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,
Switzerlandccinstitutetext: Department of Physics, McGill University, 3600 Rue
University, Montréal, H3A 2T8, QC Canadaddinstitutetext: NORDITA, Stockholm
University and KTH Royal Institute of Technology, Hannes Alfvéns väg 12,
SE-106 91 Stockholm, Swedeneeinstitutetext: Centre for Theoretical Physics,
Department of Physics and Astronomy, Queen Mary University of London, Mile End
Road, London E1 4NS, United Kingdom

# The O(N) Monolith reloaded:
Sum rules and Form Factor Bootstrap

Lucía Córdova a,b,c Miguel Correia d,e Alessandro Georgoudis b and Antoine
Vuignier<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We revisit the space of gapped quantum field theories with a global O(N)
symmetry in two spacetime dimensions. Previous works using S-matrix bootstrap
revealed a rich space in which integrable theories such as the non-linear
sigma model appear at special points on the boundary, along with an abundance
of unknown models hinting at a non conventional UV behaviour. We extend the
S-matrix set-up by including into the bootstrap form factors and spectral
functions for the stress-energy tensor and conserved O(N) currents. Sum rules
allow us to put bounds on the central charges of the conformal field theory
(CFT) in the UV. We find that a big portion of the boundary can only flow from
CFTs with infinite central charges. We track this result down to a particular
behaviour of the amplitudes in physical kinematics and discuss its physical
implications.

††preprint: NORDITA 2023-069
QMUL-PH-23-23

##  1 Introduction

The S-matrix bootstrap allows us to explore the space of consistent Quantum
Field Theories (QFTs). In practice, the imposed consistency conditions are
unitarity, analyticity and crossing symmetry of the $2\to 2$ scattering
amplitude. When establishing bounds in parameters, like couplings or Wilson
coefficients, one often finds a much larger allowed space than the one covered
by known theories Paulos:2016but ; Doroud:2018szp ; Paulos:2017fhb ;
He:2018uxa ; Cordova:2018uop ; Guerrieri:2018uew ; Homrich:2019cbt ;
EliasMiro:2019kyf ; Paulos:2018fym ; Karateev:2019ymz ; Bercini:2019vme ;
Cordova:2019lot ; Kruczenski:2020ujw ; Guerrieri:2020bto ; Guerrieri:2020kcs ;
Correia:2020xtr ; Hebbar:2020ukp ; Sinha:2020win ; Guerrieri:2021ivu ;
Correia:2021etg ; Tourkine:2021fqh ; Karateev:2022jdb ; EliasMiro:2021nul ;
He:2021eqn ; Guerrieri:2021tak ; Chowdhury:2021ynh ; Chen:2022nym ;
Miro:2022cbk ; Guerrieri:2022sod ; Haring:2022sdp ; Correia:2022dyp ;
Marucha:2023vrn ; He:2023lyy ; Acanfora:2023axz . This might mean there are
other fully consistent theories we did not know before or that we must go
beyond the $2\to 2$ set of constraints in order to discard unphysical
theories.

In this work we focus on gapped QFTs with O(N) internal symmetry in $d=2$
spacetime dimensions. The space of allowed S-matrices carved out by the $2\to
2$ set of constraints was first studied in Cordova:2018uop ; He:2018uxa ;
Cordova:2019lot  and shows a number of interesting features. In the absence of
O(N) symmetry the boundary of the allowed space can be spanned by continuous
families of integrable S-matrices Paulos:2016but ; Paulos:2017fhb ;
Chen:2021pgx ; Chen:2022nym . On the other hand, in the O(N) case the Yang-
Baxter (YB) relations become non-trivial and for $N>2$ integrability is only
achieved at isolated boundary points. For example, the O(N) non-linear sigma
model (NLSM) is found at a kink.

The space of O(N) S-matrices – the O(N) monolith – exhibits a variety of other
interesting and still mysterious features, as highlighted in figure 1. Namely,
the ‘constant’ amplitudes on which the extremal S-matrices are inelastic (a
rare observation of non-unitarity saturation at the boundary), or the ‘pre-
vertices’ where the periodic-Yang-Baxter (pYB) S-matrices are found. The
latter satisfy the Yang-Baxter relations but a corresponding physical model or
Lagrangian formulation is still unknown. The pYB amplitudes – and generically
the extremal amplitudes at the boundary – show a rich analytic structure with
infinitely many resonances arranging themselves in a periodic manner, which
raises questions about the UV nature of these putative QFTs.111Similar
periodicities are also observed in the S-matrices coming from $T\bar{T}$
deformations, which are known to be incompatible with conventional local UV-
complete QFTs Smirnov:2016lqw ; Camilo:2021gro .  Are these structures
unphysical? Is the NLSM the only non-trivial physical theory at the boundary?

These questions call for an enlargement of the subset of constraints and
observables. One option would be to include multi-particle constraints.
However, these are plagued with fundamental and technical difficulties such as
the presence of anomalous thresholds Hannesdottir:2022bmo ; Correia:2022dcu ;
Caron-Huot:2023ikn . A recent viable alternative, which allows to probe the UV
nature of these theories, is to include local operators such as the stress-
energy tensor Karateev:2019ymz . In this extended framework, form factors and
spectral densities join the S-matrix in a more powerful bootstrap setup
Karateev:2019ymz ; Karateev:2020axc ; Correia:2022dyp ; He:2023lyy .

In particular, if the local operators are chosen to be the stress tensor and
the conserved O(N) currents, information about the conformal field theory
(CFT) at the UV fixed point can be included via the so-called _sum rules_.
These relate CFT properties – the central charges $c$ and $k$,222We define $c$
and $k$ as coefficients in the two point function of the stress-energy tensor
and the O(N) conserved currents in the CFT, respectively. If the currents
define an affine Lie algebra, the current central charge $k$ defines the
‘level’ of the algebra. respectively – to the spectral density of these
operators in the QFT Karateev:2020axc . These CFT quantities can now be
targeted by the QFT bootstrap, and be used to probe the O(N) monolith. By
minimizing $c$ and $k$ across the O(N) monolith we can constrain the possible
CFTs from which these S-matrices in the infrared (IR) could flow from.

In simple terms, this framework let us see explicitly how conditions in the IR
trickle up along the flow onto the UV, allowing us to constrain the UV CFTs
via the S-matrix principles. So, we may also ask concretely, what is the
allowed space of central charges $c$ and $k$ that is compatible with the
unitarity, crossing symmetry and analyticity of the S-matrix? Where do Wess-
Zumino-Witten (WZW) models – for which $c$ and $k$ are related by the Sugawara
construction – sit inside this space? What differentiates the amplitudes
leading to finite/infinite central charges?

The goal of this work is to implement this framework and address these
questions. The organization of this paper is as follows. In section 2 we
briefly overview the O(N) monolith space of S-matrices and review how form
factors and spectral densities can be included into the O(N) bootstrap setup.
In section 3 we show our numerical results for the bounds on the central
charges $c$ and $k$. In section 4 we use analytical form factor bootstrap
methods to argue for a relation between the behaviour of the amplitudes in
physical kinematics and the finiteness of the central charges. We then discuss
our results in section 5. Our conventions, derivations of dual optimization
problems, numerical implementation and various analytical results are
collected in appendices A, B, C, D and E.

Figure 1: Two-dimensional slice of the O(N) monolith for $N=7$ taken from
Cordova:2019lot . The shaded blue region are the allowed values for the
amplitudes at the crossing symmetric point, the axes given by
$\sigma_{1,2}=\sigma_{1,2}(s=2m^{2})$. The highlighted points are different
exact S-matrices as explained in the main text. The lines connecting them are
the sections used in 3.1.

##  2 Review of O(N) monolith and Form Factor Bootstrap setup

In this work we will be dealing with three different types of observables:
$2\to 2$ scattering amplitudes of identical massive particles $S(s)$, two
particles form factors of the trace of the stress-energy tensor
$F_{\Theta}(s)$ and of the O(N) conserved currents $F_{J}(s)$ and their
spectral densities, $\rho_{\Theta}(s)$ and $\rho_{J}(s)$.

In the first part of this section we review some properties and known
bootstrap results for amplitudes of particles transforming in the vector
representation of O(N). In the second part we explain how to include the
stress tensor and O(N) currents and set up the bootstrap problem of minimizing
the associated central charges.

###  2.1 Pure S-matrix bootstrap and O(N) monolith

The particles we are scattering are states transforming in the vector
representation of O(N) and carry a label $i=1,2,...,N$. The scattering
amplitude is defined as

$\displaystyle{}_{kl}^{out}\langle
p_{1},p_{2}|k_{1},k_{2}\rangle^{in}_{ij}\equiv(2\pi)^{2}\delta^{(2)}(p_{1}+p_{2}-k_{1}-k_{2})\mathcal{N}_{2}(s)S_{ij}^{kl}(s)\,,$
(1)

where $\mathcal{N}_{2}(s)\equiv 2\sqrt{s}\sqrt{s-4m^{2}}$ and $s$ is the
center of mass energy squared $s=-(p_{1}+p_{2})^{2}$. Following group
theoretic considerations, the scattering amplitude is decomposed into the
singlet ($\bullet$), antisymmetric ($A$) and symmetric ($S$) irreducible
representations as follows333The projectors are given by  $\displaystyle
T^{\bullet}_{ij,kl}\equiv\frac{1}{N}\delta_{ij}\delta_{kl},\qquad
T^{A}_{ij,kl}\equiv\frac{\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}}{2},\qquad
T^{S}_{ij,kl}\equiv\frac{\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}}{2}-\frac{1}{N}\delta_{ij}\delta_{kl}.$

$\displaystyle
S_{ij}^{kl}(s)=S_{\bullet}(s)\,T^{\bullet}_{ij,kl}+S_{A}(s)\,T^{A}_{ij,kl}\
+S_{S}(s)\,T^{S}_{ij,kl}\,,$ (2)

The three different channel amplitudes $S_{a}(s)$ ($a=\bullet,A,S$) obey the
usual unitarity constraint $|S_{a}(s)|\leq 1$, for $s\geq 4m^{2}$, but get
mixed under crossing:

$S_{a}(4m^{2}-s)=C_{ab}S_{b}(s)\,,\qquad
C_{ab}=\left(\begin{array}[]{ccc}\frac{1}{N}&-\frac{N}{2}+\frac{1}{2}&\frac{N}{2}+\frac{1}{2}-\frac{1}{N}\\\
-\frac{1}{N}&\frac{1}{2}&\frac{1}{2}+\frac{1}{N}\\\
\frac{1}{N}&\frac{1}{2}&\frac{1}{2}-\frac{1}{N}\end{array}\right)\,,$ (3)

where $C_{ab}$ is the crossing matrix. There is an alternative basis
$S_{ij}^{kl}(s)=\sigma_{1}(s)\,\delta_{ij}\delta_{kl}+\sigma_{2}(s)\,\delta_{ik}\delta_{jl}+\sigma_{3}(s)\,\delta_{il}\delta_{jk}\,,$
in which crossing symmetry is more straightforward:
$\sigma_{1}(s)=\sigma_{3}(4m^{2}-s)\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \sigma_{2}(s)=\sigma_{2}(4m^{2}-s).$ The map
between the bases is

$\displaystyle
S_{\bullet}(s)=\sigma_{2}(s)+\sigma_{3}(s)+N\sigma_{1}(s)\,,\quad
S_{A}(s)=\sigma_{2}(s)-\sigma_{3}(s)\,,\quad
S_{S}(s)=\sigma_{2}(s)+\sigma_{3}(s)\,.$ (4)

As for the analytic properties, we will focus on theories without bound
states, so that each amplitude $S_{a}(s)$ is an analytic function of $s$ away
from the multi-particle thresholds starting at $s=4m^{2}$ and $s=0$ (from
crossing).

The space of amplitudes satisfying the above requirements was studied in
Cordova:2019lot . We review next some of the findings there. Using functionals
of the form $\mathcal{F}=\sum_{a}n_{a}S_{a}(s_{*})$ one can get slices of this
infinite dimensional space of allowed amplitudes $S_{a}(s)$. Figure 1 shows a
two-dimensional slice of this space we call the monolith, constructed by
putting bounds on the value of the amplitude at the crossing symmetric points
$\sigma_{1,2}(s_{*}=2m^{2})$, which one can see as effective quartic
couplings. A simple feature of these bounds is that they are symmetric under
the simultaneous change of sign for all channels $S_{a}\rightarrow-
S_{a}$.444This goes back to the fact that we are considering theories without
bound states poles, whose residue should be positive in unitary theories. As
usual in these optimization problems, one has a unique solution when the
bounds are saturated, so that one can read off the amplitudes at the boundary
of the monolith.

The picture for $N>2$ is the following, there are several special amplitudes
sitting at special points of the boundary (the explicit expressions can be
found in appendix A):

  * •

The free boson theory where $S_{a}(s)=1$ (+Free in figure 1);

  * •

The free fermion theory where $S_{a}(s)=-1$ ($-$Free);

  * •

The integrable O(N) non-linear sigma model (+NLSM);

  * •

The ‘negative’ NLSM S-matrix, in which one changes sign on all NLSM amplitudes
(-NLSM);

  * •

The periodic Yang-Baxter S-matrices ($\pm$pYB) which are periodic in rapidity
$S_{a}^{\text{pYB}}(\theta)=S_{a}^{\text{pYB}}(\theta+\tau^{\text{pYB}})$;555The
relation between the center of mass energy $s$ and the rapidity is
$s=4m^{2}\cosh^{2}\left(\theta/2\right)$.

  * •

The constant S-matrices ($\pm$const), inelastic in the symmetric channel given
by $S_{\bullet,A,S}=\pm\,\left(1,\,-1,\,-\frac{N-2}{N+2}\right)$.

For the S-matrices of the last three points it is currently not known if a UV
completion exists or if they correspond to any physical model. Nonetheless,
except for the constant inelastic S-matrices, all of the listed S-matrices are
integrable, i.e. they satisfy the Yang-Baxter relations.666For explicit
solutions of Yang-Baxter’s equations in theories with O(N) symmetry see
Zamolodchikov:1978xm .

Generically, the S-matrices on the boundary (even if not at a kink) enjoy
several common features. The first one is that they saturate unitarity
$|S(s)|=1$, even if they are not integrable (except for the constant
S-matrices on the last point). We expect however that these amplitudes are
good approximations to the physical ones –at least at low energies– in which
some particle production is present (see e.g. discussion section in
Cordova:2018uop ). The second one is that they exhibit a very rich structure
of infinite resonances (seen as zeros in the physical sheet) arranging
themselves in periodic fashion. That is, a generic point at the boundary in
figure 1 will obey $S^{\text{bdy}}(\theta)=S^{\text{bdy}}(\theta+\tau)$ for a
given period $\tau$ that changes along the boundary (see figure 8 in
Cordova:2019lot ). The last point is quite puzzling since it gives us an
observable which is periodic in a given parametrization of energy
$\theta\sim\ln s$. In fact, one of the motivations for this work is to
understand if such S-matrices can be compatible with a unitary UV completion.

###  2.2 Adding currents: sum rules and central charge minimization

In order to extract more information about the UV behaviour of the theories
found with the S-matrix bootstrap, we will now include into the setup the
existence of operators such as the stress tensor and O(N) currents. As we
explain in the following, the inclusion of these operators will give us access
to the central charges of the ultraviolet conformal theories. For clarity we
discuss first the setup for theories without global symmetry and a generic
operator $\mathcal{O}(x)$ and later generalize to the O(N) case.

As proposed in the pioneering work Karateev:2019ymz , the main ingredient is
the matrix of inner products of three different type of states: the first two
are the two-particle in and out states (as in the previous section) and the
third one is the state given by the operator $\mathcal{O}(x)$ acting on the
vacuum

$\ket{\psi_{1}}=\omega\ket{p_{1},p_{2}}^{in},\qquad\ket{\psi_{2}}=\omega\ket{p_{1},p_{2}}^{out},\qquad\ket{\psi_{3}}=\int
d^{2}x\,e^{ip\cdot x}\,\mathcal{O}(x)\ket{0}\,.$ (5)

where $p=p_{1}+p_{2}$, $s=-p^{2}$ and
$\omega=\left(2\sqrt{s}\sqrt{s-4}\right)^{-1/2}$.777The latter kinematic
factor $\omega(s)$ is included in order to absorb the
$\mathcal{N}_{2}(s)=\omega(s)^{-2}$ appearing in the normalization (1).
Unitarity translates to the positive semi-definiteness of the matrix of inner
products $B$

$\bra{\psi_{i}}\ket{\psi_{j}}=B_{ij}(2\pi)^{2}\delta^{2}(p-p^{\prime})\,,\qquad
B=\begin{pmatrix}1&S^{*}(s)&\omega F^{*}(s)\\\ S&1&\omega F(s)\\\ \omega
F(s)&\omega F^{*}(s)&2\pi\rho(s)\end{pmatrix}\succeq 0\,,\quad s\geq
4m^{2}\,.$ (6)

The overlap between the in and out states gives the usual two-particle
amplitude $S(s)$. If we take the inner product of two-particle in or out state
with the state created by the operator we get the two-particle form factor

$F(s)=\langle 0|\mathcal{O}(0)|p_{1},p_{2}\rangle^{in}\,.$ (7)

As a function of the complex variable $s$, the two particle form factor $F(s)$
is an analytic function except for the right hand cut at $s\geq 4m^{2}$ and
possible bound state poles as depicted in figure 2.

Lastly the overlap $\langle\psi_{3}|\psi_{3}\rangle$ gives the Fourier
transform of Wightman two point function of the operator, known as the
spectral density of the operator $\mathcal{O}$

$2\pi\,\rho(s)=\int d^{2}x\,e^{-ip\cdot
x}\bra{0}\mathcal{O}(x)\mathcal{O}(0)\ket{0}\,,$ (8)

which is a real positive function.

