# Operator product expansion coefficients from the nonperturbative functional
renormalization group

Félix Rose  Max Planck Institute of Quantum Optics, Hans-Kopfermann-Straße 1,
85748 Garching, Germany  Munich Center for Quantum Science and Technology
(MCQST), Schellingstraße 4, 80799 Munich, Germany  Carlo Pagani  Université
Grenoble Alpes, Centre National de la Recherche Scientifique, Laboratoire de
Physique et Modélisation des Milieux Condensés, 38000 Grenoble, France
Institute of Physics, Johannes Gutenberg University, Staudingerweg 7, 55099
Mainz, Germany  Nicolas Dupuis  Sorbonne Université, CNRS, Laboratoire de
Physique Théorique de la Matière Condensée, LPTMC, F-75005 Paris, France

(April 1, 2022)

###### Abstract

Using the nonperturbative functional renormalization group (FRG) within the
Blaizot-Méndez-Galain-Wschebor approximation, we compute the operator product
expansion (OPE) coefficient $c_{112}$ associated with the operators ${\cal
O}_{1}\sim\varphi$ and ${\cal O}_{2}\sim\varphi^{2}$ in the three-dimensional
$\mathrm{O}(N)$ universality class and in the Ising universality class ($N=1$)
in dimensions $2\leq d\leq 4$. When available, exact results and estimates
from the conformal bootstrap and Monte Carlo simulations compare extremely
well to our results, while FRG is able to provide values across the whole
range of $d$ and $N$ considered.

###### Contents

  1. I Introduction
  2. II OPE coefficients in the effective action formalism
    1. II.1 Correlation functions in momentum space
    2. II.2 $\mathrm{O}(N)$ model and Wilson-Fisher fixed point
    3. II.3 Effective action
    4. II.4 FRG formalism and BMW approximation
  3. III Numerical results
    1. III.1 Ising university class in dimensions $2\leq d\leq 4$
    2. III.2 Three-dimensional $\mathrm{O}(N)$ model
  4. IV Conclusion
  5. A $c_{112}$ in the free case
  6. B $c_{112}$ in the large-$N$ limit

##  I Introduction

The nonperturbative functional renormalization group (FRG) provides us with a
versatile technique to study strongly correlated systems. It has been used in
many models of quantum and statistical field theory ranging from statistical
physics and condensed matter to high-energy physics and quantum gravity [1, 2,
3]. Besides the interest in models where perturbative approaches or numerical
methods are difficult for various reasons, there is an ongoing effort to
characterize and quantify the efficiency and the accuracy of the FRG approach
by considering well-known models of statistical physics. It is now proven that
the FRG yields very accurate values of the critical exponents associated with
the Wilson-Fisher fixed point of $\mathrm{O}(N)$ models [4, 5], comparable
with the best estimates from field-theoretical perturbative RG [6, 7], Monte
Carlo simulations [8, 9, 10, 11, 12, 13] or conformal bootstrap [14, 15, 16,
17]. The FRG also allows the computation of universal quantities defined away
from the critical point, such as universal scaling functions [18, 19, 20, 21]
or universal amplitude ratios [22], again in remarkable agreement with Monte
Carlo simulations when available.

On the other hand, the operator product expansion (OPE) has received little
attention in the framework of the FRG until recently [23, 24, 25, 26, 27, 28,
29]. Wilson and Kadanoff suggested independently that in a quantum field
theory the product of two operators in the short distance limit is equivalent
to an infinite sum of operators multiplied by possibly singular functions when
inserted in any correlation function [30, 31, 32, 33]. The validity of the OPE
has been proven to all orders in pertubation theory [34] and can be
established in full generality in the case of conformal field theories [35].
Indeed, the OPE has been fundamental in the study of conformal field theories
in two and higher dimensions [36, 37]. In this context, the conformal
bootstrap program [38, 39, 40, 37] has lead to a large number of precise
results. The OPE has been instrumental as well in studies regarding quantum
chromodynamics [41] and condensed matter, where it has been used to derive the
thermodynamic properties of quantum gases [42, 43].

Despite the fact that both the FRG formalism and the OPE offer non-
perturbative approaches to quantum field theory, it is not yet clear to what
extent these two aspects can be usefully combined to extract information
regarding the non-perturbative regime of a field theory.

From the perspective of perturbation theory, the FRG provides a useful
framework that allows one to prove the existence of the OPE perturbatively
[23, 24, 25, 26, 27]. Moreover, by following the proposal of Cardy relating
the OPE coefficients to the second order terms in the expansion of the beta
functions around a fixed point [44], the standard perturbative renormalization
group has been used to derive certain OPE coefficients within the $\epsilon$
expansion [45]; we refer to [46] for an FRG perspective on these issues based
on a geometric approach to theory space.

In principle, one may reconstruct from the FRG the full operator product and
express the latter as an OPE [28, 29]. However, this may be rather cumbersome
in practice. For a conformally invariant fixed point theory 111Note that
conformal invariance, and in particular its relation to scale invariance, has
been discussed in a number of recent works based on the FRG, see Refs. [85,
86, 87, 88, 89]., a further possibility explored in [29] consists in
extracting the OPE coefficient from three-point functions. It has been shown
that within this approach it is possible to calculate the OPE coefficients in
the epsilon expansion.

The main quantities of interest in the FRG are the effective action, defined
as the Legendre transform of the free energy, and the one-particle irreducible
(1PI) vertices. Taking the Wilson-Fisher fixed point of the $\mathrm{O}(N)$
model as an example, we show how the OPE coefficient $c_{112}$ associated with
the operators ${\cal O}_{1}\sim\varphi$ and ${\cal O}_{2}\sim\varphi^{2}$ can
be deduced from a small number of low-order 1PI vertices. One difficulty in
the computation of OPE coefficients is that the latter are determined by the
full momentum dependence of the vertices in the critical regime. For this
reason, one has to go beyond the derivative expansion in order to accurately
determine the OPE coefficients. The latter can be computed in the so-called
Blaizot-Méndez-Galain-Wschebor (BMW) approximation that enables the
determination of the momentum dependence of the correlation functions [48, 49,
50]. This approximation scheme has been used in the past to obtain the
spectral function of the “Higgs” amplitude mode in the $(2+1)$-dimensional
$\mathrm{O}(N)$ model [51] providing an estimate of the Higgs mass that has
been confirmed by subsequent numerical simulations of lattice models [52, 53,
54].

