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### Hierarchy problems and why electrons don’t have infinite mass

One of the reasons why physics is hard to learn in high school is that sometimes it just seems so fraught with inconsistencies. One such inconsistency that always bothered me was the energy of the electron.

Point charges don’t make sense in classical physics

Classical electromagnetism tell us that the energy of a configuration of charges increases as the distance between them decreases. To put it simply, the energy increases the closer together you bring the charges. For two charges q1 and q2 separated by a distance R, the electrostatic energy of the configuration is given by: The first fraction is just an overall constant number, sometimes it’s just called k. Don’t be put off by the fancy pi and epsilon, this number isn’t so important. It’s the second fraction that actually tells us about physics. For constant charges q1 and q2, what we see is that the energy increases the closer we bring them together.

This should make sense without having to look at any equations: the closer you bring two like charges the harder they push each other apart.

Something is fishy here, though. If I bring two charges arbitrarily close together, does this mean that I end up with a configuration with arbitrarily large energy?  (Could I produce a black hole from the electrostatic energy of trying to force two charged particles together?)

In fact, the fishiness is even more insidious: we are told that electrons are point particles that carry some charge density. If we imagine the electron to be a charged spherical shell (like the skin of an orange) and shrink its size down to zero, doesn’t this mean that we end up with an electron of infinite electrostatic energy?

If you do the calculation (this is a common homework problem for undergraduates), the equation for a spherical shell of charge e (the electron’s charge) and radius R is almost the same as the equation above for just two charges: The net effect of the shell is an additional factor of one half. We’re not going to nitpick about overall numbers, the point is that as we take the radius of the shell to zero (i.e. to a “point particle”) the energy seems to go to infinity! Let us call this the electron’s self-energy, i.e. the energy of the electron wanting to push itself apart. Alternately it’s the energy of the electron trying to escape its own electric field.

Something is seriously wrong with classical electromagnetism in a way that is plain to anyone familiar with introductory physics.

Electron self-energy vs. mass

Now let’s try to pull in some things that we’ve learned before. In a previous post I explained Einstein’s relation between energy and mass. You can review it now if you’ve forgotten it, but the punchline is that mass can be thought of as a kind of potential energy belonging to a particle.

This, however, is exactly what the electron self-energy is: it’s a potential energy associated with the electron’s charge. This can be thought of as a contribution to the electron’s mass. Some of you might object: we said that mass is a kind of energy, but this doesn’t mean that all energy can be thought of as mass. Good, you’re thinking like a scientist! The point here is that gravity feels energy, not mass. Usually this is interchangeable because mass energy is so much larger than other types of energy (by factors of the speed of light), but in this case we’re talking about a potentially infinite electrostatic energy, so this should certainly be included in the mass.

We thus end up with the following equation for the “effective” electron mass: We’ve just written out energy using E=mc2. Here m is the observed “effective” mass, M is the “intrinsic” mass, and we’ve included the contribution from the electrostatic energy. Note that m is the mass that we measure, while M is some “non-electrostatic” contribution to the mass energy that is never directly measured. (This is a “deep idea” in quantum physics that I won’t go into right now.)

We know what the electron mass is because we’ve measured it (via experiments by Thompson and Millikan); the left-hand side of this equation is 511 KeV. Experimental results also limit the electron characteristic size to be less than 10-17 cm, so just for the heck of it let’s use that as a limit for the radius R above.

Now if one crunches the numbers, one ends up with the electrostatic term being something like 10 GeV = 10 000 000 KeV.  This makes our equation look really fishy, we end up with:

511 KeV = Mc2 + 10 000 000 KeV

This seems to imply that the Mc2 must cancel out the 10 000 000 KeV at the part-per-mille level to give 511 KeV. (The fact that it has to give a negative contribution isn’t actually too weird since M is never measured.)

Ridiculous fine-tuning: the hierarchy problem

This is an example of a kind of problem that has been at the back of theorists’ minds for over 30 years, it’s called a hierarchy problem. The problem is that the mass of the electron seems to depend on the cancellation of two numbers that are much, much bigger. This might not seem like a big problem, and indeed it took a long time before physicists identified this as something very undesirable in our theory.

