A number of features and applications of subleading-color amplitudes of ๐’ฉ=4 SYM theory are reviewed. Particular attention is given to the IR divergences of the subleading-color amplitudes, the relationships of ๐’ฉ=4 SYM theory to ๐’ฉ=8 supergravity, and to geometric interpretations of one-loop subleading-color and ๐‘๐‘˜MHV amplitudes of ๐’ฉ=4 SYM theory.

1. Introduction

Planar amplitudes of ๐’ฉ=4 SYM theory have been extensively studied by a variety of methods, see, for example, [1โ€“23]. For a recent overview, see [24] and the special issue of Journal of Physics A, devoted to โ€œScattering amplitudes in gauge theories." Subleading-color (i.e., nonplanar) amplitudes, however, usually receive less attention [25โ€“33]. Nevertheless interesting insights are available from various applications of subleading color amplitudes. One case in point is a possible weak/weak duality between ๐’ฉ=4 SYM theory and ๐’ฉ=8 supergravity [15, 34โ€“46]. Since nonplanar graphs appear on an equal footing with planar graphs in ๐’ฉ=8 supergravity, one needs to understand the nonplanar graphs in ๐’ฉ=4 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 ๐’ฉ=4 SYM theory and amplitudes of ๐’ฉ=8 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 1/๐‘ expansion.

At tree level, we can decompose the amplitudes ๐’œ๐‘› of ๐’ฉ=4 SYM into color-ordered tree amplitudes ๐ด๐‘›๐’œtree๐‘›(12โ€ฆ๐‘›)=๐‘”๐‘›โˆ’2๎“๐œŽโˆˆ๐‘†๐‘›/๐‘๐‘›Tr(๐‘‡๐‘Ž๐œŽ(1)โ€ฆ๐‘‡๐‘Ž๐œŽ(๐‘›))๐ดtree๐‘›(๐œŽ(1)โ€ฆ๐œŽ(๐‘›))=๐‘”๐‘›โˆ’2๎“๐‘ƒ(23โ€ฆ๐‘›)Tr(๐‘‡๐‘Ž1๐‘‡๐‘Ž๐‘ƒ(2)โ€ฆ๐‘‡๐‘Ž๐‘ƒ(๐‘›))๐ดtree๐‘›(1๐‘ƒ(2)โ€ฆ๐‘ƒ(๐‘›)),(1.1) where in the second line, 1 is fixed and ๐‘ƒ(23โ€ฆ๐‘›) is a permutation of 2,3,โ€ฆ,๐‘› and ๐‘‡๐‘Ž are SU(๐‘) generators in the fundamental representation, normalized according to Tr(๐‘‡๐‘Ž๐‘‡๐‘)=๐›ฟ๐‘Ž๐‘. The color-ordered amplitudes ๐ด๐‘› depend on the momenta and polarizations of the external particles.

The color-ordered amplitudes are not independent. For ๐‘›-point amplitudes, there is a basis of (๐‘›โˆ’2)! amplitudes out of the total ๐‘›!, 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] ๐ด๐‘›(1,{๐›ผ},๐‘›,{๐›ฝ})=(โˆ’1)๐‘›๐›ฝ๎“{๐œŽ}๐‘–โˆˆOP๎€ท๎€ฝ๐›ฝ{๐›ผ},๐‘‡๎€พ๎€ธ๐ด๐‘›๎€ท1,{๐œŽ}๐‘–๎€ธ,,๐‘›(1.2) where ๐œŽ๐‘– are ordered permutations, that is, ones that keep the order of {๐›ผ} and of {๐›ฝ๐‘‡} inside ๐œŽ๐‘–. Thus the KK basis is ๐ด๐‘›(1,๐’ซ(2,โ€ฆ,๐‘›โˆ’1),๐‘›), where ๐’ซ are arbitrary permutations. All the other ๐ด๐‘›โ€™s can be recovered from it by the use of the KK relations and cyclicity and reflection invariance ๐ด๐‘›(12โ€ฆ๐‘›)=(โˆ’1)๐‘›๐ด๐‘›(๐‘›โ€ฆ21).(1.3)

At one loop, we can write a similar expansion in color-ordered amplitudes ๐’œ1โˆ’loop๐‘›(12โ€ฆ๐‘›)=๐‘”๐‘›[]๐‘›/2+1๎“๐‘—=1๎“๐œŽโˆˆ๐‘†๐‘›/๐‘†๐‘›;๐‘—๐บ๐‘Ÿ๐‘›;๐‘—(๐œŽ)๐ด๐‘›;๐‘—(๐œŽ(1)โ€ฆ๐œŽ(๐‘›)),๐บ๐‘Ÿ๐‘›;1(1)=๐‘Tr(๐‘‡๐‘Ž1โ€ฆ๐‘‡๐‘Ž๐‘›),๐บ๐‘Ÿ๐‘›;๐‘—(1)=Tr(๐‘‡๐‘Ž1โ€ฆ๐‘‡๐‘Ž๐‘—โˆ’1)Tr(๐‘‡๐‘Ž๐‘—โ€ฆ๐‘‡๐‘Ž๐‘›).(1.4) However, the subleading piece in the 1/๐‘ expansion can be obtained from the leading piece by ๐ด๐‘›;๐‘—(12โ€ฆ,๐‘—โˆ’1,๐‘—,๐‘—+1,โ€ฆ๐‘›)=(โˆ’1)๐‘—โˆ’1๎“๐œŽโˆˆCOP{๐›ผ},{๐›ฝ}๐ด๐‘›;1(๐œŽ),(1.5) where COP are cyclically ordered permutations, again keeping the order of {๐›ผ} and {๐›ฝ} fixed up to cyclic permutations.

At arbitrary loops, the decomposition of the four-gluon amplitude takes a form with only single and double traces ๐’œ4(1234)=๐‘”2๎“๐œŽโˆˆ๐‘†4/โ„ค4Tr(๐‘‡๐‘Ž๐œŽ(1)๐‘‡๐‘Ž๐œŽ(2)๐‘‡๐‘Ž๐œŽ(3)๐‘‡๐‘Ž๐œŽ(4))๐‘๐ด4;1(๐œŽ(1)๐œŽ(2)๐œŽ(3)๐œŽ(4))+๐‘”2๎“๐œŽโˆˆ๐‘†4/โ„ค32Tr(๐‘‡๐‘Ž๐œŽ(1)๐‘‡๐‘Ž๐œŽ(2))Tr(๐‘‡๐‘Ž๐œŽ(3)๐‘‡๐‘Ž๐œŽ(4))๐ด4;3(๐œŽ(1)๐œŽ(2)๐œŽ(3)๐œŽ(4)).(1.6) We also define an explicit basis [48] of single and double traces: ๐’ž[1]=Tr(๐‘‡๐‘Ž1๐‘‡๐‘Ž2๐‘‡๐‘Ž3๐‘‡๐‘Ž4),๐’ž[4]=Tr(๐‘‡๐‘Ž1๐‘‡๐‘Ž3๐‘‡๐‘Ž2๐‘‡๐‘Ž4),๐’ž[7]=Tr(๐‘‡๐‘Ž1๐‘‡๐‘Ž2)Tr(๐‘‡๐‘Ž3๐‘‡๐‘Ž4),๐’ž[2]=Tr(๐‘‡๐‘Ž1๐‘‡๐‘Ž2๐‘‡๐‘Ž4๐‘‡๐‘Ž3),๐’ž[5]=Tr(๐‘‡๐‘Ž1๐‘‡๐‘Ž3๐‘‡๐‘Ž4๐‘‡๐‘Ž2),๐’ž[8]=Tr(๐‘‡๐‘Ž1๐‘‡๐‘Ž3)Tr(๐‘‡๐‘Ž2๐‘‡๐‘Ž4๐’ž),[3]=Tr(๐‘‡๐‘Ž1๐‘‡๐‘Ž4๐‘‡๐‘Ž2๐‘‡๐‘Ž3),๐’ž[6]=Tr(๐‘‡๐‘Ž1๐‘‡๐‘Ž4๐‘‡๐‘Ž3๐‘‡๐‘Ž2),๐’ž[9]=Tr(๐‘‡๐‘Ž1๐‘‡๐‘Ž4)Tr(๐‘‡๐‘Ž2๐‘‡๐‘Ž3),(1.7) in terms of which the four-gluon amplitude can be expanded as ๐’œ4(1234)=๐‘”29๎“๐‘–=1๐ด[๐‘–]๐’ž[๐‘–].(1.8)

The loop expansion of color-ordered amplitudes ๐ด[๐‘–]=โˆž๎“๐ฟ=0๐‘Ž๐ฟ๐ด[๐‘–](๐ฟ),๐‘๐ด4;1=โˆž๎“๐ฟ=0๐‘Ž๐ฟ๐ด(๐ฟ)4;1,๐ด4;3=โˆž๎“๐ฟ=0๐‘Ž๐ฟ๐ด(๐ฟ)4;3(1.9) is in terms of the natural โ€˜t Hooft loop expansion parameter [7] ๐‘”๐‘Žโ‰ก2๐‘8๐œ‹2(4๐œ‹eโˆ’๐›พ)๐œ–,(1.10) where ๐›พ is Eulerโ€™s constant and ๐œ–=(4โˆ’๐ท)/2. Note that at ๐ฟ loops, the amplitude is at most of order ๐‘๐ฟ, which means that ๐ด(๐ฟ)[๐‘–] starts at ๐’ช(๐‘0).

For a general ๐‘›-point amplitude, we will have an expansion in an arbitrary number of multitrace color-ordered amplitudes ๐ด๐‘›;๐‘—1,๐‘—2,โ€ฆ,๐‘—๐‘˜.

Besides the loop expansion in the โ€˜t Hooft parameter ๐‘Ž, we still have a 1/๐‘ expansion of the amplitudes, which can be understood in โ€˜t Hooftโ€™s double line notation as an expansion in the topology of the diagrams. For ๐’œ4, the expansion in single-trace ๐ด4;1 and double-trace ๐ด4;3 amplitudes corresponds to the topology of the outside lines, forming boundaries of the diagrams. For example, at one-loop, the contribution in ๐ด4;1 to the amplitude is leading, that is, of order ๐‘ (thus ๐ด4;1 of order 1), coming from a diagram with the topology of 4 external lines and a boundary, whereas the contribution of ๐ด4;3 is subleading, that is, of order ๐‘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 โ€˜t 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 1/๐‘, corresponding to the number of boundaries of the diagram.

