Contents

UTTG-12-11

TCC-014-11

The Gluonic Field of a Heavy Quark in

Conformal Field Theories at Strong Coupling

Mariano Chernicoff111mchernicoff@ub.edu, Alberto Güijosa222alberto@nucleares.unam.mx and Juan F. Pedraza333jpedraza@physics.utexas.edu

Departament de Física Fonamental and Institut de Ciències del Cosmos,

Universitat de Barcelona,

Marti i Franquès 1, E-08028 Barcelona, Spain

Departamento de Física de Altas Energías, Instituto de Ciencias Nucleares,

Universidad Nacional Autónoma de México,

Apartado Postal 70-543, México D.F. 04510, México

Theory Group, Department of Physics and Texas Cosmology Center,

University of Texas, 1 University Station C1608, Austin, TX 78712, USA

Abstract

We determine the gluonic field configuration sourced by a heavy quark undergoing arbitrary motion in super-Yang-Mills at strong coupling and large number of colors. More specifically, we compute the expectation value of the operator in the presence of such a quark, by means of the AdS/CFT correspondence. Our results for this observable show that signals propagate without temporal broadening, just as was found for the expectation value of the energy density in recent work by Hatta et al. We attempt to shed some additional light on the origin of this feature, and propose a different interpretation for its physical significance. As an application of our general results, we examine when the quark undergoes oscillatory motion, uniform circular motion, and uniform acceleration. Via the AdS/CFT correspondence, all of our results are pertinent to any conformal field theory in dimensions with a dual gravity formulation.

1 Introduction and Summary

1.1 Motivation

When a charge moves, it produces a propagating disturbance in the associated gauge field. The problem of determining the spacetime profile of this disturbance for an arbitrary charge trajectory was solved long ago for classical electrodynamics [1], but, under various guises, remains of significant interest today in the context of quantum non-Abelian gauge theories. In recent years, gauge/gravity duality [2, 3, 4] has given us a useful handle on this and many other problems for a varied class of gauge theories in the previously inaccessible regime of strong coupling, via a drastic and surprising rewriting in terms of string-theoretic degrees of freedom living on a curved higher-dimensional geometry.

Among the known examples of the duality, the best understood subclass is that of conformal field theories (CFTs), where the relevant curved geometry is asymptotically anti-de Sitter (AdS). In this paper we will use this AdS/CFT correspondence to study the propagation of disturbances in the gluonic field produced by a moving heavy quark in a strongly-coupled conformal gauge theory, in the limit where the number of colors is large. For concreteness, we will phrase our analysis in terms of super-Yang-Mills (SYM), even though the results we will obtain are equally relevant to other CFTs in dimensions, can easily be extended to CFTs in other dimensions, and might also be expected to apply at a qualitative level in some of the non-conformal examples of the gauge/gravity correspondence.

The AdS/CFT correspondence states that a heavy quark moving in the vacuum of SYM is dual to a string moving on a pure AdS geometry. More precisely, the quark corresponds to the endpoint of a string, whose body codifies the profile of the non-Abelian (near and radiation) fields sourced by the quark. When the quark/endpoint moves, it generally produces a wave running along the body of the string, which corresponds to the wave in the gluonic field whose profile we are interested in determining.

The translation between string and gauge theory disturbances was first explored in [5], which used tools developed in [3, 6] to study the dilaton waves given off by fluctuations on an otherwise static, radial string in AdS, and infer from them the profile of the dual gluonic field observable in the presence of an oscillating quark. Under the assumption that these oscillations are small, the authors of [5] treated the string dynamics in a linearized approximation. Their results painted an interesting picture of wave propagation in SYM: in contrast with the standard Lienard-Wiechert story of the classical Abelian case, where signals propagate strictly at the speed of light, the waves in were found to display significant temporal broadening, just as one would expect given that points on the non-Abelian field arbitrarily far from the quark can themselves reradiate.

Figure 1: Schematic representation of emission of dilaton waves by a string in AdS. The radial direction of AdS is denoted by and runs vertically, with the AdS boundary () located at the bottom of the figure. Any given point on the boundary receives a dilaton wave from each point along the string. As a result, the gauge theory wave is a superposition of components with all possible time delays.

