# Applications of Subleading-Color Amplitudes in𝒩=4 SYM Theory

Authors: Stephen G. Naculich; Horatiu Nastase; Howard J. Schnitzer
Journal: Advances in High Energy Physics
 (2011)

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## Abstract

A number of features and applications of subleading-color amplitudes of𝒩=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 NkMHV amplitudes of 𝒩=4 SYM theory.


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## Body

## 1. Introduction

Planar amplitudes of𝒩=4 SYM theory have been extensively studied by a variety of methods, see, for example, [1–23]. For a recent overview, see [24] and the special issue of Journal of Physics A, devoted to “Scattering amplitudes in gauge theories." Subleading-color (i.e., nonplanar) amplitudes, however, usually receive less attention [25–33]. 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, 34–46]. Since nonplanar graphs appear on an equal footing with planar graphs in 𝒩=8 supergravity, one needs to understand the nonplanar graphs in 𝒩=4 SYM if the weak/weak duality is to be explored.This paper will cover three significant topics. Section2 discusses the IR divergences of various subleading-color amplitudes. In Section 3 the interplay between subleading-color amplitudes of 𝒩=4 SYM theory and amplitudes of 𝒩=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 the1/N expansion.At tree level, we can decompose the amplitudes𝒜n of 𝒩=4 SYM into color-ordered tree amplitudes An(1.1)Antree(12…n)=gn-2∑σ∈Sn/ZnTr⁡(Taσ(1)…Taσ(n))Antree(σ(1)…σ(n))=gn-2∑P(23…n)Tr⁡(Ta1TaP(2)…TaP(n))Antree(1P(2)…P(n)),
where in the second line, 1 is fixed and P(23…n) is a permutation of 2,3,…,n and Ta are SU(N) generators in the fundamental representation, normalized according to Tr(TaTb)=δab. The color-ordered amplitudes An depend on the momenta and polarizations of the external particles.The color-ordered amplitudes are not independent. Forn-point amplitudes, there is a basis of (n-2)! amplitudes out of the total n!, 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] (1.2)An(1,{α},n,{β})=(-1)nβ∑{σ}i∈OP({α},{βT})An(1,{σ}i,n),
where σi are ordered permutations, that is, ones that keep the order of {α} and of {βT} inside σi. Thus the KK basis is An(1,𝒫(2,…,n-1),n), where 𝒫 are arbitrary permutations. All the other An’s can be recovered from it by the use of the KK relations and cyclicity and reflection invariance (1.3)An(12…n)=(-1)nAn(n…21).At one loop, we can write a similar expansion in color-ordered amplitudes(1.4)An1-loop(12…n)=gn∑j=1[n/2]+1∑σ∈Sn/Sn;jGrn;j(σ)An;j(σ(1)…σ(n)),Grn;1(1)=NTr⁡(Ta1…Tan),Grn;j(1)=Tr⁡(Ta1…Taj-1)Tr⁡(Taj…Tan).
However, the subleading piece in the 1/N expansion can be obtained from the leading piece by (1.5)An;j(12…,j-1,j,j+1,…n)=(-1)j-1∑σ∈COP{α},{β}An;1(σ),
where COP are cyclically ordered permutations, again keeping the order of {α} and {β} fixed up to cyclic permutations.At arbitrary loops, the decomposition of the four-gluon amplitude takes a form with only single and double traces(1.6)A4(1234)=g2∑σ∈S4/Z4Tr⁡(Taσ(1)Taσ(2)Taσ(3)Taσ(4))NA4;1(σ(1)σ(2)σ(3)σ(4))+g2∑σ∈S4/Z23Tr⁡(Taσ(1)Taσ(2))Tr⁡(Taσ(3)Taσ(4))A4;3(σ(1)σ(2)σ(3)σ(4)).
We also define an explicit basis [48] of single and double traces: (1.7)C[1]=Tr⁡(Ta1Ta2Ta3Ta4),C[4]=Tr⁡(Ta1Ta3Ta2Ta4),C[7]=Tr⁡(Ta1Ta2)Tr⁡(Ta3Ta4),C[2]=Tr⁡(Ta1Ta2Ta4Ta3),C[5]=Tr⁡(Ta1Ta3Ta4Ta2),C[8]=Tr⁡(Ta1Ta3)Tr⁡(Ta2Ta4),C[3]=Tr⁡(Ta1Ta4Ta2Ta3),C[6]=Tr⁡(Ta1Ta4Ta3Ta2),C[9]=Tr⁡(Ta1Ta4)Tr⁡(Ta2Ta3),
in terms of which the four-gluon amplitude can be expanded as
(1.8)A4(1234)=g2∑i=19A[i]C[i].The loop expansion of color-ordered amplitudes(1.9)A[i]=∑L=0∞aLA[i](L),NA4;1=∑L=0∞aLA4;1(L),A4;3=∑L=0∞aLA4;3(L)
is in terms of the natural ‘t Hooft loop expansion parameter [7] (1.10)a≡g2N8π2(4πe-γ)ϵ,
where γ is Euler’s constant and ϵ=(4-D)/2. Note that at L loops, the amplitude is at most of order NL, which means that A[i](L) starts at 𝒪(N0).For a generaln-point amplitude, we will have an expansion in an arbitrary number of multitrace color-ordered amplitudes An;j1,j2,…,jk.Besides the loop expansion in the ‘t Hooft parametera, 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 𝒜4, the expansion in single-trace A4;1 and double-trace A4;3 amplitudes corresponds to the topology of the outside lines, forming boundaries of the diagrams. For example, at one-loop, the contribution in A4;1 to the amplitude is leading, that is, of order N (thus A4;1 of order 1), coming from a diagram with the topology of 4 external lines and a boundary, whereas the contribution of A4;3 is subleading, that is, of order N0, 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 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/N, corresponding to the number of boundaries of the diagram.On top of that, we also have an expansion in integer powers of1/N2, independently for A4;1 and A4;3, corresponding to nonplanar diagrams with handles (a handle gives a factor of 1/N2). The expansion terminates at order 𝒪(N0) for the amplitude, since in the amplitude the powers of N can only be positive. Thus at L-loops, we have A4;1(L)=𝒪(1) to 𝒪(1/NL) and A4;3(L)=𝒪(1/N) to 𝒪(1/NL). Taken together, we will say that the gluon amplitudes have a 1/N expansion.


## 2. IR Divergences for Subleading𝒩=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=4-2ϵ dimensions, in which they appear as poles in a Laurent expansion in ϵ.In gluon-gluon scattering in𝒩=4 SYM, IR divergences arise both from soft gluons and from collinear gluons, each of which gives rise to an 𝒪(1/ϵ) pole at one loop, leading to an 𝒪(1/ϵ2) pole at that order. At L loops, the leading IR divergence of the scattering amplitude is therefore 𝒪(1/ϵ2L), arising from multiple soft gluon exchanges.Subleading-color amplitudesA(L,k), that is, those suppressed by 1/Nk relative to the leading-color amplitude at L loops, have less severe IR divergences, being only of 𝒪(1/ϵ2L-k) at L-loops.In this section, we review the derivation of a compact all-loop-order expression for the IR-divergent part of the𝒩=4 SYM four-gluon amplitude given in [41, 49]. This result is expressed in terms of the soft (cusp) anomalous dimension γ(a), the collinear anomalous dimension 𝒢0(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 [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 amplitudesA[i] defined in (1.8) into a vector in color space [25, 26]
(2.1)|A〉=(A[1],A[2],A[3],A[4],A[5],A[6],A[7],A[8],A[9])T,
where (…)T denotes the transposed vector. The vector of color-ordered amplitudes factorizes into [27, 29]
(2.2)|A(sijμ2,a,ϵ)〉=J(Q2μ2,a,ϵ)S(sijQ2,Q2μ2,a,ϵ)|H(sijQ2,Q2μ2,a,ϵ)〉,
where |H〉, which is IR-finite as ϵ→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 sij is the kinematic invariant (ki+kj)2, μ is a renormalization scale, and Q is an arbitrary factorization scale which serves to separate the long- and short-distance behavior.Because𝒩=4 SYM theory is conformally invariant, the product of jet functions J may be explicitly evaluated as [7]
(2.3)J(Q2μ2,a,ϵ)=exp⁡[-12∑l=1∞al(μ2Q2)lϵ(γ(l)(lϵ)2+2G0(l)lϵ)],
where γ(ℓ) and 𝒢0(ℓ) 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]: (2.4)γ(a)=∑l=1∞alγ(l)=4a-4ζ2a2+22ζ4a3+⋯,G0(a)=∑l=1∞alG0(l)=-ζ3a2+(4ζ5+103ζ2ζ3)a3+⋯.
The soft function S is given by [27, 29] (2.5)S(sijQ2,Q2μ2,a,ϵ)=Pexp⁡⁡[-12∫0Q2dμ̃2μ̃2Γ(sijQ2,a¯(μ2μ̃2,a,ϵ))],
where (2.6)Γ(sijQ2,a)=∑l=1∞alΓ(l),a¯(μ2μ̃2,a,ϵ)=(μ2μ̃2)ϵa,
suppressing the explicit dependence of Γ(ℓ) on sij/Q2 to lighten the notation.At this point, we make the assumption that the soft anomalous dimension matricesΓ(ℓ) 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 Γ(2)=(1/4)γ(2)Γ(1), as shown in [28, 29]. In [32], it was established that Γ(3)=(1/4)γ(3)Γ(1) for the nonpure gluon contributions. Further, Γ(L)=(1/4)γ(L)Γ(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 Casimir’s terms [31, 32, 51, 52].) Therefore, the path ordering in (2.5) becomes irrelevant, allowing us to explicitly integrate it, obtaining(2.7)S(sijQ2,Q2μ2,a,ϵ)=exp⁡⁡[12∑l=1∞al(μ2Q2)lϵΓ(l)lϵ].
Combining the exponents of the jet and soft functions into [27, 41] (2.8)G(l)(ϵ)=Nl2(μ2Q2)ϵ[-(γ(l)ϵ2+2G0(l)ϵ)1+1ϵΓ(l)],
we may express the four-gluon amplitude in the compact form(2.9)|A(ϵ)〉=exp⁡⁡[∑l=1∞alNlG(l)(lϵ)]|H(ϵ)〉,
or equivalently (2.10)|H(ϵ)〉=∑L=0∞aL|H(Lf)(ϵ)〉=(1-∑l=1∞alNlF(l)(ϵ))|A(ϵ)〉.
where the matrices F(ℓ)(ϵ) will be defined below. (Henceforth we suppress sij, Q, μ, and a in the arguments of the amplitudes.) Expanding (2.10) through three loops, we obtain the expressions given in [27, 41] (2.11)|A(1)(ϵ)〉=1NF(1)(ϵ)|A(0)〉+|H(1f)(ϵ)〉,|A(2)(ϵ)〉=1N2F(2)(ϵ)|A(0)〉+1NF(1)(ϵ)|A(1)(ϵ)〉+|H(2f)(ϵ)〉,|A(3)(ϵ)〉=1N3F(3)(ϵ)|A(0)〉+1N2F(2)(ϵ)|A(1)〉+1NF(1)(ϵ)|A(2)(ϵ)〉+|H(3f)(ϵ)〉,
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(ϵ), we will need to know positive powers of ϵ in the expansion of lower loop amplitudes to obtain all the IR-divergent contributions to the L-loop amplitude A(ℓ).)The equivalence of (2.9) and (2.10) follows if the matrices F(ℓ)(ϵ) are defined through the equation (2.12)(1-∑l=1∞alNlF(l)(ϵ))exp⁡⁡[∑l=1∞alNlG(l)(lϵ)]=1.
First define the functional X[M] via [7] (2.13)1+∑l=1∞alM(l)≡exp⁡⁡[∑l=1∞al(M(l)-X(l)[M])]
so that X(1)[M]=0,X(2)[M]=(1/2)[M(1)]2,X(3)[M]=-(1/3)[M(1)]3+M(1)M(2), and so forth. 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 (2.12). Thus (2.14)(1-∑l=1∞alNlF(l)(ϵ))=exp⁡[∑L=0∞alNl(-F(l)(ϵ)-X(l)[-F])],
and so (2.12) is equivalent to (2.15)F(l)(ϵ)=-X(l)[-F]+G(l)(lϵ)
which defines F(ℓ) recursively in terms of G(ℓ) and F(ℓ′) with ℓ′<ℓ. The explicit expressions up through three loops
(2.16)F(1)(ϵ)=G(1)(ϵ),F(2)(ϵ)=-12[F(1)(ϵ)]2+G(2)(2ϵ),F(3)(ϵ)=-13[F(1)(ϵ)]3-F(1)(ϵ)F(2)(ϵ)+G(3)(3ϵ).
agree (up to rescaling by a factor of NL) with the expressions given in [27] when specialized to the case of gg→gg in 𝒩=4 SYM theory.


### 2.2.1/N 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 of1/N.TheL-loop color-ordered amplitudes may be written in a 1/N expansion as (2.17)|A(L)(ϵ)〉=∑k=0L1Nk|A(L,k)(ϵ)〉,
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 (2.9): (2.18)|A(ϵ)〉=∑L=0∞∑k=0LaLNk|A(L,k)(ϵ)〉=∏l=1∞∑{nl}1nl!(alG(l)(lϵ)Nl)nl∑l0=0∞∑k0=0l0al0Nk0|H(l0,k0)(ϵ)〉.
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 (2.19)Γ(l)=γ(l)4Γ(1)(assumption)
as has been conjectured (see footnote 1). This allows us to rewrite (2.8) as (2.20)G(l)(lϵ)Nl=12(μ2Q2)lϵ[-(γ(l)(lϵ)2+2G0(l)lϵ)1+γ(l)4lϵΓ(1)].
The one-loop soft anomalous dimension matrix can be written [29]: (2.21)Γ(1)=-1N∑i=14∑j≠i4Ti⋅Tjlog⁡⁡(-sijQ2),
where Ti·Tj=TiaTja with Tia the SU(N) generators in the adjoint representation. On the basis of (1.7), it has the explicit form [48], (2.22)Γ(1)=2(α00δ)+2N(0βγ0),
where (2.23)α=(S+T000000S+U000000T+U000000T+U000000S+U000000S+T),β=(T-U0S-UU-TS-T00T-SU-S0T-SU-SU-TS-T0T-U0S-U),γ=(S-US-T00S-TS-U0U-TU-SU-SU-T0T-U0T-ST-S0T-U),δ=(2S0002U0002T)
with (2.24)S=log⁡⁡(-sQ2),T=log⁡⁡(-tQ2),U=log⁡⁡(-uQ2).
If the assumption (2.19) is valid, then the 1/N expansion of G(ℓ)(ℓϵ)/Nℓ terminates after two terms (2.25)G(l)(lϵ)Nl=gl+1Nfl,
where gℓ and fℓ can be read from (2.20) and (2.22). We rewrite (2.18) as (2.26)|A(ϵ)〉=∑L=0∞∑k=0LaLNk|A(L,k)(ϵ)〉=∏l=1∞∑{nl}1nl!(algl+alNfl)nl∑l0=0∞∑k0=0l0al0Nk0|H(l0,k0)(ϵ)〉
making all N dependence explicit.We now determine the power of the leading IR pole of|A(L,k)(ϵ)〉. Consider an individual term on the right-hand side of (2.26). By counting powers of a and 1/N, one sees that this term contributes to |A(L,k)(ϵ)〉, with (2.27)L=l0+∑l=1∞lnl,k=k0+k1,
where k1 is the number of factors fℓ present in the term. From (2.20) and (2.22), it is apparent that gℓ has a double pole in ϵ, but fℓ only has a single pole. The leading IR pole in the term under consideration is therefore 1/ϵp, where (2.28)p=2∑l=1∞nl-k1.
Combining (2.27) and (2.28), we find (2.29)p=2L-k-[2∑l=1∞(l-1)nl+2l0-k0].
Since k0≤ℓ0, the term in square brackets is nonnegative, and we conclude that (2.30)|A(L,k)(ϵ)〉~O(1ϵ2L-k).
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 |A(L,k)(ϵ)〉. Terms in (2.26) contribute to the leading IR pole only when the expression in square brackets in (2.29) vanishes, which occurs when nℓ=0 for ℓ≥2, and ℓ0=k0=0 (with n1 unconstrained). In other words, the leading IR divergences are given by [41, 49] (2.31)|A(ϵ)〉~exp⁡⁡[aG(1)(ϵ)N]|A(0)〉(leading IR divergence).Recalling that(2.32)G(1)(ϵ)N=(μ2Q2)ϵ[-2ϵ21+1ϵ(α00δ)+1Nϵ(0βγ0)],
we use (2.31) to obtain the coefficient of the leading IR pole(2.33)|A(L,k)(ϵ)〉=(-2)L-kk!(L-k)!1ϵ2L-k(0βγ0)k|A(0)〉+O(1ϵ2L-k-1),
where the tree-level amplitudes are
(2.34)|H(0,0)〉=|A(0)〉=-4iKstu(u,t,s,s,t,u,0,0,0)T,
where s=(k1+k2)2,t=(k1+k4)2 and u=(k1+k3)2 are the usual Mandelstam variables, obeying s+t+u=0 for massless external gluons. The factor K, 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(2.35)|A(L,0)(ϵ)〉=(-2)LL!ϵ2L|A(0)〉+O(1ϵ2L-1).
The remaining IR divergences, from 𝒪(1/ϵ2L-1) to 𝒪(1/ϵ), are all proportional to |A(0)〉 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 ofsubleading-color amplitudes. First we use (2.34) and (2.23) to show (2.36)γ(utsstu)=2(sY-tX)(111),γβ(111)=2(X2+Y2+Z2)(111)
with (2.37)X=log⁡⁡(tu),Y=log⁡⁡(us),Z=log⁡⁡(st).
Hence, the leading IR divergence of the subleading-color amplitudes is given by (2.38)|A(L,2m+1)(ϵ)〉=(-4iKstu)(-1)L-12L-m(X2+Y2+Z2)m(sY-tX)(2m+1)!(L-2m-1)!ϵ2L-2m-1(000000111)+O(1ϵ2L-2m-2),(2.39)|A(L,2m+2)(ϵ)〉=(-4iKstu)(-1)L2L-m-1(X2+Y2+Z2)m(sY-tX)(2m+2)!(L-2m-2)!ϵ2L-2m-2(X-YZ-XY-ZY-ZZ-XX-Y000)+O(1ϵ2L-2m-3).
The results (2.38) and (2.39) were derived in [49], generalizing expressions derived in [41].