On top of that, we also have an expansion in integer powers of 1/๐‘2, independently for ๐ด4;1 and ๐ด4;3, corresponding to nonplanar diagrams with handles (a handle gives a factor of 1/๐‘2). The expansion terminates at order ๐’ช(๐‘0) for the amplitude, since in the amplitude the powers of ๐‘ can only be positive. Thus at ๐ฟ-loops, we have ๐ด(๐ฟ)4;1=๐’ช(1) to ๐’ช(1/๐‘๐ฟ) and ๐ด(๐ฟ)4;3=๐’ช(1/๐‘) to ๐’ช(1/๐‘๐ฟ). Taken together, we will say that the gluon amplitudes have a 1/๐‘ expansion.

2. IR Divergences for Subleading ๐’ฉ=4 Four-Gluon Amplitudes

2.1. General Formalism

๐’ฉ=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 ๐ท=4โˆ’2๐œ– dimensions, in which they appear as poles in a Laurent expansion in ๐œ–.

In gluon-gluon scattering in ๐’ฉ=4 SYM, IR divergences arise both from soft gluons and from collinear gluons, each of which gives rise to an ๐’ช(1/๐œ–) pole at one loop, leading to an ๐’ช(1/๐œ–2) pole at that order. At ๐ฟ loops, the leading IR divergence of the scattering amplitude is therefore ๐’ช(1/๐œ–2๐ฟ), arising from multiple soft gluon exchanges.

Subleading-color amplitudes ๐ด(๐ฟ,๐‘˜), that is, those suppressed by 1/๐‘๐‘˜ relative to the leading-color amplitude at ๐ฟ loops, have less severe IR divergences, being only of ๐’ช(1/๐œ–2๐ฟโˆ’๐‘˜) at ๐ฟ-loops.

In this section, we review the derivation of a compact all-loop-order expression for the IR-divergent part of the ๐’ฉ=4 SYM four-gluon amplitude given in [41, 49]. This result is expressed in terms of the soft (cusp) anomalous dimension ๐›พ(๐‘Ž), the collinear anomalous dimension ๐’ข0(๐‘Ž), and the soft anomalous dimension matrices ฮ“(โ„“) and relies on the assumption that the soft anomalous dimension matrices are mutually commuting, which follows if they are all proportional to ฮ“(1), 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 ๐ด[๐‘–] defined in (1.8) into a vector in color space [25, 26] ||๎€ท๐ด๐ดโŸฉ=[1],๐ด[2],๐ด[3],๐ด[4],๐ด[5],๐ด[6],๐ด[7],๐ด[8],๐ด[9]๎€ธ๐‘‡,(2.1) where (โ€ฆ)๐‘‡ denotes the transposed vector. The vector of color-ordered amplitudes factorizes into [27, 29] ||||๐ด๎‚ต๐‘ ๐‘–๐‘—๐œ‡2๎‚ต๐‘„,๐‘Ž,๐œ–๎‚ถ๎ƒข=๐ฝ2๐œ‡2๎‚ถ๐’๎‚ต๐‘ ,๐‘Ž,๐œ–๐‘–๐‘—๐‘„2,๐‘„2๐œ‡2๎‚ถ||||๐ป๎‚ต๐‘ ,๐‘Ž,๐œ–๐‘–๐‘—๐‘„2,๐‘„2๐œ‡2,,๐‘Ž,๐œ–๎‚ถ๎ƒข(2.2) where |๐ปโŸฉ, which is IR-finite as ๐œ–โ†’0, characterizes the short-distance behavior of the amplitude and where the prefactors ๐ฝ and ๐’ encapsulate the long-distance IR-divergent behavior. The soft function ๐’ is written in boldface to denote that it is a matrix acting on the vector |๐ปโŸฉ. Also ๐‘ ๐‘–๐‘— is the kinematic invariant (๐‘˜๐‘–+๐‘˜๐‘—)2, ๐œ‡ is a renormalization scale, and ๐‘„ is an arbitrary factorization scale which serves to separate the long- and short-distance behavior.

Because ๐’ฉ=4 SYM theory is conformally invariant, the product of jet functions ๐ฝ may be explicitly evaluated as [7] ๐ฝ๎‚ต๐‘„2๐œ‡2๎‚ถ๎ƒฌโˆ’1,๐‘Ž,๐œ–=exp2โˆž๎“โ„“=1๐‘Žโ„“๎‚ต๐œ‡2๐‘„2๎‚ถโ„“๐œ–๎ƒฉ๐›พ(โ„“)(โ„“๐œ–)2+2๐’ข0(โ„“),โ„“๐œ–๎ƒช๎ƒญ(2.3) where ๐›พ(โ„“) and ๐’ข0(โ„“) 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]: ๐›พ(๐‘Ž)=โˆž๎“โ„“=1๐‘Žโ„“๐›พ(โ„“)=4๐‘Žโˆ’4๐œ2๐‘Ž2+22๐œ4๐‘Ž3๐’ข+โ‹ฏ,0(๐‘Ž)=โˆž๎“โ„“=1๐‘Žโ„“๐’ข0(โ„“)=โˆ’๐œ3๐‘Ž2+๎‚€4๐œ5+103๐œ2๐œ3๎‚๐‘Ž3+โ‹ฏ.(2.4) The soft function ๐’ is given by [27, 29] ๐’๎‚ต๐‘ ๐‘–๐‘—๐‘„2,๐‘„2๐œ‡2๎‚ถ=,๐‘Ž,๐œ–P๎ƒฌโˆ’1exp2๎€œ๐‘„20๐‘‘๎‚๐œ‡2๎‚๐œ‡2๐šช๎ƒฉ๐‘ ๐‘–๐‘—๐‘„2,๐‘Ž๎ƒฉ๐œ‡2๎‚๐œ‡2,,๐‘Ž,๐œ–๎ƒช๎ƒช๎ƒญ(2.5) where ๐šช๎‚ต๐‘ ๐‘–๐‘—๐‘„2๎‚ถ=,๐‘Žโˆž๎“โ„“=1๐‘Žโ„“๐šช(โ„“),๐‘Ž๎ƒฉ๐œ‡2๎‚๐œ‡2๎ƒช=๎ƒฉ๐œ‡,๐‘Ž,๐œ–2๎‚๐œ‡2๎ƒช๐œ–๐‘Ž,(2.6) suppressing the explicit dependence of ฮ“(โ„“) on ๐‘ ๐‘–๐‘—/๐‘„2 to lighten the notation.

At this point, we make the assumption that the soft anomalous dimension matrices ฮ“(โ„“) 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 ฮ“(2)=(1/4)๐›พ(2)ฮ“(1), as shown in [28, 29]. In [32], it was established that ฮ“(3)=(1/4)๐›พ(3)ฮ“(1) for the nonpure gluon contributions. Further, ฮ“(๐ฟ)=(1/4)๐›พ(๐ฟ)ฮ“(1) 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 Casimirโ€™s terms [31, 32, 51, 52].) Therefore, the path ordering in (2.5) becomes irrelevant, allowing us to explicitly integrate it, obtaining๐’๎‚ต๐‘ ๐‘–๐‘—๐‘„2,๐‘„2๐œ‡2๎‚ถ๎ƒฌ1,๐‘Ž,๐œ–=exp2โˆž๎“โ„“=1๐‘Žโ„“๎‚ต๐œ‡2๐‘„2๎‚ถโ„“๐œ–๐šช(โ„“)๎ƒญ.โ„“๐œ–(2.7) Combining the exponents of the jet and soft functions into [27, 41] ๐†(โ„“)๐‘(๐œ–)=โ„“2๎‚ต๐œ‡2๐‘„2๎‚ถ๐œ–๎ƒฌโˆ’๎ƒฉ๐›พ(โ„“)๐œ–2+2๐’ข0(โ„“)๐œ–๎ƒช1๐Ÿ™+๐œ–๐šช(โ„“)๎ƒญ,(2.8) we may express the four-gluon amplitude in the compact form||๎ƒฌ๐ด(๐œ–)โŸฉ=expโˆž๎“โ„“=1๐‘Žโ„“๐‘โ„“๐†(โ„“)๎ƒญ||(โ„“๐œ–)๐ป(๐œ–)โŸฉ,(2.9) or equivalently ||๐ป(๐œ–)โŸฉ=โˆž๎“๐ฟ=0๐‘Ž๐ฟ||๐ป(๐ฟ๐‘“)๎ฌ=๎ƒฉ(๐œ–)๐Ÿ™โˆ’โˆž๎“โ„“=1๐‘Žโ„“๐‘โ„“๐…(โ„“)๎ƒช||(๐œ–)๐ด(๐œ–)โŸฉ.(2.10) where the matrices ๐…(โ„“)(๐œ–) will be defined below. (Henceforth we suppress ๐‘ ๐‘–๐‘—, ๐‘„, ๐œ‡, and ๐‘Ž in the arguments of the amplitudes.) Expanding (2.10) through three loops, we obtain the expressions given in [27, 41] ||๐ด(1)๎ฌ=1(๐œ–)๐‘๐…(1)||๐ด(๐œ–)(0)๎ฌ+||๐ป(1๐‘“)๎ฌ,||๐ด(๐œ–)(2)๎ฌ=1(๐œ–)๐‘2๐…(2)||๐ด(๐œ–)(0)๎ฌ+1๐‘๐…(1)||๐ด(๐œ–)(1)๎ฌ+||๐ป(๐œ–)(2๐‘“)๎ฌ,||๐ด(๐œ–)(3)๎ฌ=1(๐œ–)๐‘3๐…(3)||๐ด(๐œ–)(0)๎ฌ+1๐‘2๐…(2)||๐ด(๐œ–)(1)๎ฌ+1๐‘๐…(1)||๐ด(๐œ–)(2)๎ฌ+||๐ป(๐œ–)(3๐‘“)๎ฌ,(๐œ–)(2.11) 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 ๐…(๐œ–), we will need to know positive powers of ๐œ– in the expansion of lower loop amplitudes to obtain all the IR-divergent contributions to the ๐ฟ-loop amplitude ๐ด(โ„“).)

