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Absorption of angular momentum by black holes and D-branes

We consider the absorption of higher angular momentum modes of scalars into black holes, at low energies, and ask if the resulting cross sections are reproduced by a D-brane model. To get the correct dependence on the volume of the compactified dimensions, we must let the absorbing element in the brane model have a tension that is the geometric mean of the tensions of the D-string and an effective stringlike tension obtained from the D-5-brane; this choice is also motivated by T-duality. In a dual model we note that the correct dependence on the volume of the compact dimensions and the coupling arise if the absorbing string is allowed to split into many strings in the process of absorbing a higher angular momentum wave. We obtain the required energy dependence of the cross section by carrying out the integrals resulting from partitioning the energy of the incoming quantum into vibrations of the string.


Introduction.
With the development of string theory and the ideas of duality, there has been considerable progress in our understanding of black holes. Following suggestions of Susskind, the number of string theory states at weak coupling have been found to agree with the number of states expected from the Bekenstein entropy of the hole that would form at strong coupling [1] [2][3] [4]. Further, it was found that the rates of absorption and emission of minimal scalars computed at weak coupling matched the Hawking radiation rates expected from the black hole at strong coupling [5] [6]. This calculation has been extended to emission of charged quanta [7], to higher orders in the energy of the incident quantum under certain conditions [8], and to nonminimal 'fixed' scalars [9].
To make contact with the black hole information paradox, we need to understand the absorption of quanta that are small in size compared to the horizon, so that they can fall into the horizon through a reasonably localised direction. This implies that we understand the absorption of higher angular momentum modes, since a wavepacket that is to be localised in the angular directions must be composed of several components of angular momentum l. It is also important to understand the absorption at wavelengths small compared to the size of the horizon, since that too is required to well localise the infalling quantum.
In this paper we discuss the absorption of low energy higher l modes for minimal scalars, by the classical black hole and by the D-brane model of the hole. For the case l = 0 it had turned out to be adequate to use a model where a D-string absorbed an incoming scalar by converting its energy into vibration modes on the D-string [5]. An equivalent result was obtained in the S-dual model where the absorbing element was an elementary string, and the absorption amplitude was computed using standard perturbative string theory [6].
But there are difficulties with naively extending these models to the absorption of quanta with higher l. Let us consider the model of the 4+1 dimensional black hole introduced in [4]. The spacetime has total dimension 10, of which 5 space dimensions are compactified on a 5-torus T 5 = T 4 × S 1 . The S 1 is the direction in which the absorbing string is wound, while the 5-branes in the model wrap all over the T 5 . Let the volume of the T 4 be V 4 , and the length of the S 1 be L.
The incident quantum is expected to convert its energy into modes that travel in the direction of the circle S 1 . As the angular momentum l of the incident quantum is increased, we expect that more and more such (fermionic) modes will be created. But if these modes are vibrations of the D-string, then the absorption calculation will be confined to the vicinity of this string, and will not be sensitive to the volume V 4 that is available transverse to the D-string. Thus the V 4 dependence of the cross section will not change with l. But the classical cross section does depend on V 4 ; this dependence is ∼ V −1−l 4 .
If we use a dual model where the absorbing element is an elementary string, and consider the absorption process as a fundamental string interaction diagram, then we see that the incoming massless quantum can create no more than 4 new fermions in the final state, if we use the three point tree vertex. This is because the world sheet conformal theory is a free theory, and the massless scalar has at most two fermionic oscillator to contribute to each of the left and right sides. But we need a number of fermions that increases without bound with increasing l. If we allow loops, then we get additional powers of g 4 in the cross section for every extra loop, while the classical cross section is seen to increase by powers of g 2 as l increases by one unit.
In this paper we do the following: (a) We observe that in the D-brane model we get the correct dependence of the absorption cross section on V 4 if we let the absorbing element be a long string with a tension that is the geometric mean of the tension of the D-string and the effective tension obtained for vibrations of the 5-D-brane which travels in the long direction S 1 . We note that such a choice is also T-duality symmetric.
We also note that in the dual elementary string model, the correct dependence on V 4 and g is obtained if we allow the initial string to split into l + 1 strings when absorbing a quantum of angular momentum l. The details of the amplitude calculation are however not very clear in such a model.
(b) The energy of the incoming quantum is expected to be shared between a pair of bosonic quanta and 2l fermionic quanta, travelling in the direction of the circle S 1 . We carry out the integrals over momenta, and obtain the energy dependence that is required by the classical cross section.
Recently the absorption of higher angular modes has been considered for 3-branes [10] and for the 4+1 dimensional black hole with three charges through an effective conformal theory [11]. There now exist a large number of results pertaining to the black hole -Dbrane comparison. The behavior of the D-brane as a black body was discussed in [12], where it was shown that emissions of quanta are proportional to the classically expected emissions. The issue of higher orders in coupling was discussed in [13]. Comparisons of brane and classical absorption were discussed in [14] [15].
The plan of this paper is as follows. In section 2 we discuss the classical cross section.
In section 3 we discuss the issue of V 4 dependence in the D-brane model. Section 4 discusses dependence on other parameters of the model. In section 5 we discuss a possible description in the dual model where we use the elementary string Polyakov amplitudes. In section 6 we discuss the energy dependence of the amplitudes. Section 7 is a discussion.

