All-loop-orders relation between Regge limits of ${\cal N}=4$ SYM and ${\cal N}=8$ supergravity four-point amplitudes

We examine in detail the structure of the Regge limit of the (nonplanar) ${\cal N}=4$ SYM four-point amplitude. We begin by developing a basis of color factors $C_{ik}$ suitable for the Regge limit of the amplitude at any loop order, and then calculate explicitly the coefficients of the amplitude in that basis through three-loop order using the Regge limit of the full amplitude previously calculated by Henn and Mistlberger. We compute these coefficients exactly at one loop, through ${\cal O} (\epsilon^2)$ at two loops, and through ${\cal O} (\epsilon^0)$ at three loops, verifying that the IR-divergent pieces are consistent with (the Regge limit of) the expected infrared divergence structure, including a contribution from the three-loop correction to the dipole formula. We also verify consistency with the IR-finite NLL and NNLL predictions of Caron-Huot et al. Finally we use these results to motivate the conjecture of an all-orders relation between one of the coefficients and the Regge limit of the ${\cal N} =8$ supergravity four-point amplitude.


Introduction
In this paper, we examine in detail the structure of the Regge limit of the (nonplanar) N = 4 SYM four-point amplitude. We begin by developing a basis of color factors C ik suitable for the Regge limit of the amplitude at any loop order, and then calculate explicitly the coefficients of the amplitude in that basis through three-loop order using the Regge limit of the full amplitude previously calculated by Henn and Mistlberger [25]. We compute these coefficients exactly at one loop, through O(ǫ 2 ) at two loops, and through O(ǫ 0 ) at three loops, verifying that the IR-divergent pieces are consistent with (the Regge limit of) the expected infrared divergence structure as determined in refs. [14-16, 26-35, 45-50], including a contribution from the three-loop correction to the dipole formula [35,50]. We also verify consistency 1 with the IR-finite NLL and NNLL predictions of Caron-Huot et al. [49,50]. Finally we use these results to motivate the conjecture of an all-orders relation between one of the coefficients and the Regge limit of the N = 8 supergravity four-point amplitude.
Color-ordered n-gluon amplitudes (i.e. the coefficients of the amplitude in a trace basis) of an SU(N) gauge theory are not independent but obey relations derived from group theory [1,[77][78][79][80][81][82]. For four-gluon amplitudes, there are 3ℓ − 1 independent color-ordered amplitudes at ℓ ≥ 2 loops (with two at tree level, and three at one loop). In the Regge limit, each of these is reduced by one. Inspired by refs. [46,[48][49][50], we define a set of color factors C ik by C 00 =f a 4 a 1 ef a 2 a 3 e , where T 2 t and T 2 s−u are operators (cf. eq. (4.7)) that act on the tree-level t-channel color factor C 00 to generate higher-loop color factors. Each color factor C ik in eq. (1.1) contains exactly i operators T 2 t or T 2 s−u with k factors of T 2 s−u . These color factors have well-defined signature (either + or −) under the crossing symmetry operation that exchanges external legs 2 and 3 (i.e., s ↔ u). As we will see in sec. 5, this basis is particularly well-suited to describe the IR divergences of the amplitude in the Regge limit.
Having defined the Regge basis of color factors, we write the N = 4 SYM four-point amplitude in the Regge limit as whereã is the loop expansion parameter, and the range of k is 0, when i = 0, 1, when i = 1, 1, 2, when i = 2, 1, i − 1, i, when i ≥ 3.

