# Perturbative Gravity and Gauge Theory Relations: A Review

Authors: Thomas Søndergaard
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

This paper is dedicated to the amazing Kawai-Lewellen-Tye relations, connecting perturbative gravity and gauge theories at tree level. The main focus is onn-point derivations and general properties both from a string theory and pure field theory point of view. In particular, the field theory part is based on some very recent developments.


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## Body

## 1. Introduction

In 1985, Kawai, Lewellen, and Tye (KLT) derived an amazing relation between gravity and gauge theory tree-level amplitudes [1–19]. This was done by factorizing a closed string into a sum of products between two open strings. As such it was a relation satisfied to all orders in α′, even when taking the field theory limit α′→0 [20, 21]. In particular, the validity in this limit has been a major puzzle for many years. At the Lagrangian level any connection between Einstein gravity and Yang-Mills theory seems highly unlikely. Expanding the Einstein-Hilbert Lagrangian perturbatively leads to an infinite series of more and more complicated interaction terms, while Yang-Mills only involve three- and four-point interactions. Nevertheless, we will in this paper see how the KLT relations can be understood from a field theoretical point of view.The new light that has recently been shed on these relations is only a small part of a remarkable progress that is currently happening in our understanding of scattering amplitudes, see for example, [22–24] for reviews on some of these developments. Indeed, as we will see, our new knowledge about KLT is to a large extent directly built upon several of the great discoveries that have been made within recent years.Inspired by Witten’s famous 2004 paper [25], Britto, Cachazo and Feng (BCF) uncovered an on-shell recursion relation for tree amplitudes from which one could construct higher-point amplitudes from lower-point amplitudes [26]. Together with Witten they soon after proved these BCFW recursion relations [27], which will play an important role for us. By now such recursion relations have also been extended to string theory [28, 29] and even to the integrand of multiloop amplitudes in planar 𝒩=4 SYM [30], see also [31].Another interesting structure, that is going to be essential in this paper, appeared in 2008, when Bern, Carrasco, and Johansson (BCJ) found a curiouscolor-kinematic duality for gauge theory amplitudes [32]. By means of this duality they were able to write down new relations, reducing the number of independent color-ordered gauge theory amplitudes from (n-2)!, as given by the Kleiss-Kuijf relations [33, 34], down to (n-3)!. At this stage these BCJ relations, and their supersymmetric extension [35], had not been proven for general n-point amplitudes. Interestingly, the first proof came from string theory where Bjerrum-Bohr et al. [36] and Stieberger [37] used monodromy to derive the (n-3)! basis for color-ordered open string amplitudes. In the field theory limit these monodromy relations reduce exactly to Kleiss-Kuijf and BCJ relations. Nearly one year passed before the BCJ relations were also proven from pure field theory [38], see also [39, 40], and most recently investigations of similar structures appearing at loop level have been made [41, 42].In mid 2010 all the tools needed for the first purely field theoretical proof of the KLT relations were at hand [15, 16]. In the process several new ways of writing the relations were found (and needed) along with the introduction of a momentum kernel which nicely captured all of these forms, including the one conjectured in [12, 14]. This momentum kernel turned out to have a lot of nice properties which was investigated in [18] along with its extension to string theory, and made it much easier to handle and express KLT relations for general n points. It was also realized that KLT relations could lead to pure gauge theory identities when nonmatching helicities were chosen [43, 44], see [17, 45, 46] on the extension to supersymmetric theories and its connection to SU(8)-violating gravity amplitudes. In addition, it became clear how closely related the BCJ and KLT relations actually are, although they appear very different.The paper is structured as follows, in Section2 we show how to derive the n-point KLT relation from string theory. This will be done in quite some detail since the original paper is a bit brief concerning some of the more technical issues. We will also arrive at a more explicit n-point form that nicely incorporates the freedom one has in different ways of expressing the relation. In Section 3 we look at its field theory limit and properties in this regime. We also review some field theoretical tools that will be needed later on, and comment some more on the close connection between BCJ and KLT relations mentioned above. In Section 4 we utilize all of the structures presented in Section 3 in order to give a field theoretical proof of the KLT relations. In Section 5 we review the color-kinematic duality approach and its connection to gravity, both at tree and loop level. Finally, in Section 6 we present our conclusions.


## 2. Factorization of Closed String Amplitudes

We begin the review with a derivation of the KLT relations from string theory. We take the same path as in the original paper [1, 18], factorizing the closed string into the product of open strings—glued together by appropriate phase factors.Then-point tree-level closed string amplitude is given by(2.1)Mn=(i2πα′)n-3∫∏i=2n-2d2zi|zi|2α′k1⋅ki|zi-1|2α′kn-1⋅ki∏i<j≤n-2|zj-zi|2α′ki⋅kjf(zi)g(z¯i),
where we have fixed the three points z1=0, zn-1=1, and zn=∞. The f(zi) and g(z¯i) functions come from the operator product expansion of vertex operators and depend on the type of external states we are considering, but since they are without any branch cuts they will not be important for the following argument. We will denote zi=vi1+ivi2, such that(2.2)|zi|2α′k1⋅ki=[(vi1)2+(vi2)2]α′k1⋅ki,|zi-1|2α′kn-1⋅ki=[(vi1-1)2+(vi2)2]α′kn-1⋅ki,|zj-zi|2α′ki⋅kj=[(vj1-vi1)2+(vj2-vi2)2]α′ki⋅kj.
By analytically continuing the vi2 variables to the complex plane, we can rotate the integration contour for these variables from the real axis to (almost) the imaginary axis(2.3)vi2⟶ie-2iϵvi2≃i(1-2iϵ)vi2,
without changing the value of the amplitude. Here ϵ>0 is some small number making sure we avoid the branch points. This changes the expressions in the integrand (to linear order in ϵ)(2.4)[(vi1)2+(vi2)2]α′k1⋅ki⟶[(vi1)2-(vi2)2+4iϵ(vi2)2]α′k1⋅ki,[(vi1-1)2+(vi2)2]α′kn-1⋅ki⟶[(vi1)2-(vi2)2-2vi1+1+4iϵ(vi2)2]α′kn-1⋅ki,[(vj1-vi1)2+(vj2-vi2)2]α′ki⋅kj⟶[(vj1-vi1)2-(vj2-vi2)2(1-4iϵ)]α′ki⋅kj.
If we now make the transformation of variables(2.5)vi±≡vi1±vi2,
and define δi≡vi+-vi-, it is easy to verify that the expression on the right-hand side of line (2.4) is given by(2.6)(vi+-iϵδi)α′k1⋅ki(vi-+iϵδi)α′k1⋅ki,(vi+-1-iϵδi)α′kn-1⋅ki(vi--1+iϵδi)α′kn-1⋅ki,(vi+-vj+-iϵ(δi-δj))α′ki⋅kj(vi--vj-+iϵ(δi-δj))α′ki⋅kj,
respectively.In total, this brings (2.1) into the form
(2.7)Mn=(i2)n-3(i2πα′)n-3∫-∞+∞∏i=2n-2dvi+dvi-f(vi-)g(vi+)×(vi+-iϵδi)α′k1⋅ki(vi-+iϵδi)α′k1⋅ki(vi+-1-iϵδi)α′kn-1⋅ki(vi--1+iϵδi)α′kn-1⋅ki×∏i<j≤n-2(vi+-vj+-iϵ(δi-δj))α′ki⋅kj(vi--vj-+iϵ(δi-δj))α′ki⋅kj,
where the additional factor of (i/2)n-3 is due to the Jacobian when changing variables in the integral and from the rotation of the vi2 contours.First we note the following, assume that at least onevi+∈]-∞,0[ and look at the contribution from vi-, that is,
(2.8)∫-∞+∞dvi-f(vi-)(vi-+iϵδi)α′k1⋅ki(vi--1+iϵδi)α′kn-1⋅ki∏i<j≤n-2(vi--vj-+iϵ(δi-δj))α′ki⋅kj.
The behaviour of the imaginary ϵ terms near the branch points is(2.9)vi-~0⟹δi~vi+<0,vi-~1⟹δi~vi+-1<0,vi-~vj-⟹δi-δj~vi+-vj+<0whenvi+<vj+.
The requirement vi+<vj+ in the last line is not a problem, since we can just choose to look at the vi- integral corresponding to the “smallest’’ vi+ variable, which has to lie in the range ]-∞,0[ due to our first assumption. This means that we can close the integral of vi- in the lower half of the complex vi--plane (again by analytical continuation), and since the closed contour does not contain any poles the integral vanishes. In general, when vi+<vj+ we avoid the branch point vi-=vj- below the real axis, and when vi+>vj+ we avoid it above the real axis. From this kind of argument we see that whenever one of the vi+-variables is in the range of ]-∞,0[ or ]1,∞[, at least one of the vi- contours can be completely closed either below or above the real axis. Hence, only when all vi+ lie between 0 and 1 will there be a contribution to (2.7).By splitting up thevi+-integration region we can write the n-point closed string amplitude as
(2.10)Mn=∑σMnσ(σ(2),…,σ(n-2)),
where Mnσ(σ(2),…,σ(n-2)) is the “ordered amplitude’’ defined such that vσ(2)+<vσ(3)+<⋯<vσ(n-2)+. For instance, at five points this corresponds to splitting the integration over the (v2+,v3+)-plane into an integral over the region “above’’ the v3+=v2+ line (i.e., v2+<v3+) and an integral “below’’ this line (i.e., v3+<v2+). Together with the above restriction on the vi+-integration range, the vi+ part of Mnσ in (2.10) becomes(2.11)∫0<vσ(2)+<⋯<vσ(n-2)+<1∏i=2n-2dvi+g(vi+)(vi+)α′k1⋅ki(1-vi+)α′kn-1⋅ki∏i<j≤n-2(vσ(j)+-vσ(i)+)α′kσ(i)⋅kσ(j),
where we have omitted the infinitesimal ϵ terms. We recognize (2.11) as the expression exactly corresponding to the color-ordered open string amplitude 𝒜n(1,σ(2,…,n-2),n-1,n). Note that, compared to (2.7), we have written (1-vi+)α′kn-1·ki instead of (vi+-1)α′kn-1·ki and (vj+-vi+)α′ki·kj such that vj+-vi+>0 always. This is needed in order to make the identification with a color-ordered open string amplitude; however, we are of course only allowed to do this if we make a similar change in the vi- part, otherwise we would get wrong phase factors.For simplicity we will from now on fix the ordering to{2,3,…,n-2}, that is, we are considering the Mnσ(2,3,…,n-2) contribution in (2.10). The remaining terms can simply be obtained through permutation of labels.We have just seen that thevi+ part is nothing but the color-ordered 𝒜n(1,2,…,n) amplitude. We therefore turn our attention to the vi- part, investigating which contours the imaginary ϵ terms dictate for the integrals. Near vi-~0 the quantity iϵδi~iϵvi+ is a positive imaginary number (remember that vi+∈]0,1[), so the contour is above the real axis here. For vi-~1 we have iϵδi~iϵ(vi+-1) which is a negative imaginary number, hence the contour lies below the real axis. Finally, for vi-~vj- we see that iϵ(δi-δj)~iϵ(vi+-vj+), meaning that the contour for vi-+iϵδi lies below the contour of vj-+iϵδj for i<j. See Figure 1 for an illustration of this nested structure.Figure 1
The nested structure of the contours of integration for thevi- variables corresponding to the ordering 0<v2+<v3+<⋯<vn-2+<1 of the vi+ variables.The next step is to deform the contours forvi-+iϵδi into expressions corresponding to color-ordered amplitudes. That is, we are going to close the contours either to the left, turning the contour below the real axis, or to the right, turning the contour above the real axis. Besides having the correct integration region, in order to identify the integrals with amplitudes, we also need to make sure that the integrand is correct. This implies that we sometime need to pull out phase factors. However, in order not to cross a branch cut we do this in the following way: for zc with Re(z)<0(2.12)zc={eiπc(-z)c,Im⁡(z)≥0,e-iπc(-z)c,Im⁡(z)<0.
We have added some additional details concerning this subtle, but important, point in Appendix A.Furthermore, there is a freedom in the number of contours one can close to the left or the right. For a given2≤j≤n-1, we can pull the contours from 2 up to j-1 to the left, and the set from j to n-2 to the right (j=2 or j=n-1 means all to the right or all to the left, resp.). Let us illustrate this in details for the five-point case (the four-point case is a bit to simple in order to capture all the features of the general argument).


