# Critical exponent $\eta$ at $O(1/N^{3})$ in the chiral XY model using the
large $N$ conformal bootstrap

J.A. Gracey,
Theoretical Physics Division,
Department of Mathematical Sciences,
University of Liverpool,
P.O. Box 147,
Liverpool,
L69 3BX,
United Kingdom

Abstract. We compute the $O(1/N^{3})$ correction to the critical exponent
$\eta$ in the chiral XY or chiral Gross-Neveu model in $d$-dimensions. As the
leading order vertex anomalous dimension vanishes, the direct application of
the large $N$ conformal bootstrap formalism is not immediately possible. To
circumvent this we consider the more general Nambu-Jona-Lasinio model for a
general non-abelian Lie group. Taking the abelian limit of the exponents of
this model produces those of the chiral XY model. Subsequently we provide
improved estimates for $\eta$ in the three dimensional chiral XY model for
various values of $N$.

LTH 1251

##  1 Introduction.

The Gross-Neveu model, [1, 2], is one of the core quantum field theories that
is on a similar level to scalar $\phi^{4}$ theory since it too has a quartic
self-interaction but is purely fermionic. By contrast although it is
renormalizable in two space-time dimensions rather than four, more
interestingly it is asymptotically free, [1]. It shares this feature with four
dimensional non-abelian gauge theories such as Quantum Chromodynamics (QCD)
[3, 4] and, moreover, has the property of dynamical symmetry breaking leading
to non-perturbative mass generation, [1]. In more recent years the original
Gross-Neveu model of [1] and its extensions have been studied in the context
of ultraviolet completion, [5, 6]. This is where the two dimensional theory is
extended to become a renormalizable one in four dimensions which is achieved
in several stages. The first is to introduce a Hubbard-Stratonovich
transformation which produces an auxiliary scalar field. In two dimensions
this field becomes dynamical and corresponds to a bound state of two fermions,
[1]. In the ultraviolet completion to four dimensions, however, this auxiliary
field develops a fundamental propagator, [5]. So to ensure four dimensional
renormalizability it is necessary for the scalar field to have a quartic self-
interaction. This in essence describes the structure of a Gross-Neveu-Yukawa
theory which is the ultimate point of the ultraviolet completion exercise,
[5]. Indeed this particular theory has been of interest for many years due to
its potential to model a composite Higgs field in extensions of the Standard
Model, [7], for example.

Our description of this connection between two and four dimensions is
primarily for the original Gross-Neveu model which has a discrete chiral
symmetry. In this context it is sometimes referred to as the Ising Gross-Neveu
model. It is not the only version of a Gross-Neveu model in that alternatively
one can endow the basic quartic fermion interaction with a continuous chiral
symmetry. In two dimensions this is known as the chiral Gross-Neveu model,
[1], and that has an ultraviolet completion to what is termed the chiral XY
model, [6]. This model as well as the Ising case have been the subject of
considerable interest in the last decade or so due to its underlying
connection with models of phase transitions in graphene. Stretching a sheet of
carbon atoms can change the electrical properties of the material from a semi-
conductor to a Mott insulator, [8, 9]. Indeed the problem of evaluating the
critical exponents describing the phase transition in the dimension of
interest, which is three, has generated an immense amount of activity. The
main theoretical approaches include the application of the functional
renormalization group, [10, 11], conformal bootstrap, [12, 13], Monte Carlo
methods, [14], the $\epsilon$ expansion, [15, 16], and the large $N$
expansion, [17]. We note that this is a representative set of recent articles
rather than a definitive one. The connection between the $\epsilon$ expansion
and the large $N$ approach is quite close. For the former the renormalization
group functions are calculated to high loop order in the ultraviolet completed
four dimensional model. Then the critical exponents at the $d$ $=$ $4$ $-$
$2\epsilon$ dimensional Wilson-Fisher fixed point are computed before the
$\epsilon$ expansion of an exponent is summed up prior to setting $\epsilon$
$=$ $\frac{1}{2}$. In some Gross-Neveu universality classes the same exercise
has been additionally performed in the original two dimensional theory. In
that case the exponents are determined at the Wilson-Fisher fixed point in an
$\bar{\epsilon}$ expansion near $d$ $=$ $2$ $+$ $2\bar{\epsilon}$ dimensions
even though the theory is only perturbatively renormalizable in strictly two
dimensions. Despite these respective theories not being in the neighbourhood
of three dimensions, accurate estimates for exponents have been extracted in
certain cases. For instance, the current situation in the Ising Gross-Neveu
model has been summarized in [16].

By contrast in the large $N$ critical point approach pioneered in scalar
$O(N)$ models in [18, 19, 20], the first three terms of critical exponents
have been computed in $d$-dimensional space-time as a function of $d$. The
large $N$ expansion is renormalizable, [21, 22], primarily due to $1/N$ being
a dimensionless parameter. Using the first three terms in the series for
relatively low $N$, estimates of exponents in three dimensions have been shown
to be competitive with those of the other techniques mentioned. It is these
large $N$ exponents depending on $d$ that drive the close connection with the
exponents determined from the $\epsilon$ expansion of the explicit
perturbative renormalization group functions. This is due to the general
observation that the exponents depend on both variables $N$ and $\epsilon$. So
expanding those in powers of $1/N$ and $\epsilon$ the expressions derived
using either technique separately should be in agreement in the area where
terms in the double expansion overlap. Indeed this is the case in many models.
Moreover the large $N$ exponents can be expanded to all orders in $\epsilon$
at each order in $1/N$ thereby providing non-trivial checks on future
perturbative results.

While the main focus so far has been on the Ising and chiral Gross-Neveu
models and their connections with graphene phase transitions, the chiral XY
model itself has been explored in relation to certain phases in matter. For
instance, the correlated symmetry protected phases were investigated in depth
in [23]. As this model clearly is of importance to these areas it is worth
summarizing the current status of the analytic evaluation of the exponents.
For instance, the renormalization group functions of the chiral XY model are
available to four loops, [6, 15]. Equally the $\epsilon$ expansion of the
large $N$ critical exponents are known to be in agreement with those high
order results. More specifically the exponents $\eta$, $\eta_{\phi}$ and
$1/\nu$ corresponding to the fermion and scalar field dimensions as well as
the correlation length exponent respectively are each known to $O(1/N^{2})$,
[17, 24]. However, for $\eta$ this means only the first two terms in $1/N$ are
available since the fermion has zero canonical dimension. However the large
$N$ conformal bootstrap formalism of [20] allows one to determine the
$O(1/N^{3})$ term of $\eta$. This has been achieved for the Ising Gross-Neveu
model in [25, 26, 27], as well as a variant known as the chiral Heisenberg
Gross-Neveu model in [28] but not yet for the chiral XY case. This is due to a
technical reason. In the large $N$ conformal bootstrap technique, that is
founded on the early work of [29, 30, 31, 32], the expansion is in powers of
the vertex anomalous dimensions. In particular in one of the amplitude
variables the vertex anomalous dimension appears in the denominator. However
in the chiral XY Gross-Neveu case the leading order large $N$ vertex exponent
is zero. So one cannot immediately effect the formalism at the outset. In
reality one has a mathematically ill-defined quantity akin to $0/0$. However
since the application of the original formalism of [28, 29] to the chiral XY
model can indeed produce non-singularly derived expressions for $\eta$ to
$O(1/N^{2})$, [24], there ought to be a way of applying the general large $N$
conformal bootstrap technology of [20] as well.

Given the need for having as accurate as possible data on the phase transition
exponents for the chiral XY model we have found an indirect method to
determine $\eta$ at $O(1/N^{3})$ in $d$-dimensions. This is the main topic of
the article and it is worth outlining our strategy at the outset. Instead of
considering the chiral XY model itself we will carry out a computation in a
generalized theory which possesses a non-abelian symmetry. In essence this is
the Nambu-Jona-Lasinio model, [33]. Critical exponents have been computed in
the large $N$ expansion to high order in several cases previously. For
instance, the exponents of the fields for the group $SU(N_{\\!c})$ have been
determined at $O(1/N^{2})$ as well as that for the correlation length, [24,
34]. A calculation of $\eta$ at $O(1/N^{3})$ for the $SU(2)$ Nambu-Jona-
Lasinio model was given in [35]. However we will consider the case of a
general Lie group rather than specific unitary ones. The reason for this is
that then the exponents will depend on the general colour Casimir group
invariants such as the standard ones of $C_{F}$ and $C_{A}$ as well as higher
rank ones. In this way previous results should be accessible in various
limits. En route though we will identify several errors in earlier results.
Unlike the present situation, when the earlier exponents were determined no
high order $\epsilon$-expansion perturbative results were available to check
against. That is not the case now since four loop chiral XY model results are
available, [15]. Large $N$ exponents for that model are subsequently deduced
from the general non-abelian Nambu-Jona-Lasinio model by simply taking the
abelian limit. It will be evident from the expression for $\eta$ at
$O(1/N^{3})$ that there are no singularities meaning that a smooth abelian
limit can be taken. Indeed we will be able to show the origin of the problem
in the direct large $N$ conformal bootstrap application to the chiral XY
model. As a corollary we will provide an improved estimate for $\eta$ in three
dimensions to the same number of terms as $\beta_{\phi}$ and $\nu$.

The article is organized as follows. We discuss the Lagrangians of the various
theories we will consider for the large $N$ analysis in Section $2$ where the
large $N$ critical point formalism is briefly reviewed. This is applied in the
following section to determine the fermion critical exponent at $O(1/N^{2})$
in the Nambu-Jona-Lasinio model. Subsequently we complete the evaluation of
both scalar field anomalous dimensions at $O(1/N^{2})$ in Section $4$ by
computing the vertex anomalous dimensions. These and the other exponents are
necessary for the $O(1/N^{2})$ evaluation of the correlation length exponent
$1/\nu$ which is carried out in Section $5$. The focus then alters to the
large $N$ conformal bootstrap formalism with the derivation of the formal
underlying equations provided in Section $6$. The details are necessary due to
the presence of more than one vertex unlike previous large $N$ applications of
the technique. The actual evaluation of $\eta$ at $O(1/N^{3})$ is given in
Section $7$ prior to extracting the corresponding expression for the chiral XY
model in the abelian limit in Section $8$. Estimates of the exponents in three
dimensions are also provided there. Finally, we provide concluding remarks in
Section $9$.

##  2 Background.

To begin with we introduce the theories we will focus on. First the four
dimensional chiral XY model has the renormalizable Lagrangian

$L^{\mbox{\footnotesize{XY}}}~{}=~{}i\bar{\psi}^{i}\partial\\!\\!\\!/\psi^{i}~{}+~{}\frac{1}{2}\left(\partial_{\mu}\tilde{\sigma}\right)^{2}~{}+~{}\frac{1}{2}\left(\partial_{\mu}\tilde{\pi}\right)^{2}~{}+~{}g_{1}\bar{\psi}^{i}\left(\tilde{\sigma}+i\pi\gamma^{5}\right)\psi^{i}~{}+~{}\frac{1}{24}g_{2}^{2}\left(\tilde{\sigma}^{2}+\tilde{\pi}^{2}\right)^{2}$
(2.1)

where $1$ $\leq$ $i$ $\leq$ $N$ introduces our expansion parameter. The
renormalization group functions of (2.1) are available in [15] and the
$\epsilon$ expansion of the associated critical exponents at the Wilson-Fisher
fixed point can be summed up to obtain estimates of the corresponding theory
in three dimensions. Here our convention for $\epsilon$ is defined by $d$ $=$
$4$ $-$ $2\epsilon$. Underlying the Wilson-Fisher fixed point is a universal
theory which is driven by a core interaction that is relevant at criticality
in all dimensions. It is this universal theory that one applies the large $N$
formalism of [18, 19, 20] to as the expansion parameter $1/N$ is dimensionless
in all space-time dimensions. In the case of (2.1) the universal Lagrangian is

$L^{\mbox{\footnotesize{XY}}}_{\mbox{\footnotesize{U}}}~{}=~{}i\bar{\psi}^{i}\partial\\!\\!\\!/\psi^{i}~{}+~{}\sigma\bar{\psi}^{i}\psi^{i}~{}+~{}i\pi\bar{\psi}^{i}\gamma^{5}\psi^{i}~{}-~{}\frac{1}{2g^{2}}\left(\sigma^{2}+\pi^{2}\right)~{}.$
(2.2)

The scalar fields $\sigma$ and $\pi$ are related to $\tilde{\sigma}$ and
$\tilde{\pi}$ of (2.1) respectively by a coupling constant rescaling. This is
because at criticality the coupling constant is in effect fixed and plays no
role due to the scaling and conformal symmetry. In (2.2) each field has a
dimension that comprises a canonical part and an anomalous piece. The former
is determined from ensuring the action is dimensionless in $d$-dimensions and
we define the full dimensions of the three fields by

$\alpha_{\psi}~{}=~{}\mu~{}+~{}\mbox{\small{$\frac{1}{2}$}}\eta~{}~{}~{},~{}~{}~{}\beta_{\sigma}~{}=~{}1~{}-~{}\eta~{}-~{}\chi_{\sigma}~{}~{}~{},~{}~{}~{}\beta_{\pi}~{}=~{}1~{}-~{}\eta~{}-~{}\chi_{\pi}$
(2.3)

where $\eta$ is the fermion anomalous dimension with $\chi_{\sigma}$ and
$\chi_{\pi}$ corresponding to the anomalous dimensions of the respective
$\sigma$ and $\pi^{a}$ vertex operators. At this stage we assume that the
latter two are unequal prior to their determination.

