Droplet-mediated long-range interfacial correlations. Exact results for
entropic repulsion effects.
Alessio Squarcini1,2,♮ and Antonio Tinti3,♭
1Max-Planck-Institut für Intelligente Systeme,
Heisenbergstr. 3, D-70569, Stuttgart, Germany
2IV. Institut für Theoretische Physik, Universität Stuttgart,
Pfaffenwaldring 57, D-70569 Stuttgart, Germany
3Dipartimento di Ingegneria Meccanica e Aerospaziale,
Sapienza Università di Roma, via Eudossiana 18, 00184 Rome, Italy

We consider near-critical two-dimensional statistical systems at phase
coexistence on the half plane with boundary conditions leading to the
formation of a droplet separating coexisting phases. General low-energy
properties of two-dimensional field theories are used in order to find exact
analytic results for one- and two-point correlation functions of both the
energy density and order parameter fields. The subleading finite-size
corrections are also computed and interpreted within an exact probabilistic
picture in which interfacial fluctuations are characterized by the probability
density of a Brownian excursion. The explicit analysis of the closed-form
expression for order parameter correlations reveals the long-ranged character
of interfacial correlations and their confinement within the interfacial
region. The analysis of correlations is then carried out in momentum space
through the notion of interface structure factor, which we extend to the case
of systems bounded by a flat wall. The presence of the wall and its associated
entropic repulsion leads to a specific term in the interface structure factor
which we identify.

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##  1 Introduction

The study of interfacial phenomena at boundaries is one of the cornerstones of
statistical mechanics [1, 2, 3, 4, 5, 6, 7]. The problem of phase separation
and fluctuating interfaces in two dimensions is a particularly interesting
one. For long time exact results have been available only for the Ising model
[8]. This fact is originated by the possibility to find exact diagonalizations
of transfer matrices in certain lattice geometries with boundary conditions
leading to the formation of interfaces. In this regard, it has to be mentioned
the exact solution of the wetting transition with a flat boundary [9], a
milestone in the field. These exact results have been crucial for the
consolidation of phenomenological interpretations for interfacial behavior in
terms of random walks in restricted geometries [10] and coarse-grained
descriptions treated within path-integral techniques [11, 12].

Despite the wealth of results available for interfaces in the Ising model [8],
such an extended degree of knowledge about phase separation for other models
has not been achieved until recent times. The principal reason is the
obstruction posed by the difficulty of finding exact results for most of the
lattice models in certain geometries. On the other hand, it is known that upon
approaching a continuous phase transition point the divergence of the bulk
correlation length leads to universal behavior and that field theory has
proven to be a versatile language for its description in the continuum setting
[13, 14]; see [15, 16] for boundary field theories in near-critical systems.
As a result, the systematic investigation of the tangled scenario of phase
separation for arbitrary models must inevitably be formulated within a
language that is able to encompass the different bulk universality classes
jointly with boundary data; this is the language of field theory.

It has been shown how general results for interface profiles and correlation
functions within the interfacial region separating coexisting bulk phases are
completely codified by low-energy properties of field theories [17, 18]. The
exact theory of phase separation developed in [17, 19, 20, 21, 22, 18] has
provided a unified framework for the study of interfacial phenomena exhibited
by general universality classes in various geometries. Among the various
findings, it has been possible to describe boundary wetting transitions [19],
wedge filling transitions [20], interfacial wetting [21, 23], and many-body
correlation functions [24, 25, 26] for the scaling limit of those models (and
boundary conditions) which are not yet solved on the lattice. Moreover, the
role of bulk and boundary integrability in these exact findings has been also
clarified [17, 21, 22].

The concept of interface and the possibility of conferring to it certain
fluctuation properties is at the core of effective formulations such as the
capillary wave model [27, 28], Weeks’ columnar model [29] and subsequent
elaborations thereof [30]. We refer to [31, 32] for a recent account on the
subject. It is known since long time from theory of inhomogeneous fluids that
density fluctuations within the interfacial region separating coexisting
phases exhibit long-range correlations in the direction parallel to the
interface [33]. This situation happens to be in sharp contrast with the
exponential decay of correlations exhibited within pure phases. Going beyond
effective formulations, it has been shown [17, 19, 20, 21, 22, 18] how a
fundamental description of phase separation and interfacial phenomena has to
be inevitably formulated in terms of the degrees of freedom of the bulk field
theory, which are the asymptotic particle states, as illustrated in this
paper.

In this paper, we examine the exact form of correlation functions of both the
order parameter and energy density fields for a system bounded on the half-
plane. Suitable boundary conditions along the wall are used in order to
enforce phase separation in the half-plane through a droplet with pinned
endpoints on the wall. Contrary to effective modelings, the approach presented
in this paper does not rely on the introduction of the notion of interface but
rather follows from the fundamental degrees of freedom of the underlying field
theory corresponding to the scaling limit of the statistical system with the
appropriate boundary conditions.

More technically, our approach follows as an amalgamation of the field-
theoretic formalism developed in [19] for the calculation of interface
profiles on the half-plane, with the techniques developed in [18] for the
calculation of two-point correlation functions. One of the key results of this
paper is the analytic expression for the order parameter correlation function
in the presence of the fluctuating droplet. The leading asymptotic behavior of
the order parameter correlation function in the direction parallel to the
interface reads

$\displaystyle\langle\sigma(x,y)\sigma(x,-y)\rangle_{B_{bab}}$
$\displaystyle=\left(\frac{\langle\sigma\rangle_{a}+\langle\sigma\rangle_{b}}{2}\right)^{2}+\frac{\langle\sigma\rangle_{a}^{2}-\langle\sigma\rangle_{b}^{2}}{2}\biggl{[}-1-4x\sqrt{\frac{2m}{\pi
R}}\textrm{e}^{-\frac{2m}{R}x^{2}}+2\textrm{erf}\left(\sqrt{\frac{2m}{R}}x\right)\biggr{]}$
(1.1)
$\displaystyle+\left(\frac{\langle\sigma\rangle_{a}-\langle\sigma\rangle_{b}}{2}\right)^{2}\biggl{[}1-\frac{32mx^{2}}{\pi
R}\sqrt{\frac{2y}{R}}\,\textrm{e}^{-\frac{2m}{R}x^{2}}\biggr{]}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}((y/R)^{3/2})$

for separations $\xi_{\rm b}\ll y\ll R/2$. In the above, $R$ is the distance
between the interface endpoints, $m=1/(2\xi_{\rm b})$, and $\xi_{\rm b}$ is
the bulk correlation length. The notation $\langle\cdots\rangle_{B_{bab}}$
stands for statistical averages in the half plane ($x>0$) with boundary
conditions enforcing a droplet, and $\langle\cdots\rangle_{a/b}$ stands for
expectation values in pure phase $a$ or $b$.

The interface separating phases $a$ and $b$ is characterized by midpoint
fluctuations of order $R^{1/2}$ along the $x$-axis. For $R\rightarrow\infty$
the result (1.1) reduces to

$\displaystyle\lim_{R\rightarrow\infty}\langle\sigma(x,y)\sigma(x,-y)\rangle_{B_{bab}}$
$\displaystyle=\langle\sigma\rangle_{a}^{2}\,,$ (1.2)

meaning that unbounded interfacial fluctuations yield an averaging over the
phase $a$ enclosed by the droplet. On the other hand, for finite $R$ the most
significant variations are localized in the region where it is most probable
to find the interface. The term proportional to $\sqrt{y/R}$ is the
characteristic signature of the long-range character of density correlations
within the interfacial region. Moreover, the $x$-dependent part is amenable of
interpretations. The presence of the quadratic factor $x^{2}$, which is also
responsible of the entropic repulsion, penalizes correlations in the proximity
of the boundary while the Gaussian factor suppresses correlations far away
from the boundary. The fact that interfacial fluctuations are long ranged and
confined to the interfacial region – as pointed out by Wertheim [33] – is
neatly realized by inspection of the analytic result (1.1).

Among the various analytic results in real space, we also investigate the
long-range character of interfacial correlations in momentum space by
extending the notion of interface structure factor to bounded systems. This
analysis will allow us to identify in the interface structure factor a
specific correction stemming from the entropic repulsion of the interface from
the wall.

