# Wiener-Hopf solution of the free energy TBA problem and instanton sectors in
the O(3) sigma model

Zoltán<EMAIL_ADDRESS>János<EMAIL_ADDRESS>and István<EMAIL_ADDRESS>
*, †, ‡ Holographic QFT Group, Institute for Particle and Nuclear Physics,
HUN-REN Wigner Research Centre for Physics
H-1525 Budapest 114, P.O.B. 49, Hungary
‡ MTA-ELTE ”Momentum” Integrable Quantum Dynamics Research Group,
Department of Theoretical Physics, Eötvös Loránd University
Pázmány Péter sétány 1/A, 1117 Budapest, Hungary

###### Abstract:

Perturbation theory in asymptotically free quantum field theories is
asymptotic. The factorially growing perturbative coefficients carry
information about non-perturbative corrections, which can be related to
renormalons and instantons. Using the Wiener-Hopf technique we determine the
full analytic solution for the free energy density in the two dimensional
O($N$) sigma models. For $N>3$ there are no instantons, and we found that the
perturbative series carries all the information about the non-perturbative
corrections. However, in the O(3) case, we identify several non-perturbative
sectors that are not related to the asymptotics of the perturbative series.
The number of sectors depends on the observables: for the ground-state energy
density we identify three sectors, which we attribute to instantons. For the
free energy density in the running perturbative coupling we found infinitely
many sectors.

##  1 Introduction

Perturbation theory works very well for weakly interacting quantum field
theories including the electro-weak theory, which describes the weak and the
electromagnetic fundamental interactions [1]. In the case of the strong
interaction, however, quantum chromodynamics (QCD) suffers from the asymptotic
nature of the perturbative expansion [2, 3, 4]. This manifests itself in the
factorially growing coefficients, which renders the perturbative series
naively meaningless. The path integral formulation naturally leads to the use
of Borel resummation [5], but singularities on the positive real line imply
non-perturbative corrections and prevent us to obtain a unique result. These
singularities in many cases are encoded in the asymptotic behaviour of the
perturbative series and are related to renormalons or instantons [6, 7].
Unfortunately QCD is a very complicated theory and we do not have access to
many perturbative coefficients, nor the analytical structure of the
observables on the Borel plane, although partial results have been already
obtained from a first few coefficients [8]. It is thus advantageous to analyse
these questions in simplified models, which shares many features with QCD, but
otherwise tractable.

The two dimensional non-linear O($N$) sigma models are of this type. On the
one hand, they are asymptotically free in perturbation theory and the mass of
the particles is generated dynamically. On the other hand, they are integrable
and the ground-state energy density can be exactly calculated, via a linear
TBA integral equation, which is the thermodynamic limit of the Bethe ansatz
equation. The non-linear O($N$) sigma models were providing useful toy models
in which various new phenomena were exactly calculated for the first time. The
integrable scattering matrix of the particles were calculated in [9], the
perturbative $\Lambda$ scale was related to the massgap in [10, 11], the O(6)
sigma model contributed to our understanding to the AdS/CFT duality [12], and
recently resurgence theory, relating perturbative and non-perturbative
corrections, were tested in [13, 14, 15, 16, 17, 18, 19]. The O($N$) models
and their integrable cousins have proven to be very useful, exactly soluble
examples, where resurgence theory [20, 21] could be developed and tested. For
modern reviews about resurgence, see [22, 23, 24].

Indeed, there has been significant activity and relevant progress recently in
the perturbative and non-perturbative investigations of TBA type linear
integral equations. The motivation and results came from the statistical [25,
26, 27, 28, 29, 30, 31, 32] as well as from the particle physics side [16, 17,
18, 33, 19, 34, 35, 36, 37, 38]. The first step in these investigations
originated from the O($N$) models, where Volin developed an efficient
algebraic method to calculate a high number of perturbative coefficients [39,
40]. Using resurgence theory the leading exponentially suppressed corrections
were extracted from the asymptotics of the perturbative coefficients [14, 15,
17]. These are related to instantons or renormalons, which were further
confirmed in particular cases by large $N$ calculations [41, 42] and in the
O(3) model by introducing a $\theta$-term [33]. A systematic expansion of the
Wiener-Hopf integral equation led to the structure of all the non-perturbative
corrections with explicit results for their first few coefficients [16]. The
result is a trans-series, i.e. a series in the perturbative coupling (and in
its logarithm) as well as in the non-perturbative, exponentially suppressed
contributions. Each non-perturbative correction is multiplied by an asymptotic
perturbative series. The convergence properties of the truncated trans-series
were analysed in [35]. The Wiener-Hopf analysis was later extended to higher
orders and the logarithmic coupling dependence was removed by introducing the
running coupling [19]. Finally, in [43] we presented a way to determine the
full trans-series, i.e. all the non-perturbative terms together with their
perturbative expansions. In the present paper we elaborate and extend the
method for the O($N$) models and investigate the resurgence properties of
various observables. The main question we are asking is to what extent the
perturbative series determines the non-perturbative corrections and how many
independent non-perturbative sectors we have.

##  2 Summary of the Wiener-Hopf method and the main results

As explained in the Introduction, 2-dimensional integrable models are useful
toy models in which non-perturbative field theory features, otherwise
unavailable, can be studied in detail. In this paper our aim is to study
problems related to resurgence theory. These include behaviour of very high
orders of perturbation theory, trans-series, Borel resummation, cancellation
of ambiguities, etc.

The particular physical quantity we will examine is the free energy in the
presence of an external field in the 2-dimesional O($N$) nonlinear sigma
model. The study of this quantity provides us an almost unique situation: all
ingredients are explicitly calculable. On the one hand, the free energy is
exactly calculable because it is described by a linear TBA integral equation
based on the bootstrap S-matrix. On the other hand, it can be expanded
perturbatively to very high orders efficiently [39, 40].

The study of the free energy was initiated in [10, 11] and developed further
in [39, 40, 13, 16, 33, 19, 43]. According to our philosophy the main result
of [10, 11] is not the free energy itself, but that by comparing the results
of the bootstrap and perturbative calculations, it became possible to
determine exactly the ratio of the mass parameters of the two approaches,
$m/\Lambda$, where $m$ is the mass of the particles and $\Lambda$ is the usual
Lambda parameter of asymptotically free perturbation theory.

We will use the Wiener-Hopf (WH) method to “solve” the TBA integral equation
in a sense explained below (see section 5). The WH method replaces the
original TBA integral equation by an apparently much more complicated WH
integral equation. The latter, however, has the advantage that it can be
systematically expanded in two small parameters: the bootstrap running
coupling $v$ and an exponentially small non-perturbative (NP) expansion
parameter $\nu$.

The next step is to find the perturbative solution. In principle, this can be
done by iteratively solving (as in [10, 11] to second order) the WH integral
equation but it is much more efficient to use Volin’s method [39, 40], who
invented an algorithm to calculate the perturbative part of the energy density
to arbitrary order algebraically.

Finally we write the complete solution in a form of a trans-series: a series
in powers of the NP expansion parameter $\nu$ where the coefficient of each
power of $\nu$ is a perturbative series in the running coupling $v$. We
derived differential equations (see section 3) which allow us to find all
perturbative coefficient series if we know any one of them. Using Volin’s
result all other series are calculable algebraically [43].

At this point we found the complete solution of our problem in the sense that
it is given as a trans-series, which can be calculated to any order in $\nu$
and the coefficient functions can be given as PT series to any desired order
in the running coupling. In this paper we do not discuss the convergence of
the trans-series [35] but note that each power of $\nu$ is multiplied by a
coefficient function, which is not a convergent series but only asymptotic in
PT.

To give definite meaning to our result, we assume (as is customary in this
field) that the physical result is obtained by Borel resummation of the
coefficient functions (see section 7). We cannot prove the correctness of this
fundamental assumption but, first of all, it can be checked numerically to
very high precision (since the original TBA equation itself can be solved very
precisely and we can compare). Secondly, it can be shown (see section 6) that
the ambiguities (related to the fact that our coefficient functions are not
Borel summable due to singularities on the positive real axis of the Borel
plane) are cancelled by similar ambiguities encountered in the derivation of
the WH integral equations. Moreover, we can study the consequences of our
fundamental assumption and we find that physical quantities are given by the
median resummation [44, 45, 23] of the purely PT part of the given quantity.
More precisely, this is the case for all O($N$) models with $N\geq 4$. In
these models the purely perturbative part $f_{0}$ “knows” about all higher
terms of the trans-series and the Borel resummation of all terms of the trans-
series is just the median sum of $f_{0}$. For O(3), which differs from all
higher $N$ models basically because it is the only model with stable
instantons, the situation is somewhat different (see section 8). Here we have
to consider three basic functions $f_{0},f_{1},f_{2}$, where $f_{0}$ is the
purely perturbative part while $f_{1}$ and $f_{2}$ can be interpreted as
quantum fluctuations around the 1 and 2 instanton solutions, respectively. In
this case the knowledge of $f_{0}$ alone is not sufficient to reconstruct the
full trans-series because physical quantities are obtained as the median sum
of $f_{0}+\nu f_{1}+\nu^{2}f_{2}$. The instanton interpretation of $f_{1}$ and
$f_{2}$ is consistent with the results of [33] where it was found that $f_{1}$
changes sign (but $f_{2}$ does not) if one includes the $\theta$ term in the
O(3) action and puts $\theta=\pi$.

At the end of the paper (sections 10,11) we introduce a physical running
coupling (which can also be defined in a field theory, Lagrangian based study
of the same model) and show that the trans-series in the language of the
physical coupling have similar properties as above except that in the O(3)
case now we have $i$-instanton sectors for any $i$ (not just $i=0,1,2$ as was
the case with the bootstrap coupling $v$).

##  3 Integral equations, differential relations

In this paper we discuss the solution of the TBA integral equations for the
free energy problem in the two-dimensional O($N$) sigma models [19]. However,
our methods are also applicable for similar asymptotically free 2-dimensional
integrable models [13, 16, 33, 43]. The equations are of the form

$\chi_{n}(\theta,B)-\int_{-B}^{B}{\rm
d}\theta^{\prime}K(\theta-\theta^{\prime})\chi_{n}(\theta^{\prime},B)=\cosh
n\theta\qquad\quad|\theta|\leq B.$ (1)

Here the integral kernel is the logarithmic derivative of the S-matrix,

$K(\theta)=\frac{1}{2\pi i}\partial_{\theta}\log S(\theta).$ (2)

In the O($N$) model

$S(\theta)=-\frac{\Gamma(\frac{1}{2}-\frac{i\theta}{2\pi})\Gamma(\Delta-\frac{i\theta}{2\pi})\Gamma(1+\frac{i\theta}{2\pi})\Gamma(\frac{1}{2}+\Delta+\frac{i\theta}{2\pi})}{\Gamma(\frac{1}{2}+\frac{i\theta}{2\pi})\Gamma(\Delta+\frac{i\theta}{2\pi})\Gamma(1-\frac{i\theta}{2\pi})\Gamma(\frac{1}{2}+\Delta-\frac{i\theta}{2\pi})}.$
(3)

The parameter $\Delta$ is related to $N$ as $1/\Delta=N-2$.

Important quantities are the boundary values

$\rho_{n}(B)=\chi_{n}(B,B)$ (4)

and the densities

$O_{n,m}=\frac{1}{2\pi}\int_{-B}^{B}{\rm d}\theta\chi_{n}(\theta,B)\cosh
m\theta,\qquad\qquad O_{n,m}(B)=O_{m,n}(B).$ (5)

From now on we suppress the $B$-dependence for all variables. These variables
satisfy the following differential relations (see appendix A).

${\dot{O}}_{n,m}=\frac{1}{\pi}\rho_{n}\rho_{m},$ (6)
$(n^{2}-m^{2})O_{n,m}=\frac{1}{\pi}(\rho_{m}\dot{\rho}_{n}-\rho_{n}\dot{\rho}_{m}),$
(7) ${\ddot{\rho}}_{n}-n^{2}\rho_{n}=F\rho_{n},$ (8)

where $F$ is $n$-independent:

$F=A^{2}+\dot{A},\qquad A=\frac{\dot{\rho}_{0}}{\rho_{0}}.$ (9)

Combining the above relations we can write the alternative form

${\ddot{O}}_{n,m}-\frac{2\dot{\rho}_{n}}{\rho_{n}}\dot{O}_{n,m}=(m^{2}-n^{2})O_{n,m}.$
(10)

Above we used the notation

$\dot{\cal F}=\frac{{\rm d}{\cal F}}{{\rm d}B}.$ (11)

The derivation of the differential relations (6)-(8) are based on the results
of [29, 30].

##  4 Properties of the integral kernels

The two physical quantities we need are the particle density ($\rho$) and the
energy density ($\varepsilon(\rho)$) given by

$\rho=mO_{0,1},\qquad\qquad\varepsilon(\rho)=m^{2}O_{1,1}.$ (12)

Here $m$ is the mass of the particles (there are no bound states in these
models).

###  4.1 The integral kernels

We will use the Wiener-Hopf (WH) method to solve the TBA integral equations
[10, 11, 16, 19] and for this method we need the Fourier transform of the
kernel and the corresponding $G_{\pm}(\omega)$ decomposition:

$\tilde{K}(\omega)=\int_{-\infty}^{\infty}{\rm d}\theta\,{\rm
e}^{i\omega\theta}K(\theta),\qquad\frac{1}{1-\tilde{K}(\omega)}=G_{+}(\omega)G_{-}(\omega),\qquad
G_{-}(\omega)=G_{+}(-\omega).$ (13)

We decompose the inverse of $1-\tilde{K}(\omega)$ as a product of two
functions, where $G_{+}(\omega)$ is analytic in the upper, and $G_{-}(\omega)$
in the lower half-plane. In our subsequent analysis we will need
$G_{-}(\omega)$ also in the upper half-plane. It is actually also analytic
there, except for the positive imaginary axis, where it has a cut and
infinitely many poles.

The generic form for bosonic models is

$G_{+}(\omega)=\frac{1}{\sqrt{-i\omega}}H(-i\omega){\rm
e}^{-\frac{ia}{2}\omega\ln(-i\omega)-\frac{ib}{2}\omega}.$ (14)

For the O($N$) models we have (with the notation $\gamma=1-2\Delta$)

$\begin{split}{\rm O}(N):\quad
H(\kappa)&=\frac{1}{\sqrt{\Delta}}\frac{\Gamma(1+\Delta\kappa)}{\Gamma(1/2+\kappa/2)}\qquad
a=\gamma\qquad b=2\Delta(1-\ln\Delta)-(1+\ln 2)\\\ {\rm O}(4):\quad
H(\kappa)&=\sqrt{2}\frac{\Gamma(1+\kappa/2)}{\Gamma(1/2+\kappa/2)}\qquad
a=b=0\\\ {\rm O}(3):\quad
H(\kappa)&=\frac{1}{\sqrt{\pi}}\Gamma(1+\kappa/2)\qquad a=-1\qquad b=1+\ln
2\end{split}$ (15)

###  4.2 Discontinuities

For the Wiener-Hopf analysis we will need the quantities

$\sigma_{\pm}(\kappa)=\frac{G_{-}(i\kappa\pm\epsilon)}{G_{+}(i\kappa)},\qquad\qquad\sigma_{+}(\kappa)-\sigma_{-}(\kappa)=-2i\beta(\kappa).$
(16)

They are of the form

$\sigma_{\pm}(\kappa)=\frac{H(-\kappa)}{H(\kappa)}{\rm
e}^{-a\kappa\ln\kappa-b\kappa}\left\\{\mp
i\cos\frac{a\pi\kappa}{2}-\sin\frac{a\pi\kappa}{2}\right\\}.$ (17)

and

$\beta(\kappa)=\frac{H(-\kappa)}{H(\kappa)}{\rm
e}^{-a\kappa\ln\kappa-b\kappa}\cos\frac{a\pi\kappa}{2}$ (18)

###  4.3 Poles and residues

In the WH analysis we will also need the positions and residues of the poles
of $\sigma_{\pm}(\kappa)$ and $\beta(\kappa)$. We have

$\sigma_{\pm}(\kappa)\approx\frac{\mp
iS_{j}}{\kappa-\kappa_{j}},\qquad\beta(\kappa)\approx\frac{S_{j}}{\kappa-\kappa_{j}}\quad(\kappa\sim\kappa_{j}).$
(19)

For our models the Stokes constants $S_{j}$ are real and the set of poles is
$\\{\kappa_{\ell}\\}$, $\ell=1,2,\dots$, where

$\kappa_{\ell}=2\xi_{o}\ell,\qquad\qquad\xi_{o}=\left\\{\begin{split}\frac{N-2}{2}\quad&N\
{\rm even}\\\ N-2\quad&N\ {\rm odd}\end{split}\right.$ (20)

($\xi_{o}=1$ for $N=3$, $4$.) We note that for all $N$ $\beta(1)=0$, and for
all $N$, except for $N=3$, $\sigma_{\pm}(1)=0$. For the O(3) model
$\sigma_{\pm}(1)=1/{\rm e}$.

###  4.4 Running coupling

The running coupling $v$ is defined by

$2B=\frac{1}{v}-a\ln v-b-\tilde{b},$ (21)

where the (arbitrary) constant $\tilde{b}$ is part of the definition of $v$.
For convenience we will also use the notation $L=-b-\tilde{b}$. The advantage
of using the running coupling is the formula for $\kappa=vx$

${\rm e}^{-2B\kappa}\beta(\kappa)={\rm e}^{-x}{\cal A}(x,v),$ (22)

where ${\cal A}(x,v)$ is perturbative,

${\cal A}(x,v)={\rm e}^{vx(\tilde{b}-a\ln
x)}\frac{H(-vx)}{H(vx)}\cos\frac{axv\pi}{2}.$ (23)

It has the expansion

${\cal A}(x,v)=\sum_{k=0}^{\infty}L_{k}(\ln x)(vx)^{k},$ (24)

where

$L_{0}(z)=1,\qquad L_{k}(z):\quad{\rm polynomial\ of\ degree}\ k.$ (25)

##  5 Wiener-Hopf solutions, trans-series representation of densities

###  5.1 Reduced densities

Let us define $W_{n,m}$ and $w_{n}$ by

$\begin{split}O_{n,m}&=\frac{1}{4\pi}G_{+}(in)G_{+}(im){\rm
e}^{(n+m)B}W_{n,m}\quad(n,m>0),\\\ \rho_{n}&=\frac{1}{2}G_{+}(in){\rm
e}^{nB}w_{n}\quad(n>0).\end{split}$ (26)

In terms of the reduced variables the differential relations (6-8) read

$(n+m)W_{n,m}+\dot{W}_{n,m}=w_{n}w_{m},$ (27)
$(n^{2}-m^{2})W_{n,m}=(n-m)w_{n}w_{m}+w_{m}\dot{w}_{n}-w_{n}\dot{w}_{m},$ (28)
$\ddot{w}_{n}+2n\dot{w}_{n}=Fw_{n}.$ (29)

Further definitions:

$O_{0,n}=\frac{1}{2\pi}G_{+}(in){\rm e}^{nB}U_{n}\quad(n>0),$ (30)
$O_{0,0}=\frac{1}{\pi}D.$ (31)

The corresponding differential relations read

$nU_{n}+\dot{U}_{n}=\rho_{0}w_{n},$ (32)
$n^{2}U_{n}=n\rho_{0}w_{n}+\rho_{0}\dot{w}_{n}-\dot{\rho}_{0}w_{n},$ (33)
$\ddot{\rho}_{0}=F\rho_{0},$ (34)

and finally

$\dot{D}=\rho_{0}^{2}.$ (35)

###  5.2 WH solution of the $\chi_{n}$ problem

The exact WH solution of the $\chi_{n}$ TBA integral equation [19] leads to
the integral equation444From now on we suppress the explicit $v$-dependence of
${\cal A}(x,v)$. Similarly, the $v$-dependence of all asymptotic perturbative
series will not be explicitly indicated.

$Q_{n}(y)+\frac{1}{\pi}\int_{{\cal C}_{+}}\frac{{\rm e}^{-x}{\cal
A}(x)Q_{n}(x)}{x+y}{\rm
d}x+\sum_{j}\frac{d_{j}q_{n,j}}{\kappa_{j}+vy}+\frac{\nu^{n}\sigma_{+}(n)}{n+vy}=\frac{1}{n-vy}.$
(36)

Here

$\nu={\rm e}^{-2B},\qquad d_{j}=iS_{j}\nu^{\kappa_{j}},\qquad
q_{n,j}=Q_{n}\left(\frac{\kappa_{j}}{v}\right)$ (37)

and the contour ${\cal C}_{+}$ runs slightly above the real line. There is an
ambiguity here: we could have used ${\cal C}_{-}$, which runs below the real
line. This ambiguity is related (and cancelled) by a similar ambiguity: the
choice of the Borel resummations $S^{\pm}$ (see section 6). The above WH
solution is valid for $n>0$, $n\not=\kappa_{j}$. In terms of this solution the
physical quantities are given by

$w_{n}=1+\frac{v}{\pi}\int_{{\cal C}_{+}}{\rm e}^{-x}{\cal A}(x)Q_{n}(x){\rm
d}x+\sum_{j}d_{j}q_{n,j}+\nu^{n}\sigma_{+}(n)$ (38)

and

$W_{n,m}=\frac{1}{n+m}+\frac{v}{\pi}\int_{{\cal C}_{+}}\frac{{\rm e}^{-x}{\cal
A}(x)Q_{n}(x)}{m-vx}{\rm
d}x+\sum_{j}\frac{d_{j}q_{n,j}}{m-\kappa_{j}}+\frac{\nu^{n}\sigma_{+}(n)}{m-n}+\nu^{m}\sigma_{+}(m)Q_{n}\left(\frac{m}{v}\right).$
(39)

This last formula is valid for $m,n>0$, $m,n\not=\kappa_{j}$, $m\not=n$.

