# How Special Are Black Holes?
Correspondence with saturons in generic theories

Gia Dvali 1,2<EMAIL_ADDRESS>Oleg Kaikov 1,2
<EMAIL_ADDRESS>and Juan Sebastián Valbuena Bermúdez 1,2
1Arnold Sommerfeld Center, Ludwig-Maximilians-Universität,
Theresienstraße 37, 80333 München, Germany
2Max-Planck-Institut für Physik,
Föhringer Ring 6, 80805 München, Germany
<EMAIL_ADDRESS>
###### Abstract

Black holes are considered to be exceptional due to their time evolution and
information processing. However, it was proposed recently that these
properties are generic for objects, the so-called _saturons_ , that attain the
maximal entropy permitted by unitarity. In the present paper, we verify this
connection within a renormalizable $SU(N)$ invariant theory. We show that the
spectrum of the theory contains a tower of bubbles representing bound states
of $SU(N)$ Goldstones. Despite the absence of gravity, a saturated bound state
exhibits a striking correspondence with a black hole: Its entropy is given by
the Bekenstein-Hawking formula; semi-classically, the bubble evaporates at a
thermal rate with a temperature equal to its inverse radius; the information
retrieval time is equal to Page’s time. The correspondence goes through a
trans-theoretic entity of Poincaré Goldstone. The black hole/saturon
correspondence has important implications for black hole physics, both
fundamental and observational.

##  1 Introduction

Black holes are considered to be exceptional objects. This exceptionality
usually refers to their time-evolution, information storage, and retrieval.
However, it has been proposed recently [1] (see [2] for a summary) that these
properties are not specific to black holes. Rather, they are the universal
features of objects saturating a certain bound on the micro-state entropy that
shall be discussed below. This bound is imposed by unitarity.

This proposal heavily relied on the evidence gathered in two earlier papers
[4, 3]. There, objects such as solitons, instantons, and various bounds states
with high degeneracy of micro-states, were constructed. This gave a
possibility of increasing the micro-state entropy in a controllable way while
simultaneously monitoring its correlation with unitarity. In all cases, this
correlation pointed to a universal bound on entropy.

The bound can be presented in two equivalent forms [1]. The first form states
that for an arbitrary self-sustained object of size $R$ in $d$ space-time
dimensions, the unitarity imposes the following bound on entropy,

$S\leqslant\dfrac{\mathcal{A}rea}{G_{\text{Gold}}}\,,$ (1)

where ${\mathcal{A}rea}\sim R^{d-2}$ is the area of the sphere within which
the object is contained, and $G_{\text{Gold}}$ is the coupling of the
Goldstone field of a spontaneously broken symmetry. Such spontaneously broken
symmetries are always present whenever we are dealing with a macroscopic
object of high entropy. In particular, any “device” capable of storing quantum
information breaks the Poincaré symmetry spontaneously. Therefore, no
ambiguity exists in defining $G_{\text{Gold}}$. Note, the bound is valid in
arbitrary dimensions, including $d=2$, since the quantity “${\mathcal{A}rea}$”
is well defined.

The second form of the entropy bound (1) is written in terms of an effective
coupling $\alpha$ of the theory. In terms of this coupling, the bound imposed
by unitarity can be written in the following form,

$S\leqslant\dfrac{1}{\alpha}\,,$ (2)

where $\alpha$ has to be understood as a running coupling evaluated at the
scale $1/R$. The above bound is equivalent to (1). This can already be noticed
from the fact that the quantity

$\alpha_{\text{Gold}}\equiv\dfrac{G_{\text{Gold}}}{\mathcal{A}rea}$ (3)

represents an effective dimensionless coupling of the Goldstone boson,
evaluated at the scale $1/R$. Both bounds are saturated simultaneously [4].
Correspondingly the maximal entropy compatible with unitarity can be written
in the combined form,

$S_{\text{max}}=\dfrac{1}{\alpha}=\dfrac{\mathcal{A}rea}{G_{\text{Gold}}}\,,$
(4)

The coupling $G_{\text{Gold}}$ can be expressed in terms of the canonically
normalized Goldstone decay constant $f$ as,

$G_{\text{Gold}}\equiv f^{-2}\,.$ (5)

In this normalization, in $d=4$, in which we shall work in the present paper,
$f$ has the dimensionality of mass. For any self-sustained bound-states of $N$
quanta of wavelengths $R$, the coupling and the decay constant of a Poincaré
Goldstone are given by,

$G_{\text{Gold}}=f^{-2}=\dfrac{R^{2}}{N}\,.$ (6)

The objects saturating the entropy bounds (1) and (2) shall be referred to as
“saturons”. Now, the point of [1] is that saturons share the following
universal properties:

  * •

Of course, as already said, their entropy satisfies the area law, given by
(4).

  * •

If unstable, they decay with the rate which, up to $1/S$ corrections, gives
the thermal rate of temperature

$T\sim 1/R\,.$ (7)

  * •

In semi-classical treatment, they exhibit an information horizon.

  * •

The minimal time-scale required for the start of the information retrieval is
bounded from below by,

$t_{\text{min}}=\dfrac{{\mathcal{V}olume}}{G_{\text{Gold}}}=\dfrac{R}{\alpha}=S_{\text{max}}R\,,$
(8)

where ${\mathcal{V}olume}\sim R^{d-1}$.

It is obvious that the above properties are in full correspondence with the
properties of black holes, under the mapping,

$G_{\text{Gold}}\,\rightarrow\,G_{\text{N}}\,,$ (9)

where $G_{\text{N}}$ is Newton’s gravitational constant. The fundamental
meaning of this mapping is that in the case of a black hole, the Goldstone
boson of spontaneously broken Poincaré symmetry comes from the graviton, which
has the coupling $G_{\text{N}}$.

It is clear that under the transformation (9) the entropy (4) becomes the
Bekenstein-Hawking entropy [5, 6] of a black hole of radius $R$,

$S_{\text{BH}}\sim\dfrac{\mathcal{A}rea}{G_{\text{N}}}\,.$ (10)

We also notice that the same entropy can be written as [1, 7],

$S_{\text{BH}}\sim\dfrac{1}{\alpha_{\text{gr}}}\,,$ (11)

where $\alpha_{\text{gr}}=G_{\text{N}}/{\mathcal{A}rea}$ is the quantum
gravitational coupling in $d$ space-time dimensions, evaluated at the scale of
the momentum transfer $1/R$.

Next, the connection between (7) and Hawking radiation is very transparent.
Finally, we notice that under the mapping (9), the time-scale (8) gives a so-
called Page’s time for a black hole [8]. For a black hole, this time was
suggested to be the minimal time-scale of the start of information recovery.

The proposal of [1] says that none of the above properties are specific either
to black holes or to gravity. Rather, they are generic features of saturons in
arbitrary theories. Various aspects of this connection have been demonstrated
for solutions in spaces both with Lorentzian and Euclidean signatures, such as
solitons, instantons, and other bound states [4, 3, 1], including [9] bound
states in the Gross-Neveu model [10] on which we shall comment below.
Recently, it has also been argued [11] that a so-called “color glass
condensate” state of ordinary QCD [12] exhibits the same correspondence.

The correspondence between black holes and other saturons should not be
understood as the correspondence between different theories. Rather, the
correspondence is between the states in different theories. The starting point
which these states share is that they all saturate the entropy bounds (1) and
(2), imposed by unitarity of corresponding theories. The rest of the features
follow. In other words, saturation is a trans-theoretic notion.

The correspondence between black holes and generic saturons offers an obvious
benefit of “demystifying” black hole physics. Indeed, on the one hand,
saturons exist in theories that are renormalizable and calculable. On the
other hand, they exhibit all of the key properties of a black hole. Hence, by
studying saturons, we may explain the origin of known black hole properties
and, at the same time, predict new phenomena that can take place in black
holes. One such effect is the phenomenon of “memory burden” [13, 14]. The
essence of it is that quantum information can stabilize the system that
carries this information.

In the present paper, we shall provide further evidence of the black
hole/saturon correspondence. We shall analyze a simple four-dimensional theory
of saturons, constructed in [1]. The model exhibits a global $SU(N)$ symmetry
and has several degenerate vacuum states with various patterns of spontaneous
symmetry breaking. The theory is renormalizable and can be studied at
arbitrarily weak coupling and large $N$. The spectrum of the theory can be
reliably analyzed. It contains solitonic vacuum bubbles separating the vacua
with different patterns of broken symmetry.

In particular, there exists an infinite set of bubbles stabilized by the
memory burden effect. Each bubble represents a bound state of a large number
of Goldstone bosons. The occupation number of the Goldstone quanta in the
bound state will be denoted by $N_{G}$. These quanta cannot propagate in the
asymptotic vacuum outside a bubble since the symmetry is unbroken there. Due
to this, the Goldstones can only exist within a bubble. Therefore, if a bubble
decays, the conserved $SU(N)$ quantum numbers carried by the Goldstone modes
must be released in the form of massive quanta. This creates an energy barrier
that stabilizes the bubble in the semi-classical theory.

The levels of the spectrum can be labeled by $N_{G}$. Each level is
exponentially degenerate. Correspondingly, the bound states carry a high
micro-state entropy. We shall see that the bubbles with occupation number
$N_{G}\sim N$ are saturons. In particular, the entropy of such a bubble
saturates the bounds (1) and (2). Correspondingly, they have maximal entropy
(4) compatible with unitarity. We shall show that such a bound state assumes
all properties of a black hole listed above. We thus confirm the black hole /
saturon correspondence proposed in [1].

The fact that a saturated bound state of $N$ Goldstone bosons exhibits a
correspondence with a black hole represents supporting evidence for the idea
of the black hole $N$-portrait [7]. In this theory, the black hole is
described as a saturated bound state of $N$ gravitons. The features predicted
by this theory of gravitons are explicitly reproduced in a calculable
renormalizable theory of the Goldstone bound states presented in the present
paper. In particular, we observe how the information is carried away by $1/S$
($1/N$) corrections to Hawking radiation. The same corrections violate the
self-similarity of the evaporation process.

Finally, we wish to say that in a parallel paper [9], the black hole saturon
correspondence was demonstrated within the Gross-Neveu model [10, 15] (for a
review, see [17, 16]). This theory contains a set of fermions-bound states
with growing mass and degeneracy. It was shown [9] that the Gross-Neveu bound
state of the maximal degeneracy is a saturon and exhibits all properties of a
black hole discussed above. This is remarkable. Despite the fact that the
Gross-Neveu model is two-dimensional, its saturon captures all properties of
saturons (and correspondingly of black holes) of higher dimensional theories.
This includes the saturons in the four-dimensional $SU(N)$ symmetric model
discussed in the present paper. This fact illustrates the universality of
large-$N$ physics at the point of saturation. This is the reason why saturated
fermion bound states in Gross-Neveu and saturated Goldstone bound states in
the present model share strong similarities with each other and with black
holes. This fact provides non-trivial supporting evidence to the idea that
black hole properties are not specific to gravity but instead are defined by
the generic physics of saturation [1].

##  2 A model of a saturon

###  2.1 The model

Following [1], we consider a theory of a scalar field $\phi$ in the adjoint
representation of $SU(N)$, with $N\geq 3$. That is, $\phi_{\alpha}^{\beta}$ is
an $N\times N$ traceless Hermitian matrix with $\alpha,\beta=1,\dotsc,N$. The
Lagrangian density is

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\text{tr}\left[\left(\partial_{\mu}\phi\right)\left(\partial^{\mu}\phi\right)\right]-V\left[\phi\right]\,,$
(12) $\displaystyle{\rm where},\quad V\left[\phi\right]$ $\displaystyle=$
$\displaystyle\dfrac{\alpha}{2}\text{tr}\left[\left(f\phi-\phi^{2}+\dfrac{I}{N}\text{tr}\left[\phi^{2}\right]\right)^{2}\right]\,.$

Here, $I$ is the unit $N\times N$ matrix, $\alpha$ is a dimensionless coupling
and $f$ is the scale of symmetry breaking.

Notice, the theory is renormalizable. We shall restrict ourselves to the
regime in which the fundamental quantum coupling $\alpha$ is weak. However,
even for arbitrarily-weak $\alpha$, the unitarity imposes the following upper
bound on the strength of the coupling,

$\alpha\lesssim\dfrac{1}{N}\,.$ (13)

This bound is fully non-perturbative. The physical reason is that the
parameter that controls unitarity, both perturbatively as well as non-
perturbatively, is the collective coupling $\alpha N$. In the context of gauge
theories, such coupling is often referred to as the ’t Hooft coupling [18].

Regardless of smallness of $\alpha$, the unitarity is saturated when the
collective coupling $\alpha N$ becomes order one. This is signalled both by
the breakdown of loop-expansion as well as by saturation of unitarity by the
scattering amplitudes. For the detailed discussion we refer the reader to [1].
As shown in this work, the constraint (13) plays a fundamental role in
enforcing the upper bound on entropy (4). We shall present an explicit example
of saturation of this bound in the $SU(N)$ theory (12).

We shall work in the regime of weak $\alpha$ and large $N$. Interesting things
happen when the collective coupling $\alpha N$ approaches the unitarity bound
(13) from below. As we shall see, in this limit certain solitons saturate the
entropy bounds (1) and (2) and start behaving like black holes.

