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N>= 4 Supergravity Amplitudes from Gauge Theory at Two Loops

We present the full two-loop four-graviton amplitudes in N=4,5,6 supergravity. These results were obtained using the double-copy structure of gravity, which follows from the recently conjectured color-kinematics duality in gauge theory. The two-loop four-gluon scattering amplitudes in N=0,1,2 supersymmetric gauge theory are a second essential ingredient. The gravity amplitudes have the expected infrared behavior: the two-loop divergences are given in terms of the squares of the corresponding one-loop amplitudes. The finite remainders are presented in a compact form. The finite remainder for N=8 supergravity is also presented, in a form that utilizes a pure function with a very simple symbol.


I. INTRODUCTION
It is well known that pure Einstein gravity is ultraviolet (UV) divergent at two loops [1].
This result, along with general power-counting arguments, has led to the widespread belief that a UV finite pointlike theory of gravity cannot be constructed. However, explicit calculations of scattering amplitudes in maximally supersymmetric (N = 8) supergravity have displayed an ultraviolet behavior that is much better than prior expectations, showing that the theory in four dimensions is finite up to at least four loops. Furthermore, N = 8 supergravity exhibits the same UV behavior, when continued to higher spacetime dimensions, as does N = 4 super-Yang-Mills (sYM) [2][3][4]. Surprising cancellations are also visible at lower loop orders [5][6][7][8][9][10], and even at tree level where the amplitudes are nicely behaved at large (complex) momenta [9,11].
In pure supergravity theories (where all states are related by supersymmetry to the graviton) no counterterm can be constructed below three loops. This is because the only possible two-loop counterterm, R 3 ≡ R λρ µν R µν στ R στ λρ , where R µν στ is the Riemann tensor, generates non-zero four-graviton amplitudes with helicity assignment (±, +, +, +) [12][13][14]. Such amplitudes are forbidden by the Ward identities for the minimal N = 1 supersymmetry [15].
The counterterm denoted by R 4 is allowed by supersymmetry and could appear at three loops [13,16]. However, as mentioned earlier, N = 8 supergravity was found to be finite at this order [6]. It was recently understood that the R 4 counterterm is forbidden [17,18] by the nonlinear E 7(7) symmetry realized by the 70 scalars of the theory [19,20]. In fact, E 7 (7) should delay the divergence in N = 8 supergravity to at least seven loops, where the first E 7(7) -invariant counterterm can be constructed [21][22][23]. Non-maximal (N < 8) supergravity does not have this extra E 7(7) symmetry, and may therefore diverge at only three loops in four dimensions.
Recently, the constraints that the smaller duality symmetries of non-maximal supergravities impose on potential counterterms have also been investigated [22,23]. In four dimensions, N = 6 supergravity is expected to be finite at three and four loops, and N = 5 supergravity should be finite at three loops [22]. These results still allow for a three-loop divergence in N ≤ 4 supergravities. In particular, for N = 4 supergravity, although the volume of superspace vanishes on shell, it has been argued that the usual three-loop R 4 counterterm can appear [23]. The finiteness results for N = 5, 6 could in principle be checked, and potential divergences for N ≤ 4 investigated, via explicit three-loop amplitude calculations in non-maximal supergravities. Because the same situation, in which the superspace volume vanishes on shell, and yet a counterterm appears to be allowed, holds for N = 8 supergravity at seven loops, as for N = 4 supergravity at three loops, this latter case may be of particular interest.
On the other hand, relatively few loop amplitudes have been computed for any nonmaximal supergravities. At one loop, the four-point amplitudes with N ≤ 8 supersymmetries were presented in ref. [24], while the N = 6 supergravity all-point maximallyhelicity-violating (MHV) and six-point non-MHV amplitudes were first obtained in ref. [25].
In the following, we present expressions for the two-loop four-graviton amplitudes in N = 4, 5, 6 supergravity. The calculations were performed using the gravity "squaring" relations [27,28], or double-copy property, which follows from the color-kinematics, or Bern-Carrasco-Johansson (BCJ), duality obeyed by gauge-theory amplitudes at the loop level [29].
