What Is the Equation for the Strong Nuclear Force?

Previously, I described how the weak nuclear force really is a force even though it’s almost never described as one. Instead of a simple inverse square law like gravity and electromagnetism, it decays exponentially so that it weakens over a very short distance.

But there’s one more piece to this puzzle. At the popular science level, no one ever explains what the equation for the strong nuclear force is, either. It does have one, and it’s simple enough to explain, but no one ever mentions it, which is a shame because it’s actually kind of cool.

This equation is called the “Cornell potential” or the “funnel potential”:

Here, α and σ are constants associated with the strong force. The main function of the strong force is that it holds the quarks inside a proton or neutron together. It holds atomic nuclei together, too, but that’s a side effect. The Cornell potential tells us how this works when we take a derivative to convert it to a force:

The first part of this is an inverse square law. In other words, inside a proton or neutron meson, quarks actually undergo a gluon-mediated scattering interaction governed by an inverse square law just like Coulomb’s law. The difference is that there’s a second term in the equation, and it’s a constant. Regardless of the distance between them, two “unpaired” quarks will be attracted to each other with a constant force on top of the inverse square law of about 10,000 newtons[1] (which is a weirdly normal-sounding number equal to about one ton of force).

Now, there are several caveats here. First is that this equation isn’t really for two quarks. It only works for the simplest system of an unflavored meson—a quark paired with its own antiquark. Other mesons, which have charge, flavor, or mass differences between the quarks, and baryons, which are three-body systems, have more complicated interactions.

Second, there’s a sense in which this equation is not “exact” like Coulomb’s law is, arising directly from the geometry of the system. It was the best of several guesses based on the full quantum theory (Eichten et al., 1978), which was first supported by experiments and was only put on a rigorous mathematical footing much later (Sumino, 2003). We now understand that the Cornell potential is an approximation to the quantum chromodynamics (QCD) that govern the strong force just like Coulomb’s law is ultimately an approximation of quantum electrodynamics (QED), but it doesn’t have the same physical intuition behind it.

Indeed, it’s not obvious why the force between quarks should include a constant. It just seems that the constant force term is what happens when the interaction stop behaving like a “weak” force (and in this context, electromagnetism counts a “weak” force) that attenuates over distance and starts doing its own thing instead. For this reason, the gluons that carry the strong force are sometimes described like springs that connect the quarks together over any distance,[2] and this part of the force is called the “string tension.” (Not to be confused with string theory.)

Note that the gluon is massless, so the strong force technically has an unlimited range (not the residual nuclear force that holds protons and neutrons together, but the strong force proper). The trouble is that there’s so much energy bound up in the strong force that it will create more quark-antiquark pairs to fill the space if they’re separated by more than the width of an atomic nucleus. This is called quark confinement. The only time we see a quark on its own is with the top quark, which decays so fast that there’s not enough time for gluons to “connect” it to other quarks, even at the speed of light.


Now, with all this talk of the strong force, we haven’t explained how protons and neutrons are attracted to each other to form atomic nuclei—just how they’re structured internally. They’re attracted to each other because of something called the residual nuclear force that happens because a quark in one proton momentarily bounces closer to a quark in a different proton, and the quarks attract one another through the strong force. Before we figured out quarks, this was simply called the nuclear force. This is the same kind of interaction that causes van der Waals forces between molecules—although those are, weirdly, an inverse seventh power force, while the residual nuclear force is a Yukawa potential.

Hold that thought.

The other thing that can happen with the strong force is, because there’s so much energy involved, is that you can have virtual quark-antiquark pairs form and interact with the “real” quarks (also called valence quarks). Inside a proton or neutron, this interaction can swap the “color charges” on two quarks. But between a proton and a neutron, you can have a quark-antiquark pair called a pion that basically swaps an up and a down quark between them. This reaction changes the proton into a neutron and the neutron into a proton. And this happens all the time in atomic nuclei.

At the top left of the figure is a circle labeled neutron. Within it are three smaller circles labeled d, u, d. At the top right corner is a circle labeled proton. Within it are three circles labeled u, d, u. At the bottom right is a circle labeled neutron. Within it are three circles labeled d, d, u. At the bottom left is a circle labeled proton. Within it are three circles labeled d, u, u. Lines from d, and u in the bottom left proton connect to the d and u in the top left neutron. Lines from the d and u in the bottom right neutron connect to those in the top right proton. A line from the u in the bottom left proton connects to the u in the top right proton. In the middle of this connecting line, the u pairs with another circle, which is labeled d bar. This pair is labeled pi plus. Pointing to the circle labeled d bar from the left is an arrow, whose base is labeled d plus d bar created. A line from the base of the arrow connects to the d in the top left neutron. To the right of the circle labeled d bar is a line, the endpoint of which is labeled d plus d bar annihilate. A line connects the d in the bottom right neutron to it. Wavy lines are shown between all connecting lines.
Credit: Physics LibreTexts (CC BY-NC-SA 3.0).

The equation for the residual nuclear force is our old friend the Yukawa potential from our analysis of the weak force. This is an inverse square law multiplied by an exponential decay with distance. Unlike the weak force, though, the residual nuclear force decays over a much longer distance about the width of an atomic nucleus.

There are two ways you can analyze the residual nuclear force to see why this Yukawa potential shows up. The first way physicists formulated it was based on those pions I mentioned. Unlike the gluon, pions do have mass and decay over a short time (and distance), just like the W and Z bosons.

The second way to look at it, looking at the actual quarks, is to notice that the Yukawa potential is also the equation for a screened Coulomb potential. If you have a charged particle like a positive ion in a hot plasma, it will naturally collect oppositely-charged particles (electrons) around it, even though they’re too hot to bind into their usual orbitals. These electrons will partially balance out the positive charge as seen from farther away, so the Coulomb force it exerts will weaken faster with distance. If you do the math, this screening factor turns out to be an exponential decay, and you get the Yukawa potential again.

So, putting all this together, a more accurate version of the XKCD comic might be this:

“Of these four forces, there’s one we don’t really understand. It’s still gravity.”

[1] This sounds weird, but we can think of familiar examples. A ball rolling in a funnel (a straight-sided conical funnel, not the curved ones people sometimes roll coins in) will experience a constant force toward the center of the funnel. This is why the Cornell potential is also called the “funnel potential.”

[2] Although this isn’t quite right; springs exert more force as they are stretched, not the same amount.

About Alex R. Howe

I'm a full-time astrophysicist and a part-time science fiction writer.
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