$s/m^{2}$$s/m^{2}$$-2$$0$$2$$4$$6$$-2$$0$$2$$4$$6$$S(s)$$F(s)$$s$$s$ Figure 2:
Analytic structure of the scattering amplitude $S(s)$ and two-particle form
factor $F(s)$. The light gray crosses are possible simple poles due to bound
states, set to zero in the rest of this paper, and red dots are branch points
with the cuts along the real axis, coming from multi-particle thresholds.

The generalization to our O(N) setup is straightforward. Now, because
unitarity constrains each channel separately, we need to consider three
different matrices $B_{\bullet,A,S}$, one for each representation. The size of
these matrices depend on which operator(s) we want to include. One of the
operators we study is the trace of the stress tensor
$\Theta(x)=T^{\mu}_{\mu}(x)$ which is a singlet of O(N) and hence appears in
$B_{\bullet}$. The second operator we consider is the O(N) global current
$J^{\mu}(x)$ which transforms in the adjoint of the group and therefore
appears in the antisymmetric channel $B_{A}$. The explicit states we use are
(for more details see B.1):

$\displaystyle\ket{\psi_{1}}_{a}=\omega\ket{p_{1},p_{2}}_{a}^{in},\qquad\ket{\psi_{2}}_{a}=\omega\ket{p_{1},p_{2}}_{a}^{out},$
(9) $\displaystyle\ket{\psi_{3}}_{\bullet}=\int d^{2}x\,e^{ip\cdot
x}\,\Theta(x)\ket{0},\qquad\ket{\psi_{3}}_{A}=\int d^{2}x\,e^{ip\cdot
x}\,\frac{q_{\mu}J^{\mu}(x)}{q^{2}}\ket{0},\qquad\ket{\psi_{3}}_{S}=0,$

where again $a$ labels the representation and $q\equiv p_{1}-p_{2}$. As
before, unitarity becomes the positive semi-definiteness of the following
matrices

$B_{\bullet}=\begin{pmatrix}1&S_{\bullet}^{*}&\omega F_{\Theta}^{*}\\\
S_{\bullet}&1&\omega F_{\Theta}\\\ \omega F_{\Theta}&\omega
F_{\Theta}^{*}&2\pi\rho_{\Theta}\end{pmatrix}\succeq 0,\quad
B_{A}=\begin{pmatrix}1&S_{A}^{*}&i\omega F_{J}^{*}\\\ S_{A}&1&i\omega F_{J}\\\
-i\omega F_{J}&-i\omega F_{J}^{*}&\frac{2\pi
s}{s-4m^{2}}\rho_{J}\end{pmatrix}\succeq 0,\quad
B_{\textbf{S}}=\begin{pmatrix}1&S_{S}^{*}\\\ S_{S}&1\end{pmatrix}\succeq 0,$

(10)

where all functions are evaluated at physical values of the energy $s\geq
4m^{2}$. Details on the derivation of the different elements in the matrices
are given in appendix B.1.

The stress tensor and O(N) currents are natural operators to consider since we
are studying QFTs with global O(N) symmetry. Moreover, the fact that they are
conserved currents gives rise to sum rules that allow us to probe the central
charges $c$ and $k$ of the conformal theory in the ultraviolet, defined as the
coefficients appearing in the CFT two point functions888We have used the
complex coordinates in Euclidean signature $z=x+iy$ and $T\equiv T_{zz}$, and
$J\equiv J_{z}$. If the holomorphic $J$ and antiholomorphic $\bar{J}\equiv
J_{\bar{z}}$ currents are separately conserved, the current central charge $k$
is the level of the associated affine Lie algebra.

$\displaystyle\langle
T(z)T(0)\rangle_{\text{CFT}}=\frac{c/2}{z^{4}},\qquad\langle
J(z)J(0)\rangle_{\text{CFT}}=\frac{k}{z^{2}}.$ (11)

Conservation of the stress energy tensor gives Zamolodchikov’s famous
c-theorem Zamolodchikov:1986gt  which we can express as a sum rule integral
for the spectral density of $\Theta(x)=T^{\mu}_{\mu}(x)$

$c=12\pi\int_{0}^{\infty}ds\,\frac{\rho_{\Theta}(s)}{s^{2}}\,.$ (12)

In the left hand side of the equation above we have only the UV central charge
$c=c_{UV}$ and not the difference between UV and IR since we are dealing with
gapped theories for which $c_{IR}=0$.

Similarly, there exists a $k$-theorem for the central charge of the global
symmetry currents $J^{\mu}(x)$ Vilasis-Cardona:1994oke . As reviewed in
appendix B.2, conservation of the currents implies the following sum rule

$k=\frac{\pi}{2}\int_{0}^{\infty}ds\,\rho_{J}(s)\,.$ (13)

The above sum rules allow us to extract the central charges $c$ and $k$ from
the spectral densities in the QFT. Since these central charges characterize
the conformal theory in the ultraviolet, they are a natural target for
bootstrap bounds. Moreover, for a fixed finite number of flavors $N$, we
expect that typical physical theories have finite central charges. With these
considerations in mind, the bootstrap problem we will solve is the following:

$\textsf{Minimize $c$, $k$ subject to }B_{a}\succeq 0\,,$ (14)

where the amplitude, form factor and spectral density entering $B_{a}$ have
analyticity and crossing built in (see appendix D for explicit ansätze). The
bounds we find when performing the above minimizations are meaningful since
the normalization of the operators we are considering is not arbitrary.
Indeed, the normalization of the stress tensor and O(N) currents is fixed by
the fact that there are conserved charges with canonical normalization
associated to them. We show how to fix their normalization in appendix B.3.

#### Numerics

To solve the optimization problem (14) we resort to numerical methods. There
are two different complimentary approaches we use. One is the primal method in
which one explores the space of functions consistent with all our assumptions
and was put forward in Paulos:2016but ; Paulos:2017fhb . In practice one
writes ansätze for $S_{a}(s)$, $F_{\mathcal{O}}(s)$ and
$\rho_{\mathcal{O}}(s)$ that are compatible with the required analyticity and
crossing, and then one solves unitarity (10) numerically using SDPB Simmons-
Duffin:2015qma ; Landry:2019qug . We explain the numerical implementation in
appendix D.1. In the dual approach we introduce dual variables which play the
role of Lagrange multipliers for the constraints and solve an optimization
problem for these Cordova:2019lot ; Guerrieri:2020kcs ; He:2021eqn ;
Guerrieri:2021tak . In appendix C we show how to construct the dual
optimization problems and in D.2 how to implement them numerically. The dual
method has the important advantage of giving rigorous bounds, whereas we have
found easier to understand analytic properties of the optimal functions using
the primal approach.

#### Saturation of unitarity constraints

Before we present our results in the next section, let us mention a generic
feature of the optimal solutions. The inequalities coming from the unitarity
constraints $B_{a}\succeq 0$ in (10) are usually saturated. The implication is
that the spectral densities and central charges we find are given exactly by
their two-particle contributions. To see this, recall that the positive semi-
definiteness of $B_{a}$ is equivalent through Sylvester’s criterion to the
positivity of all principal minors. In particular the 2x2 upper left minor
yields

$|S_{a}|\leq 1\,.$ (15)

When above inequality is saturated we have the usual unitarity saturation for
the two-particle amplitude $S_{a}$ found in pure S-matrix bootstrap
calculations. The full 3x3 determinant gives

$\begin{gathered}\omega^{2}{S_{\bullet}^{*}}{F_{\Theta}}^{2}+\omega^{2}{S_{\bullet}}{F_{\Theta}^{*}}^{2}-2\omega^{2}|F_{\Theta}|^{2}+2\pi\rho_{\Theta}(1-|S_{\bullet}|^{2})\geq
0\,,\\\
\omega^{2}{S_{A}^{*}}{F_{J}}^{2}+\omega^{2}{S_{A}}{F_{J}^{*}}^{2}-2\omega^{2}|F_{J}|^{2}+\frac{s}{s-4m^{2}}2\pi\rho_{J}(1-|S_{A}|^{2})\geq
0\,,\end{gathered}$ (16)

which reduces to Watson’s equation $F(s)=S(s)F^{*}(s)$ Watson:1954uc  when
unitarity is saturated. The latter equation allows one to construct the form
factor from the S-matrix as explained in section 4. The positivity of the 2x2
lower right minor implies

$2\pi\rho_{\Theta}\geq\omega^{2}|F_{\Theta}|^{2},\qquad\frac{s}{s-4m^{2}}2\pi\rho_{J}\geq\omega^{2}|F_{J}|^{2}\,,$
(17)

which follows from inserting a complete set of states in (8), so that the
spectral density is given by the sum of all $n$-particle form factors.
Schematically we have
$\rho_{\mathcal{O}}``="|F_{\mathcal{O}}|^{2}+\sum_{n>2}|F^{(n)}_{\mathcal{O}}|^{2}=\rho^{(2)}_{\mathcal{O}}+\sum_{n>2}\rho^{(n)}_{\mathcal{O}}$
where $F^{(n)}_{\mathcal{O}}$ is the $n$-particle form factor and the
inequality (17) arises when truncating the right hand side to the two-particle
form factors. Therefore when we are solving our optimization problem and (17)
is saturated we are really finding the two particle contribution to the
spectral density only
$\rho^{\text{opt}}_{\mathcal{O}}=\rho^{(2)}_{\mathcal{O}}$.999Whereas the
equation $|S^{(2)}(s)|=1$ for all physical energies can be consistent in
integrable theories which have factorized scattering, the condition
$\rho_{\mathcal{O}}(s)=\rho^{(2)}_{\mathcal{O}}(s)$ is only true in free
theories. That is, the $n$-particle form factors are in general non trivial in
interacting theories, even in integrable ones. Similarly the central charges
can be decomposed into their n-particle contributions
$c=c^{(2)}+\sum_{n>2}c^{(n)}$. The bounds we find are again given by the two-
particle contribution only $c^{\text{opt}}=c^{(2)}$, which is of course a
valid bound since $c^{(n)}\geq 0$. All this traces back to the fact that we
are only considering two-particle in and out states in our setup.

The bounds found by solving the optimization problem (14) shed light on the
conformal theories that can flow to the gapped O(N) QFTs described in the
previous section. It should be noted that the $c$-minimization problem was
already considered in Karateev:2019ymz  for two lines in the monolith of
figure 1. As shown in the next section, in this work we optimize both $c$ and
$k$ for each point inside the monolith, which allow us to have a more fine-
grained picture of the QFTs and associated UV fixed points.

##  3 Bounds on stress tensor and current central charges

In this section we discuss the different minimization problems we solve.
First, in section 3.1, we focus on the $N=7$ monolith shown on figure 1 and
minimize the central charges $c$ and $k$ at each point. We find that for some
of the points at the boundary the minimum central charges diverge. Then in 3.2
we find the global minima for $c$ and $k$ as a function of the number of
flavors $N$. Finally in 3.3 we do a mixed bootstrap in which we fix one of the
central charges, $c$, and minimize the other one, $k$.

###  3.1 Minimum $c$ and $k$ on the O(N) monolith

Figure 3: Temperature plot for minimum central charge $c$ on the $N=7$
monolith. The dashed lines at the boundary are the amplitudes for which $c$
diverges, they include the periodic Yang-Baxter solutions. The blue cross
indicates the position of the global minimum. Results obtained from dual
numerics with parameters $N_{\text{max}}=50$ and $N_{\text{grid}}=100$.
Figure 4: Temperature plot for minimum current central charge $k$ on the $N=7$
monolith. The two diamonds indicate the points where $k$ diverges (free boson
and -constant solution). The pink cross marks the position of the global
minimum. Results obtained from dual numerics with parameters
$N_{\text{max}}=50$ and $N_{\text{grid}}=100$.

Our first application is to study the lower bound on the central charges $c$
and $k$ for the space of gapped, O(N) symmetric QFTs with no bound states
described in the previous section. When $N=7$, the numerical primal and dual
minimization procedures yield the global minima:

$\displaystyle c_{\text{min}}^{primal}$ $\displaystyle=2.916110...\,,\qquad
c_{\text{min}}^{dual}=2.756025...\,,$ (18) $\displaystyle
k_{\text{min}}^{primal}$ $\displaystyle=0.971155...\,,\qquad
k_{\text{min}}^{dual}=0.904049...\,.$ (19)

We then generalize this minimization procedure to every point covering the
monolith of figure 1, parametrized by the value of the amplitudes at the
crossing symmetric point $\sigma_{1}(2m^{2})$ and $\sigma_{2}(2m^{2})$. The
results are shown on the temperature plots of figures 3 and 4. In each plot we
mention the values of $N_{\text{max}}$ and $N_{\text{grid}}$, giving
respectively the size of the numerical ansatz and the number of points where
we evaluate the unitarity constraints.

A notable feature of the obtained bounds for $c$ is the rapid growth of the
minimum central charges near the boundary. Most importantly, we find no lower
bound for the dashed regions on figure 3, lying on the boundary of the
monolith between NLSM and Free theories. This suggests that no unitary CFT
with finite central charge $c$ can flow to a gapped phase described by those
S-matrices, which in particular include pYB. As we argue in section 4,
amplitudes which give rise to infinite central charges exhibit a particular
decrease of the phase shift at large energies.

As for $k$, the only places where there is no lower bound are free boson and
one of the constant solutions ($-$const). The fact that both models give the
same result is obvious from the equality of the amplitudes in the
antisymmetric channel $S_{A}=1$. One can understand the divergence in $k$ near
these points in different ways. One is with a direct computation from the free
boson Lagrangian as in E.2. Another way is to recall that the UV fixed point
we are dealing with describes $N$ free non-compact bosons $\phi$ which are not
proper primary fields since their two-point functions are logarithmic. The
O(N) current is $J\propto\phi\partial\phi$ so its two-point function has also
logarithmic divergences. On the other hand, the finiteness of $c$ can be
traced to the fact that the stress tensor is proportional to
$T\propto\partial\phi\partial\phi$ and $\partial\phi$ is a well defined
primary.

(a)

(b)

(c)

(d)

Figure 5: Minimum UV central charge $c$ for different sections of the monolith
connecting known amplitudes at the boundary, namely: (a) $\pm$ Free, (b) $\pm$
NLSM, (c) $\pm$ pYB and (d) $\pm$ constant (c.f. figure 1). The solid (dashed)
lines are the lower bounds obtained with the dual (primal) methods, both with
$N_{\text{max}}=50$ and $N_{\text{grid}}=300$. The shaded regions are the
allowed values for $c$. The endpoints are the two particle contribution to the
central charge computed with the analytic form factor bootstrap from the known
S-matrices as explained in A.

(a)

(b)

(c)

(d)

Figure 6: Minimum UV current central charge $k$ for different sections of the
monolith: (a) $\pm$ Free, (b) $\pm$ NLSM, (c) $\pm$ pYB and (d) $\pm$ constant
(see figure 1). The solid (dashed) lines are the lower bounds obtained with
the dual (primal) methods, both with $N_{\text{max}}=50$ and
$N_{\text{grid}}=300$. The shaded regions are the allowed values for $k$. The
endpoints are the analytic values computed in A.

In figures 5 and 6 we show various sections of the temperature plots
connecting special points on the boundary of the monolith, namely: Free, NLSM,
pYB and constant solutions. They are the same sections depicted by dashed
lines in figure 1, with the same color coding.

In the plots we show the bounds obtained by primal (dashed lines) and dual
numerics (solid). By construction the dual (rigorous) bounds are below the
primal ones, so that the optimal one lies between the two.101010As we increase
the numerical precision by considering larger ansätze (parametrized by
$N_{\text{max}}$) the gap between dual and primal bounds closes, as
exemplified later in figure 8. For most sections we find that the convergence
of the bounds close to the boundary is challenging. This is particularly
evident for the primal bounds, which might even stop converging before
reaching the actual boundary.111111This is what happens in the constant
section of figure 5 (d), where the dual endpoint seems to be above the primal
result, but the primal bound is actually infinity at that point. In many cases
one can trace back this difficulty to the rapid growth of the minimum central
charges as one gets close to the boundary from a radial direction. However, a
clear understanding of which parameters in the numerical ansäzte are most
important for good numerical convergence eludes us at this point. In this
sense we see the clear advantage of using both primal and dual numerics to
bracket the optimal bound.

The endpoints of these sections can be computed from the analytic S-matrices
as explained in appendix A (see table 1 for explicit values). Note for
instance that the endpoints of the free and constant sections are related
since $S_{\bullet}^{\pm\text{Free}}=S_{\bullet}^{\pm\text{const}}$ and
$S_{A}^{\pm\text{Free}}=-S_{A}^{\pm\text{const}}$ (see appendix A.1 for all
the endpoint amplitudes).

###  3.2 Minimum $c$ and $k$ as a function of $N$

Now we consider the problem of finding the global minimal central charges $c$
and $k$ as we vary the number of flavors $N$ (i.e. without fixing
$S_{a}(s=2m^{2})$). The results of our optimization procedures are shown on
figure 7. At large $N$, the bound for the central charge $c$ becomes
linear.121212It is expected in general that the central charge $c$ scales with
$N$ since we have states in an $N$-dimensional representation, with each
component contributing to the stress tensor. In $d=2$ it can also be argued
from Cardy’s formula Cardy:1986ie . The slope is

$\displaystyle c_{\text{min}}^{primal}(N)\sim 0.3221N\,,\qquad
c_{\text{min}}^{dual}(N)\sim 0.2925N\,,$ (20)

which is compatible from the previous primal results of Karateev:2019ymz .
Instead for $k$ the bounds are saturated by a constant

$\displaystyle k_{\text{min}}^{primal}(N)\sim 1.0250\,,\qquad
k_{\text{min}}^{dual}(N)\sim 0.9513\,.$ (21)

(a)

(b)

Figure 7: Global minima for central charges $c$ and $k$ as a function of the
number of flavors $N$. The dual (primal) bounds are given by the solid
(dashed) lines. The allowed values for the central charges appear in the
shaded regions. In (a) we see the linear dependence with $N$ as in (20). In
(b) the current central charge $k$ remains constant at large $N$ and close to
the free fermion value $k=1$. Both primal and dual numerics were done with
$N_{\text{max}}=50$ and $N_{\text{grid}}=300$.

