# What is $\sigma_{i\eta i\eta}$?

In LHC experiments like, CMS there are variables like ieta which are used for designating the eta (pseudo rapidity) of each of the sub detector components.

There are several plots which I came across which are distributed over $\sigma_{i\eta i\eta}$. For example the following, I came across the definition of $\sigma_{i\eta i\eta}$ which is as follows.

$$\sigma_{i\eta i\eta}=\left(\frac{\sum(\eta_i-\bar{\eta})^2\omega_i}{\sum\omega_i}\right)^{\frac 1 2}$$

$$\bar{\eta}=\frac{\sum\eta_i\omega_i}{\sum\omega_i}$$

$$\omega_i=max\left(0\,,\,4.7+log\frac{E_i}{E_{5\times 5}}\right)$$

I can't understand it's meaning. Please help me out.

• If you compare it with the "standard" statistical equations, you might be able to intuit its meaning. – Guill Nov 8 '17 at 7:55

## 1 Answer

I did some research and finally found a answer which is easy to understand: sigmaIetaIeta is a shower shape variable: The energy weighted standard deviation of single crystal eta within the 5 × 5 crystals centered at the crystal with maximum energy.

In the equation I gave above, the sum runs over the 5 × 5 crystal matrix around the most energetic crystal in the SuperCluster, and the eta distances are measured in units of the crystal size in the eta direction. This variable represents the second moment of the energy distribution along the eta coordinate.

I still don't know how the value 4.7 comes in the weights and why we measure the weights in such a way. Please comment if you do.