aainstitutetext: Faculty of Physics, University of Warsaw, Pasteura 5, 02-093
Warsaw, Polandbbinstitutetext: School of Mathematics, Statistics & Physics,
Newcastle University, Newcastle NE1 7RU, U.K.ccinstitutetext: STAG Research
Center & Mathematical Sciences, University of Southampton, Highfield,
Southampton SO17 1BJ, U.K.

# A handbook of holographic 4-point functions

Adam Bzowski b Paul McFadden c and Kostas Skenderis<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present a comprehensive discussion of tree-level holographic $4$-point
functions of scalar operators in momentum space. We show that each individual
Witten diagram satisfies the conformal Ward identities on its own and is thus
a valid conformal correlator. When the $\beta=\Delta-d/2$ are half-integral,
with $\Delta$ the dimensions of the operators and $d$ the spacetime dimension,
the Witten diagrams can be evaluated in closed form and we present explicit
formulae for the case $d=3$ and $\Delta=2,3$. These correlators require
renormalization, which we carry out explicitly, and lead to new conformal
anomalies and beta functions. Correlators of operators of different dimension
may be linked via weight-shifting operators, which allow new correlators to be
generated from given ‘seed’ correlators. We present a new derivation of
weight-shifting operators in momentum space and uncover several subtleties
associated with their use: such operators map exchange diagrams to a linear
combination of exchange and contact diagrams, and special care must be taken
when renormalization is required.

##  Introduction

Witten diagrams have become an indispensable tool in analyzing CFTs at strong
coupling. Yet explicit closed formulae are rare. This is even more the case
when the correlators require renormalization. It is the purpose of this paper
to show how to obtain such formulae for tree-level holographic $4$-point
functions of operators of dimension $\Delta$, with $\Delta$ such that the
combinations $\beta=\Delta-d/2$ are half-integral, where $d$ is the spacetime
dimension.

The simplest cases, which are still of considerable interest, are those of
4-point functions of scalar operators of dimension $\Delta=2,3$ in three-
dimensional CFTs. For these cases, we discuss and present explicit formulae
for all holographic CFT $4$-point functions that can be constructed from tree-
level contact and exchange Witten diagrams, with exchanged scalar operators of
dimension $\Delta=2,3$. The methodology we discuss extends to other operators
and spacetime dimensions (provided the $\beta$ are half-integral) and the
Mathematica file we supply may be adapted to such cases.

Our focus on $4$-point functions of scalar operators with dimensions
$\Delta=2,3$ in three spacetime dimensions is motivated in part by recent
works on the cosmological bootstrap Arkani-Hamed:2015bza ; Arkani-
Hamed:2018kmz ; Baumann:2019oyu ; Baumann:2020dch ; Baumann:2021fxj ;
Sleight:2019mgd ; Sleight:2019hfp ; Sleight:2020obc ; Sleight:2021plv ;
DiPietro:2021sjt ; Meltzer:2021zin , in which late-time correlators in four-
dimensional de Sitter spacetime are constructed by solving three-dimensional
conformal Ward identities. These operators are dual to conformally coupled and
massless bulk scalars, with the latter modelling the inflaton. The same
approach was also explored at the level of the $3$-point function in the
earlier holographic cosmology literature Maldacena:2011nz ; Bzowski:2012ih ;
Mata:2012bx ; McFadden:2013ria ; Anninos:2014lwa . The correlation functions
of these operators require renormalization and a key question is whether the
divergences affecting the corresponding $4$-point correlators in anti-de
Sitter spacetime, as studied here, also play a role in de Sitter. This
question will be addressed in a follow-up paper toappear  using the results
for renormalized AdS/CFT correlators developed here.

Witten diagrams Witten:1998qj  were historically computed in position space,
see Freedman:1998tz ; Liu:1998ty ; DHoker:1999kzh ; Arutyunov:2000py ;
Dolan:2004iy  for a sample of computations from the early days of AdS/CFT.
However, the correlators we discuss here contain divergences as tabulated in
Table 1, and one of the central aims of this work is to provide a complete,
yet readable, account of their renormalization. The extraction of divergences
is best performed in momentum space Bzowski:2015pba ; Bzowski:2017poo ;
Bzowski:2018fql , and as such we work throughout in momentum space. The
momentum-space results are also of current interest for many applications
including inflationary cosmology. As the Fourier transform is well-defined
after renormalization, our results can in principle also be transformed back
to position space.

The framework we use for CFT in momentum space was introduced in
Bzowski:2013sza . The general structure of 4-point functions in momentum space
was given in Bzowski:2019kwd ; Bzowski:2020kfw  and a number of special cases
were discussed in Arkani-Hamed:2018kmz ; Isono:2018rrb ; Albayrak:2018tam ;
Albayrak:2019asr ; Maglio:2019grh ; Isono:2019wex ; Coriano:2019nkw ;
Albayrak:2020isk .111For a sample of other recent developments in momentum-
space CFT, see Anand:2019lkt ; Gillioz:2020wgw ; Jain:2021wyn ; Jain:2021vrv ;
Armstrong:2020woi ; Albayrak:2020fyp ; Gomez:2021qfd ; Coriano:2021nvn ;
Coriano:2020ees . Our hope is that this handbook of renormalized holographic
$4$-point functions, which we believe is the most comprehensive of its kind,
will be of use to a wide range of researchers. To this end, we provide full
documentation including a set of accompanying Mathematica notebooks.
Researchers interested in the renormalization of CFT correlators will find a
complete discussion of tree-level $4$-point functions, while those seeking
specific results need only consult the relevant listings or notebooks.

External operator | Contact |  $s$-channel exchange |  $s$-channel exchange
---|---|---|---
dimensions | diagram | with $\Delta_{x}=2$ | with $\Delta_{x}=3$
$[22,22]$ | 0 | 0 | 0
$[32,22]$ | 1 | 2 | 1
$[33,22]$ | 1 | 1 | 2
$[32,32]$ | 1 | 2 | 1
$[32,33]$ | 1 | 2 | 2
$[33,33]$ | 1 | 1 | 2
Table 1: Degree of divergence $n$ for Witten diagrams dimensionally regulated
according to the scheme (1). Each diagram diverges as $\epsilon^{-n}$ for
$\epsilon\rightarrow 0$, and is labelled by the dimensions
$[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}]$ of the external operators and
that of the exchanged operator $\Delta_{x}$ where present.

Aside from their applications, an additional reason for focusing on
correlators with $\Delta=2,3$ in $d=3$ is the relative simplicity of their
construction: the corresponding Witten diagrams feature propagators involving
Bessel functions with half-integer indices. The latter reduce to elementary
functions enabling all diagrams to be easily evaluated in terms of
dilogarithms. It is desirable to preserve this simplicity in the regulated
theory, and we show this is possible by regulating the spacetime and operator
dimensions as

$\displaystyle
d\longmapsto\hat{d}=d+2\epsilon,\qquad\Delta_{j}\longmapsto\hat{\Delta}_{j}=\Delta_{j}+\epsilon,$
(1)

where $j$ runs over all external and exchanged (‘$x$’) operators $j=1,2,3,4,x$
and $\epsilon$ is the regulator. This ‘half-integer’ scheme preserves the
indices on all Bessel functions, which are given by
$\Delta_{j}-d/2=\hat{\Delta}_{j}-\hat{d}/2$, and is sufficient to regulate all
the divergences we encounter here. Renormalization can then be achieved by
constructing suitable counterterms directly within the boundary CFT.

For completeness, we also explain how to change to a fully general dimensional
regularization scheme

$\displaystyle
d\longmapsto\hat{d}=d+2u\epsilon,\qquad\Delta_{j}\longmapsto\hat{\Delta}_{j}=\Delta_{j}+(u+v_{j})\epsilon,$
(2)

where $u$ and $v_{j}$, for $j=1,2,3,4,x$, are real constants parametrizing the
scheme. This can be achieved by expanding the Bessel functions in their
indices, for which several results are available. An alternative to both the
above approaches would be to employ holographic renormalization
Skenderis:2002wp , however in practice the resulting cut-off radial integrals
are less convenient to evaluate than those encountered in dimensional
regularization.

Usually, in quantum field theory, individual Feynman diagrams are not
physically meaningful and they do not satisfy Ward identities on their own.
Instead, one must sum all relevant diagrams to construct observable
quantities, such as correlators, that satisfy the relevant Ward identities. An
interesting general feature of Witten diagrams is that they individually
satisfy the conformal Ward identities. This follows from that the fact that
there is a choice of bulk AdS Lagrangian such that any given Witten diagram is
the only diagram contributing to a corresponding correlator. We illustrate
this in the cases we analyze, and we expect this to be a general feature. The
existence of a bulk Lagrangian implies that the diagram must satisfy the
conformal Ward identities, as these follow from the bulk isometries, and also
that we should be able to renormalize at the level of individual Witten
diagrams. This does not necessarily mean however that there exists an actual
CFT with the chosen spectrum and interaction. Where such a CFT does not exist
(for example, via bootstrap arguments) then the corresponding bulk AdS
Lagrangian would be in the swampland.

An interesting approach to the construction of correlators is to use weight-
shifting operators. These operators were introduced in position space in
Karateev:2017jgd  (see also Costa:2018mcg ), and by construction, map CFT
correlators of operators of given conformal dimensions to those of operators
with different (shifted) conformal dimensions. One may thus start from ‘seed’
correlators and obtain others using weight-shifting operators. These were
applied in momentum space in Arkani-Hamed:2018kmz ; Baumann:2019oyu  and we
further develop and analyze this approach here. There are however a number of
subtleties that need to be taken into account when using this method. First,
the application of weight-shifting operators to exchange diagrams generally
produces a linear combination of shifted exchange and contact diagrams, rather
than a pure exchange diagram. So if one wants to obtain exchange diagrams
using weight-shifting operators one needs to know first all the contact
diagrams. Second, care is needed when renormalization is required:
renormalized correlators obey inhomogeneous conformal Ward identities
containing anomalies and beta functions, whereas these operators only connect
solutions of the homogeneous conformal Ward identities. In particular, one
cannot construct a renormalized correlator, which depends on the RG scale
$\mu$, by acting with a ($\mu$-independent) weight-shifting operator on a
finite correlator, since the latter has no $\mu$-dependence. One may apply
them at the level of the regulated theory, but only a subset of weight-
shifting operators yield useful relations as sometimes weight-shifting
operators map regulated exchange diagrams to shifted contact diagrams only.
For the cases we analyze, one may use weight-shifting operators to reduce all
exchange diagrams to a single diagram per exchanged scalar ($\Delta_{x}=2$ and
$\Delta_{x}=3$), plus all contact diagrams. One may further link the two
master exchange integrals via a different relation, so that at the end all
exchange integrals may be reduced to the exchange diagram with all external
dimensions, and the dimension of the exchanged scalar, equal to three (plus
all contact diagrams). However, while all obstacles have been avoided and the
regulated amplitudes constructed, the resulting scheme is little simpler than
evaluating the diagrams directly, at least for the cases we analyze.

The layout of this paper is as follows. In the next section we present the
objects of interest: the regulated Witten diagrams. Section 3 proceeds to
evaluate these diagrams in the ‘half-integer’ scheme (1), after which they are
renormalized as described in Section 4. Section 5 discusses how to change to
the general regularization scheme (2), and provides a complete list of such
results. The discussion of the change of scheme is the most complicated part
of this paper. Readers interested only in results may skip this part. The
action of weight-shifting operators is discussed in Section 6. We present a
simple momentum-space derivation of these operators based on the shadow
transform, then give a new formula for their action on exchange diagrams
before discussing their limitations when applied to renormalized correlators.
Section 7 summarises the Mathematica packages accompanying this paper, which
contain a complete record of all our results. We discuss the outlook in
Section 8. The paper contains four appendices. In Appendix A we summarize our
conventions and definitions for momenta. Appendix B lists the conformal Ward
identities and Appendix C various useful mathematical results. Finally, in
Appendix D we present additional results on weight-shifting operators.

##  Definitions for regulated amplitudes

Our goal is to calculate the _amplitudes_ , i.e., the $3$\- and $4$-point
_Witten diagrams_ , as presented in Figures 1 and 2. As usual, external and
internal lines correspond to bulk-to-boundary and bulk-to-bulk propagators in
pure Euclidean AdS, while each interior point requires integration over its
radial position. All our conventions for QFT and the momenta are summarized in
Appendix A. At this point, we will not make reference to any specific bulk
action: rather, we will simply focus on the individual amplitudes defined by
the expressions below, postponing consideration of their relation to the bulk
action.

$\mathcal{O}_{1}(\bm{k}_{1})$$\mathcal{O}_{3}(\bm{k}_{3})$$\mathcal{O}_{2}(\bm{k}_{2})$$\operatorname{\mathcal{K}}_{[\Delta_{1}]}$$\operatorname{\mathcal{K}}_{[\Delta_{3}]}$$\operatorname{\mathcal{K}}_{[\Delta_{2}]}$
Figure 1: Witten diagram representing the 3-point amplitude
$i_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$ in (8), with bulk-to-boundary
propagators as given in (3).

  * •

The scalar bulk-to-boundary propagator is

$\displaystyle\operatorname{\mathcal{K}}_{d,\Delta}(z,k)=\frac{k^{\Delta-\frac{d}{2}}z^{\frac{d}{2}}K_{\Delta-\frac{d}{2}}(kz)}{2^{\Delta-\frac{d}{2}-1}\Gamma\left(\Delta-\frac{d}{2}\right)}$
(3)

while the scalar bulk-to-bulk propagator is

$\displaystyle\operatorname{\mathcal{G}}_{d,\Delta}(z,k;\zeta)=\left\\{\begin{array}[]{ll}(z\zeta)^{\frac{d}{2}}I_{\Delta-\frac{d}{2}}(kz)K_{\Delta-\frac{d}{2}}(k\zeta)&\text{
for }z<\zeta,\\\
(z\zeta)^{\frac{d}{2}}K_{\Delta-\frac{d}{2}}(kz)I_{\Delta-\frac{d}{2}}(k\zeta)&\text{
for }z>\zeta.\end{array}\right.$ (6)

where $I_{\beta}$ and $K_{\beta}$ are the modified Bessel functions.

To avoid clutter, we will use $\hat{\operatorname{\mathcal{K}}}_{[\Delta]}$
and $\hat{\operatorname{\mathcal{G}}}_{[\Delta]}$ to denote the propagators
for the regulated parameters $\hat{d}$ and $\hat{\Delta}$ defined in (2),
leaving the specific scheme implicit.

  * •

For convenience, regulated 2-point amplitudes are normalized so as to match
the holographic 2-point functions:

$\displaystyle\hat{i}_{[\Delta\Delta]}(k)=(2\hat{\Delta}-\hat{d})\times\text{coefficient
of }z^{\hat{\Delta}}\text{ in
}\hat{\operatorname{\mathcal{K}}}_{[\Delta]}(z,k).$ (7)

All non-diagonal 2-point amplitudes $\hat{i}_{[\Delta\Delta^{\prime}]}$ with
$\Delta\neq\Delta^{\prime}$ vanish.

  * •

We define the regulated 3-point amplitudes as

$\displaystyle\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}(k_{1},k_{2},k_{3})=\int_{0}^{\infty}\mathrm{d}z\,z^{-\hat{d}-1}\,\hat{\operatorname{\mathcal{K}}}_{[\Delta_{1}]}(z,k_{1})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{2}]}(z,k_{2})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{3}]}(z,k_{3}).$
(8)

The corresponding Witten diagram is presented in Figure 1.

  * •

We use $i_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}$ to denote the contact
diagram with four external scalars of dimensions
$\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}$, as presented in Figure 2. The
regulated expression is

$\displaystyle\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}(k_{1},k_{2},k_{3},k_{4})$
$\displaystyle\qquad=\int_{0}^{\infty}\mathrm{d}z\,z^{-\hat{d}-1}\hat{\operatorname{\mathcal{K}}}_{[\Delta_{1}]}(z,k_{1})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{2}]}(z,k_{2})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{3}]}(z,k_{3})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{4}]}(z,k_{4}).$
(9)

$\mathcal{O}_{1}(\bm{k}_{1})$$\mathcal{O}_{2}(\bm{k}_{2})$$\mathcal{O}_{3}(\bm{k}_{3})$$\mathcal{O}_{4}(\bm{k}_{4})$$\operatorname{\mathcal{K}}_{[\Delta_{1}]}$$\operatorname{\mathcal{K}}_{[\Delta_{2}]}$$\operatorname{\mathcal{K}}_{[\Delta_{3}]}$$\operatorname{\mathcal{K}}_{[\Delta_{4}]}$

$\mathcal{O}_{1}(\bm{k}_{1})$$\mathcal{O}_{2}(\bm{k}_{2})$$\mathcal{O}_{3}(\bm{k}_{3})$$\mathcal{O}_{4}(\bm{k}_{4})$$\operatorname{\mathcal{K}}_{[\Delta_{1}]}$$\operatorname{\mathcal{K}}_{[\Delta_{2}]}$$\operatorname{\mathcal{K}}_{[\Delta_{3}]}$$\operatorname{\mathcal{K}}_{[\Delta_{4}]}$$\operatorname{\mathcal{G}}_{[\Delta_{x}]}$

Figure 2: Witten diagrams representing the contact and exchange 4-point
amplitudes $i_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}$ and
$i_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}$ given in (• ‣ 2)
and (• ‣ 2).

  * •

We use $i_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}$ to denote
the exchange diagram $\Delta_{1}\Delta_{2}\mapsto\Delta_{3}\Delta_{4}$ with
the exchange scalar of dimension $\Delta_{x}$, as shown in Figure 2. The
regulated expression is

$\displaystyle\hat{i}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}(k_{1},k_{2},k_{3},k_{4},s)$
$\displaystyle\qquad=\int_{0}^{\infty}\mathrm{d}z\,z^{-\hat{d}-1}\hat{\operatorname{\mathcal{K}}}_{[\Delta_{1}]}(z,k_{1})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{2}]}(z,k_{2})\times$
$\displaystyle\qquad\qquad\times\int_{0}^{\infty}\mathrm{d}\zeta\,\zeta^{-\hat{d}-1}\hat{\operatorname{\mathcal{G}}}_{[\Delta_{x}]}(z,s;\zeta)\hat{\operatorname{\mathcal{K}}}_{[\Delta_{3}]}(\zeta,k_{3})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{4}]}(\zeta,k_{4}).$
(10)

  * •

Many of the regularized amplitudes defined above exhibit divergences as
$\epsilon\rightarrow 0$. The 3-point amplitudes
$\hat{i}_{[222]},\hat{i}_{[322]},\hat{i}_{[332]},\hat{i}_{[333]}$ all have a
single pole $\epsilon^{-1}$. For the 4-point amplitudes, the degrees of
divergence are summarized in Table 1.

##  Regulated amplitudes in the half-integer scheme

The idea of the half-integer scheme (1) is that all the Bessel functions
appearing in the propagators for operators of dimensions $\Delta=2$ and
$\Delta=3$ reduce to elementary functions. Explicitly, the bulk-to-boundary
propagators (3) are

$\displaystyle\hat{\operatorname{\mathcal{K}}}_{[2]}(z,k)=\operatorname{\mathcal{K}}_{3+2\epsilon,2+\epsilon}(z,k)$
$\displaystyle=z^{1+\epsilon}e^{-kz},$ (11)
$\displaystyle\hat{\operatorname{\mathcal{K}}}_{[3]}(z,k)=\operatorname{\mathcal{K}}_{3+2\epsilon,3+\epsilon}(z,k)$
$\displaystyle=z^{\epsilon}e^{-kz}(1+kz)$ (12)

while the bulk-to-bulk propagators (6) reduce to

$\displaystyle\hat{\operatorname{\mathcal{G}}}_{[2]}(z,k;\zeta)$
$\displaystyle=\operatorname{\mathcal{G}}_{3+2\epsilon,2+\epsilon}(z,k;\zeta)$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{(z\zeta)^{1+\epsilon}}{k}e^{-k\zeta}\sinh(kz)&\text{
for }z<\zeta,\\\ \frac{(z\zeta)^{1+\epsilon}}{k}e^{-kz}\sinh(k\zeta)&\text{
for }z>\zeta,\end{array}\right.$ (15)
$\displaystyle\hat{\operatorname{\mathcal{G}}}_{[3]}(z,k;\zeta)$
$\displaystyle=\operatorname{\mathcal{G}}_{3+2\epsilon,3+\epsilon}(z,k;\zeta)$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{(z\zeta)^{\epsilon}}{k^{3}}e^{-k\zeta}(1+k\zeta)\left(kz\cosh(kz)-\sinh(kz)\right)&\text{
for }z<\zeta,\\\
\frac{(z\zeta)^{\epsilon}}{k^{3}}e^{-kz}(1+kz)\left(z\zeta\cosh(k\zeta)-\sinh(k\zeta)\right)&\text{
for }z>\zeta.\end{array}\right.$ (18)

The radial integrals in the 3- and 4-point contact diagrams (8) and (• ‣ 2)
can then be evaluated in terms of gamma functions and rational functions.