### 2.3. IR Divergences ofA(L,1)

In this subsection, we consider the subleading-color amplitude|A(L,1)〉 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 ≤2: (2.40)|A(L)(ϵ)〉=1L!(g1+1Nf1)L|A(0)〉+1N(L-1)!(g1+1Nf1)L-1|H(1,1)(ϵ)〉+1(L-2)!(g1+1Nf1)L-2(g2+1Nf2)|A(0)〉+1(L-1)!(g1+1Nf1)L-1|H(1,0)(ϵ)〉+1N2(L-2)!(g1+1Nf1)L-2|H(2,2)(ϵ)〉+⋯(three leading IR poles),
where we use (2.20) and (2.22) to write
(2.41)g1=(μ2Q2)ϵ[-2ϵ21+1ϵ(α00δ)],f1=1ϵ(μ2Q2)ϵ(0βγ0),g2=(μ2Q2)2ϵ[-(γ(2)8ϵ2+G0(2)2ϵ)1+γ(2)8ϵ(α00δ)],f2=γ(2)8ϵ(μ2Q2)2ϵ(0βγ0).
To extract the |A(L,1)〉 amplitude, we employ the identity(2.42)(g1+1Nf1)L|1/Npiece=Lg1L-1f1+(L2)g1L-2[f1,g1]+(L3)g1L-3[[f1,g1],g1]+[⋯[[[f1,g1],g1],g1]⋯],
in which the first term on the right-hand side has an expansion that starts with 1/ϵ2L-1, the second term has an expansion that starts with 1/ϵ2L-2, and so forth. Thus, keeping only the terms proportional to 1/N in (2.40), and only the first three terms in the Laurent expansion, we obtain(2.43)|A(L,1)〉=1(L-1)!g1L-1f1|A(0)〉+12(L-2)!g1L-2[f1,g1]|A(0)〉+1(L-1)!g1L-1|H(1,1)(ϵ)〉+16(L-3)!g1L-3[[f1,g1],g1]|A(0)〉+1(L-2)!g1L-2f2|A(0)〉+1(L-3)!g1L-3f1g2|A(0)〉+1(L-2)!g1L-2f1|H(1,0)(ϵ)〉+O(1ϵ2L-4),as obtained in [49].


### 2.4. IR Divergences ofA(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 (2.26) that contribute to |A(L,L)〉 are those with as many factors of 1/N as of a. Thus, only f1 and |H(ℓ0,ℓ0)〉 can contribute, giving(2.44)|A(L,L)(ϵ)〉=∑l0=0L1(L-l0)!f1L-l0|H(l0,l0)(ϵ)〉,wheref1=1ϵ(μ2Q2)ϵ(0βγ0)
exact to all orders in the ϵ expansion. Keeping just the first two terms in the Laurent expansion, we find (2.45)|A(L,L)(ϵ)〉=1(L-1)!f1L-1[1Lf1|A(0)〉+|H(1,1)(ϵ)〉]+O(1ϵL-2)=1(L-1)!1ϵL-1(0βγ0)L-1|A(1,1)(Lϵ)〉+O(1ϵL-2).
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

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 et al. [54] (see also [36]). Explicit expressions for these IR-divergent integrals as the Laurent expansions in ϵ 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/N expansion of these divergences.Recall from (2.17) that A[i](L,k) denotes the L-loop color-ordered amplitude which is subleading by a factor of 1/Nk in the 1/N expansion. Single-trace amplitudes are denoted by i=1,…,6 and double-trace amplitudes by i=7,…,9 (see (1.7)).At one loop, the single-trace amplitudes are given by [34] (2.46)A[1](1,0)=M(1)(s,t)A[1](0)=2iKI4(1)(s,t)
with the other single-trace amplitudes A[2](1,0) and A[3](1,0) obtained by letting t↔u, and s↔u, respectively. The identities A[1](L)=A[6](L), A[2](L)=A[5](L), and A[3](L)=A[4](L) are satisfied at all loop orders. In (2.46), I4(1)(s,t) denotes the scalar box integral (2.47)M(1)(s,t)=-12stI4(1)(s,t),I4(1)(s,t)=I4(1)(t,s)=-iμ2ϵeϵγπ-D/2∫dDpp2(p-k1)2(p-k1-k2)2(p+k4)2,
an explicit expression for which is given, for example, in [7].The one-loop double-trace amplitudes are given by [34] (2.48)A[7](1,1)=A[8](1,1)=A[9](1,1)=2(A[1](1,0)+A[2](1,0)+A[3](1,0))(2.49)=4iK[I4(1)(s,t)+I4(1)(t,u)+I4(1)(u,s)].
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] (2.50)A[1](2,0)=M(2)(s,t)A[1](0)=-iK[sI4(2)P(s,t)+tI4(2)P(t,s)],
where I4(2)P(s,t) denotes the scalar double-box (planar) integral (2.51)M(2)(s,t)=14st[sI4(2)P(s,t)+tI4(2)P(t,s)],I4(2)P(s,t)=(-iμ2ϵeϵγπ-D/2)2∫dDpdDqp2(p+q)2q2(p-k1)2(p-k1-k2)2(q-k4)2(q-k3-k4)2,
an explicit expression for which is given, for example, in [7]. The double-trace amplitude is [54]
(2.52)A[7](2,1)=-2iK[s(3I4(2)P(s,t)+2I4(2)NP(s,t)+3I4(2)P(s,u)+2I4(2)NP(s,u))-t(I4(2)NP(t,s)+I4(2)NP(t,u))-u(I4(2)NP(u,s)+I4(2)NP(u,t))],
and the subleading-color single-trace amplitude is [54] (2.53)A[1](2,2)=-2iK[s(I4(2)P(s,t)+I4(2)NP(s,t)+I4(2)P(s,u)+I4(2)NP(s,u))+t(I4(2)P(t,s)+I4(2)NP(t,s)+I4(2)P(t,u)+I4(2)NP(t,u))-2u(I4(2)P(u,s)+I4(2)NP(u,s)+I4(2)P(u,t)+I4(2)NP(u,t))],
where I4(2)NP(s,t) denotes the two-loop nonplanar integral
(2.54)I4(2)NP(s,t)=(-iμ2ϵeϵγπ-D/2)2∫dDpdDqp2(p+q)2q2(p-k2)2(p+q+k1)2(q-k3)2(q-k3-k4)2,
an explicit expression for which is given in [56]. All the other single- and double-trace amplitudes A[i](2,k) 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 nonplanar integral I4(2)NP(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]: (2.55)A[7](2,1)=2(A[1](2,0)+A[2](2,0)+A[3](2,0))-A[3](2,2),A[8](2,1)=2(A[1](2,0)+A[2](2,0)+A[3](2,0))-A[1](2,2),A[9](2,1)=2(A[1](2,0)+A[2](2,0)+A[3](2,0))-A[2](2,2)
and may be easily verified using (2.50), (2.52), and (2.53). In addition, we have
(2.56)A[1](2,2)+A[2](2,2)+A[3](2,2)=0,
also easily verified using (2.53). Together these equations imply(2.57)6∑i=13A[i](2,0)-∑i=79A[i](2,1)=0
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 (2.58)6∑i=13A[i](L)-N∑i=79A[i](L)=0,L≤2,
which is valid through two loops.At one loop, we also saw that one can relate all the subleading-color amplitudesAn;j to the leading amplitude An;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
(2.59)M(1)(s,t)=-1ϵ2(μ2-s)ϵ-1ϵ2(μ2-t)ϵ+12log⁡2(st)+2π23+O(ϵ),
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(2.60)|A(1,1)(ϵ)〉=(-8iKstu)[(μ2-u)ϵ(sY-tX)ϵ-(s+t)XY](111)+O(ϵ),
where we have only included the [7–9] components of A[i](1,1) as the others vanish.At two loops, the planar amplitude is given by (2.50) with [60]
(2.61)M(2)(ϵ)=12[M(1)(ϵ)]2-(ζ2+ϵζ3+ϵ2ζ4)M(1)(2ϵ)-π472+O(ϵ).
The two-loop double trace amplitude has an IR divergence given by the general formula (2.38), which yields (2.62)|A(2,1)(ϵ)〉=(-8iKstu)(-2)(sY-tX)ϵ3(111)+O(1ϵ2).
Finally, the subleading-color single-trace amplitude is given by (2.45) which in this case yields (2.63)|A(2,2)(ϵ)〉=1ϵ(X-YZ-XY-ZY-ZZ-XX-Y)A[7](1,1)(2ϵ)+O(ϵ0).
Only the [1] through [6] components are listed, as the [7] through [9] components vanish.


## 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=4-2ϵ dimensions, in which they appear as poles in a Laurent expansion in ϵ.In gluon-gluon scattering in𝒩=4 SYM, IR divergences arise both from soft gluons and from collinear gluons, each of which gives rise to an 𝒪(1/ϵ) pole at one loop, leading to an 𝒪(1/ϵ2) pole at that order. At L loops, the leading IR divergence of the scattering amplitude is therefore 𝒪(1/ϵ2L), arising from multiple soft gluon exchanges.Subleading-color amplitudesA(L,k), that is, those suppressed by 1/Nk relative to the leading-color amplitude at L loops, have less severe IR divergences, being only of 𝒪(1/ϵ2L-k) at L-loops.In this section, we review the derivation of a compact all-loop-order expression for the IR-divergent part of the𝒩=4 SYM four-gluon amplitude given in [41, 49]. This result is expressed in terms of the soft (cusp) anomalous dimension γ(a), the collinear anomalous dimension 𝒢0(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 [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 amplitudesA[i] defined in (1.8) into a vector in color space [25, 26]
(2.1)|A〉=(A[1],A[2],A[3],A[4],A[5],A[6],A[7],A[8],A[9])T,
where (…)T denotes the transposed vector. The vector of color-ordered amplitudes factorizes into [27, 29]
(2.2)|A(sijμ2,a,ϵ)〉=J(Q2μ2,a,ϵ)S(sijQ2,Q2μ2,a,ϵ)|H(sijQ2,Q2μ2,a,ϵ)〉,
where |H〉, which is IR-finite as ϵ→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 sij is the kinematic invariant (ki+kj)2, μ is a renormalization scale, and Q is an arbitrary factorization scale which serves to separate the long- and short-distance behavior.Because𝒩=4 SYM theory is conformally invariant, the product of jet functions J may be explicitly evaluated as [7]
(2.3)J(Q2μ2,a,ϵ)=exp⁡[-12∑l=1∞al(μ2Q2)lϵ(γ(l)(lϵ)2+2G0(l)lϵ)],
where γ(ℓ) and 𝒢0(ℓ) 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]: (2.4)γ(a)=∑l=1∞alγ(l)=4a-4ζ2a2+22ζ4a3+⋯,G0(a)=∑l=1∞alG0(l)=-ζ3a2+(4ζ5+103ζ2ζ3)a3+⋯.
The soft function S is given by [27, 29] (2.5)S(sijQ2,Q2μ2,a,ϵ)=Pexp⁡⁡[-12∫0Q2dμ̃2μ̃2Γ(sijQ2,a¯(μ2μ̃2,a,ϵ))],
where (2.6)Γ(sijQ2,a)=∑l=1∞alΓ(l),a¯(μ2μ̃2,a,ϵ)=(μ2μ̃2)ϵa,
suppressing the explicit dependence of Γ(ℓ) on sij/Q2 to lighten the notation.At this point, we make the assumption that the soft anomalous dimension matricesΓ(ℓ) 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 Γ(2)=(1/4)γ(2)Γ(1), as shown in [28, 29]. In [32], it was established that Γ(3)=(1/4)γ(3)Γ(1) for the nonpure gluon contributions. Further, Γ(L)=(1/4)γ(L)Γ(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 Casimir’s terms [31, 32, 51, 52].) Therefore, the path ordering in (2.5) becomes irrelevant, allowing us to explicitly integrate it, obtaining(2.7)S(sijQ2,Q2μ2,a,ϵ)=exp⁡⁡[12∑l=1∞al(μ2Q2)lϵΓ(l)lϵ].
Combining the exponents of the jet and soft functions into [27, 41] (2.8)G(l)(ϵ)=Nl2(μ2Q2)ϵ[-(γ(l)ϵ2+2G0(l)ϵ)1+1ϵΓ(l)],
we may express the four-gluon amplitude in the compact form(2.9)|A(ϵ)〉=exp⁡⁡[∑l=1∞alNlG(l)(lϵ)]|H(ϵ)〉,
or equivalently (2.10)|H(ϵ)〉=∑L=0∞aL|H(Lf)(ϵ)〉=(1-∑l=1∞alNlF(l)(ϵ))|A(ϵ)〉.
where the matrices F(ℓ)(ϵ) will be defined below. (Henceforth we suppress sij, Q, μ, and a in the arguments of the amplitudes.) Expanding (2.10) through three loops, we obtain the expressions given in [27, 41] (2.11)|A(1)(ϵ)〉=1NF(1)(ϵ)|A(0)〉+|H(1f)(ϵ)〉,|A(2)(ϵ)〉=1N2F(2)(ϵ)|A(0)〉+1NF(1)(ϵ)|A(1)(ϵ)〉+|H(2f)(ϵ)〉,|A(3)(ϵ)〉=1N3F(3)(ϵ)|A(0)〉+1N2F(2)(ϵ)|A(1)〉+1NF(1)(ϵ)|A(2)(ϵ)〉+|H(3f)(ϵ)〉,
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(ϵ), we will need to know positive powers of ϵ in the expansion of lower loop amplitudes to obtain all the IR-divergent contributions to the L-loop amplitude A(ℓ).)The equivalence of (2.9) and (2.10) follows if the matrices F(ℓ)(ϵ) are defined through the equation (2.12)(1-∑l=1∞alNlF(l)(ϵ))exp⁡⁡[∑l=1∞alNlG(l)(lϵ)]=1.
First define the functional X[M] via [7] (2.13)1+∑l=1∞alM(l)≡exp⁡⁡[∑l=1∞al(M(l)-X(l)[M])]
so that X(1)[M]=0,X(2)[M]=(1/2)[M(1)]2,X(3)[M]=-(1/3)[M(1)]3+M(1)M(2), and so forth. 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 (2.12). Thus (2.14)(1-∑l=1∞alNlF(l)(ϵ))=exp⁡[∑L=0∞alNl(-F(l)(ϵ)-X(l)[-F])],
and so (2.12) is equivalent to (2.15)F(l)(ϵ)=-X(l)[-F]+G(l)(lϵ)
which defines F(ℓ) recursively in terms of G(ℓ) and F(ℓ′) with ℓ′<ℓ. The explicit expressions up through three loops
(2.16)F(1)(ϵ)=G(1)(ϵ),F(2)(ϵ)=-12[F(1)(ϵ)]2+G(2)(2ϵ),F(3)(ϵ)=-13[F(1)(ϵ)]3-F(1)(ϵ)F(2)(ϵ)+G(3)(3ϵ).
agree (up to rescaling by a factor of NL) with the expressions given in [27] when specialized to the case of gg→gg in 𝒩=4 SYM theory.


