Making sense of binding energy: how to reconcile a potential with mediators?

Question

I recently read that stable nuclei are lighter than the sum of their constituents. For instance, a helium nucleus is lighter than two neutrons and two protons, and hence it is not energetically favorable for it to undergo fission, making it stable.

For me, this makes sense when we thing about the potential energy for the interaction between nucleons. Since the nuclear force is attractive and follows a Yukawa potential, I expect it to have a potential well, and thus for the energy of the bound state to be lower. So far so good.

Now let us consider a different scenario. The proton is (very loosely speaking) composed of three quarks. Now these three quarks add to a mass that is much smaller than the proton mass, and the mass excess in the proton is usually attributed to the energy it takes to keep the quarks together. So the resulting composite object is heavier than its constituents.

I'm curious about how can we reconcile these two points of view. From a very naive point of view, I could expect the helium nucleus to be heavier than four nucleons, because we also need to account for the pions that keep the nucleons bound. In the case of the proton, maybe the energy of the quarks kept apart is much larger (due to confinement and the strong force growing with distance), so it is still energetically unfavorable for them to split.

In a more objective sense: why don't pion masses (for example) contribute to the mass of helium so that it end up being heavier than four nucleons? How can we understand the helium nucleus being lighter than four nucleons from a "mediator particle" point of view?

Answer 1

I'm curious about how can we reconcile these two points of view

This is actually a very interesting question that I had once asked when learning nuclear physics. The answer is that the residual strong force (a.k.a. the nuclear force, the thing that holds the nucleons together) works very differently from the regular strong force (a.k.a. the color force, the things that holds hadrons together).

The nuclear force works a little like electromagnetism and (Newtonian) gravity. Things trapped in gravity wells have negative total mechanical energy with a reference point set at infinity; if you supply enough energy to make the TME positive, the “bond” between attractor and attractee is broke and one component can be separated from the other to infinity. This is because the gravitational (and nuclear) force decay over distance to near-zero, so a finite amount of binding energy exists for such systems.

If the binding energy of the products in a fission reaction is less than the binding energy of the reactants, then that energy has to be released somehow during the reaction, but if the reverse is true and the product energy is greater than the reactant energy, then that energy must be externally supplied to cause the reaction to happen.

The color force works very differently. I will preface this by saying that I have only done cursory studies of quantum chromodynamics (QCD, the study of the color force) and do not directly specialize in quantum mechanics. The best explanation I’ve heard for the color force’s properties is that the force doesn’t decay over distance; in fact it can even be seen to increase over distance in some cases. As a result, when trying to separate two quarks to infinity, the binding energy increases as distance goes up from some minimum, and eventually, the binding energy is so great that it is more energetically favorable to do energy-to-mass conversion of binding energy into a new pair of quarks that each bind to the two you’re trying to pull apart. This is the case for all quarks, and the property is known as color confinement: quarks with color charge are never observed on their own, because it would require infinite energy to break the binding.

Incidentally that’s how particle accelerators work: if you throw two protons at each other with high enough energy, then their mass-energy can get converted into new quarks - often much-heavier and much more-exotic particles.

In other words, comparing the nuclear and color forces is a category error. They are manifestations of the same property of QCD, but they work in completely different ways and trying to understand quark binding in terms of nuclear binding or vice versa will not make any sense.