# OPE coefficients and the mass-gap from the integrable scattering description
of 2D CFT’s

Zoltan Bajnoka, Romuald A. Janikb

a Wigner Research Centre for Physics
H-1525 Budapest 114, P.O.B. 49, Hungary
b Institute of Theoretical Physics
Jagiellonian University
ul. Łojasiewicza 11, 30-348 Kraków, Poland e-mail<EMAIL_ADDRESS>mail<EMAIL_ADDRESS>
###### Abstract

Many two-dimensional conformal field theories have an alternative integrable
scattering description, which reproduces their spectrum of conformal weights.
Taking as an example the case of the Lee-Yang nonunitary CFT and the 3-state
Potts minimal model, we derive formulas, in terms of their integrable
description, for the OPE coefficients of a certain specific primary operator
and two identical but otherwise essentially arbitrary operators. As a side
result we also obtain a novel formula for the mass-gap relation for the
integrable massive deformation of the CFT. These results are obtained through
expressing the first nontrivial coefficient in the UV expansion of the energy
in terms of the integrable CFT data, i.e. the kink and anti-kink TBA
solutions.

##  1 Introduction

A conformal field theory is completely specified through the knowledge of the
spectrum of its conformal dimensions and of the OPE coefficients. In two
dimensions, due to the Virasoro algebra we have a thorough knowledge of both
ingredients for a wide range of nontrivial theories [1]. Thus the 2D CFTs form
a very interesting arena of exactly solvable quantum field theories which may
be used as a theoretical laboratory for exploring various issues.

In higher number of dimensions, conformal symmetry is not powerful enough,
although constraints derived from numerical bootstrap yield quite nontrivial
information (see [2] for a review). In recent years, remarkable progress has
been made in solving the four-dimensional $\mathcal{N}=4$ SYM theory due to
its properties of integrability [3]. This theory is of the utmost interest due
to its equivalence with string theory in $AdS_{5}\times S^{5}$ [4], and indeed
the progress in integrability involved insights coming from both sides of the
AdS/CFT correspondence. Currently the spectral problem for the $\mathcal{N}=4$
SYM has been essentially solved through the so-called Quantum Spectral Curve
[5, 6], a significant development on top of the Y-system formulation [7]. Huge
progress has also been done in the outstanding problem of finding the OPE
coefficients within the hexagon framework [8], although there, gluing and
wrapping poses significant technical difficulties and requires a “triple”
resummation.

Looking back at the developments in solving the $\mathcal{N}=4$ SYM spectral
problem, the community could leverage a huge body of knowledge on finding the
energy spectrum of two dimensional integrable relativistic quantum field
theories on a cylinder. This was far from trivial, as the $AdS_{5}\times
S^{5}$ worldsheet string theory has very peculiar idiosyncrasies, but the
general concepts and structures from the known relativistic setting could
serve as guiding principles. In the case of OPE coefficients, however, we
basically lack an analogous general framework, and the methods had to be
developed from scratch.

The goal of the present paper is to explore how OPE coefficients are encoded
in an _integrable_ scattering description of 2D CFTs, which is often also
given by a Y-system. On the one hand, the hope is that ultimately we may
uncover some interesting way that OPE coefficients are enmeshed within the
integrability structures of the CFT which may be generalizable to the
$\mathcal{N}=4$ SYM case. On the other hand, even if this does not happen, we
believe that the problem is of interest for its own sake.

A further, more technical, motivation for considering this problem in 2D CFTs
is that the integrable description of those theories neccessarily requires a
resummation of wrapping corrections to all orders. Thus the understanding of
potential formulas for OPE coefficients in such a regime is very interesting.

Since we are motivated by the possible generalization to $\mathcal{N}=4$ SYM,
we would like to formulate everything _only_ in terms of the integrable
description of the 2D CFT (i.e. the theory which we are studying) and _not_ in
terms of its integrable massive deformation, which is much more familiar.

In this respect, an important result of our paper is a formula for the first
nontrivial term in the UV expansion of the ground state energy expressed
completely in terms of the integrable CFT data i.e. the so-called kink and
anti-kink TBA solutions.

In this paper we restrict ourselves to OPE coefficients involving the
deformation operator and two identical operators. We study the non-unitary
Lee-Yang CFT as well as the 3-state Potts minimal model CFT. In addition, we
also derive similar formulas for the mass-gap relation coefficient $\kappa$,
which is of crucial interest when considering the massive integrable
deformations of the CFTs, and also appears in our formula for the OPE
coefficients.

The plan of this paper is as follows. In section 2, we recall the integrable
description of some 2D CFTs and formulate our key question. We discuss their
relation to the corresponding integrable massive deformations and recall the
relation between the OPE coefficient and the UV expansion of the energy. In
section 3 we derive our formula for the Lee-Yang CFT (with a generalization to
the 3-state Potts model given in Appendix E). We also present an extension to
excited states and illustrate it on the $SL(2)$ invariant vacuum and the mass-
coupling relation coefficient. In section 4, we look at the same problem from
the perspective of an auxillary linear dressing equation and Smirnov’s
formulation. The goal is here again to express the results purely in terms of
the integrable description of the CFT. We close the paper with a discussion
and a number of appendices containing technical details.

##  2 Preliminaries

Let us take an Euclidean CFT on the cylinder of circumference ${\cal R}$.
Dimensionful quantities on the cylinder can be written in terms of
dimensionless quantities on the plane. Energies depend on the volume as

$E_{\mathcal{O}}^{\mathrm{CFT}}({\cal R})=\frac{2\pi}{{\cal
R}}\left[h_{\mathcal{O}}+\bar{h}_{\mathcal{O}}-\frac{c}{12}\right]$ (2.1)

where $c$ is the central charge and $h_{\mathcal{O}}$ and
$\bar{h}_{\mathcal{O}}$ are the left/right conformal dimensions of the primary
state corresponding to the operator $\mathcal{O}$. Expectation values can be
expressed in terms of CFT 3-point functions as

$\langle{\cal O}_{1}|{\cal O}|{\cal O}_{2}\rangle=\left(\frac{2\pi}{{\cal
R}}\right)^{h_{\cal O}+{\bar{h}}_{\mathcal{O}}}C_{{\cal O}_{1}{\cal O}{\cal
O}_{2}}$ (2.2)

We would like to describe the quantities, $h,\bar{h}$ and $C$ in terms of an
integrable formulation.

###  2.1 Integrable description of 2D CFT’s – the Lee-Yang model

Let us illustrate how the integrable formulation of 2D CFT’s works for the
spectrum in the simplest setting of the Lee-Yang nonunitary CFT with
$c=-\frac{22}{5}$. In this nonunitary CFT, the ground state on a cylinder
corresponds to the primary state with conformal weights $(-1/5,-1/5)$. The
right- and left- conformal weights can be extracted from separate Y-systems
corresponding to the right-moving and left-moving sectors111We use $s$ for the
rapidity in the CFT description with no volume dependence. We will use
$\theta$ for the rapidities in the massive integrable deformation of the CFT.
The relations between these parameters depend on the sector. In the right
moving sector $\theta=s-\log\frac{r}{2}$, while in the left moving sector
$\theta=s+\log\frac{r}{2}$. [9, 10]

$Y_{R,L}(s+i\pi/3)Y_{R,L}(s-i\pi/3)=1+Y_{R,L}(s)$ (2.3)

with the asymptotic conditions

$Y_{R}(s)\sim_{s\to+\infty}\exp(e^{s})\quad\quad\quad\quad
Y_{L}(s)\sim_{s\to-\infty}\exp(e^{-s})$ (2.4)

Writing $Y_{R,L}(s)=e^{\varepsilon_{R,L}(s)}$ this can be translated into TBA
equations for the kink/anti-kink:

$\varepsilon_{R,L}(s)=e^{\pm
s}-\phi\star\log\left(1+e^{-\varepsilon_{R,L}}\right)$ (2.5)

with the upper sign corresponding to the kink (right moving sector), see Fig.
1, and the lower to the antikink (left moving sector). Note that there is no
volume dependence in (2.5), in accordance with the conformal symmetry of the
theory. Modifying the exponential source terms like $e^{\pm
s}\to\frac{r}{2}e^{\pm s}$ can be always reabsorbed by a translation in $s$
and would give the trivial $1/{\cal R}$ volume dependence of the energy levels
on the cylinder.

Figure 1: $Y_{R}^{-1}(s)$ as the function of $s$.

The dimensionless energy $E_{\mathcal{O}}^{\mathrm{CFT}}({\cal R}){\cal R}$
containing the conformal dimensions of the ground state together with the
central charge is given by the sum of the contributions of the two sectors

$E_{\mathcal{O}}^{\mathrm{CFT}}({\cal R}){\cal
R}=2\pi\left(h_{0}+\bar{h}_{0}-\frac{c}{12}\right)=E_{R}+E_{L}$ (2.6)

where

$E_{R}=-\int_{-\infty}^{\infty}\frac{ds}{2\pi}e^{s}\log\left(1+e^{-\varepsilon_{R}}\right)\quad\quad
E_{L}=-\int_{-\infty}^{\infty}\frac{ds}{2\pi}e^{-s}\log\left(1+e^{-\varepsilon_{L}}\right)$
(2.7)

The outcome of this computation, which can done analytically using the so-
called dilogarithm trick, of course reproduces the well known Lee-Yang answer
[11]

$E_{R}+E_{L}=-\frac{2\pi}{30}$ (2.8)

In the following we will need some more detailed properties of the CFT
solutions $Y_{R,L}(s)$ shown on Figure 1. The nonzero constant asymptotic
values of the two solutions is given by a constant solution of the Y-system
(2.3), which for the ground state, corresponding to the
$(h_{0},\bar{h}_{0})=(-1/5,-1/5)$ primary, is

$Y_{0}=\frac{1}{2}\left(1+\sqrt{5}\right)$ (2.9)

The precise way of how the CFT kink/antikink approach this constant value will
be of crucial importance for our further considerations. There is a systematic
expansion around the asymptotic value in powers of $e^{\pm(1-h)s}$, where
$h=-1/5$:

$Y_{R,L}^{-1}(s)Y_{0}=1-\tilde{c}\,\underbrace{e^{\pm(1-h)s}}_{e^{\pm\frac{6}{5}s}}+\ldots\quad\quad\text{as}\quad
s\to\mp\infty$ (2.10)

where the upper signs correspond to the right-moving (kink-like) TBA solution.
The expansion coefficients, in particular $\tilde{c}$ are related to the non-
local integrals of motions. In general, the coefficient $\tilde{c}$ can be
determined numerically, but for the Lee-Yang CFT there exists an analytical
expression which follows from the formulas derived222Note, however, that even
in [9] this coefficient has not been determined purely from TBA
considerations. in [9].