The problem, however, is that at leading order in $1/N$ both $\chi_{\sigma}$
and $\chi_{\pi}$ are zero, [24]. As the large $N$ conformal bootstrap
programme is based on an expansion in the vertex anomalous dimension, for
(2.2) the expansion cannot begin. This is because the factor associated with
the Polyakov conformal triangle, that is at the core of each vertex in the
large $N$ bootstrap formalism, depends on $1/\chi_{\sigma}$ and $1/\chi_{\pi}$
which we will demonstrate later. While previous work produced various large
$N$ exponents to $O(1/N^{2})$, [24, 27], and showed that the vertex exponents
are non-zero at $O(1/N^{2})$ for (2.2), a strategy needs to be devised to
avoid the problem of this singularity in the bootstrap construction at leading
order. The direction we have chosen to go in is to consider a more general
theory which contains the universality class (2.2) in a specific limit and
this is the non-abelian Nambu-Jona-Lasinio model which has the universal
Lagrangian, [34],

$L^{\mbox{\footnotesize{NJL}}}_{\mbox{\footnotesize{U}}}~{}=~{}i\bar{\psi}^{iI}\partial\\!\\!\\!/\psi^{iI}~{}+~{}\sigma\bar{\psi}^{iI}\psi^{iI}~{}+~{}i\pi^{a}\bar{\psi}^{iI}\gamma^{5}T^{a}_{IJ}\psi^{iJ}~{}-~{}\frac{1}{2g^{2}}\left({\sigma}^{2}+{\pi^{a}}^{2}\right)~{}.$
(2.4)

Here the fermion field has a non-abelian symmetry which we take to be any Lie
group, classical or exceptional, with generators $T^{a}$ and structure
constants $f^{abc}$ while the index ranges are $1$ $\leq$ $a$ $\leq$
$N_{\\!A}$ and $1$ $\leq$ $I$ $\leq$ $N_{\\!c}$ where $N_{\\!c}$ and
$N_{\\!A}$ are the respective dimensions of the fundamental and adjoint
representations and $\pi^{a}$ is a vector in group space. The four dimensional
equivalent to (2.1) is

$\displaystyle L^{\mbox{\footnotesize{NJL}}}$ $\displaystyle=$ $\displaystyle
i\bar{\psi}^{iI}\partial\\!\\!\\!/\psi^{iI}~{}+~{}\frac{1}{2}\left(\partial_{\mu}\tilde{\sigma}\right)^{2}~{}+~{}\frac{1}{2}\left(\partial_{\mu}\tilde{\pi}^{a}\right)^{2}$
(2.5)
$\displaystyle+~{}g_{1}\bar{\psi}^{iI}\left(\tilde{\sigma}\delta_{IJ}+i\tilde{\pi}^{a}\gamma^{5}T^{a}_{IJ}\right)\psi^{iJ}~{}+~{}\frac{1}{24}g_{2}^{2}\left(\tilde{\sigma}^{2}+\tilde{\pi}^{a\,2}\right)^{2}$

which has a similar coupling structure to (2.1).

Comparing (2.2) and (2.4) the main difference is in the decoration of fields
with indices leading to Feynman rules with Lie group dependence. Structurally
in terms of Feynman graphs that will arise the only difference is the
appearance of group factors. These will depend on the usual group Casimirs
which are defined by

$\mbox{Tr}\left(T^{a}T^{b}\right)~{}=~{}T_{F}\delta^{ab}~{}~{},~{}~{}T^{a}T^{a}~{}=~{}C_{F}I_{N_{\\!c}}~{}~{},~{}~{}f^{acd}f^{bcd}~{}=~{}C_{A}\delta^{ab}$
(2.6)

where $I_{N_{\\!c}}$ is the $N_{\\!c}$ $\times$ $N_{\\!c}$ unit matrix. From
the equations defining $C_{F}$ and $C_{A}$ we have the relation

$C_{F}N_{\\!c}~{}=~{}T_{F}N_{\\!A}~{}.$ (2.7)

However at high loop order it is known that new higher rank Lie group Casimirs
can appear. For instance, these first occur in a four dimensional gauge theory
in the four loop $\beta$-function of QCD, [36]. For (2.4) it transpires that
the same feature is present in higher order large $N$ exponents and in
particular the square of the fully symmetric tensor $d_{F}^{abcd}$ arises
where

$d_{F}^{abcd}~{}=~{}\frac{1}{6}\mbox{Tr}\left(T^{a}T^{(b}T^{c}T^{d)}\right)~{}.$
(2.8)

As will be evident from our final expressions the large $N$ exponents of (2.4)
will depend on the group entities $T_{F}$, $C_{F}$, $C_{A}$, $N_{\\!c}$,
$N_{\\!A}$ and $d_{F}^{abcd}d_{F}^{abcd}$. In light of this and observing that
it is possible to extract (2.2) from (2.4) by formally setting $T^{a}$ $\to$
$1$ then the critical exponents of (2.2) should emerge from those of (2.4) in
the abelian limit

$C_{F}~{}\to~{}1~{}~{}~{},~{}~{}~{}T_{F}~{}\to~{}1~{}~{}~{},~{}~{}~{}N_{\\!c}~{}\to~{}1~{}~{}~{},~{}~{}~{}C_{A}~{}\to~{}0~{}~{}~{},~{}~{}~{}d_{F}^{abcd}d_{F}^{abcd}~{}\to~{}1~{}.$
(2.9)

This then completes our computational strategy. We will focus in the main
throughout on determining the large $N$ critical exponents of (2.4) before
finding those of (2.2) as a corollary via (2.9).

Having justified why our main focus will be on (2.4) we recall the basic
formalism for the large $N$ critical point computations, [18, 19]. It is
centred on the asymptotic behaviour of the propagators in the limit to the
Wilson-Fisher fixed point in $d$-dimensions where $d$ does not play the role
of a regulator. Representing the asymptotic scaling form of each propagator by
the name of its field, the coordinate space forms are, [37],

$\displaystyle\psi(x)$ $\displaystyle\sim$
$\displaystyle\frac{A_{\psi}x\\!\\!\\!/}{(x^{2})^{\alpha_{\psi}}}\left[1+A_{\psi}^{\prime}(x^{2})^{\lambda}\right]~{}~{},~{}~{}\sigma(x)~{}\sim~{}\frac{B_{\sigma}}{(x^{2})^{\beta_{\sigma}}}\left[1+B_{\sigma}^{\prime}(x^{2})^{\lambda}\right]$
$\displaystyle\pi(x)$ $\displaystyle\sim$
$\displaystyle\frac{C}{(x^{2})^{\beta_{\pi}}}\left[1+B_{\pi}^{\prime}(x^{2})^{\lambda}\right]$
(2.10)

where $A_{\psi}$, $B_{\sigma}$ and $B_{\pi}$ are $x$-independent but
$d$-dependent amplitudes. These will appear in two combinations which we
define by

$z~{}=~{}A_{\psi}^{2}B_{\sigma}~{}~{}~{},~{}~{}~{}y~{}=~{}A_{\psi}^{2}B_{\pi}$
(2.11)

and together with $\eta$ and the other exponents can be expanded in powers of
$1/N$ through

$\eta(\mu)~{}=~{}\sum_{n=1}^{\infty}\frac{\eta_{n}(\mu)}{N^{n}}~{}~{}~{},~{}~{}~{}z(\mu)~{}=~{}\sum_{n=1}^{\infty}\frac{z_{n}(\mu)}{N^{n}}~{}~{}~{},~{}~{}~{}y(\mu)~{}=~{}\sum_{n=1}^{\infty}\frac{y_{n}(\mu)}{N^{n}}$
(2.12)

for example. In addition to the leading terms of (2.10) we have included
corrections to scaling corresponding to the terms with $(x^{2})^{\lambda}$ and
associated amplitudes $A_{\psi}^{\prime}$, $B_{\sigma}^{\prime}$ and
$B_{\pi}^{\prime}$. The exponent $\lambda$ is usually used to determine the
exponent $1/\nu$ which is related to the correlation length and we will
consider it here too. In that case the canonical value of $\lambda$ is
$(\mu-1)$. In order to find the first few orders of each exponent requires the
solution of skeleton Schwinger-Dyson $2$-point functions which require the
asymptotic scaling counterparts to the propagators of (2.10). They were
derived in [37] by inverting their momentum space forms using the Fourier
transform which is given by, [18],

$\frac{1}{(x^{2})^{\alpha}}~{}=~{}\frac{a(\alpha)}{2^{2\alpha}}\int_{k}\frac{e^{ikx}}{(k^{2})^{\mu-\alpha}}$
(2.13)

in general to set notation where

$a(\alpha)~{}=~{}\frac{\Gamma(\mu-\alpha)}{\Gamma(\alpha)}~{}.$ (2.14)

Consequently we have

$\displaystyle\psi^{-1}(x)$ $\displaystyle\sim$
$\displaystyle\frac{r(\alpha_{\psi}-1)x\\!\\!\\!/}{A_{\psi}(x^{2})^{2\mu-\alpha_{\psi}+1}}\left[1-A_{\psi}^{\prime}s(\alpha_{\psi}-1)(x^{2})^{\lambda}\right]$
$\displaystyle\sigma^{-1}(x)$ $\displaystyle\sim$
$\displaystyle\frac{p(\beta_{\sigma})}{B_{\sigma}(x^{2})^{2\mu-\beta_{\sigma}}}\left[1-B_{\sigma}^{\prime}q(\beta_{\sigma})(x^{2})^{\lambda}\right]$
$\displaystyle\pi^{-1}(x)$ $\displaystyle\sim$
$\displaystyle\frac{p(\beta_{\pi})}{B_{\pi}(x^{2})^{2\mu-\beta_{\pi}}}\left[1-B_{\pi}^{\prime}q(\beta_{\pi})(x^{2})^{\lambda}\right]$
(2.15)

where the various functions are defined by

$\displaystyle p(\beta)$ $\displaystyle=$
$\displaystyle\frac{a(\beta-\mu)}{a(\beta)}~{}~{}~{}~{}~{}~{},~{}~{}~{}~{}~{}~{}r(\alpha)~{}=~{}\frac{\alpha
p(\alpha)}{(\mu-\alpha)}$ $\displaystyle q(\beta)$ $\displaystyle=$
$\displaystyle\frac{a(\beta-\mu+\lambda)a(\beta-\lambda)}{a(\beta-\mu)a(\beta)}~{}~{}~{},~{}~{}~{}s(\alpha)~{}=~{}\frac{\alpha(\alpha-\mu)q(\alpha)}{(\alpha-\mu+\lambda)(\alpha-\lambda)}$
(2.16)

for arbitrary $\alpha$, $\beta$ and $\lambda$.

##  3 Evaluation of $\eta_{2}$.

The asymptotic scaling forms of the propagators are necessary in order to
algebraically represent the behaviour of the Schwinger-Dyson equations in the
critical region. To determine $\eta$ at $O(1/N^{2})$ we consider the $2$-point
functions of the three fields of (2.2) and to the order we are interested the
relevant graphs that contribute are provided in Figures 1, 2 and 3. In each
there are no self-energy corrections on any of the propagators. This is
because the propagator powers include the non-zero anomalous dimensions
$\eta$, $\chi_{\sigma}$ and $\chi_{\pi}$ that account for such effects. In
terms of counting with respect to the ordering parameter $1/N$, each closed
fermion loop contributes a power of $N$ whereas a $\sigma$ or $\pi^{a}$ field
has a factor of $1/N$ associated with it. This is accounted for through the
$N$ dependence of the amplitude combination variables $z$ and $y$ defined in
(2.12). So, for example, all the two loop graphs differ from the respective
one loop graphs of their Schwinger-Dyson equations by a power of $1/N$. For
the scalar $O(N)$ theory of [18, 19] various three loop graphs also contribute
at $O(1/N^{2})$ but the corresponding graphs are absent here. This is because
the extra diagrams have closed fermion loops with three scalars external to
that loop. Such graphs are zero because of the trace over an odd number of
$\gamma$-matrices. Substituting the propagators into the skeleton Schwinger-
Dyson equation for $\psi$ equates algebraically to

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
r(\alpha-1)~{}+~{}zZ_{V_{\sigma}}^{2}(x^{2})^{\chi_{\sigma}+\Delta}~{}-~{}C_{F}yZ_{V_{\pi}}^{2}(x^{2})^{\chi_{\pi}+\Delta}$
(3.1)
$\displaystyle+~{}\left[z^{2}(x^{2})^{2\chi_{\sigma}+2\Delta}~{}-~{}2C_{F}zy(x^{2})^{\chi_{\sigma}+\chi_{\pi}+2\Delta}~{}+~{}C_{F}(C_{F}-\mbox{\small{$\frac{1}{2}$}}C_{A})y^{2}(x^{2})^{2\chi_{\pi}+2\Delta}\right]\Sigma_{1}$
$\displaystyle+~{}O\left(\frac{1}{N^{3}}\right)$

where we have included the various group theory factors associated with the
non-abelian symmetry, as well as the vertex renormalization constants
$Z_{V_{\sigma}}$ and $Z_{V_{\pi}}$, and effected the $\gamma$-algebra. These
will have poles in the analytic regularization $\Delta$ that is introduced
through the replacement, [18, 19],

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

Such a regulator is required since the actual two loop corrections themselves
represented by $\Sigma_{1}$ for the $\psi$ equation are divergent.

Figure 1: Skeleton Schwinger-Dyson $2$-point function for $\psi$ at
$O(1/N^{2})$.

Once the group and amplitude dependence are factored off the remaining
contribution is the Feynman integral itself. Its value is the same
irrespective of which of the two interactions are involved. Expressions for
both $\Sigma_{1}$ and $\Pi_{1}$ are provided in [37] where the latter arises
in Figures 2 and 3. The representation of the respective skeleton Schwinger-
Dyson equations analogous to (3.1) are

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
p(\beta_{\sigma})~{}+~{}NN_{\\!c}zZ_{V_{\sigma}}^{2}(x^{2})^{\chi_{\sigma}+\Delta}$
(3.3)
$\displaystyle-~{}\frac{N}{2}\left[N_{\\!c}z^{2}(x^{2})^{2\chi_{\sigma}+2\Delta}~{}-~{}C_{F}N_{\\!c}yz(x^{2})^{\chi_{\sigma}+\chi_{\pi}+2\Delta}\right]\Pi_{1}+~{}O\left(\frac{1}{N^{2}}\right)$

and

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
p(\beta_{\pi})~{}-~{}NT_{F}yZ_{V_{\pi}}^{2}(x^{2})^{\chi_{\pi}+\Delta}$ (3.4)
$\displaystyle-~{}\frac{N}{2}T_{F}\left[y^{2}(x^{2})^{2\chi_{\pi}+2\Delta}~{}-~{}(C_{F}-\mbox{\small{$\frac{1}{2}$}}C_{A})yz(x^{2})^{\chi_{\sigma}+\chi_{\pi}+2\Delta}\right]\Pi_{1}+~{}O\left(\frac{1}{N^{2}}\right)~{}.$

In all three cases we have not included vertex renormalization constants in
the higher order terms since those counterterms would contribute to the next
order in the respective expansions. Also we have cancelled off a common factor
of $x^{2}$ whose power is related to the canonical dimension. What remains are
factors of $x^{2}$ whose exponent involves a linear combination of the
regulator and the vertex anomalous dimensions. These cannot be neglected
despite being small. This is because in the approach to criticality there is
no overall scale. For instance as $x^{2}$ $\to$ $0$

$(x^{2})^{\chi_{\sigma}+\Delta}~{}=~{}1~{}+~{}\frac{\chi_{\sigma}}{N}\ln(x^{2})~{}+~{}\Delta\ln(x^{2})~{}+~{}O\left(\Delta;\frac{1}{N^{2}}\right)$
(3.5)

to the order of interest. If the $\ln(x^{2})$ terms remain then they would
give singularities in the limit as $x^{2}$ $\to$ $0$ or $\infty$. However the
simple pole in each of $\Sigma_{1}$ and $\Pi_{1}$ is removed by the vertex
counterterm. Once this is fixed the remaining terms involve $\chi_{\sigma\,1}$
or $\chi_{\pi\,1}$ as well as a term dependent on the counterterm but where
each is multiplied by $\ln(x^{2})$. These terms are removed by defining the
respective vertex anomalous dimensions. A consistency check on this is that
the same solution for each of $\chi_{\sigma\,1}$ and $\chi_{\pi\,1}$ emerges
from all three equations.

