Two-loop supersymmetric QCD and half-maximal supergravity amplitudes

Journal of High Energy Physics, Sep 2017

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 ≤ 6 dimensions, where the upper bound corresponds to six-dimensional chiral \( \mathcal{N}=\left(1,0\right) \) SYM theory. By exploiting a close connection with \( \mathcal{N}=4 \) SYM theory — and, equivalently, six-dimensional \( \mathcal{N}=\left(1,1\right) \) 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 \( \mathcal{N}=\left(1,1\right) \) and \( \mathcal{N}=\left(2,0\right) \) 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.

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Two-loop supersymmetric QCD and half-maximal supergravity amplitudes

HJE Two-loop supersymmetric QCD and half-maximal supergravity amplitudes Henrik Johansson 0 1 2 3 5 6 Gregor Kalin 0 1 2 5 6 Gustav Mogull 0 1 2 4 5 6 Gauge Symmetry 0 The University of Edinburgh 1 Roslagstullsbacken 23 , 10691 Stockholm , Sweden 2 75108 Uppsala , Sweden 3 Nordita, Stockholm University and KTH Royal Institute of Technology 4 Higgs Centre for Theoretical Physics, School of Physics and Astronomy 5 Department of Physics and Astronomy, Uppsala University 6 Edinburgh EH9 3FD , Scotland , U.K Using the duality between color and kinematics, we construct two-loop fourpoint scattering amplitudes in N = 2 super-Yang-Mills (SYM) theory coupled to Nf fundamental hypermultiplets. Our results are valid in D bound corresponds to six-dimensional chiral N close connection with N = 4 SYM theory | and, equivalently, six-dimensional N = (1; 1) SYM theory | we nd 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 Nf 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 N = (1; 1) and N = (2; 0) supersymmetry. Finally, we extract the potential ultraviolet divergence of half-maximal supergravity Scattering Amplitudes; Supergravity Models; Supersymmetric Gauge Theory - = (1; 0) SYM theory. By exploiting a in D = 5 2 and show that it non-trivially cancels out as expected. Keywords: Scattering Amplitudes, Supergravity Models, Supersymmetric Gauge Theory, Gauge Symmetry 1 Introduction Review 2 3 Four dimensions Six dimensions 3.2.1 3.2.2 Cuts 3.3.1 3.3.2 2.1 2.2 3.1 3.2 3.3 4.1 4.2 4.3 4.4 4.5 4.6 5.1 5.2 4 Calculation of N = 2 SQCD numerators Four-dimensional correspondence Six-dimensional amplitudes One-loop example Two-loop cuts Master numerators Ansatz construction Symmetries and unitarity cuts Additional constraints 4.4.1 4.4.2 4.4.3 Matter-reversal symmetry Two-term identities Matching with the N = 4 limit All-chiral solutions MHV solutions, bubbles and tadpoles N = 0 5.1.1 5.1.2 N = 2 5.2.1 5.2.2 One loop Two loops One loop Two loops N = 4 construction N = 2 construction Color-kinematics duality with fundamental matter Pure supergravities from the double-copy prescription Trees and cuts in N = 2 SQCD { 1 { Introduction It is by now well established that a variety of gauge and gravity theories are perturbatively related through the so-called double copy. In terms of the asymptotic states a squaring relation between gauge theory and gravity follows readily from representation theory; that such a structure is preserved by the interactions is a remarkable fact rst brought to light by the Kawai-Lewellen-Tye (KLT) relations [1] between open and closed string amplitudes. More recently, the double copy has been understood to arise due to the Bern-CarrascoJohansson (BCJ) duality between color and kinematics [2, 3] that is present in many familiar gauge theories. This realization has opened the path towards constructions of scattering amplitudes in a multitude of di erent gravity theories [3{10] starting from the much simpler gauge-theory amplitudes. The duality between color and kinematics refers to the observation that many familiar gauge theories have a hidden kinematic structure that mirrors that of the gauge-group color structure [2, 3]. Amplitudes in, for example, pure Yang-Mills or super-Yang-Mills (SYM) theories in generic dimensions can be brought to forms where the kinematic numerators of individual diagrams obey Jacobi relations in complete analogue with the color factors of the same diagrams. A natural expectation is that one (or several) unknown kinematic Lie algebras underlie the duality [11, 12]. At tree level the duality is known to be equivalent to the existence of BCJ relations between partial amplitudes [2], which in turn have been proven for pure SYM theories through a variety of di erent techniques [13{17]. As of yet there is no proof of the duality at loop level, although a number of calculations have established its presence up to four loops in the maximally-supersymmetric N = 4 SYM theory [3, 18{23], up to two loops for pure YM [24], and at one loop in matter-coupled YM theories with reduced supersymmetry [4{7, 25, 26]. Color-kinematics duality has been shown to be present in weakly-coupled quantum chromodynamics (QCD) and generalizations thereof [6, 27]; this includes Yang-Mills theory coupled to massive quarks in any dimension, and corresponding supersymmetric extensions. The BCJ amplitude relations for QCD were worked out in ref. [27] and proven in ref. [28]. Using the duality for practical calculations in QCD phenomenology is a promising avenue as it poses strong constraints on the diagrammatic form of an amplitude, interweaving planar and non-planar contributions to the point of trivializing certain steps of the calculation [29{31]. In the limit of massless quarks the (super-)QCD amplitudes are a crucial ingredient in the double-copy construction of pure N = 0; 1; 2; 3 supergravities in four dimensions [6], as well as for the symmetric, magical and homogeneous N = 2 supergravities [9], and twin supergravities [ 10 ]. The double copy is most straightforwardly understood as replacing the color factors in a gauge-theory amplitude by corresponding kinematic numerators [2, 3]. When color-kinematics duality is present this replacement respects the Lie-algebraic structure of the amplitude, and furthermore enhances the gauge symmetry to di eomorphism symmetry [32{34]. Additional enhanced global symmetries typically also arise [35{39]. The resulting amplitudes are expressed in terms of Feynman-like diagrams with two copies of kinematical numerators, and describe scattering of spin-2 states in some gravitational the{ 2 { ory. The double copy gives valid gravity amplitudes even if the kinematic numerators are drawn from two di erent gauge theories [2], as long as the two theories satisfy the same type of kinematic Lie algebra [7] (with at least one copy manifestly so [3, 40]). This exibility of combining pairs of di erent gauge theories has given rise to a cornucopia of double-copy constructions [3{10, 19, 41{46]. More recently the double copy and color-kinematics duality has been observed to extend to e ective theories such as the non-linear-sigma model (chiral Lagrangian), (Dirac-) Born-Infeld, Volkov-Akulov and special galileon theory [12, 47{51]. Likewise, the double copy and duality show up in novel relations involving and interconnecting stringand eld-theory amplitudes and e ective theories [51{61]. The double copy has been extended beyond perturbation theory to classical solutions involving black holes [62{65] and gravitational-wave radiation [66{69]. In the work by Ochirov and one of the current authors [6] a detailed prescription was given for removing unwanted axion-dilaton-like states that appear in a naive double-copy construction of loop-level amplitudes of pure N = 0; 1; 2; 3 supergravities in D = 4 dimensions. The prescription calls for the introduction of compensating ghost states that are obtained by tensoring fundamental matter multiplets of the two gauge theories entering the double copy. For consistency of the double copy, the kinematic numerators of the gauge theories need to obey Jacobi identities and commutation relations mirroring the adjointand fundamental-representation color algebra. The gauge theories are thus di erent varieties of massless (super-)QCD theories with a tunable parameter Nf corresponding the number of quark avors. Speci cally, the gravitational matter is turned into ghosts by choosing Nf = 1 and Nf = 1 for the two gauge theories, respectively. The prescription was shown to correctly reproduce the one-loop amplitudes in pure N = 0; 1; 2; 3; 4 supergravity. In this paper, we obtain the two-loop amplitudes in N = 2 super-QCD (SQCD) which are needed for the computation of pure supergravity amplitudes at this loop order. Using color-kinematics duality we compute the four-vector amplitude at two loops with contributions from Nf internal hypermultiplets transforming in the fundamental representation. The amplitudes are valid for any gauge group G and for any dimension D 6. Dimensional regularization, in D = 4 2 dimensions, requires that the integrands are correct even for D > 4 dimensions, and D = 6 is the maximal uplift where the theory exists. To de ne the six-dimensional