Of Particular Significance

Protons and Charm Quarks: A Lesson From Virtual Particles

Picture of POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 09/09/2022

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.

Feynman diagram suggesting a photon is sometimes an electron-positron pair.

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!


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.

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23 Responses

  1. FWIW, I think that saying that a photon has “gravitational mass” is confusing. It is more apt to say that photon has mass-energy and that gravity couples to mass-energy and not merely to mass.

    1. Yet another storm about absolutely nothing. I was at Harvard last week and nobody even mentioned it. As far as I can tell, the article you linked is correct on the science. I can’t speak to the sociology and the personalities, but it sounds reasonable. One has to wonder about this age we live in, where fundamental ideas about the universe are reportedly overturned every other week. It doesn’t happen like that.

      The real threat to the current way of thinking about the Big Bang (not to the basic idea, but its details) is known as the Hubble Tension — two different ways of measuring the expansion rate of the universe are getting two slightly different answers. This tension has been building for at least five years, maybe more, I forget. It’s rare that things happen suddenly, and it’s basically impossible for JWST, or any other device or measurement, to overturn the Big Bang overnight. It would take years of puzzlement first.

      Incidentally, there is an enormous amount of evidence in favor of the Big Bang, including this fit of theory to measurement (https://sci.esa.int/web/planck/-/51555-planck-power-spectrum-of-temperature-fluctuations-in-the-cosmic-microwave-background) and the abundance of light elements (https://w.astro.berkeley.edu/~mwhite/darkmatter/bbn.html). Both of these could have been wildly, insanely off from the prediction of the Big Bang. But they’re not.

  2. Interesting! I had never thought of things like loop integrals being “effectively” over variable rest mass! And I’d studied many many textbooks. I tried to find it in those books, and finally did … just in one place. That’s obscure! (Its in the text at Peskin and Schoeder 6.44).

    1. It is pretty standard, though. You’ll probably find more under “Dispersion relations”, which unfortunately aren’t covered in most beginning textbooks but are reviewed in the professional literature.

      I haven’t checked it carefully, but if you can handle the math, you might find this one illuminating: https://arxiv.org/abs/1610.06090

  3. Thanks for your explanation. So, to summarize, I think you are saying that the experiment claims to find more charm/anti-charm virtual particles in the proton than could be explained by gluon to charm/anti-charm virtual particle loops.

    Is the theoretical calculation of gluon to charm/anti-charm virtual particle loops accurate enough to be inconsistent with the experiment? I thought many high-order loops would need to be calculated since the strong coupling constant is of order one and that high-order calculations are difficult to do.

    So, if there really is a theory-experiment inconsistency, what kind of new physics could explain this discrepancy? What kind of “new” particle could be added to make theory and experiment consistent?

    1. As you suggested yourself, the issue here is a lack of calculability within the Standard Model, not the need for some new physics beyond the Standard Model. It’s by no means easy to compute any of this. Nobody thinks new particles are needed; we would have already observed them in other experiments.

      But the current story is less about computation and more about *measurement* of the effect. Does it show that there’s a lot more charm and anti-charm with relatively high momentum compared to what gluon splitting to virtual charm could possibly do? The claim is “yes”, and that it’s insensitive to the details of the calculation. [This is something I will have to present later when I myself have all the details straight.]

      Also, as I alluded, this whole story is controversial. The methods used by those making the claim have been questioned by their competitors who disagree with the conclusion.

  4. My physical chemistry teacher always explained it as particles having ‘shadows’ in other fields, so that they had a presence essentially in all fields, whose ‘strength’ related to how strongly the two fields interacted. He even had a nice demonstration of this; a tank filled with water and oil, if the top layer is stirred, the bottom layer will respond without being directly touched. I’ve always seen it this way, that a photon doesn’t ‘flip’ between a photon and a particle pair, but exists as a collection of disturbances in all fields at once. But is the ‘flipping’ model more accurate?

    1. You are correct that it’s more of existing as a collection of disturbances all at once, but then again, that depends in part on how you interpret quantum physics. What I can say is that the math doesn’t need to view it as flipping. It’s really an interpretational question.

      But this is an even more advanced topic. Maybe I can address it sometime more precisely.

  5. Dr. Strassler, thank you for this paper is it fascinating. And I have a lot of questions, but I would like to ask a few if I may. From your paper; “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. ” Since we are talking disturbances in fields, why can’t it be said that any and/or all quarks, from the charm to the top quarks can exist in a proton? My first thoughts as an answer to this question would be energy constraints. But ultimately we are looking at a zero energy balance, and if a little bit of “extra” energy can be found for the charm quark why can’t a lot of extra energy be found for a top quark? The second question, in reading about the quark and anti-quark match up one of the first things that came to my mind was Hawking Radiation. How is a guaranteed, sort-a-speak, that one of the virtual quarks doesn’t “pop out of the proton?”

    1. Caution: ***NO*** extra energy is needed to create this virtual charm quark/anti-quark pair. NONE. Very important!!!! Any article that claims that energy is violated by the uncertainty principle in this process has flipped things on their heads: it’s the rest mass which changes from what you’d expect, not the energy. Energy is conserved; rest mass is not. No extra energy is needed, nor could any be obtained, simply through the uncertainty principle.

      Now, about your question: the answer is yes, but again, you have to look carefully at the last few paragraphs of the article.

      1) A photon spends some of its time as a virtual electron/positron pair, less time as a virtual charm quark/anti-quark pair, even less time as a virtual W+ W- pair, and still less as a virtual top quark/anti-quark pair. And so on for all the other particles with electric charge. It even spends some time as a virtual photon and a virtual Higgs boson, or as two virtual electron/positron pairs. But there are rules (which is what Feynman diagrams encode) as to how much of the time a photon spends as one or the other. And because of its larger mass, the top quark is much less likely to appear than an electron.

