Jet quenching in hot strongly coupled gauge theories simplified

Theoretical studies of jet stopping in strongly-coupled QCD-like plasmas have used gauge-gravity duality to find that the maximum stopping distance scales like E^{1/3} for large jet energies E. In recent work studying jets that are created by finite-size sources in the gauge theory, we found an additional scale: the typical (as opposed to maximum) jet stopping distance scales like (EL)^{1/4}, where L is the size of the space-time region where the jet is created. In this paper, we show that the results of our previous, somewhat involved computation in the gravity dual, and the (EL)^{1/4} scale in particular, can be very easily reproduced and understood in terms of the distance that high-energy particles travel in AdS_5-Schwarzschild space before falling into the black brane. We also investigate how stopping distances depend on the conformal dimension of the source operator used to create the jet.


I. INTRODUCTION AND RESULTS
Various authors [1][2][3][4] have made use of gauge-gravity duality to study the stopping distance of massless, high-energy jets in a strongly-coupled plasma of N =4 supersymmetric Yang Mills theory (with and without the addition of fundamental-charge matter). All have found that the furthest that such a jet penetrates through the plasma scales with energy as E 1/3 . Most of these methods specified the initial conditions of the problem in the gravity description of the problem, and it is not completely clear exactly what these initial conditions correspond to in the gauge theory. However, we recently showed [4] one possible way to set up the entire problem directly in the gauge theory, only then translating to the gravity description using the conventional elements of the AdS/CFT dictionary. We specifically studied jets that carried R charge, and we measured how far that charge traveled before stopping and thermalizing. Though we did find that the furthest charge would travel through the plasma scaled as E 1/3 , we also found that, on average, almost all of our jet's charge stopped and thermalized at a shorter distance that scales as (EL) 1/4 , where L is the size of the space-time region where our jet was created. Fig. 1 shows a qualitative picture of our result for, on average, how much of our jet's charge was deposited as a function of distance traveled x 3 . (Our convention here is to write 4-dimensional space-time position as x µ and take our jets to be created near the origin, traveling in the x 3 direction.) Between the (EL) 1/4 scale and the E 1/3 scale, the distribution falls algebraically like (x 3 ) −9 for jets created by the source used in Ref. [4]. We will work in units where 2πT = 1, and in those units the specific formula we derived for Fig. 1 was and where Ψ(y) is a source-dependent function that suppresses |y| ≫ 1, causing suppression of x 3 ≪ (EL) 1/4 above. The c's are constants given by c 1 ≃ 0.927 , and c 2 ≃ 3.2 . The calculation that produced (1.1) was long and not particularly enlightening as to the origin of the (EL) 1/4 scale. The purpose of the current paper is to show how that scale, and then the precise result (1.1a) for the case x 3 ≪ E 1/3 , can be derived from a very simple calculation of how far a classical massless particle travels in AdS 5 -Schwarzschild space before falling into the black brane. In the process, we will learn more about exactly what feature of the source determines the (EL) 1/4 scale. We will see that it is not directly the size but the typical "virtuality" q 2 of the source that matters (where q 2 ≡ q µ η µν q ν is squared 4-momentum).
Our analysis of the distance traveled by falling particles in AdS 5 -Schwarzschild will be essentially the same as an earlier analysis by Gubser et al. [1] and Chesler et al. [3], who used it in a discussion of the falling endpoint of a classical string. The difference here will be one of context and application: Our analysis of jets [4] does not involve classical strings, and we will use the falling particles to explain the (EL) 1/4 scale. In our earlier work [4], we created the jet by turning on a small-amplitude source whose space-time dependence had the form of (i) a high-energy plane wave e ik·x times (ii) a slowly varying envelope function Λ L (x) that localizes the source to within a distance L of the origin in both space and time. We tookk to be light-like:k µ = (E, 0, 0, E). (1.4) In addition, for the sake of simplicity, we took the source to be translation invariant in the two transverse directions. So, for example, Λ L (x) = e − 1 2 (x 0 /L) 2 e − 1 2 (x 3 /L) 2 . (1.5) The Fourier transform of the source (1.3) is non-negligible in the region of momentum space depicted in Fig. 2a: a region centered onk with width of order L −1 . We take L −1 ≪ E. Note that this source covers a range of values of q 2 , from 0 to order ±E/L, and the typical size of |q 2 | is order E/L. In the gravity description, this source causes a localized perturbation on the boundary of AdS 5 -Schwarzschild space-time, which then propagates as a wave into the fifth dimension, eventually falling into the black brane horizon. The analysis of jet stopping in Refs. [2,4] was based on the analysis of such 5-dimensional waves. Now imagine instead a source wherek is slightly off the light-cone, with ǫ ≪ E, and where the envelope size L is wide enough that the picture in momentum space looks like Fig. 2b instead of Fig. 2a, with the spread 1/L in momenta small compared to ǫ. In this case, the q 2 of the source is approximately well defined, with q 2 ≃k 2 ≃ −4Eǫ. We will show that in this case the wave created by the boundary perturbation is localized into a small wave packet, whose motion may be approximated by that of a Qualitative picture of momenta contributing to the source (1.3) used to generate jets (a) for the calculation originally used to find (1.1), with L −1 ≪ E, and (b) in the case L −1 ≪ ǫ ≪ E of (1.6). Figure (c) depicts (a) as a superposition of distributions of type (b). The cells in (c) that are extremely close to the light cone cannot be treated in particle approximation, but