$\tilde{c}=\pi^{-\frac{11}{10}}\left(\sqrt{5}-5\right)\Gamma\left(-\frac{4}{5}\right)\Gamma\left(\frac{11}{10}\right)\left(\frac{\Gamma\left(\frac{2}{3}\right)}{\Gamma\left(\frac{1}{6}\right)}\right)^{\frac{6}{5}}=0.785438261\dots$
(2.11)

The other primary field in the Lee-Yang CFT is the $SL(2)$ invariant vacuum
with conformal weights $(h_{1},\bar{h}_{1})=(0,0)$. In the integrable
description, it is given again by solutions of the same Y-system (2.3)-(2.4)
but with different analyticity conditions, which can be equivalently
formulated either as a zero of $Y_{R,L}$ on the real axis at a certain
$s=\alpha$ [9]

$Y_{R,L}(\alpha)=0$ (2.12)

or as a pair of “singularities” at $s=\alpha\pm i\pi/3$ [12]

$Y_{R,L}\left(\alpha\pm i\frac{\pi}{3}\right)=-1$ (2.13)

The asymptotic value of the kink/anti-kink solutions is now given by the
second solution of the constant Y-system

$Y_{0}=\frac{1}{2}\left(1-\sqrt{5}\right)$ (2.14)

and the approach to this asymptotic value has a different exponential scaling

$Y_{R,L}^{-1}(s)Y_{0}=1-\tilde{c}^{\prime}\,e^{\pm\frac{12}{5}s}+\ldots\quad\quad\text{as}\quad
s\to\mp\infty$ (2.15)

In fact, the coefficient $\tilde{c}^{\prime}$ is related to the _second_
nonlocal conserved charge, as the first nonlocal conserved charge in this case
vanishes. This different exponential scaling will be important for our
considerations. Let us note that the specific power $\gamma$ of the exponent
in

$Y_{R}^{-1}Y_{0}=1-\tilde{c}\,e^{\gamma s}+\ldots$ (2.16)

can be determined by substitution into the Y-system (2.3) and depends on the
asymptotic value $Y_{0}$. We get the relation

$\cos\frac{\pi}{3}\gamma=(2Y_{0})^{-1}$ (2.17)

which reproduces $\gamma=\frac{6}{5}$ for the $(-1/5,-1/5)$ primary state and
$\gamma=\frac{12}{5}$ for the $(0,0)$ $SL(2)$ invariant vacuum.

###  2.2 The key question

The key question that we would like to investigate is to find how to express
the remaining fundamental CFT quantities – the OPE coefficients directly in
terms of the integrable data defining the relevant states like $Y_{R,L}$ given
above.

In this paper we would like to concentrate on the simplest case of symmetric
OPE coefficients of the form

$C_{\mathcal{O}\Phi\mathcal{O}}$ (2.18)

where $\Phi$ is a CFT operator which generates a massive integrable
deformation with a massive TBA description. The OPE coefficients are also
related to 1-point functions on a cylinder (2.2). In the specific case of
$C_{\mathcal{O}\Phi\mathcal{O}}$, these expectation values can in turn be
related to the expansion of the energy of the state $\mathcal{O}$ at small
volume in the massive integrable deformation of the CFT by the operator
$\Phi$, which we review below.

###  2.3 Massive integrable deformations

Consider perturbing the CFT with a spinless relevant primary field of
dimension $(h,\bar{h}=h)$333Operators normalized canonically in the CFT:
$\Phi(z,\bar{z})|\Phi(0,0)=z^{-2h}{\bar{z}}^{-2\bar{h}}+\dots$.:

$S=S_{\mathrm{CFT}}+\lambda\int_{-\infty}^{\infty}dy\int_{0}^{\cal
R}dx\,\Phi(x,y)$ (2.19)

such a way that the theory remains integrable, i.e. there remains infinitely
many higher spin conserved charges. For generic minimal models there are many
inequivalent ways to do this, however for the Lee-Yang model the only choice
is $h=-\frac{1}{5}$.

####  2.3.1 UV energy expansion and the OPE coefficient

The change in the energy levels of the massive deformation of the CFT can be
calculated perturbatively

$E_{\mathcal{O}}({\cal R})=E_{\mathcal{O}}^{\mathrm{CFT}}({\cal
R})+\frac{2\pi}{{\cal
R}}\left[\sum_{n=1}^{\infty}c_{n}\lambda^{n}\left(\frac{{\cal
R}}{2\pi}\right)^{n(2-2h)}\right]$ (2.20)

where $c_{1}$ is related to the expectation value of the operator as

$c_{1}=2\pi C_{\mathcal{O}\Phi\mathcal{O}}$ (2.21)

while higher $c_{n}$-s are related to integrated multipont correlation
functions.

For the $SL(2)$ invariant CFT vacuum state, $c_{1}$ vanishes and the leading
correction comes from integrating the universal two-point function, which
results in a model-independent expression:

$c_{2}(h)=2\pi\int_{|z|<1}d^{2}z(z\bar{z})^{h-1}\langle
0|\Phi(1,1)\Phi(z,\bar{z})|0\rangle=\pi^{2}\frac{\Gamma(h)^{2}\Gamma(1-2h)}{\Gamma(1-h)^{2}\Gamma(2h)}$
(2.22)

For the integrable deformation of the non-unitary Lee-Yang CFT discussed above
(the Scaling Lee-Yang Model), the resulting expansion takes the following
forms for the states corresponding to the two primary fields of the CFT.

The lowest energy state corresponds to $\Phi$ and the leading energy
correction comes from $c_{1}=2\pi C_{\Phi\Phi\Phi}$ in the form

$E_{\Phi}({\cal R})=\frac{2\pi}{{\cal R}}\left(-\frac{1}{30}+2\pi\lambda
C_{\Phi\Phi\Phi}\left(\frac{{\cal
R}}{2\pi}\right)^{\frac{12}{5}}+\dots\right)$ (2.23)

The first excited state at small volumes corresponds to the $SL(2)$ invariant
CFT state, whose leading correction goes as

$E_{I}({\cal R})=\frac{2\pi}{{\cal
R}}\left(\frac{11}{30}+\lambda^{2}c_{2}(-1/5)\left(\frac{{\cal
R}}{2\pi}\right)^{\frac{24}{5}}+\dots\right)$ (2.24)

The expansion of the energy levels goes in powers of the dimensionful coupling
constant $\lambda$, which sets the scale in the massive scattering theory.

####  2.3.2 UV energy expansion from TBA

As the massive deformation of the CFT was chosen to be integrable, the energy
of the states can also be computed from the Thermodynamic Bethe Ansatz (TBA),
which we review below. We also identify the precise way to extract the OPE
coefficient of interest from the TBA expansion, taking into account the mass-
coupling relation.

When the perturbation is integrable, the scattering matrix, which connects
infinite volume asymptotic states, factorizes into two-particle scatterings.
In the simplest case of one single particle type of mass $m$, the two-particle
S-matrix is a simple function of the rapidity differences $S(\theta)$, which
satisfies unitarity $S(\theta)S(-\theta)=1$ and crossing symmetry
$S(i\pi-\theta)=S(\theta)$. In the Scaling Lee-Yang model it takes the form
[13]

$S(\theta)=\frac{\sinh\theta+i\sin(\pi/3)}{\sinh\theta-i\sin(\pi/3)}$ (2.25)

The pole at $\theta=2\pi i/3$ signals a boundstate, which is the particle
itself. This S-matrix can be used to calculate the finite volume
groundstate444For operators $\mathcal{O}$ corresponding to excited states, one
has to add additional terms as described below. energy [11]

$E^{TBA}_{\mathcal{O}}(r)=-m\int\frac{d\theta}{2\pi}\cosh\theta\,\log(1+e^{-\varepsilon(\theta)})$
(2.26)

where the pseudo-energy satisfies the TBA equation

$\varepsilon(\theta)=r\cosh\theta-\int\frac{d\theta^{\prime}}{2\pi}\phi(\theta-\theta^{\prime})\log(1+e^{-\varepsilon(\theta^{\prime})})=r\cosh\theta-\phi\star\log(1+e^{-\varepsilon})$
(2.27)

with $\phi(\theta)$ being the logarithmic derivative of the scattering matrix
$\phi(\theta)=-i\partial_{\theta}\log S(\theta)$ and $r=m{\cal R}$ is the
dimensionless volume. This formula can be extended for excited states by
analytical continuation [12, 14], which results in source terms in the TBA
equation. These can be constant twists, or sums of the logarithms of
S-matrices. The same equations could be derived from the integrable structure
via functional relations [9] or from the continuum limit of an integrable
lattice regularization [10].

Let us make a few remarks here. First, the TBA energy is written in terms of
the mass of the scattering particles, while the perturbed CFT energy is
expressed in terms of the coupling constant $\lambda$. On dimensional grounds
the mass and the coupling can be related as

$\lambda=\kappa m^{2-2h}$ (2.28)

where $\kappa$ is a very crucial model-dependent constant555In multiparameter
deformations it can be a non-trivial function of the dimensionless coupling
[15, 16]., which connects the UV and IR descriptions. It is in fact quite
nontrivial – in the Lee-Yang theory its value is [17]

$\kappa=\frac{5^{\frac{3}{4}}\left(\Gamma\left(\frac{2}{3}\right)\Gamma\left(\frac{5}{6}\right)\right)^{\frac{12}{5}}}{8\
2^{\frac{4}{5}}\pi^{\frac{6}{5}}\Gamma\left(\frac{3}{5}\right)\Gamma\left(\frac{4}{5}\right)}$
(2.29)

We will show shortly how it can be extracted from our approach. Formula (2.28)
is also called the mass-gap relation as it expresses the physical mass in
terms of the Lagrangian coupling.

Second, the TBA groundstate energy $E^{TBA}_{\mathcal{O}}(r)$ and the
perturbed CFT energy $E_{\mathcal{O}}({\cal R})$ are given in different
schemes: $E^{TBA}_{\mathcal{O}}(r)$ is chosen to vanish for large volumes,
while $E_{\mathcal{O}}({\cal R})$ has a nonzero bulk energy density
$\epsilon_{B}$. They are related to each other as

$E_{\mathcal{O}}({\cal R})=\epsilon_{B}{\cal
R}+E^{TBA}_{\mathcal{O}}(r)\quad\quad\quad\quad\epsilon_{B}=-\frac{m^{2}}{4\sqrt{3}}\quad\text{(for
SLYM)}$ (2.30)

A key quantity of interest for the present paper is the first nontrivial term
$\epsilon_{UV}$ in the UV expansion of the TBA energy

${\cal
R}E^{TBA}_{\mathcal{O}}(r)=2\pi\left(h_{\mathcal{O}}+\bar{h}_{\mathcal{O}}-\frac{c}{12}\right)-\frac{\epsilon_{B}}{m^{2}}r^{2}+\epsilon_{UV}r^{\alpha}+\ldots$
(2.31)

In the TBA scheme, the bulk energy appears in the small volume expansion of
the energy with an opposite sign and called, for this reason, the anti-bulk
energy.

Depending on the TBA state in question, the numerical coefficient
$\epsilon_{UV}$ can be related either to the OPE coefficient
$C_{\mathcal{O}\Phi\mathcal{O}}$ through

$C_{\mathcal{O}\Phi\mathcal{O}}=\frac{1}{\kappa(2\pi)^{2h}}\cdot\epsilon_{UV}$
(2.32)

when $\alpha=2-2h$ in (2.31), or to the mass-coupling relation $\kappa$:

$\kappa^{2}=\frac{(2\pi)^{2(2-2h)}}{2\pi c_{2}(h)}\cdot\epsilon_{UV}$ (2.33)

when the state is the $SL(2)$ invariant vacuum $\mathcal{O}\equiv 1$ (in this
case $\alpha=2(2-2h)$). We will provide a “master formula” for $\epsilon_{UV}$
purely in terms of the CFT integrable data. Consequently, this will provide
formulas for the above OPE coefficient and the mass-coupling relation
$\kappa$.

##  3 A formula for the UV energy expansion, the OPE coefficient and the mass-
gap

As we reviewed in the previous section, the OPE coefficient of the deforming
operator and the mass-coupling relation can be read off from the first
nontrivial coefficient $\epsilon_{UV}$ of the small volume expansion of the
energy. In this section we will derive our main result, a formula for this
coefficient expressed purely in terms of the integrable CFT data described in
section 2.1. The _analytical_ determination of this coefficient directly from
TBA remains in fact an outstanding problem. We hope that our formula may be a
useful step in this direction, but the possibility of an analytical evaluation
is not really that important for our purposes.

Here we perform our calculations for the simplest case of the Scaling Lee-Yang
model. In Appendix E, we show that a similar formula holds in the two-
component TBA of the (scaling) 3-state Potts model666In the 3-state Potts
model we consider the nontrivial $\sigma$ state instead of the vacuum, which
thus does not trivially reduce to the Scaling Lee-Yang model TBA. It is also
this state which is interesting from the point of view of OPE coefficients..