## 2.2.1/N 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 of1/N.TheL-loop color-ordered amplitudes may be written in a 1/N expansion as (2.17)|A(L)(ϵ)〉=∑k=0L1Nk|A(L,k)(ϵ)〉,
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 (2.9): (2.18)|A(ϵ)〉=∑L=0∞∑k=0LaLNk|A(L,k)(ϵ)〉=∏l=1∞∑{nl}1nl!(alG(l)(lϵ)Nl)nl∑l0=0∞∑k0=0l0al0Nk0|H(l0,k0)(ϵ)〉.
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 (2.19)Γ(l)=γ(l)4Γ(1)(assumption)
as has been conjectured (see footnote 1). This allows us to rewrite (2.8) as (2.20)G(l)(lϵ)Nl=12(μ2Q2)lϵ[-(γ(l)(lϵ)2+2G0(l)lϵ)1+γ(l)4lϵΓ(1)].
The one-loop soft anomalous dimension matrix can be written [29]: (2.21)Γ(1)=-1N∑i=14∑j≠i4Ti⋅Tjlog⁡⁡(-sijQ2),
where Ti·Tj=TiaTja with Tia the SU(N) generators in the adjoint representation. On the basis of (1.7), it has the explicit form [48], (2.22)Γ(1)=2(α00δ)+2N(0βγ0),
where (2.23)α=(S+T000000S+U000000T+U000000T+U000000S+U000000S+T),β=(T-U0S-UU-TS-T00T-SU-S0T-SU-SU-TS-T0T-U0S-U),γ=(S-US-T00S-TS-U0U-TU-SU-SU-T0T-U0T-ST-S0T-U),δ=(2S0002U0002T)
with (2.24)S=log⁡⁡(-sQ2),T=log⁡⁡(-tQ2),U=log⁡⁡(-uQ2).
If the assumption (2.19) is valid, then the 1/N expansion of G(ℓ)(ℓϵ)/Nℓ terminates after two terms (2.25)G(l)(lϵ)Nl=gl+1Nfl,
where gℓ and fℓ can be read from (2.20) and (2.22). We rewrite (2.18) as (2.26)|A(ϵ)〉=∑L=0∞∑k=0LaLNk|A(L,k)(ϵ)〉=∏l=1∞∑{nl}1nl!(algl+alNfl)nl∑l0=0∞∑k0=0l0al0Nk0|H(l0,k0)(ϵ)〉
making all N dependence explicit.We now determine the power of the leading IR pole of|A(L,k)(ϵ)〉. Consider an individual term on the right-hand side of (2.26). By counting powers of a and 1/N, one sees that this term contributes to |A(L,k)(ϵ)〉, with (2.27)L=l0+∑l=1∞lnl,k=k0+k1,
where k1 is the number of factors fℓ present in the term. From (2.20) and (2.22), it is apparent that gℓ has a double pole in ϵ, but fℓ only has a single pole. The leading IR pole in the term under consideration is therefore 1/ϵp, where (2.28)p=2∑l=1∞nl-k1.
Combining (2.27) and (2.28), we find (2.29)p=2L-k-[2∑l=1∞(l-1)nl+2l0-k0].
Since k0≤ℓ0, the term in square brackets is nonnegative, and we conclude that (2.30)|A(L,k)(ϵ)〉~O(1ϵ2L-k).
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 |A(L,k)(ϵ)〉. Terms in (2.26) contribute to the leading IR pole only when the expression in square brackets in (2.29) vanishes, which occurs when nℓ=0 for ℓ≥2, and ℓ0=k0=0 (with n1 unconstrained). In other words, the leading IR divergences are given by [41, 49] (2.31)|A(ϵ)〉~exp⁡⁡[aG(1)(ϵ)N]|A(0)〉(leading IR divergence).Recalling that(2.32)G(1)(ϵ)N=(μ2Q2)ϵ[-2ϵ21+1ϵ(α00δ)+1Nϵ(0βγ0)],
we use (2.31) to obtain the coefficient of the leading IR pole(2.33)|A(L,k)(ϵ)〉=(-2)L-kk!(L-k)!1ϵ2L-k(0βγ0)k|A(0)〉+O(1ϵ2L-k-1),
where the tree-level amplitudes are
(2.34)|H(0,0)〉=|A(0)〉=-4iKstu(u,t,s,s,t,u,0,0,0)T,
where s=(k1+k2)2,t=(k1+k4)2 and u=(k1+k3)2 are the usual Mandelstam variables, obeying s+t+u=0 for massless external gluons. The factor K, 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(2.35)|A(L,0)(ϵ)〉=(-2)LL!ϵ2L|A(0)〉+O(1ϵ2L-1).
The remaining IR divergences, from 𝒪(1/ϵ2L-1) to 𝒪(1/ϵ), are all proportional to |A(0)〉 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 ofsubleading-color amplitudes. First we use (2.34) and (2.23) to show (2.36)γ(utsstu)=2(sY-tX)(111),γβ(111)=2(X2+Y2+Z2)(111)
with (2.37)X=log⁡⁡(tu),Y=log⁡⁡(us),Z=log⁡⁡(st).
Hence, the leading IR divergence of the subleading-color amplitudes is given by (2.38)|A(L,2m+1)(ϵ)〉=(-4iKstu)(-1)L-12L-m(X2+Y2+Z2)m(sY-tX)(2m+1)!(L-2m-1)!ϵ2L-2m-1(000000111)+O(1ϵ2L-2m-2),(2.39)|A(L,2m+2)(ϵ)〉=(-4iKstu)(-1)L2L-m-1(X2+Y2+Z2)m(sY-tX)(2m+2)!(L-2m-2)!ϵ2L-2m-2(X-YZ-XY-ZY-ZZ-XX-Y000)+O(1ϵ2L-2m-3).
The results (2.38) and (2.39) were derived in [49], generalizing expressions derived in [41].


## 2.3. IR Divergences ofA(L,1)

In this subsection, we consider the subleading-color amplitude|A(L,1)〉 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 ≤2: (2.40)|A(L)(ϵ)〉=1L!(g1+1Nf1)L|A(0)〉+1N(L-1)!(g1+1Nf1)L-1|H(1,1)(ϵ)〉+1(L-2)!(g1+1Nf1)L-2(g2+1Nf2)|A(0)〉+1(L-1)!(g1+1Nf1)L-1|H(1,0)(ϵ)〉+1N2(L-2)!(g1+1Nf1)L-2|H(2,2)(ϵ)〉+⋯(three leading IR poles),
where we use (2.20) and (2.22) to write
(2.41)g1=(μ2Q2)ϵ[-2ϵ21+1ϵ(α00δ)],f1=1ϵ(μ2Q2)ϵ(0βγ0),g2=(μ2Q2)2ϵ[-(γ(2)8ϵ2+G0(2)2ϵ)1+γ(2)8ϵ(α00δ)],f2=γ(2)8ϵ(μ2Q2)2ϵ(0βγ0).
To extract the |A(L,1)〉 amplitude, we employ the identity(2.42)(g1+1Nf1)L|1/Npiece=Lg1L-1f1+(L2)g1L-2[f1,g1]+(L3)g1L-3[[f1,g1],g1]+[⋯[[[f1,g1],g1],g1]⋯],
in which the first term on the right-hand side has an expansion that starts with 1/ϵ2L-1, the second term has an expansion that starts with 1/ϵ2L-2, and so forth. Thus, keeping only the terms proportional to 1/N in (2.40), and only the first three terms in the Laurent expansion, we obtain(2.43)|A(L,1)〉=1(L-1)!g1L-1f1|A(0)〉+12(L-2)!g1L-2[f1,g1]|A(0)〉+1(L-1)!g1L-1|H(1,1)(ϵ)〉+16(L-3)!g1L-3[[f1,g1],g1]|A(0)〉+1(L-2)!g1L-2f2|A(0)〉+1(L-3)!g1L-3f1g2|A(0)〉+1(L-2)!g1L-2f1|H(1,0)(ϵ)〉+O(1ϵ2L-4),as obtained in [49].


## 2.4. IR Divergences ofA(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 (2.26) that contribute to |A(L,L)〉 are those with as many factors of 1/N as of a. Thus, only f1 and |H(ℓ0,ℓ0)〉 can contribute, giving(2.44)|A(L,L)(ϵ)〉=∑l0=0L1(L-l0)!f1L-l0|H(l0,l0)(ϵ)〉,wheref1=1ϵ(μ2Q2)ϵ(0βγ0)
exact to all orders in the ϵ expansion. Keeping just the first two terms in the Laurent expansion, we find (2.45)|A(L,L)(ϵ)〉=1(L-1)!f1L-1[1Lf1|A(0)〉+|H(1,1)(ϵ)〉]+O(1ϵL-2)=1(L-1)!1ϵL-1(0βγ0)L-1|A(1,1)(Lϵ)〉+O(1ϵL-2).
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

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 et al. [54] (see also [36]). Explicit expressions for these IR-divergent integrals as the Laurent expansions in ϵ 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/N expansion of these divergences.Recall from (2.17) that A[i](L,k) denotes the L-loop color-ordered amplitude which is subleading by a factor of 1/Nk in the 1/N expansion. Single-trace amplitudes are denoted by i=1,…,6 and double-trace amplitudes by i=7,…,9 (see (1.7)).At one loop, the single-trace amplitudes are given by [34] (2.46)A[1](1,0)=M(1)(s,t)A[1](0)=2iKI4(1)(s,t)
with the other single-trace amplitudes A[2](1,0) and A[3](1,0) obtained by letting t↔u, and s↔u, respectively. The identities A[1](L)=A[6](L), A[2](L)=A[5](L), and A[3](L)=A[4](L) are satisfied at all loop orders. In (2.46), I4(1)(s,t) denotes the scalar box integral (2.47)M(1)(s,t)=-12stI4(1)(s,t),I4(1)(s,t)=I4(1)(t,s)=-iμ2ϵeϵγπ-D/2∫dDpp2(p-k1)2(p-k1-k2)2(p+k4)2,
an explicit expression for which is given, for example, in [7].The one-loop double-trace amplitudes are given by [34] (2.48)A[7](1,1)=A[8](1,1)=A[9](1,1)=2(A[1](1,0)+A[2](1,0)+A[3](1,0))(2.49)=4iK[I4(1)(s,t)+I4(1)(t,u)+I4(1)(u,s)].
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] (2.50)A[1](2,0)=M(2)(s,t)A[1](0)=-iK[sI4(2)P(s,t)+tI4(2)P(t,s)],
where I4(2)P(s,t) denotes the scalar double-box (planar) integral (2.51)M(2)(s,t)=14st[sI4(2)P(s,t)+tI4(2)P(t,s)],I4(2)P(s,t)=(-iμ2ϵeϵγπ-D/2)2∫dDpdDqp2(p+q)2q2(p-k1)2(p-k1-k2)2(q-k4)2(q-k3-k4)2,
an explicit expression for which is given, for example, in [7]. The double-trace amplitude is [54]
(2.52)A[7](2,1)=-2iK[s(3I4(2)P(s,t)+2I4(2)NP(s,t)+3I4(2)P(s,u)+2I4(2)NP(s,u))-t(I4(2)NP(t,s)+I4(2)NP(t,u))-u(I4(2)NP(u,s)+I4(2)NP(u,t))],
and the subleading-color single-trace amplitude is [54] (2.53)A[1](2,2)=-2iK[s(I4(2)P(s,t)+I4(2)NP(s,t)+I4(2)P(s,u)+I4(2)NP(s,u))+t(I4(2)P(t,s)+I4(2)NP(t,s)+I4(2)P(t,u)+I4(2)NP(t,u))-2u(I4(2)P(u,s)+I4(2)NP(u,s)+I4(2)P(u,t)+I4(2)NP(u,t))],
where I4(2)NP(s,t) denotes the two-loop nonplanar integral
(2.54)I4(2)NP(s,t)=(-iμ2ϵeϵγπ-D/2)2∫dDpdDqp2(p+q)2q2(p-k2)2(p+q+k1)2(q-k3)2(q-k3-k4)2,
an explicit expression for which is given in [56]. All the other single- and double-trace amplitudes A[i](2,k) 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 nonplanar integral I4(2)NP(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]: (2.55)A[7](2,1)=2(A[1](2,0)+A[2](2,0)+A[3](2,0))-A[3](2,2),A[8](2,1)=2(A[1](2,0)+A[2](2,0)+A[3](2,0))-A[1](2,2),A[9](2,1)=2(A[1](2,0)+A[2](2,0)+A[3](2,0))-A[2](2,2)
and may be easily verified using (2.50), (2.52), and (2.53). In addition, we have
(2.56)A[1](2,2)+A[2](2,2)+A[3](2,2)=0,
also easily verified using (2.53). Together these equations imply(2.57)6∑i=13A[i](2,0)-∑i=79A[i](2,1)=0
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 (2.58)6∑i=13A[i](L)-N∑i=79A[i](L)=0,L≤2,
which is valid through two loops.At one loop, we also saw that one can relate all the subleading-color amplitudesAn;j to the leading amplitude An;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
(2.59)M(1)(s,t)=-1ϵ2(μ2-s)ϵ-1ϵ2(μ2-t)ϵ+12log⁡2(st)+2π23+O(ϵ),
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(2.60)|A(1,1)(ϵ)〉=(-8iKstu)[(μ2-u)ϵ(sY-tX)ϵ-(s+t)XY](111)+O(ϵ),
where we have only included the [7–9] components of A[i](1,1) as the others vanish.At two loops, the planar amplitude is given by (2.50) with [60]
(2.61)M(2)(ϵ)=12[M(1)(ϵ)]2-(ζ2+ϵζ3+ϵ2ζ4)M(1)(2ϵ)-π472+O(ϵ).
The two-loop double trace amplitude has an IR divergence given by the general formula (2.38), which yields (2.62)|A(2,1)(ϵ)〉=(-8iKstu)(-2)(sY-tX)ϵ3(111)+O(1ϵ2).
Finally, the subleading-color single-trace amplitude is given by (2.45) which in this case yields (2.63)|A(2,2)(ϵ)〉=1ϵ(X-YZ-XY-ZY-ZZ-XX-Y)A[7](1,1)(2ϵ)+O(ϵ0).
Only the [1] through [6] components are listed, as the [7] through [9] components vanish.


## 3. Subleading-Color Amplitudes of𝒩=4 SYM and Amplitudes of 𝒩=8 Supergravity

TheAdS5/CFT4 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, 34–38, 41–46, 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 𝒩=8 supergravity, it seems important to understand nonplanar 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 subleading𝒩=4 SYM and those of 𝒩=8 supergravity, at one and two loops. For more than two loops, 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. Others are possible relations between 𝒩=4 subleading-color amplitudes and 𝒩=8 sugra for n≥5.


