Generalized Gross–Neveu Universality Class with Non-Abelian Symmetry

Generalized Gross–Neveu Universality Class
with Non-Abelian Symmetry††This paper is a contribution to the Special Issue
on Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk
Kreimer for his 60th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Kreimer.html

John A. GRACEY

J.A. Gracey

Theoretical Physics Division, Department of Mathematical Sciences, University
of Liverpool,
P.O. Box 147, Liverpool, L69 3BX, UK<EMAIL_ADDRESS>
Received February 26, 2021, in final form June 18, 2021; Published online June
29, 2021

We use the large $N$ critical point formalism to compute $d$-dimensional
critical exponents at several orders in $1/N$ in an Ising Gross–Neveu
universality class where the core interaction includes a Lie group generator.
Specifying a particular symmetry group or taking the abelian limit of the
final exponents recovers known results but also provides expressions for any
Lie group or fermion representation.

critical exponents; large $N$ expansion; renormalization

81T17; 81T18; 81V25; 82B27

##  1 Introduction

One aspect of quantum field theory that has important applications to Nature
is the study of fixed points of the renormalization group functions. These are
defined to be the non-trivial zeros of the $\beta$-function. Using the
location of a fixed point one can compute the values of the renormalization
group functions there to produce renormalization scheme independent
expressions known as critical exponents [48, 49, 50, 51]. These quantities
govern the dynamics of the phase transitions in a material. Indeed accurate
measurements of the exponents experimentally as well as the symmetry
properties of a material, can equally guide one to the underlying quantum
field theory or spin model describing the dynamics. One property is that more
than one theory can be a valid description of a phase transition. For
instance, both a continuum quantum field theory as well as a discrete spin
model with a common symmetry can be valid tools to provide numerical exponent
estimates. The equivalence of both theoretical techniques at a fixed point is
known as universality [50, 51]. Away from a phase transition each theory will
have different properties and be inequivalent.

One theory that has risen to the fore in this context in recent years is that
of the Ising Gross–Neveu model [1, 19]. This is primarily due to the belief
that it underpins a particular phase transition in graphene. This material is
made up of a sheet of carbon atoms arranged in a hexagonal lattice. When the
two dimensional sheet is stretched it can undergo a transition from an
electrical conductor to what is termed a Mott-insulating phase. On the
theoretical side the Ising Gross–Neveu model can be supplemented with quantum
electrodynamics (QED) to describe aspects of other phase transitions. Equally
if the basic or Ising Gross–Neveu model is endowed with extra symmetries it
contributes to the understanding of transitions in other materials. For
example, what is termed the chiral Heisenberg Gross–Neveu model is an
extension of the Gross–Neveu model to include an ${\rm SU}(2)$ symmetry. It is
the effective theory for electrons on a half-filled honeycomb lattice where
there is a phase transition between an anti-ferromagnetic insulating phase and
a semi-metallic one [2, 20, 36]. Its criticality properties were studied in
[22]. More recently a variation of this version of the Gross–Neveu model,
called the fractionalized Gross–Neveu model, has been developed [35]. It has a
novel spectrum that differs from those of other Gross–Neveu models and has an
associated ${\rm SO}(3)$ symmetry. What is clear from this set of emerging
variations of the Gross–Neveu universality class is the common theme of the
core interaction being modified to include a non-abelian symmetry. In this
respect it is completely parallel to the extension of QED to include a non-
abelian symmetry that equates to quantum chromodynamics (QCD) or Yang–Mills
theory with fermions in the fundamental representation of the Lie group
responsible for colour symmetry. The only difference with the Gross–Neveu
class of theories is in the Lorentz structure of the core interaction. As QCD
has been studied at length using a general Lie group symmetry rather than
specifying ${\rm SU}(3)$ at the outset, which is the group that governs the
strong interactions, it seems sensible to develop a programme to calculate in
the Gross–Neveu model with a parallel non-abelian symmetry. Then the
properties of the various physical applications can be deduced by specifying
the appropriate group parameters.

This is the main task here. We will consider a generalized Gross–Neveu
universality class with a non-abelian symmetry and calculate the critical
exponents of the theory. This will be achieved by using the critical point
large $N$ formalism pioneered in [42, 43, 44] for the nonlinear $O(N)$
$\sigma$ model. Here $N$ would correspond to the number of quark flavours in
the analogy with QCD. The elegance of the approach is such that we can deduce
the critical exponents in $d$ spacetime dimensions as a function of the non-
abelian symmetry group Casimirs. The fixed point associated with the formalism
is the Wilson–Fisher fixed point in $d$-dimensions [50]. As the exponents are
renormalization group invariants their $\epsilon$ expansion near $2$ and $4$
dimensions, where $\epsilon$ measures the difference in these values from $d$,
will agree with the perturbative evaluation of the same exponents in the
respective quantum field theories of the universality class. More usefully
since the dimension of interest for the materials application is three, one
can determine several terms of the $1/N$ series for each exponent. These
provide reasonably accurate estimates for relatively low $N$ when compared to
results from other techniques. However, the benefit of taking the general non-
abelian universality class approach is that estimates will be readily
available if a phase transition with a new symmetry is discovered.

The article is organized as follows. Relevant background concerning the
generalized Gross–Neveu universality class is given in Section 2 together with
the basic critical point large $N$ formalism. Subsequently in Section 3 we
solve the Schwinger–Dyson equations at criticality at $O\big{(}1/N^{2}\big{)}$
to produce the fermion anomalous dimension. Various calculational tools that
are necessary for this are reviewed as well. To provide the groundwork for
finding the next order of this exponent, the anomalous dmension exponent of
the bosonic field is determined at $O\big{(}1/N^{2}\big{)}$ in Section 4. One
of the other basic exponents in critical systems is that relating to the
correlation length behaviour and we determine it at $O\big{(}1/N^{2}\big{)}$
in Section 5. Equipped with these results, the large $N$ conformal bootstrap
formalism at criticality is applied in Section 6 to deduce the fermion
anomalous dimension at $O\big{(}1/N^{3}\big{)}$. We review our results in
Section 7 and provide concluding remarks in Section 8.

##  2 Background

To begin with we recall the Lagrangian of the chiral Heisenberg
Gross–Neveu–Yukawa theory is [20]

$\displaystyle L^{\text{cHGNY}}={\rm
i}\bar{\psi}^{iI}\partial\\!\\!\\!/\psi^{iI}+\frac{1}{2}\partial_{\mu}\tilde{\pi}^{a}\partial^{\mu}\tilde{\pi}^{a}+g_{1}\tilde{\pi}^{a}\bar{\psi}^{iI}T^{a}_{IJ}\psi^{iJ}+\frac{1}{24}g_{2}^{2}\big{(}\tilde{\pi}^{a}\tilde{\pi}^{a}\big{)}^{2}$
$\displaystyle\hphantom{L^{\text{cHGNY}}=}{}+\frac{1}{24}g_{3}^{2}\tilde{\pi}^{a}\tilde{\pi}^{b}\tilde{\pi}^{c}\tilde{\pi}^{d}\mathop{\rm
Tr}\big{(}T^{a}T^{b}T^{c}T^{d}\big{)},$ (2.1)

which is renormalizable in four dimensions, where the three couplings $g_{1}$,
$g_{2}$ and $g_{3}$ are dimensionless. This is a generalization of the
Lagrangian studied in [53] and is in the chiral Heisenberg Gross–Neveu model
universality class. The renormalizability dimension is also termed the
critical dimension of the theory. The scalar-fermion interaction includes the
group generator $T^{a}$ of the Lie algebra and the indices take values in the
ranges $1$ $\leq$ $i$ $\leq$ $N$, $1$ $\leq$ $I$ $\leq$ $N_{\\!c}$ and $1$
$\leq$ $a$ $\leq$ $N_{\\!A}$, where $N$ is the number of flavours of massless
fermions and $N_{\\!c}$ and $N_{\\!A}$ are the respective dimensions of the
fundamental and adjoint representations of the symmetry group. We note that in
[20] the specific group considered was ${\rm SU}(2)$. Within our ultimate
critical exponents the generators will manifest themselves through various
colour Casimirs such as $C_{F}$ and $C_{A}$ defined by

$\displaystyle T^{a}T^{a}=C_{F},\qquad f^{acd}f^{bcd}=C_{A}\delta^{ab},$ (2.2)

where $f^{abc}$ are the structure constants. The scalar field
$\tilde{\pi}^{a}$ plays a subtle role in the construction of the large $N$
expansion but in four dimensions it corresponds to a fundamental propagating
field. The main aspect of the large $N$ critical point formalism of [42, 43,
44] is that in the approach to criticality at the Wilson–Fisher fixed point
the dynamics are driven by the core interaction of the universal quantum field
theory. For (2.1) this is the cubic interaction together with the fermion
kinetic term. These two terms determine the canonical dimensions of both
fields by ensuring the action is dimensionless in $d$-dimensions. In effect
the universal Lagrangian at criticality is

$L^{\mbox{\footnotesize{cHGN}}}={\rm
i}\bar{\psi}\partial\\!\\!\\!/\psi+g\tilde{\pi}^{a}\bar{\psi}T^{a}\psi-\frac{1}{2}\tilde{\pi}^{a}\tilde{\pi}^{a},$

where the quadratic term in $\tilde{\pi}^{a}$ is necessary for large $N$
renormalizability. We will omit the labels $i$ and $I$ for brevity from now
on. We say in effect since at criticality there is no coupling constant in the
sense it is conventionally used in perturbation theory. So the critical point
universal Lagrangian that will be the foundation for applying the large $N$
critical point formalism of [42, 43] is

$L^{\mbox{\footnotesize{univ}}}={\rm
i}\bar{\psi}\partial\\!\\!\\!/\psi+\pi^{a}\bar{\psi}T^{a}\psi-\frac{1}{2g}\pi^{a}\pi^{a},$
(2.3)

where we have rescaled the scalar field $\tilde{\pi}^{a}$ to introduce
$\pi^{a}$. This Lagrangian (2.3) is renormalizable in $d$-dimensions in the
large $N$ formalism [33, 40, 41, 53], where $1/N$ is the dimensionless
ordering parameter since the perturbative coupling constant is absent at
criticality. Ensuring that the $d$-dimensional Lagrangian (2.3) has a
dimensionless action means that $\psi$ has canonical dimension
$\mbox{\small{$\frac{1}{2}$}}(d-1)$ while that of $\pi^{a}$ is unity. For
(2.3) this implies that $g$ is dimensionless in two dimensions after
eliminating the auxiliary field $\pi^{a}$ producing

$L={\rm
i}\bar{\psi}^{iI}\partial\\!\\!\\!/\psi^{iI}+\frac{g^{2}}{2}\big{(}\bar{\psi}^{iI}T^{a}_{IJ}\psi^{iJ}\big{)}^{2}.$
(2.4)

This is similar to the Ising Gross–Neveu model discussed in [34] and to see
the equivalence, one takes the abelian limit of (2.4) by replacing the group
generators with the unit matrix. This is completely parallel to taking the
abelian limit of QCD to produce QED. The dimensionality of $\pi^{a}$ at
criticality plays a key role in the connection of the universal theory and the
Lagrangian of (2.1). In the latter the three couplings are dimensionless in
four dimensions similar to the effective coupling of the $3$-point interaction
of (2.3). Therefore (2.3) would be strictly non-renormalizable in four
dimensions and the quadratic term in $\pi^{a}$ would have a dimensionful
coupling which would be the mass. Instead the standard kinetic term and
quartic $\pi^{a}$ interactions of (2.1) would be relevant. In other words we
term the quartic interactions to be spectator interactions that would be
active solely in four dimensions. Moreover underlying the first two terms of
the universal Lagrangian (2.3) there are an infinite number of Lorentz scalar
operators built from combinations of $\pi^{a}$ and its derivatives. A finite
subset of these extra operators would become relevant in even integer
dimensions and act as interim spectators in the infinite tower of
renormalizable quantum field theories that connect to the universal theory in
the neighbourhood of their respective critical dimensions. In this context it
is worth noting that the quartic spectator interactions of (2.1) play a role
in determining the full fixed point structure of the four dimensional theory.
One of these fixed points though will correspond to the Wilson–Fisher fixed
point of the generalized Gross–Neveu universality class considered here.
However, only a perturbative evaluation of the four dimensional theory’s
renormalization group functions would determine which one it is. While this is
beyond the scope of the present article it is likely to be a solution where
the critical values of all three couplings are non-zero. One non-trivial check
in establishing such a connection though rests in the agreement of the
$\epsilon$ expansion of the various critical exponents with their large $N$
counterparts that will be determined here.

