Bootstrapping 2d $\phi^{4}$ Theory

with Hamiltonian Truncation Data

Hongbin Chen1, A. Liam Fitzpatrick1, Denis Karateev2

1Department of Physics, Boston University, Boston, MA 02215, USA
2Philippe Meyer Institute, Physics Department École Normale Supérieure (ENS),
Université PSL24 rue Lhomond, F-75231 Paris, France

We combine the methods of Hamiltonian Truncation and the recently proposed
generalisation of the S-matrix bootstrap that includes local operators to
determine the two-particle scattering amplitude and the two-particle form
factor of the stress tensor at $s>0$ in the 2d $\phi^{4}$ theory. We use the
form factor of the stress tensor at $s\leq 0$ and its spectral density
computed using Lightcone Conformal Truncation (LCT), and inject them into the
generalized S-matrix bootstrap set-up. The obtained results for the scattering
amplitude and the form factor are fully reliable only in the elastic regime.
We independently construct the “pure” S-matrix bootstrap bounds (bootstrap
without including matrix elements of local operators), and find that the sinh-
Gordon model and its analytic continuation the “staircase model” saturate
these bounds. Surprisingly, the $\phi^{4}$ two-particle scattering amplitude
also very nearly saturates these bounds, and moreover is extremely close to
that of the sinh-Gordon/staircase model.

###### Contents

  1. 1 Introduction
    1. 1.1 Models in 2d
    2. 1.2 Summary of Main Results
  2. 2 Basic Definitions and Notation
  3. 3 Analytic Results
    1. 3.1 Sinh-Gordon Model
    2. 3.2 Staircase Model
    3. 3.3 $\phi^{4}$ model
    4. 3.4 2d $O(N)$ model in the large $N$ limit
    5. 3.5 $T\overline{T}$ deformation of the 2d Ising
  4. 4 Pure S-matrix bootstrap
    1. 4.1 Set-up
    2. 4.2 Numerical Results
  5. 5 S-matrix and Form Factor Bootstrap
    1. 5.1 Set-up
    2. 5.2 Numerical Results
      1. 5.2.1 Infinite Precision Example
      2. 5.2.2 $\phi^{4}$ model
    3. 5.3 Comparison of the sinh-Gordon model and $\phi^{4}$ model
  6. 6 Discussion and Future Directions
  7. A Kinematics of 2d Scattering
  8. B $O(N)$ model
  9. C Perturbative Computations
    1. C.1 Feynman Diagrams
      1. C.1.1 $\phi^{4}$ theory
      2. C.1.2 2d $O(N)$ model in the large $N$ limit
    2. C.2 Dispersion Relations
    3. C.3 Nonrelativistic Limit
  10. D Sinh-Gordon Form Factors and $C$-function

##  1 Introduction

There is a set of powerful non-perturbative techniques to study quantum field
theories (QFTs) commonly referred to as “bootstrap” methods. Such methods
attempt to bound the space of QFTs using only basic principles such as
symmetries, unitarity, crossing, etc. The most famous bootstrap technique is
the numerical conformal bootstrap pioneered in Rattazzi:2008pe . It allows one
to derive precise bounds on the space of conformal field theories (CFTs), see
Poland:2018epd  for a review. Another bootstrap technique which allows one to
study QFTs with a mass gap was pioneered in Paulos:2016but ; Paulos:2017fhb .
In this paper we will refer to it as the numerical $S$-matrix bootstrap. The
$S$-matrix bootstrap gained further attention in recent years, see
Paulos:2016fap ; Doroud:2018szp ; He:2018uxa ; Cordova:2018uop ;
Guerrieri:2018uew ; Homrich:2019cbt ; EliasMiro:2019kyf ; Cordova:2019lot ;
Bercini:2019vme ; Correia:2020xtr ; Bose:2020shm ; Guerrieri:2020bto ;
Hebbar:2020ukp ; He:2021eqn ; Guerrieri:2021ivu ; Miro:2021rof ;
Guerrieri:2020kcs ; Guerrieri:2021tak . The recent work Karateev:2019ymz ;
Karateev:2020axc  made a concrete proposal for how to extended the $S$-matrix
bootstrap to accommodate form factors and spectral densities. We will refer to
this approach as the numerical $S$-matrix/form factor bootstrap.

A simultaneous advantage and disadvantage of bootstrap methods is their model-
independent nature. If one wants to study some particular model one
generically has to inject additional model specific information. The amount of
this additional information highly depends on the situation. For example one
can solve numerically the 3d Ising model using the conformal bootstrap method
by simply specifying that there are only two relevant operators in the
spectrum, one is $Z_{2}$ even and one is $Z_{2}$ odd, see ElShowk:2012ht .
Another example is the work Guerrieri:2018uew , where the authors attempted to
study the 4d QCD by using the $S$-matrix bootstrap injecting some known
information from chiral perturbation theory. Another notable example in this
spirit is the study of the 2d Ising Field Theory in Gabai:2019ryw , where the
authors injected the S-matrix of the theory in one kinematic regime to learn
about its behavior more generally.

A great class of tools for obtaining non-perturbative results in a particular
model are Hamiltonian Truncation methods, which involve numerically
diagonalizing the Hamiltonian in a finite dimensional subspace of the full
Hilbert space. This approach is a special case of more general variational
methods, so all else being equal the larger the truncation subspace, the more
accurate the approximation to the eigenstates of the Hamiltonian. There are
various different ways one can try to implement Hamiltonian truncation for
continuum QFT, the most well-known probably being the Truncated Conformal
Space Approximation (TCSA) of Zamolodchikov and Yurov yurov1990truncated ;
yurov1991correlation , see james2018non  for a recent review and guide to the
literature. One immediate output of such methods is the mass spectrum, which
is just the set of eigenvalues of the Hamiltonian. Because one also obtains
the eigenvectors of the Hamiltonian, one can compute spectral densities of
local operators quite straightforwardly. By contrast, constructing multi-
particle asymptotic states with Hamiltonian methods is much more subtle, since
these are not just eigenstates of the Hamiltonian. Thus the computation of
observables like the scattering amplitudes requires a more involved approach.

The main goal of this paper is to study non-perturbatively the 2d $\phi^{4}$
model (in the unbroken phase) and to compute as many observables as we can. In
a companion paper truncffsd , we used the Lightcone Conformal Truncation (LCT)
method to compute the two-particle form factor of the stress tensor in the
unphysical regime ($s\leq 0$) and its spectral density. In this paper, we will
inject this data into the S-matrix/form factor bootstrap program, and obtain
the form factor at $s>0$ and also the elastic 2-to-2 scattering amplitude in
the $\phi^{4}$ model.

The rest of the introduction is organized as follows. In section 1.1, we
discuss a wide class of scalar field theories in 2d, where we precisely define
the $\phi^{4}$ model and discuss its relation with other models. In section
1.2, we provide an extended summary of our main results.

###  1.1 Models in 2d

Let us consider the class of quantum field theories in 2d which consists of a
single real scalar field $\phi(x)$ and is defined as the deformation of the
free scalar field theory in the UV by a potential $V(\phi)$. The corresponding
action reads

$S_{UV}=\int d^{2}x\left(-{1\over 2}(\partial\phi)^{2}-V(\phi)\right).$ (1.1)

Notice that the field $\phi(x)$ has the mass dimension zero, $[\phi]=0$. This
situation is special to 2d and allows for complicated potentials $V(\phi)$ not
present in higher dimensions. In this paper we will further restrict our
attention to potentials which are invariant under the following $Z_{2}$
transformation $\phi(x)\rightarrow-\phi(x)$. The most generic potential then
has the following form

$V(\phi)={1\over
2}m_{0}^{2}\phi^{2}+\sum_{n=2}^{\infty}g_{2n}\phi^{2n}+\text{counterterms},$
(1.2)

where $m_{0}$ is the mass-like parameter and $g_{2n}$ is an infinite set of
coupling constants. We focus on the case when $m_{0}^{2}>0$ (unbroken phase).
Below we will define and discuss several potentials $V(\phi)$. We will take
the operators to be normal-ordered in order to remove divergences in the
theory; this choice is equivalent to a hard cutoff with a particular choice
for the counterterms above.

We start with two integrable models called the sine-Gordon and the sinh-Gordon
models. They are given by the following potentials respectively

$\displaystyle V_{\text{sine-Gordon}}(\phi)$ $\displaystyle\equiv-
m_{0}^{2}\beta^{-2}\left(\cos(\beta\phi)-1\right)+\text{counterterms},$ (1.3)
$\displaystyle V_{\text{sinh-Gordon}}(\phi)$
$\displaystyle\equiv+m_{0}^{2}\beta^{-2}\left(\cosh(\beta\phi)-1\right)+\text{counterterms}.$
(1.4)

Here $\beta$ is the single dimensionless parameter which specifies the models.
Expanding these potentials around $\phi=0$ one can bring them to the form
given by (1.2), and thus express all the $g_{2n}$ coefficients in terms of
$\beta$. The two models are formally related by the replacement
$\beta\leftrightarrow i\beta$. These two models have been extensively studied
in the literature. For a summary of the sine-Gordon results see, for example,
section 4.1 in Karateev:2019ymz  and references therein. We will summarize the
results for the sinh-Gordon model in section 3.1.

Another interesting model is the $\phi^{4}$ model. It will play the central
role in this paper. It is defined by the potential (1.2) with
$g_{4}=\lambda/4!$ and $g_{2n}=0$ for all $n\geq 3$. Here $\lambda\geq 0$ is
the quartic coupling constant. This is possibly the simplest quantum field
theory model one can think of. Let us write out its potential explicitly, it
reads

$V_{\phi^{4}}\equiv{1\over 2}m_{0}^{2}\,\phi^{2}+{\lambda\over
4!}\phi^{4}+{1\over 2}\delta_{m}\phi^{2}.$ (1.5)

No counterterms are required for the coupling constant $\lambda$ in $d=2$.
Normal-ordering the interaction and setting $\delta_{m}=0$ is equivalent to
choosing a hard cutoff $\Lambda_{\rm cutoff}$ and setting
$\delta_{m}=-{\lambda\over 8\pi}\log\Lambda_{\rm cutoff}^{2}/m_{0}^{2}$. The
quartic coupling $\lambda$ has mass dimension $[\lambda]=2$; we define the
dimensionless quartic coupling $\overline{\lambda}$ as111We caution the reader
that this convention for $\overline{\lambda}$ differs from that in
Anand:2020gnn ;
$\overline{\lambda}_{\text{here}}=4\pi\overline{\lambda}_{\text{there}}$.

$\overline{\lambda}\equiv m_{0}^{-2}\lambda.$ (1.6)

The $\phi^{4}$ model is non-integrable, and one needs numerical non-
perturbative techniques in order to compute observables in this theory.222See
e.g. Chabysheva:2015ynr ; Burkardt:2016ffk ; Schaich:2009jk ; Milsted:2013rxa
; Bosetti:2015lsa ; Rychkov:2014eea ; Rychkov:2015vap ; Bajnok:2015bgw ;
Elliott:2014fsa ; Chabysheva:2016ehd ; Serone:2018gjo ; Tilloy:2021hhb  for
various recent nonperturbative works on this model. It was shown in
Anand:2017yij  that the $\phi^{4}$ model in lightcone quantization333It is
important to note that, due to the contribution from zero modes, the critical
value of the coupling differs in equal-time and lightcone quantization. See
Burkardt ; Burkardt2 ; Fitzpatrick:2018xlz  for details. in the unbroken phase
is in the following range

$\overline{\lambda}\in[0,\,23.1].$ (1.7)

The critical value $\overline{\lambda}\approx 23.1$ leads to the conformal IR
fixed point given by the free massless Majorana fermion (which is the 2d Ising
model).

Finally, we consider the 2d $O(N)$ model, which is the case when instead of a
single field $\phi$, we have $N$ fields $\phi_{1}$, $\phi_{2}$, $\ldots$,
$\phi_{N}$ with the same mass. Requiring the $O(N)$ symmetry we can write the
following analogue of the pure $\phi^{4}$ theory

$V_{\phi^{4}}^{O(N)}(\phi)\equiv{1\over
2}m_{0}^{2}(\phi_{i}\phi_{i})+{\lambda\over
8N}(\phi_{i}\phi_{i})(\phi_{j}\phi_{j})+{1\over
2}\delta_{m}(\phi_{i}\phi_{i}),$ (1.8)

where there is an implicit summation over the repeated indices. In the large
$N$ limit when $N\rightarrow\infty$ the model becomes integrable. We will
mostly use it in this paper to check our numerical procedures.

To conclude our brief discussion of 2d models let us clarify an important
point. One can consider an infinite class of potentials (1.2) with
$g_{4}=\lambda/4!$ and

$m_{0}^{-2}g_{2n}\ll 1,\qquad n\geq 3.$ (1.9)

All such models will lead to observables very similar to the pure $\phi^{4}$
model (1.5). Using non-perturbative techniques we can compute in practice
observables only at finite precision and thus we will never be able to
distinguish the pure $\phi^{4}$ model from this infinite class of models. In
order to be pedantic we say that we compute observables for $\phi^{4}$-like
theories. In this sense, the sinh-Gordon model belongs to the class of
$\phi^{4}$-like models if $\beta^{2}=\overline{\lambda}$ at very small values
of the coupling, $\overline{\lambda}\ll 4\pi$.

###  1.2 Summary of Main Results

Given a model there are various observables one would like to compute. In this
paper we will focus on three different observables: the two-particle form
factor of the trace of the stress tensor $\mathcal{F}_{2,0}^{\Theta}(s)$, the
spectral density of the trace of the stress tensor $\rho_{\Theta}(s)$ and the
two-to-two scattering amplitude $\mathcal{S}(s)$. For their precise
definitions see section 2. Given the definition of the $\phi^{4}$ model in
(1.5), one would like to obtain the above observables in terms of the bare
coupling $\overline{\lambda}$ and the mass-like parameter $m_{0}$ which simply
sets the scale.444In practice we provide all our final expressions in terms of
the physical mass $m$ which can also be computed in terms of $m_{0}$ for a
given value of $\overline{\lambda}$. In the $\overline{\lambda}\ll 4\pi$
regime, one can use perturbation theory to do that. For
$\overline{\lambda}\gtrsim 4\pi$, one can use Hamiltonian Truncation methods
instead. In the companion paper truncffsd , we have computed the spectral
density of the trace of the stress tensor $\rho_{\Theta}(s)$ and two-particle
form factor $\mathcal{F}_{2,0}^{\Theta}(s)$ at $s\leq 0$ for various values of
$\overline{\lambda}$, see figure 4 and 12 therein. The main goal of this paper
is to compute $\mathcal{F}_{2,0}^{\Theta}(s)$ for $s>0$ and the scattering
amplitude $\mathcal{S}(s)$ given the input of truncffsd . Below we outline the
main results.

We start by employing the pure $S$-matrix bootstrap to study the space of
scattering amplitudes of a single $Z_{2}$ odd particle with the physical mass
$m$. One can characterize such amplitudes for example by their value (and the
value of their derivatives) at the crossing symmetric point $s=2m^{2}$. See
(2.26) and (2.27) for details. Using crossing, analyticity and unitarity we
construct a non-perturbative bound on a two-dimensional subspace of these
parameters. The bound is presented in figure 1. We discover that the left tip
describes the scattering of free bosons and the right tip describes the
scattering of free Majorana fermions (i.e. the 2d Ising model). The two tips
are connected by the lower and upper edges. The lower edge is saturated by the
sinh-Gordon model and its analytic continuation the “staircase model”.

We then propose a strategy which allows one to inject the Hamiltonian
Truncation data of truncffsd  into the $S$-matrix/form factor bootstrap. This
allows one to isolate a specific theory, instead of constructing generic
bounds on the space of allowed theories. Using this strategy we numerically
obtain the form factor of the trace of the stress tensor
$\mathcal{F}_{2,0}^{\Theta}(s)$ with $s>0$ and the scattering amplitude
$\mathcal{S}(s)$. The results are presented in figures 9 \- 11 for various
values of $\overline{\lambda}$. In the regime $\overline{\lambda}\ll 4\pi$
they agree with perturbative expressions. Due to the limitations of the
$S$-matrix/form factor bootstrap restricted to two-particle scattering states,
we expect that our results are fully accurate only in the “elastic” regime for
$s\leq 16m^{2}$. Given the numerical scattering amplitudes in the $\phi^{4}$
model, we can determine the position of the $\phi^{4}$ model with respect to
the generic bound given in figure 1. It turns out that the $\phi^{4}$ model
lies very close to the lower edge of this plot, see figure 7. This means that
the $\phi^{4}$ model is very similar to the sinh-Gordon/staircase model (which
is exactly on the lower edge) if one only looks at the two-dimension subspace
shown in this plot. This fact calls for further investigation which we
summarize in the next paragraph. Leaving this issue aside for a moment we
observe that the $\phi^{4}$ model starts close to the free boson theory for
small values of $\overline{\lambda}$ and monotonically moves towards the free
fermion theory (2d Ising) along the lower edge when we increase
$\overline{\lambda}$. This behaviour is in agreement with the fact that there
is a critical value of $\overline{\lambda}$ when the $\phi^{4}$ theory flows
to the 2d Ising fixed point.

The $\phi^{4}$ and the sinh-Gordon models are inherently different. The former
has particle production and the latter does not. In practice this difference
will become evident for example if we look at the scattering amplitude
$\mathcal{S}(s)$ in the “non-elastic” regime $s\geq 16m^{2}$. For small values
of $\overline{\lambda}\ll 4\pi$, the $\phi^{4}$ model is expected to be very
similar to the sinh-Gordon model with $\beta^{2}=\overline{\lambda}$. To our
surprise we have discovered that at strong coupling, the $\phi^{4}$ model
still gives very similar observables in the “elastic” regime as the sinh-
Gordon model with some value $\beta_{*}^{2}$. Notice however that
$\beta_{*}^{2}\neq\overline{\lambda}$. For large values of
$\overline{\lambda}$, the value of $\beta_{*}^{2}$ is allowed to become
complex (describing the “staircase model”). The comparison of the observables
in the $\phi^{4}$ model and in the sinh-Gordon model is given in figure 16, 16
and 17. There, one sees a striking similarity of the two models in the
“elastic” regime and their small deviation in the non-elastic regime.

##### Outline of the paper

We summarize basic definitions and set up the notation in section 2. We
summarize various analytic results in section 3. For instance we discuss the
sinh-Gordon model and its analytic continuation (the “staircase model”), the
$\phi^{4}$ model in perturbation theory, and the 2d $O(N)$ in the large $N$
limit. In section 4, we construct a generic bound on the space of scattering
amplitudes of $Z_{2}$ odd particles. In section 5, we show how one can inject
the LCT data into the $S$-matrix/form factor bootstrap and apply this strategy
to the 2d $\phi^{4}$ model. We discuss open questions and further directions
in section 6.