The equivalence of (2.9) and (2.10) follows if the matrices ๐…(โ„“)(๐œ–) are defined through the equation ๎ƒฉ๐Ÿ™โˆ’โˆž๎“โ„“=1๐‘Žโ„“๐‘โ„“๐…(โ„“)๎ƒช๎ƒฌ(๐œ–)expโˆž๎“โ„“=1๐‘Žโ„“๐‘โ„“๐†(โ„“)๎ƒญ(โ„“๐œ–)=๐Ÿ™.(2.12) First define the functional ๐‘‹[๐‘€] via [7] 1+โˆž๎“โ„“=1๐‘Žโ„“๐‘€(โ„“)๎ƒฌโ‰กexpโˆž๎“โ„“=1๐‘Žโ„“๎€ท๐‘€(โ„“)โˆ’๐‘‹(โ„“)[๐‘€]๎€ธ๎ƒญ(2.13) so that ๐‘‹(1)[๐‘€]=0,๐‘‹(2)[๐‘€]=(1/2)[๐‘€(1)]2,๐‘‹(3)[๐‘€]=โˆ’(1/3)[๐‘€(1)]3+๐‘€(1)๐‘€(2), and so forth. This functional was defined for scalar functions ๐‘€(โ„“), but we can also use it for commuting matrices. We have assumed that ฮ“(โ„“) and therefore ๐†(โ„“) are mutually commuting, and thus so are ๐…(โ„“), as a result of (2.12). Thus ๎ƒฉ๐Ÿ™โˆ’โˆž๎“โ„“=1๐‘Žโ„“๐‘โ„“๐…(โ„“)๎ƒช๎ƒฌ(๐œ–)=expโˆž๎“๐ฟ=0๐‘Žโ„“๐‘โ„“๎€ทโˆ’๐…(โ„“)(๐œ–)โˆ’๐‘‹(โ„“)[]๎€ธ๎ƒญ,โˆ’๐…(2.14) and so (2.12) is equivalent to ๐…(โ„“)(๐œ–)=โˆ’๐‘‹(โ„“)[]โˆ’๐…+๐†(โ„“)(โ„“๐œ–)(2.15) which defines ๐…(โ„“) recursively in terms of ๐†(โ„“) and ๐…(โ„“๎…ž) with โ„“โ€ฒ<โ„“. The explicit expressions up through three loops ๐…(1)(๐œ–)=๐†(1)๐…(๐œ–),(2)1(๐œ–)=โˆ’2๎€บ๐…(1)๎€ป(๐œ–)2+๐†(2)๐…(2๐œ–),(3)1(๐œ–)=โˆ’3๎€บ๐…(1)๎€ป(๐œ–)3โˆ’๐…(1)(๐œ–)๐…(2)(๐œ–)+๐†(3)(3๐œ–).(2.16) agree (up to rescaling by a factor of ๐‘๐ฟ) with the expressions given in [27] when specialized to the case of ๐‘”๐‘”โ†’๐‘”๐‘” in ๐’ฉ=4 SYM theory.

2.2. 1/๐‘ 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 1/๐‘.

The ๐ฟ-loop color-ordered amplitudes may be written in a 1/๐‘ expansion as ||๐ด(๐ฟ)๎ฌ=(๐œ–)๐ฟ๎“๐‘˜=01๐‘๐‘˜||๐ด(๐ฟ,๐‘˜)๎ฌ,(๐œ–)(2.17) where |๐ด(๐ฟ,0)โŸฉ are the leading-color amplitudes, arising from planar diagrams and |๐ด(๐ฟ,๐‘˜)โŸฉ, 1โ‰ค๐‘˜โ‰ค๐ฟ, are the subleading-color amplitudes, which include contributions from nonplanar diagrams as well. The single-trace amplitudes (๐‘–=1,โ€ฆ,6) only contain even powers of 1/๐‘ (relative to the leading-color amplitude), while the double-trace amplitudes (๐‘–=7,โ€ฆ,9) only contain odd powers of 1/๐‘.

We begin by expanding (2.9): ||๐ด(๐œ–)โŸฉ=โˆž๎“๐ฟ๐ฟ=0๎“๐‘˜=0๐‘Ž๐ฟ๐‘๐‘˜||๐ด(๐ฟ,๐‘˜)๎ฌ=(๐œ–)โˆž๎‘โ„“=1๎“๎€ฝ๐‘›โ„“๎€พ1๐‘›โ„“!๎‚ต๐‘Žโ„“๐†(โ„“)(โ„“๐œ–)๐‘โ„“๎‚ถ๐‘›โ„“โˆž๎“โ„“0โ„“=00๎“๐‘˜0=0๐‘Žโ„“0๐‘๐‘˜0||๐ป(โ„“0,๐‘˜0)๎ฌ.(๐œ–)(2.18) 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 ๐šช(โ„“)=๐›พ(โ„“)4๐šช(1)(assumption)(2.19) as has been conjectured (see footnote 1). This allows us to rewrite (2.8) as ๐†(โ„“)(โ„“๐œ–)๐‘โ„“=12๎‚ต๐œ‡2๐‘„2๎‚ถโ„“๐œ–๎ƒฌโˆ’๎ƒฉ๐›พ(โ„“)(โ„“๐œ–)2+2๐’ข0(โ„“)๎ƒช๐›พโ„“๐œ–๐Ÿ™+(โ„“)๐šช4โ„“๐œ–(1)๎ƒญ.(2.20) The one-loop soft anomalous dimension matrix can be written [29]: ๐šช(1)1=โˆ’๐‘4๎“4๐‘–=1๎“๐‘—โ‰ ๐‘–๐“๐‘–โ‹…๐“๐‘—๎‚ตlogโˆ’๐‘ ๐‘–๐‘—๐‘„2๎‚ถ,(2.21) where ๐“๐‘–โ‹…๐“๐‘—=๐‘‡๐‘Ž๐‘–๐‘‡๐‘Ž๐‘— with ๐‘‡๐‘Ž๐‘– the SU(๐‘) generators in the adjoint representation. On the basis of (1.7), it has the explicit form [48], ๐šช(1)โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ +2=2๐›ผ00๐›ฟ๐‘โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ ,0๐›ฝ๐›พ0(2.22) where โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽ ๐›ผ=๐’ฎ+๐’ฏ000000๐’ฎ+๐’ฐ000000๐’ฏ+๐’ฐ000000๐’ฏ+๐’ฐ000000๐’ฎ+๐’ฐ000000๐’ฎ+๐’ฏ,๐›ฝ=๐’ฏโˆ’๐’ฐ0๐’ฎโˆ’๐’ฐ๐’ฐโˆ’๐’ฏ๐’ฎโˆ’๐’ฏ00๐’ฏโˆ’๐’ฎ๐’ฐโˆ’๐’ฎ0๐’ฏโˆ’๐’ฎ๐’ฐโˆ’๐’ฎ๐’ฐโˆ’๐’ฏ๐’ฎโˆ’๐’ฏ0๐’ฏโˆ’๐’ฐ0๐’ฎโˆ’๐’ฐ๐›พ=๐’ฎโˆ’๐’ฐ๐’ฎโˆ’๐’ฏ00๐’ฎโˆ’๐’ฏ๐’ฎโˆ’๐’ฐ0๐’ฐโˆ’๐’ฏ๐’ฐโˆ’๐’ฎ๐’ฐโˆ’๐’ฎ๐’ฐโˆ’๐’ฏ0๐’ฏโˆ’๐’ฐ0๐’ฏโˆ’๐’ฎ๐’ฏโˆ’๐’ฎ0๐’ฏโˆ’๐’ฐ,๐›ฟ=2๐’ฎ0002๐’ฐ0002๐’ฏ(2.23) with ๎‚ตโˆ’๐‘ ๐’ฎ=log๐‘„2๎‚ถ๎‚ตโˆ’๐‘ก,๐’ฏ=log๐‘„2๎‚ถ๎‚ตโˆ’๐‘ข,๐’ฐ=log๐‘„2๎‚ถ.(2.24) If the assumption (2.19) is valid, then the 1/๐‘ expansion of ๐†(โ„“)(โ„“๐œ–)/๐‘โ„“ terminates after two terms ๐†(โ„“)(โ„“๐œ–)๐‘โ„“=๐‘”โ„“+1๐‘๐‘“โ„“,(2.25) where ๐‘”โ„“ and ๐‘“โ„“ can be read from (2.20) and (2.22). We rewrite (2.18) as ||๐ด(๐œ–)โŸฉ=โˆž๎“๐ฟ๐ฟ=0๎“๐‘˜=0๐‘Ž๐ฟ๐‘๐‘˜||๐ด(๐ฟ,๐‘˜)๎ฌ=(๐œ–)โˆž๎‘โ„“=1๎“๎€ฝ๐‘›โ„“๎€พ1๐‘›โ„“!๎‚ต๐‘Žโ„“๐‘”โ„“+๐‘Žโ„“๐‘๐‘“โ„“๎‚ถ๐‘›โ„“โˆž๎“โ„“0โ„“=00๎“๐‘˜0=0๐‘Žโ„“0๐‘๐‘˜0||๐ป(โ„“0,๐‘˜0)๎ฌ(๐œ–)(2.26) making all ๐‘ dependence explicit.