The outline of the paper is as follows. In Section II we recall the relation
between the OPE coefficients and the two- and three-point functions in
momentum space, focusing on the coefficient $c_{112}$ in the $d$-dimensional
$\mathrm{O}(N)$ model. We then show how to relate $c_{112}$ to the 1PI
vertices. Finally we briefly describe the nonperturbative FRG formalism and
the BMW approximation. In Section III the results obtained from a numerical
solution of the flow equations are discussed for the three-dimensional
$\mathrm{O}(N)$ model and the Ising university class ($N=1$) in dimensions
$2\leq d\leq 4$, and compared with exact values in some particular cases and
estimates from conformal bootstrap and Monte Carlo as well as $\epsilon$ and
large-$N$ expansions.

##  II OPE coefficients in the effective action formalism

###  II.1 Correlation functions in momentum space

We consider a critical, conformally invariant, theory. For fields ${\cal
O}_{a}(x)$ (be them composite or not) with scaling dimensions $\Delta_{a}$,
the two- and three-point correlation functions are given by

$\langle{\cal O}_{a}(x){\cal O}_{a}(y)\rangle=\frac{1}{|x-y|^{2\Delta_{a}}}$
(1)

and

$\langle{\cal O}_{a}(x_{1}){\cal O}_{b}(x_{2}){\cal O}_{c}(x_{3})\rangle=\\\
\frac{c_{abc}}{x_{12}^{\Delta_{a}+\Delta_{b}-\Delta_{c}}x_{23}^{\Delta_{b}+\Delta_{c}-\Delta_{a}}x_{13}^{\Delta_{a}+\Delta_{c}-\Delta_{b}}}$
(2)

where $x_{12}=|x_{1}-x_{2}|$, etc. Equation 1 assumes a proper normalization
of the fields and the coefficient $c_{abc}$ in 2 can be identified with the
OPE coefficient [55]. Since in practice we shall work in momentum space, it is
convenient to consider the Fourier transformed correlation functions. For the
two-point one,

$\displaystyle\langle{\cal O}_{a}(p){\cal
O}_{a}(-p)\rangle=\int_{x}\frac{e^{-ipx}}{|x|^{2\Delta_{a}}}=\frac{A_{d}(\Delta_{a})}{|p|^{d-2\Delta_{a}}},$
(3)

where

$A_{d}(\Delta)=4^{d/2-\Delta}\pi^{d/2}\frac{\Gamma(d/2-\Delta)}{\Gamma(\Delta)}$
(4)

with $\Gamma(x)$ the gamma function and $d$ the dimension. The Fourier
transform $\langle{\cal O}_{a}(p_{1}){\cal O}_{b}(p_{2}){\cal
O}_{c}(p_{3})\rangle$ is given by a complicated expression but it is
sufficient to consider the limit $|p_{1}|\gg|p_{2}|$, where

$\langle{\cal O}_{a}(p_{1}){\cal O}_{b}(p_{2}){\cal
O}_{c}(-p_{1}-p_{2})\rangle\simeq\\\
\frac{c_{abc}A_{d}((\Delta_{a}-\Delta_{b}+\Delta_{c})/2)A_{d}(\Delta_{b})}{|p_{1}|^{d-\Delta_{a}+\Delta_{b}-\Delta_{c}}|p_{2}|^{d-2\Delta_{b}}},$
(5)

to extract the coefficient $c_{abc}$ [29]. Equation 5 entails that the OPE
coefficient $c_{abc}$ can be deduced from the three-point function 5 once the
fields have been properly normalized in order to satisfy 3.

###  II.2 $\mathrm{O}(N)$ model and Wilson-Fisher fixed point

In the following we consider the $\mathrm{O}(N)$ model in $d$ dimensions
defined by the action

$S[\varphi]=\int_{x}\left\\{\frac{1}{2}({\partial_{\mu}}\varphi)^{2}+\frac{r_{0}}{2}\varphi^{2}+\frac{u_{0}}{4!N}{(\varphi^{2})}^{2}\right\\}$
(6)

and regularized by a UV momentum cutoff $\Lambda$.
$\varphi=(\varphi_{1},\cdots,\varphi_{N})$ is a $N$-component field. The model
can be tuned to its critical point by varying $r_{0}$. The correlation
functions are then scale and conformal invariant in the momentum range $|p|\ll
p_{G}$ where $p_{G}\sim u_{0}^{1/(d-4)}$ is the Ginzburg scale. In the
following we shall only be interested in the critical point and the scaling
limit $|p|\ll p_{G}$; we refer to [56] for an overview of the various regimes
of the $O\left(N\right)$ model.

We focus on the operators

$\begin{split}{\cal O}_{1}(x)&={\cal N}_{1}\varphi_{i}(x),\\\ {\cal
O}_{2}(x)&={\cal N}_{2}\frac{\varphi(x)^{2}}{2}\end{split}$ (7)

(the index $i$ is arbitrary) and the OPE coefficient $c_{112}$. Note that the
correlation functions $\langle{\cal O}_{1}{\cal O}_{1}\rangle$ and
$\langle{\cal O}_{1}{\cal O}_{1}{\cal O}_{2}\rangle$ are independent of $i$ at
criticality and in the whole disordered phase. ${\cal N}_{1}$ and ${\cal
N}_{2}$ are normalization constants that ensure that $\langle{\cal
O}_{1}(p){\cal O}_{1}(-p)\rangle$ and $\langle{\cal O}_{2}(p){\cal
O}_{2}(-p)\rangle$ are given by 3 in the scaling limit $|p|\ll p_{G}$. Even
though ${\cal O}_{2}$ is not, strictly speaking, a scaling operator, it can be
expressed as a linear combination of scaling operators, among which that
associated with $\Delta_{2}$. In the scaling limit, corrections due to higher
scaling dimension operators are suppressed and we neglect them; a more
detailed explanation regarding this point is provided at the end of Section
II.4. The scaling dimension

$\Delta_{1}=[\varphi_{i}]=\frac{d-2+\eta}{2}$ (8)

is related to the anomalous dimension $\eta$ while

$\Delta_{2}=[\varphi^{2}]=d-\frac{1}{\nu}$ (9)

where $\nu$ is the correlation-length exponent.

