# In and out-of-equilibrium features of hadronic layers in the low-energy
limit of quantum chromodynamics

Fabrizio Canfora<EMAIL_ADDRESS>Universidad San Sebastián, sede
Valdivia, General Lagos 1163, Valdivia 5110693, Chile  Centro de Estudios
Científicos (CECs), Casilla 1469, Valdivia, Chile  Marcela Lagos
<EMAIL_ADDRESS>Facultad de Ciencias, Universidad Arturo Prat, Avenida
Arturo Prat Chacón 2120, 1110939, Iquique, Chile  Instituto de Ciencias
Exactas y Naturales, Universidad Arturo Prat, Playa Brava 3256, 1111346,
Iquique, Chile  Pablo Pais<EMAIL_ADDRESS>Instituto de Ciencias
Físicas y Matemáticas, Universidad Austral de Chile, Casilla 567, Valdivia,
Chile  IPNP - Faculty of Mathematics and Physics, Charles University, V
Holešovičkách 2, 18000 Prague 8, Czech Republic  Aldo Vera<EMAIL_ADDRESS>Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile,
Casilla 567, Valdivia, Chile

###### Abstract

The entanglement entropy dynamics of baryonic layers in $3+1$ dimensions in
the low-energy limit of quantum chromodynamics is computed. The evolution
after quantum quenches can be described explicitly. In particular, it is shown
analytically that the von Neumann and Rényi entropies may display undamped
oscillations in time, whose frequencies can be calculated in terms of the
baryonic distribution of the hadronic matter in the cavity. Moreover, the
Loschmidt amplitude and the fidelity and work distribution can be derived as
well. These results, which are entirely unexpected in non-integrable theories
such as the Skyrme model, are achieved thanks to a remarkable mapping from any
$(3+1)$-dimensional hadronic distribution of matter in a cavity close to its
boundary to the sine-Gordon theory in $1+1$ dimensions. In the attractive
regime, such results are valid for low-enough energy scales.

Keywords— Skyrme model; Nuclear Physics; Non-equilibrium QFT

##  1 Introduction

In condensed matter physics, a deeper understanding of the phase diagram, as
well as of the entanglement and its evolution in many bodies, is one of the
most important topics, both theoretically and experimentally [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15](detailed reviews are [16, 17, 18, 19, 20,
21, 22, 23]). This issue lies at the crossroad between statistical physics,
quantum computation and quantum field theory. In low-dimensional systems, many
powerful exact results have been derived. For instance, the references
mentioned above indicate that the asymptotic entanglement of a large subsystem
has been related to thermodynamic entropy in a stationary state. It has also
been possible to connect the growth of entanglement with the capability of a
classical computer to simulate non-equilibrium quantum systems with matrix
product states. Integrable models in $1+1$ dimensions offer a unique window on
interacting systems, allowing a detailed understanding of the time evolution
of entanglement [9, 10, 11, 12, 13, 14].

These concepts are essential in quantum field theory (QFT) as well, with
implications from black hole physics to scattering amplitudes and quantum
chromodynamics (QCD) [24, 25, 26, 27]. However, except for conformal field
theory and integrable models [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43], at first glance, it looks impossible to obtain exact results
either on the phase diagram (such as the ones in [28, 29, 35, 36]) or on
entanglement dynamics (such as the ones in [9, 10, 11]) of strongly
interacting QFT. A theoretical dream is to derive those results in the low-
energy limit of QCD, where perturbation theory is useless.

This work shows that one can derive results similar to the ones in
$(3+1)$-dimensional QCD [9, 10, 11, 28, 29, 35, 36], allowing a direct
computation of the entanglement entropy, as well as of its evolution after a
quench of a baryonic distribution in a three-dimensional cavity. Since only
refined numerical techniques are commonly employed in analyzing the phase
diagram of QCD at low temperatures and finite baryon densities [44, 45, 46,
47], the present analytic results are highly relevant and produce novel sharp
predictions which can, in principle, be tested.

The starting point is the Skyrme theory, which (at leading order in the ’t
Hooft expansion [48, 49, 50]) represents the low-energy limit of QCD. The
dynamical field of the Skyrme action [51] is a $SU(N)$-valued scalar field $U$
(here, we will consider the two-flavors case). This action possesses both
small excitations describing pions and topological solitons describing baryons
[52, 53, 54, 55, 56], being the baryonic charge a topological invariant.
Skyrme theory has always been considered the prototype of non-integrable
models where the powerful non-perturbative results available in quantum many-
body physics [57, 58, 59, 60, 61] cannot be applied. However, the techniques
developed in [62, 63, 64, 65, 66, 67] allowed for the first time a successful
analytic description of non-homogeneous baryonic condensates at finite baryon
density, in good qualitative agreement with the available phenomenological
results in [68] and references therein. This framework allows describing
$(3+1)$-dimensional baryonic layers confined in a cavity in terms of the sine-
Gordon theory (SGT), which is the prototype of a relativistic integrable field
theory in $1+1$ dimensions. Hence, the results in [30, 31, 32, 33, 34, 35, 36,
37] developed for the SGT at finite temperatures and in [9, 10, 11, 12, 13,
14] for SGT out-of-equilibrium produce novel analytic results in the low-
energy limit of QCD at low temperatures. In particular, this method allows
computing the entanglement entropy, the time-oscillations of the von Neumann
and Rényi entropies, the Loschmidt amplitude and so on of any hadronic
distribution confined in a cavity whose energy and baryon densities are
homogeneous in two spatial directions (but not necessarily in the third).

##  2 Summary of the results

The action for the $SU(2)$-Skyrme model is given by

$\displaystyle I[U]\ $ $\displaystyle=\
\frac{K}{4}\,\int_{\mathcal{M}}d^{4}x\,\sqrt{-g}\
\text{Tr}\left(R_{\mu}R^{\mu}+\frac{\lambda}{8}G_{\mu\nu}G^{\mu\nu}\right)\;,$
(1) $\displaystyle R_{\mu}$
$\displaystyle=U^{-1}\nabla_{\mu}U=R_{\mu}^{a}t_{a}\ ,\quad
G_{\mu\nu}=[R_{\mu},R_{\nu}]\;,$

where $U(x)\in SU(2)$, $g$ is the metric determinant, $\nabla_{\mu}$ is the
partial derivative, and $t_{a}=i\sigma_{a}$ are the generators of the $SU(2)$
Lie group, being $\sigma_{a}$ the Pauli matrices. The space-time manifold is
split in $\mathcal{M}=\mathds{R}\times\Sigma$, and the Skyrme couplings $K$
and $\lambda$ are positive constants fixed experimentally.

The topological current $J_{\mu}$ and the baryonic charge $Q_{B}$ are defined,
respectively, as

$\displaystyle J^{\mu}\ $ $\displaystyle=\
\epsilon^{\mu\nu\alpha\beta}\text{Tr}(R_{\nu}R_{\alpha}R_{\beta})\;,$
$\displaystyle Q_{B}\ $ $\displaystyle=\
\frac{1}{24\pi^{2}}\int_{\Sigma}J^{0}\;.$ (2)

where the last integral is performed on $\Sigma$, the boundary of
$\mathcal{M}$. Geometrically, a non-vanishing $J^{\mu}$ measures the “genuine
three-dimensional nature” of the configuration since (in order to have
$J^{\mu}\neq 0$, at least locally) $U$ must encode three independent degrees
of freedom. For instance, if two of the three degrees of freedom needed to
describe $U$ depend on the same coordinate, then $J^{\mu}$ vanishes
identically.

We want to analyze the intriguing phenomena which occur when a finite amount
of baryons live within a cavity of finite spatial volume
$V=8\pi^{2}L^{2}L_{x}$, therefore we consider the metric of a box

$ds^{2}=-dt^{2}+dx^{2}+L^{2}(d\mathfrak{y}^{2}+d\mathfrak{z}^{2})\;,$ (3)

where $L_{x}$ and $L$ are constants representing the size of the box in the
directions longitudinal and orthogonal to $x$, respectively. The coordinates
have the following ranges (see [62, 63]):

$0\leq x\leq L_{x}\,,\quad\ 0\leq\mathfrak{y}\leq 2\pi\,,\quad\
0\leq\mathfrak{z}\leq 4\pi\;.$ (4)

Note that the coordinate $x$ has a length dimension, while the other two
coordinates, $\mathfrak{z}$ and $\mathfrak{z}$, are dimensionless (since the
length scale $L$ has been explicitly shown in the metric). This helps to
analyze the interplay between the two scales, $L$ and $L_{x}$, in the
following computations. In order to apply the known results on SGT mentioned
above, the limit $L_{x}\gg L$ has to be considered. In this case, one would
describe a sort of “hadronic wire” (a cavity much longer in one spatial
direction than in the other two). We will choose $\mathfrak{y}$ and
$\mathfrak{z}$ as the “homogeneous coordinates”. When $L_{x}$ is not large,
one has to use the exact available results on SGT on a finite interval [38,
39].

The Skyrme action and the corresponding field equations can be written
explicitly in terms of the $SU(2)$-valued field (as any element of $SU(2)$ can
be written in the Euler representation)

$U=\exp\left(t_{3}\,F\right)\exp\left(t_{2}\,H\right)\exp\left(t_{3}\,G\right)\
,$ (5)

where $F=F\left(x^{\mu}\right)$, $G=G\left(x^{\mu}\right)$ and
$H=H\left(x^{\mu}\right)$ are the three scalar degrees of freedom of the
Skyrme field (traditionally, in this parametrization, the field $H$ is called
profile).

It is well known that many baryonic distributions, especially at low energies,
possess a sharp boundary. Namely, the energy and baryon densities decay
exponentially fast to zero. Thus, in practice, one can define a surface that
separates the region where the energy and baryon densities are different from
zero from the region where these densities vanish. In a neighbourhood of any
point close to such a surface, the energy and baryon densities can only depend
on the spatial coordinate orthogonal to the surface and on time (we will
comment more on this point in the following sections). This is why it is so
interesting to study hadronic distributions that are homogeneous in two
spatial directions. The analysis of the energy-momentum tensor shows that the
only Ansatz able to describe energy and baryon densities homogeneous in two
spatial directions [64] is

$H(x^{\mu})=H(t,x)\ ,\quad F(x^{\mu})=\frac{q}{2}\,\mathfrak{y}\ ,\quad
G(x^{\mu})=\frac{p}{2}\,\mathfrak{z}\ .$ (6)

The above Ansatz has several remarkable properties. First, the three coupled
non-linear Skyrme field equations reduce consistently to just one partial
differential equation (PDE) for the profile; such PDE is the sine-Gordon
equation in $1+1$ dimensions. Second, the topological charge density is non-
trivial, leading to arbitrarily high baryon number. Third, suppose the
energy/temperature scale is less than $1/L$. In that case, the only relevant
degrees of freedom are fluctuations $\delta H(t,x)$ of $H(t,x)$, which only
depend on $t$ and $x$, as all the other possible fluctuations of the field $U$
have energies larger than $1/L$. These features of hadronic layers will be the
key to deriving novel properties and sharp predictions on their behavior at
finite density using known results in the literature on SGT.

###  2.1 Effective sine-Gordon theory

We will consider $p=q$, with $B\equiv p^{2}>0$, to reduce the complexity of
the formulas. Nevertheless, all the present results generalize easily to cases
where $p$ and $q$ are arbitrary integers. The on-shell Lagrangian and the
baryon density, $\rho_{B}\,\equiv\,J_{0}$, corresponding to the Ansatz in
Equation 6, give rise to the following effective SGT (we dropped out a
constant term in the action, which does not affect the theory)

$I_{\text{SG}}\ =\ \int
L_{\text{SG}}\,dtdx\,=\,\int\left(-\frac{1}{2}\partial^{\mu}\varphi\partial_{\mu}\varphi+M_{0}\left(\cos(\beta\,\varphi)-1\right)\right)\,dtdx\
,\qquad\varphi\ =\frac{4}{\beta}H\;,$ (7)
$\rho_{B}=-\frac{3\beta}{4}B\,\sin\left(\frac{\beta}{2}\varphi\right)\partial_{x}\varphi\;,$
(8)

with

$\displaystyle\quad M_{0}=\frac{\pi^{2}K}{8L^{2}}B^{2}\lambda\;,\quad\beta\ =\
\frac{2}{\pi\left[K\left(2L^{2}+B\,\lambda\right)\right]^{1/2}}\;,$ (9)

where constant terms have been discarded. From the above, the complete set of
Skyrme field equations are reduced to the sine-Gordon equation for the
$\varphi$ field,

$\Box\varphi-M_{0}\beta\sin(\beta\varphi)\ =0\
,\qquad\Box\equiv-\partial_{t}^{2}+\partial_{x}^{2}\;,$ (10)

which, in the static case, can be reduced to a quadrature

$dx=\frac{d\varphi}{\sigma(E_{0},\varphi)}\
,\qquad\sigma(E_{0},\varphi)=\pm\biggl{(}\frac{E_{0}}{L^{2}}-2M_{0}\cos(\beta\varphi)\biggl{)}^{\frac{1}{2}}\;,$
(11)

where $E_{0}>2L^{2}M_{0}$ is an integration constant.

Hence, one can deduce many intriguing analytic results on this
$(3+1)$-dimensional distribution of baryonic matter confined in a cavity by
using the classic SGT results in [9, 10, 11, 12, 13, 14], provided the
energy/temperatures scale is less than $1/L$. The boundary conditions for $H$
(or $\varphi$) are fixed by requiring that the baryonic charge in Section 2 is
the integer $B$,

$\begin{array}[]{l}H(t,0)=n\frac{\pi}{2}\\\
H(t,L_{x})=(m+\frac{n+1}{2})\,\pi\end{array}\ \ \ \ \ \Leftrightarrow\ \ \ \ \
\begin{array}[]{l}\varphi(t,0)=2n\frac{\pi}{\beta}\\\
\varphi(t,L_{x})=\frac{2}{\beta}\,\left(n+(2m+1)\right)\,\pi\end{array}\;,$
(12)

where $n$ and $m$ are integers.

If $H$ satisfies either $H(t,L_{x})=H(t,0)+n\,\pi$, or
$H(t,L_{x})+H(t,0)=m\,\pi$, for $n,m\in\mathds{Z},\quad\forall\,t$, the total
baryonic charge vanishes. Despite this, such configurations are interesting
anyway, since the baryonic density is non-trivial and one can have bound
states of two layers.