The BCJ relations allow us to combine the N = 4 sYM amplitude [30] with the N = 0, 1, 2 sYM amplitudes [31] in order to obtain the corresponding amplitudes in supergravity.
Although they have been tested now in several loop-level amplitude computations [4,10,27,29], the underlying mechanism or symmetry behind the general loop-level BCJ relations is still not well understood. (In the self-dual sector at tree level, a diffeomorphism Lie algebra appears to play a key role. [32].) Therefore it is important to validate results obtained using BCJ duality. We will verify the expected infrared divergences and forward-scattering behavior for the two-loop amplitudes that we compute. This paper is organized as follows. In section II we review BCJ duality and the squaring relations for gravity. In section III we illustrate the method for N = 8 supergravity at two loops. In section IV we present our main formula for the two-loop amplitudes in N = 4, 5, 6 supergravity. In section V we expand the (dimensionally regulated) amplitudes for D = 4 − 2ǫ around ǫ = 0. We discuss the infrared (IR) pole structure, which agrees with general expectations, thus providing a cross check on the construction. We present the finite remainders in the two independent kinematic channels. In section VI we examine the behavior of the amplitudes in the limit of forward scattering. In section VII, we present our conclusions and suggestions for future research directions. An appendix provides some one-loop results that are required for extracting the two-loop finite remainders.

II. REVIEW OF THE BCJ DUALITY AND SQUARING RELATIONS
We now briefly review BCJ duality and the gravity squaring relations that follow from it. For a more complete treatment see, for example, the recent reviews [33,34]. Here, we will focus solely on applications to loop amplitudes.
We can write any m-point L-loop-level gauge-theory amplitude, where all particles are in the adjoint representation, as and the remaining structure constant factors in the triplet of graphs are identical. The relations c i = c j + c k mean that the representation (2.1) is not unique; terms can be shuffled from one graph to others, in a kind of generalized gauge transformation [27].
A representation (2.1) is said to satisfy the BCJ duality if the three associated kinematic numerators are also related via Jacobi identities. Namely, we must have: where the left-hand side follows directly from group theory, while the right-hand side is the highly non-trivial requirement of the duality. Moreover, we demand that the numerator factors have the same antisymmetry property as the color factors under the interchange of two legs attached to a cubic vertex, The relations (2.3) were found long ago for the case of four-point tree amplitudes [35]; the idea that the relations should hold for arbitrary amplitudes is more recent [27,29].
As remarked earlier, the representation (2.1) is not unique. Work is often required in order to find a BCJ-satisfying representation of a given amplitude in a particular gauge theory.
At loop level, such representations were found initially at four points through three loops for N = 4 sYM, and through two loops for identical-helicity pure Yang-Mills amplitudes [27].
A BCJ-satisfying representation was recently obtained at five points through three loops in N = 4 sYM [10]. Very recently, a four-point four-loop representation was found in the same theory [4].
As a remarkable consequence of the BCJ duality, one can combine two gauge-theory amplitudes in the form (2.1), in order to obtain a gravity amplitude, as long as one of the two gauge-theory representations manifestly satisfies the duality [27,28]. We have, where either the n j or theñ j must satisfy eqs. (2.3) and (2.4). Here κ is the gravitational coupling constant, which is related to Newton's constant G N and the Planck mass M Planck by κ 2 = 32πG N = 32π/M 2 Planck . The proof of eq. (2.5) at tree level is inductive, and uses onshell recursion relations [36] for the gauge and gravity theories, which are based on the same complex momentum shift [28]. The extrapolation to loop level is based on reconstructing loop amplitudes from tree amplitudes using (generalized) unitarity.
