One-loop Amplitudes for N=2 Homogeneous Supergravities

We compute one-loop matter amplitudes in homogeneous Maxwell-Einstein supergravities with N=2 supersymmetry using the double-copy construction. We start from amplitudes of N=2 super-Yang-Mills theory with matter that obey manifestly the duality between color and kinematics. Taking advantage of the fact that amplitudes with external hypermultiplets have kinematical numerators which do not present any explicit dependence on the loop momentum, we find a relation between the one-loop divergence of the supergravity amplitudes and the beta function of the non-supersymmetric gauge theory entering the construction. Two distinct linearized counterterms are generated at one loop. The divergence corresponding to the first is nonzero for all homogeneous supergravities, while the divergence associated to the second vanishes only in the case of the four Magical supergravities.


Introduction and summary of results
In the past decade, the duality between color and kinematics and the related double-copy construction [1,2] have produced a breath-taking increase in our ability to conduct multiloop calculations in gravity and supergravity theories. The double-copy method expresses loop-level gravity integrands in terms of gauge-theory building blocks, which are typically far easier to obtain. Taking advantage of this construction, calculations have been performed up to five loops in maximal supergravity [3,4,5] and up to four loops in half-maximal supergravity [6,7,8]. The studies of ultraviolet (UV) behaviors which have been made possible by these calculations have uncovered a set of so-called enhanced cancellations [9,10] and revealed a connection between UV-divergences and U(1) quantum anomalies [11,12].
In the case of gravitational theories that can be directly obtained from string theory, insight about color/kinematics duality can be obtained from string monodromy relations [43,44,45,46,47,48]. Duality-satisfying structures can be obtained from the low-energy limit of string theory amplitudes [49,50,51,52]. New sets of double-copy relations that combine string and field-theory building blocks have also become a powerful tool for understanding the structure of various string theory amplitudes [53,54,55,56]. In addition, the double copy is an intrinsic feature of various modern approaches to scattering amplitudes, including the CHY formalism [57,58,59,60,61,62,48] and ambitwistor strings [63,64].
Given the considerable progress in establishing double-copy structures for wider and wider classes of theories, it is natural to seek additional examples in which the double copy can be employed for shedding some light on the loop-level properties of supergravities with matter. A related issue is that the double-copy is by nature a construction at the integrand level, a fact that makes it more difficult to establish a direct relation between the physical properties of a gravity theory and the ones of the gauge theories entering the construction, which become manifest after integration.
In this paper we focus on the double-copy construction for Maxwell-Einstein supergravities with N = 2 supersymmetry in four and five dimensions and homogeneous target spaces, which was first presented in ref. [17]. These theories have been explicitly classified in the supergravity literature [65] and provide a natural testing ground for amplitude methods. Their double-copy construction involves a N = 2 super-Yang-Mills (sYM) theory with one half-hypermultiplet in a pseudo-real representation as one gauge-theory factor, and a non-supersymmetric YM theory with adjoint scalars and matter fermions in the same representation as the other gauge-theory factor.
Loop amplitudes with four external hypermultiplets that obey the duality between color and kinematics were first obtained in ref. [66]. These amplitudes have the striking property that the loop momentum does not appear in the kinematical numerators. Focusing on supergravity amplitudes between four vector multiplets in which the vectors are constructed as the product of two spin-1/2 asymptotic states, the above property allows us to find an explicit relation between supergravity amplitudes and gauge-theory physical quantities which holds at the integrated level. Our main result is that the one-loop divergence for the (super)amplitudes we are inspecting can be directly related to a linear combination of various parts of the one-loop beta function of the non-supersymmetric gauge theory entering the construction, 2 Double-copy construction for homogeneous supergravities The double-copy construction relies on organizing gauge theory amplitudes at L loops and n points as sums over a set of cubic graphs, where D i are products of inverse scalar propagators and S i are symmetry factors. Each cubic graph is associated to a color factor C i , which is constructed out of gauge-group invariant tensors (i.e., structure constants, representation matrices and Clebsch-Gordan coefficients with three matter-representation indices). Since the relevant group-theoretical objects obey algebraic relations (e.g. commutation relations and Jacobi identities), there exist triplets of graphs {i, j, k} such that C i + C j + C k = 0. Graphs in the above amplitude presentation are further associated to numerator factors, which contain theory-specific kinematical information and involve external and loop momenta, as well as Grassmann variables in case of superamplitudes. A theory obeys color/kinematics duality if the amplitude numerators possess the same algebraic properties as the corresponding color factors, The double-copy construction utilizes duality-satisfying numerators to write (super)gravity amplitudes by replacing the color factors with a second set of numerators, where κ is the gravity coupling. The numerators n i ,ñ i may come from different gauge theories which share the same set of color factors. The formula requires that at least one of the two sets of numerators satisfies manifestly color/kinematics duality [1,2]. If this requirement is satisfied, it is possible to show that the double-copy amplitudes obey the Ward identities corresponding to linearized diffeomorphisms and hence have the interpretation of amplitudes from some gravitational theory [34]. In general, identifying the gravity theory produced by a given double-copy construction (i.e. reconstructing the corresponding Lagrangian) is a highly-nontrivial problem. However, symmetry considerations, combined with minimal information on the supergravity interactions, have been instrumental in applying the doublecopy method to very large classes of theories.
