aainstitutetext: CERN, Theoretical Physics Department,
CH-1211 Geneva 23, Switzerlandbbinstitutetext: Fields and Strings Laboratory,
Institute of Physics, Ecole Polytechnique Federale de Lausanne (EPFL),
CH-1015 Lausanne, Switzerland

# Injecting the UV into the Bootstrap: Ising Field Theory

Miguel Correia b João Penedones b and Antoine Vuignier

###### Abstract

We merge together recent developments in the S-matrix bootstrap program to
develop a dual setup in 2 space-time dimensions incorporating scattering
amplitudes of massive particles and matrix elements of local operators. In
particular, the stress energy tensor allows us to input UV constraints on IR
observables in terms of the central charge $c_{UV}$ of the UV Conformal Field
Theory. We consider two applications: (1) We establish a rigorous lower bound
on $c_{UV}$ of a class of $\mathbb{Z}_{2}$ symmetric scalar theories in the IR
(including $\phi^{4}$); (2) We target Ising Field Theory by, first, minimizing
$c_{UV}$ for different values of the magnetic field and, secondly, by
determining the allowed range of cubic coupling and one-particle form-factor
for fixed $c_{UV}=1/2$ and magnetic field.

Figure 1: _In this pyramid lies Ising Field Theory_

##  1 Introduction

The S-matrix bootstrap aims at constraining scattering amplitudes in quantum
field theory (QFT) based on general principles, such as unitarity, causality
and Lorentz invariance. The S-matrix program arose in the 60s as a way to
tackle the problem of the strong interactions. However, imposing all the
constraints in a predictive way proved difficult and, eventually, Quantum
Chromodynamics (QCD) rose to prominence with the observation of deep inelastic
scattering and the discovery of asymptotic freedom, which allowed the use of
Feynman perturbation theory at high energies.

Today, finding efficient ways to compute observables in low energy QCD and,
more generally, in non-perturbative QFTs, remains an unsolved problem at
large. Besides the S-matrix bootstrap, which has seen a revival in recent
years, other modern techniques include Monte-Carlo simulations of lattice
discretizations, Hamiltonian truncation and tensor networks. The latter
techniques typically involve an UV cutoff whose extrapolation to infinity is a
non-trivial computational problem, whereas the S-matrix bootstrap is directly
set up in the continuum, as required by Poincaré invariance.

The primal approach to the S-matrix bootstrap constructs an explicitly
analytic and crossing-symmetric ansatz for the amplitude and then constrains
its parameters by imposing unitarity on the physical scattering region
Paulos:2016but ; Doroud:2018szp ; Paulos:2017fhb ; He:2018uxa ;
Cordova:2018uop ; Guerrieri:2018uew ; Homrich:2019cbt ; EliasMiro:2019kyf ;
Paulos:2018fym ; Bercini:2019vme ; Guerrieri:2020bto ; Hebbar:2020ukp ;
Guerrieri:2021ivu ; Chen:2021pgx ; a_anomaly ; Chen:2022nym ; Miro:2022cbk .
In practice, the infinite-dimensional space of amplitudes is truncated to
finite dimension $N$ (the number of parameters in the ansatz). By maximizing a
given observable (e.g. the residue of a pole) one directly explores the space
of amplitudes from the “inside” with the true boundary of the allowed space
for the observable (presumably) reached asymptotically, i.e. as $N$ is taken
to infinity. This means that the primal approach is incapable of producing
rigorous bounds at finite $N$.

Conversely, the dual approach, as the name suggests, approaches the boundary
of the allowed space from the “outside” by excluding disallowed regions of
parameter-space using a finite number $N$ of parameters. Therefore, for each
$N$, there is a rigorous bound on the observable. As $N$ is increased the
excluded region becomes larger, and the bound becomes tighter. Dual
formulations of the S-matrix bootstrap were first developed in 70s
LOPEZ1975358 ; Lopez:1975wf ; LOPEZ1975437 ; BONNIER197563 ; Lopez:1976zs  and
recently revived in $d=2$ Cordova:2019lot ; Guerrieri_2020 ;
Kruczenski:2020ujw ; EliasMiro:2021nul  and in $d=4$ He:2021eqn ;
Guerrieri:2021tak . Reference Guerrieri:2021tak , in particular, showed that
the dual approach can be formulated as a linear optimization problem amenable
to implementation in SDPB simmonsduffin2015semidefinite ; landry2019scaling .

The S-matrix, which dictates how asymptotic one-particle states scatter, is an
IR observable. Nonetheless, if the S-matrix originates from a UV complete
QFT,111A UV complete QFT can be defined non-perturbatively as a CFT in the UV
deformed by relevant deformations that trigger a renormalization group (RG)
flow to the IR. The CFT in the IR is assumed to be empty, such that the QFT
has a mass gap and the S-matrix is a well-defined object. knowledge of the UV
conformal field theory (CFT) from which the QFT flows from should further
constrain the S-matrix. This was the idea behind Karateev_2020  which, besides
scattering states, considered states given by the action of local operators,
such as the stress-energy tensor, on the vacuum. Concretely, in $d=2$, UV
information can be included via the $c$-sum rule PhysRevLett.60.2709  which
relates the spectral density of the trace of the stress-energy tensor to
$c_{UV}$, the central charge of the UV CFT,

$c_{UV}=12\pi\int_{m^{2}\,>\,0}^{\infty}\frac{\rho(s)}{s^{2}}\;ds\,.$ (1.1)

Here, we make the next logical step in this story. We merge these ideas
together to develop a dual bootstrap approach in $d=2$ that encompasses
S-matrix elements, form factors and spectral densities. Our method, which is
described in section 2, produces a linear optimization problem that can be
tackled with SDPB and which, moreover, converges appreciably faster than the
primal approach of Karateev_2020 .

With our method we can address the following question: _Given a gapped unitary
QFT with no bound states how small can the central charge $c_{UV}$ of the UV
CFT be?_ Our method outputs $c_{UV}^{(min)}=1/2$, the central charge of the
Ising CFT, which is the smallest among the unitary conformal minimal models
PhysRevLett.52.1575 . Indeed, our optimal S-matrix is given by $S(s)=-1$
corresponding to a free massive Majorana fermion in the IR, which can
originate from the pure thermal deformation of the Ising CFT.

We can refine the previous question by fixing a parameter $\Lambda$ in the IR.
For example: the amplitude at the crossing-symmetric point,
$T(s=2m^{2})=-\Lambda$, which plays the role of a ‘quartic coupling’. In this
case we find a minimal $c_{UV}$ for a given $\Lambda$. The result is plotted
below in fig. 2.

Figure 2: Rigorous lower bound on the UV central charge in $\mathbb{Z}_{2}$
symmetric QFTs with a single stable particle, which is $\mathbb{Z}_{2}$ odd.
The parameter $\Lambda$ is defined as (minus) the value of the amplitude at
the crossing-symmetric point $s=2m^{2}$. The lower bound on $c_{UV}$ goes from
$c_{UV}^{(min)}=1$ at $\Lambda=0$, where the S-matrix, form factor and
spectral density become that of a free massive boson, to $c_{UV}^{(min)}=1/2$
at the other end $\Lambda=8$, where the matrix elements are that of a free
massive fermion.

In section 3.1, we further refine the previous studies. Following Chen_2022 ,
we parameterize the theory space by $(\Lambda,\Lambda^{(2)})$, where
$\Lambda^{(2)}$ is the second derivative of the amplitude at the crossing
symmetric point. As in the case above, we assume no cubic coupling, i.e. no
poles on the physical sheet for the amplitude nor for the form factor. In this
way, we target a class of $\mathbb{Z}_{2}$ symmetric theories, in which
$\phi^{4}$ theory is included. Using the numerical dual approach, we
rigorously bound $c_{UV}$ across the allowed range of couplings
$(\Lambda,\Lambda^{(2)})$ generating the 3-dimensional plots in fig. 5.
Remarkably, we also found analytical expressions (described in appendix A)
that match the extremal solutions (see e.g. figs. 10 to 12 for a comparison).

In section 3.2 we consider the Ising Field Theory (IFT) which can be thought
of as the QFT describing the $d=2$ Ising model near the critical point. Above
the critical temperature $T>T_{c}$ and for zero magnetic field $h=0$ IFT
reduces to the theory of a free massive Majorana fermion. For $h>0$ the theory
becomes fully interacting making it an appropriate playground for non-
perturbative methods.222The parameter space of IFT is very rich (see e.g.
Delfino:2003yr  for a short review). For $T>T_{c}$ and as $h$ increases the
number of stable particles goes from 1 to 3 Zamolodchikov:1989fp ; Zamo_2013 ;
gabai2019smatrix , and then jumps to 8 (5 resonances become stable) at the
$h\to\infty$ integrable point where IFT becomes equivalent to the $E_{8}$
affine Toda theory Zamolodchikov:1989fp ; 1989PhLB..226…73H ; Braden:1989bu .
For $T<T_{c}$ the spectrum of IFT consists of a tower of “mesons”
Fonseca:2006au . Further interesting phenomena occur when $h$ is taken to be
complex, namely the existence of Lee-Yang edge singularities Fonseca:2001dc .

We have at our disposal several pieces of information that we can use to
target IFT:

  1. 1.

The UV central charge of IFT has the value $c_{UV}=1/2$ given that IFT is, by
definition, a deformation of the Ising CFT. Specifically, both relevant
deformations are turned on, namely the operators conjugate to the temperature
$(T-T_{c})$ and the magnetic field $h$.

  2. 2.

At sufficiently low $h$ (but still outside the perturbative regime) there is
only one stable particle in the spectrum Zamo_2013 . This particle self-
interacts via a cubic interaction, meaning that both S-matrix and the two-
particle form factor contain a pole at the location of the particle’s own
mass.

  3. 3.

As shown in Zamo_2013 ; gabai2019smatrix , the S-matrix has a real zero which
slides towards the two-particle threshold as $h$ is increased.333If $h$ is
increased past a certain value the zero makes its way across the two-particle
branch cut and the associated pole pops up on the physical sheet, as the
second lightest particle becomes stable. See e.g. gabai2019smatrix .  The
position of the zero provides a non-perturbative IR handle on the value of the
magnetic field $h$.

We first implement points 2. and 3. and minimize the central charge $c_{UV}$
over a range of magnetic fields $h$ (parameterized by the position of the
zero). We find that the lower bound on $c_{UV}$ drops below $c_{UV}=1/2$ for
non-zero $h$ (see fig. 13). We then fix $c_{UV}=1/2$ and find the allowed
range of cubic coupling and one-particle form factors for a range of values of
the magnetic field. We carve a 3-dimensional ‘pyramid’ in these parameters
inside which IFT must lie (see fig. 18). We conclude with section 4 where we
discuss our results in further detail and point out some potential future
directions.

Let us now briefly outline the remaining appendices. In appendix B we present
a further application of our dual method targeting the Sine-Gordon model where
we noticeably improve on the primal result of Karateev_2020 . Technical and
numerical details regarding the dual optimization problems are collected in
appendices C and D. Appendix E contains the perturbative computations of the
one particle form factor and cubic coupling used to place IFT within the dual
bounds (fig. 18). Finally, appendices F and G respectively review the c-sum
rule and the normalization of the 2 particle form factor of the stress energy
tensor.

##  2 Dual S-matrix and form factor Bootstrap

In this section we develop our setup by recalling some definitions and results
from standard massive QFT. We will be complete but concise, more details can
be found in Karateev_2020 ; karateev2020twopoint .

The first ingredient in our setup is the 2 to 2 scattering amplitude defined
by

${}_{out}\innerproduct{p_{1}p_{2}}{k_{1}k_{2}}_{in}\equiv(2\pi)^{2}\delta^{(2)}(k_{1}+k_{2}-p_{1}-p_{2})\mathcal{N}_{2}S(s),\qquad\mathcal{N}_{2}\equiv
2\sqrt{s}\sqrt{s-4m^{2}},$ (2.1)

with $s=-(p_{1}+p_{2})^{2}$. We also define $\mathcal{T}(s)$, the interacting
part of the scattering amplitude, by

$S(s)\equiv 1+i\mathcal{N}_{2}^{-1}\mathcal{T}(s).$ (2.2)

We are also interested in some scalar local operator $\mathcal{O}(x)$, which
leads us to consider its $n$ particles form factors

$\mathcal{F}_{n}^{\mathcal{O}}(p_{1},p_{2},..,p_{n})\equiv_{out}\bra{p_{1}p_{2}...p_{n}}\mathcal{O}(0)\ket{0},$
(2.3)

and its spectral density

$2\pi\rho_{\mathcal{O}}(s)\equiv\int
d^{2}xe^{-ipx}\bra{0}\mathcal{O}^{\dagger}(x)\mathcal{O}(0)\ket{0},$ (2.4)

where we used Lorentz invariance to write
$\rho_{\mathcal{O}}=\rho_{\mathcal{O}}(s=-p^{2})$.

We focus on massive QFTs whose Hilbert space $\mathcal{H}$ is spanned by
asymptotic multi particle states $\ket{p_{1},...,p_{n}}$ with completeness
relation

$\begin{split}\mathbb{1}_{\mathcal{H}}&=\sum_{n=0}^{\infty}\frac{1}{n!}\int\frac{d\bm{p}_{1}}{(2\pi)2E_{\bm{p}_{1}}}...\frac{d\bm{p}_{n}}{(2\pi)2E_{\bm{p}_{n}}}\ket{p_{1}...p_{n}}_{in}\
{}_{in}\bra{p_{1}...p_{n}}\equiv\sumint\ket{p_{1}...p_{n}}_{in}\
{}_{in}\bra{p_{1}...p_{n}},\end{split}$ (2.5)

where $E_{\bm{p}}\equiv\sqrt{\bm{p}^{2}+m^{2}}$ and $\bm{p}$ is the spatial
part of the 2-momentum $p$. The complete set of states can be inserted in the
two point function (2.4) to get the relation between the spectral density and
the form factors

$2\pi\rho_{\mathcal{O}}(s)=\sumint(2\pi)^{2}\delta^{(2)}(p-p_{n})|\mathcal{F}_{n}^{\mathcal{O}}|^{2},$
(2.6)

where $p_{n}\equiv p_{1}+...+p_{n}$ is the total momentum of the form factor.
Note that the first contribution is a delta function at $s=m^{2}$ and the
second contribution starts at $s=4m^{2}$. Explicitly

$\rho_{\mathcal{O}}(s)=|\mathcal{F}_{1}^{\mathcal{O}}|^{2}\delta(s-m^{2})+\frac{|\mathcal{F}_{2}^{\mathcal{O}}(s)|^{2}}{2\pi\mathcal{N}_{2}}\theta(s-4m^{2})+...$
(2.7)

In this work we are interested in the trace of the stress energy tensor, so we
introduce the simplified notation

$\mathcal{F}(s)\equiv\mathcal{F}_{2}^{\Theta}(s),\qquad\rho(s)\equiv\rho_{\Theta}(s),$
(2.8)

where the $s$ dependence of the 2 particle form factor comes from Lorentz
invariance. The normalization of the stress energy tensor acting on one
particle states implies the following normalization of the 2 particle form
factor (see appendix G)

$\mathcal{F}(s=0)=-2m^{2}.$ (2.9)

The value of the UV central charge can be computed from the spectral density
of $\Theta$ as Karateev_2020  (in alternative, see appendix F)

$c_{UV}=12\pi\left(m^{-4}|\mathcal{F}_{1}^{\Theta}|^{2}+\int_{4m^{2}}^{\infty}ds\frac{\rho(s)}{s^{2}}\right).$
(2.10)

Let us now turn to the analytic structure of these functions depicted on
Figure 3. Assuming the existence of only one massive particle of mass $m$, the
scattering amplitude is analytic except for a pole at $s=m^{2}$ and a cut
starting from the two particles production threshold $s=4m^{2}$, and also
crossing symmetric. This can be written through the dispersion relation444In
writing this dispersion relation, we assumed that
$\lim_{|s|\to\infty}\frac{\mathcal{T}(s)}{s}=0$. Our dual optimization problem
will however not depend on this assumption, i.e. the behavior of the amplitude
at infinity is unconstrained. See equation (2.30) and footnote 9.

$\begin{split}\mathcal{T}(s)-\mathcal{T}(2m^{2})&=-g^{2}\left(\frac{1}{s-m^{2}}+\frac{1}{3m^{2}-s}-\frac{2}{m^{2}}\right)\\\
&+\int_{4m^{2}}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{T}(z)\left(\frac{1}{z-s}+\frac{1}{z+s-4m^{2}}-\frac{2}{z-2m^{2}}\right).\end{split}$
(2.11)

The 2-particle form factor $\mathcal{F}(s)$ has a similar analytic structure
except that it does not satisfy crossing. The dispersion relation reads555We
have used the fact that both functions are real analytic functions, i.e.
$\mathcal{T}^{*}(s)=\mathcal{T}(s^{*})$ and
$\mathcal{F}^{*}(s)=\mathcal{F}(s^{*})$ , which can be traced back to LSZ and
CPT invariance olive .

$\mathcal{F}(s)-\mathcal{F}(0)=-g_{F}\left(\frac{1}{s-m^{2}}+\frac{1}{m^{2}}\right)+\int_{4m^{2}}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{F}(z)\left(\frac{1}{z-s}-\frac{1}{z}\right).$
(2.12)

As shown in Karateev_2020 , the form factor residue $g_{F}$ is given by

$g_{F}=g\mathcal{F}_{1}^{\Theta},$ (2.13)

where $g$ is the (square root of the) residue of the scattering amplitude.

Figure 3: Complex planes of the two particles form factor
$\mathcal{F}_{2}^{\mathcal{O}}(s)$ (right) and the scattering amplitude $S(s)$
(left) in the presence of an asymptotic particle of mass $m=1$. The red
crosses symbolize the presence of a pole and the red lines represent the
branch cuts.

For future convenience, we define the analyticity and crossing constraints for
the S-Matrix and form factor, denoted respectively by
$\mathcal{A}_{\mathcal{T}}(s)=0$ and $\mathcal{A}_{\mathcal{F}}(s)=0$, with

$\begin{split}\mathcal{A}_{\mathcal{T}}(s)&\equiv\mathcal{T}(s)-\mathcal{T}(2m^{2})+g^{2}\left(\frac{1}{s-m^{2}}+\frac{1}{3m^{2}-s}-\frac{2}{m^{2}}\right)\\\
&-\int_{4m^{2}}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{T}(z)\left(\frac{1}{z-s}+\frac{1}{z+s-4m^{2}}-\frac{2}{z-2m^{2}}\right),\\\
\mathcal{A}_{\mathcal{F}}(s)&\equiv\mathcal{F}(s)-\mathcal{F}(0)+g_{F}\left(\frac{1}{s-m^{2}}+\frac{1}{m^{2}}\right)-\int_{4m^{2}}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{F}(z)\left(\frac{1}{z-s}-\frac{1}{z}\right).\end{split}$
(2.14)

Following Karateev_2020 , the unitarity constraints in presence of local
operators such as $\Theta(x)$ can be implemented by defining 3 states

$\begin{split}\ket{\Psi_{1}}&\equiv\ket{p_{1},p_{2}}_{in},\\\
\ket{\Psi_{2}}&\equiv\ket{p_{1},p_{2}}_{out},\\\ \ket{\Psi_{3}}&\equiv
m^{-1}\int d^{2}xe^{i(p_{1}+p_{2})x}\Theta(x)\ket{0}.\end{split}$ (2.15)

We now define a $3\times 3$ matrix by taking inner products between those
states and extracting an overall delta function, which reads

$B_{ij}\times(2\pi)^{2}\delta^{(2)}(p_{1}+p_{2}-p_{1}^{\prime}-p_{2}^{\prime})\equiv\innerproduct{\Psi_{i}^{\prime}}{\Psi_{j}}.$
(2.16)

Using equations (2.1), (2.3), (2.4) and the definition of $B$ as inner
products between states, the unitarity of the theory imposes the semi positive
definite constraint666Here we used the relation
${}_{in}\\!\bra{p_{1},p_{2}}\Theta(0)\ket{0}=\bra{0}\Theta^{\dagger}(0)\ket{p_{1},p_{2}}_{out}=\mathcal{F}^{*}(s)$
that comes from CPT invariance Karateev_2020 .

