## How Is the Weak Nuclear Force an Actual Force? “Of these four forces, there’s one we don’t really understand.” “Is it the weak force or the strong—” “It’s gravity.”

It’s time for another physics explainer. In an earlier post, I explained Lagrangian mechanics and why it uses the weird (to physicists) equation L=T-V. I thought I might do some more posts like that, and I found a topic that should make a good one. So, let’s jump over to nuclear physics and figure out what on Earth is going on with the weak nuclear force.

Physics says there are four fundamental forces of nature. There’s gravity, which pulls masses toward one another. There’s electromagnetism, which pulls opposite electric charges toward each other (and pushes like charges apart). There’s the strong nuclear force, which holds the quarks together inside protons and neutrons. And then, there’s the weak nuclear force, which…causes radioactive decay?

♫♪ One of these things is not like the others. ♫♪

In any popular science book or really even undergraduate textbooks (to my memory), you’ll see the four forces described this way, and you may be thinking: “Wait a minute, one of those isn’t actually a force.” Three of the fundamental forces are described how forces are always supposed to be described: by how they push or pull on particles. But the weak force isn’t. It causes this other process to happen, and no one ever mentions it pushing or pulling.

Imagine if someone said, “Gravity is a force that makes things change shape.” And it’s true; gravity makes stars and planets form into spheres, and it makes a raindrop splatter into a pancake on the ground. But that sentence is still completely missing the point (even if it doesn’t sound like total nonsense). It would be silly to talk about gravity just making things change shape and not even mention the main aspect of it—the actual force part of it—which is that it pulls masses together.

That’s how weird it sounds to me to say, “The weak nuclear force causes radioactive decay.” So, what gives? Well, that’s what I’m hoping to explain with this post.

(And I’ll get to the strong force in the next post. Ironically, that one is more complicated, even though it sounds simpler.)

First off, the weak force really is a force. You can see it in the rare cases where a neutrino bounces off an electron or an atomic nucleus. It bounces because a force was exerted on it by the weak interaction, mediated by a Z-boson. (For comparison, for charged particles, this usually happens via the electromagnetic force, mediated by a photon.) This is called a neutral current reaction. If a W-boson (which has an electric charge) is involved instead, this is a charged current reaction, and this is what can cause radioactive (beta) decay.

Second, it’s something of an oversimplification to say the weak force causes radioactive decay. It’s more accurate to say that weak interactions cause quarks (and leptons) to change flavor—for example from a down quark to an up quark. And it’s not the only force that does this kind of thing. After all, the strong force causes quarks to change color.

(For the other two forces, it’s less clear, but the electromagnetic force can make particles change charge when a pair of photons turns into an electron-positron pair, and vice versa. And gravity is just weird because of relativity.)

So yes, the weak force can push and pull just like any other force. It’s just that the other things it does are usually more interesting and noticeable. But this is only half of the question in the XKCD comic. The weak force is a force, but is there a nice, neat equation for it like there is for electromagnetism and gravity?

There is…sort of. There’s an equation that works (with some caveats), but you almost never hear it talked about in this context. You hear about it in the unified model of electroweak theory, which combines the weak force and electromagnetism. You also don’t hear it described as a force, but as a potential, just like it’s often more useful to talk about electric potential (volts) than about electric force directly for many applications. But this is no problem; with potential energy, all you have to do is take a derivative, and you get the force. It’s just that the potential equation is simpler.

The equation for the weak force is the Yukawa potential, which is just the Coulomb potential multiplied by an exponential decay.

Here, g is a measure of the strength of the weak force, and αm is the range of the weak force, which is proportional to the mass of the W and Z bosons. The actual force to match the equations in the comic is this:

which is somewhat more complicated, but at small distances, it’s approximately this:

and we’re back to Coulomb’s force law times an exponential.

Note that this equation is an approximation. It’s not quite as “clean” as the more familiar ones, but even that’s nothing new. Coulomb’s law is an approximation to quantum electrodynamics, after all, and Newton’s law is an approximation to general relativity. This is just more complicated. The weak force decays exponentially with distance because the particles that carry it are very massive, which means they can decay into other particles. Like other forms of radioactive decay, this follows an exponential function, which means the force is weakened exponentially with time, and thus with the distance the particle travels. The more massive the particle, the faster it decays and the shorter than range of the force.

The W and Z-bosons that carry the weak force are so short-lived that even at the speed of light, they will decay before they can travel the width of a single proton. And since they’re so massive, they don’t travel that fast, so the decay happens over a distance a thousand times smaller than an atomic nucleus. At distances this small, everything is more wave than particle, and the actual distance has more to do with how likely particles are to interact if they “collide.”

This is why the weak force doesn’t really look like a force. It operates over such tiny distances that you can’t really see it push or pull anything, except in those super-rare neutral current reactions, where, by quantum uncertainty, particles happen to interact when most of the time they wouldn’t.

And just as a final note, the Yukawa potential actually applies to photons, too! It’s just that because photons have no mass, the exponential is just 1. In other words, they don’t decay, so their range is unlimited.

Stay tuned for an explanation of the strong nuclear force, which also needs a proper equation.

 More accurately, the weak force causes beta decay and electron capture. All other forms of radioactivity are caused by the strong and electromagnetic forces. Many books do make this distinction correctly.

 Theoretically, a neutral current reaction can also cause radioactive decay (e.g. Chekanov et al., 2003). The super-heavy top quark could emit a Z-boson to decay directly into a charm quark or an up quark without changing its charge, but this has never been observed in the laboratory.

 This “color” is of course not a real color that we can see, but a weird quantum property that we call “color” for convenience. 