$\displaystyle=\int\frac{d^{2}z}{z_{0}^{2}}G_{\Delta_{E}}(w,z)K_{\Delta_{2}}(x_{2},z)K_{\Delta_{3}}(x_{3},z).$
(D.61)

The integral $I(z_{a},x_{a};x_{2},x_{3})$ is a boundary-boundary-to-bulk
three-point function and has conformal symmetry.

$a$$\Delta_{E}$$x_{2}$$x_{3}$

$\mathcal{L}_{a}a$$\Delta_{E}$$L_{3}x_{2}$$L_{4}x_{3}$

Figure D.3: The system is invariant under an infinitesimal transformation
(D.62).

$I(w,x_{2},x_{3})$ in (D.61) is invariant under global transformations
generated by $\mathbf{\mathcal{L}}_{a}+\vec{L}_{2}+\vec{L}_{3}$, where the
first term generates the isometries of $\text{AdS}_{2}$ and the other two
generate the conformal transformation of the boundary.222We write the
transformations under the conformal group as vectors
$\vec{L}_{2}=(L_{-1},L_{0},L_{1})$ acting on point 2. As such, we can write

$\displaystyle(\mathbf{\mathcal{L}}_{a}+\vec{L}_{2}+\vec{L}_{3})I(z_{a},x_{a};x_{2},x_{3})=0,$
(D.62)

and can therefore relate the Casimirs of the generators

$\displaystyle-\frac{1}{2}\mathbf{\mathcal{L}}_{a}^{2}I(z_{a},x_{a};x_{2},x_{3})$
$\displaystyle=-\frac{1}{2}(\vec{L}_{2}+\vec{L}_{3})^{2}I(z_{a},x_{a};x_{2},x_{3})$
(D.63) $\displaystyle=C^{(23)}_{2}I(z_{a},x_{a};x_{2},x_{3}).$ (D.64)

In one dimension, the quadratic Casimir of the $\text{AdS}_{2}$ isometries is
the Laplacian of the bulk D.2. This Laplacian will allow us to eliminate the
bulk-to-bulk propagator through the equation of motion (1.40). Linking the
previous elements together, we obtain

$\displaystyle(C_{(23)}^{(2)}-m^{2}_{E})I(z_{a},x_{a};x_{2},x_{3})$
$\displaystyle=(\nabla_{a}^{2}-m_{E}^{2})I(z_{a},x_{a};x_{2},x_{3})$ (D.65)
$\displaystyle=K_{\Delta_{\phi}}(z_{a},x_{a};x_{2})K_{\Delta_{\phi}}(z_{a},x_{a};x_{3}).$
(D.66)

The quadratic Casimir acting on points 3 and 4 commutes with the other
coordinates so that we can write a differential equation relating the full
exchange diagram to the contact term

$\displaystyle(C_{(23)}^{(2)}-m^{2}_{E})J(x_{1},x_{2},x_{2},x_{3})$
$\displaystyle=\int\frac{dz_{a}dx_{a}}{z_{a}^{2}}\Pi_{i=1}^{4}K_{\Delta_{\phi}}(z_{a},x_{a};x_{i})$
(D.67)
$\displaystyle=\frac{A}{(x_{13}x_{24})^{2\Delta_{\phi}}}\bar{D}_{\Delta_{\phi}\Delta_{\phi}\Delta_{\phi}\Delta_{\phi}}(\chi).$
(D.68)

The same analysis holds for any two legs attached to a bulk-to-bulk
propagator, though the final differential equation might depend on many
variables.

##### Polyakov Blocks and Exchange Witten Diagrams

The computation of the exchange Witten diagram in the main text (2.3.1) is the
sum of solutions to second-order differential equations. The three-point
function and the symmetry of the correlator provide the integration constants.
The example given in the main text, corresponding to the Polyakov block of
external dimension $\Delta=1$ and exchanged dimension $\Delta_{E}=1$, can be
computed with the toy model of a massless scalar theory in AdS2 with a
$\phi^{3}$ interaction. This corresponds to the action

$\displaystyle
S_{\phi^{3}}=\int\frac{dtdz}{z^{2}}\left(\partial_{\mu}\phi\partial^{\mu}\phi-\frac{\lambda}{3!}\phi^{3}\right).$
(D.69)

GFF gives the leading order contribution. The NLO ($O(\lambda)$) correlators
come from the constant vertex, giving the three-point function

$\displaystyle<\phi(x_{1})\phi(x_{2})\phi(x_{3})>$
$\displaystyle=\frac{\lambda}{\pi^{3}}\frac{\pi^{2}}{2x_{12}x_{23}x_{13}},$
(D.70)

and the OPE coefficient

$\displaystyle c_{111}$ $\displaystyle=\frac{\lambda}{2\pi}.$ (D.71)

The first sub-leading term ($O(\lambda^{2})$) in the four-point function is
generated by the exchange diagram, which contributes

$\displaystyle<\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})>$
$\displaystyle=\sum_{s,t,u}\lambda^{2}J_{\Delta_{E}=1,\Delta=1}(x_{1},...,x_{4})$
(D.72)
$\displaystyle=\frac{\lambda^{2}}{\pi^{5}}\frac{\pi}{4}\frac{1}{(x_{13}x_{24})^{2}}P^{(0)}_{(1,1)}(\chi),$
(D.73)

where $P^{(0)}_{(1,1)}(\chi)$ is written explicitly in equation (2.3.1) and
has a small $\chi$ expansion

$\displaystyle<\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})>$
$\displaystyle=\frac{1}{(x_{13}x_{24})^{2}}\frac{\lambda^{2}}{4\pi^{4}}\left(\frac{c_{1}-6\zeta(3)}{\chi^{2}}+\frac{2\left(\pi^{2}-3c_{2}\right)}{3\chi}\right)+O\left(\chi^{0}\right),$
(D.74)

where we have kept the integration constants from the differential equation.
The first term in the expansion is set to zero since it corresponds to a
correction to the weight of the identity operator. The second term
corresponds, in the conformal s-channel OPE, to

$\displaystyle\frac{1}{(x_{13}x_{24})^{2}}\sum_{h}c_{11h}^{2}\chi^{h-2}{}_{2}F_{1}(h,h,2h,\chi)$
$\displaystyle=\frac{1}{(x_{13}x_{24})^{2}}\left(\frac{c_{111}^{2}}{\chi}+O\left(\chi^{0}\right)\right),$
(D.75)

where by equating the expansions in the conformal and Polyakov blocks, this
corresponds to $c_{\Delta_{E}\Delta\Delta}$ in equation (A.86). Equating
equations D.75 and D.74 fixes the integration constants

$\displaystyle c_{1}=6\zeta(3)$ (D.76) $\displaystyle
c_{2}=-\frac{\pi^{2}}{6}.$ (D.77)

This provides the correct numerical factor for the Polyakov block computed in
2.3.1 and the maximally symmetric and Regge-bounded function.

##  Appendix E Mellin Transform

The Mellin transform is a topic in itself so the following Appendix starts
with an introduction in section E.1, before using the 1d Mellin transform
defined in the main text in subsection 5.2.1 to consider the case of GFF in
section E.2. Section E.3 then tests the sum rules presented in 5.2.4 and
addresses some of the issues linked to perturbations and poles. The final
section E.4 lists the expressions for the anomalous dimensions obtained in the
model described in 5.2.5.

###  E.1 3 Computations to Understand Mellin Formalism

The simple Mellin transform is the following transform, closely related to the
Laplace transform

$\displaystyle M[g](s)$ $\displaystyle=\int_{0}^{\infty}\frac{dz}{z}g(z)z^{s}$
(E.1) $\displaystyle=\int_{0}^{\infty}du\,g(e^{u})\,e^{su},$ (E.2)

where the anti-Mellin transform is done through a contour integral parallel to
the imaginary axis. As the Fourier and Laplace transform, the Mellin transform
reveals and highlights specific theory elements, making it sometimes simpler
to spot patterns and solve correlators. From its inception, it has revealed
itself to be very useful in hypergeometric integrals and their
generalisations. Given the omnipresence of hypergeometric functions in the
context of CFTs, one could expect there to be applications of the Mellin
transform to CFT. The founding work in [110] revealed the power of Mellin
transforms for dimension-independent formulations of conformal correlators.
Since then, it has been a major tool and a very active area of research whose
focus has been perturbative and non-perturbative computations. Further work
[109, 102, 83] developed the tools for efficient strong-coupling computations
in this setting, as well as motivation to consider the Mellin transform of
correlators as amplitudes by linking them to flat-space scattering amplitudes.
The literature over the past decade and more is quite extensive and detailed,
so the goal of this section is to be specific rather than extensive to give a
feel for the Mellin transform.111I refer the interested reader to [72, 109,
216] for a more exhaustive treatment of the subject.

In this line, I will present three key computations in the Mellin formalism,
these are:

  * •

The Mellin amplitude of contact Witten diagram

  * •

From Mellin to position space

  * •

Scattering amplitudes

From this, one can take away the key idea that Mellin formalism is useful
because the quantities’ simplicity underlines the theory’s important aspects.
At strong coupling, contact interactions in AdS give constant amplitudes, and
exchanges operators give poles, which allow for a ‘Feynman Rules’ approach.
The main obstacle to the development of this formalism is that going from
position to Mellin space and vice versa is relatively hard, even for the
simplest quantities. This motivates the quest to understand how to compute
quantities directly in Mellin space and relate them to physical observables,
such as flat-space scattering amplitudes.

