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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{d\sigma}{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) (~tf_pwa.breit_wigner.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.