aainstitutetext:  Départment de Physique Théorique, Université de Genève,
24 quai Ernest-Ansermet, 1211 Genève 4, Switzerlandbbinstitutetext: Fields and
Strings Laboratory, Institute of Physics
École Polytechnique Fédéral de Lausanne (EPFL)
Route de la Sorge, CH-1015 Lausanne, Switzerland

# Bootstrapping the $a$-anomaly in $4d$ QFTs

Denis Karateev b Jan Marucha b João Penedones b and Biswajit Sahoo

###### Abstract

We study gapped 4d quantum field theories (QFTs) obtained from a relevant
deformation of a UV conformal field theory (CFT). For simplicity, we assume
the existence of a $\mathbb{Z}_{2}$ symmetry and a single $\mathbb{Z}_{2}$-odd
stable particle and no $\mathbb{Z}_{2}$-even particles at low energies. Using
unitarity, crossing and the assumption of maximal analyticity we compute
numerically a lower bound on the value of the $a$-anomaly of the UV CFT as a
function of various non-perturbative parameters describing the two-to-two
scattering amplitude of the particle.

††dedication:  We condemn the current war in Ukraine and dedicate this paper
to its people.
We also express gratitude to the independent journalists who are reporting the
truth in Russia
and to all the people who are trying to stop this war.

##  1 Introduction and summary

The dream of a _bootstrapper_ is to map out the space of consistent Quantum
Field Theories (QFTs). Given the vastness of this space, in practice, one can
only plot its projection onto a few cleverly chosen coordinates in theory
space. Ideally, these coordinates correspond to physical observables that
characterize the QFT across all length scales, from the UV Conformal Field
Theory (CFT) to the IR, which we assume to be gapped. On the one hand, the
S-matrix bootstrap focuses on IR observables like ratios of masses of stable
particles and coupling constants defined in terms of scattering amplitudes,
for recent progress see Paulos:2016fap ; Paulos:2016but ; Paulos:2017fhb ;
Doroud:2018szp ; He:2018uxa ; Cordova:2018uop ; Guerrieri:2018uew ;
Paulos:2018fym ; Homrich:2019cbt ; EliasMiro:2019kyf ; Cordova:2019lot ;
Bercini:2019vme ; Gabai:2019ryw ; Bose:2020shm ; Bose:2020cod ;
Correia:2020xtr ; Kruczenski:2020ujw ; Guerrieri:2020bto ; Hebbar:2020ukp ;
Karateev:2019ymz ; Karateev:2020axc ; Guerrieri:2020kcs ; Tourkine:2021fqh ;
Guerrieri:2021ivu ; He:2021eqn ; EliasMiro:2021nul ; Guerrieri:2021tak ;
Chen:2021pgx ; Cordova:2022pbl ; Albert:2022oes ; Sinha:2020win ;
Chowdhury:2021ynh ; Chen:2022nym ; Miro:2022cbk .111For an overview of recent
results and discussion of some future directions, see Kruczenski:2022lot . On
the other hand, the conformal bootstrap focuses on UV data like scaling
dimensions and Operator Product Expansion (OPE) coefficients of the UV CFT,
for a review see Poland:2018epd .222For an overview of the most recent
progress and discussion of some further directions see also Poland:2022qrs .
In this work, we extend the S-matrix bootstrap method in four spacetime
dimensions to gain access to the $a$-anomaly of the UV CFT, which is a precise
measure of its degrees of freedom. This is similar to the spirit of
Karateev:2019ymz  in two spacetime dimensions, that incorporated the central
charge of the UV CFT into the S-matrix bootstrap framework.

Our strategy is to follow Komargodski and Schwimmer Komargodski:2011vj  and
probe the QFT with an external massless scalar field $\varphi(x)$ which
creates a massless particle $B$ from the vacuum, usually called the
_dilaton_.333The dilaton $B$ in this paper should not be confused with the
Nambu–Goldstone boson of spontaneously broken conformal symmetry which is also
called the dilaton. As explained in section 2.2 the better name for the former
dilaton would be the compensator particle. However in this paper we keep the
commonly used terminology. As we review in section 2, this construction does
not affect the scattering amplitudes of the original QFT but it generates non-
trivial scattering between dilaton particles. In particular, the $a$-anomaly
$a^{\text{UV}}$ of the UV CFT can be read off from the low energy behavior

$\widetilde{\mathcal{T}}_{BB\rightarrow
BB}=a^{\text{UV}}\,(s^{2}+t^{2}+u^{2})+\dots\,,$ (1)

where $s,t,u$ are the standard Mandelstam invariants for two to two
scattering.

For simplicity, we study QFTs with a $\mathbb{Z}_{2}$ symmetry and a single
stable scalar particle of mass $m$ and $\mathbb{Z}_{2}$ odd, which we shall
denote by $A$. The combined system QFT + _dilaton_ has two asymptotic states:
the original scalar particle $A$ and the new massless particle describing the
dilaton $B$. We study the complete set of 2 to 2 scattering amplitudes in this
system, see figure 1. Then we write all crossing equations and unitarity
conditions such a system must satisfy. All the details of this setup are given
in section 3. In addition, we impose the universal soft behavior of
$\widetilde{\mathcal{T}}_{AB\rightarrow AB}$ derived in section 4. In order to
obtain concrete results we employ the numerical approach of Paulos:2017fhb ;
Homrich:2019cbt . Let us now briefly summarize our main findings.

Figure 1: The complete system of scattering amplitudes of the $\mathbb{Z}_{2}$
odd particle $A$ with mass $m$ and the massless dilaton $B$.

We define non-perturbative couplings in terms of the physical scattering
amplitude. In this work we focus on

$\mathcal{T}_{AA\rightarrow
AA}(s_{0},\,t_{0},\,u_{0}),\qquad\partial_{s}^{2}\mathcal{T}_{AA\rightarrow
AA}(s_{0},\,t_{0},\,u_{0}),$ (2)

where $(s_{0},t_{0},u_{0})$ is a point inside the Mandelstam triangle defined
by $0\leq s_{0},t_{0},u_{0}\leq 4m^{2}$. We will consider two choices. The
first choice is the crossing symmetric point $s_{0}=t_{0}=u_{0}=4m^{2}/3$
which leads to the definition of the parameters $\lambda_{0}$ and
$\lambda_{2}$, namely

$\displaystyle\lambda_{0}$
$\displaystyle\equiv\frac{1}{32\pi}\mathcal{T}_{AA\rightarrow
AA}(4m^{2}/3,\,4m^{2}/3,\,4m^{2}/3),$ (3) $\displaystyle\lambda_{2}$
$\displaystyle\equiv\frac{1}{32\pi}m^{4}\partial_{s}^{2}\mathcal{T}_{AA\rightarrow
AA}(4m^{2}/3,\,4m^{2}/3,\,4m^{2}/3).$

The second choice is the “forward” point $s_{0}=u_{0}=2m^{2}$ and $t=0$ which
leads to the definition of the parameters $\Lambda_{0}$ and $\Lambda_{2}$,
namely

$\Lambda_{0}\equiv\frac{1}{32\pi}\mathcal{T}_{AA\rightarrow
AA}(2m^{2},\,0,\,2m^{2}),\qquad\Lambda_{2}\equiv\frac{1}{32\pi}m^{4}\partial_{s}^{2}\mathcal{T}_{AA\rightarrow
AA}(2m^{2},\,0,\,2m^{2}).$ (4)

Crossing, unitarity and analyticity put strong bounds on the above parameters.
For instance we found that444These numerical bounds are a rough estimate based
on our numerical results described in section 5. The exception is the upper
bound on $\lambda_{0}$ which can be determined quite precisely Paulos:2017fhb
. For the recent more detailed study of these observables see Chen:2022nym .

$\displaystyle-6\lesssim$ $\displaystyle\,\lambda_{0}\lesssim+2.6613,\qquad$
$\displaystyle 0\leq\lambda_{2}\lesssim+2,$ (5) $\displaystyle-3\lesssim$
$\displaystyle\,\Lambda_{0}\lesssim+3,\qquad$ $\displaystyle
0\leq\Lambda_{2}\lesssim+0.7.$ (6)

The minimum of the $a$-anomaly as a function of the above parameters is given
in figures 2 and 3. All consistent QFTs must live in the allowed region which
we shaded in blue. We mark the absolute minimum of the a-anomaly with a red
dot in these figures. Our best numerical estimate is

$a/a_{\text{free}}\gtrsim 0.3\,,$ (7)

with $a_{\text{free}}$ the $a$-anomaly of a free scalar field. We refer the
reader to section 5 for a detailed discussion of the numerical uncertainties
of these results. We do not know if there is any 4d QFT that saturates the
lower bound (7). In fact, we do not know of any theory with an $a$-anomaly
smaller than $a_{\text{free}}$. We conclude in section 6 with a discussion of
open questions and future work.

Figure 2: Minimum of the $a$-anomaly of the UV CFT as a function of the
parameters $\lambda_{0}$ and $\lambda_{2}$ defined in (3). The red dot marks
the absolute minimum. The red vertical lines indicate the boundaries of the
allowed regions for $\lambda_{0}$ and $\lambda_{2}$.

Figure 3: Minimum of the $a$-anomaly of the UV CFT as a function of the
parameters $\Lambda_{0}$ and $\Lambda_{2}$ defined in (4). The red dot marks
the absolute minimum. The red vertical lines indicate the boundaries of the
allowed regions for $\Lambda_{0}$ and $\Lambda_{2}$.

##  2 Review of classic results

In this paper we work in $4d$ Minkowski flat space with metric:

$\eta_{\mu\nu}=\eta^{\mu\nu}=\text{diagonal}\\{-1,+1,+1,+1\\}.$ (8)

All the quantum field theories have a very special operator called the stress-
tensor $T^{\mu\nu}(x)$. It is symmetric in its two Lorentz indices and obeys
conservation law, namely

$T^{\mu\nu}(x)=T^{\nu\mu}(x),\qquad\partial_{\mu}T^{\mu\nu}(x)=0.$ (9)

###  2.1 Stress tensor and trace anomaly in CFTs

Let us start the discussion by considering conformally invariant quantum field
theory. The conformal symmetry puts severe constraints on the form of
correlation functions. In Osborn:1993cr  it was shown that the most general
two- and three-point functions of the stress tensor in CFTs have the following
form

$\displaystyle\langle 0|T^{\mu\nu}(x_{1})T^{\rho\sigma}(x_{2})|0\rangle$
$\displaystyle=\frac{C_{T}}{x_{12}^{8}}\times\mathbf{T}^{\mu\nu;\rho\sigma}_{0},$
(10) $\displaystyle\langle
0|T^{\mu\nu}(x_{1})T^{\rho\sigma}(x_{2})T^{\alpha\beta}(x_{3})|0\rangle$
$\displaystyle=\frac{1}{x_{12}^{4}x_{23}^{4}x_{31}^{4}}\Big{(}\mathbb{A}\mathbf{T}^{\mu\nu;\rho\sigma;\alpha\beta}_{1}+\mathbb{B}\mathbf{T}^{\mu\nu;\rho\sigma;\alpha\beta}_{2}+\mathbb{C}\mathbf{T}^{\mu\nu;\rho\sigma;\alpha\beta}_{3}\Big{)}.$
(11)

Here the objects $\mathbf{T}_{0}$, $\mathbf{T}_{1}$, $\mathbf{T}_{2}$ and
$\mathbf{T}_{3}$ take care of the correct behaviour of the correlation
functions under conformal transformations. They are called tensor structures.
The standard basis for these tensor structures is defined in appendix A. The
coefficient $C_{T}\geq 0$ is usually referred to as the central charge. The
coefficients $\mathbb{A}$, $\mathbb{B}$ and $\mathbb{C}$ are called the OPE
coefficients since they appear in the OPE expansion of the stress-tensor with
itself. All of them are real quantities. Due to the conformal Ward
identities555All the generators of the conformal transformation can be written
as certain integral of the stress-tensor, see for example Osborn:1993cr . By
performing appropriate integrals over one stress-tensor in (11) and using the
properties of the generators we effectively obtain the two-point function
(10). the following relation holds

$C_{T}\ =\
\frac{\pi^{2}}{3}\Big{(}14\mathbb{A}-2\mathbb{B}-5\mathbb{C}\Big{)}.$ (12)

Summarizing, there are three independent parameters describing the 2- and the
3-point function of the stress-tensor. One can choose these three parameters
to be for example $\\{C_{T},\;\mathbb{A},\;\mathbb{B}\\}$. It is standard to
also define the following quantities

$a\equiv\frac{\pi^{4}}{64\times
90}\Big{(}9\mathbb{A}-2\mathbb{B}-10\mathbb{C}\Big{)},\qquad
c\equiv\frac{\pi^{4}}{64\times
30}\Big{(}14\mathbb{A}-2\mathbb{B}-5\mathbb{C}\Big{)}=\frac{\pi^{2}}{64\times
10}C_{T}.$ (13)

Finally we recall that in CFTs the trace of the stress-tensor vanishes, namely

$T^{\mu}{}_{\mu}(x)=0.$ (14)

Let us now discuss CFTs on the curved background which is described by the
metric $g_{\mu\nu}(x)$. Conformal invariance on a curved background is
achieved by requiring $\text{diff}\times\text{Weyl}$ invariance. We recall
that the Weyl transformation is defined as

$g_{\mu\nu}(x)\rightarrow
e^{2\sigma(x)}g_{\mu\nu}(x),\qquad\mathcal{O}(x)\rightarrow
e^{-\Delta_{\mathcal{O}}\sigma(x)}\mathcal{O}(x),$ (15)

where $\sigma(x)$ is an arbitrary scalar function, $\mathcal{O}(x)$ is a local
scalar operator and $\Delta_{\mathcal{O}}$ is the scaling dimension of the
operator $\mathcal{O}(x)$. Contrary to the flat space-time where (14) holds,
for CFTs on the curved background we instead have

$\langle 0|T^{\mu}{}_{\mu}(x)|0\rangle_{g}=\ -a\times E_{4}+c\times W^{2},$
(16)

where $E_{4}$ is the Euler density defined in (177) and $W^{2}$ is the square
of Weyl tensor defined in (178). The subscript $g$ in the left-hand side of
(16) indicates that the CFT is on the curved background rather than on the
flat one. The coefficients $a$ and $c$ are exactly the ones introduced in
(13). They are called the Weyl anomalies as well as trace anomalies. The name
Weyl anomaly is appropriate because exact Weyl invariance implies $\langle
0|T^{\mu}{}_{\mu}(x)|0\rangle_{g}=0$.

###  2.2 Compensator field and the dilaton particle

Let us define a generic quantum field theory as the renormalization group flow
from the UV to the IR fixed points which are described by the UV and the IR
conformal field theories. Such a theory in curved background can be described
by the action

$A(g,M_{i})\equiv A_{\text{UV\;CFT}}(g)+A_{\text{deformation}}(g,M_{i}),$ (17)

where the deformation of the UV CFT has the form

$A_{\text{deformation}}(g,M_{i})=\sum_{i}\int
d^{4}x\sqrt{-g}\left(\lambda_{i}M_{i}^{4-\Delta_{i}}\mathcal{O}_{i}(x)\right).$
(18)

Here $\mathcal{O}_{i}(x)$ are relevant scalar UV CFT operators (operators
obeying $\Delta_{i}<4$), $\lambda_{i}$ are dimensionless coefficients and
$M_{i}$ are the mass scales which control when the deformation due to a
particular operator becomes important. The explicit dependence on
$g_{\mu\nu}(x)$ indicates that we work on a generic curved background. The QFT
in flat space-time is recovered by setting $g_{\mu\nu}(x)$ to the flat metric
(8). The determinant of the metric is defined as follows

$g\equiv\text{det}\ g_{\mu\nu}(x).$ (19)

The action (18) is diff invariant by construction.

In curved background the stress-tensor is defined as

$T^{\mu\nu}(x)=\frac{2}{\sqrt{-g}}\,\frac{\delta A(g,M_{i})}{\delta
g_{\mu\nu}(x)}.$ (20)

Under the Weyl transformation (15), the trace of the stress-tensor can be
defined as a variation of the action with respect to the infinitesimal Weyl
transformation parameter $\sigma$ in the following way,

$T^{\mu}_{\mu}(x)\equiv\frac{1}{\sqrt{-g}}\,\frac{\delta_{W}A(g,M_{i})}{\delta\sigma(x)}.$
(21)

Performing the Weyl transformation (15) in (17) and focusing on flat space-
time we obtain the trace of the stress-tensor using the above definition,
which reads

$g_{\mu\nu}=\eta_{\mu\nu}:\qquad
T^{\mu}{}_{\mu}(x)=\sum_{i}\lambda_{i}(4-\Delta_{i})M_{i}^{4-\Delta_{i}}\mathcal{O}_{i}(x).$
(22)

This is the standard result in QFT, namely the trace is proportional to the
deforming operators.

The correlation functions of the stress-tensor both in the UV and IR are
described by (10) and (11) where the coefficients $C_{T}$, $\mathbb{A}$,
$\mathbb{B}$ and $\mathbb{C}$ have an additional label UV and IR respectively.
Out of all the above coefficients the $a$ trace anomaly is the most
interesting. In Cardy:1988cwa  it was conjectured that

$a^{\text{UV}}-a^{\text{IR}}\geq 0,$ (23)

where the equality can hold only if there is no flow and the theory remains
conformal, in other words if $A_{\text{deformation}}=0$ in (17). The
inequality (23) is known as the $a$-theorem. It was shown to hold in
perturbation theory in Osborn:1989td ; Jack:1990eb . It was proven non-
perturbatively in Komargodski:2011vj , for further discussion see also
Komargodski:2011xv ; Luty:2012ww ; Komargodski:2015grt . The proof of
Komargodski:2011vj  gives also the prescription on how to probe/compute the
difference $(a^{\text{UV}}-a^{\text{IR}})$ in a given QFT. One of the main
ingredients of this proof is the compensator field and the associated particle
which we call the dilaton. In the rest of this section we will define the
compensator field and the dilaton particle.

Let us work with the action (17) on a curved background. It is diff invariant
but not Weyl invariant. There are two sources which break the Weyl symmetry,
namely the trace anomaly of the UV and IR CFTs given by (16) and the
deformation part of the action $A_{\text{deformation}}$ in (17) which
explicitly depends on the scale. The latter breaking can be compensated for in
a modified theory with the following action

$A^{\prime}(g,M_{i},\Omega)\equiv
A(g,M_{i}\Omega(x))+A_{\text{dynamics}}(g,\Omega),$ (24)

where $\Omega(x)$ is a real scalar field called the compensator field and the
action $A$ was defined in (17). Both the metric $g_{\mu\nu}(x)$ and the
compensator field $\Omega(x)$ are non-dynamical fields. We can however promote
them to dynamical probe fields by adding a kinetic term
$A_{\text{dynamics}}(g,\Omega)$. We will discuss possible convenient choices
for this term in the end of this subsection. The compensator field $\Omega(x)$
can be represented in the following two ways

$\Omega(x)=e^{-\tau(x)}=1-\frac{\varphi(x)}{\sqrt{2}f}.$ (25)

We refer to the real scalar fields $\tau(x)$ and $\varphi(x)$ also as the
compensator or the dilaton fields interchangeably. Here $f$ is a new parameter
with mass dimension one. The following relation holds

$\tau(x)=\frac{\varphi(x)}{\sqrt{2}f}+O\left(\frac{1}{f^{2}}\right).$ (26)

Let us now emphasize that the action (24) can be made invariant under the Weyl
transformation (15) given that the dilaton transforms in the following way

$\tau(x)\rightarrow\tau(x)+\sigma(x)$ (27)

and that the term $A_{\text{dynamics}}$ is chosen appropriately. The particle
created by the compensator (or the dilaton) field $\varphi(x)$ from the vacuum
is called the dilaton particle. It will be denoted by $B$ throughout this
paper.

The simplest choice for $A_{\text{dynamics}}(g,\Omega)$, already used in
Komargodski:2011vj , reads as

$A_{\text{dynamics}}(g,\Omega)=\frac{1}{6}\,f^{2}\int d^{4}x\
\sqrt{-\widehat{g}}\,R(\widehat{g}),$ (28)

where we have defined

$\widehat{g}_{\mu\nu}\equiv e^{-2\tau}g_{\mu\nu}.$ (29)

The action (28) is Weyl invariant at the classical level but not at the
quantum level. One can simply see this for instance by taking the
$f\rightarrow\infty$ limit and focusing on the flat space. The action (28)
then reduces to the standard kinetic term describing free massless scalar.
Free massless scalar gives a particular example of a CFT with trace anomalies
$a$ and $c$ reported in (182). The latter break Weyl invariance via (16).
There are many other possible choices of $A_{\text{dynamics}}(g,\Omega)$. Let
us stress, that these choices do not have to be Weyl invariant even
classically.