Among the double copies formulated to date, the construction for N = 2 homogeneous Maxwell-Einstein supergravities plays a key role. Supergravities in this family have been explicitly classified by the supergravity literature [65] and hence give a natural testing ground for the double-copy method. Their double-copy construction has the following schematic form [17]: The right gauge theory entering the construction is a YM theory in D spacetime dimensions with P additional fermions which transform in a pseudo-real gauge group representation. The corresponding Lagrangian is where field strengths and covariant derivatives are defined as Here µ, ν = 0, . . . , D − 1 are spacetime indices and we use a mostly-minus metric. The D-dimensional gamma matrices obey the Clifford algebra relation where all gamma matrices except Γ 0 are taken to be antihermitian (our conventions are summarized in Appendix A).â,b,ĉ are adjoint indices of the gauge group. 1 The fermions' indices D n F (D, P,Ṗ ) conditions flavor group 16 n F (k, P,Ṗ ) as for k as for k where C is a matrix acting on spinor and flavor indices and V is a matrix acting on the gauge-group representation indices which is taken to be unitary and antisymmetric. The representation matrices for the fermions obey i.e. the representation R is pseudo-real. The spinor λ includes P flavors of irreducible SO(D − 1, 1) spinors, which obey reality (R) or pseudo-reality (PR) conditions, according to the choice of C in equation (2.9). Specifically, we have where C D is the SO(D − 1, 1) charge-conjugation matrix and Ω is an antisymmetric real matrix acting on the flavor indices. The dimensions in which R and PR conditions are appropriate are listed in Table 1 together with the corresponding flavor symmetry. For D = 6, 10 (mod 8), Weyl conditions are compatible with R and PR conditions. Since representations with opposite chiralities are inequivalent, we need to introduce two distinct integers P andṖ giving the number of irreducible D-dimensional spinors for each chirality.
Results of this paper are more naturally expressed in terms of the number of four-dimensional fermions n F , which is also listed in Table 1.
The left gauge theory entering the double-copy construction is a N = 2 sYM theory with a single half-hypermultiplet transforming in the same pseudo-real representation R which appears in the other gauge theory. This choice is motivated by the fact that a pseudo-real half-hypermultiplet is CPT self-conjugate and hence does not need to be augmented to a full hypermultiplet to have a sensible theory. It is convenient to consider a sYM theory in D = 6 spacetime dimensions with Lagrangian where the covariant derivatives are 14) The spin-1/2 field ψâ transforms in the adjoint representation and is the supersymmetric partner of the gluon. It obeys a chirality condition of the form The complex scalar ϕ and the spinor χ are the components of the half-hypermultiplet. They obey the conditions,χ = χ t CV , Amplitudes in this theory can be conveniently organized into superamplitudes with manifest N = 2 supersymmetry. While the Lagrangian in this section can be used for Feynman-rule computations, the quickest way to obtain amplitudes for the left theory is to start from N = 4 amplitudes and use an orbifold procedure, as explained in the next section.
