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Absorption of partial waves by three-branes

We study the absorption of a class of fields in the geometry produced by extremal three-branes. We consider fields that do not mix with the ten-dimensional graviton. For these fields we solve the wave equations and find the absorption probabilities for all partial waves at leading order in the energy. We note that in some of these cases one needs an `intermediate' region which interpolates between flat Minkowski space at infinity and the AdS geometry near the branes.


Introduction
There has been renewed interest in the study of the background produced by 3branes, due to their conjectured connection to large N Yang-Mills theories in four spacetime dimensions [1][2] [3][4] [5]. Close to the branes, the geometry reduces to an an anti de-Sitter (AdS 5 ) metric times an S 5 , with a self-dual five form field strength producing the required curvature. It is conjectured that 10 dimensional IIB supergravity on this compactification is dual to large N Yang-Mills on the four dimensional boundary of the 5-dimensional AdS space.
Away from the immediate vicinity of the branes, the metric is not of the AdS form, and in fact far from the branes the spacetime approaches the Minkowski vacuum. The following process is of interest in this complete geometry. One imagines a low energy quantum incident on the branes, starting at spatial infinity (r → ∞). Some part of this wave tunnels through to the 'horizon region' close to the branes r → 0, where it then propagates freely inwards towards the branes, and this part of the wave may be considered as absorbed by the branes. The remainder of the wave is scattered back to r → ∞. One can compute the absorption probability for each spherical partial wave around the branes, or convert this to a cross section for absorption of different fields by the branes.
The above calculation is closely related to the connection between supergravity and Yang-Mills. In this connection one needs to find the coupling between supergravity modes and operators in the Yang-Mills, such that the former act as sources for the operators in the latter. Where would this set of couplings come from? Some constraint may arise from superconformal symmetry [6], but this is not likely to be enough by itself. In earlier work it had been conjectured that the absorption of supergravity modes by the three branes could be obtained from coupling the branes to their background through a Born-Infeld type action. Such a coupling gives exact agreement for the three-charge black hole [7], and for some modes of the dilaton in the case of 3-branes [8][9] [10].
In carrying out the classical calculation for absorption one notes that there are three regions of spacetime that one encounters in general. The region of small r is the AdS region, but this may not connect directly to the free propagation on flat space at r → ∞.
The calculation of the absorption probability in some cases requires matching the solutions for small r and for large r by using an 'intermediate region'. We say that this region is 'needed' in those cases where the solutions in the 'outer' and 'inner' (AdS) regions do not match on to each other as functions of r.
In this paper we carry out this calculation for a class of massless fields of 10dimensional supergravity propagating in the background produced by 3-branes. We consider fields that do not mix with the perturbations of the 10-dimensional gravitational field. For the fields that we study we consider all partial wave components, and construct solutions to the wave equation at leading order in energy. In all cases but one the fields separate out from each other, and for these cases we calculate the absorption probability for the incident quanta.
It is hoped that this analysis will shed light on the connection between supergravity and Yang-Mills, and also on how one might make a microscopic model to study absorption by the black 3-brane as well as by other black branes and black holes. This issue has been studied recently in [11][12].

Field equations and separation of variables
We will follow for the most part the notations in [13]. In this paper the authors had studied linearised perturbations of type IIB supergravity about AdS 5 × S 5 . While our geometry has this form only very close to the 3-branes, we have spherical symmetry at all r and the eigenvalues of the Hodge-de Rham operator on S 5 can be taken from this reference. The computation of these eigenvalues is carried out in detail in [14].
The field equations are In terms of the fields of 10 dimensional supergravity, The field B is a complex scalar describing the dilaton and the RR scalar. We study linearized perturbations around the background given by a collection of coincident 3-branes.
The perturbations can be separated into two classes: those that involve or couple to the fluctuations of the 10 dimensional metric, and those that decouple from these metric perturbations. We will study only the latter perturbations in this paper. We expand the fields around their background values. In the following the background value of the fields is denoted by a dot above the symbol giving the field. The coordinates on the S 5 surrounding the branes are called y α . The spatial coordiantes long the brane are called x a , a = 1, 2, 3.
The coordinates t, r, x a are collectively denoted by x µ . (Here r is a radial coordinate; the branes are located at r = 0.) These coordinates x µ describe the AdS 5 geometry in the region close to the 3-branes. The 10-dimensional coordinates carry a 'hat'. The ǫ symbol is a tensor with with αβγδǫ giving a frame with positive orientation on S 5 .
Note: We will call a field a p-form if it has p indices of the kind µ. This follows the language that is appropriate to studying the AdS 5 × S 5 geometry in the region close to the branes. 3 The metric is negative for the coordinate t and positive for the other nine.
Here Y I 1 are the spherical harmonics appropriate to scalar functions. Similarly Y I 5 α are the harmonics with one vector index along the S 5 , and Y I 10 [αβ] are the harmonics for an antisymmetric tensor on the S 5 .
The part a I 5 in (2.11) and the part a I 1 µ in (2.10) can be set to zero after a gauge transformation. The gauge transformation gives and we set 14) The background has the metric The background also has a 5-form field strength of the 4-form antisymmetric tensor potential: We have a corresponding field strength for all five indices of F on S 5 , given through (2.2).
We will need to compute the the following quantity Herer is the proper radius of the sphere on which the operator △ y acts: The functions Y k [αβ] can be split into two sets under the action of * DY k,± where ǫ αβγδǫ is the volume form on the unit 5-sphere.

