Applications of Subleading-color Amplitudes in N=4 SYM Theory

A number of features and applications of subleading color amplitudes of N=4 SYM theory are reviewed. Particular attention is given to the IR divergences of the subleading-color amplitudes, the relationships of N=4 SYM theory to N=8 supergravity, and to geometric interpretations of one-loop subleading color and N^k MHV amplitudes of N=4 SYM theory.


Introduction
Planar amplitudes of N = 4 SYM theory have been extensively studied by a variety of methods, see e.g. refs. . For a recent overview see ref. [24] and the special issue of Journal of Physics A, devoted to "Scattering Amplitudes in Gauge Theories." Subleading color (i.e., non-planar) amplitudes, however, usually receive less attention [25][26][27][28][29][30][31][32][33]. Nevertheless interesting insights are available from various applications of subleading color amplitudes. One case in point is a possible weak/weak duality between N = 4 SYM theory and N = 8 supergravity [15,[34][35][36][37][38][39][40][41][42][43][44][45][46]. Since non-planar graphs appear on an equal footing with planar graphs in N = 8 supergravity, one needs to understand the non-planar graphs in N = 4 SYM if the weak/weak duality is to be explored. This review will cover three significant topics. Section 2 discusses the IR divergences of various subleading-color amplitudes. In section 3 the interplay between subleadingcolor amplitudes of N = 4 SYM theory and amplitudes of N = 8 supergravity will be considered. Section 4 presents various geometric interpretations of one-loop subleadingcolor amplitudes, primarily using the tools of momentum twistors and the accompanying polytope interpretation.
In the remainder of this section we define the notation for the color decomposition, the loop expansion, and the 1/N expansion.
Besides the loop expansion in the 't Hooft parameter a, we still have a 1/N expansion of the amplitudes, which can be understood in 't Hooft's double line notation as an expansion in the topology of the diagrams. For A 4 , the expansion in single-trace A 4;1 and doubletrace A 4;3 amplitudes corresponds to the topology of the outside lines, forming boundaries of the diagrams. For example, at one-loop, the contribution in A 4;1 to the amplitude is leading, i.e. of order N (thus A 4;1 of order 1), coming from a diagram with the topology of 4 external lines and a boundary, whereas the contribution of A 4;3 is subleading, i.e. of order N 0 , and comes from a nonplanar diagram with 4 external lines, but arranged on two boundaries. It can be obtained by taking two twists of the 't Hooft double lines on opposite sides of the box, or twists on all 4 sides. Thus the multi-trace expansion comes as an expansion in the topology associated with the external lines (number of boundaries for them), and is an expansion in integer powers of 1/N , corresponding to the number of boundaries of the diagram.
On top of that, we also have an expansion in integer powers of 1/N 2 , independently for A 4;1 and A 4;3 , corresponding to nonplanar diagrams with handles (a handle gives a factor of 1/N 2 ). The expansion terminates at order O(N 0 ) for the amplitude, since in the amplitude the powers of N can only be positive. Thus at L-loops, we have A . Taken together, we will say that the gluon amplitudes have a 1/N expansion.
2 IR divergences for subleading N = 4 four-gluon amplitudes 2.1 General formalism N = 4 SYM is a UV-finite theory, but IR divergences arise due to the exchange of soft and collinear gluons. These divergences can be regulated using dimensional regularization in D = 4 − 2ǫ dimensions, in which they appear as poles in a Laurent expansion in ǫ.
In gluon-gluon scattering in N = 4 SYM, IR divergences arise both from soft gluons and from collinear gluons, each of which gives rise to an O(1/ǫ) pole at one loop, leading to an O(1/ǫ 2 ) pole at that order. At L loops, the leading IR divergence of the scattering amplitude is therefore O(1/ǫ 2L ), arising from multiple soft gluon exchanges.
Subleading-color amplitudes A (L,k) , that is, those suppressed by 1/N k relative to the leading-color amplitude at L loops, have less severe IR divergences, being only of O(1/ǫ 2L−k ) at L-loops.
In this section, we review the derivation of a compact all-loop-orders expression for the IR-divergent part of the N = 4 SYM four-gluon amplitude given in ref. [41,49]. This result is expressed in terms of the soft (cusp) anomalous dimension γ(a), the collinear anomalous dimension G 0 (a), and the soft anomalous dimension matrices Γ (ℓ) , and relies on the assumption that the soft anomalous dimension matrices are mutually commuting, which follows if they are all proportional to Γ (1) , as has been conjectured in refs. [30,31,33,50]. This compact expression is then used to obtain the coefficient of the leading IR pole (and some subleading poles) of all the subleading-color amplitudes. Explicit values for the anomalous dimensions can be obtained by comparison with various exact results.