(1.3)
By using the decomposition of the color factors C ik into a trace basis, the coefficients B (ℓ) ik can be expressed as linear combinations of color-ordered amplitudes. By Bose symmetry of the full amplitude, the coefficients B (ℓ) ik will have the same signature under crossing symmetry as their associated color factors C ik , which has implications [50] for the reality properties of B (ℓ) ik , as we will see in this paper. By examining the Regge limit of the structure of IR divergences for N = 4 SYM theory, we determine that the B (ℓ) ik are polynomials of order ℓ − k in L ≡ log |s/t| − 1 2 iπ. Thus the leading logarithmic (LL) behavior of the amplitude at ℓ loops is entirely contained in the coefficient B (ℓ) 00 , with the other coefficients contributing to subleading logarithmic behavior. The full amplitude A can be factored into an infrared-divergent prefactor times an infraredfinite hard function H, whose coefficients in the Regge basis of color factors are denoted h (ℓ) ik , analogous to eq. (1.2). Using the Regge limit of the N = 4 SYM color-ordered amplitudes given in ref. [25], we compute the IR-finite coefficients h (ℓ) ik exactly at one loop, through O(ǫ 2 ) at two loops, and through O(ǫ 0 ) at three loops. The NLL and NNLL contributions to h (ℓ) ik are verified to agree with the predictions of refs. [49,50].
Finally, using our results through three loops, we observe that one of the coefficients, B (ℓ) ℓℓ , which contains no log |s/t| dependence, is equal (up to a sign) to the Regge limit of the ℓ-loop N = 8 supergravity four-point amplitude. At one and two loops, this equality follows from the exact relations presented in refs. [23]. Moreover, this relation is consistent with the leading IR divergences of the amplitudes for all ℓ. We therefore conjecture that this SYM-supergravity relation holds to all loop orders in the Regge limit. This paper is structured as follows. We review the group theory constraints on four-gluon amplitudes in sec. 2, and the Regge limits of the tree-level and one-loop N = 4 SYM amplitudes in sec. 3. In sec. 4, we develop a basis of color factors C ik suitable for the Regge limit of the N = 4 SYM amplitude at any loop order. We then obtain expressions (through three loops) for the coefficients B (ℓ) ik in this basis as linear combinations of colorordered amplitudes (and for all loops for B (ℓ) ℓℓ ). In sec. 5, after reviewing the structure of (the Regge limit of) the infrared divergences of the N = 4 SYM amplitude through three loops, we compute the infrared-finite hard function using the results of ref. [25]. In sec. 6, we review the Regge limit of the N = 8 supergravity four-graviton amplitude, and conjecture its equivalence to B (ℓ) ℓℓ . In sec. 7, we offer some concluding remarks.

Group-theory constraints on four-point amplitudes
In this section, we briefly review the group-theory constraints on the ℓ-loop color-ordered four-gluon amplitudes in an SU(N) gauge theory that were derived in ref. [80]. These will play an important role in the following sections when we define a new basis of color factors for four-gluon amplitudes in the Regge limit.
The color-ordered amplitudes of a gauge theory are the coefficients of the full amplitude in a basis using traces of generators T a in the fundamental representation of the gauge group 2 . Color-ordered amplitudes have the advantage of being individually gauge-invariant. Fourgluon amplitudes of SU(N) gauge theories can be expressed in terms of a six-dimensional basis c [λ] of single and double traces [78] c [1] All other possible trace terms vanish in SU(N) since Tr(T a ) = 0. The ℓ-loop color-ordered amplitudes can be further decomposed [2] in powers of N as where A (ℓ,0) λ are leading-order-in-N (planar) amplitudes, and A (ℓ,k) λ , k = 1, · · · , ℓ, are subleadingorder, yielding (3ℓ + 3) color-ordered amplitudes at ℓ loops. The 1/N expansion in eq. (2.2) suggests an enlargement of the trace basis to an extended (3ℓ + 3)-dimensional trace basis t in terms of which eq. (2.2) becomes (2.4) λ are not independent but are related by various group-theory constraints. These constraints can be conveniently expressed [80] in terms of a set of null 2 Our conventions are Tr(T a T b ) = δ ab so that [T a , vectors in the space spanned by t λ gives rise to a linear constraint on the components A (ℓ) (2.5) At tree level, the single null vector corresponds to the constraint [1,77] At one loop, there are three null vectors The corresponding constraints imply that the three one-loop subleading-color amplitudes are equal and proportional to the sum of leading-color amplitudes [78] A (1) There are exactly four null vectors for every loop level two and above [80]. The four null vectors at two loops are which give rise to four two-loop group-theory relations [79]. Consequently, the two-loop amplitude may be written in terms of five independent components A 1 , A 2 , A 3 , A 9 , with the four remaining color-ordered amplitudes given by (2.11) The four null vectors at three loops are (2.12) As a result, the three-loop amplitude can be written in terms of eight independent components A 3 , A 6 , A 9 , with the four remaining color-ordered amplitudes given by (2.13) For all even loop levels beyond two (ℓ = 2m+2), the four null vectors are given by prepending 6m zeros to each of the two-loop null vectors (2.10), which give rise to the constraints 3ℓ+3 . (2.14) For all odd loop levels beyond three (ℓ = 2m+3), the four null vectors are given by prepending 6m zeros to each of the three-loop null vectors (2.12) giving rise to the constraints Using eq. (2.4), these constraints can be rewritten in terms of the color-ordered amplitudes A (ℓ,k) λ , in which case it becomes evident that they involve only the two or three mostsubleading-color amplitudes (i.e., subleading in the 1/N expansion).