Let us now focus on flat space-time $g_{\mu\nu}=\eta_{\mu\nu}$. Using (18),
(22), (25) and (28) we can rewrite the modified action (24) in the following
equivalent way666If in the UV CFT there exists more than one relevant operator
which can be used to deform the theory (i.e. for $i>1$), the neglected terms
starting from order $O(f^{-2})$ in equation (30) can not be expressed in terms
of the trace of the stress-tensor in general. Indeed, the order $O(f^{-2})$
contribution in $A^{\prime}(M_{i})$ turns out to be
$\frac{1}{4f^{2}}\sum\limits_{i}(4-\Delta_{i})(3-\Delta_{i})\int d^{4}x\
\varphi^{2}(x)\left(\lambda_{i}M_{i}^{4-\Delta_{i}}\mathcal{O}_{i}(x)\right)$,
which is not expressible in terms of the trace of the stress tensor given in
equation (22) for $i>1$.

$A^{\prime}(M_{i},\varphi)\equiv A(M_{i})-\frac{1}{\sqrt{2}f}\int
d^{4}x\;T^{\mu}{}_{\mu}(x)\varphi(x)-\int
d^{4}x\left(\frac{1}{2}\partial_{\mu}\varphi(x)\partial^{\mu}\varphi(x)\right)+O\left(\frac{1}{f^{2}}\right).$
(30)

From (30) it becomes obvious that in the limit $f\rightarrow\infty$ the
interaction between the dilaton and the rest of the system disappears and the
action (24) simply becomes the original one plus the freely propagating
dilaton field. The dilaton field $\varphi(x)$ should be seen as a probe for a
given QFT which does not disturb it in the limit $f\rightarrow\infty$.

###  2.3 $a$-theorem

Let us now take the UV and IR limits of the action (24). They can be written
as777Let us emphasize that we have made here a very non-trivial statement that
the IR dilaton EFT action completely decouples from the IR CFT even though
dilaton self interaction is present. One can argue for this at least in the
limit of flat space-time: by construction (24), dilaton couples only to mass
parameters, IR CFT instead does not have dimensionful parameters.

$\displaystyle A^{\prime}_{\text{UV}}(g,\Omega)$ $\displaystyle\equiv
A_{\text{UV CFT}}(g)+A_{\text{dynamics}}(g,\Omega),$ (31) $\displaystyle
A^{\prime}_{\text{IR}}(g,\Omega)$ $\displaystyle\equiv A_{\text{IR
CFT}}(g)+A_{\text{dilaton EFT}}(g,\Omega)+A_{\text{dynamics}}(g,\Omega).$

Here $A_{\text{dilaton EFT}}$ is the effective field theory (EFT) action
describing the dilaton interaction at low energy. In order to obtain it in
some explicit QFT model one needs to integrate out all the “massive” degrees
of freedom throughout the RG flow which is almost impossible in practice.
Luckily there is a model independent way to compute $A_{\text{dilaton EFT}}$
which we will now review.

Consider the action (24). It breaks Weyl invariance in a very special way. The
Weyl symmetry breaking is coming only from the UV and IR fixed points (31).
Taking into account (16) we get

$\displaystyle\delta_{W}A^{\prime}_{\text{UV}}(g,\Omega)$ $\displaystyle=\int
d^{4}x\sqrt{-g}\sigma(x)\,\Big{(}-a^{\text{UV}}\times
E_{4}+c^{\text{UV}}\times
W^{2}\Big{)}+\delta_{W}A_{\text{dynamics}}(g,\Omega),$ (32)
$\displaystyle\delta_{W}A^{\prime}_{\text{IR}}(g,\Omega)$ $\displaystyle=\int
d^{4}x\sqrt{-g}\sigma(x)\,\Big{(}-a^{\text{IR}}\times
E_{4}+c^{\text{IR}}\times W^{2}\Big{)}$
$\displaystyle+\delta_{W}A_{\text{dilaton
EFT}}(g,\Omega)+\delta_{W}A_{\text{dynamics}}(g,\Omega).$

Here $\delta_{W}$ stands for the infinitesimal Weyl variation.

Let us now assume that the Weyl anomaly in the UV matches the Weyl anomaly in
the IR, in other words

$\delta_{W}A^{\prime}_{\text{UV}}(g,\Omega)=\delta_{W}A^{\prime}_{\text{IR}}(g,\Omega).$
(33)

Notice that contrary to the ’t Hooft anomaly matching, there is no proof for
the Weyl anomaly matching (33) and it might not be true in general.888In fact
the authors of Niarchos:2020nxk , see also Andriolo:2022lcb , found an
apparent mismatch of the $c$-anomaly on the Higgs branch of $\mathcal{N}=2$
super-conformal field theory where conformal symmetry is spontaneously broken.
For further discussion on Weyl anomaly matching and its consequences see
karateev:2023 . Plugging (32) in (33) we obtain the following variational
equation

$\delta_{W}A_{\text{dilaton EFT}}(g,\Omega)=\int
d^{4}x\sqrt{-g}\sigma(x)\,\Big{(}-\left(a^{\text{UV}}-a^{\text{IR}}\right)\times
E_{4}+\left(c^{\text{UV}}-c^{\text{IR}}\right)\times W^{2}\Big{)}.$ (34)

The most general solution for this equation can be written in the following
form

$A_{\text{dilaton EFT}}(g,\Omega)=-(a^{\text{UV}}-a^{\text{IR}})\times
A_{a}(g,\Omega)+(c^{\text{UV}}-c^{\text{IR}})\times A_{c}(g,\Omega)+\
A_{\text{invariant}}(g,\Omega),$ (35)

where the two newly introduced terms $A_{a}(g,\Omega)$ and $A_{c}(g,\Omega)$
behave in the following way under the infinitesimal Weyl transformation

$\delta_{W}A_{a}(g,\Omega)=\int
d^{4}x\sqrt{-g}\sigma(x)\,E_{4},\qquad\delta_{W}A_{c}(g,\Omega)=\int
d^{4}x\sqrt{-g}\sigma(x)\,W^{2}.$ (36)

The term $A_{\text{invariant}}$ instead remains completely invariant. The
solution to the above requirement was found in Schwimmer:2010za ;
Komargodski:2011vj , it reads

$\displaystyle A_{a}(g,\Omega)$ $\displaystyle=\int d^{4}x\sqrt{-g}\left(\tau
E_{4}+4\left(R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R\right)(\partial_{\mu}\tau)(\partial_{\nu}\tau)-4(\partial\tau)^{2}(\partial^{2}\tau)+2(\partial\tau)^{4}\right),$
$\displaystyle A_{c}(g,\Omega)$ $\displaystyle=\int
d^{4}x\sqrt{-g}\tau(x)\,W^{2}.$ (37)

This solution is not easy to obtain but it is easy to check that it satisfies
(36). It is also important to stress that even though (33) might not hold for
every QFT, the weaker Wess-Zumino consistency condition exists, see
karateev:2023 , which implies that at the very least the first line in (37)
always holds true.

The most general Weyl invariant action can be parametrized as follows

$A_{\text{invariant}}(g,\Omega)=\int d^{4}x\
\sqrt{-\widehat{g}}\Bigg{(}M^{4}\lambda+M^{2}r_{0}\widehat{R}\ +\
r_{1}\widehat{R}^{2}+r_{2}\widehat{W}^{2}+r_{3}\widehat{E}_{4}+\ldots\Bigg{)}.$
(38)

Here the Ricci scalar, Weyl tensor and the Euler density are built out of the
metric (29). The real dimensionless parameters $\lambda$, $r_{0}$, $r_{1}$,
$r_{2}$ and $r_{3}$ depend on a particular QFT. The EFT cut-off scale $M$ can
be chosen to be the lowest deformation energy scale of the UV CFT. In
spontaneously broken QFTs $\lambda=0$, but in generic QFTs $\lambda\neq 0$.
However, when defining the action (24), if needed, one can fine tune the
counterterms in such a way that $\lambda=0$.

In flat space the solution (35) together with (37) and (38) simply leads to

$A_{\text{dilaton EFT}}(\varphi)=\frac{M^{4}\lambda}{4f^{4}}\int
d^{4}x\big{(}\varphi(x))^{4}\ +\frac{6M^{2}r_{0}}{f^{2}}\int
d^{4}x\Big{(}-\frac{1}{2}\partial_{\mu}\varphi(x)\partial^{\mu}\varphi(x)\Big{)}\\\
+36r_{1}\int
d^{4}x\Big{(}\frac{1}{2f^{2}}+\frac{\varphi(x)}{\sqrt{2}f^{3}}+\frac{3\varphi(x)^{2}}{4f^{4}}\Big{)}(\partial^{2}\varphi(x))^{2}\\\
+\frac{a^{\text{UV}}-a^{\text{IR}}}{2f^{4}}\times\int d^{4}x\
\big{(}\partial\varphi(x)\big{)}^{4}+\ O\big{(}f^{-5}\partial^{4}\varphi^{5}\
,\ f^{-4}\partial^{6}\varphi^{4}\big{)}.$ (39)

The term proportional to $r_{0}$ gives an $O(f^{-2})$ correction to the
dilaton kinetic term coming from $A_{\text{dynamics}}(g,\Omega)$. In the limit
$f\rightarrow\infty$ it should be neglected. The interacting part of the
dilaton scattering process $B(p_{1})B(p_{2})\rightarrow B(p_{3})B(p_{4})$ at
low energy is described by the effective action (39) and has the following
form

$\lim_{f\rightarrow\infty}f^{4}\mathcal{T}_{BB\rightarrow
BB}(s,t,u)=6M^{4}\lambda+(a^{\text{UV}}-a^{\text{IR}})\times(s^{2}+t^{2}+u^{2})+O(s^{3}).$
(40)

where $s=-(p_{1}+p_{2})^{2}\ ,\ t=-(p_{1}-p_{3})^{2}\ ,\ u=-(p_{1}-p_{4})^{2}$
with $s+t+u=0$. Note, that the term proportional to $r_{1}$ in (39) does not
contribute to this scattering amplitude, since it vanishes under the dilaton
equation of motion. Using the standard approach one can write the following
dispersion relation in the $f\rightarrow\infty$ limit

$\displaystyle\displaystyle a^{\text{UV}}-a^{\text{IR}}\ $ $\displaystyle=$
$\displaystyle\ \frac{f^{4}}{2}\ \frac{1}{2\pi i}\oint_{0}\frac{ds}{s^{3}}\
\mathcal{T}_{BB\rightarrow BB}(s,0,-s)$ (41) $\displaystyle=$ $\displaystyle\
\frac{f^{4}}{\pi}\ \int_{0}^{\infty}\frac{ds}{s^{3}}\
\text{Im}\mathcal{T}_{BB\rightarrow BB}(s,0,-s).$

Since $\text{Im}\mathcal{T}_{BB\rightarrow BB}(s,0,-s)=s\ \sigma(s)\geq 0$
where $\sigma(s)$ is the total cross section for the scattering of
$BB\rightarrow\text{anything}$, $(a^{\text{UV}}-a^{\text{IR}})$ is non-
negative. This proves the $a$-theorem.

##### Application in free scalar theory

As an application of the $a$-theorem consider the UV CFT which is generated by
the free massless field $\Phi(x)$, namely we have the action

$A_{\text{UV\;CFT}}=-\int
d^{4}x\left(\frac{1}{2}\partial_{\mu}\Phi(x)\partial^{\mu}\Phi(x)\right).$
(42)

It is straightforward to compute then the two- and tree-point correlation
functions of the stress-tensor with itself. One obtains (10) and (11) with

$C_{T}^{\text{UV}}=\frac{1}{3\pi^{4}},\qquad\mathbb{A}^{\text{UV}}=\frac{1}{27\pi^{6}}\
,\qquad\mathbb{B}^{\text{UV}}=-\frac{4}{27\pi^{6}}\
,\qquad\mathbb{C}^{\text{UV}}=-\frac{1}{27\pi^{6}}.$ (43)

Let us now add the following deformation

$A_{\text{deformation}}(m)=-\frac{1}{2}m^{2}\Phi(x)^{2},$ (44)

where $m$ becomes the mass of the field $\Phi$. This triggers the flow to an
empty IR fixed point, thus in the deep IR we simply have

$C_{T}^{\text{IR}}=0,\qquad\mathbb{A}^{\text{IR}}=0,\qquad\mathbb{B}^{\text{IR}}=0,\qquad\mathbb{C}^{\text{IR}}=0.$
(45)

As a result according to (13) we get the following UV an IR a-anomaly

$a^{\text{UV}}=\frac{1}{5760\pi^{2}},\qquad
c^{\text{UV}}=\frac{1}{1920\pi^{2}},\qquad a^{\text{IR}}=0,\qquad
c^{\text{IR}}=0.$ (46)

Using the the modified action (30) we can also compute the dilaton scattering
at low energies. We get

$\lim_{f\rightarrow\infty}f^{4}\mathcal{T}_{BB\rightarrow
BB}(s,t,u)=\frac{1}{5760\pi^{2}}\times(s^{2}+t^{2}+u^{2})+O(s^{3}).$ (47)

The details of the computation are given in appendix B. The result (47) is in
a perfect agreement with the $a$-theorem sum-rule (40) and (46).

##  3 Non-perturbative S-matrix bootstrap setup

In this section we explain our framework in details. We define scattering and
partial amplitudes of particles $A$ and $B$ in section 3.1. We write down all
the crossing equations for the scattering amplitudes in section 3.2. We
construct unitarity conditions on the partial amplitudes in section 3.3
assuming that $B$ is a generic massless particle. In section 3.4 we discuss
further restrictions on the scattering amplitudes and modifications of the
unitarity conditions in the case when $B$ is the dilaton in the
$f\rightarrow\infty$ limit.

###  3.1 Scattering and partial amplitudes

Let us consider a QFT which contains two different scalar particles $A$ and
$B$ with the masses

$m_{A}=m,\qquad m_{B}=0.$ (48)

Here and in the next two section we will not impose that $B$ is the dilaton
particle, it will be enough to treat it as a generic massless scalar particle.
The requirement that $B$ is the dilaton will be imposed only in section 3.4.
The particles $A$ and $B$ are described by the following asymptotic states

$|A\rangle_{in}\equiv|m_{A},\vec{p}\,\rangle_{in},\quad|A\rangle_{out}\equiv|m_{A},\vec{p}\,\rangle_{out},\quad|B\rangle_{in}\equiv|m_{B},\vec{p}\,\rangle_{in},\quad|B\rangle_{out}\equiv|m_{B},\vec{p}\,\rangle_{out}.$
(49)

For simplicity we assume the presence of the $\mathbb{Z}_{2}$ symmetry. We
require that the particle of type $A$ is odd and the particle of type $B$ is
even under this symmetry. In other words

$\mathbb{Z}_{2}:\qquad|A\rangle_{in}\rightarrow-|A\rangle_{in},\quad|B\rangle_{in}\rightarrow+|B\rangle_{in}.$
(50)

The same transformation properties hold for the out states.

In this work we will be interested in the following scattering amplitudes

$AA\rightarrow AA,\qquad AA\rightarrow BB,\qquad AB\rightarrow AB,\qquad
BB\rightarrow BB.$ (51)

The presence of the $\mathbb{Z}_{2}$ symmetry required above forbids all the
two to two scattering process with an odd number of particles of type $A$. We
define the scattering amplitudes describing the processes (51) as

$\displaystyle\mathcal{S}_{AA\rightarrow
AA}(s,t,u)\times(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-p_{3}-p_{4})$
$\displaystyle\equiv{}_{out}\langle A_{3},A_{4}|A_{1},A_{2}\rangle_{in},$ (52)
$\displaystyle\mathcal{S}_{AA\rightarrow
BB}(s,t,u)\times(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-p_{3}-p_{4})$
$\displaystyle\equiv{}_{out}\langle B_{3},B_{4}|A_{1},A_{2}\rangle_{in},$
$\displaystyle\mathcal{S}_{AB\rightarrow
AB}(s,t,u)\times(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-p_{3}-p_{4})$
$\displaystyle\equiv{}_{out}\langle A_{3},B_{4}|A_{1},B_{2}\rangle_{in},$
$\displaystyle\mathcal{S}_{BB\rightarrow
BB}(s,t,u)\times(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-p_{3}-p_{4})$
$\displaystyle\equiv{}_{out}\langle B_{3},B_{4}|B_{1},B_{2}\rangle_{in}.$

Here we used the following short-hand notation for describing the two-particle
in and out asymptotic states

$\displaystyle|A_{1},A_{2}\rangle_{in}$
$\displaystyle\equiv\frac{1}{\sqrt{2}}\Big{(}|m_{A},\vec{p}_{1}\,\rangle_{in}\otimes|m_{A},\vec{p}_{2}\,\rangle_{in}+|m_{A},\vec{p}_{2}\,\rangle_{in}\otimes|m_{A},\vec{p}_{1}\,\rangle_{in}\Big{)},$
(53) $\displaystyle|A_{1},B_{2}\rangle_{in}$
$\displaystyle\equiv|m_{A},\vec{p}_{1}\,\rangle_{in}\otimes|m_{B},\vec{p}_{2}\,\rangle_{in},$
$\displaystyle|B_{1},B_{2}\rangle_{in}$
$\displaystyle\equiv\frac{1}{\sqrt{2}}\Big{(}|m_{B},\vec{p}_{1}\,\rangle_{in}\otimes|m_{B},\vec{p}_{2}\,\rangle_{in}+|m_{B},\vec{p}_{2}\,\rangle_{in}\otimes|m_{B},\vec{p}_{1}\,\rangle_{in}\Big{)},$

where $\otimes$ denotes the ordered tensor product. Notice that the two-
particle states which consist of particles of type $A$ or $B$ only are totally
symmetric. The factor $\sqrt{2}$ is introduced to comply with the standard
normalization conditions which will be provided below. The Mandelstam
variables are defined as

$s\equiv-(p_{1}+p_{2})^{2},\quad t\equiv-(p_{1}-p_{3})^{2},\quad
u\equiv-(p_{1}-p_{4})^{2}.$ (54)

As usual these are not linearly independent due to the following relations

$s+t+u=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2},$ (55)

where $m_{i}$ are the masses of particles participating in the two to two
scattering. This constraint is thus different for each process in (52). Let us
write here this constraint precisely for each process separately

$\displaystyle AA\rightarrow AA:$ $\displaystyle\quad s+t+u=4m^{2},$ (56)
$\displaystyle AA\rightarrow BB:$ $\displaystyle\quad s+t+u=2m^{2},$
$\displaystyle AB\rightarrow AB:$ $\displaystyle\quad s+t+u=2m^{2},$
$\displaystyle BB\rightarrow BB:$ $\displaystyle\quad s+t+u=0.$

It is useful to write the four-momentum $p^{\mu}$ in the spherical
coordinates. One has

$p^{\mu}=\\{p^{0},\vec{p}\,\\},\qquad\vec{p}=\\{\boldsymbol{p}\cos\phi\sin\theta,\,\boldsymbol{p}\sin\phi\sin\theta,\,\boldsymbol{p}\cos\theta\\},\qquad\boldsymbol{p}\equiv|\vec{p}\,|.$
(57)

Let us discuss the normalization of states. The same normalizations holds for
both in and out states. As a result below we write explicitly only the
normalization of in states. The one-particle states are normalized in the
following way

$\displaystyle{}_{in}\langle m_{A},\vec{p}_{1}|m_{A},\vec{p}_{2}\rangle_{in}$
$\displaystyle=2\,\sqrt{m^{2}+\boldsymbol{p}_{1}^{2}}\times(2\pi)^{3}\delta^{(3)}(\vec{p}_{1}-\vec{p}_{2}),$
(58) $\displaystyle{}_{in}\langle
m_{A},\vec{p}_{1}|m_{B},\vec{p}_{2}\rangle_{in}$ $\displaystyle=0,$
$\displaystyle{}_{in}\langle m_{B},\vec{p}_{1}|m_{B},\vec{p}_{2}\rangle_{in}$
$\displaystyle=2\boldsymbol{p}_{1}\times(2\pi)^{3}\delta^{(3)}(\vec{p}_{1}-\vec{p}_{2}).$

