Effective theory calculation of resonant high-energy scattering

TL;DR

This work develops an effective field theory framework for resonant high-energy scattering involving unstable particles by exploiting the hierarchy $\Gamma\ll M$ to organize calculations in a controlled expansion in $g^2$ and $\Gamma/M$. The authors construct a heavy-soft effective theory for the unstable scalar (HSET) and soft-collinear EFT (SCET), introduce resonant production/decay operators, and perform explicit matching to the full theory to obtain the leading and next-to-leading order line shapes in a toy abelian model. They demonstrate gauge invariance at each step and compute the NLO line shape, including the two-loop hard self-energy contributions ($\Delta^{(1)}$, $\Delta^{(2)}$) and one-loop production-decay vertex corrections ($C^{(1)}$), along with the forward-scattering amplitude decomposed into hard, soft, and collinear parts. The results show a Breit-Wigner-like peak at LO that acquires sizable, calculable NLO corrections, with manageable scheme dependencies and potential NNLO extensions outlined, illustrating the EFT's power to systematically improve unstable-particle predictions and preserve gauge invariance. The approach provides a scalable path toward more realistic non-abelian and multi-resonance scenarios, including pair production and jet-associated final states.

Abstract

Tests of the standard model and its hypothetical extensions require precise theoretical predictions for processes involving massive, unstable particles. It is well-known that ordinary weak-coupling perturbation theory breaks down due to intermediate singular propagators. Various pragmatic approaches have been developed to deal with this difficulty. In this paper we construct an effective field theory for resonant processes utilizing the hierarchy of scales between the mass of the unstable particle, M, and its width, Gamma. The effective theory allows calculations to be systematically arranged into a series in g^2 and Gamma/M, and preserves gauge invariance in every step. We demonstrate the applicability of this method by calculating explicitly the inclusive line shape of a scalar resonance in an abelian gauge-Yukawa model at next-to-leading order in Gamma/M and the weak couplings. We also discuss the extension to next-to-next-to-leading order and compute an interesting subset of these corrections.

Figures (12)

• Figure 1: Reduced diagram topologies in $2\to2$ scattering near resonance. Left: resonant scattering. Right: non-resonant scattering. See text for explanations.
• Figure 2: Left panel: line shape (in $\hbox{GeV}^{-2}$) in the pole (solid) and $\overline{\rm MS}$ scheme (dashed) as a function of the center-of-mass energy (in GeV). Right panel: Leading-order line shape (in $\hbox{GeV}^{-2}$) as a function of center-of-mass energy (in GeV) in the effective theory (solid) and the cross section off resonance in the full theory (dashed). The thick grey curve shows the leading-order line-shape with the two curves matched appropriately.
• Figure 3: Hard contributions to the next-to-leading order amplitude. Left: Insertion of $[\Delta^{(1)}]^2/4$ and $-\hat{M}\Delta^{(2)}$. Right: Insertion of $C^{(1)}$, the symmetric diagram is understood.
• Figure 4: Soft contributions to the next-to-leading order amplitude. A diagram with a soft correction at the decay vertex is understood.
• Figure 5: Contributions to the next-to-leading order amplitude involving collinear photons.