Energy Loss of an Infinitely Massive Half-Bogomol’nyi-Prasad-Sommerfeld Particle
by Radiation to All Orders in 1=N
Bartomeu Fiol
*
and Blai Garolera
†
Departament de Fı
´
sica Fonamental i Institut de Cie
`
ncies del Cosmos, Universitat de Barcelona,
Martı
´
i Franque
`
s 1, 08028 Barcelona, Catalonia, Spain
(Received 8 July 2011; published 3 October 2011)
We use the AdS/CFT correspondence to compute the energy radiated by an infinitely massive half-
Bogomol’nyi-Prasad-Sommerfeld particle charged under N ¼ 4 super Yang-Mills theory, transforming
in the symmetric or antisymmetric representation of the gauge group, and moving in the vacuum, to all
orders in 1=N and for large ’t Hooft coupling. For the antisymmetric case we consider D5-branes reaching
the boundary of five-dimensional anti–de Sitter space (AdS
5
) at arbitrary timelike trajectories, while for
the symmetric case, we consider a D3-brane in AdS
5
that reaches the boundary at a hyperbola. We
compare our results to the one obtained for the fundamental representation, deduced by considering a
string in AdS
5
.
DOI: 10.1103/PhysRevLett.107.151601 PACS numbers: 11.25.Tq, 11.25.Hf, 11.25.Uv
Introduction.—Given a gauge theory, one of the basic
questions one can address is the energy loss of a particle
charged under such gauge fields, as it follows arbitrary
trajectories. For classical electrodynamics this is a settled
question, with many practical applications [1]. Much less is
known for generic quantum field theories, especially in
strongly coupled regimes. This state of affairs has started
to improve with the advent of the AdS/CFT correspon-
dence [2], which has allowed us to explore the strongly
coupled regime of a variety of field theories. Within this
framework, the particular question of the energy radiated
by a particle charged under a strongly coupled gauge
theory—either moving in a medium or in the vacuum
with nonconstant velocity—has received a lot of attention
(see [3] for relevant reviews). The motivations are mani-
fold, from the more phenomenological ones, such as mod-
eling the energy loss of quarks in the quark-gluon plasma
[4] to the more formal ones, such as the study of the Unruh
effect [5]. In most of these studies the heavy particle
transforms in the fundamental representation of the gauge
group, and the dual computation is in terms of a string
moving in an asymptotically AdS space. The main purpose
of this note is to extend this prescription to other represen-
tations of the gauge group, which will amount to replacing
the fundamental string by D3 and D5-branes (see [6] for a
previous appearance of this idea), in complete analogy to
the prescription developed for the computation of Wilson
loops [7–11].
Besides the intrinsic interest of this generalization, our
main motivation in studying it is that, as it happens in the
computation of certain Wilson loops, the results for the
energy loss obtained with D-branes give an all-orders series
in 1=N. Given the paucity of such results for large N 4d
gauge theories, this by itself justifies its consideration.
Furthermore, these 1=N terms might shed some light on
some recent results in the study of radiation using the AdS/
CFT correspondence. Let us briefly review them.
The case of an infinitely massive particle transforming in
the fundamental representation and following an arbitrary
timelike trajectory was addressed by Mikhailov [12], who
quite remarkably found a string solution in AdS
5
that
solves the Nambu-Goto equations of motion and reaches
the boundary at any given particle worldline. Working in
Poincare
´
coordinates,
ds
2
AdS
5
¼
L
2
y
2
ðdy
2
þ
dx
dx
Þ (1)
it was furthermore shown that the energy of that string with
respect to the Poincare
´
time is given by
E ¼
ffiffiffiffi
p
2
Z
dt
~
a
2
j
~
a ^
~
vj
2
ð1 v
2
Þ
3
þ
1
y
y¼0
; (2)
where the integral is with respect to the worldline time
coordinate, and ¼ g
2
YM
N is the ’t Hooft coupling. The
second (divergent) term corresponds to the (infinite) mass
and is the Lorentz factor. The first term corresponds to
the radiated energy, so in the supergravity regime the total
radiated power by a particle in the fundamental represen-
tation is
P
F
¼
ffiffiffiffi
p
2
a
a
; (3)
which is essentially Lienard’s formula for radiation in
classical electrodynamics [1] with the substitution e
2
!