$B(s)\equiv\begin{pmatrix}1&S^{*}&m^{-1}\omega\mathcal{F}^{*}\\\
S&1&m^{-1}\omega\mathcal{F}\\\
m^{-1}\omega\mathcal{F}&m^{-1}\omega\mathcal{F}^{*}&2\pi m^{-2}\rho\\\
\end{pmatrix}\succcurlyeq 0,\qquad\omega\equiv\mathcal{N}_{2}^{-1/2}.$ (2.17)

The problem we want to solve is the minimization of some parameter, say the
central charge $c_{UV}$, under the bootstrap constraints which can be
implemented via Lagrange multipliers. More precisely, we write

$c_{UV}\geq\underset{\mathcal{T},\mathcal{F},\rho}{\rm inf}\
\underset{w,\mathbb{\Lambda}\succcurlyeq 0}{\rm
sup}\mathcal{L}\geq\underset{w,\mathbb{\Lambda}\succcurlyeq 0}{\rm sup}\
\underset{\mathcal{T},\mathcal{F},\rho}{\rm inf}\mathcal{L}$ (2.18)
$\mathcal{L}=c_{UV}(\mathcal{F}_{1}^{\Theta},\rho)+\int_{4m^{2}}^{\infty}ds\left[w_{\mathcal{T}}(s)\mathcal{A}_{\mathcal{T}}(s)+w_{\mathcal{F}}(s)\mathcal{A}_{\mathcal{F}}(s)-\operatorname{Tr}\mathbb{\Lambda}(s)B(s)\right]+...$
(2.19)

where $\mathcal{A}_{\mathcal{T}}(s)$ and $\mathcal{A}_{\mathcal{F}}(s)$ are
the analyticity and crossing constraints, $w_{\mathcal{T}}$ and
$w_{\mathcal{F}}$ are Lagrange multipliers, $\mathbb{\Lambda}$ is a hermitian
and positive semidefinite $3\times 3$ matrix of Lagrange multipliers and ”…”
stands for any other constraint we would like to implement (e.g. fixing some
parameter).

The positive semidefiniteness of $\mathbb{\Lambda}$ is the direct
generalisation of positiveness in the pure S- Matrix (non-linear) dual problem
from Guerrieri_2020 . Here we are searching for the lower bound instead of the
upper bound, which accounts for the sign difference in the trace term.

We parametrize $\mathbb{\Lambda}$ as

$\mathbb{\Lambda}\equiv\begin{pmatrix}\lambda_{1}&\lambda_{4}&\lambda_{6}\\\
\lambda_{4}^{*}&\lambda_{2}&\lambda_{5}\\\
\lambda_{6}^{*}&\lambda_{5}^{*}&\lambda_{3}\end{pmatrix}.$ (2.20)

With the dual approach, we will first extremize (2.19) over the primal
variables $\mathcal{T},\mathcal{F}$ and $\rho$ analytically to get the dual
Lagrangian, and then over the dual variables $w_{\mathcal{T}},w_{\mathcal{F}}$
and $\lambda_{i}$, $i=1,...,8$, numerically. The problem being linear in the
dual variables can be tackled down with SDPB simmonsduffin2015semidefinite ;
landry2019scaling .

Having the solution for the optimal dual variables we can get optimal primal
variables if the duality gap closes, which is guaranteed because the primal
problem is convex.777Since the primal problem can be implemented in SDPB
simmonsduffin2015semidefinite ; landry2019scaling  both the objective and the
constraints must be convex. To ensure strong duality, i.e. that the duality
gap closes, we must further require that there exists an interior point, i.e.
a non-optimal primal solution. This is known as Slater’s condition
RePEc:cwl:cwldpp:80  (see appendix A of Guerrieri_2020  for a proof in the
S-matrix language). Slater’s condition can be shown to be satisfied by
explicitly constructing a non-optimal primal solution, as it usually happens
in the primal bootstrap at finite $N$ Karateev_2020 . Physically, moreover,
the existence of such an interior point is guaranteed since we can always have
inelastic S-matrices and form factors which do not saturate unitarity.
Therefore, at the optimum we have $\operatorname{Tr}\mathbb{\Lambda}(s)B(s)=0$
and, since $\mathbb{\Lambda}$ and $B$ are both semipositive definite, the
product must vanish by itself888Indeed we have
$\begin{split}\operatorname{Tr}\mathbb{\Lambda}\cdot
B&=\operatorname{Tr}\sqrt{\mathbb{\Lambda}}^{\dagger}\sqrt{\mathbb{\Lambda}}\cdot\sqrt{B}\sqrt{B}^{\dagger}=\operatorname{Tr}(\sqrt{\mathbb{\Lambda}}\sqrt{B})^{\dagger}(\sqrt{\mathbb{\Lambda}}\sqrt{B})\equiv\lVert\sqrt{\mathbb{\Lambda}}\sqrt{B}\rVert_{F},\end{split}$
(2.21) where $\lVert\cdot\rVert_{F}$ is the Frobenius norm. It follows that
$\operatorname{Tr}\mathbb{\Lambda}\cdot B=0\Rightarrow\mathbb{\Lambda}B=0$.

$\mathbb{\Lambda}(s)B(s)=0.$ (2.22)

Solving this equation we get constraints between dual variables

$\lambda_{5}=\lambda_{6}^{*},\qquad\lambda_{1}=\lambda_{2},\qquad
2\lambda_{1}|\lambda_{6}|^{2}-\lambda_{3}\lambda_{1}^{2}-2\operatorname{Re}(\lambda_{4}^{*}\lambda_{6}^{2})+\lambda_{3}|\lambda_{4}|^{2}=0.$
(2.23)

We would like to stay with entries that are linear in the dual variables in
the matrix $\mathbb{\Lambda}$, and therefore we do not use the last constraint
to eliminate one of the variables.

Since the duality gap closes we can find the optimal primal solution:

$S=\frac{\lambda_{6}^{*}\lambda_{1}-\lambda_{6}\lambda_{4}^{*}}{\lambda_{6}\lambda_{1}-\lambda_{6}^{*}\lambda_{4}},\qquad\omega\mathcal{F}=\frac{|\lambda_{4}|^{2}-\lambda_{1}^{2}}{\lambda_{6}\lambda_{1}-\lambda_{6}^{*}\lambda_{4}},\qquad
2\pi\rho=\frac{(|\lambda_{4}|^{2}-\lambda_{1}^{2})^{2}}{|\lambda_{6}\lambda_{1}-\lambda_{6}^{*}\lambda_{4}|^{2}}.$
(2.24)

Note that we automatically saturate the unitarity bounds

$|S|^{2}-1=0,\qquad 2\pi\rho-|\omega\mathcal{F}|^{2}=0,\qquad
2\pi\rho(1-|S|^{2})-2|\omega\mathcal{F}|^{2}+2\operatorname{Re}(\omega^{2}\mathcal{F}^{2}S^{*})=0.$
(2.25)

Watson’s equation $\mathcal{F}/\mathcal{F}^{*}=S$ follows from using the first
equation on the last equation.

###  2.1 Example: Minimization of $c_{UV}$ for fixed quartic coupling
$\Lambda$

We will now present an explicit and detailed example of a dual linear
bootstrap formulation. We want to find the lower bound on the central charge
$c_{UV}$ when there is only one particle that is $\mathbb{Z}_{2}$ odd and the
scattering amplitude obeys $\mathcal{T}(2)=-\Lambda$. For simplicity we work
in units where $m=1$. The Lagrangian reads

$\begin{split}\mathcal{L}=12\pi\int_{4}^{\infty}ds\frac{\rho(s)}{s^{2}}&+\int_{4}^{\infty}dsw_{\mathcal{T}}(s)\left[\mathcal{T}(s)-\mathcal{T}(2)-\int_{4}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{T}(z)\left(\frac{1}{z-s}+\frac{1}{z+s-4}-\frac{2}{z-2}\right)\right]\\\
&+\int_{4}^{\infty}dsw_{\mathcal{F}}(s)\left[\mathcal{F}(s)-\mathcal{F}(0)-\int_{4}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{F}\left(\frac{1}{z-s}-\frac{1}{z}\right)\right]\\\
&-\int_{4}^{\infty}ds\operatorname{Tr}\mathbb{\Lambda}(s)\cdot
B(s).\end{split}$ (2.26)

The subtractions are chosen so that we can easily implement the constraint
$\mathcal{T}(2)=-\Lambda$ and the normalization $\mathcal{F}(0)=-2$.

Following Guerrieri_2020  we define the dual scattering function

$\begin{split}&W_{\mathcal{T}}(s)\equiv\int_{4}^{\infty}\frac{dz}{\pi}w_{\mathcal{T}}(z)\left(\frac{1}{z-s}-\frac{1}{z+s-4}+\frac{2}{s-2}\right),\\\
&\operatorname{Im}W_{\mathcal{T}}(s)=w_{\mathcal{T}}(s),\qquad\operatorname{Re}W_{\mathcal{T}}(s)=-P\int_{4}^{\infty}\frac{dz}{\pi}w_{\mathcal{T}}(z)\left(\frac{1}{s-z}+\frac{1}{s-(4-z)}-\frac{2}{s-2}\right),\end{split}$
(2.27)

and the dual form factor function

$W_{\mathcal{F}}(s)\equiv\int_{4}^{\infty}\frac{dz}{\pi}w_{\mathcal{F}}(z)\left(\frac{1}{z-s}+\frac{1}{s}\right).$
(2.28)

The analyticity constraints can then be considerably simplified. We get999Eq.
(2.30) can be derived directly without introducing $w_{\mathcal{T}}$ or
$w_{\mathcal{F}}$. The analyticity constraint for $\mathcal{T}(s)$, say, can
be written as $\oint W_{\mathcal{T}}(s)\mathcal{T}(s)ds=0$ where
$W_{\mathcal{T}}(s)$ is an analytic function, for an arbitrary closed cycle
enclosing no singularity. Now, we blow up this contour and, seeing
$W_{\mathcal{T}}(s)$ as infinitely many Lagrange multipliers (one for each
$s$), we are free to choose $W_{\mathcal{T}}(s)$ so that the contour only
picks up the contributions of $\mathcal{T}(s)$ we wish to constrain, which is
the unitarity cut $s\geq 4$ and the value at $\mathcal{T}(2)=-\Lambda$.
Therefore, we let $W_{\mathcal{T}}$ have a cut for $s\geq 4$, a pole at $s=2$,
and sufficiently fast decay at infinity (faster than $\mathcal{T}(s)$).
Blowing up the contour will then lead to
$0=\oint\mathcal{T}W_{\mathcal{T}}\,ds=4i\int_{4}^{\infty}\mathrm{Im}(\mathcal{T}W_{\mathcal{T}})ds-2\pi
i\left[\mathrm{Res}_{s=2}W_{\mathcal{T}}\right]\mathcal{T}(2).$ (2.29) Note
that the residue on the pole is fixed by blowing up $0=\oint
W_{\mathcal{T}}ds=-2\pi
i\mathrm{Res}_{s=2}W_{\mathcal{T}}+4i\int_{4}^{\infty}\text{Im}\,W_{\mathcal{T}}ds$.
Including crossing symmetry, repeating the argument for the form factor will
lead us to eq. (2.30).

$\begin{split}\mathcal{L}=12\pi\int_{4}^{\infty}ds\frac{\rho(s)}{s^{2}}&+\int_{4}^{\infty}ds\left[\operatorname{Im}(\mathcal{T}W_{\mathcal{T}})+\operatorname{Im}(\mathcal{F}W_{\mathcal{F}})+2\operatorname{Im}W_{\mathcal{F}}+\Lambda\operatorname{Im}W_{\mathcal{T}}-\operatorname{Tr}\mathbb{\Lambda}\cdot
B\right]\end{split}$ (2.30)

We are now ready to eliminate the primal variables. Varying with respect to
$\mathcal{F},\mathcal{T}$ and $\rho$ we get

$\lambda_{4}=-\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}},\qquad\lambda_{6}=-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}},\qquad\lambda_{3}=\frac{6}{s^{2}}.$
(2.31)

The dual Lagrangian then reads

$\mathcal{L}=\int_{4}^{\infty}ds\left(-2\lambda_{1}+2\operatorname{Im}W_{\mathcal{F}}-\mathcal{N}_{2}\operatorname{Re}W_{\mathcal{T}}+\Lambda\operatorname{Im}W_{\mathcal{T}}\right).$
(2.32)

All the primal variables are eliminated and we are ready to extremize over
dual variables. We can finally formulate the dual problem that can be
implemented in SDPB:

Dual Problem ($c_{UV}$ minimization)

$\underset{\\{\lambda_{1},W_{\mathcal{T}},W_{\mathcal{F}}\\}}{\text{Maximize}}\left[\int_{4}^{\infty}ds\left(-2\lambda_{1}+2\operatorname{Im}W_{\mathcal{F}}-\mathcal{N}_{2}\operatorname{Re}W_{\mathcal{T}}+\Lambda\operatorname{Im}W_{\mathcal{T}}\right)\right]$
(2.33)

Subject to

$\begin{pmatrix}\lambda_{1}&\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}&i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}\\\
\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}^{*}&\lambda_{1}&-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}\\\
-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}&i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}&\frac{6}{s^{2}}\end{pmatrix}\succcurlyeq
0,\quad\forall s\in[4,\infty).$ (2.34)

The result is given in figure 2.

##  3 Applications

###  3.1  $\mathbb{Z}_{2}$ symmetric theories

A first application of our dual formalism is to explore the allowed region in
the subspace spanned by $(\Lambda,\Lambda^{(2)},c_{UV})$ where $c_{UV}$ is the
UV central charge and

$\Lambda\equiv-\mathcal{T}(2),\qquad\Lambda^{(2)}\equiv\lim_{s\to
2}\frac{\partial^{2}}{\partial s^{2}}\mathcal{T}(s).$ (3.1)

Before considering the central charge we look at the space spanned by
$\Lambda$ and $\Lambda^{(2)}$, which is commonly called the ”leaf”, that was
already explored using a primal approach in Chen_2022  and also analytically
in https://doi.org/10.48550/arxiv.2207.12448 . Exploring the same region with
the dual formalism we get Figure 4 on which we also plotted the analytical
bounds derived in the appendix A and given by (A.24) that we repeat here for
convenience

$\Lambda^{(2)}_{-}(\Lambda)=\frac{1}{32}\Lambda^{2},\qquad\Lambda^{(2)}_{+}(\Lambda)=\frac{1}{32}(16\Lambda-\Lambda^{2}),\qquad\Lambda\in[0,8],$
(3.2)

where $\Lambda^{(2)}_{+}$ and $\Lambda^{(2)}_{-}$ denote respectively the
analytical upper and lower bounds for $\Lambda^{(2)}$ for a given $\Lambda$.
We reproduce these bounds numerically with our dual approach purely applied to
the S-matrix sector (see figure 4 below).

Figure 4: Allowed region in the $(\Lambda,\Lambda^{(2)}$) plane. The boundary
is obtained from the dual bootstrap problem and matches perfectly the
analytical bounds, with the color gradient corresponing to the value of the
minimal central charge discussed below. Using the non linear dual approach
given by eq.(C.11) to (C.15), the numerics for the points on the boundary were
simple enough to be done on Mathematica with only $N=2$ parameters in the
Ansatz.

We now minimize the UV central charge $c_{UV}$ for a given value of
$(\Lambda,\Lambda^{(2)})$ , which we do over the allowed region. The minimal
$c_{UV}$ is represented in color in fig. 4 over the allowed parameter space
for $(\Lambda,\Lambda^{(2)})$ and also in 3D in fig. 5 below.

Figure 5: Allowed region in the $(\Lambda,\Lambda^{(2)},c_{UV})$ space from
different angles. We used $N=50$. We find perfect agreement with the
analytical bootstrap result which assumes that the optimal S-matrix minimizes
$\Lambda^{(4)}=\lim_{s\to 2}\frac{\partial^{4}}{\partial s^{4}}\mathcal{T}(s)$
(see appendix A).

On the edge corresponding to the lower bound $\Lambda^{(2)}_{-}$, the
numerical bounds on the central charge $c^{(min)}_{UV}(\Lambda,\Lambda^{(2)})$
at the tips are

$c_{UV}^{(min)}(0,0)=0.99999...,\qquad c_{UV}^{(min)}(8,2)=0.49985..,$ (3.3)

corresponding to the central charges of the free boson and free fermion that
are respectively 1 and 1/2. We now check that we also recover the scattering
amplitudes, form factors and spectral densities of these theories. The
analytical results for the free Majorana fermion can be found in
Mussardo:1281256  and those for the free boson can be found in Karateev_2020 .
We repeat them here for convenience, they are

$\begin{split}&S(s)=1,\qquad\mathcal{F}(s)=-2,\qquad\rho(s)=\frac{1}{\pi\sqrt{s}\sqrt{s-4}},\qquad\text{(free
boson)},\\\
&S(s)=-1,\qquad\mathcal{F}(s)=-\sqrt{4-s},\qquad\rho(s)=\frac{\sqrt{s-4}}{4\pi\sqrt{s}},\qquad\text{(free
fermion)}.\end{split}$ (3.4)

The numerical results for these quantities at the tips of the leaf are plotted
on Figure 6 (at $\Lambda=0$) and Figure 7 (at $\Lambda=8)$. We observe a nice
convergence of our numerics toward the analytical results.

Figure 6: Scattering amplitude $S(s)$ (on the left), form factor
$\mathcal{F}(s)$ (in the middle) and spectral density $\rho(s)$ (on the right)
at the free boson point ($\Lambda=0,\Lambda^{(2)}=0)$). The blue lines are the
real parts, the orange lines are the imaginary parts, and the dashed red line
is the analytical solution for the free boson. We used $N=50$. Figure 7:
Scattering amplitude $S(s)$ (on the left), form factor $\mathcal{F}(s)$ (in
the middle) and spectral density $\rho(s)$ (on the right) at the free fermion
point ($\Lambda=8,\Lambda^{(2)}=2)$). The blue lines are the real parts, the
orange lines are the imaginary parts, and the dashed red line is the
analytical solution for the free fermion. We used $N=50$.

Even with $N=50$ we observe some wiggles in the functions, even if the optimal
bound for $c_{UV}$ is reached quickly with a lower $N$ (see next section for a
convergence study and comparison with the primal). We believe that it is due
to the non linear map between primal variables and dual variables (2.24) and
the fact that our ansätze for the dual functions might not be optimal.

It is interesting to compute the allowed region with a primal approach and to
compare with the dual bounds. We do it101010We use the numerical primal setup
from Karateev_2020  and the ansatz for the scattering amplitude given by eq.
(4.4) and (4.5) in Chen_2022 . in Fig. 8, and various sections of this 3D plot
are shown on Figs. 9. It is reassuring to see that the primal lower bound is
always greater than the dual, and that the two shapes seem to converge toward
each other. We believe that the reason for the gap is the slow convergence of
the primal formalism, which we also observe on Figure 13, where the dual
reaches the analytical bound.

Figure 8: Minimal central charge in the $(\Lambda,\Lambda^{(2)},c_{UV}$) space
computed with the dual approach (in orange) and the primal (in blue). We used
$N=50$.

Figure 9: Minimal central charge in the $(\Lambda,c_{UV}$) space (left) and in
the $(\Lambda^{(2)},c_{UV}$) space (right) computed with the dual approach (in
orange) and the primal (in blue). The value of $\Lambda^{(2)}$ is defined by
$\Lambda^{(2)}=\frac{1}{5}\frac{\Lambda^{2}}{32}+\frac{1}{5}\Lambda$ (left)
and on the one of $\Lambda$ by $\Lambda=3.2$ (right). We used $N=50$.

We now compare our numerical results with the analytical results from the
appendix A where we assume that $c_{UV}$ minimization for fixed
$(\Lambda,\Lambda^{(2)})$ minimizes $\Lambda^{(4)}=\lim_{s\to
2}\frac{\partial^{4}}{\partial s^{4}}\mathcal{T}(s)$. For the scattering
amplitude on the edges the analytical result is given by (A.6) and (A.22). The
form factor on the upper edge is given by (A.34) and (A.35), and on the lower
edge by (A.27). Figure 10 shows a point on the upper edge of the leaf, Figure
11 shows one on the lower edge and Figure 12 shows a point in the middle. The
agreement is perfect.