To deal with the composite field ${\cal O}_{2}$, in addition to the linear
source $J$ we introduce in the partition function a source $h$ coupled to
$\varphi^{2}$,

${\cal Z}[J,h]=\int{\cal
D}[\varphi]\,e^{-S[\varphi]+\int_{x}(J\varphi+h\varphi^{2})}.$ (10)

The correlation functions of interest, besides the propagator
$G(x-y)=\langle\varphi_{i}(x)\varphi_{i}(y)\rangle_{c}$ for $h=0$
($\langle\cdots\rangle_{c}$ stands for the connected correlation function),
are the scalar susceptibility

$\chi_{s}(x-y)=\langle\varphi(x)^{2}\varphi(y)^{2}\rangle_{c}=\frac{\delta^{2}\ln{\cal
Z}[J,h]}{\delta h(x)\delta h(y)}\biggr{|}_{J=h=0}$ (11)

and the three-point function

$\displaystyle\chi(x,y,z)$
$\displaystyle=\langle\varphi_{i}(x)\varphi_{i}(y)\varphi(z)^{2}\rangle_{c}$
$\displaystyle=\frac{\delta^{3}\ln{\cal Z}[J,h]}{\delta J_{i}(x)\delta
J_{i}(y)\delta h(z)}\biggr{|}_{J=h=0}.$ (12)

Here and in the following, there is no implicit summation over the index $i$.
The computation of $G(p)$ and $\chi_{s}(p)$ allows us to determine the
normalization constants ${\cal N}_{1}$ and ${\cal N}_{2}$ since at criticality

$\begin{split}G(p)&=\frac{1}{{\cal
N}_{1}^{2}}\frac{A_{d}(\Delta_{1})}{|p|^{d-2\Delta_{1}}},\\\
\chi_{s}(p)&=\frac{4}{{\cal
N}_{2}^{2}}\frac{A_{d}(\Delta_{2})}{|p|^{d-2\Delta_{2}}}\end{split}$ (13)

for $p\to 0$. The knowledge of $\chi(p_{1},p_{2},-p_{1}-p_{2})$ then yields
the OPE coefficient $c_{112}$ using 5.

###  II.3 Effective action

The effective action

$\Gamma[\phi,h]=-\ln{\cal Z}[J,h]+\int_{x}\sum_{i}J_{i}\phi_{i}$ (14)

is defined as the Legendre transform of the free energy [57]. The source $J$
and the order parameter field $\phi$ are related by

$\phi_{i}(x)=\frac{\delta\ln{\cal Z}[J,h]}{\delta J_{i}(x)}\quad\mbox{or}\quad
J_{i}(x)=\frac{\delta\Gamma[\phi,h]}{\delta\phi_{i}(x)}.$ (15)

All correlation functions for $h=0$ can be obtained from the one-particle
irreducible (1PI) vertices

$\Gamma^{(n,m)}_{i_{1}\cdots i_{n}}(x_{1}\cdots x_{n},y_{1}\cdots y_{m})=\\\
\frac{\delta^{n+m}\Gamma[\phi,h]}{\delta\phi_{i_{1}}(x_{1})\cdots\delta\phi_{i_{n}}(x_{n})\delta
h(y_{1})\cdots\delta h(y_{m})}\biggr{|}_{\phi=h=0}$ (16)

where, assuming the absence of spontaneously broken symmetry, we have set
$\phi=0$. In particular, the propagator

$G(p)=\Bigl{[}\Gamma^{(2,0)}_{ii}(p)\Bigr{]}^{-1}$ (17)

is related to the inverse of the two-point vertex computed with a vanishing
source $h=0$. The other two correlation functions of interest are given by
[51]

$\begin{split}\chi_{s}(p)&=-\Gamma^{(0,2)}(p),\\\
\chi(p_{1},p_{2})&=-G(p_{1})\Gamma^{(2,1)}_{ii}(p_{1},p_{2})G(p_{2}),\end{split}$
(18)

where we have used the fact that $\Gamma^{(1,1)}_{i}$ vanishes when evaluated
for $\phi=0$. To alleviate the notations we do not write the last argument of
the three-point vertices, e.g.,
$\Gamma^{(2,1)}_{ii}(p_{1},p_{2})\equiv\Gamma^{(2,1)}_{ii}(p_{1},p_{2},-p_{1}-p_{2})$.

We are now in a position to relate the OPE coefficient $c_{112}$ to the 1PI
vertices at criticality. From Eqs. (13) we obtain the normalization constants

$\displaystyle{\cal N}_{1}^{2}$ $\displaystyle=A_{d}(\Delta_{1})\lim_{p\to
0}\frac{\Gamma^{(2,0)}_{ii}(p)}{|p|^{d-2\Delta_{1}}},$ (19)
$\displaystyle{\cal N}_{2}^{2}$ $\displaystyle=-4A_{d}(\Delta_{2})\lim_{p\to
0}\frac{|p|^{2\Delta_{2}-d}}{\Gamma^{(0,2)}(p)}.$ (20)

Considering 5 in the limit $p_{2}=0$ and $p_{1}\to 0$, we finally deduce

$c_{112}=-\frac{{\cal N}_{2}}{2{\cal
N}_{1}^{2}}\frac{A_{d}(\Delta_{1})}{A_{d}(\Delta_{2}/2)}\lim_{p\to
0}\frac{\Gamma^{(2,1)}_{ii}(p,0)}{|p|^{\Delta_{2}-2\Delta_{1}}}.$ (21)

Equations 19, 20 and 21 are the basic ingredients to determine the OPE
coefficient $c_{112}$ in the effective action formalism. In Appendices A and
B, we show that they yield the known results

$c_{112}=\sqrt{\frac{2}{N}}$ (22)

for the free theory ($u_{0}=0$), and

$c_{112}=\sqrt{\frac{2}{N}}\left[\frac{\Gamma(d-2)}{(d/2-2)}\frac{\sin(\pi
d/2)}{\pi}\right]^{1/2}\frac{1}{\Gamma(d/2-1)}$ (23)

in the large-$N$ limit to the leading order and for $d<4$, in agreement with
the literature [58, 59].