This paper is structured as follows. As a warmup, in Sec. 2 the calculation of
the order parameter profile for a droplet is reviewed. In Sec. 3 the field-
theoretical method for the calculation of pair correlation functions is
presented in its simplest form, namely the energy density correlations. The
calculation of spin-spin correlations is then addressed in Sec. 4 and the
connection with a probabilistic interpretation is also illustrated. The
interface structure factor for a droplet is defined and calculated in Sec. 5.
Concluding remarks are collected in Sec. 6. A series of appendices collects
some mathematical details involved in the calculations reported in the main
body of the paper.

##  2 Magnetization profile

We begin by reviewing the calculation of the magnetization profile for a
statistical system at a first order phase transition with boundary conditions
enforcing a droplet as the one shown in Fig. 1. The derivation follows closely
the original calculation given in [19], which in the present exposition is
extended in order to include finite-size corrections originated by interface
structure. We consider a two-dimensional system on the first order phase
transition line close to a second order phase transition point. The scaling
limit in the near-critical region can be described in terms of a Euclidean
field theory obtained by analytic continuation to imaginary time of a
relativistic quantum field theory in a $1+1$ dimensional space-time.
Elementary excitations in $1+1$ dimensional quantum field theories are
topological particles (kinks) which interpolate between different ground
states and whose propagation in space-time corresponds to boundaries between
different coexisting phases.

We study phase separation on the half-plane geometry with boundary conditions
along the $y$-axis leading to the formation of a droplet, as illustrated in
Fig. 1. The system is considered for temperatures $T$ such that the bulk
correlation length $\xi_{\rm b}$ is much smaller than the separation $R$
between interface endpoints; hence, $\xi_{\rm b}\ll R$. Moreover, we also
assume that $\xi_{\rm b}$ is much larger than any microscopic scale111$a_{0}$
is of the order of the lattice spacing in a Monte Carlo simulation., i.e.,
$\xi_{\rm b}\gg a_{0}$. This last assumption allows us to adopt a field-
theoretical language for the description of the system in the regime
$a_{0}\ll\xi_{\rm b}\ll R$.

$x$$y$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{0}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{a}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{b}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{b}}$$(0,-R/2)$$(0,R/2)$${\sigma_{1}(x_{1},y_{1})}$${\sigma_{2}(x_{2},y_{2})}$
Figure 1: The half-plane geometry with boundary conditions $B_{bab}$ leading
to the formation of a droplet in the halfplane $x>0$ with pinning points in
$(0,\pm R/2)$. Green circles indicate the order parameter fields $\sigma_{1}$
and $\sigma_{2}$ appearing in correlation functions considered in this paper.

The partition function of the system with $B_{bab}$ boundary conditions can be
written as follows

$\mathcal{Z}={}_{B_{b}}\langle
0|\mu_{ba}(0,R/2)\mu_{ab}(0,-R/2)|0\rangle_{B_{b}}\,,$ (2.1)

where $|0\rangle_{B_{b}}$ is the vacuum state in which the boundary has fixed
boundary condition with spin in state $b$ and $\mu_{ab}(0,\pm R/2)$ is the
boundary condition changing operator which implements the switch of boundary
condition from $b$ to $a$ in the point $(0,\pm R/2)$. The matrix element of
$\mu_{ba}$ between the vacuum and the single-kink state takes the form

${}_{B_{b}}\langle
0|\mu_{ba}(0,y)|K_{ab}(\theta)\rangle=\textrm{e}^{-my}\mathcal{F}_{\mu}(\theta)\,,$
(2.2)

where $m$ is the kink mass and $\mathcal{F}_{\mu}(\theta)$ is the form factor
of the boundary condition changing operator $\mu_{ba}$. Form factors of
boundary operators have been studied extensively in the framework of massive
integrable quantum field theories [34, 35, 36, 37]. The non-vanishing of the
matrix element of $\mu_{ba}$ with the one-kink state corresponds to the
pinning of a single222The pinning of a double kink corresponds to the
formation of an intermediate phase [21, 22]. domain wall in those points in
which the boundary condition switches from $a$ to $b$ [19]. By inserting a
resolution of the identity between the operators $\mu_{ba}$ and $\mu_{ab}$ in
(2.1) and using the normalization of states, it follows that

$\mathcal{Z}=\int_{0}^{\infty}\frac{\textrm{d}\theta}{2\pi}\,|\mathcal{F}_{\mu}(\theta)|^{2}\textrm{e}^{-mR\cosh\theta}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}(\textrm{e}^{-2mR})\,.$
(2.3)

The restriction of the integration over positive rapidities is due to the
wall. The single-kink state appearing in the first term in the right hand side
of (2.3) gives the dominant contribution to the partition function in the
regime $mR\gg 1$, which is the one we are interested in. In the low-
temperature phase, $m=1/(2\xi_{\rm b})$, with $\xi_{\rm b}$ the bulk
correlation length. The aforementioned relationship is a form of Widom’s
scaling relation that for the Ising model is exact for all subcritical
temperatures which follows from duality [38, 39, 8]; see also [40, 41]. The
regime $R\gg\xi_{\rm b}$ projects the integrand in (2.3) at small rapidities
and the boundary form factor expands as follows

$\mathcal{F}_{\mu}(\theta)={\rm
i}\mathfrak{a}\theta+\mathfrak{b}\theta^{2}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}(\theta^{3})\,,$
(2.4)

where the model-dependent (real) coefficients $\mathfrak{a}$ and
$\mathfrak{b}$ are known for boundary integrable field theories [35, 37]. It
has been already noticed how the linear behavior at low rapidities exhibited
by $\mathcal{F}_{\mu}(\theta)$ is ultimately responsible for the entropic
repulsion of the interface from the wall [19]. By inserting (2.4) into (2.3) a
saddle-point calculation yields

$\mathcal{Z}=\frac{\mathfrak{a}^{2}\textrm{e}^{-mR}}{\sqrt{\pi}(2mR)^{3/2}}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}(R^{-5/2})\,.$
(2.5)

The magnetization profile for $|y|<R/2$ is defined by

$\langle\sigma(x,y)\rangle_{B_{bab}}=\frac{1}{\mathcal{Z}}{}_{B_{b}}\langle
0|\mu_{ba}(0,R/2)\sigma(x,y)\mu_{ab}(0,-R/2)|0\rangle_{B_{b}}\,,$ (2.6)

the subscript $B_{bab}$ stands for the expectation value with the boundary
conditions illustrated in Fig. 1. For $mx\gg 1$ the spin field entering in
(2.6) can be treated as a bulk field, for which one can use translation
invariance on the plane

$\sigma(x,y)=\textrm{e}^{{\rm i}xP+yH}\sigma(0,0)\textrm{e}^{-{\rm i}xP-yH}$
(2.7)

in order to bring the field to the origin; $H$ and $P$ are the Hamiltonian and
momentum operators in field theory. The expectation value (2.6) becomes

$\displaystyle\langle\sigma(x,y)\rangle_{B_{bab}}$
$\displaystyle=\frac{1}{\mathcal{Z}}\int_{\mathbb{R}^{2}}\frac{\textrm{d}\theta_{1}\textrm{d}\theta_{2}}{(2\pi)^{2}}\mathcal{F}_{\mu}(\theta_{1})\mathcal{M}_{ab}^{\sigma}(\theta_{1}|\theta_{2})\mathcal{F}_{\mu}^{*}(\theta_{2})U(\theta_{1},\theta_{2})\,,$
(2.8)

with $\mathcal{M}_{ab}^{\sigma}(\theta_{1}|\theta_{2})=\langle
K_{ab}(\theta_{1})|\sigma(0,0)|K_{ba}(\theta_{2})\rangle$ and

$U(\theta_{1},\theta_{2})=\exp\biggl{[}-m\left(\frac{R}{2}-y\right)\cosh\theta_{1}-m\left(\frac{R}{2}+y\right)\cosh\theta_{2}+{\rm
i}mx(\sinh\theta_{1}-\sinh\theta_{2})\biggr{]}\,.$ (2.9)

The connected part of the one-point correlation function (2.8), denoted
$\langle\sigma(x,y)\rangle_{B_{bab}}^{\rm CP}$, is determined by the connected
part of the matrix element $\mathcal{M}_{ab}^{\sigma}(\theta_{1}|\theta_{2})$,
the latter reads

