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

# Long-Range Vector Models at Large N

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

We calculate various CFT data for the $O(N)$ vector model with the long-range
interaction, working at the next-to-leading order in the $1/N$ expansion. Our
results provide additional evidence for the existence of conformal symmetry at
the long-range fixed point, as well as the continuity of the CFT data at the
long-range to short-range crossover point $s_{\star}$ of the exponent
parameter $s$. We also develop the $N>1$ generalization of the recently
proposed IR duality between the long-range and the deformed short-range
models, providing further evidence for its non-perturbative validity in the
entire region $d/2<s<s_{\star}$.

##  1 Introduction

Long-range spin models provide a simple generalization of the usual Ising spin
chain with the nearest neighbor interactions. However, they are known to
display a much richer phase structure. The long-range Ising (LRI) model was
introduced by Dyson more than fifty years ago to describe spontaneous symmetry
breaking and long-range order in a $1d$ ferromagnetic spin chain Dyson:1968up
. It features a spin-spin interaction that decays as a power law, controlled
by a positive-valued exponent $s$. In a general dimension $d$, the Hamiltonian
for the LRI model is given by

$\displaystyle H=-J\sum_{i,j}\frac{s_{i}s_{j}}{|i-j|^{d+s}}\,,$ (1)

where $J>0$ governs the strength of interaction and $s_{i}=\pm 1$ are the
Ising spins. This model has been subjected to numerous tests, both analytical
and numerical, and was found to exhibit a second-order phase transition in
low-dimensional spin chains Fisher:1972zz ; PhysRevLett.37.1577 ; Aizenman1988
. The corresponding long-range critical regime was found in the window
$d/2<s<s_{\star}$ for the parameter $s$. For the range $s\leq d/2$, the model
is described by a Gaussian mean-field theory (MFT), whereas for $s\geq
s_{\star}$, the model transitions to its short-range counterpart, given by the
usual Ising CFT.

It is easy to identify these phases in the parameter space of $s$ in the
continuum description of (1), obtained using the Landau-Ginzburg technique,

$\displaystyle
S\propto-\int\,d^{d}x\int\,d^{d}y\frac{\phi(x)\phi(y)}{|x-y|^{d+s}}\,.$ (2)

Using (2), the scaling dimension of the field $\phi$ is found to be
$\Delta_{\phi}=(d-s)/2$. It follows that we need $s\leq 2$, in order for the
long-range model to be unitary. One can subsequently deform the above action
by adding to it a usual local quartic interaction term Fisher:1972zz ,

$\displaystyle
S\propto-\int\,d^{d}x\int\,d^{d}y\frac{\phi(x)\phi(y)}{|x-y|^{d+s}}+g\,\int\,d^{d}x\,\phi(x)^{4}\,.$
(3)

When $s\leq d/2$, the quartic interaction is irrelevant, and the model is
described by the Gaussian MFT. When $s>d/2$, the quartic interaction becomes
relevant, and triggers an RG flow, taking the theory away from the Gaussian
fixed point. One can perform a systematic perturbative expansion near $s=d/2$
to identify a non-trivial interacting IR fixed point of this flow. This fixed
point has been studied extensively in the literature (see Brydges:2002wq ;
Abdesselam:2006qg ; Slade:2016yer ; mitter ; Benedetti:2020rrq for a rigorous
renormalization group derivation of this fixed point), and much is known about
the corresponding critical exponents. The scaling dimension of the operator
$\phi$ at this long-range fixed point is exact, being protected by the bi-
local kinetic term in (3) from receiving anomalous contributions. This fixed
point is also characterized by the lack of a local stress-energy tensor. The
latter fact makes it rather difficult to ascertain whether the long-range
fixed point in fact enjoys the full conformal symmetry. Strong support in
favor of existence of conformal symmetry at the long-range fixed point has
recently been provided in Paulos:2015jfa ; Behan:2017dwr ; Behan:2017emf ;
Behan:2018hfx .

As we reviewed above, the long-range critical regime is found at the end of an
RG flow in (3) for the exponent parameter $s$ taking values in the range
$d/2<s<s_{\star}$. The model crosses over to the short-range CFT regime when
$s\geq s_{\star}$.111The short-range CFT at $s>s_{\star}$ is the critical
short-range Ising model plus a decoupled sector consisting of a generalized
free field Behan:2017emf ; Behan:2018hfx , as we review below. While the
dimension of $\phi$ in the long range CFT is $(d-s)/2$, its corresponding
dimension in the short-range CFT is $d/2-1+\gamma_{\hat{\phi}}$, with
$\gamma_{\hat{\phi}}$ being the anomalous dimension of the short-range field
$\hat{\phi}$ (in this paper we will typically put a hat on top of the letter
standing for the short-range field, whenever the same letter without a hat is
used to denote the long-range field). Continuity at $s_{\star}$ then implies
that $s_{\star}=2-2\gamma_{\hat{\phi}}$ Sak1973 ; PhysRevB154344 . Notice that
since $\gamma_{\hat{\phi}}>0$ in the usual short-range CFT, then
$s_{\star}<2$, implying that interactions tighten the upper bound on $s$. In
fact, it follows that all CFT data, not just the conformal dimension of
$\phi$, are continuous across the crossover point $s_{\star}$ Sak1973 ;
PhysRevB154344 . In this paper we will illustrate such continuity by carrying
out explicit calculations of CFT data in the $O(N)$ generalization of the
model (3), working at the next-to-leading order in $1/N$ expansion.

We begin by considering an $O(N)$ version of the non-local MFT, and perturb it
by a quartic interaction. To access the strongly-interacting IR regime of this
model away from the weakly-coupled behavior near $s=d/2$, we study it in the
$1/N$ expansion. By employing the Hubbard-Stratonovich formalism we can write
down the corresponding action of the model as

$\displaystyle
S\propto-\int\,d^{d}x\int\,d^{d}y\frac{\phi_{i}(x)\phi_{i}(y)}{|x-y|^{d+s}}+\int\,d^{d}x\,\left(-\frac{1}{4g}\sigma^{2}+\frac{1}{\sqrt{N}}\sigma\phi^{2}\right)\,,$
(4)

where $\sigma$ is the Hubbard-Stratonovich field. The model (4) has been
studied previously in the literature, where critical exponents such as the
scaling dimensions of $\phi$ and $\sigma$ have been calculated Brezin_2014 ;
Gubser:2017vgc ; Giombi:2019enr . However, no other CFT data in the $1/N$
expansion is available for the long-range fixed point. One of the goals of
this paper is to fill this gap, by calculating some large $N$ CFT data,
including anomalous dimensions of composite operators, and various OPE
coefficients. We perform most of our calculations at the next-to-leading order
in the $1/N$ expansion. Our results lend a strong support to the statement
that the long-range fixed point enjoys the full conformal symmetry, even for
the values of $s$ that are beyond the scope of a perturbative regime in the
vicinity of $s=d/2$.

Following our calculations of CFT data in the long-range critical $O(N)$
vector model, we proceed to establish explicitly that the obtained anomalous
dimensions and OPE coefficients are continuous at $s=s_{\star}$. In fact, at
$s=s_{\star}$ the long-range critical vector model crosses over to the short-
range critical vector model plus a generalized free field $\chi$ Behan:2017dwr
; Behan:2017emf . The role of the decoupled field $\chi$ becomes important
when one matches the complete spectrum across the crossover point $s_{\star}$.
As it was pointed out in Behan:2017dwr ; Behan:2017emf , some operators in the
long-range model, such as $\phi^{3}$, exhibit an apparently discontinuous
behavior at $s_{\star}$, having no counterpart in the spectrum of the short-
range critical vector model. Introducing the generalized free field $\chi$
allows one to construct such operators when $s>s_{\star}$, thereby allowing
for the complete match of the spectrum. Note that there has been some debate
regarding the smooth transition of CFT data across this long-range–short-range
crossover point Picco:2012ak ; Blanchard:2012xv . However, our calculation
provides complementary support to the numerous theoretical calculations and
Monte-Carlo simulations which predicted such a smooth transition at the long-
range–short-range crossover point Honkonen:1988fq ; Honkonen:1990mr ;
Luijten2002BoundaryBL .222In this paper, whenever we refer to the long-
range–short-range crossover point $s_{\star}$, we mean the point of crossover
between the long-range critical vector model, and the short-range critical
vector model with the decoupled generalized free field.

While the long-range CFT exists in the perturbative regime near the lower
bound $s=d/2$ of the long-range window $d/2<s<s_{\star}$, making it amenable
to the Wilson-Fisher kind of $\epsilon$-expansion for $s=(d+\epsilon)/2$, its
behavior near the upper bound $s=s_{\star}$ is strongly-interacting. It was
proposed in PhysRevB154344 that the long-range CFT near $s=s_{\star}$ can be
accessed by perturbing the short-range CFT with a bi-local kinetic term for
the field $\hat{\phi}$. Such a kinetic term crosses over from being irrelevant
when $s>s_{\star}$, to being relevant when $s<s_{\star}$. Consequently, as $s$
is lowered past $s_{\star}$, an RG flow brings the model to a long-range
regime in the IR. While such a bi-local kinetic term appears to manifest a
perturbative behavior in the vicinity of $s_{\star}$, it is unclear how to
carry out this perturbative expansion. This is primarily due to a lack of
proper understanding of how to perform conformal perturbation theory involving
non-local perturbations.

Recently a new weakly-coupled description near the long-range–short-range
crossover point $s_{\star}$ was proposed in Behan:2017dwr ; Behan:2017emf . In
this description the non-local perturbation of PhysRevB154344 has been traded
for a local one, at the cost of introducing a generalized free field $\chi$
with scaling dimension $\Delta_{\chi}=(d+s)/2$ into the model. The
corresponding action is given by Behan:2017dwr ; Behan:2017emf

$\displaystyle S=S_{\text{CFT}}+\lambda\int\,d^{d}x\,\hat{\phi}\chi+\int
d^{d}x\int d^{d}y\,\frac{\chi(x)\chi(y)}{|x-y|^{d-s}}\,,$ (5)

where $\hat{\phi}$ is the spin field and $S_{\text{CFT}}$ is the action of the
original short-range CFT model, and $\chi$ is a generalized free field coupled
to it. The interacting term $\hat{\phi}\chi$ with the coupling $\lambda$ has
the leading order scaling dimension $d-\delta$, where
$\delta=(s_{\star}-s)/2$. This perturbation is therefore slightly relevant for
small positive $\delta$, that is, for $s$ slightly below the long-range–short-
range crossover point $s_{\star}$. The flow triggered by this perturbation
ends at a long-range fixed point in the IR, near $s=s_{\star}$. Although two
different UV descriptions (3) and (5) flow to the long-range fixed points in
two non-overlapping perturbative regimes, near $s=d/2$ and $s=s_{\star}$
respectively, it has been argued in Behan:2017dwr ; Behan:2017emf that these
are in fact the same long-range fixed points for all $d/2<s<s_{\star}$. This
furnishes a non-trivial example of an IR duality. As part of matching the
spectra of these two models, the new d.o.f. $\chi$ on the deformed short-range
side of the duality has been suggested to be dual to the operator $\phi^{3}$
in the long-range CFT Behan:2017dwr ; Behan:2017emf .

One can further take advantage of the perturbative behavior of the coupling
$\lambda$ near $s=s_{\star}$ to find the anomalous dimension of the stress-
energy tensor $T_{\mu\nu}$ within conformal perturbation theory. Importantly,
a non-trivial anomalous dimension of $T_{\mu\nu}$ supplies further evidence
that the fixed point at the end of an RG flow triggered by the coupling
$\lambda$ is indeed a long-range CFT. Additional evidence for the proposed IR
duality is given by matching the ratios of the three-point function amplitudes
(OPE coefficients) $\langle\phi{\cal O}_{1}{\cal O}_{2}\rangle$,
$\langle\phi^{3}{\cal O}_{1}{\cal O}_{2}\rangle$ in the long-range model, and
$\langle\hat{\phi}{\cal O}_{1}{\cal O}_{2}\rangle$, $\langle\chi{\cal
O}_{1}{\cal O}_{2}\rangle$ in the deformed short-range model, for arbitrary
operators ${\cal O}_{1,2}$ Behan:2017dwr ; Behan:2017emf .

Inspired by the construction of Behan:2017dwr ; Behan:2017emf , we propose
that the long-range $O(N)$ critical vector model for $d/2<s<s_{\star}$ admits
a dual description in terms of a deformed short-range CFT with the action

$\displaystyle S=a\,\int d^{d}x\int
d^{d}y\,\frac{\chi^{i}(x)\chi^{i}(y)}{|x-y|^{d-s}}+\frac{1}{\sqrt{N}}\,\int\,d^{d}x\,\Sigma\,\Phi^{2}+\int\,d^{d}x\,\Phi^{i}\chi^{i}\,,$
(6)

where $\Phi^{i}$, $i=1,\dots,N$ is the spin field, $\Sigma$ is the Hubbard-
Stratonovich field, and $\chi^{i}$, $i=1,\dots,N$ is a generalized free vector
field.333The coefficient $a$ reflects a particular choice of conventions
regarding normalization of $\chi^{i}$, and will be fixed below. We will argue
that the models (6) and (4) are equivalent, up to an identification of the
d.o.f.

$\displaystyle\Phi\leftrightarrow\phi,\hskip
28.45274pt\Sigma\leftrightarrow\sigma\,.$ (7)

These two models, therefore, provide dual descriptions of the same long-range
IR critical regime, for all values of $d/2<s<s_{\star}$.

It then follows that the composite operator $\sigma\phi$ is dual to the field
$\chi$ on the other side of the duality. While this can be seen as a direct
consequence of the relations (7) and the e.o.m., one can additionally carry
out a consistency check of the proposed duality, by matching the amplitudes of
the three-point functions involving the operators $\phi$, $\sigma\phi$ in the
long-range model, and $\hat{\phi}$, $\chi$ Behan:2017dwr ; Behan:2017emf . As
we move to $s>s_{\star}$, beyond the scope of the long-range critical regime,
the field $\chi$ decouples into a separate generalized free field sector. Its
existence however guarantees continuity of spectrum across the crossover point
$s_{\star}$ Behan:2017dwr ; Behan:2017emf .

Following Behan:2017dwr ; Behan:2017emf , we further use our newly proposed
dual IR long-range CFT description to calculate the anomalous dimension and
trace of the stress-energy tensor. This calculation additionally reveals that
near $s=s_{\star}$ our dual long-range CFT is perturbatively in $1/N$ close to
the short-range model in the UV, by relating anomalous dimension of the
stress-energy tensor to the anomalous dimension of the spin field $\phi$.

This paper is organized as follows. In section 2, we define the action for the
long-range critical $O(N)$ vector model at large $N$, and set up our
conventions that will subsequently be used throughout the rest of the paper.
In section 3, we compute the anomalous dimensions and amplitude corrections of
the lowest scalar primaries $\phi$ and $\sigma$, up to the next-to-leading
order in $1/N$ expansion. In section 4, we begin by calculating the anomalous
dimensions of composite scalar primaries $\sigma^{n}$, at the next-to-leading
order in $1/N$, followed by the calculation of the cross-correlator
$\langle\sigma\sigma^{2}\rangle$. We establish that such a cross-correlator in
fact vanishes, unlike its short-range counterpart. In particular, this
calculation provides additional evidence that the long-range fixed point
enjoys the full conformal symmetry. In section 5, we calculate the OPE
coefficients $C_{\phi\phi\sigma}$ and $C_{\sigma\sigma\sigma}$, and the
corresponding conformal triangles, working at the next-to-leading order in
$1/N$. In section 6, we explicitly demonstrate the continuity of all the
calculated long-range CFT data at the long-range–short-range crossover point
$s_{\star}$. In section 7, we study a short-range critical $O(N)$ vector model
coupled to a generalized free field $\chi^{i}$, which we propose to be dual to
the long-range critical vector model in the IR. We discuss our results and
outline some future research directions in section 8.

##  2 Set-up

In this section we review the basic setup of the long-range critical $O(N)$
vector model that we will be studying in this paper. This model describes
dynamics of a multiplet of $N$ scalar fields $\phi_{i}$, $i=1,\dots,N$ in the
fundamental representation of the $O(N)$ symmetry group, featuring a self-
interaction with the quartic coupling constant $g$. We will be interested in
the interacting critical regime of this model. In the free regime, $g=0$, the
model sits at its Gaussian fixed point, where it is described by the MFT, with
the bi-local kinetic action term analogous to (2),

$\displaystyle S_{0}=C(s)\int d^{d}x\,\int
d^{d}y\,\frac{\phi_{i}(x)\phi_{i}(y)}{|x-y|^{d+s}}\,.$ (8)

This class of models is parametrized by the exponent $s$, characterizing the
power-law long-range kinetic term in the action (8). The model (8) is defined
for $s<2$, which ensures that the bi-local term is more relevant than the
local kinetic term $\frac{1}{2}\partial\phi_{i}\partial\phi_{i}$. The latter
dominates for $s>2$, when the theory crosses over to the local model (free
short-range $O(N)$ vector model). In fact, when $s=2$, the model (8) is
equivalent to the short-range $O(N)$ vector model, as it is easiest to see in
momentum space. For $s=1$, the action (8) reduces, interestingly, to a
boundary CFT (bCFT) with a free scalar field in the bulk, see Giombi:2019enr
for some recent developments.

