On one-loop corrections in the standard model effective field theory; the $Γ(h \rightarrow γ\, γ)$ case

TL;DR

This work computes the complete one-loop finite terms for $h \to \gamma\gamma$ arising from dimension-six operators in the linear SMEFT, using a Background Field Method in $R_\xi$ gauge. It demonstrates that vev running modifies $\log\mu$ terms, reveals ghost interactions sourced by SMEFT via gauge fixing, and shows pure finite one-loop contributions proportional to operators like $O_{HW}$ that are not captured by RG analyses alone. The results establish that all three Wilson coefficients $C_{HB}$, $C_{HW}$, and $C_{HWB}$ contribute at one loop beyond RG logs, and that gauge- and renormalization-scheme subtleties must be carefully handled to obtain a finite, gauge-invariant amplitude. Consequently, RG-based expectations are insufficient for precise SMEFT predictions at next-to-leading order, underscoring the need for full one-loop SMEFT calculations to interpret Higgs data and constrain new physics accurately, especially for $\Lambda$ in the TeV range.$

Abstract

We calculate one loop contributions to $Γ(h \rightarrow γ\, γ)$ from higher dimensional operators, in the Standard Model Effective Field Theory (SMEFT). Some technical challenges related to determining Electroweak one loop "finite terms" are discussed and overcome. Although we restrict our attention to $Γ(h \rightarrow γ\, γ)$, several developments we report have broad implications. Firstly, the running of the vacuum expectation value modifies the $\log(μ)$ dependence of processes in a manner that is not captured in some past SMEFT Renormalization Group (RG) calculations. Secondly, higher dimensional operators can source ghost interactions in $R_ξ$ gauges due to a modified gauge fixing procedure. Lastly, higher dimensional operators can contribute with pure finite terms at one loop in a manner that is not anticipated in a RG analysis. These results cast recent speculation on the nature of one loop corrections in the SMEFT in an entirely new light.

Figures (4)

• Figure 1: Diagrams contributing to $H \rightarrow g g$ decay from $\mathcal{O}_{GG}$.
• Figure 2: One loop diagrams contributing to $H \rightarrow \gamma \gamma$ decay through interactions in $\mathcal{L}_{eff}(v^0,v)$. Arrows on propagators indicate charge flow. The insertion of the Effective Lagrangian in the diagram is indicated with a black square. Diagrams (f-i) have mirror diagrams that are not shown, where the photons are exchanged in a less trivial manner than in diagrams (a-e). Diagram (j) corresponds to the insertion of the one loop counterterms present in $\mathcal{L}_{eff}$. Here, $h$ is the Higgs field, $\phi_{0,\pm}$ are the Goldstone bosons and $W$, $Z$ and $\gamma$ are the gauge fields.
• Figure 3: Diagrams contributing to $H \rightarrow \gamma \gamma$ decay due to $\mathcal{L}_{eff}(v^2)$. The divergent terms exactly cancel in this class of contributions, as expected. $u^{\pm}$ are ghost fields.
• Figure 5: Diagrams contributing to the Higgs self energy.