The relations (2.5) are similar in spirit to the KLT relations [37]. Both types of relations express gravity amplitudes as the "square" of gauge-theory amplitudes, or more generally as the product of two different types of gauge-theory amplitudes, as the n i andñ j numerator factors may come from two different Yang-Mills theories. However, the KLT relations only hold at tree level, which means that at loop level they can only be used on the (generalized) unitarity cuts. Although the gravity cuts can be completely determined by the KLT relations in terms of local Yang-Mills integrands, the gravity integrand found in this way is not manifestly local. That is, it does not manifestly have the form of numerator factors multiplied by scalar propagators for some set of φ 3 graphs. Reconstructing a local representation can be a significant task [2,6].
In contrast, eq. (2.5) is a loop-level relation, and furnishes directly a local integrand for gravity. Most of the applications of this formula to date have been to maximal N = 8 supergravity, viewed as the tensor product of two copies of maximal N = 4 super-Yang-Mills theory. The squaring relations were shown to reproduce the N = 8 supergravity four-point amplitudes through four loops [4,27] and the five-point amplitudes through two loops [10].
Quite recently, in the first loop-level applications for N < 8, the one-loop four-and fivepoint N ≤ 8 supergravity amplitudes were shown to satisfy the double-copy property [26].
In this paper, we would like to extend this kind of analysis for N < 8 supergravity to two loops. First, however, we briefly review the N = 8 case.

III. TWO-LOOP N = 8 SUPERGRAVITY
In this section we review the construction of the two-loop four-graviton amplitude in N = 8 supergravity based on squaring relations, as preparation for a similar construction for N = 4, 5, 6 supergravity in the next section.
As mentioned previously, a manifestly BCJ-satisfying representation of the four-gluon N = 4 sYM amplitude is known at two loops [27,30], and "+ cyclic" instructs one to add the two cyclic permutations of (2,3,4). The tree-level partial amplitude is where j and k label the two negative-helicity gluons. The two-loop planar and nonplanar scalar double-box integrals are, respectively, It is easy to see that the two-loop expression (3.1) satisfies the duality [27]. For instance, let's look at the diagrams related by a Jacobi identity applied to a four-point tree-level Having verified that eq. (3.1) satisfies the BCJ relations, we may combine two copies of (3.1) following prescription (2.5) to obtain the two-loop four-graviton N = 8 amplitude. We which is precisely the known result [5]. We also recall that the four-graviton and four-gluon tree-level partial amplitudes are related to each other by   In ref. [38], it was shown that one could write a color decomposition of any one-loop fullcolor all-adjoint gauge-theory amplitude in terms of color factors called "ring diagrams".
The diagrammatic representation of these color factors have all the external legs connected directly to the loop. Other conceivable color factors, in which nontrivial trees are attached to the loop, can be removed systematically by using Jacobi identities, in favor of ring graphs with different cyclic orderings of the external legs. This decomposition is independent of the (adjoint) particle content in the loop. In the same way, we can use the Jacobi identities at two loops to rewrite any full-color four-gluon amplitude in a theory with only adjoint particles, in terms of only the color factors c For super-Yang-Mills theory with N = N YM supersymmetries, we write 1234 is the integrated color-ordered subamplitude associated with the color factor c In summary, we obtain the N = 4, 5, 6 supergravity amplitudes by first expressing the N = 0, 1, 2 sYM helicity amplitudes from ref. [31] in terms of the color basis (4.2) 1 . We then replace g 6 → i(κ/2) 6 and perform the following additional replacements (plus their In order to preserve supersymmetry, we use the four-dimensional helicity variant of dimensional regularization [39] for both copies of the gauge-theory amplitudes. The results (4.3) can be expressed in terms of master integrals for the two-loop planar and nonplanar doublebox topologies, plus various other integrals with fewer propagators present. However, in this form the results are rather lengthy. Instead of presenting them here, we expand the dimensionally-regulated results, for D = 4 − 2ǫ, around ǫ = 0, as discussed in the next section.

