A Wave–packet evolution in the vacuum

Jet evolution in the SYM plasma at strong coupling

Abstract:

Within the framework of the AdS/CFT correspondence, we study the time evolution of an energetic –current propagating through a finite temperature, strongly coupled, SYM plasma and propose a physical picture for our results. In this picture, the current splits into a pair of massless partons, which then evolve via successive branchings, in such a way that energy is quasi–democratically divided among the products of a branching. We point out a duality between the transverse size of the partonic system produced through branching and the radial distance traveled by the dual Maxwell wave in the AdS geometry. For a time–like current, the branching occurs already in the vacuum, where it gives rise to a system of low–momentum partons isotropically distributed in the transverse plane. But at finite temperature, the branching mechanism is modified by the medium, in that the rate for parton splitting is enhanced by the transfer of transverse momentum from the partons to the plasma. This mechanism, which controls the parton energy loss, is sensitive to the energy density in the plasma, but not to the details of the thermal state. We compute the lifetime of the current for various kinematical regimes and provide physical interpretations for other, related, quantities, so like the meson screening length, the drag force, or the trailing string, that were previously computed via AdS/CFT techniques.

1 Introduction

Motivated by some experimental results from the heavy ion program at RHIC, which suggest that the deconfined, ‘quark–gluon’, matter produced in the early stages of an ultrarelativistic nucleus–nucleus collision might be strongly interacting (see, e.g., the review papers [1, 2] and references therein), there was recently an abondance of applications of the AdS/CFT correspondence to problems involving a strongly–coupled gauge plasma at finite temperature and/or finite quark density (for a recent review see [3]). While the early such applications have focused on the long–range and large–time properties of the plasma, so like hydrodynamical flow and transport coefficients, more recent studies have been also concerned with the response of the plasma to a ‘hard probe’ — an energetic ‘quark’ or ‘current’ which probes the plasma on very short time and distance scales, much shorter than the thermal scale (with being the temperature). Although the relevance of such applications to actual hard probes in QCD is perhaps not so clear (since, by asymptotic freedom, QCD should be weakly coupled on such short space–time separations), the results that have been obtained in this way are conceptually interesting, in that they shed light on a new physical regime — that of a gauge theory with strong interactions — which for long time precluded all first–principles theoretical investigations other than lattice gauge theory. For these results to be accompanied by conceptual clarifications, a physical interpretation for them is strongly needed, but this seems to be difficult without direct calculations in the original gauge theory. A different strategy, which is less rigorous, is to propose a physical picture based on general arguments and then demonstrate that this picture is consistent with all the available results (to the extent that comparisons are possible). This will be our strategy in this paper.

The physical problem that we shall consider — the propagation of an ‘electromagnetic’ current (actually, an –current) through the plasma — is particularly well suited for our purposes since, first, it has a strong overlap with several other problems previously considered in the literature and, second, it does not require any extension of the AdS/CFT correspondence (like the introduction of additional D7–branes) beyond the ‘minimal’ framework of its original formulation [4, 5, 6] — so it avoids any potential artifact due to such extensions. Besides, this problem has another useful feature: for the gauge theory of interest — namely, the conformally invariant supersymmetric Yang–Mills (SYM) theory — the current–current correlator in the vacuum (or ‘vacuum polarization tensor’) is protected by supersymmetry [7], in such a way that the full result in the strong ’t Hooft coupling limit is exactly the same as the corresponding result in lowest–order perturbation theory (i.e., the one–loop approximation). This property will allow us to recognize — via a comparison between the space–time picture of the one–loop process in the gauge theory and the ‘supergravity’ picture of the dual AdS/CFT calculation — an interesting ‘duality’ between the physical transverse size of the partonic system into which the current is evolving and the radial distance in . In turn, this duality will be a key ingredient of our proposal for a physical interpretation.

Specifically, we shall follow the time–evolution of an –current which in the plasma rest frame propagates like a plane–wave in the direction, with a large longitudinal momentum and frequency , and which at has zero transverse size. At weak coupling, this current would develop a partonic fluctuation involving just two partons — massless fields from SYM which carry –charge and transform in the adjoint representation of the color group SU. For a space–like current, this pair would grow in transverse space up to a maximal size and live for a relatively long time , thus acting like a ‘color dipole’ which can mediate the current interactions with an external target. (Here is the virtuality of the current, and is positive in the space–like case.) For a time–like current (), the pair can dissociate after a time and thus give rise to two free partons which separate from each other, so like the quark and antiquark jets produced in annihilation. This perturbative, one–loop, picture formally applies also to the full vacuum polarization tensor at strong coupling, because of the non–renormalization property alluded to above. This explains the polyvalence of the –current as a ‘hard probe’, including at strong coupling1 : by tuning the virtuality and the momentum of the current, we can mimic a color dipole, a ‘meson’, or a pair of jets with the desired values for the system size and rapidity, and then study the interactions between this partonic system and the plasma (or any other target).

