Light-like mesons and deep inelastic scattering in finite-temperature AdS/CFT with flavor

We use the holographic dual of a finite-temperature, strongly-coupled, gauge theory with a small number of flavors of massive fundamental quarks to study meson excitations and deep inelastic scattering (DIS) in the low-temperature phase, where the mesons are stable. We show that a high-energy flavor current with nearly light-like kinematics disappears into the plasma by resonantly producing mesons in highly excited states. This mechanism generates the same DIS structure functions as in the high temperature phase, where mesons are unstable and the current disappears through medium-induced parton branching. To establish this picture, we derive analytic results for the meson spectrum, which are exact in the case of light-like mesons and which corroborate and complete previous, mostly numerical, studies in the literature. We find that the meson levels are very finely spaced near the light-cone, so that the current can always decay, without a fine-tuning of its kinematics.


Introduction
Motivated by some experimental results at RHIC, which suggest that the deconfined matter produced in the intermediate stages of a ultrarelativistic nucleus-nucleus collision might be strongly interacting, there is currently a large interest towards understanding the properties of strongly coupled field theories at finite temperature within the framework of the AdS/CFT correspondence (see the review papers [1,2,3,4,5,6] for details and more references). A substantial part of this effort has been concentrated on studying the response of such a plasma to energetic, 'hard', probes, so like heavy quarks [7,8,9,10,11,12,13,14,15,16], mesons (or quark-antiquark pairs) [17,18,19,20,21,22,23,24,25], or photons [26,27,28,29,30,31], in an attempt to elucidate some intriguing RHIC data, so like the unexpectedly large 'jet quenching', or to provide alternative signatures of a strongly-coupled matter.
In particular, AdS/CFT calculations of deep inelastic scattering (DIS) [32,33,26,27,31,34,35,36,37,38] have given access to the structure of strongly coupled matter at high energy and for small space-time separations and thus revealed an interesting picture, which is quite different from the corresponding picture in a gauge theory at weak coupling. One has thus found that there are no point-like 'partons' at strong coupling, that is, no constituents carrying a sizeable fraction x ∼ O(1) of the total longitudinal momentum of a 'hadron' [32,33] or plasma [26,27] at high energy. This has been interpreted as the result of a very efficient branching process through which all partons have fallen at the smallest values of x which are consistent with energy conservation. This interpretation is consistent with arguments based on the operator product expansion at strong coupling [32,39] and is further supported by the fact that the DIS structure functions were found to be large at small values of x ≪ 1, where they admit a natural interpretation in terms of partons [33,26,3]. The central scale in this picture is the saturation momentum Q s (x), which defines the borderline between the large-x and large virtuality (Q > Q s (x)) domain which is void of partons, and the small-x, small virtuality (Q Q s (x)) domain where parton exist, with occupation numbers of order 1 (a situation somewhat reminiscent of, but more extreme than, parton saturation in QCD at weak coupling [40]). This scale Q s plays an essential role also for other high energy processes, of direct relevance to heavy ion collisions, like the energy loss and the momentum broadening of a heavy quark [15,16]. It can furthermore be related to the dissociation length for a large, semiclassical, 'meson' [18,19,20,21].
So far, most studies of DIS at finite temperature and strong coupling were concerned with the N = 4 supersymmetric Yang-Mills (SYM) theory, where the role of the virtual photon is played by the R-current -a conserved current associated with a U(1) global symmetry which couples to massless fields in the adjoint representation of the color group SU(N c ). Very recently, the same problem has been addressed [31] within the context of a N = 2 supersymmetric plasma obtained by adding N f hypermultiplets of (generally massive) fundamental fields to the N = 4 SYM plasma, in the 'probe' limit where N f ≪ N c . In that case, the 'photon' is a flavor current which couples to a pair of fields (fermions and scalars) in the fundamental representation of SU(N c ), that we shall refer to as 'quarks'. In order to describe the results in Ref. [31] and also our subsequent results in this paper, it is useful to briefly recall the structure of the holographic dual of the N = 2 plasma at strong coupling λ ≡ g 2 N c ≫ 1, also known as the 'D3/D7 model' [41,42,43,4] (see also Sect. 2 below).
The supergravity fields (in particular, the Abelian gauge field dual to the flavor current) live in the worldvolume of one of the N f D7-branes that have been inserted in the AdS 5 × S 5 Schwarzschild background geometry dual to the N = 4 SYM plasma. For the case where the fundamental fields are massive, the D7-branes are separated in the radial direction from the N c D3-branes located at the 'center' of AdS 5 × S 5 . The distance between the two systems of brane then fixes the 'bare' mass of the fundamental 'quarks', represented by Nambu-Goto strings stretching from a D7-brane to a D3-brane. Flavorless 'mesons', or quark-antiquark bound states, can be described either as strings with both endpoints on a D7-brane, or (at least for sufficiently small meson masses and spins) as normal modes of the supergravity fields propagating in the worldvolume of the D7-brane. In this paper, we shall adapt the second point of view, that of the normal modes.  [23]. The solid blue curve corresponds to a pseudo-scalar meson, whereas the red dashed curve corresponds to a scalar meson. The solid black line corresponds to ω = k.
For zero or sufficiently low temperatures, the mesons are strongly bound [43,23]: their binding energy almost compensates the large quark masses (proportional to the string tension), so that the meson masses remain finite in the strong coupling limit λ → ∞ (the 'supergravity approximation'), to which we shall restrict ourselves. In that limit, the quarks become infinitely massive and then the mesons are stable: they form an infinite, discrete, tower of (scalar and vector) modes, distinguished by their quantum numbers. A remarkable property of the meson spectrum that will play an essential role in what follows is that, at finite temperature, the dispersion relation for a given mode changes virtuality, from time-like to space-like, with increasing momentum [23]. This is illustrated in Fig. 1. In particular, there exists an intermediate value of the momentum at which the mode becomes light-like.
This situation persists for sufficiently low temperatures, so long as the D7-brane, although deformed by the attraction exerted by the black hole, remains separated from the latter. But with increasing temperature, one finds a first-order phase transition (from the 'Minkowski embedding' to the 'black hole embedding') at some critical temperature T c , at which the tip of the D7-brane suddenly jumps into the black hole (BH) horizon [44,23]. For T ≥ T c , all the mesons 'melt' : their dispersion relations acquire large imaginary parts (comparable to their real parts), showing that the bound states are now highly unstable [24,25].
One should stress that this peculiar 'meson melting' phase transition is specific to this model and has no analog in QCD. The corresponding situation in QCD is not yet fully clear 1 , For this process to be possible, the kinematics of the current should match with the dispersion relations for the vector mesons. For this interaction to qualify as 'deep inelastic scattering', the associated structure functions -determined by the coupling of the current to the mesons and computed as the imaginary part of the current-current correlator -must be non-zero in a continuous domain of the phase-space, and not only at discrete values of the energy. In this paper, we shall demonstrate that these conditions are indeed satisfied at sufficiently high energy. Our final result is that the structure functions for flavor DIS are exactly the same in this low temperature-phase as in the high-temperature phase, although the respective physical pictures are quite different. This result is in fact natural, as we shall later explain.
To develop our arguments, we shall perform a detailed study of the meson excitations in the high-energy, space-like, kinematics relevant for DIS off a strongly coupled plasma; that is, ω ≫ Q ≫ T , with Q 2 ≡ k 2 − ω 2 > 0. We shall focus on vector mesons with transverse polarizations, which provide the dominant contribution to DIS at high energy [26], but we expect similar results to apply for other types of excitations (longitudinal vector mesons, scalar and pseudoscalar ones). Also, for technical reasons, we shall limit ourselves to the case of very heavy mesons, or very low temperature, M gap ≫ T , which however captures all the salient features of the general situation.
Concerning the kinematics, we shall find that a space-like current can excite mesons only for high enough energies and relatively small virtualities, such that the current and the mesons are nearly light-like. This is so because a current with large space-like virtuality encounters a potential barrier near the Minkowski boundary (associated with energy-momentum conservation) and thus cannot penetrate in the inner region of the D7-brane, where mesons could be created. However, for high enough energy ω Q 3 /T 2 , this barrier is overcome by the gravitational attraction due to the black hole (i.e., by the mechanical work done by the plasma [26,27]), and then the current can penetrate inside the bulk and thus excite mesons. The corresponding kinematics being nearly light-like, ω ≃ k, we shall focus our attention on the respective region of the meson dispersion relation in Fig. 1, but we shall provide analytic approximations also for the other regions (in Sect. 4). Our results are as follows.
For the strictly light-like mesons (ω = k), we shall construct in Sect. 5 exact, analytic, solutions for the spectrum and the wavefunctions, which take particularly simple forms for large quantum numbers n ≫ 1. We shall thus find an infinite tower of equally spaced levels, with high energies and a large level spacing: (For comparison, at zero momentum, the energy of the mode n is ω n (k = 0) ∼ nM gap .) Similarly, for the gauge field dual to a light-like flavor current, we shall find exact 'nonnormalizable' solutions, from which we shall compute the retarded current-current correlator in the high energy limit ω ≫ T (M gap /T ) 3 . As expected, this propagator exhibits poles at the energies of the light-like mesons, so its imaginary part is an infinite sum over deltalike resonances. The coefficient of each delta-function represents the probability for the resonant production of a meson by a current whose energy is exactly ω n . Conversely, they also describe the rate for the decay of a vector meson into an on-shell photon, a mechanism recently proposed as a possible signal of strong coupling behaviour in heavy ion collisions [30].
The resonant production of mesons remains possible also for slightly space-like kinematics, and, of course, for any time-like kinematics, but this is perhaps not the most interesting physical situation, as it requires the energy of the current to be finely tuned to that of a meson mode. Given the large level spacing ∆ω indicated above, it looks at a first sight unlikely that a small uncertainty δω ≪ ω in the energy of the current -as inherent in any scattering experiment (even a Gedanken one !), where the 'photon' is not a plane wave, but a wave packet -could help reducing the need for the fine-tuning. If that was true, it would mean that for the whole high-energy region in phase-space, except for a set of zero measure (as defined by the dispersion relations for the meson modes), the current survives in the plasma for arbitrarily long time. However, as we shall argue now, that conclusion would be a bit naive, as it underestimates the consequences of a small fluctuation in the energy, or the virtuality, of the current for the problem at hand.
The main point is that the energy uncertainty δω should not be compared to the (relatively large) level spacing ∆ω between two successive resonances, but rather to the change in energy which is necessary to cross from one meson level to another at a fixed value k of the momentum. Indeed, one should not forget that the dispersion relation in the relevant kinematics is nearly light-like, that is, ω n ≃ k n for the nth mode. Hence, when increasing the energy by ∆ω to move from one level to a neighboring one, one is simultaneously increasing the momentum k by the same, large, amount -one moves along the light-cone. But in a scattering problem, the momentum k of the current is fixed and its energy ω has generally an uncertainty δω, related to the fact that the source producing the current has been acting over a finite period of time δt: δω ∼ 1/δt. Before we discuss this time δt, let us make the crucial observation that the meson levels are very finely spaced in energy when probed at a fixed value of k. This is a general feature of the high-energy kinematics, which is further amplified by the peculiar shape of the meson dispersion relation near the light-cone.
As a simpler example, recall first the situation at zero temperature [43], where the meson dispersion relation reads, schematically, ω n (k) = (k 2 + n 2 M 2 gap ) 1/2 ≃ k + n 2 M 2 gap /2k, with the approximate equality holding when k ≫ nM gap . Hence the energy jump δω n (k) ≡ ω n+1 (k) − ω n (k) needed to cross from one mode to another at fixed k is δω n (k) ≃ nM 2 gap /k and becomes smaller and smaller when increasing k. As anticipated, the modes are very finely spaced at large k. Returning to the finite-T case of interest, it turns out that the respective dispersion relation is even more sensitive to small variations in the virtuality of the mode near the light-cone. Specifically, we shall find in Sects. 4.3 and 6 that the level spacing at fixed k defined as above scales with k as δω n (k) ∼ T (T /k) 1/3 when k ≃ k n = n∆ω (see Fig. 2). As anticipated, this is considerably smaller than the energy spacing ∆ω ∼ T (M gap /T ) 3 at fixed virtuality : indeed, ∆ω/δω n ∼ n 1/3 (M gap /T ) 4 ≫ 1.
To understand the typical energy uncertainty δω of the current, one needs an estimate for its interaction time in the plasma t int . Indeed, the source producing the current should act over a comparatively shorter time δt t int in order for the subsequent dynamics to be observable. Via a time-dependent analysis of the dynamics of the dual gauge field in Sect. 6, we shall find that t int is controlled by the progression of the gauge field within the D7-brane, which yields t int ∼ (k/T ) 1/3 /T (similarly to the R-current [27]). This estimate implies a lower limit δω 1/t int on the energy uncertainty of the current which is of the order of the level spacing δω n indicated above. This justifies performing an average over neighboring levels in the calculation of the imaginary part of the current-current correlator. This averaging smears out the meson resonances and produces our main result in this paper, Eq. (5.18). As anticipated, this result is identical to the DIS structure functions in the high-temperature phase, which shows that the current is completely absorbed by the plasma in both cases.
The analysis in Sect. 6 also allows us to deduce a space-time picture for the nearly lightlike mesons in the semiclassical regime at large quantum numbers n ≫ 1, where the notion of a classical orbit makes sense. We thus find that the period for one orbit is ∆t n ∼ (ω n /T ) 1/3 /T , where we recall that the energy of the bound state is ω n = n∆ω. Furthermore, we find that the meson spends the major part of this time far away from the tip of the D7-brane, at relatively large radial distances ∼ ω 1/3 n . This is so because its orbital velocity is much higher near the tip than at larger radial distances. It is finally interesting to notice that, in this light-like kinematics, the period ∆t n of the bound state has the same parametric dependence upon its energy as the interaction time t int of the current, and similarly for the typical radial location of the meson versus the saturation momentum Q s (k) ∼ k 1/3 for the current.

