(By Alessia Saggio)

I’ve already talked a little bit about the Matrix Element Method (MEM) in my third post on this blog, but since now I want to treat it more in detail, I would like to reintroduce it briefly.

The probability to observe a set of experimental events (labeled with the reconstructed four-momenta $p^{vis}$), given a certain theoretical hypothesis $\alpha$, is given by the following:

$\displaystyle P(p^{vis}|\alpha) = \dfrac{1}{\sigma_{\alpha}} \int dx_1 dx_2 f_1(x_1) f_2(x_2) \int d\Phi |M_{\alpha}(\textbf p)|^2 W (\textbf p, \textbf{p}^{vis})$,

where the probability $P$ is called weight, $\sigma$ is the cross-section of the considered process, $\Phi$ is the phase-space measure, $f_1$ and $f_2$ are the parton distribution functions, $M$ is the matrix element of the process and $W$ is the transfer function (i.e. a bi-dimensional histogram that correlates the reconstructed energy with the energy generated at the simulation level). Apart from the cross-section (which is just a normalization factor), we then need to compute this long integral to retrieve the weight.

The integration is performed by a numerical integrator (e.g. VEGAS or CUBA), basically by “launching” points ranging from 0 to 1 in the whole phase-space. It is always useful to help the integrator to do its job properly (sometimes it’s mandatory in terms of time requirements). So what we need to do is to align the many peaks coming from the functions in the integrand by mapping them onto the phase-space measure with a single variable each. While the peaks from the transfer functions are already aligned, those coming from the propagator enhancements in the matrix element are not aligned at all. Therefore, a strategy is needed!

To work this issue out, we can distribute the variables into different subsets of variables, so-called blocks, to which a transformation is applied. This means that each block is then characterized by its own change of variables.

Let’s consider for instance the process

$\displaystyle pp \rightarrow (W^+ \rightarrow l^+ \nu)(W^- \rightarrow l^- \bar{\nu})$,

in which our final state is given by two charged leptons (indicated with p3 and p4 in fig. 1) and two neutrinos (indicated with p1 and p2).

For this process the following parametrization of the phase-space has been chosen:

$\displaystyle dq_1 dq_2 \dfrac{d^3 p_1}{(2 {\pi})^3 2 E_1} \dfrac{d^3 p_2}{(2 {\pi})^3 2 E_2} (2 {\pi})^4 {\delta}^4 (P_{in} - P_{fin})$,

where $q_1$ and $q_2$ are the Bjorken fractions of the partons.

As I said before, we need to align the peaks coming from the matrix element. We have therefore to integrate out the $\delta$ function (so that we enforce the energy-momentum conservation) and to perform a suitable change of variables.

In this case, we can pass from the momenta of the neutrinos to the invariant masses of the two W bosons, $s_{13}$ and $s_{24}$:

$\displaystyle \dfrac{1}{16 {\pi}^2 E_1 E_2} dq_1 dq_2 ds_{13} ds_{24} \times J$,

where J is the Jacobian associated to this transformation.

The integration variables are therefore the two Bjorken fractions and the two invariant masses. We can now proceed with the integration and get our weight!

The framework I’m working on is called MoMEMta and it is basically a modular tool to compute easily this probability or, in other words, to make the Matrix Element Method easier to use… but I’m not gonna talk about that in this post, since the topic is really complex and I think that to many information in one shot can result in a bad understanding and can be confusing.

For sure, I will spend a lot of words about MoMEMta in my future posts so, if you are interested, stay tuned!