###  II.4 FRG formalism and BMW approximation

The nonperturbative FRG allows one to compute the effective action beyond
standard perturbation theory [1, 2, 3]. Fluctuations are regularized by the
infrared regulator term

$\Delta
S_{k}[\varphi]=\frac{1}{2}\int_{q}\sum_{i}\varphi_{i}(-q)R_{k}(q)\varphi_{i}(q),$
(24)

where the momentum scale $k$ varies from the UV cutoff $\Lambda$ down to zero.
A possible choice for the cutoff function $R_{k}$ is

$\displaystyle R_{k}(q)=Z_{k}q^{2}r\left(\frac{q^{2}}{k^{2}}\right),$ (25)

with the function $r(y)$ taken to be for instance

$\displaystyle r_{\mathrm{W}}(y)=\frac{\alpha}{e^{y}-1}\quad\text{or}\quad
r_{\mathrm{E}}(y)=\alpha e^{-y}/y.$ (26)

$r_{\mathrm{W}}(y)$ and $r_{\mathrm{E}}(y)$ define respectively the so-called
Wetterich and exponential regulators. In either case, $\alpha$ is a constant
of order one and $Z_{k}$ a field renormalization factor which varies as
$k^{-\eta}$ at criticality 222We do not introduce a source renormalization
factor, see Ref. [29] for a FRG approach that introduces explicitly the
anomalous dimension of the composite operator. . Thus the regulator suppresses
fluctuations with momenta $|q|\lesssim k$ but leaves unaffected those with
$|q|\gtrsim k$. The partition function

${\cal Z}_{k}[J,h]=\int{\cal D}[\varphi]\,e^{-S[\varphi]-\Delta
S_{k}[\varphi]+\int_{x}(J\varphi+h\varphi^{2})}$ (27)

is now $k$ dependent. The scale-dependent effective action

$\Gamma_{k}[\phi,h]=-\ln{\cal Z}_{k}[J,h]+\int_{x}\sum_{i}J_{i}\phi_{i}-\Delta
S_{k}[\phi]$ (28)

is defined as a slightly modified Legendre transform which includes the
subtraction of $\Delta S_{k}[\phi]$. Assuming that for $k=\Lambda$ the
fluctuations are completely frozen by the regulator term,

$\Gamma_{\Lambda}[\phi,h]=S[\phi]-\int_{x}h\phi^{2}.$ (29)

On the other hand, the effective action of the $\mathrm{O}(N)$ model 6 is
given by $\Gamma_{k=0}$ since $R_{k=0}$ vanishes. The FRG approach aims at
determining $\Gamma_{k=0}$ from $\Gamma_{\Lambda}$ using Wetterich’s equation
[61, 62, 63]

$\partial_{k}\Gamma_{k}[\phi,h]=\frac{1}{2}{\rm
Tr}\left\\{\partial_{k}R_{k}(\Gamma^{(2,0)}_{k}[\phi,h]+R_{k})^{-1}\right\\}.$
(30)

The infinite hierarchy of flow equations satisfied by the $k$-dependent 1PI
vertices $\Gamma_{k}^{(n,m)}$ can be obtained from 30 by taking functional
derivatives wrt $\phi$ and $h$. The presence of the source $h$ in addition to
the field $\phi$ allows one to follow the flow of composite fields, an
approach which proved to be useful in tackling a wide range of issues [64, 65,
66, 51, 67, 68, 69, 70, 71, 72].

Figure 1: Diagrammatic representation of the RG equations of
$\Gamma^{(2,0)}_{k}$, $\Gamma^{(0,2)}_{k}$ and $\Gamma^{(1,1)}_{k}$. Signs and
symmetry factors are not shown. The vertex $\Gamma^{(n,m)}_{k}$ is represented
by a black dot with $n$ solid lines and $m$ wavy lines and the solid lines
connecting vertices stand for the propagator
$G_{k}=(\Gamma_{k}^{(2,0)}+R_{k})^{-1}$. The cross stands for
$\partial_{k}R_{k}$.

In the BMW approximation [48, 49, 50], one considers the flow equations of the
1PI vertices in a uniform field $\phi$ even if one is eventually interested in
the vanishing field configuration. These equations are shown diagrammatically
in Fig. 1 for $\Gamma^{(2,0)}_{k}$, $\Gamma^{(0,2)}_{k}$ and
$\Gamma_{k}^{(1,1)}$. Since the regulator $\partial_{k}R_{k}$ in Eq. 30
restricts the loop momentum to small values $|q|\lesssim k$, whereas the
regulator term $\Delta S_{k}$ ensures that the vertices are regular functions
of $p_{i}^{2}/k^{2}$, one can set $q=0$ in the vertices $\Gamma_{k}^{(n,m)}$.
Noting then that a vertex with a vanishing momentum can be related to a lower-
order vertex, e.g.,

$\begin{split}\Gamma^{(3,0)}_{k,ijl}({\bf p},-{\bf
p},0;\phi)&=\frac{\partial\Gamma^{(2,0)}_{k,ij}({\bf
p};\phi)}{\partial\phi_{l}},\\\ \Gamma^{(2,1)}_{k,ij}({\bf p},0,-{\bf
p};\phi)&=\frac{\partial\Gamma^{(1,1)}_{k,i}({\bf
p};\phi)}{\partial\phi_{j}},\end{split}$ (31)

we obtain a closed set of equations satisfied by $\Gamma_{k}^{(2,0)}({\bf
p},\phi)$, $\Gamma_{k}^{(0,2)}({\bf p},\phi)$ and $\Gamma_{k}^{(1,1)}({\bf
p},\phi)$; see Ref. [51] for the explicit expressions. These equations,
together with the expression 31 of $\Gamma^{(2,1)}_{k}({\bf p},0,-{\bf
p};\phi)$ are sufficient to obtain the vertices necessary to determine the
normalization constants ${\cal N}_{1}$, ${\cal N}_{2}$ and the OPE coefficient
$c_{112}$.

The reader may wonder if the simple choice of bare operators made in (7) is
suitable to address our objectives. Thus, let us expound on some fundamental
properties of composite operators within the FRG formalism. In the FRG
framework a composite operator can be defined by differentiating the effective
action $\Gamma_{k}$ wrt an external source. For instance, the running
composite operator corresponding to ${\cal O}_{2}$ is defined as
$\left[\phi^{2}\right]\left(k\right)\equiv-\frac{\delta\Gamma_{k}}{\delta
h}[\phi,h]|_{h=0}$. $\left[\phi^{2}\right]\left(k\right)$ depends on the RG
scale $k$ and, if evaluated at the UV cutoff scale, satisfies
$\left[{\phi^{2}}\right]\left(\Lambda\right)=\phi^{2}$. As soon as one lowers
the scale $k$ from $\Lambda$, the bare term gets renormalized through its
coupling to the 1PI vertices. This implies that the flow of the composite
operator generates mixing with other operators in the sense that
$\left[\phi^{2}\right]\left(k\right)=Z_{22,k}\phi^{2}+Z_{24,k}\phi^{4}+\cdots$.