chiral theory, N = (1; 0) SYM coupled to hypers, we make use of its close relationship to the maximally supersymmetric N = (1; 1) SYM theory, whose tree amplitudes are conveniently written down using the six-dimensional version of spinor-helicity notation [70]. Using the two-loop amplitudes in N = 2 SQCD, which obey color-kinematics duality, we compute the (N = 2; Nf ) (N = 2; Nf = 1) double copy and obtain two-loop amplitudes in N = 4 supergravity coupled to 2(Nf + 1) vector multiplets. For the choice Nf = 1 we obtain pure N = 4 supergravity amplitudes. It should be noted that pure N = 4 supergravity amplitudes can alternatively be obtained from a (N = 4) (N = 0) double copy [19, 41, 43, 45, 71{74], without the need of removing extraneous states. Indeed, the two-loop four-point N = 4 supergravity amplitude was rst computed this way in ref. [41], and with additional vector multiplets in ref. [73]. { 3 { Our construction of the half-maximal supergravity amplitudes has several crucial virtues compared to previous work. Obtaining the N = 2 SQCD amplitude in the process is an obvious bonus; this amplitude is a stepping stone for computing the pure N = 2; 3 supergravity two-loop amplitudes, as explained in ref. [6]. Furthermore, in D = 6 the chiral nature of the N = (1; 0) SYM theory allows for the double-copy construction of two inequivalent half-maximal supergravity amplitudes: N = (1; 1) and N = (2; 0) supergravity. Only the former is equivalent to the (N = 4) (N = 0) construction. The latter theory, pure N = (2; 0) supergravity, has a chiral gravitational anomaly that shows up at one loop, thus rendering the double-copy construction of the two-loop amplitude potentially inconsistent in D = 6. However, it is known that the anomaly can be canceled by adding 21 self-dual N = (2; 0) tensor multiplets to the theory [75]. This corresponds to Nf = 10 in the current context. We extract the ultraviolet (UV) divergences of the half-maximal supergravity two-loop integrals in D = 4 and D = 5. In D = 4 the integrals are manifestly free of divergences by inspection of the power counting, whereas in D = 5 the individual integrals do diverge but after summing over all contributions to the amplitude divergences cancel out; thus exhibiting so-called enhanced UV cancellations [44, 74]. This non-trivially agrees with the calculations of the UV divergence of the same amplitudes in ref. [73], and thus provides an independent cross-check of our construction. While we do not compute the D = 6 2 already at one loop, thus the two-loop amplitude has both 1= 2 and 1= poles [73]. divergences, we note that the pure N = (1; 1) theory should have a divergence starting This paper is organized as follows: in section 2, we review color-kinematics duality and the double copy with special attention on the fundamental matter case, and pure supergravities. In section 3, we introduce the tree amplitudes and needed two-loop cuts for N = 2 SQCD both in four and six dimensions. In section 4, the details of the calculation of the two-loop numerators of SQCD are presented. In section 5, the corresponding supergravity amplitudes are assembled and the UV divergence in ve dimensions is computed. Conclusions are presented in section 6 and the explicit N = 2 SQCD numerators and color factors are given in appendix A. 2 Review Here we review color-kinematics duality in supersymmetric Yang-Mills (SYM) theory and its extension to fundamental matter, following closely the discussion in ref. [6]. We also show how the duality may be used to compute amplitudes in a wide variety of (super)gravity theories using the double copy prescription. 2.1 Color-kinematics duality with fundamental matter We begin by writing L-loop (supersymmetric) Yang-Mills amplitudes with fundamental matter as sums of trivalent graphs: Am (L) = iL 1gYmM+2L 2 X cubic graphs i Z dLD` 1 nici ; (2 )LD Si Di (2.1) { 4 { where gYM is the coupling. The structure of the gauge group G allows two types of trivalent vertices: pure-adjoint, and those with a particle in each of the adjoint, fundamental tr(T aT b) = ab. Diagrammatically, these can be represented as and anti-fundamental representations. The color factors ci may thus be expressed as products of structure constants f~abc = tr([T a; T b]T c) and generators Tia|, normalized such that f~abc = c c ; Tia| = c a ¯  i : Si and Di are the usual symmetry factors and products of propagators respectively. Finally, kinematic numerators ni collect kinematic information about the graphs. HJEP09(217) The algebraic structure of the gauge group gives rise to linear relationships between the color factors. In the adjoint representation these are Jacobi identities between the structure constants,1 f~a1a2bf~ba3a4 = f~a4a1bf~ba2a3 f~a2a4bf~ba3a1 ; 4 3 1 2 c 4 3 and in the fundamental representation they are commutation relations, Ti1{2 f~ba3a4 = Tia13|Tja{42 b Tia14|Tja{32 = [T a3 ; T a4 ]i1{2 ; 1 2 : 1 2 ; c c 4 3 4 3 a b 1 2 = c 4 3 1 2 = c 1 2 c 4 3 ci = cj ck: The identities lead to relationships between the color factors of the form Furthermore, the ip of two legs at a pure-adjoint vertex leads to an overall sign ip, f~abc = f~acb. The same behavior can be implemented when swapping fundamental with anti-fundamental legs by de ning new generators T{aj = Tja{. Kinematic numerators ni satisfying color-kinematics duality | we shall refer to these as color-dual | obey the same linear relations as their respective color factors. We demand that ni = nj ni ! nk ni , ci = cj ci ! ci: ck; The second identity incorporates a sign change for a ip of two legs at a given cubic vertex. The existence of a set of color-dual numerators is generally non-trivial. At tree level, their existence has been proven in D-dimensional pure (super-)Yang-Mills theories using an inversion formula between numerators and color-ordered amplitudes [76, 77]. This relies on the partial amplitudes satisfying BCJ relations [2, 13{15]. For (super-)Yang-Mills theories with arbitrary fundamental matter, where less is known, the corresponding BCJ relations [6] have been proven for QCD [28]. 1This may also be viewed as a commutation relation in the adjoint representation (Taadj)bc = f~bac. { 5 { (2.2) (2.3) (2.4) (2.5) (2.6) When including matter multiplets it is also useful to consider the two-term identity c 4 3 1 2 =? c 4 3 1 2 (2.7) for indistinguishable matter multiplets. Depending on the gauge-group representation, this may or may not hold. Although fundamental representations of generic gauge groups do not obey this relation, the generator of U(1) does, as well as generators of particular tensor representations of U(Nc). It turns out that it is possible to nd color-dual numerators that ful ll these two-term identities. However, the two-term identities are not necessary for the double-copy prescription [3] that we will now discuss. 2.2 Pure supergravities from the double-copy prescription Our main motivation for nding color-dual numerators of gauge-theory amplitudes is the associated double-copy prescription: this o ers a simple way to obtain gravity amplitudes in a variety of (supersymmetric) theories. The inclusion of fundamental-matter multiplets widens the class of gravity amplitudes obtainable in this way; in particular, it enables the double-copy construction of pure N < 4 supergravity amplitudes, including N = 0 Einstein gravity [6]. In this paper we will mostly be interested in pure N = 4 (half-maximal) supergravity at two loops; however, the approach that we now review also applies more generally. The basic setup is as follows. Take a pair of N ; M 2 SYM multiplets, VN and VM0, both of which are non-chiral | for instance, the former should include 2N chiral and 2N anti-chiral on-shell states. Then analyze their double copy: one recovers a factorizable graviton multiplet of the form eq. (2.8) one assigns opposite statistics to X, X in eq. (2.9) are ghosts. N (but not to 0M), which in turn means that In this paper N = M = 2; the gauge-theory matter multiplets are hypermultiplets (hypers). The complex matter N (and the conjugate N ) belong to the fundamental (antifundamental) representation of the gauge group G. In order to cancel the matter in The usual numerator double copy is modi ed when concealing unwanted matter states. Starting from a color-dual presentation of the SYM amplitude (2.1) one replaces the color factors ci by numerators of the second copy as usual. However, one also now includes an appropriate sign that enforces the opposite statistics of internal hyper loops, and the numerator is conjugated (reversing matter arrows), ci ! ( 1)jijn0i ; { 6 { where jij is the number of matter loops in the given diagram. It is useful to generalize the above replacement by promoting ( 1) be a tunable parameter, Nf , which counts the number of fundamental matter multiplets in the rst SYM copy (setting Nf = 1 in the second copy), ci ! (Nf )jijn0i : In the supergravity theory, the number of complexi ed matter multiplets