How can the strong force, which is conservative, not follow the inverse square law?

Question

In terms, which someone with a background in chemical physics & quantum chemistry might understand, what is the evidence that the strong force, across whatever its range is, follows something other than an inverse square law?

a) Specifically, as when the strong force is is said to be "~137" times the strength of the electromagnetic force, and the "~137" factor derives from the expression for the fine structure constant, which 1) explicitly is a function of electric charge squared, and 2) explicitly comprises the constant for the electromagnetic inverse law expression.

b) All fundamental are conservative and the only $1/r^n$ force function that is conservative is the $1/r^2$ force function, which is the inverse square law.

c) Do not a) and b) imply that over whatever range the strong force occurs that it obeys the inverse force law, and as regardless of charges of interacting particles, the strong force is attractive, that the strong force then is proportionate to (-)absolute value of the product of the electric charge of the interacting particles?

Answer 1

$1/r^2$ is not the only conservative force

Maybe your confusion stems from Bertrand's theorem, which states that $f(r) \propto 1/r^2$ and $f(r) \propto r$ are the only central forces for which all bound orbits are closed.

In general every spherically symmetric central force is conservative.

Yukawa interaction between nucleons

Even if the fundamental strong interaction happens between quarks and is mediated by gluons, also nucleons (protons and neutrons) feel the strong interaction, despite being color neutral. An analogy could be the Van der Waals force, which is electromagnetic, but can happen between globally neutral molecules.

Strong interaction between nucleons can described as if mediated by pions, which (unlike the photon or the gluon) are massive particles. An interaction mediate by massive particles can be described by the Yukawa potential

$$V(r) \propto \frac{e^{-r /\lambda}}{r}$$

where $\lambda = {\hbar \over mc}$ is the De Broglie wavelength of the pion (m is the mass of the pion). This is a generalization of the inverse square law. Indeed, if you put $m=0$, you obtain the electromagnetic potential, which is correct, because the photon is massless. Instead, if you use the mass of the pion, you see that the potential goes to zero way faster than the $1/r$ electromagnetic potential, hence the short range.

Strong interaction between quarks

The interaction between quarks is different. It is mediated by gluons, which are massless and therefore should be long range, but we need to consider the effects of asymptotic freedom and confinement that make the range of the interaction very short.

The bottom line is that the strong interaction at its most fundamental level must be treated with the QCD formalism and cannot be described just as a classical force $f(r)$.