Relevant interactions in quantum field theory

[spaghetti3451]

For a ##\phi^{3}## quantum field theory, the interaction term is ##\displaystyle{\frac{g}{3!}\phi^{3}}##, where ##g## is the coupling constant.

The mass dimension of the coupling constant ##g## is ##1##, which means that ##\displaystyle{\frac{g}{E}}## is dimensionless.

Therefore, ##\displaystyle{\frac{g}{3!}\phi^{3}}## is a small pertubation at high energies ##E \gg g##, but a large perturbation at low energies ##E \ll g##.

Terms with this behavior are called relevant because they’re most relevant at low energies.

However, I do not understand why the interaction term is called relevant if we cannot use perturbation theory at low energies (where the term ##\displaystyle{\frac{g}{3!}\phi^{3}}## is a large pertubation). Is it because quantum field theory is only applicable in the relativistic limit, where ##E \gg g## and the perturbation is small ?

 

[Demystifier]

The philosophy behind this terminology is that all field theories are only effective theories, which means that they are valid only at large distances (low energies). If a contribution is relevant, it means that it is not negligible. But it does not mean that it is not sufficiently small for application of perturbation theory. For instance, ##\alpha=1/137## is small but not negligible.