#### Abstract

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

#### 1. Introduction

Planar amplitudes of SYM theory have been extensively studied by a variety of methods, see, for example, [1โ23]. For a recent overview, see [24] and the special issue of Journal of Physics A, devoted to โScattering amplitudes in gauge theories." Subleading-color (i.e., nonplanar) amplitudes, however, usually receive less attention [25โ33]. Nevertheless interesting insights are available from various applications of subleading color amplitudes. One case in point is a possible weak/weak duality between SYM theory and supergravity [15, 34โ46]. Since nonplanar graphs appear on an equal footing with planar graphs in supergravity, one needs to understand the nonplanar graphs in SYM if the weak/weak duality is to be explored.

This paper will cover three significant topics. Section 2 discusses the IR divergences of various subleading-color amplitudes. In Section 3 the interplay between subleading-color amplitudes of SYM theory and amplitudes of supergravity will be considered. Section 4 presents various geometric interpretations of one-loop subleading-color 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 expansion.

At tree level, we can decompose the amplitudes of SYM into *color-ordered* tree amplitudes
where in the second line, 1 is fixed and is a permutation of and are generators in the fundamental representation, normalized according to . The color-ordered amplitudes depend on the momenta and polarizations of the external particles.

The color-ordered amplitudes are not independent. For -point amplitudes, there is a basis of amplitudes out of the total , called the Kleiss-Kuijf (KK) basis [47], and we can find the others easily in terms of it [40]. It is based on the existence of the Kleiss-Kuijf relations [47] where are ordered permutations, that is, ones that keep the order of and of inside . Thus the KK basis is , where are arbitrary permutations. All the other โs can be recovered from it by the use of the KK relations and cyclicity and reflection invariance

At one loop, we can write a similar expansion in color-ordered amplitudes However, the subleading piece in the expansion can be obtained from the leading piece by where COP are cyclically ordered permutations, again keeping the order of and fixed up to cyclic permutations.

At arbitrary loops, the decomposition of the four-gluon amplitude takes a form with only single and double traces We also define an explicit basis [48] of single and double traces: in terms of which the four-gluon amplitude can be expanded as

The loop expansion of color-ordered amplitudes is in terms of the natural โt Hooft loop expansion parameter [7] where is Eulerโs constant and . Note that at loops, the amplitude is at most of order , which means that starts at .

For a general -point amplitude, we will have an expansion in an arbitrary number of multitrace color-ordered amplitudes .

Besides the loop expansion in the โt Hooft parameter , we still have a expansion of the amplitudes, which can be understood in โt Hooftโs double line notation as an expansion in the topology of the diagrams. For , the expansion in single-trace and double-trace amplitudes corresponds to the topology of the outside lines, forming boundaries of the diagrams. For example, at one-loop, the contribution in to the amplitude is leading, that is, of order (thus of order 1), coming from a diagram with the topology of 4 external lines and a boundary, whereas the contribution of is subleading, that is, of order , 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 โt Hooft double lines on opposite sides of the box, or twists on all 4 sides. Thus the multitrace 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 , corresponding to the number of boundaries of the diagram.

On top of that, we also have an expansion in integer powers of , independently for and , corresponding to nonplanar diagrams with handles (a handle gives a factor of ). The expansion terminates at order for the amplitude, since in the amplitude the powers of can only be positive. Thus at -loops, we have to and to . Taken together, we will say that the gluon amplitudes have a expansion.

#### 2. IR Divergences for Subleading Four-Gluon Amplitudes

##### 2.1. General Formalism

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 dimensions, in which they appear as poles in a Laurent expansion in .

In gluon-gluon scattering in SYM, IR divergences arise both from soft gluons and from collinear gluons, each of which gives rise to an pole at one loop, leading to an pole at that order. At loops, the leading IR divergence of the scattering amplitude is therefore , arising from multiple soft gluon exchanges.

Subleading-color amplitudes , that is, those suppressed by relative to the leading-color amplitude at loops, have less severe IR divergences, being only of at -loops.