In the following, instead of considering the TBA $\varepsilon(\theta)$, it is
more convenient to employ $Y(\theta)=e^{\varepsilon(\theta)}$ (or more
precisely $Y^{-1}(\theta)=e^{-\varepsilon(\theta)}$, which has better
behaviour on the real line). The TBA equation of the massive integrable
deformation has the form

$\log
Y=\frac{r}{2}e^{\theta}+\frac{r}{2}e^{-\theta}-\phi\star\log\left(1+Y^{-1}\right)$
(3.1)

The asymptotic solution for small $r$ can be written as

$Y_{\mathrm{as}}=Y_{K}Y_{A}Y_{0}^{-1}$ (3.2)

where $Y_{K,A}$ are the standard kink and antikink solutions defined by

$\log
Y_{K,A}=\frac{r}{2}e^{\pm\theta}-\phi\star\log\left(1+Y_{K,A}^{-1}\right)$
(3.3)

Both of them can be expressed in terms of the volume independent CFT solutions
(2.5) as

$Y_{K}(\theta)\equiv
Y_{R}\left(\theta+\log\frac{r}{2}\right)\quad\quad\quad\quad
Y_{A}(\theta)\equiv Y_{L}\left(\theta-\log\frac{r}{2}\right)$ (3.4)

The kink and antikink thus inherit the asymptotics (2.10). As we will see
shortly, it is exactly this exponential tail of the kink in the region of the
antikink (and vice versa) which generates a correction of order
$r^{\frac{12}{5}}$ which is responsible for the energy correction (2.32) that
we are after.

Figure 2: $Y^{-1}(\theta)$ as the function of $\theta$ for dimensionless
volume $r=0.01$ in blue. Dashed green shows the kink solutions. The difference
between $Y^{-1}-Y^{-1}_{\mathrm{as}}=\delta Y^{-1}$ is of order $10^{-7}$ and
is not visible. Thus for demonstration, we draw $r^{-\frac{12}{5}}\delta
Y^{-1}$ with yellow.

###  3.1 Subleading TBA solution

We would now like to derive an equation for the first subleading correction to
$Y_{\mathrm{as}}^{-1}$ which is of order $r^{2-2h}$, $r^{\frac{12}{5}}$ in the
Lee-Yang model. Inserting

$Y^{-1}=Y_{\mathrm{as}}^{-1}+\delta Y^{-1}$ (3.5)

into the TBA equation we obtain a linear integral equation for $\delta
Y^{-1}$:

$-Y_{\mathrm{as}}\delta
Y^{-1}=\underbrace{\frac{r}{2}e^{\theta}+\frac{r}{2}e^{-\theta}-\log
Y_{\mathrm{as}}-\phi\star\log(1+Y_{\mathrm{as}}^{-1})}_{\mathrm{source}}-\phi\star\frac{\delta
Y^{-1}}{1+Y_{\mathrm{as}}^{-1}}$ (3.6)

Using the definition (3.2) and the kink/antikink equations (3.3) we can
rewrite the source term in the following form

$\displaystyle\mathrm{source}$ $\displaystyle=$
$\displaystyle-\phi\star\left[\log(1+Y_{\mathrm{as}}^{-1})-\log(1+Y_{K}^{-1})-\log(1+Y_{A}^{-1})+\log
Y_{0}\right]$ (3.7) $\displaystyle\equiv$ $\displaystyle-\phi\star G$

We need now to extract the leading small $r$ behaviour of the source $G$,
focusing separately on the two asymptotic (kink/antikink) regions of $\theta$.
Moreover, we aim to rewrite the source directly in terms of the volume-
independent CFT solutions $Y_{R,L}$. We want, therefore, to calculate the
following quantities which are concentrated in the right moving and left
moving regions

$G_{R,L}(s)=\lim_{r\to 0}r^{2h-2}G\left(s\mp\log\frac{r}{2}\right)$ (3.8)

Let us consider $G_{R}(s)$ i.e. the source in the kink asymptotic region. In
this region it is convenient to rewrite $G$ as

$G=\log\frac{1+Y_{K}^{-1}Y_{A}^{-1}Y_{0}}{1+Y_{K}^{-1}}-\log\frac{1+Y_{0}^{-1}Y_{A}^{-1}Y_{0}}{1+Y_{0}^{-1}}$
(3.9)

Now using (3.4) and (2.10) we see the appearance of the $r^{2-2h}$ scaling

$Y_{A}^{-1}Y_{0}-1\sim-\tilde{c}e^{(h-1)\left(s-2\log\frac{r}{2}\right)}=-\tilde{c}\left(\frac{r}{2}\right)^{2-2h}e^{(h-1)s}$
(3.10)

We can simply denoted the tail of the anti-kink solution in the kink region as

$y_{L}(s)=\lim_{r\to
0}r^{2h-2}(Y_{A}^{-1}(s-\log\frac{r}{2})Y_{0}-1)=-\tilde{c}\,2^{2h-2}e^{(h-1)s}$
(3.11)

such that the rescaled source in the kink region takes the form

$G_{R}=\left(\frac{1}{1+Y_{R}}-\frac{1}{1+Y_{0}}\right)y_{L}$ (3.12)

This source behaves nicely in the $s\to+\infty$ limit, but seems at first
glance dangerous for large negative $s$ (in the above formula $h=-1/5$).
Fortunately, there the term in parenthesis vanishes exponentially with just
the appropriate asymptotics to cancel this growth and the $s\to-\infty$ limit
is a constant:

$\lim_{s\to-\infty}G_{R}(s)=\frac{Y_{0}}{(1+Y_{0})^{2}}\,\tilde{c}^{2}\,2^{2-2h}$
(3.13)

The behaviour of $G_{L}(s)$ is clearly analogous.

We can now write the linearized equation for the leading $\delta Y^{-1}$
correction separately in the kink (right-moving) sector and antikink (left-
moving) sector. Extracting further the overall $r^{2-2h}$ scaling as above we
obtain

$Y_{R}\,\delta Y_{R}^{-1}=\phi\star\left[G_{R}+\frac{\delta
Y_{R}^{-1}}{1+Y_{R}^{-1}}\right]$ (3.14)

where the subscript on $\delta Y_{R}^{-1}$ denotes the appropriate sector. All
quantities in (3.14) are expressed in terms of the CFT rapidity $s$.

It is important to emphasize, that the source term in the kink sector $G_{R}$
involves not only the right-moving kink solution, but includes also the
asymptotic tail of the left-moving anti-kink solution. Hence it involves in a
nontrivial way the interplay between both sectors of the theory.

###  3.2 Evaluation of the UV energy correction coefficient $\epsilon_{UV}$

In order to compute the energy ${\cal R}E^{TBA}_{\mathcal{O}}(r)$, we need to
evaluate the integral

$-r\int\frac{d\theta}{2\pi}\cosh\theta\log\left(1+Y_{\mathrm{as}}^{-1}+\delta
Y^{-1}\right)=\underbrace{E_{R}+E_{L}}_{2\pi\left(h_{0}+\bar{h}_{0}-\frac{c}{12}\right)}-\frac{\epsilon_{B}}{m^{2}}r^{2}+\epsilon_{UV}r^{2-2h}+\dots$
(3.15)

where in the Lee-Yang model $h_{0}=h=-\frac{1}{5}$ for the ground state. We
are interested in the first nontrivial $r\to 0$ contribution of order
$r^{2-2h}$. Expanding the above integral we get two contributions

$-r\int\frac{d\theta}{2\pi}\cosh\theta\log\left(1+Y_{\mathrm{as}}^{-1}\right)-r\int\frac{d\theta}{2\pi}\cosh\theta\frac{\delta
Y^{-1}}{1+Y_{\mathrm{as}}^{-1}}$ (3.16)

The second term will already contribute at order $r^{2-2h}$. To see this we
split the $\cosh\theta$ into exponentials and integrate $e^{\theta}$ in the
kink region (and $e^{-\theta}$ in the anti-kink region):

$-\int\frac{d\theta}{2\pi}\frac{r}{2}e^{\theta}\frac{\delta
Y^{-1}}{1+Y_{\mathrm{as}}^{-1}}\quad\longrightarrow\quad-\int\frac{ds}{2\pi}e^{s}\frac{\delta
Y_{R}^{-1}\cdot r^{2-2h}}{1+Y_{R}^{-1}}$ (3.17)

Recall that we factored out $r^{2-2h}$ when defining $\delta Y_{R}^{-1}$.

We need to be careful as a contribution of the same order will also arise from
the first term in (3.16), where the tail of the anti-kink appearing in
$Y_{\mathrm{as}}$ in the kink region will lead to this scaling similarly to
(3.10). In order to isolate this contribution, it is convenient to use the
definition of the source $G$ from (3.7) and rewrite

$\log\left(1+Y_{\mathrm{as}}^{-1}\right)=\log(1+Y_{K}^{-1})+\log(1+Y_{A}^{-1})-\log(1+Y_{0}^{-1})+G$
(3.18)

In order to isolate the $r$ scaling, we have to again split the $r\cosh\theta$
into exponentials and take into account the resulting powers of $r$ arising
from moving to the kink or anti-kink regions. E.g. plugging the first term in
(3.18) into the energy formula with $\frac{r}{2}e^{\theta}$ and evaluating it
in the kink region

$-\int\frac{d\theta}{2\pi}\frac{r}{2}e^{\theta}\log\left(1+Y_{\mathrm{K}}^{-1}\right)=-\int\frac{ds}{2\pi}e^{s}\log\left(1+Y_{\mathrm{R}}^{-1}\right)$
(3.19)

we obtain the right conformal energy, which can be calculated using the
dilogarithm trick.

In the second and third term, again with the exponential
$\frac{r}{2}e^{\theta}$, we can make an integration by parts, together with a
shift in the integral into its antikink domain

$-\left(\frac{r}{2}\right)^{2}\int\frac{ds}{2\pi}e^{s}\frac{\partial\log
Y_{L}}{1+Y_{L}}$ (3.20)

This integral gives the anti bulk energy constant in (2.30), which can be
evaluated by inspecting the asymptotics of the derivative of the anti-kink
equation.

Finally, $G$ will give a contribution at the desired subleading order
$r^{2-2h}$, which we have to evaluate together with the contribution coming
from $\delta Y^{-1}$. In doing so we can again split the subleading correction
into the kink and antikink regions, which can be computed in a similar manner.
We will now focus on the kink regime. We thus obtain

$-\int\frac{ds}{2\pi}e^{s}\left[G_{R}+\frac{\delta
Y_{R}^{-1}}{1+Y_{R}^{-1}}\right]$ (3.21)

Note that the term in brackets is exactly the combination appearing in the
linearized equation for $\delta Y_{R}^{-1}$ in the kink region (3.14).
Fortunately, we do not need to find $\delta Y_{R}^{-1}$ explicitly, but we can
use an analog of the dilogarithm trick to evaluate (3.21) purely in terms of
the properties of the asymptotic CFT solutions $Y_{R,L}$. The details of this
computation are given in Appendix A and the resulting formula involves just
the UV limit CFT solutions $Y_{R,L}$ of section 2.1.

