††institutetext: The Racah Institute of Physics, The Hebrew University of
Jerusalem,
Jerusalem 91904, Israel

# Conformal correlators in the critical $O(N)$ vector model

Noam Chai Mikhail Goykhman and Ritam Sinha<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We calculate a set of conformal correlators in the critical $O(N)$ vector
model in $2<d<6$ dimensions. We focus on the correlators involving the
Hubbard-Stratonovich field $s$, and its composite form $s^{2}$. In the
process, we report a number of new calculations of diagrams involving the
composite $s^{2}$ operator. Through the calculation of the $\langle
s^{2}s^{2}s\rangle$ three-point function, we shed new light on a conjectured
$s\rightarrow-s$ symmetry in the $s$ sector of the critical $O(N)$ vector
model in $d=3$.

##  1 Introduction

The critical $O(N)$ vector model with quartic interaction and the critical
$U(n)$ Gross-Neveu model are some of the most well-studied interacting
conformal field theories. These CFTs are free in even space-time dimensions,
which allows one to study them perturbatively via the $\epsilon$-expansion in
the Wilson-Fisher regime Wilson:1971dc ; Gross:1974jv ; Brezin:1976qa ;
ZinnJustin:1991yn ; Fei:2014yja ; Fei:2014xta . In three dimensions, these
models are strongly coupled and are not accessible to the perturbative
treatment, but prove to be of great interest from the standpoint of
understanding the behavior of quantum systems at criticality. Other methods,
such as the $1/N$ expansion Vasiliev:1981yc ; Vasiliev:1981dg ; Gracey:1990wi
; Gracey:1992cp ; Derkachov:1993uw ; Vasiliev:1992wr ; Vasiliev:1993pi ;
Gracey:1993kb ; Gracey:1993kc ; Manashov:2016uam ; Manashov:2017rrx and the
conformal bootstrap Parisi:1972zm ; Polyakov:1974gs ; Ferrara:1973yt ;
ElShowk:2012ht ; Simmons-Duffin:2016gjk are frequently used to study such
CFTs, and have recently been undergoing an active development.

The interest in these critical models in $d=3$ dimensions is greatly amplified
by their relevance in the context of the $3d/4d$ holographic duality, where
they are described by the dual higher-spin theories in the AdS bulk
Klebanov:2002ja .111See also Petkou:2003zz ; Sezgin:2003pt ; Giombi:2009wh and
references therein for some of the earlier works. When coupled to the Chern-
Simons field, the fundamental scalar and fermionic matter exhibit interesting
Bose/Fermi dualities, and its holographic dual, in turn, has been conjectured
to be an interpolation between type A and type B Vasiliev higher-spin theories
Giombi:2011kc ; Aharony:2011jz ; Aharony:2012nh .

In the deep UV regime, the Gross-Neveu model in $2\leq d\leq 4$ dimensions
reaches a fixed point. This fixed point can be studied perturbatively in the
vicinity of $d=2$ dimensions (where the model is asymptotically free
Gross:1974jv ), as well as in the vicinity of $d=4$ dimensions (where the
model is critically equivalent to the Gross-Neveu-Yukawa model
ZinnJustin:1991yn ). In general $d$, this interacting CFT describes dynamics
of the fermions $\psi^{i}$ and the Hubbard-Stratonovich scalar field $\sigma$,
and its action is given by

$S_{\textrm{G.N}}=\int
d^{d}x\,\left(\bar{\psi}^{i}\gamma^{\mu}\partial_{\mu}\psi^{i}+\frac{1}{\sqrt{N}}\,\sigma\bar{\psi}^{i}\psi^{i}\right)\,.$
(1)

The Gross-Neveu model action (1) is invariant w.r.t. the discrete
$\mathbb{Z}_{2}$ symmetry Gross:1974jv ; ZinnJustin:1991yn ; Moshe:2003xn

$\displaystyle(x^{1},\dots,x^{a-1},x^{a},x^{a+1},\dots,x^{d})\rightarrow(x^{1},\dots,x^{a-1},-x^{a},x^{a+1},\dots,x^{d})\,,$
$\displaystyle\sigma\rightarrow-\sigma\,,\quad\psi\rightarrow\gamma_{a}\psi\,,\quad\bar{\psi}\rightarrow-\bar{\psi}\gamma_{a}\,,$
(2)

for any given $a=1,\dots,d$. In particular, (2) implies
$\bar{\psi}\psi\rightarrow-\bar{\psi}\psi$, and as a result the interaction
term $\sigma\bar{\psi}^{i}\psi^{i}$ in the action (1) is left invariant.

Furthermore, the symmetry (2) uniquely fixes the structure of some of the
correlation functions in the critical GN model. For instance, while conformal
symmetry allows two possible structures of the three-point correlation
functions involving one scalar and two fermionic fields Weinberg:2010fx ;
Goykhman:2018ihr , the $\mathbb{Z}_{2}$ symmetry (2) can further select which
one of these two structures is allowed, as can be seen on the following
example Goykhman:2020ffn :

$\displaystyle\langle\bar{\psi}(x_{1})\psi(x_{2})\sigma(x_{3})^{2k+1}\rangle$
$\displaystyle\sim\frac{\gamma_{\mu}\,x_{13}^{\mu}\,\gamma_{\nu}\,x_{32}^{\nu}}{|x_{12}|^{d-2-2k}(|x_{13}||x_{23}|)^{2k+2}}\,,$
(3)
$\displaystyle\langle\bar{\psi}(x_{1})\psi(x_{2})\sigma(x_{3})^{2(k+1)}\rangle$
$\displaystyle\sim\frac{\gamma_{\mu}\,x_{12}^{\mu}}{|x_{12}|^{d-2(k+1)}(|x_{13}||x_{23}|)^{2(k+1)}}\,,$
(4)

where $k=0,1,2,\dots$. While (3), (4) are leading order in $1/N$, the same
structure will hold exactly to all orders in the $1/N$ expansion.

One can also study manifestations of the symmetry (2) in the singlet sector of
the critical GN model. This is particularly relevant from the standpoint of
the holographic duality, which provides a prescription for evaluation of the
correlation function of the $U(n)$ singlets via the dual description of the
gauge degrees of freedom in the AdS bulk. An immediate consequence of the
symmetry (2) is that the correlation functions involving an odd number of the
Hubbard-Stratonovich fields $\sigma$ vanish. The simplest example of this
statement is given by the triviality of the three-point function

$\langle\sigma(x_{1})\sigma(x_{2})\sigma(x_{3})\rangle=0\,.$ (5)

The relation (5) can be explicitly verified by evaluating the corresponding
Feynman diagrams Manashov:2016uam ; Goykhman:2020ffn .

In this paper, we study the critical $O(N)$ vector model with the action

$S=\int
d^{d}x\,\left(\frac{1}{2}\,\partial_{\mu}\phi^{i}\,\partial^{\mu}\phi_{i}+\frac{1}{\sqrt{N}}\,s\,\phi^{i}\phi^{i}\right)\,,$
(6)

describing dynamics of the fundamental fields $\phi^{i}$ and the Hubbard-
Stratonovich field $s$. At first glance, the model (6) does not seem to
possess any such remarkable discrete $\mathbb{Z}_{2}$ symmetry due to the
purely bosonic nature of the fields $\phi_{i}$. However, a conjecture was put
forth in Petkou:1994ad regarding the existence of such a symmetry in the
$O(N)$ vector models in $d=3$, based on conformal bootstrap calculation of the
three-point function $\langle sss\rangle$ at the leading order in $1/N$
expansion. It is our aim to shed further light on the fate of this conjecture
at both the leading and next-to leading order in $1/N$ in the model (6).
Inspired by the GN model, we will search for manifestations of this symmetry
in certain conformal correlators. We will be mostly interested in the $d=3$
case, but majority of our calculations will be carried out in general $d$.

The CFT action (6) describes the critical behavior of the $O(N)$ vector model
with quartic interaction at its IR fixed point, and the non-linear sigma model
at its UV fixed point, for $2<d<4$ ZinnJustin:1991yn . When $4<d<6$ the model
(10) describes a UV fixed point of the $O(N)$ vector model with quartic
interaction, and is conjectured to describe the IR fixed point of certain
vector model with cubic coupling. The latter statement is supported by a
perturbative calculation in $d=6-\epsilon$ dimensions up to quartic order
Fei:2014yja ; Fei:2014xta ; Gracey:2015tta .

The counterpart of the transformation (2) in the $O(N)$ vector model (6) would
only act on the Hubbard-Stratonovich field $s$ Petkou:1994ad ,

$s\rightarrow-s\,.$ (7)

While the transformation (7) is clearly not a symmetry of the action (6), it
has been suggested in Petkou:1994ad that such a symmetry might emerge in the
$d=3$ dimensional quantum theory among the correlation functions involving
only the $s$ fields. Interestingly, Petkou:1994ad pointed out that the three-
point function vanishes at the leading order in the $1/N$ expansion,

$\langle s(x_{1})s(x_{2})s(x_{3})\rangle\Bigg{|}_{d=3}=0+{\cal
O}\left(\frac{1}{N^{3/2}}\right)\,,$ (8)

and suggested to explain it by conjecturing the symmetry (7). Notice that (8)
is not valid when $d\neq 3$, in stark contrast with the GN case, where the
$\mathbb{Z}_{2}$ symmetry holds for any $d$. In Goykhman:2019kcj the $\langle
sss\rangle$ correlation function was calculated at the next-to-leading order
in the $1/N$ expansion. Remarkably, Goykhman:2019kcj demonstrated that the
sub-leading correction to the three-point function $\langle sss\rangle$ also
vanishes in $d=3$,

$\langle s(x_{1})s(x_{2})s(x_{3})\rangle\Bigg{|}_{d=3}=0+{\cal
O}\left(\frac{1}{N^{5/2}}\right)\,.$ (9)

This further raises the question of whether the symmetry (7) is valid also
upto the first sub-leading order in $1/N$.

The proposed symmetry, however, is fundamentally different than its
counterpart (2) in the GN model for a number of reasons. Primary among them is
that the symmetry is present only in $d=3$. However, more importantly, the
conjectured symmetry transformation (7) is not respected by any correlation
functions involving the fundamental scalar $\phi_{i}$. As a simple example,
one can notice that the correlation function $\langle\phi\phi s\rangle$ is
non-vanishing in $d=3$ Petkou:1994ad , although it is odd w.r.t. the
transformation (7). In fact, it was originally suggested in Petkou:1994ad that
the symmetry (7), if established, would have to be confined to the sub-sector
of the theory, involving only the $s$ fields. This is very unlike the GN model
where correlations involving the fundamental $\psi_{i}$ field also respect the
discrete symmetry (2), as we reviewed above on the example of the correlation
functions (3), (4).

One might ask if such a limited scope of applicability of the conjectured
symmetry transformation (7) can be explained by the holographic dual of the
three-dimensional vector model, that describes only the singlet sector of the
theory. To test such an assumption, one can consider the three-point function
$\langle\phi^{2}\phi^{2}s\rangle$. At the leading order in $1/N$, this
correlation function is given by the diagram222Our notations and Feynman rules
will be explained in section 2.

$\phi(x_{1})^{2}$$\phi(x_{2})^{2}$$s(x_{3})$$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$$=-\frac{1}{8\pi^{5}\sqrt{N}}\,\frac{1}{(|x_{13}||x_{23}|)^{2}}\qquad\textrm{(in
d=3)}$

which we evaluated in $3d$, obtaining a non-zero result. This quick
calculation further excises the scope of the proposed symmetry transformation
(7), removing the $O(N)$ singlet composite operators involving the field
$\phi_{i}$.

We then intend to study the transformation of the three-point correlation
functions involving the Hubbard-Stratonovich field $s$ only. To test the
proposed symmetry transformation (7) one needs to study correlation functions
involving an odd number of the fields $s$. Above we have discussed that the
three-point function of the $s$ field vanishes up to the next-to-leading order
in $1/N$ expansion. A natural next step is to study three-point functions
involving five of the fields $s$, which requires one to deal with the
composite operators $s^{2}$ or $s^{3}$.

Specifically, the objective of this paper is to calculate the three-point
correlation function $\langle s^{2}s^{2}s\rangle$. While we establish that
this correlator vanishes at the leading order in the $1/N$ expansion, our main
result is that its next-to-leading order correction is in fact non-zero, and
therefore does not respect the conjectured symmetry (7).

