Scalar-Graviton Amplitudes

Using the CHY-formalism and its extension to a double cover we provide covariant expressions for tree-level amplitudes with two massive scalar legs and an arbitrary number of gravitons in D dimensions. Using unitarity methods, such amplitudes are needed inputs for the computation of post-Newtonian and post-Minkowskian expansions in classical general relativity.


Introduction
Recently it has been realized that modern methods for amplitude computations at loop level may provide a powerful new way to compute post-Newtonian and post-Minkowskian expansions in classical general relativity [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. This builds on the observation that the quantum mechanical scattering matrix for matter interacting gravitationally contains classical pieces at arbitrarily high order in the loop expansion [19][20][21] and the fact that the sought-for long-distance contributions are non-analytic in the exchanged momentum [22,23], thus making them straightforwardly accessible through unitarity cuts. All needed contributions being classical, one would not expect it to be necessary to regularize the loops dimensionally. However, since infrared 'super-classical' (see, e.g., ref. [21]) terms appear at intermediate steps it is nevertheless convenient to use dimensional regularization.
For the scattering of two massive objects at large distances the needed tree-level amplitudes are those of two massive scalars and, at n-loop order, (n + 1) on-shell gravitons. Using the Kawai-Lewellen-Tye (KLT) relations [24][25][26][27] these can conveniently be constructed from the corresponding amplitudes with the (n + 1) gravitons replaced by gluons, amplitudes that are given in the literature on the basis of recursion relations [28,29] in four space-time dimensions, using the spinor-helicity formalism. More recently, Naculich [30] has suggested an alternative and more direct method for the computation of such amplitudes based on the Cachazo-He-Yuan (CHY) formalism [31,32]. One advantage of using the CHY-formalism is that it immediately provides the amplitudes 'covariantly', in terms of general polarization tensors for the gravitons, and hence not restricted to four space-time dimensions.
From a practical point of view, it suffices to evaluate amplitudes with two scalar legs and (n + 1) gluons and subsequently turning them into scalar-graviton amplitudes by KLT-squaring. This is our approach here. One key point of the present calculation is the computation of a factorized expression for the amplitudes of two massive scalars coupled to Yang-Mills theory, expressing them as sums over lower-point amplitudes which are combinations of scalar-gluon amplitudes and pure gluon amplitudes. Each of these has one gluon leg off-shell and an associated polarization vector of both transverse and longitudinal components. In this way, we can iteratively construct amplitudes of an arbitrarily high order. Crucial for this factorized form is the insight gained from the double-cover version [33][34][35][36][37] of the CHY-formalism. This double-cover description naturally splits amplitudes into two lower-point amplitudes, each with one leg off-shell. These vector currents, contracted with polarization vectors, are glued together by the polarization sum. A subtlety here is the contribution from longitudinal modes that need to be dealt with carefully. Useful relations that short-cut the evaluations of some of the color-ordered amplitudes needed for the recursive evaluation of higher n-point amplitudes are provided by simple identities [38][39][40] among these partly massive amplitudes.
The outline of this paper is as follows. In sections 2 and 3 we show how to compute amplitudes with two scalars and n gluons using different methods. In section 4 we briefly discuss the straightforward application of Kawai-Lewellen-Tye relations to replace the gluons with gravitons. Some technical details and a proof of an important theorem regarding vanishing longitudinal contributions are provided in appendices.

Prelude: Two massive scalars and n gluons
We first present a simple way to obtain explicit expressions for the scattering amplitudes of two massive scalars and n gluons. Since our method is based on the CHY approach, we give a very brief review of this formalism. We then apply the factorization method developed in [37] to obtain, up to six-point, analytical expressions for the scattering of gluons where two of them, suitably defined, are massive. Next, we turn the two massive gluons into massive scalars, thus providing the scattering amplitudes for two massive scalars and in principle any number of massless gluons.