The optimal bound for $k$ at large $N$ is compatible with 1, which is the
value taken by the free fermion. The rigorous dual bound sits slightly below
1, but since the numerics only consider 2 particles contributions, we expect
that this bound will increase when including $n>3$ contributions (see the
discussion below (17)). Therefore we believe that the true lower bound should
be saturated by the free fermion.

###  3.3 Bounds on the $(c,k)$ plane

Our last application is to consider the lower bound on the current central
charge $k$ for fixed central charge $c$. Analyzing both the stress tensor and
O(N) currents at the same time gives us a more detailed idea of which CFTs can
flow to the gapped theories described by the monolith.

In figure 8 we show these bounds for $N=7$. In 8(a) we see the numerical
primal and dual bounds for different size of the ansätze $N_{\text{max}}$
along with a $N_{\text{max}}\rightarrow\infty$ extrapolation. In 8(b) we show
the path followed by the lower bound curve inside the monolith.

(a)

(b)

Figure 8: (a) Allowed values for $c$ and $k$ central charges for $N=7$. The
red (blue) points are obtained using dual (primal) numerics for different
$N_{\text{max}}$. The black line is a simple linear extrapolation in
$1/N_{\text{max}}$ and the purple dashed line are the values obtained with the
Sugawara construction. The curve starts at the global $c$ minimum (blue cross
outside plot) and ends at the global $k$ minimum (pink cross). We expect that
compact unitary CFTs lie to the right of this Sugawara line. (b) The path
followed by the curve in (a) inside the monolith.

Let us discuss known unitary conformal theories which might live in the
allowed region in figure 8. The purple dashed line is given by the following
equation

$c_{\text{Suga}}(k)=\frac{k\,N(N-1)/2}{k+N-2}\,.$ (22)

This relation between $c$ and $k$ follows from the Sugawara construction which
we briefly review next. Recall that the conservation equation for the current
of a global symmetry group $g$ reads
$\partial_{z}\bar{J}+\partial_{\bar{z}}J=0$. If the conformal theory is
compact (i.e. with discrete spectrum) it is expected that the holomorphic and
antiholomorphic currents are separately conserved
$\partial_{z}\bar{J}=\partial_{\bar{z}}J=0$. In this case the OPE of the
currents defines an affine Kac-Moody algebra whose level is given by $k$

$J^{a}(z)J^{b}(w)\sim\frac{k\delta^{ab}}{(z-w)^{2}}+\frac{if_{abc}}{z-w}J^{c}(w)\,,$
(23)

with the group indices $a$ taking values $a=1,\ldots,\text{dim}g$ and
$f_{abc}$ are the structure constants of the algebra. Importantly, one can
construct the stress-energy tensor of the theory from the normal ordered
product of these currents

$T(z)=\frac{1}{k+h^{\vee}}\sum_{a}:J^{a}(z)J^{a}(z):\,,$ (24)

where the prefactor is fixed by symmetry and $h^{\vee}$ is the dual Coxeter
number of the group. The above construction is known as Sugawara construction
and it gives a relation between the central charge $c$ and the level $k$ of
the algebra

$c_{\text{Suga}}(\hat{g}_{k})=\frac{k\,\text{dim}\,g}{k+h^{\vee}}\,,$ (25)

which reduces to (22) for $g=so(N)$. Notably this affine algebra structure can
be realized by the Wess-Zumino-Witten (WZW) models, whose Lagrangian is
defined by a nonlinear sigma model and a topological term. These models have
also free field representations in terms of free fermions or free bosons and
ghosts. For instance, for $so(N)$ at level one we have $N$ real free fermions
giving $c=N/2$. This is indeed the point $(c,k)=(7/2,1)$ being approached from
different directions by our primal and dual bounds in figure 8.131313Of course
we see as well the amplitude, form factor and spectral density approaching
those of the free fermion.

So far we have looked at possible models lying exactly at the Sugawara line in
figure 8. Starting from this line one can populate the right hand side of the
plot by taking the direct sum of WZW models and CFTs whose content is given by
O(N) singlets. In this way one increases the value of the central charge $c$
without modifying $k$. Note that a priori there is no reason to assume our
gapped theory should arise from a compact CFT. As such, the allowed region of
figure 8 could include non compact CFTs, in particular the region to the left
of Sugawara line. In fact, the simplest amplitude we can write $S_{a}=1$ for
free boson leads to a CFT lying in this left region for $N>3$ since
$(c,k)=(N,\infty)$.

Finally there is also a small region below the Sugawara line with $k<1$. As
explained before, we believe that once we include multi-particle contributions
into the bootstrap this region will no longer be allowed, so that the true
minimum for $k$ coincides with the free fermion value.141414We expect this to
be true for $N\geq 3$ since for $N=2$ the minimum $k$ sits at the boundary of
the monolith and corresponds to one of the $\gamma\rightarrow\infty$ sine-
Gordon amplitudes.

##  4 Infinite central charge S-matrices

Given that some regions on the monolith have infinite central charges, a
natural question to ask is what characterizes the amplitudes which lead to
divergent sum rules? In this section we use analytic bootstrap methods to
derive conditions on the $2\to 2$ amplitudes giving rise to infinite central
charges.

Let us define the dangerous and safe points as the positions $s_{0}$ in the
physical region $s\geq 4m^{2}$ where the amplitude obeys

dangerous: $\displaystyle s_{0}\text{ such that }S(s_{0})=-1\text{ and
}\text{Im }S^{\prime}(s_{0})>0\,,$ (26) safe: $\displaystyle s_{0}\text{ such
that }S(s_{0})=-1\text{ and }\text{Im }S^{\prime}(s_{0})<0\,.$ (27)

Denoting by $\textbf{d}_{a}$ ($\textbf{s}_{a}$) the number of dangerous (safe)
points in the representation $a$, the conditions leading to infinite $c$ or
$k$ central charges read

$\displaystyle\textbf{d}_{\bullet}\,-\textbf{s}_{\bullet}\,>1\implies
c\rightarrow\infty\,,$ (28)
$\displaystyle\textbf{d}_{A}-\textbf{s}_{A}>0\implies k\rightarrow\infty\,.$
(29)

These conditions offer a very simple explanation for why the dashed regions at
boundary of the monolith in figure 3 have infinite $c$ but finite $k$. As
illustrated in figure 9 for periodic Yang-Baxter, the singlet channel has
infinite dangerous points whereas the antisymmetric amplitude has infinite
safe points.

(a)

(b)

Figure 9: Periodic Yang-Baxter amplitude in the (a) singlet and (b)
antisymmetric channels for physical kinematics in the rapidity variable
$\theta$ (related to $s$ as $s=4m^{2}\cosh(\theta/2)$). We have set $N=2000$
which gives a period in $\theta$ of $\tau^{\text{pYB}}\approx 2.6$. In (a) we
see the singlet channel having infinite dangerous points highlighted by the
vertical red lines leading to $c\rightarrow\infty$. Instead, for the
antisymmetric amplitude in (b) we have only safe points marked with blue
vertical lines and $k$ finite.

In the rest of this section we first review in 4.1 how to construct the form
factors and spectral densities starting from a known S-matrix, and then in 4.2
we present an argument for the conditions in (28) and (29). In 4.3 we show how
dual and primal numerics behave close to a point on the monolith with infinite
central charge and how these extremal S-matrices with large but finite central
charges smartly evade these criteria.

###  4.1 Review of analytic Form Factor Bootstrap

There is a well-known procedure to compute n-particles form factors in
integrable theories known as Form Factor Bootstrap (see for instance
Smirnov:1992vz  or Omnes:1958hv ). Here we will restrict to the two-particle
form factor and its contribution to the spectral density and follow this
procedure also for non-integrable amplitudes. The main ingredient is Watson’s
equation $F(s)=S(s)F^{*}(s)$ which gives a direct relation between the two
particle form factor and amplitude for energies where there is no particle
production (for pedagogical purposes we ignore the representation indices). As
reviewed earlier, the optimization problem we are considering leads to
functions saturating the inequality constraints, so that we have unitarity
saturation and Watson’s equation for all energies. Therefore, even for non
integrable theories, we can perform the optimization (14) analytically if we
know the amplitude.

We start by writing the two particle form factor as

$F(s)=B(s)\,e^{\alpha(s)-\alpha(0)}\,,$ (30)

where $B(s)$ is a real analytic function and $\alpha(s)$ takes care of the
discontinuity at $s\in[4m^{2},\infty)$. We have chosen the difference
$\alpha(s)-\alpha(0)$ in the exponent so that the normalization of the form
factor as in (115) and (122) is fixed by $B(0)$. Given a unitarity saturating
amplitude, we can write it as a pure phase $S(s)=e^{i\varphi(s)}$ for physical
energies $s\in[4m^{2},\infty)$. Using Watson’s equation, we see that the
imaginary part of $\alpha(s)$ is proportional to the argument of $S(s)$:

$F(s)=S(s)F^{*}(s)\implies\text{Im}\,\alpha(s)=\varphi(s)/2=-i\log
S(s)/2\,,\qquad s\in[4m^{2},\infty)\,.$ (31)

Now $\alpha(s)$ should not grow as $s\to\infty$ otherwise $F(s)$ in equation
(30) would have an essential singularity. We therefore assume $\alpha(s)$ to
obey a once-subtracted dispersion relation,

$\alpha(s)-\alpha(0)=\frac{s}{\pi}\int\limits_{4m^{2}}^{\infty}ds^{\prime}\,\frac{\text{Im}\,\alpha(s^{\prime})}{s^{\prime}(s^{\prime}-s)}\,.$
(32)

which via equation (31) ‘solves’ the form factor in terms of the amplitude up
to the freedom in the function $B(s)$ which drops out of Watson’s equation
(31).151515This is a similar trick to the one often used in integrable models,
where one finds an integral representation in the rapidity $\theta$ for the
S-matrix first
$S(\theta)=\int\limits_{0}^{\infty}\frac{dt}{t}f(t)\sinh\left(\frac{t\theta}{i\pi}\right)$
and then writes the ”minimum” two-particle form factor as
$F_{\text{min}}(\theta)=\mathcal{N}\int\limits_{0}^{\infty}\frac{dt}{t}f(t)\,\csch
t\,\sin^{2}\left(\frac{t\theta}{2\pi}\right)$ (see also appendix A). In our
notation the latter is the exponential factor in (30)
$F_{\text{min}}(\theta)=e^{\alpha(\theta)-\alpha(0)}$.

We may then insert into the sum rules for $c$ and $k$ equations (12) and (13),
making use of the relation (17) between $\rho(s)$ and $F(s)$ when unitarity is
saturated in

$\displaystyle
c=3\int_{4m^{2}}^{\infty}ds\,\frac{|F_{\Theta}(s)|^{2}}{s^{2}\sqrt{s}\sqrt{s-4m^{2}}},\qquad
k=\frac{1}{8}\int_{4m^{2}}^{\infty}ds\,\frac{\sqrt{s-4m^{2}}}{s\sqrt{s}}|F_{J}(s)|^{2}.$
(33)

#### Free theory examples

To illustrate how to fix $F(s)$, let us show how to compute the free fermion
and boson form factors for both $c$ and $k$ (for direct computation from the
free Lagrangians see appendices E.1,E.2). Starting from free boson $(+)$ we
have (in all representations) $S^{+}(s)=1\implies\varphi=0$, which sets
$\alpha^{+}=0$, so all we have left to fix is $B^{+}(s)$. In principle this
could be any polynomial with fixed normalization $B(0)=F(0)$. However, as we
will now see, the degree of this polynomial is bounded from above in order for
the central charge sum rules to converge. Take for instance $c$ in (12)

$c^{+}=12\pi\int\limits_{4m^{2}}^{\infty}ds\,\frac{\rho^{+}_{\Theta}(s)}{s^{2}}=3\int\limits_{4m^{2}}^{\infty}ds\,\frac{{B^{+}_{\bullet}}^{2}(s)}{\sqrt{s}\sqrt{s-4m^{2}}\,s^{2}}\,.$
(34)

In order for the integral to converge at infinity the degree of such
polynomial should be zero. Hence
$B_{\bullet}^{+}(s)=B_{\bullet}^{+}(0)=-2\sqrt{N}m^{2}$ which leads to the
expected central charge $c^{+}=N$ and matches the result from E.2.

From (13) for $k$ we have instead

$k^{+}=\frac{\pi}{2}\int\limits_{4m^{2}}^{\infty}ds\,\rho^{+}_{J}(s)=\frac{1}{8}\int\limits_{4m^{2}}^{\infty}ds\,\frac{\sqrt{s-4m^{2}}}{s^{3/2}}\,{B^{+}_{A}}^{2}(s)\,.$
(35)

Note that if $B_{A}^{+}(s)$ is a constant ($B_{A}^{+}(s)=B_{A}^{+}(0)=2$) we
get a divergence since the integral does not decay fast enough at infinity.
Moreover, considering $B_{A}(s)$ a higher degree polynomial would only make
the divergence worse. Therefore we have $k^{+}=\infty$ in accordance to direct
computation from the Lagrangian in E.2

Now let us discuss the free fermion case. For $S^{-}=e^{i\varphi^{-}}=-1$ we
have $\varphi^{-}=\pi$ (we choose the usual principal branch
$\varphi\in(-\pi,\pi]$). This leads to $e^{\alpha^{-}}=(1-s/4m^{2})^{-1/2}$.
For the central charge $c$ to be finite we must have
$B_{\bullet}^{-}(s)=\sqrt{N}(4m^{2}-s)/2$ so that
${|F_{\bullet}^{-}}(s)|^{2}=m^{2}N(s-4m^{2})$ and we find the expected result,

$c^{-}=3\int\limits_{4m^{2}}^{\infty}ds\,\frac{m^{2}N(s-4m^{2})}{\sqrt{s}\sqrt{s-4m^{2}}\,s^{2}}=\frac{N}{2}\,.$
(36)

For the current central charge $k$ instead we have $B_{A}^{-}(s)=-2$ and we
find

$k^{-}=\frac{1}{8}\int\limits_{4m^{2}}^{\infty}ds\frac{16m^{2}}{s^{3/2}\sqrt{s-4m^{2}}}=1\,,$
(37)

which matches the computation in appendix E.1.

###  4.2 Derivation of conditions on $S(s)$

Now that we have understood how to fix the two-particle form factor starting
from the amplitude in some simple examples, let us try to derive the
conditions (28) and (29).

Recall that in order to solve Watson’s equation we have
$\mathrm{Im}\,\alpha(s)=-\frac{i}{2}\log S(s)$ as in (30). We take the usual
convention in which the logarithm has the branch cut along the negative real
axis so that whenever $S(s_{0})=-1$ we go on top of the branch cut. Close to
this point we can expand161616Smoothness of $\mathrm{Im}\,S(s)$ for $s>4m^{2}$
is guaranteed due to elastic unitarity.

$\displaystyle S(s\sim
s_{0})=-1+i(s-s_{0})\,\mathrm{Im}\,S^{\prime}(s_{0})+\dots$ (38)

Depending on the sign of $\mathrm{Im}\,S^{\prime}(s_{0})$, we either cross the
branch cut from below or above. If $\mathrm{Im}\,S^{\prime}(s_{0})>0$ the
branch cut is crossed from below, which we denote as a dangerous point

$\displaystyle s_{0}\text{ {dangerous}
point:}\qquad\mathrm{Log}\,S(s_{0}^{+})=\mathrm{Log}\,S(s_{0}^{-})-2\pi i\,.$
(39)

if instead $\mathrm{Im}\,S^{\prime}(s_{0})<0$ the branch cut is crossed from
above and we have

$\displaystyle s_{0}\text{ {safe}
point:}\qquad\mathrm{Log}\,S(s_{0}^{+})=\mathrm{Log}\,S(s_{0}^{-})+2\pi i,$
(40)

where the ’$\mathrm{Log}\,$’ is the principal branch logarithm whose imaginary
part is restricted to $(-\pi,\pi]$. Hence, as we take $s\to\infty$, and we go
over $\mathbf{s}$ safe points and $\mathbf{d}$ dangerous points, we must have
$\mathrm{Log}\,S(s)\to\mathrm{Log}\,S(s)-2\pi i(\mathbf{d}-\mathbf{s})$ or

$\displaystyle\mathrm{Im}\,\alpha(s)=\mathrm{Im}\,\alpha_{0}(s)-\pi(\mathbf{d}-\mathbf{s}),\qquad
s\to\infty\,,$ (41)

where $\mathrm{Im}\,\alpha_{0}(s)\in[-\pi/2,\pi/2)$ is on the principal branch
of the logarithm.

Plugging into the dispersion relation (32) we then have

$\displaystyle\alpha(s)=\alpha_{0}(s)+(\mathbf{d}-\mathbf{s})\log(-s)+\text{constant},\qquad
s\to\infty.$ (42)

Similarly for $\alpha_{0}(s)$ given the upper bound
$\imaginary\alpha_{0}(s)\leq\frac{\pi}{2}$ we observe the lower
bound171717This bound can be shown to apply in the case without flavour
symmetry where $\imaginary\alpha_{0}(s)$ is smooth at infinity and admits a
Taylor series (see appendix A of Correia:2022dyp ). In this case we have
$\displaystyle\imaginary\alpha_{0}(s)=\sum_{n=0}^{\infty}\frac{a_{n}}{s^{n}},\qquad
s\to\infty,$ (43) and also
$\displaystyle\imaginary\alpha_{0}(s)\leq\frac{\pi}{2}\implies
a_{0}\leq\frac{\pi}{2}\,,$ (44) Plugging into the dispersion relation (32) we
find  $\displaystyle\alpha_{0}(s)=-\frac{a_{0}}{\pi}\log(-s),\qquad
s\to\infty$ (45) Taking the real part and making use of (44) yields the bound
on the real part in equation (46). See e.g. PhysRev.137.B720 ; Bronzan:1974jh
; 1987CzJPh..37..297F  for similar arguments. With O(N) symmetry the behavior
(43) is not necessarily true. For example for pYB we have
$\imaginary\alpha_{0}(s)\to-\frac{\mathrm{arccosh}(N/2)}{2\pi}\log(s)$, but we
still observe that the bound (46) applies. It would be interesting to extend
this proof for generic O(N) amplitudes.

