††institutetext: C.N. Yang Institute for Theoretical Physics, State University
of New York, Stony Brook, NY 11794-3840, USA

YITP-SB-2023-05

#  Ising Field Theory in a magnetic field:
$\varphi^{3}$ coupling at $T>T_{c}$

Hao-Lan Xu  and Alexander Zamolodchikov<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the "three particle coupling" $\Gamma_{11}^{1}(\xi)$, in $2d$ Ising
Field Theory in a magnetic field, as the function of the scaling parameter
$\xi:=h/(-m)^{15/8}$, where $m\sim T_{c}-T$ and $h\sim H$ are scaled deviation
from the critical temperature and scaled external field, respectively. The
"$\varphi^{3}$ coupling" $\Gamma_{11}^{1}$ is defined in terms of the residue
of the $2\to 2$ elastic scattering amplitude at its pole associated with the
lightest particle itself. We limit attention to the High-Temperature domain,
so that $m$ is negative. We suggest "standard analyticity":
$(\Gamma_{11}^{1})^{2}$, as the function of $u:=\xi^{2}$, is analytic in the
whole complex $u$-plane except for the branch cut from $-\infty$ to
$-u_{0}\approx-0.03585$, the latter branching point $-u_{0}$ being associated
with the Yang-Lee edge singularity. Under this assumption, the values of
$\Gamma_{11}^{1}$ at any complex $u$ are expressed through the discontinuity
across the branch cut. We suggest approximation for this discontinuity which
accounts for singular expansion of $\Gamma_{11}^{1}$ near the Yang-Lee
branching point, as well as its known asymptotic at $u\to+\infty$. The
resulting dispersion relation agrees well with known exact data, and with
numerics obtained via Truncated Free Fermion Space Approach. This work is part
of extended project of studying the S-matrix of the Ising Field Theory in a
magnetic field.

##  1 Introduction

In this work we report the next step in the extended project of studying the
analytic properties of the thermodynamic and correlation characteristics of
the 2d Ising Field Theory (IFT) in a magnetic field, as the functions of
complex scaling parameter. We understand the IFT as the Ising conformal field
theory (CFT) perturbed by its two relevant operators, the energy density
$\varepsilon(x)$ and the spin density $\sigma(x)$, as defined by the formal
action

$\displaystyle\mathcal{A}_{\text{IFT}}=\mathcal{A}_{\text{CFT}}+\frac{m}{2\pi}\,\int\varepsilon(x)d^{2}x+h\,\int\sigma(x)d^{2}x\,.$
(1)

In fonseca2003ising  the analyticity of the free energy was analyzed in both
High- and Low-Temperature regimes. Analyticity of the correlation length
$R_{c}:=M_{1}^{-1}$ ($M_{1}$ stands for the mass of the lightest particle of
the theory) was addressed in more recent work Xu:2022mmw , albeit only in the
High-T regime (The Low-T regime turns out to be more difficult; we plan to
return to this problem in the future.) Here we study the analyticity of the
$\varphi^{3}$ coupling, again limiting attention to the High-T domain. Our
approach is very similar to the analysis in Xu:2022mmw , and we often borrow
notations from that paper.

###  $\boldsymbol{\varphi}^{3}$ coupling

We denote $A_{1}$ the lightest stable particle of the theory, and $M_{1}$ its
mass. Also, as in Xu:2022mmw , we use the notation $S_{11}(\theta)$ the
$S$-matrix element of the $A_{1}+A_{1}\to A_{1}+A_{1}$ elastic scattering. It
is a function of the rapidity difference $\theta$. In IFT $S_{11}(\theta)$
always has a pole at $\theta=2i\pi/3$ corresponding to the particle $A_{1}$
itself in the direct channel. (There is of course the associated cross-channel
pole at $\theta=i\pi/3$.) The $\varphi^{3}$ coupling $\Gamma_{11}^{1}$ is
usually defined in terms of the residue

$\displaystyle(\Gamma^{1}_{11})^{2}=-i\,\underset{\theta=\frac{2i\pi}{3}}{\text{Res}}S_{11}(\theta)\,;$
(2)

by this definition $(\Gamma^{1}_{11})^{2}$ is positive in a unitary theory.
The vertex $\Gamma^{1}_{11}$ is dimensionless. It depends on the scaling
parameter, the dimensionless ratio of the coupling constants $m$ and $h$ in
(1). As in Refs.fonseca2003ising ; Xu:2022mmw , we use here the ratio
$\xi=h/|m|^{15/8}$ (in the High-T domain $m$ is negative). We also use the
related parameter $\eta=m/|h|^{8/15}=-\xi^{-8/15}$ when convenient. Because in
the High-T regime IFT is unsensitive to the change of the sign of $\xi$ (the
change $h\to-h$ in (1) can be compensated by the field transformation
$\sigma\to-\sigma$), $(\Gamma^{1}_{11})^{2}$ in (2) is an even function of
$\xi$ 111To the contrary, the vertex $\Gamma^{1}_{11}$ itself is an odd
function, $\Gamma^{1}_{11}(-\xi)=-\Gamma^{1}_{11}(\xi)$. The sign of
$\Gamma^{1}_{11}$ is fixed by the condition that $\Gamma^{1}_{11}$ is positive
at positive $\xi$., and it is convenient to regard it as function of the
variable $u:=\xi^{2}=h^{2}/(-m)^{15/8}$. In what follows we use abbreviated
notation $\Gamma:=\Gamma_{11}^{1}$, and study the closely related function

$\displaystyle\kappa(u):=-\frac{\sqrt{3}}{2}\Gamma^{2}(\xi)=i\,\frac{\sqrt{3}}{2}\,\underset{\theta=\frac{2i\pi}{3}}{\text{Res}}S(\theta)\,.$
(3)

Here and below $S(\theta)$ is the short-hand for $S_{11}(\theta)$.

The function $\kappa(u)$ can be analytically continued to complex values of
$u$, and understanding the analytic properties of this continuation is the
primary goal of the present work. As in Xu:2022mmw , our analysis here will be
based on the combination of exact results at the integrable points, as well as
on the numerical data obtained via Truncated Free Fermion Space Approach
(TFFSA) fonseca2003ising .

It is natural to expect that analytic properties of $\kappa(u)$ in the complex
$u$-plane are similar to those of the vacuum energy density
$G_{\text{high}}(u)$ fonseca2003ising , and of the mass $M_{1}(u)$ of the
lightest particle $A_{1}$ Xu:2022mmw . Namely, one can argue that $\kappa(u)$
is analytic in the whole $u$-plane with the branch cut from $-\infty$ to
$-u_{0}=-\xi_{0}^{2}\approx-0.03585\dots$, as shown in Fig.1. We refer to it
as the "standard analyticity". The branching point $u=-u_{0}$ represents the
Yang-Lee edge singularity resulting from accumulation of Yang-Lee zeros in the
thermodynamic limityang1952statistical ; lee1952statistical . The argument is
as follows. Generally, the positions and the residues of the poles of
$S(\theta)$ may have, besides the singularity at $-u_{0}$, additional
algebraic singularities (square root branching points) signifying collisions
of different poles at special values of $u$, with two branches of the square
root representing interchange of the colliding poles. However, the particular
pole at $\theta=2\pi i/3$ which we are interested in here stays put at all
values of $u$. There is an interesting movements of other poles and associated
zeros under changes of $u$, as explained in Ref.zamolodchikov2013ising . The
choreography of those movements is such that the residue (2) never does evolve
singularities. Every time another pole hits the pole at $2\pi i/3$, there is a
zero hitting it simultaneously, so that the residue remains regular.

$-u_{0}$$u$ Figure 1: Conjectured "standard analyticity" of $\kappa(u)$ in the
complex $u$-plane: $\kappa(u)$ is analytic everywhere except for the branch
cut $[-\infty:-u_{0}]$. The branching point $-u_{0}$ is the YL edge
singularity, and the branch cut is the line of first-order phase transitions.

Below we verify this analyticity conjecture using a combination of exact and
numerical data. We build an approximation for the discontinuity across the
above branch cut. Then the analyticity assumption leads to the dispersion
relation, and one can check it against special points where $\kappa(u)$ is
known exactly, as well as the numerical estimates obtained using TFFSA. We
find reasonable agreement (see Sec.4 and Sec.5). Alternative numerical data
for $\kappa(u)$ at real positive $u$ was previously obtained in Gabai:2019ryw
by rather different approach. Our results from the dispersion relation are in
good agreement with that numerics as well. All these results support the blue
standard analyticity conjecture formulated above.

##  2 Special points

The theory (1) is exactly integrable at two special points. At $u=0$ it
reduces to a theory of free fermi particles with $S(\theta)=-1$, so that the
residue (2) vanishes. Away from this point $\kappa(u)$ admits expansion in
powers of $u$,

$\displaystyle\kappa(u)=\sum_{n=1}^{\infty}\,\kappa^{(n)}\,u^{n}\,,$ (4)

with finite radius of convergence. The leading term in this expansion was
obtained in Zamolodchikov:2011wd  via form factor perturbation theory,

$\displaystyle\kappa^{(1)}=-18\sqrt{3}\,{\bar{s}}^{2}=-57.48165545\dots\,,$
(5)

where ${\bar{s}}=2^{1/12}e^{-1/8}\,A_{G}^{3/2}=1.35783834...$, and $A_{G}$ is
Glaisher’s constantmccoy2014two .