The classical absorption cross section.
The metric of the 5-dimensional hole is We will be in the region of parameter space where r p ∼ r 0 << r 1 , r 5 . Thus only the momentum-antimomentum excitations on the string are excited; the excitations of stringsantistrings and 5-branes-anti 5-branes is suppressed.
We will consider the absorption of a graviton that is a scalar from the 5-dimensional point of view. This is a minimally coupled scalar, and satisfies the free wave equation on the 5-dimensional black hole metric. The absorption probability for a spherical wave of angular momentum l was computed in [16] [11]. In the limit where r 1 , r 5 >> r 0 , r p this probability is is the area of the horizon. The temperature of the black hole is The left and right temperatures are [8] T L = r 0 e σ p 2πr 1 r 5 , T R = r 0 e −σ p 2πr 1 r 5 (2.7) The absorption cross section for angular momentum l is (see Appendix C for details) We have, for l odd For l even As a check we note that for l = 0, ω → 0, we obtain σ = A H in accordance with the universal form of the low energy cross section for minimal scalars [17].

The behavior of the classical cross section.
In terms of microscopic variables, we have for the D-brane model [18]: Here n 1 , n 5 are the numbers of D-strings and D-5-branes respectively. V 4 is the volume of the 4-torus transverse to the direction in which the D-string is wound. L (S) is the string length, defined so that under T-duality, a circle of circumference AL (S) goes to a circle of circumference A −1 L (S) . g = e φ is the elementary string coupling. The tension of the elementary string is T (S) = 2πL (S) −2 . The tension of the D-string is The tension of the 5-D-brane is The cross section (2.9), (2.10) is seen to depend on V 4 as Suppose we have a bound state of 5-D-branes and D-strings. In the effective string model of [19] the effect of the 5-D-branes can be taken into account through a fractionation of the tension of the D-string: This model was motivated by performing a duality on the case studied in [20]. In the latter case it was shown by using S-duality that the momentum modes that travel on a D-string in the cross section, which is indeed appropriate to the l = 0 cross section: where we have noted that But for the case of l > 0 we would continue to find the V −1 4 dependence, since the quanta created on the D-string would not see the size of the transverse T 4 . This is in contradiction with (3.5).