We now determine the power of the leading IR pole of |๐ด(๐ฟ,๐‘˜)(๐œ–)โŸฉ. Consider an individual term on the right-hand side of (2.26). By counting powers of ๐‘Ž and 1/๐‘, one sees that this term contributes to |๐ด(๐ฟ,๐‘˜)(๐œ–)โŸฉ, with ๐ฟ=โ„“0+โˆž๎“โ„“=1โ„“๐‘›โ„“,๐‘˜=๐‘˜0+๐‘˜1,(2.27) where ๐‘˜1 is the number of factors ๐‘“โ„“ present in the term. From (2.20) and (2.22), it is apparent that ๐‘”โ„“ has a double pole in ๐œ–, but ๐‘“โ„“ only has a single pole. The leading IR pole in the term under consideration is therefore 1/๐œ–๐‘, where ๐‘=2โˆž๎“โ„“=1๐‘›โ„“โˆ’๐‘˜1.(2.28) Combining (2.27) and (2.28), we find ๎ƒฌ2๐‘=2๐ฟโˆ’๐‘˜โˆ’โˆž๎“โ„“=1(โ„“โˆ’1)๐‘›โ„“+2โ„“0โˆ’๐‘˜0๎ƒญ.(2.29) Since ๐‘˜0โ‰คโ„“0, the term in square brackets is nonnegative, and we conclude that ||๐ด(๐ฟ,๐‘˜)๎ฌ๎‚€1(๐œ–)โˆผ๐’ช๐œ–2๐ฟโˆ’๐‘˜๎‚.(2.30) 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 |๐ด(๐ฟ,๐‘˜)(๐œ–)โŸฉ. Terms in (2.26) contribute to the leading IR pole only when the expression in square brackets in (2.29) vanishes, which occurs when ๐‘›โ„“=0 for โ„“โ‰ฅ2, and โ„“0=๐‘˜0=0 (with ๐‘›1 unconstrained). In other words, the leading IR divergences are given by [41, 49] ||๎‚ธ๐‘Ž๐†๐ด(๐œ–)โŸฉโˆผexp(1)(๐œ–)๐‘๎‚น||๐ด(0)๎ฌ(leadingIRdivergence).(2.31)

Recalling that๐†(1)(๐œ–)๐‘=๎‚ต๐œ‡2๐‘„2๎‚ถ๐œ–โŽกโŽขโŽขโŽฃโˆ’2๐œ–21๐Ÿ™+๐œ–โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ +1๐›ผ00๐›ฟโŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ โŽคโŽฅโŽฅโŽฆ,๐‘๐œ–0๐›ฝ๐›พ0(2.32) we use (2.31) to obtain the coefficient of the leading IR pole||๐ด(๐ฟ,๐‘˜)๎ฌ=(๐œ–)(โˆ’2)๐ฟโˆ’๐‘˜1๐‘˜!(๐ฟโˆ’๐‘˜)!๐œ–2๐ฟโˆ’๐‘˜โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ 0๐›ฝ๐›พ0๐‘˜||๐ด(0)๎ฌ๎‚€1+๐’ช๐œ–2๐ฟโˆ’๐‘˜โˆ’1๎‚,(2.33) where the tree-level amplitudes are ||๐ป(0,0)๎ฌ=||๐ด(0)๎ฌ=โˆ’4๐‘–๐พ๐‘ ๐‘ก๐‘ข(๐‘ข,๐‘ก,๐‘ ,๐‘ ,๐‘ก,๐‘ข,0,0,0)๐‘‡,(2.34) where ๐‘ =(๐‘˜1+๐‘˜2)2,๐‘ก=(๐‘˜1+๐‘˜4)2 and ๐‘ข=(๐‘˜1+๐‘˜3)2 are the usual Mandelstam variables, obeying ๐‘ +๐‘ก+๐‘ข=0 for massless external gluons. The factor ๐พ, defined in (7.4.42)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 ||๐ด(๐ฟ,0)๎ฌ=((๐œ–)โˆ’2)๐ฟ๐ฟ!๐œ–2๐ฟ||๐ด(0)๎ฌ๎‚€1+๐’ช๐œ–2๐ฟโˆ’1๎‚.(2.35) The remaining IR divergences, from ๐’ช(1/๐œ–2๐ฟโˆ’1) to ๐’ช(1/๐œ–), are all proportional to |๐ด(0)โŸฉ 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 ๐›พโŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘ข๐‘ก๐‘ ๐‘ ๐‘ก๐‘ขโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽ111โŽžโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽ111โŽžโŽŸโŽŸโŽŸโŽŸโŽ ๎€ท๐‘‹=2(๐‘ ๐‘Œโˆ’๐‘ก๐‘‹),๐›พ๐›ฝ=22+๐‘Œ2+๐‘2๎€ธโŽ›โŽœโŽœโŽœโŽœโŽ111โŽžโŽŸโŽŸโŽŸโŽŸโŽ (2.36) with ๎‚€๐‘ก๐‘‹=log๐‘ข๎‚๎‚€๐‘ข,๐‘Œ=log๐‘ ๎‚๎‚€๐‘ ,๐‘=log๐‘ก๎‚.(2.37) Hence, the leading IR divergence of the subleading-color amplitudes is given by ||๐ด(๐ฟ,2๐‘š+1)๎ฌ=๎‚€(๐œ–)โˆ’4๐‘–๐พ๎‚๐‘ ๐‘ก๐‘ข(โˆ’1)๐ฟโˆ’12๐ฟโˆ’๐‘š๎€ท๐‘‹2+๐‘Œ2+๐‘2๎€ธ๐‘š(๐‘ ๐‘Œโˆ’๐‘ก๐‘‹)(2๐‘š+1)!(๐ฟโˆ’2๐‘šโˆ’1)!๐œ–2๐ฟโˆ’2๐‘šโˆ’1โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ000000111โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ๎‚€1+๐’ช๐œ–2๐ฟโˆ’2๐‘šโˆ’2๎‚,(2.38)||๐ด(๐ฟ,2๐‘š+2)๎ฌ=๎‚€(๐œ–)โˆ’4๐‘–๐พ๎‚๐‘ ๐‘ก๐‘ข(โˆ’1)๐ฟ2๐ฟโˆ’๐‘šโˆ’1๎€ท๐‘‹2+๐‘Œ2+๐‘2๎€ธ๐‘š(๐‘ ๐‘Œโˆ’๐‘ก๐‘‹)(2๐‘š+2)!(๐ฟโˆ’2๐‘šโˆ’2)!๐œ–2๐ฟโˆ’2๐‘šโˆ’2โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ000โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ๎‚€1๐‘‹โˆ’๐‘Œ๐‘โˆ’๐‘‹๐‘Œโˆ’๐‘๐‘Œโˆ’๐‘๐‘โˆ’๐‘‹๐‘‹โˆ’๐‘Œ+๐’ช๐œ–2๐ฟโˆ’2๐‘šโˆ’3๎‚.(2.39) The results (2.38) and (2.39) were derived in [49], generalizing expressions derived in [41].

2.3. IR Divergences of ๐ด(๐ฟ,1)

In this subsection, we consider the subleading-color amplitude |๐ด(๐ฟ,1)โŸฉ 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 โ‰ค2: ||๐ด(๐ฟ)๎ฌ=1(๐œ–)๎‚€๐‘”๐ฟ!1+1๐‘๐‘“1๎‚๐ฟ||๐ด(0)๎ฌ+1๎‚€๐‘”๐‘(๐ฟโˆ’1)!1+1๐‘๐‘“1๎‚๐ฟโˆ’1||๐ป(1,1)๎ฌ+1(๐œ–)๎‚€๐‘”(๐ฟโˆ’2)!1+1๐‘๐‘“1๎‚๐ฟโˆ’2๎‚€๐‘”2+1๐‘๐‘“2๎‚||๐ด(0)๎ฌ+1๎‚€๐‘”(๐ฟโˆ’1)!1+1๐‘๐‘“1๎‚๐ฟโˆ’1||๐ป(1,0)๎ฌ+1(๐œ–)๐‘2๎‚€๐‘”(๐ฟโˆ’2)!1+1๐‘๐‘“1๎‚๐ฟโˆ’2||๐ป(2,2)๎ฌ((๐œ–)+โ‹ฏthreeleadingIRpoles),(2.40) where we use (2.20) and (2.22) to write ๐‘”1=๎‚ต๐œ‡2๐‘„2๎‚ถ๐œ–โŽกโŽขโŽขโŽฃโˆ’2๐œ–21๐Ÿ™+๐œ–โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ โŽคโŽฅโŽฅโŽฆ๐›ผ00๐›ฟ,๐‘“1=1๐œ–๎‚ต๐œ‡2๐‘„2๎‚ถ๐œ–โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ ,๐‘”0๐›ฝ๐›พ02=๎‚ต๐œ‡2๐‘„2๎‚ถ2๐œ–โŽกโŽขโŽขโŽฃโˆ’๎ƒฉ๐›พ(2)8๐œ–2+๐’ข0(2)๎ƒช๐›พ2๐œ–๐Ÿ™+(2)โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ โŽคโŽฅโŽฅโŽฆ8๐œ–๐›ผ00๐›ฟ,๐‘“2=๐›พ(2)๎‚ต๐œ‡8๐œ–2๐‘„2๎‚ถ2๐œ–โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ .0๐›ฝ๐›พ0(2.41) To extract the |๐ด(๐ฟ,1)โŸฉ amplitude, we employ the identity๎‚€๐‘”1+1๐‘๐‘“1๎‚๐ฟ||||1/๐‘piece=๐ฟ๐‘”1๐ฟโˆ’1๐‘“1+โŽ›โŽœโŽœโŽ๐ฟ2โŽžโŽŸโŽŸโŽ ๐‘”1๐ฟโˆ’2๎€บ๐‘“1,๐‘”1๎€ป+โŽ›โŽœโŽœโŽ๐ฟ3โŽžโŽŸโŽŸโŽ ๐‘”1๐ฟโˆ’3๐‘“๎€บ๎€บ1,๐‘”1๎€ป,๐‘”1๎€ป+๎€บโ‹ฏ๐‘“๎€บ๎€บ๎€บ1,๐‘”1๎€ป,๐‘”1๎€ป,๐‘”1๎€ปโ‹ฏ๎€ป,(2.42) in which the first term on the right-hand side has an expansion that starts with 1/๐œ–2๐ฟโˆ’1, the second term has an expansion that starts with 1/๐œ–2๐ฟโˆ’2, and so forth. Thus, keeping only the terms proportional to 1/๐‘ in (2.40), and only the first three terms in the Laurent expansion, we obtain||๐ด(๐ฟ,1)๎ฌ=1๐‘”(๐ฟโˆ’1)!1๐ฟโˆ’1๐‘“1||๐ด(0)๎ฌ+12๐‘”(๐ฟโˆ’2)!1๐ฟโˆ’2๎€บ๐‘“1,๐‘”1๎€ป||๐ด(0)๎ฌ+1๐‘”(๐ฟโˆ’1)!1๐ฟโˆ’1||๐ป(1,1)๎ฌ+1(๐œ–)6๐‘”(๐ฟโˆ’3)!1๐ฟโˆ’3๐‘“๎€บ๎€บ1,๐‘”1๎€ป,๐‘”1๎€ป||๐ด(0)๎ฌ+1๐‘”(๐ฟโˆ’2)!1๐ฟโˆ’2๐‘“2||๐ด(0)๎ฌ+1๐‘”(๐ฟโˆ’3)!1๐ฟโˆ’3๐‘“1๐‘”2||๐ด(0)๎ฌ+1๐‘”(๐ฟโˆ’2)!1๐ฟโˆ’2๐‘“1||๐ป(1,0)๎ฌ๎‚€1(๐œ–)+๐’ช๐œ–2๐ฟโˆ’4๎‚,(2.43)as obtained in [49].