$-\int\frac{ds}{2\pi}G_{R}\cdot\partial\log Y_{R}$ (3.22)

The source term $G_{R}$ given by (3.12) involves the right moving solution
$Y_{R}$ together with the _exponential tail_ of the left moving one $Y_{L}$.
We will find that this overall structure is in fact quite generic. Adding an
analogous contribution from the antikink side of the energy integral, we get
our final formula for the key energy correction coefficient $\epsilon_{UV}$:

$\epsilon_{UV}=-\int\frac{ds}{2\pi}G_{R}\cdot\partial\log
Y_{R}-\int\frac{ds}{2\pi}G_{L}\cdot\partial\log Y_{L}$ (3.23)

###  3.3 Formula for the OPE coefficient

Using the relation (2.32) from section 2, we obtain the formula for the OPE
coefficient

$C_{\Phi\Phi\Phi}=-\frac{(2\pi)^{\frac{2}{5}}}{\kappa}\cdot\left[\int\frac{ds}{2\pi}G_{R}\cdot\partial\log
Y_{R}+\int\frac{ds}{2\pi}G_{L}\cdot\partial\log Y_{L}\right]$ (3.24)

Although we cannot evaluate the above formula analytically, we can compute it
very precisely numerically, which is close to the exact answer

$C_{\Phi\Phi\Phi}=1.91131...=\sqrt{\frac{2}{1+\sqrt{5}}}\frac{\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{6}{5}\right)}{\Gamma\left(\frac{3}{5}\right)\Gamma\left(\frac{4}{5}\right)}$
(3.25)

In Appendix E we show that a direct analog of the above formulas also exists
for the 3-state Potts model, where the energy correction is related to the
$C_{\sigma\varepsilon\sigma}$ OPE coefficient.

Several comments are in order here. First, it is encouraging that the OPE
coefficient can be expressed purely in terms of the integrable CFT description
without the need to explicitly go away from the UV point using the massive
integral deformation. Indeed, the integrand in (3.24) involves quantities
defined directly through the Y-system of the CFT. Second, the expression
involves an interplay between the right-moving and left-moving sector through
the tail of the antikink appearing in $G_{R}$. Hence it is a much more
nontrivial quantity than the energy levels (conformal weights). Third, the
link of the energy formula with the OPE coefficient involves a nontrivial
normalization factor incorporating the mass-gap coefficient $\kappa$. We will
show in the next subsection, however, that this coefficient can also be
expressed in terms of the CFT integrable data, but for the $SL(2)$-invariant
vacuum state.

Finally, let us discuss the freedom of changing the operators in the OPE
coefficient. In the Lee-Yang CFT, an analogous computation could be done
starting with the TBA solution of the identity operator with
$(h_{1},\bar{h}_{1})=(0,0)$. Clearly

$C_{1\Phi 1}=0$ (3.26)

We would like now to understand how this vanishing is recovered in the
integrable framework. The difference lies in the asymptotics of the right- and
left-moving CFT TBA solutions. Apart from an inessential change of the
asymptotic value which is now equal to $Y_{1}=(1-\sqrt{5})/2$, the crucial
modification is the change in the exponential approach to the constant value.
Indeed, in contrast to (2.10) we have

$Y_{R,L}^{-1}(s)Y_{1}=1-\tilde{c}^{\prime}\,e^{\pm\frac{12}{5}s}+\ldots\quad\quad\text{as}\quad
s\to\mp\infty$ (3.27)

so the term with $e^{\pm\frac{6}{5}s}$ is absent, which is associated to the
vanishing of the first nonlocal conserved charge in the language of [9]. This
implies that due to the limit in (3.8), the source terms $G_{R,L}$ vanish and
we recover $C_{1\Phi 1}=0$.

Note, however, that if we extract instead $r^{2(2-2h)}=r^{\frac{24}{5}}$ from
the source term, the computation of the energy correction (3.23) goes through.
The result should not be interpreted as an OPE coefficient, but rather it
provides the mass-coupling relation coefficient $\kappa$. Let us now
investigate this state in detail.

###  3.4 The $SL(2)$ invariant excited state and the mass-coupling formula

The excited state TBA equation for a one-component system has extra source
terms of the form

$\log Y=r\cosh\theta+\sum_{i}\eta_{i}\log
S(\theta-\theta_{i})-\phi\star\log(1+Y^{-1})$ (3.28)

where the roots $\theta_{i}$ are located at the zeros of the logarithm

$Y(\theta_{i})=-1$ (3.29)

The energy also gets correction from the roots as

$E_{I}^{TBA}=-im\sum_{i}\eta_{i}\sinh\theta_{i}-m\int\frac{d\theta}{2\pi}\cosh\theta\log(1+Y^{-1})$
(3.30)

These extra terms can be interpreted by chosing an alternative integration
contour in the groundstate equations, which surrounds the $1+Y=0$
singularities. Moving back the contour to the real line and picking up the
residues result in the excited state formulas [12].

In the Lee-Yang model, a single particle can be described by a pair of complex
conjugated roots, $\theta_{1}^{*}=\theta_{2}$ and $\eta_{1}=1=-\eta_{2}$. See
[12, 9] for details. At large volumes the roots behave as
$\theta_{1}=\theta+i\frac{\pi}{6}+i\delta$, where $\theta$ is the rapidity of
the particle and $\delta$ is exponentially small. The $SL(2)$ invariant state
corresponds to a standing particle ($\theta=0$) with $\log
Y(\theta_{1})=i\pi$. When the volume is decreased, the root $\theta_{1}$ moves
to $i\frac{\pi}{3}$, where it splits into two, which stay on the
$i\frac{\pi}{3}$ line as
$\theta_{1}=\theta_{1}^{\pm}=i\frac{\pi}{3}\pm\alpha$, with
$\eta_{1}^{\pm}=\frac{1}{2}=-\eta_{2}^{\pm}$. In the deep UV $\alpha$ scales
as $\alpha=-\log\frac{r}{2}+O(1)$, i.e. one root is in the kink domain, while,
symmetrically, the other one is in the anti-kink domain. In Appendix B we
calculate the $\epsilon_{UV}$ coefficient in the small volume expansion of the
energy (3.30)

${\cal
R}E_{I}^{TBA}=2\pi\frac{11}{30}-\frac{\epsilon_{B}}{m^{2}}r^{2}+\epsilon_{UV}r^{\frac{24}{5}}+\dots$
(3.31)

The outcome of the calculation is that in addition to the usual integral term
we also have to take its residues

$\frac{1}{2}\epsilon_{UV}=-i\sum_{i}\eta_{i}y_{L}(s_{i})-\int\frac{ds}{2\pi}G_{R}\partial\log
Y_{R}$ (3.32)

where

$G_{R}(s)=\text{$\left\\{\frac{1}{1+Y_{R}(s)}-\frac{1}{1+Y_{0}}\right\\}$
}y_{L}\qquad;\quad y_{L}(s)=-\tilde{c}2^{-\frac{24}{5}}e^{-\frac{12}{5}s}$
(3.33)

The first term in (3.32) is just the tail of the anti-kink evaluated at the
roots $s_{i}$ up to a factor of $i$. The $Y_{R}$ satisfies the right-moving
kink equation with the source terms

$\log Y_{R}=e^{s}+\sum_{i}\eta_{i}\log
S(s-s_{i})-\phi\star\log(1+Y_{R}^{-1})\quad;\qquad Y_{R}(s_{i})=-1$ (3.34)

and $Y_{0}=\frac{1}{2}(1-\sqrt{5})$ is its asymptotic value. This case is,
however, somewhat subtle for numerical analysis as the the Y-functions develop
zeros on the real line at $\alpha=0.495773177$ (see [9]) which leads to a
singularity in the convolution term. In order to overcome this difficulty, one
can parametrize the $Y_{R}$ function as
$Y_{R}(s)=\tanh(\frac{3}{4}(s-\alpha))\,e^{\epsilon_{R}(s)}$ and solve the
modified equation for $\epsilon_{R}$. See [9] for details.

A numerical evaluation of (3.32) yields $\epsilon_{UV}=-0.000825156$, which
leads to the following result for the mass-gap coefficient from (2.33):

$\kappa^{2}=\frac{(2\pi)^{\frac{24}{5}}}{2\pi
c_{2}\left(-\frac{1}{5}\right)}\epsilon_{UV}=0.0094185..$ (3.35)

which agrees well with the exact answer

$\kappa^{2}=\left(\frac{5^{\frac{3}{4}}\left(\Gamma\left(\frac{2}{3}\right)\Gamma\left(\frac{5}{6}\right)\right)^{\frac{12}{5}}}{8\
2^{\frac{4}{5}}\pi^{\frac{6}{5}}\Gamma\left(\frac{3}{5}\right)\Gamma\left(\frac{4}{5}\right)}\right)^{2}=0.0094184...$
(3.36)

##  4 A Smirnov formula perspective – resumming
the LeClair-Mussardo series

Summarizing the preceding results, the modification of the two external
operators $\mathcal{O}$ in the OPE coefficient which correspond to the
particular TBA state is straightforward. It is much less clear how to change
the inserted operator $\Phi$ in

$C_{\mathcal{O}\Phi\mathcal{O}}$ (4.1)

from the one generating the massive integrable deformation. Indeed, as our
method involves primarily the computation of the energy correction, it
requires _by construction_ for $\Phi$ to be equal to the operator appearing in
the massive integrable deformation. But even examining _a-posteriori_ the
ingredients entering (3.24), we do not see a natural place for modifying the
formula to accommodate for changing777Of course in the Lee-Yang CFT we do not
have other operators, but the same kind of formula appears also in the 3-state
Potts model – see Appendix E. $\Phi$. In this section we will provide a
different perspective on the formula (3.24) which has the potential to address
this issue, although we do not see its resolution at the current stage.

Returning to the massive integrable deformation, one can rewrite the energy
correction related to the OPE coefficient as a 1-point function of the
deformation operator. This 1-point function could be obtained from the
LeClair-Mussardo series [18] which can be resummed, in this case, through a
linear “dressing” integral equation [19]. In this section we will show that
the resulting formulas at the appropriate order in the UV expansion can also
be evaluated directly in terms of the CFT integrable data $Y_{R,L}$.

A key motivation for adopting this route is that it has a close structural
similarity to Smirnov’s approach to ratios of 1-point functions through an
underlying fermionic construction [20, 21, 22, 23]. There, the modification of
the operator $\Phi$ in $C_{\mathcal{O}\Phi\mathcal{O}}$ can be essentially
implemented through a modification of the kernel appearing in the linear
“dressing” integral equation. We leave the investigation of this issue for
future work. In the present paper we will still concentrate just on the case
of the deforming operator which is universal and does not depend on any
underlying microscopic fermionic construction.

The quickest way to reformulate the energy correction in terms of an auxiliary
linear dressing equation does not pass through the LeClair-Mussardo formula,
but rather starts with rewriting the vacuum expectation value of the trace of
the stress tensor $\Theta=T_{00}+T_{11}$ into the combination888Here we used
the perturbed CFT normalization as we would to extract the CFT data.