Massive Yang-Mills Amplitudes
We start by presenting a simple recursive formula that computes pure Yang-Mills amplitudes with up to three massive gluons. The method we will use was developed by one of us in a different context [33,37]. We shall show explicit expressions up to six points but it is straightforward to extend the method to any higher number of external legs. In the following, we will denote massive particles with the capital letter "P α " and the massless ones with the lower-case letter "k a ". Unless otherwise mentioned we will work under the assumption of implicit momentum conservation, Let us first recall how to extend the CHY approach to the massive case following the method of Naculich [30]. We have {P 1 , ..., P i } as momenta of the massive particles (P 2 α = 0) and {k i+1 , ..., k n } as momenta the massless gluons (k 2 a = 0). A generic momentum vector is thus K A ∈ {P 1 , ..., P i , k i+1 , ..., k n }. We define as well (2. 2) The modified CHY scattering equations are then given by where the matrix ∆ AB is still to be determined. In order to guarantee SL(2, C) invariance, i.e., n A=1 σ m A S A = 0 for m = 0, 1, 2, the matrix ∆ AB must be symmetric, ∆ AB = ∆ BA , and it must satisfy the conditions Since we are interested in at most up to three massive gluons of momenta {P 1 , P 2 , P 3 }, it is sufficient to consider only ∆ 12 , ∆ 13 , ∆ 23 . Therefore, we therefore have the simple conditions ∆ 12 + ∆ 13 = P 2 1 , ∆ 12 + ∆ 23 = P 2 2 , (2.5) that have a unique solution given by (2.6) When two masses are degenerate, e.g. , P 2 1 = P 2 2 = 0 and P 2 3 = 0, it is straightforward to see from (2.6) that ∆ 12 = P 2 1 and ∆ 13 = ∆ 23 = 0, which, not surprisingly, is in agreement with the one-loop scattering equations formulated in refs. [34,[41][42][43]. On the other hand, when only one of the legs is massive, e.g. P 2 1 = 0 and P 2 2 = P 2 3 = 0, then ∆ 12 = P 2 1 /2, ∆ 13 = P 2 1 /2 and ∆ 23 = −P 2 1 /2, i.e., in order to describe one massive particle it is necessary to use at least three ∆ AB parameters.
After having described the massive scattering equations let us now remind that the CHY prescription for color ordered amplitudes of the scattering of gluons at tree-level is given by [30,31,44] A n (P 1 , ..., P i , i + 1, ..., n) = dµ n PT(1, 2, ..., n) × Pf Ψ n , where dµ n is the usual CHY measure and PT(1, ..., n) and Pf Ψ n are the usual Parke-Taylor and reduced Pfaffian factors The 2n × 2n matrix, Ψ n , is defined as with, and The matrix, (Ψ n ) AB AB , denotes the reduced matrix obtained by removing the rows and columns A, B from Ψ n , where 1 ≤ A < B ≤ n.
Since we are interested in the case of at most three massive particles of momenta {P 1 , P 2 , P 3 } we can avoid dealing with the ∆ AB -matrix in the scattering equations altogether by choosing the labels {j, k, l} and {m, r, s} in (2.8) to match with the massive ones, i.e., {j, k, l} = {m, r, s} = {1, 2, 3}.
It is useful to recall that the reduced Pfaffian (Pf is independent of the choice of A and B, and that the SL(2, C) symmetry is guaranteed by the transversality of the external polarization vectors, ( C · K C ) = 0. However, we note that the terms C AA and C BB do not appears in the reduced matrix, (Ψ n ) AB AB . It follows that the transversality conditions on A and B are not needed to obtain an integrand invariant under the action of SL(2, C) [45]. We can therefore consistently define the integral with these two legs being off mass-shell and with arbitrary polarization vectors for ( A · K A ) = 0 and ( B · K B ) = 0. We now use the double-cover method ref. [37] to obtain compact recursive expressions for these massive and/or off-shell scattering amplitudes as defined above. The results clearly reduce to the usual expressions when all external legs are massless and on-shell.
First, let us consider the basic building block of three legs. We take all three particles to be massive and choose the polarization vectors 1 and 2 as not necessarily transverse so that we do not impose ( 1 · P 1 ) = 0 = ( 2 · P 2 ). We are going to denote with a bold source in the amplitude (as in. [30,31,37]), e.g.
the rows/columns that are removed from its reduced Pfaffian. In the above amplitude the reduced Pfaffian is given by, Particles P α and P β can thus be off-shell, so that ( α · P α ) = 0 and ( β · P β ) = 0. Therefore, using the CHY prescription given in (2.7) one has where we have used due to the momentum conservation constraint P 1 + P 2 + P 3 = 0 and the transversality condition ( 3 · P 3 ) = 0. Although the amplitude itself is independent of the choice of rows/columns that are removed in the Pfaffian, the intermediate expressions do depend on the choice and we have therefore introduced a notation where we indicate which rows and columns are removed.
We consider next a computation with three massive gluons of momenta {P 1 , P 2 , P 3 } and one massless gluon of momentum {k 4 }. Using the factorization method described in [35,37], this four-point calculation can be expressed in terms of the A 3 (P a , P b , P c ) building-blocks, where the notation P M i (P L i ) means the particle with momentum P i has as polarization vector M i ( L i ). The sums over the polarizations are given by the relations The unusual normalization factor of the longitudinal modes is precisely what is needed to recover the correct four-point amplitude [36,46]. The polarization vectors of all massive on-shell legs of course still satisfy i · P i = 0. Using that condition it is easy to see that the last term in (2.16) evaluates to The full four-point amplitude is thus remarkably simple.
Finally, in order to calculate higher-point amplitudes we will also need A 4 (P 1 , P 2 , P 3 , 4).
Using the BCJ-like identity [38][39][40], it is straightforward to deduce The calculation of higher-point amplitudes with massive gluons now proceeds recursively. We illustrate a few cases in the appendix.