Another important point is $u=\infty$ (i.e. $m=0$), where the (1) becomes an
integrable theory with eight stable particles and rich factorizable S-matrices
zamolodchikov1989integrals 222The masses of the eight particles $A_{p}$,
$p=1,2,...,8$ at this integrable point are proportional to the components of
Frobenius vector of the Cartan matrix of the Lie algebra $E_{8}$, and the
structure of the S-matrices reflects in many ways the properties of the
$E_{8}$ root system zamolodchikov1989integrals . For this reason we refer to
this integrable theory as the ”$E_{8}$ theory”.. The value of $\kappa(u)$ at
$u=\infty$ can be straightforwardly extracted from the S-matrix
$S_{11}(\theta)$ (Eq.(72) in Appendix A),

$\displaystyle\kappa(\infty)=-3\,\frac{\tan(7\pi/15)\tan(11\pi/30)}{\tan(2\pi/15)\tan(3\pi/10)}=-104.6154448\dots\,.$
(6)

In the vicinity of this point $\kappa(u)$ expands in a convergent series in
integer powers of $\eta=-u^{-4/15}$,

$\displaystyle\kappa(u)=\kappa_{0}+\kappa_{1}\,\eta+\kappa_{2}\,\eta^{2}+\kappa_{3}\,\eta^{3}+\dots$
(7)

where $\kappa_{0}:=\kappa(\infty)$, see Eq.(6). In principle, the higher
coefficients can be obtained via form factor perturbation theory around the
integrable point $u=\infty$. In practice, this can be done for $\kappa_{1}$
(see Appendix A),

$\displaystyle\kappa_{1}=-202.04...$ (8)

while exact calculations of the higher coefficients are rather difficult. We
will estimate some coefficients using numerical data in Sec.4 and Sec.5.

Interesting point where IFT (1) is "infrared integrable" is the Yang-Lee edge
singularity which is located at pure imaginary magnetic field $h$, i.e. at
real negative $u=-u_{0}$. The position $u_{0}$ was estimated in
fonseca2003ising ; BazhanovYL ; Xu:2022mmw , $u_{0}\approx 0.035846(4)$. This
point is critical in usual sense - the analytic continuation of the mass
$M_{1}(u)$ vanishes at $u=-u_{0}$, i.e. the correlation length diverges
fisher1978yang . This means that the deep infrared asymptotic of the theory
(1) at $u=-u_{0}$ is described by CFT, which was identified in
cardy1985conformal  as the non-unitary minimal CFT $\mathcal{M}_{2/5}$ (in
what follows we refer to it as YLCFT). Away from the YL point the theory is
gapped, with the mass $M_{1}(u)\sim(u+u_{0})^{5/12}$, while its deep IR
asymptotic is described by the so called Yang-Lee QFT (henceforth referred to
as YLQFT),

$\displaystyle\mathcal{A}_{\text{YLQFT}}=\mathcal{A}_{\text{YLCFT}}+i\lambda(u)\,\int\varphi(x)d^{2}x\,,$
(9)

where $\varphi(x)$ is the only nontrivial scalar primary with the conformal
dimensions $(-\frac{1}{5},-\frac{1}{5})$ of the YLCFT. The coupling constant
$\lambda=\lambda(u)$ turns to zero at the YL point, $\lambda(-u_{0})=0$, in a
regular way $\lambda(u)\sim(u+u_{0})$. The YLQFT (9) itself is known to be
integrable, and its factorizable S-matrix is determined by the $2\to 2$
elastic amplitude cardy1985conformal

$\displaystyle
S_{11}^{(\text{YLQFT})}(\theta)=\frac{\sinh\theta+i\sin(2\pi/3)}{\sinh\theta-i\sin(2\pi/3)}\,.$
(10)

However, the full theory (1) is not integrable even in small vicinity of the
YL point, not even at the YL point $u=-u_{0}$ itself. While at $u$ close to
$-u_{0}$ (9) describes deep IR asymptotic of the theory (1), in the full
theory the YLQFT is dressed with an infinite tower of irrelevant operators,

$\displaystyle\mathcal{A}_{\text{IFT}}\ \to\
\mathcal{A}_{\text{YLQFT}}+\sum_{i}\,a^{i}\int O_{i}(x)d^{2}x\,,$ (11)

which become visible at intermediate scales $R\lesssim M_{1}^{-1}$. Here
$O_{i}$ are scalar operators of YLCFT with $\Delta_{i}=\bar{\Delta}_{i}>1$;
contributions of these operators generally break integrability Xu:2022mmw  (We
will say more about these operators in Sec.3 below). While $M_{1}$ measured in
the units of $|m|$ goes to zero when $u\to-u_{0}$, the couplings
$a^{i}=a^{i}(u)$ remain finite in this limit. Equivalently, if measured in the
units of $M_{1}$, these couplings tend to zero when $u\to u_{0}$. Therefore,
the value of $\kappa$ at $u=-u_{0}$ can be read out of the S-matrix (10),

$\displaystyle\kappa(-u_{0})=i\frac{\sqrt{3}}{2}\,\underset{\theta=\frac{2i\pi}{3}}{\text{Res}}\,S_{11}^{\text{YLQFT}}(\theta)=3.$
(12)

The negative value of $\Gamma^{2}$ at this point reflects non-unitary nature
of YLQFT.

##  3 Singular expansion and TTbar deformation

In the vicinity of the YL point $\kappa(u)$ admits singular expansion in
fractional powers of $u+u_{0}$, generated by the irrelevant operators $O_{i}$
in the effective action (11). Schematically, the structure of the expansion is

$\displaystyle\kappa(u)=\sum_{n=1}^{\infty}\
\sum_{i_{1},...,i_{n}}\,H_{i_{1},i_{2},...,i_{n}}\,a^{i_{1}}(u)...a^{i_{n}}(u)\,\left[M_{1}(u)\right]^{\delta_{i_{1}}+...+\delta_{i_{n}}}$
(13)

where $-\delta_{i}=2(1-\Delta_{i})<0$ are the mass dimensions of the couplings
$a^{i}$, and $H_{i_{1},...,i_{n}}$ are dimensionless coefficients. While the
couplings $a^{i}(u)$ are regular at $u=-u_{0}$,

$\displaystyle
a^{i}(u)=a_{0}^{i}+a_{1}^{i}\,(u+u_{0})+a_{2}^{i}\,(u+u_{0})^{2}+...$ (14)

the mass $M_{1}(u)$ turns to zero at $u=-u_{0}$ in a singular manner,
$M_{1}(u)\sim[\lambda(u)]^{5/12}\sim(u+u_{0})^{5/12}$, and further admits
expansion in fractional powers of $(u+u_{0})$ around the YL point, see
Xu:2022mmw , and Eq.(28) below. The terms in (13) can be interpreted as the
contributions from the perturbative expansion in the couplings $a^{i}(u)$. Of
course, the perturbation theory in the irrelevant operators is non-
renormalizable, and by itself it does not unambiguously define the
coefficients in (13) beyond the linear order in the couplings $a^{i}$.
(Important exception is the "least irrelevant" operator $T{\bar{T}}$ whose
contributions are unambiguous to all orders in the associated coupling
$\alpha$; we discuss its role below.) Nonetheless, (13) shows the nature of
the singular expansion of $\kappa(u)$ around the YL singularity.

The list of the scalar irrelevant operators entering the expansion (11) can be
extracted from the known operator content of YLCFT, aka the minimal CFT
$\mathcal{M}_{2/5}$. We arrange them in order of growing dimensions
$\Delta_{i}$, and exclude total derivatives as they do not contribute to the
action (11). Then the lowest of the irrelevant operators is

$\displaystyle(T{\bar{T}}):=L_{-2}{\bar{L}}_{-2}I\,,\qquad\ \ \
(\Delta,{\bar{\Delta}})=(2,2)\,.$ (15)

It is well known to generate the so-called TTbar deformation of the YLQFT,
which makes it possible to treat its contributions to all orders (see
smirnov2017space ; Cavaglia:2016oda ), as we will discuss shortly. Let us also
display the next three operators,

$\displaystyle\Xi:=L_{-4}{\bar{L}}_{-4}\varphi\,,\qquad\qquad(\Delta,{\bar{\Delta}})=(3.8,3.8)\,,$
(16)
$\displaystyle\Xi_{6}=L_{-6}{\bar{L}}_{-6}\varphi\,,\qquad\qquad(\Delta,{\bar{\Delta}})=(5.8,5.8)\,,$
(17) $\displaystyle(T{\bar{T}})^{3}:=L_{-2}^{3}{\bar{L}}_{-2}^{3}I\,,\
\qquad(\Delta,{\bar{\Delta}})=(6,6)\,.$ (18)

Thus, the least irrelevant terms in (11) are

$\displaystyle\eqref{aeff0}=\mathcal{A}_{\text{YL
QFT}}+\frac{\alpha(u)}{\pi^{2}}\int(T{\bar{T}})(x)d^{2}x+\frac{\beta(u)}{2\pi}\,\int\Xi(x)d^{2}x+$
$\displaystyle\qquad\qquad\qquad\frac{\gamma(u)}{2\pi}\int\Xi_{6}(x)d^{2}x+\frac{\alpha_{5}(u)}{\pi^{2}}\int(T{\bar{T}})^{3}(x)d^{2}x\
+\ ...$ (19)

where we used individual notations for the couplings. Like all the couplings
in (11), the couplings $\alpha,\beta,\gamma$ in (19) are functions of $u$,
regular at the YL point; they admit convergent expansions around the YL point,

$\displaystyle\alpha(u)=\alpha_{0}+\alpha_{1}\,(u+u_{0})+\alpha_{2}\,(u+u_{0})^{2}+...\,,\qquad\qquad-\delta_{\alpha}=-2\,,$
$\displaystyle\beta(u)=\beta_{0}+\beta_{1}\,(u+u_{0})+\beta_{2}\,(u+u_{0})^{2}+...\,,\
\qquad\qquad-\delta_{\beta}=-5.6\,,$ (20)
$\displaystyle\gamma(u)=\gamma_{0}+\gamma_{1}\,(u+u_{0})+\gamma_{2}\,(u+u_{0})^{2}+...\,,\qquad\qquad\
\,-\delta_{\gamma}=-9.6\,,$
$\displaystyle\alpha_{5}(u)=\alpha_{5,0}+\alpha_{5,1}\,(u+u_{0})+\alpha_{5,2}\,(u+u_{0})^{2}+...\,,\qquad-\delta_{\alpha_{5}}=-10\,,$

and similarly for further couplings not displayed in (19). In (20) we also
show the mass dimensions of the couplings. The leading coefficients for the
first two coupling parameters were determined in Xu:2022mmw  (see also
fonseca2003ising ),

$\displaystyle\alpha_{0}\,|m|^{2}=-1.32(5)\,,\qquad\frac{\beta_{0}}{2\pi}\,|m|^{\frac{28}{5}}=+0.72(6)\,.$
(21)

In Sec.5 below we use TFFSA to obtain numerical estimate (not very precise)
$\alpha_{1}|m|^{2}=18.0(3.5)$ of the first sub-leading coefficient in (20).
Estimates of higher coefficients in (20) are not yet available.

