On the relevance of fermion loops for $W^+W^-$ scattering

TL;DR

This paper investigates the impact of fermion loops, especially top and bottom quarks, on $W^+W^-$ scattering within effective theories, revealing scenarios where fermion contributions can be significant or dominant at TeV energies.

Contribution

It provides the first detailed calculation of fermion-loop effects on $W^+W^-$ scattering in HEFT, highlighting their potential importance relative to boson loops at intermediate energies.

Findings

Fermion loops can produce significant imaginary parts in scattering amplitudes.

Boson loops generally dominate, but fermion loops are relevant in certain parameter regions.

Results are applicable to the Minimal Composite Higgs Model and similar theories.

Abstract

We study the one-loop corrections to Vector Boson Scattering (in particular $W^+W^-$ elastic scattering) within the framework of effective theories. Re-scattering via intermediate electroweak would-be-Goldstone bosons dominate at high energies, as the corresponding loop diagrams with these intermediate bosons scale like $\mathcal{O}(s^2/v^4)$ in the chiral effective counting. In the present article, we focus our attention on fermion-loop corrections which scale like $\mathcal{O}(M_{\rm Fer}^2 s/v^4)$ in the Higgs Effective Field Theory (HEFT). Although this dependency is formally suppressed for $s\to\infty$ with respect to that from boson loops, the large top mass can lead to a numerical competition between fermion and boson loops at intermediate energies of the order of a few TeV. For the study of these fermion effects we have calculated the imaginary part induced by loops of top and bottom quarks in $W^+ W^-\to W^+W^-$ elastic scattering and compared it to the loop contributions from purely bosonic loops. We have examined the dependence of both amplitudes on the effective couplings, allowing an $\mathcal{O}(10 \%)$ deviation from the SM. In some cases, boson loops dominate over top and bottom corrections, as expected. However, we find that there are regions in the space of effective parameters that yield a significant -- and even dominant -- imaginary contribution from fermion loops. In addition to our conclusions for the general HEFT, we also provide analyses particularized to some benchmark points in the $SO(5)/SO(4)$ Minimal Composite Higgs Model.