Figure 1 shows the energy density of a configuration with baryonic charge
$B=16$ describing baryonic layers in a cavity, where we have considered $n=0$
and $m=2$ for the boundary conditions in Equation 12.

Figure 1: Energy density of baryonic layers in a cavity.

For static configurations, $\varphi(t,x)=\varphi(x)$, the integration constant
$E_{0}$, according to Equations 4, 9, 11 and 12, is fixed by

$\pm\int_{0}^{2\pi/\beta}\frac{d\varphi}{[E_{0}-2L^{2}\,M_{0}\,\cos(\beta\varphi)]^{\frac{1}{2}}}=\frac{L_{x}}{L}\
\;,$ (13)

where we have taken, for simplicity, $n=m=0$ in Equation 12. It is easy to see
that the above equation always has a solution if $L_{x}$ is finite. Indeed, if
$L_{x}$ is small (compared to $L$) one can take a large $E_{0}$ to make the
left-hand side of Equation 13 small as well. If $L_{x}$ is large (but not
divergent) one can have the left-hand side of Equation 13 large by choosing

$E_{0}=2L^{2}\,M_{0}+\varepsilon\ ,\qquad 0<\varepsilon\ll 1\ ,$ (14)

so that the denominator comes close to have a zero when $\varphi=0$, or
$\varphi=2\pi\,\beta$. The $\frac{L_{x}}{L}\rightarrow\infty$ case corresponds
to the limit in which $\varepsilon=0$.

The results in [9, 10, 11, 12, 13, 14, 22, 23, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39] can now be used. The presence of both the two directions
$\mathfrak{y}$ and $\mathfrak{z}$ orthogonal to $x$ and the baryonic number
manifests itself in the effective sine-Gordon couplings $M_{0}$ and $\beta$.
Thus, according to Equation 9, whether or not the effective SGT describing the
hadronic distribution is in the attractive phase ($\mathbf{\mathit{AP}}$),
repulsive phase ($\mathbf{\mathit{RP}}$) or critical free-Fermion phase
($\mathbf{\mathit{FFP}}$), can be deduced from the factor
$\frac{\beta^{2}}{4\pi}$, explicitly

$\mathbf{\mathit{AP}}:\frac{1}{\pi^{3}K\left(2L^{2}+B\,\lambda\right)}<1\ ,\ \
\mathbf{\mathit{RP}}:\frac{1}{\pi^{3}K\left(2L^{2}+B\,\lambda\right)}>1\ ,\ \
\mathbf{\mathit{FFP}}:\frac{1}{\pi^{3}K\left(2L^{2}+B\,\lambda\right)}=1\ .$
(15)

Since $B$ is a positive integer, $K\lambda$ is around $1/6$ (see Ref. [54])
and $\pi^{3}>27$, then the effective theory is always in the attractive regime
and, therefore, the results in Ref. [10, 11] can be applied here. In practice,
the number of breathers

$\frac{1}{\xi}=\frac{1-\beta^{2}}{\beta^{2}}\ ,$ (16)

which is either $\frac{1}{\xi}-1$ or $\left[\frac{1}{\xi}\right]$ depending on
whether $\frac{1}{\xi}\in\mathbb{Z}$ or not, is always bigger than two,
already for $B\geq 4$.

###  2.2 In and out-of-equilibrium implications

The description of $(3+1)$-dimensional hadronic layers in a cavity presented
above in terms of the SGT in $1+1$ dimensions offers unprecedented
possibilities. When the energy/temperatures scale is low enough, the
equilibrium and out-of-equilibrium properties of these configurations can be
computed using the effective SGT with coupling constants in Equations 7 and 9.
It is worth reminding that, if one is close enough to the boundary of a
baryonic distribution of matter (such that, in one spatial direction, the
energy and baryon densities drop very rapidly to a very small value while, in
the two orthogonal directions, the energy and baryon densities are almost
homogeneous), then the present description in terms of SGT is actually
generic.

The first type of exact results, which can be “imported” from SGT in the
analysis of baryonic distributions in a cavity, has to do with the mass
spectrum of the theory, with the equilibrium correlation functions at finite
temperatures (but smaller than $1/L$) [28, 29] and with the phase diagram [4,
5, 18, 19, 20, 22, 23]. If $L_{x}/L$ is large enough, all such analytic
results can be applied directly to the effective SGT in Equations 7 and 9. In
this way, one can get the exact excitations’ spectrum of the hadronic
distribution as well as the corresponding low temperatures’ correlation
functions.

The second type is the computation of the entanglement entropy [12, 28, 29].
In particular, the above references together with the present mapping imply
that the entanglement entropy $S_{E}$ of hadronic layers confined in a cavity
is $S_{E}=\frac{c_{\text{eff}}}{3}\ln\left(\frac{l}{a}\right)$; where $a$ is
the UV cut-off beyond which the Skyrme model is not valid anymore, and $l$ is
the size of the finite interval in the $x$ direction, of which we are
computing the corresponding entanglement entropy. The effective central charge
$c_{\text{eff}}$ of the theory can be estimated following Refs. [28, 29].

The third (and, perhaps, most surprising) type has to do with out-of-
equilibrium properties, such as the dynamics of entanglement entropy after a
quantum quench and the Loschmidt Echo. These quantities are entirely out of
reach of any standard perturbative approach based on QCD in $3+1$ dimensions,
especially at finite baryon density. Nevertheless, the present mapping allows
using directly the results in Refs. [9, 10, 11, 12, 13, 14, 21, 22, 23]. These
results, together with the present mapping, allow concluding that one can
obtain exact analytical predictions for the time evolution of the entanglement
in generic hadronic layers confined in a three-dimensional cavity. Even more,
following Ref. [9] (as in the present case, the effective SGT is always in the
attractive regime with more than two breathers), the von Neumann and Renyi
entropies display undamped oscillations in time, whose frequencies can be
taken exactly from Refs. [10, 11]. For instance, the quantum quench can be
realized by either changing $B$ or $L$ (moreover, the number of oscillatory
modes grows with $B$). Finally, following Ref. [22], the Loschmidt amplitude,
the fidelity and work distribution can be computed explicitly.

##  3 Technical details

###  3.1 Euler angles parametrization and energy-density

The field equations of the system, obtained varying the action in Equation 1
with respect to the $U$ field, are

$\nabla_{\mu}\left(R^{\mu}+\frac{\lambda}{4}[R_{\nu},G^{\mu\nu}]\right)=0\;.$
(17)

Equations (17) are generally a set of three coupled non-linear partial
differential equations. The energy-momentum tensor of the theory is given by

$T_{\mu\nu}=-\frac{K}{2}\text{Tr}\left[R_{\mu}R_{\nu}-\frac{1}{2}g_{\mu\nu}R^{\alpha}R_{\alpha}+\frac{\lambda}{4}\left(g^{\alpha\beta}G_{\mu\alpha}G_{\nu\beta}-\frac{1}{4}g_{\mu\nu}G_{\sigma\rho}G^{\sigma\rho}\right)\right]\;.$
(18)

We are interested in finite density effects: in particular, we want to
describe hadronic layers confined to a cavity, where the Euler angles
parametrization for the Skyrme field is particularly convenient. The wording
“hadronic layers” refers to distributions of energy density $T_{00}$ and
baryon density $J^{0}$, which are homogeneous in two spatial coordinates.
Still, they depend non-trivially on the third spatial coordinate and on time.
The interest in such configurations lies (at the very least) in the following
facts.

First, it is possible to define the “boundary” of the distribution of nuclear
matter as the surface in space where the energy density and baryon density
drop exponentially fast to zero (or to a very small constant value). In a
small enough neighborhood of a point on such boundary, the energy density and
baryon density will not depend on the two spatial coordinates tangent to the
boundary. In contrast, they will depend very sensitively on the spatial
coordinate orthogonal to the boundary (since both $T_{00}$ and $J^{0}$ drop
rapidly to zero along this spatial direction). Hence, the configurations
discussed here are quite generic and useful close to the boundary of any
distribution of hadronic matter (see Figure 2).

Figure 2: A schematic representation of the baryonic layer in the coordinates
$(x,\mathfrak{y},\mathfrak{z})$ of (4). Taking a specific solution, if we make
a zoom-in, we find a surface perpendicular to the direction where $\rho_{B}$
drops very rapidly to zero, in this case, $\hat{x}$, separating two distinct
regions: $\rho_{B}\neq 0$ (interior) and $\rho_{B}=0$ (exterior). As the
profile $H(x,t)$ does not depend on the spatial dimensions $\mathfrak{y}$ and
$\mathfrak{z}$, at point $p$, the tangent space $T_{p}$ (where the coordinate
basis
$\\{\frac{\partial}{\partial_{\mathfrak{y}}},\frac{\partial}{\partial_{\mathfrak{z}}}\\}$
belongs to) is _approximately_ the baryonic layer.

Second, such structures are known to appear in numerical simulations at finite
baryon density, and there is robust phenomenological evidence supporting their
presence in neutron stars (see [68] and references therein). Consequently, the
starting point is the metric of a cavity in Equation 3 and Equation 4.

In the present setting, to apply the powerful results on sine-Gordon theory
(SGT) in and out of equilibrium mentioned above, the limit $L_{x}\gg L$ has to
be considered. This choice represents a sort of “hadronic wire”: a cavity
which is much longer in one spatial direction than in the other two (see
Figure 3).

Figure 3: The range of the coordinates $(x,\mathfrak{y},\mathfrak{z})$ of (4)
represents a “hadronic wire”. The longer edge has a length of $L_{x}$, while
the other two have a size of $2\pi\,L$ and $4\pi\,L$. The origin $O$ is on one
of its corners, and the coordinates axes are represented in light green.

We will choose $\mathfrak{y}$ and $\mathfrak{z}$ as the “homogeneous
coordinates” (namely, the coordinates which do not appear explicitly in
$T_{00}$ and $J^{0}$). When $L_{x}$ is not large compared to $L$, one must use
the exact available results on SGT either on a finite interval or on $S^{1}$
developed in [38, 39] and references therein (we will come back to this case
in a future publication).

As previously mentioned, the Skyrme field can be written explicitly in the
Euler angles parametrization as in Equation 5. A direct computation shows
that, in terms of this parametrization, the Skyrme action reads

$\displaystyle I(H,F,G)=$ $\displaystyle-\frac{K}{2}\int
d^{4}x\sqrt{-g}\biggl{\\{}(\nabla H)^{2}+(\nabla F)^{2}+(\nabla
G)^{2}+2\cos(2H)(\nabla F\cdot\nabla G)$
$\displaystyle-\lambda\bigr{(}2\cos(2H)((\nabla H\cdot\nabla F)(\nabla
H\cdot\nabla G)-(\nabla H)^{2}(\nabla F\cdot\nabla G))$
$\displaystyle+4\sin^{2}(H)\cos^{2}(H)((\nabla F\cdot\nabla G)^{2}-(\nabla
F)^{2}(\nabla G)^{2})$ $\displaystyle+(\nabla H\cdot\nabla F)^{2}+(\nabla
H\cdot\nabla G)^{2}-(\nabla H)^{2}(\nabla F)^{2}-(\nabla H)^{2}(\nabla
G)^{2}\bigr{)}\biggl{\\}}\ .$

The energy-momentum tensor in this parametrization reads

$\begin{split}T_{\mu\nu}=&-\frac{K}{2}\,\bigg{\\{}-2[\nabla_{\mu}F\nabla_{\nu}F+\nabla_{\mu}H\nabla_{\nu}H+\nabla_{\mu}G\nabla_{\nu}G+\cos(2H)(\nabla_{\mu}F\,\nabla_{\nu}G+\nabla_{\mu}G\,\nabla_{\nu}F)]\\\
&+g_{\mu\nu}\,[(\nabla F)^{2}+(\nabla H)^{2}+(\nabla G)^{2}+2\cos(2H)(\nabla
F\cdot\nabla G)]\\\ &+2\lambda\biggl{[}\nabla_{\mu}F(\nabla H\cdot\nabla
F)\nabla_{\nu}H-\nabla_{\mu}F(\nabla H)^{2}\nabla_{\nu}F-\nabla_{\mu}H(\nabla
F)^{2}\nabla_{\nu}H+\nabla_{\mu}H(\nabla F\cdot\nabla H)\nabla_{\nu}F\\\
&+\nabla_{\mu}G(\nabla H\cdot\nabla G)\nabla_{\nu}H-\nabla_{\mu}G(\nabla
H)^{2}\nabla_{\nu}G-\nabla_{\mu}H(\nabla
G)^{2}\nabla_{\nu}H+\nabla_{\mu}H(\nabla G\cdot\nabla H)\nabla_{\nu}G\\\
&+\cos(2H)\,\big{(}\nabla_{\mu}F(\nabla H\cdot\nabla
G)\nabla_{\nu}H-\nabla_{\mu}F(\nabla H)^{2}\nabla_{\nu}G-\nabla_{\mu}H(\nabla
F\cdot\nabla G)\nabla_{\nu}H\\\ &+\nabla_{\mu}H(\nabla F\cdot\nabla
H)\nabla_{\nu}G+\nabla_{\mu}G(\nabla H\cdot\nabla
F)\nabla_{\nu}H-\nabla_{\mu}G(\nabla H)^{2}\nabla_{\nu}F-\nabla_{\mu}H(\nabla
G\cdot\nabla F)\nabla_{\nu}H\\\ &+\nabla_{\mu}H(\nabla G\cdot\nabla
H)\nabla_{\nu}F\big{)}+4\cos^{2}(H)\,\sin^{2}(H)\,(\nabla_{\mu}F(\nabla
G\cdot\nabla F)\nabla_{\nu}G-\nabla_{\mu}F(\nabla G)^{2}\nabla_{\nu}F\\\
&-\nabla_{\mu}G(\nabla F)^{2}\nabla_{\nu}G+\nabla_{\mu}G(\nabla F\cdot\nabla
G)\nabla_{\nu}F)\biggl{]}\\\ &+\lambda\,g_{\mu\nu}\,\big{[}(\nabla
F)^{2}(\nabla H)^{2}-(\nabla F\cdot\nabla H)^{2}+(\nabla G)^{2}(\nabla
H)^{2}-(\nabla G\cdot\nabla H)^{2}\\\ &+2\cos(2H)\,((\nabla F\cdot\nabla
G)(\nabla H)^{2}-(\nabla F\cdot\nabla H)(\nabla G\cdot\nabla H))\\\
&+4\cos^{2}(H)\,\sin^{2}(H)\,((\nabla F)^{2}(\nabla G)^{2}-(\nabla
F\cdot\nabla G)^{2})\big{]}\bigg{\\}}\;.\end{split}$