We organize the 4-point color-ordered amplitudes A [i] defined in eq. (1.8) into a vector in color space [25,26] |A = A [1] , A [2] , A [3] , A [4] , A [5] , A [6] , A [7] , A [8] , A [9] T (2.1) where (· · · ) T denotes the transposed vector. The vector of color-ordered amplitudes factorizes into [27,29] where |H , which is IR-finite as ǫ → 0, characterizes the short-distance behavior of the amplitude, and where the prefactors J and S encapsulate the long-distance IR-divergent behavior. The soft function S is written in boldface to denote that it is a matrix acting on the vector |H . Also s ij is the kinematic invariant (k i + k j ) 2 , µ is a renormalization scale, and Q is an arbitrary factorization scale which serves to separate the long-and short-distance behavior.
Because N = 4 SYM theory is conformally invariant, the product of jet functions J may be explicitly evaluated as [7] where γ (ℓ) and G (ℓ) 0 are the coefficients of the soft (or Wilson line cusp) and collinear anomalous dimensions of the gluon respectively. The explicit values for these anomalous dimensions may be obtained from the exact expressions for the planar four-gluon amplitude [7]: The soft function S is given by [27,29] S s ij Q 2 , suppressing the explicit dependence of Γ (ℓ) on s ij /Q 2 to lighten the notation.
At this point, we make the assumption that the soft anomalous dimension matrices Γ (ℓ) all commute with one another 1 so that the path ordering in eq. (2.5) becomes irrelevant, allowing us to explicitly integrate it, obtaining Combining the exponents of the jet and soft functions into [27,41] we may express the four-gluon amplitude in the compact form 2 or equivalently where the matrices F (ℓ) (ǫ) will be defined below. Expanding eq. (2.10) through three loops, we obtain the expressions given in refs. [27,41] |A (1) which will be useful in extracting the IR-divergent terms of leading-and subleading-color amplitudes in the following section. (Note that, because of the presence of poles in F(ǫ), we will need to know positive powers of ǫ in the expansion of lower loop amplitudes to obtain all the IR-divergent contributions to the L-loop amplitude A (ℓ) .) The equivalence of eqs. (2.9) and (2.10) follows if the matrices F (ℓ) (ǫ) are defined through the equation (2) , etc. This functional was defined for scalar functions M (ℓ) , but we can also use it for commuting matrices. We have assumed that Γ (ℓ) and therefore G (ℓ) are mutually commuting, and thus so are F (ℓ) , as a result of eq. (2.12). Thus (2.14) and so eq. (2.12) is equivalent to which defines F (ℓ) recursively in terms of G (ℓ) and F (ℓ ′ ) with ℓ ′ < ℓ. The explicit expressions up through three loops agree (up to rescaling by a factor of N L ) with the expressions given in ref. [27] when specialized to the case of gg → gg in N = 4 SYM theory.

1/N expansion of IR divergences
In this subsection, we will use the results of the previous subsection to expand the IRdivergent contributions of the four-gluon amplitude in powers of 1/N .
The L-loop color-ordered amplitudes may be written in a 1/N expansion as where |A (L,0) are the leading-color amplitudes, arising from planar diagrams, and |A (L,k) , 1 ≤ k ≤ L, are the subleading-color amplitudes, which include contributions from nonplanar diagrams as well. The single-trace amplitudes (i = 1, . . . , 6) only contain even powers of 1/N (relative to the leading-color amplitude), while the double-trace amplitudes (i = 7, . . . , 9) only contain odd powers of 1/N .
We begin by expanding eq. (2.9): (2.18) In the derivation of eq. (2.18), we assumed that the soft-anomalous dimension matrices are mutually commuting. We now assume further that the higher-loop soft-anomalous dimension matrices are all proportional to the one-loop soft-anomalous dimension matrix as has been conjectured (see footnote 1). This allows us to rewrite eq. (2.8) as The one-loop soft anomalous dimension matrix can be written [29] where T i · T j = T a i T a j with T a i the SU(N ) generators in the adjoint representation. In the basis (1.7), it has the explicit form [48] If the assumption (2.19) is valid, then the 1/N expansion of G (ℓ) (ℓǫ)/N ℓ terminates after two terms where g ℓ and f ℓ can be read from eqs. (2.20) and (2.22). We rewrite eq. (2.18) as We now determine the power of the leading IR pole of |A (L,k) (ǫ) . Consider an individual term on the r.h.s. of eq. (2.26). By counting powers of a and 1/N , one sees that this term contributes to |A (L,k) (ǫ) , with where k 1 is the number of factors f ℓ present in the term. From eqs. (2.20) and (2.22), it is apparent that g ℓ has a double pole in ǫ, but f ℓ only has a single pole. The leading IR pole in the term under consideration is therefore 1/ǫ p , where Combining eqs. (2.27) and (2.28), we find Since k 0 ≤ ℓ 0 , the term in square brackets is non-negative, and we conclude that (2.30) This behavior was previously conjectured in ref. [41] and shown in ref. [49] (subject to the assumptions stated above).