the contribution of cells that can be treated so dominates when L ≪ the maximum stopping distance scale E 1/3 . classical, massless particle which starts at the boundary, traveling in the x 3 direction, with 4-momentum proportional to q. The trajectory of such a particle is shown qualitatively in Fig. 3a. By a simple calculation, we will find that the particle falls into the horizon after covering a distance where the constant c is given by (1.2). As measured by boundary time x 0 , the particle takes an infinite amount of time to fall into the horizon. As it gets closer and closer to the horizon, the boundary distortion that the particle creates (see Fig. 3b) becomes weaker and more spread out, which corresponds to charge diffusion in the boundary theory after the jet stops and thermalizes. This qualitative picture is similar to the qualitative picture of the effects of a classical string falling into the horizon given in Refs. [1,6]. 1 Note that the stopping distance (1.7) only makes sense for q 2 < 0 (i.e. ǫ > 0). The q 2 >0 components of a source do not create an excitation of the system that persists after the source turns off and so are not relevant [2,4]. Now consider the original source of Fig. 2a as a superposition of sources like Fig. 2b, as depicted in Fig. 2c. Since sources with different values of ǫ have different stopping distances (1.7), we might guess that the different pieces of this superposition do not interfere and so the source of Fig. 2a simply produces a distribution of stopping distances, weighted by independent probabilities that the source produces a jet with a particular q 2 . That is, where P(q 2 ) is the probability density for the source to produce a jet with a given q 2 , and 1 See in particular the discussion surrounding Fig. 2 of Ref. [6], which inspired our Fig. 3b. See also Ref.
[5]. The presence of the particle (the large dot) perturbs the boundary theory in a manner that spreads out diffusively as the particle approaches the horizon for x 0 → ∞.
where the stopping distance x 3 stop (q 2 ) is given by (1.7). We will verify that this formula precisely reproduces the x 3 ≪ E 1/3 case (1.1a) of our previous result.
We can now see where the (EL) 1/4 scale comes from. It is only time-like source momenta q 2 < 0 that produce jets. The typical value of time-like q 2 for the source of Fig. 2a is Putting this into (1.7), the typical stopping distance in this case is therefore Note that it is the q 2 of the source that determines the stopping distance, and that the typical value of q 2 is determined by L in the case of Fig. 2a.
The estimate (1.9) of the stopping distance ceases to make sense if the size L of the source becomes as large as the stopping distance itself. This happens when which gives We will see later that this is precisely the case where the wave packet in AdS 5 -Schwarzschild can no longer be approximated as a particle. The moral is that the simple particle picture gives us not only the (EL) 1/4 scale but also, simply by estimating where it breaks down, the E 1/3 scale as well.
In the next section, we will briefly review the trajectories of massless particles in AdS 5 -Schwarzschild and derive the corresponding stopping distance (1.7). In section III, we discuss the conditions for being able to approximate the 5-dimensional wave problem with particle trajectories and verify that they apply in the case of interest. Then we use the particle picture in section IV to simply reproduce our original result (1.1a) for charge deposition for x 3 ≪ E 1/3 . In section V, we generalize our results to jets created by other types of source operators than those originally considered in Ref. [4]. We will see that Fig. 1 is modified to Fig. 4. Finally, we offer our conclusions in section VI. max ∼ (E/∆) 1/3 indicated along the vertical axis assume that ∆ is held fixed when taking the limit of large energy E (as well as large coupling g 2 N c and large N c ). The parametric scaling with ∆ indicated for x 3 typical assumes a Gaussian source envelope (1.5), but the other features shown in the figure are independent of the details of the source envelope.
As an aside, some readers may be curious how the stopping distance scales (EL) 1/4 and E 1/3 generalize to other dimensions. On the gravity side, it is very easy to generalize the results of this paper to different space-time dimensions d of the boundary, but it is less certain what strongly coupled field theories these classical gravity theories correspond to. (See Refs. [7] for proposals.) Ignoring the question of interpretation, we show in Appendix A that (EL) 1/4 and E 1/3 generalize to (EL) (d−2)/2d and E (d−2)/(d+2) respectively for d > 2.
In this paper, we will use the convention that Greek indices run over the 4 space-time dimensions (µ = 0, 1, 2, 3) of the boundary theory and capital roman indices run over all five dimensions (I = 0, 1, 2, 3, 5) of the AdS 5 -Schwarzschild space-time. The symbol q 2 will refer to the squared 4-momentum, q 2 ≡ q µ η µν q ν = −ω 2 + |q| 2 . When we use light-cone coordinates, our conventions will be for any 4-vector V . Throughout this paper. the adjective "transverse" will refer to the spatial directions 1 and 2 orthogonal to q. 2

II. REVIEW OF FALLING MASSLESS PARTICLES
Null geodesics in a 5-dimensional space with 4-dimensional translation invariance are given by (see Appendix B): where g is the 5-dimensional metric and q I is a constant of motion for I = 0, 1, 2, 3. We will work in coordinates where the metric is 3 where f ≡ 1 − u 2 , and R is the AdS 5 radius. The boundary is at u=0 and the horizon at u=1. If we take the 3-momentum q to point in the x 3 direction, writing q µ = (−ω, 0, 0, |q|), then (2.1) gives the total distance x 3 traveled in falling from the boundary to the horizon to be where q 2 ≡ q µ η µν q ν is the flat-space square of the 4-momentum. This is the same result as Refs. [1,3]. 4 Now let us apply this result to the case |q 2 | ≪ |q| 2 ≃ E 2 relevant to the source of Fig. 2b. For small −q 2 , the integral of (2.3) is dominated by small u, and so we may approximate which gives (1.7).

