The role of universal and non universal Sudakov logarithms in four fermion processes at TeV energies: the one-loop approximation revisited

TL;DR

This work analyzes weak Sudakov logarithms in high-energy four-fermion e+e− processes within the xi=1 gauge, separating universal (angle-independent) and non-universal (angle-dependent) one-loop contributions. By decomposing the invariant amplitude into four gauge-invariant form factors and evaluating universal $U_W(q^2)$ and $U_Z(q^2)$ terms alongside non-universal theta-dependent box contributions, the authors assess the impact on LC and CLIC observables. They find that theta-dependent non-universal Sudakov terms can dominate the one-loop corrections for many observables at LC, while at CLIC strong cancellations between universal and non-universal terms complicate predictions and motivate a two-loop or resummed treatment for the angular-dependent piece. Forward-backward asymmetries exhibit negligible angular-independent corrections and are largely box-driven, whereas top-quark production shows a distinct pattern where angular-dependent effects are small. Overall, the study highlights the need for higher-order analyses to ensure reliable TeV-scale predictions for certain observables and energies at future linear colliders.

Abstract

We consider the separate effects on four fermion processes, in the TeV energy range, produced at one loop by Sudakov logarithms of universal and not universal kind, working in the 't Hooft xi=1 gauge. Summing the various vertex and box contributions allows to isolate two quite different terms.The first one is a combination of vertex and box quadratic and linear logarithms that are partially universal and partially not universal and independent of the scattering angle theta. The second one is theta-dependent, not universal, linearly logarithmic and only produced by weak boxes. We show that for several observables, measurable at future linear e+e- colliders (LC, CLIC), the role of the latter term is dominant and we discuss the implications of this fact for what concerns the reliability of a one-loop approximation.

Figures (13)

• Figure 1: SM diagrams contributing in the asymptotic regime. Note that for the $WW$ box contributions (d), diagram (1) contributes for $I_{3f}=-{1\over2}$, whereas diagram (2) contributes for $I_{3f}=+{1\over2}$; for the $ZZ$ box contributions (e), both diagrams contribute.
• Figure 2: Separate asymptotic contributions to $\sigma_\mu$ as functions of the energy. The solid line (RG) is the linear Renormalization Group logarithm. The dotted line (U) is the $\theta$ independent term proportional to the combination $3\log\frac{q^2}{M_Z^2}-\log^2\frac{q^2}{M_Z^2}$. The dashed line ($\theta S$) is the angular dependent linear logarithm. Finally, the thick dot-dashed line (Full) is the sum of the three contributions. The same captions applies to all the following figures showing the effects in all the other considered observables.
• Figure 3: Separate asymptotic contributions to $A_{FB,\mu}$ as functions of the energy. Same captions as in Fig.2.
• Figure 4: Separate asymptotic contributions to $A_{LR,\mu}$ as functions of the energy. Same captions as in Fig.2.
• Figure 5: Separate asymptotic contributions to $\sigma_b$ as functions of the energy. Same captions as in Fig.2.