A fairly simple unsolved physics problem, but the solution could be vitally important to our understanding. Does the electron actually have infinite charge? Surely not…

I have to say, I do agree with you, you’re right, but hear me out.

If we follow the rules of the standard model, then the electron should in fact have infinite charge.

Still don’t believe me? Read on and find out…

### The Fine Structure Constant

When talking about electric charge, we are referring to a particle’s interaction via the electromagnetic interaction.

The strength of this interaction, or force, is characterised by the **Fine Structure Constant**. This is the ‘coupling constant’ for the electromagnetic force.

There are also coupling constants for the weak and strong interactions, but these aren’t relevant for this discussion.

This constant is defined by:

$$\alpha_{\text{EM}} = \dfrac{e^2}{4 \pi \epsilon_0 \hbar c} \approx \frac{1}{137}$$

So, as you can see, $\alpha_{\text{EM}} \propto e^2$, so if we want to measure $e$, we can equivalently measure $\alpha_{\text{EM}}$.

So, what actually is the ‘coupling constant’? Well, for the electromagnetic interaction, it describes how strongly a charged particle interacts with a photon.

This is because the boson, or ‘force carrier’ for the EM interaction is the massless photon.

### Measuring $\alpha_{\text{EM}}$

So, we now know that, if we want to measure the electric charge, we can measure $\alpha_{\text{EM}}$, for which we need to look at electron interactions.

One way to do this is to look at electron-positron scattering, where, you guessed, an electron scatters off of a positron.

A positron is just the anti-particle of the electron; it has a positive charge.

One such scattering is known as Bhabha scattering, and has the following Feynman diagram:

The labels on each particle are just their momentum.

If you’re not sure what a Feynman Diagram is, it’s basically a pictorial representation of a particle interaction. For a basic, but well explained introduction, have a look at the Quantum Diaries.

### The Running of $\alpha_{\text{EM}}$

What is found when looking at these scatters is that $\alpha_{\text{EM}}$ is not constant.

In fact, as $q$, the momentum of the virtual photon, increases, so does the coupling strength, $\alpha_{\text{EM}}$.

What does this actually mean?

Well, if you remember a thing called the de Broglie wavelength (pronounced ‘de Broy’? Who knows?), we have that $\lambda \propto \frac{1}{q}$.

This means that, at large photon momentum, the de Broglie wavelength will be small.

What we’re actually saying is that the coupling strength increases as we get closer to the particles.

So, if we keep going closer to the particles, will the coupling, and hence the charge, keep increasing?

### Charge Screening

In classical physics, an electron goes along in a vacuum without interacting.

However, in QED (Quantum Electrodynamics), it’s a different story.

As the electron passes through space, it is constantly emitting and absorbing photons.

These photons then pair produce electron-positron pairs, and these then annihilate to form photons again.

The effect of this is to essentially ‘screen’ the electron charge.

If we are far from the electron, we see the bare electron charge **and **the effects of the environment; the total charge of all of those electron-positron pairs.

So, to get to the bare electron charge, we need to get closer and closer to the electron, to filter out all of this other charge.

i.e., we need to find $\alpha_{\text{EM}}$ at infinite $q$.

If you’re having trouble visualising this, it might help to think about it classically.

Imagine you put a charge, $q$, into a dielectric; you polarise it.

The result is a sphere of dipoles around the charge which screen it.

You would calculate the electric field as:

$$E = \dfrac{1}{4\pi\epsilon_0} \dfrac{q}{r^2} \dfrac{1}{\kappa}$$

where $\kappa$ is the dielectric constant.

But if you didn’t know the dielectric were there, you would measure the electric field as normal, as:

$$E = \dfrac{1}{4\pi\epsilon_0} \dfrac{\frac{q}{\kappa}}{r^2}$$

i.e., thinking that the charge had a value of $\frac{q}{\kappa}$.

### The Standard Model Prediction

The Standard Model gives a complicated equation for this, the nature of which isn’t important.

The importance is this: it predicts that $\alpha_{\text{EM}} \rightarrow \infty$ as $q \rightarrow \infty$.

So, it would appear that yes, the electron does have infinite charge!

### Shortcomings of the Standard Model

Ok, well maybe it doesn’t. This wouldn’t make physical sense, but it is actually an important result.

It tells us that the Standard Model is not complete.

At very high energies, we need a different, modified theory to explain such phenomena.

This theory has not yet been developed, and is an unsolved problem in physics!

Maybe you could be the one who solves this problem, by developing a theory for high energies?

### Thanks for Reading

Thanks for reading this post, I hope you enjoyed.

Thanks to Dr Steve Boyd, my particle physics lecturer, for teaching this stuff to me.

And thanks once again to freesvg for the featured image.

If you have any questions, or didn’t understand something, feel free to leave a comment or contact me here.

Also let me know if you liked this sort of thing, or if there’s something in particular you’d like to read next time.

Take a look around at the blog, I’m sure you’ll find something interesting.