The field equations, obtained varying the action with respect to the degrees
of freedom $F$, $H$ and $G$, are

$\displaystyle 0$
$\displaystyle=\nabla_{\mu}\,\bigg{\\{}\cos(2G)\,\sin(2H)\,\nabla^{\mu}F-\sin(2G)\,\nabla^{\mu}H$
$\displaystyle-\lambda\,\sin(2G)\biggl{(}(\nabla F)^{2}\nabla^{\mu}H-(\nabla
F\cdot\nabla H)\nabla^{\mu}F+(\nabla G)^{2}\nabla^{\mu}H-(\nabla G\cdot\nabla
H)\nabla^{\mu}G$ $\displaystyle+\cos(2H)\,(2(\nabla F\cdot\nabla
G)\nabla^{\mu}H-(\nabla F\cdot\nabla H)\nabla^{\mu}G-(\nabla H\cdot\nabla
G)\nabla^{\mu}F)\biggl{)}$
$\displaystyle-\lambda\,\cos(2G)\,\sin(2H)\biggl{(}(\nabla F\cdot\nabla
G)\nabla^{\mu}G+(\nabla F\cdot\nabla H)\nabla^{\mu}H-(\nabla
G)^{2}\nabla^{\mu}F$ $\displaystyle-(\nabla
H)^{2}\nabla^{\mu}F-\cos(2H)\,\left((\nabla F\cdot\nabla
G)\nabla^{\mu}F-(\nabla F)^{2}\nabla^{\mu}G\right)\biggl{)}\bigg{\\}}\ ,$ (19)
$\displaystyle 0$
$\displaystyle=\nabla_{\mu}\,\biggl{\\{}\sin(2G)\,\sin(2H)\,\nabla^{\mu}F+\cos(2G)\,\nabla^{\mu}H$
$\displaystyle+\lambda\,\cos(2G)\left((\nabla G)^{2}\nabla^{\mu}H-(\nabla
G\cdot\nabla H)\nabla^{\mu}G+(\nabla F)^{2}\nabla^{\mu}H-(\nabla F\cdot\nabla
H)\nabla^{\mu}F\right.$ $\displaystyle\left.+\cos(2H)\,\left(2(\nabla
F\cdot\nabla G)\nabla^{\mu}H-(\nabla F\cdot\nabla H)\nabla^{\mu}G-(\nabla
G\cdot\nabla H)\nabla^{\mu}F\right)\right)$
$\displaystyle+\lambda\,\sin(2G)\,\sin(2H)((\nabla H)^{2}\nabla^{\mu}F-(\nabla
H\cdot\nabla F)\nabla^{\mu}H+(\nabla G)^{2}\nabla^{\mu}F$
$\displaystyle-(\nabla G\cdot\nabla F)\nabla^{\mu}G+\cos(2H)((\nabla
F\cdot\nabla G)\nabla^{\mu}F-(\nabla F)^{2}\nabla^{\mu}G))\biggl{\\}}\ ,$ (20)
$\displaystyle 0$
$\displaystyle=\nabla_{\mu}\,\bigg{\\{}\cos(2H)\,\nabla^{\mu}F+\nabla^{\mu}G-\lambda\,\sin^{2}(2H)\,((\nabla
F\cdot\nabla G)\nabla^{\mu}F-(\nabla F)^{2}\nabla^{\mu}G)$
$\displaystyle+\lambda\,\cos(2H)\,((\nabla H)^{2}\nabla^{\mu}F-(\nabla
H\cdot\nabla F)\nabla^{\mu}H)+\lambda\,((\nabla H)^{2}\nabla^{\mu}G-(\nabla
H\cdot\nabla G)\nabla^{\mu}H)\bigg{\\}}\;.$ (21)

The only way to have an energy density homogeneous in $\mathfrak{y}$ and
$\mathfrak{z}$ is to require that $F$ and $G$ are linear functions of these
coordinates, and $H$ depends on the coordinate $x$ (transverse to the layer)
and on time. The only Ansatz satisfying these properties is the one in
Equation 6. Indeed, if $H$ would depend either on $\mathfrak{y}$ or on
$\mathfrak{z}$, then the energy density would depend on these coordinates as
well. Hence, the profile $H$ carries the physical information on when and
where $T_{00}$ and $J^{0}$ vanish and when they do not (that is why it makes
sense to call $H$ “profile”, as it encodes information on the spacetime
variations of $T_{\mu\nu}$ and $J^{\mu}$). Furthermore, the above Ansatz is
actually generic if one is close enough to the boundary of any baryonic
distribution (as it has been already emphasized). Consequently, the Ansatz
here above describe locally any baryonic configuration close to one of its
boundaries.

The above choice has several remarkable properties (see [62, 63, 69, 70]).
First, the three coupled non-linear Skyrme field equations reduce consistently
to just one PDE for the profile $H(t,x)$; the sine-Gordon equation in $1+1$
dimensions (as one can check directly in Sections 3.1, 20 and 21). Second,
this choice keeps alive the topological density.

In fact, by using the Ansatz in Equation 6, the field equations in Sections
3.1, 20 and 21, are reduced to

$\partial_{t}^{2}H-\partial_{x}^{2}H+\frac{B^{2}\,\lambda}{8L^{2}\,(2L^{2}+B\,\lambda)}\sin(4H)=0\;,$
(22)

where we have considered for simplicity $p=q$, and $B=p^{2}>0$. Also, the on-
shell Lagrangian density $\mathcal{L}_{on-shell}$ (apart from a constant term
$-K\frac{B}{4L^{2}}$), the energy-density $T_{00}$ (apart from a constant term
$K\frac{B}{4L^{2}}$), and the baryon density $\rho_{B}=J_{0}$ are,
respectively,

$\mathcal{L}_{on-
shell}=\frac{K}{64L^{4}}\,\left(16L^{2}\,\left(2L^{2}+\left|B\right|\,\lambda\right)\,\left[\left(\partial_{t}H\right)^{2}-\left(\partial_{x}H\right)^{2}\right]-B^{2}\,\lambda\,\left(1-\cos\left(4H\right)\right)\right)\;,$
(23)
$T_{00}=\frac{K}{64L^{4}}\,\left(16L^{2}\left(2L^{2}+\left|B\right|\,\lambda\right)\,\left[\left(\partial_{t}H\right)^{2}+\left(\partial_{x}H\right)^{2}\right]+B^{2}\,\lambda\left(1-\cos(4H)\right)\right)\;,$
(24) $\ \rho_{B}=J_{0}=-3\,B\,\left(\partial_{x}H\right)\,\sin(2H)\;.$ (25)

The boundary conditions for $H$ are fixed by requiring that the baryonic
charge is $\pm B$, as we have mentioned. For the case $Q_{B}$ to be zero,
should be $\cos(2H(t,L_{x}))=\cos(2H(t,0))$, $\forall\,t$. This implies

$H(t,L_{x})=H(t,0)+n\,\pi,\quad\mbox{\emph{or}}\quad
H(t,L_{x})+H(t,0)=m\,\pi,\quad n,m\in\mathds{Z},\quad\forall\,t\;.$ (26)

Such configurations, of $Q_{B}=0$, are interesting anyway since the baryonic
density is non-trivial and one can have bound states of two layers (in
breather-like style).

For static configurations $H(t,x)=H(x)$ one can reduce the field equation to a
simple quadrature

$\left(\partial_{x}H\right)=\pm\left[E_{0}-\frac{B^{2}\,\lambda}{16L^{2}\left(2L^{2}+B\,\lambda\right)}\cos
4H\right]^{1/2}\ ,$ (27)

where the integration constant $E_{0}$ is fixed by

$4\int_{0}^{\pi/2}\frac{dH}{\left[16L^{2}E_{0}-\frac{B^{2}\lambda}{\left(2L^{2}+B\,\lambda\right)}\cos
4H\right]^{1/2}}=\frac{L_{x}}{L}\;,$ (28)

where we have taken, for simplicity, $n=m=0$ in Equation 26. It is easy to see
that the above equation for $E_{0}$ always has a solution if $L_{x}$ is
finite. Indeed, if $L_{x}$ is small (compared to $L$) one can take a large
$E_{0}$ to make the left-hand side of Equation 28 small as well. If $L_{x}$ is
large (but not divergent) one can have the left-hand side of Equation 28 large
by choosing

$E_{0}=\frac{B^{2}\lambda}{16L^{2}\left(2L^{2}+B\lambda\right)}+\varepsilon\
,\ \ 0<\varepsilon\ll 1\ ,$

so that the denominator of the left-hand side of Equation 28 comes close to
have a zero when $H=0$, or $H=\pi/2$. The $L_{x}\rightarrow\infty$ case
corresponds to the limit in which $\varepsilon=0$.

The proper normalization of the Skyrme profile $H$ to define the effective
sine-Gordon Lagrangian $L_{SG}$ and the corresponding energy-density
$\widetilde{T}_{00}$ can be achieved by requiring that the integral along the
coordinate $x$ of $\widetilde{T}_{00}$ should give the actual total energy of
the hadronic layers (with a similar condition for the effective action
$I_{SG}$). Hence, $\widetilde{T}_{00}$ is the integral in the transverse
coordinates $\mathfrak{y}$ and $\mathfrak{z}$ of $T_{00}$, so that the
integral along $x$ of $\widetilde{T}_{00}$ will give the total energy of the
Skyrmionic system

$\widetilde{T}_{00}=\frac{\pi^{2}K}{8L^{2}}\left\\{16L^{2}\left(2L^{2}+\left|B\right|\lambda\right)\,\left[\left(\partial_{t}H\right)^{2}+\left(\partial_{x}H\right)^{2}\right]+B^{2}\,\lambda\left(1-\cos(4H)\right)\right\\}\;,$
(29)

where factor $8\pi^{2}L^{2}$ comes from the integral in $\mathfrak{y}$ and
$\mathfrak{z}$. The same is true for the effective lagrangian $L_{SG}$:

$L_{SG}=\frac{\pi^{2}K}{8L^{2}}\left\\{16L^{2}\left(2L^{2}+B\,\lambda\right)\left[\left(\partial_{t}H\right)^{2}-\left(\partial_{x}H\right)^{2}\right]-B^{2}\lambda\left(1-\cos\left(4H\right)\right)\right\\}\
.$ (30)

The proper normalization of the kinetic term can be achieved normalizing $H$
as follows:

$\varphi=\frac{4}{\beta}H\
,\quad\beta=\frac{2}{\pi\left[K\left(2L^{2}+B\,\lambda\right)\right]^{1/2}}\
,\qquad M_{0}=\frac{\pi^{2}K}{8L^{2}}B^{2}\lambda\ ,$ (31)

so that the effective sine-Gordon coupling $\beta$ and the effective
dimensionless sine-Gordon action become, respectively,

$\beta=\frac{2}{\pi\left[K\left(2L^{2}+B\,\lambda\right)\right]^{1/2}}\ ,$
(32) $I_{SG}=\int
L_{SG}dtdx=\,\,\int\left(-\frac{1}{2}\partial^{\mu}\varphi\partial_{\mu}\varphi+M_{0}\left(\cos(\beta\,\varphi)-1\right)\right)dt\,dx\
,$ (33)

where constant terms have been discarded.

###  3.2 Perturbations on the solutions

An important technical part of the present work is to show that, with the
Ansatz defined in Equation 6, not only the field equations and the energy-
density reduce to the corresponding quantities in SGT in $1+1$ dimensions in a
sector with non-vanishing baryonic charge, but also that the lowest energy
perturbations of these configurations are precisely perturbations of the sine-
Gordon effective field which only depend on $t$ and $x$ (this is the reason
why it is convenient to take $L_{x}\gg L$: in this case, the energy needed to
excite modes which depend non-trivially on the transverse coordinates is much
higher than the energy needed to excite “sine-Gordon modes”). This issue is
relevant since, if we want to use the available results on equilibrium and
non-equilibrium SGT (in particular, [71, 30, 31, 72, 73, 32, 74, 33, 75, 34,
35, 36, 37, 38, 39, 9, 76]), then we must identify a regime in which also the
low energy fluctuations are of “sine-Gordon type”.