In the process we obtain a number of new results, which can be used for other
calculations in the $O(N)$ vector model. Some of the new expressions obtained
in this paper involve the $s^{2}ss$ conformal triangle at the next-to-leading
order in $1/N$ in general $d$, the $\langle s^{2}ss\rangle$ three-point
function in $d=3$, and several self-energy diagrams, which we believe have not
been reported in the literature before. The rest of this paper is organized as
follows. In section 2 we set up our notations and review known results in the
critical $O(N)$ vector model that will be relevant for the purposes of this
paper. We also reformulate the result of Goykhman:2019kcj for the $\langle
sss\rangle$ correlation function in terms of the $sss$ conformal triangle,
representing the cubic effective vertex at the next-to-leading order in $1/N$.
In section 3 we derive the $s^{2}ss$ conformal triangle in general $d$, and
calculate the $\langle s^{2}ss\rangle$ three-point function in $d=3$ at the
next-to-leading order in $1/N$. In section 4 we derive the $\langle
s^{2}s^{2}s\rangle$ correlation function in $d=3$. We demonstrate that while
the leading-order $3d$ correlation function vanishes, its $1/N$ correction is
non-trivial. Motivated by this result, in section 5 we then explore whether
the conjectured symmetry (7) is an artifact of the large-$N$ limit. We discuss
our results in section 6.

##  2 Set-up

In this paper we will be studying the critical $O(N)$-invariant vector model
with the action

$S=\int
d^{d}x\,\left(\frac{1}{2}\,(\partial\phi)^{2}+\frac{1}{\sqrt{N}}\,s\,\phi^{2}\right)+S_{\textrm{c.t.}}\,,$
(10)

describing dynamics of the multiplet $\phi^{i}$, $i=1,\dots,N$ of the real-
valued scalar fields, and the Hubbard-Stratonovich field $s$.333Here and in
what follows we skip keeping track of the $O(N)$ indices where it does not
cause a confusion. In the action (10) we have also incorporated the
counterterm induced by wave-function renormalization of the fields $\phi$,
$s$, see Goykhman:2019kcj for a recent review.

The $\phi$ field propagator is given by

$\langle\phi(x)\phi(0)\rangle=\frac{C_{\phi}\,(1+A_{\phi})\,\mu^{-2\gamma_{\phi}}}{|x|^{2(\Delta_{\phi}+\gamma_{\phi})}}\,,$
(11)

where $\mu$ is an arbitrary RG scale, $\Delta_{\phi}$ is the free scaling
dimension, and $\gamma_{\phi}$ is the anomalous dimension, given by
Vasiliev:1981yc ; Vasiliev:1981dg ; Petkou:1994ad

$\displaystyle\Delta_{\phi}$ $\displaystyle=\frac{d}{2}-1\,,$ (12)
$\displaystyle\gamma_{\phi}$
$\displaystyle=\frac{1}{N}\,\frac{2^{d}\sin\left(\frac{\pi
d}{2}\right)\Gamma\left(\frac{d-1}{2}\right)}{\pi^{3/2}(d-2)d\Gamma\left(\frac{d}{2}-2\right)}+{\cal
O}\left(\frac{1}{N^{2}}\right)\,,$ (13)

leading in $1/N$ amplitude is given by

$C_{\phi}=\frac{\Gamma\left(\frac{d}{2}-1\right)}{4\pi^{\frac{d}{2}}}\,,$ (14)

and sub-leading correction to the amplitude is Derkachov:1997ch

$A_{\phi}=\left(\frac{d}{2-d}-\frac{2}{d}\right)\,\gamma_{\phi}+{\cal
O}\left(\frac{1}{N^{2}}\right)\,.$ (15)

The Feynman rule corresponding to the propagator (11) is

$2\Delta_{\phi}$$=\frac{C_{\phi}}{|x|^{2\Delta_{\phi}}}$$0$$x$

In a general conformal graph we will also use the following Feynman rule for
internal lines with unit amplitudes

$2a$$=\frac{1}{|x|^{2a}}$$0$$x$

Since the action (10) is quadratic in $\phi$, the corresponding path integral
is Gaussian, and can be performed explicitly, resulting in the effective
action for $s$ formally written as

$S_{\textrm{eff}}=\frac{N}{2}\,\int
d^{d}x\,\textrm{Tr}\log\left(\partial^{2}-\frac{2}{\sqrt{N}}\,s\right)\,.$
(16)

Expanding the logarithm, we obtain

$S_{\textrm{eff}}=-C_{\phi}^{2}\int
d^{d}x_{1,2}\,\frac{s(x_{1})s(x_{2})}{|x_{12}|^{2(d-2)}}+\frac{4C_{\phi}^{3}}{3\sqrt{N}}\,\int
d^{d}x_{1,2,3}\,\frac{s(x_{1})s(x_{2})s(x_{3})}{(|x_{12}||x_{13}||x_{23}|)^{d-2}}+\cdots\,,$
(17)

where ellipsis stand for vertices of higher order in $s$.

From the quadratic term in the action (17) we obtain the propagator for the
Hubbard-Stratonovich field $s$,

$\langle s(x)s(0)\rangle=\frac{C_{s}}{|x|^{2\Delta_{s}}}\,,$ (18)

where

$\displaystyle\Delta_{s}$ $\displaystyle=2\,,$ (19) $\displaystyle C_{s}$
$\displaystyle=\frac{2^{d}\,\Gamma\left(\frac{d-1}{2}\right)\sin\left(\frac{\pi
d}{2}\right)}{\pi^{\frac{3}{2}}\Gamma\left(\frac{d}{2}-2\right)}\,.$ (20)

The corresponding Feynman rule is

$2\Delta_{s}$$=\frac{C_{s}}{|x|^{2\Delta_{s}}}$$0$$x$

The loop corrections to the Hubbard-Stratonovich propagator result in

$\langle
s(x)s(0)\rangle=\frac{C_{s}\,(1+A_{s})\,\mu^{-2\gamma_{s}}}{|x|^{2(\Delta_{s}+\gamma_{s})}}\,,$
(21)

where Vasiliev:1981yc ; Vasiliev:1981dg ; Petkou:1994ad ; Derkachov:1997ch

$\displaystyle\gamma_{s}$
$\displaystyle=4\left(d+\frac{6}{d-4}+1\right)\,\gamma_{\phi}+{\cal
O}\left(\frac{1}{N^{2}}\right)\,,$ (22) $\displaystyle A_{s}$
$\displaystyle{=}2\gamma_{\phi}\left(\frac{d(d{-}3){+}4}{4{-}d}\,\left(H_{d{-}3}{+}\pi\cot\left(\frac{\pi
d}{2}\right)\right){+}\frac{8}{(d{-}4)^{2}}{+}\frac{2}{d{-}2}{+}\frac{2}{d}{-}2d{-}1\right){+}{\cal
O}\left(\frac{1}{N^{2}}\right)\,.$ (23)

We will also use the Feynman rules corresponding to the dressed propagator

$s$$=\frac{C_{s}\,(1+A_{s})\,\mu^{-2\gamma_{s}}}{|x|^{2(\Delta_{s}+\gamma_{s})}}$$0$$x$$s$$=\frac{C_{s}\,\mu^{-2\gamma_{s}}}{|x|^{2(\Delta_{s}+\gamma_{s})}}$$0$$x$

Higher order terms in the action (17) can be represented by conformal graphs
with internal $\phi$ lines and the interaction vertex

$s$$\phi$$\phi$$=-\frac{2}{\sqrt{N}}$

In particular, second term in the r.h.s. of (17) gives the leading ${\cal
O}(1/N^{1/2})$ order $sss$ vertex. Sub-leading corrections to this vertex can
be written down in terms of the corresponding Polyakov’s Polyakov:1974gs
conformal triangle

$S_{\textrm{eff}}\supset\frac{Z_{sss}}{\sqrt{N}}\,\int
d^{d}x_{1,2,3}\,\frac{s(x_{1})s(x_{2})s(x_{3})}{(|x_{12}||x_{13}||x_{23}|)^{d-2-\gamma_{s}}}\,\mu^{3\gamma_{s}}\,.$
(24)

Here the amplitude of the conformal triangle $Z_{sss}$ and the anomalous
dimension $\gamma_{s}$ are assumed to be expanded to the desired power in
$1/N$,

$Z_{sss}=Z_{sss}^{(0)}\,(1+\delta Z_{sss})\,.$ (25)

The Feynman rule corresponding to the vertex/conformal triangle (24) is given
by

$=-\frac{6Z_{sss}}{\sqrt{N}}\times$$\alpha$$\alpha$$\alpha$$s$$s$$s$$s$$s$$s$

where we denoted

$\alpha=d-2-\gamma_{s}\,.$ (26)

We will also use the following notation for the $sss$ conformal triangle with
the leading order amplitude only:

$=-\frac{6Z_{sss}^{(0)}}{\sqrt{N}}\times$$\alpha$$\alpha$$\alpha$$s$$s$$s$$s$$s$$s$

Comparing the leading order coefficients of the cubic vertex in (17) and (24),
we obtain the leading order amplitude $Z_{sss}^{(0)}$ of the conformal
triangle (25)

$Z_{sss}^{(0)}=\frac{4}{3}\,C_{\phi}^{3}\,.$ (27)

To calculate the sub-leading contribution $\delta Z_{sss}$ to the $sss$
conformal triangle, we use the expression for the three-point function

$\langle s(x_{1})s(x_{2})s(x_{3})\rangle=\frac{C_{sss}^{(0)}(1+\delta
C_{sss})}{(|x_{12}||x_{13}||x_{23}|)^{\Delta_{s}+\gamma_{s}}}\,\mu^{-3\gamma_{s}}\,,$
(28)

where Petkou:1994ad 444See App. A for our conventions.

$\displaystyle C_{sss}^{(0)}$
$\displaystyle=N\left(-\frac{2}{\sqrt{N}}\right)^{3}C_{\phi}^{3}C_{s}^{3}U\left(\frac{d}{2}-1,\frac{d}{2}-1,2\right)^{2}U(1,2,d-3)$
(29) $\displaystyle=-\frac{1}{\sqrt{N}}\frac{8^{d-1}\sin^{3}\left(\frac{\pi
d}{2}\right)\Gamma\left(3-\frac{d}{2}\right)\Gamma\left(\frac{d-1}{2}\right)^{3}}{\pi^{9/2}\Gamma(d-3)}\,,$
(30)

and $\delta C_{sss}$ was found in Goykhman:2019kcj at the next-to-leading
order in $1/N$555 In notations of Goykhman:2019kcj we have $\delta
C_{sss}=W_{s^{3}}+3A_{s}/2$. This is because $W_{s^{3}}$ was defined in
Goykhman:2019kcj as a relative correction to the amplitude of the three-point
function of the normalized fields $s$, related to the non-normalized fields by
the transformation $s\rightarrow\sqrt{C_{s}(1+A_{s})}\,s$. Appearance of the
additional term $3A_{s}/2$ is clear from such a transformation.

$\displaystyle\delta C_{sss}$ $\displaystyle=3W_{\phi\phi
s}+f+W_{3}+W_{4}+\frac{3A_{s}}{2}\,,$ (31) $\displaystyle W_{\phi\phi s}$
$\displaystyle=\gamma_{\phi}\,\left(\frac{d(d-3)+4}{4-d}\Big{(}H_{d-3}+\pi\cot\big{(}\frac{\pi
d}{2}\big{)}\Big{)}+\frac{16}{(d-4)^{2}}+\frac{6}{d-4}+\frac{2}{d-2}-2d+3\right)\,,$
$\displaystyle f$
$\displaystyle=\gamma_{\phi}\,\frac{6(d-1)(d-2)}{d-4}\,\left(H_{d-4}-\frac{2}{d-4}+\pi\cot\big{(}\frac{\pi
d}{2}\big{)}\right)\,,$ $\displaystyle W_{3}$
$\displaystyle=\gamma_{\phi}{2d(d{-}2)(d{-}3)\over(d{-}4)^{2}}\left(6\psi^{(1)}\Big{(}\frac{d}{2}{-}1\Big{)}{-}\psi^{(1)}(d{-}3){-}{\pi^{2}\over
6}{-}H_{d-4}\Big{(}H_{d{-}4}{+}2\pi\cot\big{(}{\pi d\over
2}\big{)}\Big{)}\right)\,,$ $\displaystyle W_{4}$
$\displaystyle=\gamma_{\phi}\,\frac{3d(d-2)\left(\pi^{2}-6\psi^{(1)}\left(\frac{d}{2}-1\right)\right)}{4(d-4)}\,,$

where $\psi^{(1)}$ is the first derivative of the digamma function, and
$H_{n}$ is the $n$the harmonic number. On the other hand, the $\langle
sss\rangle$ three-point function can be obtained by attaching three $s$ legs
to the $sss$ conformal triangle, and integrating over its vertices:

$=\frac{C_{sss}^{(0)}(1+\delta
C_{sss})}{(|x_{12}||x_{13}||x_{23}|)^{\Delta_{s}+\gamma_{s}}}$$s$$s$$s$$x_{3}$$x_{1}$$x_{2}$

Then we can express

$\displaystyle\delta Z_{sss}=\delta C_{sss}-3A_{s}-R_{sss}\,,$ (32)

where $R_{sss}$ is obtained by expansion of the factor

$\displaystyle
U\left(2+\gamma_{s},\frac{d-\gamma_{s}}{2}-1,\frac{d-\gamma_{s}}{2}-1\right)^{2}$
$\displaystyle
U\left(2+\gamma_{s},1+\frac{\gamma_{s}}{2},d-3-\frac{3\gamma_{s}}{2}\right)$
$\displaystyle=u_{sss}^{(0)}\left(1+R_{sss}+{\cal
O}\left(\frac{1}{N^{2}}\right)\right)$ (33)

originating from taking the integrals over the vertices of the $sss$ conformal
triangle,

$\displaystyle u_{sss}^{(0)}$
$\displaystyle=\frac{8\pi^{\frac{3d}{2}}\Gamma\left(3-\frac{d}{2}\right)}{(d-4)^{4}\Gamma(d-4)}\,,$
(34) $\displaystyle R_{sss}$
$\displaystyle=\frac{1}{N}\frac{6\sin\left(\frac{\pi
d}{2}\right)\Gamma(d)\left((d-4)H_{d-4}-2d+\pi(d-4)\cot\left(\frac{\pi
d}{2}\right)+10\right)}{\pi(d-4)\Gamma\left(\frac{d}{2}-1\right)\Gamma\left(\frac{d}{2}+1\right)}\,.$
(35)

Finally notice that

$Z_{sss}^{(0)}=-\frac{C_{sss}^{(0)}}{6C_{s}^{3}u_{sss}^{(0)}}\,,$ (36)

which agrees with (27) .

