(1232)^{++}, \frac{1}{2}, 0} & = & 1 \\
\mathcal{H}^\mathrm{decay}_{\Delta(1232)^{++}, - \frac{1}{2}, 0} & = & 1 \\
\end{array}\end{split}\]
\[\begin{split}\displaystyle \begin{array}{rcl}
\mathcal{H}^\mathrm{production}_{K^{*}(892)^{0}, 0, - \frac{1}{2}} & = & 1 \\
\mathcal{H}^\mathrm{production}_{K^{*}(892)^{0}, -1, - \frac{1}{2}} & = & 1.0 - 1.0 i \\
\mathcal{H}^\mathrm{production}_{K^{*}(892)^{0}, 1, \frac{1}{2}} & = & -3.0 - 3.0 i \\
\mathcal{H}^\mathrm{production}_{K^{*}(892)^{0}, 0, \frac{1}{2}} & = & -1.0 - 4.0 i \\
\mathcal{H}^\mathrm{production}_{K_{0}^{*}(1430)^{0}, 0, - \frac{1}{2}} & = & 1 \\
\mathcal{H}^\mathrm{production}_{K_{0}^{*}(1430)^{0}, -1, - \frac{1}{2}} & = & 1.0 - 1.0 i \\
\mathcal{H}^\mathrm{production}_{K_{0}^{*}(1430)^{0}, 1, \frac{1}{2}} & = & -3.0 - 3.0 i \\
\mathcal{H}^\mathrm{production}_{K_{0}^{*}(1430)^{0}, 0, \frac{1}{2}} & = & -1.0 - 4.0 i \\
\mathcal{H}^\mathrm{production}_{K_{2}^{*}(1430)^{0}, 0, - \frac{1}{2}} & = & 1 \\
\mathcal{H}^\mathrm{production}_{K_{2}^{*}(1430)^{0}, -1, - \frac{1}{2}} & = & 1.0 - 1.0 i \\
\mathcal{H}^\mathrm{production}_{K_{2}^{*}(1430)^{0}, 1, \frac{1}{2}} & = & -3.0 - 3.0 i \\
\mathcal{H}^\mathrm{production}_{K_{2}^{*}(1430)^{0}, 0, \frac{1}{2}} & = & -1.0 - 4.0 i \\
\mathcal{H}^\mathrm{production}_{\Lambda(1520), \frac{1}{2}, 0} & = & 1.5 \\
\mathcal{H}^\mathrm{production}_{\Lambda(1520), - \frac{1}{2}, 0} & = & 0.3 \\
\mathcal{H}^\mathrm{production}_{\Lambda(1670), \frac{1}{2}, 0} & = & -0.5 + 1.0 i \\
\mathcal{H}^\mathrm{production}_{\Lambda(1670), - \frac{1}{2}, 0} & = & -0.3 - 0.1 i \\
\mathcal{H}^\mathrm{production}_{\Delta(1232)^{++}, \frac{1}{2}, 0} & = & -13.0 + 5.0 i \\
\mathcal{H}^\mathrm{production}_{\Delta(1232)^{++}, - \frac{1}{2}, 0} & = & -7.0 + 3.0 i \\
\end{array}\end{split}\]
Intensity expression#
Incoherent sum of the amplitudes defined by Aligned amplitude:
\[\displaystyle \sum_{\lambda=-1/2}^{1/2} \sum_{\nu=-1/2}^{1/2}{\left|{\sum_{\lambda^{\prime}=-1/2}^{1/2} \sum_{\nu^{\prime}=-1/2}^{1/2}{A^{K}_{\nu^{\prime}, \lambda^{\prime}} d^{\frac{1}{2}}_{\lambda^{\prime},\lambda}\left(\zeta^1_{1(1)}\right) d^{\frac{1}{2}}_{\nu,\nu^{\prime}}\left(\zeta^0_{1(1)}\right) + A^{\Delta}_{\nu^{\prime}, \lambda^{\prime}} d^{\frac{1}{2}}_{\lambda^{\prime},\lambda}\left(\zeta^1_{3(1)}\right) d^{\frac{1}{2}}_{\nu,\nu^{\prime}}\left(\zeta^0_{3(1)}\right) + A^{\Lambda}_{\nu^{\prime}, \lambda^{\prime}} d^{\frac{1}{2}}_{\lambda^{\prime},\lambda}\left(\zeta^1_{2(1)}\right) d^{\frac{1}{2}}_{\nu,\nu^{\prime}}\left(\zeta^0_{2(1)}\right)}}\right|^{2}}\]
Remaining free_symbols are indeed the specific amplitudes as defined by Decay chain amplitudes:
