The Plasma Puddle as a Perturbative Black Hole
arXiv:0704.1146v2  [hep-th]  7 Jun 2007
Preprint typeset in JHEP style - HYPER VERSION
The Plasma Puddle as a Perturbative Black Hole
Clifford Cheung and Jared Kaplan
Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138
E-mail: cwcheung@fas.harvard.edu, kaplan@physics.harvard.edu
Abstract:We argue that the weak coupling regime of a large N gauge theory in the Higgs
phase contains black hole-like objects. These so-called “plasma puddles” are meta-stable
lumps of hot plasma lying in locally un-Higgsed regions of space. They decay via O(1/N)
thermal radiation and, perhaps surprisingly, absorb all incident matter. We show that
an incident particle of energy E striking the plasma puddle will shower into an enormous
number of decay products whose multiplicity grows linearly with E, and whose average
energy is independent of E. Once these ultra-soft particles reach the interior they are
thermalized by the plasma within, and so the object appears “black.” We determine some
gross properties like the size and temperature of the the plasma puddle in terms of funda-
mental parameters in the gauge theory. Interestingly, demanding that the plasma puddle
emit thermal Hawking radiation implies that the object is black (i.e. absorbs all incident
particles), which implies classical stability, which implies satisfaction of the Bekenstein en-
tropy bound. Because of the AdS/CFT duality and the many similarities between plasma
puddles and black holes, we conjecture that black objects are a robust feature of quantum
gravity.
http://arxiv.org/abs/0704.1146v2
mailto:cwcheung@fas.harvard.edu
mailto:kaplan@physics.harvard.edu
1. Introduction
The AdS/CFT correspondence [1], [2] [3] has greatly improved our understanding of both
gravity and gauge theory by providing a concrete realization of the holographic principle.
For example, much work has been devoted to studying strongly coupled quasi-CFT dynam-
ics using perturbative gravity. Conversely, CFTs have been useful for illuminating aspects
of black hole physics, including the unitarity of Hawking evaporation.
It has been argued [4] that there exist black holes that can be localized in the IR
of asymptotically AdS geometries, and that these solutions are dual to “plasma balls” in
a confining CFT (for related work, see [5], [6], [7]). These plasma balls are meta-stable
lumps of hot gluon plasma, and like black holes, they absorb all incoming matter and
radiate thermally. Interestingly, since the AdS/CFT duality maps quantum effects to
classical effects and vice versa, Hawking radiation is nontrivial from the gravitational point
of view but straightforward in terms of the dual plasma ball description. Conversely, the
“blackness” of black holes – their ability to absorb all incoming particles – is not obvious
in the confining CFT picture.
In particular, consider the CFT dual of a particle thrown into a black hole: a glue ball
thrown into a plasma ball. Naively, it seems that with sufficiently high energy such a glue
ball would blast through, in the same way that an extremely high energy proton might
barrel through the RHIC fireball. From this point of view, the most fundamental property
of black holes – that they absorb all incoming particles – appears to be violated.
In [4], this problem was beautifully solved by taking into account the parton substruc-
ture of the incident particles [8], in accordance with Susskind’s ideas [9, 10]. In the dual
gauge theory, the only available objects outside the plasma ball are mesons and glue balls,
and at large ’t Hooft coupling these highly boosted hadrons contain a huge number of soft
partons. Thus it is simply impossible to fire a high energy parton into the plasma ball –
instead, an incoming glue ball fragments into many low energy partons which are promptly
absorbed.
The purpose of the present paper is to explore the possibility that a weakly coupled
gauge theory might furnish a perturbative dual to a black hole. At first this might seem
like an unlikely prospect, particularly since the dual of a weakly coupled CFT is a strongly
coupled gravitational theory. Indeed, it is not unreasonable to expect a phase transition
between the weak and strong coupling regimes of the gravitational theory, corresponding to
immensely different CFT physics in the two regimes. Despite these expectations, however,
we will argue that “plasma puddles” in a weakly coupled gauge theory are qualitatively
very similar to plasma balls/black holes.
Our setup is similar to [4], except that we consider a perturbative gauge theory in
the Higgs phase, rather than a strongly coupled gauge theory in the confining phase.
The low energy theory is comprised of photons and W bosons, rather than mesons and
glue balls. Specifically, we will study an N = 4 SU(N) SYM theory Higgsed down to
U(1)N−1. By heating up a region of space we can locally un-Higgs the gauge group (see
figure 1), creating a spatially varying Higgs vev. This means that W bosons have a position
dependent mass that will act as an effective potential for the enclosed plasma. We will see
– 1 –
that for sufficiently large puddles, the plasma has a temperature much less than the height
of the enclosing potential, and so it is kinematically trapped. Thus, classical stability is
ensured. That said, the plasma puddle does emit radiation in a thermal spectrum, but we
find that its lifetime is
τ ∼ NR, (1.1)
where R is the radius of the puddle, so plasma puddles are infinitely long lived in the large
N (classical) limit.
Since our theory is weakly coupled, it is possible to analyze the absorptive properties
of a plasma puddle using standard perturbation theory. Unlike glue balls, photons and W
bosons have no partonic substructure – since they are elementary point particles, arbitrarily
large boosts involve no partonic subtleties. Thus it would seem that at sufficiently high en-
ergies, a plasma puddle can be penetrated. However, incident particles actually experience
something very similar to parton showering – they decay near the boundary of the plasma
puddle, bifurcating into a large number of ultra-soft daughter particles whose ensemble
energy is smeared over the surface of the puddle and eventually thermalized. Qualitatively,
this matches the process of gravitational “hair removal” in which an infalling particle is
delocalized over the surface of a black hole as it is absorbed. In fact, this showering can
only occur at the plasma puddle boundary (the effect requires momentum nonconserva-
tion), which dovetails nicely with the notion that black hole absorption is a local effect at
the event horizon, independent of the interior.
