The Ultraviolet Structure of Half-Maximal Supergravity with Matter Multiplets at Two and Three Loops

TL;DR

Using the BCJ color-kinematics duality and the gravity double-copy construction, the authors compute two- and three-loop four-point amplitudes of half-maximal supergravity with abelian matter multiplets in $D=4$–$6$, revealing new ultraviolet divergences in $D=4$ and $D=5$ that conflict with conjectures based on hypothetical off-shell harmonic superspaces. They connect these divergences to the corresponding divergences in nonsupersymmetric Yang–Mills theory with scalars and provide a comprehensive catalog of one- and two-loop counterterms to constrain future explanations of observed finiteness in pure half-maximal supergravity. The results demonstrate that the UV structure with matter is governed by gauge-theory divergences and show that the proposed superspace mechanism cannot account for the observed divergences, including a full $D=4$ three-loop analysis with $ abla^6R^4$-type structures and $\

Abstract

Using the duality between color and kinematics, we construct the two- and three-loop amplitudes of half-maximal supergravity with matter multiplets and show that new divergences occur in D=4 and D=5. Bossard, Howe and Stelle have recently conjectured the existence of 16-supercharge off-shell harmonic superspaces in order to explain the ultraviolet finiteness of pure half-maximal supergravity with no matter multiplets in D=4 at three loops and in D=5 at two loops. By assuming the required superspace exists in D=5, they argued that no new divergences should occur at two loops even with the addition of abelian-vector matter multiplets. Up to possible issues with the SL(2,R) global anomaly of the theory, they reached a similar conclusion in D=4 for two and three loops. The divergences we find contradict these predictions based on the existence of the desired off-shell superspaces. Furthermore, our D=4 results are incompatible with the new divergences being due to the anomaly. We find that the two-loop divergences of half-maximal supergravity are directly controlled by the divergences appearing in ordinary nonsupersymmetric Yang-Mills theory coupled to scalars, explaining why half-maximal supergravity develops new divergences when matter multiplets are added. We also provide a list of one- and two-loop counterterms that should be helpful for constraining any future potential explanations of the observed vanishings of divergences in pure half-maximal supergravity.

Figures (7)

• Figure 1: The basic Jacobi relation for either color or numerator factors given in Eq. (\ref{['BCJDuality']}). These three diagrams can be embedded in a larger diagram, including loops.
• Figure 2: The one-loop box diagram. The one-loop color factor $c^{{(1)}}_{1234}$ is obtained by dressing each vertex with an $\tilde{f}^{abc}$.
• Figure 3: The planar and nonplanar double-box graphs. The $c^{\rm P}_{1234}$ and $c^{\rm NP}_{1234}$ color factors are obtained by dressing each vertex with an $\tilde{f}^{abc}$.
• Figure 4: Diagrams with triangle and bubble subgraphs at (a) one loop and (b) two loops. These do not contribute to terms proportional to the needed color tensors in Yang-Mills and therefore do not contribute to the supergravity divergences.
• Figure 5: A Feynman diagram appearing in the calculation of the gauge-theory amplitude $\mathcal{A}^{{(2)}}(1_\phi,2_\phi,3_g,4_g)$ and supergravity amplitude $\mathcal{M}^{{(2)}}(1_\mathsmaller{\rm{V}},2_\mathsmaller{\rm{V}},3_\mathsmaller{\rm{H}},4_\mathsmaller{\rm{H}})$. The closed loop on the right is that of a ghost.