# Why do the proton and electron have equal and opposite electric charge?

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• arivero
arivero
Gold Member
The most common explanation I know is that anomaly cancelation implies the sum of electric charges of each particle must cancel generation-wise, so 3 Q(Up) + 3 Q(Down) + Q(electron) = 0, and electroweak doublets imply Q(Up) - Q(Down) = Q(neutrino) - Q(electron), so with Q(neutrino) = 0 it solves do Q(Down) = 1/3 Q(electron).

Is this explanation satisfactory enough?

Some problems I see:
• It needs to assume the SM group and the SM particle content
• It needs to assume that the electric (and colour?) charge of right and left particles is the same
• It needs to make sure that the nucleus is just protons and neutrons
• Some extra comment should explain why the cancelation must be generation-wise, and not just across all the particles.
The second point is more or less the definition of electric charge, say the parity invariant part of the electroweak force or something so, thus it can be explained via the higgs mechanism; but I am not sure if it is a complete explanation, particularly if it needs to include colour.

The first and third points are connected, an interesting secondary question is what does it happen if we have an arbitrary SU(N) colour group.

As you know, a U(1) can have any charge. Charge is just a number. In SU(N) charge is a matrix. Because $Q = I_3 + Y/2$, the question can be recast as why hypercharge is what it is.

Weak isospin is an SU(2) and so has values ±½, so your anomalies cancel so long as left-handed fermions appear in doublets. The right handed partners are singlets and have weak isospin of zero.

So the real question is why weak hypercharge is what it is. For leptons, the left handed fields are at -1. The right handed at 0 for neutrinos and 02 for electrons. Left-handed u-type quarks are +1/3, right handed +4/3. D-type quarks are +1/3 and -2/3 respectively.

"I want my theory to be anomaly free" is not an argument I find compelling. As I said on another thread, "otherwise our theories wouldn;t work or be ugly" - well, I don't think the particles care. But if you split it up this way you see:
• Left-handed and right-handed fields balance separately.
• The U(1) and SU(2) fields balance separately, the SU(2) more or less by construction.
• The factors of 3 in the quarks suggest SU(3) as the color gauge group.
Now, all those fractions look like Clebsch-Gordon coefficients. Is there some symmetry where these can be calculated? The answer is yes, and infinite number, and the smallest is SU(5). I am not the first person to have notived this.

dextercioby, DeBangis21, ohwilleke and 1 other person
Weak isospin is an SU(2) and so has values ±½, so your anomalies cancel so long as left-handed fermions appear in doublets.
Here's a tidbit of information I picked up when we were writing a paper long ago: While this is true for the regular gauge anomalies, the Witten anomaly requires that there be an even number of doublets.

First impressions
arivero said:
The most common explanation I know . . .

Is this explanation satisfactory enough?
Asking "why" the laws of physics are what they are is always problematic.

When you say "common explanation" this rings two bells in my mind.

First, it isn't the only possible way that it could be explained.

Second, it is an "explanation" not a law of physics or mathematics that can be established from anything but the example it seeks to explain.

Some laws of physics are derived from other laws of physics and then explaining why they are so isn't circular. But this doesn't appear to be one of them.

If you want to explain why must massless particles travel at the speed of light and not experience the passage of time in their own reference frame, you can point to Special Relativity, which can be established from multiple theoretical arguments unrelated to this property of massless particles, and also from experimental evidence tending to confirm this physical law in various circumstances to the limits of observational precision.

In contrast, "anomaly cancelation implies the sum of electric charges of each particle must cancel generation" is an explanation of why electric charges must cancel out with the observation that this is what happens in real life as its sole observational support of example.

Within the theoretical framework of the Standard Model and its possible extensions, you are really just trying to use an axiom of the Standard Model to explain something that you assumed in the first place.

Why must there logically be anomaly cancelation at each generation?

One could imagine a universe in which, for example, anomalies must cancel globally and not at each generation, if they didn't cancel at each generation. That universe wouldn't be logically impossible, it would just be a universe with particles and laws of physics a bit different from our own. But we don't live there.

Thus, I agree with Vanadium that:

"I want my theory to be anomaly free" is not an argument I find compelling.

There are plenty of circumstances in physics where anomalies of some kind do arise, and we deal with that and use it as a tool to do physics. For example, we calculate anomalous dimensions in effective QED, and the world doesn't come to an end.