Figure 2: Skeleton Schwinger-Dyson $2$-point function for $\sigma$ at
$O(1/N^{2})$. Figure 3: Skeleton Schwinger-Dyson $2$-point function for $\pi$
at $O(1/N^{2})$.

Eliminating $z$ and $y$ at successive orders in $1/N$ from (3.1), (3.3) and
(3.4) and expanding the scaling functions for the $2$-point functions of
(2.16), we find

$\eta_{1}~{}=~{}-~{}\frac{2[T_{F}+C_{F}N_{\\!c}]\Gamma(2\mu-1)}{\mu\Gamma(\mu-1)\Gamma(1-\mu)\Gamma^{2}(\mu)N_{\\!c}T_{F}}$
(3.6)

at leading order. From the $O(1/N^{2})$ parts of the equation, after
renormalization we deduce

$\chi_{\sigma\,1}~{}=~{}-~{}\frac{\mu[C_{F}N_{\\!c}-T_{F}]\eta_{1}}{[T_{F}+C_{F}N_{\\!c}][\mu-1]}~{}~{},~{}~{}\chi_{\pi\,1}~{}=~{}-~{}\frac{\mu[2T_{F}-2C_{F}N_{\\!c}+C_{A}N_{\\!c}]\eta_{1}}{2[\mu-1][T_{F}+C_{F}N_{\\!c}]}$
(3.7)

to ensure no $\ln(x^{2})$ terms remain resulting in

$\displaystyle\eta_{2}$ $\displaystyle=$
$\displaystyle\left[[2(2\mu-1)(C_{F}^{2}N_{\\!c}^{2}+T_{F}^{2})-4C_{F}N_{\\!c}T_{F}-\mu
C_{F}C_{A}N_{\\!c}^{2}]\frac{\Psi(\mu)}{2[\mu-1]}\right.$ (3.8)
$\displaystyle\left.~{}+\frac{(2\mu-1)(4\mu-1)}{2\mu[\mu-1]^{2}}[C_{F}^{2}N_{\\!c}^{2}+T_{F}^{2}]+\frac{(2\mu^{2}-6\mu+1)}{\mu[\mu-1]^{2}}C_{F}N_{\\!c}T_{F}\right.$
$\displaystyle\left.~{}-\frac{3\mu}{4[\mu-1]^{2}}C_{F}C_{A}N_{\\!c}^{2}\right]\frac{\eta_{1}^{2}}{[T_{F}+C_{F}N_{\\!c}]^{2}}$

from the finite part of the consistency equations. Here we use the shorthand
notation

$\Psi(\mu)~{}=~{}\psi(2\mu-1)-\psi(1)+\psi(2-\mu)-\psi(\mu)$ (3.9)

where $\psi(z)$ is the Euler $\psi$ function.

##  4 Vertex anomalous dimensions at $O(1/N^{2})$.

One of the key ingredients to proceeding to higher order in the large $N$
expansion is the provision of the vertex anomalous dimensions. The general
approach is similar to conventional perturbation theory in that at successive
loop orders the wave function renormalization constants are needed first prior
to renormalizing the mass and coupling constants. While the fermion anomalous
dimension is associated with the critical exponent $\eta$ those for the scalar
fields require $\chi_{\sigma}$ and $\chi_{\pi}$ and we summarize their
determination in this section. While $\chi_{\sigma\,1}$ and $\chi_{\pi\,1}$
were deduced as a corollary to finding $\eta_{2}$ by excluding $\ln(x^{2})$
terms thereby ensuring the scaling limit could be taken smoothly, the vertex
dimensions can also be computed directly from the two vertex $3$-point
functions. The leading order $1/N$ corrections for the $\sigma\bar{\psi}\psi$
vertex are shown in Figure 4. Those for the
$\pi^{a}\bar{\psi}\gamma^{5}T^{a}\psi$ are virtually the same but with the
modification that the external $\sigma$ line is replaced by one representing
$\pi^{a}$. It is for this reason that we have not illustrated them.

Figure 4: Leading order corrections to $\sigma\bar{\psi}\psi$ $3$-point
function.

To extract expressions for $\chi_{\sigma\,1}$ and $\chi_{\pi\,1}$ from Figure
4 we follow the method of [18, 19, 38, 39] which is the large $N$ critical
point renormalization formalism. One uses the asymptotic scaling forms of the
propagators (2.10) but $\Delta$ regularized. This produces an expansion in
powers of $1/N$ where at each order the terms are a Laurent series in
$\Delta$. The poles corresponding to the divergences are absorbed into the
vertex renormalization constants $Z_{V_{\sigma}}$ and $Z_{V_{\pi}}$ in general
although the $O(1/N)$ terms of each are already available. Once the
divergences are removed the remaining finite part does not have a non-singular
limit to criticality. By this we mean that in coordinate space $\ln(x^{2})$
parts are present. Equally carrying out the computation in momentum space the
partner $\ln(p^{2})$ dependence remains. In each case there are terms which
involve $\chi_{\sigma\,1}\ln(x^{2})$ and $\chi_{\pi\,1}\ln(x^{2})$. Thus to
ensure a scale free finite Green’s function these unknowns are chosen to
remove the residual $x^{2}$ or $p^{2}$ dependence. In the way we have
presented the larger formalism in the previous section, the values that we
extract for $\chi_{\sigma\,1}$ and $\chi_{\pi\,1}$ from the graphs of Figure 4
fully agree with (3.7).

Figure 5: Vertex corrections to $\sigma\bar{\psi}\psi$ $3$-point function at
$O(1/N^{2})$.

At $O(1/N^{2})$ there are a considerably larger number of diagrams to
evaluate. For instance, each of the three vertices in both graphs of Figure 4
will gain vertex corrections. These are displayed in Figure 5 where again we
will only illustrate the graphs for the $\sigma\bar{\psi}\psi$ $3$-point
function. Those for the $\pi^{a}\bar{\psi}\gamma^{5}T^{a}\psi$ are obtained by
swapping the external scalar in each graph. In addition to these there will be
contributions from expanding the graphs of Figure 4 out to $O(1/N^{2})$ due to
the $N$ dependence in the anomalous part of the propagator exponents in each
graph. More significantly the set of graphs in Figure 6 need to be included.
These are shown separately, partly for a reason that will become evident
later, but mainly because they can be regarded as primitive in the sense that
the highest order of pole in $\Delta$ is simple. One aspect of the evaluation
of the graphs of Figure 6 requiring care is that of determining the group
theory factor associated with the three loop light-by-light graphs. This is
particularly more involved for the $\pi^{a}\bar{\psi}\gamma^{5}T^{a}\psi$
vertex function when all the scalar fields are $\pi^{a}$ ones. In this
instance one has to rationalize traces of the form
$\mbox{Tr}\left(T^{a}T^{b}T^{c}T^{d}\right)$ for a general Lie group. To
handle this we have used the color.h module that is written in the symbolic
manipulation language Form, [40, 41]. Indeed we have used Form to handle the
algebra associated with solving all the various Schwinger-Dyson self-
consistency equations. The color.h routine is an encoding of the Lie group
Casimir analysis of [42] that goes beyond the rank $2$ ones of (2.6). In
particular the earlier trace can be rewritten in terms of
$d_{F}^{abcd}d_{F}^{abcd}$ and $d^{abc}d^{abc}$ where

$d^{abc}~{}=~{}\frac{1}{2}\mbox{Tr}\left(T^{a}T^{(b}T^{c)}\right)$ (4.1)

is fully symmetric. Though for all the light-by-light graphs of Figure 6 the
combination $d^{abc}d^{abc}$ cancels in the final expressions for not only
$\chi_{\sigma\,2}$ and $\chi_{\pi\,2}$ but also all the $O(1/N^{2})$ and
higher order exponents where $d_{F}^{abcd}d_{F}^{abcd}$ occurs.

Figure 6: Additional corrections to $\sigma\bar{\psi}\psi$ $3$-point function
at $O(1/N^{2})$.

Assembling the values of the various diagrams with their associated group
theory factor and extracting the piece associated with the separate vertex
exponents we find

$\displaystyle\chi_{\sigma\,2}$ $\displaystyle=$
$\displaystyle\left[\frac{3\mu^{2}}{[\mu-1]}[T_{F}^{2}-C_{F}N_{\\!c}T_{F}]\Theta(\mu)-\frac{\mu}{[\mu-1]^{2}}C_{F}^{2}N_{\\!c}^{2}-\frac{\mu^{2}(6\mu^{2}-17\mu+14)}{[\mu-1]^{3}}C_{F}N_{\\!c}T_{F}\right.$
(4.2)
$\displaystyle\left.~{}-\frac{\mu(2\mu-1)(\mu^{2}-\mu-1)}{[\mu-1]^{3}}T_{F}^{2}-\frac{\mu\Psi(\mu)}{2[\mu-1]^{2}}\left[2(2\mu-1)[C_{F}^{2}N_{\\!c}^{2}-T_{F}^{2}]-\mu
C_{F}C_{A}N_{\\!c}^{2}\right]\right]$
$\displaystyle\times\frac{\eta_{1}^{2}}{[T_{F}+C_{F}N_{\\!c}]^{2}}$

from the graphs of Figure 5 and 6 where

$\Theta(\mu)~{}=~{}\psi^{\prime}(\mu)~{}-~{}\psi^{\prime}(1)~{}.$ (4.3)

Similarly we find

$\displaystyle\chi_{\pi\,2}$ $\displaystyle=$
$\displaystyle\left[\frac{\mu\Psi(\mu)}{4[\mu-1]^{2}}\left[4(2\mu-1)(C_{F}^{2}N_{\\!c}^{2}-T_{F}^{2})-2((3\mu-1)C_{F}N_{\\!c}-T_{F})C_{A}N_{\\!c}+\mu
C_{A}^{2}N_{\\!c}^{2}\right]\right.$ (4.4)
$\displaystyle\left.~{}+\frac{\mu^{2}\Theta(\mu)}{8[\mu-1]}\left[24\frac{d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}}{C_{F}T_{F}}-24T_{F}^{2}-C_{A}^{2}N_{\\!c}^{2}\right]+\frac{\mu(2\mu-1)}{[\mu-1]^{3}}C_{F}^{2}N_{\\!c}^{2}+\frac{3\mu^{2}}{[\mu-1]^{3}}C_{F}N_{\\!c}T_{F}\right.$
$\displaystyle\left.~{}-\frac{\mu^{2}(2\mu-1)}{[\mu-1]^{2}}\left[\frac{d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}}{C_{F}T_{F}}-T_{F}^{2}\right]-\frac{\mu(7\mu-2)}{4[\mu-1]^{3}}C_{F}C_{A}N_{\\!c}^{2}-\frac{\mu^{2}(2\mu^{2}-6\mu+1)}{12[\mu-1]^{3}}C_{A}^{2}N_{\\!c}^{2}\right.$
$\displaystyle\left.~{}-\frac{\mu(8\mu^{3}-20\mu^{2}+19\mu-1)}{[\mu-1]^{3}}T_{F}^{2}-\frac{\mu(5\mu-2)}{4[\mu-1]^{3}}C_{A}N_{\\!c}T_{F}\right]\frac{\eta_{1}^{2}}{[T_{F}+C_{F}N_{\\!c}]^{2}}~{}.$

Both (4.2) and (4.4) are more involved than the expression for $\eta_{2}$
primarily due to the light-by-light contributions with the term involving
$\Theta(\mu)$ arising from the non-planar ones. We note though that similar to
[24] the vertex anomalous dimensions are equal at $O(1/N^{2})$ for the $SU(2)$
group.

Figure 7: Graphs for $O(1/N^{2})$ correction to the $\sigma$ skeleton
Schwinger-Dyson $2$-point function to determine $\lambda_{2}$ using shorthand
representation for the internal scalar fields.

##  5 Computation of $\lambda_{2}$.

Having (4.2) and (4.4) now means the scalar wave function anomalous dimensions
are known at $O(1/N^{2})$. This was achieved by using the leading terms of the
asymptotic scaling forms of the propagators (2.10). To deduce other exponents
one considers the correction to scaling terms and to find $\nu$ in particular
at $O(1/N^{2})$ we take $\lambda$ $=$ $1/(2\nu)$ where the canonical value of
$\lambda$ is $(\mu-1)$. For (2.4) we therefore include the correction term on
the propagators in the graphs of Figures 1, 2 and 3 retaining only those
contributions with one insertion only, [19]. Unlike the scalar $O(N)$ model of
[18, 19] for the Gross-Neveu universality class there is a reordering of the
diagrams with corrections on the $\sigma$ and $\pi^{a}$ lines in both their
self-consistency equations, [25, 26]. This stems from the correction term of
the $2$-point functions of (2.16) and in particular $q(\beta_{\sigma})$ and
$q(\beta_{\pi})$. Specifically the leading order large $N$ term of
$a(\beta-\mu+\lambda)$ of (2.16) is $O(N)$ for both $\beta_{\sigma}$ and
$\beta_{\pi}$ rather than $O(1)$ for the analogous function in the scalar
theory of [19]. Consequently to find $\nu$ at $O(1/N)$ the two graphs of
Figures 2 and 3 with corrections on the $\sigma$ and $\pi^{a}$ internal
propagators have to be included.

