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Physics Letters B

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Graviton as a pair of collinear gauge bosons

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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Received 23 September 2014 Accepted 24 October 2014 Available online 30 October 2014 Editor: L. Alvarez-Gaume

A B S T R A C T

We show that the mixed gravitational/gauge superstring amplitudes describing decays of massless closed strings - gravitons or dilatons - into a number of gauge bosons, can be written at the tree (disk) level as linear combinations of pure open string amplitudes in which the graviton (or dilaton) is replaced by a pair of collinear gauge bosons. Each of the constituent gauge bosons carry exactly one half of the original closed string momentum, while their ±1 helicities add up to ±2 for the graviton or to 0 for the dilaton.

© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

Quantization of gravitational waves yields gravitons: massless spin 2 particles with two polarized degrees of freedom (helicity +2 = ++ and —2 = —) in four dimensions. While the existence of gravitational waves is well established, the detection of individual gravitons may be impossible due to extremely low cross sections. Nevertheless, theoretical understanding of gravitons and their interactions is a prerequisite for constructing a viable theory of quantum gravity.

Superstring theory offers an interesting insight into gravitons. In this framework, they appear as zero modes of closed strings. On the other hand, it is known that zero modes of open strings give rise to spin 1 gauge bosons. With the closed string seen as a loop of two open strings connected at both ends, graviton appears to be a "bound state" of two vector bosons. This is also suggested by the form of graviton vertex operator: in type II superstring theory, it is a product of two spin 1 vertex operators (from the left- and right-moving sectors of world-sheet excitations). Helicity ++ appears as a superposition of two helicity + states while helicity —— comes as a superposition of two helicity — states. In addition, the products +— and —+ create two degrees of freedom of the scalar (complex) superstring dilaton.

In 1985, Kawai, Lewellen and Tye (KLT) [1] derived a formula which expresses any closed string tree amplitude in terms of a sum of the products of appropriate open string tree amplitudes. At the level of zero modes, KLT relations allow expressing the graviton and dilaton amplitudes in terms of products of gauge boson amplitudes. The existence of such relations means that, at least in the leading order of perturbation theory, the content of Einstein's

* Corresponding author.

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

gravity is encoded in Yang-Mills (YM) theory. The quadratic form of KLT relations is perfectly consistent with the heuristic picture of a closed string as a loop of two open strings. In fact, string field theory suggests a similar description [2]. This does not help, however, in answering the question of whether the graviton can be considered as a pair of gauge bosons beyond the world-sheet, as an actual bound state in physical space-time. One alternative description has been developed in [3,4], by constructing closed superstring amplitudes through the "single-valued" projection of open superstring amplitudes. This projection yields linear relations between the functions encompassing effects of massive closed and open superstring excitations, to all orders in the inverse string tension a'. They reveal a deeper connection between gauge and gravity string amplitudes than what is implied by the KLT relations, but they do not provide new insight into their a' ^ 0 field theory limit.

In this Letter, we present a linear relation between the amplitude for the decay of one massless closed string state, i.e. a gravi-ton or a dilaton, into an arbitrary number N — 2 of gauge bosons and a sum of purely open string amplitudes involving N gauge bosons. The sum involves so-called partial amplitudes associated to particular gauge group factors. The original closed string state is replaced by two vector bosons, each of them carrying exactly one half of its momentum, and its helicity is split in the same way as in string vertex operators. In the forthcoming publication [5], we will show that in all open and closed string amplitudes, gravitons and dilatons can be replaced by pairs of such collinear vectors bosons.

Although our derivation utilizes full-fledged Type II super-string theory, it is instructive to discuss the field theory limit (i.e. the zero slope a' = 0 limit of Regge trajectories) of mixed

http://dx.doi.org/10.1016/j.physletb.2014.10.057

0370-2693/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

gravitational/gauge interactions. This limit is described by Einstein-Yang-Mills (EYM) theory coupled to the dilaton.1 All tree level amplitudes can be constructed by using the recursion relations derived by Britto, Cachazo, Feng and Witten (BCFW) [8], with the basic building blocks provided by the following three-point amplitudes:

Л (1-A(1-

, 2 ,3++ =

, 2+ , 3-+) =

(23}2(31}2: (12)2(13)2

A(1 + , 2—, 3—) = , V ' ' ! -12}2'

A(1 — 2—, 3+-) = -12}2,

where we used superscripts to label helicity states, with +— and —+ assigned to the dilaton and its complex conjugate, respectively. We are using standard notation of the helicity formalism, see [9]. The mass dimension (—1) gravitational coupling л/К is implied by the above expressions. In addition, three gauge bosons interact with the well known Yang-Mills amplitude

Л(1-, 2-, 3+ =

(23>(31)'

where we omitted the (dimensionless) gauge coupling constant.