Regge limit of the N = 4 four-point amplitude
In this section, we briefly review the Regge limit of the N = 4 four-point amplitude at tree and one-loop level to establish conventions and prepare for the generalization to higher loops in subsequent sections.
The tree-level four-gluon amplitude is with color-ordered amplitudes 3 The polarization-dependent factor K is defined in eq. (7.4.42) of ref. [83] and the Mandelstam variables are The components in eq. (3.2) manifestly obey eq. (2.7) by virtue of massless momentum conservation s + t + u = 0.
In the Regge limit s ≫ −t (so that u ≈ −s), one has A where we define It is straightforward (using the conventions of footnote 2) to show that C 00 corresponds to the tree-level t-channel color factor C 00 =f a 4 a 1 ef a 2 a 3 e (3.6) We digress slightly to discuss the behavior of the tree-level amplitude under crossing symmetry which exchanges external legs 2 and 3. From their definitions, we can see that the trace basis color factors (2.3) obey under a 2 ↔ a 3 and thus the color factor (3.5) is odd: C 00 → −C 00 . In the Regge limit, u ≈ −s, so the Mandelstam variables transform under k 2 ↔ k 3 as while the factor K is invariant under all permutations of external legs. Thus the tree-level color-ordered amplitude A 1 is also odd: 1 . The full tree-level amplitude (3.4), being the product of two odd objects, is thus invariant, as expected from Bose symmetry.
Next, we turn to the one-loop N = 4 SYM four-gluon amplitude [3] 3 Anticipating the need to dimensionally regularize loop amplitudes, we take the spacetime dimension as D = 4 − 2ǫ. Choosing g to be a dimensionless parameter, the coupling constant in D dimensions is g D = gµ ǫ , where µ is a renormalization scale.
where [84] The one-loop color factors c Inserting these into eq. (3.9), we obtain one-loop color-ordered amplitudes where we define the dimensionless parameters (3.14) We analytically continue to positive s using −s = e −iπ s, so that (−t)/(−s) = e iπ x.
To take the Regge limit, one expands the hypergeometric functions in x, keeping only the Two-and higher-loop amplitudes for N = 4 four-gluon amplitudes are known in terms of linear combinations of planar and nonplanar integrals [2][3][4][5][6][7][8]. The two-loop planar integrals were evaluated in Laurent expansions through O(ǫ 2 ), and the three-loop planar integrals through O(ǫ 0 ), in ref. [14]. More recently, the Laurent expansions of the corresponding nonplanar integrals have been evaluated to the same accuracy [19][20][21][22]. Using these, the color-ordered four-point amplitudes through three loops have been evaluated in ref. [25] with results given in ancillary files. We have used these results to derive the Regge limits of these amplitudes. As at tree level and one-loop, the color-ordered amplitude A (ℓ) 2 is suppressed relative to the other color-ordered amplitudes, and this holds to all orders. Once we have developed a suitable basis of higher-loop color factors in sec. 4, and discussed the infrared divergent structure of the higher-loop amplitudes in sec. 5, we will be able to present our results in a compact way.