##  3 $\langle s^{2}ss\rangle$

At the leading order the $\langle s^{2}s^{2}\rangle$ propagator is given by

$0$$x$$2\Delta_{s^{2}}$$=$$2\Delta_{s}$$2\Delta_{s}$$=\frac{C_{s^{2}}}{|x|^{2\Delta_{s^{2}}}}$$0$$x$

Here the leading order scaling dimension of the composite operator $s^{2}$ is
given by

$\Delta_{s^{2}}=4\,,$ (37)

and the leading order propagator amplitude is

$C_{s^{2}}=\frac{1}{2}\,2^{2}\,C_{s}^{2}=2C_{s}^{2}\,,$ (38)

where $1/2$ is the symmetry factor, and each factor of 2 comes from the
degeneracy of each of the two $s^{2}$ insertions.

The loop corrections to the $s^{2}$ propagator result in

$\langle
s(x)^{2}s(0)^{2}\rangle=\frac{C_{s^{2}}\,(1+A_{s^{2}})\,\mu^{-2\gamma_{s^{2}}}}{|x|^{2(\Delta_{s^{2}}+\gamma_{s^{2}})}}\,,$
(39)

where Lang:1992zw

$\displaystyle\gamma_{s^{2}}$ $\displaystyle=-4(d-1)^{2}\,\gamma_{\phi}+{\cal
O}\left(\frac{1}{N^{2}}\right)\,.$ (40)

Due to the conformal symmetry the $\langle s^{2}ss\rangle$ three-point
function takes the form

$\langle s(x_{1})s(x_{2})s^{2}(x_{3})\rangle=\frac{C_{s^{2}ss}^{(0)}(1+\delta
C_{s^{2}ss})}{|x_{12}|^{2\gamma_{s}-\gamma_{s^{2}}}(|x_{13}||x_{23}|)^{4+\gamma_{s^{2}}}}\,\mu^{-2\gamma_{s}-\gamma_{s^{2}}}\,.$
(41)

Here the amplitude of the leading order diagram

$2\Delta_{s}$$2\Delta_{s}$$x_{3}$$x_{1}$$x_{2}$$=\frac{C_{s^{2}ss}^{(0)}}{(|x_{13}||x_{23}|)^{4}}$

is given by

$C_{s^{2}ss}^{(0)}=2C_{s}^{2}\,,$ (42)

where again the factor of 2 comes from the degeneracy of the $s^{2}$
insertion.

At the next-to-leading order the $\langle s^{2}ss\rangle$ three-point function
is composed of the $s$ propagator corrections to the leading-order $\langle
s^{2}ss\rangle$ diagram,

(43)

and three 1PI vertex corrections

(44)

We will denote the total of three diagrams in (44) as

$x_{3}$$x_{1}$$x_{2}$$=\frac{C_{s^{2}ss}^{(0)}(1+\delta
C_{s^{2}ss}-2A_{s})\,\mu^{2\gamma_{s}-\gamma_{s^{2}}}}{(|x_{13}||x_{23}|)^{4+\gamma_{s^{2}}-2\gamma_{s}}|x_{12}|^{2\gamma_{s}-\gamma_{s^{2}}}}-\frac{C_{s^{2}ss}^{(0)}}{(|x_{13}||x_{23}|)^{4}}$

where the amplitude and the exponents have been determined from the
requirement that the total adds up to (41), and we also subtracted the leading
order $\langle s^{2}ss\rangle$ three-point function to ensure that the result
is purely of the next-to-leading order in $1/N$. We can also re-write this
relation as

(45)

We will use the Feynman rule corresponding to the dressed $s^{2}$ propagator

$s^{2}$$=\frac{C_{s^{2}}\,(1+A_{s^{2}})\,\mu^{-2\gamma_{s^{2}}}}{|x|^{2(\Delta_{s^{2}}+\gamma_{s^{2}})}}$$0$$x$

###  3.1 $s^{2}ss$ conformal triangle

Consider the $s^{2}ss$ conformal triangle:

$=-2Z_{s^{2}ss}\times$$2a$$2a$$2b$$s^{2}$$s$$s$$s^{2}$$s$$s$

Here we defined

$a=\frac{d-\gamma_{s^{2}}}{2}-2\,,\qquad
b=\frac{d+\gamma_{s^{2}}}{2}-\gamma_{s}\,.$ (46)

We can expand the amplitude of the conformal triangle in $1/N$ as

$Z_{s^{2}ss}=Z_{s^{2}ss}^{(0)}(1+\delta Z_{s^{2}ss})\,.$ (47)

The conformal triangle is defined so that when we attach full propagators of
the $s^{2}$ and $s$ and integrate over the three internal vertices, we obtain
the three-point function:

$=\frac{C_{s^{2}ss}^{(0)}(1+\delta
C_{s^{2}ss})\,\mu^{-\gamma_{s^{2}}-2\gamma_{s}}}{(|x_{13}||x_{23}|)^{4+\gamma_{s^{2}}}|x_{12}|^{2\gamma_{s}-\gamma_{s^{2}}}}$$s^{2}$$s$$s$$x_{3}$$x_{1}$$x_{2}$

Expanding the integral over the three vertices of the conformal triangle as

$\displaystyle
U\left(4+\gamma_{s^{2}},\frac{d-\gamma_{s^{2}}}{2}-2,\frac{d-\gamma_{s^{2}}}{2}-2\right)U\left(2+\frac{\gamma_{s^{2}}}{2},2+\gamma_{s},d-4-\gamma_{s}-\frac{\gamma_{s^{2}}}{2}\right)$
$\displaystyle\times
U\left(\frac{d+\gamma_{s^{2}}}{2}-\gamma_{s},\frac{d-\gamma_{s^{2}}}{2}-2,2+\gamma_{s}\right)=u_{s^{2}ss}^{(0)}\,\left(1+R_{s^{2}ss}+{\cal
O}\left(\frac{1}{N^{2}}\right)\right)\,,$ (48)

where

$\displaystyle u_{s^{2}ss}^{(0)}$
$\displaystyle=-N\frac{\pi^{\frac{3d}{2}+2}\csc^{2}\left(\frac{\pi
d}{2}\right)\Gamma\left(\frac{d}{2}+1\right)}{3(d-8)(d-3)\Gamma(d-4)\Gamma(d+1)}\,,$
(49) $\displaystyle R_{s^{2}ss}$
$\displaystyle=\frac{1}{N}\frac{d\Gamma(d)}{6\pi(d-8)(d-6)(d-4)(d-2)\Gamma\left(\frac{d}{2}+1\right)^{2}}$
(50)
$\displaystyle\times\left(\left(-3(d-8)(d-6)(d-4)(d-2)((d-7)d+8)H_{d-5}\right.\right.$
$\displaystyle+\left.\left.d(d(d(d(d(5d-143)+1550)-8252)+23096)-32768)+19200\right)\sin\left(\frac{\pi
d}{2}\right)\right.$
$\displaystyle-\left.3\pi(d-8)(d-6)(d-4)(d-2)((d-7)d+8)\cos\left(\frac{\pi
d}{2}\right)\right)\,.$

We then obtain

$\displaystyle Z_{s^{2}ss}^{(0)}$
$\displaystyle=-\frac{C_{s^{2}ss}^{(0)}}{2C_{s^{2}}C_{s}^{2}u_{s^{2}ss}^{(0)}}\,,$
(51) $\displaystyle\delta Z_{s^{2}ss}$ $\displaystyle=\delta
C_{s^{2}ss}-A_{s^{2}}-2A_{s}-R_{s^{2}ss}\,.$ (52)

Up to the next-to-leading order in $1/N$, the $\langle s^{2}s^{2}\rangle$
propagator is given by

$0$$x$$+$$2(\Delta_{s}+\gamma_{s})$$2(\Delta_{s}+\gamma_{s})$$=\frac{C_{s^{2}}(1+A_{s^{2}})\mu^{-2\gamma_{s^{2}}}}{|x|^{2(\Delta_{s^{2}}+\gamma_{s^{2}})}}$$0$$x$

Here the second term stands for dressing one of the $s^{2}ss$ sub-diagrams of
the leading-order $\langle s^{2}s^{2}\rangle$ diagram, i.e., incorporating
three 1PI diagrams introduced in (44). In fact, using the $s^{2}ss$ conformal
triangle we can re-write the total of the diagrams contributing to $\langle
s^{2}s^{2}\rangle$ up to the next-to-leading order as666Such a method of
calculation of conformal triangles was first proposed in Goykhman:2020tsk ,
where it was applied to determine the $s^{2}ss$ conformal triangle in the
Gross-Neveu model.

$8+2\gamma_{s^{2}}$$d-4-\gamma_{s^{2}}$$d-4-\gamma_{s^{2}}$$d+\gamma_{s^{2}}-2\gamma_{s}$$d+\gamma_{s^{2}}-2\gamma_{s}$$4+2\gamma_{s}+\delta$$4+2\gamma_{s}+\delta$$d-4-\gamma_{s^{2}}$$d-4-\gamma_{s^{2}}$$8+2\gamma_{s^{2}}$$\frac{1}{2}\times$$+\frac{1}{2}\times$

Notice that here the first diagram in fact contains two $s^{2}ss$ conformal
triangles. To compensate for such a double counting, however, we multiplied it
by the factor of $1/2$. This diagram already contains the leading order
$\langle s^{2}s^{2}\rangle$ diagram, as well as its $1/N$ corrections obtained
by dressing of the internal $s$ lines. However, since it is multiplied by the
factor of $1/2$, we need to add another one-half of the leading diagram with
the corrected propagators. Finally, notice that the first diagram is
divergent. To regularize it, we introduced a small shift $\delta$ to the
dressed internal $s$ lines Vasiliev:1975mq .

