2016-09-01 07:31:47 8 Comments

I'm writing a piece about the electron, and I'm having trouble finding evidence to back up the claim that the evidence is pointlike.

People tend to say *the observation of a single electron in a Penning trap shows the upper limit of the particle's radius to be 10 ^{-22} meters*. But when you look at Hans Demelt’s Nobel lecture you read about an

*extrapolation*from the measured g value, which relies upon

*"a plausible relation given by Brodsky and Drell (1980) for the simplest composite theoretical model of the electron"*. This extroplation yields an electron radius of R ≈ 10

^{-20}cm, but it isn't a measurement. Especially when

*"the electron forms a 1 μm long wave packet, 30 nm in diameter"*.

It's similar when you look at The anomalous magnetic moment and limits on fermion substructure by Brodsky and Drell. You can read things like this: *"If the electron or muon is in fact a composite system, it is very different from the familiar picture of a bound state formed of elementary constituents since it must be simultaneously light in mass and small in spatial extension".* The conclusion effectively says *if an electron is composite it must be small*. But there's no actual evidence for a pointlike electron.

Can anybody point me at some evidence that the electron is pointlike?

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## 5 comments

## @TwoBs 2016-09-09 17:01:36

In order to answer this question we have to agree on the meaning of point-like. (This is not so obvious since nature happens to be quantum rather than classical) In practice, one has to specify a framework where the definition can be operationally, at least in principle, tested against the experimental evidence.

The tentative definition that I will adopt in the following is: a particle is point-like if every physical process (say a scattering), at any energy scale (or kinematical configuration) above a certain threshold, agrees with the prediction made by a perturbative renormalizable quantum field theory where the particle is elementary. An equivalent definition could be that the action for such a particle is dominated by its free kinetic energy at all scales above a certain threshold (or, again, that the theory is always around a Gaussian fixed point). In practice I am trading point-like for elementary which is a (slightly) better defined concept.

I had to include the notion of perturbativity to speak of particles in the first place, that is of (presumably effective) field theories that are close to a gaussian fixed point in at least a finite energy range. This definition is not perfect, but it makes clear that a theory of particles strongly interacting at all scales isn't in fact a theory of particles after all.

The proton isn't elementary because its interactions at or above the confinement scale are strongly coupled and, moreover, the theory would require infinitely many terms in the lagrangian making it non-renormalizable too. The pions, on the other hand would seem to be elementary at small energy (essentially because of their Goldstone boson nature and Adler's theorem) but the interactions become strong again at $E\sim\Lambda_\textrm{QCD}$. The interactions are non-renormalizable too. In fact, the requirement of non-renormalizability and the strong interactions usually go together in concrete realizations of compositeness.

Buying this tentative definition for point-like, we can ask whether the electron is so. The answer is yes: it is point-like, to the best of our present knowledge. In other words, up to the energy scale of the order of few tens of $\mathrm{TeV}$'s that we have been able to explore experimentally (the precise number depends on various things that would take us very far), there is no sign that the electron isn't described by the renormalizable weakly coupled quantum field theory known as the Standard Model at all scales above the $\Lambda_\mathrm{QCD}$. In such a theory the electron is an elementary field.

Various caveats are in order. First, I am neglecting gravity which makes the SM non-renormalizable (and gravity may becomes strong at $M_\textrm{Planck}$). In the leading quantum theory of gravity that explains the dynamics at the Planck scale, string theory, the electron isn't quite a particle nor point-like. The Planck lenght is however so small that we can safely ignore this point for most of the questions. Second, the gauge coupling for the hypercharge in the Standard Model is believed to have a Landau pole that may break the theory at even larger energy scales than Planck. Hence, one can safely neglect the Landau pole too (quantum gravity effects kick-in much earlier).

Say one day we discover a discrepancy between the predictions of the Standard Model (SM) concerning the electrons and the experimental data. To be concrete, imagine one day we discover a 5~$\sigma$ discrepancy in the $g_{e}-2$ of the electron. Would that mean that the electron is composite? No, at least non-necessarily. In fact, the extra corrections $\delta_\textrm{BSM}$ in $(g_{e}-2)=\delta_\textrm{SM}+\delta_\textrm{BSM}$ could be accounted for a new weakly coupled renormalizable field theory valid above a new threshold (the mass of the new particles involved in producing $\delta_\textrm{BSM}$) where the electron is still an elementary field. There exist several models beyond the SM where this is the case: they go beyond the SM coupling new weakly interacting particles to the electron, changing some of its low energy properties; however, above the mass of these new states the electron is still accounted as an elementary particle coupled weakly to the old fields and few new ones. On the other hand, the $\delta_{BSM}$ could be explained by the electron being compositeness, i.e. non point-like. This would be the correct explanation should the new weakly coupled renormalizable theory expressed in terms of other fields than the electron. One could still insist to use the electron above the compositeness scale but the theory would be strongly interacting and non-renormalizable, in such a variable.