$\displaystyle\real\alpha_{0}(s)\gtrsim-\frac{1}{2}\log(s),\qquad s\to\infty.$
(46)

Finally, since $B(s)$ is a polynomial, we find the following lower bound on
the growth of the form factor $F(s)=B(s)\,e^{\alpha(s)-\alpha(0)}$,

$\displaystyle|F(s)|\gtrsim s^{\mathbf{d}-\mathbf{s}-\frac{1}{2}},\qquad
s\to\infty.$ (47)

Plugging the above behaviour into the sum rules for $c$ and $k$ in equation
(33) and demanding that the integrand falls off faster than $1/s$ as
$s\to\infty$ leads to the conditions (28) and (29).

(a)

(b)

Figure 10: (a) Example of dangerous and safe points for an elastic amplitude
in physical kinematics. Whenever $S(s_{0})=-1$ we have a dangerous (red) or
safe (blue) point according to (27). (b) At each of these points, the
imaginary part of $\alpha(s)$ crosses $\pi/2$ (mod $\pi$). As explained in the
main text, depending on the limiting value $\imaginary\alpha_{\infty}$, the
form factor either grows or decays with $s$. As a consequence, the sum rules
for $k$ and $c$ diverge if $\imaginary\alpha_{\infty}\leq 0$ and
$\imaginary\alpha_{\infty}\leq\pi$, respectively. The solid brown line is
$\imaginary\alpha(s)$ corresponding to the amplitude in (a). We also show
curves for free boson (black solid) with $k\to\infty$ and $c$ finite, free
fermion (black dashed) with both $k$ and $c$ finite, and periodic Yang-Baxter
in the singlet channel with $c\to\infty$.

In figure 10 we illustrate these dangerous/safe points in an example. The
conditions (28) and (29) make up for a quick test as to why the dashed regions
on the monolith of figure 3 have infinite $c$ and suggest why they have finite
$k$. Counting the number of dangerous versus safe points provides a practical
bookkeeping tool to understand which amplitudes lead to infinite central
charges. However, it is important to note that the conditions (28) and (29)
are only sufficient and that some limiting cases may have infinite central
charges without having enough dangerous points. One clear example is the free
boson for which $k=\infty$ and has $\mathbf{d}_{A}-\mathbf{s}_{A}=0$.

Let us derive a more general criteria for the divergence of the central
charges which also includes the free boson. Suppose the amplitude is such that
$\imaginary\alpha(s)\to\imaginary\alpha_{\infty}$ as $s\to\infty$. Repeating
the same steps as before we now have

$\displaystyle\alpha(s)\to-\frac{\imaginary\alpha_{\infty}}{\pi}\log(-s)\,,\qquad
s\to\infty\,,$ (48)

and therefore $|F(s)|\to B(s)\,s^{-\imaginary\alpha_{\infty}/\pi}$.

Requiring that the integrand in the sum rules (33) falls off faster than $1/s$
puts again bounds on $\imaginary\alpha_{\infty}$. These read

$\displaystyle\imaginary\alpha_{\infty}\leq-\pi\implies c\to\infty\,,$ (49)
$\displaystyle\imaginary\alpha_{\infty}\leq 0\implies k\to\infty\,.$ (50)

The above rules now explain the free boson case where
$\imaginary\alpha_{\infty}=0\implies k\to\infty$. They are of course
compatible with the previous conditions (28) and (29), which imply
$\imaginary\alpha_{\infty}\leq-3\pi/2$ for the first and
$\imaginary\alpha_{\infty}\leq-\pi/2$ for the latter. In figure 10(b) we show
the different regions in $\imaginary\alpha$ leading to divergent central
charges along with various examples.

To finish this section, note that if there are bound states in the theory they
would show up as poles in the form factor below $s<4m^{2}$. These are
therefore included into $B(s)$ that can now decay as $B(s)\gtrsim s^{-n_{B}}$,
where $n_{B}$ is the number of bound states. So the condition for diverging
central charge (28) changes to

$\displaystyle\mathbf{d}_{\bullet}-\mathbf{s}_{\bullet}-n_{B}>1\implies
c\to\infty,$ (51) $\displaystyle\mathbf{d}_{A}-\mathbf{s}_{A}-n_{B}>0\implies
k\to\infty$ (52)

where for $c$ it is understood that $n_{B}$ is the number of bound states in
the singlet channel while for $k$, $n_{B}$ is the number of bound states in
the antisymmetric channel.

Interestingly, we appear to observe that (see also discussion at the end) that
in the absence of O(N) symmetry where the CDD solution for the S-matrix holds
exactly, the above conditions are never satisfied, as CDD zeros only give rise
to safe points, whereas CDD poles only give rise to dangerous points whose
effect is canceled out by the fact that a bound state pole behaves as a safe
point in the conditions above. This seems to explain why diverging central
charges were not observed e.g. in the work Correia:2022dyp  which only dealt
with flavourless scattering.

###  4.3 An example from numerics

It is interesting to understand the mechanisms used by the dual and primal
numerics to produce large but finite central charges without contradicting the
conditions (28) and (29) derived above. For instance, in the vicinity of the
regions on the monolith in which $c$ diverges (dashed lines in figure 3) there
is a tension between having finite central charge, i.e. not too many dangerous
points, and S-matrices close to those at the boundary with an infinite number
of them. As we will see explicitly in an example, this tension gets resolved
by the emergence of new safe points in physical kinematics, related to
resonances in the complex $s$ or $\theta$ plane.

Figure 11:  Dual numerical amplitudes close to periodic Yang-Baxter point for
physical values of the rapidity $\theta$ and $N=2000$. In (a) we see an
amplitude similar to pYB which however has a finite central charge $c\approx
5.38N$ due to new –as compared to pYB– safe points. In (b) we zoom into the
first safe point in the singlet channel around $\theta=3.18$, marked with a
vertical blue line. Numerics done with $N_{\text{grid}}=400$ and
$N_{\text{max}}=240$.  Figure 12:  Primal numerical amplitudes close to
periodic Yang-Baxter point for physical values of the rapidity $\theta$ and
$N=2000$. The central charge minimization gives a large but finite value
$c\approx 7.06N$. In (a) we see an amplitude similar to pYB with an extra
’jump’ in the singlet channel around $\theta=3.18$. In (b) we zoom in around
this point for which the amplitude exhibits some inelasticity that gets
localized as we increase $N_{\text{max}}$. Numerics done with
$N_{\text{grid}}=400$ and $N_{\text{max}}=240$ for (a) and the different
specified values in (b).

First, a word of caution about numerics. As it is very often the case in
S-matrix bootstrap problems, the optimization objectives ($c$ and $k$ in our
case) are not very sensitive to changes in the large energy behaviour of the
observables. In particular, the integral sum rule for $c$ (12) has a $s^{-2}$
damping factor which makes it very difficult to extract reliable information
about the optimal functions at large energies. This is usually not too
worrisome since the objective rapidly converges when increasing
$N_{\text{max}}$. Moreover, when working with both primal and dual approaches
we can get a good idea to how far we are from the true optimal bound. As for
the observables like amplitudes, at small enough energies we see saturation of
unitarity for the primal observables, so that primal and dual functions agree
and describe the true optimal function in this range of energies.

For concreteness we focus on $c$-minimization for a point close to close to
pYB in the $\sigma_{2}=0$ section. We choose a large number of flavors
$N=2000$ so that we have various dangerous points at low enough energies and
fix $\sigma_{1}=0.02227$.181818Recall that the pYB period in rapidity is
$\tau_{\text{pYB}}=2\pi^{2}/\text{arccosh}(N/2)$. In figures 11 and 12 we show
the amplitudes obtained from the dual and primal numerics, respectively. The
left panels show an amplitude very similar to pYB (c.f. figure 9), except for
some new structure in the singlet channel close to $\theta=3.18$. The
minimization gives a finite central charge of $c\approx 10,753\approx 5.38N$
for the dual and $c\approx 13,043\approx 6.52N$ for the primal.

As evident in figures 11 and 12, the dual and primal numerics evade conditions
(28) with different mechanisms. Namely, the former clearly shows the existence
of new safe points whereas the latter does not saturate unitarity at all
energies and therefore is in no contradiction with (28). Nonetheless, both
mechanisms point at the same physics.

In the dual approach, the amplitudes are constructed from the dual variables
assuming unitarity saturation (see appendix C), so that the jump in the
singlet amplitude close to $\theta=3.18$ is a true safe point (see figure
11(b)). In the primal numerics what we see instead is that at finite $N_{max}$
unitarity is not saturated at high energies so that the assumptions made to
derive (28) do not hold. In figure 12(a) we see a similar jump of the singlet
amplitude around $\theta=3.18$. However, the jump is not a clear safe point
since unitarity is not saturated around this value, as figure 12(b) shows.
This inelasticity is a finite $N_{\text{max}}$ artifact. As we increase the
size of our ansatz the inelasticity is sharper and more localized around a
given point, which often signals the presence of a resonance close to the real
line.

Figure 13:  Dual numerical amplitude in the complex rapidity plane featuring a
resonance close to the real line. This resonance is responsible for the safe
point of figure 11.

The resonance is apparent as we plot the dual amplitude in the complex
rapidity $\theta$ plane as in figure 13.191919In principle the dual amplitudes
are only defined on the real line, but one can write a dispersion relation
from the imaginary part and the known subtraction constants. There we see a
definite zero sitting very close to the real line, which explains the trouble
of the primal numerics to correctly reproduce the resonance. At higher
energies where the numerics are less reliable we expect the periodic nature of
the amplitude to continue, with many new safe points and resonances countering
the infinite dangerous points.

In this example we see how the existence of new safe points, necessary to
achieve a finite central charge, have a direct physical implication of
including more resonances in the theory.

##  5 Discussion

In this work we studied gapped, O(N) symmetric QFTs using numerical and
analytical bootstrap approaches focused on $2\rightarrow 2$ scattering
amplitudes and two-particle form factors of the trace of the stress tensor
$\Theta(x)$ and of the O(N) conserved currents $J_{A}^{\mu}(x)$. We now
summarize our results and discuss some implications and future directions.

#### Minimum central charges and WZW models.

In section 3.1 we found the minimum of the central charges $c$ and $k$ of the
UV CFT across the O(N) monolith space of S-matrices. The results are shown in
figures 3 and 4 for $N=7$ (we expect similar plots for any integer $N>2$). In
figures 5 and 6 we plot the minimum $c$ and $k$ across several sections
connecting special points of the O(N) monolith. We observe rapid growth of the
central charges in several directions towards the boundary of the O(N)
monolith, with $c$ becoming infinite over the majority of the boundary. This
severely constrains the CFTs from which these putative QFTs can flow from
(note that $N$ is _finite_). See further discussion below.

In section 3.2 we studied how the global minimum for $c$ and $k$ change with
$N$. As shown in figure 7 we find that $c$ grows linearly while $k$ goes to a
constant as $N\to\infty$. Another observation is that the global minimum
central charges $c$ and $k$ for $N>2$ are not on the boundary but inside the
bulk of the O(N) monolith. This means that the S-matrix which minimizes the
central charge is _not_ the same S-matrix that extremizes some coupling in
S-matrix space.202020At least none of the ‘quartic’ couplings that span the
O(N) monolith. This is in sharp contrast with e.g. the single scalar exchange
bootstrap of Correia:2022dyp  where the extremal S-matrices coincided (and
allowed for an analytical derivation of the minimum $c$).

In section 3.3 we studied the allowed space of central charges $c$ and $k$
given the bootstrap constraints on the S-matrix, form factors and spectral
densities. The result is shown in figure 8. We find that the Sugawara
construction realized by WZW models lies insides the bounds, but cannot cover
the full allowed space. As always, it is entirely possible that adding more
bootstrap constraints would rule out this extra space. However, it is
important to note that the Sugawara construction assumes conservation of not
only the current but also the individual holomorphic and antiholomorphic
components. It would be interesting to see how this constraint could be
included in our bootstrap setup or to repeat the same exercise of bounding $c$
and $k$ but via the numerical conformal bootstrap Chester:2019wfx , which to
our knowledge has not been done yet. It would be interesting to compare these
two approaches as it would give us an understanding of how constraining really
are the S-matrix principles in the UV, especially with the extra assumptions
taken here on the RG flow (in that we are restricting ourselves to CFTs which
admit O(N) preserving deformations).

#### Infinite central charges, dangerous points and resonances.

As already mentioned, we observe that the minimum central charge $c$ diverges
on most of the boundary of the O(N) monolith (see figure 3). The corresponding
extremal S-matrices share the following feature: they exhibit the presence of
dangerous points on the physical region where $S(s_{0})=-1$ and
$\mathrm{Im}\,S^{\prime}(s_{0})>0$. In section 4 we argued that if the
amplitudes have enough dangerous points, then the associated central charges
diverge. This criterion is satisfied by the pYB S-matrices, which are located
on this portion of the boundary. Curiously, the divergent central charge is
not a consequence of the periodicity in the amplitude, as amplitudes can be
periodic and lack dangerous points, but to how the phase shift decreases at
large energies (see figure 10) which is guaranteed by the existence of a
sufficient number of dangerous points.

Moving away from pYB slightly into bulk of the O(N) monolith, where the
central charge $c$ is large but finite, we observe that the extremal S-matrix
deviates from pYB around the second dangerous point (which is located at
relatively high energies). The S-matrix coming from primal optimization
avoided the creation of dangerous points by generating some inelasticity
around that location, see figure 12. The S-matrix coming from dual
optimization, which is elastic by construction, instead creates a safe point
(where $\mathrm{Im}\,S^{\prime}(s_{0})$ has the opposite sign), see figure 11.
This has repercussions on the analytic structure, as seen in figure 13, where
a close-by resonance is observed. There seems to be a relation between the
presence of dangerous and safe points on physical kinematics and the existence
of bound states and resonances on the complex plane. In the case of scattering
without flavour, where the CDD solution is exact, we see that CDD zeros (i.e.
resonances) only lead to safe points, and that bound states are needed in
order to have dangerous points.212121This was tested with up to six CDD zeros
arbitrarily located in the complex plane. This suggests that the necessary
condition (28) for infinite central charges is never met in the case of
scattering without internal symmetry, which is consistent with the findings in
Correia:2022dyp . In the O(N) case, on the other hand, dangerous points can be
present without a corresponding bound state, which leads to the observed
diverging central charges. Moreover, regardless of the existence of safe
points, for the O(N) S-matrices an infinite amount of resonances across the
infinitely many sheets is generically present due to crossing and unitarity
Cordova:2018uop ; Cordova:2019lot . It thus remains to be seen if a sharp
connection between resonances/bound states and safe/dangerous points can be
made.

#### Inserting more of the UV and targeting specific deformations.

In this work we did not target a specific theory whose UV CFT is known. On the
contrary, we constrained the class of CFTs from which a generic gapped theory
in the IR with O(N) symmetry would flow from. Possible theories which could be
targeted are deformation of WZW models.222222For instance, it would be
interesting to understand the connection to the integrable deformation known
as lambda model and the RG flows studied in Appadu:2017bnv . In this case $c$
and $k$ would be known from the Sugawara construction in equation (25) and
could be fixed in the bootstrap setup. Then, the space of couplings of the
S-matrix and form factors in the IR could be carved out, similarly to how
Ising Field Theory was targeted in Correia:2022dyp .

A still open problem is of course how to include more information about the UV
CFT, for example the dimensions of the deforming operators that give rise to
the RG flow. Could we put a bound on these dimensions using the S-matrix
bootstrap? The practical aspect of the central charges is that they obey sum
rules, which are _integral_ relations directly on the spectral density. There
exist sum rules for the scaling dimension of operators DELFINO1996327 , but
they involve their vacuum expectation value appearing as a new parameter. A
recent development He:2023lyy  targeting gauge theories, have used so-called
QCD sum rules, which relates the very high energy behavior of the two point
function at some large $s_{0}$ to an integral over the spectral
density.232323There is however a slight inconvenience in that the choice of
$s_{0}$ is somewhat arbitrary and it seems hard to estimate the error (see
e.g. Caron-Huot:2023tpw ). This provides a new way to introduce information
about the CFT and the deforming operator(s) using form factor perturbation
theory.

#### Non-unitarity CFTs, loop models and $N<2$.

There are various hints that the space of theories spanned by the S-matrix
bootstrap should go outside the usual UV complete paradigm, where we have a
unitary CFT in the UV. The first hint we already encountered is the free boson
case, where the central charge $k$ diverges due to the logarithmic nature of
free massless non-compact bosons. A more interesting example comes from the
well studied O(N) loop model, which gives rise to a family of conformal
theories in the window $-2\leq N\leq 2$. For $N=1$ we recover the critical
Ising model and $N=2$ is the Kosterlitz-Thouless fixed point. For generic non-
integer $N$ the conformal theories are logarithmic (see Gorbenko:2020xya  and
references therein).242424It would be interesting to compare our method of
computing $k$ to the results recently obtained in Jacobsen:2023vdf  for these
theories. By perturbing with the energy density operator one triggers an RG
flow leading to a gapped integrable QFT. The associated S-matrix given in
Zamolodchikov:1990dg  sits at the boundary of the $|N|\leq 2$ monolith, in
what becomes (-)pYB for $N>2$. In Gorbenko:2018dtm ; Gorbenko:2018ncu  it was
argued that as we continue in $N$ the fixed point above becomes complex.
Moreover in Gorbenko:2020xya  it was shown that – at least for $N\gtrsim 2$ –
the pYB solution reproduces the expected RG walking behaviour close to the
complex fixed point.