### 2.1.  Five-Point KLT Relations

Starting withj=4, pulling the contour for v2- to the left, only showing the piece involving v2-, we get(2.13)∫C2dv2-(v2-)α′k1⋅k2(1-v2-)α′k4⋅k2(v3--v2-)α′k3⋅k2f(v2-)=(eiπα′k1⋅k2-e-iπα′k1⋅k2)∫-∞0dv2-(-v2-)α′k1⋅k2(1-v2-)α′k4⋅k2(v3--v2-)α′k3⋅k2f(v2-)=2isin⁡⁡(πα′k1⋅k2)∫-∞0dv2-(-v2-)α′k1⋅k2(1-v2-)α′k4⋅k2(v3--v2-)α′k3⋅k2f(v2-).
Note that in the previous first line we write (1-v2-)α′k4·k2 instead of (v2--1)α′k4·k2 and (v3--v2-)α′k3·k2 instead of (v2--v3-)α′k3·k2 in order to compensate for the same swapping of order in the vi+ integral of (2.11). Now, as illustrated in the bottom of Figure 2, we also close the contour for v3- to the left (only showing the part involving the v3- variable)(2.14)∫C3dv3-(v3-)α′k1⋅k3(1-v3-)α′k4⋅k3(v3--v2-)α′k3⋅k2f(v3-)=(eiπα′(k1+k2)⋅k3-e-iπα′(k1+k2)⋅k3)∫-∞v2-dv3-(-v3-)α′k1⋅k3(1-v3-)α′k4⋅k3(v2--v3-)α′k2⋅k3f(v3-)+(eiπα′k1⋅k3-e-iπα′k1⋅k3)∫v2-0dv3-(-v3-)α′k1⋅k3(1-v3-)α′k4⋅k3(v3--v2-)α′k2⋅k3f(v3-)=2isin⁡⁡(πα′(k1+k2)⋅k3)∫-∞v2-dv3-(-v3-)α′k1⋅k3(1-v3-)α′k4⋅k3(v2--v3-)α′k2⋅k3f(v3-)+2isin⁡⁡(πα′k1⋅k3)∫v2-0dv3-(-v3-)α′k1⋅k3(1-v3-)α′k4⋅k3(v3--v2-)α′k2⋅k3f(v3-).
We see that the total integration over v2- and v3- exactly correspond to color-ordered open string amplitudes, which we will denote 𝒜̃5 to distinguish them from the ones following from the vi+ part, such that the whole vi- contribution in M5σ(2,3) can be written as(2.15)∝sin⁡⁡(πα′k1⋅k2)sin⁡⁡(πα′(k1+k2)⋅k3)Ã5(3,2,1,4,5)+sin⁡⁡(πα′k1⋅k2)sin⁡⁡(πα′k1⋅k3)Ã5(2,3,1,4,5).
Together with (2.11) for n=5, and (2.10), we have obtained the following relation between the five-point closed string amplitude ℳ5 and the color-ordered open string amplitudes 𝒜5, 𝒜̃5(2.16)M5=-14π2α′2[sin⁡⁡(πα′k1⋅k2)sin⁡⁡(πα′(k1+k2)⋅k3)A5(1,2,3,4,5)Ã5(3,2,1,4,5)+sin⁡⁡(πα′k1⋅k2)sin⁡(πα′k1⋅k3)A5(1,2,3,4,5)Ã5(2,3,1,4,5)]+(2⟷3).
If we take the other extreme, that is, closing both contours to the right (j=2), we get(2.17)2isin⁡⁡(πα′k4⋅k3)∫1+∞dv3-(v3-)α′k1⋅k3(v3--1)α′k4⋅k3(v3--v2-)α′k2⋅k3f(v3-),
for the v3- integration, and(2.18)2isin⁡⁡(πα′k4⋅k2)∫1v3-dv2-(v2-)α′k1⋅k2(v2--1)α′k4⋅k2(v3--v2-)α′k3⋅k2f(v2-)+2isin⁡⁡(πα′(k4+k3)⋅k2)∫v3-+∞dv2-(v2-)α′k4⋅k2(v2--1)α′k4⋅k2(v2--v3-)α′k3⋅k2f(v2-),
for the v2- integration, see the top case of Figure 2, that is,
(2.19)M5=-14π2α′2[sin⁡⁡(πα′k4⋅k3)sin⁡⁡(πα′k4⋅k2)A5(1,2,3,4,5)Ã5(1,4,2,3,5)+sin⁡⁡(πα′k4⋅k3)sin⁡⁡(πα′(k4+k3)⋅k2)A5(1,2,3,4,5)Ã5(1,4,3,2,5)]+(2⟷3).
Finally, we could have closed v2- to the left and v3- to the right, as also illustrated in Figure 2, resulting in (j=3)(2.20)M5=-14π2α′2sin⁡⁡(πα′k1⋅k2)sin⁡⁡(πα′k4⋅k3)A5(1,2,3,4,5)Ã5(2,1,4,3,5)+(2⟷3).Figure 2
The three different ways of flipping contours in the five-point case.All of these different forms can be nicely collected into one compact formula if we introduce themomentum kernel(2.21)Sα′[i1,…,ik∣j1,…,jk]p≡(πα′2)-k∏t=1ksin⁡⁡(πα′(p⋅kit+∑q>tkθ(it,iq)kit⋅kiq)),
where θ(it,iq) equals 1 if the ordering of it and iq is opposite in {i1,…,ik} and {j1,…,jk}, and 0 if the ordering is the same. Here we have defined 𝒮α′ for a general number of legs, so, for instance(2.22)Sα′[2∣2]k1=(πα′2)-1sin⁡⁡(πα′k1⋅k2),Sα′[23∣23]k1=(πα′2)-2sin⁡⁡(πα′k1⋅k2)sin⁡⁡(πα′k1⋅k3),Sα′[23∣32]k1=(πα′2)-2sin⁡⁡(πα′(k1+k3)⋅k2)sin⁡⁡(πα′k1⋅k3),
and so on. We will also define 𝒮α′[∅∣∅]p=1 for empty sets. With this 𝒮α′ function we can collect (2.16), (2.19), and (2.20) into one formula as(2.23)M5=(-i4)2∑σ∑γ,βSα′[γ(σ(2),…,σ(j-1))∣σ(2),…,σ(j-1)]k1×Sα′[σ(j),…,σ(3)∣β(σ(j),…,σ(3))]k4×A5(1,σ(2,3),4,5)Ã5(γ(σ(2),…,σ(j-1)),1,4,β(σ(j),…,σ(3)),5),
with j={2,3,4}. Note that when j=2 one should read the set {σ(2),…,σ(j-1)} as being empty, and likewise for j=4 the set {σ(j),…,σ(3)} is empty.


### 2.2.  Generaln-Point KLT Relations

Equation (2.23) not only collects all the different five-point forms, due to different ways of closing the contours, in one nice expression, but going through the same procedure for the n-point case, it directly generalizes to(2.24)Mn=(-i4)n-3∑σ∑γ,βSα′[γ(σ(2),…,σ(j-1))∣σ(2,…,j-1)]k1×Sα′[σ(j,…,n-2)∣β(σ(j),…,σ(n-2))]kn-1×An(1,σ(2,…,n-2),n-1,n)×Ãn(γ(σ(2),…,σ(j-1)),1,n-1,β(σ(j),…,σ(n-2)),n),
with 2≤j≤n-1.Expression (2.24) shows how to factorize an n-point closed string amplitude ℳn into the product of n-point color-ordered open string amplitudes 𝒜n and 𝒜̃n, “glued’’ together by kinematic factors contained in the 𝒮α′ function. We note that the expression is a sum over (n-3)!×(j-2)!×(n-1-j)! terms, taking its maximum value (n-3)!×(n-3)! for j=2 or j=n-1, and its minimum (n-3)!×(⌈n/2⌉-2)!×(⌊n/2⌋-1)! for j=⌈n/2⌉. (The floor and ceiling functions are defined on half-integers as follows: ⌊n/2⌋=(n-1)/2 if n is odd, or n/2 if n is even. ⌈n/2⌉=(n+1)/2 if n is odd, or n/2 if n is even.)Although, being one of the expressions having most terms, choosingj=n-1 the relation takes a particularly nice n-point form(2.25)Mn=(-i4)n-3∑σ,γSα′[γ(2,…,n-2)∣σ(2,…,n-2)]k1×An(1,σ(2,…,n-2),n-1,n)Ãn(n-1,n,γ(2,…,n-2),1),
only involving one 𝒮α′ function and having a greater symmetry between the sums over different ordered 𝒜n and 𝒜̃n amplitudes. An equally nice form occurs with j=2.


## 2.1.  Five-Point KLT Relations

Starting withj=4, pulling the contour for v2- to the left, only showing the piece involving v2-, we get(2.13)∫C2dv2-(v2-)α′k1⋅k2(1-v2-)α′k4⋅k2(v3--v2-)α′k3⋅k2f(v2-)=(eiπα′k1⋅k2-e-iπα′k1⋅k2)∫-∞0dv2-(-v2-)α′k1⋅k2(1-v2-)α′k4⋅k2(v3--v2-)α′k3⋅k2f(v2-)=2isin⁡⁡(πα′k1⋅k2)∫-∞0dv2-(-v2-)α′k1⋅k2(1-v2-)α′k4⋅k2(v3--v2-)α′k3⋅k2f(v2-).
Note that in the previous first line we write (1-v2-)α′k4·k2 instead of (v2--1)α′k4·k2 and (v3--v2-)α′k3·k2 instead of (v2--v3-)α′k3·k2 in order to compensate for the same swapping of order in the vi+ integral of (2.11). Now, as illustrated in the bottom of Figure 2, we also close the contour for v3- to the left (only showing the part involving the v3- variable)(2.14)∫C3dv3-(v3-)α′k1⋅k3(1-v3-)α′k4⋅k3(v3--v2-)α′k3⋅k2f(v3-)=(eiπα′(k1+k2)⋅k3-e-iπα′(k1+k2)⋅k3)∫-∞v2-dv3-(-v3-)α′k1⋅k3(1-v3-)α′k4⋅k3(v2--v3-)α′k2⋅k3f(v3-)+(eiπα′k1⋅k3-e-iπα′k1⋅k3)∫v2-0dv3-(-v3-)α′k1⋅k3(1-v3-)α′k4⋅k3(v3--v2-)α′k2⋅k3f(v3-)=2isin⁡⁡(πα′(k1+k2)⋅k3)∫-∞v2-dv3-(-v3-)α′k1⋅k3(1-v3-)α′k4⋅k3(v2--v3-)α′k2⋅k3f(v3-)+2isin⁡⁡(πα′k1⋅k3)∫v2-0dv3-(-v3-)α′k1⋅k3(1-v3-)α′k4⋅k3(v3--v2-)α′k2⋅k3f(v3-).
We see that the total integration over v2- and v3- exactly correspond to color-ordered open string amplitudes, which we will denote 𝒜̃5 to distinguish them from the ones following from the vi+ part, such that the whole vi- contribution in M5σ(2,3) can be written as(2.15)∝sin⁡⁡(πα′k1⋅k2)sin⁡⁡(πα′(k1+k2)⋅k3)Ã5(3,2,1,4,5)+sin⁡⁡(πα′k1⋅k2)sin⁡⁡(πα′k1⋅k3)Ã5(2,3,1,4,5).
Together with (2.11) for n=5, and (2.10), we have obtained the following relation between the five-point closed string amplitude ℳ5 and the color-ordered open string amplitudes 𝒜5, 𝒜̃5(2.16)M5=-14π2α′2[sin⁡⁡(πα′k1⋅k2)sin⁡⁡(πα′(k1+k2)⋅k3)A5(1,2,3,4,5)Ã5(3,2,1,4,5)+sin⁡⁡(πα′k1⋅k2)sin⁡(πα′k1⋅k3)A5(1,2,3,4,5)Ã5(2,3,1,4,5)]+(2⟷3).
If we take the other extreme, that is, closing both contours to the right (j=2), we get(2.17)2isin⁡⁡(πα′k4⋅k3)∫1+∞dv3-(v3-)α′k1⋅k3(v3--1)α′k4⋅k3(v3--v2-)α′k2⋅k3f(v3-),
for the v3- integration, and(2.18)2isin⁡⁡(πα′k4⋅k2)∫1v3-dv2-(v2-)α′k1⋅k2(v2--1)α′k4⋅k2(v3--v2-)α′k3⋅k2f(v2-)+2isin⁡⁡(πα′(k4+k3)⋅k2)∫v3-+∞dv2-(v2-)α′k4⋅k2(v2--1)α′k4⋅k2(v2--v3-)α′k3⋅k2f(v2-),
for the v2- integration, see the top case of Figure 2, that is,
(2.19)M5=-14π2α′2[sin⁡⁡(πα′k4⋅k3)sin⁡⁡(πα′k4⋅k2)A5(1,2,3,4,5)Ã5(1,4,2,3,5)+sin⁡⁡(πα′k4⋅k3)sin⁡⁡(πα′(k4+k3)⋅k2)A5(1,2,3,4,5)Ã5(1,4,3,2,5)]+(2⟷3).
Finally, we could have closed v2- to the left and v3- to the right, as also illustrated in Figure 2, resulting in (j=3)(2.20)M5=-14π2α′2sin⁡⁡(πα′k1⋅k2)sin⁡⁡(πα′k4⋅k3)A5(1,2,3,4,5)Ã5(2,1,4,3,5)+(2⟷3).Figure 2
The three different ways of flipping contours in the five-point case.All of these different forms can be nicely collected into one compact formula if we introduce themomentum kernel(2.21)Sα′[i1,…,ik∣j1,…,jk]p≡(πα′2)-k∏t=1ksin⁡⁡(πα′(p⋅kit+∑q>tkθ(it,iq)kit⋅kiq)),
where θ(it,iq) equals 1 if the ordering of it and iq is opposite in {i1,…,ik} and {j1,…,jk}, and 0 if the ordering is the same. Here we have defined 𝒮α′ for a general number of legs, so, for instance(2.22)Sα′[2∣2]k1=(πα′2)-1sin⁡⁡(πα′k1⋅k2),Sα′[23∣23]k1=(πα′2)-2sin⁡⁡(πα′k1⋅k2)sin⁡⁡(πα′k1⋅k3),Sα′[23∣32]k1=(πα′2)-2sin⁡⁡(πα′(k1+k3)⋅k2)sin⁡⁡(πα′k1⋅k3),
and so on. We will also define 𝒮α′[∅∣∅]p=1 for empty sets. With this 𝒮α′ function we can collect (2.16), (2.19), and (2.20) into one formula as(2.23)M5=(-i4)2∑σ∑γ,βSα′[γ(σ(2),…,σ(j-1))∣σ(2),…,σ(j-1)]k1×Sα′[σ(j),…,σ(3)∣β(σ(j),…,σ(3))]k4×A5(1,σ(2,3),4,5)Ã5(γ(σ(2),…,σ(j-1)),1,4,β(σ(j),…,σ(3)),5),
with j={2,3,4}. Note that when j=2 one should read the set {σ(2),…,σ(j-1)} as being empty, and likewise for j=4 the set {σ(j),…,σ(3)} is empty.