Some supplementary material is provided in appendices. We review the details
of the 2d kinematics in appendix A. We discuss the 2d $O(N)$ models and their
large $N$ limit in appendix B. We provide details of perturbative and large
$N$ computations in the $\phi^{4}$ model and the $O(N)$ model in appendix C.
Finally, in appendix D, we discuss two- and four-particle form factors in the
sinh-Gordon model.

##  2 Basic Definitions and Notation

Let us start by carefully defining the most important objects for our work. We
will first work with scalars and general number of dimensions $d$, and focus
on $d=2$ in the second half of this section. We also define the scattering
amplitudes for 2d Majorana fermions (which is used in later sections) at the
end of this section. We will use the “mostly plus” Lorentzian metric

$\eta^{\mu\nu}=\\{-1,+1,+1,\ldots\\}.$ (2.1)

We require that our quantum field theory contains the local stress tensor
$T^{\mu\nu}(x)$ which obeys the following conditions

$T^{\mu\nu}(x)=T^{\nu\mu}(x),\qquad\partial_{\mu}T^{\mu\nu}(x)=0.$ (2.2)

We denote its trace by

$\Theta(x)\equiv\eta_{\mu\nu}T^{\mu\nu}(x).$ (2.3)

One of the simplest observables of any theory is two-point function of the
trace of the stress tensor. One can distinguish the Wightman two-point
function

$\langle{\rm vac}|\Theta(x_{1})\Theta(x_{2})|{\rm
vac}\rangle_{W}\equiv\lim_{\epsilon\rightarrow 0^{+}}\langle{\rm
vac}|\Theta(x_{1}^{0}-i\epsilon,\vec{x}_{1})\Theta(x_{2})|{\rm vac}\rangle$
(2.4)

and the time-ordered two-point function

$\langle{\rm vac}|\Theta(x_{1})\Theta(x_{2})|{\rm
vac}\rangle_{T}\equiv\theta(x_{1}^{0}-x_{2}^{0})\langle{\rm
vac}|\Theta(x_{1})\Theta(x_{2})|{\rm
vac}\rangle_{W}+\theta(x_{2}^{0}-x_{1}^{0})\langle{\rm
vac}|\Theta(x_{2})\Theta(x_{1})|{\rm vac}\rangle_{W}.$ (2.5)

Let us introduce the spectral density of the trace of the stress tensor
$\rho_{\Theta}$ as the Fourier transformed Wightman two-point function. In the
notation of Weinberg:1995mt  we have

$2\pi\theta(p^{0})\rho_{\Theta}(-p^{2})\equiv\int d^{d}xe^{-ip\cdot
x}\langle{\rm vac}|\Theta(x)\Theta(0)|{\rm vac}\rangle_{W}.$ (2.6)

One can express $\rho_{\Theta}$ also in terms of the time-ordered two-point
function as follows555For a simple derivation see for example the beginning of
section 3.2 in Karateev:2020axc .

$2\pi\theta(p^{0})\rho_{\Theta}(-p^{2})=2\,\text{Re}\int d^{d}xe^{-ip\cdot
x}\langle{\rm vac}|\Theta(x)\Theta(0)|{\rm vac}\rangle_{T}.$ (2.7)

In quantum field theories with a “mass gap” there exist one-particle
asymptotic in and out states denoted by

$|m,\vec{p}\,\rangle_{\text{in}}\qquad|m,\vec{p}\,\rangle_{\text{out}},$ (2.8)

where $m$ and $\vec{p}$ stand for the physical mass and the $(d-1)$-momentum
of the asymptotic particles. The standard normalization choice for them reads
as

${}_{\text{in}}\langle
m,\vec{p}_{1}\,|m,\vec{p}_{2}\,\rangle_{\text{in}}={}_{\text{out}}\langle
m,\vec{p}_{1}\,|m,\vec{p}_{2}\,\rangle_{\text{out}}=2\sqrt{m^{2}+\vec{p}_{1}^{\,2}}\times(2\pi)^{d-1}\delta^{d-1}(\vec{p}_{1}-\vec{p}_{2}).$
(2.9)

Using the one-particle asymptotic states one can construct general
$n$-particle asymptotic states with $n\geq 2$, for details see for example
section 2.1.2 in Karateev:2019ymz . Let us denote the two-particle in and out
asymptotic states by

$|m,\vec{p}_{1};m,\vec{p}_{2}\,\rangle_{\text{in}}\quad\text{and}\quad|m,\vec{p}_{1};m,\vec{p}_{2}\,\rangle_{\text{out}}.$
(2.10)

They are constructed in such a way that they obey the following normalization

${}_{\text{in}}\langle
m,\vec{k}_{1};m,\vec{k}_{2}|m,\vec{p}_{1};m,\vec{p}_{2}\,\rangle_{\text{in}}={}_{\text{out}}\langle
m,\vec{k}_{1};m,\vec{k}_{2}|m,\vec{p}_{1};m,\vec{p}_{2}\,\rangle_{\text{out}}=\\\
4\sqrt{m^{2}+\vec{p}_{1}^{\,2}}\sqrt{m^{2}+\vec{p}_{2}^{\,2}}\,(2\pi)^{2(d-1)}\delta^{(d-1)}(\vec{p}_{1}-\vec{k}_{1})\delta^{(d-1)}(\vec{p}_{2}-\vec{k}_{2})+(\vec{p}_{1}\leftrightarrow\vec{p}_{2}).$
(2.11)

Using the asymptotic states one can define another set of observables called
the form factors. In this work we will use the following form factors of the
trace of the stress tensor

$\displaystyle\mathcal{F}^{\Theta}_{1,1}(t)$
$\displaystyle\equiv{}_{\text{out}}\langle
m,\vec{p}_{1}\,|\Theta(0)|m,\vec{p}_{2}\rangle_{\text{in}},$ (2.12)
$\displaystyle\mathcal{F}^{\Theta}_{2,0}(s)$
$\displaystyle\equiv{}_{\text{out}}\langle
m,\vec{p}_{1}\,;m,\vec{p}_{2}\,|\Theta(0)|{\rm vac}\rangle.$

Here we have introduced the analogues of the Mandelstam variables for the form
factors which are defined as

$s\equiv-(p_{1}+p_{2})^{2},\qquad t\equiv-(p_{1}-p_{2})^{2},\qquad
s+t=4m^{2}.$ (2.13)

The latter relation simply follows from the definitions of $s$ and $t$. The
two form factors in (2.12) are related by the crossing symmetry as
follows666The crossing symmetry is the condition that the matrix elements in
(2.12) remain invariant under the following change of the $d$-momenta
$p^{\mu}_{2,\text{out}}\rightarrow-p^{\mu}_{2,\text{in}}$.

$\mathcal{F}^{\Theta}_{2,0}(s)=\mathcal{F}^{\Theta}_{1,1}(s).$ (2.14)

The Ward identity imposes the following normalization condition

$\lim_{s\rightarrow 0}\mathcal{F}^{\Theta}_{2,0}(s)=-2m^{2}.$ (2.15)

See for example appendix G in Karateev:2020axc  for its derivation. The
condition (2.15) can be seen as the definition of the physical mass.

The last observable in which we are interested is the scattering amplitude
$\mathcal{S}(s)$. We define it via the following matrix element

$\mathcal{S}(s,t)\times(2\pi)^{(d-1)}\delta^{(d)}(p_{1}+p_{2}-k_{1}-k_{2})\equiv{}_{\text{out}}\langle
m,\vec{k}_{1};m,\vec{k}_{2}|m,\vec{p}_{1};m,\vec{p}_{2}\,\rangle_{\text{in}}.$
(2.16)

In this case, the usual Mandelstam variables are defined as

$s\equiv-(p_{1}+p_{2})^{2},\quad t\equiv-(p_{1}-k_{1})^{2},\quad
u\equiv-(p_{1}-k_{2})^{2},\quad s+t+u=4m^{2}.$ (2.17)

The difference between (2.13) and (2.17) should be understood from the
context. Instead of using the full scattering amplitude $\mathcal{S}(s)$, it
is often very convenient to define the interacting part of the scattering
amplitude $\mathcal{T}(s,t)$ as follows

$i\mathcal{T}(s,t)\times(2\pi)^{(d-1)}\delta^{(d)}(p_{1}+p_{2}-k_{1}-k_{2})\equiv\\\
{}_{\text{out}}\langle
m,\vec{k}_{1};m,\vec{k}_{2}|m,\vec{p}_{1};m,\vec{p}_{2}\,\rangle_{\text{in}}-{}_{\text{out}}\langle
m,\vec{k}_{1};m,\vec{k}_{2}|m,\vec{p}_{1};m,\vec{p}_{2}\,\rangle_{\text{out}}.$
(2.18)

We have defined the observables in general dimensions up to now. In the rest
of this section, we focus on $d=2$. In the special case of $d=2$, the
scattering amplitude takes a particularly simple form since it depends only on
the single Mandelstam variable $s$. In our convention, $u=0$. This restriction
is imposed via the Heaviside step function $\theta$ (not to confuse with the
rapidity $\theta$ that is used in later sections) in the equations below. We
provide details of 2d kinematics in appendix A.777See also the end of section
2 in Zamolodchikov:1978xm  (in particular equations (2.9) - (2.11) and the
surrounding discussion). In $d=2$ we define the scattering amplitude of
identical particles as888Note that although we use the vector notation
$\vec{p}$ for the momentum, it is really just a single number, since are are
in 2d, and the step functions make sense.,999In the left-hand of this equation
and all similar equations below it is understood that the scattering amplitude
implicitly contains the appropriate step functions. This is is because all our
amplitudes are required to have $u=0$.

$\mathcal{S}(s)\times(2\pi)\delta^{(2)}(p_{1}+p_{2}-k_{1}-k_{2})\equiv\\\
{}_{\text{out}}\langle
m,\vec{k}_{1};m,\vec{k}_{2}|m,\vec{p}_{1};m,\vec{p}_{2}\,\rangle_{\text{in}}\times\theta(\vec{p}_{1}-\vec{p}_{2})\theta(\vec{k}_{2}-\vec{k}_{1}).$
(2.19)

The interacting part of the scattering amplitude in $d=2$ is then defined as

$\mathcal{T}(s)\times(2\pi)\delta^{(2)}(p_{1}+p_{2}-k_{1}-k_{2})\equiv\Big{(}{}_{\text{out}}\langle
m,\vec{k}_{1};m,\vec{k}_{2}|m,\vec{p}_{1};m,\vec{p}_{2}\,\rangle_{\text{in}}\\\
-{}_{\text{out}}\langle
m,\vec{k}_{1};m,\vec{k}_{2}|m,\vec{p}_{1};m,\vec{p}_{2}\,\rangle_{\text{out}}\Big{)}\times\theta(\vec{p}_{1}-\vec{p}_{2})\theta(\vec{k}_{2}-\vec{k}_{1}).$
(2.20)

In $d=2$, it is straightforward to rewrite (2.11) in the following way

${}_{\text{out}}\langle
m,\vec{k}_{1};m,\vec{k}_{2}|m,\vec{p}_{1};m,\vec{p}_{2}\rangle_{\text{out}}=\mathcal{N}_{2}\times(2\pi)^{2}\delta^{(2)}(p_{1}+p_{2}-k_{1}-k_{2}),$
(2.21)

where we have defined

$\mathcal{N}_{2}\equiv 2\sqrt{s}\sqrt{s-4m^{2}}.$ (2.22)

Combining (2.16), (2.18) and (2.21), we obtain the following simple relation
between the full amplitude and its interacting part

$\mathcal{S}(s)=\mathcal{N}_{2}+i\mathcal{T}(s).$ (2.23)

It is also convenient to introduce the following amplitude (which can be seen
as the analogue of the partial amplitudes in higher dimensions)

$\widehat{\mathcal{S}}(s)\equiv\mathcal{N}_{2}^{-1}\mathcal{S}(s)=1+i\,\mathcal{N}_{2}^{-1}\mathcal{T}(s).$
(2.24)

Unitarity imposes the following constraint:

$\left|\widehat{\mathcal{S}}(s)\right|^{2}\leq 1,\quad\text{for }s\geq
4m^{2}.$ (2.25)

One can define the non-perturbative quartic coupling $\Lambda$ via the
interacting part of the physical amplitude as

$\Lambda\equiv-\lim_{s\rightarrow 2m^{2}}\mathcal{T}(s).$ (2.26)

We can also define the following set of non-perturbative parameters

$\Lambda^{(n)}\equiv\lim_{s\rightarrow 2m^{2}}\partial_{s}^{n}\mathcal{T}(s).$
(2.27)

Note that there is no minus sign in the definition of $\Lambda^{(n)}$, which
we find to be convenient. Due to the crossing symmetry $s\leftrightarrow
4m^{2}-s$, all the odd derivatives in $s$ at the crossing symmetric point
vanish. The infinite set of physical non-perturbative parameters $\Lambda$,
$\Lambda^{(2)}$, $\Lambda^{(4)}$, $\ldots$ can be chosen to fully describe any
scattering process.

According to Zamolodchikov:1986gt ; Cardy:1988tj , in $d=2$, one can define
the $C$-function as

$C(s_{0})\equiv 12\pi\int_{0}^{s_{0}}ds\,{\rho_{\Theta}(s)\over s^{2}}.$
(2.28)

The UV central charge $c_{UV}$ is related to the $C$-function in the following
simple way101010Here we use the standard conventions for the central charge
$c_{UV}$ in which the theory of a free scalar boson has $c_{UV}=1$. For a
summary of the standard conventions see for example the end of appendix A in
Karateev:2020axc .,111111For the derivation and further discussion see also
Cappelli:1990yc  and section 5 of Karateev:2020axc .

$c_{UV}=C(\infty).$ (2.29)

The full spectral density can be written as a sum as follows

$\rho_{\Theta}(s)=\rho^{(2)}_{\Theta}(s)\theta(s-4m^{2})+\rho^{(4)}_{\Theta}(s)\theta(s-16m^{2})+\rho^{(6)}_{\Theta}(s)\theta(s-36m^{2})+\ldots$
(2.30)

Here the superscript $(2)$, $(4)$, $(6)$ stand for 2-, 4- and 6-particle part
of the spectral density and $\ldots$ represent higher-particle contributions.
Note that $\forall n$, $\rho^{n}_{\Theta}(s)\geq 0$. In writing (2.30) we
assumed the absence of odd-number particle states due to the $Z_{2}$ symmetry.
The two-particle part of the spectral density is related to the two-particle
form factor as

$\rho^{(2)}_{\Theta}(s)=(2\pi\mathcal{N}_{2})^{-1}|\mathcal{F}^{\Theta}_{2,0}(s)|^{2}.$
(2.31)

In the “elastic” regime $s\in[4m^{2},16m^{2}]$ we also have Watson’s equation
which reads

$\widehat{\mathcal{S}}(s)={\mathcal{F}^{\Theta}_{2,0}(s)\over\mathcal{F}^{*}{}^{\Theta}_{2,0}(s)},\qquad\text{for
}s\in[4m^{2},16m^{2}].$ (2.32)

For the derivation of these relations and their analogues in higher dimensions
see Karateev:2019ymz ; Karateev:2020axc .

##### Majorana fermions

Consider the case of a single Majorana fermion in 2d with mass $m$.
Analogously to the bosonic case we can construct the two-particle fermion
states (2.10). However, now the two particle state must be anti-symmetric
under the exchange of the two fermions, thus instead of the normalization
condition (2.11) we get

$\displaystyle{}_{\text{free fermions}}$ $\displaystyle\langle
m,\vec{k}_{1};m,\vec{k}_{2}|m,\vec{p}_{1};m,\vec{p}_{2}\,\rangle_{\text{free
fermions}}$
$\displaystyle=4\sqrt{m^{2}+\vec{p}_{1}^{\,2}}\sqrt{m^{2}+\vec{p}_{2}^{\,2}}\,(2\pi)^{2}\delta^{(1)}(\vec{p}_{1}-\vec{k}_{1})\delta^{(1)}(\vec{p}_{2}-\vec{k}_{2})-(\vec{p}_{1}\leftrightarrow\vec{p}_{2}).$
(2.33)

The scattering amplitude for the Majorana fermion reads as

$\mathcal{S}_{\text{fermions}}(s)\times(2\pi)^{2}\delta^{(2)}(p_{1}+p_{2}-p_{3}-p_{4})\equiv\\\
{}_{\text{out fermions}}\langle
m,\vec{k}_{1};m,\vec{k}_{2}|m,\vec{p}_{1};m,\vec{p}_{2}\,\rangle_{\text{in
fermions}}\times\theta(\vec{p}_{1}-\vec{p}_{2})\theta(\vec{k}_{2}-\vec{k}_{1}).$
(2.34)

In the case of free fermions the scattering amplitude is simply given by the
normalization condition (2.33). Due to the presence of theta functions only
the second term in (2.33) contributes and using the change of variables, one
gets

$\mathcal{S}_{\text{free
fermions}}(s)=-\mathcal{N}_{2},\qquad\widehat{\mathcal{S}}_{\text{free
fermions}}(s)=-1,$ (2.35)

where we use the hatted amplitude for fermions is defined as in (2.24).
Analogously to (2.24) we can extract the interacting part of the fermion
scattering

$\widehat{\mathcal{S}}_{\text{fermions}}(s)=-1+i\mathcal{N}_{2}^{-1}\mathcal{T}_{\text{fermions}}(s).$
(2.36)

We finally notice that the scattering amplitude for free Majorana fermions is
equivalent to the scattering of bosons with the following interacting part

$\mathcal{T}_{\text{bosons}}(s)=2i\mathcal{N}_{2}.$ (2.37)

This can be seen by simply plugging (2.37) into (2.24). One trivially recovers
(2.35).

##  3 Analytic Results

In this section we provide analytic results for the sinh-Gordon, $\phi^{4}$
and 2d $O(N)$ models defined in section 1.1. The main objects we would like to
compute are the $2\rightarrow 2$ scattering amplitudes, the two-particle form
factor of the trace of the stress-tensor and its spectral density defined in
section 2. In $d=2$ all these observables are functions of a single variable
$s$. It will be often more convenient to use the rapidity variable $\theta$
related to the $s$ variable by

$\theta\equiv 2\text{ArcCosh}\left({\sqrt{s}\over
2m}\right)\quad\Leftrightarrow\quad s=4m^{2}\cosh^{2}\left({\theta\over
2}\right).$ (3.1)

Under crossing, we have $s\rightarrow 4m^{2}-s$, which corresponds to
$\theta\rightarrow i\pi-\theta$.