We will use the conventional choice of the factor $C(s)$ such that the
propagator of $\phi_{i}$ in momentum space is normalized to unity (from now on
we skip keeping track of the $O(N)$ indices and the associated Kronecker
symbol whenever it does not cause an ambiguity),

$\langle\phi(p)\phi(q)\rangle=(2\pi)^{d}\,\delta^{(d)}(p+q)\,\frac{1}{(p^{2})^{\frac{s}{2}}}\,.$
(9)

Using the Fourier transform relation

$\int\frac{d^{d}k}{(2\pi)^{d}}\,e^{ik\cdot
x}\,\frac{1}{(k^{2})^{\frac{d}{2}-\Delta}}=\frac{2^{2\Delta-d}}{\pi^{\frac{d}{2}}}\,\frac{1}{A(\Delta)}\,\frac{1}{|x|^{2\Delta}}\,,$
(10)

where we defined

$A(\Delta)=\frac{\Gamma\left(\frac{d}{2}-\Delta\right)}{\Gamma(\Delta)}\,,$
(11)

we obtain444Notice that naively setting $s=2$ renders a singularity in $C(s)$.
This artifact is lifted in momentum space.

$\displaystyle
C(s)=\frac{2^{s-1}}{\pi^{d/2}}\frac{\Gamma\left(\frac{d+s}{2}\right)}{\Gamma\left(-\frac{s}{2}\right)}\,.$
(12)

The field $\phi$ has the dimension

$\Delta_{\phi}=\frac{d-s}{2}\,,$ (13)

and its propagator in coordinate space has the form

$\langle\phi(x)\phi(0)\rangle=\frac{C_{\phi}}{|x|^{2\Delta_{\phi}}}\,,$ (14)

where we defined

$C_{\phi}=\frac{1}{2^{s}\pi^{\frac{d}{2}}}\,\frac{\Gamma\left(\frac{d-s}{2}\right)}{\Gamma\left(\frac{s}{2}\right)}\,.$
(15)

Perturbing the free theory (8) by a quartic self-interaction we obtain an
interacting field theory with the action

$\displaystyle S=C(s)\int d^{d}x\,\int
d^{d}y\,\frac{\phi(x)\phi(y)}{|x-y|^{d+s}}+\frac{g}{N}\int\,d^{d}x\,(\phi^{2})^{2}\,.$
(16)

Setting $g=0$, we recover the free action (8). The quartic coupling constant
$g$ is relevant when $s>d/2$, and it triggers a flow to a fixed point in the
IR. This can be established perturbatively by a Wilson-Fisher kind of
$\epsilon$-expansion for $s=d/2+\epsilon$. Together with the non-locality
constraint $s<2$, this restricts us to consider $d<4$. When $s<d/2$, the
quartic coupling $g$ is irrelevant, and the theory flows to a fixed point in
the UV limit, albeit the resulting model suffers from instabilities. This
situation is analogous to the Wilson-Fisher fixed point of the short-range
$O(N)$ vector model in $2<d<4$ and $4<d<6$ dimensions, respectively. When
$s=1$, a UV-completion of the theory can be constructed Giombi:2019enr ,
analogously to the higher-dimensional short-range $O(N)$ vector model
Fei:2014yja .

It has been argued that the IR fixed point for $s>d/2$ is reached in the
entire range $1<d<4$, not merely in the perturbative regime around
$d=2s-2\epsilon$. Importantly, one is interested in the physically relevant
integer dimension $d=3$ (where the spectrum of the theory is expected to be
unitary), and general allowed values for the exponent $s$. Some preliminary
evidence in favor of this conclusion is devised in the large-$N$ limit using
the Hubbard-Stratonovich transformation, analogously to the argument given in
Gubser:2002vv for the short-range $O(N)$ vector models. Specifically, at the
Gaussian f.p. the composite singlet field $\phi^{2}$ is a primary operator of
dimension $\Delta_{\phi^{2}}=2\Delta_{\phi}=d-s$. As we will review
momentarily, employing the Hubbard-Stratonovich transformation one can
demonstrate that the quartic interaction brings the theory to a new regime
where $\phi^{2}$ exhibits a scaling behavior with the (leading order in $1/N$)
exponent $\Delta_{\phi^{2}}=\Delta_{\sigma}=s$, where $\sigma$ is the Hubbard-
Stratonovich field.

Using the Hubbard-Stratonovich transformation, the action (16) is equivalently
re-written as

$\displaystyle S=C(s)\int d^{d}x\,\int
d^{d}y\frac{\phi(x)\phi(y)}{|x-y|^{d+s}}+\int\,d^{d}x\left(-\frac{1}{4g}\sigma^{2}+\frac{1}{\sqrt{N}}\sigma\phi^{2}\right)$
(17)

The path integral over $\phi$ is now Gaussian, and integrating it out results
in the effective action for $\sigma$

$S=\frac{N}{2}\int d^{d}x\int
d^{d}y\,\textrm{Tr}\,\log\left(\frac{C(s)}{|x-y|^{d+s}}+\frac{1}{\sqrt{N}}\,\sigma(x)\delta^{(d)}(x-y)\right)-\frac{1}{4g}\int
d^{d}x\,\sigma^{2}\,.$ (18)

Using the inverse propagator relation

$\int
d^{d}y\,\frac{1}{|y|^{2a}|x-y|^{2(d-a)}}=\pi^{d}A(a)A(d-a)\,\delta^{(d)}(x)\,,$
(19)

and expanding the logarithm we obtain

$S=-C_{\phi}^{2}\int d^{d}x\int
d^{d}y\,\frac{\sigma(x)\sigma(y)}{|x-y|^{2(d-s)}}-\frac{1}{4g}\int
d^{d}x\,\sigma^{2}+{\cal O}\left(\frac{1}{\sqrt{N}}\right)\,,$ (20)

where we also used the definition (15).

For $s>d/2$, and in the long-distance IR limit, the first term in the r.h.s.
of (20) dominates over the second term.555Analogously, when $s<d/2$ the first
term dominates over the second term in the UV limit. The second term then
drops out, and the $\langle\sigma\sigma\rangle$ propagator can be found from
the first term using (19). As a result we obtain

$\langle\sigma(x)\sigma(0)\rangle=\frac{C_{\sigma}}{|x|^{2\Delta_{s}}}\,,$
(21)

where the scaling dimension of the Hubbard-Stratonovich field $\sigma$ at the
strongly-coupled regime in the IR is,

$\Delta_{\sigma}=s\,,$ (22)

and the propagator amplitude is found to be

$\displaystyle
C_{\sigma}=-\frac{2^{2s-1}\Gamma\left(d-s\right)\Gamma\left(s\right)\Gamma\left(\frac{s}{2}\right)^{2}}{\Gamma\left(\frac{d}{2}-s\right)\Gamma\left(s-\frac{d}{2}\right)\Gamma\left(\frac{d-s}{2}\right)^{2}}\,.$
(23)

As a consistency check, notice that for the scaling dimension (22) the second
term in the r.h.s. of (20) is irrelevant.

In the effective action (20) we also omitted higher-order vertices for
$\sigma$, which are suppressed in $1/N$. We will encounter these vertices in
our calculation of CFT data in this paper, where they will be explicitly
represented diagrammatically via polygon graphs with internal $\phi$
propagator lines. These are constructed using the Feynman rules:

$=\frac{C_{\phi}}{|x|^{2\Delta_{\phi}}}$$=\frac{C_{\sigma}}{|x|^{2\Delta_{\sigma}}}$$x$$0$$2\Delta_{\phi}$$0$$x$$2\Delta_{\sigma}$$=-\frac{2}{\sqrt{N}}$

A propagator line with a generic exponent will be assumed to have a unit
amplitude

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

In appendix A we collect some well-known identifies for conformal graphs in
position space, that have been used for calculations in this paper.

##  3 Two-point functions

In this section we will calculate the $\langle\phi\phi\rangle$ and
$\langle\sigma\sigma\rangle$ two-point functions of the fundamental field
$\phi$ and the Hubbard-Stratonovich scalar field $\sigma$. Conformal
invariance ensures that these two-point functions have the form

$\displaystyle\langle\phi(x)\phi(0)\rangle$
$\displaystyle=\frac{C_{\phi}(1+A_{\phi})\mu^{-2\gamma_{\phi}}}{|x|^{2(\Delta_{\phi}+\gamma_{\phi})}}\,,$
(24) $\displaystyle\langle\sigma(x)\sigma(0)\rangle$
$\displaystyle=\frac{C_{\sigma}(1+A_{\sigma})\mu^{-2\gamma_{\sigma}}}{|x|^{2(\Delta_{\sigma}+\gamma_{\sigma})}}\,,$
(25)

where $\mu$ is an arbitrary mass scale, and $\Delta_{\phi,\sigma}$ are leading
order dimensions at the long-range critical point, defined in section 2. In
this section we will calculate the two-point functions (24), (25) at the next-
to-leading order in $1/N$ expansion. In particular, we will reproduce the
known expression for the anomalous dimensions $\gamma_{\sigma}$, and obtain
new results for the relative corrections $A_{\phi,\sigma}$ to the amplitudes
of these two-point functions.

We will also demonstrate that the anomalous dimension $\gamma_{\phi}$ vanishes
at the next-to-leading order in $1/N$. As discussed in section 2, in fact we
expect the engineering dimensions $\Delta_{\phi}$ of the field $\phi$ to be
exact to all orders in $1/N$, being fixed by the non-local kinetic term.
However, our calculation of the two-point function $\langle\phi\phi\rangle$ is
still useful, because we extract from it the relative amplitude correction
$A_{\phi}$. While the propagator amplitude corrections are not observables,
together with its counterpart $A_{\sigma}$, it will play an important part in
our calculation in section 5 of the observable CFT data, such as the
amplitudes of the $\langle\phi\phi\sigma\rangle$ and
$\langle\sigma\sigma\sigma\rangle$ three-point functions and the related
$\phi\phi$ and $\sigma\sigma$ OPE coefficients.

###  3.1 $\langle\phi\phi\rangle$

It is easy to see that the only diagram contributing to the
$\langle\phi\phi\rangle$ two-point function at the next-to-leading order in
$1/N$ is given by

$0$$x$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\sigma}+\delta$$2\Delta_{\phi}$$=\frac{C_{\phi}\,\mu^{-\delta}}{|x|^{2\Delta_{\phi}+\delta}}\,\left(\frac{2\gamma_{\phi}}{\delta}+A_{\phi}+{\cal
O}(\delta)\right)$

Here we have introduced an auxiliary regulator $\delta$ to the internal
$\sigma$ line. Evaluating the diagram gives666In this paper we utilize the
supset sign $\supset$ to denote some of the terms which contribute to the
expression on the l.h.s. of this sign.

$\displaystyle\langle\phi(x)\phi(0)\rangle\supset\frac{4}{N}\frac{C_{\phi}^{3}C_{\sigma}}{|x|^{2\Delta_{\phi}}}U\left(\frac{d{-}s}{2},\frac{d{+}s{+}\delta}{2},{-}\frac{\delta}{2}\right)U\left(\frac{d{+}\delta}{2},\frac{d{-}s}{2},\frac{s{-}\delta}{2}\right)\frac{1}{(\mu|x|)^{\delta}}\,.$
(26)

Expanding this expression around $\delta=0$ shows that it is in fact finite
and $\gamma_{\phi}=0$,777A simple explanation of the vanishing anomalous
dimension, $\gamma_{\phi}=0$, is that only local divergences can appear from
the loops, which in turn are to be cancelled by local counter-terms. However,
since the fields $\phi_{i}$ have a bi-local action, no local counter-terms can
be obtained by the wave function renormalization of the $\phi_{i}$.
Correspondingly, no divergences can appear in the loops of this renormalizable
theory. In other words, the loops cannot induce wave-function renormalization
of $\phi_{i}$. while

$\displaystyle
A_{\phi}=\frac{1}{N}\frac{2^{d-s}(2s-d)\Gamma(s)\sin\left(\frac{1}{2}\pi(d-2s)\right)\Gamma\left(\frac{1}{2}(d-s+1)\right)}{\sqrt{\pi}s\Gamma\left(\frac{d+s}{2}\right)\sin\left(\frac{\pi
s}{2}\right)}+{\cal O}\left(\frac{1}{N^{2}}\right)\,.$ (27)

###  3.2 $\langle\sigma\sigma\rangle$

There are three diagrams that contribute to the $\langle\sigma\sigma\rangle$
correlation function at the next-to-leading order in $1/N$, which we will
label as $C_{\sigma\sigma}^{(a)}$, $a=1,2,3$. Just as in the case of
$\langle\phi\phi\rangle$ two-point function, these diagrams are analogous to
their counterparts in the short-range critical $O(N)$ vector model, and
therefore calculation sequences in these two models parallel each other. We
refer the reader to Goykhman:2019kcj for a recent detailed description of such
a calculation in the $O(N)$ vector model, while here we provide only a brief
outline and a summary of results.

The total values of the anomalous dimension and the propagator amplitude
correction at the next-to-leading order in $1/N$ are obtained by summing the
individual contributions from each of the three diagrams,

$\gamma_{\sigma}=\sum_{a=1}^{3}\gamma_{\sigma}^{(a)}\,,\qquad
A_{\sigma}=\sum_{a=1}^{3}A_{\sigma}^{(a)}\,.$ (28)

The first diagram contributing to $\langle\sigma\sigma\rangle$ is given by

$C_{\sigma\sigma}^{(1)}:$$=\frac{C_{\sigma}\,\mu^{-\delta}}{|x|^{2\Delta_{\sigma}+\delta}}\,\left(\frac{2\gamma_{\sigma}^{(1)}}{\delta}+A_{\sigma}^{(1)}+{\cal
O}(\delta)\right)$$2\Delta_{\sigma}$$2\Delta_{\sigma}$$2\Delta_{\sigma}+\delta$$2\Delta_{\phi}-\eta$$2\Delta_{\phi}+\eta$$2\Delta_{\phi}+\eta$$2\Delta_{\phi}-\eta$$0$$x$

Here an auxiliary regulator $\eta={\cal O}(\delta)$ does not affect the value
of $C_{\sigma\sigma}^{(1)}$ in the $\delta\rightarrow 0$ limit Vasiliev:1981yc
; Vasiliev:1981dg ; Gubser:2017vgc . Choosing $\eta=\delta/2$ makes the
diagram integrable via the uniqueness relation, rendering

$\displaystyle\langle\sigma(x)\sigma(0)\rangle$
$\displaystyle\supset\frac{8}{N}\frac{C_{\phi}^{4}C_{\sigma}^{3}}{|x_{12}|^{2\Delta_{\sigma}}}U\left(\Delta_{\phi}-\frac{\delta}{4},\Delta_{\phi}-\frac{\delta}{4},\Delta_{\sigma}+\frac{\delta}{2}\right)U\left(\frac{d+\delta}{2},\frac{d+\delta}{2},-\delta\right)$
$\displaystyle\times
U\left(\Delta_{\sigma},2\Delta_{\phi}+\frac{\delta}{2},-\frac{\delta}{2}\right)U\left(\Delta_{\sigma},\frac{d+\delta}{2},\frac{d-\delta}{2}-\Delta_{\sigma}\right)\,\frac{1}{(\mu|x|)^{\delta}}\,.$
(29)

Expanding around $\delta=0$ one obtains the corresponding contributions of the
diagram $C_{\sigma\sigma}^{(1)}$ to the $\gamma_{\sigma}$ and
$A_{\sigma}$,888In this paper we denote $n$th derivative of the digamma
function as $\psi^{(n)}(x)$.