$\left(\mathcal{M}_{ab}^{\sigma}(\theta_{1}|\theta_{2})\right)^{\rm
CP}=F_{aba}^{\sigma}(\theta_{12}+{\rm i}\pi)\,,$ (2.10)

where $F_{aba}^{\sigma}(\theta_{12})=\langle
0|\sigma(0,0)|K_{ab}(\theta_{1})K_{ba}(\theta_{2})\rangle$ is the two-particle
(bulk) form factor of the operator $\sigma$ [19]. By virtue of relativistic
invariance the bulk form factor depends on the rapidities $\theta_{1}$ and
$\theta_{2}$ through the difference $\theta_{12}\equiv\theta_{1}-\theta_{2}$
[42]. On the other hand, boundary form factors do not exhibit such a symmetry
[35, 36, 37]. The low-rapidity expansion of the bulk form factor reads

$F_{aba}^{\sigma}(\theta_{12}+{\rm
i}\pi)=\sum_{k=-1}^{\infty}c_{k}\theta_{12}^{k}\,.$ (2.11)

The term with $k=-1$ is due to the kinematical pole exhibited by the form
factor [42]. The above expansion is then combined with the corresponding low-
rapidity behavior of boundary form factors (2.4). Hence, one writes

$\displaystyle\mathcal{F}_{\mu}(\theta_{1})F_{aba}^{\sigma}(\theta_{12}+{\rm
i}\pi)\mathcal{F}_{\mu}^{*}(\theta_{2})$
$\displaystyle=c_{-1}\mathfrak{a}^{2}\biggl{[}\frac{\theta_{1}\theta_{2}}{\theta_{12}}-{\rm
i}\omega\theta_{1}\theta_{2}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}(\theta^{3})\biggr{]}\,,$
(2.12)

where $\omega=\mathfrak{b}/\mathfrak{a}+{\rm i}c_{0}/c_{-1}$. The notation
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}(\theta^{3})$
stands for homogeneous terms with total degree $3$ in the rapidity variables
(e.g., $\theta_{1}^{2}\theta_{2}$). By rescaling rapidities through
$\theta_{i}\rightarrow\sqrt{2/(mR)}\theta_{i}$ and organizing the result in
the form of a power series in the small parameter $(mR)^{-1/2}$, we find

$\langle\sigma(x,y)\rangle_{B_{bab}}^{\rm
CP}=\frac{2c_{-1}}{\pi^{3/2}}\biggl{[}\\{\\{\frac{\theta_{1}\theta_{2}}{\theta_{12}}\\}\\}-{\rm
i}\frac{\sqrt{2}\omega}{\sqrt{mR}}\\{\\{\theta_{1}\theta_{2}\\}\\}\biggr{]}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}\left((mR)^{-1}\right)\,;$
(2.13)

the notation $\\{\\{g(\theta_{1},\theta_{2})\\}\\}$ stands for the
contribution of $g(\theta_{1},\theta_{2})$ to the magnetization profile, with
$g(\theta_{1},\theta_{2})$ the generic term of the expansion (2.12). The
double curly bracket notation is defined as follows:

$\displaystyle\\{\\{g(\theta_{1},\theta_{2})\\}\\}$
$\displaystyle=\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\mathbb{R}^{2}}\textrm{d}\theta_{1}\textrm{d}\theta_{2}\,g(\theta_{1},\theta_{2})Y(\theta_{1},\theta_{2})\,,$
(2.14) $\displaystyle Y(\theta_{1},\theta_{2})$
$\displaystyle=\exp\biggl{[}-\frac{1-\tau}{2}\theta_{1}^{2}-\frac{1+\tau}{2}\theta_{2}^{2}+{\rm
i}\eta\theta_{12}\biggr{]}\,.$

The symbol
$\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int$
stands for the principal value of the integral, which is actually needed since
spin field matrix elements exhibit a kinematical pole [17]. Then, $\eta$ and
$\tau$ are the rescaled coordinates defined by

$\eta=x/\lambda\,,\qquad\lambda=\sqrt{R/(2m)}\,,\qquad\tau=2y/R\,.$ (2.15)

A simple calculation yields

$\displaystyle\\{\\{\frac{\theta_{1}\theta_{2}}{\theta_{12}}\\}\\}$
$\displaystyle=\frac{{\rm
i}\pi^{3/2}}{2}\mathcal{D}(\chi)\,,\qquad\\{\\{\theta_{1}\theta_{2}\\}\\}$
$\displaystyle=\frac{\pi^{3/2}\lambda}{2}P_{1}(x,y)\,,$ (2.16)

where

$\displaystyle\mathcal{D}(\chi)$
$\displaystyle=-\frac{2}{\sqrt{\pi}}\chi\textrm{e}^{-\chi^{2}}+\textrm{erf}(\chi)\,,\qquad
P_{1}(x,y)$
$\displaystyle=\frac{4\chi^{2}}{\sqrt{\pi}\kappa\lambda}\textrm{e}^{-\chi^{2}}\,,$
(2.17)

and

$\kappa=\sqrt{1-\tau^{2}}\,,\qquad\qquad\chi=x/(\kappa\lambda)\,.$ (2.18)

In order to compute $\\{\\{\theta_{1}\theta_{2}/\theta_{12}\\}\\}$ it is
convenient to remove the kinematical pole singularity $1/\theta_{12}$ by
taking the first derivative with respect to $\eta$ and thus compute
$\\{\\{\theta_{1}\theta_{2}\\}\\}$. Then, by integrating back to $\eta$, we
find the desired result. The integration constant generated by such a
procedure can be set to zero and eventually it can be reabsorbed into the
disconnected term of the matrix element. Such term originates the offset for
the profile which is uniquely fixed by the asymptotic boundary conditions:
$\langle\sigma(x\rightarrow+\infty,y)\rangle_{B_{bab}}=\langle\sigma\rangle_{b}$.

Thanks to the above results, and using the known expression for the residue of
the two-particle form factor of the spin field, $c_{-1}={\rm
i}(\langle\sigma\rangle_{a}-\langle\sigma\rangle_{b})$ [43, 17], the
magnetization profile (2.13) becomes

$\langle\sigma(x,y)\rangle_{B_{bab}}=\langle\sigma\rangle_{a}-\Bigl{[}\langle\sigma\rangle_{a}-\langle\sigma\rangle_{b}\Bigr{]}\mathcal{D}(\chi)+\omega\Bigl{[}\langle\sigma\rangle_{a}-\langle\sigma\rangle_{b}\Bigr{]}P_{1}(x;y)/m+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}(R^{-1})\,,\quad
mx\gg 1\,.$ (2.19)

Since the term proportional to $\omega$ is of order $(\xi_{\rm b}/R)^{1/2}$,
it is a subleading correction to the magnetization profile. The droplet
profile $\mathcal{D}(\chi)$ is a monotonous function which interpolates
between $0$ and $+1$ as $x$ varies from $x=0$ to $x\rightarrow+\infty$ with
$|y|<R/2$. Since the leading order term in (2.19) depends on $x$ and $y$
through $\chi$, it follows that constant values of $\chi$ give the contour
lines of the magnetization profile in the plane. As a result, these contour
lines are arcs of ellipses with implicit equation

$\frac{x^{2}}{cR\xi_{\rm b}}+\frac{4y^{2}}{R^{2}}=1\,,$ (2.20)

corresponding to $\chi=c$. The theoretical prediction (2.20), which is known
for the Ising model [9], has been found also within SOS models [11, 12] and
tested again numerical simulations [44]. The result (2.20) implies that the
contour line $\chi=c$ crosses the horizontal axis in $x=\sqrt{cR\xi_{\rm b}}$.
The proportionality to $\sqrt{R}$ admits a physical interpretation in terms of
a one-dimensional random walk on the half-line, as we are going to show in a
while.

For lather convenience, we also introduce the droplet profile
$\Upsilon(\chi)=2\mathcal{D}(\chi)-1$,

$\Upsilon(\chi)=-1-\frac{4}{\sqrt{\pi}}\chi\textrm{e}^{-\chi^{2}}+2\textrm{erf}(\chi)\,,$
(2.21)

which interpolates between $-1$ and $+1$; both the profiles are plotted in
Fig. 2.

Figure 2: The droplet profiles $\mathcal{D}(\chi)$ and $\Upsilon(\chi)$.