Figure 10: Comparison of the real part (in blue) and imaginary part (in
orange) of the S-Matrix (left) and the form factor (right) between the
numerical (plain line) and analytical (black dashed line) results, on the
upper edge of the leaf with $\Lambda=2.4$. We used $N=50$. The numerical and
analytical central charges are $c_{UV}^{(num,min)}=0.514419...$,
$c_{UV}^{(an,min)}=0.514451...$ Figure 11: Comparison of the real part (in
blue) and imaginary part (in orange) of the S-Matrix (left) and the form
factor (right) between the numerical (plain line) and analytical (black dashed
line) results, on the lower edge of the leaf with $\Lambda=2.4$. We used
$N=50$. The numerical and analytical central charges are
$c_{UV}^{(num,min)}=0.993662...$, $c_{UV}^{(an,min)}=0.993812...$ Figure 12:
Comparison of the real part (in blue) and imaginary part (in orange) of the
S-Matrix (left) and the form factor (right) between the numerical (plain line)
and analytical (dashed line) results, in the interior of the leaf with
$\Lambda=4.56$ and $\Lambda^{(2)}=0.85$. We used $N=50$. The numerical and
analytical central charges are $c_{UV}^{(num,min)}=0.526354...$,
$c_{UV}^{(an,min)}=0.526591...$.

###  3.2 Ising Field Theory

Our second application is to target the Ising Field Theory. We use the fact
that the S-Matrix has a zero whose position varies with the magnetic field

$S(m^{2}(1-x))=0,\qquad x=x(\eta),\qquad\eta=\frac{m}{h^{8/15}}.$ (3.5)

The limit $x\to 0$ corresponds to $h\to 0$ and then should give back the free
massive Majorana fermion. When $x\to 1$ we are in the region just before the
second lightest particle emerges from the two particle cut.

####  3.2.1 Minimization of $c_{UV}$ for fixed magnetic field

The first exercise is to minimize the UV central charge for different values
of $x$ and the result is shown on Figure 13. We performed this analysis with
both primal111111We use the primal numerical implementation described in
Karateev_2020  and dual formulations and for different $N$ to compare them. We
also plotted the analytical central charge given by (A.53) which assumes that
optimal S-matrix maximizes the cubic coupling $g$ (and has the zero (3.5)).

As a simple check, when $x\to 0$ the zero cancels the pole, so we are back to
asking for the minimal $c_{UV}$ for an S-matrix with no poles. With the dual
for $N=30$ we get

$x=0.01,\qquad\text{min}c_{UV}=0.498814...,$ (3.6)

which numerically approaches the result for the free Majorana fermion where
$c_{UV}=1/2$.

We now compare primal and dual methods in more detail. First, it is clear that
the duality gap closes as $N\to\infty$. We also note the difference in the
rate of convergence between the two formalisms. For the primal we parametrize
the three functions $\mathcal{T},\mathcal{F}$ and $\rho$ while for the dual we
parametrize $\lambda_{1},W_{\mathcal{T}}$ and $W_{\mathcal{F}}$. Therefore the
number of numerical parameters scales like $3N$ for both problems and it makes
sense to compare them for a given $N$. Hence we claim that the rate of
convergence for the dual formalism is much better than the one for the primal.

We can now look at the observables. The scattering amplitude, the two
particles form factor and the spectral density are plotted respectively on
Figures $\ref{S0v2},\ref{FFv2}$, and 16.

Figure 13: Lower bound on the UV central charge as function of $x$, the
position of the zero of the scattering amplitude $S(m^{2}(1-x))=0$. The blue
region is allowed. For $x=0.01$ and $N=30$ with the dual we obtained
$c_{UV}^{(min)}=0.498814...$. The black line corresponds to the analytical
result (A.53) assuming that the S-matrix has maximal cubic coupling.  Figure
14: Scattering amplitude $S(s)$ for different values of $x$ when there is a
zero at $S(m^{2}(1-x))$. The imaginary part (in orange) and the real part (in
blue) tend toward the free fermion (in dashed gray) when $x\to 0$. The dashed
black line is the analytical result given by (A.40). We used $N$=50. Figure
15: Two particles form factor $\mathcal{F}^{\Theta}_{2}(s)$ for different
values of $x$ when there is a zero at $S(m^{2}(1-x))$. The imaginary part (in
orange) and the real part (in blue) tend toward the free fermion (in dashed
gray) when $x\to 0$. The dashed black line is the analytical result given by
(A.50) and (A.53). We used $N$=50. Figure 16: Spectral density
$\rho_{\Theta}(s)$ for different values of $x$ when there is a zero at
$S(m^{2}(1-x))$. The bootstrap result (in blue) tends toward the free fermion
(in dashed gray) when $x\to 0$. We also observe the delta function appearing
at $s=4m^{2}$ when $x\to 1$ which corresponds to the second lightest particle
that will become stable. The dashed black line is the analytical result given
by (A.50) and (A.53) and the elasticity relation (A.1). We used $N$=50.

We also observe perfect agreement between the dual numerical results and the
analytical results derived in appendix A. They are given by (A.40), (A.50) and
(A.53).

There is an interesting observation for the spectral density when $x\to 1$. We
know that $x=1$ is exactly the point where the pole associated to the second
lightest particle enters the first sheet and exchanges place with the zero. We
also know that the spectral density has delta functions at the positions of
the masses of the stable asymptotic particle states. On the figure we see a
peak forming $s=4m^{2}$, which is exactly where the zero and the pole exchange
their places. This is illustrated more precisely in Figure 17, where we see
the position of the peak moving toward $4m^{2}$.

Figure 17: Spectral density $\rho_{\Theta}(s)$ for different values of $x$
when there is a zero at $S(m^{2}(1-x))$. We used $N$=50.

Note however that the central charge of this theory is $c_{UV}\approx 0.34$
and therefore it can not be the IFT. Nevertheless it is still interesting to
see that those two theories share the presence of the zero in the S-Matrix and
also seem to share the resonance associated to an unstable particle when $x\to
1$.

####  3.2.2 Carving out IFT parameter space: $c_{UV}=1/2$

We now target the IFT by fixing the central charge to $c_{UV}=1/2$ and explore
the allowed region in the $(g\mathcal{F}^{\Theta}_{1},g^{2},x)$ 3-dimensional
space, where we recall

$\mathcal{T}(s)=-\frac{g^{2}}{s-m^{2}}+...,\qquad\mathcal{F}(s)=-\frac{\mathcal{F}^{\Theta}_{1}g}{s-m^{2}}+...$
(3.7)

We compute the bounds for $g^{2}$ and $g\mathcal{F}_{1}^{\Theta}$ for
$x\in(0,1)$, which corresponds to the range in magnetic field with only one
stable particle. The result is the pyramid shown in Figure 18 inside which IFT
must lie.

Figure 18: Allowed region (in green) in the
$(g\mathcal{F}_{1}^{\Theta},x,g^{2})$ space. The blue line is the perturbative
trajectory for $g^{2}(x)$ and $g(x)\mathcal{F}_{1}^{\Theta}(x)$ in IFT
(computed in appendix E), the black lines are the analytical result (A.54) and
the dashed red line is the minimal cubic coupling discussed in Figure 19.

We noticed that the lower edge of the pyramid, corresponding to the lowest
bounds on $g^{2}$ for different $x$, is slightly in-curved toward the negative
$g\mathcal{F}_{1}^{\Theta}$. The effect is tiny and shown on Figure 19.

Figure 19: Lower edge of the pyramid corresponding to the lowest bounds on
$g^{2}$. This is computed with $N=70$.

The slice at $x=1/10$ of the pyramid is shown in Figure 20, where we also
place IFT using form factor perturbation theory: the cubic coupling $g$ was
computed in Zamolodchikov_2011  and $\mathcal{F}_{1}^{\Theta}$ is an original
computation (both calculations are in appendix E).

Figure 20: Allowed region in the $(g\mathcal{F}_{1}^{\Theta},g^{2})$ plane
when the central charge is $c_{UV}=1/2$ and there is a zero in the S-Matrix at
$x=1/10$. The red and orange dashed line are the values of
$\mathcal{F}_{1}^{\Theta}$ and $g^{2}$ computed with perturbation theory. This
is computed with $N=70$. The black dots are the analytical solutions (A.54).

The figure is generated by first computing the bounds on
$g\mathcal{F}_{1}^{\Theta}$ for which we got

$g\mathcal{F}_{1}^{\Theta,(min)}=-0.0110843...,\qquad
g\mathcal{F}_{1}^{\Theta,(max)}=0.0047159...,$ (3.8)

and then computing the bounds on $g^{2}$ with fixed values of
$g\mathcal{F}^{\Theta}_{1}$.

Note that the perturbative result for $g^{2}$ saturates the higher bound for
the cubic coupling allowed by the bootstrap. The perturbative result for
$\mathcal{F}_{1}^{\Theta}$ however does not saturate the bounds but still is
inside the allowed region. At the points where the red and orange lines meet
we can compare the optimal scattering amplitude to the one known in
perturbation theory from Zamolodchikov_2011 . This is shown in Figure 21. We
observe that we match perfectly with the elastic contribution of the
perturbative S-Matrix. This is expected since our bootstrap setup saturates
unitary at all energies and does not account for theories with particle
production. Therefore we are not surprised that inelastic contributions are
not obtained.

Figure 21: Scattering amplitude obtained from the bootstrap (in blue) at the
point matching the perturbation theory results compared to one from
perturbation theory for $x=0.1$. The green dashed line is the full
perturbative S-Matrix while the orange dashed line is only the elastic
contribution. We used $N=20$.

We note the absence of wiggles in this numerical optimal function, showing
that the slow convergence of the optimal dual functions is not a systematic
feature of our formalism. This supports the idea that our ansätze are not
optimal and can only easily reproduce a certain subclass behaviours.

In Figure 22 we perform a convergence study for the upper corners of the
triangle with respect to the analytical roots given by (A.54). The corners of
the allowed region coming from the dual bootstrap approach the analytical
roots

Figure 22: Convergence study (on a logarithmic scale) for the upper corners of
the triangle approaching the analytical roots $g_{F}^{(\pm)}$. Figure 23:
Numerical scattering amplitude and form factor on the upper edge of the
triangle (in blue) compared to the analytical result (in dashed). We took the
point $g\mathcal{F}_{1}^{\Theta}=0$ for reference but the data looks identical
on the whole upper edge. We used $N=50$.

Since the perturbative result for $\mathcal{F}_{1}^{\Theta}$ does not saturate
the bootstrap bounds we can investigate what happens when the position of the
zero (or equivalently the magnetic field) is changed. We plot the perturbative
result and the bootstrap bounds for different magnetic fields in Figure 24. We
observe that both perturbation theory and bootstrap bounds scale with $x$ or,
equivalently, with $h^{2}$, but at a different rate,

$\displaystyle{g}_{F}^{(\pm)}\;=\;\left(\frac{1}{2\sqrt{3}}-\frac{1}{\pi}\pm{1\over\sqrt{3}\pi}\mp{1\over
9}\right)x\;\approx\;(-\,0.1023,+\,0.0430\,)\,x\;\approx\;(-\,11.7601,+\,4.9466\,)\,h^{2}.$
(3.9)

whereas

$\displaystyle g_{F}^{(P.T)}$
$\displaystyle=-\frac{1}{3\sqrt{2}}\left(\frac{15}{8}+\frac{1}{2\pi}-\sqrt{3}\right)x\approx-0.0712x\approx-8.1857h^{2}.$
(3.10)

These numbers were computed in appendices A and E, respectively.

Figure 24: Evolution of the upper boundary of the allowed region (in blue) via
the numerical bootstrap compared to the evolution of the perturbative result
(in orange) in the perturbative regime for small values of $x$. We observe
that the bootstrap bounds and perturbation theory both scale linearly with
$x$, or quadratically with $h$. We used $N=70$.

In Figure 25 we show the bounds on $g^{2}$ and $g\mathcal{F}_{1}^{\Theta}$ in
the strongly interacting regime $x=0.7$. The upper boundary is again saturated
by the analytical results (A.54) and (A.43). In particular the roots of
eq.(A.54) correspond to the corners.

Figure 25: Allowed region in the $(g\mathcal{F}_{1},g^{2})$ plane when the
central charge is $c_{UV}=1/2$ and there is a zero in the S-Matrix at
$x=7/10$. This is computed with $N=70$. The black dots are the analytical
roots (A.54).

##  4 Conclusion

In this work we have merged the primal S-Matrix and form factor bootstrap
proposed in Karateev_2020  with the dual S-Matrix bootstrap developed in
Guerrieri_2020 . With our method we can relate UV data, namely the central
charge $c_{UV}$ of the UV CFT, with couplings of the gapped QFT in the IR, and
place rigorous bounds on any of these parameters. We observe a vastly improved
numerical convergence compared to the primal bootstrap of Karateev_2020 .

We established a lower bound on $c_{UV}$ in a class of $\mathbb{Z}_{2}$
symmetric QFTs (fig. 5) and also for QFTs for which the particle self-
interacts with a cubic coupling and where the S-matrix has a real zero (a dial
for the magnetic field in IFT) - see fig. 13. In all these cases we find that
the optimal S-matrix extremizes some coupling. This allows us to find the
optimal S-matrix via pure analytical S-matrix bootstrap (viz. CDD bootstrap
Castillejo:1955ed ). Then, in appendix A, we showed how the form factor and
the minimal $c_{UV}$ can be obtained analytically if the S-matrix is known. We
find perfect agreement between the dual optimization and the analytical
bootstrap.

To specifically target Ising Field Theory we fix $c_{UV}=1/2$ and find the
allowed space in cubic couplings and magnetic field, see fig 18. We insert IFT
inside this ‘pyramid’ for small magnetic field - close to the tip of the
pyramid - using form factor perturbation theory (see appendix E). For larger
magnetic fields we enter the non-perturbative regime of IFT and this is where
our results can be useful: IFT _must_ lie somewhere inside the pyramid.

We know however that IFT cannot lie on the boundary of the pyramid because the
boundary is elastic: the optimal S-matrix does not have inelasticity, and we
know that IFT has particle production Zamolodchikov_2011 . But even for small
enough magnetic field $h$ (where particle production is negligible) IFT is
still somewhat away from the boundary, as the bounds on the one particle form
factor go to zero as $h^{2}$ with a different numerical coefficient than the
perturbative form factor (see fig. 24).

We can think of several ways to improve our setup and ‘shrink’ the pyramid
wherein IFT must lie. One way is to include more UV information. So far we are
only fixing the value of the central charge, but we are not ‘telling’ the
bootstrap that the RG flow is being triggered by the thermal and magnetic
deformations, $\epsilon$ and $\sigma$, respectively. The central charge is
straightforward to include in $d=2$ due to the $c$-sum rule which gives an
_integrated_ relation over the spectral density of the trace of the stress
energy tensor $\Theta$, and not just via some asymptotic constraint at
$s\to\infty$, which is difficult to implement numerically.121212 Moreover, the
normalization of the two particle form factor of $\Theta$ is fixed (see
appendix G), which is one less parameter to vary over in the bootstrap. So,
can we find similar integral relations that involve somehow $\epsilon$ or
$\sigma$ (besides the combination $\Theta$)? One possibility is the sum rule
derived in Eq. (29) of Delfino_1996  to fix the scaling dimension of an
operator. However, this sum rule requires knowing the vacuum expectation value
of that operator. For the case of $\sigma$, its expectation value should be
related to the magnetic field and therefore to the zero of the S-matrix, but
the precise relation is unknown to us. It should be possible to measure it
using other methods like the lattice or Hamiltonian truncation.

Another possibility is to use the twist property of the order field $\sigma$.
Twist fields are reviewed in Cardy_2007 ; Castro_Alvaredo_2018  and references
therein. The fact that $\sigma$ acts as a twist field is derived in
SCHROER197880 . This could be coupled to the input from lattice measurements,
for example using tensor networks. In order to measure the two point function
$\langle\sigma\Theta\rangle$ on the lattice, one needs to map the continuum
stress tensor to the lattice stress tensor. This might be done by generalizing
some ideas developed in Bastiaansen_1998  and Giusti_2015 . We plan to explore
this direction in the near future.

In addition, we can include more IR information. IFT is known to interpolate
between the free massive Majorana fermion and the $E_{8}$ Toda theory, which
has 8 particles. While we are studying the region close to the free fermion,
where there is only one particle, it is natural to expect, due to analyticity,
that the other particles exist in other ‘sheets’, i.e. as resonances
(corresponding to zeros on the physical sheet). We already impose one zero,
which eventually becomes the second lightest particle as the magnetic field is
increased. It is possible that the other particles imply the presence of
further zeros on the physical sheet gabai2019smatrix .

If we are feeling brave, we can dial the magnetic field a bit higher and go to
the regime where more particles are stable. If we want to go beyond lightest
particle scattering we would have to import the multiple amplitude bootstrap
of Homrich:2019cbt  and extend it in several directions. Namely, to dualize
it, to include form factors and, on a more fundamental level, to understand
the singularity structure of the scattering amplitude of heavier particles,
where anomalous thresholds are expected Coleman:1978kk ; Correia:2022dcu
.131313Anomalous thresholds are indeed present at the $E_{8}$ integrable
point, where in $d=2$ they are known as Coleman-Thun poles Delfino:2003yr .

In a similar vein, we can also consider multi-particle scattering, i.e. to not
only consider 2-particle states but also states of 3- or more particles. In
this way we could try target the inelasticity of IFT. However, a systematic
understanding of the analyticity structure of multi-particle amplitudes
remains a big open problem dating back to the 60s.

The latter directions we have mentioned are of course relevant to many other
QFTs besides IFT. We have put a lower bound on the $c_{UV}$ of a class of
$\mathbb{Z}_{2}$ symmetric theories but we could have chosen to be more
specific. For example, it would be interesting to target $\phi^{4}$ which has
been the focus of many recent nonperturbative studies Coser:2014lla ;
Rychkov:2014eea ; Bajnok:2015bgw ; Bosetti:2015lsa ; Anand:2017yij ;
Serone:2018gjo ; Chen:2021pgx ; Chen:2021bmm . This theory is a deformation of
the free massless boson, so we could fix $c_{UV}=1$ in our $\mathbb{Z}_{2}$
symmetric setup to target it. Moreover, we could place $\phi^{4}$ inside our
bounds, not only using perturbation theory but also with Hamiltonian
truncation data Chen:2021pgx .

Finally, it would be interesting to generalize our setup to higher dimensions.
A primal approach incorporating UV input via the $a$-anomaly was developed in
a_anomaly . Can we reproduce these bounds more rigorously and more efficiently
with a dual method?

Acknowledgements We are grateful to Denis Karateev and Kamran Salehi Vaziri
for collaboration at the early stages of this project. We thank Lucía Córdova,
Barak Gabai, Alessandro Georgoudis, Kelian Häring, Yifei He, Alexandre Homrich
and Pedro Vieira for useful discussions. M.C. is grateful to the Institute for
Advanced Study in Princeton and to the Perimeter Institute in Waterloo for the
kind hospitality while this work was being completed. This project has
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement number
949077). The authors are also supported by the Simons Foundation grant 488649
(Simons Collaboration on the Nonperturbative Bootstrap) and the Swiss National
Science Foundation through the project 200020_197160 and through the National
Centre of Competence in Research SwissMAP.

##  Appendix A Analytical Bootstrap

There are three objects we want to constrain. The $2\to 2$ scattering matrix
$S(s)$ or amplitude $\mathcal{T}(s)$, the 2-particle form factor
$\mathcal{F}(s)$ and the spectral density $\rho(s)$. The first two are real
analytic functions,141414A real analytic function $f(s)$ satisfies
$f^{*}(s)=f(s^{*})$. and the S-matrix further satisfies crossing symmetry,
$S(s)=S(4m^{2}-s)$. There is still the constraint of unitarity which relates
all of them. We observe from the optimization problem that unitarity is
“saturated”, eq. (2.25), meaning

$\displaystyle|S(s)|^{2}=1,\qquad{\frac{\mathcal{F}(s)}{\mathcal{F}^{*}(s)}}=S(s),\qquad\rho(s)={\frac{|\mathcal{F}(s)|^{2}}{4\pi\sqrt{s(s-4m^{2})}}},$
(A.1)

for $s\geq 4m^{2}$ slightly above the real axis.

Counting separately for real and imaginary parts we have five variables and
three equations, which implies that there is two-fold freedom left over from
eqs. (A.1). The general solution for (A.1) should possess this freedom.