Finally, a generic homogeneous supergravity in four-dimensions has a U-duality algebra which can be decomposed into grade 0, 1, 2 generators, with the grade-zero part where S(P,Ṗ ) is the flavor algebra listed in Table 1. A so(D − 4) ⊕ S(P,Ṗ ) subalgebra is already manifest at the level of the gauge theories entering the construction.

One-loop gauge amplitudes with hypermultiplets
Field-theory orbifolds [67] are constructed from a parent theory, which in this case is N = 4 sYM, by projecting out states that are not invariant under the transformation where Φ is a generic field of the theory, g i is an element of the gauge group G and R i is a matrix acting on the R-symmetry indices of the theory. Γ is taken to be a discrete abelian subgroup of G × SU(4). Orbifold amplitudes can be conveniently obtained by inserting the projectors in the propagators of the parent theory, where |Γ| is the order of the orbifold group. As a consequence of the invariance of the Feynman-rule vertices under gauge and R-symmetry, projectors can be moved around the diagram past individual vertices. For tree-level amplitudes, this observation implies that orbifold amplitudes are identical to the ones of the parent theory, provided that the external states are chosen to be invariant under the orbifold group. One-loop amplitudes that preserve color/kinematics duality for general abelian orbifolds of N = 4 sYM theory were obtained in ref. [66]. They are generally organized in a presentation based on cubic graphs in which the internal line labeled by the loop momentum is dressed with an extra phase factor, which is in turn expressed in terms of the entries of a diagonal matrix encoding the action of the R-symmetry part of the orbifold generators on fundamental SU(4) indices.
Without repeating the derivation in ref. [66] and referring to Appendix C for details, we focus on amplitudes with four external hypermultiplets. Hypermultiplet asymptotic states can be conveniently organized in an on-shell superfield Q, together with the CPT-conjugate superfield Q. When the representation is pseudo-real, we can identify Q = Q and χ ± =χ ± . The superamplitude between four hypermultiplets is then written as Color and numerator factors are listed in Table 2. While we have obtained these amplitudes with a particular choice of gauge group and representation matrices, amplitudes with any other choice of gauge group and complex representation have the same formal expression.
In later sections, we will simply use the numerators without any explicit reference to the orbifold procedure we have employed in their derivation.
It is interesting to look at the color and numerator relations obeyed by the above amplitude. Some color relations stem from the representation-matrices commutation relations [Tâ,Tb] =fâbĉTĉ: 2 The reader can verify that these relations are obeyed by the corresponding numerator factors. Table 2: Color and numerator factors for the 1-loop superamplitude with four external hypermultiplets in case of a complex representation. Curly lines denote adjoint fields, while solid lines denote hypermultiplet matter fields. Arrows are assigned according to the representation and we use the short-hand notation δ 4 Q = δ 4 i η r i |i . The numbering of the graphs follows the one in ref. [66]. Gauge-group representation indicesα,β are written explicitly.
Other relations stem from the two-term identity Tâγ α Tâδ β = Tâδ α Tâγ β : These relations are also obeyed by the corresponding numerator factors. If we chose a different complex representation with respect to the one from orbifolding, the color relations would no longer hold, but numerator relations would be unaltered. For later convenience, color factors in Table 2 are rewritten in terms of the representation's index T (R) and quadratic Casimir C(R), While the numerators obtained so far are appropriate for complex representations, the construction outlined in the previous section notably uses pseudo-real representations as a key ingredient. It is quite instructive to see how the construction is modified in this case. On general grounds, using a reality condition allows us to relate fermions with their Dirac conjugates. For four-fermion amplitudes, this implies that a (fermionic) permutation symmetry needs to be introduced on all external legs. This symmetry extends to superamplitudes with four-external half-hypermultiplets, taking into account that hypermultiplet on-shell superfields anticommute with each other (since their lowest component is Grassmann-odd). In short, duality-satisfying numerators can be written down for the pseudo-real case as the unique set of numerators with the following two properties: 1. Numerators are invariant under the permutation of all external legs up to a sign which is assigned according to the signature of the permutation.
2. Numerators corresponding to color factors which are non-zero in the case of a complex representation reproduce the ones listed in the Table 2.