It is instructive to specialize the general result (2.19) to the explicit case
of the Ising model. From the known expression of the boundary form factor
[35], we extract $\omega_{\rm Ising}=-1/2$ and therefore

$\langle\sigma(x,y)\rangle_{B_{+-+}}=M\Upsilon(\chi)+\frac{2^{5/2}M}{\sqrt{\pi
mR}\kappa}\chi^{2}\textrm{e}^{-\chi^{2}}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}(R^{-1})\,,$
(2.22)

where $M=\langle\sigma\rangle_{+}=-\langle\sigma\rangle_{-}>0$ is the
spontaneous magnetization. The leading-order term $\propto\Upsilon(\chi)$
shown in (2.22) coincides with scaling limit of exact results obtained for the
square lattice Ising model [9, 45, 8] and from the path-integral formulation
of Solid-On-Solid models [12, 11].

Within a probabilistic interpretation [19, 20] the magnetization profile can
be derived by summing over interfacial configurations weighted with a certain
passage probability density $P_{1}(x,y)$. Thus, the magnetization profile is
computed as follows

$\langle\sigma(x,y)\rangle_{B_{bab}}=\int_{0}^{\infty}\textrm{d}u\,P_{1}(u,y)\sigma_{ab}(x|u)\,,$
(2.23)

where

$\sigma_{ab}(x|u)=\langle\sigma\rangle_{a}-\left(\langle\sigma\rangle_{a}-\langle\sigma\rangle_{b}\right)\theta(x-u)+A_{ab}\delta(x-u)+\dots\,,$
(2.24)

gives the magnetization at point $x$ when the interface, regarded as a sharp
curve, passes through $(u,u+\textrm{d}u)$ at ordinate $y$. The first two terms
in (2.24) corresponds to a description in which the coexisting phases are
sharply separated. The correction to this picture is achieved by endowing the
structureless sharp profile with interface structure effects $\propto A_{ab}$
whose effect is to produce subleading finite-size corrections [17]. The
identification of the passage probability with the expression for $P_{1}(x,y)$
given in (2.17) is established by matching the probabilistic construction
(2.23) with the field-theoretic result (2.19). The following identities prove
to be useful

$\displaystyle\int_{0}^{x}\textrm{d}u\,P_{1}(u,y)$
$\displaystyle=\mathcal{D}(\chi)\,,$ (2.25)
$\displaystyle\int_{0}^{\infty}\textrm{d}u\,P_{1}(u,y)\textrm{sign}(x-u)$
$\displaystyle=\Upsilon(\chi)\,,$

jointly with the fact that $\int_{0}^{\infty}\textrm{d}u\,P_{1}(u,y)=1$, i.e.,
the passage probability density $P_{1}(x,y)$ is normalized. The above passage
probability characterizes the so called _Brownian excursion_ and its
expression is given by (2.17). By pushing the comparison at the next-to-
leading order, we can identify the structure amplitude

$\displaystyle A_{ab}$
$\displaystyle=\frac{\omega}{m}\Bigl{[}\langle\sigma\rangle_{a}-\langle\sigma\rangle_{b}\Bigr{]}$
(2.26)
$\displaystyle=\frac{c_{0}}{m}+\frac{\mathfrak{b}}{\mathfrak{a}}\frac{\Delta\langle\sigma\rangle}{m}\,.$

The application of the above reasoning to the profile of the energy density
field on the half plane follows _mutatis mutandis_ (we refer to [26, 24] for
the calculation on the strip geometry). It is worth noticing that on the strip
$A_{ab}^{\rm(strip)}=c_{0}/m$ vanishes for the Ising model while the
corresponding result for the half-plane geometry given by (2.26) is actually
non zero. Thus, $A_{ab}$ is geometry-dependent.

##  3 Energy density correlations

The connected part of energy density correlation functions contains
information on the passage probability. More precisely, the connected energy
density correlation function is proportional to the joint passage probability.
Such a feature, which has been established for the strip geometry [26], is
valid also for the half-plane, as we are going to show. Let
$P_{2}(x_{1},y_{1};x_{2},y_{2})$ be the joint passage probability density,
therefore $P_{2}(x_{1},y_{1};x_{2},y_{2})\textrm{d}x_{1}\textrm{d}x_{2}$ is
the probability of the interface to pass through the intervals
$(x_{1},x_{1}+{\rm d}x_{1})$ _and_ $(x_{2},x_{2}+{\rm d}x_{2})$ at ordinates
$y_{1}$ and $y_{2}$, respectively.

In this section, we extract $P_{2}(x_{1},y_{1};x_{2},y_{2})$ from the two-
point correlation function of the energy density field computed in field
theory. The correlation function we are interested in reads

$\langle\varepsilon(x_{1},y_{1})\varepsilon(x_{2},y_{2})\rangle_{B_{bab}}^{\rm
CP}=\frac{1}{\mathcal{Z}}{}_{B_{b}}\langle
0|\mu_{ba}(0,R/2)\varepsilon(x_{1},y_{1})\varepsilon(x_{2},y_{2})\mu_{ab}(0,-R/2)|0\rangle_{B_{b}}^{\rm
CP}\,;$ (3.1)

with $y_{1}-y_{2}\gg\xi_{\rm b}$ and both energy density fields far from the
boundaries in the following sense: $(R/2)-y_{1}\gg\xi_{\rm b}$ and
$y_{2}+R/2\gg\xi_{\rm b}$. The superscript CP means that we retain the
connected part. By employing the decomposition over intermediate states, (3.1)
becomes

$\displaystyle\langle\varepsilon(x_{1},y_{1})\varepsilon(x_{2},y_{2})\rangle_{B_{bab}}^{\rm
CP}$
$\displaystyle=\frac{1}{\mathcal{Z}}\int\frac{\textrm{d}\theta_{1}\textrm{d}\theta_{2}\textrm{d}\theta_{3}}{(2\pi)^{3}}\biggl{[}\mathcal{F}_{\mu}(\theta_{1})\left(\mathcal{M}_{ab}^{\varepsilon}(\theta_{1}|\theta_{2})\mathcal{M}_{ab}^{\varepsilon}(\theta_{2}|\theta_{3})\right)^{\rm
CP}\mathcal{F}_{\mu}^{*}(\theta_{3})U_{+}(\\{\theta\\})$ (3.2)
$\displaystyle+\mathcal{F}_{\mu}(\theta_{1})\left(\mathcal{M}_{ab}^{\varepsilon}(\theta_{1}|-\theta_{2})\mathcal{R}_{aba}(\theta_{2})\mathcal{M}_{ab}^{\varepsilon}(\theta_{2}|\theta_{3})\right)^{\rm
CP}\mathcal{F}_{\mu}^{*}(\theta_{3})U_{-}(\\{\theta\\})\biggr{]}\,,$

where

$\displaystyle U_{\pm}(\\{\theta\\})$
$\displaystyle=\exp\biggl{[}-m\left(\frac{R}{2}-y_{1}\right)\cosh\theta_{1}-m(y_{1}-y_{2})\cosh\theta_{2}-m\left(y_{2}+\frac{R}{2}\right)\cosh\theta_{3}$
(3.3) $\displaystyle+{\rm i}mx_{1}(\sinh\theta_{1}\mp\sinh\theta_{2})+{\rm
i}mx_{2}(\sinh\theta_{2}-\sinh\theta_{3})\biggr{]}\,.$

The second term appearing in the right hand side of (3.2) contains the
boundary $S$-matrix333See [46, 47, 48] for boundary $S$-matrices in integrable
field theories. $\mathcal{R}_{aba}(\theta_{2})$ which gives the amplitude for
the scattering of a kink state off the vertical wall with fixed boundary
conditions $a$ [46]; the latter admits the pictorial representation of (3.4).