###  3.1 Sinh-Gordon Model

We have defined the sinh-Gordon model in (1.4). In what follows we will review
its scattering amplitude and the stress-tensor form factor.121212For a recent
extensive study of the sinh-Gordon model see Konik:2020gdi . For aesthetic
purposes let us define the following parameter

$b\equiv{\beta\over\sqrt{8\pi}}.$ (3.2)

The spectrum of the sinh-Gordon model consists of a single $Z_{2}$ odd
particle with mass $m$. The $2\rightarrow 2$ scattering amplitude was found in
Arinshtein:1979pb . It reads131313Under analytic continuation this amplitude
maps to the scattering of the lightest breathers in the sine-Gordon model. See
for example equation (4.18) in Karateev:2019ymz . Notice however the slight
clash of notation, namely $\gamma_{\text{here}}$ is equivalent to
$\gamma_{\text{there}}/8$.

$\widehat{\mathcal{S}}(\theta)={\sinh\theta-i\sin\gamma\over\sinh\theta+i\sin\gamma},\qquad\gamma\equiv{\pi
b^{2}\over 1+b^{2}},$ (3.3)

which is crossing symmetric, since $\sinh\theta$ is invariant when
$\theta\rightarrow i\pi-\theta$. It also possesses the following non-trivial
symmetry $b\leftrightarrow b^{-1}$. We can thus restrict our attention on the
following parameter range

$b\in[0,1]\quad\Leftrightarrow\quad\gamma\in[0,\pi/2].$ (3.4)

Using the definition of the non-perturbative quartic coupling (2.26), we
conclude that

$\Lambda=8m^{2}\left(1-{1\over 1+\sin\gamma}\right).$ (3.5)

Due to (3.4), the non-perturbative quartic coupling $\Lambda$ in the sinh-
Gordon model has the following range

$\Lambda\in[0,4m^{2}].$ (3.6)

One can use the relation (3.5) to eliminate the parameter $\gamma$ and rewrite
the scattering amplitude (3.3) as

$\widehat{\mathcal{S}}(\theta)=1+{2i\Lambda\over(\Lambda-8m^{2})\sinh\theta-i\Lambda}.$
(3.7)

The form factor of a scalar local operator in the sinh-Gordon model was
computed in Fring:1992pt . Adjusting the normalization of their result
according to (2.15), we can write the following expression for 2-particle form
factor for the trace of the stress-tensor

$\mathcal{F}^{\Theta}_{2,0}(\theta)=-2m^{2}\times\\\
\exp\left(8\,\int_{0}^{\infty}{dx\over x}{\sinh\left({x\gamma\over
2\pi}\right)\sinh\left({x(\pi-\gamma)\over 2\pi}\right)\sinh\left({x\over
2}\right)\over\sinh^{2}(x)}\sin^{2}\left({x(i\pi-\theta)\over
2\pi}\right)\right).$ (3.8)

In section 5, in order to compare the spectral densities of the sinh-Gordon
model and the $\phi^{4}$ model above $s=16m^{2}$, we will also need the
4-particle form factor for $\Theta$, which we review in appendix D.

Let us now notice that the actual expressions for the scattering amplitude
(3.3) and for the form factor (3.8) are analytic functions of the parameter
$\gamma$. They can be thus analytically continued away from the original range
of $\gamma$ given by (3.4). The resulting amplitude and the form factor are
the ones of the so-called staircase model, which we review next.

###  3.2 Staircase Model

Because of the strong-weak duality $b\leftrightarrow b^{-1}$ in the sinh-
Gordon model, it is effectively impossible to increase the coupling beyond
$b=1$ and as a result $\Lambda$ is restricted to be $\leq 4m^{2}$. However, by
analytically continuing the coupling to complex values, it is formally
possible to obtain larger values of $\Lambda$. The Staircase Model
zamolodchikov2006resonance  is the analytic continuation

$\gamma={\pi\over 2}+i\theta_{0}$ (3.9)

with $\theta_{0}$ real. Although the Lagrangian is no longer real and it is
not clear why such a deformation should correspond to an underlying unitary
theory,141414It would be interesting to interpret the theory as a “Complex
CFT” along the lines of Gorbenko:2018ncu ; Gorbenko:2018dtm . In particular,
the large amount of RG time that the Staircase Model spends near each minimal
model suggests a form of “walking” near the minimal model fixed points. in
zamolodchikov2006resonance  Zamolodchikov showed that the $c$-function of the
theory, defined using the thermodynamic Bethe Ansatz, flows from a free scalar
in the UV to the Ising model in the IR and moreover approaches very close to
each of the $c<1$ minimal models along this RG flow. The amount of RG time
spent near each minimal model is proportional to $\theta_{0}$, so that at
large $\theta_{0}$ the $c$-function resembles a staircase.

Substituting equation (3.9) into (3.5), the $\sin\gamma$ in the denominator
becomes $\cosh\theta_{0}$, and now the maximum value that $\Lambda$ can reach
is $8m^{2}$. Solving for $\theta_{0}$ in terms of $\Lambda/m^{2}$, we get

$\theta_{0}=\log\left({2+\sqrt{{\Lambda\over m^{2}}-4}\over
2-\sqrt{{\Lambda\over m^{2}}-4}}\right),$ (3.10)

which grows logarithmically like $\theta_{0}\approx\log{16m^{2}\over
8m^{2}-\Lambda}$ as $\Lambda$ approaches the upper limit $8m^{2}$.
Parametrized in terms of $\Lambda$, the $S$ matrix of the staircase model is
given precisely by (3.7) with $\Lambda\in[4m^{2},8m^{2}]$. Similarly one
obtains the form factor of the trace of the stress tensor in the staircase
model by using the analytic continuation (3.9) and (3.10) in the expression
(3.8). It is interesting to notice that using (3.9), we can read off the value
of $\beta^{2}$ from (3.2) and (3.3). Since
$\beta=\sqrt{8\pi{\gamma\over\pi-\gamma}}$, one thus has
$\left|\beta^{2}\right|=8\pi$ for $m^{-2}\Lambda\geq 4$.

Expanding the analytically continued amplitude (3.7) around $\Lambda=8m^{2}$,
one finds

$\mathcal{T}_{\text{bosons}}=2i\mathcal{N}_{2}+{st\over
M^{2}}+{\mathcal{O}}\left({1\over M^{4}}\right),$ (3.11)

where ${1\over M^{2}}\equiv{8m^{2}-\Lambda\over 4m^{4}}$. According to the
discussion around (2.37), the interacting part of the boson scattering
amplitude (3.11) is equivalent to the scattering of Majorana fermions with the
following interacting part

$\mathcal{T}_{\text{fermions}}={st\over M^{2}}+{\mathcal{O}}\left({1\over
M^{4}}\right).$ (3.12)

In other words at $\Lambda=8m^{2}$,
$\mathcal{T}_{\text{bosons}}=2i\mathcal{N}_{2}$, which is equivalent to
$\mathcal{T}_{\text{fermions}}=0$, so the S-matrix approaches that of a free
massive fermion (the Ising model). Moreover, at $\Lambda$ slightly below
$8m^{2}$, the S-matrix additionally has contributions from irrelevant
deformations that should capture the approach to Ising from the UV (in this
case, from the next minimal model up, i.e. the tricritical Ising model). In
section 3.5 we will explicitly check this leading correction.

###  3.3 $\phi^{4}$ model

The $\phi^{4}$ model defined by (1.5) allows for the presence of one-particle
asymptotic states (2.8) which are $Z_{2}$ odd. Due to the presence of the
$Z_{2}$ symmetry the “elastic” regime in the $\phi^{4}$ model is extended to
$s\in[4m^{2},16m^{2}]$. The relation between the lightcone quantization bare
mass $m_{0}$ and the physical mass $m$ is given by

$m=m_{0}\left(1-{\overline{\lambda}^{2}\over 768}+{\overline{\lambda}^{3}\over
3072\pi}+O(\overline{\lambda}^{4})\right).$ (3.13)

For higher order corrections see equation (2.14) in Fitzpatrick:2018xlz .

Using perturbation theory we compute the two-particle form-factor and the
spectral density of the trace of the stress-tensor. The form factor reads

$m^{-2}\mathcal{F}^{\Theta}_{2,0}(s)=-2+\left({\overline{\lambda}\over
4\pi}\right)\,\Delta(s)+\\\ {1\over 2}\left({\overline{\lambda}\over
4\pi}\right)^{2}\left({\pi^{2}s\over
8(s-4m^{2})}-\Delta(s)\left(\Delta(s)/2+1\right)\right)+\mathcal{O}(\overline{\lambda}^{3}),$
(3.14)

where the function $\Delta(s)$ is defined as

$\Delta(s)\equiv-1+\lim_{\epsilon\rightarrow
0^{+}}{4m^{2}\text{ArcTan}\left({\sqrt{s}\over\sqrt{4m^{2}-s-i\epsilon}}\right)\over\sqrt{s}\sqrt{4m^{2}-s-i\epsilon}}.$
(3.15)

The expression (3.14) is valid for any complex value of $s$. The function
$\Delta(s)$ has a single branch cut along the horizontal axis in the $s$
complex plane for $s\in[4m^{2},\infty)$. The infinitesimally small $\epsilon$
is present in order to specify the correct side of the branch cut. At times,
it is more convenient to use the rapidity variable $\theta$ defined in (3.1),
which opens up this branch cut. In this variable, the $\epsilon$ prescription
translates to taking the $\theta>0$ branch for $s>4m^{2}$, and the function
$\Delta(s)$ is simply

$\Delta(s(\theta))={i\pi-\theta\over\sinh\theta}-1.$ (3.16)

The following limits hold true

$\Delta(0)=0,\qquad\Delta(2m^{2})={\pi\over 2}-1,\qquad\Delta(4m^{2})=\infty.$
(3.17)

The first entry in (3.17) implies (3.14) satisfies the normalization condition
(2.15).

The spectral density of the trace of the stress tensor reads as

${2\pi\mathcal{N}_{2}\over
4m^{4}}\times\rho_{\Theta}(s)=1+{\overline{\lambda}\over
4\pi}\times\left(1+4m^{2}\mathcal{N}_{2}^{-1}\log\left({\sqrt{s}+\sqrt{s-4m^{2}}\over\sqrt{s}-\sqrt{s-4m^{2}}}\right)\right)+\mathcal{O}(\overline{\lambda}^{2}).$
(3.18)

It is defined in the region $s\in[4m^{2},\infty]$. Fully computing the next
correction to the spectral density is quite difficult. We notice however that
in the “elastic” regime $s\in[4m^{2},16m^{2}]$ with no particle production the
next correction to the spectral density simply follows from (2.30) and (3.14).
We derive (3.14) and (3.18) in appendix C. We are not aware of any literature
where these results were previously presented, though in principle it should
be possible to obtain them from small $\beta$ expansions of results for the
corresponding observables in the sinh-Gordon model.

For completeness, let us also provide the textbook result for the interacting
part of the scattering amplitude. It reads

$m^{-2}\mathcal{T}(s)=-\overline{\lambda}\times\left(1-{1\over
2}\left({\overline{\lambda}\over
4\pi}\right)\times\big{(}1+\Delta(s)+\Delta(4m^{2}-s)\big{)}+O(\overline{\lambda}^{3})\right).$
(3.19)

It is straightforward to check that (3.14) and (3.19) obey Watson’s equation
(2.32) in the “elastic” regime. Using (3.19) and the second entry in (3.17) we
can relate the quartic coupling $\lambda$ and the non-perturbative quartic
coupling $\Lambda$ defined in (2.26) as follows

$m^{-2}\Lambda=\overline{\lambda}-{\overline{\lambda}^{2}(\pi-1)\over
8\pi}+\mathcal{O}(\overline{\lambda}^{3}).$ (3.20)

Another thing that is important to emphasize is that the perturbative results
diverge at the two-particle threshold $s=4m^{2}$. This divergence is an
artifact of perturbation theory and does not appear in the non-perturbative
amplitude.

By inspecting the perturbative results (3.14), (3.18) and (3.19), we notice
that the perturbative expansion parameter is more accurately
${\overline{\lambda}\over 4\pi}$ rather than $\overline{\lambda}$. Thus, we
expect that the strong coupling regime where perturbation theory breaks down
is

$\overline{\lambda}\sim 4\pi\sim 12.$ (3.21)

###  3.4 2d $O(N)$ model in the large $N$ limit

Let us now consider the generalization of the $\phi^{4}$ model given by (1.5)
where the field $\phi(x)$ has $N$ components and the action is invariant under
$O(N)$ symmetry. In such a theory there are three different two-particle
states transforming in the trivial, symmetric and antisymmetric
representations of the $O(N)$ group. For details see appendix B.

Let us consider here the large $N$ limit $N\rightarrow\infty$. In this limit
it is enough to only consider the two-particle states in the trivial
representation. In what follows we compute the spectral density and the form
factor of the trace of the stress tensor together with the scattering
amplitude for the two-particle states in the trivial representation. Our
results are valid to all orders of perturbation theory. The details of all the
computations are given in appendix C.

In the large $N$ limit the relation between the physical mass $m$ and
lightcone quantization bare mass $m_{0}$ is extremely simple, namely

$m=m_{0}.$ (3.22)

The two-particle form factor of the stress-tensor in the large $N$ limit reads
as

$m^{-2}\mathcal{F}^{\Theta}_{2,0}(s)=-2+{2\overline{\lambda}\Delta(s)\over
8\pi+\overline{\lambda}\,(1+\Delta(s))}.$ (3.23)

It is important to notice that this form factor does not have a singularity at
$s=4m^{2}$. Using the third entry in (3.17) we conclude that
$\mathcal{F}^{\Theta}_{2,0}(4m^{2})=0$. In the large $N$ limit there is no
particle production. As a result the full spectral density is simply given by
the two-particle form factor (3.23) via (2.30). Concretely speaking

$2\pi\mathcal{N}_{2}\rho_{\Theta}(s)=|\mathcal{F}^{\Theta}_{2,0}(s)|^{2}.$
(3.24)

The full scattering amplitude in the large $N$ limit reads

$\widehat{\mathcal{S}}(s)=-{\overline{\lambda}\,(\pi-i\theta)+8\pi
i\,\text{Sinh}(\theta)\over\overline{\lambda}\,(\pi+i\theta)-8\pi
i\,\text{Sinh}(\theta)}.$ (3.25)

Note that this S-matrix is not crossing symmetric, contrary to the other
models that we consider in this paper. It is straightforward to check that
(3.23) and (3.25) satisfy Watson’s equation (2.32) for the whole range of
energies $s\in[4m^{2},\infty)$. Moreover, there is no divergence at the two-
particle threshold $s=4m^{2}$, where $\widehat{\mathcal{S}}(4m^{2})=-1$.

Using the definition of the non-perturbative quartic coupling $\Lambda$, the
relation between the scattering amplitude and the interacting part of the
scattering amplitude and the explicit solution (3.25), we can evaluate
precisely the non-perturbative coupling $\Lambda$ in terms of $\lambda$. It
takes the following simple form

$m^{-2}\Lambda={16\overline{\lambda}\over 16+\overline{\lambda}}.$ (3.26)

One can see that for real positive $\overline{\lambda}$, we have
$\Lambda\in[0,16]$.

###  3.5 $T\overline{T}$ deformation of the 2d Ising

In the vicinity of the critical point, both $\phi^{4}$ theory and the
Staircase Model flow to the Ising model with a $\mathbb{Z}_{2}$ symmetry that
forbids the magnetic deformation $\sigma$. In that case, the lowest-dimension
deformation around Ising is the thermal operator $\epsilon$, which is just the
fermion mass term of the free Majorana fermion description of the Ising model.
The next-lowest-dimension scalar operator is $T\overline{T}$, which in terms
of the left- and right-moving components $\psi$ and $\widetilde{\psi}$ of the
fermion is

$\delta\mathcal{L}={1\over
M^{2}}\psi\partial_{-}\psi\widetilde{\psi}\partial_{+}\widetilde{\psi}.$
(3.27)

Here, $M$ is the scale of the UV cut-off of the low-energy expansion. In the
limit that $M$ is much larger than the mass gap $m$, the contributions to the
S-matrix from all other higher dimension operators are suppressed by higher
powers of $m/M$, so near $\Lambda/m^{2}=8$ the S-matrix is well-approximated
by the tree-level contribution from (3.27). The leading contribution to the
scattering amplitude is most easily computed in lightcone coordinates, where
each $\psi$ contraction with an external fermion produces a factor of
$\sqrt{p_{-}}$ for that fermion, and each $\widetilde{\psi}$ contraction
produces a factor of $\sqrt{p_{-}}{\sqrt{2}p_{+}\over m}$ (the extra factor
follows from the fermion equation of motion
$\sqrt{2}i\partial_{+}\psi=m\widetilde{\psi}$). So, the full tree-level
contribution is simply ${2\over
m^{2}}\sqrt{p_{1-}p_{2-}p_{3-}p_{4-}}p_{2-}p_{3+}p_{4+}^{2}$, anti-symmetrized
on all permutations of $p_{1},p_{2},-p_{3},$ and $-p_{4}$. Finally, there are
only two solutions to the kinematic constraint $p_{1}+p_{2}=p_{3}+p_{4}$;
either $p_{1}=p_{3},p_{2}=p_{4}$ or $p_{1}=p_{4},p_{2}=p_{3}$. Taking the
former, and using $p_{+}={m^{2}\over 2p_{-}}$, we obtain

$\mathcal{T}_{\text{fermions}}={m^{4}(p_{1-}-p_{2-})^{2}(p_{1-}+p_{2-})^{2}\over
M^{2}p_{1-}^{2}p_{2-}^{2}}={st\over M^{2}},$ (3.28)

in agreement with (3.12). The Ising model S-matrix has
$(\Lambda/m^{2},m^{2}\Lambda^{(2)})$ = $(8,2)$, and from the above expression
we can read off that the leading correction which gives

$(\Lambda/m^{2},m^{2}\Lambda^{(2)})=\left(8-{4m^{2}\over M^{2}},2-{2m^{2}\over
M^{2}}\right).$ (3.29)

##  4 Pure S-matrix bootstrap

In this section we will construct general non-perturbative bounds on the space
of 2d scattering amplitudes of $Z_{2}$ odd particles (assuming there is no
bound state pole). We will define the exact optimization problem in section
4.1 and present our numerical results in section 4.2. The main result of this
section is presented in figure 1. The amplitudes in the sinh-Gordon and its
analytic continuation (the staircase model) saturate the lower edge of this
bound.

