Effective Field Theory Amplitudes the On-Shell Way: Scalar and Vector Couplings to Gluons

TL;DR

The paper advances an on-shell program to derive tree-level EFT amplitudes for new SM-singlet scalars ($h$) and vectors ($Z'$) coupled to gluons, without resorting to EFT Lagrangians. By enforcing Lorentz symmetry, unitarity, and Bose statistics, it fixes the allowed kinematic structures and uses factorization to relate higher-point amplitudes to lower-point ones, enabling a direct counting of independent EFT operators from polynomials in Mandelstam invariants $s_{ij}$. The authors compute ${\\cal M}(h;gg)$, ${\\cal M}(h;ggg)$, ${\\cal M}(hh;gg)$, and ${\\cal M}(Z';ggg)$ (including massive and massless limits) and show how these amplitudes decompose into massless-vector plus scalar pieces in the high-energy limit, reproducing operator counts in line with Henning et al. The work demonstrates a powerful bridge between on-shell amplitudes and EFT operator structures, with clear implications for Higgs- and Z$'$-related phenomenology at the LHC and for systematic EFT counting at high dimensions.

Abstract

We use on-shell methods to calculate tree-level effective field theory (EFT) amplitudes, with no reference to the EFT operators. Lorentz symmetry, unitarity and Bose statistics determine the allowed kinematical structures. As a by-product, the number of independent EFT operators simply follows from the set of polynomials in the Mandelstam invariants, subject to kinematical constraints. We demonstrate this approach by calculating several amplitudes with a massive, SM-singlet, scalar ($h$) or vector ($Z^\prime$) particle coupled to gluons. Specifically, we calculate $hggg$, $hhgg$ and $Z^\prime ggg$ amplitudes, which are relevant for the LHC production and three-gluon decays of the massive particle. We then use the results to derive the massless-$Z^\prime$ amplitudes, and show how the massive amplitudes decompose into the massless-vector plus scalar amplitudes. Amplitudes with the gluons replaced by photons are straightforwardly obtained from the above.

Figures (3)

• Figure 1: Factorization of the $+++$ amplitude on $s_{12}$. The direction of the arrow indicates the direction of momentum flow.
• Figure 2: Possible factorizations of the $++-$ amplitude.
• Figure 3: Possible factorizations of the $2h \to 2g$ amplitude.