Regge basis of color factors
As noted in sec. 2, the ℓ-loop color-ordered amplitudes A (ℓ) λ , coefficients in the extended trace basis t (ℓ) λ with λ = 1, · · · , 3ℓ + 3, are not independent due to group-theory constraints. Moreover, A (ℓ) 2 is suppressed in the Regge limit relative to the other color-ordered amplitudes, so taking into account the group-theory constraints, there remain in the Regge limit one independent amplitude at tree level, two independent amplitudes at one loop, and 3ℓ − 2 independent amplitudes for ℓ ≥ 2.
In this section, we develop an alternative basis C ik of color factors that will be useful for characterizing independent amplitudes in the Regge limit. In particular, the color factors C ik are chosen to have definite signature under the exchange of external legs 2 and 3. We then express the coefficients B (ℓ) ik of the ℓ-loop amplitude in this basis as linear combinations of the color-ordered amplitudes A (ℓ) λ . First, let's count the number of independent even and odd signature amplitudes. As noted in sec. 3, the exchange of legs 2 and 3 acts on the 3ℓ + 3 elements of the extended trace basis (2.3) according to eq. (3.7), so one can form linear combinations with 2ℓ + 2 even elements t 3+3j . The null vectors that correspond to group-theory constraints can also be chosen to have definite signature. The tree-level null vector (2.6) is even, there are two even and one odd one-loop null vectors (2.8), and for two loops and above, there are three even and one odd null vectors, cf. eqs. (2.10) and (2.12). In the Regge limit, the coefficient of t (ℓ) 2 is suppressed, eliminating one more even element. The upshot is that there remain in the Regge limit one odd independent amplitude at tree level, one odd and one even independent amplitude at one loop, and ℓ odd and 2ℓ − 2 even independent amplitudes for ℓ ≥ 2. Our choice of basis must reflect this.
The Regge basis of color factors C ik will be built from operators acting on the odd tree-level color factor C 00 given by eq. (3.5). In the color-space formalism introduced in refs. [15,85,86], the color operator T i acts on a color factor by inserting a generator in the adjoint representation of SU(N) on the ith external leg, specifically (T a i ) bc = 1 √ 2f bac , while acting as the identity on all the other external legs. Color conservation implies that when acting on a four-point amplitude [15].
Applying the operator T i · T j ≡ a T a i T a j to a given color factor acts to attach a rung between external legs i and j. For example, the operators T 1 · T 2 and T 1 · T 3 convert the tree-level t-channel color factor C 00 to a one-loop box and crossed box color factor, respectively, but C 00 is an eigenstate of T 1 · T 4 , sincef daef ebgf gca = Nf abc . More precisely, for any i, so that eq. (4.1) implies Next, following refs. [45,46,87], we define color operators associated with color flow in each channel A key observation is that the operators are even and odd, respectively, under the exchange of external legs 2 and 3 (which implies s ↔ u and t → t).

One-loop basis
At one loop, we have one independent odd amplitude and one independent even amplitude.
Recalling that both C 00 and T 2 s−u are odd under 2 ↔ 3, and defining we form a one-loop basis of color factors {NC 00 , C 11 } whose elements have signature {−, +}. Their components in the one-loop trace basis t can be obtained by using eqs. (3.12) and (4.2). Expressing the amplitude in this one-loop basis 00 NC 00 + B (1) 11 C 11 (4.10) we obtain The coefficients B (1) 00 and B (1) 11 inherit the same signature under crossing symmetry (which takes A 3 ) as their associated color factors, leaving the whole amplitude Bose symmetric.
Using eq. (3.15) in eq. (4.11), we obtain where we define [50] L ≡ − log x − 1 2 iπ . (4.13) Note that L is the Regge limit of the even-signature combination of logarithms As noted in ref. [50], when expressed in terms of the natural variable L, the coefficients of odd-signature color factors are real, whereas the coefficients of even-signature color factors are imaginary. We will see this at higher loops as well.

Two-loop basis
At two loops, we have two independent odd amplitudes and two independent even amplitudes. Thus, to the previous two color factors, we add two more to form a two-loop basis of color factors {N 2 C 00 , NC 11 , C 21 , C 22 } whose elements have signature {−, +, +, −}. In order to obtain explicit expressions for these color factors, it is convenient to express the operators T 2 t and T 2 s−u as matrices 4 (4.15) in the original six-dimensional trace basis (2.1). Using these, we can obtain the components of the elements of the two-loop basis in the extended trace basis t ,  Expressing the amplitude in this two-loop basis 00 N 2 C 00 + B 3 ) , 9 ) . (4.18) As we saw previously, the coefficients B ik inherit the same signature under crossing symmetry (which takes A