### 3.1. One and Two-Loop Relations

In this subsection, we demonstrate the existence of some exact relations between𝒩=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/ϵL) leading IR divergences. Other subleading-color SYM amplitudes A(L,k) have 𝒪(1/ϵ2L-k) leading IR divergences and consequently satisfy relations involving lower-loop supergravity amplitudes. The normalization of ASYM(L,2m+1)(s,t) 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(3.1)ASYM(L,2m)(s,t)=aLA[1](L,2m),ASYM(L,2m+1)(s,t)=-aL2A[8](L,2m+1),
noting that the other components A[i](L,k) 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.Factor out the tree amplitude to define(3.2)MSYM(L,k)(s,t)=ASYM(L,k)(s,t)ASYM(0)(s,t),
so that the coupling constant aL is now included in the definition of MSYM(L,k)(s,t), and where. (3.3)ASYM(0)(s,t)=-4iKst.In what follows we denoteASYMtree(ij…k)=A(ij…k) (see also (2.34)). Recall that the one-loop N-independent SYM four-gluon amplitude is given by (2.47), obtaining (3.4)ASYM(1,1)=-22iK[g2N8π2(4πe-γ)ϵ][I4(1)(s,t)+I4(1)(t,u)+I4(1)(u,s)].
The one-loop supergravity four-graviton amplitude may be expressed as [34, 36] (3.5)ASG(1)=8iK2[(κ/2)28π2(4πe-γ)ϵ][I4(1)(s,t)+I4(1)(t,u)+I4(1)(u,s)].after stripping off a factor of (κ/2)2 for a four-point amplitude. The supergravity amplitude is proportional to K2 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 as follows:(3.6)ASG(1)=ASG(0)MSG(1)=(16iK2stu)MSG(1).
Defining λSYM=g2N and λSG=(κ/2)2, one observes that the one-loop SYM and supergravity amplitudes are related by(3.7)MSYM(1,1)(s,t)=2λSYMλSGuMSG(1).
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 λSGu for supergravity because λSG is dimensionful.Finally, rewrite (3.7) in the manifestly permutation-symmetric form(3.8)13[(λSGu)MSYM(1,1)(s,t)+c.p.]=2λSYMMSG(1),
(where c.p. denotes cyclic permutations of s, t, and u) even though uMSYM(1,1)(s,t) is already symmetric under permutations. A similar symmetrized relation can be written for the one-loop leading-color amplitude (3.9)(λSGu)MSYM(1,0)(s,t)+c.p.=-λSYMMSG(1)
obtained from the one-loop decoupling relation (2.48) together with (3.7).Now turn totwo 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 ASYM(2,2)(s,t), since, from (2.63), (3.10)ASYM(2,2)(s,t)=-2aX-YϵASYM(1,1)(2ϵ)+O(ϵ0).
This can be rewritten as(3.11)MSYM(2,2)(s,t)=-2aλSYMλSGu(X-Yϵ)MSG(1)(2ϵ)+O(ϵ0),
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 relationMSG(2)(ϵ)=(1/2)[MSG(1)(ϵ)]2+𝒪(ϵ0) between the one- and two-loop supergravity amplitudes [41, 58, 64, 65], we can write this as(3.12)13[(λSGu)2MSYM(2,2)(s,t)+c.p.]=2λSYM2MSG(2),
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) (3.13)MSYM(2,2)(s,t)=a2st2[s(I4(2)P(s,t)+I4(2)NP(s,t)+I4(2)P(s,u)+I4(2)NP(s,u))+t(I4(2)P(t,s)+I4(2)NP(t,s)+I4(2)P(t,u)+I4(2)NP(t,u))-2u(I4(2)P(u,s)+I4(2)NP(u,s)+I4(2)P(u,t)+I4(2)NP(u,t))]
and that for the two-loop supergravity amplitude [36] (3.14)MSG(2)=-s3tu4[(κ/2)28π2(4πe-γ)ϵ]2[I4(2)P(s,t)+I4(2)NP(s,t)+I4(2)P(s,u)+I4(2)NP(s,u)]+c.p.,
where I4(2)P and I4(2)NP are the two-loop planar and nonplanar 4-point functions.Now consider the two-loop subleading-color amplitudeMSYM(2,1). The two-loop decoupling relation (2.57) can be rewritten as(3.15)-2[uMSYM(2,1)(s,t)+c.p.]=6[uMSYM(2,0)(s,t)+c.p.].
Using the ABDK relation [60] (3.16)MSYM(2,0)(ϵ)=12[MSYM(1,0)(ϵ)]2+af(2)(ϵ)MSYM(1,0)(2ϵ)+O(ϵ),f(2)(ϵ)=-(ζ2+ϵζ3+ϵ2ζ4),
together with (3.9), we can rewrite (3.15) as(3.17)13[(λSGu)MSYM(2,1)(s,t)+c.p.]+12{(λSGu)[MSYM(1,0)(s,t)]2+c.p.}=2λSYM28π2(4πe-γ)ϵf(2)(ϵ)MSG(1)(2ϵ)+O(ϵ).
Unlike (3.12), however, (3.17) only holds through 𝒪(ϵ0), 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 (3.18)13[(λSGu)LMSYM(L,L)(s,t)+c.p.]=(2λSYM)LMSG(L)
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.


### 3.2. Three or More Loops

On the supergravity side, there is an exact exponentiation formula [64, 65], which implies(3.19)MSG(L)=1L![MSG(1)]L+O(1ϵL-2)=1L![-λSG(sY-tX)8π2ϵ]L+O(1ϵL-1).
Since the leading IR divergences of A(L,L) is 𝒪(1/ϵL), one can show that the following relations hold: (3.20)[λSG2s2+t2+u23]k13[(λSGu)MSYM(2k+1,2k+1)(s,t;ϵ)+c.p.]=λSYM2k+122k+1/2(2k+1)![MSG(2)(ϵ)+16(λSG8π2)2(sX+tY+uZϵ)2]kMSG(1)(ϵ)+O(1ϵ2k),
for L=2k+1(3.21)[λSG2s2+t2+u23]k13[(λSGu)2MSYM(2k+2,2k+2)(s,t;ϵ)+c.p.]=λSYM2k+222k+2(2k+2)![MSG(2)(ϵ)+16(λSG8π2)2(sX+tY+uZϵ)2]kMSG(2)(ϵ)+O(1ϵ2k+1)
for L=2k+2 (where k=0,1,2,…).That is, we have an exact relation atL-loops for the leading IR divergence ~𝒪(1/ϵL), with an untested relation for the subleading divergence of 𝒪(1/ϵL-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 ‘t Hooft string picture of the 1/N expansion. Thus at least in the case of L=1,2, (3.18) and (3.17) still do, so one can hope that there is a correct relation at higher L 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 (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, 66–68] (among older works see also [37, 69, 70]). Recent work applying the KLT relations includes [71–74]. In our quest for SYM-supergravity relations, we first review previous KLT relations; we then note that A5;3 and the 1-loop supergravity amplitude both have 1/ϵ 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/ϵ IR divergence, we explore the possibility that the 1-loop 5-point supergravity amplitude can be expressed as a linear combination of the A5;3 SYM amplitudes. In particular a linear relation is proposed among the 1/ϵ IR divergences of the two theories.At tree level, the KLT relations are quadratic relations between then-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 sij 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 theα′→0 limit [35, 63] and can be expressed as (we use the notation of [37])
(3.22)An,sugratree(12…n)=(-1)n+1[An(12…n)∑permsf(i1…ij)f¯(l1…lj′)×An(i1,…,ij,1,n-1,l1,…,lj′,n)∑perms]+P(2,…,n-2)f(i1,…,ij)=s(1,ij)∏m=1j-1(s(1,im)+∑k=m+1jg(im,ik)),f¯(l1,…,lj′)=s(l1,n-1)∏m=2j′(s(lm,n-1)+∑k=1m-1g(lk,lm)),
where “perms’’ are (i1,…,ij)∈𝒫(2,…,n/2),  (l1,…,lj′∈𝒫(n/2+1,…,n-2),  j=n/2-1,j′=n/2-2, and gi,j=sij if i>j and zero otherwise.In [44, 45], new forms of the KLT relations for any n-point function were found. They are both written in terms of the functions:(3.23)S[i1…ik∣j1…jk]=∏t=1k(sit1+∑q>tkθ(it,iq)sitiq),S̃[i1…ik∣j1…jk]=∏t=1k(sjtn+∑q<tkθ(jq,jt)sjqjt),
where θ(it,iq) is zero in (it,iq) has the same order in both sets ℐ={i1,…,ik} and 𝒥={j1,…,jk} and is 1 otherwise, and similarly for θ(jq,jt).A form of KLT relations was found in [44], but needs to be regularized, due to a singular denominator (3.24)An,sugratree=(-1)n∑γ,βÃn(n,γ2,n-1,1)S[γ2,n-1,β2,n-1]p1An(1,β2,n-1,n)s12…n-1,An,sugratree=(-1)n∑β,γAn(n,β2,n-1,1)S̃[β2,n-1,γ2,n-1]pnÃn(1,γ2,n-1,n)s23…n.
However they have a large (n-2)! manifest symmetry. Another set was proven in [45] which is nonsingular (3.25)An,sugratree=(-1)n+1∑σ∈Sn-3∑α∈Sj-1∑β∈Sn-j-2An(1,σ2,j,σj+1,n-2,n-1,n)S[ασ(2),σ(j)∣σ2,j]p1×S̃[σj+1,n-2∣βσ(j+1),σ(n-2),n]pnÃn(ασ(2),σ(j),1,n-1,βσ(j+1),σ(n-1),n)
but with only (n-3)! manifest symmetry.The original KLT relation for the 5-point function is(3.26)A5,sugratree=s12s34A(12345)Ã(21435)+s13s24A(13245)Ã(31425)
and has (n-3)!=2! symmetry, whereas the KLT relations (3.25) become, explicitly, (3.27)A5,sugratree=∑σ,σ̃∈S2Ã(45,σ̃23,1)A(1,σ23,45)S[σ̃2,3∣σ2,3]p1=s12s13(A(45231)A(12345)+A(45321)A(13245))+s13(s12+s23)A(45231)A(13245)+s12(s13+s23)A(45321)A(12345),A5,sugratree=∑σ,σ̃∈S2Ã(14,σ̃23,5)A(1,σ23,45)S̃[σ2,3∣σ̃2,3]p4=s24s34[A(12345)A(14235)+A(13245)A(14325)]+s34(s24+s23)A(12345)A(14325)+s24(s34+s23)A(13245)A(14235)
and have (n-3)!=2! symmetry.We now derive another KLT relation for 5-point amplitudes using information about subleading one-loop amplitudes.As we saw in (1.5), the An;j are related to the An;1 via group theory. In particular, for 5-point amplitudes, one has a single-trace amplitude A5;1 and a double-trace amplitude A5;3 related by [76] (3.28)A5;3(45123)=∑σ∈COP4123A5;1(σ(1),…,σ(4),5).The single-trace amplitude is given by(3.29)A5(1,0)(12345)≡A5;1(12345)=-14A(12345)∑cyclicF(1)(s,t,m2),
where(3.30)F(1)(s,t,m2)=stI5(1)(s,t,m2)
is the dimensionless one-mass box, and I(1)(s,t,m2) is the 1-loop scalar box integral (2.47) with momenta 3,4 in the same corner and m2=P2=(p3+p4)2.Substituting in (3.28), we find (3.31)A5;3(fg;hij)=∑abcde∈30fixed termsF(cde;ab)[sabcde;+;fghijA(abcde)+sabcde;-;fghijA(abedc)].
Here sabcde;±;fghij 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-loop𝒩=8 supergravity amplitude is [62], written in terms of the scalar 1 m box I(123,45) (with momenta 4,5 on the same corner of the box), and the dimensionless box F(123;45) is(3.32)A5,sugra1-loop(12q3q2q1)=-12∑30permssq2q1s12s2q3A(12q3q2q1)A(12q3q1q2)F(12q3;q2q1)
or(3.33)A5,sugra1-loop(12345)=-12∑30permsF(cde;ab)scdsdesabA(cdeab)A(cdeba).The IR behavior of the 1-loop 1 m scalar box is(3.34)I4,1m(s,t,m2)=rΓs12s23{2ϵ2[(-s12)-ϵ+(-s23)-ϵ-(-s45)-ϵ]+finite}⟹F(cde;ab)≃rΓϵ2[(-scd)-ϵ+(-sde)-ϵ-(-sab)-ϵ]+finiterΓ=Γ(1+ϵ)Γ2(1-ϵ)Γ(1-2ϵ),
where D=4-2ϵ.


#### 3.3.1. IR Behavior of the Double-Trace 1-Loop SYM AmplitudeA5;3

Using (3.34), we find(3.35)A5;3(fg;hij)=∑abcd∈30termsF(cde;ab)[sabcde;+;fghijA(abcde)+sabcde;-;fghijA(abedc)]~rΓϵ2∑abcd∈30terms[scd-ϵ+sde-ϵ-sab-ϵ][sabcde;+;fghijA(abcde)+sabcde;-;fghijA(abedc)].
Organizing the coefficients of each divergence, we find(3.36)A5;3(fg;lmn)≃rΓϵ2∑i<j(-sij)-ϵ∑abc≠i,jϵlmn[A(ijabc)],
where ϵlmn[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.The leading (1/ϵ2) divergence of A5;3(45;123), given by
(3.37)∑i<j∑abc≠i,jϵ123[A(ijabc)],
vanishes by explicit computation, so that the leading IR divergence of A5;3 is 1/ϵ, as expected from a generalization of the subleading-color amplitude of the 4-gluon amplitude [39, 49]. (The vanishing of the 1/ϵ2 IR divergence of (3.37) is also a consequence of (4.7).)


#### 3.3.2. IR Behavior of𝒩=8 Supergravity One-Loop Amplitudes and KLT Relations

Using (3.32), we obtain(3.38)A5,sugra1-loop(12345)=-12∑30permsF(cde;ab)scdsdesabA(cdeab)A(cdeba)≃-12ϵ2∑30perms[scd-ϵ+sde-ϵ-sab-ϵ]scdsdesabA(cdeab)A(cdeba).
Organizing the terms by IR divergences, we obtain (3.39)A5,sugra1-loop≃1ϵ2∑i<jsij1-ϵ×[∑dscdsdeA(ijcde)A(ijedc)+∑csicsabA(ijabc)A(ijbac)+∑csjcsabA(ijcba)A(ijcab)].On the other hand, we know that the IR behavior of the 1-loopn-point supergravity amplitude is [77] (3.40)An,sugra1-loop(1…n)≃1ϵ2An,sugratree(1…n)∑i<jsij1-ϵ
which means that we must have the KLT relation (3.41)A5,sugratree(12345)=∑dscdsdeA(ijcde)A(ijedc)+∑csicsabA(ijabc)A(ijbac)+∑csjcsabA(ijcba)A(ijcab),∀(ij).
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(3.42)An,SYMtree(1+2+…i-…j-…n+)=〈ij〉4〈12〉〈23〉⋯〈n1〉,
or for our case, for instance choosing 1-2-, (3.43)A(1-2-3+4+5+)=〈12〉4〈12〉〈23〉〈34〉〈45〉〈51〉.
A similar formula exists for the supergravity amplitude [37] (3.44)A5,sugratree(1-2-3+4+5+)=〈12〉8ϵ(1234)N(5),
where(3.45)ϵ(ijkl)=4iϵμνρσkiμkjνkkρklσ,N(5)=∏i=14∏j=i+15〈ij〉.A specific case of (3.41) is proved, namely,(3.46)A5,sugratree(12345)=s34s45A(12345)A(12543)+s53s34A(12534)A(12435)+s45s53A(12453)A(12354)+s23s45A(12345)A(12354)+s24s35A(12435)A(12453)+s25s34A(12534)A(12543)+s13s45A(21345)A(21354)+s14s35A(21435)A(21453)+s15s34A(21534)A(21543).
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.


#### 3.3.3. Relation betweenA5,sugraone-loop and A5;3

Motivated by the fact that the leading IR divergence of then=5-point supergravity amplitude and that of A5;3 are both of order 1/ϵ at 1 loop, one investigates whether A5,sugra1-loop can be expressed as a linear combination of A5;3 amplitudes. One uses information from (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 Section3.1, we want to find A5,sugra1-loop as a linear combination of the A5;3 amplitudes.In analogy with the 4-point function, we would like to find a relation of the type(3.47)A5,sugra1-loop(12345)=∑i∈fghijβiA5;3(i)=∑abcde∈30fixedtermsF(cde;ab)∑i∈fghijβi[sabcde;+;iA(abcde)+sabcde;-;iA(abedc)].
On the other hand, (3.48)A5,sugra1-loop(12345)=∑abcde∈30fixedtermsF(cde;ab)αabcde,
where from (3.33), (3.49)αabcde=-12scdsdesabA(cdeab)A(cdeba)
which means that we need (3.50)αabcde=∑i∈fghijβi[sabcde;+;iA(abcde)+sabcde;-;iA(abedc)]
to be satisfied, which are 30 equations for 10 unknowns (βi), so (3.50) is not guaranteed to have solutions.The 30 equations can then be rewritten, using the explicit form ofαabcde, and a new notation that will prove useful, as (3.51)-12sabsbcsdeA(abcde)A(abced)=∑fg;hijβ(fg)ϵhij[A(abcde)](1-ϵhij(de)A(abced)A(abcde)),
where ϵhij(de) is plus if both d,e belong to h,i,j, and minus otherwise.In order to see if a unique solution for theβi 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)1ϵ2A5,sugratree(12345)∑i<jsij(-sij)-ϵ=rΓϵ2∑k∈fg;lmnβk∑i<j(-sij)-ϵ∑abc≠i,jϵlmn[A(ijabc)]
which means that one requires, using the vanishing of the 1/ϵ2 IR divergence, (3.53)A5,sugratree(12345)sij=∑k∈fg;lmnβk∑abc≠i,jϵlmn[A(ijabc)].
If we denote the A5,sugratree(12345) by just M5, then the lhs is a vector column of (ij),M5sij. Also denote ∑abc≠i,jϵlmn[A(ijabc)] as N(ij),(fg), so that(3.54)N(ij),(fg)β(fg)=M5sij⟹[β(fg)]=[N(ij),(fg)]-1M5sij.
Note that the index (fg) on the matrix N has 10 values, and these values can also be identified by the lmn of ϵlmn[A(ijabc)], since it corresponds to the same 10 terms, picking out a group (fg) 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 (3.55)∑(ij)N(ij),(fg)=0,
that is, that the matrix N has rank 9 instead of 10. One then needs to work with the corresponding 9×9 reduced matrix Nred;(ij),(fg) and give the 10th coefficient β(fg) 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(fg) are unique. For that, one must calculate the rank of Nred. 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(fg) 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(fg) 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. (As (3.18) exemplifies for L=1,n=4, (3.47) may not be the only equation relating 𝒩=8 supergravity to 𝒩=4 SYM for L=1,n=5.)In principle the strategy described above can be applied to highern-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 et al. [38] are suitable for this purpose.