In order to identify such solitons, let us notice that the vacuum equations,

$f\phi^{\beta}_{\alpha}-(\phi^{2})^{\beta}_{\alpha}+\dfrac{\delta^{\beta}_{\alpha}}{N}\text{tr}\left[\phi^{2}\right]=0,$
(14)

admit several solutions corresponding to vacua with different unbroken
symmetries. These include the vacuum with unbroken $SU(N)$ symmetry, $\phi=0$,
and the vacua with spontaneous symmetry breaking (SSB) patterns
$SU(N)\rightarrow SU(N-K)\times SU(K)\times U(1)$, with $0<K<N$. By
construction, all of these vacua are degenerate in energy. For definiteness,
we focus on the vacuum $\phi=0$ and the one with $K=1$. In the second vacuum,
only the following component

$\phi_{\alpha}^{\beta}=\dfrac{\phi(x)}{\sqrt{N(N-1)}}\text{diag}\left((N-1),-1,\dotsc,-1\right)\,,$
(15)

has a non-zero expectation value

$\braket{\phi}=f\dfrac{\sqrt{N(N-1)}}{(N-2)}\simeq f\,,$ (16)

(since we shall be working at large $N$, when appropriate, we shall
approximate parameters by their leading order values). In this vacuum, the
symmetry group is $SU(N-1)\times U(1)_{Y}$, where the generator of $U(1)_{Y}$
is,

$\hat{Y}=\dfrac{1}{\sqrt{2N(N-1)}}\text{diag}\left((N-1),-1,\dotsc,-1\right)\,.$
(17)

In the vacuum with unbroken $SU(N)$ symmetry, there exist no massless
excitations, as the theory in this vacuum exhibits a mass gap,

$m=\sqrt{\alpha}f\,.$ (18)

In contrast, in the broken symmetry vacuum there exist

$N_{\text{Gold}}=2(N-1)$ (19)

different species (“flavors”) of massless Goldstone bosons, which we denote by
$\theta^{a}(x_{\mu})$ with $a=1,2,\dotsc,N_{\text{Gold}}$. They correspond to
broken generators $T^{a}$. It is convenient to represent these broken
generators as off-diagonal Pauli matrices. We can take

$(T^{a})_{\alpha}^{\beta}=\dfrac{1}{2}(\delta_{\alpha}^{1}\delta_{k+1}^{\beta}+\delta_{\alpha}^{k+1}\delta_{1}^{\beta})\,\quad\text{for}\leavevmode\nobreak\
a=2k-1\,,$ (20)

and

$(T^{a})_{\alpha}^{\beta}=\dfrac{-i}{2}(\delta_{\alpha}^{1}\delta_{k+1}^{\beta}-\delta_{\alpha}^{k+1}\delta_{1}^{\beta})\,\quad\text{for}\leavevmode\nobreak\
a=2k\,,$ (21)

where $k=1,\dotsc,N-1$.

The combinations, $T_{k}^{\pm}=T^{2k-1}\pm iT^{2k}$ form $(N-1)$-dimensional
fundamental and anti-fundamental representations under the unbroken $SU(N-1)$
symmetry group. Their charges $Y_{\pm}=\pm\frac{N}{\sqrt{2N(N-1)}}$ under the
$U(1)_{Y}$-symmetry are fixed by

$[\hat{Y},T_{k}^{\pm}]=Y_{\pm}T_{k}^{\pm}\,.$ (22)

Correspondingly, the combinations of Goldstones,
$\theta_{k}^{\pm}\equiv\theta^{2k-1}\pm i\theta^{2k}$, transform under the
same representations as the generators $T_{k}^{\pm}$. Obviously,
$\sum\limits_{a}\theta_{a}T^{a}=\frac{1}{2}\sum\limits_{k}\theta_{k}^{\mp}T_{k}^{\pm}$.

The effective low energy theory of the Goldstone modes can be obtained in the
standard way, by parametrizing the field as

$\phi_{\alpha}^{\beta}=\left(U^{\dagger}\braket{\phi}U\right)_{\alpha}^{\beta}$
(23)

with

$\quad U=\exp\left[-i\theta^{a}T^{a}\right]\,,$ (24)

and subtituting into the original action. Taking into account the commutation
relations and keeping terms up to second order in $\theta^{a}$, we obtain the
following effective Goldstone Lagrangian

$\mathcal{L}_{\text{eff}}=\dfrac{N^{2}}{4(N-2)^{2}}f^{2}\sum_{a}\left(\partial_{\mu}\theta^{a}\right)\left(\partial^{\mu}\theta^{a}\right)\simeq\dfrac{1}{4}f^{2}\sum_{k}\left(\partial_{\mu}\theta_{k}^{+}\right)\left(\partial^{\mu}\theta_{k}^{-}\right)\,.$
(25)

Now, due to gapless Goldstone excitations, an “island” of a broken symmetry
vacuum, embedded into a symmetric one, can carry a very high micro-state
entropy. Such islands are realized in the form of the vacuum bubbles which we
shall now consider.

###  2.2 Vacuum bubble

Since the vacua $SU(N)$ and $SU(N-1)\times U(1)_{Y}$ are degenerate, there
exist domain walls that separate the two. A planar infinite wall is static.
For such a wall, the component (15) of the adjoint field has the following
form,

$\phi(x)=\dfrac{f}{2}\left[1\pm\tanh\left(\dfrac{mx}{2}\right)\right]\,,$ (26)

where $x$ is a coordinate that is perpendicular to the wall. The thickness of
the wall is

$\delta_{\text{w}}\sim m^{-1}\,.$ (27)

Likewise, we can consider a closed bubble of radius $R\gg\delta_{\text{w}}$,
inside of which the $SU(N)$ symmetry is broken down to $SU(N-1)\times
U(1)_{Y}$. Outside of the bubble, the $SU(N)$-symmetry is unbroken. The exact
analytic solution for a finite radius bubble is not known. Moreover, such a
bubble experiences the force of tension directed inwards. If not counteracted
by an equal force, the finite size bubble will collapse.

For a slow-moving bubble wall, in the regime $R\gg\delta_{\text{w}}$, the
profile of the $\phi$ field can be approximated by

$\phi(r)=\dfrac{f}{2}\left[1+\tanh\left(\dfrac{m(R-r)}{2}\right)\right]\,.$
(28)

Adopting the standard terminology, we shall refer to this approximation as the
“thin-wall” approximation.

Due to SSB of the $SU(N)$ symmetry to $SU(N-1)\times U(1)$, the vacuum in the
interior of the bubble contains $N_{\text{Gold}}=2(N-1)$ gapless Goldstone
bosons. These Goldstone modes are “trapped” inside the bubble and cannot
propagate outside. This is because in an asymptotic vacuum outside of the
bubble, $r\gg R$, the symmetry is unbroken. There, the theory has a mass gap
(18) and no Goldstone excitations can penetrate this region. The existence of
gapless species in the bubble interior, endows the bubble with high capacity
of information storage [1]. The information can be stored in the excitations
of the Goldstone modes. This results in a memory burden effect that
counteracts the collapse.

##  3 Bubble stabilization by memory burden

We now wish to study how the effect of memory burden influences the evolution
of the bubble. The quantum information can be encoded in excitations of
Goldstone modes at relatively low energy cost. This quantum information can be
parameterized in form of sequences of excitation levels (occupation numbers
$n^{a}$) of the Goldstone bosons with distinct $SU(N)$ flavor quantum numbers.
We shall refer to distinct arrangements as the “memory patterns”.

When the bubble collapses and decays, this information must be carried away by
the outgoing quanta. However, the information pattern, released in the
$SU(N)$-symmetric vacuum, is much more costly in energy than when it is stored
in $S(N-1)\times U(1)$-symmetric vacuum inside the bubble. This is because no
gapless excitations exist in the $SU(N)$-symmetric vacuum. In fact, as we
shall see, the energy cost of the pattern outside the bubble can be larger
than the entire energy of the system. This creates a conflict.

Correspondingly, the information encoded in Goldstone modes must back react
and create a resistance against the decay. That is, the occupied Goldstone
modes should prevent the bubble from collapsing and decaying. The memory
burden effect is measured by the overall excitation of the Goldstone modes. We
may choose to excite a large diversity of modes to low levels each, or highly
excite a single mode. In the latter case, we can approximate a corresponding
Goldstone mode with a classical field. Below, we shall encounter both regimes.

Let us again consider the model (12) with the field given by

$\phi_{\alpha}^{\beta}=\left(U^{\dagger}\Phi U\right)_{\alpha}^{\beta}\,,$
(29)

where

$\Phi_{\alpha}^{\beta}=\dfrac{\varphi(\vec{x},t)}{\sqrt{N(N-1)}}\text{diag}\left((N-1),-1,\dotsc,-1\right)\,,$
(30)

and

$\quad U=\exp\left[-i\theta^{a}(\vec{x},t)T^{a}\right]\,.$ (31)

Substituting the above ansatz into (12), we obtain

$\mathcal{L}=\dfrac{1}{2}\left(\partial_{\mu}\varphi\right)\left(\partial^{\mu}\varphi\right)+\dfrac{N}{4(N-1)}\varphi^{2}\left(\partial_{\mu}\theta^{a}\right)\left(\partial^{\mu}\theta^{a}\right)-\dfrac{\alpha}{2}\varphi^{2}\left(\varphi-f\right)^{2}\,,$
(32)

where we have absorbed the $1/N$-corrections into the redefinitions of
parameters $f$ and $\alpha$.

We assume that the field $\varphi$ takes the initial configuration of a
finite-size bubble of the broken symmetry vacuum embedded in the symmetric
one. For definiteness, we shall assume spherical symmetry. That is, at some
initial time, the field is described by a configuration $\varphi(0,r)$ which
interpolates from a non-zero value at $r=0$ to zero at $r=\infty$. As already
said, such a bubble can carry a large amount of information stored in
excitations of the Goldstone fields $\theta^{a}$. These form a memory pattern.
If none of the Goldstone modes are excited, the information pattern is empty
and cannot affect the dynamics of the bubble. In such a situation, the bubble
collapses and after some oscillations decays into the asymptotic quanta of the
$SU(N)$-symmetric vacuum.

We now wish to study how a non-empty information pattern changes the story.
Depending on the occupation numbers of the individual Goldstone flavors, we
can distinguish the following two regimes.

When the individual occupation numbers are large, we can use Bogoliubov
approximation and replace the corresponding field operators by $c$-numbers.
This regime can be studied classically, by treating the highly excited
Goldstone modes $\theta^{a}$ as classical fields. In the opposite case, when
the individual occupation numbers are small, such a replacement is not
possible and we need to study the Goldsone modes in a quantum regime.

Notice that the quantum regime by no means implies that the memory burden is
weak. Although the individual occupation numbers are small, the diversity of
excited Goldstone flavors can be very high for large $N$. As a result, even a
classical bubble can be stabilized by the memory burden that is fully quantum.
As we shall see, this can happen if the bubble saturates the entropy bounds
(1) and (2). We shall discuss the classical regime first.

###  3.1 Classical regime

We shall look for a localized stationary solution in which we give the
Goldstone modes a time-dependence. To start with, we use the ansatz,

$\varphi=\varphi(r)\,,\leavevmode\nobreak\ \theta^{a}=\delta^{a1}\omega t\,,$
(33)

where $\omega$ is the frequency of rotation in the internal space. Only one of
the Goldstone modes is involved in this rotation. In other words, as
previously mentioned, we occupy macroscopically a single Goldstone flavor.

Since the Goldstones with labels $a\neq 1$ are all set to zero, the effective
Lagrangian (32) describes a full non-linear classical theory for the ansatz
(33). Notice, this is legitimate because all other components enter bi-
linearly (or in higher powers) in the Lagrangian. It is therefore consistent
to set them to zero without affecting the validity of the ansatz (33) 111 For
example, the effective Lagrangian obtained by the ansatz (29) in which the
matrix
$U=\begin{pmatrix}\cos(\frac{\theta^{1}}{2})\mathrm{e}^{-i\alpha}&-\sin(\frac{\theta^{1}}{2})\mathrm{e}^{i\beta}\\\
\sin(\frac{\theta^{1}}{2})\mathrm{e}^{-i\beta}&\cos(\frac{\theta^{1}}{2})\mathrm{e}^{i\alpha}\end{pmatrix}$
represents a generic $SU(2)$-transformation of the subgroup $SU(2)\times
SU(N-2)\times U(1)\subset SU(N)$, would give an effective Lagrangian (with the
same rescaling of $f$ and $\alpha$ as before),
$\mathcal{L}=\dfrac{1}{2}\left(\partial_{\mu}\varphi\right)\left(\partial^{\mu}\varphi\right)+\dfrac{N}{4(N-1)}\varphi^{2}\left(\partial_{\mu}\theta^{1}\right)\left(\partial^{\mu}\theta^{1}\right)+\dfrac{N\sin^{2}(\theta^{1})}{4(N-1)}\varphi^{2}\left(\partial_{\mu}(\alpha+\beta)\right)\left(\partial^{\mu}(\alpha+\beta)\right)-\dfrac{\alpha}{2}\varphi^{2}\left(\varphi-f\right)^{2}\,,$
for which $\alpha+\beta=0$ is a solution. .

Correspondingly, the only non-trivial equation is,

$\mathrm{d}_{r}^{2}\varphi+\dfrac{2}{r}\mathrm{d}_{r}\varphi+\varphi\left(\omega^{2}-\alpha(\varphi-f)(2\varphi-f)\right)=0\,.$
(34)

Here we have again absorbed the $N$-dependent pre-factors into the
redefinitions of $f$, $\alpha$ and

$\sqrt{\dfrac{N}{2(N-1)}}\omega\to\omega\,.$ (35)

A stationary bubble is described by a solution with the following boundary
conditions,

$\varphi(0)\neq 0\,,\leavevmode\nobreak\ \varphi(\infty)=0\,.$ (36)

Notice, the Goldstone fields of primary symmetry breaking, $SU(N)\rightarrow
SU(N-1)\times U(1)_{Y}$, transform non-trivially under $SU(N-1)\times
U(1)_{Y}$. Due to this, the ansatz (33) induces a secondary symmetry breaking
down to $SU(N-2)\times U(1)_{X}$, where the generator of $U(1)_{X}$ is given
by,

$\hat{X}=\dfrac{1}{\sqrt{N(N-2)}}\text{diag}\left((N-2)/2,(N-2)/2,-1,\dotsc,-1\right)\,.$
(37)

Due to this secondary symmetry breaking, there emerges an additional set of
Goldstone modes. Their number is $2N-3$. Note, however, that there is a mixing
between some internal Goldstones with the Goldstone boson of broken time-
translation symmetry. This is an important phenomenon but does not influence
the entropy count for large $N$.

Let us focus on the stationary classical configuration. If we mentally think
of $r$ as a time coordinate, the above equation is equivalent to the one
satisfied by a “coordinate” $\varphi$ of a particle moving in the external
potential

$V(\varphi)=\dfrac{1}{2}\varphi^{2}\left(\omega^{2}-\alpha(\varphi-f)^{2}\right)\,.$
(38)

For

$\dfrac{\omega^{2}}{\alpha f^{2}}\,<\,1\,,\leavevmode\nobreak\ \,\text{or
equivalently},\,\leavevmode\nobreak\ \omega^{2}\,<\,m^{2}\,,$ (39)

the above potential has two maxima, $\varphi_{0}=0,\,\leavevmode\nobreak\
\varphi_{\text{max}}=\frac{f}{4}(3+\sqrt{1+\frac{8\omega^{2}}{\alpha f^{2}}})$
separated by a minimum at
$\varphi_{\text{min}}=\frac{f}{4}(3-\sqrt{1+\frac{8\omega^{2}}{\alpha
f^{2}}})$. In addition to $\varphi_{0}=0$, the potential also becomes zero at
the points $\varphi_{\pm}=f\pm\frac{\omega}{\sqrt{\alpha}}$.