Hence, let us consider a general solution $U_{0}$ of the form in Equation 5,
where $F$, $H$ and $G$ fulfilling Sections 3.1, 20 and 21. Let us now take a
perturbation of $U_{0}$ as

$U\ =U_{0}\left(\mathbb{1}+\chi\right)\;,$ (34)

where $\chi$ is a $2\times 2$ matrix with the conditions

$\chi^{\dagger}=-\chi\;,\qquad\mbox{Tr}\,\chi=0\,.$ (35)

These conditions ensure that $\chi$ is an arbitrary element of the
$\mathfrak{su}(2)$ algebra, i.e.,

$\chi=\epsilon\,\chi_{1}\,t_{1}\,+\epsilon\,\chi_{2}\,t_{1}\,+\epsilon\,\chi_{3}\,t_{3}\;,$
(36)

where $\left|\epsilon\right|\ll 1$ is the perturbation parameter. Observe that
$\chi_{i}$ are real functions of the coordinates $t$, $x$, $\mathfrak{y}$ and
$\mathfrak{z}$.111Notice that we could have equally defined $U\
=U_{0}+\chi^{\prime}$, where $\chi^{\prime}=U_{0}\chi$. Therefore, as $U_{0}$
is invertible, all the results for $\chi$ are equivalent to $\chi^{\prime}$.
In this way, we see that Equation 34 is directly related to the usual
prescription to write a perturbation as the solution $U_{0}$ plus something.
Introducing this expansion in Equation 17, we get

$\displaystyle[R_{0\mu}+\frac{\lambda}{4}[R_{0}^{\nu},G_{0\mu\nu}],\nabla^{\mu}\chi]+\nabla^{\mu}\bigg{(}\nabla_{\mu}\chi+\frac{\lambda}{4}[\nabla^{\nu}\chi,G_{0\mu\nu}]$
$\displaystyle\bigg{)}$
$\displaystyle+\frac{\lambda}{4}\nabla^{\mu}\bigg{(}[R_{0}^{\nu},[\nabla_{\mu}\chi,R_{0\nu}]]+[R_{0}^{\nu},[R_{0\mu},\nabla_{\nu}\chi]]$
$\displaystyle\bigg{)}=0\;,$ (37)

up to order $O(\epsilon^{2})$. Now, let us consider a generic perturbation (in
the present context, the wording “generic perturbation” means that we allow
the perturbation to depend on all the four space-time coordinates). A natural
Ansatz for the perturbation is

$\displaystyle\chi^{1}\ =\ $
$\displaystyle\zeta_{1}(x)\,\cos(H(x))\,\cos(p\,\mathfrak{z})\,e^{\mathrm{i}(\omega\,t+k_{1}\,\mathfrak{y}+k_{2}\,\mathfrak{z})}\;,$
$\displaystyle\chi^{2}\ =\ $
$\displaystyle\zeta_{2}(x)\,\cos(H(x))\,\sin(p\,\mathfrak{z})\,e^{\mathrm{i}(\omega\,t+k_{1}\,\mathfrak{y}+k_{2}\,\mathfrak{z})}\;,$
$\displaystyle\chi^{3}\ =\ $
$\displaystyle\zeta_{3}(x)\,\sin(H(x))\,e^{\mathrm{i}(\omega\,t+k_{1}\,\mathfrak{y}+k_{2}\,\mathfrak{z})}\;,\qquad
k_{1}\neq 0\ ,\ k_{2}\neq 0\ ,$

where we are taking into account the fact that the energy and baryon densities
do not depend on the transverse coordinates. The profiles $\zeta_{j}(x)$
($j=1,2,3$) of the perturbations only depend on $x$. The first conclusion
which arises analyzing the linearized field equations is that, actually, only
one of these three profiles is independent. Namely, one can choose222The fact
that only one radial function is necessary in the Ansatz can be easily
verified for the non-linear sigma model case, that is, when $\lambda=0$.
Moreover, for small values of $\lambda$ this could still be true by analytic
continuation. It can be also checked that the linearized equations for
perturbations of the form in Section 3.2 are always consistent (namely, for
any value of $\lambda$ one always gets as many equations as unknown
functions).: $\zeta_{1}(x)=\zeta_{2}(x)=\zeta(x)=-\zeta_{3}(x)$. Now,
considering

$\displaystyle\chi^{1}\ =\ $
$\displaystyle\zeta(x)\,\cos(H(x))\,\cos(p\,\mathfrak{z})\,e^{\mathrm{i}(\omega\,t+k_{1}\,\mathfrak{y}+k_{2}\,\mathfrak{z})}\;,$
$\displaystyle\chi^{2}\ =\ $
$\displaystyle\zeta(x)\,\cos(H(x))\,\sin(p\,\mathfrak{z})\,e^{\mathrm{i}(\omega\,t+k_{1}\,\mathfrak{y}+k_{2}\,\mathfrak{z})}\;,$
(38) $\displaystyle\chi^{3}\ =\ $
$\displaystyle-\zeta(x)\,\sin(H(x))\,e^{\mathrm{i}(\omega\,t+k_{1}\,\mathfrak{y}+k_{2}\,\mathfrak{z})}\;,$

one can check that the complete set of linearized Skyrme equations (3.2) are
satisfied if $k_{2}=\frac{q}{p}\,k_{1},$ (i.e., $k_{2}=k_{1}\equiv k$, because
we are taken $p=q$), and if $\zeta$ satisfies a linear ordinary differential
equation (ODE) of the form
$\zeta^{\prime\prime}(x)+A(x)\,\zeta^{\prime}(x)+B(x)\,\zeta(x)\ =\ 0$, (where
the functions $A(x)$ and $B(x)$ can be computed explicitly in terms of the
background solution). In order to use the Sturm-Liouville theory, it is
convenient the following change of variables $\zeta(x)=\alpha(x)\,\xi(x)$,
choosing $\alpha$ in such a way to eliminate the first derivative term. In
this way, we get

$-\xi^{\prime\prime}(x)+\frac{Q(x)}{L^{2}}\,\xi(x)=\omega^{2}\,W(x)\,\xi(x)\;,$
(39)

where the functions $Q(x)$ and $W(x)$ are

$\displaystyle Q(x)=$
$\displaystyle\frac{1}{32\,(B\uplambda+2)\,(B\,\uplambda\,\cos^{2}(H(x))+2)^{2}}\,\times$
$\displaystyle\bigg{\\{}(B\uplambda\cos(2H(x))+B\uplambda+4)\,\bigg{(}B^{3}\,\uplambda^{2}\,\cos(6H(x))+B^{2}\uplambda\,(3B\,\uplambda+8)\,\cos(2H(x))$
$\displaystyle+2(B\uplambda+2)\,(B^{2}\,\uplambda\,\cos(4H(x))+B^{2}\,\uplambda+32k^{2})\bigg{)}$
$\displaystyle-\bigg{(}B\,\uplambda\bigg{(}4\cos(2H(x))\,(\uplambda(B-k^{2})+3)+B\,\uplambda\,\cos(4H(x))\bigg{)}$
$\displaystyle+B\,\uplambda^{2}\,(3B-4k^{2})+8\uplambda\,(B-2k^{2})+8\bigg{)}\,\bigg{(}16(B\,\uplambda+2)\,E_{0}+B^{2}\,\uplambda\,\cos(4H(x))\bigg{)}\bigg{\\}}\;,$
$\displaystyle W(x)=$
$\displaystyle\frac{4\uplambda\,E_{0}+B\,\uplambda\left(\frac{B\uplambda\,\cos(4H(x))}{4B\,\uplambda+8}+\cos(2H(x))\right)+B\,\uplambda+4}{(2B\,\uplambda\,\cos^{2}(H(x))+4)}\;.$

Here we have defined $\uplambda\equiv\frac{\lambda}{L^{2}}$, and used
Equations 22 and 27. Also,

$\alpha(x)=\frac{C_{1}}{\sqrt{4+B\,\uplambda(1+\cos(H(x)))}}\;,$ (40)

being $C_{1}$ an arbitrary dimensionless constant. Equation 39 can be written
in a slightly different manner by the change of variable $\rho=\frac{x}{L}$,

$-\xi^{\prime\prime}+Q\,\xi=\upomega^{2}\,W\,\xi\;,$ (41)

where the substitution of $x$ as a function of $\rho$ is carried out whatever
is necessary, the prime now denotes $\frac{\partial}{\partial\rho}$, and we
have defined $\upomega\equiv\omega\,L$.

The ODE in Equation 41 is a particular case of the Sturm-Liouville problem
333Usually, the SLP is written as
$-\xi^{\prime\prime}+q\,\xi=\lambda\,W\,\xi$, being $\lambda$ the eigenvalues
to be found for particular boundary conditions [77]. Therefore, Equation 41)
is a particular case with $p=1$, $q=Q$, $w=W$, and eigenvalues
$\lambda=\upomega^{2}$. (SLP), it can hardly be solved analytically and even
numerically, it is not simple. For us it is enough to have some sufficient
stability conditions: we require that the function $\xi$ does not diverge
inside the range where $x$ is defined, i.e., the interval $[0,L_{x}]$.
Therefore, we find the eigenvalues $\upomega^{2}$ using the method of _simple
centred differences_ [77], and the minimum $\upomega^{2}_{\mbox{min}}$ should
be positive for stability, with boundary conditions $\xi(0)=0$ and
$\xi(L_{x})=0$. The function $H(x)$ as a solution of Equation 22, with
boundary conditions in Equation 12, is well-behaved. Also, the function in
Equation 40 is well-behaved, strictly positive and without singularities
inside the integration range if $\lambda\geq 0$ and $L\neq 0$. In Figure 4, we
show the stabilitpy regions for different values of $B$ and $\frac{L_{x}}{L}$
(see details in the caption).

(a) $B=0$ and $\frac{L_{x}}{L}=1$

(b) $B=0$ and $\frac{L_{x}}{L}=5$

(c) $B=0$ and $\frac{L_{x}}{L}=10$

(d) $B=1$ and $\frac{L_{x}}{L}=1$

(e) $B=1$ and $\frac{L_{x}}{L}=5$

(f) $B=1$ and $\frac{L_{x}}{L}=10$

(g) $B=4$ and $\frac{L_{x}}{L}=1$

(h) $B=4$ and $\frac{L_{x}}{L}=5$

(i) $B=4$ and $\frac{L_{x}}{L}=10$

(j) $B=9$ and $\frac{L_{x}}{L}=1$

(k) $B=9$ and $\frac{L_{x}}{L}=5$

(l) $B=9$ and $\frac{L_{x}}{L}=10$

Figure 4: Plots of the stability regions at the $(k\,\lambda/L^{2})$ plane for
$m=0,2,4,6,8$ in (12) for different values of $B$ and $\frac{L_{x}}{L}$. For
example, plot (j) shows that above the separation line of $m=8$ (in violet)
the region is stable, whilst below such separation line is unstable; for
$m=6$, above the separation line (in red) is stable, whilst below it is
unstable; same for the separation line $m=4$ (in green) and $m=2$ (in yellow).
Plot (b) shows that, for those values of $\frac{L_{x}}{L}$ and $B$, the region
is stable for all values of $m\leq 8$. Notice that the $k$ values are
representative, as by boundary condition in the cavity, the allowed values of
$k$ are, in fact, integers.

By analyzing the energy scale of the fluctuations of $F$, $G$ and $H$, it is
observed that the minimum of positive frequencies $\omega_{\mbox{min}}$ goes
as $1/L$ due to the scale normalization of $\upomega$ in Equation 41, at least
for the set of parameters analyzed. This can be seen in Figure 5 for some
values of $B$ and $\frac{L_{x}}{L}$.

(a) $B=0$ and $\frac{L_{x}}{L}=1$

(b) $B=0$ and $\frac{L_{x}}{L}=5$

(c) $B=1$ and $\frac{L_{x}}{L}=1$

(d) $B=1$ and $\frac{L_{x}}{L}=5$

(e) $B=4$ and $\frac{L_{x}}{L}=1$

(f) $B=4$ and $\frac{L_{x}}{L}=5$

(g) $B=9$ and $\frac{L_{x}}{L}=1$

(h) $B=9$ and $\frac{L_{x}}{L}=5$

Figure 5: Plots $k$ vs. $\lambda/L^{2}$ for $m=0$ taking different values of
$B$ and $\frac{L_{x}}{L}$. The dimensionless quantities
$\upomega^{2}_{\mbox{min}}=\omega^{2}_{\mbox{min}}\,L^{2}$ are shown in the
color palette on the right of each plot. As the values of $\upomega^{2}$ are
not much tiny, $\omega$ goes as $L^{-1}$, at least for the set of parameters
analyzed. The $k$ values are representative (see the caption of Figure 4).

On the other hand, the lowest energy perturbations are perturbations $\delta
H$ of $H(t,x)$ which depend only on $t$ and $x$. In particular, these
perturbations of static profile $H(t,x)=H(x)$ are gapless, as in SGT (when
$L_{x}\gg L$). One can readily see this as follows. The Skyrme field equations
for a static profile $H(t,x)=H(x)$ corresponding to the Ansatz in Equation 6
reduces to Equation 22, which, in its turn, reduces to Equation 27, where the
condition in Equation 28 fixes the integration constant $E_{0}$. Given a
solution $H_{0}(x)$ of Equation 27 one can always find a solution $\delta H$
of the linearized field equation with zero energy as $\delta
H=\partial_{x}H_{0}(x)$. With the appropriate choice of $E_{0}$,
$\partial_{x}H_{0}(x)$ never changes sign, so $\partial_{x}H_{0}(x)$ is a
nodeless zero mode. Moreover, as it has been shown in Ref. [78], SGT possesses
gapless modes.

Summarizing, the above arguments show that at energy and/or temperatures less
than $1/L$, the only modes which are energetically available in the full
Skyrme theory in the cavity, like Figure 3, are the sine-Gordon modes
associated with perturbations $\delta H$ of $H(t,x)$ which depend only on $t$
and $x$. Consequently, not only does the Skyrme model reduce to SGT for
baryonic layers configurations in such a cavity, but also perturbations in
this regime are, in fact, the lowest energy perturbations of SGT because
generic perturbations like the ones shown above always possess higher energy.

##  4 Conclusions

In this article, it has been constructed an explicit mapping between hadronic
layers in a three-dimensional cavity in the low-energy limit of QCD and an
effective sine-Gordon theory in $1+1$ dimensions. This mapping allows deriving
exact non-perturbative results on these baryonic layers’ low-
energy/temperature behaviour using well-known results in SGT. These analytic
results (especially the out-of-equilibrium one) are entirely out of reach of
the other available theoretical methods in the low-energy sector of QCD. The
sharp predictions on the oscillations of von Neumann and Renyi entropies can
be, at least in principle, tested experimentally. The consequences of the
existence of this mapping are far-reaching and will be further investigated in
forthcoming papers.

Finally, the implications of our results are especially intriguing in the
analysis of neutron stars. Indeed, configurations such as hadronic tubes and
layers of baryons (known as nuclear pasta states) appear [79, 80]. Our
framework allows us to determine, among other things, the transport properties
of these inhomogeneous baryonic distributions with such beautiful shapes,
which are challenging to compute using numerical simulations.

## Acknowledgements

F. C. has been funded by Fondecyt Grant No. 1200022. M. L. is funded by ANID,
Convocatoria Nacional Subvención a la Instalación en la Academia Convocatoria
Año 2022, Folio SA85220027. P. P. is supported by Fondo Nacional de Desarrollo
Científico y Tecnológico–Chile (Fondecyt Grant No. 3200725) and by Charles
University Research Center (UNCE/SCI/013). A. V. is funded by FONDECYT post-
doctoral Grant No. 3200884. The Centro de Estudios Científicos (CECs) is
funded by the Chilean Government through the Centers of Excellence Base
Financing Program of ANID.