#### Mellin Amplitudes of Contact Witten Diagrams

We will approach this from the point of view of boundary correlators of
operators propagating in a bulk theory on AdSd+1. We introduce a redundant
coordinate system called embedding lightcone coordinates
($X_{\mu}X^{\mu}=-R^{2}$)

$\displaystyle X^{\mu}$
$\displaystyle=R\left(\frac{1-x^{2}-z^{2}}{2z},\frac{x^{\mu}}{z},\frac{1+x^{2}+z^{2}}{2z}\right)$
$\displaystyle P^{\mu}=\lim_{z\rightarrow 0}zX^{\mu}$ (E.3)

which map AdSd+1 to flat space in d+2 dimensions. Note that $P^{\mu}$
parametrises the coordinates of the boundary CFT. These lead to easier
integrals and simpler propagators. In these coordinates, for example, the
bulk-to-boundary propagator is simply

$\displaystyle G_{\Delta}(X,P)=\frac{C_{\Delta}}{\left(-2P\cdot
X\right)^{\Delta}}$ (E.4)

The $n-$point contact integral in this context is

$\displaystyle
I_{\Delta_{1},...,\Delta_{n}}(P_{1}^{\mu},...,P_{m}^{\mu})=\int_{\text{AdS}_{d+1}}dX\prod_{i=1}^{n}\frac{C_{\Delta_{i}}}{\left(-2P\cdot
X\right)^{\Delta_{i}}}$ (E.5)

which we can write as

$\displaystyle
I_{\Delta_{1},...,\Delta_{n}}(P_{1}^{\mu},...,P_{m}^{\mu})=\prod_{i=1}^{n}\left(\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\right)\int_{0}^{\infty}[dt]\prod_{i}^{n}t_{i}^{\Delta_{i}-1}\int
dXe^{2Q\cdot X}$ (E.6)

where

$\displaystyle[dt]=\prod_{i=0}^{n}dt_{i}Q^{\mu}=\sum_{i=0}^{n}t_{i}P_{i}^{\mu}$
(E.7)

and we use the Schwinger parametrisation

$\displaystyle\frac{1}{A^{p_{i}}}=\frac{1}{\Gamma(p_{i})}\int_{0}^{\infty}dt_{i}t_{i}^{p_{i}-1}e^{-t_{i}A}.$
(E.8)

The boundary integral of AdSd+1 is now easy:

$\displaystyle\int dXe^{2Q\cdot
X}=\pi^{\frac{d}{2}}\int\frac{dz}{z}z^{-\frac{d}{2}}e^{-z+\tfrac{Q^{2}}{z}}$
(E.9)

where

$\displaystyle Q^{2}=(\sum_{i}P_{i}t_{i})^{2}=\sum_{i>j}-(-2P_{i}\cdot
P_{j})t_{i}t_{j}.$ (E.10)

The total integral is now

$\displaystyle
I_{\Delta_{1},...,\Delta_{n}}(P_{1}^{\mu},...,P_{m}^{\mu})=\pi^{\tfrac{d}{2}}\prod_{i=1}^{n}\left(\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\right)\int_{0}^{\infty}[dt]\prod_{i}^{n}t_{i}^{\Delta_{i}-1}\int\frac{dz}{z}z^{-\frac{d}{2}}e^{-z-\sum_{i>j}P_{ij}\tfrac{t_{i}t_{j}}{z}}$
(E.11)

where $P_{ij}=-2P_{i}\cdot P_{j}=|x_{i}-x_{j}|^{2}$. By rescaling the
variables $s_{i}=\frac{t_{i}}{\sqrt{z}}$, the $z-$integral is simply a gamma
function.

$\displaystyle I_{\Delta_{1},...,\Delta_{n}}(P_{1}^{\mu},...,P_{m}^{\mu})$
$\displaystyle=\pi^{\tfrac{d}{2}}\prod_{i=1}^{n}\left(\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\right)\int_{0}^{\infty}[ds]\prod_{i}^{n}s_{i}^{\Delta_{i}-1}e^{-\sum_{i>j}P_{ij}s_{i}s_{j}}\int\frac{dz}{z}z^{\frac{\sum\Delta_{i}-d}{2}}e^{-z}$
(E.12)
$\displaystyle=A\int_{0}^{\infty}[ds]\prod_{i}^{n}s_{i}^{\Delta_{i}-1}e^{-\sum_{i>j}P_{ij}s_{i}s_{j}}$
(E.13)

where

$\displaystyle
A=\pi^{\tfrac{d}{2}}\Gamma\left(\tfrac{\sum\Delta_{i}-d}{2}\right)\prod_{i=1}^{n}\left(\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\right)$
(E.14)

Since the conformal symmetry fixes the integral up to $n=3$, we separate the
cases $n=1,2,3$ from the others to make some symmetry explicit. This is also
where the Mellin transform comes into play; we introduce the Mellin transform
of the exponential function

$\displaystyle\exp^{-P_{ij}s_{i}s_{j}}=\int\frac{d\gamma_{ij}}{2\pi
i}\frac{\Gamma(\gamma_{ij})}{(P_{ij}s_{i}s_{j})^{\gamma_{ij}}}$ (E.15)

to write

$\displaystyle I$
$\displaystyle=A\int_{0}^{\infty}[ds]\prod_{i}^{n}s_{i}^{\Delta_{i}-1}e^{-s_{i}\sum_{j>1}P_{1j}s_{j}+s_{2}s_{3}P_{23}}e^{-\sum_{j>3,i\geq
2}P_{ij}s_{i}s_{j}}$ (E.16)
$\displaystyle=A\int_{0}^{\infty}\prod_{i=1}^{n}\left(\frac{ds_{i}}{s_{i}}s_{i}^{\Delta_{i}}\right)\exp^{-\sum_{j>1}s_{1}s_{j}P_{1j}-s_{2}s_{3}P_{23}}\prod_{i\geq
2,j>3}^{n}\int\frac{d\gamma_{ij}}{2\pi
i}\frac{\Gamma(\gamma_{ij})}{\left(P_{ij}s_{i}s_{j}\right)^{\gamma_{ij}}}$
(E.17)

Collecting the $s_{i}$-dependent factor for $j\geq 4$, we can integrate them

$\displaystyle\int_{0}^{\infty}\tfrac{ds_{i}}{s_{i}}s_{i}^{\Delta_{i}}\exp^{-s_{1}s_{i}P_{1i}}s_{i}^{-\sum_{j=2}^{i-1}\gamma_{j,i}-\sum_{j=i+1}^{n}\gamma_{i,j}}$
$\displaystyle=\int_{0}^{\infty}\tfrac{ds_{i}}{s_{i}}\exp^{-s_{1}s_{i}P_{1i}}s_{i}^{\gamma_{1i}}$
(E.18) $\displaystyle=\frac{\Gamma(\gamma_{1i})}{(s_{1}P_{1i})^{\gamma_{1i}}}$
(E.19)

where we have defined redundant variables through the following

$\displaystyle\gamma_{ii}=-\Delta_{i}$ (E.20)
$\displaystyle\gamma_{ij}=\gamma_{ji}$ (E.21)
$\displaystyle\sum_{j=1}^{n}\gamma_{ij}=0.$ (E.22)

Therefore, integrating $s_{j}$ ($j\geq 4$), we obtain

$\displaystyle I$
$\displaystyle=A\int\prod_{j>i>3}^{n}\frac{d\gamma_{ij}\Gamma(\gamma_{ij})}{\left(P_{ij}\right)^{\gamma_{ij}}}\prod_{j=4}^{n}\frac{\Gamma(\gamma_{1j})}{\left(P_{1j}\right)^{\gamma_{1j}}}\frac{\Gamma(\gamma_{2j})}{\left(P_{2j}\right)^{\gamma_{2j}}}\frac{\Gamma(\gamma_{3j})}{\left(P_{3j}\right)^{\gamma_{3j}}}I_{s}$
(E.23) $\displaystyle I_{s}$
$\displaystyle=\int\frac{ds_{1}ds_{2}ds_{3}}{s_{1}s_{2}s_{3}}s_{1}^{\gamma_{12}+\gamma_{13}}s_{2}^{\gamma_{12}+\gamma_{23}}s_{3}^{\gamma_{13}+\gamma_{23}}\exp^{-s_{1}s_{2}P_{12}-s_{1}s_{3}P_{13}-s_{2}s_{3}P_{23}}$
(E.24)
$\displaystyle=\int\frac{dt_{1}dt_{2}dt_{3}}{t_{1}t_{2}t_{3}}t_{1}^{\gamma_{23}}t_{2}^{\gamma_{13}}t_{3}^{\gamma_{12}}\exp^{-t_{3}P_{12}-t_{2}P_{13}-t_{1}P_{23}}$
(E.25)
$\displaystyle=\frac{\Gamma(\gamma_{12})}{\left(P_{12}\right)^{\gamma_{12}}}\frac{\Gamma(\gamma_{13})}{\left(P_{13}\right)^{\gamma_{13}}}\frac{\Gamma(\gamma_{23})}{\left(P_{23}\right)^{\gamma_{23}}}$
(E.26)

This gives us the final result

$\displaystyle
I=\pi^{\tfrac{d}{2}}\Gamma\left(\tfrac{\sum\Delta_{i}-d}{2}\right)\prod_{i=1}^{n}\left(\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\right)\int\prod_{j>i}^{n}\frac{[d\gamma_{ij}]}{2\pi
i}\frac{\Gamma(\gamma_{ij})}{\left(P_{ij}\right)^{\gamma_{ij}}}.$ (E.27)

Here, the integral is over the non-redundant variables $\gamma_{ij}$
satisfying the relations in E.20. If one defines a truncated Mellin transform
or Mellin amplitude where

$\displaystyle f(P_{ij})=\int\prod_{j>i}^{n}\frac{[d\gamma_{ij}]}{2\pi
i}\frac{\Gamma(\gamma_{ij})}{\left(P_{ij}\right)^{\gamma_{ij}}}M(\gamma_{ij})$
(E.28)

this is the claim that contact Witten diagrams give a constant Mellin
amplitude.