$\mathcal{R}_{aba}(\theta)=\leavevmode\hbox to62.83pt{\vbox
to102.83pt{\pgfpicture\makeatletter\hbox{\hskip
56.90552pt\lower-51.41504pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{-51.21504pt}{0.0pt}\pgfsys@lineto{0.0pt}{51.21504pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}{{
{\pgfsys@beginscope\pgfsys@setlinewidth{0.32pt}\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{}
{}{}{} {}{}{}
\pgfsys@moveto{-1.19998pt}{1.59998pt}\pgfsys@curveto{-1.09998pt}{0.99998pt}{0.0pt}{0.09999pt}{0.29999pt}{0.0pt}\pgfsys@curveto{0.0pt}{-0.09999pt}{-1.09998pt}{-0.99998pt}{-1.19998pt}{-1.59998pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{-51.21504pt}{0.0pt}\pgfsys@lineto{-25.93256pt}{25.28204pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7071}{0.7071}{-0.7071}{0.7071}{-25.93256pt}{25.28204pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{-51.21504pt}{0.0pt}\pgfsys@lineto{0.0pt}{-51.21504pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-51.21504pt}\pgfsys@lineto{-25.28204pt}{-25.93256pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.7071}{0.7071}{-0.7071}{-0.7071}{-25.28204pt}{-25.93256pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-44.14328pt}{31.99045pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{a}}$
}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-44.14328pt}{-36.29599pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{a}}$
}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-43.64616pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{b}}$
}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.4366pt}{-40.46088pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\theta}}$
}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-12.48209pt}{33.51645pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{-\theta}}$
}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\definecolor[named]{tikz@axis@top}{rgb}{1,1,1}\definecolor{tikz@axis@middle}{rgb}{1,1,1}\definecolor[named]{tikz@axis@bottom}{rgb}{.5,.5,.5}\definecolor{tikz@axis@middle}{rgb}{0.75,0.75,0.75}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{-56.90552pt}{-51.21504pt}\pgfsys@moveto{-56.90552pt}{-51.21504pt}\pgfsys@lineto{-56.90552pt}{51.21504pt}\pgfsys@lineto{-51.21504pt}{51.21504pt}\pgfsys@lineto{-51.21504pt}{-51.21504pt}\pgfsys@closepath\pgfsys@moveto{-51.21504pt}{51.21504pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.06027pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{2.04121}{0.0}{0.0}{0.11339}{0.0pt}{0.0pt}\pgfsys@invoke{
}\definecolor[named]{pgf@tempcol}{rgb}{.5,.5,.5}\definecolor[named]{pgf@tempcol}{rgb}{.5,.5,.5}\definecolor[named]{pgf@tempcol}{rgb}{0.75,0.75,0.75}\definecolor[named]{pgf@tempcol}{rgb}{1,1,1}\definecolor[named]{pgf@tempcol}{rgb}{1,1,1}\definecolor{pgfshadetemp}{rgb}{0.5,0.5,0.5}\definecolor{pgfshadetemp}{rgb}{1,1,1}\definecolor{pgfshadetemp}{rgb}{0.5,0.5,0.5}\definecolor{pgfshadetemp}{rgb}{0.75,0.75,0.75}\definecolor{pgfshadetemp}{rgb}{1,1,1}\definecolor{pgfshadetemp}{rgb}{1,1,1}{}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44499pt}{-69.44499pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (3.4)

The two terms appearing in the square brackets of (3.2) admit the diagrammatic
representation provided in (3.5).

$\displaystyle\mathfrak{M}_{D}^{\varepsilon,{\rm CP}}=\leavevmode\hbox
to74.01pt{\vbox to114.21pt{\pgfpicture\makeatletter\hbox{\hskip
45.52458pt\lower-57.10551pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.8875,0.4}\pgfsys@color@rgb@fill{1}{0.8875}{0.4}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.8875,0.4}\pgfsys@color@rgb@fill{1}{0.8875}{0.4}\pgfsys@invoke{
}{}\pgfsys@moveto{13.57156pt}{22.76228pt}\pgfsys@curveto{13.57156pt}{27.11494pt}{10.04312pt}{30.64339pt}{5.69046pt}{30.64339pt}\pgfsys@curveto{1.3378pt}{30.64339pt}{-2.19064pt}{27.11494pt}{-2.19064pt}{22.76228pt}\pgfsys@curveto{-2.19064pt}{18.40962pt}{1.3378pt}{14.88118pt}{5.69046pt}{14.88118pt}\pgfsys@curveto{10.04312pt}{14.88118pt}{13.57156pt}{18.40962pt}{13.57156pt}{22.76228pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{22.76228pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.35887pt}{20.60951pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\varepsilon$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.8875,0.4}\pgfsys@color@rgb@fill{1}{0.8875}{0.4}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.8875,0.4}\pgfsys@color@rgb@fill{1}{0.8875}{0.4}\pgfsys@invoke{
}{}\pgfsys@moveto{2.19064pt}{-22.76228pt}\pgfsys@curveto{2.19064pt}{-18.40962pt}{-1.3378pt}{-14.88118pt}{-5.69046pt}{-14.88118pt}\pgfsys@curveto{-10.04312pt}{-14.88118pt}{-13.57156pt}{-18.40962pt}{-13.57156pt}{-22.76228pt}\pgfsys@curveto{-13.57156pt}{-27.11494pt}{-10.04312pt}{-30.64339pt}{-5.69046pt}{-30.64339pt}\pgfsys@curveto{-1.3378pt}{-30.64339pt}{2.19064pt}{-27.11494pt}{2.19064pt}{-22.76228pt}\pgfsys@closepath\pgfsys@moveto{-5.69046pt}{-22.76228pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.02205pt}{-24.91505pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\varepsilon$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{3.7306pt}{14.9225pt}\pgfsys@lineto{-3.7306pt}{-14.9225pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{5.69046pt}{30.84338pt}\pgfsys@lineto{5.69046pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{-5.69046pt}{-30.84338pt}\pgfsys@lineto{-5.69046pt}{-56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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$\displaystyle=F_{aba}^{\varepsilon}(\theta_{12}+{\rm
i}\pi)F_{aba}^{\varepsilon}(\theta_{23}+{\rm i}\pi)$ (3.5)
$\displaystyle\mathfrak{M}_{R}^{\varepsilon,{\rm CP}}=\leavevmode\hbox
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$\displaystyle=F_{aba}^{\varepsilon}(\widehat{\theta}_{12}+{\rm
i}\pi)\mathcal{R}_{aba}(\theta_{2})F_{aba}^{\varepsilon}(\theta_{23}+{\rm
i}\pi)\,,\qquad\widehat{\theta}_{ij}=\theta_{i}+\theta_{j}\,.$

The kink with rapidity $\theta_{2}$ appearing in the diagram
$\mathfrak{M}_{D}^{\varepsilon,{\rm CP}}$ connects the two energy density
fields in a “direct” ($D$) fashion while in
$\mathfrak{M}_{R}^{\varepsilon,{\rm CP}}$ the kink with rapidity $\theta_{2}$
is reflected ($R$) off the boundary before being connected to the other field.
As a result, $\mathfrak{M}_{R}^{\varepsilon,{\rm CP}}$ picks up a boundary
reflection factor $\mathcal{R}_{aba}$. The leading-order form of the
correlation function (3.1) emerges from the behavior of matrix elements at
small momenta. Since
$\mathcal{R}_{aba}(\theta_{2})=-1+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}(\theta_{2})$
for $\theta_{2}\rightarrow 0$, the matrix elements in (3.5) tend to
$\pm(F_{aba}^{\varepsilon}({\rm i}\pi))^{2}$, respectively. The calculation
involved in (3.1) requires the evaluation of Gaussian integrations. Leaving
the details in Appendix A and C, we find

$\displaystyle\langle\varepsilon(x_{1},y_{1})\varepsilon(x_{2},y_{2})\rangle_{B_{bab}}^{\rm
CP}$ $\displaystyle=\frac{\left(F_{aba}^{\varepsilon}({\rm
i}\pi)\right)^{2}}{m^{2}}P_{2}(x_{1},y_{1};x_{2},y_{2})\,,$ (3.6)

where

$P_{2}(x_{1},y_{1};x_{2},y_{2})=\frac{8\chi_{1}\chi_{2}}{\rho\kappa_{1}\kappa_{2}\lambda^{2}}\bigl{[}\Pi_{2}(\sqrt{2}\chi_{1},\sqrt{2}\chi_{2}|\rho)-\Pi_{2}(\sqrt{2}\chi_{1},\sqrt{2}\chi_{2}|-\rho)\bigr{]}\,;$
(3.7)

in the above, $\Pi_{2}(u_{1},u_{2}|\rho)$ is a normal Gaussian bivariate
distribution (see Appendix C) with correlation coefficient

$\rho=\sqrt{\frac{1-\tau_{1}}{1+\tau_{1}}\frac{1+\tau_{2}}{1-\tau_{2}}}\,,\qquad\tau_{j}=2y_{j}/R\,,$
(3.8)

and the notation

$\chi_{j}=\eta_{j}/\kappa_{j}\,,\qquad\eta_{j}=x_{j}/\lambda\,,\qquad\kappa_{j}=\sqrt{1-\tau_{j}^{2}}$
(3.9)

has been adopted. Notice that $0<\rho<1$, however, since the two fields are
both far from each other and far from the boundaries the extremal values
($\rho=0$ and $\rho=1$) are never reached.