We start by solving the first equation of eq. (A.1). It will be particularly
useful to us here to exploit a well-known trick used for solving elastic
unitarity in $d>2$ (see e.g. Correia:2020xtr ). We write $S(s)$ in terms of
the amplitude $\mathcal{T}(s)$,

$\displaystyle S(s)=1+{\mathcal{T}(s)\over 2\sqrt{s(4m^{2}-s)}}.$ (A.2)

Then, elastic unitarity $|S(s)|^{2}=1$ reads

$\displaystyle\text{Im}\,\mathcal{T}(s)={|\mathcal{T}(s)|^{2}\over
4\sqrt{s(s-4m^{2})}}.$ (A.3)

Noting that
$\text{Im}\,\left[1/\mathcal{T}(s)\right]=-\text{Im}\,\mathcal{T}(s)\,/\,|\mathcal{T}(s)|^{2}$
we can write the above as

$\displaystyle\text{Im}\,\left({1\over\mathcal{T}(s)}\right)=-{1\over
4\sqrt{s(s-4m^{2})}}$ (A.4)

The general solution to the above is then

$\displaystyle{1\over\mathcal{T}(s)}=-{1\over 4\sqrt{s(4m^{2}-s)}}+{A(s)\over
4}$ (A.5)

where $A(s)$ is some crossing symmetric real function on the real line.

We then have

$\displaystyle\mathcal{T}(s)={4\over A(s)-1/\sqrt{(4m^{2}-s)s}},\qquad\text{
or }\qquad S(s)={A(s)\sqrt{(4m^{2}-s)s}+1\over A(s)\sqrt{(4m^{2}-s)s}-1}.$
(A.6)

Now, since $\mathcal{T}(s)$ is analytic, $A(s)$ must be meromorphic in the
physical sheet.

Poles of $A(s)$ translate into zeros of $\mathcal{T}(s)$. Note that if
$A(s)=\text{const}$ the solution for $S(s)$ is precisely the so-called
Castillejo-Dalitz-Dyson (CDD) factor. In fact, any solution to elastic
unitarity, crossing and analyticity can be written as a product of CDD
factors. It is straightforward to show that this amounts to (A.6) with
rational $A(s)$.

The function $A(s)$ encondes the large freedom that the first equation of
(A.1) allows for. We may further fix $A(s)$ at certain values if further
constraints are involved, e.g. bound-state poles or zeros.

With solution (A.6) in hand we can now try to solve for the form factor using
the second equation in (A.1). If we write

$\displaystyle\mathcal{F}(s)=-2m^{2}B(s)\,e^{\alpha(s)-\alpha(0)},$ (A.7)

where $B(s)$ is real, Watson’s equation will only fix $\alpha(s)$,

$\displaystyle{\mathcal{F}(s)\over\mathcal{F}^{*}(s)}=S(s)\implies\text{Im}\,\alpha(s)=-{i\over
2}\log S(s+i\epsilon),\qquad\text{ for }s\geq 4m^{2}$ (A.8)

Making use of the solution for the S-matrix (A.6) we have

$\displaystyle\text{Im}\,\alpha(s)=\mathrm{arccot}\left(A(s)\sqrt{(s-4m^{2})s}\right)\Theta(s-4m^{2})$
(A.9)

It is difficult to explicitly find a function which gives such imaginary part
for a generic $A(s)$. We may still find numerically $\alpha(s)$ through a
dispersion relation,

$\displaystyle\alpha(s)-\alpha(0)={s\over\pi}\int_{4m^{2}}^{\infty}{\mathrm{arccot}\left(A(s^{\prime})\sqrt{(s^{\prime}-4m^{2})s^{\prime}}\right)\over
s^{\prime}(s^{\prime}-s)}\,ds^{\prime}.$ (A.10)

Note that any analytic function could be added to the right hand side of the
above. Such analytic function would drop out upon taking the imaginary part
meaning that eq. (A.9) would still be satisfied. The presence of a non-trivial
analytic function,151515By trivial we mean a constant piece which can be
simply absorbed into $B(s)$. i.e. a polynomial, will lead to an essential
singularity for $\mathcal{F}(s)$ so we discard this possibility.

The real function $B(s)$ is not fixed by Watson’s equation. It plays a similar
role to $A(s)$ in eq. (2.2), i.e. $B(s)$ is analytic in a neighbourhood of the
physical region. These properties are of course fixed by whatever analyticity
assumption we may have on $\mathcal{F}(s)$. Causality requirements indicate
that $B(s)$ should at most be meromorphic, with poles signalling potential
bound-states.

We now impose the normalization

$\displaystyle\mathcal{F}(0)=-2m^{2}\implies B(0)=1.$ (A.11)

With $\mathcal{F}(s)$ in hand, via eqs. (A.7) and (A.10), we can now simply
solve for the spectral density from eq. (A.1). The two-fold freedom left over
from eq. (A.1) is encoded into the meromorphic functions $A(s)$ and $B(s)$.
$A(s)$ can be determined if some coupling is being extremized over in the
S-matrix space, which appears to occur when $c_{UV}$ is minimized. $B(s)$ is
determined by minimizing over $c_{UV}$ explicitly,

$\displaystyle c_{UV}=12\pi\int_{4m^{2}}^{\infty}{\rho(s)\over
s^{2}}ds=12m^{4}\int_{4m^{2}}^{\infty}{B^{2}(s)\,e^{2\bar{\alpha}(s)}\over
s^{2}\sqrt{s(s-4m^{2})}}\,ds,$ (A.12)

where we made use of the relation between $\rho(s)$ and $\mathcal{F}(s)$, eq.
(A.1), and eq. (A.7) for $\mathcal{F}(s)$, and where $\bar{\alpha}(s)$ is the
real part of $\alpha(s)-\alpha(0)$, i.e. the principal value of (A.10),

$\displaystyle\bar{\alpha}(s)=\text{Re}\,[\alpha(s)-\alpha(0)]={s\over\pi}-\hskip-11.66573pt\int_{4m^{2}}^{\infty}{\mathrm{arccot}\left(A(s^{\prime})\sqrt{(s^{\prime}-4m^{2})s^{\prime}}\right)\over
s^{\prime}(s^{\prime}-s)}\,ds^{\prime}.$ (A.13)

As we will see in more detail, the polynomial degree of $B(s)$ must be bounded
from above so that the integral (A.12) for $c_{UV}$ is convergent. In fact, we
will see that for every case considered here $B(s)$ is at most linear in $s$
(apart from a very special case described in A.2.2 where it can be quadratic).
The constant coefficient in $B(s)$ is fixed by the normalization
$\mathcal{F}(0)=-2m^{2}$, so we only need to minimize $c_{UV}$ over the free
linear coefficient, which can be done analytically.

###  A.1 $\mathbb{Z}_{2}$ symmetric theories

####  A.1.1 Finding the leaf

We are interested in finding the allowed space of the following parameters

$\displaystyle\Lambda\equiv-\mathcal{T}(2),\qquad\Lambda^{(2)}\equiv\lim_{s\to
2}\frac{\partial^{2}}{\partial s^{2}}\mathcal{T}(s).$ (A.14)

where $S$ and $\mathcal{T}$ are related by eq. (2.2),

$\displaystyle S(s)=1+{\mathcal{T}(s)\over 2\sqrt{s(4m^{2}-s)}}.$ (A.15)

Elastic unitarity for the S-matrix reads

$\displaystyle|S(s)|^{2}=1,\;\;\text{ for }\;\;s\geq 4m^{2}\vee s\leq 0.$
(A.16)

Crossing symmetry further implies

$\displaystyle S(s)=S(4m^{2}-s).$ (A.17)

Now, we start by assuming that the S-matrix that extremizes the bounds is
elastic and has the minimal number of CDD zeros. We have two parameters that
we need to fix, $\Lambda$ and $\Lambda^{(2)}$, so we need at least two CDD
factors,

$\displaystyle S=S_{a_{-}}S_{a_{+}},$ (A.18)

with

$\displaystyle S_{a}={a-\sqrt{s(4m^{2}-s)}\over a+\sqrt{s(4m^{2}-s)}}.$ (A.19)

Let us now set $m=1$ as choice of units. From eqs. (A.15) and (A.18) we then
solve for the amplitude (A.6) to find

$\displaystyle A(s)=-{1+{a_{+}a_{-}\over s(4-s)}\,\over a_{+}+a_{-}},$ (A.20)

We now fix $a_{-}$ and $a_{+}$ in terms of (A.14) (or (3.1) in the main text):

$\displaystyle a_{\pm}=-\frac{2\Lambda^{2}\pm
8\sqrt{\Lambda^{3}-32\Lambda\Lambda^{(2)}+64(\Lambda^{(2)})^{2}}}{(\Lambda-16)\Lambda+32\Lambda^{(2)}},$
(A.21)

so that

$\displaystyle A(s)=\left({1\over
4}-{4\over\Lambda}+{8\left(\Lambda^{(2)}\right)\over\Lambda^{2}}\right)\;+\;{1-32\Lambda^{(2)}/\Lambda^{2}\over
s(4-s)}.$ (A.22)

Figure 26: Allowed space for $(\Lambda,\Lambda^{(2)})$ such that the amplitude
(A.20) has no poles on the physical sheet. The blue region corresponds to the
case where both CDD zeros are real, and is given by
$\left(0<\Lambda<4\land\left(\frac{\Lambda^{2}}{32}<\Lambda^{(2)}\leq\frac{\Lambda}{4}-\frac{1}{8}\sqrt{4\Lambda^{2}-\Lambda^{3}}\lor\frac{1}{8}\sqrt{4\Lambda^{2}-\Lambda^{3}}+\frac{\Lambda}{4}\leq\Lambda^{(2)}<\frac{1}{32}\left(16\Lambda-\Lambda^{2}\right)\right)\right)\lor\left(4\leq\Lambda<8\land\frac{\Lambda^{2}}{32}<\Lambda^{(2)}<\frac{1}{32}\left(16\Lambda-\Lambda^{2}\right)\right)$.
The red region corresponds to the case where the CDD zeros are complex, and is
given by
$0<\Lambda<4\land\frac{\Lambda}{4}-\frac{1}{8}\sqrt{4\Lambda^{2}-\Lambda^{3}}<\Lambda^{(2)}<\frac{1}{8}\sqrt{4\Lambda^{2}-\Lambda^{3}}+\frac{\Lambda}{4}$.
Their union is given by eq. (A.24).

What remains to impose is the analyticity constraint. Concretely, for what
values of $\Lambda$ and $\Lambda^{(2)}$ does $\mathcal{T}(s)$ or $S(s)$ not
develop any other singularities on the physical sheet besides the physical
cuts for $s\geq 4m^{2}$ and $s\leq 0$? We want $a_{-}$ and $a_{+}$ to be such
that the denominator in eq. (2.2) is never zero on the physical sheet, i.e.
that no pole develops and that the CDD factors remain CDD zeros.

Note that the square root function only covers half of the complex plane. On
the principal sheet it covers the right half plane, i.e. the real part of the
square root function is always positive. This means that if

$\displaystyle\mathrm{Re}\,a_{-}>0\qquad\text{and}\qquad\mathrm{Re}\,a_{+}>0$
(A.23)

we are sure that the denominator of (A.6) is never zero. Therefore, imposing
(A.23) on (A.21) we find the allowed range for $\Lambda$ and $\Lambda^{(2)}$.

So here we can split the problem in two. If the discriminant of (A.21) is
positive then $a_{\pm}$ is real, meaning that if $a_{\pm}>0$ the S-matrix will
have two real zeros on the physical sheet. This region is represented in blue
in fig. 26.

If the discriminant of (A.21) is negative then $a_{\pm}$ will be complex.
However, if $\text{Re}\,a_{\pm}>0$, i.e.
$(\Lambda-16)\Lambda+32\Lambda^{(2)}<0$, then the denominator of (A.6) will
never be zero on the physical sheet. Instead, the numerator will have
(complex) zeros. This region is represented in red in fig. 26.

The union of these regions is the leaf in the main text, fig. 4. Explicitly,
in agreement with eqs. (A.8) and (A.10) of Chen:2022nym ,

$\displaystyle
0<\Lambda<8\;\;\land\;\;\frac{\Lambda^{2}}{32}<\Lambda^{(2)}<{16\Lambda-\Lambda^{2}\over
32}$ (A.24)

The optimal amplitude on the lower bound, i.e. when
$\Lambda^{(2)}={\Lambda^{2}\over 32}$, and on the upper bound, i.e. when
$\Lambda^{(2)}={16\Lambda-\Lambda^{2}\over 32}$ is given by eq. (A.20) with,
respectively,

$\displaystyle\text{Lower bound: }A(s)={1\over
2}\left(1-{8\over\Lambda}\right),\qquad\text{Upper bound: }A(s)={2\over
s(4-s)}\left(1-{8\over\Lambda}\right)$ (A.25)

It is straightforward to see that multiplying the 2-CDD ansatz (A.18) by any
further CDD zero will lead to values of $\Lambda$ and $\Lambda^{(2)}$ inside
the leaf, and the same holds true if inelasticity is considered. So we
conclude that the bounds are optimal.

####  A.1.2 $c_{UV}$ at the edges of the leaf

We can now find the two-particle stress tensor form factor $\mathcal{F}(s)$ on
the edges of the leaf. We plug eqs. (A.25) for $A(s)$ at the edges of the leaf
into eqs. (A.7) and (A.10).

We want minimize $c_{UV}$ over $B(s)$. First note that $B(s)$ must not have
any poles outside the real cut, as it would be inconsistent with the
analyticity assumptions for the form factor $\mathcal{F}(s)$. In principle it
may have poles, e.g. at $s\to 4m^{2}$ or somewhere along the real axis, as
long as as the integral (A.12) converges, which can only be the case if
$e^{2\bar{\alpha}(s)}$ cancels such poles. While we could not find an explicit
form for $\bar{\alpha}(s)$ for either of the lower or uppers bounds (A.25) we
do observe numerically that $e^{2\bar{\alpha}(s)}$ is never zero, indicating
that $B(s)$ cannot have poles and therefore be holomorphic, i.e. a
polynomial.161616 The only way we could make $e^{2\bar{\alpha}(s)}$ have a
zero was to have $A(s)$ be zero at some point, making the $\arccot$ function
jump by $\pi$. However, $A(s)$ is sign definite at the edges of the leaf
(A.25).  Finally, we can constrain the polynomial degree of $B(s)$ according
to the convergence of the integral (A.12).

Lower edge. On the lower edge, we find that, according to eq. (A.9),
$\text{Im}\,\alpha(\infty)=\mathrm{arccot}\left(-\infty\right)=0$, meaning
that $\alpha(s\to\infty)\to\text{const}$ and $e^{2\bar{\alpha}(s)}\to 1$. We
conclude that the integrand of (A.12) will go as $\sim B^{2}(s)s^{-3}$ and,
therefore, $B(s)$ must be a constant. Otherwise, the integral will not
converge. Since $B(0)=1$ we are led to the following unique choice on the
lower edge,

$\displaystyle\text{Lower edge:}\qquad B(s)=1.$ (A.26)

So the form factor is entirely specified by $A(s)$,

$\displaystyle\mathcal{F}(s)=-2me^{\alpha(s)-\alpha(0)}.$ (A.27)

As a simple check we may explicitly compute $\mathcal{F}(s)$ and $c_{UV}$ at
the tips of the leaf, $\Lambda=0$ and $\Lambda=8$. In the former case we have
$A(s)={1\over 2}\left(1-{8\over\Lambda}\right)\to-\infty$, for which
$\mathrm{arccot}(-\infty)=0$ and $\text{Im}\,\alpha(s)=0$. In this case eq.
(A.10) gives $\alpha(s)-\alpha(0)=0$ and we have $\mathcal{F}(s)=-2m^{2}$ and
$c_{UV}=1$.

For $\Lambda=8$ instead we have $A(s)=0^{-}$, meaning
$\mathrm{arccot}(0^{-})=-{\pi\over 2}$. From eq. (A.10) it follows that

$\displaystyle\alpha(s)-\alpha(0)={1\over 2}\log\left(1-{s\over
4m^{2}}\right)$ (A.28)

and

$\displaystyle\mathcal{F}(s)=-m\sqrt{4m^{2}-s},\qquad
c_{UV}=12m^{4}\int_{4m^{2}}^{\infty}\sqrt{s-4m^{2}\over s}{ds\over
s^{2}}={1\over 2}.$ (A.29)

For $0<\Lambda<8$ we perform both integrals (A.13) and (A.10) numerically. In
fig. 11 we show the optimal $S(s)$ and $\mathcal{F}(s)$ at $\Lambda=2.4$.

Upper edge. Let us now consider the upper edge of the leaf. In this case we
have $A(s)={2\over s(4-s)}\left(1-{8\over\Lambda}\right)>0$ across the
integration region $s>4m^{2}$ in (A.10). Now we find that
$e^{2\bar{\alpha}(s)}\sim s^{-1}$ when $s\to\infty$ so that the integrand of
(A.10) goes as $\sim B^{2}(s)s^{-4}$. We see that $B(s)$ can at most be linear
in $s$ to ensure convergence of the integral. Therefore,

$\displaystyle\text{Upper edge:}\qquad B(s)=1-b\,s,$ (A.30)

where $b$ is a free real parameter.

Plugging into (A.12) we have

$\displaystyle c_{UV}(b)=I_{0}-2b\,I_{1}+b^{2}\,I_{2},\qquad\text{ with
}\qquad I_{n}\equiv
12m^{4}\int_{4m^{2}}^{\infty}{s^{n}\,e^{2\bar{\alpha}(s)}\over
s^{2}\sqrt{s(s-4m^{2})}}\,ds,$ (A.31)

We now minimize the central charge with respect to $b$,

$\displaystyle{dc_{UV}\over db}=0\qquad\implies\qquad b_{min}={I_{1}\over
I_{2}},\qquad c_{UV}(b_{min})=I_{0}-{I_{1}^{2}\over I_{2}}.$ (A.32)

We may again test these expressions for the trivial cases $\Lambda=0,8$. For
the former we have $A(s)\to\infty$, for which again $\text{Im}\,\alpha(s)=0$
and $\alpha(s)-\alpha(0)=0$. In this case $I_{2}\to\infty$ and we’ll trivially
get $b_{min}\to 0$ and $c_{UV}(b_{min})=I_{0}=1$. For $\Lambda\to 8$ we have
$A(s)\to 0^{+}$, for which $\mathrm{arccot}(0^{+})=+{\pi\over 2}$. From eq.
(A.10) it follows that

$\displaystyle\alpha(s)-\alpha(0)=-{1\over 2}\log\left(1-{s\over
4m^{2}}\right)$ (A.33)

and

$\displaystyle e^{2\bar{\alpha}(s)}=\left(1-{s\over
4m^{2}}\right)^{-1},\qquad\mathcal{F}(s)=-4m^{2}{1-bs\over\sqrt{4m^{2}-s}}$
(A.34)

Plugging into (A.32) we find171717Technically, in this limiting case, $I_{n}$
does not converge due to the end-point singularity at $s\to 4m^{2}$. However,
upon regulating, the ratio $I_{1}/I_{2}$ is finite and is given by $1/4m^{2}$.

$\displaystyle b_{min}={1\over 4m^{2}},$ (A.35)

i.e. that the polynomial $B(s)$ softens the singularity of integrand at $s\to
4m^{2}$, and we recover eqs. (A.29).

For $0<\Lambda<8$ we find $\bar{\alpha}(s)$ numerically eq. (A.13). Then, we
plug the result into eq. (A.32) and compute the $I_{n}$’s numerically. In fig.
10 we show the optimal $S(s)$ and $\mathcal{F}(s)$ for $\Lambda=2.4$.

####  A.1.3 $c_{UV}$ in the interior of the leaf

To find $c_{UV}$ in the interior of the leaf we have to first find the optimal
S-matrix. At the edges of the leaf the S-matrix is uniquely fixed because
$\Lambda^{(2)}$ is either maximal (upper edge) or minimal (lower edge). In the
interior of the leaf we consider extremizing over $\Lambda^{(4)}$, with

$\displaystyle\Lambda\equiv-\mathcal{T}(2),\qquad\Lambda^{(2)}\equiv\lim_{s\to
2}\frac{\partial^{2}}{\partial
s^{2}}\mathcal{T}(s),\qquad\Lambda^{(4)}\equiv\lim_{s\to
2}\frac{\partial^{4}}{\partial s^{4}}\mathcal{T}(s),$ (A.36)

under the hypothesis that $c_{UV}$ minimization for fixed
$(\Lambda,\Lambda^{(2)})$ extremizes $\Lambda^{(4)}$.

We can proceed as in section A.1.1 and consider three CDD zeros which can be
fixed in terms of the three parameters
$(\Lambda,\Lambda^{(2)},\Lambda^{(4)})$. Then explore the parameter-space for
which these zeros remain zeros and do not turn into poles. Avoiding further
technical details we find that

$\displaystyle\frac{3\left(-\Lambda^{3}+24\Lambda^{2}-128\Lambda\Lambda^{(2)}+256(\Lambda^{(2)})^{2}\right)}{128(8-\Lambda)}\leq\Lambda^{(4)}\leq\frac{3\left(-\Lambda^{3}+128\Lambda\Lambda^{(2)}-256(\Lambda^{(2)})^{2}\right)}{128\Lambda},$
(A.37)

in agreement with eq. (A.12) of Chen:2022nym  which determined these bounds
using the Schwarz-Pick theorem.