Three-loop basis
At three loops, we have three independent odd amplitudes and four independent even amplitudes. To the previous four color factors, we add three more ,  5 It is easiest to obtain these results (and similarly at higher loops) by expressing t

Higher-loop basis
For each additional loop, the number of independent odd amplitudes increases by one and the number of independent even amplitudes increases by two. Thus to obtain the ℓ-loop basis for independent amplitudes we add to the (ℓ − 1)-loop basis three additional color factors where each color factor contains exactly ℓ operators T 2 . The operator C ℓk contains k factors of T 2 s−u and thus has odd signature if k is even, and even signature if k is odd. Thus C ℓ1 has even signature, and C ℓ,ℓ−1 and C ℓℓ have opposite signatures, as required. This choice of basis is motivated by the Regge limit of the structure of IR divergences as studied in refs. [46,[48][49][50].
The ℓ-loop amplitude can then be expressed in this basis as where the range of k is ik grow increasingly complicated at higher loops, but the general expression for one of them, B (ℓ) ℓℓ , can be easily guessed from explicit results (obtained through ℓ = 9). The expressions differ depending on whether ℓ is even or odd. In the former case, for even ℓ . In the latter case, we have for ℓ = 1, and for odd ℓ > 1, we have (4.28) where we have expressed these in terms of color-ordered amplitudes in both the (3ℓ + 3)dimensional extended trace basis (2.2) and also the original six-dimensional trace basis (2.4).
In refs. [23,56], we showed that the leading IR divergence of the color-ordered amplitude A (ℓ,k) is 1/ǫ 2ℓ−k , with planar amplitudes A (ℓ,0) λ having the most severe 1/ǫ 2ℓ divergences, and the most-subleading-color amplitudes A (ℓ,ℓ) λ having at most a 1/ǫ ℓ divergence. At the end of sec. 5, we will show that B (ℓ) ℓℓ also has at most a 1/ǫ ℓ divergence. For even ℓ, this is manifest from eq. (4.26), where B (ℓ) ℓℓ is expressed in terms of color-ordered amplitudes that are most-subleading in the 1/N expansion. For odd ℓ, however, eq. (4.28) shows that this is not the case, so the 1/ǫ ℓ behavior of B

IR-divergence structure of the N = 4 SYM amplitude
In this section we first briefly review the known structure of infrared divergences of the N = 4 SYM four-point amplitude through three loops [14-16, 26-35, 45-50], focusing in particular on the Regge limit. We then take the Regge limit of known results for the four-point amplitude at one, two, and three loops [25] to confirm the expected IR divergences and to extract the IR-finite part of the amplitude in this limit, writing the result in terms of the Regge basis of color factors introduced in sec. 4. Finally we compare these to results obtained via an effective Hamiltonian approach based on Balitsky-Fadin-Kuraev-Lipatov theory in refs. [49,50].
The amplitude may be factored into jet, soft, and hard functions [16,26,27] A s ij where the factors J and S characterize the long-distance IR-divergent behavior, and H, which is IR-finite, characterizes the short-distance behavior. Here s ij = (k i + k j ) 2 , µ is the renormalization scale, and Q is an arbitrary factorization scale that serves to separate the long-and short-distance behavior. Since we are interested in the Regge limit s ≫ −t, we choose the factorization scale as Q 2 = −t in this paper.
Because N = 4 SYM theory is conformally invariant, the jet function may be explicitly evaluated as [14] J −t where γ (ℓ) and G The soft function S depends on the coefficients Γ (ℓ) of the anomalous dimension matrix [16]. The one-loop anomalous dimension matrix is [26,27] which simplifies in the Regge limit to where in the last line we used eqs. (4.7) and (4.13). On the assumption that the matrices Γ (ℓ) commute with one another, one may explicitly evaluate S for N = 4 SYM theory as 6 [56] Commutativity is guaranteed if we assume the anomalous dimension matrix is given by the dipole formula [29][30][31][32][33][34] which is valid through two loops [26,27], but receives corrections at three [35] and four [49] loops. If the dipole formula were valid for all loops, then using eqs. (5.5) and (5.7) in eq. (5.6), one finds that the Regge limit of the soft function can be written in the compact form [45,46] The three-loop correction to the dipole formula [35] persists in the Regge limit, and has the effect of modifying eq. (5.8) to [50] Since the operators in the exponent of eq. (5.10) do not commute with one another, it is useful [46] to employ a variant of Campbell-Baker-Haussdorf known as the Zassenhaus formula Putting all the pieces together, we write the Regge limit of the amplitude through O(ã 3 ) as This equation exhibits all of the IR-divergent contributions to the amplitude through three loops, with the IR-finite part encoded in the hard function H. To obtain H, we need to compare eq. (5.14) with known expressions for the amplitude at one, two, and three loops. This we now proceed to do.

