Ambiguities and Subtleties in Fermion Mass Terms in Practical Quantum Field Theory

TL;DR

This review clarifies complex subtleties in fermion mass terms within quantum field theory, emphasizing the equivalence of Dirac and Majorana states and highlighting ambiguities in neutrino mass descriptions in the Standard Model.

Contribution

It provides a detailed pedagogical analysis of fermion mass structures, clarifying misconceptions and ambiguities often overlooked in standard textbooks.

Findings

Dirac fermion mass eigenstates are equivalent to two degenerate Majorana states.

Ambiguities exist in describing neutrino masses as Dirac or Majorana.

Terminology depends on broken SU(2) charges and is model-dependent.

Abstract

This is a review on structure of the fermion mass terms in quantum field theory, under the perspective of its practical applications in the real physics of Nature -- specifically, we discuss fermion mass structure in the Standard Model of high energy physics, which successfully describes fundamental physics up to the TeV scale. The review is meant to be pedagogical, with detailed mathematics presented beyond the level one can find any easily in the textbooks. The discussions, however, bring up important subtleties and ambiguities about the subject that may be less than well appreciated. In fact, the naive perspective of the nature and masses of fermions as one would easily drawn from the presentations of fermion fields and their equations of motion from a typical textbook on quantum field theory leads to some confusing or even wrong statements which we clarify here. In particular, we illustrate clearly that a Dirac fermion mass eigenstate is mathematically equivalent to two degenerated Majorana fermion mass eigenstates at least so long as the mass terms are concerned. There are further ambiguities and subtleties in the exact description of the eigenstate(s). Especially, for the case of neutrinos, the use of the Dirac or Majorana terminology may be mostly a matter of choice. The common usage of such terminology is rather based on the broken $SU(2)$ charges of the related Weyl spinors hence conventional and may not be unambiguously extended to cover more complicate models.