Mesons in the D3/D7 brane model at finite temperature
According to the AdS/CFT correspondance [47,48,49], the four-dimensional N = 4 super-Yang-Mills (SYM) gauge theory with 'color' gauge group SU(N c ) is dual to a type IIB string theory living in the ten-dimensional curved space-time AdS 5 × S 5 , which describes the decoupling limit of N c black D3-branes. By further adding a black brane to this geometry, one obtains the holographic dual of the finite-temperature, plasma, phase of the N = 4 SYM [50]. The ensuing metric reads (see, e.g., [1]) where f (u) = 1 − u 4 0 /u 4 , with u 0 = πL 2 T the radial position of the black hole horizon and T the common temperature of the N = 4 SYM plasma and of the black hole. The curvature radius L is defined in terms of the string coupling constant g s and the string length scale ℓ s via L 4 = 4πg s N c ℓ 4 s . The holographic dictionary relates the gauge and string theory coupling constants as g 2 = 4πg s . In the "strong coupling limit" of the gauge theory, defined as N c → ∞, λ ≡ g 2 N c → ∞, with g fixed and small (g ≪ 1), the string theory reduces to classical supergravity theory in the AdS 5 × S 5 Schwarzschild geometry with metric (2.1).
All fields in the N = 4 SYM theory transform in the adjoint representation of SU(N c ). Fields transforming in the fundamental representation of the gauge group can be introduced in the gravity dual by inserting a second set of D-branes in the supergravity background [41,4]. In particular, we consider the decoupling limit of the intersection of N c black D3-branes and N f D7-branes as described by the array: where the first four dimensions (0, 1, 2, and 3) correspond to the Minkowski coordinates {t, x i } and the last six ones (from 5 to 9) to the six-dimensional space with coordinates {u, Ω 5 }. The dual field theory is now an N = 2 gauge theory consisting of the original SYM theory coupled to N f fundamental hypermultiplets which consists of two Weyl fermions and their superpartner, complex, scalars (see e.g. [51]). For brevity, we shall globally refer to these fundamental fields as 'quarks'. In the limit where the number of flavors is relatively small, N f ≪ N c , the D7-branes may be treated as probes in the black D3-brane geometry (2.1). That is, the D7-branes are generally deformed by their gravitational interactions with the D3-branes and the black hole, but one can neglect their back reaction on the ambient geometry, Eq. (2.1). The ensuing geometry is dual to a N = 2 plasma at finite temperature in which the effects of the fundamental degrees of freedom (say, on thermodynamical quantities) represent only small corrections, of relative order g 2 N f = λ(N f /N c ) ≪ 1 (see e.g. [23]).
Although both the D3-branes and the D7 ones fill the Minkowski space, these two types of branes need not overlap with each other, as they can be separated in the 89-directions, which are orthogonal to both of them. When this happens, the conformal symmetry is explicitly broken already at classical level 2 and then the fundamental fields in the dual gauge theory become massive: their 'bare' mass is proportional to the radial separation u m between the two sets of branes at zero temperature. Indeed, a fundamental field is 'dual' to an open string connecting a D7-brane to a D3-brane, so its 'bare' mass is equal to the string length u m times the string tension: To render such geometrical considerations more suggestive, it is helpful to perform some changes of coordinates [44,23]. First, we introduce a new, dimensionless, radial coordinate ρ, related to the coordinate u via 3 Quantum mechanically, the conformal symmetry is broken by the D7-branes even when they overlap with the D3-branes, i.e., when um = 0. But the β-function for the 't Hooft coupling λ = g 2 Nc is of order N f /Nc and thus is suppressed in the probe limit N f /Nc → 0. 3 We notice that ρ is related to the Fefferman-Graham [52] radial coordinate z via z/ √ 2 = L 2 /(u0ρ).
Note that that the BH horizon corresponds to ρ 0 = 1 and the Minkowski boundary to ρ → ∞, with u 0 ρ ≃ √ 2u when u ≫ u 0 (i.e., ρ ≫ 1). Then the background metric (2.1) becomes It is furthermore useful to adapt the metric on the five-sphere to the D7-brane embedding.
Since the D7-brane spans the 4567-directions, we introduce spherical coordinates {r, Ω 3 } in this space and {R, φ} in the orthogonal 89-directions. Denoting by θ the angle between these two spaces, we have (see also Fig. 3) and therefore Note that, on the D7-brane, the Minkowski boundary lies at r → ∞.
To specify the D7-brane (background) embedding, we require translational symmetry in the 0123-space and rotational symmetry in the 4567-directions, and fix φ = 0. Then the embedding can be described as the profile function R = R v (r). The subscript 'v' on R v stays for the 'meson vacuum': the small fluctuations of the D7-brane around its stationary geometry are dual to low-lying 'mesons' in the boundary gauge theory, i.e., (colorless and flavorless) bound states which involve a pair of fields from a fundamental hypermultiplet -say, a quark-antiquark pair. Such mesons are represented by strings with both ends on the D7-branes and thus can be studied (at least for small enough meson sizes and masses; see below) by examining the small fluctuations of the worldvolume fields on the D7-branes. These include the fluctuations δφ and δR in the shape of the D7-brane -which give rise to pseudo-scalar and scalar mesons, respectively -, and also fluctuations of the worldvolume gauge fields, which describe vector mesons. The 'vacuum' profile function R v (r) and the spectrum of the various type of fluctuations have been systematically studied in the literature, via analytic methods in the zero-temperature case [43], and via mostly numerical methods at non-zero temperature [42,44,23,24,25]. In what follows, we shall collect the previous results which are relevant for the present analysis, with a minimum of formulae.
The dynamics of the D7-brane is described the Dirac-Born-Infeld (DBI) action 4 . The profile function R v (r) for the 'vacuum' embedding is obtained by solving the equation of motion for R(r) which follows from this action. The meson spectrum is then obtained by solving the linearized equations of motion (EOM) which follow after expanding the DBI action to quadratic order in small fluctuations around the 'vacuum' embedding.
At zero temperature, one finds that the 'vacuum' profile is trivial, i.e., independent of r : where we recall that u m is the separation between the two types of brane in the original radial coordinate u. (At T = 0, Eq. (2.4) reduces to ρ ≡ √ 2(u/u 0 ) where u 0 is an arbitrary reference scale, which drops out from the final results.) The EOM for the small fluctuations have been solved exactly, in terms of hypergeometric functions [43]. At zero temperature, both Lorentz symmetry and supersymmetry are manifest. Accordingly, for a meson with fourmomentum q µ = (ω, 0, 0, k), the dispersion relation ω(k) involves only the 'invariant mass' combination M 2 ≡ ω 2 − k 2 . Besides, this relation depends upon two 'quantum numbers': a 'radial' number n = 0, 1, 2, ..., which counts the number of zeroes of the corresponding wavefunction in the interval 0 < r < ∞, and an 'angular' number ℓ, with ℓ = 0, 1, 2, ..., which refers to rotations along the S 3 component of the D7-brane. (In the dual gauge theory, ℓ represents a charge under the internal symmetry group SO(4) which is dual to rotations on S 3 .) Supersymmetry together with the global SO(4) symmetry imply additional degeneracies for the meson spectrum, as discussed in [43]. Specifically, Ref. [43] found the following dispersion relation L 4 4(n + ℓ + 1)(n + ℓ + 2) (2.10) for both (pseudo)scalar and vector mesons. Note the presence of a mass gap in the spectrum: the mass of the lightest mesons is non-zero, namely, Note also that the meson masses are much smaller, by a factor 1/ √ λ, than the bare quark mass, Eq. (2.3). This shows that in this strong coupling limit the mesons are tightly bound: in the total energy, the binding energy almost cancels the mass of the quarks.
At finite temperature, the D7-brane feels the attraction exerted by the black hole and thus is deflected towards the latter -the stronger the deviation, the shorter is the radial separation u (or ρ) between the two. This deflection becomes negligible towards the Minkowski boundary (r → ∞), where the profile function R v (r) approaches the value R 0 that it would have (at any r) at T = 0. More precisely, for asymptotically large r one finds [42,44,23] R with R 0 related to the 'bare' quark mass, as in Eq. (2.3), and c a positive number proportional to the quark condensate.
On the other hand, closer to the black hole horizon (ρ ∼ 1), one finds two different types of behaviour -corresponding to two thermodynamically distinct phases separated by a firstorder phase transition -, depending upon the ratio M gap /T = 2πR 0 between the (zero-T ) mass gap and the temperature: (i) for relatively large values of R 0 , larger than a critical value numerically found as R c ≃ 1.306 [23], the D7-branes close off above the black hole horizon ('low-temperature', or 'Minkowski embeddings'); (ii) for R 0 < R c , the D7-branes extend through the horizon ('high-temperature', or 'black hole embeddings') 5 . In the gauge theory, the most striking feature of this transition is the change in the meson spectrum [44] : in the low temperature phase, the spectrum of mesons has a mass gap and the bound states are stable, so like at T = 0 [23]; in the high temperature phase, there is no mass gap and the mesonic excitations are unstable and characterized by a discrete spectrum of quasinormal modes (i.e., they have dispersion relations with non-zero, and large, imaginary parts) [24,25].
For the reasons explained in the Introduction, in this paper we shall restrict ourselves to the low-temperature phase, in which the mesons are stable. The corresponding dispersion relations have been numerically computed in Ref. [23], at least within restricted regions of the phase space. As expected, the spectrum shows deviations from both Lorentz symmetry and supersymmetry, and these deviations become more and more important with increasing temperature (for a given M gap ). What was perhaps less expected and, in any case, remarkable is the pattern of the violation of the Lorentz symmetry by the spectrum: when increasing the momentum k of a given mode (i.e., , for fixed values of n and ℓ, which remain good 'quantum numbers' also at finite temperature), the 'virtuality' −Q 2 ≡ ω 2 nℓ (k) − k 2 of that mode is continuously decreasing, from time-like values (−Q 2 > 0) at relatively low k to space-like values (−Q 2 < 0) for sufficiently large k, in such a way that, for asymptotically large k, the dispersion relation approaches a limiting velocity which is strictly smaller than one: (2.13) It has been furthermore noticed in the numerical analysis in Ref. [23] that, with increasing k, the mode wavefunction becomes more and more peaked near the bottom (r → 0) of the D7-brane. This led to the interesting suggestion, which was furthermore confirmed by the numerical results, that the limiting velocity v 0 coincides with the local velocity of light at (2.14) As we shall later argue in Sect. 4, this identification follows indeed from the respective EOM.
Our analytic study will also clarify other aspects of the dispersion relation, like the precise conditions for the onset of the asymptotic behaviour (2.13) and the subleading corrections to it, which in particular contain the dependence upon the quantum numbers n and ℓ. More generally, we shall be able to construct piecewise analytic approximations for the dispersion relation ω nℓ (k) and also for the wavefunctions of the modes, which will confirm the numerical findings in Ref. [23] and provide further, analytic, insight into these results. Although, in our analysis, we shall cover all kinematical domains in k and thus provide a global picture for the meson spectrum, our main focus will be on the nearly light-light mesons with ω ≃ k. Indeed, as we shall explain in the next section, this regime is the only one to be relevant for the deep inelastic scattering of the flavor current.