is then NX = (1+ Nf ) where, depending on the amount of supersymmetry, X is either a complexi ed vector, hyper, chiral multiplet or complex scalar. For instance, when Nf = 0 the factorizable double copy is recovered (we take 00 = 1). After the double copy is performed, we can assemble terms on common denominators and thus consider (super-)gravity numerators given by where all numerators sharing the same propagator Di are summed over, i.e. all possible routings of internal vector and complex matter multiplets. The bar on top of the second numerator implies a reversal of hyper loop directions, i.e. conjugating the hypermultiplets. It is su cient that only one of the two numerator copies, ni or n0i, is color-dual. We are left with complete (super-)gravity amplitudes: Mm (L) = iL 1 2 m+2L 2 X consists of N = 2 SYM coupled to supersymmetric matter multiplets (hypermultiplets) in the fundamental representation of a generic gauge group G. For simplicity, we consider the limit of massless hypers.2 N = 2 SQCD's particle content is similar to that of N = 4 SYM, so we refrain from describing its full Lagrangian; instead, we project its tree amplitudes out of the well-known ones in N = 4 SYM. We then proceed to calculate generalized unitarity cuts. As our intention is to evaluate cuts in D = 4 2 dimensions, we also consider six-dimensional trees. These are carefully extracted from the N = (1; 1) SYM theory | the six-dimensional uplift of four-dimensional N = 4 SYM. The on-shell vector multiplet of N = 4 SYM contains 2N = 16 states: VN =4( I ) = A+ + I I+ + 2 1 I J 'IJ + 1 3! IJKL I J K L + 1 2 3 4A ; (3.1) where we have introduced chiral superspace coordinates I with SU(4) R-symmetry indices fI; J; : : :g. The N = 4 multiplet is CPT self-conjugate and thus it is not chiral. The 2Amplitudes with massive hypers can be recovered from the massless case by considering the D = 6 version of the theory and reinterpreting the extra-dimensional momenta of the hypers as a mass. { 7 { VN =2( I ) = VN =2 + 3 4 V N =2; VN =2( I ) = A+ + I I+ + 1 2'12; V N =2( I ) = '34 + I34J I J + 1 2A ; VN =4 = VN =2 + N =2 + N =2; N =2( I ) = ( 3+ + I 'I3 + 1 2 4 ) 3; N =2( I ) = ( 4+ + I 'I4 where the SU(2) indices I; J = 1; 2 are inherited from SU(4). As is obvious, this is a subset of the full N = 4 multiplet: where we have introduced a hypermultiplet and its conjugate parametrized as apparent chirality is an artifact of the notation; it can equally well be expressed in terms of anti-chiral superspace coordinates I : VN =4( I ) = 1 2 3 4A+ + + A : The on-shell spectrum of N = 2 SYM contains 2 2N = 8 states, which can be packaged into one chiral VN =2 and one anti-chiral V N =2 multiplet, each with four states. As explained in ref. [6], it is convenient to combine them into a single non-chiral multiplet VN =2 using the already-introduced superspace coordinates: One can easily check that these elements make up the full N = 4 spectrum (3.1). The particle content of N = 2 SQCD is the same as that of ordinary N = 4 SYM, except that the hypermultiplet N =2 ( N =2) transforms in the fundamental (antifundamental) representation of the gauge group G. We also generalize to Nf = avors by attaching avor indices f ; ; : : :g to the hypers, i.e. ( N =2) , ( N =2) . The di erent representation a ects the color structure of the amplitudes and the avor adds more structure; however, when considering color-ordered tree amplitudes with identically- avored hypers the distinction between N = 2 SQCD and N = 4 SYM is insigni cant. The former amplitudes are obtainable from the latter by decomposing according to the above N = 2 multiplets, as we shall now demonstrate. For multi- avor N = 2 SQCD tree amplitudes the procedure to extract them from the N = 4 ones is more complicated, although some tricks have been developed for this purpose [78, 79]. 3.1 Four dimensions SYM [80, 81]:3 To calculate four-dimensional tree amplitudes we start from the color-stripped ParkeTaylor formula for maximally-helicity-violating (MHV) scattering of n states in N = 4 hVN =4VN =4 VN =4iMHV = i 8(Q) h12ih23i hn1i ; (3.6) 3Here we adopt the usual spinor-helicity notation, see for example [82, 83]. { 8 { (3.2) (3.3a) (3.3b) (3.3c) (3.4) (3.5a) (3.5b) where for N supersymmetries the supersymmetric delta function is 2N (Q) = Y Xhi ji iI jI : N n I=1 i<j By inspection of the hypers (3.5), N =2 states are identi ed by a single 3 factor and states carry 4. In the full N = 2 multiplet VN =2 (3.3) a factor of 3 4 identi es a state belonging to V N =2; anything else belongs to V N =2. This enables us to project out the relevant MHV amplitudes in N = 2 SQCD, giving color-ordered amplitudes where the ellipsis represents insertions of VN =2. We will only use tree amplitudes with up to ve external legs, so MHV and MHV are su cient. 3.2 Six dimensions The six-dimensional uplift of four-dimensional N = 4 SYM is N = (1; 1) SYM; an on-shell superspace for this theory was introduced by Dennen, Huang and Siegel (DHS) [84]. The DHS formalism uses a pair of Grassmann variables, a and ~a_ , carrying little-group indices in the two factors of SU(2) SU(2) SO(4). The full non-chiral on-shell multiplet is VN =(1;1)( a; ~a_ ) = + a a + ~a_ ~a_ + 0( )2 + gaa_ a ~a_ + 0(~)2 + ~a_ ( )2 ~a_ + a a(~)2 + ( )2(~)2; (3.9) change the handedness of (anti-)chiral spinors a ( ~a_ ). where ( )2 = 12 a a, (~)2 = 12 ~a_ ~a_ . Note that in six dimensions CPT conjugation does not 3.2.1 Four-dimensional correspondence The correspondence between four-dimensional N = 4 SYM and six-dimensional N = (1; 1) SYM is most easily seen using a non-chiral superspace construction of the N = 4 multiplet [85]. Two chiral and two anti-chiral superspace coordinates are used to capture N = 4 SYM's 16 on-shell states: we choose 1 superspaces is done with half-Fourier transforms: , 2 , 3 and 4 . Switching between the Z chiral( I ) = d 2d 3 e 2 2+ 3 3 non-chiral( I=1;4; I=2;3): In terms of the non-chiral superspace, a valid mapping to six dimensions is [86] a $ ( 1; 2); ~ a_ $ ( 3; 4): { 9 { (3.7) N =2 (3.8a) (3.8b) (3.8c) (3.10) (3.11) A+ ψ+ I ϕIJ ψI − A − η I ηJ ηK ηL η˜ g21˙ χ˜ ˙ 1 ¯ χ˜ ˙ 1 HJEP09(217) (3.12) and ~ act as raising and lowering operators in the weight space [87] as denoted by solid black arrows. Red-colored states belong to the vector multiplet; green- and bluecolored states belong to the two parts of the D = 6, N = (1; 0) hypermultiplet. The dotted gray arrows show how 6D states are projected to the four-dimensional weight space. This allows us to identify on-shell states between the four-dimensional N = 4 and sixdimensional N = (1; 1) multiplets. We do this for the N = 2 vector (3.3) and matter (3.5) multiplets separately; the results are illustrated in gure 1. Starting with the N = 2 vector multiplet (3.3), we switch to the four-dimensional non-chiral superspace and then use the coordinate mapping (3.11): Z VN =2( a; ~a_ ) = d 3d 2e 2 2+ 3 3 VN =2( I ) = ~1_ ( 2+ 1'12 + 2A+ ( 1; 2)! a;( 3; 4)!~a_ _ 1 2 1+) + ~2( 1 1A + 2'34 This clearly forms part of VN =(1;1): for instance, 2+ = ~1_ and '12 = g11_ . As it contains a six-dimensional vector and a pair of anti-chiral spinors, we identify it as the on-shell multiplet of N = (1; 0) SYM [88]: Hypermultiplets are transformed by the same procedure: VN =(1;0)( a; ~a_ ) = ~a_ ~a_ + gaa_ a ~a_ + ~a_ ( )2 ~a_ : N =2( a; ~a_ ) = '23 + 1 N =2( a; ~a_ ) = ~1_ ~2_ ('24 4 1 + Each of these contains a chiral spinor and a pair of scalars, so we recognize them as the two parts of the N = (1; 0) hypermultiplet: N =(1;0)( a; ~a_ ) = N =(1;0)( a; ~a_ ) = 0(~)2 + + a a + 0( )2; a a(~)2 + ( )2(~)2: The three multiplets make up VN =(1;1) (3.9) as expected. Notice that the above procedure was not unique: we could instead have chosen to identify VN =2 with VN =(0;1), which carries the chiral spinors a and a . The choice was made when we picked the speci c mapping between four- and six-dimensional states (3.11).4 Amplitudes in this theory are given in terms of the six-dimensional spinor-helicity formalism [70]. For each particle i they depend on the chiral and anti-chiral spinors, i and ~i, and the supermomenta, q iA = i A;a i;a; ~qi;A = ~i;A;a_ ~i : a_ (no sum over i) (3.16) HJEP09(217) The indices fA; B; : : :g belong to SU*(4) | the spin group of SO(1,5) Lorentz symmetry. For notational convenience we de ne the two four-brackets: hia; jb; kc; ldi [ia_ ; jb_ ; kc_; ld_] ABCD i j A;a B;b C;c D;d ; k l ABCD ~i;A;a_ ~j;B;b_ ~k;C;c_ ~l;D;d_ : The four-point color-ordered tree amplitude of N = (1; 1) SYM is [84] hVN =(1;1)VN =(1;1)VN =(1;1)VN =(1;1)i D=6 = i 4(Pi qi) 4(Pi ~qi) ; st where s = (p1 + p2)2 and t = (p2 + p3)2; the fermionic delta functions are 4 4 X qi = X ~qi = i i 1 1 X 4! i;j;k;l X 4! i;j;k;l hia; jb; kc; ldi i;a j;b k;c l;d; ia_ ; jb_ ; kc_; ld_ ~ia_ ~jb_ ~kc_ ~ld_: This amplitude is invariant under qi $ ~qi | a re ection of the non-chiral nature of N = (1; 1) SYM. In six dimensions there is no notion of helicity sectors (like NkMHV sectors in four dimensions) because rotations of the little group SO(4) SU(2) SU(2) mix them. The above superamplitude is therefore a single expression that uni es all the four-dimensional helicity sectors. We can now write down six-dimensional equivalents of the four-dimensional N = 2 SQCD trees (3.8). States are identi ed using the ~a_ variables: we project these out of 4(Pi ~qi) in the four-point amplitude (3.18). With four external legs the color-ordered amplitudes are hV1V2V3V4i h 1 2; V3V4i h 1 2; 3; D=6 = D=6 = 4i D=6 = i i i st st 4(Pi qi) [1a_ ; 2b_ ; 3c_; 4d_]~1a_ ~ 2b_ ~3c_ ~4d_; 4(Pi qi) [21_ ; 22_ ; 3c_; 4d_]~21 ~2 ~ _ 2_ 3c_ ~4d_; st 4(Pi qi) [21_ ; 22_ ; 31_ ; 32_ ]~21 ~2 ~ _ _ _ _ 2 31 ~32; 4For instance, in ref. [85] the states are instead identi ed as a $ ( 1; 4), ~a_ $ ( 3; 2). (3.17a) (3.17b) (3.18) (3.19a) (3.19b) (3.20a) (3.20b) (3.20c) where f ; ; : : :g are avor indices. These amplitudes are not invariant under qi $ ~qi: our use of the VN =(1;0) multiplet implies that ~qi encodes information about the external states. In principle, we will also need ve-point amplitudes when computing three-particle cuts of the two-loop amplitude. However, the ve-point N = (1; 1) SYM tree amplitude [84] is less compact and, as we will discuss in section 4.3, it turns out that these cuts do not provide any new physical information compared to the four-dimensional ones given the constraints on our ansatze. When the momenta are taken in a four-dimensional subspace the six-dimensional trees should reproduce their four-dimensional counterparts (3.8). This is easily con rmed: write the six-dimensional spinors in terms of four-dimensional ones [70], a = m = `4 i `5; me = `4 + i `5: 5For a good review of both four- and six-dimensional unitarity techniques see ref. [90]. where here the Greek letters denote SL(2,C) spinor indices (not to be confused with the previously-introduced avor indices, the distinction should be clear from the context). Then substitute into the six-dimensional tree amplitudes (3.20). Finally, switch to fourdimensional fermionic coordinates using the half-Fourier transform (3.10) and the state mapping (3.11) on each of the external legs. A helpful intermediate relationship is [85] Z 4 i=1 Y d i;2 exp 4 X i=1 i;2 i 2 2 4 4 X qi i 3 a!( 1; 2)5 = ~a_ !( 3; 4) h34i which exchanges the fermionic delta functions. This procedure gives all external helicity sectors in four dimensions, but at four points only the MHV is non-zero. 3.3 Cuts In the next section we will use ansatze to construct color-dual numerators for N = 2 SQCD. The physical input for these will come from unitarity cuts. In four dimensions the cuts are calculable using the MHV tree amplitudes (3.8) | the techniques involved are fairly standard and we will not repeat them here. In six dimensions we follow ref. [89]; however, the tree amplitudes (3.20) create some new subtleties that we will now address.5 As we always take external states in a four-dimensional subspace, the key to this approach is nding six-dimensional spinor solutions for the on-shell loop momenta in terms of four-dimensional external spinors. We write the six-dimensional massless on-shell condition as a four-dimensional massive condition: `2 = ` 2 (`4)2 (`5)2 = ` 2 m me = 0; where ` is the part of ` living in the four-dimensional subspace. The complex masses are related to the fth and sixth components of `: (3.21) (3.22) (3.23) (3.24) Now considering ` as a massive on-shell four-momentum we write it in terms of two pairs of four-dimensional spinors, ( ; ~) and ( ; ~), as: ` _ = ~ _ + mm e 2 ~ _ : Appropriate six-dimensional spinors are then [87] A a = im e h i ~ _ By taking `4 = `5 = 0, and therefore m = m = 0, we recover the four-dimensional e We now illustrate the cutting method by calculating the one-loop quadruple cut in gure 2. Without loss of generality we set Nf = 1; powers of Nf can be restored at any point by counting the number of hyper loops. The quadruple cut integrand is then Cut 1 2 = Z 2 i=1 Y d2 li d2 ~li (l2 p3)2A(0) L (l1 p1)2A(R0): (3.27) 4 3 1 2 i i s (3.25) (3.26) HJEP09(217) (3.28a) (3.28b) (3.29) A(0) = L A(R0) = s(l2 s(l1 4(Pi2L qi) 4(Pi2R qi) p3)2 [(l1)1_ ; (l1)2_ ; 3c_; 4d_]~l1 l1 3c_ ~4d_; 1_ ~2_ ~ p1)2 [(l2)1_ ; (l2)2_ ; 1a_ ; 2b_ ]~l2 l2 1a_ ~2b_ ; 1_ ~2_ ~ and, with ` = l1 = p1 + p2 + l2, the on-shell conditions are `2 = (` p1)2 = (` p1 p2)2 = (` + p4)2 = 0. Putting the pieces together, and summing over the supersymmetric states, we obtain the cut for internal hypers: Cut 4 3 = 4(Pi qi) [(l2) 1_; (l2) 2_; 1a_ ; 2b_ ][(l1)1_ ; (l1)2_ ; 3c_; 4d_]~1a_ ~ 2b_ ~3c_ ~4d_: This expression can now be converted to external spinors and loop momenta using the previously-given expressions. The full details on how to work this out may be found in ref. [89]; we simply quote the nal results below. The external states are purely four-dimensional, and thus through the half-Fourier transform (3.10) we may split up the cut into the contributions from sectors of di erent powers of the four-dimensional Grassmann variables iI . Unlike the maximally-supersymmetric case, the N = 2 SYM loop amplitudes will not have automatically-vanishing integrands when all external states are picked from the chiral vector multiplet VN =2 | thus, besides the MHV sector we will also have an \all-chiral" helicity sector (as well as an \all-antichiral" sector that is trivially obtained by CPT conjugation). A further sector would be Instead of starting from the (rather complicated) three-point six-dimensional amplitudes we study factorization limits of the simpler four-point amplitudes. Using (3.20b), taken on shell. To compute it, we rst 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 nite. the double copy. The cut (3.29) then becomes the one-chiral-vector case (and its CPT conjugate), but we will not consider it here. All sectors are consistent with N = 2 supersymmetry, but the all-chiral integrand will only give rise to O( ) contributions in D = 4 2 dimensions. Nevertheless, the integrand of the all-chiral sector is important as it gives non-vanishing supergravity amplitudes through We will come back to the all-chiral case shortly, but for now consider the MHV sector. 4 CutMHV 3 1 2 = mm e s where we have introduced the variables ij = h12ih34i to identify contributions from external legs i, j belonging to the anti-chiral vector multiplet V N =2. The six-dimensional cut contains all the information to be formally re-interpreted as a D-dimensional cut. Strictly speaking, it is only valid for D 6 since the theory does not exist beyond six dimensions, but we may write it in a dimensionally-agnostic form. Parametrizing the dimension as D = 4 2 , the loop momentum is conveniently split into the four- and extra-dimensional parts as ` = ` + . As external momenta are pi are purely four-dimensional, we expect that appears in the Lorentz-invariant combination agrees with the D-dimensional fundamental box numerator found in ref. [6]. 2 = > 0. Indeed, we may eliminate m and me in favor of 2 = mem. The cut then Now consider the all-chiral sector, which is identi ed by all external states belonging to the chiral VN =2 multiplet. In this case the cut (3.29) becomes (3.30) (3.31) (3.32) 4 Cutall-chiral 3 1 2 = [12][34] h12ih34i m2 4(Q): propagators are understood as being taken on shell. Interestingly, rotational symmetry in the extra-dimensional direction has been broken. Naively, we cannot eliminate the complex variable m2 in favor of the rotational invariant 2 (as we did in the MHV sector). This is because the external states probe the extradimensional space and pick out a chiral direction. Indeed, the component amplitudes in the all-chiral sector always have either scalars or fermions (of the vector multiplet) on the external legs, and these particles are secretly six dimensional (e.g. the scalars are the extra-dimensional gluons). There is no all-gluon amplitude in the all-chiral sector, and thus there is no component amplitude in this sector that involves only four-dimensional external states. We can avoid the issue with complex momenta by instead choosing to work with a vedimensional embedding corresponding to m = me. This should be su cient to parametrize the physical content of the dimensionally-regulated theory. In ve dimensions the spinors belong to USp(2,2) SO(1,4), which means that they can be made to satisfy a reality condition (see e.g. recent work [91]): 2 is an element of the six-dimensional Cli ord algebra. This e ectively identi es the little-group SU(2) labels a and a_. One can then write 2 = m2 = me2, which The two-loop cuts that we wish to calculate are displayed in gures 3 and 4. As the oneloop feature of complex extra-dimensional momenta related to chiral states remains, at two loops we choose to keep all the degrees of freedom of the six-dimensional momenta. The integrands presented will therefore be unavoidably chiral. We will continue to use the complex m and me variables, which we generalize to mi and mei for di erent loop momenta `i = `i + i. These are related to the extra-dimensional parts of the loop momenta by ij = me(imj); ( i; j ) = i me[imj]; (3.34) where ij = i j > 0. Again, for the four-point amplitude the issue of complex momenta is only relevant to the all-chiral sector, while the MHV sector will also have a chiral dependence introduced through the antisymmetric combination ( 1; 2). As we exposed propagators are understood as being taken on shell. Notice that cuts (b) and (c) di er by the routing of fundamental hypermultiplets on the right-hand side. shall see, a non-chiral D-dimensional amplitude will emerge in half-maximal supergravity when the double copy is performed between the amplitudes of a chiral and an anti-chiral gauge theory. The procedure for calculating the two-loop cuts is exactly the same as we have outlined at one loop. A minor distinction is that we avoid excessive analytical manipulations of the cuts: for the purposes of tting an ansatze in the next section we found it su cient to compute the cuts numerically on various phase-space points. 