3
ffiffiffiffi
p
=4. This
ffiffiffiffi
p
dependence also appears—and has the
same origin—in the computation of the vacuum expecta-
tion value (VEV) of Wilson loops at strong coupling [8,13].
Having computed the total radiated power, a more re-
fined question is to determine its angular distribution. For a
particle moving in the vacuum, this has been done in
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[14,15], who found that this angular distribution is essen-
tially like that of classical electrodynamics. This is a some-
what counterintuitive result, as one might have expected
that the strong coupling of the gauge fields would tend to
broaden the radiating pulses and make radiation more
isotropic. In particular, the authors of [15] argue that these
results are an artifact of the supergravity approximation,
and might go away once stringy effects are taken into
account (see [16] for alternative interpretations). Here is
where considering particles in other representations might
be illuminating, since the 1=N expansion of the radiated
power we find can be interpreted as capturing string loop
corrections [17].
The plan of the present note is as follows: in the next
section we introduce D5-branes dual to particles in the
antisymmetric representation following arbitrary timelike
trajectories, and evaluate the corresponding energy loss.
We then consider a D3-brane dual to a particle in the
symmetric representation following hyperbolic motion,
and compute its energy loss. We end by discussing the
possible connection of this result with the similar one for
particles in the fundamental representation, and mention-
ing possible extensions of this work.
D5-branes and the antisymmetric representation.—
Given a string worldsheet that solves the Nambu-Goto
action in an arbitrary manifold M, there is a quite general
construction due to Hartnoll [18] that provides a solution
for the D5-brane action in M S
5
, of the form S
4
where ,! M is the string worldsheet and S
4
,! S
5
. The
evaluation of the respective renormalized actions gives
then a universal relation between the VEV of Wilson loops
in the antisymmetric and fundamental representations, al-
ready observed, in particular, examples [9,19]. More re-
cently, this construction has been used to evaluate the
energy loss of a particle in the antisymmetric representa-
tion, moving with constant speed in a thermal medium [6].
In this section we combine Mikhailov’s string worldsheet
solution [12] with Hartnoll’s D5-brane construction [18]to
compute the radiated power for a particle in the antisym-
metric representation.
For a given timelike trajectory, we consider a D5-brane
in AdS
5
S
5
, with worldvolume S
4
where is the
corresponding Mikhailov worldsheet [12]. On there is in
addition an electric Dirac-Born-Infeld (DBI) field strength
with k units of charge [18]. This D5-brane is identified as
the dual to a particle transforming in the kth antisymmetric
representation, and following the given timelike trajectory.
As shown in [18] the equations of motion force the angle of
S
4
in S
5
to be
sin
0
cos
0
0
¼
k
N
1
: (4)
We now proceed to compute the energy with respect to the
Poincare
´
time coordinate and the radiated power of such
particle. The energy density for the D5 -brane is
E
D5
¼ T
D5
L
2
y
2
j þ Fj
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j þ Fj
p
¼ T
D5
L
2
y
2
j
j
s
sin
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j
j
p
ffiffiffiffiffiffiffiffiffiffi
j
S
4
j
q
;
where the subscript s means that the determinant is re-
stricted to the spatial directions of the D5-brane or the
fundamental string. We have used that in Hartnoll’s solu-
tion the DBI field strength is purely electric and the DBI
determinant is block diagonal. Integrating over the S
4
part
of the worldvolume one immediately obtains up to con-
stants the energy density of the fundamental string, so
E
D5
¼
2N
3
sin
3
0
E
F1
:
This is the same relation as the one found between the
renormalized actions of the D5-brane and the fundamental
string [18], and in [6] for the relation of drag forces in a
thermal medium. In the regime of validity of supergravity,
the radiated power of a particle in the kth antisymmetric
representation is therefore related to the radiated power of
a particle in the fundamental representation (3)by
P
A
k
¼
2N
3
sin
3
0
P
F
: (5)
The range of validity of this computation is determined by
demanding that backreaction of the D5-brane can be ne-
glected and its size is large in string units, yielding
g
2
s
Nsin
3
0
1 and
1=4
sin
0
1, respectively. For
comparison with the symmetric case it is convenient to
write these conditions as N
2
=
2
Nsin
3
0
N=
3=4
.In
particular, this implies that the result cannot be trusted
when k=N is very close to 0 or 1.