III. WAVE PACKETS AND GEOMETRIC OPTICS
In this section, we will discuss the conditions necessary for making the particle approximation. A wave packet behaves like a particle when it is wide enough to contain many phase oscillations of the field yet small enough that the properties of the background do not vary significantly across its width, as depicted in Fig. 5b at a particular moment in time.
We can arrange such a width provided the background properties do not vary significantly over one wavelength of the phase oscillation. This is the geometric optics limit, which we referred to in our earlier work [4] as a WKB approximation. To check the geometric optics limit, one may focus as in Fig. 5c on a wave with a single, generic value of q typical of the wave packet, and investigate how much things change over one phase oscillation.
To assess whether a wave packet is adequately particle-like to use a particle-based calculation of the stopping distance, it will be helpful to understand what the important scale for u is in determining the stopping distance (2.4). The integral in (2.4) is dominated by u of order Our formulas in this paper would be a little tidier (fewer square roots) if we worked with the coordinate z ≡ 2 √ u instead of u. We will stick with u in order to facilitate comparison with our previous work [4]. 4 Our (2.4) corresponds to the first part of Eq. (5.3) of Gubser et al. [1], where their y is our √ u, their z H is 2 in our units 2πT =1, their p 1 /p 0 is replaced by the q 3 /q 0 of the momentum q µ typical of our source, and their y UV is set to zero. It also corresponds to Eq. (4.28) of Chesler et al. [3], where their u is our 2 √ u, their u h = 2 in our units, their ξ is replaced by our q 0 /q 1 , and their u * is set to zero.
FIG. 5: (a,b) A snapshot in time x 0 of waves in the fifth dimension u for times after the boundary source has turned off but early enough that u u ⋆ ≪ 1 (that is, before the wave gets very close to the horizon). (a) shows the type of wave generated by a localized source that superposes a range of q 2 values such as The relevant question is then whether the background varies significantly across one phase oscillation for u ∼ u ⋆ . By (1.7), the distances x 3 stop E 1/3 relevant to jet stopping correspond to ǫ E −1/3 in the particle picture and so to u ⋆ E −2/3 ≪ 1. So we may focus on small u in what follows.

A. Geometric Optics
For a massless 5-dimensional field with definite 4-momentum q µ , the exponential in the WKB approximation is where the q µ are constant and q 5 (x 5 ) is determined by the 5-dimensional massless condition q I g IJ q J = 0, giving 5 For the metric (2.2), this is For the geometric optics limit, we need the wavelength to vary insignificantly over one wavelength. 6 For the important values u ∼ u ⋆ ≪ 1 of u, the wavelength λ(u) ∼ 1/q 5 (u) in the fifth dimension satisfies this condition if Using the size (3.1) of u ⋆ , this condition is (−q 2 ) 3 /|q| 2 ≫ 1, or equivalently Referring to the stopping distance (1.7), we then see that the approximation of geometric optics, necessary for a particle interpretation, breaks down unless Within the context of our approach to jet stopping, a proper analysis of what happens at distances E 1/3 requires a wave rather than particle description of the problem, as in Refs. [2,4]. The wave analysis gives exponential fall-off for propagation beyond E 1/3 , as described by (1.1b).
More on the geometric optics approximation can be found in Appendix C.

B. Wave Packets
The geometric optics limit for (3.8) allows us to create localized wave packets. Here we will see how wide those wave packets are in u for sources of the form of Fig. 2b. We are primarily interested in the case where the center of the wave packet is at the critical scale u ∼ u ⋆ in the fifth dimension. However, the presentation will be a little more straightforward if we first make parametric estimates for earlier times, when the center of the wave packet is at u ≪ u ⋆ , and then extrapolate those parametric estimates to u ∼ u ⋆ .
For u ≪ u ⋆ , the distance traveled (2.1) for a massless particle is Turning this around, the location of the particle in the fifth dimension is (3.10) When the particle is replaced by a wave packet, there are two sources of uncertainty. The size L of the source introduces an uncertainty in the initial position of the excitation of ∆x 3 ∼ L. It also introduces an uncertainty in the 4-momentum q, and so in ǫ, of ∆ǫ ∼ 1/L as in Fig. 2b. From (3.10), the combined uncertainty ∆u in u is then of order Extrapolating this parametric estimate to the case u ∼ u ⋆ of interest, (3.1) and (3.11) give ∆u The wave packet will then be localized provided (i) L ≫ 1/ǫ, as in Fig. 2b, and (ii) L ≪ (E/ǫ) 1/4 . By (1.7), the last condition is just the condition that L be much less than the stopping distance x 3 stop .