The normalization of two-particle states (53) follows immediately and read

$\displaystyle{}_{in}\langle A_{3},A_{4}|A_{1},A_{2}\rangle_{in}$
$\displaystyle=4\,(2\pi)^{6}\,\sqrt{m^{2}+\boldsymbol{p}_{1}^{2}}\sqrt{m^{2}+\boldsymbol{p}_{2}^{2}}$
$\displaystyle\times\Big{(}\delta^{(3)}(\vec{p}_{1}-\vec{p}_{3})\delta^{(3)}(\vec{p}_{2}-\vec{p}_{4})+\delta^{(3)}(\vec{p}_{1}-\vec{p}_{4})\delta^{(3)}(\vec{p}_{2}-\vec{p}_{3})\Big{)},$
$\displaystyle{}_{in}\langle B_{3},B_{4}|A_{1},A_{2}\rangle_{in}$
$\displaystyle=0,$ $\displaystyle{}_{in}\langle
A_{3},B_{4}|A_{1},B_{2}\rangle_{in}$
$\displaystyle=4\,(2\pi)^{6}\,\boldsymbol{p}_{2}\,\sqrt{m^{2}+\boldsymbol{p}_{1}^{2}}\times\delta^{(3)}(\vec{p}_{1}-\vec{p}_{3})\delta^{(3)}(\vec{p}_{2}-\vec{p}_{4}),$
(59) $\displaystyle{}_{in}\langle B_{3},B_{4}|B_{1},B_{2}\rangle_{in}$
$\displaystyle=4\,(2\pi)^{6}\,\boldsymbol{p}_{1}\boldsymbol{p}_{2}\times\Big{(}\delta^{(3)}(\vec{p}_{1}-\vec{p}_{3})\delta^{(3)}(\vec{p}_{2}-\vec{p}_{4})+\delta^{(3)}(\vec{p}_{1}-\vec{p}_{4})\delta^{(3)}(\vec{p}_{2}-\vec{p}_{3})\Big{)}.$

We can finally define the interacting part of the scattering amplitudes. This
is done by subtracting the trivial normalization terms (which describe
particles propagating without interacting), more precisely

$\displaystyle i\mathcal{T}_{AA\rightarrow
AA}(s,t,u)\times(2\pi)^{4}\delta^{4}(p)$ $\displaystyle\equiv{}_{out}\langle
A_{3},A_{4}|A_{1},A_{2}\rangle_{in}-{}_{in}\langle
A_{3},A_{4}|A_{1},A_{2}\rangle_{in},$ (60) $\displaystyle
i\mathcal{T}_{AA\rightarrow BB}(s,t,u)\times(2\pi)^{4}\delta^{4}(p)$
$\displaystyle\equiv{}_{out}\langle
B_{3},B_{4}|A_{1},A_{2}\rangle_{in}-{}_{in}\langle
B_{3},B_{4}|A_{1},A_{2}\rangle_{in},$ $\displaystyle
i\mathcal{T}_{AB\rightarrow AB}(s,t,u)\times(2\pi)^{4}\delta^{4}(p)$
$\displaystyle\equiv{}_{out}\langle
A_{3},B_{4}|A_{1},B_{2}\rangle_{in}-{}_{in}\langle
A_{3},B_{4}|A_{1},B_{2}\rangle_{in},$ $\displaystyle
i\mathcal{T}_{BB\rightarrow BB}(s,t,u)\times(2\pi)^{4}\delta^{4}(p)$
$\displaystyle\equiv{}_{out}\langle
B_{3},B_{4}|B_{1},B_{2}\rangle_{in}-{}_{in}\langle
B_{3},B_{4}|B_{1},B_{2}\rangle_{in},$

where in order to make the formulas more compact we have introduced the
following short-hand notation

$\delta^{4}(p)\equiv\delta^{4}(p_{1}+p_{2}-p_{3}-p_{4}).$ (61)

Notice the presence of the imaginary unit in the left-hand side which is
introduced in order to match the standard conventions.

###  3.2 Crossing equations

The crossing equations for the process $AA\rightarrow AA$ and $BB\rightarrow
BB$ are extremely simple. They require that the associated amplitudes are
fully symmetric under any permutations of the Mandelstam variables, namely

$\displaystyle\mathcal{T}_{AA\rightarrow AA}(s,t,u)$
$\displaystyle=\mathcal{T}_{AA\rightarrow
AA}(t,s,u)\,=\mathcal{T}_{AA\rightarrow AA}(u,t,s),$ (62)
$\displaystyle\mathcal{T}_{BB\rightarrow BB}(s,t,u)$
$\displaystyle=\mathcal{T}_{BB\rightarrow
BB}(t,s,u)=\mathcal{T}_{BB\rightarrow BB}(u,t,s).$ (63)

Things are more complicated for the processes $AA\rightarrow BB$ and
$AB\rightarrow AB$. They are related by crossing. Recall that the exchange of
particles $14$ and $23$ lead to the s-t crossing equations, instead the
exchange of particles $13$ and $24$ lead to the s-u crossing equations. In
other words

$\displaystyle\mathcal{T}_{AB\rightarrow AB}(s,t,u)$
$\displaystyle=\mathcal{T}_{AA\rightarrow
BB}(t,s,u)=\mathcal{T}_{BB\rightarrow AA}(t,s,u),$ (64)
$\displaystyle\mathcal{T}_{AB\rightarrow AB}(s,t,u)$
$\displaystyle=\mathcal{T}_{AB\rightarrow AB}(u,t,s).$ (65)

From the relation (64) it follows immediately that

$\mathcal{T}_{BB\rightarrow AA}(s,t,u)=\mathcal{T}_{AA\rightarrow BB}(s,t,u).$
(66)

###  3.3 Unitarity

The two-particle states (53) do not transform in the irreducible
representation of the Little group $SO(3)$. We can however decompose them into
irreducible representations, see for example Karateev:2019ymz ; Hebbar:2020ukp
.

To begin with let us evaluate the states (53) in the center of mass frame,
namely when the directions of particles are aligned and opposite to each other

$\vec{p}_{1}=+\vec{p},\qquad\vec{p}_{2}=-\vec{p}.$ (67)

In spherical coordinates the momentum $\vec{p}$ reads as

$+\vec{p}=(\boldsymbol{p},\,\theta,\,\phi),\quad-\vec{p}=(\boldsymbol{p},\,\pi-\theta,\,\pi+\phi),\quad\theta\in[0,\pi],\quad\phi\in[0,2\pi].$
(68)

Here $\boldsymbol{p}\equiv|\vec{p}\,|$. We denote the states (53) in this
particular frame by

$|A_{1},A_{2}\rangle_{in}^{com},\qquad|A_{1},B_{2}\rangle_{in}^{com},\qquad|B_{1},B_{2}\rangle_{in}^{com}.$
(69)

These states depend on the masses $m_{A}$, $m_{B}$ and the spherical
coordinates $\boldsymbol{p}$, $\theta$ and $\phi$. The same notation holds for
the out states. According to equations (2.29) and (2.30) in Hebbar:2020ukp  we
can write

$\displaystyle|A_{1},A_{2}\rangle_{in}^{com}$
$\displaystyle=\sqrt{2}\sum_{\ell,\lambda}C_{\ell}^{AA}(\boldsymbol{p})e^{-i\lambda\phi}d^{(\ell)}_{\lambda
0}(\theta)|\sqrt{s},\vec{0};\ell,\lambda\rangle_{in}^{AA},$ (70)
$\displaystyle|A_{1},B_{2}\rangle_{in}^{com}$
$\displaystyle=\quad\;\,\sum_{\ell,\lambda}C_{\ell}^{AB}(\boldsymbol{p})\,e^{-i\lambda\phi}d^{(\ell)}_{\lambda
0}(\theta)|\sqrt{s},\vec{0};\ell,\lambda\rangle_{in}^{AB},$
$\displaystyle|B_{1},B_{2}\rangle_{in}^{com}$
$\displaystyle=\sqrt{2}\sum_{\ell,\lambda}C_{\ell}^{BB}(\boldsymbol{p})e^{-i\lambda\phi}d^{(\ell)}_{\lambda
0}(\theta)|\sqrt{s},\vec{0};\ell,\lambda\rangle_{in}^{BB},$

where $d_{\lambda_{1}\lambda_{2}}^{(\ell)}(\theta)$ is the small Wigner d
matrices and $|\sqrt{s},\vec{0};\ell,\lambda\rangle$ are states in the
irreducible representation of the $SO(3)$ Little group. Here we have also
introduced the object $C_{\ell}$ which is defined as

$\displaystyle C_{\ell}(\boldsymbol{p})^{2}\equiv
4\pi\,(2\ell+1)\times\frac{\sqrt{s}}{\boldsymbol{p}},$ (71)

where $\sqrt{s}$ is the total energy of the two-particle state which is

$\displaystyle AA:\quad\sqrt{s}$
$\displaystyle=2\sqrt{m^{2}+\boldsymbol{p}^{2}},$ (72) $\displaystyle
AB:\quad\sqrt{s}$
$\displaystyle=\sqrt{m^{2}+\boldsymbol{p}^{2}}+\boldsymbol{p},$ $\displaystyle
BB:\quad\sqrt{s}$ $\displaystyle=2\boldsymbol{p}.$

The small Wigner $d$-matrices are defined for example in equation (2.9) of
Hebbar:2020ukp . The states in the right-hand side of (70) transform in the
irreducible representation of the Lorentz group. They have the Little group
spin $\ell$ and helicity $\lambda=-\ell,\ldots,+\ell$. They also have the zero
total spatial momentum $\vec{0}$. Their normalization is completely fixed by
(70) and (59).

By using the orthogonality of the small Wigner d-matrices we can invert the
decompositions (70) and write

$\displaystyle|\sqrt{s},\vec{0};\ell,\lambda\rangle_{in}^{AA}$
$\displaystyle=\Pi_{\ell}^{AA}|A_{1},A_{2}\rangle_{in}^{com},$ (73)
$\displaystyle|\sqrt{s},\vec{0};\ell,\lambda\rangle_{in}^{AB}$
$\displaystyle=\Pi_{\ell}^{AB}|A_{1},B_{2}\rangle_{in}^{com},$
$\displaystyle|\sqrt{s},\vec{0};\ell,\lambda\rangle_{in}^{BB}$
$\displaystyle=\Pi_{\ell}^{BB}|B_{1},B_{2}\rangle_{in}^{com},$

where $\Pi_{\ell}$ are the projectors to the definite Little group spin. Their
explicit form reads

$\displaystyle\Pi_{\ell}^{AA}$
$\displaystyle\equiv\frac{2\ell+1}{4\pi\sqrt{2}C^{AA}_{\ell}(\boldsymbol{p})}\int_{0}^{2\pi}d\phi\,e^{i\lambda\phi}\int_{-1}^{+1}d\cos\theta\;d^{(\ell)}_{\lambda
0}(\theta),$ (74) $\displaystyle\Pi_{\ell}^{AB}$
$\displaystyle\equiv\;\;\frac{2\ell+1}{4\pi
C^{AB}_{\ell}(\boldsymbol{p})}\;\;\;\int_{0}^{2\pi}d\phi\,e^{i\lambda\phi}\int_{-1}^{+1}d\cos\theta\;d^{(\ell)}_{\lambda
0}(\theta),$ $\displaystyle\Pi_{\ell}^{BB}$
$\displaystyle\equiv\frac{2\ell+1}{4\pi\sqrt{2}C^{BB}_{\ell}(\boldsymbol{p})}\int_{0}^{2\pi}d\phi\,e^{i\lambda\phi}\int_{-1}^{+1}d\cos\theta\;d^{(\ell)}_{\lambda
0}(\theta).$

Let us define the partial amplitudes $\mathcal{S}_{\ell}(s)$ as the following
matrix elements

$\displaystyle\mathcal{S}_{AA\rightarrow
AA}^{\ell}(s)\times\delta_{\ell^{\prime}\ell}\delta_{\lambda^{\prime}\lambda}(2\pi)^{4}\delta(\sqrt{s}^{\prime}-\sqrt{s})\delta^{3}(\vec{0})$
$\displaystyle\equiv{}_{out}^{AA}\langle\sqrt{s}^{\prime},\vec{0};\ell^{\prime},\lambda^{\prime}|\sqrt{s},\vec{0};\ell,\lambda\rangle_{in}^{AA},$
(75) $\displaystyle\mathcal{S}_{AA\rightarrow
BB}^{\ell}(s)\times\delta_{\ell^{\prime}\ell}\delta_{\lambda^{\prime}\lambda}(2\pi)^{4}\delta(\sqrt{s}^{\prime}-\sqrt{s})\delta^{3}(\vec{0})$
$\displaystyle\equiv{}_{out}^{BB}\langle\sqrt{s}^{\prime},\vec{0};\ell^{\prime},\lambda^{\prime}|\sqrt{s},\vec{0};\ell,\lambda\rangle_{in}^{AA},$
$\displaystyle\mathcal{S}_{AB\rightarrow
AB}^{\ell}(s)\times\delta_{\ell^{\prime}\ell}\delta_{\lambda^{\prime}\lambda}(2\pi)^{4}\delta(\sqrt{s}^{\prime}-\sqrt{s})\delta^{3}(\vec{0})$
$\displaystyle\equiv{}_{out}^{AB}\langle\sqrt{s}^{\prime},\vec{0};\ell^{\prime},\lambda^{\prime}|\sqrt{s},\vec{0};\ell,\lambda\rangle_{in}^{AB},$
$\displaystyle\mathcal{S}_{BB\rightarrow
BB}^{\ell}(s)\times\delta_{\ell^{\prime}\ell}\delta_{\lambda^{\prime}\lambda}(2\pi)^{4}\delta(\sqrt{s}^{\prime}-\sqrt{s})\delta^{3}(\vec{0})$
$\displaystyle\equiv{}_{out}^{BB}\langle\sqrt{s}^{\prime},\vec{0};\ell^{\prime},\lambda^{\prime}|\sqrt{s},\vec{0};\ell,\lambda\rangle_{in}^{BB}.$

Using (73) and (74) we obtain the explicit relations between the partial
amplitudes and the scattering amplitudes. The detailed steps in this
derivation are explained in section 2.5 in Hebbar:2020ukp . They read

$\displaystyle\mathcal{S}_{AA\rightarrow AA}^{\ell}(s)$
$\displaystyle=\frac{1}{32\pi}\left(1-4m^{2}/s\right)^{1/2}\int_{-1}^{+1}d\cos\theta
P_{\ell}(\cos\theta)\mathcal{S}_{AA\rightarrow AA}(s,t,u),$ (76)
$\displaystyle\mathcal{S}_{AA\rightarrow BB}^{\ell}(s)$
$\displaystyle=\frac{1}{32\pi}\left(1-4m^{2}/s\right)^{1/4}\int_{-1}^{+1}d\cos\theta
P_{\ell}(\cos\theta)\mathcal{S}_{AA\rightarrow BB}(s,t,u),$
$\displaystyle\mathcal{S}_{AB\rightarrow AB}^{\ell}(s)$
$\displaystyle=\frac{1}{16\pi}(1-m^{2}/s)\int_{-1}^{+1}d\cos\theta
P_{\ell}(\cos\theta)\mathcal{S}_{AB\rightarrow AB}(s,t,u),$
$\displaystyle\mathcal{S}_{BB\rightarrow BB}^{\ell}(s)$
$\displaystyle=\frac{1}{32\pi}\int_{-1}^{+1}d\cos\theta
P_{\ell}(\cos\theta)\mathcal{S}_{BB\rightarrow BB}(s,t,u),$

where $P_{\ell}(\cos\theta)$ are the Legendre polynomials. They are related to
the small Wigner d matrices as $P_{\ell}(\cos\theta)=d_{00}^{(\ell)}(\theta)$.
To complete these equations it is important to specify the relation between
the Mandelstam variables and the scattering angle for each process. One has

$\displaystyle AA\rightarrow AA:$ $\displaystyle\quad
t=\frac{1}{2}(4m^{2}-s)(1-\cos\theta),\quad
u=\frac{1}{2}(4m^{2}-s)(1+\cos\theta),$ $\displaystyle AA\rightarrow BB:$
$\displaystyle\quad
t=m^{2}-\frac{s}{2}+\frac{1}{2}\sqrt{s(s-4m^{2})}\cos\theta,\quad
u=m^{2}-\frac{s}{2}-\frac{1}{2}\sqrt{s(s-4m^{2})}\cos\theta,$ $\displaystyle
AB\rightarrow AB:$ $\displaystyle\quad
t=\frac{(s-m^{2})^{2}(\cos\theta-1)}{2s},\quad
u=\frac{2m^{4}-(s-m^{2})^{2}(\cos\theta+1)}{2s},$ $\displaystyle BB\rightarrow
BB:$ $\displaystyle\quad t=-\frac{s}{2}\,(1-\cos\theta),\quad
u=-\frac{s}{2}\,(1+\cos\theta).$ (77)

In (76) we found the relations between the partial amplitudes and the full
scattering amplitudes. We can further rewrite them by splitting the scattering
amplitudes into their trivial and their interacting parts given by (59) and
(60). Performing the explicit integral over the trivial part we obtain the
final expressions

$\displaystyle\mathcal{S}_{AA\rightarrow AA}^{\ell}(s)$
$\displaystyle=1+i\mathcal{T}_{AA\rightarrow AA}^{\ell}(s),$ (78)
$\displaystyle\mathcal{S}_{AA\rightarrow BB}^{\ell}(s)$
$\displaystyle=0+i\mathcal{T}_{AA\rightarrow BB}^{\ell}(s),$
$\displaystyle\mathcal{S}_{AB\rightarrow AB}^{\ell}(s)$
$\displaystyle=1+i\mathcal{T}_{AB\rightarrow AB}^{\ell}(s),$
$\displaystyle\mathcal{S}_{BB\rightarrow BB}^{\ell}(s)$
$\displaystyle=1+i\mathcal{T}_{BB\rightarrow BB}^{\ell}(s),$

where we have defined

$\displaystyle\mathcal{T}_{AA\rightarrow AA}^{\ell}(s)$
$\displaystyle\equiv\frac{1}{32\pi}\left(1-4m^{2}/s\right)^{1/2}\int_{-1}^{+1}d\cos\theta
P_{\ell}(\cos\theta)\mathcal{T}_{AA\rightarrow AA}(s,t,u),$ (79)
$\displaystyle\mathcal{T}_{AA\rightarrow BB}^{\ell}(s)$
$\displaystyle\equiv\frac{1}{32\pi}\left(1-4m^{2}/s\right)^{1/4}\int_{-1}^{+1}d\cos\theta
P_{\ell}(\cos\theta)\mathcal{T}_{AA\rightarrow BB}(s,t,u),$
$\displaystyle\mathcal{T}_{AB\rightarrow AB}^{\ell}(s)$
$\displaystyle\equiv\frac{1}{16\pi}(1-m^{2}/s)\int_{-1}^{+1}d\cos\theta
P_{\ell}(\cos\theta)\mathcal{T}_{AB\rightarrow AB}(s,t,u),$
$\displaystyle\mathcal{T}_{BB\rightarrow BB}^{\ell}(s)$
$\displaystyle\equiv\frac{1}{32\pi}\int_{-1}^{+1}d\cos\theta
P_{\ell}(\cos\theta)\mathcal{T}_{BB\rightarrow BB}(s,t,u),$

We are now in position to address the unitarity constraints. Let us first
consider the $\mathbb{Z}_{2}$ odd two-particle in and out states projected to
the definite spin, namely

$|\sqrt{s},\vec{0};\ell,\lambda\rangle_{in}^{AB},\qquad|\sqrt{s},\vec{0};\ell,\lambda\rangle_{out}^{AB}.$
(80)

We take all possible inner products of these states. Unitarity requires that
such a matrix is semipositive definite. Using the definition (75) we conclude
that

$\forall\ell\geq 0,\quad\forall
s\in[m^{2},\infty):\qquad\begin{pmatrix}1&\mathcal{S}_{AB\rightarrow
AB}^{*\ell}(s)\\\ \mathcal{S}_{AB\rightarrow
AB}^{\ell}(s)&1\end{pmatrix}\succeq 0.$ (81)

Let us consider now the $\mathbb{Z}_{2}$ even two-particles in and out states
projected to the definite spin, namely

$|\sqrt{s},\vec{0};\ell,\lambda\rangle_{in}^{AA},\qquad|\sqrt{s},\vec{0};\ell,\lambda\rangle_{in}^{BB},\qquad|\sqrt{s},\vec{0};\ell,\lambda\rangle_{out}^{AA},\qquad|\sqrt{s},\vec{0};\ell,\lambda\rangle_{out}^{BB}.$
(82)

Analogously we obtain the following unitarity constraint

$\begin{aligned} \forall\ell=0,2,4,\ldots\\\ \forall
s\in[4m^{2},\infty)\end{aligned}:\qquad\begin{pmatrix}1&0&\mathcal{S}_{AA\rightarrow
AA}^{*\ell}(s)&\mathcal{S}_{AA\rightarrow BB}^{*\ell}(s)\\\
0&1&\mathcal{S}_{BB\rightarrow AA}^{*\ell}(s)&\mathcal{S}_{BB\rightarrow
BB}^{*\ell}(s)\\\ \mathcal{S}_{AA\rightarrow
AA}^{\ell}(s)&\mathcal{S}_{BB\rightarrow AA}^{\ell}(s)&1&0\\\
\mathcal{S}_{AA\rightarrow BB}^{\ell}(s)&\mathcal{S}_{BB\rightarrow
BB}^{\ell}(s)&0&1\end{pmatrix}\succeq 0.$ (83)

For the energy range $s\in[0,4m^{2})$ the two particle state $AA$, do not
exist, however the two particle state $BB$ do. Considering the states

$|\sqrt{s},\vec{0};\ell,\lambda\rangle_{in}^{BB},\qquad|\sqrt{s},\vec{0};\ell,\lambda\rangle_{out}^{BB},$
(84)

we conclude that the following unitarity constraint also holds

$\forall\ell=0,2,4,\ldots,\quad\forall
s\in[0,4m^{2}):\qquad\begin{pmatrix}1&\mathcal{S}_{BB\rightarrow
BB}^{*\ell}(s)\\\ \mathcal{S}_{BB\rightarrow
BB}^{\ell}(s)&1\end{pmatrix}\succeq 0.$ (85)

###  3.4 Dilaton scattering

So far the particle $B$ was simply a generic massless scalar particle. We
would like now to identify it with the dilaton. First, let us define rescaled
amplitudes that remain finite in the probe limit $f\to\infty$,

$\displaystyle\widetilde{\mathcal{T}}_{AA\rightarrow
BB}(s,t,u)=\lim_{f\to\infty}f^{2}\,\mathcal{T}_{AA\rightarrow BB}(s,t,u)\,,$
(86) $\displaystyle\widetilde{\mathcal{T}}_{AB\rightarrow
AB}(s,t,u)=\lim_{f\to\infty}f^{2}\,\mathcal{T}_{AB\rightarrow AB}(s,t,u)\,,$
$\displaystyle\widetilde{\mathcal{T}}_{BB\rightarrow
BB}(s,t,u)=\lim_{f\to\infty}f^{4}\,\mathcal{T}_{BB\rightarrow BB}(s,t,u)\,.$

The main result of Komargodski:2011vj  which we use in this paper is the
relation between the $a$-anomaly in the UV with the dilaton field scattering.
More precisely

$\lim_{f\rightarrow\infty}f^{4}\mathcal{T}_{BB\rightarrow
BB}(s,t,u)=a^{\text{UV}}\times(s^{2}+t^{2}+u^{2})+O(s^{3}).$ (87)

The limit $f\rightarrow\infty$ corresponds to the full decoupling of the
dilaton from the system. Nevertheless it still carries some non-trivial
information about the system.