Next we review the derivation [41,49] of the coefficient of the leading IR pole of |A (L,k) (ǫ) . Terms in eq. (2.26) contribute to the leading IR pole only when the expression in square brackets in eq. (2.29) vanishes, which occurs when n ℓ = 0 for ℓ ≥ 2, and ℓ 0 = k 0 = 0 (with n 1 unconstrained). In other words, the leading IR divergences are given by [41,49] Recalling that we use eq. (2.31) to obtain the coefficient of the leading IR pole where the tree-level amplitudes are where s = (k 1 +k 2 ) 2 , t = (k 1 +k 4 ) 2 , and u = (k 1 +k 3 ) 2 are the usual Mandelstam variables, obeying s + t + u = 0 for massless external gluons. The factor K, defined in eq. (7.4.42) of ref. [53], depends on the momenta and helicity of the external gluons, and is totally symmetric under permutations of the external legs.
The leading IR pole of the planar amplitude is simply and are given by the (generalized) ABDK equation [7] (see appendix A of ref. [49] ).
We now write an explicit expression for the coefficients of the leading IR poles of subleading-color amplitudes. First we use eqs. (2.34) and (2.23) to show Hence, the leading IR divergence of the subleading-color amplitudes is given by The results (2.38) and (2.39) were derived in ref. [49], generalizing expressions derived in ref. [41].

IR divergences of A (L,1)
In this subsection, we consider the subleading-color amplitude |A (L,1) , and derive the first three 3 terms in the Laurent expansion. Consider all terms in eq. (2.26) for which the expression in square brackets in eq. (2.29) is ≤ 2: where we use eqs. (2.20) and (2.22) to write To extract the |A (L,1) amplitude, we employ the identity in which the first term on the r.h.s. has an expansion that starts with 1/ǫ 2L−1 , the second term has an expansion that starts with 1/ǫ 2L−2 , and so forth. Thus, keeping only the terms proportional to 1/N in eq. (2.40), and only the first three terms in the Laurent expansion, we obtain (2.43) as obtained in ref. [49],

IR divergences of A (L,L)
In this subsection, we derive an expression for the coefficient of the IR divergences of the first two terms in the Laurent expansion of the most subleading-color amplitude |A (L,L) .
The only terms in eq. (2.26) that contribute to |A (L,L) are those with as many factors of 1/N as of a. Thus, only f 1 and |H (ℓ 0 ,ℓ 0 ) can contribute, giving (2.44) exact to all orders in the ǫ expansion. Keeping just the first two terms in the Laurent expansion, we find (2. 45) This was derived in ref. [49], and confirms the conjecture made in eqs. (4.45) and (4.46) of ref. [41].

Exact expressions at one-and two-loops
N = 4 SYM scattering amplitudes may be expressed in terms of planar and nonplanar scalar loop integrals. The two-loop four-gluon scattering amplitude was first computed by Bern, Rosowsky, and Yan [54] (see also ref. [36]). Explicit expressions for these IR-divergent integrals as Laurent expansions in ǫ were later obtained by Smirnov in the planar case [55], and by Tausk in the non-planar case [56]. In this subsection, we review these results and some formulas for the 1/N expansion of these divergences.

(s, t) denotes the scalar box integral
an explicit expression for which is given, e.g., in ref. [7].
At two loops, the leading-color single-trace amplitude is given by [54] A (2,0) an explicit expression for which is given, e.g., in ref. [7]. The double-trace amplitude is [54] A (2,1) [7] = −2iK s 3I and the subleading-color single-trace amplitude is [54] A (2,2) where I (2)N P 4 (s, t) denotes the two-loop non-planar integral (2.54) an explicit expression for which is given in ref. [56]. All the other single-and double-trace amplitudes A are obtained by making the appropriate permutations of s, t, and u in these expressions.