For exchange diagrams, the inner integral in (• ‣ 2) is expressible in terms
of incomplete gamma functions and rational functions. Depending on which end
of the integration is fixed, we have

$\displaystyle\gamma(\alpha,x)$
$\displaystyle=\int_{0}^{x}\mathrm{d}t\,e^{-t}t^{\alpha-1},$
$\displaystyle\Gamma(\alpha,x)$
$\displaystyle=\int_{x}^{\infty}\mathrm{d}t\,e^{-t}t^{\alpha-1}.$ (19)

To calculate the outer integral in (• ‣ 2) it then suffices to use the
following integrals,

$\displaystyle\int_{0}^{\infty}\mathrm{d}x\,x^{\mu-1}e^{-\beta
x}\gamma(\nu,\alpha x)$
$\displaystyle=\frac{\alpha^{\nu}\Gamma(\mu+\nu)}{\nu(\alpha+\beta)^{\mu+\nu}}\,{}_{2}F_{1}\left(1,\mu+\nu;\nu+1;\frac{\alpha}{\alpha+\beta}\right),$
(20) $\displaystyle\int_{0}^{\infty}\mathrm{d}x\,x^{\mu-1}e^{-\beta
x}\Gamma(\nu,\alpha x)$
$\displaystyle=\frac{\alpha^{\nu}\Gamma(\mu+\nu)}{\mu(\alpha+\beta)^{\mu+\nu}}\,{}_{2}F_{1}\left(1,\mu+\nu;\mu+1;\frac{\beta}{\alpha+\beta}\right).$
(21)

Since for a specific amplitude the variables $\mu$ and $\nu$ depend on the
regulator $\epsilon$, one must series expand the hypergeometric function with
respect to its parameters. In all cases the parameters reduce to integer
values plus terms of order $O(\epsilon)$ allowing use of the excellent
Mathematica package `HypExp` Huber:2005yg .

In the remainder of this section, we list the $2$-, $3$\- and $4$-point
amplitudes defined by expressions (7), (8), (• ‣ 2) and (• ‣ 2), all regulated
in the half-integer scheme (1).

###  2-point amplitudes

The 2-point amplitudes are

$\displaystyle\hat{i}_{[22]}=-k,\qquad\qquad\qquad\hat{i}_{[33]}=k^{3},$ (22)

where the normalization matches that of the holographic 2-point functions. In
the half-integer scheme these amplitudes are finite and independent of the
regulator $\epsilon$.

###  3-point amplitudes

For later use, we will need the regulated 3-point amplitudes expanded to order
$\epsilon$. Alternatively, we can simply keep the full solutions evaluated to
all orders in the regulator:

$\displaystyle\hat{i}_{[222]}$ $\displaystyle=\operatorname{Div}_{1}(k_{t}),$
(23) $\displaystyle\hat{i}_{[322]}$
$\displaystyle=-\operatorname{Div}_{1}(k_{t})\left[(k_{2}+k_{3})+\frac{\epsilon
k_{t}}{1-\epsilon}\right],$ (24) $\displaystyle\hat{i}_{[332]}$
$\displaystyle=\frac{1}{2}\operatorname{Div}_{1}(k_{t})\left[(k_{3}^{2}-k_{1}^{2}-k_{2}^{2})+\frac{\epsilon
k_{t}}{(1-\epsilon)(2-\epsilon)}\left((3-\epsilon)k_{3}-(1-\epsilon)(k_{1}+k_{2})\right)\right],$
(25) $\displaystyle\hat{i}_{[333]}$
$\displaystyle=\frac{1}{3}\operatorname{Div}_{1}(k_{t})\Big{[}(k_{1}^{3}+k_{2}^{3}+k_{3}^{3})+\frac{\epsilon
k_{t}}{(1-\epsilon)(3-\epsilon)}\left((4-\epsilon)(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.-(1-\epsilon)(k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3})\right)\Big{]}.$
(26)

Here and throughout, we use the conventions listed in appendix A. In
particular, $\operatorname{Div}_{a}(k)=\Gamma(a\epsilon)k^{-a\epsilon}$ as
given in (• ‣ A.1) and $k_{t}$ is the total 3-point momentum magnitude as
defined in (461). Note that all these 3-point amplitudes exhibit a single pole
in the regulator.

###  Contact diagrams

The regulated contact 4-point amplitudes are

$\displaystyle\hat{i}_{[2222]}$ $\displaystyle=\frac{1}{k_{T}}+O(\epsilon),$
(27) $\displaystyle\hat{i}_{[3222]}$
$\displaystyle=\operatorname{Div}_{2}(k_{T})+\frac{k_{1}}{k_{T}}+O(\epsilon),$
(28) $\displaystyle\hat{i}_{[3322]}$
$\displaystyle=-(k_{3}+k_{4})\operatorname{Div}_{2}(k_{T})+\frac{k_{1}k_{2}}{k_{T}}-k_{T}+O(\epsilon),$
(29) $\displaystyle\hat{i}_{[3332]}$
$\displaystyle=\frac{k_{4}^{2}-k_{1}^{2}-k_{2}^{2}-k_{3}^{2}}{2}\operatorname{Div}_{2}(k_{T})+\frac{k_{1}k_{2}k_{3}}{k_{T}}+k_{T}\left(k_{4}-\frac{k_{T}}{4}\right)+O(\epsilon),$
(30) $\displaystyle\hat{i}_{[3333]}$
$\displaystyle=\frac{k_{1}^{3}+k_{2}^{3}+k_{3}^{3}+k_{4}^{3}}{3}\operatorname{Div}_{2}(k_{T})+\frac{k_{1}k_{2}k_{3}k_{4}}{k_{T}}+k_{T}\left(-\sigma_{(2)1234}+\frac{4}{9}k_{T}^{2}\right)+O(\epsilon).$
(31)

where $k_{T}$ is the total 4-point momentum magnitude (461) and
$\sigma_{(2)1234}$ is defined in (467).

###  Exchange diagrams

We split the results for exchange diagrams into two parts,

$\displaystyle\hat{i}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}=i^{\text{div}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}+i^{\text{fin}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]},$
(32)

where $i^{\text{div}}$ contains both the divergences and logarithmic terms,
while the finite part $i^{\text{fin}}$ contains the remainder of the
amplitude. This split is non-unique, but is convenient for organizing the
amplitude and will be particularly useful in the context of renormalization
and change of scheme as we will discuss later.

####  Exchange diagrams with $\Delta_{x}=2$

For $\Delta_{x}=2$, we find:

  * •

$i_{[22,22x2]}$ is finite,

$\displaystyle i^{\text{div}}_{[22,22x2]}$ $\displaystyle=0+O(\epsilon),$ (33)
$\displaystyle i^{\text{fin}}_{[22,22x2]}$
$\displaystyle=-\frac{1}{2s}\mathcal{D}^{(+)},$ (34)

where $s$ is the usual Mandelstam variable (460) and $\mathcal{D}^{(+)}$ is
defined in (473).

  * •

$i_{[32,22x2]}$ exhibits a double pole,

$\displaystyle i^{\text{div}}_{[32,22x2]}$
$\displaystyle=\frac{1}{2}\operatorname{Div}_{1}^{2}(l_{34+})+\operatorname{Div}_{2}(l_{34+})+O(\epsilon),$
(35) $\displaystyle i^{\text{fin}}_{[32,22x2]}$
$\displaystyle=\frac{k_{2}}{2s}\mathcal{D}^{(+)}+\frac{1}{2}\mathcal{D}^{(-)}+\frac{1}{4}\left[2-\log^{2}\left(\frac{l_{34+}}{l_{12+}}\right)\right],$
(36)

where $\mathcal{D}^{(-)}$ is defined in (474) and $l_{ij+}$ is defined in
(470).

  * •

$i_{[33,22x2]}$ exhibits a single pole,

$\displaystyle i^{\text{div}}_{[33,22x2]}$
$\displaystyle=\frac{k_{3}+k_{4}}{2}\operatorname{Div}_{2}(k_{T})+O(\epsilon),$
(37) $\displaystyle i^{\text{fin}}_{[33,22x2]}$
$\displaystyle=\frac{k_{1}^{2}+k_{2}^{2}-s^{2}}{4s}\mathcal{D}^{(+)}+\frac{k_{1}+k_{2}}{2}\left[\log\left(\frac{l_{34+}}{k_{T}}\right)+1\right]+\frac{7}{8}(k_{3}+k_{4}).$
(38)

  * •

$i_{[32,32x2]}$ exhibits a double pole,

$\displaystyle i^{\text{div}}_{[32,32x2]}$
$\displaystyle=-\frac{k_{2}+s}{2}\left(\operatorname{Div}_{1}^{2}(l_{12+})+6\operatorname{Div}_{2}(l_{12+})\right)-\frac{k_{4}+s}{2}\left(\operatorname{Div}_{1}^{2}(l_{34+})+6\operatorname{Div}_{2}(l_{34+})\right)$
$\displaystyle\qquad\qquad+(s-2k_{1})\operatorname{Div}_{2}(l_{12+})+(s-2k_{3})\operatorname{Div}_{2}(l_{34+})+O(\epsilon),$
(39) $\displaystyle i^{\text{fin}}_{[32,32x2]}$
$\displaystyle=\frac{s^{2}-k_{2}k_{4}}{2s}\mathcal{D}^{(+)}+\frac{k_{2}-k_{4}}{2}\mathcal{D}^{(-)}+\frac{k_{2}+k_{4}+2s}{4}\log^{2}\left(\frac{l_{12+}}{l_{34+}}\right)$
$\displaystyle\qquad\qquad+(k_{3}+k_{4})\log\left(\frac{k_{T}}{l_{34+}}\right)+(k_{1}+k_{2})\log\left(\frac{k_{T}}{l_{12+}}\right)$
$\displaystyle\qquad\qquad-3k_{T}-\frac{k_{2}+k_{4}}{2}-3s.$ (40)

  * •

$i_{[32,33x2]}$ exhibits a double pole,

$\displaystyle i^{\text{div}}_{[32,33x2]}$
$\displaystyle=\frac{s^{2}-k_{3}^{2}-k_{4}^{2}}{4}\operatorname{Div}_{1}^{2}(l_{34+})+\left[s(k_{3}+k_{4})-(k_{3}^{2}+k_{3}k_{4}+k_{4}^{2})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\frac{5s^{2}+k_{1}^{2}-k_{2}^{2}}{4}\right]\operatorname{Div}_{2}(l_{34+})+O(\epsilon),$
(41) $\displaystyle i^{\text{fin}}_{[32,33x2]}$
$\displaystyle=\frac{s^{2}-k_{3}^{2}-k_{4}^{2}}{4}\left[\frac{k_{2}}{s}\mathcal{D}^{(+)}+\mathcal{D}^{(-)}\right]-\frac{s^{2}-k_{3}^{2}-k_{4}^{2}}{8}\log^{2}\left(\frac{l_{34+}}{l_{12+}}\right)$
$\displaystyle\qquad\qquad+\frac{1}{4}\left[k_{1}^{2}-k_{2}^{2}+(2s-k_{3}-k_{4})(k_{3}+k_{4})\right]\log\left(\frac{l_{34+}}{l_{12+}}\right)$
$\displaystyle\qquad\qquad+(s+k_{2})(k_{3}+k_{4})\log\left(\frac{k_{T}}{l_{12+}}\right)$
$\displaystyle\qquad\qquad+\frac{1}{4}\left[k_{1}^{2}-(k_{4}+k_{3}-k_{2})^{2}+4s(k_{3}+k_{4})\right]\log\left(\frac{l_{12+}}{k_{T}}\right)$
$\displaystyle\qquad\qquad+\frac{1}{16}\left[17s^{2}+5k_{1}^{2}-4k_{1}k_{2}-9k_{2}^{2}+4(5s+k_{1}-k_{2})\sigma_{(1)34}\right.$
$\displaystyle\qquad\qquad\qquad\quad\left.-12\sigma_{(1)34}^{2}+8\sigma_{(2)34}\right],$
(42)

where $\sigma_{(m)J}$ is defined in (462).

  * •

$i_{[33,33x2]}$ exhibits a single pole,

$\displaystyle i^{\text{div}}_{[33,33x2]}$
$\displaystyle=-\frac{k_{1}^{3}+k_{2}^{3}+k_{3}^{3}+k_{4}^{3}}{6}\operatorname{Div}_{2}(k_{T})+O(\epsilon),$
(43) $\displaystyle i^{\text{fin}}_{[33,33x2]}$
$\displaystyle=-\frac{(s^{2}-k_{1}^{2}-k_{2}^{2})(s^{2}-k_{3}^{2}-k_{4}^{2})}{8s}\mathcal{D}^{(+)}$
$\displaystyle\qquad\qquad+\frac{1}{4}(s^{2}-k_{1}^{2}-k_{2}^{2})(k_{3}+k_{4})\log\left(\frac{l_{12+}}{k_{T}}\right)$
$\displaystyle\qquad\qquad+\frac{1}{4}(s^{2}-k_{3}^{2}-k_{4}^{2})(k_{1}+k_{2})\log\left(\frac{l_{34+}}{k_{T}}\right)$
$\displaystyle\qquad\qquad+\frac{1}{72}\left(18s^{2}\sigma_{(1)1234}-25\sigma_{(1)1234}^{3}+63\sigma_{(1)1234}\sigma_{(2)1234}-27\sigma_{(3)1234}\right)$
$\displaystyle\qquad\qquad-\frac{s}{4}(k_{1}+k_{2})(k_{3}+k_{4}).$ (44)

####  Exchange diagrams with $\Delta_{x}=3$

For $\Delta_{x}=3$, we find:

  * •

$i_{[22,22x3]}$ is finite,

$\displaystyle i^{\text{div}}_{[22,22x3]}$ $\displaystyle=0+O(\epsilon),$ (45)
$\displaystyle i^{\text{fin}}_{[22,22x3]}$
$\displaystyle=\frac{(k_{1}+k_{2})(k_{3}+k_{4})}{2s^{3}}\mathcal{D}^{(+)}-\frac{k_{1}+k_{2}}{s^{2}}\log\left(\frac{l_{12+}}{k_{T}}\right)-\frac{k_{3}+k_{4}}{s^{2}}\log\left(\frac{l_{34+}}{k_{T}}\right)+\frac{1}{s}.$
(46)

  * •

$i_{[32,22x3]}$ exhibits a single pole,

$\displaystyle i^{\text{div}}_{[32,22x3]}$
$\displaystyle=\frac{1}{2}\operatorname{Div}_{2}(l_{12+})+O(\epsilon),$ (47)
$\displaystyle i^{\text{fin}}_{[32,22x3]}$
$\displaystyle=\frac{(s^{2}+k_{1}^{2}-k_{2}^{2})(k_{3}+k_{4})}{4s^{3}}\mathcal{D}^{(+)}$
$\displaystyle\qquad\qquad+\frac{k_{1}^{2}-k_{2}^{2}}{2s^{2}}\log\left(\frac{k_{T}}{l_{12+}}\right)+\frac{(k_{1}-k_{2})(k_{3}+k_{4})}{2s^{2}}\log\left(\frac{k_{T}}{l_{34+}}\right)$
$\displaystyle\qquad\qquad+\frac{3s+4(k_{1}-k_{2})}{8s}.$ (48)

  * •

$i_{[33,22x3]}$ exhibits a double pole,

$\displaystyle i^{\text{div}}_{[33,22x3]}$
$\displaystyle=-\frac{k_{3}+k_{4}}{6}\operatorname{Div}_{1}^{2}(l_{34+})-\frac{6s+7(k_{3}+k_{4})}{9}\operatorname{Div}_{2}(l_{34+})+O(\epsilon),$
(49) $\displaystyle i^{\text{fin}}_{[33,22x3]}$
$\displaystyle=-\frac{k_{3}+k_{4}}{6}\left[\frac{k_{1}^{3}+k_{2}^{3}}{s^{3}}\mathcal{D}^{(+)}+\mathcal{D}^{(-)}\right]$
$\displaystyle\qquad\qquad+\frac{k_{3}+k_{4}}{12}\log^{2}\left(\frac{l_{12+}}{l_{34+}}\right)+\frac{s^{3}+k_{1}^{3}+k_{2}^{3}}{3s^{2}}\log\left(\frac{l_{12+}}{l_{34+}}\right)$
$\displaystyle\qquad\qquad-\frac{k_{T}(k_{1}^{2}-k_{1}k_{2}+k_{2}^{2})}{3s^{2}}\log\left(\frac{k_{T}}{l_{34+}}\right)$
$\displaystyle\qquad\qquad-\frac{1}{3}\left[\frac{k_{1}^{2}-k_{1}k_{2}+k_{2}^{2}}{s}+(k_{1}+k_{2})+\frac{7}{3}s+\frac{43}{18}(k_{3}+k_{4})\right].$
(50)

  * •

$i_{[32,32x3]}$ exhibits a single pole,

$\displaystyle i^{\text{div}}_{[32,32x3]}$
$\displaystyle=-\frac{k_{2}+k_{4}}{2}\operatorname{Div}_{2}(k_{T})+O(\epsilon),$
(51) $\displaystyle i^{\text{fin}}_{[32,32x3]}$
$\displaystyle=\frac{(s^{2}+k_{1}^{2}-k_{2}^{2})(s^{2}+k_{3}^{2}-k_{4}^{2})}{8s^{3}}\mathcal{D}^{(+)}$
$\displaystyle\qquad\qquad+\frac{(s^{2}+k_{1}^{2}-k_{2}^{2})(k_{3}-k_{4})}{4s^{2}}\log\left(\frac{k_{T}}{l_{12+}}\right)$
$\displaystyle\qquad\qquad+\frac{(s^{2}+k_{3}^{2}-k_{4}^{2})(k_{1}-k_{2})}{4s^{2}}\log\left(\frac{k_{T}}{l_{34+}}\right)$
$\displaystyle\qquad\qquad+\frac{1}{8s}\left[2(k_{1}-k_{2})(k_{3}-k_{4})-2s(k_{1}+k_{3})-5s(k_{2}+k_{4})\right].$
(52)

  * •

$i_{[32,33x3]}$ exhibits a double pole,

$\displaystyle i^{\text{div}}_{[32,33x3]}$
$\displaystyle=-\frac{s^{2}+k_{1}^{2}-k_{2}^{2}}{12}\operatorname{Div}_{1}^{2}(l_{12+})$
$\displaystyle\qquad\qquad-\frac{1}{36}\left[11s^{2}+12s(k_{1}-k_{2})+4(2k_{1}-5k_{2})(k_{1}+k_{2})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+9(k_{3}^{2}+k_{4}^{2})\right]\operatorname{Div}_{2}(l_{12+})+O(\epsilon),$
(53) $\displaystyle i^{\text{fin}}_{[32,33x3]}$
$\displaystyle=\frac{(s^{2}+k_{1}^{2}-k_{2}^{2})}{12}\left[-\frac{k_{3}^{3}+k_{4}^{3}}{s^{3}}\mathcal{D}^{(+)}+\mathcal{D}^{(-)}\right]$
$\displaystyle\qquad\qquad+\frac{s^{2}+k_{1}^{2}-k_{2}^{2}}{24}\log^{2}\left(\frac{l_{12+}}{l_{34+}}\right)-\frac{(k_{1}-k_{2})(s^{3}+k_{3}^{3}+k_{4}^{3})}{6s^{2}}\log\left(\frac{l_{12+}}{l_{34+}}\right)$
$\displaystyle\qquad\qquad-\frac{k_{T}}{12}\left[k_{1}+k_{2}-k_{3}-k_{4}+\frac{2(k_{1}-k_{2})(k_{3}^{2}-k_{3}k_{4}+k_{4}^{2})}{s^{2}}\right]\log\left(\frac{k_{T}}{l_{12+}}\right)$
$\displaystyle\qquad\qquad-\frac{(k_{1}-k_{2})(k_{3}^{2}-k_{3}k_{4}+k_{4}^{2})}{6s}+\frac{1}{432}\left[-52k_{1}^{2}+292k_{2}^{2}+108k_{2}(k_{3}+k_{4})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.-9(11k_{3}^{2}+4k_{3}k_{4}+11k_{4}^{2})+12k_{1}(20k_{2}-3k_{3}-3k_{4})\right]$
$\displaystyle\qquad\qquad-\frac{s(11k_{1}-17k_{2})}{36}-\frac{85s^{2}}{432}.$
(54)

  * •

$i_{[33,33x3]}$ exhibits a double pole,

$\displaystyle i^{\text{div}}_{[33,33x3]}$
$\displaystyle=\frac{s^{3}+k_{1}^{3}+k_{2}^{3}}{18}\operatorname{Div}_{1}^{2}(l_{12+})+\frac{s^{3}+k_{3}^{3}+k_{4}^{3}}{18}\operatorname{Div}_{1}^{2}(l_{34+})$
$\displaystyle\qquad\qquad+\frac{1}{9}\left[3\sigma_{(1)12s}^{3}-7\sigma_{(1)12s}\sigma_{(2)12s}+\sigma_{(3)12s}-\frac{s^{3}}{3}\right]\operatorname{Div}_{2}(l_{12+})$
$\displaystyle\qquad\qquad+\frac{1}{9}\left[3\sigma_{(1)34s}^{3}-7\sigma_{(1)34s}\sigma_{(2)34s}+\sigma_{(3)34s}-\frac{s^{3}}{3}\right]\operatorname{Div}_{2}(l_{34+})+O(\epsilon),$
(55) $\displaystyle i^{\text{fin}}_{[33,33x3]}$
$\displaystyle=-\frac{2s^{3}+k_{1}^{3}+k_{2}^{3}+k_{3}^{3}+k_{4}^{3}}{36}\log^{2}\left(\frac{l_{12+}}{l_{34+}}\right)$
$\displaystyle\qquad\qquad+\frac{1}{9}\left[s^{2}(k_{1}+k_{2})+s(k_{1}^{2}-k_{1}k_{2}+k_{2}^{2}-k_{3}^{2}+k_{3}k_{4}-k_{4}^{2})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\frac{(k_{1}+k_{2})^{3}}{3}-\frac{(k_{1}^{3}+k_{2}^{3})(k_{3}^{2}-k_{3}k_{4}+k_{4}^{2})}{s^{2}}\right]\log\left(\frac{l_{12+}}{k_{T}}\right)$
$\displaystyle\qquad\qquad+\frac{1}{9}\left[s^{2}(k_{3}+k_{4})+s(k_{3}^{2}-k_{3}k_{4}+k_{4}^{2}-k_{1}^{2}+k_{1}k_{2}-k_{2}^{2})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\frac{(k_{3}+k_{4})^{3}}{3}-\frac{(k_{3}^{3}+k_{4}^{3})(k_{1}^{2}-k_{1}k_{2}+k_{2}^{2})}{s^{2}}\right]\log\left(\frac{l_{34+}}{k_{T}}\right)$
$\displaystyle\qquad\qquad+\frac{(k_{1}^{3}+k_{2}^{3})(k_{3}^{3}+k_{4}^{3})-s^{6}}{18s^{3}}\mathcal{D}^{(+)}+\frac{k_{3}^{3}+k_{4}^{3}-k_{1}^{3}-k_{2}^{3}}{18}\mathcal{D}^{(-)}$
$\displaystyle\qquad\qquad+\frac{1}{27}\Big{[}14s^{3}+11s^{2}k_{T}+s\left(8(k_{1}^{2}+k_{2}^{2}+k_{3}^{2}+k_{4}^{2})-5(k_{1}k_{2}+k_{3}k_{4})\right)$
$\displaystyle\qquad\qquad\qquad\qquad+\frac{1}{6}\left(61\sigma_{(1)1234}^{3}-129\sigma_{(1)1234}\sigma_{(2)1234}+15\sigma_{(3)1234}\right)$
$\displaystyle\qquad\qquad\qquad\qquad+5\left(2\sigma_{(1)34}\sigma_{(2)12}+2\sigma_{(1)12}\sigma_{(2)34}-\sigma_{(1)12}\sigma_{(1)34}k_{T}\right)$
$\displaystyle\qquad\qquad\qquad\qquad+\frac{3(k_{1}^{2}-k_{1}k_{2}+k_{2}^{2})(k_{3}^{2}-k_{3}k_{4}+k_{4}^{2})}{s}\Big{]}.$
(56)

##  Renormalization

While the amplitudes listed in the previous section are naturally associated
with AdS Witten diagrams, we do not necessarily have to invoke any form of
holography. In principle, it is sufficient simply to write down the
expressions (7)-(• ‣ 2) and verify that they solve the regulated conformal
Ward identities (CWIs). This check is performed explicitly in the Mathematica
notebook `RegulatedAmplitudes.nb` using the Ward identities listed in Appendix
B. Since every amplitude satisfies the Ward identities, each potentially
represents a valid correlation function.

