For massive Proca fields with quartic self-interaction, the massless limit is not smooth: classical hyperbolicity loss persists at arbitrarily small mass, and the Vainshtein mechanism does not restore unitarity at scales comparable to the mass.
All-order Finiteness of the Higgs Boson Mass in the Dynamical Gauge-Higgs Unification
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In the dynamical gauge-Higgs unification, it is shown that the mass of the Higgs boson (4D scalar field) in U(1) gauge theory in $M^4 \times T^n$ ($n=1,2,3,...$) is finite to all order in perturbation theory as a consequence of the large gauge invariance. It is conjectured that the Higgs boson mass is finite in non-Abelian gauge theory in $M^4 \times S^1$, $M^4 \times (S^1/Z_2)$ and the Randall-Sundrum warped spacetime to all order in the rearranged perturbation theory where the large gauge invariance is maintained.
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Massless limit of massive self-interacting vector fields
For massive Proca fields with quartic self-interaction, the massless limit is not smooth: classical hyperbolicity loss persists at arbitrarily small mass, and the Vainshtein mechanism does not restore unitarity at scales comparable to the mass.