IV. REPRODUCING THE DISTRIBUTION OF STOPPING DISTANCES
Now we will show that the formula (1.1a) found in our earlier work [4] for the average distribution of charge deposition can be understood as a convolution (1.8) of the particle stopping distance with the probability density P(q 2 ) for the source to create a jet with a given q 2 .
We will later give in section V a very general argument, based on dimensional analysis, for determining the P(q 2 ) associated with different choices of source operator. This argument will also require a discussion of massive fields and massive particles in the gravity dual. For the moment, we will be less general, stick to the specific type of source operator that we used in previous work, and show how to extract P(q 2 ) from a result for the average total charge produced by the operator. Readers who would prefer to just see the more general argument may skip section IV A below and instead wait for section V B.
A. Extracting P(q 2 ) from results in the literature In Ref. [4], we used a source involving R-current operators j a µ . Specifically, we modified the 4-dimensional gauge theory Lagrangian by with a localized background field where N A is an arbitrarily small source amplitude,ε is a transverse linear polarization, and τ i are Pauli matrices for any SU(2) subgroup of the SU(4) R-symmetry. We then measured the response j (3)µ (x) of the R charge current associated with τ 3 /2. The gravity dual to the R charge current operators is a massless 5-dimensional SU(4) gauge field. We chose our source (4.2) to be translationally invariant in the transverse spatial directions (x 1 , x 2 ) to simplify the calculation. In what follows, we will refer to the R charge associated with τ 3 /2 as simply "the charge." For an arbitrarily small source amplitude N A , the source will usually have no effect at all on the system. On rare occasions, with probability proportional to N 2 A , the source creates an excitation (in our case a "jet") with the same quantum numbers as the source operator. In our case (4.2), that means it creates a jet with total charge equal to 1. The creation of an excitation with different quantum numbers would be even higher-order in N A and so negligible. Since an excitation (if any is created) has charge 1, the average total charge Q created by the source is then equal to the probability of the source creating a jet. We will see that from the previously calculated result for Q we can then extract the probability density P(q 2 ).
In Ref. [4], 8 we showed how to use the field theory Ward identity to make a simple calculation of the average charge Q. Here we will just quote the result, which was where g 2 SG = 4π/N c (N c →∞ is the number of colors) and θ is the step function. Thẽ Our source only has support for q =k + ∆q with ∆q small compared tok, in which case q 2 ≃ 4Eq + . Eq. (4.3) may then be approximated as of the source envelope and V ⊥ is the area of transverse space (x 1 and x 2 ). Since (4.4) involves an integral over q + , and since Q is the total probability of creating a jet, it is natural to interpret (4.4) as giving a probability density for producing a jet with a given value of q + . If a jet is produced, the probability distribution for its q + is then the relative probability Here we've put a bar over the argument of P + just to distinguish it from the q + integration variable in the denominator on the right-hand side. Now package the source dependence into the definition This is the definition we made in our earlier work [4] for the Ψ that appears in the charge deposition result (1.1). With this definition, Since q 2 ≃ 4Eq + , the probability distribution P(q 2 ) for q 2 is related to the probability distribution P + (q + ) of (4.9) for q + by However, at this point it will be easier to just stick with q + and P + . Note that in each cell of Fig. 2c, the typical value of q + is just what we have previously called −ǫ for that cell. The distribution (1.8) of stopping distances based upon the picture of massless falling particles in 5 dimensions can then be written as As promised, in the case x 3 ≪ E 1/3 where we have argued that the particle picture should work, this exactly reproduces our earlier result (1.1a) that came from a full, much more complicated calculation.

