PDFs after all these years
I've only just realized that for years I've misunderstood, or perhaps had the wrong mental model, for the proton parton distribution functions (PDFs) and the structure of the proton. The PDF for a parton species tells us the number density of partons of that species with a fraction \(x\) of the longitudinal (or strictly speaking light-cone) momentum of the proton.
There are various sum rules that PDFs must satisfy. For example, the fractions must sum (in expectation) to one, $$ \sum_i \int_0^1 x f_i(x) \,\text{d}x = 1, $$ and there should be 2 up valence-quarks in the proton, $$ \int_0^1 \left(f_u(x) - f_{\bar u}(x)\right)\,\text{d}x = 2 $$ Now what's that formula about? Well, we don't distinguish between valence and sea quarks in the PDFs. The up quark could be created in the quark sea, popping in and out in pairs \(u\bar u\). The above formula says that the number of sea + valence up quarks minus the number of sea anti-up quarks equals 2.
The PDFs depend on a scale, however, called \(\mu\), and we write them as \(f(x, \mu)\). What's the meaning of that scale? I'd always thought of this scale as the scale at which we probed the proton. That we were observing the proton in some way that interfered with it or made it reveal different structure or that the proton was physically changing because of our probing it.
This probably won't lead to any mathematical errors, but it did leave me confused about what the structure of the proton really was. I think I almost saw it quantum mechanically, that you can't look at it without disturbing it, and that it reacted to the scale that you probed, \(\mu\).
The idea of the proton changing as we prod it is a terrible mental model; let's build a better one. What actually matters here is what we mean by a parton. We know that inside a proton, as well as the valence quarks \(uud\), there's a soup of partons popping into and out of existence; these are the sea partons. The partons are surrounded by this mess all the time. We can simplify things by saying that any mess around a parton, at energy less than \(\mu\), is just part of that parton; they are its dressing. Any mess above that scale counts as extra partons. The PDFs depend on that choice; of course they do! We are just saying here that parton distribution functions depend on how we define a parton.
Thus the hard scale is just something we pick to define what we mean by a parton. In order to make our calculations as accurate as possible, we are best to pick it to be a scale that minimizes higher-order logarithms in perturbative calculations. Formally, though, our calculations, which involve PDFs and hard scattering elements, shouldn't depend on our choice of scale, up to higher-order corrections. The DGLAP equations ensure that the scale-dependence of PDFs cancels with that of the hard-scattering elements up to higher-order corrections.