system
Given the following post, respond as if you're commenting on Reddit:
human
What is the exact relation between mass and electromagnetic charge in the scope of modern QFT?

Years ago I actually had several courses on GR and QFT (alas I forget most of it over the years) but why is it that all electromagnetically charged particles are massive. Is this a direct consequence of the symmetry breaking introduced by the Higgs field?
I speak Lagrangians and Hamiltonians, you don't have to be gentle.
gpt
The mass and electric charge of a given particle* are independent numbers, arising from independent considerations. In the context of QFT, an elementary particle is something that transforms under an irreducible representation of the Poincare group. Such a representation turns out to be completely specified by two numbers: a non-negative real number (mass), and a non-negative half-integer (spin). Electric charge, on the other hand, is a number measuring how much a field's complex phase changes under a certain U(1) transformation, which is an internal symmetry not related to the Poincare group.

When you write the Lagrangian for a general particle, you include the mass and electric charge as free parameters (that are then fixed by experiment). In principle you could make the mass zero and charge non-zero and nothing would go wrong. It just happens that in our universe there doesn't seem to be any massless charged particles. We can be quite sure of this - if such a particle existed, it would be very easy to produce and detect, and massive charged particles like the electron would no longer be stable.

\*Although interestingly, the masses of particles in our universe depend crucially on the Higgs being uncharged. The photon is massless precisely because the Higgs is uncharged, and the Yukawa interactions that give rise to the masses of other particles would not be possible if the Higgs were charged. But there is no relation between a particle's mass and *its own* charge.