####  5.2.1 Perturbative solution

Let us denote the purely perturbative part of $Q_{n}(x)$ by $P_{n}(x)$. It
satisfies

$P_{n}(y)+\frac{1}{\pi}\int_{{\cal C}_{+}}\frac{{\rm e}^{-x}{\cal
A}(x)P_{n}(x)}{x+y}{\rm d}x=\frac{1}{n-vy}.$ (40)

$P_{n}(x)$, as an asymptotic series in $v$, is well-defined for all $n\not=0$.
We define the asymptotic series

$P_{n,m}=\frac{v}{\pi}\int_{{\cal C}_{+}}\frac{{\rm e}^{-x}{\cal
A}(x)P_{n}(x)}{m-vx}{\rm d}x,$ (41)

which is perturbatively well-defined for all $n,m\not=0$. The quantity

$A_{n,m}=\frac{1}{n+m}+P_{n,m}\qquad(n,m,n+m\not=0)$ (42)

will play an important role in our considerations. First of all,

$W_{n,m}\longmapsto A_{n,m}$ (43)

meaning that it is the purely perturbative part of $W_{n,m}$. Similarly

$w_{n}\longmapsto a_{n},$ (44)

where

$a_{n}=1+\frac{v}{\pi}\int_{{\cal C}_{+}}{\rm e}^{-x}{\cal A}(x)P_{n}(x){\rm
d}x.$ (45)

The perturbative series satisfy the differential relations

$\begin{split}(n+m)A_{n,m}+\dot{A}_{n,m}&=a_{n}a_{m},\\\
(n^{2}-m^{2})A_{n,m}&=(n-m)a_{n}a_{m}+a_{m}\dot{a}_{n}-a_{n}\dot{a}_{m},\\\
\ddot{a}_{n}+2n\dot{a}_{n}&=fa_{n},\\\
1+\dot{P}_{n,-n}&=a_{n}a_{-n}.\end{split}$ (46)

Here $f$ is the perturbative part of the $n$-indepenedent function $F$.

Further perturbative limits are

$U_{n}\longmapsto V_{n}\ (n\not=0),\qquad\rho_{0}\longmapsto s,\qquad
D\longmapsto D_{0}$ (47)

satisfying

$\begin{split}nV_{n}+\dot{V}_{n}&=sa_{n},\\\
n^{2}V_{n}&=nsa_{n}+s\dot{a}_{n}-\dot{s}a_{n},\\\ \ddot{s}&=fs,\\\
\dot{D}_{0}&=s^{2}.\end{split}$ (48)

Let us calculate the leading (constant) term of $P_{n}(x)$:

$P_{n}(x)=P_{n}^{(0)}(x)+{\rm O}(v).$ (49)

This is proportional to the function $\psi_{0}(x)$, calculated explicitly in
[19]:

$P_{n}^{(0)}(x)=\frac{1}{n}\psi_{0}(x).$ (50)

Here we need only the integral of this function,

$\frac{1}{\pi}\int_{0}^{\infty}{\rm e}^{-x}\psi_{0}(x){\rm d}x=\frac{1}{4}$
(51)

to calculate

$P_{n,m}=\frac{v}{4nm}+{\rm O}(v^{2}),\qquad
A_{n,m}=\frac{1}{n+m}+\frac{v}{4nm}+{\rm O}(v^{2}),$ (52)
$a_{n}=1+\frac{v}{4n}+{\rm O}(v^{2}),\qquad f=-v^{2}+{\rm O}(v^{3}),$ (53)
$s=\frac{\lambda}{\sqrt{v}}\left[1+{\rm O}(v)\right],\qquad
D_{0}=\frac{\lambda^{2}}{4v^{2}}\left[1+{\rm O}(v)\right].$ (54)

We see that (nearly) all perturbative building blocks are completely
determined if we know just one of them. For example, using Volin’s method [39,
40], we can start from $A_{1,1}$ and calculate successively:

$A_{1,1}\longrightarrow a_{1}\longrightarrow
f\longrightarrow\left\\{\begin{split}&a_{n}\longrightarrow A_{n,m},\
P_{n,-n}\\\ &s\ ({\rm up\ to\ overall\
}\lambda)\end{split}\right\\}\longrightarrow V_{n},D_{0}$ (55)

This way only the overall constant $\lambda$ in $s$ and the integration
constant in $D_{0}$ is not determined. ($V_{n}$ is also proportional to
$\lambda$ and $D_{0}$ is proportional to $\lambda^{2}$.)

####  5.2.2 Trans-series representation

Inspecting the form of the WH integral equation (36), we can write the
solution in the form of a trans-series: an expansion in powers of the non-
perturbative parameter $\nu$, where the expansion coefficients are asymptotic
power series in the running coupling $v$. We indicate the trans-series nature
of the solution by a tilde.

$\tilde{Q}_{n}(x)=P_{n}(x)+\sum_{j}d_{j}\tilde{q}_{n,j}P_{-\kappa_{j}}(x)+\nu^{n}\sigma_{+}(n)P_{-n}(x).$
(56)

Using

$\tilde{Q}_{n}\left(\frac{\tau}{v}\right)=A_{n,-\tau}+\sum_{j}d_{j}\tilde{q}_{n,j}A_{-\kappa_{j},-\tau}+\nu^{n}\sigma_{+}(n)A_{-n,-\tau}$
(57)

we can calculate

$\tilde{q}_{n,j}=\tilde{Q}_{n}\left(\frac{\kappa_{j}}{v}\right)$ (58)

and

$\tilde{w}_{n}=a_{n}+\sum_{j}d_{j}\tilde{q}_{n,j}a_{-\kappa_{j}}+\nu^{n}\sigma_{+}(n)a_{-n}$
(59)

and finally

$\tilde{W}_{n,m}=A_{n,m}+\sum_{j}d_{j}\tilde{q}_{n,j}A_{-\kappa_{j},m}+\nu^{n}\sigma_{+}(n)A_{-n,m}+\nu^{m}\sigma_{+}(m)\tilde{Q}_{n}\left(\frac{m}{v}\right).$
(60)

Matrix-vector notation

Our formulas become more transparent if we introduce the following compact
notation. We introduce the (infinite) vectors $q^{(n)}$, $p$, $R$, $Y^{(n)}$
with components

$q^{(n)}_{j}=\tilde{q}_{n,j},\qquad p_{j}=a_{-\kappa_{j}},\qquad
R_{j}=V_{-\kappa_{j}},\qquad Y^{(n)}_{j}=A_{n,-\kappa_{j}}$ (61)

and the (infinite) matrices ${\cal N}$, ${\cal D}$, $X$ with entries

${\cal N}_{js}=A_{-\kappa_{j},-\kappa_{s}},\qquad{\cal
D}_{js}=d_{j}\delta_{js},\qquad X_{js}=\kappa_{j}\delta_{js}.$ (62)

Moreover, we define

${\cal M}={\cal N}{\cal D},\qquad\bar{\cal M}=(1-{\cal M})^{-1}=1+{\cal
M}+{\cal M}^{2}+\dots$ (63)

and finally the symmetric matrix ${\cal L}$:

${\cal L}={\cal D}\bar{\cal M}={\cal D}+{\cal D}{\cal N}{\cal D}+{\cal D}{\cal
N}{\cal D}{\cal N}{\cal D}+\dots$ (64)

With this notation (58) can be compactly written as

$q^{(n)}=Y^{(n)}+\nu^{n}\sigma_{+}(n)Y^{(-n)}+{\cal M}q^{(n)}$ (65)

and it has solution

$q^{(n)}=\bar{\cal M}Y^{(n)}+\nu^{n}\sigma_{+}(n)\bar{\cal M}Y^{(-n)}.$ (66)

The compact form of (59) is

$\tilde{w}_{n}=a_{n}+\nu^{n}\sigma_{+}(n)a_{-n}+p^{\rm T}{\cal
L}Y^{(n)}+\nu^{n}\sigma_{+}(n)p^{\rm T}{\cal L}Y^{(-n)}$ (67)

and finally (60) becomes

$\begin{split}\tilde{W}_{n,m}&=A_{n,m}+\nu^{n}\sigma_{+}(n)A_{-n,m}+\nu^{m}\sigma_{+}(m)A_{n,-m}\\\
&+\nu^{n+m}\sigma_{+}(n)\sigma_{+}(m)A_{-n,-m}+Y^{(m){\rm T}}{\cal
L}Y^{(n)}+\nu^{n}\sigma_{+}(n)Y^{(m){\rm T}}{\cal L}Y^{(-n)}\\\
&+\nu^{m}\sigma_{+}(m)Y^{(-m){\rm T}}{\cal
L}Y^{(n)}+\nu^{n+m}\sigma_{+}(n)\sigma_{+}(m)Y^{(-m){\rm T}}{\cal
L}Y^{(-n)}.\end{split}$ (68)

This form is manifestly symmetric in $n,m$, which was not obvious in some
intermediate forms of this quantity.

The most important physical quantities are $\tilde{w}_{1}$ and the energy
density $\tilde{W}_{1,1}$. The latter must be treated as a limiting case
because of the $n\not=m$ restriction for $\tilde{W}_{n,m}$. To calculate the
limit we introduce

$\sigma_{+}(1)=\kappa,\qquad\frac{{\rm d}}{{\rm
d}\varepsilon}\sigma_{+}(1+\varepsilon)\Big{|}_{\varepsilon=0}=-\frac{M}{2},\qquad
Y^{(1)}=Y,\qquad Y^{(-1)}=Z.$ (69)

We find

$\tilde{w}_{1}=a_{1}+\nu\kappa a_{-1}+p^{\rm T}{\cal L}Y+\nu\kappa p^{\rm
T}{\cal L}Z$ (70)

and

$\begin{split}\tilde{W}_{1,1}=\lim_{\varepsilon\to
0}\tilde{W}_{1,1+\varepsilon}&=\frac{M}{2}\nu+2B\nu\kappa+2\nu\kappa
P_{1,-1}+A_{1,1}+\nu^{2}\kappa^{2}A_{-1,-1}\\\ &+Y^{\rm T}{\cal L}Y+2\nu\kappa
Y^{\rm T}{\cal L}Z+\nu^{2}\kappa^{2}Z^{\rm T}{\cal L}Z.\end{split}$ (71)

(We recall that for all models except O(3) $\kappa=0$ and $\kappa=1/{\rm e}$
for O(3).)

Derivatives

The differential relations satisfied by the perturbative objects can be
compactly written in matrix form as follows.

$\begin{split}\dot{\cal N}&={\cal N}X+X{\cal N}+pp^{\rm T},\\\ \dot{\cal
D}&=-2X{\cal D},\\\ \dot{\cal M}&=X{\cal M}-{\cal M}X+pp^{\rm T}{\cal D},\\\
\dot{\bar{\cal M}}&=X\bar{\cal M}-\bar{\cal M}X+\bar{\cal M}pp^{\rm T}{\cal
L},\\\ \dot{\cal L}&=-X{\cal L}-{\cal L}X+{\cal L}pp^{\rm T}{\cal L},\\\
\dot{Y}&=XY-Y+a_{1}p,\\\ \dot{Z}&=XZ+Z+a_{-1}p,\\\ \dot{R}&=XR+sp.\end{split}$
(72)

With the help of the above formulas one can verify the

${\dot{\tilde{W}}}_{1,1}+2\tilde{W}_{1,1}=\tilde{w}_{1}^{2}$ (73)

relation.

###  5.3 WH solution of the $\chi_{0}$ problem

The exact WH equations are more complicated in this case:

$\ell(y)+\frac{1}{\pi}\int_{{\cal C}_{+}}\frac{{\rm e}^{-x}{\cal
A}(x)\ell(x)}{x+y}{\rm d}x=\frac{g_{o}}{\pi v\sqrt{v}}\int_{{\cal
C}_{+}}\frac{{\rm e}^{-x}{\cal A}(x)-1}{(x+y)x\sqrt{x}}{\rm
d}x-\sum_{j}\frac{d_{j}y_{j}}{\kappa_{j}+vy},$ (74)
$\begin{split}L(\tau)&=-\frac{v}{\pi}\int_{{\cal C}_{+}}\frac{{\rm
e}^{-x}{\cal A}(x)\ell(x)}{\tau+vx}{\rm
d}x-\frac{g_{o}\sqrt{v}}{\pi\tau}\int_{{\cal C}_{+}}\frac{{\rm e}^{-x}{\cal
A}(x)}{(\tau+vx)\sqrt{x}}{\rm d}x\\\
&+\frac{g_{o}}{\pi\tau\sqrt{v}}\int_{{\cal C}_{+}}\frac{{\rm e}^{-x}{\cal
A}(x)-1}{x\sqrt{x}}{\rm
d}x-\sum_{j}\frac{d_{j}y_{j}}{\tau+\kappa_{j}},\end{split}$ (75)
$y_{j}=L(\kappa_{j}),$ (76)

where $g_{o}=H(0)$. The physical quantities are

$\begin{split}U_{n}&=\frac{v}{\pi}\int_{{\cal C}_{+}}\frac{{\rm e}^{-x}{\cal
A}(x)\ell(x)}{n-vx}{\rm d}x-\frac{g_{o}\sqrt{v}}{\pi n}\int_{{\cal
C}_{+}}\frac{{\rm e}^{-x}{\cal A}(x)}{(n-vx)\sqrt{x}}{\rm d}x\\\
&-\frac{g_{o}}{\pi n\sqrt{v}}\int_{{\cal C}_{+}}\frac{{\rm e}^{-x}{\cal
A}(x)-1}{x\sqrt{x}}{\rm
d}x+\sum_{j}\frac{d_{j}y_{j}}{n-\kappa_{j}}+\nu^{n}\sigma_{+}(n)L(n)\end{split}$
(77)

and

$\rho_{0}=\frac{v}{\pi}\int_{{\cal C}_{+}}{\rm e}^{-x}{\cal A}(x)\ell(x){\rm
d}x-\frac{g_{o}}{\pi\sqrt{v}}\int_{{\cal C}_{+}}\frac{{\rm e}^{-x}{\cal
A}(x)-1}{x\sqrt{x}}{\rm d}x+\sum_{j}d_{j}y_{j}.$ (78)

####  5.3.1 Perturbative solution

The perturbative part of the problem requires to solve

$q(y)+\frac{1}{\pi}\int_{{\cal C}_{+}}\frac{{\rm e}^{-x}{\cal
A}(x)q(x)}{x+y}{\rm d}x=\frac{1}{\pi}\int_{{\cal C}_{+}}\frac{{\rm
e}^{-x}{\cal A}(x)-1}{(x+y)x\sqrt{x}}{\rm d}x$ (79)

and calculate

$\begin{split}V_{n}&=\frac{g_{o}}{\sqrt{v}}\Bigg{\\{}\frac{1}{\pi}\int_{{\cal
C}_{+}}\frac{{\rm e}^{-x}{\cal A}(x)q(x)}{n-vx}{\rm d}x-\frac{v}{\pi
n}\int_{{\cal C}_{+}}\frac{{\rm e}^{-x}{\cal A}(x)}{(n-vx)\sqrt{x}}{\rm d}x\\\
&-\frac{1}{\pi n}\int_{{\cal C}_{+}}\frac{{\rm e}^{-x}{\cal
A}(x)-1}{x\sqrt{x}}{\rm d}x\Bigg{\\}},\end{split}$ (80)
$s=\frac{g_{o}}{\sqrt{v}}\Bigg{\\{}\frac{1}{\pi}\int_{{\cal C}_{+}}{\rm
e}^{-x}{\cal A}(x)q(x){\rm d}x-\frac{1}{\pi}\int_{{\cal C}_{+}}\frac{{\rm
e}^{-x}{\cal A}(x)-1}{x\sqrt{x}}{\rm d}x\Bigg{\\}}.$ (81)

####  5.3.2 Trans-series representation

The trans-series solution is of the form

$\tilde{\ell}(x)=\frac{g_{o}}{v\sqrt{v}}q(x)+\sum_{j}d_{j}\tilde{y}_{j}P_{-\kappa_{j}}(x),$
(82)
$\tilde{L}(\tau)=\sum_{j}d_{j}\tilde{y}_{j}A_{-\kappa_{j},-\tau}+V_{-\tau},$
(83) $\tilde{y}_{j}=\tilde{L}(\kappa_{j}).$ (84)

The physical quantities are given by

$\tilde{U}_{n}=V_{n}+\sum_{j}Y^{(n)}_{j}d_{j}\tilde{y}_{j}+\nu^{n}\sigma_{+}(n)\left[V_{-n}+\sum_{j}Y^{(-n)}_{j}d_{j}\tilde{y}_{j}\right],$
(85) $\tilde{\rho}_{0}=s+\sum_{j}p_{j}d_{j}\tilde{y}_{j}.$ (86)

It is useful to use the compact matrix-vector notation here too. Introducing
the vector $y$ with components

$y_{j}=\tilde{y}_{j}$ (87)

(84) can be compactly written and solved as

$y=R+{\cal M}y,\qquad y=\bar{\cal M}R.$ (88)

Then

$\tilde{U}_{n}=V_{n}+R^{\rm T}{\cal
L}Y^{(n)}+\nu^{n}\sigma_{+}(n)\left[V_{-n}+R^{\rm T}{\cal L}Y^{(-n)}\right],$
(89) $\tilde{\rho}_{0}=s+R^{\rm T}{\cal L}p$ (90)

and the physical particle density is

$\tilde{U}_{1}=V_{1}+\nu\kappa V_{-1}+R^{\rm T}{\cal L}Y+\nu\kappa R^{\rm
T}{\cal L}Z.$ (91)

Using the differential relations (72) the relation

$\tilde{U}_{1}+{\dot{\tilde{U}}}_{1}=\tilde{\rho}_{0}\tilde{w}_{1}$ (92)

can be verified.

The leading term

The leading (constant) term of the perturbative solution satisfies

$q_{0}(y)+\frac{1}{\pi}\int_{0}^{\infty}\frac{{\rm e}^{-x}q_{0}(x)}{x+y}{\rm
d}x=\frac{1}{\pi}\int_{0}^{\infty}\frac{{\rm e}^{-x}-1}{(x+y)x\sqrt{x}}{\rm
d}x.$ (93)

In [19] we calculated the constant

$s_{0}=\frac{1}{\pi}\int_{0}^{\infty}{\rm e}^{-x}q_{0}(x){\rm
d}x-\frac{1}{\pi}\int_{0}^{\infty}\frac{{\rm e}^{-x}-1}{x\sqrt{x}}{\rm
d}x=\frac{\sqrt{\pi}}{2}.$ (94)

This determines the overall constant in

$s=\frac{\lambda}{\sqrt{v}}\left(1+{\rm O}(v)\right),\qquad
V_{n}=\frac{\lambda}{n\sqrt{v}}\left(1+{\rm
O}(v)\right),\qquad\lambda=\frac{g_{o}\sqrt{\pi}}{2}.$ (95)

In [43] the first few terms of $s$ are given:

$s=\frac{\lambda}{\sqrt{v}}\left(1+\frac{av}{2}-\frac{5a^{2}v^{2}}{8}+\dots\right).$
(96)

Using

${\dot{D}}_{0}=-\frac{2v^{2}}{1+av}D_{0}^{\prime}=s^{2}$ (97)

(with the notation $f^{\prime}={\rm d}f/{\rm d}v$) we find

$D_{0}^{\prime}=-\frac{\lambda^{2}}{2v^{3}}-\frac{\lambda^{2}a}{v^{2}}+{\rm
O}(1)$ (98)

and

$D_{0}=\frac{\lambda^{2}}{4v^{2}}+\frac{a\lambda^{2}}{v}+{\rm const.}+{\rm
O}(v).$ (99)

We see that there is no log term here, $D_{0}$ is fully perturbative. To
complete the calculation of all perturbative quantities it remains to find the
integration constant in $D_{0}$.

In our family of models we have

$a=1-2\Delta,\qquad
g_{o}=\frac{1}{\sqrt{\pi\Delta}},\qquad\lambda=\frac{1}{2\sqrt{\Delta}}.$
(100)

In [19] we calculated the NLO coefficient $r_{1}$ in

$s=\frac{1}{2\sqrt{v\Delta}}\left(1+r_{1}v+{\rm O}(v^{2})\right)$ (101)

and found

$r_{1}=\left(2\Delta-\frac{1}{2}\right)\ln 2-\Delta\ln\Delta+\frac{L}{2},$
(102)

where the constant $L$ occurs in the definition of the running coupling:

$2B=\frac{1}{v}-a\ln v+L.$ (103)

Comparing (102) to (96) we determine the $L$ value corresponding to the
running coupling used in [43]:

$L=1-2\Delta+2\Delta\ln\Delta+(1-4\Delta)\ln 2.$ (104)

(This corresponds to $\tilde{b}=-b-L=4\Delta\ln 2$.)