But, of course, all that applies to a current propagating through the vacuum, and there is a priori no guarantee that a similar strategy should also work in the thermal bath. In a previous analysis [9], we have considered the case of a space–like current — i.e., the problem of deep inelastic scattering off the plasma — and shown (via the appropriate AdS/CFT calculation) that this strategy still works, but only for not too high energies. A current with relatively low energy, such that , propagates through the plasma essentially without interacting, so like a ‘small meson’ in the approach of Refs. [10, 11, 12, 13, 14, 15], where the ‘meson’ was made with two ‘heavy quarks’ attached to a D7–brane. Note that the above condition on can be rewritten as a lower bound on the current virtuality, , with the plasma saturation momentum, and also as an upper bound on the transverse size of the effective dipole, , which is then consistent with the respective bound (the ‘meson screening length’) found in Refs. [10, 11, 12].

But for higher energies , or, equivalently, lower virtualities , the analysis of Ref. [9] shows that the current is very rapidly absorbed into the plasma, over a time which is much shorter than the period required for the formation of a nearly on–shell partonic fluctuation. That is, for such a high energy, the current cannot be assimilated with a color dipole anymore (not even over a finite period of time), as it disappears before a dipole can form. The above result on can be restated by saying that the current propagates through the plasma over a longitudinal distance before it disappears. Interestingly, the same parametric estimate has been very recently found for the penetration length of an effective gluon [16]. In view of the physical picture that we shall develop, this similarity is not just a coincidence, but rather it reflects the universality of the dissipation mechanism in the strongly–coupled plasma.

In Ref. [9], the current lifetime has been inferred from physical considerations, but no temporal evolution (on top of the usual phase from the definition of the plane–wave) was explicitly considered. In this paper we shall extend that analysis by addressing the time–dependent problem, for both space–like and time–like currents, with the purpose of elucidating the dynamics of the current and the mechanism responsible for its dissipation. As already mentioned, we shall consider initial conditions such that the current has zero transverse size at . In the dual string calculation, this is represented by a vector field propagating in the –Schwarzschild metric which at is localized at the Minkowski boundary of . For , this perturbation propagates inside the bulk of the ‘5th dimension’ as a Maxwell wave, i.e., according to the Maxwell equations in the –Schwarzschild geometry. For the kinematics of interest, these equations can be suggestively rewritten in the form of time–dependent Schrödinger equations, to be presented in Sect. 2.

We first analyze the zero–temperature case (in Sect. 3), where the relevant geometry is purely . By solving the ‘Schrödinger equations’ in the approximations of interest and comparing the results to the space–time picture of the quantum fluctuation of the current into a pair of massless fields, we find that the two pictures match indeed with each other after provided one identifies the radial dimension in with the inverse of the transverse size of the partonic fluctuation: more precisely, , wit the curvature radius for . For instance, on the supergravity side, the temporal scale appears as the time after which the wave has penetrated into the bulk up to a distance . This corresponds, on the gauge theory side, to the fact that the partonic fluctuation requires a formation time in order to grow up to a transverse size . Notice that this formation time is a genuine quantum effect, which reflects the uncertainty principle. This suggests that, for the problem at hand, the 5th dimension of somehow mimics, within the context of the classical supergravity calculation, the phase–space for physical quantum fluctuations in transverse space.

This duality helps us identify a physical picture for the current dynamics in the vacuum. The naive one–loop picture (2–parton fluctuation) cannot be right at strong coupling, where any of the two partons produced in the original splitting of the current can further radiate, or scatter off the vacuum fluctuations. Quantum field are known for their strong propension to radiation. When the coupling is weak, the radiation is concentrated in specific corners of the phase space (collinear and soft radiation for a gauge theory), where the smallness of the coupling is compensated by the phase–space available to radiation. This leads, e.g, to the celebrated DGLAP and BFKL evolution equations in perturbative QCD. But at strong coupling, there is no reason why radiation should be restricted to small corners of phase–space; this should most naturally proceed via the ‘democratical’ branching of the original quanta into two daughter partons with more or less equal shares of their parent energy and momentum. We therefore propose a physical picture for current (or parton) evolution a strong coupling (and in the vacuum) in terms of successive branchings, such that the energy and the virtuality are divided by two (for simplicity) at each individual branching, and that the lifetime of a generation is determined by the formation time for the next generation, in agreement with the uncertainty principle. As we show in Sect. 5.2, this simple picture reproduces indeed the results of the respective AdS/CFT calculation, at a qualitative level.

Moving now to the finite–temperature case, we find (by solving the appropriate ‘Schrödinger equations’, in Sect. 4) that, for low and moderate energies, such that , and for not too large values of time, the dynamics remains essentially as in the vacuum. For a space–like current, this confirms the previous results in Ref. [9]: a ‘small meson’, with size , survives almost unaltered in the plasma. (The ‘meson’ can decay via tunnel effect, but the decay rate is exponentially suppressed [9].) But for a time–like current, the vacuum–like evolution proceeds only up to a time , with the Lorentz factor of the current. At this time , the ‘current’ — or, more precisely, the partonic system produced via its successive branching — has propagated inside the plasma over a longitudinal distance and has extended in transverse space up to a size . Note that this value is precisely of the order of the meson screening length computed in Refs. [10, 11, 12], and more commonly written as , with the velocity of the meson. More detailed comparisons between our approach and previous results in the literature will be performed in Sect. 5.1.