We observe that the optimal amplitude coming from $c_{UV}$ minimization is the
one for which $\Lambda^{(4)}$ is minimal, where $\Lambda^{(4)}$ assumes the
lower bound of (A.37). The optimal amplitude at the lower bound is specified
by

$\displaystyle
A(s)=-\frac{4(8-\Lambda)^{2}}{64\Lambda+\Lambda^{2}\left(s^{2}-4s-4\right)-32\Lambda^{(2)}(s-2)^{2}},$
(A.38)

for which the S-matrix $S(s)$ follows via (A.6). For $A(s)$ given by eq.
(A.38) we have the same asymptotics of the upper edge of the leaf,
$e^{2\bar{\alpha}(s)}\sim s^{-1}$, so $B(s)$ is at most linear: $B(s)=1-bs$.
As in the previous section, the form factor $\mathcal{F}(s)$ and the minimal
$c_{UV}$ follow from eqs. (A.7), (A.10), (A.31) and (A.32). In fig. 12 we
compare the analytical result with the output of dual optimization for
$\Lambda=4.56$ and $\Lambda^{(2)}=0.85$.

###  A.2 Ising Field Theory

Here the analyticity assumptions consist of a zero for the S-matrix $S(s)$ at
$s=m^{2}(1-x)$, where $x$ is related to the magnetic field, and a pole at
$s=m^{2}$, and their crossing-symmetric counterparts. Similarly, the form
factor $\mathcal{F}(s)$, has a pole at $s=m^{2}$.

####  A.2.1 Minimization of the central charge $c_{UV}$

The first problem consists of minimizing the central charge $c_{UV}$,

$\displaystyle
c_{UV}=12\pi\left(m^{-4}|\mathcal{F}_{1}^{\Theta}|^{2}+\int_{4m^{2}}^{\infty}ds\frac{\rho(s)}{s^{2}}\right)$
(A.39)

where $\mathcal{F}_{1}^{\Theta}$ is the 1-particle form-factor, which is a
constant. The residue of the 2-particle form factor $F(s)$ at $s\to m^{2}$ is
given by $g\mathcal{F}_{1}^{\Theta}$, where $g$ is the cubic coupling, i.e.
$-g^{2}$ is the residue of the S-matrix pole at $s\to m^{2}$.

The minimal choice for an S-matrix with the above properties, and which is
also purely elastic, is the product of one CDD pole at $s=m^{2}$ and one CDD
zero at $s=m^{2}(1-x)$,

$\displaystyle S_{\pm}(s)=\pm{m^{2}\sqrt{(1-x)(3+x)}-\sqrt{s(4m^{2}-s)}\over
m^{2}\sqrt{(1-x)(3+x)}+\sqrt{s(4m^{2}-s)}}\cdot{\sqrt{3}m^{2}+\sqrt{s(4m^{2}-s)}\over\sqrt{3}m^{2}-\sqrt{s(4m^{2}-s)}}.$
(A.40)

Now, near the pole we must have $S(s)\sim-{g^{2}\over s-m^{2}}$, i.e. $S(s)$
must have a negative residue. Only $S_{-}(s)$ satisfies this requirement. We
again observe that $S_{-}(s)$ matches with the optimal S-matrix in fig. 14
coming from minimizing the central charge.

By writing $S_{-}(s)$ in the form (A.6) we find (in units where $m=1$)

$\displaystyle{}A(s)=-\frac{\sqrt{3}-\sqrt{(1-x)(x+3)}}{(s-4)s+\sqrt{3(1-x)(x+3)}},$
(A.41)

From eq. (A.6) we see that, near the pole, we have

$\displaystyle T(s\to m^{2})\sim-{g^{2}\over s-m^{2}},$ (A.42)

with

$\displaystyle
g^{2}=\frac{4\sqrt{3}\left(3-\sqrt{3(1-x)(x+3)}\right)^{2}}{x(x+2)}>0$ (A.43)

Now, given $A(s)$, the form factor $F(s)$ follows from eqs. (A.9) and (A.7).
The function $B(s)$ must contain the simple pole of $F(s)$ when $s\to m^{2}=1$
but no other singularity. We must therefore have

$\displaystyle B(s)={P(s)\over 1-s},$ (A.44)

where $P(s)$ is a polynomial, for which $B(0)=1$ implies $P(0)=1$. If,
furthermore,

$\displaystyle F(s\to 1)\sim-{g_{F}\over s-1},$ (A.45)

where $g_{F}\equiv g\mathcal{F}_{1}^{\Theta}$, we must have, from (A.7),

$\displaystyle P(1)=-{g_{F}\,e^{\alpha(0)-\alpha(1)}\over 2}.$ (A.46)

To constrain further $P(s)$ we consider the central charge,

$\displaystyle c_{UV}=12\pi{g_{F}^{2}\over
g^{2}}+12\int_{4}^{\infty}\frac{P^{2}(s)\,e^{2\bar{\alpha}(s)}}{(1-s)^{2}\,s^{2}\sqrt{s(s-4)}}\,ds.$
(A.47)

Similarly to the previous section, we see that convergence of the above
integral bounds the degree of $P(s)$. According to eq. (A.9), we see that
$\text{Im}\,\alpha(s\to\infty)=\arccot(0^{-})=-{\pi\over 2}$ meaning that
$e^{2\bar{\alpha}(s)}\sim s$, when $s\to\infty$. The integrand in (A.47) will
then go like $\sim P^{2}(s)s^{-4}$. In order for the integral to converge, we
see that $P(s)$ must be at most linear,

$\displaystyle P(s)=1-bs$ (A.48)

where $b$ is some real constant parameter, which can be fixed in terms of
$g_{F}$ via equation (A.46),

$\displaystyle b=1+{g_{F}\,e^{\alpha(0)-\alpha(1)}\over 2},$ (A.49)

for which the form factor reads,

$\displaystyle\mathcal{F}(s)=-2e^{\alpha(s)-\alpha(0)}+{g_{F}s\over
1-s}e^{\alpha(s)-\alpha(1)}$ (A.50)

and the central charge (A.47) can be expressed as

$\displaystyle
c_{UV}=I_{0}+g_{F}\left(e^{-\bar{\alpha}(1)}I_{1}\right)+g_{F}^{2}\left({12\pi\over
g^{2}}+{e^{-2\bar{\alpha}(1)}\over 4}I_{2}\right),$ (A.51)

with

$\displaystyle I_{n}\equiv
12\int_{4}^{\infty}{e^{2\bar{\alpha}(s)}\over(s-1)^{n}s^{2-n}\sqrt{s(s-4)}}ds.$
(A.52)

We now minimize the central charge $c_{UV}$ with respect to $g_{F}$,

$\displaystyle{dc_{UV}\over dg_{F}}=0\quad\implies\quad
g_{F,min}=-{2I_{1}\over
I_{2}\,e^{-\bar{\alpha}(1)}+48\pi\,g^{-2}\,e^{\bar{\alpha}(1)}},\qquad
c_{UV,min}=I_{0}-{I_{1}^{2}\over I_{2}+48\pi\,g^{-2}\,e^{2\bar{\alpha}(1)}},$
(A.53)

with $g^{2}$ given by eq. (A.43), and $\bar{\alpha}(s)$ given by eqs. (A.13)
and (A.41).

####  A.2.2 Bounds on $g_{F}$ for fixed central charge $c_{UV}=1/2$ and
maximal $g$

For a fixed central charge $c_{UV}=1/2$ we find two possible roots for $g_{F}$
from eq. (A.51),

$\displaystyle
g_{F}^{(\pm)}={-e^{-\bar{\alpha}(1)}I_{1}\pm\sqrt{e^{-2\bar{\alpha}(1)}I_{1}^{2}-(48\pi
g^{-2}+e^{-2\bar{\alpha}(1)}I_{2})(I_{0}-{1\over 2})}\over 2I_{0}-1},$ (A.54)

where $g^{2}$ is maximal and given by (A.43) in terms of the zero $x$. For
$x=1/10$ we find

$\displaystyle g_{F}^{(-)}=-0.011028...,\qquad g_{F}^{(+)}=0.004658....$
(A.55)

These correspond to the leftmost and rightmost corners in fig. 20.

Note that fig. 20 seems to indicate that there exists a line of solutions with
the maximal coupling connecting the two extrema (A.54), whereas our previous
analysis only provides two isolated points for an S-matrix with maximal
coupling. The solution to this apparent paradox lies in the possibility of
having an S-matrix with cubic coupling arbitrarily close to the maximal one,
but with different asymptotics, such that $P(s)$ in (A.47) can now be
quadratic with a free parameter, instead of just linear as in (A.48).

Take for example

$\displaystyle S(s)=S_{-}(s)\,{a_{1}+a_{2}\,s(4-s)-\sqrt{s(4-s)}\over
a_{1}+a_{2}\,s(4-s)+\sqrt{s(4-s)}},$ (A.56)

where $S_{-}(s)$ is the maximal S-matrix given by eq. (A.40). Now, for
$a_{1}\to\infty$ we have $S(s)\to S_{-}(s)$. On the other hand, it is not hard
to see that solving for $A(s)$ using eq. (A.6) leads to the following
asymptotics

$\displaystyle
A(s\to\infty)\to-\frac{\sqrt{3}-3\sqrt{31}/10-1/a_{2}}{s^{2}}+O\left(\frac{1}{s^{3}}\right)$
(A.57)

which only depends on $a_{2}$. In particular, if we take $a_{2}$ large enough
we get back the previous case, for which $A(s\to\infty)\to 0^{-}$. Now,
however, $a_{2}$ can be tuned such that $A(s\to\infty)\to 0^{+}$ while also
keeping $S(s)$ free of further poles, i.e. for
$0<a_{2}<(\sqrt{3}-3\sqrt{31}/10)^{-1}\approx 16.2$. In this case we have
$\text{Im}\,\alpha(s\to\infty)=\arccot(0^{+})=+{\pi\over 2}$ so that
$e^{2\bar{\alpha}(s)}\sim s^{-1}$. The integrand in (A.47) will then go like
$\sim P^{2}(s)s^{-6}$, which allows for a quadratic degree polynomial $P(s)$.
Two of the coefficients of $P(s)$ are fixed by the normalization $P(0)=1$ and
by the residue of the form factor $g_{F}$, the remaining coefficient is free,
i.e. it allows for a line of solutions with near-maximal cubic coupling (by
letting $a_{1}$ be arbitrarily large) as observed in fig. 20.

####  A.2.3 Small magnetic field limit $h\to 0$: Bounds on $g_{F}$ and extra
zeros

We first find how the analytical $g_{F}$ scales in the small magnetic field
limit,

$\displaystyle g_{F}(x\to 0)=x\,\tilde{g}_{F}.$ (A.58)

We assume this behavior for now, but will then confirm it by explicit
computation. We start by noting that in the limit $x\to 0$, the cubic coupling
$g^{2}$ (A.43) goes as

$\displaystyle g^{2}(x\to 0)=2\sqrt{3}x$ (A.59)

Then, the central charge (A.47) reads

$\displaystyle
c_{UV}\;=\;{6\pi\over\sqrt{3}}\tilde{g}_{F}^{2}\,x\;+\;12\int_{4}^{\infty}\frac{P^{2}(s)\,e^{2\bar{\alpha}(s)}}{(1-s)^{2}\,s^{2}\sqrt{s(s-4)}}\,ds\;+\;O(x^{2}).$
(A.60)

We still have to determine the $x\to 0$ limit of $P(s)$ and $\bar{\alpha}(s)$.
Let us first consider $\bar{\alpha}(s)$. From (A.41) we have

$\displaystyle A(s)=-{x/\sqrt{3}\over 3+s(s-4)}+O(x^{2})$ (A.61)

Plugging into (A.9) and expanding at small $x$

$\displaystyle\text{Im}\,\alpha(s)=\left[-{\pi\over
2}\;+\;x\,{\sqrt{s(s-4)/3}\over 3+s(s-4)}\;+\;O(x^{2})\right]\Theta(s-4)$
(A.62)

We find

$\displaystyle\alpha(s)-\alpha(0)\;=\;{1\over 2}\log\left(1-{s\over
4}\right)\;+\;x\,\big{[}\alpha_{1}(s)-\alpha_{1}(0)\big{]}\;+\;O(x^{2})$
(A.63)

with

$\displaystyle\alpha_{1}(s)={\beta(s)\over(s-3)(s-1)}-{\beta(3)\over
2(s-3)}+{\beta(1)\over 2(s-1)},\qquad\beta(s)=-{2\over\pi}\sqrt{s(4-s)\over
3}\,\arctan\sqrt{s\over 4-s}.$ (A.64)

Let us now take the limit $x\to 0$ of the polynomial $P(s)=1-bs$ in (A.60),
where $b$ is given by (A.49). We have

$\displaystyle b=1\;+\;x\,{\tilde{g}_{F}\,e^{\alpha(0)-\alpha(1)}\over
2}\;+\;O(x^{2})\;=\;1\;+\;x\,{\tilde{g}_{F}\over\sqrt{3}}\;+\;O(x^{2})$ (A.65)

where we made use of (A.63).

We then have

$\displaystyle{P^{2}(s)\,e^{2\bar{\alpha}(s)}\over(1-s)^{2}}={s-4\over
4}\;+\;x\,(s-4)\left[{\sqrt{3}\tilde{g}_{F}s\over
6(s-1)}\;+\;{\mathrm{Re}[\alpha_{1}(s)-\alpha_{1}(0)]\over
2}\right]\;+\;O(x^{2}).$ (A.66)

Now, replacing the above formulas into (A.60) and performing the simple
integrals, we find

$\displaystyle c_{UV}-{1\over
2}\;=\;x\left[\,2\sqrt{3}\pi\,\tilde{g}_{F}^{2}\>+\;(4\sqrt{3}-2\pi)\,\tilde{g}_{F}\;-\;\frac{126-23\sqrt{3}\pi}{81}-{2\over\sqrt{3}\pi}I_{t}\right]\;+\;O(x^{2}).$
(A.67)

where

$\displaystyle
I_{t}\;=\;\int_{4}^{\infty}\frac{s-4}{s^{2}\left(s^{2}-4s+3\right)}\log\left({\sqrt{s-4}+\sqrt{s}\over
2}\right)ds\;=\;{23\over 648}\pi^{2}-{1\over 3}.$ (A.68)

Notice that setting $x=0$ in (A.67) leads necessarily to $c_{UV}=1/2$, the
central charge of the free fermion. Since we are studying IFT we set
$c_{UV}=1/2$ regardless, which leads to a quadratic equation for the rate
$\tilde{g}_{F}$ with the following roots

$\displaystyle\tilde{g}_{F}^{(\pm)}\;=\;\frac{1}{2\sqrt{3}}-\frac{1}{\pi}\pm{1\over\sqrt{3}\pi}\mp{1\over
9}\;=\;(-\,0.1023000,+\,0.0430304\,)$ (A.69)

To express this in terms of the magnetic field $h$ we make use of the relation
$x=36\sqrt{3}(\mathcal{F}_{1}^{\sigma})^{2}\,h^{2}$, where
$\mathcal{F}_{1}^{\sigma}=1.3578$, to find the rates,

$\displaystyle
g_{F}^{(\pm)}=x\,\tilde{g}_{F}^{(\pm)}\;=\;(-\,11.7601,+\,4.9466\,)\,h^{2}.$
(A.70)

Extra zeros. Let us now see what we can say about potential extra zeros of the
IFT S-matrix in the small magnetic field limit $h\to 0$.

For the case of IFT at lowest order in $h^{2}$ we have Zamolodchikov_2011
(see also eqs. (E.51) and (E.52))

$\displaystyle
S_{\text{IFT}}(s)=-1-ih^{2}\sqrt{(s-4)s}\left(\frac{72(\mathcal{F}^{\sigma}_{1})^{2}}{s^{2}-4s+3}+\int_{4}^{\infty}\frac{(s^{\prime}-2)\,\sigma_{2\to
3}(s^{\prime})\,ds^{\prime}}{\pi\sqrt{(s^{\prime}-4)s^{\prime}}(s^{\prime}-s)(s+s^{\prime}-4)}\right)+O(h^{4})$
(A.71)

with $\sigma_{2\to 3}(s)$ given by eq. (E.54).

Let us now match eq. (A.71) with the general nonperturbative solution to
unitarity and crossing symmetry. In $d=2$ unitarity reads

$\displaystyle|S(s)|^{2}=1-\sigma_{inel}(s)$ (A.72)

where $\sigma_{inel}(s)>0$ is the particle production cross-section. The
general solution to this equation and crossing symmetry $S(s)=S(4m^{2}-s)$
reads Tourkine:2021fqh ; PhysRevD.6.2763 ; Paulos_2017_I

$\displaystyle
S(s)=S_{CDD}(s)\,\exp\left[\int_{4m^{2}}^{\infty}{ds^{\prime}\over 2\pi
i}\log(1-\sigma_{inel}(s^{\prime}))\sqrt{s(s-4m^{2})\over
s^{\prime}(s^{\prime}-4m^{2})}\left({1\over s^{\prime}-s}+{1\over
s^{\prime}-(4m^{2}-s)}\right)\right]$ (A.73)

where $|S_{CDD}(s)|^{2}=1$. Now, $S_{CDD}(s)$ can be any product of CDD
factors. In particular for IFT, it should include the CDD pole at $s=m^{2}=1$
and the CDD zero (3.5). So we take

$\displaystyle
S_{CDD}(s)=S_{-}(s)\,{\sqrt{a(4-a)}-\sqrt{s(4-s)}\over\sqrt{a(4-a)}+\sqrt{s(4-s)}}\,{\sqrt{b(4-b)}-\sqrt{s(4-s)}\over\sqrt{b(4-b)}+\sqrt{s(4-s)}}\,\cdots$
(A.74)

with $S_{-}(s)$ given by eq. (A.40) and the remaining zeros $a,b,\dots$ we
wish to constrain as $h\to 0$.

Letting $\sigma_{inel}=h^{2}\sigma_{2\to 3}+O(h^{4})$ and expanding (A.73) at
small $h$, assuming $a=a_{0}h^{-2}+O(h^{-4})$, and likewise for $b$ and etc,
we find

$\displaystyle
S(s)-S_{\text{IFT}}(s)=2h^{2}\sqrt{s(s-4)}\,\big{(}|a_{0}|+|b_{0}|+\dots\big{)}+O(h^{4})$
(A.75)

Since $S(s)$ and $S_{\text{IFT}}(s)$ must match at order $h^{2}$ we see that
$a_{0}=b_{0}=\dots=0$.

In case the zeros are complex we must have $a=b^{*}$ so that $S_{CDD}$ is
real. Letting $a_{0}=\alpha+i\beta$, and $b_{0}=\alpha-i\beta$ we have

$\displaystyle|a_{0}|+|b_{0}|=\sqrt{a_{0}^{2}}+\sqrt{b_{0}^{2}}=\sqrt{\alpha^{2}-\beta^{2}+2i\alpha\beta}+\sqrt{\alpha^{2}-\beta^{2}-2i\alpha\beta}=2|\alpha|$
(A.76)

Meaning that $\alpha=0$, but the imaginary part $\beta$ of the zeros is
unconstrained.

Therefore, the (real part of the) extra zeros must go to infinity at least as
$\sim h^{-4}$ when $h\to 0$.

##  Appendix B Dual Bootstrap of the Sine-Gordon model

In this appendix we reproduce one of the results of Karateev_2020 , namely the
one presented on their Figure 3, but with the dual setup. The assumed spectrum
consists of 2 particles of mass $m_{1}=1$ and $m_{2}=\sqrt{3}$ to target the
Sine-Gordon theory. Only one pole in the scattering amplitude is assumed, the
one corresponding to an exchange of the particle $m_{2}$. This residue $g$ is
fixed and the central charge is minimized, which leads to Figure 27.

Figure 27: Lower bound on the central charge of the UV CFT for different
values of the cubic coupling $g$ between two particles of mass $m_{1}=1$ and
one particle of mass $m_{2}=\sqrt{3}$. We used $N=50$ and the bound extends
until $g^{2}=12\sqrt{3}$ after which the problem is unfeasible. The red line
at $c_{UV}=1/2$ is added for convenience.