The complete list of numerators is given in Table 3.  Table 3: Color and numerator factors for the 1-loop superamplitude with four external halfhypermultiplets in case of a pseudo-real representation. Note that all bubbles have symmetry factor S i = 2. Representation matrices with two low indices are defined as Tâ αβ = (TâV −1 )αβ, where V is the antisymmetric matrix entering the pseudo-reality condition.
Numerators with different assignments of external legs are related by permutation symmetry, as it can be verified using the identities s 12 34 = − u 13 24 = t 14 23 . (3.48) The identity (TâTb) t = −TbTâ , (3.49) which relies on the antisymmetry of V , is also useful. An important difference between amplitudes in the complex and pseudo-real cases is that the symmetry factor for the bubble integrals with internal half-hypermultiplets needs to be changed to take into account that the corresponding graphs no longer carry arrows. The additional nonzero color and numerator factors obey identities of the form which stem from the gauge-group generators commutation relations. However, the extra identities in which gauge-group generators are contracted by an adjoint index require a more detailed discussion. In contrast to the complex case, color factors will no longer obey identities of the form Tâγ α Tâδ β = Tâδ α Tâγ β . While it would be natural to impose additional three-term identities for both color and numerator factors in the pseudo-real case, the numerators listed in Table 3 still obey the same set of two-term identities as in the complex case. It should be noted that these identities are not necessary for the consistency of the theory from the double-copy, i.e. the numerator identities stemming from the gauge-group generator commutation relations are sufficient for ensuring that the double-copy amplitudes obey the relevant Ward identities. However, it might be possible to find different amplitude presentations which apply specifically to hypermultiplets in pseudo-real representations and obey additional three-term identities in place of the two-term identities. As a consequence of the numerator relations and permutation symmetry, all numerators can be obtained in terms of a single master box numerator.
Finally, we note that the amplitude presentation in Table 3 can be extended to the case of an arbitrary number n H of half-hypermultiplets by dressing the numerator factors with Kronecker deltas with indices running over the number of half-hypermultiplets. For example, the first numerator is modified as Additionally, bubble numerators with a matter loop acquire an extra factor of n H . This procedure leaves the numerator relations corresponding to (3.5) unaltered, but the extra two-term relations are lost in the generic case. 3

One-loop supergravity amplitudes from the double copy
We now focus on four-dimensional supergravity superamplitudes with four vector multiplets 4 where a, b, c, d are global indices running over the number of supergravity vectors that are realized as the product of two spin-1/2 asymptotic states. Note that these vectors transform as SO(D − d) spinors under the global symmetry. These amplitudes can be obtained taking the double-copy of the amplitudes from the previous section with a four-fermion amplitude in the non-supersymmetric theory discussed in Section 2, The choice of chiralities for the external fermions determines whether a given external leg in (4.1) is associated to a N = 2 vector on-shell superfield or its conjugate. We now focus on divergent contributions to the amplitude (4.2). Since the non-supersymmetric theory is renormalizable, there is no UV-divergent box integral, and all the amplitude's divergences are linked to the divergences of three-and two-point Green functions. In particular, we write the one-loop corrections to vertices and inverse propagators as Here we have split the D-dimensional gluons into d-dimensional gluons and d-dimensional scalars according to the values of the spacetime indices µ, ν. This is done by introducing the d-dimensional metric η d defined as η d is generated by the integral reduction identities collected in Appendix D, since the loop momentum is taken to be in d = 4 − 2ǫ dimensions. We further split the tree-level, four-point amplitude in contributions corresponding to the three channels, where the vector and scalar channels have been written separately. Taking into account the color factors in Table 3, the integrated numerator factors of the non-supersymmetric theory are related to the part of the one-loop corrections to vertices and propagators which is proportional to the indices of the adjoint and matter representations: where analogous relations for the other diagrams can be obtained by permutation. Using the numerators in Table 3, we have the following expression for the UV-divergent part of the supergravity amplitude, We recognize that the combination of one-loop vertex and propagator corrections in (4.6) corresponds to a piece of the beta function of the non-supersymmetric theory. The beta function β(T (R), T (G), C(R)) will formally depend on the index and quadratic Casimir of the representation, which are independent as long as no particular assumption is made on R. We can then rewrite the result above as where β φ T (G) denotes the part of the beta