More concretely we now summarize the key aspects of the large $N$ critical
point formalism for (2.3). In the approach to the fixed point the propagators
have the following asymptotic behaviour in coordinate space [11]

$\psi(x)\sim\frac{Ax\\!\\!\\!/}{\big{(}x^{2}\big{)}^{\alpha}}\big{[}1+A^{\prime}\big{(}x^{2}\big{)}^{\lambda}\big{]},\qquad\pi(x)\sim\frac{C}{\big{(}x^{2}\big{)}^{\gamma}}\big{[}1+C^{\prime}\big{(}x^{2}\big{)}^{\lambda}\big{]},$
(2.5)

where the name of the corresponding field is used. The dimensionless
quantities $A$ and $C$ are the coordinate independent amplitudes that will
always occur in the combination $y=A^{2}C$ from the $3$-point interaction. The
next to leading order terms in (2.5) that involve the exponent $\lambda$ are
called the corrections to scaling. Here $\lambda$ will be identified with the
correlation length exponent $\nu$ through $1/\nu=2\lambda$. In addition to the
canonical dimension the two fields have anomalous contributions and the
respective full dimensions of $\psi$ and $\pi^{a}$ are

$\alpha=\mu+\frac{1}{2}\eta,\qquad\gamma=1-\eta-\chi_{\pi},$ (2.6)

where we use $d=2\mu$ for shorthand [43] and $\eta$ and $\chi$ are the fermion
field and vertex anomalous dimensions respectively. For applications in
condensed matter problems the dimension of $\pi^{a}$ that is conventionally
used is

$\eta_{\pi}=4-2\mu-2(\eta+\chi_{\pi}).$

When $\lambda$ corresponds to the correlation length exponent its canonical
dimension will then be taken to be $(\mu-1)$. In this respect the leading and
next to leading order terms of (2.5) then have different dimensions which is
the reason for the second set of independent dimensionless amplitudes
$A^{\prime}$ and $C^{\prime}$. Each of the exponents that we will compute as
well as $y$ will depend on $N$, $\mu$ and the Lie group Casimir invariants.
The dependence on the former means that each entity has a Taylor series in
powers of $1/N$ that is formally given by

$\eta(\mu)=\sum_{n=1}^{\infty}\frac{\eta_{n}(\mu)}{N^{n}},\qquad
y(\mu)=\sum_{n=1}^{\infty}\frac{y_{n}(\mu)}{N^{n}}$ (2.7)

for $\eta$ and $y$ for example and we will determine the first three terms of
$\eta$ for (2.3).

These general considerations cover the basic formalism for the technique
introduced in [42, 43, 44]. To determine all bar $\eta_{3}$ we can apply the
original method [42, 43] that was used for the Ising Gross–Neveu universality
class in [8, 13, 14, 15, 39, 45]. This required solving the skeleton
Schwinger–Dyson equations for the $\psi$ and $\pi^{a}$ $2$-point functions. So
the scaling forms of these are needed given that (2.5) represents the critical
point behaviour of the propagator. However one definition of the $2$-point
function is that it is the inverse of the propagator in momentum space and
this mapping can be carried out through the Fourier transform. Using the
convention given in [42, 43] which is

$\frac{1}{\big{(}x^{2}\big{)}^{\alpha}}=\frac{a(\alpha)}{2^{2\alpha}}\int_{k}\frac{{\rm
e}^{{\rm i}kx}}{\big{(}k^{2}\big{)}^{\mu-\alpha}},$ (2.8)

where $\int_{k}=\int\frac{{\rm d}^{d}k}{(2\pi)^{d}}$ and

$a(\alpha)\equiv\frac{\Gamma(\mu-\alpha)}{\Gamma(\alpha)},$ (2.9)

we transform (2.5) to momentum space carry out the inversion and then apply
the inverse Fourier transform. This results in the coordinate space $2$-point
function asymptotic scaling forms which are [11]

$\displaystyle\psi^{-1}(x)\sim\frac{r(\alpha-1)x\\!\\!\\!/}{A\big{(}x^{2}\big{)}^{2\mu-\alpha+1}}\big{[}1-A^{\prime}s(\alpha-1)\big{(}x^{2}\big{)}^{\lambda}\big{]},$
$\displaystyle\pi^{-1}(x)\sim\frac{p(\gamma)}{C\big{(}x^{2}\big{)}^{2\mu-\gamma}}\big{[}1-C^{\prime}q(\gamma)\big{(}x^{2}\big{)}^{\lambda}\big{]}.$
(2.10)

The presence of the function $a(\alpha)$ in the Fourier transform produces a
complicated dependence on $\mu$ and the exponents, leading to the compact
functions

$\displaystyle p(\gamma)=\frac{a(\gamma-\mu)}{a(\gamma)},\qquad
r(\alpha)=\frac{\alpha p(\alpha)}{(\mu-\alpha)},$ $\displaystyle
q(\gamma)=\frac{a(\gamma-\mu+\lambda)a(\gamma-\lambda)}{a(\gamma-\mu)a(\gamma)},\qquad
s(\alpha)=\frac{\alpha(\alpha-\mu)q(\alpha)}{(\alpha-\mu+\lambda)(\alpha-\lambda)}.$
(2.11)

While the large $N$ conformal bootstrap formalism of [44] has its origins in
these asymptotic scaling functions and was applied to the Ising Gross–Neveu
model in [8, 15, 39], the extraction of an expression for $\eta_{3}$ derives
from the scaling behaviour of the $3$-point function. We defer to a later
section for the required technicalities of that formalism.

##  3 2-point Schwinger–Dyson equation

Equipped with the asymptotic scaling forms of the full propagators, (2.5) and
(2.10), which represent the behaviour at criticality, we use them to solve the
Schwinger–Dyson equations. In conventional perturbation theory one
systematically renormalizes the divergent $n$-point Green’s functions in a
renormalizable theory order by order in perturbation theory. This principle is
respected in the large $N$ technique of [42, 43] except that the ordering of
graphs in the $n$-point functions is achieved by the variable $1/N$ which is
dimensionless across all spacetime dimensions unlike the perturbative coupling
constant. For (2.3) the first few terms in the respective $2$-point functions
of the fields are given in Figure 1, where the dotted line represents the
fermion and the wiggly line denotes the $\pi^{a}$ field. The two loop graphs
are the $O\big{(}1/N^{2}\big{)}$ corrections to the one loop ones. The
counting of powers of $N$ arise from closed fermion loops giving a factor of
$N$ and the $\pi^{a}$ field. The expansion of the amplitude variable begins at
$O(1/N)$ and this translates into each $\pi^{a}$ line in Figure 1 carrying a
power of $1/N$. One key point worth noting concerns the lack of dressing of
lines with self-energy corrections. Contributions from such graphs are already
accounted for in the inclusion of a non-zero anomalous dimension in the power
of the asymptotic scaling forms.

$0\ =\ \psi^{-1}\ \ +\ \
\raisebox{-17.07164pt}{\includegraphics[scale={1.5}]{ihes20procfigsde1aa.pdf}}\
\ +\ \
\raisebox{-17.07164pt}{\includegraphics[scale={1.5}]{ihes20procfigsde1ab.pdf}}\,,\put(-15.0,-10.0){\makebox(0.0,0.0)[lb]{$\Sigma_{1}$}}$

$0\ =\ \pi^{-1}\ \ +\ \
\raisebox{-17.07164pt}{\includegraphics[scale={1.5}]{ihes20procfigsde1ba.pdf}}\
\ +\ \
\raisebox{-17.07164pt}{\includegraphics[scale={1.5}]{ihes20procfigsde1bb.pdf}}\,.\put(-15.0,-17.0){\makebox(0.0,0.0)[lb]{$\Pi_{1}$}}$

Figure 1: $O\big{(}1/N^{2}\big{)}$ corrections to the skeleton Schwinger–Dyson
$2$-point functions.

At leading order the two equations of Figure 1 equate to

$\displaystyle 0=r(\alpha-1)+C_{F}y+O\bigg{(}\frac{1}{N^{2}}\bigg{)},$
$\displaystyle 0=p(\gamma)+T_{F}Ny+O\bigg{(}\frac{1}{N}\bigg{)},$ (3.1)

where we have included the respective group theory factors which derive from
the properties of $T^{a}$ and

$\mathop{\rm Tr}\big{(}T^{a}T^{b}\big{)}=T_{F}\delta^{ab}.$

In this coordinate space representation the one loop graphs require no
evaluation. This is because one integrates over the coordinate of the internal
vertices. As the one loop graphs have external vertices the corresponding
terms of (3.1) are the products of the propagators. In this leading order
instance any integration has been effected in the derivation of the scaling
forms for the full $2$-point functions. The algebraic representation of Figure
1 is (3.1) and to $O(1/N)$ it contains two unknowns. Using (2.7) and the $1/N$
expansion of $r(\alpha-1)$ and $p(\gamma)$ these are $\eta_{1}$ and $y_{1}$
and moreover at this order they occur linearly. Thus eliminating $y_{1}$
between the equations of (3.1) produces

$\eta_{1}=-\frac{2\Gamma(2\mu-1)C_{F}}{\mu\Gamma(1-\mu)\Gamma(\mu-1)\Gamma^{2}(\mu)T_{F}}.$

At next order the situation is not as straightforward due to the additional
graphs of Figure 1 being divergent which necessitates the introduction of
renormalization constants. The formalism for this was provided in [40, 41] and
requires the introduction of a regularization which is achieved through the
shift [42, 43]

$\chi_{\pi}\rightarrow\chi_{\pi}+\Delta,$ (3.2)

where $\Delta$ is a small parameter. In effect it equates to an analytic
regularization of the propagators and we emphasize that the spacetime
dimension $d$ does not play any role in the regularization in contrast to
coupling constant perturbation theory. Consequently the extension of (3.1) to
the next order has to account for this and so the algebraic representation of
Figure 1 becomes