It is well-known [7] that planar amplitudes have the property of uniform transcendentality. It is less obvious but nevertheless true [41] that subleading-color N = 4 amplitudes at one and two loops (and presumably beyond) also have uniform transcendentality. What makes this surprising is that the non-planar integral I (2)N P 4 (s, t) that contributes to A (2,1) and A (2,2) does not have uniform transcendentality [39,58]. The subleading transcendentality parts, however, cancel out in the expressions (2.52) and (2.53). (The same thing happens for the two-loop four-point amplitude of N = 8 supergravity [39,58].) The two-loop amplitudes obey the following group theory relations [59] A (2,1) (2.55) and may be easily verified using eqs. (2.50), (2.52), and (2.53). In addition, we have also easily verified using eq. (2.53). Together these equations imply which is the two-loop generalization of the U(1) decoupling relation (2.48). Both eqs. (2.56) and (2.57) are encapsulated in the equation which is valid through two loops.
At one-loop, we also saw that one can relate all the subleading-color amplitudes A n;j to the leading amplitude A n;1 via the group theory relation (1.5).
We now list some explicit formulas for the IR-divergent pieces of one and two-loop amplitudes that will be of use in the following section. At one-loop, the leading 4-point amplitude is given by eq. (2.46) with while the exact relation (2.48) can be used to write both the IR-divergent and IR-finite contributions to the double-trace subleading-color amplitude where we have only included the [7], [8], and [9] components of A as the others vanish.
At two loops, the planar amplitude is given by eq. (2.50) with [60] The two-loop double trace amplitude has an IR divergence given by the general formula (2.38), which yields Finally, the subleading-color single-trace amplitude is given by eq. (2.45) which in this case yields Only the [1] through [6] components are listed, as the [7] through [9] components vanish.
3 Subleading color amplitudes of N = 4 SYM and amplitudes of N = 8 supergravity The AdS 5 /CF T 4 correspondence provides a strong/weak duality between N = 4 SYM and N = 8 supergravity. These relationships have been extensively explored and exploited. There are also numerous indications of a weak/weak duality between the two theories, although this latter possibility is much less developed. Nevertheless this may be a very fruitful approach in attempts to understand relationships between the two theories. A lot of work has been done to relate the perturbation expansions of these two theories [15,[34][35][36][37][38][41][42][43][44][45][46]61,62]. Part of this program is the extension of the tree-level KLT theories, but many relations have been found at loop level as well. Since this work is extensive, we will not attempt to review it all here. Since non-planar graphs appear on an equal footing with planar graphs in N = 8 supergravity, it seems important to understand non-planar graphs in N = 4 SYM if a weak-weak duality is to be explored. This is the focus of this section.
We will review the known exact relations between the 4-point functions of subleading N = 4 SYM and those of N = 8 supergravity, at one and two-loops. For more than twoloops, the known relation for n = 4 is for the leading IR singularity only. One application of these ideas for n = 5 at one-loop is a new form of (tree level) KLT relations. Another are possible relations between N = 4 subleading-color amplitudes and N = 8 sugra for n ≥ 5.

One and two-loop relations
In this subsection, we demonstrate the existence of some exact relations between N = 4 SYM amplitudes and N = 8 supergravity amplitudes at the one-and two-loop levels. The L-loop N -independent SYM amplitude A (L,L) may be expected to be related to the L-loop supergravity amplitude, as both have O(1/ǫ L ) leading IR divergences. Other subleadingcolor SYM amplitudes A (L,k) have O(1/ǫ 2L−k ) leading IR divergences, and consequently satisfy relations involving lower-loop supergravity amplitudes. 4 In this section we use the notation noting that the other components A are obtained by permutations of s, t, and u. However, we omit the argument (s, t) for functions that are completely symmetric under permutations of s, t, and u.
Recall that the one-loop N -independent SYM four-gluon amplitude is given by (2.47), obtaining 4 (s, t) + I The one-loop supergravity four-graviton amplitude 6 may be expressed as [34,36] A (1) The supergravity amplitude is proportional to K 2 rather than K due to the KLT relations [63] (a manifestation of the relation "closed string = (open string) 2 "). This factor is also present in the tree-level supergravity amplitude, so we can factor it out Defining λ SYM = g 2 N and λ SG = (κ/2) 2 , one observes that the one-loop SYM and supergravity amplitudes are related by In other words, the ratio of the one-loop subleading-color SYM and the one-loop supergravity amplitudes (after factoring out the tree amplitudes) is simply proportional to the ratio of coupling constants, where we encounter the effective dimensionless coupling λ SG u for supergravity because λ SG is dimensionful.
Finally, rewrite eq. (3.7) in the manifestly permutation-symmetric form  In what follows we denote A tree SY M (ij...k) = A(ij...k). 6 after stripping off a factor of (κ/2) 2 for a four-point amplitude obtained from the one-loop decoupling relation (2.48) together with eq. (3.7). Now turn to two loops. There are some relations between SYM and supergravity amplitudes that hold only for the IR-divergent terms. The easiest case to analyze is the two-loop N -independent SYM amplitude A (2,2) SYM (s, t), since, from (2.63) This can be rewritten as where X = log(t/u), Y = log(u/s), Z = log(s/t), as in (2.37), thus obtaining a relation to the one-loop supergravity amplitude.