###  4.1 Set-up

Let us start by discussing the unitarity constraint. In 2d the scattering
amplitude $\widehat{\mathcal{S}}(s)$ must obey the following positive
semidefinite condition

$\begin{pmatrix}1&\widehat{\mathcal{S}}^{*}(s)\\\
\widehat{\mathcal{S}}(s)&1\end{pmatrix}\succeq 0,\qquad\text{for
}s\in[4m^{2},\infty).$ (4.1)

Due to Sylvester’s criterion, this condition is equivalent to the more
familiar one (2.25). To see that, one can simply evaluate the determinant of
(4.1).

It was proposed in Paulos:2016but ; Paulos:2017fhb  how to use the constraint
(4.1) in practice. One can write the following ansatz for the scattering
amplitude which automatically obeys maximal analyticity and crossing

$\widehat{\mathcal{S}}(s)-1=-{\Lambda\over
4m^{2}}+\sum_{n=1}^{N_{\text{max}}}a_{n}\times\left({\mathfrak{r}}(s;2m^{2})^{n}+{\mathfrak{r}}(4m^{2}-s;2m^{2})^{n}\right),$
(4.2)

where $\Lambda$ is the non-perturbative quartic coupling defined in (2.26),
$a_{n}$ are some real coefficients and the ${\mathfrak{r}}$ variable is
defined as

${\mathfrak{r}}(s;s_{0})\equiv\lim_{\epsilon\rightarrow
0^{+}}{\sqrt{4m^{2}-s_{0}}-\sqrt{4m^{2}-s-i\epsilon}\over\sqrt{4m^{2}-s_{0}}+\sqrt{4m^{2}-s-i\epsilon}}.$
(4.3)

Here $s_{0}$ is a free parameter which can be chosen at will. For scattering
amplitudes it is convenient to choose $s_{0}=2m^{2}$; this guaranties that at
the crossing symmetric point $s=2m^{2}$, the ${\mathfrak{r}}(s;2m^{2})$
variable vanishes. In theory, one should take $N_{\text{max}}=\infty$. This is
impossible in practice however, and one is thus has to choose a large enough
but finite value of $N_{\text{max}}$ which leads to stable numerical results
(stable under the change of $N_{\text{max}}$). Alternatively to (4.2), one
could also parametrize only the interacting part of the scattering amplitude,
namely

$\mathcal{T}(s)=-\Lambda+\sum_{n=1}^{N_{\text{max}}}\widetilde{a}_{n}\times\left({\mathfrak{r}}(s;2m^{2})^{n}+{\mathfrak{r}}(4m^{2}-s;2m^{2})^{n}\right).$
(4.4)

In this ansatz we denote the unknown paramters by $\widetilde{a}$ in order to
distinguish them from the parameters $a$ entering in (4.2). Depending on the
situation sometimes this choice is more convenient than (4.2).

Using SDPB Simmons-Duffin:2015qma ; Landry:2019qug  we can scan the parameter
space $(\Lambda,a_{0},a_{1},a_{2},\ldots)$ of the ansatz (4.2) (or
alternatively
$(\Lambda,\widetilde{a}_{0},\widetilde{a}_{1},\widetilde{a}_{2},\ldots)$ of
the ansatz (4.4)) by looking for amplitudes with the largest or smallest value
of $\Lambda$ which obey (4.1). Once the allowed range of $\Lambda$ is
determined, we can look for example for amplitudes for each allowed value of
$\Lambda$ with the largest or smallest value of the parameter $\Lambda^{(2)}$
defined in (2.27). Using this definition we can express $\Lambda^{(2)}$ in
terms of the paramenters of the ansatz as

$\Lambda^{(2)}={\Lambda+m^{2}\,(2a_{1}+a_{2})\over
4m^{4}}\qquad\text{or}\qquad\Lambda^{(2)}={2\widetilde{a}_{1}+\widetilde{a}_{2}\over
16m^{4}}.$ (4.5)

###  4.2 Numerical Results

Solving the optimization problem for $\Lambda$ defined in section 4.1 we
obtain the following bound

$\Lambda\in[0,\,8m^{2}].$ (4.6)

For each $\Lambda$ in this range, we can now minimize and maximize the
parameter $\Lambda^{(2)}$. As a result we obtain a 2d plot of allowed values
which is given in figure 1. On the boundary of the allowed region in figure 1,
we can extract the numerical expressions of the scattering amplitudes. For
instance the scattering amplitudes extracted from the lower edge are presented
in figures 3 and 3. Remarkably they coincide with the analytic expression
(3.7) which describes the sinh-Gordon model and its analytic continuation (the
staircase model). In particular, notice that the amplitudes extracted in the
vicinity of the tips of the allowed region in figure 1 approach the following
expressions

$\widehat{\mathcal{S}}_{\text{left
tip}}(s)=+1\qquad\text{and}\qquad\widehat{\mathcal{S}}_{\text{right
tip}}(s)=-1.$ (4.7)

These are the amplitudes of the free boson (lower left corner with
$\Lambda=0$) and of the free Majorana fermion (upper right corner with
$\Lambda=8m^{2}$). Let us also make a fun observation that the amplitudes
extracted from the upper edge of the bound in figure 1 are related to the ones
extracted from the lower edge by

$\widehat{\mathcal{S}}_{\text{upper
edge}}(s;\,\Lambda)=-\widehat{\mathcal{S}}_{\text{lower
edge}}(s;\,8m^{2}-\Lambda).$ (4.8)

Using (2.36), we can interpret $\widehat{\mathcal{S}}_{\text{upper
edge}}(s;\,\Lambda)$ as complex conjugated amplitudes of Majorana fermions
with the interacting part exactly as in the sinh-Gordon expression with
$\Lambda\rightarrow 8m^{2}-\Lambda$. We do not know any UV complete model from
where such amplitudes could originate.

Figure 1: Bound on the parameters $\Lambda$ and $\Lambda^{(2)}$. The allowed
region is depicted in blue.

Figure 2: Real part of the scattering amplitude obtained by minimizing
$\Lambda^{(2)}$ for various values of $\Lambda$.

Figure 3: Imaginary part of the scattering amplitude obtained by minimizing
$\Lambda^{(2)}$ for various values of $\Lambda$.

Finally, let us relax the requirement that the amplitude $\mathcal{S}(s)$ is
crossing invariant. This means the when writing down the ansatz for the
S-matrix, we do not have the second term in the parentheses in equation (4.2).
In this case, we obtain the following bound

$\Lambda\in[0,16m^{2}].$ (4.9)

We notice that the large $N$ limit of the 2d $O(N)$ model with $\phi^{4}$
potential populates this interval, see (3.26). We can then minimize
$\Lambda^{(2)}$ by fixing the value of $\Lambda$ in the interval (4.9). The
resulting numerical amplitudes correspond precisely to the large $N$ analytic
solution (3.25). In practice, for this optimization problem it was important
to parametrize the interacting part of the scattering amplitude $\mathcal{T}$
instead of $\widehat{\mathcal{S}}$.

##  5 S-matrix and Form Factor Bootstrap

In this section we will define the numerical optimisation which allows one to
compute the two-to-two scattering amplitude and the two-particle form factor
of the trace of the stress tensor at $s>0$ in the 2d $\phi^{4}$ model given
the LCT input obtained in a companion paper truncffsd . We begin in section
5.1 by quickly reviewing the generalization of the S-matrix bootstrap proposed
in Karateev:2019ymz  which allows one to include local operators. We then
explain how one can define an optimization problem which allows one to compute
the scattering amplitude and the form factor at $s>0$. In section 5.2, we
present our numerical findings. The main results are given in figures 9 \- 11.
Also in section 5.2, we will observe that the $\phi^{4}$ model is very similar
to the sinh-Gordon model in the elastic regime. We will investigate this
similarity further in section 5.3.

###  5.1 Set-up

In Karateev:2019ymz , it was shown that unitarity allows to write a more
complicated constraint than (4.1) which entangles the scattering amplitude
with the two-particle form factor and the spectral density of a scalar local
operator $\mathcal{O}$. In this paper we consider the case where the local
operator is the trace of the stress tensor $\Theta$. The following positive
semidefinite condition can be written151515Various entries in this matrix have
different mass dimensions. This positivity condition is equivalent however to
the one which is obtained from (5.1) by the following rescaling
$\mathcal{F}_{\Theta}\rightarrow
m^{-1}\mathcal{F}_{\Theta},\qquad\rho_{\Theta}\rightarrow
m^{-2}\rho_{\Theta}.$ The unitarity condition in the latter form was
originally presented in Karateev:2019ymz . It contains only dimensionless
quantities.

$\begin{pmatrix}1&\widehat{\mathcal{S}}^{*}(s)&\mathcal{N}_{2}^{-1/2}\,\mathcal{F}^{*}{}^{\Theta}_{2,0}(s)\\\
\widehat{\mathcal{S}}(s)&1&\mathcal{N}_{2}^{-1/2}\,\mathcal{F}^{\Theta}_{2,0}(s)\\\
\mathcal{N}_{2}^{-1/2}\,\mathcal{F}^{\Theta}_{2,0}(s)&\mathcal{N}_{2}^{-1/2}\,\mathcal{F}^{*}{}^{\Theta}_{2,0}(s)&2\pi\rho(s)\end{pmatrix}\succeq
0,\qquad\text{for }s\in[4m^{2},\infty].$ (5.1)

Analogously to section 4.1, one can define various numerical optimization
problems which utilize (5.1) instead of (4.1). For that we should write an
ansatz for all the ingredients entering in (5.1). For the scattering amplitude
we use the ansatz (4.2) or (4.4). In practice, we will use (4.4) in this
section. For the form factor we can write instead

$\mathcal{F}^{\Theta}_{2,0}(s)=-2m^{2}+\sum_{n=1}^{N_{\text{max}}}b_{n}\times{\mathfrak{r}}(s;0)^{n},$
(5.2)

where $b_{n}$ are some real parameters. By construction it is an analytic
function in $s$ with a single branch cut on the real axis between $4m^{2}$ and
$+\infty$. The ${\mathfrak{r}}$ variable was defined in (4.3). Here, we have
chosen the parameter $s_{0}$ to be 0, such that ${\mathfrak{r}}(s;0)$ vanishes
at $s=0$. This is convenient since this ansatz automatically satisfies the
normalization condition (2.14) at $s=0$. If we also write an ansatz for the
spectral density (which is simply a real function), one can then bound for
example the UV central charge (2.29) for various values of $\Lambda$. Although
this may be an interesting problem, we do not pursue it in this paper.

Instead of parametrizing the spectral density in this section, we will use its
explicit form in the 2d $\phi^{4}$ model found in the companion paper
truncffsd , see figure 4 there. We use the superscript LCT in order to denote
these spectral densities, namely

$s\in[4m^{2},s_{\text{max}}]:\qquad\rho_{\Theta}^{\text{LCT}}(s).$ (5.3)

Here $s_{\text{max}}$ is the maximal value of $s$ for which we trust the
results of truncffsd . In truncffsd , see figure 12, we have also computed the
two-particle form factor of the trace of the stress tensor at $s\leq 0$. We
also use the LCT superscript to denote these form factors, namely

$s\in[s_{\text{min}},0]:\qquad\mathcal{F}^{\Theta\text{LCT}}_{2,0}(s).$ (5.4)

Here $s_{\text{min}}<0$ is the minimal value of $s$ for which we trust the
results of truncffsd . We refer to (5.3) and (5.4) as the input data.

Let us now precisely define our optimization problem. Given the value of the
physical mass $m$ (which is obtained by the LCT method), determine the unknown
coefficients $\Lambda$, $a_{n}$ and $b_{n}$ in the ansatze (4.4) and (5.2),
such that $\Lambda$ has the maximal/minimal value and the following
constraints are satisfied

$\displaystyle\begin{pmatrix}1&\widehat{\mathcal{S}}^{*}(s)&\mathcal{N}_{2}^{-1/2}\,\mathcal{F}^{*}{}^{\Theta}_{2,0}(s)\\\
\widehat{\mathcal{S}}(s)&1&\mathcal{N}_{2}^{-1/2}\,\mathcal{F}^{\Theta}_{2,0}(s)\\\
\mathcal{N}_{2}^{-1/2}\,\mathcal{F}^{\Theta}_{2,0}(s)&\mathcal{N}_{2}^{-1/2}\,\mathcal{F}^{*}{}^{\Theta}_{2,0}(s)&2\pi\rho_{\Theta}^{\text{LCT}}(s)\end{pmatrix}\succeq
0,\quad s\in[(4+\sigma)m^{2},s_{\text{max}}],$ (5.5)
$\displaystyle\begin{pmatrix}1&\widehat{\mathcal{S}}^{*}(s)\\\
\widehat{\mathcal{S}}(s)&1\end{pmatrix}\succeq 0,\quad
s\in[4m^{2},(4+\sigma)m^{2})\cup(s_{\text{max}},\infty)$ (5.6)

The first constraint (5.5) allows one to inject information about the LCT
spectral density (5.3) in the set-up. The second constraint (5.6) can be seen
as the reduced version of the first one in the region where no information
about the spectral density is available. In the above equations we have
introduced an addition small parameter $\sigma\ll 1$. The numerical bootstrap
set-up is sensitive to numerical errors in the LCT data, and the presence of
$\sigma$ mitigates the effect of these errors in the spectral density near
threshold $s=4m^{2}$ and the uncertainty in the value of the physical mass
itself. In addition to equation (5.5) and (5.6), we require that the Ansatz
for the form factor match the one obtained by the LCT method. We can impose
this by demanding

$|\mathcal{F}^{\Theta}_{2,0}(s)-\mathcal{F}^{\Theta\text{LCT}}_{2,0}(s)|\leq\epsilon,\quad\text{for
}s\in[s_{\text{min}},\,0],$ (5.7)

where $\epsilon\geq 0$ is a small positive parameter. We have introduced the
$\epsilon$ parameter in the set-up in order to accommodate the numerical
errors in the LCT input data. The constraint (5.7) can be equivalently
rewritten in the semi-positive form as

$\begin{pmatrix}\epsilon&\mathcal{F}^{*\Theta}_{2,0}(s)-\mathcal{F}^{*\Theta\text{LCT}}_{2,0}(s)\\\
\mathcal{F}^{\Theta}_{2,0}(s)-\mathcal{F}^{\Theta\text{LCT}}_{2,0}(s)&\epsilon\end{pmatrix}\succeq
0,\quad\text{for }s\in[s_{\text{min}},\,0].$ (5.8)

In practice, we parameterize $\epsilon$ in terms of the exponent $\delta$
defined in the following way

$m^{-2}\epsilon=10^{-\delta}.$ (5.9)

The larger the value of $\delta$, the stronger the constraint (5.8) becomes.

When we present our numerical results in section 5.2, we will see that given a
large enough value of $\delta$ in (5.9), we find a unique solution to the
optimisation problem described in this section, namely the upper and lower
bounds lead to almost the same result. Moreover, we will see that the
unitarity conditions (5.5) and (5.6) tend to get saturated in the “elastic”
regime $s\in[4m^{2},16m^{2}]$. As a result, the obtained form factors obey
equation (2.31) and the scattering amplitudes obey equation (2.32) as they
should. In the non-elastic regime $s\geq 16m^{2}$, the LCT spectral density
contains four- and higher- particle contributions, however we do not include
four- and higher-particle form factors in the set-up. Therefore,
conservatively speaking, this means that for $s\geq 16m^{2}$ the behaviour of
the obtained scattering amplitude and the form factor has nothing to do with
the $\phi^{4}$ model.

Formulating the above paragraph in different words, one can roughly say that
the above optimization procedure determines the coefficients of the form
factor ansatz in equation (5.2) given two constraints: that the Ansatz matches
the LCT form factor result (5.4) for $s\leq 0$, and the square of its norm
saturates the LCT spectral density (5.3) for $4m^{2}\leq s\leq 16m^{2}$ via
(2.31). The scattering amplitude is then obtained by solving Watson’s equation
(2.32).

###  5.2 Numerical Results

We present now the solutions of the optimization problem defined in section
5.1. As a demonstration of our approach, in section 5.2.1, instead of using
the LCT input data (which obviously contains numerical errors), we use the
input data obtained from the analytic solution for the 2d $O(N)$ model in the
large $N$ limit given in section 3.4. We stress however that we use only the
part of the analytic data which is computable with the LCT methods. The reason
for this exercise is to show how the optimization problem works in the
presence of high accuracy data. We present our optimization for the $\phi^{4}$
model using the LCT data in section 5.2.2. For small values of the quartic
coupling constant $\overline{\lambda}$, our results are in agreement with
perturbation theory. For large values of $\overline{\lambda}$ our results are
novel.

In order to proceed, let us provide some details on the choice of the
optimization parameters used in SDPB. We use the following range for the input
data

$s_{\text{min}}=-80m^{2},\qquad s_{\text{max}}=100m^{2}.$ (5.10)

We use the following size of the ansatzes in equation (4.4) and equation (5.2)

$N_{\text{max}}=50,$ (5.11)

which is large enough in practice. We impose the conditions (5.5), (5.6) and
(5.8) at a finite number of points $s$. Let us denote by $N_{\mathcal{F}}$ the
number of $s$ values picked in the interval $[s_{\text{min}},0]$ where the
condition (5.8) is imposed, and by $N_{\rho}$ the number of $s$ values picked
in the interval $[(4+\sigma)m^{2},s_{\text{max}}]$ where the condition (5.5)
is imposed. For the choice of $N_{\text{max}}$ in (5.11), we chose the
following values

$N_{\mathcal{F}}=1000,\qquad N_{\rho}=2500.$ (5.12)

We impose the condition (5.6) at about 100 points in the range
$s>s_{\text{max}}$. We use the Chebyshev grid to distribute the above points.
In practice for the LCT data with $\overline{\lambda}\leq 13$ we use
$\sigma=0.001$ and for $\overline{\lambda}>13$ we use $\sigma=0.01$. This
indicates that the LCT data for higher values of $\overline{\lambda}$ contains
larger errors. For smaller values of $\sigma$ the optimization problem often
simply does not converge.

Our strategy is then as follows. We run the optimization routine for different
values of $\epsilon$ or equivalently $\delta$, see (5.9). For small values of
$\delta$, the problem is not constraining enough. For too large values of
$\delta$, the problem becomes unfeasible. In order to find the optimal value
for $\delta$ we perform the binary scan in the range

$\delta\in[0,10].$ (5.13)

We then pick the largest value of $\delta$ where the optimization problem is
still feasible. The binary scan is performed until the difference between the
feasible and unfeasible values of $\delta$ drops below some threshold value.
We pick this threshold value to be 0.1.

