Physics Letters B 744 (2015) 160-162

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Physics Letters B

www.elsevier.com/locate/physletb

Graviton amplitudes from collinear limits of gauge amplitudes

Stephan Stiebergera *. Tomasz R. Taylorb

a Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, 80805 München, Germany b Department of Physics, Northeastern University, Boston, MA 02115, USA

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A R T I C L E I N F 0

Article history: Received 9 March 2015 Accepted 24 March 2015 Available online 28 March 2015 Editor: L. Alvarez-Gaume

A B S T R A C T

We express all tree-level graviton amplitudes in Einstein's gravity as the collinear limits of a linear combination of pure Yang-Mills amplitudes in which each graviton is represented by two gauge bosons, each of them carrying exactly one half of graviton's momentum and helicity.

© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

The amplitudes describing scattering processes of many gravitons in quantized Einstein's general relativity are related to the amplitudes describing vector gauge boson scattering in Yang-Mills theory. As discovered by Kawai, Lewellen and Tye (KLT) [1] in 1985, gravitational amplitudes can be written as "squares" of gauge amplitudes, more precisely as bilinear forms of partial gauge amplitudes weighted by the kinematic coefficients presently known as the "KLT kernel." Kawai, Lewellen and Tye discovered these relations in the framework of string theory, as a connection between closed and open string amplitudes. Their quadratic form is a direct consequence of the factorization of the graviton vertex operator into the operators creating non-interacting left- and right-moving fluctuations of the world-sheet. They support an intuitive picture of the closed string as a loop of two open strings connected at the ends [2]. More recently, Cachazo, He and Yuan [3-5] developed a novel representation of graviton amplitudes, by utilizing the so-called scattering equations. Here again, gravity appears as a "square" of gauge theory.

In a recent work [6], we expressed tree-level Einstein-Yang-Mills amplitudes involving one graviton and an arbitrary number of gauge bosons as linear combinations of pure Yang-Mills amplitudes in which the graviton appears as a pair of collinear vector bosons, each of them carrying exactly one-half of its momentum and helicity. This result is a low energy, field theory manifestation of a much broader class of linear relations between closed and open string amplitudes, which will be discussed elsewhere [7]. It indicates that, in some way, the graviton can be considered as a pair of gauge bosons beyond the world-sheet, as a bound state in physical space-time. Although Weinberg-Witten theorem [8] rules

* Corresponding author.

E-mail address: stephan.stieberger@mpp.mpg.de (S. Stieberger).

out a massless spin 2 graviton emergent from pure gauge dynam-ics,1 it is possible that such linear relations reflect something more subtle.

In the present work, we express the N -graviton amplitude as a collinear limit of a particular linear combination of pure Yang-Mills (partial) 2N-gluon2 amplitudes. The paper is organized in a simple way. First, we establish notation. Then we will state the result and prove it by showing agreement with the KLT formula. As an illustration, we will work out explicitly the three-graviton case.

The momenta ki and helicities Xi =±2 of N — 1 gravitons will be split into momenta pi, qi and helicities ¡ii, vi of 2(N — 1) gluons in the following way:

i = 1, 2,...,N - 1.

Pi = qi = ^ , fli = vi = 2

We will also introduce one additional pair of gluons, with the momenta p and q, and opposite helicities, ^ = +1 and v = —1, respectively. With all momenta assumed to be incoming into the scattering process, the momentum conservation is N — 1 N — 1 N — 1

Y^pi + Y qi+p+q=Y ki+p+q=0- (2)

i=1 i=1 i=1

All momenta are on-shell (light-like). It is convenient to represent p and q as matrices which factorize into helicity spinor variables in the following way:

p = apcfp , q = OqOq - (3)

1 This is due to the existence of a Lorentz-covariant energy-momentum tensor in Yang-Mills theory.

2 For brevity vector gauge bosons are called "gluons" below. All Yang-Mills amplitudes are partial, associated to a single trace of (arbitrary) gauge group generators in the fundamental representation.

http://dx.doi.Org/10.1016/j.physletb.2015.03.053

0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativec0mm0ns.0rg/licenses/by/4.0/). Funded by SCOAP3.