## @Lehs 2016-09-02 10:08:58

This question is about the energy of an electron. Since the energy stored in the electromagnetic field of an electron $$u_{EM}=\frac{\varepsilon}{2}|\mathbb E|^2+\frac{1}{2\mu}|\mathbb B|^2$$ must be a significant part of the energy of the electron, even the field must be regarded as a part of the electron. Which thus not is a "point".

But that was the classic model. In QED the electron is defined to be pointlike and that works well, as far as it has been possible to calculate and measure. But also astronomical calculations give good results for pointlike stars and planets. Also, I think it is a disadvantage that a "single" electron is considered to be field-less. In reality, however, no electrons are totally single, so one might wonder how close two electrons have to be before they are not single.

## @anna v 2016-09-02 10:14:26

you are talking of this en.wikipedia.org/wiki/Classical_electron_radius . But the electron is a quantum mechanical entity and has to be treated with quantum mechanical mathematical tools.

## @Lehs 2016-09-02 10:17:58

@annav, no I'm not talking of any radius at all, even if I saw that calculation 45 years ago. But I guess that also the QED fields is supposed to store energy?

## @anna v 2016-09-02 13:45:16

the QED field creation operator has zero energy if there is no electron there, and one electron is generated by the electron creation operator if it is there, all at one point. Thst is what "point particles" means in the standard model.

## @ACuriousMind 2016-09-02 14:40:15

I don't see how this answers the question. If you regard the electric field of an electron as "part" of it, the electron is infinitely extended, which is a patently useless notion of size.

## @John Duffield 2016-09-02 16:08:47

@ACuriousMind: you cannot separate an electron from its electromagnetic field - there are no neutral electrons. The electron’s electromagnetic field is part of what it is. In fact the electron’s electromagnetic field

iswhat it is.## @dmckee --- ex-moderator kitten 2016-09-03 18:45:14

Electrons scatter as if they were point-like in experiments sensitive to sizes around an attometer. This is an experimental fact. You might not be able to "separate an electron from its field", but their scattering behavior is not ruled by the distances at which they effect the field. The rest is just words.

## @Constantine Black 2016-09-04 08:55:22

Hi. How does one then counts two electrons if they are infinite in size but have the same energy density (even classically)?

## @Lehs 2016-09-04 09:20:14

@dmckee: looking at the classical model almost all energy is concentrated near the mass center, while the non-point-like part of the energy is very small. I don't want to argue for the classical model, but evidently electromagnetic fields exists and store energy - that somehow belongs to the rest mass of the electron.

## @John Duffield 2016-09-04 11:58:27

@dmckee : re

electrons scatter as if they were point-like in experiments sensitive to sizes around an attometer. What experiments?## @user104372 2016-09-04 13:23:42

@dmckee, yes, could we please have some links describing these experiments? and how you get from an attometre to zero?

## @dmckee --- ex-moderator kitten 2016-09-04 13:29:56

LEP. Where else? There are other approaches to setting the limits, but they are more subtle. At LEP they collided beams of electrons and positrons at lab frame energies above 100 GeV each. Setting straight Rutherford-style limits is hard because the cross-section for elastic scattering falls like a stone with increasing momentum transfer, so most of the papers talk about searches for resonances representing excitations of electrons. The 2015 RPP suggests the the LHC sets tighter limits still, but the hadronic context means the analysis will be more model dependent.

## @flippiefanus 2016-09-02 14:35:03

Before addressing the pointlike nature of the electron, let's consider the proton. It was found that when the energy with which particles (such as electrons) scatter off a proton exceeds a certain level (about 1 GeV), it starts to resolve the proton. What we mean by that is that, below this energy the scattering cross-section seems to follow a scale invariant curve (a pure power law), while at this scale, the curve for the scattering cross-section as a function of energy changes its behaviour, indicating the presence of a specific scale. This scale (1 GeV) is called the QCD scale, because it turns out that quantum chromodynamics (QCD - the underlying theory that binds together the constituent in the proton) becomes confined at this scale (below the QCD scale the QCD interactions become invisible; above it one sees the effects of this interaction because one can peer inside the proton).

The energy with which a scattering experiment is performed determines the

resolutionof the experiment. What this means is that the energy translates to a distance [see clarification below]. For higher energies, one can observe smaller distances. Above the QCD scale the resolution is small enough that one can observe distances smaller than size of the proton.One other very important thing to notice is that the mass of the proton is also roughly equal to the QCD scale (using Eintein's famous equation $E=mc^2$). This is important because that means that the same scale is responsible both for the size of the proton and the proton's mass.

