Mass of an Electron

Factual Questions
1

The rest mass of an electron has been measured with quite a bit of accuracy. Here one link among many to that quantity. CODATA Value: electron mass But being a measured value, there is still some uncertainty.

Suppose I were to hypothesize that the rest masses of all electrons are exactly the same. What observations could be made to prove or disprove this hypothesis?

I am mostly interested in physical observations and not mathematical proofs but a discussion of the latter might also be interesting.

2

Mathematical proof is not really a thing in science. Instead, you have things like “all electrons have a mass of 510,998.950,69 eV +/- 0.000,16 eV”. That is a measurement with error bounds.

Edited to add:
To expand on that, the above measurement is consistent with multiple “flavors” of electrons with different masses, with those differences within the error bounds. If your experiment is precise enough, you’ll need to account for that in your calibration or results.

Further edit: I see you linked to the same page I did. The theme I use doesn’t make in-line links very visible. Sorry about that.

3

A more mathematical argument is John Wheeler’s idea that all electrons have the same mass because they are all the same electron. There seems to be many of them because this one electron is zigzagging forward and backward in time. When traveling forward in time it looks like an electron, and when traveling backward in time it looks like a positron (an anti-electron).

4

Are there any particles whose invariant mass ever changes?* Is there any theory that allows this? True, the Standard Model has no explanation for why masses are what they are but that’s not quite the same.

*I know neutrinos change during flight, but then they are called by different names, so are not the same particle.

5

For an individual particle, the conservation of energy requires the mass remain constant across time. But that kind of gets the cart before the horse, since we believe conservation of energy holds because individual particles don’t have time-varying mass.

I don’t think that there’s a fundamental reason that all electrons are the same mass, other than we measure them all to have the same mass (for a certain value of same).

6

If electrons on Mondays should be heavier than those on Tuesdays, then you have a hypothesis you can design around: compare Monday masses to Tuesday masses. But if the idea is that they are just sort of randomly different – a mixed bag of electrons of different masses – then the first hint would be when the spread in mass measurements is inexplicably higher than the calculated experimental uncertainties. That is, if you can’t explain the observed spread of results, there must be another source of spread – either experimental oversight or Nobel-Prize-worthy discovery.

This ontology doesn’t have much grip anymore, despite it’s role in history. However, the foundational concept today that electrons are excitations of a single “field” is in the same spirit. The field has properties, and electrons that spawn from that field echo those properties.

The answer is either a simple “no” if you can keep your footing atop a slippery slope or else “this gets messy very fast” if you slip down. And then you hit the bottom and still find that the answer is best said as “no”. The main two tricky bits:

Many particles live for a short amount of time, and so any one instance of such a particle will have a measured mass that is random with a spread call its “width”. This is just Heisenberg uncertainty, but it means that if you make, say, a Z boson and measure that specific Z boson’s mass super well, you won’t get 91.2 GeV but instead you’ll get something random with a bell-shaped spread around 91.2 GeV with width 2.5 GeV.

Separately, what’s in the bare theory in the Standard Model for, say, an electron mass is unobservable. Work is needed to convert that into an observable quantity that feels like a mass. For an electron, this is called the “pole mass”, and it has two wonderful properties: it’s independent of how the conversion calculation is done, and it corresponds to the effective mass of the thing we want to talk about – a long-lived, non-bound, at-rest electron. Great! But…

Quarks suffer from color confinement, so you can’t calculate a pole mass at all. Fundamentally you have to define mass in terms of conventions that include arbitrary calculational schemes and energy scales. For instance, the typically tabulated “up quark” mass is based on (*inhale*) “a modified minimal subtraction renormalization scheme evaluated at a renormalization energy scale of 2 GeV”. Even if you ignore all the technobabble, the “energy scale of 2 GeV” might jump out at you, as that energy is three orders of magnitude higher than the tabulated up quark mass itself. That’s because trying to talk about a mass-like parameter for the up quark at energy scales around its value is ill-defined, pointing to the fact that “the invariant mass” of “this specific quark” is ill-defined. But if you push through the noise and accept the need for arbitrary conventions to even talk about quarks as “particles” with “mass parameters”, then the answers from those schemes are at least static. But quarks are so intrinsically dynamical beasts that one has to take care with what is meant by mass in a given context.

A side note that isn’t responsive to the question of “are invariant masses always the same” but connects the two above ideas in a neat way: The pole mass for an unstable particle – and importantly for very short lived particles like the Z boson – is fundamentally a complex number, and the reported mass is sometimes the real part of the pole mass and is other times an effective mass that folds in the fact that the spectrum of decays of the particle varies subtly across its practical mass-width. That is, not every Z boson can be the same in terms of its decay properties, and that influences mass inferences.