In this regime, the existence of a stationary bubble solution is rather
transparent. The “particle” starts at the “time” $r=0$ with some initial
position $\varphi(0)$. Notice, in order to tame the friction term, which
becomes singular at $r=0$, the initial velocity is fixed to be zero,
$\mathrm{d}_{r}\varphi(r)|_{r=0}=0$. This leaves the initial position
$\varphi(0)$ as the only adjustable initial condition. In order for the
particle to reach $\varphi_{0}=0$ in asymptotic future $r=\infty$, the initial
coordinate must satisfy $\varphi_{-}<\varphi(0)\leqslant\varphi_{\text{max}}$.
The initial acceleration is then determined from,

$\mathrm{d}_{r}^{2}\varphi|_{r=0}=\varphi(0)\left(-\omega^{2}+\alpha(\varphi(0)-f)(2\varphi(0)-f)\right)\,.$
(40)

It is clear that the solution with $\varphi(\infty)=0$ can be obtained by
continuously varying $\varphi(0)$. Indeed, taking $\varphi(0)=\varphi_{-}$ the
particle always undershoots, since it inevitably loses energy due to a non-
zero friction. On the contrary, by taking $\varphi(0)$ sufficiently close to
$\varphi_{\text{max}}$, we can make the initial acceleration arbitrarily
small, which will give the particle enough time not to move until the friction
becomes negligible. In such a case, the particle would reach $\varphi_{0}=0$
with non-zero kinetic energy within a finite time. Obviously, between the two
extremes there exists an initial $\varphi(0)$ which makes the friction just
right for reaching $\varphi_{0}=0$ for $r=\infty$.

The study of the bubble solution simplifies in thin-wall approximation, which
is given by,

$\dfrac{\omega^{2}}{\alpha f^{2}}\,\ll\,1\,,\leavevmode\nobreak\ \,\text{or
equivalently},\,\leavevmode\nobreak\ \omega^{2}\,\ll\,m^{2}\,.$ (41)

The above implies that the Goldstone frequency $\omega$ is much smaller than
the mass gap $m$ in the symmetric vacuum. The quantum meaning of this regime
shall become more transparent later.

In the above regime, the energy splitting between the maxima
$\varphi_{\text{max}}\simeq f(1+\frac{\omega^{2}}{\alpha f^{2}})$ and
$\varphi_{0}=0$ is approximately given by

$V_{\text{max}}\simeq\dfrac{1}{2}\omega^{2}f^{2}\left(1+\dfrac{\omega^{2}}{\alpha
f^{2}}\right)\,.$ (42)

In the thin-wall approximation, for a given $\omega$ the radius $R$ of the
bubble can be found by extremizing the action

$S=4\pi\int\limits_{0}^{\infty}r^{2}\,\mathrm{d}r\,\dfrac{1}{2}\left(\mathrm{d}_{r}\varphi(r)\right)^{2}-V\left(\varphi(r)\right)\,,$
(43)

evaluated on the bubble solution $\varphi(r)$. In the thin-wall regime, the
particle spends the time interval $0<r<R$ near the initial value close to the
maximum $\varphi_{\text{max}}$. During this time, $\varphi(r)$ is essentially
constant and the main contribution to the action comes from the potential
energy at the maximum. This portion of the action is therefore given by
$S_{\text{int}}\simeq-\frac{4\pi}{3}V\left(\varphi_{\text{max}}\right)R^{3}\simeq-\frac{2\pi}{3}\omega^{2}\frac{m^{2}}{\alpha}R^{3}$.
The bulk of the transition from $\varphi_{\text{max}}$ to $\varphi_{0}=0$ is
completed during the time interval $\Delta r=m^{-1}$, which sets the thickness
of the bubble wall. Correspondingly, this part of the action is given by,
$S_{\text{wall}}\simeq\frac{2\pi}{3}\frac{m^{3}}{\alpha}R^{2}$. Thus, the
total action on thin-wall bubble solution is,

$S_{\text{bubble}}=\dfrac{2\pi}{3}\frac{m^{3}}{\alpha}\left(R^{2}-\dfrac{\omega^{2}}{m}R^{3}\right)\,.$
(44)

Extremizing this expression with respect to $R$, gives the following value of
the radius of a stationary bubble,

$R=\dfrac{2}{3}\frac{m}{\omega^{2}}$ (45)

Notice that for $\omega\ll m$, we have,

$R\gg\frac{1}{m}\,.$ (46)

Of course, this matches the fact that in the thin-wall approximation the
bubble radius is much larger than the thickness of the wall, which is given by
$m^{-1}$.

The solution $\varphi(r)$ obtained in this way, describes a stationary
spherically-symmetric bubble in Minkowski space. The energy of such a bubble
consists of the energy of the wall tension and the energy of the interior
(both positive),

$E_{\text{bubble}}=E_{\text{int}}+E_{\text{wall}}=\dfrac{\omega}{\alpha}\,\frac{m^{5}}{\omega^{5}}\,\left(\frac{40\pi}{81}\right)\,,$
(47)

where,

$E_{\text{int}}=\dfrac{2}{3}E_{\text{wall}}\,.$ (48)

###  3.2 Quantum picture of classical stability

The wall of the bubble is predominantly made of the radial excitations of the
$\varphi$ field of mass $m$, whereas the interior is predominantly made of
Goldstone excitations of frequency $\omega$. It is therefore convenient to
write the two energies in terms of the occupation numbers of the corresponding
quanta,

$E_{\text{int}}=\omega N_{G}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\text{where},\,\leavevmode\nobreak\ \leavevmode\nobreak\
N_{G}\equiv\dfrac{1}{\alpha}\dfrac{m^{5}}{\omega^{5}}\left(\dfrac{16\pi}{81}\right)\,,$
(49)

and

$E_{\text{wall}}=mN_{\varphi}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\text{where},\,\leavevmode\nobreak\ \leavevmode\nobreak\
N_{\varphi}\equiv\dfrac{1}{\alpha}\,\dfrac{m^{4}}{\omega^{4}}\,\left(\dfrac{8\pi}{27}\right)\,.$
(50)

We see that,

$\dfrac{N_{G}}{N_{\varphi}}=\frac{2}{3}\frac{m}{\omega}\gg 1\,.$ (51)

That is, the thin-wall bubble contains much higher occupation number of the
Goldstones than of the massive $\varphi$-quanta.

Thus, we arrive to the conclusion that a stationary bubble is obtained thanks
to the excitations of the Goldstone modes. Let us try to understand this
effect in the quantum language. In the thin wall regime (41), the Goldstone
frequency $\omega$ in the bubble interior is much less than the mass of the
quanta $m$ in the asymptotic vacuum outside of the bubble. Now, the interior
energy (49), which amounts to $2/3$ of the wall energy, is given by the
Goldstone excitations.

The bubble is stable because of two factors: 1) The fact that the Goldstone
$SU(N)$ charge is conserved; and 2) the fact that the same amount of charge in
the exterior vacuum would cost much higher energy.

If the bubble would decay, the $SU(N)$ charge, stored in the Goldstone modes
in the interior of the bubble, must be released into the asymptotic
$SU(N)$-symmetric vacuum. But, this vacuum contains no gapless excitations and
the lowest energy cost per particle is $m$. Carrying the entire charge in the
form of massive particles would cost more energy than the energy of the
bubble. This is the source of stability.

##  4 Closer look at Goldstones

###  4.1 Goldstones of internal symmetry

The spontaneous breaking of global $SU(N)$ symmetry by the solution (33)
results into the appearance of Goldstone species localized in the bubble
interior. As already discussed, at large $N$, their number scales as $N_{\rm
Gold}\simeq 4N$. These Goldstone species possess the tower of eigenmodes
$\theta^{a}_{\epsilon}$ of various eigenfrequencies $\epsilon$. In the limit
of an infinitely large bubble, these eigenmodes reproduce the momentum modes
of free Goldstone plane waves.

The exact form of mode functions can be obtained by the standard mode analysis
of linearized fluctuations in the background of the classical solution. This
will not be presented here. Instead, we shall focus on the resulting
eigenmodes of two specific frequencies.

The modes $\theta_{0}^{a}$ corresponding to $\epsilon=0$ have zero
frequencies. These modes account for the degeneracy of the bubble “vacuum”
with respect to broken $SU(N)$ transformations. In other words, by acting on a
bubble with a global $SU(N)$ transformation, we excite a set of corresponding
gapless Goldstone modes. Of course, since these modes are gapless, their
excitations cost no energy. This accounts for strict degeneracy of bubbles
that are related by global $SU(N)$ transformations. Such bubbles form
irreducible representations of $SU(N)$ symmetry.

The second set of harmonics $\theta_{\omega}^{a}$ correspond to
eigenfrequencies $\epsilon=\omega$. These harmonics represent the actual
constituents of the bubble. On the bubble solution, at least some of them have
non-zero occupation numbers. When we act on a bubble by an $SU(N)$
transformation, we change the occupation numbers of $\theta_{\omega}^{a}$
modes without affecting the energy of the state.

To summarize, various Goldstone modes contribute into the portrait of the
stationary bubble as follows. The energy of the interior of the bubble (49)
represents the energy of a state of Goldstone bosons $\theta_{\omega}^{a}$ of
frequencies $\omega$ and a total mean occupation number, $N_{G}$. In the
original ansatz (33), this entire occupation number was attributed to a single
mode $a=1$. However, it can be arbitrarily redistributed among the Goldstones
of different flavors due to $SU(N)$ symmetry of the theory. All the solutions
obtained by the $SU(N)$ transformation are degenerate in energy. This
degeneracy is accounted for by the presence of gapless Goldstone modes
$\theta_{0}^{a}$.

The occupation numbers can be redistributed arbitrarily among the excited
primary Goldstones of frequency $\omega$ subject to the constraint (we use
large-$N$ approximation),

$\sum\limits_{a}^{2N}n^{a}=N_{G}\,.$ (52)

Each sequence of such numbers $n^{a}$, represents a memory pattern,

$\ket{Pattern}=\ket{n_{\omega}^{1},n_{\omega}^{2},\,\dotsc}\,,$ (53)

All of them amount to the classical solutions that are degenerate in energy.
All such information patterns cost the energy given by (49). That is,
$E_{\text{int}}$ is the energy of the information pattern encoded in the
$SU(N)$ charge via the excitations of Goldstones of frequency $\omega$.

Now, it is important to understand the following distinction. Let us consider
the two patterns, $\ket{n_{\omega}^{1},n_{\omega}^{2},\,\dotsc}$ and
$\ket{n_{\omega}^{\prime 1},n_{\omega}^{\prime 2},\,\dotsc}$ for which the
differences $\Delta n_{\omega}^{a}=|n_{\omega}^{a}-n_{\omega}^{\prime a}|$
among some occupation numbers are large. To be more precise, let us say that
for some $a$-s, the ratio $\frac{\Delta n_{\omega}^{a}}{N_{G}}$ is nonzero for
$N_{G}\rightarrow\infty$. Then, such patterns are distinguishable in classical
theory.

In addition to these, the set of all possible degenerate patterns contains
subsets which have small differences. Such patterns are not distinguishable
classically. Of course, they nevertheless contribute into the count of quantum
micro-states. Correspondingly, the memory burden effect has a part that is
classically observable and a part that can only be resolved in a quantum
theory.

Notice, as it is clear from (49) in the thin-wall approximation (41) we have,

$N_{G}\sim\dfrac{1}{\alpha}\dfrac{m^{5}}{\omega^{5}}\gg\dfrac{1}{\alpha}\sim
N_{\text{Gold}}\,.$ (54)

Now remembering that the number of Goldstone flavors scales as
$N_{\text{Gold}}\sim N$, and that unitarity puts the upper bound $N\sim
1/\alpha$, we get that for a thin-wall bubble,

$N_{G}\gg N_{\text{Gold}}\,.$ (55)

That is, in thin-wall regime the occupation number of the Goldstone modes is
much larger than the number of their species. Due to this, the main
stabilizing force is the classical part of the memory burden. For smaller
bubbles, the situation changes dramatically and the quantum part of the memory
burden becomes as important as the classical one.

If the bubble decays, the information stored in the memory pattern (53) must
be released in the form of asymptotic quanta of mass $m$. Since the $SU(N)$
charges are conserved, the minimal energy cost of such an asymptotic state is

$E_{\text{pattern}}=mN_{G}\,,$ (56)

where $N_{G}$ is given by (49). This can be rewritten as,

$E_{\text{pattern}}\,=\,mN_{G}\,=\,m\,\dfrac{1}{\alpha}\,\dfrac{m^{5}}{\omega^{5}}\left(\dfrac{16\pi}{81}\right)\,=\,\dfrac{2m}{5\omega}\,E_{\text{bubble}}\,.$
(57)

Thus, the energy cost of the information pattern released in case of a bubble
decay would exceed the energy of the bubble by a factor $2m/(5\omega)$. This
is of course impossible. It is therefore energetically more favourable to
store this amount of charge in the form of the bubble, rather than to release
it in the form of the asymptotic quanta.

One may wonder whether instead of free quanta the Goldstone charge can be
released in the form of the smaller bound states with more effective mass-to-
charge ratio. However, this does not appear plausible. Due to the weak
coupling $\alpha$ it takes at least $\sim\frac{1}{\alpha}$ particles to make
up a bound state of size $R\sim 1/m$. This is the smallest possible size for a
bound state due to the energy scale in the problem. Such a bound state will
have energy $\sim m/\alpha$ and a charge capacity $\sim 1/\alpha$. In fact, it
would represent a smallest size vacuum bubble. It is clear that the charge of
the large bubble cannot be released in the form of such objects. Basically, a
bubble cannot decay into smaller bubbles. The stationary bubble is
energetically the optimal configuration for the given charge, at least among
the spherically symmetric ones. This is also confirmed by our numerical
analysis.