As shown in Fig. 1, this feature emerges naturally in the gravity side of the correspondence, because motion of the string endpoint generates waves that move up along the body of the string, and each point on the string then emits a dilaton wave that travels back down to the observation point on the AdS boundary, where, via the AdS/CFT recipe for correlation functions [3], the value of is deduced by assembling together all such contributions. The dilaton waves originating from points on the string that are further away from the AdS boundary give rise to components of the SYM wave that have a larger time delay. This makes perfect physical sense, because, through the UV/IR connection [7], such points are known to be dual to regions of the gluonic field further away from the quark. So, even when treating the string waves at the linearized level, the very fact that we are dealing with a string leads to gauge theory disturbances that display the nonlinear propagation expected from the non-Abelian character of the strongly-coupled SYM fields.

The simple observable that was the focus of [5] served well to exhibit the main features of propagating disturbances in the gluonic field, but did not allow a definite identification of waves with the characteristic falloff associated with radiation, i.e., contributions that transport energy to infinity. Of course, in the case of an accelerating quark, fluctuations in the near-fields are fully expected to be accompanied by radiation proper. The unambiguous detection of the latter calls for examination of the SYM energy-momentum tensor, , which in the gravity side of the correspondence requires a determination of the gravitational waves emitted by the string. Indeed, some time after [5], the falloff was established in the same setup through a calculation of the (time-averaged) energy density, [8].

In more recent years, motivated by potential contact with the phenomenology of the quark-gluon plasma [9], analyses of both and have been carried out in the case of a heavy quark moving at constant velocity through a thermal plasma (where an exact solution for the corresponding string embedding is available [10, 11]), in a large body of work that includes [12] and has been reviewed in [13]. The results are again compatible with the expected nonlinear dynamics of the (in this case, finite-temperature) SYM medium.

Given these antecedents, it came as a surprise when, back at zero temperature, additional calculations going beyond the linearized string approximation found, first for special cases [14, 15] and then for an arbitrary quark trajectory [16], that the ensuing energy density displays no temporal broadening, and is in fact as sharply localized in spacetime as the corresponding classical profile. As shown in [15, 16], the fact that disturbances in propagate strictly at the speed of light does not actually conflict with the UV/IR argument presented three paragraphs above, because the full gluonic profile can be understood as arising purely from a contribution of the string endpoint, and this is the reason why it features a single time delay.

Our work was motivated by the tension between these two sets of results for the gluonic fields in vacuum. How can it be that the profile obtained in [5] displays significant temporal broadening, while the pattern deduced in [16] does not? A natural strategy to further explore and attempt to resolve this tension is to carry out the calculation beyond the linearized string approximation, and for an arbitrary quark trajectory, to put it on a par with the computation in [16]. This is what we set out to do in this paper.

1.2 Outline and main results

We begin in Section 2 by presenting the ingredients and recipe for our calculation. The desired one-point function of (or, more precisely, of the Lagrangian density operator (6)), in the presence of a moving quark, must be extracted via (7) from the leading near-boundary behavior of the dilaton field (12) sourced by the string profile (16). The latter is the unique string embedding codifying the gluonic fields generated by a quark undergoing arbitrary motion, under the assumption that such fields are retarded or purely outgoing, i.e., they propagate outwards from the quark to infinity. This solution was obtained in [17] for an infinitely heavy quark and generalized in [18, 19, 20] to the case of a quark that has a finite mass, and, consequently, a finite size (i.e., Compton wavelength) given by (2).

The actual computation is carried out in Section 3. A key feature is that, when expressed in terms of appropriately geometric variables (the quark proper time and the invariant AdS distance (9)), a total derivative is found to appear in the worldsheet integrand (23), allowing one of the two integrals to be done trivially. This feature results from a nontrivial cancelation (noted below (21)) that in turn originates from the retarded structure of the string embedding (14). An analogous cancelation was found in the energy density computation in [16], arising there from an interplay between the various components of the string energy-momentum tensor. Since our analysis involves only scalar quantities, it becomes clear that the cancelations in question are not intrinsically tied to the tensorial nature of the derivation.

Our final result for the gluonic profile in the case of a quark with finite mass is somewhat involved: it is written in (3) in terms of auxiliary (tilde) variables whose explicit dependence on the quark’s velocity, acceleration, jerk and snap, as well as on the external force (and the first and second derivatives thereof) it is subjected to, is given in (32)-(34), (36)-(37). It is easy to check that, when the quark is static, our general expression correctly reproduces the known [21] finite-mass result (42). For arbitrary motion, the gluonic field at any given observation point is found to depend only on dynamical data evaluated at a single retarded time along the quark trajectory, specified by (38) or, in non-covariant form, (39). This is the same surprising characteristic demonstrated for the radiation component of the field in [15, 16].