V. INFRARED POLES AND FINITE REMAINDERS
At two loops, all pure supergravity amplitudes are ultraviolet finite [12][13][14]. Therefore all of their divergences are infrared in nature, either soft or possibly collinear. As two massless external particles become collinear, gravitational tree amplitudes have singularities only in phase, not in magnitude. The same universal "splitting amplitude" that controls the phase behavior governs loop amplitudes as well as tree amplitudes [7]. Correspondingly, there are no virtual divergences from purely collinear regions of integration [40]. Soft divergences were studied long ago and found to exponentiate [41]. More recent, explicit analyses can be found in refs. [40,42,43]. At one loop, the IR pole behavior is [24,41,44,45], At L loops, the leading divergence is at order 1/ǫ L . We first checked that the leading divergence of our two-loop N = 4, 5, 6 supergravity amplitudes is indeed at order 1/ǫ 2 .
Moreover, the exponentiation of soft divergences implies that the full two-loop IR behavior can be expressed in terms of the one-loop amplitude as follows: where F (2) 4 is the finite remainder in the limit ǫ → 0. This infrared behavior was checked explicitly for the four-point N = 8 supergravity amplitude [44,45], and was conjectured to hold for all supersymmetric gravity amplitudes [43]. We have checked that our expressions indeed satisfy eq. (5.2). We remark that the lack of any additional (ultraviolet) poles in ǫ confirms the absence of UV divergences for N = 4, 5, 6 supergravity in four dimensions at two loops [12][13][14].
In order to verify eq. (5.2) and extract F It is convenient to express the remainders for N < 8 supergravity in terms of the N = 8 remainder plus an additional term. The result for N = 8 supergravity was first presented in refs. [44,45]. We always consider the helicity configuration (1 − , 2 − , 3 + , 4 + ). There are three separate physical kinematic regions: the s channel, with s > 0 and t, u < 0; the t channel (t > 0 and s, u < 0); and the u channel (u > 0 and s, t < 0). The s channel is singled out by the fact that it has identical-helicity incoming gravitons. For all the supergravity theories, the (1 − , 2 − , 3 + , 4 + ) helicity configuration chosen is symmetric under 3 ↔ 4. Therefore we do not have to present results separately for the u channel; they can be obtained from the t-channel results by relabeling t ↔ u. In the case of N = 8 supergravity, an N = 8 supersymmetric Ward identity implies that the results in the t channel (normalized by the tree amplitude) can be obtained simply by relabeling s ↔ t. For N < 8, this property no longer holds, and we will have to quote the s-and t-channel results separately.
The N = 8 finite remainder was expressed in refs. [44,45] partly in terms of Nielsen polylogarithms S n,p (x). Here we give a representation similar to ref. [44], and a second representation entirely in terms of classical polylogarithms Li n , for consistency with the forms we present below for N < 8. The finite remainder is It was noted previously [44,45] that F (2), N =8 4 has a uniform maximal transcendentality.
That is, all functions appearing are degree-four combinations of polylogarithms, logarithms, and transcendental constants. A pure function is a function with a uniform degree of transcendentality, having only constants (rational numbers) multiplying the combinations of polylogarithms, etc. A pure function f has a well-defined symbol, S(f ), which can be obtained by an iterated differentiation procedure [46][47][48][49]. In the representation (5.3), the functions f 1 , f 2 and f 3 are pure functions with very simple, one-term symbols: We have shuffled terms slightly with respect to refs. [44,45] in order to make this property Curiously, the symbol of f 1 obeys a certain "final entry" condition recently observed to appear in the context of the remainder function for planar N = 4 sYM amplitudes or Wilson loops [50,51]. Furthermore, f 1 (x) obeys the generalization of this condition to functions, where p(x) is also a pure function, in this case When the finite remainder of the four-graviton amplitude in N = 8 supergravity becomes available at three loops (for example by computing the integrals for one of the three available expressions for it [6,27]), it will be very interesting to see whether it can also be expressed in terms of pure functions of degree six with simple symbols. Perhaps the functions will even obey a relation like eq. (5.11).