2.4. IR Divergences of ๐ด(๐ฟ,๐ฟ)

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 |๐ด(๐ฟ,๐ฟ)โŸฉ.

The only terms in (2.26) that contribute to |๐ด(๐ฟ,๐ฟ)โŸฉ are those with as many factors of 1/๐‘ as of ๐‘Ž. Thus, only ๐‘“1 and |๐ป(โ„“0,โ„“0)โŸฉ can contribute, giving||๐ด(๐ฟ,๐ฟ)๎ฌ=(๐œ–)๐ฟ๎“โ„“0=01๎€ท๐ฟโˆ’โ„“0๎€ธ!๐‘“๐ฟโˆ’โ„“01||๐ป(โ„“0,โ„“0)๎ฌ,(๐œ–)where๐‘“1=1๐œ–๎‚ต๐œ‡2๐‘„2๎‚ถ๐œ–โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ 0๐›ฝ๐›พ0(2.44) exact to all orders in the ๐œ– expansion. Keeping just the first two terms in the Laurent expansion, we find ||๐ด(๐ฟ,๐ฟ)๎ฌ=1(๐œ–)๐‘“(๐ฟโˆ’1)!1๐ฟโˆ’1๎‚ƒ1๐ฟ๐‘“1||๐ด(0)๎ฌ+||๐ป(1,1)๎ฌ๎‚„๎‚€1(๐œ–)+๐’ช๐œ–๐ฟโˆ’2๎‚=11(๐ฟโˆ’1)!๐œ–๐ฟโˆ’1โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ 0๐›ฝ๐›พ0๐ฟโˆ’1||๐ด(1,1)๎ฌ๎‚€1(๐ฟ๐œ–)+๐’ช๐œ–๐ฟโˆ’2๎‚.(2.45) 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

๐’ฉ=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 et al. [54] (see also [36]). Explicit expressions for these IR-divergent integrals as the Laurent expansions in ๐œ– 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 1/๐‘ expansion of these divergences.

Recall from (2.17) that ๐ด(๐ฟ,๐‘˜)[๐‘–] denotes the ๐ฟ-loop color-ordered amplitude which is subleading by a factor of 1/๐‘๐‘˜ in the 1/๐‘ expansion. Single-trace amplitudes are denoted by ๐‘–=1,โ€ฆ,6 and double-trace amplitudes by ๐‘–=7,โ€ฆ,9 (see (1.7)).

At one loop, the single-trace amplitudes are given by [34] ๐ด[1](1,0)=๐‘€(1)(๐‘ ,๐‘ก)๐ด[1](0)=2๐‘–๐พ๐ผ4(1)(๐‘ ,๐‘ก)(2.46) with the other single-trace amplitudes ๐ด(1,0)[2] and ๐ด(1,0)[3] obtained by letting ๐‘กโ†”๐‘ข, and ๐‘ โ†”๐‘ข, respectively. The identities ๐ด(๐ฟ)[1]=๐ด(๐ฟ)[6], ๐ด(๐ฟ)[2]=๐ด(๐ฟ)[5], and ๐ด(๐ฟ)[3]=๐ด(๐ฟ)[4] are satisfied at all loop orders. In (2.46), ๐ผ4(1)(๐‘ ,๐‘ก) denotes the scalar box integral ๐‘€(1)1(๐‘ ,๐‘ก)=โˆ’2๐‘ ๐‘ก๐ผ4(1)๐ผ(๐‘ ,๐‘ก),4(1)(๐‘ ,๐‘ก)=๐ผ4(1)(๐‘ก,๐‘ )=โˆ’๐‘–๐œ‡2๐œ–e๐œ–๐›พ๐œ‹โˆ’๐ท/2๎€œ๐‘‘๐ท๐‘๐‘2๎€ท๐‘โˆ’๐‘˜1๎€ธ2๎€ท๐‘โˆ’๐‘˜1โˆ’๐‘˜2๎€ธ2๎€ท๐‘+๐‘˜4๎€ธ2,(2.47) an explicit expression for which is given, for example, in [7].

The one-loop double-trace amplitudes are given by [34] ๐ด[7](1,1)=๐ด[8](1,1)=๐ด[9](1,1)๎‚€๐ด=2[1](1,0)+๐ด[2](1,0)+๐ด[3](1,0)๎‚(2.48)๎‚ƒ๐ผ=4๐‘–๐พ4(1)(๐‘ ,๐‘ก)+๐ผ4(1)(๐‘ก,๐‘ข)+๐ผ4(1)๎‚„.(๐‘ข,๐‘ )(2.49) Relation (2.48) follows from the one-loop U(1) decoupling identity [57].

At two loops, the leading-color single-trace amplitude is given by [54] ๐ด[1](2,0)=๐‘€(2)(๐‘ ,๐‘ก)๐ด[1](0)๎‚ƒ=โˆ’๐‘–๐พ๐‘ ๐ผ4(2)๐‘ƒ(๐‘ ,๐‘ก)+๐‘ก๐ผ4(2)๐‘ƒ๎‚„,(๐‘ก,๐‘ )(2.50) where ๐ผ4(2)๐‘ƒ(๐‘ ,๐‘ก) denotes the scalar double-box (planar) integral ๐‘€(2)1(๐‘ ,๐‘ก)=4๎‚ƒ๐‘ ๐‘ก๐‘ ๐ผ4(2)๐‘ƒ(๐‘ ,๐‘ก)+๐‘ก๐ผ4(2)๐‘ƒ๎‚„,๐ผ(๐‘ก,๐‘ )4(2)๐‘ƒ๎€ท(๐‘ ,๐‘ก)=โˆ’๐‘–๐œ‡2๐œ–e๐œ–๐›พ๐œ‹โˆ’๐ท/2๎€ธ2๎€œ๐‘‘๐ท๐‘๐‘‘๐ท๐‘ž๐‘2(๐‘+๐‘ž)2๐‘ž2๎€ท๐‘โˆ’๐‘˜1๎€ธ2๎€ท๐‘โˆ’๐‘˜1โˆ’๐‘˜2๎€ธ2๎€ท๐‘žโˆ’๐‘˜4๎€ธ2๎€ท๐‘žโˆ’๐‘˜3โˆ’๐‘˜4๎€ธ2,(2.51) an explicit expression for which is given, for example, in [7]. The double-trace amplitude is [54] ๐ด[7](2,1)๎‚ƒ๐‘ ๎‚€=โˆ’2๐‘–๐พ3๐ผ4(2)๐‘ƒ(๐‘ ,๐‘ก)+2๐ผ4(2)๐‘๐‘ƒ(๐‘ ,๐‘ก)+3๐ผ4(2)๐‘ƒ(๐‘ ,๐‘ข)+2๐ผ4(2)๐‘๐‘ƒ๎‚๎‚€๐ผ(๐‘ ,๐‘ข)โˆ’๐‘ก4(2)๐‘๐‘ƒ(๐‘ก,๐‘ )+๐ผ4(2)๐‘๐‘ƒ๎‚๎‚€๐ผ(๐‘ก,๐‘ข)โˆ’๐‘ข4(2)๐‘๐‘ƒ(๐‘ข,๐‘ )+๐ผ4(2)๐‘๐‘ƒ,(๐‘ข,๐‘ก)๎‚๎‚„(2.52) and the subleading-color single-trace amplitude is [54] ๐ด[1](2,2)๎‚ƒ๐‘ ๎‚€๐ผ=โˆ’2๐‘–๐พ4(2)๐‘ƒ(๐‘ ,๐‘ก)+๐ผ4(2)๐‘๐‘ƒ(๐‘ ,๐‘ก)+๐ผ4(2)๐‘ƒ(๐‘ ,๐‘ข)+๐ผ4(2)๐‘๐‘ƒ๎‚๎‚€๐ผ(๐‘ ,๐‘ข)+๐‘ก4(2)๐‘ƒ(๐‘ก,๐‘ )+๐ผ4(2)๐‘๐‘ƒ(๐‘ก,๐‘ )+๐ผ4(2)๐‘ƒ(๐‘ก,๐‘ข)+๐ผ4(2)๐‘๐‘ƒ๎‚๎‚€๐ผ(๐‘ก,๐‘ข)โˆ’2๐‘ข4(2)๐‘ƒ(๐‘ข,๐‘ )+๐ผ4(2)๐‘๐‘ƒ(๐‘ข,๐‘ )+๐ผ4(2)๐‘ƒ(๐‘ข,๐‘ก)+๐ผ4(2)๐‘๐‘ƒ,(๐‘ข,๐‘ก)๎‚๎‚„(2.53) where ๐ผ4(2)๐‘๐‘ƒ(๐‘ ,๐‘ก) denotes the two-loop nonplanar integral ๐ผ4(2)๐‘๐‘ƒ๎€ท(๐‘ ,๐‘ก)=โˆ’๐‘–๐œ‡2๐œ–e๐œ–๐›พ๐œ‹โˆ’๐ท/2๎€ธ2๎€œ๐‘‘๐ท๐‘๐‘‘๐ท๐‘ž๐‘2(๐‘+๐‘ž)2๐‘ž2๎€ท๐‘โˆ’๐‘˜2๎€ธ2๎€ท๐‘+๐‘ž+๐‘˜1๎€ธ2๎€ท๐‘žโˆ’๐‘˜3๎€ธ2๎€ท๐‘žโˆ’๐‘˜3โˆ’๐‘˜4๎€ธ2,(2.54) an explicit expression for which is given in [56]. All the other single- and double-trace amplitudes ๐ด(2,๐‘˜)[๐‘–] are obtained by making the appropriate permutations of ๐‘ ,๐‘ก, and ๐‘ข 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 ๐’ฉ=4 amplitudes at one and two loops (and presumably beyond) also have uniform transcendentality. What makes this surprising is that the nonplanar integral ๐ผ4(2)๐‘๐‘ƒ(๐‘ ,๐‘ก) that contributes to ๐ด(2,1) and ๐ด(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 ๐’ฉ=8 supergravity [39, 58].)