## 2.2.  Generaln-Point KLT Relations

Equation (2.23) not only collects all the different five-point forms, due to different ways of closing the contours, in one nice expression, but going through the same procedure for the n-point case, it directly generalizes to(2.24)Mn=(-i4)n-3∑σ∑γ,βSα′[γ(σ(2),…,σ(j-1))∣σ(2,…,j-1)]k1×Sα′[σ(j,…,n-2)∣β(σ(j),…,σ(n-2))]kn-1×An(1,σ(2,…,n-2),n-1,n)×Ãn(γ(σ(2),…,σ(j-1)),1,n-1,β(σ(j),…,σ(n-2)),n),
with 2≤j≤n-1.Expression (2.24) shows how to factorize an n-point closed string amplitude ℳn into the product of n-point color-ordered open string amplitudes 𝒜n and 𝒜̃n, “glued’’ together by kinematic factors contained in the 𝒮α′ function. We note that the expression is a sum over (n-3)!×(j-2)!×(n-1-j)! terms, taking its maximum value (n-3)!×(n-3)! for j=2 or j=n-1, and its minimum (n-3)!×(⌈n/2⌉-2)!×(⌊n/2⌋-1)! for j=⌈n/2⌉. (The floor and ceiling functions are defined on half-integers as follows: ⌊n/2⌋=(n-1)/2 if n is odd, or n/2 if n is even. ⌈n/2⌉=(n+1)/2 if n is odd, or n/2 if n is even.)Although, being one of the expressions having most terms, choosingj=n-1 the relation takes a particularly nice n-point form(2.25)Mn=(-i4)n-3∑σ,γSα′[γ(2,…,n-2)∣σ(2,…,n-2)]k1×An(1,σ(2,…,n-2),n-1,n)Ãn(n-1,n,γ(2,…,n-2),1),
only involving one 𝒮α′ function and having a greater symmetry between the sums over different ordered 𝒜n and 𝒜̃n amplitudes. An equally nice form occurs with j=2.


## 3. The Field Theory Limit

Now that we have obtained the KLT relations in string theory, we will take a closer look at the field theory limit of (2.24), that is, letting α′→0. The amplitudes just go to their corresponding field theory expressions(3.1)Mn⟶Mn,An⟶An,Ãn⟶Ãn,
and 𝒮α′→𝒮0≡𝒮, where it follows from (2.21) that(3.2)S[i1,…,ik∣j1,…,jk]p=∏t=1k(spit+∑q>tkθ(it,iq)sitiq),
with sij≡(ki+kj)2=2ki·kj, or more generally sij…k=(pi+pj+⋯+pk)2, which will be used later on. Let us at this point also make a slight change in the overall constant such that it fits with a more commonly used normalization of field theory amplitudes. The n-point KLT relations in field theory then take the final form(3.3)Mn=(-1)n+1∑σ∑γ,βS[γ(σ(2),…,σ(j-1))∣σ(2,…,j-1)]k1×S[σ(j),…,σ(n-2)∣β(σ(j,…,n-2))]kn-1×An(1,σ(2,…,n-2),n-1,n)×Ãn(γ(σ(2),…,σ(j-1)),1,n-1,β(σ(j),…,σ(n-2)),n),
again with the freedom 2≤j≤n-1. We have not been very explicit about which theories the amplitudes belong to, and as one might have anticipated from the string theory derivation they are actually valid for very general classes of amplitudes. We could, for instance, have an 𝒩=8 supergravity amplitude on the l.h.s. and 𝒩=4 SYM amplitudes on the r.h.s., or even express the full tree-level gauge theory amplitude as products between color-ordered gauge theory and a color-scalar theory [13, 19]. However, for simplicity, we will for the rest of this paper just consider it as a relation between pure graviton and pure gluon amplitudes.Since it will be relevant for a later section, and because of its simple expression, let us explicitly write out the forms one obtains withj=n-1 and j=2, and at the same time introduce a short-hand notation that will be used when expressions get large. With j=n-1 or j=2 (3.3) becomes(3.4)Mn=(-1)n+1∑σ,σ̃∈Sn-3Ãn(n-1,n,σ̃2,n-2,1)S[σ̃2,n-2∣σ2,n-2]k1An(1,σ2,n-2,n-1,n),
or(3.5)Mn=(-1)n+1∑σ,σ̃∈Sn-3An(1,σ2,n-2,n-1,n)S[σ2,n-2∣σ̃2,n-2]kn-1Ãn(1,n-1,σ̃2,n-2,n),
respectively, see also Figure 3. Here we have introduced the short-hand notation σ2,n-2≡σ(2,…,n-2), for a σ permutation over leg {2,…,n-2}, and likewise for σ̃. Note that σ and σ̃ are two unrelated permutations, the tilde is just to remind us which one is going with Ãn, and we are trying to be economic with our use of Greek letters.Figure 3
Diagrammatic representation of the KLT relation withj=n-1.These relations are totally crossing symmetric, which they of course have to be since they equal a gravity amplitude. However, the r.h.s. is onlymanifest crossing symmetric in legs 2,3,…,n-2. The crossing symmetry between, for instance, legs n and n-1 in (3.4) can also easily be seen by use of the reflection symmetry of color-ordered amplitudes An(1,2,…,n)=(-1)nAn(n,n-1,…,1), and the following identity:(3.6)S[i1,…,ik∣j1,…,jk]p=S[jk,…,j1∣ik,…,i1]p,
but in general the crossing symmetry is not obvious at all. These comments also apply in the string theory case.Although the calculations in Section2 were a bit involved, at least one had an intuitive picture of breaking up a closed string into two open strings glued together by phase factors. It was also clear how one could get different expressions for this factorization by choosing different closures for the contours. The field theory limit then follows naturally when taking α′→0. However, it would be quite unsatisfactory if we could not understand this field theory expression without going through string theory first. This will be our main focus for the remaining of this paper; how to see the KLT relations from a purely field theoretical point of view, including the possibility of going between different expressions without having contours to deform. In order to get a better feel for the task that lies ahead let us start by looking at some explicit lower-point examples and make some comments.


### 3.1.  Lower-Point Examples

The connection between gravity and gauge theory already starts at three points. It is well known that for real momenta an on-shell three-point amplitude must vanish. However, if we go to complex momenta this is no longer the case. Interestingly, it turns out that Lorentz invariance, and the spin of the external particles, uniquely fix the three-point amplitude, see for example, [47]. In detail, the spin 2 three-point amplitude (gravitons) is the product of two spin 1 three-point amplitudes (gluons)(3.7)M3(1,2,3)=A3(1,2,3)Ã3(1,2,3).
This can also be directly seen from Feynman rules when legs are put on-shell, something that in particular simplifies the expression for the gravity three-point vertex. Although we did not have the complex three-point amplitude in mind when we wrote down (3.3), it does give the correct result even in this case. At first it might seem a bit strange to consider amplitudes with complex momenta, but recent progress in amplitude calculations has largely been inspired by such an extension. Indeed, the BCFW recursion relation, that will be reviewed below, relies on the deformation to complex momenta, and so will our proof of (3.3) in Section 4.The four point case also looks very simple(3.8)M4(1,2,3,4)=-s12A4(1,2,3,4)Ã4(1,2,4,3),
however, its field theoretical origin already becomes a bit unclear here. Compared to (3.7) we see the appearance of a kinematic factor that makes sure to cancel one of the s12 poles present in both gauge theory amplitudes. We also see that the total crossing symmetry of the r.h.s. has already been well hidden, although we know it has to be there.When we go to five points, even expressed in the form with fewest terms (j=3),(3.9)M5=s12s34A5(1,2,3,4,5)Ã5(2,1,4,3,5)+s13s24A5(1,3,2,4,5)Ã5(3,1,4,2,5),
not only is the total crossing symmetry of the r.h.s. by no means obvious, but also the correct cancellation of poles begins to get more complicated to see. These properties only get more and more disguised as we increase the number of external particles.From the above examples it might seem almost impossible to identify, in all generality, the r.h.s. of (3.3) with a gravity amplitude from a purely field theoretical/analytical point of view. Even the simplest features of gravity amplitudes have become very nontrivial statements about the gauge theory side. Before we can attack this problem we therefore need to review some important concepts and properties. These will not only lead to a better understanding of KLT relations in field theory, but are by themselves amazing structures of scattering amplitudes. The first thing we will introduce is a rather unusual way of expressing the KLT relations. This form will have a higher degree of manifest crossing symmetry, compared to (3.3), but it requires a regularization. It has been proven separately by pure field theory, and will turn out to be important even in the proof of (3.3). Secondly, we introduce the BCJ relations, providing identities between color-ordered tree-level amplitudes, and explain its connection to the j-independence of (3.3). Third, we look at what happens with the r.h.s. of (3.3) when An and Ãn belong to different helicity sectors. Fourth, we review the BCFW recursion relation, exploiting very general analytic properties of tree-level amplitudes to recursively construct higher-point amplitudes from lower-point ones. All of these structures will be important for Section 4, where we will utilize them in order to give a purely field theoretical proof of (3.3).