$\frac{1}{m^{2}}\langle
0|\Theta|0\rangle_{r}=\partial_{r}E(r)+\frac{1}{r}E(r)+\frac{2\epsilon_{B}}{m^{2}}$
(4.2)

which captures the relevant term in the $r\to 0$ energy expansion. Here
$E(r)=E^{TBA}(r)/m$. The first term can be rewritten as

$\partial_{r}E(r)=-\partial_{r}\int\frac{d\theta}{2\pi}\cosh\theta\log(1+Y^{-1})=\int\frac{d\theta}{2\pi}\frac{e^{\theta}\partial_{r}\log
Y}{1+Y}$ (4.3)

where we also used the $\theta\to-\theta$ symmetry of the problem. The second
term can be rewritten using integration by parts as

$\frac{1}{r}E(r)=-\frac{1}{r}\int\frac{d\theta}{2\pi}\frac{e^{\theta}\partial_{\theta}\log
Y}{1+Y}$ (4.4)

Putting the two expressions together we get

$\partial_{r}E(r)+\frac{1}{r}E(r)=\int\frac{d\theta}{2\pi}\frac{e^{\theta}H(\theta)}{1+Y}$
(4.5)

with

$H(\theta)=\partial_{r}\log Y-\frac{1}{r}\partial_{\theta}\log Y$ (4.6)

The key property of $H(\theta)$ is that it solves a _linear_ integral equation
with the same kernel as the TBA equation:

$H(\theta)=e^{-\theta}+\int\frac{d\theta^{\prime}}{2\pi}\phi(\theta-\theta^{\prime})\frac{H(\theta^{\prime})}{1+Y(\theta^{\prime})}$
(4.7)

which follows by taking the $r$ and $\theta$ derivatives of the standard
massive TBA equation (2.27). Denoting the convolution together with the
LeClair-Mussardo measure factor $1/(1+Y)$ by $\circ$, we can rewrite it as

$H=e^{-\theta}+\phi\circ H$ (4.8)

The iterative solution for $H(\theta)$ indicates that for large volume999In
(4.8), the dependence on the volume sits implicitly in the $1/(1+Y)$ factor
appearing in the $\circ$ convolution. $H(\theta)$ is simply $e^{-\theta}$,
while for smaller volumes it is dressed up like

$H(\theta)=e^{-\theta}+e^{-\theta}\phi(\theta-\theta^{\prime})\circ
e^{-\theta^{\prime}}+\dots=\frac{1}{1-\phi(\theta-\theta^{\prime})\circ}e^{-\theta^{\prime}}\equiv(e^{-\theta})^{\mathrm{dr}}$
(4.9)

The expression (4.5) can then be rewritten as

$\partial_{r}E(r)+\frac{1}{r}E(r)=\int\frac{d\theta}{2\pi}\frac{e^{\theta}H(\theta)}{1+Y}\equiv
e^{\theta}\circ(e^{-\theta})^{\mathrm{dr}}\equiv\omega_{1,-1}$ (4.10)

where we used the notation
$\omega_{n,m}=e^{n\theta}\circ(e^{m\theta})^{\mathrm{dr}}$ of [22, 23] .
Plugging in the UV expansion of the TBA energy (2.31) we get the following UV
expansion of $\omega_{1,-1}$ and its link with $\epsilon_{UV}$:

$\omega_{1,-1}=-\frac{2\epsilon_{B}}{m^{2}}+\underbrace{\alpha\epsilon_{UV}}_{\omega_{UV}}r^{\alpha-2}+\dots$
(4.11)

The formulas (4.8) and (4.10) are indeed a TBA analog of the NLIE formula
given in [20]. The general approach of [20, 22, 10] to ratios of 1-point
functions involves exactly equations of the form (4.8), but with the kernel
deformed away from the TBA one. We provide further comments on this relation
in Appendix C.

Yet another interpretation of (4.10) with the explicit expansion (4.9), as
already indicated above, is that it represents the LeClair-Mussardo series of
the trace of the stress tensor as in [19].

It is actually useful to develop a method to expand directly $\omega_{1,-1}$
for small volumes and express the coefficient $\omega_{UV}$ purely in terms of
integrable data of the CFT. This calculation, which we present in Appendix D
can be generalized for other $\omega$’s, too. Here we summarize the outcome.
The small volume expansion of $\omega_{1,-1}$ in the Scaling Lee-Yang model
becomes

$\omega_{1,-1}=\frac{1}{2\sqrt{3}}+r^{\frac{2}{5}}\omega_{UV}+\dots$ (4.12)

In Appendix D, the quantity $\omega_{UV}$ was rewritten purely in terms of CFT
quantities as

$\omega_{UV}=\int_{-\infty}^{\infty}\frac{ds}{2\pi}g_{R}(s)f_{R}(s)$ (4.13)

where $g_{R}$ is the source term governing the deviation of $H$ from its anti-
kink counterpart $H_{A}$ as

$g_{R}(s)=\tilde{c}\frac{6}{5}\left(\frac{1}{2}\right)^{\frac{7}{5}}e^{-\frac{6}{5}s}\left[\frac{1}{1+Y_{R}(s)}-\frac{1}{1+Y_{0}}\right]+\dots$
(4.14)

This term actually contains the tail of the anti-kink $H_{A}$ in the kink
domain. It turns out to be simply related to the source $G_{R}$ (see (3.12))
by

$g_{R}(s)=-\frac{6}{5}\cdot 2\cdot G_{R}(s)$ (4.15)

The remaining ingredient of (4.13), $f_{R}$ is the dressed version of $e^{s}$
in the right domain

$f_{R}(s)=e^{s}+\frac{\phi(s-s^{\prime})}{1+Y_{R}(s^{\prime})}\star
f_{R}(s^{\prime})$ (4.16)

This equation is in fact solved by $f_{R}(s)=\partial\log Y_{R}(s)$, hence we
finally arrive at

$\omega_{UV}=\int_{-\infty}^{\infty}\frac{ds}{2\pi}g_{R}(s)\partial\log
Y_{R}(s)$ (4.17)

Comparing (4.11) with (4.12) and using (4.15), we recover our previous formula
(3.23).

##  5 Discussion and outlook

In this paper we initiated a novel approach in which the OPE coefficients of
2D CFTs can be expressed in terms of the integrable characteristics of the
CFT, namely in terms of their Y-functions. We focused on the diagonal matrix
elements of the primary operators, which define _integrable_ massive
perturbations. By expanding the energy levels at small volumes we managed to
go beyond the central charge and bulk energy constant terms. We derived a
formula for the leading nontrivial coefficient in the UV expansion from the
TBA approach. This is the first term in the perturbative expansion, which goes
in powers of the volume which contain the conformal dimension of the
perturbing operator. We expressed this coefficient in terms of the kink (and
anti-kink) function – the solution of the TBA equation in the CFT limit. An
essential ingredient of our formula is the tail of the kink function, i.e. the
way how it approaches its asymptotic value. The coefficient describing this
behaviour is related to the leading nonvanishing non-local charge [9].
Actually, what matters for our formula is how the anti-kink function behaves
in the kink domain and the kink solution itself. In this sense our formula for
the OPE coefficient involves crucially the interaction between the kink and
anti-kink domains, in contrast to the conformal weights for which the two
domains are clearly separated and independent.

Depending on the state for which we calculate the small volume expansion, our
formula can provide very valuable quantities. If the state is the $SL(2)$
invariant vacuum state, then our formula provides the mass-coupling (mass-gap)
relation coefficient. If we choose some other state then our formula can
describe CFT 3-point functions, i.e. OPE coefficients.

We first derived our formula for the ground-state in the Lee-Yang theory, then
we also extended it for the simplest excited state, which corresponds to the
$SL(2)$ invariant vacuum state with $h=\bar{h}=0$. These results together
determine the mass-coupling relation and the only non-trivial OPE coefficient
$C_{\Phi\Phi\Phi}$. We also generalized our approach to the 3-state Potts
minimal model CFT and its two-component TBA system [24]. We investigated both
the $SL(2)$ invariant vacuum and the primary field
$\sigma\;(\frac{1}{15},\frac{1}{15})$. Again these results together determine
the mass-coupling relation and a non-trivial OPE coefficient
$C_{\sigma\varepsilon\sigma}$.

The formulas allow for a straightforward extension to the case of arbitrary
excited states in a manner consistent with [12]. Consequently, one can readily
change the operator $\mathcal{O}$ in the OPE coefficient
$C_{\mathcal{O}\Phi\mathcal{O}}$. Modifying the operator $\Phi$ remains,
however, an outstanding problem, and our original formula does not seem to
have a natural place for performing such a modification. As a first step in
this direction, we rewrote our formula in terms of an auxiliary linear
dressing equation, akin to the approach of Smirnov et.al. [20, 21, 22, 23],
where deformations of the kernel in the dressing equation can accommodate for
modifying the operator. Unfortunately, it is not clear how this works in the
generic case which is not related to sine-Gordon theory, like the two
component TBA of the (Scaling) Potts model. We leave this investigation for
further study.

From a more general perspective, we believe, that an in-depth study of various
integrable descriptions of 2D CFTs, including but not stopping at the level of
the spectrum, may lead to interesting insights and a deeper understanding of
the scope of integrability.

Acknowledgments. RJ would like to thank the Wigner Research Centre for Physics
and KITP (within the program Integrability in String, Field, and Condensed
Matter Theory), while ZB the Jagiellonian University, for hospitality where
part of this work was done. This research was supported in part by the
National Science Foundation under Grant No. NSF PHY-1748958 and by the NKFIH
grant K134946.

##  Appendix A An analog of the dilogarithm trick for the UV energy correction

The subleading energy correction in the kink regime is given by (3.21) which
we quote here for convenience:

$-\int\frac{ds}{2\pi}e^{s}\left[G_{R}+\frac{\delta
Y_{R}^{-1}}{1+Y_{R}^{-1}}\right]$ (A.1)

This expression involves a subleading solution of the massive TBA equation
$\delta Y_{R}^{-1}$. In this section we will show that one can evaluate the
above integral without the need to explicitly find $\delta Y_{R}^{-1}$.
Indeed, let us now follow some initial steps in the dilogarithm trick
computation. We differentiate the TBA equation for the kink

$\log Y_{R}=e^{s}-\phi\star\log(1+Y_{R}^{-1})$ (A.2)

and integrate by parts the convolution term:

$\int ds^{\prime}\partial_{s}\phi(s-s^{\prime})f(s^{\prime})=-\int
ds^{\prime}\partial_{s^{\prime}}\phi(s-s^{\prime})f(s^{\prime})=\int
ds^{\prime}\phi(s-s^{\prime})\partial_{s^{\prime}}f(s^{\prime})$ (A.3)

We then find an expression for $e^{s}$

$e^{s}=\phi\star\frac{\partial Y_{R}^{-1}}{1+Y_{R}^{-1}}-\frac{\partial
Y_{R}^{-1}}{Y_{R}^{-1}}$ (A.4)

Plugging this into (A.1), we get

$-\int\frac{dsds^{\prime}}{(2\pi)^{2}}\phi(s-s^{\prime})\frac{\partial
Y_{R}^{-1}}{1+Y_{R}^{-1}}(s^{\prime})\left[G_{R}+\frac{\delta
Y_{R}^{-1}}{1+Y_{R}^{-1}}\right](s)+\int\frac{ds}{2\pi}\frac{\partial
Y_{R}^{-1}}{Y_{R}^{-1}}\left[G_{R}+\frac{\delta
Y_{R}^{-1}}{1+Y_{R}^{-1}}\right]$ (A.5)

Performing first the integral over $s$ in the first term and using the linear
integral equation (3.14), we get

$-\int\frac{ds^{\prime}}{2\pi}\frac{\partial
Y_{R}^{-1}}{1+Y_{R}^{-1}}\frac{\delta
Y_{R}^{-1}}{Y_{R}^{-1}}+\int\frac{ds}{2\pi}\frac{\partial
Y_{R}^{-1}}{Y_{R}^{-1}}\left[G_{R}+\frac{\delta
Y_{R}^{-1}}{1+Y_{R}^{-1}}\right]$ (A.6)

We see that the terms involving $\delta Y_{R}^{-1}$ cancel out and we are left
with a formula involving just the UV limit CFT solutions $Y_{R,L}$.

$\int\frac{ds}{2\pi}G_{R}\frac{\partial
Y_{R}^{-1}}{Y_{R}^{-1}}\equiv-\int\frac{ds}{2\pi}G_{R}\cdot\partial\log Y_{R}$
(A.7)

The computation in the anti-kink region is analogous.