#### From Mellin to Position Space

As expected, doing the Mellin integral (or anti-Mellin transform) is trivial
for the cases $n=1,2,3$:

  * •

$n=1$

$\displaystyle
I_{n=1,\Delta}=\pi^{\tfrac{d}{2}}\Gamma\left(\tfrac{\Delta-d}{2}\right)\left(\frac{C_{\Delta}}{\Gamma(\Delta)}\right)\delta_{\Delta,0}$
(E.29)

where $\sum_{i}\gamma_{1i}=-\Delta=0$ was used

  * •

$n=2$

$\displaystyle
I_{n=2}(\Delta_{1},\Delta_{2},P_{12}=|\vec{x}_{1}-\vec{x}_{2}|^{2})=\delta_{\Delta_{1},\Delta_{2}}\pi^{\tfrac{d}{2}}\Gamma\left(\tfrac{2\Delta-d}{2}\right)\left(\frac{C_{\Delta}}{\Gamma(\Delta)}\right)^{2}\frac{\Gamma{(\Delta)}}{\left(P_{12}\right)^{\Delta}}$
(E.30)

where $\gamma_{12}=\Delta_{1}=\Delta_{2}$

  * •

$n=3$

$\displaystyle
I=\pi^{\tfrac{d}{2}}\Gamma\left(\tfrac{\sum\Delta_{i}-d}{2}\right)\prod_{i=1}^{n}\left(\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\right)\frac{\Gamma(\gamma_{12})\Gamma(\gamma_{13})\Gamma(\gamma_{23})}{\left(P_{12}\right)^{\gamma_{12}}\left(P_{13}\right)^{\gamma_{13}}\left(P_{23}\right)^{\gamma_{23}}}$
(E.31)

where

$\displaystyle\gamma_{12}$ $\displaystyle=\Delta_{1}+\Delta_{2}-\Delta_{3}$
(E.32) $\displaystyle\gamma_{13}$
$\displaystyle=\Delta_{1}+\Delta_{3}-\Delta_{2}$ (E.33)
$\displaystyle\gamma_{23}$ $\displaystyle=\Delta_{2}+\Delta_{3}-\Delta_{1}$
(E.34)

For $n=4$, we obtain our first non-trivial integrals since solving the
redundancy equations E.20 leaves two variables free

$\displaystyle\gamma_{ij}=\begin{pmatrix}-\Delta_{1}&\gamma_{12}&\Delta_{1}-\gamma_{12}-\gamma_{14}&\gamma_{14}\\\
&-\Delta_{2}&\gamma_{14}+\tfrac{1}{2}\Delta_{23,14}&\tfrac{1}{2}\Delta_{124,3}-\gamma_{12}-\gamma_{14}\\\
&&-\Delta_{3}&\gamma_{12}-\frac{1}{2}\Delta_{12,34}\\\
&&&-\Delta_{4}\end{pmatrix}$ (E.35)

where

$\displaystyle\Delta_{a_{1}a_{2}..,b_{1}b_{2}..}=\sum_{i}\Delta_{a_{i}}-\sum_{j}\Delta_{b_{j}}$
(E.36) $\displaystyle I_{n=4,\Delta_{i}}(P_{ij})$
$\displaystyle=\pi^{\tfrac{d}{2}}\Gamma\left(\tfrac{\sum\Delta_{i}-d}{2}\right)\prod_{i=1}^{n}\left(\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\right)P_{13}^{-\Delta_{1}}P_{23}^{\frac{1}{2}\Delta_{14,23}}P_{24}^{-\frac{1}{2}\Delta_{124,3}}P_{34}^{\frac{1}{2}\Delta_{12,34}}\int\frac{d\gamma_{12}d\gamma_{14}}{(2\pi
i)^{2}}M(\gamma_{12},\gamma_{14})$ (E.37) $\displaystyle
M(\gamma_{12},\gamma_{14})$
$\displaystyle=\left(\frac{P_{14}P_{23}}{P_{13}P_{24}}\right)^{-\gamma_{14}}\left(\frac{P_{12}P_{34}}{P_{13}P_{24}}\right)^{-\gamma_{12}}\prod_{j>i=1}^{4}\Gamma\left(\gamma_{ij}\right)$
(E.38)

where the conformal cross-ratios

$\displaystyle u$ $\displaystyle=\frac{P_{12}P_{34}}{P_{13}P_{24}}$ (E.39)
$\displaystyle v$ $\displaystyle=\frac{P_{14}P_{23}}{P_{13}P_{24}}$ (E.40)

and conformal prefactors appear naturally in this formalism. Additionally, the
crossing of operators is related to the exchange of the Mellin variables
$\gamma_{ij}$. Additionally, each cross-ratio is in 1-1 correspondence with a
non-redundant Mellin variable.

For simplicity, we will keep to the case where $\Delta_{i}=\Delta$, so we have

$\displaystyle I_{n=4,\Delta_{i}}(P_{ij})$
$\displaystyle=\pi^{\tfrac{d}{2}}\Gamma\left(\tfrac{4\Delta-d}{2}\right)\left(\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\right)^{4}P_{13}^{-\Delta}P_{24}^{-\Delta}\int\frac{d\gamma_{12}d\gamma_{14}}{(2\pi
i)^{2}}M(\gamma_{12},\gamma_{14})$ (E.41) $\displaystyle
M(\gamma_{12},\gamma_{14})$
$\displaystyle=v^{-\gamma_{14}}u^{-\gamma_{12}}\Gamma\left(\gamma_{12}\right)^{2}\Gamma\left(\gamma_{14}\right)^{2}\Gamma\left(\Delta-\gamma_{12}-\gamma_{14}\right)^{2}$
(E.42)

To solve this integral, one can relate the Mellin integral to an Euler-type
integral through Hypergeometric functions (see [93]):

$\displaystyle{}_{2}F_{1}(a,b,c,z)$
$\displaystyle=\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\int\frac{ds}{2\pi
i}\frac{\Gamma(s)\Gamma(1-s)\Gamma(b-s)}{\Gamma(c-s)}(-z)^{-s}$ (E.43)
$\displaystyle=\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\int\frac{ds}{2\pi
i}\frac{\Gamma(s)\Gamma(a-s)\Gamma(b-s)\Gamma(c-a-b+s)}{\Gamma(c-a)\Gamma(c-b)}(1-z)^{-s}$
(E.44)
$\displaystyle=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_{0}^{1}\frac{dt}{t(1-t)}t^{b}(1-t)^{c-b}(1-zt)^{-a}.$
(E.45)

We can therefore rewrite the integral

$\displaystyle\int\frac{d\gamma_{12}d\gamma_{14}}{(2\pi i)^{2}}$
$\displaystyle M_{n=4,\Delta}(\gamma_{12},\gamma_{14})$
$\displaystyle=\int\frac{d\gamma_{12}d\gamma_{14}}{(2\pi
i)^{2}}\Gamma\left(\gamma_{12}\right)^{2}\Gamma\left(\gamma_{14}\right)^{2}\Gamma\left(\Delta-\gamma_{12}-\gamma_{14}\right)^{2}u^{-\gamma_{12}}v^{-\gamma_{14}}$
(E.46) $\displaystyle=\int\frac{\gamma_{14}}{2\pi
i}\Gamma\left(\gamma_{14}\right)^{2}\Gamma\left(\Delta-\gamma_{14}\right)^{2}\int_{0}^{1}\frac{dt}{t(1-t)}\left(\frac{t(1-t)}{1-(1-u)t}\right)^{\Delta-\gamma_{14}}$
(E.47)
$\displaystyle=\frac{\Gamma(\Delta)^{4}}{\Gamma(2\Delta)v^{\Delta}}\int_{0}^{1}\frac{dt}{t(1-t)}\left(\frac{vt(1-t)}{1-(1-u)t}\right)^{\Delta}{}_{2}F_{1}(\Delta,\Delta,2\Delta,1-\frac{vt(1-t)}{1-(1-u)t})$
(E.48)
$\displaystyle=\frac{\Gamma(\Delta)^{4}}{\Gamma(2\Delta)v^{\Delta}}\int_{0}^{1}\frac{dt}{t(1-t)}X^{\Delta}{}_{2}F_{1}(\Delta,\Delta,2\Delta,1-X).$
(E.49)

In the familiar case $\Delta=1$ this is

$\displaystyle\int\frac{d\gamma_{12}d\gamma_{14}}{(2\pi
i)^{2}}M_{n=4,\Delta}(\gamma_{12},\gamma_{14})$
$\displaystyle=\int_{0}^{1}dt\left(\frac{1}{1-(1-u)t-vt(1-t)}\right)\log\left(\frac{vt(1-t)}{1-(1-u)t}\right)$
(E.50)
$\displaystyle=\int_{0}^{1}dt\left(\frac{1}{(1-t(1-\chi))(1-t(1-\bar{\chi}))}\right)\log\left(\frac{(1-t)t(1-\chi)(1-\bar{\chi})}{1-t(1-\chi\bar{\chi})}\right)$
(E.51)

where

$\displaystyle u$ $\displaystyle=\chi\bar{\chi}$ $\displaystyle v$
$\displaystyle=(1-\chi)(1-\bar{\chi})$ (E.52)

and the remaining integral can be evaluated easily using partial fractions to
give to usual box integral result

$\displaystyle\int\frac{d\gamma_{12}d\gamma_{14}}{(2\pi
i)^{2}}M_{n=4,\Delta=1}(\gamma_{12},\gamma_{14})=\frac{1}{2(\chi-\bar{\chi})}\left(\log(\chi\bar{\chi})\log(\frac{1-\chi}{1-\bar{\chi}})+2\textup{Li}(\chi)-2\textup{Li}(\bar{\chi})\right)$
(E.53)

Explicitely computing the explicit Mellin and Anti-Mellin transform is often
impractical, especially when considering more complex cases. Instead, using
Mellin amplitudes directly to find quantities such as CFT data and scattering
amplitudes is more convenient.

#### Scattering Amplitudes

The conditions on the Mellin variables in (E.20) can be written as the on-
shell conserved momenta of $n$ particles of mass $m_{i}^{2}=\Delta_{i}$:

$\displaystyle\gamma_{ij}=p_{i}\cdot p_{j}$ (E.54) $\displaystyle
p_{j}\cdot\sum p_{i}=0$ (E.55) $\displaystyle p_{i}^{2}=-m^{2}$ (E.56)

The Mellin amplitudes can then be expressed in terms of Mandelstam variables

$\displaystyle-(p_{1}+p_{2})^{2}=\Delta_{1}+\Delta_{2}-2\gamma_{12}=s$ (E.57)
$\displaystyle-(p_{1}+p_{3})^{2}=\Delta_{1}+\Delta_{3}-2\gamma_{13}=t$ (E.58)
$\displaystyle-(p_{1}+p_{4})^{2}=\Delta_{1}+\Delta_{4}-2\gamma_{14}=u$ (E.59)
$\displaystyle s+t+u=\sum_{i}\Delta_{i}$ (E.60)

In [109, 84], the Mellin amplitudes for Witten diagrams were related to flat-
space amplitudes in a certain limit:

$\displaystyle\frac{\mathcal{T}_{n}(l_{s},k_{i})}{l_{s}^{n\tfrac{d-1}{2}-d-1}}=\lim_{\theta\rightarrow\infty}\frac{1}{\mathcal{N}}\int\frac{d\alpha}{2\pi
i}\alpha^{\frac{d-\sum\Delta_{i}}{2}}\text{e}^{\alpha}M_{n}(\gamma_{ij}=\frac{\theta^{2}}{2\alpha}l_{s}^{2}k_{i}\cdot
k_{j})\theta^{n\frac{d-1}{2}-d-1}$ (E.61)

where $l_{s}$ is a scattering length intrinsic to the theory, $k_{i}$ are the
external momenta, $\mathcal{N}$ is a normalisation factor to match with flat-
space amplitudes. The limit in $\theta=\frac{R}{l_{s}}$ controls the flat-
space limit and suppresses the disconnected diagram contribution naturally,
finally the integral in $\alpha$, like the anti-Mellin transform is also on a
contour parallel to the imaginary axis. We will illustrate this formula in the
following examples.

