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To my knowledge, all field interactions are carried out through force-mediating particles. For example, electromagnetic interactions are carried out through exchanging photons. However, under the entry "Quantum Field Theory" in Wikipedia, it is written:

The notion of a force-mediating particle comes from perturbation theory, and does not make sense in the context of non-perturbative approaches to QFT, such as with bound states.

If this is true, then how are field interactions carried out in the context of non-perturbative approach to QFT? Besides, interactions among quarks, which are non-perturbative, are carried out through exchanging gluons. Isn't this a case of force-mediating particles in the context of non-perturbative approach?

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The answer is no, if you refer "gauge boson" as "force-mediating particle".

For example, $\phi ^4$ theory for scalar bosons don't need any other gauge bosons to carry the interaction. In fact, they can interact with themselves. See David Tong's lecture notes on QFT .

When it comes to the "strong" interaction regime, the perturbative approach does not work, because the higher order perturbation cannot be omitted. Under this circumstance, it's no longer useful to interpret the theory with "particle" (i.e. quantized field) to describe the interaction any more.

As far as I understand, one can of course say that QCD is mediated by gluons, but meanwhile there are many of them and they interact strongly with themselves, which does not look so easily described as the case for QED and photons.

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