For larger times , this partonic system starts interacting with the plasma, although its transverse size is still much smaller than . In this regime, the dual string calculation shows that the Maxwell wave feels the attraction of the black hole and thus undergoes an accelerated fall towards the horizon, that it gets close to after a time of order . When this radial dynamics is translated to transverse space, via the duality mentioned before, it implies that the partons feel a constant, decelerating, force in the transverse directions,

(1.1)

whose precise physical meaning remains a little mysterious to us (see the discussion in Sect. 5.2). By including the action of this force on the dynamics of branching, we will be able to show that medium–induced branching is a likely physical scenario, which is qualitatively consistent with the results of the AdS/CFT calculation. By ‘medium–induced branching’ we mean that the change in virtuality over the lifetime of a partonic generation is due to the transfer of transverse momentum to the plasma, at a constant rate (see Sect. 5.2 for more details).

The same physical scenario — an accelerated fall of the Maxwell wave into the black hole, which physically corresponds to medium–induced branching — holds also in the high–energy regime at , for both space–like and time–like currents, but in this regime, this scenario starts to apply much earlier, after a time which is much shorter than the formation time for an on–shell partonic fluctuation. (We recall that is the plasma saturation momentum.) Thus, such an energetic current disappears in the plasma before having the time to create ‘jets’ (on–shell partons). Within this scenario, we find a natural physical explanation for the ‘drag force’ experienced by partons in the plasma (originally computed for a heavy quark probe in Refs. [17, 18]), and also for the ‘trailing string solution’ (the string attached to an energetic heavy quark propagating through the plasma –Schwarzschild geometry [17, 18]), that we recognize as the dual of the enveloping curve of the spatial distribution of partons produced through medium–induced branching. It remains as an interesting open problem to understand whether this scenario can naturally accommodate also other quantities computed within AdS/CFT, so like the jet quenching [19], or the transverse momentum broadening [20, 21].

One can succinctly summarize the previous discussion as follows: While a highly virtual (and hence very small) space–like ‘meson’, with virtuality , can survive in the strongly–coupled plasma (essentially without feeling the latter), this is not also the case for the jets produced by the decay of a highly virtual time–like current, which can only propagate over a longitudinal distance before disappearing in the plasma. Less virtual (or more energetic) partonic systems, for which , cannot form in the first place : the virtual partons melt in the plasma over a time , which is too short to get on–shell. The dissipation mechanism is universal, i.e., the same for all kind of partons — off–shell or on–shell, massive or massless, quarks, scalars, or gluons — and it consists in medium–induced branching.

Let us conclude this Introduction with a remark concerning the spatial distribution of the hadrons produced via the decay of a time–like current in the vacuum. For the concept of hadron to make sense, we supplement the theory with an infrared cutoff and assume that the current is ‘hard’ relative to this cutoff : . Then the decay of the current via successive branchings will produce a system of hadrons with small transverse momenta , which are isotropically distributed in the transverse plane. (In the rest frame of the current, this distribution would be spherically symmetric.) This observation is the ‘time–like’ counterpart of a previous result that, at strong coupling, there are no large– partons (i.e., no partons carrying a sizeable fraction of the hadron longitudinal momentum) in the wavefunction of an energetic hadron [22, 23, 9]. Rather, all partons have fallen down at very small values of (smaller than in the case of the SYM plasma [9] and, respectively, for a hadron [23]), via successive branchings within space–like cascades. Because of that, a high–energy ‘nucleus–nucleus’ collision which would liberate these partons would produce no jets, but only a multitude of particles at central rapidity which are isotropically distributed in transverse space, with small momenta . This picture is similar, and possibly related, to a very recent result [24] showing that the energy distribution of the particle produced in the strong–coupling analog of the annihilation exhibits spherical symmetry. On the other hand, this picture looks quite different from the corresponding one in perturbative QCD and also from the respective results for nucleus–nucleus collisions at RHIC.

2 General equations

We would like to address the following physical problem: at time , an –current with momentum oriented along the axis and energy acts in the SYM plasma, producing a system of SYM quanta which then propagate through the plasma until they finally disappear. In the dual gravity problem, this current dynamics is represented by the propagation of a Maxwell–like gauge field in the background geometry of the black hole (representing the SYM plasma) [3]. The corresponding metric reads

(2.1)

where is the temperature of the black hole (the same as for the SYM plasma), is the curvature radius of , and are the time and, respectively, spatial coordinates of the physical Minkowski world, is the radial coordinate on , and . Note that our radial coordinate has been rescaled in such a way to be dimensionless: in terms of the more standard, dimensionfull, coordinate , it reads , with . Hence, in our conventions, the black hole horizon lies at and the Minkowski boundary at .

Since the radial component vanishes at , it is possible and convenient to work in the gauge where is identically zero. Then, the non–zero components , with , are of the form:

(2.2)

where has a relatively weak time dependence, such that , and obeys the initial condition that, for , is localized near . (The precise structure of the initial condition is not really needed.) The relevant equations of motion are the Maxwell equations in the Schwarzschild geometry. They are most conveniently written as a set of equations for the transverse components , with , and, respectively, the longitudinal component (the component is not independent, but it is related to via the equations of motion), and read [3]

(2.3)

and, respectively,

(2.4)

where a prime on a field indicates a –derivative and we have introduced the following, dimensionless, variables

(2.5)

(The variables and will be used later on.) The above equations can be further simplified for our present purposes. First, as already mentioned, the field is slowly varying in time; hence we can write

(2.6)

where we have neglected the second–order time derivative of . Furthermore, as we shall see, the interesting dynamics happens near to the boundary at (and hence far away from the horizon at ), so we can replace in the coefficients of the above equations everywhere except in the term : indeed, in that term, the small quantity can be amplified by the longitudinal momentum of the incoming current, which becomes very large at high energy. Notice that this term represents the potential for the long–range (in ) gravitational interaction between the current and the black hole. Since we keep only this particular medium effect in our equations, but neglect those which would signal the black–hole singularity at (the gravity–dual hallmark of a thermal system), we expect our subsequent results to apply to matter distributions more general than a finite–temperature plasma: similar results should hold for any matter distribution which is infinite and homogeneous (say, a cold matter), after replacing the energy density of the finite–temperature plasma by the corresponding quantity for the cold matter.

