Appendix B Derivation of the flux factor $\mathcal{F}$

Many particle physics experiments are scattering experiments where we take two particles and collide them. In analogy to classical scattering, we define the cross section of the scattering. Since we are working in a quantum theory rather than a classical one, the cross section describes a probability rather than a physical size.

We start with the relation between the scattering probability and the amplitude $\mathcal{M}$

The squared delta function is a problem because we only need one of them to solve our integration; the other delta function will then automatically lead to yet another $\delta^{(4)}(0)$. However, we have dealt with this problem before and know to write $(2\pi)^{4}\delta^{(4)}(0)=V$ with the volume of spacetime $V$. Therefore, we instead consider the probability per volume $P/V$. We also need to keep in mind that the states require a normalisation $1/\sqrt{E}$. This leads to the probability density

Consider a cloud of particles of type $a$ at rest with number density $\rho_{a}$. Now we shoot a bunch of particles of (a potentially different) type $b$ at the cloud (cf. Figure 4). Along the axis of collision, we have a cross-sectional area $A$ and bunch lengths $l_{a}$ and $l_{b}$. The cross section of the scattering is defined through the number $N$ of scattering events as

Here we have also defined the total number of $a$ ($b$) particles $N_{a}$ ($N_{b}$). The combination $N_{a}N_{b}/A$ is called the luminosity and it is the reason that the cross section $\sigma$ is a useful quantity. If we were to repeat our $a$-$b$ scattering experiment at a different collider which has e.g. more particles in its beams, we would see more events even though the underlying process has not changed. $\sigma$ encodes the physics, $L$ the parameters of the experiment. This allows us to focus on two-particle scattering and set $\rho_{a}=\rho_{b}=1$ even if the real beams may contain as many as $10^{11}$ particles (the beam intensity of the LHC beams).

For a $2\to f$ process of momenta $p_{a},p_{b}\to p_{1},...,p_{f}$, we have from (223)

Keep in mind that the volume here is the spacetime volume of the scattering, i.e. $V=t\cdot A\cdot l_{a}$. For a single scattering, the probability $P$ is the number of scattered particles. Therefore the cross section

Identifying $l_{b}/t$ as the velocity of the beam relative to our cloud of particles $a$, we can now write

We now need to convince ourselves that ${\rm d}\sigma$ is Lorentz invariant since we could otherwise stop a process from happening simply by moving relative to it. We have already seen that the measure ${\rm d}^{3}p/(2E)$ is invariant, making the entire phase space ${\rm d}\Phi$ Lorentz invariant. The matrix element $\mathcal{M}$ is also fine so that the remaining part is the flux factor $E_{a}E_{b}|\vec{v}|$. We an rewrite this in terms of invariants³³ 3 Note that technically this is only invariant for boosts along the beam axis. For boosts along any other axis, it is not invariant which fits well with our intuition of cross-sectional areas.