Deep inelastic scattering off the N = 2 plasma
The N = 2 theory with N f flavors of equal mass has a global U (N f ) ≃ SU (N f ) × U (1) q symmetry (describing flavor rotations of the fields in the fundamental hypermultiplets), to which one can associate N 2 f conserved currents bilinear in the 'quark' operators (see Appendix A in [25] for explicit expressions). In particular, the current J µ q corresponding to the diagonal subgroup U (1) q is associated with the conservation of the net 'quark' number (i.e., the number of fundamental quarks and scalars minus the number of antiquarks and hermitean conjugate scalars). By adding to the theory a U(1) e.m. gauge field A µ minimally coupled to this J µ q current (with an 'electromagnetic' coupling which is arbitrarily small), one can construct a model for the electromagnetic interactions and thus set up a Gedanken deep inelastic scattering experiment which measures the distribution of the fundamental fields inside the plasma. One can visualise this process as the exchange of a virtual, space-like, 'photon' (as described by the field A µ ) between the strongly coupled N = 2 plasma at finite temperature and a hard lepton propagating through the plasma.

Equations of motion in the D3/D7 brane model
Within the D3/D7 brane model, the flavor current J µ q is dual to an abelian gauge field A m living in the worldvolume of the D7-brane, whose dynamics is encoded in the DBI action. According to the gauge/gravity duality, the correlation functions of the operator J µ q are obtained from the 'non-renormalizable' modes of the field A m , that is, the solutions to the classical EOM in the bulk of the D7-brane which obey non-trivial (Dirichlet) boundary conditions at the Minkowski boundary: as r → ∞, the solution A m must approach the U(1) e.m. gauge field A µ which acts as a source for the current J µ q . This should be contrasted to the 'normalizable' modes dual to vector mesons, which must vanish sufficiently fast when approaching the Minkowski boundary (see below for details).
In particular, the DIS cross-sections (or 'structure functions') are obtained from the (retarded) current-current correlator where the brackets · · · T denote the thermal expectation value in the N = 2 plasma. To compute this two-point function, it is enough to study the linearized EOM for the bulk field A m , i.e., the same equations which determine the spectrum of the low-lying vector mesons, but with different boundary conditions at r → ∞. Specifically, the polarization tensor (3.1) can be given the following tensorial decomposition (in a generic frame) : where Π 1 and Π 2 are scalar functions, n µ is the four-velocity of the plasma in the considered frame, Q 2 = q µ q µ > 0 is the (space-like) virtuality of the current, and is the Bjorken variable for DIS off the plasma. Via the optical theorem, the DIS structure functions are obtained as In what follows, it will be convenient to compute the (boost-invariant) structure functions by working in the plasma rest frame, where n µ = (1, 0, 0, 0) and q µ = (ω, 0, 0, k), and therefore Q 2 = k 2 − ω 2 and x = Q 2 /2ωT . However, one should keep in mind that the physical interpretation of the results is most transparent in the plasma 'infinite momentum frame', i.e., a frame in which the plasma is boosted at a large Lorentz factor γ ≫ 1. Then, the kinematic invariants Q 2 and x specify the transverse area (∼ 1/Q 2 ) and, respectively, the longitudinal momentum fraction (equal to x) of the plasma constituent ('parton') which has absorbed the space-like 'photon', and the structure functions represent parton distributions. The piece of the DBI action which is quadratic in the gauge fields reads (see e.g. [25]) where T D7 = 2π/(2πℓ s ) 8 g s , the space-time indices m, n, p, q run over the eight directions in the worldvolume of the D7-brane, g mn is the induced metric on the D7-brane, and F mn = ∂ m A n − ∂ n A m . As already mentioned, the EOM must be solved with the following boundary conditions which together with the fact that the equations are linear and homogeneous in all the worldvolume directions but r imply that the solution A m is such that A r = A S 3 = 0 (i.e., the radial and S 3 -components of the gauge field are identically zero) and the remaining, four, components A µ (t, x, r), with µ = t, x, y, z, are plane-wave in the Minkowski directions with r-dependent coefficients. Since the gauge fields are independent of the coordinates on S 3 , one can reduce Eq. (3.5) to an effective action in the relevant five dimensions. The induced metric in these directions, that we denote asg mn , follows from Eqs. (2.5)-(2.8) as is the profile of the D7-brane embedding. After integrating over the coordinates on S 3 , the action (3.5) reduces to where Ω 3 = 2π 2 , m, n, · · · = t, x, y, z, r, and Clearly, the EOM generated by the action (3.7) read The propagation of the virtual photon along the z axis introduces an anisotropy axis in the problem, so the equations of motion look different for the longitudinal (µ = t, z) and respectively transverse (µ = x, y) components of the gauge field. In what follows we shall focus on the transverse fields, A i with i = x, y, since from the experience with the R-current [26,27] we expect these fields to provide the dominant contributions to the structure functions F 1,2 in the high energy limit. Moreover, our final argument in Sect. 5 will allow us to also reconstruct the flavor longitudinal structure function F L = F 2 − 2xF 1 from the corresponding one for the R-current. The relevant components of the field strength tensor are