Contribution of the second diagram is given by

$\langle s^{2}(x)s^{2}(0)\rangle\supset
C_{s^{2}}\,\frac{1}{|x|^{8}}\,\left(\frac{1}{2}+A_{s}-2\gamma_{s}\log(\mu|x|)\right)$
(53)

while contribution of the first diagram is

$\displaystyle\langle s^{2}(x)s^{2}(0)\rangle$
$\displaystyle\supset\frac{1}{2}(C_{s^{2}}(1+A_{s^{2}})C_{s}(1+A_{s})(-2)Z_{s^{2}ss}^{(0)}(1+\delta
Z_{s^{2}ss}))^{2}\,V(\delta)\,,$ (54)

where $V(\delta)$ is obtained by integrating over the internal vertices of the
diagram. To find the latter we first integrate over the left-most and the
right-most vertices, resulting in

$4+\gamma_{s^{2}}+\eta$$4+\gamma_{s^{2}}-\eta$$2d-8-\gamma_{s^{2}}-2\gamma_{s}$$2d-8-\gamma_{s^{2}}-2\gamma_{s}$$4+2\gamma_{s}+\delta$$4+2\gamma_{s}+\delta$$4+\gamma_{s^{2}}-\eta$$4+\gamma_{s^{2}}+\eta$

Here we have introduced an auxiliary parameter $\eta$ Vasiliev:1981yc ;
Vasiliev:1981dg ; Gubser:2017vgc shifting exponents of some of the lines. One
can easily see that the diagram is an even function of $\eta$,777One can see
this by noticing that $\eta\rightarrow-\eta$ can be undone by swapping
vertices of integration related by mirror reflection in the horizontal axes.
and as a result choosing $\eta={\cal O}(\delta)$ will not affect the value of
the diagram in the limit $\delta\rightarrow 0$. We will take advantage of this
fact by setting $\eta=\delta$, which will render two of the vertices unique.
Integrating over those vertices we obtain the diagram:

$8-d+\gamma_{s^{2}}+2\gamma_{s}+\eta^{\prime}$$d+\gamma_{s^{2}}-2\gamma_{s}-\eta^{\prime}$$2d-8-2\gamma_{s^{2}}+2\delta$$8-d+\gamma_{s^{2}}+2\gamma_{s}-\eta^{\prime}$$d+\gamma_{s^{2}}-2\gamma_{s}+\eta^{\prime}$

Here we introduced yet another auxiliary shift $\eta^{\prime}$, such that the
resulting diagram is an even function of $\eta^{\prime}$.888This can be seen
by renaming the vertices of integration $x_{1,2}$ as $x_{1}\rightarrow
x-x_{2}$, $x_{2}\rightarrow x-x_{1}$. We refer the reader to Gubser:2017vgc
for the detailed explanation of this method of calculating similar diagrams.
Consequently choosing $\eta^{\prime}=\delta$ we will not change the value of
the diagram in the $\delta\rightarrow 0$ limit, while this will make the
topmost vertex unique. Completing the last two integrals we obtain for the
total:

$\displaystyle V(\delta)$
$\displaystyle{=}\frac{1}{2}U\left(4{+}\gamma_{s^{2}},\frac{d{-}\gamma_{s^{2}}}{2}{-}2,\frac{d{-}\gamma_{s^{2}}}{2}{-}2\right)^{2}U\left(2{+}\gamma_{s}{+}\frac{\delta}{2},d{-}4{-}\gamma_{s}{-}\frac{\gamma_{s^{2}}}{2},2{+}\frac{\gamma_{s^{2}}{-}\delta}{2}\right)^{2}$
(55) $\displaystyle\times
U\left(d{-}4{-}\gamma_{s^{2}}{+}\delta,\frac{d{+}\gamma_{s^{2}}{-}\delta}{2}{-}\gamma_{s},\frac{\gamma_{s^{2}}{-}d{-}\delta}{2}{+}4{+}\gamma_{s}\right)U\left(\frac{d}{2}{+}\delta,\frac{d}{2}{+}\delta,{-}2\delta\right)\frac{\mu^{-2\delta}}{|x|^{8+2\gamma_{s^{2}}+2\delta}}\,,$

where $1/2$ is the symmetry factor of the diagram. Expanding the product of
the $U$ functions around $\delta=0$ and $N=\infty$ we obtain

$\displaystyle V(\delta)$
$\displaystyle=v_{0}\left(1+\frac{\gamma_{s^{2}}-2\gamma_{s}}{\delta}+\delta
v\right)\,\frac{\mu^{-2\delta}}{|x|^{8+2\gamma_{s^{2}}+2\delta}}$ (56)
$\displaystyle=v_{0}\,\left(1+\delta
v+(4\gamma_{s}-4\gamma_{s^{2}})\,\log(\mu|x|)\right)\,\frac{1}{|x|^{8}}\,,$

where we subtracted divergent part using $s^{2}ss$ counterterm, and999Here
$\gamma$ is Euler’s constant.

$\displaystyle v_{0}$
$\displaystyle=\frac{C_{s^{2}}}{\left(C_{s^{2}}\,C_{s}\,(-2)Z_{s^{2}ss}^{(0)}\right)^{2}}\,,$
(57) $\displaystyle\delta v$
$\displaystyle=2\gamma_{s}(\gamma-1)+\gamma_{s^{2}}-\frac{8\gamma_{s^{2}}}{3}+\frac{2(2\gamma_{s}+\gamma_{s^{2}})}{d-8}-\frac{4(d-7)(2\gamma_{s}-\gamma_{s^{2}})}{(d-8)(d-6)}$
$\displaystyle+\pi(2\gamma_{s}+\gamma_{s^{2}})\cot\left(\frac{\pi
d}{2}\right)+(2\gamma_{s}+\gamma_{s^{2}})\psi^{(0)}(d-4)\,.$ (58)

The corresponding contribution to the two-point function is then

$\langle s^{2}(x)s^{2}(0)\rangle\supset
C_{s^{2}}\,\left(\frac{1}{2}+A_{s^{2}}+A_{s}+\delta Z_{s^{2}ss}+\frac{\delta
v}{2}+(2\gamma_{s}-2\gamma_{s^{2}})\,\log(\mu|x|)\right)\,\frac{1}{|x|^{8}}\,,$
(59)

Combining (53), (59) we obtain

$\langle s^{2}(x)s^{2}(0)\rangle=C_{s^{2}}\,\left(1+A_{s^{2}}+2A_{s}+\delta
Z_{s^{2}ss}+\frac{\delta v}{2}\right)\,\frac{1}{|x|^{8+2\gamma_{s^{2}}}}\,,$
(60)

Consequently

$\boxed{\delta Z_{s^{2}ss}=-2A_{s}-\frac{\delta v}{2}}$ (61)

For the purpose of calculating correction $\delta C_{s^{2}ss}$ to the
amplitude of the three-point function (41), we can now use eq. (52). Expanding
around $d=3$, we notice that the singular parts of two terms $\delta
Z_{s^{2}ss}$ and $R_{s^{2}ss}$ contributing to $\delta C_{s^{2}ss}$ cancel
each other out:

$\displaystyle\delta Z_{s^{2}ss}$
$\displaystyle=-\frac{1}{N}\frac{64}{3\pi^{2}(d-3)}+{\cal O}((d-3)^{0})\,,$
(62) $\displaystyle R_{s^{2}ss}$
$\displaystyle=\frac{1}{N}\frac{64}{3\pi^{2}(d-3)}+{\cal O}((d-3)^{0})\,.$
(63)

Below in section 3.2 we will calculate the value of $\delta C_{s^{2}ss}$ in
$d=3$. Using (52), we then can solve for the value of the $A_{s^{2}}$ in $3d$,

$A_{s^{2}}=\delta C_{s^{2}ss}-\delta Z_{s^{2}ss}-2A_{s}-R_{s^{2}ss}\,.$ (64)

###  3.2 $\langle s^{2}ss\rangle$ in $3d$

In this subsection we will calculate the $\langle s^{2}ss\rangle$ three-point
function in $d=3$ dimensions. The key simplification of considering
specifically the three-dimensional case is that in $3d$ the third 1PI vertex
correction diagram in (44) does not contribute to the three-point function. In
fact, in three dimensions it exhibits a vanishing contribution both to the
anomalous dimensions exponents of the three-point function (41), as well as to
its overall amplitude.

We have established this as follows. First, one can easily extract only
divergent contributions of the diagrams (44) in any dimension $d$.101010Since
these are divergent diagrams, ordinarily one proceeds by regularizing them via
a small correction $\delta$ added to the exponent of the internal $s$ field
propagators Vasiliev:1975mq . However, for the purpose of extracting the
singular contributions / anomalous dimensions only, a simpler calculation can
be performed, see, e.g., Chai:2020hnu ; Goykhman:2020ffn . In such an approach
one carries out all of the integrals explicitly, and replaces the
logarithmically divergent integral with $S_{d}\log\mu$, where
$S_{d}=2\pi^{d/2}/\Gamma(d/2)$ is the surface area of $d-1$-dimensional
sphere. The total of the anomalous dimensions exponents of the considered
diagram is then read off from the coefficient in front of the $\log\mu$ term.
Setting then $d=3$ one can see that the third diagram in (44) does not
contribute any divergence in three dimensions. Independently, one can verify
that the contributions of the diagram (43) and the first two diagrams in (44)
to the total anomalous dimensions, entirely account for the anomalous
dimensions structure of the three-point function (41). With this result in
mind, we conclude that the third diagram in (44) is finite in $3d$, and
proceed to its evaluation without the need to introduce regulators:

$x_{4}$$x_{5}$$1$$1$$1$$1$$4$$4$$4$$4$$4$$1$$1$$x_{3}$$x_{1}$$x_{2}$

By taking the unique integrals over $x_{4,5}$ one can see that the resulting
diagram becomes proportional to
$\delta^{(3)}(x_{13})\,\delta^{(3)}(x_{23})$111111One can see that using the
inverse propagator relation  $\int
d^{d}x_{3}\frac{1}{|x_{13}|^{2\Delta}|x_{23}|^{2(d-\Delta)}}=\pi^{d}A(\Delta)A(d-\Delta)\delta^{(d)}(x_{12})\,,$
(65) applied for $d=3$, $\Delta=1$., and is therefore identically zero for the
three-point function, that is defined for non-coincident points only.

We proceed to the calculation of the contributions of the diagram (43) and the
first two diagrams in (44) to the $\langle s^{2}ss\rangle$. We performed this
calculation in any $d$, so we keep the dimension general, and set $d=3$ in the
very end. We have the following equation:

$(\ref{s2ss general})=(\ref{first correction to s2ss})+(\ref{second correction
to s2ss})\,.$ (66)

Following Goykhman:2020tsk we replace the $s^{2}ss$ sub-diagram in the first
two diagrams in (44) with the $s^{2}ss$ conformal triangle. Such a procedure
creates two internal $s$ propagators, which we regularize by adding a small
shift $\delta$ to their exponent:

(67)

We proceed by integrating both sides of (66) over $x_{3}$. Due to (41) on the
l.h.s. of (66) we obtain

$\displaystyle\textrm{L.H.S.}=C_{s^{2}ss}^{(0)}(1+\delta
C_{s^{2}ss})U\left(2+\frac{\gamma_{s^{2}}}{2},2+\frac{\gamma_{s^{2}}}{2},d-4-\gamma_{s^{2}}\right)\,\frac{\mu^{-\gamma_{s^{2}}-2\gamma_{s}}}{|x_{12}|^{8-d+2\gamma_{s}+\gamma_{s^{2}}}}$
$\displaystyle=2C_{s}^{2}U(2,2,d-4)(1+\delta
C_{s^{2}ss}+h_{1})\,\frac{\mu^{-\gamma_{s^{2}}-2\gamma_{s}}}{|x_{12}|^{8-d+2\gamma_{s}+\gamma_{s^{2}}}}\,,$
(68) $\displaystyle h_{1}$
$\displaystyle=-\frac{1}{N}\frac{2^{d+3}(d-1)\sin\left(\frac{\pi
d}{2}\right)\Gamma\left(\frac{d+1}{2}\right)\left(-\frac{2}{d-6}+\pi\cot\left(\frac{\pi
d}{2}\right)+\psi^{(0)}(d-4)+\gamma-1\right)}{\pi^{3/2}(d-2)d\Gamma\left(\frac{d}{2}-2\right)}\,.$
(69)

Next, using (43) we obtain the following contribution due to the tree-level
diagram with dressed $s$ propagators:

$\displaystyle\int
d^{d}x_{3}\frac{C_{s^{2}ss}^{(0)}(1+2A_{s})\mu^{-4\gamma_{s}}}{(|x_{13}||x_{23}|)^{4+2\gamma_{s}}}=2C_{s}^{2}(1+2A_{s})U(2+\gamma_{s},2+\gamma_{s},d-4-2\gamma_{s})\frac{\mu^{-4\gamma_{s}}}{|x_{12}|^{8-d+4\gamma_{s}}}$
$\displaystyle=2C_{s}^{2}(1+2A_{s}+h_{2})U(2,2,d-4)\frac{\mu^{-4\gamma_{s}}}{|x_{12}|^{8-d+4\gamma_{s}}}\,,$
(70) $\displaystyle h_{2}$
$\displaystyle=\frac{1}{N}\frac{8\sin\left(\frac{\pi
d}{2}\right)\Gamma(d)\left((d-6)H_{d-5}-d+\pi(d-6)\cot\left(\frac{\pi
d}{2}\right)+4\right)}{\pi(d-6)\Gamma\left(\frac{d}{2}-1\right)\Gamma\left(\frac{d}{2}+1\right)}\,.$
(71)

Integrating diagrams (a), (b) in figure (67) over $x_{3}$ we obtain

$\int d^{d}x_{3}\frac{\langle
s(x_{3})^{2}s(x_{1})s(x_{2})\rangle}{2C_{s}^{2}U(2,2,d-4)}\supset\left(h_{a,b}-\omega_{a,b}\log(\mu|x_{12}|)\right)\frac{1}{|x_{12}|^{8-d}}\,.$
(72)

Analogously, integrating the third diagram in (44) we obtain

$\int d^{d}x_{3}\frac{\langle
s(x_{3})^{2}s(x_{1})s(x_{2})\rangle}{2C_{s}^{2}U(2,2,d-4)}\supset\left(h_{c}-\omega_{c}\log(\mu|x_{12}|)\right)\frac{1}{|x_{12}|^{8-d}}\,.$
(73)

