Jet quenching in hot strongly coupled gauge theories simplified

0.48 MB·English

by  Peter Arnold

Jet quenching in hot strongly coupled gauge theories simplified
Peter Arnold and Diana Vaman
Department of Physics, University of Virginia,
Box 400714, Charlottesville, Virginia 22904, USA
(Dated: January 17, 2011)
Abstract
Theoreticalstudiesofjetstoppinginstrongly-coupledQCD-likeplasmashaveusedgauge-gravity
1 duality to find that the maximum stopping distance scales like E1/3 for large jet energies E. In
1
recent work studying jets that are created by finite-size sources in the gauge theory, we found an
0
2 additional scale: the typical (as opposed to maximum) jet stopping distance scales like (EL)1/4,

n where L is the size of the space-time region where the jet is created. In this paper, we show that
a the results of our previous, somewhat involved computation in the gravity dual, and the (EL)1/4
J
3 scale in particular, can be very easily reproduced and understood in terms of the distance that
1 high-energy particles travel in AdS -Schwarzschild space before falling into the black brane. We
5

also investigate how stopping distances depend on the conformal dimension of the source operator
]
h
used to create the jet.
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-
p
e
h
[

1
v
9
8
6
2
.
1
0
1
1
:
v
i
X
r
a
I. INTRODUCTION AND RESULTS
Various authors [1–4] have made use of gauge-gravity duality to study the stopping dis-
tance of massless, high-energy jets in a strongly-coupled plasma of =4 supersymmetric
N
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
E1/3. 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 con-
ditions 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 specif-
ically 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 E1/3, 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
3
distance traveled x . (Our convention here is to write 4-dimensional space-time position as
xµ and take our jets to be created near the origin, traveling in the x3 direction.) Between
the (EL)1/4 scale and the E1/3 scale, the distribution falls algebraically like (x3)−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
(4c4EL)2 c4EL
Prob(x3) 2 Ψ for x3 E1/3 (1.1a)
≃ (2x3)9 −(2x3)4 ≪
(cid:16) (cid:17)
and
(c L)2 2c x3
Prob(x3) 4 2 Ψ(0) exp 1 for x3 E1/3, (1.1b)
≃ E − E1/3 ≫
(cid:18) (cid:19)
where Ψ(y) is a source-dependent function that suppresses y 1, causing suppression of
x3 (EL)1/4 above. The c’s are constants given by | | ≫
≪
Γ2(1)
c 4 , (1.2)
≡ (2π)1/2
c 0.927, and c 3.2.
1 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 x3 E1/3, can be derived from a very simple
≪
calculation of how far a classical massless particle travels in AdS -Schwarzschild space before
5
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” q2 of the source that matters (where q2 q ηµνq is squared
µ ν
≡
4-momentum).
Our analysis of the distance traveled by falling particles in AdS -Schwarzschild will be
5
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
2
d ∝(x3)−9
e
t
si
o
p
e
d
e exponential
g
r fall−off
a
h
c
x3
1/4 −1 1/3 −4/3
(EL) T E T
3
FIG.1: Theaveragedepositionofchargeasafunctionofx forjetscreatedbythesourcedescribed
in Sec. IVA and in Ref. [4].
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
source(x) eik¯·xΛ (x) (1.3)
L
∼
of (i) a high-energy plane wave eik¯·x times (ii) a slowly varying envelope function Λ (x) that
L
¯
localizes the source to within a distance L of the origin in both space and time. We took k
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,
ΛL(x) = e−12(x0/L)2e−12(x3/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 on k¯ with width of order L−1. We take L−1 E.
≪
Note that this source covers a range of values of q2, from 0 to order E/L, and the typical
±
size of q2 is order E/L.
| |
In the gravity description, this source causes a localized perturbation on the boundary of
AdS -Schwarzschild space-time, which then propagates as a wave into the fifth dimension,
5
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 where k is slightly off the light-cone,
k¯µ = (E +ǫ,0,0,E ǫ) (1.6)
−
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 q2 of the source is approximately well defined, with q2 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
3
q0 q0 q0
L−1 L−1 L−1
E E ~ε E
q3 q3 q3
E E E
(a) (b) (c)
FIG. 2: Qualitative picture of momenta contributing to the source (1.3) used to generate jets (a)
for thecalculation originally usedto find(1.1), with L−1 E, and(b) inthe case L−1 ǫ E of
≪ ≪ ≪
(1.6). Figure (c) depicts (a) as a superpositionof distributionsof type (b). Thecells 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 E1/3.
≪
3
classical, massless particle which starts at the boundary, traveling in the x 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
c q 2 1/4 c E 1/4
3
x | | . (1.7)
stop ≃ √2 q2 ≃ 2 ǫ
(cid:18)− (cid:19) (cid:18) (cid:19)
0