$\displaystyle\gamma_{\sigma}^{(1)}$
$\displaystyle=-\frac{1}{N}\,\frac{4\Gamma\left(\frac{s}{2}\right)^{2}\Gamma(d-s)}{\Gamma\left(\frac{d}{2}\right)\Gamma\left(\frac{d-s}{2}\right)^{2}\Gamma\left(s-\frac{d}{2}\right)}\,,$
(30) $\displaystyle A_{\sigma}^{(1)}$
$\displaystyle=\gamma_{\sigma}^{(1)}\left(\psi^{(0)}\left(\frac{d{-}s}{2}\right)-\psi^{(0)}(d{-}s)\psi^{(0)}\left(s{-}\frac{d}{2}\right)+\psi^{(0)}\left(\frac{s}{2}\right)\right)\,.$
(31)

Next, we consider contribution of the following diagram to
$\langle\sigma\sigma\rangle$,

$C_{\sigma\sigma}^{(2)}:$$2\Delta_{\sigma}$$2\Delta_{\sigma}$$2\Delta_{\phi}+\eta$$2\Delta_{\phi}-\eta$$2\Delta_{\phi}+\eta$$2\Delta_{\phi}-\eta$$2\Delta_{\sigma}+\delta/2$$2\Delta_{\sigma}+\delta/2$$2\Delta_{\phi}$$2\Delta_{\phi}$$0$$x$$=\frac{C_{\sigma}\,\mu^{-\delta}}{|x|^{2\Delta_{\sigma}+\delta}}\,\left(\frac{2\gamma_{\sigma}^{(2)}}{\delta}+A_{\sigma}^{(2)}+{\cal
O}(\delta)\right)$

Here we again introduced an auxiliary regulator $\eta={\cal O}(\delta/2)$
without affecting the value of this diagram in the $\delta\rightarrow 0$
limit. We choose $\eta=\delta/2$, which renders some of the vertices unique
and therefore integrable. Integrating over those vertices will result in a
graph of the same topology to $C_{\sigma\sigma}^{(1)}$, which will require
introducing yet again an auxiliary regulator $\eta^{\prime}={\cal O}(\delta)$,
in such a manner that the final answer in the $\delta\rightarrow 0$ limit will
remain unaffected, while the uniqueness of the diagram becomes restored
Vasiliev:1981yc ; Vasiliev:1981dg ; Gubser:2017vgc . At the end, we obtain

$\displaystyle\langle\sigma(x)\sigma(0)\rangle$
$\displaystyle\supset\frac{32}{N}\frac{C_{\phi}^{6}C_{\sigma}^{4}}{|x_{12}|^{2\Delta_{\sigma}}}U\left(\Delta_{\phi}-\frac{\delta}{4},\Delta_{\phi},\Delta_{\sigma}+\frac{\delta}{4}\right)^{2}U\left(d-\frac{3\Delta_{\sigma}}{2}-\frac{\delta}{4},\frac{\Delta_{\sigma}}{2}-\frac{\delta}{4},\Delta_{\sigma}+\frac{\delta}{2}\right)$
(32) $\displaystyle\times
U\left(\frac{d{+}\delta}{2},\frac{d{+}\delta}{2},{-}\delta\right)U\left(2\Delta_{\phi}{+}\frac{\delta}{2},\Delta_{\sigma},{-}\frac{\delta}{2}\right)U\left(\Delta_{\sigma},\frac{d{+}\delta}{2},\frac{d{-}\delta}{2}{-}\Delta_{\sigma}\right)\frac{1}{(\mu|x|)^{\delta}}\,.$

The contributions to anomalous dimension and the propagator amplitude are then
given by

$\displaystyle\gamma_{\sigma}^{(2)}$
$\displaystyle=\frac{1}{N}\,\frac{16\Gamma\left(\frac{s}{2}\right)^{3}\Gamma\left(\frac{d}{2}-s\right)\Gamma(d-s)^{2}\Gamma\left(\frac{3s}{2}-\frac{d}{2}\right)}{\Gamma\left(\frac{d}{2}\right)\Gamma(s)\Gamma\left(d-\frac{3s}{2}\right)\Gamma\left(\frac{d-s}{2}\right)^{3}\Gamma\left(s-\frac{d}{2}\right)^{2}}\,,$
(33) $\displaystyle A_{\sigma}^{(2)}$
$\displaystyle{=}\frac{\gamma_{\sigma}^{(2)}}{4}\,\left(3\psi^{(0)}\left(\frac{d{-}s}{2}\right){+}\psi^{(0)}\left(\frac{3s}{2}{-}\frac{d}{2}\right){+}\psi^{(0)}\left(d{-}\frac{3s}{2}\right){-}2\psi^{(0)}\left(\frac{d}{2}{-}s\right)\right.$
(34)
$\displaystyle{-}\left.2\psi^{(0)}(d{-}s){-}2\left(\psi^{(0)}\left(s{-}\frac{d}{2}\right){+}\psi^{(0)}(s)\right){+}3\psi^{(0)}\left(\frac{s}{2}\right)\right)\,.$

Finally, the third diagram contributing to $\langle\sigma\sigma\rangle$ at the
next-to-leading order in $1/N$ is obtained by incorporating $1/N$ correction
to the $\langle\phi\phi\rangle$ sub-diagram of the leading-order $\phi$
bubble:

$C_{\sigma\sigma}^{(3)}:$$=\frac{C_{\sigma}\,\mu^{-\delta}}{|x|^{2\Delta_{\sigma}+\delta}}\,\left(\frac{2\gamma_{\sigma}^{(3)}}{\delta}+A_{\sigma}^{(3)}+{\cal
O}(\delta)\right)$$0$$2\Delta_{\sigma}$$2\Delta_{\phi}$$2\Delta_{\phi}$$x$$2\Delta_{\sigma}$$2\Delta_{\sigma}+\delta$$2\Delta_{\phi}$$2\Delta_{\phi}$

Such a correction to the $\phi$ propagator sub-diagram has been discussed in
section 3.1, where it was established to be finite. Specifically, we obtain

$\displaystyle\langle\sigma(x)\sigma(0)\rangle$
$\displaystyle\supset\frac{16}{N}\frac{C_{\phi}^{4}C_{\sigma}^{3}}{|x|^{2\Delta_{\sigma}}}U\left(\Delta_{\phi},\Delta_{\phi}+\Delta_{\sigma}+\frac{\delta}{2},-\frac{\delta}{2}\right)U\left(\Delta_{\phi},\frac{d+\delta}{2},\frac{d-\delta}{2}-\Delta_{\phi}\right)$
$\displaystyle
U\left(2\Delta_{\phi}+\frac{\delta}{2},\Delta_{\sigma},-\frac{\delta}{2}\right)U\left(\Delta_{\sigma},\frac{d+\delta}{2},\frac{d-\delta}{2}-\Delta_{\sigma}\right)\frac{1}{(\mu|x|)^{\delta}}\,,$
(35)

expanding which around $\delta=0$ gives

$\displaystyle\gamma_{\sigma}^{(3)}$ $\displaystyle=0\,,$ (36) $\displaystyle
A_{\sigma}^{(3)}$
$\displaystyle{=}\frac{1}{N}\,\frac{2^{d-s+1}(d-2s)\csc\left(\frac{\pi
s}{2}\right)\Gamma(s)\sin\left(\frac{1}{2}\pi(d-2s)\right)\Gamma\left(\frac{1}{2}(d-s+1)\right)}{\sqrt{\pi}s\Gamma\left(\frac{d+s}{2}\right)}\,.$
(37)

Combining (28), (30), (31), (33), (34), (36), (37) we obtain the final answer
for $\gamma_{\sigma}$, $A_{\sigma}$. Our result for $\gamma_{\sigma}$,

$\displaystyle\gamma_{\sigma}$
$\displaystyle{=}-\frac{1}{N}\frac{4\Gamma\left(\frac{s}{2}\right)^{2}\Gamma(d{-}s)}{\Gamma\left(\frac{d}{2}\right)\Gamma(s)\Gamma\left(d{-}\frac{3s}{2}\right)\Gamma\left(\frac{d{-}s}{2}\right)^{3}\Gamma\left(s{-}\frac{d}{2}\right)^{2}}\left(\Gamma(s)\Gamma\left(d{-}\frac{3s}{2}\right)\Gamma\left(\frac{d{-}s}{2}\right)\Gamma\left(s{-}\frac{d}{2}\right)\right.$
$\displaystyle-\left.2\Gamma\left(\frac{s}{2}\right)\Gamma\left(\frac{d}{2}-s\right)\Gamma(d-s)\Gamma\left(\frac{3s}{2}-\frac{d}{2}\right)\right)\,,$
(38)

agrees with Gubser:2017vgc ; Giombi:2019enr , while the expressions we
obtained for the relative corrections $A_{\phi,\sigma}$ to the propagators of
the $\phi$ and $\sigma$ are, to the best of our knowledge, new. While these
are not observables, in combination with the $1/N$ corrections to the
effective cubic interaction vertices (conformal triangles) such as
$\phi\phi\sigma$ and $\sigma\sigma\sigma$, they give amplitudes of the three-
point functions (e.g., $\langle\phi\phi\sigma\rangle$ and
$\langle\sigma\sigma\sigma\rangle$), which determine the OPE coefficients, and
are a part of CFT data.

##  4 Conformal invariance and composite operators

In this section, we will discuss conformal invariance of the long-range fixed
point at the level of two-point correlation functions of composite operators
such as $\sigma^{n}$, $n>1$, and $\sigma\phi$. Recall that in the short-range
critical $O(N)$ vector model composite operators of the kind
$\hat{\sigma}^{n}$, $n>1$999We use $\hat{\sigma}$ to denote the Hubbard-
Stratonovich field in the short-range critical $O(N)$ vector model, while its
counterpart in the long-range model is denoted as $\sigma$. are usually
scaling operators Derkachov:1997gc ; Derkachov:1998js .101010See Ma:1974qh for
an extensive earlier work on the formalism of scaling operators. These
operators, therefore, have a fixed scaling dimension, and further acquire an
anomalous dimension at the fixed point. However, such operators are usually
not conformal primaries in the theory. In fact, since $\hat{\sigma}$ has a
fixed scaling dimension, $\Delta_{\hat{\sigma}}=2$ for any $2<d<4$, it mixes
with other descendent operators of the same dimension, created by replacing
some of the $\hat{\sigma}$ fields with derivatives. The resulting operators
furnish true conformal primaries in the theory Derkachov:1997gc ;
Derkachov:1998js .

The spectrum of long-range CFTs is slightly different. The Hubbard-
Stratonovich field $\sigma$ in such a theory has a leading scaling dimension
$\Delta_{\sigma}=s$, which is generally a non-integer number. It implies
immediately that one cannot create local descendent operators with leading-
order scaling dimensions equal to that of $\sigma^{n}$, by simply replacing
some of the $\sigma$ fields with derivatives. Such composite operators,
therefore, do not mix with any other operators, and are expected to be
conformal primaries by themselves.111111The case of $s=1$ needs a separate
discussion. Below, we will test these statements at the level of two-point
correlators of some composite operators. If the composite operators are indeed
conformal primaries in the long-range CFT, we will find that the two point
function of any composite operator with itself exhibits a power-law scaling
behavior, while its correlators with other operators of different scaling
dimensions are exactly zero.

###  4.1 $\langle\sigma^{2}\sigma^{2}\rangle$

We begin by calculating the two-point correlation function of the composite
operator $\sigma^{2}$. Starting from the assumption that $\sigma^{2}$ is a
scaling operator, as we will explicitly establish below at the next-to-leading
order in $1/N$ expansion, its two-point function is expected to have the form

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

where we separated the amplitude into the leading part $C_{\sigma^{2}}$ and
its relative $1/N$ corrections $A_{\sigma^{2}}$. At the same time,
$\Delta_{\sigma^{2}}$ stands for the leading order scaling dimension, while
$\gamma_{\sigma^{2}}$ is the anomalous dimension contribution. In the
large-$N$ limit the $\langle\sigma^{2}\sigma^{2}\rangle$ propagator is
determined by the diagram

$\sigma(0)^{2}$$\sigma(x)^{2}$$2\Delta_{\sigma^{2}}$$=$$2\Delta_{\sigma}$$2\Delta_{\sigma}$$=\frac{C_{\sigma^{2}}}{|x|^{2\Delta_{\sigma^{2}}}}$$\sigma(0)^{2}$$\sigma(x)^{2}$

from which one easily finds

$\displaystyle
C_{\sigma^{2}}=2C_{\sigma}^{2}\,,\qquad\Delta_{\sigma^{2}}=2\Delta_{\sigma}=2s\,.$
(40)

There are four diagrams that contribute to the
$\langle\sigma^{2}\sigma^{2}\rangle$ propagator at the next-to-leading order
in $1/N$. The first contribution is simply due to the $1/N$ corrections to the
propagators of $\sigma$ in the leading order diagram. We calculated these
corrections in section 3.2, and will denote their total contribution with a
gray blob:

$\sigma(0)^{2}$$\sigma(x)^{2}$

This diagram includes the leading-order $\langle\sigma^{2}\sigma^{2}\rangle$
propagator, as well as some contributions to $\gamma_{\sigma^{2}}$ and
$A_{\sigma^{2}}$,

$\displaystyle\langle\sigma^{2}(x)\sigma^{2}(0)\rangle\supset\frac{C_{\sigma^{2}}(1+A_{\sigma})^{2}}{|x|^{4\Delta_{\sigma}}}\left(1-4\gamma_{\sigma}\log(|x|\mu)\right)\,.$
(41)

Here, and in what follows, we are going to ignore the finite terms such as
$2A_{\sigma}$ in (41), and focus only on the singular terms, that contribute
to the anomalous dimension $\gamma_{\sigma^{2}}$. This will greatly simplify
the calculation, since we can avoid introducing the regulator $\delta$, and
therefore easily take all of the unique integrals. The last logarithmically
divergent integral is simply replaced with
$(2\pi^{\frac{d}{2}}/\Gamma(d/2))\,\log(\mu)$, where $\mu$ is IR mass scale
Goykhman:2020ffn ; Chai:2020hnu .

The second contribution to $\langle\sigma^{2}\sigma^{2}\rangle$ at the next-
to-leading order in $1/N$ is due to insertions of two cubic $\sigma$ vertices
into the leading order diagram:121212To lighten up the notation, we skip
labeling exponents on diagrams in the rest of this subsection.

$\sigma(0)^{2}$$\sigma(x)^{2}$

Evaluating this diagram gives

$\displaystyle\langle\sigma^{2}(x)\sigma^{2}(0)\rangle$
$\displaystyle\supset\frac{128}{N}\frac{C_{\phi}^{6}C_{\sigma}^{5}}{|x|^{2\Delta_{\sigma^{2}}}}U\left(s,\frac{d-s}{2},\frac{d-s}{2}\right)^{2}U\left(\frac{s}{2},s,d-\frac{3s}{2}\right)$
(42) $\displaystyle\times
U\left(\frac{d-s}{2},d-\frac{3s}{2},2s-\frac{d}{2}\right)U\left(s,s,d-2s\right)\frac{4\pi^{d/2}}{\Gamma\left(d/2\right)}\log(|x|\mu)\,.$

The third and fourth diagrams are obtained due to insertion of a single
quartic vertex into the propagator. The corresponding planar diagram is given
by

$\sigma(0)^{2}$$\sigma(x)^{2}$

and contributes

$\displaystyle\langle\sigma^{2}(x)\sigma^{2}(0)\rangle\supset\frac{32}{N}\frac{C_{\phi}^{4}C_{\sigma}^{4}}{|x|^{2\Delta_{\sigma^{2}}}}U\left(s,\frac{d{-}s}{2},\frac{d{-}s}{2}\right)^{2}U\left(\frac{s}{2},\frac{3s}{2},d{-}2s\right)\frac{4\pi^{d/2}}{\Gamma\left(d/2\right)}\log(|x|\mu)\,,$
(43)

while the non-planar diagram

$\sigma(0)^{2}$$\sigma(x)^{2}$

contributes

$\displaystyle\langle\sigma^{2}(x)\sigma^{2}(0)\rangle\supset\frac{16}{N}\frac{C_{\phi}^{4}C_{\sigma}^{4}}{|x|^{2\Delta_{\sigma^{2}}}}U\left(s,\frac{d{-}s}{2},\frac{d{-}s}{2}\right)^{2}U\left(s,s,d{-}2s\right)\frac{4\pi^{d/2}}{\Gamma\left(d/2\right)}\log(|x|\mu)\,.$
(44)

Combining (41), (42), (43) and (44) we obtain the anomalous dimension of
$\sigma^{2}$ at the next-to-leading order in $1/N$,

$\displaystyle\gamma_{\sigma^{2}}=-\frac{1}{N}\frac{4\Gamma\left(\frac{s}{2}\right)^{2}\Gamma(d-s)}{\Gamma\left(\frac{d}{2}\right)\Gamma(s)^{2}\Gamma\left(\frac{3s}{2}\right)\Gamma(d-2s)\Gamma\left(d-\frac{3s}{2}\right)^{2}\Gamma\left(\frac{d-s}{2}\right)^{4}\Gamma\left(s-\frac{d}{2}\right)^{3}}$
$\displaystyle\times\Bigg{(}\Gamma\left(\frac{s}{2}\right)\Gamma(d-s)\Gamma\left(s-\frac{d}{2}\right)\Gamma\left(2s-\frac{d}{2}\right)\Gamma\left(d-\frac{3s}{2}\right)^{2}$
$\displaystyle\times\left(\Gamma\left(\frac{s}{2}\right)\Gamma\left(\frac{3s}{2}\right)\Gamma\left(\frac{d}{2}-s\right)^{2}+2\Gamma(s)^{2}\Gamma\left(\frac{1}{2}(d-3s)\right)\Gamma\left(\frac{d-s}{2}\right)\right)$
$\displaystyle+2\Gamma\left(\frac{3s}{2}\right)\Gamma(d-2s)\Bigg{(}\Gamma(s)^{2}\Gamma\left(d-\frac{3s}{2}\right)^{2}\Gamma\left(\frac{d-s}{2}\right)^{2}\Gamma\left(s-\frac{d}{2}\right)^{2}$
$\displaystyle-2\Gamma\left(\frac{s}{2}\right)\Gamma(s)\Gamma\left(d-\frac{3s}{2}\right)\Gamma\left(\frac{d}{2}-s\right)\Gamma\left(\frac{d-s}{2}\right)\Gamma(d-s)\Gamma\left(\frac{3s}{2}-\frac{d}{2}\right)\Gamma\left(s-\frac{d}{2}\right)$
$\displaystyle-2\Gamma\left(\frac{s}{2}\right)^{2}\Gamma\left(\frac{d}{2}-s\right)^{2}\Gamma(d-s)^{2}\Gamma\left(\frac{3s}{2}-\frac{d}{2}\right)^{2}\Bigg{)}\Bigg{)}$
(45)

A quick consistency check of this expression is given by
$\gamma_{\sigma^{2}}|_{s=d/2}=0$, since at $s=d/2$ the long-range CFT becomes
free. Another check is given by continuity at the long-range–short-range
crossover point $s=s_{\star}$, which demands
$\gamma_{\sigma^{2}}|_{s=s_{\star}}=\gamma_{\hat{\sigma}^{2}}$, where
$\gamma_{\hat{\sigma}^{2}}$ is anomalous dimension of the composite operator
$\hat{\sigma}^{2}$ in short-range $O(N)$ vector model CFT. This crossover will
be discussed in more detail in section 6.