### 3.2.  Regularized KLT Relations

Consideringn-point amplitudes we start by making the following deformation of momenta p1 and pn:(3.10)p1⟶p1′=p1-xq,pn⟶pn′=pn+xq,
where x is just some arbitrary parameter, and q is a four-vector satisfying q·p1=q2=0 and q·pn≠0. This preserves conservation of momentum and keeps p1′2=0, but makes pn′2=s1′2…(n-1)≠0.The gravity amplitudeMn can then be written as [15](3.11)Mn=(-1)nlim⁡x→0∑σ,σ̃∈Sn-2Ãn(n′,σ̃2,n-1,1′)S[σ̃2,n-1∣σ2,n-1]p1′An(1′,σ2,n-1,n′)s1′2…(n-1).
We note that as x→0 the denominator goes to zero, but as we will see from (3.16) below, so does the numerator. However, the whole expression has a limit which is exactly equal to a gravity amplitude. This can also be seen by taking the soft limit of leg n in (3.4) [18], and comparing against the well-known soft-limit behaviour of gravity amplitudes [12, 48] (see [49] for an alternative approach). This might make (3.11) seem less strange, but we do stress that this expression for a gravity amplitude can be proven without knowledge of (3.4) (or in general (3.3)).If we make a deformation withp1′ being off-shell instead of pn′, we can write down the “dual’’ expression to (3.11)(3.12)Mn=(-1)nlim⁡y→0∑σ,σ̃∈Sn-2An(1′,σ2,n-1,n′)S[σ2,n-1∣σ̃2,n-1]pn′Ãn(n′,σ̃2,n-1,1′)s23…n′,
where we have called the deformation parameter y.


### 3.3.  BCJ Relations

The story of relations among color-ordered tree-level amplitudes is in itself very interesting. However, to not lose focus of our current goal we will here only remind the reader about the Bern-Carrasco-Johansson (BCJ) relations, which will play an important role for us.There are several different ways of presenting the BCJ relations, which all have their own advantages and disadvantages. Here we choose one of the simplest general representations. Forn-point amplitudes these read(3.13)0=s12An(1,2,3,…,n)+(s12+s23)An(1,3,2,4,…,n)+(s12+s23+s24)An(1,3,4,2,5,…,n)+⋯+(s12+s23+s24+⋯+s2(n-1))An(1,3,4,…,n-1,2,n),
along with all relations obtained by permutation of labels in the above expression. Note how leg 2 moves one position to the right in each term and picks up an additional factor of s2i, where i is the leg just passed through. This particular form of the BCJ relations actually contains all the information, in the sense that once you have these you can go from the (n-2)! color-ordered amplitudes, provided by the Kleiss-Kuijf relations, down to (n-3)! independent amplitudes [38].Some simple explicit examples of (3.13) are(3.14)0=s12A4(1,2,3,4)+(s12+s23)A4(1,3,2,4),
for four points, and(3.15)0=s12A5(1,2,3,4,5)+(s12+s23)A5(1,3,2,4,5)+(s12+s23+s24)A5(1,3,4,2,5),
at five points. We could of course use momentum conservation to write some of the kinematic invariant factors simpler, but keeping them like this make the relations easy to remember.It turns out, that there is a very nice way of rephrasing the above BCJ relations in terms of our𝒮 function. This is achieved through(3.16)0=∑σ∈Sn-2S[β(2,…,n-1)∣σ(2,…,n-1)]k1An(1,σ(2,…,n-1),n),
where β is just some arbitrary permutation of the legs {2,3,…,n-1}. This is nothing but a sum of relations in the form of (3.13). To see this, let us first write out the five-point case with β(2,3,4)=(2,3,4) explicitly, that is,
(3.17)∑σ∈S3S[2,3,4∣σ(2,3,4)]k1A5(1,σ(2,3,4),5)=s13s14[s12A5(1,2,3,4,5)+(s12+s23)A5(1,3,2,4,5)+(s12+s23+s24)A5(1,3,4,2,5)]+s14(s13+s34)×[s12A5(1,2,4,3,5)+(s12+s24)A5(1,4,2,3,5)+(s12+s23+s24)A5(1,4,3,2,5)].
The two [⋯] is exactly zero due to (3.13).The general argument is to divide the sum ofσ(2,…,n-1) into a sum of groups where all, except the first leg in the β ordering, call it β(2), have fixed ordering, and then insert β(2) at any place. For each group all factors from 𝒮 will be the same except for the factor contributing from β(2). This will give a BCJ relation and thereby vanish. In the above five-point example we had β(2)=2 and the two groups we summed over were the one with ordering {3,4} and {4,3}, respectively. From this we see that (3.16) is a consequence of (3.13).


#### 3.3.1.  Thej-Independence

With the BCJ relations we can now address thej-independence of (3.3) from a field theory point of view, which in string theory followed naturally due to the freedom in how one closes the contours. Indeed, it is useful to first establish the equivalence between all forms only differing in the j-value chosen. In this way, if we can prove (3.3) for just one specific form, we have proven them all.The form of BCJ relation that implies thej-independence is given by
(3.18)∑α,βS[αi2,ij∣i2,…,ij]p1S[ij+1,…,in-2∣βij+1,in-2]pn-1Ãn(αi2,ij,1,n-1,βij+1,in-2,n)=∑α′,β′S[αi2,ij-1′∣i2,…,ij-1]p1S[ij,ij+1,…,in-2∣βij,in-2′]pn-1Ãn(αi2,ij-1′,1,n-1,βij,in-2′,n).
Although, (3.18) looks rather complicated it is straightforward to prove using only BCJ relations in the form of (3.13), and momentum conservation. Since the proof is not very enlightening we will not repeat it here, but simply refer to the appendix of [16] for details. Instead we will illustrate how to use (3.18) in the simple case of five points.Forn=5 and j=2 (3.18) reads (where we have taken ik=k)(3.19)S[2∣2]p1S[3∣3]p4Ã5(2,1,4,3,5)=∑β′S[2,3∣β′(2,3)]p4Ã5(1,4,β′(2,3),5),
and since (3.3) for n=5 and j=2 is given by(3.20)M5=∑σ∑βA5(1,σ(2,3),4,5)S[σ(2,3)∣β(2,3)]k4Ã5(1,4,β(2,3),5)=∑σA5(1,σ(2,3),4,5)∑βS[σ(2,3)∣β(2,3)]k4Ã5(1,4,β(2,3),5),
using (3.19) we immediately get(3.21)M5=∑σA5(1,σ(2,3),4,5)S[σ(2)∣σ(2)]p1S[σ(3)∣σ(3)]p4Ã5(σ(2),1,4,σ(3),5),
which is just (3.3) with j=3. Likewise we could now use (3.18) with j=3 to rewrite (3.21) into (3.3) with j=4.This procedure generalizes ton points; by repeated use of (3.18) the j-independence of (3.3) directly follows.It is interesting to note that the BCJ relations could have been discovered much earlier. Indeed, equating different expressions for the KLT relations leads to pure gauge theory amplitude relations, which, as we have just seen, are directly related to the BCJ relations. This was already apparent in [1] but at that time not really appreciated. We also see that writing both the KLT and BCJ relations in terms of the 𝒮 function, their close connection becomes much more manifest than one would have anticipated from the original papers.


### 3.4.  Vanishing Identities

The next surprising property of (3.3) we want to introduce is that if we take An and Ãn to live in different helicity sectors the r.h.s. of (3.3) is identically zero.To be more specific let us denote ann-point color-ordered tree amplitude with helicity configuration belonging to NkMHV as Ank, that is, Ank has 2+k negative helicity gluons, with k∈{0,1,…,n-4} for nonvanishing amplitudes. We are not interested in the exact helicity configuration, just which helicity sector it belongs to. We then have [43, 44](3.22)0=∑σ∑γ,βS[γ(σ2,j-1)∣σ2,j-1]k1S[σj,n-2∣β(σj,n-2)]kn-1Anh(1,σ2,j-1,σj,n-2,n-1,n)×Ãnk≠h(γ(σ2,j-1),1,n-1,β(σj,n-2),n),
which is nothing but the r.h.s. of (3.3), written in our short-hand notation, with the color-ordered amplitudes living in different helicity sectors.At four points these relations are trivial, in the sense that we always get at least one amplitude that vanishes all by itself; however, already at five points nontrivial cancellations start to occur. For instance, in the form withj=3, we have(3.23)0=s12s34A5(1-,2-,3+,4+,5+)Ã5(2-,1-,4+,3-,5+)+s13s24A5(1-,3+,2-,4+,5+)Ã5(3-,1-,4+,2-,5+).We also note that just as the KLT relations could be written in the form of (3.11) and (3.12) with a regularization, so can these vanishing relations. Equation (3.22) can be directly proven from the analytic properties of tree-level scattering amplitudes, but there is also a more physical understanding of these identities. Looking at the KLT relations from an 𝒩=8 supergravity point of view, the vanishing of the r.h.s. of (3.3), when An and Ãn belong to different helicity sectors, correspond to SU(8)-violating gravity amplitudes and must therefore vanish [17, 45, 46] (see also [22, 50]).


### 3.5. BCFW Recursion Relation

Contrary to the properties and relations reviewed above, this section is not directly related to the KLT relations, but will be essential for Section4. Not only will the BCFW expansion of amplitudes in itself be important, but also the method from which it can be derived [27].Start by deforming two of the external momenta, saypj and pl, as(3.24)pj⟶p̂j=pj-zq,pl⟶p̂l=pl+zq,
where z is a complex variable and q is a four-vector satisfying q2=q·pj=q·pl=0. This preserves conservation of momentum and on-shellness. At tree-level An is a rational function of external momenta, implying that the deformed amplitude An(z) is a rational function of z. Assuming An(z)→0 when z→∞, Cauchy’s theorem tells us that(3.25)0=12πi∮dzzAn(z)=An(0)+∑poleszp≠0Resp(An(z),zp)zp,
where An(0) is the undeformed amplitude, coming from the residue of the z=0 pole. The remaining poles (those different from z=0) come from An(z), which we know can only follow from Feynman propagators going on-shell, that is, when Pk,m(z)2 vanishes, where(3.26)Pk,m(z)≡pk+pk+1+⋯+p̂j+⋯+pm,
with j∈{k,…,m} and l∉{k,…,m} (or vice versa). We do not get poles in z if j,l∈{k,…,m} since such a Pk,m is independent of z.Let us denote the value ofz where Pk,m(z) is going on-shell by zk,m, which can be found by solving Pk,m(zk,m)2=0. Since the poles of An(z) will be simple poles, the residues are given by(3.27)Resp(An(z),zk,m)zk,m=-lim⁡z→zk,m[Pk,m(z)2An(z)]Pk,m2,
where Pk,m=Pk,m(0). Using the general factorization property of gluon amplitudes(3.28)An→P2→0∑h=±Ar+1(k,…,m,-P-h)1P2An-r+1(Ph,m+1,…,k-1),
we see that(3.29)Resp(An(z),zk,m)zk,m=-∑h=±Ar+1(k,…,m,-P̂k,m-h)1Pk,m2An-r+1(P̂k,mh,m+1,…,k-1),
with P̂k,m≡Pk,m(zk,m). Hence, combined with (3.25), we get the BCFW recursion relation [26, 27](3.30)An=∑r∑h=±Ar+1(k,…,m,-P̂k,m-h)1Pk,m2An-r+1(P̂k,mh,m+1,…,k-1),
where the sum, denoted with r, states that we must sum over all internal momenta affected by the deformation.This derivation was performed for color-ordered tree amplitudes; however, the recursion relation is also valid for tree-level gravity amplitudes, we just need to sum overall different combinations of momenta where the pole includes one of the deformed legs [51, 52]. It has been shown that the fall-off at z→∞ for the graviton amplitude is even stronger than one could naively have guessed [53–62].