In order to check that (3.7) is indeed the joint passage probability, we
primarily observe that (3.7) satisfies the following properties:

$\displaystyle\int_{0}^{\infty}\textrm{d}x_{2}\,P_{2}(x_{1},y_{1};x_{2},y_{2})$
$\displaystyle=P_{1}(x_{1},y_{1})\,,$ (3.10)
$\displaystyle\int_{0}^{\infty}\textrm{d}x_{1}\int_{0}^{\infty}\textrm{d}x_{2}\,P_{2}(x_{1},y_{1};x_{2},y_{2})$
$\displaystyle=1\,,$

thus, $P_{2}$ is correctly normalized and its marginal reduces to the passage
probability $P_{1}$, as consistency requires for passage probabilities.

Figure 3: The rescaled passage probability
$\lambda^{2}P_{2}(x_{1},y;x_{2},-y)$ as function of the rescaled coordinates
$\eta_{1}=x_{1}/\lambda$, $\eta_{2}=x_{2}/\lambda$ for $2y/R=0.1$.

Moreover, by applying the above properties the result (3.6) for the energy
density correlation function follows from the probabilistic reconstruction

$\langle\varepsilon(x_{1},y_{1})\varepsilon(x_{2},y_{2})\rangle_{B_{bab}}=\int_{0}^{\infty}\textrm{d}u_{1}\int_{0}^{\infty}\textrm{d}u_{2}\,P_{2}(u_{1},y_{1};u_{2},y_{2})\varepsilon(x_{1}|u_{1})\varepsilon(x_{2}|u_{2})\,,$
(3.11)

with the energy density profile
$\varepsilon(x_{i}|u_{i})=\langle\varepsilon\rangle+A_{\varepsilon}\delta(x_{i}-u_{i})+\dots$
constructed by following the same guidelines which lead us to the
magnetization profile. By matching the connected part of (3.11) with the
field-theoretical result (3.6), we identify the structure coefficient
$A_{\varepsilon}=F_{aba}^{\varepsilon}({\rm i}\pi)/m$. The latter perfectly
coincides with the corresponding result already obtained for $n$-point point
correlation functions on the strip geometry444We refer to [24, 25] for
analytical and numerical results for $n$-point correlation functions with
$n=1,2,3$, and $4$. [26].

##  4 Pair correlation function of the spin field

By following the method illustrated in the previous section, we compute the
pair correlation function of the spin field. The latter is defined by

$\langle\sigma_{1}(x_{1},y_{1})\sigma_{2}(x_{2},y_{2})\rangle_{B_{bab}}=\frac{1}{\mathcal{Z}}{}_{B_{b}}\langle
0|\mu_{ba}(0,R/2)\sigma_{1}(x_{1},y_{1})\sigma_{2}(x_{2},y_{2})\mu_{ab}(0,-R/2)|0\rangle_{B_{b}}\,.$
(4.1)

We can replace $\varepsilon$ with $\sigma_{j}$ into (3.2) and use an analogous
decomposition for spin fields matrix elements

$\displaystyle\mathcal{M}^{\sigma_{1}}(\theta_{1}|\theta_{2})\mathcal{M}^{\sigma_{2}}(\theta_{2}|\theta_{3})$
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\definecolor[named]{tikz@axis@top}{rgb}{1,1,1}\definecolor{tikz@axis@middle}{rgb}{1,1,1}\definecolor[named]{tikz@axis@bottom}{rgb}{.5,.5,.5}\definecolor{tikz@axis@middle}{rgb}{0.75,0.75,0.75}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{-45.52458pt}{-56.90552pt}\pgfsys@moveto{-45.52458pt}{-56.90552pt}\pgfsys@lineto{-45.52458pt}{56.90552pt}\pgfsys@lineto{-39.83368pt}{56.90552pt}\pgfsys@lineto{-39.83368pt}{-56.90552pt}\pgfsys@closepath\pgfsys@moveto{-39.83368pt}{56.90552pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-42.67912pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{2.26802}{0.0}{0.0}{0.1134}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44499pt}{-69.44499pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{\mathfrak{M}_{R}^{\rm
CP}}\quad+\quad\textrm{disconnected}\,.$ (4.2)

The leading low-energy behavior of the above diagrams is given by

$\displaystyle\mathfrak{M}_{D}^{\sigma,{\rm CP}}$ $\displaystyle=$
$\displaystyle({\rm
i})^{2}\Delta\langle\sigma_{1}\rangle\Delta\langle\sigma_{2}\rangle\frac{1}{\theta_{12}\theta_{23}}\,,$
(4.3) $\displaystyle\mathfrak{M}_{R}^{\sigma,{\rm CP}}$ $\displaystyle=$
$\displaystyle({\rm
i})^{2}\Delta\langle\sigma_{1}\rangle\Delta\langle\sigma_{2}\rangle\frac{\mathcal{R}_{aba}(0)}{\widehat{\theta}_{12}\theta_{23}}\,,$
(4.4)

respectively for the first and second diagrams in (4.2). In the above,
$\mathcal{R}_{aba}(0)$ is the boundary $S$-matrix evaluated at zero rapidity.
It has to be noticed that within the low-rapidity regime pertinent to $mR\gg
1$ the boundary $S$-matrix reduces to $\mathcal{R}_{aba}(0)=-1$. Such a
limiting behavior suffices for the determination of correlation functions at
leading order in powers of $(\xi_{\rm b}/R)^{1/2}$.

The calculation of (4.1) proceeds as follows. Let us indicate with
$\\{\mathfrak{M}\\}$ the contribution of the diagram $\mathfrak{M}$ to the
pair correlation function. The two diagrams appearing in (4.2) give the
following contribution to the spin-spin correlation function

$\\{\mathfrak{M}_{D}^{\sigma,{\rm CP}}+\mathfrak{M}_{R}^{\sigma,{\rm
CP}}\\}=-\frac{\Delta\langle\sigma_{1}\rangle\Delta\langle\sigma_{2}\rangle}{\pi^{5/2}}\biggl{[}\Lbag\frac{\theta_{1}\theta_{3}}{\theta_{12}\theta_{23}}\Rbag_{+}-\Lbag\frac{\theta_{1}\theta_{3}}{\widehat{\theta}_{12}\theta_{23}}\Rbag_{-}\biggr{]}\,,$
(4.5)

where $\Lbag\cdots\Rbag_{\pm}$ is the notation defined in (A.3). By taking the
first derivatives with respect to $x_{1}$ and $x_{2}$ and using the identity
(A.7), which relates the calculation of matrix elements to the joint passage
probability, it follows that

$\displaystyle\partial_{x_{1}}\partial_{x_{2}}\\{\mathfrak{M}_{D}^{\sigma,{\rm
CP}}+\mathfrak{M}_{R}^{\sigma,\rm CP}\\}$
$\displaystyle=\frac{\Delta\langle\sigma_{1}\rangle\Delta\langle\sigma_{2}\rangle}{\pi^{5/2}\lambda^{2}}\biggl{[}\Lbag\theta_{1}\theta_{3}\Rbag_{+}-\Lbag\theta_{1}\theta_{3}\Rbag_{-}\biggr{]}\,,$
(4.6)
$\displaystyle=\Delta\langle\sigma_{1}\rangle\Delta\langle\sigma_{2}\rangle
P_{2}(x_{1},y_{1};x_{2},y_{2})\,.$

Thus, by integrating back with respect to $x_{1}$ and $x_{2}$, we have

$\\{\mathfrak{M}_{D}^{\sigma,{\rm CP}}+\mathfrak{M}_{R}^{\sigma,{\rm
CP}}\\}=\Delta\langle\sigma_{1}\rangle\Delta\langle\sigma_{2}\rangle\int_{0}^{x_{1}}\textrm{d}u_{1}\int_{0}^{x_{2}}\textrm{d}u_{2}\,P_{2}(u_{1},y_{1};u_{2},y_{2})+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}(R^{-1/2})\,,$
(4.7)

up to integration terms555The integration constants are actually functions
which are annihilated by the differential operator
$\partial_{x_{1}}\partial_{x_{2}}.$ which, without loss of generality, can be
reabsorbed in the disconnected diagrams; for example, by fixing their
contribution for $x_{j}\rightarrow+\infty$.

