# Is there an equation for the strong nuclear force?

The equation describing the force due to gravity is $$F = G \frac{m_1 m_2}{r^2}.$$ Similarly the force due to the electrostatic force is $$F = k \frac{q_1 q_2}{r^2}.$$

1. Is there a similar equation that describes the force due to the strong nuclear force?

2. What are the equivalent of masses/charges if there is?

3. Is it still inverse square or something more complicated?

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## 5 Answers

From the study of the spectrum of quarkonium (bound system of quark and antiquark) and the comparison with positronium one finds as potential for the strong force

$V(r) = - \frac{4}{3} \frac{\alpha_s(r) \hbar c}{r} + kr$

where the constant $k$ determines the field energy per unit length and is called string tension. For short distances this resembles the Coulomb law, while for large distances the $kr$ factor dominates (confinement). It is important to note that the coupling $\alpha_s$ also depends on the distance between the quarks.

This formula is valid and in agreement with theoretical predictions only for the quarkonium system and its typical energies and distances. For example charmonium: $r \approx 0,4 fm$. So it is not as universal as eg. the gravity law in Newtonian gravity.

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. +1 This potential makes more physical sense for quarks since it includes both the QED-like $-1/r$ and the confining $+kr$. – dbrane Apr 11 '11 at 20:48

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Nice. Of course, the "breaking of the flux tube" has no classical or semi-classical analogue, making this formulation better for hand waving than calculation. – dmckee♦ Apr 11 '11 at 21:19

No, there is none such equation. Reason is that these equations are highly classical and invalid in both relativistic (there is an action at a distance, incompatible with finite speed of light) and quantum mechanical regime (distances strong force is important at are quite microscopic). Also, strong force is confining, meaning you can't ever observe individual color charged particles (color is a property associated with strong force), so there can't really be a macroscopic equation for them.

You obviously need at least quantum mechanics to account for strong force, because distances are so tiny (on the scale of nucleus or smaller). But it turns out you need relativity too. The complete theory which incorporates both QM and relativity is called quantum field theory and individual forces are described by QFT Lagrangians which essentially tell you which particles interact with which other particles (e.g. photons with electrically charged particles, gluons with color charged particles, etc.). This is the fundamental theory and the electric force equation you described can be derived from it in classical (both non-QM and nonrelativistic) limit. As for gravitation law, that too can be derived but from different theory, namely general relativity.

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Thanks, I guess that the goal is to formulate gravity in the same language of quantum field theory? That is the goal of stringtheory and other such unification theories? – ergodicsum Apr 11 '11 at 18:21
@ergodicsum: yep, pretty much. (Either that, or formulate the standard model in the language of GR, or formulate both in some new theoretical framework yet to be discovered) – David Zaslavsky♦ Apr 11 '11 at 18:28
@ergodicsum: that would be the intuitive proposition, right. But it turns out gravity doesn't play well with QFT in the way other forces do. So the language will probably be of some other theory (e.g. string theory) from which both QFT and GR can be derived in some limits. – Marek Apr 11 '11 at 18:28

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Well, it's still true that at short distances, much shorter than a Fermi, and in the non-relativistic limit, the strong force is still governed by the Coulomb's law. It's not a terribly useful limit for the strong force but it is misleading to suggest that the strong force is something "entirely different". – Luboš Motl Apr 11 '11 at 18:38
@Lubos Motl thanks for your clarification. My intuition told me something like that should be true, but my intuition is often wrong :). – anna v Apr 11 '11 at 19:11
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At the level of quantum hadron dynamics (i.e. the level of nuclear physics, not the level of particle physics where the real strong force lives) one can talk about a Yukawa potential of the form

$$V(r) = - \frac{g^2}{4 \pi c^2} \frac{e^{-mr}}{r}$$

where $m$ is roughly the pion mass and $g$ is an effective coupling constant. To get the force related to this you would take the derivative in $r$.

This is a semi-classical approximation, but it is good enough that Walecka uses it breifly in his book.

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Let me add one obvious thing: There is an exact equation for the strong force. It is what Gross, Politzer and Wilczek got the Nobel prize for. It is called quantum chromodynamics (QCD). Google it or look it up in Wikipedia, and you can see the Lagrangian for QCD, and compare it to the Lagrangian for electrodynamics.

Of course, you could argue about the similarities and differences of a Lagrangian, and a force equation, such as your two examples.

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The strong force as seen in nuclear matter

The nuclear force, is now understood as a residual effect of the even more powerful strong force, or strong interaction, which is the attractive force that binds particles called quarks together, to form the nucleons themselves. This more powerful force is mediated by particles called gluons. Gluons hold quarks together with a force like that of electric charge, but of far greater power.

Marek is talking of the strong force that binds the quarks within the protons and neutrons. There are charges, called colored charges on the quarks, but protons and neutrons are color neutral. Nuclei are bound by the interplay between the residual strong force , the part that is not shielded by the color neutrality of the nucleons, and the electromagnetic force due to the charge of the protons. That also cannot be simply described. Various potentials are used to calculate nuclear interactions.

Simplicity and similarity of form for all forces comes not in the formalism of forces, but as Marek said the formalism of quantum field theory.

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