In Section 4, we apply the general result (3) to three specific examples. The first is the oscillating quark that was the focus of [5] and the main motivation for our work. We start by performing a numerical analysis of the case (not covered by [5]) where the quark has a finite mass, and display the resulting gluonic profile in Fig. 4. Moving on to the case of infinite mass, we show explicitly that the integrated expression obtained in [5], which manifestly displays the gluonic field as a sum of contributions with all possible time delays, actually gives the same result (within its range of validity) as the linearized and infinitely-massive version of our final formula (3) for the gluonic profile, which incorporates a single time delay. This proves that, despite appearances, there is in fact no conflict between [5] and [15, 16]: it is just that the choice of worldsheet coordinates and the lack of an exact solution for the string embedding prevented the authors of [5] from being able to carry out the integral explicitly to find the net retardation pattern.

Our second example is uniform circular motion, studied in Section 4.2 to complement the energy density analysis of [14]. Our third and final example, examined in Section 4.3, is uniform acceleration, for which [22, 23, 24] had already made some interesting physical inferences based on the structure of the string embedding.

In our final Section 5 we go back to our result for arbitrary quark motion, and discuss its physical implications. We consider first the case of an infinitely heavy quark, which was the only one contemplated by the computation in [5] and the calculations in [14, 15, 16]. In this limit, the quark becomes pointlike (), and our result simplifies drastically, taking the form (43). For an arbitrary spacetime trajectory of the quark, this is manifestly just the boosted Coulomb profile associated with a uniformly translating quark with the retarded position and velocity inferred by projecting back along the past lightcone of the observation point. The full profile agrees (up to an overall constant) with the corresponding Lienard-Wiechert result in classical electrodynamics [1]. The observable determined in this paper is thus seen to yield an image of the gluonic profile for a pointlike quark that is in complete consonance with the one obtained via in [16], with the main difference being that, just like in electrodynamics, the former is sensitive only to disturbances in the near field of the quark, whereas the latter also explicitly incorporates radiation. Both here and in [16] signals in the gluonic field are found to propagate strictly at the speed of light (as indicated below (43)), with no temporal (or, equivalently, radial) broadening.

Our more general result (3), relevant for a quark with finite mass, differs from (43) and from the energy density obtained in [16] in two ways. First, the retarded quark data are read off at a source point that is related to the observation point through the timelike interval (38). In other words, signals are found to propagate at a (variable) subluminal speed, which as seen in (63) can in fact be arbitrarily small. Second, the field profile depends on more data than just the position and velocity of the quark. As explained in Section 5, both of these features are naturally associated with the fact that the quark is no longer pointlike.

Even with these differences, it is still true that in our general result (3) the field at a given observation point is controlled by data at a single retarded event, as in [15, 16]. The discrepancy between this pattern and the one reported in [5] was discussed in [15] in terms of a fundamental distinction between the quark’s near field (which is all that is visible in ) and its radiation field (which dominates at long distances), but our results show that there is in fact no such distinction, because both components display exactly the same unbroadened propagation. The reason on the gravity side is the same in both cases: the fact that (in accord with the UV/IR connection) the result can be expressed in terms of a contribution arising purely from the string endpoint. Moreover, we know from Section 4.1 that the discrepancy between [5] and [15, 16] is only apparent, and that, in spite of the fact that the gluonic field emerges as a superposition of contributions with all possible time delays, the net result evidences only the smallest of these delays.

In [15, 16] it was argued that the lack of broadening is unphysical and points to a deficiency of the AdS/CFT calculation. The argument visualizes the radiation process in terms of emission of gluons that are themselves able to reemit, and would be expected to do so profusely at strong coupling. This would bring into play large quantities of off-shell gluons, which would naturally be expected to propagate at subluminal speeds and therefore to lead inevitably to temporal/radial broadening of the emitted field. This ‘parton branching’ picture is essentially a perturbative rephrasing of the discussion in [5] (recalled above) of non-Abelian reemission by the gluonic field at all different length scales, and has been shown to be consistent with several other AdS/CFT results [25]. Given the apparent incompatibility of this picture with the lack of broadening, the authors of [15, 16] went on to suggest that, contrary to widespread belief, the supergravity approximation to physics on the gravity side does not capture the full quantum dynamics of the large and large limit, and tried to identify the missing element as arising from longitudinal fluctuations in a heuristic lightcone gauge calculation.

