# How to measure (missing) transverse energy

There is traditionally a bit of confusion between missing transverse energy, and missing transverse momentum. I've seen both used interchangably, and sometimes even things like "$\not E_T = -|\sum \vec p_T|$".

Just to clarify before my question, if both quantities are used, then missing transverse momentum usually refers to the sum of all tracks' or reconstructed objects' $p_T$:

$$\not \vec p_T = - \!\!\!\sum_{i \;\in\; \mathrm{tracks}}\!\!\! \vec p_T(i)$$ and missing transverse energy refers to something calculated using mainly the calorimeter, right?

So, how do you calculate (missing) transverse energy from calorimeter readings? Especially, how do you do a vectorial sum, even though calorimeter energies are scalar values? You don't have particle momentum vectors or similar, but only the readings of calorimeter cells. (Reconstructed objects come into play later, for corrections, but are not involved at first order.)

Naively, I'd say you construct a vector from the position of each calorimeter cell, and give it a length proportional to its energy: $$\qquad\quad \vec E_\mathrm{cell} = \frac{\vec x_\mathrm{cell}}{|\vec x_\mathrm{cell}|} E_\mathrm{cell} \qquad (?)$$ and then just sum those vectors up. But you want to have transverse energy. How do you do that? Just by using 2-d vectors (only $x$ and $y$ coordinates)? And finally, is the geometry of the cells a factor in the calculation?

(I'm looking for a general answer, but where it is experiment-specific, I'm interested in ATLAS and the way it was done at DZero. I can imagine the answer would be very different for CMS with their special calorimeter. And sorry, I tried to read the design documents, but I couldn't really understand how it is calculated. I'd be happy if someone would point me to some clear documentation though.

There are a couple of similar questions, but they don't hit the point I'm wondering about. This question basically asks "why use transverse quantities", and the answer conflates transverse energy and momentum (I'm interested in the one determined by the calorimeter where you don't have $\vec p_T$ vectors!). And this question comes from the other side and asks how to calculate $\not E_T$ from four-vectors, not from experimental readings.)

• This would need a long answer. Here is a CMS paper with all the definitions in use: arxiv.org/pdf/1106.5048v1.pdf . The direction of the energy deposits is used by the way. – anna v Jan 15 '14 at 14:30
• "How do you do that? Just by using 2-d vectors (only x and y coordinates)?" Yes. – Matt Reece Jan 15 '14 at 16:49
• And the reason you use calorimeter deposits instead of just tracks is that you don't want to omit things like photons and $\pi^0$'s. – Matt Reece Jan 15 '14 at 16:49

In high energy physics momentum and energy are sloppily conflated because they approximate each other extremely well in the relativistic limit (E = $\sqrt{m^2c^4+p^2c^2}$ $\approx$ pc, or E = p in natural units ). If you had top quarks decaying in the calorimeter then the approximation would be invalid, but you don't: massive particles decay quickly into relativistic decay products (whose momenta are roughly equal to their energy) before they reach the calorimeter. The transverse energy is calculated from the calorimeter cells by exploiting the fact that the calorimeter cells are by design organized into towers that point roughly towards the collision point. Thus a transverse component can be defined. It would be difficult to argue that it is a "precision measurement." More generally the conversion of calorimeter energy deposits into a measured energy and direction associated with an originating parton is a messy business, involving clustering algorithms (which energy deposit is associated with which jet?) and the jet energy scale (how does the jet's true energy and direction relate to the measured signals in the calorimeter?).