Using the relation M SG (ǫ)] 2 + O(ǫ 0 ) between the one-and two-loop supergravity amplitudes [41,58,64,65], we can write this as where this relation is exact (!), as may be easily verified by using the exact expression for the N -independent SYM amplitude [54] and from (2.53)  for L = 0, 1, and 2. Can this relation be valid at higher loops? It turns out not to be the case, but we can still find some relations valid for L ≥ 3.

Three or more loops
On the supergravity side, there is an exact exponentiation formula [64,65], which implies Since the leading IR divergences of A (L,L) is O(1/ǫ L ), one can show that the following relations hold:

New KLT relations
One of the pioneering connections between SYM and supergravity theories are the KLT relations [63], originally proved using string theory methods [35,63]. More recently, alternate versions of KLT relations have been presented based on field theoretic techniques at the tree level [44,45]. One form of these new relations has manifest (n − 3)! permutation symmetry for the n-point functions, and another has (n − 2)! symmetry, but requires regularization as a consequence of singularities. They are part of a flurry of recent activity relating N = 4 SYM and N = 8 supergravity, including [40,42,43,46,61,[66][67][68] (among older works see also [37,69,70]). Recent work applying the KLT relations include [71][72][73][74]. In our quest for SYM-supergravity relations, we first review previous KLT relations; we then note that A 5;3 and the 1-loop supergravity amplitude both have 1/ǫ IR divergences. We present here a tree-level KLT relation for the n = 5-point amplitudes derived in [75], using information from one-loop SYM and supergravity amplitudes and their IR divergences. This results in a KLT relation for 5-point functions with 2(n − 2)! manifest symmetry, without the need for regularization. These KLT relations are proved explicitly using the helicity spinor formalism and the Parke-Taylor formula. In analogy with section 3.1 on 4-point functions of N = 8 supergravity and subleading-color N = 4 SYM theories, both with the 1/ǫ IR divergence, we explore the possibility that the 1-loop 5-point supergravity amplitude can be expressed as a linear combination of the A 5;3 SYM amplitudes. In particular a linear relation is proposed among the 1/ǫ IR divergences of the two theories.
At tree level, the KLT relations are quadratic relations between the n-point amplitudes of N = 4 SYM and those of N = 8 supergravity. In these relations, the helicity information is all contained within the amplitudes, and the coefficients are all function of the kinematic invariants s ij only.
In [44] and [45], new forms of the KLT relations for any n-point function were found.
They are both written in terms of the functions where θ(i t , i q ) is zero in (i t , i q ) has the same order in both sets I = {i 1 , ..., i k } and J = {j 1 , ..., j k } and is 1 otherwise, and similarly for θ(j q , j t ).
A form of KLT relations was found in [44], but need to be regularized, due to a singular denominator, however they have a large (n − 2)! manifest symmetry. Another set was proven in [45] which is non-singular, We now derive another KLT relation for 5-point amplitudes using information about subleading one-loop amplitudes.
As we saw in (1.5), the A n;j are related to the A n;1 via group theory. In particular, for 5-point amplitudes, one has a single-trace amplitude A 5;1 and a double-trace amplitude A 5;3 related by [76]  The single-trace amplitude is given by where is the dimensionless one-mass box, and I (1) (s, t, m 2 ) is the 1-loop scalar box integral (2.47) with momenta 3,4 in the same corner and m 2 = P 2 = (p 3 + p 4 ) 2 .
Substituting in (3.28), we find (3.31) Here s abcde;±;f ghij are signs, defined as follows. The relative sign is plus if ab belong to hij, and minus otherwise, and the overall sign is plus if the permutation of hij inside abcde is even, and minus if it is odd. The 1-loop N = 8 supergravity amplitude is [62], written in terms of the scalar 1m box I(123, 45) (with momenta 4,5 on the same corner of the box) and the dimensionless box F (123; 45) is The IR behavior of the 1-loop 1m scalar box is Organizing the coefficients of each divergence we find where ǫ lmn [A(ijabc)] means A(ijabc) is multiplied by the sign of the permutation of l, m, n inside i, j, a, b, c, and the sum over a, b, c contains all the 6 terms of the arbitrary permutation of the a, b, c = i, j. Organizing the terms by IR divergences, we obtain On the other hand, we know that the IR behavior of the 1-loop n−point supergravity amplitude is [77] The vanishing of the 1/ǫ 2 IR divergence of (3.37) is also a consequence of (4.7).
which means that we must have the KLT relation Note that it has the larger manifest symmetry of 2 × (n − 2)! = 2 × 3!, and has no need to be regularized.