The disconnected terms appearing in the right hand side of (4.2) are depicted
as follows:

$\displaystyle\mathfrak{M}^{\sigma_{1},{\rm disc}}$
$\displaystyle=\quad\leavevmode\hbox to75.61pt{\vbox
to114.21pt{\pgfpicture\makeatletter\hbox{\hskip
45.52458pt\lower-57.10551pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.7,1,0.7}\pgfsys@color@rgb@fill{0.7}{1}{0.7}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.7,1,0.7}\pgfsys@color@rgb@fill{0.7}{1}{0.7}\pgfsys@invoke{
}{}\pgfsys@moveto{15.60379pt}{25.6073pt}\pgfsys@curveto{15.60379pt}{31.08234pt}{11.1655pt}{35.52063pt}{5.69046pt}{35.52063pt}\pgfsys@curveto{0.21542pt}{35.52063pt}{-4.22287pt}{31.08234pt}{-4.22287pt}{25.6073pt}\pgfsys@curveto{-4.22287pt}{20.13226pt}{0.21542pt}{15.69397pt}{5.69046pt}{15.69397pt}\pgfsys@curveto{11.1655pt}{15.69397pt}{15.60379pt}{20.13226pt}{15.60379pt}{25.6073pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{25.6073pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.4334pt}{24.35675pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\sigma_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.7,1,0.7}\pgfsys@color@rgb@fill{0.7}{1}{0.7}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.7,1,0.7}\pgfsys@color@rgb@fill{0.7}{1}{0.7}\pgfsys@invoke{
}{}\pgfsys@moveto{4.22287pt}{-25.6073pt}\pgfsys@curveto{4.22287pt}{-20.13226pt}{-0.21542pt}{-15.69397pt}{-5.69046pt}{-15.69397pt}\pgfsys@curveto{-11.1655pt}{-15.69397pt}{-15.60379pt}{-20.13226pt}{-15.60379pt}{-25.6073pt}\pgfsys@curveto{-15.60379pt}{-31.08234pt}{-11.1655pt}{-35.52063pt}{-5.69046pt}{-35.52063pt}\pgfsys@curveto{-0.21542pt}{-35.52063pt}{4.22287pt}{-31.08234pt}{4.22287pt}{-25.6073pt}\pgfsys@closepath\pgfsys@moveto{-5.69046pt}{-25.6073pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.94753pt}{-26.85785pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\sigma_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{-5.69046pt}{-14.22638pt}\pgfsys@lineto{-5.69046pt}{-7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{28.45276pt}{7.11319pt}\pgfsys@lineto{28.45276pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{-5.69046pt}{-7.11319pt}\pgfsys@lineto{28.45276pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{-5.69046pt}{-35.72063pt}\pgfsys@lineto{-5.69046pt}{-56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-31.33963pt}{-27.76007pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{a}}$
}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.83661pt}{-29.07951pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{b}}$
}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.92569pt}{45.7996pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\theta_{1}}}$
}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.14615pt}{1.69774pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\theta_{2}}}$
}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.99117pt}{-48.09457pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\theta_{3}}}$
}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\definecolor[named]{tikz@axis@top}{rgb}{1,1,1}\definecolor{tikz@axis@middle}{rgb}{1,1,1}\definecolor[named]{tikz@axis@bottom}{rgb}{.5,.5,.5}\definecolor{tikz@axis@middle}{rgb}{0.75,0.75,0.75}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{-45.52458pt}{-56.90552pt}\pgfsys@moveto{-45.52458pt}{-56.90552pt}\pgfsys@lineto{-45.52458pt}{56.90552pt}\pgfsys@lineto{-39.83368pt}{56.90552pt}\pgfsys@lineto{-39.83368pt}{-56.90552pt}\pgfsys@closepath\pgfsys@moveto{-39.83368pt}{56.90552pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-42.67912pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{2.26802}{0.0}{0.0}{0.1134}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44499pt}{-69.44499pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad=2\pi{\rm
i}\langle\sigma_{1}\rangle_{a}\Delta\langle\sigma_{2}\rangle\frac{\delta(\theta_{12})}{\theta_{23}}$
(4.8) $\displaystyle\mathfrak{M}^{\sigma_{2},{\rm disc}}$
$\displaystyle=\quad\leavevmode\hbox to79.7pt{\vbox
to114.21pt{\pgfpicture\makeatletter\hbox{\hskip
45.52458pt\lower-57.10551pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.7,1,0.7}\pgfsys@color@rgb@fill{0.7}{1}{0.7}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.7,1,0.7}\pgfsys@color@rgb@fill{0.7}{1}{0.7}\pgfsys@invoke{
}{}\pgfsys@moveto{15.60379pt}{25.6073pt}\pgfsys@curveto{15.60379pt}{31.08234pt}{11.1655pt}{35.52063pt}{5.69046pt}{35.52063pt}\pgfsys@curveto{0.21542pt}{35.52063pt}{-4.22287pt}{31.08234pt}{-4.22287pt}{25.6073pt}\pgfsys@curveto{-4.22287pt}{20.13226pt}{0.21542pt}{15.69397pt}{5.69046pt}{15.69397pt}\pgfsys@curveto{11.1655pt}{15.69397pt}{15.60379pt}{20.13226pt}{15.60379pt}{25.6073pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{25.6073pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.4334pt}{24.35675pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\sigma_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.7,1,0.7}\pgfsys@color@rgb@fill{0.7}{1}{0.7}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.7,1,0.7}\pgfsys@color@rgb@fill{0.7}{1}{0.7}\pgfsys@invoke{
}{}\pgfsys@moveto{-7.1585pt}{-25.6073pt}\pgfsys@curveto{-7.1585pt}{-20.13226pt}{-11.59679pt}{-15.69397pt}{-17.07182pt}{-15.69397pt}\pgfsys@curveto{-22.54686pt}{-15.69397pt}{-26.98515pt}{-20.13226pt}{-26.98515pt}{-25.6073pt}\pgfsys@curveto{-26.98515pt}{-31.08234pt}{-22.54686pt}{-35.52063pt}{-17.07182pt}{-35.52063pt}\pgfsys@curveto{-11.59679pt}{-35.52063pt}{-7.1585pt}{-31.08234pt}{-7.1585pt}{-25.6073pt}\pgfsys@closepath\pgfsys@moveto{-17.07182pt}{-25.6073pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-21.32889pt}{-26.85785pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\sigma_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{5.69046pt}{36.98866pt}\pgfsys@lineto{5.69046pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{5.69046pt}{-56.90552pt}\pgfsys@lineto{5.69046pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-19.9587pt}{23.45453pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{a}}$
}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}{{}}{}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
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\definecolor[named]{tikz@axis@top}{rgb}{1,1,1}\definecolor{tikz@axis@middle}{rgb}{1,1,1}\definecolor[named]{tikz@axis@bottom}{rgb}{.5,.5,.5}\definecolor{tikz@axis@middle}{rgb}{0.75,0.75,0.75}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{-45.52458pt}{-56.90552pt}\pgfsys@moveto{-45.52458pt}{-56.90552pt}\pgfsys@lineto{-45.52458pt}{56.90552pt}\pgfsys@lineto{-39.83368pt}{56.90552pt}\pgfsys@lineto{-39.83368pt}{-56.90552pt}\pgfsys@closepath\pgfsys@moveto{-39.83368pt}{56.90552pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
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i}\langle\sigma_{2}\rangle_{a}\Delta\langle\sigma_{1}\rangle\frac{\delta(\theta_{23})}{\theta_{12}}\,.$

The calculation of the above diagrams retraces the same arguments already
followed for the one point function; thus,

$\displaystyle\\{\mathfrak{M}^{\sigma_{1},{\rm disc}}\\}$
$\displaystyle=-\langle\sigma_{1}\rangle_{a}\Delta\langle\sigma_{2}\rangle\mathcal{D}(\chi_{2})+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}(R^{-1/2})\,,$
(4.9) $\displaystyle\\{\mathfrak{M}^{\sigma_{2},{\rm disc}}\\}$
$\displaystyle=-\langle\sigma_{2}\rangle_{a}\Delta\langle\sigma_{1}\rangle\mathcal{D}(\chi_{1})+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}(R^{-1/2})\,.$

Collecting all the results obtained so far and expressing the scaling function
$\mathcal{D}(\chi_{j})$ as an integral involving the passage probability, with
the aid of (2.25), we find