In order for a plasma puddle to absorb even the highest energy particles, an incident
particle of energy E has to shower into a large number of decay products that are too soft
to escape the enclosing potential. In particular, it is necessary that the average energy
of the final decay products, Eavg, does not increase with increasing E. If this is not the
case, then an arbitrarily large E implies a commensurately large Eavg, allowing the decay
products to blast through the plasma unharmed. Thus, total absorption is only possible if
the rate1 of showering increases at least linearly with energy. For E less than a particular
(large) threshold we show that this is the case: the rate of decay of a gauge boson to two
W ’s is
ΓA→WW ∼ λE, (1.2)
where λ is the ’t Hooft coupling. Moreover, even though the decay rate goes to a constant
for incident particles above this threshold energy, we find that they are still absorbed. In
particular, these high energy particles shower promptly inside the plasma puddle into a
large number of decay products at the threshold energy, which in turn decay at a rate
λE. Thus, a high energy incident particle will shower into O(λER) decay products each of
energy Eavg ∼ (λR)−1. We confirm this reasoning with a more precise argument in section
3.2. As long as Eavg is less than the height of the enclosing potential, these decay products
are trapped inside the plasma puddle, and so the puddle appears black.
1For a detailed calculation of the dimensionless probability of showering, see appendix B
– 2 –
Figure 1: A plasma puddle cross section (above) and energy profile (below). The latter depicts
the W mass as it asymptotes to a value of m0 at infinity and vanishes within an un-Higgsed region
of radius R. The plasma within settles into a hot puddle of temperature T . Both diagrams show
a high energy incident particle on the left showering in the atmosphere, as well as some Hawking
radiation escaping on the right.
Interestingly, as we show in section 4, if we demand that the plasma puddle emit ther-
mal Hawking radiation, then it automatically also absorbs all incident matter, is classically
stable, and satisfies the Bekenstein entropy bound [11]. During the process of Hawking
evaporation, a stable plasma puddle will lose each of these properties sequentially, until it
eventually becomes a free gas of gauge bosons. Since the plasma puddle is so similar to a
black hole, our hope is that the large λ plasma ball/black hole duality established in [4]
interpolates at small λ to a correspondence between the plasma puddle and some black
object of a strongly coupled gravitational theory.
Note that for our purposes we will focus entirely on the gauge degrees of freedom
of the N = 4 SYM, ignoring gaugino partners and scalar moduli. Moreover, as we are
interested only in parametric scalings, we will be largely ignoring numerical factors. The
outline of the paper is as follows. In section 2 we determine the general properties of our
– 3 –
setup: gross properties of the plasma puddle, Hawking radiation, and classical stability. In
section 3, we give a nice physical estimate of the rate of particle showering in the plasma
puddle atmosphere. In section 4 we show that demanding thermal Hawking radiation
immediately implies other black hole-like properties, and we conclude in section 5. The
appendices contain a more formal derivation of the probability of decay.
2. The Plasma Puddle
In this section we give a detailed account of what a plasma puddle is and how it forms.
Our setup is as follows. Consider an N = 4 SU(N) SYM at large N and weak ’t Hooft
coupling λ that is Higgsed down to U(1)N−1 (note that due to the Higgsing, the theory
is not conformal and so particles and S-matrices are well defined). The spectrum of the
theory consists of N − 1 massless photons and N2−N massive W bosons. Now, let us fire
an ensemble of high energy W ’s into a small region of space2. The influx of W ’s heats up
the region and locally un-Higgses the gauge group. Once the W ’s thermalize, the resulting
meta-stable object is a plasma puddle.
The local un-Higgsing can be parameterized by a spatially varying Higgs vev which
induces a spatially varying mass, m(x), for the W bosons. The W mass profile vanishes
inside the plasma puddle but asymptotes to some nonzero mass m0 outside (the precise
mass of individual W s depends on the Higgsing pattern, but the details will not be impor-
tant). Let us define the atmosphere to be the region near the plasma puddle boundary in
which the mass profile varies. Again, we emphasize that the W ’s are confined by the m(x)
potential, while the photons are actually massless everywhere.
If we ignore all interactions, it is straightforward to see what happens to free streaming
W ’s as they collapse into a plasma puddle. Those W ’s with E > m0 escape to infinity
while those with E < m0 settle at the basin of the m(x) potential. Due to the potential
barrier, this puddle of W ’s can never escape.
The story is similar if we include gauge interactions, except that the puddle of W ’s
thermalizes into a puddle of photons and W bosons. To see this, we compute the mean
free path d of a gauge boson Aij traversing the hot plasma. From color conservation, Aij
can only scatter off of some Ajk, leaving Ail and Alk in the final state, where k and l are
free indices. Summing the amplitude squared, g4, over phase space, k and l contribute N2
to the cross section, yielding
σ ∼ λ2T−2, (2.1)
since the temperature sets the energy scale of the interaction. The number density n of a
gauge boson with a given pair of color indices is T 3, so the mean free path is
d ∼ (nσ)−1 ∼ (λ2T )−1. (2.2)
2Note that an ensemble of W bosons that are initially at rest will naturally collapse due to the attractive
force of dilaton gravity, as in described in [12]. In the nonrelativistic limit, this behaves exactly like true
gravitational collapse.
– 4 –
For sufficiently large ’t Hooft coupling the mean free path is smaller than the size of the
plasma puddle and thus gauge bosons cannot go very far without scattering (see figure 2).
Let us denote this as the highly thermalized regime. In this case the plasma puddle quickly
thermalizes into a hot, homogenous soup of photons and W bosons at a temperature T .
The plasma puddle is classically stable as long as its temperature is less than m0, so that
the interior plasma is kinematically trapped. Without this condition, nothing prevents the
plasma from simply escaping to infinity, and so the relation
T < m0, (2.3)
implies classical stability of the plasma puddle. In addition, since the plasma puddle is
highly thermalized, only particles at the very surface have any hope of escape. We argue
in section 2.2 that this phenomenon is an O(1/N) area effect that is thermal and dual to
Hawking radiation. For these reasons (along with their absorptive properties, which we
discuss in section 3) we claim that plasma puddles in the d < R regime are black hole-like
objects.