We note that in the case of the O(3) model the coupling $v$ used here is
different from the $\beta$ coupling of [19]. They are related by

$2B=\frac{1}{v}+\ln v-1-\ln 8=\frac{1}{\beta}+\ln\beta+1-\ln 8.$ (105)

##  6 Resummation and the leading NP corrections

In this section we show that the lateral Borel resummation of the perturbative
series naturally cancels the imaginary part of the leading NP corrections. We
will restrict our attention in this section to the $N\geq 4$ cases.

For these models the parameter $a$, appearing in the relation

$\nu=v^{a}{\rm e}^{-\frac{1}{v}}{\rm e}^{-L}$ (106)

is restricted to the interval

$0\leq a<1.$ (107)

We recall that the parameters are

$a=1-2\Delta,\qquad b+L=-4\Delta\ln 2,$ (108) $M\nu=-v^{a}{\rm
e}^{-\frac{1}{v}}\frac{\Gamma(1-\Delta)}{\Gamma(1+\Delta)}\,2^{4\Delta}{\rm
e}^{i\pi\Delta}.$ (109)

The spectrum of poles is equidistant:

$\kappa_{j}=kj,\qquad{\rm where}\qquad k=\left\\{\begin{split}N-2\ \ \
\quad&{\rm(even\ }N)\\\ 2(N-2)\ \quad&{\rm(odd\ }N)\end{split}\right.$ (110)

We see that $a\kappa_{j}$ is always an even integer, hence for these models
all cuts are logarithmic, except the one starting at $t=1$ and inspecting (17)
we see that all Stokes constants are indeed real.

For all models the leading NP corrections to the densities are of the form

$\hat{\epsilon}=2\tilde{W}_{1,1}=2A_{1,1}+M\nu+2iS_{1}A_{1,-k}^{2}\nu^{k}+{\rm
O}(\nu^{2k}),\quad\hat{\rho}=\tilde{U}_{1}=V_{1}+iS_{1}V_{-k}A_{1,-k}\nu^{k}+{\rm
O}(\nu^{2k}).$ (111)

###  6.1 Cancellation of the leading NP term

The (reduced) energy density in these models (for $N\geq 4$) is

$S^{+}(2\tilde{W}_{1,1})=S^{+}(\Psi)+M\nu+2S^{+}(Y^{\rm T}{\cal L}Y).$ (112)

Note that we are using $S^{+}$ here, which corresponds to the choice of the
contour ${\cal C}_{+}$. In (112) $\Psi=2A_{1,1}$, $M=M_{0}+iM_{1}$, and the
leading NP contributions coming from the last term are

$2Y^{\rm T}{\cal L}Y=2iS_{1}\nu^{k}A_{1,-k}^{2}+{\rm O}(\nu^{2k}),$ (113)

where

$k=\kappa_{1}\geq 2$ (114)

is the position of the nearest pole. Since the energy density is real,
$S^{+}(\Psi)$ must compensate the imaginary part of the leading NP correction,

$iM_{1}v^{a}{\rm e}^{-\frac{1}{v}}{\rm e}^{-L}.$ (115)

Using (390) we can write

$\Psi=F+K\Psi^{(-a)},$ (116)

where

$K=-\frac{M_{1}}{\pi}{\rm e}^{-L}$ (117)

and the Borel transform $\hat{F}(t)$ has no singularity at $t=1$ and must be
regular between $1$ and $k$. $\hat{\Psi}(t)$ itself was analytic between $-1$
and $1$ and has cuts starting there. $\hat{\Psi}^{(-a)}(t)$ is analytic for
$t<1$ and has a cut starting there. From (116) we see that $\hat{F}(t)$ is
analytic between $-1$ and $k$. We have numerically verified this behaviour of
$\hat{F}(t)$ for $N=6$ and $N=8$ to many digits.

We emphasize once more that our choice of ${\cal C}_{+}$ and the lateral Borel
resummation $S^{+}$ are related. Had we chosen ${\cal C}_{-}$ instead, $M_{1}$
would change sign and we would need $S^{-}$. We will stick to ${\cal C}_{+}$
and $S^{+}$ throughout.

##  7 The main conjecture

In this section we again restrict our attention to the $N\geq 4$ cases not
including the O(3) model, where the reduced energy density is of the form

$\hat{\epsilon}=2\tilde{W}_{1,1}=\Psi+M\nu+2Y^{\rm T}{\cal L}Y.$ (118)

Here $\Psi=2A_{1,1}$ is the perturbative part.

The main conjecture is about reconstructing the physical value from the trans-
series $\hat{\epsilon}$. In the rest of the paper we will assume that the
physical value (given by the solution of the original TBA equation) can be
obtained by the lateral Borel sum of the trans-series:

$\epsilon_{\rm ph}=\epsilon_{\rm TBA}=S^{+}(\hat{\epsilon})$

We will analyse and study the consequences of this fundamental assumption.

First of all, we take care of the leading NP correction, $M\nu$. As explained
in section 6

${\rm Im}\,S^{+}\left(K\Psi^{(-a)}\right)=-M_{1}\nu$ (119)

so we write

$\epsilon_{\rm ph}=M_{0}\nu+{\rm Re}\,S^{+}(\Psi)+i{\rm Im}\,S^{+}(\Psi_{\rm
sub})+2S^{+}(Y^{\rm T}{\cal L}Y),$ (120)

where

$\Psi_{\rm sub}=\Psi-K\Psi^{(-a)}.$ (121)

All remaining singularities are of logarithmic type and we can use the theory
of alien derivatives and use the explicit form of the Stokes automorphism, see
also Appendix B.1.

###  7.1 Stokes automorphism

The Stokes automorphism is related to the alien derivative sum

${\cal R}=\sum_{\omega}{\rm e}^{-\omega z}\Delta^{({\rm
st})}_{\omega}=\sum_{\omega}\nu^{\omega}\Delta_{\omega},$ (122)

where

$\omega=jk,\qquad j=1,2,\dots$ (123)

The Stokes automorphism, its inverse and its square root (for later use) are
given respectively as

$\mathfrak{S}={\rm e}^{-{\cal R}},\qquad{\mathfrak{S}}^{-1}={\rm e}^{{\cal
R}},\qquad{\mathfrak{S}}^{-1/2}={\rm e}^{{\cal R}/2}.$ (124)

The meaning of the Stokes automorphism is that it is a map between the two
lateral Borel resummations:

$S^{-}(X)=S^{+}({\mathfrak{S}}^{-1}X).$ (125)

For a real asymptotic series we have the relation

$2i{\rm Im}\,S^{+}(X)=S^{+}\left[(1-{\mathfrak{S}}^{-1})X\right].$ (126)

Using this relation, we write

$i{\rm Im}\,S^{+}(\Psi_{\rm
sub})=\frac{1}{2}S^{+}\left[(1-{\mathfrak{S}}^{-1})\Psi_{\rm
sub}\right]=\frac{1}{2}S^{+}\left[(1-{\mathfrak{S}}^{-1})\Psi\right]=S^{+}\left[(1-{\mathfrak{S}}^{-1})A_{1,1}\right].$
(127)

This relation holds because as explained in section 6 $\hat{\Psi}^{(-a)}$ has
no singularity at $jk$. This observation leads to the relation

$\epsilon_{\rm ph}=M_{0}\nu+{\rm
Re}\,S^{+}(2A_{1,1})+S^{+}\left[(1-{\mathfrak{S}}^{-1})A_{1,1}\right]+2S^{+}(Y^{\rm
T}{\cal L}Y).$ (128)

Let us spell out the above relation in the first few NP orders. Starting from

$1-{\mathfrak{S}}^{-1}=-{\cal R}-\frac{1}{2}{\cal R}^{2}-\frac{1}{6}{\cal
R}^{3}+\dots$ (129)

and

$Y^{\rm T}{\cal L}Y=Y^{\rm T}{\cal D}Y+Y^{\rm T}{\cal D}{\cal N}{\cal
D}Y+Y^{\rm T}{\cal D}{\cal N}{\cal D}{\cal N}{\cal D}Y+\dots$ (130)

we write the first few NP orders as

$\epsilon_{\rm ph}=M_{0}\nu+{\rm Re}\,S^{+}(2A_{1,1})+S^{+}(p{\cal
H}_{1}+p^{2}{\cal H}_{2}+p^{3}{\cal H}_{3}+\dots),$ (131)

where

$\begin{split}{\cal H}_{1}&=-\Delta_{k}A_{1,1}+2iS_{1}A_{1,-k}^{2},\\\ {\cal
H}_{2}&=-\Delta_{2k}A_{1,1}+2iS_{2}A_{1,-2k}^{2}-\frac{1}{2}\Delta_{k}^{2}A_{1,1}-2S_{1}^{2}A_{1,-k}^{2}A_{-k,-k},\\\
{\cal
H}_{3}&=-\Delta_{3k}A_{1,1}+2iS_{3}A_{1,-3k}^{2}-\frac{1}{2}(\Delta_{k}\Delta_{2k}+\Delta_{2k}\Delta_{k})A_{1,1}-\frac{1}{6}\Delta_{k}^{3}A_{1,1}\\\
&-4S_{1}S_{2}A_{1,-k}A_{1,-2k}A_{-k,-2k}-2iS_{1}^{3}A_{1,-k}^{2}A_{-k,-k}^{2}\end{split}$
(132)

and we introduced the shorthand

$p=\nu^{k}.$ (133)

Let us start our considerations at O$(p)$. Since, according to our conjecture,
$\epsilon_{\rm ph}$ is real, the purely imaginary ${\cal H}_{1}$ must vanish
and the following relation must hold:

$\Delta_{k}A_{1,1}=2iS_{1}A_{1,-k}^{2}.$ (134)

This relation is of the form of the alien derivative lemma (428) and using
(439) ${\cal H}_{2}$ simplifies to

${\cal
H}_{2}=-\Delta_{2k}A_{1,1}+2iS_{2}A_{1,-2k}^{2}+2S_{1}^{2}A_{1,-k}^{2}A_{-k,-k}.$
(135)

Here the third term is real so we find the relation

$\Delta_{2k}A_{1,1}=2iS_{2}A_{1,-2k}^{2}.$ (136)

If we go up to third NP order, the lateral Borel sum of the remaining real
asymptotic series has also an imaginary part:

$S^{+}(p^{2}{\cal H}_{2})=2S_{1}^{2}p^{2}{\rm
Re}\,S^{+}(A_{1,-k}^{2}A_{-k,-k})-p^{3}S_{1}^{2}S^{+}(\Delta_{k}A_{1,-k}^{2}A_{-k,-k}).$
(137)

Using also this relation and the alien derivative lemma with (134) and (136),
the O($p^{3}$) contribution simplifies to:

$\bar{\cal
H}_{3}=-\Delta_{3k}A_{1,1}+2iS_{3}A_{1,-3k}^{2}+4S_{1}S_{2}A_{1,-k}A_{1,-2k}A_{-k,-2k}.$
(138)

Cancellation of the imaginary part gives

$\Delta_{3k}A_{1,1}=2iS_{3}A_{1,-3k}^{2}$ (139)

and finally, up to O($p^{3}$) order, the physical density is

$\epsilon_{\rm ph}=M_{0}\nu+{\rm
Re}\,S^{+}(2A_{1,1}+2p^{2}S_{1}^{2}A_{1,-k}^{2}A_{-k,-k}+4p^{3}S_{1}S_{2}A_{1,-k}A_{1,-2k}A_{-k,-2k}+\dots)$
(140)

It can be shown order by order (see appendix D) that for all higher alien
derivatives the reality condition is equivalent to the relation

$\Delta_{jk}A_{1,1}=2iS_{j}A_{1,-jk}^{2}.$ (141)

###  7.2 Median resummation

The relations (141) and the alien derivative lemma together allow to write the
physical densities in an elegant compact form.

We start from the identity

$\begin{split}({\cal R}Y)_{s}&={\cal
R}A_{1,-sk}=\sum_{j=1}^{\infty}p^{j}\Delta_{jk}A_{1,-sk}=\sum_{j=1}^{\infty}2iS_{j}p^{j}A_{1,-jk}A_{-sk,-jk}\\\
&=2\sum_{j=1}^{\infty}{\cal N}_{sj}{\cal D}_{jj}Y_{j}=2({\cal
M}Y)_{s}.\end{split}$ (142)

This can be written in matrix form as

${\cal R}Y=2{\cal M}Y$ (143)

and for the transpose we have

${\cal R}Y^{\rm T}{\cal D}=2Y^{\rm T}{\cal M}^{\rm T}{\cal D}=2Y^{\rm T}{\cal
D}{\cal N}{\cal D}=2Y^{\rm T}{\cal D}{\cal M}.$ (144)

Similarly,

${\cal R}{\cal M}=2{\cal M}^{2},\qquad{\cal R}R=2{\cal M}R,\qquad{\cal
R}R^{\rm T}{\cal D}=2R^{\rm T}{\cal D}{\cal M}.$ (145)

We now calculate

$\begin{split}{\cal
R}&A_{1,1}=\sum_{j=1}^{\infty}p^{j}\Delta_{jk}A_{1,1}=\sum_{j=1}^{\infty}2iS_{j}p^{j}A_{1,-jk}^{2}=2Y^{\rm
T}{\cal D}Y,\\\ {\cal R}^{2}&A_{1,1}=8Y^{\rm T}{\cal D}{\cal M}Y,\\\
&\vdots\\\ {\cal R}^{k}&A_{1,1}=2^{k}k!Y^{\rm T}{\cal D}{\cal
M}^{k-1}Y\end{split}$ (146)

and find

${\mathfrak{S}}^{-1/2}A_{1,1}=A_{1,1}+\sum_{k=1}^{\infty}\frac{1}{k!}\left(\frac{\cal
R}{2}\right)^{k}A_{1,1}=A_{1,1}+\sum_{k=1}^{\infty}Y^{\rm T}{\cal D}{\cal
M}^{k-1}Y=A_{1,1}+Y^{\rm T}{\cal L}Y.$ (147)

This way we have proved the relation

$\hat{\epsilon}=M\nu+{\mathfrak{S}}^{-1/2}\Psi.$ (148)

Using the definition of the median resummation,

$S^{\rm med}(X)=S^{+}({\mathfrak{S}}^{-1/2}X)$ (149)

we see that the physical density can compactly be written as

$\epsilon_{\rm ph}=M\nu+S^{\rm med}(\Psi).$ (150)

We see that in this family of models the perturbative series $\Psi$ “knows”
about all NP corrections (apart from the bulk energy constant term $M\nu$).

The case of the particle density is completely analogous:

${\cal R}V_{1}=2Y^{\rm T}{\cal D}R,\qquad{\cal R}^{k}V_{1}=2^{k}k!Y^{\rm
T}{\cal D}{\cal M}^{k-1}R,\qquad{\mathfrak{S}}^{-1/2}V_{1}=V_{1}+Y^{\rm
T}{\cal L}R$ (151)

and finally we have

$\hat{\rho}=\tilde{U}_{1}=V_{1}+R^{\rm T}{\cal L}Y={\mathfrak{S}}^{-1/2}V_{1}$
(152)

and

$\rho_{\rm ph}=S^{+}(\hat{\rho})=S^{\rm med}(V_{1}).$ (153)

Finally for the last density we have

${\cal R}D_{0}=2R^{\rm T}{\cal
D}R,\qquad{\mathfrak{S}}^{-1/2}D_{0}=D_{0}+R^{\rm T}{\cal L}R=\tilde{D},$
(154) $D_{\rm ph}=S^{+}(\tilde{D})=S^{\rm med}(D_{0}).$ (155)

##  8 The O(3) model

The O(3) model is more complicated than the $N\geq 4$ O($N$) models but it is
also the most interesting one.

In this model

$\kappa_{j}=2j,\qquad M=M_{0}+iM_{1}=-\frac{i\pi}{{\rm
e}}+2\hat{k}(1-\gamma_{E}-\ln 2).$ (156)

Here $\hat{k}=1/{\rm e}$ is the same as $\kappa$ defined by (69). The running
coupling $v$ is introduced by

$2B=\frac{1}{v}+\ln v-1-3\ln 2.$ (157)

This is the O(3) case of the standard choice of [43]. The trans-series
corresponding to the densities are found in section 5:

$\begin{split}\hat{\epsilon}&=2{\tilde{W}}_{1,1}=M\nu+4B\nu\hat{k}+2A_{1,1}+4\nu\hat{k}P_{1,-1}+2\nu^{2}{\hat{k}}^{2}A_{-1,-1}\\\
&+2Y^{\rm T}{\cal L}Y+4\nu\hat{k}Y^{\rm T}{\cal
L}Z+2\nu^{2}{\hat{k}}^{2}Z^{\rm T}{\cal L}Z,\\\
\hat{\rho}&={\tilde{U}}_{1}=V_{1}+\nu\hat{k}V_{-1}+R^{\rm T}{\cal
L}Y+\nu\hat{k}R^{\rm T}{\cal L}Z,\\\ \tilde{D}&=D_{0}+R^{\rm T}{\cal
L}R.\end{split}$ (158)

Here

$Y_{j}=A_{1,-2j},\qquad Z_{j}=A_{-1,-2j},\qquad R_{j}=V_{-2j}.$ (159)

We also introduce

$\tilde{M}=M+4B\hat{k}=iM_{1}+{\tilde{M}}_{0},\quad{\tilde{M}}_{0}=2\hat{k}(2B+1-\gamma_{E}-\ln
2)=2\hat{k}\Big{(}\frac{1}{v}+\ln v-\gamma_{E}-4\ln 2\Big{)}.$ (160)

Later we will use the definitions

$S^{\pm}(\ln v)=\ln v,\qquad\Delta_{\omega}\ln v=0.$ (161)

The trans-series $\hat{\epsilon}$ given in (158) contains all integer powers
of $\nu$:

$\hat{\epsilon}=\sum_{m=0}^{\infty}\epsilon_{m}\nu^{m}.$ (162)

Let us expand this up to quartic order explicitly:

$\begin{split}\epsilon_{0}&=2A_{1,1},\\\
\epsilon_{1}&=iM_{1}+{\tilde{M}}_{0}+4\hat{k}P_{1,-1},\\\
\epsilon_{2}&=2{\hat{k}}^{2}A_{-1,-1}+2iS_{1}A_{1,-2}^{2},\\\
\epsilon_{3}&=4i\hat{k}S_{1}A_{1,-2}A_{-1,-2},\\\
\epsilon_{4}&=2iS_{2}A_{1,-4}^{2}-2S_{1}^{2}A_{1,-2}^{2}A_{-2,-2}+2i{\hat{k}}^{2}S_{1}A_{-1,-2}^{2}.\end{split}$
(163)

###  8.1 The main conjecture

In this subsection we derive the consequences of our main conjecture, which is
the same as for the $N\geq 4$ models:

$\epsilon_{\rm TBA}=\epsilon_{\rm ph}=S^{+}(\hat{\epsilon}).$ (164)

We will use the alien derivative operators

${\cal R}=\sum_{j=1}^{\infty}\nu^{j}\Delta_{j},\qquad{\mathfrak{S}}^{-1}={\rm
e}^{\cal R},\qquad{\mathfrak{S}}^{-1/2}={\rm e}^{{\cal R}/2}$ (165)

and the relation (valid for real $X$)

$S^{+}(X)={\rm
Re}\,S^{+}(X)+\frac{1}{2}S^{+}\left\\{(1-{\mathfrak{S}}^{-1})X\right\\}.$
(166)

With the help of the above relation we write (working up to quartic order)

$\begin{split}S^{+}(\epsilon_{0})&={\rm
Re}\,S^{+}(2A_{1,1})-S^{+}\Big{\\{}\big{[}\nu\Delta_{1}+\nu^{2}\Delta_{2}+\nu^{3}\Delta_{3}+\nu^{4}\Delta_{4}+\frac{\nu^{2}}{2}\Delta_{1}^{2}\\\
&+\frac{\nu^{3}}{2}(\Delta_{1}\Delta_{2}+\Delta_{2}\Delta_{1})+\frac{\nu^{4}}{2}(\Delta_{1}\Delta_{3}+\Delta_{3}\Delta_{1})+\frac{\nu^{4}}{2}\Delta_{2}^{2}+\frac{\nu^{3}}{6}\Delta_{1}^{3}\\\
&+\frac{\nu^{4}}{6}(\Delta_{1}^{2}\Delta_{2}+\Delta_{1}\Delta_{2}\Delta_{1}+\Delta_{2}\Delta_{1}^{2})+\frac{\nu^{4}}{24}\Delta_{1}^{4}\big{]}A_{1,1}\Big{\\}}+{\rm
O}(\nu^{5}),\\\ \nu
S^{+}(\epsilon_{1})&=iM_{1}\nu+{\tilde{M}}_{0}\nu+4\nu\hat{k}{\rm
Re}\,S^{+}(P_{1,-1})-2\nu\hat{k}S^{+}\Big{\\{}\big{[}\nu\Delta_{1}+\nu^{2}\Delta_{2}+\nu^{3}\Delta_{3}+\frac{\nu^{2}}{2}\Delta_{1}^{2}\\\
&+\frac{\nu^{3}}{2}(\Delta_{1}\Delta_{2}+\Delta_{2}\Delta_{1})+\frac{\nu^{3}}{6}\Delta_{1}^{3}\big{]}P_{1,-1}\Big{\\}}+{\rm
O}(\nu^{5}),\\\
\nu^{2}S^{+}(\epsilon_{2})&=2\nu^{2}S^{+}\big{\\{}{\hat{k}}^{2}A_{-1,-1}+iS_{1}A_{1,-2}^{2}\big{\\}}.\end{split}$
(167)

