Two-Loop Sudakov Form Factor in a Theory with a Mass Gap

The two-loop Sudakov form factor is computed in a $\mathrm{U}(1)$ model with a massive gauge boson and a $\mathrm{U}(1)\times \mathrm{U}(1)$ model with mass gap. We analyze the result in the context of hard and infrared evolution equations and establish a matching procedure which relates the theories with and without mass gap, setting the stage for the complete calculation of the dominant two-loop corrections to electroweak processes at high energy.

Figure 1

The two-loop correction to the form factor ${\mathcal{F}}_{\alpha }(M,Q)$ in LL (including ${\alpha }^2{\mathcal{L}}^4$), NLL (including ${\alpha }^2{\mathcal{L}}^3$), ${\mathrm{N}}^2\mathrm{L}\mathrm{L}$ (including ${\alpha }^2{\mathcal{L}}^2$), and ${\mathrm{N}}^3\mathrm{L}\mathrm{L}$ (including ${\alpha }^2{\mathcal{L}}^1$) approximations and the complete two-loop correction as functions of the momentum transfer for $M=80\mathrm{G}\mathrm{e}\mathrm{V}$, $\alpha /(4\pi )=3\times {10}^{-3}$.

Figure 2

The same as Fig. 1 after changing the argument of the logarithm.