A good example of an amplitude involving both gravitational and gauge couplings is the amplitude for the graviton decay into three gauge bosons. In this case,

Л(1 + , 2+, 3-; q--) =

(12)(23)(31)

which can be obtained either by using BCFW recursion relations or by a straightforward Feynman diagram calculation. In this Letter, we focus on the amplitudes similar to (3), describing gravitons and dilatons decaying into an arbitrary number of gauge bosons. In string theory, these are disk amplitudes with one closed string vertex insertion on the world-sheet and a number of open strings attached at the boundary.

In order to compute the amplitudes, it is convenient to use the "doubling trick," to convert disk correlators to the standard holo-morphic ones by extending the fields to the entire complex plane [10]. Furthermore, the integration over positions of world-sheet symmetric closed string states (such as graviton or dilaton) can be extended from the half-plane covering the disk to the full complex plane. Open string vertices representing N — 2 gauge bosons with momenta p,, i = 1, ..., N — 2 (in an arbitrary helicity configuration) are inserted on the real axis at x,, while a single closed string vertex operator, which represents the graviton or dilaton with momentum q, is inserted at complex z. All momenta are restricted to four dimensions, with p2 = q2 = 0 (although the following derivation is independent on the space-time dimension and can be adjusted to massive states). The amplitudes involve integrals of the form

FN = Va<C/4) £ Pi + q

r N—2

X / П dXi f] |Xr - Xs|2a'PrPs (Xr - Xs)nrs J i=1 1<r<s<N-2

1 For early work on EYM scattering amplitudes, see Ref. [6]; for more recent work,

see [7].

d2z(z - z)nW |Xi - z| { i=1

2a'piq (xi - z)ni (xi - z)n', (4)

where we included the momentum-conserving delta function and divided by the volume VCKG of the conformal Killing group. The powers nrs, n,, n,, n are some integer numbers. To be specific, we focus on the amplitude associated to one particular Chan-Paton factor (partial amplitude), Tr(T1T2 ... TN—2), with the integral over ordered x1 < x2 < ... < xN—2.

The techniques for evaluating generic disk integrals involving both open and closed strings have been developed in [11]. For the concrete case (4), we write the complex integral as an integral over holomorphic and anti-holomorphic coordinates, by following the method proposed in [1]. After writing z = z1 + iz2, the integrand becomes an analytic function of z2 with 2(N — 2) branch points at ±i(x, — z1). We then deform the z2-integral along the real axis Im(z2) = 0 to the pure imaginary axis Re(z2) = 0. In this way, the variables

f = z1 + iz2 = z,

Г = zt - z = z

become real. After changing the integration variables (z1, z2) ^ (%, n) (with the Jacobian deta(Z(%%'n2>' = 2), Eq. (4) becomes an integral over N real positions x,, %, n

Fn = v-cwe kiutldxijdfj

\ \ — i = l — rvi

— CO —CO

Ц |Xr - xs|2a'krks (Xr - Xs)nrs

1<r<s<N-2

x - (f - ri)nft n(xi ,f,n)\Xi - f \2a'kikN-1 ixi - n|2a kikN

x (Xi - f)ni (Xi - r)ni,

with the open string momenta kr = pr, r = 1, ..., N — 2 and the closed string momentum split in half:

kN-1 = kN = 1 q.

Eq. (6) resembles a generic open string integral involving N open strings with external momenta ki supplemented by the extra phase factors

n(Xi ,f, r) = e2n ia'kikN e [-(Xi-f)(Xi-

where 0 denotes the Heaviside step function. These monodromy factors (8) account for the correct branch of the integrand, making the integral well defined. Note that the phases, which are independent on the integers nrs, n,, n, n do not depend on the particular values of integration variables, but only on the ordering of % and n with respect to the original N — 2 vertex positions. In this way, the original integral becomes a weighted (by phase factors) sum of integrals, each of them having the same form as the integrals appearing in N -point (partial) open string amplitudes, with the vertices inserted at xi, l = 1, ..., N, where we identified xN—1 = % and xN = n. Note that the order of the original N — 2 positions remains unchanged. Since the graviton as well as dilaton vertices factorize into two gauge bosons inserted at z = % = xN—1 and z = n = xN, we conclude that the amplitude A(1, 2,..., N — 2; q) describing graviton (or dilaton) decays into N — 2 gauge bosons can be written as a weighted sum of pure open string amplitudes with the graviton (or dilaton) replaced by a pair of collinear gauge bosons, each of them carrying exactly one-half of its momentum, cf. Eq. (7).

g—27TQ2 g—27ra3 g-îîraj V

Xl+1.....XN~3 XN-Î

. . . '-t-l .-.-.-

c ......