Although we did not provide a proof, it is rather natural that for a bubble
with zero angular momentum, the departure from the spherical symmetry cannot
lower the energy, since there is no parameter in the theory that would set the
measure of asymmetry for the lowest energy configuration, which must exist due
to impossibility of the decay into the free quanta.

The important thing is that the bubble is an existence proof of the bound
state that stores the $SU(N)$ charge in a more energy-efficient way than the
collection of the asymptotic quanta. Thus, there exist bound states stabilized
by the memory burden effect.

In this sense, the stabilized bubbles can be considered as sort of non-
topological solitons, or $Q$-balls [19, 20]. There exists a large literature
on the subject. In particular, $Q$-balls with arbitrarily small classical
charges were analyzed in [21]. The physical effect, such as catalysis of
proton decay, due to symmetry breaking inside the $Q$-ball interior were also
studied [22]. In this sense, it can be said that the present paper also sheds
a very different light on the mechanism of $Q$-ball stability, by interpreting
it as a sort of memory burden effect.

At the same time, the vacuum bubbles considered here are qualitatively
different from the previously studied cases of $Q$-balls. First of all,
bubbles carry a large micro-state entropy which can saturate the unitarity
bound. Due to this, the information encoded in the Goldstone modes that
stabilizes the bubble, can be intrinsically quantum.

This form of stabilization takes place in thick-wall bubbles, for which
$\omega\sim m\sim 1/R$, where the quantum contribution into the memory burden
becomes as important as the classical one. This is linked with the fact that
such bubbles saturate the entropy bounds (1) and (2). As saturons, they
satisfy $N_{G}\sim N_{\text{Gold}}\sim N\sim 1/\alpha$.

###  4.2 Goldstones of broken Poincaré symmetry

Let us now focus on the Goldstone bosons of broken Poincaré symmetry. They
consist of broken space and time translations. Of course, correspondingly, the
Lorentz boosts are also broken. Let us consider a thin-wall bubble first.

The different constituents of the bubble share the breaking of space-time
symmetries as follows. The spontaneous breaking of space translations comes
predominantly from the bubble walls. The main weight of breaking is carried by
the $\varphi$-quanta of occupation number $N_{\varphi}$, each contributing $m$
in the order parameter. Therefore, up to order-one numerical factors, the
couplings of the space-translation Goldstone bosons are,

$G_{\text{Gold}}^{(\text{s})}=\dfrac{R}{mN_{\varphi}}\,.$ (58)

In contrast with space translations, the dominant contribution into the
breaking of time translation symmetry comes from the interior of the bubble.
This is due to non-zero frequency $\omega$ of Goldstone modes of broken
$SU(N)$-symmetry. Their total occupation number is $N_{G}$. Each occupied
quantum contributes $\omega$ into the breaking of time-translations. The
coupling of the time translation Goldstone mode therefore is,

$G_{\text{Gold}}^{(\text{t})}=\dfrac{R}{\omega N_{G}}\,.$ (59)

From the relation (51) between the occupation numbers we have,

$\omega N_{G}\sim mN_{\varphi}\,.$ (60)

It is therefore clear that the couplings of Goldstones of broken space and
time translations are of the same order,

$G_{\text{Gold}}^{(\text{s})}\sim
G_{\text{Gold}}^{(\text{t})}\sim\frac{R}{\omega N_{G}}\,.$ (61)

This is not surprising, since the bubble walls that break space translations
are stabilized due to breaking of time translation by its interior.

The relation (60), and correspondingly (61), holds for an arbitrary
classically-stable bubble. This includes the thick-wall regime. Therefore, the
coupling of a generic Goldstone of broken Poincaré symmetry can be expressed
in terms of the occupation number of $SU(N)$ Goldstones as,

$G_{\text{Gold}}^{(\text{P})}\sim\dfrac{R}{\omega}\dfrac{1}{N_{G}}\sim\sqrt{\dfrac{R^{3}}{m}}\dfrac{1}{N_{G}}\,,$
(62)

where we took into account the relation (45) between $\omega$, $m$ and $R$.

Now, the dimensionless effective coupling of the Poincaré Goldstone,
$\alpha_{\text{Gold}}^{(\text{P})}$, evaluated at the energy scale $1/R$
(corresponding to the size of bubble) is,

$\alpha_{\text{Gold}}^{(\text{P})}\equiv
G_{\text{Gold}}^{(\text{P})}\dfrac{1}{R^{2}}\,=\,\dfrac{G_{\text{Gold}}^{(\text{P})}}{{\mathcal{A}rea}}\,=\,\dfrac{1}{\sqrt{mR}}\dfrac{1}{N_{G}}\,=\dfrac{\omega}{m}\dfrac{1}{N_{G}}\,,$
(63)

where, as before, ${\mathcal{A}rea}\sim R^{2}$ is the area of the bubble.

Using (62) and (63) we can present the entropy bounds (1) and (2) for a bubble
in terms of the occupation number of the $SU(N)$ Goldstone modes,

$S_{\text{max}}\,\sim\,\dfrac{1}{\alpha_{\text{Gold}}^{(\text{P})}}\sim\dfrac{{\mathcal{A}rea}}{G_{\text{Gold}}^{(\text{P})}}\,\sim\,N_{G}\,\sqrt{mR}\,\sim\,N_{G}\,\dfrac{m}{\omega}\,.$
(64)

The above represents the maximal entropy that can be attained by a bubble in
an unitary theory. As we shall see, only the thick-wall bubbles are able to
saturate this bound.

##  5 Spectrum of bubbles

Our previous analysis shows that bubbles represent the bound-states of
Goldstone bosons, with occupation number $N_{G}$. The energy spectrum of
bubbles can be expressed as a function of $N_{G}$. For thin-wall bubbles we
have,

$E_{N_{G}}\,=\,\frac{5}{2}\,\omega\,N_{G}\,=\,\dfrac{5}{\sqrt{6}}\,\sqrt{\dfrac{m}{R}}\,N_{G}\,.$
(65)

Up to an order-one coefficient, the same relation holds for the thick-wall
bubbles for which $m\sim 1/R$. As we shall see later, only thick-wall bubbles
can saturate the unitarity bound on entropy. This happens in the regime,
$N_{G}\sim N\sim 1/\alpha$. The energy of a saturon bubble is thus given by,

$E_{N}\,\sim\,\frac{N}{R}\,\sim\,\frac{1}{\alpha}\,\frac{1}{R}\,\sim\,\frac{m}{\alpha}\,.$
(66)

Thus, a saturon represents a bound state of $N_{G}\sim N$ Goldstone bosons and
of the same order $N_{\varphi}\sim N$ radial modes.

##  6 Entropy of a bubble

Let us estimate the entropy of a bubble. The number of degenerate micro-states
is given by all possible sequences (53) satisfying the constraint (52). Using
Stirling approximation, and working at leading order in large $N_{G}$ and $N$,
this number can be written as

$n_{\text{st}}\sim\left(1+\dfrac{2N}{N_{G}}\right)^{N_{G}}\left(1+\dfrac{N_{G}}{2N}\right)^{2N}\,.$
(67)

As explained in [3, 4, 1], the above expression for the number of degenerate
micro-states is common for solitons that spontaneously break global symmetry.
It can be understood in the following two complementary languages.

Internal view:

From the point of view of the world-volume theory of the soliton, the
degeneracy (67) accounts for the degeneracy of the Goldstone vacuum. An
observer located in the interior of the soliton (in the present case, a
bubble), sees a spontaneous breaking of global $SU(N)$ symmetry with the
resulting gapless Goldstone modes. Just as any other Goldstone vacuum, this
vacuum is degenerate. However, since the bubble has a finite size, the number
of independent orthogonal vacuum states is finite rather than infinite. This
degeneracy becomes infinite in the limit of a bubble of an infinite extent. In
this limit, the bubble vacuum becomes an ordinary Goldstone vacuum of four-
dimensional theory.

To summarize, according to internal view, the different micro-states counted
in (67) are Goldstone vacua that are obtained from one another by the action
of $SU(N)$ transformations.

External view:

From the point of view of an external observer, the same degeneracy is
explained differently. This observer lives in the asymptotic vacuum for which
$SU(N)$ is a good symmetry. Correspondingly, all the states can be classified
according to representations of the $SU(N)$ group.

Thus, the external observer sees the bubble as an object transforming under
one of such representations. This representation is exponentially large. This
is because the bubble represents a bound state of large number of quanta each
transforming under the adjoint representation of $SU(N)$. Let the total
occupation number of quanta be $N_{T}$. Then, the bubble transforms as a
tensor product of $N_{T}$ adjoints. The wave function of the bubble can be
written as a tensor,

${\mathcal{B}}_{\alpha_{1},\alpha_{2},\,\dotsc\,,\alpha_{N_{T}}}^{\beta_{1},\beta_{2},\,\dotsc\,,\beta_{N_{T}}}\,,$
(68)

which is totally symmetric under lower and upper indices and has a zero trace
with respect to each conjugated pair. To the leading order, the dimensionality
of such a tensor is given by the square of the binomial coefficient. Using the
Stirling approximation it can be written as,

$n_{\text{st}}\sim\left(1+\dfrac{2N}{N_{T}}\right)^{N_{T}}\left(1+\dfrac{N_{T}}{2N}\right)^{2N}\,.$
(69)

Now, our previous analysis shows that $N_{T}$ is well approximated by $N_{G}$.
Indeed, as it is clear from (51), for the thin-wall bubble (41), the
occupation number of Goldstones $N_{G}$ dominates over the number of other
constituents. So, in this regime we have $N_{T}\simeq N_{G}$. For the thick-
wall regime, the occupation number of the non-Goldstone quanta, $N_{\varphi}$,
becomes comparable to $N_{G}$. Thus, without loss of generality we can write
$N_{T}\sim N_{G}$. It is then clear that the expression (69) gives the same
count of the micro-states as (67).

The corresponding micro-state entropy is given by,

$S=\ln(n_{\text{st}})\simeq
2N\,\ln\left[\left(1+\lambda\right)^{\frac{1}{\lambda}}\left(1+\dfrac{1}{\lambda}\right)\right]\,.$
(70)

where we have introduced a notation $\lambda\equiv 2N/N_{T}\sim 2N/N_{G}$.
This quantity represents the ratio of the number of primary Goldstone species
to their occupation number in the bubble state. This ratio decides how far is
the entropy of the system from saturating the bounds imposed by unitarity. We
shall study this question next.

##  7 Saturation

We now wish to investigate, in which regime a bubble saturates the bounds on
entropy (1) and (2). Naively, it may appear that the entropy (70) can be made
arbitrarily large, at the expense of increasing $N$. However, this is not the
case, since $N$ and $\alpha$ are “locked” in the unitarity constraint (13).
Due to this, $N$ reaches its maximum when the ’t Hooft coupling of the theory
is taken at its unitarity upper bound,

$N\alpha\sim 1\,.$ (71)

As explained in [1], any further growth of the ’t Hooft coupling beyond this
point, must lead to the change of the regime. This is signalled both by the
breakdown of the loop expansion, as well as, by saturation of unitarity by
various scattering amplitudes.

Now, since the number of Goldstone species $N_{\text{Gold}}$ is of order $N$,
the condition (71) implies,

$N_{\text{Gold}}\,\sim\,N\,\sim\,\dfrac{1}{\alpha}\,.$ (72)

The entropy of the bubble must be evaluated subject to this constraint. In
what follows, we shall perform this evaluation in two different regimes.

###  7.1 Entropy of thin-wall bubbles

Let us start by considering the thin-wall regime. From (49) it is clear that
for a thin-wall bubble we have,

$N_{G}\,\gg\,\frac{1}{\alpha}\,.$ (73)

Taking into account the expressions (71) and (72), it becomes obvious that in
a thin-wall bubble we have $\lambda\ll 1$. That is, the thin-wall bubble
satisfies,

$N_{G}\gg N_{\text{Gold}}\sim N\,\sim\dfrac{1}{\alpha}\,.$ (74)

Therefore, in this regime the expression (70) becomes,

$S\,\simeq\,2N\,\ln\left(\dfrac{\mathrm{e}}{\lambda}\right)\,\sim\dfrac{1}{\alpha}\,\ln\left(\dfrac{m^{10}}{\omega^{10}}\right)\,.$
(75)

In the last term we took into account (71), expressed $N_{G}$ as function of
$\alpha,\omega,m$ via (49) and ignored order-one numerical factors
($8\mathrm{e}\pi/81\sim 1$) in the logarithm.

The actual entropy of the bubble (75) is much smaller than the upper bound (1)
applied to the same bubble. Indeed, expressing $N_{G}$ via (49), the maximal
entropy (64) of a thin-wall bubble, permitted by the Poincaré Goldstone is,

$S_{\text{max}}\sim\dfrac{1}{\alpha}\dfrac{m^{6}}{\omega^{6}}\,.$ (76)

Obviously, for a thin-wall bubble ($m/\omega\gg 1$) this is much larger than
the actual entropy (75), which scales only logarithmically with $m/\omega$.
Thus, a thin-wall bubble represents an under-saturated state in terms of its
entropy capacity.

The same conclusion can be reached by examining the entropy bounds posed by
$SU(N)$ Goldstones. One can see this, by repeating the logic of [1]. Let us
consider an effective dimensionless coupling of $SU(N)$ Goldstones, evaluated
at the scale $1/R$. It is equal to,

$\alpha_{\text{Gold}}^{(SU(N))}\,=\,\dfrac{1}{(fR)^{2}}\,\sim\,\alpha\dfrac{\omega^{4}}{m^{4}}\,.$
(77)

This coupling controls the scattering processes of Goldstones with momentum-
transfer $\sim 1/R$. Let us now put ourselves in the position of an observer
living in the effective low energy theory of Goldstones with the cutoff
$\Lambda_{\text{cutoff}}\sim 1/R$.

For the consistency of this limited sector, the observer only needs to make
sure that the effective theory is unitary up to the scale $\sim 1/R$. With
this minimal requirement, the observer has the “luxury” of introducing as many
as $N_{\text{Gold}}\sim 1/\alpha_{\text{Gold}}^{(SU(N))}$ species of
Goldstones. Such a large diversity of Goldstone flavors would endow the bubble
with the micro-state entropy,

$S_{\text{max}}\sim\dfrac{1}{\alpha}\,\dfrac{m^{4}}{\omega^{4}}\,.$ (78)

Therefore, a low energy observer would conclude that the bubble of size $R$
can attain the above entropy. However, this would certainly run the full
theory into a problem which is beyond the knowledge of the low energy
observer.

