While working on a review of a book that talks about the fortuitously bio-friendly constants of nature (review forthcoming, don't worry), I mistyped "ratio of proton to electron masses" and "ratio of proton to electron charges." Which is, of course, 1, and thus not a terribly interesting ratio.
But that got me wondering: is there a solid fundamental reason why that ratio is one? This is, could you have a self-consistent universe in which the electron and proton had different charges-- say, a proton having twice the charge of an electron? That wouldn't be a useful sort of universe, of course, as the lightest element would be chemically more like helium than hydrogen, and that pretty effectively screws up all of chemistry, but that doesn't necessarily mean it would be mathematically impossible.
I suspect it must be impossible, just because it wouldn't make much sense otherwise, but I'm not sure what the argument would be. And I'm too lazy to Google it up this morning, so instead I'll make myself look like an idiot, in hopes of getting an answer in comments.
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The obvious way to do this would be to have the \Delta^{++} = {u, u, u} baryon be stable, rather than the proton {u, u, d}. My recollection, from reading it many years ago, is that Harald Fritzsch discusses this thought experiment in his popular book _Quarks_ (1983), and even says something about how QCD would have to be modified to make it happen. But if I pursue this any further I'll start having flash-backs to my field theory classes and that will not be pretty.
The obvious follow-up question would be why the electric charge of quarks is 1/3 of a lepton.
I am sure superstring theory provides a good answer...
Cosma is right; you can make the Delta^++ the lightest baryon by making the down quark much heavier than it is. People sometimes talk about this in an anthropic context: if the Higgs VEV were significantly larger and every other constant in the Standard Model is kept fixed, the Delta^++ would be the lightest baryon. The usual claim is that then there would be no interesting chemistry, so anthropically this can't happen.
The cancellation of various anomalies in the standard model requires the quark charges to be 2/3 and -1/3. Hence the charges of the proton and the electron have to be equal (and the Ward identity of QED ensures that they stay equal under renormalisation)
It's not an explanation for electrons and protons having the same charge, but electroweak theory constrains the charge on the electron to be the difference between the up and down quark charges, doesn't it? (Because downs can convert into ups by emiting a W- which can then decay into an electron and an electron antineutrino, and charge is conserved.)
Off the top of my head, I don't think there is even a reason that ratio should be a rational number (but maybe things like anomaly cancellation are strong enough constraints). Any proposed structure that relates leptons and hadrons, for example grand unification, will be a putative explanation.
Looking at the answers (and I was going to give a similar one) a meta-question comes up - what kind of explanation is acceptable? Is it ok to say that the charges of the electron and proton have to be the same because we have several symmetry relations in the standard model that allow these particles to transform such that the charge on one becomes the charge one the other? We do see those transformations so it isn't quite a tautology directly, but certainly charge conservation symmetry was "built in" the standard model.
This is where I always feel itchy. Physical models are supposed to explain what we see in the world, but it seems like a lot of people now are going - wow we have these wonderful models - they must define the world - lets forget experiments, try to find a beautiful model and check it for consistency and we think with the right axioms it MUST look like the real world - everything will be constrained. I think all theories beyond the GR and SM are doing this. I guess it is ok, but it seems like it is throwing out the physical part of physics. As long as some people are still looking at what we can't explain in nature (or haven't yet) for clues also I guess it is ok, I would put money on the second group to find new changes to our current models first.
Okay, after a bit more thought... I seem to recall that the condition for the anomaly cancellation is that the charges for each generation sum to zero. (But I can barely recall any quantum field theory, and was never much good at it anyway.) If this is the case then it's a simple matter to deduce the equality of magnitudes of the proton and electron charge from that condition and the electroweak one I mentioned earlier.
From electroweak theory, the difference between the up and down quark charges has to be the charge on the electron, so
Q_e = Q_d - Q_u (1)
From anomaly cancellation, the sum of the charges in a generation has to be zero, and there are three quark colours so
Q_e + 3 ( Q_u + Q_d ) = 0
That implies that Q_u + Q_d = - Q_e / 3 (2)
Add the two equations:
2 Q_d = 2 Q_e / 3
so Q_d = Q_e / 3
Subtract them:
2 Q_u = - 4 Q_e / 3
so Q_u = - 2 Q_e / 3
A proton is two ups and a down so
Q_p = 2 ( - 2 Q_e / 3 ) + Q_e / 3 = - Q_e
The ratio of charges is actually -1. heh.