The goal of this section is to formulate the bootstrap setup in terms of
amplitudes (86) and take explicitly the limit $f\rightarrow\infty$.

##### Analyticity and soft behavior

We start by noticing that dilatons in the limit $f\rightarrow\infty$ cannot
contribute to the scattering amplitudes as intermediate states. In practice
this means that in all the scattering amplitudes there are no branch cuts
$s\in[0,\infty]$ due to $n$-dilaton particle states, where $n=1,2,3,4,\ldots$.
In other words

$s\in[4m^{2},\infty]:\qquad\begin{aligned}
&\text{disc}_{s}\mathcal{T}_{AA\rightarrow AA}(s,t,u)\neq 0,\\\
&\text{disc}_{s}\widetilde{\mathcal{T}}_{AA\rightarrow BB}(s,t,u)\neq 0,\\\
&\text{disc}_{s}\widetilde{\mathcal{T}}_{BB\rightarrow BB}(s,t,u)\neq
0,\end{aligned}$ (88)

where the non-zero discontinuity is due to the $n$-particle states of type A,
where $n\geq 2$ is only allowed to be even due to the $\mathbb{Z}_{2}$
symmetry. Analogously we conclude that

$s\in[9m^{2},\infty]:\qquad\text{disc}_{s}\widetilde{\mathcal{T}}_{AB\rightarrow
AB}(s,t,u)\neq 0,$ (89)

where the discontinuity appears due to the $n$-particle states of type A,
where $n\geq 3$ is only allowed to be odd due to the $\mathbb{Z}_{2}$
symmetry. It is important also to discuss the contribution of the $n=1$
particle state of type A to the scattering amplitudes describing the processes
$AB\rightarrow AB$ and $AA\rightarrow BB$. It appears as an s-channel pole in
the amplitude, namely

$\displaystyle\widetilde{\mathcal{T}}_{AB\rightarrow AB}(s,t,u)$
$\displaystyle=-\frac{h^{2}}{s-m^{2}}+\ldots,$ (90)
$\displaystyle\widetilde{\mathcal{T}}_{AA\rightarrow BB}(s,t,u)$
$\displaystyle=\qquad 0\quad\;\,+\ldots,$

where $h$ is the real number which describes the interaction strength between
the particle $A$ and the dilaton. The second amplitude in (90) does not have
an s-channel pole due to the $\mathbb{Z}_{2}$ symmetry. Imposing the crossing
symmetry given by equations (64) and (65) we conclude that

$\displaystyle\widetilde{\mathcal{T}}_{AB\rightarrow AB}(s,t,u)$
$\displaystyle=-\frac{h^{2}}{s-m^{2}}-\frac{h^{2}}{u-m^{2}}+g(s,t,u)$ (91)
$\displaystyle\widetilde{\mathcal{T}}_{AA\rightarrow BB}(s,t,u)$
$\displaystyle=-\frac{h^{2}}{t-m^{2}}-\frac{h^{2}}{u-m^{2}}+g(t,s,u),$

where the function $g$ describes the finite part of the amplitudes. No poles
in the other amplitudes are allowed either due to the $\mathbb{Z}_{2}$
symmetry or due to the limit $f\rightarrow\infty$. In appendix C we give a
non-perturbative argument that the residue $h$ has the following value

$h^{2}=2m^{4}.$ (92)

In the next section we will obtain a stronger result which not only fixes the
residue of the poles but also the constant piece of the $AB\rightarrow AB$ and
$AA\rightarrow BB$ amplitudes in the soft limit. The result reads

$\displaystyle\displaystyle\widetilde{\mathcal{T}}_{AA\rightarrow BB}(s,t,u)\
=-\frac{2m^{4}}{t-m^{2}}-\frac{2m^{4}}{u-m^{2}}-m^{2}+\ O(u-m^{2},t-m^{2}).$
(93)

##### Unitarity constraints

The final step is to write the unitarity constraints (81), (83) and (85) in
the limit $f\rightarrow\infty$. Let us start with the constraint (81). We use
the relations (86), compute the matrix eigenvalues and expand them in the
inverse-powers of $f$ to the sub-leading order. Semipositive definiteness
requires that all the eigenvalues are non-negative. Explicitly we get

$\displaystyle 0+f^{-2}\text{Im}\widetilde{\mathcal{T}}^{\ell}_{AB\rightarrow
AB}(s)+O(f^{-4})\geq 0,$ (94) $\displaystyle
2-f^{-2}\text{Im}\widetilde{\mathcal{T}}^{\ell}_{AB\rightarrow
AB}(s)+O(f^{-4})\geq 0.$

In the limit $f\rightarrow\infty$ the second condition is satisfied
automatically since the second term gives a negligible contribution. The
leading term in the first condition vanishes however and we are forced to
study the sub-leading contribution. Clearly in the limit $f\rightarrow\infty$
the condition (81) simply reduces to

$\forall\ell\geq 0,\quad\forall
s\in[9m^{2},\infty):\qquad\text{Im}\widetilde{\mathcal{T}}^{\ell}_{AB\rightarrow
AB}(s)\geq 0.$ (95)

Here we have also used (89). Analogous reasoning holds for the condition (85).
However due to the conditions (88) the imaginary part of
$\widetilde{\mathcal{T}}^{\ell}_{BB\rightarrow BB}(s)$ vanishes in the
interval $s\in[0,4m^{2}]$. As a consequence the condition (85) is trivially
satisfied.

The best way to analyse the condition (83) is to cast it into a different form
where the limit $f\rightarrow\infty$ is obvious. Most of the manipulations
bellow were already presented in appendix B in Homrich:2019cbt . First, due to
(78) we can write (83) as

$\begin{pmatrix}\mathbb{I}&\mathbb{I}-i\mathbb{T}^{\dagger}(s)\\\
\mathbb{I}+i\mathbb{T}(s)&\mathbb{I}\end{pmatrix}\succeq 0,$ (96)

where we have defined

$\mathbb{I}\equiv\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\qquad\mathbb{T}(s)\equiv\begin{pmatrix}\mathcal{T}_{AA\rightarrow
AA}^{\ell}(s)&\mathcal{T}_{BB\rightarrow AA}^{\ell}(s)\\\
\mathcal{T}_{AA\rightarrow BB}^{\ell}(s)&\mathcal{T}_{BB\rightarrow
BB}^{\ell}(s)\end{pmatrix}.$ (97)

Second, we notice that the condition (96) is equivalent to

$2\text{Im}\mathbb{T}-\mathbb{T}^{\dagger}(s)\mathbb{T}(s)\succeq 0.$ (98)

Third, we rewrite this condition in another equivalent form

$\begin{pmatrix}\mathbb{I}&\mathbb{T}^{\dagger}(s)\\\
\mathbb{T}(s)&2\text{Im}\mathbb{T}\end{pmatrix}\succeq 0.$ (99)

The matrix appearing in the left-hand side of (99) is 4x4. Let us write it
explicitly using (97) and (86). We get

$\begin{pmatrix}1&0&\mathcal{T}_{AA\rightarrow
AA}^{*\ell}(s)&\frac{1}{f^{2}}\widetilde{\mathcal{T}}_{AA\rightarrow
BB}^{*\ell}(s)\\\ 0&1&\frac{1}{f^{2}}\widetilde{\mathcal{T}}_{BB\rightarrow
AA}^{*\ell}(s)&\frac{1}{f^{4}}\widetilde{\mathcal{T}}_{BB\rightarrow
BB}^{*\ell}(s)\\\ \mathcal{T}_{AA\rightarrow
AA}^{\ell}(s)&\frac{1}{f^{2}}\widetilde{\mathcal{T}}_{BB\rightarrow
AA}^{\ell}(s)&2\text{Im}\mathcal{T}_{AA\rightarrow
AA}^{\ell}(s)&\frac{2}{f^{2}}\text{Im}\widetilde{\mathcal{T}}_{BB\rightarrow
AA}^{\ell}(s)\\\ \frac{1}{f^{2}}\widetilde{\mathcal{T}}_{AA\rightarrow
BB}^{\ell}(s)&\frac{1}{f^{4}}\widetilde{\mathcal{T}}_{BB\rightarrow
BB}^{\ell}(s)&\frac{2}{f^{2}}\text{Im}\widetilde{\mathcal{T}}_{AA\rightarrow
BB}^{\ell}(s)&\frac{2}{f^{4}}\text{Im}\widetilde{\mathcal{T}}_{BB\rightarrow
BB}^{\ell}(s)\end{pmatrix}\succeq 0.$ (100)

We notice that multiplying the last row and subsequently the last column of
the matrix in the left-hand side of (100) by $f^{2}$ does not change its
semidefinite positive property.999This can be proven in several different
ways. For instance one can multiply the condition (100) by the diagonal matrix
$\\{1,1,1,f^{2}\\}$ to the left and to the right. Alternatively one can use
the Sylvester’s criterion. Considering all the principal minors of the
original matrix and the one with the last row and column multiplied by $f^{2}$
we see that their non-negativity conditions are equivalent as long as
$f^{2}>0$.

Taking the limit $f\rightarrow\infty$ we obtain the unitarity condition (83)
in its final form

$\begin{aligned} \forall\ell=0,2,4,\ldots\\\ \forall
s\in[4m^{2},\infty)\end{aligned}:\qquad\begin{pmatrix}1&\mathcal{T}_{AA\rightarrow
AA}^{*\ell}(s)&\widetilde{\mathcal{T}}_{AA\rightarrow BB}^{*\ell}(s)\\\
\mathcal{T}_{AA\rightarrow AA}^{\ell}(s)&2\text{Im}\mathcal{T}_{AA\rightarrow
AA}^{\ell}(s)&2\text{Im}\widetilde{\mathcal{T}}_{BB\rightarrow
AA}^{\ell}(s)\\\ \widetilde{\mathcal{T}}_{AA\rightarrow
BB}^{\ell}(s)&2\text{Im}\widetilde{\mathcal{T}}_{AA\rightarrow
BB}^{\ell}(s)&2\text{Im}\widetilde{\mathcal{T}}_{BB\rightarrow
BB}^{\ell}(s)\end{pmatrix}\succeq 0.$ (101)

Notice that in going from (100) to (101) after taking the limit
$f\rightarrow\infty$ we dropped the second row and the second column which
were simply $(0,1,0,0)$, did not affect the semidefinite positiveness.

Summarizing, the unitarity conditions which must be satisfied for the system
of a particle and the dilaton are (95) and (101). We can use the Sylvester’s
criterion in order to rewrite the latter condition as a set of inequalities.
As a result the conditions (95) and (101) can be written as

$\displaystyle\text{Im}\widetilde{\mathcal{T}}^{\ell}_{AB\rightarrow
AB}(s)\geq 0,$ $\displaystyle 2\text{Im}\mathcal{T}_{AA\rightarrow
AA}^{\ell}(s)\geq\left|\mathcal{T}_{AA\rightarrow AA}^{\ell}(s)\right|^{2}\geq
0,\qquad 2\text{Im}\widetilde{\mathcal{T}}_{BB\rightarrow
BB}^{\ell}(s)\geq\left|\widetilde{\mathcal{T}}_{AA\rightarrow
BB}^{\ell}(s)\right|^{2}\geq 0,$ (102)
$\displaystyle\text{Im}\mathcal{T}_{AA\rightarrow
AA}^{\ell}(s)\times\text{Im}\widetilde{\mathcal{T}}_{BB\rightarrow
BB}^{\ell}(s)\geq\left(\text{Im}\widetilde{\mathcal{T}}_{AA\rightarrow
BB}^{\ell}(s)\right)^{2}$

together with

$2\text{Im}\widetilde{\mathcal{T}}_{BB\rightarrow
BB}^{\ell}(s)\times\left(2\text{Im}\mathcal{T}_{AA\rightarrow
AA}^{\ell}(s)-\left|\mathcal{T}_{AA\rightarrow
AA}^{\ell}(s)\right|^{2}\right)+\\\
2\text{Im}\widetilde{\mathcal{T}}_{AA\rightarrow
BB}^{\ell}(s)\times\left(\mathcal{T}_{AA\rightarrow
AA}^{\ell}(s)\widetilde{\mathcal{T}}_{AA\rightarrow
BB}^{*\ell}(s)+\mathcal{T}_{AA\rightarrow
AA}^{*\ell}(s)\widetilde{\mathcal{T}}_{AA\rightarrow
BB}^{\ell}(s)\right)\geq\\\ 2\text{Im}\mathcal{T}_{AA\rightarrow
AA}^{\ell}(s)\times\left|\widetilde{\mathcal{T}}_{AA\rightarrow
BB}^{\ell}(s)\right|^{2}+4\left(\text{Im}\widetilde{\mathcal{T}}_{AA\rightarrow
BB}^{\ell}(s)\right)^{2}.$ (103)

The last condition comes from the determinant of the matrix (95).

##### Sum-rules for the $a$-anomaly

Let us start by recalling the formula (41). In the notation of this section it
reads

$a^{\text{UV}}=\frac{1}{\pi}\ \int_{4m^{2}}^{\infty}\frac{ds}{s^{3}}\
\text{Im}\widetilde{\mathcal{T}}_{BB\rightarrow BB}(s,0,-s).$ (104)

Inverting the last equation in (79) and plugging it to the above sum-rule we
obtain

$\displaystyle a^{\text{UV}}=16\sum_{\ell=0,2,4,\ldots}(2\ell+1)\
\int_{4m^{2}}^{\infty}\frac{ds}{s^{3}}\
\text{Im}\widetilde{\mathcal{T}}^{\ell}_{BB\rightarrow BB}(s).$ (105)

Using this formula and the conditions (102) we conclude that

$a^{\text{UV}}\geq 8\sum_{\ell=0,2,4,\ldots}(2\ell+1)\
\int_{4m^{2}}^{\infty}\frac{ds}{s^{3}}\
\left|\widetilde{\mathcal{T}}_{AA\rightarrow BB}^{\ell}(s)\right|^{2}.$ (106)

Equivalently using (79), the completeness relation of Legendre polynomials and
changing integration variables from $(s,\cos\theta)$ to $(t,u)$ the above
equation can be written as

$a^{\text{UV}}\geq\frac{1}{32\pi^{2}}\int_{tu>m^{4}\atop
t,u<0}\frac{dtdu}{(2m^{2}-t-u)^{4}}\
\left|\widetilde{\mathcal{T}}_{AA\rightarrow BB}(s,t,u)\right|^{2}.$ (107)

In appendix B, in particular see equation (196), we will show that in the case
when particle $A$ is a free boson, the matter - dilaton scattering amplitude
has the following simple form

$\widetilde{\mathcal{T}}^{\text{free}}_{AA\rightarrow
BB}(s,t,u)=-m^{2}-\frac{2m^{4}}{t-m^{2}}-\frac{2m^{4}}{u-m^{2}}.$ (108)

Plugging it into (107) we obtain $a^{\text{UV}}\geq a_{\text{free}}$ and
conclude that the inequality (107) is saturated in this case. Equivalently
plugging (108) into (106) we obtain

$a/a_{\text{free}}=0.83864+0.14908+0.01048+0.00140+\ldots=0.99960+\ldots,$
(109)

where the numbers indicated correspond to $\ell=0$, $2$, $4$ and $6$ and
$\ldots$ represent all the higher spin contributions to the $a$-anomaly of a
free scalar field.

##  4 Matter - dilaton scattering at low energy

In Komargodski:2011vj  the authors derived the most general low energy
effective action of the dilaton field in a curved background. As reviewed in
section 2.3 this action is given in (35). It is found by solving the ’t Hooft
anomaly matching conditions which can be written as a system of differential
equations (36). In flat space this action reduces to (39) and leads to the
explicit expression of the low energy $BB\rightarrow BB$ amplitude given by
equation (40).

Here we perform a similar analysis for the $AB\rightarrow AB$ amplitude. We
start in section 4.1 by writing the most general low energy effective action
in curved background which describes this process. In section 4.2 we evaluate
it in flat space and derive the resulting scattering amplitude at low energy.
The final result turns out to be universal (independent of a particular model)
and is given by equation (143).

###  4.1 The most general effective action

The assumption of this paper is the presence of a single $\mathbb{Z}_{2}$
asymptotic particle $A$. Let us denote by $\Phi(x)$ the effective field
associated to this particle. We would like to find the most general low energy
effective action $A_{\text{eff}}[\Phi,\varphi]$, using which we can compute
the S-matrix for the scattering process $AB\rightarrow AB$.101010One can think
of the effective action $A_{\text{eff}}[\Phi,\varphi]$ as the Legendre
transform of the generating function $W[\zeta,\varphi]$ where $\zeta(x)$ is
the source that couples to a local operator $\mathcal{O}_{\Delta}(x)$ that can
create particle $A$ from the vacuum. We remind that the filed $\varphi(x)$
creates the dilaton particle $B$ from the vacuum.

Following the discussion of section 2.2 we work with the diff and Weyl
invariant action (24). We would like to rewrite this action in terms of low
energy degrees of freedom $\Phi(x)$ and $\varphi(x)$ using only diff and Weyl
symmetry. The main ingredients for doing this are the Weyl invariant metric
$\widehat{g}_{\mu\nu}(x)$ and scalar $\widehat{\Phi}(x)$, defined in terms of
background metric $g_{\mu\nu}(x)$ and scalar field $\Phi(x)$ in the following
way

$\widehat{g}_{\mu\nu}(x)=e^{-2\tau(x)}\
g_{\mu\nu}(x),\qquad\widehat{\Phi}(x)=e^{\Delta\tau(x)}\Phi(x),$ (110)

where $\tau(x)$ is the dilaton field related to $\varphi(x)$ according to (25)
and $\Delta$ is some effective scaling dimension of the field $\Phi(x)$. Now
we want to write the most generic form of the effective action
$A^{g}_{\text{eff}}[\Phi,\varphi]$ in the curved background, which should be
general coordinate invariant in the metric $\widehat{g}_{\mu\nu}(x)$ and
should contain the scalar $\widehat{\Phi}(x)$. Finally we need to substitute
$g_{\mu\nu}=\eta_{\mu\nu}$ in $A^{g}_{\text{eff}}[\Phi,\varphi]$ to get the
flat space effective action $A_{\text{eff}}[\Phi,\varphi]$ which describes the
scattering process $AB\rightarrow AB$. For the construction of
$A^{g}_{\text{eff}}[\Phi,\varphi]$ we adapt a variation of the
covariantization prescription originally developed in Sen:2017nim ;
Chakrabarti:2017ltl  for the soft gravitational background.

