IC/97/161;TIFR/TH/97-59;IMSc-97/11/40;hep-th/9801194 Planckian Scattering of D-Branes

We consider the gravitational scattering of point particles in four dimensions, at Planckian centre of mass energy and low momentum transfer, or the eikonal approximation. The scattering amplitude can be exactly computed by modelling point particles by very generic metrics. A class of such metrics are black hole solutions obtained from dimensional reduction of p-brane solutions with one or more Ramond-Ramond charges in string theory. At weak string coupling, such black holes are replaced by a collection of wrapped D-branes. Thus, we investigate eikonal scattering at weak coupling by modelling the point particles by wrapped D-branes and show that the amplitudes exactly match the corresponding amplitude found at strong coupling. We extend the calculation for scattering of charged particles.


Introduction
It is expected that the quantum gravity effects would become important at energies compared to the Planck Scale. Since the gravitational coupling constant G is not dimensionless, one can construct two independent dimensionless coupling constants, which for example in four space-time dimensions can be defined as G ' G s and G ' G t. Here G is the four dimensional Newton's constant, and s,t are the Mandelstam variables. Remarkably, it was shown in w x 1 that the full theory of quantum gravity can be split up into two independent theories with these 1 E-mail: saurya@imsc.ernet.in. 2  former signifies Planckian centre-of-mass energies, the latter implies Planckian momentum transfers. The first quantum gravitational scenario is easier to deal with because although s is large, t can be held fixed at a relatively small value such that srt ™`. The impact parameter of scattering, in this case, is very large and the scattering is almost forward. This is the so called eikonal approximation where the exact two particle scattering amplitude can be computed. In practice, it is advantageous to view one of Ž . the particles as static say A and the other moving Ž . say B at almost luminal velocity past it with a large impact parameter. A being static, can be suitably modelled by a metric, whose gravitational field B is supposed to experience. Then one can solve the ( ) wave equation for B in this given background and obtain the scattering amplitude. Of course, the reverse process is equally valid, when the 'shock-wave' space time produced by the A is obtained by Lorentz boosting the metric and analysing the wave function of the slow particle in this shock-wave background. As expected, the two pictures yield the same result w x 2 .
In the above picture, the point particles are usually modelled by Schwarzschild or Reissner-Nordstrom metrics, depending on whether the particles are neutral or charged. Although this seems natural in the framework of general relativity, these specific choices are certainly not mandatory. We show here that the results can be extended for a large class of generic spherically symmetric metrics. As an example, a large class of metrics arise as solutions of low energy string theory and one could model the particles by these metrics as well. The black holes which carry the NS-NS charges have already been considered in Planckian scattering which lead to w x interesting consequences 3 . It is shown here that these black hole metrics do not fall into the class of metrics we consider here, except in the extremal limit. Recently a new class of black hole solutions of low energy effective action of superstring theory Ž have been found, whose weak coupling in terms of . the string coupling g description consists of certain configurations of solitonic string states or D-branes, wrapped on suitable compact manifolds. Several pieces of evidence have emerged supporting this identification, the most notable being the fact that the degeneracy associated with the D-brane configuration exactly reproduces the Bekenstein-Hawking enw x tropy 4 and open string interactions on the D-brane reproduce the Hawking radiation spectrum of these w x black holes 5 . Thus, while the black hole description can be used at large coupling, the D-brane w x description is appropriate at small coupling 4-6 . w x D-brane scattering has been considered in 7-12 , w x and in particular in 7-10 , it has been shown scattering amplitudes of R-R charged p-branes agree with appropriate D-brane scattering in ten dimensions. There are black hole solutions with a singular horizon obtained by wrapping these R-R charged pbranes on compact spaces. The entropy for these black holes is zero, and hence the appropriate process to check for D-brane black hole correspondence is to look at scattering amplitudes. In this paper, we show that indeed the exact eikonal scattering amplitude can be computed for wrapped D-branes at weak coupling. Moreover, this amplitude agrees with that found in the black hole picture. The agreement per-Ž . sists when the particles carry U 1 charges also.
In the following section, we calculate the eikonal scattering phase shift in the strong coupling regime by modelling the particles by a general spherically symmetric black hole metric in four space-time dimensions. We also consider the special cases of R-R and NS-NS charged black holes. In the next section, we calculate the corresponding phase shift at weak coupling using D-p-branes wrapped on tori. The D-brane result is found to be independent of the brane dimensions as long as they are completely wrapped on the internal tori. Significantly, the scattering phase shift is dominated by graviton exchanges at ultrarelativistic velocities, as anticipated w x earlier in the calculations of 2 . Finally, we extend the calculations to include particles carrying electric charges, where too the results continue to agree.