function for the Yukawa couplings which is proportional to T (G). Contributions from the wave-function renormalization of the external fermions are proportional to C(R) and hence do not appear in the above expression. At one loop, the corrections to the vertices for the right-hand YM theory are obtained from the Feynman diagrams in Figure 1 (see Appendix B for the Feynman rules employed in the calculation), (4.8) This expression can be further simplified by employing the integral reduction identities collected in Appendix D. We obtain: where we have omitted a finite part proportional to the triangle integral I 3 (p 1 , p 2 ). I 2 (k) denote bubble integrals. Bubble-on-external-leg integrals I 2 (p 1 ) and I 2 (p 2 ) have dropped out of the part of the vertex corrections which is proportional to T (G). This implies that the 1/ǫ divergence of the vertex corrections that we will use in (4.6) is interpreted as a genuine UV divergence without any infrared contamination. The calculation for the propagator corrections is performed along similar lines. The final result is:Π where n F is the number of four-dimensional fermions. Equation where c Γ = i/(4π) 2 . By setting D = 4 + n S , where n S is the number of four-dimensional scalars, we obtain our master formula which expresses the value of the superamplitude's one-loop divergence in terms of the parameters of the construction for homogeneous supergravities: (4.12)

Examples
To simplify the expression (4.12) further we specialize on amplitudes between four supergravity vectors, where the global indices of the last two vectors are raised with the inverse charge-conjugation matrix. With this assignment of external polarizations and after using spinor-helicity identities which are collected in Appendix A, we get the following expressions for tree amplitudes in the three channels, where we have written the higher-dimensional Dirac and charge-conjugation matrices as Γ I = γ 5 ⊗Γ I and C = C 4 ⊗C. In this case, the expression (4.12) can be simplified as (4.17) We now consider some interesting particular cases. n S = 0 (D = 4) corresponds to the socalled CP(n) or Luciani model [68]. Supergravities in this family do not uplift to dimension higher than four and have symmetric scalar manifold , (4.18) where n is the number of vector multiplets. The corresponding matter amplitudes between four vectors have one-loop divergence Another important example is the so-called Generic Jordan Family [69,69], which can be obtained by setting D = 6 and keeping the number of 4D fermions arbitrary (with the relation n = n F + 3). 5 This theory has symmetric target space Focusing on amplitudes between two identical vectors and their CPT-conjugate states, we get the expression which reproduces the earlier result in ref. [15]. By inspecting the two terms contributing to (4.12), we see that the contribution corresponding to the vector exchange never vanishes. 6  These are precisely the four Magical supergravities [70,71]. Additionally, the contribution from the remaining channel matches the earlier computation in ref. [17].

Discussion
In this paper, we have calculated the one-loop divergence for selected amplitudes between four vector multiplets in N = 2 homogeneous Maxwell-Einstein supergravities with the double-copy construction, focusing on amplitudes between vector constructed as the double copy of two spin-1/2 fields. Supergravity amplitudes are constructed using as building blocks gauge-theory amplitudes between four hypermultiplets in a presentation that obeys color/kinematics duality. Such amplitudes were first obtained in ref. [66] in terms of kinematical numerators which do not possess any explicit dependence on the loop momentum. Because of this property, the supergravity divergence is directly linked to the beta function of the non-supersymmetric gauge theory. In a sense, our calculation presents analogies with the one in ref. [7], where the absence of some one-and two-loops divergences in half-maximal supergravity was linked to the renormalizability of the non-supersymmetric gauge theory entering the construction thanks to the absence of loop-momenta dependence in N = 4 sYM numerators at one and two loops.
Among the homogeneous supergravities, we do not find any matter amplitude which remains finite at one loop. An open question is how robust is this finding with respect to modifications of the construction. For example, we can generalize the construction by including n φ complex matter scalars in the non-supersymmetric gauge theory. If the scalars are in the same pseudo-real representation as the half-hypermultiplets, the resulting supergravity theory will contain hypermultiplets in addition to the vector multiplets which are already present in the basic construction. The contribution to the supergravity divergence is modified by hypermultiplet loops and eq. (4.12) becomes (5.1) Hence, the additional contribution increases the divergence with respect to the Maxwell-Einstein case. Another possible modification is to add adjoint fermions. Since adjoint and matter representation contributions to (4.6) come with opposite sign, adjoint fermions alleviate the UV divergence. However, they can be introduced in a way that is consistent with color/kinematics duality only if the gauge theory becomes supersymmetric [66].