The specific amplitudes from Decay chain amplitudes need to be formulated for each value of \(\nu, \lambda\), so that they can be substituted in the top expression:
\[\begin{split}\displaystyle \begin{array}{rcl}
A^{K}_{- \frac{1}{2}, - \frac{1}{2}} & = & \sum_{\tau=-1}^{1}{- \delta_{- \frac{1}{2}, \tau + \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{K^{*}(892)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{K^{*}(892)^{0}, \tau, - \frac{1}{2}} \mathcal{R}\left(\sigma_{1}\right) d^{1}_{\tau,0}\left(\theta_{23}\right)} \\
A^{K}_{- \frac{1}{2}, \frac{1}{2}} & = & \sum_{\tau=-1}^{1}{\delta_{- \frac{1}{2}, \tau - \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{K^{*}(892)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{K^{*}(892)^{0}, \tau, \frac{1}{2}} \mathcal{R}\left(\sigma_{1}\right) d^{1}_{\tau,0}\left(\theta_{23}\right)} \\
A^{K}_{\frac{1}{2}, - \frac{1}{2}} & = & \sum_{\tau=-1}^{1}{- \delta_{\frac{1}{2}, \tau + \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{K^{*}(892)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{K^{*}(892)^{0}, \tau, - \frac{1}{2}} \mathcal{R}\left(\sigma_{1}\right) d^{1}_{\tau,0}\left(\theta_{23}\right)} \\
A^{K}_{\frac{1}{2}, \frac{1}{2}} & = & \sum_{\tau=-1}^{1}{\delta_{\frac{1}{2}, \tau - \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{K^{*}(892)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{K^{*}(892)^{0}, \tau, \frac{1}{2}} \mathcal{R}\left(\sigma_{1}\right) d^{1}_{\tau,0}\left(\theta_{23}\right)} \\
A^{\Lambda}_{- \frac{1}{2}, - \frac{1}{2}} & = & \sum_{\tau=-3/2}^{3/2}{- \delta_{- \frac{1}{2} \tau} \mathcal{H}^\mathrm{decay}_{\Lambda(1520), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1520), \tau, 0} \mathcal{R}\left(\sigma_{2}\right) d^{\frac{3}{2}}_{\tau,\frac{1}{2}}\left(\theta_{31}\right)} + \sum_{\tau=-1/2}^{1/2}{- \delta_{- \frac{1}{2} \tau} \mathcal{H}^\mathrm{decay}_{\Lambda(1670), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1670), \tau, 0} \mathcal{R}\left(\sigma_{2}\right) d^{\frac{1}{2}}_{\tau,\frac{1}{2}}\left(\theta_{31}\right)} \\
A^{\Lambda}_{- \frac{1}{2}, \frac{1}{2}} & = & \sum_{\tau=-3/2}^{3/2}{\delta_{- \frac{1}{2} \tau} \mathcal{H}^\mathrm{decay}_{\Lambda(1520), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1520), \tau, 0} \mathcal{R}\left(\sigma_{2}\right) d^{\frac{3}{2}}_{\tau,- \frac{1}{2}}\left(\theta_{31}\right)} + \sum_{\tau=-1/2}^{1/2}{\delta_{- \frac{1}{2} \tau} \mathcal{H}^\mathrm{decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1670), \tau, 0} \mathcal{R}\left(\sigma_{2}\right) d^{\frac{1}{2}}_{\tau,- \frac{1}{2}}\left(\theta_{31}\right)} \\