## 3.1.  Lower-Point Examples

The connection between gravity and gauge theory already starts at three points. It is well known that for real momenta an on-shell three-point amplitude must vanish. However, if we go to complex momenta this is no longer the case. Interestingly, it turns out that Lorentz invariance, and the spin of the external particles, uniquely fix the three-point amplitude, see for example, [47]. In detail, the spin 2 three-point amplitude (gravitons) is the product of two spin 1 three-point amplitudes (gluons)(3.7)M3(1,2,3)=A3(1,2,3)Ã3(1,2,3).
This can also be directly seen from Feynman rules when legs are put on-shell, something that in particular simplifies the expression for the gravity three-point vertex. Although we did not have the complex three-point amplitude in mind when we wrote down (3.3), it does give the correct result even in this case. At first it might seem a bit strange to consider amplitudes with complex momenta, but recent progress in amplitude calculations has largely been inspired by such an extension. Indeed, the BCFW recursion relation, that will be reviewed below, relies on the deformation to complex momenta, and so will our proof of (3.3) in Section 4.The four point case also looks very simple(3.8)M4(1,2,3,4)=-s12A4(1,2,3,4)Ã4(1,2,4,3),
however, its field theoretical origin already becomes a bit unclear here. Compared to (3.7) we see the appearance of a kinematic factor that makes sure to cancel one of the s12 poles present in both gauge theory amplitudes. We also see that the total crossing symmetry of the r.h.s. has already been well hidden, although we know it has to be there.When we go to five points, even expressed in the form with fewest terms (j=3),(3.9)M5=s12s34A5(1,2,3,4,5)Ã5(2,1,4,3,5)+s13s24A5(1,3,2,4,5)Ã5(3,1,4,2,5),
not only is the total crossing symmetry of the r.h.s. by no means obvious, but also the correct cancellation of poles begins to get more complicated to see. These properties only get more and more disguised as we increase the number of external particles.From the above examples it might seem almost impossible to identify, in all generality, the r.h.s. of (3.3) with a gravity amplitude from a purely field theoretical/analytical point of view. Even the simplest features of gravity amplitudes have become very nontrivial statements about the gauge theory side. Before we can attack this problem we therefore need to review some important concepts and properties. These will not only lead to a better understanding of KLT relations in field theory, but are by themselves amazing structures of scattering amplitudes. The first thing we will introduce is a rather unusual way of expressing the KLT relations. This form will have a higher degree of manifest crossing symmetry, compared to (3.3), but it requires a regularization. It has been proven separately by pure field theory, and will turn out to be important even in the proof of (3.3). Secondly, we introduce the BCJ relations, providing identities between color-ordered tree-level amplitudes, and explain its connection to the j-independence of (3.3). Third, we look at what happens with the r.h.s. of (3.3) when An and Ãn belong to different helicity sectors. Fourth, we review the BCFW recursion relation, exploiting very general analytic properties of tree-level amplitudes to recursively construct higher-point amplitudes from lower-point ones. All of these structures will be important for Section 4, where we will utilize them in order to give a purely field theoretical proof of (3.3).


## 3.2.  Regularized KLT Relations

Consideringn-point amplitudes we start by making the following deformation of momenta p1 and pn:(3.10)p1⟶p1′=p1-xq,pn⟶pn′=pn+xq,
where x is just some arbitrary parameter, and q is a four-vector satisfying q·p1=q2=0 and q·pn≠0. This preserves conservation of momentum and keeps p1′2=0, but makes pn′2=s1′2…(n-1)≠0.The gravity amplitudeMn can then be written as [15](3.11)Mn=(-1)nlim⁡x→0∑σ,σ̃∈Sn-2Ãn(n′,σ̃2,n-1,1′)S[σ̃2,n-1∣σ2,n-1]p1′An(1′,σ2,n-1,n′)s1′2…(n-1).
We note that as x→0 the denominator goes to zero, but as we will see from (3.16) below, so does the numerator. However, the whole expression has a limit which is exactly equal to a gravity amplitude. This can also be seen by taking the soft limit of leg n in (3.4) [18], and comparing against the well-known soft-limit behaviour of gravity amplitudes [12, 48] (see [49] for an alternative approach). This might make (3.11) seem less strange, but we do stress that this expression for a gravity amplitude can be proven without knowledge of (3.4) (or in general (3.3)).If we make a deformation withp1′ being off-shell instead of pn′, we can write down the “dual’’ expression to (3.11)(3.12)Mn=(-1)nlim⁡y→0∑σ,σ̃∈Sn-2An(1′,σ2,n-1,n′)S[σ2,n-1∣σ̃2,n-1]pn′Ãn(n′,σ̃2,n-1,1′)s23…n′,
where we have called the deformation parameter y.


## 3.3.  BCJ Relations

The story of relations among color-ordered tree-level amplitudes is in itself very interesting. However, to not lose focus of our current goal we will here only remind the reader about the Bern-Carrasco-Johansson (BCJ) relations, which will play an important role for us.There are several different ways of presenting the BCJ relations, which all have their own advantages and disadvantages. Here we choose one of the simplest general representations. Forn-point amplitudes these read(3.13)0=s12An(1,2,3,…,n)+(s12+s23)An(1,3,2,4,…,n)+(s12+s23+s24)An(1,3,4,2,5,…,n)+⋯+(s12+s23+s24+⋯+s2(n-1))An(1,3,4,…,n-1,2,n),
along with all relations obtained by permutation of labels in the above expression. Note how leg 2 moves one position to the right in each term and picks up an additional factor of s2i, where i is the leg just passed through. This particular form of the BCJ relations actually contains all the information, in the sense that once you have these you can go from the (n-2)! color-ordered amplitudes, provided by the Kleiss-Kuijf relations, down to (n-3)! independent amplitudes [38].Some simple explicit examples of (3.13) are(3.14)0=s12A4(1,2,3,4)+(s12+s23)A4(1,3,2,4),
for four points, and(3.15)0=s12A5(1,2,3,4,5)+(s12+s23)A5(1,3,2,4,5)+(s12+s23+s24)A5(1,3,4,2,5),
at five points. We could of course use momentum conservation to write some of the kinematic invariant factors simpler, but keeping them like this make the relations easy to remember.It turns out, that there is a very nice way of rephrasing the above BCJ relations in terms of our𝒮 function. This is achieved through(3.16)0=∑σ∈Sn-2S[β(2,…,n-1)∣σ(2,…,n-1)]k1An(1,σ(2,…,n-1),n),
where β is just some arbitrary permutation of the legs {2,3,…,n-1}. This is nothing but a sum of relations in the form of (3.13). To see this, let us first write out the five-point case with β(2,3,4)=(2,3,4) explicitly, that is,
(3.17)∑σ∈S3S[2,3,4∣σ(2,3,4)]k1A5(1,σ(2,3,4),5)=s13s14[s12A5(1,2,3,4,5)+(s12+s23)A5(1,3,2,4,5)+(s12+s23+s24)A5(1,3,4,2,5)]+s14(s13+s34)×[s12A5(1,2,4,3,5)+(s12+s24)A5(1,4,2,3,5)+(s12+s23+s24)A5(1,4,3,2,5)].
The two [⋯] is exactly zero due to (3.13).The general argument is to divide the sum ofσ(2,…,n-1) into a sum of groups where all, except the first leg in the β ordering, call it β(2), have fixed ordering, and then insert β(2) at any place. For each group all factors from 𝒮 will be the same except for the factor contributing from β(2). This will give a BCJ relation and thereby vanish. In the above five-point example we had β(2)=2 and the two groups we summed over were the one with ordering {3,4} and {4,3}, respectively. From this we see that (3.16) is a consequence of (3.13).


### 3.3.1.  Thej-Independence

With the BCJ relations we can now address thej-independence of (3.3) from a field theory point of view, which in string theory followed naturally due to the freedom in how one closes the contours. Indeed, it is useful to first establish the equivalence between all forms only differing in the j-value chosen. In this way, if we can prove (3.3) for just one specific form, we have proven them all.The form of BCJ relation that implies thej-independence is given by
(3.18)∑α,βS[αi2,ij∣i2,…,ij]p1S[ij+1,…,in-2∣βij+1,in-2]pn-1Ãn(αi2,ij,1,n-1,βij+1,in-2,n)=∑α′,β′S[αi2,ij-1′∣i2,…,ij-1]p1S[ij,ij+1,…,in-2∣βij,in-2′]pn-1Ãn(αi2,ij-1′,1,n-1,βij,in-2′,n).
Although, (3.18) looks rather complicated it is straightforward to prove using only BCJ relations in the form of (3.13), and momentum conservation. Since the proof is not very enlightening we will not repeat it here, but simply refer to the appendix of [16] for details. Instead we will illustrate how to use (3.18) in the simple case of five points.Forn=5 and j=2 (3.18) reads (where we have taken ik=k)(3.19)S[2∣2]p1S[3∣3]p4Ã5(2,1,4,3,5)=∑β′S[2,3∣β′(2,3)]p4Ã5(1,4,β′(2,3),5),
and since (3.3) for n=5 and j=2 is given by(3.20)M5=∑σ∑βA5(1,σ(2,3),4,5)S[σ(2,3)∣β(2,3)]k4Ã5(1,4,β(2,3),5)=∑σA5(1,σ(2,3),4,5)∑βS[σ(2,3)∣β(2,3)]k4Ã5(1,4,β(2,3),5),
using (3.19) we immediately get(3.21)M5=∑σA5(1,σ(2,3),4,5)S[σ(2)∣σ(2)]p1S[σ(3)∣σ(3)]p4Ã5(σ(2),1,4,σ(3),5),
which is just (3.3) with j=3. Likewise we could now use (3.18) with j=3 to rewrite (3.21) into (3.3) with j=4.This procedure generalizes ton points; by repeated use of (3.18) the j-independence of (3.3) directly follows.It is interesting to note that the BCJ relations could have been discovered much earlier. Indeed, equating different expressions for the KLT relations leads to pure gauge theory amplitude relations, which, as we have just seen, are directly related to the BCJ relations. This was already apparent in [1] but at that time not really appreciated. We also see that writing both the KLT and BCJ relations in terms of the 𝒮 function, their close connection becomes much more manifest than one would have anticipated from the original papers.


## 3.3.1.  Thej-Independence

With the BCJ relations we can now address thej-independence of (3.3) from a field theory point of view, which in string theory followed naturally due to the freedom in how one closes the contours. Indeed, it is useful to first establish the equivalence between all forms only differing in the j-value chosen. In this way, if we can prove (3.3) for just one specific form, we have proven them all.The form of BCJ relation that implies thej-independence is given by
(3.18)∑α,βS[αi2,ij∣i2,…,ij]p1S[ij+1,…,in-2∣βij+1,in-2]pn-1Ãn(αi2,ij,1,n-1,βij+1,in-2,n)=∑α′,β′S[αi2,ij-1′∣i2,…,ij-1]p1S[ij,ij+1,…,in-2∣βij,in-2′]pn-1Ãn(αi2,ij-1′,1,n-1,βij,in-2′,n).
Although, (3.18) looks rather complicated it is straightforward to prove using only BCJ relations in the form of (3.13), and momentum conservation. Since the proof is not very enlightening we will not repeat it here, but simply refer to the appendix of [16] for details. Instead we will illustrate how to use (3.18) in the simple case of five points.Forn=5 and j=2 (3.18) reads (where we have taken ik=k)(3.19)S[2∣2]p1S[3∣3]p4Ã5(2,1,4,3,5)=∑β′S[2,3∣β′(2,3)]p4Ã5(1,4,β′(2,3),5),
and since (3.3) for n=5 and j=2 is given by(3.20)M5=∑σ∑βA5(1,σ(2,3),4,5)S[σ(2,3)∣β(2,3)]k4Ã5(1,4,β(2,3),5)=∑σA5(1,σ(2,3),4,5)∑βS[σ(2,3)∣β(2,3)]k4Ã5(1,4,β(2,3),5),
using (3.19) we immediately get(3.21)M5=∑σA5(1,σ(2,3),4,5)S[σ(2)∣σ(2)]p1S[σ(3)∣σ(3)]p4Ã5(σ(2),1,4,σ(3),5),
which is just (3.3) with j=3. Likewise we could now use (3.18) with j=3 to rewrite (3.21) into (3.3) with j=4.This procedure generalizes ton points; by repeated use of (3.18) the j-independence of (3.3) directly follows.It is interesting to note that the BCJ relations could have been discovered much earlier. Indeed, equating different expressions for the KLT relations leads to pure gauge theory amplitude relations, which, as we have just seen, are directly related to the BCJ relations. This was already apparent in [1] but at that time not really appreciated. We also see that writing both the KLT and BCJ relations in terms of the 𝒮 function, their close connection becomes much more manifest than one would have anticipated from the original papers.


