What is a structure function (or proton structure function)?

There is one thing that comes to my mind when thinking of the connection between the strong coupling constant $alpha_{S}$ and the proton structure functions $F_{1}left( x, Q^{2}right)$ and $F_{2}left(x, Q^{2} right)$.

When you studied elastic electron-proton scattering (such as in Mott or Rutherford scattering), you might have come across the term form factors $G_{E}$ and $G_{M}$, where $G_{E}$ represents the a priori unknown charge distribution of the proton, and $G_{M}$ the magnetic moment distribution of the proton.

In deep inelastic scattering experiments, one calls the structure functions, though they do not have the same physical meaning as the form factors do in elastic electron-proton scattering. For example, the structure function $F_{2}left( x, Q^{2}right)$ (compared to elastic scattering, we need two independent variables to describe them, here chosen as $Q^{2} := -q^{2}$, where $q$ describes the four-momentum of the photon, and $x := frac{Q^{2}}{2qP}$ describes the Björken scaling $x$) describes the probability to find a quark with the momentum fraction $x$ in the proton (this is the meaning of the Björken scaling $x$, it gives you the momentum fraction the quark participating in the scattering takes from the total proton momentum $P$).

One can now define relations for $F_{1}left( x, Q^{2}right)$ and $F_{2}left(x, Q^{2} right)$ (I do not want to elaborate on this here, please take a look at Mark Thomson's "Modern Particle Physics", chapters 8.2.1 and 8.3 if you are interested) , and in doing so, one assumes that the electron scatters off from a quark in the proton elastically. This is very important, so let me repeat: The scattering of the electron with the proton is inelastic, but we assume that it is elastic with a (point-like) quark inside the proton.

However, why do we assume this? The reason for that is the running of the coupling constant $alpha_{S}$. Now, at high values of $Q^{2}$ ($= -q^{2}$), we know that $alpha_{S}$ rapidly decreases, and thus we talk about asymptotic freedom, which basically means that the quarks inside the proton can be described as quasi-free particles in the proton for high enough $Q^{2}$.

This plot might help in understanding (taken from (1)):

(1) https://dispatchesfromturtleisland.blogspot.com/2018/06/measuring-strong-force-coupling-constant.html, accessed 14/02/2021