## References

  * [1] Norbert Schuch, Michael M Wolf, Frank Verstraete and J Ignacio Cirac “Entropy scaling and simulability by matrix product states” In _Physical review letters_ 100.3 APS, 2008, pp. 030504
  * [2] Álvaro Perales and Guifré Vidal “Entanglement growth and simulation efficiency in one-dimensional quantum lattice systems” In _Physical Review A_ 78.4 APS, 2008, pp. 042337
  * [3] Philipp Hauke et al. “Can one trust quantum simulators?” In _Reports on Progress in Physics_ 75.8 IOP Publishing, 2012, pp. 082401
  * [4] Pasquale Calabrese and John Cardy “Quantum quenches in extended systems” In _Journal of Statistical Mechanics: Theory and Experiment_ 2007.06 IOP Publishing, 2007, pp. P06008
  * [5] Pasquale Calabrese, Fabian HL Essler and Giuseppe Mussardo “Introduction to ‘quantum integrability in out of equilibrium systems”’ In _Journal of Statistical Mechanics: Theory and Experiment_ 2016.6 IOP Publishing, 2016, pp. 064001
  * [6] JM Deutsch, Haibin Li and Auditya Sharma “Microscopic origin of thermodynamic entropy in isolated systems” In _Physical Review E_ 87.4 APS, 2013, pp. 042135
  * [7] Wouter Beugeling, Alexei Andreanov and Masudul Haque “Global characteristics of all eigenstates of local many-body Hamiltonians: participation ratio and entanglement entropy” In _Journal of Statistical Mechanics: Theory and Experiment_ 2015.2 IOP Publishing, 2015, pp. P02002
  * [8] Adam M Kaufman et al. “Quantum thermalization through entanglement in an isolated many-body system” In _Science_ 353.6301 American Association for the Advancement of Science, 2016, pp. 794–800
  * [9] Vincenzo Alba and Pasquale Calabrese “Entanglement and thermodynamics after a quantum quench in integrable systems” In _Proceedings of the National Academy of Sciences_ 114.30 National Acad Sciences, 2017, pp. 7947–7951
  * [10] Olalla Castro-Alvaredo and David Horváth “Branch point twist field form factors in the sine-Gordon model I: Breather fusion and entanglement dynamics” In _SciPost Physics_ 10.6, 2021, pp. 132
  * [11] David Horváth, Pasquale Calabrese and Olalla Castro-Alvaredo “Branch point twist field form factors in the sine-Gordon model II: composite twist fields and symmetry resolved entanglement” In _SciPost Physics_ 12.3, 2022, pp. 088
  * [12] Pinaki Banerjee, Atanu Bhatta and B Sathiapalan “Sine-Gordon theory: entanglement entropy and holography” In _Physical Review D_ 96.12 APS, 2017, pp. 126014
  * [13] Elisa Ercolessi, Stefano Evangelisti and Francesco Ravanini “Exact entanglement entropy of the XYZ model and its sine-Gordon limit” In _Physics Letters A_ 374.21 Elsevier, 2010, pp. 2101–2105
  * [14] A Alonso Izquierdo, J Mateos Guilarte and NG Almeida “Solitons and entanglement in the double sine-Gordon model” In _Journal of Physics B: Atomic, Molecular and Optical Physics_ 48.1 IOP Publishing, 2014, pp. 015501
  * [15] Anatoli Polkovnikov, Krishnendu Sengupta, Alessandro Silva and Mukund Vengalattore “Colloquium: Nonequilibrium dynamics of closed interacting quantum systems” In _Reviews of Modern Physics_ 83.3 APS, 2011, pp. 863
  * [16] Christian Gogolin and Jens Eisert “Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems” In _Reports on Progress in Physics_ 79.5 IOP Publishing, 2016, pp. 056001
  * [17] Pasquale Calabrese, Fabian HL Essler and Giuseppe Mussardo “Introduction to ‘quantum integrability in out of equilibrium systems”’ In _Journal of Statistical Mechanics: Theory and Experiment_ 2016.6 IOP Publishing, 2016, pp. 064001
  * [18] Daniel J Amit, Yadin Y Goldschmidt and S Grinstein “Renormalisation group analysis of the phase transition in the 2D Coulomb gas, Sine-Gordon theory and XY-model” In _Journal of Physics A: Mathematical and General_ 13.2 IOP Publishing, 1980, pp. 585
  * [19] K. Babu Joseph and V.C. Kuriakose “Effective potential in sine-Gordon theory at finite temperature” In _Physics Letters A_ 88.9, 1982, pp. 447–449 DOI: https://doi.org/10.1016/0375-9601(82)90537-0
  * [20] WH Kye, SI Hong and Jae Kwan Kim “Phase transition of the sine-Gordon theory at finite temperature” In _Physical Review D_ 45.8 APS, 1992, pp. 3006
  * [21] David Horváth, Spyros Sotiriadis, Márton Kormos and Gabor Takacs “Inhomogeneous quantum quenches in the sine-Gordon theory” In _SciPost Physics_ 12.5, 2022, pp. 144
  * [22] Colin Rylands and Natan Andrei “Loschmidt amplitude and work distribution in quenches of the sine-Gordon model” In _Phys. Rev. B_ 99 American Physical Society, 2019, pp. 085133 DOI: 10.1103/PhysRevB.99.085133
  * [23] Pinaki Banerjee, Atanu Bhatta and B Sathiapalan “Sine-Gordon theory: entanglement entropy and holography” In _Physical Review D_ 96.12 APS, 2017, pp. 126014
  * [24] Edward Witten “APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory” In _Reviews of Modern Physics_ 90.4 APS, 2018, pp. 045003
  * [25] Tatsuma Nishioka “Entanglement entropy: holography and renormalization group” In _Reviews of Modern Physics_ 90.3 APS, 2018, pp. 035007
  * [26] Robi Peschanski and Shigenori Seki “Entanglement entropy of scattering particles” In _Physics Letters B_ 758 Elsevier, 2016, pp. 89–92
  * [27] Alex Kovner, Michael Lublinsky and Mirko Serino “Entanglement entropy, entropy production and time evolution in high energy QCD” In _Physics Letters B_ 792 Elsevier, 2019, pp. 4–15
  * [28] Alexander B Zamolodchikov and Alexey B Zamolodchikov “Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models” In _Annals of Physics_ 120.2, 1979, pp. 253–291 DOI: https://doi.org/10.1016/0003-4916(79)90391-9
  * [29] Al B Zamolodchikov “Mass scale in the sine–Gordon model and its reductions” In _International Journal of Modern Physics A_ 10.08 World Scientific, 1995, pp. 1125–1150
  * [30] Fedor Aleksandrovich Smirnov “Form factors in completely integrable models of quantum field theory” World Scientific, 1992
  * [31] H Babujian, A Fring, M Karowski and A Zapletal “Exact form factors in integrable quantum field theories: The sine-Gordon model” In _Nuclear Physics B_ 538.3 Elsevier, 1999, pp. 535–586
  * [32] H Babujian and M Karowski “Towards the construction of Wightman functions of integrable quantum field theories” In _International Journal of Modern Physics A_ 19.supp02 World Scientific, 2004, pp. 34–49
  * [33] M Jimbo, T Miwa and F Smirnov “Fermionic screening operators in the sine–Gordon model” In _Physica D: Nonlinear Phenomena_ 241.23-24 Elsevier, 2012, pp. 2122–2130
  * [34] Francesco Buccheri and Gabor Takacs “Finite temperature one-point functions in non-diagonal integrable field theories: the sine-Gordon model” In _Journal of High Energy Physics_ 2014.3 Springer, 2014, pp. 1–32
  * [35] A Leclair and Giuseppe Mussardo “Finite temperature correlation functions in integrable QFT” In _Nuclear Physics B_ 552.3 Elsevier, 1999, pp. 624–642
  * [36] Rebekka Koch and Alvise Bastianello “Exact Thermodynamics and Transport in the Classical Sine-Gordon Model”, 2023 arXiv:2303.16932 [cond-mat.stat-mech]
  * [37] Balázs Pozsgay “Mean values of local operators in highly excited Bethe states” In _Journal of Statistical Mechanics: Theory and Experiment_ 2011.01 IOP Publishing, 2011, pp. P01011 DOI: 10.1088/1742-5468/2011/01/p01011
  * [38] Michael Pawellek “Quantization of sine-Gordon solitons on the circle: Semiclassical vs. exact results” In _Nuclear Physics B_ 810.3 Elsevier, 2009, pp. 527–541
  * [39] David Horváth, Spyros Sotiriadis, Márton Kormos and Gabor Takacs “Inhomogeneous quantum quenches in the sine-Gordon theory” In _SciPost Physics_ 12.5, 2022, pp. 144
  * [40] J M Kosterlitz and D J Thouless “Ordering, metastability and phase transitions in two-dimensional systems” In _Journal of Physics C: Solid State Physics_ 6.7 IOP Publishing, 1973, pp. 1181–1203 DOI: 10.1088/0022-3719/6/7/010
  * [41] J M Kosterlitz “The critical properties of the two-dimensional xy model” In _Journal of Physics C: Solid State Physics_ 7.6 IOP Publishing, 1974, pp. 1046–1060 DOI: 10.1088/0022-3719/7/6/005
  * [42] Satoshi Takada, Toshifumi Sakaguchi and Susumu Misawa “The Kosterlitz-Thouless Transition in the Quantum Sine-Gordon Model and the Corresponding Classical Film System” In _Progress of Theoretical Physics_ 66.3 Oxford University Press, 1981, pp. 820–830
  * [43] Takao Ohta “Phase Transition in 2-D Sine-Gordon System” In _Progress of Theoretical Physics_ 60.4 Oxford University Press, 1978, pp. 968–974
  * [44] Yoichiro Nambu and Giovanni Jona-Lasinio “Dynamical model of elementary particles based on an analogy with superconductivity. I” In _Physical review_ 122.1 APS, 1961, pp. 345
  * [45] Krishna Rajagopal and Frank Wilczek “The Condensed Matter Physics of QCD” In _At The Frontier of Particle Physics_ , pp. 2061–2151 DOI: 10.1142/9789812810458˙0043
  * [46] Mark Alford, Jeffrey A Bowers and Krishna Rajagopal “Crystalline color superconductivity” In _Physical Review D_ 63.7 APS, 2001, pp. 074016
  * [47] Roberto Casalbuoni and Giuseppe Nardulli “Inhomogeneous superconductivity in condensed matter and QCD” In _Reviews of Modern Physics_ 76.1 APS, 2004, pp. 263
  * [48] G.’t Hooft “A planar diagram theory for strong interactions” In _Nuclear Physics B_ 72.3, 1974, pp. 461–473 DOI: https://doi.org/10.1016/0550-3213(74)90154-0
  * [49] Gabriele Veneziano “Some aspects of a unified approach to gauge, dual and Gribov theories” In _Nuclear Physics B_ 117.2 Elsevier, 1976, pp. 519–545
  * [50] Edward Witten “Baryons in the 1N expansion” In _Nuclear Physics B_ 160.1 Elsevier, 1979, pp. 57–115
  * [51] Tony Hilton Royle Skyrme “A non-linear field theory” In _Selected Papers, with Commentary, of Tony Hilton Royle Skyrme_ World Scientific, 1994, pp. 195–206
  * [52] A˙P Balachandran et al. “Doubly strange dibaryon in the chiral model” In _Physical Review Letters_ 52.11 APS, 1984, pp. 887
  * [53] Edward Witten “Global aspects of current algebra” In _Nuclear Physics B_ 223.2 Elsevier, 1983, pp. 422–432
  * [54] Gregory S Adkins, Chiara R Nappi and Edward Witten “Static properties of nucleons in the Skyrme model” In _Nuclear Physics B_ 228.3 Elsevier, 1983, pp. 552–566
  * [55] Enore Guadagnini “Baryons as solitons and mass formulae” In _Nuclear Physics B_ 236.1 Elsevier, 1984, pp. 35–47
  * [56] Alfred S Goldhaber and NS Manton “Maximal symmetry of the Skyrme crystal” In _Physics Letters B_ 198.2 Elsevier, 1987, pp. 231–234
  * [57] J. Eisert, M. Friesdorf and C. Gogolin “Quantum many-body systems out of equilibrium” In _Nature Phys._ 11, 2015, pp. 124 eprint: 1408.5148
  * [58] Marton Kormos, Mario Collura, Gabor Takács and Pasquale Calabrese “Real-time confinement following a quantum quench to a non-integrable model” In _Nature Physics_ 13.3 Springer ScienceBusiness Media LLC, 2016, pp. 246–249 DOI: 10.1038/nphys3934
  * [59] Carolyn Meldgin et al. “Probing the Bose glass–superfluid transition using quantum quenches of disorder” In _Nature Physics_ 12.7 Springer ScienceBusiness Media LLC, 2016, pp. 646–649 DOI: 10.1038/nphys3695
  * [60] Tianci Zhou and Adam Nahum “Entanglement Membrane in Chaotic Many-Body Systems” In _Phys. Rev. X_ 10.3, 2020, pp. 031066
  * [61] A. A. Michailidis et al. “Slow Quantum Thermalization and Many-Body Revivals from Mixed Phase Space” In _Phys. Rev. X_ 10.1, 2020, pp. 011055
  * [62] Eloy Ayón-Beato, Fabrizio Canfora and Jorge Zanelli “Analytic self-gravitating Skyrmions, cosmological bounces and AdS wormholes” In _Physics Letters B_ 752 Elsevier, 2016, pp. 201–205
  * [63] Eloy Ayón-Beato et al. “Analytic self-gravitating 4-Baryons, traversable NUT-AdS wormholes, flat space-time multi-Skyrmions at finite volume and a novel transition in the SU (3)-Skyrme model” In _The European Physical Journal C_ 80.5 Springer, 2020, pp. 1–8
  * [64] PD Alvarez, F Canfora, N Dimakis and A Paliathanasis “Integrability and chemical potential in the (3+ 1)-dimensional Skyrme model” In _Physics Letters B_ 773 Elsevier, 2017, pp. 401–407
  * [65] Fabrizio Canfora “Nonlinear superposition law and Skyrme crystals” In _Physical Review D_ 88.6 APS, 2013, pp. 065028
  * [66] Fabrizio Canfora “Ordered arrays of Baryonic tubes in the Skyrme model in $(3+1)$ dimensions at finite density” In _The European Physical Journal C_ 78.11 Springer, 2018, pp. 1–11
  * [67] Fabrizio Canfora, Marcela Lagos and Aldo Vera “Crystals of superconducting Baryonic tubes in the low energy limit of QCD at finite density” In _The European Physical Journal C_ 80.8 Springer, 2020, pp. 1–18
  * [68] Jorge A López, Claudio O Dorso and Guillermo Frank “Properties of nuclear pastas” In _Frontiers of Physics_ 16.2 Springer, 2021, pp. 1–55
  * [69] Pedro D Alvarez, Sergio L Cacciatori, Fabrizio Canfora and Bianca L Cerchiai “Analytic SU (N) Skyrmions at finite baryon density” In _Physical Review D_ 101.12 APS, 2020, pp. 125011
  * [70] Fabrizio Canfora et al. “Infinite conformal symmetry and emergent chiral fields of topologically nontrivial configurations: From Yang-Mills-Higgs theory to the Skyrme model” In _Phys. Rev. D_ 106.10, 2022, pp. 105016
  * [71] FA Smirnov “A general formula for soliton form factors in the quantum sine-Gordon model” In _Journal of Physics A: Mathematical and General_ 19.10 IOP Publishing, 1986, pp. L575
  * [72] H Babujian and Michael Karowski “The exact quantum sine-Gordon field equation and other non-perturbative results” In _Physics Letters B_ 471.1 Elsevier, 1999, pp. 53–57
  * [73] H Babujian and M Karowski “Exact form factors in integrable quantum field theories: the sine-Gordon model (II)” In _Nuclear Physics B_ 620.3 Elsevier, 2002, pp. 407–455
  * [74] Michio Jimbo, Tetsuji Miwa and Fedor Smirnov “Fermionic structure in the sine-Gordon model: Form factors and null-vectors” In _Nuclear Physics B_ 852.2 Elsevier, 2011, pp. 390–440
  * [75] Michio Jimbo, Tetsuji Miwa and Fedor Smirnov “Hidden Grassmann structure in the XXZ model V: sine-Gordon model” In _Letters in Mathematical Physics_ 96.1 Springer, 2011, pp. 325–365
  * [76] Vincenzo Alba and Pasquale Calabrese “Entanglement dynamics after quantum quenches in generic integrable systems” In _SciPost Physics_ 4.3, 2018, pp. 017
  * [77] J.D. Pryce and L.S.E.G.J.D. Pryce “Numerical Solution of Sturm-Liouville Problems”, Monographs on numerical analysis Clarendon Press, 1993 URL: https://books.google.cl/books?id=bTDvAAAAMAAJ
  * [78] K. Takayama and M. Oka “Bosonic excitations in a crystal of sine-Gordon kinks” In _Nuclear Physics A_ 551.4, 1993, pp. 637–656 DOI: https://doi.org/10.1016/0375-9474(93)90270-8
  * [79] William G. Newton “A taste of pasta?” In _Nature Physics_ 9.7, 2013, pp. 396–397 DOI: 10.1038/nphys2663
  * [80] Jos´e A. Pons, Daniele Viganò and Nanda Rea “A highly resistive layer within the crust of X-ray pulsars limits their spin periods” In _Nature Phys._ 9, 2013, pp. 431–434


