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Surprising one-loop finiteness of 6D half-maximal supergravities

Yu-tin Huang, Henrik Johansson, Michele Santagata, Congkao Wen

TL;DR

This work reveals an unexpected one-loop finiteness in certain 6D half-maximal supergravities with matter: the amplitudes are finite precisely at $n_T=21$ for $ obreak ext{N}=(2,0)$ and $n_V=20$ for $ obreak ext{N}=(1,1)$, despite symmetry-allowed counterterms. The authors combine unitarity-based discontinuity bootstrap with BCJ double-copy techniques to compute the one-loop four-matter amplitudes and their UV structures, showing the logarithmic UV terms vanish at the critical values. The finiteness coincides with Type II string theory compactifications on K3, hinting at a duality-based origin and suggesting deeper cancellations beyond naive power counting. The results motivate further exploration of these cancellations at higher loops, higher-point amplitudes, and in AdS contexts, with potential implications for understanding the UV structure of quantum gravity and string dualities.

Abstract

In four dimensions, it has long been established that gravity coupled to matter exhibits ultraviolet divergences at one loop, irrespective of supersymmetry. Notably, the four-matter one-loop amplitudes of half-maximal supergravity coupled to Maxwell multiplets were shown in the 1970s to be divergent. Surprisingly, we demonstrate in this work that half-maximal theories can nevertheless become one-loop finite when uplifted to higher dimensions, contrary to naive expectations. Specifically, we study the ultraviolet properties of the four-matter and two-matter two-graviton amplitudes in six-dimensional $\mathcal{N}=(2,0)$ and $\mathcal{N}=(1,1)$ supergravities, coupled to $n_T$ tensor and $n_V$ vector multiplets, respectively. We find that the one-loop amplitudes are finite for $n_T=21$ and $n_V=20$. This finiteness is unexpected, as symmetry-preserving counterterms do exist. Interestingly, both values exactly correspond to low-energy limits of type II string theories compactified on K3, which hints at possible origins to the surprising cancellations.

Surprising one-loop finiteness of 6D half-maximal supergravities

TL;DR

This work reveals an unexpected one-loop finiteness in certain 6D half-maximal supergravities with matter: the amplitudes are finite precisely at for and for , despite symmetry-allowed counterterms. The authors combine unitarity-based discontinuity bootstrap with BCJ double-copy techniques to compute the one-loop four-matter amplitudes and their UV structures, showing the logarithmic UV terms vanish at the critical values. The finiteness coincides with Type II string theory compactifications on K3, hinting at a duality-based origin and suggesting deeper cancellations beyond naive power counting. The results motivate further exploration of these cancellations at higher loops, higher-point amplitudes, and in AdS contexts, with potential implications for understanding the UV structure of quantum gravity and string dualities.

Abstract

In four dimensions, it has long been established that gravity coupled to matter exhibits ultraviolet divergences at one loop, irrespective of supersymmetry. Notably, the four-matter one-loop amplitudes of half-maximal supergravity coupled to Maxwell multiplets were shown in the 1970s to be divergent. Surprisingly, we demonstrate in this work that half-maximal theories can nevertheless become one-loop finite when uplifted to higher dimensions, contrary to naive expectations. Specifically, we study the ultraviolet properties of the four-matter and two-matter two-graviton amplitudes in six-dimensional and supergravities, coupled to tensor and vector multiplets, respectively. We find that the one-loop amplitudes are finite for and . This finiteness is unexpected, as symmetry-preserving counterterms do exist. Interestingly, both values exactly correspond to low-energy limits of type II string theories compactified on K3, which hints at possible origins to the surprising cancellations.

Paper Structure

This paper contains 8 sections, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Tree-level and on-shell setup
  3. One-loop discontinuities
  4. One-loop amplitudes
  5. UV divergences from the double copy
  6. Conclusions
  7. Acknowledgments
  8. Phase-space and Feynman integrals

Figures (2)

  • Figure 1: The BCJ diagram topologies for 6D SQCD with external hypermultipets. Arrows carry $USp(n_\Phi)$ flavor charge and numerators map $N_i\to-N_i$ under arrow flip.
  • Figure 2: The BCJ diagram topologies for 6D SQCD with external mixed gauge and hypermultipets.