On the other hand, if λ is sufficiently small, then d > R and a typical gauge boson can
traverse the extent of the plasma puddle without ever scattering (see figure 2). However,
given a time of order d, the plasma will eventually thermalize, yielding a collection of
nearly free photons and W ’s at a temperature T . The W bosons travel in straight lines
through the puddle until they reach the potential barrier from m(x), after which they
roll back towards the interior, and repeat. As a result the W ’s form a relativistic plasma
at the bottom of the m(x) potential, but they become nonrelativistic in near the m(x)
barrier wall, simply because they have less kinetic energy there. In contrast, the photons
are massless everywhere and can free stream outwards. As we discuss in section 2.2, since
plasma puddles in the d > R regime do not Hawking radiate in the traditional sense, we
do not identify them as dual black holes.
2.1 Gross Properties and Relations
Thus far our discussion has included T and R as a priori attributes of the plasma puddle.
However, as we show in this section, these two variables are fixed in terms of the “universal”
quantities N , λ, m0, and the total energy of the plasma puddle, M .
To begin we note that unlike Schwarzchild black holes, plasma balls/puddles do not
obey the relation T ∼ 1/R for the following reason. Since the plasma puddle has an entropy
S = N2T 3R3, the relation T ∼ 1/R would imply that the entropy is independent of the
size of the object. However, this is not the case – the entropy of a plasma puddle/ball
increases with size. In the case of the strongly coupled plasma balls of [4], the temperature
of large plasma balls is set by the confinement scale. Analogously, one might expect a
similar situation for plasma puddles, i.e. that the temperature is given by the Higgs vev.
We will see that this is not the case.
Once formed, a plasma puddle is an isolated system, so its total energy is conserved.
However, its entropy should be maximized subject to this constraint, so we treat the plasma
3Here we use the symbol M in anticipation of matching this total energy of the plasma puddle to the
mass of a dual black hole.
– 5 –
puddle with the micro-canonical ensemble. In N = 4 SYM, the scalar fields Φij are in the
adjoint representation of the gauge group. When these fields Higgs SU(N) to U(1)N−1,
their vevs can be written as diagonal N ×N matrices with spatially varying components.
The total energy of a plasma puddle arises from two contributions: the gradient kinetic
energy from the (spatially varying) Higgs mechanism, and the thermal energy of the plasma
within. Because these contributions are boundary and volume effects, respectively, we are
essentially balancing the pressure against the surface tension. Neglecting O(1) factors, the
total energy of the plasma puddle is thus
d3xTr(∇Φ)2 +N2T 4. (2.4)
The second term only accounts for the relativistic (T ≫ m(x)) region of the plasma. The
nonrelativistic (T ≪ m(x)) region has an energy density N2T 2m(x)2 exp(−m(x)/T ), and
this is always less than or equal to N2T 4. Thus neglecting this contribution merely amounts
to a rescaling of the plasma energy by an O(1) factor.
Each diagonal entry of Φ gets a vev that asymptotes to ∼ m0/g at infinity but vanishes
in some region of size R. Moreover, let us define L to be the thickness of the atmosphere,
i.e. the region in which the Higgs vev varies. Estimating the energy we find that
M = N2
m20(R+ L)
+ T 4R3
. (2.5)
Before maximizing the entropy at fixed energy, let us motivate why this fixes the radius
of the plasma puddle. Consider what happens as we increase the radius of the puddle at
fixed m0. This increases the gradient Φ energy, but since the total energy is fixed, this
forces the temperature to diminish. Since the entropy goes as T 3R3, the increase in radius
and decrease in temperature are competing effects. Hence, there is an extremal value for
R such that the entropy is maximized. Note that this is what fixes the size of a balloon
full of gas.
Fixing M allows us to solve for T , yielding an entropy
S = N2T 3R3 (2.6)
NR3/4
M − N
2m20(R+ L)
. (2.7)
The entropy is maximized for L ∼ R, because this maximizes the term in large parentheses.
Maximizing the entropy with respect to R, we find that
R ∼ λM
N2m20
, (2.8)
and also that
T ∼ Nm
λ3/4M1/2
. (2.9)
Thus we see that while strongly coupled plasma balls obey the relation R ∼ M1/3 [4],
perturbative plasma puddles have that R ∼ M .
– 6 –
Finally, let us add some remarks about the precise shape of the Higgs vev, Φ. Near
the boundary of the plasma puddle it is nontrivial to determine Φ, because the plasma
back-reacts on the Higgs vev and vice versa. However, outside R the Φ field is free, so it
obeys Laplace’s equation, and thus
for r > R, (2.10)
obtains. Thus the scalar field atmosphere has a 1/r tail, so again we see that the thickness
L of the plasma puddle is of order R. In d ≥ 4 spacetime dimensions (in the gauge theory),
the Higgs profile would go as
Φ ∼ 1−
, (2.11)
so in higher dimensions there are no IR subtleties. For this reason we do not expect that
this tail is qualitatively important for black hole/plasma puddle physics.
2.2 Hawking Radiation
In the gravitational picture, the total absorption of incident particles by a black hole is
obvious at the classical level, while Hawking radiation is a nontrivial quantum effect. In
contrast, in our CFT setup we will see that plasma puddle absorption is nontrivial but
Hawking radiation is manifest. In this section we consider the latter.
To begin, let us consider the regime in which d < R and the enclosed plasma is highly
thermalized. In this case the gauge bosons cannot go very far without scattering, and
thus a particle has no hope of escape unless it is at the very surface of the plasma puddle.
