Perturbative Gravity and Gauge Theory Relations: A Review

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 on 𝑛-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.

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 curious color-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 Section 2 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.

The 𝑛-point tree-level closed string amplitude is given byℳ_𝑛=𝑖2𝜋𝛼^{′𝑛−3𝑛−2}_𝑖=2𝑑²𝑧_𝑖𝑧_𝑖^{2𝛼′𝑘1⋅𝑘𝑖}𝑧_𝑖−1^{2𝛼′𝑘𝑛−1⋅𝑘𝑖}_{𝑖<𝑗≤𝑛−2}𝑧_𝑗−𝑧_𝑖^{2𝛼′𝑘𝑖⋅𝑘𝑗}𝑓𝑧_𝑖𝑔𝑧_𝑖, (2.1) where we have fixed the three points 𝑧₁=0, 𝑧_𝑛−1=1, and 𝑧_𝑛=∞. The 𝑓(𝑧_𝑖) 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 𝑧_𝑖=𝑣¹_𝑖+𝑖𝑣²_𝑖, such that𝑧_𝑖^{2𝛼′𝑘1⋅𝑘𝑖}=𝑣¹_𝑖²+𝑣²_𝑖^{2𝛼′𝑘1⋅𝑘𝑖},𝑧_𝑖−1^{2𝛼′𝑘𝑛−1⋅𝑘𝑖}=𝑣¹_𝑖−1²+𝑣²_𝑖^{2𝛼′𝑘𝑛−1⋅𝑘𝑖},𝑧_𝑗−𝑧_𝑖^{2𝛼′𝑘𝑖⋅𝑘𝑗}=𝑣¹_𝑗−𝑣¹_𝑖²+𝑣²_𝑗−𝑣²_𝑖^{2𝛼′𝑘𝑖⋅𝑘𝑗}. (2.2) By analytically continuing the 𝑣²_𝑖 variables to the complex plane, we can rotate the integration contour for these variables from the real axis to (almost) the imaginary axis𝑣²_𝑖⟶𝑖𝑒^−2𝑖𝜖𝑣²_𝑖≃𝑖(1−2𝑖𝜖)𝑣²_𝑖, (2.3) 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)𝑣¹_𝑖²+𝑣²_𝑖^{2𝛼′𝑘1⋅𝑘𝑖}⟶𝑣¹_𝑖²−𝑣²_𝑖²+4𝑖𝜖𝑣²_𝑖^{2𝛼′𝑘1⋅𝑘𝑖},𝑣¹_𝑖−1²+𝑣²_𝑖^{2𝛼′𝑘𝑛−1⋅𝑘𝑖}⟶𝑣¹_𝑖²−𝑣²_𝑖²−2𝑣¹_𝑖+1+4𝑖𝜖𝑣²_𝑖^{2𝛼′𝑘𝑛−1⋅𝑘𝑖},𝑣¹_𝑗−𝑣¹_𝑖²+𝑣²_𝑗−𝑣²_𝑖^{2𝛼′𝑘𝑖⋅𝑘𝑗}⟶𝑣¹_𝑗−𝑣¹_𝑖²−𝑣²_𝑗−𝑣²_𝑖²(1−4𝑖𝜖)^{𝛼′𝑘𝑖⋅𝑘𝑗}. (2.4) If we now make the transformation of variables𝑣^±_𝑖≡𝑣¹_𝑖±𝑣²_𝑖, (2.5) and define 𝛿_𝑖≡𝑣⁺_𝑖−𝑣⁻_𝑖, it is easy to verify that the expression on the right-hand side of line (2.4) is given by𝑣⁺_𝑖−𝑖𝜖𝛿_𝑖^{𝛼′𝑘1⋅𝑘𝑖}𝑣⁻_𝑖+𝑖𝜖𝛿_𝑖^{𝛼′𝑘1⋅𝑘𝑖},𝑣⁺_𝑖−1−𝑖𝜖𝛿_𝑖^{𝛼′𝑘𝑛−1⋅𝑘𝑖}𝑣⁻_𝑖−1+𝑖𝜖𝛿_𝑖^{𝛼′𝑘𝑛−1⋅𝑘𝑖},𝑣⁺_𝑖−𝑣⁺_𝑗−𝑖𝜖𝛿_𝑖−𝛿_𝑗^{𝛼′𝑘𝑖⋅𝑘𝑗}𝑣⁻_𝑖−𝑣⁻_𝑗+𝑖𝜖𝛿_𝑖−𝛿_𝑗^{𝛼′𝑘𝑖⋅𝑘𝑗}, (2.6) respectively.

In total, this brings (2.1) into the form ℳ_𝑛=𝑖2^𝑛−3𝑖2𝜋𝛼^{′𝑛−3+∞}_−∞^𝑛−2_𝑖=2𝑑𝑣⁺_𝑖𝑑𝑣⁻_𝑖𝑓𝑣⁻_𝑖𝑔𝑣⁺_𝑖×𝑣⁺_𝑖−𝑖𝜖𝛿_𝑖^{𝛼′𝑘1⋅𝑘i}𝑣⁻_𝑖+𝑖𝜖𝛿_𝑖^{𝛼′𝑘1⋅𝑘𝑖}𝑣⁺_𝑖−1−𝑖𝜖𝛿_𝑖^{𝛼′𝑘𝑛−1⋅𝑘𝑖}𝑣⁻_𝑖−1+𝑖𝜖𝛿_𝑖^{𝛼′𝑘𝑛−1⋅𝑘𝑖}×_{𝑖<𝑗≤𝑛−2}𝑣⁺_𝑖−𝑣⁺_𝑗−𝑖𝜖𝛿_𝑖−𝛿_𝑗^{𝛼′𝑘𝑖⋅𝑘𝑗}𝑣⁻_𝑖−𝑣⁻_𝑗+𝑖𝜖𝛿_𝑖−𝛿_𝑗^{𝛼′𝑘𝑖⋅𝑘𝑗}, (2.7) 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 𝑣²_𝑖 contours.

First we note the following, assume that at least one 𝑣⁺_𝑖∈]−∞,0[ and look at the contribution from 𝑣⁻_𝑖, that is, ^+∞_−∞𝑑𝑣⁻_𝑖𝑓𝑣⁻_𝑖𝑣⁻_𝑖+𝑖𝜖𝛿_𝑖^{𝛼′𝑘1⋅𝑘𝑖}𝑣⁻_𝑖−1+𝑖𝜖𝛿_𝑖^{𝛼′𝑘𝑛−1⋅𝑘𝑖}_{𝑖<𝑗≤𝑛−2}𝑣⁻_𝑖−𝑣⁻_𝑗+𝑖𝜖𝛿_𝑖−𝛿_𝑗^{𝛼′𝑘𝑖⋅𝑘𝑗}. (2.8) The behaviour of the imaginary 𝜖 terms near the branch points is𝑣⁻_𝑖∼0⟹𝛿_𝑖∼𝑣⁺_𝑖𝑣<0,_−𝑖∼1⟹𝛿_𝑖∼𝑣⁺_𝑖𝑣−1<0,_−𝑖∼𝑣⁻_𝑗⟹𝛿_𝑖−𝛿_𝑗∼𝑣⁺_𝑖−𝑣⁺_𝑗<0when𝑣⁺_𝑖<𝑣⁺_𝑗. (2.9) The requirement 𝑣⁺_𝑖<𝑣⁺_𝑗 in the last line is not a problem, since we can just choose to look at the 𝑣⁻_𝑖 integral corresponding to the “smallest’’ 𝑣⁺_𝑖 variable, which has to lie in the range ]−∞,0[ due to our first assumption. This means that we can close the integral of 𝑣⁻_𝑖 in the lower half of the complex 𝑣⁻_𝑖-plane (again by analytical continuation), and since the closed contour does not contain any poles the integral vanishes. In general, when 𝑣⁺_𝑖<𝑣⁺_𝑗 we avoid the branch point 𝑣⁻_𝑖=𝑣⁻_𝑗 below the real axis, and when 𝑣⁺_𝑖>𝑣⁺_𝑗 we avoid it above the real axis. From this kind of argument we see that whenever one of the 𝑣⁺_𝑖-variables is in the range of ]−∞,0[ or ]1,∞[, at least one of the 𝑣⁻_𝑖 contours can be completely closed either below or above the real axis. Hence, only when all 𝑣⁺_𝑖 lie between 0 and 1 will there be a contribution to (2.7).