S. Stieberger, T.R. Taylor / Physics Letters B 744 (2015) 160-162

For real light-like momenta p and q, the limit spq = (p + q)2 = (pq)[qp] ^ 0 at finite p and q constraints the respective three-momenta to a collinear configuration, i.e. both pointing in the same direction.3 Here, however, we will be considering complex momenta, which allows two ways of reaching spq = 0:

[pq] ^ 0 with (pq) = 0 :

Up ^ xa„

where x is an arbitrary number, and similarly,

(pq) ^ 0 with [pq] = 0 :

a„ ^ xa„

We will be using the following 2N -gluon Yang-Mills partial amplitudes

A[ p, N - 1,1,n(2, 3,...,N - 2), 1,p(2,...,N - 2), N - 1, q] = A[p, f = +1; pn-i,hN-I; ...; p(N-2)n, f^(N-2)n;

qi,vi;...; qN-1,VN-1; q,v = -1]

where n, p e SN—3 denote permutations of N — 3 elements and in = n(i), jp = p(j). We will also need the KLT kernel S[n\p] introduced in [1,10,11]

S [n\p] = S [ n(2,...,N - 2) \ p(2,...,N - 2) ]

N-2 i-1

= n (/W + J2 9(iП, jn) sin,jn i=2 j=2

where sitj = (pi + pj)2 and 6(in, jn) = 1 if the ordering of the legs (in, jn) and (ip, jp) is the same for n(2,..., N — 2) and p(2,..., N — 2), and zero otherwise.4

Theorem. The N-graviton amplitude in Einstein's gravity is given at the tree level by:

Ae[kt,X1;...; 1<n—1 ,Xn—V; kn = p + q,^N = +2]

[pq]^0\2xj (pq)

= lim I —

x Y^ S [n\p]A[p, N - 1,1,n(2, 3,...,N - 2),

n,pcSN-3 1, p(2,...,N - 2), N - 1, q] ,

where the limit is defined in Eq. (4). Similarly, Ae[kt,X1 ;... ; In-1 ,Xn-V; kn = p + q,^N = -2]

= lim (2x) -s

4 ( pq) 2 -s

[pq]spq

x S [n\p] A[p, N - 1,1,n(2,3,

k,P^Sn-3 1, p(2,...,N - 2), N - 1, q] ,

with the limit defined5 in Eq. (5).

..,N - 2),

2 s3 , there-

pq * pq'

Proof. In order to prove Eq. (8), we note that [pq]s fore the limit [pq] ^ 0 pushes the Yang Mills amplitude on the r.h.s. of (8) into a highly singular (triple "factorization pole") kinematic configuration. In the first step, we factorize on the pole in

3 We are using standard notation of the helicity formalism [9].

4 Note, that the kernel (7) does not depend explicitly on the momenta p and q.

5 In the above relations, we omit constant factors involving Yang-Mills and gravitational coupling constants.

(N—2)n 1

Fig. 1. Factorization in the N-gluon channel.

(N—2).

Fig. 2. Factorization of subamplitudes.

the N -gluon channel shown in Fig. 1, with the total momentum of:

n—1 p q s

1 = £ pi + p = p-——q , l2 = —s-f. (10)

Furthermore, the subamplitude on the left side of Fig. 1 develops a pole in the two-gluon channel with

pn = p - l =

p2 — b_pq pN = 4

as shown in Fig. 2. Similarly, the subamplitude on the right side of Fig. 1 develops a pole in the two-gluon channel with

p + q__l<N_ = 2

qN = q + l =

q2 — qN = 4

It is easy to show that in the limit (4), there is a unique helicity configuration contributing to the triple pole

A[ p, N — 1,1,n(2, 3,...,N — 2), 1, p(2,...,N — 2), N — 1, q] 4 "

x A[p+, —l—, — pN]