# New probe of inflationary gravitational waves: cross-correlations of lensed
primary CMB B-modes with large-scale structure

Toshiya Namikawa  Center for Data-Driven Discovery, Kavli Institute for the
Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo,
Kashiwa, 277-8583, Japan  Blake D. Sherwin  Department of Applied Mathematics
and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge
CB3 0WA, United Kingdom  Kavli Institute for Cosmology, University of
Cambridge, Madingley Road, Cambridge CB3 OHA, United Kingdom

###### Abstract

We propose a new probe of inflationary gravitational waves (IGWs): the cross-
correlation of the lensing of inflationary $B$-mode polarization with a large-
scale structure (LSS) tracer, which can also be a cosmic microwave background
(CMB) lensing map. This is equivalent to measuring a three-point function of
two CMB $B$-modes and an LSS tracer. We forecast expected $1\,\sigma$
constraints on the tensor-to-scalar ratio $r$, albeit with a simplistic
foreground treatment, and find constraints of $\sigma_{r}\simeq 7\times
10^{-3}$ from the correlation of CMB-S4-Deep $B$-mode lensing and LSST
galaxies, $\sigma_{r}\simeq 5\times 10^{-3}$ from the correlation of
CMB-S4-Deep $B$-mode lensing and CMB-S4-Deep CMB lensing, and
$\sigma_{r}\simeq 10^{-2}$ from the correlation of LiteBIRD $B$-mode lensing
and CMB-S4-Wide lensing. Because this probe is inherently non-Gaussian, simple
Gaussian foregrounds will not produce any biases to the measurement of $r$.
While a detailed investigation of non-Gaussian foreground contamination for
different cross-correlations will be essential, this observable has the
potential to be a useful probe of IGWs, which, due to different sensitivity to
many potential sources of systematic errors, can be complementary to standard
methods for constraining $r$.

Introduction.— Measurements of the cosmic microwave background (CMB)
anisotropies have played a key role in developing the current picture in
cosmology [1, 2]. In the coming decades, measuring the polarization of the CMB
will be at the forefront of observational cosmology. In particular,
measurements of the parity-odd component – the $B$-modes – in the CMB
polarization will be of great importance [3, 4], as these provide us with a
unique avenue to test the presence of gravitational waves predicted by cosmic
inflation [5, 6, 7, 8] and gain new insights into the early Universe (see
reviews [9, 10] and references therein). Observations have not yet confirmed
the presence of these inflationary gravitational waves (IGWs) but have placed
upper bounds on the IGW amplitude, which is parametrized by the tensor-to-
scalar ratio $r$, as $r\lesssim 0.03$ ($2\,\sigma$) [11, 12, 13] at a pivot
scale of $0.05\,$Mpc-1, using data from BICEP/Keck Array, Planck, and WMAP.
These measurements have already ruled out several inflationary models. Several
ongoing and upcoming CMB experiments, including the BICEP Array [14], Simons
Array [15], Simons Observatory (SO) [16], LiteBIRD [17], and CMB-S4 [18], are
targeting detections of, or much tighter constraints on, IGW $B$-modes over
the next decade.

A high-precision measurement of the large-scale $B$-mode polarization has the
potential to tightly constrain $r$ at the level of $\sigma_{r}\lesssim
10^{-3}$ in future CMB experiments [19, 17]. However, precise measurements of
IGW $B$-modes must overcome bias and noise arising from other $B$-mode
sources. One of the most challenging issues is Galactic foregrounds (e.g.,
[20, 21]). The large-scale $B$-modes are dominated by polarized Galactic
foregrounds, so that we need to mitigate Galactic foregrounds very accurately.
Multiple studies have presented increasingly powerful techniques for modeling
or mitigating foregrounds (see, e.g., [22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36]), but no observations have yet demonstrated bias-free
foreground cleaning at the level of $r<10^{-2}$ in data. Foreground cleaning
can also lead to a significant increase of statistical uncertainties in a
$B$-mode measurement, especially if the foregrounds are very complex [17].

Here, we propose a new way to search for IGWs, which has an entirely different
– and potentially lower – sensitivity to Galactic foregrounds. The method
utilizes the fact that the IGW $B$-modes are affected by the gravitational
lensing effect induced by the large-scale structure (LSS), while the polarized
foregrounds are not. More specifically, we propose to estimate a lensing
signal that scales with $r$ from two observed $B$-modes, using the standard
lensing quadratic estimator [37] but optimized for IGW $B$-modes. Cross-
correlations of this $r$-dependent lensing signal with the LSS then provide a
constraint on $r$. This correlation is equivalent to the bispectrum between
two $B$-modes and a LSS tracer. The correlation is immune to Galactic
foregrounds if either (i) the measured LSS tracer does not contain significant
imprints of Galactic foregrounds or (ii) the Galactic foregrounds can be well
approximated as Gaussian since the bispectrum observable is fundamentally non-
Gaussian. The LSS tracer can correspond to a CMB lensing map reconstructed
from small-scale data or to a galaxy number density map, both of which may
only have minimal contamination by Galactic foregrounds. While the foregrounds
are, of course, known to be non-Gaussian, we note that any remaining non-
Gaussian foreground bias can be mitigated with a suitable choice of LSS
tracers or CMB lensing maps as we will outline in Discussion.

Method.— We first begin with a brief reminder of the properties of CMB
polarization anisotropies. CMB observations measure the Stokes $Q$ and $U$
parameters in each line-of-sight direction $\hat{\bm{n}}$ on the unit sphere.
We then define the scalar and pseudo-scalar $E$\- and $B$-modes of the CMB
polarization from the Stokes parameters as [38]

$\displaystyle E_{\ell m}\pm{\rm i}\hskip 0.50003ptB_{\ell
m}=-\int\\!\\!\,{\rm d}^{2}\hat{\bm{n}}\,\,(Y_{\ell m}^{\pm
2}(\hat{\bm{n}}))^{*}[Q\pm{\rm i}\hskip 0.50003ptU](\hat{\bm{n}})\,,$ (1)

where $Y_{\ell m}^{2}(\hat{\bm{n}})$ are spin-2 spherical harmonics.

The path of CMB photons is affected by the gravitational potential of the LSS
along the geodesic. The lensing distorts the polarization map by remapping it
with a deflection angle $\bm{d}(\hat{\bm{n}})$ (see [39] for a review). This
lensing distortion mixes the $E$\- and $B$-modes at different angular scales.
Introducing the lensing potential $\phi$ so that $\bm{d}=\bm{\nabla}\phi$, the
$B$-modes are modified as follows [40]:

$\displaystyle\tilde{B}_{\ell m}$ $\displaystyle=B_{\ell
m}+\sum_{LM\ell^{\prime}m^{\prime}}(-1)^{m}\begin{pmatrix}\ell&L&\ell^{\prime}\\\
-m&M&m^{\prime}\end{pmatrix}$ $\displaystyle\qquad\times\phi_{LM}(-{\rm
i}\hskip 0.50003ptE_{\ell^{\prime}m^{\prime}}W^{-}_{\ell
L\ell^{\prime}}+B_{\ell^{\prime}m^{\prime}}W^{+}_{\ell L\ell^{\prime}})\,,$
(2)

where $\phi_{LM}$ is the spherical harmonic coefficients of the lensing
potential, and we ignore higher-order terms in $\phi$. We define

$\displaystyle W^{\pm}_{\ell L\ell^{\prime}}$
$\displaystyle\equiv-q^{\pm}_{\ell
L\ell^{\prime}}\sqrt{\frac{L(L+1)(2\ell+1)(2L+1)(2\ell^{\prime}+1)}{4\pi}}$
$\displaystyle\qquad\times\frac{1}{2}\bigg{[}\sqrt{(\ell^{\prime}-2)(\ell^{\prime}+3)}\begin{pmatrix}\ell&L&\ell^{\prime}\\\
2&1&-3\end{pmatrix}$
$\displaystyle\qquad\qquad+\sqrt{(\ell^{\prime}+2)(\ell^{\prime}-1)}\begin{pmatrix}\ell&L&\ell^{\prime}\\\
-2&1&1\end{pmatrix}\bigg{]}\,,$ (3)

with $q^{\pm}_{\ell L\ell^{\prime}}\equiv[1\pm(-1)^{\ell+L+\ell^{\prime}}]/2$.

We now propose our new approach to detect the IGWs. We first note that lensing
breaks the statistical isotropy of the $B$-modes, leading to correlations
between the lensed primary CMB $B$-modes at different angular scales:

$\displaystyle\langle\tilde{B}_{\ell
m}\tilde{B}_{\ell^{\prime}m^{\prime}}\rangle=\sum_{LM}\begin{pmatrix}\ell&\ell^{\prime}&L\\\
m&m^{\prime}&M\end{pmatrix}f^{\psi}_{\ell L\ell^{\prime}}r\phi^{*}_{LM}\,,$
(4)

where the ensemble average is taken over CMB realizations with a fixed $\phi$,
and we introduce a response function $f^{\psi}_{\ell
L\ell^{\prime}}=W^{+}_{\ell
L\ell^{\prime}}C^{BB,r=1}_{\ell^{\prime}}+(-1)^{\ell+L+\ell^{\prime}}W^{+}_{\ell^{\prime}L\ell}C^{BB,r=1}_{\ell}$
(see also Ref. [41] for the non-perturbative response in the flat-sky
approximation). Here, $C_{\ell}^{BB,r=1}$ is the IGW $B$-mode power spectrum
with $r=1$. We therefore can use this correlation to reconstruct $\psi\equiv
r\phi$. The reconstructed $\psi$ fields from the large-scale lensed $B$-modes
alone are then cross-correlated with an external LSS tracer, $x$. This is
equivalent to measuring a cross-bispectrum between two CMB $B$-modes and the
LSS tracer, $x$. More specifically, for the idealistic full-sky case, we
follow closely the construction of quadratic CMB lensing estimators in Ref.
[42] and define an estimator to reconstruct $\psi$ from the observed
$B$-modes, $\hat{B}_{\ell m}$, as

$\displaystyle\hat{\psi}^{*}_{LM}=\frac{1}{2}A_{L}\sum_{\ell\ell^{\prime}mm^{\prime}}\begin{pmatrix}\ell&\ell^{\prime}&L\\\
m&m^{\prime}&M\end{pmatrix}f^{\psi}_{\ell L\ell^{\prime}}\frac{\hat{B}_{\ell
m}}{\hat{C}^{BB}_{\ell}}\frac{\hat{B}_{\ell^{\prime}m^{\prime}}}{\hat{C}^{BB}_{\ell^{\prime}}}\,,$
(5)

where $\hat{C}^{BB}_{\ell}$ is the best estimate of the observed power
spectrum from theoretical computation or simulation. $A_{L}$ is the estimator
normalization and is given in the idealistic full-sky case by

$\displaystyle
A_{L}^{-1}=\frac{1}{2L+1}\sum_{\ell\ell^{\prime}}\frac{(f^{\psi}_{\ell
L\ell^{\prime}})^{2}}{2\hat{C}^{BB}_{\ell}\hat{C}^{BB}_{\ell^{\prime}}}\,.$
(6)

As defined above, $\langle\hat{\psi}\rangle=r\phi$ is proportional to $r$. The
cross-power spectrum between $\hat{\psi}$ and an LSS tracer is therefore also
proportional to $r$. Thus, we can constrain $r$ by measuring the amplitude of
the cross-power spectrum $C_{L}^{\psi x}=rC_{L}^{\phi x}$ and, provided that
$C_{L}^{\phi x}$ is well determined by other measurements, finding the best
fit value of $r$.