Indeed, since the number of Goldstone species inhabiting the bubble is set by
$N$, attaining (78) would require an increase of $N$ up to
$N\,\sim\,\frac{1}{\alpha}\frac{m^{4}}{\omega^{4}}$. However, this would
violate the unitarity constraint (13) of the full theory. This of course is
not possible.

The unitarity constraint (13) precludes the introduction of Goldstone species
with the number above (72). This limits the actual entropy capacity of the
bubble by (75). This limiting entropy is much smaller than (78) which is
mistakenly presumed by the effective low energy theory with the cutoff
$\Lambda_{\text{cutoff}}\sim 1/R$.

We thus arrive to the following conclusion. In the thin-wall regime, the
entropy of the bubble (75) is far from saturating the bounds (1) and (2).

###  7.2 Entropy of thick-wall bubbles

The situation changes drastically for thick-wall bubbles, for which $R\sim
1/m$. As already explained, as long as condition (39) is satisfied, (or
equivalently, if $\omega<m$), the existence of a classical solution of a
stationary bubble, described by (33) and (36) is guaranteed. For definiteness,
we shall assume that the condition (39) is satisfied and we shall focus on the
case $\omega\sim m$.

Obviously, for such a bubble, all three scales are of the same order,

$\omega\sim m\sim\dfrac{1}{R}\,.$ (79)

Correspondingly, we have,

$N_{G}\,\sim\,\dfrac{1}{\alpha}\,.$ (80)

Taking into account (71), from (70) it follows that the entropy is,

$S\sim\dfrac{1}{\alpha}\,.$ (81)

Thus, the entropy saturates the unitarity bound (2).

The area-law bound (1) is also saturated. Indeed, since the size of the bubble
satisfies (79), the coupling of the Poincaré Goldstone (62) becomes,

$G_{\text{Gold}}^{(\text{P})}\,\sim\,\dfrac{1}{N}\,{\mathcal{A}rea}\,\sim\,\alpha\,{\mathcal{A}rea}\,.$
(82)

It is clear that the entropy of the bubble (81) saturates the bound (1).

It is also obvious that for a saturated bubble, the couplings of the Poincaré
and internal Goldstones are equal. So are their decay constants. We shall
therefore drop the subscripts and denote these quantities universally by
$G_{\text{Gold}}$ and $f$ respectively. Also, from now on, we shall denote the
dimensionless coupling of Goldstones by $\alpha_{G}$.

For both Poincaré and internal Goldstones, the expressions for the couplings
and the decay constants are given by (6), which we shall rewrite as,

$G_{\text{Gold}}\,=\,f^{-2}\,=\,\dfrac{1}{N}\,{\mathcal{A}rea}\,\sim\,\alpha\,{\mathcal{A}rea}\,.$
(83)

Similarly, the dimensionless couplings $\alpha_{G}$ of both types of
Goldstones, evaluated at the scale $1/R$, are equal to the fundamental
coupling $\alpha$. That is, in the thick-wall regime we have

$\alpha_{G}=\dfrac{G_{\text{Gold}}}{{\mathcal{A}rea}}=\dfrac{1}{(fR)^{2}}=\alpha\,.$
(84)

Correspondingly, the bounds (1) and (2) are universally saturated in terms of
all couplings and all the decay constants of the theory.

Interestingly, as already noticed in [1], the entropy of the saturon bubble
obeys,

$S\,\sim\,E_{\rm bubble}\,R\,.$ (85)

This is easy to see from (66), (81) and (79). The above expression agrees with
the bound on entropy suggested by Bekenstein [23], despite the fact that a
priori the latter bound has no information about the coupling of the system.
However, for generic bubbles the bounds (1) and (2) appear more stringent, as
they exclude an excessive entropy of a bubble even if the Bekenstein bound is
formally satisfied.

In summary, we observe that the thick-wall bubbles saturate the entropy bounds
(1) and (2), when the theory saturates unitarity. That is, a thick-wall bubble
becomes a saturon when the ’t Hooft coupling is critical (13). At the same
time, the thin-wall bubbles are under-saturated. That is, their entropy is
much less than what would be permitted by the effective low-energy theory of
Goldstones, with the cutoff $\sim 1/R$. The thin-wall bubble cannot achieve
such entropy, without invalidating the full theory. The results are summarized
in table 1. 222 Interestingly, the same conclusion was reached in [1] for thin
and thick wall bubbles that were not stabilized by memory burden of the
Goldstone charge but instead were assumed to freely oscillate (and eventually
decay).

Bubble Entropy | Bound on $S$ | Actual $S$
---|---|---
thin-wall, $\frac{m}{\omega}\gg 1$ | $S\simeq\frac{1}{\alpha}\frac{m^{6}}{\omega^{6}}$ | $S\simeq\frac{1}{\alpha}\ln\left(\frac{m^{10}}{\omega^{10}}\right)$
thick-wall, $\frac{m}{\omega}\sim 1$ | $S\sim\frac{1}{\alpha}$ | $S\sim\frac{1}{\alpha}$
Table 1: Entropy bounds and the actual entropies for thin-wall and thick-wall
bubbles. It is clear that thin-wall bubbles are under-saturated. The saturons
originate from thick-wall bubbles.

The above reveals a deep connection between the saturation of the system and
the nature of the stabilizing memory burden effect. For under-saturated
bubbles, the occupation number of Goldstones $N_{G}$ is much higher than the
number of the Goldstone species $N_{\text{Gold}}\sim N$. Due to this, the
memory burden that stabilizes the under-saturated bubble is classical. In
contrast, a saturated bubble can be stabilized by the memory burden that is
fully quantum in its nature. The physical meaning of this statement shall be
explained below.

##  8 Stabilization by quantum memory burden

We have seen that when the collective coupling of the theory hits the
unitarity bound (71), the thin-wall bubble becomes a saturon. In this regime
we are dealing with an interesting phenomenon. Despite the fact that the
stationary bubble solution is well described classically, the memory burden
stabilizing it can be fully quantum.

In order to see this, let us start with the ansatz (33). As we already
discussed, in the quantum language this classical solution corresponds to the
state in which a single Goldstone mode, say $\theta^{1}$, is macroscopically
occupied. The occupation number is given by (80). Since our analysis becomes
only better at weak coupling $\alpha$, the occupation number $N_{G}\sim N$ can
be made arbitrarily large by taking $\alpha$ small. Such a state can be
treated classically up to corrections $\sim 1/N$.

Now, we should remember that in the current regime the number of Goldstone
species $N_{\text{Gold}}$ is of the same order as the total occupation number
$N_{G}$. Both of these numbers are of order $N$. In the classical bubble,
corresponding to the ansatz (33), the entire occupation number is “credited”
to a single Goldstone flavor, $\theta^{1}$. The corresponding memory pattern
(53) has the form $n_{\omega}^{a}=\delta^{a1}N_{G}$.

However, as we already discussed, due to $SU(N)$ symmetry, there exist bubbles
in which the number $N_{G}$ is distributed among various Goldstone flavors,
subject to the constraint (52). Such states are related to each other by
$SU(N)$ symmetry transformations. Due to this, they are all strictly
degenerate in energy.

Among the patterns (53), we wish to pay special attention to the ones in which
the number is distributed approximately equally among all of the Goldstone
flavors. Since for the saturated bubble $N_{G}\sim N_{\text{Gold}}\sim N$, in
such states all occupation numbers are small. The corresponding memory
patterns (53) have $n_{\omega}^{a}\sim 1$ for all $a$. Since the individual
occupation numbers are small, we can say that the Goldstone modes are in
intrinsically-quantum states.

By power of symmetry, it is obvious that the corresponding memory burden has
the same stabilizing effect as the one $n_{\omega}^{a}=\delta^{a1}N_{G}$,
which is described by the classical solution (33). Thus, although the states
(53) with $n_{\omega}^{a}\sim 1$ are quantum, they possess the same
stabilizing power as the “classical” state $n_{\omega}^{a}=\delta^{a1}N_{G}$
to which they are related by the $SU(N)$ symmetry. In other words, in such
states, the stabilization happens due to a quantum memory burden.

We thus observe that for saturon bubbles, the stabilization by the memory
pattern can be fully quantum. Such a bubble is similar to a black hole.

##  9 Information horizon

As, already pointed out in [1], an universal property of the saturons is that
in semi-classical limit, they possess a strict information horizon. That is,
the information stored in the interior of the saturon cannot be extracted in
any form. The general physical reason is that in this limit the memory modes,
that carry quantum information, decouple. Correspondingly, the information
stored in these modes becomes permanently hidden.

In the present construction we can demonstrate this feature explicitly. The
memory modes are represented by the Goldstone modes of spontaneously broken
$SU(N)$ symmetry. Their interaction strength is suppressed by their decay
constant $f$. The effective coupling of a Goldstone mode of frequency
$\epsilon$ is,

$\alpha_{G}=\dfrac{\epsilon^{2}}{f^{2}}\,.$ (86)

In the semi-classical limit this coupling goes to zero for any finite
$\epsilon$.

In order to see this, we must specify the correct semi-classical limit. This
is the limit in which the classical bubble solution with finite radius $R$ and
frequency $\omega$ experiences no back reaction from quantum fluctuations.
Such a limit is uniquely given by

$\alpha\rightarrow 0\,,\leavevmode\nobreak\
\,R=\text{finite}\,,\leavevmode\nobreak\
\omega=\text{finite}\,,\leavevmode\nobreak\ \,\alpha N=\text{finite}\,.$ (87)

Notice, this implies that the mass $m$ stays finite, whereas the Goldstone
decay constant $f$ becomes infinite. Therefore, at the same time, in the above
limit we have

$f\rightarrow\infty\,,\leavevmode\nobreak\ \alpha_{G}\rightarrow 0\,,$ (88)

for any finite $\epsilon$. Thus, the coupling of a Goldstone mode of
arbitrarily high frequency vanishes. This applies to the Goldstones of
frequencies higher than the mass $m$ of the quanta outside of the bubble.
Correspondingly, in the zero back reaction limit, there is no way of
transmitting any information from the interior of the bubble to the exterior.

Even if the energy of a Goldstone perturbation is much higher than the mass
gap in the asymptotic vacuum $r\gg R$, it is impossible for such a mode to
communicate with the ones propagating in the outside vacuum. The bubble
literally possesses an information horizon, analogous to that of a black hole.

Another side of the story is the existence of the Goldstone horizon which
manifests itself already at finite $f$. Goldstone waves of frequencies
$\epsilon\ll m$ cannot propagate outside the bubble, even though at finite $f$
the coupling among the modes is finite. Here we can distinguish two cases.

The first case is when the energy of an internal perturbation is less than
$m$. In such a case propagation is simply impossible due to the finite energy
gap.

In the second case, although the frequency $\epsilon$ of the Goldstone
perturbation is smaller than $m$, its energy can exceed the mass gap at the
expense of the occupation number of Goldstone quanta in the perturbation. In
the classical language, the amplitude of the Goldstone wave may be
sufficiently high to reach the energy $m$, even if the frequency is very low.

Even in such a case, the propagation is highly suppressed. The way to
understand this is to think of a process as the transition from the initial
state of high occupation number $n_{\epsilon}$ of soft quanta of frequency
$\epsilon\ll m$, into a state of few quanta of frequencies $m$ or higher. For
example, at the threshold, number $n_{\epsilon}=m/\epsilon$ of Goldstones of
frequencies $\epsilon$ can transit into a single exterior particle of mass
$m$. Such a process is exponentially suppressed by a factor
$\mathrm{e}^{-n_{\epsilon}}$ [1]. This is a general feature of the transition
processes between high and low occupation numbers.

##  10 Hawking evaporation

Despite their classical stability, the saturated vacuum bubbles can decay
through quantum processes. As it is the case for generic saturons [1], this
decay is strikingly similar to Hawking evaporation of a black hole. Just like
the case of a black hole, the emission has a thermal-like spectrum with a
characteristic temperature $T$ given by the inverse radius of the bubble,

$T\sim\dfrac{1}{R}\,.$ (89)

In the semi-classical limit (87), the quantum information carried by the
radiated quanta is not resolvable. In full quantum theory, the resolution time
is given by (8) which is very similar to Page’s time for a black hole.

In case of a black hole, the asymptotic theory is gapless, as it contains a
massless field in the form of the graviton. Therefore, in order to make the
connection between the saturated bubbles and black holes maximally
transparent, we shall introduce some additional massless fields that can
propagate in the asymptotic vacuum.

For definiteness, let us introduce a scalar $\xi_{\alpha}$ transforming in the
fundamental representation of the $SU(N)$ group. We assume that the Lagrangian
is completed to a most general $SU(N)$-invariant renormalizable theory of
$\phi_{\alpha}^{\beta}$ and $\xi_{\alpha}$. The only restriction we put is
that in $SU(N)$-invariant vacuum $\xi_{\alpha}$ is massless.

For making our point, it suffices to consider the following terms,

$L_{\xi}=(\partial_{\mu}\xi^{\dagger\alpha})(\partial^{\mu}\xi_{\alpha})\,-\,\alpha_{\xi}\xi^{\dagger\alpha}(\phi^{2})_{\alpha}^{\beta}\xi_{\beta}\,,$
(90)

where $\alpha_{\xi}>0$ is a coupling constant, which is subject to the
unitarity constraint,

$N\alpha_{\xi}\lesssim 1\,,$ (91)

similar to the bound (13) obeyed by the coupling $\alpha$. We shall choose
$\alpha_{\xi}$ to be near the saturation point of the above bound,
$\alpha_{\xi}\sim 1/N$.

Now, as said, in the vacuum with unbroken $SU(N)$ symmetry, the field $\xi$ is
massless. In the $SU(N-1)\times U(1)$-symmetric vacuum, its components get
positive mass terms of order $m_{\xi}^{2}\sim\alpha_{\xi}f^{2}\sim m^{2}$. Let
us now consider the $\xi$-field in the background of a stationary vacuum
bubble of radius $R$.