####  5.2.1 Infinite Precision Example

We study here the 2d $O(N)$ model in the large $N$ limit where the exact
analytic solution exists, see section 3.4. We pick the following value of the
quartic coupling

$\overline{\lambda}=10$ (5.14)

as an example. At large $N$ we keep only the singlet component of the
scattering amplitude, which therefore loses its crossing symmetry
$s\leftrightarrow 4m^{2}-s$, see appendix B. As a result, in the ansatz (4.4),
we relax crossing symmetry by dropping the last term in the sum.

Figure 4: Lower and upper bounds on the non-perturbative quartic coupling
constant $\Lambda$. Blue dots are the numerical data. The blue vertical lines
indicate the allowed region for $\Lambda$ for each value of $\delta$. The
horizontal dashed line indicates the analytic value of $\Lambda\approx
6.1538$, which the numerical lower and upper bounds are expected to converge
to. Here we use the analytic large $N$ data of the 2d $O(N)$ model to mimic
the LCT input data.

Figure 5: The real and imaginary part of the interacting part of the
scattering amplitude $\mathcal{T}(s)$. The solid red line is the numerical
result. The dashed black line is the expected result from perturbation theory
(3.25). Here we use the analytic large $N$ data of the 2d $O(N)$ model to
mimic the LCT input data.

Figure 6: The real and imaginary part of the form factor of the trace of the
stress tensor $\mathcal{F}^{\Theta}_{2,0}(s)$. The solid red line is the
numerical result. The dashed black line is the expected result from
perturbation theory (3.23). Here we use the analytic large $N$ data of the 2d
$O(N)$ model to mimic the LCT input data.

The bound on the non-perturbative quartic coupling $\Lambda$ is given on
figure 6. We see that the upper and lower bounds quickly converge to the
analytic value of $\Lambda$ and starting from $\delta\gtrsim 9$ basically
coincide. We pick the “lower bound” solution with the largest value of
$\delta$ and extract the interacting part of the scattering amplitude and the
form factor. The result is given in figures 6 and 6 respectively. The
optimization problem result is given by the red solid line and the analytic
results are given by the black dashed line. Both are in a perfect agreement.

####  5.2.2 $\phi^{4}$ model

Let us now address the optimization problem with the $\phi^{4}$ LCT data as an
input. In what follows we will denote the data obtained by maximization of
$\Lambda$ by the subscript “upper” and the data obtained by minimization of
$\Lambda$ by the subscript “lower”. The obtained numerical values of $\Lambda$
and $\Lambda^{(2)}$ are presented in table 1. Looking at this table one can
see that both optimization problems lead to almost the same numerical values.
This indicates that our procedure converges to the unique solution. The
relative difference between $\Lambda_{\text{upper}}$ and
$\Lambda_{\text{lower}}$ can be taken as a rough error estimate. It is
illuminating to place the data of table 1 on figure 1. We display the result
in figure 7. Remarkably the $\phi^{4}$ model lies very close to the boundary
of the allowed region and almost coincides with the sinh-Gordon/staircase
model. We address the similarity between the two models in detail in the next
section.

$\overline{\lambda}$ | 1 | 3 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---
$m^{-2}\Lambda_{\text{upper}}$ | 0.903 | 2.102 | 3.292 | 3.608 | 3.909 | 4.196
$m^{-2}\Lambda_{\text{lower}}$ | 0.878 | 2.093 | 3.290 | 3.604 | 3.904 | 4.189
$m^{+2}\Lambda^{(2)}_{\text{upper}}$ | 0.029 | 0.140 | 0.340 | 0.409 | 0.479 | 0.551
$m^{+2}\Lambda^{(2)}_{\text{lower}}$ | 0.027 | 0.140 | 0.340 | 0.408 | 0.478 | 0.550
$1-\Lambda_{\text{lower}}/\Lambda_{\text{upper}}$ | 0.028 | 0.004 | 0.0004 | 0.001 | 0.001 | 0.002

$\overline{\lambda}$ | 10 | 11 | 12 | 13 | 16 | 18
---|---|---|---|---|---|---
$m^{-2}\Lambda_{\text{upper}}$ | 4.465 | 4.773 | 5.062 | 5.347 | 5.974 | 6.681
$m^{-2}\Lambda_{\text{lower}}$ | 4.462 | 4.753 | 5.018 | 5.310 | 5.941 | 6.635
$m^{+2}\Lambda^{(2)}_{\text{upper}}$ | 0.624 | 0.713 | 0.804 | 0.897 | 1.149 | 1.479
$m^{+2}\Lambda^{(2)}_{\text{lower}}$ | 0.623 | 0.708 | 0.792 | 0.887 | 1.146 | 1.472
$1-\Lambda_{\text{lower}}/\Lambda_{\text{upper}}$ | 0.001 | 0.004 |  0.009 | 0.007 | 0.006 | 0.007

Table 1: The numerical values of the non-perturbative couplings $\Lambda$ and
$\Lambda^{(2)}$ describing the $\phi^{4}$ model computed for various values of
$\overline{\lambda}$. We present the numerical values for both the upper and
the lower bound. We also indicate a relative difference between the upper and
the lower values of $\Lambda$. Analogous values are shown for the sinh-Gordon
model and its analytic continuation (staircase model) in table 2. Figure 7:
Bound on the parameters $\Lambda$ and $\Lambda^{(2)}$. The allowed region is
depicted in blue. The obtained numerical values for the $\phi^{4}$ model using
the LCT data from table 1 are indicated by red and purple crosses. These
crosses correspond to lower and upper bounds respectively.

As a solution of our optimization problem we obtain not only the data of table
1 but also all the coefficients in the ansatz (4.4) and (5.2). Taking these
coefficients as averages between the upper and lower bound results we obtain
numerical expressions for the interacting part of the scattering amplitude and
the form factor of the trace of the stress tensor. The results are presented
in figures 9 \- 11 for different values of $\overline{\lambda}$. For
$\overline{\lambda}=1$ we can compare our result with the perturbative
amplitude (3.19). It is depicted by the red dashed lines in figure 9 and 9. We
find an excellent agreement. For completeness we provide here the perturbative
value of $m^{-2}\Lambda$ for $\overline{\lambda}=1$ using equation (3.20). It
reads

$m^{-2}\Lambda=0.914\pm 0.006.$ (5.15)

This value is rather close to the one of table 1 for the upper bound which is
$0.903$. For $\overline{\lambda}=18$ we could try to compare our result with
the scattering amplitude of the deformed 2d Ising model. It is given by
equation (2.37) and (3.28) and reads

$\mathcal{T}(s)=4i\sqrt{s}\sqrt{s-4m^{2}}+{s(4m^{2}-s)\over M^{2}}.$ (5.16)

The value of $M^{2}$ can be estimated from equation (3.29) by plugging there
the value $m^{-2}\Lambda=6.681$ found in table 1. The amplitude (5.16) is
depicted in figure 9 and 9 by the black dashed line. We see that the
$\overline{\lambda}=18$ result has a similar shape to the amplitude (5.16).
Notice however that this comparison is rather crude since the
$\overline{\lambda}=18$ amplitude is still far away from the critical point
(its mass gap in unit of $m_{0}$ is $m/m_{0}\simeq 0.6186$).

In the “elastic” regime $s\in[4m^{2},16m^{2}]$, one can reconstruct the
spectral density from the obtained two particle form factor using equation
(2.31). For $\overline{\lambda}=10$ we explicitly compare the reconstructed
two-particle part of the spectral density with the LCT result (which was used
as part of the input data to the optimization problem) in figure 12. We see
that in the elastic regime they basically coincide. We present relative error
between the reconstructed two-particle part of the spectral density and the
LCT result for different values of $\overline{\lambda}$ in figure 14. The
relative errors become large at the threshold $s=4m^{2}$ (since
$\rho_{\Theta}^{\text{LCT}}$ is approaching 0 as $s$ goes to $4m^{2}$, and a
small uncertainty in the form factor can cause a somewhat large relative
error), but stay relatively low in the “elastic” region. This provides a solid
check for our bootstrap results for the form factor. The obtained scattering
amplitude must obey Watson’s equation (2.32) in the “elastic” regime. As
presented in figure 14, our bootstrap results indeed satisfy it well. This
provides validation of our bootstrap results for the obtained scattering
amplitudes. Outside of the “elastic” regime the presence of four- and higher-
particle form factors becomes necessary for the bounds from unitarity to be
tight. Since we do not have them in our bootstrap set-up, a potential concern
is that the bootstrap algorithm tries to saturate unitarity in this regime by
letting the two-particle form factor grow larger than it should be. So it is
not clear if our results for the form factor and the scattering amplitude in
the $s\geq 16m^{2}$ regime are relevant to the $\phi^{4}$ model. From figure
14 and 14, one can also see that generally for larger $\overline{\lambda}$,
the relative errors are larger. Therefore, we expect the uncertainties in the
results from the S-matrix/form factor bootstrap in figure 9 \- 11 to be
relatively larger for larger $\overline{\lambda}$.

Figure 8: Real part of the interacting scattering amplitude in the $\phi^{4}$
theory computed using the LCT data as an input to the S-matrix/form factor
bootstrap problem for various values of $\overline{\lambda}$. As a consistency
check, we also plotted the real part of the perturbative two-loop scattering
amplitude (equation (3.19)) with $\overline{\lambda}=1$ (red dotted line).

Figure 9: Imaginary part of the interacting scattering amplitude in the
$\phi^{4}$ theory computed using the LCT data as an input to the S-matrix/form
factor bootstrap problem for various values of $\overline{\lambda}$. As a
consistency check, we also plotted the imaginary part of the perturbative two-
loop scattering amplitude (equation (3.19)) with $\overline{\lambda}=1$ (red
dotted line).

Figure 10: Real part of the form factor of the trace of the stress tensor in
the $\phi^{4}$ theory computed using the LCT data as an input to the
S-matrix/form factor bootstrap problem for various values of
$\overline{\lambda}$. As a consistency check, we also plotted the real part of
the perturbative two-loop form factor (equation (3.14)) with
$\overline{\lambda}=1$ (red dotted line).

Figure 11: Imaginary part of the form factor of the trace of the stress tensor
in the $\phi^{4}$ theory computed using the LCT data as an input to the
S-matrix/form factor bootstrap problem for various values of
$\overline{\lambda}$. As a consistency check, we also plotted the imaginary
part of the perturbative two-loop form factor (equation (3.14)) with
$\overline{\lambda}=1$ (red dotted line). Figure 12: Comparison between the
LCT spectral density and the spectral density reconstructed from the obtained
form factor for $\overline{\lambda}=10$

Figure 13: Relative error between the LCT spectral density and the spectral
density obtained from the obtained two-particle form factor for various values
of $\overline{\lambda}$.

Figure 14: Check of Watson’s equation (2.32) using the obtained expressions of
the form factor and the spectral density for various values of
$\overline{\lambda}$. The vertical axis is given in the log scale.

###  5.3 Comparison of the sinh-Gordon model and $\phi^{4}$ model

Figure 7 nicely summarises the results of sections 4.2 and 5.2. It provides
the allowed region in the space of consistent quantum field theories (blue
region) and indicates the position of the $\phi^{4}$ model in this region (red
and purple crosses). Remarkably, the $\phi^{4}$ model lies super close to the
lower boundary of the allowed region where the sinh-Gordon model and its
analytic continuation (the staircase model) lie. In this section we discuss
the plausibility of this result.

To begin with, let us write explicitly the parameters $\Lambda$ and
$\Lambda^{(2)}$ in the sinh-Gordon model (and its analytic continuation) for
various values of $\beta^{2}$. The results are summarised in table 2. We chose
the values of $\beta^{2}$ in these tables in such a way that the sinh-Gordon
(and its analytic continuation) has the same values of $\Lambda$ as in table
1. As already expected from figure 7, the values of $\Lambda^{(2)}$ of the
$\phi^{4}$ model and the (analytically continued) sinh-Gordon model are almost
the same. Tables 1 and 2 quantify this similarity.

The comparison of tables 1 and 2 can be summarized as follows: given some
value of $\overline{\lambda}$ in the $\phi^{4}$ model, there is always some
value $\beta^{2}$ in the (analytically continued) sinh-Gordon model which
results in the $\Lambda$ and $\Lambda^{(2)}$ values similar to the ones in the
$\phi^{4}$ model. Only in the preturbative regime when $\overline{\lambda}\ll
4\pi$ we have $\beta^{2}\approx\overline{\lambda}$. (For example, the
$\phi^{4}$ model with $\overline{\lambda}=1$ is similar to the sinh-Gordon
model with $\beta^{2}=1.043$.)

Using the values of $\Lambda$ in table 1, one can compute the scattering
amplitude, the form factor of the trace of the stress tensor and spectral
density in the sinh-Gordon model (and its analytic continuation) using the
results of section 3.1. We compare them with our LCT expressions for the form
factor at $s\leq 0$ and the spectral density in figure 16. We observe that the
two models have a very similar behaviour in a wide range of values $s$ even at
strong coupling. Since the $\phi^{4}$ LCT data is so close to the sinh-Gordon
model, it is not surprising that the form factor at $s>0$ and the scattering
amplitudes we obtain from the numerical optimization will also be similar to
those of the sinh-Gordon model. To be concrete, we compare the form factors at
$s>0$ in the two models in figure 16 and the scattering amplitudes in figure
17. Notice especially that the form factors and scattering amplitudes for
these two theories are almost the same at $0<s<4m^{2}$, even for large
$\overline{\lambda}$. This also explains what we saw in figure 7.

It is important to stress that the amplitude and the form factor in the
$\phi^{4}$ and sinh-Gordon models must differ in the non-elastic regime
$s>16m^{2}$, however our bootstrap method does not allow us to compute the
$\phi^{4}$ observables in this regime reliably to see the difference. It is
interesting that there exist amplitudes belonging to different models which
are very similar in the elastic regime and differ significantly in the non-
elastic regime. See Tourkine:2021fqh  for a related discussion, where the
authors studied the question of how sensitive the elastic part of the
amplitude is to the inelastic regime, if one regards the latter as an input to
the S-matrix bootstrap and the former as an output. In particular, it would
likely shed light on the similarity of the $\phi^{4}$ and sinh-Gordon elastic
amplitudes by studying how much our S-matrix bounds vary under changes of the
inelastic amplitudes, using the framework of Tourkine:2021fqh .

$\beta^{2}$ | 1.043 | 3.298 | 8.219 | 11.1466 | 17.183
---|---|---|---|---|---
$m^{-2}\Lambda$ | 0.908 | 2.102 | 3.292 | 3.610 | 3.912
$m^{2}\Lambda^{(2)}$ | 0.026 | 0.138 | 0.339 | 0.407 | 0.478

$\beta^{2}$ | $8\pi\,e^{-0.560i}$ | $8\pi\,e^{-0.850i}$ | $8\pi\,e^{-1.081i}$ | $8\pi\,e^{-1.255i}$ | $8\pi\,e^{-1.402i}$ | $-8\pi\,e^{+1.466i}$ | $-8\pi\,e^{+1.196i}$
---|---|---|---|---|---|---|---
$m^{-2}\Lambda$ | 4.197 | 4.466 | 4.772 | 5.062 | 5.346 | 5.974 | 6.681
$m^{2}\Lambda^{(2)}$ | 0.551 | 0.623 | 0.712 | 0.801 | 0.893 | 1.115 | 1.395

Table 2: The numerical values of non-perturbative couplings $\Lambda$ and
$\Lambda^{(2)}$ describing the sinh-Gordon model and its analytic continuation
(staircase model) computed for various values $\beta^{2}$. Analogous values
are shown for the $\phi^{4}$ model in table 1.

Figure 15: Comparison of the form factor of the stress tensor at $s\leq 0$
(left plot) and its spectral density (right plot) in the $\phi^{4}$ model
(solid lines) and the (analytically continued) sinh-Gordon model (dotted
lines) for $\Lambda=\\{0.903,2.102,3.292,3.909,4.465,5.347,5.974,6.681\\}$
which correspond to $\overline{\lambda}=\\{1,3,6,8,10,13,16,18\\}$ in the
$\phi^{4}$ model according to table 1. The solid lines for the $\phi^{4}$
model are from light-cone truncation computation, while the dotted lines for
the sinh-Gordon model are from the analytic form factor formulas (3.8) and
(D.4). Note that for the sinh-Gordon spectral densities, we included the four-
particle form factor contribution, so that the result shown above is exact up
to $s=36m^{2}$.

Figure 16: Comparison of the real part (left plots) and imaginary part (right
plot) of the form factor of the stress tensor at $s>0$ in the $\phi^{4}$ model
(solid lines) and the (analytically continued) sinh-Gordon model (dotted
lines) for $\Lambda=\\{0.903,2.102,3.292,3.909,4.465,5.347,5.974,6.681\\}$
which correspond to $\overline{\lambda}=\\{1,3,6,8,10,13,16,18\\}$ in the
$\phi^{4}$ model according to table 1. The solid lines for the $\phi^{4}$
model are from the S-matrix/form factor bootstrap, while the dotted lines for
sinh-Gordon are from the analytic two-particle form factor formula (3.8).

Figure 17: Comparison of the two-to-two scattering amplitudes in the
$\phi^{4}$ model (solid lines) and the (analytically continued) sinh-Gordon
model (dotted lines) for
$\Lambda=\\{0.903,2.102,3.292,3.909,4.465,5.347,5.974,6.681\\}$, which
correspond to $\overline{\lambda}=\\{1,3,6,8,10,13,16,18\\}$ in the $\phi^{4}$
model according to table 1. The solid lines for the $\phi^{4}$ model are from
the S-matrix/form factor bootstrap, while the dotted lines for the sinh-Gordon
model are from the analytic S-matrix formula (3.3)

##  6 Discussion and Future Directions

The main purpose of this paper was to start with some nonperturbative data for
a specific model, in this case for $\phi^{4}$ theory in 2d, and to inject that
data into the S-matrix/form factor bootstrap in order to compute additional
observable quantities. Ideally, one might hope that with a finite amount of
such data, the constraints of crossing, analyticity, and unitarity completely
determine the rest of the theory. Less ambitiously, the S-matrix
bootstrap/form factor might simply provide a robust method to extract
additional results from some initial data. In our specific application, our
‘initial data’ was the spectral density of the stress tensor, and its form
factor with two-particle states in a certain kinematic regime $s\leq 0$,
computed using lightcone Hamiltonian truncation methods from our companion
paper truncffsd . Roughly, the bootstrap can take this data and obtain the
form factor in a different kinematic regime, at $s>0$, after which the form of
the elastic scattering amplitude follows from Watson’s theorem. An important
part of the challenge was that the input data itself is determined
numerically, so that simply analytically continuing between different
kinematic regimes is not straightforward.161616In a system at finite volume,
Luscher’s method Luscher:1985dn ; Luscher:1986pf  provides another handle on
the elastic scattering amplitudes, which could be used to verify or improve
the S-matrix bootstrap results. The work bajnok2016truncated  applied
Luscher’s method to equal-time Hamiltonian truncation in finite volume in the
broken phase of $\phi^{4}$ theory, and it should be possible to repeat their
analysis in the unbroken phase that we have studied in this work. One could
instead try to obtain the finite volume spectrum from lattice Monte Carlo
rather than from truncation methods.