With the account of the three lowest irrelevant operators displayed in (19)
the expansion (13) takes the form

$\displaystyle\kappa(u)=3+H_{1}\,\alpha(u)M_{1}^{2}(u)+H_{2}\,\alpha^{2}(u)M_{1}^{4}(u)+$
$\displaystyle
H_{1}^{\Xi}\,\beta(u)M_{1}^{\frac{28}{5}}(u)+H_{3}\,\alpha^{3}(u)M_{1}^{6}(u)+H_{4}\,\alpha^{4}(u)M_{1}^{8}(u)+$
(22)
$\displaystyle\left({H}_{2}^{\Xi}\,\alpha(u)\beta(u)+H_{1}^{\Xi_{6}}\,\gamma(u)\right)\,M_{1}^{\frac{48}{5}}(u)+\left(H_{5}\,\alpha^{5}(u)+{\tilde{H}}_{1}\,\alpha_{5}(u)\right)M_{1}^{10}(u)+...$

where the leading term comes from the exact number (12). The omitted terms
have the powers of $M_{1}(u)$ 11.6 and higher. Note that apart from the terms
linear in the couplings, (22) involves higher powers of $\alpha$, as well as
the term $\alpha\beta$. These are associated with the higher order
perturbation theory in $\alpha$. Generally, such terms can not be determined
within the perturbation theory itself, as they depend on short-distance
behavior of the full IFT. However, due to special properties of the operator
$(T{\bar{T}})$ many such terms are uniquely defined. As was already mentioned,
this operator generates the TTbar deformation, which allows to collect
explicitly the higher order terms in $\alpha$. We will not explain it here,
referring the reader to smirnov2017space ; Cavaglia:2016oda ; Dubovsky:2017cnj
. In particular, the TTbar deformation leads to simple "dressing" of the two-
particle elastic S-matrix smirnov2017space ; Dubovsky:2017cnj ,

$\displaystyle S^{(\alpha)}(\theta)=\exp\left\\{-i\alpha
M_{1}^{2}\,\sinh\theta\right\\}\,S^{(0)}(\theta)\,,$ (23)

where $S^{(0)}(\theta)$ is the S-matrix of the undeformed theory333For ”TTbar
dressing” of the full S-matrix see Dubovsky:2017cnj .. If $S^{(0)}(\theta)$
has a pole at $\theta=2\pi i/3$, so does the deformed $S^{(\alpha)}(\theta)$,
with the residue determined by (23). Therefore the "TTBar deformed"
$\kappa^{(\alpha)}$ relates to $\kappa^{(0)}$ at $\alpha=0$ as

$\displaystyle\kappa^{(\alpha)}=\exp\left\\{\frac{\sqrt{3}}{2}\,\alpha\,M_{1}^{2}\right\\}\,\kappa^{(0)}\,.$
(24)

In our context only five leading terms of the expansion of the exponential in
(23) and (24) are useful. This is because the higher order terms interfere
with the contributions of higher level descendants of the identity operator in
the effective action (19). Thus, the term $\sim\alpha^{5}$ mixes with the
leading contribution from $(T{\bar{T}})^{3}$ ($\alpha^{5}$ has the same mass
dimension as $\alpha_{5}$)444From the point of view of straightforward
perturbation theory, the TTbar deformation itself generates an infinite tower
of higher descendants of $I$, which mix with the higher terms in (19).
Alternatively, one can think of the term $\alpha(T{\bar{T}})$ as the ”germ” of
the full TTbar deformation; with this convention (23) is exact to all orders
in $\alpha$. Likewise, the term $\alpha_{5}(T{\bar{T}})^{3}$, if understood as
the germ of generalized TTbar deformation smirnov2017space , leads to exact
dressing by the factor  $\exp\left\\{-i\alpha
M_{1}^{2}\sinh(\theta)-i\alpha_{5}M_{1}^{10}\sinh(5\theta)+...\right\\}$ . For
the purpose of this work, the expansion

$\displaystyle\kappa^{(\alpha)}=\left(1+\frac{\sqrt{3}}{2}\alpha
M_{1}^{2}+\frac{3}{8}\alpha^{2}M_{1}^{4}+\frac{\sqrt{3}}{16}\alpha^{3}M_{1}^{6}+\frac{3}{128}\alpha^{4}M_{1}^{8}\right)\,\kappa^{(0)}+O(\alpha^{5})$
(25)

is more than sufficient. Here $\kappa^{(0)}$ is given by the expansion (22)
with $\alpha(u)$ set to zero,

$\displaystyle\kappa^{(0)}(u)=3+H_{1}^{\Xi}\,\beta(u)\,M_{1}^{\frac{28}{5}}(u)+H_{1}^{\Xi_{6}}\,\gamma(u)\,M_{1}^{\frac{48}{5}}(u)+O(M_{1}^{10})\,.$
(26)

The second and the third terms in the r.h.s. of (26) are linear in $\beta$ and
$\gamma$; these terms express the leading order contributions from the
operators $\Xi$ and $\Xi_{6}$ in (19). As usual, unlike the higher orders,
these coefficients are determined unambiguously using the perturbation theory.
The calculation involves the four particle form factors of the operators $\Xi$
and $\Xi_{6}$, and is rather bulky. We will present detailed calculations
elsewhere. Here we just quote the result XuEtAl2022 ,

$\displaystyle
H^{\Xi}=-\frac{164480625}{100352}\frac{5^{\frac{1}{4}}\,\pi^{\frac{21}{5}}\,\Gamma(\frac{3}{5})\,\Gamma(\frac{2}{3})^{\frac{38}{5}}\,\Gamma(\frac{4}{5})\,\Gamma(\frac{8}{3})^{2}}{2^{\frac{1}{5}}\,\Gamma(\frac{1}{6})^{10}\,\Gamma(\frac{5}{6})^{\frac{12}{5}}\,\Gamma(\frac{25}{6})^{2}}=-0.004876389\dots\,.$
(27)

One can note that this coefficient is exceedingly small numerically, and it
does not have any noticeable effect on the analysis in Sec.4 and Sec.5 below.
We also have reasons to believe that $H^{\Xi_{6}}$ is yet smaller. Because of
this, in what follows we ignore the $\beta$\- and $\gamma$-contributions in
(26).

Important ingredient in (13) is the mass $M_{1}(u)$ which enjoys the singular
expansion around the YL point Xu:2022mmw ,

$\displaystyle\frac{M_{1}(u)}{|m|}=(u+u_{0})^{\frac{5}{12}}\left(b(u)+(u+u_{0})^{\frac{5}{6}}\,c(u)+(u+u_{0})^{\frac{5}{3}}\,e(u)+(u+u_{0})^{\frac{7}{3}}\,d(u)+\dots\right)\,,\quad$
(28)

where $b(u),c(u),d(u),e(u)...$ are regular at $u=-u_{0}$,

$\displaystyle b(u)=b_{0}+b_{1}\,(u+u_{0})+b_{2}\,(u+u_{2})^{2}+...$ (29)
$\displaystyle c(u)=c_{0}+c_{1}\,(u+u_{0})+...$ (30)

and similarly for $e(u)$, $d(u)$ and the higher coefficient functions. Some
lower coefficients in (29),(30) were determined in Xu:2022mmw ,

$\displaystyle b_{0}=4.228(5),\qquad b_{1}=21.9(9)\,,\qquad c_{0}=-14.4(6)\,.$
(31)

For the Ising effective action near the YL point, Eq.(19), the expansion (13)
combined with (20), generates expansion of $\kappa(u)$ in fractional powers of
$u+u_{0}$

$\displaystyle\kappa(u)=3+k_{1}(u+u_{0})^{\frac{5}{6}}+k_{2}(u+u_{0})^{\frac{5}{3}}+\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\qquad
k_{3}(u+u_{0})^{\frac{11}{6}}+k_{4}(u+u_{0})^{\frac{5}{2}}+k_{5}(u+u_{0})^{\frac{8}{3}}+...\,,$
(32)

where the coefficients are expressed in terms of the coefficients of the
expansion of $\alpha(u)$ in (20), as well as the coefficients in (28), e.g.

$\displaystyle
k_{1}=\frac{3\sqrt{3}}{2}\,b_{0}^{2}\,\alpha_{0}\approx-61.3\,,$ (33)
$\displaystyle k_{2}=\frac{9}{8}\,b_{0}^{2}\,\alpha_{0}^{2}\approx 35.0\,,$
(34) $\displaystyle
k_{3}=\frac{3\sqrt{3}}{2}\,(2b_{0}b_{1}\,\alpha_{0}+\alpha_{1})\approx-681\,,$
(35)

where the estimate of $\alpha_{0}$, Eq.(21), is used to produce numerics for
$k_{1}$ and $k_{2}$. Higher $k_{n}$ involve higher coefficients in (21) and
(28). We give numerical estimate of more $k_{n}$ in Sec.4 and Sec.5.