V. MASSIVE PARTICLES IN 5 DIMENSIONS
In the preceding sections, we have assumed that the 5-dimensional bulk field dual to the source which creates the jet is massless, such as the 5-dimensional gauge field dual to R current operators. One may wonder what results change if we choose different types of source operators that are instead dual to massive bulk fields. In this section, we will see that the basic qualitative picture of Fig. 1 of the distribution of stopping distances remains the same, except that the exponent of the (x 3 ) −9 power-law tail changes, depending on the conformal dimension of the source operator.
When we wish to make contact with a particular example, we will for simplicity restrict attention to scalar BPS (e.g. chiral primary [8]) operators. 9 In that case, the mass m of the 5-dimensional field is related to the scaling dimension ∆ of the operator by [9] (Rm) 2 where R is the radius of AdS 5 , and d=4 is the dimension of ordinary space-time. The possible values of ∆ are bounded below by Examples of scalar BPS operators include the Lagrangian density and the symmetrized trace tr(φ (i1 φ i2 · · · φ in) ), where φ 1 , φ 2 , and φ 3 are the three complex scalar fields of N =4 supersymmetric Yang Mills.
In what follows, we will hold m fixed when we consider the limit of large jet energy E. However, it will be interesting to consider the case ∆ ≫ 1 (i.e. Rm ≫ 1) in addition to the case where ∆ is of order one. We will find that the typical and maximum stopping distances decrease for larger ∆.
We will study the propagation of excitations of a massive bulk field by studying the propagation of massive particles in the bulk, similar to the massless case studied earlier in this paper. (An alternative discussion directly in terms of a wave analysis is sketched in Appendix D.) We should emphasize that the term "massive" refers only to the bulk fields and corresponding bulk "particles" in our discussion, and so to the conformal dimension of the source operators in the boundary theory. We have not introduced any masses in the 4-dimensional strongly-coupled field theory: the theory is still just N =4 supersymmetric Yang Mills theory.
A. Stopping distance of massive particles For a particle of mass m in 5 dimensions, the stopping distance integrals (2.1) and (2.3) are modified to As in the massless case, we shall see below that the stopping distance will be dominated by u ∼ u ⋆ ≪ 1. So we will be able to approximate f ≃ 1 above: For ∆ > d (in which case m 2 is positive), there is an issue with the lower limit of integration in (5.4): our classical particle with 4-momentum q µ cannot exist in the region where the square root in (5.4) is imaginary. For u ≪ u * (and focusing on q 2 < 0), this condition allows for a classical particle when u ≥ u min with How to interpret this? The wave equation is not well described by geometric optics near the turning point u min . However, as long as u min ≪ u ⋆ , the calculation of the stopping distance will be dominated by much larger u (where m is ignorable), and so we may still use the particle picture to approximate which is the same as the massless particle result (2.4). In this respect, the mass can be ignored.
What happens at u ∼ u min can be made more concrete by returning to the wave problem and looking at the solution to the massive scalar wave equation in the limit u ≪ u ⋆ ≪ 1. In this limit, one is close enough to the boundary that AdS 5 -Schwarzschild is approximately just AdS 5 , and the equation becomes the zero-temperature wave equation of a massive scalar in AdS 5 . The retarded solution to this equation is where H (1) ν is the Hankel function, d=4 is the space-time dimension of the boundary theory, N q is an overall normalization, and The solution behaves like Φ ≃ N q z d−∆ (5.10) in the boundary limit z→0, where z ≡ 2 √ u. The divergence of (5.10) as z → 0 for ∆ > d (i.e. m 2 > 0) reflects the renormalization required of the corresponding operators in the 4-dimensional gauge theory. In our discussion, we will be able to ignore the details of holographic renormalization prescriptions and simply summarize that (5.8) should approach z d−∆ φ b (q) as z approaches the (regulated) boundary, where φ b is the (renormalized) boundary source. Choosing φ b (q) = 1 defines the bulk-to-boundary propagator, which corresponds to (5.8) with N q ≃ 1. 10 The Hankel function goes through many oscillations, and so is well approximated by the geometric optics limit, when its argument is large compared to both 1 and ν. In our case, this condition is parametrically equivalent to which may be also be written as This generalizes the condition u ≫ 1/(−q 2 ) previously discussed for the massless case. If we convolve (5.8) with a high-energy source (1.3), we will not be able to use the particle 10 In more detail, follow Ref. [10] and normalize the bulk-to-boundary propagator to be z d−∆

B
at z=z B , where z B is arbitrarily small. Then and one takes z B → 0 at the very end of the calculation. In yet more detail, a systematic method for holographic renormalization is described in Refs. [11][12][13].
approximation to figure out the details of what is happening at u ∼ u min , but we will be able to use it when the resulting wave packet propagates to u ∼ u ⋆ provided u * ≫ u match , and so we may then use the particle approximation to calculate the stopping distance. Because we get the same stopping distance as for the massless case, we can take over (1.7): The geometric optics approximation at u ∼ u ⋆ (and so this result for the stopping distance) will fail unless u match ≪ u ⋆ . Using (3.1) and (5.11), that condition requires (5.14) One might suspect that the particle approximation breaks down at the maximum possible stopping distance, in which case (5.13) then gives that maximum to be We give a more detailed argument for this result in Appendix D.
Eq. (5.15) implies that the maximum stopping distance decreases as the conformal dimension ∆ of the BPS source operator is increased. This qualitative feature is not novel to the strongly-coupled theory: it is true for the weakly-coupled theory as well. For the BPS operators, large ∆ corresponds to an operator with roughly ∆ powers of scalar fields, such as tr(φ ∆ ), where φ is one of the three complex scalar fields in the theory. In weak coupling, if we inject total energy E with such an operator, it will typically generate ∆ particles that each have energy of order E/∆. In weak coupling, the stopping distance of a particle with energy E scales as E 1/2 (up to logarithms), and so the stopping distance of the ∆ particles each with energy E/∆ will scale as x 3 max ∼ (E/∆) 1/2 .

B. Generalizing the power-law tail
In this section, we investigate how the (x 3 ) −9 power-law tail in Fig. 1 generalizes to other choices of source operator. We will take the source term in the gauge-theory action to be of the form where O(x) is a scalar BPS operator with dimension ∆. We will see, however, that our result also applies to the case of O being a transverse-polarized R current, which was the case discussed in section IV. Since the dependence (5.13) of the stopping distance on q 2 is the same as in the massless case, the only significant qualitative difference in the distribution of stopping distances will come from the distribution P(q 2 ) of q 2 created by the source operator. The shape of this distribution is determined by the dimension ∆ of the source operator, as we now describe. As discussed in Refs. [2,4], temperature does not affect the initial creation of the jet, and so we can simplify the analysis by evaluating P(q 2 ) at zero temperature. Consider the probability density in q associated with a source operator O acting on the vacuum: (5.17) where G > is the Wightman correlator of O. At zero temperature, it is related to the spectral density ρ of the operator by 11 iG > (q) = θ(q 0 ) ρ(q).
The distribution of jet 4-momenta is therefore given by the spectral density ρ(q). At zero temperature, Lorentz and scaling invariance allow us to use simple dimensional analysis to know how ρ scales with q: 12 where ν = ∆ − 1 2 d as in (5.9). The θ(−q 2 ) appears because only sources with time-like q 2 produce persistent excitations at zero temperature.
If O were a vector operator V µ , like an R current, the dimensional analysis would be complicated by the fact that one could get factors of q µ associated with the vector index (rather than only factors of the virtuality q 2 ). However the transverse spatial components q ⊥ of q µ vanish, by definition. So this complication does not arise for the transverse R current operator j ⊥ that we discussed earlier, and (5.19) can also be used in that case.
So far, we have only looked at the operator O and not the other factors in the source term (5.16). We can rewrite (5.16) in q space as qΛ * L (q −k) O(q). (5.20) Correspondingly attaching a factor of Λ * L (q −k) to each O(q) in (5.17), we get a probability distribution for q proportional to For a transverse-translation invariant source, the relative probability distribution for creating a jet with a given q + , where q 2 ≃ 4Eq + , is then which generalizes (4.7). We will repackage this as has been normalized so that Ψ ν (0) = 1 in the case of a Gaussian source envelope (1.5). Following (4.11), the distribution of stopping distances is then So the power-law tail in Fig. 1 has generalized to (x 3 ) −(5+4ν) = (x 3 ) 3−4∆ , as shown in Fig. 4.
For the transverse R current operator, ∆ = 3, which recovers our previous result of (x 3 ) −9 in that case.