The two-loop amplitudes obey the following group theory relations [59]: ๐ด[7](2,1)๎‚€๐ด=2[1](2,0)+๐ด[2](2,0)+๐ด[3](2,0)๎‚โˆ’๐ด[3](2,2),๐ด[8](2,1)๎‚€๐ด=2[1](2,0)+๐ด[2](2,0)+๐ด[3](2,0)๎‚โˆ’๐ด[1](2,2),๐ด[9](2,1)๎‚€๐ด=2[1](2,0)+๐ด[2](2,0)+๐ด[3](2,0)๎‚โˆ’๐ด[2](2,2)(2.55) and may be easily verified using (2.50), (2.52), and (2.53). In addition, we have ๐ด[1](2,2)+๐ด[2](2,2)+๐ด[3](2,2)=0,(2.56) also easily verified using (2.53). Together these equations imply63๎“๐‘–=1๐ด[๐‘–](2,0)โˆ’9๎“๐‘–=7๐ด[๐‘–](2,1)=0(2.57) which is the two-loop generalization of the U(1) decoupling relation (2.48). Both (2.56) and (2.57) are encapsulated in the equation 63๎“๐‘–=1๐ด[๐‘–](๐ฟ)โˆ’๐‘9๎“๐‘–=7๐ด[๐‘–](๐ฟ)=0,๐ฟโ‰ค2,(2.58) which is valid through two loops.

At one loop, we also saw that one can relate all the subleading-color amplitudes ๐ด๐‘›;๐‘— to the leading amplitude ๐ด๐‘›;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 (2.46) with ๐‘€(1)1(๐‘ ,๐‘ก)=โˆ’๐œ–2๎‚ต๐œ‡2๎‚ถโˆ’๐‘ ๐œ–โˆ’1๐œ–2๎‚ต๐œ‡2๎‚ถโˆ’๐‘ก๐œ–+12log2๎‚€๐‘ ๐‘ก๎‚+2๐œ‹23+๐’ช(๐œ–),(2.59) 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||๐ด(1,1)๎ฌ=๎‚€(๐œ–)โˆ’8๐‘–๐พ๎‚๐œ‡๐‘ ๐‘ก๐‘ข๎‚ธ๎‚ต2๎‚ถโˆ’๐‘ข๐œ–(๐‘ ๐‘Œโˆ’๐‘ก๐‘‹)๐œ–๎‚นโŽ›โŽœโŽœโŽœโŽœโŽ111โŽžโŽŸโŽŸโŽŸโŽŸโŽ โˆ’(๐‘ +๐‘ก)๐‘‹๐‘Œ+๐’ช(๐œ–),(2.60) where we have only included the [7โ€“9] components of ๐ด(1,1)[๐‘–] as the others vanish.

At two loops, the planar amplitude is given by (2.50) with [60] ๐‘€(2)1(๐œ–)=2๎€บ๐‘€(1)๎€ป(๐œ–)2โˆ’๎€ท๐œ2+๐œ–๐œ3+๐œ–2๐œ4๎€ธ๐‘€(1)๐œ‹(2๐œ–)โˆ’472+๐’ช(๐œ–).(2.61) The two-loop double trace amplitude has an IR divergence given by the general formula (2.38), which yields ||๐ด(2,1)๎ฌ=๎‚€(๐œ–)โˆ’8๐‘–๐พ๎‚๐‘ ๐‘ก๐‘ข(โˆ’2)(๐‘ ๐‘Œโˆ’๐‘ก๐‘‹)๐œ–3โŽ›โŽœโŽœโŽœโŽœโŽ111โŽžโŽŸโŽŸโŽŸโŽŸโŽ ๎‚€1+๐’ช๐œ–2๎‚.(2.62) Finally, the subleading-color single-trace amplitude is given by (2.45) which in this case yields ||๐ด(2,2)๎ฌ=1(๐œ–)๐œ–โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ๐ด๐‘‹โˆ’๐‘Œ๐‘โˆ’๐‘‹๐‘Œโˆ’๐‘๐‘Œโˆ’๐‘๐‘โˆ’๐‘‹๐‘‹โˆ’๐‘Œ[7](1,1)๎€ท๐œ–(2๐œ–)+๐’ช0๎€ธ.(2.63) Only the [1] through [6] components are listed, as the [7] through [9] components vanish.

3. Subleading-Color Amplitudes of ๐’ฉ=4 SYM and Amplitudes of ๐’ฉ=8 Supergravity

The AdS5/CFT4 correspondence provides a strong/weak duality between ๐’ฉ=4 SYM and ๐’ฉ=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, 34โ€“38, 41โ€“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 nonplanar graphs appear on an equal footing with planar graphs in ๐’ฉ=8 supergravity, it seems important to understand nonplanar graphs in ๐’ฉ=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 ๐’ฉ=4 SYM and those of ๐’ฉ=8 supergravity, at one and two loops. For more than two loops, the known relation for ๐‘›=4 is for the leading IR singularity only. One application of these ideas for ๐‘›=5 at one loop is a new form of (tree level) KLT relations. Others are possible relations between ๐’ฉ=4 subleading-color amplitudes and ๐’ฉ=8 sugra for ๐‘›โ‰ฅ5.

3.1. One and Two-Loop Relations

In this subsection, we demonstrate the existence of some exact relations between ๐’ฉ=4 SYM amplitudes and ๐’ฉ=8 supergravity amplitudes at the one- and two-loop levels. The ๐ฟ-loop ๐‘-independent SYM amplitude ๐ด(๐ฟ,๐ฟ) may be expected to be related to the ๐ฟ-loop supergravity amplitude, as both have ๐’ช(1/๐œ–๐ฟ) leading IR divergences. Other subleading-color SYM amplitudes ๐ด(๐ฟ,๐‘˜) have ๐’ช(1/๐œ–2๐ฟโˆ’๐‘˜) leading IR divergences and consequently satisfy relations involving lower-loop supergravity amplitudes. The normalization of ๐ด(๐ฟ,2๐‘š+1)SYM(๐‘ ,๐‘ก) 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 notation๐ด(๐ฟ,2๐‘š)SYM(๐‘ ,๐‘ก)=๐‘Ž๐ฟ๐ด[1](๐ฟ,2๐‘š),๐ด(๐ฟ,2๐‘š+1)SYM๐‘Ž(๐‘ ,๐‘ก)=โˆ’๐ฟโˆš2๐ด[8](๐ฟ,2๐‘š+1),(3.1) noting that the other components ๐ด(๐ฟ,๐‘˜)[๐‘–] are obtained by permutations of ๐‘ , ๐‘ก, and ๐‘ข. However, we omit the argument (๐‘ ,๐‘ก) for functions that are completely symmetric under permutations of ๐‘ , ๐‘ก, and ๐‘ข.

Factor out the tree amplitude to define๐‘€(๐ฟ,๐‘˜)SYM๐ด(๐‘ ,๐‘ก)=(๐ฟ,๐‘˜)SYM(๐‘ ,๐‘ก)๐ด(0)SYM,(๐‘ ,๐‘ก)(3.2) so that the coupling constant ๐‘Ž๐ฟ is now included in the definition of ๐‘€(๐ฟ,๐‘˜)SYM(๐‘ ,๐‘ก), and where. ๐ด(0)SYM(๐‘ ,๐‘ก)=โˆ’4๐‘–๐พ.๐‘ ๐‘ก(3.3)

In what follows we denote ๐ดtreeSYM(๐‘–๐‘—โ€ฆ๐‘˜)=๐ด(๐‘–๐‘—โ€ฆ๐‘˜) (see also (2.34)). Recall that the one-loop ๐‘-independent SYM four-gluon amplitude is given by (2.47), obtaining ๐ด(1,1)SYMโˆš=โˆ’2๎‚ธ๐‘”2๐‘–๐พ2๐‘8๐œ‹2(4๐œ‹eโˆ’๐›พ)๐œ–๎‚น๎‚ƒ๐ผ4(1)(๐‘ ,๐‘ก)+๐ผ4(1)(๐‘ก,๐‘ข)+๐ผ4(1)๎‚„.(๐‘ข,๐‘ )(3.4) The one-loop supergravity four-graviton amplitude may be expressed as [34, 36] ๐ด(1)SG=8๐‘–๐พ2๎‚ธ(๐œ…/2)28๐œ‹2(4๐œ‹eโˆ’๐›พ)๐œ–๎‚น๎‚ƒ๐ผ4(1)(๐‘ ,๐‘ก)+๐ผ4(1)(๐‘ก,๐‘ข)+๐ผ4(1)๎‚„.(๐‘ข,๐‘ )(3.5)after stripping off a factor of (๐œ…/2)2 for a four-point amplitude. The supergravity amplitude is proportional to ๐พ2 rather than ๐พ due to the KLT relations [63] (a manifestation of the relation โ€œclosed string = (openstring)2โ€). This factor is also present in the tree-level supergravity amplitude, so we can factor it out as follows:๐ด(1)SG=๐ด(0)SG๐‘€(1)SG=๎‚ต16๐‘–๐พ2๎‚ถ๐‘€๐‘ ๐‘ก๐‘ข(1)SG.(3.6) Defining ๐œ†SYM=๐‘”2๐‘ and ๐œ†SG=(๐œ…/2)2, one observes that the one-loop SYM and supergravity amplitudes are related by๐‘€(1,1)SYMโˆš(๐‘ ,๐‘ก)=2๐œ†SYM๐œ†SG๐‘ข๐‘€(1)SG.(3.7) 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 ๐œ†SG๐‘ข for supergravity because ๐œ†SG is dimensionful.