##### Disconnected Contributions

For disconnected contributions, the Mellin amplitude has no integration
variables, so the replacement conditions for the Mellin variables in equations
E.20 give the momentum conservation condition expected from flat-space
scattering.

##### Contact Interactions

The Mellin transform of a four-point function with a contact interaction
$-\lambda_{4}\phi_{\Delta}^{4}$is

$\displaystyle
M_{n=4,\Delta}=\lambda_{4}\pi^{\tfrac{d}{2}}\Gamma\left(\tfrac{4\Delta-d}{2}\right)\left(\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\right)^{4}P_{13}^{-\Delta}P_{24}^{-\Delta}\Gamma\left(\gamma_{12}\right)^{2}\Gamma\left(\gamma_{14}\right)^{2}\Gamma\left(\Delta-\gamma_{12}-\gamma_{14}\right)^{2}$
(E.62)

The associated ‘truncated’ Mellin amplitude is

$\displaystyle\hat{M}_{n=4,\Delta}$
$\displaystyle=\lambda_{4}\pi^{\tfrac{d}{2}}\Gamma\left(\tfrac{\sum\Delta-d}{2}\right)\left(\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\right)^{4}$
(E.63) $\displaystyle\hat{M}_{n,\Delta}$
$\displaystyle=\lambda_{n}\mathcal{N}\Gamma\left(\tfrac{\sum\Delta-d}{2}\right)$
(E.64)

However, the interaction constant $\lambda_{4}$ is a dimensionful parameter.
To make it dimensionless, we define a new interaction constant through
dimensional analysis by using the only dimensionful parameter of our theory:
the radius $R$ of AdS:

$\displaystyle\tilde{\lambda_{n}}$
$\displaystyle=\lambda_{n}R^{d+1+n\frac{1-d}{2}}$ (E.65)
$\displaystyle=\lambda_{n}\left(\theta l_{s}\right)^{d+1+n\frac{1-d}{2}}$
(E.66)

The scattering amplitude formula (E.74) then gives

$\displaystyle\frac{\mathcal{T}_{n}(l_{s},k_{i})}{l_{s}^{n\tfrac{d-1}{2}-d-1}}$
$\displaystyle=\lim_{\theta\rightarrow\infty}\frac{1}{\mathcal{N}}\int\frac{d\alpha}{2\pi
i}\alpha^{\frac{d-\sum\Delta_{i}}{2}}\text{e}^{\alpha}\tilde{\lambda}_{n}\mathcal{N}\Gamma\left(\tfrac{\sum\Delta-d}{2}\right)\left(\theta
l_{s}\right)^{d+1+n\frac{1-d}{2}}\theta^{n\frac{d-1}{2}-d-1}$ (E.67)
$\displaystyle\mathcal{T}_{n}(l_{s},k_{i})$
$\displaystyle=\tilde{\lambda}_{n}\int\frac{d\alpha}{2\pi
i}\alpha^{\frac{d-\sum\Delta_{i}}{2}}\text{e}^{\alpha}\Gamma\left(\tfrac{\sum\Delta-d}{2}\right)$
(E.68) $\displaystyle=\tilde{\lambda}_{n}$ (E.69)

##### Derivative Interactions

Though the preceding example may seem slightly artificial, derivative
interactions are where this formalism starts showing some of its power (as
well as exchange diagrams). Using the identity

$\displaystyle\partial_{\mu}K_{\Delta_{1}}\partial^{\mu}K_{\Delta_{2}}=\Delta_{1}\Delta_{2}(K_{\Delta_{1}}K_{\Delta_{2}}-2P_{12}K_{\Delta_{1}+1}K_{\Delta_{2}+1}),$
(E.70)

the diagram corresponding to the Wick contraction such that the derivatives of
an interaction such as
$\lambda_{4,2}(\partial\phi_{\Delta})^{2}\phi_{\Delta}^{2}$ are acting on the
legs $1$ and $2$ gives the conformal integral

$\displaystyle\lambda_{4}\Delta_{1}\Delta_{2}(D_{\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}}-2P_{12}D_{\Delta_{1}\Delta_{2}\Delta_{3}+1\Delta_{4}+1}).$
(E.71)

Which gives the Mellin integral

$\displaystyle\lambda_{4,2}\Delta_{1}\Delta_{2}\int[d\gamma_{ij}]\prod_{i<j}\frac{\Gamma(\gamma{ij})}{(P_{ij})^{\gamma_{ij}}}\left(1-2P_{12}\frac{\Gamma(\gamma_{12}+1)\Gamma\left(\tfrac{\sum\Delta+2-d}{2}\right)\Gamma(\Delta_{1})\Gamma(\Delta_{2}))}{\Gamma(\gamma_{12})P_{12}\Gamma\left(\tfrac{\sum\Delta-d}{2}\right)\Gamma(\Delta_{1}+1)\Gamma(\Delta_{2}+1)}\right)$
(E.72)

and the truncated Mellin amplitude

$\displaystyle M_{\partial^{2},n=4}$
$\displaystyle=\tilde{\lambda_{4,2}}\Delta_{1}\Delta_{2}\left(\theta
l_{s}\right)^{d+1+n\frac{1-d}{2}-2}\left(1-2\frac{\gamma_{12}}{\Delta_{1}\Delta_{2}}\tfrac{\sum\Delta+2-d}{2}\right)\mathcal{N}\Gamma\left(\tfrac{\sum\Delta-d}{2}\right)$
(E.73)

Notice that the effective interaction term $\tilde{\lambda}_{4,2}$ is
additionally suppressed due to the derivative interactions, so the constant
term above will vanish in the $\theta\rightarrow\infty$ limit. The other term
gives

$\displaystyle\frac{\mathcal{T}_{\partial^{2},n=4}(l_{s},k_{i})}{l_{s}^{n\tfrac{d-1}{2}-d-1}}$
$\displaystyle=-2k_{1}\cdot
k_{2}\tilde{\lambda_{4,2}}\Gamma\left(\tfrac{\sum\Delta-d}{2}+1\right)l_{s}^{d+1+n\frac{1-d}{2}}\int\frac{d\alpha}{2\pi
i}\alpha^{\frac{d-\sum\Delta_{i}}{2}}\text{e}^{\alpha}\frac{1}{2\alpha}$
(E.74) $\displaystyle\mathcal{T}_{\partial^{2},n=4}(l_{s},k_{i})$
$\displaystyle=-\tilde{\lambda_{4,2}}k_{1}\cdot k_{2}$ (E.75)

Which corresponds to the contribution from the equivalent Wick contractions on
a Feynman diagram with the same interaction term. It is easy to see that the
$\theta$-suppression means that for a generic interaction term, only the term
with the highest power of $\gamma_{ij}$ survives so that for a product of
fields $\phi_{i}$ pairwise contracted with a derivative power $\alpha_{i,j}$,
the resulting flat-space amplitude will be

$\displaystyle\mathcal{T}_{n}=\lambda_{n}\prod_{i,j}\left(k_{i}\cdot
k_{j}\right)^{\alpha_{i,j}}$ (E.76)

###  E.2 Generalized Free Field Theory

Let us consider the simplest possible example of 1d CFT. The GFF correlator
for four identical scalars of dimension $\Delta_{\phi}$ reads

$f^{\text{GFF}}(t)=1+t^{2\Delta_{\phi}}+\left(\frac{t}{1+t}\right)^{2\Delta_{\phi}}.$
(E.77)

Of course, in this case, we know very well the spectrum of exchanged
operators, which includes the identity, “two-particle” operators of the
schematic form

$\displaystyle[\phi\phi]_{n}\sim\phi\,\square^{n}\phi,$ (E.78)

and their conformal descendants. The conformal primary operators have
dimension $\Delta_{n}=2\Delta_{\phi}+2n$ for $n\geq 0$. As usual, the fact
that the dimensions of these operators are separated by integers creates huge
degeneracies. In particular, we should be worried by the coincidence of left
and right poles in (5.79) and (5.80). This happens for $s=0$, where the first
right pole (associated to the exchange of the identity in the $s$-channel)
coincides with the second left pole (associated to the exchange of $\phi^{2}$
in the $t$-channel) and at $s=2\Delta_{\phi}$ for the crossing symmetric case.
The situation is depicted in Figure E.1.

$s$$2\Delta_{\phi}$

$s$$\mathcal{C}$$\epsilon$$2\Delta_{\phi}$$2\Delta_{\phi}\\!\\!+\\!\\!\epsilon$

Figure E.1: Left: The configuration of the poles for GFF. The left and right
poles coincide in $s=0$ and $s=2\Delta_{\phi}$. Right: the right poles are
slightly shifted to the right so that the contour $\mathcal{C}$ can run in the
middle.