##  Appendix B Small volume expansion of the excited state TBA

In this appendix we explain how excited states TBA equations of the form

$\log Y=r\cosh\theta+\sum_{i}\eta_{i}\log
S(\theta-\theta_{i})-\phi\star\log(1+Y^{-1})\quad;\quad Y(\theta_{i})=-1$
(B.1)

can be expanded at small volume. We assume that for $r\to 0$ half of the roots
$\\{\theta_{i}^{+}\\}$ goes to the kink domain and, symmetrically, the other
half $\\{\theta_{i}^{-}=-\theta_{i}^{+}\\}$ goes to the anti-kink domain. We
introduce the kink equations

$\log Y_{K}=\frac{r}{2}e^{\theta}+\sum_{i}\eta_{i}\log
S(\theta-\theta_{i}^{+})-\phi\star\log(1+Y_{K}^{-1})\quad;\quad
Y_{K}(\theta_{i}^{+})=-1$ (B.2)

and define the volume independent right quantities as

$Y_{R}(s)=Y_{K}(s-\log\frac{r}{2})\quad;\qquad
s_{i}=\theta_{i}^{+}+\log\frac{r}{2}$ (B.3)

We then expand the $Y$-function and the roots around the asymptotic solution

$Y=Y_{\mathrm{as}}+\delta Y=Y_{K}Y_{A}Y_{0}^{-1}+\delta
Y\quad;\qquad\theta_{i}=\theta_{i}^{+}+\delta\theta_{i}$

Following similar steps we had for the groundstate, we can write the
linearized equations

$Y_{\mathrm{as}}^{-1}\delta
Y=-i\sum_{i}\eta_{i}(\phi(\theta-\theta_{i}^{+})+\phi(\theta+\theta_{i}^{+}))\delta\theta_{i}-\phi\star\left[G-\frac{Y_{\mathrm{as}}^{-1}\delta
Y}{1+Y_{\mathrm{as}}}\right]$ (B.4)

where

$G=\log\frac{1+Y_{K}^{-1}Y_{A}^{-1}Y_{0}}{1+Y_{K}^{-1}}-\log\frac{1+Y_{0}^{-1}Y_{A}^{-1}Y_{0}}{1+Y_{0}^{-1}}$
(B.5)

The quantization condition is modified as

$Y_{A}(\theta_{i}^{+})Y_{0}^{-1}-1=\delta Y(\theta_{i}^{+})+\partial
Y_{K}(\theta_{i}^{+})\delta\theta_{i}$ (B.6)

By exploiting that

$y_{L}(s)=\lim_{r\to
0}r^{-\frac{24}{5}}(Y_{A}^{-1}(s-\log\frac{r}{2})Y_{0}-1)=-\tilde{c}2^{-\frac{24}{5}}e^{-\frac{12}{5}s}+\dots$
(B.7)

we can see that the magnitudes of each term is of order of $r^{\frac{24}{5}}$.
We can thus project out the contribution at this order by focusing on the kink
domain and writing only right quantities

$Y_{R}^{-1}\delta Y_{R}=-i\sum_{i}\eta_{i}\phi(s-s_{i})\delta
s_{i}-\phi\star\left[G_{R}-\frac{Y_{R}^{-1}\delta Y_{R}}{1+Y_{R}}\right]$
(B.8)

and

$-y_{L}(s_{i})=\delta Y_{R}(s_{i})+\partial Y_{R}(s_{i})\delta s_{i}$ (B.9)

where $\delta s_{i}=\delta\theta_{i}$. Fortunately, we do not have to solve
these equations in order to calculate the energy correction containing
$\epsilon_{UV}$. The modification compared to the previous calculation is the
contribution of the extra source terms:

$-i\sum_{i}\eta_{i}e^{s_{i}}\delta
s_{i}-\int\frac{ds}{2\pi}e^{s}\left[G_{R}-\frac{Y_{R}^{-1}\delta
Y_{R}}{1+Y_{R}}\right]$ (B.10)

Here we can again use the adapted dilogarithm trick, namely to differentiate
the $Y_{R}$ kink TBA equation to get $e^{s}$ and plug back to the energy
integral. We then transpose the symmetric kernel and perform the integral in
the second argument. In doing so we exploit the linearized equations for
$Y_{R}^{-1}\delta Y_{R}$ and for $\delta s_{i}$. On the way we also evaluate
the linearized equations and the derivative of the kink equation at the root
positions. Putting everything together we arrive at the final formula

$-i\sum_{i}\eta_{i}y_{L}(s_{i})-\int\frac{ds}{2\pi}G_{R}\,\partial\log Y_{R}$
(B.11)

Observe that the integral term is our generic result, while the discrete part
is merely its residue term, which can be easily seen from

$G_{R}=\text{$\left\\{\frac{1}{1+Y_{R}}-\frac{1}{1+Y_{0}}\right\\}$ }y_{L}$
(B.12)

by taking the residue in the first term at $s=s_{i}$.

##  Appendix C Comments on the relation with Smirnov’s approach

In this section we make a more direct connection with Smirnov’s approach,
which calculates the vacuum expectation values of exponential vertex operators
in terms of $\omega$’s, and evaluate their small volume expansion.

As a first step we need to normalize vacuum expectation values the same way.
In the sinh-Gordon theory the ratio of the vacuum expectation values of the
exponential fields can be written in terms of the building blocks
$\omega_{n,m}=e^{n\theta}\circ(e^{m\theta})^{\mathrm{dr}}$. The simplest of
these, $\omega_{1,-1}$, describes the VEV of the exponential operator,
$e^{b\phi}$, in the following way

$\frac{\langle 0|e^{b\phi}|0\rangle_{r}}{\langle
0|e^{b\phi}|0\rangle_{\infty}}=1+4\sin(\pi p)\omega_{1,-1}$ (C.1)

where the subscript indicates the volume. As the $S$-matrix of the sinh-Gordon
model coincides with the Lee-Yang one for $p=-\frac{2}{3}$ we expect that the
formula above describes under this continuation the VEV of $\Phi$. We can
confirm this by recalling that $\Theta$ is proportional to the perturbing
field $\Phi$, so

$\frac{\langle 0|\Phi|0\rangle_{r}}{\langle
0|\Phi|0\rangle_{\infty}}=\frac{\langle 0|\Theta|0\rangle_{r}}{\langle
0|\Theta|0\rangle_{\infty}}$ (C.2)

Writing (4.2) as

$\frac{1}{m^{2}}\langle
0|\Theta|0\rangle_{r}=\omega_{1,-1}+\frac{2\epsilon_{B}}{m^{2}}$ (C.3)

and noting that $\omega_{1,-1}=\partial_{r}E(r)+\frac{1}{r}E(r)$ vanishes at
large volume, we see that $\langle 0|\Theta|0\rangle_{\infty}=2\epsilon_{B}$,
hence putting the two equations together we get

$\frac{\langle 0|\Phi|0\rangle_{r}}{\langle
0|\Phi|0\rangle_{\infty}}=1+\frac{m^{2}}{2\epsilon_{B}}\omega_{1,-1}$ (C.4)

in agreement with (C.1) for $p=-\frac{2}{3}$. For small volumes the VEV has to
approach the finite volume CFT 3-point function as

$\lim_{{r}\to 0}\langle
0|\Phi|0\rangle_{{r}}=\left(\frac{2\pi}{r}\right)^{2h}C_{\Phi\Phi\Phi}+\dots$
(C.5)

##  Appendix D Small volume expansion of the linear dressing formula

We would like now to express the appropriate UV contribution to
$\omega_{1,-1}$ in terms of the CFT integrable data. We consider thus the
small volume expansion of $\omega_{1,-1}$

$\omega_{1,-1}=\frac{1}{2\sqrt{3}}+r^{\frac{2}{5}}\omega_{UV}+\dots$ (D.1)

and look for an expression for $\omega_{UV}$.

We start by investigating the small volume expansion of $H(\theta)$ defined in
(4.7). Since $H(\theta)$ is the dressed version of $e^{-\theta}$ it behaves
qualitatively similarly, being large in the anti-kink domain. This large
dominating part can be captured by its anti-kink analogue, which satisfies the
following equation

$H_{A}(\theta)=e^{-\theta}+\int_{-\infty}^{\infty}\frac{d\theta^{\prime}}{2\pi}\frac{\phi(\theta-\theta^{\prime})H_{A}(\theta^{\prime})}{1+Y_{A}(\theta^{\prime})}$
(D.2)

Due to $Y_{A}$ this is a volume-dependent quantity, which can be written in
terms of its volume-independent counterpart as

$H_{A}\left(\theta\right)=\frac{2}{r}H_{L}(\theta-\log\frac{r}{2})$ (D.3)

where $H_{L}(\theta)$ satisfies the same equation as $H_{A}(\theta)$, but with
$Y_{L}(\theta)$ instead of $Y_{A}(\theta)$. This equation is actually the same
as the equation we have for $-\partial_{\theta}\log Y_{L}(\theta)$, thus
$H_{L}=-Y_{L}^{-1}\partial Y_{L}$. The difference between $H(\theta)$ and
$H_{A}(\theta)$ is small for small volumes and satisfies the linear integral
equation:

$\delta
H(\theta)=H\left(\theta\right)-H_{A}(\theta)=\text{source($\theta)$+$\phi(\theta-\theta^{\prime})\circ\delta
H(\theta^{\prime})$}$ (D.4)

where the source term is

$\text{source($\theta)$}=\phi(\theta-\theta^{\prime})\star{\cal
G}(\theta^{\prime})\quad;\quad{\cal
G}(\theta)=H_{A}(\theta)\left[\frac{1}{1+Y(\theta)}-\frac{1}{1+Y_{A}(\theta)}\right]$
(D.5)

To understand the magnitude of ${\cal G}(\theta)$ we point out that at leading
order

$\frac{1}{1+Y(\theta)}-\frac{1}{1+Y_{A}(\theta)}=\frac{1}{1+Y_{K}(\theta)}-\frac{1}{1+Y_{0}}+\dots$
(D.6)

which is non-trivial in the kink domain only. We thus need the behaviour of
the anti-kink $H_{A}(\theta)$ in the kink domain, which, being the interaction
between the two domains, is small. Since $H_{L}=-Y_{L}^{-1}\partial Y_{L}$ we
can extract its large $\theta$ asymptotics as

$Y_{L}(\theta)=Y_{0}(1+\tilde{c}e^{-\frac{6}{5}\theta}+\dots)\to
H_{L}(\theta)=\tilde{c}\frac{6}{5}e^{-\frac{6}{5}\theta}+\dots$ (D.7)

which implies, in the variable centered around the kink domain, that

$H_{A}(\theta-\log\frac{r}{2})=\frac{2}{r}H_{L}(\theta-2\log\frac{r}{2})\sim\tilde{c}\frac{6}{5}\left(\frac{r}{2}\right)^{\frac{7}{5}}e^{-\frac{6}{5}\theta}+\dots$
(D.8)

Thus $\delta H$ starts at order $r^{\frac{7}{5}}$. We might focus on the first
subleading correction by defining the kink part of the source as

$g_{R}(\theta)=\lim_{r\to 0}r^{-\frac{7}{5}}{\cal
G}(\theta-\log\frac{r}{2})=\tilde{c}\frac{6}{5}\left(\frac{1}{2}\right)^{\frac{7}{5}}e^{-\frac{6}{5}\theta}\left[\frac{1}{1+Y_{R}(\theta)}-\frac{1}{1+Y_{0}}\right]+\dots$
(D.9)

The corresponding $\delta H_{R}$ satisfies

$\delta
H_{R}(\theta)=\phi(\theta-\theta^{\prime})\star\left\\{g_{R}(\theta)+\frac{\delta
H_{R}(\theta)}{1+Y_{R}(\theta)}\right\\}$ (D.10)

and is written purely in terms of CFT right moving quantities.

Eventually we would like to calculate

$\omega_{1,-1}=\int_{-\infty}^{\infty}\frac{d\theta}{2\pi}\frac{e^{\theta}(H_{A}(\theta)+\delta
H(\theta))}{1+Y(\theta)}=\int_{-\infty}^{\infty}\frac{d\theta}{2\pi}e^{\theta}\left\\{\frac{H_{A}(\theta)}{1+Y_{A}}+\frac{\delta
H(\theta)}{1+Y(\theta)}+{\cal G}(\theta)\right\\}$ (D.11)

The first, leading order term can be calculated from the large $\theta$
asymptotics of $H_{A}(\theta)$. Indeed, taking the integral equation defining
$H_{A}(\theta)$ and using the large $\theta$ asymptotics of the kernel
$\lim_{\theta\to\infty}\phi(\theta)=-2\sqrt{3}e^{-\theta}+\dots.$ we can
obtain

$\lim_{\theta\to\infty}H_{A}(\theta)=e^{-\theta}\left(1-2\sqrt{3}\int_{-\infty}^{\infty}\frac{d\theta^{\prime}}{2\pi}\frac{e^{\theta^{\prime}}H_{A}(\theta^{\prime})}{1+Y_{A}(\theta^{\prime})}\right)+\dots$
(D.12)

Since asymptotically $H_{A}(\theta)$ behaves as $H_{A}(\theta)\sim
e^{-\frac{6}{5}\theta}$, term proportional to $e^{-\theta}$ has to be
cancelled. This implies that the leading, volume-independent term in
$\omega_{1,-1}$ is

$\int_{-\infty}^{\infty}\frac{d\theta}{2\pi}\frac{e^{\theta}H_{A}(\theta)}{1+Y_{A}}=\frac{1}{2\sqrt{3}}$
(D.13)

This is related to the bulk energy constant and simply cancels the $1$ in
(C.1).