###  4.2 $\langle\sigma^{n}\sigma^{n}\rangle$

The $\langle\sigma^{n}\sigma^{n}\rangle$ propagator of the composite field
$\sigma^{n}$ (where $n\in\mathbb{N}$) at the next-to-leading order in $1/N$ is
determined by the diagrams

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Here we used gray blobs to denote dressed $\langle\sigma\sigma\rangle$ and
$\langle\sigma^{2}\sigma^{2}\rangle$ propagators. The anomalous dimension
$\gamma_{\sigma^{n}}$ is therefore determined only by $\gamma_{\sigma}$ and
$\gamma_{\sigma^{2}}$. Further corrections contribute only at higher orders in
$1/N$. The value of $\gamma_{\sigma^{n}}$ is then found to be,

$\gamma_{\sigma^{n}}=n(2-n)\gamma_{\sigma}+\frac{n(n-1)}{2}\gamma_{\sigma^{2}}$
(46)

The combinatorics involved is simple: we first account for the anomalous
dimension corresponding to $n$ propagators of $\sigma$, the second term then
computes the propagator corrections for $\sigma^{2}$ (there are $n(n-1)/2$
such corrections). To avoid over-counting, we subtract $2\gamma_{\sigma}$ from
each $\sigma^{2}$ correction, which ultimately gives the first term.

###  4.3 The $\langle\sigma^{2}\sigma\rangle$ correlator and conformal
invariance

At the short-range IR fixed point of the $O(N)$ vector model in $2<d<4$
dimensions the composite operator $\hat{\sigma}^{2}$ is in general not a
conformal primary Derkachov:1997gc ; Derkachov:1998js . For instance, in
general $d$ the correlator $\langle\hat{\sigma}^{2}\hat{\sigma}\rangle$ is
non-vanishing, while the scaling dimensions of $\hat{\sigma}$ and
$\hat{\sigma}^{2}$ are manifestly different. However, the leading order
scaling dimensions of the operators $\hat{\sigma}^{2}$ and
$\partial^{2}\hat{\sigma}$ are the same, and therefore due to the non-
vanishing $\langle\hat{\sigma}^{2}\hat{\sigma}\rangle$ these operators can
mix. In fact they do, and result in a primary operator of the form
$\hat{\sigma}^{2}+\alpha\,\partial^{2}\hat{\sigma}$ for a certain
$\alpha={\cal O}(1/\sqrt{N})$ Derkachov:1997gc ; Derkachov:1998js .131313Note,
however, that the mixing coefficient $\alpha$ vanishes for the integer-valued
$d=2,3,4$.

However, in the long-range CFT, due to the non-integer scaling dimension of
$\sigma^{2}$, we do not expect it to mix with any other descendent operator.
In fact, we expect it to be a conformal primary in the theory with $1\leq
d<4$, and for any allowed value of $s$ in the long-range CFT region. One
immediate manifestation of this fact should be borne out in the correlator
$\langle\sigma^{2}\sigma\rangle$, which is expected to vanish. In this
section, we perform this check explicitly, working at the leading order in
$1/N$.

To the leading order, the $\langle\sigma^{2}\sigma\rangle$ correlator is
determined by the kite diagram with a $\phi$ loop in the middle:

$\sigma^{2}$$\sigma$

One way to compute it is to replace the $\sigma^{2}\sigma\sigma$, and
$\sigma\sigma\sigma$ vertices with their respective conformal
triangles:141414It should be noted that we introduce the conformal triangles
to regulate the diagram even though it is finite. However, the diagram is
superficially divergent, and in order to calculate it, we choose to re-write
it in an equivalent form that possesses internal $\sigma$ lines. The diagram
is then regularized by a small shift $\delta$ of the exponents of internal
$\sigma$ propagators, even though the $\delta\rightarrow 0$ limit of such a
diagram is finite.

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

Integrating over the internal vertices, while introducing auxiliary regulators
wherever necessary (such as choosing $\eta=\delta/2$ in the diagram above), we
arrive at the final result given by

$\displaystyle\langle\sigma^{2}(x)\sigma(0)\rangle=\frac{1}{2}(-6Z_{\sigma\sigma\sigma}^{(0)})(-2Z_{\sigma^{2}\sigma\sigma}^{(0)})C_{\sigma^{2}}C_{\sigma}^{3}U\left(\frac{d{-}\gamma_{\sigma^{2}}}{2}{-}s{-}\frac{\delta}{4},s{+}\gamma_{\sigma}{+}\frac{\delta}{4},\frac{d{+}\gamma_{\sigma^{2}}}{2}{-}\gamma_{\sigma}\right)$
$\displaystyle\times
U\left(\frac{d-s}{2},\frac{d-s}{2}-\frac{\delta}{4},s+\frac{\delta}{4}\right)U\left(d-\frac{3s}{2},\frac{3s}{2}+\frac{\delta}{2},-\frac{\delta}{2}\right)U\left(s,\frac{d+s+\delta}{2},\frac{d-3s-\delta}{2}\right)$
$\displaystyle\times
U\left(d-2s,\frac{3s+\delta}{2},\frac{s-\delta}{2}\right)U\left(2s,\frac{d-s+\delta}{2},\frac{d-3s-\delta}{2}\right)\,\frac{\mu^{-\delta}}{|x|^{3s+\delta}}\,.$
(47)

Here the large-$N$ amplitudes of the $\sigma^{2}\sigma\sigma$ and
$\sigma\sigma\sigma$ conformal triangles are given by151515See section 5 for
detailed discussion of conformal triangles in the long-range critical vector
model.

$\displaystyle
Z_{\sigma^{2}\sigma\sigma}^{(0)}=-\frac{1}{2}\frac{C_{\sigma^{2}\sigma\sigma}^{(0)}}{C_{\sigma^{2}}C_{\sigma}^{2}u_{\sigma^{2}\sigma\sigma}^{(0)}}=-\frac{1}{C_{\sigma^{2}}u_{\sigma^{2}\sigma\sigma}^{(0)}},\hskip
28.45274ptZ_{\sigma\sigma\sigma}^{(0)}=\frac{4C_{\phi}^{3}}{3}\,,$ (48)

where we have used the leading-order amplitude of the
$\langle\sigma^{2}\sigma\sigma\rangle$ three-point function

$C_{\sigma^{2}\sigma\sigma}^{(0)}=2C_{\sigma}^{2}\,,$ (49)

as well as the propagator amplitude (40). The expression for
$u_{\sigma^{2}\sigma\sigma}^{(0)}$ is obtained by integrating over the
vertices of the conformal triangle for $\sigma^{2}\sigma\sigma$, and then
keeping only the leading large $N$ terms,161616See Chai:2021uhv for analogous
calculation in the short-range $O(N)$ vector model, where the corresponding
$s^{2}ss$ conformal triangle is calculated at the next-to-leading order in
$1/N$ expansion.

$\displaystyle
u_{\sigma^{2}\sigma\sigma}^{(0)}=\frac{2}{2\gamma_{\sigma}-\gamma_{\sigma^{2}}}\frac{\pi^{3d/2}\Gamma\left(\frac{d}{2}-2s\right)\Gamma\left(2s-\frac{d}{2}\right)}{\Gamma\left(\frac{d}{2}\right)\Gamma(2s)\Gamma(d-2s)}\,.$
(50)

The factor of $2\gamma_{\sigma}-\gamma_{\sigma^{2}}$ is cancelled precisely by
a leading large $N$ expansion of the only factor in (47) whose dependence on
the anomalous dimensions we retain for the sake of regularizing the
corresponding $U$ function, namely

$\displaystyle
U\left(\frac{d-\gamma_{\sigma^{2}}}{2}-s-\frac{\delta}{4},s+\gamma_{\sigma}+\frac{\delta}{4},\frac{d+\gamma_{\sigma^{2}}}{2}-\gamma_{\sigma}\right)=\frac{2\pi^{d/2}}{\left(2\gamma_{\sigma}-\gamma_{\sigma^{2}}\right)\Gamma\left(\frac{d}{2}\right)}+\mathcal{O}(N^{0})\,.$
(51)

Combining everything together and taking the limit $\delta\rightarrow 0$, we
obtain

$\displaystyle\langle\sigma^{2}(x)\sigma(0)\rangle=0+\mathcal{O}\left(\frac{1}{N^{3/2}}\right)\,.$
(52)

The vanishing of the leading order correlator between $\sigma$ and
$\sigma^{2}$ provides further evidence that $\sigma^{2}$ is indeed a primary
at the long-range fixed point, and that the fixed point in fact enjoys the
full conformal symmetry. As an additional check, one can compute the first
sub-leading order correction in the $1/N$ expansion, and show that it also
vanishes for any $1\leq d<4$, and the corresponding allowed values of $s$ for
the long-range CFT region.

###  4.4 $\langle\sigma\phi\,\sigma\phi\rangle$

In this section we will discuss the operator $\sigma\phi$, which will play an
important role in section 7, where we will develop the $O(N)$ generalization
of the short-range–long-range duality proposal of Behan:2017dwr . This
composite operator has two constituents, one of which, $\sigma$, possesses an
anomalous dimension. However, interestingly enough, the composite operator
$\sigma\phi$ itself does not acquire anomalous dimension to all orders in
$1/N$. This exact result follows immediately from the $\phi$ e.o.m. due to the
action (17), relating dimension of $\sigma\phi$ to the dimension of $\phi$:

$\sigma\phi=-\sqrt{N}\,C(s)\,\int d^{d}y\,\frac{\phi(y)}{|x-y|^{d+s}}\,.$ (53)

Using the $\langle\phi\phi\rangle$ two-point function (24), while taking into
account $\gamma_{\phi}=0$, we obtain

$\langle\sigma\phi(x)\sigma\phi(0)\rangle=N\,C(s)^{2}C_{\phi}\,(1+A_{\phi})\,\pi^{d}A\left(\frac{d-s}{2}\right)A\left(\frac{d+s}{2}\right)\,\frac{1}{|x|^{d+s}}\,,$
(54)

where we also took advantage of the identity (19). We will be ignoring the
$1/N$ corrections to the propagator amplitude, while noticing that the scaling
dimension of the composite operator $\sigma\phi$ is given by

$\Delta_{\sigma\phi}=\frac{d+s}{2}\,,$ (55)

exactly to all orders in $1/N$, despite the fact that the constituent $\sigma$
has a non-trivial anomalous dimension.

As a consistency check, the relation (55) can be quickly verified at the next-
to-leading order in $1/N$ expansion. The first contribution is obtained by
dressing of the internal $\sigma$ and $\phi$ lines of the leading-order
diagram:

$2(\Delta_{\sigma}+\gamma_{\sigma})$$2\Delta_{\phi}$

Its contribution to the anomalous dimension is then given by

$\gamma_{\sigma\phi}^{(1)}=\gamma_{\sigma}\,.$ (56)

Next, we have the contribution due to the effective $\sigma\sigma\sigma$
vertex inserted into the leading-order diagram:

$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\sigma}$$2\Delta_{\sigma}$$2\Delta_{\phi}$$2\Delta_{\phi}$$2\Delta_{\sigma}$

This diagram is divergent. Since we are only interested in anomalous
dimensions, we can regularize it by simply extracting the logarithmically
divergent term of the last integral.171717This can be done by using the
relation  $\int d^{d}y\,\frac{1}{|y|^{d}|x-y|^{d}}\supset
2\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d}{2}\right)}\,\log(\mu)\,\frac{1}{|x|^{d}}\,.$
(57) As a result we obtain contribution to the anomalous dimension given by

$\gamma_{\sigma\phi}^{(2)}=-\frac{16}{N}C_{\phi}^{4}C_{\sigma}^{2}\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d}{2}\right)}U\left(\frac{d-s}{2},\frac{d-s}{2},s\right)^{2}U\left(s,\frac{s}{2},d-\frac{3s}{2}\right)\,.$
(58)

Finally, we have contribution from the following diagram:

$2\Delta_{\phi}$$2\Delta_{\sigma}$$2\Delta_{\sigma}$$2\Delta_{\phi}$$2\Delta_{\phi}$

Regularizing it using (57) we obtain

$\gamma_{\sigma\phi}^{(3)}=-\frac{8}{N}C_{\phi}^{2}C_{\sigma}\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d}{2}\right)}U\left(\frac{d-s}{2},\frac{d-s}{2},s\right)\,.$
(59)

Combining (56), (58), (59), while taking into account $\gamma_{\sigma}$ given
by (38),181818Alternatively, this calculation can be viewed as a consistency
check for the value of $\gamma_{\sigma}$. we arrive at

$\gamma_{\sigma\phi}=\gamma_{\sigma\phi}^{(1)}+\gamma_{\sigma\phi}^{(2)}+\gamma_{\sigma\phi}^{(3)}=0+{\cal
O}\left(\frac{1}{N^{2}}\right)\,,$ (60)

in agreement with the general argument given above.

##  5 OPE coefficients

We now proceed to calculating various OPE coefficients in the long-range
$O(N)$ critical vector model. In section 5.1 we calculate the
$\langle\phi\phi\sigma\rangle$ three-point function, and determine its
amplitude at the next-to-leading order in $1/N$ expansion. Such a calculation
requires knowing the next-to-leading order corrections to the $\phi\phi\sigma$
interaction vertex. We calculate this non-local effective vertex using the
background field method Goykhman:2020ffn . In section 5.2 we calculate the
$\langle\sigma\sigma\sigma\rangle$ three-point function at the leading order
in $1/N$ expansion.

###  5.1 $\langle\phi\phi\sigma\rangle$

In this section we will calculate the $\langle\phi\phi\sigma\rangle$
correlation function at the next-to-leading order in $1/N$ expansion. Due to
conformal symmetry, this three-point function has the form

$\langle\phi(x_{1})\phi(x_{2})\sigma(x_{3})\rangle=\frac{C_{\phi\phi\sigma}^{(0)}(1+\delta
C_{\phi\phi\sigma})\mu^{-\gamma_{\sigma}}}{(|x_{13}||x_{23}|)^{\Delta_{\sigma}+\gamma_{\sigma}}|x_{12}|^{2\Delta_{\phi}-\Delta_{\sigma}-\gamma_{\sigma}}}\,,$
(61)

where $\mu$ is an arbitrary mass scale. The amplitude of the three-point
function (61) was separated into the leading order factor of
$C_{\phi\phi\sigma}^{(0)}$ and the relative $1/N$ corrections to it, $\delta
C_{\phi\phi\sigma}={\cal O}(1/N)$. We also took into account that the
anomalous dimension of $\phi$ vanishes, and therefore its total dimension is
given by $\Delta_{\phi}$.