and the equation satisfied by A i (r) reads
This equation must be solved with the Dirichelet boundary condition (3.6) at r → ∞ together with the condition of regularity at r = 0. The solution to this boundary-value problem is a 'non-normalizable' mode, as opposed to the 'normalizable' modes which describe vector meson excitations of the D7-brane: the latter are the solutions Eq. (3.11) which vanish sufficiently fast (namely, like A i (r) ∼ 1/r 2 ) when r → ∞ [43,23]. Once the 'non-normalizable' solution is known as a (linear) function of the boundary value A (0) i , the current-current correlator (3.1) is obtained, roughly speaking, by taking the second derivative of the classical action (i.e., the action (3.8) evaluated with that particular solution) with respect to A i . This procedure is unambiguous in so far as the euclidean (i.e., imaginary-time) correlators are concerned, but it misses the imaginary part for the real-time correlators. Rather, the correct prescription for computing the retarded polarization tensor (3.1) reads (for the case of Π xx = Π yy = Π 1 ) [53,54] where i is either x or y. The overall normalization factor reflects the fact that the flavor current couples to N c N f fundamental fields. When using the above formula, the precise normalization at r → ∞, i.e., the boundary value A (0) i , becomes irrelevant, as it cancels in the ratio. Note that, in order to make use of Eq. (3.12), it is enough to know the solution in the vicinity of the the Minkowski boundary. But to that aim, one generally needs to solve the EOM for arbitrary values of r, since the second boundary condition is imposed at r = 0.
In general, the coefficients in Eq. (3.11) are rather complicated functions, as visible on Eqs. (3.7) and (3.9), and this complication hinders the search for analytic solutions. However, how we now explain, they can be considerably simplified without loosing any salient feature by restricting ourselves to the very low temperature, or very heavy meson, case R 0 ≫ 1, or M gap ≫ T . This restriction entails two important types of simplifications. The first one refers to the 'vacuum' profile R v (r), which in the general case is known only numerically [23], but which becomes essentially flat when R 0 ≫ 1. Indeed, in that case, the maximal deviation from the asymptotic value R 0 , namely (see App. A in Ref. [23]), is truly negligible, so one can use R v (r) ≃ R 0 (and henceṘ v = 0) at any r.
The second type of simplifications refer to the BH horizon at ρ 0 = 1 : when R 0 ≫ 1, the condition ρ ≫ 1 is automatically satisfied at any point within the worldvolume of the D7-brane. Then, the thermal effects encoded in f andf , which scale like 1/ρ 4 , cf. Eq. (2.6), can be safely neglected in all the terms in Eq. (3.11) except for the last one: indeed, within that term, the finite-T deviations 1−f and 1−f are potentially amplified by the large energy factor ω. Specifically (with ρ 2 = R 2 0 + r 2 ) (3.14) where we have also used To summarize, under the assumption that R 0 ≫ 1, the EOM for the transverse gauge fields A i (r) takes a particularly simple form: whereȦ i = dA i /dr, ρ 2 = R 2 0 + r 2 , and we have introduced the dimensionless variables One should emphasize here that this condition R 0 ≫ 1 introduces no loss of generality, neither for a study of the DIS process (in which case we are anyway interested in ω,Q ≫ T , and then the dominant dynamics takes place at large radial distances ρ ≫ 1 [26,27]), nor for that of the meson spectrum (for which we shall find results which are consistent with the numerical analysis in Ref. [23], although that analysis was performed for R 0 ∼ O(1)).
Although considerably simpler than the original equation (3.11), the above equation is still too complicated to be solved exactly, except in the special caseQ = 0, to be discussed in Sect. 5. For more general situations, related to either the meson spectrum or the problem of DIS, we shall later construct analytic approximations. In preparation for that and in order to gain more insight into the role of the various terms in Eq. (3.15), it is useful to first consider a different but related problem, whose solution is already known : this is the DIS of the R-current [32,33,26,27].

Some lessons from the R-current
The R-current is a conserved current associated with one of the U(1) subgroups of a global SU(4) symmetry of the N = 4 SYM theory. The respective operator is bilinear in the massless, adjoint, fields of N = 4, and remains conserved even in the presence of the fundamental hypermultiplets (i.e., in N = 2 theory), because of the probe limit g 2 N f ≪ 1. The supergravity field dual to the R-current is, once again, a gauge field A µ , whose dynamics however is not anymore restricted to the worldvolume of the D7-brane -rather, this field can propagate everywhere in the AdS 5 × S 5 Schwarzschild space-time, in particular, it can fall into the black hole. Because of that, the D7-brane plays no role in the case of the R-current, so the following discussion applies to both N = 4 and N = 2 theories (with N f ≪ N c , of course).
For a space-like R-current with high virtuality Q ≫ T and for large radial coordinates ρ ≫ ρ 0 , the dynamics of the dual R-field A i (r) is described by an equation similar to Eq. (3.15), but where the variables ρ and r are now identified with each other (since R 0 plays no role in this case). That is, where nowȦ i = dA i /dρ and it is understood that ρ ≫ 1. The dynamics is driven by the competition between the two terms inside the brackets in Eq. (3.17). The first term, proportional to Q 2 , acts as a potential barrier which opposes to the progression of the field towards the interior of AdS 5 : by itself, this would confine the field near the Minkowski boundary, at large radial distances ρ ρ Q ≡Q. The second term, proportional to ω 2 , is present only at finite temperature (as manifest from its derivation in Eq. (3.14)) and it represents the gravitational attraction between the gauge field and the BH. For sufficiently small values of ρ, smaller than ρ c ≡ (2ω/Q) 1/2 , this attraction overcomes the repulsive barrier ∝ Q 2 , and then the overall potential becomes attractive. However, unless the energyω is high enough, this change in the potential has no dynamical consequences 6 , because the field is anyway stuck near the Minkowski boundary and thus cannot feel the attraction. Clearly a change in the dynamics will occur when the energy is so high that ρ c ρ Q , which requiresω Q 3 , or, in physical units, ω Q 3 /T 2 . When this happens, the potential barrier at ρ → ∞ cannot prevent the gauge field to penetrate (through diffusion; see the discussion in Sect. 6) down to the attractive part of the potential at ρ ρ c , and from that point on, the potential barrier plays no role anymore. Hence, the dynamics at radial distances 1 ≪ ρ ρ c is controlled by the even simpler equation valid for (Q/ω 1/3 ) ξ ≪ω 2/3 . This involves one unknown coefficient c which can be fixed, in principle, by matching onto the corresponding solution at smaller distances ρ ∼ O(1), which in particular obeys the appropriate boundary condition at ρ = ρ 0 = 1. This boundary condition is rather clear on physical grounds: the gauge field can be only absorbed by the BH, but not also reflected, hence the near-horizon solution must be a infalling wave [53,2], i.e., a field which with increasing time approaches the horizon. The solution near ρ = 1 obeying this boundary condition can be explicitly computed, and its matching onto Eq. (3.19) can indeed be done [26], but it turns out that this actually not needed for the purpose of computing the DIS structure function: the problem of the Rcurrent offers an important simplification, which is worth emphasizing here, since the same simplification appears for the flavor current in the high-temperature case (the 'black hole embedding') [31], but not also in the low-temperature, or 'Minkowski', embedding of interest for us here. Namely, the infalling boundary condition can be enforced not only near the BH horizon, but also at much larger values of ρ, where Eq. (3.19) applies. This is so since there is no qualitative change in the shape of the potential at any intermediate point in the range 1 < ρ ≪ ρ c which could give rise to a reflected wave.
The last observation allows us to identify c = i in Eq. (3.19): indeed, consider this approximate solution for ρ ≪ω 1/3 , or ξ ≫ 1, where one can resort on the asymptotic expansions for the Airy functions. Using Eq. (3.19) with c = i, one obtains which is indeed an infalling wave. Now that the coefficient c has been fixed, one can use the approximate solution (3.19) for relatively small values of ξ and compute the current-current correlator according to Eq. (3.12).
(One can adapt Eq. (3.12) to the R-current by multiplying its r.h.s by a factor N c /4N f .) Specifically, Eq. (3.19) is still correct for ξ ∼Q/ω 1/3 ≪ 1, where one can use the small-ξ expansions for the Airy functions, and thus deduce [26] where in the last estimate we indicated the parametric dependencies of the structure functions upon the variables relevant for DIS 7 . As it should be clear from the previous discussion, these results hold for sufficiently high energy, ω Q 3 /T 2 , a condition which can be rewritten in terms of the DIS variables x and Q 2 as On the other hand, for larger values of Bjorken-x, x ≫ T /Q, or higher virtualities Q ≫ Q s , the structure functions are exponentially small (since generated through tunelling). This strong suppression of the structure functions at large values of x and/or Q 2 implies the absence of point-like constituents in the strongly coupled plasma [32,33,26,27]. The critical value Q s (x) ∼ T /x is known as the saturation momentum, since Eq. (3.21) is consistent with a parton picture in which partons occupy the phase space at Q Q s (x) with occupation numbers of O(1) [26].
Returning to the flavor current of interest here, let us now identify the similarities and the differences with respect to the problem of the R-current, that we have just discussed.
In the high-temperature case, where the tip of the D7-brane enters the BH horizon, there are no serious conceptual differences with respect to the R-current. For r ≫ 1, Eq. (3.15) is still valid, so the large-r dynamics is exactly the same as discussed in relation with Eq. (3.19). At smaller r ∼ O(1), the EOM becomes more complicated (in particular because of the rdependence of the profile function R v (r), which is non-trivial in that high-temperature case), but there is no ingredient in the dynamics which could prevent the fall of the flavor field A i into the BH. Hence, the appropriate boundary condition at ρ = 1 (the tip of the D7-brane) is still the infalling one, and moreover this condition can again be enforced ar large r ≫ 1, where Eq. (3.19) applies. As before, this condition fixes c = i, thus finally yielding the same result for the DIS structure function as in Eq. (3.21), except for the overall normalization: (flavor current in the BH embedding) .

(3.23)
This is indeed the result found in [31]. In particular, the saturation momentum for the flavor current (in this high-temperature regime, at least) is exactly the same as for the R-current, cf. Eq. (3.22), since fully determined by the current interactions with the BH.
Consider now the low-temperature phase, which is the most interesting case for us here. For the DIS problem, it is natural to assume that Q M gap ≫ T , orQ R 0 ≫ 1. The situation near the Minkowski boundary will be quite similar to that for the R-current: At relatively low energiesω ≪Q 3 , there is a potential barrier at ρ Q > R 0 , which however disappears at larger energiesω Q 3 . When this happens, the flavor field can penetrate all the way within the worldvolume of the D7-brane. However, this worldvolume ends up at ρ = R 0 ≫ ρ 0 , so there is clearly no possibility for this field to fall into the BH. Accordingly, the infalling boundary conditions do not apply here, but rather must be replaced by the condition of regularity at ρ = R 0 (the tip of the D7-brane). In order to establish the fate of the high-energy current, we therefore cannot make the economy of actually solving Eq.