      2) Within the proton, there are gluons, and a gluon too can be a virtual charm quark/anti-quark pair or a virtual top quark/anti-quark pair and so on; but the more mass the quark has, the more suppressed is the production of the virtual pair. In other words, the required disturbance in the top quark field with zero rest mass is harder to produce than the corresponding disturbance in the charm quark field. However, this is the *non-interesting* way of putting charm in the proton — we knew decades ago that such pairs are in the proton.

      3) All the hullabaloo this month is a claim (not proven yet) that there are *other* sources of virtual charm-anticharm pairs which do not come from individual gluons; it’s not clear where they come from. These *other* sources of quark/anti-quark pairs are going to be very suppressed for top quark/anti-quark pairs, though the degree of suppression depends on what the dominant source is, which isn’t known.

  6. So what do people mean when they say “A photon can be an electron-positron pair part of the time”?
    Gödel showed that the “#Augmented (enhanced Penrose stairs connection) axiomatic system will allow the construction of a new, true formula ‘Gʹ that can’t be proved within the new, augmented system (Universe). In striving for a complete mathematical system, you can never catch your own tail.

    In the “Penrose stairs connection”, you get a logic (relativity) only by connecting the ascending connecting point to the descending connecting point by reducing the enhancement (#ImaginaryMass) with Dirac equation (G and ~G).

    Can G (augmented Mass or charm mass) be proved?
    The opposite of G says no such proof exists (No parity invariant, No chirality, No electron-positron ?).
    Opposite statements, G and ~G, can’t both be true in a consistent axiomatic system. So the truth of G must be undecidable. However, although G is undecidable, it’s clearly true (in a #Equation like E = mc^2)?

  7. That is interesting stuff. I had always thought of “real” particle decays as the particle shoving its rest mass energy into other fields. In other words, there were never any particles hiding inside the decaying particle, it just happens (based on probability I would imagine) to shove its energy into “other” fields. With that said, isn’t this the reason the rest mass of the decaying particle must be GREATER than the rest mass of the particles it decays into? In other words, the rest mass energy of the decaying particle must equal the rest mass of the particles it decays into….plus any kinetic energy of the new particles?
    With the virtual particles, It’s almost as if nature “mistakenly” shoves the energy into the quark field, realized it’s mistake, as there isn’t enough energy to make a “real” quark, and quickly withdraws it before anyone notices.

    1. You’re not wrong about decaying particles. As for your last remark (“It’s almost as if nature “mistakenly” shoves the energy into the quark field, realized it’s mistake,…and quickly withdraws it before anyone notices.”) it’s a nice way of putting it, but of course it’s putting too much intention into nature. Because the electromagnetic field interacts with the electron field, it’s impossible for this effect not to occur. Also, the even more correct story involves absorbing this effect into the *definition* of what a photon is — because you’ll never observe one that doesn’t do this. So rather than it being a photon thinking about decaying to an electron and positron, realizing that it can’t, and given up, it’s more that *it’s a photon being a photon*, and it’s you who were naive about what a photon actually is.

  8. “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.”

    I bet there’s an interesting history of initial confusion that lead to this, incredible for me, realization. What other quantum fields can carry quantum particle disturbances associated with other quantum fields?

    1. In a sense, all of them. An electron can become a joint disturbance in the electron and electromagnetic field. A neutrino can become a disturbance in the neutrino and the Z boson field. And the Higgs boson can become a joint disturbance in the bottom quark field. Any field that interacts with other fields will do this; it’s a central and unavoidable feature of quantum field theory. (And there are Feynman diagrams for each of these processes.) The article on “virtual particles” that I linked to above may be useful to you.

  9. > 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.

    Could you please go into a bit more detail about what this means? (If you don’t have an answer targeting a lay audience, I’d still appreciate some links to explanations that do involve some mathematics.)

    1. This is easier to see for just a single particle, rather than a particle anti-particle pair. The issue is directly related to why the electromagnetic force is long-range while the weak nuclear force is short range.

      The electromagnetic force between two electrons comes from the electric field, and in this language it is via a virtual photon — i.e., a general disturbance in the electromagnetic field; that disturbance, in this case, *is* the electrostatic potential that we calculate in first-year physics. (The force is 1/r^2, where r is the distance between the electrrons, but the potential — the disturbed electric field — is 1/r.)

      Now, the weak nuclear force between two electrons comes from the Z boson field. Because the Z boson has a large mass, the equations which governs the Z-based force are different from those of electrostatics, such that the Z-static potential isn’t just 1/r ; it is has an exponential suppression

      Exp[- c M_Z r / hbar ] / r,

      where M_Z is the mass of the Z particle, hbar is Planck’s constant and c is the cosmic speed limit. Now what that tells you is that at distances larger than hbar/(c M_Z) which is about 10^(-17) meters, the effect of the weak force falls like a rock and is basically negligible beyond that point.

      ***Correspondingly, the virtual Z can only exist for the time it takes to travel that distance — about 10^-26 seconds.*** This is the answer to your question.

      The equations we’re solving in figuring this out are similar to the Klein-Gordon equation with and without a mass, namely https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation for the overall equation and http://www.physics.usu.edu/Wheeler/ClassicalMechanics/CMYukawaPotential.pdf for the solving of the equation.

      Does that help with the intuition?

    1. I thought I replied to this, but it seems the reply disappeared.

      It plays no role. The disturbances in question happen just the same for both chiralities of the electron (which is the opposite of that of the positron.) By contrast the mass (or equivalently, the constant Higgs field) flips the chiralities.

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