## 3.1. One and Two-Loop Relations

In this subsection, we demonstrate the existence of some exact relations between𝒩=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/ϵL) leading IR divergences. Other subleading-color SYM amplitudes A(L,k) have 𝒪(1/ϵ2L-k) leading IR divergences and consequently satisfy relations involving lower-loop supergravity amplitudes. The normalization of ASYM(L,2m+1)(s,t) 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(3.1)ASYM(L,2m)(s,t)=aLA[1](L,2m),ASYM(L,2m+1)(s,t)=-aL2A[8](L,2m+1),
noting that the other components A[i](L,k) 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.Factor out the tree amplitude to define(3.2)MSYM(L,k)(s,t)=ASYM(L,k)(s,t)ASYM(0)(s,t),
so that the coupling constant aL is now included in the definition of MSYM(L,k)(s,t), and where. (3.3)ASYM(0)(s,t)=-4iKst.In what follows we denoteASYMtree(ij…k)=A(ij…k) (see also (2.34)). Recall that the one-loop N-independent SYM four-gluon amplitude is given by (2.47), obtaining (3.4)ASYM(1,1)=-22iK[g2N8π2(4πe-γ)ϵ][I4(1)(s,t)+I4(1)(t,u)+I4(1)(u,s)].
The one-loop supergravity four-graviton amplitude may be expressed as [34, 36] (3.5)ASG(1)=8iK2[(κ/2)28π2(4πe-γ)ϵ][I4(1)(s,t)+I4(1)(t,u)+I4(1)(u,s)].after stripping off a factor of (κ/2)2 for a four-point amplitude. The supergravity amplitude is proportional to K2 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 as follows:(3.6)ASG(1)=ASG(0)MSG(1)=(16iK2stu)MSG(1).
Defining λSYM=g2N and λSG=(κ/2)2, one observes that the one-loop SYM and supergravity amplitudes are related by(3.7)MSYM(1,1)(s,t)=2λSYMλSGuMSG(1).
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 λSGu for supergravity because λSG is dimensionful.Finally, rewrite (3.7) in the manifestly permutation-symmetric form(3.8)13[(λSGu)MSYM(1,1)(s,t)+c.p.]=2λSYMMSG(1),
(where c.p. denotes cyclic permutations of s, t, and u) even though uMSYM(1,1)(s,t) is already symmetric under permutations. A similar symmetrized relation can be written for the one-loop leading-color amplitude (3.9)(λSGu)MSYM(1,0)(s,t)+c.p.=-λSYMMSG(1)
obtained from the one-loop decoupling relation (2.48) together with (3.7).Now turn totwo 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 ASYM(2,2)(s,t), since, from (2.63), (3.10)ASYM(2,2)(s,t)=-2aX-YϵASYM(1,1)(2ϵ)+O(ϵ0).
This can be rewritten as(3.11)MSYM(2,2)(s,t)=-2aλSYMλSGu(X-Yϵ)MSG(1)(2ϵ)+O(ϵ0),
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 relationMSG(2)(ϵ)=(1/2)[MSG(1)(ϵ)]2+𝒪(ϵ0) between the one- and two-loop supergravity amplitudes [41, 58, 64, 65], we can write this as(3.12)13[(λSGu)2MSYM(2,2)(s,t)+c.p.]=2λSYM2MSG(2),
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) (3.13)MSYM(2,2)(s,t)=a2st2[s(I4(2)P(s,t)+I4(2)NP(s,t)+I4(2)P(s,u)+I4(2)NP(s,u))+t(I4(2)P(t,s)+I4(2)NP(t,s)+I4(2)P(t,u)+I4(2)NP(t,u))-2u(I4(2)P(u,s)+I4(2)NP(u,s)+I4(2)P(u,t)+I4(2)NP(u,t))]
and that for the two-loop supergravity amplitude [36] (3.14)MSG(2)=-s3tu4[(κ/2)28π2(4πe-γ)ϵ]2[I4(2)P(s,t)+I4(2)NP(s,t)+I4(2)P(s,u)+I4(2)NP(s,u)]+c.p.,
where I4(2)P and I4(2)NP are the two-loop planar and nonplanar 4-point functions.Now consider the two-loop subleading-color amplitudeMSYM(2,1). The two-loop decoupling relation (2.57) can be rewritten as(3.15)-2[uMSYM(2,1)(s,t)+c.p.]=6[uMSYM(2,0)(s,t)+c.p.].
Using the ABDK relation [60] (3.16)MSYM(2,0)(ϵ)=12[MSYM(1,0)(ϵ)]2+af(2)(ϵ)MSYM(1,0)(2ϵ)+O(ϵ),f(2)(ϵ)=-(ζ2+ϵζ3+ϵ2ζ4),
together with (3.9), we can rewrite (3.15) as(3.17)13[(λSGu)MSYM(2,1)(s,t)+c.p.]+12{(λSGu)[MSYM(1,0)(s,t)]2+c.p.}=2λSYM28π2(4πe-γ)ϵf(2)(ϵ)MSG(1)(2ϵ)+O(ϵ).
Unlike (3.12), however, (3.17) only holds through 𝒪(ϵ0), 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 (3.18)13[(λSGu)LMSYM(L,L)(s,t)+c.p.]=(2λSYM)LMSG(L)
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.


## 3.2. Three or More Loops

On the supergravity side, there is an exact exponentiation formula [64, 65], which implies(3.19)MSG(L)=1L![MSG(1)]L+O(1ϵL-2)=1L![-λSG(sY-tX)8π2ϵ]L+O(1ϵL-1).
Since the leading IR divergences of A(L,L) is 𝒪(1/ϵL), one can show that the following relations hold: (3.20)[λSG2s2+t2+u23]k13[(λSGu)MSYM(2k+1,2k+1)(s,t;ϵ)+c.p.]=λSYM2k+122k+1/2(2k+1)![MSG(2)(ϵ)+16(λSG8π2)2(sX+tY+uZϵ)2]kMSG(1)(ϵ)+O(1ϵ2k),
for L=2k+1(3.21)[λSG2s2+t2+u23]k13[(λSGu)2MSYM(2k+2,2k+2)(s,t;ϵ)+c.p.]=λSYM2k+222k+2(2k+2)![MSG(2)(ϵ)+16(λSG8π2)2(sX+tY+uZϵ)2]kMSG(2)(ϵ)+O(1ϵ2k+1)
for L=2k+2 (where k=0,1,2,…).That is, we have an exact relation atL-loops for the leading IR divergence ~𝒪(1/ϵL), with an untested relation for the subleading divergence of 𝒪(1/ϵL-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 ‘t Hooft string picture of the 1/N expansion. Thus at least in the case of L=1,2, (3.18) and (3.17) still do, so one can hope that there is a correct relation at higher L 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 (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, 66–68] (among older works see also [37, 69, 70]). Recent work applying the KLT relations includes [71–74]. In our quest for SYM-supergravity relations, we first review previous KLT relations; we then note that A5;3 and the 1-loop supergravity amplitude both have 1/ϵ 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/ϵ IR divergence, we explore the possibility that the 1-loop 5-point supergravity amplitude can be expressed as a linear combination of the A5;3 SYM amplitudes. In particular a linear relation is proposed among the 1/ϵ IR divergences of the two theories.At tree level, the KLT relations are quadratic relations between then-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 sij 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 theα′→0 limit [35, 63] and can be expressed as (we use the notation of [37])
(3.22)An,sugratree(12…n)=(-1)n+1[An(12…n)∑permsf(i1…ij)f¯(l1…lj′)×An(i1,…,ij,1,n-1,l1,…,lj′,n)∑perms]+P(2,…,n-2)f(i1,…,ij)=s(1,ij)∏m=1j-1(s(1,im)+∑k=m+1jg(im,ik)),f¯(l1,…,lj′)=s(l1,n-1)∏m=2j′(s(lm,n-1)+∑k=1m-1g(lk,lm)),
where “perms’’ are (i1,…,ij)∈𝒫(2,…,n/2),  (l1,…,lj′∈𝒫(n/2+1,…,n-2),  j=n/2-1,j′=n/2-2, and gi,j=sij if i>j and zero otherwise.In [44, 45], new forms of the KLT relations for any n-point function were found. They are both written in terms of the functions:(3.23)S[i1…ik∣j1…jk]=∏t=1k(sit1+∑q>tkθ(it,iq)sitiq),S̃[i1…ik∣j1…jk]=∏t=1k(sjtn+∑q<tkθ(jq,jt)sjqjt),
where θ(it,iq) is zero in (it,iq) has the same order in both sets ℐ={i1,…,ik} and 𝒥={j1,…,jk} and is 1 otherwise, and similarly for θ(jq,jt).A form of KLT relations was found in [44], but needs to be regularized, due to a singular denominator (3.24)An,sugratree=(-1)n∑γ,βÃn(n,γ2,n-1,1)S[γ2,n-1,β2,n-1]p1An(1,β2,n-1,n)s12…n-1,An,sugratree=(-1)n∑β,γAn(n,β2,n-1,1)S̃[β2,n-1,γ2,n-1]pnÃn(1,γ2,n-1,n)s23…n.
However they have a large (n-2)! manifest symmetry. Another set was proven in [45] which is nonsingular (3.25)An,sugratree=(-1)n+1∑σ∈Sn-3∑α∈Sj-1∑β∈Sn-j-2An(1,σ2,j,σj+1,n-2,n-1,n)S[ασ(2),σ(j)∣σ2,j]p1×S̃[σj+1,n-2∣βσ(j+1),σ(n-2),n]pnÃn(ασ(2),σ(j),1,n-1,βσ(j+1),σ(n-1),n)
but with only (n-3)! manifest symmetry.The original KLT relation for the 5-point function is(3.26)A5,sugratree=s12s34A(12345)Ã(21435)+s13s24A(13245)Ã(31425)
and has (n-3)!=2! symmetry, whereas the KLT relations (3.25) become, explicitly, (3.27)A5,sugratree=∑σ,σ̃∈S2Ã(45,σ̃23,1)A(1,σ23,45)S[σ̃2,3∣σ2,3]p1=s12s13(A(45231)A(12345)+A(45321)A(13245))+s13(s12+s23)A(45231)A(13245)+s12(s13+s23)A(45321)A(12345),A5,sugratree=∑σ,σ̃∈S2Ã(14,σ̃23,5)A(1,σ23,45)S̃[σ2,3∣σ̃2,3]p4=s24s34[A(12345)A(14235)+A(13245)A(14325)]+s34(s24+s23)A(12345)A(14325)+s24(s34+s23)A(13245)A(14235)
and have (n-3)!=2! symmetry.We now derive another KLT relation for 5-point amplitudes using information about subleading one-loop amplitudes.As we saw in (1.5), the An;j are related to the An;1 via group theory. In particular, for 5-point amplitudes, one has a single-trace amplitude A5;1 and a double-trace amplitude A5;3 related by [76] (3.28)A5;3(45123)=∑σ∈COP4123A5;1(σ(1),…,σ(4),5).The single-trace amplitude is given by(3.29)A5(1,0)(12345)≡A5;1(12345)=-14A(12345)∑cyclicF(1)(s,t,m2),
where(3.30)F(1)(s,t,m2)=stI5(1)(s,t,m2)
is the dimensionless one-mass box, and I(1)(s,t,m2) is the 1-loop scalar box integral (2.47) with momenta 3,4 in the same corner and m2=P2=(p3+p4)2.Substituting in (3.28), we find (3.31)A5;3(fg;hij)=∑abcde∈30fixed termsF(cde;ab)[sabcde;+;fghijA(abcde)+sabcde;-;fghijA(abedc)].
Here sabcde;±;fghij 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-loop𝒩=8 supergravity amplitude is [62], written in terms of the scalar 1 m box I(123,45) (with momenta 4,5 on the same corner of the box), and the dimensionless box F(123;45) is(3.32)A5,sugra1-loop(12q3q2q1)=-12∑30permssq2q1s12s2q3A(12q3q2q1)A(12q3q1q2)F(12q3;q2q1)
or(3.33)A5,sugra1-loop(12345)=-12∑30permsF(cde;ab)scdsdesabA(cdeab)A(cdeba).The IR behavior of the 1-loop 1 m scalar box is(3.34)I4,1m(s,t,m2)=rΓs12s23{2ϵ2[(-s12)-ϵ+(-s23)-ϵ-(-s45)-ϵ]+finite}⟹F(cde;ab)≃rΓϵ2[(-scd)-ϵ+(-sde)-ϵ-(-sab)-ϵ]+finiterΓ=Γ(1+ϵ)Γ2(1-ϵ)Γ(1-2ϵ),
where D=4-2ϵ.


### 3.3.1. IR Behavior of the Double-Trace 1-Loop SYM AmplitudeA5;3

Using (3.34), we find(3.35)A5;3(fg;hij)=∑abcd∈30termsF(cde;ab)[sabcde;+;fghijA(abcde)+sabcde;-;fghijA(abedc)]~rΓϵ2∑abcd∈30terms[scd-ϵ+sde-ϵ-sab-ϵ][sabcde;+;fghijA(abcde)+sabcde;-;fghijA(abedc)].
Organizing the coefficients of each divergence, we find(3.36)A5;3(fg;lmn)≃rΓϵ2∑i<j(-sij)-ϵ∑abc≠i,jϵlmn[A(ijabc)],
where ϵlmn[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.The leading (1/ϵ2) divergence of A5;3(45;123), given by
(3.37)∑i<j∑abc≠i,jϵ123[A(ijabc)],
vanishes by explicit computation, so that the leading IR divergence of A5;3 is 1/ϵ, as expected from a generalization of the subleading-color amplitude of the 4-gluon amplitude [39, 49]. (The vanishing of the 1/ϵ2 IR divergence of (3.37) is also a consequence of (4.7).)


### 3.3.2. IR Behavior of𝒩=8 Supergravity One-Loop Amplitudes and KLT Relations

Using (3.32), we obtain(3.38)A5,sugra1-loop(12345)=-12∑30permsF(cde;ab)scdsdesabA(cdeab)A(cdeba)≃-12ϵ2∑30perms[scd-ϵ+sde-ϵ-sab-ϵ]scdsdesabA(cdeab)A(cdeba).
Organizing the terms by IR divergences, we obtain (3.39)A5,sugra1-loop≃1ϵ2∑i<jsij1-ϵ×[∑dscdsdeA(ijcde)A(ijedc)+∑csicsabA(ijabc)A(ijbac)+∑csjcsabA(ijcba)A(ijcab)].On the other hand, we know that the IR behavior of the 1-loopn-point supergravity amplitude is [77] (3.40)An,sugra1-loop(1…n)≃1ϵ2An,sugratree(1…n)∑i<jsij1-ϵ
which means that we must have the KLT relation (3.41)A5,sugratree(12345)=∑dscdsdeA(ijcde)A(ijedc)+∑csicsabA(ijabc)A(ijbac)+∑csjcsabA(ijcba)A(ijcab),∀(ij).
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(3.42)An,SYMtree(1+2+…i-…j-…n+)=〈ij〉4〈12〉〈23〉⋯〈n1〉,
or for our case, for instance choosing 1-2-, (3.43)A(1-2-3+4+5+)=〈12〉4〈12〉〈23〉〈34〉〈45〉〈51〉.
A similar formula exists for the supergravity amplitude [37] (3.44)A5,sugratree(1-2-3+4+5+)=〈12〉8ϵ(1234)N(5),
where(3.45)ϵ(ijkl)=4iϵμνρσkiμkjνkkρklσ,N(5)=∏i=14∏j=i+15〈ij〉.A specific case of (3.41) is proved, namely,(3.46)A5,sugratree(12345)=s34s45A(12345)A(12543)+s53s34A(12534)A(12435)+s45s53A(12453)A(12354)+s23s45A(12345)A(12354)+s24s35A(12435)A(12453)+s25s34A(12534)A(12543)+s13s45A(21345)A(21354)+s14s35A(21435)A(21453)+s15s34A(21534)A(21543).
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.