Duality considerations, and the 'mean string'.
Let us take one 5-D-brane bound to one D-string. Consider a situation where the length L of the circle where the D-string is wrapped is very long compared to the sides of the compact torus perpendicular to the D-string, which we take to be of order V The 5-brane is wrapped on T 4 × S 1 , so it also sees the length L. Further, in this case we can consider excitations of wavelength There are two kinds of such excitations that we can naively see in this system. If we oscillate the D-string, we get vibrations on a string with tension: If we oscillate the 5-D-brane, then we expect that this will behave as a string with tension If we perform four T-dualities, in the four directions of the torus T 4 , then the D-string will become a 5-D-brane, and the 5-D-brane will become a D-string. The new coupling will be and the new volume of T 4 will be The 5-D-brane becomes a D-string, which has the tension which agrees with (3.11) as it should. The original D-string becomes a 5-D-brane with tension so the effective tension for the long wavelength modes considered here is Thus the two tensions T 1 , T 2 get interchanged under T-duality. From the point of view of the noncompact spacetime, both tensions T 1 , T 2 that appear are on a symmetrical footing.
Thus it is not natural to choose either as the effective tension for the vibrations that are excited on the system. We take instead the geometric mean of the two tensions We will have 4n 1 n 5 bosonic degrees of freedom on this string, together with their 4n 1 n 5 fermionic superpartners. Let us call the string with this tension the 'mean' string to differentiate it from the string with tension given by (3.6), which is usually termed the 'effective string'.
We can still have the case that the ∼ n 1 n 5 degrees of freedom in certain domain of parameters give 4 bosonic and 4 fermionic degrees of freedom on a circle of length n 1 n 5 L, just as was the case for the effective string model. But note that the tension T m does not give in any simple way the mass of either the D-strings or the 5-D-branes in the system.
It is an effective parameter for the excitations of the D-string -5-D-brane bound state.

Disc diagram calculations.
When the incoming scalar is absorbed in the brane model, we expect that there is one bosonic excitation created on each of the left and right sides; these bosons carry the spins of the scalar. there are also l fermions on each side, for absorption of angular momentum l.
(Some details of the group theoretic structure of partial waves is given in Appendix B. The above kinds of excitations were also involved in the effective conformal theory description used in [11].) Again note that the l = 0 case is not altered, since the change in the normalisation of the two open strings is compensated by the change in the volume of the interaction region, which also appears in the amplitude.

Obtaining the V 4 dependence.
In any of the above ways of taking into account the effective tension for the vibrations, Thus in each case we get the desired V 4 dependence.
In the classical cross section, we note that with regard to g, L dependence, Further with regard to the dependence on the number of 1-branes and 5-branes, (r 1 r 5 ) 2 ∼ n 1 n 5 , σ l ∼ (n 1 n 5 ) l+1 (4.4) Thus σ l ∼ g 2l+2 (n 1 n 5 ) l+1 (4.5) All these dependences are seen to result if we assume that we have ∼ n 1 n 5 degrees of freedom on a very long string. The local nature of the interaction says that the cross section is not sensitive to the length L of the string, apart from the l independent factors that were found in [6]  , which gives ∼ g 2l+2 closed in the cross section. (These dependences on g and n 1 n 5 were noted in [11].) Thus note that here we seem to need that the different strands of the string interact locally with each other, and the essential physics is not contained in just the vibrations of one long effective string.