One of the surprises of our analysis is that the elastic 2-to-2 S-matrix in
$\phi^{4}$ theory is extremely close to that of the sinh-Gordon model and its
analytic continuation (the staircase model), after the couplings of both
models are adjusted to have the same value of $\Lambda$ (the interacting part
of the scattering amplitude value at the crossing-symmetric point $s=2m^{2}$).
The fact that the scattering amplitudes in both models are are somewhat close
is perhaps not very surprising. As we have emphasized, the 2-to-2 S-matrices
for the two theories are identical in perturbation theory around $\Lambda=0$
until $\mathcal{O}(\Lambda^{4})$, which is the first order in perturbation
theory where $\phi^{4}$ has particle production. Moreover, both theories reach
a critical point at the upper limit $\Lambda=8m^{2}$ where they describe the
Ising model S-matrix, and perturbation theory around this upper limit is
described at $\mathcal{O}(8m^{2}-\Lambda)$ by the leading irrelevant
deformation $T\overline{T}$, so the first difference between the theories
arises at $\mathcal{O}((8m^{2}-\Lambda)^{2})$. So one could reasonably expect
the S-matrices to be quite similar in between these two limits. Nevertheless,
the degree to which they agree even at intermediate strongly coupled values is
still remarkable. One might worry that this agreement is an artifact of the
S-matrix bootstrap itself, which tends to push theories to saturate unitarity
conditions and therefore tends to find integrable models. In fact, we have
shown that a pure S-matrix bootstrap analysis, without any injection of
dynamical data from LCT, exactly finds the sinh-Gordon/staircase model
S-matrix. However, we emphasize that our $\phi^{4}$ S-matrix bootstrap
analysis used a different optimization condition from our pure S-matrix
bootstrap analysis. In the former, we fixed the data from LCT and maximized
$\Lambda$, whereas in the latter we fixed $\Lambda$ and maximized the second
derivative $\Lambda^{(2)}$ of the S-matrix at the crossing-symmetric point.
Moreover, $\phi^{4}$ theory really should saturate unitarity in the elastic
regime $4m^{2}<s<16m^{2}$ due to kinematics, so one cannot think of this
saturation as an artifact of the S-matrix bootstrap. Rather, in practical
terms it appears that the origin of this close agreement is that even at
strong coupling, the stress tensor form factor at $s\leq 0$ and the spectral
density at $4m^{2}<s<16m^{2}$, which we compute in LCT, is very similar to
that of sinh-Gordon/staircase model.171717We also compute the stress tensor
spectral density at $s>16m^{2}$, and here we do see a significant deviation
between $\phi^{4}$ and sinh-Gordon. However, the S-matrix bootstrap result for
the elastic scattering amplitude does not seem to be very sensitive to the
detailed behavior of the spectral density in this regime.

Although 2-to-2 elastic scattering appears to be very similar in $\phi^{4}$
theory and the sinh-Gordon model, we do not expect it to be similar at large
$s$ and it certainly cannot be similar for 2-to-$2+n$ since particle
production exactly vanishes in sinh-Gordon. The S-matrix bootstrap with both
two- and four-particle external states would therefore be particularly
illuminating in this case since it would uncover more of the qualitative
difference between the two models. In $d>2$, including higher multiplicities
in the S-matrix bootstrap is likely quite challenging due to the large
kinematic parameter space, but in $d=2$ we are optimistic that it would be
practical. If one wanted to use the S-matrix bootstrap in combination with UV
CFT operators, as we have done in this work, then the inclusion of four-
particle external states would necessitate the appearance of four-particle
form factors $\mathcal{F}_{4,0}^{\Theta}$ in the unitarity condition which is
very hard to compute in the LCT framework. Perhaps, one could simply
parameterize it and try to obtain it as one of the outputs of the S-matrix
bootstrap.

Finally, we end by mentioning possible generalizations of the method. There
are many other models in 2d that would be interesting to analyze using this
approach. LCT can be applied to theories with more general field content in
2d, including gauge fields and fermions, and 2d QCD at finite $N_{c}$ would be
a particularly interesting application.181818See e.g. Dempsey:2021xpf ;
Katz:2014uoa ; Katz:2013qua  for recent LCT and DLCQ applications to 2d QCD.
Our approach here is similar in spirit to that of Gabai:2019ryw , which
studied Ising Field theory with both a $\sigma$ and $\epsilon$ deformation
using TFFSA and Luscher’s method Luscher:1985dn ; Luscher:1986pf , but it
would be interesting to see if any more mileage could be gained by also
including form factors and spectral densities in a generalized unitarity
condition as we did in this paper. More ambitiously, our method in principle
can be applied to higher dimensions, the main challenge being that it is
difficult to obtain the input data. LCT has been applied to the $\phi^{4}$
model in 3d, and the stress tensor spectral density was obtained in
Anand:2020qnp .191919Both lightcone and equal-time Hamiltonian truncation have
seen important recent progress for $\phi^{4}$ theory in $d>2$
Hogervorst:2014rta ; Katz:2016hxp ; Elias-Miro:2020qwz ; Anand:2020qnp . One
of the main challenges has been dealing with state-dependent counterterms for
divergences. The recent works Elias-Miro:2020qwz ; Anand:2020qnp  developed
systematic methods to handle this issue and specifically applied their work in
the context of 3d $\phi^{4}$ theory. One would have to generalize our
treatment of form factors to 3d, but the basic idea would be the same. In
$d>2$ there are two stress-tensor two-particle form factors
$\mathcal{F}_{2,0}^{\Theta}(s)$ and $\mathcal{F}_{2,0}^{(2)}(s)$, as well as
two spectral densities $\rho_{\Theta}(s)$ and $\rho_{2}(s)$, and the
scattering amplitude $\mathcal{S}(s,t)$ can be decomposed into partial
amplitudes $\mathcal{S}_{j}(s)$ with $j=0,2,4,\ldots$. The generalization of
the unitarity constraint (5.1) was worked out in Karateev:2020axc .202020See
equations (6.36) and (6.41) there. So although generalizing our work to 3d
would involve significant work, at least all the pieces have already been
assembled and are waiting to be used.

### Acknowledgments

We thank Ami Katz, Alexander Monin, Giuseppe Mussardo, João Penedones, Balt
van Rees, Matthew Walters, for helpful conversations, and in particular Ami
Katz and Matthew Walters for comments on a draft. ALF and HC were supported in
part by the US Department of Energy Office of Science under Award Number DE-
SC0015845 and the Simons Collaboration Grant on the Non-Perturbative
Bootstrap, and ALF in part by a Sloan Foundation fellowship.

##  Appendix A Kinematics of 2d Scattering

Consider the scattering of two identical scalar particles in two space-time
dimensions. We denote the initial two-momenta of two particles (before the
scattering) by $p_{1}^{\mu}$ and $p_{2}^{\mu}$ and the final two-momenta of
two particles (after the scattering) by $k_{1}^{\mu}$ and $k_{2}^{\mu}$. The
two particles obey the mass-shell condition

$p_{i}^{2}=k_{i}^{2}=-m^{2},$ (A.1)

where $i=1,2$. The conservation of two-momenta leads to the requirement

$p_{1}^{\mu}+p_{2}^{\mu}-k_{1}^{\mu}-k_{2}^{\mu}=0.$ (A.2)

Due to the mass-shell condition (A.1), there are only two different solutions
for the two momenta after the scattering, namely

$\vec{k}_{1}=\vec{p}_{1},\quad\vec{k}_{2}=\vec{p}_{2}\qquad\text{or}\qquad\vec{k}_{1}=\vec{p}_{2},\quad\vec{k}_{2}=\vec{p}_{1}.$
(A.3)

Let us recall that the Mandelstam variables are defined as

$\displaystyle s\equiv-(p_{1}+p_{2})^{2},\qquad
t\equiv-(p_{1}-k_{1})^{2},\qquad u\equiv-(p_{1}-k_{2})^{2}.$ (A.4)

Plugging the two solutions (A.3) into the definition of the Mandelstam
variables we see that they correspond to two different situation

$t=0\qquad\text{or}\qquad u=0.$ (A.5)

The two solutions (A.3) are related by the discrete $Z_{2}$ symmetry
$\vec{k}_{1}\leftrightarrow\vec{k}_{2}$. The scattering in $d=2$ happens on
the line. It is standard to work with the convention when two particle states
are defined in such a way that particle 1 (with momentum $\vec{p}_{1}$) is to
the left of particle 2 (with momentum $\vec{p}_{2}$) on the line. Then the in
two-particle states are required to obey $\vec{p}_{1}>\vec{p}_{2}$. This
condition forces the trajectories of two particles to cross as time goes by.
Instead the out two-particle states are required to obey
$\vec{p}_{1}<\vec{p}_{2}$ condition which ensures that the particles will
never meet in the future. When considering the scattering process
$p_{1}p_{2}\rightarrow k_{1}k_{2}$, the above convention is imposed by adding
the following product of step-functions

$\theta(\vec{p}_{1}-\vec{p}_{2})\theta(\vec{k}_{2}-\vec{k}_{1})$ (A.6)

into the definition of 2d scattering amplitudes. Plugging here the solution
(A.3) we see that in this convention the $t=0$ solution vanishes and we are
left only with the $u=0$ solution.

Let us now derive a very useful relation. Consider the Dirac $\delta$-function
which encodes the conservation condition (A.2), namely

$\delta^{2}(p_{1}+p_{2}-k_{1}-k_{2})=\delta(p_{1}^{0}+p_{2}^{0}-k_{1}^{0}-k_{2}^{0})\delta(\vec{p}_{1}+\vec{p}_{2}-\vec{k}_{1}-\vec{k}_{2}).$
(A.7)

Here the energies $p_{i}^{0}$ and $k_{i}^{0}$ are fixed in term of the momenta
$\vec{p}_{i}$ and $\vec{k}_{i}$ due to the mass-shell condition (A.1). Given
the initial values of $\vec{p}_{i}$, this Dirac $\delta$-function restricts
the values of $\vec{k}_{i}$ to their allowed range, in 2d this restriction is
severe and leads only to two possibilities (A.3). Let us now imagine that we
would like to integrate (A.7) with some kernel over all possible values of
$\vec{k}_{i}$, namely

$\int_{-\infty}^{+\infty}d\vec{k}_{1}\int_{-\infty}^{+\infty}d\vec{k}_{2}\,f(\vec{k}_{1},\vec{k}_{2})\delta^{2}(p_{1}+p_{2}-k_{1}-k_{2}).$
(A.8)

In order to perform this integration we need to perform several steps which we
explain below.

Due to the second Dirac $\delta$-function in the right-hand side of (A.7), we
have $\vec{k}_{2}=\vec{p}_{1}+\vec{p}_{2}-\vec{k}_{1}$. Thus, we can fully
eliminate the integral over $\vec{k}_{2}$. Plugging this restriction back into
(A.7) and using the mass-shell condition (A.1), we get

$\delta(p_{1}^{0}+p_{2}^{0}-k_{1}^{0}-k_{2}^{0})=\delta\Big{(}g(\vec{k}_{1})\Big{)},$
(A.9)

where we have defined

$g(\vec{k}_{1})\equiv
p_{1}^{0}+p_{2}^{0}-\sqrt{m^{2}+\vec{k}_{1}^{\,2}}-\sqrt{m^{2}+(\vec{p}_{1}+\vec{p}_{2}-\vec{k}_{1})^{\,2}}.$
(A.10)

Let use now use the standard property of the Dirac $\delta$-functions and the
fact that $g(\vec{k}_{1})=0$ has only two solutions given by (A.3). We have
then

$\delta(p_{1}^{0}+p_{2}^{0}-k_{1}^{0}-k_{2}^{0})={\delta(\vec{k}_{1}-\vec{p}_{1})\over\Big{|}g^{\prime}(\vec{p}_{1})\Big{|}}+{\delta(\vec{k}_{1}-\vec{p}_{2})\over\Big{|}g^{\prime}(\vec{p}_{2})\Big{|}}.$
(A.11)

Evaluating the derivatives we finally obtain

$\delta(p_{1}^{0}+p_{2}^{0}-k_{1}^{0}-k_{2}^{0})={p_{1}^{0}p_{2}^{0}\over\left|\vec{p}_{1}p_{2}^{0}-\vec{p}_{2}p_{1}^{0}\right|}\times\left(\delta(\vec{k}_{1}-\vec{p}_{1})+\delta(\vec{k}_{1}-\vec{p}_{2})\right).$
(A.12)

Plugging (A.12) into (A.7), we get the final relation

$\delta^{2}(p_{1}+p_{2}-k_{1}-k_{2})=\\\
{p_{1}^{0}p_{2}^{0}\over\left|\vec{p}_{1}p_{2}^{0}-\vec{p}_{2}p_{1}^{0}\right|}\times\left(\delta(\vec{k}_{1}-\vec{p}_{1})\delta(\vec{k}_{2}-\vec{p}_{2})+\delta(\vec{k}_{1}-\vec{p}_{2})\delta(\vec{k}_{2}-\vec{p}_{1})\right).$
(A.13)

Equivalently we could write it as

$4\left|\vec{p}_{1}p_{2}^{0}-\vec{p}_{2}p_{1}^{0}\right|\times\delta^{2}(p_{1}+p_{2}-k_{1}-k_{2})=\\\
4p_{1}^{0}p_{2}^{0}\times\left(\delta(\vec{k}_{1}-\vec{p}_{1})\delta(\vec{k}_{2}-\vec{p}_{2})+\delta(\vec{k}_{1}-\vec{p}_{2})\delta(\vec{k}_{2}-\vec{p}_{1})\right).$
(A.14)

Let us now evaluate the expression (A.14) in the center of mass frame defined
as $\vec{p}_{2}=-\vec{p}_{1}$. Plugging this condition into (A.4) we conclude
that in the center of mass frame

$|\vec{p}_{1}|={1\over 2}\,\sqrt{s-4m^{2}},\qquad p_{1}^{0}={1\over
2}\,\sqrt{s}.$ (A.15)

Plugging these into the left-hand side of (A.14) we get

$4\left|\vec{p}_{1}p_{2}^{0}-\vec{p}_{2}p_{1}^{0}\right|=2\sqrt{s}\sqrt{s-4m^{2}}.$
(A.16)

We then notice that the quantity
$4\left|\vec{p}_{1}p_{2}^{0}-\vec{p}_{2}p_{1}^{0}\right|$ is Lorentz
invariant, thus (A.16) holds in a generic frame! The result (A.14) together
with (A.16) gives precisely (2.21).

##  Appendix B $O(N)$ model

Let us consider the case when the system has a global $O(N)$ symmetry. We will
require our asymptotic states to transform in the vector representation of
$O(N)$. They will thus carry an extra label $a=1\ldots N$. The one particle
states are normalized as before with an addition of the Kronecker delta due to
the presence of the $O(N)$ vector indicies

$\displaystyle{}_{b}\langle
m,\vec{p}_{2}|m,\vec{p}_{1}\rangle_{a}=2p^{0}\delta_{ab}\times
2\pi\delta(\vec{p}_{2}-\vec{p}_{1}).$ (B.1)

The full scattering amplitude can be decomposed into three independent
scattering amplitudes $\sigma_{i}(s)$, $i=1,2,3$. In the notation of
Zamolodchikov:1978xm  we have

$\displaystyle{}_{cd}\langle
m,\vec{p}_{3};m,\vec{p}_{4}|S|m,\vec{p}_{1};m,\vec{p}_{2}\rangle_{ab}=$
$\displaystyle(2\pi)^{2}\delta^{(2)}(p_{1}+p_{2}-p_{3}-p_{4})\times$
$\displaystyle\big{(}\sigma_{1}(s)\delta_{ab}\delta_{cd}+\sigma_{2}(s)\delta_{ac}\delta_{bd}+\sigma_{3}(s)\delta_{ad}\delta_{bc}\big{)}.$
(B.2)

Crossing $1\leftrightarrow 3$ implies the following relations

$\sigma_{1}(s)=\sigma_{3}(4m^{2}-s),\quad\sigma_{2}(s)=\sigma_{2}(4m^{2}-s).$
(B.3)

Let us discuss unitarity now. The two-particle states transform in the
reducible $O(N)$ representation and can be further decomposed into three
irreducible representations as

$|m,\vec{p}_{1};m,\vec{p}_{2}\rangle_{ab}={\delta_{ab}\over\sqrt{N}}|m,\vec{p}_{1};m,\vec{p}_{2}\rangle^{\bullet}+|m,\vec{p}_{1};m,\vec{p}_{2}\rangle^{\textbf{S}}_{(ab)}+|m,\vec{p}_{1};m,\vec{p}_{2}\rangle^{\textbf{A}}_{[ab]},$
(B.4)

where we have defined

$\displaystyle|m,\vec{p}_{1};m,\vec{p}_{2}\rangle^{\bullet}$
$\displaystyle\equiv{1\over\sqrt{N}}\,\sum_{a=1}^{N}|m,\vec{p}_{1};m,\vec{p}_{2}\rangle_{aa},$
(B.5) $\displaystyle|m,\vec{p}_{1};m,\vec{p}_{2}\rangle^{\textbf{S}}_{(ab)}$
$\displaystyle\equiv{1\over
2}\,\Big{(}|m,\vec{p}_{1};m,\vec{p}_{2}\rangle_{ab}+|m,\vec{p}_{1};m,\vec{p}_{2}\rangle_{ba}\Big{)}-{\delta_{ab}\over\sqrt{N}}|m,\vec{p}_{1};m,\vec{p}_{2}\rangle^{\bullet},$
(B.6) $\displaystyle|m,\vec{p}_{1};m,\vec{p}_{2}\rangle^{\textbf{A}}_{[ab]}$
$\displaystyle\equiv{1\over
2}\,\Big{(}|m,\vec{p}_{1};m,\vec{p}_{2}\rangle_{ab}-|m,\vec{p}_{1};m,\vec{p}_{2}\rangle_{ba}\Big{)}.$
(B.7)

The labels $\bullet$, S and A stand for trivial, symmetric traceless and
antisymmetric representations. Using the normalization condition (B.1) we find
that

$\displaystyle{}^{\bullet}\langle
m,\vec{p}_{3};m,\vec{p}_{4}|m,\vec{p}_{1};m,\vec{p}_{2}\rangle^{\bullet}$
$\displaystyle=\mathcal{N}_{2}\times(2\pi)^{2}\delta^{(2)}(p_{1}+p_{2}-p_{3}-p_{4}),$
(B.8) $\displaystyle{}_{(cd)}^{\textbf{S}}\langle
m,\vec{p}_{3};m,\vec{p}_{4}|m,\vec{p}_{1};m,\vec{p}_{2}\rangle^{\textbf{S}}_{(ab)}$
$\displaystyle=\mathcal{N}_{2}\,T^{ab,cd}_{\textbf{S}}\times(2\pi)^{2}\delta^{(2)}(p_{1}+p_{2}-p_{3}-p_{4}),$
(B.9) $\displaystyle{}_{[cd]}^{\textbf{A}}\langle
m,\vec{p}_{3};m,\vec{p}_{4}|m,\vec{p}_{1};m,\vec{p}_{2}\rangle^{\textbf{A}}_{[ab]}$
$\displaystyle=\mathcal{N}_{2}\,T^{ab,cd}_{\textbf{A}}\times(2\pi)^{2}\delta^{(2)}(p_{1}+p_{2}-p_{3}-p_{4}).$
(B.10)

Notice that the normalization condition for the trivial representation is
exactly the one used in the main text, see (2.21).