Our results can be understood in terms of counterterm analysis [72,73,74,75,76,77,78,79,80,81,82,83,84]. The observed divergences correspond to the appearance of the linearized counterterms 2) together with their supersymmetric completions. In the above equation, self-dual and antiself-dual components of the vector field strengths in four dimensions are written using the two-component spinor notation as F αβ = 1 2 F µν σ µν αβ . The second counterterm does not appear in the case of the Magical supergravities, which signals symmetry enhancement corresponding to the enlarged U-duality groups of these theories.
Finally, it would be interesting to see if the relation between supergravity divergences and physical quantities of the non-supersymmetric theory entering the construction (i.e. beta functions) can carry over to other matter amplitudes or higher loops. When supergravity vectors constructed as vector times scalar are taken into account, the numerators for the supersymmetric gauge theory contain explicit dependence on the loop momentum, which makes it difficult to observe a relation between integrated quantities. As for extending the computation to higher loops, amplitudes which manifestly satisfy color/kinematics duality at two loops have recently become available [85] and are likely to trigger further progress.

A Conventions
In this appendix we collect the conventions employed throughout this paper. Our notation can be obtained from the one of Elvang and Huang [86] by replacing η µν → −η µν . Our metric has mostly-minus signature and the Clifford algebra relation is Γ 0 is hermitian while the other gamma matrices are antihermitian. The four-dimensional gamma matrices γ µ are Four-dimensional charge-conjugation and B matrix are taken to be with ǫ 12 = −ǫ 12 = +1. t 1 is a sign to be assigned according to the value of the parameter D in the construction. The charge conjugation matrix obeys the conditions C t 4 = −C 4 and (γ µ ) t = −t 1 C −1 4 γ µ C 4 . Higher dimensional gamma matrices are written as We introduce indices I, J running over the internal dimensions.C andΓ I denote the components of the charge-conjugation matrix acting on the spinor indices corresponding to the internal (D − d) dimensions. The sign t 1 is fixed by the requirement thatCΓ I is always symmetric or, alternatively, that CΓ µ is always antisymmetric. Introducing the spinor-helicity variables as the Majorana condition is rewritten as

B Feynman Rules
In this appendix, we collect the Feynman rules for the non-supersymmetric gauge theory entering the double-copy construction, which are obtained from the Lagrangian (2.5). All momenta are taken as in-going.

C Details on the orbifold numerators
To obtain an explicit presentation of the one-loop (super)amplitude with four external hypermultiplets in a complex representation, it is convenient to start from the amplitude specified in eq. (5.46) of ref. [66] and perform the summation over the orbifold group elements. We consider a Z 3 orbifold taking SU(3N) as the gauge group for the parent theory. The orbifold action is given by where Φ is a generic field of the parent theory. Representation matrices of the R representation are then given byTâβ α = −(P R )Â α (P R )βB(P G )âĈfÂBĈ .

(C.4)
At four points, supersymmetry implies that amplitudes with four external hypermultiplet fields can be organized in superamplitudes which can be directly obtained from the amplitudes between two identical scalars and their conjugates given in ref. [66], It should be noted that the amplitudes in [66] were obtained with a procedure that is not sensitive to bubble-on-external-leg graphs. In principle, it is possible to add back these graphs in a way that preserves color/kinematics duality by adding to all numerators terms proportional to the squares of the external momenta p 2 i . When the external momenta are put on-shell, i.e. the limit p 2 i → 0 is taken, the additional contributions drop out of the final expression in all graphs except the ones with bubbles on one external leg, which have a 1/p 2 i factor in the propagators. However, these graphs can be safely ignored in the present context as they do not contribute to the gravity amplitudes because each of the two numerators entering the double-copy formula is proportional to p 2 i . At the level of gauge-theory amplitudes, bubbles-on-external-legs integrals vanish in dimensional regularization. However, they can lead to non-vanishing contributions if particular kinematical limits (UV or infrared) are inspected.