Nielsen identities of the SM and the definition of mass

In a generic gauge theory the gauge parameter dependence of individual Green functions is controlled by the Nielsen identities, which originate from an enlarged BRST symmetry. We give a practical introduction to the Nielsen identities of the standard model (SM) and to their renormalization and illustrate the power of this elegant formalism in the case of the problem of the definition of mass. We prove to all orders in perturbation theory the gauge independence of the complex pole of the propagator for all physical fields of the SM, in the most general case with mixing and $\mathrm{CP}$ violation. At the amplitude level, the formalism provides an intuitive and general understanding of the gauge recombinations which makes it particularly useful at higher orders. We also include in an Appendix the explicit expressions for the fermionic two-point functions in a generic $R_{\xi }$ gauge.