However, the fact that a given expression satisfies the CWIs does not by
itself imply that a CFT with such a 4-point function actually exists. Other
constraints, for example unitarity or the bootstrap, could rule out the
existence of such a CFT. Nevertheless, we will show in Section 4.2 that, for
each amplitude listed in the previous section, a holographic CFT can be
constructed for which the corresponding correlation function is equal to this
amplitude, i.e., a bulk action exists such that the Witten diagram is the
entire correlator. If it turns out that a CFT with the desired spectrum and
correlators does not exist, then the corresponding bulk theory would be in the
swampland.

Next, we will discuss the renormalization of the regulated amplitudes listed
in the previous section. In a textbook approach, it is usually the correlation
functions themselves that are renormalized. However, as every regulated
amplitude here is equal to a valid regulated correlator in some CFT, every
amplitude must be individually renormalizable. This means, for example, that
its divergences must be of an appropriately local form, so that eventually
they can be removed by a suitable local counterterm. In particular, the
divergences of an $n$-point function must take the form of lower-point
functions of the same operators, plus possible additional ultralocal pieces.
Our central focus will therefore be on renormalizing individual Witten
diagrams. We will be able to write down counterterm actions and counterterm
contributions that can be used in the renormalization procedure once the
theory is specified. The renormalized amplitudes are listed in Section 4.3.

As an illustration of the complete renormalization procedure, we will consider
two holographic theories which we call _symmetric_ and _asymmetric_. The
asymmetric theory, defined in Section 4.2, contains five operators of
dimension two or three and is the least symmetric theory in the sense that
there are no non-vanishing 4-point functions containing identical operators.
Even when some operators have the same dimension, the correlators are not
symmetric under exchange of these operators as they are not identical. The
symmetric theory, on the other hand, contains only two operators, one of
dimension two and one of dimension three, and there are non-vanishing 4-point
function containing identical operators. These correlators are symmetric under
exchange of the operators of the same dimension. The dual bulk theory contains
bulk scalars corresponding to the boundary operators (five and two scalars,
respectively) plus suitable $3$\- and $4$-point interaction terms. In Sections
4.4 and 4.5, we carry out the complete renormalization procedure for these two
theories. For the symmetric theory, we further derive the beta functions and
anomalies and verify the Callan-Symanzik equation.

From a holographic perspective, the renormalization procedure we adopt is a
hybrid one: boundary correlation functions are derived from bulk amplitudes at
generic dimensions $d$ and $\Delta_{j}$, and the subsequent renormalization is
performed entirely within the boundary theory. Thus, instead of regulating the
bulk theory by cutting off the radial direction as per holographic
renormalization, we employ dimensional regularization in the boundary theory
as, e.g., in Schwimmer:2000cu ; Schwimmer:2003eq . The resulting counterterms
are an integral part of the bulk theory (dimensionally regularized as in
Bzowski:2016kni ; Bzowski:2019xri ) as the counterterms are also uniquely
determined by requiring the bulk variational problem to be well-posed
Papadimitriou:2005ii .

###  Conventions and definitions

In this section, we use the following conventions and definitions:

  * •

The counterterm contribution $i^{\text{ct}}$ is added to the regulated
amplitude $\hat{i}$, to yield the finite renormalized amplitude
$i^{\text{ren}}$

$\displaystyle i^{\text{ren}}=\lim_{\epsilon\rightarrow
0}\left[\hat{i}+i^{\text{ct}}\right].$ (57)

  * •

For the exchange amplitudes, we will list only their _logarithmic terms_
defined by

$\displaystyle
i^{\text{log}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}=\lim_{\epsilon\rightarrow
0}\left[i^{\text{div}}_{[\Delta_{1}\Delta_{2},\Delta_{3},\Delta_{4}x\Delta_{x}]}+i^{\text{ct}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}\right].$
(58)

The full amplitude then reads

$\displaystyle
i^{\text{ren}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}=i^{\text{log}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}+i^{\text{fin}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]},$
(59)

where the finite parts $i^{\text{fin}}$ are as listed in Sections 3.4.1 and
3.4.2.

  * •

As previously, the Euclidean $\text{AdS}_{4}$ metric in Poincaré coordinates
is

$\displaystyle\mathrm{d}s^{2}=\frac{\mathrm{d}z^{2}+\mathrm{d}\bm{x}^{2}}{z^{2}}.$
(60)

The AdS/CFT correspondence in the half-integer regularization scheme then
gives

$\displaystyle W[\phi_{j}]=-S_{\text{on-shell}}[\phi_{j}],$
$\displaystyle\langle\mathcal{O}\rangle_{s,\text{reg}}=-(2\Delta-d)\phi_{(\Delta+\epsilon)}.$
(61)

In particular, the combination $2\Delta-d$ remains unchanged in the half-
integer scheme.

For every operator $\mathcal{O}_{j}$, we denote its source by $\phi_{j}$. If
the operator has dimension $\Delta_{j}$, its source has dimension
$d-\Delta_{j}$ and thus $\mathcal{O}_{j}^{[\Delta_{j}]}$ couples to
$\phi_{j}^{[d-\Delta_{j}]}$. In particular $\mathcal{O}^{[2]}$ is sourced by
$\phi^{[1]}$, while $\mathcal{O}^{[3]}$ is sourced by $\phi^{[0]}$.

  * •

All counterterms appear with the renormalization scale $\mu$, and multiplied
by constants that we collectively denote as $\mathfrak{a}$. These constants
are series in $\epsilon$,

$\displaystyle\mathfrak{a}=1+\epsilon\,\mathfrak{a}^{(1)}+\epsilon^{2}\,\mathfrak{a}^{(2)}+O(\epsilon^{3}),$
(62)

and the $\epsilon$-dependent part is generally scheme-dependent. As we show,
one may choose a scheme such that the scheme-dependent part depends only on
the dimensions of the operators under consideration, and not on any other
labels of the operators.

###  Every amplitude is a correlator

Here we consider the holographic theory that realizes each amplitude (Witten
diagram) as a stand-alone correlation function. We consider five scalar fields
$\Phi_{j}$ with $j=1,2,3,4,x$ dual to operators $\mathcal{O}_{j}$ of arbitrary
dimensions $\Delta_{j}$. The (unregulated) bulk action, up to quartic order,
is

$\displaystyle S^{\text{asym}}$
$\displaystyle=\frac{1}{2}\int\mathrm{d}^{d}x\sqrt{g}\sum_{j=1,2,3,4,x}\left[\partial_{\mu}\Phi_{j}\partial^{\mu}\Phi_{j}+m^{2}_{\Delta_{j}}\Phi_{j}^{2}\right]$
$\displaystyle\qquad+\int\mathrm{d}^{d}x\sqrt{g}\left[\lambda_{12x}\Phi_{1}\Phi_{2}\Phi_{x}+\lambda_{34x}\Phi_{x}\Phi_{3}\Phi_{4}-\lambda_{1234}\Phi_{1}\Phi_{2}\Phi_{3}\Phi_{4}\right],$
(63)

where

$\displaystyle m^{2}_{\Delta}=\Delta(\Delta-d)$ (64)

and $\lambda_{12x},\lambda_{34x},\lambda_{1234}$ are arbitrary AdS couplings.
We refer to this theory as the _asymmetric theory_ , since the resulting
correlators have as few discrete symmetries as possible. Later on, in Section
4.5, we will introduce the symmetric theory where all correlators exhibit some
form of crossing symmetry.

####  Regulated correlators

Let us now derive the regulated correlators of the asymmetric theory (4.2),
following Skenderis:2002wp .

  * •

Since the action (4.2) is symmetric under the exchange of the fields
$\Phi_{1}$ and $\Phi_{2}$, their dynamics is identical. The same conclusion
holds for $\Phi_{3}$ and $\Phi_{4}$. The equations of motion for the bulk
fields read

$\displaystyle(-\Box_{AdS}+\hat{m}^{2}_{\Delta_{1}})\Phi_{1}=-\lambda_{12x}\Phi_{2}\Phi_{x}+\lambda_{1234}\Phi_{2}\Phi_{3}\Phi_{4},$
(65)
$\displaystyle(-\Box_{AdS}+\hat{m}^{2}_{\Delta_{3}})\Phi_{3}=-\lambda_{34x}\Phi_{4}\Phi_{x}+\lambda_{1234}\Phi_{1}\Phi_{2}\Phi_{4},$
(66)
$\displaystyle(-\Box_{AdS}+\hat{m}^{2}_{\Delta_{x}})\Phi_{x}=-\lambda_{12x}\Phi_{1}\Phi_{2}-\lambda_{34x}\Phi_{3}\Phi_{4}.$
(67)

  * •

The equations of motion can now be solved perturbatively to obtain terms with
up to and including three sources. Let us denote by $\Phi_{n\\{j\\}}$ the term
in the solution depending on any $(j+1)$ sources. Thus, for example,
$\Phi_{n\\{0\\}}$ is the solution to the free field equation. Concentrating on
$\Phi_{1}$, in position space we find

$\displaystyle\Phi_{1\\{0\\}}$
$\displaystyle=\operatorname{\mathcal{K}}_{[\Delta_{1}]}\ast\phi_{1},$ (68)
$\displaystyle\Phi_{1\\{1\\}}$
$\displaystyle=-\lambda_{12x}\operatorname{\mathcal{G}}_{[\Delta_{1}]}\ast(\Phi_{2\\{0\\}}\Phi_{x\\{0\\}}),$
(69) $\displaystyle\Phi_{1\\{2\\}}$
$\displaystyle=-\lambda_{12x}\operatorname{\mathcal{G}}_{[\Delta_{1}]}\ast\left[\Phi_{2\\{1\\}}\Phi_{x\\{0\\}}+\Phi_{2\\{0\\}}\Phi_{x\\{1\\}}\right]$
$\displaystyle\qquad\qquad+\lambda_{1234}\operatorname{\mathcal{G}}_{[\Delta_{1}]}\ast(\Phi_{2\\{0\\}}\Phi_{3\\{0\\}}\Phi_{4\\{0\\}}),$
(70)

where $\ast$ indicates the convolution of the position-space variables, e.g.,

$\displaystyle\Phi_{1\\{0\\}}(x)=\int\mathrm{d}^{d+1}x^{\prime}\,\sqrt{g(x^{\prime})}\operatorname{\mathcal{K}}_{[\Delta_{1}]}(x,x^{\prime})\phi_{1}(x^{\prime}).$
(71)

  * •

The holographic 2-point functions are diagonal and given by the standard
expression

$\displaystyle\langle\\!\langle\mathcal{O}_{j}(\bm{k})\mathcal{O}_{j}(-\bm{k})\rangle\\!\rangle=(2\Delta-d)\times\text{
coefficient of }z^{\Delta}\text{ in
}\operatorname{\mathcal{K}}_{\Delta}(z,k).$ (72)

In Section 3.1, we defined the 2-point amplitudes $i_{[\Delta\Delta]}$ in (7)
in such a way that they are equal to the holographic 2-point function,

$\displaystyle\langle\\!\langle\mathcal{O}_{j}(\bm{k})\mathcal{O}_{j}(-\bm{k})\rangle\\!\rangle=i_{[\Delta_{j}\Delta_{j}]}.$
(73)

  * •

To go to the boundary for higher-point functions, we use the fact that

$\displaystyle\operatorname{\mathcal{G}}_{[\Delta]}(z,k;\zeta)=\frac{z^{\Delta}}{2\Delta-d}\operatorname{\mathcal{K}}_{[\Delta]}(\zeta,k)+O(z^{\Delta+2}).$
(74)

Thus, from (61) we see that regulated 1-point functions with sources reads

$\displaystyle\langle\mathcal{O}_{1}\rangle_{\\{1\\}s,\text{reg}}$
$\displaystyle=\lambda_{12x}\operatorname{\mathcal{K}}_{[\Delta_{1}]}\ast(\operatorname{\mathcal{K}}_{[\Delta_{2}]}\ast\phi_{2})(\operatorname{\mathcal{K}}_{[\Delta_{x}]}\ast\phi_{x}),$
(75) $\displaystyle\langle\mathcal{O}_{1}\rangle_{\\{2\\}s,\text{reg}}$
$\displaystyle=-\lambda_{12x}\lambda_{34x}\operatorname{\mathcal{K}}_{[\Delta_{1}]}\ast\left[(\operatorname{\mathcal{K}}_{[\Delta_{2}]}\ast\phi_{2})\operatorname{\mathcal{G}}_{[\Delta_{x}]}\ast(\operatorname{\mathcal{K}}_{[\Delta_{3}]}\ast\phi_{3})(\operatorname{\mathcal{K}}_{[\Delta_{4}]}\ast\phi_{4})\right]$
$\displaystyle\qquad\qquad-\lambda_{12x}^{2}\operatorname{\mathcal{K}}_{[\Delta_{1}]}\ast\left[(\operatorname{\mathcal{K}}_{[\Delta_{x}]}\ast\phi_{x})\operatorname{\mathcal{G}}_{[\Delta_{2}]}\ast(\operatorname{\mathcal{K}}_{[\Delta_{1}]}\ast\phi_{1})(\operatorname{\mathcal{K}}_{[\Delta_{x}]}\ast\phi_{x})\right]$
$\displaystyle\qquad\qquad-\lambda_{1234}\operatorname{\mathcal{K}}_{[\Delta_{1}]}\ast(\operatorname{\mathcal{K}}_{[\Delta_{2}]}\ast\phi_{2})(\operatorname{\mathcal{K}}_{[\Delta_{3}]}\ast\phi_{3})(\operatorname{\mathcal{K}}_{[\Delta_{4}]}\ast\phi_{4})$
(76)

The $(n+1)$-point function is then obtained by taking $n$ functional
derivatives with the factor $(-1)^{n}$.

These expressions represent the 3- and 4-point Witten diagrams, but with
sources present. For example,

$\displaystyle\langle\mathcal{O}_{1}(x_{1})\rangle_{\\{1\\}s,\text{reg}}=\lambda_{12x}\int\mathrm{d}^{d+1}x\,\sqrt{g(x)}\operatorname{\mathcal{K}}_{[\Delta_{1}]}(x_{1},x)\times$
$\displaystyle\qquad\times\int\mathrm{d}^{d+1}x_{2}\,\sqrt{g(x_{2})}\operatorname{\mathcal{K}}_{[\Delta_{2}]}(x,x_{2})\phi_{2}(x_{2})\int\mathrm{d}^{d+1}x_{3}\,\sqrt{g(x_{3})}\operatorname{\mathcal{K}}_{[\Delta_{x}]}(x,x_{3})\phi_{x}(x_{3})$
$\displaystyle=\lambda_{12x}\int\mathrm{d}^{d+1}x_{2}\sqrt{g(x_{2})}\int\mathrm{d}^{d+1}x_{3}\sqrt{g(x_{3})}\,i_{[\Delta_{1}\Delta_{2}\Delta_{3}]}(x_{1},x_{2},x_{3})\phi_{2}(x_{2})\phi_{x}(x_{3}),$
(77)

where here $i_{[\Delta_{1}\Delta_{2}\Delta_{3}]}(x_{1},x_{2},x_{3})$ denotes
the position space expression for the 3-point amplitude
$i_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$. We will not work in position space in
this paper any further.

$\mathcal{O}_{1}(x_{1})$$\phi_{3}(x_{3})$$\phi_{2}(x_{2})$$\operatorname{\mathcal{K}}_{[\Delta_{1}]}$$\operatorname{\mathcal{K}}_{[\Delta_{x}]}$$\operatorname{\mathcal{K}}_{[\Delta_{2}]}$$\lambda_{12x}$

$\mathcal{O}_{1}(x_{1})$$\phi_{2}(x_{2})$$\phi_{3}(x_{3})$$\phi_{4}(x_{4})$$\operatorname{\mathcal{K}}_{[\Delta_{1}]}$$\operatorname{\mathcal{K}}_{[\Delta_{2}]}$$\operatorname{\mathcal{K}}_{[\Delta_{3}]}$$\operatorname{\mathcal{K}}_{[\Delta_{4}]}$$\operatorname{\mathcal{G}}_{[\Delta_{x}]}$$\lambda_{34x}$$\lambda_{12x}$

Figure 3: 3- and 4-point Witten diagrams corresponding to (75) and the first
line of (76).

  * •

Directly from the action (4.2), we see that there are two non-vanishing
3-point functions,
$\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{x}\rangle$ and
$\langle\mathcal{O}_{x}\mathcal{O}_{3}\mathcal{O}_{4}\rangle$. From (69), we
see that

$\displaystyle\langle\\!\langle\mathcal{O}_{i}(\bm{k}_{1})\mathcal{O}_{j}(\bm{k}_{2})\mathcal{O}_{x}(\bm{k}_{3})\rangle\\!\rangle$
$\displaystyle=\lambda_{ijx}\,i_{[\Delta_{i}\Delta_{j}\Delta_{x}]}(k_{1},k_{2},k_{3}),$
(78)

where $(ij)=(12)$ or $(34)$.

  * •

Among the non-vanishing 4-point functions, the most important is
$\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle$.
This contains two contributions: a single exchange diagram and a single
contact diagram. From (70),

$\displaystyle\langle\\!\langle\mathcal{O}_{1}(\bm{k}_{1})\mathcal{O}_{2}(\bm{k}_{2})\mathcal{O}_{3}(\bm{k}_{3})\mathcal{O}_{4}(\bm{k}_{4})\rangle\\!\rangle$
$\displaystyle\qquad\qquad=\lambda_{12x}\lambda_{34x}\,i_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}+\lambda_{1234}\,i_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}.$
(79)

If we want a bulk theory where the contact diagram
$i_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}$ is a correlator on its own,
it therefore suffices to consider a bulk action with $\lambda_{12x}=0$ or
$\lambda_{34x}=0$. On the other hand, if we want the exchange diagram to be a
correlator on its own, then we need $\lambda_{1234}=0$.

Note that $i_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}$ and
$i_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}$ only depend on
the dimensions on the operators involved – not on any other labels that the
operators may have, or whether the operators are identical or not – and so we
have just demonstrated that each is on its own a correlator in a specific bulk
theory.

  * •

For completeness, we note that the asymmetric theory has additional non-
vanishing 4-point functions:

$\displaystyle\langle\\!\langle\mathcal{O}_{i}(\bm{k}_{1})\mathcal{O}_{j}(\bm{k}_{2})\mathcal{O}_{j}(\bm{k}_{3})\mathcal{O}_{i}(\bm{k}_{4})\rangle\\!\rangle$
$\displaystyle\quad=\lambda_{ijk}^{2}\,\left[i_{[\Delta_{i}\Delta_{j},\Delta_{j}\Delta_{i}x\Delta_{k}]}(\bm{k}_{1},\bm{k}_{2};\bm{k}_{3},\bm{k}_{4})+i_{[\Delta_{i}\Delta_{j},\Delta_{j}\Delta_{i}x\Delta_{k}]}(\bm{k}_{1},\bm{k}_{3};\bm{k}_{2},\bm{k}_{4})\right],$
(80)

where $(ijk)=(12x)$ or $(34x)$ or any permutation within each bracket (see
Figure 4 on page 4).

In conclusion, all $3$\- and $4$-point Witten diagrams are valid $3$\- and
$4$-point functions as they can be realized in the holographic theory given by
(4.2) for specific choices for couplings. Additional symmetry factors may
appear if correlation functions of the same operator are considered. Similar
conclusions can be reached for all higher-point diagrams.

###  Renormalized amplitudes

####  3-point functions

Since every amplitude is a correlator, we can now list the counterterm actions
that renormalize the $3$\- and $4$-point functions in (78) - (• ‣ 4.2.1). We
set $\lambda_{12x}=\lambda_{34x}=\lambda_{1234}=1$, so the counterterm
contributions derived from the actions cure the divergences of the amplitudes
only. We will restore the $\lambda$ dependence when we discuss specific bulk
actions. Let us start with the 3-point function to give some more details on
the procedure.