As explained in Section 5, even though we find in this paper that displays exactly the same retardation pattern as , we do not subscribe to the point of view of [15, 16]. As seen in the calculation for arbitrary quark trajectory in Section 3 (as well as in [16]), and also when we make contact in Section 4.1 between [5] and our general result (3), the supergravity description assembles the gluonic field precisely in the physically expected manner, by summing over contributions reradiated from all possible length scales (as depicted schematically in Fig. 1), associated with all possible time delays. It is therefore not true that the absence of net temporal broadening in the final AdS/CFT result implies that the supergravity approximation is somehow leaving out the expected non-Abelian rescattering. Rather, we interpret the no-broadening result as a prediction of the AdS/CFT correspondence for the net pattern of propagation in the CFT at strong coupling and with a large number of colors.

We stress in particular that the appearance in (23) of a total derivative in the worldsheet integrand, which enables us to present the resulting dilaton field as a pure endpoint contribution, does not mean that points on the body of the string do not contribute, but only that their aggregated contribution can be reexpressed in terms of the behavior at the edge of the integration region. The same applies then on the SYM side: the fact that the gluonic field at the observation point can be reported purely in terms of the behavior of the quark at a single retarded time does not indicate that only that instant contributes, but only that the cumulative effect of summing over the contributions from all previous emission events (arising via non-Abelian rescattering at all possible length scales) can be reexpressed in terms of the aforementioned behavior. This is explicitly shown by our calculations, by those of [14, 15, 16], and also by a recent surprising reformulation of the latter as a superposition of gravitational shock waves emitted by each point along the string [26].

It is important to keep in mind that the form of our result depends crucially on the retarded structure of the worldsheet embedding (16), which in turn follows from the assumption of a purely outgoing condition for the gluonic field generated by the quark. For any choice other than the purely outgoing (or purely ingoing) SYM configuration, we would expect not to obtain a total derivative on the worldsheet, and this would then lead to a final gluonic profile showing finite temporal/radial broadening. At least in retrospect, it is natural for the purely outgoing condition to impose a restriction on the overall retardation pattern of the field, because arbitrary reradiation from all points on the non-Abelian medium would result in wave scattering back towards the quark. That the final gluonic profile at any given observation point, which receives contributions from the entire quark trajectory, can be reexpressed in terms of data at a single retarded time, is also not surprising per se. At the calculational level, that happens every time we evaluate an integral in terms of the behavior at the integration endpoint. What is remarkable about the restriction predicted by AdS/CFT is that it involves only a finite number of quark data at the relevant instant.

In short, then, we believe our results resolve the apparent conflict between [5] and [15, 16], and support an interpretation of the lack of temporal broadening that does not challenge the ability of the supergravity approximation of AdS/CFT to capture the full dynamics in the strong-coupling limit.

2 Ingredients of the Computation

We are interested in studying the gluonic field sourced by a heavy quark in a strongly-coupled gauge theory. Gauge/gravity duality grants us access to many different setups, and in particular, to conformal field theories (CFTs) in any dimension, but for concreteness we will focus on super-Yang-Mills (SYM) with gauge group . This is a conformally invariant theory with a vector field, 6 real scalars and 4 Weyl fermions, all in the adjoint representation of the gauge group. The AdS/CFT correspondence [2] asserts that this theory, on -dimensional Minkowski spacetime, is fully equivalent to Type IIB string theory on the Poincaré patch of the AdS geometry,

(1)

(with a constant dilaton and units of Ramond-Ramond five-form flux through the five-sphere). The radius of curvature is related to the SYM ’t Hooft coupling through

where denotes the string length. In more detail, the state of IIB string theory described by (1) corresponds to the (symmetry-preserving) vacuum of the gauge theory, and the closed string sector describing small or large fluctuations on top of it encodes the gluonic ( adjoint scalar and fermionic) physics. The coordinates parallel to the AdS boundary are directly identified with the gauge theory spacetime coordinates, the radial direction is mapped to a variable length (or, equivalently, inverse energy) scale in SYM [7], and the five-sphere coordinates are associated with the global internal (R-) symmetry of SYM. These angular coordinates will play no role in our analysis, so our results will hold equally well in the more general case where the is replaced by a different compact five-dimensional space , which corresponds to replacing SYM with a different -dimensional CFT.