## 3.4.  Vanishing Identities

The next surprising property of (3.3) we want to introduce is that if we take An and Ãn to live in different helicity sectors the r.h.s. of (3.3) is identically zero.To be more specific let us denote ann-point color-ordered tree amplitude with helicity configuration belonging to NkMHV as Ank, that is, Ank has 2+k negative helicity gluons, with k∈{0,1,…,n-4} for nonvanishing amplitudes. We are not interested in the exact helicity configuration, just which helicity sector it belongs to. We then have [43, 44](3.22)0=∑σ∑γ,βS[γ(σ2,j-1)∣σ2,j-1]k1S[σj,n-2∣β(σj,n-2)]kn-1Anh(1,σ2,j-1,σj,n-2,n-1,n)×Ãnk≠h(γ(σ2,j-1),1,n-1,β(σj,n-2),n),
which is nothing but the r.h.s. of (3.3), written in our short-hand notation, with the color-ordered amplitudes living in different helicity sectors.At four points these relations are trivial, in the sense that we always get at least one amplitude that vanishes all by itself; however, already at five points nontrivial cancellations start to occur. For instance, in the form withj=3, we have(3.23)0=s12s34A5(1-,2-,3+,4+,5+)Ã5(2-,1-,4+,3-,5+)+s13s24A5(1-,3+,2-,4+,5+)Ã5(3-,1-,4+,2-,5+).We also note that just as the KLT relations could be written in the form of (3.11) and (3.12) with a regularization, so can these vanishing relations. Equation (3.22) can be directly proven from the analytic properties of tree-level scattering amplitudes, but there is also a more physical understanding of these identities. Looking at the KLT relations from an 𝒩=8 supergravity point of view, the vanishing of the r.h.s. of (3.3), when An and Ãn belong to different helicity sectors, correspond to SU(8)-violating gravity amplitudes and must therefore vanish [17, 45, 46] (see also [22, 50]).


## 3.5. BCFW Recursion Relation

Contrary to the properties and relations reviewed above, this section is not directly related to the KLT relations, but will be essential for Section4. Not only will the BCFW expansion of amplitudes in itself be important, but also the method from which it can be derived [27].Start by deforming two of the external momenta, saypj and pl, as(3.24)pj⟶p̂j=pj-zq,pl⟶p̂l=pl+zq,
where z is a complex variable and q is a four-vector satisfying q2=q·pj=q·pl=0. This preserves conservation of momentum and on-shellness. At tree-level An is a rational function of external momenta, implying that the deformed amplitude An(z) is a rational function of z. Assuming An(z)→0 when z→∞, Cauchy’s theorem tells us that(3.25)0=12πi∮dzzAn(z)=An(0)+∑poleszp≠0Resp(An(z),zp)zp,
where An(0) is the undeformed amplitude, coming from the residue of the z=0 pole. The remaining poles (those different from z=0) come from An(z), which we know can only follow from Feynman propagators going on-shell, that is, when Pk,m(z)2 vanishes, where(3.26)Pk,m(z)≡pk+pk+1+⋯+p̂j+⋯+pm,
with j∈{k,…,m} and l∉{k,…,m} (or vice versa). We do not get poles in z if j,l∈{k,…,m} since such a Pk,m is independent of z.Let us denote the value ofz where Pk,m(z) is going on-shell by zk,m, which can be found by solving Pk,m(zk,m)2=0. Since the poles of An(z) will be simple poles, the residues are given by(3.27)Resp(An(z),zk,m)zk,m=-lim⁡z→zk,m[Pk,m(z)2An(z)]Pk,m2,
where Pk,m=Pk,m(0). Using the general factorization property of gluon amplitudes(3.28)An→P2→0∑h=±Ar+1(k,…,m,-P-h)1P2An-r+1(Ph,m+1,…,k-1),
we see that(3.29)Resp(An(z),zk,m)zk,m=-∑h=±Ar+1(k,…,m,-P̂k,m-h)1Pk,m2An-r+1(P̂k,mh,m+1,…,k-1),
with P̂k,m≡Pk,m(zk,m). Hence, combined with (3.25), we get the BCFW recursion relation [26, 27](3.30)An=∑r∑h=±Ar+1(k,…,m,-P̂k,m-h)1Pk,m2An-r+1(P̂k,mh,m+1,…,k-1),
where the sum, denoted with r, states that we must sum over all internal momenta affected by the deformation.This derivation was performed for color-ordered tree amplitudes; however, the recursion relation is also valid for tree-level gravity amplitudes, we just need to sum overall different combinations of momenta where the pole includes one of the deformed legs [51, 52]. It has been shown that the fall-off at z→∞ for the graviton amplitude is even stronger than one could naively have guessed [53–62].


## 4. A Purely Field Theoretical View

We are finally in a position to see why (3.3) must be true for all n from a purely field theoretical point of view. This will be obtained in terms of an induction proof, and since we have already shown that the r.h.s. of (3.3) is equivalent for all j-values, we are free to choose any of the versions we like the most. For us this will be the ones shown in (3.4) and (3.5).


### 4.1.  Ann-Point Proof

We assume that we have checked (3.3) up to n-1 points, that is, we have checked that the expression on the r.h.s. is equal to gravity. Then we write down the n-point expression for the r.h.s., let us denote it Rn. Our goal is to show, only based on our knowledge of lower point cases, that this is equal to the n-point gravity amplitude, that is Rn=Mn.In the same way as the BCFW recursion relation was derived, we start out by deforming two momenta in our expression forRn and consider the contour integral(4.1)B=12πi∮dzzRn(z)=Rn(0)+∑poleszp≠0Resp(Rn(z),zp)zp,
where Rn(0) is just the undeformed n-point expression and we have included a potential boundary term B on the left-hand side. Let us first argue that B=0. If we make a deformation in p1 and pn the 𝒮 function in (3.5) will be independent of z. The vanishing of the boundary term is then guaranteed by the large z behaviour of the gauge theory amplitudes An and Ãn, this was also used in [19]. In the proof we will present below it is more convenient to make a deformation in p1 and pn-1. However, such a deformation cannot have a boundary term either since in Section 3 we already argued for the crossing symmetry between pn and pn-1 in Rn. The large z behaviour must therefore be equally good for this deformation.Now that we have establishedB=0, the goal is to show that the sum of residues exactly makes up the BCFW-expansion of an n-point gravity amplitude, and hence(4.2)Mn~∑poleszp≠0Resp(Rn(z),zp)zp.
If we can show this, using only lower-point cases and the properties/relations reviewed in the last section, we are done.We begin by considering the residues that follow from poles of the forms12…k, and we will be using Rn in the form of (3.4). Like in (3.27) we can compute these from limz→z12…k[s1̂2…k(z)Rn(z)]/s12…k, where z12…k is the z-value that makes s1̂2…k go on-shell.There are basically only two classes of terms which have the possibility of contributing to a residue.(a)
The pole appears only inone of the amplitudes Ãn or An.(b)
The pole appears inbothÃn and An.First we investigate (a), and we will consider the case where the pole only appears inÃn. The case with the pole appearing in An can be handled in a similar way. The terms contributing to this class must all involve an Ãn amplitude with the set of legs {1,2,…,k} always next to each other, that is, the terms from (3.4) that make up this contribution is(4.3)∑σ,σ̃,αÃn(n-1̂,n,σ̃k+1,n-2,α2,k,1̂)S[σ̃k+1,n-2α2,k∣σ2,n-2]p̂1An(1̂,σ2,n-2,n-1̂,n),
where we have omitted the overall factor of (-1)n+1, which can easily be reinstated into the proof. We emphasis that since we are considering (a) we have excluded all σ permutations that would lead to a s12…k pole in An. From this we get the residue(4.4)∑σ,σ̃,α∑hÃ(n-1̂,n,σ̃k+1,n-2,P̂h)Ã(-P̂-h,α2,k,1̂)s12…kS[σ̃k+1,n-2α2,k∣σ2,n-2]p̂1×An(1̂,σ2,n-2,n-1̂,n),
where P̂≡p̂1+p2+⋯+pk, and we have used the factorization property of color-ordered gauge theory amplitudes in (3.28). Also note that the pole s1̂2…k, from the factorization of Ãn, has been replaced with s12…k, that is, without the hat on 1, from the calculation of the residue. Furthermore, following from the definition of 𝒮 in (3.2), we can write(4.5)S[σ̃k+1,n-2α2,k∣σ2,n-2]p̂1=S[α2,k∣ρ2,k]p̂1×(afactorindependentofα),
where ρ denotes the relative ordering of leg 2,3,…,k in the σ set. Collecting everything in (4.4) that involves the α permutation we have something of the form(4.6)∑σ∑h(∑αÃ(-P̂-h,α2,k,1̂)S[α2,k∣ρ2,k]p̂1)︸0×∑σ̃[⋯],
where, as we have seen above, the quantity inside (⋯) vanishes when P̂ is on-shell (to get it in the exact same form as (3.16) just use the reflection symmetry and (3.6)). We therefore conclude that contributions from (a) vanish altogether. All the nonvanishing stuff must come from (b). Let us see how this comes about.Since bothÃn and An now contain the pole s12…k, they must both have the set of legs {1,2,…,k} collected next to each other. The contributing terms, from (3.4), then take the form(4.7)∑σ,σ̃,α,βÃn(n-1̂,n,σ̃k+1,n-2,α2,k,1̂)S[σ̃k+1,n-2α2,k∣β2,kσk+1,n-2]p̂1An(1̂,β2,k,σk+1,n-2,n-1̂,n).
Like in (4.5) we will again make use of a factorization property of the 𝒮 function. It is not hard to convince oneself that in the limit where P̂=p̂1+p2+⋯+pk goes on-shell, we can write(4.8)S[σ̃k+1,n-2α2,k∣β2,kσk+1,n-2]p̂1=S[α2,k∣β2,k]p̂1×S[σ̃k+1,n-2∣σk+1,n-2]P̂.
The residue for s12…k can then be expressed as(4.9)1s12…k∑hlim⁡z→z12…k∑α,βÃ(-P̂h,α2,k,1̂)S[α2,k∣β2,k]p̂1A(1̂,β2,k,-P̂h)s1̂2…k×∑σ,σ̃Ã(n-1̂,n,σ̃k+1,n-2,P̂-h)S[σ̃k+1,n-2∣σk+1,n-2]P̂A(P̂-h,σk+1,n-2,n-1̂,n)+(mixedhelicityterms),
where the “mixed helicity terms’’ are expressions of the same form but with products between amplitudes with, for instance, (P̂h,P̂-h) instead of (-P̂h,-P̂h) or (P̂-h,P̂-h) like in line one and two of (4.9). However, as we discussed in Section 3.4, such terms are identically zero.What is left is a sum overα and β, which precisely makes up the regularized KLT form in (3.11), that is, it is just Mk+1(1̂,2,…,k,-P̂h), and a sum over σ and σ̃ which is an n-k+1 point version of (3.4) and hence, by induction, equal to Mn-k+1(k+1,…,P̂-h). Altogether (4.9) is(4.10)∑hMk+1(1̂,2,…,k,-P̂h)Mn-k+1(k+1,…,P̂-h)s12…k.
This is exactly the BCFW contribution to the n-point gravity amplitude from a s12…k pole, and due to the manifest (n-3)! symmetry we have also already obtained the contributions for all poles related to these by a permutation of legs 2,3,…,n-2.We are still not completely done, since the above analysis does not cover the pole contributions involving both 1 andn, that is, poles of the form s12…kn=sk+1…n-1. To investigate these we use the form in (3.5). It is well suited for this case since here legs 1 and n are always next to each other, and as we have already mentioned, we are free to use whichever form we want to calculate a residue, they are just different ways of writing the same quantity. The following calculations are very similar to what we have already seen so we go through it a bit more briefly.The (a) part of the residue for poles12…kn takes the form(4.11)∑σ,σ̃,αAn(1̂,σ2,n-2,n-1̂,n)S[σ2,n-2∣σ̃k+1,n-2α2,k]p̂n-1∑hÃ(n-1̂,σ̃k+1,n-2,P̂h)Ã(-P̂-h,α2,k,n,1̂)s12…kn,
where we again assume the pole appears in the Ãn amplitude. Using the factorization property(4.12)S[σ2,n-2∣σ̃k+1,n-2α2,k]p̂n-1=S[ρk+1,n-2∣σ̃k+1,n-2]p̂n-1×(afactorindependentofσ̃),
where ρ denotes the relative ordering of leg k+1,…,n-2 in the σ set, we once again find that these contributions contain a factor of(4.13)∑σ̃Ã(n-1̂,σ̃k+1,n-2,P̂h)S[ρk+1,n-2∣σ̃k+1,n-2]p̂n-1=0,
that vanishes due to BCJ relations. There is therefore no contribution from (a).Considering part (b) for thes12…kn pole, the contributing terms from (3.5) are(4.14)∑σ,σ̃,α,βAn(1̂,β2,k,σk+1,n-2,n-1̂,n)S[β2,kσk+1,n-2∣σ̃k+1,n-2α2,k]p̂n-1Ãn(1̂,n-1̂,σ̃k+1,n-2,α2,k,n),
with 𝒮 satisfying the factorization property (when P̂ goes on-shell)(4.15)S[β2,kσk+1,n-2∣σ̃k+1,n-2α2,k]p̂n-1=S[σk+1,n-2∣σ̃k+1,n-2]p̂n-1×S[β2,k∣α2,k]P̂.
Hence the residue can be written(4.16)1s12…kn∑h∑α,βA(1̂,β2,k,P̂h,n)S[β2,k∣α2,k]P̂Ã(1̂,P̂h,α2,k,n)×lim⁡z→zk+1…n-1∑σ̃,σA(-P̂-h,σk+1,n-2,n-1̂)S[σk+1,n-2∣σ̃k+1,n-2]p̂n-1Ã(n-1̂,σ̃k+1,n-2,-P̂-h)sk+1…n-1̂,
where we have used s1̂2…kn=sk+1…n-1̂, and already removed the vanishing mixed helicity terms. The first part is just a lower-point version of (3.5), and the second part the regularized dual KLT form, that is, we get(4.17)∑hMk+2(1̂,2,…,k,n,P̂h)Mn-k(k+1,…,n-1̂,-P̂-h)s12…kn.
Once again we obtain the correct BCFW expansion for all s12…kn poles, and all poles related to these by a permutation of 2,3,…,n-2.We have now covered all residues in (4.1), and see that they indeed make up the full BCFW expansion of the n-point gravity amplitude, and therefore Rn=Mn, see Figure 4. Notice how all of the properties/relations from Section 3 played an important role; the BCFW method was the main tool for the whole proof, the BCJ relations were needed not only for showing the equivalence between all j-values, but also to argue for the vanishing of contributions from (a), and both the vanishing identities and regularized KLT relations were important for identifying contributions from (b) with terms from a BCFW expansion of Mn. We also stress that no other crossing symmetry than that which was already manifest has been used, so the identification with a gravity amplitude is also an indirect proof of the total crossing symmetry of the r.h.s. of (3.3).Figure 4
Schematic outline of field theory proof.Maybe the most important thing to take from this proof is that it very explicitly illustrates how amazingly constrained scattering amplitudes are just from their very general analytical properties. These constraints are so strong that they basically force perturbative gravity and gauge theories to be related through the KLT relations, although a priori these theories seem completely unrelated.