We return now to two loops and N < 8 supergravity. We present the finite remainder for N = 6 supergravity, first in the s channel: where f 6,s (x) = f 6,s;4 (x) + f 6,s;3 (x) (5.14) gives the decomposition into a degree-four function, and a degree-three one, x ln x ln 2 x + 3 ln 2 (1 − x) + π 2 It has been observed [45] that at one loop the four-graviton amplitude in N = 6 supergravity has maximal transcendentality (degree two). This result extends to one-loop amplitudes with more gravitons, thanks to the absence of bubble integrals [25,26]. However, the degree-three nature of eq. (5.16) shows that this property is broken at two loops. The breaking comes from both the two-loop amplitude M 4 , but also from the square of the one-loop amplitude M (1) 4 , which has to be subtracted in eq. (5.2). As can be seen from eqs. (A5) and (A7), the one-loop N = 6 amplitude has degree-two terms as well as degree-three terms at O(ǫ); the former terms multiply the 1/ǫ degree-one terms from the IR pole shown in eq. (5.1) to generate degree-three contributions to eq. (5.16). On the other hand, these contributions are purely logarithmic; the polylogarithmic terms in eq. (5.16) can be traced to M (2) 4 . The complexity of the expressions (5.15) and (5.16), in terms of their power-law dependence on x, makes it unprofitable to try to separate the N < 8 finite remainders into pure functions and to compute their symbols.
Because of the helicity assignment (1 − , 2 − , 3 + , 4 + ), the s-channel remainder is always symmetric under t ↔ u. However, in the t channel there is no such symmetry. The N = 6 remainder in this channel is, where the degree-four part is (5.18) and the degree-three part is In the s channel, the finite remainder for N = 5 supergravity at two loops is given by, gives the decomposition into a degree-four function, a degree-three function, and a degree-two function, The N = 5 remainder function in the t channel is, where the degree-four part is the degree-three part is 27) and the degree-two part is The results for N = 4 supergravity are the lengthiest of all. In the s channel, the finite remainder for N = 4 supergravity at two loops is given by, gives the decomposition into a degree-four function, a degree-three function, a degree-two function, a degree-one function, and a rational part, The N = 4 remainder function in the t channel is, where the degree-four part is the degree-three part is the degree-two part is the degree-one part is 40) and the rational part is

VI. FORWARD-SCATTERING LIMIT OF THE AMPLITUDES
We now inspect the behavior of the two-loop supergravity amplitudes in the limit of smallangle, forward scattering, i.e. small momentum transfer at fixed center-of-mass energy. In particular, we want to verify the contributions from matter exchange, versus graviton exchange, in the forward-scattering limit. The results are sensitive to the helicity configuration, or for fixed helicity configuration, to which invariant is time-like and which of the two space-like invariants is becoming small.
We first consider configurations, or channels, for which the associated tree-level amplitudes have a pole at small momentum transfer. These configurations are dominated by the exchange of soft gravitons, and require helicity conservation along the forward-going graviton line. (They also require helicity conservation along the backward-going line, but this second condition follows automatically from the first one, for the MHV amplitudes that we study.) To see the helicity conservation explicitly, we rewrite the tree amplitude as, where the quantity in brackets is a pure phase. Expanding eq. (6.1) for small t at fixed s in the physical s channel (s > 0 kinematics), one gets a leading term of O(s 2 /t) as t → 0.
Because the s-channel amplitude is symmetric under t ↔ u, one could also have taken the small u limit and gotten a pole-dominated behavior. However, in the physical t channel, one has to take u small in order to conserve helicity at both vertices. Then the leading tree-level behavior is O(t 2 /u) as u → 0. In contrast, the limit of small s in the t channel violates helicity conservation, and the tree amplitude is heavily power-law suppressed with respect to the dominant pole behavior, having a leading term of O(s 3 /t 2 ) as s → 0.
Interestingly, in the helicity-conserving channels described above, the two-loop remainders, F and at two loops we have Both equations hold for any number of supersymmetries. We also verified the analogous equations in the limit u → 0 in the physical t channel (t > 0 kinematics).