At lowest order, O$(\nu)$, the imaginary contribution vanishes if

$S^{+}\\{-\Delta_{1}A_{1,1}\\}+iM_{1}={\rm O}(\nu).$ (168)

This means that for these purely imaginary quantities

$\Delta_{1}A_{1,1}=iM_{1}$ (169)

must hold and since $M_{1}$ is constant, from the considerations in appendix C
we conclude that

$\Delta_{1}X=0\quad(X\not=A_{1,1}).$ (170)

Similarly, at O$(\nu^{2})$, we have to require that

$S^{+}\big{\\{}-\Delta_{2}A_{1,1}+2iS_{1}A_{1,-2}^{2}\big{\\}}={\rm O}(\nu),$
(171)

but since $S^{+}$ acts here on a purely imaginary quantity, the relation

$\Delta_{2}A_{1,1}=2iS_{1}A_{1,-2}^{2}$ (172)

follows. Now from the alien derivative relations derived in appendix C we have

$\Delta_{2}A_{n,m}=2iS_{1}A_{n,-2}A_{m,-2}\quad(n,m\not=2)$ (173)

and analogous relations for other quantities. Using cancellations and
simplifications following from (170) and (173) we find

$\begin{split}S^{+}(\epsilon_{0})&\Rightarrow{\rm
Re}\,S^{+}(2A_{1,1})-S^{+}\Big{\\{}\big{[}\nu^{3}\Delta_{3}+\nu^{4}\Delta_{4}+\frac{\nu^{4}}{2}\Delta_{1}\Delta_{3}\big{]}A_{1,1}\Big{\\}}\\\
&+4S_{1}^{2}\nu^{4}S^{+}(A_{1,-2}^{2}A_{-2,-2})+{\rm O}(\nu^{5}),\\\ \nu
S^{+}(\epsilon_{1})&\Rightarrow\nu{\tilde{M}}_{0}+4\nu\hat{k}{\rm
Re}\,S^{+}(P_{1,-1})-2\nu^{4}\hat{k}S^{+}\big{(}\Delta_{3}P_{1,-1}\big{)}\\\
&-4iS_{1}\hat{k}\nu^{3}S^{+}(A_{1,-2}A_{-1,-2})+{\rm O}(\nu^{5}),\\\
\nu^{2}S^{+}(\epsilon_{2})&\Rightarrow 2\nu^{2}{\hat{k}}^{2}{\rm
Re}\,S^{+}(A_{-1,-1})-2i{\hat{k}}^{2}S_{1}\nu^{4}S^{+}(A_{-1,-2}^{2})+{\rm
O}(\nu^{5}).\end{split}$ (174)

After these simplifications we see that at O$(\nu^{3})$ we have to require

$S^{+}\\{-\Delta_{3}A_{1,1}\\}={\rm O}(\nu),$ (175)

which is satisfied only if

$\Delta_{3}A_{1,1}=0$ (176)

and consequently

$\Delta_{3}X=0\quad(X\not=A_{3,3}).$ (177)

Finally, at O$(\nu^{4})$, we have

$S^{+}\big{\\{}-\Delta_{4}A_{1,1}+2iS_{2}A_{1,-4}^{2}\big{\\}}={\rm O}(\nu).$
(178)

Again, this means that the purely imaginary quantity itself vanishes:

$\Delta_{4}A_{1,1}=2iS_{2}A_{1,-4}^{2}$ (179)

and consequently

$\Delta_{4}A_{n,m}=2iS_{2}A_{n,-4}A_{m,-4}\quad(n,m\not=2)$ (180)

and $\Delta_{4}$ acts similarly on all other quantities.

We have thus demonstrated that our main conjecture implies555So far we have
used only the fact that from the main conjecture the reality of $\epsilon_{\rm
ph}=S^{+}(\hat{\epsilon})$ follows. the relations (170), (173), (177), and
(180). We see that up to quartic order $\epsilon_{ph}$ is manifestly real:

$S^{+}(\hat{\epsilon})=\nu{\tilde{M}}_{0}+{\rm
Re}\,S^{+}\big{\\{}2A_{1,1}+4\nu\hat{k}P_{1,-1}+2\nu^{2}{\hat{k}}^{2}A_{-1,-1}+2\nu^{4}S_{1}^{2}A_{1,-2}^{2}A_{-2,-2}\big{\\}}+{\rm
O}(\nu^{5}).$ (181)

Going to higher orders we can show, step by step, that for all $j=1,2,\dots$

$\Delta_{2j}A_{1,1}=2iS_{j}A_{1,-2j}^{2}\quad\Rightarrow\quad\Delta_{2j}A_{n,m}=2iS_{j}A_{n,-2j}A_{m,-2j}\quad(n,m\not=2j)$
(182)

and

$\Delta_{2j-1}A_{1,1}=iM_{1}\delta_{j,1}\quad\Rightarrow\quad\Delta_{2j-1}X=0\quad(X\not=A_{2j-1,2j-1}).$
(183)

($\Delta_{2j}$ acts on other quantities similarly to (182).)

###  8.2 Median resummation

In this subsection we will use and generalize the results of subsection 7.2.
The even part of the alien derivative operator ${\cal R}$ will be denoted by
$\bar{\cal R}$ in this subsection:

$\bar{\cal R}=\sum_{j=1}^{\infty}\nu^{2j}\Delta_{2j}.$ (184)

The action of the total ${\cal R}$ on $A_{1,1}$ is

${\cal R}A_{1,1}=iM_{1}\nu+\bar{\cal R}A_{1,1}=iM_{1}\nu+2Y^{\rm T}{\cal D}Y$
(185)

and for higher powers we have

${\cal R}^{n}A_{1,1}={\bar{\cal R}}^{n}A_{1,1}\quad n\geq 2.$ (186)

Thus we find

${\mathfrak{S}}^{-1/2}A_{1,1}=\frac{iM_{1}\nu}{2}+A_{1,1}+Y^{\rm T}{\cal L}Y.$
(187)

For $P_{1,-1}$ we have

${\cal R}^{n}P_{1,-1}={\bar{\cal R}}^{n}P_{1,-1}\quad
n=1,2,\dots,\quad\bar{\cal R}P_{1,-1}=2Y^{\rm T}{\cal D}Z$ (188)

and using the relation

$\bar{\cal R}Z=2{\cal M}Z$ (189)

we can show that

${\mathfrak{S}}^{-1/2}P_{1,-1}=P_{1,-1}+Y^{\rm T}{\cal L}Z.$ (190)

Similarly for $A_{-1,-1}$ we find

$\bar{\cal R}A_{-1,-1}=2Z^{\rm T}{\cal
D}Z,\qquad{\mathfrak{S}}^{-1/2}A_{-1,-1}=A_{-1,-1}+Z^{\rm T}{\cal L}Z.$ (191)

Let us introduce

$f^{(0)}=2A_{1,1},\qquad f^{(1)}={\tilde{M}}_{0}+4\hat{k}P_{1,-1},\qquad
f^{(2)}=2{\hat{k}}^{2}A_{-1,-1}.$ (192)

With this notation

$\hat{\epsilon}={\mathfrak{S}}^{-1/2}\left(f^{(0)}+\nu
f^{(1)}+\nu^{2}f^{(2)}\right).$ (193)

This relation can be used to prove (182) and (183) recursively. We can copy
the proof given in appendix D after noting that starting from order $m-1$ only
$f^{(0)}$ contains new quantities at the next order because $\nu f^{(1)}$ and
$\nu^{2}f^{(2)}$ are already of the required form up to order $m$ and $m+1$,
respectively.

The case of the particle density is analogous. Here we have

$\bar{\cal R}V_{-1}=2R^{\rm T}{\cal
D}Z,\qquad{\mathfrak{S}}^{-1/2}V_{-1}=V_{-1}+R^{\rm T}{\cal L}Z.$ (194)

Introducing

$\rho^{(0)}=V_{1},\qquad\rho^{(1)}=\hat{k}V_{-1}$ (195)

we find

$\hat{\rho}={\mathfrak{S}}^{-1/2}\left(\rho^{(0)}+\nu\rho^{(1)}\right).$ (196)

Finally

$\epsilon_{\rm ph}=S^{+}(\hat{\epsilon})=S^{\rm med}\left(f^{(0)}+\nu
f^{(1)}+\nu^{2}f^{(2)}\right)$ (197)

and

$\rho_{\rm ph}=S^{+}(\hat{\rho})=S^{\rm
med}\left(\rho^{(0)}+\nu\rho^{(1)}\right).$ (198)

In this model the perturbative solution, $f^{(0)}$, in itself is not
sufficient to reproduce the full physical result because also $f^{(1)}$ and
$f^{(2)}$ are needed. Since the O(3) model is exceptional also because it is
the only one that has stable instanton solutions, it is tempting to regard
$f^{(i)}$ as the $i$-instanton contribution to the density. If this is indeed
the case then we can say that the perturbative solution plus the perturbative
expansions around the one and two instanton solutions together determine the
physical energy density completely. We do not understand why only these two
instanton sectors matter and also do not understand why only the one instanton
sector contributes for the particle density $\rho_{\rm ph}$. See, however,
(311), which shows that infinitely many instanton sectors are present in the
free energy.

Interestingly, for the capacity the formulas (154) and (155) are also valid
here (with ${\cal R}\to\bar{\cal R}$) so there is no separate instanton
contribution for the capacity.

##  9 Checking the trans-series for ${O}_{1,1}$ in the O(3) model

The derivation of our results for the O(3) alien derivatives (182) was based
on the main conjecture. Although there is little doubt about the correctness
of these formulas: they lead to elegant and compact formulas for the full
trans-series and in addition (see section 12) we have numerically checked them
to a lot of decimal places. Nevertheless, since the original definition of
alien derivatives is based on the study of the singularity structure of the
Borel transform of the original perturbative series, it is desirable to verify
(182) directly by studying the asymptotic behaviour of the coefficients of the
perturbative series.

To this end, in this section we investigate the analytical structure of the
energy density

${O}_{1,1}=\frac{1}{4\pi}G_{+}^{2}(i)e^{2B}2W_{1,1}$ (199)

on the Borel plane based on the perturbative expansion of $W_{1,1}(v)$:

$A_{1,1}(v)=\sum_{n=0}^{\infty}\alpha_{n}v^{n}$ (200)

in the running coupling $2B=\frac{1}{v}+\ln v-1-3\ln 2$.

These perturbative coefficients can be obtained by implementing Volin’s
algorithm [39, 40] to the O(3) model. In the paper [18] we obtained 336
perturbative coefficients numerically with high precisions and now we use
these data to check the following part of the trans-series solution

$\mathfrak{S}^{-1/2}(A_{1,1})=A_{1,1}+i\frac{M_{1}}{2}\nu+iS_{1}A_{1,-2}^{2}\nu^{2}+(iS_{2}A_{1,-4}^{2}-S_{1}^{2}A_{1,-2}^{2}A_{-2,-2})\nu^{4}+\dots$
(201)

where $\nu=\frac{8e}{v}e^{-\frac{1}{v}}$ and
$S_{n}=2e^{-2n}n^{2n-1}\Gamma_{n}^{-2}$. We thus check the alien
derivatives666Note that the coefficients of a series might grow faster than
$\Gamma(n)$ as assumed in the definition (385) of the standard alien
derivative, that is, as $\Gamma(n+\gamma)$, where $\gamma>0$ is an integer.
One has to use (386) and (390) to extend the formula (385) to these cases.
Then the alien derivative will contain negative powers of $v$, which is
exactly the case here.

$\displaystyle\Delta^{(\text{st})}_{1}A_{1,1}$
$\displaystyle=iM_{1}(8e/v)\quad;\qquad\Delta^{(\text{st})}_{2}A_{1,1}=2iS_{1}(8e/v)^{2}A_{1,-2}^{2}$
(202) $\displaystyle\Delta^{(\text{st})}_{4}A_{1,1}$
$\displaystyle=2iS_{2}(8e/v)^{4}A_{1,-4}^{2}\quad;\quad\Delta^{(\text{st})}_{2}\Delta^{(\text{st})}_{2}A_{1,1}=-8(8e/v)^{4}S_{1}^{2}A_{1,-2}^{2}A_{-2,-2}$
(203)

This can be done by investigating the asymptotic behaviour of the perturbative
coefficients

$\displaystyle\alpha_{n}=\alpha\Gamma_{n+1}$
$\displaystyle+2^{-n}(b_{0}\Gamma_{n+2}+2b_{1}\Gamma_{n+1}+\dots+2^{k}b_{k}\Gamma_{n+2-k}+\dots)$
$\displaystyle+4^{-n}(c_{0}\Gamma_{n+4}+4c_{1}\Gamma_{n+3}+\dots+4^{k}c_{k}\Gamma_{n+4-k}+\dots)$
(204)

The leading coefficient $\alpha$ is related to
$\Delta^{(\text{st})}_{1}A_{1,1}$ as $\Delta^{(\text{st})}_{1}A_{1,1}=-2\pi
i\alpha/v$, while the coefficients $-8\pi ib_{n}$ and $-512\pi ic_{n}$ are the
perturbative expansions of $\Delta^{(\text{st})}_{2}A_{1,1}$ and
$\Delta^{(\text{st})}_{4}A_{1,1}$, respectively. Typically the coefficients
$b_{n}$ also grow as

$\frac{b_{n}}{b_{0}}=2^{-n}(d_{0}\Gamma_{n+2}+2d_{1}\Gamma_{n+1}+\dots+2^{k}d_{k}\Gamma_{n+2-k}+\dots)+\dots$
(205)

and the coefficients $-64\pi^{2}b_{0}d_{n}$ correspond to the expansion of
$\Delta_{2}^{2\,(\text{st})}A_{1,1}$. Since the asymptotic contribution of the
$b_{n}$s are mixing with the $c_{n}$s it is highly non-trivial to separate
$\Delta^{(\text{st})}_{4}A_{1,1}$ from $\Delta_{2}^{2\,(\text{st})}A_{1,1}$.
We are going to do it by manipulating various Borel transformed function.

In the following we first introduce the first few terms of the building
blocks, $A_{n,m}$. We then perform an asymptotic analysis and extract the
$\alpha$ and $b_{n}$ coefficients using high order Richardson transforms.
Finally, we construct various Borel transformed functions and extract the
$b_{n},c_{n}$ and $d_{n}$ coefficients from their analytical properties. We
find complete agreement in all the cases.

###  9.1 The basis of perturbative functions

In this subsection we recall how the perturbative basis $A_{n,m}$ can be
constructed explicitly. For demonstrations we present the result for the first
four orders in $v$ only, but it can be easily extended to higher orders.

We start from the perturbative part of the energy density $A_{1,1}(v)$, which
has the expansion

$2A_{1,1}=1+\frac{v}{2}+\frac{19v^{2}}{8}+\frac{649v^{3}}{48}+\left(\frac{33277}{384}-\frac{27\zeta_{3}}{16}\right)v^{4}+O\left(v^{5}\right)$
(206)

We then use the differential equation

$(n+m)A_{n,m}+\dot{A}_{n,m}=(n+m)A_{n,m}+\frac{2v^{2}}{1-v}\frac{dA_{n,m}}{dv}=a_{n}a_{m}$
(207)

for $n=m=1$ to obtain the perturbative part of $w_{1}$:

$a_{1}=1+\frac{v}{4}+\frac{29v^{2}}{32}+\frac{1501v^{3}}{384}+\left(\frac{116953}{6144}-\frac{27\zeta_{3}}{32}\right)v^{4}+O\left(v^{5}\right)$
(208)

The perturbative part of the universal function $f$ follows from

$f=2n\frac{\dot{a}_{n}}{a_{n}}+\frac{\ddot{a}_{n}}{a_{n}}$ (209)

once used it for $n=1$:

$f=-v^{2}-6v^{3}-26v^{4}+\frac{1}{2}(27\zeta_{3}-193)v^{5}+O\left(v^{6}\right)$
(210)

By using again eq. (209) for generic $n$ we can determine the perturbative
part of $w_{n}$:

$\displaystyle a_{n}=1$
$\displaystyle+\frac{v}{4n}+\frac{(20n+9)v^{2}}{32n^{2}}+\frac{\left(640n^{2}+636n+225\right)v^{3}}{384n^{3}}$
$\displaystyle+\frac{v^{4}\left(-576(9\zeta_{3}-47)n^{3}+43696n^{2}+35160n+11025\right)}{6144n^{4}}+O\left(v^{5}\right)$
(211)

Finally, using the differential equation for $A_{n,m}$ we can obtain

$\displaystyle A_{n,m}$
$\displaystyle=\frac{1}{m+n}+\frac{v}{4mn}+\frac{v^{2}(20mn+9m+9n)}{32m^{2}n^{2}}+$
$\displaystyle\frac{v^{3}\left(m^{2}\left(640n^{2}+636n+225\right)+6mn(106n+39)+225n^{2}\right)}{384m^{3}n^{3}}$
(212)
$\displaystyle+\frac{v^{4}\left(m^{3}\left(8n\left(72(47-9\zeta_{3})n^{2}+5462n+4395\right)+11025\right)\right)}{6144m^{4}n^{4}}$
$\displaystyle+\frac{v^{4}\left(m^{2}n(16n(2731n+2277)+11475)+15mn^{2}(2344n+765)+11025n^{3}\right)}{6144m^{4}n^{4}}+O\left(v^{5}\right)$

In checking the first few alien derivatives we need

$A_{1,-2}^{2}=1+\frac{v}{4}+\frac{25v^{2}}{32}+\frac{1159v^{3}}{384}+\left(\frac{84577}{6144}-\frac{27\zeta_{3}}{32}\right)v^{4}+O\left(v^{5}\right)$
(213)

$A_{1,-4}^{2}=\frac{1}{9}+\frac{v}{24}+\frac{55v^{2}}{384}+\frac{5431v^{3}}{9216}+\left(\frac{90931}{32768}-\frac{9\zeta_{3}}{64}\right)v^{4}+O\left(v^{5}\right)$
(214)

$A_{1,-2}^{2}A_{-2,-2}=-\frac{1}{4}-\frac{3v^{2}}{32}-\frac{73v^{3}}{128}-\frac{381v^{4}}{128}+O\left(v^{5}\right)$
(215)

together with the structure constants:

$M_{1}=-\frac{\pi}{e}\quad;\quad S_{1}=\frac{2}{e^{2}}\quad;\quad
S_{2}=\frac{16}{e^{4}}$ (216)

###  9.2 Asymptotic analysis

Using high order Richardson transformation one can investigate the large $n$
behaviour of the $\alpha_{n}/\Gamma_{n+1}$ series and extract that

$\alpha=4$ (217)

with 95 digits precision. Taking into account the value of $M$ with $\nu$ we
can see complete agreement, i.e. $\Delta^{(\text{st})}_{1}A_{1,1}=-2\pi
i\alpha$.

By subtracting the leading term and inspecting the asymptotics of
$(\alpha_{n}-\alpha\Gamma_{n+1})2^{n}/\Gamma_{n+2}$ we can again use high
order Richardson transform to extract

$b_{0}=-\frac{32}{\pi}\quad;\quad\frac{b_{1}}{b_{0}}=\frac{1}{4}\quad;\quad\frac{b_{2}}{b_{0}}=\frac{25}{32}\quad;\quad\frac{b_{3}}{b_{0}}=\frac{1159}{384}\quad;\quad\frac{b_{4}}{b_{0}}=\frac{84577}{6144}-\frac{27\zeta_{3}}{32}$
(218)

with precisions of ($39-29$) digits. We could go to higher orders, but for
demonstrative reasons it is enough to stop here. Already at this level we can
see the match with the alien derivative
$\Delta^{(\text{st})}_{2}A_{1,1}=256iA_{1,-2}^{2}/v^{2}$.

Technically, it is very difficult to extract enough $b_{n}$ coefficients to
investigate their asymptotic behaviour. Nevertheless, by performing an
asymptotic analysis on $A_{1,-2}^{2}$ we can determine the coefficients
$d_{n}$ as

$d_{0}=\frac{16}{\pi}\quad;\quad\frac{d_{1}}{d_{0}}=0\quad;\quad\frac{d_{2}}{d_{0}}=\frac{3}{8}\quad;\quad\frac{d_{3}}{d_{0}}=\frac{73}{32}\quad;\quad\frac{d_{4}}{d_{0}}=\frac{381}{32}$
(219)

Their contribution is actually mixing with the subleading $4^{-n}$ behaviour
preventing to extract the $c_{n}$ coefficients at all. In order to avoid this
complication we switch to work directly on the Borel plane where we can
disentangle the contributions of the various sub-divergent behaviours.