For our purposes, we only need to know

$h_{c}(d=3)=0\,,\qquad\omega_{c}(d=3)=0\,,$ (74)

as we established above. Integrating over vertices of the $s^{2}ss$ conformal
triangle in the diagrams (a), (b) in figure (67), and collecting the
$\delta$-dependent factors, we obtain

$\displaystyle
A\left(2{+}\frac{\delta}{2}\right)A\left(\frac{d{-}\delta}{2}{-}2\right)U\left(2{+}\frac{\delta}{2},2{+}\frac{\delta}{2},d{-}4{-}\delta\right){=}A(2)A\left(\frac{d}{2}{-}2\right)U(2,2,d{-}4)(1{+}r\delta)\,,$
$\displaystyle r$ $\displaystyle=-\frac{2}{d-6}+\pi\cot\left(\frac{\pi
d}{2}\right)+\psi^{(0)}(d-4)+\gamma-1\,.$ (75)

In the process we dropped the $1/N$ corrections due to the anomalous
dimensions, which are sub-leading in $1/N$.121212The $U$ functions generated
due to integrals over the vertices of the $s^{2}ss$ conformal triangle,
contain the factor $A\left(\frac{d+\gamma_{s^{2}}}{2}-\gamma_{s}\right)$. We
have expanded it in $1/N$ and kept the leading ${\cal O}(N)$ term. At the same
time, using (51), (42), the leading order factors can be assembled into the
leading order amplitude $C_{s^{2}ss}^{(0)}=2C_{s}^{2}$. The factor of
$1+r\,\delta$ will be important below for the calculation of the finite
correction $\delta C_{s^{2}ss}$ to the amplitude of the three-point function
$\langle s^{2}ss\rangle$. Integrating over the remaining four vertices of the
(a) in figure (67) we obtain

$\displaystyle\int d^{d}x_{3}\frac{\langle
s(x_{3})^{2}s(x_{1})s(x_{2})\rangle}{2C_{s}^{2}U(2,2,d-4)}\supset
NC_{s}^{2}C_{\phi}^{4}(1+r\delta)\left(-\frac{2}{\sqrt{N}}\right)^{4}U\left(3+\delta,\frac{d}{2}-1,\frac{d}{2}-2-\delta\right)$
$\displaystyle
U\left(\frac{d}{2}-1,2+\delta,\frac{d}{2}-1-\delta\right)U\left(\frac{d}{2}+\delta,2,\frac{d}{2}-2-\delta\right)U(2,2+\delta,d-4-\delta)\frac{\mu^{-2\delta}}{x^{8-d+2\delta}}\,.$
(76)

Expanding it in $\delta$ we obtain

$\displaystyle\omega_{a}$
$\displaystyle=\frac{d-4}{2(d-1)}(2\gamma_{s}-\gamma_{s^{2}})\,,$ (77)
$\displaystyle h_{a}$
$\displaystyle=\frac{1}{N}\frac{4(d-3)\Gamma\left(4-\frac{d}{2}\right)\Gamma(d-2)\sin^{2}\left(\frac{\pi
d}{2}\right)}{\pi^{2}(d-6)^{2}(d-2)\Gamma\left(\frac{d}{2}\right)}\left(d(5d-32)-4\pi(d-6)(d-2)\cot\left(\frac{\pi
d}{2}\right)\right.$ $\displaystyle+\left.28-4(d-6)(d-2)H_{d-5}\right)\,.$
(78)

Integrating over the remaining four vertices of the (b) in figure (67) we
obtain

$\displaystyle\int d^{d}x_{3}\frac{\langle
s(x_{3})^{2}s(x_{1})s(x_{2})\rangle}{2C_{s}^{2}U(2,2,d-4)}{\supset}\frac{1}{2}NC_{s}^{2}C_{\phi}^{4}(1{+}r\delta)\left({-}\frac{2}{\sqrt{N}}\right)^{4}U\left(\frac{d}{2}{-}1,\frac{d}{2}{-}1,2\right)^{2}\frac{\textrm{sk}\left(\frac{d}{2}{+}\delta\right)\mu^{-2\delta}}{|x_{12}|^{8-d+2\delta}}\,,$
(79)

where $1/2$ is the symmetry factor, and we used the special kite diagram

$x_{1}$$x_{2}$$2$$2$$2a$$2$$2$$=\frac{\textrm{sk}(a)}{|x_{12}|^{8+2a-2d}}$

For general $a$, the value of the special kite diagram can be found in
Kotikov:2000yd .131313See eq. (22) therein; notice that each integral over
internal vertex in Kotikov:2000yd is multiplied by $1/(2\pi)^{d}$, so we need
to multiply that expression by $(2\pi)^{2d}$ to adjust it to our conventions.
However, since we are only interested in expansion around $\delta=0$ and
retaining only singular and finite terms, it is sufficient to use expression
(133) for $\alpha_{1,2,3,4}=1$, $\alpha_{5}=d/2+\delta$, and expand around
$\delta=0$. While the 2nd and 3rd diagrams in the r.h.s. of (133) are
straightforward to calculate using the propagator merging relation, the first
diagram can be calculated by inserting a point into the diagonal propagator,
splitting it into two propagators with the exponents $2d-4$ and $2+2\delta$.
Taking the unique integral we will obtain the diagram equal to
$F\left(\frac{d}{2}-1,\frac{d}{2}-1\right)$, where we dropped corrections
linear in $\delta$, since the diagram is finite. Assembling everything
together, we obtain

$\displaystyle\textrm{sk}\left(\frac{d}{2}+\delta\right)=\frac{f_{1}}{\delta}+f_{2}+{\cal
O}(\delta)\,,$ (80) $\displaystyle f_{1}$
$\displaystyle=\frac{2(d-6)\pi^{d+1}\csc\left(\frac{\pi
d}{2}\right)}{(d-2)\Gamma(d-3)}\,,$ (81) $\displaystyle f_{2}$
$\displaystyle{=}\frac{\pi^{d}}{2}\left(\frac{4\pi\csc\left(\frac{\pi
d}{2}\right)}{(d-2)^{2}\Gamma(d-3)}\left({-}2(d{-}5)d{+}\pi(d{-}6)(d{-}2)\cot\left(\frac{\pi
d}{2}\right){-}4{+}(d{-}6)(d{-}2)H_{d{-}4}\right)\right.$
$\displaystyle+\left.(d-4)\cos\left(\frac{\pi
d}{2}\right)\Gamma(3-d)\left(\pi^{2}-6\psi^{(1)}\left(\frac{d}{2}-1\right)\right)\right)\,.$
(82)

Using it in (79) and expanding it in $\delta$ we obtain

$\displaystyle\omega_{b}$
$\displaystyle=\frac{d-6}{2(d-1)}(2\gamma_{s}-\gamma_{s^{2}})\,,$ (83)
$\displaystyle h_{b}$
$\displaystyle=\frac{1}{N}\frac{4^{d-3}\sin\left(\frac{\pi
d}{2}\right)\Gamma\left(\frac{d-1}{2}\right)^{2}}{\pi^{2}(d-2)^{2}\Gamma(d-3)\Gamma(d-1)}\left(-4(3-d)(d-2)^{2}\Gamma(d-3)\left(\gamma(d-6)-d\right.\right.$
$\displaystyle+\left.\left.\pi(d-6)\cot\left(\frac{\pi
d}{2}\right)+(d-6)\psi^{(0)}(d-4)+4\right)+\Gamma(d-1)\left(-8(d-5)d-16\right.\right.$
$\displaystyle+\left.\left.4(d-6)(d-2)H_{d-4}+4\pi(d-6)(d-2)\cot\left(\frac{\pi
d}{2}\right)\right.\right.$
$\displaystyle+\left.\left.\frac{(d-4)(d-2)^{2}\sin(\pi
d)\Gamma(3-d)\Gamma(d-3)\left(\pi^{2}-6\psi^{(1)}\left(\frac{d}{2}-1\right)\right)}{2\pi}\right)\right)\,.$
(84)

Using (70), (77), (83) in $d=3$ we obtain the total anomalous dimensions term
$-(\gamma_{s^{2}}+2\gamma_{s})\log(\mu|x_{12}|)$, in agreement with (66),
(3.2). At the same time, using (66), (69), (71), (78), (84) we obtain

$\delta C_{s^{2}ss}=-h_{1}+h_{2}+2A_{s}+h_{a}+h_{b}+h_{c}\,.$ (85)

Evaluating in $d=3$ and using (64) gives141414Recall that $h_{c}(d=3)=0$, due
to (74).

$\boxed{A_{s^{2}}(d=3)=\delta
C_{s^{2}ss}(d=3)=\frac{1}{N}\,\left(\frac{176}{9\pi^{2}}-1\right)}$ (86)

##  4 $\langle s^{2}s^{2}s\rangle$

In this section we are going to calculate the $\langle s^{2}s^{2}s\rangle$
three-point function at the next-to-leading order in the $1/N$ expansion.
Conformal symmetry demands it to have the form

$\langle
s^{2}(x_{1})s^{2}(x_{2})s(x_{3})\rangle=\frac{C_{s^{2}s^{2}s}^{(0)}(1+\delta
C_{s^{2}s^{2}s})\,\mu^{-\gamma_{s}-2\gamma_{s^{2}}}}{|x_{12}|^{6+2\gamma_{s^{2}}-\gamma_{s}}(|x_{13}||x_{23}|)^{2+\gamma_{s}}}\,.$
(87)

Our goal is to find the amplitude correction $\delta C_{s^{2}s^{2}s}$.

At the leading ${\cal O}(1/\sqrt{N})$ order in $1/N$ the $\langle
s^{2}s^{2}s\rangle$ three-point function is determined by the diagram

$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{s}$$2\Delta_{s}$$2\Delta_{s}$$x_{3}$$x_{1}$$x_{2}$$=\frac{C_{s^{2}s^{2}s}^{(0)}}{|x_{12}|^{6}(|x_{13}||x_{23}|)^{2}}$

where the leading order amplitude is given by151515In such diagrams the factor
of $2^{2}$ is a degeneracy factor due to the composite operators $s^{2}$.

$\boxed{C_{s^{2}s^{2}s}^{(0)}=2^{2}C_{sss}^{(0)}C_{s}}$ (88)

This amplitude vanishes in $3d$, owing to the fact that
$C_{sss}^{(0)}(d=3)=0$.

### Next-to-leading order

We now proceed to calculation of the $\langle s^{2}s^{2}s\rangle$ three-point
function at the next-to-leading order in the $1/N$ expansion. Let us begin by
considering the contributing diagrams which are obtained by incorporating the
$1/N$ corrections to the $s$ propagators and the and the $sss$ sub-diagram of
the leading order $\langle s^{2}s^{2}s\rangle$ diagram:

(89)

Here we have used the $sss$ conformal triangle to integrate the diagram

(90)

obtaining

$v_{1}=4A_{s}+\delta Z_{sss}+R_{sss}=\delta C_{sss}+A_{s}\,,$ (91)

where in the last equality we used (32).

Next we consider $1/N$ corrections to the $s^{2}ss$ sub-diagrams of the the
leading order $\langle s^{2}s^{2}s\rangle$. We will demonstrate below that
these diagrams have the form

(92)

We proceed to calculating diagrams involving corrections to the left-hand and
the right-hand $s^{2}ss$ sub-diagrams separately. The total of these
contributions is additive, due to linearization of the next-to-leading $1/N$
corrections. Using (45) we can express

$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{s}$$x_{3}$$x_{1}$$x_{2}$$=C_{s^{2}ss}^{(0)}(1+\delta
C_{s^{2}ss}-2A_{s})\times$$2\gamma_{s}-\gamma_{s^{2}}$$4+\gamma_{s^{2}}$$4+\gamma_{s^{2}}$$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{s}$$x_{3}$$x_{1}$$x_{2}$$-$$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{s}$$x_{3}$$x_{1}$$x_{2}$

By linearizing over the $1/N$ corrections to the exponents of propagators of
diagrams with identical leading-order skeleton structure, we can further re-
write this expression as

$-\frac{8C_{\phi}^{3}}{\sqrt{N}}C_{s^{2}ss}^{(0)}(1+\delta
C_{s^{2}ss}-2A_{s})\times$$2\gamma_{s}-\gamma_{s^{2}}$$4+\gamma_{s^{2}}$$4+\gamma_{s^{2}}$$d-2-\gamma_{s}$$d-2-\gamma_{s}$$d-2-\gamma_{s}$$x_{3}$$x_{1}$$x_{2}$$s$$s$$-$$s$$s$$s$$s$$x_{3}$$x_{1}$$x_{2}$