$\displaystyle
0=r(\alpha-1)+C_{F}yZ_{V}^{2}\big{(}x^{2}\big{)}^{\chi_{\pi}+\Delta}+\frac{1}{2}C_{F}[2C_{F}-C_{A}]y^{2}Z_{V}^{4}\Sigma_{1}\big{(}x^{2}\big{)}^{2\chi_{\pi}+2\Delta}+O\bigg{(}\frac{1}{N^{3}}\bigg{)},$
$\displaystyle
0=p(\gamma)+T_{F}NyZ_{V}^{2}\big{(}x^{2}\big{)}^{\chi_{\pi}+\Delta}+\frac{1}{2}T_{F}[2C_{F}-C_{A}]Ny^{2}Z_{V}^{4}\Pi_{1}\big{(}x^{2}\big{)}^{2\chi_{\pi}+2\Delta}+O\bigg{(}\frac{1}{N^{2}}\bigg{)}$
(3.3)

in coordinate space where $Z_{V}$ is the vertex renormalization constant. It
has the Laurent expansion

$Z_{V}=1+\sum_{l=1}^{\infty}\sum_{n=1}^{l}\frac{m_{ln}}{\Delta^{n}},$

where the residues are

$m_{ln}=\sum_{i=1}^{\infty}\frac{m_{ln,i}}{N^{i}},$

after expanding in powers of $1/N$. We follow [40, 41] and restrict to the
$\overline{\mbox{MS}}$ scheme. In (3.3) the $x^{2}$ dependence arises from the
dimensionality of the integrals in the regularized Lagrangian. As it stands
the various factors prevent the limit to criticality from being taken
smoothly. Moreover the factors associated with the one loop graphs of Figure 1
will give contributions at $O\big{(}1/N^{2}\big{)}$ from the expansion of
$\big{(}x^{2}\big{)}^{\chi_{\pi}}$. This will produce problematic logarithms
but these are connected with the simple poles of the values of the two loop
graphs denoted by $\Sigma_{1}$ and $\Pi_{1}$. In particular they have the
formal structure

$\Sigma_{1}=\frac{K_{1}}{\Delta}+\Sigma_{1}^{\prime},\qquad\Pi_{1}=\frac{L_{1}}{\Delta}+\Pi_{1}^{\prime},$

where $\Sigma^{\prime}_{1}$ and $\Pi^{\prime}_{1}$ are finite. They were
computed previously in [11], where the explicit $d$-dependent values are
available. We note our trace conventions at this stage are the same as [11]
and we use $2$ $\times$ $2$ $\gamma$-matrices. To adjust for higher
dimensional $\gamma$-matrix representations one simply redefines $N$ using

$N=\frac{1}{2}d_{\gamma}N,$

where $d_{\gamma}$ is the dimension of the $\gamma$-matrix representation.

Figure 2: Uniqueness rule for scalar-fermion vertex for arbitrary exponents
$\alpha$, $\beta$ and $\gamma$.

The key integration tool for evaluating the graphs in [11] is shown in Figure
2 and is termed uniqueness or conformal integration for the scalar-Yukawa
interaction. There are several ways to establish the relation provided in
Figure 2. One is to use Feynman parameters. In that derivation the final
integration is over a hypergeometric function and it cannot proceed unless the
uniqueness condition of

$\alpha+\beta+\gamma=2\mu+1$

is fulfilled. Setting this allows the final integration to be completed which
produces the factor on the right side of Figure 2. A more elegant alternative
is to apply a conformal transformation to the integral which is

$x_{\mu}\to\frac{x_{\mu}}{x^{2}},\qquad y_{\mu}\to\frac{y_{\mu}}{y^{2}},\qquad
z_{\mu}\to\frac{z_{\mu}}{z^{2}},$ (3.4)

which implies the mapping

$(x\\!\\!\\!/-y\\!\\!\\!/)\to-\frac{y\\!\\!\\!/(x\\!\\!\\!/-y\\!\\!\\!/)x\\!\\!\\!/}{x^{2}y^{2}}=-\frac{x\\!\\!\\!/(x\\!\\!\\!/-y\\!\\!\\!/)y\\!\\!\\!/}{x^{2}y^{2}}$
(3.5)

for instance. The consequence is that when applied to strings of contracted
$\gamma$-matrices the transformation does not alter the initial string of
$\gamma$-matrices. In the application to the left hand side of the equation of
Figure 2 the exponent of the scalar becomes $(2\mu+1-\alpha-\beta-\gamma)$.
Setting this to zero allows the $z$-integration to proceed resulting in the
expression on the right hand side after undoing the initial conformal
transformation.

With the availability of the counterterm from the vertex renormalization
constant the divergences are removed minimally. However
$\ln\big{(}x^{2}\big{)}$ terms remain through two contributions in each
Schwinger–Dyson equation. One is from the power of
$\big{(}x^{2}\big{)}^{2\Delta}$ in the $O\big{(}1/N^{2}\big{)}$ correction.
The other arises from the factor $\big{(}x^{2}\big{)}^{\chi_{\pi}}$ that is
present in the one loop graph. Expanding this in powers of $1/N$ the $O(1/N)$
term involves $\ln\big{(}x^{2}\big{)}$. To be able to take the $x^{2}$ $\to$
$0$ limit safely means that the as yet undefined $\chi_{\pi\,1}$ has to be
suitably chosen. Doing so to ensure there are no $\ln\big{(}x^{2}\big{)}$
terms in each Schwinger–Dyson equation at $O\big{(}1/N^{2}\big{)}$ requires

$\chi_{\pi\,1}=\frac{(2C_{F}-C_{A})\mu}{2(\mu-1)C_{F}}\eta_{1}$ (3.6)

from the explicit values for the residues which satisfy $L_{1}=-$ $2K_{1}$ and
implies the same value results for both equations. This finally renders the
algebraic representation of Figure 1 finite as well as ensuring that it is
scale free. Since the two equations have two unknown variables $\eta_{2}$ and
$y_{2}$ that appear linearly, eliminating the latter leads to

$\eta_{2}=\bigg{[}\frac{(2\mu-1)C_{F}}{(\mu-1)}\Psi(\mu)-\frac{\mu
C_{A}}{2(\mu-1)}\Psi(\mu)+\frac{(4\mu-1)(2\mu-1)C_{F}}{2\mu(\mu-1)^{2}}-\frac{3\mu
C_{A}}{4(\mu-1)^{2}}\bigg{]}\frac{\eta_{1}^{2}}{C_{F}},$

where we use the shorthand

$\Psi(\mu)=\psi(2\mu-1)-\psi(1)+\psi(2-\mu)-\psi(\mu)$

which involves the Euler $\psi$ function defined by $\psi(z)={\rm
d}\ln\Gamma(z)/{\rm d}z$.

##  4 $\boldsymbol{\pi^{a}}$ critical exponent at
$\boldsymbol{O\big{(}1/N^{2}\big{)}}$

Having established the fermion critical exponent at $O\big{(}1/N^{2}\big{)}$
the next stage in the large $N$ formalism is to determine the same quantity
for the boson field. In this instance from the definition (2.6) this requires
the vertex anomalous dimension at $O\big{(}1/N^{2}\big{)}$. While
$\chi_{\pi\,1}$ followed as a corollary to ensuring the $2$-point function was
finite in the approach to criticality, in order to proceed to the next order
to find $\chi_{\pi\,2}$ by the same method is too intractable. Indeed
evaluating the analogous exponent in other models has not been achieved that
way. Instead a more direct approach suffices which is to examine the scaling
behaviour of the $3$-point vertex in the critical limit. In other words the
$O\big{(}1/N^{2}\big{)}$ graphs illustrated in Figures 3 and 4 are computed.

Figure 3: $O(1/N)$ corrections to vertex function.

$\raisebox{-28.45274pt}{\includegraphics{ihes20procfigchi2aa.pdf}}\quad+\quad\raisebox{-28.45274pt}{\includegraphics{ihes20procfigchi2ab.pdf}}\quad+\quad\raisebox{-28.45274pt}{\includegraphics{ihes20procfigchi2ac.pdf}}\quad+\quad\raisebox{-28.45274pt}{\includegraphics{ihes20procfigchi2ad.pdf}}$

$+\quad\raisebox{-28.45274pt}{\includegraphics{ihes20procfigchi2ba.pdf}}\quad+\quad\raisebox{-28.45274pt}{\includegraphics{ihes20procfigchi2bb.pdf}}\quad+\quad\raisebox{-28.45274pt}{\includegraphics{ihes20procfigchi2bc.pdf}}$

Figure 4: $O\big{(}1/N^{2}\big{)}$ corrections to vertex function.

In practical terms the diagrams are more straightforward to evaluate in
momentum space than in coordinate space. This is primarily due to
simplifications in the application of the uniqueness rule of [11]. However one
can connect the values of graphs in both the coordinate and momentum space
evaluations through the Fourier transform (2.8). Underlying this one needs to
use the momentum space forms of the asymptotic scaling functions which are

$\psi(p)\sim\frac{\tilde{A}p\\!\\!\\!/}{(p^{2})^{\mu-\alpha+1}},\qquad\pi(p)\sim\frac{\tilde{C}}{(p^{2})^{\mu-\gamma}},$
(4.1)

where we have new momentum independent amplitudes $\tilde{A}$ and $\tilde{C}$
with the associated variable $\tilde{y}=\tilde{A}^{2}\tilde{C}$. The explicit
value for the latter can be related to the known expansion of $y$ either via
the Fourier transform relation of (2.5) and (4.1) or by repeating the exercise
of the previous section by setting up the formalism in momentum space at the
outset. Both approaches lead to the same expression for $\tilde{y}$ to
$O\big{(}1/N^{2}\big{)}$ as well as $\eta_{2}$ as a check. In relation to this
momentum space $2$-point function approach the asymptotic scaling forms of the
$2$-point functions are

$\psi^{-1}(p)\sim\frac{p\\!\\!\\!/}{\tilde{A}(p^{2})^{\alpha-\mu}},\qquad\pi^{-1}(p)\sim\frac{1}{\tilde{C}(p^{2})^{\gamma-\mu}}.$

It is the inverse Fourier transform of these that produce the leading terms of
(2.10).