A^{\Lambda}_{\frac{1}{2}, - \frac{1}{2}} & = & \sum_{\tau=-3/2}^{3/2}{- \delta_{\frac{1}{2} \tau} \mathcal{H}^\mathrm{decay}_{\Lambda(1520), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1520), \tau, 0} \mathcal{R}\left(\sigma_{2}\right) d^{\frac{3}{2}}_{\tau,\frac{1}{2}}\left(\theta_{31}\right)} + \sum_{\tau=-1/2}^{1/2}{- \delta_{\frac{1}{2} \tau} \mathcal{H}^\mathrm{decay}_{\Lambda(1670), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1670), \tau, 0} \mathcal{R}\left(\sigma_{2}\right) d^{\frac{1}{2}}_{\tau,\frac{1}{2}}\left(\theta_{31}\right)} \\
A^{\Lambda}_{\frac{1}{2}, \frac{1}{2}} & = & \sum_{\tau=-3/2}^{3/2}{\delta_{\frac{1}{2} \tau} \mathcal{H}^\mathrm{decay}_{\Lambda(1520), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1520), \tau, 0} \mathcal{R}\left(\sigma_{2}\right) d^{\frac{3}{2}}_{\tau,- \frac{1}{2}}\left(\theta_{31}\right)} + \sum_{\tau=-1/2}^{1/2}{\delta_{\frac{1}{2} \tau} \mathcal{H}^\mathrm{decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1670), \tau, 0} \mathcal{R}\left(\sigma_{2}\right) d^{\frac{1}{2}}_{\tau,- \frac{1}{2}}\left(\theta_{31}\right)} \\
A^{\Delta}_{- \frac{1}{2}, - \frac{1}{2}} & = & \sum_{\tau=-3/2}^{3/2}{\delta_{- \frac{1}{2} \tau} \mathcal{H}^\mathrm{decay}_{\Delta(1232)^{++}, - \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1232)^{++}, \tau, 0} \mathcal{R}\left(\sigma_{3}\right) d^{\frac{3}{2}}_{\tau,- \frac{1}{2}}\left(\theta_{12}\right)} \\
A^{\Delta}_{- \frac{1}{2}, \frac{1}{2}} & = & \sum_{\tau=-3/2}^{3/2}{\delta_{- \frac{1}{2} \tau} \mathcal{H}^\mathrm{decay}_{\Delta(1232)^{++}, \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1232)^{++}, \tau, 0} \mathcal{R}\left(\sigma_{3}\right) d^{\frac{3}{2}}_{\tau,\frac{1}{2}}\left(\theta_{12}\right)} \\
A^{\Delta}_{\frac{1}{2}, - \frac{1}{2}} & = & \sum_{\tau=-3/2}^{3/2}{\delta_{\frac{1}{2} \tau} \mathcal{H}^\mathrm{decay}_{\Delta(1232)^{++}, - \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1232)^{++}, \tau, 0} \mathcal{R}\left(\sigma_{3}\right) d^{\frac{3}{2}}_{\tau,- \frac{1}{2}}\left(\theta_{12}\right)} \\
A^{\Delta}_{\frac{1}{2}, \frac{1}{2}} & = & \sum_{\tau=-3/2}^{3/2}{\delta_{\frac{1}{2} \tau} \mathcal{H}^\mathrm{decay}_{\Delta(1232)^{++}, \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1232)^{++}, \tau, 0} \mathcal{R}\left(\sigma_{3}\right) d^{\frac{3}{2}}_{\tau,\frac{1}{2}}\left(\theta_{12}\right)} \\
\end{array}\end{split}\]
Polarization sensitivity#
We introduce the polarimeter vector field (polarization sensitivity) of the \(\Lambda_c\) decay. It is defined by three quantities \((\alpha_x,\alpha_y,\alpha_z)\) forming a three-dimensional vector \(\vec\alpha\) dependent on just two decay variables, \(\sigma_1=m_{K\pi}^2\), and \(\sigma_2=m_{pK}^2\).