Notice that the first diagram is completely integrable, giving

$\langle
s(x_{1})^{2}s(x_{2})^{2}s(x_{3})\rangle\supset\frac{C_{s^{2}s^{2}s}^{(0)}(1+\delta
C_{s^{2}ss}-2A_{s}+v_{2})\mu^{-\gamma_{s^{2}}-3\gamma_{s}}}{|x_{12}|^{6+\gamma_{s^{2}}+\gamma_{s}}|x_{13}|^{2-\gamma_{s}+\gamma_{s^{2}}}|x_{23}|^{2-\gamma_{s^{2}}+3\gamma_{s}}}\,,$
(93)

where

$\displaystyle
v_{2}{=}\frac{1}{N}\,\frac{\Gamma(d)\left(\left(3(d{-}4)(d{-}2)H_{d{-}4}{+}d((d{-}15)d{+}60){-}60\right)\sin\left(\frac{\pi
d}{2}\right){+}3\pi(d{-}4)(d{-}2)\cos\left(\frac{\pi
d}{2}\right)\right)}{\pi(d{-}4)\Gamma\left(\frac{d}{2}{+}1\right)\Gamma\left(\frac{d}{2}\right)}\,.$
(94)

is obtained by the expansion of the $U$ functions generated during the
integration over the unique internal vertices

$\displaystyle
U\left(\frac{d-\gamma_{s}}{2}-1,\frac{d-\gamma_{s}}{2}-1,2+\gamma_{s}\right)$
$\displaystyle
U\left(1+\frac{\gamma_{s}}{2},2+\gamma_{s},d-3-\frac{3\gamma_{s}}{2}\right)$
(95) $\displaystyle\times
U\left(\frac{d-\gamma_{s}}{2}-1,2+\frac{\gamma_{s^{2}}}{2},\frac{d+\gamma_{s}-\gamma_{s^{2}}}{2}-1\right)\,,$

while the second diagram is given by (90), for the total of

$\langle
s(x_{1})^{2}s(x_{2})^{2}s(x_{3})\rangle\supset\frac{C_{s^{2}s^{2}s}^{(0)}(1+\delta
C_{s^{2}ss}-2A_{s}+v_{2}-R_{sss})\mu^{-\gamma_{s^{2}}+2\gamma_{s}}}{|x_{12}|^{6+\gamma_{s^{2}}-2\gamma_{s}}|x_{13}|^{2-2\gamma_{s}+\gamma_{s^{2}}}|x_{23}|^{2-\gamma_{s^{2}}+2\gamma_{s}}}-\frac{C_{s^{2}s^{2}s}^{(0)}}{|x_{12}|^{6}(|x_{13}||x_{23}|)^{4}}\,,$
(96)

Analogously, correcting the right-hand $s^{2}ss$ sub-diagram in (92) we obtain

$\langle
s(x_{1})^{2}s(x_{2})^{2}s(x_{3})\rangle\supset\frac{C_{s^{2}s^{2}s}^{(0)}(1+\delta
C_{s^{2}ss}-2A_{s}+v_{3}-R_{sss})\mu^{-\gamma_{s^{2}}+2\gamma_{s}}}{|x_{12}|^{6+\gamma_{s^{2}}-2\gamma_{s}}|x_{13}|^{2-\gamma_{s^{2}}+2\gamma_{s}}|x_{23}|^{2-2\gamma_{s}+\gamma_{s^{2}}}}-\frac{C_{s^{2}s^{2}s}^{(0)}}{|x_{12}|^{6}(|x_{13}||x_{23}|)^{4}}\,,$
(97)

where

$v_{3}=v_{2}\,.$ (98)

Combining (89), (96), (97), we obtain

$\langle
s(x_{1})^{2}s(x_{2})^{2}s(x_{3})\rangle\supset\frac{C_{s^{2}s^{2}s}^{(0)}(1+w_{0})\mu^{-\gamma_{s}-2\gamma_{s^{2}}}}{|x_{12}|^{6+2\gamma_{s^{2}}-\gamma_{s}}(|x_{13}||x_{23}|)^{2+\gamma_{s}}}\,,$
(99)

where we denoted

$w_{0}=v_{1}+2v_{2}+2\delta C_{s^{2}ss}-4A_{s}-2R_{sss}\,.$ (100)

Using (91) we obtain

$w_{0}=\delta C_{sss}-3A_{s}+2(\delta C_{s^{2}ss}+v_{2}-R_{sss})\,.$ (101)

Notice that (99) already has the structure required by the conformal symmetry
of the $\langle s^{2}s^{2}s\rangle$ three-point function (87). This means that
the rest of the diagrams which contribute to the $\langle s^{2}s^{2}s\rangle$
are finite, as we will confirm explicitly below in this section. Expanding
around $d=3$, we obtain

$\displaystyle v_{2}=\frac{16}{\pi^{2}}\frac{1}{d-3}+{\cal
O}((d-3)^{0})\,,\qquad R_{sss}=\frac{16}{\pi^{2}}\frac{1}{d-3}+{\cal
O}((d-3)^{0})\,.$ (102)

We also know that $\langle sss\rangle$ vanishes in $d=3$, and therefore
$\delta C_{sss}={\cal O}((d-3)^{0})$. In addition, we know that $A_{s}={\cal
O}((d-3)^{0})$. Finally, while we haven’t calculated $\delta C_{s^{2}ss}$, we
know that $C_{s^{2}ss}^{(0)}\delta C_{s^{2}ss}$ should be finite in $3d$, and
therefore, since $C_{s^{2}ss}^{(0)}$ is finite, we conclude that $\delta
C_{s^{2}ss}$ must be finite in $3d$.161616Actually we calculated $\delta
C_{s^{2}ss}$ in $3d$ and showed explicitly that it is finite. Then from (101)
we conclude that

$w_{0}={\cal O}((d-3)^{0})\,.$ (103)

Consider the following diagram

$2\Delta_{s}$$2\Delta_{s}$$2\Delta_{s}$$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{s}$$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$$x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$$=\frac{w_{1}}{|x_{12}|^{6}(|x_{13}||x_{23}|)^{2}}$

Integrating over all of the internal vertices except for $x_{4,5}$, and
denoting

$w_{1}=2^{2}\,N^{2}\left(-\frac{2}{\sqrt{N}}\right)^{6}C_{sss}^{(0)}C_{\phi}^{6}C_{s}^{4}U\left(\frac{d}{2}-1,\frac{d}{2}-1,2\right)^{2}U(1,2,d-3)^{2}\,\hat{w}_{1}\,,$
(104)

we obtain

$x_{1}$$x_{2}$$x_{3}$$d-2$$d-2$$d-2$$d-2$$2$$2$$2$$=\frac{\hat{w}_{1}}{|x_{12}|^{2d-6}(|x_{13}||x_{23}|)^{2}}$

Integrating both sides of the last diagrammatic equation w.r.t. $x_{2}$ we
obtain

$\hat{w}_{1}=\frac{\textrm{ChT}(1,1)}{U(1,2,d-3)}\,.$ (105)

Combining (104), (105), we obtain

$w_{1}{=}{-}\frac{1}{N^{3/2}}2^{4d{-}3}(d{-}3)^{6}\pi^{{-}\frac{d}{2}{-}9}\sin^{7}\left(\frac{\pi
d}{2}\right)\Gamma\left(3{-}\frac{d}{2}\right)^{3}\Gamma\left(\frac{d{-}3}{2}\right)^{4}\left(\pi^{2}{-}6\psi^{(1)}\left(\frac{d}{2}{-}1\right)\right)\,.$
(106)

Next, consider the diagram

$x_{1}$$x_{2}$$x_{3}$$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{s}$$2\Delta_{s}$$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{s}$$=\frac{w_{2}}{|x_{12}|^{6}(|x_{13}||x_{23}|)^{2}}$

It is straightforward to find $w_{2}$ by integrating both sides of this
diagrammatic equation over $x_{3}$:

$w_{2}=2^{2}N\left({-}\frac{2}{\sqrt{N}}\right)^{4}C_{sss}^{(0)}C_{\phi}^{4}C_{s}^{3}U\left(\frac{d}{2}{-}1,\frac{d}{2}{-}1,2\right)^{3}U(2,2,d{-}4)\,.$
(107)

Simplifying this expression, we obtain

$w_{2}=\frac{1}{N^{3/2}}\frac{4^{3d-2}\sin^{6}\left(\frac{\pi
d}{2}\right)\Gamma\left(2-\frac{d}{2}\right)\Gamma\left(4-\frac{d}{2}\right)\Gamma\left(\frac{d-1}{2}\right)^{6}}{\pi^{9}\Gamma(d-3)^{2}}\,.$
(108)

The other contributing diagram is given by

$x_{1}$$x_{2}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{s}$$=\frac{w_{3}}{|x_{12}|^{6}(|x_{13}||x_{23}|)^{2}}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{s}$$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{s}$$2\Delta_{s}$$x_{3}$$x_{4}$$x_{5}$

To calculate this diagram, we proceed by integrating over the $sss$ sub-
diagram, followed by the integral over the $x_{4,5}$ vertices. Integrating
both sides of the resulting diagrammatic equation over $x_{1}$, we obtain

$w_{3}=2^{2}N\left({-}\frac{2}{\sqrt{N}}\right)^{4}C_{sss}^{(0)}C_{\phi}^{4}C_{s}^{3}U\left(\frac{d}{2}{-}1,\frac{d}{2}{-}1,2\right)^{2}\,\hat{w}_{3}\,,$
(109)

where $\hat{w}_{3}$ is determined by

$x_{2}$$x_{3}$$2$$2$$d-2$$2$$4$$=\frac{\hat{w}_{3}}{|x_{23}|^{8-d}}$

To calculate this self-energy diagram, we split the diagonal propagator into
two merging propagators, with the exponents $2d-4$ and $2$, while dividing the
diagram by $U\left(\frac{d}{2}{-}1,\frac{d}{2}{-}1,2\right)$. Integrating over
the unique topmost vertex produces the factor of
$U\left(\frac{d}{2}{-}1,\frac{d}{2}{-}1,2\right)$, resulting in

$x_{2}$$x_{3}$$d-2+\delta$$d-2+\delta$$2$$2$$4$$=\frac{\hat{w}_{3}(\delta)}{|x_{23}|^{4+2\delta}}$

Here we introduced an auxiliary regulator $\delta$, which we will eventually
set to zero.171717The regulator is auxiliary because the diagram is in fact
finite in the $\delta\rightarrow 0$ limit. Applying the integration by parts
identity to the topmost vertex (with the exponents $\alpha_{1}=1$,
$\alpha_{2}=\alpha_{3}=\frac{d+\delta}{2}-1$), we can solve for
$\hat{w}_{3}(\delta)$ by calculating four diagrams, three of which can be
easily calculated using the propagator merging relations, while the fourth one
is the Fourier dual of the ChT diagram. Assembling everything together, and
taking the limit $\delta\rightarrow 0$, we obtain

$\hat{w}_{3}=\frac{\pi^{d+1}\left(-\pi^{2}(d-4)^{2}+\frac{8(d((d-4)d-8)+36)}{(d-2)^{2}}+6(d-4)^{2}\psi^{(1)}\left(\frac{d}{2}\right)\right)}{4(d-4)\Gamma(d-2)\sin\left(\frac{\pi
d}{2}\right)}\,.$ (110)

Plugging this into (109) we obtain

$\displaystyle w_{3}$
$\displaystyle=\frac{1}{N^{3/2}}\frac{2^{5d-7}\sin^{5}\left(\frac{\pi
d}{2}\right)\Gamma\left(3-\frac{d}{2}\right)\Gamma\left(\frac{d-1}{2}\right)^{5}}{\pi^{15/2}\Gamma(d-3)\Gamma\left(\frac{d}{2}\right)^{2}}\left(\pi^{2}\left(d^{2}-6d+8\right)^{2}-6\left(d^{2}-6d+8\right)^{2}\psi^{(1)}\left(\frac{d}{2}\right)\right.$
$\displaystyle-\left.8(d((d-4)d-8)+36)\right)\,.$ (111)

Next, consider the first non-planar pentagon-based diagram

$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{s}$$2\Delta_{s}$$2\Delta_{s}$$2\Delta_{s}$$2\Delta_{s}$$=\frac{w_{4}}{|x_{12}|^{6}(|x_{13}||x_{23}|)^{2}}$$x_{3}$$x_{1}$$x_{2}$$x_{4}$$x_{5}$$x_{6}$

To calculate this diagram, we integrate over the internal vertex $x_{4}$, and
then integrate both sides of this diagrammatic equation over $x_{3}$, followed
by integration over the vertices $x_{5,6}$. This will give

$w_{4}=2^{2}N\left({-}\frac{2}{\sqrt{N}}\right)^{5}C_{\phi}^{5}C_{s}^{5}U\left(\frac{d}{2}{-}1,\frac{d}{2}{-}1,2\right)^{3}\,\hat{w}_{4}\,,$
(112)

where $\hat{w}_{4}$ is determined by

$x_{1}$$x_{2}$$2$$4$$d-4$$4$$4$$=\frac{\hat{w}_{4}}{|x_{12}|^{10-d}}$

To find $\hat{w}_{4}$, we split the diagonal propagator with the exponent
$d-4$ into two propagators with the exponents $2d-6$ and $2$, while dividing
the diagram by $U(1,2,d-3)$. This will make the topmost vertex unique,
integrating over which will produce the factor of $U(1,2,d-3)$, resulting in