With (4.1) it is straightforward to evaluate the graph of Figure 3 and
determine an expression for $\chi_{\pi\,1}$ from the scaling behaviour. It is
equivalent to (3.6). The key point is that the same procedure of [40, 41] can
now be applied to determine $\chi_{\pi\,2}$. This requires in part the
evaluation of the $O\big{(}1/N^{2}\big{)}$ corrections illustrated in Figure
4, where the fermions can be routed around the closed loop in both directions.
The values of the diagrams in the absence of any group theory considerations
were given in [13]. With the presence of the group generator in the vertex of
(2.3), obtaining the associated group factor of each graph is straightforward
in most cases using (2.2) for example and the definition of the Lie algebra
which in our conventions is

$\big{[}T^{a},T^{b}\big{]}={\rm i}f^{abc}T^{c}.$

However for the graphs where there is a closed fermion loop with four
$\pi^{a}$ fields attached a higher group Casimir is present. In particular the
group factor associated with each graph will contain the fully symmetric rank
$4$ tensor

$d_{F}^{abcd}=\frac{1}{6}\mathop{\rm Tr}\big{(}T^{a}T^{(b}T^{c}T^{d)}\big{)}$

among others. To treat these so called light-by-light graphs we have used the
color.h routine that accompanies the symbolic manipulation language Form [46,
37]. The package allows one to manipulate group theory factors associated with
Feynman graphs and write them in terms of Casimirs. It is based on the
comprehensive analysis provided in [38]. However, rather than use color.h to
determine the group factor solely for these light-by-light graphs we have
applied it to all the graphs treated throughout this article for consistency.
We note that the rank $3$ fully symmetric colour tensor $d^{abc}$ given by

$\displaystyle d^{abc}=\frac{1}{2}\mathop{\rm
Tr}\big{(}T^{a}T^{(b}T^{c)}\big{)}$

also occurs in graphs that contain $d_{F}^{abcd}$. In the final expressions
for the exponents, however, these are absent as a result of a cancellation
between graphs where the fermions are routed around the closed loop in
different directions. In addition to the $O\big{(}1/N^{2}\big{)}$ corrections
of Figure 4 there are contributions to $\chi_{\pi\,2}$ from the graph of
Figure 3. These arise from the correction to the variable $\tilde{y}$ which is
$\tilde{y}_{2}$ as well as the parts from the $1/N$ expansion of the exponents
of the propagators in (4.1). In addition one has to include the vertex
renormalization constant $Z_{V}$ in the $O(1/N)$ graph as this cancels the
subgraph divergences in the first three graphs of the first row in Figure 4.
This cancellation is necessary in order to ensure the approach to criticality
is smooth. Assemblying all these components allows us to deduce the vertex
critical exponent at the next order which is

$\displaystyle\chi_{\pi\,2}=\bigg{[}\frac{\mu(2\mu-1)}{(\mu-1)^{3}}C_{F}^{2}-\frac{\mu(7\mu-2)}{4(\mu-1)^{3}}C_{F}C_{A}-\frac{\mu^{2}(2\mu^{2}-6\mu+1)}{12(\mu-1)^{3}}C_{A}^{2}$
$\displaystyle\hphantom{\chi_{\pi\,2}=\bigg{[}}{}-\frac{\mu^{2}(2\mu-1)}{(\mu-1)^{2}}\frac{d_{F}^{abcd}d_{F}^{abcd}}{N_{A}T_{F}^{2}}+\frac{\mu(C_{A}-2C_{F})(C_{A}\mu-2(2\mu-1)C_{F})}{4(\mu-1)^{2}}\Psi(\mu)$
$\displaystyle\hphantom{\chi_{\pi\,2}=\bigg{[}}{}-\frac{\mu^{2}(C_{A}^{2}T_{F}^{2}N_{A}-24d_{F}^{abcd}d_{F}^{abcd})}{8(\mu-1)N_{A}T_{F}^{2}}\Theta(\mu)\bigg{]}\frac{\eta_{1}^{2}}{C_{F}^{2}},$
(4.2)

where

$\displaystyle\Theta(\mu)=\psi^{\prime}(\mu)-\psi^{\prime}(1),$

and the contributions from the light-by-light graphs is evident. We close this
section by noting that the $d_{F}^{abcd}$ tensor will appear in the same
combination as it does in (4.2) in the perturbative renormalization group
functions of (2.1) from Feynman diagrams involving interactions with the
couplings $g_{1}$ and $g_{3}$.

##  5 Correlation length exponent at $\boldsymbol{O\big{(}1/N^{2}\big{)}}$

Having established the dimensions of the two fields at
$O\big{(}1/N^{2}\big{)}$ using the leading term of the asymptotic forms of the
propagators in the approach to criticality, it is possible to study the
corrections to scaling. These are contained in both (2.5) and (2.10)
corresponding to the terms involving the coordinate independent additional
dimensionless parameters $A^{\prime}$ and $C^{\prime}$. The extra exponent
$\lambda$ can be regarded as any exponent but to access the correlation length
exponent $\nu$ we set $\lambda=1/(2\nu)$ which has the canonical dimension of
$(\mu-1)$. To determine the $1/N$ corrections in $d$-dimensions to this
exponent requires a consistency equation that extends (3.1) and (3.3). To find
$\lambda_{1}$ we substitute the various asymptotic scaling functions into the
Schwinger–Dyson equations for the $2$-point function which produces the
representation

$\displaystyle
0=r(\alpha-1)\big{[}1-A^{\prime}s(\alpha-1)\big{(}x^{2}\big{)}^{\lambda}\big{]}+C_{F}yZ_{V}^{2}\big{(}x^{2}\big{)}^{\chi_{\pi}+\Delta}\big{[}1+(A^{\prime}+C^{\prime})\big{(}x^{2}\big{)}^{\lambda}\big{]}$
$\displaystyle\hphantom{0=}{}+\frac{1}{2}C_{F}[2C_{F}-C_{A}]y^{2}\big{(}x^{2}\big{)}^{2\chi_{\pi}+2\Delta}\big{[}\Sigma_{1}+\big{(}\Sigma_{1A}A^{\prime}+\Sigma_{1C}C^{\prime}\big{)}\big{(}x^{2}\big{)}^{\lambda}\big{]}+O\bigg{(}\frac{1}{N^{3}}\bigg{)}$
(5.1)

and

$\displaystyle
0=p(\gamma)\big{[}1-C^{\prime}q(\gamma)\big{(}x^{2}\big{)}^{\lambda}\big{]}+T_{F}NyZ_{V}^{2}\big{(}x^{2}\big{)}^{\chi_{\pi}+\Delta}\big{[}1+2A^{\prime}\big{(}x^{2}\big{)}^{\lambda}\big{]}$
$\displaystyle\hphantom{0=}{}-\frac{1}{2}T_{F}[2C_{F}-C_{A}]Ny^{2}\big{(}x^{2}\big{)}^{2\chi_{\pi}+2\Delta}\big{[}\Pi_{1}+\big{(}\Pi_{1A}A^{\prime}+\Pi_{1C}C^{\prime}\big{)}\big{(}x^{2}\big{)}^{\lambda}\big{]}+O\bigg{(}\frac{1}{N^{2}}\bigg{)},$
(5.2)

where we have omitted the factor of $Z_{V}^{4}$ in the respective
$O\big{(}1/N^{2}\big{)}$ and $O(1/N)$ correction terms. The counterterms from
these only come into effect at the next order.

Unlike the computation of $\eta_{1}$ we have included the two loop graphs of
Figure 1, where the correction to scaling is included. These are denoted by
$\Sigma^{\prime}_{1\phi}$ and $\Pi^{\prime}_{1\phi}$, where $\phi$ indicates
that the insertion is on either a $\psi$ or $\pi^{a}$ line according to
whether it is $A$ or $C$ respectively. The reason why these all have to be
included in principle resides in the leading order $N$ dependence of the
$2$-point scaling functions. For the two key combinations that appear in the
correction to scaling Schwinger–Dyson equation we note that this dependence is
[8, 45]

$r(\alpha-1)s(\alpha-1)=O(1),\qquad
p(\gamma)q(\gamma)=O\bigg{(}\frac{1}{N}\bigg{)}.$ (5.3)

In fact the constant of proportionality of the latter is the combination
$(\lambda_{1}-\eta_{1}-\chi_{\pi\,1})$. As $\eta_{1}$ and $\chi_{\pi\,1}$ are
both available this leaves $\lambda_{1}$ as the unknown we seek. These terms
will form part of the consistency equation that determines $\lambda$ to
$O\big{(}1/N^{2}\big{)}$ and emerges from decoupling the
$\big{(}x^{2}\big{)}^{\lambda}$ terms in (5.1) and (5.2) which follows on
dimensional grounds. The resulting two equations are

$\displaystyle
0=-r(\alpha-1)s(\alpha-1)A^{\prime}+C_{F}yZ_{V}^{2}\big{(}x^{2}\big{)}^{\chi_{\pi}+\Delta}\big{[}A^{\prime}+C^{\prime}\big{]}$
$\displaystyle\hphantom{0=}{}+\frac{1}{2}[2C_{F}-C_{A}]y^{2}\big{(}x^{2}\big{)}^{2\chi_{\pi}+2\Delta}\big{[}\Sigma_{1A}A^{\prime}+\Sigma_{1C}C^{\prime}\big{]}+O\bigg{(}\frac{1}{N^{3}}\bigg{)}$

and

$\displaystyle
0=-p(\gamma)q(\gamma)C^{\prime}+T_{F}NyZ_{V}^{2}\big{(}x^{2}\big{)}^{\chi_{\pi}+\Delta}A^{\prime}$
$\displaystyle\hphantom{0=}{}-\frac{1}{2}C_{F}[2C_{F}-C_{A}]Ny^{2}\big{(}x^{2}\big{)}^{2\chi_{\pi}+2\Delta}\big{[}\Pi_{1A}A^{\prime}+\Pi_{1C}C^{\prime}\big{]}+O\bigg{(}\frac{1}{N^{2}}\bigg{)}.$

Alternatively the relevant terms that produce an expression for $\lambda_{1}$
with respect to the large $N$ expansion due to (5.3) can be written as a
matrix ${\cal M}$, where

${\cal M}=\begin{pmatrix}-r(\alpha-1)s(\alpha-1)C_{F}y\\\\[4.30554pt]
T_{F}Ny-p(\gamma)q(\gamma)-\frac{1}{2}C_{F}[2C_{F}-C_{A}]Ny^{2}\Pi^{\prime}_{1C}\end{pmatrix}\\!.$

Examining the $(2,2)$ element both terms are the same order in $1/N$ as $y$ is
$O(1/N)$. Setting $\det{\cal M}=0$ produces the consistency equation for
$\lambda_{1}$ which can be solved to give

$\lambda_{1}=-(2\mu-1)\eta_{1}.$ (5.4)

To proceed to the next stage and find $\lambda_{2}$ the higher order $1/N$
correction graphs have to be added in to the two Schwinger–Dyson equations
that produced the $O(1/N)$ consistency equations. However the ones we
neglected to determine (5.4), since their $N$ dependence in the determinant is
an order lower that the contribution from $\Pi_{1C}^{\prime}$, now have to be
included. These are $\Sigma_{1A}^{\prime}$, $\Sigma_{1C}^{\prime}$ and
$\Pi_{1A}^{\prime}$ while the graph corresponding to $\Pi_{1C}^{\prime}$ has
to be expanded to the next order in $1/N$ since there is $N$ dependence in the
propagator exponents. To ease the extraction of the expansion of the
consistency equation determinant we formally set

$\Pi^{\prime}_{1C}=\Pi^{\prime}_{1C1}+\Pi^{\prime}_{1C2}\frac{1}{N}+O\bigg{(}\frac{1}{N^{2}}\bigg{)}$

to clarify this. The main work however resides in including the final
contributions to find $\lambda_{2}$ which are illustrated in Figure 5. While
the individual $d$-dependent values have been recorded in [14] for instance,
we have had to append the respective group theory factors. Again we have used
the Form color.h routine due to the presence of the light-by-light diagrams.
Repeating the exercise of setting the determinant of the consistency equations
to zero at the next order in $1/N$ produces the expression