Coincident poles generate a problem in the realization of the contour
$\mathcal{C}$, which was defined precisely in such a way as to separate the
left and right poles. To avoid this issue, one can slightly split the poles by
shifting the right (or left) poles by a small amount, as shown in Figure E.1.
The result of the procedure outlined in subsection 5.2.1 leads to a Mellin
transform which depends on the shift $\epsilon$

$M^{\text{GFF}}_{\epsilon}(s)=\frac{1}{\Gamma(s)\Gamma(2\Delta_{\phi}-s)}\left(\frac{\Gamma(s)\Gamma(2\Delta_{\phi}+\epsilon-s)}{\Gamma(2\Delta_{\phi})}+\frac{1}{s-\epsilon}+\frac{1}{s-2\Delta_{\phi}-\epsilon}-\frac{1}{s}-\frac{1}{s-2\Delta_{\phi}}\right)\,.$
(E.79)

Taking the inverse Mellin transform (5.75) with the contour $\mathcal{C}$
depicted in Figure E.1 and then taking the limit $\epsilon\to 0$, one recovers
(E.77). Taking the limit $\epsilon\to 0$ before integrating leads to the
straightforward result $M_{0}(s)=\frac{1}{\Gamma(2\Delta_{\phi})}$; however,
the contour $\mathcal{C}$ is not well-defined and one cannot consistently take
the inverse Mellin transform.222However, the correction to this is of order
$M_{\epsilon}(s)-M_{0}(s)=\frac{\epsilon}{s^{2}}+\frac{\epsilon}{(s-2\Delta_{\phi})^{2}}$
which is only non-negligible in the $s\rightarrow 0$ and $s\rightarrow
2\Delta_{\phi}$ limit. Therefore, for all other purposes (for example, for
comparisons at perturbative level or sum rule results), we can effectively
take the $M^{GFF}_{0}(s)$ result. Notice that this procedure is simply the
limit in the weak sense normally considered in distribution theory. In
particular, the identity

$\lim_{\epsilon\to 0}\int_{p-i\infty}^{p+i\infty}\frac{ds}{2\pi
i}\left(\frac{1}{s-p-\epsilon}-\frac{1}{s-p+\epsilon}\right)t^{s}=t^{p}$
(E.80)

tells us that the function $\frac{1}{s-p-\epsilon}-\frac{1}{s-p+\epsilon}$ in
the limit $\epsilon\to 0$ defines a delta function $\delta(s-p)$ for Mellin
integration.333Attentive readers will probably have noticed that the
individual terms in $\lim_{\epsilon\to
0}\int_{p-i\infty}^{p+i\infty}\frac{ds}{2\pi i}(\frac{1}{s-p-\epsilon})$
diverge and one cannot use the residue theorem. However, the combination does
not have this problem since
$\left(\frac{1}{s-p-\epsilon}-\frac{1}{s-p+\epsilon}\right)=\frac{2\epsilon}{(s-p)^{2}-\epsilon^{2}}$
which ensures convergence at large $s$. In this sense, we can say that the
Mellin transform for GFF is a distribution rather than a function. This issue
is caused by the exchange of the identity operator and by the degeneracy
typical of GFF. An alternative route for obtaining the Mellin transform of the
GFF correlator would be to use equations (5.77) and (5.78) with the well-known
expressions of the GFF OPE coefficients

$c^{(0)}_{n}=\frac{2\Gamma(2n+2\Delta_{\phi})^{2}\Gamma(2n+4\Delta_{\phi}-1)}{\Gamma(2\Delta_{\phi})^{2}\Gamma(2n+1)\Gamma(4n+4\Delta_{\phi}-1)}$
(E.81)

and summing over $\Delta_{n}=2\Delta_{\phi}+2n$. Doing this, one finds the
result $M(s)=\frac{1}{\Gamma(2\Delta_{\phi})}$ which is consistent with the
fact that equations (5.77) and (5.78) were obtained under the assumption of
non-coincident poles. Also, in this case, one could slightly shift the
position of the poles in $\psi_{0}(s)$ (or $\psi_{\infty}(s)$) and recover an
expression which coincides with $M_{\epsilon}^{GFF}$ in the weak limit
$\epsilon\to 0$.

###  E.3 Sum Rules for the 1d Mellin Transform

Testing the sum rules (5.106) or (5.104) on a fully nonperturbative spectrum
is a task which is momentarily out of reach. Therefore, we start by testing it
on the simplest possible 1d CFT, i.e. generalised free field theory and
perturbations thereof.

#### Generalized free theories

One may immediately raise several objections to our attempt to apply the sum
rules (5.104) to GFF theories. First of all, as discussed in section E.2, the
definition of the Mellin amplitude for GFF requires the inclusion of a cut-off
to regulate the exchange of the identity operator. Furthermore, a crucial
assumption in deriving the sum rules was that $M(s)$ must have zeros for
$s=-k$ and $s=2\Delta_{\phi}+k$. Still, GFF is precisely the example where
two-particle operators with dimension $2\Delta_{\phi}+2n+k$ are exchanged;
therefore, those zeros are absent. We will see that these issues can be
avoided by considering the function

$\displaystyle
F_{p}(s)=\frac{1}{(2\Delta_{\phi}+2p-s)(2\Delta_{\phi}+2p-s+1)}\,.$ (E.82)

Since this function has two poles and no residue at infinity, we have the
property

$\displaystyle\text{Res}_{s=2\Delta_{\phi}+2p}(F_{p}(s))+\text{Res}_{s=2\Delta_{\phi}+2p+1}(F_{p}(s))=0\,.$
(E.83)

Furthermore, one can easily check from (E.79) that
$M_{\epsilon}^{\text{GFF}}(2\Delta_{\phi}+2p)=M_{\epsilon}^{\text{GFF}}(2\Delta_{\phi}+2p+1)$
for $p\in\mathbb{N}$. Combining these two properties, it is clear that the
last term in (5.102) vanishes even though $M(s)$ has no zeros at the positions
of the poles of $F_{p}(s)$. We are then left with a sum over the residues of
$M(s)$. For GFF, the position of the poles in principle depends on the
regulator $\epsilon$, but the regulator’s role in (E.79) is to separate the
left and right poles. This is precisely what we have done to go from (5.102)
to (5.103). Therefore, equation (5.106) can be used with $\epsilon\to 0$ and,
inserting the GFF spectrum $\Delta=2\Delta_{\phi}+2n$ we end up with the
following sum rule

$\displaystyle\sum_{n,k}c^{(0)}_{n}\tfrac{(-1)^{k+1}\Gamma(4(\Delta_{\phi}+n))\Gamma(2\Delta_{\phi}+2n+k)}{\Gamma(2\Delta_{\phi}+2n)^{2}\Gamma(4(\Delta_{\phi}+n)+k)\Gamma(-2n-k)\Gamma(k+1)}\tfrac{2(2\Delta_{\phi}+4p+1)(\Delta_{\phi}+k+2n)}{(k+2n-2p-1)(k+2n-2p)(2\Delta_{\phi}+k+2n+2p+1)(k+2(\Delta_{\phi}+n+p))}=0\,.$
(E.84)

Notice that the $\Gamma(-2n-k)$ factor in the denominator kills all the terms
in this sum that are not compensated by a pole in the second ratio. The sum
over $k$ then receives only two contributions at $k=2p-2n$ and $k=2p-2n-1$,
which are present only for $p\geq n$. We are then left with a finite sum over
$n$

$\displaystyle\sum_{n=0}^{p}c_{n}^{(0)}\frac{\Gamma(2p+1)\Gamma(4(n+\Delta_{\phi}))\Gamma(2(p+\Delta_{\phi}))\left(\Delta_{\phi}-2n^{2}-4\Delta_{\phi}n+n+2\Delta_{\phi}p\right)}{\Gamma(2(n+\Delta_{\phi}))^{2}\Gamma(2(p-n+1))\Gamma(2(n+p+2\Delta_{\phi}+1))}=0\,.$
(E.85)

Each value of $p$ leads to an equation for $c_{p}^{(1)}$ in terms of all the
other OPE coefficients with a lower index. In other words, this sum rule can
be solved recursively for $c^{(0)}_{n}$ determining everything in terms of
$c^{(0)}_{0}$, which sets the overall normalization. Doing so, one easily
finds that the unique solution to this sum rule is provided by (E.81). The
same strategy can be used to determine the OPE coefficients for a fermionic
GFF and a GFF with $O(N)$ symmetry. For these theories, sum rules are obtained
inserting (E.82) into (5.100) and using the GFF spectrum, and read

$\displaystyle\\!\\!\\!\\!\sum_{\Delta,k}c_{\Delta}\frac{(-1)^{k+1}\Gamma(2\Delta)\Gamma(\Delta+k)}{\Gamma(\Delta)^{2}\Gamma(2\Delta+k)\Gamma(2\Delta_{\phi}-\Delta-k)\Gamma(k+1)}(\frac{1}{\Delta+k+p}-\frac{1}{2\Delta_{\phi}-\Delta-k+p})=0\,.$
(E.86)

The GFF spectrum $\Delta=2\Delta_{\phi}+n$ for exchanged operators has the
effect of truncating the sum above because of the factor of
$\Gamma(2\Delta_{\phi}-\Delta-k)$ in the denominator. The example of bosonic
GFF in section E.3 illustrates the case of even $n$, namely
$\Delta=2\Delta_{\phi}+2n$. Here we extend this to odd integers, namely
$\Delta=2\Delta_{\phi}+2n+1$, in covering the free fermionic model. We also
consider the free bosonic and fermionic models with O($N$) symmetry.

In the case of a free fermion theory, the spectrum
$\Delta=2\Delta_{\phi}+2n+1$ of exchanged operators leads to the sum rule

$\displaystyle\sum_{n=0}^{p}c_{n}\frac{2\Gamma(2p+1)\Gamma(2(p+\Delta_{\phi}))\left(\Delta_{\phi}-2\Delta_{\phi}p+2n^{2}+4\Delta_{\phi}n+n\right)\Gamma(4n+4\Delta_{\phi}+2)}{\Gamma(2p-2n+1)\Gamma(2n+2\Delta_{\phi}+1)^{2}\Gamma(2(p+n+2\Delta_{\phi}+1))}=0.$
(E.87)

Just like in the bosonic case, see (E.85), this is a recursive relation for
the OPE coefficients, whose general solution is

$\displaystyle
c_{n}^{(0)}=\frac{2\Gamma(2n+2\Delta_{\phi}+1)^{2}\Gamma(2n+4\Delta_{\phi})}{\Gamma(2\Delta_{\phi})\Gamma(2\Delta_{\phi}+1)\Gamma(2n+2)\Gamma(4n+4\Delta_{\phi}+1)}\,.$
(E.88)

This is confirmed by the vanishing of (E.87). As usual, this is true up to an
overall scaling, and the choice in (E.88) is set by requiring $c_{0}^{(0)}=1$.

