How are high-energy detectors able to to distinguish between $m_{e}$ and $m_{\mu}$? I had a teacher pose this interesting question yesterday:
Suppose you're running a high-energy scattering experiment at the LHC. For concreteness, let's suppose it's a 2 to 2 scattering event which involves electrons and/or muons.
The theorist uses QFT to compute some cross-section which comes from the amplitude
$$
\mathcal{A}_{p_{1} p_{2} \to p_{3} p_{4} } = F(s,t,u,m_{e},m_{\mu},\ldots)
$$
The amplitude is a function of the Mandelstam variables $s \equiv (p_1+p_2)^2$, $t \equiv (p_1-p_3)^2$ and $u \equiv (p_1-p_4)^2$, as well as the mass of the electron $m_{e}$ and muon $m_{\mu}$ (and some other stuff).
Because we're running a high-energy experiment we obviously have that $s,t,u \gg m_e,m_\mu$, and for this reason the theorist makes the approximation $m_{e} \approx 0$ and $m_{\mu} \approx 0$.
The Question: How is the LHC able to distinguish between an electron and a muon if the theorist makes the approximation that the electron and muon are both massless? 
For some reason, the approximation $m_{e} \approx m_{\mu} \approx 0$ is a bad one and the question is why this is. One idea that a colleague had was that the tracks of the electron and muon look different; because of the cyclotron radius $r \sim \frac{cm}{qB} \propto m$ the magnetic fields used in the machine to track the particles coming out of the collision will see the electron spiral more dramatically than the muon.
Any ideas as to other reasons why?
 A: Identification of muons and electrons is indeed a hard problem for the experiments in the LHC. For example, one needs to distinguish electrons from fake muons that leave a electron-like signal even if there are two different parts that these particles can deposit their energies.
At the LHC, there are two general purpose detectors, namely CMS and ATLAS, which are designed as onion-like structure from inside to outside. Both have specialized parts in order to detect muons, which are basically some chambers at the very outer layer.
First of all, the concept of a "particle" in the detector is not obvious or natural due to the physical limitation of the resolution and data acquisition.
The raw detector data is essentially an array of energy deposits in the pixels of the calorimeters or strips of the detectors which does not give any information about which particle is which. The physicists need to reconstruct the particles using some algorithms (like Particle-Flow) and cuts (selection of a value range for energy, momentum, angles, pseudo-rapidity, etc.). A bunch of energy deposits needs to be associated to a particle but it could be also contaminated by background noise or other interfering energy deposits of other particles.
So, there are several tags for each muon candidate which needs to be consistent with each other. For instance, there are calo muons which are reconstructed particle signals that are detected in the electromagnetic calorimeter (where electrons also show up). There are standalone muons which are only detected in the muon chambers. There are tracker muons which are detected in the trackers and matching with the muon hits in the chambers. Also there are fake muons that are actually charged hadrons seen in the calorimeter but misleadingly matching to the muon hits in the chambers. 
So, one needs to make appropriate cuts in the momentum, etc., in order to eliminate those mismatchings which depends on the detector, type of identification (standalone, calo, tracker, etc) and energy scale. There are loose cuts and tight cuts that could be used according to which analysis or topology of the process is in question.
A: The usual way to identify an electron vs a muon in a high-energy detector is via their interactions with matter:
 - an electron will dump all its energy quickly in an "electromagnetic shower" and quickly stop
 - a muon will interact minimally and go far through the material.
In pictures, these look like:

Note that we don't try to measure the rest mass of the electron or muon this way. We know what those are. We just try to identify which kind of particle a particular track is, and then supply the correct mass for the computations.
A: Whatever approximations theorist make has no affect on detectors.
The curvature of a track in a magnetic field measures charge divided by momentum, not mass:
$$ \frac 1 r \propto \frac q p $$
Of course:
$$ p = \gamma m v $$
but everything is moving near $c$. In fact, as far as track timing and position triggers (time-of-flight), everything is moving at the speed of light.
Nevertheless, there are velocity threshold effects, like the Cherenkov effect and transition radiation that can take 2 equal momentum particles and distinguish electrons from muons; however, at LHC energies that is not always practical--since both may have velocities exceeding the threshold.
Enter the calorimeter. This is a bunch of leaded glass (c.f., transparent lead). Bob Jacobsen's answer explains these. The only additional info is that the leptons interact via bremsstrahlung and pair production, whose cross sections contains powers of $1/m$, such the electrons dump all their energy and muons penetrate. 
A: Particles as expected to be seen in the CMS experiment at LHC

So electrons and muons leave completely different signatures although because of high energy they may leave a  similar track in the tracking detector. Electrons are absorbed in the electromagnetic calorimeter, and muons go through,their electromagnetic interactions minimal due to their large mass,  leaving a track signal that can be fitted. 