In the asymptotic vacuum outside the bubble, $r\gg R$, the $\xi$-field
propagates as massless. At the same time, in the interior of the bubble it
gets a positive mass term. Thus, effectively, $\xi$ sees the bubble as a
potential barrier.

For a thin-wall bubble (41), due to the positive mass2 inside the bubble, the
wavefunctions of the $\xi$ modes of momenta $q\ll m$ are exponentially
suppressed (“screened”) in the bubble center. The screening factor is
$\sim\mathrm{e}^{-mR}$.

For the saturated bubble, the above suppression is not significant. Such
bubbles are in thick-wall regime and satisfy (79). Due to this, the radius of
the bubble $R$ is of the same order as the screening depth, which is given by
$1/m$. In addition, $\omega\sim m$. Due to this, the $\xi$ modes in general
have order-one overlap with the interior region of the saturated bubble.

The presence of a massless field $\xi$ in the theory does not affect the
classical stability of the bubble. However, it opens up a new quantum decay
channel. In the semi-classical approximation, the decay rate can be derived by
quantization of $\xi$ in the background of the classical bubble solution (33).
However, since the saturation takes place in a thick-wall regime, where the
solution is not known analytically, such analysis will be purely qualitative
and shall not be displayed here.

Instead, we shall take a shortcut directly into a full quantum theory and try
to exploit the power of Goldstone universality. Namely, we shall use the fact
that the couplings of Goldstones are governed by the symmetry breaking order
parameter. Also, as already explained, most of the quantum information of the
bubble, is carried by the Goldstone modes. Due to this, the decays of the
bubble through the Goldstone processes are the most interesting from the
perspective of the retrieval of quantum information by an outside observer.

The couplings between the Goldstone modes and the modes of $\xi$-quanta, are
uniquely defined by the Goldstone theorem and are controlled by a single
scale. In the broken-symmetry vacuum, in the language of a full
(3+1)-dimensional theory, this coupling is local and has the form,

$i\partial_{\mu}\theta^{a}\,\left(\partial^{\mu}\xi^{\dagger}T^{a}\xi\,-\,\xi^{\dagger}T^{a}\partial^{\mu}\xi\right)\,+\,...\,,$
(92)

where the dimensionless phases are related with the canonically normalized
Goldstone fields ($a^{a}$) through $\theta^{a}\equiv\frac{a^{a}}{f}$.

In the $SU(N-1)\times U(1)$-symmetric vacuum of infinite extent, the
Goldstones are well defined everywhere. The mode-expansions for both
Goldstones and $\xi$-particles are given by plane-waves. Of course, the finite
size of the bubble affects the mode expansion and the profiles of various
momentum eigenstates. In particular, the Goldstones exist only in the bubble
interior and become ill-defined outside. The effective coupling that controls
the scattering between the Goldstone modes of frequency $\omega$ and the $\xi$
quanta is given by $\alpha_{G}(\omega)=\frac{\omega}{f_{\omega}}$. The entire
information about the mode profiles and other factors is encoded in the
effective scale $f_{\omega}$. For the saturated bubble, $f_{\omega}$ is of the
same order as the fundamental scale $f$. This is because for the saturated
bubble all the scales are of the same order (79). We thus have,
$\alpha_{G}(\omega)=\frac{\omega}{f}$. This coupling controls all the
Goldstone processes of our interest.

Since $\xi$-quanta are massless, the conservation of energy and of the $SU(N)$
charge are not an obstacle for the bubble decay. However, each elementary
vertex involving $\xi$, is suppressed by the Goldstone coupling $\alpha_{G}$.
Due to this, an explosive decay of the bubble into a large number of $\xi$
quanta is highly improbable. Such an “explosion” does not represent a
significant channel of the bubble-decay. This is similar to the suppression of
the explosive decay of a black hole into a large number of soft gravitons.

The leading order processes contributing to the bubble evaporation are of the
following two types. The first category includes the decays of Goldstones of
frequency $\omega$ into pairs of the asymptotic $\xi$ quanta. The second
category includes the re-scatterings of the pairs of Goldstones into $\xi$-s.
The rates of these processes are given by,

$\Gamma_{1\rightarrow 2}\sim\omega\alpha_{G}N_{G}\sim 1/R\,,$ (93)

and

$\Gamma_{2\rightarrow 2}\sim\omega\alpha_{G}^{2}N_{G}^{2}\sim 1/R\,,$ (94)

respectively. Here we took into account that the saturated bubbles satisfy the
relation (79), as well as, $N_{G}\sim N_{\text{Gold}}\sim 1/\alpha_{G}$. The
re-scattering processes involving higher number of Goldstones are more
suppressed. Thus, the decay rate of the saturated bubble is given by,

$\Gamma_{\text{decay}}\sim 1/R\,.$ (95)

This is nothing but the familiar Hawking rate of a black hole decay. Just like
a black hole, on average, the saturon bubble emits a quantum of energy $\Delta
E\sim 1/R$ per time $\Delta t\sim R$. Correspondingly, the power of the
emitted radiation,

$P\,\sim\,\frac{1}{R^{2}}\,\sim\,T^{2}\,,$ (96)

is similar to the power of radiation from a black hole of temperature (89).

It is clear that the emission of $\xi$ quanta of energies $E\gg 1/R$, requires
a re-scattering of a larger number $n$ of Goldstones into a small number of
$\xi$-s. Such processes are highly suppressed. For example, production of
$\xi$ quanta of energy $E=n/R$ with $n\gg 1$, requires re-scattering of $\sim
n$ Goldstones. This brings an extra suppression factor, $\mathrm{e}^{-n}$,
characteristic for the processes $n\rightarrow 2$ at large $n$ [1].

Thus, the rate of emission of quanta of energy $E\gg 1/R$, is suppressed by an
exponential factor of the type,

$\Gamma_{E\gg T}\sim\mathrm{e}^{-ER}\,.$ (97)

For an observer unfamiliar with the microscopic origin of the suppression, the
factor (97) is naturally interpreted and the Boltzmann factor

$\Gamma_{E\gg T}\sim\mathrm{e}^{-\frac{E}{T}}\,,$ (98)

characteristic of thermal radiation with the effective temperature $T\sim
1/R$, as given by (89). Such an observer would naturally conclude that the
saturon state is thermal, despite the fact that the microscopic theory clearly
tells us otherwise. The secret is in $1/N$ ($1/S$) corrections, which carry
the information about the purity of the state. Only after clearly resolving
these corrections, can an observer identify in which pure state the saturon
is. However, such a resolution takes a rather long time, given by (8). This
aspect of radiation shall be quantified below.

##  11 Information in Hawking radiation

Let us now discuss how the information is extracted from the bubble by the
Hawking radiation. We start by noticing that in the semi-classical limit (87)
the quantum information carried by the radiated quanta is strictly
unresolvable. This is obvious from the fact that in this limit all the quantum
couplings vanish, $\alpha_{G},\alpha_{\xi},\alpha\,\rightarrow\,0$. It is
therefore impossible for an external observer to recognize the $SU(N)$ quantum
numbers of the outgoing radiation. At the same time, in the limit (87) the
evaporation rate remains finite and so does the power of the emitted radiation
(96). This is at the expense of infinite $N_{G}$.

Thus, an observer sees a persistent radiation, with an effective temperature
(89), but is unable to resolve its information content within any finite time.
Hence, in the semi-classical limit (87), the information processing by a
saturon bubble is not any different from that of a black hole.

Let us discuss this crucial point more carefully. For this, we compare the
evaporation of two saturated bubbles that carry distinct information patterns,
$\ket{n_{\omega}^{1},n_{\omega}^{2},\,\dotsc}$ and $\ket{n_{\omega}^{\prime
1},n_{\omega}^{\prime 2},\,\dotsc}$, which are related by
$SU(N)$-transformation. The patterns satisfy the constraint (52).

Notice, despite the fact that $SU(N)$ commutes with the Hamiltonian, the two
bubbles carry the distinct quantum information messages. Since the information
patterns are $SU(N)$-transformed relative to one another, they will interact
differently with a fixed reference probe.

Let us recall that for a saturated bubble, the parameters satisfy the
following important relations,

$\alpha\sim\alpha_{G}\sim\alpha_{\xi}\,\sim\frac{1}{N_{\text{Gold}}}\sim\frac{1}{N_{G}}\sim\frac{1}{N}\,.$
(99)

That is, for a saturated bubble, the total occupation number of Goldstone
modes $N_{G}$ is of order the number of the Goldstone species
$N_{\text{Gold}}$, and both numbers are of order $N$. At the same time, all of
the collective couplings are critical.

If the bubble is saturated, the above relations hold regardless whether we
work in the semi-classical approximation or in the full quantum theory. The
difference is that in quantum theory, although the numbers are large (and all
$\alpha$-couplings are weak), they are finite. In the exact semi-classical
limit, all three couplings become infinitely weak, while numbers become
infinitely large.

The two regimes (semi-classical versus quantum) of the saturated bubble can be
summarized as,

$\alpha\sim\alpha_{G}\sim\alpha_{\xi}\,\sim\frac{1}{N_{\text{Gold}}}\sim\frac{1}{N_{G}}\sim\frac{1}{N}\,=\,\begin{cases}\neq
0\ll 1&\text{in\leavevmode\nobreak\ full\leavevmode\nobreak\
quantum\leavevmode\nobreak\ theory},\\\ =0&\text{in\leavevmode\nobreak\ semi-
classical\leavevmode\nobreak\ theory}.\end{cases}$ (100)

Now, let us compare the radiations coming from the above two bubbles. Since
the two patterns are related by $SU(N)$-transformation, all the
$SU(N)$-invariant characteristics of the decay process are identical for the
two bubbles. For example, the two bubbles have the same decay rates, given by
(95).

In order to distinguish between the two bubbles, the observer must perform a
measurement that can be sensitive to differences $\Delta n_{\omega}^{a}$
between the individual occupation numbers of the two patterns. In the semi-
classical limit, this is only possible if some of the ratios $\Delta
n_{\omega}^{a}/N_{G}$ remain finite. In other words, the observer can only
differentiate among the patterns that are classically distinct. The relative
fraction of such patterns vanishes in the limit of infinite $N_{G}$. The bulk
of the quantum information is carried by the patterns in which the differences
are small. Within a semi-classical treatment, such patterns cannot be told
apart. This explains why in the semi-classical theory the radiation from the
bubble carries no quantum information. Or to be more precise, the information
is there but is unresolvable within any finite time.

Now, in full quantum treatment, the radiations coming out of the two bubbles
with distinct information patterns, can be distinguished. Even if the
differences $\Delta n_{\omega}^{a}$ are of order one, they are detectable. The
distinction is encoded in the corrections of order $1/N_{G}$, or equivalently,
of order $1/S$.

The process of the information emission is very transparent in the effective
description of the bubble as of a composite state transforming as $SU(N)$
tensor representation (68). The effective coupling between the bubble and the
$\xi$ field has the following form,

${\mathcal{B}}_{\alpha_{1},\alpha_{2},\,\dotsc\,,\alpha_{N_{T}-1},\,\alpha_{N_{T}}}^{\beta_{1},\beta_{2},\,\dotsc\,,\beta_{N_{T}-1},\,\beta_{N_{T}}}\,\xi^{\dagger\alpha_{N_{T}}}\xi_{\beta_{N_{T}}}\,\tilde{{\mathcal{B}}}^{\alpha_{1},\alpha_{2},\,\dotsc\,,\alpha_{N_{T}-1}}_{\beta_{1},\beta_{2},\,\dotsc\,,\beta_{N_{T}-1}}\,,$
(101)

where
$\tilde{{\mathcal{B}}}^{\alpha_{1},\alpha_{2},\,\dotsc\,,\alpha_{N_{T}-1}}_{\beta_{1},\beta_{2},\,\dotsc\,,\beta_{N_{T}-1}}$
is the operator describing a bubble that transforms under a smaller
representation of $SU(N)$. The emission process is given by

${\mathcal{B}}_{\alpha_{1},\alpha_{2},\,\dotsc\,,\alpha_{N_{T}-1},\,\alpha_{N_{T}}}^{\beta_{1},\beta_{2},\,\dotsc\,,\beta_{N_{T}-1},\,\beta_{N_{T}}}\rightarrow\,\xi_{\alpha_{N_{T}}}\,+\,\xi^{\dagger\beta_{N_{T}}}\,+\,\tilde{{\mathcal{B}}}_{\alpha_{1},\alpha_{2},\,\dotsc\,,\alpha_{N_{T}-1}}^{\beta_{1},\beta_{2},\,\dotsc\,,\beta_{N_{T}-1}}\,,$
(102)

We can describe the process as a planar diagram in ’t Hooft’s notation in
which the oppositely directed arrows describe the flow of global $SU(N)$
“color” and “anti-color” (see figure 1). This diagrammatic language makes the
process of the outflow of information from the decaying bubble very clear. In
particular, it is obvious that a single emission carries away a negligible
fraction of the information stored in the bubble.

$\xi$$\xi^{\dagger}$$\vdots$$\mathcal{B}\leavevmode\nobreak\
$$\leavevmode\nobreak\ \tilde{\mathcal{B}}$ Figure 1: The decay
$\mathcal{B}\to\tilde{\mathcal{B}}+\xi+\xi^{\dagger}$ as a planar diagram in
’t Hooft notation. The opposite arrows indicate the flow of $SU(N)$ “color”
and anti-color. The red and blue lines indicate the quantum numbers taken away
by the emitted $\xi$ particles.

The diagrammatic language makes the origin of the exponential suppression
factor (98) very transparent. A typical process contributing in emission of
highly energetic $\xi$-s is given by figure 2.

$\xi$$\xi^{\dagger}$ Figure 2: A diagram describing many $\rightarrow 2$
scattering, resulting in the emission of highly energetic $\xi$-s.

It is also clear that the back-reaction experienced by the bubble during the
emission is of order $1/N$. Obviously, in the semi-classical limit (87) the
initial (${\mathcal{B}}$) and final ($\tilde{{\mathcal{B}}}$) bubble states
are indistinguishable. At finite $N$, the difference is non-zero. However, the
resolution takes time that scales with $N$. This determines the required time-
scale for the information read-out, which we shall discuss next.

