Electroweak Sudakov at two loop level

TL;DR

This paper addresses whether electroweak double-logarithmic Sudakov corrections to a fermion form factor exponentiate in the spontaneously broken $SU(2)\times U(1)$ theory. It performs an explicit two-loop calculation in the Feynman gauge, treating $M_W \sim M_Z \equiv M$ and introducing a photon regulator $\lambda$ in the regime $s \gg M^2 \gg \lambda^2$, and classifies contributions from ladder, crossed ladder, and triple gauge boson–coupling diagrams using the pinch technique. The main finding is that all non-exponentiating terms cancel, so the two-loop result matches the second term in the expansion of an exponentiated Sudakov form factor: $\Gamma^{(2)} = 1 - \frac{1}{16\pi^2}( g^2 C_2(R) + g'^2 Y^2 - e^2 Q^2 ) \ln^2 \frac{s}{M^2} - \frac{1}{16\pi^2} e^2 Q^2 \ln^2 \frac{s}{\lambda^2} + \frac{1}{2!}\left[ \frac{1}{16\pi^2}( g^2 C_2(R) + g'^2 Y^2 - e^2 Q^2 ) \ln^2 \frac{s}{M^2} + \frac{1}{16\pi^2} e^2 Q^2 \ln^2 \frac{s}{\lambda^2} \right]^2$. The work clarifies how the electroweak mixing and non-Abelian structure influence soft infrared physics and demonstrates a QCD-like exponentiation in a realistic electroweak setting. This has implications for high-energy collider phenomenology and the understanding of IR behavior in spontaneously broken gauge theories, highlighting both similarities and unique electroweak features.

Abstract

We investigate the Sudakov double logarithmic corrections to the form factor of fermion in the SU(2)XU(1) electroweak theory. We adopt the familiar Feynman gauge and present explicit calculations at the two loop level. We show that the leading logarithmic corrections coming from the infrared singularities are consistent with the "postulated" exponentiated electroweak Sudakov form factor. The similarities and differences in the "soft" physics between the electroweak theory and the unbroken non-abelian gauge theory (QCD) will be clarified.

Figures (3)

• Figure 1: The ladder and crossed ladder diagrams. The dashed (wavy) line represents the photon ($W$ and/or $Z$) with the mass $\lambda$ ($M$).
• Figure 2: The diagrams which have the triple point couplings. The meaning of lines is the same as in Fig.\ref{['diag1']}.
• Figure 3: The reduced Feynman diagrams for Fig.\ref{['diag2']}. The blob stands for the vertex $\Gamma^F$ defined in Eq.\ref{['nwt']}. The wavy lines represent both the photon and $W , Z$.