Turning massive gluons into scalars
Now, using the prescriptions of Naculich [30] and Cachazo, He, and Yuan [32] we can compute the amplitudes of interest which also involve massive scalar legs. The basic idea is to consider the massive gluon theory in one extra dimension (i.e., in D + 1 dimensions) with "polarizations" and momenta of massive scalars chosen to be Massless gluons (a = 2, ..., n − 2, and k 2 a = 0) .
As a first step we note that when P 1 and P 2 are associated with scalar legs the threepoint amplitude reads We illustrate our method by evaluating the four-point function of two massive scalars and two gluons. Using the above conditions and the cyclicity property (2.23) we im-mediately infer this amplitude from eqs. (2.16): where the superscript (or subscript) "ϕ" refers to one of the massive scalars. We note that the term (2. 19) does not contribute at all, (we shall return to this point later).
Remark: Since µ 1 = µ 4 = ( 0, 1) the contraction relation for the second term in (2.25), is non-vanishing only when V µ has a non-zero projection on 4 . Therefore, it is equivalent to choosing M µ P 34 = M µ P 12 = ( 0, 1), i.e., the internal lines corresponding to momenta P 12 and P 34 turn out to be propagating scalars as expected due to current conservation. In other words, The same phenomenon occurs for higher n-point amplitudes. Let us now introduce some convenient notation: as well as where V µ a and W ν b are two generic vectors. From (2.17) and (2.14) it is straightforward to compute as well as The four-point covariant amplitude of two massive scalars and two gluons is thus given by the simple expression Specializing to four dimensions, this is in agreement with the result found in the literature on the basis of the spinor-helicity formalism [29].
In an analogous way, the five-point amplitude becomes where eq. (A.1) has been used. As in the four-point case, the purely longitudinal contributions vanish on account of the orthogonality conditions for the polarization vectors associated with external scalar legs, ( 1 · 3 ) = ( 5 · 2 ) = 0. In appendix B, we prove the vanishing of these longitudinal contributions for any number of external gluons.

Kleiss-Kuijf decomposition
While the method described in the previous section is straightforward and immediately generalizable to any number of gluons n, we wish to point out that an alternative track based on an expansion with analytically computed BCJ-numerators is of comparable simplicity. The trick is to compute the scattering of two massive scalar fields with massless gluons (eventually gravitons) by decomposing the reduced Pffafian in terms of a Kleiss-Kuijf (KK) basis [47] by using the Bern-Carrasco-Johansson (BCJ) numerators [48] for Yang-Mills theory. This useful technique was developed in 1 [44,49,50].
To illustrate, let us consider the four-point amplitude A 4 (1 ϕ , 2 g , 3 g , 4 ϕ ). From (3.3) we arrive at Now applying the method of ref. [50], the BCJ numerators are readily found to be given by where we have fixed the reference ordering to be (1, 2, 3, 4). The massive integrals obtained in (3.5) are straightforward to do using the Λ-algorithm [33]. We find For the four-point amplitude we therefore get which agrees with the result we found in equation (2.32).