Taking into account (B.4) alternatively to (B.2) we can rewrite the full
scattering amplitude in terms of independent scattering amplitudes
$S_{\bullet}(s)$, $S_{\textbf{S}}(s)$ and $S_{\textbf{A}}(s)$, as

$\displaystyle{}_{cd}\langle
m,\vec{p}_{3};m,\vec{p}_{4}|S|m,\vec{p}_{1};m,\vec{p}_{2}\rangle_{ab}=$
$\displaystyle(2\pi)^{2}\delta^{(2)}(p_{1}+p_{2}-p_{3}-p_{4})\times$
$\displaystyle\big{(}S_{\bullet}(s)T^{ab,cd}_{\bullet}+S_{\textbf{S}}(s)T^{ab,cd}_{\textbf{S}}+S_{\textbf{A}}(s)T^{ab,cd}_{\textbf{A}}\big{)},$
(B.11)

where the tensor structures associated to the three irreducible
representations are defined as

$T^{ab,cd}_{\bullet}\equiv{1\over N}\delta_{ab}\delta_{cd},\quad
T^{ab,cd}_{\textbf{S}}\equiv{\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc}\over
2}-{1\over N}\delta_{ab}\delta_{cd},\quad
T^{ab,cd}_{\textbf{A}}\equiv{\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc}\over
2}.$ (B.12)

The relation between two sets of amplitudes $\sigma_{1}$, $\sigma_{2}$,
$\sigma_{3}$ and $S_{\bullet}$, $S_{\textbf{S}}$, $S_{\textbf{A}}$ simply
reads as

$\displaystyle S_{\bullet}(s)$
$\displaystyle=\sigma_{2}(s)+\sigma_{3}(s)+N\sigma_{1}(s),$ $\displaystyle
S_{\textbf{S}}(s)$ $\displaystyle=\sigma_{2}(s)+\sigma_{3}(s),$ (B.13)
$\displaystyle S_{\textbf{A}}(s)$ $\displaystyle=\sigma_{2}(s)-\sigma_{3}(s),$

In section 3.2 of Karateev:2019ymz  it was shown that using the states in the
irreducible representation of the $O(N)$ one can formulate the unitarity
constraints in the simple form. For the trivial representation we have

$\begin{pmatrix}1&\mathcal{N}_{2}^{\,-1}{\mathcal{S}}_{\bullet}^{*}(s)&\mathcal{N}_{2}^{\,-1/2}\,\mathcal{F}_{2,0}^{*\Theta}(s)\\\
\mathcal{N}_{2}^{\,-1}{\mathcal{S}}_{\bullet}(s)&1&\mathcal{N}_{2}^{\,-1/2}\,\mathcal{F}_{2,0}^{\Theta}(s)\\\
\mathcal{N}_{2}^{\,-1/2}\,\mathcal{F}_{2,0}^{\Theta}(s)&\mathcal{N}_{2}^{\,-1/2}\,\mathcal{F}_{2,0}^{*\Theta}(s)&2\pi\rho_{\Theta}(s)\end{pmatrix}\succeq
0,$ (B.14)

where the form factor is defined as

$\displaystyle\mathcal{F}_{2,0}^{\Theta}(s)$ $\displaystyle\equiv\langle
0|\Theta(0)|m,\vec{p}_{1};m,\vec{p}_{2}\rangle^{\bullet}$ (B.15)
$\displaystyle=\sqrt{N}\;\langle
0|\Theta(0)|m,\vec{p}_{1};m,\vec{p}_{2}\rangle_{11}.$

For the symmetric and antisymmetric representations we have instead

$\begin{pmatrix}1&\mathcal{N}_{2}^{\,-1}{\mathcal{S}}_{\textbf{S}}^{*}(s)\\\
\mathcal{N}_{2}^{\,-1}{\mathcal{S}}_{\textbf{S}}(s)&1\end{pmatrix}\succeq
0,\qquad\begin{pmatrix}1&\mathcal{N}_{2}^{\,-1}{\mathcal{S}}_{\textbf{A}}^{*}(s)\\\
\mathcal{N}_{2}^{\,-1}{\mathcal{S}}_{\textbf{A}}(s)&1\end{pmatrix}\succeq 0.$
(B.16)

The crossing equations (B.3) in the new basis read as

$\begin{pmatrix}{\mathcal{S}}_{\bullet}(s)\\\ {\mathcal{S}}_{\textbf{S}}(s)\\\
{\mathcal{S}}_{\textbf{A}}(s)\end{pmatrix}=\begin{pmatrix}{1\over N}&&{1\over
2}-{1\over N}+{N\over 2}&&{1\over 2}-{N\over 2}\\\ {1\over N}&&{1\over
2}-{1\over N}&&{1\over 2}\\\ -{1\over N}&&{1\over 2}+{1\over N}&&{1\over
2}\end{pmatrix}\begin{pmatrix}{\mathcal{S}}_{\bullet}(4m^{2}-s)\\\
{\mathcal{S}}_{\textbf{S}}(4m^{2}-s)\\\
{\mathcal{S}}_{\textbf{A}}(4m^{2}-s)\end{pmatrix}.$ (B.17)

Let us now consider the 2d $O(N)$ model with $\phi^{4}$ potential. In the
large $N$ limit $N\rightarrow\infty$ limit using perturbation theory it is
straightforward to show that

$\sigma_{i}(s)={\overline{\sigma}_{i}(s)\over N}+O(N^{-2}),\quad i=1,2,3,$
(B.18)

where $\overline{\sigma}_{i}(s)$ is the finite part in the large $N$ limit.
Using (B.13) we conclude that

$S_{\bullet}(s)=\overline{\sigma}_{1}(s),\qquad
NS_{\textbf{S}}(s)=\overline{\sigma}_{2}(s)+\overline{\sigma}_{3}(s),\qquad
NS_{\textbf{A}}(s)=\overline{\sigma}_{2}(s)-\overline{\sigma}_{3}(s).$ (B.19)

Using these we can read off from (B.17) the crossing equation for the trivial
scattering amplitude. It reads

$S_{\bullet}(s)=\overline{\sigma}_{1}(s)=\overline{\sigma}_{3}(4m^{2}-3).$
(B.20)

Clearly this crossing equation does not close if we consider only the trivial
scattering amplitude.

##  Appendix C Perturbative Computations

In this appendix, we will detail various analytic computations of the form
factors and scattering amplitudes in solvable limits (large $N$, non-
relativistic, and perturbative $\lambda$) that we use throughout the paper.

###  C.1 Feynman Diagrams

Figure 18: Feynman diagrams for the 2-to-2 $S$-matrix up to one loop (plus
crossed diagrams).

####  C.1.1 $\phi^{4}$ theory

Figure 19: Feynman diagrams for the two-particle form factor of $\Theta$ up to
two loops.

Figure 20: Feynman diagrams for the $T_{--}$ two-point function up to two
loops.

We begin with the form factors and amplitudes in a loop expansion, in powers
of the coupling $\lambda$. The leading order $\mathcal{O}(\lambda^{0})$ free
theory expressions are

$\widehat{{\mathcal{S}}}=1,\quad\mathcal{F}_{2,0}^{\Theta}=-2m^{2},\quad\pi\rho_{\Theta}=2m^{4}\omega^{2}\theta(s-4m^{2}),$
(C.1)

where $\omega\equiv{1\over 2\sqrt{s(s-4m^{2})}}={\mathcal{N}}_{2}^{-1}$.

To compute the form factors and spectral densities of $\Theta$, it is in
general easier to compute those of $T_{--}$ first and then use the Ward
identity than it is to compute those of $\Theta$ directly. The reason is that
$T_{--}$ is simply $(\partial_{-}\phi)^{2}$, independent of the interaction
and mass terms, and so involves fewer Feynman diagrams. At tree-level,

$\langle m^{2},p_{1};m^{2},p_{2}|T_{--}(0)\rangle=2p_{1-}p_{2-},$ (C.2)

The Ward identity implies

$\langle m^{2},p_{1};m^{2},p_{2}|\Theta(0)\rangle=-{s\over p_{-}^{2}}\langle
m^{2},p_{1};m^{2},p_{2}|T_{--}(0)\rangle$ (C.3)

where

$s=(p_{1}+p_{2})^{2}=m^{2}{p_{-}^{2}\over p_{1-}p_{2-}},\qquad p_{-}\equiv
p_{1-}+p_{2-},$ (C.4)

so at $\mathcal{O}(\lambda^{0})$ we obtain

$\langle m^{2},p_{1};m^{2},p_{2}|\Theta(0)\rangle=-2m^{2},$ (C.5)

as claimed, and as can easily be verified by a direct computation with
$\Theta$.

At the next order, $\mathcal{O}(\lambda)$, the S-matrix is given by a tree
diagram, the form factor involves an one-loop computation, and the spectral
density involves a two-loop diagram, as shown in the corresponding diagrams in
figure 18, 19, and 20. The S-matrix is simply

$\widehat{{\mathcal{S}}}=1-i\lambda\omega^{2}.$ (C.6)

The form factor one-loop diagram (top right diagram in figure 19) can be
computed by standard methods,

$\langle m^{2},p_{1};m^{2},p_{2}|T_{--}(0)\rangle=-{\lambda\over
4\pi}\int_{0}^{1}dx{x(1-x)p_{-}^{2}\over m_{0}^{2}-x(1-x)s}.$ (C.7)

The integral over the Feynman parameter $x$ can be done in closed form to
obtain the expression given in equation (3.14), which for reference we write
here as

$\langle m^{2},p_{1};m^{2},p_{2}|\Theta(0)\rangle=-2m^{2}+{\lambda\over
4\pi}\Delta(s)+\mathcal{O}(\lambda^{2}),\quad\Delta(m^{2}x)\equiv-1+\lim_{\epsilon\rightarrow
0^{+}}{4\text{ArcTan}\left({\sqrt{x}\over\sqrt{4-x-i\epsilon}}\right)\over\sqrt{x(4-x-i\epsilon)}}.$
(C.8)

We have used $m_{0}=m+\mathcal{O}(\lambda^{2})$, so $m$ and $m_{0}$ are
interchangeable at this order.

The $\mathcal{O}(\lambda)$ (i.e. two-loop) diagram (second diagram in figure
20) for the $T_{--}$ time-ordered two-point function factors into a product of
two one-loop diagrams. The Ward identity can again be used to obtain the
correlator with $T_{--}$s replaced by $\Theta$s, so by evaluating a couple of
one-loop diagrams we obtain

$\pi\rho_{\Theta}(s)=\textrm{Re}\int d^{2}xe^{-ip\cdot x}\langle{\rm
vac}|\Theta(x)\Theta(0)|{\rm
vac}\rangle_{T}=\theta(s-4m^{2})\left[2m^{4}\omega^{2}+{\lambda\over(4\pi)^{2}}\textrm{Im}(\Delta^{2}(s))+\mathcal{O}(\lambda^{2})\right].$
(C.9)

At $s>4m^{2}$, the result for $\rho_{\Theta}$ can be written a bit more
explicitly with the following expressions for the real and imaginary parts of
$\Delta(s)$:

$\Delta(s)\stackrel{{\scriptstyle s>4m^{2}}}{{=}}-\left[4m^{2}{{\rm
ArcCosh}\left(\sqrt{{s\over
4m^{2}}}\right)\over\sqrt{s(s-4m^{2})}}\right]-4i\pi m^{2}\omega^{2}.$ (C.10)

One can also perform these computations directly with $\Theta$; in that case,
it is crucial to include a subtle contribution $\propto\lambda\phi^{2}$ in the
definition of $\Theta$ itself:

$\Theta=m^{2}\phi^{2}+{\lambda\over 12}\phi^{4}+{\lambda\over 8\pi}\phi^{2},$
(C.11)

see e.g. Anand:2017yij  for details.212121One way to “discover” the
contribution ${\lambda\over 8\pi}\phi^{2}$ to $\Theta$ is that the relation
$\mathcal{F}_{2,0}^{\Theta}(0)=-2m^{2}$ is not satisfied at one-loop if it is
not included.

At the next order, $\mathcal{O}(\lambda^{2})$, the perturbative diagrams for
$\mathcal{F}_{2,0}^{\Theta}$ and $\rho_{\Theta}$ become more difficult to
evaluate, involving a two-loop and three-loop computation, respectively. Here
we will only derive the $\mathcal{O}(\lambda^{2})$ contribution to
$\mathcal{F}_{2,0}^{\Theta}$. As a check, in the next subsection we will
rederive the $\mathcal{O}(\lambda^{2})$ contribution to
$\mathcal{F}^{\Theta}_{2,0}$ using dispersion relations. Since particle
production is kinematically forbidden for $s<16m^{2}$, we can actually obtain
the three-loop spectral density in this regime from the two-loop form factor.

First, the $\mathcal{O}(\lambda^{2})$ contribution to the S-matrix involves
only an one-loop diagram (second diagram in figure 18) that can be easily
evaluated:

${\mathcal{T}}=-\lambda+{\lambda^{2}\over 8\pi m^{2}}\left(1+4\pi
im^{2}\omega^{2}\right)+\mathcal{O}(\lambda^{3}).$ (C.12)

To compute the $\mathcal{O}(\lambda^{2})$ correction to
$\mathcal{F}_{2,0}^{\Theta}$, we again compute $\langle
m^{2},p_{1};m^{2},p_{2}|T_{--}(0)\rangle$ and use the Ward identity. There are
two two-loop diagrams that must be evaluated, as shown in Fig. 19. The first
is a simple product of two one-loop diagrams, and is easily evaluated to be

$F_{2,0}^{\Theta}\supset-{\lambda^{2}\over
2(4\pi)^{2}m^{2}}\Delta(s)(\Delta(s)+1).$ (C.13)

The second two-loop diagram in Fig. 19 involves the integral222222See e.g.
(10.57) in Peskin:1995ev , which is easily generalized to the diagram we are
considering.

${\mathcal{I}}\equiv\int_{0}^{1}dx\int_{0}^{1}dy\int_{0}^{1}dw\int{d^{2}k\over(2\pi)^{2}}{(1-w)k_{-}(k+p)_{-}\over(w[x(1-x)(k+p_{1})^{2}]+(1-w)[k^{2}+2yk\cdot
p+yp^{2}]+m^{2})^{3}},$ (C.14)

(where $p=p_{1}+p_{2}$) plus a symmetric contribution with
$p_{1}\leftrightarrow p_{2}$. With some effort, these integrals can be
evaluated and massaged into the closed form result in equation (3.14). In
(3.14), we have also had to adjust for an $\mathcal{O}(\lambda^{2})$
wavefunction renormalization Serone:2018gjo ,232323The wavefunction
renormalization factor is given by $b_{2}^{(1)}$ from Table 8 of
Serone:2018gjo ; we have used the fact that their numeric value for
$b_{2}^{(1)}$ is equal to ${9\over 2\pi^{2}}-{3\over 8}$.

$Z^{-1}=1-\left({\lambda\over 4!m_{0}^{2}}\right)^{2}\left({9\over
2\pi^{2}}-{3\over 8}\right)+\mathcal{O}(\lambda^{2}),$ (C.15)

after which the tree-level contribution to $\langle
m^{2},p_{1};m^{2},p_{2}|\Theta(0)\rangle$ becomes

$\langle
m^{2},p_{1};m^{2},p_{2}|\Theta(0)\rangle\supset-2m^{2}Z^{-1}=-2m^{2}\left(1-\left({\lambda\over
4!m^{2}}\right)^{2}\left({9\over 2\pi^{2}}-{3\over
8}\right)+\mathcal{O}(\lambda^{2})\right).$ (C.16)

This wavefunction renormalization contribution has the effect of canceling out
the $s=0$ contribution from the other two-loop diagrams, so that the Ward
identity $\mathcal{F}_{2,0}^{\Theta}(0)=-2m^{2}$ is preserved.

####  C.1.2 2d $O(N)$ model in the large $N$ limit

The S-matrix, form factor $\mathcal{F}_{2,0}^{\Theta}$, and spectral density
$\rho_{\Theta}$ in the $O(N)$ theory at large $N$ simply involve diagrams we
have just computed, together with a standard resummation of higher loop
diagrams that factorize and form a geometric series.

The $\Theta$ form factor is, in units with $m=1$,

$\mathcal{F}_{2,0}^{\Theta}(s)=-2\left(1-{\lambda\Delta(s)\over
8\pi+\lambda(1+\Delta(s))}\right),$ (C.17)

where $\Delta(s)$ is the function given in (3.15). The S-matrix is simplest in
the rapidity variable $\theta$:

$S=-{(-i\theta+\pi)\lambda\operatorname{csch}(\theta)+8i\pi\over(i\theta+\pi)\lambda\operatorname{csch}(\theta)-8i\pi}.$
(C.18)

Finally, the time-ordered two-point function $\mathbf{\Delta}_{\Theta}(s)$ is

$\mathbf{\Delta}_{\Theta}(s)=4i{{s\over 6}-\Delta(s)+{\lambda\over
8\pi}\left({s\over 6}+\Delta(s)\left({s\over 6}-1\right)\right)\over
8\pi+\lambda(\Delta(s)+1)},$ (C.19)

and the spectral density $\rho_{\Theta}(s)$ can be obtained either by taking
the real part of $\mathbf{\Delta}_{\Theta}(s)$ or by using the form factor
together with the fact that the large $N$ limit theory saturates the
inequality (2.30).

###  C.2 Dispersion Relations

As a check of our previous two-loop formulas for the $\phi^{4}$ model, we will
see how to rederive the one- and two-loop contributions using unitarity,
Watson’s equation, and dispersion relations.