The tree-level KLT formula (3.41) has been derived using the IR behavior of 1-loop computations. However it can be proved explicitly. To do so, use the helicity spinor formalism and the Parke-Taylor formula [78], which states that A similar formula exists for the supergravity amplitude [37] A tree A specific case of (3.41) is proved, namely The others follow from permutations and symmetry.
One makes use of helicity spinor identities to verify that the right hand side of (3.46) is equal to (3.44), proving the KLT relation.
Relation between A one−loop 5,sugra and A 5;3 . Motivated by the fact that the leading IR divergence of the n = 5-point supergravity amplitude and of A 5;3 are both of order 1/ǫ at 1-loop, one investigates whether A 1−loop 5,sugra can be expressed as a linear combination of A 5;3 amplitudes. One uses information from eqs. (3.34) to (3.38), and finds a relation valid for IR divergences, and then one conjectures how one possibly could extend to a relation for the full amplitudes.
Based on what happened at 4−points at 1− and 2−loops, as discussed in section 3.1, we want to find A 1−loop 5,sugra as a linear combination of the A 5;3 amplitudes. In analogy with with the 4-point function, we would like to find a relation of the type The 30 equations can then be rewritten, using the explicit form of α abcde , and a new notation that will prove useful, as where ǫ hij (de) is plus if both d, e belong to h, i, j, and minus otherwise.
In order to see if a unique solution for the β i is possible, one can match the IR behaviors on the two sides of (3.47). Expressing the IR behaviors of the lhs and the rhs, (3.52) which means that one requires, using the vanishing of the 1/ǫ 2 IR divergence, If we denote the A tree 5,sugra (12345) by just M 5 , then the lhs is a vector column of (ij), M 5 s ij . Also denote abc =i,j ǫ lmn [A(ijabc)] as N (ij),(f g) , so that Note that the index (f g) on the matrix N has 10 values, and these values can also be identified by the lmn of ǫ lmn [A(ijabc)], since it corresponds to the same 10 terms, picking out a group (f g) or (lmn) out of 1, 2, 3, 4, 5.
At this point however note that the vanishing of the leading IR divergence in (3.37) means that i.e., that the matrix N has rank 9 instead of 10. One then needs to work with the corresponding 9 × 9 reduced matrix N red;(ij),(f g) and give the 10th coefficient β (f g) an arbitrary value.
Therefore one has found a linear relation, (3.47), with coefficients obtained from (3.54), which is satisfied by the IR divergences, and containing an arbitrary parameter. Of course, it is still not clear that the remaining β (f g) are unique. For that, one must calculate the rank of N red . If its rank is less than 9, the solution is parametrized by more than one parameter, since then some of the remaining β's will be undetermined. As the algebra is quite involved, this is a project for further work.
In order to see if (3.47) is also satisfied for the full amplitude, one must substitute the solution for β (f g) back in (3.51) and see if these equations are satisfied, since now one needs to check whether the 30 equations are satisfied by substituting the 10 unknowns β (f g) solved as in (3.54). The verification of (3.47) for n = 5 is analogous with that for the (successful) relation (3.9) for n = 4. Therefore, it would be interesting if (3.47), (3.50) were true. 8 In principle the strategy described above can be applied to higher n−point amplitudes. Namely one can analyze the IR behavior of the results for N = 4 SYM and N = 8 supergravity at 1-loop, and compare these to the known behavior, which would imply a relation among tree amplitudes from SYM, and a KLT-type relation from the supergravity. Finally, one can relate the subleading-color SYM and supergravity amplitudes, and use the consistency of the IR behavior to fix the proposed relation. For n = 6, L = 1, the results of Bjerrum-Bohr, Dunbar, and Ita [38] are suitable for this purpose. In [79], a simple picture was found for the 1-loop color ordered leading amplitudes of N = 4 SYM theory, in terms of the volume of a closed polytope in AdS 5 . In [80], it was generalized to subleading-color amplitudes. The picture for the leading M HV amplitude was obtained as follows. We start by writing the amplitude in a space dual to momenta, thus trivializing momentum conservation i p i = 0, by p i = x i − x i+1 . Then, e.g., the 1loop dimensionless massless box function (in 4 dimensions, which is of course IR divergent) F 0m (1234) = − 1 2 stI We then construct x αα = x µ (σ µ ) αα and finally map Here the X's, satisfying are coordinate patches on the quadric X · X = 0 in RP 5 , with X AB ∼ λX AB their homogeneous coordinates. These X's are considered as vertices situated at the boundary of an AdS 5 and are simple bitwistors living in twistor space, i.e. ∃ twistors A A and B B such that Consider a box function characterized by vertices X 1 , X 2 , X 3 , X 4 . Then the following function of the Feynman parameters α i ∈ (0, 1) with α i = 1, is a map to RP 5 , but such that X(α) · X(α) = 0, and in fact they vary over a tetrahedron in RP 5 . After normalizing by one obtains Y (α) · Y (α) = 1, which means that Y (α) lies in Euclidean AdS 5 . Since straight lines X(α) are mapped to geodesics in AdS 5 , the edges and faces of the tetrahedron in AdS 5 are geodesic, which by definition makes the tetrahedron ideal.