Moreover, since the m(x) potential kinematically bounds W bosons, a surface particle can
only escape if it is a photon. Since only 1/N of the particles is a photon (due to the
Higgsing pattern SU(N) → U(1)N−1), the plasma puddle radiates photons in a thermal
spectrum at temperature T . Since this is a O(1/N) effect, it is natural to associate this
radiation to O(~) Hawking radiation in a gravitational dual.
Next, let us compute the rate at which a plasma puddle loses energy as the result of
Hawking radiation. In an infinitesimal time interval dt, an order one fraction of photons
that are within dt of the surface of the plasma puddle will exit. If nγ is the number density
of photons in the plasma, then there are nγR
2dt photons in this region near the surface.
Since each has an energy of order T ,
= −nγR2T. (2.12)
Assuming that these photons are in thermal equilibrium, we find
= −NR2T 4
= −Nm
, (2.13)
– 7 –
so we see that at fixed temperature the energy loss is an area effect, while at fixed m0 the
energy loss is completely independent of size and temperature. The lifetime is given by the
time it takes for the total energy M to be radiated
τ = −M
∼ NR, (2.14)
so plasma puddles are completely stable in the classical limit N → ∞.
The situation is more complicated if d > R, so let us be more careful. We consider
scattering inside the plasma puddle for arbitrary d and R. If the overall number density of
particles is ntot, then there is a total of ntotR
3 particles, and so the total rate of scattering
processes is ntotR
3/d. Of these processes, an O(1/N) fraction create additional photons.
Likewise, there are nγR
3 photons in the plasma, so the rate for scattering events involving
photons is nγR
3/d. Almost all of these scattering events destroy photons, because AW →
AW is suppressed by a factor N compared to AW → WW . Thus scattering events change
the number of photons in the plasma at a rate
d(nγR
scattering
ntot − nγ
. (2.15)
However, radiation leaving the plasma puddle also decreases the number of photons at the
d(nγR
radiation
= −nγR2. (2.16)
These two processes balance when
. (2.17)
Thus we see that as d becomes larger than R, the number of photons becomes less than
1/N of the total number of particles, and so we leave thermal equilibrium. In particular,
once d > R, the rate of Hawking radiation begins to decrease by the significant factor R/d.
Thus, in the d > R regime the plasma puddle does not radiate a thermal spectrum, and
exiting photons can free stream from anywhere within the plasma, not just the surface.
For this reason we do not consider such a plasma puddle to be the dual of a black hole.
We have glossed over a subtlety in our treatment of Hawking radiation. As we argue
in section 3, a high energy photon striking the plasma puddle atmosphere will shower
into many other gauge bosons, until all of the decay products have a tiny energy of order
(λR)−1. But since it is possible that T > (λR)−1 (in fact, an even stronger condition is
required to be in the highly thermalized regime), we might worry that an outgoing photon
will simply shower in the plasma puddle atmosphere. If this happens, its low energy decay
products are likely to be reabsorbed by the plasma puddle, leaving no reason to expect
Hawking radiation, let alone a thermal distribution of Hawking radiation.
The resolution to this puzzle is that once an outgoing photon makes it to the outer
atmosphere of the plasma, it does not have the 2m(x) of energy necessary to shower into
– 8 –
Figure 2: Here we show the typical trajectory of a gauge boson in the highly thermalized regime,
d < R, and otherwise. Hawking radiation is only thermal when d < R.
two W bosons, since it only has energy T . Thus, kinematical constraints allow photons
at temperature T to escape from the plasma puddle as thermal Hawking radiation, even
though high energy incident photons typically shower as they fall in.
Finally, let us discuss briefly the possibility of W boson Hawking radiation. While
most of the W bosons are of course kinematically trapped, there is a Boltzmann tail in the
thermal distribution which allows an e−m(x)/T fraction of the W ’s to escape. While this
effect is exponentially suppressed, it is not suppressed by powers of 1/N . This is puzzling
from the standpoint of the gravity dual, in which all Hawking radiation should be an O(~)
effect. Understanding the true N dependence of W boson Hawking radiation will require
further study.
3. Plasma Puddle Absorption
The primary feature of the plasma puddle that distinguishes it from more conventional
objects that are hot and stable (such as stars) is its ability to absorb all incident matter.
In this section we verify that the plasma puddle is black by showing that a high energy
incident particle will shower into numerous soft decay products that thermalize with the
interior plasma. While a formal calculation of the decay probability is given in appendix
A, we present a more direct physical argument in the following section.
3.1 Estimation of the Showering Rate
To begin, let us consider the propagation of a single photon through free space. The
photon is absolutely stable due to kinematics and phase space – momentum conservation
– 9 –
Figure 3: An incoming gauge boson is surrounded by a cloud of virtual particles. Only when the
gauge boson strikes the plasma puddle atmosphere can these virtual particles become real. The
characteristic length scale of quantum fluctuations is given by (λE)−1.
only allows decays into exactly collinear, massless decay products and since this is a set
of measure zero in phase space the decay rate vanishes. However, in the presence of
momentum violation (such as the plasma puddle atmosphere), noncollinearities are allowed
and so the decay rate is nonzero.
To estimate this rate, we invoke the uncertainty principle, which tells us that a photon
in the vacuum is immersed in a O(λ) cloud of virtual gauge bosons constantly coming in
and out of existence. Since the only relevant time scale for a massless particle in free space
is its energy E, the photon produces virtual particles of energy ∼ E at a rate Γ ∼ λE.
Indeed, once the photon strikes the plasma puddle atmosphere, the resulting momentum
violation allows for these virtual particles to become real4. Since Γ is the rate at which
photons are produced, and in the momentum violating background they can simply escape
to infinity, we expect that
ΓA→WW ∼ λE. (3.1)
The reader may suspect that this answer breaks down at large energy, and this is in
fact the case. There are several reasons why we might expect this to happen. First of
all, if the photon is arbitrarily boosted (relative to the rest frame of the plasma puddle),
it eventually reaches the threshold energy for plasma puddle creation, and so the correct
4This is a common effect, and is detailed in standard textbooks on quantum field theory [14]. For
example, consider the electron, which carries with it a number density of virtual photons given by Nγ ∼ α.
