All one-loop scalar vertices in the effective potential approach

TL;DR

This paper derives a comprehensive analytic formula for all derivatives of the one-loop effective potential in renormalizable theories, enabling calculation of scalar vertex functions with any number of external legs.

Contribution

It provides the first general analytic expression for all derivatives of the one-loop effective potential with respect to scalar fields, applicable to theories with multiple scalars, fermions, and gauge bosons.

Findings

Derived a general formula for all one-loop scalar vertices.

Applied the formula to scalar singlet extensions of the Standard Model.

Calculated dominant contributions to scalar triple couplings.

Abstract

Using the one-loop Coleman-Weinberg effective potential, we derive a general analytic expression for all the derivatives of the effective potential with respect to any number of classical scalar fields. The result is valid for a renormalisable theory in four dimensions with any number of scalars, fermions or gauge bosons. This result corresponds to the zero-external momentum contribution to a general one-loop diagram with N scalar external legs. We illustrate the use of the general result in two simple scalar singlet extensions of the Standard Model, to obtain the dominant contributions to the triple couplings of light scalar particles under the zero external momentum approximation