Carrying out the calculation of (61) in this section, we will reproduce the
anomalous dimension $\gamma_{\sigma}$, and derive the leading order amplitude
$C_{\phi\phi\sigma}^{(0)}$ and the next-to-leading order contribution to
$\delta C_{\phi\phi\sigma}$. In the process, we will introduce and calculate
the $\phi\phi\sigma$ conformal triangle, at the next-to-leading order in
$1/N$, representing the corresponding non-local vertex in the effective
action.

It is customary to use conventions in which position-space propagators of the
fields in the considered correlation function are normalized to unity. This
can be achieved by the corresponding rescaling of the fields,

$\phi\rightarrow\sqrt{C_{\phi}(1+A_{\phi})}\,\phi\,,\qquad\sigma\rightarrow\sqrt{C_{\sigma}(1+A_{\sigma})}\,\sigma\,.$
(62)

In particular, such a normalization provides a renormalization scheme choice
related to the freedom of redefining the scale $\mu$ by a constant factor. We
will denote the amplitudes of the correlation functions with so-normalized
fields with a bar, e.g., $\bar{C}_{\phi\phi\sigma}^{(0)}$ and
$\delta\bar{C}_{\phi\phi\sigma}$ for the three-point function (61) in which
all of the fields have been normalized according to (62).

The leading order amplitude $C_{\phi\phi s}^{(0)}$ can be easily found from
the tree-level diagram

$\sigma(x_{3})$$\phi(x_{1})$$\phi(x_{2})$$=-\frac{2}{\sqrt{N}}C_{\phi}^{2}C_{\sigma}U\left(\Delta_{\phi},\Delta_{\phi},\Delta_{\sigma}\right)\frac{1}{(|x_{13}||x_{23}|)^{\Delta_{\sigma}}|x_{12}|^{2\Delta_{\phi}-\Delta_{\sigma}}}$

For the fields normalized according to (62) we obtain the leading-order
amplitude

$\displaystyle\bar{C}_{\phi\phi\sigma}^{(0)}$
$\displaystyle=-\frac{2}{\sqrt{N}}C_{\phi}C_{\sigma}^{\frac{1}{2}}U\left(\Delta_{\phi},\Delta_{\phi},\Delta_{\sigma}\right)$
(63)
$\displaystyle=-\frac{1}{\sqrt{N}}\frac{\Gamma\left(\frac{s}{2}\right)^{2}\Gamma\left(\frac{d}{2}-s\right)\sqrt{(d-2s)\Gamma(s)\sin\left(\frac{1}{2}\pi(d-2s)\right)\Gamma(d-s)}}{\sqrt{\pi}\Gamma(s)\Gamma\left(\frac{d-s}{2}\right)^{2}}\,.$

To determine correction $\delta C_{\phi\phi\sigma}$ to the amplitude of the
$\langle\phi\phi\sigma\rangle$ three-point function, we begin by calculating
the corresponding $\phi\phi\sigma$ conformal triangle (see Polyakov:1970xd for
the original discussion of conformal triangles in CFTs). In terms of the
effective action, such a conformal triangle gives a diagrammatic
representation of the non-local interaction term

$S_{\textrm{eff}}\supset\frac{Z_{\phi\phi\sigma}}{\sqrt{N}}\,\mu^{\gamma_{\sigma}}\int
d^{d}x_{1,2,3}\,\frac{\phi(x_{1})\phi(x_{2})\sigma(x_{3})}{(|x_{13}||x_{23}|)^{2\alpha}|x_{12}|^{2\beta}}\,,$
(64)

where we denoted

$\alpha=\frac{d-s-\gamma_{\sigma}}{2}\,,\qquad\beta=s+\frac{\gamma_{\sigma}}{2}\,.$
(65)

The amplitude of the conformal triangle admits the $1/N$ expansion,
$Z_{\phi\phi\sigma}=Z_{\phi\phi\sigma}^{(0)}(1+\delta Z_{\phi\phi\sigma})$.
Here the leading order amplitude $Z_{\phi\phi\sigma}^{(0)}={\cal O}(1)$ is
constrained by the requirement that the classical interaction term
$\frac{1}{\sqrt{N}}\phi^{2}\sigma$ is reproduced in the large-$N$ limit.
Incorporating the next-to-leading order correction, and taking into account
the vertex counter-term contribution, we obtain the following diagrammatic
equation for the $\phi\phi\sigma$ conformal triangle:

(66)

The first term in the r.h.s. of (66) represents the tree-level contribution of
the leading-order $\phi^{2}\sigma$ interaction vertex, while the second term
stands for the counter-term contribution due to the wave-function
renormalization of the field $\sigma$. The last two terms in the r.h.s. of
(66) originate due to loop corrections to the $\phi^{2}\sigma$ vertex at the
next-to-leading order in $1/N$. In the latter diagrams, we have adjusted the
scaling dimension of the corresponding graphs by a small shift $\delta$ by
regularizing the internal $\sigma$ lines. At the end of the calculation we
will take the limit $\delta\rightarrow 0$. We will also observe explicitly
that the total $1/\delta$ pole of the second and third diagrams is cancelled
out precisely by the second (counter-term) diagram. Inversely, imposing a
cancellation of divergencies reproduces the correct value for the anomalous
dimension $\gamma_{\sigma}$ given by (38).

The $\phi\phi\sigma$ conformal triangle can be used directly to calculate
correlation functions and extract CFT data. For instance, in the previous
paragraph, we outlined how the anomalous dimension $\gamma_{\sigma}$ can be
found from such a calculation. Importantly, OPE coefficients can be calculated
as well. To this end, one needs to attach to the conformal triangle the full
(dressed) propagators, and integrate over the internal unique vertices: such a
procedure is a direct generalization of using the Feynman rule for a tree-
level vertex and integrating over an insertion of the vertex. In particular,
to calculate the $\langle\phi\phi\sigma\rangle$ three-point function we
proceed as follows:

(67)

where we denoted the factor obtained from integration over three unique
vertices of the conformal triangle as

$\displaystyle{\cal U}_{\phi\phi\sigma}$
$\displaystyle=U\left(\frac{d-s-\gamma_{\sigma}}{2},\frac{d-s-\gamma_{\sigma}}{2},s+\gamma_{\sigma}\right)U\left(\frac{d-s}{2},\frac{d-\gamma_{\sigma}}{2},\frac{s+\gamma_{\sigma}}{2}\right)$
$\displaystyle\times
U\left(\frac{d-s}{2},\frac{d-s-\gamma_{\sigma}}{2},s+\frac{\gamma_{\sigma}}{2}\right)\,.$
(68)

Expanding (67) in $1/N$ to the next-to-leading order, and comparing the result
with (61) we obtain

$\displaystyle\bar{C}_{\phi\phi\sigma}^{(0)}$
$\displaystyle=-\frac{2Z_{\phi\phi\sigma}^{(0)}}{\sqrt{N}}\,C_{\phi}\,C_{\sigma}^{\frac{1}{2}}\,{\cal
U}_{\phi\phi\sigma}^{(0)}\,,$ (69)
$\displaystyle\delta\bar{C}_{\phi\phi\sigma}^{(0)}$ $\displaystyle=\delta
Z_{\phi\phi\sigma}+\delta{\cal
U}_{\phi\phi\sigma}+A_{\phi}+\frac{A_{\sigma}}{2}\,.$ (70)

where we denoted the $1/N$ expansion of (68) as

$\displaystyle{\cal U}_{\phi\phi\sigma}$ $\displaystyle={\cal
U}_{\phi\phi\sigma}^{(0)}(1+\delta{\cal U}_{\phi\phi\sigma})\,,$
$\displaystyle{\cal U}_{\phi\phi\sigma}^{(0)}$
$\displaystyle=\frac{2\pi^{\frac{3d}{2}}\Gamma\left(\frac{s}{2}\right)^{4}\Gamma\left(\frac{d}{2}-s\right)^{2}}{\Gamma\left(\frac{d}{2}\right)\Gamma(s)^{2}\Gamma\left(\frac{d-s}{2}\right)^{4}\,\gamma_{\sigma}}\,,$
(71) $\displaystyle\delta{\cal U}_{\phi\phi\sigma}$
$\displaystyle{=}\frac{\gamma_{\sigma}}{2}\left(2\psi^{(0)}\left(\frac{d{-}s}{2}\right){-}3\psi^{(0)}\left(\frac{d}{2}{-}s\right){+}\psi^{(0)}\left(\frac{d}{2}\right){+}2\psi^{(0)}\left(\frac{s}{2}\right){-}3\psi^{(0)}(s){-}\gamma\right)\,.$

Using (63), (69) we can solve for the leading-order amplitude of the conformal
triangle

$Z_{\phi\phi\sigma}^{(0)}=\frac{U\left(\Delta_{\phi},\Delta_{\phi},\Delta_{\sigma}\right)}{{\cal
U}_{\phi\phi\sigma}^{(0)}}=\frac{\Gamma\left(\frac{d}{2}\right)\Gamma(s)\Gamma\left(\frac{d-s}{2}\right)^{2}\,\gamma_{\sigma}}{2\pi^{d}\,\Gamma\left(\frac{s}{2}\right)^{2}\Gamma\left(\frac{d}{2}-s\right)}\,.$
(72)

At the same time, (70) gives a prescription to calculate the $1/N$ correction
to the OPE coefficient. To finish that calculation we need to determine first
the next-to-leading correction $\delta Z_{\phi\phi\sigma}$ to the
$\phi\phi\sigma$ conformal triangle.

In Goykhman:2020ffn it was proposed to use the background field method to
calculate conformal triangles. Such a method can be applied for the purpose of
determining the values of $Z_{\phi\phi\sigma}^{(0)}$, $\delta
Z_{\phi\phi\sigma}$ in our case as well. Following Goykhman:2020ffn , we set
the field $\sigma$ to a non-dynamical background value
$\sigma\equiv\bar{\sigma}$, and attach the full $\phi$ propagators to each
term on both sides of (66). The resulting diagrammatic equation for the
propagator $\langle\phi\phi\rangle|_{\bar{\sigma}}$ in the $\bar{\sigma}$
background is given by

(73)

Every diagram in (73) can be readily calculated simply by applying the
propagator merging relation, while taking the integrals starting from the
topmost vertex. In particular, the l.h.s. of (73) gives

$\displaystyle\textrm{l.h.s. \;\; of\;\;\;\;}(\ref{Dressed scalar propagator
in s background
vm})\textrm{\;\;\;\;=\;\;}-\frac{2}{\sqrt{N}}\,Z_{\phi\phi\sigma}\,(C_{\phi}(1+A_{\phi}))^{2}\,{\cal
U}_{\phi\phi\sigma}\,\frac{\bar{\sigma}\,\mu^{\gamma_{\sigma}}}{|x_{12}|^{d-2s-\gamma_{\sigma}}}\,,$
(74)

while we denote the four terms contributing on the r.h.s. of (73) as

$\textrm{r.h.s. \;\; of\;\;\;\;}(\ref{Dressed scalar propagator in s
background vm})\textrm{\;\;\;\;=\;\;}v_{0}+v_{\textrm{c.t.}}+v_{1}+v_{2}\,.$
(75)

Here the tree level diagram and the vertex counter-term contribute

$\displaystyle v_{0}$
$\displaystyle=-\frac{2}{\sqrt{N}}\,(C_{\phi}(1+A_{\phi}))^{2}\,U\left(\frac{d-s}{2},\frac{d-s}{2},s\right)\,\frac{\bar{\sigma}}{|x_{12}|^{d-2s}}\,,$
(76) $\displaystyle v_{\textrm{c.t.}}$
$\displaystyle=-\frac{2}{\sqrt{N}}\,(C_{\phi}(1+A_{\phi}))^{2}\,U\left(\frac{d-s}{2},\frac{d-s}{2},s\right)\frac{\gamma_{\sigma}}{\delta}\,\frac{\bar{\sigma}\,\mu^{\gamma_{\sigma}}}{|x_{12}|^{d-2s-\gamma_{\sigma}}}\,.$
(77)

At the same time, for the vertex correction diagrams in the second line of
(73) we obtain

$\displaystyle v_{1}$
$\displaystyle=\left(-\frac{2}{\sqrt{N}}\right)^{3}C_{\phi}^{4}C_{\sigma}(1+A_{\phi})^{2}U\left(\frac{d-s}{2},\frac{d-s}{2},s\right)U\left(\frac{d-s}{2},\frac{d+\delta}{2},\frac{s-\delta}{2}\right)$
$\displaystyle\times
U\left(\frac{d-s}{2},\frac{d-s+\delta}{2},s-\frac{\delta}{2}\right)\,\frac{\bar{\sigma}\,\mu^{\gamma_{\sigma}}}{|x_{12}|^{d-2s+\delta}}\,,$
(78) $\displaystyle v_{2}$
$\displaystyle=4C_{\phi}^{2}C_{\sigma}U\left(d{-}\frac{3s}{2},s{+}\frac{\delta}{4},\frac{s}{2}{-}\frac{\delta}{4}\right)U\left(s{+}\frac{\delta}{4},\frac{d{-}s}{2}{+}\frac{\delta}{4},\frac{d{-}s{-}\delta}{2}\right)\,v_{1}\,.$
(79)

Comparing (74), (75) we first of all cancel the common factor of
$-\frac{2}{\sqrt{N}}\,(C_{\phi}(1+A_{\phi}))^{2}\,\bar{\sigma}$. Matching the
leading order terms, we reproduce $Z_{\phi\phi\sigma}^{(0)}$ given by (72).
Expanding the sub-leading contributions (5.1), (79) around $\delta=0$ we
observe that the $1/\delta$ poles are exactly cancelled out by the counter-
term (77) for $\gamma_{\sigma}$ given by (38). At the same time, the anomalous
dimension $\log|x_{12}|$ term in (5.1), (79) matches its counterpart on the
l.h.s. (74) of the conformal triangle equation. Finally, matching the finite
terms on both sides of this equation, we obtain the next-to-leading order
correction to the conformal triangle amplitude

$\displaystyle\delta Z_{\phi\phi\sigma}$
$\displaystyle=-\frac{1}{N}\,\frac{4\Gamma\left(\frac{s}{2}\right)^{2}\Gamma(d-s)}{\Gamma\left(\frac{d}{2}\right)\Gamma(s)\Gamma\left(d-\frac{3s}{2}\right)\Gamma\left(\frac{d-s}{2}\right)^{3}\Gamma\left(s-\frac{d}{2}\right)^{2}}$
(80)
$\displaystyle\times\left(\Gamma(s)\Gamma\left(d{-}\frac{3s}{2}\right)\Gamma\left(\frac{d{-}s}{2}\right)\Gamma\left(s{-}\frac{d}{2}\right){-}3\Gamma\left(\frac{s}{2}\right)\Gamma\left(\frac{d}{2}{-}s\right)\Gamma(d{-}s)\Gamma\left(\frac{3s}{2}{-}\frac{d}{2}\right)\right)$
$\displaystyle\times\left(-\psi^{(0)}\left(\frac{d-s}{2}\right)+\psi^{(0)}\left(\frac{d}{2}-s\right)-\psi^{(0)}\left(\frac{s}{2}\right)+\psi^{(0)}(s)\right)+{\cal
O}\left(\frac{1}{N^{2}}\right)\,.$

With all the ingredients in place, we can calculate the $1/N$ correction to
the amplitude of the $\langle\phi\phi\sigma\rangle$ three-point function (70).
Our result for $\delta C_{\phi\phi\sigma}$ satisfies several consistency
checks. When $s=\frac{d}{2}$ the long-range fixed point becomes free, and
consequently $\delta C_{\phi\phi\sigma}|_{s=d/2}=0$, as can be established by
substituting $\sigma\sim\phi^{2}$ and performing Wick contractions. This
agrees with the $s\rightarrow d/2$ limit of (70).

When $s=1$, and $2<d<4$, the UV fixed point of the long-range $O(N)$ vector
model was argued in Giombi:2019enr to be critically equivalent to the IR fixed
point of an interacting ‘mixed $\sigma\phi$ theory’ with local kinetic terms
for $\phi$ and $\sigma$, cubic interaction $\phi^{2}\sigma$ and quartic
interaction $\sigma^{4}$. The critical coupling in the latter model,
calculated at the leading order in the $\epsilon$-expansion performed around
$d=4$ dimensions, gives Giombi:2019enr

$g_{1}^{\star}=8\pi\,\sqrt{\frac{2\epsilon}{N-32}}+{\cal
O}(\epsilon^{3/2})\,.$ (81)

This implies that in the ‘mixed $\sigma\phi$’ theory,

$\delta\bar{C}_{\phi\phi\sigma}|_{s=1,d=4-\epsilon}=\frac{16}{N}+{\cal
O}(\epsilon^{3/2},1/N^{2})\,,$ (82)

which agrees with (70) for $s=1$ expanded in $d=4-\epsilon$ dimensions.