Finally, rewrite (3.7) in the manifestly permutation-symmetric form13๎‚ƒ๎€ท๐œ†SG๐‘ข๎€ธ๐‘€(1,1)SYM(๐‘ ,๐‘ก)+c.p.๎‚„=โˆš2๐œ†SYM๐‘€(1)SG,(3.8) (where c.p. denotes cyclic permutations of ๐‘ , ๐‘ก, and ๐‘ข) even though ๐‘ข๐‘€(1,1)SYM(๐‘ ,๐‘ก) is already symmetric under permutations. A similar symmetrized relation can be written for the one-loop leading-color amplitude ๎€ท๐œ†SG๐‘ข๎€ธ๐‘€(1,0)SYM(๐‘ ,๐‘ก)+c.p.=โˆ’๐œ†SYM๐‘€(1)SG(3.9) 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 ๐‘-independent SYM amplitude ๐ด(2,2)SYM(๐‘ ,๐‘ก), since, from (2.63), ๐ด(2,2)SYMโˆš(๐‘ ,๐‘ก)=โˆ’2๐‘Ž๐‘‹โˆ’๐‘Œ๐œ–๐ด(1,1)SYM๎€ท๐œ–(2๐œ–)+๐’ช0๎€ธ.(3.10) This can be rewritten as๐‘€(2,2)SYM๐œ†(๐‘ ,๐‘ก)=โˆ’2๐‘ŽSYM๐œ†SG๐‘ข๎‚€๐‘‹โˆ’๐‘Œ๐œ–๎‚๐‘€(1)SG๎€ท๐œ–(2๐œ–)+๐’ช0๎€ธ,(3.11) where ๐‘‹=log(๐‘ก/๐‘ข),๐‘Œ=log(๐‘ข/๐‘ ),๐‘=log(๐‘ /๐‘ก), as in (2.37), thus obtaining a relation to the one-loop supergravity amplitude.

Using the relation ๐‘€(2)SG(๐œ–)=(1/2)[๐‘€(1)SG(๐œ–)]2+๐’ช(๐œ–0) between the one- and two-loop supergravity amplitudes [41, 58, 64, 65], we can write this as13๎‚ƒ๎€ท๐œ†SG๐‘ข๎€ธ2๐‘€(2,2)SYM(๐‘ ,๐‘ก)+c.p.๎‚„=2๐œ†2SYM๐‘€(2)SG,(3.12) where this relation is exact (!), as may be easily verified by using the exact expression for the ๐‘-independent SYM amplitude [54] and from (2.53) ๐‘€(2,2)SYM๐‘Ž(๐‘ ,๐‘ก)=2๐‘ ๐‘ก2๎‚ƒ๐‘ ๎‚€๐ผ4(2)๐‘ƒ(๐‘ ,๐‘ก)+๐ผ4(2)๐‘๐‘ƒ(๐‘ ,๐‘ก)+๐ผ4(2)๐‘ƒ(๐‘ ,๐‘ข)+๐ผ4(2)๐‘๐‘ƒ๎‚๎‚€๐ผ(๐‘ ,๐‘ข)+๐‘ก4(2)๐‘ƒ(๐‘ก,๐‘ )+๐ผ4(2)๐‘๐‘ƒ(๐‘ก,๐‘ )+๐ผ4(2)๐‘ƒ(๐‘ก,๐‘ข)+๐ผ4(2)๐‘๐‘ƒ๎‚๎‚€๐ผ(๐‘ก,๐‘ข)โˆ’2๐‘ข4(2)๐‘ƒ(๐‘ข,๐‘ )+๐ผ4(2)๐‘๐‘ƒ(๐‘ข,๐‘ )+๐ผ4(2)๐‘ƒ(๐‘ข,๐‘ก)+๐ผ4(2)๐‘๐‘ƒ(๐‘ข,๐‘ก)๎‚๎‚„(3.13) and that for the two-loop supergravity amplitude [36] ๐‘€(2)SG๐‘ =โˆ’3๐‘ก๐‘ข4๎‚ธ(๐œ…/2)28๐œ‹2(4๐œ‹eโˆ’๐›พ)๐œ–๎‚น2๎‚ƒ๐ผ4(2)๐‘ƒ(๐‘ ,๐‘ก)+๐ผ4(2)๐‘๐‘ƒ(๐‘ ,๐‘ก)+๐ผ4(2)๐‘ƒ(๐‘ ,๐‘ข)+๐ผ4(2)๐‘๐‘ƒ๎‚„+(๐‘ ,๐‘ข)c.p.,(3.14) where ๐ผ4(2)๐‘ƒ and ๐ผ4(2)๐‘๐‘ƒ are the two-loop planar and nonplanar 4-point functions.

Now consider the two-loop subleading-color amplitude ๐‘€(2,1)SYM. The two-loop decoupling relation (2.57) can be rewritten asโˆ’โˆš2๎‚ƒ๐‘ข๐‘€(2,1)SYM(๐‘ ,๐‘ก)+c.p.๎‚„๎‚ƒ=6๐‘ข๐‘€(2,0)SYM(๐‘ ,๐‘ก)+c.p.๎‚„.(3.15) Using the ABDK relation [60] ๐‘€(2,0)SYM1(๐œ–)=2๎‚ƒ๐‘€(1,0)SYM๎‚„(๐œ–)2+๐‘Ž๐‘“(2)(๐œ–)๐‘€(1,0)SYM(2๐œ–)+๐’ช(๐œ–),๐‘“(2)๎€ท๐œ(๐œ–)=โˆ’2+๐œ–๐œ3+๐œ–2๐œ4๎€ธ,(3.16) together with (3.9), we can rewrite (3.15) as13๎‚ƒ๎€ท๐œ†SG๐‘ข๎€ธ๐‘€(2,1)SYM(๐‘ ,๐‘ก)+c.p.๎‚„+1โˆš2๎‚ป๎€ท๐œ†SG๐‘ข๎€ธ๎‚ƒ๐‘€(1,0)SYM๎‚„(๐‘ ,๐‘ก)2+c.p.๎‚ผ=โˆš2๐œ†2SYM8๐œ‹2(4๐œ‹eโˆ’๐›พ)๐œ–๐‘“(2)(๐œ–)๐‘€(1)SG(2๐œ–)+๐’ช(๐œ–).(3.17) Unlike (3.12), however, (3.17) only holds through ๐’ช(๐œ–0), 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 13๎‚ƒ๎€ท๐œ†SG๐‘ข๎€ธ๐ฟ๐‘€(๐ฟ,๐ฟ)SYM(๐‘ ,๐‘ก)+c.p.๎‚„=๎‚€โˆš2๐œ†SYM๎‚๐ฟ๐‘€(๐ฟ)SG(3.18) for ๐ฟ=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 ๐ฟโ‰ฅ3.

3.2. Three or More Loops

On the supergravity side, there is an exact exponentiation formula [64, 65], which implies๐‘€(๐ฟ)SG=1๎‚ƒ๐‘€๐ฟ!(1)SG๎‚„๐ฟ๎‚€1+๐’ช๐œ–๐ฟโˆ’2๎‚=1๎‚ธ๐ฟ!โˆ’๐œ†SG(๐‘ ๐‘Œโˆ’๐‘ก๐‘‹)8๐œ‹2๐œ–๎‚น๐ฟ๎‚€1+๐’ช๐œ–๐ฟโˆ’1๎‚.(3.19) Since the leading IR divergences of ๐ด(๐ฟ,๐ฟ) is ๐’ช(1/๐œ–๐ฟ), one can show that the following relations hold: ๎‚ธ๐œ†2SG๐‘ 2+๐‘ก2+๐‘ข23๎‚น๐‘˜13๎‚ƒ๎€ท๐œ†SG๐‘ข๎€ธ๐‘€(2๐‘˜+1,2๐‘˜+1)SYM(๐‘ ,๐‘ก;๐œ–)+c.p.๎‚„=๐œ†2๐‘˜+1SYM22๐‘˜+1/2๎ƒฌ๐‘€(2๐‘˜+1)!(2)SG1(๐œ–)+6๎‚ต๐œ†SG8๐œ‹2๎‚ถ2๎‚€๐‘ ๐‘‹+๐‘ก๐‘Œ+๐‘ข๐‘๐œ–๎‚2๎ƒญ๐‘˜๐‘€(1)SG๎‚€1(๐œ–)+๐’ช๐œ–2๐‘˜๎‚,(3.20) for ๐ฟ=2๐‘˜+1๎‚ธ๐œ†2SG๐‘ 2+๐‘ก2+๐‘ข23๎‚น๐‘˜13๎‚ƒ๎€ท๐œ†SG๐‘ข๎€ธ2๐‘€(2๐‘˜+2,2๐‘˜+2)SYM(๐‘ ,๐‘ก;๐œ–)+c.p.๎‚„=๐œ†2๐‘˜+2SYM22๐‘˜+2๎ƒฌ๐‘€(2๐‘˜+2)!(2)SG1(๐œ–)+6๎‚ต๐œ†SG8๐œ‹2๎‚ถ2๎‚€๐‘ ๐‘‹+๐‘ก๐‘Œ+๐‘ข๐‘๐œ–๎‚2๎ƒญ๐‘˜๐‘€(2)SG๎‚€1(๐œ–)+๐’ช๐œ–2๐‘˜+1๎‚(3.21) for ๐ฟ=2๐‘˜+2 (where ๐‘˜=0,1,2,โ€ฆ).