4 Calculation of N = 2 SQCD numerators In this section we describe the computation of color-dual N = 2 SQCD numerators. All external states are taken from the vector multiplet VN =2 (3.3). Without loss of generality, we consider color-dual numerators with a single hypermultiplet avor (Nf = 1); powers of Nf are straightforward to restore at the end by counting the number of hyper loops in a given diagram. We compute both the all-chiral-vector and MHV external-helicity sectors, and as the discussion for the two sectors is mostly the same, we present them alongside each other. The full list of diagrams involved in our all-chiral and MHV solutions is given in gure 5. Expressions for their numerators are presented in section 4.5 and appendix A respectively. The solutions are also provided in ancillary les attached to the arXiv version of this paper. In the MHV sector we provide two alternative solutions. The rst of these includes nonzero numerators corresponding to bubble-on-external-leg and tadpole diagrams | diagrams (17){(24) in gure 5. All of these diagrams have propagators of the form 1=p2 1=0 that are ill-de ned in the on-shell limit (unless amputated away). Their appearance in the solution follow from certain desirable but auxiliary relations that we choose to impose on the color-dual numerators, making them easier to nd through an ansatz. As we shall (10) (15) (20) HJEP09(217) (1) (6) (11) (2) (7) (12) (16) (17) (18) (19) (21) (22) (23) (24) The eight master graphs that we choose to work with are (1){(5), (13), (19) and (22). tadpoles and external bubbles are dropped from the nal amplitudes it is useful to consider them at intermediate steps of the calculation. see, contributions from these diagrams either vanish upon integration or can be dropped because the physical unitarity cuts are insensitive to them. However, as they potentially can give non-vanishing contributions to the ultraviolet (UV) divergences in N = 2 SQCD, we also provide an alternative color-dual solution in which such terms are manifestly absent. The two solutions may be found in separate ancillary les. To aid the discussion we generally represent numerators pictorially (as we have already done for color factors and cuts). For example, the double-box numerator is represented as n1(1234; `1; `2) = n 4 ←ℓ2ℓ1→ 1 3 2 : (4.1) Similar correspondences are made for the other diagrams in gure 5. (3) (8) (13) 4.1 We begin by identifying a basis of master numerators for two-loop four-point diagrams and for external vector multiplets. All other numerators, called descendants, are expressed as linear combinations of the masters using Jacobi identities. For instance, double-triangle numerators are given as di erences of double boxes: Considering the combined equation system of all all Jacobi (2.3) and commutation (2.4) relations, it is straightforward to nd eight masters: two with no internal matter content, three with a single matter loop and three with two matter loops. The same choice is made for both the all-chiral and MHV sectors: these are speci ed in gure 5. Our task is to calculate expressions for the numerators of these diagrams. The reduction of a given numerator to a linear combination of masters is generally not unique. Additional numerator relations therefore give a rst set of consistency constraints for the masters. Further constraints come from (i) the requirement that all numerators satisfy the symmetries of their color factors and (ii) that the overall integrand matches the unitarity cuts given in gures 3 and 4. In the MHV sector, we satisfy these requirements by nding suitable ansatze for the master numerators and solving for constraints on the relevant coe cients. In the all-chiral sector this is unnecessary: the system is su ciently simple that the solution can simply be postulated by knowledge of the maximal cuts. 4.2 Ansatz construction Our procedure for constructing ansatze for the eight MHV-sector master numerators is a straightforward two-loop extension of the discussion in ref. [6]. The ansatz for each master numerator is of the same form. The basic building blocks include the Mandelstam invariants sij = (pi + pj )2 together with contractions of loop momenta `i (i = 1; 2) and external momenta pi (i = 1; : : : ; 4). As in section 3.3.1 we decompose D-dimensional loop momenta into their four- and extra-dimensional parts as `i = `i + i. The following set of kinematic variables is then su cient: M = sij ; `is; `it; `iu; `i `j ; ij ; where ` t i ij = i j : ` s i 2`i (p1 + p2); u i 2`i (p2 + p3); ` 2`i (p3 + p1); Note that M contains 14 independent objects after accounting for u = s t, where s = s12, t = s23 and u = s13. We also include contractions with the four-dimensional Levi-Civita tensor: (4.3) (4.4) = (1; 2; 3; `1); (1; 2; 3; `2); (1; 2; `1; `2); (2; 3; `1; `2); (3; 1; `1; `2) ; (4.5) which acts only on the four-dimensional parts of the loop momenta. To be speci c about normalizations, given some momenta ki, we de ne (k1; k2; k3; k4) Det(ki ). The last ingredient is the antisymmetric combination ( 1; 2) | we saw earlier how it arises from the chiral nature of the N extra-dimensional echo of the six-dimensional Levi-Civita tensor: = (1; 0) theory (see section 3.3.2). It is an where vi are some four-dimensional vectors, such as external momenta and polarizations (their speci c form is not important since they cancel out in the above ratio). It satis es HJEP09(217) We label external states using four-dimensional variables: in section 3 we introduced variables ij that carry the correct helicity weight for all MHV helicity con gurations (3.31). The indices i and j label the legs belonging to the anti-chiral part of the vector multiplet V N =2, i.e. those containing negative-helicity gluons. With these building blocks, su cient ansatze for the master numerators are ni(1234; `1; `2) = X The objects cim;jk, dim;jnk and eim;jk are free parameters to be solved for; M (N) denotes the set of monomials of engineering dimension 2N built from the set M in eq. (4.3): M (N) = ( N Y z i=1 i zi 2 M ) : Note that there exist non-linear relations between the monomials in M (N) and the LeviCivita objects in . In principle, we could use these to slightly reduce the size of the ansatz; however, the bene t in doing so is marginal. Furthermore, working with an over-determined set of building blocks is helpful when searching for a compact form of the integrand. Finally, we should comment on the fact that the kinematic numerators we choose to work with have poles of the form jk=sj2k. Because we choose to work with gauge-invariant building blocks (as opposed to polarization-dependent numerators) the price to pay is some non-locality in the numerators.6 While the speci c form of the non-locality is not a priori obvious we nd that the observation of ref. [6] carries over to two loops: the MHV numerators have at worst jk=sj4k N poles, where N is the number of supersymmetries. 