## 4.1.  Ann-Point Proof

We assume that we have checked (3.3) up to n-1 points, that is, we have checked that the expression on the r.h.s. is equal to gravity. Then we write down the n-point expression for the r.h.s., let us denote it Rn. Our goal is to show, only based on our knowledge of lower point cases, that this is equal to the n-point gravity amplitude, that is Rn=Mn.In the same way as the BCFW recursion relation was derived, we start out by deforming two momenta in our expression forRn and consider the contour integral(4.1)B=12πi∮dzzRn(z)=Rn(0)+∑poleszp≠0Resp(Rn(z),zp)zp,
where Rn(0) is just the undeformed n-point expression and we have included a potential boundary term B on the left-hand side. Let us first argue that B=0. If we make a deformation in p1 and pn the 𝒮 function in (3.5) will be independent of z. The vanishing of the boundary term is then guaranteed by the large z behaviour of the gauge theory amplitudes An and Ãn, this was also used in [19]. In the proof we will present below it is more convenient to make a deformation in p1 and pn-1. However, such a deformation cannot have a boundary term either since in Section 3 we already argued for the crossing symmetry between pn and pn-1 in Rn. The large z behaviour must therefore be equally good for this deformation.Now that we have establishedB=0, the goal is to show that the sum of residues exactly makes up the BCFW-expansion of an n-point gravity amplitude, and hence(4.2)Mn~∑poleszp≠0Resp(Rn(z),zp)zp.
If we can show this, using only lower-point cases and the properties/relations reviewed in the last section, we are done.We begin by considering the residues that follow from poles of the forms12…k, and we will be using Rn in the form of (3.4). Like in (3.27) we can compute these from limz→z12…k[s1̂2…k(z)Rn(z)]/s12…k, where z12…k is the z-value that makes s1̂2…k go on-shell.There are basically only two classes of terms which have the possibility of contributing to a residue.(a)
The pole appears only inone of the amplitudes Ãn or An.(b)
The pole appears inbothÃn and An.First we investigate (a), and we will consider the case where the pole only appears inÃn. The case with the pole appearing in An can be handled in a similar way. The terms contributing to this class must all involve an Ãn amplitude with the set of legs {1,2,…,k} always next to each other, that is, the terms from (3.4) that make up this contribution is(4.3)∑σ,σ̃,αÃn(n-1̂,n,σ̃k+1,n-2,α2,k,1̂)S[σ̃k+1,n-2α2,k∣σ2,n-2]p̂1An(1̂,σ2,n-2,n-1̂,n),
where we have omitted the overall factor of (-1)n+1, which can easily be reinstated into the proof. We emphasis that since we are considering (a) we have excluded all σ permutations that would lead to a s12…k pole in An. From this we get the residue(4.4)∑σ,σ̃,α∑hÃ(n-1̂,n,σ̃k+1,n-2,P̂h)Ã(-P̂-h,α2,k,1̂)s12…kS[σ̃k+1,n-2α2,k∣σ2,n-2]p̂1×An(1̂,σ2,n-2,n-1̂,n),
where P̂≡p̂1+p2+⋯+pk, and we have used the factorization property of color-ordered gauge theory amplitudes in (3.28). Also note that the pole s1̂2…k, from the factorization of Ãn, has been replaced with s12…k, that is, without the hat on 1, from the calculation of the residue. Furthermore, following from the definition of 𝒮 in (3.2), we can write(4.5)S[σ̃k+1,n-2α2,k∣σ2,n-2]p̂1=S[α2,k∣ρ2,k]p̂1×(afactorindependentofα),
where ρ denotes the relative ordering of leg 2,3,…,k in the σ set. Collecting everything in (4.4) that involves the α permutation we have something of the form(4.6)∑σ∑h(∑αÃ(-P̂-h,α2,k,1̂)S[α2,k∣ρ2,k]p̂1)︸0×∑σ̃[⋯],
where, as we have seen above, the quantity inside (⋯) vanishes when P̂ is on-shell (to get it in the exact same form as (3.16) just use the reflection symmetry and (3.6)). We therefore conclude that contributions from (a) vanish altogether. All the nonvanishing stuff must come from (b). Let us see how this comes about.Since bothÃn and An now contain the pole s12…k, they must both have the set of legs {1,2,…,k} collected next to each other. The contributing terms, from (3.4), then take the form(4.7)∑σ,σ̃,α,βÃn(n-1̂,n,σ̃k+1,n-2,α2,k,1̂)S[σ̃k+1,n-2α2,k∣β2,kσk+1,n-2]p̂1An(1̂,β2,k,σk+1,n-2,n-1̂,n).
Like in (4.5) we will again make use of a factorization property of the 𝒮 function. It is not hard to convince oneself that in the limit where P̂=p̂1+p2+⋯+pk goes on-shell, we can write(4.8)S[σ̃k+1,n-2α2,k∣β2,kσk+1,n-2]p̂1=S[α2,k∣β2,k]p̂1×S[σ̃k+1,n-2∣σk+1,n-2]P̂.
The residue for s12…k can then be expressed as(4.9)1s12…k∑hlim⁡z→z12…k∑α,βÃ(-P̂h,α2,k,1̂)S[α2,k∣β2,k]p̂1A(1̂,β2,k,-P̂h)s1̂2…k×∑σ,σ̃Ã(n-1̂,n,σ̃k+1,n-2,P̂-h)S[σ̃k+1,n-2∣σk+1,n-2]P̂A(P̂-h,σk+1,n-2,n-1̂,n)+(mixedhelicityterms),
where the “mixed helicity terms’’ are expressions of the same form but with products between amplitudes with, for instance, (P̂h,P̂-h) instead of (-P̂h,-P̂h) or (P̂-h,P̂-h) like in line one and two of (4.9). However, as we discussed in Section 3.4, such terms are identically zero.What is left is a sum overα and β, which precisely makes up the regularized KLT form in (3.11), that is, it is just Mk+1(1̂,2,…,k,-P̂h), and a sum over σ and σ̃ which is an n-k+1 point version of (3.4) and hence, by induction, equal to Mn-k+1(k+1,…,P̂-h). Altogether (4.9) is(4.10)∑hMk+1(1̂,2,…,k,-P̂h)Mn-k+1(k+1,…,P̂-h)s12…k.
This is exactly the BCFW contribution to the n-point gravity amplitude from a s12…k pole, and due to the manifest (n-3)! symmetry we have also already obtained the contributions for all poles related to these by a permutation of legs 2,3,…,n-2.We are still not completely done, since the above analysis does not cover the pole contributions involving both 1 andn, that is, poles of the form s12…kn=sk+1…n-1. To investigate these we use the form in (3.5). It is well suited for this case since here legs 1 and n are always next to each other, and as we have already mentioned, we are free to use whichever form we want to calculate a residue, they are just different ways of writing the same quantity. The following calculations are very similar to what we have already seen so we go through it a bit more briefly.The (a) part of the residue for poles12…kn takes the form(4.11)∑σ,σ̃,αAn(1̂,σ2,n-2,n-1̂,n)S[σ2,n-2∣σ̃k+1,n-2α2,k]p̂n-1∑hÃ(n-1̂,σ̃k+1,n-2,P̂h)Ã(-P̂-h,α2,k,n,1̂)s12…kn,
where we again assume the pole appears in the Ãn amplitude. Using the factorization property(4.12)S[σ2,n-2∣σ̃k+1,n-2α2,k]p̂n-1=S[ρk+1,n-2∣σ̃k+1,n-2]p̂n-1×(afactorindependentofσ̃),
where ρ denotes the relative ordering of leg k+1,…,n-2 in the σ set, we once again find that these contributions contain a factor of(4.13)∑σ̃Ã(n-1̂,σ̃k+1,n-2,P̂h)S[ρk+1,n-2∣σ̃k+1,n-2]p̂n-1=0,
that vanishes due to BCJ relations. There is therefore no contribution from (a).Considering part (b) for thes12…kn pole, the contributing terms from (3.5) are(4.14)∑σ,σ̃,α,βAn(1̂,β2,k,σk+1,n-2,n-1̂,n)S[β2,kσk+1,n-2∣σ̃k+1,n-2α2,k]p̂n-1Ãn(1̂,n-1̂,σ̃k+1,n-2,α2,k,n),
with 𝒮 satisfying the factorization property (when P̂ goes on-shell)(4.15)S[β2,kσk+1,n-2∣σ̃k+1,n-2α2,k]p̂n-1=S[σk+1,n-2∣σ̃k+1,n-2]p̂n-1×S[β2,k∣α2,k]P̂.
Hence the residue can be written(4.16)1s12…kn∑h∑α,βA(1̂,β2,k,P̂h,n)S[β2,k∣α2,k]P̂Ã(1̂,P̂h,α2,k,n)×lim⁡z→zk+1…n-1∑σ̃,σA(-P̂-h,σk+1,n-2,n-1̂)S[σk+1,n-2∣σ̃k+1,n-2]p̂n-1Ã(n-1̂,σ̃k+1,n-2,-P̂-h)sk+1…n-1̂,
where we have used s1̂2…kn=sk+1…n-1̂, and already removed the vanishing mixed helicity terms. The first part is just a lower-point version of (3.5), and the second part the regularized dual KLT form, that is, we get(4.17)∑hMk+2(1̂,2,…,k,n,P̂h)Mn-k(k+1,…,n-1̂,-P̂-h)s12…kn.
Once again we obtain the correct BCFW expansion for all s12…kn poles, and all poles related to these by a permutation of 2,3,…,n-2.We have now covered all residues in (4.1), and see that they indeed make up the full BCFW expansion of the n-point gravity amplitude, and therefore Rn=Mn, see Figure 4. Notice how all of the properties/relations from Section 3 played an important role; the BCFW method was the main tool for the whole proof, the BCJ relations were needed not only for showing the equivalence between all j-values, but also to argue for the vanishing of contributions from (a), and both the vanishing identities and regularized KLT relations were important for identifying contributions from (b) with terms from a BCFW expansion of Mn. We also stress that no other crossing symmetry than that which was already manifest has been used, so the identification with a gravity amplitude is also an indirect proof of the total crossing symmetry of the r.h.s. of (3.3).Figure 4
Schematic outline of field theory proof.Maybe the most important thing to take from this proof is that it very explicitly illustrates how amazingly constrained scattering amplitudes are just from their very general analytical properties. These constraints are so strong that they basically force perturbative gravity and gauge theories to be related through the KLT relations, although a priori these theories seem completely unrelated.