### 3.3.3. Relation betweenA5,sugraone-loop and A5;3

Motivated by the fact that the leading IR divergence of then=5-point supergravity amplitude and that of A5;3 are both of order 1/ϵ at 1 loop, one investigates whether A5,sugra1-loop can be expressed as a linear combination of A5;3 amplitudes. One uses information from (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 Section3.1, we want to find A5,sugra1-loop as a linear combination of the A5;3 amplitudes.In analogy with the 4-point function, we would like to find a relation of the type(3.47)A5,sugra1-loop(12345)=∑i∈fghijβiA5;3(i)=∑abcde∈30fixedtermsF(cde;ab)∑i∈fghijβi[sabcde;+;iA(abcde)+sabcde;-;iA(abedc)].
On the other hand, (3.48)A5,sugra1-loop(12345)=∑abcde∈30fixedtermsF(cde;ab)αabcde,
where from (3.33), (3.49)αabcde=-12scdsdesabA(cdeab)A(cdeba)
which means that we need (3.50)αabcde=∑i∈fghijβi[sabcde;+;iA(abcde)+sabcde;-;iA(abedc)]
to be satisfied, which are 30 equations for 10 unknowns (βi), so (3.50) is not guaranteed to have solutions.The 30 equations can then be rewritten, using the explicit form ofαabcde, and a new notation that will prove useful, as (3.51)-12sabsbcsdeA(abcde)A(abced)=∑fg;hijβ(fg)ϵhij[A(abcde)](1-ϵhij(de)A(abced)A(abcde)),
where ϵhij(de) is plus if both d,e belong to h,i,j, and minus otherwise.In order to see if a unique solution for theβi 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)1ϵ2A5,sugratree(12345)∑i<jsij(-sij)-ϵ=rΓϵ2∑k∈fg;lmnβk∑i<j(-sij)-ϵ∑abc≠i,jϵlmn[A(ijabc)]
which means that one requires, using the vanishing of the 1/ϵ2 IR divergence, (3.53)A5,sugratree(12345)sij=∑k∈fg;lmnβk∑abc≠i,jϵlmn[A(ijabc)].
If we denote the A5,sugratree(12345) by just M5, then the lhs is a vector column of (ij),M5sij. Also denote ∑abc≠i,jϵlmn[A(ijabc)] as N(ij),(fg), so that(3.54)N(ij),(fg)β(fg)=M5sij⟹[β(fg)]=[N(ij),(fg)]-1M5sij.
Note that the index (fg) on the matrix N has 10 values, and these values can also be identified by the lmn of ϵlmn[A(ijabc)], since it corresponds to the same 10 terms, picking out a group (fg) 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 (3.55)∑(ij)N(ij),(fg)=0,
that is, that the matrix N has rank 9 instead of 10. One then needs to work with the corresponding 9×9 reduced matrix Nred;(ij),(fg) and give the 10th coefficient β(fg) 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(fg) are unique. For that, one must calculate the rank of Nred. 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(fg) 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(fg) 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. (As (3.18) exemplifies for L=1,n=4, (3.47) may not be the only equation relating 𝒩=8 supergravity to 𝒩=4 SYM for L=1,n=5.)In principle the strategy described above can be applied to highern-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 et al. [38] are suitable for this purpose.


## 3.3.1. IR Behavior of the Double-Trace 1-Loop SYM AmplitudeA5;3

Using (3.34), we find(3.35)A5;3(fg;hij)=∑abcd∈30termsF(cde;ab)[sabcde;+;fghijA(abcde)+sabcde;-;fghijA(abedc)]~rΓϵ2∑abcd∈30terms[scd-ϵ+sde-ϵ-sab-ϵ][sabcde;+;fghijA(abcde)+sabcde;-;fghijA(abedc)].
Organizing the coefficients of each divergence, we find(3.36)A5;3(fg;lmn)≃rΓϵ2∑i<j(-sij)-ϵ∑abc≠i,jϵlmn[A(ijabc)],
where ϵlmn[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.The leading (1/ϵ2) divergence of A5;3(45;123), given by
(3.37)∑i<j∑abc≠i,jϵ123[A(ijabc)],
vanishes by explicit computation, so that the leading IR divergence of A5;3 is 1/ϵ, as expected from a generalization of the subleading-color amplitude of the 4-gluon amplitude [39, 49]. (The vanishing of the 1/ϵ2 IR divergence of (3.37) is also a consequence of (4.7).)


## 3.3.2. IR Behavior of𝒩=8 Supergravity One-Loop Amplitudes and KLT Relations

Using (3.32), we obtain(3.38)A5,sugra1-loop(12345)=-12∑30permsF(cde;ab)scdsdesabA(cdeab)A(cdeba)≃-12ϵ2∑30perms[scd-ϵ+sde-ϵ-sab-ϵ]scdsdesabA(cdeab)A(cdeba).
Organizing the terms by IR divergences, we obtain (3.39)A5,sugra1-loop≃1ϵ2∑i<jsij1-ϵ×[∑dscdsdeA(ijcde)A(ijedc)+∑csicsabA(ijabc)A(ijbac)+∑csjcsabA(ijcba)A(ijcab)].On the other hand, we know that the IR behavior of the 1-loopn-point supergravity amplitude is [77] (3.40)An,sugra1-loop(1…n)≃1ϵ2An,sugratree(1…n)∑i<jsij1-ϵ
which means that we must have the KLT relation (3.41)A5,sugratree(12345)=∑dscdsdeA(ijcde)A(ijedc)+∑csicsabA(ijabc)A(ijbac)+∑csjcsabA(ijcba)A(ijcab),∀(ij).
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(3.42)An,SYMtree(1+2+…i-…j-…n+)=〈ij〉4〈12〉〈23〉⋯〈n1〉,
or for our case, for instance choosing 1-2-, (3.43)A(1-2-3+4+5+)=〈12〉4〈12〉〈23〉〈34〉〈45〉〈51〉.
A similar formula exists for the supergravity amplitude [37] (3.44)A5,sugratree(1-2-3+4+5+)=〈12〉8ϵ(1234)N(5),
where(3.45)ϵ(ijkl)=4iϵμνρσkiμkjνkkρklσ,N(5)=∏i=14∏j=i+15〈ij〉.A specific case of (3.41) is proved, namely,(3.46)A5,sugratree(12345)=s34s45A(12345)A(12543)+s53s34A(12534)A(12435)+s45s53A(12453)A(12354)+s23s45A(12345)A(12354)+s24s35A(12435)A(12453)+s25s34A(12534)A(12543)+s13s45A(21345)A(21354)+s14s35A(21435)A(21453)+s15s34A(21534)A(21543).
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.


## 3.3.3. Relation betweenA5,sugraone-loop and A5;3

Motivated by the fact that the leading IR divergence of then=5-point supergravity amplitude and that of A5;3 are both of order 1/ϵ at 1 loop, one investigates whether A5,sugra1-loop can be expressed as a linear combination of A5;3 amplitudes. One uses information from (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 Section3.1, we want to find A5,sugra1-loop as a linear combination of the A5;3 amplitudes.In analogy with the 4-point function, we would like to find a relation of the type(3.47)A5,sugra1-loop(12345)=∑i∈fghijβiA5;3(i)=∑abcde∈30fixedtermsF(cde;ab)∑i∈fghijβi[sabcde;+;iA(abcde)+sabcde;-;iA(abedc)].
On the other hand, (3.48)A5,sugra1-loop(12345)=∑abcde∈30fixedtermsF(cde;ab)αabcde,
where from (3.33), (3.49)αabcde=-12scdsdesabA(cdeab)A(cdeba)
which means that we need (3.50)αabcde=∑i∈fghijβi[sabcde;+;iA(abcde)+sabcde;-;iA(abedc)]
to be satisfied, which are 30 equations for 10 unknowns (βi), so (3.50) is not guaranteed to have solutions.The 30 equations can then be rewritten, using the explicit form ofαabcde, and a new notation that will prove useful, as (3.51)-12sabsbcsdeA(abcde)A(abced)=∑fg;hijβ(fg)ϵhij[A(abcde)](1-ϵhij(de)A(abced)A(abcde)),
where ϵhij(de) is plus if both d,e belong to h,i,j, and minus otherwise.In order to see if a unique solution for theβi 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)1ϵ2A5,sugratree(12345)∑i<jsij(-sij)-ϵ=rΓϵ2∑k∈fg;lmnβk∑i<j(-sij)-ϵ∑abc≠i,jϵlmn[A(ijabc)]
which means that one requires, using the vanishing of the 1/ϵ2 IR divergence, (3.53)A5,sugratree(12345)sij=∑k∈fg;lmnβk∑abc≠i,jϵlmn[A(ijabc)].
If we denote the A5,sugratree(12345) by just M5, then the lhs is a vector column of (ij),M5sij. Also denote ∑abc≠i,jϵlmn[A(ijabc)] as N(ij),(fg), so that(3.54)N(ij),(fg)β(fg)=M5sij⟹[β(fg)]=[N(ij),(fg)]-1M5sij.
Note that the index (fg) on the matrix N has 10 values, and these values can also be identified by the lmn of ϵlmn[A(ijabc)], since it corresponds to the same 10 terms, picking out a group (fg) 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 (3.55)∑(ij)N(ij),(fg)=0,
that is, that the matrix N has rank 9 instead of 10. One then needs to work with the corresponding 9×9 reduced matrix Nred;(ij),(fg) and give the 10th coefficient β(fg) 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(fg) are unique. For that, one must calculate the rank of Nred. 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(fg) 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(fg) 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. (As (3.18) exemplifies for L=1,n=4, (3.47) may not be the only equation relating 𝒩=8 supergravity to 𝒩=4 SYM for L=1,n=5.)In principle the strategy described above can be applied to highern-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 et al. [38] are suitable for this purpose.


## 4. Geometric Interpretations of Subleading-Color Amplitudes

### 4.1. Polytope Picture

#### 4.1.1. Polytopes forMHV Leading Amplitudes

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 AdS5. In [80], it was generalized to subleading-color amplitudes. The picture for the leading MHV amplitude was obtained as follows. We start by writing the amplitude in a space dual to momenta, thus trivializing momentum conservation ∑ipi=0, by pi=xi-xi+1. Then, for example, the 1-loop dimensionless massless box function (in 4 dimensions, which is of course IR divergent) F0m(1234)=-(1/2)stI4(1)(s,t), with I4(1)(s,t) in (2.47) becomes(4.1)F0m(1,2,3,4)=i∫d4x02π2(x1-x3)2(x2-x4)2(x0-x1)2(x0-x2)2(x0-x3)2(x0-x4)2.We then constructxαα̇=xμ(σμ)αα̇ and finally map(4.2)xαα̇⟶XAB=(-12ϵαβx2ixβ̇α-ixα̇βϵα̇β̇).
Here the X’s, satisfying (4.3)X2≡12ϵABCDXABXCD=0,Xi⋅Xj=-(xi-xj)2
are coordinate patches on the quadric X·X=0 in RP5, with XAB~λXAB being their homogeneous coordinates. These X’s are considered as vertices situated at the boundary of an AdS5 and are simple bitwistors living in twistor space, that is, there exist twistors AA and BB such that XAB=A[ABB] (a twistor AA is made of (Aα,Aα̇)).Consider a box function characterized by verticesX1,X2,X3,X4. Then the following function of the Feynman parameters αi∈(0,1) with ∑αi=1, (4.4)X(α)=α1X1+α2X2+α3X3+α4X4,
is a map to RP5, but such that X(α)·X(α)≠0, and in fact they vary over a tetrahedron in RP5. After normalizing by (4.5)Y(α)=X(α)X(α)⋅X(α),
one obtains Y(α)·Y(α)=1, which means that Y(α) lies in the Euclidean AdS5. Since straight lines X(α) are mapped to geodesics in AdS5, the edges and faces of the tetrahedron in AdS5 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 AdS5. The IR-divergent lower mass functions need to be regularized, either in dimensional regularization, or using a mass regularization as in [79], modifying X2=0 to X·X=μ2(X·I), with I being a fixed point (a useful choice of I is Xi·I=1,foralli).The one-loopMHV  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 verticesxi 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.


#### 4.1.2. Polytopes forMHV  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-loopMHV to the tree level MHV amplitudes is the volume of a boundary of a 4-simplex, (4.6)A5;1MHV(12345)A5MHV(12345)≡M5MHV(12345)=∑cyclicI(x1,x2,x3,x4,(x5))≡V(x1,x2,x3,x4,x5).
Here I(x1,x2,x3,x4,(x5)) is the volume of the tetrahedron with vertices x1,x2,x3,x4, equal to F(1,2,3,4), and the missing vertex (x5) is added in brackets since the cyclicity involves all 5 points; V(x1,x2,x3,x4,x5) is the volume of the boundary of the 4-simplex in twistor space, with (y)i→(Y)i; that is, 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 (4.7)A5;3(12345)=A5(12345)[(M5(12345)-M5(41235))+(M5(43125)-M5(31245))]+A5(12435)[(M5(12435)-M5(31245))+(M5(34125)-M5(41235))]+A5(14235)[(M5(14235)-M5(31425))+(M5(34125)-M5(41235))]+A5(13245)[(M5(23145)-M5(42315))+(M5(43125)-M5(31245))]+A5(13425)[(M5(23145)-M5(31425))+(M5(43125)-M5(24315))]+A5(14325)[(M5(23145)-M5(31425))+(M5(34125)-M5(23415))].
We see that for each KK basis member we have the sums of two terms which are some simple differences of M5’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 A5(12345) in (4.7) is(4.8)(M5MHV(12345)-M5MHV(41235))+(M5MHV(43125)-M5MHV(31245))=[V(x1,x2,x3,x4,x5)-V((x4-x5+x1),x1,x2,x3,x4)]+[V(x4,(x1+x4-x5),x1,(x1-x3+x4),(x2-x3+x4))-V(x1,(x1-x3+x4),(x2-x3+x4),x4,x5)].
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 then=6 case [80] as well, we can understand the general pattern for An;3. The general formula is (4.9)An;3MHV(n-1,n,1,2,….,n-2)=∑{σ}i∈OP({α},{βT})∣jmax⁡AnMHV(1,{σ}i,n)×∑n-1∈{α},{β};jmax⁡∈{α},{β}(-)nβMnMHV({β},1,{α},n)
with MnMHV being the volume of a closed polytope and pairs of opposite sign Mn’s adding up to another closed polytope. It is obtained as follows. In the above, just from (1.5), we have Mn({β},1,{α},n), where n-1 is either in {α} or in {β}, and otherwise {α} contains 2,3,…,k and {β} contains k+1,…,n-2.For the tree amplitude prefactors, when using the KK relations (1.2), from {α} and {β} we form the permutation {σ}i which contains {α} and {βT}, keeping the ordering, that is, in the KK basis amplitude, we have A(1,{σ},n). Here if we extract the n-1, then {α}=2,…,jmax is ordered; that is, it goes from left to right in the permutation, and then {β}=jmax+1,…,n-2 is transposed and still ordered; that is, it goes from right to left. The same jmax (extracted from the resulting KK basis member) is obtained from either k or k+1. That means that there are exactly 4 terms corresponding to the same KK basis member, corresponding to both jmax and n-1 belonging to either {α} or {β}.The sign of the terms is obtained from the sign in the KK relations (1.2), that is, (-1)nβ, where here {β} refers to the individual Mn({β},1,{α},n) term. The fact that jmax belongs to either {α} or {β} means that in Mn({β},1,{α},n) we have jmax  either at the end of {α}, or at the beginning of {β}; that is, we have a flip of jmaxn versus njmax in between terms with different signs, hence a different nβ (with or without jmax). The exception is when actually (n-1) is at the end of {α}, and not jmax, in which case the same flip is now (n-1)n versus 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 samen-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 toAn;j also, obtaining(4.10)An;jMHV(n-j+2,…,n,1,…,n-j+1)=∑{σ}i∈OP({α},{βT})∣jmax⁡∈{1,…,n-j+1},lmax⁡∈{n-j+2,…,n-1}AnMHV(1,{σ}i,n)×∑{n-1,…,n-j+2}∈{α},{β};jmax⁡∈{α},{β}(-)nβ+j-1MnMHV({β},1,{α},n),
where again we have pairs of MnMHV’s of different signs and with n-1 common vertices adding up to give other closed polytopes (of n+1 vertices).The new features with respect to theAn;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 jmax 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 {βT}, which means that n-1,…,lmax+1 is cyclic (i.e., towards the right), and n-j+2,…,lmax is also cyclic (i.e., we change the direction of the cyclicity at lmax).The number of terms multiplying a KK basis member is even, corresponding to havingjmax 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 jmax 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+nβ. 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 Mn come in alternating pairs. Each of these Mn has leading IR singularity 1/ϵ2+𝒪(1/ϵ) and therefore at one-loop An,j has only a 1/ϵ IR singularity.


