# How many gluons on average are there in a proton at once?

• B
Gold Member

## Main Question or Discussion Point

A proton is made of quarks and gluons bound by the strong force in a confined system. Further, all protons are basically interchangeable parts for the purposes of this question. Each one is identical in all material respects.

Scientists know a lot about gluon energy density in protons. We have, in principle, the exact equations that govern strong force interactions, although, in practice we use good approximations of them. In particular, we know to moderate accuracy (always, always less than we'd like) the value of the strong force coupling constant that among other things, tells us the likelihood that a color charged particle with emit or absorb a gluon in a particular time frame.

Admittedly, what we know is probabilistic. But, it seems to me that it ought to be possible to determine how many individual gluons are present, on average, in any given proton (and indeed, the answer should be almost identical for a neutron), at any given point in time.

I have looked for articles (technical or layman oriented) doing this calculation, and haven't found one. But, surely somebody must have done it, because we seem to know everything we need to know to do that calculation, and it seems like an obvious, foundational thing to want to calculate. Even if it doesn't change the math used to calculate physical observables, this kind of number is helpful in conceptualizing what is going on in your head, like an illustration for a story problem in algebra.

Is this perhaps something that everyone does as a homework exercise in their first graduate school QCD course (I recall doing derivations of Kepler's laws, e.g., when I was in first year physics), and so it isn't considered worth publishing?

Indeed, if we knew that, we could also easily extrapolate to the total number of gluons in existence at any one time in the universe (determining the total mass-energy attributable to gluons in the universe is trivial, even without the number of gluons in existence), which doesn't have a lot of practical value, but would nonetheless be interesting to know as just sort of a basic fact about the universe.

To the extent it is relevant, I would be looking for the "ground state" of minimum energy and temperature. Basically, a hydrogen-1 atom's nucleus in a vacuum, although factors that would impact this would be nice to know.

Extreme precision isn't a priority. An order of magnitude estimate or +/- 30% or something like that , would be fine. I'm not asking for a three year long NNNLO calculation unless one is already readily available in the literature.

So, what is the answer, or if this hasn't been determined, is there a reason that this is harder than it seems to determine?

(I struggled with the difficulty level for this question. Ultimately, I classified this as "basic" level because the ideal answer should be a simple number plus or minus a margin of error, and would still make sense with only the barest sense of the quark-gluon model of the proton. But, I recognize that coming up with the answer is no double an advanced matter requiring graduate school level knowledge, and that even having a sense of what goes into the calculation and why it should or shouldn't be possible to do the calculation is probably an intermediate level matter at minimum.)

FWIW, my intuition is that the number ought to be in the double digits or high single digits, but that doesn't really have a well thought out basis and I recognize that my intuition may be and indeed probably is, totally wrong.

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mfb
Mentor
As far as I remember: If you integrate over the gluon PDFs the integral diverges. Does that mean there are "infinite" gluons? I would say it means the question doesn't have a well-defined answer. If you assign a larger weight to gluons with higher momentum there is some way to get a finite answer, but the answer depends on what exactly you do.

• ohwilleke
Staff Emeritus
2019 Award
The proton is not in an eigenstate of gluon number. If you counted gluons - i.e. forced it into such an eigenstate - it would no longer be a proton.

Section 2 of "High Gluon Densities in Heavy Ion Collisions" (2016), entitled "The wave function of a hadron at high energy", includes the following remarks:
I use here the words “wave function” in a rather loose sense. The notion of a wave-function at high energy suffers indeed from well-known ambiguities: it depends on the frame where it is defined, on the gauge chosen, with the parton picture emerging more naturally in the light-cone gauge and in the infinite momentum frame. A further ambiguity arises in high order calculations of a given process in the separation of the constituents of the hadrons from the probe that is used to measure them. For instance, when two colliding hadrons exchange gluons, one may choose to regard these gluons as being part of the wave function of the projectile, or of that of the target, or as part of the interaction mechanism...

Because gluons can themselves emit gluons, as well as quark-antiquark pairs, the structure of the wave function in terms of Fock states becomes gradually more and more complicated. Fortunately, we rarely need the details of this structure, but only a more inclusive information. Thus for instance, one may ask how many gluons are present in the wave function at a given energy. This is expressed in terms of a so-called integrated gluon distribution function...
... and some details follow. I do not see a numerical result anywhere, but the righthand graph in Figure 14 (page 32) looks like it is the function you have to integrate in order to get a number.

• ohwilleke
Staff Emeritus
2019 Award
<sigh> It's going to be one of those threads, I see.

One cannot simply pluck a plot from the Blaizot paper (which isn't really about protons, anyway). Earlier in the paper he discusses factorization and the factorization scale. That is the scale where the person doing the calculation draws the distinction between a quark with its color field and a quark with an associated gluon. Because this is a calculational distinction, not a physical distinction, there is no way to answer the question - "number of gluons" is simply undefined. A proton does not have a fixed number of gluons in it. It doesn't even have an average number of gluons in it.

• nikkkom and weirdoguy
Gold Member
The proton is not in an eigenstate of gluon number. If you counted gluons - i.e. forced it into such an eigenstate - it would no longer be a proton.
Accepting your statement that you can't count gluons in a proton, is there any physical system for which is it possible to determine an average number of gluons in it (I certainly recognize that it wouldn't be a fixed number)?

Staff Emeritus
2019 Award
No.

The picture of exchange of individual virtual gauge bosons works well for perturbative processes. E.g. two electrons scattering off each other at low energy are well described by the exchange of one photon. At high energy, you need to add loops, so you have an exchange of "on average" more than one photon, say 1.1 photon.

This breaks down if perturbative treatment of the process is not working - i.e. when higher-order loops contributions do not tend to zero. Perturbative series diverge in bad non-renormalizable ways, and corresponding "number of average gluons" is formally infinite. (I see it as a failure of our current math we have to tackle the process in question. Need better math...)

For color force, this happens for low energy processes, such as internals of proton.

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• ohwilleke
vanhees71
Gold Member
2019 Award
Hm, I'm a bit puzzled whether "gluon number" is a well-defined quantity at all. Already for the much simpler photons, which are Abelian rather than non-Abelian gauge bosons, it's hard to define photon number properly. It's possible for (asymptotically) free photon states although it's not a Lorentz invariant quantity, i.e., it's frame dependent. A better measure is the total energy of free photon states, which can together with the total momentum be defined as a Lorentz-covariant quantity.

Since there are, as far as we know, no asymptotic free gluons, the idea of a "gluon number" is at least a very difficult notion.

• ohwilleke
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