Two-Loop Electroweak Logarithms in Four-Fermion Processes at High Energy

TL;DR

This work delivers a complete analytic treatment of two-loop electroweak logarithms in the high-energy Sudakov regime for vector form factors and neutral-current four-fermion processes within a spontaneously broken SU(2) framework. By combining expansion by regions with evolution-equation techniques, it isolates universal Sudakov logs from QED infrared effects and systematically incorporates W–Z mass splitting through NNLL accuracy. The authors provide explicit expressions for form-factor and four-fermion amplitudes, along with detailed numerical estimates showing that two-loop logarithms can amount to percent-level corrections with controlled uncertainties. The developed methodology, including QED separation and mass-gap matching, offers a practical path to precise predictions for future colliders and can be extended to Standard Model extensions and gauge-boson production scenarios.

Abstract

We present the complete analytical result for the two-loop logarithmically enhanced contributions to the high energy asymptotic behavior of the vector form factor and the four-fermion cross section in a spontaneously broken SU(2) gauge model. On the basis of this result we derive the dominant two-loop electroweak corrections to the neutral current four-fermion processes at high energies. Previously neglected effects of the gauge boson mass difference are included through the next-to-next-to-leading logarithmic approximation.

Figures (3)

• Figure 1: Separate logarithmic contributions to $R(e^+e^-\to q\bar{q})$ in % to the Born approximation: (a) the one-loop LL $(\ln^2(s/M^2)$, long-dashed line), NLL $(\ln^1(s/M^2)$, dot-dashed line) and N$^2$LL $(\ln^0(s/M^2)$, solid line) terms; (b) the two-loop LL $(\ln^4(s/M^2)$, short-dashed line), NLL $(\ln^3(s/M^2)$, long-dashed line), NNLL $(\ln^2(s/M^2)$, dot-dashed line) and N$^3$LL $(\ln^1(s/M^2)$, solid line) terms.
• Figure 2: The total logarithmic corrections to $R(e^+e^-\to Q\bar{Q})$ (dashed line), $R(e^+e^-\to q\bar{q})$ (dot-dashed line) and $R(e^+e^-\to \mu^+\mu^-)$ (solid line) in % to the Born approximation: (a) the one-loop correction up to N$^2$LL term; (b) the two-loop correction up to N$^3$LL term.
• Figure 3: (a) The total logarithmic two-loop corrections to the forward-backward asymmetry $R^{FB}(e^+e^-\to Q\bar{Q})$ (dashed line), $R^{FB}(e^+e^-\to q\bar{q})$ (dot-dashed line) and $R^{FB}(e^+e^-\to \mu^+\mu^-)$ (solid line) in % to the Born approximation. (b) The same as (a) but for the left-right asymmetry $\tilde{R}^{LR}$.