Manifesting enhanced cancellations in supergravity: integrands versus integrals

TL;DR

The paper investigates enhanced ultraviolet cancellations in half-maximal supergravity and argues these cancellations cannot be exposed at the integrand level. By analyzing one- and two-loop cases and employing BCJ double-copy constructions, unitarity cuts, and integration-by-parts reorganizations, the authors show that finiteness often emerges only after integration. They develop a systematic framework using vacuum expansions and IBP identities to relate divergences and demonstrate that Lorentz invariance and SL(L) relabeling generate the necessary integral identities at large loop momenta. This leads to a conjecture that such symmetry-generated IBP relations suffice to reveal enhanced cancellations at L loops when the theory is finite up to L−1 loops, offering a potential all-order mechanism for these mysterious cancellations.

Abstract

Examples of "enhanced ultraviolet cancellations" with no known standard-symmetry explanation have been found in a variety of supergravity theories. By examining one- and two-loop examples in four- and five-dimensional half-maximal supergravity, we argue that enhanced cancellations in general cannot be exhibited prior to integration. In light of this, we explore reorganizations of integrands into parts that are manifestly finite and parts that have poor power counting but integrate to zero due to integral identities. At two loops we find that in the large loop-momentum limit the required integral identities follow from Lorentz and SL(2) relabeling symmetry. We carry out a nontrivial check at four loops showing that the identities generated in this way are a complete set. We propose that at $L$ loops the combination of Lorentz and SL($L$) symmetry is sufficient for displaying enhanced cancellations when they happen, whenever the theory is known to be ultraviolet finite up to $(L-1)$ loops.

Figures (5)

• Figure 1: The three box diagrams contributing to the one-loop four-point amplitude of maximal ${\cal N} = 4$ super-Yang--Mills theory and half-maximal supergravity.
• Figure 2: The planar and nonplanar double-box diagrams that contribute to the four-point amplitudes of ${\cal N} = 4$ supergravity.
• Figure 3: The three sunset integrals. These are ultraviolet divergent in $D=4$ and $D=5$.
• Figure 4: The contributing diagrams of the three-particle cut of the half-maximal supergravity two-loop four-point amplitude. The shaded (red) dashed lines indicated the legs which are cut.
• Figure 5: A four-loop relation between ultraviolet divergences of vacuum integrals in $D=11/2$ dimensions, matching identity 22 from Table I in Appendix C of Ref. Simplifying. Where a black dot appears, the propagator is raised to a squared power.