# Is there some special case where a fermion can mediate a force?

Looking at the comments of this questions Does the gravitino contribute to the gravitational interaction? and even considering that the answers here in this other question Why are all force particles bosons? do explain why a force carrier needs to be bosonic, I still wonder if there are some particular cases where a fermion similar to a gaugino could mediate a force.

A case in mind, that could avoid the issues on angular momentum conservation, is zero-range interactions ("contact" interactions). Still, I can not see how such beast could be described with a Lorentz invariant Lagrangian.

I think this would be tricky, since any force mediator (at least from conventional thinking) must have a three-valent vertex, two of which are the charged object and one of them is the force carrier. If the force carrier is a fermion, I don't think this combination can be Lorentz invariant (spin zero combination).

• Yep, it seems that a Lorentz invariant single term in the lagrangian is not possible. And still I am a bit uneasy about a definite no. – arivero Oct 19 '14 at 19:31
• @arivero, I understand and for the same reason, my answer is measured. – Siva Oct 20 '14 at 2:03

It depends on your definition of force. Force means a change in momentum, ~dp/dt , so any change in momentum in a Feynman diagram is a force. For example this diagram for compton scattering

says yes.

If one is talking of gauge theories and exchanged bosons , because those are the ones that build up the three, electromagnetic, weak, strong ( maybe four if gravity is unified) forces , then no, by the construction.

It depends what you would accept as a valid answer but two bosons could feel a small force resulting from a virtual fermion loop for example. Does that count?

• I wonder if the fermion loop is equivalent to a composite bosonic operator $\langle \bar{\psi} \psi \rangle$ mediating the force. – Siva Oct 22 '14 at 15:25
• Hmm that looks as an exchange of a meson, does it? – arivero Oct 22 '14 at 16:35
• Oops, I didn't mean to put a vev $\langle \rangle$ above. @arivero: Yes, that's what I was wondering -- the composite force mediating degree-of-freedom is again bosonic, now! – Siva Oct 23 '14 at 0:02
• I would expect that some regularisation technique for the four-leg fermi interaction could be using this kind of fermion loops. But by becoming a boson, they already allow to substitute the zero-range by a finite range. A regularisation keeping the property of "zero range" would be a lot more interesting. – arivero Apr 11 '16 at 10:56

Besides the need of having a Lorentz invariant term, there is another "against" that comes only when we consider classic fields. If we restore units, a Planck constant appears in the kinetic term of the fermion field, telling that it will disappear in the limit $\hbar \to 0$. But of course this "non-go" argument is bypassed by any macroscopic electrical current.