  * •

The form of the counterterms depends on the dimensions of the operators
involved, and read

$\displaystyle S_{[222]}^{\text{ct}\,(3)}$
$\displaystyle=\mathfrak{s}_{[222]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\phi^{[1]}_{1}\phi^{[1]}_{2}\phi^{[1]}_{3},$
(81) $\displaystyle S_{[322]}^{\text{ct}\,(3)}$
$\displaystyle=\mathfrak{s}_{[322]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\phi^{[0]}_{1}(\phi^{[1]}_{2}\mathcal{O}^{[2]}_{3}+\phi^{[1]}_{3}\mathcal{O}^{[2]}_{2}),$
(82) $\displaystyle S_{[332]}^{\text{ct}\,(3)}$
$\displaystyle=\mathfrak{s}_{[332]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\phi^{[1]}_{3}\partial_{\mu}\phi^{[0]}_{1}\partial^{\mu}\phi^{[0]}_{2},$
(83) $\displaystyle S_{[333]}^{\text{ct}\,(3)}$
$\displaystyle=\mathfrak{s}_{[333]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,(\phi^{[0]}_{1}\phi^{[0]}_{2}\mathcal{O}^{[3]}_{3}+\phi^{[0]}_{2}\phi^{[0]}_{3}\mathcal{O}^{[3]}_{1}+\phi^{[0]}_{3}\phi^{[0]}_{1}\mathcal{O}^{[3]}_{2}),$
(84)

where the values of the constants are

$\displaystyle\mathfrak{s}_{[222]}=-\operatorname{Div}_{1}(\mu)\mathfrak{a}_{[222]},$
$\displaystyle\mathfrak{s}_{[322]}=\operatorname{Div}_{1}(\mu)\mathfrak{a}_{[322]},$
(85)
$\displaystyle\mathfrak{s}_{[332]}=\operatorname{Div}_{1}(\mu)\mathfrak{a}_{[332]},$
$\displaystyle\mathfrak{s}_{[333]}=\frac{1}{3}\operatorname{Div}_{1}(\mu)\mathfrak{a}_{[333]}.$
(86)

The scheme-dependent part of the constants $\mathfrak{a}$ could in principle
be different for different operators, but one may choose to work in a scheme
where they depend only on the dimensions of the operators involved and are the
same regardless of the permutation of the dimensions, e.g.,
$\mathfrak{a}_{[322]}=\mathfrak{a}_{[232]}=\mathfrak{a}_{[223]}$. In all
cases, the constants have an $\epsilon$-expansion given by (62).

  * •

Given a counterterm action
$S_{[\Delta_{1}\Delta_{2}\Delta_{3}]}^{\text{ct}\>(3)}$, its contribution
$i^{\text{ct}}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$ to the 3-point amplitude in
(78) is obtained by taking three functional derivatives, with respect to the
sources, of the generating functional incorporating the counterterm action:

$\displaystyle\frac{-\delta^{3}}{\delta\phi_{1}(\bm{x}_{1})\delta\phi_{2}(\bm{x}_{2})\delta\phi_{3}(\bm{x}_{3})}\langle\exp\left(-S_{[\Delta_{1}\Delta_{2}\Delta_{3}]}^{\text{ct}\>(3)}\right)\rangle_{s,\text{reg}}=\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}+i^{\text{ct}}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}.$
(87)

We view $\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$ as following from the
differentiation of the regulated generating functional. The expression on the
right-hand side should now possesses a finite limit as $\epsilon\rightarrow
0$.

  * •

From the counterterm actions above, the counterterm contributions to the
3-point amplitudes are

$\displaystyle i^{\text{ct}}_{[222]}$ $\displaystyle=\mathfrak{s}_{[222]},$
(88) $\displaystyle i^{\text{ct}}_{[322]}$
$\displaystyle=-\mathfrak{s}_{[322]}\times\left(\hat{i}_{[22]}(k_{2})+\hat{i}_{[22]}(k_{3})\right),$
(89) $\displaystyle i^{\text{ct}}_{[332]}$
$\displaystyle=-\mathfrak{s}_{[332]}\times(\bm{k}_{1}\cdot\bm{k}_{2}),$ (90)
$\displaystyle i^{\text{ct}}_{[333]}$
$\displaystyle=-\mathfrak{s}_{[333]}\times\left(\hat{i}_{[33]}(k_{1})+\hat{i}_{[33]}(k_{2})+\hat{i}_{[33]}(k_{3})\right).$
(91)

It is easy to see they match perfectly the divergences in (23) - (26).

  * •

A brief comment on notation is in order. The form of the counterterms (81) -
(84) depends on the dimensions of the operators involved, as indicated by the
subscript in square brackets. Here, we assume that all three operators and
their sources featuring in the counterterms are distinct, as indicated by the
indices. In particular, the counterterms functionally depend on the sources
and operators, so more accurately, listing explicitly the functional
dependence, we have

$\displaystyle S_{[\Delta_{1}\Delta_{2}\Delta_{3}]}^{\text{ct}\,(3)}$
$\displaystyle=S_{[\Delta_{1}\Delta_{2}\Delta_{3}]}^{\text{ct}\,(3)}(\phi_{1}^{[3-\Delta_{1}]},\phi_{2}^{[3-\Delta_{2}]},\phi_{3}^{[3-\Delta_{3}]};\mathcal{O}_{1}^{[\Delta_{1}]},\mathcal{O}_{2}^{[\Delta_{2}]},\mathcal{O}_{3}^{[\Delta_{3}]})$
$\displaystyle=S_{[\Delta_{1}\Delta_{2}\Delta_{3}]}^{\text{ct}\,(3)}(\phi_{1},\phi_{2},\phi_{3}).$
(92)

We will not list the arguments explicitly however unless some of them
coincide. We will also drop the operators and dimensions, as shown in the
second line, assuming that the pairing of the sources and operators is fixed.
Note that the conformal dimensions in
$S_{[\Delta_{1}\Delta_{2}\Delta_{3}]}^{\text{ct}\,(3)}$ cannot be dropped, as
they indicate which form of the counterterm we consider.

If two or more fields coincide, additional symmetry factors may be required.
If $n$ sources in a given counterterm coincide, its counterterm contribution
equals $n!\,i^{\text{ct}}$. For example,
$S_{[222]}^{\text{ct}\,(3)}(\phi_{1},\phi_{2},\phi_{3})$ produces
$i^{\text{ct}}_{[222]}$ when the three sources are different,
$S_{[222]}^{\text{ct}\,(3)}(\phi_{1},\phi_{1},\phi_{3})$ produces
$2i^{\text{ct}}_{[222]}$, while with the identical sources
$S_{[222]}^{\text{ct}\,(3)}(\phi,\phi,\phi)$ gives $6i^{\text{ct}}_{[222]}$.
Note also that some counterterms contain several terms, which would coincide
when some sources are equal.

####  4-point functions

In order to write the remaining counterterms for $4$-point functions, we
notice two facts. First, the $3$-point counterterms (81) - (84) will generally
contribute to 4-point amplitudes, and this contribution must be taken into
account before the 4-point counterterm is added. Second, the form of the
4-point counterterms cannot depend on the operator in the exchange channel,
because the counterterm action can only depend on the four external sources
and/or operators, $\phi_{j}$ and $\mathcal{O}_{j}$ for $j=1,2,3,4$.

  * •

It is convenient to list first all the types of counterterms contributing to
the 4-point amplitudes of interest, as given in Table 2.

Amplitude | Singularity type | Counterterm | Type | Contributes to
---|---|---|---|---
$2222$ | – | – | – | –
$3222$ | ultralocal | $\phi^{[0]}\phi^{[1]}\phi^{[1]}\phi^{[1]}$ | anomaly | 4-pt
| 3-pt | $\phi^{[0]}\phi^{[1]}\mathcal{O}^{[2]}$ | beta for $\phi^{[1]}$ | 3, 4-pt
| 3-pt | $\phi^{[0]}\phi^{[1]}\mathcal{O}^{[2]}$ | beta for $\phi^{[1]}$ | 3, 4-pt
$3322$ | 3-pt | $\phi^{[0]}\phi^{[0]}\mathcal{O}^{[3]}$ | beta for $\phi^{[0]}$ | 3, 4-pt
| 2-pt | $\phi^{[0]}\phi^{[0]}\phi^{[1]}\mathcal{O}^{[2]}$ | beta for $\phi^{[1]}$ | 4-pt
| ultralocal | $\phi^{[0]}\phi^{[0]}\phi^{[0]}\phi^{[1]}\partial^{2}$ | anomaly | 4-pt
$3332$ | 3-pt | $\phi^{[0]}\phi^{[1]}\mathcal{O}^{[2]}$ | beta for $\phi^{[1]}$ | 3, 4-pt
| 3-pt | $\phi^{[0]}\phi^{[0]}\mathcal{O}^{[3]}$ | beta for $\phi^{[0]}$ | 3, 4-pt
$3333$ | 3-pt | $\phi^{[0]}\phi^{[0]}\mathcal{O}^{[3]}$ | beta for $\phi^{[0]}$ | 3, 4-pt
| 2-pt | $\phi^{[0]}\phi^{[0]}\phi^{[0]}\mathcal{O}^{[3]}$ | beta for $\phi^{[0]}$ | 4-pt
Table 2: Singularity and counterterm types for various 4-point amplitudes. The
first column specifies the type of amplitude: only the dimensions of the
external operators as listed are relevant. The second column presents the
general form of the singularities arising in the given amplitudes. Where more
than a single type of singularity arises, we list the most nonlocal. The third
column shows the general form of the counterterm curing the divergence. Each
counterterm induces either an anomaly or a beta function for the couplings, as
shown by the fourth column. Finally, the fifth column tells us which $n$-point
functions the counterterm contributes to.

  * •

Starting with the counterterm actions (81) - (84), one can take four
functional derivatives similar to (87) and obtain their contributions to the
4-point amplitudes. For $\Delta_{x}=2$ these are as follows,

$\displaystyle i^{\text{ct}({3})}_{[22,22x2]}$ $\displaystyle=0,$ (93)
$\displaystyle i^{\text{ct}({3})}_{[32,22x2]}$
$\displaystyle=-\mathfrak{s}_{[322]}\hat{i}_{[222]}(s,k_{3},k_{4}),$ (94)
$\displaystyle i^{\text{ct}({3})}_{[33,22x2]}$ $\displaystyle=0,$ (95)
$\displaystyle i^{\text{ct}({3})}_{[32,32x2]}$
$\displaystyle=-\mathfrak{s}_{[322]}\left[\hat{i}_{[322]}(k_{1},k_{2},s)+\hat{i}_{[322]}(k_{3},k_{4},s)\right]+\mathfrak{s}_{[322]}^{2}\hat{i}_{[22]}(s),$
(96) $\displaystyle i^{\text{ct}({3})}_{[32,33x2]}$
$\displaystyle=-\mathfrak{s}_{[322]}\hat{i}_{[332]}(k_{3},k_{4},s),$ (97)
$\displaystyle i^{\text{ct}({3})}_{[33,33x2]}$ $\displaystyle=0$ (98)

and for $\Delta_{x}=3$ one finds,

$\displaystyle i^{\text{ct}({3})}_{[22,22x3]}$ $\displaystyle=0,$ (99)
$\displaystyle i^{\text{ct}({3})}_{[32,22x3]}$ $\displaystyle=0,$ (100)
$\displaystyle i^{\text{ct}({3})}_{[33,22x3]}$
$\displaystyle=-\mathfrak{s}_{[333]}\hat{i}_{[322]}(s,k_{3},k_{4}),$ (101)
$\displaystyle i^{\text{ct}({3})}_{[32,32x3]}$ $\displaystyle=0,$ (102)
$\displaystyle i^{\text{ct}({3})}_{[32,33x3]}$
$\displaystyle=-\mathfrak{s}_{[333]}\hat{i}_{[332]}(s,k_{1},k_{2}),$ (103)
$\displaystyle i^{\text{ct}({3})}_{[33,33x3]}$
$\displaystyle=-\mathfrak{s}_{[333]}\left[\hat{i}_{[333]}(k_{1},k_{2},s)+\hat{i}_{[333]}(s,k_{3},k_{4})\right]+\mathfrak{s}_{[333]}^{2}\hat{i}_{[33]}(s).$
(104)

  * •

We can now write down the most general local terms consistent with the
symmetries that are not fixed by the renormalization of the 3-point functions.
These are

$\displaystyle S^{\text{ct}\,(4)}_{[22,22x\Delta_{x}]}$ $\displaystyle=0,$
(105) $\displaystyle S^{\text{ct}\,(4)}_{[32,22x\Delta_{x}]}$
$\displaystyle=\mathfrak{s}_{[32,22x\Delta_{x}]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\phi^{[0]}_{1}\phi^{[1]}_{2}\phi^{[1]}_{3}\phi^{[1]}_{4},$
(106) $\displaystyle S^{\text{ct}\,(4)}_{[33,22x\Delta_{x}]}$
$\displaystyle=\mathfrak{s}_{[33,22x\Delta_{x}]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\phi^{[0]}_{1}\phi^{[0]}_{2}\left(\phi^{[1]}_{3}\mathcal{O}^{[2]}_{4}+\phi^{[1]}_{4}\mathcal{O}^{[2]}_{3}\right),$
(107) $\displaystyle S^{\text{ct}\,(4)}_{[32,32x\Delta_{x}]}$
$\displaystyle=\mathfrak{s}_{[32,32x\Delta_{x}]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\phi^{[0]}_{1}\phi^{[0]}_{3}\left(\phi^{[1]}_{2}\mathcal{O}^{[2]}_{4}+\phi^{[1]}_{4}\mathcal{O}^{[2]}_{2}\right),$
(108) $\displaystyle S^{\text{ct}\,(4)}_{[32,33x\Delta_{x}]}$
$\displaystyle=\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\left[\mathfrak{s}^{\\{3+4\\}}_{[33,23x\Delta_{x}]}\left(\phi^{[0]}_{1}\phi^{[1]}_{2}(\partial^{2}\phi^{[0]}_{3})\,\phi^{[0]}_{4}+\phi^{[0]}_{1}\phi^{[1]}_{2}\phi^{[0]}_{3}(\partial^{2}\phi^{[0]}_{4})\right)\right.$
(109)
$\displaystyle\qquad\qquad+\mathfrak{s}^{\\{1\\}}_{[32,33x\Delta_{x}]}(\partial^{2}\phi^{[0]}_{1})\,\phi^{[1]}_{2}\phi^{[0]}_{3}\phi^{[0]}_{4}+\mathfrak{s}^{\\{2\\}}_{[32,33x\Delta_{x}]}\phi^{[0]}_{1}(\partial^{2}\phi^{[1]}_{2})\,\phi^{[0]}_{3}\phi^{[0]}_{4}$
$\displaystyle\qquad\qquad\left.+\mathfrak{s}^{\\{34\\}}_{[32,33x\Delta_{x}]}\phi^{[0]}_{1}\phi^{[1]}_{2}\partial_{\mu}\phi^{[0]}_{3}\partial^{\mu}\phi^{[0]}_{4}+\mathfrak{s}^{\\{12\\}}_{[32,33x\Delta_{x}]}\partial_{\mu}(\partial^{\mu}\phi^{[0]}_{1}\,\phi^{[1]}_{2})\phi^{[0]}_{3}\phi^{[0]}_{4}\right],$
$\displaystyle S^{\text{ct}\,(4)}_{[33,33x\Delta_{x}]}$
$\displaystyle=\mathfrak{s}_{[33,33x\Delta_{x}]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\left(\phi^{[0]}_{1}\phi^{[0]}_{2}\phi^{[0]}_{3}\mathcal{O}^{[3]}_{4}+\phi^{[0]}_{2}\phi^{[0]}_{3}\phi^{[0]}_{4}\mathcal{O}^{[3]}_{1}\right.$
$\displaystyle\qquad\qquad\qquad\left.+\phi^{[0]}_{3}\phi^{[0]}_{4}\phi^{[0]}_{1}\mathcal{O}^{[3]}_{2}+\phi^{[0]}_{4}\phi^{[0]}_{1}\phi^{[0]}_{2}\mathcal{O}^{[3]}_{3}\right),$
(110)

where the labels in curly brackets in
$\mathfrak{s}^{\\{\ldots\\}}_{[32,33x\Delta_{x}]}$ indicate on which sources
the $\partial^{2}$ acts. For example, $\\{3+4\\}$ indicates it acts on
$\phi^{[0]}_{3}$ and $\phi^{[0]}_{4}$, $\\{1\\}$ that it acts on
$\phi^{[0]}_{1}$, $\\{34\\}$ in the product $\phi^{[0]}_{3}\phi^{[0]}_{4}$ and
so on, with the precise structure as listed above. Note the representation
(109) is over-complete: only four of the terms listed are independent, and the
fifth can always be obtained through integration by parts. In particular, we
can always set one of the four constants
$\mathfrak{s}^{\\{\ldots\\}}_{[32,33x\Delta_{x}]}$ to zero. We will retain
this over-complete representation, however, as it will prove convenient to use
different non-vanishing structures for $\Delta_{x}=2$ and $\Delta_{x}=3$.

  * •

The contributions from the counterterm actions (105) - (110) are

$\displaystyle i^{\text{ct}({4})}_{[22,22x\Delta_{x}]}$ $\displaystyle=0,$
(111) $\displaystyle i^{\text{ct}({4})}_{[32,22x\Delta_{x}]}$
$\displaystyle=-\mathfrak{s}_{[32,22x\Delta_{x}]},$ (112) $\displaystyle
i^{\text{ct}({4})}_{[33,22x\Delta_{x}]}$
$\displaystyle=\mathfrak{s}_{[33,22x\Delta_{x}]}\left[\hat{i}_{[22]}(k_{3})+\hat{i}_{[22]}(k_{4})\right],$
(113) $\displaystyle i^{\text{ct}({4})}_{[32,32x\Delta_{x}]}$
$\displaystyle=\mathfrak{s}_{[32,32x\Delta_{x}]}\left[\hat{i}_{[22]}(k_{2})+\hat{i}_{[22]}(k_{4})\right],$
(114) $\displaystyle i^{\text{ct}({4})}_{[32,33x\Delta_{x}]}$
$\displaystyle=\mathfrak{s}^{\\{1\\}}_{[32,33x\Delta_{x}]}k_{1}^{2}+\mathfrak{s}^{\\{2\\}}_{[32,33x\Delta_{x}]}k_{2}^{2}+\mathfrak{s}^{\\{3+4\\}}_{[32,33x\Delta_{x}]}(k_{3}^{2}+k_{4}^{2})$
$\displaystyle\qquad\qquad+\frac{1}{2}\mathfrak{s}^{\\{12\\}}_{[32,33x\Delta_{x}]}(s^{2}+k_{1}^{2}-k_{2}^{2})+\frac{1}{2}\mathfrak{s}^{\\{34\\}}_{[32,33x\Delta_{x}]}(s^{2}-k_{3}^{2}-k_{4}^{2}),$
(115) $\displaystyle i^{\text{ct}({4})}_{[33,33x\Delta_{x}]}$
$\displaystyle=\mathfrak{s}_{[33,33x\Delta_{x}]}\left[\hat{i}_{[33]}(k_{1})+\hat{i}_{[33]}(k_{2})+\hat{i}_{[33]}(k_{3})+\hat{i}_{[33]}(k_{4})\right],$
(116)

where the values of the constants $\mathfrak{s}$ are listed in Table 3.

| $\Delta_{x}=2$ | $\Delta_{x}=3$
---|---|---
$\mathfrak{s}_{[32,22x\Delta_{x}]}$ | $-\tfrac{1}{2}\operatorname{Div}_{1}^{2}(\mu)\mathfrak{a}_{[32,22x2]}$ | $\tfrac{1}{2}\operatorname{Div}_{2}(\mu)\mathfrak{a}_{[32,22x3]}$
$\mathfrak{s}_{[33,22x\Delta_{x}]}$ | $\tfrac{1}{2}\operatorname{Div}_{2}(\mu)\mathfrak{a}_{[33,22x2]}$ | $\tfrac{1}{6}\operatorname{Div}_{1}^{2}(\mu)\mathfrak{a}_{[33,22x3]}$
$\mathfrak{s}_{[32,32x\Delta_{x}]}$ | $\tfrac{1}{2}\operatorname{Div}_{1}^{2}(\mu)\mathfrak{a}_{[32,32x2]}$ | $-\tfrac{1}{2}\operatorname{Div}_{2}(\mu)\mathfrak{a}_{[32,32x3]}$
$\mathfrak{s}^{\\{12\\}}_{[32,33x\Delta_{x}]}$ | $0$ | $-\tfrac{1}{6}\operatorname{Div}_{1}^{2}(\mu)\mathfrak{a}_{[32,33x3]}^{\\{12\\}}$
$\mathfrak{s}^{\\{34\\}}_{[32,33x\Delta_{x}]}$ | $\tfrac{1}{2}\operatorname{Div}_{1}^{2}(\mu)\mathfrak{a}_{[32,33x2]}^{\\{34\\}}$ | $0$
$\mathfrak{s}^{\\{3+4\\}}_{[32,33x\Delta_{x}]}$ | $\tfrac{3}{4}\operatorname{Div}_{2}(\mu)\mathfrak{a}_{[32,33x2]}^{\\{3+4\\}}$ | $\tfrac{1}{4}\operatorname{Div}_{2}(\mu)\mathfrak{a}_{[32,33x3]}^{\\{3+4\\}}$
$\mathfrak{s}^{\\{1\\}}_{[32,33x\Delta_{x}]}$ | $-\tfrac{1}{4}\operatorname{Div}_{2}(\mu)\mathfrak{a}_{[32,33x2]}^{\\{1\\}}$ | $-\tfrac{1}{12}\operatorname{Div}_{2}(\mu)\mathfrak{a}_{[32,33x3]}^{\\{1\\}}$
$\mathfrak{s}^{\\{2\\}}_{[32,33x\Delta_{x}]}$ | $\tfrac{1}{4}\operatorname{Div}_{2}(\mu)\mathfrak{a}_{[32,33x2]}^{\\{2\\}}$ | $\tfrac{1}{12}\operatorname{Div}_{2}(\mu)\mathfrak{a}_{[32,33x3]}^{\\{2\\}}$
$\mathfrak{s}_{[33,33x\Delta_{x}]}$ | $\tfrac{1}{6}\operatorname{Div}_{2}(\mu)\mathfrak{a}_{[33,33x2]}$ | $\tfrac{1}{18}\operatorname{Div}_{1}^{2}(\mu)\mathfrak{a}_{[33,33x3]}$
Table 3: The values of the constants $\mathfrak{s}$ for exchange diagrams with
exchange $\Delta_{x}=2$ and $\Delta_{x}=3$.