###  5.2 $\langle\sigma\sigma\sigma\rangle$

In this section we will calculate the $\langle\sigma\sigma\sigma\rangle$
three-point function at the leading order in the $1/N$ expansion. Expanding
the trace log in the large-$N$ effective action for $\sigma$ (18) to ${\cal
O}(1/N^{3/2})$ we obtain the following cubic term

$S_{\textrm{eff}}\supset-\frac{1}{3!}\left(-\frac{2}{\sqrt{N}}\right)^{3}\,C_{\phi}^{3}\int
d^{d}x_{1,2,3}\frac{\sigma(x_{1})\sigma(x_{2})\sigma(x_{3})}{(|x_{12}||x_{13}||x_{23})^{d-s}}\,.$
(83)

Such a non-local cubic interaction vertex can be represented diagrammatically
using the $\sigma\sigma\sigma$ conformal-triangle. The corresponding Feynman
rule is then given by

$d-s$$d-s$$d-s$$\sigma(x_{1})$$\sigma(x_{2})$$\sigma(x_{3})$$=-\frac{6Z_{\sigma\sigma\sigma}}{\sqrt{N}}\,\frac{\sigma(x_{1})\sigma(x_{2})\sigma(x_{3})}{(|x_{12}||x_{13}||x_{23})^{d-s}}$

Here the leading order amplitude of the conformal triangle is given by

$Z_{\sigma\sigma\sigma}^{(0)}=\frac{4C_{\phi}^{3}}{3}\,.$ (84)

Attaching $\sigma$ propagators to the conformal triangle and integrating over
three unique vertices we obtain the three-point function for normalized
$\sigma$,

$\langle\sigma(x_{1})\sigma(x_{2})\sigma(x_{3})\rangle\Bigg{|}_{\textrm{normalized}}=\frac{\bar{C}_{\sigma\sigma\sigma}^{(0)}}{(|x_{12}||x_{13}||x_{23}|)^{s}}\,,$
(85)

where we denoted the normalized amplitude as

$\displaystyle\bar{C}_{\sigma\sigma\sigma}^{(0)}$
$\displaystyle=-\frac{8}{\sqrt{N}}C_{\phi}^{3}C_{\sigma}^{\frac{3}{2}}U\left(\frac{d-s}{2},\frac{d-s}{2},s\right)^{2}U\left(s,\frac{s}{2},d-\frac{3s}{2}\right)$
(86)
$\displaystyle=\frac{2^{d-2s}(d-2s)\sin\left(\frac{1}{2}\pi(d-2s)\right)\Gamma\left(\frac{d}{2}-s\right)^{2}\Gamma\left(\frac{1}{2}(d-s+1)\right)\Gamma\left(\frac{3s}{2}-\frac{d}{2}\right)}{\pi\Gamma\left(\frac{s+1}{2}\right)\Gamma\left(d-\frac{3s}{2}\right)}\hat{C}_{\phi\phi\sigma}^{(0)}\,.$

A simple consistency check of this result is
$\bar{C}_{\sigma\sigma\sigma}^{(0)}|_{s=1,d=4-\epsilon}=0+{\cal
O}(\epsilon^{\frac{3}{2}})$, in agreement with the behavior of its counterpart
in the dual model Giombi:2019enr .

##  6 Continuity of CFT data across long-range–short-range crossover

In previous sections we derived various CFT data for the long-range conformal
fixed point for a general exponent $s$ and space-time dimension $d$. At the
same time, we know that when $s=2$ the long-range CFT is classically
equivalent to the short-range CFT; this statement is evident by matching the
actions of these two models (for instance, in momentum space). Therefore one
might wonder if taking the limit $s\rightarrow 2$ in the expressions for the
long-range CFT data obtained above, one would recover the known CFT data of
the short-range $O(N)$ vector model. As reviewed in Introduction, it is well
known that such a naive check would not fulfill that expectation; and the
crossover between short-range and long-range CFTs in fact happens at the value
$s=s_{\star}<2$ Sak1973 ; PhysRevB154344 . Here
$s_{\star}=2-2\gamma_{\hat{\phi}}$, where $\gamma_{\hat{\phi}}$ is the
anomalous dimension of $\hat{\phi}$ in the short-range model. The value of
$s_{\star}$ is defined by requiring a continuity of CFT data across the long-
range to the short-range transition.191919Note that the definition of the
crossover point is independent of the value of $N$ (while the value of
$s_{\star}$ is dependent on $N$), and can be easily generalized to the $O(N)$
model. In this subsection, we provide explicit evidence for the existence of a
continuous transition of all CFT data obtained above using the $1/N$
expansion, 202020Analogous calculation in the perturbative
$\epsilon$-expansion, for $\epsilon=2s-d$, was first carried out in Sak1973 to
order $\epsilon^{2}$. namely the scaling dimensions of $\sigma$, $\sigma^{2}$,
$\sigma\phi$, and the OPE coefficients $C_{\phi\phi\sigma}$, and
$C_{\sigma\sigma\sigma}$, at the crossover point $s=s_{\star}$. This is a non-
trivial consistency check of our long-range CFT data computed near a strongly-
coupled point $s=s_{\star}$ in general $d$, and at the next-to-leading order
in $1/N$.

###  6.1 Continuity of dimensions of $\sigma$, $\sigma^{2}$, and $\sigma\phi$

Before proceeding onto the calculations in this section, we make the following
simple observations regarding the leading order behaviour of the scaling
dimensions and amplitudes of $\phi$ and $\sigma$,

$\displaystyle\begin{split}\Delta_{\phi}|_{s\rightarrow
2}&=\Delta_{\hat{\phi}}=\frac{d}{2}-1\,,\qquad\Delta_{\sigma}|_{s\rightarrow
2}=\Delta_{\hat{\sigma}}=2\,,\\\ C_{\phi}|_{s\rightarrow
2}&=C_{\hat{\phi}}=\frac{\Gamma\left(\frac{d}{2}-1\right)}{4\pi^{\frac{d}{2}}}\,,\;\;C_{\sigma}|_{s\rightarrow
2}=C_{\hat{\sigma}}=\frac{2^{d}\Gamma\left(\frac{d-1}{2}\right)\sin\left(\frac{\pi
d}{2}\right)}{\pi^{\frac{3}{2}}\Gamma\left(\frac{d}{2}-2\right)}\,,\end{split}$
(87)

To illustrate the continuity of scaling dimension of the Hubbard-Stratonovich
field $\sigma$ across the crossover point $s_{\star}$ we will utilize the
derivation of the anomalous dimension $\gamma_{\sigma}$ reviewed in section
3.2. The total scaling dimension of $\sigma$ is given by
$\Delta_{\sigma}+\gamma_{\sigma}$, where $\Delta_{\sigma}=s$. Notice that at
the crossover point $s_{\star}=2-2\gamma_{\hat{\phi}}$, $\Delta_{\sigma}$ gets
split into the sum of two parts, $\Delta_{\hat{\sigma}}=2$, that equals to the
leading order contribution to scaling dimension of $\hat{\sigma}$, and the
sub-leading term $-2\gamma_{\hat{\phi}}$. For future purposes, let us write
down this rearrangement of total scaling dimension of $\sigma$ in the long-
range CFT at $s_{\star}$ as

$(\Delta_{\sigma}+\gamma_{\sigma})|_{s\rightarrow
s_{\star}}=\Delta_{\hat{\sigma}}+(\gamma_{\sigma}|_{s\rightarrow
s_{\star}}-2\gamma_{\hat{\phi}})\,.$ (88)

Notice that terms in brackets in r.h.s. of (88) are sub-leading in $1/N$, and
therefore taking the limit $s\rightarrow s_{\star}=2+{\cal O}(1/N)$ in those
terms (more precisely, in $\gamma_{\sigma}$, since $\gamma_{\hat{\phi}}$ does
not depend on $s$) needs to be replaced with taking the limit $s\rightarrow
2$. In order for the scaling dimension of the $\sigma$ field to be continuous
at the crossover point $s_{\star}$,

$(\Delta_{\sigma}+\gamma_{\sigma})|_{s\rightarrow
s_{\star}}=\Delta_{\hat{\sigma}}+\gamma_{\hat{\sigma}}\,,$ (89)

it is then required that

$\gamma_{\sigma}|_{s\rightarrow
2}=\gamma_{\hat{\sigma}}+2\gamma_{\hat{\phi}}\,.$ (90)

To verify that the above expression is true, we notice that among the three
diagrams contributing to $\langle\sigma\sigma\rangle$ at the next-to-leading
order in $1/N$ (discussed in section 3.2), two of them, namely
$C^{(1,2)}_{\sigma\sigma}$, reproduce the corresponding diagrams in the short-
range critical vector model when $s\rightarrow 2$, i.e.
$C^{(1,2)}_{\sigma\sigma}|_{s\rightarrow
2}=C^{(1,2)}_{\hat{\sigma}\hat{\sigma}}$. However, the third diagram,
$C^{(3)}_{\sigma\sigma}$, does not reduce to its short-range counterpart
$C^{(3)}_{\hat{\sigma}\hat{\sigma}}$ in that limit. Its contribution to
$\gamma_{\sigma}$ is in fact absent, unlike the contribution of
$C^{(3)}_{\hat{\sigma}\hat{\sigma}}$ to $\gamma_{\hat{\sigma}}$. To find the
latter, we consider $C^{(3)}_{\hat{\sigma}\hat{\sigma}}$ diagram
explicitly:212121See Goykhman:2019kcj for a recent detailed calculation of
anomalous dimension of the Hubbard-Stratonovich field.

$C^{(3)}_{\hat{\sigma}\hat{\sigma}}=$$2\Delta_{\hat{\sigma}}$$2\Delta_{\hat{\phi}}$$2\Delta_{\hat{\phi}}$$2\Delta_{\hat{\sigma}}$$2\Delta_{\hat{\sigma}}+\delta$$2\Delta_{\hat{\phi}}$$2\Delta_{\hat{\phi}}$

As pointed out above, this diagram is divergent in the short-range model, but
its counterpart in the long-range model is finite. Importantly, the divergent
behavior of this graph can be traced back to its sub-diagram, representing the
$1/N$ correction to the propagator of the field $\hat{\phi}$. However, as
shown in section 3.1, the one-loop correction to the propagator of $\phi$ is
finite in the long-range CFT, consistently with the expectation that scaling
dimension of the field $\phi$ does not receive anomalous contributions.

We regularize $C^{(3)}_{\hat{\sigma}\hat{\sigma}}$ by adding a small shift
$\delta$ to the exponent of the internal line of the Hubbard-Stratonovich
field $\hat{\sigma}$, obtaining

$\displaystyle C^{(3)}_{\hat{\sigma}\hat{\sigma}}$
$\displaystyle{=}\frac{C_{\hat{\sigma}}\mu^{-\delta}}{|x|^{4{+}\delta}}4C_{\hat{\phi}}^{2}C_{\hat{\sigma}}\left(\frac{2\gamma_{\hat{\phi}}}{\delta}{+}A_{\hat{\phi}}\right)U\left(2,d{-}2{+}\frac{\delta}{2},{-}\frac{\delta}{2}\right)U\left(2,\frac{d{+}\delta}{2},\frac{d{-}\delta}{2}{-}2\right)\,\frac{1}{(\mu|x|)^{\delta}}$
$\displaystyle=\frac{C_{\hat{\sigma}}}{|x|^{4}}\left(4\gamma_{\hat{\phi}}\log(\mu|x|)+\dots\right)\,,$
(91)

where in the last line we took the limit $\delta\rightarrow 0$ and omitted
everything except for the contribution to the anomalous dimension.
Consequently $C^{(3)}_{\hat{\sigma}\hat{\sigma}}$ contributes
$-2\gamma_{\hat{\phi}}$ to $\gamma_{\hat{\sigma}}$. This contribution is
precisely offset by the second term in the r.h.s. of (90), consistent with the
fact that in the long-range CFT the counterpart diagram
$C^{(3)}_{\sigma\sigma}$ is finite, and therefore does not provide any
contributions to $\gamma_{\sigma}$.

Remarkably, our argument for continuity of scaling dimension of $\sigma$ does
not use the specific value of $\gamma_{\hat{\phi}}$. More importantly, this
calculation pin-points how the continuity of the scaling dimension of $\sigma$
is inherited from continuity of the scaling dimension of $\phi$. Such an
inheritance makes sense, since continuity of $\phi$ is intrinsically connected
with the irrelevance of the bi-local kinetic term for $\phi$ above the
crossover point.

Having established continuity of $\gamma_{\sigma}$, it is straightforward to
generalize our argument to demonstrate the continuity of
$\gamma_{\sigma^{2}}$. We derived the latter in section 4.1. The total
dimension of $\sigma^{2}$ is given by $2s+\gamma_{\sigma^{2}}$, which at the
crossover point $s_{\star}=2-2\gamma_{\hat{\phi}}$ can be rewritten as
$4+(\gamma_{\sigma^{2}}|_{s\rightarrow 2}-4\gamma_{\hat{\phi}})$. Continuity
at $s_{\star}$ requires this value to be equal to
$4+\gamma_{\hat{\sigma}^{2}}$, the scaling dimension of $\hat{\sigma}^{2}$ in
the short-range vector model. In other words,

$\gamma_{\sigma^{2}}|_{s\rightarrow
2}=\gamma_{\hat{\sigma}^{2}}+4\gamma_{\hat{\phi}}\,.$ (92)

One can see that almost all of the diagrams contributing to
$\gamma_{\sigma^{2}}$ have a smooth limit $s\rightarrow 2$, under which they
reduce to their short-range counterparts. The only non-trivial diagram in this
limit is the one with two $\langle\sigma\sigma\rangle$ sub-diagrams, where
each of these receive $1/N$ corrections in the $s\rightarrow 2$ limit. Then
(92) immediately follows from (90). It should be noted that since the
anomalous dimensions of all operators $\sigma^{n}$ with $n>2$, come from the
corrections to the $\sigma$ and $\sigma^{2}$ sub-diagrams, the continuity of
their scaling dimensions across the crossover point follows from the
continuity of the scaling dimensions of $\sigma$ and $\sigma^{2}$.

To close this subsection, we will discuss continuity of scaling dimension of
the composite operator $\sigma\phi$. Scaling dimension of this composite
operator was discussed in section 4.4, where we provided a quick argument due
to e.o.m. for the exact value of the anomalous dimension
$\gamma_{\sigma\phi}=0$ to all orders in $1/N$, as well as an explicit
perturbative calculation at the next-to-leading order in $1/N$ expansion.
Analogous calculation in the short-range vector model proceeds along the
similar steps, with setting $s\rightarrow 2$ in all the sub-leading diagrams.
The only adjustment which one needs to make to the long-range derivation is
that the short-range $\hat{\phi}$ acquires anomalous dimension
$\gamma_{\hat{\phi}}$ (the second and the third diagram in section 4.4
contributing $\gamma_{\sigma\phi}^{(2,3)}$ have a continuous behavior at
$s=2$). Therefore the short-range counterpart of the first diagram in section
4.4 scales as

$\displaystyle
2(\Delta_{\hat{\sigma}}+\gamma_{\hat{\sigma}})+2(\Delta_{\hat{\phi}}+\gamma_{\hat{\phi}})$
$\displaystyle=2(\Delta_{\sigma}|_{s\rightarrow
2}+\gamma_{\sigma}|_{s\rightarrow
2}-2\gamma_{\hat{\phi}})+2(\Delta_{\phi}|_{s\rightarrow
2}+\gamma_{\hat{\phi}})$
$\displaystyle=\left[2(\Delta_{\sigma}+\gamma_{\sigma})+2\Delta_{\phi}\right]|_{s\rightarrow
s_{\star}}$ (93)

where we used (90), consistently with the continuity at
$s=s_{\star}=2-2\gamma_{\hat{\phi}}$.

###  6.2 Continuity of $\langle\phi\phi\sigma\rangle$ and
$\langle\sigma\sigma\sigma\rangle$

We now proceed to demonstrate continuity of normalized three-point functions
$\langle\phi\phi\sigma\rangle$, $\langle\sigma\sigma\sigma\rangle$ at the
crossover point $s_{\star}$. Particularly, we are going to demonstrate the
continuity of the three-point function amplitude (OPE coefficient)
$\bar{C}_{\phi\phi\sigma}$ at the next-to-leading order in $1/N$ expansion, as
well as the amplitude $\bar{C}_{\sigma\sigma\sigma}$ at the leading order in
$1/N$ expansion.