The observed $B$-modes in Eq. (5) suffer from a large cosmic variance due to
the presence of $B$-modes converted from $E$-modes by lensing, i.e., the first
term in the second line of Eq. (2). To reduce this variance, we consider
applying the following template-delensing method. One can create a template of
the lensing-induced $E$-to-$B$ leakage using an estimate of lensing potential
$\hat{\phi}$ and observed $E$-modes, and then subtract this template from the
observed $B$-modes (e.g., [43, 44]) as follows111Note that $\hat{\phi}$ can
either derive from CMB lensing reconstruction or from a suitably scaled LSS
tracer as in [45].:

$\displaystyle\hat{B}^{\rm del}_{\ell m}$ $\displaystyle=\hat{B}_{\ell
m}-\sum_{LM\ell^{\prime}m^{\prime}}(-1)^{m}\begin{pmatrix}\ell&L&\ell^{\prime}\\\
-m&M&m^{\prime}\end{pmatrix}$
$\displaystyle\times\left(\frac{C^{\phi\phi}_{L}\hat{\phi}_{LM}}{C^{\phi\phi}_{L}+N^{\phi\phi}_{L}}\right)\left(\frac{-{\rm
i}\hskip
0.50003ptC^{EE}_{\ell^{\prime}}\hat{E}_{\ell^{\prime}m^{\prime}}}{C^{EE}_{\ell^{\prime}}+N^{EE}_{\ell^{\prime}}}\right)W^{-}_{\ell
L\ell^{\prime}}\,.$ (7)

Here, $N^{\phi\phi}_{L}$ and $N^{EE}_{\ell}$ are the noise power spectra of
the lensing estimate and $E$-modes, respectively. The delensed modes
$\hat{B}^{\rm del}_{\ell m}$ and their power spectra are then used for the
$\psi$-reconstruction with the estimator of Eq. (5). To leading order in
$\phi$, this delensing method only reverses lensing of $E$-modes but,
crucially, does not undo the lensing of IGW $B$-modes in Eq. (2); it hence
does not affect the mode coupling of Eq. (4) that allows us to reconstruct
$\psi$. To a good approximation, the method just reduces the residual lensing
$B$-mode power that contributes to $\hat{C}^{BB}_{\ell}$ by a factor $A_{\rm
lens}$ and thus lowers the reconstruction noise of the estimator of Eq. (5).

The expected $1\,\sigma$ uncertainty of $r$ from the cross-correlations
between $\psi$ and an LSS tracer is

$\displaystyle\sigma^{-2}_{r}$ $\displaystyle=\sum_{L}\frac{(2L+1)f_{\rm
sky}(\partial C_{L}^{\psi x}/\partial
r)^{2}}{(r^{2}C_{L}^{\phi\phi}+N_{L}^{\psi\psi})\hat{C}_{L}^{xx}+(rC_{L}^{\phi
x})^{2}}\,,$ (8)

where $f_{\rm sky}$ is the sky fraction of a CMB polarization observation,
$\hat{C}_{L}^{xx}$ is the observed power spectrum of $x$, and
$N_{L}^{\psi\psi}$ is the reconstruction noise spectrum. For small fiducial
values of $r$, the above equation can be approximated as

$\displaystyle\sigma^{-2}_{r}\simeq\sum_{L}(2L+1)f_{\rm
sky}\frac{\rho^{2}_{L}C_{L}^{\phi\phi}}{N_{L}^{\psi\psi}}\,,$ (9)

where $\rho_{L}\equiv C_{L}^{\phi x}/\sqrt{C^{\phi\phi}_{L}\hat{C}^{xx}_{L}}$
is the correlation coefficient between the true CMB lensing potential and the
observed LSS tracer. In the idealized fullsky case, the reconstruction noise
is equal to the estimator normalization and we adopt $N_{L}^{\psi\psi}\simeq
A_{L}$. Note that the cosmic variance of the IGW $B$-modes increases
$N_{L}^{\psi\psi}$ as in Eq. (6). In our analysis, we assume $r=0$ and focus
on the significance of rejecting the null hypothesis.

Forecast.— We first estimate $\sigma_{r}$ for specific experimental
specifications. We assume that the different experiments listed in Table 1
(SO-SAT, LiteBIRD, S4D-SAT) measure $\psi$. Table 2 summarizes our results,
where we assume that the experiments in the second column of Table 2 (SO-LAT,
S4W, S4D-LAT, LSST galaxies) provide the LSS tracer $x$ used for cross-
correlation with $\psi$. Note that “SO,” “LiteBIRD,” “S4W,” and “S4D” mimic
the Simons Observatory, LiteBIRD, CMB-S4 Wide Survey, and CMB-S4 Ultra-Deep
Survey, respectively. To compute $\sigma_{r}$, we evaluate $N_{L}^{\psi\psi}$
and $\rho_{L}$ in Eq. (9) as follows.

To compute $N_{L}^{\psi\psi}$, we use a CMB noise spectrum with a polarization
noise level $\sigma_{\rm P}$ and include the effect of beam deconvolution with
angular resolution $\theta$. We include in our forecasts the degradation of
the noise level due to component separation as in Ref. [46] for LiteBIRD and
S4. For SO-SAT, we use the noise curve corresponding to the goal noise level
for the optimistic $1/f$ case shown in Fig. 11 of Ref. [16]. We set the lowest
multipole of the S4D-SAT to $50$, corresponding to the $\ell_{\rm knee}$ of
the $1/f$ noise [19]. We note that the residual foregrounds increase the
variance of the large-scale $B$-modes and affect the $\psi$-reconstruction
noise. In component separation with suitable algorithms, however, the residual
foregrounds can typically be suppressed to be small compared to the noise
(e.g., [46]), and so we neglect these residuals in our forecasts except for
LiteBIRD where we include the residual foregrounds of Ref. [46]. We also
include delensing prior to the $\psi$ measurement by simply scaling the
overall amplitude of the lensing $B$-mode power by a value $A_{\rm lens}$,
which describes the appropriate level of residual $B$-modes after delensing
each experiment.

To compute $\rho_{L}$, we consider either reconstructed $\phi$ or galaxy
number density. For the $\phi$ measurement, since the large-scale $B$-modes
from a high-resolution experiment usually suffer from high atmospheric $1/f$
noise levels, we do not use $B$-modes at $\ell<500$ for computing the
$\phi$-reconstruction noise. This split also avoids any delensing bias arising
from the correlation between the $B$-modes in the $\phi$-reconstruction and
those to be delensed (e.g., [47, 48, 44]). For $E$-modes used for the
$\phi$-reconstruction, we use $\ell\geq 100$ where the CMB signal is dominant
[16]. We compute the $\phi$-reconstruction noise using the iterative approach
to mimic optimal lensing reconstruction [49].

Experiment for $\psi$ |  $\sigma_{\rm P}$ ($\mu$K′) |  $\theta$ (′) | $\ell_{\rm min}$ | $\ell_{\rm max}$ | $A_{\rm lens}$
---|---|---|---|---|---
SO-SAT |  |  | 30 | 300 | 0.3
LiteBIRD | 2.6 | 30 | 2 | 500 | 0.2
S4D-SAT | 0.5 | 20 | 50 | 500 | 0.05
Table 1: Specific experimental setup for a $\psi$ measurement used in our forecasts. For each experiment, the noise level includes degradation due to the component separation and the remaining fraction of the lensing $B$-mode power parametrized with $A_{\rm lens}$. For SO-SAT, we use the noise curve corresponding to the goal noise level for the optimistic $1/f$ case shown in Fig.11 of Ref. [16]. $A_{\rm lens}$ for SO-SAT, LiteBIRD, and S4D-SAT are derived by assuming delensing of Ref. [50], S4W, and S4D internal, respectively. $\psi$ | $x$ | $f_{\rm sky}$ | $\sigma_{r}$
---|---|---|---
SO-SAT | SO-LAT $\phi$ | 0.1 | $0.04$
LiteBIRD | S4W $\phi$ | 0.4 | $0.01$
S4D-SAT | S4D-LAT $\phi$ | 0.03 | $0.005$
S4D-SAT | LSST galaxies | 0.03 | $0.007$
Table 2: Forecasts for $1\sigma$ constraints on $r$ using $C_{L}^{\psi x}$ at
$L$ in $[10,500]$. We choose $f_{\rm sky}$ to only include the overlap regions
between experiments. We choose $\sigma_{\rm P}=6\,\mu$K′ for SO-LAT,
$1\,\mu$K′ for S4W, and $0.4\,\mu$K′ for S4D-LAT. We assume
$\theta=1^{\prime}$ for all these CMB experiments and use multipoles between
$\ell=100$ and $4000$ but remove $\ell<500$ for $B$-modes used for
reconstructing $\phi$ to be correlated with $\psi$. For the LSST galaxies, we
assume a constant correlation coefficient with lensing of $70\%$.

We find that, for S4D, $\sigma_{r}\simeq 5\times 10^{-3}$. The constraints
will be somewhat degraded if we use, as an LSS tracer, galaxies that have a
lower correlation coefficient with the CMB lensing map instead of the lensing
reconstruction map. However, this may be worthwhile because such a cross-
correlation might reduce potential concerns from foregrounds. For LSST
galaxies [51], the correlation coefficient is expected to be $\rho_{L}\gtrsim
0.7$ at $L\lesssim 1000$ (e.g., [50]); then the constraint becomes
$\sigma_{r}\simeq 7\times 10^{-3}$.

Note that the high sensitivity of our method to $r$ comes first from the fact
that the estimator is not limited by the cosmic variance until we detect
nonzero IGW $B$-modes, and second, from the fact that cross-correlations can
boost the signal-to-noise ratio if only one of the fields being correlated has
high noise.

Since the sensitivity to $r$ might be further improved by optimizing the
experimental setup, including the number of frequency bands and detectors
currently assumed in each experiment, we next discuss how $\sigma_{r}$ changes
if we vary instrumental specifications. The constraint depends on at least
four factors: (1) the observed sky fraction, (2) the correlation coefficient
of the measured LSS tracer to $\phi$, (3) the noise level of the large-scale
$B$-modes after component separation, and (4) the delensing efficiency. The
dependence on $f_{\rm sky}$ is trivial. The correlation coefficients become
close to unity if we use a CMB-S4 lensing map and are not much lower if we use
a future, high-$z$ LSS survey. The last two determine the reconstruction noise
of $\psi$, which enters into the denominator of Eq. (9).

Figure 1: The $1\sigma$ constraints on $r$ arising from our method; we
consider an experimental setup that resembles LiteBIRD+S4W but allows for a
variation of the polarization noise level of LiteBIRD,
$\sigma_{\mathrm{P}}^{\mathrm{LB}}$ and the delensing efficiency, parametrized
via $A_{\rm lens}$. We use a reconstructed lensing map from S4W as an LSS
tracer for the cross-correlation, although the results are not very sensitive
to the detailed properties of this tracer. Note that $A_{\rm lens}=0.05$
corresponds to the S4D case, although $f_{\rm sky}$ and the noise level are
allowed to vary here. We conclude that our method is capable of producing
competitive constraints on $r$, especially if efficient delensing can be
implemented.

Figure 1 shows $\sigma_{r}$ for a configuration that resembles the
LiteBIRD+S4W setup except that we now allow the post-component-separation
noise level of LiteBIRD $\sigma_{\rm P}^{\rm LB}$ and the delensing efficiency
to vary. These two parameters change the performance of the
$\psi$-reconstruction. Since the variance from the residual foregrounds is
much smaller than that from the noise, we ignore the residual foregrounds in
the calculation for simplicity. We assume the S4W $\phi$ map as an LSS tracer
for cross-correlation, but the improvements when using a perfect $\phi$ map
are negligible. On the other hand, a precise $\phi$ map is crucial for
delensing: The constraints cannot improve without delensing as the noise level
decreases below $\sigma_{\rm P}^{\rm LB}\ll 1\mu$K′ due to the lensing
$B$-mode cosmic variance.

We also explore the dependence of $\sigma_{r}$ on the multipole range of the
large-scale $B$-modes used in the $\psi$-reconstruction. We find that removing
$B$-modes below $\ell\lesssim 30$ or above $\ell\gtrsim 200$ only increases
$\sigma_{r}$ by $10\%$ for the LiteBIRD+S4W case. Even removing multipoles
below $\ell\lesssim 60$ or above $\ell\gtrsim 100$ only increases $\sigma_{r}$
by $50\%$. For S4D configurations, our findings are similar: Assuming the
standard minimum multipole for the S4D setup, $\ell_{\rm min}=50$, only
increases $\sigma_{r}$ by $\sim 20\%$ compared to the case when multipoles
$\ell<50$ are included. This is because the lensing becomes important at
higher multipoles but then the noise (and lensing noise) become dominant, so
the recombination bump ($60<\ell<100$) contains the most information on the
lensing of the IGW $B$-modes. Our method thus does not require $B$-mode
information from the largest (or smallest) angular scales that can be most
challenging for CMB experiments to observe and foreground-clean.

Discussion.— We first discuss comparison with $r$ constraints from the
standard analysis using the $B$-mode autopower spectrum. For example, if we
assume the LiteBIRD+S4W case with $A_{\rm lens}=0.2$, the statistical
uncertainty from the standard method in the same analysis setup is
$\sigma_{r}\simeq 2\times 10^{-4}$, which is more than an order of magnitude
better than our method. However, the standard method suffers from a large bias
from residual Galactic foregrounds (e.g., [52]). A more realistic estimate by
[17] reports $\sigma_{r}\simeq 10^{-3}$ with a conservative method to reduce
the level of residual foreground biases and instrumental systematics. Since
Gaussian foregrounds do not bias our estimator, this level of conservatism may
not be required for our method, so that our simple forecasts may be more
realistic. Moreover, we note that in the standard method the residual
foreground bias can depend sensitively on the assumptions in the foreground
model, so that a small mismatch between assumptions and real data can lead to
a significant bias in the $r$ estimate. Thus, our new method will provide a
valuable, independent test of constraints on $r$.

Our estimator could be biased by the foreground non-Gaussianity. However,
tracers can be chosen to minimize the non-Gaussian foregrounds. For example,
we can select galaxy catalogs to minimize the impact of extinction, or use a
$\phi$ map measured from small-scale CMB fluctuations with subdominant or
near-independent foreground fluctuations that are negligibly correlated with
large-scale foregrounds. Furthermore, different LSS tracers can be used to
cross-check foreground stability and gain confidence in the robustness of the
results. Although much further work on this subject is needed before
conclusions can be drawn, it seems plausible that foreground contamination
could be much less limiting for our method than for the standard method. Even
if foreground biases are not immediately negligible, multifrequency foreground
cleaning could certainly also be used. In any case, our method will provide a
complementary cross-check of standard results, with any sensitivity to
Galactic foregrounds entering in a qualitatively different manner.