where the constant c is given by (1.2). As measured by boundary time x , 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 q2 < 0 (i.e. ǫ > 0). The q2>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 q2. That is,
Prob(x3) d(q2) (q2)δ x3 x3 (q2) , (1.8)
≃ P − stop
Z
(cid:0) (cid:1)
where (q2) is the probability density for the source to produce a jet with a given q2, and
P
1 See in particular the discussion surrounding Fig. 2 of Ref. [6], which inspired our Fig. 3b. See also Ref.
[5].
4
stopping distance x3 stopping distance x3
stop stop
µ
j
boundary boundary
0 x3 0 x3
1 1
horizon horizon
u u
(a) (b)
3
FIG. 3: (a) A classical particle in the AdS -Schwarzschild space-time, moving in the x direction
5
as it falls from the boundary to the black brane in the fifth dimension u. (b) The presence of the
particle (the large dot) perturbs the boundary theory in a manner that spreads out diffusively as
0
the particle approaches the horizon for x .
→ ∞
where the stopping distance x3 (q2) is given by (1.7). We will verify that this formula
stop
precisely reproduces the x3 E1/3 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
q2 < 0 that produce jets. The typical value of time-like q2 for the source of Fig. 2a is
q2 E/L, corresponding to ǫ L−1. Putting this into (1.7), the typical stopping distance
∼ − ∼
in this case is therefore
x3 (EL)1/4. (1.9)
typical ∼
Note that it is the q2 of the source that determines the stopping distance, and that the
typical value of q2 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
L x3 (EL)1/4, (1.10)
∼ stop ∼
which gives
x3 E1/3. (1.11)
stop ∼
We will see later that this is precisely the case where the wave packet in AdS -Schwarzschild
5
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
E1/3 scale as well.
In the next section, we will briefly review the trajectories of massless particles in AdS -
5
Schwarzschild andderivethecorrespondingstoppingdistance(1.7). InsectionIII, wediscuss
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
x3 E1/3. 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.
5
y
t
bili ∝(x3)3 −4∆
a
b
o
r
p
g
n exponential
pi fall−off
p
o
t
s
x3
∆−1/8(EL)1/4T−1 ∆−1/3E1/3T−4/3
FIG. 4: The probability distribution of jet stopping distances for scalar or transverse BPS sources
with conformal dimension ∆. Fig. 1 corresponds to ∆=3. The scales x3 (EL/√∆)1/4 and
typical ∼
x3 (E/∆)1/3 indicated along the vertical axis assume that ∆ is held fixed when taking the
max ∼
limit of large energy E (as well as large coupling g2N and large N ). The parametric scaling with
c c
3
∆ indicated for x assumes a Gaussian source envelope (1.5), but the other features shown in
typical
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
E1/3 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 E1/3 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 -Schwarzschild space-time. The symbol q2 will
5
refer to the squared 4-momentum, q2 q ηµνq = ω2 + q 2. When we use light-cone
µ ν
≡ − | |
coordinates, our conventions will be
V± V3 V0, V 1V∓ = 1(V3 V0) (1.12)
≡ ± ± ≡ 2 2 ∓
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):
gµνq
xµ(x5) = √g55dx5 ν , (2.1)
( q gαβq )1/2
α β
Z −
2 This is as opposed to the alternative usage of “transverse” to mean all three space-time components
orthogonal to q . Note that the spatial directions transverse to q are the same for all q in our problem
µ
because we take our source (4.2) to be translationally invariant in the transverse directions.
6
where g is the 5-dimensional metric and q is a constant of motion for I = 0,1,2,3. We will
I
work in coordinates where the metric is3
R2 1 1
ds2 = ( f dt2 +dx2)+ du2 , (2.2)
4 u − u2f
(cid:20) (cid:21)
where f 1 u2, and R is the AdS radius. The boundary is at u=0 and the horizon at
5
u=1. If w≡e ta−ke the 3-momentum q to point in the x3 direction, writing q = ( ω,0,0, q ),
µ
3 − | |
then (2.1) gives the total distance x traveled in falling from the boundary to the horizon
to be
1 du
3
x = , (2.3)
stop
Z0 u(u2 q2 )
− |q|2
where q2 q ηµνq is the flat-space square qof the 4-momentum. This is the same result as
µ ν
≡
Refs. [1, 3].4
Now let us apply this result to the case q2 q 2 E2 relevant to the source of Fig. 2b.
| | ≪ | | ≃
For small q2, the integral of (2.3) is dominated by small u, and so we may approximate
−
∞ du c q 2 1/4
3
x = | | , (2.4)
stop ≃ Z0 u(u2 − |qq2|2) √2 (cid:18)−q2(cid:19)
q
which gives (1.7).
III. WAVE PACKETS AND GEOMETRIC OPTICS
In this section, we will discuss the conditions necessary for making the particle approx-
imation. 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 cal-
culation 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
q2 ǫ
u − . (3.1)
⋆ ∼ s q 2 ∼ E
| | r
3 Our formulas in this paper would be a little tidier (fewer square roots) if we workedwith 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) correspondsto the first partof 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 /p is replaced by the q /q of the momentum q typical of our source,