It must be noted, however, that scaling operators are not just mere composite
operators. A scaling operator is a particular combination of composite
operators that diagonalize the flow linearized about the fixed point. As a
consequence, in the scaling regime, $\left[\phi^{2}\right]\left(k\right)$ can
be expressed as a linear combination of scaling operators. However, by
lowering the RG scale $k$ to $0$ in such a linear combination, the scaling
operators of higher scaling dimension $\Delta_{i}$ are suppressed and only the
lowest scaling operator survives. Indeed, numerically solving the flow of e.g.
$\Gamma_{k}^{(0,2)}$ within our approximation scheme, we have been able to
check that our procedure reproduces the expected behavior of a scaling
operator in the fixed point regime 333This reasoning could not be applied if
one were interested in some other scaling operator, say the one having scaling
dimension $\Delta_{4}$..

Let us conclude by noticing that the dependence of the vertices
$\Gamma_{k}^{(2,0)}$, $\Gamma_{k}^{(0,2)}$, and $\Gamma_{k}^{(1,1)}$ on the
background field $\phi$ entails an infinite number of 1PI vertices (albeit in
a specific momentum configuration). This shows that our ansatz includes non-
trivial mixing among field monomials $\phi^{2}$, $\phi^{4}$ and so on.

##  III Numerical results

The flow equations are integrated numerically, see e.g. Refs. [51, 50] for
details. We work with dimensionless variables, $\tilde{p}=p/k$ and
$\tilde{\rho}=Z_{k}k^{2-d}\rho$. The field dependence of the potential and the
vertices is discretized on a finite and evenly spaced grid
$\tilde{\rho}\in[0,\tilde{\rho}_{\text{max}}]$ comprising $N_{\rho}$ points,
while the momentum dependence of the vertices is approximated by Chebyschev
polynomials of order $N_{p}$ defined on $[0,\tilde{p}_{\text{max}}]$. The
integration of the flow with respect to the RG scale $k$ is done with an
adaptive step integration. Convergence of the results with respect to the
parameters has been verified; their typical range are $N_{\rho}=40$–$80$,
$\tilde{\rho}_{\text{max}}=4$–$8$, $\tilde{p}_{\text{max}}=4$–$10$ and
$N_{p}=20$–$30$ with the precise value depending on $d$ and $N$.

For each universality class set by $d$ and $N$ and each choice of the cutoff
function 25 parameterized by $\alpha$, the critical point is found by tuning
the initial condition of the flow. This enables the computation of $G(p)$,
$\chi_{s}(p)$ and $\Gamma^{(2,1)}(p,0)$ [Eqs. 13 and 31] at criticality, from
which one fits the values of the critical exponents $\eta$ and $\nu$ (or
equivalently $\Delta_{1,2}$) and normalization constants $\mathcal{N}_{1,2}$,
yielding $c_{112}$ through Eq. 21.

A crucial question is that of the regulator dependence. Indeed, while Eq. 30
is exact, any approximation scheme such as BMW introduces a regulator
dependence to the results. In order to provide a meaningful prediction for a
physical quantity $Q(\alpha)$, a choice of the regulator must be made. The
usual rationale is the so-called principle of minimum sensitivity (PMS),
according to which the best value of $\alpha$ is that for which the regulator
dependence of $Q(\alpha)$ is minimal, i.e., for which
$\partial_{\alpha}Q(\alpha)=0$, or failing that for which
$|\partial_{\alpha}Q(\alpha)|$ is minimal.

Figure 2: Dependence of the critical exponent $\eta$ (top) and the OPE
coefficient $c_{112}$ (bottom) of the $3d$ Ising universality class ($N=1$) on
the coefficient $\alpha$ for the Wetterich (blue) and exponential (orange)
regulators. For reference, the conformal bootstrap estimates
$\eta_{\mathrm{CB}}=0.036308$ and $c_{112,\mathrm{CB}}=1.0518537$ are shown as
black dashed lines [14]. Applying the PMS yields optimal parameters
$\alpha_{\mathrm{E}}=1.6$ and $\alpha_{\mathrm{W}}=2.4$ for the exponential
and Wetterich regulators (respectively subscript E and W), shown by gray
vertical lines, with corresponding values $\eta_{\mathrm{E}}=0.0402$,
$c_{112,\mathrm{E}}=1.027$, $\eta_{\mathrm{W}}=0.0394$ and
$c_{112,\mathrm{W}}=1.039$. The value we retain for $c_{112}$ is
$c_{112,\mathrm{W}}$, given that it corresponds to the extremal value of
$\eta$ across both families of regulators.

However, the PMS for $c_{112}$, shown for the $3d$ Ising universality ($N=1$)
class in Fig. 2, does not provide a satisfactory result. Indeed, for a given
regulator, $c_{112}$ is a monotonous concave function of $\alpha$, with no
extremum or inflection point, varying by about $3\%$ over the range of
regulators considered. As a consequence, we choose the regulator that fulfills
the PMS for the anomalous dimension $\eta$. The value thus obtained for
$c_{112}$ depends only weakly on the family of regulators considered, with a
variation of about $1.15\%$ between the Wetterich and exponential regulators.
The regulator dependency is slightly smaller than the $1.2\%$ difference with
the conformal bootstrap estimate.

As a side note, we point out a recent proposal for an alternative way to fix
the regulator dependence for conformally invariant theories, the principle of
maximal conformality (PMC) [74]. Conformal invariance implies a set of
(modified) Ward identities associated with scale and special conformal
transformations (SCT). While scale invariance is always fulfilled at the fixed
point, invariance under SCT is broken within the derivative expansion at high
order. PMC suggests to choose the regulator that minimizes the symmetry
breaking. While in the present case it is not straightforward to implement the
PMC for BMW, because the Ward identities are either trivially fulfilled or
involve high-order vertices that cannot be computed using the BMW
approximation, its implementation for the derivative expansion shows that the
PMS for $\eta$ and PMC yield very close results, providing a further argument
in favor of our regulator choice.

###  III.1 Ising university class in dimensions $2\leq d\leq 4$

We first consider the OPE coefficients of the Ising university class ($N=1$)
for dimensions $d$ between the lower and upper critical dimensions $d=2$ and
$4$, for which the results are shown in Fig. 3 and Table 1. The FRG results
can be compared to the exact values in $d=2$ and $4$, the conformal bootstrap
[14] and Monte Carlo [75] estimates in $d=3$ and the $\epsilon$ expansion up
to fourth order [76, 77, 78, 79],

$\displaystyle c_{112}={}$
$\displaystyle\sqrt{2}\bigg{(}1-\frac{1}{6}\epsilon-\frac{77}{648}\epsilon^{2}$
$\displaystyle{}+\frac{3726\zeta(3)-1915}{34992}\epsilon^{3}+A_{4}\epsilon^{4}\bigg{)}+\mathcal{O}(\epsilon^{5})$
(32)

where $\zeta(3)$ is Apéry’s constant and $A_{4}=-0.158947\ldots$ is only known
numerically [79].