  * •

If the degree of divergence is higher than one, the subleading terms in the
$\mathfrak{a}$ constants are related to the 3-point $\mathfrak{a}$ constants
as explained in the following subsection.

This leads to the following set of identities, which it is extremely important
to impose in all subsequent calculations:

$\displaystyle\mathfrak{a}_{[32,22x2]}^{(1)}$
$\displaystyle=2\mathfrak{a}_{[322]}^{(1)}-1,$
$\displaystyle\mathfrak{a}_{[33,22x3]}^{(1)}$
$\displaystyle=2\mathfrak{a}_{[333]}^{(1)}-\tfrac{1}{3},$ (117)
$\displaystyle\mathfrak{a}_{[32,32x2]}^{(1)}$
$\displaystyle=2\mathfrak{a}_{[322]}^{(1)}-1,$
$\displaystyle\mathfrak{a}_{[32,33x3]}^{\\{12\\}(1)}$
$\displaystyle=2\mathfrak{a}_{[333]}^{(1)}-\tfrac{5}{6},$ (118)
$\displaystyle\mathfrak{a}_{[32,33x2]}^{\\{34\\}(1)}$
$\displaystyle=2\mathfrak{a}_{[322]}^{(1)}+\tfrac{1}{2},$
$\displaystyle\mathfrak{a}_{[33,33x3]}^{(1)}$
$\displaystyle=2\mathfrak{a}_{[333]}^{(1)}-\tfrac{1}{3}.$ (119)

####  Example

To understand this last point better, and as an example calculation, consider
the 4-point amplitude $\hat{i}_{[32,22x2]}$. This amplitude exhibits a double
pole at $\epsilon=0$ and its divergent terms are given by (35). While one can
work with the divergences expressed in terms of $\operatorname{Div}_{1}$ and
$\operatorname{Div}_{2}$, it is easier to understand the situation when fully
expanded in the regulator,

$\displaystyle\hat{i}_{[32,22x2]}$
$\displaystyle=\frac{1}{2\epsilon^{2}}-\frac{1}{\epsilon}\left[\log
l_{34+}+\gamma_{E}-\tfrac{1}{2}\right]+O(\epsilon^{0}).$ (120)

The renormalization of the 3-point amplitude $\hat{i}_{[322]}$ introduces the
counterterm (82), which also contributes to the 4-point amplitude under
consideration. Its contribution is given by (94). With the value of
$\mathfrak{s}_{[322]}$ given in (85), we find

$\displaystyle i^{\text{ct}({3})}_{[32,22x2]}$
$\displaystyle=-\frac{1}{\epsilon^{2}}+\frac{1}{\epsilon}\left[\log
l_{34+}+\log\mu+2\gamma_{E}-\mathfrak{a}_{[322]}^{(1)}\right]+O(\epsilon^{0}).$
(121)

Before moving to the renormalization of the 4-point functions, the
renormalization procedure for 3-point functions must be carried out. This
means that we treat $\mathfrak{a}_{[322]}^{(1)}$ as fixed at this point.

To fully renormalize the 4-point amplitude, we have to introduce the
counterterm (106). Its contribution to the 4-point function is then given by
taking four functional derivatives with respect to the sources, with the
result given by (112). Thus, the sum of (112), (120) and (121) must be finite.
With $\mathfrak{s}_{[32,22x2]}^{(n)}$ denoting terms of order $\epsilon^{n}$
in the expansion of $\mathfrak{s}_{[32,22x2]}$, we find

$\displaystyle\hat{i}_{[32,22x2]}+i^{\text{ct}({3})}_{[32,22x2]}+i^{\text{ct}({4})}_{[32,22x2]}=\frac{1}{\epsilon^{2}}\left[\mathfrak{s}_{[32,22x2]}^{(-2)}-\frac{1}{2}\right]$
$\displaystyle\qquad\qquad+\frac{1}{\epsilon}\left[\mathfrak{s}_{[32,22x2]}^{(-1)}+\gamma_{E}+\log\mu-\mathfrak{a}_{[322]}^{(1)}+\frac{1}{2}\right]+O(\epsilon^{0}).$
(122)

Thus, we must choose

$\displaystyle\mathfrak{s}_{[32,22x2]}^{(-2)}$ $\displaystyle=\frac{1}{2},$
$\displaystyle\mathfrak{s}_{[32,22x2]}^{(-1)}$
$\displaystyle=-\gamma_{E}-\log\mu+\mathfrak{a}_{[322]}^{(1)}-\frac{1}{2}.$
(123)

As we can see, both the leading $\mathfrak{s}_{[32,22x2]}^{(-2)}$ as well as
the subleading $\mathfrak{s}_{[32,22x2]}^{(-1)}$ term is completely fixed by
the renormalization. Furthermore, the subleading term
$\mathfrak{s}_{[32,22x2]}^{(-1)}$ depends on the subleading part of the
renormalization constant $\mathfrak{a}_{[322]}^{(1)}$. Only the third term,
$\mathfrak{s}_{[32,22x2]}^{(0)}$, is unrestricted and will contribute to the
finite part of the renormalized correlator.

In the discussion above we presented our results using the divergences
$\operatorname{Div}_{1}$ and $\operatorname{Div}_{2}$. These objects also keep
track of the scale-dependence, so that the $\log\mu$ term is automatically
taken care of. For example, we can rewrite

$\displaystyle\frac{\mathfrak{s}_{[32,22x2]}^{(-2)}}{\epsilon^{2}}+\frac{\mathfrak{s}_{[32,22x2]}^{(-1)}}{\epsilon}+O(\epsilon^{0})=\frac{1}{2}\left[1+\epsilon(2\mathfrak{a}_{[322]}^{(1)}-1)\right]\operatorname{Div}_{1}^{2}(\mu)+O(\epsilon^{0}).$
(124)

Thus, if we identify the term in square brackets with the counterterm constant
$\mathfrak{a}_{[32,22x2]}=1+\epsilon\mathfrak{a}_{[32,22x2]}^{(1)}+O(\epsilon^{2})$,
we must impose $\mathfrak{a}_{[32,22x2]}^{(1)}=2\mathfrak{a}_{[322]}^{(1)}-1$.
This is indeed the first relation in (117).

Finally, let us stress here that the relations (117) - (119) are essential for
the theory to be renormalizable. Indeed, if $\mathfrak{s}_{[32,22x2]}^{(-1)}$
were any different in the example above, the 4-point amplitude
$\hat{i}_{[32,22x2]}$ could not be renormalized.

####  Contact diagrams

  * •

The form of the counterterms depends on the exchange operator only through the
constants $\mathfrak{s}$. This also includes the contact diagrams, which we
can identify with exchange diagrams with $\Delta_{x}=0$, but for the sake of
clarity, let us rewrite these expressions,

$\displaystyle S^{\text{ct}\,(4)}_{[2222]}$ $\displaystyle=0,$ (125)
$\displaystyle S^{\text{ct}\,(4)}_{[3222]}$
$\displaystyle=\mathfrak{s}_{[3222]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\phi^{[0]}_{1}\phi^{[1]}_{2}\phi^{[1]}_{3}\phi^{[1]}_{4},$
(126) $\displaystyle S^{\text{ct}\,(4)}_{[3322]}$
$\displaystyle=\mathfrak{s}_{[3322]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\phi^{[0]}_{1}\phi^{[0]}_{2}\left(\phi^{[1]}_{3}\mathcal{O}^{[2]}_{4}+\phi^{[1]}_{4}\mathcal{O}^{[2]}_{3}\right),$
(127) $\displaystyle S^{\text{ct}\,(4)}_{[3332]}$
$\displaystyle=\mathfrak{s}_{[3332]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\left(\partial^{2}\phi^{[0]}_{1}\,\phi^{[0]}_{2}\phi^{[0]}_{3}\phi^{[1]}_{4}+\phi^{[0]}_{1}\partial^{2}\phi^{[0]}_{2}\,\phi^{[0]}_{3}\phi^{[1]}_{4}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\phi^{[0]}_{1}\phi^{[0]}_{2}\partial^{2}\phi^{[0]}_{3}\,\phi^{[1]}_{4}-\phi^{[0]}_{1}\phi^{[0]}_{2}\phi^{[0]}_{3}\partial^{2}\phi^{[1]}_{4}\right),$
(128) $\displaystyle S^{\text{ct}\,(4)}_{[3333]}$
$\displaystyle=\mathfrak{s}_{[3333]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\left(\phi^{[0]}_{1}\phi^{[0]}_{2}\phi^{[0]}_{3}\mathcal{O}^{[3]}_{4}+\phi^{[0]}_{2}\phi^{[0]}_{3}\phi^{[0]}_{4}\mathcal{O}^{[3]}_{1}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\phi^{[0]}_{3}\phi^{[0]}_{4}\phi^{[0]}_{1}\mathcal{O}^{[3]}_{2}+\phi^{[0]}_{4}\phi^{[0]}_{1}\phi^{[0]}_{2}\mathcal{O}^{[3]}_{3}\right).$
(129)

The values of the constants are

$\displaystyle\mathfrak{s}_{[3222]}$
$\displaystyle=\operatorname{Div}_{2}(\mu)\mathfrak{a}_{[3222]},$ (130)
$\displaystyle\mathfrak{s}_{[3322]}$
$\displaystyle=-\operatorname{Div}_{2}(\mu)\mathfrak{a}_{[3322]},$ (131)
$\displaystyle\mathfrak{s}_{[3332]}$
$\displaystyle=\frac{1}{2}\operatorname{Div}_{2}(\mu)\mathfrak{a}_{[3332]},$
(132) $\displaystyle\mathfrak{s}_{[3333]}$
$\displaystyle=-\frac{1}{3}\operatorname{Div}_{2}(\mu)\mathfrak{a}_{[3333]},$
(133)

and the counterterm contributions are

$\displaystyle i^{\text{ct}({4})}_{[2222]}$ $\displaystyle=0,$ (134)
$\displaystyle i^{\text{ct}({4})}_{[3222]}$
$\displaystyle=-\mathfrak{s}_{[3222]},$ (135) $\displaystyle
i^{\text{ct}({4})}_{[3322]}$
$\displaystyle=\mathfrak{s}_{[3322]}\left[\hat{i}_{[22]}(k_{3})+\hat{i}_{[22]}(k_{4})\right],$
(136) $\displaystyle i^{\text{ct}({4})}_{[3332]}$
$\displaystyle=\mathfrak{s}_{[3332]}(k_{1}^{2}+k_{2}^{2}+k_{3}^{2}-k_{4}^{2}),$
(137) $\displaystyle i^{\text{ct}({4})}_{[3333]}$
$\displaystyle=\mathfrak{s}_{[3333]}\left[\hat{i}_{[33]}(k_{1})+\hat{i}_{[33]}(k_{2})+\hat{i}_{[33]}(k_{3})+\hat{i}_{[33]}(k_{4})\right].$
(138)

  * •

Similarly to the case of the 3-point counterterms in (• ‣ 4.3.1), the 4-point
counterterms depend functionally on four sources and operators,

$\displaystyle
S_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}^{\text{ct}\,(4)}=S_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}^{\text{ct}\,(4)}(\phi_{1},\phi_{2},\phi_{3},\phi_{4}).$
(139)

We omit the explicit arguments, if all the sources and operators are as
listed.

  * •

In the remaining subsections, we will present two examples of how to use these
results in practice.

####  Renormalized amplitudes

When the counterterm contributions $i^{\text{ct}}$ are added to the regulated
amplitudes $\hat{i}$, the limit $\epsilon\rightarrow 0$ exists. Here we list
the _renormalized amplitudes_ $i^{\text{ren}}$ defined as

$\displaystyle i^{\text{ren}}=\lim_{\epsilon\rightarrow
0}\left[\hat{i}+i^{\text{ct}}\right].$ (140)

  * •

2-point amplitudes are finite and do not require renormalization,

$\displaystyle i^{\text{ren}}_{[22]}$ $\displaystyle=-k,$ (141) $\displaystyle
i^{\text{ren}}_{[33]}$ $\displaystyle=k^{3},$ (142) $\displaystyle
i^{\text{ren}}_{[23]}$ $\displaystyle=0.$ (143)

We do not allow for a counterterm of the form $\int\phi^{[0]}\Box\phi^{[1]}$,
which would result in a local contribution to
$\langle\mathcal{O}_{[2]}\mathcal{O}_{[3]}\rangle$. We keep the 2-point
functions diagonal.

  * •

The renormalized 3-point amplitudes are

$\displaystyle i^{\text{ren}}_{[222]}$
$\displaystyle=-\log\left(\frac{k_{t}}{\mu}\right)-\mathfrak{a}_{[222]}^{(1)},$
(144) $\displaystyle i^{\text{ren}}_{[322]}$
$\displaystyle=(k_{2}+k_{3})\left[\log\left(\frac{k_{t}}{\mu}\right)+\mathfrak{a}_{[322]}^{(1)}-1\right]-k_{1},$
(145) $\displaystyle i^{\text{ren}}_{[332]}$
$\displaystyle=\frac{k_{1}^{2}+k_{2}^{2}-k_{3}^{2}}{2}\left[\log\left(\frac{k_{t}}{\mu}\right)+\mathfrak{a}_{[332]}^{(1)}-\tfrac{3}{2}\right]+\frac{1}{2}\left[k_{1}^{2}+k_{2}^{2}-k_{1}k_{2}+k_{3}(k_{1}+k_{2})\right],$
(146) $\displaystyle i^{\text{ren}}_{[333]}$
$\displaystyle=-\frac{k_{1}^{3}+k_{2}^{3}+k_{3}^{3}}{3}\left[\log\left(\frac{k_{t}}{\mu}\right)+\mathfrak{a}_{[333]}^{(1)}-\tfrac{4}{3}\right]+\frac{1}{3}\left[k_{1}^{2}k_{2}+5\text{
perms.}-k_{1}k_{2}k_{3}\right].$ (147)

  * •

The renormalized contact diagrams are

$\displaystyle i^{\text{ren}}_{[2222]}$ $\displaystyle=\frac{1}{k_{T}},$ (148)
$\displaystyle i^{\text{ren}}_{[3222]}$
$\displaystyle=-\left[\log\left(\frac{k_{T}}{\mu}\right)+\frac{1}{2}\mathfrak{a}_{[3222]}^{(1)}\right]+\frac{k_{1}}{k_{T}},$
(149) $\displaystyle i^{\text{ren}}_{[3322]}$
$\displaystyle=(k_{3}+k_{4})\left[\log\left(\frac{k_{T}}{\mu}\right)+\frac{1}{2}\mathfrak{a}_{[3322]}^{(1)}\right]+\frac{k_{1}k_{2}}{k_{T}}-k_{T},$
(150) $\displaystyle i^{\text{ren}}_{[3332]}$
$\displaystyle=\frac{k_{1}^{2}+k_{2}^{2}+k_{3}^{2}-k_{4}^{2}}{2}\left[\log\left(\frac{k_{T}}{\mu}\right)+\frac{1}{2}\mathfrak{a}_{[3332]}^{(1)}\right]+\frac{k_{1}k_{2}k_{3}}{k_{T}}+k_{T}\left(k_{4}-\frac{k_{T}}{4}\right),$
(151) $\displaystyle i^{\text{ren}}_{[3333]}$
$\displaystyle=-\frac{k_{1}^{3}+k_{2}^{3}+k_{3}^{3}+k_{4}^{3}}{3}\left[\log\left(\frac{k_{T}}{\mu}\right)+\frac{1}{2}\mathfrak{a}_{[3333]}^{(1)}\right]$
$\displaystyle\qquad\qquad+\frac{k_{1}k_{2}k_{3}k_{4}}{k_{T}}+k_{T}\left(-\sigma_{(2)1234}+\frac{4}{9}k_{T}^{2}\right).$
(152)

  * •

For exchange diagrams we only list the _logarithmic part_ here,
$i^{\text{log}}$, defined as

$\displaystyle
i^{\text{log}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}=\lim_{\epsilon\rightarrow
0}\left[i^{\text{div}}_{[\Delta_{1}\Delta_{2},\Delta_{3},\Delta_{4}x\Delta_{x}]}+i^{\text{ct}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}\right].$
(153)

The full renormalized amplitude then reads

$\displaystyle
i^{\text{ren}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}=i^{\text{log}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}+i^{\text{fin}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]},$
(154)

where the finite parts $i^{\text{fin}}$ are listed in Section 3.

  * •

All in all, the logarithmic parts of the renormalized amplitudes with
$\Delta_{x}=2$ are

$\displaystyle i^{\text{log}}_{[22,22x2]}$ $\displaystyle=0,$ (155)
$\displaystyle i^{\text{log}}_{[32,22x2]}$
$\displaystyle=\frac{1}{2}\log^{2}\left(\frac{l_{34+}}{\mu}\right)+\log\left(\frac{l_{34+}}{\mu}\right)\left[-1+\mathfrak{a}_{[322]}^{(1)}\right]+\left[-\mathfrak{a}_{[322]}^{(2)}+\frac{1}{2}\mathfrak{a}_{[32,22x2]}^{(2)}\right],$
(156) $\displaystyle i^{\text{log}}_{[33,22x2]}$
$\displaystyle=-\frac{k_{3}+k_{4}}{2}\left[\log\left(\frac{k_{T}}{\mu}\right)+\frac{1}{2}\mathfrak{a}_{[33,22x2]}^{(1)}\right],$
(157) $\displaystyle i^{\text{log}}_{[32,32x2]}$
$\displaystyle=-\frac{k_{2}+s}{2}\log^{2}\left(\frac{l_{12+}}{\mu}\right)-\frac{k_{4}+s}{2}\log^{2}\left(\frac{l_{34+}}{\mu}\right)$
$\displaystyle\qquad\qquad+\left[(l_{12+}+k_{2})-(k_{2}+s)\mathfrak{a}_{[322]}^{(1)}\right]\log\left(\frac{l_{12+}}{\mu}\right)$
$\displaystyle\qquad\qquad+\left[(l_{34+}+k_{4})-(k_{4}+s)\mathfrak{a}_{[322]}^{(1)}\right]\log\left(\frac{l_{34+}}{\mu}\right)$
$\displaystyle\qquad\qquad+(2s+k_{T})\left[1+\mathfrak{a}_{[322]}^{(1)}\right]+(k_{2}+k_{4})\left[\mathfrak{a}_{[322]}^{(2)}-\frac{1}{2}\mathfrak{a}_{[32,32x2]}^{(2)}\right]$
$\displaystyle\qquad\qquad-s(\mathfrak{a}_{[322]}^{(1)})^{2},$ (158)
$\displaystyle i^{\text{log}}_{[32,33x2]}$
$\displaystyle=\frac{s^{2}-k_{3}^{2}-k_{4}^{2}}{4}\log^{2}\left(\frac{l_{34+}}{\mu}\right)$
$\displaystyle\qquad\qquad+\frac{1}{4}\left[-k_{1}^{2}+k_{2}^{2}+(3-2\mathfrak{a}_{[322]}^{(1)})(k_{3}^{2}+k_{4}^{2})+2k_{3}k_{4}-2s(k_{3}+k_{4})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+2s^{2}(-1+\mathfrak{a}_{[322]}^{(1)})\right]\log\left(\frac{l_{34+}}{\mu}\right)$
$\displaystyle\qquad\qquad+\frac{(k_{3}+k_{4})^{2}}{8}\left[1+2\mathfrak{a}_{[322]}^{(1)}\right]-\frac{s(k_{3}+k_{4})}{4}\left[3+2\mathfrak{a}_{[322]}^{(1)}\right]$
$\displaystyle\qquad\qquad-\frac{s^{2}}{4}\left[\frac{7}{2}+3\mathfrak{a}_{[322]}^{(1)}\right]+\frac{s^{2}-k_{3}^{2}-k_{4}^{2}}{4}\left[-2\mathfrak{a}_{[322]}^{(2)}+\mathfrak{a}_{[32,33x2]}^{\\{34\\}(2)}\right]$
$\displaystyle\qquad\qquad+\frac{1}{8}\left[-k_{1}^{2}\mathfrak{a}_{[32,33x2]}^{\\{1\\}(1)}+k_{2}^{2}\mathfrak{a}_{[32,33x2]}^{\\{2\\}(1)}+3(k_{3}^{2}+k_{4}^{2})\mathfrak{a}_{[32,33x2]}^{\\{3+4\\}(1)}\right],$
(159) $\displaystyle i^{\text{log}}_{[33,33x2]}$
$\displaystyle=\frac{k_{1}^{3}+k_{2}^{3}+k_{3}^{3}+k_{4}^{3}}{6}\left[\log\left(\frac{k_{T}}{\mu}\right)+\frac{1}{2}\mathfrak{a}_{[33,33x2]}^{(1)}\right].$
(160)