Now let's turn to the electron and the question of its pointlike nature. Obviously, we cannot do scattering experiments at infinite energies. Therefore, the resolution with which we can observe the electron is limited by the largest energy that we can produce in collider experiments. The cross-section that we observe from a pointlike particle is therefore determined by the resolution with which we observe it. With current experiments one sees that the scattering cross-section of the electron follows a scale invariant curve. Hence, no scale where the electron is resolved has yet been seen. An important observation though is that the energies with which the scattering has been done, far exceeds the mass of the electron. So if there does exist an energy scale where the electron would be resolved, such a scale would be very high above the scale set by the mass of the electron.

The thing about scales in physics is that they don't just fall out of the air. There are usually very specific dynamics involved that produce such scales. In the case of particles bound together, one would normally expect the mass of the bound particle to have roughly the same scale as that associated with the size of the bound particle. If the size of a particle is so much smaller than its mass, then there would have to be an amazingly powerful reason for that.

For this reason, although we cannot measure the electron's size to infinitely small distances, it is believed that the electron must be pointlike.

Clarification:Just to address some of the comments. To

resolvemeans that one observes something with a resolution that is smaller than the size of the object. The resolution of an observation refers to a physical distance. In particle physics, for instance, the resolution is directly related to the energy of the scattering process. (Energy gives the frequency via $\hbar$ and frequency gives wavelength via $c$. The wavelength is the physical distance that defines the resolution.) The notion of a resolution is widely applicable in observations. For example, in astronomy a telescope would be able to resolve a galaxy if the resolution of the telescope is smaller than the size of the galaxy in the image.Some comments seem to suggest that the electron should have a finite size due to the electric field that is produced by its electric charge. Unfortunately this does not work either. The electric field decays as a power law away from the electron. Such a power law does not have a scale. It is scale invariant. As a result the field cannot give a scale that one can interpret as the size of the electron.

See here for Particle Data Group information about lepton (electron) compositeness.

## @tparker 2016-09-04 03:34:00

There's never any direct experimental proof that anything does

notexist, including a nonzero electron radius. But we have a very good (you might even say "Standard") Model that describes the electron as a point particle and accurately explains all known experimental data (at least, data describing processes involving electrons). With no experimental reason to expect electrons to have a nonzero radius, Occam's Razor suggests that we should consider electrons to be pointlike until there is a concrete reason not to.Of course, it's completely possible that one day, higher-energy experiments will discover that electrons are composite or extended structures, and if that happens then we'll need to revise our assumption. There's precedent for this in the history of particle physics: the neutron, proton, and pion, among others, were all once assumed to be pointlike elementary particles, until a better model came along that described them as quark-gluon bound states.

## @tparker 2016-09-05 17:00:47

@Rococo I'm not sure if I agree that "pointlike" and "elementary" mean the same thing. In string theory, strings are elementary but extended, and their mathematical description differs from that of usual QFT. I think "pointlike" is a strictly stronger notion than "elementary," and roughly corresponds to the definition I gave in my last comment.

## @Rococo 2016-09-05 17:49:29

@tparker Oh, that is certainly not something that had occurred to me- fair enough. But it still seems to me that it is worth emphasizing that the above statements are true within the standard model and equivalently at any energy we have ever accessed, which seems relevant given that the OP's question was about what the experimental evidence says.

## @anna v 2016-09-01 11:05:14

One who is familiar with the history of particle physics, and physics in general, knows that physics is about observations fitted with mathematical models.

This review examines the limits on size we presently accept for the fundamental particles which presently are at the foundation of the present standard model of particle physics.

This analysis of what "point like " means is reasonable in my opinion.

The measurable quantities are the form factors:

From these measurable form factors one can get a limit for the size of the electron, no proof of real "point" nature can be given. Models are only validated or falsified, and the "point" nature of the electron is a model of the existing data involving electrons which has not been falsified.

The point nature works in the present standard model of physics because our experiments cannot probe smaller distances than these limits, and the SM which depends on these pointlike elementary blocks WORKS.

## @Jarek Duda 2019-05-26 07:04:04

Regarding "SM which depends on these pointlike elementary blocks WORKS", remember that the used perturbative QFT has a few infinities (also energy of electric field of perfect point charge), which are manually removed - especially ultraviolet divergence: restricting minimal distance, and divergence of perturbative series: restricting sizes of scenarios (Feynman diagrams) which could fit there. Assuming this is not just mathematical idealization, but fundamental particles are indeed perfect points, would we need these two restrictions - mathematical tricks to remove infinities?

## @anna v 2020-01-10 05:11:20

@JarekDuda This has to be answered by a theorist. Maybe if you ask it as a question. You could also try asking at physicsoverflow.org a site where many theorists answer..

## @Jarek Duda 2020-01-10 08:13:22

I have asked this kind of questions to many theoreticians and experimentalists, but did not get anything concrete. Usually "evidence" for point-like electron leads to g-factor argument, which leads to Dehmelt's 1988 paper where he literally gets it from extrapolation by fitting parabola to two points (no kidding!) - gathered materials: physics.stackexchange.com/questions/397022/…