The maximal value of $g^{2}$ for which the problem is feasible and the
corresponding minimal central charge are

$g^{2}=12\sqrt{3},\qquad c_{UV}=0.808823...$ (B.1)

and they correspond to the cubic coupling and the contribution to the central
charge of the Sine-Gordon model from the particle $m_{2}$ and states with two
particles $m_{1}$.

The advantage of our dual setup is that the bound on $c_{UV}$ is more accurate
than what was presented in Karateev_2020 . Indeed at $g=0$ we have for sure
the free Majorana fermion that is allowed and the corresponding central charge
of $1/2$ should not be excluded. This is in agreement with our bound.

The optimal scattering amplitude and form factor are presented on Figure 28,
and correspond also to the analytical solutions in the Sine-Gordon model.

Figure 28: Scattering amplitude (on the left) and form factor (on the right)
for the optimal $c_{UV}$ when $g^{2}=12\sqrt{3}$. The blue lines are the real
parts and the orange lines are the imaginary parts. They correspond to the
analytical solutions of the Sine-Gordon model with $m_{1}=1$ and
$m_{2}=\sqrt{3}$, that are plotted in dashed lines and given by eq. (4.18) and
(4.19) in Karateev_2020 . We used $N=50$.

##  Appendix C Dual optimization problems

In this appendix we derive various generalizations of the example in section 2
and present the dual optimization problems.

###  C.1 $\mathbb{Z}_{2}$ symmetric theories

Here the goal is to minimize the central charge on top of the leaf
parametrized by (3.2). We do not assume any pole in the S-matrix and form
factor. We have

$\begin{split}\mathcal{L}=12\pi\int_{4}^{\infty}ds\frac{\rho(s)}{s^{2}}&+\int_{4}^{\infty}dsw_{\mathcal{T}}(s)\left[\mathcal{T}(s)+\Lambda-\int_{4}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{T}(z)\left(\frac{1}{z-s}+\frac{1}{z+s-4}-\frac{2}{z-2}\right)\right]\\\
&+\int_{4}^{\infty}dsw_{\mathcal{F}}(s)\left[\mathcal{F}(s)-\mathcal{F}(0)-\int_{4}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{F}\left(\frac{1}{z-s}-\frac{1}{z}\right)\right]\\\
&+\int_{4}^{\infty}ds\operatorname{Tr}\mathbb{\Lambda}(s)\cdot
B(s)+b\left(\int_{4}^{\infty}\frac{dz}{\pi}\operatorname{Im}\mathcal{T}(z)\frac{4}{(z-2)^{3}}-\Lambda^{(2)}\right)\end{split}$
(C.1)

The dual scattering function is defined

$\begin{split}&W_{\mathcal{T}}(s)\equiv\int_{4}^{\infty}\frac{dz}{\pi}w_{\mathcal{T}}(z)\left(\frac{1}{z-s}-\frac{1}{z+s-4}+\frac{2}{s-2}\right),\\\
&\operatorname{Im}W_{\mathcal{T}}(s)=w_{\mathcal{T}}(s),\qquad\operatorname{Re}W_{\mathcal{T}}(s)=-P\int_{4}^{\infty}\frac{dz}{\pi}w_{\mathcal{T}}(z)\left(\frac{1}{s-z}+\frac{1}{s-(4-z)}-\frac{2}{s-2}\right),\end{split}$
(C.2)

where the last term is not eliminated as in the example in section 2 since we
do not extremize over the primal variable $\mathcal{T}(2)$ which is fixed to
be $-\Lambda$.

The elimination of the primal variable $\mathcal{T}$ constrains the dual
variables to obey

$\lambda_{4}=\frac{\mathcal{N}_{2}}{2}\left(W_{\mathcal{T}}+\frac{4b}{\pi(s-2)^{3}}\right).$
(C.3)

We therefore get

Dual Problem (Minimization of the central charge on top of the leaf)
$\underset{\\{\lambda_{1},W_{\mathcal{T}},W_{\mathcal{F}},b\\}}{\text{Maximize}}\left[\int_{4}^{\infty}ds\left(2\lambda_{1}+2\operatorname{Im}W_{\mathcal{F}}+\mathcal{N}_{2}\operatorname{Re}W_{\mathcal{T}}+\Lambda\operatorname{Im}W_{\mathcal{T}}\right)+(1-\Lambda^{(2)})b\right]$
(C.4) Constrained by
$\begin{pmatrix}\lambda_{1}&\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}+\frac{2\mathcal{N}_{2}b}{\pi(s-2)^{3}}&i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}\\\
\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}^{*}+\frac{2\mathcal{N}_{2}b}{\pi(s-2)^{3}}&\lambda_{1}&-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}\\\
-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}&i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}&-\frac{6}{s^{2}}\end{pmatrix}\preccurlyeq
0,\quad\forall s\in[4,\infty),$ (C.5)

where $\mathbb{\Lambda}\preccurlyeq 0$ means $-\mathbb{\Lambda}\succcurlyeq
0$.

#### Non-Linear dual for $\mathbb{Z}_{2}$ symmetric theories

In the pure S-Matrix context the dual problem is simpler and in particular we
can get the leaf by computing the extreme values for
$\Lambda^{(2)}\equiv\mathcal{T}^{\prime\prime}(2)$ and then the extreme values
for $\Lambda\equiv\mathcal{T}(2)$ with fixed $\Lambda^{(2)}$ within the
allowed range.

For the first problem the lagrangian is

$\mathcal{L}=\int_{4}^{\infty}\frac{dz}{\pi}\operatorname{Im}\mathcal{T}(z)\frac{4}{(z-2)^{3}}+\int_{4}^{\infty}ds\left(\lambda(s)U(s)+w(s)\mathcal{A}(s)\right),$
(C.6)

where $w$ and $\lambda\geq 0$ are Lagrange multipliers and $\mathcal{A}$ is
the usual analyticity and crossing constraint for the scattering amplitude and

$U(s)\equiv
2\operatorname{Im}\mathcal{T}(s)-\frac{|\mathcal{T}(s)|^{2}}{\mathcal{N}_{2}(s)}$
(C.7)

is the unitarity constraint in the pure S-Matrix setup. Using the equations of
motion for $\mathcal{T}$ to eliminate it, we get the dual problem

$\mathcal{L}=\int_{4}^{\infty}ds\mathcal{N}_{2}(s)\left[\sqrt{(\operatorname{Im}W(s))^{2}+\left(\operatorname{Re}W(s)+\frac{4}{\pi(s-2)^{3}}\right)^{2}}+\operatorname{Re}W(s)+\frac{4}{\pi(s-2)^{3}}\right].$
(C.8)

For the second problem we start with the lagrangian

$\mathcal{L}=\mathcal{T}(2)+b\left(\Lambda^{(2)}-\int_{4}^{\infty}\frac{dz}{\pi}\operatorname{Im}\mathcal{T}(z)\frac{4}{(z-2)^{3}}\right)+\int_{4}^{\infty}ds\left(\lambda(s)U(s)+w(s)\mathcal{A}(s)\right).$
(C.9)

Eliminating the primal variable $\mathcal{T}$ we get

$\begin{split}\mathcal{L}&=\mathcal{T}(2)\left(1-\int_{4}^{\infty}ds\operatorname{Im}W(s)\right)+b\Lambda^{(2)}\\\
&+\int_{4}^{\infty}ds\mathcal{N}_{2}(s)\left[\sqrt{(\operatorname{Im}W(s))^{2}+\left(\operatorname{Re}W(s)-\frac{4b}{\pi(s-2)^{3}}\right)^{2}}+\operatorname{Re}W(s)-\frac{4b}{\pi(s-2)^{3}}\right]\end{split}$
(C.10)

To summarize we have

Dual Problem ($\Lambda^{(2)}$ maximization)
$\begin{split}\underset{\\{W(s)\\}}{\text{Minimize}}&\left(\int_{4}^{\infty}ds\mathcal{N}_{2}(s)\left[\sqrt{(\operatorname{Im}W(s))^{2}+\left(\operatorname{Re}W(s)-\frac{4}{\pi(s-2)^{3}}\right)^{2}}\right.\right.\\\
&\left.\left.+\operatorname{Re}W(s)-\frac{4}{\pi(s-2)^{3}}\right]\right)\end{split}$
(C.11)

Dual Problem ($\Lambda^{(2)}$ minimization)
$\begin{split}\underset{\\{W(s)\\}}{\text{Minimize}}&\left(\int_{4}^{\infty}ds\mathcal{N}_{2}(s)\left[-\sqrt{(\operatorname{Im}W(s))^{2}+\left(\operatorname{Re}W(s)-\frac{4}{\pi(s-2)^{3}}\right)^{2}}\right.\right.\\\
&\left.\left.+\operatorname{Re}W(s)-\frac{4}{\pi(s-2)^{3}}\right]\right)\end{split}$
(C.12)

Dual Problem ($\Lambda$ maximization with fixed $\Lambda^{(2)}$)
$\begin{split}\underset{\\{b,W(s)\\}}{\text{Minimize}}&\left(b\Lambda^{(2)}+\int_{4}^{\infty}ds\mathcal{N}_{2}(s)\left[\sqrt{(\operatorname{Im}W(s))^{2}+\left(\operatorname{Re}W(s)-\frac{4b}{\pi(s-2)^{3}}\right)^{2}}\right.\right.\\\
&\left.\left.+\operatorname{Re}W(s)-\frac{4b}{\pi(s-2)^{3}}\right]\right)\end{split}$
(C.13) Constrained by $\int_{4}^{\infty}ds\operatorname{Im}W(s)=1$ (C.14)

Dual Problem ($\Lambda$ minimization with fixed $\Lambda^{(2)}$)
$\begin{split}\underset{\\{b,W(s)\\}}{\text{Minimize}}&\left(b\Lambda^{(2)}+\int_{4}^{\infty}ds\mathcal{N}_{2}(s)\left[-\sqrt{(\operatorname{Im}W(s))^{2}+\left(\operatorname{Re}W(s)-\frac{4b}{\pi(s-2)^{3}}\right)^{2}}\right.\right.\\\
&\left.\left.+\operatorname{Re}W(s)-\frac{4b}{\pi(s-2)^{3}}\right]\right)\end{split}$
(C.15) Constrained by $\int_{4}^{\infty}ds\operatorname{Im}W(s)=1$ (C.16)

Following Guerrieri_2020 , for the $\Lambda^{(2)}$ maximization we use the
ansatz

$W(s)=\frac{1}{s(4-s)}\sum_{n=0}^{N}a_{n}\left(\rho^{n}(s,2)-\rho^{n}(4-s,2)\right),$
(C.17)

and for the $\Lambda$ maximization we use

$W(s)=\frac{1}{s(4-s)}\left(\frac{8}{\pi(s-2)}+\sum_{n=0}^{N}a_{n}\left(\rho^{n}(s,2)-\rho^{n}(4-s,2)\right)\right),$
(C.18)

with $\rho(s,s_{0})$ defined by (D.1).

###  C.2 Ising Field Theory

#### Zero in the S-Matrix at $s=1-x$

To implement the zero in the S-matrix we can use the subtraction

$\begin{split}\mathcal{T}(s)-\mathcal{T}(1-x)&=-g^{2}\left(\frac{1}{s-1}+\frac{1}{3-s}-\frac{1}{1-x-1}-\frac{1}{3-(1-x)}\right)\\\
&-\int_{4}^{\infty}\frac{dz}{\pi}\operatorname{Im}T(z)\left(\frac{1}{z-s}+\frac{1}{z+s-4}-\frac{1}{z-1+x}-\frac{1}{z+1-x-4}\right),\end{split}$
(C.19)

and we will fix $\mathcal{T}(1-x)=i\mathcal{N}_{2}(1-x)$ so that $S(1-x)=0$.
The dual scattering function is now defined as

$W_{\mathcal{T}}(s)=\int_{4}^{\infty}\frac{dz}{\pi}w_{\mathcal{T}}(z)\left(\frac{1}{z-s}-\frac{1}{z+s-4}+\frac{1}{s+x-1}+\frac{1}{s-x-3}\right).$
(C.20)

Its ansatz therefore needs to have poles at $s=1-x$ and $s=3+x$ with residues
$\int\operatorname{Im}W_{\mathcal{T}}/\pi$ and to be odd under crossing. To
moreover satisfy the fact that the objective must be integrable we can make
the ansatz181818In principle the coefficient $a$ should be fixed by the
constraint on the residues. A direct computation gives
$\frac{1}{\pi}\int_{4}^{\infty}\operatorname{Im}W_{\mathcal{T}}=a\frac{1}{4}\left(\frac{1}{3+x}+\frac{1}{1-x}\right)=\frac{a}{(3+x)(1-x)}.$
(C.21) On the other hand the residue is given by
$\underset{s=1-x}{\operatorname{Res}W_{\mathcal{T}}(s)}=\frac{a}{(1-x)(4-1+x)},$
(C.22) and the constraint is therefore satisfied for all $a$ that can be a
free parameter.

$W_{\mathcal{T}}(s)=\frac{a}{s(4-s)}\left(\frac{1}{s+x-1}+\frac{1}{s-x-3}\right)+\frac{1}{s(4-s)}\sum_{n=1}^{N}f_{n}(\rho(s,2)^{n}-\rho(4-s,2)^{n}),$
(C.23)

where the second term is the ansatz that we make when we don’t impose the
zero, defined by (D.2).

The only modification to the lagrangian induced from the addition of the zero
comes from the subtraction of $\mathcal{T}(1-x)$ which gives the new term

$\mathcal{L}\supset\int_{4}^{\infty}dsw_{\mathcal{T}}(s)\left(-\mathcal{T}(1-x)\right)=-i\mathcal{N}_{2}(1-x)\int_{4}^{\infty}\operatorname{Im}W_{\mathcal{T}}.$
(C.24)

In summary to implement the zero in the $S$-Matrix, it suffices to derive the
problem without the zero and at the end we modify the lagrangian as

$\mathcal{L}\to\mathcal{L}-i\mathcal{N}_{2}(1-x)\int_{4}^{\infty}\operatorname{Im}W_{\mathcal{T}},$
(C.25)

and use (C.23) for $W_{\mathcal{T}}$.

#### Minimization of the central charge

An interesting problem is the minimization of the UV central charge when there
is the zero in the S-Matrix. We start with the lagrangian

$\mathcal{L}=12\pi\left(\frac{g_{F}^{2}}{g^{2}}+\int_{4}^{\infty}\frac{\rho}{s^{2}}\right)+\int_{4}^{\infty}(\mathcal{A}_{\mathcal{T}}w_{\mathcal{T}}+\mathcal{A}_{\mathcal{F}}w_{\mathcal{F}}+\operatorname{Tr}\mathbb{\Lambda}B),$
(C.26)

and primal variables $\mathcal{T}$, $\mathcal{F}$ and $\rho$ are eliminated as
before. It remains to eliminate $g_{F}$ and $g^{2}$. For the former, a
functional variation with respect to $g_{F}$ yields

$g_{F}=-\frac{W_{\mathcal{F}}(1)}{24}g^{2}.$ (C.27)

Then, variation of $g^{2}$ yields the constraint

$48W_{\mathcal{T}}(1)=W_{\mathcal{F}}(1)^{2}.$ (C.28)

It was appearing in the lagrangian as
$g^{2}(W_{\mathcal{T}}(1)-W_{\mathcal{F}}(1)^{2}/48)$, with positive $g^{2}$,
and we are minimizing over the primal variables $g^{2}$, so it can be
implemented linearly with the positive semidefinite constraint

$\begin{pmatrix}W_{\mathcal{T}}(1)&W_{\mathcal{F}}(1)\\\
W_{\mathcal{F}}(1)&48\end{pmatrix}\succcurlyeq 0.$ (C.29)

Therefore

Dual Problem (Minimization of the central charge $c_{UV}$ with a zero
$S(1-x)=0)$
$\underset{\\{\lambda_{1},W_{\mathcal{T}},W_{\mathcal{F}}\\}}{\text{Maximize}}\left[\int_{4}^{\infty}ds\left(2\lambda_{1}+2\operatorname{Im}W_{\mathcal{F}}+\mathcal{N}_{2}\operatorname{Re}W_{\mathcal{T}}-i\mathcal{N}_{2}(1-x)\operatorname{Im}W_{\mathcal{T}}\right)\right]$
(C.30) Constrained by
$\begin{pmatrix}\lambda_{1}&\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}&i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}\\\
\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}^{*}&\lambda_{1}&-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}\\\
-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}&i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}&-\frac{6}{s^{2}}\end{pmatrix}\preccurlyeq
0,\quad\forall
s\in[4,\infty),\qquad\begin{pmatrix}W_{\mathcal{T}}(1)&W_{\mathcal{F}}(1)\\\
W_{\mathcal{F}}(1)&48\end{pmatrix}\succcurlyeq 0.$ (C.31)

#### Bounds in the $(g^{2},g\mathcal{F}_{1}^{\Theta})$ plane

We will know derive the problem to get bounds on $g_{F}\equiv
g\mathcal{F}_{1}^{\Theta}$ and $g^{2}$. We start by getting absolute bounds on
$g_{F}$ and then we optimize $g^{2}$ for fixed values of $g_{F}$. We will
impose the zero in the S-Matrix at the end.

The lagrangian can be written

$\begin{split}\mathcal{L}=g_{F}&+\int_{4}^{\infty}dsw_{\mathcal{T}}(s)\left[\mathcal{T}(s)-\mathcal{T}(2)+g^{2}\left(\frac{1}{s-1}+\frac{1}{3-s}-2\right)\right.\\\
&\left.-\int_{4}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{T}(z)\left(\frac{1}{z-s}+\frac{1}{z+s-4}-\frac{2}{z-2}\right)\right]\\\
&+\int_{4}^{\infty}dsw_{\mathcal{F}}(s)\left[\mathcal{F}(s)-\mathcal{F}(0)+g_{F}\left(\frac{1}{s-1}+1\right)-\int_{4}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{F}\left(\frac{1}{z-s}-\frac{1}{z}\right)\right]\\\
&+\int_{4}^{\infty}ds\operatorname{Tr}\mathbb{\Lambda}(s)\cdot B(s)\\\
&+c_{\rho}\left[12\pi\left(\frac{g_{F}^{2}}{g^{2}}+\int_{4}^{\infty}ds\frac{\rho(s)}{s^{2}}\right)-c_{UV}\right],\end{split}$
(C.32)

where the last term constrains the UV central charge to be $c_{UV}$. The
extremization over the primal variable $\mathcal{T}(2)$ already gives

$\int_{4}^{\infty}dsw_{\mathcal{T}}(s)=0.$ (C.33)

We can now define the dual scattering function

$W_{\mathcal{T}}(s)\equiv\int_{4}^{\infty}\frac{dz}{\pi}w_{\mathcal{T}}(z)\left(\frac{1}{z-s}-\frac{1}{z+s-4}\right),$
(C.34)

and the dual form factor function

$W_{\mathcal{F}}(s)\equiv\int_{4}^{\infty}\frac{dz}{\pi}w_{\mathcal{F}}(z)\left(\frac{1}{z-s}+\frac{1}{s}\right).$
(C.35)

The analyticity constraints can then considerably be simplified. We get

$\begin{split}\mathcal{L}=g_{F}&+\int_{4}^{\infty}ds\left(\operatorname{Im}(\mathcal{T}W_{\mathcal{T}})+\operatorname{Im}(\mathcal{F}W_{\mathcal{F}})+2\operatorname{Im}W_{\mathcal{F}}+\operatorname{Tr}\mathbb{\Lambda}(s)\cdot
B(s)\right)\\\
&+c_{\rho}\left[12\pi\left(\frac{g_{F}^{2}}{g^{2}}+\int_{4}^{\infty}ds\frac{\rho(s)}{s^{2}}\right)-c_{UV}\right]+\pi
g^{2}W_{\mathcal{T}}(1)+\pi g_{F}W_{\mathcal{F}}(1)\end{split}$ (C.36)

We are now ready to eliminate the primal variables. Varying with respect to
$\mathcal{F},\mathcal{T}$ and $\rho$ we get

$\lambda_{4}=\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}},\qquad\lambda_{6}=i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}},\qquad\lambda_{3}=-\frac{6C_{\rho}}{s^{2}}.$
(C.37)

The lagrangian is reduced to

$\mathcal{L}=g_{F}+\int_{4}^{\infty}ds\left(2\lambda_{1}+2\operatorname{Im}W_{\mathcal{F}}+\mathcal{N}_{2}\operatorname{Re}W_{\mathcal{T}}\right)+c_{\rho}\left(12\pi\frac{g_{F}^{2}}{g^{2}}-c_{UV}\right)+\pi
g^{2}W_{\mathcal{T}}(1)+\pi g_{F}W_{\mathcal{F}}(1).$ (C.38)