The introduction of an open string sector associated with a stack of D7-branes in the geometry (1) is equivalent [27] to the addition, on the gauge theory side, of hypermultiplets (each composed of a Dirac fermion and 2 complex scalars) in the fundamental representation of the gauge group, that we will refer to as ‘quarks’. For , we are allowed to neglect the backreaction of the D7-branes on the geometry;111More precisely, the relevant condition is [28]. in the gauge theory this corresponds to working in a ‘quenched’ approximation that ignores quark loops. The D7-branes cover the four gauge theory directions , and extend along the radial AdS direction up from the boundary at to a position where they ‘end’ (meaning that the that they are wrapped on shrinks down to zero size), whose location is related to the mass of the quarks through

(2)

An isolated quark is dual to an open string that extends radially from the location on the D7-branes to the horizon of the Poincaré patch, . We will describe the dynamics of our string in first-quantized language, and since we take it to be heavy, we are allowed to treat it semiclassically. In gauge theory language, then, we are coupling a first-quantized quark to the gluonic ( other SYM) field(s), and carrying out the full path integral over the strongly-coupled field(s) (the result of which is codified by the AdS spacetime), but treating the path integral over the quark trajectory in a saddle-point approximation.

In the nonperturbative framework provided to us by the AdS/CFT correspondence, a quark with finite mass () is automatically not ‘bare’ but ‘composite’ or ‘dressed’. This can be inferred, for instance, from the expectation value of the gluonic field surrounding a static quark, as in [21] and the calculations we will perform in the next section, or from the deformed nature of the quark’s dispersion relation [18, 19, 20]. The characteristic thickness of the ‘gluonic cloud’ surrounding the quark is given by , which is thus the analog of the Compton wavelength for our non-Abelian source.

Most of the time we will write the quark trajectory in Lorentz-covariant notation parametrized by proper time, . In our analysis below we will have need to refer to the velocity, acceleration, jerk and snap of the quark, which will be denoted as

(3)

or, in four-vector form,

(4)

which by definition satisfy the relations

(5)

By the AdS/CFT dictionary, it is the endpoint of the string on the D7-branes that directly corresponds to the quark, whereas the body of the string codifies the profile of the gluonic field sourced by the quark, which is precisely the information we are after. Specifically, our aim is to probe the gluonic field configuration by determining the one-point function of the operator

(6)

where , , denote the scalar fields of the SYM theory. is essentially the SYM Lagrangian density. As a shorthand, from this point on we will refer to it (including the normalization constant in front) as simply . Since, as mentioned above, we work in the approximation where the quark follows a definite trajectory, its presence amounts to the addition of the corresponding Wilson line, due to which we expect the one-point function of to be nonvanishing. The latter operator is known to be dual to the IIB (s-wave) dilaton field [29, 30], and the standard GKPW recipe for correlation functions [3] at large and relates its one-point function to a variation of the supergravity action with respect to the boundary value of . The connection can be summarized as [5]

(7)

in terms of a rescaled dilaton field .

Our task, then, is to calculate the dilaton field sourced by the string dual to the quark, and to pick out the term in its expansion near the boundary of AdS. In the linearized approximation appropriate at large , is obtained simply by convolving the string source with the retarded dilaton propagator [6]

(8)

which depends only on the invariant AdS distance between the observation (unprimed) and source (primed) point,

(9)

The symbol inside the brackets of (8) denotes the Heaviside step function, which implements causality, as we will elaborate on below.

The string dynamics is prescribed as usual by the Nambu-Goto action

(10)

where () denotes the induced metric on the worldsheet. We work for the time being in the static gauge , , so the string embedding is described by . Given that the string endpoint represents the quark, the latter’s trajectory is read off from the string profile via

(11)

Since we have mentioned that for our fundamental color source is not pointlike, we should be more precise: the AdS/CFT dictionary identifies as the location of the most UV contribution to the gluonic field, and it is this that we adopt as a natural definition for the position of the quark. This definition has proven to be useful to capture the physics of a range of phenomena including Brownian motion [31] and (as we will recall below) radiation damping [19, 20], but, given the extended nature of the quark, is of course not unique (see, e.g., [32]).

Using (8), and taking into account the coupling (in the Einstein frame) between the dilaton and the string worldsheet, one finds the relation [5]

(12)

Together with (7), this will yield the result we are seeking. To put this to use, we are missing only one ingredient: we must know the explicit form of the string embedding.