That is, we have an exact relation at ๐ฟ-loops for the leading IR divergence ~๐’ช(1/๐œ–๐ฟ), with an untested relation for the subleading divergence of ๐’ช(1/๐œ–๐ฟโˆ’1); 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 โ€˜t Hooft string picture of the 1/๐‘ expansion. Thus at least in the case of ๐ฟ=1,2, (3.18) and (3.17) still do, so one can hope that there is a correct relation at higher ๐ฟ 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 (๐‘›โˆ’3)! permutation symmetry for the ๐‘›-point functions, and another has (๐‘›โˆ’2)! symmetry, but requires regularization as a consequence of singularities. They are part of a flurry of recent activity relating ๐’ฉ=4 SYM and ๐’ฉ=8 supergravity, including [40, 42, 43, 46, 61, 66โ€“68] (among older works see also [37, 69, 70]). Recent work applying the KLT relations includes [71โ€“74]. In our quest for SYM-supergravity relations, we first review previous KLT relations; we then note that ๐ด5;3 and the 1-loop supergravity amplitude both have 1/๐œ– IR divergences. We present here a tree-level KLT relation for the ๐‘›=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(๐‘›โˆ’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 ๐’ฉ=8 supergravity and subleading-color ๐’ฉ=4 SYM theories, both with the 1/๐œ– IR divergence, we explore the possibility that the 1-loop 5-point supergravity amplitude can be expressed as a linear combination of the ๐ด5;3 SYM amplitudes. In particular a linear relation is proposed among the 1/๐œ– IR divergences of the two theories.

At tree level, the KLT relations are quadratic relations between the ๐‘›-point amplitudes of ๐’ฉ=4 SYM and those of ๐’ฉ=8 supergravity. In these relations, the helicity information is all contained within the amplitudes, and the coefficients are all function of the kinematic invariants ๐‘ ๐‘–๐‘— 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 ๐›ผโ€ฒโ†’0 limit [35, 63] and can be expressed as (we use the notation of [37]) ๐ดtree๐‘›,sugra(12โ€ฆ๐‘›)=(โˆ’1)๐‘›+1๎ƒฌ๐ด๐‘›๎“(12โ€ฆ๐‘›)perms๐‘“๎€ท๐‘–1โ€ฆ๐‘–๐‘—๎€ธ๐‘“๎€ท๐‘™1โ€ฆ๐‘™๐‘—โ€ฒ๎€ธร—๐ด๐‘›๎€ท๐‘–1,โ€ฆ,๐‘–๐‘—,1,๐‘›โˆ’1,๐‘™1,โ€ฆ,๐‘™๐‘—โ€ฒ๎€ธ๎ƒญ๐‘“๎€ท๐‘–,๐‘›+๐’ซ(2,โ€ฆ,๐‘›โˆ’2)1,โ€ฆ,๐‘–๐‘—๎€ธ๎€ท=๐‘ 1,๐‘–๐‘—๎€ธ๐‘—โˆ’1โˆ๐‘š=1๎ƒฉ๐‘ ๎€ท1,๐‘–๐‘š๎€ธ+๐‘—โˆ‘๐‘˜=๐‘š+1๐‘”๎€ท๐‘–๐‘š,๐‘–๐‘˜๎€ธ๎ƒช,๐‘“๎€ท๐‘™1,โ€ฆ,๐‘™๐‘—โ€ฒ๎€ธ๎€ท๐‘™=๐‘ 1๎€ธ,๐‘›โˆ’1๐‘—โ€ฒโˆ๐‘š=2๎ƒฉ๐‘ ๎€ท๐‘™๐‘š๎€ธ+,๐‘›โˆ’1๐‘šโˆ’1โˆ‘๐‘˜=1๐‘”๎€ท๐‘™๐‘˜,๐‘™๐‘š๎€ธ๎ƒช,(3.22) where โ€œpermsโ€™โ€™ are (๐‘–1,โ€ฆ,๐‘–๐‘—)โˆˆ๐’ซ(2,โ€ฆ,๐‘›/2),โ€‰โ€‰(๐‘™1,โ€ฆ,๐‘™๐‘—๎…žโˆˆ๐’ซ(๐‘›/2+1,โ€ฆ,๐‘›โˆ’2),โ€‰โ€‰๐‘—=๐‘›/2โˆ’1,๐‘—โ€ฒ=๐‘›/2โˆ’2, and ๐‘”๐‘–,๐‘—=๐‘ ๐‘–๐‘— if ๐‘–>๐‘— and zero otherwise.

In [44, 45], new forms of the KLT relations for any ๐‘›-point function were found. They are both written in terms of the functions:๐’ฎ๎€บ๐‘–1โ€ฆ๐‘–๐‘˜โˆฃ๐‘—1โ€ฆ๐‘—๐‘˜๎€ป=๐‘˜๎‘๐‘ก=1๎ƒฉ๐‘ ๐‘–๐‘ก1+๐‘˜๎“๐‘ž>๐‘ก๐œƒ๎€ท๐‘–๐‘ก,๐‘–๐‘ž๎€ธ๐‘ ๐‘–๐‘ก๐‘–๐‘ž๎ƒช,๎‚๐’ฎ๎€บ๐‘–1โ€ฆ๐‘–๐‘˜โˆฃ๐‘—1โ€ฆ๐‘—๐‘˜๎€ป=๐‘˜๎‘๐‘ก=1๎ƒฉ๐‘ ๐‘—๐‘ก๐‘›+๐‘˜๎“๐‘ž<๐‘ก๐œƒ๎€ท๐‘—๐‘ž,๐‘—๐‘ก๎€ธ๐‘ ๐‘—๐‘ž๐‘—๐‘ก๎ƒช,(3.23) where ๐œƒ(๐‘–๐‘ก,๐‘–๐‘ž) is zero in (๐‘–๐‘ก,๐‘–๐‘ž) has the same order in both sets โ„={๐‘–1,โ€ฆ,๐‘–๐‘˜} and ๐’ฅ={๐‘—1,โ€ฆ,๐‘—๐‘˜} and is 1 otherwise, and similarly for ๐œƒ(๐‘—๐‘ž,๐‘—๐‘ก).

A form of KLT relations was found in [44], but needs to be regularized, due to a singular denominator ๐ดtree๐‘›,sugra=(โˆ’1)๐‘›๎“๐›พ,๐›ฝ๎‚๐ด๐‘›๎€ท๐‘›,๐›พ2,๐‘›โˆ’1๎€ธ๐’ฎ๎€บ๐›พ,12,๐‘›โˆ’1,๐›ฝ2,๐‘›โˆ’1๎€ป๐‘1๐ด๐‘›๎€ท1,๐›ฝ2,๐‘›โˆ’1๎€ธ,๐‘›๐‘ 12โ€ฆ๐‘›โˆ’1,๐ดtree๐‘›,sugra=(โˆ’1)๐‘›๎“๐›ฝ,๐›พ๐ด๐‘›๎€ท๐‘›,๐›ฝ2,๐‘›โˆ’1๎€ธ๎‚๐’ฎ๎€บ๐›ฝ,12,๐‘›โˆ’1,๐›พ2,๐‘›โˆ’1๎€ป๐‘๐‘›๎‚๐ด๐‘›๎€ท1,๐›พ2,๐‘›โˆ’1๎€ธ,๐‘›๐‘ 23โ€ฆ๐‘›.(3.24) However they have a large (๐‘›โˆ’2)! manifest symmetry. Another set was proven in [45] which is nonsingular ๐ดtree๐‘›,sugra=(โˆ’1)๐‘›+1๎“๐œŽโˆˆ๐‘†๐‘›โˆ’3๎“๐›ผโˆˆ๐‘†๐‘—โˆ’1๎“๐›ฝโˆˆ๐‘†๐‘›โˆ’๐‘—โˆ’2๐ด๐‘›๎€ท1,๐œŽ2,๐‘—,๐œŽ๐‘—+1,๐‘›โˆ’2๎€ธ๐’ฎ๎€บ๐›ผ,๐‘›โˆ’1,๐‘›๐œŽ(2),๐œŽ(๐‘—)โˆฃ๐œŽ2,๐‘—๎€ป๐‘1ร—๎‚๐’ฎ๎€บ๐œŽ๐‘—+1,๐‘›โˆ’2โˆฃ๐›ฝ๐œŽ(๐‘—+1),๐œŽ(๐‘›โˆ’2)๎€ป,๐‘›๐‘๐‘›๎‚๐ด๐‘›๎€ท๐›ผ๐œŽ(2),๐œŽ(๐‘—),1,๐‘›โˆ’1,๐›ฝ๐œŽ(๐‘—+1),๐œŽ(๐‘›โˆ’1)๎€ธ,๐‘›(3.25) but with only (๐‘›โˆ’3)! manifest symmetry.

The original KLT relation for the 5-point function is๐ดtree5,sugra=๐‘ 12๐‘ 34๎‚๐ด(12345)๐ด(21435)+๐‘ 13๐‘ 24๎‚๐ด(13245)๐ด(31425)(3.26) and has (๐‘›โˆ’3)!=2! symmetry, whereas the KLT relations (3.25) become, explicitly, ๐ดtree5,sugra=๎“๐œŽ,๎‚๐œŽโˆˆ๐‘†2๎‚๐ด๎€ท45,๎‚๐œŽ23๎€ธ๐ด๎€ท,11,๐œŽ23๎€ธ๐‘†๎€บ,45๎‚๐œŽ2,3โˆฃ๐œŽ2,3๎€ป๐‘1=๐‘ 12๐‘ 13(๐ด(45231)๐ด(12345)+๐ด(45321)๐ด(13245))+๐‘ 13๎€ท๐‘ 12+๐‘ 23๎€ธ๐ด(45231)๐ด(13245)+๐‘ 12๎€ท๐‘ 13+๐‘ 23๎€ธ๐ด๐ด(45321)๐ด(12345),tree5,sugra=๎“๐œŽ,๎‚๐œŽโˆˆ๐‘†2๎‚๐ด๎€ท14,๎‚๐œŽ23๎€ธ๐ด๎€ท,51,๐œŽ23๎€ธ๎‚๐‘†๎€บ๐œŽ,452,3โˆฃ๎‚๐œŽ2,3๎€ป๐‘4=๐‘ 24๐‘ 34[]๐ด(12345)๐ด(14235)+๐ด(132