To finalize the elimination of primal variables we need to do variations over
$g_{F}$ and $g^{2}$. The former gives the equation

$g_{F}=-\frac{g^{2}}{24\pi C_{\rho}}(1+\pi W_{\mathcal{F}}(1)),$ (C.39)

that yields the lagrangian

$\mathcal{L}=-\frac{g^{2}}{48\pi C_{\rho}}(1+\pi
W_{\mathcal{F}}(1))^{2}+\int_{4}^{\infty}ds\left(2\lambda_{1}+2\operatorname{Im}W_{\mathcal{F}}+\mathcal{N}_{2}\operatorname{Re}W_{\mathcal{T}}\right)-c_{UV}C_{\rho}+\pi
g^{2}W_{\mathcal{T}}(1).$ (C.40)

Now extremizing over $g^{2}$ we get the constraint

$C_{\rho}=\frac{(1+\pi W_{\mathcal{F}}(1))^{2}}{48\pi^{2}W_{\mathcal{T}}(1)}.$
(C.41)

Therefore the lagrangian becomes

$\mathcal{L}=\int_{4}^{\infty}ds\left(2\lambda_{1}+2\operatorname{Im}W_{\mathcal{F}}+\mathcal{N}_{2}\operatorname{Re}W_{\mathcal{T}}\right)-c_{UV}\frac{(1+\pi
W_{\mathcal{F}}(1))^{2}}{48\pi^{2}W_{\mathcal{T}}(1)}.$ (C.42)

All the primal variables are eliminated and we’re ready to extremize over dual
variables. The last term is not linear in those variables, and it’s convenient
to introduce a new variable $u$ such that

$\begin{pmatrix}-u&1+\pi W_{\mathcal{F}}(1)\\\ 1+\pi
W_{\mathcal{F}}(1)&-48\pi^{2}W_{\mathcal{T}}(1)\end{pmatrix}\succcurlyeq 0.$
(C.43)

We can finally add the zero in the scattering amplitude as described above and
formulate the dual problem that can be implemented in SDPB :

Dual Problem ($g_{F}$ maximization and $S(m^{2}(1-x))=0$)

$\underset{\\{\lambda_{1},W_{\mathcal{T}},W_{\mathcal{F}},u\\}}{\text{Minimize}}\left[\int_{4}^{\infty}ds\left(2\lambda_{1}+2\operatorname{Im}W_{\mathcal{F}}+\mathcal{N}_{2}\operatorname{Re}W_{\mathcal{T}}-i\mathcal{N}_{2}(1-x)\operatorname{Im}W_{\mathcal{T}}\right)-c_{UV}u\right]$
(C.44)

Constrained by

$\begin{pmatrix}\lambda_{1}&\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}&i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}\\\
\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}^{*}&\lambda_{1}&-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}\\\
-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}&i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}&-\frac{6u}{s^{2}}\end{pmatrix}\succcurlyeq
0,\quad\forall s\in[4,\infty),\quad\begin{pmatrix}-u&1+\pi
W_{\mathcal{F}}(1)\\\ 1+\pi
W_{\mathcal{F}}(1)&-48\pi^{2}W_{\mathcal{T}}(1)\end{pmatrix}\succcurlyeq 0,$
(C.45)

and

$\int_{4}^{\infty}dsW_{\mathcal{T}}(s)=0.$ (C.46)

The minimization is similar, we just have to maximize the dual objective
instead of minimizing and adjust the constraint on $u$ so that it becomes the
correct form when the dual objective is maximized. The dual matrix
$\mathbb{\Lambda}(s)$ needs also to be negative instead of positive, or
equivalently $-\mathbb{\Lambda}$ is positive. We get

Dual Problem ($g_{F}$ minimization and $S(m^{2}(1-x))=0$)

$\underset{\\{\lambda_{1},W_{\mathcal{T}},W_{\mathcal{F}},u\\}}{\text{Maximize}}\left[\int_{4}^{\infty}ds\left(2\lambda_{1}+2\operatorname{Im}W_{\mathcal{F}}+\mathcal{N}_{2}\operatorname{Re}W_{\mathcal{T}}-i\mathcal{N}_{2}(1-x)\operatorname{Im}W_{\mathcal{T}}\right)-c_{UV}u\right]$
(C.47)

Constrained by

$\begin{pmatrix}\lambda_{1}&\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}&i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}\\\
\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}^{*}&\lambda_{1}&-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}\\\
-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}&i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}&-\frac{6u}{s^{2}}\end{pmatrix}\preccurlyeq
0,\quad\forall s\in[4,\infty),\quad\begin{pmatrix}u&1+\pi
W_{\mathcal{F}}(1)\\\ 1+\pi
W_{\mathcal{F}}(1)&48\pi^{2}W_{\mathcal{T}}(1)\end{pmatrix}\succcurlyeq 0,$
(C.48)

and

$\int_{4}^{\infty}dsW_{\mathcal{T}}(s)=0.$ (C.49)

Now we have to maximize the cubic coupling $g^{2}$ with fixed $g_{F}$ and we
start with the lagrangian

$\begin{split}\mathcal{L}=g^{2}&+\int_{4}^{\infty}dsw_{\mathcal{T}}(s)\left[\mathcal{T}(s)-\mathcal{T}(2)+g^{2}\left(\frac{1}{s-1}+\frac{1}{3-s}-2\right)\right.\\\
&\left.-\int_{4}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{T}(z)\left(\frac{1}{z-s}+\frac{1}{z+s-4}-\frac{2}{z-2}\right)\right]\\\
&+\int_{4}^{\infty}dsw_{\mathcal{F}}(s)\left[\mathcal{F}(s)-\mathcal{F}(0)+g_{F}\left(\frac{1}{s-1}+1\right)-\int_{4}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{F}\left(\frac{1}{z-s}-\frac{1}{z}\right)\right]\\\
&+\int_{4}^{\infty}ds\operatorname{Tr}\mathbb{\Lambda}(s)\cdot B(s)\\\
&+c_{\rho}\left[12\pi\left(\frac{g_{F}^{2}}{g^{2}}+\int_{4}^{\infty}ds\frac{\rho(s)}{s^{2}}\right)-c_{UV}\right]+C_{g}(g_{F}-g_{F}^{*}).\end{split}$
(C.50)

The primal variables are eliminated as before, and we arrive at

$\begin{split}\mathcal{L}&=g^{2}+\int_{4}^{\infty}ds\left(2\lambda_{1}+2\operatorname{Im}W_{\mathcal{F}}+\mathcal{N}_{2}\operatorname{Re}W_{\mathcal{T}}\right)+c_{\rho}\left(12\pi\frac{g_{F}^{2}}{g^{2}}-c_{UV}\right)\\\
&+\pi g^{2}W_{\mathcal{T}}(1)+\pi
g_{F}W_{\mathcal{F}}(1)+C_{g}(g_{F}-g_{F}^{*}).\end{split}$ (C.51)

The variations with respect to $g_{F}$ and then $g^{2}$ give

$g_{F}=-\frac{g^{2}}{24\pi C_{\rho}}\left(\pi
W_{\mathcal{F}}(1)+C_{g}\right),\qquad C_{\rho}=\frac{(\pi
W_{\mathcal{F}}(1)+C_{g})^{2}}{48\pi+48\pi^{2}W_{\mathcal{T}}(1)}.$ (C.52)

Introducing a new variable $u$ to linearize the objective as before, we get
Dual Problem ($g^{2}$ maximization with fixed $g_{F}=g_{F}^{*}$ and
$S(m^{2}(1-x))=0$)
$\underset{\\{\lambda_{1},W_{\mathcal{T}},W_{\mathcal{F}},C_{g},u\\}}{\text{Minimize}}\left[\int_{4}^{\infty}ds\left(2\lambda_{1}+2\operatorname{Im}W_{\mathcal{F}}+\mathcal{N}_{2}\operatorname{Re}W_{\mathcal{T}}-i\mathcal{N}_{2}(1-x)\operatorname{Im}W_{\mathcal{T}}\right)-c_{UV}u-C_{g}g_{F}^{*}\right]$
(C.53) Constrained by
$\begin{pmatrix}\lambda_{1}&\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}&i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}\\\
\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}^{*}&\lambda_{1}&-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}\\\
-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}&i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}&-\frac{6u}{s^{2}}\end{pmatrix}\succcurlyeq
0,\quad\forall s\in[4,\infty),$ (C.54) $\displaystyle\begin{pmatrix}-u&\pi
W_{\mathcal{F}}(1)+C_{g}\\\ \pi
W_{\mathcal{F}}(1)+C_{g}&-48\pi-48\pi^{2}W_{\mathcal{T}}(1)\end{pmatrix}\succcurlyeq
0,$ (C.55) and $\int_{4}^{\infty}dsW_{\mathcal{T}}(s)=0.$ (C.56)

Again the minimization is similar and we get

Dual Problem ($g_{T}^{2}$ minimization with fixed $g_{F}=g_{F}^{*}$ and
$S(m^{2}(1-x))=0$)
$\underset{\\{\lambda_{1},W_{\mathcal{T}},W_{\mathcal{F}},C_{g},u\\}}{\text{Maximize}}\left[\int_{4}^{\infty}ds\left(2\lambda_{1}+2\operatorname{Im}W_{\mathcal{F}}+\mathcal{N}_{2}\operatorname{Re}W_{\mathcal{T}}-i\mathcal{N}_{2}(1-x)\operatorname{Im}W_{\mathcal{T}}\right)-c_{UV}u-C_{g}g_{F}^{*}\right]$
(C.57) Constrained by
$\begin{pmatrix}-\lambda_{1}&-\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}&-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}\\\
-\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}^{*}&-\lambda_{1}&i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}\\\
i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}&-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}&\frac{6u}{s^{2}}\end{pmatrix}\succcurlyeq
0,\quad\forall s\in[4,\infty),$ (C.58) $\begin{pmatrix}u&\pi
W_{\mathcal{F}}(1)+C_{g}\\\ \pi
W_{\mathcal{F}}(1)+C_{g}&48\pi+48\pi^{2}W_{\mathcal{T}}(1)\end{pmatrix}\succcurlyeq
0,$ (C.59) and $\int_{4}^{\infty}dsW_{\mathcal{T}}(s)=0.$ (C.60)

###  C.3 Two poles : minimization of the central charge

We now consider the case with two particles having masses $m_{1}=1$ and
$m_{2}=\sqrt{3}$. There is only one pole in the scattering amplitude and in
the form factor corresponding to an exchange of the second particle between
two of the lightest particles. The pole terms in the dispersion relations are
slightly modified and read

$\begin{split}\mathcal{T}(s)-\mathcal{T}(2)&=-g_{112}^{2}\left(\frac{1}{s-3}+\frac{1}{1-s}+2\right)+\int_{4}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{T}(z)\left(\frac{1}{z-s}+\frac{1}{z+s-4}-\frac{2}{z-2}\right),\\\
\mathcal{F}(s)-\mathcal{F}(0)&=-g_{F}\left(\frac{1}{s-3}+\frac{1}{3}\right)+\int_{4}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{F}\left(\frac{1}{z-s}-\frac{1}{z}\right).\end{split}$
(C.61)

We therefore have the lagrangian

$\begin{split}\mathcal{L}&=12\pi\left(\frac{g_{F}^{2}}{9g_{112}^{2}}+\int_{4}^{\infty}ds\frac{\rho(s)}{s^{2}}\right)\\\
&+\int_{4}^{\infty}dsw_{\mathcal{T}}(s)\left[\mathcal{T}(s)-\mathcal{T}(2)+g_{112}^{2}\left(\frac{1}{s-1}+\frac{1}{3-s}-2\right)\right.\\\
&\left.-\int_{4}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{T}(z)\left(\frac{1}{z-s}+\frac{1}{z+s-4}-\frac{2}{z-2}\right)\right]\\\
&+\int_{4}^{\infty}dsw_{\mathcal{F}}(s)\left[\mathcal{F}(s)-\mathcal{F}(0)+g_{F}\left(\frac{1}{s-1}+1\right)-\int_{4}^{\infty}\frac{dz}{\pi}\text{Im}\mathcal{F}\left(\frac{1}{z-s}-\frac{1}{z}\right)\right]\\\
&+\int_{4}^{\infty}ds\operatorname{Tr}\mathbb{\Lambda}(s)\cdot
B(s).\end{split}$ (C.62)

Again the steps are similar and we obtain the dual problem

Dual Problem ($c_{UV}$ minimization with $m_{1}=1,m_{2}=\sqrt{3}$ and
$g_{112}$ fixed)
$\underset{\\{\lambda_{1},W_{\mathcal{T}},W_{\mathcal{F}},u\\}}{\text{Maximize}}\left[\int_{4}^{\infty}ds\left(2\lambda_{1}+2\operatorname{Im}W_{\mathcal{F}}+\mathcal{N}_{2}\operatorname{Re}W_{\mathcal{T}}\right)-3\frac{g_{112}^{2}}{16\pi}u+\pi
g_{T}^{2}W_{\mathcal{T}}(3)\right]$ (C.63) Constrained by
$\begin{pmatrix}\lambda_{1}&\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}&i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}\\\
\frac{\mathcal{N}_{2}}{2}W_{\mathcal{T}}^{*}&\lambda_{1}&-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}\\\
-i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}^{*}&i\frac{\sqrt{\mathcal{N}_{2}}}{4}W_{\mathcal{F}}&-\frac{6u}{s^{2}}\end{pmatrix}\preccurlyeq
0,\quad\forall s\in[4,\infty),$ (C.64) $\begin{pmatrix}1&\pi
W_{\mathcal{F}}(3)\\\ \pi W_{\mathcal{F}}(3)&u\end{pmatrix}\succcurlyeq 0,$
(C.65) and $\int_{4}^{\infty}dsW_{\mathcal{T}}(s)=0.$ (C.66)

##  Appendix D Numerical implementation

We are now almost ready to use our linear dual formulation to get non
perturbative bounds on different quantities. The last step is to implement the
dual problem numerically. The dual functions $W_{\mathcal{T}}$ and
$W_{\mathcal{F}}$ have branch cuts on the segment $[4,\infty)$ and it is
therefore useful to use the new variable

$\rho(s,s_{0})\equiv\frac{\sqrt{4-s_{0}}+\sqrt{4-s}}{\sqrt{4-s_{0}}-\sqrt{4-s}}$
(D.1)

that maps the $s$ complex plane in the unit disk with the cut mapped on the
boundary and that we depict on Figure 29 for $s_{0}=2$. Then any function that
is analytic apart for some poles and cuts starting at $s=4$ can be expressed
as the sum of the pole terms to which we add a Taylor series in the variable
$\rho$.

Figure 29: Map $\rho(s)$ from the $s$ complex plane to the unit disk given in
eq.(D.1). Figure taken from Paulos:2017fhb .

Accounting for the fact that $W_{\mathcal{T}}$ is odd under crossing and
$\mathcal{N}_{2}\operatorname{Re}W_{\mathcal{T}}$ must be integrable, we make
the ansatz

$W_{\mathcal{T}}(s)=\frac{1}{s(4-s)}\sum_{n=1}^{N}f_{n}(\rho(s,2)^{n}-\rho(4-s,2)^{n}).$
(D.2)

In the case of $\mathbb{Z}_{2}$ odd theories, $W_{\mathcal{T}}$ also has a
pole as can be seen from eq.(2.27). To account for this singularity and match
the residue, the ansatz is modified as

$W_{\mathcal{T}}(s)=-\frac{8}{\pi
s(s-2)(s-4)}+\frac{1}{s(4-s)}\sum_{n=1}^{N}f_{n}(\rho(s,2)^{n}-\rho(4-s,2)^{n}).$
(D.3)

As derived in (C.2) when there is a zero in the scattering amplitude at
$s=1-x$ the ansatz becomes

$W_{\mathcal{T}}(s)=\frac{f_{0}}{s(4-s)}\left(\frac{1}{s+x-1}+\frac{1}{s-x-3}\right)+\frac{1}{s(4-s)}\sum_{n=1}^{N}f_{n}(\rho(s,2)^{n}-\rho(4-s,2)^{n}).$
(D.4)

The second dual function $W_{\mathcal{F}}$ must have an integrable imaginary
part, and the definition ((2.28) implies the existence of a pole at $s=0$ with
residue $\int\operatorname{Im}W_{\mathcal{F}}/\pi$. We therefore propose the
ansatz

$\begin{split}W_{\mathcal{F}}(s)&=\frac{1}{s-4}\sum_{n=0}^{N}g_{n}\rho(s,0)^{n}+h_{0}+\frac{1}{\pi
s}\int_{4}^{\infty}ds\operatorname{Im}W_{\mathcal{F}}(s)\\\
&=\sum_{n=0}^{N}g_{n}\left(\frac{\rho(s,0)^{n}}{s-4}-\frac{(-1)^{n}}{s}\right)+h_{0},\end{split}$
(D.5)

where we were careful to include the contribution of the pole term in the
imaginary part

$\frac{1}{4-s}=\frac{1}{4-\bar{s}-i\epsilon}={\rm
PV}\frac{1}{4-\bar{s}}+i\pi\delta(\bar{s}-4),$ (D.6)

where we wrote explicitely $s=\bar{s}+i\epsilon,\bar{s}\in\mathbb{R}$. Finally
the remaining dual variable $\lambda_{1}$ is a real function that needs to be
integrable. We use the ansatz

$\lambda_{1}(s)=\frac{1}{s\sqrt{s-4}}\left(\sum_{n=0}^{1}a_{n}\frac{\rho(s,0)^{n}+\rho(s,0)^{n,*}}{2}+\sum_{n=1}^{N}b_{n}\frac{\rho(s,0)^{n}-\rho(s,0)^{n,*}}{2i}\right).$
(D.7)

In some cases we observed the numerical convergence is improved if the ansatz
for $\lambda_{1}$ is modified as

$\tilde{\lambda}_{1}(s)=\lambda_{1}(s)+\frac{1}{s^{2}}\sum_{n=1}^{10}c_{n}\frac{\rho(s,0)^{n}-\rho(s,0)^{n,*}}{2i}.$
(D.8)

In this work we use (D.4) and (D.8) in section 3.1, and (D.2) and (D.7) in
section 3.2 .

To implement the semipositive definite constraint on the interval
$[4m^{2},\infty)$, we discretize it on a Chebyshev grid with 200 points. We
also checked, without rigorous analysis, that increasing this number does not
change the results. Then we optimize numerically over the free parameters
$a_{n},b_{n},f_{n},g_{n},h_{0}$ and eventually $c_{n}$ using SDPB to solve the
dual optimization problem.

##  Appendix E Form factor perturbation theory

In this appendix we present our perturbative calculation for the one particle
form factor $\mathcal{F}_{1}^{\Theta}$ and then we review Zamolodchikov’s
result for the perturbative scattering amplitude Zamolodchikov_2011
.191919Form factor perturbation theory was first proposed in Delfino:1996xp
and was applied in different contexts (see e.g. Delfino:1995zk ;
Delfino:2005bh ).

The Ising Field Theory has been extensively studied e.g. in Zamo_2013 ;
Fonseca:2006au ; Zamolodchikov_2011  for non integrable directions. The
starting point for perturbation theory will be the action

$S_{IFT}=S_{CFT}^{(c=1/2)}+\frac{m}{2\pi}\int d^{2}x\epsilon(x)+h\int
d^{2}x\sigma(x),$ (E.1)

where the scaling dimensions of the operators $\epsilon$ and $\sigma$ are
respectively

$\Delta_{\epsilon}=1,\qquad\Delta_{\sigma}=\frac{1}{8}.$ (E.2)

###  E.1 Perturbation theory for $\mathcal{F}_{1}^{\Theta}$

In the interacting theory the trace of the stress energy tensor can be deduced
from (E.1) and (E.2) and takes the form202020 When $h=0$, one can check that
the normalization for $\Theta$ is compatible with
$\mathcal{F}_{2}^{\Theta}(s=0)=-2m^{2}$ and the CFT normalization of the
operator $\epsilon$. On the one hand, we have
$\langle\Theta(x)\Theta(0)\rangle_{0}=\left(\frac{m}{2\pi}\right)^{2}\langle\epsilon(x)\epsilon(0)\rangle\xrightarrow[x\to
0]{}\left(\frac{m}{2\pi}\right)^{2}\frac{1}{|x|^{2}}\,.$ (E.3) On the other
hand, using that only the $n=2$ form factor
$\mathcal{F}_{2}^{\Theta}=im\sqrt{s-4m^{2}}$ is non vanishing when $h=0$ we
also have
$\langle\Theta(x)\Theta(0)\rangle_{0}=\int_{4m^{2}}^{\infty}ds\frac{|\mathcal{F}_{2}^{\Theta}(s)|^{2}}{4\pi\sqrt{s(s-4m^{2})}}\Delta_{E}(x,s)\xrightarrow[x\to
0]{}\left(\frac{m}{2\pi}\right)^{2}\frac{1}{|x|^{2}},$ (E.4) showing that the
normalizations are compatible.