This is evident when the self-consistency equation for $\lambda$ is
constructed. Clearly the correction to scaling terms in (2.10) are of
different length dimension to the leading terms. So the leading and correction
terms with regard to $x^{2}$ in the algebraic representation of the Schwinger-
Dyson equations decouple. The leading term that allowed us to deduce
$\eta_{2}$ has been dealt with but three equations are left each involving the
correction amplitudes $A_{\psi}^{\prime}$, $B_{\sigma}^{\prime}$ and
$B_{\pi}^{\prime}$. The three equations are combined into one equation
involving a $3$ $\times$ $3$ matrix containing $\lambda_{1}$ as the only
unknown at leading order. However examining the $N$ dependence of the terms in
the matrix it is evident that it is different in different rows. Omitting the
higher order graphs for the $\sigma$ and $\pi^{a}$ Schwinger-Dyson equations
would have left an inconsistent set of equations. The final step to find
$\lambda_{1}$ is to set the determinant of this matrix to zero whence we find

$\lambda_{1}~{}=~{}-~{}(2\mu-1)\eta_{1}~{}.$ (5.1)

The consequence of the reordering due to the singular nature of
$q(\beta_{\sigma})$ and $q(\beta_{\pi})$ is that at next order the higher
order graphs of Figure 7 have to be included where the dot on a scalar line
indicates the propagator with the correction to scaling term of (2.10). Here
we have introduced a shorthand notation to compress the number of graphs that
actually contribute. In Figure 7 the zigzag line represents both $\sigma$ and
$\pi^{a}$ fields. So, for instance, this means that in reality there are four
graphs from each possible choice zigzag line in the three loop graphs and
eight for the four loop ones. In total that is $48$ graphs for each scalar
field Schwinger-Dyson equation. The values of each graph independent of the
group theory factor are available in [43]. However appending the group values
the solution to the determinant of the self-consistency equation to next order
in $1/N$ gives

$\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\left[\left[8(2\mu-1)(2\mu-3)C_{F}T_{F}+4\frac{d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}}{T_{F}^{2}}+4\frac{T_{F}^{2}}{N_{\\!c}}\right.\right.$
(5.2)
$\displaystyle\left.\left.~{}~{}~{}+(3\mu^{2}-6\mu+2)\frac{C_{F}C_{A}^{2}N_{\\!c}^{2}}{6T_{F}}\right]\frac{\mu[T_{F}+C_{F}N_{\\!c}]}{[\mu-1][\mu-2]^{2}\eta_{1}}\right.$
$\displaystyle\left.~{}-\left[4C_{F}N_{\\!c}T_{F}^{2}+2T_{F}^{3}+\frac{1}{24}C_{A}^{2}C_{F}N_{\\!c}^{3}+2\frac{d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}^{2}}{T_{F}}\right]\frac{\mu^{2}(2\mu-3)}{[\mu-1][\mu-2]}\left[\Phi(\mu)+\Psi^{2}(\mu)\right]\right.$
$\displaystyle\left.~{}+\left[\mu(\mu-1)(2\mu-1)(\mu-2)^{2}C_{F}C_{A}N_{\\!c}^{2}T_{F}-(\mu+1)(\mu-2)^{2}(\mu-1)(2\mu-1)^{2}C_{F}^{3}N_{\\!c}^{3}\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}~{}~{}-(2\mu^{2}-5\mu+1)(2\mu^{4}-5\mu^{3}+4\mu^{2}-4)T_{F}^{3}\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}~{}~{}-(\mu-1)(\mu-2)^{2}(2\mu-1)(6\mu^{2}-5\mu-3)C_{F}^{2}N_{\\!c}^{2}T_{F}\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}~{}~{}-(12\mu^{6}-92\mu^{5}+251\mu^{4}-294\mu^{3}+111\mu^{2}+28\mu-12)C_{F}N_{\\!c}T_{F}^{2}\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}~{}~{}+\frac{1}{24}\mu^{2}(\mu-1)(6\mu^{2}-21\mu+20)C_{A}^{2}C_{F}N_{\\!c}^{3}+\mu(\mu-1)(\mu-2)^{2}(2\mu-1)C_{A}C_{F}^{2}N_{\\!c}^{3}\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}~{}~{}-\mu^{2}(3\mu-5)(2\mu-5)\frac{d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}^{2}}{T_{F}}\right]\frac{\Psi(\mu)}{[\mu-1]^{2}[\mu-2]^{2}}\right.$
$\displaystyle\left.~{}+\left[\frac{3}{4}\mu^{2}(2\mu-3)(\mu-2)C_{F}C_{A}N_{\\!c}^{2}T_{F}-\frac{3}{2}\mu^{2}(2\mu+1)(\mu-2)C_{F}^{3}N_{\\!c}^{3}\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}~{}~{}-\frac{3}{2}\mu^{2}(2\mu^{2}-13\mu+12)T_{F}^{3}+\frac{3}{2}\mu^{2}(2\mu-3)(\mu-2)C_{F}^{2}N_{\\!c}^{2}T_{F}\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}~{}~{}+\frac{3}{2}\mu^{2}(2\mu^{2}+5\mu-14)C_{F}N_{\\!c}T_{F}^{2}-\frac{11}{8}\mu^{2}(\mu-2)C_{A}^{2}C_{F}N_{\\!c}^{3}\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}~{}~{}+\frac{3}{4}\mu^{2}(2\mu+5)(\mu-2)C_{A}C_{F}^{2}N_{\\!c}^{3}+3\mu^{2}(5\mu-7)\frac{d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}^{2}}{T_{F}}\right]\frac{\Theta(\mu)}{[\mu-1][\mu-2]}\right.$
$\displaystyle\left.~{}+\frac{(2\mu-1)^{2}(2\mu^{3}-4\mu^{2}-2\mu+1)}{2\mu[\mu-1]^{2}}C_{F}^{3}N_{\\!c}^{3}+\frac{\mu(3\mu-1)(2\mu-3)}{4[\mu-1]^{3}}C_{F}C_{A}N_{\\!c}^{2}T_{F}\right.$
$\displaystyle\left.~{}+\frac{(8\mu^{8}-68\mu^{7}+212\mu^{6}-292\mu^{5}+145\mu^{4}+48\mu^{3}-85\mu^{2}+32\mu-4)}{2\mu[\mu-1]^{3}[\mu-2]^{2}}T_{F}^{3}\right.$
$\displaystyle\left.~{}+\frac{(24\mu^{6}-96\mu^{5}+126\mu^{4}-38\mu^{3}-30\mu^{2}+21\mu-3)}{2\mu[\mu-1]^{3}}C_{F}^{2}N_{\\!c}^{2}T_{F}\right.$
$\displaystyle\left.~{}+\frac{[24\mu^{8}-216\mu^{7}+730\mu^{6}-1178\mu^{5}+868\mu^{4}-101\mu^{3}-207\mu^{2}+96\mu-12]}{2\mu[\mu-1]^{3}[\mu-2]^{2}}C_{F}N_{\\!c}T_{F}^{2}\right.$
$\displaystyle\left.~{}-\frac{\mu^{2}(8\mu^{4}-42\mu^{3}+85\mu^{2}-75\mu+20)}{48[\mu-1]^{3}[\mu-2]^{2}}C_{A}^{2}C_{F}N_{\\!c}^{3}+\frac{3\mu(2\mu-1)}{4[\mu-1]^{2}}C_{A}C_{F}^{2}N_{\\!c}^{3}\right.$
$\displaystyle\left.~{}-\frac{\mu^{2}(4\mu^{4}-18\mu^{3}+26\mu^{2}-15\mu+7)}{2[\mu-1]^{3}[\mu-2]^{2}}\frac{d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}^{2}}{T_{F}}\right]\frac{\eta_{1}^{2}}{[T_{F}+C_{F}N_{\\!c}]^{3}}~{}.$

The contributions from the light-by-light graphs of Figure 7 are evident.

Finally we can note that the expressions for the exponents to this stage are
in agreement with previously determined exponents in the Gross-Neveu model,
[25, 26, 37, 43], in the limit

$C_{F}~{}\to~{}0~{}~{}~{},~{}~{}~{}T_{F}~{}\to~{}1~{}~{}~{},~{}~{}~{}N_{\\!c}~{}\to~{}0~{}~{}~{},~{}~{}~{}C_{A}~{}\to~{}0~{}~{}~{},~{}~{}~{}d_{F}^{abcd}d_{F}^{abcd}~{}\to~{}0$
(5.3)

since this excludes graphs involving the
$\pi^{a}\bar{\psi}\gamma^{5}T^{a}\psi$ vertex. For the Gross-Neveu XY model
taking the limit of (2.9) we also find agreement with [17, 24]. These provide
important checks on our exponents for (2.4).

Figure 8: Definition of internal indices in the Polyakov conformal triangle of
a scalar Yukawa interaction. Figure 9: Schwinger-Dyson equation for $\psi$.

##  6 Large $N$ conformal bootstrap equations.

Figure 10: Schwinger-Dyson equations for $\sigma$ and $\pi^{a}$.

Before concentrating on the $O(1/N^{3})$ evaluation of $\eta$ using the
conformal bootstrap we devote this section to recalling one of the key
features of the formalism. This is the Polyakov conformal triangle, [29, 44],
which represents the full vertex function in the theory at criticality.
Ordinarily a vertex operator has a non-zero anomalous dimension. In other
words in coordinate space the sum of the exponents of the propagators joining
to a $3$-point vertex has a non-zero part that means the overall dimension
does not equal the canonical dimension. This is in fact the reason why the
conformal integration rule referred to as uniqueness, [19, 31], cannot
directly be applied to the two loop graphs of Figures 1, 2 and 3. The
canonical dimension of each vertex in the theory matches the value of
$(2\mu+1)$ satisfied by the uniqueness rule of [37] but $\chi_{\sigma}$ and
$\chi_{\pi}$ have small but non-zero values. The Polyakov conformal triangle
construction allows one to exploit uniqueness as a computational tool by
replacing the full vertex with a one loop graph where the exponents of the
internal lines are arbitrary but are chosen to make the new internal vertices
unique. For a Yukawa vertex the most general conformal triangle is shown in
Figure 8, [20, 25, 26]. Specifically

$\displaystyle a_{1}~{}+~{}a_{2}~{}+~{}\alpha_{3}$ $\displaystyle=$
$\displaystyle 2\mu~{}+~{}1$ $\displaystyle
a_{2}~{}+~{}a_{3}~{}+~{}\alpha_{1}$ $\displaystyle=$ $\displaystyle
2\mu~{}+~{}1$ $\displaystyle a_{3}~{}+~{}a_{1}~{}+~{}\alpha_{2}$
$\displaystyle=$ $\displaystyle 2\mu~{}+~{}1$ (6.1)

where $\alpha_{i}$ are the exponents of the original bare vertex of the
underlying Lagrangian and $a_{i}$ are the arbitrary internal exponents. By way
of orientation, for example, one could have $\alpha_{1}$ $=$ $\alpha_{2}$ $=$
$\alpha$ and $\alpha_{3}$ $=$ $\beta_{\sigma}$ for the $\sigma\bar{\psi}\psi$
vertex of (2.4). The overall function $f(\alpha_{i},a_{i})$ represents the
normalization. While it is straightforward to solve the simultaneous equations
(6.1) to find the $a_{i}$ in terms of $\alpha_{i}$ in general we will leave
this to its explicit use of the construction for (2.4).

Figure 11: $\delta^{\prime}$ regularized vertex functions. Figure 12: $\delta$
regularized vertex functions.

One of the reasons for determining the various exponents for (2.4) at
$O(1/N^{2})$ using the skeleton Schwinger-Dyson equations was in part to
establish $\eta$, $\chi_{\sigma}$ and $\chi_{\pi}$ at this order for the
computation of $\eta$ at $O(1/N^{3})$. Since this will use a different
formalism knowledge of $\eta_{2}$ will serve as an independent check while
$\chi_{\sigma\,2}$ and $\chi_{\pi\,2}$ are necessary to extract $\eta_{3}$.
The main difference in the two formalisms rests in how the class of Feynman
diagrams are analysed in the critical region. In the conformal bootstrap the
key graphs are the primitive ones in the sense that not only are there no
self-energy corrections but there are no vertex corrections. In the former
case this was because the propagators are dressed but now in the bootstrap
approach the vertices are also dressed. So the focus is not on $2$-point
functions per se but instead on the vertex or $3$-point functions which we
denote by $V_{\sigma}$ and $V_{\pi}$ for the respective vertices of (2.4). The
method to construct the consistency equations for each here has been discussed
previously, [25, 26, 37], based on the early work of [29, 30, 44, 45, 46]. As
that work involved Lagrangians with a single coupling constant we have
repeated those derivations for (2.4) which has two independent vertices and
will now summarize.

Figure 13: $\delta^{\prime}$ regularized Schwinger-Dyson equation for $\psi$
consistency equation.

The starting point is the Schwinger-Dyson equations for the three $2$-point
functions of Figures 9 and 10 where the blob represents the full vertex which
will correspond to the Polyakov conformal triangle. Clearly the graphs in each
figure contain the explicit ones of Figures 1, 2 and 3. The next step in the
derivation is to use the Schwinger-Dyson equation defining each vertex
function to replace the bare vertex in each $2$-point function of Figures 9
and 10, [45, 46]. These are shown in Figures 11 and 12 where the shaded box
indicates all possible contributions to the respective $4$-point functions.
The two relations illustrated in Figure 11, for example, are used to
substitute for the bare vertices in the $\psi$ Schwinger-Dyson equation of
Figure 9. In both equations for $V_{\sigma}$ and $V_{\pi}$ one has to take
account of the two interactions in (2.2) rather than one as in previous Gross-
Neveu bootstrap computations. The appearance of the regularizing parameter
$\delta^{\prime}$ is due to the fact that in the formalism of [45, 46] there
is at least one divergent graph in the final result for the conformal
bootstrap consistency equation for each field. The regularization is
introduced in the exponent of the external line where the designation of
$2\delta$ and $2\delta^{\prime}$ appears. Unlike [20] we will denote the
regularizing parameters by $\delta$ and $\delta^{\prime}$ instead of
$\epsilon$ and $\epsilon^{\prime}$ in order to avoid confusion with the
parameter that is usually associated with the regulator of dimensional
regularization. By contrast we recall that here the regularization is
analytic.

Figure 14: Distribution of regularizing parameters $\delta$ and
$\delta^{\prime}$ in a conformal triangle.

The situation after following the procedure of [20, 45, 46] is illustrated in
Figure 13. In that equation the Lorentz index indicates that the original full
Schwinger-Dyson equation of Figure 9 has been multiplied by the vector
$x_{\mu}$, [45, 46], where $x$ is the location of one of the external vertices
with the other at the origin. In following the manipulations to arrive at this
equation in Figure 13 the contribution from the graphs involving the $4$-point
boxes have been eliminated. In terms of notation the directed solid line
represents a coordinate space vector joining the endpoints of the respective
full $3$-point vertices. At this point we divide the graphs into those with
the directed line joining the external points of one full vertex subgraph and
those where there is a directed line between two different full vertices. The
former set are divergent while the latter are finite. To see this explicitly
one replaces the full vertex by the equivalent Polyakov triangle defined by
(6.1). As each of the vertices of the resulting three loop graphs are unique
it is a simple exercise to apply the Yukawa uniqueness relation to find
explicit expressions for all the graphs. For the graphs where the evaluation
produces a finite expression then there is a cancellation due to the relative
minus sign. This is because the regularized vertex functions
$V_{\sigma}(\delta^{\prime})$ and $V_{\pi}(\delta^{\prime})$ are unity in the
limit $\delta^{\prime}$ $\to$ $0$, [20, 45, 46]. By contrast for the divergent
graphs one has to retain the next term of the Taylor series in
$\delta^{\prime}$ before taking the $\delta^{\prime}$ $\to$ $0$ limit.