The gist of the problem is that nature appears to be very sensitive to how these two big numbers happen to cancel. We say that the two big numbers have to be finely-tuned. The physical mass of the electron, 511 KeV, depends on physics at much higher scales. But this is like wondering why we’re able to calculate the trajectory of a baseball using high school physics without taking into account quantum corrections. The quantum corrections represent very short-distance physics relative to the macroscopic Newtonian physics of the baseball. As explained in a previous post, the physics at one scale should be insensitive to physics of very different scales. In the same way, it would be very surprising that 511 KeV physics would depend on very fine cancellations between quantities from 10,000,000 KeV physics.

In some sense this is an aesthetic problem. It may well be that nature is “finely-tuned” and the electron mass does come from a miraculous cancellation. But this sensitivity to much-higher-scale physics goes against the intuition that one develops from other physical examples of hierarchies.

Quantum theory: antimatter to the rescue

It turns out that quantum mechanics saves us from this “electron mass hierarchy problem. ” This is not actually so surprising: it appeared that our problem came from using classical physics to probe scales (R ~ 10-17 cm ~ 10 GeV) that are much smaller than the electron mass. We know, however, that quantum mechanics becomes effective at small scales.

One of the consequences of quantum mechanics is that one can violate energy conservation, but only for very short amounts of time. This “energy-time” uncertainty is a direct analogue to the “position-momentum” Heisenberg uncertainty relation that is now part of the popular zeitgeist. Written as an equation, At short distance scales, quantum electrodynamics predicts that virtual electron-positron pairs may pop out of the vacuum before disappearing again. Their main effect is to become polarized under the influence of the actual electron’s charge, as illustrated below: Image from Peskin and Schroeder.

These virtual electron-positron pop in and out of existence and end up smearing out the electron’s charge. As we probe the electron at shorter and shorter distances, it no longer behaves as a hard ball of charge, but rather an infinitesimal point particle with a cloud of charge. (Don’t confuse this with an atom having a quantum ‘electron cloud,’ this is a similar but different phenomenon.) In fact, as you keep probing even shorter distances the electron-positron pairs would “screen” the original electron’s charge and you would think that the original electron has a smaller charge than expected. This is an actual effect that physicists call renormalization. (It’s a fancy word that you can tell your friends.)

Thus our toy model of a charged shell breaks down and we are saved from having to think that the electron should have an infinite mass.

Have we solved our hierarchy problem? We need to figure out at what scale does the quantum picture become effective? Using the uncertainty relation above and converting into distance, Where we’ve written the energy uncertainty as the amount of energy required to create a virtual electron-positron pair; m is the physical electron mass and the h-bar is Planck’s constant (representing the scale of quantum fluctuations). Putting in actual numbers, we find that quantum mechanics becomes relevant at the scale

R ~ 10-11 cm ~ 10 KeV

So indeed we find that quantum effects cancel out the classically infinite contribution to the electron mass at roughly the scale of the electron mass itself. (To the best of my knowledge this argument was first made clear by Hitoshi Murayama in his ICTP lectures on supersymmetry.)

The Higgs Mass Hierarchy Problem and why we expect something new at the LHC

It turns out that this “electron mass” hierarchy problem is exactly analogous to what is more commonly known as The Hierarchy Problem. This is the question of why the Higgs mass is so small.

Wait a second, we haven’t even discovered the Higgs boson… why do we think its mass is small? If there is a Higgs, then it turns out have a similar self-energy as the electron example above. It would appear that the Higgs mass should be arbitrarily large. From the above example, we know what this means now: the Higgs mass should be roughly at the scale of the new physics which “completes” the previous theory.

In order for the Higgs mass not to be finely-tuned, there should be some new phenomena waiting to be discovered at the TeV scale. One proposal for new physics is called supersymmetry, which predicts a “superpartner” particle for each known particle in much the same way that “charge-parity symmetry” predicts an antimatter particle for each known particle. In the above example the virtual effects of matter-antimatter pairs smeared out the electron charge to cancel the electrostatic contribution to its mass. In exactly the same way the virtual supersymmetric particles cancel out the contribution to the divergent contributions to the Higgs mass.

This, however, is a whole different story that I’d like to tell in another blog post. The take-home message is that the “Hierarchy Problem” that physicists always mention as a motivation for new physics can be understood in terms of the classical problem of an electron’s self-energy: a problem that even high school physics students can identify from their textbooks.