Two-loop supersymmetric QCD and half-maximal supergravity amplitudes

TL;DR

This work achieves a two-loop, four-point construction of color-dual numerators for $ ext{N}=2$ SQCD with fundamental hypers in $D\le6$, enabling a robust double-copy to half-maximal supergravity and, at specific $N_f$ choices, to pure supergravity with tunable vector-multiplet content. By exploiting the $ ext{color--kinematics}$ duality and a carefully engineered ghost-statistics mechanism, the authors obtain compact integrands in both MHV and all-chiral sectors, and provide six-dimensional uplifts that regulate the loop integrations consistently. They assemble the $ ext{N}=2$ SQCD numerators into two-loop supergravity amplitudes in $D=4$ and $D=5-2\epsilon$, finding enhanced ultraviolet cancellations in the latter case and cross-checking against known $ ext{N}=4$ results. The results extend the reach of the double-copy program to matter-containing gauge theories, offering new pathways to compute pure and matter-coupled supergravities and to study their UV behavior in diverse dimensions.

Abstract

Using the duality between color and kinematics, we construct two-loop four-point scattering amplitudes in $\mathcal{N}=2$ super-Yang-Mills (SYM) theory coupled to $N_f$ fundamental hypermultiplets. Our results are valid in $D\le 6$ dimensions, where the upper bound corresponds to six-dimensional chiral $\mathcal{N}=(1,0)$ SYM theory. By exploiting a close connection with $\mathcal{N}=4$ SYM theory - and, equivalently, six-dimensional $\mathcal{N}=(1,1)$ SYM theory - we find compact integrands with four-dimensional external vectors in both the maximally-helicity-violating (MHV) and all-chiral-vector sectors. Via the double-copy construction corresponding $D$-dimensional half-maximal supergravity amplitudes with external graviton multiplets are obtained in the MHV and all-chiral sectors. Appropriately tuning $N_f$ enables us to consider both pure and matter-coupled supergravity, with arbitrary numbers of vector multiplets in $D=4$. As a bonus, we obtain the integrands of the genuinely six-dimensional supergravities with $\mathcal{N}=(1,1)$ and $\mathcal{N}=(2,0)$ supersymmetry. Finally, we extract the potential ultraviolet divergence of half-maximal supergravity in $D=5-2ε$ and show that it non-trivially cancels out as expected.

Figures (5)

• Figure 1: Weight-space diagrams for states in the (a) $D=4$, $\mathcal{N}=4$ and (b) $D=6$, $\mathcal{N}=(1,1)$ vector multiplets transforming under the little groups U(1) and $\text{SU}(2)\times\text{SU}(2)$ respectively. The Grassmann variables $\eta$ and $\tilde{\eta}$ act as raising and lowering operators in the weight space Boels:2009bv as denoted by solid black arrows. Red-colored states belong to the vector multiplet; green- and blue-colored states belong to the two parts of the $D=6$, $\mathcal{N}=(1,0)$ hypermultiplet. The dotted gray arrows show how 6D states are projected to the four-dimensional weight space.
• Figure 2: A one-loop quadruple cut with circulating hypermultiplets --- all four exposed lines are taken on shell. To compute it, we first write it as a two-particle cut involving a pair of four-point tree amplitudes --- as indicated by the transparent blobs --- and then we multiply by the inverse poles exposed inside these blobs. The quadruple cut limit is then finite.
• Figure 3: The three two-loop spanning cuts involving only pure $\mathcal{N}=2$ states. All exposed propagators are understood as being taken on shell.
• Figure 4: The eight two-loop spanning cuts involving hypermultiplets required at four points. All exposed propagators are understood as being taken on shell. Notice that cuts (b) and (c) differ by the routing of fundamental hypermultiplets on the right-hand side.
• Figure 5: A complete list of non-vanishing graphs and graphs corresponding to master numerators. The eight master graphs that we choose to work with are (1)--(5), (13), (19) and (22). While tadpoles and external bubbles are dropped from the final amplitudes it is useful to consider them at intermediate steps of the calculation.