Bremsstrahlung effects from the scattering of an electron off of a charged target can be understood as
virtual photons becoming real due to a nontrivial background.
– 10 –
calculation would involve plasma puddle/plasma puddle scattering. This is akin to firing
a particle into a black hole at trans-Planckian energies, which is really the same as black
hole/black hole scattering.
Secondly, at very high energies theW decay products of the photon are so collinear that
their splitting is completely unmeasurable (by the plasma puddle) and is thus unphysical.
Let us illustrate this soft collinear divergence more explicitly. As noted previously, the
photon cannot decay into two exactly collinear W bosons due to phase space, but if it
receives of a momentum “kick”, then it can decay into W s which split in the transverse
direction. However, it can only decay in this way in the momentum violating background.
In the plasma puddle atmosphere, where the plasma itself is negligible, the only source
of momentum violation (position dependence) is the gradient of the Higgs field, so we
expect that the kick is proportional to 1/R. The angle subtended by the outgoing W ’s
is approximately k⊥/E, where k
∼ E/R is the transverse momentum difference between
the W ’s5. Demanding that the “cone” traced out by the outgoing particles grows to a size
greater than their Compton wavelength ∼ 1/m0, we find that
⇒ E < m20R. (soft collinear bound) (3.2)
If this bound is violated, then there is a soft collinear divergence and the splitting is
unmeasurable. Note that if the energies of the decay products are different, only the lesser
of the two need satisfy this bound.
In appendix A.4 we address these high energy issues directly by calculating the rate
of decay across the extent of the plasma puddle, yielding
ΓA→WW ∼ λE, (E < m20R) (3.3)
∼ λm20R, (E > m20R,Bremsstrahlung) (3.4)
, (E > m20R, symmetric decay), (3.5)
where “Bremsstrahlung” denotes the region in phase space in which one of the outgoing
particles is much softer than the other (that is, with energies of order m20R and E −m20R)
and “symmetric decay” denotes the regime in which they are comparably energetic (with
energies of order E). Since the probability of a symmetric decay is suppressed by a factor of
m20R/E relative to Bremsstrahlung, soft emission totally dominates showering at energies
above the threshold m20R. For example, consider an incoming particle boosted to some
high energy a hundred times greater than m20R. Emission of a soft particle of energy
m20R is a hundred times more likely than decay into two particles of comparable energy.
Moreover, once this Bremsstrahlung occurs, the harder particle only loses m20R energy, and
so a consecutive soft emission is still ninety nine times more likely than a symmetric decay,
and so on. Before long, the particle emits enough Bremstrahlung that its energy drops to
m20R and its decay rate begins to scale linearly with energy. In the following section we
consider the physical consequences of these rates.
5We estimate k⊥ as follows. We begin with an incident photon with 4-momentum (E, 0, 0, E) and assume
that it receives a momentum “kick” (0, 0, 0,−1/R). This allows it to split into two massless particles with
≈ E2 − (E − 1/R)2 ∼ E/R
– 11 –
3.2 Absorption and Eavg
In section 3 we saw that PA→WW increases linearly with E up to the scale m
0R, after
which it remains constant. Next, we show that this implies that an incident particle, no
matter how energetic, will always be absorbed by the plasma puddle.
To begin, let us consider showering in the E < m20R regime. An incident particle of
energy E will decay in the plasma puddle atmosphere at a rate λE. In turn, its decay
products have a smaller rate for decaying simply because they have less energy. In fact,
after a long sequence of decays the average energy of the final decay products eventually
diminishes to Eavg ∼ (λR)−1. At this point the decay rate dips below 1/R, and the ultra-
soft daughter particles can traverse the extent of the plasma puddle without showering.
Thus, showering completely terminates once the decay products reach an energy of (λR)−1.
Since the final decay products have a Compton wavelength proportional to the radius of
the plasma puddle, showering effectively de-localizes the incident particle over the entire
puddle. This picture matches nicely with Susskind’s ideas about black hole absorption
[9, 10].
In addition, as long as this terminal energy is less than m0, the daughter particles are
kinematically trapped by the Higgs profile, and they will eventually be thermalized. This
translates into a blackness bound
< m0, (3.6)
which must be satisfied if the plasma puddle is to absorb all incident matter.
We now argue that the above conclusions are still valid even above the threshold energy,
m20R. As we argued earlier, such a high energy particle favors the emission of soft particles
of energy m20R. However, because the decay rate for Bremsstrahlung remains constant
at high energies, one might wonder whether a sufficiently energetic particle might blast
through the plasma puddle. A careful analysis shows this is not the case.
In particular, let us very roughly estimate the rate of 1 → n showering for an incident
particle of energy E = nm20R, where n is some larger integer. Noting that a 1 → n decay
includes a sequence of i → i+ 1 sub-processes, and is often dominated by on-shell regions
of phase space, we can estimate the rate of 1 → n decay by
Γ1→n =
P1→2 × P2→3 × . . .× Pn−1→n, (3.7)
(λm20R
2)n, (3.8)
where schematically P ∼ ΓR (see appendix A, where we calculate these probabilities
explicitly). As we show in section 4, classical stability immediately implies that the quantity
in parentheses is greater than unity. Thus, we argue that an incident particle of energy
nm20R will decay into n particles of energy m
0R within a region of size R. After this
sequence of Bremsstrahlung events, the decay products all have energies of order m20R,
and so we can apply our analysis for the E < m20R regime to these decay products. Since
these particles decay to a multitude of ultra-soft particles of energy (λR)−1, we find that
total absorption occurs, no matter what the incident energy.