  * •

The logarithmic parts of the renormalized amplitudes with $\Delta_{x}=3$ are

$\displaystyle i^{\text{log}}_{[22,22x3]}$ $\displaystyle=0,$ (161)
$\displaystyle i^{\text{log}}_{[32,22x3]}$
$\displaystyle=-\frac{1}{2}\left[\log\left(\frac{l_{12+}}{\mu}\right)+\frac{1}{2}\mathfrak{a}_{[32,22x3]}^{(1)}\right],$
(162) $\displaystyle i^{\text{log}}_{[33,22x3]}$
$\displaystyle=-\frac{k_{3}+k_{4}}{6}\log^{2}\left(\frac{l_{34+}}{\mu}\right)+\frac{1}{3}\left[s+\left(\frac{4}{3}-\mathfrak{a}_{[333]}^{(1)}\right)(k_{3}+k_{4})\right]\log\left(\frac{l_{34+}}{\mu}\right)$
$\displaystyle\qquad\qquad+\frac{k_{3}+k_{4}}{3}\left[1+\mathfrak{a}_{[333]}^{(1)}+\mathfrak{a}_{[333]}^{(2)}-\frac{1}{2}\mathfrak{a}_{[33,22x3]}^{(2)}\right]+\frac{s}{3}\left[1+\mathfrak{a}_{[333]}^{(1)}\right],$
(163) $\displaystyle i^{\text{log}}_{[32,32x3]}$
$\displaystyle=\frac{k_{2}+k_{4}}{2}\left[\log\left(\frac{k_{T}}{\mu}\right)+\frac{1}{2}\mathfrak{a}_{[32,32x3]}^{(1)}\right],$
(164) $\displaystyle i^{\text{log}}_{[32,33x3]}$
$\displaystyle=\frac{k_{2}^{2}-k_{1}^{2}-s^{2}}{12}\log^{2}\left(\frac{l_{12+}}{\mu}\right)+\frac{1}{6}\left[k_{1}^{2}\left(\frac{5}{6}-\mathfrak{a}_{[333]}^{(1)}\right)+k_{2}^{2}\left(-\frac{11}{6}+\mathfrak{a}_{[333]}^{(1)}\right)\right.$
$\displaystyle\qquad\qquad\left.-k_{1}k_{2}+s(k_{1}-k_{2})+\frac{3}{2}(k_{3}^{2}+k_{4}^{2})+s^{2}\left(\frac{4}{3}-\mathfrak{a}_{[333]}^{(1)}\right)\right]\log\left(\frac{l_{12+}}{\mu}\right)$
$\displaystyle\qquad\qquad-\left(\frac{1}{4}+\frac{1}{6}\mathfrak{a}_{[333]}^{(1)}\right)k_{2}(k_{1}+s)+\frac{1}{12}(1+2\mathfrak{a}_{[333]}^{(1)})sk_{1}$
$\displaystyle\qquad\qquad+\frac{k_{3}^{2}+k_{4}^{2}}{8}\mathfrak{a}_{[32,33x3]}^{\\{3+4\\}(1)}+\frac{s^{2}+k_{1}^{2}-k_{2}^{2}}{24}\left[1+2\mathfrak{a}_{[333]}^{(1)}+4\mathfrak{a}_{[333]}^{(2)}-2\mathfrak{a}_{[32,33x3]}^{\\{12\\}(2)}\right]$
$\displaystyle\qquad\qquad-\frac{k_{1}^{2}}{24}\mathfrak{a}_{[32,33x3]}^{\\{1\\}(1)}-\frac{k_{2}^{2}}{24}\left[6+4\mathfrak{a}_{[333]}^{(1)}-\mathfrak{a}_{[32,33x3]}^{\\{2\\}(1)}\right],$
(165) $\displaystyle i^{\text{log}}_{[33,33x3]}$
$\displaystyle=\frac{s^{3}+k_{1}^{3}+k_{2}^{3}}{18}\log^{2}\left(\frac{l_{12+}}{\mu}\right)+\frac{s^{3}+k_{3}^{3}+k_{4}^{3}}{18}\log^{2}\left(\frac{l_{34+}}{\mu}\right)$
$\displaystyle\qquad\quad-\frac{1}{9}\left[\left(\frac{5}{3}-\mathfrak{a}_{[333]}^{(1)}\right)\sigma_{(1)12s}^{3}+(-4+3\mathfrak{a}_{[333]}^{(1)})\sigma_{(1)12s}\sigma_{(2)12s}\right.$
$\displaystyle\qquad\quad\qquad\qquad\left.+(1-3\mathfrak{a}_{[333]}^{(1)})\sigma_{(3)12s}-\frac{s^{3}}{3}\right]\log\left(\frac{l_{12+}}{\mu}\right)$
$\displaystyle\qquad\quad-\frac{1}{9}\left[\left(\frac{5}{3}-\mathfrak{a}_{[333]}^{(1)}\right)\sigma_{(1)34s}^{3}+(-4+3\mathfrak{a}_{[333]}^{(1)})\sigma_{(1)34s}\sigma_{(2)34s}\right.$
$\displaystyle\qquad\quad\qquad\qquad\left.+(1-3\mathfrak{a}_{[333]}^{(1)})\sigma_{(3)34s}-\frac{s^{3}}{3}\right]\log\left(\frac{l_{34+}}{\mu}\right)$
$\displaystyle\qquad\quad-\frac{4+3\mathfrak{a}_{[333]}^{(1)}}{27}\left[k_{1}k_{2}(k_{1}+k_{2})+k_{3}k_{4}(k_{3}+k_{4})+s(k_{1}^{2}+k_{2}^{2}+k_{3}^{2}+k_{4}^{2})+s^{2}k_{T}\right]$
$\displaystyle\qquad\quad+\frac{s}{27}\left[1+3\mathfrak{a}_{[333]}^{(1)}\right](k_{1}k_{2}+k_{3}k_{4})-\frac{s^{3}}{81}\left[26+24\mathfrak{a}_{[333]}^{(1)}-9(\mathfrak{a}_{[333]}^{(1)})^{2}\right]$
$\displaystyle\qquad\quad-\frac{26+24\mathfrak{a}_{[333]}^{(1)}+18\mathfrak{a}_{[333]}^{(2)}-9\mathfrak{a}_{[33,33x3]}^{(2)}}{162}(k_{1}^{3}+k_{2}^{3}+k_{3}^{3}+k_{4}^{3}).$
(166)

###  Asymmetric theory

In the previous sections we listed: (i) the counterterm contributions to the
amplitudes that would render them finite, (ii) the renormalized amplitudes,
and (iii) the counterterm actions that produce the counterterm contributions.
In this section, we apply these results to completely renormalize 2-, 3-, and
4-point functions in the asymmetric theory (4.2).

The 2-point functions are finite and do not require renormalization. Thus,
they already represent the renormalized correlators,

$\displaystyle\langle\\!\langle\mathcal{O}_{j}^{[\Delta_{j}]}(\bm{k})\mathcal{O}_{j}^{[\Delta_{j}]}(-\bm{k})\rangle\\!\rangle_{\text{ren}}$
$\displaystyle=i^{\text{ren}}_{[\Delta_{j}\Delta_{j}]},$ (167)

with $i^{\text{ren}}_{[\Delta_{j}\Delta_{j}]}$ given by (141) and (142).

We will now show that the following CFT counterterm action renormalizes all 3-
and 4-point functions in the asymmetric theory,

$\displaystyle S^{\text{asym, ct}}=S^{\text{asym,
ct}\,(3)}+S_{1234}^{\text{asym, ct}\,(4)}+S_{\text{cross}}^{\text{asym,
ct}\,(4)},$ (168)

where

$\displaystyle S^{\text{asym, ct}\,(3)}$
$\displaystyle=\lambda_{12x}S^{\text{ct}\,(3)}_{[\Delta_{1}\Delta_{2}\Delta_{x}]}(\phi_{1},\phi_{2},\phi_{x})+\lambda_{34x}S^{\text{ct}\,(3)}_{[\Delta_{3}\Delta_{4}\Delta_{x}]}(\phi_{3},\phi_{4},\phi_{x}),$
(169) $\displaystyle S^{\text{asym, ct}\,(4)}_{1234}$
$\displaystyle=\lambda_{12x}\lambda_{34x}S^{\text{ct}\,(4)}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}(\phi_{1},\phi_{2},\phi_{3},\phi_{4})$
$\displaystyle\qquad\qquad+\lambda_{1234}S^{\text{ct}\,(4)}_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}(\phi_{1},\phi_{2},\phi_{3},\phi_{4}),$
(170) $\displaystyle S^{\text{asym, ct}\,(4)}_{\text{cross}}$
$\displaystyle=\frac{1}{2}\lambda_{12x}^{2}\left[S^{\text{ct}\,(4)}_{[\Delta_{1}\Delta_{2},\Delta_{2}\Delta_{1}x\Delta_{x}]}(\phi_{1},\phi_{2},\phi_{2},\phi_{1})+S^{\text{ct}\,(4)}_{[\Delta_{1}\Delta_{x},\Delta_{x}\Delta_{1}x\Delta_{2}]}(\phi_{1},\phi_{x},\phi_{x},\phi_{1})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+S^{\text{ct}\,(4)}_{[\Delta_{2}\Delta_{x},\Delta_{x}\Delta_{2}x\Delta_{1}]}(\phi_{2},\phi_{x},\phi_{x},\phi_{2})\right]$
$\displaystyle+\frac{1}{2}\lambda_{34x}^{2}\left[S^{\text{ct}\,(4)}_{[\Delta_{3}\Delta_{4},\Delta_{4}\Delta_{3}x\Delta_{x}]}(\phi_{3},\phi_{4},\phi_{4},\phi_{3})+S^{\text{ct}\,(4)}_{[\Delta_{3}\Delta_{x},\Delta_{x}\Delta_{3}x\Delta_{4}]}(\phi_{3},\phi_{x},\phi_{x},\phi_{3})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+S^{\text{ct}\,(4)}_{[\Delta_{4}\Delta_{x},\Delta_{x}\Delta_{4}x\Delta_{3}]}(\phi_{4},\phi_{x},\phi_{x},\phi_{4})\right].$
(171)

The counterterm actions in these expressions are listed in Section 4.3.

The first term in (168), $S^{\text{asym, ct}\,(3)}$, renormalizes the 3-point
functions (78). Indeed, it contributes
$\lambda_{12x}i^{\text{ct}}_{[\Delta_{1}\Delta_{2}\Delta_{x}]}$ to the
correlator $\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{x}\rangle$ and
$\lambda_{34x}i^{\text{ct}}_{[\Delta_{3}\Delta_{4}\Delta_{x}]}$ to the
correlator $\langle\mathcal{O}_{3}\mathcal{O}_{4}\mathcal{O}_{x}\rangle$. It
also contributes to the 4-point functions, contributing
$\lambda_{12x}\lambda_{34x}i^{\text{ct}({3})}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}$
to
$\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle$.
To see this more explicitly, consider an example with
$\Delta_{1}=\Delta_{2}=\Delta_{x}=3$ and $\Delta_{3}=\Delta_{4}=2$. In such a
case $S^{\text{asym, ct}\,(3)}$ reads

$\displaystyle S^{\text{asym, ct}\,(3)}$
$\displaystyle=\operatorname{Div}_{1}(\mu)\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\left[\frac{1}{3}\lambda_{12x}\mathfrak{a}_{[333]}\left(\phi_{1}\phi_{2}\mathcal{O}_{x}+\phi_{2}\phi_{x}\mathcal{O}_{1}+\phi_{x}\phi_{1}\mathcal{O}_{2}\right)\right.$
$\displaystyle\qquad\qquad\qquad\left.+\lambda_{34x}\mathfrak{a}_{[322]}\phi_{x}\left(\phi_{3}\mathcal{O}_{4}+\phi_{4}\mathcal{O}_{3}\right)\vphantom{\Big{[}}\right].$
(172)

If we are interested in its contribution to the 4-point function
$\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle$,
we can drop all sources $\phi_{x}$ of the exchange scalar leaving only the
first term. Then, we differentiate

$\displaystyle\left.\frac{\delta^{4}}{\delta\phi_{1}(\bm{x}_{1})\delta\phi_{2}(\bm{x}_{2})\delta\phi_{3}(\bm{x}_{3})\delta\phi_{4}(\bm{x}_{4})}\langle\exp\left(-\frac{1}{3}\lambda_{12x}\operatorname{Div}_{1}(\mu)\mathfrak{a}_{[333]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\phi_{1}\phi_{2}\mathcal{O}_{x}\right)\rangle\right|_{\phi_{j}=0}$
$\displaystyle=\left.\frac{\delta^{2}}{\delta\phi_{1}(\bm{x}_{1})\delta\phi_{2}(\bm{x}_{2})}\langle\mathcal{O}_{3}(\bm{x}_{3})\mathcal{O}_{4}(\bm{x}_{4})\,\exp\left(-\frac{1}{3}\lambda_{12x}\operatorname{Div}_{1}(\mu)\mathfrak{a}_{[333]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\phi_{1}\phi_{2}\mathcal{O}_{x}\right)\rangle\right|_{\phi_{j}=0}$
$\displaystyle=\left.\frac{\delta}{\delta\phi_{1}(\bm{x}_{1})}\langle\mathcal{O}_{3}(\bm{x}_{3})\mathcal{O}_{4}(\bm{x}_{4})\left(-\mathcal{O}_{2}(\bm{x}_{2})-\frac{1}{3}\lambda_{12x}\operatorname{Div}_{1}(\mu)\mathfrak{a}_{[333]}\phi_{1}(\bm{x}_{2})\mathcal{O}_{x}(\bm{x}_{2})\right)\rangle\right|_{\phi_{j}=0}$
$\displaystyle=\langle\mathcal{O}_{1}(\bm{x}_{1})\mathcal{O}_{2}(\bm{x}_{2})\mathcal{O}_{3}(\bm{x}_{3})\mathcal{O}_{4}(\bm{x}_{4})\rangle-\frac{1}{3}\lambda_{12x}\operatorname{Div}_{1}(\mu)\mathfrak{a}_{[333]}\delta(\bm{x}_{1}-\bm{x}_{2})\langle\mathcal{O}_{x}(\bm{x}_{1})\mathcal{O}_{3}(\bm{x}_{3})\mathcal{O}_{4}(\bm{x}_{4})\rangle.$
(173)

The Fourier transform of the second term yields
$\lambda_{12x}i^{\text{ct}({3})}_{[33,22x3]}$ (use (101) and (86)) as
expected. Thus, the 4-point function
$\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle$
will be renormalized by the addition of $i^{\text{ct}({4})}_{[33,22x3]}$. By
design, this is what the counterterm action
$\lambda_{12x}\lambda_{34x}S^{\text{ct}\,(4)}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}$
contributes.

We argued that the terms $S^{\text{asym, ct}\,(3)}$ and $S_{1234}^{\text{asym,
ct}\,(4)}$ renormalize all 3-point functions as well as the 4-point function
$\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle$.
The renormalized correlators are given by (78) and (• ‣ 4.2.1) with the
amplitudes on the right hand side replaced by the renormalized amplitudes,

$\displaystyle\langle\\!\langle\mathcal{O}_{i}(\bm{k}_{1})\mathcal{O}_{j}(\bm{k}_{2})\mathcal{O}_{x}(\bm{k}_{3})\rangle\\!\rangle_{\text{ren}}$
$\displaystyle=\lambda_{ijx}\,i^{\text{ren}}_{[\Delta_{i}\Delta_{j}\Delta_{x}]}(k_{1},k_{2},k_{3}),$
(174)
$\displaystyle\langle\\!\langle\mathcal{O}_{1}(\bm{k}_{1})\mathcal{O}_{2}(\bm{k}_{2})\mathcal{O}_{3}(\bm{k}_{3})\mathcal{O}_{4}(\bm{k}_{4})\rangle\\!\rangle_{\text{ren}}$
$\displaystyle=\lambda_{12x}\lambda_{34x}\,i^{\text{ren}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}+\lambda_{1234}\,i^{\text{ren}}_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}.$
(175)

As we can see, we obtain the renormalized correlators by simply replacing the
amplitudes by the renormalized amplitudes listed in Section 4.3.

$1$$2$$x$$\oplus$$3$$4$$x$$1$$x$$2$$x$$1$$1$$2$$x$$3$$4$$3$$4$$x$$3$$4$$\lambda_{12x}$$\lambda_{34x}$$\lambda_{12x}^{2}$$\lambda_{12x}\lambda_{34x}$$\lambda_{34x}^{2}$
Figure 4: Some examples of non-vanishing exchange 4-point functions in the
asymmetric theory. On the left, we have two 3-vertices present in the action
(4.2). The top and bottom diagrams on the right represent the 4-point
functions
$\langle\mathcal{O}_{1}\mathcal{O}_{x}\mathcal{O}_{x}\mathcal{O}_{1}\rangle$
and
$\langle\mathcal{O}_{3}\mathcal{O}_{4}\mathcal{O}_{3}\mathcal{O}_{4}\rangle$
obtained by using the same vertex twice. In this way, six different
correlators can be obtained. The middle line corresponds to the 4-point
function
$\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle$
obtained by combining the two vertices.

The asymmetric theory contains additional non-vanishing 4-point functions,
namely those in (• ‣ 4.2.1). To complete our analysis, we should renormalize
these as well. This is achieved by $S_{\text{cross}}^{\text{asym, ct}\,(4)}$,
the last term in (168). To see that this indeed works, first we have to
consider the contributions to the 4-point function from the 3-point
counterterms in $S^{\text{asym, ct}\,(3)}$. To do this, first notice that we
only have three correlators to consider, depending on the dimensions of the
operators involved:

$\langle\mathcal{O}_{i}^{[2]}\mathcal{O}_{j}^{[2]}\mathcal{O}_{j}^{[2]}\mathcal{O}_{i}^{[2]}\rangle,\qquad\langle\mathcal{O}_{i}^{[3]}\mathcal{O}_{j}^{[2]}\mathcal{O}_{j}^{[2]}\mathcal{O}_{i}^{[3]}\rangle,\qquad\langle\mathcal{O}_{i}^{[3]}\mathcal{O}_{j}^{[3]}\mathcal{O}_{j}^{[3]}\mathcal{O}_{i}^{[3]}\rangle.$
(176)

There are six possible correlators in each case, by choosing $(ij)$ from
$(12)$, $(1x)$, $(2x)$, $(34)$, $(3x)$, $(4x)$ as shown in Figure 4. The
analysis is similar in all cases, however, so we will concentrate on, say,
$(ij)=(12)$ and the counterterm
$\lambda_{12x}S^{\text{ct}\,(3)}_{[\Delta_{1}\Delta_{2}\Delta_{x}]}$. Finally,
for the counterterm to contribute, it must be non-vanishing when we set
$\phi_{x}=0$, since we will not be differentiating with respect to this
source. An inspection of (81)-(84) shows that there are only two cases where
the contribution to the 4-point function from the 3-point counterterm action
is non-vanishing: (i) $\Delta_{1}=3,\Delta_{2}=\Delta_{x}=2$ and (ii)
$\Delta_{1}=\Delta_{2}=\Delta_{x}=3$. The relevant form of the counterterm
actions, $S^{\text{ct}(3)}_{[322]}$ and $S^{\text{ct}(3)}_{[333]}$ with
$\phi_{x}$ set to zero, is identical for both cases up to the overall
normalization,

$\displaystyle
S^{\text{ct}(3)}_{I}=c_{I}\operatorname{Div}_{1}(\mu)\mathfrak{a}_{I}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\phi_{1}\phi_{2}\mathcal{O}_{x},$
(177)

where $c_{I}=2$ for $I=[322]$ and $c_{I}=1/3$ for $I=[333]$. By taking four
functional derivatives, we find

$\displaystyle\left.\frac{\delta^{4}}{\delta\phi_{1}(\bm{x}_{1})\delta\phi_{2}(\bm{x}_{2})\delta\phi_{2}(\bm{x}_{3})\delta\phi_{1}(\bm{x}_{4})}\langle\exp\left(-c_{I}\operatorname{Div}_{1}(\mu)\mathfrak{a}_{I}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\phi_{1}\phi_{2}\mathcal{O}_{x}\right)\rangle\right|_{\phi_{j}=0}$
$\displaystyle=\langle\mathcal{O}_{1}(\bm{x}_{1})\mathcal{O}_{2}(\bm{x}_{2})\mathcal{O}_{2}(\bm{x}_{3})\mathcal{O}_{1}(\bm{x}_{4})\rangle$
$\displaystyle\qquad\qquad-
c_{I}\operatorname{Div}_{1}(\mu)\mathfrak{a}_{I}\left[\,\delta(\bm{x}_{3}-\bm{x}_{4})\langle\mathcal{O}_{1}(\bm{x}_{1})\mathcal{O}_{2}(\bm{x}_{2})\mathcal{O}_{x}(\bm{x}_{3})\rangle\right.$
$\displaystyle\qquad\qquad\qquad\qquad+\delta(\bm{x}_{1}-\bm{x}_{2})\langle\mathcal{O}_{x}(\bm{x}_{2})\mathcal{O}_{2}(\bm{x}_{3})\mathcal{O}_{1}(\bm{x}_{4})\rangle$
$\displaystyle\qquad\qquad\qquad\qquad+\delta(\bm{x}_{2}-\bm{x}_{4})\langle\mathcal{O}_{1}(\bm{x}_{1})\mathcal{O}_{2}(\bm{x}_{3})\mathcal{O}_{x}(\bm{x}_{4})\rangle$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\delta(\bm{x}_{1}-\bm{x}_{3})\langle\mathcal{O}_{x}(\bm{x}_{1})\mathcal{O}_{2}(\bm{x}_{2})\mathcal{O}_{1}(\bm{x}_{4})\rangle\right]$
$\displaystyle\qquad\qquad+c_{I}^{2}\operatorname{Div}_{1}^{2}(\mu)\mathfrak{a}^{2}_{I}\left[\delta(\bm{x}_{1}-\bm{x}_{2})\delta(\bm{x}_{3}-\bm{x}_{4})\langle\mathcal{O}_{x}(\bm{x}_{1})\mathcal{O}_{x}(\bm{x}_{3})\rangle\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\delta(\bm{x}_{1}-\bm{x}_{3})\delta(\bm{x}_{2}-\bm{x}_{4})\langle\mathcal{O}_{x}(\bm{x}_{1})\mathcal{O}_{x}(\bm{x}_{2})\rangle\right].$
(178)

Fourier transforming this expression, we see that the counterterm contribution
from the 3-point counterterm action $S^{\text{asym, ct}\,(3)}$ to the 4-point
functions
$\langle\mathcal{O}_{i}\mathcal{O}_{j}\mathcal{O}_{j}\mathcal{O}_{i}\rangle$
reads

$\displaystyle
i^{\text{ct}({3})}_{[\Delta_{i}\Delta_{j},\Delta_{j}\Delta_{i}x\Delta_{x}]}(\bm{k}_{1},\bm{k}_{2};\bm{k}_{3},\bm{k}_{4})+i^{\text{ct}({3})}_{[\Delta_{i}\Delta_{j},\Delta_{j}\Delta_{i}x\Delta_{x}]}(\bm{k}_{1},\bm{k}_{3};\bm{k}_{2},\bm{k}_{4}).$
(179)

These two terms correspond to the two terms in the regulated correlator itself
in (• ‣ 4.2.1). Thus, what remains to be shown is that the counterterms
$S^{\text{asym, ct}\,(4)}_{\text{cross}}$ in (171) correctly produce

$\displaystyle
i^{\text{ct}({4})}_{[\Delta_{i}\Delta_{j},\Delta_{j}\Delta_{i}x\Delta_{x}]}(\bm{k}_{1},\bm{k}_{2};\bm{k}_{3},\bm{k}_{4})+i^{\text{ct}({4})}_{[\Delta_{i}\Delta_{j},\Delta_{j}\Delta_{i}x\Delta_{x}]}(\bm{k}_{1},\bm{k}_{3};\bm{k}_{2},\bm{k}_{4}).$
(180)

First, let us consider case (i) with $\Delta_{1}=3$ and
$\Delta_{2}=\Delta_{x}=2$. The counterterm action (109) becomes

$\displaystyle
S^{\text{ct}\,(4)}_{[32,23x2]}(\phi_{1},\phi_{2},\phi_{2},\phi_{1})=2\mathfrak{s}_{[32,23x2]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\phi_{1}^{2}\phi_{2}\mathcal{O}_{2}.$
(181)

By taking four functional derivatives as above, we find that the contribution
from this counterterm to the correlation function in momentum space equals
twice the expression in (180). Thus, we have to introduce the symmetry factor
of $1/2$ in (171). Similarly, for case (ii) where
$\Delta_{1}=\Delta_{2}=\Delta_{x}=3$, the counterterm action (110) reads

$\displaystyle
S^{\text{ct}\,(4)}_{[33,33x3]}(\phi_{1},\phi_{2},\phi_{2},\phi_{1})=2\mathfrak{s}_{[33,33x3]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\left[\phi_{1}^{2}\phi_{2}\mathcal{O}_{2}+\phi_{1}\phi_{2}^{2}\mathcal{O}_{1}\right].$
(182)

By taking functional derivatives we again find that the counterterm
contribution to the 4-point function is twice that of (180). Thus, the
multiplicative factor of $1/2$ in (171) is correct in all cases.