##  12 Time-scale of information retrieval

The discussion of the information carried by Hawking radiation naturally
brings us to the question of the time scale of information retrieval. In [1]
it was argued that an arbitrary saturated system is subjected to an universal
lower bound on the time-scale of information retrieval given by (8). By
applying the general argument of [1] to the present system, it is easy to see
that the saturated bubble reproduces this expression. Indeed, let us consider
such a bubble. As already discussed, the saturation is reached for the thick-
wall bubbles, (79), and they satisfy (99).

As we already know, the quantum information is stored in the form of the
memory pattern (53) formed by the Goldstone modes. Therefore, if we wish to
retrieve any quantum information from such a bubble, the interaction with the
Goldstone modes is mandatory. The precise nature of the measurement is not
important. The necessary condition for the extraction of the information is
that the Goldstone modes must interact with some “agents” that can carry the
$SU(N)$ quantum numbers outside the bubble. We can distinguish the two ways of
information retrieval to which we shall refer to as _passive_ and _proactive_.
We shall discuss them separately.

###  12.1 Passive retrieval

For the passive retrieval of information, an external observer has to analyze
the properties of the Hawking radiation coming from the bubble. As we already
discussed, to the zeroth order in $1/S$ (or $1/N$), the radiation carries no
information. In order to retrieve the information, the observer has to resolve
$1/S$-corrections. For this, the observer must detect the $SU(N)$ content of
the emitted quanta. This is a time-consuming process because of the following
two reasons.

First, the quanta emitted at the initial stage carry a tiny fraction of the
total information. For example, let us consider a decay of a particular
Goldstone mode into a pair of $\xi$-s. In the language of the information
pattern, the process can be described as a “spontaneous emission”, during
which one of the entries in the pattern (say $n_{\omega}^{1}$) decreases by
one,

$\ket{n_{\omega}^{1},n_{\omega}^{2},\,\dotsc}\rightarrow\ket{n_{\omega}^{1}-1,n_{\omega}^{2},\,\dotsc}\,+\,\xi^{\dagger}\,+\,\xi\,.$
(103)

Of course, the $SU(N)$ charge of the decaying Goldstone is carried away by
$\xi$-s. The typical wavelength of the emitted $\xi$ is $\sim R$.

Now, in order to decipher the charge carried by $\xi$, the observer must take
it through a detector with some sample particles. A maximally packed detector
has the occupation number of probe quanta $N_{\xi}\sim 1/\alpha_{\xi}$ per de
Broglie volume $\sim R^{3}$. The maximal interaction rate for an outgoing
$\xi$-quantum with such a detector is,
$\Gamma_{\xi}\sim\alpha_{\xi}^{2}N_{\xi}/R\sim\alpha_{\xi}/R$.
Correspondingly, the minimal time required for recognizing the quantum numbers
of $\xi$, is given by,

$t_{\xi}\sim\dfrac{R}{\alpha_{\xi}}\sim SR\sim R^{3}f^{2}\,.$ (104)

However, the analysis of a single particle does not provide any significant
knowledge about the information pattern carried by the bubble. In order to
start gathering any reasonable amount of information about the $SU(N)$ charge
content of the bubble, the observer needs to analyze at least of order $N_{G}$
emitted quanta.

However, the collection of such amount of quanta takes time,

$t_{\text{min}}\sim N_{G}R\,,$ (105)

which is of the same order as (104).

Thus, the picture is the following. In order to start getting any idea about
the information content of the bubble, the observer needs to collect of order
$N_{G}$ emitted quanta and also detect their $SU(N)$ quantum numbers. Both
processes take the time (104) that is of the same order as (8).

###  12.2 Proactive retrieval

An observer has an alternative option for retrieving the quantum information
stored inside the bubble. Instead of passively waiting for the bubble to
evaporate, an observer can proactively scatter some probe quanta at the bubble
and analyze the scattering products. As we shall see, also in this case the
minimal time-scale for the start of the information retrieval is given by (8).

The universal reason behind this is that $t_{\text{min}}$ cannot be shorter
that the interaction time between the external probe and the Goldstone modes
that carry the information pattern. Regardless of the setup, by Goldstone
theorem, this interaction time is controlled by the Goldstone decay constant
$f$.

The interaction rate is suppressed by the square of the Goldstone coupling
$\alpha_{G}$, evaluated at the scale $\omega\sim 1/R$. At the same time, the
rate is enhanced by the total occupation number of Goldstones, $N_{G}$.
Therefore, the rate is given by

$\Gamma_{\text{Gold}}\sim\alpha_{G}^{2}mN_{G}\sim\dfrac{1}{R^{3}f^{2}}\,.$
(106)

The corresponding time-scale is given by,

$t_{\text{min}}=\dfrac{1}{\Gamma_{\text{Gold}}}\sim R^{3}f^{2}\,.$ (107)

This sets the minimal time of the start of information recovery by the
proactive method. Due to the saturation relations, the time $t_{\text{min}}$
can be rewritten as,

$t_{\text{min}}\sim SR\sim\dfrac{R}{\alpha}\,.$ (108)

We thus discover that with both methods (passive and proactive) the minimal
time for the start of the information retrieval is given by (8).

This result confirms the general point of [1] that saturated systems obey the
universal lower bound (8) on the minimal time of information-recovery. The
black holes are by no means unique in this respect. For a non-gravitational
saturon of size $R$, the time $t_{\text{min}}$ is equal to the Page time of a
same-size black hole.

Notice, the expression (107) of information-recovery time nicely matches the
existence of the information horizon in the semi-classical limit of the
theory. From the expressions (87) and (88), it is clear that in the same limit
the information recovery time $t_{\text{min}}$ becomes infinitely long. Thus,
the expressions (107) and (108) explain why any saturated system must possess
an information horizon in the semi-classical treatment of the theory.

Finally, we wish to comment that the above-discussed origin of Hawking
radiation and of information-retrieval time-scales is very similar to the
picture provided by the black hole $N$-portrait [7]. This is no accident,
since the $N$-portrait describes a black hole as the saturated state of soft
gravitons. By power of universality, the behaviour of the two systems must be
very similar. We shall comment more on the connection with black holes later.

##  13 Numerical results

For the numerical analysis we consider the case of occupying one $\theta^{a}$
macroscopically. For this, we take $\theta^{a}=\delta^{a1}\theta(x)$. Let us
first write the Lagrangian (12) in its full form in terms of the fields
$\varphi$ and $\theta$,

$\mathcal{L}=\dfrac{1}{2}\left(\partial_{\mu}\varphi\right)\left(\partial^{\mu}\varphi\right)+\dfrac{N}{4(N-1)}\varphi^{2}\left(\partial_{\mu}\theta^{a}\right)\left(\partial^{\mu}\theta^{a}\right)-\dfrac{\tilde{\alpha}}{2}\varphi^{2}\left(\varphi-\tilde{f}\right)^{2}\,,$
(109)

where

$\tilde{\alpha}\equiv\alpha\frac{(N-2)^{2}}{N(N-1)}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \tilde{f}\equiv f\frac{\sqrt{N(N-1)}}{(N-2)}\,.$ (110)

Now, let us define

$\Psi\equiv\dfrac{1}{\sqrt{2}}\rho\text{e}^{i\chi/\tilde{f}}$ (111)

with

$\rho=\varphi\,,\quad\chi=\sqrt{\dfrac{N}{2(N-1)}}\tilde{f}\theta.$ (112)

We can now re-write the Lagrangian as

$\mathcal{L}=\left(\partial_{\mu}\Psi^{*}\right)\left(\partial^{\mu}\Psi\right)-\tilde{\alpha}\left|\Psi\right|^{2}\left(\sqrt{2}\left|\Psi\right|-\tilde{f}\right)^{2}.$
(113)

The corresponding equations of motion for $\Psi$ are

$\Box\Psi+\tilde{\alpha}\Psi\left(\sqrt{2}\left|\Psi\right|-\tilde{f}\right)\left(2\sqrt{2}\left|\Psi\right|-\tilde{f}\right)=0\,.$
(114)

Solving these for $\Psi$, we can obtain the original fields $\varphi$ and
$\theta$ as

$\varphi=\sqrt{2}|\Psi|\quad\text{and}\quad\theta=\sqrt{\dfrac{2(N-1)}{N}}\text{Arg}(\Psi)\,,$
(115)

respectively.

As initial conditions for the simulations we could consider

$\Psi(t,r)|_{t=0}=\dfrac{\tilde{f}}{2\sqrt{2}}\left[1+\tanh\left(\dfrac{m(R_{0}-r)}{2}\right)\right]$
(116)

and

$\partial_{t}\Psi(t,r)|_{t=0}=i\tilde{\omega}\dfrac{\tilde{f}}{2\sqrt{2}}\left[1+\tanh\left(\dfrac{m(R_{0}-r)}{2}\right)\right],$
(117)

where $r$ is the radial coordinate, $R_{0}$ is the initial bubble radius,
$\tilde{\omega}$ is the initial rotation frequency of the field $\chi$ and
$m=\sqrt{\tilde{\alpha}}\tilde{f}=\sqrt{{\alpha}}{f}$. These can be
equivalently expressed as

$\varphi(t,r)|_{t=0}=\dfrac{\tilde{f}}{2}\left[1+\tanh\left(\dfrac{m(R_{0}-r)}{2}\right)\right]$
(118)

and

$\dot{\theta}\equiv\partial_{t}\theta(t,r)|_{t=0}=\sqrt{\frac{2(N-1)}{N}}\tilde{\omega}\,,$
(119)

respectively.

However, the above is valid only in the thin-wall approximation. Moreover, we
are able to obtain better initial conditions for the simulation by first
solving (34) numerically. We then set this solution as the initial condition
for the simulations. This allows us firstly, to improve our numerical
analysis, and secondly, to extend our analysis to that of thick-wall bubbles.

We numerically solve the equations of motion for $\Psi(t,r)$ in (3+1) space-
time dimensions. In the simulations below we set $\tilde{\alpha}=1$ and
$\tilde{f}=1$ throughout. From here on, just as we did before, we absorb the
$N$-dependent pre-factors into the definitions of $\tilde{f}$,
$\tilde{\alpha}$ and $\tilde{\omega}$ to obtain $f$, $\alpha$ and $\omega$,
respectively. In our numerical simulations we find that for a certain
_critical frequency_ , $\omega_{c}$, the bubble stabilizes, as expected, see
figure 3. Thus, a certain critical macroscopic occupation number of the
Goldstone modes, parametrized by $\omega_{c}$, stabilizes the vacuum bubble
and impedes both its collapse and expansion. As expected, this is the same
frequency as found in (45). Below we shall re-derive this expression from the
condition of conserved charge and, additionally, extend the calculation to
obtain an estimate for the frequency of the bubble wall oscillations for a
near-perfectly stabilized bubble.

###  13.1 Critical frequency estimate

In what follows we will assume spherical symmetry. Using the thin-wall
approximation, the total energy of a vacuum bubble of radius $R\gg m^{-1}$ can
be expressed as

$E=\dfrac{2\pi}{3\alpha}m^{3}R^{2}(1-\dot{R}^{2})^{-1/2}+\dfrac{2\pi}{3\alpha}m^{2}\omega^{2}R^{3}\,,$
(120)

where we assumed $\omega=\omega(R)$ to be homogeneous in space. In order to
obtain an explicit expression for $\omega(R)$, we use the fact that the
charge,

$Q=-i\int
r^{2}\,\mathrm{d}r\,\left(\Psi^{*}\partial_{t}\Psi-\Psi\partial_{t}\Psi^{*}\right)\,,$
(121)

is conserved in time; $\dot{Q}=0$. In the thin wall approximation,

$Q=\dfrac{2\pi}{3}f^{2}\omega R^{3}=\dfrac{2\pi}{3\alpha}m^{2}\omega R^{3}\,.$
(122)

Substituting this into (120) we can re-write the total energy of the bubble as

$E=\dfrac{2\pi}{3\alpha}m^{3}R^{2}(1-\dot{R}^{2})^{-1/2}+\dfrac{2\pi}{3\alpha}m^{2}\left(\dfrac{3\alpha
Q}{2\pi m^{2}R^{3}}\right)^{2}R^{3}\,.$ (123)

We are interested in the static bubble configuration. Thus we take
$\dot{R}=0$. The total energy therefore simplifies to

$E=\dfrac{2\pi}{3\alpha}m^{3}R^{2}+\dfrac{3\alpha Q^{2}}{2\pi m^{2}R^{3}}\,,$
(124)

and solving $\frac{\mathrm{d}E}{\mathrm{d}R}=0$ for $R$, we find

$R_{0}\equiv\left[\dfrac{3}{2}\left(\dfrac{3\alpha
Q}{2\pi}\right)^{2}\right]^{1/5}m^{-1}\,.$ (125)

From here, using (122), we obtain the following estimate for the critical
frequency

$\omega_{c}=\sqrt{\dfrac{2m}{3R_{0}}}\,.$ (126)

This is the same frequency as that in (45).

We will set $R_{0}$ as the initial bubble radius as an initial condition in
the numerical simulations below. The total energy and the total charge for
this static bubble are

$E_{0}\equiv\dfrac{40\pi m^{5}}{81\alpha\omega_{c}^{4}}\quad\text{and}\quad
Q_{0}\equiv\dfrac{16\pi}{81\alpha}\left(\dfrac{m}{\omega_{c}}\right)^{5}\,,$
(127)

respectively. In what follows we test the above estimate and investigate the
response of the system in various frequency regimes.

We make a further note on the radius oscillations in the near-critical
frequency regime. Namely, we can estimate their frequency, $\omega_{0}$, in
the following way. Let us take $R=R_{0}+\delta R$. Then the energy in (120)
becomes

$E=\dfrac{10\pi m^{3}R_{0}^{2}}{9\alpha}+\left(\dfrac{\pi
m^{3}R_{0}^{2}}{3\alpha}\right)\delta\dot{R}^{2}+\left(\dfrac{10\pi
m^{3}}{3\alpha}\right)\delta R^{2}+\mathcal{O}\left(\delta R^{3},\delta
R^{2}\delta\dot{R},\delta R\,\delta\dot{R}^{2},\delta\dot{R}^{3}\right)\,.$
(128)

We can now extract the frequency of the radius oscillations from the above as

$\omega_{0}\equiv\sqrt{10}R_{0}^{-1}=\sqrt{10}\dfrac{3\omega_{c}^{2}}{2m}\,.$
(129)

###  13.2 Critical frequency regime

A certain critical macroscopic occupation number of the Goldstone modes,
parametrized by $\omega_{c}$, stabilizes the vacuum bubble and impedes both
its collapse and expansion (see figure 3). As we illustrate in the next
section, bubbles in the low frequency regime, specifically with
$\omega=0.1\omega_{c}$, collapse at $t_{\text{col}}\sim R_{0}$. Notice that
here, in the critical frequency regime, at $4t_{\text{col}}\simeq 60/f^{-1}$,
the bubbles have not collapsed.