1 0 3 0 µ
and their y is set to zero. It also corresponds to Eq. (4.28) of Chesler et al. [3], where their u is our
UV
2√u, their uh =2 in our units, their ξ is replaced by our q0/q1, and their u∗ is set to zero.
7
field field ∆u field
u u u
λ∼1/q5 λ∼1/q5 λ∼1/q5
(a) (b) (c)
0
FIG. 5: (a,b) A snapshot in time x 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 q2 values such as Fig. 2a. (b) shows the wave packet generated by a source with approximately
well-defined q2 such as Fig. 2b. (c) shows a single 4-momentum component, corresponding to a
single, definite value of 4-momentum q . Case (a) matches Fig. 7a of our earlier paper [4].
µ
The relevant question is then whether the background varies significantly across one phase
oscillationforu u . By(1.7),thedistancesx3 . E1/3 relevanttojetstoppingcorrespond
∼ ⋆ stop
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 , the exponential in the
µ
WKB approximation is
exp iqµxµ +i dx5q5(x5) , (3.2)
(cid:18) Z (cid:19)
5
where the qµ are constant and q5(x ) is determined by the 5-dimensional massless condition
q gIJq = 0, giving5
I J
5
q5(x ) = g55( qµgµνqν). (3.3)
−
q
For the metric (2.2), this is
1 u2 q 2 q2
q5(u) = | | − . (3.4)
f u
r
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/q5(u) in
∼ ≪ ∼
5 The null geodesics (2.1) can be expressed in terms of q5 as xµ(x5) = dx5(∂q5/∂qµ). This particle
−
formulaissimplythesaddlepointcondition∂qµ[iqνxν+i dx5q5(x5)]=0wRithrespecttoqµ forthewave
(3.2). Also, dx5q5(x5) was referred to as the WKB expRonent S in Ref. [4], where various expansions of
the integral may be found.
R
6 This condition can be phrased in an x5-reparametrization invariant way as 5(1/q5) 1, which is
∇ ≪
(g55)−1/2∂5[(g55)1/2/q5] 1. We givemore detail in Appendix C, in the specific contextof the particular
≪
type of source operator that we used in our original calculation.
8
the fifth dimension satisfies this condition if
u⋆q5(u⋆) 1, (3.5)
≫
which, using (3.4), is7
1
u . (3.6)
⋆ ≫ q2
−
Using the size (3.1) of u , this condition is ( q2)3/ q 2 1, or equivalently
⋆
− | | ≫
ǫ E−1/3. (3.7)
≫
Referring to the stopping distance (1.7), we then see that the approximation of geometric
optics, necessary for a particle interpretation, breaks down unless
x3 E1/3. (3.8)
stop ≪
Within the context of our approach to jet stopping, a proper analysis of what happens at
distances & E1/3 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 E1/3, 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 inthe fifthdimension. However, thepresentation will be alittle morestraightforward
⋆
∼
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
⋆
≪
u du′ u q 2 uE
3
x (u) = 2 | | . (3.9)
Z0 u′(u′2 − |qq2|2) ≃ s −q2 ≃ r ǫ
q
Turning this around, the location of the particle in the fifth dimension is
ǫ(x3)2
u . (3.10)
≃ E
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
3
∆x 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
3
∆u ∆ǫ ∆x 1 ǫ
max , max , L . (3.11)
u ∼ ǫ x3 ∼ Lǫ uE
(cid:18) (cid:19) (cid:18) r (cid:19)
7 In the language of Ref. [4], the condition u 1/( q2) is u u .
⋆ ⋆ match
≫ − ≫
9
Extrapolating this parametric estimate to the case u u of interest, (3.1) and (3.11)
⋆
∼
give
∆u 1 ǫ 1/4
max , L . (3.12)
u ∼ Lǫ E
(cid:18) (cid:19)⋆ (cid:18) (cid:19)
(cid:16) (cid:17)
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
3
stopping distance x .
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 (q2) for the source to create a jet with a
P
given q2.
Wewill later give insection Va very general argument, basedondimensional analysis, for
determining the (q2) associated with different choices of source operator. This argument
P
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 (q2) from a result for the average total
P
charge produced by the operator. Readers who would prefer to just see the more general
argument may skip section IVA below and instead wait for section VB.
A. Extracting P(q2) from results in the literature
In Ref. [4], we used a source involving R-current operators ja. Specifically, we modified
µ
the 4-dimensional gauge theory Lagrangian by
+jaAaµ, (4.1)
L → L µ cl
with a localized background field
τ+
Aµ(x) = ε¯µ eik¯·x +h.c. Λ (x), (4.2)
cl NA 2 L
h i
where is an arbitrarily small source amplitude, ε¯is a transverse linear polarization, and
A
N
τi 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 τ3/2. The gravity dual to
h i
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 (x1,x2) to
simplify the calculation. In what follows, we will refer to the R charge associated with τ3/2
as simply “the charge.”
For an arbitrarily small source amplitude , the source will usually have no effect at all
A
N
on the system. On rare occasions, with probability proportional to 2, the source creates
NA
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 and so
A
N
negligible. Since an excitation (if any is created) has charge 1, the average total charge
Q
10