# What is an accidental symmetry?

Wikipedia describes an accidental symmetry as

a symmetry which is present in a renormalizable theory only because the terms which break it have too high a dimension to appear in the Lagrangian

but I don't understand what this means: are non-accidental symmetries always present, no matter the dimension of terms that I allow in the renormalized Lagrangian? Also, I know that some examples of accidental vs non-accidental symmetries are the lepton number vs the electric charge, but I fail to see why the first one would be accidental and the second one wouldn't.

• Accidental is a synonym for "small explicit" breaking. A gauge symmetry (such as the one associated with the electric charge) is unforgiving, however: it has to be exact. Oct 20, 2021 at 13:07

Typically, when you want to require a symmetry you choose to restrict the allowed terms in your Lagrangian to a smaller subset, this subset being the terms that are singlets under the symmetry group. For example, consider the Lagrangian of two fields $$\phi_1$$ and $$\phi_2$$. If it depends only on the combinations $$\phi_1^2+\phi_2^2\,,\quad(\partial\phi_1)^2+(\partial\phi_2)^2\,,$$ then there is an $$\mathrm{SO}(2)$$ symmetry. This statement amounts to saying that terms such as $$\phi^2_1$$ or $$\phi_1\phi_2$$ are absent.
An example of this is QED which is $$\mathsf{P}$$ invariant but just because you can't write any $$\mathsf{P}$$-breaking term that has dimension below four which is made only of electrons and photons.