D3-branes and the symmetric representation.—The
computation of Wilson loops of half-BPS particles in the
symmetric representation is given by evaluating the renor-
malized action of D3-branes [10], and analogously we
propose to compute the radiated power of a half-BPS
particle in the symmetric representation by evaluating the
energy of a D3-brane that reaches the boundary of AdS at
the given timelike trajectory. Contrary to what happens for
the fundamental or the antisymmetric representations, we
currently do not have the generic D3-brane solution, so we
will focus on a particular trajectory. On the other hand,
since these D3-branes are fully embedded in AdS
5
,wedo
not use any possible transverse dimensions, so the results
should be valid for other 4d conformal theories with a
gravity dual.
The particular trajectory we will consider is one-
dimensional motion with constant proper acceleration,
which in an inertial system corresponds to
3
a ¼ 1=R.
The trajectory is hyperbolic, ðx
0
Þ
2
þðx
1
Þ
2
¼ R
2
. A rele-
vant feature is that a special conformal transformation
applied to a straight worldline (static particle) gives the
two branches of hyperbolic motion [20]. Besides its promi-
nent role in the study of radiation and the Unruh effect,
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another reason to choose this trajectory is that the relevant
D3-brane is the analytic continuation of the one already
found in [7].
The radiated energy of a particle in the fundamental
representation, Eq. (2) derived in [12], is written in terms
of the worldline of the heavy particle. At least in particular
cases, it is possible to obtain an alternative derivation that
emphasizes the presence of a horizon in the worldsheet
metric, which encodes the split between radiative and non-
radiative gluonic fields, and therefore signals the existence
of energy loss of the dual particle, even in the vacuum [21].
It is convenient to briefly rederive this result for the
particular case of hyperbolic motion, since the computa-
tion of the energy loss using a D3-brane that we will
shortly present resembles closely this second derivation.
Working in Poincare
´
coordinates, Mikhailov’s string solu-
tion for hyperbolic motion can be rewritten as y
2
¼ R
2
þ
ðx
0
Þ
2
ðx
1
Þ
2
; the Euclidean continuation of this world-
sheet is the one originally used to evaluate the VEV of a
circular Wilson loop [22] (see also [23]). This worldsheet
is locally AdS
2
and has a horizon at y ¼ R, with tempera-
ture T ¼ 1=2R, which is the Unruh temperature mea-
sured by an observer following a r
1
¼ R trajectory in
Rindler space. By integrating the energy density from the
horizon to the boundary we obtain
E ¼
ffiffiffiffi
p
2
Z
R
0
dy
y
2
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R
2
þðx
0
Þ
2
y
2
p
¼
ffiffiffiffi
p
2
x
0
R
2
þ
1
y
y¼0
: (6)
The contribution from the boundary is just the (divergent)
second term, corresponding to the mass of the particle. The
first term comes from the horizon contribution, and corre-
sponds to the radiated energy.
A. The D3-brane solution.—We are interested in a
D3-brane that reaches the boundary of AdS
5
at a single
branch of the hyperbola ðx
0
Þ
2
þðx
1
Þ
2
¼ R
2
. To find it,
we change coordinates on the ðx
0
;x
1
Þ plane of (1)to
Rindler coordinates, so the new coordinates cover only a
Rindler wedge
ds
2
¼
L
2
y
2
ðdy
2
þ dr
2
1
r
2
1
d
c
2
þ dr
2
2
þ r
2
2
d
2
Þ: (7)
In these coordinates the relevant D3-brane solution found
in [7] is given by
ðr
2
1
þ r
2
2
þ y
2
R
2
Þ
2
þ 4R
2
r
2
2
¼ 4
2
R
2
y
2
; (8)
where
¼
k
ffiffiffiffi
p
4N
:
Near the AdS
5
boundary y ¼ 0, this solution goes to
r
2
¼ 0, r
2
1
¼ R
2
, so it reaches a circle in Euclidean signa-
ture and the branch of a hyperbola in the Lorentzian one.
