Geometry of the Scalar Sector

Abstract

The $S$-matrix of a quantum field theory is unchanged by field redefinitions, and so only depends on geometric quantities such as the curvature of field space. Whether the Higgs multiplet transforms linearly or non-linearly under electroweak symmetry is a subtle question since one can make a coordinate change to convert a field that transforms linearly into one that transforms non-linearly. Renormalizability of the Standard Model (SM) does not depend on the choice of scalar fields or whether the scalar fields transform linearly or non-linearly under the gauge group, but only on the geometric requirement that the scalar field manifold $\mathcal{M}$ is flat. We explicitly compute the one-loop correction to scalar scattering in the SM written in non-linear Callan-Coleman-Wess-Zumino (CCWZ) form, where it has an infinite series of higher dimensional operators, and show that the $S$-matrix is finite. Standard Model Effective Field Theory (SMEFT) and Higgs Effective Field Theory (HEFT) have curved $\mathcal{M}$, since they parametrize deviations from the flat SM case. We show that the HEFT Lagrangian can be written in SMEFT form if and only if $\mathcal{M}$ has a $SU(2{)}_L\times U(1{)}_Y$ invariant fixed point. Experimental observables in HEFT depend on local geometric invariants of $\mathcal{M}$ such as sectional curvatures, which are of order $1/\hat{I}{›}^2$, where $\hat{I}›$ is the EFT scale. We give explicit expressions for these quantities in terms of the structure constants for a general $\mathcal{G}\to \mathcal{H}$ symmetry breaking pattern. (Full abstract in pdf)