##  4 Analyticity and Dispersion relation

According to our analyticity conjecture, the function $\kappa(u)$ admits
analytic continuation to the complex $u$-plane, with no singularities except
for the branch cut along the real axis from $-\infty$ to $-u_{0}$, as shown in
Fig.1. Under this assumption the values of $\kappa(u)$ everywhere in the
$u$-plane are expressed in terms of the discontinuity across the branch cut,
$\text{Disc}\,\kappa(u):=\kappa(u+i0)-\kappa(u-i0)=2i\,\Im m\,\kappa(u+i0)$

$\displaystyle\kappa(u)=-u\,\int_{u_{0}}^{\infty}\,\frac{dx}{\pi}\,\frac{\Im
m\,\kappa(-x)}{x(x+u)}$ (36)

where we have taken into account that $\kappa(0)=0$. The integral in (36)
converges because $\Im m\,\kappa(-x)\simeq x^{15/16}$ as $x\to+\infty$, see
(7). Here we are going to build certain approximation for the discontinuity
$\text{Disc}\,\kappa(u)$ and then check the numerics for $\kappa(u)$ obtained
via (36) against exact numbers in Sec.2 and Sec.3. In addition, in the next
section we use TFFSA to obtain numerical values of $\kappa(u)$ at generic real
$u>-u_{0}$, and test the dispersion relation against that numerics as well.

We found it convenient to to use the scaling variable

$\displaystyle\eta=-u^{-4/15}$ (37)

instead of $u$, and define

$\displaystyle{\hat{\kappa}}(\eta)=\kappa\left((-\eta)^{-15/4}\right)\,.$ (38)

The variable transformation maps the $u$-plane in Fig.1 to the wedge

$\displaystyle-4\pi/15<\text{arg}(-\eta)<4\pi/15$ (39)

in the $\eta$-plane, with the positive real $u$ mapped to the negative part of
the real axis of $\eta$, and negative real $u$ mapped to the rays
$\eta=y\,e^{\pm 4\pi i/15}$ with negative real $y$. In these variable, the
dispersion relation (36) takes the form

$\displaystyle{\hat{\kappa}}(\eta)=-\frac{15}{4\pi}\,\int_{0}^{Y_{0}}\,\frac{y^{\frac{11}{4}}\,\Delta(-y)\,dy}{\big{(}y^{\frac{15}{4}}+(-\eta)^{\frac{15}{4}}\big{)}}$
(40)

where it is assumed that $\eta$ lays in the wedge (39), and

$\displaystyle Y_{0}=u_{0}^{-4/15}=2.4293\dots\,.$ (41)

The function

$\displaystyle\Delta(y)=\frac{1}{2i}\,\left[{\hat{\kappa}}\big{(}y\,e^{\frac{4\pi
i}{15}+i0}\big{)}-{\hat{\kappa}}\big{(}y\,e^{\frac{4\pi
i}{15}-i0}\big{)}\right]=\Im m\,{\hat{\kappa}}\big{(}y\,e^{\frac{4\pi
i}{15}+i0}\big{)}$ (42)

represents the discontinuity across the branch cut in Fig.1. To make (40)
useful we need to build some approximation for $\Delta(y)$.

### Approximating $\boldsymbol{\Delta(y)}$

As was mentioned in Sec.2, the function ${\hat{\kappa}}(\eta)$ admits
convergent expansion (7) in integer powers of $\eta$. This generates expansion
of the function (42),

$\displaystyle\Delta(y)=\sum_{n=0}^{\infty}\,\kappa_{n}\,\sin\left(\frac{4\pi
n}{15}\right)\,y^{n}=\kappa_{1}\,\sin\left(\frac{4\pi}{15}\right)\,y+\kappa_{2}\,\sin\left(\frac{8\pi}{15}\right)\,y^{2}+...$
(43)

The first two coefficients are

$\displaystyle\Delta(0)=0\,,\qquad\Delta^{\prime}(0)=\kappa_{1}\,\sin\left(\frac{4\pi}{15}\right)=-150.14\dots\,,$
(44)

where we have used (8).

On the other hand, ${\hat{\kappa}}(\eta)$ enjoys the singular expansion near
the YL point $y+Y_{0}=0$,

$\displaystyle{\hat{\kappa}}(y)=3+\epsilon^{\frac{5}{6}}\,K_{1}(\epsilon)+\epsilon^{\frac{5}{3}}\,K_{2}(\epsilon)+\epsilon^{\frac{5}{2}}\,K_{3}(\epsilon)+\epsilon^{\frac{19}{6}}\,K_{4}(\epsilon)+\epsilon^{\frac{10}{3}}\,K_{5}(\epsilon)+\epsilon^{\frac{25}{6}}\,K_{6}(\epsilon)+\cdots\,,\quad$
(45)

Here and below we use the notation $\epsilon:=-(y+Y_{0})$, so that

$\displaystyle
u+u_{0}=-(-y)^{-15/4}+Y_{0}^{-15/4}=\frac{15}{4}\,Y_{0}^{-\frac{19}{4}}\,\epsilon-\frac{285}{32}Y_{0}^{-\frac{23}{4}}\,\epsilon^{2}+...$
(46)

The coefficient functions $K_{n}(\epsilon)$ in (45) are regular around the YL
point

$\displaystyle
K_{n}(\epsilon)=K^{(0)}_{n}+K^{(1)}_{n}\,\epsilon+K^{(2)}_{n}\,\epsilon^{2}+\cdots\,.$
(47)

Like in (3), the coefficients $K_{n}^{(l)}$ are related in a straightforward
way to the coefficients of the expansion of the $(T{\bar{T}})$ coupling

$\displaystyle\alpha=\alpha_{0}+{\hat{\alpha}_{1}}\,\epsilon+{\hat{\alpha}}_{2}\,\epsilon^{2}+...$
(48)

and the mass,

$\displaystyle
M_{1}/|h|^{\frac{8}{15}}=\epsilon^{\frac{5}{12}}\,({\hat{b}}_{0}+{\hat{b}}_{1}\,\epsilon+{\hat{b}}_{2}\,\epsilon^{2})+\epsilon^{\frac{5}{4}}\,({\hat{c}}_{0}+{\hat{c}}_{1}\,\epsilon)+\epsilon^{\frac{25}{12}}{\hat{e}}_{0}+\epsilon^{\frac{11}{4}}\,{\hat{d}}_{0}+O(\epsilon^{\frac{19}{6}})\,,$
(49)

which are obtained by re-expanding (20) and (28) in terms of $\epsilon$. The
coefficient $\alpha_{0}$ is the same as in (21) while

$\displaystyle{\hat{\alpha}}_{1}\,|m|^{2}=-0.85\pm 0.1$ (50)

will be determined in Sec.5. Moreover, analysis in Sec.5 suggests that
${\hat{\alpha}}_{2}$ is small, and within the  (low) accuracy of our numerics
we will neglect its contributions. The coefficients ${\hat{b}}_{0}$,
${\hat{b}}_{1}$, and ${\hat{c}_{0}}$ in (49) are expressed through the numbers
(31), see corresponding entries in Tab.1. Moreover, it was found in Xu:2022mmw
that the mass $M_{1}$ is approximated well at $y+Y_{0}\lesssim 1$ by all six
terms explicitly displayed in (49), with the numerical values of the
coefficients in Tab.1.

${\hat{b}}_{0}$ | ${\hat{b}}_{1}$ | ${\hat{b}}_{2}$ | ${\hat{c}}_{0}$ | ${\hat{c}}_{1}$ | ${\hat{d}}_{0}$ | ${\hat{e}}_{0}$
---|---|---|---|---|---|---
$3.0754$ | $0.8932$ | $-0.8618$ | $-0.9412$ | $0.7835$ | $0.3134$ | $0.2880$
Table 1: Numerical values of the expansion coefficients in Eq.(49).

It is straightforward to obtain the expansion of the discontinuity $\Delta(y)$
by termwise analytic continuation $\epsilon\to e^{i\pi}\epsilon$ in (45),

$\displaystyle\Delta(y)=(-\epsilon)^{\frac{5}{6}}\,\tilde{K}_{1}(\epsilon)+(-\epsilon)^{\frac{5}{3}}\,\tilde{K}_{2}(\epsilon)+(-\epsilon)^{\frac{5}{2}}\,\tilde{K}_{3}(\epsilon)+\cdots$
(51)

where $\epsilon=y+Y_{0}$ is now negative ($\Delta(y)$ vanishes at $y<-Y_{0}$),
and

$\displaystyle{\tilde{K}}_{n}(\epsilon)=\sin(\nu_{n}\,\pi)\,K_{n}(\epsilon)\,,$
(52)

with $\nu_{n}$ standing for the exponent in $\epsilon^{\nu}\,K_{n}(\epsilon)$
term in (45). Numerical values of the coefficients
${\tilde{K}}_{n}^{(l)}=(-)^{l}\,\sin(\pi\nu_{n})\,K_{n}^{(l)}$ at nine leading
terms of the expansion (51) obtained from the known coefficients in (49) and
(48) are collected in Tab.2

$\tilde{K}_{1}^{(0)}$ | $\tilde{K}_{2}^{(0)}$ | $\tilde{K}_{1}^{(1)}$ | $\tilde{K}_{3}^{(0)}$ | $\tilde{K}_{2}^{(1)}$ | $\tilde{K}_{1}^{(2)}$ | $\tilde{K}_{4}^{(0)}$ | $\tilde{K}_{5}^{(0)}$ | $\tilde{K}_{3}^{(1)}$
---|---|---|---|---|---|---|---|---
$2.7481$ | $7.274$ | $-1.103$ | $10.78$ | $-1.400$ | $-1.655$ | $-0.5601$ | $10.47$ | $-0.8974$
Table 2: Numerical values of the coefficients in the expansion (51) with
${\tilde{K}}_{n}(\epsilon)=\sum_{l=0}^{\infty}\,{\tilde{K}}_{n}^{l}\,\epsilon^{l}$.