Meson spectrum at low temperature
In this section we shall construct piecewise approximations to the spectrum of the meson excitations in the low temperature phase, or Minkowski embedding, with the purpose of clarifying some global properties of the spectrum numerically obtained in Ref. [23] and exhibited in Fig. 1. In particular, we shall follow the transition of the dispersion relation of a given mode from time-like to space-like with increasing momentum k, and thus identify the 'critical' momentum k n at which the mode with radial quantum number n crosses the light-cone. Also we shall recover previous analytic results in the literature which concentrated on special limits, like zero-temperature [43] or very high momentum [22]. The particular case of a light-like mode (ω n (k) = k) will be given further attention in the next section, where we shall construct the exact respective solutions for both normalizable and non-normalizable modes, with the purposes of understanding DIS.

Equation of motion in Schrödinger form
For the subsequent analysis, it is convenient to change the definition of the radial coordinate once again, in such a way that the infinite interval 0 ≤ r < ∞ be mapped into the compact interval 0 ≤ ζ ≤ 1. Here ζ is defined as where we have replaced the name of the function by Φ, for more generality: indeed, Eqs. (3.15) or (4.2) apply not only to transverse vector mesons, but also to the pseudoscalar mesons corresponding to small fluctuations in the azimuthal angle φ (the angle in the 89-plane transverse to the D7-brane; recall that the 'vacuum' embedding corresponds to φ = 0). Furthermore, we have defined where the virtualityQ 2 =k 2 −ω 2 can now take any sign (and thus the same is true for κ 2 ). It is furthermore convenient to rewrite Eq. (4.2) in the form of a Schrödinger equation, i.e., to remove the term involving the first derivative; this can be done by writing The corresponding "Schrödinger equation" reads where we have allowed for one further generalization by adding to the potential the term corresponding to a generic value ℓ, with ℓ = 0, 1, 2, . . . , for the angular 'quantum number' corresponding to rotations around the S 3 -sphere internal to the D7-brane. (Such rotations cannot be excited by either the flavor or the R-current considered in the previous section, so ℓ = 0 in the case of DIS. But modes with non-zero ℓ can be excited by other operators in the boundary gauge theory, which are charged under the global SO(4) symmetry of the fundamental hypermultiplets [43,23,25].) Since the radial coordinate ζ terminates at ζ = 1, it is understood that the potential becomes an infinite wall at that point; given the structure of Eq. (4.6), this additional constraint has no consequence except in the limiting case where ℓ = 0 and κ 2 = 0. The zero temperature case is obtained by formally taking Ω = 0 in Eq. (4.6).
Since we are interested in the normalizable modes describing mesons, we shall look for solutions ψ ℓ (ζ) to Eq. (4.5) obeying the following boundary conditions [43] : for ζ → 1. In what follows we would like to follow the change in the dispersion relation when increasing the meson momentum k for fixed quantum numbers (i.e., for a given mode). This study will drive us through different regimes in terms of the variables κ 2 and Ω 2 . The potential V (ζ) ≡ V ℓ=0 (ζ) in these various regimes is illustrated in Figs. 4 and 5.

The low momentum regime: time-like dispersion relation
When the momentum k is sufficiently small (see Eq. (4.10) below for the precise condition), the dispersion relation is time-like (κ 2 < 0) and it is such that the last term, proportional to Ω 2 , in the potential becomes negligible (so that the potential has the symmetric shape shown in Fig. 4 left). Then Eq. (4.2) is formally the same as at zero temperature and the corresponding solutions are known exactly [43]. Namely, the solution obeying the right boundary condition at ζ = 0, cf. Eq. (4.7), reads (the overall normalization is chosen for convenience) where F (a, b; c; x) is the usual hypergeometric function, also denoted as 2 F 1 (a, b; c; x), which obeys F (a, b; c; 0) = 1 (see e.g. Chapter 15 in [55]) and we have set −κ 2 ≡ µ 2 > 0. For Eq. (4.8) to also obey the correct boundary condition at ζ = 1, the hypergeometric function must be regular at that point, which for the indicated values of the parameters a, b and c = ℓ+2 requires the hypergeometric series to terminate [55]. That is, b = −n with n = 0, 1, 2, ..., and then F (a, −n; c; x) is a polynomial of degree n in ζ. This condition yields the vacuum-like (i.e., T = 0) spectrum in Eq. (2.10), that is, At finite temperature, Eq. (4.9) remains a good approximation so long as one can neglect the term ∝ Ω 2 in the potential, that is, for Ω 2 n ≪ µ 2 n or, equivalently (recall (4.3)),ω ≪ nR 3 0 . Since we also assume R 0 ≫ 1, it is clear that this condition is satisfied up to relatively high values of the momentum k, namely so long as Note that we include the radial quantum number within parametric estimates, e.g., µ n ∼ n orω n ∼ nR 0 , since we shall be also interested in large values n ≫ 1 (whereas ℓ will never be too large). Although, for definiteness, we refer to the kinematical domain (4.10) as the 'low-momentum regime', it is clear that, towards the upper end of this domain, the momenta are so large that k ≫ M n ∼ nM gap and thus the dispersion relation becomes nearly light-like. Still within this low-momentum regime, it is easy to see that the main effect of the "finite-T " term ∝ Ω 2 in the potential (4.6) is to decrease the meson virtuality as compared to its "zero-T " value (4.9). Indeed, a simple estimate for this effect is obtained by replacing µ 2 → µ 2 + 16Ω 2 in the l.h.s. of Eq. (4.9) (this procedure overestimates the correction when ζ ≃ 1, but it should be correct at least qualitatively); hence the corrected virtuality reads µ 2 (k, n, ℓ) ≈ µ 2 (0) (n, ℓ) − 16Ω 2 nℓ , with µ 2 (0) (n, ℓ) ≡ 4(n + ℓ + 1)(n + ℓ + 2) . (4.11) In particular, for non-relativistic momenta (k ≪ M n ∼ nM gap ), the above dispersion relation can be expanded out as (the quantum numbers are kept implicit) and therefore M rest M kin ≈ M 2 (0) . (M (0) denotes the "zero-T " meson mass, as given by Eq. (2.10) or (4.9).) The estimates (4.12) are in fact in agreement with the respective numerical findings in Ref. [23] (see the discussion of Eq. (4.45) there).

The intermediate momentum regime: light-like dispersion relation
With further increasing k, the energy ω n of the mode n is also increasing and the last term, proportional to Ω 2 , in the potential (4.6) becomes more and more important. Since, at the same time, the virtuality µ 2 n of the mode is decreasing, it should be clear that for sufficiently large k -namely, whenk ∼ nR 3 0 -one enters a regime where Ω 2 n ≫ µ 2 n and then the roles of the respective terms in the potential are interchanged: the term in Ω 2 becomes the dominant one, while that in µ 2 represents only a small correction. Then the mode n is nearly light-like and in fact it crosses the light cone (i.e., its virtuality µ 2 n (k) changes sign) when varying k within this domain. We have not been able to analytically follow this transition, but the fact that it actually happens is quite obvious by inspection of the shape in the potential in this regime. This is shown in Fig. 4 right for the three cases of interest: (a) κ 2 ≡ −µ 2 is negative but small, (b) κ 2 = 0, and (c) κ 2 is positive but small.
Namely, consider the genuine Schrödinger equation associated to this potential, that is, where E is the energy of a bound state. Given the shape of the potential in Fig. 4 right, it is clear that bound states with both positive and negative energies will exist for all the three cases aforementioned. It is furthermore clear that, with increasing Ω 2 at fixed κ 2 the potential becomes more and more attractive, so some of the bound states will cross from positive to negative energies. This means that, for any fixed value of κ 2 , there exist corresponding values of Ω 2 such that the respective bound states have E = 0. These are, of course, the meson modes that we are interested in. This Schrödinger argument also suggest the use of the semi-classical WKB method for computing the meson spectrum. Given the shape of the potential this should be a reasonable approximation at least for sufficiently large numbers n ≫ 1 (we set ℓ = 0 for simplicity). The Bohr-Sommerfeld quantization condition for the mode n with energy E n = 0 reads where ζ 0 is the turning point in the potential, that is, ζ 0 = 1 when κ 2 ≤ 0 and ζ 0 = 1 − κ/4Ω when κ 2 > 0. When κ = 0, the integral is straightforward and yields That is, the mode n crosses the light-cone at k = k n withk n ≈ √ 2nR 3 0 . As we shall see in Sect. 5, this is indeed the correct result when n ≫ 1.
An interesting property of the spectrum near the light-cone, which will play an important role in our subsequent study of DIS (see Sect. 5) and can be also understood on the basis of Eq. (4.14), is the extreme sensitivity of the dispersion relation to changes in the virtuality κ 2 around κ = 0 : so long as |κ 2 | ≪ Ω 2 , a small change in κ 2 entails a large change in Ω 2 . Before we explain the origin of this property, let us first use Eq. (4.14) to render it more specific. Consider the space-like case κ 2 > 0 for definiteness, and denote ε ≡ κ/4Ω ≪ 1, so that the turning point lies at ζ 0 = 1 − ε. Changing the integration variable according to x ≡ 1 − ζ, we can successively write where we have observed that, after subtracting the dominant contribution π/2 to the first integral, the subtracted integral is dominated by its lower limit x = ε; this allowed us to perform the simplifications in the second line, where we denoted x ≡ λε. The final integral multiplying ε 3/2 is clearly a positive number of O(1). One can combine together the space-like and time-like cases into the following formula where C is a positive constant and sgn(x) = Θ(x) − Θ(−x) is the sign function. This formula shows that, when moving away from the light-cone, say, towards space-like virtualities, the energy of the mode grows by a substantial amount ∆Ω n ∼ 1 for a relatively modest increase in the virtuality, from κ = 0 to κ ∼ Ω 1/3 n ≪ Ω n . In physical units, we changeω n by a large amount ∆ω n ∼ R 3 0 ≫ 1 when increasingQ from zero toQ ∼ n 1/3 R 0 ∼ω 1/3 n . This strong sensitivity of the dispersion relation to κ 2 around κ = 0 can be traced back to the behavior of the potential near the turning point ζ 0 (recall that it is this turning point which controls the κ-dependence of the integral in Eq. (4.14)). Namely, from Eq. (4.6) we deduce (for positive κ with κ ≪ Ω)

