What's the deepest reason why QCD bound states have integer electric charge, i.e. equal to an integer times the electron charge?

Given that the quarks have the fractional electric charges they do, this is a consequence of color confinement. The charges of the quarks are constrained in the context of the standard model by anomaly cancellation, and can be explained by grand unification. The GUT explanation for the charges doesn't care about the bound state spectrum of the QCD sector, so it just seems to be a coincidence that hadrons (which are composite) have integer charge, and that leptons (which are elementary) also have integer charge.

Now maybe there's some anthropic argument for why such a coincidence is useful (in the case of proton and electron, it gives us atoms as we know them). Or maybe you can argue that GUTs naturally produce fractionally charged particles and strongly coupled sectors, and it's just not much of a coincidence.

But I remain curious as to whether Seiberg duality, anyons, some UV/IR relationship... could really produce something like the lepton-hadron charge coincidence, for deeper reasons. I suppose one is looking for a theory in which properties of bound states in one sector have a direct and nontrivial relationship to properties of elementary states in another sector. Is there anything like this out there?

(This question was prompted by muster-mark's many recent questions about fractional charge, and by a remark of Ron Maimon's that the hadron-lepton charge coincidence is a "semi-coincidence", which assured me that I wasn't overlooking some obvious explanation.)

  • $\begingroup$ Seems to me that the comment of Ron you link to points to the answer: that it is QED constraints that need integer leptons and baryons? $\endgroup$
    – anna v
    Sep 24, 2012 at 3:33
  • $\begingroup$ Isn't this just group theory? Anomaly cancellation fixes the charges of quarks, and color SU(3) fixes the charges of color singlet hadrons. Also note that the hypercharge assignments that lead to anomaly cancellation are essentially unique. $\endgroup$
    – Thomas
    Sep 24, 2012 at 17:21
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    $\begingroup$ @Thomas: it doesn't fix the charges uniquely, it fixes it up to multiples, and there's a question of why it turns out that all singlets are integer charged. If you add fractionally charged stronly interacting scalars, for instance, the integer charge is out the window. $\endgroup$
    – Ron Maimon
    Sep 25, 2012 at 8:51

4 Answers 4


The universal quantization electric charge definitely isn't a coincidence; it's a necessary result of the fact that the gauge group of QED is the compact group $U(1)$ instead of the noncompact group $\mathbb{R}$, as explained in this extremely short paper by Yang: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.1.2360.

People often act like only the Lie algebra of the gauge group matters, so gauge groups $U(1)$ and $\mathbb{R}$ are equivalent (as are gauge groups $SU(2)$ and $SO(3)$). But this isn't true: the global topological structure of the gauge group matters as well. It would be perfectly logically consistent for QED to have gauge group $\mathbb{R}$ and for electric charge to not be quantized, but experimentally this isn't the case. This ambiguity doesn't come up in nonabelian gauge theory, because in this case the Lie algebra must be semisimple in order for the kinetic energy term to be positive-definite, which implies that the gauge group must be compact.

(In fact, if there were two elementary particles with incommensurate electric charges, implying that the QED gauge group were $\mathbb{R}$ rather than $U(1)$, then magnetic monopoles would be impossible. Roughly speaking, this is because Dirac's quantization argument giving the allowed magnetic monopole charges requires that electric charge be quantized.)

  • $\begingroup$ I didn't ask why charge is quantized, I asked why QCD bound states have integer charge. $\endgroup$ Jun 22, 2017 at 7:56
  • $\begingroup$ @MitchellPorter I don't understand the distinction. "Charge is quantized" means that everything has integer charge. $\endgroup$
    – tparker
    Jun 22, 2017 at 8:08
  • $\begingroup$ In the usual normalization, quarks do not. But regardless of normalization, the question is why QCD bound states have a charge which is an integer multiple of the electron charge. $\endgroup$ Jun 22, 2017 at 12:20
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    $\begingroup$ @MitchellPorter I still don't understand what you're confused about. The compactness of the QED gauge group implies that any deconfined particle, whether elementary or composite, has an electric charge which is an integer multiple of the fundamental unit of electric charge $e$. QCD bound states are deconfined particles, therefore they an electric charge which is an integer multiple of the fundamental unit of electric charge $e$. $\square$. $\endgroup$
    – tparker
    Jun 22, 2017 at 19:54
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    $\begingroup$ @MitchellPorter Perhaps you're confused about why confined particles (like quarks) are not required to have electric charges that are an integer multiple of the elementary charge. The answer is that the electric charge of a confined particle is actually a bit tricky to define, because if you try moving it around a magnetic monopole or a magnetic flux tube, the "strong strings" contribute a phase factor that exactly makes up for the "missing" charge $2/3 e$, as explained at physics.stackexchange.com/a/269932/92058. $\endgroup$
    – tparker
    Jun 22, 2017 at 19:57

The simplest answer to your question is a quite old idea, captured best I think by the Rishon model of Haim Harari, Michael Shupe, Nathan Seiberg, and others.

Their answer is the simple and rather obvious one: Hadrons and leptons have identical charge because they are composed out of the same set of more fundamental particles and anti-particles, specifically an uncharged V particle and a one-third charged T particle.

Alas, in terms of mathematical development the Rishon model is more akin to an intriguing speculation than a fully developed and predictive physics model. I do not personally think that any particle-based version of the Rishon model can ever be made to work. My suspicion is that theories like the Rishon model are best viewed as incomplete and distorted images of some far less obvious form of composition, one with components that conserve certain properties but cannot be called particles in any traditional meaning of the word.