We propose the following approximation for $\Delta(y)$ in the whole domain of
integration in (40):

$\displaystyle\Delta_{\text{approx}}(y)=\epsilon^{\frac{5}{6}}\tilde{K}_{1}^{(0)}+\epsilon^{\frac{5}{3}}\tilde{K}_{2}^{(0)}+\epsilon^{\frac{11}{6}}\tilde{K}_{1}^{(1)}+\epsilon^{\frac{5}{2}}\tilde{K}_{3}^{(0)}+\epsilon^{\frac{8}{3}}\tilde{K}_{2}^{(1)}+\epsilon^{\frac{17}{6}}\tilde{K}_{1}^{(2)}$
$\displaystyle+\epsilon^{\frac{19}{6}}\tilde{K}_{4}^{(0)}+\epsilon^{\frac{10}{3}}\tilde{K}_{5}^{(0)}+\epsilon^{\frac{7}{2}}\tilde{K}_{3}^{(1)}+\epsilon^{\frac{11}{3}}\tilde{K}_{2}^{(2)}+\epsilon^{\frac{23}{6}}\tilde{K}_{1}^{(3)}+\epsilon^{\frac{25}{6}}\tilde{K}_{6}^{(0)}\,,$
(53)

with ${\tilde{K}}_{1}^{(0)},...,K_{3}^{(1)}$ taken from Tab.2, and the last
three coefficients $\tilde{K}_{2}^{(2)}\,,\tilde{K}_{1}^{(3)}\,,$ and
$\tilde{K}_{6}^{(0)}$ determined by demanding that $\Delta_{\text{approx}}(y)$
meets the two conditions (44), and that the dispersion integral (40) evaluated
with $y=-Y_{0}$ returns returns exact value ${\hat{\kappa}}(-Y_{0})=3$ (see
(12)),

$\tilde{K}_{2}^{(2)}=-182.959\,,\quad\tilde{K}_{1}^{(3)}=+238.478\,,\quad\tilde{K}_{6}^{(0)}=-66.657\,.$
(54)

The shape of the function $\Delta_{\text{approx}}(y)$ at $y\in[-Y_{0},0]$ is
shown in Fig.2. Despite somewhat artificial way of constructing this
approximation, we believe that it is very close to actual discontinuity (42).
This expectation can be verified against exact numbers (Sec.2) and numerical
data obtained in Sec.5.

Figure 2: Approximation (53) for the discontinuity $\Delta(-y)$ in the
dispersion relation (40).

### Approximation $\boldsymbol{\Delta_{\text{approx}}(y)}$ vs Data

In constructing the approximation (53) we have used the exact slope
$\kappa_{1}$, Eq.(8). However, the leading term $\kappa_{0}=\kappa(\infty)$ in
(7) does not contribute to the discontinuity. The dispersion integral (40)
with our approximation (53), evaluated at $\eta=0$, results in

$\displaystyle\kappa_{\text{disp}}(\infty)=-\frac{15}{4\pi}\,\int_{0}^{Y_{0}}\,\frac{\Delta_{\text{approx}}(-y)}{y}\,dy=-103.206\,,$
(55)

to be compared with the exact value (6). The approximation (53) suggests
estimates for further coefficients in the expansion (4), e.g.

$\displaystyle\kappa_{2\,\text{disp}}=-213.278\,,\qquad\kappa_{3\,\text{disp}}=-192.962\,.$
(56)

Another exactly known quantity is the slope (5) of $\kappa(u)$ at $u=0$. The
dispersion relation yields

$\displaystyle\kappa^{(1)}_{\text{disp}}=-\frac{15}{4\pi}\,\int_{0}^{\infty}\,y^{\frac{11}{4}}\,\Delta_{\text{approx}}(-y)\,dy=-56.9019\,.$
(57)

Again, it agrees well with the exact value (5). Our approximation gives
predictions for several higher terms of the expansion (4),

$\displaystyle\kappa^{(n)}_{\text{disp}}=(-)^{n}\,\frac{15}{4\pi}\,\int_{0}^{\infty}\,y^{\frac{15}{4}\,n-1}\,\Delta_{\text{approx}}(-y)\,dy\,.$
(58)

Thus

$\displaystyle\kappa_{\text{disp}}^{(2)}=317.754\,,\qquad\kappa_{\text{disp}}^{(3)}=-3524.51\,.$
(59)

Figure 3: Numerical estimates of $\kappa(u)$ obtained by fitting (62) to TFFSA
data (red bullets), compared to $\kappa_{\text{disp}}(u)$ from the dispersion
relation (40) with the approximation (53).

In the next Section we will use TFFSA to develop numerics for $\kappa(u)$,
both at positive and negative $u$. Although our numerical estimates are not
very precise (we present some details of our numerical analysis in Sec.5
below), Fig.3 we compare them with $\kappa_{\text{disp}}(u)$, the integral
(40) evaluated with $\Delta_{\text{approx}}(y)$ in the integrand. The numerics
for $g^{2}_{111}(\eta):=-4\sqrt{3}\,\hat{\kappa}(\eta)$ at negative $\eta$
(positive $u$) was previously obtained in Gabai:2019ryw  using much different
analysis of the TFFSA data. We compare these results with
$\hat{\kappa}_{\text{disp}}(\eta)$ in Fig.4. The agreement seems to support
the standard analyticity conjecture.

Figure 4: Numerical data for
$g^{2}_{111}(\eta):=-4\sqrt{3}\,{\hat{\kappa}}(\eta)$ from Gabai:2019ryw
(blue dots), compared with the same quantity evaluated from the dispersion
relation (40) with the approximation (53) (red curve). For better visibility,
we plot $\log(g_{111}^{2})$ against $-\eta$.

##  5 Numerical Data from TFFSA

Additional data on $\kappa(u)$ can be obtained numerically, using the TFFSA.
As was mentioned, TFFSA was applied previously to estimate the scattering
amplitude and in particular the residue (3) in Gabai:2019ryw . We use
different approach, based on the finite-size correction to the two lowest
energy levels. TFFSA is a routine which computes the energy levels $E_{n}(R)$
of the IFT in the geometry of a cylinder, with the spatial coordinate $x$
compactified on a circle of circumference $R$. With sufficiently high
truncation level $L$ TFFSA returns very accurate estimates for $E_{n}(R)$ as
long as $R\,\lesssim\,7\,|h|^{-8/15}$, but at larger $R$ the accuracy quickly
deteriorates due to the truncation effects. All numerics quoted below was
obtained at the truncation level $L=13$ 555Here we assume the same definition
of the ”truncation level” as given in fonseca2003ising ; Xu:2022mmw ; it
differs by a factor of $1/2$ from the definition in Gabai:2019ryw ..

Since in this geometry the lowest excited state $n=1$ may be interpreted as a
single particle $A_{1}$ at rest on the circle, the gap $\Delta
E_{1}(R):=E_{1}(R)-E_{0}(R)$ is expected to converge to $M_{1}$ as
$R\to\infty$. Moreover, when $M_{1}\neq 0$ the leading finite-size corrections
are expressed exactly in terms of the residue (3) and the analytic
continuation of the amplitude $S(\theta)$ Yurov:1989yu ,

$\displaystyle\frac{\Delta
E_{1}(R)-M_{1}}{M_{1}}=\kappa\,e^{-\frac{\sqrt{3}}{2}M_{1}R}-\int\frac{d\theta}{2\pi}\cosh\theta\left[S\big{(}\theta+\frac{\pi
i}{2}\big{)}-1\right]e^{-M_{1}R\cosh\theta}+\cdots\,,\,\,$ (60)

where the dots represent further corrections which decay yet faster at large
$R$. The two terms in the r.h.s. of (60) express contributions of the particle
$A_{1}$ winding around the space-time cylinder once, as shown in Fig.5. Other
stable particles, when present (as in IFT at $u>u_{2}=0.0640(25)$), lead to
additional contributions to the gap (60). We describe some of these
contributions in Appendix B.

Figure 5: The diagrams are presenting the leading finite size corrections to
$\Delta E_{1}(R)$, see (60) and (84). Figure 6: Plots of $\Delta
E_{1}(R)/M_{1}$ at different $\eta$ (real $h$). The exponential approach to
$1$ is in accord with (60) with negative $\kappa$. Figure 7: Plots of $\Delta
E_{1}(R)/M_{1}$ at different $y$ for (imaginary $h$). The approach to $1$ is
consistent with (60) with positive $\kappa$.

The gaps $\Delta E_{1}(R)$ obtained via TFFSA with $L=13$, at few sample
values of $u$ are shown in Fig.6 and Fig.7. The exponential approach to
$M_{1}$ at large $R$ consistent with the leading term in (60) is clearly
visible, except for when $u$ is close to the YL point $-u_{0}$ where the
behavior at $R\lesssim 7\,|h|^{-8/15}$ becomes rather power-like. Rough
estimate of numerical values of $\kappa(u)$ can be obtained by fitting the
leading term in the r.h.s. of (60) to the TFFSA data.

Unfortunately, precision of such estimate is low. The reason is that
truncation effects spoil the TFFSA data for $\Delta E_{1}(R)$ at large $R$
(typically at $MR\gtrsim 20$) where the leading term in (60) dominates. On the
other hand, at smaller values of $MR$ the correction terms in the large $R$
expansion (60) become comparable to the leading exponential term. Thus, the
second term in the r.h.s. of (60) has the leading large-$R$ behavior

$\displaystyle-{\sigma}\,(M_{1}R)^{-1/2}\,e^{-M_{1}R}\,,$ (61)

where $\sigma=[S(i\pi/2)-1]/\sqrt{2\pi}$; with $\sigma\sim\kappa$ 666The ratio
$\sigma/\kappa$ is $\approx-3.2$ at $u=+\infty$. It becomes positive at
$-0.0036\lessapprox u\lessapprox 12.5$ (turning to zero at the boundaries of
this segment) and it approaches $-1.7166..$ when $u\to-u_{0}$. Obviously, this
ratio diverges at $u=0$, where $\sigma$ remains finite; in some vicinity of
this point the ration actually is very large, which makes it difficult to
estimate $\kappa$ near the point $u=0$. the first term dominates only at
$MR\gtrsim 10$. Therefore the window in which the above fitting is meaningful
is not sufficiently wide, while the fitting results were rather sensitive to
the choice of the domain of $R$ where fitting was performed. We estimated the
accuracy of this approach by the dependence on the fitting interval to be
typically about $15-25\%$. It can be somewhat improved by fitting the data to
the two-parameter expression

$\displaystyle\kappa\,e^{-\frac{\sqrt{3}}{2}M_{1}R}-{\sigma}\,(M_{1}R)^{-1/2}\,e^{-M_{1}R}\,,$
(62)

via $\kappa$ and $\sigma$. The data presented in Fig.3 were obtained through
this two-parameter fitting. We estimate the accuracy at generic $u$ as
$10-20\%$.