Explicit BCJ numerators at five points
This method easily generalizes. For two massive scalar legs and three gluons we need to evaluate are given by where we have fixed the reference ordering to be (1, 2, 3, 4, 5).
Using again the Λ-algorithm [33], it is straightforward to compute, with two massive legs, This method does have the drawback for n large that the number of BCJ numerators grow in a factorial way. For instance, to compute the six and seven-point amplitudes one needs to calculate 4! = 24 and 5! = 120 numerators, respectively.

Two massive scalars and gravitons
In the previous sections, we have shown different methods for efficient evaluation of scattering amplitudes of two massive scalar fields and (n − 2) gluons. Staying within the CHY-framework as in section 2, one could similarly express the amplitude of the scattering among two massive scalars (ϕ) and gravitons (h a ) through [44,52,53], where the gravitons are identified as, h µν a ≡ µ a ν a and using the same massive measure defined in (2.8). Similarly, one can use a KK-decomposition analogous to what we explained above for the case of gluons in (3.3), and write However, by using the Kawai-Lewellen-Tye (KLT) [24] relations at the amplitude level, it seems much more straightforward to find the scattering between two massive scalar and (n − 2) gravitons by use of the momentum kernel [26,27], i.e, A n (n ϕ , (n−1) g , β g , 1 ϕ ) .

(4.4)
Here A n is an amplitude of two massive scalars and (n − 2) gluons as defined in (2.22), and the momentum kernel S[α|β] is where Θ is the step function. For instance, for the four-point amplitude we immediately get where S[3|2] = −s 23 , thus using the result found in (2.32), one has which is the correct 4-point amplitude. Higher order amplitudes follow by KLTsquaring analogously.

Conclusion
We have presented different methods to compute the tree-level scattering amplitudes of two massive scalars and an in principle arbitrary number of gravitons in D-dimensions. These are the tree-level amplitudes needed to obtain the classical two-body scattering of two massive objects without spin in general relativity through the use of unitarity. The most economical method appears to be the one based on a new set of recursive relations that can be derived from the so-called Λ-algorithm (or double cover) in the CHY-formalism. In this method one first defines an extension of scattering amplitudes where one external leg is taken off-shell (defining, effectively, a current in the case of Yang-Mills theory) and then glues off-shell legs together by an appropriate polarization sum. We have proven a particular simplification in comparison to the pure Yang-Mills case when the amplitude contains two massive scalar legs: a sum over longitudinal polarizations cancels exactly. The resulting amplitude relations for two massive scalars and any number of on-shell gluons thus becomes surprisingly simple. Although a similar technique can be used to compute amplitudes of two massive scalars with an arbitrary number of gravitons we have found it economical to simply use KLT-squaring in order to obtain these. Again, they are then provided in D-dimensions and with arbitrary polarization tensors.
We have checked our general recursive formula up to six points with existing expressions in the literature for the case D = 4, always finding complete agreement. An interesting observation is the possibility of establishing a new on-shell set of recursion relations for these amplitudes based on BCFW-recursion combined with the double-cover analysis of the Λ-algorithm. This will be discussed elsewhere. where we have written A 5 (P 1 , P 2 , P 3 , 4, 5) in terms of the smaller amplitudes, A 3 (P a , P b , P c ), A 4 (P a , P b , P c , d) and A 4 (P a , P b , P c , d). As in the four-point case, we must use the identities in (2.17) and (2.18).
Finally, let us show how to compute the six-point amplitude, A 6 (P 1 , P 2 , P 3 , 4, 5, 6). The factorization decomposition of A 6 (P 1 , P 2 , P 3 , 4, 5, 6) is given by where 2 ↔ 3 means the changing of the two labels, α = 1, 3 and PĀ is the complement of P A (by the momentum conservation condition, P A + PĀ = 0). For example, P A is given by P 2456 , P 24 and P 245 in the last three term in (A.7), respectively, therefore, PĀ is P 13 , P 1356 and P 136 . Additionally, the identities in (2.17) and (2.18) must be used in the above factorization expansion.

B Longitudinal contributions
As we have observed in all special cases worked out in this paper, the longitudinal contributions to the factorized amplitudes with massive scalars always vanish identically. In this section we prove this important fact in all generality.