In this section, we review the derivation of a compact all-loop-order expression for the IR-divergent part of the SYM four-gluon amplitude given in [41, 49]. This result is expressed in terms of the soft (cusp) anomalous dimension , the collinear anomalous dimension , and the soft anomalous dimension matrices and relies on the assumption that the soft anomalous dimension matrices are mutually commuting, which follows if they are all proportional to , as has been conjectured in [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 defined in (1.8) into a vector in color space [25, 26] where denotes the transposed vector. The vector of color-ordered amplitudes factorizes into [27, 29] where , which is IR-finite as , characterizes the short-distance behavior of the amplitude and where the prefactors and encapsulate the long-distance IR-divergent behavior. The soft function is written in boldface to denote that it is a matrix acting on the vector . Also is the kinematic invariant , is a renormalization scale, and is an arbitrary factorization scale which serves to separate the long- and short-distance behavior.

Because SYM theory is conformally invariant, the product of jet functions may be explicitly evaluated as [7] where and are the coefficients of the soft (or the 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 is given by [27, 29] where suppressing the explicit dependence of on to lighten the notation.

At this point, we make the assumption that the soft anomalous dimension matrices all commute with one another. (This assumption was also used to simplify the IR divergences of QCD in [33]. The assumption is certainly valid through two loops, since , as shown in [28, 29]. In [32], it was established that for the nonpure gluon contributions. Further, has been conjectured to hold to all orders in [30, 31, 33, 50]. Difficulties may arise at four loops, however, due to the possibility of quartic Casimirโs terms [31, 32, 51, 52].) Therefore, the path ordering in (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 or equivalently where the matrices will be defined below. (Henceforth we suppress , , , and in the arguments of the amplitudes.) Expanding (2.10) through three loops, we obtain the expressions given in [27, 41] 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 , we will need to know positive powers of in the expansion of lower loop amplitudes to obtain all the IR-divergent contributions to the -loop amplitude .)

The equivalence of (2.9) and (2.10) follows if the matrices are defined through the equation First define the functional via [7] so that , and so forth. This functional was defined for scalar functions , but we can also use it for commuting matrices. We have assumed that and therefore are mutually commuting, and thus so are , as a result of (2.12). Thus and so (2.12) is equivalent to which defines recursively in terms of and with . The explicit expressions up through three loops agree (up to rescaling by a factor of ) with the expressions given in [27] when specialized to the case of in SYM theory.

##### 2.2. Expansion of IR Divergences

In this subsection, we will use the results of the previous subsection to expand the IR-divergent contributions of the four-gluon amplitude in powers of .

The -loop color-ordered amplitudes may be written in a expansion as where are the leading-color amplitudes, arising from planar diagrams and , , are the subleading-color amplitudes, which include contributions from nonplanar diagrams as well. The single-trace amplitudes () only contain even powers of (relative to the leading-color amplitude), while the double-trace amplitudes () only contain odd powers of .

We begin by expanding (2.9): In the derivation of (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 (2.8) as The one-loop soft anomalous dimension matrix can be written [29]: where with the generators in the adjoint representation. On the basis of (1.7), it has the explicit form [48], where with If the assumption (2.19) is valid, then the expansion of terminates after two terms where and can be read from (2.20) and (2.22). We rewrite (2.18) as making all dependence explicit.

We now determine the power of the leading IR pole of . Consider an individual term on the right-hand side of (2.26). By counting powers of and , one sees that this term contributes to , with where is the number of factors present in the term. From (2.20) and (2.22), it is apparent that has a double pole in , but only has a single pole. The leading IR pole in the term under consideration is therefore , where Combining (2.27) and (2.28), we find Since , the term in square brackets is nonnegative, and we conclude that This behavior was previously conjectured in [41] and shown in [49] (subject to the assumptions stated above).

Next we review the derivation [41, 49] of the coefficient of the leading IR pole of . Terms in (2.26) contribute to the leading IR pole only when the expression in square brackets in (2.29) vanishes, which occurs when for , and (with unconstrained). In other words, the leading IR divergences are given by [41, 49]

Recalling that we use (2.31) to obtain the coefficient of the leading IR pole where the tree-level amplitudes are where and are the usual Mandelstam variables, obeying for massless external gluons. The factor , defined in of [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 The remaining IR divergences, from to , are all proportional to and are given by the (generalized) ABDK equation [7] (see Appendix A of [49]).