Owing to its asymptotic nature, the $\epsilon$ expansion does not converge;
indeed, for $d=3$, its relative error to all estimates (FRG, conformal
bootstrap, Monte Carlo) increases from $4\%$ at order ${\cal O}(\epsilon^{2})$
to $6\%$ at order ${\cal O}(\epsilon^{3})$ and $16\%$ at order ${\cal
O}(\epsilon^{4})$. In order to make sense of the results, a resummation
procedure must be carried out, for instance by approximating
$c_{112}(\epsilon)$ by a Padé approximant of order $(n,m)$, i.e. a rational
fraction of $\epsilon$ with numerator and denominator of degree $n$ and $m$,
respectively. Following Ref. [79], we pick an approximant of order $(3,2)$,
whose coefficients are uniquely determined by imposing the ${\cal
O}(\epsilon^{4})$ expansion around $\epsilon=0$ and the exact value
$c_{112}(\epsilon=2)=1/2$. This gives $c_{112}=1.0507$ for $d=3$, in very good
agreement with the conformal bootstrap, to be compared to the $0.8\%$ error
when the approximant of order $(3,1)$ is used and the exact result for $d=2$
is not imposed. Let us mention that in this case different choices of Padé
approximants lead to somewhat different results, which may be used to estimate
an average value and its uncertainty: $c_{112}=1.048(31)$, see Table 1

Figure 3: OPE coefficient $c_{112}$ of the Ising university class as a function of the dimension $d$. The solid red crosses are obtained from FRG, with the full line a guide to the eye. The black symbols correspond to reference estimates from Monte Carlo (square) [75] and conformal bootstrap (diamond) [14, 80] and the exact values in $d=2$ and $4$. The gray lines are given by the $\epsilon$ expansion [Eq. 32] about $d=4$ to order ${\cal O}(\epsilon)$ (dashed), ${\cal O}(\epsilon^{2})$ (dashdotted) and the $(3,2)$ resummation imposing the $2d$ result (dotted) [79]. $d$ | $2$ | $3$ | $4$
---|---|---|---
FRG | $0.484$ | $1.039$ | $1.413$
$\epsilon$ exp. $(3,1)$ | $0.4259$ | $1.0432$ | $\sqrt{2}\simeq 1.414$
$\epsilon$ exp. $(2,2)$ | $-0.0698$ | $1.0200$ | $\sqrt{2}\simeq 1.414$
$\epsilon$ exp. $(1,3)$ | $0.7442$ | $1.0805$ | $\sqrt{2}\simeq 1.414$
$\epsilon$ exp.${}+2d$ (3,2) |  | $1.0507$ | $\sqrt{2}\simeq 1.414$
$\epsilon$ exp.${}+2d$ (2,3) |  | $1.0464$ | $\sqrt{2}\simeq 1.414$
MC [75] |  | $1.07(3)$ |
CB [14] |  | $1.051\,853\,7(41)$ |
Exact [55] | $1/2=0.5$ |  | $\sqrt{2}\simeq 1.414$
Table 1: OPE coefficient $c_{112}$ of the Ising universality class for
dimensions $d=2$, $3$, $4$. We compare the numerical FRG results to: (i)
various resummations of the $\epsilon$ expansion [Eq. 32] to order ${\cal
O}(\epsilon^{4})$ obtained by employing different Padé approximants and by
possibly taking into account the exactly known 2$d$ value, (ii) conformal
bootstrap and (iii) Monte Carlo estimates and (iv) the exact values for the
$2d$ and $4d$ Ising universality classes.

Compared to the best results (exact in $d=2$ and $4$, conformal bootstrap and
Monte Carlo in $2<d<4$), the FRG always has an error smaller than $3\%$, with
less than $2\%$ error in $d=3$. By contrast with the resummed $\epsilon$
expansion, which requires additional input in the form of the $2d$ data to
provide accurate results, the FRG and conformal bootstrap are able to
interpolate smoothly between dimensions $d=2$ and $4$. As the dimension is
increased, $c_{112}$ increases monotonously, with an almost linear behavior
between $d=2$ and $d=3$, which might partly explain the remarkable agreement
of the $\epsilon$ expansion resummation (when supplemented with the exact $2d$
result) with conformal bootstrap, Monte Carlo and FRG. However, if the
resummation is not supplemented with the $2d$ result then the
$\epsilon$-expansion estimate of $c_{112}$ is rather poor close to and at
$d=2$.

In $d=4$, the FRG within the BMW approximation scheme gives the exact analytic
value of $c_{112}$. The small difference ($\sim 0.1\%$) between the numerical
result and the exact value seen in Table 1 arises from the fitting of the
critical exponents and the normalization constants. This serves as an estimate
of this numerical error: in lower dimensions, it is much smaller than the
difference to the best estimates.

###  III.2 Three-dimensional $\mathrm{O}(N)$ model

We now focus on the three-dimensional $\mathrm{O}(N)$ model. Given that the
large-$N$ result is [58, 59, 81]

$c_{112}=\dfrac{2}{\pi}\dfrac{1}{\sqrt{N}}+\dfrac{24}{\pi^{3}}\dfrac{1}{N^{3/2}}+{\cal
O}\bigg{(}\dfrac{1}{N^{5/2}}\bigg{)},$ (33)

we consider rather than $c_{112}$ the rescaled OPE coefficient
$\sqrt{N}c_{112}$ that has a well-defined large-$N$ limit. FRG results and
estimates from the $\epsilon$ expansion [78], conformal bootstrap [14, 17, 82]
and Monte Carlo [75, 83] are shown in Fig. 4 and Table 2. For $N>1$, $c_{112}$
is only known up to order ${\cal O}(\epsilon^{3})$ and we resum the $\epsilon$
expansion using a $(2,1)$ Padé approximant.