$\Theta(x)=\frac{m}{2\pi}\epsilon(x)+\frac{15h}{8}\sigma(x).$ (E.5)

We can split the IFT action in the thermal deformation part $S_{0}$ and treat
the magnetic deformation as a perturbation, which reads

$S=S_{0}+h\int d^{2}x\sigma(x).$ (E.6)

#### First approach

Our goal is now to use the two point function of the trace of the stress
energy tensor to extract the one particle form factor
$\mathcal{F}_{1}^{\Theta}$. To this end we can use the Euclidean spectral
decomposition

$\begin{split}\langle\Theta(x)\Theta(0)\rangle_{c}&=\int_{0}^{\infty}ds\rho_{\Theta}(s)\Delta_{E}(x,s)\\\
&=|\mathcal{F}_{1}^{\Theta}|^{2}\Delta_{E}(x,m^{2})+\mathcal{O}(e^{-2mx}),\end{split}$
(E.7)

where we used to explicit form of the Euclidean propagator and its asymptotic
form

$\Delta_{E}(x,m^{2})=\frac{1}{2\pi}K_{0}(mx)\underset{x\to\infty}{\sim}\frac{1}{2\pi}\sqrt{\frac{\pi}{2mx}}e^{-mx}.$
(E.8)

On the other hand the two point function (E.7) can be written as a path
integral

$\langle\Theta(x)\Theta(0)\rangle=\frac{1}{Z}\int[\mathcal{D}\phi]e^{-S_{0}}\left(1-h\int
d^{2}y\sigma(y)+\frac{h^{2}}{2}\int
d^{2}yd^{2}z\sigma(y)\sigma(z)+\mathcal{O}(h^{3})\right)\Theta(x)\Theta(0).$
(E.9)

We can therefore get a perturbative expression for this two point function in
terms of correlation functions in the theory with $h=0$, which we denote as
$\langle...\rangle_{0}$. We get

$\begin{split}\langle\Theta(x)\Theta(0)\rangle_{c}&=\left(\frac{m}{2\pi}\right)^{2}\langle\epsilon(x)\epsilon(0)\rangle_{0}+\left(\frac{15h}{8}\right)^{2}\langle\sigma(x)\sigma(0)\rangle_{0}\\\
&-\frac{15mh^{2}}{8\pi}\int
d^{2}y\langle\epsilon(x)\sigma(0)\sigma(y)\rangle_{0}\\\
&+\left(\frac{m^{2}h^{2}}{8\pi^{2}}\right)\int
d^{2}yd^{2}z\langle\epsilon(x)\epsilon(0)\sigma(y)\sigma(z)\rangle_{0}+\mathcal{O}(h^{3}).\end{split}$
(E.10)

Comparing (E.7) and (E.10), the one particle form factor can be computed as

$|\mathcal{F}_{1}^{\Theta}|^{2}=h^{2}\left[\left(\frac{15}{8}\right)^{2}G^{(2)}-\frac{15m}{8\pi}G^{(3)}+\frac{m^{2}}{8\pi^{2}}G^{(4)}\right]+\mathcal{O}(h^{3}),$
(E.11)

where the coefficients $G^{(i)}$ are defined as

$\begin{split}&G^{(2)}\equiv\lim_{|x|\to\infty}\frac{1}{\Delta_{E}(x,m^{2})}\langle\sigma(x)\sigma(0)\rangle_{0},\\\
&G^{(3)}\equiv\lim_{x\to\infty}\frac{1}{\Delta_{E}(x,m^{2})}\int
d^{2}y\langle\epsilon(x)\sigma(0)\sigma(y)\rangle_{0},\\\
&G^{(3)}\equiv\lim_{x\to\infty}\frac{1}{\Delta_{E}(x,m^{2})}\int
d^{2}yd^{2}z\langle\epsilon(x)\epsilon(0)\sigma(y)\sigma(z)\rangle_{0}.\end{split}$
(E.12)

The two point function contribution is immediately obtained by using again the
spectral representation

$G^{(2)}=\lim_{x\to\infty}\frac{1}{\Delta_{E}(x,m^{2})}\int_{0}^{\infty}\rho_{\sigma}(s)\Delta_{E}(x,s)=|\mathcal{F}_{1}^{\sigma}|^{2}.$
(E.13)

To evaluate the 3 and 4 point functions in the thermal deformation we can use
the known form factors Mussardo:1281256 ; PhysRevD.19.2477

$\begin{split}&\mathcal{F}_{2}^{\epsilon}(p_{1},p_{2})=-2\pi\sqrt{4m^{2}-s_{12}},\\\
&\mathcal{F}_{3}^{\sigma}(p_{1},p_{2},p_{3})=2\mathcal{F}_{1}^{\sigma}\sqrt{\frac{4m^{2}-s_{12}}{s_{12}}}\sqrt{\frac{4m^{2}-s_{13}}{s_{13}}}\sqrt{\frac{4m^{2}-s_{23}}{s_{23}}},\\\
&\mathcal{F}_{1}^{\sigma}=2^{1/12}e^{-1/8}m^{1/8}A_{G}^{3/2}\approx
1.3578...,\end{split}$ (E.14)

where $A_{G}$ is the Glaisher’s constant and we use the $s$ Mandelstam
variable defined as

$s_{12}\equiv-(p_{1}+p_{2})^{2}=2m^{2}+2(E_{\bm{p}_{1}}E_{\bm{p}_{2}}-\bm{p}_{1}\bm{p}_{2}).$
(E.15)

We will also need the parity reversed 2-momentum $\tilde{p}$ and the rest
frame 2-momentum $\bar{k}$ defined as

$p\equiv\begin{pmatrix}E_{\bm{p}}\\\
\bm{p}\end{pmatrix},\qquad\tilde{p}\equiv\begin{pmatrix}E_{\bm{p}}\\\
-\bm{p}\end{pmatrix},\qquad\bar{k}\equiv\begin{pmatrix}m\\\ 0\end{pmatrix}.$
(E.16)

We can evaluate the correlation functions written as a time ordered vacuum
correlator. Without loss of generality, we choose $x$ to be pointing in the
Euclidean time direction and we parametrize $y$ as

$x=\begin{pmatrix}|x|\\\ 0\end{pmatrix},\qquad y=\begin{pmatrix}y^{1}\\\
y^{2}\end{pmatrix}.$ (E.17)

Then, we can write

$\begin{split}\int d^{2}y\langle\epsilon(x)\sigma(0)\sigma(y)\rangle_{0}&=\int
dy^{2}\int_{-\infty}^{0}dy^{1}\bra{0}\epsilon(x)\sigma(0)\sigma(y)\ket{0}\\\
&+\int
dy^{2}\int_{0}^{|x|}dy^{1}\bra{0}\epsilon(x)\sigma(y)\sigma(0)\ket{0}\\\
&+\int
dy^{2}\int_{|x|}^{\infty}dy^{1}\bra{0}\sigma(y)\epsilon(x)\sigma(0)\ket{0}.\end{split}$
(E.18)

Using that the operators can be translated as 212121With these conventions,
$P_{1}$ is the hamiltonian and $P_{2}$ is the spatial momentum and both are
hermitian operators on the physical Hilbert space.

$\mathcal{O}(y)=e^{P_{1}y^{1}-iP_{2}y^{2}}\mathcal{O}(0)e^{-P_{1}y^{1}+iP_{2}y^{2}},$
(E.19)

the first term in (E.18) reads

$\begin{split}G^{(3)}_{1}&\equiv\int
dy^{2}\int_{-\infty}^{0}dy^{1}\bra{0}\epsilon(x)\sigma(0)\sigma(y)\ket{0}\\\
&=\int dy^{2}\int_{-\infty}^{0}dy^{1}\int
d\Phi_{12}d\Phi_{3}\bra{0}\epsilon(0)\ket{p_{1},p_{2}}_{in}{}_{in}\bra{p_{1},p_{2}}\sigma(0)\ket{p_{3}}_{out}{}_{out}\bra{p_{3}}\sigma(0)\ket{0}\\\
&\quad\times\quad
e^{-|x|(E_{\bm{p}_{1}}+E_{\bm{p}_{2}})}e^{y^{1}E_{\bm{p}_{3}}}e^{-iy^{2}\bm{p}_{3}}+\mathcal{O}(e^{-2m|x|})\\\
&=\int
d\Phi_{12}\frac{1}{4m^{2}}e^{-|x|(E_{\bm{p}_{1}}+E_{\bm{p}_{2}})}\mathcal{F}_{2}^{\epsilon}(p_{1},p_{2})\mathcal{F}_{3}^{\sigma,*}(\bar{k},-p_{1},-p_{2})\mathcal{F}_{1}^{\sigma}+\mathcal{O}(e^{-2m|x|})=\mathcal{O}(e^{-2m|x|}).\end{split}$
(E.20)

In the first line we inserted a complete set of states between each operator,
using (2.5) and the notation

$\displaystyle d\Phi_{12\dots
n}=\frac{1}{n!}\frac{d{\bm{p}}_{1}}{(2\pi)2E_{\bm{p}_{1}}}\dots\frac{d{\bm{p}}_{n}}{(2\pi)2E_{\bm{p}_{n}}}\,.$
(E.21)

Between $\epsilon(x)$ and $\sigma(0)$ only two particle states can contribute
because $\epsilon$ can only create two particles from the vacuum. Between
$\sigma(0)$ and $\sigma(y)$, there are many states that contribute to the
correlator but only the single particle state can hope to survive the limit
(E.12). In the second line we integrated over $y^{2}$ to get a spatial delta
function that we used to integrate over $\bm{p}_{3}$, and we also integrated
over $y^{1}$. We also used Karateev_2020

$\displaystyle{}_{out}\bra{p_{1},\dots,p_{n}}\mathcal{O}(0)\ket{k_{1},\dots,k_{m}}_{in}=\mathcal{F}_{m+n}^{\mathcal{O}}(p_{1},\dots,p_{n},-k_{1},\dots,-k_{m})\,,$
(E.22)

for hermitian operators $\mathcal{O}$. This contribution does not survive the
limit (E.12).

We now treat the second term in (E.18) that reads

$\begin{split}G^{(3)}_{2}&\equiv\int
dy^{2}\int_{0}^{|x|}dy^{1}\bra{0}\epsilon(x)\sigma(y)\sigma(0)\ket{0}\\\
&=\int dy^{2}\int_{0}^{|x|}dy^{1}\int
d\Phi_{12}d\Phi_{3}\bra{0}\epsilon(0)\ket{p_{1},p_{2}}_{in}{}_{in}\bra{p_{1},p_{2}}\sigma(0)\ket{p_{3}}_{out}{}_{out}\bra{p_{3}}\sigma(0)\ket{0}\\\
&\quad\times\quad
e^{-|x|(E_{\bm{p}_{1}}+E_{\bm{p}_{2}})}e^{y^{1}(E_{\bm{p}_{1}}+E_{\bm{p}_{2}}-E_{\bm{p}_{3}})}e^{-iy^{2}(\bm{p}_{1}+\bm{p}_{2}-\bm{p}_{3})}+\mathcal{O}(e^{-2m|x|})\\\
&=\int
d\Phi_{12}\frac{1}{2E_{\bm{p}_{1}+\bm{p}_{2}}}\frac{1}{E_{\bm{p}_{1}}+E_{\bm{p}_{2}}-E_{\bm{p}_{1}+\bm{p}_{2}}}e^{-|x|E_{\bm{p}_{1}+\bm{p}_{2}}}\mathcal{F}_{2}^{\epsilon}(p_{1},p_{2})\mathcal{F}_{3}^{\sigma,*}(p_{3},-p_{1},-p_{2})\mathcal{F}_{1}^{\sigma}\\\
&+\mathcal{O}(e^{-2m|x|}).\end{split}$ (E.23)

where $p_{3}=(E_{\bm{p}_{1}+\bm{p}_{2}},\bm{p}_{1}+\bm{p}_{2})$. The steps are
identical as above and we now get a non-zero contribution to the limit (E.12).
A similar calculation shows that the third term (E.18) is

$\begin{split}G^{(3)}_{3}&\equiv\int
dy^{2}\int_{|x|}^{\infty}dy^{1}\bra{0}\sigma(y)\epsilon(x)\sigma(0)\ket{0}=\int
d\Phi_{1}\frac{1}{2m^{2}}e^{-E_{\bm{p}_{1}}|x|}(\mathcal{F}_{1}^{\sigma})^{2}\mathcal{F}_{2}^{\epsilon,*}(p_{1},-\bar{k})\end{split}$
(E.24)

To evaluate the $x\to\infty$ limit we can use that the $x$ dependence in the
integrand only comes from the exponential. If the integral is only over
$\bm{p}$ we have

$\displaystyle\lim_{x\to\infty}\frac{1}{\Delta_{E}(x,m^{2})}\int
d\bm{p}f(\bm{p})e^{-|x|E_{\bm{p}}}=$ $\displaystyle=$
$\displaystyle\lim_{x\to\infty}\frac{1}{\Delta_{E}(x,m^{2})}\int\frac{d\bm{p}}{\sqrt{|x|}}f\left(\frac{\bm{p}}{\sqrt{|x|}}\right)e^{-|x|(m+\bm{p}^{2}/(2m|x|))+\mathcal{O}(\bm{p}^{3}/\sqrt{|x|})}=4\pi
f(0),$ (E.25)

where we rescaled the integration variable by $1/\sqrt{|x|}$ and expanded the
energy with its Taylor series. When we have an integral over $\bm{p}_{1}$ and
$\bm{p}_{2}$ and the exponential damping contains $E_{\bm{p}_{1}+\bm{p}_{2}}$,
we get

$\lim_{x\to\infty}\frac{1}{\Delta_{E}(x,m^{2})}\int
d\bm{p}_{1}d\bm{p}_{2}f(\bm{p}_{1},\bm{p}_{2})e^{-|x|E_{\bm{p}_{1}+\bm{p}_{2}}}=4\pi\int
d\bm{p}f(\bm{p},-\bm{p}).$ (E.26)

This result is obtained by first changing variables to $\bm{p}_{1}+\bm{p}_{2}$
and $\bm{p}_{1}-\bm{p}_{2}$ and then rescaling as above.

Using these results we can take the limit (E.12) and get the contribution of
the 3 point function that reads

$\begin{split}G^{(3)}&=4\pi\frac{1}{8\pi
m^{3}}\mathcal{F}_{2}^{\epsilon,*}(\bar{k},-\bar{k})(\mathcal{F}_{1}^{\sigma})^{2}\\\
&-4\pi i\int d\bm{p}\frac{1}{2(2\pi
2E_{\bm{p}})^{2}}\frac{1}{2m}\frac{1}{2E_{\bm{p}}-m}\mathcal{F}_{2}^{\epsilon}(p,\tilde{p})\mathcal{F}_{1}^{\sigma}\mathcal{F}_{3}^{\sigma,*}(\bar{k},-p,-\tilde{p})\end{split}$
(E.27)

The contributions from the 4 point function are similar. We have

$\int
d^{2}yd^{2}z\langle\epsilon(x)\epsilon(0)\sigma(y)\sigma(z)\rangle_{0}=2\int
dy^{2}dz^{2}\int_{-\infty}^{\infty}dy^{1}\int_{-\infty}^{y^{1}}dz^{1}\langle\epsilon(x)\epsilon(0)\sigma(y)\sigma(z)\rangle_{0}.$
(E.28)

Splitting up the integrals in time-ordered correlators, we get four
contributing terms

$\begin{split}G^{(4)}_{1}&\equiv\int
dy^{2}dz^{2}\int_{0}^{|x|}dy^{1}\int_{-\infty}^{0}dz^{1}\bra{0}\epsilon(x)\sigma(y)\epsilon(0)\sigma(z)\ket{0}\\\
&=\int
d\Phi_{2}\frac{1}{2m^{2}}\frac{1}{E_{\bm{p}_{1}}+E_{\bm{p}_{2}}-E_{\bm{p}_{1}+\bm{p}_{2}}}e^{-|x|(E_{\bm{p}_{1}+\bm{p}_{2}})}\mathcal{F}_{2}^{\epsilon}(p_{1},p_{2})\mathcal{F}^{\sigma,*}_{3}(p_{1},p_{2},-p_{3})\mathcal{F}_{1}^{\sigma}\mathcal{F}_{2}^{\epsilon}(-p_{3},\bar{k}),\end{split}$
(E.29)

where $p_{3}=(E_{\bm{p}_{1}+\bm{p}_{2}},\bm{p}_{1}+\bm{p}_{2})$,

$\begin{split}G^{(4)}_{2}&\equiv\int
dy^{2}dz^{2}\int_{0}^{|x|}dy^{1}\int_{0}^{y^{1}}dz^{1}\bra{0}\epsilon(x)\sigma(y)\sigma(z)\epsilon(0)\ket{0}\\\
&=\int
d\Phi_{2}d\Phi_{2}^{\prime}\frac{1}{2E_{\bm{p}_{1}+\bm{p}_{2}}}\frac{1}{E_{\bm{p}_{1}}+E_{\bm{p}_{2}}-E_{\bm{p}_{1}+\bm{p}_{2}}}\frac{1}{E_{\bm{p}_{1}}+E_{\bm{p}_{2}}-E_{\bm{p}^{\prime}_{1}+\bm{p}^{\prime}_{2}}}e^{-|x|(E_{\bm{p}_{1}+\bm{p}_{2}})}\\\
&(2\pi)\delta(\bm{p}_{1}+\bm{p}_{2}-\bm{p}^{\prime}_{1}-\bm{p}^{\prime}_{2})\quad\times\quad\mathcal{F}_{2}^{\epsilon}(p_{1},p_{2})\mathcal{F}_{2}^{\epsilon,*}(p_{1}^{\prime},p_{2}^{\prime})\mathcal{F}^{\sigma}_{3}(p_{1}^{\prime},p_{2}^{\prime},-p_{3}^{\prime})\mathcal{F}^{\sigma,*}_{3}(p_{1},p_{2},-p_{3}),\end{split}$
(E.30)

where $p_{3}=(E_{\bm{p}_{1}+\bm{p}_{2}},\bm{p}_{1}+\bm{p}_{2})$ and
$p_{3}^{\prime}=(E_{\bm{p}_{1}^{\prime}+\bm{p}_{2}^{\prime}},\bm{p}_{1}^{\prime}+\bm{p}_{2}^{\prime})$,

$\begin{split}G^{(4)}_{3}&\equiv\int
dy^{2}dz^{2}\int_{|x|}^{\infty}dy^{1}\int_{-\infty}^{0}dz^{1}\bra{0}\sigma(y)\epsilon(x)\epsilon(0)\sigma(z)\ket{0}\\\
&=\int
d\Phi_{1}\frac{1}{4m^{2}}e^{-|x|E_{\bm{p}_{1}}}(\mathcal{F}_{1}^{\sigma})^{2}|\mathcal{F}_{2}^{\epsilon}(-p_{1},\bar{k})|^{2},\end{split}$
(E.31)

and finally

$\begin{split}G^{(4)}_{4}&\equiv\int
dy^{2}dz^{2}\int_{|x|}^{\infty}dy^{1}\int_{0}^{|x|}dz^{1}\bra{0}\sigma(y)\epsilon(x)\sigma(z)\epsilon(0)\ket{0}\\\
&=\int
d\Phi_{2}\frac{1}{2m^{2}}\frac{1}{E_{\bm{p}_{1}}+E_{\bm{p}_{2}}-E_{\bm{p}_{1}+\bm{p}_{2}}}e^{-|x|(E_{\bm{p}_{1}+\bm{p}_{2}})}\mathcal{F}_{2}^{\epsilon}(p_{1},p_{2})\mathcal{F}^{\sigma,*}_{3}(p_{1},p_{2},-p_{3})\mathcal{F}_{1}^{\sigma}\mathcal{F}_{2}^{\epsilon}(-p_{3},\bar{k})\\\