## 5. The BCJ Representation

In the last couple of years an exciting new structure for gauge theory amplitudes has been discovered [32], which has also offered a new way of thinking of gravity amplitudes as a “double-copy’’ of gauge theory amplitudes. One of the more surprising results of this picture is that it seems to generalize to loop level [63, 64] whereas the KLT relations by themself only make sense at tree level. In this section we will briefly review this new and interesting structure.


### 5.1.  Motivation

The approach can be motivated by making some observations at four points. We start by forcing the color-ordered amplitudes into a form corresponding to having only antisymmetric three-point vertices(5.1)A4(1,2,3,4)=n1s12+n2s14,A4(1,2,4,3)=-n3s13-n1s12,A4(1,3,2,4)=-n2s14+n3s13.
Any four-point contact terms have simply been absorbed into the numerators, using trivial relations like s12/s12=1, and the relative signs have been chosen in correspondence with anti-symmetric cubic vertices. This is quite straightforward to do at four points. In terms of this representation the full color-dressed gauge theory amplitude takes the form(5.2)A4full=c1n1s12+c2n2s14+c3n3s13,
where the color factors ci are given by contractions between the structure constants f̃abc≡i2fabc=Tr([Ta,Tb]Tc), with Ti being the generators of the gauge group, as(5.3)c1≡f̃a1a2bf̃ba3a4,c2≡f̃a2a3bf̃ba4a1,c3≡f̃a4a2bf̃ba3a1.
For simplicity we have omitted the coupling constant in (5.2), something we will do for the remainder of this section. It is well known that the color factors satisfy the Jacobi identity c2+c3-c1=0; however, what might be a bit more surprising is the fact that so do the kinematic numerators, that is, n2+n3-n1=0, see Figure 5, although this is actually a quite old result [65]. There seems to exist a kind of color-kinematic duality in the structure of four-point amplitudes.Figure 5
Diagrammatic representation of the Jacobi identity. Theci’s can be obtained by dressing the vertices with structure constants f̃abc, but the diagrams can also be thought of as representing the kinematic numerators ni.In addition, if we use the representation of amplitudes in (5.1), plug it into the four point KLT relation in (3.8) and use the Jacobi identity for the numerators, we get(5.4)M4=n1ñ1s12+n2ñ2s14+n3ñ3s13,
where ñi is just the numerators belonging to Ã4 written in a similar representation. Compared to (5.2) we have basically just replaced the color factors ci with kinematic numerators ñi. This nicely captures the statement of gravity being a “double-copy’’ of gauge theory.


### 5.2.  At Generaln-Point Tree and Loop Level

One could think that the above duality between color factors and kinematic numerators is merely a curious coincidence at four points; however, Bern, Carrasco, and Johansson promoted it to a general principle saying; it isalways possible to represent an n-point gauge theory amplitude as a sum over all distinct n-point diagrams with only anti-symmetric cubic vertices, and in such a way that the kinematic numerators ni satisfy the same Jacobi identities as the color factors ci. In detail, this is to write the full tree-level amplitude in the form(5.5)Anfull(1,2,…,n)=∑icini(∏jsj)i,
where the color-kinematic duality(5.6)cα+cβ-cγ=0⟶nα+nβ-nγ=0
is satisfied. The sum in (5.5) is over all different diagrams, only containing cubic vertices, and (∏jsj)i is the corresponding pole structure of diagram i. It should be noted, that in the four-point case the duality in (5.6) was automatically satisfied once the amplitude was forced into the form of (5.2), this is not the case for general n-point amplitudes. To write an amplitude in the form of (5.5) is not difficult, but to also have the numerators satisfy (5.6) is a highly non-trivial task [66–72]. Recently there has been some progress in understanding the duality at a deeper level, looking for an underlying kinematic group [73, 74].It is also worth mentioning that the BCJ relations reviewed in Section3.3, although having a life of thier own now, were originally found as a consequence of the color-kinematic duality. The constraint of a Jacobi structure on the kinematic numerators leads to those relations among the color-ordered amplitudes.Similarly to (5.4), once a representation satisfying the color-kinematic duality is obtained, the n-point gravity amplitude is given by(5.7)Mn(1,2,…,n)=∑iñini(∏jsj)i,
we just replace the color factors ci in (5.5) with kinematic numerators ñi from a gauge theory amplitude written in a similar representation [32, 75].Remarkably, there are strong indications that the color-kinematic duality, and the connection to gravity through “squaring’’ of numerators, has an extension to loop level [63, 64]. The statement is that the n-point gauge theory amplitude at L loops can be written in the form(5.8)(-i)LAnloop=∑i∫∏l=1LdDpl(2π)D1Sicini(∏jsj)i,
where the sum is over all n-point L-loop diagrams with cubic vertices, Si is the symmetry factor of diagram i, and the numerators ni again satisfy the color-kinematic duality in (5.6). Note that the numerators are something belonging to the integrand. The conjecture for the n-point L-loop gravity amplitude is then(5.9)(-i)L+1Mnloop=∑i∫∏l=1LdDpl(2π)D1Siñini(∏jsj)i,
with the same sum as in the gauge theory case. This is not only a theoretically interesting extension of the connection between perturbative gravity and gauge theory beyond tree level, but together with generalized unitarity cut methods [76–78] can have significant implications for multiloop calculations, see, for example, [79–84], which makes it a most interesting structure.


## 5.1.  Motivation

The approach can be motivated by making some observations at four points. We start by forcing the color-ordered amplitudes into a form corresponding to having only antisymmetric three-point vertices(5.1)A4(1,2,3,4)=n1s12+n2s14,A4(1,2,4,3)=-n3s13-n1s12,A4(1,3,2,4)=-n2s14+n3s13.
Any four-point contact terms have simply been absorbed into the numerators, using trivial relations like s12/s12=1, and the relative signs have been chosen in correspondence with anti-symmetric cubic vertices. This is quite straightforward to do at four points. In terms of this representation the full color-dressed gauge theory amplitude takes the form(5.2)A4full=c1n1s12+c2n2s14+c3n3s13,
where the color factors ci are given by contractions between the structure constants f̃abc≡i2fabc=Tr([Ta,Tb]Tc), with Ti being the generators of the gauge group, as(5.3)c1≡f̃a1a2bf̃ba3a4,c2≡f̃a2a3bf̃ba4a1,c3≡f̃a4a2bf̃ba3a1.
For simplicity we have omitted the coupling constant in (5.2), something we will do for the remainder of this section. It is well known that the color factors satisfy the Jacobi identity c2+c3-c1=0; however, what might be a bit more surprising is the fact that so do the kinematic numerators, that is, n2+n3-n1=0, see Figure 5, although this is actually a quite old result [65]. There seems to exist a kind of color-kinematic duality in the structure of four-point amplitudes.Figure 5
Diagrammatic representation of the Jacobi identity. Theci’s can be obtained by dressing the vertices with structure constants f̃abc, but the diagrams can also be thought of as representing the kinematic numerators ni.In addition, if we use the representation of amplitudes in (5.1), plug it into the four point KLT relation in (3.8) and use the Jacobi identity for the numerators, we get(5.4)M4=n1ñ1s12+n2ñ2s14+n3ñ3s13,
where ñi is just the numerators belonging to Ã4 written in a similar representation. Compared to (5.2) we have basically just replaced the color factors ci with kinematic numerators ñi. This nicely captures the statement of gravity being a “double-copy’’ of gauge theory.


## 5.2.  At Generaln-Point Tree and Loop Level

One could think that the above duality between color factors and kinematic numerators is merely a curious coincidence at four points; however, Bern, Carrasco, and Johansson promoted it to a general principle saying; it isalways possible to represent an n-point gauge theory amplitude as a sum over all distinct n-point diagrams with only anti-symmetric cubic vertices, and in such a way that the kinematic numerators ni satisfy the same Jacobi identities as the color factors ci. In detail, this is to write the full tree-level amplitude in the form(5.5)Anfull(1,2,…,n)=∑icini(∏jsj)i,
where the color-kinematic duality(5.6)cα+cβ-cγ=0⟶nα+nβ-nγ=0
is satisfied. The sum in (5.5) is over all different diagrams, only containing cubic vertices, and (∏jsj)i is the corresponding pole structure of diagram i. It should be noted, that in the four-point case the duality in (5.6) was automatically satisfied once the amplitude was forced into the form of (5.2), this is not the case for general n-point amplitudes. To write an amplitude in the form of (5.5) is not difficult, but to also have the numerators satisfy (5.6) is a highly non-trivial task [66–72]. Recently there has been some progress in understanding the duality at a deeper level, looking for an underlying kinematic group [73, 74].It is also worth mentioning that the BCJ relations reviewed in Section3.3, although having a life of thier own now, were originally found as a consequence of the color-kinematic duality. The constraint of a Jacobi structure on the kinematic numerators leads to those relations among the color-ordered amplitudes.Similarly to (5.4), once a representation satisfying the color-kinematic duality is obtained, the n-point gravity amplitude is given by(5.7)Mn(1,2,…,n)=∑iñini(∏jsj)i,
we just replace the color factors ci in (5.5) with kinematic numerators ñi from a gauge theory amplitude written in a similar representation [32, 75].Remarkably, there are strong indications that the color-kinematic duality, and the connection to gravity through “squaring’’ of numerators, has an extension to loop level [63, 64]. The statement is that the n-point gauge theory amplitude at L loops can be written in the form(5.8)(-i)LAnloop=∑i∫∏l=1LdDpl(2π)D1Sicini(∏jsj)i,
where the sum is over all n-point L-loop diagrams with cubic vertices, Si is the symmetry factor of diagram i, and the numerators ni again satisfy the color-kinematic duality in (5.6). Note that the numerators are something belonging to the integrand. The conjecture for the n-point L-loop gravity amplitude is then(5.9)(-i)L+1Mnloop=∑i∫∏l=1LdDpl(2π)D1Siñini(∏jsj)i,
with the same sum as in the gauge theory case. This is not only a theoretically interesting extension of the connection between perturbative gravity and gauge theory beyond tree level, but together with generalized unitarity cut methods [76–78] can have significant implications for multiloop calculations, see, for example, [79–84], which makes it a most interesting structure.


## 6. Conclusions

In this paper we have taken a close look at the KLT relations. The first part involved a rather detailed derivation from string theory, showing how to factorize theclosed string integrals into two separate sectors which were then deformed into expressions corresponding to products of open string amplitudes. One of the important steps was to keep track of phase factors. These depended a lot on the way we chose to close the integral contours around the branch cuts. However, we saw that this freedom could be nicely incorporated into a kinematic 𝒮 function leading to very compact n-point expressions.The second part focused on the KLT relations in the field theory limit. We investigated some of the amazing properties these relations exhibit and introduced tools that would be needed for a purely field theoretical proof. In particular, we saw how the BCJ relations connected all the different expressions obtained in string theory from closing contours in different ways. In light of the monodromy relations, the connection between deforming integral contours in string theory and the BCJ relations in field theory is now well understood. Indeed, the BCJ and KLT relations are much more closely related than one would first have anticipated. We ended the second part of the review with a proof of then-point KLT relations from a purely field theoretical point of view. Like most other properties presented it was based on very general analytical structures of scattering amplitudes.In the third and final part, we reviewed a very recently discovered way of representing gauge theory amplitudes, following from a color-kinematic duality, and we saw how this implied a new representation for gravity amplitudes as well. Probably one of the most exciting things about this approach is its generalization to all loop-order amplitudes, rendering a new way of doing loop calculations. Only the future will show what additional secrets the connection between gravity and gauge theory has in store for us.



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*Source: 726030-2012-01-19.xml*