The Ultraviolet Properties of N=4 Supergravity at Four Loops

TL;DR

It is demonstrated that pure N=4 supergravity is ultraviolet divergent at four loops, and the form of the divergence suggests that it is due to the rigid U(1) duality-symmetry anomaly of the theory.

Abstract

We demonstrate that pure N=4 supergravity is ultraviolet divergent at four loops. The form of the divergence suggests that it is due to the rigid U(1) duality-symmetry anomaly of the theory. This is the first known example of an ultraviolet divergence in a pure ungauged supergravity theory in four dimensions. We use the duality between color and kinematics to construct the integrand of the four-loop four-point amplitude, whose ultraviolet divergence is then extracted by standard integration techniques.

Figures (2)

• Figure 1: Four of the 85 diagrams with cubic vertices used to organize the ${{\cal N}=4}$ super-Yang-Mills amplitudes into a form that respects the duality between color and kinematics. The remaining diagrams are listed in Ref. ck4l.
• Figure 2: The two basic vacuum graphs.