$\displaystyle\langle\sigma_{1}(x_{1},y_{1})\sigma_{2}(x_{2},y_{2})\rangle_{B_{bab}}$
$\displaystyle=\langle\sigma_{1}\rangle_{a}\langle\sigma_{2}\rangle_{a}+\Delta\langle\sigma_{1}\rangle\Delta\langle\sigma_{2}\rangle\int_{0}^{x_{1}}\textrm{d}u_{1}\int_{0}^{x_{2}}\textrm{d}u_{2}P_{2}(u_{1},y_{1};u_{2},y_{2})$
(4.10)
$\displaystyle-\langle\sigma_{2}\rangle_{a}\Delta\langle\sigma_{1}\rangle\int_{0}^{x_{1}}\textrm{d}u_{1}\,P_{1}(u_{1},y_{1})-\langle\sigma_{1}\rangle_{a}\Delta\langle\sigma_{2}\rangle\int_{0}^{x_{2}}\textrm{d}u_{2}\,P_{1}(u_{2},y_{2})$
$\displaystyle+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}(R^{-1/2})\,.$

The first term in the right hand side of (4.10) follows by imposing the
correct boundary conditions for $x_{j}\rightarrow+\infty$. Notice also that
upon taking one of the two spins deep inside the bulk, the pair correlation
function (4.10) satisfies the clustering property

$\lim_{x_{2}\rightarrow+\infty}\langle\sigma_{1}(x_{1},y_{1})\sigma_{2}(x_{2},y_{2})\rangle_{B_{bab}}=\langle\sigma_{2}\rangle_{b}\langle\sigma_{1}(x_{1},y_{1})\rangle_{B_{bab}}\,,$
(4.11)

and analogously when $x_{1}\rightarrow+\infty$ with finite $x_{2}$.

In view of future use, we write (4.10) in the following form

$\displaystyle\langle\sigma_{1}(x_{1},y_{1})\sigma_{2}(x_{2},y_{2})\rangle_{B_{bab}}$
$\displaystyle=\widehat{\langle\sigma_{1}\rangle}\widehat{\langle\sigma_{2}\rangle}\mathcal{G}(\eta_{1},\tau_{1};\eta_{2},\tau_{2})-\widetilde{\langle\sigma_{1}\rangle}\widehat{\langle\sigma_{2}\rangle}\Upsilon(\chi_{2})-\widetilde{\langle\sigma_{2}\rangle}\widehat{\langle\sigma_{1}\rangle}\Upsilon(\chi_{1})$
(4.12)
$\displaystyle+\widetilde{\langle\sigma_{1}\rangle}\widetilde{\langle\sigma_{2}\rangle}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}(R^{-1/2})\,,$

where
$\widetilde{\langle\sigma_{j}\rangle}=(\langle\sigma_{j}\rangle_{a}+\langle\sigma_{j}\rangle_{b})/2$
is the averaged vacuum expectation value in pure phases,
$\widehat{\langle\sigma_{j}\rangle}=(\langle\sigma_{j}\rangle_{a}-\langle\sigma_{j}\rangle_{b})/2$
is the half jump of order parameter across the interface, and
$\mathcal{G}(\eta_{1},\tau_{1};\eta_{2},\tau_{2})$ is the scaling function
which encodes the connected part of the two-point correlation function, i.e.,

$\displaystyle\langle\sigma_{1}(x_{1},y_{1})\sigma_{2}(x_{2},y_{2})\rangle_{B_{bab}}^{\rm
CP}$
$\displaystyle=\widehat{\langle\sigma_{1}\rangle}\widehat{\langle\sigma_{2}\rangle}\mathcal{G}(\eta_{1},\tau_{1};\eta_{2},\tau_{2})+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}(R^{-1/2})\,,$
(4.13)

with

$\displaystyle\mathcal{G}(\eta_{1},\tau_{1};\eta_{2},\tau_{2})$
$\displaystyle=\int_{0}^{\infty}\textrm{d}u_{1}\int_{0}^{\infty}\textrm{d}u_{2}\,P_{2}(u_{1},y_{1};u_{2},y_{2})\textrm{sign}(x_{1}-u_{1})\textrm{sign}(x_{2}-u_{2})\,.$
(4.14)

The function $\mathcal{G}$ defined by (4.14) is plotted in Fig. 4 as function
of the rescaled coordinates $\eta_{1}$ and $\eta_{2}$ with fixed vertical
separation $\tau_{1}-\tau_{2}$ of spin fields.

Figure 4: The function $\mathcal{G}(\eta_{1},\tau_{1};\eta_{2},\tau_{2})$ with
$\tau_{1}=-\tau_{2}=0.1$. The solid red curves and the dashed blue curves
correspond to the droplet profiles $\pm\Upsilon(\chi_{1})$ and
$\pm\Upsilon(\chi_{2})$, respectively.

From (4.14) it is simple to establish the following clustering properties:

$\displaystyle\lim_{x_{1}\rightarrow+\infty}\mathcal{G}(\eta_{1},\tau_{1};\eta_{2},\tau_{2})$
$\displaystyle=\Upsilon(\chi_{2})\,,$ (4.15)
$\displaystyle\lim_{x_{2}\rightarrow+\infty}\mathcal{G}(\eta_{1},\tau_{1};\eta_{2},\tau_{2})$
$\displaystyle=\Upsilon(\chi_{1})\,,$

whose occurrence is visualized by means of the solid red curves at the
boundaries of the surface depicted in Fig. 4. In a completely analogous way,
one finds

$\displaystyle\lim_{x_{1}\rightarrow
0}\mathcal{G}(\eta_{1},\tau_{1};\eta_{2},\tau_{2})$
$\displaystyle=-\Upsilon(\chi_{2})\,,$ (4.16)
$\displaystyle\lim_{x_{2}\rightarrow
0}\mathcal{G}(\eta_{1},\tau_{1};\eta_{2},\tau_{2})$
$\displaystyle=-\Upsilon(\chi_{1})\,,$

which correspond to the dashed blue curves visualized in Fig. 4.

A rather more transparent expression for the spin-spin correlation function
can be obtained for spin fields arranged parallel to the interface, or, more
precisely, parallel to the line which joins the pinning points. For this
configuration, $x\equiv x_{1}=x_{2}$, $y\equiv y_{1}=-y_{2}$. By further
restricting the focus to the regime of small vertical separations between spin
fields, we obtain (1.1), which we report here for convenience

$\displaystyle\langle\sigma(x,y)\sigma(x,-y)\rangle_{B_{bab}}$
$\displaystyle=\left(\frac{\langle\sigma\rangle_{a}+\langle\sigma\rangle_{b}}{2}\right)^{2}-\frac{\langle\sigma\rangle_{a}^{2}-\langle\sigma\rangle_{b}^{2}}{2}\Upsilon(\eta)+$
(4.17)
$\displaystyle+\left(\frac{\langle\sigma\rangle_{a}-\langle\sigma\rangle_{b}}{2}\right)^{2}\biggl{[}1-\frac{16}{\pi}\sqrt{\tau}\eta^{2}\textrm{e}^{-\eta^{2}}\biggr{]}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\mathcal{O}}}}(\tau^{3/2})\,,$

for $\xi_{\rm b}\ll y\ll R$. The derivation of (4.17) is supplied in Appendix
B.

The expression for the parallel correlation function given by (4.17) is one of
the most important results of this paper. From the result (4.17) it is
immediate to appreciate how phase separation on the half plane generates long-
range correlations. The term proportional to $\sqrt{\tau}$ is the signature of
the long-range character of interfacial correlations. Quite interestingly, the
term $\sqrt{\tau}$ is multiplied by a function which is proportional to the
passage probability, $\eta^{2}\textrm{e}^{-\eta^{2}}$. The entropic factor
$\eta^{2}$ suppresses the correlations in the closeness of the wall, while the
exponential factor suppresses interfacial correlations in the bulk. The
maximum effect is thus achieved when the passage probability reaches its
maximum value. The latter is reached for $\eta=1$, corresponding to a distance
$x=\sqrt{R\xi_{\rm b}}$ from the wall. This feature – which emerges from an
exact field-theoretic calculation – could have been guessed from the very
beginning once the Brownian excursion character of the interface is
established by the calculation of the one-point correlation function.

We conclude this section by observing how the spin-spin correlation function
given by (4.10) coincides with the following expression obtained within the
probabilistic interpretation