Another indication comes from integrability. The expectation is that most
integrable amplitudes one can write down are incompatible with UV
completeness, in a similar manner to $T\bar{T}$ deformed theories
Smirnov:2016lqw ; Cavaglia:2016oda . More precisely, what happens is that the
finite size ground state energy computed from the exact S-matrix develops a
square root singularity at a size $R^{*}$ below which the energy becomes
complex. This obstacle for UV completeness was shown explicitly for when the
S-matrix is a product of two or more CDD zeros Camilo:2021gro ;
Cordova:2021fnr . Intriguingly, this exotic UV behavior goes undetected by the
central charge minimization bootstrap in Correia:2022dyp , where the space of
optimal scalar amplitudes minimizing central charges is spanned with
amplitudes of up to three CDD factors.

We believe the above points highlight the fact that the probes we are using
are not fully sensitive to potential non-trivial UV behaviours. After all, we
are only imposing positivity of the spectral functions for the stress tensor
and O(N) currents, so that we might be restricting to unitary sub-sectors of a
more complicated theory. In this light, it would be very interesting to
understand which modifications to our bootstrap approach would allow us to
test logarithmic/complex CFTs, either to include them as a more general class
of theories we should study or to definitely exclude them from our bounds.

## Acknowledgments

We would like to thank Nima Afkhami-Jeddi, Victor Gorbenko, Kelian Häring,
Yifei He, Andrea Manenti, Nafiz Ishtiaque, Denis Karateev, Shota Komatsu, Nat
Levine, João Penedones, Aninda Sinha, Emilio Trevisani, Pedro Vieira, Xi Yin
and Bernardo Zan for useful discussions. The computations were enabled by
resources provided by the National Academic Infrastructure for Supercomputing
in Sweden (NAISS) at UPPMAX (Rackham) partially funded by the Swedish Research
Council through grant agreement no. 2022-06725. Nordita is partially supported
by Nordforsk. AG is also supported by a Royal Society funding, URF\R\221015.
AV is 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. 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).

##  Appendix A Results for exact amplitudes

###  A.1 Exact amplitudes on $N>2$ monolith

We first write the exact amplitudes appearing at the boundary of the $N>2$
monolith. We use the following notation for the different channels
$S_{a}=(S_{\bullet},\,S_{A},\,S_{S})$ and the rapidity variable $\theta$,
related to the Mandelstam invariant $s$ through $s=4m^{2}\cosh(\theta/2)$.

free: $\displaystyle\quad S_{a}=\pm\,(1,1,1)\,,$ (53) NLSM:
$\displaystyle\quad
S_{a}(\theta)=\mp\left(1,\;\frac{\theta-i\pi}{\theta+i\pi},\;\frac{\theta-i\pi}{\theta+i\pi}\,\frac{\theta-i\lambda_{N}}{\theta+i\lambda_{N}}\right)F_{\pi+\lambda_{N}}(\theta)F_{2\pi}(\theta)\,,$
(54) pYB: $\displaystyle\quad
S_{a}(\theta)=\pm\,\left(\frac{\sinh\left[\nu\left(1-\frac{i\theta}{\pi}\right)\right]}{\sinh\left[\nu\left(1+\frac{i\theta}{\pi}\right)\right]},\;-1,\;1\right)\prod\limits_{n=-\infty}^{\infty}F_{\pi+\frac{in\pi^{2}}{\nu}}(-\theta)\,,$
(55) constant: $\displaystyle\quad
S_{a}=\pm\,\left(1,\,-1,\,-\frac{N-2}{N+2}\right)\,,$ (56)

where $\lambda_{N}=\frac{2\pi}{(N-2)}$, $\nu=\text{arccosh}(\tfrac{N}{2})$ and

$F_{a}(\theta)\equiv\frac{\Gamma\left(\frac{a+i\theta}{2\pi}\right)\Gamma\left(\frac{a-i\theta+\pi}{2\pi}\right)}{\Gamma\left(\frac{a-i\theta}{2\pi}\right)\Gamma\left(\frac{a+i\theta+\pi}{2\pi}\right)}={}_{2}F_{1}\left(-\frac{1}{2},\frac{i\theta}{\pi},\frac{a+i\theta}{2\pi},1\right)\,.$
(57)

Except for the constant solution (56), all other amplitudes are integrable and
saturate unitarity.

###  A.2 Integrable Form Factor Bootstrap

In the following we review some of the conventions used in the integrability
literature regarding integral representations of S-matrices and the
computation of form factors. Here we restrict to the two-particle form
factors, for a detailed review containing higher particle form factors see
e.g. Smirnov:1992vz ; Mussardo:2020rxh . Here we deal with amplitudes
saturating unitarity. Using the map $s=4m^{2}\cosh(\theta/2)$, the two
particle cuts in the $s$ complex plane get opened so that all different
Riemann sheets in $s$ are mapped to strips $n\pi\leq\text{Im}\theta<(n+1)\pi$
of the rapidity plane. Therefore the only non-analyticities of the amplitude
are poles (and zeros from their unitarity image). One can exploit this fact
and write a compact, often simpler, integral representation for the amplitude

$S(\theta)=\exp{\int\limits_{0}^{\infty}\frac{dt}{t}\,f(t)\,\sinh{\frac{t\theta}{i\pi}}}\,.$
(58)

This type of representation can be easily derived when writing the poles and
zeros of the amplitude in the following form

$\ln\left(\frac{1+i\alpha/\beta}{1-i\alpha/\beta}\right)=2i\int\limits_{0}^{\infty}\frac{dt}{t}\,e^{\beta
t}\,\sin\left(\alpha t\right)\,.$ (59)

The same function $f(t)$ appearing in (58) can then be used to compute the
“minimum” two-particle form factor

$F_{\text{min}}(\theta)=\mathcal{N}\exp{\int\limits_{0}^{\infty}\frac{dt}{t}\,\frac{f(t)}{\sinh{t}}\,\sin^{2}\left[\frac{t(i\pi-\theta)}{2\pi}\right]}\,,$
(60)

which solves Watson’s equation and is analogous to the factor
$e^{\alpha(s)}\subset F(s)$ appearing in (30). The full two-particle form
factor will be given by $F_{2}(\theta)=Q(\theta)/D(\theta)\times
F_{\text{min}}(\theta)$, where $Q$ and $D$ are polynomials in $\cosh{\theta}$.
The polynomial in the denominator $D$ is fixed by the possible poles of the
S-matrix, whereas $Q$ depends on the operator we are considering. They play
the role of our function $B(s)$ in (30).

Some examples for the functions $f(t)$ are Karowski:1978vz  252525A useful
result for the building block $F_{a}(\theta)$ in (57) is its integral
representation
$F_{a}(\theta)=\frac{\Gamma\left(\frac{a+i\theta}{2\pi}\right)\Gamma\left(\frac{a-i\theta+\pi}{2\pi}\right)}{\Gamma\left(\frac{a-i\theta}{2\pi}\right)\Gamma\left(\frac{a+i\theta+\pi}{2\pi}\right)}\,,\quad
f_{a}(t)=2\frac{e^{-at/\pi}}{1+e^{-t}}\,.$ (61)

$\begin{aligned} &S^{+}=1\,,\\\ &S^{-}=-1\,,\\\
&S^{\text{shG}}(\theta)=\dfrac{\sinh{\theta}-i\sin{\left(\pi\lambda\right)}}{\sinh{\theta}+i\sin{\left(\pi\lambda\right)}}\,,\\\
&S^{\text{NLSM}}(\theta)=\eqref{S_NLSM}\,\end{aligned}\quad\begin{aligned}
&f^{+}=0\,,\\\ &f^{-}=2\,,\\\
&f^{\text{shG}}(t)=2\left(1-\frac{\cosh\left(\frac{1}{2}\left(1-\frac{2\lambda}{\pi}\right)t\right)}{\cosh\left(\frac{t}{2}\right)}\right)\,,\\\
&f^{\text{NLSM}}(t)=2\left(\frac{e^{-\frac{2}{N-2}t}+1+e^{t}+e^{-t}}{1+e^{t}}\,,\frac{e^{-\frac{2}{N-2}t}-1}{1+e^{t}}\,,\frac{1-e^{-\frac{2}{N-2}t}}{1+e^{-t}}\right)\,.\end{aligned}$
(62)

Knowing these functions $f(t)$ for a given model, the analytic structure of
the minimum form factor can be extracted from the identity

$\ln\left(1+\frac{\alpha^{2}}{\beta^{2}}\right)=4\int\limits_{0}^{\infty}\frac{dt}{t}e^{-\beta
t}\sin^{2}\frac{\alpha t}{2}\,.$ (63)

In practice, one writes a series expansion for the ratio $f(t)/\sinh t$
appearing in (60) and uses (63) iteratively to find:

$\frac{f(t)}{\sinh t}=4\sum_{n}a_{n}e^{-\beta_{n}t}\implies
F_{\text{min}}(\theta)=\prod_{n}\left[1+\frac{(i\pi-\theta)^{2}/\pi^{2}}{\beta_{n}^{2}}\right]^{a_{n}}\,.$
(64)

###  A.3 Examples minimum Form Factors

Particularly simple examples include free fermion and the non-linear sigma
model for $N=3$

$\displaystyle F_{\text{min}}^{-}(\theta)$ $\displaystyle=$
$\displaystyle-i\sinh\frac{\theta}{2}\,,$ (65) $\displaystyle
F_{a,\text{min}}^{\text{NLSM, N=3}}(\theta)$ $\displaystyle=$
$\displaystyle\left(\frac{-\pi^{2}}{\theta(\theta-2\pi
i)}\sinh^{2}\frac{\theta}{2},\;\frac{\pi^{2}(\theta-i\pi)}{2\theta(\theta-2\pi
i)}\,\tanh\frac{\theta}{2},\;\frac{\theta-i\pi}{2}\,\tanh\frac{\theta}{2}\right)\,.$
(66)

Note that if we have $F_{\text{min}}$ for a given amplitude $S$ and we want to
compute the form factor for $-S$ it is enough to multiply by (65).

#### Periodic Yang-Baxter

To evaluate the integral representation of periodic Yang-Baxter’s S-matrix
(55) we will use the following identity

$\sum\limits_{k\in\mathbb{Z}}e^{-ikt}=2\pi\sum\limits_{p=0}^{\infty}\frac{\delta(t-2\pi
p)}{1+\delta_{p,0}}\,,\quad t\in(0,\infty)\,.$ (67)

Let us start with the prefactor in the singlet channel of(55)

$\frac{\sinh{\frac{\nu}{\pi}(\pi-i\theta)}}{\sinh{\frac{\nu}{\pi}(\pi+i\theta)}}\implies
f(t)=2e^{-t}\,\sum\limits_{k\in\mathbb{Z}}e^{ikt\pi/\nu}=4\nu
e^{-t}\sum\limits_{p=0}^{\infty}\,\frac{\delta(t-2\pi p)}{1+\delta_{p,0}}\,.$
(68)

It is a simple exercise to plug in the infinite sum over Dirac deltas in (58)
and recover the prefactor.

For the overall factor we have instead

$\prod\limits_{n=-\infty}^{\infty}F_{\pi+\frac{in\pi^{2}}{\nu}}(-\theta)\implies
f(t)=\frac{-2}{1+e^{t}}\,\sum\limits_{k\in\mathbb{Z}}e^{ikt\pi/\nu}=\frac{-4\nu}{1+e^{t}}\,\sum\limits_{p=0}^{\infty}\frac{\delta(t-2\pi
p)}{1+\delta_{p,0}}\,.$ (69)

The delta functions make it easy to perform the integration (58) and recover
the amplitudes in an alternative representation. For the singlet channel
amplitude we have

$f_{\bullet}^{\text{pYB}}(t)=\frac{4\nu\,e^{-t}}{1+e^{t}}\,\sum\limits_{p=0}^{\infty}\frac{\delta(t-2\nu
p)}{1+\delta_{p,0}}\implies
S_{\bullet}^{\text{pYB}}(\theta)=\text{exp}\left\\{-\frac{i\theta\nu}{\pi}-\sum\limits_{p=1}^{\infty}-\frac{2ie^{-2\nu
p}\sin\left(\frac{2\theta\nu p}{\pi}\right)}{p\left(e^{2\nu
p}+1\right)}\right\\}\,,$ (70)

where the last terms in the exponential are negligible for physical kinematics
$\theta\in\mathbb{R}$. Similar reasoning leads to the kernel $f_{A}(t)$ for
antisymmetric amplitude

$f_{A}^{\text{pYB}}(t)=2-\frac{4\nu\,}{1+e^{t}}\,\sum\limits_{p=0}^{\infty}\frac{\delta(t-2\nu
p)}{1+\delta_{p,0}}\implies
S_{\bullet}^{\text{pYB}}(\theta)=-\text{exp}\left\\{\frac{i\theta\nu}{\pi}-\sum\limits_{p=1}^{\infty}\frac{2i\sin\left(\frac{2\theta\nu
p}{\pi}\right)}{p\left(e^{2\nu p}+1\right)}\right\\}\,.$ (71)

For the minimum form factors we find

$\displaystyle F_{\text{min},\bullet}(\theta)$
$\displaystyle=\text{exp}\left\\{-\frac{(\pi+i\theta)^{2}\nu}{4\pi^{2}}+\sum\limits_{p=1}^{\infty}\frac{2e^{-2\nu
p}\text{csch}(2\nu p)\sin^{2}\left(\frac{(-\theta+i\pi)\nu
p}{\pi}\right)}{p\left(e^{2\nu p}+1\right)}\right\\}\,,$ (72) $\displaystyle
F_{\text{min},A}(\theta)$
$\displaystyle=i\sinh\left(\frac{\theta}{2}\right)\text{exp}\left\\{\frac{(\pi+i\theta)^{2}\nu}{4\pi^{2}}-\sum\limits_{p=1}^{\infty}\frac{2\text{csch}(2\nu
p)\sin^{2}\left(\frac{(-\theta+i\pi)\nu p}{\pi}\right)}{p\left(e^{2\nu
p}+1\right)}\right\\}\,.$ (73)

| $c_{2}$ | $k_{2}$
---|---|---
$+$free | $N$ | $\infty$
$-$free | $N/2$ | 1
$+$const | $N$ | 1
$-$const | $N/2$ | $\infty$
$+$NLSM | 5.2124 ($N$=7) | 15.6380 ($N$=7)
$-$NLSM | 3.9137 ($N$=7) | 1.1380 ($N$=7)
$+$pYB | $\infty$ | 2.1043 ($N$=7)
$-$pYB | $\infty$ | 1.6090 ($N$=7)
$(N=2)$ $+$pYB/$-$NLSM | 1.5649 | 1.4889
$(N=2)$ $-$pYB/$+$NLSM | 0.9875 | 1.6852
Table 1: Selected values of $c_{2}$ and $k_{2}$ for models with exact
amplitudes. Numerical values are approximate since the integrations are
performed numerically with a cutoff.

###  A.4 Computing $c_{2}$ and $k_{2}$

Having the exact S-matrix and the corresponding function $f(t)$ for a given
theory, we can compute the two-particle form factors and their contribution
the central charges coming from the sum rules (12) and (13). As explained
below (60), there is a polynomial freedom in constructing the full two-
particle form factor $F_{2}(\theta)=Q(\theta)/D(\theta)\times
F_{\text{min}}(\theta)$. The polynomial can be fixed by requiring that: 1) the
form factor is analytic except for the s-channel two-particle cut, and 2) the
sum rule integral converges. For most of the amplitudes treated in this
section it is enough to consider the redundancy
$\left[4m^{2}\sinh^{2}(\theta/2)\right]^{\pm 1}$ or equivalently $(s-4)^{\pm
1}$, where the only potential dangerous points happen at threshold and the
maximum degree of the polynomial in $s$ is set to 1 for convergence of the sum
rule (see section 4). We can also trace back this polynomial form to the $2\pi
n$ redundancy in the argument of the S-matrix. For the $c$ sum rule of
periodic Yang-Baxter however, it is not possible to use this freedom to get a
convergent integral, since the minimum form factor (72) grows as
$F_{\text{min},\bullet}(\theta)\sim e^{\theta^{2}}$. Except for simple cases
like free theories and $N=3$ NLSM showed above, one needs to compute the
integrals for the form factor (60) and sum rules numerically. This is what we
did to find the endpoints of the minimum $c$ and $k$ sections in figures 5 and
6. The values we find for the two particle contribution to the central charge
are given in the table 1.

(a)

(b)

(c)

Figure 14: Contribution of the two-particle form factors to the central
charges as a function of the number of flavors $N$. (a) Central charge $c_{2}$
for +NLSM (solid) and -NLSM (dashed). (b) Central charge $k_{2}$ for +NLSM
(solid) and -NLSM (dashed). (c) Central charge $k_{2}$ for +pYB (solid) and
-pYB (dashed). In the limit $N\rightarrow 2$ the amplitudes $\pm$NLSM and
$\mp$pYB coincide, so that the corresponding values $k_{2}(N=2)$ agree.

In figures 14 we show the central charges for $\pm$NLSM and $\pm$pYB for
different values of $N$.