$\displaystyle\lambda_{2}=\bigg{[}\bigg{[}\frac{\mu\big{(}3\mu^{2}-6\mu+2\big{)}C_{A}^{2}C_{F}}{6T_{F}}+4\mu\frac{d_{F}^{abcd}d_{F}^{abcd}C_{F}}{T_{F}^{3}N_{A}}\bigg{]}\frac{1}{(\mu-1)(\mu-2)^{2}\eta_{1}}$
$\displaystyle\hphantom{\lambda_{2}=}{}-\bigg{[}\frac{1}{24}\mu^{2}(2\mu-3)C_{A}^{2}+2\mu^{2}(2\mu-3)\frac{d_{F}^{abcd}d_{F}^{abcd}}{T_{F}^{2}N_{A}}\bigg{]}\frac{[\Psi^{2}(\mu)+\Phi(\mu)]}{(\mu-1)(\mu-2)}$
$\displaystyle\hphantom{\lambda_{2}=}{}+\bigg{[}{-}(2\mu-1)^{2}(\mu+1)(\mu-1)(\mu-2)^{2}C_{F}^{2}+\mu(2\mu-1)(\mu-1)(\mu-2)^{2}C_{F}C_{A}$
$\displaystyle\hphantom{\lambda_{2}=\bigg{[}}{}+\frac{1}{24}\mu^{2}(\mu\\!-1)\big{(}6\mu^{2}\\!-21\mu+20\big{)}C_{A}^{2}\\!-\mu^{2}(3\mu\\!-5)(2\mu\\!-5)\frac{d_{F}^{abcd}d_{F}^{abcd}}{T_{F}^{2}N_{A}}\bigg{]}\frac{\Psi(\mu)}{(\mu\\!-1)^{2}(\mu\\!-2)^{2}}$
$\displaystyle\hphantom{\lambda_{2}=}{}+\bigg{[}{-}\frac{3}{2}\mu^{2}(2\mu+1)(\mu-2)C_{F}^{2}+\frac{3}{4}\mu^{2}(2\mu+5)(\mu-2)C_{F}C_{A}-\frac{11}{8}\mu^{2}(\mu-2)C_{A}^{2}$
$\displaystyle\hphantom{\lambda_{2}=\bigg{[}}{}+3\mu^{2}(5\mu-7)\frac{d_{F}^{abcd}d_{F}^{abcd}}{T_{F}^{2}N_{A}}\bigg{]}\frac{\Theta(\mu)}{(\mu-1)(\mu-2)}+\frac{3\mu(2\mu-1)}{4(\mu-1)^{2}}C_{F}C_{A}$
$\displaystyle\hphantom{\lambda_{2}=}{}+\frac{(2\mu-1)^{2}\big{(}2\mu^{3}-4\mu^{2}-2\mu+1\big{)}}{2\mu(\mu-1)^{2}}C_{F}^{2}-\frac{\mu^{2}\big{(}8\mu^{4}-42\mu^{3}+85\mu^{2}-75\mu+20\big{)}}{48(\mu-1)^{3}(\mu-2)^{2}}C_{A}^{2}$
$\displaystyle\hphantom{\lambda_{2}=}{}-\frac{\mu^{2}\big{(}4\mu^{4}-18\mu^{3}+26\mu^{2}-15\mu+7\big{)}}{2(\mu-1)^{3}(\mu-2)^{2}}\frac{d_{F}^{abcd}d_{F}^{abcd}}{T_{F}^{2}N_{A}}\bigg{]}\frac{\eta_{1}^{2}}{C_{F}^{2}}.$

We have introduced the additional shorthand notation

$\Phi(\mu)=\psi^{\prime}(2\mu-1)-\psi^{\prime}(2-\mu)-\psi^{\prime}(\mu)+\psi^{\prime}(1).$

Essential in determining this was the values of $\eta_{2}$ and
$\chi_{\pi\,2}$.

$\hphantom{+\quad}\raisebox{-22.76219pt}{\includegraphics[scale={1.05}]{ihes20procfignu2craa.pdf}}\quad+\quad\raisebox{-22.76219pt}{\includegraphics[scale={1.05}]{ihes20procfignu2crab.pdf}}$

$+\quad\raisebox{-22.76219pt}{\includegraphics[scale={1.05}]{ihes20procfignu2crba.pdf}}\quad+\quad\raisebox{-22.76219pt}{\includegraphics[scale={1.05}]{ihes20procfignu2crbb.pdf}}$

$+\quad\raisebox{-22.76219pt}{\includegraphics[scale={1.05}]{ihes20procfignu2crca.pdf}}\quad+\quad\raisebox{-22.76219pt}{\includegraphics[scale={1.05}]{ihes20procfignu2crcb.pdf}}$

$+\quad\raisebox{-22.76219pt}{\includegraphics[scale={1.05}]{ihes20procfignu2crda.pdf}}\quad+\quad\raisebox{-22.76219pt}{\includegraphics[scale={1.05}]{ihes20procfignu2crdb.pdf}}$

Figure 5: Higher order graphs for corrections to the $\pi^{a}$ Schwinger–Dyson
$2$-point function needed for $\lambda_{2}$.

##  6 Large $\boldsymbol{N}$ conformal bootstrap

The final part of our study follows a different tack by applying the large $N$
conformal bootstrap programme developed in [44], based on the early insights
of [7, 28, 29, 30, 31]. In general terms the focus is initially directed
towards the $3$-point vertex of (2.3) and its behaviour in the critical
region. The fields will still obey the asymptotic scaling forms of (2.10) but
in treating the Green’s functions in the bootstrap formalism not only are
there no self-energy corrections on the propagators but there are no vertex
corrections. In effect the primitive graphs are the building blocks and are
illustrated in Figure 6. In that figure the dotted vertices do not correspond
to the vertex of the Lagrangian (2.3). Instead they denote the presence of a
Polyakov conformal triangle [30] which includes all vertex corrections at
criticality. It is defined in Figure 7 for the general Yukawa type interaction
that includes the one of (2.3). The external exponents $\alpha_{i}$ are
general and are determined from the underlying theory. For example
$\alpha_{1}=\alpha_{2}=\alpha$ and $\alpha_{3}=\gamma$ for (2.3). The values
of the internal indices $a_{i}$ are the solution to the simultaneous equations

$\displaystyle a_{1}+a_{2}+\alpha_{3}=2\mu+1,$ $\displaystyle
a_{2}+a_{3}+\alpha_{1}=2\mu+1,$ $\displaystyle a_{3}+a_{1}+\alpha_{2}=2\mu+1.$

They ensure that the internal vertices of the triangle graph in Figure 7 are
all unique unlike the vertex on the left side of the equation for (2.3). The
calculational benefit of regarding the full vertex correction as a conformal
triangle is that applying a conformal transformation, (3.4) and (3.5), the
graphs of Figure 6 are reduced to $2$-point ones which are easier to evaluate.

$\raisebox{-22.76219pt}{\includegraphics{ihes20procfigeta3sdeaa.pdf}}\quad+\quad\raisebox{-22.76219pt}{\includegraphics{ihes20procfigeta3sdeab.pdf}}$

$+\quad\raisebox{-28.45274pt}{\includegraphics{ihes20procfigeta3sdeba.pdf}}\quad+\quad\raisebox{-28.45274pt}{\includegraphics{ihes20procfigeta3sdebb.pdf}}\quad+\quad\raisebox{-28.45274pt}{\includegraphics{ihes20procfigeta3sdebc.pdf}}$

Figure 6: Primitive graphs which determine $\eta_{3}$ in the large $N$
conformal bootstrap method. Figure 7: Polyakov conformal triangle for a scalar
Yukawa interaction.

The graphs of Figure 6, however, are the lowest order contributions to the
full vertex function that we will denote by
$V(\bar{y},\alpha,\gamma;\delta,\delta^{\prime})$, where the last two
arguments are regularizing parameters. These are required in the derivation of
one of the two consistency equations defining the large $N$ bootstrap
formalism [29, 30, 31, 44]. The first equation represents Figure 6 and is

$1=V(\bar{y},\alpha,\gamma;0,0),$ (6.1)

where $\bar{y}$ is similar to the early amplitude combination $y$ but includes
the normalization of the Polyakov conformal triangle of Figure 7. Indeed (6.1)
is responsible for determining the terms in the $1/N$ expansion of $\bar{y}$
once the first few orders of $\eta$ have been reproduced in this formalism.
This is because the explicit expressions are necessary to extract $\eta_{3}$
from the third order term of the second bootstrap equation which is

$\displaystyle\frac{T_{F}Nr(\alpha-1)}{C_{F}p(\gamma)}=\frac{\big{[}1+2\chi_{\pi}\frac{\partial}{\partial\delta^{\prime}}V(\bar{y},\alpha,\gamma;\delta,\delta^{\prime})\big{]}}{\big{[}1+2\chi_{\pi}\frac{\partial}{\partial\delta}V(\bar{y},\alpha,\gamma;\delta,\delta^{\prime})\big{]}}\bigg{|}_{\delta=\delta^{\prime}=0}.$
(6.2)

We note briefly that the regularizations $\delta$ and $\delta^{\prime}$ that
appear here arise because of singularities in the $2$-point Schwinger–Dyson
equations when all the vertices are replaced by conformal triangles. In other
words it was recognised in the original work of [7] that in the absence of any
regularization the $2$-point functions with dressed vertices would be finite
overall. However each of the contributing diagrams were individually
divergent. To accommodate this, and similar to the introduction of $\Delta$
earlier, the vertex anomalous dimension has to be continued in a parallel way
to (3.2). This is illustrated in Figure 8 for the leading order contribution
to $V(\bar{y},\alpha,\gamma;\delta,\delta^{\prime})$, where we have set

$\chi_{\pi}=2\tilde{\Delta}_{\pi}$

for shorthand. This figure indicates the values of the exponents of the
internal lines of the Polyakov triangle. Moreover the appearance of both
$\delta$ and $\delta^{\prime}$ on the external legs of the graph on the left
hand side indicate the addition of the regularizations to the exponents of the
respective fields. This is reflected internally in the conformal triangles in
the right hand graph as each of the vertices have to be unique even when there
is a regularization.

Figure 8: Regularized one loop contribution to the vertex bootstrap equations.