Reduced amplitude
Expanding eq. (5.14) in powers of the loop expansion parameterã, it is apparent that the first exponential term in eq. (5.14) is responsible for the most IR-divergent terms in the Laurent expansion at ℓ loops, starting with an O(1/ǫ 2ℓ ) term. It is also apparent that the second exponential term in eq. (5.14) is responsible for the leading log behavior, causing A (ℓ) to go as log ℓ |s/t| at ℓ loops, and leading to Reggeization [45,46]. The amplitude, however, also has an intricate structure of subleading logarithms, to which the remaining terms in eq. (5.14) contribute. To isolate this subleading logarithmic behavior, it is useful [49,50] to define a reduced amplitude by factoring off the first two exponential terms: The removal of J implies that the leading Laurent coefficient ofÂ at ℓ loops will be 1/ǫ ℓ rather than 1/ǫ 2ℓ . The removal of e KLT 2 t implies that the leading logarithmic behavior ofÂ at ℓ loops will be log ℓ−1 |s/t|.
Note that the reduced amplitude (5.15) is almost, but not quite the same as, the reduced amplitude defined in refs [49,50], which multiplies A by a factor J −1 exp[−α g LT 2 t ] rather than where α g is the Regge trajectory. The differenceα g = α g − K vanishes at one-loop order (because we are expanding inã rather than a) and is IR-finite at two loops [50].
Using eq. (5.14) we may write the reduced amplitude (5.15) aŝ We then expand this in powers ofã, using eq. (5.9), to obtain where we denote the loop expansion of the hard function as We now consider the amplitude at each order inã.

One loop
The O(ã) term of eq. (5.17) isÂ Recalling that 1 C 00 and C 11 = T 2 s−u C 00 , this can be expressed in the Regge color factor basis asÂ where we define Using the exact (all orders in ǫ) one-loop amplitude from eqs. (4.10) and (4.12), one obtains the exact one-loop reduced amplitudê Comparing eqs. (5.20) and (5.22), we obtain the one-loop IR-finite contributions to all orders in ǫ :

Two loops
Next, we consider the O(ã 2 ) term in eq. (5.17)

(5.24)
All of the terms in this equation can be expressed in terms of the Regge color factor basiŝ 00 N 2 C 00 + h 22 .

Three loops
Finally we consider the O(ã 3 ) term in eq. (5.17). Most of the terms can be immediately written in terms of the three-loop Regge color basis, with the exception of T 2 s−u [T 2 t , T 2 s−u ] and also W which appears in ∆ (3) . Using eq. (4.15) and eq. (4.20), we may determine that Because we are expanding inã rather than a, the values of the Regge trajectory coefficientsα (ℓ) g differ from eqs. (D.1) and (D.2) of ref. [50]. In particular,α (1) g vanishes in our case, as noted in that reference.
We thus obtain for the three-loop amplitudê where we have boxed all the terms that come from the three-loop dipole correction term ∆ (3) . Expressing the amplitude in the three-loop Regge color factor basiŝ 32 C 32 + h 33 . (5.34) Once again, to determine the IR-finite contributions h ik , we use the ancillary files of Henn and Mistlberger [25] to extract the Regge limit of the three-loop color-ordered amplitudes λ through O(ǫ 0 ), then use eq. (5.15) to derive the corresponding reduced amplitudesÂ (3) λ , and finally use eq. (4.22) to determineB (3) ik to O(ǫ 0 ). In doing so, we recover precisely the IR-divergent terms of eq. (5.26) together with the IR-finite coefficients at O(ǫ 0 ): 31 matches the NLL prediction (4.42) of ref. [50], and the coefficients of L of h