X2 X3 ......Xl Ç

eiirai+i einaN_3 eiimN_2

xi<Ç< xi+1

Fig. 1. Complex rç-plane and contour integrations. Here ai = a'piq = 2a'kikN. g-wrai2 g-i7ra3 g-wraj V

-^V-O......... .

......*.....O " " ' '

£ Si+l.....ZjV-3 XN-2

g-Î7ro;2 g—¿7TO:3 g-ttrû:;

®2 X3....... Xl Xl<Ç< Xl+1

Fig. 2. Contour deformation in complex rç-plane.

In order to express the partial amplitude A(1, 2,..., N — 2; q) in il—3 /

terms of N -point open string amplitudes, we need to analyze the + n I sj,N—1

phase factors. For a given xi < Ç < x+ with l = 2, ..., N — 3 the i=rN1 i=l+^ J=l+1

phase factor (8) in the integrand can be accommodated by con- /1/1 t m 1 t , 1 ; ; , , iw -n

. , , ,ft fi • x aymO,...,1, n — 1,l + 1,...,j, N,i + 1,...,N — 2). sidering respective contours in the complex rç-plane. After fixing

the position of the first open string vertex at x1 = — œ we have (10) the situation depicted in Fig. 1. Quite generally, around all open

string vertex positions xi < Ç the contour goes clockwise, while for Let us 2consider some examples with a small number of external

xi > Ç anti-clockwise. In either case we can deform the contour to particles- For N = 5, 6 and N = 7 our formula (9) yields: the left or right. To obtain a minimal set of integration regions for

x2 <Ç< xrN, we move the contours to the left, cf. Fig. 2. On the A(1, 2' 3; q) = sin(ns24)A(1, 5, 2, 4, 3), (11)

other hand, for x^n 1 < Ç < xN—2 we swap the contour to the right. A(1, 2, 3, 4; q) = sin(ns25)A(1, 6, 2, 5, 3, 4)

This way for each region xi <ç < xi+1 with I = 2, ..., r J1 — 1 we + sin(n s45) A(1, 2,3, 5,4, 6), (12)

obtain a residual contour of i — 1 loops starting from x1 = — œ and

encircling the i — 1 points x2, ..., xi. On the other hand, for each A(1, ^ 3 4, 5; q~) = sin(n s26) A(1, 7, 2, 6 3, 4, 5)

region xi <ç< xi+1 with i =rN ^ ..., N — 3 we get a contour of + sin(n s36) A(1, 2, 7,3, 6, 4, 5) N — 2 — i loops starting from +œ and encircling the N — 2 — i points

xN—2,..., xi+1. In total we obtain (r J1 — 2)(L J J — 1) terms: + sin[n(s36 + s26)] A(1, 7, 2, 3, 6, 4, 5)

A(1, 2,...,N — 2; q) + sin (n s56) A(1, 2, 3,4, 6, 5, 7). (13)

r 21—1 i / i \ The first two cases have already been worked out in [11]. However,

= V^ y^ sjn i n y^ s - n 1 | let us investigate their structure in more detail.

\ -i ^ / In order to make connection with EYM theory, let us take the

= i= ]=i zero slope limit of Eq. (11), for the same helicity configuration as

x A(1,...,i — 1, N, i,...,i, N — 1, i + 1,..., N — 2) in Eq. (3):

N—3 N—2 / i \

+ ^ Y. sinl n Yi sj,N—A A(1 + , 2+, 3—; q——) = ^24Aym(1 + , 5—, 2+, 4—, 3—)

i=r21 i=i+1 V j=i+1 ' 0). (14) x A(1,...,i, N — 1, i + 1,...,i, N, i + 1,..., N — 2), (9)

The Yang-Mills amplitude is the maximally helicity violating

where sj,j = sjj = 2a'kjkj. On the r.h.s., according to (7) kN—1 =

kN = q/2, and the helicities of respective (labeled by N — 1 and N, a (1+ 5— 2+ 4— 3—) — 4 [12] (15)

respectively) gauge bosons are determined by the graviton (— or ^ , , , , ' [1 q][q3][13][2q]2 ++), or by the dilaton (+— or —+). Note that in the zero slope

a' ^ 0 limit sin(n ski ) ^ n skl all N -point open string amplitudes where we set |4] = |5] = q, cf. Eq. (7). After using s24 = ^ = 2

become pure Yang-Mills subamplitudes: and momentum conservation, we find that the graviton amplitude

agrees with Eq. (3), up to an overall factor which is necessary in

Aeym(1, 2,...,N — 2; q) order to convert string mass units into the gravitational *J~k. On

r N1—1 i / i s the other hand, at the full-fledged string level of Eq. (11), we can

2 use the expression for the five-point open superstring amplitude

= n Z^ 2-^sj,N—1 f

i=2 i=2 \ j=i '

x AYM (1,...,i — 1, N, i,...,i, N — 1, i + 1,..., N — 2) 2 A formula similar to Eq. (10) has been considered before in Ref. [12].