– 12 –
Interestingly, we are finding that large multiplicity events dominate at high energies,
which is reminiscent of Hawking evaporation – after all, the probability that a large black
hole will decay to two particles is extremely small, because it is suppressed by e−S , while
the decay to a huge number of soft particles is virtually guaranteed. Thus the necessity of
including 1 → n decays at high energies seems to indicate that high energy particles carry
a great deal of entropy.
4. Implications of d < R
In section 2.1 we argued that in the highly thermalized regime (d < R) the plasma puddle
emits thermal Hawking radiation at the surface. In this section we show that d < R also
implies total absorption, which implies classical stability, which implies satisfaction of the
Bekenstein entropy bound. 6 For this reason, we identify d < R plasma puddles as black
hole-like objects.
Applying our relations from section 2.1, we can rewrite the d < R bound, the blackness
bound (Eq. (3.6)), the classical stability bound (Eq. (2.3)), and the Bekenstein bound as
m0R > λ
−7/2, (highly thermalized) (4.1)
> λ−1, (total absorption) (4.2)
> λ−1/2, (classical stability) (4.3)
> λ1/2, (Bekenstein bound). (4.4)
The above sequence of bounds gives an interesting depiction of what happens during plasma
puddle evaporation. As the energy of the plasma puddle diminishes via Hawking radiation,
its size shrinks commensurately. First, the plasma puddle leaves the highly thermalized
regime and stops emitting thermal Hawking radiation. Next, the plasma puddle becomes
transparent to incident matter, and finally the object becomes classically unstable. After
this point, the enclosed photons and W bosons in the plasma simply stream outwards,
leaving a free gas of gauge bosons. It would be interesting to explicitly connect these
processes with the Horowitz-Polchinski transition [15], [16], [17].
It is also noteworthy that the Bekenstein bound is precisely equivalent to the condition
, (4.5)
which is necessary for the application of classical thermodynamics. As this bound is ap-
proached, the energy/size of the system begins to saturate the uncertainty principle, and
we are forced to count individual quantum mechanical microstates.
5. Discussion
In this paper we have argued for the existence of black hole-like objects living in large
N gauge theories at weak ’t Hooft coupling. Since these theories are completely pertur-
bative, we can calculate much of the physics. In particular, in the regime in which the
6The Bekenstein entropy bound gives an upper bound on the entropy of a system in terms of its gravi-
tational mass M and its size R: S < MR.
– 13 –
mean free path d is smaller than R, we find that these meta-stable puddles of plasma are
classically stable and emit radiation in a thermal spectrum. While these properties are
of course common to any hot star-like object, we moreover find that the plasma puddle
absorbs all incident matter, no matter how energetic. This occurs because high energy
particles invariably shower into ultra-soft decay products that are kinematically bound by
the effective potential from the spatially varying Higgs vev. All in all, we find this to be
compelling evidence that the plasma puddle is dual to a black object in a strongly coupled
gravitational theory.
In addition, our work gives a particularly nice picture of plasma puddle evaporation,
which may be connected to the Horowitz-Polchinski transition [15], [16], [17]. Indeed from
section 4 we saw that as the object radiates away energy, it eventually leaves the d < R
regime and stops emitting thermal Hawking radiation. After even more energy loss, the
plasma puddle stops being black, and eventually becomes classically unstable. Interestingly,
we also find that the d < R bound gives a nice lower bound on the total energy of a plasma
puddle, given by
M > λ−9/2N2m0. (5.1)
Since a plasma puddle must have at least this much energy to form a black hole-like object,
we might interpret the left-hand side as the CFT dual to the Planck mass.
Acknowledgements
We would like to thank Ofer Aharony and Nima Arkani-Hamed for helpful discussions, and
Toby Wiseman for correspondence. We also thank Jonathan Heckman for collaboration
when this work was at an early stage. Jared Kaplan is supported by a Hertz Foundation
Graduate Fellowship.
A. Detailed Absorption Computation
In this section we give a detailed derivation of the probability for the decay of a photon
to two W bosons as it passes through the plasma puddle atmosphere. Although we are
formally calculating a probability, we can re-interpret it as a decay rate via Γ ∼ P/R. To
begin, we consider a toy scalar model for simplicity. This scalar field theory has an action
∂φ2 + ∂χ2 − (m2(x3) + gφ)χ2. (A.1)
In this theory, a massless scalar field φ is coupled cubically to a scalar χ whose mass m2
is a nontrivial function of x3. Here φ and χ are scalar analogs of the photon and the W
boson, respectively, and m2 mimics the effects of a space varying Higgs vev on the W mass.
Notice that we have made the simplification that the plasma puddle is infinite in the x1
and x2 directions, which is a very good approximation if the plasma puddle is large and
the incoming particles approach from the x3 direction.
– 14 –
A.1 Propagator
Next, let us compute the χ propagator in the regime in which the incoming and outgoing
energies are much larger than the characteristic energy scale 1/R set by the mass profile.
To do this we use the WKB approximation to solve the wave equation7. The wave equation
is given by
χ = 0, (A.2)
where the space dependent mass is
m2(x3) = m
0(1−B(x3)), (A.3)
and B(x3) is a “bump” function which vanishes at infinity and peaks to unity in a compact
region of size R. From section 2.1 we know that B(r) goes as 1/r for large r, but for now
lets us take B(x3) to be a general function. Consider the following WKB ansatz solution:
χ(x) = exp(ip̃(x)x) (A.4)
p̃(x) ≡
p0, p1, p2, p3 +
f(x3)
, (A.5)
where f(x3) is a function that will be determined by plugging χ into the wave equation.
From now on, a tilde on a momentum variable will represent a nontrivial space dependence
of this kind.
If we plug this ansatz back into the wave equation, we find
−p2 +m20 +m20
−B + f + x3f ′ −
− ix3f
χ. (A.6)
If the quantity in curved parentheses vanishes, then χ is an eigenstate of the wave equation.