After these simplifications, we end up with the following equations for and , valid when :

(2.7)

where we have introduced the notation and the plus (minus) sign in front of corresponds to a space–like (time–like) current. Note that, with its above definition, is always a positive quantity.

It will be furthermore useful (especially in view of constructing approximate solutions) to rewrite these equations in a form which resembles the time–dependent Schrödinger equation. To that aim, we shall perform the following changes of variable and functions:

(2.8)

Also, we shall be mostly interested in the high energy regime where , meaning , so that we can replace in the coefficients of the equations. Then our final equations, valid for high energy and , read

(2.9)
(2.10)

To avoid a proliferation of cases, we shall mostly focus on the longitudinal case, cf. Eq. (2.10).

As anticipated, we choose initial conditions such that, at , the fields are localized near (i.e., ). We shall implement that by working in Fourier space; we thus write, e.g.,

(2.11)

where is a wave packet in peaked around . For convenience, we shall take this wave packet to be modulated by a Gaussian, that is,

(2.12)

where the width is constrained by

(2.13)

and the additional –dependence in will be fixed by Eq. (2.10). The first inequality in Eq. (2.13) () ensures that the typical energy fluctuations obey , as originally assumed. The second inequality () is necessary to allow for quasi–localized configurations in the initial condition and at early times (see Appendix A). This implies that the allowed fluctuations can be quite large, , so that the space–like, or time–like, nature of the current (depending upon the sign of ) becomes apparent only for sufficiently large times, after the effects of the initial condition have dissipated. In fact, as we shall see in the next section, the early–time behavior of the solution is independent of , and hence the same for both space–like and time–like currents.

Another boundary condition refers to the behavior of the solution in the stationary regime (meaning, for large enough times) and at relatively large values of : since a black hole is a purely absorptive medium, from which no signal can escape, the solution near the horizon located at must be a purely outgoing wave (i.e., a wave departing from the boundary and impinging in the black hole). In fact, this outgoing–wave behavior should manifest itself already for relatively low values , and hence can be used as a boundary condition on Eq. (2.10), since the potential term in this effective Schrödinger equation is monotonous everywhere except near the boundary at , and hence it cannot generate reflected waves. This will be further discussed in the forthcoming sections.

3 Jets in the vacuum

We start with the zero–temperature case, i.e., with the propagation of the Abelian current through the vacuum. The results to be obtained here will not only serve as a level of comparison for the subsequent discussion of a plasma, but they will also prepare that discussion, at two levels (at least): First, as we shall see, there are special regimes (so like early times) where the dynamics in the plasma is quite similar to that in the vacuum. Second, the physical interpretation of our results, whose understanding is the main purpose in this paper, is easier to introduce and motivate in the context of the vacuum dynamics.

3.1 The vacuum polarization tensor

We begin this discussion with the stationary case, i.e., the case where the fields are purely plane waves in the physical, 4–dimensional, space, meaning that the fields denoted with a tilde in the previous discussion (e.g., ) are independent of time. The corresponding AdS/CFT calculation will provide the retarded current–current correlator (or ‘polarization tensor’) in Fourier space (with ) :

(3.1)

and in that sense it is the analog at strong coupling of computing momentum–space Fourier diagrams in perturbation theory at weak coupling. More precisely, is obtained by differentiating the classical action with respect to boundary values of the fields at :

(3.2)

where in the notations of Eq. (2.2) and is the four–dimensional action density, which is homogeneous: , with the volume of space–time. In turn, the classical action density can be fully expressed (after using the equations of motion) in terms of the values of the field and of its first derivative at :

(3.3)

where the component is determined by the EOM as . A star on a field denotes complex conjugation: the classical solutions develop an imaginary part (in spite of obeying equations of motion with real coefficients) because of the outgoing–wave condition at large (see below). Via Eq. (3.2), this introduces an imaginary part in which physically describes the dissipation of the current in the original gauge theory. In fact, the imaginary part of the expression within the square brackets in Eq. (3.3) is independent of (as it can be checked by using the EOM) and hence it can be evaluated at any [25].

Note that, even in this zero–temperature context, we keep using the dimensionless variables and , which were previously defined with respect to the black–hole horizon . Throughout this section, it will be understood that and are arbitrary (length and, respectively, momentum) scales of reference, which are used to construct dimensionless coordinates at intermediate stages of the calculations, but which will drop out from the final, physical, results. In this context, the radial coordinates and can take on all the values from 0 to .