Figure 15: Respective $O(1/N)$ graphs $V_{1\,\sigma\sigma\sigma}$ and
$V_{1\,\sigma\pi\pi}$ contributing to $V_{\sigma}$.

To assist with the evaluation we note that the distribution of the
regularizing parameters within a conformal triangle is shown in Figure 14
where these are added or subtracted to the extant internal exponents. In
defining the regularization with a shift of $2\delta$ rather than $\delta$,
[20], we are merely avoiding the appearance of
$\mbox{\small{$\frac{1}{2}$}}\delta$ internally in the conformal triangle.
There is no ambiguity as ultimately the regularization will be lifted. We note
that in Figure 11 and other figures we only include the relevant argument of
the vertex functions for brevity. The full dependence on the various exponents
and parameters in fact is

$V_{\sigma}~{}=~{}V_{\sigma}(\bar{z},\bar{y};\alpha_{\psi},\beta_{\sigma},\beta_{\pi};\delta,\delta^{\prime})~{}~{},~{}~{}V_{\pi}~{}=~{}V_{\pi}(\bar{z},\bar{y};\alpha_{\psi},\beta_{\sigma},\beta_{\pi};\delta,\delta^{\prime})~{}.$
(6.2)

The parameters $\bar{z}$ and $\bar{y}$ are similar in origin to those of the
skeleton Schwinger-Dyson equations of the earlier approach in that they depend
on the amplitude combinations $A_{\psi}^{2}B_{\sigma}$ and
$A_{\psi}^{2}B_{\pi}$. They differ in value, however, due to the normalization
factor in the definition of the conformal triangle shown in Figure 8. The
result of the exercise is the conformal bootstrap equation for $\psi$ which is

$\displaystyle r(\alpha_{\psi}-1)$ $\displaystyle=$
$\displaystyle\bar{z}t_{\sigma}\left.\frac{\partial~{}}{\partial\delta^{\prime}}V_{\sigma}(\bar{z},\bar{y};\alpha_{\psi},\beta_{\sigma},\beta_{\pi};\delta,\delta^{\prime})\right|_{\delta=0,\delta^{\prime}=0}$
(6.3)
$\displaystyle-~{}C_{F}\bar{y}t_{\pi}\left.\frac{\partial~{}}{\partial\delta^{\prime}}V_{\pi}(\bar{z},\bar{y};\alpha_{\psi},\beta_{\sigma},\beta_{\pi};\delta,\delta^{\prime})\right|_{\delta=0,\delta^{\prime}=0}$

where we have included the group factors associated with (2.4) and

$\displaystyle
t_{\sigma}~{}=~{}\frac{a^{2}(a_{\psi|\sigma}-1)a^{2}(\alpha_{\psi}-1)a(b_{\sigma})a(\beta_{\sigma})}{(\alpha_{\psi}-1)^{2}(a_{\psi|\sigma}-1)^{2}a(\beta_{\sigma}-b_{\sigma})}$
$\displaystyle
t_{\pi}~{}=~{}\frac{a^{2}(a_{\psi|\pi}-1)a^{2}(\alpha_{\psi}-1)a(b_{\pi})a(\beta_{\pi})}{(\alpha_{\psi}-1)^{2}(a_{\psi|\pi}-1)^{2}a(\beta_{\pi}-b_{\pi})}~{}.$
(6.4)

These are related to the residue with respect to $\delta^{\prime}$ of the
earlier divergent graphs of Figure 9. In the expression for $t_{\sigma}$ the
notation is

$a_{\psi|\sigma}~{}=~{}\mu~{}+~{}\mbox{\small{$\frac{1}{2}$}}-\mbox{\small{$\frac{1}{2}$}}\beta_{\sigma}~{}~{}~{},~{}~{}~{}b_{\sigma}~{}=~{}\mu~{}+~{}\mbox{\small{$\frac{1}{2}$}}~{}-~{}\alpha_{\psi}~{}+~{}\mbox{\small{$\frac{1}{2}$}}\beta_{\sigma}$
(6.5)

while

$a_{\psi|\pi}~{}=~{}\mu~{}+~{}\mbox{\small{$\frac{1}{2}$}}-\mbox{\small{$\frac{1}{2}$}}\beta_{\pi}~{}~{}~{},~{}~{}~{}b_{\pi}~{}=~{}\mu~{}+~{}\mbox{\small{$\frac{1}{2}$}}~{}-~{}\alpha_{\psi}~{}+~{}\mbox{\small{$\frac{1}{2}$}}\beta_{\pi}$
(6.6)

for $t_{\pi}$ which are the solutions to (6.1) for the internal exponents. In
arriving at (6.3) we have also made use of another result of [20, 45, 46] in
the derivation of the bootstrap equations for the vertex functions which is

$1~{}=~{}V_{\sigma}(\bar{z},\bar{y};\alpha_{\psi},\beta_{\sigma},\beta_{\pi};0,0)~{}~{},~{}~{}1~{}=~{}V_{\pi}(\bar{z},\bar{y};\alpha_{\psi},\beta_{\sigma},\beta_{\pi};0,0)~{}.$
(6.7)

These reflect the sum of all the graphs with dressed propagators and vertices
contributing to the two $3$-point vertices. The $1/N$ leading order graphs for
both $V_{\sigma}$ and $V_{\pi}$ are shown in Figures 15 and 16 which contain
the graphs of Figures 4 and 5. The regularization is not necessary for these
bootstrap equations since none of the contributing graphs are divergent.

Figure 16: Respective $O(1/N)$ graphs $V_{1\,\pi\sigma\sigma}$ and
$V_{1\,\pi\pi\pi}$ contributing to $V_{\pi}$.

The final part of the exercise in constructing the bootstrap equations is to
repeat the procedure that gave (6.3) for the remaining two equations of Figure
10. The process is similar except that now in these graphs there is a $\delta$
regularization on the external $\sigma$ and $\pi^{a}$ fields. So the initial
part of the derivation of the equations uses the regularized results of Figure
12 where the $4$-point boxes will contain graphs with both $\sigma$ and
$\pi^{a}$ fields. This is the reason for the simpler forms compared with
Figure 11. The result of implementing the regularized vertex of Figure 12 is
shown in Figure 17 for $\sigma$ where the Lorentz index indicates that the
scaling function has been multiplied by $x_{\mu}$ too. The corresponding
graphs for $\pi^{a}$ are similar to Figure 17 with the $\sigma$ external legs
replaced with $\pi^{a}$ fields in addition to the replacement of
$\sigma_{\mu}^{-1}$ with $\pi_{\mu}^{-1}$. While the four graphs of Figure 17
differ from the corresponding ones of Figure 13 it is still the case that
those with the directed line connecting two conformal triangles are finite.
Evaluating the remaining divergent graphs one arrives at the respective
conformal bootstrap consistency equation for $\sigma$ and $\pi^{a}$ which are

$\displaystyle p(\beta_{\sigma})$ $\displaystyle=$ $\displaystyle
NN_{\\!c}\bar{z}t_{\sigma}\left.\frac{\partial~{}}{\partial\delta}V_{\sigma}(\bar{z},\bar{y};\alpha_{\psi},\beta_{\sigma},\beta_{\pi};\delta,\delta^{\prime})\right|_{\delta=0,\delta^{\prime}=0}$
$\displaystyle p(\beta_{\pi})$ $\displaystyle=$
$\displaystyle-~{}NT_{F}\bar{y}t_{\pi}\left.\frac{\partial~{}}{\partial\delta}V_{\pi}(\bar{z},\bar{y};\alpha_{\psi},\beta_{\sigma},\beta_{\pi};\delta,\delta^{\prime})\right|_{\delta=0,\delta^{\prime}=0}$
(6.8)

where the same factors $t_{\sigma}$ and $t_{\pi}$ occur. Eliminating these and
the amplitudes $\bar{z}$ and $\bar{y}$ finally produces the consistency
equation for $\eta$ which is

$\displaystyle r(\alpha_{\psi}-1)$ $\displaystyle=$
$\displaystyle\frac{p(\beta_{\sigma})}{NN_{\\!c}}\left[\left.\frac{\partial~{}}{\partial\delta^{\prime}}V_{\sigma}(\bar{z},\bar{y};\alpha_{\psi},\beta_{\sigma},\beta_{\pi};\delta,\delta^{\prime})\right|\right]\left[\left.\frac{\partial~{}}{\partial\delta}V_{\sigma}(\bar{z},\bar{y};\alpha_{\psi},\beta_{\sigma},\beta_{\pi};\delta,\delta^{\prime})\right|\right]^{-1}$
(6.9)
$\displaystyle+~{}\frac{C_{F}p(\beta_{\pi})}{NT_{F}}\left[\left.\frac{\partial~{}}{\partial\delta^{\prime}}V_{\pi}(\bar{z},\bar{y};\alpha_{\psi},\beta_{\sigma},\beta_{\pi};\delta,\delta^{\prime})\right|\right]\left[\left.\frac{\partial~{}}{\partial\delta}V_{\pi}(\bar{z},\bar{y};\alpha_{\psi},\beta_{\sigma},\beta_{\pi};\delta,\delta^{\prime})\right|\right]^{-1}~{}~{}~{}$

where the restriction indicates that both regularizations are lifted at the
end of the evaluation of the underlying graphs. At this stage (6.9) is exact
and its solution would determine $\eta$ to all orders. However to achieve this
would require the explicit values from all the contributing graphs.

Figure 17: $\delta$ regularized Schwinger-Dyson equation for $\sigma$
consistency equation.

The consistency equation (6.9) can be refined though by recalling the
observation of in [20]. For the moment we consider a general situation and
denote a regularized $n$-point function by
$\Gamma_{(n)}(\Delta_{i},\delta,\delta^{\prime})$ where each vertex is
represented by a conformal triangle, the $j$th vertex has anomalous dimension
$\chi_{j}$ $\equiv$ $2\Delta_{j}$ and two of the external legs have $\delta$
and $\delta^{\prime}$ regularizations. Then the Green’s function will have a
simple structure. In the unregularized case it will formally be
$\Gamma(\Delta_{i},0,0)/(\prod_{j=1}^{n}\Delta_{j})$ but in the regularized
case

$\Gamma_{(n)}(\Delta_{i},\delta,\delta^{\prime})~{}=~{}\frac{\Gamma(\Delta_{i},\delta,\delta^{\prime})}{(\Delta_{1}-\delta)(\Delta_{2}-\delta^{\prime})(\prod_{j=3}^{n}\Delta_{j})}$
(6.10)

where legs $1$ and $2$ are chosen to have the respective regularizations
$\delta$ and $\delta^{\prime}$ and
$\Gamma_{(n)}(\Delta_{i},\delta,\delta^{\prime})$ is analytic in all the
$\Delta_{i}$ and both regulators. Then

$\displaystyle\frac{\partial~{}}{\partial\delta}\Gamma_{(n)}(\Delta_{i},\delta,\delta^{\prime})$
$\displaystyle=$
$\displaystyle\frac{1}{\Delta_{1}}~{}+~{}\frac{1}{(\prod_{j=1}^{n}\Delta_{j})}\left.\frac{\partial~{}}{\partial\delta}\Gamma(\Delta_{i},\delta,\delta^{\prime})\right|_{\delta=0,\delta^{\prime}=0}$
$\displaystyle\frac{\partial~{}}{\partial\delta^{\prime}}\Gamma_{(n)}(\Delta_{i},\delta,\delta^{\prime})$
$\displaystyle=$
$\displaystyle\frac{1}{\Delta_{2}}~{}+~{}\frac{1}{(\prod_{j=1}^{n}\Delta_{j})}\left.\frac{\partial~{}}{\partial\delta^{\prime}}\Gamma(\Delta_{i},\delta,\delta^{\prime})\right|_{\delta=0,\delta^{\prime}=0}$
(6.11)

for instance. The first term depends only on $\Delta_{1}$ and $\Delta_{2}$
respectively since the equivalent of (6.7) has been employed and the relation
can be written more compactly as, [20],

$\displaystyle\frac{\partial~{}}{\partial\delta}\Gamma_{(n)}(\Delta_{i},\delta,\delta^{\prime})$
$\displaystyle=$
$\displaystyle\frac{1}{\Delta_{1}}\left[1~{}+~{}\Delta_{1}\left.\frac{\partial\Gamma_{(n)}}{\partial\delta}\right|_{{\mbox{\footnotesize{res}}};\,\delta=0,\delta^{\prime}=0}\right]$
$\displaystyle\frac{\partial~{}}{\partial\delta^{\prime}}\Gamma_{(n)}(\Delta_{i},\delta,\delta^{\prime})$
$\displaystyle=$
$\displaystyle\frac{1}{\Delta_{2}}\left[1~{}+~{}\Delta_{2}\left.\frac{\partial\Gamma_{(n)}}{\partial\delta^{\prime}}\right|_{{\mbox{\footnotesize{res}}};\,\delta=0,\delta^{\prime}=0}\right]$
(6.12)

where res indicates the contribution from the derivative of the residue of
$\Gamma_{(n)}(\Delta_{i},\delta,\delta^{\prime})$ only. This was for a general
scenario but returning to (2.4) and applying this procedure to (6.9) we have

$\displaystyle r(\alpha_{\psi}-1)$ $\displaystyle=$
$\displaystyle\frac{p(\beta_{\sigma})}{NN_{\\!c}}\left[1~{}+~{}\Delta_{\sigma}\left.\frac{\partial
V_{\sigma}}{\partial\delta^{\prime}}\right|_{{\mbox{\footnotesize{res}}};\,\delta=0,\delta^{\prime}=0}\right]\left[1~{}+~{}\Delta_{\sigma}\left.\frac{\partial
V_{\sigma}}{\partial\delta}\right|_{{\mbox{\footnotesize{res}}};\,\delta=0,\delta^{\prime}=0}\right]^{-1}$
(6.13)
$\displaystyle+~{}\frac{C_{F}p(\beta_{\pi})}{NT_{F}}\left[1~{}+~{}\Delta_{\pi}\left.\frac{\partial
V_{\pi}}{\partial\delta^{\prime}}\right|_{{\mbox{\footnotesize{res}}};\,\delta=0,\delta^{\prime}=0}\right]\left[1~{}+~{}\Delta_{\pi}\left.\frac{\partial
V_{\pi}}{\partial\delta}\right|_{{\mbox{\footnotesize{res}}};\,\delta=0,\delta^{\prime}=0}\right]^{-1}$

where we have now set

$\chi_{\sigma}~{}=~{}2\Delta_{\sigma}~{}~{}~{},~{}~{}~{}\chi_{\pi}~{}=~{}2\Delta_{\pi}$
(6.14)

and the arguments of the two vertex functions have been suppressed for brevity
but are the same as the corresponding terms in (6.13).