After seeing the electric charges so wonderfully rational, it's kind of a shock to find out that the "weak charge" involves the Weinberg angle. And of course the various elementary particles have very arbitrary masses. Or are they arbitrary?
I guess it wasn't quite as stupid a question as I thought...
It's not an explanation for electrons and protons having the same charge, but electroweak theory constrains the charge on the electron to be the difference between the up and down quark charges, doesn't it? (Because downs can convert into ups by emiting a W- which can then decay into an electron and an electron antineutrino, and charge is conserved.)
Of course, you could ask whether it would be sensible to construct a theory whereby the decay took place through a process that emitted two electrons, and two antineutrinos... Well, "sensible" probably isn't the right word for something that daft, but you get the idea...
Rich,
yes indeed, the ABJ anomaly is resolved by
Q_e + Q_v + 3 ( Q_u + Q_d ) = 0
for the standard model.
Very nice!
PS: But somehow I share Markk's doubt whether all this 'explains' why the standard model has to be pieced together the way it is.
I guess it depends what you allow to vary. It is clear that in the standard model the ratio is one, and that the standard model is the only model consistent with all observations. I thought Chad's question had to do with those notorious alternative universes, which would have some strong interactions (so one could distinguish between hadrons and leptons), and some electromagnetic interactions to define charge. One could define electron as the lightest charged lepton, and proton as the lightest charged baryon. I think in such cases the ratio of charges can be different than one (for example by choosing the strong interactions to be some other non-abelian theory), though perhaps it cannot be irrational.
Of course, it's always possible to look for deeper reasons for things, but given the three-generation Standard Model the ratio of lepton and quark charges are not among the 19 free parameters, so that's an explanation of sorts. The reasons for the values of those free parameters is another question entirely.
(In one formulation, the nineteen are three coupling constants, the four Cabibbo-Kobayashi-Maskawa parameters, three lepton and six quark masses, the masses of the Z and Higgs bosons, and, um something else...)
Reasoning from theory misses the point. That the lepton and proton charges are what they are is an experimental fact. Any theory which led to them being different would therefore be rejected as disproved by experiment. So all existing theories, of course, say they're the same. The question is: Can a consistent theory be constructed in which they're different, and which would be true in a not-too-terribly-different real world?
My suspicion is you'd want to construct such a theory where the charges are only slightly different, rather than where one is a multiple of the other. So basic chemistry would look similar, but everything would be charged: all atoms, all molecules, would be slightly ionized. So all matter would be conducting . . ..
Reasoning from theory misses the point. That the lepton and proton charges are what they are is an experimental fact. Any theory which led to them being different would therefore be rejected as disproved by experiment. So all existing theories, of course, say they're the same. The question is: Can a consistent theory be constructed in which they're different, and which would be true in a not-too-terribly-different real world?
Exactly.
The question came up in thinking about the "multiverse"/"landscape" theories, which posit that there are bazillions of different universes with different values of the physical constants that ours. The question is really which of those constants are allowed to vary among the various universes that arise from the same underlying mathematics as ours.
(Of course, if you take the really hard-core Max Tegmark view, then all conceivable universes exist somewhere, but I'm not talking about that extreme... It's just that, when the "string theory landscape" comes up, people always talk about different universes in which the masses of particles are different, but not the charges. The question is whether it's possible for those to vary, too.)
Jim said:
On the other hand, it would be possible in principle for our leading theory to say nothing at all about the relationships between quark and lepton charges, and for these to all be parameters which had to be set by hand to match experimental data. If this were the case then in some sense it would be "easier" to imagine a spectrum of possible universes differing from ours only in the ratios of charges of the fundamental particles. For example, this is the situation in which we find ourselves regarding the masses of most of the fundamental particles.
However, this isn't the situation in which we find ourselves: a universe having different ratios of quark and lepton charges to ours would have to vary not just in the values of physical constants but in more "fundamental" things like gauge groups.