Accuracy of the estimates obtained using (62) deteriorates even more when $u$
is close to one of the three special points mentioned in Sec.3. Since the
correlation length $M^{-1}$ diverges at the YL point, the window available for
the fitting shrinks as $u$ approaches $-u_{0}$. When $u$ is close to zero
$\kappa(u)\sim u$ becomes small, and within the acceptable range it is barely
detectable on the background of the higher finite-size corrections. In
vicinity of the $E_{8}$ point $\eta=0$ the theory involves more than one
stable particle; these additional particles give rise to additional sub-
leading terms which are significant in the interval of $R$ available for
fitting. However, near these points the accuracy can be significantly improved
with the use of exact scattering amplitudes at these points.

### Vicinity of Free Fermion point

At $u=0$ we are dealing with the free fermion theory, with trivial S-matrix
$S(\theta)=-1$ and $\kappa(0)=0$. In high-T domain, the finite size ground
state is the Neveu-Schwarz vacuum, while the first excited state is understood
as one particle $A_{1}$ at rest over the Ramond vacuum. Therefore, at $u=0$
the finite size gap is given by

$\frac{\Delta
E^{\text{FF}}_{1}(R)-M_{1}}{M_{1}}=-\int_{-\infty}^{+\infty}\frac{d\theta}{2\pi}\cosh\theta\,\log\Big{(}\frac{1-e^{-M_{1}R\cosh\theta}}{1+e^{-M_{1}R\cosh\theta}}\Big{)}\,,$
(63)

which accounts for contributions from particles $A_{1}$ winding around the
cylinder any number of times. At small $u$ the amplitude $S(\theta)$ acquires
a correction

$S(\theta)=-\Big{(}1+\frac{i\kappa A(\theta)}{\sinh\theta}\Big{)}\,,\qquad
A(\theta)=A_{\text{pole}}(\theta)+A_{\sigma}(\theta)\,,$ (64)

where:

$A_{\text{pole}}(\theta)=-\frac{2}{\sqrt{3}}\,\,\frac{\sinh^{2}\theta}{\sinh^{2}\theta+\frac{3}{4}}\,,$
(65)

and the absorptive part $A_{\sigma}(\theta)$ is determined by the inelastic
processes, see Zamolodchikov:2011wd . This leads to correction terms to the
finite size gap (63),

$\displaystyle\frac{\Delta
E_{1}(R)-M_{1}}{M_{1}}=-\int_{-\infty}^{+\infty}\frac{d\theta}{2\pi}\cosh\theta\,\log\Big{(}\frac{1-e^{-M_{1}R\cosh\theta}}{1+e^{-M_{1}R\cosh\theta}}\Big{)}+\qquad\qquad$
$\displaystyle\qquad\qquad+\kappa\,e^{-\frac{\sqrt{3}}{2}M_{1}R}-\frac{2}{\sqrt{3}}\,\kappa\int\frac{d\theta}{2\pi}\,A(i\pi/2+\theta)\,e^{-M_{1}R\cosh\theta}\,.$
(66)

When $M_{1}R\gtrsim 1$ the integral in (66) is dominated by $\theta\lesssim
1$, where $A_{\sigma}(\theta)$ is numerically small as compared to
$A_{\text{pole}}(\theta)$ (see Fig.3 in Zamolodchikov:2011wd ), and one can
neglect its contribution to the finite size gap. Therefore, we have used (66)
with $A(\theta)=A_{\text{pole}}(\theta)$ to fit TFFSA data via $\kappa$. The
best fit was obtained in the interval $3.5<M_{1}R<9.5$, and the result is
shown in Fig.8, where it is compared to $\kappa_{\text{disp}}(u)$ obtained
from the dispersion relation (38).

Figure 8: The results of the fitting (66) to TFFSA data via $\kappa$, at
$-0.02<u<+0.03$, are shown as red bullets. The blue curve is
$\kappa_{\text{disp}}(u)$, the integral (40) with $\Delta(y)$ given by (53).

### Vicinity of $E_{8}$ point

When negative $\eta$ is sufficiently close to the $E_{8}$ point $\eta=0$, IFT
involves more than one stable particle. At $\eta_{3}<\eta<0$,
$\eta_{3}\approx-0.13$, there are two heavier particles $A_{2}$ and $A_{3}$,
and at $\eta_{2}<\eta<\eta_{3}$, $\eta_{2}\approx-2.08$, only one additional
stable particle $A_{2}$ remains in the spectrum777There are eight stable
particles at $\eta=0$, see footnote 2 in Sec.2. When $\eta$ shifted from zero
five of them loose stability and become resonance states.
zamolodchikov2013ising . While the finite-size correction terms displayed in
(60) describe contributions of the particle $A_{1}$ winding once around the
cylinder, similar winding of these heavier particles generates further sub-
leading terms in the large $R$ expansion of $\Delta E_{1}(R)$,

$\displaystyle\frac{\Delta
E_{1}(R)-M_{1}}{M_{1}}=\kappa\,e^{-\frac{\sqrt{3}}{2}M_{1}R}+\kappa_{2}\,e^{-\mu^{1}_{22}R}+\kappa_{3}\,e^{-\mu^{1}_{33}R}-$
$\displaystyle\sum_{p=1}^{3}\frac{M_{p}}{M_{1}}\int\frac{d\theta}{2\pi}\cosh\theta\big{[}S_{1p}(\theta+\frac{\pi
i}{2})-1\big{]}e^{-M_{p}R\cosh\theta}+O(e^{-2M_{1}R})\,,\,\,$ (67)

where the notations are explained in Appendix B. Of course when
$\eta<\eta_{3}$ the terms associated with the particle $A_{3}$ (i.e. the
exponential term with $\kappa_{3}$ as well as the term with $p=3$ in the
second line in (67)) are to be dropped.

We used (67) as a fit of the TFFSA data for $\Delta E_{1}(R)$ at
$-0.6<\eta<0$. To make (67) useful for determining $\kappa$ one needs to know
the masses $M_{2}$ and $M_{3}$ in this interval. We have determined $M_{2}$
and $M_{3}$ directly from the TFFSA data for $M_{1}R\gg 1$. Also, $\kappa_{3}$
must be set to zero at $\eta<\eta_{3}$, and moreover one can argue that the
parameter $\kappa_{3}$ is rather small at $\eta\in[\eta_{3}..0]$. Indeed,
$\kappa_{3}$ must turn to zero at the boundaries of this narrow interval. For
that reason we set $\kappa_{3}$ to zero in the whole domain $[-0.6..0]$.
Finally, we assumed that when $\eta$ is small the S-matrix elements
$S_{1p}(\theta)$ are not too much different from these amplitudes at $\eta=0$
which are known exactly. With these approximations, $\kappa$ was determined
from the best fit to the TFFSA via two parameters $\kappa$ and $\kappa_{2}$.
In all cases some segments within the window $12.2<M_{1}R<18.8$ were taken as
the fitting domain. This procedure substantially improved the accuracy of the
estimate of $\kappa$, especially when $\eta$ is close to zero
($\eta\gtrsim-0.6$). The results are shown in Fig.9, where the error bars
reflect the dependence on the choice of the fitting interval.

Figure 9: Red bullets show the result of the best fit of $\hat{\kappa}(\eta)$
to TFFSA data for $\Delta E_{1}(R)$, using (67) with $\kappa$ and $\kappa_{2}$
($\kappa_{3}$ was set to zero, see main text). The blue curve is
${\hat{\kappa}}_{\text{disp}}(\eta)$, the integral (40) with $\Delta(y)$ given
by (53), and the magenta dashed line is showing the exact slope $\kappa_{1}$
at $E_{8}$ point, see (8) and Appendix A. Figure 10: Numerical estimates of
${\tilde{\kappa}}(y):=\kappa((-y)^{-15/4})$ obtained by fitting (68) to TFFSA
data via $\kappa$ (grey bullets). The blue line shows ${\tilde{\kappa}}(y)$
given by (24) with $\kappa^{(0)}=3$ and set
$\alpha\approx\alpha_{0}=-1.32\,|m|^{-2}$, respectively. The red line is the
plot of (24) with $\kappa^{(0)}=3$, and $\alpha$ given by (69) with
${\hat{\alpha}}_{1}=-0.85\,|m|^{-2}$. The dotted lines show the same function
with slightly different values of ${\hat{\alpha}}_{1}$.