We now turn to the actual calculation of the vacuum polarization tensor for the SYM theory in the strong coupling limit. The outcome of this calculation is already known (the case of a space–like current has been treated in detail in Ref. [9]), so our subsequent presentation will be very streamlined, with emphasis on the physical interpretation of the results. We shall give details for the longitudinal sector alone. The relevant equation reads (cf. Eq. (2.4) in which we let and )

(3.4)

where we recall that the upper (lower) sign in front of corresponds to a space–like (time–like) current. After the change of variable , this is recognized as the equation for the Bessel functions, of either real, or imaginary, argument (depending upon the sign in front of ). The solution is constrained by the boundary condition (see Ref. [9] for details)

(3.5)

together with the condition of regularity at . For the space–like case, these conditions uniquely determine the solution as

(3.6)

The other independent solution, , is rejected since it would exponentially diverge as . For the time–like case, on the other hand, the general solution is a superposition of oscillating Bessel functions,

(3.7)

so the condition of regularity at introduces no constraint. To fix the solution in this case, we shall require to be an outgoing wave at large ; then, the solution becomes imaginary, with the appropriate sign for the imaginary part to yield the retarded polarization tensor, via Eq. (3.2). This constraint implies which together with the boundary condition (3.5) completely fixes the solution as

(3.8)

where is a Hankel function encoding the desired outgoing–wave behavior at large : when .

The transverse–wave solutions can be similarly obtained, but the above solutions for the longitudinal wave are in fact sufficient to complete the calculation of : by Lorentz and gauge symmetry, the polarization tensor is transverse (with ) :

(3.9)

and the scalar function can be computed with the longitudinal waves alone. A standard calculation, which involves the removal of a logarithmic divergence in the real part at (the AdS/CFT analog of ultraviolet renormalization), finally yields

(3.10)

where is the current virtuality in physical units and is the renormalization scale. As expected, the imaginary part is non–zero only for a time–like () current, which can decay into the massless fields (adjoint scalars and Weyl fermions) of the SYM theory carrying –charge.

As anticipated in the Introduction, this result (3.10) is exactly the same as the respective perturbative result to one–loop order, so in particular its imaginary part (formally) describes the current decay into one pair of massless fields (a quark–antiquark pair, or two scalar fields). This interpretation is, of course, only formal: At strong coupling, the current can couple to arbitrarily complicated multi–particle states, but it so happens that, for the SYM theory, the total cross–section is determined by the two–particle final states alone. Our present calculation being an ‘inclusive’ one — it provides the total cross–section, but it does not discriminate between the various final states —, its result cannot give us any direct insight into the nature of these final states. We shall later try to gain such an insight based on physical considerations.

We conclude this discussion of the stationary case with another point of physical interpretation, which points towards an interesting ‘duality’ between the radial dimension in and the transverse size of the current (or, more precisely, of the partonic system into which the current has evolved) in the physical Minkowski space. Consider a space–like current, for definiteness. The modified Bessel function in Eq. (3.6) decays exponentially when , meaning that the current penetrates in the radial dimension only up to a finite distance , or (recall that ). The more virtual the current is, the closer it remains to the boundary. This is quite similar to the picture of the current in transverse space, as familiar in perturbation theory (for either QCD or SYM): the virtual current fluctuates into a quark–antiquark pair (a ‘color dipole’) whose transverse size is inversely proportional to the current virtuality: . This analogy is in fact even closer: the longitudinal wave solution in Eq. (3.6) involves the same modified Bessel function as the wavefunction describing the dipole fluctuation of a longitudinal photon in lowest order perturbation theory, which allows us to identify the respective arguments as . Recalling that for a space–like current, we deduce the correspondence . The same argument holds in the transverse sector, where is replaced by . Also, the argument can be adapted to a time–like current, since the oscillatory Bessel functions in Eq. (3.8) are rapidly decaying at radial distances .

The above writing of the correspondence between the transverse size of the partonic fluctuation and the radial coordinate in , namely, , explicitly involves the temperature and hence it might look a bit formal in the present context of the vacuum (but this writing will be natural for the subsequent discussion of a plasma). To avoid confusion, it is preferable to recall the definition with in order to rewrite this correspondence in the form , from which the temperature has dropped out.

To summarize the previous discussion, the vacuum polarization tensor in the SYM theory at strong coupling formally describes the fluctuation of the current into a pair of elementary, massless, fields, whose transverse size appears to be in a one–to–one correspondence with the radial distance for the current penetration in . As we shall see, this interpretation is comforted by the discussion of the time–dependent case, to which we now turn.

3.2 Jet evolution in the vacuum

As explained in Sect. 2, we are interested in the time evolution of vector fields which at start as a perturbation localized near (or ). Given the correspondence between and (the transverse size of the current), as argued at the end of the previous subsection, we see that this initial condition corresponds to a current which, at , is point–like in transverse space. To describe its evolution, we shall use the Schrödinger form of the equations of motion, cf. Eqs. (2.9)–(2.10), and focus on the longitudinal sector, for definiteness. The vacuum version of these equations is obtained by removing the last term in the potential; this yields, e.g.,

(3.11)

for which we shall construct solutions obeying the relevant initial and boundary conditions. To remain as simple as possible, we shall consider approximate solutions which are valid piecewise in and which are sufficient to illustrate the main points of physics. A more systematic method to construct solutions to Eq. (3.11), which is based on the wave–packet decomposition in Eqs. (2.12)–(2.13), will be described in Appendix A.

