Protons and Charm Quarks: A Lesson From Virtual Particles

There’s been a lot of chatter lately about a claim that charm quarks are found in protons. While the evidence for this claim of “intrinsic charm” (a name that goes back decades) is by no means entirely convincing yet, it might in fact be true… sort of. But the whole idea sounds very confusing. A charm quark has a larger mass than a proton: about 1.2 GeV/c2 vs. 0.938 GeV/c2. On the face of it, suggesting there are charm quarks in protons sounds as crazy as suggesting that a football could have a lead brick inside it without you noticing any difference.

What’s really going on? It’s a long story, and subtle even for experts, so it’s not surprising that most articles about it for lay readers haven’t been entirely clear. At some point I’ll write a comprehensive explanation, but that will require a longer post (or series of posts), and I don’t want to launch into that until my conceptual understanding of important details is complete.

But in the meantime, here’s a related question: how can a particle with zero mass (zero rest mass, to be precise) spend part of its time as a combination of objects that have positive mass? For instance, a photon [a particle of light, including both visible and invisible forms of light] has zero rest mass. [Note, however, that it has non-zero gravitational mass]. Meanwhile electrons and positrons [the anti-particles of electrons] both have positive rest mass. So what do people mean when they say “A photon can be an electron-positron pair part of the time”? This statement comes with a fancy “Feynman diagram”, in which the photon is shown as the wavy line, time is running left to right, and the loop represents an electron and a positron created from the photon.

Photons and Electron-Positron Pairs

This phrase is often heard in particle physics classes, such as in this one, where you’ll see it explicitly stated that “A photon can be an e+ e− pair part of the time“. It’s also common to find it mentioned in scientific journalism and in most books about particle physics.

Many non-experts have problems with this mysterious statement, and rightly so.

• A photon, like any particle with zero rest mass, has to travel at the cosmic speed limit, a.k.a. “c“, a.k.a. 300,000 kilometers/second, a.k.a. the speed of light.
• But an electron has a positive rest mass, and a positron has the same rest mass as an electron, so they must travel slower than the cosmic speed limit.

So how can a photon, traveling at exactly the speed of light, spend any time at all as a pair of particles with non-zero rest mass? Wouldn’t this force it to slow down, or something?

The well-meaning but incorrect answer, found in many books and articles, is that the electron and positron can come into existence as long as they do so only for a very short time, via the uncertainty principle. This implies that a massless particle can turn into particles with mass as long as the time involved is short enough. This would also then seem to imply that a photon, too, can travel below the speed limit, as long as it does so for a very short time. But if that were true, then either its overall speed would be slightly slower than c, or, to make up for lost time, it would occasionally have to travel above the speed limit!

Huh??!??!??!??!??!?!!

Fortunately, this distressing line of argument is wrong-headed. In fact, the statement “A photon can be an e+ e− pair part of the time” is shorthand. It’s harmless shorthand in a physics class, where the math quickly makes clear what is and isn’t meant by it, but it’s misleading otherwise. The precise statement is that, via the uncertainty principle, “a photon can be a virtual electron/virtual positron pair part of the time.” This word “virtual” makes a world of difference.

As I’ve written elsewhere, a “virtual particle” is not a particle. It is a general disturbance in a field, and abides by far fewer rules than a true particle, which is a steady, regular ripple in a field. A true particle has a definite rest mass: every real electron, and every real positron, always has a rest mass of 0.000511 GeV/c2. But a virtual electron can have any mass — in fact it can have any mass-squared. That includes negative mass-squared, in which case it would have an imaginary mass! [From pre-college math: the square root of any negative number is, by definition, an “imaginary number.”]

When the photon turns into a virtual electron and a virtual positron and back again, energy and momentum are conserved (i.e. the total amounts of momentum and energy don’t change during each stage of the process). This fact is hard-coded into Feynman diagrams, which is why professional physicists don’t get confused about this point. The conservation of these quantities would fail completely if a massless particle were to turn into two particles with positive rest mass. (In fact this is why massless particles cannot decay; see here, rule 2.) Instead, if the virtual electron had the electron’s usual mass of 0.000511 GeV/c2, the positron would have to have imaginary mass. And vice versa. More generally, the math quickly tells you that either

• both of these virtual particles have zero mass, or
• at least one has imaginary mass.

So these are not in any sense normal electrons and positrons!

Instead, a better way to understand the virtual electron/positron pair is as a general disturbance in the electron field, a disturbance whose total rest mass is zero. [Note: both electrons and positrons are ripples in the electron field; there’s no separate positron field.] The photon, usually a ripple in the electromagnetic field, can become occasionally a disturbance in the electron field, but when it does so it retains its zero rest mass, and travels always at the cosmic speed limit. It’s misleading, confusing and inaccurate to view it as a combination of a real electron and a real positron.

The uncertainty principle prohibits such non-particle disturbances from existing for very long. Only true particles are able to exist indefinitely. That’s why these virtual disturbances come and go quickly.

Protons and Charm Quark/Anti-quark Pairs

Similarly, a proton may briefly contain a virtual charm quark and a virtual charm anti-quark, while always retaining its mass of 0.938 GeV/c2. Such a disturbance in the charm quark field need not, in this case, have zero rest mass, but it must have rest mass less than that of a proton. Nature is not somehow sneaking real charm quarks into a proton. That would violate basic rules, such as energy and momentum conservation.

But is that really a big deal? The proton contains many gluons, which, like photons, have zero rest mass, and which, analogously to photons, spend a little of their time as a combination of a virtual charm quark and a virtual charm anti-quark. And so, simply because protons contain gluons, there are automatically virtual charm quark/anti-quark pairs in protons — i.e., it’s obvious that there are disturbances in the charm quark field inside a proton! So what’s up? Presumably the claim that “there is intrinsic charm in the proton” must mean something more significant than just this!

Indeed, it does. The question is whether the majority of virtual charm quark/anti-quark pairs do not originate from individual gluons. In other words, are there other sources of disturbances in the charm quark field that cannot be accounted for using simple Feynman diagrams like the one we started with? Are these disturbances more common and more important within the proton than the mere presence of gluons would have led us to expect? The claim — still to be confirmed with better data and additional analysis — is that they are.

Hopefully this gives you a better picture of what all the chatter is about. But I imagine that for some of you a question has arisen: which particles in a proton are actually real? That’s a subtle point, which I’ll have to address in a later post.