The subleading correction for $\omega_{1,-1}$ has the terms appearing on the
rhs in the integral equation for $\delta H$. Due to the $e^{\theta}$ factor
their leading contribution comes from the kink region implying

$\int_{-\infty}^{\infty}\frac{d\theta}{2\pi}e^{\theta}\left\\{\frac{\delta
H(\theta)}{1+Y(\theta)}+{\cal
G}(\theta)\right\\}=\int_{-\infty}^{\infty}\frac{d\theta}{2\pi}e^{\theta}\left\\{\frac{\delta
H_{K}(\theta))}{1+Y_{K}(\theta)}+{\cal G}_{K}(\theta)\right\\}+\dots$ (D.14)

where $\delta H_{K}(\theta)$ and ${\cal G}_{K}(\theta)$ are quantities in the
kink domain, which satisfy the equation

$\delta H_{K}(\theta)=\phi(\theta-\theta^{\prime})\star\left\\{{\cal
G}_{K}(\theta^{\prime})+\frac{\delta
H_{K}(\theta))}{1+Y_{K}(\theta)}\right\\}$ (D.15)

In order to extract the volume dependence we make the shift
$\theta=s-\log\frac{r}{2}$ and express everything in terms of the CFT right
moving quantities:

$r^{\frac{2}{5}}\int_{-\infty}^{\infty}\frac{ds}{2\pi}e^{s}\left\\{\frac{\delta
H_{R}(s))}{1+Y_{R}(s)}+g_{R}(s)\right\\}$ (D.16)

We can observe that the overall $r$ dependence is $r^{\frac{2}{5}}$ as it is
expected from an operator of dimension $h=-\frac{1}{5}$ in the UV limit on the
cylinder. Finally, we take such a quantity $f_{R}$, which satisfies the
integral equation of the form

$f_{R}(s)=e^{s}+\frac{\phi(s-s^{\prime})}{1+Y_{R}(s^{\prime})}\star
f_{R}(s^{\prime})$ (D.17)

and use the analogue of the dilogarithm trick. Clearly $f_{R}=\partial\log
Y_{R}$. After replacing $e^{s}$ from this integral equation and using the
symmetry of the kernel we obtain

$\int_{-\infty}^{\infty}\frac{ds}{2\pi}\text{$\left\\{f_{R}(s)-\phi(s-s^{\prime})\circ
f_{R}(s^{\prime})\right\\}$}\left\\{\frac{\delta
H_{R}(s))}{1+Y_{R}(s)}+g_{R}(s)\right\\}=\int_{-\infty}^{\infty}\frac{ds}{2\pi}g_{R}(s)f_{R}(s)$
(D.18)

This quantity, which is related to the CFT 3-point functions is written purely
in terms of right moving CFT quantities.

Let us finally relate this quantity to the one appearing in the small volume
expansion of the energy. The source term $g_{R}$ governing the deviation of
$H$ from its anti-kink counterpart $H_{A}$ contains the tail of the anti-kink
$H_{A}$ in the kink domain (D.9). It turns out to be simply related to the
source $G_{R}$ (see (3.12)) by

$g_{R}(s)=-\frac{6}{5}\cdot 2\cdot G_{R}(s)$ (D.19)

The remaining ingredient $f_{R}$ is the dressed version of $e^{s}$ in the
right domain, which is in fact given by $f_{R}(s)=\partial\log Y_{R}(s)$,
hence we finally arrive at

$\omega_{UV}=\int_{-\infty}^{\infty}\frac{ds}{2\pi}g_{R}(s)\partial\log
Y_{R}(s)$ (D.20)

Comparing (4.11) with (D.1) and using (D.19), we recover our previous formula
(3.23).

##  Appendix E Derivation for the 3-state Potts model

In this Appendix we extend the analysis from the one-component scaling Lee-
Yang TBA system to the two-component scaling Potts model. This integrable
model is obtained by perturbing the D-invariant unitary ${\cal M}_{5,6}$
minimal model with the primary field $\varepsilon$ of dimensions
$(\frac{2}{5},\frac{2}{5})$. Depending on the sign of the perturbation we can
be either in the paramagnetic (single groundstate) or in the ferromagnetic
(triply degenerate groundstate) regime. The scattering theory in the
paramagnetic case has two massive particles, with mass $m$ which scatter on
each other and themselves diagonally [11]. The two-component TBA equations for
the pseudo energies $Y_{1}=e^{\epsilon_{1}}$ and $Y_{2}=e^{\epsilon_{2}}$ can
be written in the compact form

$\log Y_{i}(\theta)=r\cosh\theta-\phi_{ij}\star\log(1+Y_{j}(\theta)^{-1})$
(E.1)

where the logarithmic derivatives of the scattering matrices
$\phi_{ij}(\theta)=-\partial_{\theta}\log S_{ij}(\theta)$ read explicitly as

$\phi_{11}=\phi_{22}=-\frac{\sqrt{3}}{1+2\cosh\theta}\quad;\qquad\phi_{12}=\phi_{21}=-\frac{\sqrt{3}}{-1+2\cosh\theta}$
(E.2)

These TBA equations follow from the $A_{2}$ Y-system relations [25]:

$Y_{1}(\theta-i\frac{\pi}{3})Y_{1}(\theta+i\frac{\pi}{3})=1+Y_{2}(\theta)\quad;\quad
Y_{2}(\theta-i\frac{\pi}{3})Y_{2}(\theta+i\frac{\pi}{3})=1+Y_{1}(\theta)$
(E.3)

The groundstate energy is

$E_{0}({\cal
R})=-m\sum_{i=1}^{2}\int\frac{d\theta}{2\pi}\cosh\theta\,\log(1+Y_{i}(\theta)^{-1})$
(E.4)

For the groundstate the Y-functions are equal $Y_{1}(\theta)=Y_{2}(\theta)$
and real, thus the only independent TBA equation is the same as the Lee-Yang
one (2.27), while the groundstate energy is doubled. Since the model is
unitary the perturbing operator has a zero VEV and the leading perturbative
correction is a second order one

$E_{0}({\cal R})=\frac{2\pi}{{\cal
R}}\left(-\frac{1}{15}+c_{2}(2/5)\lambda^{2}\left(\frac{{\cal
R}}{2\pi}\right)^{\frac{12}{5}}+\dots\right)$ (E.5)

which corresponds to the integrated two-point function (2.22) and is equal to
twice the formula (2.23). Observe that this formula provides the correct
central charge $c=\frac{4}{5}$ together with the correct dimension of the
perturbing operator $h=\frac{2}{5}$. By comparing

$c_{2}(2/5)=\frac{\pi\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{2}{5}\right)^{2}}{2\Gamma\left(\frac{3}{5}\right)^{2}\Gamma\left(\frac{4}{5}\right)}$
(E.6)

and the expansion of the TBA, the mass-coupling relation $\lambda=\kappa
m^{\frac{6}{5}}$ can be established, giving

$\kappa=\frac{\Gamma\left(\frac{3}{10}\right)\left(\Gamma\left(\frac{2}{3}\right)\Gamma\left(\frac{5}{6}\right)\right)^{\frac{6}{5}}\sqrt{\frac{\Gamma\left(-\frac{1}{5}\right)\Gamma\left(\frac{7}{5}\right)}{\Gamma\left(-\frac{2}{5}\right)\Gamma\left(\frac{6}{5}\right)}}}{4\sqrt[5]{2}\pi^{\frac{8}{5}}\Gamma\left(\frac{7}{10}\right)}$
(E.7)

which agrees with the literature [26].

Interestingly, the perturbing operator has a non-trivial 3-point function
between states of dimensions $(\frac{1}{15},\frac{1}{15})$. In order to
extract this coupling we need to calculate the small volume expansion of the
energy of this excited state. In the paramagnetic phase this is a genuine
excited state with moving particles [24]. In the ferromagnetic phase, however
it corresponds to one of the vacua. The ferromagnetic regime has four kinks
which scatter on each other with the same S-matrices which we had for the
paramagnetic regime and give rise to the same TBA system for the true finite
volume groundstate. The finiteness of the volume lifts the groundstate
degeneracy and the twisted vacuum, corresponding to the state
$(\frac{1}{15},\frac{1}{15})$, can be described by complex twists in the TBA,
which implies that $Y_{1}$ and $Y_{2}$ are no longer real. Since they are
complex conjugate of each other we denote $Y_{1}$ by $Y$ while $Y_{2}$ by
$\bar{Y}$. The twisted groundstate TBA equations can be written as [24]

$\log
Y=\omega+r\cosh\theta-\phi_{11}\star\log(1+Y^{-1})-\phi_{12}\star\log(1+\bar{Y}^{-1})$
(E.8)

together with its complex conjugate, where $\omega=i\frac{2\pi}{3}$. We now
follow analogous steps to the Lee-Yang model to expand the energy at small
volumes and extract the sought for 3-point function.

In the small volume limit the $Y$ function behaves similarly to the one in the
Lee-Yang model and consists of the kink, anti-kink and the middle (complex)
plateaux parts. The kink/anti-kink $Y_{K/A}$ function satisfies the
$r\cosh\to\frac{r}{2}e^{\pm\theta}$ modified equations

$\log
Y_{K}=\omega+\frac{r}{2}e^{\theta}-\phi_{11}\star\log(1+Y_{K}^{-1})-\phi_{12}\star\log(1+\bar{Y}_{K}^{-1})$
(E.9)

and $Y$ has asymptotic value

$Y_{0}=\frac{1-\sqrt{5}}{2}$ (E.10)

which can be obtained from the plateau equation

$\log
Y_{0}=\omega+\frac{1}{3}\log(1+Y_{0}^{-1})+\frac{2}{3}\log(1+\bar{Y}_{0}^{-1})$
(E.11)

The volume-independent kink functions, which are centered around the origin
are defined as

$Y_{R}(s)=Y_{K}(s-\log\frac{r}{2})\quad;\qquad
Y_{L}(s)=Y_{A}(s+\log\frac{r}{2})=Y_{R}(-s)$ (E.12)

By inspecting the numerical solution we can observe the following leading
correction to the asymptotic $\theta\to-\infty$ behaviour

$Y_{R}(s)Y_{0}^{-1}=1+i\tilde{c}e^{\frac{3}{5}s}+\dots\quad;\qquad\tilde{c}=-1.381101886..$
(E.13)

where $\tilde{c}$ is real. The real part of $Y_{R}$ approaches $Y_{0}$ as
$e^{\frac{6}{5}\theta}$ and is subleading.