### Vicinity of the YL point

When close to the YL point there is a good approximation for the amplitude
$S(i\pi/2+\theta)$ in the second term in (60). The main correction due to the
irrelevant operators in (11) to the YLQFT S-matrix (10) comes from the
operator $(T{\bar{T}})$ in (19). As was explained in Sec.3, up to the order
$\alpha^{4}M_{1}^{8}\sim(u+u_{0})^{\frac{10}{3}}$, its contribution is
identical to the TTbar deformation (23), with the $S^{(0)}$ taken to be the
YLQFT S-matrix (10). Therefore near the YL point (60) can be approximated as

$\displaystyle\frac{\Delta
E_{1}(R)-M_{1}}{M_{1}}=\kappa\,e^{-\frac{\sqrt{3}}{2}M_{1}R}-\qquad\qquad\qquad$
(68)
$\displaystyle\qquad\qquad\int\frac{d\theta}{2\pi}\cosh\theta\left[e^{-M_{1}R\cosh\theta}\,e^{\alpha
M_{1}^{2}\cosh\theta}\,S_{11}^{\text{YLQFT}}\big{(}\theta+\frac{\pi
i}{2}\big{)}-1\right]+\cdots\,\,$

with the mass $M_{1}$ given by the expansion (28). We also used two term
approximation for $\alpha$,

$\displaystyle\alpha=\alpha_{0}+\hat{\alpha}_{1}\,(y+Y_{0})\,,$ (69)

with known $\alpha_{0}=-1.32$ Xu:2022mmw , and adjustable slope $\alpha_{1}$.
With this approximations of the second term in (60) we found that the best fit
of (60) to the TFFSA data via $\kappa$ can be obtained with

$\displaystyle\hat{\alpha}_{1}|m|^{2}=-0.85\pm 0.1\,.$ (70)

When the fitting interval was taken within the segment $M_{1}R\in[4.0,7.5]$
the fitting results were stable for $-3.8<y<-2.8$. As we approached yet closer
to the YL point $-Y_{0}=-2.4293$ the quality of the fit deteriorated rapidly.
The mass $M_{1}\simeq 1.602\ \epsilon^{5/6}$ becomes small, and the domain of
$R$ where (68) is expected to give meaningful approximation is pushed beyond
the range where TFFSA data is accurate. The result is shown in Fig.10, along
with ${\tilde{\kappa}}(y):=\kappa((-y)^{-15/4})$ from the approximation (24)
with $\alpha=\alpha_{0}$ and $\alpha$ given by (69).

##  6 Discussion

In this work we have tested analyticity of the $\varphi^{3}$ coupling
$\kappa(u)$ as the function of the scaling parameter
$u=h^{2}/|m|^{\frac{15}{4}}$. Assuming maximal analyticity, we have
constructed an approximation for the discontinuity across the branch cut in
Fig.1, and compared the dispersion relation (38) with exact and numerical data
available through TFFSA in Sec.5, and previously in Gabai:2019ryw . Agreement
supports the analyticity conjecture, as well as the approximation (53) for the
discontinuity.

As in Xu:2022mmw , important tool in our analysis was application of the
$T{\bar{T}}$ deformation formulae, which allowed for control of higher order
terms in the coupling parameter $\alpha(u)$ in the effective action (19). For
$\kappa(u)$ the equation (24) reproduces exactly the $\alpha$-expansion up to
terms $\alpha^{4}$, leading to the singular expansion (25), (26). The terms
$\sim\alpha^{5}$ and higher interfere with the contributions from the
operators $(T{\bar{T}})^{3}=L_{-2}^{3}{\bar{L}}_{-2}^{3}I$ and higher
descendants of identity. We would like to note in this connection that the
operator $(T{\bar{T}})^{3}$, as well as
$(T{\bar{T}})^{4}:=L_{-2}^{4}{\bar{L}}_{-2}^{4}I$ and a string of certain
higher descendants of identity, generate the "generalized TTbar deformation"
smirnov2017space  (see also Camilo:2021gro ) with known effect on the
S-matrix. This gives access to yet higher terms in the singular expansion
(22), and potentially may throw some light on important general question of
convergence of the effective action expansion (11): Does it (or, rather the
associated singular expansions like (13)) converge in some finite domain
around the YL point? We hope to address this intriguing problem in the future.

One of practical outcomes of this work is the first estimate (50) of the sub-
leading term $\alpha_{1}$ in (48), which would help to further constrain
coefficients in singular expansions of physical quantities around the YL
point. This is important in view of potential application to the problem of
"extended analyticity" which would include the low-T domain. Unfortunately,
our estimate (50) is not too precise, with possible error of about 20%.
Further work is required to improve this result, and to determine higher
couplings in the effective action (19).

We regard this work as a step toward quantitative understanding of how the
particle spectrum (including resonance states) and S-matrix of IFT depends on
the parameters of the theory. Except of some points in the parameter space,
the theory is not integrable, and its S-matrix involve all kinds of inelastic
processes. We will address some properties of inelastic amplitudes and
resonance states in forthcoming work XuEtAl2022 . Significance of this study
is two-fold. First, IFT describes the basic universality class of second order
phase transitions (such as the liquid-vapor critical point), and its analysis
is important in understanding detailed structure of the associated critical
singularity. On the other hand, we develop tools for quantitative description
of strongly interacting QFT with rich particle/resonance spectrum and
S-matrix.

### Acknowledgments:

AZ is grateful to V.Bazhanov and F.Smirnov for interest to this work and very
useful discussions, and HLX thanks B.McCoy and R.Shrock for helpful
discussions and comments. Research of AZ was supported in part by National
Science Foundation under Grant PHY-1915093 and PHY-2210533.

Note added: In forthcoming paper BazhanovYL  the free energy of IFT in
imaginary magnetic field was computed numerically using an improved corner
transfer matrix approach. The authors managed to locate the YL singularity at
$u_{0}=0.0358536516(1)$, with the precision which seems to be substantially
better then our estimate $u_{0}=0.03586(4)$. In addition, they give what
appears to be more precise estimate of some of the parameters in the effective
action (19), particularly the coupling $\alpha_{0}$. We are grateful to the
authors of BazhanovYL  for sharing these results prior to publication.

## Appendix

##  Appendix A Derivation of $\kappa_{1}$ in (8)

Consider again the $A_{1}+A_{1}\to A_{1}+A_{1}$ elastic scattering amplitude
$S_{11}(\theta)$. This amplitude of course depends on the scaling parameter
$\eta$, but we suppress this argument to simplify notations. Analytic
continuation of this amplitude to complex $\theta$ yields the function
analytic in the $\theta$-plane with the poles manifesting the bound states,
and the branch cuts associated with the inelastic channels $A_{1}+A_{1}\ \to
X$, where $X$ stands for any states involving more than two of the particles
$A_{1}$, as well as the higher bound states $A_{p}$. The principal branch
values of this amplitude enjoy the functional relations

$\displaystyle S_{11}(\theta)S_{11}(-\theta)=1\,,\qquad
S_{11}(\theta)=S_{11}(i\pi-\theta)\,,$ (71)

which follow from unitarity and analyticity (see e.g. Zamolodchikov:2011wd
for the details).

At $\eta=0$ the theory is integrable. We denote $S_{11}^{(0)}(\theta)$ the
amplitude at $\eta=0$. At this integrable point all inelastic channels are
closed, and $S_{11}^{(0)}(\theta)$ is a meromorphic function of $\theta$. Its
exact form is

$\displaystyle
S_{11}^{(0)}(\theta):=\frac{\sinh\theta+i\sin\frac{2\pi}{3}}{\sinh\theta-i\sin\frac{2\pi}{3}}\,\frac{\sinh\theta+i\sin\frac{2\pi}{5}}{\sinh\theta-i\sin\frac{2\pi}{5}}\,\frac{\sinh\theta+i\sin\frac{\pi}{15}}{\sinh\theta-i\sin\frac{\pi}{15}}\,,$
(72)

The poles at $\theta=2\pi i/3$, $\theta=2\pi i/5$, and $\theta=i\pi/15$ are
the manifestations of the particles $A_{1}$, $A_{2}$, and $A_{3}$ in the
direct channel of the scattering, while the poles at $\theta=i\pi/3$,
$\theta=3\pi i/5$, and $\theta=14\pi i/15$ represent the same particles in the
cross channel888Although the theory at this integrable point involves eight
stable particles, only three lightest particles $A_{1}$, $A_{2}$, and $A_{3}$
appear as the poles in $S_{11}^{(0)}(\theta)$. The remaining particles show up
in as the poles in the elastic amplitudes of $S_{pp^{\prime}}^{(0)}(\theta)$
with $p$ or $p^{\prime}$ greater than one, see zamolodchikov1989integrals  for
details.. Correspondingly, the masses of the particles $A_{2}$ and $A_{3}$ at
$\eta=0$ are

$\displaystyle
M_{2}^{(0)}=2M_{1}^{(0)}\,\cos\left(\frac{\pi}{5}\right)\,,\qquad
M_{2}^{(0)}=2M_{1}^{(0)}\,\cos\left(\frac{\pi}{30}\right)\,,$ (73)

where $M_{1}^{(0)}$ is the mass of $A_{1}$ at $\eta=0$ 999Closed form
expression for the coefficient $M_{1}^{(0)}/|h|^{\frac{8}{15}}$ can be found
in e.g. Xu:2022mmw , Appendix B..

At nonzero but sufficiently small $\eta$ the elastic amplitude
$S_{11}(\theta)$ admits convergent expansion in powers of $\eta$, which we
choose to write as

$\displaystyle
S_{11}(\theta)=S_{11}^{(0)}(\theta)\,\left(1+\eta\,\,\Phi^{(1)}(\theta)+\frac{1}{2}\eta^{2}\,\,\Phi^{(2)}(\theta)+...\right)\,.$
(74)

Here we concentrate attention on the first correction term
$\eta\,\varphi^{(1)}(\theta)$, which satisfy the relations

$\displaystyle\Phi^{(1)}(\theta)=-\Phi^{(1)}(-\theta)\,,\qquad\Phi^{(1)}(\theta)=\Phi^{(1)}(i\pi-\theta)\,,$
(75)

in virtue of (71). In fact, $\Phi^{(1)}(\theta)$ can be found in a closed
form, as follows.