We first note that at early times, so long as remains smaller than a critical value (the corresponding limit on time will be determined later on), the second term, proportional to , in the potential in Eq. (3.11) can be neglected compared to the first term . Thus, this early–time dynamics is identical for both space–like and time–like currents. With the term omitted, Eq. (3.11) admits the following, exact, solution

(3.12)

This implies that the actual longitudinal wave (cf. Eq. (2.8))

(3.13)

is localized near at (although this is not exactly a delta function). With increasing time, the energy density carried by the wave (3.13) diffuses towards larger values of , so that the typical distance traveled by the corresponding wave–packet after a time is

(3.14)

This behavior holds so long as , meaning for times . In physical units, this yields , which is precisely the coherence time of the high–energy current; that is, this is the time interval which controls the Fourier transform2 in Eq. (3.1), as it can be checked by rewriting the complex exponential there as

(3.15)

where we have used at high energy.

It is interesting to compare these results with the evolution of the quark–antiquark (‘dipole’) fluctuation of a virtual photon in perturbation theory in QCD: if the photon dissociates at into a point–like pair of fermions, then with increasing time the transverse size of this pair is increasing diffusively, due to quantum dynamics, like [26, 27]

(3.16)

until it reaches a maximal size at a time . (To avoid cumbersome notations, we shall often write instead of and instead of within parametric estimates. For instance, the coherence time will be estimated as , where the modulus on is implicit for a time–like current.) For , the pair is either recombining back into a photon, or — if the photon was time–like — it splits apart, thus giving rise to two on–shell particles which move away from each other.

Clearly, the early time () evolution of the fluctuation in perturbation theory is very similar to the corresponding evolution of the vector perturbation in provided one identifies , in agreement with the discussion at the end of Sect. 3.1. As we show now, this correspondence persists also for larger times .

Indeed, for , one can heuristically estimate the time–derivative in the l.h.s. of Eq. (3.11) as (this heuristic argument will be confirmed by the wave–packet analysis in Appendix A). That is, the time–derivative term in the equation is much smaller than the last term in the potential, which becomes the dominant term when . Hence, for and , the equation simplifies to

(3.17)

(i.e., Schrödinger equation in a flat potential) with the obvious, acceptable, solutions

(3.18)

which are time–independent and coincide, as they should, with the asymptotic versions (valid at large ) of the respective stationary solutions constructed in Sect. 3.1. In the space–like case, the above solution confirms that, for times , the perturbation remains localized near the boundary, within a distance . In the time–like case, it implies that the actual wave

(3.19)

propagates with constant group velocity along the radial direction of :

(3.20)

Note that, for , we have , as it should for consistency with the previous solution at early times. Hence, after the wave packet has diffused up to a distance inside the bulk of , it furthermore propagates with constant radial velocity, so like a free particle.

This free–motion pattern at looks, of course, natural, in view of the flatness of the potential in Eq. (3.17). In Appendix B, we shall verify that Eq. (3.20) is the same as the radial part of the geodesics of a massless classical particle which propagates in with longitudinal velocity and radial velocity , which is the same as the group velocity in Eq. (3.20) (recall that ); this classical particle is massless since . On the other hand, the classical particle dynamics cannot reproduce the diffusion at early stages (a genuinely quantum effects), nor the fall of the wave into the black hole (to be later described, in Sect. 4.3).

Via the correspondence , the result in Eq. (3.20) is again consistent with the transverse dynamics expected for a time–like current which dissociates into a pair of massless particles (in lowest–order perturbation theory). Indeed, this result translates into

(3.21)

where is the common longitudinal velocity of the two massless particles, as inherited from the current, and is the modulus of their transverse velocity: the two particles move in opposite directions in the transverse plane, so the transverse distance between them increases like . The above results have been obtained by working in the high–energy regime where . However, it is easy to repeat the analysis for other regimes, with similar conclusions. For instance, for a zero–momentum current (, ), one finds that , i.e., the two particles move in the transverse plane at the speed of light ().

At this point, one should again stress that the reason why the physical picture looks so simple in the strongly–coupled SYM theory is because of the non–renormalization property of the current–current correlator, as alluded to before. The AdS/CFT calculation correctly provides the total cross–section for the current decay, but it does not capture the detailed nature of the final states. Formally, the total cross–section is saturated by the two–particle final state; therefore, it is the dynamics of this particularly simple state which emerges, via the correspondence , from the dual calculation on the gravity side.

This discussion has an interesting corollary: it shows that the current can be also viewed as a device for introducing a pair of elementary, massless, fields of the SYM theory (Weyl fermions or adjoint scalars) at a given radial distance within , which is controlled by the current virtuality: or . This is tantamount to fixing the transverse size of the partonic pair in the physical space. That is, the dynamics of the current at times larger than the coherence time is the same as that of ‘meson’, or of a ‘color dipole’. For a space–like current, this effective ‘meson’ simply sits at , meaning that its transverse size is fixed. For a time–like current, this ‘meson’ propagates with constant velocity along the radial direction , meaning that its transverse size grows at constant speed. In the next section we shall study the influence of a thermal bath on this dynamics.