C. Gaussian Source Envelope
Throughout this paper, we have discussed two different scales x 3 typical ∼ (EL) 1/4 and x 3 max ∼ E 1/3 characterizing the stopping distance. In (5.15), we generalized the latter to x 3 max ∼ (E/∆) 1/3 for the case of large ∆. Now we will discuss the similar generalization of x 3 typical . In general, we will still have but the relation between the typical q + of jets and the source envelope size L for large ∆ will depend on details of how the source envelope∆ L (q) falls off for large q + . That's because the probability distribution (5.22) for the q + of the jet involves a competition between the |q + | ν factor which favors large |q + | and the |Λ L (q + , ∆q − )| 2 factor which suppresses |q + | ≫ L −1 . The typical value of q + represents a balance between the two and will scale with ν. For the sake of a concrete example, we consider here the case of a Gaussian source envelope (1.5).
In this case, the function Ψ ν (q + L) in (5.23) and (5.25) is simply The typical values of q + from the probability distribution (5.23) then scale as  13 Alternatively, one could compute the average value of x 3 stop directly from (5.25), giving

VI. CONCLUSION
The 5-dimensional particle picture provides a relatively easy way of understanding (from the gravity side of the calculation) the appearance of the scale (EL) 1/4 in jets created by finite-size sources in strongly coupled N =4 super Yang Mills plasmas. By thinking about sources with different types of momentum distributions, such as Figs. 2a and b, we have learned that it is the range of q 2 of the source which determines the range of stopping distances. Making |q 2 | larger causes the jet to stop sooner. For any finite size L of source, the uncertainty principle implies that there will be a spread in the components of q of at least order 1/L and so a spread in q 2 of at least order E/L. As a result, almost all of the jets produced will travel distances (EL) 1/4 [of order (EL) 1/4 in the case of Fig. 2a and ≪ (EL) 1/4 in the case of Fig. 2b]. Events where a jet travels further (up to E 1/3 ) will always be rare if the source size L is small compared to the maximum stopping distance scale E 1/3 . This interpretation, based on the 5-dimensional particle picture, provides an important clarification to our original derivation of the average distribution of charge deposition shown in Fig. 1. This average includes an average over all events. From the original result, it was unclear whether on not Fig. 1 qualitatively tracks how the jet deposits its energy, momentum, and so forth on an event-by-event basis. It might have been that every single jet produced deposits some of its energy at x 3 ∼ E 1/3 and most of its energy at x 3 ∼ (EL) 1/4 . The success of the 5-dimensional particle interpretation, and in particular the success of (1.8), indicates that Fig. 1 instead reflects a probability distribution for how far the jet travels, and each individual jet dumps its energy and charge in a very localized region of x 3 . It should be possible to independently verify this conclusion by calculating correlations of the charge deposition at different distances, which we will leave to future work.
Our analysis of massive 5-dimensional fields indicated that the maximum stopping distance E 1/3 depends on the type of operator used to create the jet and that the distance decreases as the conformal dimensions of that operator increases. That is, the maximum stopping distance depends on the type of high-energy excitation created. This may shed some light on a discrepancy between (i) the stopping distances found here and in Refs. [2,4], which find x 3 max ∼ E 1/3 , and (ii) those based on the evolution of classical strings in 5 dimensions [1,3], which find the parametrically smaller result x 3 max ∼ (E/ √ λ) 1/3 , where λ ≡ N c g 2 is the large 't Hooft coupling. The gauge theory states corresponding to classical strings in the gravity dual may simply be states that are much more easily stopped by the strongly-coupled quark-gluon plasma than are the states created by the source operators considered in this paper. One could then ponder which (if either) might be more instructive for lessons about the theory of real QCD plasmas. In order to further clarify the differences between the two approaches, it would be interesting to find a 4-dimensional gauge-theory description of a source that could be precisely linked through duality to the 5-dimensional initial classical string configurations that have been used to study jet quenching.
where w = −g (4) g ⊥⊥ (C5) and g (4) = g 55 g is the determinant of the 4-dimensional part of the 5-dimensional metric. (As another simple example, one could consider the case of a source operator dual to a massless scalar field, which would corresponds instead to taking w = √ −g (4) in the equation This looks just like a one-dimensional quantum mechanics problem in ℓ with wavenumber k(ℓ) ≡ 2m(E − V (ℓ)) replaced by The WKB condition that the wavelength λ(ℓ) = 2π/k(ℓ) in such a quantum mechanics problem not change significantly over distances of one wavelength is ∂ ℓ λ(ℓ) ≪ 1.

