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Constraints on the symmetric mass generation paradigm for lattice chiral gauge theories

Within the symmetric mass generation (SMG) approach to the construction of lattice chiral gauge theories, one attempts to use interactions to render mirror fermions massive without symmetry breaking, thus obtaining the desired chiral massless spectrum. If successful, the gauge field can be turned on, and thus a chiral gauge theory can be constructed in the phase in which SMG takes place. In this paper we argue that the zeros that often replace the mirror poles of fermion two-point functions in an SMG phase should be ``kinematical'' singularities. We conjecture that the SMG interactions generate opposite-chirality bound states, which combine with the gapped elementary mirror states to form massive Dirac fermions. The propagator zeros can then be avoided by choosing an appropriate set of interpolating fields that contains both elementary and composite fields. This allows us to apply general constraints on the existence of a chiral fermion spectrum which are valid in the presence of (non-gauge) interactions of arbitrary strength, including in any SMG phase. Using a suitably constructed one-particle lattice hamiltonian describing the fermion spectrum, we formulate a generalized no-go theorem which establishes the conditions for the applicability of the Nielsen-Ninomiya theorem to this hamiltonian. If these conditions are satisfied, the massless fermion spectrum must be vector-like. We add some general observations on the strong coupling limit of SMG models. We also elaborate on the qualitative differences between four-dimensional and two-dimensional theories that limit the lessons that can be drawn from two-dimensional models. Finally, we compile a list of open questions which must be addressed in any SMG model in order to determine whether or not it is subject to the generalized no-go theorem.