#### 4.1.3. Polytope Picture for the 6-Point LeadingNMHV  Amplitude

The leading (planar) gluon amplitudesA6;1NMHV for the split-helicity configuration are [81] (4.11)A6;1NMHV(1+2+3+4-5-6-)=cΓ2(B1W6(1)+B2W6(2)+B3W6(3)),
where W6(i) are cyclic permutations of W6(1), and W6(i+3)≡W6(i), given in terms of box functions by(4.12)W6(i)=F6:i1m+F6:i+31m+F6:2;i+12mh+F6:2;i+42mh,
and the F’s are dimensionless boxes. We can write polytope interpretations for the W6(i)’s based on the fact that the F’s have polytope interpretation. Denoting for instance by (4561(23)) what was previously called I(x4,x5,x6,x1(x2,x3)), we write(4.13)W6(1)=(4561(23))+(1234(56))+(12(3)4(5)6)+(45(6)1(2)3)≡A1+A3+A2+A4,W6(2)=(5612(34))+(2345(61))+(23(4)5(6)1)+(56(1)2(3)4)≡A5+A7+A6+A8,W6(3)=(6123(45))+(3456(12))+(34(5)6(1)2)+(61(2)3(4)5)≡A9+A11+A10+A12,
where the A’s are tetrahedra defined in the order they appear in the W6(i) above, while for example for the 6-point MHV amplitude, we have(4.14)A6;1MHV(123456)=A(123456)[(12(3)45(6))+(23(4)56(1))+(34(5)61(2))+(1234(56))+(2345(61))+(3456(12))+(4561(23))+(5612(34))+(6123(45))]=A(123456)[A13+A14+A15+A3+A7+A11+A1+A5+A9],
where again the various A’s are defined in the order they appear. However, because of the spin coefficients of W6(i), we cannot find a simple polytope interpretation for the subleading-color amplitudes.


### 4.2. Momentum Twistor Representation

#### 4.2.1. Momentum Twistor Representation for LeadingNkMHV  Amplitudes

Instead, we can use a momentum twistor [17, 20] representation for the NkMHV superamplitudes in order to find a simple formula for the subleading NkMHV amplitudes.TheMHV tree-level color-ordered superamplitudes are given by the Nair formula [82], a supersymmetric generalization of the Parke-Taylor formula [78, 83], (4.15)An,2(12…n)=δ4(∑i=1nλiλ̃i)δ8(∑i=1nλiη̃i)〈12〉〈23〉⋯〈n-1,n〉〈n,1〉,
where as usual 〈ij〉≡ϵαβλα(i)λβ(j), η̃ is a spinor with an index I=1,…,4 for supersymmetries suppressed, and the 2 in An,2 refers to R-charge, since the NkMHV 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.In terms of supermomentum twistorsZi, the leading singularity of the (color-ordered, planar, i.e., leading) NkMHV  superamplitude is [19, 23] (4.16)Ln,m=δ4(∑λλ̃)δ8(∑λη̃)〈12〉〈23〉⋯〈n1〉∫dnkDVol(Gl(2))∏μ=1kδ4|4(∑i=1nDμiZi)(12…k)(23…k+1)⋯(n12k-1)=Ln,2×Rn,k,
where k=m-2. The prefactor ℒn,2 is the tree MHV amplitude (4.15), and the integral ℛn,k=ℛn,m-2, is Yangian invariant. This object is dual conformal covariant, only ℛn,k being dual conformal invariant, and the tree amplitude is covariant.The one-loop amplitudes of𝒩=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.


#### 4.2.2. SubleadingNkMHV Amplitudes in Momentum Twistor Space

The planar (leading) color-orderedNkMHV amplitude is a sum of permutations of boxes with coefficients equal to the leading singularities, (4.17)An;1(1…n)=∑σLn,k(σ)In;4(σ)=∑AnMHV(σ)Rn;k(σ)In;4(σ),
where In;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
(4.18)A6;1(1…6)=A6MHV(1…6)∑λ=cyclicR6;k(λ)∑σ/λI6;4(σ).
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 still have 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 NkMHV amplitudes(4.19)An;1(1…n)=AnMHV(1…n)∑groupsofdiagrams∑λ=cyclicRn;k(λ)∑σ/λIn;4(σ)≡AnMHV(1…n)Mn;k(1…n)
which implicitly defines Mn,k.We now finally note that we have the same formula forAn;1(1…n) in terms of AnMHV and Mn;k from the previous section on polytopes, so we can apply the same calculations we used to obtain the MHVAn;j in terms of An;1 in Section 2. We just have to change the definition of Mn;k as in (4.19) and thus also drop the polytope interpretation of Mn,k. But otherwise the same (4.10) found in the MHV case holds in the general NkMHV case as well, as can be seen from (4.9).


#### 4.2.3. Application to the 6-PointNMHV 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] (4.20)A6;1(1)NMHV(123456)=a2A6(0)MHV(123456)×[(R413+R146)W6(1)+(R524+R251)W6(2)+(R635+R362)W6(3)]≡A6(0)MHV(123456)M6(1)NMHV(123456).
From the Rn;k terms in (4.18), one gets the sum of basic dual conformal invariant R-invariants Rj,j+3,j+5 above. Here the Rj,j+3,j+5 are given by (4.21)Rrst=-〈s-1s〉〈t-1t〉δ(4)(Ξrst)xst2〈r|xrtxts|s-1〉〈r|xrtxts|s〉〈r|xrsxst|t-1〉〈r|xrsxst|t〉,Ξrst=∑tr-1ηi〈i|xtsxsr|r〉+∑rs-1ηi〈i|xstxtr|r〉,xst=xs-xt=∑i=st-1pi.As explained before, we can then perform the same combinatorics that led us to (4.9), just that now we use M6(1)NMHV(123456) instead of the M6(123456).


## 4.1. Polytope Picture

### 4.1.1. Polytopes forMHV Leading Amplitudes

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 AdS5. In [80], it was generalized to subleading-color amplitudes. The picture for the leading MHV amplitude was obtained as follows. We start by writing the amplitude in a space dual to momenta, thus trivializing momentum conservation ∑ipi=0, by pi=xi-xi+1. Then, for example, the 1-loop dimensionless massless box function (in 4 dimensions, which is of course IR divergent) F0m(1234)=-(1/2)stI4(1)(s,t), with I4(1)(s,t) in (2.47) becomes(4.1)F0m(1,2,3,4)=i∫d4x02π2(x1-x3)2(x2-x4)2(x0-x1)2(x0-x2)2(x0-x3)2(x0-x4)2.We then constructxαα̇=xμ(σμ)αα̇ and finally map(4.2)xαα̇⟶XAB=(-12ϵαβx2ixβ̇α-ixα̇βϵα̇β̇).
Here the X’s, satisfying (4.3)X2≡12ϵABCDXABXCD=0,Xi⋅Xj=-(xi-xj)2
are coordinate patches on the quadric X·X=0 in RP5, with XAB~λXAB being their homogeneous coordinates. These X’s are considered as vertices situated at the boundary of an AdS5 and are simple bitwistors living in twistor space, that is, there exist twistors AA and BB such that XAB=A[ABB] (a twistor AA is made of (Aα,Aα̇)).Consider a box function characterized by verticesX1,X2,X3,X4. Then the following function of the Feynman parameters αi∈(0,1) with ∑αi=1, (4.4)X(α)=α1X1+α2X2+α3X3+α4X4,
is a map to RP5, but such that X(α)·X(α)≠0, and in fact they vary over a tetrahedron in RP5. After normalizing by (4.5)Y(α)=X(α)X(α)⋅X(α),
one obtains Y(α)·Y(α)=1, which means that Y(α) lies in the Euclidean AdS5. Since straight lines X(α) are mapped to geodesics in AdS5, the edges and faces of the tetrahedron in AdS5 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 AdS5. The IR-divergent lower mass functions need to be regularized, either in dimensional regularization, or using a mass regularization as in [79], modifying X2=0 to X·X=μ2(X·I), with I being a fixed point (a useful choice of I is Xi·I=1,foralli).The one-loopMHV  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 verticesxi 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.


### 4.1.2. Polytopes forMHV  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-loopMHV to the tree level MHV amplitudes is the volume of a boundary of a 4-simplex, (4.6)A5;1MHV(12345)A5MHV(12345)≡M5MHV(12345)=∑cyclicI(x1,x2,x3,x4,(x5))≡V(x1,x2,x3,x4,x5).
Here I(x1,x2,x3,x4,(x5)) is the volume of the tetrahedron with vertices x1,x2,x3,x4, equal to F(1,2,3,4), and the missing vertex (x5) is added in brackets since the cyclicity involves all 5 points; V(x1,x2,x3,x4,x5) is the volume of the boundary of the 4-simplex in twistor space, with (y)i→(Y)i; that is, 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 (4.7)A5;3(12345)=A5(12345)[(M5(12345)-M5(41235))+(M5(43125)-M5(31245))]+A5(12435)[(M5(12435)-M5(31245))+(M5(34125)-M5(41235))]+A5(14235)[(M5(14235)-M5(31425))+(M5(34125)-M5(41235))]+A5(13245)[(M5(23145)-M5(42315))+(M5(43125)-M5(31245))]+A5(13425)[(M5(23145)-M5(31425))+(M5(43125)-M5(24315))]+A5(14325)[(M5(23145)-M5(31425))+(M5(34125)-M5(23415))].
We see that for each KK basis member we have the sums of two terms which are some simple differences of M5’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 A5(12345) in (4.7) is(4.8)(M5MHV(12345)-M5MHV(41235))+(M5MHV(43125)-M5MHV(31245))=[V(x1,x2,x3,x4,x5)-V((x4-x5+x1),x1,x2,x3,x4)]+[V(x4,(x1+x4-x5),x1,(x1-x3+x4),(x2-x3+x4))-V(x1,(x1-x3+x4),(x2-x3+x4),x4,x5)].
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 then=6 case [80] as well, we can understand the general pattern for An;3. The general formula is (4.9)An;3MHV(n-1,n,1,2,….,n-2)=∑{σ}i∈OP({α},{βT})∣jmax⁡AnMHV(1,{σ}i,n)×∑n-1∈{α},{β};jmax⁡∈{α},{β}(-)nβMnMHV({β},1,{α},n)
with MnMHV being the volume of a closed polytope and pairs of opposite sign Mn’s adding up to another closed polytope. It is obtained as follows. In the above, just from (1.5), we have Mn({β},1,{α},n), where n-1 is either in {α} or in {β}, and otherwise {α} contains 2,3,…,k and {β} contains k+1,…,n-2.For the tree amplitude prefactors, when using the KK relations (1.2), from {α} and {β} we form the permutation {σ}i which contains {α} and {βT}, keeping the ordering, that is, in the KK basis amplitude, we have A(1,{σ},n). Here if we extract the n-1, then {α}=2,…,jmax is ordered; that is, it goes from left to right in the permutation, and then {β}=jmax+1,…,n-2 is transposed and still ordered; that is, it goes from right to left. The same jmax (extracted from the resulting KK basis member) is obtained from either k or k+1. That means that there are exactly 4 terms corresponding to the same KK basis member, corresponding to both jmax and n-1 belonging to either {α} or {β}.The sign of the terms is obtained from the sign in the KK relations (1.2), that is, (-1)nβ, where here {β} refers to the individual Mn({β},1,{α},n) term. The fact that jmax belongs to either {α} or {β} means that in Mn({β},1,{α},n) we have jmax  either at the end of {α}, or at the beginning of {β}; that is, we have a flip of jmaxn versus njmax in between terms with different signs, hence a different nβ (with or without jmax). The exception is when actually (n-1) is at the end of {α}, and not jmax, in which case the same flip is now (n-1)n versus 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 samen-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 toAn;j also, obtaining(4.10)An;jMHV(n-j+2,…,n,1,…,n-j+1)=∑{σ}i∈OP({α},{βT})∣jmax⁡∈{1,…,n-j+1},lmax⁡∈{n-j+2,…,n-1}AnMHV(1,{σ}i,n)×∑{n-1,…,n-j+2}∈{α},{β};jmax⁡∈{α},{β}(-)nβ+j-1MnMHV({β},1,{α},n),
where again we have pairs of MnMHV’s of different signs and with n-1 common vertices adding up to give other closed polytopes (of n+1 vertices).The new features with respect to theAn;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 jmax 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 {βT}, which means that n-1,…,lmax+1 is cyclic (i.e., towards the right), and n-j+2,…,lmax is also cyclic (i.e., we change the direction of the cyclicity at lmax).The number of terms multiplying a KK basis member is even, corresponding to havingjmax 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 jmax 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+nβ. 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 Mn come in alternating pairs. Each of these Mn has leading IR singularity 1/ϵ2+𝒪(1/ϵ) and therefore at one-loop An,j has only a 1/ϵ IR singularity.


### 4.1.3. Polytope Picture for the 6-Point LeadingNMHV  Amplitude

The leading (planar) gluon amplitudesA6;1NMHV for the split-helicity configuration are [81] (4.11)A6;1NMHV(1+2+3+4-5-6-)=cΓ2(B1W6(1)+B2W6(2)+B3W6(3)),
where W6(i) are cyclic permutations of W6(1), and W6(i+3)≡W6(i), given in terms of box functions by(4.12)W6(i)=F6:i1m+F6:i+31m+F6:2;i+12mh+F6:2;i+42mh,
and the F’s are dimensionless boxes. We can write polytope interpretations for the W6(i)’s based on the fact that the F’s have polytope interpretation. Denoting for instance by (4561(23)) what was previously called I(x4,x5,x6,x1(x2,x3)), we write(4.13)W6(1)=(4561(23))+(1234(56))+(12(3)4(5)6)+(45(6)1(2)3)≡A1+A3+A2+A4,W6(2)=(5612(34))+(2345(61))+(23(4)5(6)1)+(56(1)2(3)4)≡A5+A7+A6+A8,W6(3)=(6123(45))+(3456(12))+(34(5)6(1)2)+(61(2)3(4)5)≡A9+A11+A10+A12,
where the A’s are tetrahedra defined in the order they appear in the W6(i) above, while for example for the 6-point MHV amplitude, we have(4.14)A6;1MHV(123456)=A(123456)[(12(3)45(6))+(23(4)56(1))+(34(5)61(2))+(1234(56))+(2345(61))+(3456(12))+(4561(23))+(5612(34))+(6123(45))]=A(123456)[A13+A14+A15+A3+A7+A11+A1+A5+A9],
where again the various A’s are defined in the order they appear. However, because of the spin coefficients of W6(i), we cannot find a simple polytope interpretation for the subleading-color amplitudes.


