Madgraph_Search

Installation Instructions

1. Code to automatically generate hypotheses and search for parameters using MadGraph
2. Requires Madgraph: MG5
3. Save this folder under Madgraph Directory to implement the search
4. To start the scan the user needs to run "run.sh" from terminal from the Madgraph_Search directory
5. All instructions consolidated in shell script to facilitate job submissions on cluster

Introduction to the hypothesis

The Lagrangian for the hypotheses is given by

$$ \mathcal{L} = \mathcal{L}_{SM} + y_X H \bar{X}{X} + y_Y H \bar{Y}{Y} + \frac{g'g_W}{2\sqrt{2}}\bar{X}{W}^{-}Y + h.c. $$

where $y_X$, $y_Y$ and $g'$ are the parameters to be estimated from the benchmark dataset. Typically such a multi-dimensional scan must be run on a cluster, due to the computational cost. As a proof of concept, this code searches for the parameter $y_X (M_X)$.

This model corresponds to a typical BSM hypothesis where traditional one-dimensional scans based on variables such as MET are not as effective as using all the kinematic information in an event. The final state of interest has an electron-positron pair with missing transverse energy.

The same final state is achieved in the Standard Model with neutrinos carrying the MET.

Neural Network Analysis

DNNs are implemented as binary classifiers. They are trained to discriminate between signal for a particular choice of parameters and the background. The DNNs are fully connected and use ReLu in the hidden layers, sigmoid in the output layer.

Binary-CrossEntropy is chosen as the loss function:

$$BCE[f] = -\int dx \left[p_{A}(x)\log{f(x)} + p_{B}(x)\log{(1-f(x))}\right]$$

Analytically, it can be seen that, the function that minimizes the BCE is given by:

$$ f^*(x) = \frac{p_{A}(x)}{p_{A}(x) + p_{B}(x)} $$

Bayesian Analysis (MLE)

For a given choice of parameters $\theta$, there are two hypotheses:

1. Signal Hypothesis: There are S signal events and B background events in the data-set:

$$ p(x|\theta, \mu) = \mu p_{sig}(x|\theta) + (1 - \mu) p_{bg}(x) $$

where, $\mu = S/(S + B)$ is also a parameter to be estimated.

2. Null Hypothesis:

$$ p_0(x) = p_{bg}(x) $$

If a DNN is trained to classify between signal and background classes, using Equation 3, the output of the network $f(x|\theta)$ can be used to obtain:

$$ \frac{p(x|\theta, \mu)}{p_0(x)} = (1 - \mu) + \frac{\mu f(x|\theta)}{1 - f(x|\theta)} $$

Maximum-Log-Likelihood estimation can be done using:

$$ \hat{\theta}, \hat{\mu} = \underset{\theta, \mu}{\mathrm{argmax}}\sum_i \log{\left [(1 - \mu) + \frac{\mu f(x_i|\theta)}{1 - f(x_i|\theta)} \right]} $$