Note: Block B, also referred as final block (FB), and class in the MadWeight world. Description inspired by MadWeight.
This block corresponds to an initial state with Bjorken fractions \(q_1\) and \(q_2\), and a final state with a visible particle with momenta \(p_2\), and an invisible particle with momenta \(p_1\), both coming from a resonance with mass \(s_{12}\). Extra radiation (ISR) it is also allowed.
The goal of this Block is to address the change of variables needed to pass from the standard phase-space parametrization to the \(\frac{1}{4\pi E_1} ds_{12} \times J\) parametrization. Per integration point in CUBA, this Block outputs the value of the jacobian, J, and the four momenta of the invisible particle, \(p_1\). The system of equations needed to compute \(p_1\) is described below.
Change of variables
From the standard phase-space parametrisation:
\[ dq_1 dq_2\frac{d^3 p_1}{(2\pi)^3 2E_1}(2\pi)^4 \delta^4 (P_{in}-P_{out}) \]
where \(P_{in} and P_{out}\) are the total four-momenta in the initial and final states, to the following parametrisation:
\[ \frac{1}{4\pi E_1} ds_{12} \times J \]
where the jacobian, J, is given by:
\[ J = \frac{E_1}{s} \left| p_{2z} E_1 - E_2 p_{1z} \right|^{-1} \]
Parameters
- Collision energy.
Inputs
- Visible particle, with 4-momentum \(p_2\)
- The invariant \(s_{12}\) as output from the Flatter or NarrowWidthApproximation module.
Outputs
- Invisible particle, with 4-momentum \(p_1\) (up to two solutions possible)
- Jacobian, one per solution.
System of equations to compute \(p_1\)
The integrator throws random points in the invariant ( \(s_{12}\)) while \(p_2\) is a known quantity. The equations to compute \(p_1\) are:
\begin{eqnarray} s_{12} &=& (p_1 + p_2)^2 = M_{1}^{2} + M_{2}^2 + 2 E_1 E_2 + 2 p_{1x}p_{2x} + 2p_{1y}p_{2y} + p_{1z}p_{2z} \\ p_{1x} &=& - p_{Tx} \\ p_{1y} &=& - p_{Ty} \\ p_1^2 &=& 0 \Leftrightarrow E_{1}^2 = p_{1x}^2 + p_{1y}^2 + p_{1z}^2 \end{eqnarray}
Where \(\vec{p}_T\) is the total transverse momentum of the visible particles. Using the values of \(p_{1x}, p_{1y}\) from equations (2) and (3), equation (1) can be written as \(p_{1z} = A - B E_1\), where A and B are:
\begin{eqnarray} A &=& \frac{s_{12} - M_{2}^2 + 2(p_{Tx}p_{2x} + p_{Ty}p_{2y})}{2 p_{2z}} \\ B &=& \frac{E_2}{p_{2z}} \\ \end{eqnarray}
Finally equation (4) can be written as \((1 - B) E_{1}^2 + 2AB E_1 - C = 0\), where \(C = p_{Tx}^{2} + p_{Ty}^{2}\).
Each solution of the quadratic equation with a positive value of \(E_1\) is taken.