Figure 18: $O(1/N^{2})$ graphs contributing to $V_{\sigma}$.

For practical purposes it is best to expand the two vertex functions by
setting

$V_{\sigma}~{}=~{}\sum_{i=1}^{\infty}\frac{V_{\sigma\,i}}{N^{i}}~{}~{},~{}~{}V_{\pi}~{}=~{}\sum_{i=1}^{\infty}\frac{V_{\pi\,i}}{N^{i}}$
(6.15)

although we will only need $V_{\sigma\,1}$, $V_{\sigma\,2}$, $V_{\pi\,1}$ and
$V_{\pi\,2}$ to determine $\eta_{3}$. If one focuses on the leading order
large $N$ piece of (6.13) it is equivalent to that which produced $\eta_{1}$
earlier. So, for instance, a contribution to the $O(1/N^{3})$ part from the
first term on the right hand side of (6.13) can be simplified to

$\Delta_{\sigma\,1}\left.\left[\frac{\partial
V_{\sigma\,2}}{\partial\delta^{\prime}}~{}-~{}\frac{\partial
V_{\sigma\,2}}{\partial\delta}\right]\right|_{{\mbox{\footnotesize{res}}};\,\delta=0,\delta^{\prime}=0}$
(6.16)

for the $V_{\sigma}$ vertex while the contribution from the second term
involving $V_{\pi}$ is similar.

##  7 Evaluation of $\eta_{3}$.

Having derived the key equation (6.13) satisfied by $\eta_{3}$ we now turn to
details of its determination. This entails computing the corrections to the
vertex functions of the conformal bootstrap equations (6.7) and (6.13). The
leading order contributions are shown in Figures 15 and 16 and correspond to
$V_{\sigma\,1}$ and $V_{\pi\,1}$ respectively where

$V_{\sigma\,1}~{}=~{}V_{1\,\sigma\sigma\sigma}~{}+~{}V_{1\,\sigma\pi\pi}~{}~{},~{}~{}V_{\pi\,1}~{}=~{}V_{1\,\pi\sigma\sigma}~{}+~{}V_{1\,\pi\pi\pi}~{}.$
(7.1)

At next order the graphs that comprise $V_{\sigma\,2}$ and $V_{\pi\,2}$ are
illustrated in Figures 18 and 19 and are clearly primitive. While the leading
order $1/N$ terms of both $V_{\sigma\,1}$ and $V_{\pi\,1}$ are needed to
verify $\eta_{2}$ the next term in the series is required for $\eta_{3}$. This
$N$ dependence arises through the exponents $\alpha_{\psi}$, $\beta_{\sigma}$
and $\beta_{\pi}$. Therefore what are sometimes termed $3$-gamma graphs, which
are illustrated in Figures 15 and 16, have to be computed. This is achieved
via the method developed in [20] using the properties of the Polyakov
conformal triangle.

Figure 19: $O(1/N^{2})$ graphs contributing to $V_{\pi}$.

For instance, the explicit designation of the exponents on the lines of both
$V_{1\,\sigma\sigma\sigma}$ and $V_{1\,\sigma\pi\pi}$ are shown in Figures 20
and 21 where

$\gamma~{}=~{}1~{}-~{}\eta$ (7.2)

is set for shorthand due to $\beta_{\sigma}$ and $\beta_{\pi}$ differing only
in the piece involving the respective vertex anomalous dimensions. The factor
of $2$ associated with $\Delta_{\sigma}$ and $\Delta_{\pi}$ is a consequence
of (6.5) and (6.6). The graph of Figure 20 was evaluated in [27]. If we set

$V_{1\,\sigma\sigma\sigma}~{}=~{}-~{}\frac{Q^{3}}{\Delta_{\sigma}(\Delta_{\sigma}-\delta)(\Delta_{\sigma}-\delta^{\prime})}\exp[F_{\sigma\sigma\sigma}(\delta,\delta^{\prime},\Delta_{\sigma})]$
(7.3)

which is consistent with the general form of $\Gamma_{(n)}$ then we recall
that

$\displaystyle F_{\sigma\sigma\sigma}(\delta,\delta^{\prime},\Delta_{\sigma})$
$\displaystyle=$
$\displaystyle\left[5B_{\gamma}-2B_{\alpha_{\psi}-1}-3B_{0}-\frac{2}{(\alpha_{\psi}-1)}\right]\Delta_{\sigma}-\left[B_{\gamma}-B_{0}\right]\delta$
(7.4)
$\displaystyle+~{}\left[B_{0}-B_{\alpha_{\psi}-1}-\frac{1}{(\alpha_{\psi}-1)}\right]\delta^{\prime}+\left[C_{\alpha_{\psi}-1}-\frac{1}{(\alpha_{\psi}-1)}\right]\delta\delta^{\prime}$
$\displaystyle+~{}\left[C_{\gamma}+C_{0}-2C_{\alpha_{\psi}-1}+\frac{2}{(\alpha_{\psi}-1)^{2}}\right]\Delta_{\sigma}\delta$
$\displaystyle+~{}\frac{1}{2}\left[\frac{1}{(\alpha_{\psi}-1)^{2}}-C_{\alpha_{\psi}-1}-C_{0}\right]{\delta^{\prime}}^{2}$
$\displaystyle+~{}\left[C_{0}-C_{\gamma}-2C_{\alpha_{\psi}-1}+\frac{2}{(\alpha_{\psi}-1)^{2}}\right]\Delta_{\sigma}\delta^{\prime}-\frac{1}{2}\left[C_{\gamma}+C_{0}\right]\delta^{2}$
$\displaystyle+~{}\left[C_{\alpha_{\psi}-1}-\frac{7}{2}C_{\gamma}-\frac{3}{2}C_{0}-\frac{1}{(\alpha_{\psi}-1)^{2}}\right]\Delta_{\sigma}^{2}$

with the shorthand notation, [20],

$\displaystyle B_{z}$ $\displaystyle=$
$\displaystyle\psi(\mu-z)~{}+~{}\psi(z)~{}~{}~{},~{}~{}~{}B_{0}~{}=~{}\psi(1)~{}+~{}\psi(\mu)$
$\displaystyle C_{z}$ $\displaystyle=$
$\displaystyle\psi^{\prime}(z)~{}-~{}\psi^{\prime}(\mu-z)~{}~{}~{},~{}~{}~{}C_{0}~{}=~{}\psi^{\prime}(\mu)~{}-~{}\psi^{\prime}(1)$
(7.5)

and

$Q~{}=~{}-\,\frac{\pi^{2\mu}a^{2}(\alpha_{\psi}-1)a(\gamma)}{(\alpha_{\psi}-1)^{2}\Gamma(\mu)}~{}.$
(7.6)

We note that in [27] not all the terms quadratic in combinations of the
parameters $\delta$ or $\delta^{\prime}$ were recorded. We have included them
here for completeness and as an aid to an interested reader although they do
not contribute to the consistency equation.

Figure 20: Regularized one loop $3$-gamma graph denoted by
$V_{1\,\sigma\sigma\sigma}$ contributing to $V_{\sigma}$ containing a $\sigma$
propagator and including the conformal triangles.

We have rederived (7.4) that appeared in [27] for $V_{1\,\sigma\sigma\sigma}$
in order to extend it for $V_{1\,\sigma\pi\pi}$. The latter is more general
than the former since it will depend on $\chi_{\pi}$ as well as
$\chi_{\sigma}$. This leads to a check in that formally taking the limit
$\chi_{\pi}$ $\to$ $\chi_{\sigma}$ in $V_{1\,\sigma\pi\pi}$ should produce the
expression for $V_{1\,\sigma\sigma\sigma}$. If we define

$\Gamma_{1\,\sigma\pi\pi}~{}=~{}-~{}\frac{Q^{3}}{\Delta_{\pi}(\Delta_{\sigma}-\delta)(\Delta_{\pi}-\delta^{\prime})}\exp[F_{\sigma\pi\pi}(\delta,\delta^{\prime},\Delta_{\sigma},\Delta_{\pi})]$
(7.7)

and repeat the procedure that resulted in (7.4) we find

$\displaystyle
F_{\sigma\pi\pi}(\delta,\delta^{\prime},\Delta_{\sigma},\Delta_{\pi})$
$\displaystyle=$
$\displaystyle\left[B_{0}-B_{\gamma}\right]\delta+\left[B_{0}-B_{\alpha_{\psi}-1}-\frac{1}{(\alpha_{\psi}-1)}\right]\delta^{\prime}+\left[B_{\gamma}-B_{0}\right]\Delta_{\sigma}$
(7.8)
$\displaystyle+~{}\left[4B_{\gamma}-2B_{0}-2B_{\alpha_{\psi}-1}-\frac{2}{(\alpha_{\psi}-1)}\right]\Delta_{\pi}-\frac{1}{2}\left[C_{\gamma}+C_{0}\right]\Delta_{\sigma}^{2}$
$\displaystyle+~{}\left[\frac{1}{(\alpha_{\psi}-1)^{2}}-C_{\alpha_{\psi}-1}-3C_{\gamma}-C_{0}\right]\Delta_{\pi}^{2}+\left[C_{\gamma}+C_{0}\right]\delta\Delta_{\sigma}$
$\displaystyle+~{}2\left[C_{\alpha_{\psi}-1}-\frac{1}{(\alpha_{\psi}-1)^{2}}\right]\Delta_{\sigma}\Delta_{\pi}+\left[\frac{1}{(\alpha_{\psi}-1)^{2}}-C_{\alpha_{\psi}-1}\right]\delta^{\prime}\Delta_{\sigma}$
$\displaystyle+~{}\left[C_{0}-C_{\alpha_{\psi}-1}-C_{\gamma}+\frac{1}{(\alpha_{\psi}-1)^{2}}\right]\delta^{\prime}\Delta_{\pi}-\frac{1}{2}\left[C_{\gamma}+C_{0}\right]\delta^{2}$
$\displaystyle+~{}\frac{1}{2}\left[\frac{1}{(\alpha_{\psi}-1)^{2}}-C_{\alpha_{\psi}-1}-C_{0}\right]{\delta^{\prime}}^{2}+2\left[\frac{1}{(\alpha_{\psi}-1)^{2}}-C_{\alpha_{\psi}-1}\right]\delta\Delta_{\pi}$
$\displaystyle+~{}\left[C_{\alpha_{\psi}-1}-\frac{1}{(\alpha_{\psi}-1)^{2}}\right]\delta\delta^{\prime}~{}.$

Clearly this is consistent with (7.4) since
$F_{\sigma\pi\pi}(\delta,\delta^{\prime},\Delta_{\sigma},\Delta_{\sigma})$ $=$
$F_{\sigma\sigma\sigma}(\delta,\delta^{\prime},\Delta_{\sigma})$. These two
expressions are sufficient to calculate the contribution to (6.13) from
$V_{\sigma\,1}$. For the second term of (6.13) involving $V_{\pi\,1}$ the
expressions for the corresponding $3$-gamma graphs can be deduced from those
of $V_{\sigma\,1}$ by formally swapping $\Delta_{\sigma}$ and $\Delta_{\pi}$
in (7.4) and (7.8).

Figure 21: Regularized one loop $3$-gamma graph denoted by
$V_{1\,\sigma\pi\pi}$ contributing to $V_{\sigma}$ containing a $\pi$
propagator and including the conformal triangles.

To complete the evaluation of $\eta_{3}$ requires the contribution to (6.13)
from $V_{\sigma\,2}$ and $V_{\pi\,2}$. The values of the four topologies
underlying the graphs in Figures 18 and 19 have been computed previously, [25,
26, 27]. Strictly though it is the combination of (6.16) and the $\pi^{a}$
counterpart that have been calculated. The values of the individual terms
cannot be extracted as the underlying three loop integral does not have unique
vertices to allow the integration to proceed. While it would appear from the
figures that there are three distinct topologies this would be the case if
there was no regularization. With non-zero $\delta$ and $\delta^{\prime}$
through specified external legs there are two independent contributions from
the three loop non-planar graphs. Indeed the contribution to (6.7) from these
non-planar graphs is the same but not for (6.13). For each of the graphs of
Figures 20 and 21 we have computed the associated group factor by again using
the color.h package based on [42]. The higher rank colour Casimir will arise
in the graphs of Figure 19 where four $\pi^{a}$ fields connect to the fermion
loop.

The next stage is to assemble all the components contributing to (6.7) and
(6.13). The purpose of the two equations of (6.7) is to fix the values of
$\bar{z}$ and $\bar{y}$ at $O(1/N^{2})$. While these were eliminated to arrive
at (6.13) combinations of them are present as factors in each of the graphs of
Figures 15, 16, 18 and 19. However we can now illuminate the difficulties with
a direct evaluation in the $U(1)$ case. The formal solution for $\bar{z}$ in
(2.4) at leading order is

$\bar{z}_{1}~{}=~{}\left[[2C_{F}-C_{A}]\Delta_{\sigma\,1}^{3}~{}+~{}2C_{F}\Delta_{\sigma\,1}^{2}\Delta_{\pi\,1}\right]\frac{1}{C_{A}Q^{3}}~{}.$
(7.9)

While $C_{A}$ vanishes in the abelian limit producing a singularity, when the
leading order values for the vertex dimensions are included these are
proportional to $C_{A}$. So that overall the leading order variable $\bar{z}$
is finite in the abelian limit. For a direct evaluation one could not
determine $\bar{z}_{1}$ since it would equate to the mathematically ill-
defined quantity of $0/0$. The situation for $\bar{y}_{1}$ is similar. It was
this that forced us to consider the more general Lagrangian. For (2.4) we find

$\displaystyle\bar{z}_{1}$ $\displaystyle=$
$\displaystyle-~{}\frac{(C_{F}N_{\\!c}-T_{F})^{2}\Gamma^{3}(2\mu-1)}{\Gamma^{3}(\mu-1)\Gamma^{3}(1-\mu)N_{\\!c}^{3}T_{F}^{3}}$
$\displaystyle\bar{y}_{1}$ $\displaystyle=$
$\displaystyle-~{}\frac{(2C_{F}N_{\\!c}-2T_{F}-C_{A}N_{\\!c})^{2}\Gamma^{3}(2\mu-1)}{4\Gamma^{3}(\mu-1)\Gamma^{3}(1-\mu)N_{\\!c}^{3}T_{F}^{3}}$
(7.10)

for instance which is finite in the abelian limit. With these values and those
for the next order we ultimately arrive at our main result for (2.2) which is