We now write an explicit expression for the coefficients of the leading IR poles of *subleading-color* amplitudes. First we use (2.34) and (2.23) to show
with
Hence, the leading IR divergence of the subleading-color amplitudes is given by
The results (2.38) and (2.39) were derived in [49], generalizing expressions derived in [41].

##### 2.3. IR Divergences of

In this subsection, we consider the subleading-color amplitude and derive the first three terms in the Laurent expansion. (It is straightforward to obtain further terms in the Laurent expansion as needed.) Consider all terms in (2.26) for which the expression in square brackets in (2.29) is โค2: where we use (2.20) and (2.22) to write To extract the amplitude, we employ the identity in which the first term on the right-hand side has an expansion that starts with , the second term has an expansion that starts with , and so forth. Thus, keeping only the terms proportional to in (2.40), and only the first three terms in the Laurent expansion, we obtainas obtained in [49].

##### 2.4. IR Divergences of

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 .

The only terms in (2.26) that contribute to are those with as many factors of as of . Thus, only and can contribute, giving exact to all orders in the expansion. Keeping just the first two terms in the Laurent expansion, we find This was derived in [49] and confirms the conjecture made in (4.45) and (4.46) of [41].

##### 2.5. Exact Expressions at One and Two Loops

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 et al. [54] (see also [36]). Explicit expressions for these IR-divergent integrals as the Laurent expansions in were later obtained by Smirnov in the planar case [55] and by Tausk in the nonplanar case [56]. In this subsection, we review these results and some formulas for the expansion of these divergences.

Recall from (2.17) that denotes the -loop color-ordered amplitude which is subleading by a factor of in the expansion. Single-trace amplitudes are denoted by and double-trace amplitudes by (see (1.7)).

At one loop, the single-trace amplitudes are given by [34] with the other single-trace amplitudes and obtained by letting , and , respectively. The identities , , and are satisfied at all loop orders. In (2.46), denotes the scalar box integral an explicit expression for which is given, for example, in [7].

The one-loop double-trace amplitudes are given by [34] Relation (2.48) follows from the one-loop decoupling identity [57].

At two loops, the leading-color single-trace amplitude is given by [54] where denotes the scalar double-box (planar) integral an explicit expression for which is given, for example, in [7]. The double-trace amplitude is [54] and the subleading-color single-trace amplitude is [54] where denotes the two-loop nonplanar integral an explicit expression for which is given in [56]. All the other single- and double-trace amplitudes are obtained by making the appropriate permutations of , and 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 amplitudes at one and two loops (and presumably beyond) also have uniform transcendentality. What makes this surprising is that the nonplanar integral that contributes to and 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 supergravity [39, 58].)

The two-loop amplitudes obey the following group theory relations [59]: and may be easily verified using (2.50), (2.52), and (2.53). In addition, we have also easily verified using (2.53). Together these equations imply which is the two-loop generalization of the decoupling relation (2.48). Both (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 to the leading amplitude 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 (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 [7โ9] components of as the others vanish.

At two loops, the planar amplitude is given by (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 (2.45) which in this case yields Only the through components are listed, as the through components vanish.

#### 3. Subleading-Color Amplitudes of SYM and Amplitudes of Supergravity

The correspondence provides a strong/weak duality between SYM and 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, 34โ38, 41โ46, 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 nonplanar graphs appear on an equal footing with planar graphs in supergravity, it seems important to understand nonplanar graphs in 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 SYM and those of supergravity, at one and two loops. For more than two loops, the known relation for is for the leading IR singularity only. One application of these ideas for at one loop is a new form of (tree level) KLT relations. Others are possible relations between subleading-color amplitudes and sugra for .

##### 3.1. One and Two-Loop Relations

In this subsection, we demonstrate the existence of some exact relations between SYM amplitudes and supergravity amplitudes at the one- and two-loop levels. The -loop -independent SYM amplitude may be expected to be related to the -loop supergravity amplitude, as both have leading IR divergences. Other subleading-color SYM amplitudes have leading IR divergences and consequently satisfy relations involving lower-loop supergravity amplitudes. The normalization of is arbitrary. We have chosen one that is most natural in the context of the SYM/supergravity relations presented in this subsection.