4.3 Symmetries and unitarity cuts Graph symmetries, or automorphisms, can be used to obtain relations between numerators with di erent external-leg orderings and di erently parametrized loop momenta. For 6One reason for this is that the number of gauge-invariant local functions one can write down is strongly dependent on the engineering dimension. If one wants to have access to su ciently many of these while keeping the overall dimension small one is forced to compensate with momentum factors in the denominator. ! (4.7) (4.8) and promoting them to ghosts. More generally, by using Nf hypermultiplet avors in one of the gauge-theory copies (and Nf = 1 for the other copy), i.e. considering N = 2 SQCD, this construction is generalizable to NV = 2(1 + Nf ) vector multiplets. Dimensional regularization requires us to construct the loop integrands in D > 4 dimensions. It is therefore convenient to consider a six-dimensional uplift of the fourdimensional double copy (5.15): HN =(1;1) = VN =(1;0) VN =(0;1); where HN =(1;1) is the multiplet of N = (1; 1) supergravity (its on-shell particle content is described in ref. [88]). The precise form of HN =(1;1) will not concern us here; it is enough for us to know that, upon dimensional reduction to four dimensions, it gives the desired HN =4 and VN =4 multiplets. A similar six-dimensional interpretation works for the double copy of hypermultiplets: It is well known that the uplift of N = 4 supergravity to six dimensions is not unique. Besides the N = (1; 1) theory, there exists the chiral N = (2; 0) supergravity (see ref. [88] for further details), which has the following state content in the factorizable double copy: The extra self-dual tensor multiplet TN =(2;0) can be removed (or more can be added) by exploiting that the same multiplet appears in the double copy of the matter multiplets: (5.17) (5.18) (5.19) (5.20) TN =(2;0) = N =(1;0) N =(1;0): By removing the bar on the second factor we emphasize that the half-hypers should transform in a pseudo-real representation in the gauge theories. This avoids over-counting the number of tensor multiplets that are obtained in the double copy. In terms of the six-dimensional gauge-theory numerators, double-copying to obtain pure N = (1; 1)-type supergravities is simple: multiply the N = (1; 0) SQCD numerators by those corresponding to the chiral-conjugate theory, obtained by swapping mi $ (which reverses the sign of the Levi-Civita invariant ( i; j )). For the N = (2; 0)-type supergravities simply square the N = (1; 0) SQCD numerators. For ve-dimensional loop momenta, the two alternative double copies become identical as mi = mei and ( i; j ) = 0. Since the external states of our numerators carry four-dimensional momenta, they mei can be properly identi ed using the four-dimensional double copy (5.15). The chiral and anti-chiral supergravity multiplets, H N =4 and HN =4, are associated with the chiral and anti-chiral N = 2 multiplets, VN =2 and V N =2, respectively: HN =4 = VN =2 VN =2; HN =4 = V N =2 V N =2: (5.21) The cross terms between external states VN =2 and V N =2 give the extra two VN =4 multiplets that should be manually truncated away in the pure four-dimensional theory. In terms of the helicity sectors of SQCD the double copy can be performed in each sector separately since the cross-terms between the sectors should integrate to zero. This vanishing is necessary in order for the gravitational R-symmetry SU(4) to emerge out of the R-symmetry of the two gauge-theory factors SU(2) SU(2). Thus the all-chiral and MHV sectors of N = 4 supergravity can be isolated by considering color-dual numerators of N = 2 SQCD belonging to the all-chiral and MHV sectors respectively. At two loops, these are the sets of numerators provided in the previous section. In the all-chiral sector the only non-zero one-loop numerators are boxes. The relevant N = (1; 1)-type double copy gives the following half-maximal supergravity numerator in generic dimensions: HJEP09(217) + (Ds numerator in six dimensions:8 where we have replaced Nf using NV = 2(1 + Nf ) and NV = Ds 4. The modulus-square notation of the numerators indicates multiplication between the chiral and chiral-conjugate numerators: these are related by mi $ mei and ( i; j ) ! states chiral conjugation is the same as complex conjugation). Pure theories in D = 4; 5; 6 ( i; j ) (for real external dimensions can be obtained by setting Ds = D. After plugging in the N numerators given earlier (4.15) into eq. (5.22), and using numerator (5.10) of the (N = 0) (N = 4) construction. 2 = mem, one recovers the box The N = (2; 0)-type double copy gives the following N = (2; 0) supergravity box N [N =(2;0) SG] 4 3 1 2 = n 4 3 1 2 2 + (NT 1) n 1 2 2 ; (5.23) where the modulus square is replaced by an ordinary square and the parameter NT = 1 + 2Nf counts the number of self-dual N = (2; 0) tensor multiplets in the six-dimensional theory (with NT = 0 being the pure theory). In general, for any diagram and at any loop order, we can obtain the N = (2; 0) supergravity numerators from the N = (1; 1) ones by replacing the modulus square with an ordinary square and replacing (Ds 6) with (NT 1), as exempli ed above. In the MHV sector the one-loop amplitude of half-maximal gravity obtained from SQCD has already been considered by Ochirov and one of the present authors [6]. In this case the triangle and bubble diagrams will also contribute to the amplitude. 5.2.2 Two loops At two loops there are more non-zero numerators to consider (see ancillary les for the assembled supergravity numerators). For instance, the double copy of the double-box 8The Grassmann-odd parameters iI should not be literally squared in eq. (5.23) or eq. (5.22); instead I the R-symmetry index should be shifted i ! i I+4 in one of the copies. 4 3 4 3 3 4 ←ℓ2ℓ1→ 1 2 + (Ds 2 = numerators gives given earlier (5.13) using ij = me(imj). They are found using similar double copies: 0 4 ←ℓ2ℓ1→ 1 2 4 ←ℓ2ℓ1→ 1 21 + n + n 3 2 3 2 3 2 where the last two terms are related by relabelling and reparametrization of the states and momenta. It is a straightforward exercise to show that, in the all-chiral sector, this matches the double-box numerator (5.14a) found by the (N = 0) (N = 4) construction: one assembles the combinations of mi and mei into the extra-dimensional function F1( 1; 2) HJEP09(217) In the all-chiral sector, there are only two more non-zero supergravity numerators. 2 ; (5.25) 21 A ; (5.26) (5.24) A ; (5.27) 21 A ; (5.28) n + (Ds ℓ2ւ ℓ1→ 1 3 2 2 + n 4 ℓ2ւ ℓ1→ 1 3 2 + n 4 ℓ2ւ ℓ1→ 1 3 2 2 + (Ds 6)2 n 4 տℓ2ℓ1ր 1 2 3 2 where, again, some of the terms are related by relabelling and reparametrization. In the all-chiral sector, these expressions perfectly match the (N = 0) (N = 4) double copy given in eqs. (5.14b) and (5.14c). Identical formulas apply for the MHV-sector supergravity numerators, of which there 0 ℓ1→ 1 2 2 3 + n 4ℓ2ւ ℓ1→ 1 2 2 3 + n 4ℓ2ւ ℓ1→ 1 2 3 4 3 n ℓ↓2ℓ1ր 1 2 + (Ds 6) 2 + n ℓ↓2ℓ1ր 1 2 ; are two more: ℓ1→ 1 = 2 3 2 N [N =4 SG] ℓ↓2ℓ1ր 1 2 As already explained, we drop the remaining bubble-on-external-leg and tadpole diagrams since they integrate to zero in dimensional regularization (see section 4.6). Pure half-maximal supergravity numerators (including N = (1; 1) but not N = (2; 0)) are obtained by setting Ds = D. The N = (2; 0) supergravity two-loop numerators can be obtained from the above formulas by replacing the modulus square with an ordinary square and (Ds 6) with (NT 1). Enhanced ultraviolet cancellations in D = 5 2 the two-loop pure half-maximal supergravity amplitude is known to be UV- nite for all external helicity con gurations. This was demonstrated in ref. [72] as an example of an enhanced cancellation. While a potentially valid counterterm seems to exist, recent arguments con rm that half-maximal D = 5 supergravity is nite at two loops [72, 99{101]. Using the obtained pure supergravity amplitude (5.2) we con rm this enhanced UV cancelation. In the all-chiral sector, the only UV-divergent integrals are the planar and non-planar double boxes, divergences for which may be found in ref. [72]. It is a simple exercise to show that these contributions, when substituted into the two-loop supergravity amplitude (5.2), give an overall cancellation. In the MHV sector, where there is a wider range of non-trivial integrals involved, the UV calculation provides a useful check on our results. Following the procedure outlined in ref. [102], we consider the limit of small external momenta with respect to the loop momenta jpij j`j j in the integrand (5.2). This is formally achieved by taking pi ! pi for a small parameter , keeping only the leading term. The resulting integrand is then reduced to a sum of vacuum-like integrands of the form [103] 1 1 Z dD`1dD`2 i D2 2 ( `21) 1 ( `22) 2 ( (`1 + `2 + k)2) 3 = HD( 1; 2; 3)( k ) 2 D 1 2 3 ; (5.29) where we have introduced a scale k2 to regulate intermediate IR divergences and HD( 1; 2; 3) D 2 D 2 2 Coe cients of these integrals depend only on the external momenta. When D = 5 there are no subdivergences to subtract so the leading UV behavior in each integral is at most O( 1). `i `j using9 This requires us to eliminate all loop-momentum dependence in the numerators. First we remove ( 1; 2) using ( 1; 2)2 = 212; any odd powers integrate to zero. All remaining loop-momentum dependence can then be converted to contractions of the form `i `j ! `i `j D : (5.31) Contractions `i `j can then be converted to inverse propagators, which shifts the i indices in eq. (5.29). Similar reductions hold for higher tensor ranks; the relevant identities follow from Lorentz invariance. 