The high momentum regime: space-like dispersion relation
Consider now further increasing the momentumk of the mode n, beyond the critical valuē k n ≈ √ 2nR 3 0 at which the dispersion relation crosses the light-cone. Then the virtuality of the modeQ n will increase as well, i.e., the mode becomes more and more space-like (although, as we shall see, this virtuality remains relatively small, in the sense thatω n ≃k ≫Q n ). For instance, Eq. (4.17) implies that, so long as κ ≪ Ω, the virtuality grows with the energy (or the momentum) according to Eventually, when Ω n ≫ n, κ n becomes comparable with Ω n and then the potential (4.6) has the shape shown in Fig. 5 (for ℓ = 0 and two different values of κ).
As manifest on these pictures, when increasing the ratio κ/Ω, the turning point ζ 0 = 1 − κ/4Ω in the potential moves towards ζ = 0, i.e., towards the bottom of the D7-brane. Thus, clearly, the attractive region of the potential, which can support Schrödinger bound states with energy E n = 0 (or, equivalently, space-like mesons), exists only so long as κ ≤ 4Ω, and becomes very tiny (ζ 0 ≪ 1) when κ approaches the upper limit 4Ω (cf. Fig. 5 right). What we would like to argue in what follows is that for sufficiently large momentumk ≫ nR 3 0 , the dispersion relation approaches this kinematical limit in which κ n ≃ 4Ω n : this is the 'limiting velocity' regime, previously mentioned in relation with Eq. (2.14) [23,22].
To that aim, let us compute the spectrum in the regime where κ is indeed close to, but smaller than 4Ω, in such a way that ζ 0 ≪ 1. The corresponding modes will be localized in the classically permitted region at ζ ≤ ζ 0 . It is then a good approximation to replace the potential (4.6) by its expansion near ζ = 0. The ensuing Schrödinger-like equation reads where we have denoted When deriving Eq. (4.20) and also when simplifying the expressions in Eq. (4.21) we have anticipated the fact that, for the mode n, both Ω and κ are very large but relatively close to each other, such that Ω n , κ n ≫ n , Also, we restricted ourselves to angular momenta ℓ ≪ Ω n . Eq. (4.20) is formally similar to the radial Schrödinger equation for a non-relativistic particle with mass m = 1 and electric charge e in the three-dimensional Coulomb potential V C (ζ) = (−e)/ζ, with −E playing the role of the (negative) energy of a bound state. There are however some interesting differences with respect to the genuine Coulomb problem. First, the would-be 'angular' momentum of our fictitious 'Coulomb particle' is equal to 8 ℓ/2, and hence it can also take half-integer values. Second, our radial variable ζ is restricted to ζ ≤ 1 and, moreover, the approximate equation (4.20) is valid only for ζ ≪ 1; by contrast, in the corresponding Coulomb problem the radius ζ can be arbitrarily large. Yet, this last difference should not be important for the situation at hand: given the potential barrier at ζ ≤ ζ 0 , cf. Fig. 5, it is clear that the actual meson wavefunction is exponentially decaying for ζ > ζ 0 before exactly vanishing at ζ = 1. When ζ 0 ≪ 1, there should be only a minor difference between the exact wavefunction, which is strictly zero at ζ = 1, and its Coulombic approximation, which is exponentially small there.
Hence, one can solve Eq. (4.20) by following the same steps as for the Coulomb problem in quantum mechanics [56]. The general solution which is regular at ζ = 0 reads 9 ψ ℓ (z) = z ℓ 2 +1 e −z/2 M (−ν + 1 + ℓ/2, ℓ + 2; z) , (4.23) Here M (a, b; z) is the confluent hypergeometric function, also denoted as 1 F 1 (a, b; z), which obeys M (a, b; 0) = 1 (see Chapter 13 in [55]). Note that z ≃ 4 √ 2Ωζ, cf. Eq. (4.21), hence ζ = 1 corresponds to large z ≫ 1, where the asymptotic behaviour of Eq. (4.23) becomes relevant. Specifically, for the solution to exponentially vanish at z ≫ 1, the confluent hypergeometric series must terminate, which in turn requires − ν + 1 + ℓ 2 = −n , n = 0, 1, 2 , ... (4.25) and then M (−n, ℓ + 2; z) is a polynomial in z of degree n (actually, a Laguerre polynomial L (ℓ+1) n (z), up to a numerical factor [55]). The 'quantization' condition (4.25) together with the definitions (4.21) and (4.24) can now be used to deduce the meson spectrum in this high momentum regime. The resulting dispersion relation can be written in various, equivalent, ways, either as a function of the virtualityQ, 26) or as a function of the momentumk, 27) or, finally, in physical units: (4.28) The last two equations feature the limiting velocity v 0 which has been generated here as which is indeed consistent with Eq. (2.13) (recall that we assume R 0 ≫ 1). The results (4.26)-(4.28) are in agreement with a previous analytic study of this high-momentum regime, in Ref. [22], which is more precise than ours. In any of these equations, the two terms appearing in the left hand side are large but comparable with each other, while the term in the right hand side, which expresses the deviation from the linear dispersion relation ω = v 0 k and involves the dependence upon the quantum numbers, is comparatively small. By inspection of the meson wavefunction Eq. (4.23), where we recall that z ≃ 4 √ 2Ωζ, it is clear that the mode is localized near the bottom of the D7-brane, at ζ 1/Ω n ≪ 1. This domains lies within the classically allowed region at ζ ≤ ζ 0 (indeed, ζ 0 ≡ 1 − κ/4Ω ∼ n/Ω n for a mode satisfying (4.26)), which confirms the consistency of our previous approximations. Interestingly, the higher the energy is, the stronger is the mode localized near ζ = 0 (or ρ = R 0 ), in agreement with the numerical findings in Ref. [23].
Although the limiting velocity (4.29) is very close to 1 under the present assumptions, the virtualityQ 2 =k 2 −ω 2 of the meson is nevertheless very large,Q ≫ nR 0 , because its energy and momentum are even larger:ω n ≃k ≫ nR 3 0 . Thus the mode looks nearly light-like in the sense thatω n ≃k ≫Q, yet its virtuality is too high to be resonantly excited by an incoming space-like current: indeed, to be resonant with the meson, the flavor current should have an energyω and virtualityQ obeyingω ≃ (R 2 0 /2)Q ≫ nR 3 0 , and thereforeω/Q 3 ∼ (R 0 /Q) 2 ≪ 1. According to the discussion in Sect. 3, such a current would encounter a large repulsive barrier near the Minkowski boundary and hence it would get stuck at large radial coordinates ρ Q ≫ nR 0 , far away from the region at ρ ≃ R 0 where the would-be resonant mesons could exist. We conclude that such high-energy, space-like, meson excitations cannot contribute to the DIS of a flavor current. To investigate the possibility of DIS, we therefore turn to the only potentially favorable case, that of the 'nearly light-like' mesons withω n ≃k ∼ nR 3 0 and arbitrarily small virtualities.

Resonant deep inelastic scattering off the light-like mesons
We now return to the problem of DIS off the strongly coupled N = 2 plasma at low temperature, as formulated in Sect. 3. Recall that we are interested in a relatively hard space-like flavor current, with virtualityQ > R 0 (or κ > 1). So long as the energyω of this current is relatively low,ω ≪Q 3 (or Ω ≪ κ 3 ), there is a repulsive barrier which confines the dual gauge field A i (ρ) near the Minkowski boundary, where no resonant meson states can exist. (This barrier is visible in Fig. 5 as the repulsive potential at ζ > ζ 0 .) We thus conclude that the flavor structure functions vanish whenω ≪Q 3 , so like for the R-current.
However, the situation changes when the energy of the current is sufficiently high, such thatω Q 3 (or Ω κ 3 ). Then, the repulsive barrier becomes so narrow that it plays no role anymore (this is visible as the curve 'κ 2 > 0' in Fig. 4 right). Indeed, even in the presence of this barrier, the gauge field can penetrate across the barrier, via tunneling, up to a distance ρ ∼Q, or 1 − ζ ∼ 1/κ 2 ; when Ω κ 3 , this penetration is larger then the width 1 − ζ 0 = κ/4Ω of the potential barrier, and then the field can escape in the classical allowed region at ζ < ζ 0 . Thus, the field has now the capability to excite vector mesons at any value of ρ within the wordvolume of the D7-brane. Moreover the kinematics of the high-energy current matches with that of the nearly light-like mesons discussed in Sect. 4.3. Indeed, those mesons have energiesω n ∼ nR 3 0 , cf. Eq. (4.15), which can match the energyω Q 3 > R 3 0 of the current with a suitable choice for n. Furthermore, for a given n, there are mesons at all virtualities Q n nR 0 , and in particular such thatQ 3 n ω n , so like for the current. These kinematical arguments indicate that the high-energy flavor current can disappear into the plasma by resonantly exciting nearly light-like mesons. In what follows, we shall demonstrate that this picture is indeed correct, by explicitly computing the decay rate (i.e., the imaginary part of the current-current correlator) corresponding to the resonant excitation of large-n light-like mesons. The restriction to large quantum numbers n ≫ 1, that is, to very high energiesω ≫ R 3 0 , is necessary for technical convenience, but it also has the advantage to make the physics sharper. In that case, it becomes possible to smear our the delta-like resonances associated with the individual mesons and thus obtain a spectral function which is a continuous function ofω and hence describes DIS. Remarkably, that spectral function turns out to be identical with the DIS structure function in the high-temperature phase ('black hole embedding'), Eq. (3.23), that was previously computed [31] by imposing infalling boundary conditions at large ρ ≫ R 0 (cf. the discussion in Sect. 3.2).