###  Symmetric theory

In this section we consider the most symmetric case: a bulk theory containing
only two fields, $\Phi_{[2]}$ and $\Phi_{[3]}$, dual to two operators,
$\mathcal{O}_{[2]}$ of dimension two and $\mathcal{O}_{[3]}$ of dimension
three. The AdS action contains all possible couplings between the fields,

$\displaystyle S^{\text{sym}}$
$\displaystyle=\frac{1}{2}\int\mathrm{d}^{4+2\epsilon}x\sqrt{g}\sum_{\Delta=2,3}\left[\partial_{\mu}\Phi_{[\Delta]}\partial^{\mu}\Phi_{[\Delta]}+\hat{m}^{2}_{\Delta}\Phi_{[\Delta]}^{2}\right]$
$\displaystyle\qquad+\int\mathrm{d}^{4+2\epsilon}x\sqrt{g}\left[\frac{1}{6}\lambda_{[222]}\Phi_{[2]}^{3}+\frac{1}{2}\lambda_{[322]}\Phi_{[3]}\Phi_{[2]}^{2}+\frac{1}{2}\lambda_{[332]}\Phi_{[3]}^{2}\Phi_{[2]}+\frac{1}{6}\lambda_{[333]}\Phi_{[3]}^{3}\right]$
$\displaystyle\qquad-\int\mathrm{d}^{4+2\epsilon}x\sqrt{g}\left[\frac{1}{24}\lambda_{[2222]}\Phi_{[2]}^{4}+\frac{1}{6}\lambda_{[3222]}\Phi_{[3]}\Phi_{[2]}^{3}+\frac{1}{4}\lambda_{[3322]}\Phi_{[3]}^{2}\Phi_{[2]}^{2}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.+\frac{1}{6}\lambda_{[3332]}\Phi_{[3]}^{3}\Phi_{[2]}+\frac{1}{24}\lambda_{[3333]}\Phi_{[3]}^{4}\right],$
(183)

where we inserted standard symmetry factors for convenience. The coupling
constants $\lambda_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$ and
$\lambda_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}$ depend only on the
dimensions $\Delta_{j}$ of the operators interacting, and not on the order in
which they appear. When convenient, we can thus permute the dimensions
determining the coupling, e.g.,
$\lambda_{[322]}=\lambda_{[232]}=\lambda_{[223]}$, and so on for other
couplings.

####  Regulated correlators

Let us derive the regulated correlators in the symmetric theory (4.5).

  * •

With the symmetry factors included in the action, the equations of motion read

$\displaystyle(-\Box_{AdS}+\hat{m}^{2}_{2})\Phi_{[2]}=-\frac{1}{2}\lambda_{[222]}\Phi_{[2]}^{2}-\lambda_{[322]}\Phi_{[3]}\Phi_{[2]}-\frac{1}{2}\lambda_{[332]}\Phi_{[3]}^{2}$
$\displaystyle\qquad+\frac{1}{6}\lambda_{[2222]}\Phi_{[2]}^{3}+\frac{1}{2}\lambda_{[3222]}\Phi_{[3]}\Phi_{[2]}^{2}+\frac{1}{2}\lambda_{[3322]}\Phi_{[3]}^{2}\Phi_{[2]}+\frac{1}{6}\lambda_{[3332]}\Phi_{[3]}^{3},$
(184)
$\displaystyle(-\Box_{AdS}+\hat{m}^{2}_{3})\Phi_{[3]}=-\frac{1}{2}\lambda_{[322]}\Phi_{[2]}^{2}-\lambda_{[332]}\Phi_{[3]}\Phi_{[2]}-\frac{1}{2}\lambda_{[333]}\Phi_{[3]}^{2}$
$\displaystyle\qquad+\frac{1}{6}\lambda_{[3222]}\Phi_{[2]}^{3}+\frac{1}{2}\lambda_{[3322]}\Phi_{[3]}\Phi_{[2]}^{2}+\frac{1}{2}\lambda_{[3332]}\Phi_{[2]}\Phi_{[3]}^{2}+\frac{1}{6}\lambda_{[3333]}\Phi_{[3]}^{3}.$
(185)

  * •

We now solve these equations of motion perturbatively, up to and including
terms with three sources. Let us denote by $\Phi_{[\Delta]\\{j\\}}$ the term
in the solution which depends on any $(j+1)$ sources. This means that
$\Phi_{[\Delta]\\{0\\}}$ is the solution to the free field equation. We have

$\displaystyle\Phi_{[3]\\{0\\}}$
$\displaystyle=\operatorname{\mathcal{K}}_{[3]}\ast\phi_{[3]},$ (186)
$\displaystyle\Phi_{[3]\\{1\\}}$
$\displaystyle=-\operatorname{\mathcal{G}}_{[3]}\ast\left[\frac{1}{2}\lambda_{[333]}\Phi_{[3]\\{0\\}}\Phi_{[3]\\{0\\}}+\lambda_{[332]}\Phi_{[3]\\{0\\}}\Phi_{[2]\\{0\\}}\right.$
$\displaystyle\qquad\qquad\left.+\frac{1}{2}\lambda_{[322]}\Phi_{[2]\\{0\\}}\Phi_{[2]\\{0\\}}\right],$
(187) $\displaystyle\Phi_{[3]\\{2\\}}$
$\displaystyle=-\operatorname{\mathcal{G}}_{[3]}\ast\left[\lambda_{[333]}\Phi_{[3]\\{1\\}}\Phi_{[3]\\{0\\}}+\lambda_{[322]}\Phi_{[2]\\{1\\}}\Phi_{[2]\\{0\\}}\right.$
$\displaystyle\qquad\qquad\left.+\lambda_{[332]}\left(\Phi_{[3]\\{1\\}}\Phi_{[2]\\{0\\}}+\Phi_{[3]\\{0\\}}\Phi_{[2]\\{1\\}}\right)\right]$
$\displaystyle\qquad+\operatorname{\mathcal{G}}_{[3]}\ast\left[\frac{1}{6}\lambda_{[3333]}\Phi_{[3]\\{0\\}}\Phi_{[3]\\{0\\}}\Phi_{[3]\\{0\\}}+\frac{1}{2}\lambda_{[3332]}\Phi_{[3]\\{0\\}}\Phi_{[3]\\{0\\}}\Phi_{[2]\\{0\\}}\right.$
$\displaystyle\qquad\qquad\left.+\frac{1}{2}\lambda_{[3322]}\Phi_{[3]\\{0\\}}\Phi_{[2]\\{0\\}}\Phi_{[2]\\{0\\}}+\frac{1}{6}\lambda_{[3222]}\Phi_{[2]\\{0\\}}\Phi_{[2]\\{0\\}}\Phi_{[2]\\{0\\}}\right],$
(188)

where we recall that $\ast$ indicates the convolution of the position-space
variables (see (71)). Analogous expressions with $2\leftrightarrow 3$ follow
for $\Phi_{[2]}$. Setting
$\psi_{[\Delta]}=\Phi_{[\Delta]\\{0\\}}=\operatorname{\mathcal{K}}_{[\Delta]}\ast\phi_{[\Delta]}$,
we can write the expansion of $\Phi_{[3]\\{2\\}}$ in the form

$\displaystyle\Phi_{[3]\\{2\\}}$
$\displaystyle=\operatorname{\mathcal{G}}_{[3]}\ast\left[\left(\lambda_{[333]}\psi_{[3]}+\lambda_{[332]}\psi_{[2]}\right)\operatorname{\mathcal{G}}_{[3]}\ast\left(\frac{1}{2}\lambda_{[333]}\psi_{[3]}\psi_{[3]}+\lambda_{[332]}\psi_{[3]}\psi_{[2]}+\frac{1}{2}\lambda_{[322]}\psi_{[2]}\psi_{[2]}\right)\right.$
$\displaystyle\qquad+\left(\lambda_{[332]}\psi_{[3]}+\lambda_{[322]}\psi_{[2]}\right)\operatorname{\mathcal{G}}_{[3]}\ast\left(\frac{1}{2}\lambda_{[332]}\psi_{[3]}\psi_{[3]}+\lambda_{[322]}\psi_{[3]}\psi_{[2]}+\frac{1}{2}\lambda_{[222]}\psi_{[2]}\psi_{[2]}\right)$
$\displaystyle\qquad\left.+\frac{1}{6}\lambda_{[3333]}\psi_{[3]}^{3}+\frac{1}{2}\lambda_{[3332]}\psi_{[3]}^{2}\psi_{[2]}+\frac{1}{2}\lambda_{[3322]}\psi_{[3]}\psi_{[2]}^{2}+\frac{1}{6}\lambda_{[3222]}\psi_{[2]}^{3}\right].$
(189)

  * •

All 3-point functions then follow from $\Phi_{[2]\\{1\\}}$ and
$\Phi_{[3]\\{1\\}}$. They have no additional symmetry factors and read

$\displaystyle\langle\\!\langle\mathcal{O}_{[\Delta_{1}]}(\bm{k}_{1})\mathcal{O}_{[\Delta_{2}]}(\bm{k}_{2})\mathcal{O}_{[\Delta_{3}]}(\bm{k}_{3})\rangle\\!\rangle_{\text{reg}}=\lambda_{[\Delta_{1}\Delta_{2}\Delta_{3}]}\,\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}(k_{1},k_{2},k_{3})$
(190)

for $\Delta_{1},\Delta_{2},\Delta_{3}=2,3$.

  * •

The 4-point functions follow from $\Phi_{[2]\\{2\\}}$ and $\Phi_{[3]\\{2\\}}$.
For example, starting with (• ‣ 4.5.1), we replace the external propagator
$\operatorname{\mathcal{G}}_{[3]}$ with $-\operatorname{\mathcal{K}}_{[3]}$,
multiply by $(-1)^{3}$ and take three functional derivatives with respect to
the sources. All in all, the symmetry factors are selected in such a way that,
after all symmetries of the amplitudes are used, every correlator takes the
form

$\displaystyle\langle\\!\langle\mathcal{O}_{[\Delta_{1}]}(\bm{k}_{1})\mathcal{O}_{[\Delta_{2}]}(\bm{k}_{2})\mathcal{O}_{[\Delta_{3}]}(\bm{k}_{3})\mathcal{O}_{[\Delta_{4}]}(\bm{k}_{4})\rangle\\!\rangle_{\text{reg}}$
$\displaystyle\qquad=\sum_{\Delta_{x}=2,3}\left[\lambda_{[\Delta_{1}\Delta_{2}\Delta_{x}]}\lambda_{[\Delta_{x}\Delta_{3}\Delta_{4}]}\hat{i}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}(\bm{k}_{1},\bm{k}_{2};\bm{k}_{3},\bm{k}_{4})\right.$
$\displaystyle\qquad\qquad\qquad\qquad+\lambda_{[\Delta_{1}\Delta_{3}\Delta_{x}]}\lambda_{[\Delta_{x}\Delta_{2}\Delta_{4}]}\hat{i}_{[\Delta_{1}\Delta_{3},\Delta_{2}\Delta_{4}x\Delta_{x}]}(\bm{k}_{1},\bm{k}_{3};\bm{k}_{2},\bm{k}_{4})$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\lambda_{[\Delta_{1}\Delta_{4}\Delta_{x}]}\lambda_{[\Delta_{x}\Delta_{2}\Delta_{3}]}\hat{i}_{[\Delta_{1}\Delta_{4},\Delta_{2}\Delta_{3}x\Delta_{x}]}(\bm{k}_{1},\bm{k}_{4};\bm{k}_{2},\bm{k}_{3})\right]$
$\displaystyle\qquad\qquad+\lambda_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}\,\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}(k_{1},k_{2},k_{3},k_{4}).$
(191)

Note that crossing symmetric terms appear with unit coefficients.

####  Renormalization

Let us now renormalize the symmetric theory with only two operators
$\mathcal{O}_{[2]}$ and $\mathcal{O}_{[3]}$, governed by the bulk action
(4.5). The regulated 3-point functions are given by (190), and 4-point
functions by (191). As discussed, the renormalized 3- and 4-point functions
are found by replacing the regulated amplitudes by the renormalized ones, as
listed in Section 4.3. This gives

$\displaystyle\langle\\!\langle\mathcal{O}_{[\Delta_{1}]}(\bm{k}_{1})\mathcal{O}_{[\Delta_{2}]}(\bm{k}_{2})\mathcal{O}_{[\Delta_{3}]}(\bm{k}_{3})\rangle\\!\rangle_{\text{ren}}=\lambda_{[\Delta_{1}\Delta_{2}\Delta_{3}]}\,i^{\text{ren}}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}(k_{1},k_{2},k_{3}),$
(192)

and

$\displaystyle\langle\\!\langle\mathcal{O}_{[\Delta_{1}]}(\bm{k}_{1})\mathcal{O}_{[\Delta_{2}]}(\bm{k}_{2})\mathcal{O}_{[\Delta_{3}]}(\bm{k}_{3})\mathcal{O}_{[\Delta_{4}]}(\bm{k}_{4})\rangle\\!\rangle_{\text{ren}}$
$\displaystyle\qquad=\sum_{\Delta_{x}=2,3}\left[\lambda_{[\Delta_{1}\Delta_{2}\Delta_{x}]}\lambda_{[\Delta_{x}\Delta_{3}\Delta_{4}]}i^{\text{ren}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}(\bm{k}_{1},\bm{k}_{2};\bm{k}_{3},\bm{k}_{4})\right.$
$\displaystyle\qquad\qquad\qquad\qquad+\lambda_{[\Delta_{1}\Delta_{3}\Delta_{x}]}\lambda_{[\Delta_{x}\Delta_{2}\Delta_{4}]}i^{\text{ren}}_{[\Delta_{1}\Delta_{3},\Delta_{2}\Delta_{4}x\Delta_{x}]}(\bm{k}_{1},\bm{k}_{3};\bm{k}_{2},\bm{k}_{4})$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\lambda_{[\Delta_{1}\Delta_{4}\Delta_{x}]}\lambda_{[\Delta_{x}\Delta_{2}\Delta_{3}]}i^{\text{ren}}_{[\Delta_{1}\Delta_{4},\Delta_{2}\Delta_{3}x\Delta_{x}]}(\bm{k}_{1},\bm{k}_{4};\bm{k}_{2},\bm{k}_{3})\right]$
$\displaystyle\qquad\qquad+\lambda_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}\,i^{\text{ren}}_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}(k_{1},k_{2},k_{3},k_{4}),$
(193)

thus providing explicit expressions for all 3- and 4-point functions in the
symmetric theory.

Just as we did for the asymmetric theory in Section 4.4, let us now specify
the counterterm action that renormalizes the theory and analyze its
properties. Since we have only two fields that are distinguishable by their
dimensions, we may drop all indices in the counterterm actions and define

$\displaystyle S^{\text{sym, ct}\,(3)}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$
$\displaystyle=S^{\text{ct}\,(3)}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}(\phi_{[d-\Delta_{1}]},\phi_{[d-\Delta_{2}]},\phi_{[d-\Delta_{3}]}),$
(194) $\displaystyle S^{\text{sym,
ct}\,(4)}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}$
$\displaystyle=S^{\text{ct}\,(4)}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}(\phi_{[d-\Delta_{1}]},\phi_{[d-\Delta_{2}]},\phi_{[d-\Delta_{3}]},\phi_{[d-\Delta_{4}]}).$
(195)

Note that some symmetry factors may appear, for example,

$\displaystyle S^{\text{sym,
ct}\,(3)}_{[322]}=2\mathfrak{s}_{[322]}\int\mathrm{d}^{3+2\epsilon}\bm{x}\,\phi_{[0]}\phi_{[1]}\mathcal{O}_{[2]}.$
(196)

The complete counterterm action is given by

$\displaystyle S^{\text{sym, ct}}$
$\displaystyle=\frac{1}{6}\lambda_{[222]}S^{\text{sym,
ct}\,(3)}_{[222]}+\frac{1}{2}\lambda_{[322]}S^{\text{sym,
ct}\,(3)}_{[322]}+\frac{1}{2}\lambda_{[332]}S^{\text{sym,
ct}\,(3)}_{[332]}+\frac{1}{6}\lambda_{[333]}S^{\text{sym, ct}\,(3)}_{[333]}$
$\displaystyle\quad+\frac{1}{6}\lambda_{[3222]}S^{\text{sym,
ct}\,(4)}_{[3222]}+\frac{1}{4}\lambda_{[3322]}S^{\text{sym,
ct}\,(4)}_{[3322]}+\frac{1}{6}\lambda_{[3332]}S^{\text{sym,
ct}\,(4)}_{[3332]}+\frac{1}{24}\lambda_{[3333]}S^{\text{sym,
ct}\,(4)}_{[3333]}$
$\displaystyle\quad+\sum_{\Delta_{x}=2,3}\left[\frac{1}{2}\lambda_{[32\Delta_{x}]}\lambda_{[22\Delta_{x}]}S^{\text{sym,
ct}\,(4)}_{[32,22x\Delta_{x}]}\right.$
$\displaystyle\quad\qquad\qquad+\frac{1}{4}\left(\lambda_{[33\Delta_{x}]}\lambda_{[22\Delta_{x}]}S^{\text{sym,
ct}\,(4)}_{[33,22x\Delta_{x}]}+2\lambda_{[32\Delta_{x}]}^{2}S^{\text{sym,
ct}\,(4)}_{[32,32x\Delta_{x}]}\right)$
$\displaystyle\quad\qquad\qquad+\frac{1}{6}\lambda_{[33\Delta_{x}]}\lambda_{[32\Delta_{x}]}\left(S^{\text{sym,
ct}\,(4)}_{[32,33x\Delta_{x}]}+S^{\text{sym,
ct}\,(4)}_{[33,23x\Delta_{x}]}+S^{\text{sym,
ct}\,(4)}_{[33,32x\Delta_{x}]}\right)$
$\displaystyle\quad\qquad\qquad\left.+\frac{1}{8}\lambda_{[33\Delta_{x}]}^{2}S^{\text{sym,
ct}\,(4)}_{[33,33x\Delta_{x}]}\right].$ (197)

The symmetry factors here are quite straightforward for the 3-point
counterterm actions, but more complicated for the 4-point amplitudes. Since
each 3-point function in (190) is given by a single amplitude, the
corresponding symmetry factors are equal to one over the symmetry factor of
the Witten diagram. The same argument holds for contact 4-point functions,
which leads to the values listed.

