Let's say we describe an unstable particle using perturbation theory. Then we have a non-zero decay width, which we say $\Gamma$. Now, if we define mass to be the pole of the propagator, we get $$ \mu^2=m^2-\Re(\Gamma) $$ which is sometimes referred to as the Real(Breit Wigner) mass, where we still have the $\Im(\Gamma)$ left in the propagator, which gives the decay shape. This scheme defines the mass for most of the physical particles, like $ Z,W$, etc. But from twoloop corrections, this definition suffers from gauge dependence. What I guess, if we take the entire zero of the pole, we do get $\mu=m-i\Gamma/2$ as mass (complex scheme), and by definition, gauge invariant and physical, but can not describe the resonance shape. But in the real scheme, we do not take the entire zero hence we get gauge variance. As some literature like this one
The loop-corrected Higgs boson masses are defined through the complex poles of the propagator matrix which are evaluated by using an iterative method. While this method gives precise values of the complex poles, it mixes the contributions of different orders of perturbation theory and therefore introduces a dependence on the gauge parameter.
I do not understand which mass they are talking about. If they are taking about complex mass, how can it be gauge-dependent? Precisely, why from one-loop, the gauge dependence does not start (It starts from two-loop)?