The D3-brane also supports a nontrivial Born-Infeld field-
strength on its worldvolume [7]. By a suitable change of
coordinates, its worldvolume metric can be written as [7]
ds
2
¼ L
2
ð1 þ
2
Þðd
2
sinh
2
d
c
2
Þ
þ L
2
2
ðd
2
þ sin
2
d
2
Þ (9)
so it is locally AdS
2
S
2
, with radii L
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ
2
p
and L
respectively, and it has a horizon at ¼ 0 [i.e., r
1
¼ 0 in
the coordinates of (7)]. The temperature of this horizon can
be computed by requiring that the associated Killing vector
is properly normalized at infinity; this is easily done in the
coordinates of (7) and the resulting temperature is again
T ¼
1
2R
: (10)
B. Evaluation of the energy.—To determine the total
radiated power of this solution we will evaluate the energy
with respect the Poincare
´
time coordinate x
0
. The energy
density is
E ¼ T
D3
L
2
y
2
j þ Fj
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j þ Fj
p
L
4
y
4
: (11)
After we substitute the Lorentzian continuation of the
solution of [7] in this expression, the energy density is
E ¼ T
D3
L
4
y
4
0
@
ð1 þ
2
ÞR
2
þðx
0
Þ
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
ð1 þ
2
ÞR
4
2
R
2
r
2
1
ð1 þ
2
ÞR
2
r
2
2
q
1
1
A
:
(12)
The energy is the integral of this energy density from the
boundary to the worldvolume horizon. A long computation
yields
E ¼
2N
x
0
R
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ
2
p
þ
1
y
y¼0
: (13)
Exactly as it happened for the string, Eq. (6), the boundary
contributes only the second term, which is divergent, and is
just k times the one for the fundamental string, Eq. (6). The
first term is the contribution from the horizon, and from it
we can read off the total radiated power
P
S
k
¼
2N
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ
2
p
1
R
2
¼
k
ffiffiffiffi
p
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ
k
2
16N
2
s
1
R
2
:
This result was found for a particular timelike trajectory
with a
a
¼ 1=R
2
. Nevertheless, in classical electrody-
namics the radiated power depends on the kinematics only
through the square of the 4-acceleration, a
a
and as we
have seen, the same is true in theories with gravity duals for
particles in the fundamental, Eq. (3), and antisymmetric
representations, Eq. (5). It is then natural to conjecture that
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in the regime of validity of supergravity, the radiated power
by a particle in the symmetric representation following
arbitrary timelike motion is
P
S
k
¼
k
ffiffiffiffi
p
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ
k
2
16N
2
s
a
a
: (14)
It would be interesting to check this conjecture by find-
ing D3-branes that reach the AdS boundary at arbitrary
timelike trajectories and evaluating the corresponding
energies.
We now discuss the range of validity of this result, and
its possible relevance for the case of a particle in the
fundamental representation. By demanding that the radii
of the D3-brane are much larger than l
s
and that its back-
reaction can be neglected, one can conclude [7] that this
result can be trusted when N
2
=
2
k N=
3=4
.Itis
therefore not justified a priori to set k ¼ 1 in our result,
Eq. (14). Nevertheless, the Euclidean continuation of this
D3-brane was used in [7] to compute the VEV of a circular
Wilson loop, which for k ¼ 1 is known exactly for all N
and thanks to a matrix model computation [17,24], and it
was found [7] that the D3-brane reproduces the correct
result in the large N, limit with fixed, i.e., even for
k ¼ 1. This better than expected performance (probably
due to supersymmetry) of the Euclidean counterpart of this
D3-brane in a very similar computation suggests the ex-
citing possibility that (14) might capture correctly all the
1=N corrections to the radiated power of a particle in the
fundamental representation, i.e., for k ¼ 1, in the limit of
validity of supergravity.
We are investigating whether the angular distributions of
the radiated energy obtained with this D3-brane and with
fundamental strings [14,15] differ qualitatively.
Finally, as already mentioned, the Euclidean version of
the D3-brane considered here was used in [7] to evaluate the
VEV of a circular Wilson loop. That D3-brane result is in
turn only an approximation to the exact result, available for
all N and thanks to a matrix model computation [17,24]. It
would be extremely interesting to understand whether the
radiated power of a particle coupled to a conformal gauge
theory can be similarly computed by a matrix model.
We would like to thank Mariano Chernicoff, Nadav
Drukker, Roberto Emparan, and David Mateos for helpful
conversations. The research of B. F. is supported by MEC
FPA2009-20807-C02-02, CPAN CSD2007-00042, within
the Consolider-Ingenio2010 program, and AGAUR
2009SGR00168. The research of B. G. is supported by
the ICC and by MEC FPA2009-20807-C02-02.
*bfiol@ub.edu
†
bgarolera@ffn.ub.es
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