Light-like mesons: exact solutions
In this subsection we shall concentrate on the EOM for light-like, transverse, gauge fields in the worldvolume of the D7-brane, that is, Eq. (3.15) withQ = 0, for which we shall construct exact solutions obeying the condition of regularity at r = 0 (or ρ = R 0 ). By using the asymptotic expansion of these solutions at large ρ and high energy, we shall study their behaviour near the Minkwoski boundary and thus distinguish between normalizable and nonnormalizable modes. In particular, this procedure will yield the spectrum of the light-like mesons for large quantum numbers n ≫ 1.
Once again, it is more convenient to use the variable ζ defined in Eq. (4.1) and which has a compact support. Then the relevant EOM is Eq. (4.2) with κ = 0, or, equivalently, the 'Schrödinger equation' (4.5) with ℓ = 0 and κ = 0. We shall choose the latter, that we rewrite here for convenience: Clearly, this is a particular case 10 of Eq. (4.20) that we have solved already, namely it is the limit of that equation when ℓ = 0 and 2e 2 = 2E ≡ 4Ω 2 . The solution which is regular at ζ = 0 is then obtained by adapting Eq. (4.23), and reads with Ω n the on-shell energy of a light-like meson. For generic values Ω ∼ O(1), this equation is difficult to solve except through numerical methods. A similar mathematical difficulty arises when trying to use Eq. (5.2) in order to compute the current-current correlator according to Eq. (3.12). For all such purposes, one needs the behavior of the solution near the Minkowski boundary at ζ = 1, and this is generally difficult to extract from Eq. (5.2). However this mathematical problem becomes tractable for the high energy regime of interest here, which is such thatω ≫ R 3 0 , or Ω ≫ 1. Then, one can use a special asymptotic expansion of the function M (a, b; z) with a < 0, which applies when the variables |a| and z are simultaneously large and such that z ≈ 2b − 4a ≫ 1. This last condition is truly essential, since in general, i.e., for generic values of |a| and z which are both large but uncorrelated with each other, very little is known about the asymptotic behavior of M (a, b; z). This specific limit is precisely the one that we need for our present purposes: indeed, in Eq. (5.2), we have z = 4Ωζ and 2b − 4a = 4Ω, and therefore z ∼ 2b − 4a = 4Ω ≫ 1 in the high energy limit and in the vicinity of ζ = 1. The asymptotic formula which applies to this case is formula 13.5.19 in Ref. [55] and can be formulated as follows: when In order to adapt this formula to Eq. (5.2), we shall write so that the variable ξ be positive. Then for Ω ≫ 1 and 1 − ζ ≪ 1, the solution (5.2) becomes (up to an irrelevant overall normalization) In particular, for relatively large ξ ≫ 1, one can use the asymptotic expansions of the Airy functions to deduce The solution (5.7) has the same general structure and validity range as the approximate solution shown in Eq. (3.19) -in particular, the argument ξ of the Airy functions is indeed the same in both equations, as it can be checked by using Eqs. (4.1) and (5.6) -, and this should not be a surprise: as explained in Sect. 3,Eq. (3.19) is the general form of the solution at high energy and large ρ. The whole purpose of a more complete analysis at smaller values of ρ, like the one that we have just performed here, is to fix the coefficients of the two Airy functions appearing in that equation. Note that, unlike for the solution with infalling boundary condition, i.e., Eq. (3.19) with c = i, the coefficients in Eq. (5.7) depend upon the energy variable Ω.
As a first application of Eq. (5.7), we now use it to determine the energies of the light-like meson excitations according to Eq. (5.3). To that aim, we also need [55] Ai(0) = 1 (the formulae involving the derivatives will be useful later on). Then a simple calculation shows that for ξ = 0 the right hand side of Eq. ... One may think that the constant shift Ω n − n = 1/6 in the eigenvalues is merely a tiny correction that can be safely ignored at large n, but this is generally not the case. Note first that this shift is uniquely determined by the values of the two Airy functions at ξ = 0, as it can be checked by inspection of the previous manipulations, and hence it is not affected by the approximation in Eq. (5.7). Moreover it is essential to take this shift into account whenever one is interested in the behavior of the solution near Ω = Ω n , which will be also our case in the next subsection.
The wavefunction corresponding to the mode n is obtained by replacing Ω → Ω n within the general formulae (5.2) or (5.7). For radial coordinates deeply inside the D7-brane, where the asymptotic expansion (5.7) does not apply, one can rely on the exact solution (5.2), but this is perhaps a little opaque. An approximate formula valid for intermediate values of ζ will be constructed via the WKB method in Appendix A. This WKB solution, shown in Eq. (A.2), is consistent with the asymptotic behaviour (5.8) and has the nice feature to exhibit exactly n nodes in the interval 0 < ζ < 1, as a priori expected for the nth radial excitation.

Current-current correlator and DIS
We are now prepared to compute the current-current correlator for a highly energetic, nearly light-like, current, and thus make the connection to DIS, as anticipated. To that aim, we shall use Eq.
Then a straightforward calculation using Φ(ξ, Ω) from (5.7) together with Eq. (5.9) yields with Ω related toω via Eq. (4.3). As expected, the function Π 1 (ω) exhibits poles at the energies Ω n = n + 1/6 of the light-like meson modes. These poles can be made more explicit by using the expansion of the cotangent as a series of simple functions: To extract the spectral weight associated with these poles, i.e., the imaginary part of the correlator, we use retarded boundary conditions, ω → ω + iǫ, together with the formula We thus find (recall that our energy variable is always positive) This result has been obtained here by working with a light-like current, but a similar result holds also for a highly energetic space-like current withω Q 3 , since the repulsive barrier plays no role in that case, and since the plasma can indeed sustain slightly space-like mesons which are resonant with the current. (In fact, the WKB method in Appendix A can be easily generalized to such slightly space-like mesons.) The emergence of the delta-functions in the imaginary part of the current-current correlator confirms our expectation that a high-energy flavor current can resonantly produce mesons in highly excited states (n ≫ 1) and thus disappear into the plasma. Taken literally, Eq. (5.15) would imply that the meson production by the current can only occur for a discrete set of energies which are resonant with the energiesω n = √ 2(n + The argument goes as follows: A current with a given, large, momentum k which is produced by a source acting over a finite time interval δt has an uncertainty δω ∼ 1/δt in its energy, and hence an uncertainty δQ 2 ≃ 2kδω in its virtuality. (We have used here k = ω 2 + Q 2 ≃ ω + Q 2 /2ω at high energy k ≫ Q.) As we shall demonstrate in Sect. 6, via an analysis of the time scales for the current interactions in the plasma, the typical interaction time for a nearly light-like current scales with its momentum likē t int ∼k 1/3 .
(5.16) (As usual, a bar over a kinematic variable denotes the dimensionless version of that variable measured in units of πT , e.g.t = πT t.) So, for this process to be experimentally observable, the source producing the current must act over a comparatively short period of time: δt t int .
(If δt ≫ t int , one cannot distinguish between the absorbtion of the photons in the plasma and their reabsorbtion by the source.) This in turn implies This means that the high-energy flavor current has the potential to produce meson excitations with momentumk (the momentum of the current) and virtualities within a range δQ k 1/3 aroundQ = 0. At this point one should remember the discussion towards the end of Sect. 4.3, about the high sensitivity of the nearly light-like meson dispersion relation to changes in the virtuality. Let us rephrase that discussion but from a different perspective: assume that the momentum k of the meson is now fixed, but consider changes in the mode quantum number n associated with changes in virtuality κ (near κ = 0). As it should be clear from Eq. (4.17), n varies by a number of order one when κ changes by ∆κ ∼ Ω 1/3 , that is, whenQ varies by ∆Q ∼k 1/3 . This is of the same order as the lower limit on the uncertainty (5.17) in the virtuality of the current. Thus, for a given momentumk, the current has the possibility to be resonant with several meson states, with neighboring quantum numbers.
Thus, in order to compute the total interaction rate for the current, as given by the imaginary part of the current-current correlator, we are allowed to average Eq. (5.15) over several neighboring levels and thus smear our the delta-function resonances. This averaging amounts to integrating Eq. (5.15) over an interval δΩ which contains a few neighboring resonances and dividing the result by δΩ, thus yielding the following result for the imaginary part of the retarded 2-point function of a high energy flavor current with momentumk and energyω ≃k. In particular, when the current is space-like (with relatively small virtuality, though:Q k 1/3 ), a non-zero imaginary part is synonymous of deep inelastic scattering, and the above result can be identified with the DIS structure function: F 1 = (1/2π)Im Π 1 . By comparing this result to Eq. (5.15) we see that the structure function in this low temperature phase is exactly the same as in the high-temperature phase, or 'black hole embedding'. This coincidence reflects that fact that in both cases the current is completely absorbed into the plasma, although the respective mechanisms are quite different: resonant excitation of nearly light-like mesons at low temperature and, respectively, partonic fluctuations (a quark-antiquark pair together with arbitrary many N = 4 quanta), which disappear into the plasma via successive branching, at high temperature. This physical picture will be further clarified by a discussion of the relevant time scales in the next section. Although our present calculations apply to the transverse field and structure function alone (recall that F T = 2xF 1 ), it is clear that a similar argument must be valid also in the longitudinal sector. So the corresponding, flavor, structure function F L = F 2 − F T can be deduced by simply rescaling, by a factor 4N f /N c , the respective result for the R-current [26] To summarize, for the N = 2 plasma at strong coupling, the above results for the flavor DIS structure functions are valid at either low, or high, temperatures, for high enough momentā k ≫ R 3 0 and for sufficiently low virtualitiesQ k 1/3 . In physical units, these conditions amount to k ≫ T (M gap /T ) 3 and Q Q s (x), where the saturation momentum Q s (x) ≃ T /x is the same as for the R-current.