For the exchange 4-point counterterms, we have to remember that the
expressions in (193) always contain three crossing-symmetric terms. Consider,
for example, the contribution to the 4-point function
$\langle\mathcal{O}_{[3]}\mathcal{O}_{[2]}\mathcal{O}_{[2]}\mathcal{O}_{[2]}\rangle$.
We need the counterterm contribution to each of the three crossing symmetric
terms in (193). Notice, however, that
$i^{\text{ct}({4})}_{[32,22x\Delta_{x}]}$ in (112) is a constant and thus is
trivially crossing symmetric. Hence, we need the contribution of
$3i^{\text{ct}({4})}_{[32,22x\Delta_{x}]}$ from the action $S^{\text{sym,
ct}\,(4)}_{[32,22x\Delta_{x}]}$. On the other hand, $S^{\text{sym,
ct}\,(4)}_{[32,22x\Delta_{x}]}=\mathfrak{s}_{[32,22x\Delta_{x}]}\int\phi_{[0]}\phi_{[1]}^{3}$
and thus after taking the functional derivatives, its contribution to the
4-point function equals $6i^{\text{ct}({4})}_{[3222]}$. This leads to the
factor of $1/2$ in (4.5.2) multiplying $S^{\text{sym,
ct}\,(4)}_{[32,22x\Delta_{x}]}$. Similarly, let us look at the counterterm
contribution rendering the 4-point function
$\langle\mathcal{O}_{[3]}\mathcal{O}_{[3]}\mathcal{O}_{[2]}\mathcal{O}_{[2]}\rangle$
finite. The three terms in (193) become

$\displaystyle\left(\lambda_{[33\Delta_{x}]}\lambda_{[22\Delta_{x}]}\mathfrak{s}_{[33,22x\Delta_{x}]}+2\lambda_{[32\Delta_{x}]}^{2}\mathfrak{s}_{[32,32x\Delta_{x}]}\right)\times\left[\hat{i}_{[22]}(k_{3})+\hat{i}_{[22]}(k_{4})\right].$
(198)

On the other hand, the contributions to the 4-point function from the
counterterm actions $S^{\text{sym, ct}\,(4)}_{[33,22x\Delta_{x}]}$ and
$S^{\text{sym, ct}\,(4)}_{[32,32x\Delta_{x}]}$ are
$4\mathfrak{s}_{[33,22x\Delta_{x}]}(\hat{i}_{[22]}(k_{3})+\hat{i}_{[22]}(k_{4}))$
and
$4\mathfrak{s}_{[32,32x\Delta_{x}]}(\hat{i}_{[22]}(k_{3})+\hat{i}_{[22]}(k_{4}))$
respectively. This leads to the factor of $1/4$ in (4.5.2).

####  Beta functions

From a QFT point of view, it is more convenient to organize the counterterm
action as

$\displaystyle S^{\text{sym}}_{\text{ct}}$
$\displaystyle=\int\mathrm{d}^{3+2\epsilon}\bm{x}\left[Z_{[0]}(\phi_{[0]}\mu^{-\epsilon})\phi_{[0]}\mathcal{O}_{[3]}+Z_{[1]}(\phi_{[0]}\mu^{-\epsilon})\phi_{[1]}\mathcal{O}_{[2]}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\mu^{2\epsilon}Z_{[3]}(\phi_{[0]}\mu^{-\epsilon},\phi_{[1]}\mu^{-\epsilon})\bm{1}\
\right].$ (199)

In the language of textbook QFT, $Z_{[0]}$ and $Z_{[1]}$ are multiplicative
renormalization factors inducing beta functions for the sources $\phi_{[0]}$
and $\phi_{[1]}$ respectively, while $Z_{[3]}$ is the anomaly-inducing
additive renormalization factor. This means that we identify $\phi_{[0]}$ and
$\phi_{[1]}$ as _renormalized sources_ , which depend implicitly on the
renormalization scale $\mu$. The _bare sources_ are the combinations that
couple directly to the operators in (4.5.3), i.e.,

$\displaystyle\phi^{\text{bare}}_{[0]}$
$\displaystyle=Z_{[0]}(\phi_{[0]}\mu^{-\epsilon})\phi_{[0]},$
$\displaystyle\phi^{\text{bare}}_{[1]}$
$\displaystyle=Z_{[1]}(\phi_{[0]}\mu^{-\epsilon})\phi_{[1]}.$ (200)

By re-organizing the action (4.5.2), we find the multiplicative
renormalization factors

$\displaystyle Z_{[0]}$
$\displaystyle=1+\frac{1}{2}\lambda_{[333]}\mathfrak{s}_{[333]}\phi_{[0]}$
$\displaystyle\qquad+\left[\frac{1}{6}\lambda_{[3333]}\mathfrak{s}_{[3333]}+\frac{1}{2}\sum_{\Delta_{x}=2,3}\lambda_{[33\Delta_{x}]}^{2}\mathfrak{s}_{[33,33x\Delta_{x}]}\right]\phi_{[0]}^{2}+O(\phi_{[0]}^{3}),$
(201) $\displaystyle Z_{[1]}$
$\displaystyle=1+\lambda_{[322]}\mathfrak{s}_{[322]}\phi_{[0]}+\left[\frac{1}{2}\lambda_{[3322]}\mathfrak{s}_{[3322]}\right.$
$\displaystyle\qquad\left.+\frac{1}{2}\sum_{\Delta_{x}=2,3}\left(2\lambda_{[32\Delta_{x}]}^{2}\mathfrak{s}_{[32,32x\Delta_{x}]}+\lambda_{[33\Delta_{x}]}\lambda_{[22\Delta_{x}]}\mathfrak{s}_{[33,22x\Delta_{x}]}\right)\right]\phi_{[0]}^{2}+O(\phi_{[0]}^{3}).$
(202)

It is now straightforward to compute the beta functions for the two couplings,

$\displaystyle\beta_{\phi_{[n]}}=\mu\frac{\mathrm{d}}{\mathrm{d}\mu}\phi_{[n]}.$
(203)

To do this, we invert the relations (200) in order to express the renormalized
sources in terms of the bare, scale-independent sources. In this way, we find

$\displaystyle\beta_{\phi_{[0]}}$
$\displaystyle=\frac{1}{6}\lambda_{[333]}\phi_{[0]}^{2}+\left[-\frac{1}{18}\lambda_{[3333]}+\frac{1}{12}\lambda_{[332]}^{2}-\frac{1}{54}\lambda_{[333]}^{2}\right]\phi_{[0]}^{3}+O(\phi_{[0]}^{4}),$
(204) $\displaystyle\beta_{\phi_{[1]}}$
$\displaystyle=\lambda_{[322]}\phi_{[0]}\phi_{[1]}+\left[-\frac{1}{2}\lambda_{[3322]}-\lambda_{[322]}^{2}+\frac{1}{4}\lambda_{[222]}\lambda_{[332]}-\frac{1}{2}\lambda_{[332]}^{2}\right.$
$\displaystyle\qquad\qquad\left.-\frac{1}{18}(1+3\mathfrak{a}^{(1)}_{[322]}-3\mathfrak{a}_{[333]}^{(1)})\lambda_{[322]}\lambda_{[333]}\right]\phi_{[0]}^{2}\phi_{[1]}+O(\phi_{[0]}^{3}).$
(205)

Notice that the second term in $\beta_{\phi_{[1]}}$ is scheme-dependent.
However, this is not in contradiction with the well-known statement that the
first two terms in the beta function are scheme-independent. Rather, this
happens because the beta functions are calculated for the dimensionful
regulated sources $\phi_{[0]}$ and $\phi_{[1]}$. If instead we were to look at
the dimensionless couplings $g_{[0]}$ and $g_{[1]}$, defined as

$\displaystyle g_{[0]}=\phi_{[0]}\mu^{-\epsilon},\qquad
g_{[1]}=\phi_{[1]}\mu^{-1-\epsilon},$ (206)

we would obtain beta functions

$\displaystyle\beta_{g_{[0]}}$ $\displaystyle=-\epsilon
g_{[0]}+\frac{1}{6}\lambda_{[333]}g_{[0]}^{2}+\left[-\frac{1}{18}\lambda_{[3333]}+\frac{1}{12}\lambda_{[332]}^{2}-\frac{1}{54}\lambda_{[333]}^{2}\right]g_{[0]}^{3}+O(g_{[0]}^{4}),$
(207) $\displaystyle\beta_{g_{[1]}}$
$\displaystyle=-(1+\epsilon)g_{[1]}+\lambda_{[322]}g_{[0]}g_{[1]}+\left[-\frac{1}{2}\lambda_{[3322]}-\lambda_{[322]}^{2}+\frac{1}{4}\lambda_{[222]}\lambda_{[332]}-\frac{1}{2}\lambda_{[332]}^{2}\right.$
$\displaystyle\qquad\qquad\left.-\frac{1}{18}(1+3\mathfrak{a}^{(1)}_{[322]}-3\mathfrak{a}_{[333]}^{(1)})\lambda_{[322]}\lambda_{[333]}\right]g_{[0]}^{2}g_{[1]}+O(g_{[0]}^{3}),$
(208)

where, as customary, we kept the terms of order $\epsilon$ in the classical
scaling term. The beta function for $g_{[1]}$ now contains a non-vanishing
linear term so that the first two terms are indeed scheme-independent.

####  Anomalies

The dependence of the correlation functions on the renormalization scale $\mu$
is governed by the Callan-Symanzik equation. Let
$W[\phi_{[0]},\phi_{[1]};\mu]$ be the renormalized generating functional and
$\phi_{[0]}$ and $\phi_{[1]}$ denote the renormalized sources. The chain rule
then leads to the Callan-Symanzik equation,

$\displaystyle\mu\frac{\mathrm{d}}{\mathrm{d}\mu}W=\left[\mu\frac{\partial}{\partial\mu}+\sum_{j=0,1}\int\mathrm{d}^{d}\bm{x}\,\beta_{\phi_{[j]}}\frac{\delta}{\delta\phi_{[j]}(\bm{x})}\right]W=\int\mathrm{d}^{d}\bm{x}\,\mathcal{A},$
(209)

where $\mathcal{A}$ is the anomaly. The anomaly reads

$\displaystyle\mathcal{A}=\mathcal{A}^{(3)}+\mathcal{A}^{(4)},$ (210)

where

$\displaystyle\mathcal{A}^{(3)}$
$\displaystyle=-\frac{1}{6}\lambda_{[222]}\phi_{[1]}^{3}+\frac{1}{2}\lambda_{[332]}\phi_{[1]}\partial_{\mu}\phi_{[0]}\partial^{\mu}\phi_{[0]},$
(211) $\displaystyle\mathcal{A}^{(4)}$
$\displaystyle=\phi_{[0]}\phi_{[1]}^{3}\,\left[\frac{1}{2}\lambda_{[322]}\lambda_{[222]}(1+\mathfrak{a}_{[222]}^{(1)}-\mathfrak{a}_{[322]}^{(1)})+\frac{1}{4}\lambda_{[332]}\lambda_{[322]}+\frac{1}{6}\lambda_{[3222]}\right]$
$\displaystyle\qquad+\left(\frac{1}{6}\phi_{[0]}^{3}\partial^{2}\phi_{[1]}-\frac{1}{2}\phi_{[0]}^{2}\phi_{[1]}\partial^{2}\phi_{[0]}\right)\times\left[-\frac{1}{2}\lambda_{[3332]}\right.$
$\displaystyle\qquad\qquad\left.-\frac{1}{2}\lambda_{[332]}\lambda_{[322]}(2+\mathfrak{a}_{[332]}^{(1)}-\mathfrak{a}_{[322]}^{(1)})-\frac{1}{3}\lambda_{[333]}\lambda_{[332]}(\tfrac{25}{12}+\mathfrak{a}_{[332]}^{(1)}-\mathfrak{a}_{[333]}^{(1)})\right]$
$\displaystyle\qquad+\phi_{[0]}^{3}\partial^{2}\phi_{[1]}\,\left[\frac{1}{3}\lambda_{[332]}\lambda_{[322]}+\frac{1}{9}\lambda_{[333]}\lambda_{[332]}\right].$
(212)

As we can see the anomaly is scheme-dependent, which means that some terms can
be eliminated by the suitable choice of the counterterm constants.

The Callan-Symanzik equation (209) serves as a check on our results. First, we
act with suitable functional derivatives on both sides in order to translate
it into a statement on the correlators. As an example, consider the
renormalized 4-point function
$\langle\mathcal{O}_{[3]}\mathcal{O}_{[2]}\mathcal{O}_{[2]}\mathcal{O}_{[2]}\rangle$.
The explicit derivative of (191) with respect to the scale $\mu$ reads

$\displaystyle\mu\frac{\partial}{\partial\mu}\langle\\!\langle\mathcal{O}_{[3]}(\bm{k}_{1})\mathcal{O}_{[2]}(\bm{k}_{2})\mathcal{O}_{[2]}(\bm{k}_{3})\mathcal{O}_{[2]}(\bm{k}_{4})\rangle\\!\rangle=\frac{3}{2}\lambda_{[332]}\lambda_{[322]}+\lambda_{[3222]}$
$\displaystyle\qquad\qquad+\lambda_{[322]}\lambda_{[222]}\left[-\log\left(\frac{l_{34+}l_{24+}l_{23+}}{\mu^{3}}\right)+3-3\mathfrak{a}_{[322]}^{(1)}\right].$
(213)

It is easy to see that the only relevant term in the beta functions is
$\lambda_{[322]}\phi_{[0]}\phi_{[1]}$ present in $\beta_{[1]}$. The
corresponding term in the Callan-Symanzik equation, Fourier-transformed into
momentum space, reads

$\displaystyle\left.\frac{\delta^{4}}{\delta\phi_{[0]}(\bm{x}_{1})\delta\phi_{[1]}(\bm{x}_{2})\delta\phi_{[1]}(\bm{x}_{3})\delta\phi_{[1]}(\bm{x}_{4})}\int\mathrm{d}^{3}\bm{x}\,\lambda_{[322]}\phi_{[0]}\phi_{[1]}\frac{\delta
W}{\delta\phi_{[1]}}\right|_{\text{Fourier}}$
$\displaystyle\qquad\qquad=\lambda_{[322]}\lambda_{[222]}\left[\log\left(\frac{l_{34+}l_{24+}l_{23+}}{\mu^{3}}\right)+3\mathfrak{a}_{[222]}^{(1)}\right].$
(214)

When the two terms are added together, the logarithmic terms cancel, leading
to an ultralocal expression. This is an important check on our results as the
anomaly in a local theory must be ultralocal. It is now straightforward to
check that the remaining sum is obtained by acting with the functional
derivatives on the anomaly $\mathcal{A}^{(3)}$ in (211).

##  Change of scheme

The aim of this section is to present a comprehensive analysis allowing for a
change of the regularization scheme. The amplitudes presented in Section 3 are
regulated in the half-integer scheme (1). In this section, we show how to
obtain the amplitudes regulated in the general scheme (2), i.e.,

$\displaystyle d\longmapsto\hat{d}=d+2u\epsilon,$
$\displaystyle\Delta_{j}\longmapsto\hat{\Delta}_{j}=\Delta_{j}+(u+v_{j})\epsilon,$
(215)

where $u$ and $v_{j}$ for $j=1,2,3,4,x$ are arbitrary regularization
parameters. There are several motivations for this:

  * •

The half-integer scheme we used so far is very convenient, but different
regularization schemes may be needed for other computations. In general, one
needs to shift all parameters by different amounts, as in (215), in order to
regulate all correlators. For example, in the computation of tensorial
correlators involving conserved currents and/or stress tensors, the general
scheme is typically required as discussed in Bzowski:2015pba ; Bzowski:2017poo
; Bzowski:2018fql .

  * •

Another motivation arises from applications to inflationary cosmology. As
discussed in Bzowski:2012ih ; McFadden:2013ria , a certain class of slow-roll
models can be described holographically in terms of a three-dimensional CFT
deformed by a marginally relevant scalar of dimension $\Delta=3-\lambda$. The
small parameter $\lambda\ll 1$ is related to the slow-roll parameters and the
spectral tilt, and allows one to compute cosmological correlators using
conformal perturbation theory. For such purposes, we want to know amplitudes
in the regularization scheme $u=0$ and $v_{j}=-1$, i.e., for $d=3$ and
$\Delta=3-\epsilon$, where the regulator $\epsilon$ now plays the role of
$\lambda$.

In the following, since our focus will now be on the $u$ and $v_{j}$
dependence of the regulated amplitudes, we will drop all other variables and
denote $\hat{i}=\hat{i}(u,v_{j})$.

###  2-point amplitudes

In the general regularization scheme (215), 2-point amplitudes follow from
(61) and the expansion of the bulk-to-boundary propagator (3). With
$\hat{d}=3+2u\epsilon$ and $\hat{\Delta}=\Delta+(u+v)\epsilon$, one finds

$\displaystyle\hat{i}_{[22]}$
$\displaystyle=-\frac{\Gamma\left(\frac{1}{2}-v\epsilon\right)}{4^{v\epsilon}\Gamma\left(\frac{1}{2}+v\epsilon\right)}k^{1+2v\epsilon},\qquad\quad\hat{i}_{[33]}=-\frac{\Gamma\left(-\frac{1}{2}-v\epsilon\right)}{4^{1+v\epsilon}\Gamma\left(\frac{3}{2}+v\epsilon\right)}k^{3+2v\epsilon}.$
(216)

###  3-point amplitudes

In this subsection we address the scheme-change procedure for 3-point
amplitudes, following the discussion in Bzowski:2015pba  and Bzowski:2015yxv .
In general, it is difficult to calculate directly the amplitudes in the
general regularization scheme (215) with arbitrary $u,v_{j}$. However, if we
know the regulated amplitude for some specific choice $\bar{u},\bar{v}_{j}$,
$\hat{i}(\bar{u},\bar{v}_{j})$, then it is possible to compute the additional
terms needed to obtain the regulated amplitude $\hat{i}(u,v_{j})$ for general
$u,v_{j}$. As we already know the results in the half-integer scheme with
$\bar{u}=1$ and $\bar{v}_{j}=0$ from Section 3, this will be our starting
point.

As discussed in Bzowski:2015yxv , the scheme-dependent terms are contained in
a specific part of the regulated amplitude, which also contains all divergent
terms. The idea is then to isolate this part, which we will call the
_parameter-dependent_ or _scheme-dependent_ part. The remaining part is finite
and scheme-independent, and can be obtained from the half-integer scheme. We
computed in full generality the finite scheme-dependent part of 3-point
functions in Bzowski:2015yxv , but for our purposes here we also need the
order $\epsilon$ scheme-dependent part. The reason is that this part will be
needed in the computation of finite scheme-dependent terms in 4-point exchange
diagrams.

We begin this section by reviewing the procedure for scheme change discussed
in Bzowski:2015yxv , and showing how the procedure can be simplified when the
starting point is the half-integer scheme (1). Our main goal is to derive the
3-point amplitudes, including parameter-dependent terms up to and including
terms of order $\epsilon$. This will be essential for the derivation of the
exchange 4-point functions in the general regularization scheme (215) in
Section 5.4. Furthermore, for this same reason, we also must calculate
3-point-like amplitudes with one Bessel-$K$ function in (8) replaced by a
Bessel-$I$.

####  Definitions

Let $\mathcal{I}$ denote the integrand in (8). Its power expansion around
$z=0$ reads

$\displaystyle\mathcal{I}$
$\displaystyle=\sum_{\sigma_{1},\sigma_{2},\sigma_{3}=\pm
1}\>\sum_{n_{1},n_{2},n_{3}=0}^{\infty}b_{n_{1}}(\sigma_{1}\hat{\beta}_{1})b_{n_{2}}(\sigma_{2}\hat{\beta}_{2})b_{n_{3}}(\sigma_{3}\hat{\beta}_{3})z^{\frac{\hat{d}}{2}-1+\sum_{j=1}^{3}(\sigma_{j}\hat{\beta}_{j}+2n_{j})},$
(217)

where the coefficients $b_{n}$ are defined in (490) and the regulated
$\beta_{j}$ parameters are given as usual by

$\displaystyle\hat{\beta}_{j}=\hat{\Delta}_{j}-\frac{\hat{d}}{2}=\beta_{j}+v_{j}\epsilon,$
$\displaystyle\beta_{j}=\Delta_{j}-\frac{d}{2}.$ (218)

Let us rearrange the sum as

$\displaystyle\mathcal{I}$
$\displaystyle=\sum_{\sigma_{1},\sigma_{2},\sigma_{3}=\pm
1}\sum_{n=0}^{\infty}\mathcal{I}_{(\sigma_{j},n)}z^{L(\sigma_{j},n)-1},$ (219)

where the non-vanishing coefficients and the corresponding $L$s are
parametrized by three signs, $\sigma_{1},\sigma_{2},\sigma_{3}=\pm 1$ and a
non-negative integer $n$,

$\displaystyle L(\sigma_{j},n)$
$\displaystyle=\frac{\hat{d}}{2}+\sum_{j=1}^{3}\sigma_{j}\hat{\beta}_{j}+2n,$
(220) $\displaystyle\mathcal{I}_{(\sigma_{j},n)}$
$\displaystyle=\sum_{\begin{subarray}{c}n_{1},n_{2},n_{3}\geq 0\\\
n_{1}+n_{2}+n_{3}=n\end{subarray}}b_{n_{1}}(\sigma_{1}\hat{\beta}_{1})b_{n_{2}}(\sigma_{2}\hat{\beta}_{2})b_{n_{3}}(\sigma_{3}\hat{\beta}_{3}).$
(221)

Next we analyze the $\epsilon$-dependence. From (220), we see that $L$ is
linear in the regulator,

$\displaystyle
L(\sigma_{j},n)=L_{0}(\sigma_{j},n)+\epsilon\,L_{\sigma_{1}\sigma_{2}\sigma_{3}}(u,v_{j}),$
(222)

where

$\displaystyle L_{0}(\sigma_{j},n)$
$\displaystyle=\frac{d}{2}+\sum_{j=1}^{3}\sigma_{j}\beta_{j}+2n,$
$\displaystyle
L_{\sigma_{1}\sigma_{2}\sigma_{3}}(u,v_{j})=u+\sum_{j=1}^{3}\sigma_{j}v_{j}\,,$
(223)

and all scheme-dependence in $L$ enters via
$L_{\sigma_{1}\sigma_{2}\sigma_{3}}(u,v_{j})$.

The coefficients $\mathcal{I}_{(\sigma_{j},n)}$ depend on the regulator