Logarithmic expansion of electroweak corrections to four-fermion processes in the TeV region

TL;DR

The paper analyzes high-energy electroweak corrections to four-fermion neutral-current processes within the Standard Model, using Z-peak measurements as input to define gauge-invariant one-loop quantities. It disentangles universal renormalization-group running effects from Sudakov-type non-universal logarithms, showing the latter dominate the TeV energy regime and can reach about 10% in size near 1 TeV. Explicit asymptotic expressions for cross sections and asymmetries are derived, and an simple empirical parametrization is proposed to describe TeV-region observables, calibrated to reproduce exact one-loop results down to below 1 TeV. The study highlights the need for higher-order resummations at multi-TeV scales, while offering practical, accurate formulas for energies up to ~1 TeV that can aid collider phenomenology and model comparisons.

Abstract

Starting from a theoretical representation of the electroweak component of four-fermion neutral current processes that uses as theoretical input the experimental measurements at the Z peak, we consider the asymptotic high energy behaviour in the Standard Model at one loop of those gauge-invariant combinations of self-energies, vertices and boxes that contribute all the different observables. We find that the logarithmic contribution due to the renormalization group running of the various couplings is numerically overwhelmed by single and double logarithmic terms of purely electroweak (Sudakov-type) origin, whose separate relative effects grow with energy, reaching the 10% size at about one TeV. We then propose a simple "effective" parametrization that aims at describing the various observables in the TeV region, and discuss its validity both beyond and below 1 TeV, in particular in the expected energy range of future linear electron-positron (LC) and muon-muon colliders.

Figures (9)

• Figure 1: Self-energy diagrams for $\gamma,Z$ gauge bosons. It must be understood that $W$ or $Z$ running inside the loop are accompagned by their corresponding Goldstones and ghosts states.
• Figure 2: $WW$ triangle contribution to the $\gamma-f\bar{f}$, $Z-f\bar{f}$ vertex. Here also the contribution of the corresponding Goldstones is to be added.
• Figure 3: Single $W$ or $Z$ exchange contribution to the $\gamma-f\bar{f}$, $Z-f\bar{f}$ vertex. Here also the contribution of the corresponding Goldstone is to be added.
• Figure 4: $WW$ and $ZZ$ box contributions to $e^+e^-\to f\bar{f}$. In the $WW$ case diagram (a) contributes for $I_{3f}=-{1\over2}$, whereas diagram (b) contributes for $I_{3f}=+{1\over2}$. In the $ZZ$ case both diagrams contribute.
• Figure 5: Logarithmic contributions to the asymptotic cross section $\sigma (e^+e^- \to \mu^+\mu^-)$ from eq.(\ref{['sigmu']}).