Absorption of a Quantum in a D1/D5 System

In {\em Nucl. Phys.} B {\bf 615}, 285 (2001) [arXiv:hep-th/0107113], the wave equation for a minimally coupled scalar was studied in the geometry of a D1-D5 system with non-zero angular momentum. The probability for a quantum to enter the throat was computed by taking small a parameter $\g$ which is associated with the value of the angular momentum. In the leading order in $\g$, the result was found to agree with the dual CFT result. In this note, we report on an observation that there are corrections of higher order in $\g$. Our results should be useful for determining the higher order correction terms that the dual CFT needs in order to incorporate the presence of the `capped' geometry.

A D1/D5 brane system has been a valuable asset for studying black hole physics leading to many fruitful results in the calculations of entropy and radiation rate. For this system, the microscopic consideration yielded exactly the same value as the Bekenstein entropy [1,2]. Similarly precise agreement was found between its low energy radiation rate and the Hawking radiation of the corresponding black hole [3,4].
It has also served as a stage where one tackles the black hole information puzzle. A recent attempt to resolve the information puzzle can be found in [5,6]. The system that was considered in these works is not the conventional D1/D5 brane which has an infinite throat, but rather the one with its throat 'capped' off at a finite distance in the conical singularity. The way the throat caps off maps to a dual CFT microstate.
The mapping was established in [6,8] for special family of metrics providing another test for AdS/CFT correspondence.
In [9], the wave equation for a minimally coupled scalar was studied in the geometry of these special family of metrics. The truncated throat was called a 'tube'. Most of the low energy quanta that are sent toward the throat from spatial infinity get reflected back to the infinity when they arrive at the start of the throat. But there is a small probability for them to enter the throat and travel until they reach the end of the throat. Once reaching the end, they get reflected back to the start of the throat where they have the same small probability to exit from the tube.
The probability for a quantum to enter the throat was obtained by the same authors.
They also computed the 'time delay' for two low frequency quanta that emerge from the tube successively. However for both of these computations a parameter γ which is associated with the value of the angular momentum was taken small. In the leading order in γ, agreements were found with the corresponding CFT computations. Since the presence of γ itself is tied with the fact that the geometry is capped, it is worth reanalyzing the problem in the sub-leading order in γ in order to see whether corrections occur. This is, in fact, what we will do in this note: we keep γ arbitrary, and re-compute the time delay and the probability. Also in [9] the computation was done with null momentum along the compact direction of D1 brane wrapping. We will lift this condition too since it is not difficult to do so. As we will see below, the time delay turns out to be the same. However, the probability gets corrected in γ (and in λ as well). In other words, for a generic value of the parameter γ the gravity calculation of the probability gives a different result than the one obtained from the CFT analysis although the former gets reduced to the latter in the small γ limit. The discrepancy could probably be accounted by putting correction terms, which arise from a DBI action, in the CFT to incorporate the presence of the 'capped' geometry. Our results should be useful to determine them.
We begin by reviewing [9]. Let R be the radius of the circle that is common to the D1 and D5 branes, and n 1 , n 5 the number of D1, D5 branes respectively. The metric for the rotating D1/D5 brane system [10,7] is: where The parameter a has a physical meaning that the throat ends at r ∼ a. It was rewritten in [9] in terms of a dimensionless parameter γ = 2J n 1 n 5 where J represents the angular momentum: We take R ≫ (Q 1 Q 5 ) 1/4 . Consider the wave equation for a minimally coupled scalar in this background. For a massless scalar, it is Writing the solution in the following form: Φ(t, r, θ, φ, ψ, y) = exp(−iωt + imφ + inψ + iλy)Φ(r, θ).
withΦ (r, θ) = H(r)Θ(θ), equation (4) turns into a set of two equations, one for the radial part and the other for the angular part. The existence of separation of variables for such a wave equation was shown in [11]. Denoting the eigenvalue of the angular equation by Λ, its approximate value was determined in [9]: where l is a nonnegative integer that specifies the spherical harmonic of the angular solution. By introducing a new coordinate and dimensionless parameters ω = Rω, λ = Rλ the radial equation can be put into the following form, A solution was found in [9] imposing λ = 0 and taking γ → 0. Here we relax these constraints.
As in [9], we solve (9) in two asymptotic regions and match the solutions obtained in each region. First consider the large-x region. In this region we can borrow the corresponding steps of [9] with minimal modifications. For self-containedness of this note, we enclose some details: (9) simplifies to which is the same as the corresponding equation in [9] other than the shift, w 2 → The general solution is a linear combination of Bessel functions, but now with a shifted argument, where Expanding the Bessel functions in the leading order in the region where one gets Now let's turn to the inner region. The inner region is defined by Then (9) can be approximated to: By taking where equation (17) is converted to the hypergeometric differential equation The solution that is regular at x = 0 is For further manipulation, it is convenient to re-express G(x) using one of the hypergeometric identities, In [9], the leading behavior of G(x) was obtained by first approximating the hypergeometric functions to the Bessel functions. Here we do not take these intermediate steps but rather directly deal with the hypergeometric functions.
Noting that the hypergeometric function has the following series expansion one can approximate (24): with z ≡ γ 2 x and Imagine a low frequency quantum that is sent toward r = 0 from spatial infinity. As shown in [9], it is likely to be reflected at the start of the throat. But there is a small probability, P , for the quantum to enter the throat. Once it enters the throat, it travels to the end of the throat. There it gets reflected back to the start of the throat and could leave the throat toward spatial infinity with the same probability, P . Consider a quantity R, whose absolute square is the reflection coefficient. Its explicit expression is obtained in the first equation of (4.22) of [9], which we quote here, Note that As we will see below, C 2 (−1) l C 1 −(−1) l C 2 iπǫ is small, therefore we have To obtain the matching conditions, compare the inner solution (29) with the outer solution (15). Comparison of the coefficients of the negative power of x gives Solving for C 2 , one gets The other matching condition is which gives From this and (34), one gets where γ 0 is Euler's constant and using a formula derived in the appendix, (A.6), we get where the prime on Γ represents a derivative with respect to the argument and ξ ≡ ξ 0 + ǫ ′ ξ 1 + · · · . The first two coefficients, ξ 0 and ξ 1 , are derived in the appendix. They are given by Substituting (40) into (39), one gets This, combined with (32), allows one to re-write (30), Depending on the number of trips that a quantum makes inside the tube, each of them has a different phase. To discover the phase, we rewrite the cos sin -term as where in the second identity, we have employed the formal expansion that was used in [9]. With this R becomes, From the last term which contains the exponential factor, one can read off the time delay between two adjacent quanta that emerge from the tube, This is precisely the same as the result in [9]: neither the momentum on the compact direction nor the generic value of γ affects the time delay. However, both of them do affect the probability for a quantum to go into the throat. The probability is given as a square of the coefficient of the exponential factor, This strengthens the validity of the use of the orbifold CFT for the physics inside the throat region. Meanwhile, the flat region plays a role, as one might have expected, for the probability. In the CFT analysis, the flat region is accounted by terms that break conformal symmetry. In the leading order the 'non-CFT' terms give results that agree with the gravity calculation. Since we now have more general results in the gravity side, it should be possible to match them with CFT computation by including more detailed and higher order terms that arise from expanding a DBI action. Our results should be useful for that purpose.

Appendices A Useful Formulae
Define ξ ≡ ξ 0 + ǫ ′ ξ 1 . In the first order in ǫ ′ , where the prime on Γ's represents a derivative with respect to the argument. From this one reads off The ΓΓ ΓΓ -term that appears e.g., in (34) can be approximated as follows, where in the first equality we have used , (A.4) and in the second equality a trigonometric identity. In the first order of ǫ ′ the four Γ-piece that appears in the third line of (A.3) can be written as 1 + πǫ ′ cos π(β − α + l 2 ) sin π(β − α + l 2 ) (−1) l+1