Let us start from the tangent space with locally flat metric $\eta_{ab}$. The
connection between the curved and tangent space is provided by the objects
$e_{\mu}^{a}(x)$ and $E_{a}^{\mu}(x)$ called the vierbein and inverse vierbein
respectively. They are defined via the relations

$\widehat{g}_{\mu\nu}(x)=e_{\mu}^{a}(x)e_{\nu}^{b}(x)\eta_{ab},\qquad\eta_{ab}=E_{a}^{\mu}(x)E_{b}^{\nu}(x)\widehat{g}_{\mu\nu},$
(111)

where $\eta_{ab}$ is the flat Minkowski metric. We start with the quadratic
part of the tangent space 1PI effective action for scalar field $\Phi(x)$,

$A^{\text{tangent}}=\frac{1}{2}\int d^{4}x\ \Phi(x)K(\partial_{a})\Phi(x)\ ,$
(112)

where we dropped all the terms with cubic and higher powers of $\Phi$ since
they will not contribute to the scattering process $AB\rightarrow AB$ after
covariantization. The most general form of the kinetic operator $K$ can be
written as111111The form of the kinetic operator as Taylor series expansion in
derivatives is always possible for a EFT where massless particles are absent.

$K(\partial_{a})\equiv\sum_{n=0}^{\infty}c_{n}^{a_{1}a_{2}\ldots
a_{n}}\partial_{a_{1}}\partial_{a_{2}}\ldots\partial_{a_{n}},$ (113)

where the coefficients $c_{n}$ are built out of all possible combinations of
$\eta^{ab}$ with arbitrary numerical coefficients and $c_{n}^{a_{1}\ldots
a_{n}}$ is symmetric under the exchange of tangent space indices. Now we would
like to first covariantize the tangent space action (112). This requires to
make the following replacements: $d^{4}x\rightarrow d^{4}x\sqrt{-\widehat{g}}$
together with

$\displaystyle
K(\partial_{a})\rightarrow\sum_{n=0}^{\infty}c_{n}^{a_{1}a_{2}\ldots
a_{n}}E_{a_{1}}^{\mu_{1}}E_{a_{2}}^{\mu_{2}}\cdots
E_{a_{n}}^{\mu_{n}}D_{\mu_{1}}D_{\mu_{2}}\cdots D_{\mu_{n}}\ \equiv F.$ (114)

Above $F$ is obtained only by minimally covariantizing the kinetic operator.
In principle $F$ could contain non-minimal terms involving one or more Riemann
tensors and derivatives on them. Since those non-minimal terms contain two or
more derivatives of the dilaton field, they start contributing at higher
orders in low dilaton momentum expansion of $AB\rightarrow AB$ scattering
amplitude. Since our interest is to constrain first two terms in low dilaton
momentum expansion of the scattering amplitude $AB\rightarrow AB$, we are not
including those terms under our covariantization process. Now on top of the
covariantization we also need to make the action Weyl invariant which is
achieved by replacing $\Phi(x)\rightarrow\widehat{\Phi}(x)$. Also in
$A^{g}_{\text{eff}}[\Phi,\varphi]$ we should include general coordinate
invariant scalar term purely constructed out of metric
$\widehat{g}_{\mu\nu}(x)$ given in (38) as well as the classically Weyl
invariant action (28) to make the dilaton dynamical. As a result of all these,
our curved space effective action becomes

$A^{g}_{\text{eff}}[\Phi,\varphi]=\frac{1}{2}\int d^{4}x\sqrt{-\widehat{g}}\
\widehat{\Phi}(x)F\widehat{\Phi}(x)\
+A_{\text{dynamics}}(g,\Omega)+A_{\text{invariant}}(g,\Omega)+\ \cdots$ (115)

We denote by $\ldots$ all the possible non-minimal terms appearing in the
covariantization procedure as described earlier. From here on to reduce
complexity we ignore $A_{\text{invariant}}(g,\Omega)$ part of the above action
as it always contribute terms at higher power in $f^{-1}$ compare to similar
terms coming from $A_{\text{dynamics}}(g,\Omega)$ and won’t affect our result
later on.

Now to read off the flat space effective action from
$A_{\text{eff}}[\Phi,\varphi]$ from (115) we need to set
$g_{\mu\nu}(x)=\eta_{\mu\nu}$ which in turn requires the following
substitutions in (115)

$\displaystyle\widehat{g}_{\mu\nu}(x)$ $\displaystyle=$ $\displaystyle
e^{-2\tau(x)}\ \eta_{\mu\nu}=\Big{(}1-\frac{\varphi}{\sqrt{2}f}\Big{)}^{2}\
\eta_{\mu\nu},$ $\displaystyle e_{\mu}^{a}(x)$ $\displaystyle=$ $\displaystyle
e^{-\tau(x)}\delta_{\mu}^{a}=\Big{(}1-\frac{\varphi}{\sqrt{2}f}\Big{)}\
\delta_{\mu}^{a},$ $\displaystyle E_{a}^{\mu}(x)$ $\displaystyle=$
$\displaystyle
e^{\tau(x)}\delta_{a}^{\mu}=\Big{(}1-\frac{\varphi}{\sqrt{2}f}\Big{)}^{-1}\delta_{a}^{\mu}.$
(116)

With these substitutions the flat space effective action becomes

$\displaystyle A_{\text{eff}}[\Phi,\varphi]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int d^{4}x\
\Big{(}1-\frac{\varphi(x)}{\sqrt{2f}}\Big{)}^{4-\Delta}\Phi(x)\sum_{n=0}^{\infty}c_{n}^{a_{1}a_{2}\ldots
a_{n}}\Big{(}1-\frac{\varphi(x)}{\sqrt{2f}}\Big{)}^{-n}\delta_{a_{1}}^{\mu_{1}}\delta_{a_{2}}^{\mu_{2}}\cdots\delta_{a_{n}}^{\mu_{n}}$
(117) $\displaystyle\times\ D_{\mu_{1}}D_{\mu_{2}}\cdots
D_{\mu_{n}}\Big{(}1-\frac{\varphi(x)}{\sqrt{2f}}\Big{)}^{-\Delta}\Phi(x)$
$\displaystyle+\int
d^{4}x\left(-\frac{1}{2}\partial_{\mu}\varphi(x)\partial^{\mu}\varphi(x)+\frac{1}{f}O(\partial^{4}\varphi^{3})\right)\
+\cdots$

Now we expand the effective action $A_{\text{eff}}$ in power of $f$ to the
second order around $f=\infty$. The result can be written in the following
form

$A_{\text{eff}}=A^{0}_{\text{eff}}+\ A^{1}_{\text{eff}}+\
A^{2}_{\text{eff}}+\ldots.$ (118)

where the superscript $n$ in $A^{n}_{\text{eff}}$ indicates the order in
$f^{-n}$ expansion of $A_{\text{eff}}$ around $f\rightarrow\infty$. The first
term in (118) reads as

$A^{0}_{\text{eff}}=\frac{1}{2}\int
d^{4}x\sum_{n=0}^{\infty}c_{n}^{a_{1}a_{2}\ldots
a_{n}}\Big{(}\Phi(x)\partial_{a_{1}}\partial_{a_{2}}\cdots\partial_{a_{n}}\Phi(x)\Big{)}-\frac{1}{2}\int
d^{4}x\partial_{\mu}\varphi(x)\partial^{\mu}\varphi(x).$ (119)

In the second term $A^{1}_{\text{eff}}$ we grouped all the contributions
proportional to a single dilaton field $\varphi$ up to one derivative acting
on it and also schematically kept the order of three dilaton field interaction
term. The result reads

$\displaystyle A^{1}_{\text{eff}}=\frac{1}{2\sqrt{2}f}\int d^{4}x\Bigg{(}$
$\displaystyle\sum_{n=0}^{\infty}c_{n}^{a_{1}a_{2}\ldots
a_{n}}\Big{\\{}n-(4-2\Delta)\Big{\\}}\varphi(x)\Phi(x)\partial_{a_{1}}\partial_{a_{2}}\cdots\partial_{a_{n}}\Phi(x)$
(120) $\displaystyle+$ $\displaystyle\sum_{n=2}^{\infty}c_{n}^{a_{1}\ldots
a_{n}}\sum_{\begin{subarray}{c}i,j=1\\\
i<j\end{subarray}}^{n}\Big{\\{}\delta_{a_{i}}^{\nu}\partial_{a_{j}}\varphi(x)+\delta_{a_{j}}^{\nu}\partial_{a_{i}}\varphi(x)-\eta_{a_{i}a_{j}}\partial^{\nu}\varphi(x)\Big{\\}}$
$\displaystyle\qquad\;\;\,\times\Phi(x)\partial_{a_{1}}\cdots\partial_{a_{i-1}}\partial_{a_{i+1}}\cdots\partial_{a_{j-1}}\partial_{a_{j+1}}\cdots\partial_{a_{n}}\partial_{\nu}\Phi(x)$
$\displaystyle+$ $\displaystyle\sum_{n=1}^{\infty}c_{n}^{a_{1}\ldots
a_{n}}\sum_{i=1}^{n}\Delta\partial_{a_{i}}\varphi(x)\
\Phi(x)\partial_{a_{1}}\cdots\partial_{a_{i-1}}\partial_{a_{i+1}}\cdots\partial_{a_{n}}\Phi(x)$
$\displaystyle+O(\Phi^{2}\partial^{2}\varphi,\;\partial^{4}\varphi^{3})\Bigg{)}\
.$

In the above expression the first term is linear in $\varphi(x)$. It is
obtained by replacing all the covariant derivatives in (117) by ordinary
derivatives and commuting
$\Big{(}1-\frac{\varphi(x)}{\sqrt{2}f}\Big{)}^{-\Delta}$ through the
derivatives (in other words neglecting terms containing derivatives operating
on $\varphi(x)$). Expansion of the resulting expression at linear order in
$\varphi(x)$ generates the first term. In the second and third terms in (120)
one derivative operates on $\varphi(x)$. In presence of two covariant
derivatives on $\Phi(x)$ in the first term in (117), we need to substitute
$D_{\mu_{i}}D_{\mu_{j}}\Phi(x)=\partial_{\mu_{i}}\partial_{\mu_{j}}\Phi(x)-\Gamma_{\mu_{i}\mu_{j}}^{\nu}\partial_{\nu}\Phi(x)$.
Then writing down the Christoffel connection up to linear order in
$\varphi(x)$ for any pair of such covariant derivatives and setting
$\varphi=0$ in all other places we get the second term above. On the other
hand when any one of the ordinary derivative from the set of covariant
derivatives operates on
$\Big{(}1-\frac{\varphi(x)}{\sqrt{2}f}\Big{)}^{-\Delta}$ in the first term in
(117), we get the third term above at the linear order in $\varphi(x)$ from
the expansion of the resulting expression. First non-vanishing contribution to
three dilaton interaction appears at four derivative order as schematically
written as $O(\partial^{4}\varphi^{3})$ in the above expression.

In the third term of (118), we grouped all the contributions involving
$\varphi(x)^{2}$ and no derivative on it, and the result reads

$\displaystyle A^{2}_{\text{eff}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int d^{4}x\
\Big{(}\frac{\varphi(x)}{\sqrt{2}f}\Big{)}^{2}\
\Phi(x)\sum_{n=0}^{\infty}\Bigg{(}\frac{(4-2\Delta)(3-2\Delta)}{2}-(4-2\Delta)\
n+\frac{n(n+1)}{2}\Bigg{)}$ (121) $\displaystyle\times\
c_{n}^{a_{1}a_{2}\cdots
a_{n}}\partial_{a_{1}}\partial_{a_{2}}\cdots\partial_{a_{n}}\Phi(x)\ +\
\frac{1}{f^{2}}O(\Phi^{2}\varphi\partial\varphi).$

To get the above contribution we first replace all the covariant derivatives
in (117) by ordinary derivatives and commute
$\Big{(}1-\frac{\varphi(x)}{\sqrt{2}f}\Big{)}^{-\Delta}$ through the
derivatives neglecting terms containing derivatives operating on $\varphi(x)$.
Then we expand the resulting contribution and collect terms at quadratic order
in $\varphi(x)$. We do not need to compute $A_{\text{eff}}^{n}$ for $n\geq 3$
as these terms of the effective action do not contribute to $AB\rightarrow AB$
scattering amplitude. Another important point is that once the expansion is
done in power of $f$ about $f\rightarrow\infty$ in (118), in the expressions
of $A_{\text{eff}}^{n}$ both tangent space indices $a_{1},a_{2},\cdots$ and
curved space indices $\mu_{1},\mu_{2},\cdots$ can be treated as just flat
space Lorentz indices.

##### Momentum space

It is cleaner to present further discussion in momentum space. We would like
to rewrite $A_{\text{eff}}^{0}$, $A_{\text{eff}}^{1}$ and $A_{\text{eff}}^{2}$
as integrals in momenta variables. Let us start by Fourier transforming the
object in (113) in momentum variable $q$

$\mathcal{K}(\partial_{a})\longrightarrow\mathcal{K}(q)\equiv\sum_{n=0}^{\infty}(i)^{n}\
c_{n}^{a_{1}a_{2}\ldots a_{n}}q_{a_{1}}q_{a_{2}}\ldots q_{a_{n}}.$ (122)

In appendix D we have derived some other important identities under Fourier
transformations which we use below.

Using the definition of (122) the momenta space expression of (119) becomes

$\displaystyle A_{\text{eff}}^{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\frac{d^{4}q_{1}}{(2\pi)^{4}}\frac{d^{4}q_{2}}{(2\pi)^{4}}\
(2\pi)^{4}\delta^{(4)}(q_{1}+q_{2})\ \Phi(q_{1})\mathcal{K}(q_{2})\Phi(q_{2})$
(123)
$\displaystyle-\frac{1}{2}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\frac{d^{4}k_{2}}{(2\pi)^{4}}\
(2\pi)^{4}\delta^{(4)}(k_{1}+k_{2})\varphi(k_{1})k_{2}^{2}\varphi(k_{2}).$

The kinetic operator $\mathcal{K}(q)$ should vanish on-shell as follows from
the equation of motion. Using the definition (122) and equalities from
appendix D in momentum space the expression (120) can be written as

$\displaystyle A_{\text{eff}}^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2}f}\int\frac{d^{4}q_{1}}{(2\pi)^{4}}\frac{d^{4}q_{2}}{(2\pi)^{4}}\frac{d^{4}k}{(2\pi)^{4}}\
(2\pi)^{4}\delta^{(4)}(q_{1}+q_{2}+k)\ \Phi(q_{1})\Phi(q_{2})\ \varphi(k)$
(124)
$\displaystyle\Bigg{(}\Big{\\{}-(4-2\Delta)\mathcal{K}(q_{2})+q_{2}^{\mu}\frac{\partial\mathcal{K}(q_{2})}{\partial
q_{2}^{\mu}}\Big{\\}}+\frac{1}{2}\big{\\{}\delta_{\mu}^{\nu}k_{\rho}+\delta_{\rho}^{\nu}k_{\mu}-\eta_{\mu\rho}k^{\nu}\big{\\}}$
$\displaystyle\times q_{2\nu}\frac{\partial^{2}\mathcal{K}(q_{2})}{\partial
q_{2\mu}\partial q_{2\rho}}+\Delta\
k^{\mu}\frac{\partial\mathcal{K}(q_{2})}{\partial q_{2}^{\mu}}+\
O(k^{2})\Bigg{)}.$

In the above expression we are not explicitly writing down the three dilaton
interaction part, as this term involves four power of dilaton momenta and will
not be important for the computation of $AB\rightarrow AB$ scattering
amplitude up to the order we are interested in. Using the definition (122) and
the properties (220) and (224) the expression (121) can be written in momentum
space as

$A_{\text{eff}}^{2}=\frac{1}{4f^{2}}\int\frac{d^{4}q_{1}}{(2\pi)^{4}}\frac{d^{4}q_{2}}{(2\pi)^{4}}\frac{d^{4}k_{1}}{(2\pi)^{4}}\frac{d^{4}k_{2}}{(2\pi)^{4}}(2\pi)^{4}\delta^{(4)}(q_{1}+q_{2}+k_{1}+k_{2})\Phi(q_{1})\Phi(q_{2})\varphi(k_{1})\varphi(k_{2})\\\
\Bigg{(}(2-\Delta)(3-2\Delta)\mathcal{K}(q_{2})+(-3+2\Delta)q_{2}^{\mu}\frac{\partial\mathcal{K}(q_{2})}{\partial
q_{2}^{\mu}}+\frac{1}{2}q_{2}^{\mu}q_{2}^{\nu}\frac{\partial^{2}\mathcal{K}(q_{2})}{\partial
q_{2}^{\mu}\partial q_{2}^{\nu}}+O(k_{1},k_{2})\Bigg{)}.$ (125)

###  4.2 Low energy amplitude

Having obtained the low energy effective action components given by (123) -
(125) we can finally compute the scattering amplitude $AB\rightarrow AB$. We
start by deriving the Feynman rules. The Feynman propagators are defined as

$(2\pi)^{4}\delta^{(4)}(q_{1}+q_{2})\times
D_{F}^{-1}(q_{1})\equiv-i\frac{\delta^{2}A_{\text{eff}}[\Phi,\varphi]}{\delta\Phi(q_{1})\delta\Phi(q_{2})}\Bigg{|}_{\Phi,\varphi=0},\qquad
D_{F}(q_{1})\equiv\leavevmode\hbox{{\leavevmode\includegraphics{}{}{./prerender/main-
figure3.pdf}}}$ (126)
$(2\pi)^{4}\delta^{(4)}(k_{1}+k_{2})\times\Delta_{F}^{-1}(k_{1})\equiv-i\frac{\delta^{2}A_{\text{eff}}[\Phi,\varphi]}{\delta\varphi(k_{1})\delta\varphi(k_{2})}\Bigg{|}_{\Phi,\varphi=0},\qquad\Delta_{F}(k_{1})\equiv\leavevmode\hbox{{\leavevmode\includegraphics{}{}{./prerender/main-
figure4.pdf}}}$ (127)

The cubic effective vertex $\Phi\Phi\varphi$ is defined as

$(2\pi)^{4}\delta^{(4)}(q_{1}+q_{2}+k)\times\Gamma^{(3)}(q_{1},q_{2};k)\equiv
i\frac{\delta^{3}A_{\text{eff}}[\Phi,\varphi]}{\delta\Phi(q_{1})\delta\Phi(q_{2})\delta\varphi(k)}\Bigg{|}_{\Phi,\varphi=0}\equiv\leavevmode\hbox{\lower
37.89229pt\hbox{{\leavevmode\includegraphics{}{}{./prerender/main-
figure5.pdf}}}}$ (128)

The quartic $\Phi\Phi\varphi\varphi$ vertex is defined as

$\displaystyle(2\pi)^{4}\delta^{(4)}(q_{1}+q_{2}$
$\displaystyle+k_{1}+k_{2})\times\Gamma^{(4)}(q_{1},q_{2};k_{1},k_{2})$ (129)
$\displaystyle\equiv
i\frac{\delta^{4}A_{\text{eff}}[\Phi,\varphi]}{\delta\Phi(q_{1})\delta\Phi(q_{2})\delta\varphi(k_{1})\delta\varphi(k_{2})}\Bigg{|}_{\Phi,\varphi=0}\equiv\leavevmode\hbox{\lower
35.26968pt\hbox{{\leavevmode\includegraphics{}{}{./prerender/main-
figure6.pdf}}}}$

The cubic $\varphi\varphi\varphi$ vertex is defined as

$\displaystyle(2\pi)^{4}\delta^{(4)}(k_{1}+k_{2}+k_{3})$ $\displaystyle\times
V^{(3)}(k_{1},k_{2},k_{3})$ $\displaystyle\equiv
i\frac{\delta^{3}A_{\text{eff}}[\Phi,\varphi]}{\delta\varphi(k_{1})\delta\varphi(k_{2})\delta\varphi(k_{3})}\Bigg{|}_{\Phi,\varphi=0}\equiv\leavevmode\hbox{\lower
37.89229pt\hbox{{\leavevmode\includegraphics{}{}{./prerender/main-
figure7.pdf}}}}$ (130)