Given the form of (6.2) we can rederive $\eta_{2}$ from knowledge of the value
of the graph of Figure 8. This was determined in [15], using the technique
given in [44], for the Ising Gross–Neveu model as a function of the exponents
of that theory and that result can be translated to (2.3). If we expand
$V(\bar{y},\alpha,\gamma;\delta,\delta^{\prime})$ in large $N$ in the same
notation as (2.7) then to the order that will eventually be necessary to
evaluate $\eta_{3}$ we recall [15]

$V_{1}=-\frac{Q^{3}}{\tilde{\Delta}_{\pi}\big{(}\tilde{\Delta}_{\pi}-\delta\big{)}\big{(}\tilde{\Delta}_{\pi}-\delta^{\prime}\big{)}}\exp\big{[}F(\delta,\delta^{\prime},\tilde{\Delta}_{\pi})\big{]}$

with

$Q=-\frac{a^{2}(\alpha-1)a(\gamma)}{(\alpha-1)^{2}\Gamma(\mu)}$

and

$\displaystyle
F\big{(}\delta,\delta^{\prime},\tilde{\Delta}_{\pi}\big{)}=\bigg{[}5B_{\gamma}-2B_{\alpha-1}-3B_{0}-\frac{2}{\alpha-1}\bigg{]}\tilde{\Delta}_{\pi}-[B_{\gamma}-B_{0}]\delta$
$\displaystyle\hphantom{F\big{(}\delta,\delta^{\prime},\tilde{\Delta}_{\pi}\big{)}=}{}+\bigg{[}B_{0}-B_{\alpha-1}-\frac{1}{\alpha-1}\bigg{]}\delta^{\prime}+\bigg{[}C_{\alpha-1}-\frac{1}{\alpha-1}\bigg{]}\delta\delta^{\prime}$
$\displaystyle\hphantom{F\big{(}\delta,\delta^{\prime},\tilde{\Delta}_{\pi}\big{)}=}{}+\bigg{[}C_{\gamma}+C_{0}-2C_{\alpha-1}+\frac{2}{(\alpha-1)^{2}}\bigg{]}\tilde{\Delta}_{\pi}\delta{}+\frac{1}{2}\bigg{[}\frac{1}{(\alpha-1)^{2}}-C_{\alpha-1}-C_{0}\bigg{]}{\delta^{\prime}}^{2}$
$\displaystyle\hphantom{F\big{(}\delta,\delta^{\prime},\tilde{\Delta}_{\pi}\big{)}=}+\bigg{[}C_{0}-C_{\gamma}-2C_{\alpha-1}+\frac{2}{(\alpha-1)^{2}}\bigg{]}\tilde{\Delta}_{\pi}\delta^{\prime}-\frac{1}{2}[C_{\gamma}+C_{0}]\delta^{2}$
$\displaystyle\hphantom{F(\delta,\delta^{\prime},\tilde{\Delta}_{\pi})=}{}+\bigg{[}C_{\alpha-1}-\frac{7}{2}C_{\gamma}-\frac{3}{2}C_{0}-\frac{1}{(\alpha-1)^{2}}\bigg{]}\tilde{\Delta}_{\pi}^{2}+O\big{(}\tilde{\Delta}_{\pi}^{3},\delta^{3},{\delta^{\prime}}^{3}\big{)},$
(6.3)

where the order symbol indicates that terms cubic in any combination of the
parameters are neglected. The functions $B_{z}$ and $C_{z}$ are defined by
[44]

$\displaystyle B_{z}=\psi(\mu-z)+\psi(z),\qquad B_{0}=\psi(1)+\psi(\mu),$
$\displaystyle C_{z}=\psi^{\prime}(z)-\psi^{\prime}(\mu-z),\qquad
C_{0}=\psi^{\prime}(\mu)-\psi^{\prime}(1)$

in terms of the Euler $\psi$ function. With (6.3) and setting $V=V_{1}/N$ in
(6.1) and (6.2) it is straightforward to expand both bootstrap equations to
$O(1/N)$ and verify the earlier expressions for $\eta_{1}$ and $\eta_{2}$.

Having established the formalism reproduces available results the extension to
the next order to find $\eta_{3}$ requires several steps. The first is to
compute the next term in the $1/N$ expansion of (6.3) as that will contribute
to the $O\big{(}1/N^{2}\big{)}$ part of (6.2). However contributions from all
the graphs of Figure 6 bar the one loop one have to be determined and included
as they correspond to $V_{2}$. The relevant parts of these graphs were
calculated in [15]. By this we mean that a computational shortcut was used
akin to that used in [44]. From examining the terms in the formal Taylor
expansion of (6.2) in powers of $1/N$ the part involving $V_{2}$ occurs in the
combination

$\displaystyle\bigg{[}\frac{\partial}{\partial\delta^{\prime}}V_{2}(y,\alpha,\gamma;\delta,\delta^{\prime})-\frac{\partial}{\partial\delta}V_{2}(y,\alpha,\gamma;\delta,\delta^{\prime})\bigg{]}\bigg{|}_{\delta=\delta^{\prime}=0}.$
(6.4)

In [15] the contribution to (6.4) from each of the higher order correction
graphs in Figure 6 were determined. Indeed in most cases only the value for
the difference in derivatives could be found. Appending the group theory
values from the color.h routine allows us to finally extract $\eta_{3}$. We
find

$\displaystyle\eta_{3}=\bigg{[}\frac{(2\mu-1)\big{(}35\mu^{3}-43\mu^{2}+16\mu-2\big{)}}{4\mu^{2}(\mu-1)^{4}}C_{F}^{2}-\frac{\mu^{2}\big{(}43\mu^{2}-35\mu+6\big{)}}{8\mu^{2}(\mu-1)^{4}}C_{F}C_{A}$
$\displaystyle\hphantom{\eta_{3}=\bigg{[}}{}-\frac{\mu^{4}\big{(}4\mu^{3}-5\mu^{2}-9\mu-14\big{)}}{48\mu^{2}(\mu-1)^{4}}C_{A}^{2}-\frac{\mu^{4}\big{(}4\mu^{3}-2\mu^{2}-3\mu+10\big{)}}{4\mu^{2}(\mu-1)^{4}}\frac{d_{F}^{abcd}d_{F}^{abcd}}{T_{F}^{2}N_{A}}$
$\displaystyle\hphantom{\eta_{3}=\bigg{[}}{}+\bigg{[}\frac{1}{2}(11\mu-3)(2\mu-1)^{2}-\frac{1}{4}\mu(19\mu-2)(2\mu-1)C_{F}C_{A}-\frac{1}{24}\mu^{3}\big{(}2\mu^{2}-6\mu-23\big{)}C_{A}^{2}$
$\displaystyle\hphantom{\eta_{3}=\bigg{[}+\bigg{[}}{}-\frac{\mu^{3}\big{(}2\mu^{2}-3\mu+4\big{)}d_{F}^{abcd}d_{F}^{abcd}}{2T_{F}^{2}N_{A}}\bigg{]}\frac{\Psi(\mu)}{\mu(\mu-1)^{3}}$
$\displaystyle\hphantom{\eta_{3}=\bigg{[}}{}+\frac{3(2C_{F}-4\mu C_{F}+\mu
C_{A})^{2}}{8(\mu-1)^{2}}\Psi^{2}(\mu)+\frac{(2C_{F}-4\mu C_{F}+\mu
C_{A})^{2}}{8(\mu-1)^{2}}\Phi(\mu)$
$\displaystyle\hphantom{\eta_{3}=\bigg{[}}{}+\bigg{[}(2\mu-1)(\mu+1)C_{F}^{2}-\frac{1}{2}\mu(5\mu-1)C_{F}C_{A}-\frac{1}{24}\mu^{2}(\mu-4)C_{A}^{2}$
$\displaystyle\hphantom{\eta_{3}=\bigg{[}+\bigg{[}}{}+\mu^{2}(\mu+8)\frac{d_{F}^{abcd}d_{F}^{abcd}}{T_{F}^{2}N_{A}}\bigg{]}\bigg{[}\Theta(\mu)+\frac{1}{(\mu-1)^{2}}\bigg{]}\frac{1}{4(\mu-1)^{2}}$
$\displaystyle\hphantom{\eta_{3}=\bigg{[}}{}-\bigg{[}C_{A}^{2}-24\frac{d_{F}^{abcd}d_{F}^{abcd}}{N_{A}T_{F}^{2}}\bigg{]}\bigg{[}\Theta(\mu)+\frac{1}{(\mu-1)^{2}}\bigg{]}[2\Psi(\mu)+\Xi(\mu)]\frac{\mu^{2}}{16(\mu-1)}\bigg{]}\frac{\eta_{1}^{3}}{C_{F}^{2}}.$

Again the pieces arising from the light-by-light graphs can be clearly
identified. In addition to the Euler $\psi$ function and its derivatives a new
function arises which is related to the function $I(\mu)$ defined in [44]. It
corresponds to the derivative of a two loop self-energy graph, where a
regularizing exponent of one of the propagators is differentiated. In [24]
this master graph was expressed in terms of an ${}_{4}F_{3}$ hypergeometric
function where the regularizing parameter appears in the parameter arguments
of the function. A compact expression for $I(\mu)$ that is valid in
$d$-dimensions was recorded in [4]. Here we have set

$I(\mu)=-\frac{2}{3(\mu-1)}+\Xi(\mu),$

so that $\Xi(\mu)$ has an $\epsilon$ expansion involving multiple zeta values
[3, 24, 44]. For instance,

$\Xi(1-\epsilon)=\frac{2}{3}\zeta_{3}\epsilon^{2}+\zeta_{4}\epsilon^{3}+\frac{13}{3}\zeta_{5}\epsilon^{4}+O\big{(}\epsilon^{5}\big{)}.$

In strictly three dimensions [44]

$I\bigg{(}\frac{3}{2}\bigg{)}=2\ln
2+\frac{3\psi^{\prime\prime}\big{(}\mbox{\small{$\frac{1}{2}$}}\big{)}}{2\pi^{2}}$

and the expansion around three dimensions is known up to ten terms [5].

##  7 Results

We focus in this section on general aspects of the critical exponents of (2.3)
that we have determined. One of the main reasons for considering a generalized
universality class was the fact that known results for specific models could
be extracted as well as be of use where a different Lie group underlies the
physics problem. To assist with that we have collected electronic expressions
in an attached data file. One aspect of the results that needs to be stated is
that we have checked that the $d$-dimensional exponents are in agreement with
several models where the results were determined directly. For instance, the
original Ising Gross–Neveu model of [48] corresponds to the abelian limit of
the symmetry group by specifying

$C_{F}=1,\qquad T_{F}=1,\qquad d_{F}^{abcd}d_{F}^{abcd}=1,\qquad C_{A}=0$

while the Mott insulating phase [2, 20] that corresponds to taking the
symmetry group to be ${\rm SU}(2)$ takes the values

$C_{F}=\frac{3}{4},\qquad T_{F}=\frac{1}{2},\qquad
d_{F}^{abcd}d_{F}^{abcd}=\frac{5}{64},\qquad C_{A}=2.$

For the more recent application of (2.3) to the fractionalized Gross–Neveu
model discussed in [35] the respective values are

$C_{F}=2,\qquad T_{F}=2,\qquad d_{F}^{abcd}d_{F}^{abcd}=\frac{20}{3},\qquad
C_{A}=2.$

For each of these cases the $\epsilon$ expansion of the exponents near four
dimensions with $d=4-2\epsilon$ are in full agreement with known three and
four loop perturbative results [9, 21, 23, 27, 32, 35, 52, 53]. In the case of
the Ising Gross–Neveu model exponents these also are in accord with two
dimensional perturbation theory [10, 12, 16, 18, 25, 26, 47]. Moreover taking
the limits for the three cases, all the large $N$ exponents agree with
previous work [8, 11, 13, 14, 17, 32, 34, 39, 45].