The polarimeter vector field is computed by averaging the Pauli matrices \(\vec\sigma\) contracted with the \(\Lambda_c^+\) helicity indices given the transition amplitude.
(3)#\[
\begin{align}
\vec\alpha(m_{K\pi},m_{pK}) =
\sum_{\lambda,\nu,\nu'}
A^{*}_{\nu,\lambda}\vec\sigma_{\nu,\nu'}
A_{\nu',\lambda} \,\big / \sum_{\lambda,\nu}
\left|A_{\nu,\lambda}\right|^2
\end{align}
\]
The quantities \(\vec\alpha(m_{K\pi},m_{pK})\) give the model-independent representation of the \(\Lambda_c^+\) decay process. It can be used to study \(\Lambda_c^+\) production polarization using
(4)#\[
\begin{align}
I(\alpha,\beta,\gamma,m_{K\pi},m_{pK}) =
I_0(m_{K\pi},m_{pK})\,
\left(1 + \sum_{i,j} P_i R_{ij}(\alpha,\beta,\gamma) \alpha_j(m_{K\pi},m_{pK}) \right)\,,
\end{align}
\]
where \(R_{ij}(\alpha,\beta,\gamma)\) is a three-dimensional rotation matrix:
\[
\begin{align}
R(\alpha,\beta,\gamma) = R_z(\alpha)R_y(\beta)R_z(\gamma)\,,
\end{align}
\]
and \(I_0\) is the averaged decay rate
\[
\begin{align}
I_0(m_{K\pi},m_{pK}) = \sum_{\lambda,\nu}\left|A_{\nu,\lambda}\right|^2\,.
\end{align}
\]
def to_index(helicity):
"""Symbolic conversion of half-value helicities to Pauli matrix indices."""
# https://github.com/ComPWA/compwa.github.io/pull/129#issuecomment-1096599896
return sp.Piecewise(
(1, sp.LessThan(helicity, 0)),
(0, True),
)
ν_prime = sp.Symbol(R"\nu^{\prime}")
polarimetry_exprs = tuple(
PoolSum(
formulate_aligned_amplitude(ν, λ).conjugate()
* msigma(i)[to_index(ν), to_index(ν_prime)]
* formulate_aligned_amplitude(ν_prime, λ),
(λ, [-half, +half]),
(ν, [-half, +half]),
(ν_prime, [-half, +half]),
)
/ intensity_expr
for i in (1, 2, 3)
)
Properties of the vector \(\vec\alpha\)#
The vector \(\vec \alpha\) introduced in Eq. (3) obeys the following properties:
It is a three-dimensional vector defined in the rest frame of the decaying particle. Particularly, it is transformed as a regular vector in case initial (alignment) configuration change.