Applying these definitions to (123) - (125) we conclude that the Feynman
propagators read as

$D_{F}(q)=i\big{(}\mathcal{K}(q)\big{)}^{-1}\equiv\\{q^{2}+m^{2}-i\epsilon\\}^{-1}\Xi(q),\hskip
28.45274pt\Delta_{F}(k)=-i(k^{2}-i\epsilon)^{-1}.$ (131)

where we have introduced for later convenience a new object $\Xi(q)$ which is
the numerator of the scalar field propagator. The cubic vertex
$\Phi\Phi\varphi$ reads

$\displaystyle\Gamma^{(3)}(q_{1},q_{2};k)$
$\displaystyle=\frac{i}{2\sqrt{2}f}\Bigg{(}-(4-2\Delta)\mathcal{K}(q_{2})+q_{2}^{\mu}\frac{\partial\mathcal{K}(q_{2})}{\partial
q_{2}^{\mu}}+\frac{1}{2}\big{\\{}\delta_{\mu}^{\nu}k_{\rho}+\delta_{\rho}^{\nu}k_{\mu}-\eta_{\mu\rho}k^{\nu}\big{\\}}$
(132) $\displaystyle\times
q_{2\nu}\frac{\partial^{2}\mathcal{K}(q_{2})}{\partial q_{2\mu}\partial
q_{2\rho}}+\Delta\ k^{\mu}\frac{\partial\mathcal{K}(q_{2})}{\partial
q_{2}^{\mu}}+\ O(k^{2})\Bigg{)}+(q_{1}\leftrightarrow q_{2}),$

where $q_{1}+q_{2}+k=0$. In the above expression of $\Gamma^{(3)}$ the terms
linear in $k$ actually vanishes when we explicitly write the terms under
$q_{1}\leftrightarrow q_{2}$ exchange and substitute $q_{2}=-q_{1}-k$. The
quartic vertex $\Phi\Phi\varphi\varphi$ reads as

$\displaystyle\Gamma^{(4)}(q_{1},q_{2};k_{1},k_{2})$
$\displaystyle=\frac{i}{2f^{2}}\Bigg{(}(2-\Delta)(3-2\Delta)\mathcal{K}(q_{2})$
(133)
$\displaystyle-(3-2\Delta)q_{2}^{\mu}\frac{\partial\mathcal{K}(q_{2})}{\partial
q_{2}^{\mu}}+\frac{1}{2}q_{2}^{\mu}q_{2}^{\nu}\frac{\partial^{2}\mathcal{K}(q_{2})}{\partial
q_{2}^{\mu}\partial q_{2}^{\nu}}+O(k_{1},k_{2})\Bigg{)}+(q_{1}\leftrightarrow
q_{2}),$

where $q_{1}+q_{2}+k_{1}+k_{2}=0$. Three dilaton field interaction vertex
$\varphi\varphi\varphi$ reads as

$\displaystyle\displaystyle V^{(3)}(k_{1},k_{2},k_{3})$ $\displaystyle=$
$\displaystyle\ \frac{i}{f}\ \mathcal{O}(k_{i}^{4}).$ (134)

We can now compute the $AB\rightarrow AB$ amplitude using the Feynman diagrams
depicted in figure 4.

Figure 4: The set of diagrams contributing to the $\mathcal{T}_{AB\rightarrow
AB}$ amplitudes. Here the solid lines represent scalar particles $A$ and
dashed lines represent dilatons $B$.

The total amplitude describing the scattering process $AB\rightarrow AB$
becomes

$\mathcal{T}_{AB\rightarrow
AB}=\mathcal{T}_{1}+\mathcal{T}_{2}+\mathcal{T}_{3}+\mathcal{T}_{4},$ (135)

where we have

$\displaystyle i\mathcal{T}_{1}$
$\displaystyle=\Gamma^{(3)}(p_{1},-p_{1}-p_{2};+p_{2})D_{F}(p_{1}+p_{2})\Gamma^{(3)}(p_{1}+p_{2},-p_{3};-p_{4}),$
(136) $\displaystyle i\mathcal{T}_{2}$
$\displaystyle=\Gamma^{(3)}(p_{1},-p_{1}+p_{4};-p_{4})D_{F}(p_{1}-p_{4})\Gamma^{(3)}(p_{1}-p_{4},-p_{3};+p_{2}),$
$\displaystyle i\mathcal{T}_{3}$
$\displaystyle=\Gamma^{(4)}(p_{1},-p_{3};p_{2},-p_{4}),$ $\displaystyle
i\mathcal{T}_{4}$
$\displaystyle=\Gamma^{(3)}(p_{1},-p_{3};p_{2}-p_{4})\Delta_{F}(p_{2}-p_{4})V^{(3)}(-p_{2}+p_{4},p_{2},-p_{4}).$

Plugging the explicit expressions of the Feynman propagators and the effective
vertices, expanding around $p^{\mu}_{2}=0$ and $p^{\mu}_{4}=0$ one obtains the
following

$\displaystyle f^{2}\mathcal{T}_{1}$ $\displaystyle=m^{4}(p_{1}\cdot
p_{2})^{-1}-2m^{2}(\Delta-1)-2im^{4}\Xi^{\prime}(m^{2})+O\big{(}(p_{1}\cdot
p_{2}),(p_{3}\cdot p_{4})\big{)},$ (137) $\displaystyle f^{2}\mathcal{T}_{2}$
$\displaystyle=-m^{4}(p_{1}\cdot
p_{4})^{-1}-2m^{2}(\Delta-1)-2im^{4}\Xi^{\prime}(m^{2})+O\big{(}(p_{1}\cdot
p_{2}),(p_{3}\cdot p_{4})\big{)},$ $\displaystyle f^{2}\mathcal{T}_{3}$
$\displaystyle=(4\Delta-5)m^{2}\
+4im^{4}\Xi^{\prime}(m^{2})+O\big{(}(p_{1}\cdot p_{2}),(p_{3}\cdot
p_{4})\big{)},$ $\displaystyle f^{2}\mathcal{T}_{4}$ $\displaystyle=0\
+O\big{(}(p_{1}\cdot p_{2}),(p_{3}\cdot p_{4})\big{)},$

where $\Xi^{\prime}(m^{2})\equiv\frac{\partial\Xi(q)}{\partial
q^{2}}\Big{|}_{q^{2}=-m^{2}}$.

In deriving these we have used several straightforward relations which follow
from the fact that $\mathcal{K}$ is a scalar quantity and thus can depend only
on $q^{2}$. As a consequence the same is true for $\Xi$ defined in (131).
These relations read as

$\mathcal{K}(q)=\mathcal{K}(-q),\quad\Xi(q)=\Xi(-q),\quad\frac{\partial\mathcal{K}(q)}{\partial
q^{\mu}}=2q_{\mu}\frac{\partial\mathcal{K}(q)}{\partial
q^{2}},\quad\frac{\partial\Xi(q)}{\partial
q^{\mu}}=2q_{\mu}\frac{\partial\Xi(q)}{\partial q^{2}},$ (138)

together with

$\displaystyle\frac{\partial\mathcal{K}(q)}{\partial
q^{\mu}}\Xi(q)+\mathcal{K}(q)\frac{\partial\Xi(q)}{\partial
q^{\mu}}=2iq_{\mu},$ (139)
$\displaystyle\frac{\partial^{2}\mathcal{K}(q)}{\partial q^{\mu}\partial
q^{\nu}}\Xi(q)+\frac{\partial\mathcal{K}(q)}{\partial
q^{\nu}}\frac{\partial\Xi(q)}{\partial
q^{\mu}}+\frac{\partial\mathcal{K}(q)}{\partial
q^{\mu}}\frac{\partial\Xi(q)}{\partial
q^{\nu}}+\mathcal{K}(q)\frac{\partial^{2}\Xi(q)}{\partial q^{\mu}\partial
q^{\nu}}=2i\eta_{\mu\nu}.$

At the end of the evaluation of the Feynman diagrams we use the following on-
shell conditions,

$\displaystyle\mathcal{K}(q)\Big{|}_{q^{2}=-m^{2}}=0\ ,\
\Xi(q)\Big{|}_{q^{2}=-m^{2}}=-i\ ,\ \frac{\partial\mathcal{K}(q)}{\partial
q^{\mu}}\Big{|}_{q^{2}=-m^{2}}=-2q_{\mu},$ (140)
$\displaystyle\frac{\partial^{2}\mathcal{K}(q)}{\partial q^{\mu}\partial
q^{\nu}}\Big{|}_{q^{2}=-m^{2}}=-2\eta_{\mu\nu}+8iq_{\mu}q_{\nu}\
\Xi^{\prime}(m^{2}).$ (141)

Plugging (137) into (135) we obtain

$f^{2}\mathcal{T}_{AB\rightarrow AB}=\frac{m^{4}}{(p_{1}\cdot
p_{2})}-\frac{m^{4}}{(p_{1}\cdot p_{4})}-m^{2}+O\big{(}(p_{1}\cdot
p_{2}),(p_{3}\cdot p_{4})\big{)}.$ (142)

Using the definition of the Mandelsatm variables (54) and the definition of
the tilded amplitudes (86) we obtain the final result

$\widetilde{T}_{AB\rightarrow
AB}(s,t,u)=\frac{-2m^{4}}{s-m^{2}}+\frac{-2m^{4}}{u-m^{2}}-m^{2}+\
O(u-m^{2},s-m^{2}).$ (143)

The above result is independent of $\Delta$, which is not surprising since
$\Delta$ appears in the definition of $\widehat{\Phi}(x)$ which is nothing but
a field redefinition of $\Phi(x)$. Here we want to emphasize that up to the
subleading order in the expansion parameters $s-m^{2}$ and $u-m^{2}$, our
amplitude $\widetilde{T}_{AB\rightarrow AB}(s,t,u)$ is theory independent. We
perturbatively verify the pole parts of the above result for a particular
model in appendix E. In addition, in appendix F we argue for this universal
soft behavior by analysing the worldline action of a massive particle in a
dilaton background.

At the next order in $O(u-m^{2},s-m^{2})$ the contribution to (143) will
depend on $\Xi^{\prime}(m^{2}),\Xi^{\prime\prime}(m^{2})$ and the non-minimal
interaction strength of the scalar field with the dilaton e.g. flat space
interaction follows from
$\widehat{R}^{\mu\nu}\partial_{\mu}\widehat{\Phi}\partial_{\nu}\widehat{\Phi}$.121212The
theory dependence of sub-subleading soft graviton theorem along the same line
of derivation has been worked out in Laddha:2017ygw . Our derivation is also
motivated from the literatures DiVecchia:2015jaq ; DiVecchia:2017uqn ;
Guerrieri:2017ujb , where tree level single and double soft dilaton theorems
have been studied.

##  5 Numerical bounds

In order to put numerical bounds on the $a$-anomaly we use the numerical
approach of Paulos:2017fhb ; Homrich:2019cbt . For the concise summary of this
approach see section 1 and 4.1 of Hebbar:2020ukp . In what follows we first
explain very briefly our numerical setup and then present our numerical
bounds. All the technical details of the numerical setup can be found in
appendix G.

###  5.1 Numerical setup

We start by introducing two types of the rho-variables which automatically
take care of the branch cuts discussed in (88) and (89). They read

$\displaystyle{\mathfrak{r}}_{1}(z;z_{0})$
$\displaystyle\equiv\frac{\sqrt{4m^{2}-z_{0}}-\sqrt{4m^{2}-z}}{\sqrt{4m^{2}-z_{0}}+\sqrt{4m^{2}-z}},$
(144) $\displaystyle{\mathfrak{r}}_{2}(z;z_{0})$
$\displaystyle\equiv\frac{\sqrt{9m^{2}-z_{0}}-\sqrt{9m^{2}-z}}{\sqrt{9m^{2}-z_{0}}+\sqrt{9m^{2}-z}}.$

Here $z_{0}$ is a free parameter and can be set to any convenient value for
both variables ${\mathfrak{r}}_{1}$ and ${\mathfrak{r}}_{2}$ independently.
Using these variables we can write the following ansatze

$\displaystyle\mathcal{T}_{AA\rightarrow AA}(s,t,u)$
$\displaystyle=\sum_{a=0}^{\infty}\sum_{b=0}^{\infty}\sum_{c=0}^{\infty}\alpha_{abc}\left({\mathfrak{r}}_{1}(s;4m^{2}/3)\right)^{a}\left({\mathfrak{r}}_{1}(t;4m^{2}/3)\right)^{b}\left({\mathfrak{r}}_{1}(u;4m^{2}/3)\right)^{c},$
(145) $\displaystyle\widetilde{\mathcal{T}}_{AB\rightarrow AB}(s,t,u)$
$\displaystyle=-\frac{2m^{4}}{s-m^{2}}-\frac{2m^{4}}{u-m^{2}}$
$\displaystyle+m^{2}\sum_{a=0}^{\infty}\sum_{b=0}^{\infty}\sum_{c=0}^{\infty}\beta_{abc}\left({\mathfrak{r}}_{2}(s;2m^{2}/3)\right)^{a}\left({\mathfrak{r}}_{1}(t;2m^{2}/3)\right)^{b}\left({\mathfrak{r}}_{2}(u;2m^{2}/3)\right)^{c},$
$\displaystyle\widetilde{\mathcal{T}}_{BB\rightarrow BB}(s,t,u)$
$\displaystyle=m^{4}\sum_{a=0}^{\infty}\sum_{b=0}^{\infty}\sum_{c=0}^{\infty}\gamma_{abc}\left({\mathfrak{r}}_{1}(s;0)\right)^{a}\left({\mathfrak{r}}_{1}(t;0)\right)^{b}\left({\mathfrak{r}}_{1}(u;0)\right)^{c}.$

Here $\alpha$, $\beta$ and $\gamma$ are the unknown real coefficients. Due to
the crossing symmetry discussed in section 3.2 the coefficients $\alpha$ and
$\gamma$ are fully symmetric in their indices. Instead the coefficients
$\beta$ obey the following condition

$\beta_{abc}=\beta_{cba}.$ (146)

The ansatz for the amplitude $\widetilde{\mathcal{T}}_{AA\rightarrow
BB}(s,t,u)$ is obtained from (145) by exchanging $s\leftrightarrow t$
according to the crossing equation (64). In order to simplify the ansatze and
take into account that the variable $u$ is not independent we further demand
that our coefficients obey

$(abc)\neq 0:\qquad\alpha_{abc}=\beta_{abc}=\gamma_{abc}=0.$ (147)

We will be working in the units of the physical mass $m$. Effectively this
allows us to set

$m=1$ (148)

in all the equations below.

Using the ansatze (145) one can define the following optimization problem:
numerically find the set of coefficients $\alpha$, $\beta$ and $\gamma$ which
maximizes some linear combination of these coefficients such that the
unitarity conditions (95) and (101) are satisfied for all physical values of
$s$ and spins $\ell$. Below we will explain how the coefficients $\alpha$,
$\beta$ and $\gamma$ are related to the $a$-anomaly and to the physical
observables.

In order to make the above optimization problem feasible in practice we first
truncate the ansatz (145) to a finite sum with the parameter $N_{max}$
according to

$a+b+c\leq N_{max}.$ (149)

We impose unitarity conditions for a finite number of spins $\ell$ up to some
fixed value $L_{max}$, namely

$\ell=0,1,2,\ldots,L_{max}.$ (150)

Finally, we pick a finite grid of $s$ values where the unitarity conditions
are imposed. The number of points in this grid is denoted by
$N_{\text{grid}}$. After performing these “truncations” we use SDPB software
Simmons-Duffin:2015qma ; Landry:2019qug  to solve the optimization problem
numerically.

##### Soft conditions

At low energy the $AA\rightarrow BB$ and $BB\rightarrow BB$ amplitudes have a
very particular (soft) behavior according to (93) and (87) respectively. As a
result we have additional constraints on the coefficients of the ansatze.
Plugging here the definition of $(t,u)$ variables in terms of $(s,\cos\theta)$
according to (77) and expanding around small values of $s$ keeping
$\cos\theta$ fixed we get expressions which should match (93) and (87).
Requiring this matching one obtains the following constraints

$\beta_{000}=-1-(98-40\sqrt{6})\beta_{001}+\ldots,\quad\gamma_{000}=0,\quad\gamma_{001}=\frac{1}{4}\left(512a-2\gamma_{002}+\gamma_{011}\right),$
(151)

where $a$ is the $a$-anomaly. Plugging the solutions (151) into the anstatze
(145) we effectively eliminate three coefficients $\beta_{000}$,
$\gamma_{000}$, $\gamma_{001}$ and introduce one additional coefficient $a$
(the $a$-anomaly).

##### Physical observables

As explained in the introduction we will use either the pair of observables
$(\lambda_{0},\lambda_{2})$ or $(\Lambda_{0},\Lambda_{2})$ in order to
describe the scattering amplitude $AA\rightarrow AA$.131313In the definitions
(3) and (4) one could take the derivative in $s$ is two different ways. First,
one could plug the value of $u=4m^{2}-s-t$ and then treating $s$ and $t$ as
independent variables evaluate the derivative in $s$ at the $s_{0}$ and
$t_{0}$ point. Second, one could replace both $t$ and $u$ variables in terms
of $s$ and $\cos\theta$, where $\theta$ is the scattering angle and only then
take the derivative in $s$ at the $s_{0}$ and $x_{0}$ point. In this paper we
will use the second option. At the crossing symmetric point
$\lambda_{2}^{\text{former}}=4/3\ \lambda_{2}^{\text{latter}}$. At the forward
point $\Lambda_{2}^{\text{former}}=\Lambda_{2}^{\text{latter}}$.  Applying
either the definition (3) or (4) to the ansatz (145) for the amplitude
$AA\rightarrow AA$ we obtain a linear relation between
$(\lambda_{0},\lambda_{2})$ or $(\Lambda_{0},\Lambda_{2})$ and the
coefficients $\vec{\alpha}$. We solve these relations for

$\alpha_{000}\quad\text{and}\quad\alpha_{001}$ (152)

in terms of either $(\lambda_{0},\lambda_{2})$ or $(\Lambda_{0},\Lambda_{2})$
and the rest of $\vec{\alpha}$. We substitute this solution in the ansatz.
This effectively removes two coefficients $\alpha_{000}$ and $\alpha_{001}$ in
the ansatz and instead introduce the depends either on
$(\lambda_{0},\lambda_{2})$ or on $(\Lambda_{0},\Lambda_{2})$.

##### Extension of the ansatz

The ansatz (145) is very powerful. Nevertheless since in practice we work at
finite value of $N_{max}$ the bounds depend on $N_{max}$. And in order to get
accurate bounds one needs to perform an extrapolation
$N_{max}\rightarrow\infty$. The convergence with $N_{max}$ depends on a
particular optimization problem.

It is convenient to enlarge the ansatz for the $AA\rightarrow AA$ by adding
the following term

$\mathcal{T}_{AA\rightarrow
AA}(s,t,u)\supset\alpha^{\prime}_{000}\times\left(\frac{1}{{\mathfrak{r}}_{1}(s;4/3)-1}+\frac{1}{{\mathfrak{r}}_{1}(t;4/3)-1}+\frac{1}{{\mathfrak{r}}_{1}(u;4/3)-1}\right),$
(153)

which contains a new parameter $\alpha^{\prime}_{000}$. When maximizing
$\lambda_{0}$ this term gives a major contribution and improves the
convergence with $N_{max}$ drastically Paulos:2017fhb . We also enlarge the
ansatz for $BB\rightarrow BB$ amplitude by adding the following term

$\widetilde{\mathcal{T}}_{BB\rightarrow BB}(s,t,u)\supset
c\,\widetilde{\mathcal{T}}^{\text{free}}_{BB\rightarrow BB}(s,t,u),$ (154)

where $c$ is a free parameter and
$\widetilde{\mathcal{T}}^{\text{free}}_{BB\rightarrow BB}$ is the dilaton
scattering amplitude in the free scalar theory. It is hard to compute
$\widetilde{\mathcal{T}}^{\text{free}}_{BB\rightarrow BB}$ exactly. Luckily we
do not need its explicit expression, in practice we will only need its
imaginary part and the imaginary part of the associated partial amplitude,
since only the latter enter in the unitarity constraint (101). These will be
computed in equations (199), (272) and (273).

