Spontaneous symmetry breaking from an on-shell perspective

Abstract

We show how the well known patterns of masses and interactions that arise from spontaneous symmetry breaking can be determined from an entirely on-shell perspective, that is, without reference to Lagrangians, gauge symmetries, or fields acquiring a vacuum expectation value. To do this, we review how consistent factorization of 2 → 2 tree level scattering can lead to the familiar structures of Yang-Mills theories, and extend this to find structures of Yukawa theories. Considering only spins-0, 1/2 and 1 particles, we construct all the allowed on-shell UV amplitudes under a symmetry group G, and consider all the possible IR amplitudes. By demanding that on-shell IR amplitudes match onto on-shell UV amplitudes in the high energy limit, we reproduce the Higgs mechanism and generate masses for spins-1/2 and 1, find that there is a subgroup H ⊆ G in the IR, and other interesting relations. To highlight the results, we show the breaking pattern of the Standard Model U(1)EM ⊂ SU(2)L × U(1)Y, along with the generation of the masses and interactions of the particles.