The dual model.
In [6] we had seen that the leading term in the absorption of minimal scalars could be reproduced from a calculation where the absorption of the incident scalar by the string present in the black hole model was viewed as a three point vertex of ordinary string theory. For convenience we use the S-dual model to the brane model used in the preceeding sections, though the same method could be applied to either model with suitable changes of string tensions and couplings. We wish to see if some fundamental string interaction vertex reproduces the V 4 and g dependences required by the classical cross section.
We consider the S-dual model where the black hole is composed of solitonic 5-branes, elementary strings, and momentum along the elementary strings. In this case with regard to V 4 and g dependence The dependence in (5.4) is the same as that in the D-brame model.
Let us postulate that when the incoming scalar is absorbed then the initial string bound to the 5-brane splits into a total of l + 1 strings, all bound to the 5-brane. (Thus for the case l = 0 we have just one string in the final state, as was the case in [5].) The total number of strings involved in the intertaction is l + 3, because the initial state had a masslesss scalar and the initial string bound to the 5-brane. Each string has a normalisation factor V −1/2 4 , and the amplitude also has a factor V 4 from the volume of the interaction region. The cross section contains then the square of the resulting V 4 dependence: which agrees with (5.4).
Note that we have assumed here that the volume V 4 is small, and the wavelength of the incoming scalar is large, so that there is no energy to excite momentum modes of the strings in the directions of the torus T 4 . In the opposite limit, where such momentum modes are in fact continuous, we would have a sum n ∼ V 4 d 4 k for each string in the final state, and we would obtain that σ l ∼ V −1 4 for all l, which is not in agreement with (5.4).
Now note that the amplitude depends on g as g l+1 , for a tree vertex involving l + 3 closed strings, which gives in the cross section which also agrees with (5.4). Thus we get the powers of both V 4 and g to agree at the same time, which a priori need not have been the case.

Spin dependence.
Let us see how the strings on the 5-brane world volume carry the angular momentum of the 4+1 dimensional transverse space. The rotation group of the transverse space is . The string is confined to the 5-brane, and its low energy bosonic excitations are thus vibrations in the compact directions, labelled by an index i = 6, 7, 8, 9 for the 4 directions in the 5-brane transverse to the string. If we quantise the string by an NSR prescription, we would take fermions ψ i , i = 6, 7, 8, 9, and it is not immediately clear where the angular behavior in the directions X 1 , X 2 , X 3 , X 4 would come from.
But we can rigorously prove that the ground states of a string bound to a 5-brane can carry spin for the directions X 1 , X 2 , X 3 , X 4 . By a sequence of S dualitities and T dualities in the compact directions, we can map the D 5-brane bound to a D-string to a D-string carrying say a left moving momentum mode. The 5 brane has been transformed to the D-string, and the D-string has been transformed to the momentum mode. But the momentum mode can be one of 8 bosonic and 8 fermionic states, which were described in [20]. An analysis of the spin properties of the string bound to the 5-brane can be found in [21]. (Since the string [21] was quantised while ignoring the fact that it was not a critical string, one may argue that this is not a rigorously correct deriuvation of the degrees of freedom. But the argument of duality in the above paragraph shows rigorously that the obtained spin properties of the ground state are correct.) There is indeed a bosonic vector state, and fermionic states that are spinors, for the transvese SO(4). Thus the ground states can be writen as Here |α, k L > are the two left moving bosonic ground states, while |a, k L > are the two left moving fermionic ground states. Overall we get 16 ground states, of which 8 are bosonic and 8 are fermionic, just as expected from the above argument through duality.
The vibrations of the string are the following. There are bosonic modes X i , i = 6, 7, 8, 9 that can travel left or right on the string. There are fermionic modes λ α a which travel left on the string and λβ b that travel right on the string. Note that the ground states of the string on the left side, say, carry either the index a or the index α, while the travelling fermionic modes carry two indices α, a.
To get the spin required of the final state, we postulate that each time a new closed string is produced by splitting, one left and one right moving fermion wave is produced on the initial string. The new closed string is taken to be in its ground state, with polarisations given by (a,ȧ) so that there is no spin of the transverse SO(4) carried by this string. One possible form of the interaction, for the case l = 1, is where the λ ′ refer to the polarisations of the new closed string in its ground state, and the λ are the fermionic waves that are created on the (long) initial string during the process of absorption of the scalar. (The spatial momentum p i of the incident scalar has components only in the transverse directions, since we are considering neutral scalars.) The details of such interactions are, however, not clear. In particular normalisation factors suggest that the new strings that are produced will have small winding number, also such strings will prefer to be in their ground states because they cannot support very low energy excitations. Summing over winding numbers may give rise to additional logarithms, not present in the classical cross section (2.9),(2.10).

Sources of ω dependence.
We find ω dependence of the absorption cross section from the following sources: (a) As explained in [9], the absorption cross section is not given by Γ(ω), the absorption when unit flux is incident, but by The reason is that while the systen can absorb from the incident flux, it can also radiate at the same time, and the absorption cross section only measures the net amount of absorption. Thus we will apply (6.1) to find σ(ω) after computing Γ(ω); the steps below pertain to the calculation of the latter quantity.
(b) The amplitude contains a factor of the energy |ω 1 | of each boson that is produced, and also the normalisation factor for the boson which is ω (c) The incoming scalar contributes a normalisation factor ω −1/2 in the amplitude, which gives ω −1 in the cross section.
(d) The excitations on the string have a left temperature T L = β −1 L and a right temperature T R = β −1 R . The incoming quantum interacts with the string through a vertex that involves one boson and l fermions on each side, when the angular momentum absorbed is l. These bosons and fermions can either be added to the initial state of the string or can be absorbed from the initial state. The analysis of weight factors for these two cases was carried out for bosonic excitations in [9]. We repeat such an analysis for our case here.
Consider either the left or the right set of variables, and let the inverse temperature be denoted by β. The distribution function for bosons is ρ B = (e βω − 1) −1 and for fermions is ρ F = (e βω + 1) −1 . If the boson appears in the final state then ω > 0 and the weightage factor is 1 + ρ B (ω) = −ρ B (−ω). If the boson was absorbed from the initial state then ω < 0 and the weightage factor is ρ B (−ω). Note that the weight factor from part (b) above is always positive; as a consequence the two cases of the boson being in the final and in the intial state can be combined to have an integral Similarily, a fermion in the final state has ω > 0 and a weight 1 + ρ F (ω) = ρ F (−ω). A fermion in the initial state has ω < 0 and a weight ρ F (−ω). The two cases can thus be combined to an integral We have on each of the left and right sides the energy conservation delta function where ω 1 is the energy of the boson and ω i , i = 2 . . . l + 1 are the energies of the fermions.
Note that we are considering the absorption of neutral quanta, so half the energy ω goes to left movers and half to right movers.
(f) We assume that the absorption vertex has a factor ω/2 for each pair of fermions (one left and one right) that are involved in the interaction. The factor 1/2 is added for convenience; since we are not computing the actual numerical amplitude here it is of no real significance. But the factor ω can be seen in the disc amplitude. In a Green-Schwarz formalism, the fermions either have the form ∼ S, or the form ∼ (∂X)S. The term ∂X is contracted with the factor e ikX in the incident scalar vertex, and gives a factor |k| = ω.
The absorption of angular momentum l needs l fermion vertices of each type, giving (ω/2) l in the amplitude, and thus (ω/2) 2l in the cross section.

Example: l = 1
We have one boson and one fermion on each of the left and right sides. Following where we have noted the temperature dependence of the distribution functions.
where J BF is defined in Appendix A.
Using (6.1) we get the contribution

Example: l = 2
We have one boson and two fermions on each of the left and right sides. Following steps (b)-(g) above we obtain 2T L ) (6.10) Using (6.1) we thus obtain the contribution

General form of ω dependence.
For l odd we obtain 1 4 For l even we obtain 1 4 We see that these dependences on ω agree with the dependences required by the classical cross section (2.9), (2.10).