Appendix D: Maximum Stopping Distance for High-Dimension Source Operators
In section V A, we saw that the particle picture breaks down when x 3 x 3 max with and we suggested that this x 3 max was the furthest that jets would propagate-that is, that energy or charge deposition at larger distances would be exponentially suppressed, similar to our previous ∆=3 result of (1.1b). In this appendix, we will discuss how (D1) arises in a wave analysis following the methods of Ref. [4]. We will focus on the case of ∆ ≫ 1. As in the main text, we assume that the large E limit is taken first, and only then do we consider large ∆.
There is a subtlety to the results we will find. In section D 2, we will analyze the exponential fall-off of jet charge deposition at very large x by finding the location in the complex q + plane corresponding to the first quasi-normal mode of the bulk field. For large ∆, we will find an exponential fall-off of the form where c ′ 1 is a constant. This is the large-∆ version of the exponential in (1.1b). The x 3 scale that determines the rate of exponential fall-off in (D2) is E 1/3 /∆ 4/3 . One might naturally guess that exponential suppression therefore applies whenever x 3 ≫ E 1/3 /∆ 4/3 and so guess that x 3 max ∼ E 1/3 /∆ 4/3 instead of (D1). This guess fails, however, for reasons we shall now outline.

Overview
To understand the issues involved, we briefly highlight some relevant aspects of the m=0 calculation from Ref. [4]. The main part of the calculation there involved computing the bulk response A to the high-energy source on the boundary, given by where G is the bulk-to-boundary propagator. Our result for the stopping distance came from extracting the behavior of A near the horizon, u → 1. 14 The critical part of the q integration was the integral over q + . For x 3 ≪ E 1/3 (in the m=0 case), we found that we could deform the q + integration contour in the complex plane so that the integral was everywhere exponentially suppressed except at a saddle point 15 of (3.2). This contour is depicted in Fig. 6. Parametrically far into the interior of the shaded region indicates places where the magnitude of the integrand is exponentially suppressed. Parametrically far into the unshaded regions indicates places where it is exponentially large. The dashed line depicts a line of poles of the bulk-to-boundary propagator G R , corresponding to quasi-normal modes. In the WKB approximation to that propagator, this line of poles became a cut.
14 Readers may wonder at the connection between (i) studying u → 1 and (ii) studying u ∼ u ⋆ ≪ 1 as in the particle arguments in the main text of this paper. The point is that how far the particle travels is determined by where it is when u → 1, but the integral which gives that distance is dominated by u ∼ u ⋆ ≪ 1. In the wave analysis of Ref. [4], we studied the response at late times, corresponding to u→1 for the bulk excitation. Our results for the near-horizon bulk response were determined by the E 1/4 (−q + ) 3/4 term in the WKB exponent S for G (see eq. (4.51) of Ref. [4]). But this term was generated by the u∼u ⋆ region of the integral that gave S. See, for example, eq. (D13) of Ref. [4], which is proportional to the current paper's particle stopping distance integral (2.3). 15 In Ref. [4], we expressed formulas in terms of X + ≡ x + − τ (u) instead of x 3 . As discussed in that paper, the late-time response is localized to x − ≃ −τ (u) (see eq. (4.49) of Ref. [4]), and so X + ≃ 2x 3 . . This is a slightly simplified version, appropriate for u→1, of Fig. 13b of Ref. [4]. The location of the saddle point q ⋆ + is marked by the large dot. The circles indicate different parametric scales for |q + |. For x 3 ≫ E 1/3 , the regions of exponential suppression for the integrand are shown in Fig. 7a. Saddle point methods are unreliable. Instead, choose the integration contour shown there. The piece that goes around the line of poles can be re-expressed as a sum of contributions from each pole, as shown in Fig. 7b, which depicts a magnification of the neighborhood of the origin of Fig. 7a. The e iq + ·x + piece of the e iq·x in (D3) causes the contributions from the poles to be exponentially suppressed according to their distance Im q + from the real axis. For x 3 ≫ E 1/3 , the nearest pole dominates and produces the exponential fall-off (1.1b) of the jet's charge deposition. Now we return to the massive case, with ∆ ≫ 1, and ask what happens for This is the interesting case, where (i) the geometric optics and particle arguments of section V indicate that the charge deposition is not exponentially suppressed but (ii) the guess we might make based on (D2) suggested that it is suppressed. As we shall discuss in section D 3, the mass does not significantly affect the massless picture of Fig. 6 provided which is the condition (5.14) discussed in the main text. For x 3 ≪ E 1/3 /∆ 1/3 as in (D5), the condition (D6) is satisfied at the saddle point (D4) and for larger |q + |. So, as long as we are careful to route the contour as in Fig. 8, we can take over the methods of the m=0 calculation, make a saddle point approximation to the q + integral (which corresponds to making the geometric optics approximation), and so find a result that is not exponentially suppressed. But now consider a large x 3 calculation, along the lines of Fig. 7. We shall see in section D 2 that the closest pole to the real axis has Im q pole That is, the line of poles in Fig. 8 begins at the edge of the hatched circle. For x 3 in the range of (D5), picking up the poles is not so useful. Fig. 9 shows the large-∆ analog of Fig. 7 for this x 3 range. As one looks at poles progressively further from the origin, the exp(iq · x) factors leads to suppression, as before, but the G R factor grows exponentially (as well as oscillates), and this exponential growth dominates the integrand in the unshaded region of Fig. 9. So one cannot approximate the integral by the contribution from the nearest pole, and asymptotic formulas like (D2) do not apply in this case.  Since the saddle point approximation is controlled and gives a result that is not exponentially suppressed in the range (D5), whereas the approximations that lead to (D2) are not valid there, we conclude that the maximum stopping distance is of order E 1/3 /∆ 1/3 and not E 1/3 /∆ 4/3 .
What happens as one continues to increase x 3 ? For x 3 ≫ E 1/3 /∆ 1/3 , the saddle point approximation breaks down and Fig. 9 becomes Fig. 10

