Luminosity

We assume, that a particle beam with $\dot{N}_\mathrm{a}$ particles per second of the type a hit target with $N_\mathrm{b}$ particles of type b. Each of the target particles should have the cross-section $\sigma_\mathrm{b}$, whereas the particle beam has the cross-section $A$. The probability that a particle of type a interacts with a particle of type b is then given by the ratio of both cross-sections multiplied by the cross-section of each target particle: $$p = \frac{\sigma_\mathrm{b}}{A}N_\mathrm{b}$$ Therefore, the number of interacting particles per second can then be calculated according to $$\dot{N} = p \dot{N}_\mathrm{a} =\frac{\sigma_\mathrm{b}}{A}N_\mathrm{b} \dot{N}_\mathrm{a}$$ The product $\sigma_\mathrm{b}N_\mathrm{b}/A$ is defined as the so-called luminosity: $$\boxed{\mathcal{L} = \frac{\dot{N}_\mathrm{a}}{A}N_\mathrm{b}}$$ Inserting this equation into the previous one gives the possibility to calculate the interaction rate of a particle collider from the product of its luminosity and the cross-section of the target or another beam in head-on collisions: $$\boxed{\dot{N}_\mathrm{a} = \mathcal{L}\sigma_\mathrm{b}}$$ In the case of head-on collisions, the formula derivation $\dot{N}_\mathrm{a}$ can be replaced by the product of the beam particles of type a and the beam frequency $f$. Finally, the result has to be multiplied by the number of bunches in each beam that interact with the given frequency. This leads to the following formula: $$\boxed{\mathcal{L} = \frac{nN_\mathrm{a}N_\mathrm{b}f}{A}}$$ However, the cross-section $A$ is not a constant but usually follows a Gaussian distribution with the standard deviations $\sigma_x$ in $x$-direction and $\sigma_y$ in $y$-direction. In this case, the above-mentioned formula has to be modified according to: $$\boxed{\mathcal{L} = \frac{nN_\mathrm{a}N_\mathrm{b}f}{4\pi\sigma_x\sigma_y}}$$ The higher the required statistics, the higher the value of the luminosity has to be. For instance, in the year 2017, the LHC reached a luminosity of approximately $$\mathcal{L} = 1.75\cdot 10^{34}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$$