###  9.3 The Borel plane with square root cuts

Our calculation will be based on [46, 47]. From the asymptotic coefficients
$\alpha_{n}$ (after subtracting the trivial leading behaviour) we can define
the generalized Borel transforms

$B_{k}(s)=\sum_{n=0}^{\infty}\frac{(\alpha_{n}-4\Gamma_{n+1})}{\Gamma_{n+k+1}}s^{n+k}=s^{k}\hat{B}_{k}(s)$
(220)

This function has the closest cut to the origin starting at $s=2$ and the next
closest at $s=4$. The nature of the cut depends on $k$. For $k$ integer the
cuts are logarithmic and the functions multiplying the logarithms are directly
related to the alien derivatives, see [23] for details. For numerical
investigations, however it is advantageous to take $k$ half integer as in this
case the cuts are of square root types. In the following we analyse the $s=2$
cut with $k=\frac{3}{2}$ and the $s=4$ cut with $k=\frac{7}{2}$.

####  9.3.1 The cut at $s=2$

Let us subtract the leading contribution and switch to the Borel plane with
$k=\frac{3}{2}$. The closest singularity of $B_{\frac{3}{2}}(s)$ to the origin
is a square root type cut with singular behaviour

$B_{\frac{3}{2}}(s)=\frac{4\pi}{\sqrt{2-s}}\left(\frac{b_{0}}{\Gamma_{\frac{1}{2}}}+\frac{b_{1}(s-2)}{\Gamma_{\frac{3}{2}}}+\dots+\frac{b_{k}(s-2)^{k}}{\Gamma_{k+\frac{1}{2}}}+\dots\right)+\mathrm{regular}$
(221)

where the regular piece has a Taylor expansion in $(s-2)$, i.e. it does not
have any square roots.

Following the idea of [47] we open up the cut by introducing $s(z)=2-z^{2}$
and expand around $z=0$, which leads to

$zB(2-z^{2})=4\pi\left(\frac{b_{0}}{\Gamma_{\frac{1}{2}}}-\frac{b_{1}z^{2}}{\Gamma_{\frac{3}{2}}}+\dots+\frac{(-1)^{k}b_{k}z^{2k}}{\Gamma_{k+\frac{1}{2}}}+\dots\right)+\textrm{odd
powers in }z$ (222)

Thus the singular and regular parts in $s$ can be separated by focusing on the
odd and even powers in $z$, respectively.

This is particularly useful in the practical implementation as we know only
the first $336$ perturbative coefficients numerically. We thus calculate the
Borel transform upto order $336$ and prepare the Pade approximant of the
corresponding $\hat{B}_{\frac{3}{2}}(s)$, which we denote by
$P\\!\hat{B}_{\frac{3}{2}}(s)$. We then open up the cut by introducing
$s(w)=2-(1-w)^{2}$, such that $w=0$ corresponds to $s=1$, where the original
perturbative series is convergent and the Pade approximant is reliable. After
this we expand

$(1-w)s(w)^{\frac{3}{2}}P\negthinspace\hat{B}_{\frac{3}{2}}(s(w))$ (223)

around $w=0$ in $w$ up to $336$ order and prepare its Pade approximant, which
we expand around $w=1$. As a result we could extract the same $b_{n}$
coefficients as before, but with much higher ($60-50$ digits) precisions than
in the asymptotic analysis.

####  9.3.2 The cut at $s=4$

In investigating the cut at $s=4$ it is more natural to analyse the
$B_{\frac{7}{2}}(s)$ Borel function. The two functions are related to each
other by differentiation $d^{2}B_{\frac{7}{2}}(s)/ds^{2}=B_{\frac{3}{2}}(s)$.
The closest singularity of $B_{\frac{7}{2}}(s)$ to the origin is still a
square root type cut at $s=2$ containing the contribtutions of the $b_{n}$
coefficients as

$B_{\frac{7}{2}}(s)=-4\pi\sqrt{2-s}\left(\frac{b_{0}(s-2)}{\Gamma_{\frac{5}{2}}}+\frac{b_{1}(s-2)^{2}}{\Gamma_{\frac{7}{2}}}+\dots+\frac{b_{k}(s-2)^{k+1}}{\Gamma_{k+\frac{5}{2}}}+\dots\right)+\mathrm{regular}$
(224)

where we assumed that $s$ is larger than $2$ as we analyze the contribution of
this function near $s=4$. In doing so we need the asymptotical behaviour of
the $b_{n}$ coefficient and use them in

$B_{\frac{7}{2}}(s)=-4\pi\frac{\sqrt{2-s}}{\sqrt{s-2}}\sum_{n=0}^{\infty}\frac{b_{n}(s-2)^{n+\frac{3}{2}}}{\Gamma_{n+\frac{5}{2}}}+\dots$
(225)

But this is exactly the $k=\frac{3}{2}$ Borel transformed function with
coefficients $b_{n}$ leading to the their contributions to the $s\sim 4$
behaviour as

$B_{\frac{7}{2}}(s)=-\frac{\sqrt{2-s}}{\sqrt{s-2}}\frac{16\pi^{2}b_{0}}{\sqrt{4-s}}\left(\frac{d_{0}}{\Gamma_{\frac{1}{2}}}+\frac{d_{1}(s-4)}{\Gamma_{\frac{3}{2}}}+\dots+\frac{d_{k}(s-4)^{k}}{\Gamma_{k+\frac{1}{2}}}+\dots\right)+\dots$
(226)

This is not all the square root type behaviour at $s\sim 4$ as we also have
the direct contribution of the $c_{n}$ coefficients:

$B_{\frac{7}{2}}(s)=\frac{256\pi}{\sqrt{4-s}}\left(\frac{c_{0}}{\Gamma_{\frac{1}{2}}}+\frac{c_{1}(s-4)}{\Gamma_{\frac{3}{2}}}+\dots+\frac{c_{k}(s-4)^{k}}{\Gamma_{k+\frac{1}{2}}}+\dots\right)+\dots$
(227)

The stark difference between these contributions are in their reality
properties, which enables us to disentangle the $c$ and $d$ coefficients.

In investigating the square root cut at $s=4$ we would like to do the same
trick as we did before to open up that cut. Unfortunately the cut starting at
$s=2$ overlaps with this cut, thus we should disentangle the two cuts first.
This can be done by introducing a conformal map as

$s(z)=2\frac{4z}{(1+z)^{2}}$ (228)

This map resolves the cut at $s=2$ and maps the cut at $s=4$ to $z=\pm i$. We
can rotate this cut back to the real line using

$z(w)=-iw$ (229)

in order to be able to see clearly the real and imaginary parts. (The
prefactor $\frac{\sqrt{2-s}}{\sqrt{s-2}}$ after this map for real $w$ will
give $i$). We then resolve the cut at $w=1$ by introducing $w(t)=1-t^{2}$,
expanding around $t=0$ and focusing on the odd powers.

Technically, we prepare the Pade approximant of $\hat{B}_{\frac{7}{2}}(s(z))$,
which we denote as
$P\\!\hat{B}_{\frac{7}{2}}(s(z(w)))$ and desingularize the cut at $w=1$ by
introducing

$w(t)=1-(t-2^{-1/2})^{2}$ (230)

We then expand

$(t-2^{-1/2})s(z(w(t)))^{\frac{7}{2}}P\\!\hat{B}(s(z(w(t)))$ (231)

around $t=0$ and prepare its Pade approximant, which we expand around
$t=2^{-\frac{1}{2}}$. From the even terms in the expansion we can recalculate
the corresponding $d_{n}$ and $c_{n}$ coefficients as

$c_{0}=-\frac{256}{9\pi}\quad;\quad\frac{c_{1}}{c_{0}}=\frac{3}{8}\quad;\quad\frac{c_{2}}{c_{0}}=\frac{165}{128}\quad;\quad\frac{c_{3}}{c_{0}}=\frac{5431}{1024}\quad;\quad\frac{c_{4}}{c_{0}}=\frac{818379}{32768}-\frac{81\zeta_{3}}{64}$
(232)

$d_{0}=\frac{16}{\pi}\quad;\quad\frac{d_{1}}{d_{0}}=0\quad;\quad\frac{d_{2}}{d_{0}}=\frac{3}{8}\quad;\quad\frac{d_{3}}{d_{0}}=\frac{73}{32}\quad;\quad\frac{d_{4}}{d_{0}}=\frac{381}{32}$
(233)

with ($16,13,12,9,8$) digits precision.

Clearly they completely agree with the various alien derivatives.

##  10 Physical couplings

###  10.1 Physical quantities

The main players in our considerations are the particle density $\rho$ and the
energy density $\varepsilon$. Eliminating $B$ (or equivalently $v$) we can
establish a relation $\varepsilon=E(\rho)$ between them. They can also be used
to calculate the free energy ${\cal F}$ and a related dimensionless function
$\Omega_{1}$:

${\cal F}=\varepsilon-h\rho,\qquad\Omega_{1}=-\frac{\cal F}{h^{2}},$ (234)

where

$h=E^{\prime}(\rho).$ (235)

We use the TBA integral equation to calculate

$\varepsilon=m^{2}O_{1,1}(B),\qquad\quad\rho=mO_{1,0}(B)$ (236)

and apply the Wiener-Hopf method to analyse the solution. Here we recall some
important relations.

Using the differential equations derived in section 3 we have

$h(B)=m\frac{\rho_{1}(B)}{\rho_{0}(B)}.$ (237)

In this section the quantities $O_{1,1}(B)$, $O_{1,0}(B)$, $\rho_{1}(B)$, and
$\rho_{0}(B)$ will play a role. They are parametrized as

$\begin{split}O_{1,1}(B)&=\frac{G_{+}^{2}(i)}{8\pi}\,{\rm e}^{2B}\,F(v),\qquad
F(v)=2W_{1,1}(v),\\\ O_{1,0}(B)&=\frac{G_{+}(i)}{2\pi}\,{\rm
e}^{B}\,\frac{\lambda}{\sqrt{v}}u(v),\qquad
u(v)=\frac{\sqrt{v}}{\lambda}U_{1},\qquad\lambda=\frac{g_{o}\sqrt{\pi}}{2},\qquad
g_{o}=H(0),\\\ \rho_{1}(B)&=\frac{G_{+}(i)}{2}\,{\rm e}^{B}\,w_{1}(v),\\\
\rho_{0}(B)&=\frac{\lambda}{\sqrt{v}}\,\sigma(v).\end{split}$ (238)

It is also useful to introduce the rescaled quantity $\Omega$:

$\Omega_{1}=\frac{g_{o}^{2}}{4}\Omega,\qquad\Omega=\frac{1}{2v}\left(\frac{2u\sigma}{w_{1}}-\frac{F\sigma^{2}}{w_{1}^{2}}\right).$
(239)

From now on we will use the set of variables $\\{F,u,w_{1},\sigma\\}$ and the
corresponding trans-series
$\\{\tilde{F},\tilde{u},\tilde{w}_{1},\tilde{\sigma}\\}$. The latter are
normalized so that they all are of the form $1+{\rm O}(v)$. More precisely,
excuding the O(3) case,

$\begin{split}\tilde{F}&=M\nu+2A_{1,1}+2Y^{\rm T}{\cal
L}Y=1+\frac{v}{2}+\Big{(}\frac{9}{8}-\frac{5a}{4}\Big{)}v^{2}+\dots,\\\
\tilde{u}&=\frac{\sqrt{v}}{\lambda}\big{(}V_{1}+R^{\rm T}{\cal
L}Y\big{)}=1+\Big{(}\frac{a}{2}-\frac{3}{4}\Big{)}v+\Big{(}a-\frac{15}{32}-\frac{5a^{2}}{8}\Big{)}v^{2}+\dots,\\\
\tilde{w}_{1}&=a_{1}+p^{\rm T}{\cal
L}Y=1+\frac{v}{4}+\Big{(}\frac{9}{32}-\frac{5a}{8}\Big{)}v^{2}+\dots,\\\
\tilde{\sigma}&=\frac{\sqrt{v}}{\lambda}\big{(}s+R^{\rm T}{\cal
L}p\big{)}=1+\frac{av}{2}-\frac{5a^{2}v^{2}}{8}+\dots\end{split}$ (240)

Our main conjecture is that physical quantities are obtained by laterally
Borel resumming the corresponding trans-series:

$F(v)=S^{+}(\tilde{F})(v),\quad u(v)=S^{+}(\tilde{u})(v),\quad
w_{1}(v)=S^{+}(\tilde{w}_{1})(v),\quad\sigma(v)=S^{+}(\tilde{\sigma})(v).$
(241)

For the rescaled $\Omega$ we have

$\tilde{\Omega}=\frac{1}{2v}\left(\frac{2\tilde{u}\tilde{\sigma}}{\tilde{w}_{1}}-\frac{\tilde{F}{\tilde{\sigma}}^{2}}{\tilde{w}_{1}^{2}}\right)=\frac{1}{2v}+{\rm
O}(1).$ (242)

Assuming our main conjecture (241) and using the properties of $S^{\pm}$
discussed above, we arrive at

$\Omega(v)=S^{+}(\tilde{\Omega})(v).$ (243)

At this point we introduce the physical running coupling $\beta$. Its
definition is

$\frac{1}{\beta}+\xi\ln\beta=\ln\frac{h}{\Lambda},\qquad\xi=\frac{1-a}{2},$
(244)

where $\Lambda=\Lambda_{\overline{\rm MS}}$ is the non-perturbative scale
generated via dimensional transmutation in the model. Expressed in terms of
our variables,

$\frac{1}{\beta}+\xi\ln\beta=P+\frac{1}{2v}+\xi\ln(2v)+\ln(1+S),$ (245)

where $P$ is a constant,

$P=\ln\frac{m}{\Lambda}+\ln\frac{H(1)}{\sqrt{\pi}H(0)}+\frac{b+L}{2}-\xi\ln 2$
(246)

and we have written

$\frac{w_{1}}{\sigma}=1+S$ (247)

so that the trans-series corresponding to $S$ is “small”:

$\tilde{S}=\frac{\tilde{w}_{1}}{\tilde{\sigma}}-1=\Big{(}\frac{1}{4}-\frac{a}{2}\Big{)}v+{\rm
O}(v^{2}).$ (248)

We will need the leading terms in the following expansions.

$\frac{2\tilde{u}\tilde{\sigma}}{\tilde{w}_{1}}-\frac{\tilde{F}{\tilde{\sigma}}^{2}}{\tilde{w}_{1}^{2}}=1+(a-2)v+\Big{(}2a-a^{2}-\frac{3}{2}\Big{)}v^{2}+\dots,$
(249)
$\frac{\tilde{F}}{{\tilde{u}}^{2}}=1+(2-a)v+\Big{(}2a^{2}-6a+\frac{9}{2}\Big{)}v^{2}+\dots$
(250)

We can somewhat simplify the problem of calculating the physical coupling by
writing

$\beta=b(v)=\frac{2v}{1+vI(v)},$ (251)

where $I(v)$ satisfies

$\frac{1}{2}I(v)-\xi\ln[1+vI(v)]=P+\ln(1+S(v)).$ (252)

The trans-series corresponding to $I(v)$ satisfies

$\frac{1}{2}\tilde{I}(v)-\xi\ln[1+v\tilde{I}(v)]=P+\ln(1+\tilde{S}(v)).$ (253)

The expansion of $\tilde{I}(v)$ starts as

$\tilde{I}(v)=y_{1}+y_{2}v+{\rm O}(v^{2}),\qquad y_{1}=2P,\qquad y_{2}=2\xi
y_{1}+\frac{1}{2}-a.$ (254)

Using the properties of $S^{+}$ we have

$I(v)=S^{+}(\tilde{I})(v)$ (255)

and further

$b(v)=S^{+}(\tilde{b})(v),\qquad\tilde{b}(v)=\frac{2v}{1+v\tilde{I}(v)}.$
(256)

###  10.2 Field theory

The physical quantity $\Omega_{1}$ is related to the free energy density in
the presence of an external field of strength $h$ and can, in principle,
independently be calculated from field theory by functional integration. It is
difficult to perform functional integration exactly, but, at least, the
corresponding trans-series in terms of the physical coupling $\bar{\beta}$ is
well-defined in field theory:

$\Omega_{1}=d_{o}\tilde{G}(\bar{\beta}),$ (257)

where $d_{o}$ is some overall constant introduced for later convenience and
the physical running coupling is defined in field theory by

$\frac{1}{\bar{\beta}}+\eta\ln\bar{\beta}=\ln\frac{h}{\Lambda},\qquad\eta=\frac{\beta_{1}}{2\beta_{0}^{2}},$
(258)

where $\beta_{0}$, $\beta_{1}$ are the two scheme independent coefficients of
the field theory beta function. Of course, in practice so far only a few
leading contributions have been calculated:

$\tilde{G}(\bar{\beta})=\frac{1}{\bar{\beta}}+f_{0}+f_{1}\bar{\beta}+{\rm
O}(\bar{\beta}^{2}).$ (259)

Consistency with the Wiener-Hopf calculation requires that the two definitions
of the physical coupling coincide:

$\bar{\beta}=\beta,\qquad\eta=\xi,\qquad d_{o}=\frac{g_{o}^{2}}{4}.$ (260)

Further consistency conditions follow from the comparison of the first few
expansion coefficients:

$\begin{split}{\rm(from\
FT)}\quad\tilde{\Omega}&=\frac{1}{2v}\Big{\\{}1+(y_{1}+2f_{0})v+(y_{2}+4f_{1})v^{2}+\dots\Big{\\}}\\\
{\rm(from\
WH)}\quad\tilde{\Omega}&=\frac{1}{2v}\Big{\\{}1+(a-2)v+\Big{(}2a-a^{2}-\frac{3}{2}\Big{)}v^{2}+\dots\Big{\\}}.\end{split}$
(261)

We obtain

$y_{1}=a-2-2f_{0},\qquad\quad y_{2}=2a-a^{2}-\frac{3}{2}-4f_{1}.$ (262)

The first one can be used to find the link between bootstrap (TBA) and field
theory data, $m/\Lambda$:

$\ln\frac{m}{\Lambda}=\frac{a-2-2f_{0}}{2}-\ln\frac{H(1)}{\sqrt{\pi}H(0)}-\frac{b+L}{2}+\xi\ln
2.$ (263)

The second one gives

$f_{1}=\xi f_{0}.$ (264)

We can check that both (260) and (264) are satisfied in the O($N$) case, where

$\begin{split}a&=1-2\Delta,\qquad b+L=-4\Delta\ln 2,\qquad
H(0)=\frac{1}{\sqrt{\pi\Delta}},\qquad
H(1)=\frac{\Gamma(1+\Delta)}{\sqrt{\Delta}},\\\
\beta_{0}&=\frac{1}{4\pi\Delta},\qquad\beta_{1}=\frac{1}{8\pi^{2}\Delta},\qquad\eta=\xi=\Delta,\qquad
d_{o}=\beta_{0},\\\ f_{0}&=-\frac{1}{2},\qquad
f_{1}=-\frac{\Delta}{2},\qquad\ln\frac{m}{\Lambda}=\Delta(\ln
8-1)-\ln\Gamma(1+\Delta).\end{split}$ (265)

Full consistency means that the two completely different approaches lead to
the same result:

$\tilde{G}\circ\tilde{b}=\tilde{\Omega},\qquad\quad\tilde{G}(\tilde{b}(v))=\tilde{\Omega}(v).$
(266)

In practice, in the absence of more data beyond (259), we cannot check full
consistence. However, we can calculate the trans-series $\tilde{G}(\beta)$
from (266).