$\displaystyle\eta_{3}$ $\displaystyle=$
$\displaystyle\left[\left[\mu^{2}(\mu+8)\frac{d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}^{2}}{T_{F}}+\frac{1}{2}\mu(3\mu+1)C_{A}C_{F}T_{F}N_{\\!c}^{2}-\frac{1}{2}\mu(5\mu-1)C_{F}^{2}C_{A}N_{\\!c}^{3}\right.\right.$
(7.11)
$\displaystyle\left.\left.~{}~{}~{}-\frac{1}{24}\mu^{2}(\mu-4)C_{F}C_{A}^{2}N_{\\!c}^{3}+(\mu^{3}+10\mu^{2}+\mu-1)T_{F}^{3}-(2\mu^{3}+22\mu^{2}-7\mu+3)C_{F}N_{\\!c}T_{F}^{2}\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}-(6\mu^{2}-7\mu+3)C_{F}^{2}N_{\\!c}^{2}T_{F}+(2\mu-1)(\mu+1)C_{F}^{3}N_{\\!c}^{3}\right]\frac{\Theta(\mu)}{4[\mu-1]^{2}}\right.$
$\displaystyle\left.~{}-\left[4\mu(2\mu-1)C_{F}^{2}C_{A}N_{\\!c}^{3}-4\mu
C_{A}C_{F}T_{F}N_{\\!c}^{2}-\mu^{2}C_{F}C_{A}^{2}N_{\\!c}^{3}-4(2\mu-1)^{2}T_{F}^{3}\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}~{}~{}+4(4\mu-3)C_{F}N_{\\!c}T_{F}^{2}+4(4\mu-3)C_{F}^{2}N_{\\!c}^{2}T_{F}-4(2\mu-1)^{2}C_{F}^{3}N_{\\!c}^{3}\right]\frac{\Phi(\mu)}{8[\mu-1]^{2}}\right.$
$\displaystyle\left.~{}+\left[\frac{3}{2}T_{F}^{3}-3C_{F}N_{\\!c}T_{F}^{2}-\frac{1}{16}C_{A}^{2}C_{F}N_{\\!c}^{3}+\frac{3}{2}\frac{d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}^{2}}{T_{F}}\right]\left[\Theta(\mu)+\frac{1}{[\mu-1]^{2}}\right]\frac{\mu^{2}\Xi(\mu)}{[\mu-1]}\right.$
$\displaystyle\left.~{}+\left[24d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}^{2}+24T_{F}^{4}-48C_{F}N_{\\!c}T_{F}^{3}-C_{A}^{2}C_{F}N_{\\!c}^{3}T_{F}\right]\frac{\mu^{2}\Theta(\mu)\Psi(\mu)}{8[\mu-1]T_{F}}\right.$
$\displaystyle\left.~{}-\left[4(4\mu-3)C_{F}N_{\\!c}T_{F}^{2}-4\mu
C_{A}C_{F}T_{F}N_{\\!c}^{2}-\mu^{2}C_{F}C_{A}^{2}N_{\\!c}^{3}-4(2\mu-1)^{2}T_{F}^{3}\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}~{}~{}+4\mu(2\mu-1)C_{F}^{2}C_{A}N_{\\!c}^{3}+4(4\mu-3)C_{F}^{2}N_{\\!c}^{2}T_{F}-4(2\mu-1)^{2}C_{F}^{3}N_{\\!c}^{3}\right]\frac{3\Psi^{2}(\mu)}{8[\mu-1]^{2}}\right.$
$\displaystyle\left.~{}+\left[\frac{(24\mu^{2}-52\mu+9)}{2\mu[\mu-1]^{2}}C_{F}^{2}N_{\\!c}^{2}T_{F}-\frac{(12\mu^{2}-23\mu+2)}{4[\mu-1]^{3}}C_{A}C_{F}T_{F}N_{\\!c}^{2}\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}~{}~{}-\frac{(19\mu-2)(2\mu-1)}{4[\mu-1]^{3}}C_{F}^{2}C_{A}N_{\\!c}^{3}-\frac{(2\mu^{5}-3\mu^{4}-46\mu^{3}+56\mu^{2}-23\mu+3)}{2\mu[\mu-1]^{3}}T_{F}^{3}\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}~{}~{}-\frac{(12\mu^{5}-34\mu^{4}+20\mu^{3}+76\mu^{2}-61\mu+9)}{2\mu[\mu-1]^{3}}C_{F}N_{\\!c}T_{F}^{2}-\frac{\mu^{2}(\mu+2)(\mu-5)}{12[\mu-1]^{3}}C_{F}C_{A}^{2}N_{\\!c}^{3}\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}~{}~{}+\frac{(11\mu-3)(2\mu-1)^{2}}{2\mu[\mu-1]^{3}}C_{F}^{3}N_{\\!c}^{3}-\frac{\mu^{2}(2\mu+1)(\mu-2)}{2[\mu-1]^{3}}\frac{d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}^{2}}{T_{F}}\right]\Psi(\mu)\right.$
$\displaystyle\left.~{}-\frac{3(11\mu^{2}-12\mu+2)}{8[\mu-1]^{4}}C_{A}C_{F}T_{F}N_{\\!c}^{2}-\frac{\mu^{2}(8\mu^{3}-10\mu^{2}-17\mu-32)}{96[\mu-1]^{4}}C_{F}C_{A}^{2}N_{\\!c}^{3}\right.$
$\displaystyle\left.~{}-\frac{3(4\mu-1)(2\mu-1)}{4[\mu-1]^{4}}C_{F}^{2}C_{A}N_{\\!c}^{3}-\frac{(2\mu-1)(\mu^{6}-\mu^{4}-18\mu^{3}+21\mu^{2}-8\mu+1)}{2\mu^{2}[\mu-1]^{4}}T_{F}^{3}\right.$
$\displaystyle\left.~{}-\frac{(12\mu^{7}-14\mu^{6}-24\mu^{5}+11\mu^{4}+108\mu^{3}-99\mu^{2}+30\mu-3)}{2\mu^{2}[\mu-1]^{4}}C_{F}N_{\\!c}T_{F}^{2}\right.$
$\displaystyle\left.~{}+\frac{3(15\mu^{4}-36\mu^{3}+33\mu^{2}-10\mu+1)}{2\mu^{2}[\mu-1]^{4}}C_{F}^{2}N_{\\!c}^{2}T_{F}+\frac{(3\mu-1)^{2}(2\mu-1)^{2}}{2\mu^{2}[\mu-1]^{4}}C_{F}^{3}N_{\\!c}^{3}\right.$
$\displaystyle\left.~{}-\frac{\mu^{2}(\mu+1)(2\mu-1)}{2[\mu-1]^{3}}\frac{d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}^{2}}{T_{F}}\right]\frac{\eta_{1}^{3}}{[T_{F}+C_{F}N_{\\!c}]^{3}}$

which is considerably more involved than the other exponents. We note that
electronic versions of (7.11) and all the large $N$ exponents of (2.4) are
available in an attached data file.

Like the $\eta_{3}$ result of [20] (7.11) contains the function $\Xi(\mu)$
which is related to the function $I(\mu)$ of that paper via

$I(\mu)~{}=~{}-~{}\frac{2}{3(\mu-1)}~{}+~{}\Xi(\mu)~{}.$ (7.12)

The $\epsilon$ expansion of $I(\mu)$ around $d$ $=$ $D$ $-$ $2\epsilon$ is
known for integer $D$. In the case of $D$ $=$ $2$ and $4$ this was derived in
[47] and built on the earlier expansion provided in [20]. When $D$ $=$ $3$ the
expansion was provided in [48] where the leading order term is

$I(\mbox{\small{$\frac{3}{2}$}})~{}=~{}2\ln
2~{}-~{}\frac{21}{\pi^{2}}\zeta(3)$ (7.13)

which was derived in [20] using uniqueness methods. So, for instance, we can
record the first three terms of the large $N$ expansion of each exponent in
three dimensions for (2.4) which are

$\displaystyle\left.\eta\right|_{d=3}$ $\displaystyle=$
$\displaystyle\frac{8[C_{F}N_{\\!c}+T_{F}]}{3\pi^{2}N_{\\!c}T_{F}N}~{}+~{}\left[28C_{F}^{2}N_{\\!c}^{2}-9C_{A}C_{F}N_{\\!c}^{2}-16C_{F}N_{\\!c}T_{F}+28T_{F}^{2}\right]\frac{64}{27\pi^{4}N_{\\!c}^{2}T_{F}^{2}N}$
$\displaystyle+~{}\left[2592\pi^{2}\ln(2)d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}^{2}-108\pi^{2}\ln(2)C_{A}^{2}C_{F}N_{\\!c}^{3}T_{F}-5184\pi^{2}\ln(2)C_{F}N_{\\!c}T_{F}^{3}\right.$
$\displaystyle\left.~{}~{}~{}~{}+2592\pi^{2}\ln(2)T_{F}^{4}+1134\zeta(3)C_{A}^{2}C_{F}N_{\\!c}^{3}T_{F}+54432\zeta(3)C_{F}N_{\\!c}T_{F}^{3}\right.$
$\displaystyle\left.~{}~{}~{}~{}-27216\zeta(3)d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}^{2}-27216\zeta(3)T_{F}^{4}-351\pi^{2}C_{A}^{2}C_{F}N_{\\!c}^{3}T_{F}\right.$
$\displaystyle\left.~{}~{}~{}~{}+2916C_{A}^{2}C_{F}N_{\\!c}^{3}T_{F}+1368\pi^{2}C_{A}C_{F}^{2}N_{\\!c}^{3}T_{F}-17568C_{A}C_{F}^{2}N_{\\!c}^{3}T_{F}\right.$
$\displaystyle\left.~{}~{}~{}~{}-360\pi^{2}C_{A}C_{F}N_{\\!c}^{2}T_{F}^{2}-35712C_{A}C_{F}N_{\\!c}^{2}T_{F}^{2}-2112\pi^{2}C_{F}^{3}N_{\\!c}^{3}T_{F}+33536C_{F}^{3}N_{\\!c}^{3}T_{F}\right.$
$\displaystyle\left.~{}~{}~{}~{}+1152\pi^{2}C_{F}^{2}N_{\\!c}^{2}T_{F}^{2}+94848C_{F}^{2}N_{\\!c}^{2}T_{F}^{2}-5328\pi^{2}C_{F}N_{\\!c}T_{F}^{3}-65856C_{F}N_{\\!c}T_{F}^{3}\right.$
$\displaystyle\left.~{}~{}~{}~{}+3240\pi^{2}d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}^{2}-54432d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}^{2}+1128\pi^{2}T_{F}^{4}\right.$
$\displaystyle\left.~{}~{}~{}~{}-20896T_{F}^{4}\right]\frac{8}{243\pi^{6}N_{\\!c}^{3}T_{F}^{4}N^{3}}~{}+~{}O\left(\frac{1}{N^{4}}\right)$
$\displaystyle\left.\beta_{\sigma}\right|_{d=3}$ $\displaystyle=$
$\displaystyle
1~{}+~{}\frac{32[C_{F}N_{\\!c}-2T_{F}]}{3\pi^{2}N_{\\!c}T_{F}N}$
$\displaystyle+~{}\frac{64[45C_{A}C_{F}N_{\\!c}^{2}-92C_{F}^{2}N_{\\!c}^{2}+27\pi^{2}C_{F}N_{\\!c}T_{F}-76C_{F}N_{\\!c}T_{F}-27\pi^{2}T_{F}^{2}+304T_{F}^{2}]}{27\pi^{4}N_{\\!c}^{2}T_{F}^{2}N^{2}}$
$\displaystyle+~{}O\left(\frac{1}{N^{3}}\right)$
$\displaystyle\left.\beta_{\pi}\right|_{d=3}$ $\displaystyle=$ $\displaystyle
1~{}+~{}\frac{8[3C_{A}N_{\\!c}-8C_{F}N_{\\!c}+4T_{F}]}{3\pi^{2}N_{\\!c}T_{F}N}$
$\displaystyle+~{}\left[9\pi^{2}C_{A}^{2}C_{F}N_{\\!c}^{2}T_{F}-252C_{A}^{2}C_{F}N_{\\!c}^{2}T_{F}+864C_{A}C_{F}^{2}N_{\\!c}^{2}T_{F}+936C_{A}C_{F}N_{\\!c}T_{F}^{2}\right.$
$\displaystyle\left.~{}~{}~{}~{}-1024C_{F}^{3}N_{\\!c}^{2}T_{F}-2336C_{F}^{2}N_{\\!c}T_{F}^{2}+216\pi^{2}C_{F}T_{F}^{3}+992C_{F}T_{F}^{3}\right.$
$\displaystyle\left.~{}~{}~{}~{}-216\pi^{2}d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}+3456d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}\right]\frac{8}{27\pi^{4}C_{F}N_{\\!c}^{2}T_{F}^{3}N^{2}}~{}+~{}O\left(\frac{1}{N^{3}}\right)$
$\displaystyle\left.\frac{1}{\nu}\right|_{d=3}$ $\displaystyle=$
$\displaystyle 1~{}-~{}\frac{32[C_{F}N_{\\!c}+T_{F}]}{3\pi^{2}N_{\\!c}T_{F}N}$
(7.14)
$\displaystyle+~{}\left[324C_{A}^{2}C_{F}N_{\\!c}^{3}T_{F}-27\pi^{2}C_{A}^{2}C_{F}N_{\\!c}^{3}T_{F}+108\pi^{2}C_{A}C_{F}^{2}N_{\\!c}^{3}T_{F}\right.$
$\displaystyle\left.~{}~{}~{}~{}-1260C_{A}C_{F}^{2}N_{\\!c}^{3}T_{F}-72C_{A}C_{F}N_{\\!c}^{2}T_{F}^{2}-108\pi^{2}C_{F}^{3}N_{\\!c}^{3}T_{F}+1264C_{F}^{3}N_{\\!c}^{3}T_{F}\right.$
$\displaystyle\left.~{}~{}~{}~{}+624C_{F}^{2}N_{\\!c}^{2}T_{F}^{2}+108\pi^{2}C_{F}N_{\\!c}T_{F}^{3}-240C_{F}N_{\\!c}T_{F}^{3}+162\pi^{2}d_{F}^{abcd}d_{F}^{abcd}N_{\\!c}^{2}\right.$
$\displaystyle\left.~{}~{}~{}~{}+54\pi^{2}T_{F}^{4}+1264T_{F}^{4}\right]\frac{32}{27\pi^{4}N_{\\!c}^{2}T_{F}^{3}[C_{F}N_{\\!c}+T_{F}]N^{2}}~{}+~{}O\left(\frac{1}{N^{3}}\right)$

from which we will be able to deduce the analogous expressions for the chiral
XY model. Finally we note that (7.11) agrees with known expressions in the
limit of (5.3).