Of course, just like the specification of a quark trajectory does not uniquely determine a gluonic field configuration, knowing the motion of the string endpoint is not enough to select a unique string profile. Additional information is needed, in the form of initial or boundary conditions. For a quark in vacuum, the configuration of most evident interest is the retarded one, where waves in the gluonic field move out from the quark to infinity, rather than the other way around, or some (nonlinear) superposition thereof. Luckily, the corresponding solution of the Nambu-Goto equation is known, for an arbitrary time-like quark/endpoint trajectory. In the case where the quark is infinitely massive (), this solution can be written as [17]

(13)

where the behavior of the string at a given time and radial depth is seen to be completely determined by the behavior of the quark/string endpoint at the earlier, retarded time .222Reversing the sign of the -dependent terms in (14) yields instead an advanced solution [17]. The form of the most general Nambu-Goto solution for an arbitrary endpoint trajectory is not known. The definition of implicit in (13) can be shown to follow from projecting back to the AdS boundary along a curve that is null on the string worldsheet [17], in analogy with the Lienard-Wiechert story in classical electrodynamics. In Lorentz-covariant language, the solution (13) takes the simple form

(14)

where is the quark proper time corresponding to .333To avoid confusion, we emphasize here that is defined with the Minkowski metric appropriate for the SYM theory, and therefore differs from the proper time of the string endpoint (computed with the AdS metric (1)) by a factor of (where we are currently discussing the case ). From now on we will mostly work with this parametrization of the string embedding, where the natural notion of proper time associated (modulo a rescaling) with the endpoint has been extended to the full worldsheet by following the upward null geodesics. We should stress that is associated with and is therefore still a retarded time parameter, in spite of the fact that, for brevity, we are not labeling it with the subindex .

For the case of a quark with finite mass (and therefore, finite size), , the string endpoint is at , and is subject to the boundary condition (11). As in [18, 19, 20], the associated string embeddings can be regarded as the portions of the solutions (14), which are parametrized by data at the AdS boundary . From this point on we will use tildes to label these (now merely auxiliary) data, and distinguish them from the actual physical quantities (velocity, proper time, etc.) associated with the endpoint/quark at , which will be denoted without tildes. In this notation, (14) reads

(15)

As shown in [19, 20], this can be rewritten purely in terms of physical () data as

(16)

where , with the external four-force that needs to be exerted on the quark to get it to follow the prescribed trajectory.444On the gravity side of the correspondence, this is achieved by subjecting the string endpoint to an electromagnetic field on the D7-branes, and is then the usual Lorentz four-force. Expression (16) clearly reduces to (14) as . As explained in [20], the inclusion of the force at finite is a convenient way of summarizing a dependence of the string embedding on an infinite number of higher derivatives of the quark trajectory, whose appearance is natural, given that the dressed quark is an extended object. The AdS/CFT correspondence makes it possible to deduce that the trajectory and the force are connected through the equation of motion

(17)

which incorporates the effects of radiation damping and is thus a nonlinear generalization of the classic Lorentz-Dirac equation, codifying (inside the parentheses on the left-hand side) a non-standard dispersion relation for the dressed quark, as well as (in the second term on the right-hand side) a Lorentz-covariant formula for its rate of radiation [19, 20].

In the following section, we will use the string embedding (16) in (12) to determine the resulting dilaton profile, and then extract from (7) the desired expectation value of the gluonic field generated by the quark.

3 Gluonic Profile for Arbitrary Quark Motion

We now proceed to the calculation. If we employ the more physical parametrization of the string worldsheet, with the (retarded) quark proper time, the string profile takes the form (16). Computing the induced metric on the worldsheet, one finds that

(18)

We have found it more convenient to carry out our entire computation with the worldsheet parametrized by , with the proper time of the auxiliary (and fictitious) string endpoint at , under which the embedding takes the simpler form (15), implying

(19)

(which naturally coincides with the limit of (18)), and similarly, all subsequent expressions turn out to be more compact. Once we obtain a final result for the expectation value of , we will show how to eliminate all auxiliary (tilde) variables in favor of the physical (nontilde) ones.

Given (19) and (2), to determine the dilaton field (12) we need to compute the double integral

(20)

where the invariant distance (9) with the source point taken on the embedding (15) adopts the form

(21)

Notice how the term of the numerator, present in the definition (9), has exactly canceled due to the -dependence of the retarded solution (15). This cancelation will play a crucial role in the form of our results, and we will return to it in Section 5.

The presence of the -derivative in (20) suggests a change of variables . From (21) we see that

(22)

so we are left with555Upon carrying out the integral to be left with the surface term, one might worry about the fact that the quotient in front of the Heaviside function diverges at the endpoints of the interval that the latter defines. This concern can be dispelled by noting that appears within the derivative, and upon differentiation via the Leibniz rule, would give rise to delta functions that precisely cancel these divergences.