Polarized extragalactic foregrounds – in particular, polarized radio sources –
could be an additional concern; if the sources contributed a $B$-mode signal
$S_{B}$, they could, in principle, bias our measurement by a bispectrum
contribution $S_{B}S_{B}\phi$. The delensing process also potentially
introduces a bias [53]. Fortunately, we can use standard techniques from
lensing reconstruction to mitigate potential point source biases. In
particular, we can combine aggressive source masking with modifications of the
estimator, using analogs of bias-hardening in lensing reconstruction [54, 55],
to further mitigate any foreground biases.

We note that our analysis did not include the non-Gaussian covariance of the
bispectrum. The $B$-modes are non-Gaussian due to the lensing $B$-modes even
after foreground cleaning. However, the large-scale lensing $B$-modes are well
described by a Gaussian field [56], and such non-Gaussianity should hence be
small.

We have also ignored terms arising from higher orders in $\phi$. While we
defer a detailed analysis to future work, we will discuss this issue briefly
and schematically here. By expanding the delensed $B$-modes to higher orders
in $\phi$ and substituting this expansion into our $\psi$ estimator, we find
that the leading higher-order terms in the cross-correlation are either: i)
terms such as (schematically) $\sim C_{\ell}^{EE}A^{1/2}_{\rm lens}\phi^{4}$
and $\sim C_{\ell}^{EE}A_{\rm lens}\phi^{3}$ that are proportional to $A_{\rm
lens}$ and hence should be substantially reduced by efficient delensing, or
ii) terms such as $\sim N_{\ell}^{EE}\phi^{3}$ that arise from $E$-mode noise
and can be removed by cross-correlation of different splits of the data.
Although these terms may not be entirely negligible, we therefore expect them
to be small enough that any bias they produce could simply be modeled and
subtracted.

Summary.— We have proposed a new method to measure $r$ using cross-
correlations of an LSS tracer with lensing of primary $B$-modes, reconstructed
with a quadratic estimator. We have explored the sensitivity of this cross-
correlation to IGWs for upcoming CMB experiments and have found that the
cross-correlation can constrain $r$ with $\sigma_{r}\simeq 5\times
10^{-3}-4\times 10^{-2}$. Compared to the $B$-mode power spectrum, our method
is insensitive to Gaussian foregrounds and would be less limited by Galactic
foreground uncertainties more generally. Our method can thus potentially
constrain $r$ in a manner that is complementary to standard analyses with the
$B$-mode autopower spectrum.

In closing, we note a number of interesting features of our method. Compared
to standard $B$-mode power spectrum analyses, our method is expected to have
somewhat different sensitivity to the spectral shape of the IGW power
spectrum. Furthermore, in principle, our method can also probe a redshift
dependence of $B$-mode sources by using a tomographic measurement of the
cross-correlation with LSS tracers at different redshifts. We will discuss
these points in more detail in our follow-up papers.

###### Acknowledgements.

We thank Anton Baleato-Lizancos, Anthony Challinor, William R. Coulton, Colin
Hill, Eiichiro Komatsu, Antony Lewis, Mathew Madavacheril, and Tomotake
Matsumura for helpful comments and discussions. This work is supported in part
by JSPS KAKENHI Grants No. JP20H05859 and No. JP22K03682 (T.N.). Part of this
work uses the resources of the National Energy Research Scientific Computing
Center. The Kavli IPMU is supported by World Premier International Research
Center Initiative (WPI Initiative), MEXT, Japan.

## References

  * Komatsu and Bennett [2014] E. Komatsu and C. L. Bennett (WMAP Science Team), Prog. Theor. Exp. Phys. 2014, 06B102 (2014), eprint 1404.5415.
  * Planck Collaboration [2020] Planck Collaboration, Astron. Astrophys. 641, A1 (2020), eprint 1807.06205.
  * Kamionkowski et al. [1997] M. Kamionkowski, A. Kosowsky, and A. Stebbins, Phys. Rev. Lett. 78, 2058 (1997), eprint astro-ph/9609132.
  * Seljak and Zaldarriaga [1997] U. Seljak and M. Zaldarriaga, Phys. Rev. Lett. 78, 2054 (1997), eprint astro-ph/9609169.
  * Guth [1981] A. H. Guth, Phys. Rev. D 23, 347 (1981).
  * Sato [1981] K. Sato, Mon. Not. R. Astron. Soc. 195, 467 (1981), URL https://doi.org/10.1093/mnras/195.3.467.
  * Linde [1982] A. D. Linde, Phys. Rev. B 108, 389 (1982).
  * Albrecht and Steinhardt [1982] A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982), URL https://link.aps.org/doi/10.1103/PhysRevLett.48.1220.
  * Kamionkowski and Kovetz [2016] M. Kamionkowski and E. D. Kovetz, Ann. Rev. Astron. Astrophys. 54, 227 (2016), eprint 1510.06042.
  * Komatsu [2022] E. Komatsu, Nature Rev. Phys. 4, 452 (2022), eprint 2202.13919.
  * BICEP/Keck Collaboration: P. A. R. Ade et al. [2021] BICEP/Keck Collaboration: P. A. R. Ade, Z. Ahmed, M. Amiri, D. Barkats, R. Basu Thakur, D. Beck, C. Bischoff, J. J. Bock, H. Boenish, E. Bullock, et al., Phys. Rev. Lett. 127, 151301 (2021), eprint 2110.00483.
  * Tristram et al. [2022] M. Tristram et al., Phys. Rev. D 105, 083524 (2022), eprint 2112.07961.
  * Campeti and Komatsu [2022] P. Campeti and E. Komatsu, Astrophys. J. 941, 110 (2022), eprint 2205.05617.
  * Moncelsi et al. [2020] L. Moncelsi, P. A. R. Ade, Z. Ahmed, M. Amiri, D. Barkats, R. Basu Thakur, C. A. Bischoff, J. J. Bock, V. Buza, J. R. Cheshire, et al., Proc. SPIE Int. Soc. Opt. Eng. 11453, 1145314 (2020), eprint 2012.04047.
  * Suzuki et al. [2016] A. Suzuki, P. Ade, Y. Akiba, C. Aleman, K. Arnold, C. Baccigalupi, B. Barch, D. Barron, A. Bender, D. Boettger, et al., J. Low Temp. Phys. 184, 805 (2016), eprint 1512.07299.
  * The Simons Observatory Collaboration [2019] The Simons Observatory Collaboration, J. Cosmol. Astropart. Phys. 02, 056 (2019), eprint 1808.07445.
  * LiteBIRD Collaboration [2022] LiteBIRD Collaboration, Prog. Theor. Exp. Phys. 2023, 042F01 (2022), eprint 2202.02773.
  * CMB-S4 Collaboration: K. Abazajian et al. [2019] CMB-S4 Collaboration: K. Abazajian, G. Addison, P. Adshead, Z. Ahmed, S. W. Allen, D. Alonso, M. Alvarez, A. Anderson, K. S. Arnold, C. Baccigalupi, et al. (2019), eprint 1907.04473.
  * CMB-S4 Collaboration: K. Abazajian et al. [2022] CMB-S4 Collaboration: K. Abazajian, G. E. Addison, P. Adshead, Z. Ahmed, D. Akerib, A. Ali, S. W. Allen, D. Alonso, M. Alvarez, M. A. Amin, et al., Astrophys. J. 926, 54 (2022), eprint 2008.12619.
  * BICEP2 Collaboration [2014] BICEP2 Collaboration, Phys. Rev. Lett. 112, 241101 (2014).
  * BICEP/Keck Collaboration: P. A. R. Ade et al. [2023] BICEP/Keck Collaboration: P. A. R. Ade, Z. Ahmed, M. Amiri, D. Barkats, R. B. Thakur, C. A. Bischoff, D. Beck, J. J. Bock, H. Boenish, E. Bullock, et al., Astrophys. J. 945, 72 (2023), eprint 2210.05684.
  * Eriksen et al. [2008] H. K. Eriksen, J. B. Jewell, C. Dickinson, A. J. Banday, K. M. Gorski, and C. R. Lawrence, Astrophys. J. 676, 10 (2008), eprint 0709.1058.
  * Dunkley et al. [2009] J. Dunkley, A. Amblard, C. Baccigalupi, M. Betoule, D. Chuss, A. Cooray, J. Delabrouille, C. Dickinson, G. Dobler, J. Dotson, et al., 1141, 222 (2009), eprint 0811.3915.
  * Betoule et al. [2009] M. Betoule, E. Pierpaoli, J. Delabrouille, M. Le Jeune, and J.-F. Cardoso, Astronomy and Astrophysics 503, 691 (2009), eprint 0901.1056.
  * Katayama and Komatsu [2011] N. Katayama and E. Komatsu, Astrophys. J. 737, 78 (2011), eprint 1101.5210.
  * Ichiki [2014] K. Ichiki, Prog. Theor. Exp. Phys. 06, B109 (2014).
  * Chluba et al. [2017] J. Chluba, J. C. Hill, and M. H. Abitbol, Mon. Not. R. Astron. Soc. 472, 1195 (2017), eprint 1701.00274.
  * Azzoni et al. [2021] S. Azzoni, M. H. Abitbol, D. Alonso, A. Gough, N. Katayama, and T. Matsumura, J. Cosmol. Astropart. Phys. 05, 047 (2021), eprint 2011.11575.
  * de la Hoz et al. [2020] E. de la Hoz, P. Vielva, R. B. Barreiro, and E. Martínez-González, J. Cosmol. Astropart. Phys. 06, 006 (2020), eprint 2002.12206.
  * Remazeilles et al. [2021] M. Remazeilles, A. Rotti, and J. Chluba, Mon. Not. R. Astron. Soc. 503, 2478 (2021), eprint 2006.08628.
  * Puglisi et al. [2022] G. Puglisi, G. Mihaylov, G. V. Panopoulou, D. Poletti, J. Errard, P. A. Puglisi, and G. Vianello, Mon. Not. R. Astron. Soc. 511, 2052 (2022), eprint 2109.11562.
  * Delouis et al. [2022] J. M. Delouis, E. Allys, E. Gauvrit, and F. Boulanger, Astron. Astrophys. 668, A122 (2022), eprint 2207.12527.
  * Minami and Ichiki [2022] Y. Minami and K. Ichiki, Prog. Theor. Exp. Phys. 2023, 033E01 (2022), eprint 2212.01773.
  * Rahman et al. [2022] F. Rahman, P. Chingangbam, and T. Ghosh (2022), eprint 2212.06076.
  * Vacher et al. [2023] L. Vacher, J. Chluba, J. Aumont, A. Rotti, and L. Montier, Astron. Astrophys. 669, A5 (2023), eprint 2205.01049.
  * Carones et al. [2022] A. Carones, M. Migliaccio, G. Puglisi, C. Baccigalupi, D. Marinucci, N. Vittorio, and D. Poletti (2022), eprint 2212.04456.
  * Hu and Okamoto [2002] W. Hu and T. Okamoto, Astrophys. J. 574, 566 (2002), eprint astro-ph/0111606.
  * Zaldarriaga and Seljak [1997] M. Zaldarriaga and U. Seljak, Phys. Rev. D D55, 1830 (1997), eprint astro-ph/9609170.
  * Lewis and Challinor [2006] A. Lewis and A. Challinor, Phys. Rep. 429, 1 (2006), eprint astro-ph/0601594.
  * Zaldarriaga and Seljak [1998] M. Zaldarriaga and U. Seljak, Phys. Rev. D 58, 023003 (1998), eprint astro-ph/9803150.
  * Fabbian et al. [2019] G. Fabbian, A. Lewis, and D. Beck, J. Cosmol. Astropart. Phys. 10, 057 (2019), eprint 1906.08760.
  * Okamoto and Hu [2003] T. Okamoto and W. Hu, Phys. Rev. D 67, 083002 (2003), eprint astro-ph/0301031.
  * Seljak and Hirata [2004] U. Seljak and C. M. Hirata, Phys. Rev. D 69, 043005 (2004), eprint astro-ph/0310163.
  * Baleato Lizancos et al. [2021] A. Baleato Lizancos, A. Challinor, and J. Carron, J. Cosmol. Astropart. Phys. 03, 016 (2021), eprint 2007.01622.
  * Sherwin and Schmittfull [2015] B. D. Sherwin and M. Schmittfull, Phys. Rev. D 92, 043005 (2015), eprint 1502.05356.
  * Errard et al. [2016] J. Errard, S. M. Feeney, H. V. Peiris, and A. H. Jaffe, J. Cosmol. Astropart. Phys. 03, 052 (2016).
  * Teng et al. [2011] W.-H. Teng, C.-L. Kuo, and J.-H. P. Wu (2011), eprint 1102.5729.
  * Namikawa and Nagata [2014] T. Namikawa and R. Nagata, J. Cosmol. Astropart. Phys. 09, 009 (2014), eprint 1405.6568.
  * Smith et al. [2012] K. M. Smith, D. Hanson, M. LoVerde, C. M. Hirata, and O. Zahn, J. Cosmol. Astropart. Phys. 06, 014 (2012), eprint 1010.0048.
  * Namikawa et al. [2022] T. Namikawa, A. Baleato Lizancos, N. Robertson, B. D. Sherwin, A. Challinor, et al., Phys. Rev. D 105, 023511 (2022), eprint 2110.09730.
  * LSST Dark Energy Science Collaboration [2012] LSST Dark Energy Science Collaboration (2012), eprint arXiv:1211.0310.
  * Aurlien et al. [2022] R. Aurlien et al., J. Cosmol. Astropart. Phys. 06, 034 (2022), eprint 2211.14342.
  * Baleato Lizancos and Ferraro [2022] A. Baleato Lizancos and S. Ferraro, Phys. Rev. D 106, 063534 (2022), eprint 2205.09000.
  * Namikawa and Takahashi [2014] T. Namikawa and R. Takahashi, Mon. Not. R. Astron. Soc. 438, 1507 (2014), eprint 1310.2372.
  * Sailer et al. [2023] N. Sailer, S. Ferraro, and E. Schaan, Phys. Rev. D 107, 023504 (2023), eprint 2211.03786.
  * Namikawa and Nagata [2015] T. Namikawa and R. Nagata, J. Cosmol. Astropart. Phys. 10, 004 (2015), eprint 1506.09209.