As a last example we show how to get the two-particle contributions to $c$ and
$k$ for the sine-Gordon model in the regime where the spectrum contains only
kinks $\gamma\geq 8\pi$. As shown in appendix E of Cordova:2019lot , the
kinks/antikinks amplitudes cover the boundary of the $N=2$ monolith, more
precisely the two-dimensional section spanned by $\sigma_{i}(s=2m^{2})$. Using
the kernels Karowski:1978vz

$\displaystyle f(t)^{\text{sG}}_{\bullet}$
$\displaystyle=-\frac{2\left(\sinh\left(\frac{t}{2}\right)-\sinh\left(\left(\frac{3}{2}-\frac{\gamma}{8\pi}\right)t\right)\right)}{\sinh\left(\frac{t}{2}-\frac{\gamma
t}{8\pi}\right)-\sinh\left(\frac{\gamma t}{8\pi}+\frac{t}{2}\right)}\,,$ (74)
$\displaystyle f(t)^{\text{sG}}_{A}$
$\displaystyle=\frac{2\left(-2\sinh\left(\frac{1}{8}\left(4-\frac{\gamma}{\pi}\right)t\right)+\sinh\left(\frac{(\gamma+4\pi)t}{8\pi}\right)+\sinh\left(\frac{3t}{2}\right)\right)}{\sinh\left(\frac{t}{2}-\frac{\gamma\,t}{8\pi}\right)-\sinh\left(\frac{\gamma\,t}{8\pi}+\frac{t}{2}\right)}\,.$
(75)

we can get the central charges for the boundary in the upper quadrant of the
monolith $\sigma_{1,2}>0$ as shown in figure 15. We see the interpolation from
the $N=2$ limits of +pYB and -NLSM corresponding to $\sigma_{2}=0$ and
$\gamma\rightarrow\infty$ and free bosons with $\sigma_{2}=1$ and
$\gamma=8\pi$ (c.f. table 1). The rest of the sections at the boundary can be
computed from the mappings described in Cordova:2019lot .

(a)

(b)

Figure 15: Central charges $c_{2}$ and $k_{2}$ for the sine-Gordon model with
$\gamma\geq 8\pi$. The horizontal axis is the position $\sigma_{2}$ along the
$N=2$ monolith as we vary the coupling $\gamma$ from $8\pi$ ($\sigma_{2}=1$)
to infinity ($\sigma_{2}=0$).

##  Appendix B Notation and useful formulae

###  B.1 Conventions and derivation of inner products

Our conventions for particle states follow from Karateev:2019ymz . In
particular one particle states are defined as unitary irreducible
representations of the 2 dimensional Poincaré group $ISO(1,1)$ with discrete
eigenvalues of $P^{2}$, and are normalized as

$\displaystyle{}_{a}\langle
p_{1}|p_{2}\rangle_{b}=2p_{1}^{0}\delta_{ab}2\pi\delta(\bm{p}_{1}-\bm{p}_{2}),$
(76)

where $a=1,...,N$ is a color index in the vector representation of O(N) and
$p=(p^{0},\bm{p})$. Two particle states are decomposed into singlet,
antisymmetric and traceless symmetric irreducible representations of O(N) as

$\displaystyle\ket{p_{1}p_{2}}_{ab}=\frac{\delta_{ab}}{\sqrt{N}}\ket{p_{1},p_{2}}^{\bullet}+\ket{p_{1},p_{2}}^{A}_{[a,b]}+\ket{p_{1},p_{2}}^{S}_{(a,b)}$
(77)

where

$\displaystyle\ket{p_{1},p_{2}}^{\bullet}$
$\displaystyle\equiv\frac{1}{\sqrt{N}}\ket{p_{1}p_{2}}_{a,a},$ (78)
$\displaystyle\ket{p_{1},p_{2}}^{A}_{[a,b]}$
$\displaystyle\equiv\frac{1}{2}(\ket{p_{1},p_{2}}_{a,b}-\ket{p_{1},p_{2}}_{b,a}),$
(79) $\displaystyle\ket{p_{1},p_{2}}^{S}_{(a,b)}$
$\displaystyle\equiv\frac{1}{2}(\ket{p_{1},p_{2}}_{a,b}+\ket{p_{1},p_{2}}_{b,a})-\frac{1}{\sqrt{N}}\ket{p_{1}p_{2}}_{a,b}.$
(80)

The two particle form factor of the trace of the stress energy tensor
$\Theta\equiv T^{\mu}_{\mu}$ is

$\displaystyle F_{\Theta}(s)\equiv{}_{out}\langle
0|\Theta(0)|p_{1},p_{2}\rangle_{\bullet}^{in}$ (81)

and its spectral density is

$\displaystyle\rho_{\Theta}(s)\equiv\int d^{2}xe^{-ip\cdot x}\langle
0|\Theta(x)\Theta(0)|0\rangle.$ (82)

The 2 particles form factors of conserved currents, being in the adjoint, only
take non vanishing values with states in the antisymmetric representation and
are defined as

$F^{\mu}_{ab,cd}(p_{1},p_{2})\equiv_{out}\bra{0}J^{\mu}_{[ab]}(0)\ket{p_{1},p_{2}}^{A}_{[cd],in}=iT^{A}_{ab,cd}(p_{1}^{\mu}-p_{2}^{\mu})F_{J}(s),$
(83)

where the second equality follows from Lorentz and O(N) symmetry and
conservation of the currents262626The second Lorentz vector that could appear
would be $p_{1}^{\mu}+p_{2}^{\mu}$, but the conservation equation
$\partial_{\mu}J^{\mu}=0$ implies $(p_{1}+p_{2})_{\mu}F^{\mu}_{ab,cd}=0$,
which follows from differentiating
${}_{out}\bra{0}J^{\mu}_{[ab]}(x)\ket{p_{1},p_{2}}^{A}_{[cd],in}=e^{i(p_{1}+p_{2})\cdot
x}F^{\mu}_{ab,cd}$., and we used
$T^{A}_{ab,cd}\equiv\frac{1}{2}(\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc})$.
Similarly the spectral density of those currents reads

$2\pi\rho^{\mu\nu}_{J;ab,cd}\equiv\int
d^{2}xe^{-ipx}\bra{0}J^{\mu}_{[ab]}(x)J^{\nu}_{[cd]}(0)\ket{0}=T^{A}_{ab,cd}2\pi\rho_{J}(s)(p^{\mu}p^{\nu}-\eta^{\mu\nu}p^{2}).$
(84)

We want to compute it in terms of the form factors. Inserting a complete set
of states between the currents, focusing on the two particles contribution
sector, translating the operators and changing variables with

$\displaystyle\frac{1}{2}\frac{d\bm{p}}{2p^{0}(2\pi)}\frac{d\bm{q}}{2q^{0}(2\pi)}=\frac{1}{\mathcal{N}_{2}}\frac{d^{2}(p+q)}{(2\pi)^{2}}$
(85)

we get for $s>4m^{2}$

$\displaystyle
T^{A}_{ab,cd}2\pi\rho_{J}(s)(p^{\mu}p^{\nu}-\eta^{\mu\nu}p^{2})=\frac{1}{\mathcal{N}_{2}}T^{A}_{ab,mn}T^{A}_{cd,mn}(p_{1}^{\mu}-p_{2}^{\mu})(p_{1}^{\nu}-p_{2}^{\nu})|F_{J}(s)|^{2}\theta(s-4m^{2})+...$
(86)

where $\theta$ is the Heaviside step function and the dots indicate
contributions from higher particles form factors. We will denote
$\rho_{J}^{(2)}(s)$ for the two particles contribution only. Using
$T^{A}_{ab,mn}T^{A}_{cd,mn}=T^{A}_{ab,cd}$, contracting both sides with
$q^{\nu}\equiv(p_{1}^{\nu}-p_{2}^{\nu})$ and collecting the components of
$q^{\mu}$ we get

$\displaystyle-2\pi\rho_{J}^{(2)}p^{2}=\frac{1}{\mathcal{N}_{2}}q^{2}|F_{J}|^{2}\theta(s-4m^{2}).$
(87)

We can finally use $p^{2}=-s$ and $q^{2}=s-4m^{2}$ to get

$\displaystyle
2\pi\rho_{J}^{(2)}=\frac{1}{\mathcal{N}_{2}}\frac{s-4m^{2}}{s}|F_{J}(s)|^{2}\theta(s-4m^{2})$
(88)

This is just the equivalent of Watson’s equation that can be derived by a
similar procedure for the stress tensor form factor and reads

$\displaystyle
2\pi\rho_{\Theta}^{(2)}=\frac{1}{\mathcal{N}_{2}}|F_{\Theta}|^{2}.$ (89)

We now have everything to derive the inner products for the components of the
matrices $B_{a}$ in (10). The upper 2x2 blocks and the entries of the
$B_{\bullet}$ matrix were already computed in section 3.1 in Karateev:2019ymz
. Here we focus on the remaining components of the $B_{A}$ matrix.

The entry (3,3) is

$\begin{split}{}_{[ab]}\innerproduct{\psi_{3}}{\psi_{3}}_{[cd]}&=\int
d^{2}xd^{2}ye^{-ipx+ip^{\prime}y}\frac{q_{\mu}q^{\prime}_{\nu}}{q^{4}}\bra{0}J^{\mu}_{[ab]}(x)J^{\nu}_{[cd]}(y)\ket{0}\\\
&=(2\pi)\delta^{(2)}(p-p^{\prime})T^{A}_{ab,cd}\frac{q_{\mu}q_{\nu}}{q^{4}}(2\pi)\rho^{\mu\nu}_{J}\\\
&=(2\pi)^{2}\delta^{(2)}(p-p^{\prime})T^{A}_{ab,cd}\frac{(-p^{2})}{q^{2}}(2\pi)\rho_{J}(s).\end{split}$
(90)

Note that

$s=-(p_{1}+p_{2})^{2},\qquad q^{2}=(p_{1}-p_{2})^{2}=p^{2}-4p_{1}\cdot
p_{2}=s-4m^{2}.$ (91)

We therefore have

${}_{[ab]}\innerproduct{\psi_{3}}{\psi_{3}}_{[cd]}=(2\pi)^{2}\delta^{(2)}(p-p^{\prime})T^{A}_{ab,cd}\frac{s}{s-4m^{2}}(2\pi)\rho_{J}(s)$
(92)

The off diagonal components (2,3) are

$\begin{split}{}_{[ab]}\bra{\psi_{2}}\ket{\psi_{3}}_{[cd]}&=\int
d^{2}xe^{ipx}\frac{\omega}{q^{2}}\
{}^{out}_{[ab]}\bra{p^{\prime}_{1}p^{\prime}_{2}}q_{\nu}J^{\nu}_{[cd]}(x)\ket{0}=\int
d^{2}x\frac{\omega}{q^{2}}e^{ix(p-p^{\prime})}q_{\nu}F^{\nu}_{ab,cd}(p_{1}^{\prime},p_{2}^{\prime})\\\
&=(2\pi)^{2}\delta^{2}(p-p^{\prime})iT^{A}_{ab,cd}\omega F_{J}(s).\end{split}$
(93)

Similarly we have for the entry (1,3)272727Note that the order of the
particles is important for the antisymmetric channel, and since
$(\ket{p_{1},p_{2}})^{\dagger}=\bra{p_{2},p_{1}}$ we get an extra minus sign
when permuting to get the same tensor structure.

$\begin{split}{}_{[ab]}\bra{\psi_{1}}\ket{\psi_{3}}_{[cd]}&=\int
d^{2}xe^{ipx}\frac{\omega}{q^{2}}\
{}^{in}_{[ab]}\bra{p^{\prime}_{1}p^{\prime}_{2}}q_{\nu}J^{\nu}_{[cd]}(x)\ket{0}=\int
d^{2}x\frac{\omega}{q^{2}}e^{ix(p-p^{\prime})}q_{\nu}(F^{\nu}_{ba,cd})^{*}(p_{1}^{\prime},p_{2}^{\prime})\\\
&=(2\pi)^{2}\delta^{2}(p-p^{\prime})iT^{A}_{ab,cd}\omega
F_{J}^{*}(s).\end{split}$ (94)

###  B.2 Sum rules

Using complex coordinates the conservation of the currents becomes

$\displaystyle\partial_{\mu}J^{\mu}=0\implies\bar{\partial}J+\partial\bar{J}=0,$
(95)

where $\partial\equiv\partial_{z}$, $\bar{\partial}\equiv\partial_{\bar{z}}$,
$J\equiv J_{z}$ and $\bar{J}\equiv J_{\bar{z}}$. By rotational invariance we
can then write282828The factor of $\pi^{2}$ extracted from $G(\bar{z}z)$ will
make the resulting sum rule agree with the free fermion result (and
subsequently all other analytical results).

$\displaystyle\langle
J(z,\bar{z})J(0)\rangle=\frac{F(z\bar{z})}{z^{2}},\qquad\langle\bar{J}(z,\bar{z})J(0)\rangle=\pi^{2}\frac{G(z\bar{z})}{z\bar{z}}.$
(96)

At a conformal fixed point these functions become

$\displaystyle F_{CFT}=k_{CFT},\qquad G_{CFT}=0.$ (97)

The conservation equation implies

$\displaystyle\langle\bar{\partial}J(z,\bar{z})J(0)\rangle+\langle\partial\bar{J}(z,\bar{z})J(0)\rangle=0\implies
F^{\prime}+\pi^{2}G^{\prime}=\pi^{2}\frac{G}{\bar{z}z}.$ (98)

Integrating both sides we obtain a sum rule for $k$

$\displaystyle
k_{UV}-k_{IR}=\pi^{2}\int_{0}^{\infty}dr^{2}\frac{G(r^{2})}{r^{2}}=\pi\int
d^{2}z\,\langle\bar{J}(z,\bar{z})J(0)\rangle.$ (99)

We can relate the RHS to the spectral density $\rho_{J}$ by considering the
(Euclidean) Källén-Lehmann spectral decomposition of a spin 1 field
Karateev:2020axc

$\displaystyle\langle
J_{\mu}(x)J_{\nu}(0)\rangle=\int_{0}^{\infty}ds\,\rho_{J}(s)\left(\delta_{\mu\nu}s-\partial_{\mu}\partial_{\nu}\right)\Delta_{E}(x;s),$
(100)

where the Euclidean scalar propagator is

$\displaystyle\Delta_{E}(x;s)\equiv\int\frac{d^{2}p}{(2\pi)^{2}}\frac{e^{ip\cdot
x}}{p^{2}+s}.$ (101)

Evaluating the trace in complex coordinates we get

$\displaystyle
4\langle\bar{J}(z,\bar{z})J(0)\rangle=\int_{0}^{\infty}ds\,\rho_{J}(s)\left(2s-\Box\right)\Delta_{E}(x;s).$
(102)

The sum-rule therefore becomes292929Here we used that the spectral density has
non vanishing support only from the two-particle threshold $s\geq 4m^{2}$. In
principle there is also a contribution from the stable particle at $s=m^{2}$,
but it is proportional to the one particle form factor $|F_{1}|^{2}$ that here
vanishes due to O(N) symmetry. This is also true for the $c$ sum rule.

$\displaystyle k_{UV}-k_{IR}=\frac{\pi}{4}\int_{0}^{\infty}ds\ \rho_{J}(s)\int
d^{2}p\frac{2s+p^{2}}{p^{2}+s}\int\frac{d^{2}x}{(2\pi)^{2}}e^{ip\cdot
x}=\frac{\pi}{2}\int_{4m^{2}}^{\infty}ds\,\rho_{J}(s).$ (103)

As shown in more details in Correia:2022dyp ; Karateev:2020axc  we can do the
same for the stress tensor where conservation reads

$\bar{\partial}T+\frac{\pi}{2}\partial\Theta=0,$ (104)

and rotational invariance gives

$\langle T(z,\bar{z})T(0,0)\rangle=\frac{F(z\bar{z})}{z^{4}},\qquad\langle
T(z,\bar{z})\Theta(0,0)\rangle=\frac{G(z\bar{z})}{z^{3}\bar{z}},\qquad\langle\Theta(z,\bar{z})\Theta(0,0)\rangle=\frac{H(z\bar{z})}{z^{2}\bar{z}^{2}},$
(105)

Using the UV behaviour

$F_{UV}=\frac{c}{2},\qquad G_{UV}=H_{UV}=0,$ (106)

similar steps lead to

$c=12\pi\int_{4m^{2}}^{\infty}ds\frac{\rho_{\Theta}(s)}{s^{2}},$ (107)

###  B.3 Normalization of form factors

The currents are related to conserved charges by

$Q_{[ab]}\equiv\int d\bm{x}\,J^{0}_{[ab]}(\bm{x},t).$ (108)

Those conserved charges generate O(N) transformations on the Hilbert space and
therefore act on one particle states as

$Q_{[ab]}\ket{p}_{c}=-i\delta_{bc}\ket{p}_{a}+i\delta_{ac}\ket{p}_{b}.$ (109)

Different normalizations for this symmetry will lead to different
normalizations for the form factor. In particular let us consider the matrix
element

$\mathcal{M}=\sum_{a,b}\int
d\bm{x}\tensor[_{b}]{\bra{p_{1}}J^{0}_{[ab]}(\bm{x},0)\ket{p_{2}}}{{}_{a}}.$
(110)

On the one hand $\mathcal{M}$ can be evaluated by using the conserved charge
and its action on one particle states as

$\displaystyle\mathcal{M}$ $\displaystyle=\
{}_{b}\langle{p_{1}}|Q_{[ab]}|{p_{2}}\rangle_{a}=\sum_{a,b}(-i\delta_{ab}\ \
{}_{b}\langle{p_{1}}|{p_{2}}\rangle_{a}+i\delta_{aa}\
{}_{b}\langle{p_{1}}|{p_{2}}\rangle_{b})$ (111)
$\displaystyle=i(N^{2}-N)(2\pi)(2p_{1}^{0})\delta(\bm{p}_{1}-\bm{p}_{2}).$
(112)

On the other hand we can translate the current with $J(x)=e^{-iPx}J(0)e^{iPx}$
and use the analytic continuation $p_{2}\to-p_{2}$ to get