Figure 4: Rescaled OPE coefficient $\sqrt{N}c_{112}$ of the three-dimensional $\mathrm{O}(N)$ model as a function of the inverse number of field components $1/N$. The solid red crosses are obtained from FRG. The horizontal dashed green line shows the large-$N$ result to leading order. The gray lines are given by the order ${\cal O}(\epsilon)$ (dashed), ${\cal O}(\epsilon^{2})$ (dashdotted) and the $(2,1)$ resummation (dotted) of the $\epsilon$ expansion [78]. The black symbols correspond to estimates from Monte Carlo (square) [75, 83] and conformal bootstrap (diamond) [14, 17, 82].  $N$ | FRG |  $\epsilon$ exp. | MC | CB
---|---|---|---|---
$1$ | $1.039$ |  $1.075$ [78] |  $1.07(3)$ [75] |  $1.0518537(41)$ [14]
$2$ | $0.943$ |  $1.002$ [78] |  $0.9731(14)$ [83] |  $0.971743(38)$ [17]
$3$ | $0.884$ |  $0.943$ [78] |  |  $0.908047(102)$ [82]
$4$ | $0.829$ |  $0.895$ [78] |  |
$5$ | $0.792$ |  $0.855$ [78] |  |
$6$ | $0.766$ |  $0.823$ [78] |  |
$8$ | $0.731$ |  $0.773$ [78] |  |
$10$ | $0.709$ |  $0.737$ [78] |  |
$100$ | $0.643$ |  $0.607$ [78] |  |
$1000$ | $0.638$ |  $0.603$ [78] |  |
Table 2: Rescaled OPE coefficient $\sqrt{N}c_{112}$ of the three-dimensional
$\mathrm{O}(N)$ model for different numbers of field components $N$. We
compare the FRG results to the $(2,1)$ resummation of the $\epsilon$ expansion
to order ${\cal O}(\epsilon^{3})$, conformal bootstrap and Monte Carlo
estimates. The exact large-$N$ result is
$\lim_{N\to\infty}\sqrt{N}c_{112}=2/\pi\simeq 0.637$.

For $N=1$, $2$, $3$, FRG differs from conformal bootstrap by respectively
$1.2\%$, $3.0\%$ and $2.6\%$. Furthermore FRG accurately reproduces the
large-$N$ behavior: for $N=1000$, the FRG estimate $\sqrt{N}c_{112}=0.638$
differs from the exact large-$N$ result
$\lim_{N\to\infty}\sqrt{N}c_{112}=2/\pi\simeq 0.637$ by $0.1\%$, which is
about the order of magnitude corresponding to a $1/N$ correction. This is
expected as it is known that the relevant vertices are exact in the large-$N$
limit [48, 51]. By contrast, the $(2,1)$ resummation of the $\epsilon$
expansion gives
$\lim_{N\to\infty}\sqrt{N}c_{112}=\sqrt{2}(4\zeta(3)+1)/(8\zeta(3)+4)\simeq
0.603$ with a $5.3\%$ error.

Moreover, numerically fitting the FRG data for $N\gtrsim 5$ by a law of the
form $c_{112}=2/\pi\sqrt{N}+\kappa/N^{3/2}$ yields $\kappa\simeq 0.76$, in
very good agreement with the exact value $24/\pi^{3}\simeq 0.774$.

##  IV Conclusion

We have shown how to extract the OPE coefficients of a conformal theory within
the framework of FRG, by determining three-point vertices in specific momentum
configurations. We have used our approach to determine the $c_{112}$
coefficient, corresponding to the simplest possible OPE coefficient, in the
$\mathrm{O}(N)$ universality class for various $d$ and $N$. This provides the
first non-perturbative determination of the OPE coefficients based on field
theory, aside from the lattice computations in [75, 83] and the conceptually
different conformal bootstrap.

While the accuracy of FRG can be sometimes difficult to gauge in the absence
of a small expansion parameter, the fact that the results compare extremely
well with the values, when available, obtained from Monte Carlo and conformal
bootstrap increases confidence in the validity of the method. It is a
testament to the versatility of FRG that, in this specific case, tuning such
parameters as $d$ or $N$ demands relatively little effort as they only enter
the flow equations through their explicit values.

Lastly, we note that the OPE can be used in settings very different from the
critical $\mathrm{O}(N)$ models investigated in this work, for instance in
theories away from a fixed point or at a non-equilibrium fixed point. In these
cases many methods holding for equilibrium critical theories are not
available. Our work suggests that the FRG may constitute the right framework
to tackle these issues thanks to its aforementioned versatility and to the
fact that the FRG equations can be solved without invoking further
requirements, such as conformal symmetry.

###### Acknowledgements.

The authors thank Gonzalo De Polsi, Matthieu Tissier and Nicolás Wschebor for
stimulating discussions and are extremely grateful towards Johan Henriksson
for pointing out Refs. [59, 80, 79]. C. P. and N. D. wish to thank the
organizers of the meeting FRGIM 2019 where some of the ideas developed in this
work were discussed. C. P. thanks Hidenori Sonoda for discussions and
collaboration on closely related projects and the Laboratoire de Physique
Théorique de la Matière Condenséee in Paris for hospitality. C. P.
acknowledges support by the DFG grant PA 3040/3-1. F. R. acknowledges support
from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
under Germany’s Excellence Strategy–EXC–2111–390814868.

##  Appendix A $c_{112}$ in the free case

When $u_{0}$ vanishes, the functional integral over the field $\varphi$ can be
done exactly and yields the partition function

$Z[J,h]=e^{\frac{N}{2}{\rm Tr}\ln
G[h]+\frac{1}{2}\int_{x,y}\sum_{i}J_{i}(x)G[x,y;h]J_{i}(y)},$ (34)

where $G[h]$ denotes the propagator in the presence of an arbitrary external
source $h$:

$G^{-1}[x,y;h]=\bigl{(}-\nabla^{2}_{x}+2h(x)\bigr{)}\delta(x-y).$ (35)

The expectation value of the field is given by

$\phi_{i}(x)=\int_{y}G[x,y;h]J_{i}(y),$ (36)

and the effective action is simply

$\Gamma[\phi,h]=\int_{x}\left\\{\frac{1}{2}({\partial_{\mu}}\phi)^{2}-h\phi^{2}\right\\}-\frac{N}{2}{\rm
Tr}\ln G[h].$ (37)

We thus obtain

$\displaystyle\Gamma^{(2,0)}(p)$ $\displaystyle=p^{2},$ (38)
$\displaystyle\Gamma^{(0,2)}(p)$
$\displaystyle=-2N\int_{q}\frac{1}{q^{2}(p+q)^{2}}\simeq-\frac{2NB_{d}}{|p|^{4-d}},$
(39) $\displaystyle\Gamma^{(2,1)}(p_{1},p_{2})$ $\displaystyle=-2,$ (40)

where

$B_{d}=\frac{1}{(4\pi)^{d/2}}\frac{\Gamma(2-d/2)\Gamma(d/2-1)^{2}}{\Gamma(d-2)}.$
(41)