9This prescription also works for extra-dimensional components. We write ij = ~ `i `j where ~ is the extra-dimensional part of the metric, = ~ . We present the MHV-sector UV divergences diagram by diagram. Diagrams containing tadpoles or bubbles on external legs are dropped since they vanish for dimensional reasons in gravity (the integrals evaluate to positive powers of the momentum or mass used to regulate the infrared singularities, implying that no logarithmically-dependent UV poles survive). For the remaining diagrams, considering the numerators of the rst MHV solution (see appendix A.1), we obtain the following UV divergences: (29Ds 142) 210 3 Ds) 70 N [N =4 SG] 4ℓ2ւ ℓ1→ 1 3 I I 1 1 1 (4 )5 5 2 I I (4 )5 5 2 212 + 123 + 124 + 223 + 224 + 324 + O( 0); (18 23Ds) 105 ( 122 + 324) 2(4 + 5Ds) 105 ( 123 + 124 + 223 + 224) + O( 0); 2 2 3 2 2 = = = = O( 0); = (5.32a) (5.32b) (5.32c) (5.32d) (5.32e) 212 + 123 + 124 + 223 + 224 + 324 + O( 0): These integrals cancel among themselves when substituted into the assembled amplitude (5.2), and after summing over permutations of external legs. Using the numerators of the second MHV solution (see appendix A.2), we arrive at the same vanishing result. This con rms that the UV divergence is absent in D = 5 dimensions. 6 Conclusions and outlook In this paper we have computed the two-loop integrand of four-vector scattering in N = 2 SQCD in a color-dual form. This provides the rst example of color-dual numerators in a two-loop amplitude containing Nf fundamental matter multiplets. The calculation builds on and extends the previous one-loop work by Ochirov and one of the present authors [6]. Through the double-copy construction, the color-dual N = 2 SQCD numerators can be recycled into the construction of pure and matter-coupled N = 4 supergravity amplitudes. In particular, we have considered numerators belonging to the MHV and all-chiral sectors of N = 2 SQCD, where, in the latter, all external states belong to the chiral VN =2 multiplet (3.3b). The latter are non-zero before (and zero after) integration, but they are needed as the double copy gives non-vanishing gravity amplitudes in the all-chiral sector. Indeed, this peculiar behavior of the all-chiral sector is needed in order to give correct U(1)-anomalous amplitudes in N = 4 supergravity [92], while not introducing any corresponding anomalies in the N = 2 gauge theory [95]. We found the two-loop numerators by tting ansatze to physical data from generalized unitarity cuts and utilizing kinematic Jacobi relations and graph symmetries. Furthermore, we used the fact that there is a close connection between the states of N and N = 4 SYM after accounting for simple di erences in gauge-group representation and avor. For the color-dual numerators of N = 2 SQCD we identi ed the separate contributions from the vector multiplets and the fundamental hypermultiplets. We then constrained them by demanding that appropriate linear combinations of the numerators sum up to their known N = 4 SYM counterparts. For this to work seamlessly, we amended the kinematic numerator relations with a speci c (auxiliary) two-term identity (2.7) valid in the limit Nf ! 1. This enabled us to nd unique color-dual numerators in both the MHV and all-chiral helicity sectors. Somewhat surprisingly, the two-loop MHV solution found using the above rules includes non-zero numerators corresponding to bubble-on-external-leg and tadpole diagrams. Heuristic expectations from power counting of individual diagrams suggest that these diagrams should be absent [6, 20, 25, 104, 105]; however, this expectation seems not to be compatible with the two-term identity. The appearance of cubic tadpoles and external bubbles is potentially troublesome since they have singular propagators; nevertheless, these diagrams vanish after integration in the massless theories and can thus be dropped. Since general expectations from N = 2 supersymmetry suggest that color-dual numerators should exist where the singular bubble-on-external-leg and tadpole diagrams are absent [6], we searched and found such a solution that di ers only slightly from the rst one. This second solution does not obey the auxiliary two-term identity (which is optional from the point of view of color-kinematics duality) but it has somewhat better diagramby-diagram and loop-by-loop UV power counting than the rst solution. This potentially makes it better suited for studies of UV behavior in supergravity theories constructed out of double copies involving N = 2 SQCD. The need to use dimensional regularization for the amplitudes required us to de ne the gauge and gravity theories in D = 4 2 dimensions. Speci cally, it is necessary to precisely know the integrand in D > 4 dimensions: this led us to consider the sixdimensional spinor-helicity formalism [70] and the corresponding on-shell superspace [84]. In six dimensions the N = 2 SQCD theory lifts to a chiral N = (1; 0) SYM theory with fundamental hypers. The chirality has important consequences: in the all-chiral-vector sector we encounter numerators depending on complexi ed extra-dimensional momenta, and in the MHV sector it was necessary to introduce ( 1; 2) to account for a dependence on the six-dimensional Levi-Civita tensor. However, this chiral dependence canceled upon double-coping chiral and anti-chiral sets of numerators corresponding to an N = (1; 0) N = (0; 1) construction: we successfully obtained D 6-dimensional N = 4 supergravity numerators from the N = (1; 1) supergravity theory. Alternatively, chiral N = (2; 0) supergravity amplitudes in six dimensions could be obtained from our numerators using the N = (1; 0) N = (1; 0) double SYM N = (0; 0) N = (1; 0) N = (1; 1) N = (1; 0) N = (1; 0) N = (1; 1) N = (1; 1) (0; 0) (0; 0) (0; 0) (0; 1) (1; 0) (1; 0) (1; 1) haba_ b_ Bab Ba_ b_ HN =(1;0) TN =(1;0) HN =(1;1) HN =(1;1) HN =(2;1) HN =(2;2) TN =(2;0) ( Bab Aaa_ ) TN =(1;0) VN =(1;0) VN =(1;1) TN =(2;0) { { { ) ( Ba_ b_ Aaa_ { { VN =(1;0) VN =(1;1) pure supergravities in six dimensions for various amounts of supersymmetry. The multiplets are: graviton H, tensor T , vector V, and hyper/chiral . In the non-supersymmetric case there are two natural choices for the matter double copy, leading to either vectors or tensors and scalars; however, neither perfectly matches the matter content in the V it is always possible to nd matching matter states in the V V product. For the supersymmetric cases V and products, implying the latter can be used as ghosts. For the (1,0) and (2,0) supergravities, the matter double copy can be done using a half-hypermultiplet in a pseudo-real representation [9] (instead of fundamental); this gives a self-dual tensor multiplet. copy. In ve dimensions, both constructions reduce to the same expressions corresponding to amplitudes of D = 5 half-maximal supergravity. We explicitly calculated the UV divergences of the two-loop supergravity integrals in D = 5 2 dimensions, and showed that the 1= poles non-trivially cancel out in the full amplitude. An important aspect of the double copy when moving between dimensions is that the precise details for constructing the pure supergravities is sensitive to the dimension D and chirality of the theory. In four dimensions, we have to subtract two unwanted internal matter multiplets to arrive at pure supergravity, whereas in six dimensions the pure N = (1; 1) theory is factorizable (i.e. the double copy of N = (1; 0) N = (0; 1) SYM exactly produces the pure theory). In contrast, pure N = (2; 0) supergravity is nonfactorizable, and thus to obtain it one needs to subtract out exactly one self-dual N = (2; 0) tensor multiplet from the N = (1; 0) N = (1; 0) double copy. This is straightforwardly done for our two-loop numerators using ghosts of type (and dropping the contributions of states to avoid overcount). Similarly, for pure N = (1; 0) supergravity one needs to subtract out a single N = (1; 0) tensor multiplet from the N = (1; 0) N = (0; 0) double copy. Table 1 lists the multiplet decomposition of di erent six-dimensional double copies. As can be seen from this table, except for the non-supersymmetric case, the strategy for obtaining pure six-dimensional supergravities is a direct generalization of the fourdimensional situation [6]: the states in the matter-antimatter tensor product can be used as ghosts to remove unwanted states in the vector-vector tensor product. HJEP09(217) Several extensions of this work are possible. Using the obtained integrands a next step is to complete the integration of the two-loop amplitudes in N = 2 SQCD and half-maximal supergravities in various dimensions. A particularly interesting case to study is the superconformal boundary point of SQCD, where color and avor are balanced Nf = 2Nc. Both leading and subleading Nc contributions can be obtained from the current integrand, since all non-planar diagrams are included. Previous four-point one- and two-loop results in this theory include the works of refs. [106{110]. Finding color-dual numerators with further-reduced supersymmetry is an important task| this would enable calculations of two-loop pure N < 4 supergravity amplitudes. As these supergravities are not factorizable, a conventional double-copy approach involving