Nonetheless, the Rishon model strikes me as orders of magnitude better than some of the more recent trends to explain issues such as electron-proton charge equality by invoking what amounts to anthropic self-selection gone wild. Why? Because Rishon theory at least tries to explain astonishing coincidences. If Newton had given up so easily on looking for deeper roots behind an effect as infinitely precise and in-your-face obvious as electrons and protons have identical charge magnitudes, we'd still be talking about how amazing and lovely it is that Great Angels push the planets around in patterns too lofty and subtle for humans ever to understand.

2012-09-27 Addendum

Here's a point I should make clear for the record, since I came down pretty heavy on the idea that evolving universes could create balanced sets of charges via nothing more than the anthropic principle.

The anthropic observation that the existence of life as we know it seems to require that many fundamental constants to be very tightly constrained and balanced with each other is a simply delightful observation that truly needs explanation. Simple examples include such things as the remarkably long an sharp ridge of stable isotopes that enable complex chemistry, nuclear fusion suitable for stars, and the ability of carbon (with nitrogen and other helpers) to form indefinitely long stable chains. These applications of the anthropic principle are all in effect fine-tuning issues, and I think they are entirely legitimate issues for applying your own personal favorite version of anthropic selection if you are so inclined.

Where I have deep heartburn is with the far more radical versions of the idea that essentially toss all aspects of physics into one big mysterious anthropic pot that then magically burps out whatever it is you need to make life possible. If that is true, why do physics and chemistry constantly throw unexpected structure and marvelous little symmetries in our faces, in even a cursory look? Wouldn't a true, unbiased anthropic cauldron simply toss out a universe that works fine for life, but shows no unnecessary correlations or symmetries between the resulting diverse components of its physics? Such patterns and correlations would after all represent an unnecessary, irrational, and mechanistically inexplicable "extra effort" on the part of the anthropic cauldron, an effort that goes far beyond what is needed simply to enable life. If you own a true anthropic cauldron, Occam's razor says "why bother?" with anything more in the product.

Or stated another way: I have no problem with using anthropic ideas to adjust the ratio between two tightly meshed gears, but I have a lot of trouble with using it to create the gears themselves. Nearly every finding in physics seems to be shouting at us that the bones and tendons of the universe arise from complex permutations and various degrees of breaking of symmetries, with many of details of those symmetries and their permutations being being captured at least partially in that marvelous work called the Standard Model.

So, my real message on this issue is a simple one: Extreme applications of otherwise good ideas tend to be wrong, often rather spectacularly so. Exclusion of extremes is a nicely general principle that applies to a very wide range of phenomena, and I just can't see any good reason why the anthropic principle should get a waiver from it.

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    $\begingroup$ This is not correct--- the rishon idea is not any better than just the standard model regarding this question--- you might as well ask why is it that the rishons that are not confined have the same integer multiples as the rishons that make the leptons. The rishon model does not explain this thing. $\endgroup$
    – Ron Maimon
    Sep 25, 2012 at 8:50
  • $\begingroup$ I wasn't critiquing the standard model. It's been seriously postulated that the identical charges of electrons and quark-composed protons are just the result of self-selection among evolving universes, and thus have no deeper explanation than the need for matching values to enable us to be here to observe them. If you accept the premise of physics of the 1800s and early-to-mid 1900s that there is deep and profound simplicity underlying the apparent complexity in physics (and I do), then simple composition easily beats such extreme versions of anthropic evolutionary universe selection. $\endgroup$ Sep 25, 2012 at 14:18
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    $\begingroup$ Except it's not any simpler to have integer charged hadrons, you have fractional charged hadrons in models no more complex than the usual ones. there is no real relation between the quarks and leptons other than anomaly cancellation or coming from a GUT. The major thing that would happen with a fractionally charged hadron is that there would be a stable lightest one and it would affect cosmology, it wouldn't be complicated, just wrong. $\endgroup$
    – Ron Maimon
    Sep 25, 2012 at 16:07

I would say the deepest reason is anomaly cancellation. If the charge of proton and electron were not the same (even 1 in 1000000!) then the current conservation in the standard model wouldn't be fulfilled due to anomalies.

They say that anomalies have topological roots.

Given the fact that electric charge is quantized in the EM sector (Charge Quantization, Compactness of the Gauge Group, and Flux Quantization), the aforementioned anomaly cancellation technique implies the charge quantization for hadrons that you demanded.

For a more detailed explanation take a look at these pages of Schwartz QFT book: 633 and 634.


An experimentalist's view:

I do not see the need to search further for why the three quarks add up to the electron charge than that given by the group structure of the Standard Model. The SM is very successful in organizing into beautiful symmetries the particle and resonances data gathered the last sixty years or so. There is no experimental reason to assume further layers of compositness defining a "deeper" group structure from which the "measured" SU(3)xSU(2)xU(1) should emerge. It will just introduce a lower level of unnecessary complexity.

If what intrigues you is the unit one, after all we can always say the down quark has charge -1, the up quark 2 and the electron -3. The group symmetries are the same and we will have a generic unit 1 .

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    $\begingroup$ Well... aren't questions like this what physics is all about? Explaining why certain things are the way they are. Merely describing things is rather unsatisfactory and a little bit boring. The standard model offers a lot of answers to why questions, but there are still some unanswered. One of them is why the proton charge is exactly the opposite of the electron charge. To me this startling fact is enough reason to search for something deeper. (Another example is why there exactly three generations ) $\endgroup$
    – jak
    Apr 29, 2015 at 13:17
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    $\begingroup$ @JakobH Physics ultimately does not answer "why" questions. Why questions after perusing the mathematical models end up at the axioms and postulates. They are really metaphysical. Physics answers how from axioms and simple postulates observations can be explained and future behavior predicted. $\endgroup$
    – anna v
    Apr 29, 2015 at 13:31

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