One advantage of the arbitrary group approach in $d$-dimensions means that the
structure of the exponents can be studied in various representations. For
example if we restrict the fermions to be in the adjoint representation $A$
whence $C_{F}=C_{A}$ and $T_{F}=C_{A}$. In that case we find

$\displaystyle\eta_{2}^{\rm
adj}=\bigg{[}\frac{13\mu^{2}-12\mu+2}{4\mu[\mu-1]^{2}}+\frac{3\mu-2}{2[\mu-1]}\Psi(\mu)\bigg{]}\big{(}\eta_{1}^{\rm
adj}\big{)}^{2},$ $\displaystyle\chi_{\pi\,1}^{\rm
adj}=\frac{\mu}{2[\mu-1]}\eta_{1}^{\rm adj},$ $\displaystyle\chi_{\pi\,2}^{\rm
adj}=\bigg{[}\bigg{[}\frac{3\mu^{2}}{\mu-1}\frac{d_{A}^{abcd}d_{A}^{abcd}}{C_{A}^{4}N_{A}}-\frac{\mu^{2}}{8[\mu-1]}\bigg{]}\Theta(\mu)+\frac{\mu(3\mu-2)}{4[\mu-1]^{2}}\Psi(\mu)$
$\displaystyle\hphantom{\chi_{\pi\,2}^{\rm
adj}=\bigg{[}\bigg{[}}{}-\frac{\mu(\mu+1)(\mu-3)}{6[\mu-1]^{2}}-\frac{\mu^{2}(2\mu-1)}{[\mu-1]^{2}}\frac{d_{A}^{abcd}d_{A}^{abcd}}{C_{A}^{4}N_{A}}\bigg{]}\big{(}\eta_{1}^{\rm
adj}\big{)}^{2},$ $\displaystyle\lambda_{2}^{\rm
adj}=\bigg{[}\bigg{[}\frac{1}{6}\mu\big{(}3\mu^{2}-6\mu+2\big{)}+4\mu\frac{d_{A}^{abcd}d_{A}^{abcd}}{C_{A}^{4}N_{A}}\bigg{]}\frac{1}{[\mu-1][\mu-2]^{2}\eta_{1}^{\rm
adj}}$ $\displaystyle\hphantom{\lambda_{2}^{\rm
adj}=\bigg{[}}{}-\bigg{[}\frac{1}{24}\mu^{2}(2\mu-3)+2\mu^{2}(2\mu-3)\frac{d_{A}^{abcd}d_{A}^{abcd}}{C_{A}^{4}N_{A}}\bigg{]}\frac{\Psi^{2}(\mu)+\Phi(\mu)}{[\mu-1][\mu-2]}$
$\displaystyle\hphantom{\lambda_{2}^{\rm
adj}=\bigg{[}}{}-\bigg{[}\frac{1}{24}(\mu-1)\big{(}96\mu^{5}-438\mu^{4}+549\mu^{3}+4\mu^{2}-288\mu+96\big{)}$
$\displaystyle\hphantom{\lambda_{2}^{\rm
adj}=\bigg{[}{-}\bigg{[}}{}+\mu^{2}(3\mu-5)(2\mu-5)\frac{d_{A}^{abcd}d_{A}^{abcd}}{C_{A}^{4}N_{A}}\bigg{]}\frac{\Psi(\mu)}{[\mu-1]^{2}[\mu-2]^{2}}$
$\displaystyle\hphantom{\lambda_{2}^{\rm
adj}=\bigg{[}}{}+\bigg{[}3\mu^{2}(5\mu-7)\frac{d_{A}^{abcd}d_{A}^{abcd}}{C_{A}^{4}N_{A}}-\frac{1}{8}\mu^{2}(\mu-2)(12\mu-7)\bigg{]}\frac{\Theta(\mu)}{[\mu-1][\mu-2]}$
$\displaystyle\hphantom{\lambda_{2}^{\rm
adj}=\bigg{[}}{}+\frac{192\mu^{8}-1544\mu^{7}+4770\mu^{6}-6865\mu^{5}+3951\mu^{4}+724\mu^{3}-1896\mu^{2}+768\mu-96}{48\mu[\mu-1]^{3}[\mu-2]^{2}}$
$\displaystyle\hphantom{\lambda_{2}^{\rm
adj}=\bigg{[}}{}-\frac{\mu^{2}\big{(}4\mu^{4}-18\mu^{3}+26\mu^{2}-15\mu+7\big{)}}{2[\mu-1]^{3}[\mu-2]^{2}}\frac{d_{A}^{abcd}d_{A}^{abcd}}{C_{A}^{4}N_{A}}\bigg{]}\big{(}\eta_{1}^{\rm
adj}\big{)}^{2},$ $\displaystyle\eta_{3}^{\rm
adj}=\bigg{[}\frac{(3\mu-2)^{2}}{8[\mu-1]^{2}}[\Phi(\mu)+3\Psi^{2}(\mu)]-\frac{\mu^{2}(2\mu-1)(\mu^{2}-1)}{2[\mu-1]^{4}}\frac{d_{A}^{abcd}d_{A}^{abcd}}{C_{A}^{4}N_{A}}$
$\displaystyle\hphantom{\eta_{3}^{\rm
adj}=\bigg{[}}{}-\frac{8\mu^{7}-10\mu^{6}-17\mu^{5}-1184\mu^{4}+2448\mu^{3}-1704\mu^{2}+480\mu-48}{96\mu^{2}[\mu-1]^{4}}$
$\displaystyle\hphantom{\eta_{3}^{\rm
adj}=\bigg{[}}{}-\bigg{[}\frac{1}{12}\big{(}\mu^{5}-3\mu^{4}-160\mu^{3}+267\mu^{2}-132\mu+18\big{)}$
$\displaystyle\hphantom{\eta_{3}^{\rm
adj}=\bigg{[}{-}\bigg{[}}{}+\frac{1}{2}\mu^{3}(2\mu+1)(\mu-2)\frac{d_{A}^{abcd}d_{A}^{abcd}}{C_{A}^{4}N_{A}}\bigg{]}\frac{\Psi(\mu)}{\mu[\mu-1]^{3}}$
$\displaystyle\hphantom{\eta_{3}^{\rm
adj}=\bigg{[}}{}+\bigg{[}\frac{1}{4}\mu^{2}(\mu+8)\frac{d_{A}^{abcd}d_{A}^{abcd}}{C_{A}^{4}N_{A}}-\frac{1}{96}(\mu^{3}+8\mu^{2}-36\mu+24)\bigg{]}\frac{\Theta(\mu)}{[\mu-1]^{2}}$
$\displaystyle\hphantom{\eta_{3}^{\rm
adj}=\bigg{[}}{}-\frac{\mu^{2}}{8[\mu-1]}\bigg{[}1-24\frac{d_{A}^{abcd}d_{A}^{abcd}}{C_{A}^{4}N_{A}}\bigg{]}\Theta(\mu)\Psi(\mu)-\frac{\mu^{2}\Xi(\mu)}{16[\mu-1]^{3}}\bigg{[}1-24\frac{d_{A}^{abcd}d_{A}^{abcd}}{C_{A}^{4}N_{A}}\bigg{]}$
$\displaystyle\hphantom{\eta_{3}^{\rm
adj}=\bigg{[}}{}-\frac{\mu^{2}}{16[\mu-1]}\bigg{[}1-24\frac{d_{A}^{abcd}d_{A}^{abcd}}{C_{A}^{4}N_{A}}\bigg{]}\Xi(\mu)\Theta(\mu)\bigg{]}\big{(}\eta_{1}^{\rm
adj}\big{)}^{3},$

where $d_{A}^{abcd}$ is the adjoint version of the fully symmetric rank $4$
Casimir. These exponents simplify substantially in three dimensions since

$\displaystyle\lambda\big{|}_{d=3}=1-\frac{16}{3\pi^{2}N}+\bigg{[}96\frac{d_{A}^{abcd}d_{A}^{abcd}}{C_{A}^{4}N_{A}}+\frac{5248}{\pi^{2}}-432\bigg{]}\frac{1}{27\pi^{2}N^{2}},$
$\displaystyle\eta\big{|}_{d=3}=\frac{8}{3\pi^{2}N}+\frac{1216}{27\pi^{4}N^{2}}+\big{[}\big{[}9072\zeta_{3}-864\pi^{2}\ln
2\big{]}\big{[}C_{A}^{4}N_{A}-24d_{A}^{abcd}d_{A}^{abcd}\big{]}$
$\displaystyle\hphantom{\eta\big{|}_{d=3}=}{}+\big{[}25920\pi^{2}-435456\big{]}d_{A}^{abcd}d_{A}^{abcd}+\big{[}151072-8760\pi^{2}\big{]}C_{A}^{4}N_{A}\big{]}\frac{1}{243\pi^{6}C_{A}^{4}N_{A}N^{3}}.$

The group valued coefficient of the terms involving $\zeta_{3}$ and
$\pi^{2}\ln 2$, which derive from
$I\big{(}\mbox{\small{$\frac{3}{2}$}}\big{)}$, has an interesting combination
of Casimirs. Indeed there might be instances of this factor being zero for
certain Lie groups. However, we have computed the value of
$\big{(}C_{A}^{4}N_{A}-24d_{A}^{abcd}d_{A}^{abcd}\big{)}$ for all the
classical and exceptional Lie groups and found that it is always non-zero.

##  8 Discussion

As the Ising Gross–Neveu universality class is central to a number of phase
transitions in various materials, we have examined a generalized version of
the underlying quantum field theory that incorporates the respective condensed
matter systems. The key aspect is that the core interaction is endowed with a
non-abelian symmetry that has a parallel in gauge theories. There the gauge
interaction of QED is extended from an abelian to a non-abelian one to produce
QCD by the inclusion of the generators of a Lie group thereby endowing QED
with a colour symmetry. The similar extension of the Ising Gross–Neveu model
is simpler in some respects. One obvious one is the absence of gauge symmetry.
A benefit, however, is that considering (2.3) at the outset means results for
specific phase transitions can be quickly deduced by specifying the Lie group.
Indeed if a phase transition were discovered in a material that was in the
same universality class as the Ising Gross–Neveu model but possessed a new
symmetry other than the specific examples we have noted here, then information
on the exponents can readily be deduced from our results. Throughout we have
focussed on the application of the critical point large $N$ technique
developed in [42, 43, 44] to determine $d$-dimensional critical exponents. The
advantage of this is that results are available for the renormalization group
functions of the four dimensional quantum field theories in the same
universality class too. By the same token the large $N$ exponents contain a
wealth of information on the structure of the same functions. For instance,
coefficients in the anomalous dimension beyond the first few known loop orders
can be accessed at successive orders in $1/N$. This is particularly useful in
that our $O\big{(}1/N^{2}\big{)}$ and $O\big{(}1/N^{3}\big{)}$ exponents can
reveal where the new colour group Casimirs, such as
$d_{F}^{abcd}d_{F}^{abcd}$, appear. In indicating the parallel of the QED to
QCD generalization, examining the large $N$ $O\big{(}1/N^{2}\big{)}$ exponents
in this universality class, albeit with a simpler vertex structure, does
provide useful insight into what to expect in the calculation of critical
exponents in QCD at $O\big{(}1/N^{2}\big{)}$. We have to qualify this comment
by noting that while there are similarities, in the QCD large $N$ critical
exponent computation for $\nu$ for instance, there will be more graphs to
consider than those of Figure 5. This is because in (2.3) Feynman diagrams
with subgraphs involving three $\pi^{a}$ lines connecting to a fermion loop
are zero after taking the $\gamma$-matrix trace. In QCD this would not be the
case due to each vertex adding an extra $\gamma$-matrix to the trace. While
such graphs remain to be computed the associated group theory factor that
would result should not involve any Casimir higher than
$d_{F}^{abcd}d_{F}^{abcd}$.

