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.