The length of the vector is limited by 1: \(|\vec{\alpha}| < 1\)
\(\alpha_y=0\) for the decays of a fermion to a fermions and (pseudo)scalar
Here is the prove of the second statement:
\[\begin{split}
\begin{align}
I_{\nu',\nu} = \sum_{\lambda} A_{\nu',\lambda}^* A_{\nu,\lambda} =
\begin{pmatrix}
a & c^*\\
c & b
\end{pmatrix} = \frac{a+b}{2}\left(
\mathbb{I} + (\vec{\sigma} \cdot \vec{\alpha})
\right)\,,
\end{align}
\end{split}\]
where
\[\begin{split}
\begin{align}
a &= \left|A_{+,+}\right|^2+\left|A_{+,-}\right|^2\,,\\ \nonumber
b &= \left|A_{-,+}\right|^2+\left|A_{-,-}\right|^2\,,\\ \nonumber
c &= A_{+,+}^*A_{-,+} + A_{+,-}^*A_{-,-}\,,
\end{align}
\end{split}\]
and
\[
\begin{align}
\alpha_x &= \frac{\text{Re}\,c}{a+b}\,,&
\alpha_x &= \frac{\text{Im}\,c}{a+b}\,,&
\alpha_z &= \frac{a-b}{a+b}\,,
\end{align}
\]
To constraint the length of the \(\vec\alpha\), one notices \(ab - c \ge 0\). Therefore,
\[
\begin{align}
|\vec\alpha|^2 &= \frac{(a-b)^2+c^2}{(a+b)^2} = \frac{(a+b)^2-4ab+c^2}{(a+b)^2} \leq \frac{(a+b)^2-3ab}{(a+b)^2} \leq 1\,
\end{align}
\]
since \(a,b \geq 0\).
Computations with TensorWaves#
Conversion to computational backend#
The full expression tree can be converted to a computational, parametrized function as follows. Note that identify all coupling symbols are interpreted as parameters. The remaining symbols (the angles) become arguments to the function.
free_parameters = {
symbol: value
for symbol, value in parameter_defaults.items()
if (symbol.name.startswith("m_") and symbol not in masses)
or symbol.name.startswith(R"\Gamma_")
or symbol in prod_couplings
}
fixed_parameters = {
symbol: value
for symbol, value in parameter_defaults.items()
if symbol not in free_parameters
}
intensity_func = create_parametrized_function(
unfolded_intensity_expr.xreplace(fixed_parameters),
parameters=free_parameters,
backend="jax",
)
polarimetry_funcs = tuple(
create_parametrized_function(
expr.xreplace(fixed_parameters),
parameters=free_parameters,
backend="jax",
)
for expr in unfolded_polarimetry_exprs
)
Phase space#
The \(\Lambda_c^+ \to p K \pi\) kinematics is fully described by two dynamic variables, \(m_{K\pi}\) and \(m_{pK}\) (see Phase space for a three-body decay). The third Mandelstam variable can be computed from the other two and the masses of the initial and final state:
\[\begin{split}\displaystyle \begin{array}{rcl}
\sigma_{3} & = & m_{K}^{2} + m_{p}^{2} + m_{\pi}^{2} + m_{\Lambda_c}^{2} - \sigma_{1} - \sigma_{2} \\
\end{array}\end{split}\]
Values for the angles will be computed form the Mandelstam values with a data transformer for the symbolic angle definitions:
kinematic_variables = {
symbol: expression.doit().xreplace(masses).xreplace(fixed_parameters)
for symbol, expression in angles.items()
}
kinematic_variables.update({s: s for s in [σ1, σ2, σ3]}) # include identity
transformer = SympyDataTransformer.from_sympy(kinematic_variables, backend="jax")
We now define phase space over a grid that contains the space in the Dalitz plane that is kinematically ‘available’ to the decay:
m0_val, m1_val, m2_val, m3_val = masses.values()
σ1_min = (m2_val + m3_val) ** 2
σ1_max = (m0_val - m1_val) ** 2
σ2_min = (m1_val + m3_val) ** 2
σ2_max = (m0_val - m2_val) ** 2
def generate_phsp_grid(resolution: int):
x = np.linspace(σ1_min, σ1_max, num=resolution)
y = np.linspace(σ2_min, σ2_max, num=resolution)
X, Y = np.meshgrid(x, y)
Z = compute_third_mandelstam.function(X, Y)
phsp = {"sigma1": X, "sigma2": Y, "sigma3": Z}
return X, Y, transformer(phsp)
X, Y, phsp = generate_phsp_grid(resolution=500)
Intensity distribution#
Finally, all intensities can be computed as follows:
Fit fractions#
The total decay rate for \(\Lambda_c^+ \to pK\pi\) can be broken into fractions that correspond to the different decay chains and interference terms. The total rate is computed as an integral of the intensity over decay kinematics:
\[
\begin{align}
I_\text{tot}(\{\mathcal{H}\}) = \int d m_{pK}^2 d m_{K\pi}^2\,
I_0(m_{pK}, m_{K\pi} | \{\mathcal{H}\})
\approx \frac{\Phi_0}{N_\text{MC}} \sum_{e=1}^{N_\text{MC}}\,\,I_0(m_{pK,e}, m_{K\pi,e} | \{\mathcal{H}\})\,,
\end{align}
\]
where \(\Phi_0\) is an (irrelevant) constant equal to the flat phase-space integral, \((m_{pK,e}, m_{K\pi,e})\) is a vector of the kinematic variables for the \(e\)-th point in the MC sample.