Eikonal scattering at large string coupling
We will assume that the slow target particle gives rise to the following most general spherically symmetric metric in four dimensions: In general let the ultrarelativistic particle of charge Ž . e be minimally coupled to the U 1 gauge field A m produced by the static particle. Then its wave function F satisfies the covariant Klein-Gordon equation 1 where m and E are the mass and energy of the test w x particle. In the spirit of 2 we will assume that  in our problem s ; M we get the above condition pl on the mass. Thus we will ignore all terms quadratic in M,m in subsequent discussions. We substitute a solution of the form lm r and linearize the metric components at large distances as Here M is the ADM mass of the black hole metric. Retaining terms upto order 1rr 2 , the radial equation The gauge potential is assumed to have A as the 0 only non-zero component. With its explicit spherically symmetric form 0 and the identity s s 2 ME, the above equation reduces to

Ž .
It is straightforward to obtain the phase shift from w x here and the answer is 13 Ž . The above phase shift resembles Rutherford scattering with the fine structure constant a being re-Ž placed by the effective coupling constant y G sy 4 . eK , which is attractive for large s. The phase shift can be substituted iǹ to obtain the scattering amplitude. Using the asymptotic formula for large l P cosu ™ J 2 l q 1 sinur2 , Ž .
l 0 w x and converting the sum into an integral as in 1 , we get`u where y ' lr s and t s 4 E sin ur2. Finally, one gets Ž .
The cross section follows:  Ž .

ž /
Mr where a ' Q 2 e y2 f 0 , Q being the electric charge and f the asymptotic value of the dilaton field. There is 0 a curvature singularity at the horizon r s arM, which expands without limit for vanishing masses. It Ž . Ž Ž .. can be seen that the corresponding R r Eq. 1 Ž . here cannot be linearised in the asymptotic r ™r egion, except in the extremal limit. Thus, the eikonal scattering can be computed with these metrics only w x in the extremal limit 3 .

Eikonal scattering at small string coupling
In this section, we will compute the eikonal phase shift at weak string coupling, when the D-brane picture is appropriate. We develop a general formalism for the scattering of wrapped D-branes before specialising to four dimensions.
Consider a D-p-brane moving with a relative velocity Õ with respect to a D-l-brane in 10 space-time dimensions. They are separated by a large transverse distance b. We assume l F p and that none of the coordinate directions of the D-l-brane are orthogonal to those of the D-p-brane. Apart from the direction of velocity and the time coordinate, the end points of an open string ending on the two branes satisfy Ž . Ž . either Neumann N or Dirichlet D boundary conditions. We denote as NN the number of string coordinates which satisfy N condition at both the ends. Ž .
The oscillator sum and the integral finally yields where B, J are the bosonic and fermionic contributions to the oscillator sum in the in the one loop open w x superstring amplitude given by 8 1r12 n f q s q 1 y q , 1 9 Ž . Ž . Ž .
The rapidity e is defined as tanhpe s Õ. Note that the last term in J comes from summation of the Ž . F NS y1 sector and contributes only when ND s 0.
( ) This term is due to R-R exchange, and we will concentrate on this term in the next section when we look at charged particle amplitudes. Now, to compare the D-brane results with the results on the black hole side, we have to compactify the branes on suitable compact manifolds, such that in the non-compact space-time they look like point particles. For simplicity, we compactify on a c di-Ž . mensional torus with p F c , such that the resultant Ž . noncompact space time is a 10 y c dimensional with a Lorentzian signature. We take the range of each coordinate of the torus to be L ,i s 1, . . . ,c. Ž . Now, large impact parameter b ™`, scattering is dominated by the exchange of massless closed string states, for which it is sufficient to restrict the intew x grand in the regime t ™ 0 10,18 . Using the relevant w x formulae given in 9,19 , we get Ž .