Consider a bosonic four-point function with $O(N)$ symmetry. We write the
Mellin amplitude as a sum of the singlet, antisymmetric and traceless
symmetric contributions

$\displaystyle\hat{\mathcal{M}}^{1234}(s)=\hat{\mathcal{M}}^{S}\delta^{12}\delta^{34}+\hat{\mathcal{M}}^{A}(\delta^{13}\delta^{24}-\delta^{14}\delta^{23})+\hat{\mathcal{M}}^{T}(\frac{\delta^{13}\delta^{24}+\delta^{14}\delta^{23}}{2}-\frac{\delta^{12}\delta^{34}}{N}),$
(E.89)

with scalar coefficient functions $\hat{\mathcal{M}}^{S}(s)$,
$\hat{\mathcal{M}}^{A}(s)$ and $\hat{\mathcal{M}}^{T}(s)$. In these channels,
exchanged operators will be of the form
$\phi_{i}\partial^{2n}_{x}\phi^{i}$,$\phi^{[i}\partial^{2n+1}_{x}\phi^{j]}$,$\phi^{(i}\partial^{2n}_{x}\phi^{j)}$
respectively, with same spectra of exchanged operators previously seen
($\Delta=2\Delta_{\phi}+2n$, $\Delta=2\Delta_{\phi}+2n+1$,
$\Delta=2\Delta_{\phi}+2n$ respectively). Therefore we get the corresponding
OPE coefficients

$\displaystyle c_{n}^{S}$
$\displaystyle=c^{S}_{0}\frac{2\Gamma(2n+2\Delta_{\phi})^{2}\Gamma(2n+4\Delta_{\phi}-1)}{\Gamma(2\Delta_{\phi})^{2}\Gamma(2n+1)\Gamma(4n+4\Delta_{\phi}-1)}\,,$
(E.90) $\displaystyle c_{n}^{A}$
$\displaystyle=c^{A}_{0}\frac{2\Gamma(2n+2\Delta_{\phi}+1)^{2}\Gamma(2n+4\Delta_{\phi})}{\Gamma(2\Delta_{\phi})\Gamma(2\Delta_{\phi}+1)\Gamma(2n+2)\Gamma(4n+4\Delta_{\phi}+1)}\,,$
(E.91) $\displaystyle c_{n}^{T}$
$\displaystyle=c^{T}_{0}\frac{2\Gamma(2n+2\Delta_{\phi})^{2}\Gamma(2n+4\Delta_{\phi}-1)}{\Gamma(2\Delta_{\phi})^{2}\Gamma(2n+1)\Gamma(4n+4\Delta_{\phi}-1)}\,,$
(E.92)

up to a normalisation factor which can be easily found by looking at the first
identity contribution in the different channels. This gives the known result
(see [49])

$\displaystyle c_{n}^{S}$
$\displaystyle=\frac{1}{N}\frac{2\Gamma(2n+2\Delta_{\phi})^{2}\Gamma(2n+4\Delta_{\phi}-1)}{\Gamma(2\Delta_{\phi})^{2}\Gamma(2n+1)\Gamma(4n+4\Delta_{\phi}-1)}\,,$
(E.93) $\displaystyle c_{n}^{T}$
$\displaystyle=\frac{2\Gamma(2n+2\Delta_{\phi})^{2}\Gamma(2n+4\Delta_{\phi}-1)}{\Gamma(2\Delta_{\phi})^{2}\Gamma(2n+1)\Gamma(4n+4\Delta_{\phi}-1)}\,,$
(E.94) $\displaystyle c_{n}^{A}$
$\displaystyle=-\frac{2\Gamma(2n+2\Delta_{\phi}+1)^{2}\Gamma(2n+4\Delta_{\phi})}{\Gamma(2\Delta_{\phi})^{2}\Gamma(2n+2)\Gamma(4n+4\Delta_{\phi}+1)}\,.$
(E.95)

The same procedure for free fermions with O($N$) symmetry leads to

$\displaystyle c_{n}^{S}$
$\displaystyle=\frac{1}{N}\frac{2\Gamma(2n+2\Delta_{\phi}+1)^{2}\Gamma(2n+4\Delta_{\phi})}{\Gamma(2\Delta_{\phi})\Gamma(2\Delta_{\phi}+1)\Gamma(2n+2)\Gamma(4n+4\Delta_{\phi}+1)}\,,$
(E.96) $\displaystyle c_{n}^{T}$
$\displaystyle=\frac{2\Gamma(2n+2\Delta_{\phi}+1)^{2}\Gamma(2n+4\Delta_{\phi})}{\Gamma(2\Delta_{\phi})\Gamma(2\Delta_{\phi}+1)\Gamma(2n+2)\Gamma(4n+4\Delta_{\phi}+1)}\,,$
(E.97) $\displaystyle c_{n}^{A}$
$\displaystyle=-\frac{2\Delta_{\phi}\Gamma(2n+2\Delta_{\phi})^{2}\Gamma(2n+4\Delta_{\phi}-1)}{\Gamma(2\Delta_{\phi})^{2}\Gamma(2n+1)\Gamma(4n+4\Delta_{\phi}-1)}\,.$
(E.98)

#### Perturbative Sum Rules

In this section we further test our sum rules by using known CFT data for a
class of perturbations around GFF. These perturbations, which will be treated
in great details in subsection 5.2.5, are constructed by introducing an
effective field theory in AdS2 background and considering the 1d boundary
conformal field theory through the usual holographic dictionary. In
particular, we will be interested in quartic contact interaction with
derivatives. A classification of these independent contact interactions is
given e.g. in [55], where the authors find there is a one-parameter family
labelled by $L$, where $4L$ is the number of derivatives in the schematic
interaction $(\partial^{L}\Phi)^{4}$. As we mentioned at the end of subsection
5.2.3, single terms in the perturbative expansion of the correlator may have a
worse Regge behaviour than the general bound (5.85). In particular, let us
consider a perturbed correlator

$f(t)=f^{\text{GFF}}(t)+g_{L}f_{L}^{(1)}(t)+O(g_{L}^{2}),$ (E.99)

where $L$ labels the maximum number of derivatives in the quartic interaction
(i.e. the interaction term may involve a combination of terms with $\ell\leq
L$ derivatives) and $g_{L}$ is the associated coupling. The Regge behaviour of
this correlator is determined by the term with the maximum number of
derivatives, and it reads [55]

$f_{L}^{(1)}(z)\sim z^{2\Delta+2L-1}\qquad z\to\frac{1}{2}+i\infty.$ (E.100)

According to our discussion in subsection 5.2.4, the associated Mellin
amplitude will behave as

$M^{(1)}_{L}(s)\sim|s|^{2L-1}\qquad|s|\to\infty,$ (E.101)

and we need to choose a function $F_{p}(s)$ which vanishes at infinity faster
than $|s|^{-2L}$. Here we will derive and check the sum rules for the cases
$L=0$ and $L=1$. The strategy is the following. We use equation (5.104) to
write down nonperturbative sum rules with a specific function $F_{p}(s)$,
which will be chosen to decay sufficiently fast at $|s|\to\infty$ at a given
value of $L$. We then expand the CFT data as

$\displaystyle\Delta$
$\displaystyle=2\Delta_{\phi}+2n+g_{L}\gamma_{L,n}^{(1)}+\mathcal{O}(g_{L}^{2})\,,$
(E.102) $\displaystyle c_{\Delta}$
$\displaystyle=c^{(0)}_{n}+g_{L}c^{(1)}_{L,n}+\mathcal{O}(g_{L}^{2})$ (E.103)

and derive perturbative sum rules for $\gamma^{(1)}_{L,n}$ and
$c^{(1)}_{L,n}$. We then check that these sum rules are satisfied by the
$L=0,1$ results obtained in [55], which read

$\displaystyle\gamma_{0,n}^{(1)}$
$\displaystyle=\frac{\left(\frac{1}{2}\right)_{n}\left((\Delta_{\phi})_{n}\right){}^{2}\left(2\Delta_{\phi}-\frac{1}{2}\right)_{n}}{(1)_{n}\,(2\Delta_{\phi})_{n}\left(\left(\Delta_{\phi}+\frac{1}{2}\right)_{n}\right){}^{2}}\,,$
(E.104) $\displaystyle\gamma_{1,n}^{(1)}$
$\displaystyle=A_{\Delta_{\phi}}^{-1}\gamma_{0,n}^{(1)}\frac{2n(4\Delta_{\phi}+2n-1)}{(\Delta_{\phi}+n-1)(2\Delta_{\phi}+2n+1)}(16\Delta_{\phi}^{5}-13\Delta_{\phi}^{3}-3\Delta_{\phi}^{2}+16\Delta_{\phi}n^{4}+8n^{4}+64\Delta_{\phi}^{2}n^{3}$
$\displaystyle+16\Delta_{\phi}n^{3}-8n^{3}+96\Delta_{\phi}^{3}n^{2}+8\Delta_{\phi}^{2}n^{2}-24\Delta_{\phi}n^{2}-2n^{2}+64\Delta_{\phi}^{4}n-28\Delta_{\phi}^{2}n-2\Delta_{\phi}n+2n)$
(E.105)

where the result for $\gamma^{(1)}_{1,n}$ differs from [55] by an overall
factor

$A_{\Delta_{\phi}}\equiv\frac{\Delta_{\phi}(\Delta_{\phi}+1)(\Delta_{\phi}+2)(4\Delta_{\phi}-1)(4\Delta_{\phi}+1)^{2}(4\Delta_{\phi}+3)}{(2\Delta_{\phi}+1)^{2}(2\Delta_{\phi}+3)}\,,$
(E.106)

which we introduced to normalize the anomalous dimension as
$\gamma_{1,1}^{(1)}=1$. Notice that $\gamma_{1,0}^{(1)}=0$. This is equivalent
to a choice of basis for the set of independent interactions that can be built
with up to one derivative. We discuss this issue in detail in subsection
5.2.5. The OPE coefficients $c^{(1)}_{n}$ are given by the relation

$c^{(1)}_{L,n}=\partial_{n}(\gamma^{(1)}_{L,n}c^{(0)}_{n})\,.$ (E.107)

Let us start with the case $L=0$. In that case, the Regge behaviour is better
than the GFF case, so we could even choose the function

$F_{p}(s)=\frac{1}{s+p}\,,$ (E.108)

which gives the sum rules

$\displaystyle\sum_{\Delta,k}c_{\Delta}\frac{(-1)^{k+1}\Gamma(2\Delta)\Gamma(\Delta+k)}{\Gamma(\Delta)^{2}\Gamma(2\Delta+k)\Gamma(2\Delta_{\phi}-\Delta-k)\Gamma(k+1)(\Delta+k+p)(-2\Delta+\Delta+k-p)}=0\,.$
(E.109)

Notice that we can also use the function (E.108) as a building block from
which we can construct more suppressed functions of the class (5.105), for
example