4 Jets in the plasma

We now turn to the problem of main interest for us here, which is the propagation of the current and of the associated partonic system through a strongly–coupled SYM plasma with temperature . We are interested in ‘hard probes’, so we shall choose a current with relatively high virtuality (either space–like, or time–like) : (or ), which probes the structure of the plasma on distances much shorter than the thermal wavelength . We shall mostly consider a relativistic current, for which , but the non–relativistic case () will be briefly discussed too, for completeness. In fact, the physically most interesting case — the one where the medium effects should be truly relevant — is when the coherence time of the current is much larger than , so that the current explores a relatively large longitudinal slice of the plasma, with width . This implies a lower limit on the current momentum: , which is tantamount to the condition that the Bjorken– variable3, defined as (in the plasma rest frame), be very small: .

4.1 Physical regimes

We would like to determine the characteristic time scale for the dissipation of the current in the plasma; in the dual, gravity, problem, this is the time scale for the fall–off of the Maxwell field into the black hole. As we shall see, this scale is controlled by the dynamics at relatively small , and thus is insensitive to the detailed geometry of the black hole near its horizon at . As explained in Sect. 2, the strength of the gravitational interactions between the wave and the black hole is proportional to the wave longitudinal momentum , and also to the temperature. In view of that, we shall be led to distinguish between two important physical regimes:

(i) a relatively low–energy (or low–temperature) regime at (or ), where the medium effects are strongly delayed, so that the current dynamics proceeds as in the vacuum up to time scales much larger than , and

(ii) a very high–energy (or high–temperature) regime at (or ), in which the current dissipates very fast, on a time scale much shorter than .

One can understand these various regimes by studying the potential in the Schrödinger–like equations (2.9)–(2.10). Let us first recall from the previous section that the time–dependence in these equations is important only at very early stages, when the typical values of are so small that the only relevant term in the potential is the first term, , which is independent of both the current virtuality and the properties of the medium. Hence, for such early times, the equations describe diffusion, for both space–like and time–like currents, and in the same way as in the vacuum. But for later times, where the meaning of ‘later’ is generally medium– and energy–dependent (see below), the solutions approach a stationary regime, where Eqs. (2.9)–(2.10) can be replaced by their time–independent versions, of the generic form with (since corresponds to the time–derivative, which is negligible). Hence, this late–time behavior is determined by the time–independent Schrödinger solution with zero energy in the potential .

Figure 1: The potential in Eq. (2.10) for the space–like case at relatively low energy ().
Figure 2: The potential in Eq. (2.10) for the time–like case at relatively low energy ().
Figure 3: The potential in Eq. (2.10) in the high–energy regime at , where this potential is essentially the same for both space–like and time–like currents.

In what follows we shall study this solution in the longitudinal sector alone (the corresponding discussion for the transverse sector being very similar). The respective potential is graphically illustrated in Figs. 1, 2, and 3, for the various physical regimes: low energy and space–like in Fig. 1, low energy and time–like in Fig. 2, and high energy (both space–like and time–like) in Fig. 3. Some general features of the dynamics are already clear by inspection of these figures: At sufficiently small values of , the potential is the same as in the vacuum; the medium starts to be felt only at relatively large values of , where the potential is a decreasing function which describes attraction by the black hole. Also, in the high–energy regime, the dynamics is essentially the same for both space–like and time–like currents. Finally, in the space–like case at least, there is an obvious difference between the low–energy regime, where the potential barrier constrains the wave to remain on the left of the ‘classical turning point’ at (so like in the vacuum), and the high–energy regime, where the barrier has disappeared and the wave can easily propagate towards the horizon. (The transition between the two regimes occurring at corresponds precisely to the situation where the height of the potential barrier becomes negligible.) In the time–like case, on the other hand, there is never such a barrier, so the difference between the low–energy and high–energy regimes looks perhaps less obvious. As we now explain, this difference is nevertheless important in that case too.

To that aim, it is convenient to separate the three pieces in the potential in Eq. (2.10), which play different physical roles. We thus write (for a time–like current)

(4.1)

Let denote the value of at which and become comparable with each other, and similarly for and . We have, parametrically,

(4.2)

Consider now the two physical regimes alluded to before:

(i) Low energy regime :      
It is easy to check that, in this regime, the potential admits the piecewise approximation

(4.3)

Thus, so long as the relevant values of (those where most of the wave energy is located) remain much smaller than , the dynamics is the same as in the vacuum. This is the situation at sufficiently small times, that we shall study in the next subsection. But when , the medium effects become important and entail the fall of the wave into the black hole. This fall be studied in Sect. 4.3. Note that the transition point is where the potential changes from a plateau to a rapidly decreasing function (see Fig. 2).

(ii) High energy regime :      
In this regime, the virtuality–dependent piece is never important, and the potential can be approximated as (see also Fig. 3)

(4.4)

which now holds for both space–like and time–like currents. Unlike the low–energy potential, the high–energy one has no intermediate plateau, but only a pronounced peak at . In fact, the transition between the two energy regimes, which occurs for , corresponds to the situation where the two endpoints of the plateau, and , merge with each other, and also with the position of the emerging peak.

4.2 Early–time dynamics

In this subsection, we shall consider the dynamics of the current at early times, which is insensitive to medium effects (except for its validity limits, as introduced by the plasma), and thus can be inferred from the corresponding discussion in Sect. 3.