$x_{1}$$x_{2}$$d-4$$d-2+\delta$$2$$4$$4$$=\frac{\hat{w}_{4}(\delta)}{|x_{12}|^{4+\delta}}$

Here we introduced an auxiliary regulator $\delta$, which allows us to apply
the integration by parts relation to the topmost vertex (with the exponents
$\alpha_{1}=1$, $\alpha_{2}=\frac{d}{2}-2$,
$\alpha_{3}=\frac{d+\delta}{2}-1$), resulting in

$\displaystyle\hat{w}_{4}$
$\displaystyle=\frac{\frac{d}{2}-2}{1-\frac{\delta}{2}}\left(U(2,2,d-4)U\left(\frac{d}{2}-1,\frac{d+\delta}{2}-1,2-\frac{\delta}{2}\right)-c_{1}\right)$
(113)
$\displaystyle+\frac{\frac{d+\delta}{2}-1}{1-\frac{\delta}{2}}\left(U(2,2,d-4)U\left(\frac{d}{2}-2,\frac{d+\delta}{2},2-\frac{\delta}{2}\right)-c_{2}\right)\,,$

where we denoted

$x_{1}$$x_{2}$$d-2$$d-2+\delta$$2$$2$$4$$=\frac{c_{1}}{|x_{12}|^{4+\delta}}$$x_{1}$$x_{2}$$d-4$$d+\delta$$2$$4$$2$$=\frac{c_{2}}{|x_{12}|^{4+\delta}}$

The diagram $c_{1}$ is in fact finite, so we can set $\delta=0$. In fact, it
is equal to the diagram $\hat{w}_{3}$, calculated above,

$c_{1}=\lim_{\delta\rightarrow 0}\hat{w}_{3}\,.$ (114)

To calculate $c_{2}$, we apply the integration by parts relation to the
topmost vertex (with the exponents $\alpha_{1}=1$, $\alpha_{2}=\frac{d}{2}-2$,
$\alpha_{3}=\frac{d+\delta}{2}$), which will express that diagram in terms of
a sum of four diagrams. Three of these diagrams can be easily calculated using
the propagator merging relation, while the fourth diagram is given by the
Fourier transform of the ChT diagram. Assembling everything together, we
obtain

$\displaystyle c_{2}$
$\displaystyle{=}\frac{4(d{-}6)\pi^{d+1}\csc\left(\frac{\pi
d}{2}\right)}{(d{-}2)\delta\Gamma(d{-}3)}{+}\frac{\pi^{d}}{12}\left({-}\frac{4\left(\pi^{2}(d{-}4)(d{-}2){-}6(d{-}4)d{+}24\right)\cos\left(\frac{\pi
d}{2}\right)\Gamma(4{-}d)}{(d{-}2)^{2}}\right.$
$\displaystyle+\left.\frac{\pi\csc\left(\frac{\pi
d}{2}\right)\Gamma(d-2)}{(d-4)\Gamma(d-1)^{2}}\left(-18\left(d^{2}-6d+8\right)^{2}\psi^{(1)}\left(\frac{d}{2}\right)+\pi^{2}(d-2)d(d-4)^{2}\right.\right.$
$\displaystyle-\left.\left.12(d(d((d-13)d+64)-140)+120)\right)\right)\,.$
(115)

Combining (112), (113), (114), (115), we obtain

$\displaystyle w_{4}$
$\displaystyle{=}{-}\frac{1}{N^{3/2}}\frac{32^{d-1}\Gamma\left(\frac{d-1}{2}\right)^{5}\sin^{5}\left(\frac{\pi
d}{2}\right)}{\pi^{15/2}\Gamma\left(\frac{d}{2}-2\right)^{2}\Gamma\left(\frac{d-2}{2}\right)}\left(-\frac{8\cos\left(\frac{\pi
d}{2}\right)\Gamma(7-d)}{d^{2}-9d+20}\right.$
$\displaystyle-\left.\frac{\pi(d-4)\csc\left(\frac{\pi
d}{2}\right)\left(-8d-6\psi^{(1)}\left(\frac{d}{2}-2\right)+\pi^{2}+48\right)}{\Gamma(d-2)}\right)\,.$
(116)

The other non-planar diagram based on a pentagon effective vertex for $s$ is
given by

$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{s}$$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{s}$$2\Delta_{s}$$2\Delta_{\phi}$$x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$$x_{6}$$=\frac{w_{5}}{|x_{12}|^{4}(|x_{13}||x_{23}|)^{2}}$

To calculate this diagram, we integrate over $x_{4}$, followed by integration
of both sides of this diagrammatic equation over $x_{3}$, followed by
integration over $x_{5}$, $x_{6}$. This will give

$w_{5}=2^{2}N\left({-}\frac{2}{\sqrt{N}}\right)^{5}C_{\phi}^{5}C_{s}^{5}U\left(\frac{d}{2}{-}1,\frac{d}{2}{-}1,2\right)^{2}U\left(\frac{d}{2}{-}1,\frac{d}{2}{-}2,3\right)\,\hat{w}_{5}\,,$
(117)

where $\hat{w}_{5}$ is determined by

$x_{1}$$x_{2}$$4$$4$$d-6$$4$$4$$=\frac{\hat{w}_{5}}{|x_{12}|^{10-d}}$

To determine $\hat{w}_{5}$, we split the diagonal propagator with the exponent
$d-6$ into two, with the exponents $2d-8$ and $2$, while dividing the diagram
by $U(1,3,d-4)$. This will make the top-most vertex unique, integrating over
which will produce the factor of $U(2,2,d-4)$, resulting in

$x_{1}$$x_{2}$$d-4$$d-4$$2$$4$$4$$=\frac{\tilde{w}_{5}}{|x_{12}|^{2}}$

Here we denoted

$\hat{w}_{5}=\frac{U(2,2,d-4)}{U(1,3,d-4)}\,\tilde{w}_{5}\,.$ (118)

To find $\tilde{w}_{5}$ we will apply the integration by parts relation to the
topmost vertex (with the exponents $\alpha_{1}=1$,
$\alpha_{2}=\alpha_{3}=\frac{d}{2}-2$), resulting in

$\tilde{w}_{5}=\frac{d-4}{4}\left(2U(2,2,d-4)U\left(\frac{d}{2}-1,\frac{d}{2}-2,3\right)-2c_{3}\right)\,,$
(119)

where $c_{3}$ is determined by

$x_{1}$$x_{2}$$d-4$$d-2+\delta$$2$$4$$2$$=\frac{c_{3}(\delta)}{|x_{12}|^{2+\delta}}$

Here we introduced an auxiliary regulator $\delta$, assuming that in the end
we will take the limit $c_{3}=\lim_{\delta\rightarrow 0}c_{3}(\delta)$. To
find $c_{3}(\delta)$ we apply the integration by parts relation to the top-
most vertex (with the exponents $\alpha_{1}=1$,
$\alpha_{2}=\frac{d+\delta}{2}-1$, $\alpha_{3}=\frac{d}{2}-2$), which will
express that diagram in terms of a sum of four diagrams. Three of these
diagrams can be straightforwardly integrated using the propagator merging
relation, while the fourth one is given by the Fourier transform of the ChT
diagram. Assembling everything together, we obtain

$\displaystyle c_{3}{=}-\frac{\pi^{d+1}\csc\left(\frac{\pi
d}{2}\right)\left(\pi^{2}(d-4)^{2}-8d-6(d-4)^{2}\psi^{(1)}\left(\frac{d}{2}-1\right)+24\right)}{4((d-4)\Gamma(d-2))}\,.$
(120)

Putting together (117), (118), (119), (120), we obtain

$\displaystyle
w_{5}{=}\frac{1}{N^{3/2}}\frac{2^{4d{-}1}\sin^{4}\left(\frac{\pi
d}{2}\right)\Gamma\left(\frac{d{-}1}{2}\right)^{4}\left({-}\pi^{2}(d{-}4)^{2}{+}4(d{-}3)(d{-}2){+}6(d{-}4)^{2}\psi^{(1)}\left(\frac{d}{2}{-}1\right)\right)}{\pi^{6}(d{-}4)^{2}\Gamma\left(\frac{d}{2}{-}2\right)^{4}}\,.$
(121)

The last diagram based on the pentagon effective vertex for $s$ is planar:

$x_{3}$$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{s}$$2\Delta_{\phi}$$2\Delta_{s}$$2\Delta_{s}$$x_{1}$$x_{2}$$=\frac{w_{6}}{|x_{12}|^{4}(|x_{13}||x_{23}|)^{2}}$

This diagram is straightforward to calculate by applying the uniqueness and
the propagator merging relations, as well as integrating both sides over
$x_{3}$, yielding

$\displaystyle
w_{6}{=}2^{2}N\left({-}\frac{2}{\sqrt{N}}\right)^{5}C_{\phi}^{5}C_{s}^{5}U\left(\frac{d}{2}{-}1,\frac{d}{2}{-}1,2\right)^{3}U\left(\frac{d}{2}{-}1,\frac{d}{2}{-}2,3\right)U(2,3,d{-}5)\,.$
(122)

Simplifying it, we obtain

$w_{6}=-\frac{1}{N^{3/2}}\frac{2^{5d-3}\sin^{5}\left(\frac{\pi
d}{2}\right)\Gamma\left(5-\frac{d}{2}\right)\Gamma\left(\frac{d-1}{2}\right)^{5}}{\pi^{15/2}(d-6)^{3}\Gamma\left(\frac{d}{2}-1\right)^{2}\Gamma(d-6)}\,.$
(123)

Combining everything together, we obtain

$\boxed{C_{s^{2}s^{2}s}^{(0)}\,\delta
C_{s^{2}s^{2}s}=C_{s^{2}s^{2}s}^{(0)}\,w_{0}+\sum_{a=1}^{6}w_{a}}$ (124)

The only non-vanishing contributions in $3d$ are given by

$w_{4}(d=3)=\frac{1}{N^{3/2}}\,\frac{512}{\pi^{6}}\,,\qquad
w_{5}(d=3)=\frac{1}{N^{3/2}}\,\frac{256}{\pi^{6}}\,.$ (125)

This implies

$C_{s^{2}s^{2}s}^{(0)}\,\delta
C_{s^{2}s^{2}s}(d=3)=\frac{1}{N^{3/2}}\,\frac{768}{\pi^{6}}\,.$ (126)

##  5 Fate of the emergent $\mathbb{Z}_{2}$ symmetry at large $N$

In the previous section, we demonstrated that while the leading ${\cal
O}(1/N^{1/2})$ order amplitude of the $\langle s^{2}s^{2}s\rangle$ three-point
function vanishes in $d=3$ dimensions, its next-to-leading ${\cal
O}(1/N^{3/2})$ order correction is non-vanishing. This implies that the
conjectured $s\rightarrow-s$ symmetry of correlation functions in the $s$
sector of the theory is violated at the first sub-leading order in $1/N$.

Instead, in this section we intend to discuss whether the statement of the
$s\rightarrow-s$ symmetry has a more general validity at the leading order in
$1/N$. Specifically, we are going to focus on the three-point correlation
functions $\langle s^{k}s^{m}s^{n}\rangle$, with $k+m+n=2l+3$, where $k$, $m$,
$n$ are positive integers and $l=0,1,2\dots$. This question is particularly
relevant from the standpoint of the holographic correspondence, which is
usually limited for technical reasons to the leading $1/N$ order calculations
in the bulk Klebanov:2002ja ; Petkou:2003zz ; Sezgin:2003pt ; Giombi:2009wh .

Consider a sub-set of the $\langle s^{k}s^{m}s^{n}\rangle$ three-point
functions that at the leading order in $1/N$ expansion are determined by the
diagram

$s^{k}$$s^{m}$$s^{n}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$

Here we symbolically denoted the possible contractions of the $s$ lines with
dots. Since $k+m+n-3=2l$ is even-valued, we will always be able to connect the
$s$ lines which did not go into the vertices of the $\phi$ triangle. To avoid
tadpole loops of the $s$ propagators we need to impose conditions

$\displaystyle\boxed{k\leq m+n-1\,,\quad k\geq m\geq n}$ (127)

The resulting diagram will be given by181818We skipped keeping track of
numerical symmetry and degeneracy factors, that are unimportant for our
purposes.

$\langle
s^{m}(x_{1})s^{n}(x_{2})s^{k}(x_{3})\rangle\simeq\frac{C_{sss}^{(0)}C_{s}^{l}}{|x_{12}|^{2(m+n-k)}|x_{13}|^{2(m+k-n)}|x_{23}|^{2(k+n-m)}}\,.$
(128)

Since $C_{sss}^{(0)}(d=3)=0$, this leading ${\cal O}(1/N^{1/2})$ diagram (128)
vanishes in $3d$.