### Acknowledgements

This work was fully supported by a DFG Mercator Fellowship and in part with
the STFC Consolidated ST/T000988/1. The graphs were drawn with the Axodraw
package [6]. Computations were carried out in part using the symbolic
manipulation language Form [37, 46].

## References

  * [1] Ashkin J., Teller E., Statistics of two-dimensional lattices with four components, Phys. Rev. 64 (1943), 178–184.
  * [2] Assaad F.F., Herbut I.F., Pinning the order: the nature of quantum criticality in the Hubbard model on honeycomb lattice, Phys. Rev. X 3 (2013), 031010, 8 pages, arXiv:1304.6340.
  * [3] Bierenbaum I., Weinzierl S., The massless two-loop two-point function, Eur. Phys. J. C 32 (2003), 67–78, arXiv:hep-ph/0308311.
  * [4] Broadhurst D.J., Gracey J.A., Kreimer D., Beyond the triangle and uniqueness relations: non-zeta counterterms at large $N$ from positive knots, Z. Phys. C 75 (1997), 559–574, arXiv:hep-th/9607174.
  * [5] Broadhurst D.J., Kotikov A.V., Compact analytical form for non-zeta terms in critical exponents at order $1/N^{3}$, Phys. Lett. B 441 (1998), 345–353, arXiv:hep-th/9612013.
  * [6] Collins J.C., Vermaseren J.A.M., Axodraw Version 2, arXiv:1606.01177.
  * [7] D’eramo M., Peliti L., Parisi G., Theoretical predictions for critical exponents at the $\lambda$-point of bose liquids, Lett. Nuovo Cimento 2 (1971), 878–880.
  * [8] Derkachov S.É., Kivel N.A., Stepanenko A.S., Vasil’ev A.N., On calculation in $1/n$ expansions of critical exponents in the Gross–Neveu model with the conformal technique, arXiv:hep-th/9302034.
  * [9] Fei L., Giombi S., Klebanov I.R., Tarnopolsky G., Yukawa CFTs and emergent supersymmetry, Progr. Theoret. Exp. Phys. 2016 (2016), 12C105, 32 pages, arXiv:1607.05316.
  * [10] Gracey J.A., Three loop calculations in the $O(N)$ Gross–Neveu model, Nuclear Phys. B 341 (1990), 403–418.
  * [11] Gracey J.A., Calculation of exponent $\eta$ to $O\big{(}1/N^{2}\big{)}$ in the $O(N)$ Gross Neveu model, Internat. J. Modern Phys. A 6 (1991), 395–407, Erratum, Internat. J. Modern Phys. A 6 (1991), 2755.
  * [12] Gracey J.A., Computation of the three-loop $\beta$-function of the ${\rm O}(N)$ Gross–Neveu model in minimal subtraction, Nuclear Phys. B 367 (1991), 657–674.
  * [13] Gracey J.A., Anomalous mass dimension at $O\big{(}1/N^{2}\big{)}$ in the $O(N)$ Gross–Neveu model, Phys. Lett. B 297 (1992), 293–297.
  * [14] Gracey J.A., Computation of $\beta^{\prime}(g_{c})$ at $O\big{(}1/N^{2}\big{)}$ in the $O(N)$ Gross–Neveu model in arbitrary dimensions, Internat. J. Modern Phys. A 9 (1994), 567–589, arXiv:hep-th/9306106.
  * [15] Gracey J.A., Computation of critical exponent $\eta$ at $O\big{(}1/N^{3}\big{)}$ in the four-fermi model in arbitrary dimensions, Internat. J. Modern Phys. A 9 (1994), 727–744, arXiv:hep-th/9306107.
  * [16] Gracey J.A., Four loop $\overline{\rm MS}$ mass anomalous dimension in the Gross–Neveu model, Nuclear Phys. B 802 (2008), 330–350, arXiv:0804.1241.
  * [17] Gracey J.A., Large $N$ critical exponents for the chiral Heisenberg Gross–Neveu universality class, Phys. Rev. D 97 (2018), 105009, 17 pages, arXiv:1801.01320.
  * [18] Gracey J.A., Luthe T., Schröder Y., Four loop renormalization of the Gross–Neveu model, Phys. Rev. D 94 (2016), 125028, 18 pages, arXiv:1609.05071.
  * [19] Gross D.J., Neveu A., Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D 10 (1974), 3235–3253.
  * [20] Herbut I.F., Juric̆ić V., Vafek O., Relativistic Mott criticality in graphene, Phys. Rev. B 80 (2009), 075432, 4 pages, arXiv:0904.1019.
  * [21] Ihrig B., Mihaila L.N., Scherer M.M., Critical behavior of Dirac fermions from perturbative renormalization, Phys. Rev. B 98 (2018), 125109, 20 pages, arXiv:1806.04977.
  * [22] Janssen L., Herbut I.F., Antiferromagnetic critical point on graphene’s honeycomb lattice: A functional renormalization group approach, Phys. Rev. B 89 (2014), 205403, 14 pages, arXiv:1402.6277.
  * [23] Kärkkäinen L., Lacaze R., Lacock P., Petersson B., Critical behavior of the three-dimensional Gross–Neveu and Higgs–Yukawa models, Nuclear Phys. B 415 (1994), 781–796, arXiv:hep-lat/9310020.
  * [24] Kotikov A.V., The Gegenbauer polynomial technique: the evaluation of a class of Feynman diagrams, Phys. Lett. B 375 (1996), 240–248, arXiv:hep-ph/9512270.
  * [25] Ludwig A.W.W., Critical behavior of the two-dimensional random $q$-state Potts model by expansion in $(q-2)$, Nuclear Phys. B 285 (1987), 97–142.
  * [26] Luperini C., Rossi P., Three-loop $\beta$ function(s) and effective potential in the Gross–Neveu model, Ann. Physics 212 (1991), 371–401.
  * [27] Mihaila L.N., Zerf N., Ihrig B., Herbut I.F., Scherer M.M., Gross–Neveu–Yukawa model at three loops and Ising critical behavior of Dirac systems, Phys. Rev. B 96 (2017), 165133, 6 pages, arXiv:1703.08801.
  * [28] Parisi G., On self-consistency conditions in conformal covariant field theory, Lett. Nuovo Cimento 4 (1972), 777–780.
  * [29] Parisi G., Peliti L., Calculation of critical indices, Lett. Nuovo Cimento 2 (1971), 627–628.
  * [30] Polyakov A.M., Microscopic description of critical phenomena, Sov. Phys. JETP 28 (1969), 533–539.
  * [31] Polyakov A.M., Conformal symmetry of critical fluctuations, JETP Lett. 12 (1970), 381–383.
  * [32] Ray S., Ihrig B., Kruti D., Gracey J.A., Scherer M.M., Janssen L., Fractionalized quantum criticality in spin-orbital liquids from field theory beyond the leading order, Phys. Rev. B 103 (2021), 155160, 18 pages, arXiv:2101.10335.
  * [33] Rosenstein B., Warr B.J., Park S.H., Four-fermion theory is renormalizable in 2+1 dimensions, Phys. Rev. Lett. 62 (1989), 1433–1436.
  * [34] Rosenstein B., Warr B.J., Park S.H., Dynamical symmetry breaking in four-fermion interaction models, Phys. Rep. 205 (1991), 59–108.
  * [35] Seifert U.F.P., Dong X.Y., Chulliparambil S., Vojta M., Tu H.H., Janssen L., Fractionalized fermionic quantum criticality in spin-orbital Mott insulators, Phys. Rev. Lett. 125 (2020), 257202, 7 pages, arXiv:2009.05051.
  * [36] Sorella S., Otsuka Y., Yunoki S., Absence of a spin liquid phase in the Hubbard model on the honeycomb lattice, Sci. Rep. 2 (2012), 992, 5 pages, arXiv:1207.1783.
  * [37] Tentyukov M., Vermaseren J.A.M., The multithreaded version of FORM, Comput. Phys. Comm. 181 (2010), 1419–1427, arXiv:hep-ph/0702279.
  * [38] van Ritbergen T., Schellekens A.N., Vermaseren J.A.M., Group theory factors for Feynman diagrams, Internat. J. Modern Phys. A 14 (1999), 41–96, arXiv:hep-ph/9802376.
  * [39] Vasil’ev A.N., Derkachov S.É., Kivel N.A., Stepanenko A.S., The $1/n$ expansion in the Gross–Neveu model: conformal bootstrap calculation of the index $\eta$ in order $1/n^{3}$, Theoret. and Math. Phys. 94 (1993), 127–136.
  * [40] Vasil’ev A.N., Nalimov M.Yu., Analog of dimensional regularization for calculation of the renormalization group functions in the $1/n$ expansion for arbitrary dimension of space, Theoret. and Math. Phys. 55 (1983), 423–431.
  * [41] Vasil’ev A.N., Nalimov M.Yu., The $CP^{N-1}$ model: calculation of anomalous dimensions and the mixing matrices in the order $1/N$, Theoret. and Math. Phys. 56 (1983), 643–653.
  * [42] Vasil’ev A.N., Pismak Yu.M., Honkonen J.R., $1/n$ expansion: calculation of the exponents $\eta$ and $\nu$ in the order $1/n^{2}$ for arbitrary number of dimensions, Theoret. and Math. Phys. 47 (1981), 465–475.
  * [43] Vasil’ev A.N., Pismak Yu.M., Honkonen J.R., Simple method of calculating the critical indices in the $1/n$ expansion, Theoret. and Math. Phys. 46 (1981), 104–113.
  * [44] Vasil’ev A.N., Pismak Yu.M., Honkonen J.R., $1/n$ expansion: calculation of the exponent $\eta$ in the order $1/n^{3}$ by the conformal bootstrap method, Theoret. and Math. Phys. 50 (1982), 127–134.
  * [45] Vasil’ev A.N., Stepanenko A.S., $1/n$-expansion in the Gross–Neveu model: calculation of the $1/\nu$ index to the order $1/n^{2}$ by the conformal bootstrap method, Theoret. and Math. Phys. 97 (1993), 1349–1354.
  * [46] Vermaseren J.A.M., New features of FORM, arXiv:math-ph/0010025.
  * [47] Wetzel W., Two-loop $\beta$-function for the Gross–Neveu model, Phys. Lett. B 153 (1985), 297–299.
  * [48] Wilson K.G., Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior, Phys. Rev. B 4 (1971), 3184–3205.
  * [49] Wilson K.G., Feynman graph expansion for critical exponents, Phys. Rev. Lett. 28 (1972), 548–551.
  * [50] Wilson K.G., Fisher M.E., Critical exponents in 3.99 dimensions, Phys. Rev. Lett. 28 (1972), 240–243.
  * [51] Wilson K.G., Kogut J., The renormalization group and the $\epsilon$ expansion, Phys. Rep. 12 (1974), 75–199.
  * [52] Zerf N., Mihaila L.N., Marquard P., Herbut I.F., Scherer M.M., Four-loop critical exponents for the Gross–Neveu–Yukawa models, Phys. Rev. D 96 (2017), 096010, 19 pages, arXiv:1709.05057.
  * [53] Zinn-Justin J., Four-fermion interaction near four dimensions, Nuclear Phys. B 367 (1991), 105–122.