Sum Rules in the Standard Model Effective Field Theory from Helicity Amplitudes

TL;DR

This paper develops a framework of sum rules that connect low-energy SMEFT dimension-6 coefficients to high-energy observables through dispersion relations applied to forward elastic amplitudes, using a massless, helicity-based approach. By classifying forward massless elastic SMEFT amplitudes, the authors systematically derive sum rules for Higgs-Higgs, Higgs-fermion, and fermion-fermion channels, linking operator coefficients to measurable cross-section differences and boundary terms. They illustrate the method with explicit mappings to SMEFT operators (such as O_H and O_T) and count the resulting independent relations (two for Higgs-Higgs, seven for each fermion family, and twenty for fermion-fermion sectors), comparing to prior results and highlighting new insights. The paper also discusses robustness against SM contributions, custodial symmetries that suppress certain amplitudes, and the boundary term, before applying the framework to benchmark UV completions (doubly charged scalars, triplets, Beautiful Mirror, and Zbb custodial-symmetry models) to demonstrate how sum rules constrain UV properties and EFT matching, with implications for precision measurements and direct searches.

Abstract

The dispersion relation of an elastic 4-point amplitude in the forward direction leads to a sum rule that connects the low energy amplitude to the high energy observables. We perform a classification of these sum rules based on massless helicity amplitudes. With this classification, we are able to systematically write down the sum rules for the dimension-6 operators of the Standard Model Effective Field Theory (SMEFT), some of which are absent in previous literatures. These sum rules offer distinct insights on the relations between the operator coefficients in the EFT and the properties of the full theory that generates them. Their applicability goes beyond tree level, and in some cases can be used as a practical method of computing the one loop contributions to low energy observables. They also provide an interesting perspective for understanding the custodial symmetries of the SM Higgs and fermion sectors.

Figures (6)

• Figure 1: Possible factorization channels of $\mathcal{A}^{[2]}_4$ that contain a $\mathcal{A}(V^+V^+V^+)$ part. The superscript of a particle indicates the sign of its helicity, with the all-in/all-out convention. The diagrams with $\mathcal{A}(V^-V^-V^-)$ can be obtained by flipping all the helicities. The red dot denotes an insertion of a dimension-6 operator. None of the processes are elastic in the helicity basis. Note in particular that $\mathcal{A}(V^-V^-V^+V^+)$ can not be generated by only one insertion of dimension-6 operators.
• Figure 2: A schematic plot on the interplay between precision measurements and direct searches. For simplicity, we assume only two new particles $X_1$ and $X_2$ with masses $M_1$ and $M_2$ and some universal couplings to SM. They each contribute to one of the cross sections in the sum rule, with $\sigma(ab\to X_1)$ and $\sigma(a\bar{b} \to X_2)$. The symmetry in \ref{['eq:symab']} corresponds to the diagonal line, where the contribution to $\mathcal{A}(ab\to ab)$ from dimension-6 operators vanishes, while the plus (minus) sign denotes the region in which this contribution is positive (negative). Contributions to $\mathcal{A}(ab\to ab)$ from dimension-8 operators are proportional to $\frac{1}{M^4_1} + \frac{1}{M^4_2}$, as illustrated by the orange contours.
• Figure 3: The one-loop contributions to $\mathcal{A}^{[2]} (\phi^+ \phi^0 \to \phi^+ \phi^0)$ from the BM model in \ref{['eq:Lbm']} proportional to $y^4_R$ (left) and $y^4_L$ (right). All external particles are going in.
• Figure 4: The one-loop contributions to $\mathcal{A}^{[2]} (\phi^+ \phi^0 \to \phi^+ \phi^0)$ from the BM model in \ref{['eq:Lbm']} that are proportional to $y^2_t y^2_L$. The coupling of each vertex (up to some common overall phase) is also labelled. All external particles are going in.
• Figure 5: The diagrams for the $2\to2$ cross sections corresponding to the amplitudes in \ref{['fig:bmT3']} (via the optical theorem). The two diagrams on the left contribute to $\sigma(\phi^+ \phi^0 \to \overline{\hat{B}}_R t_R)$. The two diagrams on the right contribute to $\sigma(\phi^+ \phi^{0*} \to t_L \overline{b}_L)$, but only the interference term is proportional to $y^2_t y^2_L$.