###  5.2 Results

We start in section 5.2.1 by addressing the simplest possible question: what
is the lowest value of the $a$-anomaly in the UV CFT which leads to a single
$\mathbb{Z}_{2}$ odd asymptotic state given some relevant deformation. We will
reconstruct the spin=0 partial amplitudes of our setup which lead to the
absolute minimum of the $a$-anomaly. In section 5.2.2 we discuss several
consistency checks of our numerical code. In section 5.2.3 we construct a
lower bound on the $a$-anomaly as a function of $\lambda_{0}$, $\lambda_{2}$,
$\Lambda_{0}$ and $\Lambda_{2}$. We reconstruct the spin=0 partial amplitudes
of our setup corresponding to the maximally allowed value of $\lambda_{0}$.

####  5.2.1 Absolute minimum of the $a$-anomaly

Let us start by addressing the following question: what is the lowest value of
the $a$-anomaly in the UV CFT which leads to a single $\mathbb{Z}_{2}$ odd
asymptotic state given some relevant deformation. For running the numerics we
have found the optimal size of the grid to be $N_{\text{grid}}=300$. We have
checked that $N_{\text{grid}}=350$ and $N_{\text{grid}}=400$ lead to the same
solution. For $N_{\text{grid}}=200$ the results differ significantly from the
ones found with larger grids.

Let us fix the size of the ansatz to be $N_{max}=20$. The minimum of the
$a$-anomaly for various values of $L_{max}$ is found to be

$\\{L_{max},a/a_{\text{free}}\\}=\\{(16,0.4015),(18,0.4074),(20,0.4100),(22,0.4115),(24,0.4125),\\\
(26,0.4133),(28,0.4140),(30,0.4146),(32,0.4150),(34,0.4154)\\}.$ (155)

We see the the lower bound on the $a$-anomaly is stable under the change of
$L_{max}$ and gets stronger when $L_{max}$ increases. In all the future
numerical studies we make the following conservative choice for

$L_{max}=N_{max}+10.$ (156)

Let us now investigate the dependence of the numerical solution on $N_{max}$.
We obtain the following values for the minimum of the $a$-anomaly for various
values of $N_{max}$

$\\{N_{max},a/a_{\text{free}}\\}=\\{(16,0.4401),(18,0.4223),(20,0.4146),\\\
(22,0.4058),(24,0.4001),(26,0.3897)\\}.$ (157)

Using $N_{max}=$ $16$, $18$, $20$, $22$, $24$ and $26$ we can extrapolate our
data to $N_{max}=\infty$ with the following linear function

$a/a_{\text{free}}=0.316+1.969/N_{max}.$ (158)

The dependence of the minimum of the $a$-anomaly on $N_{max}$ and its
extrapolation is given in figure 5. We conclude that the absolute minimum of
the $a$-anomaly in our setup is

$a/a_{\text{free}}\gtrsim 0.316\pm 0.015.$ (159)

Here we have also included the estimated extrapolation error.

At the absolute minimum of the $a$-anomaly we can actually reconstruct
numerically scattering and partial amplitudes of all the process of our setup.
In figures 8 \- 8 we present the spin zero partial amplitudes of the
$AA\rightarrow AA$ and $AA\rightarrow BB$ processes. In the left figure 8 we
have plotted the real part of the spin zero phase shift of the $AA\rightarrow
AA$ process. We recall that the spin $\ell$ phase shift $\delta^{\ell}$ of the
$AA\rightarrow AA$ process is defined via

$\mathcal{S}^{\ell}_{AA\rightarrow AA}(s)=e^{2i\delta^{\ell}_{AA\rightarrow
AA}(s)}.$ (160)

From the right figure 8 we see that the amplitude is fully “elastic” up to
very high energies.

In figure 10 we plot the integrand of the sum-rule (104) for various values of
$N_{max}$. Numerical integration of these functions gives the values
$a/a_{\text{free}}=0.4146$, $0.4058$, $0.4001$ and $0.3897$ which are in a
perfect agreement with (157). In figure 10 we plot the integrand of the sum-
rule (105) for spin 0, 2 and 4. Numerical integration of these functions gives
the following value of the $a$-anomaly
$a/a_{\text{free}}=0.2396+0.1605+0.0105+\ldots=0.4106+\ldots$, where the three
entries correspond to spin 0, 2 and 4 respectively and the dots indicate the
contribution due to higher spins..

Figure 5: Minimum possible value of the a-anomaly without any further
assumptions as a function of $1/N_{max}$ with $L_{max}=N_{max}+10$. The
numerical results are depicted by blue points. Linear extrapolation to
$N_{max}\rightarrow\infty$ depicted by the red line gives $0.316\pm 0.015$ for
the minimum of $a/a_{\text{free}}$.

Figure 6: Real and imaginary parts of the spin 0 interacting part of the
$AA\rightarrow AA$ partial amplitude leading to the absolute minimum of the
$a$-anomaly. It is constructed at $N_{max}=20$ and $L_{max}=30$.

Figure 7: An alternative representation of the amplitude given in figure 8.
Left plot represents the real part of the spin 0 phase shift of the
$AA\rightarrow AA$ scattering defined in (160). The apparent jump around $s=8$
is due to the periodicity $\delta\simeq\delta+\pi$. Right plot represents the
absolute value of the spin 0 partial amplitude of the $AA\rightarrow AA$
scattering. On the real axis instead of the $s$ variable we use the $\phi$
variable defined in (266). The amplitude is fully “elastic” up to very high
energies.

Figure 8: Real and imaginary parts of the spin 0 interacting part of the
$AA\rightarrow BB$ partial amplitude leading to the absolute minimum of the
$a$-anomaly. It is constructed at $N_{max}=20$ and $L_{max}=30$.

Figure 9: Integrand appearing in the sum-rule (104) for the absolute minimum
of the $a$-anomaly. Different colors indicate different values of $N_{max}$.
Numerical integration of these function leads to $a/a_{\text{free}}=0.4146$,
$0.4058$, $0.4001$ and $0.3896$ in a perfect agreement with (157).

Figure 10: Integrand appearing in the sum-rule (105) for the absolute minimum
of the $a$-anomaly for spin 0, 2 and 4. Numerical integration of this function
leads to $a/a_{\text{free}}=0.2396+0.1605+0.0105+\ldots=0.4106+\ldots$, where
the three entries correspond to spin 0, 2 and 4 respectively and the dots
indicate the contribution due to higher spins. Plots are constructed at
$N_{max}=20$ and $L_{max}=30$. Summing up these three contribution gives the
$N_{max}=20$ curve of figure 10.

####  5.2.2 Consistency checks

Before presenting more bounds let us perform several consistency checks.

First, we can set all the coefficients $\alpha$ and $\beta$ in the ansatz
(145) to zero after imposing the first condition in (151). This situation
corresponds to particle $A$ being a free massive scalar, with the
$AA\rightarrow BB$ scattering amplitude given by (108). As explained below
(108) this inevitably leads to $a/a_{\text{free}}=1$. We successfully
reproduce this theoretical outcome numerically. It is important to notice that
the ansatz (145) requires very large numbers $N_{max}$ in order to reproduce
the free theory accurately. In practice it is very non-economical to work with
such a big value of $N_{max}$. This was the motivation behind the introduction
of the additional term (154) in the $BB\rightarrow BB$ ansatz which improves
the convergence with $N_{max}$ drastically in this particular situation.

Second, let us set all the $\alpha$ coefficients to zero and leave $\beta$
coefficients to be completely free. One might expect the same outcome as
before, however solving the optimization problem we obtain

$a/a_{\text{free}}\approx 0.9071.$ (161)

This result was obtained with $N_{max}=L_{max}=20$. For comparison we get
$0.9107$ for $N_{max}=L_{max}=10$. This result is very stable under the change
of $N_{max}$ and $L_{max}$. The solution of the optimization problem gives the
coefficients of the ansatz such that only

$\beta_{00n}\neq 0.$ (162)

Under closer inspection of the unitarity conditions one observes that setting
$\alpha$ to zero forces the imaginary part of the
$\widetilde{\mathcal{T}}_{AA\rightarrow BB}^{\ell}(s)$ partial amplitude to be
zero due to the last line in (102). However, the real part can still be non-
zero. The solution (162) exactly reproduces such a situation. We, thus,
conclude that the unitarity conditions we use do not fully constraint the
behavior of the matter - dilaton scattering given the form of the matter
scattering.

It is possible to guess analytically the $AA\rightarrow BB$ scattering
amplitude which gives the solution (161), (162). Let us assume no
discontinuity in the s-channel of the amplitude $AA\rightarrow BB$ (as follows
from absence of $AA\to AA$ scattering), we can then write the general ansatz

$\widetilde{\mathcal{T}}_{AA\rightarrow
BB}=-m^{2}-\frac{2m^{4}}{t-m^{2}}-\frac{2m^{4}}{u-m^{2}}-\int_{9m^{2}}^{\infty}dwq(w)\left(\frac{m^{2}}{t-w}+\frac{m^{2}}{u-w}-\frac{2m^{2}}{m^{2}-w}\right),$
(163)

where $q(w)\geq 0$ by unitarity. For the special case when
$q(w)=Qm^{2}\delta(w-9m^{2})$ we obtain

$\displaystyle\displaystyle\widetilde{\mathcal{T}}_{AA\rightarrow
BB}=-m^{2}-\frac{2m^{4}}{t-m^{2}}-\frac{2m^{4}}{u-m^{2}}-Qm^{2}\left(\frac{m^{2}}{t-9m^{2}}+\frac{m^{2}}{u-9m^{2}}+\frac{1}{4}\right).$
(164)

Plugging it into (107) we obtain $a^{\text{UV}}/a_{\text{free}}\geq 0.905527$
for $Q=2.07869$. The amplitude (164) matches precisely the one obtained
numerically.

Finally, analogously to (104) one can write the following dispersion relation

$\Lambda_{2}=\frac{m^{4}}{8\pi^{2}}\int_{4m^{2}}^{\infty}\frac{ds}{s^{3}}\
\text{Im}\mathcal{T}_{AA\rightarrow AA}(s,0,4m^{2}-s).$ (165)

Setting $\Lambda_{2}$ to zero will force the imaginary part of the
$AA\rightarrow AA$ process to be zero since the integrand is non-zero due to
the first inequality in the second line in (102). Due to the same inequality
we see that the imaginary part in turn forces the whole amplitude to be zero.
In practice we indeed observe that by setting $\Lambda_{2}$ to zero the
numerical solution leads to all the coefficients $\alpha$ being zero. In other
words our numerics leads to (161) if $\Lambda_{2}=0$.

####  5.2.3 Lower bound on the $a$-anomaly as a function of $\lambda_{0}$,
$\lambda_{2}$, $\Lambda_{0}$ and $\Lambda_{2}$

Let us now construct a lower bound on the $a$-anomaly as a function of the
coupling constants $\lambda_{0}$, $\lambda_{2}$, $\Lambda_{0}$ and
$\Lambda_{2}$. They were precisely defined in (3) and (4).

We begin by noticing that these couplings are bounded themselves. We can use
our setup of section 5.1 to obtain upper and lower bounds which read as

$\displaystyle-6.0253\leq$ $\displaystyle\lambda_{0}\leq+2.6613,\qquad
0\leq\lambda_{2}\leq+2.2568,$ (166) $\displaystyle-2.8145\leq$
$\displaystyle\Lambda_{0}\leq+2.8086,\qquad 0\leq\Lambda_{2}\leq+0.6550.$

These are obtained by setting $a=5a_{\text{free}}$ in the setup and using
$N_{max}=20$ and $L_{max}=30$. Increasing the value of $a$ does not change the
result. Around $a/a_{\text{free}}\sim 1$ we get a non-trivial dependence of
the bounds on $a$ which will be better represented in the later plots. The
dependence of these bounds on $L_{max}$ is negligible. The dependence on
$N_{max}$ is non-trivial and should be taken into account. The correct bounds
are obtained using the extrapolation to $N_{max}\rightarrow\infty$. The
exception is the upper bound on $\lambda_{0}$, it is independent of $N_{max}$
due to the presence of the singularity term (153) which drastically improves
the convergence of this particular bound. We do not perform
$N_{max}\rightarrow\infty$ extrapolations in this section and instead we will
always work at $N_{max}=20$ and $L_{max}=30$.

We can now pick value of $\lambda_{0}$, $\lambda_{2}$, $\Lambda_{0}$ and
$\Lambda_{2}$ from the allowed ranges (166) and minimize the $a$-anomaly. The
result is presented in figures 12 and 12. The allowed area is shaded in blue.
The red dot represents the solution with the absolute minimum of the
$a$-anomaly found in section 5.2.1. The right plots in figures 12 and 12 have
a sharp peak around $\lambda_{2}=0$ and $\Lambda_{2}=0$. This point
corresponds to a freely propagating particle $A$ and gives the value of the
$a$-anomaly quoted in (161). See the explanation below (161) why this value is
not exactly one.

Another interesting point in these plots is the one with $\lambda_{0}\approx
2.66$ and $a/a_{\text{free}}=1.2$. As was done in section 5.2.1 we can plot
spin zero partial amplitudes in our setup for this solution. They are given in
figures 15 \- 16.

Figure 11: Lower bound on the $a$-anomaly as a function of $\lambda_{0}$ in
the left plot and as a function of the $\lambda_{2}$ in the right plot. The
allowed region is depicted in blue. Red dot represents the point with the
lowest value of the $a$-anomaly. The red vertical lines indicate the
boundaries of the allowed regions for $\lambda_{0}$ and $\lambda_{2}$ given in
(166). The plots are built with $N_{max}=20$ and $L_{max}=30$.

Figure 12: Lower bound on the $a$-anomaly as a function of $\Lambda_{0}$ in
the left plot and as a function of the $\Lambda_{2}$ in the right plot. The
allowed region is depicted in blue. Red dot represents the point with the
lowest value of the $a$-anomaly. The red vertical lines indicate the
boundaries of the allowed regions for $\Lambda_{0}$ and $\Lambda_{2}$ given in
(166). The plots are built with $N_{max}=20$ and $L_{max}=30$.

Figure 13: Real and imaginary parts of the spin 0 interacting part of the
$AA\rightarrow AA$ partial amplitude for $\lambda_{0}=2.66$ which leads to
$a/a_{\text{free}}=1.2002$. It is constructed for $N_{max}=20$ and
$L_{max}=30$.

Figure 14: Left plot represents the real part of the spin 0 phase shift of the
$AA\rightarrow AA$ scattering defined in (160). No resonances are present.
Right plot represents the absolute value of the spin 0 partial amplitude of
the $AA\rightarrow AA$ scattering. On the real axis instead of $s$ variable we
use $\phi$ variable defined in (266). The amplitude is fully “elastic” up to
very high energies.

Figure 15: Real and imaginary parts of the spin 0 interacting part of the
$AA\rightarrow BB$ partial amplitude for $\lambda_{0}=2.66$ which leads to
$a/a_{\text{free}}=1.2002$. It is constructed for $N_{max}=20$ and
$L_{max}=30$.

Figure 16:  Left plot: integrand appearing in the sum-rule (104) for the
absolute minimum of the $a$-anomaly. Numerical integration of this function
leads to $a/a_{\text{free}}=1.2002$. Right plot: integrand appearing in the
sum-rule (105) for the absolute minimum of the $a$-anomaly. Numerical
integration of this function leads to
$a/a_{\text{free}}=0.71146+0.45094+0.03238+\ldots=1.19477+\ldots$, where the
three entries correspond to spin 0, 2 and 4 respectively and the dots indicate
higher spin contributions. The plots are constructed at $N_{max}=20$ and
$L_{max}=30$.

##  6 Discussion

We proposed a framework to study gapped QFTs defined as relevant deformations
of an UV CFT. For simplicity we assumed the existence of a $\mathbb{Z}_{2}$
symmetry and a single stable $\mathbb{Z}_{2}$-odd particle at low energies
denoted by $A$. By studying the unitarity constraints on the system of
scattering amplitudes between the particle $A$ and the dilaton particle $B$,
as depicted in figure 1, we related the physical scattering data for the
particle $A$ to the $a$-anomaly of the UV CFT. This allowed us to derive lower
bounds on the $a$-anomaly as a function of several parameters defined in (3)
and (4) in terms of the amplitude $\mathcal{T}_{AA\to AA}$ (see figures 2 and
3).

In practice, we used the primal numerical S-matrix bootstrap method for
multiple amplitudes.141414For an example of the primal numerical setup with
multiple amplitudes in 2d see Homrich:2019cbt . This method requires
extrapolation in the number of parameters of the ansatz ($N_{max}\to\infty$)
and the number of unitarity constraints ($L_{max}\to\infty$). As usual in
S-matrix bootstrap problems with a mass gap, the extrapolation
$L_{max}\to\infty$ is easy. On the other hand, the extrapolation
$N_{max}\to\infty$ is more subtle and deserves further study. In this context,
it would be extremely useful to have a dual S-matrix bootstrap method to
approach the minimal $a$-anomaly from below. It should be possible to
generalize the methods of Lopez:1976zs ; He:2021eqn ; Guerrieri:2021tak  to
our setup.

Our main result is the derivation of a universal lower bound on the
$a$-anomaly. We estimated the bound to be $a/a_{\text{free}}>0.3$. Combining
this result with the conformal collider bound
$\frac{31}{18}\geq\frac{a}{c}\geq\frac{1}{3}$ from Hofman:2008ar , we obtain a
lower bound $c>0.17\ a_{\text{free}}$.

It would be interesting to generalize our analysis by dropping some of the
simplifying assumptions about the presence of the $\mathbb{Z}_{2}$ symmetry
and the low energy spectrum of particles. For example, we can start by
allowing more stable particles either even or odd under $\mathbb{Z}_{2}$. More
stable particles lead to extra poles in the ansatz (145) and change the branch
point in ${\mathfrak{r}}_{2}$ associated with the beginning of the continuum
in the $\mathbb{Z}_{2}$ even sector (in practice $9m^{2}\to(m+m^{\prime})^{2}$
where $m^{\prime}$ is the mass of the lightest $\mathbb{Z}_{2}$ even
particle). Dropping the assumption of $\mathbb{Z}_{2}$ symmetry leads to
similar effects: now particle $A$ gives rise to a pole in every channel of
every amplitude and all cuts start at $4m^{2}$. More ambitiously, we could
include in our setup also the processes $\mathcal{T}_{AA\to AB}$ and
$\mathcal{T}_{AB\to BB}$. This would give rise to a stronger set of unitarity
constraints. Any of these more general scenarios must give rise to a weaker
lower bound than $a/a_{\text{free}}>0.3$.151515It may be possible to derive a
stronger lower bound for QFTs with more than one stable particle by
considering $2\to 2$ scattering amplitudes with other external particles
besides the dilaton $B$ and the lightest particle $A$. This is similar to the
discussion below about promoting particle $A$ to a vector multiplet of $O(3)$.
This is intriguing because we do not know of any 4d QFT with
$a<a_{\text{free}}$. On the one hand, one may interpret our results as
pointing to the existence of an unknown QFT with $a<a_{\text{free}}$. On the
other hand, it is also possible that our setup is just not constraining enough
to approach the true minimal value of $a$ in the space of QFTs.

Another interesting generalization would be to study QFTs with a continuous
global symmetry. In practice, this is achieved by promoting particle $A$ to an
irreducible representation of the global symmetry group. For example, we could
study pions as a triplet of $O(3)$. Building on Guerrieri:2018uew , we could
impose the known value of the $a$-anomaly in QCD and study several observables
like scattering lengths, chiral zeros and resonances. Alternatively, we could
inject some experimental data about $\pi-\pi$ scattering and then minimize the
$a$-anomaly.

It would also be interesting to apply our approach to QFTs with fermionic
asymptotic states. For example, we can ask the question: what is the lowest
$a_{UV}$ that can give rise to massive fermions in the IR?161616We thank H.
Osborn for suggesting this question. In practice, this would require a
combination of the techniques of Hebbar:2020ukp  with the present paper.

Our setup can easily accommodate a massless physical particle described by
some Effective Field Theory (EFT) at low energies. This would have several
interesting applications. The simplest one is for particle $A$ to be a
Goldstone boson of spontaneous symmetry breaking. For example, particle $A$
could describe massless pions like in Guerrieri:2020bto . More challenging,
would be to take particle $A$ to be a photon. This would allows to ask the
question: what is the minimum $a$-anomaly of the UV CFT that can give rise to
a photon in the IR? Our method creates a non-perturbative bridge between low-
energy EFT and their UV completions (within the realm of QFT). Unfortunately,
such applications will be very challenging numerically for the current
methods. The main challenge is the extrapolation $L_{max}\to\infty$ in gapless
theories (see for instance Guerrieri:2021ivu ).

In this paper, the dilaton particle $B$ was an external probe. However, one
can also study physical dilatons, which are the Goldstone bosons from
spontaneous conformal symmetry breaking. In this case, the dilaton decay
constant $f$ would be finite and generically of order $m$. For this reason,
dilatons can now appear as internal massless particles in the amplitudes. Such
non-perturbative setup would be more challenging numerically.

