How much top quark is in the proton?

We know that protons are made up of two up quarks and a down quark. Each only weigh a few MeV—the rest of the proton mass comes from the strong force binding energy coming from gluon exchange. When we collider protons at high energies, these partons interact with each other to produce other particles. In fact, the LHC is essentially a gluon collider. Recently, however, physicists have been asking, “How much top quark is there in the proton?”

In fact, at first glance, this is a ridiculous question. The top quark is 175 times heavier than the proton! How does it make sense that there are top quarks “in” the proton?

The discussion is based on preliminary plans to build a 100 TeV collider, though there’s a similar story for b quarks (5 times the mass of the proton) at the LHC.

Before we define what we mean by treating the top as a parton, we should define what we mean by proton! We can describe the proton constituents by a series of parton distribution functions (pdf): these tell us the probability of that you’ll interact with a particular piece of the proton. These pdfs are energy-dependent: at high energies, it turns out that you’re more likely to interact with a gluon than any of the “valence quarks.” At sufficiently high energies, these gluons can also produce pairs of heavier objects, like charm, bottom, and—at 100 TeV—even top quarks.

But there’s an even deeper sense in which these heavy quarks have a non-zero parton distribution function (i.e. “fraction of the proton”): it turns out that perturbation theory breaks down for certain kinematic regions when a gluon splits into quarks. That is to say, the small parameters we usually expand in become large.

Theoretically, a technique to keep the expansion parameter small leads to an interpretation of this “high-energy gluon splitting into heavy quarks inside the proton” process as the proton having some intrinsic heavy quark content. This is called perturbative QCD, the key equation known as DGLAP.

In the cartoon above: physically what’s happening is that a gluon in the proton splits into a top and anti-top. When one of these is collinear (i.e. goes down the collider beamline), the expansion parameter blows up and the calculation misbehaves. In order to maintain a well behaved perturbation theory, DGLAP tells us to pretend that instead of a top/anti-top pair coming from a gluon splitting, one can treat these as a top that lives inside the high-energy proton.

This is the sense in which the top quark can be considered as a parton. It doesn’t have to do with whether the top “fits” inside a proton and whether this makes sense given the mass—it boils down to a trick to preserve perturbativity.

One can recast this as the statement that the proton (or even fundamental particles like the electron) look different when you probe them at different energy scales. One can compare this story to this explanation for why the electron doesn’t have infinite electromagnetic energy.

The authors of 1411.2588 a study of the sensitivity a 100 TeV collider to processes that are produced from fusion of top quarks “in” each proton. With any luck, such a collider may even be on the horizon for future generations.

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