$\mathcal{M}=\int
d\bm{x}e^{i\bm{x}(\bm{p_{1}}-\bm{p_{2}})}F^{0}_{ab,ab}(p_{1},-p_{2})=(2\pi)\delta(\bm{p}_{1}-\bm{p}_{2})i\frac{N(N-1)}{2}(p_{1}^{0}+p_{2}^{0})F_{J}(4m^{2}-s).$
(113)

Comparing the two results we obtain

$\delta(\bm{p}_{1}-\bm{p}_{2})\left(\frac{1}{2}F_{J}(4m^{2}-s)(p_{1}^{0}+p_{2}^{0})(N^{2}-N)-2p_{1}^{0}(N^{2}-N)\right)=0$
(114)

Evaluating this constraint in the center of mass frame, we therefore deduce

$F_{J}(s=0)=2.$ (115)

The case of the stress tensor is similar and derived by starting from

$P^{\mu}\ket{p}=\int dxT^{0\mu}(x)\ket{p}=p^{\mu}\ket{p}.$ (116)

We then consider the matrix element

$\displaystyle\mathcal{M}^{\mu}=\sum_{a,b}\int d\bm{x}\ {}_{b}\langle
p_{1}|T^{0\mu}(\bm{x},0)|p_{2}\rangle_{a}$ (117)

We then have on one hand, using the charge $P^{\mu}$,

$\displaystyle\mathcal{M}^{\mu}=N\delta(\bm{p}_{1}-\bm{p}_{2})2p_{1}^{0}p_{1}^{\mu},$
(118)

and on the other hand, using the definition of form factors,

$\displaystyle\mathcal{M}^{\mu}=2\pi\sqrt{N}\delta(\bm{p}_{1}-\bm{p}_{2})F_{T^{\mu
0}}(p_{1},-p_{2}).$ (119)

Then using that by Lorentz invariance

$F_{T^{\mu\nu}}(s)=B(s)q^{\mu}q^{\nu},\qquad F_{\Theta}(s)=(s-4m^{2})B(s),$
(120)

where we used $q^{\mu}=p_{1}^{\mu}-p_{2}^{\mu}$, we get

$\left(\sqrt{N}B(s-4m^{2})(p_{1}^{\mu}+p_{2}^{\mu})-Np_{2}^{\mu}\right)\delta(\bm{p}_{1}-\bm{p}_{2})=0,$
(121)

and therefore $B(0)=\sqrt{N}/2$ leading to

$F_{\Theta}(0)=-2\sqrt{N}m^{2}.$ (122)

##  Appendix C Dual setup and optimization problems

###  C.1 General setup

Let us now discuss the dual implementation. We begin by analyticity and
crossing. Defining the dual function $W_{a}$ to be crossing antisymmetric,
$W_{a}(4-s)C_{ab}=-W_{b}(s)$, with a cut for $s\geq 4$ and a pole at $s=2$
(and the crossing symmetrics of those singularities), and sufficiently fast
decay at infinity, we get the dispersion relation

$0=\oint\frac{1}{2\pi
i}W_{a}S_{a}=-S_{a}(2)\mathop{\mathrm{Res}}_{s=2}W_{a}(s)+\frac{2}{\pi}\int_{4}^{\infty}ds\operatorname{Im}(W_{a}S_{a}).$
(123)

A similar argument for the form factor $F\equiv F_{\Theta/J}$ gives

$0=\oint\frac{1}{2\pi
i}W_{F}F=-F_{\Theta}(0)\mathop{\mathrm{Res}}_{s=0}W_{F}(s)+\frac{1}{\pi}\int_{4}^{\infty}ds\operatorname{Im}(W_{F}F),$
(124)

where $W_{F}$ is analytic except for a pole at $s=0$ and a branch point at
$s=4$, with a cut along the positive real axis. Note that the residues on the
poles are fixed by blowing up $0=\oint W_{a}ds=-2\pi
i\mathrm{Res}_{s=2}W_{a}+2i(1+C^{T})\int_{4}^{\infty}\operatorname{Im}W_{a}ds$,
and $0=\oint W_{F}ds=-2\pi
i\mathrm{Res}_{s=0}W_{F}+2i\int_{4}^{\infty}\operatorname{Im}W_{F}ds$.

Now we discuss unitarity for which the constraint $B_{a}\succeq 0$ is written
with Lagrange multipliers packaged in matrices $\lambda_{a}$ Guerrieri:2020kcs

$\int_{4}^{\infty}ds\ \operatorname{Tr}B_{a}(s)\Lambda_{a}(s),$ (125)

where $\Lambda_{a}$ are hermitian and negative semidefinite parametrized as

$\Lambda_{\bullet}=\begin{pmatrix}\lambda_{1,\bullet}&\lambda_{4,\bullet}&\lambda_{6,\bullet}\\\
\lambda_{4,\bullet}^{*}&\lambda_{2,\bullet}&\lambda_{5,\bullet}\\\
\lambda_{6,\bullet}^{*}&\lambda_{5,\bullet}^{*}&\lambda_{3,\bullet}\end{pmatrix},\qquad\Lambda_{A}=\begin{pmatrix}\lambda_{1,A}&\lambda_{4,A}&\lambda_{6,A}\\\
\lambda_{4,A}^{*}&\lambda_{2,A}&\lambda_{5,A}\\\
\lambda_{6,A}^{*}&\lambda_{5,A}^{*}&\lambda_{3,A}\end{pmatrix},\qquad\Lambda_{\textbf{S}}=\begin{pmatrix}\lambda_{1,S}&\mu_{S}\\\
\mu_{S}^{*}&\lambda_{2,S}\end{pmatrix}.$ (126)

Some components can be eliminated by noticing that at the optimum we need to
satisfy $\operatorname{Tr}\Lambda B=0\implies\Lambda B=0$. Solving those
equations for the 3x3 matrices gives

$\begin{gathered}\lambda_{5,\bullet}=\lambda_{6,\bullet}^{*},\qquad\lambda_{1,\bullet}=\lambda_{2,\bullet}\equiv\lambda_{\bullet},\\\
\lambda_{5,A}=-\lambda_{6,A}^{*},\qquad\lambda_{1_{A}}=\lambda_{2,A}\equiv\lambda_{A},\\\
2\lambda_{1,a}|\lambda_{6,a}|^{2}-\lambda_{3,a}\lambda_{1,a}^{2}-2\operatorname{Re}(\lambda_{4,a}^{*}\lambda_{6,a}^{2})+\lambda_{3,a}|\lambda_{4,a}|^{2}=0,\qquad
a=\bullet,A.\end{gathered}$ (127)

For the 2x2 matrices the equations fix

$\lambda_{S}\equiv\lambda_{1,S}=\lambda_{2,S}=-|\mu_{S}|.$ (128)

The unitarity constrain can then be expanded

$\begin{gathered}\operatorname{Tr}\Lambda_{\bullet}B_{\bullet}=2\lambda_{\bullet}+2\operatorname{Re}\lambda_{4,\bullet}S_{\bullet}+4\omega\operatorname{Re}\lambda_{6,\bullet}F_{\Theta}+2\pi\rho_{\Theta}\lambda_{3,\bullet},\\\
\operatorname{Tr}\Lambda_{A}B_{A}=2\lambda_{A}+2\operatorname{Re}\lambda_{4,A}S_{A}+4\omega\operatorname{Im}\lambda_{6,A}F_{J}+2\pi\frac{s}{s-4}\rho_{J}\lambda_{3,A},\\\
\operatorname{Tr}\Lambda_{S}B_{S}=2\lambda_{S}+2\operatorname{Re}\mu_{S}S_{S}.\end{gathered}$
(129)

The remaining equations coming from $\Lambda B=0$ give the primal observables
as function of the optimal dual functions as

$\begin{gathered}S_{\bullet}=\frac{\lambda_{6,\bullet}^{*}\lambda_{\bullet}-\lambda_{6,\bullet}\lambda_{4,\bullet}^{*}}{\lambda_{6,\bullet}\lambda_{\bullet}-\lambda_{6,\bullet}^{*}\lambda_{4,\bullet}},\qquad\omega
F_{\Theta}=\frac{|\lambda_{4,\bullet}|^{2}-\lambda_{\bullet}^{2}}{\lambda_{6,\bullet}\lambda_{\bullet}-\lambda_{6,\bullet}^{*}\lambda_{4,\bullet}},\qquad
2\pi\rho_{\Theta}=|\omega F_{\Theta}|^{2},\\\
S_{A}=-\frac{\lambda_{6,A}^{*}\lambda_{A}+\lambda_{6,A}\lambda_{4,A}^{*}}{\lambda_{6,A}\lambda_{A}+\lambda_{6,A}^{*}\lambda_{4,A}},\qquad
i\omega(s-4)F_{J}=\frac{\lambda_{A}^{2}-|\lambda_{4,A}|^{2}}{{\lambda_{6,A}}\lambda_{A}+\lambda_{6,A}^{*}\lambda_{4,A}},\qquad
2\pi\rho_{J}=\frac{s-4}{s}|\omega F_{J}|^{2},\\\
S_{\textbf{S}}=-\frac{\mu_{\textbf{S}}^{*}}{\lambda_{\textbf{S}}}.\end{gathered}$
(130)

###  C.2 Minimization of the central charge $c$ on the monolith

To minimize the central charge $c$ on top of the monolith we start with the
constrained problem

$\begin{split}\mathcal{L}&=12\pi\int_{4}^{\infty}\frac{\rho_{\Theta}(s)}{s^{2}}-\pi
F_{\Theta}(0)\mathop{\mathrm{Res}}_{s=0}W_{F_{\Theta}}(s)-\frac{\pi}{2}S_{a}(2)\mathop{\mathrm{Res}}_{s=2}W_{a}(s)\\\
&+\int_{4}^{\infty}ds\left(\operatorname{Im}(W_{a}S_{a}+W_{F_{\Theta}}F_{\Theta})+\operatorname{Tr}\Lambda_{a}B_{a}\right).\end{split}$
(131)

The equations of motion coming from functional variations with respect to the
primal functions give

$\displaystyle\lambda_{4,\bullet}=\frac{i}{2}W_{\bullet},\qquad\lambda_{6,\bullet}=\frac{i}{4\omega}W_{F_{\Theta}},\qquad\lambda_{3,\bullet}=-\frac{6}{s^{2}},$
(132) $\displaystyle\mu_{A}=\frac{i}{2}W_{A},\qquad\mu_{S}=\frac{i}{2}W_{S}.$
(133)

In terms of the coordinates $(\sigma_{1},\sigma_{2})$ on the monolith (at
$s=2$), the Lagrangian becomes

$\begin{split}\mathcal{L}&=2\pi\sqrt{N}\mathop{\mathrm{Res}}_{s=0}W_{F_{\Theta}}-\frac{\pi}{2}((N+1)\sigma_{1}+\sigma_{2})\mathop{\mathrm{Res}}_{s=2}W_{\bullet}-\frac{\pi}{2}(\sigma_{1}+\sigma_{2})\mathop{\mathrm{Res}}_{s=2}W_{S}\\\
&-\frac{\pi}{2}(\sigma_{2}-\sigma_{1})\mathop{\mathrm{Res}}_{s=2}W_{A}+\int_{4}^{\infty}(2\lambda_{\bullet}+2\lambda_{A}+2\lambda_{S}).\end{split}$
(134)

This leads us to the dual problem ready to be implemented in SDPB :

Dual Problem (Minimization of the central charge $c$ on top of the monolith)

$\underset{\\{\lambda_{a},W_{a},W_{F_{\Theta}}\\}}{\text{Maximize}}\mathcal{L}$
(135)

Constrained by

$\begin{pmatrix}\lambda_{\bullet}&\frac{i}{2}W_{\bullet}&\frac{i}{4\omega}W_{F_{\Theta}}\\\
-\frac{i}{2}W_{\bullet}^{*}&\lambda_{\bullet}&-\frac{i}{4\omega}W_{F_{\Theta}}^{*}\\\
-\frac{i}{4\omega}W_{F_{\Theta}}^{*}&\frac{i}{4\omega}W_{F_{\Theta}}&-\frac{6}{s^{2}}\end{pmatrix}\preccurlyeq
0,\qquad\begin{pmatrix}\lambda_{A}&\frac{i}{2}W_{A}\\\
-\frac{i}{2}W_{A}^{*}&\lambda_{A}\end{pmatrix}\preccurlyeq
0,\qquad\begin{pmatrix}\lambda_{S}&\frac{i}{2}W_{S}\\\
-\frac{i}{2}W_{S}^{*}&\lambda_{S}\end{pmatrix}\preccurlyeq 0.$ (136)

###  C.3 Minimization of the current central charge $k$ on the monolith

Analogously to the $c$ case we start with

$\begin{split}\mathcal{L}&=\frac{\pi}{2}\int_{4}^{\infty}\rho_{J}(s)-\pi
F_{J}(0)\mathop{\mathrm{Res}}_{s=0}W_{F_{J}}(s)-\frac{\pi}{2}S_{a}(2)\mathop{\mathrm{Res}}_{s=2}W_{a}(s)\\\
&+\int_{4}^{\infty}ds\left(\operatorname{Im}(W_{a}S_{a}+W_{F_{J}}F_{J})+\operatorname{Tr}\Lambda_{a}B_{a}\right),\end{split}$
(137)

The equations of motion of the primal functions yield

$\displaystyle\lambda_{4,A}=\frac{i}{2}W_{A},\qquad\lambda_{6,A}=-\frac{1}{4\omega}W_{F_{J}},\qquad\lambda_{3,A}=-\frac{s-4}{4s},$
(138)
$\displaystyle\mu_{\bullet}=\frac{i}{2}W_{\bullet},\qquad\mu_{S}=\frac{i}{2}W_{S}.$
(139)

The Lagrangian becomes

$\begin{split}\mathcal{L}&=-2\pi\mathop{\mathrm{Res}}_{s=0}W_{F_{J}}-\frac{\pi}{2}((N+1)\sigma_{1}^{*}+\sigma_{2}^{*})\mathop{\mathrm{Res}}_{s=2}W_{\bullet}-\frac{\pi}{2}(\sigma_{1}^{*}+\sigma_{2}^{*})\mathop{\mathrm{Res}}_{s=2}W_{S}\\\
&-\frac{\pi}{2}(\sigma_{2}^{*}-\sigma_{1}^{*})\mathop{\mathrm{Res}}_{s=2}W_{A}+\int_{4}^{\infty}(2\lambda_{\bullet}+2\lambda_{A}+2\lambda_{S}),\end{split}$
(140)

and the dual problem is

Dual Problem (Minimization of $k$ on top of the monolith)

$\underset{\\{\lambda_{a},W_{a},W_{F_{J}}\\}}{\text{Maximize}}\mathcal{L}$
(141)

Constrained by

$\begin{pmatrix}\lambda_{\bullet}&\frac{i}{2}W_{\bullet}\\\
-\frac{i}{2}W_{\bullet}^{*}&\lambda_{\bullet}\end{pmatrix}\preccurlyeq
0,\qquad\begin{pmatrix}\lambda_{A}&\frac{i}{2}W_{A}&-\frac{1}{4\omega}W_{F_{J}}\\\
-\frac{i}{2}W_{A}^{*}&\lambda_{A}&\frac{1}{4\omega}W_{F_{J}}^{*}\\\
-\frac{1}{4\omega}W_{F_{J}}^{*}&\frac{1}{4\omega}W_{F_{J}}&-\frac{s-4}{4s}\end{pmatrix}\preccurlyeq
0,\qquad\begin{pmatrix}\lambda_{S}&\frac{i}{2}W_{S}\\\
-\frac{i}{2}W_{S}^{*}&\lambda_{S}\end{pmatrix}\preccurlyeq 0.$ (142)

###  C.4 Minimization of $k$ for fixed $c$

We start with

$\begin{split}\mathcal{L}&=\frac{\pi}{2}\int_{4}^{\infty}\rho_{J}(s)-\pi
F_{J}(0)\mathop{\mathrm{Res}}_{s=0}W_{F_{J}}(s)-\pi
F_{\Theta}(0)\mathop{\mathrm{Res}}_{s=0}W_{F_{\Theta}}(s)\\\
&+\int_{4}^{\infty}ds\left(\operatorname{Im}(W_{a}S_{a}+W_{F_{J}}F_{J}+W_{F_{\Theta}}F_{\Theta})+\operatorname{Tr}\Lambda_{a}B_{a}\right)\\\
&+C_{\rho}(c_{UV}-12\pi\int_{4}^{\infty}\frac{\rho_{\Theta}(s)}{s^{2}}),\end{split}$
(143)

The equations of motion of the primal functions are

$\displaystyle\lambda_{4,\bullet}=\frac{i}{2}W_{\bullet},\qquad\lambda_{6,\bullet}=\frac{i}{4\omega}W_{F_{\Theta}},\qquad\lambda_{3,\bullet}=-C_{\rho}\frac{6}{s^{2}},$
(144)
$\displaystyle\lambda_{4,A}=\frac{i}{2}W_{A},\qquad\lambda_{6,A}=-\frac{1}{4\omega}W_{F_{J}},\qquad\lambda_{3,A}=-\frac{s-4}{4s},$
(145) $\displaystyle\mu_{S}=\frac{i}{2}W_{S}.$ (146)

The Lagrangian becomes

$\begin{split}\mathcal{L}&=2\pi\sqrt{N}\mathop{\mathrm{Res}}_{s=0}W_{F_{\Theta}}-2\pi\mathop{\mathrm{Res}}_{s=0}W_{F_{J}}+C_{\rho}c_{UV}\\\
&-\frac{\pi}{2}((N+1)\sigma_{1}^{*}+\sigma_{2}^{*})\mathop{\mathrm{Res}}_{s=2}W_{\bullet}-\frac{\pi}{2}(\sigma_{1}^{*}+\sigma_{2}^{*})\mathop{\mathrm{Res}}_{s=2}W_{S}\\\
&-\frac{\pi}{2}(\sigma_{2}^{*}-\sigma_{1}^{*})\mathop{\mathrm{Res}}_{s=2}W_{A}+\int_{4}^{\infty}(2\lambda_{\bullet}+2\lambda_{A}+2\lambda_{S}),\end{split}$
(147)

and the dual problem is

Dual Problem (Minimization of $k$ with fixed $c$)