The last expression for $\Gamma^{(0,2)}(p)$ is obtained for $p\to 0$ and
$d<4$. From (38) and (19) we deduce $\Delta_{1}=d/2-1$ (in agreement with the
vanishing anomalous dimension in the free theory) and

${\cal N}_{1}^{2}=A_{d}(d/2-1).$ (42)

On the other hand, comparing (39) with (20) yields $\Delta_{2}=d-2$ (in
agreement with $\nu=1/2$) and

$\displaystyle\frac{1}{{\cal N}_{2}^{2}}$
$\displaystyle=\frac{N}{2}\frac{B_{d}}{A_{d}(d-2)}$
$\displaystyle=\frac{N}{32\pi^{d}}\Gamma(d/2-1)^{2}.$ (43)

From (21) and (40) we finally obtain (22).

##  Appendix B $c_{112}$ in the large-$N$ limit

Following the Appendix of Ref. [51], we introduce the field $\rho=\varphi^{2}$
and a Lagrange multiplier $\lambda$ to write the partition function ${\cal
Z}[h]\equiv{\cal Z}[J=0,h]$ of the $\mathrm{O}(N)$ model as

$\displaystyle{\cal Z}[h]={}$ $\displaystyle\int{\cal
D}[\varphi,\rho,\lambda]\exp\biggl{\\{}-\int_{x}\Bigl{[}\frac{1}{2}(\boldsymbol{\nabla}\varphi)^{2}$
$\displaystyle+\left(\frac{r_{0}}{2}-h\right)\rho+\frac{u_{0}}{4!N}\rho^{2}+i\frac{\lambda}{2}(\varphi^{2}-\rho)\Bigr{]}\biggr{\\}}$
$\displaystyle={}$ $\displaystyle\int{\cal
D}[\varphi,\lambda]\exp\biggl{\\{}\int_{x}\biggl{[}\frac{3N}{2u_{0}}(2h+i\lambda-
r_{0})^{2}$
$\displaystyle-\frac{1}{2}\left[(\nabla\varphi)^{2}+i\lambda\varphi^{2}\right]\biggr{\\}}.$
(44)

Then we split the field $\varphi$ into a $\sigma$ field and an
$(N-1)$-component field $\pi$. Integrating over the $\pi$ field, we obtain the
action

$S[\sigma,\lambda,h]=\int_{x}\biggl{[}-\frac{3N}{2u_{0}}(2h+i\lambda-
r_{0})^{2}\\\
+\frac{1}{2}\left[(\nabla\sigma)^{2}+i\lambda\sigma^{2}\right]+\frac{N-1}{2}{\rm
Tr}\ln g^{-1}[\lambda],$ (45)

where

$g^{-1}[x,x^{\prime};\lambda]=[-\nabla_{x}^{2}+i\lambda(x)]\delta(x-x^{\prime})$
(46)

is the inverse propagator of the field $\pi_{i}$ in the fluctuating $\lambda$
field. In the limit $N\to\infty$, the action becomes proportional to $N$ (if
one rescales the $\sigma$ field, $\sigma\to\sqrt{N}\sigma$); the saddle point
approximation becomes exact for the partition function ${\cal Z}[h]$ and the
Legendre transform of the free energy coincides with the action $S$ [84]. This
implies that the effective action is simply equal to $S[\sigma,\lambda,h]$:

$\Gamma[\sigma,\lambda,h]=\int_{x}\biggl{\\{}-\frac{3N}{2u_{0}}(2h+i\lambda-
r_{0})^{2}\\\
+\frac{1}{2}\left[(\nabla\sigma)^{2}+i\lambda\sigma^{2}\right]\biggr{\\}}+\frac{N}{2}{\rm
Tr}\ln g^{-1}[\lambda]$ (47)

(we use $N-1\simeq N$ for large $N$). We can eliminate the Lagrange multiplier
$\lambda$ using

$\frac{\delta\Gamma[\sigma,\lambda,h]}{\delta\lambda(x)}\biggl{|}_{\lambda=\lambda[\sigma,h]}=0$
(48)

to obtain the effective action
$\Gamma[\sigma,h]\equiv\Gamma[\sigma,\lambda[\sigma,h],h]$, which is the
starting point to compute the vertices $\Gamma^{(n,m)}$ in the large-$N$
limit.

In Ref. [51] it was shown that, at criticality,

$\begin{split}\Gamma^{(2,0)}(p)&=p^{2},\\\
\Gamma^{(0,2)}(p)&=-\frac{12N}{u_{0}}+\left(\frac{6N}{u_{0}}\right)^{2}\Gamma^{(2)}_{\lambda\lambda}(p)^{-1},\end{split}$
(49)

where

$\begin{split}\Gamma^{(2)}_{\lambda\lambda}(p)&=\frac{3N}{u_{0}}+\frac{N}{2}\Pi(p),\\\
\Pi(q)&=\int_{q}\frac{1}{q^{2}(p+q)^{2}}\simeq\frac{B_{d}}{|p|^{4-d}}\end{split}$
(50)

for $p\to 0$ and $d<4$, where $B_{d}$ is defined in (41). Calculating the
three-point function along the same lines, one finds

$\displaystyle\Gamma^{(2,1)}(p_{1},p_{2})$
$\displaystyle=-\frac{6N}{u_{0}}\Gamma^{(2)}_{\lambda\lambda}(p_{1}+p_{2})^{-1}$
$\displaystyle\simeq-\frac{12|p_{1}+p_{2}|^{4-d}}{u_{0}B_{d}}$ (51)

for $p_{1}+p_{2}\to 0$ and $d<4$. From (19) and (20) one then obtains
$\Delta_{1}=d/2-1$ and $\Delta_{2}=2$ (in agreement with the large-$N$ results
$\eta=0$ and $\nu=1/(d-2)$ to leading order) and

$\begin{split}{\cal N}_{1}^{2}&=A_{d}(d/2-1),\\\ {\cal
N}_{2}^{2}&=-\frac{u_{0}^{2}B_{d}A_{d}(2)}{18N}.\end{split}$ (52)

Equation (21) then gives

$c_{112}=\left[-\frac{2A_{d}(2)}{NB_{d}}\right]^{1/2}\frac{1}{A_{d}(1)}$ (53)

and in turn (23) using standard properties of the Gamma function.

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