(23)

where we have defined

(24)

It is now necessary to understand the physical import of the factor. Its presence in (8) implies that the propagator has the geometric structure shown in Fig. 2: a given source point induces a nonvanishing dilaton on and inside its forward lightcone, up to the event where this lightcone reflects back from the boundary. Beyond that, there are some observation points that lie in the future of the source point but are not influenced by it. The region corresponds to observation points that are in the causal past of the event or are spacelike related to it. Causality dictates that these points will be beyond the region of influence of the source. Exclusion of points at , on the other hand, is a priori unexpected, because they do obey the naive causality restriction, and are only disallowed due to conditions at the boundary of AdS. These are points that can be reached by a timelike curve originating at , but not by a timelike geodesic.

Figure 2: Penrose diagram of AdS (with boundary directions other than the time omitted), showing a Poincaré coordinate grid. The shaded yellow region indicates the observation points that can be influenced by the given source point .
Figure 3: For a static quark, this is the way in which our new coordinates cover the string worldsheet which we had originally parametrized with . A detailed explanation is given in the paragraph before (25). For ease of visualization, we have chosen the observation point at , (even though we are ultimately interested in ). The causal swath is the region between the 2 dotted (red) hyperbolae on the left (which collapses to a line as ).

In Fig. 3 we show the way in which our new coordinates cover the string worldsheet, which we had originally parametrized with . For concreteness, the coordinate grid has been plotted for the case where the quark is static; for more general motions the overall pattern is similar but somewhat distorted. As mentioned above, the constant (or equivalently, constant ) curves are null; on the plane they are simply straight lines with slope . The range of constant- curves that intersect a given constant- line runs from , where the intersection takes place at , , to , which is asymptotic to the line and only intersects it at . From this it becomes clear that, for any finite and any finite observation point, the point lies outside of the causal swath allowed by the Heaviside function. Consequently, only the endpoint contributes to (23), leaving us with

(25)

To carry out the remaining integral, it is convenient to perform a final change of variables . From (24) we can deduce, via implicit differentiation, that

(26)

but proceeding further requires explicitly inverting (24), which is impossible to do exactly for an arbitrary trajectory. We must remember from (7), however, that we are ultimately only interested in the value of in the limit , up to order . In this limit the causal swath depicted in Fig. 3 in fact collapses to a curve on the plane, but the function inside the brackets in (25) still varies wildly over the allowed range, making it necessary to really carry out the integral instead of just evaluating the integrand at any point within the interval and multiplying times the interval width. A natural strategy is then to Taylor-expand all expressions in powers of . Using (26) in (25), we see that we already have a factor of up front, so we only need to carry out the expansions to order , because terms of order and higher will not contribute to the one-point function in the limit. We start by proposing that

(27)

where are independent of . Using (27) we can deduce that

(28)

and similarly for . With this information in hand, we can Taylor-expand (24) in a power series in , match terms of the same order and solve recursively to find the coefficients . At leading order we get a condition determining ,

(29)

Proceeding to higher order one finds subsequently

(30)

where from now on it is understood that functions denoted with a prime are evaluated at the retarded time defined implicitly by (29).

Using (26)-(30) in (25), one can Taylor expand in powers of , and carry out the integral over to determine . The master formula (7) then leads to the gluonic profile we were after,

The calculation is straightforward but a bit messy, so we defer the details to the Appendix.

Now that we have our desired result, all that remains is to rewrite the auxiliary variables , , , and , which describe the motion of a fictitious endpoint at , in terms of the real physical variables , , , and , associated with the quark/endpoint at . This connection has been worked out in [20]. Equation (25) of that paper states that translates into nontilde variables according to

(32)

where is the rescaled version of the external four-force applied on the quark that we introduced below (16). From this and equation (20) of [20] it also follows that

(33)

which in turn implies, via (15),

(34)

Comparing (18) and (19) we see that

(35)

which enables us to deduce from (32) that

(36)

and

(37)

where dots over the four-force denote derivatives.

Employing (32), (33), (34), (36) and (37), we can express the profile (3) of the gluonic field directly in terms of the physical (nontilde) variables that describe the quark trajectory. This is our main result. We refrain from writing the resulting expression, because it is long and not particularly enlightening.