The value of the IR-finite 4-mass box matches twice the volume of the tetrahedron in AdS 5 . The IR divergent lower mass functions need to be regularized, either in dimensional regularization, or using a mass regularization as in [79], modifying X 2 = 0 to X · X = µ 2 (X · I), with I a fixed point (A useful choice of I is X i · I = 1, ∀i).
The one loop MHV n−point amplitudes divided by the tree MHV amplitudes are given by the sum of 1-mass and 2-mass easy box functions with coefficient one, which add up to the volume of a closed 3-dimensional polytope (without a boundary) with n vertices.
Note that by this definition, the volume of a tetrahedron comes with a sign, determined by the order of the dual space vertices x i in the box function F (i, j, k, l). That also induces an orientation (sign) for the triangular faces of the tetrahedron, determined by whether the missing vertex from (ijkl) is in an even or odd position. Faces with same vertices and different orientation (sign) can be glued together, forming a continuous object.

Polytopes for M HV subleading-color amplitudes
For subleading-color amplitudes, we want to use (1.5) to relate to the leading amplitudes, and expand in the KK basis (1.2), where we will obtain a nice geometrical interpretation.
We start with the 5-point amplitude as an example. The ratio of the leading 1-loop M HV to the tree level M HV amplitudes is the volume of a boundary of a 4-simplex, ) is the volume of the tetrahedron with vertices x 1 , x 2 , x 3 , x 4 , equal to F (1, 2, 3, 4), and the missing vertex (x 5 ) is added in brackets since the cyclicity involves all 5 points; V (x 1 , x 2 , x 3 , x 4 , x 5 ) is the volume of the boundary of the 4-simplex in twistor space, with (y) i → (Y ) i , i.e. we map the arguments of V into twistor space.
Using (1.5) and writing the tree amplitudes in terms of the KK basis, we obtain We see that for each KK basis member we have the sums of two terms which are some simple differences of M 5 's. In fact these differences can be written as the differences of two polytope volumes, which in turn can be written as the volume of a simple polytope. For instance, the coefficient of the KK basis element In the two brackets, the two volumes of opposite sign correspond to polytopes with n−1 = 4 points common out of n = 5, and the relative sign is such that we can write this as the volume of the polytope obtained by the taking the union of the two polytopes.
By examining the n = 6 case [80] as well, we can understand the general pattern for A n;3 . The general formula is we have a flip of j max n vs. n j max in between terms with different signs, hence a different n β (with or without j max ). The exception is when actually (n − 1) is at the end of {α}, and not j max , in which case the same flip is now (n − 1)n vs. n(n − 1), and the same relative minus sign applies.
Since the pair in the difference in the () bracket multiplying KK basis members has the same n − 2 permutation, and the remaining two terms are flipped, we have the difference of two n−polytopes with a common n − 1-polytope, just as in the 5-point case.
We can generalize to A n;j also, obtaining  The new features with respect to the A n;3 are as follows. The KK basis elements that we get are of a special type: If we take out n − 1, ..., n − j + 2 from the amplitude, then the situation should be like the one for n = 3, namely in the remaining permutation we go from 1 to a j max towards the right, and then towards the left. But moreover, in n−1, ..., n−j +2 we also have some ordering: some of them are in {α}, some in {β T }, which means that n − 1, ..., l max + 1 is cyclic (i.e., towards the right), and n − j + 2, ..., l max is also cyclic (i.e., we change the direction of the cyclicity at l max ).
The number of terms multiplying a KK basis member is even, corresponding to having j max in {α} or {β} and any number of the j − 2 terms {n − 1, ..., n − j + 2} in {α} and the rest in {β}. They come in pairs, the pairs corresponding to j max being just before n or just after, or otherwise one of the {n − 1, ..., n − j + 2} being either just before, or just after n, and the pairs as before having different sign. The sign of the terms is then simply (−1) j−1+n β . In terms of polytopes, the two terms of different sign correspond as before to polytopes with only a vertex differing between them, which means they again add up to another polytope with one more vertex.