Recall that we split the amplitude into the leading large-$N$ factor and the
relative $1/N$ corrections as follows:

$\bar{C}_{\phi\phi\sigma}=\bar{C}_{\phi\phi\sigma}^{(0)}(1+\delta\bar{C}_{\phi\phi\sigma})\,.$
(94)

Here the leading order amplitude $\bar{C}_{\phi\phi\sigma}^{(0)}$ originates
from the tree-level digram, while the relative sub-leading terms
$\delta\bar{C}_{\phi\phi\sigma}$ are due to next-to-leading order corrections
to the $\phi\phi\sigma$ vertex. The continuity requirement is then

$\displaystyle\begin{split}\bar{C}_{\hat{\phi}\hat{\phi}\hat{\sigma}}^{(0)}&=\bar{C}_{\phi\phi\sigma}^{(0)}\Bigg{|}_{s=2}\,,\\\
\bar{C}_{\hat{\phi}\hat{\phi}\hat{\sigma}}^{(0)}\,\delta\bar{C}_{\hat{\phi}\hat{\phi}\hat{\sigma}}&=\left(\bar{C}_{\phi\phi\sigma}^{(0)}\,\delta\bar{C}_{\phi\phi\sigma}-2\gamma_{\hat{\phi}}\,\frac{\partial\bar{C}_{\phi\phi\sigma}^{(0)}}{\partial
s}\right)\Bigg{|}_{s=2}\,.\end{split}$ (95)

Expressions for $\bar{C}_{\hat{\phi}\hat{\phi}\hat{\sigma}}^{(0)}$,
$\delta\bar{C}_{\hat{\phi}\hat{\phi}\hat{\sigma}}$ are well-known
Petkou:1994ad and therefore the continuity (95) can be readily checked. Using
our result (63), (70) we confirm that (95) is indeed satisfied.

Notice that while the first relation in (95) is a simple consequence of (87),
the meaning behind the second line in (95) is more opaque. However it does
provide an interesting relation between the leading and next-to-leading
contributions to the three-point function $\langle\phi\phi\sigma\rangle$.

To close this section, we notice that at the leading order in $1/N$,
continuity of $\langle\sigma\sigma\sigma\rangle$ immediately follows from (95)
and smooth behavior of the triangle diagram.

##  7 A dual description for the long-range CFT at large $N$

As we reviewed above, the long-range critical $O(N)$ vector model is defined
for the exponent $s$ taking values in the range $d/2<s<s_{\star}$. It crosses
over to the MFT regime for $s\leq d/2$, and to the short-range regime for
$s\geq s_{\star}$ (given by critical vector model plus a decoupled generalized
free field Behan:2017dwr ; Behan:2017emf ). The model can be studied near the
MFT transition point by setting $s=d/2+\epsilon$. This makes the quartic
interaction slightly relevant, and allows one to study the model
perturbatively in $\epsilon$ expansion. No such weakly coupled description of
the short-range crossover point near $s=s_{\star}$ was available222222With
Sak1973 providing an earlier attempt of the construction perturbative near
$s_{\star}$, as well as near $d=4$. until a completely new model was suggested
in Behan:2017dwr for the $N=1$ case (Ising model).

The proposal was to start with the action $S_{\textrm{crit}}$ for the short-
range critical Ising model ($N=1$ vector model), and couple it to a
generalized free field $\chi$,

$\displaystyle S=S_{\textrm{crit}}+\int d^{d}x\int
d^{d}y\,\frac{\chi(x)\chi(y)}{|x-y|^{d-s}}+\lambda\int\,d^{d}x\,\hat{\phi}\chi\,.$
(96)

The scaling dimension of $\chi$ is therefore fixed to be
$\Delta_{\chi}=\frac{d+s}{2}$, while the field $\hat{\phi}$ (which is the
lowest scalar primary of the critical Ising CFT) has dimension
$\frac{d}{2}-1+\gamma_{\hat{\phi}}=\frac{d-s_{\star}}{2}$, when $\lambda=0$.
The dimension of $\chi$ was tuned such that the perturbation $\hat{\phi}\chi$
is irrelevant for $s>s_{\star}$, and relevant for $s<s_{\star}$. In the latter
case, the coupling $\lambda$ triggers an RG flow that can be studied
perturbatively in the vicinity of $s_{\star}$. Interestingly, the IR fixed
point of this flow is described by a long-range CFT. Moreover, it has been
suggested in Behan:2017dwr that the resulting long-range CFT is, for all
$d/2\leq s\leq s_{\star}$, the same as the long-range critical vector model,
obtained at the end of the RG flow of the Gaussian MFT deformed by a quartic
operator. Therefore, these two UV descriptions are dual in the IR, and have
complementary perturbative regimes on the opposite ends of the range $d/2\leq
s\leq s_{\star}$. However, evidence for the duality for all intermediate
values of $s$ has so far been obtained only for certain three-point function
coefficient ratios, valid for all orders in the $\epsilon$-expansion
Behan:2017dwr .

In this section, we aim to provide a complete non-perturbative description of
the duality for all values of $s$, by reformulating the duality suggested in
Behan:2017dwr at large $N$. We begin with re-writing the action for the short-
range critical $O(N)$ vector model at large $N$ in the Hubbard-Stratonovich
formalism,

$\displaystyle
S_{\textrm{crit}}=\int\,d^{d}x\left(\frac{1}{2}(\partial_{\mu}\hat{\phi})^{2}-\frac{\hat{\sigma}^{2}}{4g}+\frac{1}{\sqrt{N}}\hat{\sigma}\hat{\phi}^{2}\right)+\cdots\,,$
(97)

where ellipsis stands for $1/N$ corrections, and we also skip explicitly
writing the counterterms. Following Behan:2017dwr , we introduce a new field
$\chi^{i}$, $i=1,\dots,N$ with a bi-local kinetic term, and scaling dimension

$\Delta_{\chi}=\frac{d+s}{2}\,,$ (98)

and couple it to the field $\hat{\phi}^{i}$ as follows:

$S=S_{\textrm{crit}}+a\,\int d^{d}x\int
d^{d}y\,\frac{\chi^{i}(x)\chi^{i}(y)}{|x-y|^{d-s}}+\lambda\,\int
d^{d}x\,\hat{\phi}^{i}\chi^{i}\,.$ (99)

Here, the scaling dimension of $\hat{\phi}^{i}$ is $(d-s_{\star})/2$. The
kinetic term normalization coefficient $a$ can be arbitrary, and reflects
conventions regarding the definition of $\chi$. A convenient choice to fix it
will be explained momentarily. To obtain a large $N$ description of the IR
CFT, we first perform the following redefinition of the fields and couplings,

$\Phi^{i}=\lambda\hat{\phi}^{i}\,,\qquad\Sigma=\frac{1}{\lambda^{2}}\hat{\sigma}\,,\qquad
g=\lambda^{4}G\,.$ (100)

This allows us to re-write the action as232323We skip keeping track of the
$O(N)$ indices.

$S_{\textrm{SR}\chi}={a}\int d^{d}x\int
d^{d}y\,\frac{\chi(x)\chi(y)}{|x{-}y|^{d{-}s}}{+}\int
d^{d}x\,\left(\frac{1}{2\lambda^{2}}\,(\partial\Phi)^{2}{-}\frac{1}{4G}\Sigma^{2}{+}\Phi\chi{+}\frac{1}{\sqrt{N}}\Sigma\Phi^{2}\right)+\cdots\,,$
(101)

where we have again omitted higher-order corrections to vertices and
propagators.

We will now argue that when both $G$ and $\lambda$ are tuned to criticality,
and $s$ assumes values in the range $d/2<s<s_{\star}$, the model (101) is
equivalent to the long-range CFT (17). The argument can be constructed by
going from the latter towards the former.

Starting from the long-range CFT model (17), we can add to it its action a
local kinetic term $\frac{1}{2\lambda^{2}}\,(\partial\phi)^{2}$, where
$\lambda$ is a dimensionful constant that we choose to be equal to the IR
critical value of the coupling constant in the model (101). This can be done
without affecting dynamics of the field $\phi$ in the range $d/2<s<s_{\star}$,
where such a term is suppressed compared to the bi-local kinetic term for the
field $\phi$.

We then insert an identity into the partition function $Z_{\textrm{LR}}$ of
the model (17), represented as a Gaussian integral over a field
$\chi$242424Our notation for this field is deliberately the same as for the
$\chi$ field in the action (101), as we intend to identify these d.o.f. on
both sides of the argued IR duality.

$\displaystyle Z_{\textrm{LR}}$ $\displaystyle\rightarrow
Z_{\textrm{LR}}\,\frac{1}{z}\,\int[D\chi]\exp\left(-a\int\frac{d^{d}x_{1,2}}{|x_{12}|^{d-s}}\left(\chi+\alpha\int
d^{d}x_{1}^{\prime}\,\frac{\phi(x_{1}^{\prime})}{|x_{1}-x_{1}^{\prime}|^{d+s}}\right)\right.$
$\displaystyle\times\left.\left(\chi+\alpha\int
d^{d}x_{2}^{\prime}\,\frac{\phi(x_{2}^{\prime})}{|x_{2}-x_{2}^{\prime}|^{d+s}}\right)\right)\,,$
(102)

accompanied by the corresponding normalization prefactor $1/z$. Choosing

$\displaystyle a$
$\displaystyle=-\frac{1}{4C(s)\pi^{d}A(\Delta_{\Phi})A(d-\Delta_{\Phi})}\,,$
(103) $\displaystyle\alpha$
$\displaystyle=\frac{1}{2a\pi^{d}A(\Delta_{\Phi})A(d-\Delta_{\Phi})}=-2C(s)\,,$
(104)

we obtain that the action (17) transforms into

$\displaystyle S_{\textrm{LR}}=a\int d^{d}x\int
d^{d}y\,\frac{\chi(x)\chi(y)}{|x-y|^{d-s}}+\int\,d^{d}x\left(\frac{1}{2\lambda^{2}}\,(\partial\phi)^{2}-\frac{1}{4g}\sigma^{2}+\phi\chi+\frac{1}{\sqrt{N}}\sigma\phi^{2}\right)\,.$
(105)

Under the identification

$\Phi\leftrightarrow\phi\,,\qquad\Sigma\leftrightarrow\sigma\,,\qquad
g\leftrightarrow G$ (106)

the models (101), (105) are indeed equivalent.

Notice that when $s>s_{\star}$ the coupling $\lambda$ in (99) becomes
irrelevant, and its critical value is a trivial $\lambda=0$, making the field
redefinition (100) ill-defined. In fact, the field $\chi$ decouples into an
independent d.o.f. with a bi-local kinetic term. Therefore while the IR
duality between the long-range vector model and the deformed short-range
critical vector model, defined for $d/2<s<s_{\star}$, is quite straightfoward
to establish, a more subtle prediction is that when $s>s_{\star}$ (outside the
scope of the long-range model, and therefore beyond the regime where one can
talk about such a duality) the system retains a decoupled generalized free
field d.o.f Behan:2017dwr ; Behan:2017emf .252525The dual description of the
critical long-range model is also useful near $s=s_{\star}$, where it gives a
perturbative handle on the calculations Behan:2017dwr ; Behan:2017emf .

Having argued for the duality between the long-range and the deformed short-
range critical vector models in the range $d/2<s<s_{\star}$, we would like to
explore the meaning of the dual of the generalized free field $\chi$ on the
long-range side, with an aim to make the duality more precise. Motivated by
this goal, for the rest of this section we are going to return to the picture
with the original field $\chi$. The resulting action of the critical regime of
the $O(N)$ vector model coupled to $\chi$ is given by

$S=-\frac{1}{4C(s)\pi^{d}A(\Delta_{\Phi})A(d-\Delta_{\Phi})}\,\int d^{d}x\int
d^{d}y\,\frac{\chi(x)\chi(y)}{|x-y|^{d-s}}+\int
d^{d}x\,\left(\Phi\chi+\frac{1}{\sqrt{N}}\Sigma\Phi^{2}\right)\,.$ (107)

It follows from the action (107) that the propagator of the field $\chi$ is

$\langle\chi(x)\chi(0)\rangle=\frac{C_{\chi}\,(1+A_{\chi})}{|x|^{2\Delta_{\chi}}}\,,\qquad
C_{\chi}=-2C(s)\,,$ (108)

where we took into account that the dimension $\Delta_{\chi}$ is exact, and
the only possible $1/N$ corrections to the $\chi$ propagator can go into the
amplitude correction $A_{\chi}$. At the same time, the $\chi$ e.o.m. following
from (107) is,

$\Phi=\frac{1}{2C(s)\pi^{d}A(\Delta_{\Phi})A(d-\Delta_{\Phi})}\int
d^{d}y\,\frac{\chi(y)}{|x-y|^{d-s}}\,,$ (109)

which, in turn, gives

$\langle\Phi(x)\Phi(0)\rangle=-\frac{1+A_{\chi}}{2\pi^{d}C(s)A(\Delta_{\Phi})A(d-\Delta_{\Phi})}\,\frac{1}{|x|^{2\Delta_{\Phi}}}\,.$
(110)

Important manifestation of a duality between two CFTs is given by matching
conformal correlation functions on both sides of the duality. When matching
amplitudes of the correlation functions of operators in different models, one
needs to ensure consistent normalization of these operators. A convenient
choice is given by normalization of the two-point functions of the considered
operators to unity.262626See Goykhman:2019kcj ; Goykhman:2020tsk ;
Chai:2021uhv for a recent discussion. Particularly, one finds it useful that
rescaling the fields $\chi$, $\Phi$ as

$\displaystyle\chi\rightarrow\frac{1}{\sqrt{-2C(s)\,(1+A_{\chi})}}\,\chi\,,\quad\Phi\rightarrow\sqrt{-\frac{2\pi^{d}C(s)A(\Delta_{\Phi})A(d-\Delta_{\Phi})}{1+A_{\chi}}}\,\Phi\,,$
(111)

obtaining unit-normalized propagators $\langle\chi\chi\rangle$,
$\langle\Phi\Phi\rangle$. The three-point functions involving the fields
$\chi$, $\Phi$ will be calculated for such fields normalized according to
(111). Similar normalization conventions can be used for their counterparts in
the long-range model.

In particular, one can establish that the ratios of coefficients of three-
point functions $\langle\chi O_{2}O_{3}\rangle$, $\langle\Phi
O_{2}O_{3}\rangle$, and $\langle\sigma\phi O_{2}O_{3}\rangle$, $\langle\phi
O_{2}O_{3}\rangle$ where $O_{2,3}$ are some conformal operators, match exactly
to all orders in $1/N$ and for all $d/2<s<s_{\star}$, analogously to
Behan:2017dwr ; Behan:2017emf . This implies the following matching of d.o.f.
on both sides of the duality:

$\chi\leftrightarrow\sigma\phi\,.$ (112)

Simply put, the relation (112) between $\chi$ and $\sigma\phi$ follows from
the relation (106) $\Phi\leftrightarrow\phi$, accompanied by the observation
that the pairs of fields $\sigma\phi,\phi$ and $\chi,\Phi$ satisfy analogous
equations of motion (53), (109).

An infra-red duality between the critical long-range model and the short-range
model deformed by a coupling to the generalized free field $\chi$, requires
one to match the spectrum of the two CFTs, as well as all of the CFT data. An
immediate observation one can make is that in the $N=1$ case the long-range
model possesses an operator $\phi^{3}$ with exact scaling dimension $(d+s)/2$,
while the short-range model has no analogous counterpart Behan:2017dwr . The
corresponding operator in the critical long-range $O(N)$ model in the Hubbard-
Stratonovich language is $\sigma\phi$, whose dimension $\Delta_{\sigma\phi}$
given by (55), as we derived in section 4.4, is fixed exactly by the e.o.m.
relating it to the dimension of $\phi$. Together with the other operator
$\phi$ of a fixed scaling dimension (13), the ‘shadow relation’ is satisfied,
analogously to the $N=1$ case of Paulos:2015jfa

$\Delta_{\phi}+\Delta_{\sigma\phi}=d\,.$ (113)

Notice that absence of the anomalous dimension contribution to the composite
operator $\sigma\phi$ in the long-range critical model is to be contrasted
with its counterpart $\hat{\sigma}\hat{\phi}$ in the short-range vector model.
This being said, the full scaling dimension of $\sigma\phi$ is continuous
across $s_{\star}$, as we demonstrated in section 6.1.