We begin with the S-matrix,

$\widehat{S}=1+i\omega^{2}\mathcal{T}$ (C.20)

By definition, up to $\mathcal{O}(\lambda)$, it is

$\mathcal{T}=-\lambda+\mathcal{O}(\lambda^{2})\qquad\omega^{2}={1\over
2\sqrt{s(s-4m^{2})}}$ (C.21)

Let us also divide $\mathcal{T}$ into real and imaginary parts as follows

$\mathcal{T}=\mathcal{T}_{R}+i\mathcal{T}_{I}$ (C.22)

From on-shell unitarity $SS^{*}=1$, we infer that at $s>4$,

$\mathcal{T}_{I}=\lambda^{2}{\omega^{2}\over 2}+\mathcal{O}(\lambda^{3})$
(C.23)

Then, we can reconstruct $\mathcal{T}$ at $\mathcal{O}(\lambda^{2})$ from its
imaginary part using dispersion relations:

$\mathcal{T}(s)=\mathcal{T}_{\infty}-{1\over\pi}\int_{4m^{2}}^{\infty}d\mu^{2}\mathcal{T}_{I}(\mu^{2})\left({1\over
s-\mu^{2}}-{1\over
s-(4m^{2}-\mu^{2})}\right)=\mathcal{T}_{\infty}+{\lambda^{2}\over
4\sqrt{(4m^{2}-s)s}}.$ (C.24)

It is easy to see that the imaginary part of $\mathcal{T}(s)$ is indeed
$\mathcal{T}_{I}(s)$. The constant “subtraction” piece $\mathcal{T}_{\infty}$
depends on the definition of the theory and cannot be determined by dispersion
relations. If we define the theory to have a bare quartic coupling
${\mathcal{L}}\supset-{\lambda\over 4!}\phi^{4}$ without additional
counterterms, then a one-loop computation shows
$\mathcal{T}_{\infty}=-\lambda+{\lambda^{2}\over 8\pi
m^{2}}+\mathcal{O}(\lambda^{3})$.

Next, we apply Watson’s equation to obtain the form factor for $\Theta$. In
the rest of this appendix, for notational convenience, we will denote
$\mathcal{F}^{\Theta}_{2,0}$ simply as $\mathcal{F}$. At
$\mathcal{O}(\lambda^{0})$, we have $\mathcal{F}=-2m^{2}$. Expanding
$\mathcal{F}$ in powers of $\lambda$,242424Note that the subscripts in
$\mathcal{F}$ in equation (C.25) have different meanings from those in other
parts of this paper.

$\mathcal{F}=-2m^{2}\left(1+{\lambda\over
m^{2}}(\mathcal{F}_{1,R}+i\mathcal{F}_{1,I})+{\lambda^{2}\over
m^{4}}(\mathcal{F}_{2,R}+i\mathcal{F}_{2,I})+\dots\right)$ (C.25)

and imposing

${\mathcal{F}(s)\over\mathcal{F}^{*}(s)}=\widehat{S}(s),$ (C.26)

we immediately find

$\mathcal{F}_{1,I}=-{1\over 2}m^{2}\omega^{2}=-{m^{2}\over
4\sqrt{s(s-4m^{2})}}$ (C.27)

Applying dispersion relations, we have

$\mathcal{F}_{1}(s)=\mathcal{F}_{1,\infty}-{1\over\pi}\int_{4}^{\infty}d\mu^{2}{\mathcal{F}_{1,I}(\mu^{2})\over
s-\mu^{2}}=\mathcal{F}_{1,\infty}-{m^{2}\sec^{-1}\left({2m\over\sqrt{4m^{2}-s}}\right)\over
2\pi\sqrt{(4m^{2}-s)s}}$ (C.28)

where $\mathcal{F}_{1,\infty}$ is another constant subtraction. We can fix its
value by demanding that $\mathcal{F}(0)=-2m^{2}$, which implies

$\mathcal{F}_{1,\infty}={1\over 8\pi}$ (C.29)

Putting this together, we obtain

$\mathcal{F}_{1}(s)=-{1\over 8\pi}\Delta(s).$ (C.30)

At the next order, using our expression for $\mathcal{T}$ up to
$\mathcal{O}(\lambda^{2})$, we find from Watson’s equation that

$\mathcal{F}_{2,I}(s)=-{m^{2}\text{csch}^{-1}\left({2m\over\sqrt{s-4m^{2}}}\right)\over
8\pi(s-4m^{2})s}-{\pi\over 32}m^{2}\delta(s-4m^{2}).$ (C.31)

There is a subtle $\delta(s-4m^{2})$ contribution here that arises from taking
the difference between ${1\over s-4m^{2}}$ and $\left({1\over
s^{-}4m^{2}}\right)^{*}$, which differ by a $\delta$ function at $s=4m^{2}$
due to the change in $i\epsilon$ prescription under complex conjugation. The
clearest way to see this difficult term is by studying the non-relativistic
limit $s\sim 4m^{2}$ directly, as we will do in the next subsection.

Finally, we can reconstruct the full form factor at this order:

$\mathcal{F}_{2}(s)=\mathcal{F}_{2,\infty}-{1\over\pi}\int_{4m^{2}}^{\infty}d\mu^{2}{\mathcal{F}_{2,I}(\mu^{2})\over
s-\mu^{2}}=-{1\over(8\pi)^{2}}\left({\pi^{2}s\over
8(s-4m^{2})}-\Delta(s)(\Delta(s)/2+1)\right),$ (C.32)

which agrees with the result (3.14). We again fixed the subtraction term
$\mathcal{F}_{2,\infty}$ by demanding that $\mathcal{F}(0)=-2m^{2}$.

###  C.3 Nonrelativistic Limit

Scattering of two particles near the threshold $s=4m^{2}$ is simply a one-
dimensional non-relativistic quantum mechanics problem that can be solved, as
we review briefly. The interaction $\lambda\phi^{4}$ is a $\delta$ function
potential in position space. We take the scattering wavefunction to be

$\psi(x)=\left\\{\begin{array}[]{cc}e^{ikx}+Se^{-ikx},&x<0\\\
Se^{ikx}+e^{-ikx},&x>0\end{array}\right.,$ (C.33)

which is even as a function of $x$ since the two particles are identical. The
Schrodinger equation is

$-{\psi^{\prime\prime}(x)\over 2(m/2)}+{\lambda\over
8}\delta(x)\psi(x)=E\psi(x)$ (C.34)

where $E={s-4m^{2}\over 4m}={k^{2}\over m}$.

Integrating the Schrodinger equation around $x=0$, we find

$S={16k-im\lambda\over
16k+im\lambda}\approx-{\lambda+8i\theta\over\lambda-8i\theta}$ (C.35)

where we have written the result in terms of rapidity $\theta$ and taken the
limit $\theta\rightarrow 0$ with $\lambda/\theta$ fixed. This is regular at
small $\theta$, but if we first take a small $\lambda$ limit then each power
in $\lambda$ is individually singular:

$S(\theta)\stackrel{{\scriptstyle\theta\rightarrow
0}}{{\approx}}1-{i\lambda\over 4\theta}-{\lambda^{2}\over
32\theta^{2}}+{i\lambda^{3}\over 256\theta^{3}}+\dots$ (C.36)

where we have kept only the most singular terms at each order. So the small
$\lambda$ limit and the small $\theta$ limit do not commute, and in fact
perturbative loop computations, which are an expansion in powers of $\lambda$,
should not be trusted below about $\theta\lesssim{\lambda\over 4}$.

From the scattering wavefunction, we can also extract the form factor in this
limit. Each individual particle has energy $m+E/2$, so the time-dependent
wavefunction is

$\psi_{2}(x_{1},x_{2},t_{1},t_{2})=e^{i(m+{E\over
2})(t_{1}+t_{2})}\psi({x_{1}-x_{2}\over 2})$ (C.37)

where $\psi(x)$ is from eq. (C.33). We will take $m=1$. In second
quantization, $\psi_{2}$ is

$\psi_{2}(x_{1},x_{2},t_{1},t_{2})=\langle\phi(x_{1},t_{1})\phi(x_{2},t_{2})|p_{1},p_{2}\rangle$
(C.38)

where $|p_{1},p_{2}\rangle$ is the two-particle state, with momentum
$p_{1}=-p_{2}=k/2$ since we are in the rest frame. Then, we can easily
calculate the overlap with $T_{--}$ by taking

$\langle T_{--}(0)|p_{1},p_{2}\rangle=\lim_{x_{i}\rightarrow
0,t_{i}\rightarrow
0}(\partial_{t_{1}}-\partial_{x_{1}})(\partial_{t_{2}}-\partial_{x_{2}})\psi_{2}(x_{1},x_{2},t_{1},t_{2}).$
(C.39)

In the nonrelativistic limit, both $E=k^{2}$ and $k$ go to zero, and we obtain

$\langle T_{--}(0)|p_{1},p_{2}\rangle=-(1+S).$ (C.40)

with $S$ given in equation (C.35). This result has the correct phase according
to Watson’s equation, since one can easily check that ${1+S\over 1+S^{*}}=S$.
Expanding in small $\lambda$ we have

$-{1\over 2}\langle T_{--}(0)|p_{1},p_{2}\rangle=1-{i\lambda\over
8\theta}-{\lambda^{2}\over 64\theta^{2}}+\dots,$ (C.41)

which agrees with the perturbative result (3.14) if we expand (3.14) in small
$\theta$. From this expression, we see that $\mathcal{F}_{2,0}^{\Theta}$ at
$\mathcal{O}(\lambda^{2})$ has a pole $\sim{\lambda^{2}\over 32(s-4)}$, which
implies that the imaginary part of $\mathcal{F}_{2,0}^{\Theta}$ at
$\mathcal{O}(\lambda^{2})$ contains a $\delta$ function of the form

$\textrm{Im}(\mathcal{F}_{2,0}^{\Theta})\supset-\lambda^{2}{\pi\over 32(s-4)}$
(C.42)

as claimed in eq (C.31).

##  Appendix D Sinh-Gordon Form Factors and $C$-function

In this appendix, we provides some details about the four-particle form factor
of the trace of the stress-tensor $\Theta$ and the computation of the spectral
density in section 5 in the sinh-Gordon/staircase model. The result of the
$2n$-particle form factor for $\Theta$ is given in Fring:1992pt  (the form
factors with an odd number of particles vanish for $\Theta$). In terms of the
minimal form factor

$\mathcal{F}_{\text{min}}\left(\theta\right)=\mathcal{N}\exp\left(8\int_{0}^{\infty}{dx\over
x}{\sinh\left({x\gamma\over 2\pi}\right)\sinh\left({x(\pi-\gamma)\over
2\pi}\right)\sinh\left({x\over
2}\right)\over\sinh^{2}(x)}\sin^{2}\left({x(i\pi-\theta)\over
2\pi}\right)\right),$ (D.1)

with the normalization constant

$\mathcal{N}=\exp\left[-4\int_{0}^{\infty}{dx\over x}{\sinh\left({x\gamma\over
2\pi}\right)\sinh\left({x\over
2}\left(1-{\gamma\over\pi}\right)\right)\sinh{x\over
2}\over\sinh^{2}x}\right],$ (D.2)

the 2-particle form factor in equation (3.8) in our convention is given by

$\mathcal{F}_{2,0}^{\Theta}\left(\theta\right)=-2m^{2}{\mathcal{F}_{\text{min}}\left(\theta\right)\over\mathcal{N}}.$
(D.3)

And the expression for the four-particle form factor is

$\mathcal{F}_{4,0}^{\Theta}(\theta_{1},\theta_{2},\theta_{3},\theta_{4})={8\pi
m^{2}\sin\gamma\over\mathcal{N}^{2}}\sigma_{1}^{(4)}\sigma_{2}^{\left(4\right)}\sigma_{3}^{(4)}\prod_{0<i<j\leq
4}{\mathcal{F}_{\text{min}}\left(\theta_{ij}\right)\over x_{i}+x_{j}},$ (D.4)

where $\theta_{ij}=\theta_{i}-\theta_{j}$, $x_{i}=e^{\theta_{i}}$, and
$\sigma_{k}^{\left(4\right)}$s are degree $k$ symmetric polynomials of
$x_{i}$. Specifically, we have

$\displaystyle\sigma_{1}^{\left(4\right)}$
$\displaystyle=x_{1}+x_{2}+x_{3}+x_{4},$
$\displaystyle\sigma_{2}^{\left(4\right)}$
$\displaystyle=x_{1}x_{2}+x_{1}x_{3}+x_{1}x_{4}+x_{2}x_{3}+x_{2}x_{4}+x_{3}x_{4},$
(D.5) $\displaystyle\sigma_{3}^{\left(4\right)}$
$\displaystyle=x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+x_{1}x_{3}x_{4}+x_{2}x_{3}x_{4}.$

As mentioned in Fring:1992pt , to numerically evaluate the form factors, it is
easier to use the following expression for $\mathcal{F}_{\text{min}}$:

$\displaystyle\mathcal{F}_{\text{min}}\left(\theta\right)$
$\displaystyle=\mathcal{N}I_{N}(\theta)$
$\displaystyle\times\prod_{k=0}^{N-1}\left[{\left(1+\left({\widehat{\theta}/2\pi\over
k+{1\over 2}}\right)^{2}\right)\left(1+\left({\widehat{\theta}/2\pi\over
k+{3\over 2}-{\gamma\over
2\pi}}\right)^{2}\right)\left(1+\left({\widehat{\theta}/2\pi\over
k+1+{\gamma\over
2\pi}}\right)^{2}\right)\over\left(1+\left({\widehat{\theta}/2\pi\over
k+{3\over 2}}\right)^{2}\right)\left(1+\left({\widehat{\theta}/2\pi\over
k+{1\over 2}+{\gamma\over
2\pi}}\right)^{2}\right)\left(1+\left({\widehat{\theta}/2\pi\over
k+1-{\gamma\over 2\pi}}\right)^{2}\right)}\right]^{k+1}$ (D.6)

where the integral $I_{N}(\theta)$ is defined as

$I_{N}(\theta)\equiv\exp\Bigg{[}8\int_{0}^{\infty}{dx\over
x}{\sinh\left({x\gamma\over 2\pi}\right)\sinh\left({x\over
2}\left(1-{\gamma\over\pi}\right)\right)\sinh{x\over
2}\over\sinh^{2}x}\times\\\
\left(N+1-Ne^{-2x}\right)e^{-2Nx}\sin^{2}\left({x\widehat{\theta}\over
2\pi}\right)\Bigg{]}.$ (D.7)

Here $\widehat{\theta}\equiv i\pi-\theta$. The integral in the exponent of
$I_{N}(\theta)$ is approaching 0 as one increases $N$. And for large $N$, the
the contribution from the integral is actually negligible. For example, for
$N=1000$, in the cases we considered in this paper, the integral is order
$\mathcal{O}\left(10^{-8}\right)$. In the actual computation for getting the
spectral density, we simply take large enough $N$ and discard the integral
part in (D.6). We then use a rational function to fit the result for
$\mathcal{F}_{\text{min}}$ for each values of $\gamma$ (which only introduces
an uncertainty of order $\mathcal{O}(10^{-8}$)), in order for Mathematica to
be able to evaluate the 4-particle form factor contribution to the spectral
density quickly later on.

The spectral density is given exactly by

$\rho_{\Theta}(s)=\rho_{\Theta,2}(s)\theta(s-4m^{2})+\rho_{\Theta,4}(s)\theta(s-16m^{2})+\rho_{\Theta,6}(s)\theta(s-36m^{2})+\ldots,$
(D.8)

where

$\rho_{\Theta,2n}\left(s\right)={1\over
2\pi}{1\over(2n)!}\int{d\theta_{1}\ldots
d\theta_{2n}\over(2\pi)^{2n}}\left|F_{2n,0}^{\Theta}\left(\theta_{1},\ldots,\theta_{2n}\right)\right|^{2}\\\
\delta\left(\sum_{i=1}^{2n}m\sinh\theta_{i}\right)\delta\left(\sum_{i=1}^{2n}m\cosh\theta_{i}-\sqrt{s}\right),$
(D.9)

In what follows we will consider only two- and four-particle contributions to
the spectral density only. This means that $\rho_{\Theta}(s)$ remains exact up
to $s=36m^{2}$ and then will start deviating from the exact answer due to six-
and higher particle states.

The comparison of the sinh-Gordon spectral density (with two- and four-
particle states) and the $\phi^{4}$ spectral density is given in figure 16.
They happen to be very similar in a wide range of values of $s$. Using (D.8)
one can also compute the $C$-function. We show the contributions to the change
of the central charge $\Delta C=12\pi\int_{0}^{\infty}ds{\rho_{\Theta}(s)\over
s^{2}}$ from the two-particle and four-particle form factors in figure 21. As
expected we reproduce the value of the free boson $\Delta C=1$ very well (at
least for $\Lambda\leq 4$). For small values of $\Lambda$, the free boson
central charge is mostly given by the two-particle part of the spectral
density. When $\Lambda$ increases, four- and then higher-particle
contributions become important.

Figure 21: Contributions to $\Delta C$ from the two-particle and four-particle
form factors in the sinh-Gordon/staircase model for various values of the non-
perturbative quartic coupling $\Lambda$. The red dashed line is $\Delta C=1$
for comparison.

## References

  * (1) R. Rattazzi, V. S. Rychkov, E. Tonni and A. Vichi, _Bounding scalar operator dimensions in 4D CFT_ , _JHEP_ 12 (2008) 031, [0807.0004].
  * (2) D. Poland, S. Rychkov and A. Vichi, _The Conformal Bootstrap: Theory, Numerical Techniques, and Applications_ , _Rev. Mod. Phys._ 91 (2019) 015002, [1805.04405].
  * (3) M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees and P. Vieira, _The S-matrix bootstrap II: two dimensional amplitudes_ , _JHEP_ 11 (2017) 143, [1607.06110].
  * (4) M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees and P. Vieira, _The S-matrix bootstrap. Part III: higher dimensional amplitudes_ , _JHEP_ 12 (2019) 040, [1708.06765].
  * (5) M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees and P. Vieira, _The S-matrix bootstrap. Part I: QFT in AdS_ , _JHEP_ 11 (2017) 133, [1607.06109].
  * (6) N. Doroud and J. Elias Miró, _S-matrix bootstrap for resonances_ , _JHEP_ 09 (2018) 052, [1804.04376].
  * (7) Y. He, A. Irrgang and M. Kruczenski, _A note on the S-matrix bootstrap for the 2d O(N) bosonic model_ , _JHEP_ 11 (2018) 093, [1805.02812].