only adjoint-representation particles is not possible. The N = 4 SYM multiplet and tree amplitudes could be further decomposed into lower-supersymmetric pieces in order to provide input for the calculation (see related work [111]). An interesting challenge would be to obtain two-loop pure Einstein gravity amplitudes using the double copy, removing the unwanted dilaton and axion trough the ghost prescription of ref. [6], in order to recalculate the UV divergence rst found by Goro and Sagnotti [112{114]. Recently, the role of evanescent operators for the two-loop UV divergence was clari ed in ref. [115] (see also refs. [116, 117]); it would be interesting to examine how such contributions enter into a double-copy construction of the two-loop amplitude. Finally, a potentially challenging but rewarding future task is to consider the threeloop amplitudes in N = 2 SQCD using the same building blocks for the numerators as used in the current work, and imposing color-kinematics duality. This calculation would be a stepping stone towards obtaining the pure N < 4 supergravity amplitudes at three loops, and a natural direction to take if one wants to elucidate the UV behavior of these theories. Acknowledgments The authors would like thank Simon Badger, Marco Chiodaroli, Oluf Engelund, Alexander Ochirov, Donal O'Connell and Tiziano Peraro for useful discussions and for collaborations on related work. The research is supported by the Swedish Research Council under grant 621-2014-5722, the Knut and Alice Wallenberg Foundation under grant KAW 2013.0235, and the Ragnar Soderberg Foundation under grant S1/16. The research of G.M. is also supported by an STFC Studentship ST/K501980/1. A Two-loop MHV solutions We present two distinct versions of the color-dual four-point two-loop N = 2 SQCD amplitude in the MHV sector, manifesting di erent properties and auxiliary constraints on the numerators. The following short-hand notations are adopted: i `i = `i + i ; 2`i (p1 + p2) ; t i ij = 2`i (p2 + p3) ; ` u i `3 2`i (p3 + p1) ; `1 + `2 ; (A.1) i j ; where pi are four-dimensional external momenta and `i are internal D-dimensional loop momenta. Without loss of generality we set Nf = 1 in the numerators; powers of Nf can be restored by counting the number of hyper loops in each diagram. The symmetry factors Si used in eq. (2.1) are given by the number of permutations of the internal lines that leaves the graph invariant. The symmetry factor is 2 for the graphs 14, 17, 20 and 23 in gure 5; all other graphs have symmetry factor 1. The following two subsections list the numerators for the two MHV solutions discussed in the paper; the same expressions, together with color and symmetry factors, are also given in machine-readable ancillary les submitted to arXiv. We also provide a similar le containing the two-loop all-chiral solution discussed in section 4.5: HJEP09(217) First two-loop MHV solution: solMHV1.txt, Second two-loop MHV solution: solMHV2.txt, All-chiral two-loop solution: solAllChiral.txt. n + n1 4 ←ℓ2ℓ1→ 1 3 2 = n2 n 4 ←ℓ2ℓ1→ 1 3 2 = A.1 First solution The rst MHV solution incorporates: matter-reversal symmetry (section 4.4.1), two-term identities (section 4.4.2) and matching with the N = 4 limit (section 4.4.3). There are 19 non-zero numerators labelled according to gure 5: 2( 13 + 22) ( 12 + 34) + 2i ( 1; 2)( 12 34) u (`s2)2 (`t2)2 + (`2u)2 + `s1`s3 `t1`t3 + `1 `3 u u `t3`2u + `t2`3u + 2s `1u`3u + (`2u)2 + 2su2 4su(`1 `3 + `22 + 13 + 22) t $ u 132+u2 24 + `t3 (1; 2; 3; `1) + `t2 (1; 2; 3; `2) + t (1; 2; `1; `2) 4i( 14t2 23) ; 12( 12 + 34) + i ( 1; 2)( 12 34) + u `s1`s2 + (`1u `t1)(`t2 + `2u) + 2s `1u`2u 2u(`1 `2 + 12) t $ u t $ u + `2u (1; 2; 3; `1) + u (1; 2; `1; `2) 22ii(( 11u342 24) 134+u2 24 2 = ℓ1→ 1 3 2 = ℓ2ւ ℓ1→ 1 3 2 = s $ u s $ t s $ u s $ t n3 n 3 2 13( 12 + 34) i ( 1; 2)( 12 34) t $ u t $ u t2 23); `s1(`t2 + `2u) (`t1 + `1u)`s2 + 4s`2 `3 12+ 34 2s `2 `3 + 41s `s1(`t2 + `2u) (`t1 + `1u)`s2 ( 12 + 34) s2 t2 i ( 1; 2)( 12 34 + 13 24 14 + 23); 2s2 + (`t1 + `1u)`s2 `s1(`t2 + `2u) 4s(`2 `3 + 13 + 22) 12+ 34 2s t $ u u2 4i( 14 23) t2 2u 2t ( 13 + 24) ( 14 + 23) ℓ2ւ 3 = n `s1(`t2 + `2u) 4s(`2 `3 + 23) t $ u 24) 23); 4t2 s2 4 12( 12 + 34) + 4i ( 1; 2)( 12 34) t $ u + u `s1`s2 + (`1u `t1)(`t2 + `2u) + 2s `1u`2u 2u(`1 `2 + 12) 13+ 24 u2 + (`2u + 2p3 `2) (1;2;3;`1) + 2p1 `1 (1;2;3;`2) + (`t2 + 2p3 `2) (1;2;3;`1) + 2p2 `1 (1;2;3;`2) t2 u2 = 2 12( 12 + 34) 2i ( 1; 2)( 12 34) `2u (1;2;3;`1) + u (1;2;`1;`2) t $ u = n 4 ←ℓ2ℓ1→ 1 3 2 t $ u u2 t2 23); n2; n13 n ℓ2ℓ1ր 1 2 ℓ2ℓ1ր 1 2 4 4 ←ℓ2րℓ1 1 3 n14 n ↓ 3 = 4`2 `3( 12 + 34 + 13 + 24 + 14 + 23); n16 n ↓ 3 = 2`2 `3( 12 + 34 + 13 + 24 + 14 + 23); n17 n 4 2 = 4`2 (`2 p4)( 12 + 34 + 13 + 24 + 14 + 23); n18 n 4 ←ℓ2րℓ1 1 2 = 2`2 (p4 `2)( 12 + 34 + 13 + 24 + 14 + 23); 3 4 3 = 4`1 `2( 12 + 34 + 13 + 24 + 14 + 23); ℓ1 1 n23 n ℓ↓2 4ր 3 2 = 8p4 `2( 12 + 34 + 13 + 24 + 14 + 23); n24 n ℓ↓2 4ր 3 2 = 4`2 p4( 12 + 34 + 13 + 24 + 14 + 23): A.2 Second solution Another possible presentation of the two-loop color-dual N = 2 SQCD amplitude in the MHV sector is found by demanding that singular diagrams corresponding to graphs with tadpoles or bubbles on external legs vanish in the on-shell limit. We keep two non-singular bubble-on-external-leg graphs (17 and 18): these numerators contain factors of p24 that cancel the singular 1=p24 propagators coming from the external lines. This solution consists of 18 non-vanishing color-dual numerators: n1 n 4 ←ℓ2ℓ1→ 1 3 2 = (`s1 `s 2`12 2`22 + s 6 13 6 22) 12+3 34 + 2i ( 1; 2)( 12 34) 2 `t2 + `2u)(`t2 + `2u) + 2u2(`s1 `s2 2`12 2`22) 75 136+u2 24 t $ u 3 146+t2 23 + (`1u + 2p3 `2) (1;2;3;`1) + (`2u + 2p1 `1) (1;2;3;`2) + s (1;3;`1;`2) 4i( 1u32 24) + `t3 (1;2;3;`1) + `t2 (1;2;3;`2) + t (1;2;`1;`2) 4i( 14t2 23); 4 ←ℓ2ℓ1→ 1 2 = +" 15u `s1`s2 + (`1u `t1)(`t2 + `2u) + 4u2 `s 2 + s 8u2 + 30`1u`2u t $ u `s1 + 2`12 + 2`22 # 1630+u224 1640+t223 t $ u 2 n4 n 4ℓ2ւ 3 2 = n5 n 4ℓ2ւ n6 n 4ℓ2ւ 2 = 2 = 2 = 4s 5`s1 `s2 + 10`12 2`22 + 30 13 12+ 34 i ( 1; 2)( 12 30 15u `s1`s3 + (`1u `t1)(`t3 + `3u) + 2u2(5`s1 + `s2 10`12 + 2`22) s 8u2 t $ u 30`1u`3u + 60u(`1 `3 + 13) `3u (1;2;3;`1) + u (1;2;`1;`2) 2i( 13 24) u2 t $ u t2 23); HJEP09(217) 3 (`t1 + `1u)`s2 `s1(`t2 + `2u) 4s(p4 `1 + 2p4 `2 + `1 `2) 12+ 34 6s ( 12 + 13 + 14 + 23 + 24 + 34); p4 (`1 `2) 2`1 `2 ( 12 + 13 + 14 + 23 + 24 + 34); + 3 (`t1 + `1u)`s2 `s1(`t2 + `2u) 4s p4 `1 + 2p4 `2) + `1 `2 12+ 34 12s `t1 (1;2;3;`2) s2 t2 s $ u s $ t s $ u s $ t 34) # 60u2 60t2 6u 6t 12u 12t 2s2 + 3 (`t1 + `1u)`s2 + 2s `s1 + `t2 + `2u 3 2 ℓ2ւ 6 +6 6 4 2 +6 4 2 +6 4 (`t1)2 + (`1u)2 + (`s2)2 (`t2)2 + (`2u)2 + `s1`s2 `t1`t2 + ` ` 1u 2u + `s1`2u u s ` ` 1 2 + 2u2 ` 2(p4 `1 + `1 `3 + `22) t $ u 6s u2 t2 34) s1 s2 + (`1u `t1)(`t2 + `2u) u2 6`s1 + 2`1u + 2`t1 + `2u (1;2;3;`1) + u (1;2;`1;`2) ℓ1→ 1 3 2 + s 8s t $ u = 3`s1 15 `s1(`t2 + `2u) (`t1 + `1u)`s2 8s 6`s1 2`t1 2`1u + 3`s2 `t2 `2u + 8`12 16`1 `2 4`22 60 12 12+ 34 + s 8u2 + 30`1u`2u t $ u 23); 5(`t1 + `1u + 2(`t2 + `2u)) 4`12 + 20`2 `3 + 60 23 15u `s2(`s3 u `1) s1 2u + `2u`3u u2 3`s1 + 5(`t2 + `2u + `t3 + `u) + 4`12 3 20`2 `3 + s 8u2 30`2u`3u + 60u(`2 `3 + 23) t $ u 3 7 13+ 24 5 60u2 60t2 i ( 1; 2) ( 13 24) 23); 3 14+ 23 60t2 12+ 34 60s s2 2 = n12 n 2 = t $ u t $ u t $ u t $ u t $ u t2 23); t2 23); 34) + u `s1`s2 + (`1u `t1)(`t2 + `2u) + 2s `1u`2u 2u(`1 `2 + 12) 13u+2 24 34) `2u (1;2;3;`1) + u (1;2;`1;`2) 44ii(( 11u342 24) 34) 13t+2 24 4i( 1u32 24) n16 n ℓ↓2ℓ1ր 14 23 = 31`12( 12 + 13 + 14 + 23 + 24 + 34); n17 n 4 ←ℓ2րℓ1 1 2 = 34p42 ( 12 + 13 + 14 + 23 + 24 + 34); 3 n18 n 4 ←ℓ2րℓ1 1 2 = 23p24 ( 12 + 13 + 14 + 23 + 24 + 34): 3 A.3 Color The color factors corresponding to the N = 2 SQCD diagrams in gure 5 are given here in terms of structure constants f~abc = tr([Ta;Tb]Tc) and traces over generators Ta, normalc1 = f~a1bcf~a2bdf~a3ef f~a4egf~df hf~gch; c2 = tr(T a1 T a2 T bT a3 T a4 T b); c3 = f~a3bcf~a4db tr(T a1 T a2 T cT d); c13 = tr(T a1 T a2 T b) tr(T a3 T a4 T b); c14 = f~a1bcf~a2dbf~a3edf~a4f ef~f ghf~gch; c15 = tr(T a1 T a2 T a3 T a4 T aT a); c4 = f~a1bcf~a2bdf~a3def~a4f gf~eghf~f ch; c16 = f~a1bcf~a2dbf~a3edf~a4f e tr(T cT f ); c5 = tr(T a1 T a2 T a3 T bT a4 T b); c6 = f~a4ab tr(T a1 T a2 T a3 T bT a); c7 = f~a1bcf~a2dbf~a3ed tr(T a4 T cT e); c17 = f~a1bcf~a2dbf~a3edf~a4f gf~ehcf~hgf ; c18 = f~a1bcf~a2dbf~a3edf~ef c tr(T a4 T f ); c19 = tr(T a1 T a2 T a3 T b) tr(T a4 T b); c8 = f~a1bcf~a2dbf~a3ef f~a4ghf~dhef~gcf ; c20 = 0; c9 = f~a3bc tr(T a1 T a2 T bT a4 T c); c10 = f~a1bcf~a2db tr(T a3 T dT a4 T c); c21 = f~a1bcf~a2dbf~a3edf~a4f ef~f gc tr(T g); c22 = tr(T a1 T a2 T a3 T a4 T b) tr(T b); c11 = f~a1bcf~a2dbf~a3ef f~a4gef~dhcf~hf g; c23 = 0; c12 = f~a3bcf~a4dbf~ecd tr(T a1 T a2 T e); c24 = f~a1bcf~a2dbf~a3edf~a4f gf~egc tr(T f ): (A.2) Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 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Henrik Johansson, Gregor Kälin, Gustav Mogull. Two-loop supersymmetric QCD and half-maximal supergravity amplitudes, Journal of High Energy Physics, 2017, 19, DOI: 10.1007/JHEP09(2017)019