Quasi-normal modes
To find the poles of G R in the complex q + plane for large E, we follow the method of Ref. [4]. For the sake of simplicity, we will focus on the case of a massive bulk scalar field, whose equation of motion is (5.7). As discussed in Ref. [4], the pole positions at high energy are determined (up to small corrections) by the nature of the equation of motion for u ≪ 1. It will be more convenient to work with the variable z = 2 √ u instead of u. Writing and a defined in terms of q + as in Ref. [4]: Solving (D16) with the desired boundary conditions is equivalent to setting a to be twice the bound-state energies associated with the potential V(z). For M ≫ 1, these can be well approximated by treating V(z) in harmonic-oscillator approximation around its minimum. The result is 1 2 a n = 3 8 M 4/3 + (n + 1 2 for n = 0, 1, 2, · · · . 18 Using (D18), we find that the first pole in q + is a distance of order ∆ 4/3 /E 1/3 from the real axis, but the spacing between successive poles in Fig. 10b is only of order ∆ 1/3 /E 1/3 . The specific result for a 0 determines for the exponential fall-off (D2) in the case ∆ ≫ 1.

Saddle point analysis
In section D 1, we claimed that a large mass m for the bulk field does not qualitatively change the massless saddle-point picture of Fig. 6 except inside the (avoidable) hatched region of Fig. 8. Here, we will briefly sketch why. For Rm ≫ 1, the condition for the validity of the WKB approximation 19 is satisfied in the small u → 0 regime u ≪ u min as well as in the oscillatory regime u ≫ u min . (The turning point u ∼ u min can be avoided simply by analytically continuing around it, as in the textbook discussion of WKB in Ref. [15]). So we may use WKB all the way to the boundary u = u B : where, for the sake of simplicity of presentation, we will suppress showing the WKB prefactor. For the massive case, The integrand in (D3) then has exponential dependence e iq·x G R ∝ e iS ≡ e i(q·x+S) , and the saddle point of its integral is determined by which gives Together with (D22), this simply reproduces the particle-based formula (5.3) for the stopping distance. Approximating q − ≃ E and solving for q + in terms of x 3 will then give the saddle point q ⋆ + for the q + integration in (D3). We've already discussed the effect of the mass on the particle stopping formula (5.3) back in section V A. Tracing the discussion of section V A backward, x 3 ≪ (E/∆) 1/3 corresponds to −q + given by ǫ ≫ (E/∆ 4 ) −1/3 , which corresponds in turn to u min ≪ u ⋆ . That's precisely the case where the mass had a negligible effect on the relationship between x 3 and q + . In consequence, the mass m will not have a significant effect on the determination of the saddle point q * + for x 3 ≪ (E/∆) 1/3 . What about the behavior of the integrand elsewhere along the contour in Fig. 6? A discussion of the WKB exponent S of (D21) is complicated by the divergence (5.10) of the bulk-to-boundary propagator on the boundary, which shows up as a logarithmic divergence (∝ ln u B ) of the integral in (D21). We will briefly indicate in section D 4 how one can do a WKB analysis that avoids this divergence, but such details lose the forest for the trees. More simply, the ln u B divergence of du q 5 is independent of q + and so does not affect the q + dependence of the integrand in (D3), and so it will only affect the result by overall factors. To focus on the question of whether the mass makes a significant effect on the q + dependence, look at the effect of the mass on ∂S/∂q + instead of on S. So look at The first term is mass independent, and the second term is just once again our integral for the particle stopping distance as a function of q + , given by the right-hand side of (5.3), though with an imaginary part even for real negative q + due to integrating over u < u min . However, if |q + | ≫ ∆ 4/3 /E 1/3 , then the effect of the mass on this integral will have negligible relative magnitude, just as in the previous discussion concerning the location of the saddle point.

Avoiding WKB exponent divergences
Finally, we sketch how one could set up a finite WKB integral if one wanted to carry through the analysis of this appendix in more detail than we have given. To get the normalization (5.10) appropriate for the bulk-to-boundary propagator G, we want Φ to give u where we have again suppressed showing the WKB prefactor, other than the overall power of u B . We can trade the divergent u (d−∆)/2 B normalization factor for a finite u (d−∆)/2 by introducing a compensating change in the exponent: