Subleading Sudakov logarithms in electroweak high energy processes to all orders

TL;DR

The paper develops an all-orders resummation framework for subleading Sudakov logarithms in electroweak processes at energies well above the weak scale. By deriving the next-to-leading kernel from virtual Altarelli-Parisi splitting functions and solving a generalized infrared evolution equation, it treats both transverse and longitudinal (via the Goldstone equivalence) sectors, including Z-γ mixing, and provides explicit region-matching across the electroweak scale. The results show that subleading terms exponentiate and can be interpreted probabilistically in the massless limit, with validation against known one-loop high-energy results for processes like $e^+e^-\to W^+W^-$. Limitations include angular logarithms and Yukawa effects, which are left for future refinement but the framework already offers a robust tool for high-precision predictions at future TeV-scale colliders.

Abstract

In future collider experiments at the TeV scale, large logarithmic corrections originating from massive boson exchange can lead to significant corrections to observable cross sections. Recently double logarithms of the Sudakov-type were resummed for spontaneously broken gauge theories and found to exponentiate. In this paper we use the virtual contributions to the Altarelli-Parisi splitting functions to obtain the next to leading order kernel of the infrared evolution equation in the fixed angle scattering regime at high energies where particle masses can be neglected. In this regime the virtual corrections can be described by a generalized renormalization group equation with infrared singular anomalous dimensions. The results are valid for virtual electroweak corrections to fermions and transversely polarized vector bosons with an arbitrary number of external lines. The subleading terms are found to exponentiate as well and are related to external lines, allowing for a probabilistic interpretation in the massless limit. For $Z$-boson and $γ$ final states our approach leads to exponentiation with respect to each amplitude containing the fields of the unbroken theory. For longitudinal degrees of freedom it is shown that the equivalence theorem can be used to obtain the correct double logarithmic asymptotics. At the subleading level, corrections to the would be Goldstone bosons contribute which should be considered separately. Explicit comparisons with existing one loop calculations are made.

Figures (5)

• Figure 1: In an axial gauge, all collinear logarithms come from corrections to a particular external line (depending on the choice of the four vector $n^\nu$ satisfying $n^\nu A^a_\nu=0$) as illustrated in the figure. In a covariant gauge, the sum over all possible insertions is reduced to a sum over all $n$-external legs due to Ward identities. Overall, these corrections factorize with respect to the Born amplitude.
• Figure 2: Feynman diagrams contributing to the infrared evolution equation (\ref{['eq:mrg']}) for a process with $n$ external legs. In a general covariant gauge the virtual gluon with the smallest value of ${\hbox{\boldmath $k$}}_{\perp}$ is attached to different external lines. The inner scattering amplitude is assumed to be on the mass shell.
• Figure 3: The two counterterms contributing to the quark anomalous dimension $\gamma_{q\overline{q}} = \frac{\partial}{\partial \log \overline{\mu}^2} \left( -\delta_{q \overline{q}}+\delta_2 \right)$. Here $\overline{\mu}$ denotes the $\overline{\rm MS}$ dimensional regularization mass parameter. Due to divergences in loop corrections there are scaling violations also in the massless theory.
• Figure 4: The schematic corrections to external gauge boson emissions in terms of the fields in the unbroken phase of the electroweak theory. There are no mixing terms between the $W^3_\nu$ and $B_\nu$ fields for massless fermions. We denote $\cos \theta_{\rm w}$ by $c_{\rm w}$ and $\sin \theta_{\rm w}$ by $s_{\rm w}$. For $W^\pm$ final states, the corrections factorize with respect to the physical amplitude. In general, one has to sum over all fields of the unbroken theory with each amplitude being multiplied by the respective mixing coefficient.
• Figure 5: The pictorial Goldstone boson equivalence theorem for $W$-pair production in $e^+e^-$ collisions. The correct DL-asymptotics for longitudinally polarized bosons are obtained by using the quantum numbers of the charged would be Goldstone scalars at high energies.