## 4.1.1. Polytopes forMHV Leading Amplitudes

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 AdS5. In [80], it was generalized to subleading-color amplitudes. The picture for the leading MHV amplitude was obtained as follows. We start by writing the amplitude in a space dual to momenta, thus trivializing momentum conservation ∑ipi=0, by pi=xi-xi+1. Then, for example, the 1-loop dimensionless massless box function (in 4 dimensions, which is of course IR divergent) F0m(1234)=-(1/2)stI4(1)(s,t), with I4(1)(s,t) in (2.47) becomes(4.1)F0m(1,2,3,4)=i∫d4x02π2(x1-x3)2(x2-x4)2(x0-x1)2(x0-x2)2(x0-x3)2(x0-x4)2.We then constructxαα̇=xμ(σμ)αα̇ and finally map(4.2)xαα̇⟶XAB=(-12ϵαβx2ixβ̇α-ixα̇βϵα̇β̇).
Here the X’s, satisfying (4.3)X2≡12ϵABCDXABXCD=0,Xi⋅Xj=-(xi-xj)2
are coordinate patches on the quadric X·X=0 in RP5, with XAB~λXAB being their homogeneous coordinates. These X’s are considered as vertices situated at the boundary of an AdS5 and are simple bitwistors living in twistor space, that is, there exist twistors AA and BB such that XAB=A[ABB] (a twistor AA is made of (Aα,Aα̇)).Consider a box function characterized by verticesX1,X2,X3,X4. Then the following function of the Feynman parameters αi∈(0,1) with ∑αi=1, (4.4)X(α)=α1X1+α2X2+α3X3+α4X4,
is a map to RP5, but such that X(α)·X(α)≠0, and in fact they vary over a tetrahedron in RP5. After normalizing by (4.5)Y(α)=X(α)X(α)⋅X(α),
one obtains Y(α)·Y(α)=1, which means that Y(α) lies in the Euclidean AdS5. Since straight lines X(α) are mapped to geodesics in AdS5, the edges and faces of the tetrahedron in AdS5 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 AdS5. The IR-divergent lower mass functions need to be regularized, either in dimensional regularization, or using a mass regularization as in [79], modifying X2=0 to X·X=μ2(X·I), with I being a fixed point (a useful choice of I is Xi·I=1,foralli).The one-loopMHV  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 verticesxi 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.


## 4.1.2. Polytopes forMHV  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-loopMHV to the tree level MHV amplitudes is the volume of a boundary of a 4-simplex, (4.6)A5;1MHV(12345)A5MHV(12345)≡M5MHV(12345)=∑cyclicI(x1,x2,x3,x4,(x5))≡V(x1,x2,x3,x4,x5).
Here I(x1,x2,x3,x4,(x5)) is the volume of the tetrahedron with vertices x1,x2,x3,x4, equal to F(1,2,3,4), and the missing vertex (x5) is added in brackets since the cyclicity involves all 5 points; V(x1,x2,x3,x4,x5) is the volume of the boundary of the 4-simplex in twistor space, with (y)i→(Y)i; that is, 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 (4.7)A5;3(12345)=A5(12345)[(M5(12345)-M5(41235))+(M5(43125)-M5(31245))]+A5(12435)[(M5(12435)-M5(31245))+(M5(34125)-M5(41235))]+A5(14235)[(M5(14235)-M5(31425))+(M5(34125)-M5(41235))]+A5(13245)[(M5(23145)-M5(42315))+(M5(43125)-M5(31245))]+A5(13425)[(M5(23145)-M5(31425))+(M5(43125)-M5(24315))]+A5(14325)[(M5(23145)-M5(31425))+(M5(34125)-M5(23415))].
We see that for each KK basis member we have the sums of two terms which are some simple differences of M5’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 A5(12345) in (4.7) is(4.8)(M5MHV(12345)-M5MHV(41235))+(M5MHV(43125)-M5MHV(31245))=[V(x1,x2,x3,x4,x5)-V((x4-x5+x1),x1,x2,x3,x4)]+[V(x4,(x1+x4-x5),x1,(x1-x3+x4),(x2-x3+x4))-V(x1,(x1-x3+x4),(x2-x3+x4),x4,x5)].
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 then=6 case [80] as well, we can understand the general pattern for An;3. The general formula is (4.9)An;3MHV(n-1,n,1,2,….,n-2)=∑{σ}i∈OP({α},{βT})∣jmax⁡AnMHV(1,{σ}i,n)×∑n-1∈{α},{β};jmax⁡∈{α},{β}(-)nβMnMHV({β},1,{α},n)
with MnMHV being the volume of a closed polytope and pairs of opposite sign Mn’s adding up to another closed polytope. It is obtained as follows. In the above, just from (1.5), we have Mn({β},1,{α},n), where n-1 is either in {α} or in {β}, and otherwise {α} contains 2,3,…,k and {β} contains k+1,…,n-2.For the tree amplitude prefactors, when using the KK relations (1.2), from {α} and {β} we form the permutation {σ}i which contains {α} and {βT}, keeping the ordering, that is, in the KK basis amplitude, we have A(1,{σ},n). Here if we extract the n-1, then {α}=2,…,jmax is ordered; that is, it goes from left to right in the permutation, and then {β}=jmax+1,…,n-2 is transposed and still ordered; that is, it goes from right to left. The same jmax (extracted from the resulting KK basis member) is obtained from either k or k+1. That means that there are exactly 4 terms corresponding to the same KK basis member, corresponding to both jmax and n-1 belonging to either {α} or {β}.The sign of the terms is obtained from the sign in the KK relations (1.2), that is, (-1)nβ, where here {β} refers to the individual Mn({β},1,{α},n) term. The fact that jmax belongs to either {α} or {β} means that in Mn({β},1,{α},n) we have jmax  either at the end of {α}, or at the beginning of {β}; that is, we have a flip of jmaxn versus njmax in between terms with different signs, hence a different nβ (with or without jmax). The exception is when actually (n-1) is at the end of {α}, and not jmax, in which case the same flip is now (n-1)n versus 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 samen-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 toAn;j also, obtaining(4.10)An;jMHV(n-j+2,…,n,1,…,n-j+1)=∑{σ}i∈OP({α},{βT})∣jmax⁡∈{1,…,n-j+1},lmax⁡∈{n-j+2,…,n-1}AnMHV(1,{σ}i,n)×∑{n-1,…,n-j+2}∈{α},{β};jmax⁡∈{α},{β}(-)nβ+j-1MnMHV({β},1,{α},n),
where again we have pairs of MnMHV’s of different signs and with n-1 common vertices adding up to give other closed polytopes (of n+1 vertices).The new features with respect to theAn;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 jmax 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 {βT}, which means that n-1,…,lmax+1 is cyclic (i.e., towards the right), and n-j+2,…,lmax is also cyclic (i.e., we change the direction of the cyclicity at lmax).The number of terms multiplying a KK basis member is even, corresponding to havingjmax 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 jmax 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+nβ. 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 Mn come in alternating pairs. Each of these Mn has leading IR singularity 1/ϵ2+𝒪(1/ϵ) and therefore at one-loop An,j has only a 1/ϵ IR singularity.


## 4.1.3. Polytope Picture for the 6-Point LeadingNMHV  Amplitude

The leading (planar) gluon amplitudesA6;1NMHV for the split-helicity configuration are [81] (4.11)A6;1NMHV(1+2+3+4-5-6-)=cΓ2(B1W6(1)+B2W6(2)+B3W6(3)),
where W6(i) are cyclic permutations of W6(1), and W6(i+3)≡W6(i), given in terms of box functions by(4.12)W6(i)=F6:i1m+F6:i+31m+F6:2;i+12mh+F6:2;i+42mh,
and the F’s are dimensionless boxes. We can write polytope interpretations for the W6(i)’s based on the fact that the F’s have polytope interpretation. Denoting for instance by (4561(23)) what was previously called I(x4,x5,x6,x1(x2,x3)), we write(4.13)W6(1)=(4561(23))+(1234(56))+(12(3)4(5)6)+(45(6)1(2)3)≡A1+A3+A2+A4,W6(2)=(5612(34))+(2345(61))+(23(4)5(6)1)+(56(1)2(3)4)≡A5+A7+A6+A8,W6(3)=(6123(45))+(3456(12))+(34(5)6(1)2)+(61(2)3(4)5)≡A9+A11+A10+A12,
where the A’s are tetrahedra defined in the order they appear in the W6(i) above, while for example for the 6-point MHV amplitude, we have(4.14)A6;1MHV(123456)=A(123456)[(12(3)45(6))+(23(4)56(1))+(34(5)61(2))+(1234(56))+(2345(61))+(3456(12))+(4561(23))+(5612(34))+(6123(45))]=A(123456)[A13+A14+A15+A3+A7+A11+A1+A5+A9],
where again the various A’s are defined in the order they appear. However, because of the spin coefficients of W6(i), we cannot find a simple polytope interpretation for the subleading-color amplitudes.


## 4.2. Momentum Twistor Representation

### 4.2.1. Momentum Twistor Representation for LeadingNkMHV  Amplitudes

Instead, we can use a momentum twistor [17, 20] representation for the NkMHV superamplitudes in order to find a simple formula for the subleading NkMHV amplitudes.TheMHV tree-level color-ordered superamplitudes are given by the Nair formula [82], a supersymmetric generalization of the Parke-Taylor formula [78, 83], (4.15)An,2(12…n)=δ4(∑i=1nλiλ̃i)δ8(∑i=1nλiη̃i)〈12〉〈23〉⋯〈n-1,n〉〈n,1〉,
where as usual 〈ij〉≡ϵαβλα(i)λβ(j), η̃ is a spinor with an index I=1,…,4 for supersymmetries suppressed, and the 2 in An,2 refers to R-charge, since the NkMHV 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.In terms of supermomentum twistorsZi, the leading singularity of the (color-ordered, planar, i.e., leading) NkMHV  superamplitude is [19, 23] (4.16)Ln,m=δ4(∑λλ̃)δ8(∑λη̃)〈12〉〈23〉⋯〈n1〉∫dnkDVol(Gl(2))∏μ=1kδ4|4(∑i=1nDμiZi)(12…k)(23…k+1)⋯(n12k-1)=Ln,2×Rn,k,
where k=m-2. The prefactor ℒn,2 is the tree MHV amplitude (4.15), and the integral ℛn,k=ℛn,m-2, is Yangian invariant. This object is dual conformal covariant, only ℛn,k being dual conformal invariant, and the tree amplitude is covariant.The one-loop amplitudes of𝒩=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.


### 4.2.2. SubleadingNkMHV Amplitudes in Momentum Twistor Space

The planar (leading) color-orderedNkMHV amplitude is a sum of permutations of boxes with coefficients equal to the leading singularities, (4.17)An;1(1…n)=∑σLn,k(σ)In;4(σ)=∑AnMHV(σ)Rn;k(σ)In;4(σ),
where In;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
(4.18)A6;1(1…6)=A6MHV(1…6)∑λ=cyclicR6;k(λ)∑σ/λI6;4(σ).
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 still have 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 NkMHV amplitudes(4.19)An;1(1…n)=AnMHV(1…n)∑groupsofdiagrams∑λ=cyclicRn;k(λ)∑σ/λIn;4(σ)≡AnMHV(1…n)Mn;k(1…n)
which implicitly defines Mn,k.We now finally note that we have the same formula forAn;1(1…n) in terms of AnMHV and Mn;k from the previous section on polytopes, so we can apply the same calculations we used to obtain the MHVAn;j in terms of An;1 in Section 2. We just have to change the definition of Mn;k as in (4.19) and thus also drop the polytope interpretation of Mn,k. But otherwise the same (4.10) found in the MHV case holds in the general NkMHV case as well, as can be seen from (4.9).


### 4.2.3. Application to the 6-PointNMHV 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] (4.20)A6;1(1)NMHV(123456)=a2A6(0)MHV(123456)×[(R413+R146)W6(1)+(R524+R251)W6(2)+(R635+R362)W6(3)]≡A6(0)MHV(123456)M6(1)NMHV(123456).
From the Rn;k terms in (4.18), one gets the sum of basic dual conformal invariant R-invariants Rj,j+3,j+5 above. Here the Rj,j+3,j+5 are given by (4.21)Rrst=-〈s-1s〉〈t-1t〉δ(4)(Ξrst)xst2〈r|xrtxts|s-1〉〈r|xrtxts|s〉〈r|xrsxst|t-1〉〈r|xrsxst|t〉,Ξrst=∑tr-1ηi〈i|xtsxsr|r〉+∑rs-1ηi〈i|xstxtr|r〉,xst=xs-xt=∑i=st-1pi.As explained before, we can then perform the same combinatorics that led us to (4.9), just that now we use M6(1)NMHV(123456) instead of the M6(123456).


## 4.2.1. Momentum Twistor Representation for LeadingNkMHV  Amplitudes

Instead, we can use a momentum twistor [17, 20] representation for the NkMHV superamplitudes in order to find a simple formula for the subleading NkMHV amplitudes.TheMHV tree-level color-ordered superamplitudes are given by the Nair formula [82], a supersymmetric generalization of the Parke-Taylor formula [78, 83], (4.15)An,2(12…n)=δ4(∑i=1nλiλ̃i)δ8(∑i=1nλiη̃i)〈12〉〈23〉⋯〈n-1,n〉〈n,1〉,
where as usual 〈ij〉≡ϵαβλα(i)λβ(j), η̃ is a spinor with an index I=1,…,4 for supersymmetries suppressed, and the 2 in An,2 refers to R-charge, since the NkMHV 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.In terms of supermomentum twistorsZi, the leading singularity of the (color-ordered, planar, i.e., leading) NkMHV  superamplitude is [19, 23] (4.16)Ln,m=δ4(∑λλ̃)δ8(∑λη̃)〈12〉〈23〉⋯〈n1〉∫dnkDVol(Gl(2))∏μ=1kδ4|4(∑i=1nDμiZi)(12…k)(23…k+1)⋯(n12k-1)=Ln,2×Rn,k,
where k=m-2. The prefactor ℒn,2 is the tree MHV amplitude (4.15), and the integral ℛn,k=ℛn,m-2, is Yangian invariant. This object is dual conformal covariant, only ℛn,k being dual conformal invariant, and the tree amplitude is covariant.The one-loop amplitudes of𝒩=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.


## 4.2.2. SubleadingNkMHV Amplitudes in Momentum Twistor Space

The planar (leading) color-orderedNkMHV amplitude is a sum of permutations of boxes with coefficients equal to the leading singularities, (4.17)An;1(1…n)=∑σLn,k(σ)In;4(σ)=∑AnMHV(σ)Rn;k(σ)In;4(σ),
where In;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
(4.18)A6;1(1…6)=A6MHV(1…6)∑λ=cyclicR6;k(λ)∑σ/λI6;4(σ).
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 still have 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 NkMHV amplitudes(4.19)An;1(1…n)=AnMHV(1…n)∑groupsofdiagrams∑λ=cyclicRn;k(λ)∑σ/λIn;4(σ)≡AnMHV(1…n)Mn;k(1…n)
which implicitly defines Mn,k.We now finally note that we have the same formula forAn;1(1…n) in terms of AnMHV and Mn;k from the previous section on polytopes, so we can apply the same calculations we used to obtain the MHVAn;j in terms of An;1 in Section 2. We just have to change the definition of Mn;k as in (4.19) and thus also drop the polytope interpretation of Mn,k. But otherwise the same (4.10) found in the MHV case holds in the general NkMHV case as well, as can be seen from (4.9).


## 4.2.3. Application to the 6-PointNMHV 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] (4.20)A6;1(1)NMHV(123456)=a2A6(0)MHV(123456)×[(R413+R146)W6(1)+(R524+R251)W6(2)+(R635+R362)W6(3)]≡A6(0)MHV(123456)M6(1)NMHV(123456).
From the Rn;k terms in (4.18), one gets the sum of basic dual conformal invariant R-invariants Rj,j+3,j+5 above. Here the Rj,j+3,j+5 are given by (4.21)Rrst=-〈s-1s〉〈t-1t〉δ(4)(Ξrst)xst2〈r|xrtxts|s-1〉〈r|xrtxts|s〉〈r|xrsxst|t-1〉〈r|xrsxst|t〉,Ξrst=∑tr-1ηi〈i|xtsxsr|r〉+∑rs-1ηi〈i|xstxtr|r〉,xst=xs-xt=∑i=st-1pi.As explained before, we can then perform the same combinatorics that led us to (4.9), just that now we use M6(1)NMHV(123456) instead of the M6(123456).


## 5. Summary

We have reviewed a number of features of subleading-color amplitudes of𝒩=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 are to be understood, as nonplanar graphs appear on an equal footing in 𝒩=8 supergravity.After presenting a detailed description of the IR divergences of𝒩=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 NkMHV 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 of𝒩=4 SYM to 𝒩=8 supergravity to extend (3.20) and (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 paper should provide many opportunities for future work.



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*Source: 190587-2011-12-10.xml*