Discussion.
We have seen that to have the classical absorption agree with the brane models, we need to obtain a dependence on V 4 (the volume of the compact torus perpendicular to the string) in the brane model. While this may show up in different ways in different treatments of the string dynamics, it is possible that these differences are due to the different coupling regimes appropriate to these calculations, and not to an error in either description. Recently it has been shown that there is 'stringy dynamics' in all higher branes, in some domain of parameters [22].
The issue of V 4 dependence may appear in other calculations, for example that for the fixed scalars in [9]. In this calculation it was assumed that r 1 = r 5 , which is equivalent to choosing a particular value for V 4 . It would be interesting to see the details of the agreement when r 1 = r 5 .
Regardless of the details of the absorption, we note that the desired ω dependence arises from a partitioning of the energy of the incoming scalar into the energy of a certain number of momentum modes, with this number being determined by the angular momentum of the partial wave that is absorbed. The calculation also provides naturally the factorials present in the relative cross sections for different l (though since we have not computed the actual disc amplitudes themselves, we cannot know that there will be no other factorials from that source.) The argument using an 'effective conformal theory' carried out in [11] yielded an ω dependence that was the desired one, but there was no known way to normalise the amplitude. The energy dependence calculation of the effective conformal theory calculation is plausibly equivalent to summing over ways of sharing energy between l + 1 quanta of the left and right sides when the quanta are at temperatures T L , T R respectively, since one has to evaluate correlators of free fields at the appropriate temperatures.
In [16] it was noted that when we are working in a domain r 5 >> r 1 , r p , r 0 then the absorption of the lth partial wave becomes significant when ωr 5 = l + 1. In particular the l = 1 partial wave is significantly absorbed starting at the energy where the first massive mode of the effective string can be created. Creation of a new string state would bring in the required factor of V 4 as we have seen. It would be interesting to connect the present calculations to this domain of parameters where r 1 is also small, and so winding modes and momentum modes play a more symmetric role.
It may be thought that the agreement of cross sections for D-branes at low energy with the black hole cross sections implies that for low energy quanta we understand the mechanism by which the Hawking paradox is to be resolved in string theory. This is not the case. A black hole geometry takes an incident low energy quantum and, in the Schwarzschild coordinate system, converts it to a high frequency mode close to the horizon. This high frequency mode is then eaten by the hole. Starting with a quantum of even lower energy just means that we have to follow the mode closer to the horizon before we see it attaining a short wavelength. (Of course most of the low energy quanta escape falling into the hole, but the cross section we compute relates to those that are in fact swallowed by the hole in the above fashion.) Equivalently, if we study the deflection of a particle trajectory by the gravitational field of the hole, and consider the deflection to be expanded in powers of the mass of the hole, then we would see a divergence of the series when the impact parameter approaches the value where the particle will be swallowed by the hole. When we study the D-brane calculation we need to understand whether or not such a divergence occurs, when the number of branes and the coupling are increased to a value large enough to give a classical sized black hole. In the classical calculation taking low energy while keeping other parameters fixed simply pushes the growth of the perturbation series towards higher terms in the series; thus there may be a similar phenomenon for the D-brane calculation as well.
If string theory is to resolve the Hawking paradox, then we either need to see that the effective size of the solitonic bound state is comparable to the horizon size, so that there is really no black hole, or we need to see that loops of virtual quanta in a theory with strings and higher dimensional branes are quite different from loops in particle theory, and give nonlocal effects that take information from near the singularity and send it out with the Hawking radiation. The latter is equivalent to finding a length scale in string theory that is not plank length but is a length that grows with the number of branes involved.
Thus it appears that the agreements found between cross sections of branes and for black holes are to some extent both mysterious and interesting, and provide a strong suggestion that the black hole paradox may actually be resolved in string theory. One needs to better understand the bound states of many branes at strong coupling; for the non-BPS interactions that are involved the values of moduli also affect the nature of energy levels and interaction properties [23] [16].

Appendix A. The basic integrals.
Let us define the following two basic integrals (The subscripts B and F stand for bose and fermi type distributions respectively.) These integrals can be calculated in the following way. We take a contour in the complex x plane, running along the real x-axis, and backwards along the line x + 2πi.
Consider the integralÎ The segments at ℜx = ±∞ do not contribute. So we havê where now x runs along the real line. There is one pole in the contour, at x = iπ − log A.