$\displaystyle F_{p,p+1}(s)=F_{p}(s)-F_{p+1}(s)=\frac{1}{(s+p)(s+p+1)}\,.$
(E.110)

As we discussed below (5.105), the crossing-antisymmetric part of this
function decays as $\frac{1}{s^{3}}$ at large $s$ and for this reason, we will
also use it for the case $L=1$. Inserting the expansions (E.102) and (E.103)
into the sum rule (E.109), we get two contributions: a finite sum from the
terms where $k=p-2n$

$\displaystyle\tilde{\omega}_{p}$
$\displaystyle=\sum_{n=0}^{[\frac{p}{2}]}c_{n}^{(1)}\frac{\Gamma(p+1)\Gamma(4(n+\Delta_{\phi}))\Gamma(p+2\Delta_{\phi})}{(p-2n)!\,\Gamma(2(n+\Delta_{\phi}))^{2}\Gamma(2n+p+4\Delta_{\phi})}$
(E.111)
$\displaystyle+c_{n}^{(0)}\gamma_{n}^{(1)}\frac{\Gamma(p+1)\Gamma(4(n+\Delta_{\phi}))\Gamma(p+2\Delta_{\phi})}{2(\Delta_{\phi}+p)(p-2n)!\,\Gamma(2(n+\Delta_{\phi}))^{2}\Gamma(2n+p+4\Delta_{\phi})}\eta(\Delta_{\phi},n,p)\,,$
(E.112)

where

$\displaystyle\eta(\Delta_{\phi},n,p)=1+2(\Delta_{\phi}+p)(-2\psi_{2n+2\Delta_{\phi}}+2\psi_{4n+4\Delta_{\phi}}-2\psi_{2n+p+4\Delta_{\phi}}+\psi_{p+2\Delta_{\phi}}+\psi_{p+1})$
(E.113)

with $\psi_{n}=\frac{\Gamma^{\prime}(n)}{\Gamma(n)}$, and an infinite sum

$\tilde{\omega}^{\prime}_{p}=\sum_{n,k\neq
p-2n}c_{n}^{(0)}\gamma_{n}^{(1)}\frac{2(k+2n)!(\Delta_{\phi}+k+2n)\Gamma(4(n+\Delta_{\phi}))\Gamma(k+2n+2\Delta_{\phi})}{k!(k+2n-p)\Gamma(2(n+\Delta_{\phi}))^{2}\Gamma(k+4(n+\Delta_{\phi}))(2\Delta_{\phi}+k+2n+p)}\,.$
(E.114)

This last term does not allow us to follow the same strategy we used for GFF
to extract the value of $\gamma^{(1)}_{L,n}$. This is consistent with the fact
that these sum rules are not as constraining as in the GFF case, and we may
expect more than one solution. Therefore, instead of using the sum rules to
derive the CFT data, we limit ourselves to numerically checking the validity
of the equation

$\omega_{p}=\tilde{\omega}_{p}+\tilde{\omega}^{\prime}_{p}=0$ (E.115)

by inserting the data (E.104), (E.3) and (E.107). In Figure E.2, we show our
results for the case $p=0$ both for the functional $\omega_{0}$ and for the
combination (E.110), i.e. the functional $\omega_{0}-\omega_{1}$. It is very
clear from the plot that the latter shows a faster convergence reflecting the
better asymptotic behaviour at $|s|\to\infty$. In Figure E.3, we show the
analogous plot for the $L=1$ case, where we used the function (E.110) since we
needed a large $s$ behaviour at least $\frac{1}{s^{3}}$.

Figure E.2: $L=0$ sum rules for external dimensions
$\Delta_{\phi}=1,\Delta_{\phi}=2$ and different functionals $\omega_{0}$ and
$\omega_{0}-\omega_{1}$. The plot shows the value of the finite sum truncated
after $N$ terms in a logarithmic scale. The better convergence is seen for the
$\omega_{0}-\omega_{1}$ functional, due to a better large $s$ behaviour.
Figure E.3: Sum rule applied for the $L=1$ case for $\Delta_{\phi}=1,2,3$,
truncated after summing over N terms seen on a logarithmic scale. The slower
convergence compared to $L=0$ is to be expected from the worse Regge
behaviour.

We conclude with a couple of remarks. Since the finite sum in (E.111) always
ranges up to $[\frac{p}{2}]$, we can always find a combination of functionals
$\omega_{k}$ such that the contributions proportional to $c^{(1)}_{L,n}$ are
all cancelled, and we can write down a sum rule which only involves the
anomalous dimensions $\gamma^{(1)}_{L,n}$

$\sum_{k=0}^{2p+1}\frac{(-1)^{1+k}\,\Gamma(2p+2)}{\Gamma(k+1)\,\Gamma(2p+3-k)}\,\omega_{k}=0\,.$
(E.116)

The simplest case is $p=0$ where (E.116) gives $\omega_{0}-2\omega_{1}=0$.
Even after doing that, however, we are left with an infinite sum with
alternating signs, which we did not manage to use constructively to extract a
solution. Nevertheless, notice that once the anomalous dimensions are known,
we can use our sum rules at fixed $p$ to determine recursively the value of
the OPE coefficients $c^{(1)}_{L,n}$ following the same strategy as in the GFF
case. This provides a very convoluted way to rederive the relation (E.107).

#### Poles and Series

In this Appendix, we address some subtleties in the perturbative expansion of
the Mellin amplitudes in the context of the analytic sum rules. In all the
following, we will consider a function $\hat{F}(s)$ defined nonperturbatively
and its expansion in a small parameter $\lambda$, as well as the functional

$\displaystyle\omega_{K}[\hat{F}]=\oint_{\mathbb{C}|_{\infty}}\frac{ds}{2\pi
i}\hat{F}(s)K(s),$ (E.117)

defined as the contour integral of $\hat{F}(s)$ over the circle at $\infty$ in
the complex plane with an integration kernel $K(s)$. This functional is well
defined and vanishing for meromorphic functions $\hat{F}(s)$ and $K(s)$ such
that

$\displaystyle\hat{F}(s)K(s)\underset{|s|\rightarrow\infty}{\sim}s^{-1-\alpha}\qquad\alpha>0.$
(E.118)

#### Nonperturbative Zeros and Perturbative Poles

The nonperturbative Mellin function $\hat{M}(s)$ has zeros at positions
$s=2\Delta+n$, while the perturbative expansion has poles in those positions.
The reason for this and the subtleties of evaluating the sum rule will be
illustrated with an example.

Let $f(s)$ be a well-behaved function444By this, we mean that $f$ is
meromorphic, has no essential singularities and behaves as $s^{-\alpha}$ at
large s, where $\alpha$ is a positive parameter. For simplicity, we take
$f(s)$ to be regular without poles. However, including these poles (different
to those of $\hat{F}$) is straightforward and does not change the
conclusions.. We define the $\lambda$-dependent function

$\displaystyle\hat{F}(s)=\frac{(s-1)f(s)}{s-1-\lambda}.$ (E.119)

Upon expanding $F(s)$ in the parameter $\lambda$, we obtain the geometric
series

$\displaystyle\hat{F}(s)=\sum_{k=0}^{\infty}\left(\frac{\lambda}{s-1}\right)^{k}f(s),$
(E.120)

whose radius of convergence is $\lambda<|s-1|$. This perturbative series
cannot be evaluated at finite $\lambda$ in $s=1$. However, several features
are noteworthy. At $s=1$, we have poles of increasing degree at each order in
the $\lambda$ expansion, while the nonperturbative expression has a simple
zero at that point. Let us evaluate the contour integral

$\displaystyle\omega_{1}[\hat{F}]=\oint\frac{ds}{2\pi
i}\frac{\hat{F}(s)}{s-1}$ (E.121)

in three different ways. Firstly, by inserting the nonperturbative $\hat{F}$

$\displaystyle\oint\frac{ds}{2\pi i}\frac{\hat{F}(s)}{s-1}=\oint\frac{ds}{2\pi
i}\frac{f(s)}{s-1-\lambda}=f(1+\lambda).$ (E.122)

This method is the most intuitive way of evaluating (E.121) and is perfectly
well-defined. However, it requires the knowledge of the full nonperturbative
function, which is generally unknown.

Secondly, we can separately consider the poles of the integration kernel
$\frac{1}{s-1}$ from those of the function $\hat{F}$:

$\displaystyle\oint\frac{ds}{2\pi i}\frac{\hat{F}(s)}{s-1}$
$\displaystyle=\textrm{Res}(\frac{1}{s-1})\hat{F}(1)+\sum_{s^{*}}\frac{\textrm{Res}(\hat{F})(s^{*})}{s^{*}-1}=f(1+\lambda).$
(E.123)

This is not identical to the previous method since it assumes more (notably
that the poles of $K(s)$ and those of $\hat{F}$, $s^{*}$ above, are distinct)
and requires less information from the nonperturbative function; only the
residues of $\hat{F}(s)$ and the value of $\hat{F}(s_{i})$ at a finite number
of points are required to evaluate (E.121).

Finally, one can use the series expansion of $\hat{F}(s)$

$\displaystyle\omega_{1}[\hat{F}]$ $\displaystyle=\oint\frac{ds}{2\pi
i}\left(\sum_{k=0}^{\infty}\frac{\lambda^{k}}{(s-1)^{k+1}}f(s)\right)=\sum_{k=0}^{\infty}\frac{\lambda^{k}f^{(k)}(1)}{k!},$
(E.124)

which gives the $\lambda$-expansion of $f(1+\lambda)$. Under the assumption
that $f$ is analytic near 1, we can then resum the Taylor series to obtain
$f(1+\lambda)$ and obtain the same result as the previous two methods. In this
specific example, the only operation which is not allowed is to evaluate
$\hat{F}(1)$ using the nonperturbative series since the latter has a vanishing
radius of convergence for $s=1$. In the next subsection, we will explore a
setting where the perturbative evaluation of the functional is problematic and
requires the truncation of the series at a fixed order.

#### Bad Regge Behaviour

We now consider another pathological case where the perturbative expansion of
the correlator makes the large $s$ behaviour worse than that of the
nonperturbative expression. To illustrate this, we take the example

$\displaystyle\hat{F}(s)$ $\displaystyle=\frac{f(s)}{1-\lambda s^{2}},$
(E.125)

and evaluate the functional

$\displaystyle\omega_{3}[\hat{F}]$
$\displaystyle=\oint_{\mathbb{C}|_{\infty}}\frac{ds}{2\pi
i}\frac{\hat{F}(s)}{s(s^{2}-1)}.$ (E.126)

The convergence of the integral (E.126) imposes a bound on the large $s$
behaviour of $f(s)$ in (E.125)555As in the previous case, we consider that
$f(s)$ is meromorphic in $s$ and ignore its poles since they act as spectators
in the comparison between the perturbative and nonperturbative case.