The asymptotic solution is

$Y_{\mathrm{as}}=Y_{K}Y_{A}Y_{0}^{-1}$ (E.14)

and its correction satisfies the equation

$\displaystyle\log Y_{\mathrm{as}}+Y_{\mathrm{as}}^{-1}\delta Y$
$\displaystyle=\omega+\frac{r}{2}e^{\theta}+\frac{r}{2}e^{-\theta}-\phi_{11}\star\log(1+Y_{\mathrm{as}}^{-1})+\phi_{11}\star\frac{Y_{\mathrm{as}}^{-1}\delta
Y}{1+Y_{\mathrm{as}}}$
$\displaystyle\qquad\qquad-\phi_{12}\star\log(1+\bar{Y}_{\mathrm{as}}^{-1})+\phi_{12}\star\frac{\bar{Y}_{\mathrm{as}}^{-1}\delta\bar{Y}}{1+\bar{Y}_{\mathrm{as}}}$
(E.15)

together with is complex conjugate. This can be brought into a two-component
linear integral equation of the form

$\left(\begin{array}[]{c}\frac{\delta Y}{Y_{\mathrm{as}}}\\\
\frac{\delta\bar{Y}}{\bar{Y}_{\mathrm{as}}}\end{array}\right)=\left(\begin{array}[]{c}\mathrm{source}\\\
\overline{\mathrm{source}}\end{array}\right)+\left(\begin{array}[]{cc}\phi_{11}\star\frac{1}{1+Y_{\mathrm{as}}}&\phi_{12}\star\frac{1}{1+\bar{Y}_{\mathrm{as}}}\\\
\phi_{21}\star\frac{1}{1+Y_{\mathrm{as}}}&\phi_{22}\star\frac{1}{1+\bar{Y}_{\mathrm{as}}}\end{array}\right)\left(\begin{array}[]{c}\frac{\delta
Y}{Y_{\mathrm{as}}}\\\
\frac{\delta\bar{Y}}{\bar{Y}_{\mathrm{as}}}\end{array}\right)$ (E.16)

where the convolutions acts on the unknown vectors as well. The source terms
can be written as

$\displaystyle\mathrm{source}$ $\displaystyle=-\phi_{11}\star
G-\phi_{12}\star\bar{G}$ (E.17)

with

$\displaystyle G$
$\displaystyle=\log(1+Y_{K}^{-1}Y_{A}^{-1}Y_{0})-\log(1+Y_{K}^{-1})-\log(1+Y_{A}^{-1})+\log(1+Y_{0}^{-1})$
$\displaystyle=\log\frac{1+Y_{K}^{-1}Y_{A}^{-1}Y_{0}}{1+Y_{K}^{-1}}-\log\frac{1+Y_{0}^{-1}Y_{A}^{-1}Y_{0}}{1+Y_{0}^{-1}}$
(E.18)

Let us define the volume-independent leading right kink form of the source as

$G_{R}(s)=\lim_{r\to 0}r^{-\frac{6}{5}}\mathrm{source}(s-\log\frac{r}{2})$
(E.19)

Using the asymptotics of the anti-kink $Y_{A}$ function in the kink domain

$y_{L}(s)=\lim_{r\to
0}r^{-\frac{6}{5}}(Y_{A}^{-1}(s-\log\frac{r}{2})Y_{0}-1)=-i\tilde{c}\;2^{-\frac{6}{5}}e^{-\frac{3}{5}s}+\dots$
(E.20)

we obtain

$G_{R}=\text{$\left\\{\frac{1}{1+Y_{R}}-\frac{1}{1+Y_{0}}\right\\}$ }y_{L}$
(E.21)

From the $s\to-\infty$ asymptotics of the kink solution we can see that

$\lim_{s\to-\infty}G_{R}(s)=\frac{Y_{0}^{-1}}{(1+Y_{0}^{-1})^{2}}\tilde{c}^{2}(2)^{-\frac{6}{5}}$
(E.22)

We can now proceed to calculate the energy correction. We follow the
calculation in the Lee-Yang model and evaluate

$\int\frac{d\theta}{2\pi}\cosh\theta\,\log(1+(Y_{\mathrm{as}}^{-1}+\delta
Y^{-1}))+\mathrm{cc.}$ (E.23)

By using that

$\log(1+Y_{\mathrm{as}}^{-1})=G+\log(1+Y_{K}^{-1})+\log(1+Y_{A}^{-1})-\log(1+Y_{0}^{-1})$
(E.24)

we can calculate the contributions of the kink/anti-kinks as the central
charge and bulk energy constants. The remaining terms can be evaluated in the
two kink regions

$\int\frac{ds}{2\pi}\left[G_{R}-\frac{Y_{R}^{-1}\delta
Y_{R}}{1+Y_{R}}\right]e^{\theta}+\mathrm{cc}.$ (E.25)

Let us now differentiate the kink TBA

$e^{s}=Y_{R}^{-1}\partial Y_{R}-\phi_{11}\star\frac{Y_{R}^{-1}\partial
Y_{R}}{1+Y_{R}}-\phi_{12}\star\frac{\bar{Y}_{R}^{-1}\partial\bar{Y}_{R}}{1+\bar{Y}_{R}}$
(E.26)

and plug back to the above form. We then transpose the kernels and use the
linearized integral equations to arrive at

$\int\frac{ds}{2\pi}G_{R}\,\partial\log Y_{R}+{\rm cc.}$ (E.27)

This result is written purely in terms of the CFT right moving quantities and
does not require to solve the linearized equation. There is a similar
contribution form the left moving particles. By adding their contributions we
arrive at the 3-point functions

$C_{\sigma\Phi\sigma}\equiv
C_{\sigma\varepsilon\sigma}=-\frac{1}{\kappa(2\pi)^{\frac{4}{5}}}\left(\int\frac{ds}{2\pi}G_{L}\,\partial\log
Y_{L}+\int\frac{ds}{2\pi}G_{R}\,\partial\log Y_{R}+{\rm cc.}\right)$ (E.28)

where we also used the conventional notation $\varepsilon$ for the operator
with $(\frac{2}{5},\frac{2}{5})$ in the 3-state Potts minimal model. A
numerical evaluation of this quantity, using (E.7), yields

$C_{\sigma\varepsilon\sigma}=0.54617761809..$ (E.29)

while the exact answer is $0.54617761825..$.

Clearly, one can repeat the same steps for the $SL(2)$ vacuum (i.e. TBA with
the twist $\omega=0$), taking into account different asymptotics of the anti-
kink, and recover the mass-gap coefficient (E.7). However, the numerics are
essentially identical to the Lee-Yang computation for the reasons explained at
the beginning of this section.

## References

  * [1] A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333–380.  doi:10.1016/0550-3213(84)90052-X.
  * [2] D. Poland, S. Rychkov, A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys. 91 (2019) 015002.  arXiv:1805.04405, doi:10.1103/RevModPhys.91.015002.
  * [3] N. Beisert, et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3–32.  arXiv:1012.3982, doi:10.1007/s11005-011-0529-2.
  * [4] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231–252.  arXiv:hep-th/9711200, doi:10.1023/A:1026654312961.
  * [5] N. Gromov, V. Kazakov, S. Leurent, D. Volin, Quantum Spectral Curve for Planar $\mathcal{N}=4$ Super-Yang-Mills Theory, Phys. Rev. Lett. 112 (1) (2014) 011602\.  arXiv:1305.1939, doi:10.1103/PhysRevLett.112.011602.
  * [6] N. Gromov, V. Kazakov, S. Leurent, D. Volin, Quantum spectral curve for arbitrary state/operator in AdS5/CFT4, JHEP 09 (2015) 187.  arXiv:1405.4857, doi:10.1007/JHEP09(2015)187.
  * [7] N. Gromov, V. Kazakov, P. Vieira, Exact Spectrum of Anomalous Dimensions of Planar N=4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 103 (2009) 131601\.  arXiv:0901.3753, doi:10.1103/PhysRevLett.103.131601.
  * [8] B. Basso, S. Komatsu, P. Vieira, Structure Constants and Integrable Bootstrap in Planar N=4 SYM Theory (5 2015).  arXiv:1505.06745.
  * [9] V. V. Bazhanov, S. L. Lukyanov, A. B. Zamolodchikov, Integrable quantum field theories in finite volume: Excited state energies, Nucl. Phys. B 489 (1997) 487–531.  arXiv:hep-th/9607099, doi:10.1016/S0550-3213(97)00022-9.
  * [10] Z. Bajnok, O. el Deeb, P. A. Pearce, Finite-Volume Spectra of the Lee-Yang Model, JHEP 04 (2015) 073.  arXiv:1412.8494, doi:10.1007/JHEP04(2015)073.
  * [11] A. B. Zamolodchikov, Thermodynamic Bethe Ansatz in Relativistic Models. Scaling Three State Potts and Lee-yang Models, Nucl. Phys. B 342 (1990) 695–720.  doi:10.1016/0550-3213(90)90333-9.
  * [12] P. Dorey, R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 639–659.  arXiv:hep-th/9607167, doi:10.1016/S0550-3213(96)00516-0.
  * [13] J. L. Cardy, G. Mussardo, S Matrix of the Yang-Lee Edge Singularity in Two-Dimensions, Phys. Lett. B 225 (1989) 275–278.  doi:10.1016/0370-2693(89)90818-6.
  * [14] P. Dorey, R. Tateo, Excited states in some simple perturbed conformal field theories, Nucl. Phys. B 515 (1998) 575–623.  arXiv:hep-th/9706140, doi:10.1016/S0550-3213(97)00838-9.
  * [15] Z. Bajnok, J. Balog, K. Ito, Y. Satoh, G. Z. Tóth, Exact mass-coupling relation for the homogeneous sine-Gordon model, Phys. Rev. Lett. 116 (18) (2016) 181601.  arXiv:1512.04673, doi:10.1103/PhysRevLett.116.181601.
  * [16] Z. Bajnok, J. Balog, K. Ito, Y. Satoh, G. Z. Toth, On the mass-coupling relation of multi-scale quantum integrable models, JHEP 06 (2016) 071.  arXiv:1604.02811, doi:10.1007/JHEP06(2016)071.
  * [17] A. B. Zamolodchikov, Mass scale in the sine-Gordon model and its reductions, Int. J. Mod. Phys. A 10 (1995) 1125–1150.  doi:10.1142/S0217751X9500053X.
  * [18] A. Leclair, G. Mussardo, Finite temperature correlation functions in integrable QFT, Nucl. Phys. B 552 (1999) 624–642.  arXiv:hep-th/9902075, doi:10.1016/S0550-3213(99)00280-1.
  * [19] H. Saleur, A Comment on finite temperature correlations in integrable QFT, Nucl. Phys. B 567 (2000) 602–610.  arXiv:hep-th/9909019, doi:10.1016/S0550-3213(99)00665-3.
  * [20] M. Jimbo, T. Miwa, F. Smirnov, Fermionic structure in the sine-Gordon model: Form factors and null-vectors, Nucl. Phys. B 852 (2011) 390–440.  arXiv:1105.6209, doi:10.1016/j.nuclphysb.2011.06.016.
  * [21] H. Boos, M. Jimbo, T. Miwa, F. Smirnov, Hidden Grassmann Structure in the XXZ Model IV: CFT limit, Commun. Math. Phys. 299 (2010) 825–866.  arXiv:0911.3731, doi:10.1007/s00220-010-1051-6.
  * [22] S. Negro, F. Smirnov, On one-point functions for sinh-Gordon model at finite temperature, Nucl. Phys. B 875 (2013) 166–185.  arXiv:1306.1476, doi:10.1016/j.nuclphysb.2013.06.023.
  * [23] Z. Bajnok, F. Smirnov, Diagonal finite volume matrix elements in the sinh-Gordon model, Nucl. Phys. B 945 (2019) 114664.  arXiv:1903.06990, doi:10.1016/j.nuclphysb.2019.114664.
  * [24] M. Lencsés, G. Takács, Excited state TBA and renormalized TCSA in the scaling Potts model, JHEP 09 (2014) 052.  arXiv:1405.3157, doi:10.1007/JHEP09(2014)052.
  * [25] A. B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991) 391–394.  doi:10.1016/0370-2693(91)91737-G.
  * [26] V. A. Fateev, The Exact relations between the coupling constants and the masses of particles for the integrable perturbed conformal field theories, Phys. Lett. B 324 (1994) 45–51.  doi:10.1016/0370-2693(94)00078-6.