Amplitude

Helicity Formula

Each Decay has Amplitude like

\[A^{A \rightarrow B+C}_{\lambda_{A},\lambda_{B},\lambda_{C}} = H_{\lambda_{B},\lambda_{C}}^{A \rightarrow B+C} D^{J_{A}\star}_{\lambda_{A},\lambda_{B}-\lambda_{C}}(\phi,\theta,0)\]

For a chain decay, amplitude can be combined as

\[A^{A \rightarrow R+B,R \rightarrow C+D}_{\lambda_{A},\lambda_{B},\lambda_{C},\lambda_{D}} = \sum_{\lambda_{R}}A^{A \rightarrow R+B}_{\lambda_{A},\lambda_{R},\lambda_{B}} \color{red}{R(m_{R})}\color{black} A^{R \rightarrow C+D} _{\lambda_{R},\lambda_{C},\lambda_{D}}\]

with angle aligned

\[{\hat{A}}^{A \rightarrow R+B,R \rightarrow C+D}_{\lambda_{A},\lambda_{B},\lambda_{C},\lambda_{D}} = \sum_{\lambda_{B}',\lambda_{C}',\lambda_{D}'}A^{A \rightarrow R+B,R \rightarrow C+D}_{\lambda_{A},\lambda_{B}',\lambda_{C}',\lambda_{D}'} D^{J_{B}\star}_{\lambda_{B}',\lambda_{B}}(\alpha_{B},\beta_{B},\gamma_{B}) D^{J_{C}\star}_{\lambda_{C}',\lambda_{C}}(\alpha_{C},\beta_{C},\gamma_{C}) D^{J_{D}\star}_{\lambda_{D}',\lambda_{D}}(\alpha_{D},\beta_{D},\gamma_{D})\]

the sum of resonances

\[A_{\lambda_{A},\lambda_{B},\lambda_{C},\lambda_{D}}^{total} = \sum_{R_{1}} {\hat{A}}^{A \rightarrow R_{1}+B,R_{1} \rightarrow C+D}_{\lambda_{A},\lambda_{B},\lambda_{C},\lambda_{D}} + \sum_{R_{2}} {\hat{A}}^{A \rightarrow R_{2}+C,R_{2} \rightarrow B+D}_{\lambda_{A},\lambda_{B},\lambda_{C},\lambda_{D}} + \sum_{R_{3}} {\hat{A}}^{A \rightarrow R_{3}+D,R_{3} \rightarrow B+C}_{\lambda_{A},\lambda_{B},\lambda_{C},\lambda_{D}}\]

then the differential cross-section

\[\frac{\mathrm{d}\sigma}{\mathrm{d}\Phi} = \frac{1}{N}\sum_{\lambda_{A}}\sum_{\lambda_{B},\lambda_{C},\lambda_{D}}|A_{\lambda_{A},\lambda_{B},\lambda_{C},\lambda_{D}}^{total}|^2\]

Amplitude Combination Rules

For a decay process A -> R B, R -> C D, we can get different part of amplitude:

1. Particle:

   1. Initial state: \(1\)

   2. Final state: \(D(\alpha, \beta, \gamma)\)

   3. Propagator: \(R(m)\)

2. Decay:

   Two body decay (A -> R B): \(H_{\lambda_R,\lambda_B} D_{\lambda_A, \lambda_R - \lambda_B} (\varphi, \theta,0)\)

Now we can use combination rules to build amplitude for the whole process.

Probability Density:

\(P = |\tilde{A}|^2\) (modular square)

Decay Group:

\(\tilde{A} = a_1 A_{R_1} + a_2 A_{R_2} + \cdots\) (addition)

Decay Chain:

\(A_{R} = A_1 \times R \times A_2 \cdots\) (multiplication)

Decay: \(A_i = HD(\varphi, \theta, 0)\)

Particle: \(R(m)\)

The indices part is quantum number, and it can be summed automatically.

Default Amplitude Model

The defalut model for Decay is helicity amplitude

\[A^{A \rightarrow B C}_{\lambda_A,\lambda_B, \lambda_C} = H_{\lambda_B,\lambda_C}^{A \rightarrow B C} D^{J_{A}*}_{\lambda_A,\lambda_B - \lambda_C}(\phi, \theta, 0).\]

The LS coupling formula is used

\[H_{\lambda_{B},\lambda_{C}}^{A \rightarrow B+C} = \sum_{ls} g_{ls} \sqrt{\frac{2l+1}{2 J_{A}+1}} \langle l 0; s \delta|J_{A} \delta\rangle \langle J_{B} \lambda_{B} ;J_{C} -\lambda_{C} | s \delta \rangle q^{l} B_{l}'(q, q_0, d)\]

\(g_{ls}\) are the fit parameters, the first one is fixed. \(q\) and \(q_0\) is the momentum in rest frame for invariant mass and resonance mass.

\(B_{l}'(q, q0, d)\) (Bprime) is Blatt-Weisskopf barrier factors. \(d\) is \(3.0 \mathrm{GeV}^{-1}\) by default.

Resonances model use Relativistic Breit-Wigner function

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

with running width

\[\Gamma(m) = \Gamma_0 \left(\frac{q}{q_0}\right)^{2L+1}\frac{m_0}{m} B_{L}'^2(q,q_0,d).\]

By using the combination rules, the amplitude is built automatically.