# Why there is no 3-body (or more generally $N$-body) fundamental force?

Fundamental forces are believed to be two body interactions. However, I found myself if there is no opportunity for a 3-body or more generally $N$-body "fundamental" force. Is there a proof that any multiparticle interactions being reducible to 2-body interactions of particles?

Comment: In other words, is every $N$-body interaction reducible to 2-body interaction?

• In a lesson I have been to about hartree fock methods they made no such claim, just wrote ... for higher order terms. Also, it might be good enough in many cases. – Emil Feb 15 '18 at 23:01
• I am not sure if you'd accept this as a counterargument to the 2-body hypothesis... but consider the case of particle-antiparticle pair production. How many particles are interacting there? I'd say it's hard to argue that this comes from 2-particle interactions. Then consider lagrangian interactions with higher n-point terms, like gluons in QCD. – secavara Feb 15 '18 at 23:08

1. Not all fundamental interactions are mediated by 2-body interactions. E.g. the fundamental strong force has a 3-body force that is responsible for the stability of the ${}^3{\rm He}$ nucleus. See also e.g. this related answer by Ron Maimon.

2. If the theory is a non-trivial integrable model, then all interactions are elastic 2-to-2 body scatterings in 2D (in pertinent variables).

• The Wikipedia article cites an article from 1967 as direct evidence of a 3-nucleon force in helium-3. But a posterior article as The Three-Nucleon Force Is Not Made by Nature says those three-body interactions are only theoretical artifacts. – juanrga Apr 13 '19 at 12:03
• @juanrga: Thanks. – Qmechanic May 7 '19 at 7:57

Fields have positive mass dimension, and local $n$-body interactions are suppressed by higher powers of the UV scale.

Indeed, in relativistic field theories in four dimensions the only renormalizable local 2-body interaction are scalar $\phi^4$ interactions, and the $A_\mu^4$ term hidden in the non-abelian Yang-Mills action. All other terms are local interactions of matter fields with Yang-Mills fields or scalars that lead to finite rage $n$-body interactions.

Note that non-abelian Yang-Mills fields not only lead to a Coulomb-like two-body interaction, but also to three and four-body forces (due to three and four gluon vertices). Indeed, at higher order in perturbation theory there are also higher n-body forces.

In non-relativistic theories the typical long range forces is mediated by a $U(1)$ gauge boson, which does not directly mediate n-body forces (although loop effects can mediate n-body forces). Finite range forces are subject to the power counting argument mentioned above. A standard example is nuclear physics, where the two-body interaction between neutrons and protons dominates, but three and four-body forces exist.

A beautiful non-relativistic system with some relevance to nuclear physics that can now be studied in cold atoms is the Efimov effect: A two-body force tuned to infinite scattering length (in a systems of bosons, or fermions with at least three degrees of freedom) requires a three-body force to be renormalizable.

That is a very good question.

I believe that complex interactions can be reduced to 2-body interactions due to the additivity of energy. This additivity is inherited to the driving forces. Ultimately, the total energy can be broken down to contributions between at most two particles.

For example, consider three particles in gravitational interaction. The potential energy is the sum of the 3 different 2-particle gravitational potentials.

From a modeling point of view, there are some exceptions: For example, there are "embedded atom models" that account for increased electron density of clusters of atoms, additionally to the pair interactions. But this is in a sense an averaging of pair interactions.

Fundamental n-body interactions would lead to very strange behavior. Assume that you need at least an A, B and C particle for an interaction. If you have many particles of A and B- they would not interact. But if you add 1 particle of C, they would start to interact, releasing or binding a large amount of energy.

• Well, what if energy were NOT additive? After all, we have non-extensive entropy, why not non-extensive energies or non-linearity in the additivity of energies, keeping it as a constant of motion? – riemannium Mar 10 '18 at 13:47