All loop results
We wish to characterize the logarithmic dependence of each coefficient B (ℓ) ik in the expansion In the IR-divergent prefactors in eq. (5.14), we observe that each power of L requires the presence of a factor of T 2 t . The Regge color factor C ik contains i − k factors of T 2 t , and N ℓ−i can correspond to ℓ − i additional factors of T 2 t (since T 2 t acting on C 00 produces a factor of N). Hence we anticipate that B ik is sometimes milder than eq. (5.37) because we have stripped off some of the logarithms in eq. (5.15).) By eq. (5.37), the coefficient B (ℓ) ℓℓ should have no logarithmic dependence whatsoever. In fact, the only term in eq. (5.14) that contributes to B and thus where the O(ǫ 3−ℓ ) corrections come from H (2) , H (3) , · · · . This is consistent with the explicit one-, two-, and three-loop results 10 In sec. 6, we will observe that the coefficients B

SYM/supergravity relation in the Regge limit
In this section, we review the Regge limit of the N = 8 four-point amplitude. This will allow us to make an all-loop-orders conjecture between the N = 4 SYM and N = 8 supergravity four-point amplitudes in the Regge limit.
The tree-level four-graviton amplitude is 11 [3] ℓℓ because the prefactors in eq. (5.15) cannot contribute to this coefficient. 11 In this paper, we take G to be the four-dimensional Newton's constant, with G D = Gµ 2ǫ its counterpart in D = 4 − 2ǫ dimensions.
In particular, eq. (6.7) gives (in the Regge limit) We observe that (up to signs) these are precisely the one-, two-, and three-loop values of the coefficients B relating the Regge limits of the N = 8 supergravity amplitude and the N = 4 SYM amplitude. Comparing the leading infrared-divergent contribution (5.40) with eq. (6.7) confirms that eq. (6.11) is valid for any ℓ to at least the first three orders in the Laurent expansion in ǫ. Based on all this, we conjecture that eq. (6.11) holds to all orders in ǫ for any ℓ.
We can express the SYM/supergravity relation (6.11) directly in terms of color-ordered amplitudes using eqs. where we have expressed both relations using the notation of the current paper. Both eqs. (6.13) and (6.14) are proved by expressing the amplitudes in terms of planar and nonplanar integrals [2,3,23]. Although previous attempts at finding exact SYM/supergravity relations beyond two loops were not successful, there may nevertheless exist exact relations of which eq. (6.12) is the Regge limit.

Conclusions
We presented in this paper an all-loop-order basis of color factors C ik suitable for writing the Regge limit of the N = 4 SYM four-gluon amplitude. These color factors have well-defined signature under crossing symmetry u ↔ s; specifically, C ik has negative/positive signature for k even/odd.
We found that the coefficients B i2 at NNLL, etc. The coefficients of color factors with negative/positive signature are real/imaginary respectively (when expressed in terms of L), as shown on general grounds in ref. [50]. Using results from ref. [25], we computed these coefficients explicitly through three-loop order, verifying consistency with all the expected IR divergences [14-16, 26-35, 45-50], as well as with certain IR-finite NLL and NNLL predictions [49,50].
Based on our explicit results, we conjectured an all-loop-orders equivalence (up to sign) between the coefficients B (ℓ) ℓℓ and the Regge limit of the ℓ-loop N = 8 supergravity four-point amplitude. This equivalence was proven to be valid to all orders in ǫ at one and two loops, through O(ǫ 0 ) at three loops, and for the first three terms in the Laurent expansion in ǫ at ℓ loops.
Naturally, it would be nice to establish the validity of the conjectured SYM/supergravity relation more generally, perhaps via an eikonal exponentiation approach [75,76], or via known representations of these amplitudes in terms of planar and nonplanar integrals [2-7, 7, 8, 37].
It would also be interesting to know if the SYM/supergravity relations for ℓ > 2 are the Regge limits of more general exact SYM/supergravity relations, as is the case for ℓ = 1 and ℓ = 2.
Finally, it would be intriguing to discover all-orders-in-ǫ expressions for the other coefficients B