# The Higgs Boson

The hunt that could make or break the standard model.

The confirmation of the existence of the Higgs boson is probably one of the most important findings made by the LHC. Not only did it complete the Standard model, but also gave insight into the early cosmological universe by having a better understanding of how the processes could function.

So not only does the LHC play an important role in furthering particle physics, but its findings are of high significance for astrophysicists who seek to understand the early universe some of whose properties are best emulated in a particle accelerator.

## Finding the Higgs

Finding the Higgs boson is not easy. Not only does it decay about an order of magnitude faster than other elementary particles making it impossible to detect directly in ATLAS, or other detectors, while it does decay into a lot of things, all of those are very common making isolating the Higgs boson very difficult.

One common decay pattern is for the Higgs boson to decay to two photons ($H\rightarrow \gamma_1 \gamma_2$) with the 4 momntum vectors $\textbf{p}^\alpha_{\gamma_1}= \frac{E_1}{c} \cdot (1,\hat{n}_1)$ and $\textbf{p}^\alpha_{\gamma_2}= \frac{E_2}{c} \cdot (2,\hat{n}_2)$. Furthermore, we know the angle between the two photon beams is given by $\theta$, and $\cos(\theta)=\hat{n}_1\cdot\hat{n}_2$.

Since $\textbf{p}^2=-m_0^2c^2$, and photons don't have a invariant mass, we know that $\textbf{p}_1^2 = \textbf{p}_2^2 = 0$.

We therefore find that $(\textbf{p}^\alpha_{\gamma_1}+\textbf{p}^\alpha_{\gamma_2})^2=2\textbf{p}^\alpha_{\gamma_1}\cdot\textbf{p}^\alpha_{\gamma_2}$ which gives us

$(\textbf{p}^\alpha_{\gamma_1}+\textbf{p}^\alpha_{\gamma_2})^2=-2 \frac{E_1E_2}{c^2} (1-cos(\theta)).$

Remembering that $\textbf{p}^2=-m_0^2c^2$, we can find the invariant mass of the parent particle (the Higgs boson)

$m_H = \frac{1}{c^2} \sqrt{2E_1E_2(1-\cos(\theta))}.$

Since $E_1,E_2,\theta$ are all measureable by the detector, we can therefore get a value of the mass of the Higgs boson. Exactly this was done on data from 2011 and 2012 where it was found that the mass of the Higgs boson was around $m_H=126.8 \text{GeV}$.

While the above algebra may seem easy, it's not quite as simple to isolate $H\rightarrow \gamma \gamma$ events. In order to find such events a likelihood function must be developed based on statistical knowledge as well as the physical understanding of the problem. It took most of the first run of the LHC to tune to likelihood function, due to the many tunable parameters, and it's still limiting how many collisions are possible in each batch due to the added noise of more collisions.

These are some of the reasons why experiments in using machine learning algoirthms such as neural networks to let computer automatically find an optimal likelihood function have begun.

Below has a simple neural network been implemented to classify Higgs decay based on the Higgs ML dataset from 2014.

You can use the two features to compete against the neural network in detecting Higgs decays. (The feature values have been normalised, and the $y$-axis is a kernel density estimation of the value. The features have been crafted and chosen from the detector data as those with the highest predictive power)

You can learn more about a similar network to the one above here, or you can learn more about neural networks in general here.