By splitting up the 𝑣⁺_𝑖-integration region we can write the 𝑛-point closed string amplitude as ℳ_𝑛=_𝜎𝑀^𝜎_𝑛(𝜎(2),…,𝜎(𝑛−2)), (2.10) where 𝑀^𝜎_𝑛(𝜎(2),…,𝜎(𝑛−2)) is the “ordered amplitude’’ defined such that 𝑣⁺_𝜎(2)<𝑣⁺_𝜎(3)<⋯<𝑣⁺_{𝜎(𝑛−2)}. For instance, at five points this corresponds to splitting the integration over the (𝑣⁺₂,𝑣⁺₃)-plane into an integral over the region “above’’ the 𝑣⁺₃=𝑣⁺₂ line (i.e., 𝑣⁺₂<𝑣⁺₃) and an integral “below’’ this line (i.e., 𝑣⁺₃<𝑣⁺₂). Together with the above restriction on the 𝑣⁺_𝑖-integration range, the 𝑣⁺_𝑖 part of 𝑀^𝜎_𝑛 in (2.10) becomes_{0<𝑣+𝜎(2)<⋯<𝑣+𝜎(𝑛−2)<1}^𝑛−2_𝑖=2𝑑𝑣⁺_𝑖𝑔𝑣⁺_𝑖𝑣⁺_𝑖^{𝛼′𝑘1⋅𝑘𝑖}1−𝑣⁺_𝑖^{𝛼′𝑘𝑛−1⋅𝑘𝑖}_{𝑖<𝑗≤𝑛−2}𝑣⁺_𝜎𝑗−𝑣⁺_𝜎(𝑖)^{𝛼′𝑘𝜎(𝑖)⋅𝑘𝜎(𝑗)}, (2.11) where we have omitted the infinitesimal 𝜖 terms. We recognize (2.11) as the expression exactly corresponding to the color-ordered open string amplitude 𝒜_𝑛(1,𝜎(2,…,𝑛−2),𝑛−1,𝑛). Note that, compared to (2.7), we have written (1−𝑣⁺_𝑖)^{𝛼′𝑘𝑛−1⋅𝑘𝑖} instead of (𝑣⁺_𝑖−1)^{𝛼′𝑘𝑛−1⋅𝑘𝑖} and (𝑣⁺_𝑗−𝑣⁺_𝑖)^{𝛼′𝑘𝑖⋅𝑘𝑗} such that 𝑣⁺_𝑗−𝑣⁺_𝑖>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 𝑣⁻_𝑖 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 𝑀^𝜎_𝑛(2,3,…,𝑛−2) contribution in (2.10). The remaining terms can simply be obtained through permutation of labels.

We have just seen that the 𝑣⁺_𝑖 part is nothing but the color-ordered 𝒜_𝑛(1,2,…,𝑛) amplitude. We therefore turn our attention to the 𝑣⁻_𝑖 part, investigating which contours the imaginary 𝜖 terms dictate for the integrals. Near 𝑣⁻_𝑖∼0 the quantity 𝑖𝜖𝛿_𝑖∼𝑖𝜖𝑣⁺_𝑖 is a positive imaginary number (remember that 𝑣⁺_𝑖∈]0,1[), so the contour is above the real axis here. For 𝑣⁻_𝑖∼1 we have 𝑖𝜖𝛿_𝑖∼𝑖𝜖(𝑣⁺_𝑖−1) which is a negative imaginary number, hence the contour lies below the real axis. Finally, for 𝑣⁻_𝑖∼𝑣⁻_𝑗 we see that 𝑖𝜖(𝛿_𝑖−𝛿_𝑗)∼𝑖𝜖(𝑣⁺_𝑖−𝑣⁺_𝑗), meaning that the contour for 𝑣⁻_𝑖+𝑖𝜖𝛿_𝑖 lies below the contour of 𝑣⁻_𝑗+𝑖𝜖𝛿_𝑗 for 𝑖<𝑗. See Figure 1 for an illustration of this nested structure.

Figure 1 

The nested structure of the contours of integration for the 𝑣−𝑖 variables corresponding to the ordering 0<𝑣+2<𝑣+3<⋯<𝑣+𝑛−2<1 of the 𝑣+i variables.

The next step is to deform the contours for 𝑣⁻_𝑖+𝑖𝜖𝛿_𝑖 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 𝑧^𝑐 with Re(𝑧)<0𝑧^𝑐=𝑒^𝑖𝜋𝑐(−𝑧)^𝑐(𝑧)𝑒,Im≥0,_{−𝑖𝜋𝑐}(−𝑧)^𝑐,Im(𝑧)<0. (2.12) 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 given 2≤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 with 𝑗=4, pulling the contour for 𝑣⁻₂ to the left, only showing the piece involving 𝑣⁻₂, we get_𝐶2𝑑𝑣⁻₂𝑣⁻₂^{𝛼′𝑘1⋅𝑘2}1−𝑣⁻₂^{𝛼′𝑘4⋅𝑘2}𝑣⁻₃−𝑣⁻₂^{𝛼′𝑘3⋅𝑘2}𝑓𝑣⁻₂=𝑒^{𝑖𝜋𝛼′𝑘1⋅𝑘2}−𝑒^{−𝑖𝜋𝛼′𝑘1⋅𝑘20}_−∞𝑑𝑣⁻₂−𝑣⁻₂^{𝛼′𝑘1⋅𝑘2}1−𝑣⁻₂^{𝛼′𝑘4⋅𝑘2}𝑣⁻₃−𝑣⁻₂^{𝛼′𝑘3⋅𝑘2}𝑓𝑣⁻₂=2𝑖sin𝜋𝛼^′𝑘₁⋅𝑘₂⁰_−∞𝑑𝑣⁻₂−𝑣⁻₂^{𝛼′𝑘1⋅𝑘2}1−𝑣⁻₂^{𝛼′𝑘4⋅𝑘2}𝑣⁻₃−𝑣⁻₂^{𝛼′𝑘3⋅𝑘2}𝑓𝑣⁻₂. (2.13) Note that in the previous first line we write (1−𝑣⁻₂)^{𝛼′𝑘4⋅𝑘2} instead of (𝑣⁻₂−1)^{𝛼′𝑘4⋅𝑘2} and (𝑣⁻₃−𝑣⁻₂)^{𝛼′𝑘3⋅𝑘2} instead of (𝑣⁻₂−𝑣⁻₃)^{𝛼′𝑘3⋅𝑘2} in order to compensate for the same swapping of order in the 𝑣⁺_𝑖 integral of (2.11). Now, as illustrated in the bottom of Figure 2, we also close the contour for 𝑣⁻₃ to the left (only showing the part involving the 𝑣⁻₃ variable)_𝐶3𝑑𝑣⁻₃𝑣⁻₃^{𝛼′𝑘1⋅𝑘3}1−𝑣⁻₃^{𝛼′𝑘4⋅𝑘3}𝑣⁻₃−𝑣⁻₂^{𝛼′𝑘3⋅𝑘2}𝑓𝑣⁻₃=𝑒^{𝑖𝜋𝛼′(𝑘1+𝑘2)⋅𝑘3}−𝑒^{−𝑖𝜋𝛼′(𝑘1+𝑘2)⋅𝑘3𝑣−2}_−∞𝑑𝑣⁻₃−𝑣⁻₃^{𝛼′𝑘1⋅𝑘3}1−𝑣⁻₃^{𝛼′𝑘4⋅𝑘3}𝑣⁻₂−𝑣⁻₃^{𝛼′𝑘2⋅𝑘3}𝑓𝑣⁻₃+𝑒^{𝑖𝜋𝛼′𝑘1⋅𝑘3}−𝑒^{−𝑖𝜋𝛼′𝑘1⋅𝑘30}_𝑣−2𝑑𝑣⁻₃−𝑣⁻₃^{𝛼′𝑘1⋅𝑘3}1−𝑣⁻₃^{𝛼′𝑘4⋅𝑘3}𝑣⁻₃−𝑣⁻₂^{𝛼′𝑘2⋅𝑘3}𝑓𝑣⁻₃=2𝑖sin𝜋𝛼^′𝑘₁+𝑘₂⋅𝑘₃^𝑣−2_−∞𝑑𝑣⁻₃−𝑣⁻₃^{𝛼′𝑘1⋅𝑘3}1−𝑣⁻₃^{𝛼′𝑘4⋅𝑘3}𝑣⁻₂−𝑣⁻₃^{𝛼′𝑘2⋅𝑘3}𝑓𝑣⁻₃+2𝑖sin𝜋𝛼^′𝑘₁⋅𝑘₃⁰_𝑣−2𝑑𝑣⁻₃−𝑣⁻₃^{𝛼′𝑘1⋅𝑘3}1−𝑣⁻₃^{𝛼′𝑘4⋅𝑘3}𝑣⁻₃−𝑣⁻₂^{𝛼′𝑘2⋅𝑘3}𝑓𝑣⁻₃. (2.14) We see that the total integration over 𝑣⁻₂ and 𝑣⁻₃ exactly correspond to color-ordered open string amplitudes, which we will denote 𝒜^∼₅ to distinguish them from the ones following from the 𝑣⁺_𝑖 part, such that the whole 𝑣⁻_𝑖 contribution in 𝑀^𝜎₅(2,3) can be written as∝sin𝜋𝛼^′𝑘₁⋅𝑘₂sin𝜋𝛼^′𝑘₁+𝑘₂⋅𝑘𝒜₃^∼₅(3,2,1,4,5)+sin𝜋𝛼^′𝑘₁⋅𝑘₂sin𝜋𝛼^′𝑘₁⋅𝑘𝒜₃^∼₅(2,3,1,4,5). (2.15) 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, 𝒜^∼₅ℳ₅=−14𝜋²𝛼′²sin𝜋𝛼^′𝑘₁⋅𝑘₂sin𝜋𝛼^′𝑘₁+𝑘₂⋅𝑘₃𝒜₅(1,2,3,4,5)𝒜^∼₅(3,2,1,4,5)+sin𝜋𝛼^′𝑘₁⋅𝑘₂sin𝜋𝛼^′𝑘₁⋅𝑘₃𝒜₅(1,2,3,4,5)𝒜^∼₅(+().2,3,1,4,5)2⟷3 (2.16) If we take the other extreme, that is, closing both contours to the right (𝑗=2), we get2𝑖sin𝜋𝛼^′𝑘₄⋅𝑘₃₁^+∞𝑑𝑣⁻₃𝑣⁻₃^{𝛼′𝑘1⋅𝑘3}𝑣⁻₃−1^{𝛼′𝑘4⋅𝑘3}𝑣⁻₃−𝑣⁻₂^{𝛼′𝑘2⋅𝑘3}𝑓𝑣⁻₃, (2.17) for the 𝑣⁻₃ integration, and2𝑖sin𝜋𝛼^′𝑘₄⋅𝑘₂^𝑣−3₁𝑑𝑣⁻₂𝑣⁻₂^{𝛼′𝑘1⋅𝑘2}𝑣⁻₂−1^{𝛼′𝑘4⋅𝑘2}𝑣⁻₃−𝑣⁻₂^{𝛼′𝑘3⋅𝑘2}𝑓𝑣⁻₂+2𝑖sin𝜋𝛼^′𝑘₄+𝑘₃⋅𝑘_2𝑣^+∞₋₃𝑑𝑣⁻₂𝑣⁻₂^{𝛼′𝑘4⋅𝑘2}𝑣⁻₂−1^{𝛼′𝑘4⋅𝑘2}𝑣⁻₂−𝑣⁻₃^{𝛼′𝑘3⋅𝑘2}𝑓𝑣⁻₂, (2.18) for the 𝑣⁻₂ integration, see the top case of Figure 2, that is, ℳ₅=−14𝜋²𝛼′²sin𝜋𝛼^′𝑘₄⋅𝑘₃sin𝜋𝛼^′𝑘₄⋅𝑘₂𝒜₅(1,2,3,4,5)𝒜^∼₅(1,4,2,3,5)+sin𝜋𝛼^′𝑘₄⋅𝑘₃sin𝜋𝛼^′𝑘₄+𝑘₃⋅𝑘₂𝒜₅(1,2,3,4,5)𝒜^∼₅(+().1,4,3,2,5)2⟷3 (2.19) Finally, we could have closed 𝑣⁻₂ to the left and 𝑣⁻₃ to the right, as also illustrated in Figure 2, resulting in (𝑗=3)ℳ₅=−14𝜋²𝛼^′2sin𝜋𝛼^′𝑘₁⋅𝑘₂sin𝜋𝛼^′𝑘₄⋅𝑘₃𝒜₅(1,2,3,4,5)𝒜^∼₅()+().2,1,4,3,52⟷3 (2.20)

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 the momentum kernel 𝒮_𝛼′𝑖₁,…,𝑖_𝑘∣𝑗₁,…,𝑗_𝑘𝑝≡𝜋𝛼^′2^𝑘−𝑘_𝑡=1sin𝜋𝛼^′𝑝⋅𝑘_𝑖𝑡+^𝑘_𝑞>𝑡𝜃𝑖_𝑡,𝑖_𝑞𝑘_𝑖𝑡⋅𝑘_𝑖𝑞, (2.21) where 𝜃(𝑖_𝑡,𝑖_𝑞) equals 1 if the ordering of 𝑖_𝑡 and 𝑖_𝑞 is opposite in {𝑖₁,…,𝑖_𝑘} and {𝑗₁,…,𝑗_𝑘}, and 0 if the ordering is the same. Here we have defined 𝒮_𝛼′ for a general number of legs, so, for instance𝒮_𝛼′[2∣2]_𝑘1=𝜋𝛼^′2⁻¹sin𝜋𝛼^′𝑘₁⋅𝑘₂,𝒮_𝛼′[23∣23]_𝑘1=𝜋𝛼^′2⁻²sin𝜋𝛼^′𝑘₁⋅𝑘₂sin𝜋𝛼^′𝑘₁⋅𝑘₃,𝒮_𝛼′[23∣32]_𝑘1=𝜋𝛼^′2⁻²sin𝜋𝛼^′𝑘₁+𝑘₃⋅𝑘₂sin𝜋𝛼^′𝑘₁⋅𝑘₃, (2.22) and so on. We will also define 𝒮_𝛼′[∅∣∅]_𝑝=1 for empty sets. With this 𝒮_𝛼′ function we can collect (2.16), (2.19), and (2.20) into one formula asℳ₅=−𝑖4²_{𝜎𝛾,𝛽}𝒮_𝛼′𝛾𝜎(2),…,𝜎𝑗−1∣𝜎(2),…,𝜎𝑗−1_𝑘1×𝒮_𝛼′𝜎𝑗,…,𝜎(3)∣𝛽𝜎𝑗,…,𝜎(3)_𝑘4×𝒜₅(1,𝜎(2,3),4,5)𝒜^∼₅𝛾𝜎(2),…,𝜎𝑗−1,1,4,𝛽𝜎𝑗,…,𝜎(3),5, (2.23) 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. General 𝑛-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 toℳ_𝑛=−𝑖4^𝑛−3_{𝜎𝛾,𝛽}𝒮_𝛼′𝛾𝜎(2),…,𝜎𝑗−1∣𝜎2,…,𝑗−1_𝑘1×𝒮_𝛼′𝜎𝑗,…,𝑛−2∣𝛽𝜎𝑗,…,𝜎(𝑛−2)_𝑘𝑛−1×𝒜_𝑛(())×1,𝜎2,…,𝑛−2,𝑛−1,𝑛_∼𝒜_𝑛𝛾𝜎(2),…,𝜎𝑗−1,1,𝑛−1,𝛽𝜎𝑗,…,𝜎(𝑛−2),𝑛, (2.24) with 2≤𝑗≤𝑛−1.

Expression (2.24) shows how to factorize an 𝑛-point closed string amplitude ℳ_𝑛 into the product of 𝑛-point color-ordered open string amplitudes 𝒜_𝑛 and 𝒜^∼_𝑛, “glued’’ together by kinematic factors contained in the 𝒮_𝛼′ 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, choosing 𝑗=𝑛−1 the relation takes a particularly nice 𝑛-point formℳ_𝑛=−𝑖4^𝑛−3_𝜎,𝛾𝒮_𝛼′𝛾(2,…,𝑛−2)∣𝜎(2,…,𝑛−2)_𝑘1×𝒜_𝑛(1,𝜎(2,…,𝑛−2),𝑛−1,𝑛)𝒜^∼_𝑛𝑛−1,𝑛,𝛾(2,…,𝑛−2),1, (2.25) only involving one 𝒮_𝛼′ function and having a greater symmetry between the sums over different ordered 𝒜_𝑛 and 𝒜^∼_𝑛 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 expressionsℳ_𝑛⟶𝑀_𝑛,𝒜_𝑛⟶𝐴_𝑛,𝒜^∼_𝑛⟶𝐴^∼_𝑛, (3.1) and 𝒮_𝛼′→𝒮₀≡𝒮, where it follows from (2.21) that𝒮𝑖₁,…,𝑖_𝑘∣𝑗₁,…,𝑗_𝑘𝑝=^𝑘_𝑡=1𝑠_𝑝𝑖𝑡+^𝑘_𝑞>𝑡𝜃𝑖_𝑡,𝑖_𝑞𝑠_{𝑖𝑡𝑖𝑞}, (3.2) with 𝑠_𝑖𝑗≡(𝑘_𝑖+𝑘_𝑗)²=2𝑘_𝑖⋅𝑘_𝑗, or more generally 𝑠_{𝑖𝑗…𝑘}=(𝑝_𝑖+𝑝_𝑗+⋯+𝑝_𝑘)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 form𝑀_𝑛=(−1)^𝑛+1_{𝜎𝛾,𝛽}𝒮𝛾𝜎(2),…,𝜎𝑗−1∣𝜎2,…,𝑗−1_𝑘1×𝒮𝜎𝑗,…,𝜎(𝑛−2)∣𝛽𝜎𝑗,…,𝑛−2_𝑘𝑛−1×𝐴_𝑛(())×1,𝜎2,…,𝑛−2,𝑛−1,𝑛_∼𝐴_𝑛𝛾𝜎(2),…,𝜎𝑗−1,1,𝑛−1,𝛽𝜎𝑗,…,𝜎(𝑛−2),𝑛, (3.3) 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 with 𝑗=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) becomes𝑀_𝑛=(−1)^𝑛+1_𝜎,𝜎_{∈𝑆𝑛−3}^∼𝐴_𝑛𝑛−1,𝑛,𝜎_2,𝑛−2,1𝒮𝜎_2,𝑛−2∣𝜎_{2,𝑛−2𝑘1}𝐴_𝑛1,𝜎_2,𝑛−2,𝑛−1,𝑛, (3.4) or𝑀_𝑛=(−1)^𝑛+1_𝜎,𝜎_{∈𝑆𝑛−3}𝐴_𝑛1,𝜎_2,𝑛−2,𝑛−1,𝑛𝒮𝜎_2,𝑛−2∣𝜎𝐴_{2,𝑛−2𝑘𝑛−1}^∼_𝑛1,𝑛−1,𝜎_2,𝑛−2,𝑛, (3.5) respectively, see also Figure 3. Here we have introduced the short-hand notation 𝜎_2,𝑛−2≡𝜎(2,…,𝑛−2), for a 𝜎 permutation over leg {2,…,𝑛−2}, and likewise for 𝜎. Note that 𝜎 and 𝜎 are two unrelated permutations, the tilde is just to remind us which one is going with 𝐴^∼_𝑛, and we are trying to be economic with our use of Greek letters.

Figure 3 

Diagrammatic representation of the KLT relation with 𝑗=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 only manifest 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 𝐴_𝑛(1,2,…,𝑛)=(−1)^𝑛𝐴_𝑛(𝑛,𝑛−1,…,1), and the following identity:𝒮𝑖₁,…,𝑖_𝑘∣𝑗₁,…,𝑗_𝑘𝑝=𝒮𝑗_𝑘,…,𝑗₁∣𝑖_𝑘,…,𝑖_1𝑝, (3.6) but in general the crossing symmetry is not obvious at all. These comments also apply in the string theory case.

Although the calculations in Section 2 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)𝑀₃(1,2,3)=𝐴₃(1,2,3)𝐴^∼₃(1,2,3). (3.7) 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 simple𝑀₄(1,2,3,4)=−𝑠₁₂𝐴₄(1,2,3,4)𝐴^∼₄(1,2,4,3), (3.8) 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),𝑀₅=𝑠₁₂𝑠₃₄𝐴₅(1,2,3,4,5)𝐴^∼₅(2,1,4,3,5)+𝑠₁₃𝑠₂₄𝐴₅(1,3,2,4,5)𝐴^∼₅(3,1,4,2,5), (3.9) 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 𝐴_𝑛 and 𝐴^∼_𝑛 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

Considering 𝑛-point amplitudes we start by making the following deformation of momenta 𝑝₁ and 𝑝_𝑛:𝑝₁⟶𝑝^′₁=𝑝₁𝑝−𝑥𝑞,_𝑛⟶𝑝^′_𝑛=𝑝_𝑛+𝑥𝑞, (3.10) where 𝑥 is just some arbitrary parameter, and 𝑞 is a four-vector satisfying 𝑞⋅𝑝₁=𝑞²=0 and 𝑞⋅𝑝_𝑛≠0. This preserves conservation of momentum and keeps 𝑝₁^′2=0, but makes 𝑝_𝑛^′2=𝑠_{1′2…(𝑛−1)}≠0.

The gravity amplitude 𝑀_𝑛 can then be written as [15]𝑀_𝑛=(−1)^𝑛lim_{𝑥→0𝜎,}𝜎_{∈𝑆𝑛−2}^∼𝐴_𝑛𝑛^′,𝜎_2,𝑛−1,1^′𝒮𝜎_2,𝑛−1∣𝜎_{2,𝑛−1𝑝′1}𝐴_𝑛1^′,𝜎_2,𝑛−1,𝑛^′𝑠_{1′2…(𝑛−1)}. (3.11) 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 with 𝑝^′₁ being off-shell instead of 𝑝^′_𝑛, we can write down the “dual’’ expression to (3.11)𝑀_𝑛=(−1)^𝑛lim_{𝑦→0𝜎,}𝜎_{∈𝑆𝑛−2}𝐴_𝑛1^′,𝜎_2,𝑛−1,𝑛^′𝒮𝜎_2,𝑛−1∣𝜎𝐴_{2,𝑛−1𝑝′𝑛}^∼_𝑛𝑛^′,𝜎_2,𝑛−1,1^′𝑠_23…𝑛′, (3.12) 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. For 𝑛-point amplitudes these read0=𝑠₁₂𝐴_𝑛(1,2,3,…,𝑛)+𝑠₁₂+𝑠₂₃𝐴_𝑛()+𝑠1,3,2,4,…,𝑛₁₂+𝑠₂₃+𝑠₂₄𝐴_𝑛()+𝑠1,3,4,2,5,…,𝑛+⋯₁₂+𝑠₂₃+𝑠₂₄+⋯+𝑠_2(𝑛−1)𝐴_𝑛(1,3,4,…,𝑛−1,2,𝑛), (3.13) 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 𝑠_2𝑖, 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) are0=𝑠₁₂𝐴₄(1,2,3,4)+𝑠₁₂+𝑠₂₃𝐴₄(1,3,2,4), (3.14) for four points, and0=𝑠₁₂𝐴₅(1,2,3,4,5)+𝑠₁₂+𝑠₂₃𝐴₅(1,3,2,4,5)+𝑠₁₂+𝑠₂₃+𝑠₂₄𝐴₅(1,3,4,2,5), (3.15) 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 our 𝒮 function. This is achieved through0=_{𝜎∈𝑆𝑛−2}𝒮𝛽(2,…,𝑛−1)∣𝜎(2,…,𝑛−1)_𝑘1𝐴_𝑛(1,𝜎(2,…,𝑛−1),𝑛), (3.16) 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, _𝜎∈𝑆3𝒮[2,3,4∣𝜎(2,3,4)]_𝑘1𝐴₅(1,𝜎(2,3,4),5)=𝑠₁₃𝑠₁₄𝑠₁₂𝐴₅(1,2,3,4,5)+𝑠₁₂+𝑠₂₃𝐴₅(1,3,2,4,5)+𝑠₁₂+𝑠₂₃+𝑠₂₄𝐴₅(1,3,4,2,5)+𝑠₁₄𝑠₁₃+𝑠₃₄×𝑠₁₂𝐴₅(1,2,4,3,5)+𝑠₁₂+𝑠₂₄𝐴₅(1,4,2,3,5)+𝑠₁₂+𝑠₂₃+𝑠₂₄𝐴₅(1,4,3,2,5). (3.17) The two [⋯] is exactly zero due to (3.13).

The general argument is to divide the sum of 𝜎(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. The 𝑗-Independence

With the BCJ relations we can now address the 𝑗-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 the 𝑗-independence is given by _𝛼,𝛽𝒮𝛼_{𝑖2,𝑖𝑗}∣𝑖₂,…,𝑖_𝑗𝑝1𝒮𝑖_𝑗+1,…,𝑖_𝑛−2∣𝛽𝐴_{𝑖𝑗+1,𝑖𝑛−2𝑝𝑛−1}^∼_𝑛𝛼_{𝑖2,𝑖𝑗},1,𝑛−1,𝛽_{𝑖𝑗+1,𝑖𝑛−2}=,𝑛_{𝛼′,𝛽′}𝒮𝛼^′_{𝑖2,𝑖𝑗−1}∣𝑖₂,…,𝑖_{𝑗−1𝑝1}𝒮𝑖_𝑗,𝑖_𝑗+1,…,𝑖_𝑛−2∣𝛽𝐴^′_{𝑖𝑗,𝑖𝑛−2𝑝𝑛−1}^∼_𝑛𝛼^′_{𝑖2,𝑖𝑗−1},1,𝑛−1,𝛽^′_{𝑖𝑗,𝑖𝑛−2},𝑛. (3.18) 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.

For 𝑛=5 and 𝑗=2 (3.18) reads (where we have taken 𝑖_𝑘=𝑘)𝒮[2∣2]_𝑝1𝒮[3∣3]𝐴_𝑝4^∼₅(2,1,4,3,5)=_𝛽′𝒮2,3∣𝛽^′(2,3)𝐴_𝑝4^∼₅1,4,𝛽^′(2,3),5, (3.19) and since (3.3) for 𝑛=5 and 𝑗=2 is given by𝑀₅=_𝜎𝛽𝐴₅(1,𝜎(2,3),4,5)𝒮𝜎(2,3)∣𝛽(2,3)𝐴_𝑘4^∼₅()=1,4,𝛽2,3,5_𝜎𝐴₅(1,𝜎(2,3),4,5)_𝛽𝒮𝜎(2,3)∣𝛽(2,3)𝐴_𝑘4^∼₅1,4,𝛽(2,3),5, (3.20) using (3.19) we immediately get𝑀₅=_𝜎𝐴₅(1,𝜎(2,3),4,5)𝒮[𝜎(2)∣𝜎(2)]_𝑝1𝒮[𝜎(3)∣𝜎(3)]𝐴_𝑝4^∼₅(𝜎(2),1,4,𝜎(3),5), (3.21) 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 to 𝑛 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 𝐴_𝑛 and 𝐴^∼_𝑛 to live in different helicity sectors the r.h.s. of (3.3) is identically zero.

To be more specific let us denote an 𝑛-point color-ordered tree amplitude with helicity configuration belonging to 𝑁^𝑘MHV as 𝐴^𝑘_𝑛, that is, 𝐴^𝑘_𝑛 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]0=_{𝜎𝛾,𝛽}𝒮𝛾𝜎_2,𝑗−1∣𝜎_{2,𝑗−1𝑘1}𝒮𝜎_{𝑗,𝑛−2}∣𝛽𝜎_{𝑗,𝑛−2𝑘𝑛−1}𝐴^ℎ_𝑛1,𝜎_2,𝑗−1,𝜎_{𝑗,𝑛−2}×,𝑛−1,𝑛_∼𝐴_𝑛^𝑘≠ℎ𝛾𝜎_2,𝑗−1,1,𝑛−1,𝛽𝜎_{𝑗,𝑛−2},𝑛, (3.22) 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 with 𝑗=3, we have0=𝑠₁₂𝑠₃₄𝐴₅1⁻,2⁻,3⁺,4⁺,5𝐴^+∼₅2⁻,1⁻,4⁺,3⁻,5⁺+𝑠₁₃𝑠₂₄𝐴₅1⁻,3⁺,2⁻,4⁺,5𝐴^+∼₅3⁻,1⁻,4⁺,2⁻,5⁺. (3.23)

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 𝐴_𝑛 and 𝐴^∼_𝑛 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 Section 4. 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, say 𝑝_𝑗 and 𝑝_𝑙, as𝑝_𝑗⟶𝑝_𝑗=𝑝_𝑗𝑝−𝑧𝑞,_𝑙⟶𝑝_𝑙=𝑝_𝑙+𝑧𝑞, (3.24) where 𝑧 is a complex variable and 𝑞 is a four-vector satisfying 𝑞²=𝑞⋅𝑝_𝑗=𝑞⋅𝑝_𝑙=0. This preserves conservation of momentum and on-shellness. At tree-level 𝐴_𝑛 is a rational function of external momenta, implying that the deformed amplitude 𝐴_𝑛(𝑧) is a rational function of 𝑧. Assuming 𝐴_𝑛(𝑧)→0 when 𝑧→∞, Cauchy’s theorem tells us that10=2𝜋𝑖𝑑𝑧𝑧𝐴_𝑛(𝑧)=𝐴_𝑛(0)+_{poles𝑧𝑝≠0}Res_𝑝𝐴_𝑛(𝑧),𝑧_𝑝𝑧_𝑝, (3.25) where 𝐴_𝑛(0) is the undeformed amplitude, coming from the residue of the 𝑧=0 pole. The remaining poles (those different from 𝑧=0) come from 𝐴_𝑛(𝑧), which we know can only follow from Feynman propagators going on-shell, that is, when 𝑃_𝑘,𝑚(𝑧)² vanishes, where𝑃_𝑘,𝑚(𝑧)≡𝑝_𝑘+𝑝_𝑘+1+⋯+𝑝_𝑗+⋯+𝑝_𝑚, (3.26) 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 𝑃_𝑘,𝑚 is independent of 𝑧.

Let us denote the value of 𝑧 where 𝑃_𝑘,𝑚(𝑧) is going on-shell by 𝑧_𝑘,𝑚, which can be found by solving 𝑃_𝑘,𝑚(𝑧_𝑘,𝑚)²=0. Since the poles of 𝐴_𝑛(𝑧) will be simple poles, the residues are given byRes_𝑝𝐴_𝑛(𝑧),𝑧_𝑘,𝑚𝑧_𝑘,𝑚=−lim_{𝑧→𝑧𝑘,𝑚}𝑃_𝑘,𝑚(𝑧)²𝐴_𝑛(𝑧)𝑃²_𝑘,𝑚, (3.27) where 𝑃_𝑘,𝑚=𝑃_𝑘,𝑚(0). Using the general factorization property of gluon amplitudes𝐴−−−−−→_𝑛^𝑃2→0_ℎ=±𝐴_𝑟+1𝑘,…,𝑚,−𝑃^−ℎ1𝑃²𝐴_{𝑛−𝑟+1}𝑃^ℎ,𝑚+1,…,𝑘−1, (3.28) we see thatRes_𝑝𝐴_𝑛(𝑧),𝑧_𝑘,𝑚𝑧_𝑘,𝑚=−_ℎ=±𝐴_𝑟+1𝑘,…,𝑚,−𝑃^−ℎ_𝑘,𝑚1𝑃²_𝑘,𝑚𝐴_{𝑛−𝑟+1}𝑃^ℎ_𝑘,𝑚,𝑚+1,…,𝑘−1, (3.29) with 𝑃_𝑘,𝑚≡𝑃_𝑘,𝑚(𝑧_𝑘,𝑚). Hence, combined with (3.25), we get the BCFW recursion relation [26, 27]𝐴_𝑛=_𝑟ℎ=±𝐴_𝑟+1𝑘,…,𝑚,−𝑃^−ℎ_𝑘,𝑚1𝑃²_𝑘,𝑚𝐴_{𝑛−𝑟+1}𝑃^ℎ_𝑘,𝑚,𝑚+1,…,𝑘−1, (3.30) 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 over all 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. An 𝑛-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 𝑅_𝑛. 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 𝑅_𝑛=𝑀_𝑛.

In the same way as the BCFW recursion relation was derived, we start out by deforming two momenta in our expression for 𝑅_𝑛 and consider the contour integral1𝐵=2𝜋𝑖𝑑𝑧𝑧𝑅_𝑛(𝑧)=𝑅_𝑛(0)+_{poles𝑧𝑝≠0}Res_𝑝𝑅_𝑛(𝑧),𝑧_𝑝𝑧_𝑝, (4.1) where 𝑅_𝑛(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 𝑝_𝑛 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 𝐴_𝑛 and 𝐴^∼_𝑛, this was also used in [19]. In the proof we will present below it is more convenient to make a deformation in 𝑝₁ and 𝑝_𝑛−1. However, such a deformation cannot have a boundary term either since in Section 3 we already argued for the crossing symmetry between 𝑝_𝑛 and 𝑝_𝑛−1 in 𝑅_𝑛. The large 𝑧 behaviour must therefore be equally good for this deformation.

Now that we have established 𝐵=0, the goal is to show that the sum of residues exactly makes up the BCFW-expansion of an 𝑛-point gravity amplitude, and hence𝑀_𝑛∼_{poles𝑧𝑝≠0}Res_𝑝𝑅_𝑛(𝑧),𝑧_𝑝𝑧_𝑝. (4.2) 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 form 𝑠_12…𝑘, and we will be using 𝑅_𝑛 in the form of (3.4). Like in (3.27) we can compute these from lim_{𝑧→𝑧12…𝑘}[𝑠_12…𝑘(𝑧)𝑅_𝑛(𝑧)]/𝑠_12…𝑘, where 𝑧_12…𝑘 is the 𝑧-value that makes 𝑠_12…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 in one of the amplitudes 𝐴^∼_𝑛 or 𝐴_𝑛.(b)The pole appears in both 𝐴^∼_𝑛 and 𝐴_𝑛.

First we investigate (a), and we will consider the case where the pole only appears in 𝐴^∼_𝑛. The case with the pole appearing in 𝐴_𝑛 can be handled in a similar way. The terms contributing to this class must all involve an 𝐴^∼_𝑛 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_𝜎,𝜎^∼_,𝛼𝐴_𝑛𝑛−1,𝑛,𝜎_{𝑘+1,𝑛−2},𝛼_2,𝑘,1𝒮𝜎_{𝑘+1,𝑛−2}𝛼_2,𝑘∣𝜎_{2,𝑛−2𝑝1}𝐴_𝑛1,𝜎_2,𝑛−2,𝑛−1,𝑛, (4.3) 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 𝑠_12…𝑘 pole in 𝐴_𝑛. From this we get the residue_𝜎,𝜎_,𝛼∑𝐴_ℎ^∼𝑛−1,𝑛,𝜎_{𝑘+1,𝑛−2},𝑃𝐴^ℎ∼−𝑃^−ℎ,𝛼_2,𝑘,1𝑠_12…𝑘𝒮𝜎_{𝑘+1,𝑛−2}𝛼_2,𝑘∣𝜎_{2,𝑛−2𝑝1}×𝐴_𝑛1,𝜎_2,𝑛−2,𝑛−1,𝑛, (4.4) where 𝑃≡𝑝₁+𝑝₂+⋯+𝑝_𝑘, and we have used the factorization property of color-ordered gauge theory amplitudes in (3.28). Also note that the pole 𝑠_12…𝑘, from the factorization of 𝐴^∼_𝑛, has been replaced with 𝑠_12…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 write𝒮𝜎_{𝑘+1,𝑛−2}𝛼_2,𝑘∣𝜎_{2,𝑛−2𝑝1}=𝒮𝛼_2,𝑘∣𝜌_2,𝑘𝑝1×afactorindependentof𝛼, (4.5) 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_𝜎ℎ𝛼𝐴^∼(−𝑃^−ℎ,𝛼_2,𝑘,1)𝒮𝛼_2,𝑘∣𝜌_2,𝑘𝑝10𝜎×[⋯], (4.6) where, as we have seen above, the quantity inside (⋯) vanishes when 𝑃 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 both 𝐴^∼_𝑛 and 𝐴_𝑛 now contain the pole 𝑠_12…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_𝜎,𝜎^∼_,𝛼,𝛽𝐴_𝑛𝑛−1,𝑛,𝜎_{𝑘+1,𝑛−2},𝛼_2,𝑘,1𝒮𝜎_{𝑘+1,𝑛−2}𝛼_2,𝑘∣𝛽_2,𝑘𝜎_{𝑘+1,𝑛−2𝑝1}𝐴_𝑛1,𝛽_2,𝑘,𝜎_{𝑘+1,𝑛−2},𝑛−1,𝑛. (4.7) 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 𝑃=𝑝₁+𝑝₂+⋯+𝑝_𝑘 goes on-shell, we can write𝒮𝜎_{𝑘+1,𝑛−2}𝛼_2,𝑘∣𝛽_2,𝑘𝜎_{𝑘+1,𝑛−2𝑝1}=𝒮𝛼_2,𝑘∣𝛽_2,𝑘𝑝1×𝒮𝜎_{𝑘+1,𝑛−2}∣𝜎_{𝑘+1,𝑛−2𝑃}. (4.8) The residue for 𝑠_12…𝑘 can then be expressed as1𝑠_12…𝑘ℎlim_{𝑧→𝑧12…𝑘}^∼_𝛼,𝛽𝐴−𝑃^ℎ,𝛼_2,𝑘,1𝒮𝛼_2,𝑘∣𝛽_2,𝑘𝑝1𝐴1,𝛽_2,𝑘,−𝑃^ℎ𝑠_12…𝑘×_𝜎,𝜎^∼𝐴𝑛−1,𝑛,𝜎_{𝑘+1,𝑛−2},𝑃^−ℎ𝒮𝜎_{𝑘+1,𝑛−2}∣𝜎_{𝑘+1,𝑛−2𝑃}𝐴𝑃^−ℎ,𝜎_{𝑘+1,𝑛−2},+,𝑛−1,𝑛mixedhelicityterms (4.9) where the “mixed helicity terms’’ are expressions of the same form but with products between amplitudes with, for instance, (𝑃^ℎ,𝑃^−ℎ) instead of (−𝑃^ℎ,−𝑃^ℎ) or (𝑃^−ℎ,𝑃^−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 over 𝛼 and 𝛽, which precisely makes up the regularized KLT form in (3.11), that is, it is just 𝑀_𝑘+1(1,2,…,𝑘,−𝑃^ℎ), and a sum over 𝜎 and 𝜎 which is an 𝑛−𝑘+1 point version of (3.4) and hence, by induction, equal to 𝑀_{𝑛−𝑘+1}(𝑘+1,…,𝑃^−ℎ). Altogether (4.9) is_ℎ𝑀_𝑘+11,2,…,𝑘,−𝑃^ℎ𝑀_{𝑛−𝑘+1}𝑘+1,…,𝑃^−ℎ𝑠_12…𝑘. (4.10) This is exactly the BCFW contribution to the 𝑛-point gravity amplitude from a 𝑠_12…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 and 𝑛, that is, poles of the form 𝑠_{12…𝑘𝑛}=𝑠_{𝑘+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 pole 𝑠_{12…𝑘𝑛} takes the form_𝜎,𝜎_,𝛼𝐴_𝑛1,𝜎_2,𝑛−2,𝑛−1,𝑛𝒮𝜎_2,𝑛−2∣𝜎_{𝑘+1,𝑛−2}𝛼_{2,𝑘𝑝𝑛−1ℎ}^∼𝐴𝑛−1,𝜎_{𝑘+1,𝑛−2},𝑃𝐴^ℎ∼−𝑃^−ℎ,𝛼_2,𝑘,𝑛,1𝑠_{12…𝑘𝑛}, (4.11) where we again assume the pole appears in the 𝐴^∼_𝑛 amplitude. Using the factorization property𝒮𝜎_2,𝑛−2∣𝜎_{𝑘+1,𝑛−2}𝛼_{2,𝑘𝑝𝑛−1}=𝒮𝜌_{𝑘+1,𝑛−2}∣𝜎_{𝑘+1,𝑛−2𝑝𝑛−1}×afactorindependentof𝜎, (4.12) 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 of𝜎^∼𝐴𝑛−1,𝜎_{𝑘+1,𝑛−2},𝑃^ℎ𝒮𝜌_{𝑘+1,𝑛−2}∣𝜎_{𝑘+1,𝑛−2𝑝𝑛−1}=0, (4.13) that vanishes due to BCJ relations. There is therefore no contribution from (a).

Considering part (b) for the 𝑠_{12…𝑘𝑛} pole, the contributing terms from (3.5) are_𝜎,𝜎_,𝛼,𝛽𝐴_𝑛1,𝛽_2,𝑘,𝜎_{𝑘+1,𝑛−2},𝑛−1,𝑛𝒮𝛽_2,𝑘𝜎_{𝑘+1,𝑛−2}∣𝜎_{𝑘+1,𝑛−2}𝛼𝐴_{2,𝑘𝑝𝑛−1}^∼_𝑛1,𝑛−1,𝜎_{𝑘+1,𝑛−2},𝛼_2,𝑘,𝑛, (4.14) with 𝒮 satisfying the factorization property (when 𝑃 goes on-shell)𝒮𝛽_2,𝑘𝜎_{𝑘+1,𝑛−2}∣𝜎_{𝑘+1,𝑛−2}𝛼_{2,𝑘𝑝𝑛−1}=𝒮𝜎_{𝑘+1,𝑛−2}∣𝜎_{𝑘+1,𝑛−2𝑝𝑛−1}×𝒮𝛽_2,𝑘∣𝛼_2,𝑘𝑃. (4.15) Hence the residue can be written1𝑠_{12…𝑘𝑛ℎ𝛼,𝛽}𝐴1,𝛽_2,𝑘,𝑃^ℎ,𝑛𝒮𝛽_2,𝑘∣𝛼𝐴_2,𝑘𝑃^∼1,𝑃^ℎ,𝛼_2,𝑘,𝑛×lim_{𝑧→𝑧𝑘+1…𝑛−1}𝜎_,𝜎𝐴−𝑃^−ℎ,𝜎_{𝑘+1,𝑛−2},𝑛−1𝒮𝜎_{𝑘+1,𝑛−2}∣𝜎𝐴_{𝑘+1,𝑛−2𝑝𝑛−1}^∼𝑛−1,𝜎_{𝑘+1,𝑛−2},−𝑃^−ℎ𝑠_{𝑘+1…𝑛−1}, (4.16) where we have used 𝑠_{12…𝑘𝑛}=𝑠_{𝑘+1…𝑛−1}, 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_ℎ𝑀_𝑘+21,2,…,𝑘,𝑛,𝑃^ℎ𝑀_𝑛−𝑘𝑘+1,…,𝑛−1,−𝑃^−ℎ𝑠_{12…𝑘𝑛}. (4.17) Once again we obtain the correct BCFW expansion for all 𝑠_{12…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 𝑅_𝑛=𝑀_𝑛, 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 𝑀_𝑛. 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 vertices𝐴₄()=𝑛1,2,3,4¹𝑠₁₂+𝑛₂𝑠₁₄,𝐴₄()𝑛1,2,4,3=−³𝑠₁₃−𝑛₁𝑠₁₂,𝐴₄()𝑛1,3,2,4=−²𝑠₁₄+𝑛₃𝑠₁₃. (5.1) 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 form𝐴^full₄=𝑐₁𝑛₁𝑠₁₂+𝑐₂𝑛₂𝑠₁₄+𝑐₃𝑛₃𝑠₁₃, (5.2) where the color factors 𝑐_𝑖 are given by contractions between the structure constants 𝑓^{∼𝑎𝑏𝑐}≡𝑖√2𝑓^𝑎𝑏𝑐=Tr([𝑇^𝑎,𝑇^𝑏]𝑇^𝑐), with 𝑇^𝑖 being the generators of the gauge group, as𝑐₁≡𝑓^∼𝑓^{𝑎1𝑎2𝑏∼𝑏𝑎3𝑎4},𝑐₂≡𝑓^∼𝑓^{𝑎2𝑎3𝑏∼𝑏𝑎4𝑎1},𝑐₃≡𝑓^∼𝑓^{𝑎4𝑎2𝑏∼𝑏𝑎3𝑎1}. (5.3) 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. The 𝑐𝑖’s can be obtained by dressing the vertices with structure constants ̃‌𝑓𝑎𝑏𝑐, 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 get𝑀₄=𝑛𝑛₁^∼₁𝑠₁₂+𝑛𝑛₂^∼₂𝑠₁₄+𝑛𝑛₃^∼₃𝑠₁₃, (5.4) where 𝑛^∼_𝑖 is just the numerators belonging to 𝐴^∼₄ written in a similar representation. Compared to (5.2) we have basically just replaced the color factors 𝑐_𝑖 with kinematic numerators 𝑛^∼_𝑖. This nicely captures the statement of gravity being a “double-copy’’ of gauge theory.

5.2. At General 𝑛-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 is always 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 𝑛_𝑖 satisfy the same Jacobi identities as the color factors 𝑐_𝑖. In detail, this is to write the full tree-level amplitude in the form𝐴^full_𝑛(1,2,…,𝑛)=_𝑖𝑐_𝑖𝑛_𝑖∏_𝑗𝑠_𝑗𝑖, (5.5) where the color-kinematic duality𝑐_𝛼+𝑐_𝛽−𝑐_𝛾=0⟶𝑛_𝛼+𝑛_𝛽−𝑛_𝛾=0 (5.6) is satisfied. The sum in (5.5) is over all different diagrams, only containing cubic vertices, and (∏_𝑗𝑠_𝑗)_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 Section 3.3, 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 by𝑀_𝑛(1,2,…,𝑛)=_𝑖^∼𝑛_𝑖𝑛_𝑖∏_𝑗𝑠_𝑗𝑖, (5.7) we just replace the color factors 𝑐_𝑖 in (5.5) with kinematic numerators 𝑛^∼_𝑖 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(−𝑖)^𝐿𝐴^loop_𝑛=_𝑖^𝐿_𝑙=1𝑑^𝐷𝑝_𝑙(2𝜋)^𝐷1𝑆_𝑖𝑐_𝑖𝑛_𝑖∏_𝑗𝑠_𝑗𝑖, (5.8) where the sum is over all 𝑛-point 𝐿-loop diagrams with cubic vertices, 𝑆_𝑖 is the symmetry factor of diagram 𝑖, and the numerators 𝑛_𝑖 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(−𝑖)^𝐿+1𝑀^loop_𝑛=_𝑖^𝐿_𝑙=1𝑑^𝐷𝑝_𝑙(2𝜋)^𝐷1𝑆_𝑖^∼𝑛_𝑖𝑛_𝑖∏_𝑗𝑠_𝑗𝑖, (5.9) 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 the closed 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 the 𝑛-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.

Appendix

A. The Complex Power Function and Branch Cuts

The complex power function 𝑧^𝑐 is in general a multivalued function. As a simple example, consider 𝑧^1/2 and take 𝑧=|𝑧|𝑒^𝑖𝜃=|𝑧|𝑒^{𝑖𝜃+2𝜋𝑖}, then𝑧^1/2=|𝑧|𝑒^𝑖𝜃1/2=|𝑧|^1/2𝑒^𝑖𝜃/2,𝑧^1/2=|𝑧|𝑒^{𝑖𝜃+2𝜋𝑖1/2}=|𝑧|^1/2𝑒^𝑖𝜃/2𝑒^𝑖𝜋=−|𝑧|^1/2𝑒^𝑖𝜃/2, (A.1) which is obviously two different results for the same 𝑧.

In order to have a well-defined function, here meaning single-valued, we need to impose a branch cut on the complex plane. Most often, one chooses the branch cut to lie on the negative real axis, that is we restrict the power function to𝑧^𝑐=|𝑧|^𝑐𝑒^𝑐𝑖𝜃,−𝜋<𝜃≤𝜋. (A.2) What this means is (1) take any complex number 𝑧, (2) write it up in polar coordinates, that is, 𝑧=|𝑧|𝑒^𝑖𝜃, with 𝜃∈]−𝜋,𝜋], (3) then the function gives you back the complex number |𝑧|^𝑐𝑒^𝑐𝑖𝜃.

It might seem like we are being overly cautions, but it is so easy to just start using rules we know from ordinary power functions, like (𝑧₁𝑧₂)^𝑐=𝑧^𝑐₁𝑧^𝑐₂, which is not always true in the complex case.

A.1. Phase Factors

Let us see how the above branch cut dictates the phase factors in (2.12). That is, we are considering (A.2) for some 𝑧₀ with Re(𝑧₀)<0. If Im(𝑧₀)≥0 we have𝑧₀^𝑐=𝑧₀^𝑐𝑒^{𝑐𝑖𝜃0},𝜋2<𝜃₀≤𝜋. (A.3) If we now look at the power of −𝑧₀ we get−𝑧₀^𝑐=𝑧₀^𝑐𝑒^{𝑐𝑖(𝜃0−𝜋)}=𝑒^{−𝑖𝜋𝑐}𝑧₀^𝑐𝑒^{𝑐𝑖𝜃0}=𝑒^{−𝑖𝜋𝑐}𝑧₀^𝑐. (A.4) Note that we have to take the argument for (−𝑧₀) as 𝜃₀−𝜋 in order not to fall outside the ]−𝜋,𝜋] range. We can then conclude that𝑧₀^𝑐=𝑒^𝑖𝜋𝑐−𝑧₀^𝑐. (A.5) In particular, we see that 𝑒^{−𝑖𝜋𝑐}(−𝑧₀)^𝑐=𝑒^{−2𝑖𝜋𝑐}(𝑧₀)^𝑐≠(𝑧₀)^𝑐 in general, and we therefore have to pick the phase 𝑒^𝑖𝜋𝑐, like in (2.12).

Likewise, if Im(𝑧₀)<0 we have𝑧₀^𝑐=𝑧₀^𝑐𝑒^{𝑐𝑖𝜃0},−𝜋<𝜃₀𝜋<−2, (A.6) and for −𝑧₀−𝑧₀^𝑐=𝑧₀^𝑐𝑒^{𝑐𝑖(𝜃0+𝜋)}=𝑒^𝑖𝜋𝑐𝑧₀^𝑐𝑒^{𝑐𝑖𝜃0}=𝑒^𝑖𝜋𝑐𝑧₀^𝑐, (A.7) hence𝑧₀^𝑐=𝑒^{−𝑖𝜋𝑐}−𝑧₀^𝑐. (A.8) Again we see that (𝑧₀)^𝑐≠𝑒^𝑖𝜋𝑐(−𝑧₀)^𝑐, so our branch cut implies the phases used in (2.12).

The careful reader might also have noticed that in (2.6) we actually did use (𝑧₁𝑧₂)^𝑐=𝑧^𝑐₁𝑧^𝑐₂, so how do we know this is allowed here without any modifications in terms of phase factors? The reason is that in this case 𝑧₁ and 𝑧₂ have opposite sign on the imaginary part. For instance, say Im(𝑧₁)<0 and Im(𝑧₂)>0, then 𝜃₁∈]−𝜋,0[ and 𝜃₂∈]0,𝜋[, which imply (𝜃₁+𝜃₂)∈]−𝜋,𝜋[, and therefore in this situation 𝑧^𝑐₁𝑧^𝑐₂ is equal to (𝑧₁𝑧₂)^𝑐.

Acknowledgments

The authors thanks Zvi Bern, Emil Bjerrum-Bohr, Poul Henrik Damgaard, Bo Feng, Pierre Vanhove, and Cristian Vergu for discussions and comments. Special thanks go to Kemal Ozeren for his comments on an early draft. This paper was supported in part by the National Science Foundation under Grant no. NSF PHY05-51164.