In case when the triangle inequalities (127) are not satisfied, the leading
order behavior of the $\langle s^{k}s^{m}s^{n}\rangle$ three-point correlation
functions can be further suppressed than ${\cal O}(1/N^{1/2})$. For instance,
the naive ${\cal O}(1/N^{1/2})$ diagram contributing to $\langle
s^{3}ss\rangle$ is given by:191919 To lighten up the notation, we skip
labeling exponents of the propagators; all of the internal lines are the $s$
lines, except for the $\phi$ lines in the polygons.

$s^{3}$$s$$s$

This diagram is in fact identically zero in any $d$. While this can be
established by an explicit calculation, a faster way to arrive at the same
conclusion is by noticing that the diagram above contains the $\langle
s^{2}s\rangle$ sub-diagram, which is zero due to conformal symmetry. Along
with the diagrams involving tadpoles of $s$, these kinds of diagrams do not
make any physical contributions to the corresponding correlation function.

As a result, the leading order contributions to the $\langle s^{3}ss\rangle$
will instead be of the order ${\cal O}(1/N^{3/2})$. Among the contributing
diagrams there are planar and non-planar graphs consisting of the pentagon of
the $\phi$ lines, such as

$s^{3}$$s$$s$

Notice, that this diagram needs to be regularized. It is an interesting open
problem to finish the calculation of the three-point function $\langle
s^{3}ss\rangle$, which is non-trivial even at the leading ${\cal
O}(1/N^{3/2})$ order. Similar unsolved polygon diagrams appear in the leading
order contributions to the three-point functions $\langle
s^{k}s^{m}s^{n}\rangle$ that do not satisfy the triangle inequalities (127).
Therefore, an explicit proof of whether the $s\rightarrow-s$ symmetry persists
throughout the entire set of these three-point correlators at the leading
order in $1/N$ remains an open problem.

##  6 Discussion

In this paper, we set out to calculate the three-point correlation function
$\langle s^{2}s^{2}s\rangle$ in the critical $O(N)$ vector model at the next-
to-leading order in the $1/N$ expansion. In the process, we computed the
$s^{2}ss$ conformal triangle, following the technique developed in
Goykhman:2020tsk for the analogous calculation in the Gross-Neveu model.
Additionally, we determined the finite correction $A_{s^{2}}$ to the amplitude
of the $\langle s^{2}s^{2}\rangle$ two-point function. This involved
calculating an extra diagram, absent in the GN model, that is hard to find in
general $2<d<6$ using conventional techniques, and is still an open problem.
However, it conveniently vanishes in $d=3$,202020This can be traced back to
the triviality of the $\langle sss\rangle$ sub-diagram, which appears in that
calculation, at the leading order in $1/N$. allowing us to compute the
amplitude correction $A_{s^{2}}$ in $3d$. The computation of the $\langle
s^{2}s^{2}s\rangle$ three-point function required evaluating a new set of
self-energy diagrams that were unknown in the literature. We have outlined the
steps of all these calculations in detail. Assembling all these components
together, we were able to compute the $\langle s^{2}s^{2}s\rangle$ correlator
in $d=3$ dimensions, that turned out to have a non-zero value.

As was discussed in detail in section 5, we explored the $\mathbb{Z}_{2}$
symmetry $s\rightarrow-s$, that was originally conjectured to emerge in the
large $N$ limit of the $O(N)$ vector model in $d=3$. In Goykhman:2019kcj , the
symmetry seemed to surprisingly persist in the three-point function $\langle
sss\rangle$ up to the next-to-leading order in the $1/N$ expansion. The non-
zero value at the next-to-leading order of the $\langle s^{2}s^{2}s\rangle$
correlator, obtained in this paper, suggests that the emergent
$\mathbb{Z}_{2}$ symmetry is lifted at the sub-leading orders in $1/N$.
Besides this result, the calculation of the conformal triangle $s^{2}ss$ as
well as the sub-diagrams contributing to the $\langle s^{2}s^{2}s\rangle$ are
important in themselves for other conformal correlators in vector models.

Regarding the emergent symmetry, our results are supportive of the statement
that it is exact at large $N$, as we have illustrated in section 5, where we
demonstrated its presence for an entire set of correlators involving composite
operators in $s$. This may have important implications for the AdS/CFT
correspondence, which states that the critical $O(N)$ vector model in $3d$ is
dual to a higher-spin Vasiliev theory on $AdS_{4}$. The AdS/CFT mapping
suggests that the operator $s$ is dual to the spin zero field $A_{0}$ in AdS,
and correspondingly all its polynomial powers correspond to the polynomial
powers of $A_{0}$. Thus, any statement we make about large $N$ three-point
correlators of $s^{n}$ operators in the boundary CFT, has a direct implication
for the three-point correlators of $A_{0}^{n}$ in the bulk theory.
Correlations of composite currents in the AdS bulk, are hitherto largely
undiscussed in the literature to the best of our knowledge. However,
symmetries of the bulk action alone, known to exist at least at the classical
level, can demand some boundary correlators to vanish. For instance, the cubic
interaction $A_{0}^{3}$ is absent in the Vasiliev’s theory in the bulk
Petkou:2003zz ; Sezgin:2003pt , which translates to the statement that the
boundary CFT possesses the $s\rightarrow-s$ symmetry. Next, the calculation of
sub-leading corrections to such correlators in the large $N$ CFT has a direct
implication on one-loop corrections in the AdS bulk, which are otherwise very
hard to compute Aharony:2016dwx . At the same time, the symmetry
considerations might again indicate whether certain correlators can become
non-zero at sub-leading orders, due to an anomalous symmetry breaking
mechanism in the bulk. An example of the latter is furnished by the possible
anomalous torsion term generated by loops in the bulk.212121See, e.g.,
Leigh:2008tt for a holographic description of parity-breaking system via
torsion deformation of the AdS bulk. We thank A. Petkou for the discussion and
drawing our attention to relevant references.

## Acknowledgements

We thank J. Gracey, A. Petkou, and especially M. Smolkin for helpful
discussions, comments on the draft, and correspondence. Our work is partially
supported by the Binational Science Foundation (grant No. 2016186), the
Israeli Science Foundation Center of Excellence (grant No. 2289/18), and by
the Quantum Universe I-CORE program of the Israel Planning and Budgeting
Committee (grant No. 1937/12). The work of NC is partially supported by Yuri
Milner scholarship.

##  Appendix A Some useful identities

In this appendix we review some useful expressions and identities.

Loop diagram in the position space are additive:

$=$$2b$$2a$$2(a+b)$

The propagator merging relation has the form

$\int
d^{d}x_{3}\,\frac{1}{(x_{3}^{2})^{a}((x_{3}-x_{12})^{2})^{b}}=U(a,b,d-a-b)\,\frac{1}{(x_{12}^{2})^{a+b-\frac{d}{2}}}\,,$
(129)

where we defined

$\displaystyle U(a,b,c)$ $\displaystyle=\pi^{\frac{d}{2}}A(a)A(b)A(c)\,,$
(130) $\displaystyle A(a)$
$\displaystyle=\frac{\Gamma\left(\frac{d}{2}-a\right)}{\Gamma(a)}\,.$ (131)

This can be graphically represented as

$2a$$2b$$=$$2(a+b)-d$$\times U(a,b,d-a-b)$

Uniqueness relation for $a_{1}+a_{2}+a_{3}=d$ is written as DEramo:1971hnd ;
Symanzik:1972wj

$\int
d^{d}x\,\frac{1}{|x_{1}-x|^{2a_{1}}|x_{2}-x|^{2a_{2}}|x_{3}-x|^{2a_{3}}}=\frac{U(a_{1},a_{2},a_{3})}{|x_{12}|^{d-2a_{3}}|x_{13}|^{d-2a_{2}}|x_{23}|^{d-2a_{1}}}\,,$
(132)

This can be diagrammatically represented as

$2a_{1}$$2a_{2}$$2a_{3}$$=$$\alpha$$\beta$$\gamma$$\times\left(-\frac{2}{\sqrt{N}}\right)U\left(a_{1},a_{2},a_{3}\right)$

Here we denoted $\alpha=d-2a_{3}$, $\beta=d-2a_{2}$, $\gamma=d-2a_{1}$.

Integration by parts relation Chetyrkin:1981qh ; Kazakov:1983ns :

$2\alpha_{1}$$2\alpha_{2}$$2\alpha_{3}$$(d-2\alpha_{1}-\alpha_{2}-\alpha_{3})\times$$2(\alpha_{1}-1)$$2(\alpha_{2}+1)$$2\alpha_{3}$$=\;\;\alpha_{2}\times\;\;$$2\alpha_{1}$$2(\alpha_{2}+1)$$2\alpha_{3}$$-\;\;\alpha_{2}\times\;\;$$-2$$2(\alpha_{1}-1)$$2\alpha_{2}$$2(\alpha_{3}+1)$$+\;\;\alpha_{3}\times\;\;$$2\alpha_{1}$$2\alpha_{2}$$2(\alpha_{3}+1)$$-\;\;\alpha_{3}\times\;\;$$-2$

We will also find useful the following relation Gracey:2018ame 222222See fig.
15 therein.

(133)

Here

$\displaystyle y_{1}$
$\displaystyle=\frac{(d-s_{1})(d-s_{2})}{(d-t_{2})(t_{2}-d/2-1)}\,,\;\;y_{2}=\frac{(d-s_{2})(D+\alpha_{5}-3d/2-1)}{(d-t_{2})(t_{2}-d/2-1)}\,,$
(134) $\displaystyle y_{3}$
$\displaystyle=\frac{(d-s_{1})(D+\alpha_{5}-3d/2-1)}{(d-t_{2})(t_{2}-d/2-1)}\,,$
(135)

and

$\displaystyle s_{1}$ $\displaystyle=\alpha_{1}+\alpha_{2}+\alpha_{5}\,,\qquad
s_{2}=\alpha_{3}+\alpha_{4}+\alpha_{5}\,,$ (136) $\displaystyle t_{1}$
$\displaystyle=\alpha_{1}+\alpha_{4}+\alpha_{5}\,,\qquad
t_{2}=\alpha_{2}+\alpha_{3}+\alpha_{5}\,,\quad D=\sum_{i=1}^{5}\alpha_{i}\,.$
(137)

##  Appendix B Some useful diagrams

Using the integration by parts relation, we can derive Chetyrkin:1981qh ;
Kazakov:1983ns ; Goykhman:2019kcj

$x_{1}$$x_{2}$$2$$2$$2$$2\alpha$$2\beta$$=\frac{F(\alpha,\beta)}{|x_{12}|^{6-2d+2\alpha+2\beta}}$

where

$\displaystyle F(\alpha,\beta)$
$\displaystyle{=}\frac{U(d{-}2,1,1)}{d{-}2{-}\alpha{-}\beta}\left(\alpha\left(U(\alpha{+}1,\beta,d{-}\alpha{-}\beta{-}1){-}U\left(\alpha{+}1,\beta{+}2{-}\frac{d}{2},\frac{3d}{2}{-}\alpha{-}\beta{-}3\right)\right)\right.$
$\displaystyle{+}\left.\beta\left(U(\beta{+}1,\alpha,d{-}\alpha{-}\beta{-}1){-}U\left(\beta{+}1,\alpha{+}2{-}\frac{d}{2},\frac{3d}{2}{-}\alpha{-}\beta{-}3\right)\right)\right)\,.$
(138)

Performing the Fourier transform we can derive the ChT diagram
Chetyrkin:1981qh ; Kazakov:1983ns

$x_{1}$$x_{2}$$d-2$$2\alpha$$d-2$$d-2$$2\beta$$=\frac{\textrm{ChT}(\alpha,\beta)}{|x_{12}|^{d-6+2\alpha+2\beta}}$

where

$\displaystyle\textrm{ChT}(\alpha,\beta)$
$\displaystyle=\frac{\pi^{d}\Gamma\left(2-\frac{d}{2}\right)}{\Gamma\left(\frac{d}{2}-1\right)\Gamma(d-2)}\left(\frac{\Gamma\left(\frac{d}{2}-\alpha\right)\Gamma\left(\frac{d}{2}+\alpha-2\right)}{(1-\beta)(\alpha+\beta-2)\Gamma(2-\alpha)\Gamma(\alpha)}\right.$
(139)
$\displaystyle+\left.\frac{\Gamma\left(\frac{d}{2}-\beta\right)\Gamma\left(\frac{d}{2}+\beta-2\right)}{(1-\alpha)(\alpha+\beta-2)\Gamma(2-\beta)\Gamma(\beta)}+\frac{\Gamma\left(\frac{d}{2}-\alpha-\beta+1\right)\Gamma\left(\frac{d}{2}+\alpha+\beta-3\right)}{(\alpha-1)(\beta-1)\Gamma(-\alpha-\beta+3)\Gamma(\alpha+\beta-1)}\right)\,.$

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