As discussed in refs. [52], in the physical s channel only the s-channel ladder and crossedladder diagrams (shown in fig. 1 with s flowing horizontally) contribute to the eikonal limit t → 0. The limit is dominated by graviton exchanges because the coupling of a particle It is also interesting to consider the helicity-violating limit in which s → 0 for t > 0 kinematics (u ≃ −t). As mentioned before, the associated tree-level amplitude is powersuppressed in this limit with respect to the dominant pole behavior; its leading behavior is O(s 3 /t 2 ). In this limit, many of the finite-remainder expressions naively appear to blow up (see for instance eq. (5.18) as x → 1). However, one can check in all cases that these spurious singularities cancel, and the leading behavior of the ratio of the one-and two-loop amplitudes to the tree amplitude is of O(t L ), L = 1, 2. Thus the one-and two-loop amplitudes never have a power (1/s) enhancement over the tree amplitude in the helicity-violating limit, but are of the same order in s. (There is a ln(s) enhancement, but only in the pure N = 8 supergravity terms, not in any of the matter contributions.)

VII. CONCLUSIONS
In this paper, we have computed the full four-graviton two-loop amplitudes in N = 4, 5, 6 supergravity. As expected, their IR divergences can be expressed in terms of the square of the corresponding one-loop amplitudes. The finite remainders were presented in a simple form. We also noted that the finite remainder in N = 8 supergravity can be expressed in terms of permutations of a pure function f 1 (x) possessing a simple, one-term symbol.
The N = 4, 5, 6 supergravity results were obtained using the double-copy property of gravity, which is a consequence of the recently-conjectured BCJ duality. The former property allowed us to combine the BCJ-satisfying N = 4 sYM representation with known N = 0, 1, 2 sYM gauge-theory amplitudes, in order to obtain the corresponding supergravity amplitudes, including all loop integrations.
Our task was vastly simplified by the fact that both sets of Yang-Mills amplitudes entering the double-copy formula were known, as well as by the lack of loop-momentum dependence for the N = 4 sYM amplitudes in this case. As mentioned in the introduction, generic N < 8 supergravity theories are expected to diverge at three loops (but not N = 5 or 6 [22,23]), because the counterterm R 4 is allowed by supersymmetry. It would thus be very interesting to compute explicit three-loop non-maximal supergravity amplitudes. If one computes in N ≥ 4 supergravity, then one can use the double-copy formula, because a BCJ-satisfying form exists for one of the two copies, namely the three-loop N = 4 sYM amplitude [27].
However, for the other gauge-theory copy, N < 4 sYM, the three-loop amplitudes are not known. Full-color amplitudes (including nonplanar terms) are required, and they should be known at the level of the integrand, because the BCJ form for the three-loop N = 4 sYM amplitude contains loop-momentum dependence in its numerator factors. BCJ duality for N < 4 sYM could help simplify these gauge-theory calculations. For instance, for the three-loop four-point N = 4 sYM amplitude, the duality reduced the computation of the full amplitude to the evaluation of the maximal cut [54] of a single diagram [27]. Nonmaximal amplitude calculations are not expected to be as simple, however. More powers of loop momentum will appear in the numerator factors, and graphs containing triangle and bubble subgraphs will also arise. It would be interesting nonetheless to investigate the simplifications that may be provided by BCJ duality in these cases. N = 5 case is trivially related to N = 6 at one loop). Expressions valid to all orders in ǫ, in terms of box, triangle and bubble integrals, can be found in ref. [26].
We write for each theory in these two channels.
For N = 8 supergravity in the s channel we have, where g s (x) = 2 x (ln x + iπ) ln(1 − x) The t-channel result for the N = 8 supergravity amplitude, divided by the tree, is obtained by exchanging s and t in the corresponding s-channel result. (This is not quite the case for F (1), N =8 4 , due to the explicit factor of |s| −ǫ extracted in eq. (A1).) We express the finite remainders for N < 8 supergravities in terms of the one for N = 8 supergravity. For N = 6 supergravity we find, in the s channel, where The t-channel result is where The corresponding one-loop results for N = 5 supergravity are trivially related to those for N = 6, because the difference in field content from N = 8 is due to the same matter multiplet, just three copies instead of two. Therefore we have, The s-channel one-loop finite remainder for N = 4 supergravity is given by The t-channel expression is where