Time dependence and physical picture
In what follows we would like to provide a space-time picture for the interactions of the flavor current in the strongly-coupled N = 2 plasma, and thus in particular clarify the energy averaging over neighboring resonances performed in the previous section. To that aim, we need to assume that the source producing the current has acted over a finite interval of time δt, which is much shorter than the typical interaction time in the plasma, t int , that we shall compute. Accordingly, the current is not a simple plane-wave anymore, but rather a wave-packet in energy. To study the dynamics of this wave-packet, it is convenient to first reformulate the respective EOM as a time-dependent Schrödinger equation. This will also give us insight into the typical time scales for meson excitations in the plasma. A current produced over a finite period of time δt can be described as a wave packet in energy, with a width δω ∼ 1/δt which is much smaller than the central value ω 0 ≃ k. (We assume that the current has a sharply defined longitudinal momentum k and we consider the high energy kinematics where ω ≃ k.) To study the evolution of this wave packet with time, we need to restore the time-dependence in the respective equations of motion, as written down in Sect. 3.1. For the problem at hand, this can be readily done by replacing in equations like (3.15). Indeed, for the (relatively narrow) wave packet under consideration, the time dependence will be represented by a wave e −iω 0 t modulated by a relatively slowly varying function (|i∂ t | ≪ ω 0 ). Starting with (3.15), we thus obtain where we recall thatt = πT t,k = k/πT , ρ 2 = R 2 0 + r 2 , and a dot denotes a derivative w.r.t. r. In writing this equation, we have replaced ω 0 by k everywhere except in the virtuality termQ 2 =k −ω 2 0 (which in what follows will be treated as a small correction). Also, we have neglected a subleading term involving the time derivative which is proportional to 1/ρ 8 (recall that ρ ≥ R 0 ≫ 1). We would like to rewrite this equation as a time-dependent Schrödinger equation, since then we can rely on the techniques and intuition developed within quantum mechanics. The first step is to eliminate the term ∝Ȧ i , by writing A i (t, r) ≡ Φ(t, r)/r 3/2 : Given the r-dependent factor 1/ρ 4 multiplying the time derivative in the l.h.s., this equation is not yet in Schrödinger form. It turns out that the canonical, Schrödinger, form of the equation can be achieved by changing the radial coordinate one more time, namely, by using the angle θ introduced in Sect. 2, cf. Eq. (2.7), to that purpose. Specifically, after writinḡ Formally, these equations describe the quantum dynamics of a non-relativistic particle with mass Ω and Hamiltonian H defined by the r.h.s. of Eq. (6.5). Note that the term involving the virtuality κ within the potential is independent of θ and thus merely acts as a constant shift in the total energy, in the same way as the time derivative. This is as expected: the time-dependence in the problem arises because of the uncertainty δω in the energy of the wave-packet; for a fixed momentum k, this corresponds to an uncertainty δQ in the virtuality, such that δω ≃ δQ 2 /2k. In the units of Eq. (6.5), this amounts to i∂ τ ∼ δκ 2 /2Ω, so the time-derivative and the (central value of the) virtuality act indeed on the same footing. This being said, in what follows we shall often ignore the last term κ 2 /2Ω in Eq. (6.6), precisely because we are interested in situations where the fluctuations in virtuality associated with the uncertainty principle are larger than the central value Q 2 . That is, we shall assumeQ ≪k 1/3 (the high-energy kinematics where the DIS process becomes possible), whereas we shall see that δQ k 1/3 .
The potential in Eq. (6.6) is displayed in Fig. 6 for the interesting case Ω ≫ 1 and κ = 0. The Minkowski boundary lies at θ = 0 and the bottom of the D7-brane at θ = π/2. Other remarkable points are the two classical turning points, θ 1 and θ 3 , and the minimum of the potential at θ 2 . One finds Figure 6: The potential V (θ) in Eq. (6.6) for Ω = 5 and κ = 0. so that θ 1 ≪ 1, while θ 2 and θ 3 are close to π/2. The potential has a rather deep minimum: To estimate the interaction time for the flavor current, we shall evaluate the time that the wave packet takes to travel from the boundary to the interior of the D7-brane, where it can excite mesons. A similar analysis for the case of the R-current was presented in Ref. [27]. Near the boundary, where θ ≪ 1, the equation becomes with the following, exact, solution (C is a constant) which describes diffusion : at early times, the penetration θ of the wave packet in the radial dimension grows with τ like Here τ 1 is the time the wave takes to reach the point θ 1 , cf. Eq. (6.7), where the potential becomes attractive; in physical units,t 1 ∼ R 0 τ 1 ∼k 1/3 . For τ > τ 1 , and so long as θ lies in between θ 1 and the minimum of the potential at θ 2 , the wave packet falls in the potential, essentially by following the classical equation of motion: For most of this travel, θ is still small, so sin 4 θ ≃ θ 4 and . (6.12) So, the penetration θ becomes of O(1) at τ ∼ τ 2 , with That is, it takes (parametrically) as much time to the wave to fall in the potential up to large distances θ ∼ O(1) as it takes to diffusively reach the attractive part of the potential, although the respective distances are very different: θ 2 ≫ θ 1 . This is so because along the second part of the trajectory, from θ 1 to θ 2 , the wave has un accelerated motion under the influence of the BH. This timet 2 ∼k 1/3 is already a realistic estimate for the interaction time, since the current can resonantly produce mesons when θ is of O(1). Moreover, this estimate would not change if the meson was to be produced further down in the worldvolume of the D7-brane (say, near its lower tip at θ ≃ π/2), since the final part of the fall in the potential is very rapid because the potential is so attractive around θ 2 . We thus conclude thatt int ∼k 1/3 , as anticipated in Eq. (5.16). This also confirms that the typical uncertainties in the energy and the virtuality of the wave packet satisfy δω 1/k 1/3 and, respectively, δQ k 1/3 , in agreement with our original assumptions. Since determined by the dynamics relatively close to the Minkowski boundary, this interaction time is independent of R 0 , and thus is parametrically the same as for the R-current [27]. As we shall shortly see, a similar conclusion holds also for the light-like meson excitations with large quantum numbers n ≫ 1 : such a meson has a period (the interval of time corresponding to one rotation around a semiclassical orbit) ∆t n ∼ω 1/3 n and spends most of this time at radial locations far away from R 0 , namely around ρ ∼ n 1/3 R 0 . The subsequent calculation will also shed light on an interesting property of the meson spectrum, which played an important role in our previous argument: the strong sensitivity of the dispersion relation to variations in the virtualityQ around the light-cone (Q = 0).
To that aim, we resort again on the Bohr-Sommerfeld quantization formula, valid for large n. We thus write nπ = θ 3 θ 1 dθ p(θ ; E n , Ω) , p(θ ; E, Ω) ≡ 2Ω(E − V (Ω)) , (6.14) where E n are the energy levels associated with the Schrödinger Hamiltonian in Eq. (6.5). These energies are defined by the usual eigenvalue problem HΨ n = E n Ψ n and depend upon the two parameters Ω and κ within H. They should not be confused with the meson energies Ω n , which rather correspond to the special values of Ω (at a given κ) for which the homogeneous equation HΨ = 0 has non-trivial, normalizable, solutions. One clearly has E n (Ω, κ) = 0 for Ω = Ω n (κ) . (6.15) This last equation has a unique solution for any n = 0, 1, 2, ..., as it can be easily recognized by inspection of the potential in Eq. (6.6) and Fig. 6. By choosing E n = 0 in Eq. (6.14), one could work out the corresponding integral and thus recover our previous result that, e.g., Ω n (κ) = n, cf. Eq. (4.15). However, for the present purposes, we shall rather use Eq. (6.14) to compute the level spacing of the Schrödinger energies near E = 0, that is ∆E(Ω n ) ≡ E n+1 (Ω n ) − E n (Ω n ) = E n (Ω n ) (κ = 0) . (6.16) Using Eq. (6.14), this is obtained as (we anticipate that ∆E ≪ 1) π = Indeed, this quantity ∆E(Ω n ) provides the answer to the two questions that we are interested in, as we explain now: (i) The last integral in Eq. (6.17) is the same as half of the period for a round trip around a semiclassical orbit: indeed, p = Ω∂ τ θ, hence (Ω/p)dθ = dτ .
(ii) The quantity ∆E(Ω n ) characterizes the response of the meson dispersion relation to variations in the meson virtuality near the light-cone. This can be understood by recalling the discussion below Eq. (6.6), about the last term, κ 2 /2Ω, in the potential. More precisely, ∆E(Ω n ) is a measure of the increase in the virtuality which is needed to jump from one meson level (n + 1) to the neighboring one (n) at fixed energy and in the vicinity of the light-cone. In formulae, ∆E(Ω n ) = κ 2 n /2Ω n with κ n defined by Ω n+1 (0) = Ω n (κ n ), or E n (Ω n (κ n ), κ n ) = E n+1 (Ω n (κ n ), 0) = 0 .  where in the second line we have used the fact that the integral is dominated by its lower limit θ 1 ≪ 1, and B(1/6, 1/2) is the respective Beta function. We thus find that the period for motion of a meson around the semiclassical orbit with quantum number n is ∆t n ∼ 1/θ 1 ∼ω 1/3 n . Moreover, the fact that the above integral is dominated by θ ∼ θ 1 also means that a quantum particle in the bound state with energy E n ≃ 0 spends most of its time at relatively large radial distances ρ ∼ω 1/3 n ∼ n 1/3 R 0 . This implies a similar property for the light-like meson with energyω n ≃ √ 2nR 3 0 . Of course, the meson wavefunction has support everywhere in the range R 0 ρ n 1/3 R 0 , but the radial velocity ∂ τ θ is smaller towards the upper end of this range (as it should be clear by inspection of the potential in Fig. 6), therefore there is a larger probability to find the meson in that region than towards the bottom of the D7-brane. Furthermore, the result for ∆E in Eq. (6.19) implies orQ n ∼ω 1/3 n . This is in agreement with our previous discussion of Eq. (4.17) in Sect. 4.3 : there is enough to make a rather small change ∆Q n ∼ n 1/3 R 0 in the meson virtuality in order to jump from one mode to another at fixed energy, whereas one needs a substantially larger increase in the energy of the meson, namely ∆ω n ∼ R 3 0 , in order to make that jump at fixed virtuality. The above result forQ n is moreover of the same order as the fluctuations δQ in the virtuality of the flavor current due to its energy uncertainty, thus justifying the energy averaging performed in our previous calculation of the spectral weight, in Eq. (5.18).
The shift α in the argument of the sine function is not accurately determined by the WKB method, but will be later fixed by matching onto the exact solution near ζ = 1. With the potential V (ζ) in Eq. (5.1), the integral in Eq. (A.1) is straightforward and yields This solution is not reliable very close to the end points at ζ = 0 (where the potential becomes singular) and ζ = 1 (where the potential is not differentiable), but it should be a reasonable approximation at the intermediate points.
A similar WKB solution can be constructed on the right of ζ = 0. The condition that the two solutions match with each other at intermediate points leads to the Bohr-Sommerfeld quantization condition (4.14), which in turn implies Ω n ≈ n when n ≫ 1. With Ω n = n, the WKB solution (A.2) has exactly n nodes in between 0 and 1. When approaching the end point at ζ = 1, we expect Eq. (A.2) to remain a good approximation so long as [56] ( where −dV /dζ = 4Ω 2 is the left derivative of the potential at ζ = 1. This range of validity, which in terms of the variable ξ introduced in Eq. (5.6) amounts to ξ ≫ 1, is wide enough to allow for a matching between the WKB solution and the exact solution near ζ = 1. The latter is the following linear combination of Airy functions