So far, we restricted ourselves to 4 spacetime dimensions. However, our setup
can be easily adapted to 6d. The main difference with 4d is the low energy
behavior of the dilaton scattering amplitude which reads in 6d as
Elvang:2012st

$\widetilde{\mathcal{T}}_{BB\rightarrow
BB}(s,t,u)=\frac{b}{8}\times(s^{2}+t^{2}+u^{2})+\frac{9}{8}(a^{\text{UV}}-a^{\text{IR}})\times
stu+O(s^{4}).$ (167)

Here $b$ is some parameter of the theory. It is an outstanding open question
to prove the $a$-theorem in 6d. The paper Kundu:2019zsl  made an interesting
proposal: consider the RG flow on a fixed Anti-de Sitter background and
formulate the $a$-theorem as a conformal bootstrap problem for its boundary
correlation functions. Unfortunately, this approach has not yet succeeded. In
flat space, there seems to be no positive sum rule for
$(a^{\text{UV}}-a^{\text{IR}})$ in 6d Elvang:2012st . Nevertheless, we can try
to run our numerical approach and minimize $a^{\text{UV}}$ in QFTs with a mass
gap. We leave this exploration for the future.

## Acknowledgements

The authors thank Hongbin Chen, Liam Fitzpatrick, Andrea Guerrieri, Brian
Henning, Riccardo Rattazzi, Sébastien Reymond, Matt Walters and Pedro Vieira
for useful discussions. We also thank Zohar Komargodski for comments on the
draft.

The work of DK is supported by the SNSF Ambizione grant PZ00P2_193411. The
work of JM, BS and JP is supported by the Simons Foundation grant 488649
(Simons Collaboration on the Nonperturbative Bootstrap) and by the Swiss
National Science Foundation through the project 200020_197160 and through the
National Centre of Competence in Research SwissMAP.

##  Appendix A Correlation functions of the stress tensor

In this appendix we summarize results for the two- and three-point correlation
functions of the stress-tensor in four spacetime dimension derived in a
seminal paper Osborn:1993cr  by Osborn and Petkou. Let us start by defining
the short-hand notation

$x_{ij}^{\mu}\equiv x_{i}^{\mu}-x_{j}^{\nu}.$ (168)

All the tensor structures for the stress-tensor correlators are built out of
these elementary tensors

$\displaystyle I^{\mu\nu}(x)$
$\displaystyle\equiv\delta^{\mu\nu}-2\,\frac{x^{\mu}x^{\nu}}{x^{2}},$ (169)
$\displaystyle\mathcal{I}^{\mu\nu,\rho\sigma}(x)$
$\displaystyle\equiv\frac{1}{2}I^{\mu\rho}(x)I^{\nu\sigma}(x)+\frac{1}{2}I^{\mu\sigma}(x)I^{\nu\rho}(x)-\frac{1}{4}\eta^{\mu\nu}\eta^{\rho\sigma},$
$\displaystyle X_{3,12}^{\mu}$
$\displaystyle\equiv\frac{x_{13}^{\mu}}{x_{13}^{2}}-\frac{x_{23}^{\mu}}{x_{23}^{2}}.$

According to Osborn:1993cr  we have

$\displaystyle\langle T^{\mu\nu}(x_{1})T^{\rho\sigma}(x_{2})\rangle$
$\displaystyle=C_{T}\,\frac{1}{x_{12}^{8}}\
\mathbf{T}^{\mu\nu;\rho\sigma}_{0},$ (170) $\displaystyle\langle
T^{\mu\nu}(x_{1})T^{\rho\sigma}(x_{2})T^{\alpha\beta}(x_{3})\rangle$
$\displaystyle=\frac{1}{x_{12}^{4}x_{23}^{4}x_{31}^{4}}\Big{[}\mathbb{A}\
\mathbf{T}^{\mu\nu;\rho\sigma;\alpha\beta}_{1}+\mathbb{B}\
\mathbf{T}^{\mu\nu;\rho\sigma;\alpha\beta}_{2}+\mathbb{C}\
\mathbf{T}^{\mu\nu;\rho\sigma;\alpha\beta}_{3}\Big{]}.$ (171)

Here $C_{T}$ is the central charge and $\mathbb{A}$, $\mathbb{B}$ and
$\mathbb{C}$ are the stress-tensor OPE coefficients with itself. These obey
the following relation

$\displaystyle\displaystyle C_{T}\ =\
\frac{\pi^{2}}{3}\Big{(}14\mathbb{A}-2\mathbb{B}-5\mathbb{C}\Big{)}.$ (172)

The four tensor structures written above are defined as

$\displaystyle\mathbf{T}^{\mu\nu;\rho\sigma}_{0}$
$\displaystyle\equiv\mathcal{I}_{\mu\nu,\rho\sigma}(x_{12}),$ (173)
$\displaystyle\mathbf{T}^{\mu\nu;\rho\sigma;\alpha\beta}_{1}$
$\displaystyle\equiv\mathcal{I}^{\mu\nu,\mu^{\prime}\nu^{\prime}}(x_{13})\mathcal{I}^{\rho\sigma,\rho^{\prime}\sigma^{\prime}}(x_{23})t^{1}_{\mu^{\prime}\nu^{\prime},\rho^{\prime}\sigma^{\prime}}\
{}^{\alpha\beta}(X_{3,12}),$
$\displaystyle\mathbf{T}^{\mu\nu;\rho\sigma;\alpha\beta}_{2}$
$\displaystyle\equiv\mathcal{I}^{\mu\nu,\mu^{\prime}\nu^{\prime}}(x_{13})\mathcal{I}^{\rho\sigma,\rho^{\prime}\sigma^{\prime}}(x_{23})t^{2}_{\mu^{\prime}\nu^{\prime},\rho^{\prime}\sigma^{\prime}}\
{}^{\alpha\beta}(X_{3,12}),$
$\displaystyle\mathbf{T}^{\mu\nu;\rho\sigma;\alpha\beta}_{3}$
$\displaystyle\equiv\mathcal{I}^{\mu\nu,\mu^{\prime}\nu^{\prime}}(x_{13})\mathcal{I}^{\rho\sigma,\rho^{\prime}\sigma^{\prime}}(x_{23})t^{3}_{\mu^{\prime}\nu^{\prime},\rho^{\prime}\sigma^{\prime}}\
{}^{\alpha\beta}(X_{3,12}),$

where

$\displaystyle\displaystyle t^{1}_{\mu\nu,\rho\sigma,\alpha\beta}(X)$
$\displaystyle\equiv$ $\displaystyle
h^{5}_{\mu\nu,\rho\sigma,\alpha\beta}(X)-2h^{4}_{\mu\nu,\rho\sigma,\alpha\beta}(X)-2h^{4}_{\rho\sigma,\mu\nu,\alpha\beta}(X)+24h^{2}_{\mu\nu,\rho\sigma}(X)h^{1}_{\alpha\beta}(X)$
$\displaystyle-16h^{1}_{\rho\sigma}(X)h^{2}_{\mu\nu,\alpha\beta}(X)-16h^{1}_{\mu\nu}(X)h^{2}_{\rho\sigma,\alpha\beta}(X)+64h^{1}_{\mu\nu}(X)h^{1}_{\rho\sigma}(X)h^{1}_{\alpha\beta}(X),$
$\displaystyle t^{2}_{\mu\nu,\rho\sigma,\alpha\beta}(X)$ $\displaystyle\equiv$
$\displaystyle
h^{4}_{\alpha\beta,\mu\nu,\rho\sigma}(X)-h^{4}_{\mu\nu,\rho\sigma,\alpha\beta}(X)-h^{4}_{\rho\sigma,\mu\nu,\alpha\beta}(X)+6h^{2}_{\mu\nu,\rho\sigma}(X)h^{1}_{\alpha\beta}(X)$
$\displaystyle-2h^{1}_{\rho\sigma}(X)h^{2}_{\mu\nu,\alpha\beta}(X)-2h^{1}_{\mu\nu}(X)h^{2}_{\rho\sigma,\alpha\beta}(X)+32h^{1}_{\mu\nu}(X)h^{1}_{\rho\sigma}(X)h^{1}_{\alpha\beta}(X),$
$\displaystyle t^{3}_{\mu\nu,\rho\sigma,\alpha\beta}(X)$ $\displaystyle\equiv$
$\displaystyle
h^{3}_{\mu\nu,\rho\sigma}(X)h^{1}_{\alpha\beta}(X)+h^{1}_{\mu\nu}(X)h^{3}_{\rho\sigma,\alpha\beta}(X)+h^{1}_{\rho\sigma}(X)h^{3}_{\mu\nu,\alpha\beta}(X)$
(174)
$\displaystyle-6h^{2}_{\mu\nu,\rho\sigma}(X)h^{1}_{\alpha\beta}(X)+4h^{1}_{\rho\sigma}(X)h^{2}_{\mu\nu,\alpha\beta}(X)+4h^{1}_{\mu\nu}(X)h^{2}_{\rho\sigma,\alpha\beta}(X)$
$\displaystyle-16h^{1}_{\mu\nu}(X)h^{1}_{\rho\sigma}(X)h^{1}_{\alpha\beta}(X).$

and

$\displaystyle\displaystyle h^{1}_{\mu\nu}(X)$ $\displaystyle\equiv$
$\displaystyle\frac{X_{\mu}X_{\nu}}{X^{2}}-\frac{1}{4}\eta_{\mu\nu},$
$\displaystyle h^{2}_{\mu\nu,\rho\sigma}(X)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{X^{2}}\Big{[}X_{\mu}X_{\rho}\eta_{\nu\sigma}+X_{\mu}X_{\sigma}\eta_{\nu\rho}+X_{\nu}X_{\rho}\eta_{\mu\sigma}+X_{\nu}X_{\sigma}\eta_{\mu\rho}-X_{\mu}X_{\nu}\eta_{\rho\sigma}$
$\displaystyle\
-X_{\rho}X_{\sigma}\eta_{\mu\nu}\Big{]}+\frac{1}{4}\eta_{\mu\nu}\eta_{\rho\sigma},$
$\displaystyle h^{3}_{\mu\nu,\rho\sigma}(X)$ $\displaystyle\equiv$
$\displaystyle\
\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho}-\frac{1}{2}\eta_{\mu\nu}\eta_{\rho\sigma},$
$\displaystyle h^{4}_{\mu\nu,\rho\sigma,\alpha\beta}(X)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{X^{2}}\Big{[}h^{3}_{\mu\nu,\rho\alpha}X_{\sigma}X_{\beta}+h^{3}_{\mu\nu,\sigma\alpha}X_{\rho}X_{\beta}+h^{3}_{\mu\nu,\rho\beta}X_{\sigma}X_{\alpha}+h^{3}_{\mu\nu,\sigma\beta}X_{\rho}X_{\alpha}\Big{]}$
$\displaystyle-\frac{1}{2}\eta_{\rho\sigma}h^{2}_{\mu\nu,\alpha\beta}(X)-\frac{1}{2}\eta_{\alpha\beta}h^{2}_{\mu\nu,\rho\sigma}(X)-\frac{1}{2}\eta_{\rho\sigma}\eta_{\alpha\beta}h^{1}_{\mu\nu}(X),$
$\displaystyle h^{5}_{\mu\nu,\rho\sigma,\alpha\beta}(X)$ $\displaystyle\equiv$
$\displaystyle\
\eta_{\mu\rho}\eta_{\nu\alpha}\eta_{\sigma\beta}+\eta_{\nu\rho}\eta_{\mu\alpha}\eta_{\sigma\beta}+\eta_{\mu\sigma}\eta_{\nu\alpha}\eta_{\rho\beta}+\eta_{\nu\sigma}\eta_{\mu\alpha}\eta_{\rho\beta}+\eta_{\mu\rho}\eta_{\nu\beta}\eta_{\sigma\alpha}$
(175)
$\displaystyle+\eta_{\nu\rho}\eta_{\mu\beta}\eta_{\sigma\alpha}+\eta_{\mu\sigma}\eta_{\nu\beta}\eta_{\rho\alpha}+\eta_{\nu\sigma}\eta_{\mu\beta}\eta_{\rho\alpha}-\frac{1}{2}\eta_{\mu\nu}\eta_{\rho\sigma}\eta_{\alpha\beta}$
$\displaystyle-\eta_{\mu\nu}h^{3}_{\rho\sigma,\alpha\beta}(X)-\eta_{\rho\sigma}h^{3}_{\mu\nu,\alpha\beta}(X)-\eta_{\alpha\beta}h^{3}_{\mu\nu,\rho\sigma}(X).$

In curved spacetime one-point function of the trace trace of the stress-tensor
is non-zero. It has the following form

$\displaystyle\displaystyle\langle T^{\mu}_{\mu}\rangle_{g}\ $
$\displaystyle=$ $\displaystyle\ -a\times\ E_{4}\ +\times\ c\ W^{2},$ (176)

where $E_{4}$ is the Eular density and $W^{2}$ is the square of the Weyl
tensor. They have the following expressions in $4d$,

$\displaystyle\displaystyle E_{4}\ $ $\displaystyle=$ $\displaystyle\
R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}-4R^{\alpha\beta}R_{\alpha\beta}+R^{2},$
(177) $\displaystyle W^{2}\ $ $\displaystyle=$ $\displaystyle\
R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}-2R^{\alpha\beta}R_{\alpha\beta}+\frac{1}{3}R^{2}.$
(178)

The coefficients $a$ and $c$ are called the Weyl anomalies. They are related
to the OPE coefficients $\mathbb{A}$, $\mathbb{B}$ and $\mathbb{C}$ as

$\displaystyle\displaystyle a\ =\ \frac{\pi^{4}}{64\times
90}\Big{(}9\mathbb{A}-2\mathbb{B}-10\mathbb{C}\Big{)},\quad c\ =\
\frac{\pi^{4}}{64\times
30}\Big{(}14\mathbb{A}-2\mathbb{B}-5\mathbb{C}\Big{)}.$ (179)

As an example let us consider a theory of a free massless (conformally
coupled) scalar $\Phi(x)$ which is described by the free CFT. According to
CALLAN197042  it has the following stress-tensor

$\displaystyle\displaystyle
T_{\mu\nu}=\partial_{\mu}\Phi\partial_{\nu}\Phi-\frac{1}{12}\Big{[}2\partial_{\mu}\partial_{\nu}\Phi^{2}+\eta_{\mu\nu}\partial^{2}\Phi^{2}\Big{]}.$
(180)

Such a CFT has the following parameters

$\mathbb{A}=\frac{1}{27\pi^{6}},\quad\mathbb{B}=-\frac{4}{27\pi^{6}},\quad\mathbb{C}=-\frac{1}{27\pi^{6}},\quad
C_{T}=\frac{1}{3\pi^{4}},$ (181) $a=\frac{1}{5760\pi^{2}},\quad
c=\frac{1}{1920\pi^{2}}.$ (182)

##  Appendix B Example of the free scalar theory

In this appendix we consider the theory of a free massive scalar field
$\Phi(x)$ which has the following action

$A_{\text{free}}(m)=\ \int d^{4}x\
\Big{[}-\frac{1}{2}\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{1}{2}m^{2}\Phi^{2}\Big{]}.$
(183)

It can be interpreted as a free massless CFT in the UV deformed by the mass
term. As reviewed in sections 2.2 and 2.3 one can define the following
modified action

$A^{\prime}_{\text{free}}(m)=\int
d^{4}x\Big{[}-\frac{1}{2}\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{1}{2}m^{2}\Phi^{2}-\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi+\frac{m^{2}}{\sqrt{2}f}\
\varphi\Phi^{2}-\frac{m^{2}}{4f^{2}}\varphi^{2}\Phi^{2}\Big{]}.$ (184)

In what follows using this action we will compute the $BB\rightarrow BB$
scattering amplitude, where $B$ is the dilaton particle created by the dilaton
field $\varphi(x)$ from the vacuum. We will show that in this particular model
this amplitude at low energy is given by equation (47). The particle created
by the field $\Phi(x)$ from the vacuum is referred to as the particle $A$. We
will do the computation in two different ways.

All the Feynman rules needed for the computation of scattering amplitudes in
the model (184) read as

$\displaystyle\feynmandiagram[horizontal=atob,inline=(a.base)]{a--b};\ $
$\displaystyle=\frac{-i}{p^{2}+m^{2}-i\epsilon}$ (185)
$\displaystyle\feynmandiagram[horizontal=atob,inline=(a.base)]{a--[dashed]b};\
$ $\displaystyle=\frac{-i}{p^{2}-i\epsilon}$ (186)
$\displaystyle=\frac{i\sqrt{2}m^{2}}{f}$ (187)
$\displaystyle=-\frac{im^{2}}{f^{2}}$ (188)

Here solid lines represent the field $\Phi(x)$ and dashed lines represent the
dilaton field $\varphi(x)$.

##### Direct computation

The $BB\rightarrow BB$ dilaton scattering at the order $\mathcal{O}(f^{-4})$
is described by the Feynman diagram depicted in figure 17. We compute these
diagrams one by one using the standard Feynman parametrization. We will then
expand these expression at the leading order in energy and the perform the
Feynman integrals.

momentum/arrow distance=1mm momentum/label distance=-0.15em

Figure 17: We consider the momenta of incoming dilatons be $k_{1}$ and $k_{2}$
and the momenta of outgoing dilatons be $k^{\prime}_{1}$ and $k^{\prime}_{2}$.
In the total amplitude contribution we also need to add the contributions of
two topologically in-equivalent diagrams for the above individuals which we
can easily read off using crossing symmetry.

The amplitude described by the first diagram in figure 17 together with the
ones obtained from it by using crossing symmetry has the following form

$\displaystyle i\mathcal{A}_{I}(s,t)$ $\displaystyle=$ $\displaystyle\
\frac{m^{4}}{2f^{4}}\int\frac{d^{4}\ell}{(2\pi)^{4}}\frac{1}{\ell^{2}+m^{2}-i\epsilon}\
\frac{1}{(k_{1}+k_{2}-\ell)^{2}+m^{2}-i\epsilon}+\text{(cross-sym)}$
$\displaystyle=$ $\displaystyle-\frac{im^{4}}{32\pi^{2}f^{4}}\int_{0}^{1}dx\
\Bigg{[}\ln\Bigg{(}\frac{m^{2}-sx(1-x)}{\Lambda^{2}}\Bigg{)}+\ln\Bigg{(}\frac{m^{2}-tx(1-x)}{\Lambda^{2}}\Bigg{)}+\ln\Bigg{(}\frac{m^{2}-ux(1-x)}{\Lambda^{2}}\Bigg{)}\Bigg{]}.$

Here $\Lambda$ is the Pauli-Villars regularisation parameter representing the
$UV$ cut-off. Contribution from the second diagram in figure 17 together the
ones obtained from it by corssing symmetry has the form

$\displaystyle i\mathcal{A}_{II}(s,t)$ $\displaystyle=$ $\displaystyle\
-\frac{2m^{6}}{f^{4}}\int\frac{d^{4}\ell}{(2\pi)^{4}}\frac{1}{\ell^{2}+m^{2}-i\epsilon}\frac{1}{(\ell-k^{\prime}_{1})^{2}+m^{2}-i\epsilon}\frac{1}{(k_{1}+k_{2}-\ell)^{2}+m^{2}-i\epsilon}+\text{(cross-
sym)}$ $\displaystyle=$ $\displaystyle\
-\frac{im^{6}}{4\pi^{2}f^{4}}\int_{0}^{1}dx\ dy\ dz\
\delta(x+y+z-1)\Bigg{[}\frac{1}{m^{2}-sxy}+\frac{1}{m^{2}-txy}+\frac{1}{m^{2}-uxy}\Bigg{]}.$

Contribution from the third diagram in figure 17 together the ones obtained
from it by corssing symmetry has the form

$\displaystyle i\mathcal{A}_{III}(s,t)$ $\displaystyle=$
$\displaystyle+\frac{im^{8}}{4\pi^{2}f^{4}}\int_{0}^{1}dxdydzdw\
\delta(x+y+z+w-1)\
\Bigg{[}\frac{1}{\big{[}m^{2}-\\{sy(z+w)+tyz+uz(1-z-w)\\}\big{]}^{2}}$
$\displaystyle\
+\frac{1}{\big{[}m^{2}-\\{ty(z+w)+uyz+sz(1-z-w)\\}\big{]}^{2}}+\frac{1}{\big{[}m^{2}-\\{uy(z+w)+syz+tz(1-z-w)\\}\big{]}^{2}}\Bigg{]}.$