###  10.3 Example: O(4) model expansion to $1^{\rm st}$ order

In this (simplest) case

$a=b=0,\quad L=-2\ln 2,\quad M=-2i,\quad y_{1}=-1,\quad\nu={\rm e}^{-2B}=4{\rm
e}^{-\frac{1}{v}},$ (267)

and (245) can be rewritten as

$\frac{\nu}{v(1+S)^{2}}=\frac{8}{{\rm e}\beta}{\rm e}^{-\frac{2}{\beta}}.$
(268)

Taking care of the exceptional $M\nu$ term we write

$\Omega=\frac{i\nu}{v}\,\frac{1}{(1+S)^{2}}+\Omega_{\rm
reg},\qquad\tilde{\Omega}_{\rm
reg}(v)=\sum_{s=0}^{\infty}\Omega_{s}(v)\nu^{2s}$ (269)

and analogously

$\tilde{G}(\beta)=\frac{8i}{{\rm e}\beta}{\rm
e}^{-\frac{2}{\beta}}+\tilde{G}_{\rm reg}(\beta),\qquad\tilde{G}_{\rm
reg}(\beta)=\sum_{r=0}^{\infty}{\cal G}_{r}(\beta){\rm
e}^{-\frac{4r}{\beta}}.$ (270)

For later convenience we introduce

${\cal G}_{r}(\beta)=G_{r}(\beta)\left(\frac{8}{{\rm e}\beta}\right)^{2r}$
(271)

and write

$\tilde{G}_{\rm
reg}(\tilde{b}(v))=\sum_{r=0}^{\infty}G_{r}(\tilde{b}(v))\left[\frac{\nu}{v(1+\tilde{S}(v))^{2}}\right]^{2r}=\tilde{\Omega}_{\rm
reg}(v).$ (272)

We will use the trans-series expansions

$\tilde{b}(v)=\sum_{r=0}^{\infty}b_{r}(v)\nu^{2r},\qquad\tilde{I}(v)=\sum_{r=0}^{\infty}I_{r}(v)\nu^{2r},\qquad$
(273)

and (for convenience) two representations of $\tilde{S}(v)$:

$\tilde{S}(v)=\sum_{r=0}^{\infty}S_{r}(v)\nu^{2r},\qquad[1+\tilde{S}(v)]^{2}=\bar{S}_{0}(v)\left\\{1+\sum_{r=1}^{\infty}\bar{S}_{r}(v)\nu^{2r}\right\\}.$
(274)

The leading (purely perturbative) part of (253) is

$I_{0}(v)-\ln[1+vI_{0}(v)]=-1+\ln(1+S_{0}(v))^{2}$ (275)

and after having solved (275) for $I_{0}(v)$ as an asymptotic series, we write

$b_{0}(v)=\frac{2v}{1+vI_{0}(v)}$ (276)

and all higher trans-series components can be calculated algebraically from
linear equations like

$\begin{split}-\frac{2b_{1}}{b_{0}^{2}}&+\frac{b_{1}}{b_{0}}=\bar{S}_{1},\\\
-\frac{2b_{2}}{b_{0}^{2}}&+\frac{b_{2}}{b_{0}}+\frac{2b_{1}^{2}}{b_{0}^{3}}-\frac{1}{2}\frac{b_{1}^{2}}{b_{0}^{2}}=\bar{S}_{2}-\frac{1}{2}\bar{S}_{1}^{2}.\end{split}$
(277)

Next we can calculate $G_{0}(\beta)$, $G_{1}(\beta)$, $G_{2}(\beta)$ etc. from
the implicit relations

$\begin{split}\Omega_{0}&=G_{0}(b_{0}),\\\
\Omega_{1}&=G_{0}^{\prime}(b_{0})b_{1}+\frac{G_{1}(b_{0})}{(v\bar{S}_{0})^{2}},\\\
\Omega_{2}&=G_{0}^{\prime}(b_{0})b_{2}+\frac{1}{2}G_{0}^{\prime\prime}(b_{0})b_{1}^{2}+\frac{G_{2}(b_{0})}{(v\bar{S}_{0})^{4}}+\frac{G_{1}^{\prime}(b_{0})b_{1}}{(v\bar{S}_{0})^{2}}-2\bar{S}_{1}\frac{G_{1}(b_{0})}{(v\bar{S}_{0})^{2}},\end{split}$
(278)

and so on.

In section 7 we established the relations

$\tilde{W}_{1,1}={\mathfrak{S}}^{-1/2}A_{1,1},\qquad\quad\tilde{U}_{1}={\mathfrak{S}}^{-1/2}V_{1},\qquad\quad\tilde{D}={\mathfrak{S}}^{-1/2}D_{0}.$
(279)

From the second one it follows

$\tilde{u}={\mathfrak{S}}^{-1/2}u_{0},$ (280)

where

$\tilde{u}(v)=\sum_{r=0}^{\infty}u_{r}(v)\nu^{2r}.$ (281)

Since ${\cal R}$ commutes with $\frac{{\rm d}}{{\rm d}B}$ and this property is
inherited by ${\mathfrak{S}}^{-1/2}$, we can write

$\tilde{w}_{1}^{2}={\dot{\tilde{W}}}_{1,1}+2\tilde{W}_{1,1}={\mathfrak{S}}^{-1/2}(\dot{A}_{1,1}+2A_{1,1})={\mathfrak{S}}^{-1/2}a_{1}^{2}=\left({\mathfrak{S}}^{-1/2}a_{1}\right)^{2}$
(282)

and conclude that

$\tilde{w}_{1}={\mathfrak{S}}^{-1/2}a_{1}.$ (283)

Similarly

${\dot{\tilde{D}}}=\tilde{\rho}_{0}^{2}\quad\Longrightarrow\quad\tilde{\rho}_{0}={\mathfrak{S}}^{-1/2}s$
(284)

and after rescaling

$\tilde{\sigma}={\mathfrak{S}}^{-1/2}\sigma_{0},\qquad\quad\tilde{\sigma}=\sum_{r=0}^{\infty}\sigma_{r}\nu^{2r}.$
(285)

This structure is inherited by the composite objects:

$1+\tilde{S}={\mathfrak{S}}^{-1/2}(1+S_{0}),\quad\tilde{\Omega}_{\rm
reg}={\mathfrak{S}}^{-1/2}\Omega_{0},\quad\tilde{I}={\mathfrak{S}}^{-1/2}I_{0},\quad\tilde{b}=\frac{2v}{1+v\tilde{I}}={\mathfrak{S}}^{-1/2}b_{0}.$
(286)

At leading order we have

$\Omega_{1}=\frac{1}{2}\Delta_{2}\Omega_{0},\qquad
b_{1}=\frac{1}{2}\Delta_{2}b_{0},$ (287)

where

$\Delta_{\omega}=4^{-\omega}\Delta_{\omega}^{\rm(st)}.$ (288)

Using this in (410) we first write

$\Omega_{1}(v)=\frac{1}{2}\,\left[\frac{{\rm
e}b_{0}(v)}{8v\bar{S}_{0}(v)}\right]^{2}(D_{2}G_{0})(b_{0}(v))+G^{\prime}_{0}(b_{0}(v))b_{1}(v)$
(289)

and comparison with (278) then gives

$G_{1}(\beta)=\frac{1}{2}\left(\frac{{\rm
e}\beta}{8}\right)^{2}(D_{2}G_{0})(\beta),\qquad{\cal
G}_{1}(\beta)=\frac{1}{2}(D_{2}G_{0})(\beta).$ (290)

###  10.4 Conjecture A

It seems natural to conjecture that, analogously to the $\tilde{\Omega}_{\rm
reg}$ relation (286), $\tilde{G}_{\rm reg}$ satisfies

$\tilde{G}_{\rm reg}(\beta)=\left(\Sigma^{-1/2}G_{0}\right)(\beta),$ (291)

where

$\Sigma^{-1/2}={\rm e}^{\frac{1}{2}{\cal T}},\qquad{\cal T}=\sum_{\omega}{\rm
e}^{-\omega x}D_{\omega}=\sum_{r=1}^{\infty}{\rm e}^{-2rx}D_{2r},\qquad
x=\frac{2}{\beta}.$ (292)

Expanding (291) to leading order

$G_{0}+\frac{1}{2}{\rm e}^{-2x}D_{2}G_{0}+\dots=G_{0}+{\cal G}_{1}{\rm
e}^{-2x}+\dots$ (293)

we see that our result (290) shows that Conjecture A is satisfied, at least to
first order.

###  10.5 Conjecture B

From our main conjecture, using the properties of $S^{+}$ and
${\mathfrak{S}}^{-1/2}$ it follows that

$\Omega_{\rm reg}=S^{+}(\tilde{\Omega}_{\rm reg})(v)=S^{\rm
med}(\Omega_{0})(v),\qquad\beta=S^{+}(\tilde{b})(v)=S^{\rm med}(b_{0})(v)$
(294)

and therefore it is natural to conjecture that also

$G_{\rm reg}=S^{+}(\tilde{G}_{\rm reg})(\beta)$ (295)

holds. In a more abstract notation we conjecture that

$S^{+}\left(\tilde{G}_{\rm reg}\circ\tilde{b}\right)(v)=S^{+}(\tilde{G}_{\rm
reg})\left(S^{+}(\tilde{b})(v)\right).$ (296)

In appendix E the validity of conjectures A and B is discussed in the special
case of the O(4) model.

##  11 Conjecture A: the O(3) model

In this section we discuss the O(3) model analogue of conjecture A. Before we
start, here are some O(3) specific definitions and relations.

$a=-1,\qquad L=-1-\ln 8,\qquad\nu=\frac{8{\rm e}}{v}{\rm e}^{-1/v},$ (297)
$u=2\sqrt{v}U_{1},\qquad
r=2\sqrt{v}\rho_{0},\qquad\bar{V}_{n}=2\sqrt{v}V_{n},\qquad
s_{o}=2\sqrt{v}s,\qquad$ (298) ${\overset{\circ}{\Delta}}_{\omega}={\rm
e}^{-\omega/v}\Delta_{\omega}^{({\rm st})}=\nu^{\omega}\Delta_{\omega},$ (299)
$\Omega=-\frac{4\pi}{h^{2}}(\varepsilon-h\rho)=\frac{1}{2v}\left(\frac{2ur}{w_{1}}-\frac{Fr^{2}}{w_{1}^{2}}\right),\qquad
F=2W_{1,1},$ (300) $F=-\frac{i\pi}{{\rm e}}\nu+F_{\rm reg},$ (301) ${\cal
L}=\frac{r^{2}}{vw_{1}^{2}},\qquad\Omega=\frac{i\pi}{2{\rm e}}\nu{\cal
L}+\Omega_{\rm reg},\qquad\Omega_{\rm
reg}=\frac{1}{2v}\left(\frac{2ur}{w_{1}}-\frac{F_{\rm
reg}r^{2}}{w_{1}^{2}}\right).$ (302)

The regular part777In this section ${\mathfrak{S}}$ denotes the regular part
of the Stokes automorphism (not including the $\Delta_{1}$ part). of the
Stokes automorphism in this model is

${\mathfrak{S}}=\exp\left\\{-\sum_{r=1}^{\infty}\nu^{2r}\Delta_{2r}\right\\}$
(303)

and for the trans-series we have

$\tilde{F}_{\rm reg}={\mathfrak{S}}^{-1/2}\left(f_{0}+\nu
f_{1}+\nu^{2}f_{2}\right),$ (304)

where888Here $\kappa=1/{\rm e}$ again.

$\begin{split}f_{0}&=2A_{1,1},\\\ f_{1}&=4\kappa
P_{1,-1}+2\kappa\left(\frac{1}{v}+\ln v-\gamma_{E}-4\ln 2\right),\\\
f_{2}&=2\kappa^{2}A_{-1,-1}.\end{split}$ (305)

Further,

$\tilde{u}={\mathfrak{S}}^{-1/2}\left(\bar{V}_{1}+\nu\kappa\bar{V}_{-1}\right),\qquad\tilde{r}={\mathfrak{S}}^{-1/2}s_{o},\qquad\tilde{w}_{1}={\mathfrak{S}}^{-1/2}\left(a_{1}+\nu\kappa
a_{-1}\right).$ (306)

This structure is inherited by $\tilde{\cal L}$:

$\tilde{\cal L}={\mathfrak{S}}^{-1/2}{\cal K},$ (307)

where

${\cal K}=\frac{s_{o}^{2}}{v(a_{1}+\nu\kappa a_{-1})^{2}}.$ (308)

Note that

${\cal K}=\sum_{r=1}^{\infty}{\cal K}_{r}\nu^{r}$ (309)

is purely real.

Similarly,

$\tilde{\Omega}_{\rm reg}={\mathfrak{S}}^{-1/2}X,$ (310)

where

$X=\frac{1}{2v}\left[\frac{2s_{o}(\bar{V}_{1}+\nu\kappa\bar{V}_{-1})}{(a_{1}+\nu\kappa
a_{-1})}-\frac{s_{o}^{2}(f_{0}+\nu f_{1}+\nu^{2}f_{2})}{(a_{1}+\nu\kappa
a_{-1})^{2}}\right].$ (311) $X=\sum_{r=1}^{\infty}X_{r}\nu^{r}$ (312)

is also purely real.

###  11.1 Physical running coupling

In the O(3) case the running coupling is defined by

$\frac{1}{\beta}+\ln\beta=\ln\frac{h}{\Lambda}.$ (313)

Equivalently, it is the solution of

$\frac{2}{\beta}+2\ln\beta=\ln 4-2+\frac{1}{v}+\ln v-\ln{\cal L}.$ (314)

This last equation can also be formulated as

$\frac{1}{\beta^{2}}{\rm e}^{-2/\beta}=\frac{{\rm e}\nu}{32}{\cal L}.$ (315)

It is also useful to introduce the variable $I$ by

$\beta=\frac{2v}{1+vI}.$ (316)

This satisfies

$\frac{1}{2}I-\ln[1+vI]=-1-\frac{1}{2}\ln(v{\cal L})$ (317)

and the corresponding trans-series relation is

$\frac{1}{2}\tilde{I}-\ln[1+v\tilde{I}]=-1-\frac{1}{2}\ln(v\tilde{\cal L}).$
(318)

The above equation is solved by

$\tilde{I}={\mathfrak{S}}^{-1/2}Y,$ (319)

where $Y$ is the solution of

$\frac{1}{2}Y-\ln[1+vY]=-1-\frac{1}{2}\ln(v{\cal K}).$ (320)

Since $Y$ is purely real,

$b=\frac{2v}{1+vY}=\sum_{r=0}^{\infty}b_{r}\nu^{r}$ (321)

is also real and

$\tilde{\beta}={\mathfrak{S}}^{-1/2}b.$ (322)

The defining equation in terms of $b$ becomes

$\frac{1}{b}+\ln b=\frac{1}{2}\ln\frac{32}{\rm e}+B-\frac{1}{2}\ln{\cal K}.$
(323)

We note, for later purposes, that this implies the relations

$\left(\frac{1}{b_{0}}-\frac{1}{b_{0}^{2}}\right)b_{1}=-\frac{1}{2}\frac{{\cal
K}_{1}}{{\cal
K}_{0}},\qquad\left(\frac{1}{b_{0}}-\frac{1}{b_{0}^{2}}\right)\dot{b}_{0}=1-\frac{1}{2}\frac{\dot{\cal
K}_{0}}{{\cal K}_{0}}.$ (324)

Also for later purposes we note that the relation (410) in the O(3) special
case reads

$\Delta_{\omega}[q(b_{0})]=\left(\frac{{\rm e}b_{0}^{2}}{32}{\cal
K}_{0}\right)^{\omega}(D_{\omega}q)(b_{0})+q^{\prime}(b_{0})\Delta_{\omega}b_{0}.$
(325)

##  12 Numerical results

To back up our analytical calculations, we present some high-precision
numerical evidence that our trans-series indeed reproduces the exact solution
of the integral equation, way beyond the leading non-perturbative correction.
That is, we demonstrate that if one truncates the trans-series at some given
non-perturbative level, and takes its lateral Borel resummation, the result’s
difference to the (numerically evaluated) TBA is of the order of the next non-
perturbative correction coming from the same trans-series. The figures 2-7
below show exactly this: each coloured line corresponds to such a truncation,
and each of them is proportional essentially to some integer power of
$e^{-2B}$, which would appear as a straight line with negative slope on a
logarithmic plot. We analyse the O(3) model, and as before, our other example
from the O($N\geq 4$) family will be O(4) itself. In each case, we compared
our TBA and trans-series data at integer values of $B$, in the range $1\leq
B\leq 20$, however, in some cases this interval was restricted by the
numerical errors.

To calculate the physical quantities such as $\rho,\varepsilon$ or
$\mathcal{F}$ numerically directly from the integral equation (1) one may
expand its solution on the basis of even Chebyshev polynomials. As higher-
order polynomials encode finer details of the solution, one may truncate the
series at a given level to reach a certain precision. Evaluating the integral
equation at a finite number of specific rapidities then translates it to a
matrix equation (the kernel being represented as a square-matrix), which one
can then feed to a linear solver [15]. This method works well for any model,
however, for the O(3) kernel (due to its specific form) one can even rephrase
the problem as a recursion [48], which may be solved significantly faster,
practically leading to a higher precision. The method described here is
essentially a small $B$ expansion, therefore the error of the TBA solution
increases with the size of the integration domain - see Figure 5 for the O(4)
model.

On the other side of the comparison, one needs to calculate the trans-series
of the above-mentioned quantities for relatively high perturbative orders (at
every non-perturbative order) in the running couplings $v$ or $\beta$. To
achieve this, at first one needs to determine the perturbative building blocks
as described in (55) in terms of the bootstrap running coupling $v$.

Then, the trans-series of the different quantities like
$w_{1},W_{1,1},\rho_{0}$ or $U_{1}$ \- see (70), (71), (90) and (91) - may be
calculated in a recursive way, order-by-order in $\nu$. These contain terms
expressed via the infinite matrix-vector notation introduced in Subsection
5.2.2. To calculate the truncation of such a term (in other words, that of
matrix $\mathcal{L}$) at order $\nu^{\kappa_{1}M}$ one may set up a two-
dimensional recursion for $v$-dependent (asymptotic) power serieses $q_{s,M}$
(these are truncated at some power $v^{L}$ as well):

$\displaystyle q_{s,M}(v)$
$\displaystyle=\sum_{l=1}^{M}iS_{l}\,A_{-\kappa_{s},-\kappa_{l}}(v)q_{l,M-l}(v)+\mathrm{O}(v^{L+1}),$
$\displaystyle q_{s,0}(v)$
$\displaystyle=f_{-\kappa_{s}}(v)+\mathrm{O}(v^{L+1})$ (326)

where at each step one has to multiply two (truncated) power series together -
namely a previous result $q_{l,M-l}$ and the building block
$A_{-\kappa_{s},-\kappa_{l}}$. This can be efficiently performed by a discrete
convolution of their coefficients.999Note that this works only if
$\kappa_{j}=\kappa_{1}j$, i.e. the Wiener-Hopf kernel has equally spaced
poles. Also, notice that $\kappa_{1}=2$ for both the O(3) and O(4) models. The
starting point $f_{-\kappa_{s}}$ of the recursion is another perturbative
building block, based on which vector-matrix-vector product (mentioned above)
we would like to calculate. In the end, one needs to ”close” the recursion by
yet another perturbative building block $g_{-\kappa_{l}}$:

$\langle
g,f\rangle_{M}(v)=\sum_{l=1}^{M}iS_{l}\,g_{-\kappa_{l}}(v)q_{l,M-l}(v)+\mathrm{O}(v^{L+1}).$
(327)

For example, to calculate the coefficient of $\nu^{\kappa_{1}M}$ in the term
$p^{T}\mathcal{L}Y$ in (70) one needs to choose $f_{n}=A_{1,n}$ and
$g_{n}=a_{n}$ as starting- and end-points.101010Due to the building blocks
$A_{n,m}$ being symmetric in their indices $n\leftrightarrow m$, expression
(327) is also symmetric for $f\leftrightarrow g$.

Next, the lateral resummation is done similarly to the Borel-Padé resummation
method explained in [14, 15].111111In this section we avoided the conformal
mapping method. The reason is, that although it improves the resummations’
accuracy significantly (even by several orders of magnitude), yet for our
purposes - that is, comparing exponentially small contributions plotted on a
logarithmic scale - it proved to have about the same performance as the Borel-
Padé method. After the Borel transformation of our asymptotic series we apply
a Padé-approximation, and finally integrate it numerically along a contour
tilted slightly above the positive real axis. The lateral resummation needs to
be performed on each of the non-perturbative levels of the trans-series
individually (themselves being asymptotic serieses in $v$), then combined via
the appropriate power of $\nu$. The resulting quantity at a finite truncation
of the trans-series is in general complex: the real part approximates the TBA,
whereas the imaginary part will decrease and eventually approach zero when
including more and more terms from the trans-series (see Figure 4).

The analysis in terms of the physical coupling requires the assembly of the
trans-series of the free energy density. A naive but working approach is to
take an ansatz for $\tilde{G}(\beta)$ where the coefficients of the asymptotic
serieses $\mathcal{G}_{r}(\beta)$ in (270) and (492) are a priori
undetermined. Then, the trans-series $\tilde{b}(v)$ of the running coupling
needs to be determined by solving the equation (253) order-by-order (both in
terms of variable $v$ and in $\nu$, formally treated as independent expansion
parameters) for $\tilde{I}(v)$. Finally, the composition of the two serieses,
namely our ansatz for $\tilde{G}(\beta)$ and $\tilde{b}(v)$, needs to be
compared to $\tilde{\Omega}(v)$ as in (266). After a $v,\nu$ expansion of both
sides one arrives at realations fixing the coefficients of the ansatz, and one
can calculate their value iteratively again. The lateral resummations in
$\beta$ were performed in a very similar manner to the analysis in the
bootstrap coupling $v$.

###  12.1 Particle and energy density

Instead of the physical densities $\rho,\varepsilon$ we analyse their
normalized variants $u(v),F(v)$ as defined in (238). For our models, these
look as

$\displaystyle\frac{\rho}{m}$
$\displaystyle=\frac{1}{4}e^{B}\frac{u(v)}{\sqrt{\pi
v}}\cdot\,\begin{cases}\sqrt{\frac{e}{2\pi}}&\text{O}(3)\\\
1&\text{O}(4),\end{cases}\quad$ $\displaystyle\frac{\varepsilon}{m^{2}}$
$\displaystyle=\frac{1}{16}e^{2B}F(v)\cdot\begin{cases}\frac{e}{\pi}&\text{O}(3)\\\
1&\text{O}(4)\end{cases}$ (328)

where - as we have seen previously - the bootstrap couplings are defined via
(104) as

$2B=\frac{1}{v}+\begin{cases}\ln v-3\ln 2-1,\quad\quad 0.2171\gtrsim
v>0&\text{O}(3)\\\ \quad\quad-2\ln 2,\quad\quad\quad\;\;\,0.7213\gtrsim
v>0&\text{O}(4),\end{cases}$ (329)

for the two cases. The numeric bounds indicate the largest possible values of
these couplings when approaching $B\simeq 0$.