Ž .
Note that the integral in the above expression is independent of p and l and depends only on c. That is, it is the same for branes of arbitrary dimensions for a given compactification. Thus, scattering phase shifts for all p,l can be calculated from the above expression provided the branes completely wrap on Ž . the internal torus. Also, as expected, K e s 0 s 0 for p s l or p y l s 4. This is the familiar no-force condition for BPS states.
However, for our present purposes, we specialise to the case of c s 6, i.e. scattering in 4 dimensional non-compact space time. Then the integral over t in X ' Ž . Ž . 28 simply yields a factor y2ln br 2pa . In addition, we make the ultrarelativistic approximation Õ ™ 1 , e ™`, such that Ž . Ž .
The integral is elementary, and when expressed in terms of the p-brane mass and the Newton's con-Ž . Ž . stant given Eqs. 32 and 33 respectively, we get Ž .

Discussions
We have shown that the eikonal scattering amplitude obtained by modelling the point particles as black holes is exactly reproduced by eikonal scattering of wrapped D-branes. In this regime only the Ž gravitational field at infinity is probed as the black . hole metric is linearised and the details of the metric are not realised. So, one should study the corrections to the eikonal phase shift, as the impact parameter and velocity are tuned to smaller values, and see w x whether the details begin to emerge 12,21 . Our results are independent of the dimensions of the Ž . brane but the kinematical factor k e , and hence Ž . d b ceases to be independent of p and l once one relaxes the condition e ™`.
We know that the type II B string theory has Ž . S-duality group as SL 2,Z which relates fundamental strings to the D-strings. Under this S-duality operation, gravitons are left invariant and NS-NS charged fields become R-R charged fields. Our D-Ž brane calculation for gravitational exchange domi-. Ž nant term matches the leading order term eikonal . limit in the scattering amplitude for fundamental w x strings 22 . This confirms the S-duality symmetry. Moreover, we have obtained the subdominant term Ž . eikonal limit due to R-R charged field exchange between the two D-p-branes. Invoking the S-duality, we can say that this must be also be the amplitude for NS-NS gauge field exchange in fundamental w x string scattering. In 22 , corrections to the eikonal fundamental string-string graviton exchange ampli-2 tude were calculated and shown to be order 1rl . It will be interesting to see whether the D-brane scattering amplitude gives the same.
Though the fundamental string and the D-brane scattering amplitudes are same, the relevant energy scales for eikonal scattering are different for the two. For the D-p-branes to be relativistic, the condition p must be imposed on their energy. Using the expression for the brane mass, we get As is evident, in 10-dimensions, the mass of the D-0 brane is much larger than the Planck mass M and pl X ' the string mass m ; 1r a . Hence for condition s Ž . 45 to be realised, energies relevant for D-0 brane eikonal scattering has to be much larger than both M and m . In other words for D-0 branes to pl s become relativistic, we need to consider regimes 2 X Ž 2 . Ž where sg a 4 1, In the c.m. frame s s E . Note that for non-perturbative effects of M-theory to be-2 X w x come important, we need to have tg a ™ 1 18 , and . we are not probing that regime. On the other hand, for fundamental strings, sa X 4 1 gives the eikonal w x w x limit, as considered in 22 . Following 23 we take X ' 1rg a as the energy scale in our problem. and the condition on the compactification volume, from Eq. Ž . 44 is p X ' V < a , p ( ) which implies that the compactification radii should be sufficiently small compared to the string scale. X ' w x Note that if we had used 1r a , as in 22 the conditions on compactification lengths would have become g dependent, and difficult to interpret.
Another interesting observation is that though the D-brane scattering amplitude includes all long range closed string exchanges, only the graviton exchange dominates in the above kinematical regime 5 . This was anticipated in the black hole calculation by 't Hooft and the weak coupling calculation vindicates this. Our weak coupling calculations can be generalised to higher dimensions. For example, in five non-compact dimensions, it is easy to see that the phase shift goes as 1rb, which is the Green's function for three transverse dimensions.
One can try to examine more sophisticated compactifications e.g. on K 3 to see whether similar w x conclusions hold for those situations also 25 . We hope to report on it in the near future.