A(\, 5, 2, 4, 3) [13], and take its collinear limit, i.e. s12 = s, s23 = u, s34 = 2, s45 = 0 and s51 = cf. Eq. (7), to obtain

t is u

A(1, 2,3; q) = n-Aym(1, 5, 2,4,3)sv W -, -

where3

F (s, u) =

F(1 + s)r(1 + u) r(1 + s + u)

is the four-point open superstring formfactor and sv is the single-valued projection,4 previously discussed in the string context in [3,4]. Alternatively, we can use the well-known relation [14]

s25Aym(1, 5, 2,4,3) = — s12Aym(1, 2,3,4, 5)

— (s12 + s23)Aym(1, 3, 2,4, 5), (18)

to rewrite (16) as:

A(1, 2,3; q) = — n[sAYM(1, 2, 3,4, 5)

— tAYM(1, 3, 2,4, 5)]sv|F^2, 0 J. (19)

Note that in (16) and (19) the single-valued projection eliminates all powers of f2 in the a'-expansion of the amplitude A(1, 2, 3; q). This is special for final states with two or three gauge bosons; with more gauge bosons in the final state, the amplitudes will start receiving contributions from the Zi(F^v)a effective interaction terms.

Next, let us discuss the five-point amplitude (12). Here, we use the expressions for six-point open superstring amplitudes A(1, 2, 3, 5, 4, 6) and A(1, 6, 2, 5, 3, 4) [13], and take their collinear limit. Six-point string functions depend on nine kinematic invariants: si,i+1 = a'(ki + ki+1)2, i = 1, ..., 6 mod 6, and t1 = a'(k1 + k2 + k3 )2, t2 = a'(k2 + k3 + k4)2, t3 = a'(k3 + k4 + k5)2. In the collinear limit of Eq. (7), s12 = s1, s23 = s2, s34 = s3, s45 = s4, s56 = 0, s61 = f and t1 = s4, t2 = s5, t3 = 2 + 2 (cf. [15]), where si = si i+1, i = 1, ..., 5 mod 5, are the five-point kinematic invariants. In this way, we obtain

A(1, 2,3,4; q) = n{FeaAvMO, 6, 2, 5,3,4) + F6b[Aym(1, 6, 5, 2,3,4) + Aym(1, 5, 6, 2, 3,4)]} + (1 ^ 3), (20) with the a' expansions:

1 s5(sj — s3 — s4) [ Z2

F6a = —-{ 1 (2s1 s2 — s1 s4 + s3s4 — s4s5)

2 s4 2

+ O(a'4),

F 6b = 1 {(s4 + s5) — Z2(s2 s3s4 + s1s2 s5 — s2 s4 s5) }

+ O (a'4).

The collinear limits of Yang-Mills amplitudes have been studied for a long time [9]. Partial amplitudes with adjacent (in the gauge group trace factor) gauge bosons, like number 4 and 5 on the r.h.s. of Eqs. (19) and (18), contain collinear divergences and, at the leading order, factorize into a divergent factor times the amplitude with the collinear pair replaced by a single particle [9]. These leading divergences cancel in Eq. (19), as it is clear from Eq. (18).

The collinear limits on the r.h.s. of Eq. (9) do not contain singularities because the relevant gauge bosons are not adjacent. It would be very useful to have some compact formulas for such limits. They would require understanding the case of adjacent collinear gauge bosons at the subleading level.5 For full-fledged string amplitudes, one also needs collinear limits of string formfactors, as in Eq. (21), to all orders in a'.

It is tempting to think of the two gauge bosons - that substitute for the graviton or dilaton in the scattering amplitudes - as their constituent particles. The idea that gravity may be induced by some other interactions was contemplated long ago by Andrei Sakharov [18] (see also [19]), but it has never been implemented in a satisfactory theoretical framework. It is clear that WeinbergWitten theorem [20] represents a significant (but hopefully surmountable) obstacle to graviton compositeness, so it would be interesting to see how it works in the context of amplitude relations derived in this work. In order to seriously consider gravitons as bound states of gauge bosons, one would have to understand the monodromy factors of Eq. (9) in terms of two-particle wave functions of the underlying gauge (open superstring) theory.

Acknowledgements

We are grateful to CERN Theory Unit, where a substantial portion this work was performed, for its hospitality and financial support. This material is based in part upon work supported by the National Science Foundation under Grants Nos. PHY-0757959 and 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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