(a) $R_{0}=12/f^{-1}$

(b) $R_{0}=1.01611/f^{-1}$

Figure 3: The frequency for both simulations is $\omega=\omega_{c}$.

For a frequency slightly distinct from $\omega_{c}$, the bubble performs
oscillations around its mean value $R_{m}$. We find at least two frequencies
that contribute to the oscillations. Figure 4 illustrates this.

(a) $R_{0}=12/f^{-1}$

(b) $R_{0}=1.01611/f^{-1}$

Figure 4: The frequency for both simulations is $\omega=1.1\omega_{c}$. The
bubble radius, determined by the $r$-coordinate of the maximum of the energy
density for each time step, is shown in red. The corresponding numerical fit
$R(t)=R_{m}+A_{0}\cos(\omega_{0}t)+A_{1}\cos(\omega_{1}t)$ is shown in black.

Below we list the fit parameters of the fit function
$R(t)=R_{m}+A_{0}\cos(\omega_{0}t)+A_{1}\cos(\omega_{1}t)$ in figure 4. For
the large bubble these are: $R_{m}=13.730\pm 0.002$, $A_{0}=-0.464\pm 0.003$,
$\omega_{0}=0.1537\pm 0.0002$, $A_{1}=-0.042\pm 0.002$ and
$\omega_{1}=0.370\pm 0.002$. The coefficient of determination
$\overline{R}^{2}$ and the unbiased root-mean-square error (RMSE), both
adjusted for the number of fit-model parameters, are $0.999995$ and $0.0301$,
respectively. For the small bubble these are: $R_{m}=2.5297\pm 0.0004$,
$A_{0}=-0.0519\pm 0.0005$, $\omega_{0}=1.4287\pm 0.0003$, $A_{1}=-0.0237\pm
0.0005$ and $\omega_{1}=0.2023\pm 0.0006$, with $\overline{R}^{2}=0.999994$
and $\text{RMSE}=0.00599$. Four outlier data points were excluded from the fit
for the small bubble. Both fit results of $\omega_{0}$ are close to our
analytical estimate of $\omega_{0}=\sqrt{10}R_{0}^{-1}$ from before; $0.264$
and $3.11$ for the large and the small bubbles, respectively.

###  13.3 Low frequency regime

For sufficiently low frequencies $\omega$ the bubble collapses. This is not
surprising, as in this case the pressure due to the $\chi$ field’s rotation is
insufficient to counteract to the tension of the bubble wall. Thus, the vacuum
bubble decays. For $\omega=0$ we numerically obtain that the bubble collapses.
This regime has already been extensively studied in the literature. Here we
consider the case $\omega=0.1\omega_{c}$ and observe that the bubble collapses
also in this case. From figure 5 we are able to read off the collapse time of
the large bubble as $t_{\text{col}}\simeq 1.25R_{0}=15/f^{-1}$. We estimate
the collapse time of the small bubble also as $t_{\text{col}}\sim R_{0}$.

(a) $R_{0}=12/f^{-1}$

(b) $R_{0}=1.01611/f^{-1}$

Figure 5: Time evolution of the bubbles with different initial radii $R_{0}$.
The frequency for both simulations is $\omega=0.1\omega_{c}$. The color bar
corresponds to $\sqrt{2}|\Psi(t,r)|/f=\varphi(t,r)/f$. The arrows represent
the vector $\sqrt{2}\Psi(t,r)$ on the complex plane and thus indicate the
value of $\theta(t,r)$.

###  13.4 High frequency regime

In the high frequency regime, the bubble initially expands: see figure 6. This
is clear as the corresponding $\omega$ is above the critical value. Again,
this should be intuitive, as in this case the pressure from the internal
rotation is higher than the bubble wall tension. Once the bubble has expanded
to a radius where the internal pressure is counterbalanced by the wall
tension, it starts shrinking again. After this the process repeats itself.
However, we additionally observe that a part of the energy is lost to the
surroundings. This is by no means unexpected. In fact, this is a consistency
check, which our numerical analysis fulfills. In the semi-classical limit the
energy of the bubble is infinite. Therefore, any finite mass gap should result
in waves propagating away from the oscillating bubble. This is exactly what we
observe.

(a) $R_{0}=12/f^{-1}$

(b) $R_{0}=1.01611/f^{-1}$

Figure 6: Time evolution of the bubbles with different initial radii $R_{0}$.
The frequency for both simulations is $\omega=4\omega_{c}$. The color bar
corresponds to $\sqrt{2}|\Psi(t,r)|/f=\varphi(t,r)/f$. The arrows represent
the vector $\sqrt{2}\Psi(t,r)$ on the complex plane. Note the increased domain
for the large bubble in panel (a).

###  13.5 Information horizon

To illustrate the notion of the information horizon of the vacuum bubble, for
the initial $r$-profile of $\theta$ we consider a perturbation,

$p(r)=\exp\left[\dfrac{i\pi}{2}\dfrac{f(r)}{f(0)}\right]\,,$ (130)

with the probability density function $f(r)$ of a normal distribution,

$f(r)=\dfrac{1}{\sigma\sqrt{2\pi}}\mathrm{e}^{-\frac{1}{2}\left(r/\sigma\right)^{2}}\,,$
(131)

where we set the size of the perturbation as $\sigma\equiv 5/m\gg 1/m$.

Thus, the initial conditions are mapped from the old to the new ones in the
following way: $\Psi|_{t=0}\to\Psi|_{t=0}p(r)$ and
$\partial_{t}\Psi|_{t=0}\to\partial_{t}\Psi|_{t=0}p(r)$. Figure 7 shows the
corresponding simulation results. We observe that the total energy and the
total charge are both conserved throughout the time evolution. Additionally,
no waves are propagating out, in contrast to the high frequency regime.
Moreover, the perturbation of $\theta$ is contained within the bubble.
Additionally, note that $\theta$ is not homogeneous inside the bubble (cf.
stabilized bubble in figure 3). The spatially dependent directions of the
vector arrows clearly indicate the various values of $\theta$ within the
bubble.

(a) $R_{0}=12/f^{-1}$

(b) $R_{0}=1.01611/f^{-1}$

Figure 7: Time evolution of the bubbles with different initial radii $R_{0}$,
with a perturbation of $\theta$. The frequency for both simulations is
$\omega=\omega_{c}$. The color bar corresponds to
$\sqrt{2}|\Psi(t,r)|/f=\varphi(t,r)/f$. The arrows represent the vector
$\sqrt{2}\Psi(t,r)$ on the complex plane.

##  14 Correspondence to black holes

We have seen that the saturated bubble exhibits a remarkable correspondence
with a black hole, as suggested in [1]. The direct correspondence is
established via the coupling $G_{\text{Gold}}^{(\text{P})}$ of the Goldstone
boson of spontaneously broken Poincaré symmetry. When expressed through this
parameter, the saturon of $SU(N)$ theory and a black hole in gravity share the
same characteristics. Since the spontaneous breaking of Poincaré symmetry by
any self-sustained object is an universal phenomenon, the coupling
$G_{\text{Gold}}^{(\text{P})}$ is unambiguously defined in both cases. For a
black hole, this quantity is equal to Newton’s constant, whereas the decay
constant $f$ is equal to Planck mass $M_{\text{P}}$. Once we understand this,
it becomes clear that all the characteristics of a black hole are the same as
of a generic saturon. This correspondence is summarized in table 2.

Quantity Object | Saturons | Black holes
---|---|---
$S$ | $(fR)^{2}=\alpha^{-1}$ | $(RM_{\text{P}})^{2}$
$T$ | $R^{-1}$ | $R^{-1}$
$t_{\text{min}}$ | $R^{3}f^{2}=SR$ | $R^{3}M_{\text{P}}^{2}=SR$
Table 2: Correspondence between properties of saturons and their specific
realizations - black holes.

Using the above correspondence as the guiding principle, we can learn some
useful lessons about black holes. First, we observe that $1/S$ ($1/N$)
corrections to thermal radiation play the crucial role in purification of the
state and in information recovery from it. Due to the same effects, the black
hole evaporation is expected not to be self-similar, as opposed to the
standard view. The picture also makes it clear that during its decay the black
hole state must be subjected to an inner entanglement, as it is suggested by
the microscopic picture [7]. We also observe the phenomenon of the memory
burden [13, 14] at work. We see that the saturated bubble is stable due to the
quantum information that it carries.

All of the above indicates that a black hole is likely a saturated bound state
of gravitons, as this was suggested by the quantum $N$-portrait [7]. The fact
that we can formulate properties both for black holes and other saturons in
terms of universal language of the Poincaré Goldstone, provides an additional
support to this idea. The Goldstone boson of Poincaré symmetry shares one
fundamental similarity with the graviton: It is universally coupled to
everything that carries energy-momentum. In case of a black hole, the two
objects are the same. That is, the Poincaré Goldstone comes from the
collective excitations of the graviton “condensate”.

##  15 Conclusions and outlook

In the present paper we have verified the following points, proposed in [1].
First, the unitarity upper bound on the entropy of a self-sustained object is
given by (1), or equivalently, by (2). Secondly, the objects saturating this
bound, called “saturons”, share the key properties with a black hole. The
correspondence is briefly summarized in the table 2. The various aspects of
this correspondence were already observed on a number of examples, [3, 4, 9].

The present work represents a continuation of this program. At the same time,
we have incorporated the study of another universal phenomenon exhibited by
generic systems of enhanced capacity of information storage. This is a so-
called “memory burden” effect [13, 14]. The essence of it is that the quantum
information carried by an object tends to stabilize the “host”. It has been
suggested that due to its universal nature the same effect must take place in
black holes. In the present paper we have detected this phenomenon in a
saturon of $SU(N)$ theory.

In order to address the above points, in the present work, we have studied
(both analytically and numerically) the correspondence between black holes and
the saturons of a specific model, originally discussed in [1]. This is a
renormalizable theory with $SU(N)$ global symmetry and coupling $\alpha$,
which we took arbitrarily small. The theory has a set of degenerate vacua in
which the $SU(N)$ symmetry is spontaneously broken down to various maximal
subgroups. As an asymptotic vacuum we chose the $SU(N)$-invariant one. In this
theory we considered bubbles of $SU(N-1)\times U(1)$ symmetric vacuum. Due to
spontaneous symmetry breaking, the bubble interior houses $N_{\rm Gold}\sim N$
Goldstone species.

We showed that the spectrum of the theory contains a tower of stationary
bubbles. In quantum description, a bubble represents a bound state of
Goldstone modes with occupation number $N_{G}$. Bubbles that minimize the
energy cost of the Goldstone charge are stable in classical theory. The
stability can be understood as a particular case of the “memory burden”
effect. The bubble is stable due to the quantum information encoded in the
$SU(N)$ flavor quantum numbers of the Goldstone modes that are “locked up” in
the bubble interior.

The stabilized bubbles can be viewed as generalized versions of non-
topological solitons, or $Q$-balls [19, 20]. However, the special property of
the presented bound state is a very high entropy carried by the Goldstone
modes 333 A different version of a high entropy $Q$-ball was studied in [24]..
This entropy reaches the limit (1), equivalently (2), when the theory
saturates the bound on unitarity. A saturated bubble, acquires the properties
of a black hole. These include: The area form of entropy; the information
horizon; the thermal decay with an effective temperature $T\sim 1/R$; and the
time-scale of information-retrieval similar to Page time. All these
characteristics, expressed in theory-invariant quantities $G_{\rm Gold}$ or
$\alpha_{G}$, have similar forms for the saturated bubble and a black hole.

Not surprisingly, close to saturation, the bubbles of $d=4$ dimensional
$SU(N)$ theory considered in the present work exhibit the close similarities
with the recently found [9] saturon bound states in $d=2$ dimensional Gross-
Neveu theory [10]. This close similarity is the manifestation of the
universality of the phenomenon of saturation.

We have observed a certain deep connection between the level of quantumness of
the memory burden effect and the phenomenon of saturation. For the stability
of the bubble, only the total occupation number of Goldstone modes $N_{G}$
matters. The memory burden effect depends only on $N_{G}$ and is insensitive
to the specific occupation number distribution among the various Goldstone
species. For example, a single Goldstone mode can be occupied macroscopically,
as opposed to occupying multiple Goldstone modes microscopically.

When the number $N_{G}$ exceeds the number of Goldstone species $N_{\rm
Gold}$, the memory burden effect has a valid classical description. This is
because for $N_{G}\gg N_{\rm Gold}$, necessarily, some of the modes are
occupied macroscopically. We have observed that such bubbles always have less
entropy than given by (1) and (2). That is, the bubbles stabilized by a
classical memory burden are undersaturated.

Conversely, the bubbles with $N_{G}\sim N_{\rm Gold}$ can attain the maximal
entropy (1). For such bubbles we get a curious situation. They form an $SU(N)$
multiplet in which the bubbles with classical memory burden are fully
degenerate with the quantum ones.

The construction of the present paper can be straightforwardly generalized to
models with other field content and symmetry groups. For instance, $SO(N)$
with symmetric representation has almost identical structure. Also, it can be
easily supersymmetrized.

In all known examples, the saturons represent the bound states in which the
occupation number of quanta, their inverse couplings and entropy are of the
same order. Thus, the correspondence between the black holes and saturons
naturally hints towards the composite picture of a black hole [7]. The bound
state of $N$ gravitons describing a black hole in this picture is very similar
to the saturated bound state of $N$ Goldstones of $SU(N)$ theory. Many
properties predicted for black holes by $N$-portrait [7] are explicitly seen
in the presented theory of saturons. In particular, we saw how the information
is carried away by $1/S$ effects and how from these effects the time-scale
(108) of information-retrieval emerges.

The discussed correspondence also reinforces some previous suggestions for
black holes. In particular, the reality of the memory burden effect. It is
also evident that the standard assumption about self-similar evaporation of a
black hole deserves to be reconsidered.

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