Indeed. For concreteness: the familiarly named neutrinos (“electron neutrino”, “muon neutrino”, “tau neutrino”) do not have definable masses. They are each quantum superpositions of a different three things that do have definable masses. The latter are simply labeled by the numbers 1, 2, and 3: \nu_1, \nu_2, and \nu_3. Those have masses m_1, m_2, and m_3. But there is no such thing as an “electron neutrino mass”. (One can talk about an effective electron neutrino mass, but its a quantity calculated from the m_i values and aspects of the superposition; it’s not a fundamental quantity of any sort.)

7

Summary of the long answer to “Are there any particles whose invariant mass ever changes?”: The decay width thing is very real and could mean a “yes” to your question, but the center of the invariant mass distribution is fixed and thus could mean a “no” to your question. The quark mass stuff points to a need for more context when talking about individual quark invariant masses at all; that is, the first question isn’t “is the invariant mass the same” but rather “what even is mass here?”

8

It’s a cute idea, but it doesn’t actually work. You can generate an electron-positron pair from high-energy gamma rays, do various measurements on the electron and positron, and then re-annihilate them. You can say that that electron and positron are the “same particle” as each other, but they’re not the same as any other electron.

But fundamentally, it’s almost never possible to prove that anything in science (such as the difference in mass between two electrons) is exactly zero. All you can do is to place successively smaller upper bounds on it.

While we’re at it with the electron mass: Electrons have charge. Pack some quantity of charge into some space, and you have electric potential energy. The smaller the space you squeeze the charge into, the greater the potential energy. Well, we don’t know how big electrons are (we usually model them as point particles, but see above about never knowing anything’s exactly zero), but we do know an upper bound on their size. And if you pack one electron’s worth of charge into a sphere that size, the electrical potential energy you calculate is actually much higher than the observed mass of the electron.

This is generally reconciled by saying that the electron has both a “bare mass” and a mass due to its charge, with the sum of the two being its total mass… but the “bare mass” is negative, and might even be infinite.

9

The sign of the bare mass does actually depend on the calculational scheme. Some schemes imply negative bare mass, others positive (and asymptotically zero), but all such questions fortunately carry no observable consequences at any scale for which the theory has any justifiable application.

10

Tell me more. In the schemes with a positive bare mass, is there some other (negative) contributor to the total mass that cancels out the electrostatic energy (some Weak Interaction effect, maybe)? Or is there some way of making the electrostatic energy smaller? And what do you mean by “asymptotically zero”: It approaches zero as… something? approaches infinity?

11

The classical picture of shrinking a ball of fixed charge smaller and smaller leading to more and more self-energy is a few paradigm shifts away from the concept of “bare mass”, so there isn’t any reason a priori that such self-energy has a bearing on bare masses in renormalized QFTs.

If we bridge the gap partway, we would need at a minimum to fold in corrections as the length scales get smaller (energy scales get higher). That substantially softens the divergence by a logarithm, at which point it’s not even divergent at any conceivably relevant scale. (The correction becomes of the order of the electron mass only at scales hundreds of orders of magnitude past the Planck scale.)

This isn’t entirely accidentally because the fundamental QED interaction vertex can’t flip chirality, and mass terms do flip chirality, so in QED it’s rather hard to generate something that acts like mass in the Lagrangian just from self-interaction. Taking this further, if the renormalization scheme respects this chiral symmetry, the connection between bare and physical masses is fundamentally multiplicative, not additive. The bare mass is positive but scaled down to the observable mass.

The phenomenology of renormalization is typically presented in terms of additive counter-terms that pull on the bare mass. That is usually a truncated perturbative treatment, and so it isn’t valid to use the approximate form at arbitrarily high scales. If you sum the whole series, it would restore the multiplicate treatment, which in the MS-bar scheme is ultimately an exponential correction that is always positive and gets smaller and smaller at higher energy scales.

In certain types of non-perturbative lattice calculations, the bare mass can be negative and divergent as the inverse of the lattice spacing, but that negativeness is unrelated still to the “shrinking ball of classical charge” intuition.

12

When you do statistical mechanics on electrons, you find that it only gives the correct answers if you assume that electrons are absolutely indistinguishable. Thus all electrons must have precisely the same rest mass and electric charge. The resulting statistical properties are called Fermi-Dirac statistics.