We close this subsection by pointing out that using e.o.m. (53) we can also
calculate the following cross-correlator

$\langle\sigma\phi(x)\phi(0)\rangle=-\frac{C(s)}{\sqrt{N}}\,C_{\phi}\,\pi^{d}A\left(\frac{d-s}{2}\right)A\left(\frac{d+s}{2}\right)\,\delta^{(d)}(x)\,,$
(114)

which vanishes for non-coincident points. The counterpart of this correlator
in the deformed short-range model is given by
$\langle\chi(x)\hat{\phi}(0)\rangle\simeq\delta^{(d)}(x)$. Notice that the
result (114) is exact to all orders in $1/N$, but can be seen explicitly at
the leading order in $1/N$ from the diagram

$2\Delta_{\sigma}$$2\Delta_{\phi}$$2\Delta_{\phi}$

###  7.1 Anomalous dimension of the stress-energy tensor

In this section, we continue studying the $\hat{\phi}\chi$ perturbation (99)
of the short-range $O(N)$ vector model (97) that brings it to a long-range
critical regime in the IR. Our focus will be on the fate of the stress-energy
tensor operator $T_{\mu\nu}$,

$T_{\mu\nu}=\partial_{\mu}\hat{\phi}\partial_{\nu}\hat{\phi}-\delta_{\mu\nu}\,\left(\frac{1}{2}\,(\partial\hat{\phi})^{2}-\frac{1}{4g_{\star}}\,\hat{\sigma}^{2}+\frac{1}{\sqrt{N}}\,\hat{\sigma}\hat{\phi}^{2}\right)\,$
(115)

of the original short-range vector model (97) under such a deformation. In
(115) we have substituted the critical value of the coupling $g_{\star}$. One
anticipates that conservation of the composite operator (115) is broken at the
long-range fixed point of the model (99) Behan:2017dwr ; Behan:2017emf , where
it no longer has the status of a conserved stress-energy tensor. In agreement
with such an expectation, working at the first order in $1/N$ expansion, we
will derive anomalous dimension and trace of the operator (115).

Unlike calculation of the previous section (that was carried out in the entire
range $d/2<d<s_{\star}$), in this section we will stay perturbatively close to
the cross-over point $s_{\star}$, working at the linear order in
$\delta=(s_{\star}-s)/2$. Since our starting point is the short-range critical
vector model, we find it convenient for the purposes of this section to work
in terms of the original conventions $\hat{\phi}$, $\hat{\sigma}$ for the
degrees of freedom of this model.

While in the original theory, $\lambda=0$, the stress-energy tensor (115) is
conserved on shell, conservation law of that operator is broken at the new
fixed point, where it no longer plays the role of stress-energy tensor. This
can manifest in the anomalous dimension $\gamma_{T}$ and trace $\hat{c}$, in
the ansatz for the correlation function 272727Here we skipped $2\gamma_{T}$ in
the exponent in the second term in the r.h.s., anticipating linearization in
$\delta$, while taking into account that both $\gamma_{T}$ and $\hat{c}$
vanish when $\delta=0$.

$\displaystyle\langle\partial_{\mu}T^{\mu\nu}(x_{1})\partial_{\lambda}T^{\lambda\rho}(x_{2})\rangle=\frac{\partial}{\partial
x_{1}^{\mu}}\frac{\partial}{\partial
x_{2}^{\lambda}}\left(C_{T}\frac{I^{\mu\nu,\lambda\rho}(x_{12})}{|x_{12}|^{2d+2\gamma_{T}}}+\hat{c}\,\delta^{\mu\nu}\,\delta^{\lambda\rho}\,\frac{1}{|x_{12}|^{2d}}\right)\,,$
(116)

where $\gamma_{T}$ is the anomalous dimension, $C_{T}$ is the central change
of the short-range theory,

$C_{T}=N\,\left(\frac{d}{(d-1)S_{d}^{2}}+{\cal
O}\left(\frac{1}{N}\right)\right)\,,\qquad
S_{d}=\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d}{2}\right)}\,,$ (117)

We also defined tensor structures

$\displaystyle I_{\mu\nu}(x)$
$\displaystyle=\delta_{\mu\nu}-2\frac{x_{\mu}x_{\nu}}{x^{2}}\,,$ (118)
$\displaystyle I_{\mu\nu,\lambda\rho}(x)$
$\displaystyle=\frac{1}{2}\left(I_{\mu\lambda}(x)I_{\nu\rho}(x)+I_{\mu\rho}(x)I_{\nu\lambda}(x)-\frac{2}{d}\delta_{\mu\nu}\delta_{\lambda\rho}\right)\,,$
(119)

and the $\hat{c}$ term in (116) accounts for a possible non-vanishing
trace.282828We thank M. Smolkin for discussion of this point. Simplifying
(116), while linearizing in $\delta$, we obtain

$\langle\partial_{\mu}T^{\mu\nu}(x)\partial_{\lambda}T^{\lambda\rho}(0)\rangle=\frac{C_{T}(d-1)(d+2)\gamma_{T}}{d}\,I^{\nu\rho}\,\frac{1}{|x|^{2d+2}}+2d\hat{c}\,\left(I^{\nu\rho}-2d\frac{x^{\nu}x^{\rho}}{|x|^{2}}\right)\,\frac{1}{|x|^{2d+2}}\,.$
(120)

We are going to compare the general ansatz (120) with what we obtain
specifically in the model (99). Working at the second order in conformal
perturbation theory in $\lambda_{\star}$ we derive

$\displaystyle\langle\partial_{\mu}T^{\mu\nu}(x)\partial_{\lambda}T^{\lambda\rho}(0)\rangle=\frac{\lambda_{\star}^{2}}{2}\,\int
d^{d}x_{1,2}\,\langle\partial_{\mu}T^{\mu\nu}(x)\partial_{\lambda}T^{\lambda\rho}(0)\hat{\phi}(x_{1})\chi(x_{1})\hat{\phi}(x_{2})\chi(x_{2})\rangle+{\cal
O}(\lambda_{\star}^{3})$ $\displaystyle=\frac{\lambda_{\star}^{2}}{2}\,\int
d^{d}x_{1,2}\,\langle\chi(x_{1})\chi(x_{2})\rangle\,\langle\partial_{\mu}T^{\mu\nu}(x)\partial_{\lambda}T^{\lambda\rho}(0)\hat{\phi}(x_{1})\hat{\phi}(x_{2})\rangle+{\cal
O}(\lambda_{\star}^{3})$
$\displaystyle=\lambda_{\star}^{2}\,\langle\chi(x)\chi(0)\rangle\,\langle\partial^{\nu}\hat{\phi}(x)\partial^{\rho}\hat{\phi}(0)\rangle+{\cal
O}(\lambda_{\star}^{3})\,,$ (121)

where in the last line we used Ward identity for the divergence of the stress-
energy tensor in the original short-range sector of the model. From (121) we
then obtain

$\langle\partial_{\mu}T^{\mu\nu}(x)\partial_{\lambda}T^{\lambda\rho}(0)\rangle=N\lambda_{\star}^{2}(d-s_{\star})C_{\hat{\phi}}C_{\chi}\,\frac{1}{|x|^{2d+2}}\,\left(I^{\nu\rho}-(d-s_{\star})\,\frac{x^{\nu}x^{\rho}}{|x|^{2}}\right)\,,$
(122)

where we substituted $s=s_{\star}$ at the short-range fixed point.

Comparing (122), (120) while demanding that tensor structures match, we arrive
at

$\displaystyle\hat{c}$
$\displaystyle=N\lambda_{\star}^{2}\frac{(d-s_{\star})^{2}C_{\hat{\phi}}C_{\chi}}{4d^{2}}+{\cal
O}(\lambda_{\star}^{3})\,,$ (123) $\displaystyle\gamma_{T}$
$\displaystyle=N\lambda_{\star}^{2}\frac{(d+s_{\star})(d-s_{\star})C_{\hat{\phi}}C_{\chi}}{2C_{T}(d-1)(d+2)}+{\cal
O}(\lambda_{\star}^{3})\,.$ (124)

In the perturbative regime, near the short-range fixed point, we have
Behan:2017dwr

$\lambda_{\star}^{2}=\alpha(d)\delta+{\cal O}(\delta^{2})\,,$ (125)

where $\delta=(s_{\star}-s)/2$. Furthermore, plugging
$s_{\star}=2-2\gamma_{\hat{\phi}}$ in (123), (124) and expanding in
$\gamma_{\hat{\phi}}$ (which in the large $N$ language means expanding in
$1/N$), we obtain

$\displaystyle\hat{c}$
$\displaystyle=N\,\alpha\,\delta\,\frac{(d-2)\Gamma\left(\frac{d}{2}+1\right)^{2}}{d^{3}\pi^{d}}\,\gamma_{\hat{\phi}}+{\cal
O}(\delta^{2},\gamma_{\hat{\phi}}^{2})\,,$ (126) $\displaystyle\gamma_{T}$
$\displaystyle=\frac{2\,\alpha\,\delta}{\Gamma\left(\frac{d}{2}\right)^{2}}\,\gamma_{\hat{\phi}}+{\cal
O}(\delta^{2},\gamma_{\hat{\phi}}^{2})\,.$ (127)

While the anomalous dimension and trace (126), (127) have been derived
perturbatively close to the short-range fixed point, the operator
$\partial_{\mu}T^{\mu\nu}$ can be studied in the entire range
$d/2<s<s_{\star}$. We can decompose this operator in terms of the primary
Behan:2017dwr ; Giombi:2016hkj

$V^{\nu}=\hat{\phi}\partial^{\nu}\chi-\frac{\Delta_{\chi}}{\Delta_{\phi}}\,\chi\partial^{\nu}\hat{\phi}\,,$
(128)

and the descendant $\partial^{\nu}(\hat{\phi}\chi)$. Notice that the latter
has a vanishing (leading order) cross-correlator with (128). To find
coefficients of such a decomposition, we will be working in the conformal
perturbation theory at linear order in $\lambda_{\star}$:

$\displaystyle\langle{\partial}_{\mu}T^{\mu\nu}(x)\partial^{\rho}(\hat{\phi}\chi)(0)\rangle$
$\displaystyle=\lambda_{\star}\,\int
d^{d}x_{1}\,\langle{\partial}_{\mu}T^{\mu\nu}(x)\,\partial^{\rho}(\hat{\phi}\chi)(0)\,\hat{\phi}\chi(x_{1})\rangle$
$\displaystyle=\lambda_{\star}\langle\partial^{\nu}\hat{\phi}\,\chi(x)\,\partial^{\rho}(\hat{\phi}\chi)(0)\rangle\,,$
(129)

where in the last line we used Ward identity. Similarly, we obtain

$\displaystyle\langle{\partial}_{\mu}T^{\mu\nu}(x)V^{\rho}(0)\rangle=\lambda_{\star}\langle\partial^{\nu}\hat{\phi}\,\chi(x)\,V^{\rho}(0)\rangle\,.$
(130)

Therefore

${\partial}_{\mu}T^{\mu\nu}(x)=\lambda_{\star}\partial^{\nu}\hat{\phi}\,\chi+{\cal
O}(\lambda_{\star}^{2})\,,$ (131)

which we can rewrite as

$\partial_{\mu}T^{\mu\nu}=-\lambda_{\star}\,\frac{d-s}{2d}\,V^{\nu}-\frac{d-s}{2d}\,\lambda_{\star}\,\partial^{\nu}(\Phi\chi)\,.$
(132)

A short-cut to the calculation of $\gamma_{T}$, providing a faster alternative
derivation of the result (124), is afforded by the decomposition (132). One
can easily see that focusing only on the $V^{\nu}$ contribution to the
divergence of the stress-energy tensor (132), while simultaneously ignoring
that trace contribution $\hat{c}$ in the general ansatz (120), reproduces the
final expression $(\ref{gamma T result})$ for the anomalous dimension
$\gamma_{T}$ Behan:2017dwr ; Behan:2017emf .292929In particular, one can see
that the coefficient in front of the $V^{\nu}$ term in r.h.s. of (132) matches
with its counterpart $b(g_{\star})=b_{1}g_{\star}+{\cal O}(g_{\star}^{2})$
introduced in Behan:2017dwr ; Behan:2017emf . Indeed, following such a
strategy we obtain from (132)

$\langle\partial_{\mu}T^{\mu\nu}(x)\partial_{\lambda}T^{\lambda\rho}(0)\rangle=\lambda_{\star}^{2}\,\frac{(d-s)^{2}}{4d^{2}}\,\langle
V^{\nu}(x)V^{\rho}(0)\rangle+\dots\,.$ (133)

Substituting here

$\langle
V^{\nu}(x)V^{\rho}(0)\rangle=N\,\frac{2dC_{\hat{\phi}}C_{\chi}(d+s)}{d-s}\,\frac{1}{|x|^{2d+2}}\,I^{\nu\rho}\,,$
(134)

and comparing the result with the $\gamma_{T}$ term in (120) we recover (124).

##  8 Discussion

The main focus of this paper was to study the long-range critical $O(N)$
vector model in the large-$N$ limit. This model appears in the IR regime of
the Gaussian MFT for a generalized free field, deformed by a local quartic
interaction. Working within the Hubbard-Stratonovich formalism, we calculated
several new scaling dimensions and OPE coefficients of various primary
operators in this model, performing most of our calculations at the next-to-
leading order in the $1/N$ expansion. The CFT data we obtained furnishes a
non-trivial consistency check for the existence of the full conformal symmetry
at the long-range fixed point.

In particular, we determined the leading order contribution to the anomalous
dimension of the composite operator $\sigma^{n}$, $n\geq 2$, and calculated
the three-point functions $\langle\phi\phi\sigma\rangle$,
$\langle\sigma\sigma\sigma\rangle$. We also established that the cross-
correlator $\langle\sigma^{2}\sigma\rangle$ vanishes, at least at the leading
order in $1/N$, unlike its short-range counterpart discussed in
Derkachov:1997gc ; Derkachov:1998js . The vanishing of this correlator
simultaneously indicates that the long-range fixed point is indeed a CFT, and
that $\sigma^{2}$ is in fact a primary operator in this CFT.

While in the short-range CFT certain composite operators, such as
$\hat{\sigma}^{2}$, can mix with the descendants, such as
$\partial^{2}\hat{\sigma}$, such a mixing is impossible in the long-range CFT
for the Hubbard-Stratonovich field with a leading-order scaling dimension $s$,
taking a general value in the range $d/2<s<s_{\star}$ (see Behan:2018hfx for a
bootstrap analysis of the long-range Ising model). The case of $s=1$ requires
a special attention, since it allows for the possibility of operator mixing,
at least in principle, such as a mixing between $\sigma^{4}$ and
$\partial^{2}\sigma^{2}$. Notice that $s=1$ in fact has the physical
interpretation of a boundary CFT (BCFT), i.e., a free field theory in the
$d+1$-dimensional bulk perturbed by a quartic interaction localized on its
$d$-dimensional boundary (or defect).303030For the quartic operator to be
relevant in the IR one needs to consider $d<2$. When $d>2$ the theory arrives
at the critical regime in the UV, however it suffers from usual instabilities
typically found in such cases Giombi:2019enr . We therefore expect the
spectrum of conformal primaries to shrink in the BCFT, as contrasted to a
generic long-range CFT. It would be interesting to explore this direction
further.

We have also investigated the interplay between long-range and short-range
critical vector models. While the former is expected to occupy the region
$d/2<s<s_{\star}$ of the exponent parameter $s$, the latter exists in the
region $s>s_{\star}$. We have performed an explicit consistency check of our
results for the calculated CFT data, by demonstrating its continuity at the
long-range–short-range crossover point $s_{\star}$. Furthermore, generalizing
the construction of Behan:2017dwr ; Behan:2017emf to the large-$N$ case, we
have argued for the existence of an exact IR duality between the long-range
vector model, and the short-range model deformed by the coupling to a
generalized free field. Performing a simultaneous perturbative expansion to
the linear order in $s_{\star}-s$ and to the next-to-leading order in $1/N$ we
also obtained expression for the anomalous dimension and trace of the stress-
energy tensor of the short-range vector model, which it acquires at the long-
range fixed point.

To close this discussion, we would like to notice that long-range critical
vector models has recently received attention from the perspective of
persistent symmetry breaking at all temperatures Chai:2021djc , adding extra
motivation for expanding our understanding of CFT data of these models.

## Acknowledgements

We thank M. Smolkin, S. Rychkov for helpful discussions. 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 collect some useful expressions and identities.

Loop diagram in the position space are simply additive:

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

The propagator merging relation is given by

$\int
d^{d}x_{2}\,\frac{1}{|x_{2}|^{2a}|x_{1}-x_{2}|^{2b}}=U(a,b,d-a-b)\,\frac{1}{|x_{1}|^{2a+2b-d}}\,,$
(135)

where we introduced

$\displaystyle U(a,b,c)$ $\displaystyle=\pi^{\frac{d}{2}}A(a)A(b)A(c)\,.$
(136)

This relation can be diagrammatically 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

$\int
d^{d}x\,\frac{1}{|x_{1}-x|^{2a_{1}}|x_{2}-x|^{2a_{2}}|x_{3}-x|^{2a_{3}}}=\frac{U(a_{1},a_{2},a_{3})}{|x_{12}|^{d-2a_{3}}|x_{13}|^{d-2a_{2}}|x_{23}|^{d-2a_{1}}}\,,$
(137)

and is graphically 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 defined $\alpha=d-2a_{3}$, $\beta=d-2a_{2}$, $\gamma=d-2a_{1}$.

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