In the regime where p3 ≫ 1/R, the terms containing 1/p3 are small, since they go as inverse
powers of p3R, so we drop them. Now it is easy to solve for f in the resulting differential
equation
−B + f + x3f ′ = 0, (A.7)
yielding
p̃(x) =
p0, p1, p2, p3 +
2p3x3
B(x′3)dx
. (A.8)
Thus in the WKB approximation, the χ propagator is
Gχ(x, y) =
p2 −m20
× ei(p̃(x)x−p̃(y)y), (A.9)
to zeroth order in 1/(ER) and all orders in m20.
7If we include m0 as a mass insertion in Feynman diagrams, then our answer is necessarily a series
expansion in m0. Using the WKB approximation, we will be neglecting terms suppressed by higher powers
of 1/(ER), but keeping terms to all orders in m0.
– 15 –
A.2 Amplitude
Next, let us consider the amplitude for a φ particle of momentum p to decay into two χ
particles of momenta k and q in the spatially varying background. To do so we write down
the corresponding time ordered 3-point correlator in coordinate space
〈χ(y)χ(z)φ(w)〉 ∼ g
Gχ(y − x)Gχ(z − x)Gφ(x− w)d4x. (A.10)
Next, we Fourier transform to the variables p, k and q. In accordance with the LSZ
reduction formula, a scattering amplitude corresponds to the Fourier transform of the
appropriate time ordered correlator with the pole from each external leg stripped off.
Amputating the legs, we find that the Feynman amplitude is
M(k, q, p) = g ×
d4x ei(k̃(x)+q̃(x)−p)x, (A.11)
= g × δ(012)(k + q − p)F (k3, q3, p3) (A.12)
F (k3, q3, p3) =
dx3 e
i(k̃3(x3)+q̃3(x3)−p3)x3 . (A.13)
where g is the dimensionful coupling strength. Notice that in the m20 → 0 limit, F reverts
to a full 4-momentum conserving delta function, as expected. Next, let us evaluate F .
By separating a factor of exp i(k3 + q3 − p3)x in the integrand of F , it is clear that F
is simply the Fourier transform of the quantity
B(x′3)dx
. (A.14)
Let us consider the case in which the Higgs profile is a square bump function and so B is
unity for |x3| < R and 0 otherwise. Given this simplification the integral is easy to evaluate
piece-wise and F takes the simple form
F ∼ lim
sin(a/ǫ+Rb)
sinR(a+ b)
(a+ b)
, (A.15)
∼ πδ(a) − b
sinR(a+ b)
(a+ b)
, (A.16)
where we are using ǫ to regulate the δ function, and for convenience we have defined
a = k3 + q3 − p3, (A.17)
b = m20
. (A.18)
Since R is the largest length scale in the problem, we can actually simplify F even further,
writing
F ∼ δ(a) − b
δR(a+ b), (A.19)
where δR denotes an R-regulated delta function with width 1/R and height R. Physically,
F takes this form because it receives two contributions, corresponding to exact momentum
conservation and deviations from momentum conservation, set by the scale 1/R.
– 16 –
Also, note that our results are parametrically correct even if the Higgs profile differs
from the square bump function form which we have assumed. Looking at other forms for
B, we find it is only really necessary that B ≈ 1 in the region x3 ∈ [−R,R].
Next, let us integrate over phase space and compute the total probability of decay
using this amplitude.
A.3 Probability
The decay rate for a 1 → 2 process is given by
Γ ∼ 1
d3kd3q
|M|2. (A.20)
In translationally invariant theories, M is proportional to a 4-momentum conserving delta
function, so |M|2 is a product of squares of delta functions. While naively this introduces a
divergent δ(0) term, we normally divide Γ by the volume of spacetime, hence removing these
factors. However, in our case, F has a component that exactly conserves 4-momentum and
a component that violates momentum in the x3 direction by an amount 1/R. From basic
kinematics, we know that the contribution to Γ from the 4-momentum conserving piece
will not contribute, since a massless particle cannot decay into two massive particles in free
space. For this reason, we will only need to compute the momentum violating contribution
to |F |2. Since we are not dividing by the size of the x3 direction, we are actually computing
the decay rate integrated over all of x3, i.e. the total probability of decay (see appendix B
for details).
For an incoming momentum
p = (E, 0, 0, E), (A.21)
the (dimensionless) decay probability in our toy model is
Pφ→χχ ∼
δ(k0 + q0 − E)δ(k1 + q1)δ(k2 + q2)|F (k3, q3, E)|2, (A.22)
where the energies are
~k2 +m20, (A.23)
~q2 +m20. (A.24)
It is trivial to integrate over the transverse q momenta, after which we parameterize the
two transverse k momenta in polar coordinates as
(k1, k2) = k⊥(cos θ, sin θ) (A.25)
d3k = k⊥dk⊥dθdk3. (A.26)
From the energy conservation delta function we obtain the useful expressions
(E2 + k23 − q23), (A.27)
(E2 + q23 − k23). (A.28)
– 17 –
Applying these formulae, the delta function becomes
δ(k0 + q0 − E) =
× δ(k⊥ −K⊥), (A.29)
where K⊥ is defined as
(k3 + q3 − E)(k3 − q3 − E)(k3 + q3 + E)(k3 − q3 + E)−m2. (A.30)
The factor multiplying the delta function cancels with most of the integral, eliminating
all k⊥ dependence except for the delta function! Note that for a completely 4-momentum
conserving interaction, k3+ q3 = E, and so K
= −m2 < 0 and the k⊥ integral yields zero,
corresponding to the fact that a massless particle cannot decay into two massive particles
if 4-momentum is conserved. Since we lack momentum conservation in the 3-direction,
> 0 and the k⊥ integral instead gives unity. Consequently, we obtain the probability
of a φ particle to decay into two χ particles
Pφ→χχ ∼
dk3dq3|F (k3, q3, E)|2. (A.31)
The domain of integration is the compact region
k23 +m
q23 +m
0 ≤ E, (A.32)
as derived from energy conservation. This inequality is saturated when k⊥ is zero. Note
that we have made no approximations in evaluating this phase space integral, so the result,
written in terms of F , is correct up to numerical coefficients.