We aim to compare the resummed trans-series of the quantities in (328) to
their exact value. The latter gets very precisely approximated when directly
evaluating the integral equation in the Chebyshev basis as explained in the
introductory part of the present section, and then taking the moments of the
solution to arrive at the quantities (328). That is, one has an independent
way to calculate the physical result at a certain $B$ value and also full
control over its accuracy. We denote the densities obtained like this as
$u_{\text{TBA}}(v)$ and $F_{\text{TBA}}(v)$. To get rid of the prefactor
$1/\sqrt{v}$ in $\rho$ one needs to invert (329) as well.

To quantify the discrepancy between the TBA and the trans-series taken into
account up to a given order in $\nu$ we define the differences

$\displaystyle\delta u^{(k)}(v)$ $\displaystyle\equiv
u_{\text{TBA}}(v)-\text{Re}\,S^{+}(\tilde{u}^{(k)})(v),$ $\displaystyle\delta
F^{(k)}(v)$ $\displaystyle\equiv
F_{\text{TBA}}(v)-\text{Re}\,S^{+}(\tilde{F}^{(k)})(v),$ (330)

where the upper index indicates at which NP term we truncate the trans-series:

$\displaystyle\tilde{u}^{(k)}$ $\displaystyle=\sum_{r=0}^{k}u_{r}\nu^{r},$
$\displaystyle\tilde{F}^{(k)}$ $\displaystyle=\sum_{r=0}^{k}F_{r}\nu^{r}.$
(331)

Here $u_{r}$ and $F_{r}$ are the formal power serieses at order $\nu^{r}$ in
the full trans-series, and $S^{+}$ acts on them individually.

####  12.1.1 The O(3) model

We spell out these trans-serieses for the O(3) model below, giving the first
two non-vanishing perturbative orders for each NP order we analyse in this
section. For the particle density, this looks as

$\displaystyle\tilde{u}(v)$
$\displaystyle\overset{\text{\tiny\text{O(3)}}}{=}\Bigg{\\{}1-\frac{5v}{4}+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}-\frac{\nu}{e}\Bigg{\\{}1+\frac{v}{4}+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}+i\frac{\nu^{2}}{e^{2}}\Bigg{\\{}1+\frac{v^{2}}{8}+\mathrm{O}\left(v^{3}\right)\Bigg{\\}}$
$\displaystyle+i\frac{\nu^{3}}{e^{3}}\Bigg{\\{}\frac{1}{3}-\frac{1}{6}v+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}+\frac{\nu^{4}}{e^{4}}\Bigg{\\{}\left(\frac{1}{2}+\frac{4i}{3}\right)-\left(\frac{1}{8}+\frac{i}{6}\right)v+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}$
$\displaystyle+\frac{\nu^{5}}{e^{5}}\Bigg{\\{}\left(\frac{1}{6}+\frac{4i}{5}\right)-\left(\frac{1}{8}+\frac{i}{2}\right)v+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}$
$\displaystyle+\frac{\nu^{6}}{e^{6}}\Bigg{\\{}\left(\frac{20}{9}+\frac{19i}{5}\right)-\left(\frac{11}{18}+\frac{11i}{20}\right)v+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}+\mathrm{O}(\nu^{7}),$
(332)

while for the energy density, it is

$\displaystyle\tilde{F}(v)$
$\displaystyle\overset{\text{\tiny\text{O(3)}}}{=}\Bigg{\\{}1+\frac{v}{2}+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}+\frac{\nu}{e}\Bigg{\\{}\frac{1}{v}+\left(\ln
v-i\pi-2\gamma_{E}-4\ln 2+2\right)-v+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}$
$\displaystyle-\frac{\nu^{2}}{e^{2}}\Bigg{\\{}(1-4i)-\left(\frac{1}{2}+i\right)v+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}+i\frac{\nu^{3}}{e^{3}}\Bigg{\\{}\frac{8}{3}-\frac{2v}{3}+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}$
$\displaystyle+\frac{\nu^{4}}{e^{4}}\Bigg{\\{}(2+4i)+iv+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}+\frac{\nu^{5}}{e^{5}}\Bigg{\\{}\left(\frac{4}{3}+\frac{64i}{15}\right)-\left(\frac{2}{3}+\frac{8i}{15}\right)v+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}$
$\displaystyle+\frac{\nu^{6}}{e^{6}}\Bigg{\\{}\left(\frac{22}{3}+10i\right)+\left(\frac{2}{3}+\frac{7i}{2}\right)v+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}+\mathrm{O}(\nu^{7}).$
(333)

Clearly, we only know a finite number of coefficients in an asymptotic series
like $u_{r}$ or $F_{r}$ for a given $r$. From these, we may only approximate
the exact value of the lateral sum. A Padé-approximant of the truncated Borel-
transform captures the analytic structure of the Borel-plane already from
these coefficients, and it allows us to reconstruct the Borel-transform beyond
its convergence radius in the presence of singularities. In this way, we
approximate the integrand of the inverse transform (383), which in principle
should be the analytically continued Borel-transform if we knew all its
coefficients and were able to sum it up.

Our procedure (Borel-Padé resummation) to evaluate $S^{+}$ of a formal
asymptotic series $\Psi(v)$ from the finitely many numerical coefficients then
looks

$\displaystyle\Psi(v)\;\overset{\text{Borel-
transform}}{\longrightarrow}\;\mathcal{B}_{\Psi}(t)\;\overset{\text{diagonal
Pad\'{e}}}{\longrightarrow}\;\big{[}\tfrac{N}{2}\big{/}\tfrac{N}{2}\big{]}_{\mathcal{B}_{\Psi}}(t)\;\overset{\begin{subarray}{c}\text{Laplace-
tf. over}\\\ \text{tilted
contour}\end{subarray}}{\longrightarrow}\;S^{+}_{N}(\Psi)(v)$ (334)

where the series is known up to a certain order $N$:

$\displaystyle\Psi(v)$
$\displaystyle=\sum_{n=0}^{N}\psi_{n}v^{n}+\mathrm{O}(v^{N+1}),$
$\displaystyle\mathcal{B}_{\Psi}(t)$
$\displaystyle=\sum_{n=0}^{N}\frac{\psi_{n}}{n!}t^{n}+\mathrm{O}(t^{N+1}).$
(335)

Note that in this section the Borel transform (and its inverse) are defined
slightly differently from, but equivalently to (382) and (383). Now the
diagonal Padé-approximant of $\mathcal{B}_{\Psi}(t)$ is a ratio of two
polynomials of order $\frac{N}{2}$, i.e. $P_{\frac{N}{2}}(t)$ and
$Q_{\frac{N}{2}}(t)$

$\displaystyle\big{[}\tfrac{N}{2}\big{/}\tfrac{N}{2}\big{]}_{\mathcal{B}_{\Psi}}(t)$
$\displaystyle=\frac{P_{\frac{N}{2}}(t)}{Q_{\frac{N}{2}}(t)},$ such that
$\displaystyle\mathcal{B}_{\Psi}(t)$
$\displaystyle=\frac{P_{\frac{N}{2}}(t)}{Q_{\frac{N}{2}}(t)}+\mathrm{O}(t^{N+1}),$
(336)

which we integrate along a contour above the positive real line

$\displaystyle S^{+}_{N}(\Psi)(v)$
$\displaystyle=v^{-1}\int\limits_{0}^{e^{i\varphi}\infty}\mathrm{d}t\,e^{-t/v}\big{[}\tfrac{N}{2}\big{/}\tfrac{N}{2}\big{]}_{\mathcal{B}_{\Psi}}(t),$
(337)

where $\varphi$ is some acute angle between the real axis and the contour of
integration to perform the numerical integrals.

The technique described above was used to evaluate the trans-series of the
O(3) model’s densities (328).121212All the integrations were performed at
$\varphi=3/5$. There were $N_{\max}=336$ terms at our disposal for the
perturbative sector from the data obtained by Volin’s method [40] previously
in [18, 19], and thus at every NP order we had the same amount of trans-series
coefficients.131313See Appendix D of [19] on how to re-express the $B^{-1},\ln
B$ expansion of the original algorithm [40] in terms of the running coupling
$v$. See also the remark above (105).

For a rough estimate of the magnitude of the resummation’s error, we compared
the Borel-Padé sum evaluated for only $\approx 90\%$ of these coefficients
(i.e. $N=300$) to the same resummation performed for all known data.141414See
the discussion on this choice later. Based on the left of Figure 1, this
comparison is an upper estimate for the error, as we expect the difference of
the $\text{Re}\,S^{+}_{N_{\max}}$ sum to the exact value orders of magnitude
smaller than the difference of $\text{Re}\,S^{+}_{N}$ to
$\text{Re}\,S^{+}_{N_{\max}}$ for $N<N_{\text{max}}$. Also, when assessing the
error, we only have to consider the leading, perturbative sector as our main
source of inaccuracies. This is because the error of the resummation is
approximately the same for each asymptotic series (e.g. $S^{+}(F_{r})$ for
each $r$), yet these uncertainties are suppressed by the exponential factor
$\nu^{r}$ (this is shown on the right of Figure 1).

Figure 1:  Left: Convergence of the $\text{Re}\,S^{+}_{N}(F_{0})(v)$ sums for
increasing $N$ to the most precise resummation available at $N=N_{\max}$;
plotted for three different $B$ values. On a logarithmic scale, the difference
shows a qualitatively exponential decay for large $N$. Right: Comparison of
the relevance of the different $\text{Re}\,S^{+}_{N}(F_{r})\nu^{r}$ terms in
the resummed trans-series as error sources. The solid line shows that the
estimated error from the perturbative sector dominates.

The estimated relative precision of the TBA result for the O(3) model was
$\simeq 10^{-257}$ even at the largest $B=20$ value. To reach this accuracy we
used the recursive method in [48], and a basis of even Chebyshev polynomials
up to the order 1400 was chosen. The relative magnitudes of the last and first
expansion coefficients of the solution in this finite Chebyshev basis
indicated the above-mentioned relative error. As this value is several orders
of magnitude smaller than the last subtractions plotted in Figures 2 and 3
(lines labeled as $g$), it does not affect our analysis over the range $1\leq
B\leq 20$.

Figure 2: The difference of the TBA and the various truncations of the resummed trans-series of the normalized particle density $u(v)$ for the O(3) model as a function of $B$ (see Table 1 for the notations a-g of the coloured lines). The black line shows the TBA value itself. The same data is plotted on the inset as on the main plot, but the lines on the latter are rescaled by the appropriate inverse power of $\nu$ (see Table 1), to have a constant large $B$ asymptotics. The red dashed line for $\delta u^{(k=10)}(v)$ is added to show how the error of the resummation would influence further subtractions.  Figure 3: Similar demonstration as in Figure 2 for the normalized energy density $F(v)$ of the O(3) model. See the explanation for the labeling in Table 1. The line $a$ of the inset shows linear growth in $B$ (on a log scale) due to the $1/v+\ln v$ term in $F_{1}$ \- see (12.1.1).  |  |  | a | b | c | d | e | f | g
---|---|---|---|---|---|---|---|---|---
|  | $k$ | 0 | 1 | 2 | 3 | 4 | 5 | 6
O(3) | $u(v)$ | $\tilde{u}^{(k)}-\tilde{u}^{(k-1)}$ | $\mathbb{R}\nu^{0}$ | $\mathbb{R}\nu^{1}$ | $i\mathbb{R}\nu^{2}$ | $i\mathbb{R}\nu^{3}$ | $\mathbb{C}\nu^{4}$ | $\mathbb{C}\nu^{5}$ | $\mathbb{C}\nu^{6}$
$\text{Re}\,S^{+}(u_{k})\nu^{k}$ | $+\nu^{0}$ | $-\nu^{1}$ | $-\nu^{4}$ | $-\nu^{5}$ | $+\nu^{4}$ | $+\nu^{5}$ | $+\nu^{6}$
$\delta u^{(k)}$ | $-\nu^{1}$ | $-\nu^{4}$ | $+\nu^{4}$ | $+\nu^{4}$ | $+\nu^{5}$ | $+\nu^{6}$ | $+\nu^{7}$
| $\text{Im}\,S^{+}(u_{k})$ | $-\nu^{2}$ | $-\nu^{3}$ | $+\nu^{2}$ | $+\nu^{3}$ | $+\nu^{4}$ | $+\nu^{5}$ | $+\nu^{6}$
| $\text{Im}\,S^{+}(\tilde{u}^{(k)})$ | $-\nu^{2}$ | $-\nu^{2}$ | $-\nu^{3}$ | $-\nu^{4}$ | $-\nu^{5}$ | $-\nu^{6}$ | $-\nu^{7}$
| $F(v)$ | $\tilde{F}^{(k)}-\tilde{F}^{(k-1)}$ | $\mathbb{R}\nu^{0}$ | $iM_{1}\nu^{1}+\mathbb{R}\nu^{1}$ | $\mathbb{C}\nu^{2}$ | $i\mathbb{R}\nu^{3}$ | $\mathbb{C}\nu^{4}$ | $\mathbb{C}\nu^{5}$ | $\mathbb{C}\nu^{6}$
| $\text{Re}\,S^{+}(F_{k})\nu^{k}$ | $+\nu^{0}$ | $+\nu^{1}$ | $-\nu^{2}$ | $-\nu^{5}$ | $+\nu^{4}$ | $+\nu^{5}$ | $+\nu^{6}$
| $\delta F^{(k)}$ | $+\nu^{1}$ | $-\nu^{2}$ | $+\nu^{4}$ | $+\nu^{4}$ | $+\nu^{5}$ | $+\nu^{6}$ | $+\nu^{7}$
|  | $k$ | 0 | 2 | 4 | 6 | 8 |  |
O(4) | $u(v)$ | $\tilde{u}^{(k)}-\tilde{u}^{(k-2)}$ | $\mathbb{R}\nu^{0}$ | $i\mathbb{R}\nu^{2}$ | $\mathbb{C}\nu^{4}$ | $\mathbb{C}\nu^{6}$ | $\mathbb{C}\nu^{8}$ |  |
$\text{Re}\,S^{+}(u_{k})\nu^{k}$ | $+\nu^{0}$ | $-\nu^{4}$ | $+\nu^{4}$ | $+\nu^{6}$ | $+\nu^{8}$ |  |
$\delta u^{(k)}$ | $-\nu^{4}$ | $+\nu^{4}$ | $+\nu^{6}$ | $+\nu^{8}$ | $+\nu^{10}$ |  |
| $F(v)$ | $\tilde{F}^{(k)}-\tilde{F}^{(k-2)}$ | $\mathbb{R}\nu^{0}$ | $iM_{1}\nu^{1}+i\mathbb{R}\nu^{2}$ | $\mathbb{C}\nu^{4}$ | $\mathbb{C}\nu^{6}$ | $\mathbb{C}\nu^{8}$ |  |
| $\text{Re}\,S^{+}(F_{k})\nu^{k}$ | $+\nu^{0}$ | $-\nu^{4}$ | $+\nu^{4}$ | $+\nu^{6}$ | $+\nu^{8}$ |  |
| $\delta F^{(k)}$ | $-\nu^{4}$ | $+\nu^{4}$ | $+\nu^{6}$ | $+\nu^{8}$ | $+\nu^{10}$ |  |
Table 1: Explanation for the alphabetical notations in Figures 2,3,5. For each
model O(3),O(4) and both quantities $u(v),F(v)$ the table presents three
different rows (except for $u(v)$ of O(3), where two extra rows were added for
the imaginary parts). In every case, the upper one indicates the magnitudes
(in powers of $\nu$) and the reality properties ($\mathbb{R}:\text{real}$,
$i\mathbb{R}:\text{imaginary}$, $\mathbb{C}:\text{complex}$) of the last
term(s) included in the truncated trans-serieses
$\tilde{u}^{(k)},\tilde{F}^{(k)}$ (i.e. $u_{k}\nu^{k},F_{k}\nu^{k}$, except
for the $iM_{1}\nu^{1}$ terms). The middle one shows the order of the
magnitude and the sign for the real part of the lateral resummation of the
same terms - that is
$\text{Re}\,S^{+}(u_{k})\nu^{k},\;\text{Re}\,S^{+}(F_{k})\nu^{k}$. Finally,
the lower row tells us the sign and the order of magnitude for the differences
$\delta u^{(k)},\delta F^{(k)}$ themselves.

Our results for the O(3) model i.e. $\delta u^{(k)}(v),\delta F^{(k)}(v)$ for
$k=0,1,\ldots,6$ are shown on Figures 2,3. These differences are all
exponentially small in $B$ and each tends to a straight line on a logarithmic
scale, corresponding to some integer power of $\nu$. This power can be easily
determined (the insets of these plots show the same data divided by exactly
this power of $\nu$), and it is always larger or equal151515 Note that for the
asymptotic series where $u_{r},F_{r}$ itself has only purely imaginary
coefficients, the scaling of the real part of their resummation is
exponentially suppressed even compared to $\nu^{r}$. This means an additional
power of $\nu^{p}$ where $p>0$ is the position $t=p$ of the closest
singularity to the origin on the transformed series’ Borel plane along the
positive real axis (see Table 1). to the one for the previous subtraction. The
importance of this is that if an NP correction term from the resummed trans-
series would be incorrect, then we were stuck at a given $\nu$ power endlessly
when subtracting more and more trans-series terms (being orders of magnitude
smaller) from the TBA result. Moreover, it turns out that the exponential
order of $\delta u^{(k)}(v),\delta F^{(k)}(v)$ is the same as that of the next
term in the trans-series which would have a leading exponential scaling after
taking the real part of its Borel-Padé resummation (see Table 1).

Although it is easy to generate the coefficients $u_{r},F_{r}$ for $r>0$, we
stopped at $k=6$ when plotting the subtractions. (See some demonstration of
the accuracy for this truncation in Table 2.) We opted to do this, since over
the chosen range of $B$ values, the $\delta u^{(k)}(v),\delta F^{(k)}(v)$
curves for $k>6$ would reach the magnitude of the resummation’s error (see the
dotted lines on Figures 2,3) way below the upper end $B=20$.

| $B=1$ | $B=10$ | $B=20$
---|---|---|---
$u_{\text{TBA}}(v)$ |  $0.69095550$602 | $0.94918\ldots 943253$265 |  $0.97235\ldots 201968$450
$\text{Re}\,S^{+}_{N=336}(\tilde{u}^{(k=6)})$ |  $0.69095550$496 | $0.94918\ldots 943253$128 |  $0.97235\ldots 201968$231
digits in agreement | $8$ | $63$ | $124$
Table 2: The accuracy of our best numerics in the case of the particle density
of the O(3) model. The agreeing number of digits after the decimal point is
shown, together with the following three digits in smaller font.

Our experience is that indeed, when reaching this approximate error level, the
sofar straight lines start to diverge. To demonstrate this, we added the red
dashed line corresponding to $\delta u^{(k=10)}(v)$ to Figure 2 which stops
behaving in the expected way at around $B=8$. Comparing its $B>8$ part (which
is clearly influenced by the error) to the black dotted line it is also clear
that our upper estimate for the error was slightly pessimistic. We could have
used a much tighter estimation on the error by comparing the real parts of
$S^{+}_{N_{\max}-2}$ and $S^{+}_{N_{\max}}$ (see the black dashed line on
Figure 2). Although this seems to predict the magnitude at which our
subtraction method will fail more precisely in most cases, it also seems to
underestimate the error for certain $B$ values. Therefore we opted for the
more conservative estimate in case of Figures 3-7, to be on the safe side.

In the end, the closest approximation to the TBA we were able to obtain over
this domain from the perturbative data we had is then for $B=20$ and $B=19$,
and is of the order $\simeq 10^{-125}$ and $\simeq 10^{-118}$ respectively for
$\delta u^{(k=6)}(v)$ and $\delta F^{(k=6)}(v)$, as can be deduced from the
plots.

For a consistency check, we also calculated the imaginary parts
$\text{Im}\,S^{+}(\tilde{u}^{(k)})$ for the same truncations $k=0,1,\ldots,6$.
As expected, for increasing $k$, the sums are decreasing in magnitude, i.e.
they show a $\nu^{p},\;p\in\mathbb{Z}^{+}$ behaviour with increasing values of
the power $p$, just as the differences $\delta u^{(k)}(v)$ did. See the extra
rows for the particle density of the O(3) model in Table 1, and the
corresponding data on Figure 4. Throughout this section, similarily to the
above case, we followed the cancellation of the imaginary parts of the
resummed trans-series terms for every other quantity in Tables 1, 3. In each
case, we confirmed the reduction in the sum’s magnitude (scaling as some
integer power in the corresponding NP parameter of the problem) when
truncating the trans-series at higher orders.