In this section we use the notation noting that the other components are obtained by permutations of , , and . However, we omit the argument for functions that are completely symmetric under permutations of , , and .

Factor out the tree amplitude to define so that the coupling constant is now included in the definition of , and where.

In what follows we denote (see also (2.34)). Recall that the one-loop -independent SYM four-gluon amplitude is given by (2.47), obtaining The one-loop supergravity four-graviton amplitude may be expressed as [34, 36] after stripping off a factor of for a four-point amplitude. The supergravity amplitude is proportional to rather than due to the KLT relations [63] (a manifestation of the relation โclosed string = โ). This factor is also present in the tree-level supergravity amplitude, so we can factor it out as follows: Defining and , 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 for supergravity because is dimensionful.

Finally, rewrite (3.7) in the manifestly permutation-symmetric form (where denotes cyclic permutations of , , and ) even though is already symmetric under permutations. A similar symmetrized relation can be written for the one-loop leading-color amplitude obtained from the one-loop decoupling relation (2.48) together with (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 -independent SYM amplitude , since, from (2.63),
This can be rewritten as
where , as in (2.37), thus obtaining a relation to the one-loop supergravity amplitude.

Using the relation 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 -independent SYM amplitude [54] and from (2.53) and that for the two-loop supergravity amplitude [36] where and are the two-loop planar and nonplanar 4-point functions.

Now consider the two-loop subleading-color amplitude . The two-loop decoupling relation (2.57) can be rewritten as Using the ABDK relation [60] together with (3.9), we can rewrite (3.15) as Unlike (3.12), however, (3.17) only holds through , which relates to the one-loop supergravity amplitude rather than the two-loop one.

Note that (3.8) and (3.12) can be written as for , 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 .

##### 3.2. Three or More Loops

On the supergravity side, there is an exact exponentiation formula [64, 65], which implies Since the leading IR divergences of is , one can show that the following relations hold: for for (where ).

That is, we have an exact relation at -loops for the leading IR divergence ~, with an untested relation for the subleading divergence of ; see also (2.45).

An interesting fact is that either (3.18) or (3.20) and (3.21) without the extra term, and also the relation (3.17), have a possible interpretation in terms of the โt Hooft string picture of the expansion. Thus at least in the case of , (3.18) and (3.17) still do, so one can hope that there is a correct relation at higher yet to be determined.

##### 3.3. 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 permutation symmetry for the -point functions, and another has symmetry, but requires regularization as a consequence of singularities. They are part of a flurry of recent activity relating SYM and supergravity, including [40, 42, 43, 46, 61, 66โ68] (among older works see also [37, 69, 70]). Recent work applying the KLT relations includes [71โ74]. In our quest for SYM-supergravity relations, we first review previous KLT relations; we then note that and the 1-loop supergravity amplitude both have IR divergences. We present here a tree-level KLT relation for the -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 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 supergravity and subleading-color SYM theories, both with the IR divergence, we explore the possibility that the 1-loop 5-point supergravity amplitude can be expressed as a linear combination of the SYM amplitudes. In particular a linear relation is proposed among the IR divergences of the two theories.

At tree level, the KLT relations are quadratic relations between the -point amplitudes of SYM and those of supergravity. In these relations, the helicity information is all contained within the amplitudes, and the coefficients are all function of the kinematic invariants only.

These relations relate graviton tree amplitudes with sums of squares (products) of gauge tree amplitudes. The original KLT relations were derived from string theory in the limit [35, 63] and can be expressed as (we use the notation of [37]) where โpermsโโ are ,โโ,โโ, and if and zero otherwise.

In [44, 45], new forms of the KLT relations for any -point function were found. They are both written in terms of the functions: where is zero in has the same order in both sets and and is 1 otherwise, and similarly for .

A form of KLT relations was found in [44], but needs to be regularized, due to a singular denominator However they have a large manifest symmetry. Another set was proven in [45] which is nonsingular but with only manifest symmetry.

The original KLT relation for the 5-point function is and has symmetry, whereas the KLT relations (3.25) become, explicitly,