A.4 From Toy Scalars to Gauge Bosons
Given Pφ→χχ, it is straightforward to obtain PA→WW , the probability of a photon to
decay to two W ’s. The only parametric difference in the two calculations is that in the
gauge theory calculation, g becomes a dimensionless coupling and the three gauge boson
interaction has an extra factor of momentum due to the derivative coupling. This introduces
an extra factor of E2 into the decay probability. Moreover, since the outgoing W ’s can be
any of N gauge bosons, we also get a factor of N , yielding
PA→WW ∼ λ
dk3dq3|F (k3, q3, E)|2. (A.33)
Before we plug in for F , let us pause to note a high energy subtlety which was mentioned
earlier in terms of soft collinear divergences. Naively, we would be tempted to simply set
b = −a in F because of the delta function of a + b. However, since b is only fixed to be
equal −a up to a 1/R width, this replacement is only valid if b is of order 1/R or more,
i.e. if
, (A.34)
– 18 –
which is precisely the soft collinear bound derived in Eq. (3.2). As we will show, the energy
scaling of the decay probability is quite different above and below this bound.
First, let us consider the E < m20R regime. Setting b = −a and integrating |F |2 over
k3 and q3 (ignoring contributions from the 4-momentum conserving delta function), we
find that
PA→WW ∼ λ
dk3dq3 |δR (a+ b)|2 (A.35)
∼ λδR(0)
dk3, (A.36)
∼ λER, (E < m20R) (A.37)
which is our final answer for the probability of a photon to decay to two W bosons at
energies E < m20R.
Next, let us determine the probability of decay in the regime E > m20R, which we
will divide into two phenomenologically distinct regions of phase space. First, consider
the regime where k3 is less than m
0R but q3 is large enough that their sum, E, exceeds
m20R (obviously our result is symmetric under k3 ↔ q3). We will denote this as the
“Bremsstrahlung” regime because one of the outgoing particles is much softer than the
other. Since Eq. (A.34) obtains in this limit, the entire discussion of the previous para-
graphs applies except that the k3 integral is bounded by k3 < m
0R, and so
PA→WW ∼ λm20R2. (E > m20R, Bremsstrahlung) (A.38)
If we are instead interested in a “symmetric decay,” defined by k3, q3 > m
0R, then it
becomes necessary to include the b/a in our expression for F , and the answer becomes
PA→WW ∼
λm40R
. (E > m20R, symmetric decay) (A.39)
Note the crucial difference between these two regions of phase space – the first corre-
sponds to decay products of energy m20R and E − m20R while the second corresponds to
two decay products with energies roughly of order E.
A.5 Higher Order Effects and Approximations
Thus far we have only taken into account the role of three gauge boson interactions mediat-
ing binary decays in the plasma puddle atmosphere. However, it is fair to ask whether these
contribution necessarily dominate over the four gauge boson interactions, which mediate
trinary decays. In this section compare the relative sizes of PA→WWW and PA→WW→WWW,
and argue that the former contribution is subdominant.
Repeating our procedure from section A.2, we find the contribution to the decay am-
plitude from the four gauge boson interaction,
M(k, q, r, p) = g2 ×
d4x ei(k̃(x)+q̃(x)+r̃(x)−p)x. (A.40)
– 19 –
Again, this amplitude simplifies to the form δR(a+ b), where this time
a = k3 + q3 + r3 − p3, (A.41)
b = m20
. (A.42)
Summing over N2 possible final states, we find that
PA→WWW ∼ λ2ER. (A.43)
Despite multiple phase space integrals, the form of this answer is expected because |F |2 ∼
|δR(a + b)|2 yields precisely one factor of R, and factors of E make up the rest of the
expression.
Comparing, we see that the probability of one trinary decay PA→WWW is much less
than that of two consecutive binary decays
PA→WW→WWW ∼ (λER)2, (A.44)
so it is the latter that dominates the 1 → 3 decay. Thus, the dominant contribution to the
decay is binary and our expression from section 3.2 is valid. Similarly, loop effects are not
large even though the 1 → n rate is large at high energies, because only initial and final
states involve propagators that are almost on-shell.
B. Rates and Cross Sections
Normally, we are interested quantities such as the cross section and decay rate. However,
in our setup the “decay rate” is position dependent, and so to avoid this complication we
simply compute the total probability that the incident particle will decay. This requires
a slight revision of the standard formulas for converting S-matrix elements into physical
observables.
Let us begin with Weinberg’s [13] formula Eq. 3.4.9. Putting the universe in a box of
size L and in a time interval of size T , we have that
dP (α → β) ∼ 1
|Sβα|2dβ. (B.1)
In a 4-momentum conserving theory, the S-matrix element is defined as
Sβα ∼ iδ0123(pβ − pα)Mβα, (B.2)
where we identify δ(0) ∼ L. However, in our case momentum is violated in the x3 direction,
so we have that
Sβα ∼ iδ012(pβ − pα)F (pβ3 − pα3). (B.3)
Thus the cancellation of volume factors will not be complete. Instead
dP (α → β) ∼ T
|Mβα|2δ012(pβ − pα) |F (pβ3 − pα3)|2 dβ. (B.4)
– 20 –
Normally we would divide by T and take the L → ∞ limit, but in our case this sends the
probability to zero. This is just the statement that a particle in infinite volume will, on
average, take an infinite time to hit the domain wall. Similarly, given a finite volume, if
T → ∞ then probability will blow up.
Thus we should take T,L → ∞ with T/L held constant. It seems that this leads to an
arbitrary factor, but physically this factor should be one, since it just corresponds to the
number of times the particle crosses the domain wall.
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– 21 –