As a simple application of this analysis, we note that (4.7), (4.9) and (4.10) show that the amplitudes M n come in alternating pairs. Each of these M n has leading IR singularity 1/ǫ 2 + O(1/ǫ), and therefore at one-loop A n,j has only a 1/ǫ IR singularity.

Momentum twistor representation
Momentum twistor representation for leading N k M HV amplitudes Instead, we can use a momentum twistor [17,20] representation for the N k M HV superamplitudes in order to find a simple formula for the subleading N k M HV amplitudes.
The MHV tree-level color-ordered superamplitudes are given by the Nair formula [82], a supersymmetric generalization of the Parke-Taylor formula [78,83], where as usual ij ≡ ǫ αβ λ β ,η is a spinor with an index I = 1, ..., 4 for supersymmetries suppressed, and the 2 in A n,2 refers to R-charge, since the N k M HV amplitude has m = k + 2 R-charge.
The leading singularities of an amplitude are the discontinuities of the amplitude over the singularities where we put a maximum number of propagators on-shell, as explained in [18], where a conjecture for these leading singularities was proposed.
The one-loop amplitudes of N = 4 SYM can be reduced to just boxes via the van Neerven and Vermaseren procedure, with some coefficients. The leading singularities also coincide with the coefficients of these box functions [18]. For one-loop MHV, the coefficients of the boxes are known to be just the MHV tree amplitudes, agreeing with the result above.
Subleading N k M HV amplitudes in momentum twistor space The planar (leading) color-ordered N k M HV amplitude is a sum of permutations of boxes with coefficients equal to the leading singularities, A n;1 (1...n) = σ L n,k (σ)I n;4 (σ) = A M HV n (σ)R n;k (σ)I n;4 (σ) (4.17) where I n;4 are boxes. At 6-points, the permutations σ combine such that we can organize the sum as a sum over cyclic permutations, with several boxes having the same coefficient [18]. For this coefficient we can factorize the tree MHV amplitude, which is cyclically invariant, so that it appears as a common factor At higher n-point, the situation is slightly more complicated. The box diagrams are ordered in groups that can be cyclically permuted, for each group having a given formula for the residue, but unlike 6-point, the residue is not universal for all the groups [18]. However, all the diagrams have still the external legs in the original order, which means, since the MHV tree amplitude is cyclically invariant, that we can again factorize the MHV tree amplitude, obtaining for planar N k M HV amplitudes which implicitly defines M n,k .
We now finally note that we have the same formula for A n;1 (1...n) in terms of A M HV n and M n;k from the previous section on polytopes, so we can apply the same calculations we used to obtain the M HV A n;j in terms of A n;1 in section 2. We just have to change the definition of M n;k as in (4.19), thus also drop the polytope interpretation of M n,k . But otherwise the same (4.10) found in the MHV case holds in the general N k M HV case as well, as can be seen from (4.9).
Application to the 6-point N M HV amplitude For the superamplitude, we use an explicit form of the twistor formula (4.18), doing the twistor space integrals over the 1-loop NMHV contours. The result is [84,85]  From the R n;k terms in (4.18), one gets the sum of basic dual conformal invariant Rinvariants R j,j+3,j+5 above. Here the R j,j+3,j+5 are given by As explained before, we can then perform the same combinatorics that led us to (4.9), just that now we use M

Summary
We have reviewed a number of features of subleading-color amplitudes of N = 4 SYM theory, a subject considerably less developed than that of the leading (planar) amplitudes. Nevertheless this topic should not be ignored if the structure of perturbative N = 8 supergravity and its relationship to N = 4 SYM theory is to be understood, as non-planar graphs appear on an equal footing in N = 8 supergravity.
After presenting a detailed description of the IR divergences of N = 4 SYM theory, we obtained the leading (and some subleading) IR divergences of subleading-color amplitudes at L loops, and tested these against known exact results for one-and two-loop four-point functions. These ideas applied to the one-loop five-point function led to a new KLT relation, as well as possible new relations between N = 4 SYM and N = 8 supergravity amplitudes. A geometric interpretation of the one-loop subleading and N k M HV amplitudes of N = 4 SYM was presented in the last section.
Since reformulations and extensions of known results frequently lead to new insights, we advocate that continued study of subleading-color amplitudes is likely to be fruitful. In particular, it would be important for our understanding of the relation of N = 4 SYM to N = 8 supergravity to extend (3.20), (3.21) to subleading IR divergences, and to higher n-point functions. An example of the latter is the speculative (3.47) for L = 1 and n = 5. However, (3.8) and (3.17) remind us that (3.47) may not be the only way to relate the two theories, so that the subjects discussed in this review should provide many opportunities for future work.