Figure 4: The imaginary part of the resummed trans-series at different
truncations (see Table 1) as a function of $B$, in case of the O(3) model’s
particle density. When including more terms from the trans-series $\tilde{u}$,
the sum decreases. The inset shows the same data divided by the corresponding
$\nu$ power taken from Table 1. The error was estimated in a similar way, as
for the real part.

At this point we ought to make a remark about the constants at which the lines
on the insets of Figures 2,3 tend to, for large values of $B$. Each is equal
to the real part of the leading perturbative coefficient of the first such NP
term in the trans-series, which was not included in the truncation yet - and
whose leading coefficient has a real part at all.161616By Table 1, after the
first few $\nu$ orders, each NP term has complex perturbative coefficents.
Both for $u(v)$ and $F(v)$ the magnitude of these constants tend to decrease,
with increasing order of the NP parameter. This is true even for the O(4)
model (see Figure 5). For the free energy of both models in terms of the
physical running coupling, from the insets of Figures 6,7 we see that it is
plausible, that the same leading coefficients do not grow with the NP order
too fast. That is, for a small enough value of the NP parameter the resummed
trans-serieses of these quantities seem to have good convergence properties.
This is consistent with the findings of [35] where numerical evidence for the
convergence of the free energy’s Borel-resummed trans-series was presented in
the case of the O($N\geq 4$) models. The argument in the latter work was based
on the convergence properties of the LO and NLO perturbative coefficients of
each NP term in the trans-series (trans-asymptotics). By knowing the full
trans-series up to arbitrarily high perturbative orders in principle, the
convergence can be tested by direct evaluation of the Borel-resummed trans-
series. We plan to publish such investigations in our following paper [49].

####  12.1.2 The O(4) model

Now let us turn to the case of the O(4) model. The trans-series of the
particle density starts as

$\displaystyle\tilde{u}(v)$
$\displaystyle\overset{\text{\tiny\text{O(4)}}}{=}\Bigg{\\{}1-\frac{3v}{4}+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}+i\frac{\nu^{2}}{2^{2}}\Bigg{\\{}1+\frac{v}{2}+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}$
$\displaystyle+\frac{\nu^{4}}{2^{4}}\Bigg{\\{}\left(\frac{1}{2}+\frac{3i}{4}\right)+\left(\frac{1}{8}+\frac{9i}{32}\right)v+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}$
$\displaystyle+\frac{\nu^{6}}{2^{6}}\Bigg{\\{}\left(\frac{5}{4}+i\right)+\left(\frac{9}{32}+\frac{5i}{12}\right)v+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}$
$\displaystyle+\frac{\nu^{8}}{2^{8}}\Bigg{\\{}\left(\frac{103}{32}+\frac{103i}{64}\right)+\left(\frac{293}{384}+\frac{827i}{1024}\right)v+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}+\mathrm{O}(\nu^{10}),$
(338)

while that of the energy density as

$\displaystyle\tilde{F}(v)$
$\displaystyle\overset{\text{\tiny\text{O(4)}}}{=}\Bigg{\\{}1+\frac{v}{2}+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}-2i\nu+i\frac{\nu^{2}}{2^{2}}\Bigg{\\{}4+v+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}$
$\displaystyle+\frac{\nu^{4}}{2^{4}}\Bigg{\\{}(2+2i)+\frac{3iv}{4}+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}$
$\displaystyle+\frac{\nu^{6}}{2^{6}}\Bigg{\\{}(4+2i)+\left(\frac{1}{2}+\frac{3i}{2}\right)v+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}$
$\displaystyle+\frac{\nu^{8}}{2^{8}}\Bigg{\\{}\left(\frac{37}{4}+\frac{9i}{4}\right)+\left(\frac{33}{16}+\frac{207i}{64}\right)v+\mathrm{O}\left(v^{2}\right)\Bigg{\\}}+\mathrm{O}(\nu^{10}).$
(339)

For previous works [14, 15] we took advantage of Volin’s method to generate
2000 of the coefficients for the perturbative sector. In principle, this would
allow us to determine the same amount of coefficients for all the NP
corrections. For the present work we restricted ourselves to half of them,
that is, we performed the Borel-Padé resummations as described in (334) for
$N_{\max}=1000$ coefficients at each NP order171717With $\varphi=1/10$
integration angle.. To get a picture of the error of this procedure we made a
comparison to $S^{+}_{N}$ taken at $N=900$. In this case, however, the error
source limiting our comparison’s domain of applicability was the accuracy of
the numeric TBA solution as can be seen from Figure 5. This happened since due
to the more complicated functional form of the O(4) model’s TBA kernel, we had
to stick to the less efficient method in [15], and thus the highest Chebyshev
polynomial involved in our expansion was only the $200^{\text{th}}$.

Note that as the NP orders in the trans-series for the O(4) case are
suppressed by even powers in $\nu$ (with the exception of the $iM_{1}\nu$
term), we were able to check fewer terms from the trans-series compared to the
O(3) case, as the smallest ones (lines labeled $e$ on Figure 5) have already
higher order in $\nu$ than the last ones, i.e. lines $g$ on Figures 2,3.

The best approximation of the TBA by the trans-series we could reach this way
was of the order $\simeq 10^{-63}$ at $B=7$ for both $\delta u^{(k=8)}(v)$ and
$\delta F^{(k=8)}(v)$.

Figure 5:  Comparison of the TBA result and the truncated and resummed trans-
series for the normalized particle and energy densities of the O(4) model
(dashed vs solid lines for $u(v)$ and $F(v)$ respectively).

###  12.2 Free energy density

When analysing the physical free energy density, for notational consistency we
define and denote its normalized version as $G(\beta)$ in terms of the
physical running coupling:181818Here $\Delta=1$ for O(3) and $\Delta=1/2$ for
O(4) as usual.

$\displaystyle\mathcal{F}$ $\displaystyle=-\frac{h^{2}}{4\pi\Delta}G(\beta),$
$\displaystyle G(\beta)=\frac{g(\beta)}{\beta}.$ (340)

However, due to its leading $1/\beta$ behaviour - see (259) - it is more
convenient to work with the rescaled quantity $g(\beta)$ defined above, whose
trans-series $\tilde{g}(\beta)$ starts with unity - just like those of
$\tilde{F}(v)$ and $\tilde{u}(v)$.

The physical couplings are defined as:

$\displaystyle\frac{1}{\beta}\;+\;\ln\beta$
$\displaystyle=\ln\left(\frac{h}{m}\right)\;+\;\ln\left(\frac{8}{e}\right)$
$\displaystyle\begin{matrix}0.6881&\gtrsim\beta>&0\quad\;\\\ \quad
1&<\frac{h}{m}<&\infty\quad\quad\\\ \quad 0&<B<&\infty\quad\quad\end{matrix}$
$\displaystyle\text{for }\text{O}(3)$ (341)
$\displaystyle\frac{1}{\beta}+\frac{1}{2}\ln\beta$
$\displaystyle=\ln\left(\frac{h}{m}\right)+\frac{1}{2}\ln\left(\frac{32}{e\pi}\right)$
$\displaystyle\begin{matrix}\quad\quad\;2&>\beta>&0\quad\quad\\\
1.2045&\lesssim\frac{h}{m}<&\infty\quad\quad\\\ 0.6688&\lesssim
B<&\infty\quad\quad\end{matrix}$ $\displaystyle\text{for }\text{O}(4),$ (342)

where the approximate bound on $B$ for O(4) was determined from the TBA result
of $h/m$.

Analogously to the differences introduced in (330), we compare the TBA vs the
real part of the Borel resummation via

$\displaystyle\delta g^{(k)}(\beta)$
$\displaystyle=g_{\text{TBA}}(\beta)-\text{Re}\,S_{+}(\tilde{g}^{(k)})(\beta),$
$\displaystyle\tilde{g}^{(k)}$ $\displaystyle=\sum_{r=0}^{k}g_{r}\mu^{r},$
(343)

where for notational simplicity we introduced the NP parameter $\mu$ as
$\mu\equiv e^{-2/\beta}$ for O(4) while $\mu\equiv\frac{2}{\beta}e^{-2/\beta}$
for O(3). In the latter case, the extra $\frac{2}{\beta}$ factor was
introduced to shift the $g_{r}$ asymptotic serieses such that each (except
$g_{1}$) starts with a constant instead of some negative power of $\beta$.

####  12.2.1 The O(3) model

The coefficients of the O(3) model up to the first $\zeta_{3}$-containing term
in $\beta$ at each analyzed NP order are given below:

$\displaystyle\tilde{g}(\beta)$
$\displaystyle\overset{\text{\tiny\text{O(3)}}}{=}1-\frac{\beta}{2}-\frac{\beta^{2}}{2}-\left(\frac{7}{8}-\frac{3}{32}\zeta_{3}\right)\beta^{3}+\mathrm{O}(\beta^{4})$
$\displaystyle+\left(\frac{32}{e^{2}\beta}\right)e^{-2/\beta}\Bigg{\\{}-\frac{2}{\beta}+\left(\frac{i\pi}{2}+\gamma_{E}+5\ln
2-3-\ln\beta\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-\frac{\beta}{2}+\frac{1}{32}(4-3\zeta_{3})\beta^{3}+\mathrm{O}(\beta^{4})\Bigg{\\}}$
$\displaystyle+\left(\frac{32}{e^{2}\beta}\right)^{2}e^{-4/\beta}\Bigg{\\{}\frac{1}{2}\left(1-i\right)-\frac{1}{8}\left(1-\frac{3i}{2}\right)\beta+\frac{1}{8}\left(1-\frac{i}{32}\right)\beta^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad+\frac{1}{32}\left[\left(3-\frac{129i}{64}\right)-\frac{3}{2}\left(1-i\right)\zeta_{3}\right]\beta^{3}+\mathrm{O}(\beta^{4})\Bigg{\\}}$
$\displaystyle+\left(\frac{32}{e^{2}\beta}\right)^{3}e^{-6/\beta}\Bigg{\\{}\frac{1}{4}\left(1-2i\right)-\frac{1}{24}\left(1-\frac{13i}{2}\right)\beta+\frac{1}{12}\left(1-\frac{83i}{64}\right)\beta^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad+\frac{1}{32}\left[\left(5-\frac{317i}{64}\right)-3\left(\frac{1}{2}-i\right)\zeta_{3}\right]\beta^{3}+\mathrm{O}(\beta^{4})\Bigg{\\}}+\mathrm{O}(e^{-8/\beta}).$
(344)

The Borel-Padé resummation was performed in a similar manner as we defined the
Borel-resummation (383) for an asymptotic series starting from some negative
power of the perturbative variable. That is, we separated the
$\beta^{-p},\;p\geq 1$ terms from the power series $g_{r}(2/\beta)^{r}$, and
used the method as explained in (336) for the rest of the coefficients (now in
$\beta$). Finally, we added the separated terms to this sum thus obtaining the
total value. We evaluated the $\ln\beta$ term in $g_{1}$ in the same way.

To carry out the numerics we used only $N_{\max}=100$ coefficients ($N=90$ to
estimate the error), and were calculating $\delta g^{(k)}$ up to $k=3$ only,
as obtaining more coefficients proved to be time-consuming with the method
explained in the introductory part of the present section (i.e. fixing an
ansatz for $\tilde{G}(\beta)$ and obtaining its coefficients from
$\tilde{G}(\tilde{b}(v))\overset{!}{=}\Omega(v)$).

The TBA data was the same that we used for Figures 2,3. The definition (234)
and the normalization (340) was used to determine $g_{\text{TBA}}(\beta)$ at a
given $B$ value. The $\beta$ values to $B=1,2,\ldots,20$ were obtained from
numerically solving (341) using $h/m$ as an input from the boundary values of
the TBA solutions as in (237).

The results are shown in Figure 6. With this data, we could verify the
agreement to the TBA in the physical coupling up to $\simeq 10^{-57}$ at
$B=16$ based on line $d$.

Figure 6: The comparison plot for the normalized free energy density
$g(\beta)$ of the O(3) model in terms of $B$. The upper scale shows the
corresponding $\beta$ and $h/m$ values simultaneously. See Table 3 for
explanations of the labeling. The inset shows the same data as the main plot
divided by the appropriate power of $\mu=\frac{2}{\beta}e^{-2/\beta}$ to have
a constant behaviour for large $B$, as given in the table. Line $a$ of the
inset grows linearly (on a log-scale) in terms of $B$ due to the
$-\frac{2}{\beta}-\ln\beta$ term appearing in the second line of (12.2.1).

####  12.2.2 The O(4) model

For our last numerical investigation, we took the free energy density of the
O(4) model under scrutiny. The coefficients of the trans-series (up to the
first terms containing $\zeta_{3}$) are presented below:

$\displaystyle\tilde{g}(\beta)$
$\displaystyle\overset{\text{\tiny\text{O(4)}}}{=}\Bigg{\\{}1-\frac{\beta}{2}-\frac{\beta^{2}}{4}-\left(\frac{5}{16}-\frac{3\zeta_{3}}{32}\right)\beta^{3}+\mathrm{O}(\beta^{4})\Bigg{\\}}+\frac{8i}{e}e^{-2/\beta}$
$\displaystyle-i\left(\frac{2}{e}\right)^{2}e^{-4/\beta}\Bigg{\\{}2-\frac{3}{4}\beta+\frac{13}{64}\beta^{2}-\left(\frac{29}{512}+\frac{3\zeta_{3}}{16}\right)\beta^{3}+\mathrm{O}(\beta^{4})\Bigg{\\}}$
$\displaystyle-\left(\frac{2}{e}\right)^{4}e^{-8/\beta}\Bigg{\\{}\left(2+\frac{i}{2}\right)-\left(\frac{7}{4}+\frac{15i}{32}\right)\beta+\left(\frac{9}{8}+\frac{237i}{1024}\right)\beta^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-\left[\left(\frac{51}{128}-\frac{73i}{16384}\right)+\left(\frac{9}{16}+\frac{9i}{64}\right)\zeta_{3}\right]\beta^{3}+\mathrm{O}(\beta^{3})\Bigg{\\}}$
$\displaystyle-\left(\frac{2}{e}\right)^{6}e^{-12/\beta}\Bigg{\\{}(3-5i)-\left(\frac{67}{16}-\frac{151i}{24}\right)\beta+\left(\frac{1703}{512}-\frac{705i}{128}\right)\beta^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-\left[\left(\frac{45}{32}-\frac{75i}{32}\right)\zeta_{3}+\left(\frac{10159}{8192}-\frac{8711i}{3072}\right)\right]\beta^{3}+\mathrm{O}(\beta^{4})\Bigg{\\}}+\mathrm{O}(e^{-16/\beta}).$
(345)

At each NP order, we used $N_{\max}=600$ coefficients for the Borel-Padé
resummation ($N=540$ to estimate its error), except for the last term $k=6$ in
Table 3 (line $d$ of Figure 7), where we obtained the expansion only up to
$N_{\max}=150$ in $\beta$. The same TBA data were used as those for the
analysis in the bootstrap coupling $v$. Compared to the latter case (see
Figure 5) here we managed to subtract one less NP term in the trans-series
from the TBA, as the error of the resummation was higher due to using fewer
(600 instead of 1000) coefficients to resum the perturbative level $k=0$.
Based on line $c$ of Figure 7 we managed to verify the trans-series
$\tilde{g}(\beta)$ up to $\simeq 10^{-53}$ at $B=10$ for the O(4) model.

Figure 7: The truncated trans-series vs TBA plot for the normalized free energy density $g(\beta)$ of the O(4) model (see also Table 3). The inset shows the main plot’s data divided by the correct power of $\mu=e^{-2/\beta}$. The shaded region indicates that the $\beta$ coupling is defined - and so the lateral resummation of the trans-series makes sense - above $B_{*}\simeq 0.6688$ or below $\beta<2$ according to (342). Both the resummation’s error and the TBA uncertainty limit the lines shown. The first even restricts the domain from below, that is why the data was plotted only for $B\geq 2$. |  | a | b | c | d
---|---|---|---|---|---
| $k$ | 0 | 1 | 2 | 3
O(3) | $\tilde{g}^{(k)}-\tilde{g}^{(k-1)}$ | $\mathbb{R}\mu^{0}$ | $-iM_{1}\frac{8}{e}\mu^{1}+\mathbb{R}\mu^{1}$ | $\mathbb{C}\mu^{2}$ | $\mathbb{C}\mu^{3}$
$\text{Re}\,S^{+}(g_{k})\mu^{k}$ | $+\mu^{0}$ | $-\mu^{1}$ | $+\mu^{2}$ | $+\mu^{3}$
$\delta g^{(k)}$ | $-\mu^{1}$ | $+\mu^{2}$ | $+\mu^{3}$ | $+\mu^{4}$
| $k$ | 0 | 2 | 4 | 6
O(4) | $\tilde{g}^{(k)}-\tilde{g}^{(k-2)}$ | $\mathbb{R}\mu^{0}$ | $-iM_{1}\frac{4}{e}\mu^{1}+i\mathbb{R}\mu^{2}$ | $\mathbb{C}\mu^{4}$ | $\mathbb{C}\mu^{6}$
$\text{Re}\,S^{+}(g_{k})\mu^{k}$ | $+\mu^{0}$ | $+\mu^{4}$ | $-\mu^{4}$ | $-\mu^{6}$
$\delta g^{(k)}$ | $+\mu^{4}$ | $-\mu^{4}$ | $-\mu^{6}$ | $+\mu^{8}$
Table 3: Explanation for the labels in Figures 6,7. The NP parameter $\mu$ is
defined as stated below (343), and the meaning of the rows is similar to what
was explained in Table 1: the upper row shows the magnitude and reality of the
last trans-series term $g_{k}\mu^{k}$ subtracted, the middle one tells us the
sign and $\mu$-order of its resummation’s real part, and lastly, the lower one
indicates the sign and order of $\delta g^{(k)}(\beta)$.

##  13 Conclusions

We used the Wiener-Hopf technique to solve the integral equation governing the
groundstate energy density in the two dimensional non-linear O($N$) $\sigma$
model. This linear integral equation is the thermodynamic limit of the Bethe
ansatz equation, it has a finite support $[-B,B]$ and its kernel is the
logarithmic derivative of the scattering matrix. We obtained the solution in a
trans-series form, which was a double series in the perturbative running
coupling $v$ defined by $2B=\frac{1}{v}-a\log v+const$, and in the non-
perturbative, exponentially small quantity $\nu=e^{-2B}\sim
v^{a}e^{-\frac{1}{v}}$. The structure of the non-perturbative corrections was
governed by the pole structure of the Wiener-Hopf kernel. Every power of the
non-perturbative parameter $\nu$ was multiplied by an asymptotic perturbative
series in $v$. The latter were expressed in terms of the basic quantities
$A_{n,m}$. The $A_{n,m}$ are asymptotic series, which are the perturbative
parts of the generalized moments of the generalized energy densities
$W_{n,m}$, such that $A_{1,1}$ is related to the energy density, while
$A_{1,0}$ to the particle density. We derived a system of differential
equations relating these observables to each other, which enabled us to
calculate all of them in terms of just a single one, say from $A_{1,1}$, which
was already known from Volin’s method. When deriving our solution, we paid
particular attention to the degenerate cases $O_{n,0}$, which included the
calculation of the particle density itself.

In order to connect our trans-series solution to the physical observables all
the asymptotic series have to be summed up. This is understood via lateral
Borel resummations. Although there were many indications along the derivation
that this is the right resummation prescription it does not follow directly
from our solution. We thus assumed that this is the case and thoroughly
investigated its consequences. In particular, the ambiguity free nature of the
result requires that the asymptotic behaviour of the asymptotic series in $v$
of a given power of $\nu$ is related to the perturbative expansion in $v$ of
other powers of $\nu$. We investigated these _resurgence_ relations separately
for the $N>3$ and $N=3$ cases, and determined the Stokes automorphism, which
relates the two lateral resummations. We found that for $N>3$ the perturbative
series contains all the information about the non-perturbative terms and its
median resummation reproduces the result of the numerical solution of the TBA
equation. However, in the $N=3$ case we could identify three sectors: in
addition to the perturbative sector the $\nu$ and $\nu^{2}$ sectors. We called
them the perturbative, the one-instanton and the two-instanton sector,
respectively. We found that the physical result is the median resummation of
the sum of these three sectors. By carefully investigating the asymptotics of
the perturbative sector we showed that the other two sectors are indeed
indepent. We further supported our trans-series solution with very high
precision numerical investigations, in which we compared the numerical
solution of the TBA equations with the resummation of various orders of the
trans-series. In doing so we truncated the trans-series at some given non-
perturbative level, and evaluated its lateral Borel resummation. Since the
result’s difference to the (numerically evaluated) TBA was always of the order
of the next non-perturbative correction coming from the same trans-series, we
confirmed the correctness of all tested non-perturbative order. Any
discrepancy of a given non-perturbative order would prevent us from getting
more and more precise results as we were stuck at the wrong non-perturbative
order. Considering how the values of the individual resummed asymptotic
serieses at each non-perturbative level grow, we also found some indication
that the trans-series will be convergent.

We then switched to the physical couplings, which appears in the Lagrangean
formulation of the O($N$) models, when one of the global charges is coupled to
a magnetic field. The observable of interest is the free energy density, which
is the Legrendre transform of the energy density. Since the physical coupling