The conditional argument \(\{\mathcal{H}\}\) indicates dependence of the rate on the value of the couplings. The individual fractions are found by computing the total rate for a subset of couplings set to zero,
\[\begin{split}
\begin{align}
I_\text{tot}^{K} &= I_\text{tot}\left(\{\mathcal{H}^{\Lambda_c^+\to\Delta^{**} K}, \mathcal{H}^{\Lambda_c^+\to\Lambda^{**} \pi} = 0\}\right)\,,\\
I_\text{tot}^{\Delta} &= I_\text{tot}\left(\{\mathcal{H}^{\Lambda_c^+\to K^{**} p}, \mathcal{H}^{\Lambda_c^+\to\Lambda^{**} \pi} = 0\}\right)\,,\\
I_\text{tot}^{\Lambda} &= I_\text{tot}\left(\{\mathcal{H}^{\Lambda_c^+\to\Delta^{**} K}, \mathcal{H}^{\Lambda_c^+\to K^{**} p} = 0\}\right)\,,\\
I_\text{tot}^{K/\Lambda} &= I_\text{tot}\left(\{\mathcal{H}^{\Lambda_c^+\to\Delta^{**} K} = 0\}\right) - I_\text{tot}^{K} - I_\text{tot}^{\Lambda}\,,\\
& \dots\,,
\end{align}
\end{split}\]
where the terms with a single chain index are the rate of the decay chain. The sum of all fractions should give the total rate:
\[
\begin{align}
I_\text{tot}\left(\{\mathcal{H}\}\right)
= \sum_{R} I_\text{tot}^{R} + \sum_{R < R'} I_\text{tot}^{R/R'}
\end{align}
\]
\[\begin{split}\displaystyle \begin{array}{crr}
& I_\mathrm{sub}\,/\,I \\
\hline K^{**} & 0.371 \\
\Lambda^{**} & 0.051 \\
\color{gray}{\Lambda(1520)} & \color{gray}{0.031} \\
\color{gray}{\Lambda(1670)} & \color{gray}{0.020} \\
\Delta^{**} & 0.582 \\
\Delta/\Lambda & 0.013 \\
K/\Delta & -0.018 \\
K/\Lambda & 0.001 \\
\hline \mathrm{total} & 1.000 \\
\end{array}\end{split}\]
Polarimetry distributions#
\[\begin{split}\displaystyle \begin{array}{cccc}
& \bar{|\alpha|} & \bar\alpha_x & \bar\alpha_y & \bar\alpha_z \\
K^{**} & 0.873 \pm 0.040 & 0.000 \pm 0.554 & 0.000 \pm 0.396 & +0.100 \pm 0.538 \\
\Lambda^{**} & 0.906 \pm 0.101 & -0.529 \pm 0.214 & 0.000 \pm 0.190 & -0.338 \pm 0.596 \\
\Delta^{**} & 0.540 \pm 0.000 & +0.320 \pm 0.153 & -0.000 \pm 0.000 & -0.324 \pm 0.246 \\
\end{array}\end{split}\]