Similarly, a pion is often called a "pseudo-Goldstone boson", because it is almost a Goldstone boson, but rather than involving an exact symmetry, it involves a symmetry that real world physics closely approximates, but anomalously, real world physics deviates slightly from.

Isn't assumption one circular?

As the OP notes, for this explanation to work one has to "assume the SM group and the SM particle content". But, if the outcome you are trying to explain is compelled by one of the axioms that you assume in the first place, then you're engaged in circular reasoning and the exercise is meaningless.

What does assumption three do? And isn't it false anyway?

I also don't see that it is important one way or the other to the explanation that "It needs to make sure that the nucleus is just protons and neutrons". This seems like a non sequitur and I don't see where you are going with it.

This also isn't an assumption that is really true. A nucleus doesn't just have protons and neutrons. It also has electromagnetically neutral mesons like pions and kaons that are carrying the residual strong force that binds the nucleus in atoms together between the protons and neutrons in that nucleus. And, of course, atomic hydrogen-1 doesn't have any neutrons in its nucleus. So you have things that aren't protons or neutrons in a nucleus, and you don't have to have both of them in a nucleus. This doesn't even begin to investigate whether the sea quarks of a proton or a neutron count or not, with the theory behind "not" being that they are part of particular protons and neutrons and not directly a part of the nucleus itself. And, the reason that there are no other stable hadrons in a nucleus is rather obviously because protons and bound neutrons are the only stable hadrons. The electron is a stable particle, but if any particle in a nucleus was a lepton, then the residual strong force wouldn't bind to it because leptons don't have color charge, so it wouldn't stay in the nucleus or be part of a structure bound by the residual strong force, and so, it wouldn't really be a part of the nucleus.

In sum, this question is unanswerable

Ultimately, the only real answer to "Why do proton and electron have equal and opposite electric charge?" is "because that's what we observe them to be to high precision," therefore a theory makes the logical leap of faith of assuming that it is true exactly, because this keeps the math clean and simple and seems to work.

While some other answer than "that's how it is" could hypothetically be possible for some laws of physics, in this particular example, I don't see how any elaboration or analysis that you could add to "that's how it is" would add any meaningful explanatory power or understanding to the question.

Last edited:
DeBangis21
Well I guess a mathematician could do a better formulation of the question, such as a verbose expansion of "what axioms grant that the most stable baryon has the same electric charge that the charged leptons".

arivero said:
what axioms
This is a bit of an issue, as if we had a set of axioms that didn't lead to atiomic neutrality, the theory would be immediately discarded.

As gar as "why" questions being unanswerable, I think this one has an answer. We just don't know what it is. I suspect we are seeing some symmetry principle in play. Maybe it's SU(5). Maybe it's not. But I don't think its an accident - not with such tight constraints on atomic neutrality. "Gotta be something. Why not exactly zero?" does not strike me as a good argument,

arivero
Not really; you can have a set of axioms that doesn't imply atom neutrality, and add an extra axiom "atoms must be neutral". Of course, if this produces a set of incompatible axioms, then the theory must be discarded.

By the way I find amusing that mathmos call "theory" to the axioms and "model" to the representation of the axioms. Famous theorem, "If the axioms are a first-order theory then we have Goedel's Completeness Theorem which states that a theory is consistent if and only if there is a model of this theory"

Demystifier and ohwilleke
Well, you can certainly take as an axion - although I would say ansatz - that quarks and leptons are part of SU(5) multiplets. Then atoms are automatically neutral.

Of course, then protons decay. I can fix that too, with another ansatz. But theory-whack-a-mole tends not to be very convincing.

arivero and ohwilleke
Just a reference to anomalies, to remember the lore:
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.39.693
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.41.715
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.41.717
Also I like how Gell-Mann Ramond Slansky fix colour to be SU(3) and still have exotic atomic numbers for quark and leptons. So it is not only than one must explain equal charge; one also must explain barion number. (of course Ramond also has other paper where even with SU(N) he attempts A=1/3 quarks)

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Last edited:
fresh_42
I don't find the anomaly argument very convincing. At best, it says "we have constructed our theories to do this". I don't know about those particular papers (but am pleased to see Pierre Ramond's name - he was the guy who first got me seriously thinking about these problems) but as I understand it, even if GUTs accommodate atomic neutrality, they do not require it.

arivero

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