We start with a simple case, which has not been explicitly covered by the previous discussion, but which can be viewed as a special limit of the low–energy case: a non–relativistic, time–like, current with and . The corresponding ‘Schrödinger’ equations are obtained from Eqs. (2.9)–(2.10) by omitting the last, medium–dependent, term in the potential (since this term is comparatively small for any ) and replacing and in all the other terms. The ensuing equations are the same as in the vacuum, and therefore so is also the current dynamics, until it eventually dissipates into the plasma. Specifically, at very early times , the associated Maxwell wave diffusively penetrates into , up to a distance (meaning that the current diffusively spreads out in the physical space, up to a size ). Then, for larger times , the wave propagates inside with constant group velocity , meaning that the current has decayed into a pair of massless particles which move away from each other at the speed of light: , or . This behavior continues until the wave hits the horizon ( or ) and is thus absorbed by the black hole. Physically, this means that, for times , the separation between the products of the current decay has become as large as the thermal wavelength, , so these products cannot be anymore distinguished from the plasma fluctuations — they get ‘lost’ in the plasma.

We now turn to a relativistic current with , for which we have to distinguish between two physical regimes, as previously explained:

(i) At moderate energies, such that , and for a time–like current, the upper limit for the vacuum–like behavior is much larger than , which is the maximal penetration length through diffusion. Hence, in this regime, one can observe both stages of the evolution identified in Sect. 3 : a ‘diffusive’ stage at early times , during which the wave–packet progresses according to Eq. (3.14), and a ‘free motion’ stage at , during which the wave has constant group velocity . This second stage lasts until the position of the wave–packet becomes comparable to  ; this happens at a time (the subscript on stands for “free motion”), which represents the upper time limit for this vacuum–like dynamics. In physical units, this yields a time scale

(4.5)

which, as indicated above, is much larger than the coherence time.

The physical interpretation of this result, as deduced via the correspondence , is as follows: the time–like current develops a partonic fluctuation with size over a time of the order of the coherence time , and then decays into a system of massless particles which move freely (without feeling the plasma) up to a time . At this stage, the transverse extent of the partonic system has increased up to a value

(4.6)

which is still small as compared to the thermal wavelength in the medium. For even later times , the dynamics is driven by the medium and will be analyzed in the next subsection.

It is interesting to notice at this point that, although it influences the dynamics at late times , the plasma has no effect on the polarization tensor, which is still given by the same expression as in the vacuum, namely Eqs. (3.9)–(3.10) with . Indeed, if one considers a (time–like) plane–wave solution, so like in Sect. 3.1, then for this solution is again given by Eq. (3.8), which describes an outgoing wave for any within the range . The change in the solution at larger has no incidence on the calculation of , which is determined by the behavior of the solution near , cf. Eq. (3.2). Physically, this is so since the current disappears by decaying into a pair of massless fields, so like in the vacuum, and the subsequent fate of these fields is irrelevant for the calculation of the total decay rate (the imaginary part of ). This is similar to the calculation of the total cross–section for annihilation in lowest–order perturbation theory: this cross–section is fully given by the annihilation into a pair, although the quark and the antiquark are not the actual final states in the experiments. The ‘late stages’ radiations or interactions involving the quark and the antiquark, although essential for hadronisation and the composition of the final state, do not affect the total cross–section.

(ii) At ultrarelativistic energies , the medium effects (as enhanced by the energy) become important already at the very short radial distance , meaning that the current has no time to fully develop its partonic fluctuation before getting absorbed. At very early times, such a fluctuation starts to develop via diffusion, but this process is interrupted after the relatively short period , or

(4.7)

(which is still bigger than though), when the wave–packet has diffused up to . Physically, this means that the growth of the fluctuation is stopped at a transverse size

(4.8)

which is much smaller than the natural size for the same fluctuation in the vacuum. Accordingly, the dynamics is very different from the vacuum case, in the sense that the dissipation of the current and the respective polarization tensor are now controlled by the medium. This polarization tensor, which is now identical for space–like and time–like currents, has been computed in Ref. [9], and the result was used to deduce a partonic interpretation for the structure of the plasma in its infinite momentum frame. In the next subsection, we shall address this problem from a different perspective, by following the time evolution of the Maxwell wave during its fall towards the black hole. The insight that we shall gain in this way will allow us to propose, in Sect. 5, a physical picture for the current dissipation in the strongly coupled plasma.

4.3 The fall of the wave into the black hole

We now come to the most interesting physical situation, which refers to the (relatively) late stages of the current evolution in the plasma and the mechanism for current dissipation. In this situation, the dynamics is controlled by the medium–dependent piece in the potential (the piece in Eq. (4.1)), which yields the following equation of motion in Schrödinger form

(4.9)

As previously explained, for a time–like current this equation holds both at moderate energies, , and in the high–energy limit , but in ranges for which are different in the two cases (cf. Eqs. (4.3)–(4.4)). For a space–like current, it holds only for and .

We have not been able to find an exact solution to this equation, but we shall construct a WKB approximation to it. To that aim, we use the wave–packet representation in Eq. (2.12). The corresponding WKB solution reads then

(4.10)

where we have kept only the outgoing wave and is either , or , depending upon the physical regime under consideration (and similarly for ). At this point, we notice that the interesting values of are large enough for the typical energy to be much smaller than the potential . For instance, when , we are interested in together with . Then we can expand the square root within to linear order in and perform the ensuing Gaussian integration, to obtain

(4.11)

We have also used here

(4.12)

The modulus tells us where the energy of the wave–packet is located at time . Clearly, with increasing time, the peak of the energy distribution moves along a trajectory