$\displaystyle f(s)$
$\displaystyle\underset{s\rightarrow\infty}{\sim}s^{4-\alpha}$
$\displaystyle\alpha$ $\displaystyle>0.$ (E.127)

The functional (E.126) can then be evaluated explicitly as

$\displaystyle\omega_{3}[\hat{F}]=-f(0)+\frac{f(1)+f(-1)}{2(1-\lambda)}-\frac{\lambda}{2(1-\lambda)}\left(f\left(\frac{1}{\sqrt{\lambda}}\right)+f\left(-\frac{1}{\sqrt{\lambda}}\right)\right).$
(E.128)

The perturbative expansion of $\hat{F}(s)$ is

$\displaystyle\hat{F}(s)=f(s)+\lambda
s^{2}f(s)+\lambda^{2}s^{4}f(s)+O(\lambda^{3}).$ (E.129)

In this perturbative setting, even with a stricter condition on the large $s$
behaviour of $f(s)$

$\displaystyle f(s)$
$\displaystyle\underset{s\rightarrow\infty}{\sim}s^{-\alpha}$
$\displaystyle\alpha$ $\displaystyle>0,$ (E.130)

we must truncate the series at order $\lambda$ in order to evaluate the
functional

$\displaystyle\omega_{3}[f(s)+\lambda s^{2}f(s)]$
$\displaystyle=-f(0)+\frac{f(1)+f(-1)}{2}+\lambda\frac{f(1)+f(-1)}{2}.$
(E.131)

Comparing (E.131) to the small-$\lambda$ expansion of (E.128)

$\displaystyle\omega_{3}[\hat{F}]$
$\displaystyle=-f(0)+\frac{f(1)+f(-1)}{2}+\lambda\frac{f(1)+f(-1)}{2}-\frac{\lambda}{2}(f(\frac{1}{\sqrt{\lambda}})+f(-\frac{1}{\sqrt{\lambda}}))+o(\lambda),$
(E.132)

we see that the condition of convergence (E.130) is exactly what is needed to
get rid of the final terms and find agreement between the results

$\displaystyle-\frac{\lambda}{2}(f(\frac{1}{\sqrt{\lambda}})+f(-\frac{1}{\sqrt{\lambda}}))$
$\displaystyle\underset{\lambda\rightarrow
0}{\sim}\lambda^{1+\frac{\alpha}{2}}\underset{\lambda\rightarrow
0}{=}o(\lambda).$ (E.133)

We therefore have agreement between the functional of the truncated
$\lambda$-expansion of $\hat{F}(s)$ and the truncated $\lambda$-expansion of
the functional of $\hat{F}(s)$

$\displaystyle\omega_{3}[\hat{F}|_{\lambda}]=\left(\omega_{3}[\hat{F}]\right)|_{\lambda},$
(E.134)

where the order at which we are required to truncate is controlled by the
large $s$ behaviour of $K$ (which can always be chosen to satisfy the
convergence condition (E.118) at a given order of expansion).

###  E.4 Anomalous Dimensions for Higher Derivative Interactions

This section lists the various results for the polynomial part of the
anomalous dimension in equation (5.137). The Mathematica notebook attached to
the arXiv submission of [2] has values of $L$ ranging from $L=0$ to $L=8$ as
well as a function FindBootstrapPolynomial[L,$\Delta$,n] to compute
$\mathcal{P}_{L,n}(\Delta_{\phi})$ for arbitrary $L$ (the function gets slower
and slower at higher $L$, but in principle, it works for any $L$).

$\displaystyle\mathcal{P}_{0,n}(\Delta)$ $\displaystyle=1$ (E.135)
$\displaystyle\mathcal{P}_{1,n}(\Delta)$
$\displaystyle=8(2\Delta+1)n^{4}+8\left(8\Delta^{2}+2\Delta-1\right)n^{3}+2(2\Delta-1)(2\Delta+1)(12\Delta+1)n^{2}$
$\displaystyle+\left(64\Delta^{4}-28\Delta^{2}-2\Delta+2\right)n+\Delta^{2}\left(16\Delta^{3}-13\Delta-3\right)$
(E.136) $\displaystyle\mathcal{P}_{2,n}(\Delta)$
$\displaystyle=64(2\Delta+3)(2\Delta+5)n^{8}+128(2\Delta+3)(2\Delta+5)(4\Delta-1)n^{7}$
$\displaystyle+32(2\Delta+3)(2\Delta+5)\left(56\Delta^{2}-22\Delta-1\right)n^{6}$
$\displaystyle+32(2\Delta+3)(2\Delta+5)(4\Delta-1)(28\Delta^{2}-5\Delta-5)n^{5}$
$\displaystyle+4(2\Delta+3)\left(2240\Delta^{5}+4800\Delta^{4}-2924\Delta^{3}-2156\Delta^{2}+246\Delta-415\right)n^{4}$
$\displaystyle+8(2\Delta+3)(4\Delta-1)(224\Delta^{5}+576\Delta^{4}-158\Delta^{3}-572\Delta^{2}-243\Delta-160)n^{3}$
$\displaystyle+4(2\Delta-1)(2\Delta+3)(448\Delta^{6}+1392\Delta^{5}+84\Delta^{4}-2183\Delta^{3}-2091\Delta^{2}-1134\Delta-105)n^{2}$
$\displaystyle+4(2\Delta+3)(4\Delta-1)(64\Delta^{7}+208\Delta^{6}-36\Delta^{5}-605\Delta^{4}-554\Delta^{3}-30\Delta^{2}+243\Delta+90)n$
$\displaystyle+(\Delta-2)(\Delta-1)\Delta^{2}(\Delta+1)^{2}(4\Delta+3)(4\Delta+5)(4\Delta+7)(4\Delta+9)$
(E.137) $\displaystyle\mathcal{P}_{3,n}(\Delta)$
$\displaystyle=512(2\Delta+5)(2\Delta+7)(2\Delta+9)n^{12}$
$\displaystyle+1536(2\Delta+5)(2\Delta+7)(2\Delta+9)(4\Delta-1)n^{11}$
$\displaystyle+128(2\Delta+5)(2\Delta+7)(2\Delta+9)(264\Delta^{2}-102\Delta+5)n^{10}$
$\displaystyle+640(2\Delta+5)(2\Delta+7)(2\Delta+9)(4\Delta-1)(44\Delta^{2}-7\Delta-3)n^{9}$
$\displaystyle+96(2\Delta+5)(2\Delta+7)(5280\Delta^{5}+22080\Delta^{4}-8610\Delta^{3}-4790\Delta^{2}-798\Delta-2931)n^{8}$
$\displaystyle+96(2\Delta+5)(2\Delta+7)(4\Delta-1)(2112\Delta^{5}+9792\Delta^{4}+268\Delta^{3}-5448\Delta^{2}-4628\Delta-5493)n^{7}$
$\displaystyle+8(2\Delta+5)(2\Delta+7)\left(118272\Delta^{7}+556416\Delta^{6}-8736\Delta^{5}-656280\Delta^{4}-661308\Delta^{3}-560400\Delta^{2}\right.$
$\displaystyle\quad\left.+371392\Delta+17415\right)n^{6}$
$\displaystyle+24(2\Delta+5)(2\Delta+7)(4\Delta-1)\left(8448\Delta^{7}+45504\Delta^{6}+22128\Delta^{5}-79708\Delta^{4}-143680\Delta^{3}-114082\Delta^{2}\right.$
$\displaystyle\quad\left.+52985\Delta+27645\right)n^{5}$
$\displaystyle+4(2\Delta+5)\left(253440\Delta^{10}+2327040\Delta^{9}+5816640\Delta^{8}-1506240\Delta^{7}-22985970\Delta^{6}-33151830\Delta^{5}\right.$
$\displaystyle\quad\left.-9079800\Delta^{4}+25792815\Delta^{3}+10370477\Delta^{2}-446534\Delta+2131794\right)n^{4}$
$\displaystyle+8(2\Delta+5)(4\Delta-1)\left(14080\Delta^{10}+142080\Delta^{9}+423840\Delta^{8}+8160\Delta^{7}-2172753\Delta^{6}-4187481\Delta^{5}\right.$
$\displaystyle\quad\left.-1812050\Delta^{4}+3606930\Delta^{3}+3965596\Delta^{2}+1661325\Delta+791091\right)n^{3}$
$\displaystyle+6(2\Delta-1)(2\Delta+5)\left(11264\Delta^{11}+125184\Delta^{10}+437120\Delta^{9}+118880\Delta^{8}-2771604\Delta^{7}-6808095\Delta^{6}\right.$
$\displaystyle\quad\left.-4248981\Delta^{5}+6860955\Delta^{4}+13140919\Delta^{3}+9496058\Delta^{2}+4002384\Delta+360360\right)n^{2}$
$\displaystyle+2(2\Delta+5)(4\Delta-1)\left(3072\Delta^{12}+36096\Delta^{11}+132224\Delta^{10}-3360\Delta^{9}-1214676\Delta^{8}-2926395\Delta^{7}\right.$
$\displaystyle\quad\left.-970776\Delta^{6}+6196080\Delta^{5}+10143424\Delta^{4}+5128059\Delta^{3}-1542528\Delta^{2}-3028860\Delta-907200\right)n$
$\displaystyle+(\Delta-3)(\Delta-2)(\Delta-1)\Delta^{2}(\Delta+1)^{2}(\Delta+2)^{2}(4\Delta+5)(4\Delta+7)(4\Delta+9)(4\Delta+11)(4\Delta+13)(4\Delta+15)$
(E.138)

## Translations

To pay hommage to the four languages I have used during this PhD, there is
some text in English, French, Italian and German. The English translations are
as follows:

‘Will there be sky above me,//and will the stars // enfold my eyes//with their
maternal light?’ (T.1) ’By other routes, by other ports // you will reach the
shore, not here for crossing lighter ship must carry you’ (T.2) ‘A thousand-
beat waltz//alone gives lovers// three hundred and thirty-three times// the
time to write a story.’ (T.3)

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