All dynamical quantities in the final result are understood to be evaluated at the retarded proper time defined by translating (29) into physical variables, i.e.,

(38)

This equation describes a two-sheeted hyperboloid about the observation point , which is intersected by the quark worldline twice, once on each sheet. By causality, the root of interest is of course the one in the sheet to the past of , which, in noncovariant notation, corresponds to the retarded time

(39)

This past hyperboloid describes all the points that can influence the given observation point. We can of course read (38) the other way around, as a statement that the events that can be influenced by a given source point on the quark worldline, lie on the future hyperboloid at constant timelike interval from .

Before moving on to applications and a discussion on the physical content of our result, let us perform a check on it, by examining what it implies for a free quark. In the absence of external forcing, the equation of motion (17) naturally implies that the quark moves at constant velocity (although, interestingly, the converse is not true [20]). From (32)-(37), we have , , , and therefore (3) reduces to

(40)

By Poincaré invariance, it suffices to evaluate this in the frame where the quark is at rest at the origin. Using (39), we then have , so we are left with

(41)

This can be further massaged into the form

(42)

which precisely coincides with the gluonic profile determined in [21] for a static quark with finite mass.

For future use we also note that in the limit , where the quark becomes pointlike (as well as infinitely massive), our general result (3) simplifies drastically, and we are left with the single term

(43)

which according to (38) is to be evaluated at such that . In other words, in this case signals propagate purely along null intervals: the hyperboloid we had for converges to the lightcone, and the retarded time (39) is now at the point of intersection between the quark worldline and the past lightcone of , i.e., .

4 Some Applications

4.1 Harmonic motion

As a first application of our general result (3), we will study the example of a quark undergoing one-dimensional harmonic motion. This is the setup where, under a linearized approximation to the string dynamics, the authors of [5] obtained a propagation pattern with a broad tail, seemingly in conflict with the no-broadening result of [15, 16] and the present paper, but, on the other hand, in consonance with general expectations for a strongly-coupled non-Abelian system [5, 15, 16]. In what follows, we will show in particular that, despite appearances, the result reported in [5] correctly reproduces the linearized limit of our own result.

Let denote the direction of oscillation. The trajectory is then given by

(44)

with corresponding four-velocity

(45)

We will consider first the case (not examined by [5, 15, 16]) where the quark has a finite mass. According to (3), determining the gluonic profile in this case requires, beyond the position, velocity, acceleration and jerk of the quark, knowledge of the total external force applied to it, together with its first and second derivatives, sometimes called yank and tug. The generalized Lorentz-Dirac equation (17) specifies the four-force that corresponds to any given quark trajectory. For one-dimensional harmonic motion, this equation demands that satisfy

(46)

Notice that the nonlinearity of this equation implies that the force is in general not harmonic, as a consequence both of the extended nature of the quark and of the damping effect due to the emitted radiation.

In the textbook analysis of the forced and damped harmonic oscillator, the forcing is prescribed to be harmonic and the oscillator motion is synchronized with this forcing only after the decay of a transient component. Conversely, purely harmonic motion would be associated with a force that is initially not harmonic. Numerical exploration of (46) reveals similar behavior: if we prescribe the motion of the quark to be purely harmonic, as in (44), then for generic initial condition, the force contains a transient component that dies down in a time interval of order . At late times, the force is found to be purely oscillatory, but not quite harmonic. Now, as shown in [19, 20] and recalled in Section 2, the dependence of the string embedding on the external force codifies an infinite number of higher derivatives of the quark’s trajectory, so the initial condition supplied to the numerical integration of (46) at any finite time should in fact be deduced from the behavior of the quark at previous times. If we truly want to explore the situation where the quark has been undergoing harmonic motion at all times, then we are automatically forced to work in the late-time force regime where there is no transient. Equivalently, we must fine-tune the initial condition so that the force is purely oscillatory from the start.

To use equation (3), we also need to solve (38) or its noncovariant equivalent (39) to obtain the retarded time or at which all dynamical data are to be evaluated. For harmonic quark motion with arbitrary oscillation amplitude, this equation is transcendental, and must also be solved numerically. Using the information from both numerical integrations in our general expression (3), we have obtained the gluonic field profile shown in Fig. 4, which shows that the oscillatory motion generates a non-linear wave, with crests splitting off from the gluonic cloud of the quark every half-cycle. As stipulated by (3), the overall pattern decays very fast (), showing no sign of the characteristic radiation falloff. The waves seen in Fig. 4 are thus fluctuations in the near-field of the quark.