X A[ pN, fiN = +1; N — 1,1,n(2, 3,...,N — 2)]

x A[1, p(2,...,N — 2), N — 1; qN ,vN = +1]

x A[q—, l+, —q— ], (13)

where we used ± superscripts to indicate the respective ±1 vector boson helicities. In this limit, the three-gluon amplitudes reduce to:

A[ p+, —l—, — pN ] = - (pq) , A[q—, l+, —qN ] = -(pq) . (14)

After inserting Eqs. (13) and (14) into Eq. (8), we obtain Ae№1,^1;... ; kN-1,^N-1'; IN= +2]

= J2 S [n\p]

n,p^SN-3

x A[ pN, fN = +1; N - 1,1, n(2, 3,...,N - 2)] x A[1, p(2,...,N - 2), N - 1; qN ,vN = +1] ,

S. Stieberger, T.R. Taylor/ Physics Letters B 744 (2015) 160-162

which is the KLT formula for the N-graviton amplitude [1,10]. The proof of Eq. (9) proceeds in a similar way.6

Example (Three-graviton amplitude). For N = 3, the KLT kernel is trivial and Eq. (8) reads

Ae [ki Д i; /<2 Д2; k_3 = p + д,Хз = +2]

= lim f^) Ms2pqA[p, 2, {1,1}, 2, q] , [pq]^o\2xJ <pq> pq

where we used curly brackets to indicate symmetrization in {p1, q1} before setting p1 = q1. The symmetrization removes the collinear singularity at p1 = q1; it is necessary in the N = 3 case only. The r.h.s. of Eq. (16) can be rewritten by using the BCJ relation [12]

sjqA[p, 2, {1,1}, 2, q]

= spq (- spqA[p, 1, 2,1, 2, q] + 2 sq1 A[p, 2,1, 2, q, 1])

which is manifestly finite. The combination on the r.h.s. of the above equation appears in the zero string slope limit of the four-particle open-closed string disk amplitude involving two gravitons and two gauge bosons, cf. Eq. (3.40) of Ref. [13]:

spqA[p, 2, {1, 1}, 2, q] = A[k\ ; /<2 Д2; p,ß; q, v] .

In this way, we find that the limit of Eq. (16) amounts to factorizing this Einstein-Yang-Mills amplitude in the s channel, on the graviton pole. In Ref. [7], we will show that the same conclusion holds for higher N: the triple pole limits of Yang-Mills amplitudes in Eqs. (8) and (9) correspond to the degeneration limit of disk amplitudes involving N — 1 gravitons (closed strings) and two gauge bosons, i.e. open strings attached to the disk boundary. In this limit, the boundary shrinks to a point and the amplitude factor-izes into N-graviton amplitude on the sphere times the amplitude for one of the gravitons to decay into a pair of gauge bosons. Now returning to the case of N = 3, we see from Eq. (17) that a non-vanishing amplitude requires two gravitons to carry opposite helicities, thus we set Xi = +2, X2 = —2. The amplitude (18) can be computed by substituting the well-known six-point NMHV Yang-Mills amplitudes [9] into the r.h.s. of Eq. (17), giving

A[k1,X1 = +2; k2,X2 = —2; p, /л = +1; q,v = —1] [1 p][1q] <2q}4

<1p><1q> spq

Next, we substitute it to Eqs. (18) and (16), and take the limit by using

[1 p] [2 p] , , , , fnn,

x = —r = —r , k3 = p + q = —k1 — l<2 , (20)

[1q] [2q]

which yields the correct result:

Ae[k1,X1 =+2; l<2,X2 = —2; ks,Xs = +2]= [12]2[23]2 - (21)

To summarize, we proposed a new representation of gravitational amplitudes at the tree level. It would be interesting to learn whether it can be extended beyond the tree level, to improve our understanding of loop corrections in quantum gravity.

Acknowledgements

This material is based in part upon work supported by the National Science Foundation under Grant No. PHY-1314774. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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6 The amplitudes involving string dilatons can be obtained in a similar way, as collinear limits of Yang-Mills amplitudes involving two additional gauge bosons, but carrying identical helicities, i.e. ц. = v = ±1.