Some mathematical relations
We consider perturbations of the form We will construct the solutions for φ(r) by dividing the radial coordinate r into three regions: (a) The outer region, given by r > r 1 for some choice of r 1 satisfying r 1 R >> 1 but also ωr 1 << 1. [We are working to leading order in ω, so we can always assume that ω is as small as we wish.] In the 'outer part' of this outer region, we will have ωr >> 1, while in the 'inner part' of this outer region we will have ωr << 1.
In this region we can ignore ω in the equation.
(c) The inner region, given by r < r 2 . The region r → 0 corresponds to v → ∞. A wave propagating to v → ∞ is considered to be absorbed into the branes. The boundary condition we impose in the computation of cross sections is that there be no wave propagating from v = ∞ towards smaller v.
In the outer region the wave equations turn out to take the form The solution is In the inner region the equations take the form which is equivalent to The solutions are Thus to compute the absorption probability we will take in (2.31) a solution with In the outer region at r → ∞, we have Thus from a solution of the form (2.28) we find for the probability of absorption of the spherical wave

Scattering matrices
We will write out the solution for each field in each of the three regions mentioned above. In may cases it is not essential to introduce the intermediate region to solve the equation, but we introduce this region in all the cases since one of our goals is to compare the behavior in this region of the different fields. We also wish to see the relation between the power law solutions that arise here with the dimensions of operators in the Yang-Mills or Born-Infeld descriptions of the 3-branes.
If one wants to compute the absorption probabilities, one can simplify the calculation by using a comparison of fluxes at r → ∞ and r → 0 [15]. In this case one does not need to compute both the solutions to the equation in each region, if one wants the result at leading order in the energy. We choose to instead to compute both solutions in each region, since from this we can extract the result to the following more general question.
We can imagine the wave to be incident either from r → ∞ or from r → 0. The final waveform will have components that travel in each of these two directions. Thus there is a 2 × 2 scattering matrix describing the scattering/tunneling through the intermediate region.
We do not write this matrix explicitly, but it can be read off from the solutions in each case.

Absorption of the dilaton-axion
The equation (2.4) with the ansatz (2.24) becomes

Outer region
Here the equation is The solution is

Intermediate region
Dropping the ω 2 term, we have The solution is Thus in this case we see that the equation does not have a nontrivial intermediate region.

Inner region
Here the equation reduces to The solutions are

Scalar from the antisymmetric two form
The field equation arises from (2.3), with the free indices taken to lie along the S 5 : Here the sign ± corresponds to the sign in the spherical harmonic Y k,± [αβ] (eq. (2.23)).

Outer region
The equation is The solutions are

Intermediate region
The equation is The solution is Thus in this case we have a nontrivial intermediate region.

Vector from the two-form field
The wave equation in this case arises from (2.3) with one free index on the S 5 and one free index ρ in the remaining 5 directions. For ρ = 0 we get For ρ = r we get We find that a 0 and a r can be algebraically determined from each other. For a r we have the equation Then we find the following equation for a r : and for a 0 we have the equation

Intermediate region
The equations are 1 The solutions are We find that a r vanishes if ω is set to zero, but it is still helpful to write it down since one can match either a r or a 0 :

Inner region
The equations are The solutions are (5.20)

Antisymmetric tensor from the 4-form
In (2.2) we take two of the free indices to be of type αβ and the rest to be of type µ.

Intermediate region
The equation is The solutions are b 12 = C 3 r k+2 + C 4 r −(k+2) (6.9) Thus the field equation does not have a nontrivial intermediate region.

Two form from the antisymmetric tensor
The wave equation follows from (2.3) + 2iF µ 1 µ 2 µ 3 µ 4 µ 5 ∂ µ 3 a µ 4 µ 5 = 0 The solution is We can then get a 03 from (7.6) and we find The equation for a 12 decouples from the other fields in the outer region: 1 r 5 ∂ r [r 5 a 12,r ] + ω 2 a 12 − k(k + 4) r 2 a 12 = 0 (7.12) The solution is Then we get The solutions are

Inner region
Here we use the factorized form of the equations. In this region the equation can be written as sets the sign of the volume form in the AdS 5 .
Assume that the second factor vanishes; the analysis is similar when the first factor vanishes. Then we get which reads 2(k + 4) R a µν + iǫ µν µ 1 µ 2 µ 3 ∂ µ 1 a µ 2 µ 3 = 0 (7.24) The r3 component is The solution is From this we then get If in (7.20) we assume that the first factor vanishes (The two factors commute in the inner region) we get the solutions The case with k = 0 is special because in this case we have a gauge freedom given by a function Λ µ (x) which was not fixed by the choices (2.13), (2.14) [13]. This case was studied in [16]. To make the solutions (7.18), (7.19) in the intermediate region agree with the corresponding solutions given in this reference, we have to note that this gauge freedom allows us to add arbitrary constants to a 12 and a 03 . This freedom removes two of the four constants that appeared in the solution for general k, and corresponds to the fact that only one propagating degree of freedom exists for k = 0 in contrast to two degrees of freedom for higher k. Thus we can solve the dynamical equations for a 12 and then note that for k = 0 we can find a 03 algebraically from the equations of motion. For all these fields the absorption probability (2.35) is

Absorption probabilities, structure of solutions
Now we consider the scalar in the x space arising from the 2-form field A αβ . The spherical harmonics for this case split into two classes denoted by Y k,± [αβ] . The absorption probabilities for these two classes are given by For the 2-form field in the x space arising from A µν we have seen that the longitudinal and transverse components do not decouple, except for k = 0 when the longitudinal component is not dynamical. It appears to not be helpful to try to define cross sections in this case, so we present instead the relation between the solutions at r → 0 and the solutions at r → ∞.
In the outer region we have found in (7.13)(7.10)(7.11) In Table 1 Table 1: Scattering matrix for the two form from the antisymmetric tensor

Analysis of solutions
Let us comment on the nature of the calculation and its results. We note that in the outer and the inner regions the waveform attains the form of a freely traveling wave. The intermediate region acts as a barrier that the solution has to tunnel through, and at small ω the tunneling is small. We can imagine having the wave incident from either side of this barrier. If we throw in a particle from infinity and have it absorbed by the brane, then the particle is incident from the outer region, and the boundary condition on the solution is that there be no outgoing wave at r → 0 in the inner region. Perhaps for emission of a quantum from the branes we should take the particle to be incident from the inner region.
The results on the absorption probabilities are summarized in Table 2. The first column gives the the field and its expansion in spherical harmonics. The second column gives the range of the index for the harmonics. The third column lists the cases that occur in the supergravity multiplet of the 5-dimensional AdS space. (These fields are labelled by ⊙ in [13], and we have used the same notation here.) The fourth column gives the absorption probabilities. For the antisymmetric tensor with no indices along the S 5 the components mix in a way that does not make it helpful to define an absorption probability. One can choose combinations of fields incident on the branes such that the reflected fields are a multiple of the combination that was incident. In this case one can define an absorption probability for the incident combination, but it turns out that in this basis the two power laws of ω that arise during the calculation mix with each other, so the diagonalization obscures the ω dependence of the solution.
For the case k = 0 however we have seen that only one degree of freedom propagates, and the absorption probability can be computed for this mode. The result from [16] can be written as which agrees with the general pattern of results in Table 1.
In the case of the scalar field B the above computation was carried out for all partial waves in [8] [10], and our result for this agrees with the results presented there.
The probability for passage from one side of the barrier to the other is the same, whichever side the particle is incident from, as would be expected on general grounds. In Here we have assumed that a ≥ 0, c ≥ 0. Then the absorption probability has the dependence P ∼ ω 2a+2c (9.4) The parameters a and c are the indices of the Bessel function J in the inner and outer regions respectively.
We summarize some information about the solutions in Table 3. In the first column we list the fields that we study. Here the symbols a + , a − correspond to the two kinds of spherical harmonics that we have for this field, Y k,+ The fourth column gives the dimension ∆ of the field in the Yang-Mills theory which corresponds to the mode of supergravity having the given power law behaviors in the inner region. This dimension is given by [4] [17] m 2 = (∆ − p)(∆ + p − 4) (9.5) wherem 2 is the eigenvalue of the Maxwell operator on the appropriate p-form in the 5dimensional AdS geometry in the inner region. Thesem 2 values are also the same as those listed in [13]. Note that ∆ is connected to the power a + b of r in the solution of the field equation in the outer part of the inner region [4]; this is the power that decays as we go towards smaller r values. The relation is ∆ = a + b − p + 4 (9.6) Where p is the rank of the form in the AdS space.
The last column gives the ω dependence of the absorption probability, which is given by (9.4). For the antisymmetric tensor with no indices along the S 5 the components mix and we do not get a statement analogous to (9.4). For the case k = 0, the absorption probability (9.1) agrees with the pattern of results in Table 2 for the ∆ = 6.

Discussion
In the geometry produced by the 3-branes we have the asymptotic regions of large r and small r, in each of which we have the maximal number of supersymmetries (32 real components). In the intermediate region the number of supersymmetries is only half that be extended to higher partial waves. From Table 2 we observe that if ∆ = k + m, then the power of ω is 4k + 2m. Thus the case k = 0 might not exhibit the full structure of the relation between the classical calculation and the microscopic theory. We hope to return to this issue in the future.