At nonzero $\eta$ the integrability of IFT is broken, and $S_{11}(\theta)$
develops branching points at the inelastic thresholds. However, the
discontinuities of this amplitude across the associated branch cuts are
$O(\eta^{2})$, and hence $\varphi^{(1)}(\theta)$ in (74) remains a meromorphic
function of $\theta$. Indeed, the discontinuities across the branch cuts are
related in a simple way to the inelastic cross sections, which in turn are
proportional to the squares of the absolute values of the inelastic
amplitudes. Since at small $\eta$ the inelastic amplitudes are $O(\eta)$, the
discontinuities are $O(\eta^{2})$. It is then possible to argue that, up to
terms $\eta^{2}$ and higher, the amplitude (74) can be written as

$\displaystyle
S_{11}(\theta)=\frac{\sinh\theta+i\sin\frac{2\pi}{3}}{\sinh\theta-i\sin\frac{2\pi}{3}}\,\frac{\sinh\theta+i\sin\alpha_{2}}{\sinh\theta-i\sin\alpha_{2}}\,\frac{\sinh\theta+i\sin\alpha_{3}}{\sinh\theta-i\sin\alpha_{3}}+O(\eta^{2})\,,$
(76)

where

$\displaystyle\alpha_{2}=\frac{2\pi}{5}+\alpha_{2}^{\prime}\,\eta\,,\qquad\alpha_{3}=\frac{\pi}{15}+\alpha_{3}^{\prime}\,\eta\,,$
(77)

and $\alpha_{p}^{\prime}\,,\ p=2,3$ are numbers, to be determined shortly. In
other words, the leading $\sim\eta$ correction to $S_{11}(\theta)$ is
determined entirely by the shifts of the positions of the poles (and related
zeros) of $S_{11}(\theta)$ associated with the particles $A_{2}$ and $A_{3}$.
Expanding (76) to the first order in $\eta$ one finds

$\displaystyle\Phi^{(1)}(\theta)=\frac{2i\,\alpha_{2}^{\prime}\,\cos\frac{2\pi}{5}\,\sinh\theta}{\sinh^{2}\theta+\sin^{2}\frac{2\pi}{5}}+\frac{2i\,\alpha_{3}^{\prime}\,\cos\frac{\pi}{15}\,\sinh\theta}{\sinh^{2}\theta+\sin^{2}\frac{\pi}{15}}\,.$
(78)

and hence

$\displaystyle\kappa_{1}=\sqrt{3}\,\left[\frac{\alpha_{2}^{\prime}\,\cos\frac{2\pi}{5}}{\sin^{2}\frac{2\pi}{3}-\sin^{2}\frac{2\pi}{5}}+\frac{\alpha_{3}^{\prime}\,\cos\frac{\pi}{15}}{\sin^{2}\frac{2\pi}{3}-\sin^{2}\frac{\pi}{15}}\right]\,\kappa(\infty)\,,$
(79)

where $\kappa_{0}=\kappa(\infty)$, see Eq.(6).

The slopes $\alpha_{p}^{\prime}$ in (80) are related in a simple way

$\displaystyle\alpha_{p}^{\prime}=\frac{2}{\sqrt{\big{(}2M_{1}^{(0)}/M_{p}^{(0)}\big{)}^{2}-1}}\,\left(\frac{M_{1}^{(1)}}{M_{1}^{(0)}}-\frac{M_{p}^{(1)}}{M_{p}^{(0)}}\right)\,,$
(80)

to the leading coefficients in the expansions of the masses $M_{p}$ of the
particles $A_{p}$,

$\displaystyle M_{p}=M_{p}^{(0)}+M_{p}^{(1)}\,\eta+O(\eta^{2})\,.$ (81)

While $M_{p}^{(0)}$ are known exactly from the integrable theory, the leading
corrections $M^{(1)}_{p}$ were obtained via the form factor perturbation
theory in Delfino:1996xp , leading to

$\displaystyle\alpha_{2}^{\prime}=0.378325...\,,\qquad\alpha_{3}^{\prime}=1.35226...\,.$
(82)

With this, (79) leads to (8).

One could rise two objections to the proposed expression (78). First, (78)
seems to ignore the possibility of appearance of new poles, not directly
related to the particles $A_{2}$ and $A_{3}$. It is easy to argue though that
such new poles can appear (and indeed do appear) only in the order
$O(\eta^{2})$, i.e. in $\Phi^{(k)}(\theta)$ with $k\geq 2$ in (74). Assume
that a new pole appears in $\Phi^{(1)}(\theta)$ at $\theta=\theta_{\text{new
pole}}$ within the strip $-\pi\leq\Im m\,\theta\leq 0$ in the complex
$\theta$-plane. Obviously, it has to be accompanied by a new zero at
$\theta=\theta_{\text{new zero}}$, close to the new pole, $\theta_{\text{new
zero}}-\theta_{\text{new pole}}\sim\eta$, so that the pole disappears at
$\eta=0$. The position of the new pole can’t be deep inside the above strip,
because then the associated zero would lead to a pole within the physical
strip $0<\Im m\,\theta<\pi$, as dictated by the first equation in
(71)101010For more details on possible locations of poles and zeroes of the
elastic 2 $\to$ 2 amplitude see e.g. Zamolodchikov:2011wd ;
zamolodchikov2013ising ; Gabai:2019ryw . This would contradict general
S-matrix requirement of macro casuality. Therefore, the new pole can only
appear close to the boundary $\Im m\,\theta=0$ or $\pi$, so that the
associated new zero would be located in the physical strip $0<\Im
m\,\theta<\pi$. In this case the pole $\theta_{\text{new pole}}$ would signal
a conventional resonance state, with the width proportional to the separation
between the new pole and and the associated new zero. This interpretation
immediately rules out possibility of emergence of the new poles in the order
$\eta$, since then by continuation to the opposite sign of $\eta$ one could
make the width negative, in contradiction with unitarity 111111Resonance poles
can appear in $S_{11}(\theta)$, and indeed do appear, in the order $\eta^{2}$,
see e.g.Delfino:2005bh .. To summarize, no additional poles in the finite part
of the $\theta$-plane can appear in $\Phi^{(1)}(\theta)$.

Another possible objection concerns asymptotic $\theta\to\infty$ behavior of
$S_{11}(\theta)$ in the order $\sim\eta$. Without violation of crossing and
unitarity one can add to (76) an entire function of $\theta$, subject to the
conditions (71). Since those conditions suggest $2\pi i$ periodicity of the
amplitude, such additional entire function could be either a constant
(impossible, as it would violate the condition $S_{11}(0)=-1$), or grow at
least as $\sinh\theta$ as $\theta\to\infty$. However, analysis of the form
factor perturbation theory suggested that the corrections $\Phi^{(k)}(\theta)$
decay as $\theta^{k-1}\,e^{-\theta}$ as $\theta\to+\infty$; in particular
$\Phi^{(1)}(\theta)$ behaves as $e^{-\theta}$ in this limit, as (78) obviously
does. Indeed, as $S_{11}(\infty)=+1$, at high energies the particles $A_{1}$
behave much like free bosons. In particular, the form factors $\langle
A_{1}(\theta_{1})...A_{1}(\theta_{N})\mid O_{R}(0)\mid
A_{1}(\theta_{1}^{\prime})...A_{1}(\theta_{M}^{\prime})\rangle$ of any
relevant field operator $O_{R}(x)$ are bounded by a constant; the above
asymptotic behavior of $\Phi^{(n)}(\theta)$ follows.

##  Appendix B Finite-Size Corrections to Mass Gap

Consider a generic 2d massive QFT, and denote $A_{a}$ the stable particles of
the theory, and $M_{a}$ their masses. We assume that $A_{1}$ is the lightest
particle in the spectrum. If one puts this theory in the geometry of a
cylinder $x\sim x+R$, the energy spectrum acquires the finite size
corrections. Thus, the leading large-$R$ correction to the ground state energy
is

$\displaystyle
E_{0}(R)=FR-\sum_{a}M_{a}\int\frac{d\theta}{2\pi}e^{-M_{a}R\cosh\theta}\cosh\theta+\cdots\,,$
(83)

where $F$ is the bulk vacuum energy density. Furthermore, if $E_{1}(R)$ is the
energy of the state of one particle $A_{1}$ at rest (See e.g. Klassen:1990ub )

$\displaystyle
E_{1}(R)=FR+M_{1}+\sum_{b,c}M_{1}\kappa^{1}_{bc}e^{-\mu^{1}_{bc}R}-\sum_{a}M_{a}\int\frac{d\theta}{2\pi}e^{-M_{a}R\cosh\theta}S_{1a}(\theta+\frac{\pi
i}{2})\cosh\theta\,,\quad$ (84)

The sum in the first line here includes particles $a,b$ whose masses satisfy
the inequality $M^{2}_{1}\geq|M^{2}_{a}-M^{2}_{b}|$, and

$\displaystyle\mu^{a}_{bc}=\frac{M_{b}M_{c}}{M_{a}}\sin
u^{a}_{bc}\,,\qquad\kappa^{a}_{bc}=-(\Gamma^{a}_{bc})^{2}\,\mu^{a}_{bc}/M_{a}\,,$
(85)

Here $iu_{ab}^{c}$ is the rapidity position of the pole associated with
$A_{c}$ in the amplitude $S_{ab}(\theta)$ of the elastic process
$A_{a}+A_{b}\to A_{a}+A_{b}$,

$\displaystyle M^{2}_{c}=M^{2}_{a}+M^{2}_{b}+2M_{a}M_{b}\cos u^{c}_{ab}\,,$
(86)

and

$\displaystyle\big{(}\Gamma^{c}_{ab}\big{)}^{2}=-i\underset{\theta\to
iu^{c}_{ab}}{\text{Res}}S_{ab}(\theta)\,.$ (87)

In the $E_{8}$ theory in Sec.5 we have $M_{2}=2M_{1}\cos\frac{\pi}{5}$ and
$M_{3}=2M_{1}\cos\frac{\pi}{30}$, and

$\displaystyle\mu^{1}_{22}=\frac{\sqrt{5+2\sqrt{5}}}{2}M_{1}\approx
1.53884\,M_{1}\,,\quad\mu^{1}_{33}=\frac{1}{2}\sqrt{7+8\cos\frac{\pi}{15}}M_{1}\approx
1.92517\,M_{1}\,.\quad$ (88)

This leads to the fitting formula (67).

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