$\underset{\\{\lambda_{a},W_{a},W_{F},C_{\rho}\\}}{\text{Maximize}}\mathcal{L}$
(148)

Constrained by

$\displaystyle\begin{pmatrix}\lambda_{\bullet}&\frac{i}{2}W_{\bullet}&\frac{i}{4\omega}W_{F_{\Theta}}\\\
-\frac{i}{2}W_{\bullet}^{*}&\lambda_{\bullet}&-\frac{i}{4\omega}W_{F_{\Theta}}^{*}\\\
-\frac{i}{4\omega}W_{F_{\Theta}}^{*}&\frac{i}{4\omega}W_{F_{\Theta}}&-\frac{6C_{\rho}}{s^{2}}\end{pmatrix}\preccurlyeq
0,\qquad\begin{pmatrix}\lambda_{A}&\frac{i}{2}W_{A}&-\frac{1}{4\omega}W_{F_{J}}\\\
-\frac{i}{2}W_{A}^{*}&\lambda_{A}&\frac{1}{4\omega}W_{F_{J}}^{*}\\\
-\frac{1}{4\omega}W_{F_{J}}^{*}&\frac{1}{4\omega}W_{F_{J}}&-\frac{s-4}{4s}\end{pmatrix}\preccurlyeq
0,$ (149) $\displaystyle\begin{pmatrix}\lambda_{S}&\frac{i}{2}W_{S}\\\
-\frac{i}{2}W_{S}^{*}&\lambda_{S}\end{pmatrix}\preccurlyeq 0.$ (150)

##  Appendix D Numerical implementation

To implement this problem we need ansätze for the variables. We use the disc
variable

$\displaystyle\rho(s,s_{0})\equiv\frac{\sqrt{4-s_{0}}-\sqrt{4-s}}{\sqrt{4-s_{0}}+\sqrt{4-s}}.$
(151)

###  D.1 Primal

We follow the same conventions as Karateev:2019ymz , were we make the
following ansätze for the S-matrix, the two-particle form factor FΘ and the
spectral density $\rho_{\Theta}$:

$\displaystyle S_{a}(s)$
$\displaystyle=S_{a}(2)+\sum_{n=1}^{N_{max}}(a_{a}^{(n)}\rho(s,2)^{n}+C_{ab}a_{b}^{(n)}\rho(4-s,2)^{n})$
$\displaystyle F_{\Theta}(s)$
$\displaystyle=F^{Norm}_{\Theta}+\sum_{n=1}^{N_{max}}b^{(n)}\rho(s,0)^{n}$
$\displaystyle\rho_{\Theta}(s)$
$\displaystyle=\sum_{n=0}^{1}c^{(n)}\frac{\rho(s)^{n}+\rho(s)^{n,*}}{2}+\sum_{n=1}^{N_{max}}d^{(n)}\frac{\rho(s)^{n}-\rho(s)^{n,*}}{2i}\,.$
(152)

The condition (3) fixes $S_{a}(2)$ in terms of two unknown values
$\sigma_{1,2}$ that characterize the position in the monolith that we want to
target. For obtaining the global minimal value of $c$ we can left them as an
unfixed parameter.The values $F^{Norm}_{\Theta}$ can be fixed by normalization
to be $-2\sqrt{N}$.

The optimization problem then can then be cast using (D.1) as an input in (12)
as

$c_{\text{min}}=6\pi c^{(0)}-\frac{3}{2}\pi^{2}d^{(1)}$ (153)

In the case of the anti-symmetric current we need to modify (D.1) to account
for a different normalization for the two-particle form factor
$F^{Norm}_{J}=2$ and re-scale the spectral density by

$\rho_{J}=\frac{\rho_{\Theta}(s)}{s^{2}}\,,$ (154)

to obtain a finite constrain for the sum-rule. Again to obtain the minimal
value of $\eqref{eq:ksumrule}$ we can write the minimization requirement as

$k_{\text{min}}=\frac{\pi}{4}c^{(0)}-\frac{\pi^{2}}{12}d^{(1)}$ (155)

We can now also fix a value of $c$, using (153) to fix one of the two
coefficients in the stress-enrgy spectral density and again obtain the minimal
value of $k(c)$.

###  D.2 Dual

The crossing constraints on the dual functions are
$W_{a}(4-s)C_{ab}=-W_{b}(s)$. If we call $g_{a}$ the residue at $s=2$, those
constraints give

$g_{a}=\frac{1}{2}(g_{a}+C_{ba}g_{b})\Leftrightarrow g=\frac{1+C^{T}}{2}g,$
(156)

in particular $g$ is an eigenvector of $C^{T}$ with eigenvalue 1. A convenient
basis of such eigenvectors is

$v_{1}=\begin{pmatrix}-\frac{1}{N-1}\\\ 1\\\
0\end{pmatrix},v_{2}=\begin{pmatrix}\frac{1}{N-1}\\\ 0\\\
1\end{pmatrix}\implies g=\begin{pmatrix}\frac{g_{S}-g_{A}}{N-1}\\\ g_{A}\\\
g_{S}\end{pmatrix}.$ (157)

Therefore we propose the ansätze 303030The large $s$ behaviour compatible with
the non linear constraints in (127) and (128).

$W_{a}(s)=g_{a}\frac{2}{(s-2)\sqrt{s}\sqrt{4-s}}+\frac{1}{(s-2)\sqrt{s}\sqrt{4-s}}\sum_{n=1}^{N_{max}}\left(a_{a}^{(n)}\rho^{n}(s,2)+C_{ba}a_{b}^{(n)}\rho^{n}(4-s,2)\right),$
(158)
$W_{F_{\Theta,J}}(s)=\frac{1}{s^{2}\sqrt{4-s}}\sum_{n=1}^{N_{max}}b^{(n)}_{\bullet,A}\rho^{n}(s,0),$
(159)

$\lambda_{a}(s)=\frac{1}{s\sqrt{s}\sqrt{s-4}}\left(\sum_{n=0}^{1}c^{(n)}_{a}(\rho(s)^{n}+\rho(s)^{n,*})+\sum_{n=1}^{N_{max}}d^{(n)}_{a}i(\rho(s,0)^{n}-\rho(s,0)^{n,*})\right),$
(160)

Therefore the optimization problems over the infinitely many parameters
characterizing the functions is truncated to a finite dimensional optimization
problem over the variables $g_{a},a_{a}^{(n)},b_{a}^{(n)},c_{a}^{(n)}$ and
$d_{a}^{(n)}$.

##  Appendix E Central charges for free boson and free fermion

###  E.1 Free Majorana fermion

This theory can be described by the Lagrangian

$\displaystyle\mathcal{L}=2im\psi_{+}^{a}\psi_{-}^{a}+\psi_{+}^{a}\partial_{+}\psi_{+}^{a}+(\partial_{-}\psi_{-}^{a})\psi_{-}^{a},$
(161)

where $\partial_{\pm}=\partial_{0}\pm\partial_{1}$ and $a=1,...,N$ is the O(N)
index in the vector representation There is a O(N) symmetry under the
simultaneous rotation of the two components $\psi_{\pm}$. The components of
the Noether current are

$\displaystyle
J^{0}_{[ab]}=\frac{1}{2}T_{[ab]}^{mn}(\psi_{+}^{m}\psi_{+}^{n}+\psi_{-}^{n}\psi_{-}^{m}),\qquad
J^{1}_{[ab]}=\frac{1}{2}T_{[ab]}^{mn}(\psi_{+}^{m}\psi_{+}^{n}+\psi_{-}^{m}\psi_{-}^{n}),$
(162)

where
$T_{[ab]}^{mn}=(\delta_{am}\delta_{bn}-\delta_{an}\delta_{bm})=2T^{A}_{ab,mn}$
are the generators of the adjoint representation. We compute the form factor
by expanding the Majorana fields in creation and annihilation operators. The
expansion comes from plane wave solutions of the Dirac equation313131We work
in a basis of $\gamma$ matrices
$\displaystyle\gamma^{0}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\qquad\gamma^{1}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}.$
(163) and reads

$\displaystyle\psi^{a}_{\pm}(x)=\int\frac{d\bm{p}}{2p^{0}(2\pi)}\left(a_{p}^{a}\sqrt{p_{-}}e^{ip\cdot
x}\pm a_{p}^{\dagger,a}\sqrt{p_{+}}e^{-ip\cdot x}\right),$ (164)

where $p_{\pm}=p^{0}\pm\bm{p}$. For the spatial component of the form factor
we obtain

$\displaystyle
F^{1}_{ab,cd}=\frac{1}{4}[T_{[ab]}]^{mn}\int\frac{d\bm{p}}{2p^{0}(2\pi)}\frac{d\bm{q}}{2q^{0}(2\pi)}(\sqrt{p_{-}q_{-}}+\sqrt{p_{+}q_{+}})\bra{0}a_{p}^{m}a_{q}^{n}(a_{p_{1}}^{\dagger,c}a_{p_{2}}^{\dagger,d}-a_{p_{1}}^{\dagger,d}a_{p_{2}}^{\dagger,c})\ket{0}.$
(165)

The matrix element is computed by using the anti-commutator relations

$\displaystyle\bra{0}a_{p}^{m}a_{q}^{n}a_{p_{1}}^{\dagger,c}a_{p_{2}}^{\dagger,d}\ket{0}=\\{a_{p}^{m},a_{p_{2}}^{\dagger,d}\\}\\{a_{q}^{n},a_{p_{1}}^{\dagger,c}\\}-\\{a_{p}^{m},a_{p_{1}}^{\dagger,c}\\}\\{a_{q}^{n},a_{p_{2}}^{\dagger,d}\\},$
(166)

and

$\displaystyle\\{a_{p}^{a},a_{q}^{\dagger,n}\\}=i2\pi
2p^{0}\delta(\bm{p}-\bm{q})\delta^{ab}.$ (167)

It yields

$\displaystyle
F^{1}_{ab,cd}=-\frac{1}{2}[T_{[ab]}]^{mn}(\delta^{nc}\delta^{md}-\delta^{mc}\delta^{nd})\left(\sqrt{p_{1}^{+}p_{2}^{+}}+\sqrt{p_{1}^{-}p_{2}^{-}}\right).$
(168)

To proceed we evaluate the last parenthesis in the center of mass frame (COM).
We get

$\displaystyle F^{1}_{ab,cd}=-4mT^{A}_{ab,cd}.$ (169)

This is to compare with the generic expression

$\displaystyle
F^{1}_{ab,cd}=iT^{A}_{ab,cd}(\bm{p}_{1}-\bm{p}_{2})F_{J}(s)=2iT^{A}_{ab,cd}\bm{p}F_{J}(s),$
(170)

where the second equality holds in the COM. The expressions match if

$\displaystyle F_{J}(s)=2i\frac{m}{\bm{p}}=i\frac{4m}{\sqrt{s-4m^{2}}}.$ (171)

As there isn’t any non-vanishing $n$ particles form factor for $n>2$, it is
then straightforward to obtain the spectral density as

$\displaystyle
2\pi\rho_{J}=\frac{1}{\mathcal{N}_{2}}\frac{s-4}{s}|F_{J}(s)|^{2}\theta(s-4m^{2})=\frac{8m^{2}}{s\sqrt{s}\sqrt{s-4m^{2}}}\theta(s-4m^{2}),$
(172)

and the corresponding value for $k$

$\displaystyle k=\frac{\pi}{2}\int_{4m^{2}}^{\infty}ds\,\rho_{J}(s)=1.$ (173)

For the trace of the stress energy tensor we get

$\Theta(x)=T^{0}_{0}+T^{1}_{1}=im\psi^{a}_{+}\psi^{a}_{-},$ (174)

which following identical steps as above yields

$F_{\Theta}(p_{1},p_{2})=\sqrt{N}m\left(\sqrt{p_{1-}p_{2+}}-\sqrt{p_{2-}p_{1+}}\right)=im\sqrt{N}\sqrt{s-4m^{2}},$
(175)

where the last equality comes fromes evaluating the expression in the COM
frame. Again there is no $n>2$ form factors, so this gives the spectral
density

$\rho_{\Theta}(s)=\frac{|F_{\Theta}|^{2}}{4\pi\sqrt{s}\sqrt{s-4m^{2}}}\theta(s-4m^{2})=\frac{Nm^{2}\sqrt{s-4m^{2}}}{4\pi\sqrt{s}}\theta(s-4m^{2}).$
(176)

We therefore deduce the central charge $c$

$\displaystyle
c=\int_{4m^{2}}^{\infty}ds\frac{\rho_{\Theta}}{s^{2}}=\frac{N}{2}.$ (177)

Let us comment about the large $s$ behaviour. For a generic operator of
conformal dimension $\Delta$ we expect the high energy regime to be dictated
by the CFT, for which we obtain Karateev:2019ymz

$\displaystyle\lim_{s\to\infty}\rho_{\mathcal{O}}(s)=\text{const}\times
s^{\Delta-d/2},\qquad\lim_{s\to\infty}F_{\mathcal{O}}(s)\lesssim
s^{1+\frac{\Delta-d}{2}}.$ (178)

The spectral densities (172) and (176) seem to be in contradiction with this
result. It is because our operators $J^{\mu}$ and $\Theta$ are not generic
operators. For $J$, it is a spin 1 conserved current for which the "const" in
the above equation, being proportional to $\Delta_{J}-1$, vanishes
Karateev:2020axc , and indeed our result $\rho_{J}$ vanishes when
$s\to\infty$.323232As explained in Karateev:2020axc , in general spectral
densities of vector operators also have a spin 0 component, which for us was
set to zero due to the conservation of the currents. This is compatible with
equation (4.10) in Karateev:2020axc  where ”const” is proportional to
$\Delta_{J}-d+1$ which vanishes for conserved currents in $d=2$. As for
$\Theta$, since it is the trace of the stress energy tensor, its value is just
zero in the CFT and the result (178) does not apply.

###  E.2 Free boson

The free boson theory is described by the Lagrangian

$\displaystyle\mathcal{L}=-\frac{1}{2}\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{a}-\frac{1}{2}\phi^{a}\phi^{a}.$
(179)

The Noether currents are

$\displaystyle J^{\mu}_{[ab]}=[T_{[ab]}]^{mn}\partial^{\mu}\phi^{m}\phi^{n}.$
(180)

The mode expansion of the bosonic field reads

$\displaystyle\phi^{a}(x)=\int\frac{d\bm{p}}{2p^{0}(2\pi)}\left(a_{p}^{a}e^{ip\cdot
x}+a_{p}^{\dagger,a}e^{-ip\cdot x}\right).$ (181)

Plugging it in the definition of the form factor we get

$F^{\mu}_{ab,cd}=[T_{[ab]}]^{mn}\frac{i}{4}\int\frac{d\bm{p}}{2p^{0}(2\pi)}\frac{d\bm{q}}{(2\pi)}\bra{0}a_{p}^{m}a_{q}^{n}(a_{p_{1}}^{\dagger,c}a_{p_{2}}^{\dagger,d}-a_{p_{1}}^{\dagger,d}a_{p_{2}}^{\dagger,c})\ket{0}$
(182)

The matrix element can be evaluated by using the commutator relations

$\displaystyle\bra{0}a_{p}^{m}a_{q}^{n}a_{p_{1}}^{\dagger,c}a_{p_{2}}^{\dagger,d}\ket{0}=[a_{p}^{m},a_{p_{1}}^{\dagger,c}][a_{q}^{n},a_{p_{2}}^{\dagger,d}]+[a_{p}^{m},a_{p_{2}}^{\dagger,d}][a_{q}^{n},a_{p_{1}}^{\dagger,c}],$
(183)

and

$\displaystyle[a_{p}^{a},a_{q}^{\dagger,n}]=2\pi
2p^{0}\delta(\bm{p}-\bm{q})\delta^{ab}.$ (184)

We get

$\displaystyle
F^{\mu}_{ab,cd}=[T_{[ab]}]^{mn}\frac{i}{2}(\delta^{nd}\delta^{mc}-\delta^{nc}\delta^{md})(p_{1}^{\mu}-p_{2}^{\mu})=2iT_{A}^{ab,cd}(p_{1}^{\mu}-p_{2}^{\mu}).$
(185)

Comparing to the general decomposition for the form factor

$\displaystyle
F^{\mu}_{ab,cd}=iT^{A}_{ab,cd}(p_{1}^{\mu}-p_{2}^{\mu})F_{J}(s),$ (186)

we identify

$\displaystyle F_{J}(s)=2.$ (187)

We deduce the spectral density

$\displaystyle\rho_{J}(s)=\frac{\sqrt{s-4}}{\pi s\sqrt{s}}\theta(s-4m^{2}).$
(188)

We get $k$ by using the sum rule yielding

$\displaystyle k=\frac{\pi}{2}\int_{4m^{2}}^{\infty}ds\,\rho_{J}=\infty,$
(189)

which can be understood by noticing that the free boson $\phi$ is not a
primary and the current two point function has a $\operatorname{log}(z)/z^{2}$
leading singularity. Moving to the stress energy tensor we get

$\displaystyle\Theta(x)=-m^{2}\phi^{a}\phi^{a},$ (190)

leading to the form factor and spectral density

$F_{\Theta}(p_{1},p_{2})=-2\sqrt{N}m^{2},\qquad\rho_{\Theta}(s)=\frac{Nm^{4}}{\pi\sqrt{s}\sqrt{s-4m^{2}}}\theta(s-4m^{2}).$
(191)

The central charge $c$ is therefore

$\displaystyle c=12\pi\int_{4m^{2}}^{\infty}ds\frac{\rho_{\Theta}}{s^{2}}=N.$
(192)

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