# Why don't we call the fermions in the standard model force carriers?

Maybe this is a chicken-and-egg problem, but couldn't we call all the bosons fundamental and treat the fermions as force carriers between them?

EDIT: After all we never see the asymptotic states of tree level Feynman diagrams. We can only measure an electron if it interacts with our measuring apparatus again, producing a photon...

EDIT2: Still not satisfied with the answers. Fermions always appear up to quadratic order in the Lagrangian of the standardmodel. We could easily integrate them out in the Path integral and describe our world solely with interacting bosons.

If we identify a force as a scattering process, i.e. with a mediator of some interactions, then this need not be a vector boson of course. One can speak of "Higgs" force for instance if the process under consideration is mediated by the the Higgs (which is a scalar). There are also numerous cases where the interaction is mediated by a fermion. Therefore, it's not an either/or situation as your question seems to imply.

Let's take the following toy model:

\begin{equation} \mathcal{L}_{toy} = \,(\lambda\,\overline{X}\,F\,S +\lambda^\star\overline{F}\,X\,S) + M_X \overline{X} X+ M_F \overline{F} F +\frac{1}{2}M_S^2 S^2\end{equation} Where $X$ and $F$ are fermions, and $S$ a real scalar. All the fields are assumed to be pure gauge singlets.

Assuming the hierarch $M_F\gg M_X \gtrsim M_S$, then this lagrangian leads to the interactions $\bar X X \to S S$, which are precisely mediated by the heavy fermion $F$. The cross-section scales as (at energies $\sim M_X$): $$\sigma \sim \frac{|\lambda|^4}{M_F^2}\,.$$

(side note: we can imagine that this lagrangian is a low energy manifestation of some more complete model at higher energies -which would explain the absence of some terms-. The interaction above could be for instance a process leading to the production of dark matter in the early Universe).

With the same toy model, for different hierarchies, we can have the 'standard' interactions mediated by the scalar $S$.

Such kind of processes are either absent of suppressed in the SM, but beyond it they are quite common.

The situation is not symmetric at all: This diagram describes a force between two fermions, but a diagram such as just doesn't exist (in the Standard Model).

Fermions can in fact mediate a force between bosons, like in: Such diagrams are highly suppressed loop diagrams though, and the one above would after renormalization be seen as just one contribution to the resummed 4-boson vertex.

• what about compton scattering? – anna v Apr 8 '16 at 17:06
• we never see the asymptotic states of Feynman diagrams. We can only measure an electron if it interacts with our measuring apparatus again, producing a photon... – tonydo Apr 11 '16 at 11:05
• @tonydo ...which will then interact with the electrons and protons in the measuring apparatus, which will then produce/scatter a photon which will interact with the electrons and protons in your eye... What we "see" is dependent on the apparatus/object separation. – OON May 4 '16 at 16:54

I think that this is more about the historical construction of the theory than about the actual interactions. In a lagrangian, two fields A, B interact when there is a product term of both such as AB. So, I see no real fundamental distinction there, even with more complicated expressions.

But when one introduces the interaction bosons, it's by the mean of gauge theories. Basically we start with "a particle" (a field describing this kind of particle), for which we impose a local symmetry. For the Lagrangian to be invariant, we must introduce new fields, which are those of the interaction bosons. (That construction leads to the term for the photon, the Z, the W's)

Why don't we call the fermions in the standard model force carriers?

Because the Standard Model is what it is. By the way, I think it's far less complete than people make it out to be, and that it comes with some unfortunate baggage. Consider for example an electron and a positron interacting. People say they interact via virtual-photon force carriers: CCASA image by Manticorp/Rubber Duck, see Wikipedia

But see anna's answer here and note this: "virtual particles exist only in the mathematics of the model". The electron and the positron are not throwing photons at one another. The only particles present are the electron and the positron. So in truth these fermions are the "force carriers".

Maybe this is a chicken-and-egg problem, but couldn't we call all the bosons fundamental and treat the fermions as force carriers between them?

You could say that photons are "more fundamental" than other particles, in that you can reduce fermions down to photons: Image credit CSIRO, see The Big Bang & the Standard Model of the Universe

But photons interact with photons, and with electrons, and electrons interact with electrons etc, and neutrinos are more like photons than electrons. So it wouldn't be right to treat the fermions as force carriers between bosons.

EDIT: After all we never see the asymptotic states of tree level Feynman diagrams. We can only measure an electron if it interacts with our measuring apparatus again, producing a photon...

Don't forget electron and positron tracks in a magnetic field. Or that Feynman diagrams shouldn't be taken literally.

EDIT2: Still not satisfied with the answers. Fermions always appear up to quadratic order in the Lagrangian of the Standard Model. We could easily integrate them out in the Path integral and describe our world solely with interacting bosons.

Sounds like a plan. After all, what does pair production really do? Remember that in atomic orbitals, electrons "exist as standing waves". What do you think they exist as outside of an orbital? What with electron spin and magnetic moment and the Einstein de Haas effect and the Poynting vector, and the wave nature of matter and the wave in the box, you don't have to be the Brain of Britain to work out that the electron is just a 511keV photon in a closed chiral spinor path. And that electrons and positrons interact the way that they do because of the way that they are: