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 -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 [119]. 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 , even when taking the field theory limit [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, [2224] 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 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 , as given by the Kleiss-Kuijf relations [33, 34], down to . At this stage these BCJ relations, and their supersymmetric extension [35], had not been proven for general -point amplitudes. Interestingly, the first proof came from string theory where Bjerrum-Bohr et al. [36] and Stieberger [37] used monodromy to derive the 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 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 -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 -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 -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 -point tree-level closed string amplitude is given by where we have fixed the three points , , and . The and 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 , such that By analytically continuing the variables to the complex plane, we can rotate the integration contour for these variables from the real axis to (almost) the imaginary axis without changing the value of the amplitude. Here is some small number making sure we avoid the branch points. This changes the expressions in the integrand (to linear order in ) If we now make the transformation of variables and define , it is easy to verify that the expression on the right-hand side of line (2.4) is given by respectively.

In total, this brings (2.1) into the form where the additional factor of is due to the Jacobian when changing variables in the integral and from the rotation of the contours.

First we note the following, assume that at least one and look at the contribution from , that is, The behaviour of the imaginary terms near the branch points is The requirement in the last line is not a problem, since we can just choose to look at the integral corresponding to the “smallest’’ variable, which has to lie in the range due to our first assumption. This means that we can close the integral of in the lower half of the complex -plane (again by analytical continuation), and since the closed contour does not contain any poles the integral vanishes. In general, when we avoid the branch point below the real axis, and when we avoid it above the real axis. From this kind of argument we see that whenever one of the -variables is in the range of or , at least one of the contours can be completely closed either below or above the real axis. Hence, only when all lie between 0 and 1 will there be a contribution to (2.7).

By splitting up the -integration region we can write the -point closed string amplitude as where is the “ordered amplitude’’ defined such that . For instance, at five points this corresponds to splitting the integration over the -plane into an integral over the region “above’’ the line (i.e., ) and an integral “below’’ this line (i.e., ). Together with the above restriction on the -integration range, the part of in (2.10) becomes where we have omitted the infinitesimal terms. We recognize (2.11) as the expression exactly corresponding to the color-ordered open string amplitude . Note that, compared to (2.7), we have written instead of and such that 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 part, otherwise we would get wrong phase factors.

For simplicity we will from now on fix the ordering to , that is, we are considering the contribution in (2.10). The remaining terms can simply be obtained through permutation of labels.

We have just seen that the part is nothing but the color-ordered amplitude. We therefore turn our attention to the part, investigating which contours the imaginary terms dictate for the integrals. Near the quantity is a positive imaginary number (remember that ), so the contour is above the real axis here. For we have which is a negative imaginary number, hence the contour lies below the real axis. Finally, for we see that , meaning that the contour for lies below the contour of for . See Figure 1 for an illustration of this nested structure.

The next step is to deform the contours for 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 with 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 , we can pull the contours from 2 up to to the left, and the set from to to the right ( or 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 , pulling the contour for to the left, only showing the piece involving , we get Note that in the previous first line we write instead of and instead of in order to compensate for the same swapping of order in the integral of (2.11). Now, as illustrated in the bottom of Figure 2, we also close the contour for to the left (only showing the part involving the variable) We see that the total integration over and exactly correspond to color-ordered open string amplitudes, which we will denote to distinguish them from the ones following from the part, such that the whole contribution in can be written as Together with (2.11) for , and (2.10), we have obtained the following relation between the five-point closed string amplitude and the color-ordered open string amplitudes , If we take the other extreme, that is, closing both contours to the right (), we get for the integration, and for the integration, see the top case of Figure 2, that is, Finally, we could have closed to the left and to the right, as also illustrated in Figure 2, resulting in ()

All of these different forms can be nicely collected into one compact formula if we introduce the momentum kernel where equals 1 if the ordering of and is opposite in and , and 0 if the ordering is the same. Here we have defined for a general number of legs, so, for instance and so on. We will also define for empty sets. With this function we can collect (2.16), (2.19), and (2.20) into one formula as with . Note that when one should read the set as being empty, and likewise for the set is empty.

2.2. General -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 -point case, it directly generalizes to with .

Expression (2.24) shows how to factorize an -point closed string amplitude into the product of -point color-ordered open string amplitudes and , “glued’’ together by kinematic factors contained in the function. We note that the expression is a sum over terms, taking its maximum value for or , and its minimum for . (The floor and ceiling functions are defined on half-integers as follows: if is odd, or if is even. if is odd, or if is even.)

Although, being one of the expressions having most terms, choosing the relation takes a particularly nice -point form only involving one function and having a greater symmetry between the sums over different ordered and amplitudes. An equally nice form occurs with .

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 . The amplitudes just go to their corresponding field theory expressions and , where it follows from (2.21) that with , or more generally , 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 -point KLT relations in field theory then take the final form again with the freedom . 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 supergravity amplitude on the l.h.s. and 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 and , and at the same time introduce a short-hand notation that will be used when expressions get large. With or (3.3) becomes or respectively, see also Figure 3. Here we have introduced the short-hand notation , for a permutation over leg , and likewise for . Note that and are two unrelated permutations, the tilde is just to remind us which one is going with , and we are trying to be economic with our use of Greek letters.

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 . The crossing symmetry between, for instance, legs and in (3.4) can also easily be seen by use of the reflection symmetry of color-ordered amplitudes , and the following identity: 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 . 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) 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 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 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 (), 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 -independence of (3.3). Third, we look at what happens with the r.h.s. of (3.3) when and 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 -point amplitudes we start by making the following deformation of momenta and : where is just some arbitrary parameter, and is a four-vector satisfying and . This preserves conservation of momentum and keeps , but makes .

The gravity amplitude can then be written as [15] We note that as 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 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 being off-shell instead of , we can write down the “dual’’ expression to (3.11) where we have called the deformation parameter .

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 -point amplitudes these read 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 , where 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 color-ordered amplitudes, provided by the Kleiss-Kuijf relations, down to independent amplitudes [38].

Some simple explicit examples of (3.13) are for four points, and 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 function. This is achieved through where is just some arbitrary permutation of the legs . 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 explicitly, that is, The two is exactly zero due to (3.13).

The general argument is to divide the sum of into a sum of groups where all, except the first leg in the ordering, call it , have fixed ordering, and then insert at any place. For each group all factors from will be the same except for the factor contributing from . This will give a BCJ relation and thereby vanish. In the above five-point example we had and the two groups we summed over were the one with ordering and , respectively. From this we see that (3.16) is a consequence of (3.13).

3.3.1. The -Independence

With the BCJ relations we can now address the -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 -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 -independence is given by 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 and (3.18) reads (where we have taken ) and since (3.3) for and is given by using (3.19) we immediately get which is just (3.3) with . Likewise we could now use (3.18) with to rewrite (3.21) into (3.3) with .

This procedure generalizes to points; by repeated use of (3.18) the -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 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 and to live in different helicity sectors the r.h.s. of (3.3) is identically zero.

To be more specific let us denote an -point color-ordered tree amplitude with helicity configuration belonging to as , that is, has negative helicity gluons, with for nonvanishing amplitudes. We are not interested in the exact helicity configuration, just which helicity sector it belongs to. We then have [43, 44] 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 , we have

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 supergravity point of view, the vanishing of the r.h.s. of (3.3), when and belong to different helicity sectors, correspond to -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 and , as where is a complex variable and is a four-vector satisfying . This preserves conservation of momentum and on-shellness. At tree-level is a rational function of external momenta, implying that the deformed amplitude is a rational function of . Assuming when , Cauchy’s theorem tells us that where is the undeformed amplitude, coming from the residue of the pole. The remaining poles (those different from ) come from , which we know can only follow from Feynman propagators going on-shell, that is, when vanishes, where with and (or vice versa). We do not get poles in if since such a is independent of .

Let us denote the value of where is going on-shell by , which can be found by solving . Since the poles of will be simple poles, the residues are given by where . Using the general factorization property of gluon amplitudes we see that with . Hence, combined with (3.25), we get the BCFW recursion relation [26, 27] where the sum, denoted with , 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 for the graviton amplitude is even stronger than one could naively have guessed [5362].

4. A Purely Field Theoretical View

We are finally in a position to see why (3.3) must be true for all 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 -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 -Point Proof

We assume that we have checked (3.3) up to points, that is, we have checked that the expression on the r.h.s. is equal to gravity. Then we write down the -point expression for the r.h.s., let us denote it . Our goal is to show, only based on our knowledge of lower point cases, that this is equal to the -point gravity amplitude, that is .

In the same way as the BCFW recursion relation was derived, we start out by deforming two momenta in our expression for and consider the contour integral where is just the undeformed -point expression and we have included a potential boundary term on the left-hand side. Let us first argue that . If we make a deformation in and the function in (3.5) will be independent of . The vanishing of the boundary term is then guaranteed by the large behaviour of the gauge theory amplitudes and , this was also used in [19]. In the proof we will present below it is more convenient to make a deformation in and . However, such a deformation cannot have a boundary term either since in Section 3 we already argued for the crossing symmetry between and in . The large behaviour must therefore be equally good for this deformation.

Now that we have established , the goal is to show that the sum of residues exactly makes up the BCFW-expansion of an -point gravity amplitude, and hence 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 , and we will be using in the form of (3.4). Like in (3.27) we can compute these from , where is the -value that makes 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 or .(b)The pole appears in both and .

First we investigate (a), and we will consider the case where the pole only appears in . The case with the pole appearing in can be handled in a similar way. The terms contributing to this class must all involve an amplitude with the set of legs always next to each other, that is, the terms from (3.4) that make up this contribution is where we have omitted the overall factor of , which can easily be reinstated into the proof. We emphasis that since we are considering (a) we have excluded all permutations that would lead to a pole in . From this we get the residue where , and we have used the factorization property of color-ordered gauge theory amplitudes in (3.28). Also note that the pole , from the factorization of , has been replaced with , that is, without the hat on 1, from the calculation of the residue. Furthermore, following from the definition of in (3.2), we can write where denotes the relative ordering of leg in the set. Collecting everything in (4.4) that involves the permutation we have something of the form where, as we have seen above, the quantity inside vanishes when 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 and now contain the pole , they must both have the set of legs collected next to each other. The contributing terms, from (3.4), then take the form Like in (4.5) we will again make use of a factorization property of the function. It is not hard to convince oneself that in the limit where goes on-shell, we can write The residue for can then be expressed as where the “mixed helicity terms’’ are expressions of the same form but with products between amplitudes with, for instance, instead of or 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 and , which precisely makes up the regularized KLT form in (3.11), that is, it is just , and a sum over and which is an point version of (3.4) and hence, by induction, equal to . Altogether (4.9) is This is exactly the BCFW contribution to the -point gravity amplitude from a pole, and due to the manifest symmetry we have also already obtained the contributions for all poles related to these by a permutation of legs .

We are still not completely done, since the above analysis does not cover the pole contributions involving both 1 and , that is, poles of the form . To investigate these we use the form in (3.5). It is well suited for this case since here legs 1 and 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 takes the form where we again assume the pole appears in the amplitude. Using the factorization property where denotes the relative ordering of leg in the set, we once again find that these contributions contain a factor of that vanishes due to BCJ relations. There is therefore no contribution from (a).

Considering part (b) for the pole, the contributing terms from (3.5) are with satisfying the factorization property (when goes on-shell) Hence the residue can be written where we have used , 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 Once again we obtain the correct BCFW expansion for all poles, and all poles related to these by a permutation of