Exponent | Padé | $N_{t}~{}=~{}2$ | $N_{t}~{}=~{}4$ | $N_{t}~{}=~{}6$ | $N_{t}~{}=~{}8$
---|---|---|---|---|---
$\eta^{\mbox{\footnotesize{XY}}}$ | [1,1] | 0.087176 | 0.038058 | 0.024343 | 0.017894
| [1,2] | 0.078821 | 0.037198 | 0.024104 | 0.017797
$\eta_{\phi}^{\mbox{\footnotesize{XY}}}$ | [0,1] | 0.880984 | 0.936726 | 0.956909 | 0.967330
| [1,1] | 0.902301 | 0.943303 | 0.960064 | 0.969176
| [0,2] | 0.907741 | 0.944124 | 0.960325 | 0.969290
$\frac{1}{\nu^{\mbox{\footnotesize{XY}}}}$ | [0,1] | 0.787284 | 0.880984 | 0.917378 | 0.936726
| [1,1] | 0.902616 | 0.928416 | 0.943409 | 0.953210

Table $1$. Padé approximants for exponents in chiral XY model for $N_{t}$ $=$
$2$, $4$, $6$ and $8$.

Exponent | Reference | Method | $N_{t}~{}=~{}2$
---|---|---|---
$\eta^{\mbox{\footnotesize{XY}}}$ | [15] |  $[2,2]$ Padé | 0.117
| [15] |  $[3,1]$ Padé | 0.108
| [11] | functional RG | 0.062
$\eta_{\phi}^{\mbox{\footnotesize{XY}}}$ | [15] |  $[2,2]$ Padé | 0.810
| [15] |  $[3,1]$ Padé | 0.788
| [11] | functional RG | 0.88
$\frac{1}{\nu^{\mbox{\footnotesize{XY}}}}$ | [15] |  $[2,2]$ Padé | 0.840
| [15] |  $[3,1]$ Padé | 0.841
| [11] | functional RG | 0.862

Table $2$. Exponent estimates in chiral XY model for $N_{t}$ $=$ $2$ from
$[2,2]$ and $[3,1]$ Padé approximants of four dimensional $\epsilon$ expansion
[15] and the functional renormalization group [11].

##  8 XY model exponents.

Having established the critical exponents for the non-abelian Nambu-Jona-
Lasinio model, (2.4), we can now derive those for the chiral XY model in the
abelian limit. It is clear that there are no singular colour group factors for
the results of (3.6), (3.7), (3.8), (4.2), (4.4), (5.1), (5.2) and (7.11).
Therefore it is safe to take the abelian limit of (2.9) giving

$\displaystyle\eta_{1}^{\mbox{\footnotesize{XY}}}$ $\displaystyle=$
$\displaystyle-~{}\frac{4\Gamma(2\mu-1)}{\mu\Gamma(1-\mu)\Gamma(\mu-1)\Gamma^{2}(\mu)}$
$\displaystyle\chi_{\sigma\,1}^{\mbox{\footnotesize{XY}}}$ $\displaystyle=$
$\displaystyle\chi_{\pi\,1}^{\mbox{\footnotesize{XY}}}~{}=~{}0$
$\displaystyle\lambda_{1}^{\mbox{\footnotesize{XY}}}$ $\displaystyle=$
$\displaystyle-~{}(2\mu-1)\eta_{1}^{\mbox{\footnotesize{XY}}}$ (8.1)

at leading order which clearly illustrates the issue surrounding the vanishing
leading order vertex anomalous dimension for both Yukawa interactions. At next
order we have

$\displaystyle\eta_{2}^{\mbox{\footnotesize{XY}}}$ $\displaystyle=$
$\displaystyle\left[\frac{(5\mu-1)}{2\mu(\mu-1)}+\Psi(\mu)\right]{\eta_{1}^{\mbox{\footnotesize{XY}}}}^{2}$
$\displaystyle\chi_{\sigma\,2}^{\mbox{\footnotesize{XY}}}$ $\displaystyle=$
$\displaystyle\chi_{\pi\,2}^{\mbox{\footnotesize{XY}}}~{}=~{}-~{}\frac{\mu^{2}(4\mu^{2}-10\mu+7)}{2(\mu-1)^{3}}{\eta_{1}^{\mbox{\footnotesize{XY}}}}^{2}$
$\displaystyle\lambda_{2}^{\mbox{\footnotesize{XY}}}$ $\displaystyle=$
$\displaystyle\left[\frac{8\mu(\mu-1)}{(\mu-2)^{2}\eta_{1}^{\mbox{\footnotesize{XY}}}}+\frac{(8\mu^{6}-52\mu^{5}+108\mu^{4}-68\mu^{3}-21\mu^{2}+24\mu-4)}{2\mu(\mu-1)(\mu-2)^{2}}\right.$
(8.2)
$\displaystyle\left.~{}-\frac{2(2\mu^{6}-13\mu^{5}+31\mu^{4}-32\mu^{3}+9\mu^{2}+6\mu-2)}{(\mu-1)^{2}(\mu-2)^{2}}\Psi(\mu)\right.$
$\displaystyle\left.~{}-\frac{\mu^{2}(2\mu-3)}{(\mu-1)(\mu-2)}\left[\Phi(\mu)+\Psi^{2}(\mu)\right]+\frac{3\mu^{2}(3\mu-4)}{2(\mu-1)(\mu-2)}\Theta(\mu)\right]{\eta_{1}^{\mbox{\footnotesize{XY}}}}^{2}~{}.$

We have checked that these expressions are in agreement with earlier work,
[17, 24]. Finally we obtain the main result for this model which is

$\displaystyle\eta_{3}^{\mbox{\footnotesize{XY}}}$ $\displaystyle=$
$\displaystyle\left[\frac{1}{2}\left[\Phi(\mu)+3\Psi^{2}(\mu)\right]-\frac{(2\mu^{4}-3\mu^{3}-15\mu^{2}+18\mu-3)}{2\mu(\mu-1)^{2}}\Psi(\mu)-\frac{1}{4}\Theta(\mu)\right.$
(8.3)
$\displaystyle\left.~{}-\frac{(4\mu^{7}-4\mu^{6}-7\mu^{5}-26\mu^{4}+84\mu^{3}-68\mu^{2}+20\mu-2)}{4\mu^{2}(\mu-1)^{4}}\right]{\eta_{1}^{\mbox{\footnotesize{XY}}}}^{3}~{}.$

The expression is considerably simpler than its non-abelian counterpart
similar to the $O(1/N^{2})$ exponents. One aspect of this is that the function
$\Xi(\mu)$ is absent which is not unrelated to the vanishing of
$\chi_{\sigma}$ and $\chi_{\pi}$ at leading order.

Figure 22: Padé approximants of $\eta^{\mbox{\footnotesize{XY}}}$ for $N_{t}$
$=$ $2$, $4$, $6$ and $8$.

While the exponents to $O(1/N^{2})$ agree with previous derivations, it is
possible to check them against perturbative computations in four dimensions.
In [15] the $\beta$-function and anomalous dimensions of the chiral XY model
were determined to four loops in the $\overline{\mbox{MS}}$ scheme. Moreover
the $\epsilon$ expansion of the corresponding critical exponents were computed
to $O(\epsilon^{4})$. Therefore if we expand our expressions in the same
$\epsilon$ expansion they ought to be in agreement. Setting $d$ $=$ $4$ $-$
$2\epsilon$ in (8.1), (8.2) and (8.3) we find

$\displaystyle\left.\eta^{\mbox{\footnotesize{XY}}}\right|_{d=4-2\epsilon}$
$\displaystyle=$
$\displaystyle\left[\epsilon-\frac{3}{2}\epsilon^{2}-\frac{3}{4}\epsilon^{3}+\left[2\zeta_{3}-\frac{3}{8}\right]\epsilon^{4}+O(\epsilon^{5})\right]\frac{1}{N_{t}}$
$\displaystyle+~{}\left[-~{}\epsilon+\frac{23}{4}\epsilon^{2}-\frac{43}{8}\epsilon^{3}-\left[8\zeta_{3}+\frac{19}{4}\right]\epsilon^{4}+O(\epsilon^{5})\right]\frac{1}{N_{t}^{2}}$
$\displaystyle+~{}\left[\epsilon-\frac{43}{4}\epsilon^{2}+\frac{77}{4}\epsilon^{3}+\left[\frac{39}{2}\zeta_{3}+\frac{581}{16}\right]\epsilon^{4}+O(\epsilon^{5})\right]\frac{1}{N_{t}^{3}}~{}+~{}O\left(\frac{1}{N_{t}^{4}}\right)$
$\displaystyle\left.\eta_{\phi}^{\mbox{\footnotesize{XY}}}\right|_{d=4-2\epsilon}$
$\displaystyle=$ $\displaystyle
2\epsilon+\left[-~{}2\epsilon+3\epsilon^{2}+\frac{3}{2}\epsilon^{3}+\left[\frac{3}{4}-4\zeta_{3}\right]\epsilon^{4}+O(\epsilon^{5})\right]\frac{1}{N_{t}}$
$\displaystyle+\left[2\epsilon+\frac{1}{2}\epsilon^{2}-\frac{101}{4}\epsilon^{3}+\left[16\zeta_{3}+\frac{51}{2}\right]\epsilon^{4}+O(\epsilon^{5})\right]\frac{1}{N_{t}^{2}}~{}+~{}O\left(\frac{1}{N_{t}^{3}}\right)$
$\displaystyle\left.\frac{1}{\nu^{\mbox{\footnotesize{XY}}}}\right|_{d=4-2\epsilon}$
$\displaystyle=$ $\displaystyle
2-2\epsilon+\left[-~{}6\epsilon+13\epsilon^{2}-\frac{3}{2}\epsilon^{3}-\left[\frac{3}{4}+12\zeta_{3}\right]\epsilon^{4}+O(\epsilon^{5})\right]\frac{1}{N_{t}}$
(8.4)
$\displaystyle+\left[78\epsilon-\frac{497}{2}\epsilon^{2}+\left[\frac{453}{4}-120\zeta_{3}\right]\epsilon^{3}\right.$
$\displaystyle\left.~{}~{}~{}+\left[240\zeta_{5}-180\zeta_{4}+744\zeta_{3}+\frac{205}{2}\right]\epsilon^{4}+O(\epsilon^{5})\right]\frac{1}{N_{t}^{2}}~{}+~{}O\left(\frac{1}{N_{t}^{3}}\right)$

where the combination

$\eta_{\phi}~{}=~{}4~{}-~{}2\mu~{}-~{}2\eta~{}-~{}2\chi_{\sigma}$ (8.5)

tallies with the definition used in [15, 16]. We have also set $N$ $=$
$4N_{t}$ to adjust for the different trace and group conventions used here in
the underlying master integrals for the skeleton Dyson-Schwinger and conformal
bootstrap equations. So $N_{t}$ corresponds to the $N$ used in [15, 16]. It is
straightforward to verify that each expansion is in full agreement with [15]
after allowing for a different convention in the definition of $d$ in terms of
$\epsilon$. In light of this agreement with high order perturbation theory we
can obtain expressions for the exponents in three dimensions. Analytically we
have

$\displaystyle\left.\eta^{\mbox{\footnotesize{XY}}}\right|_{d=3}$
$\displaystyle=$
$\displaystyle\frac{4}{3\pi^{2}N_{t}}~{}+~{}\frac{160}{27\pi^{4}N_{t}^{2}}~{}-~{}\frac{80[3\pi^{2}+20]}{243\pi^{6}N_{t}^{3}}~{}+~{}O\left(\frac{1}{N_{t}^{4}}\right)$
$\displaystyle\left.\eta_{\phi}^{\mbox{\footnotesize{XY}}}\right|_{d=3}$
$\displaystyle=$ $\displaystyle
1~{}-~{}\frac{8}{3\pi^{2}N_{t}}~{}+~{}\frac{544}{27\pi^{4}N_{t}^{2}}~{}+~{}O\left(\frac{1}{N_{t}^{3}}\right)$
$\displaystyle\left.\frac{1}{\nu^{\mbox{\footnotesize{XY}}}}\right|_{d=3}$
$\displaystyle=$ $\displaystyle
1~{}-~{}\frac{16}{3\pi^{2}N_{t}}~{}+~{}\frac{[216\pi^{2}+2912]}{27\pi^{6}N_{t}^{2}}~{}+~{}O\left(\frac{1}{N_{t}^{3}}\right)$
(8.6)

or

$\displaystyle\left.\eta^{\mbox{\footnotesize{XY}}}\right|_{d=3}$
$\displaystyle=$ $\displaystyle
0.135095\frac{1}{N_{t}}~{}+~{}0.060835\frac{1}{N_{t}^{2}}~{}-~{}0.016988\frac{1}{N_{t}^{3}}~{}+~{}O\left(\frac{1}{N_{t}^{4}}\right)$
$\displaystyle\left.\eta_{\phi}^{\mbox{\footnotesize{XY}}}\right|_{d=3}$
$\displaystyle=$ $\displaystyle
1~{}-~{}0.270190\frac{1}{N_{t}}~{}+~{}0.206840\frac{1}{N_{t}^{2}}~{}+~{}O\left(\frac{1}{N_{t}^{3}}\right)$
$\displaystyle\left.\frac{1}{\nu^{\mbox{\footnotesize{XY}}}}\right|_{d=3}$
$\displaystyle=$ $\displaystyle
1~{}-~{}0.540380\frac{1}{N_{t}}~{}+~{}1.917775\frac{1}{N_{t}^{2}}~{}+~{}O\left(\frac{1}{N_{t}^{3}}\right)$
(8.7)

numerically.

Figure 23: Padé approximants of $\eta_{\phi}^{\mbox{\footnotesize{XY}}}$ for
$N_{t}$ $=$ $2$, $4$, $6$ and $8$.

The next stage is to extract estimates for the three dimensional exponents for
the case of interest in [15] which corresponds to $N_{t}$ $=$ $2$ here. To
improve the convergence of the expansion we have followed a similar approach
to [28] and evaluated several Padé approximants for each of the exponents.
These have been plotted in Figures 22, 23 and 24 for $N_{t}$ $=$ $2$, $4$, $6$
and $8$ for $2$ $<$ $d$ $<$ $4$ for $\eta^{\mbox{\footnotesize{XY}}}$,
$\eta_{\phi}^{\mbox{\footnotesize{XY}}}$ and
$1/\nu^{\mbox{\footnotesize{XY}}}$ respectively with numerical values for
three dimensions recorded in Table $1$. The latter three values of $N_{t}$ are
selected to gauge the effect the higher order $1/N$ corrections have. In the
case of $\eta$ it is worth noting that we can plot the behaviour of $\eta_{3}$
unlike [28]. This is because $\Xi(\mu)$ is absent from $\eta_{3}$ and it was
not possible to write the function in an analytic form that could be passed to
the plotting routine in that case. For $\eta^{\mbox{\footnotesize{XY}}}$ the
$[2,1]$ Padé had a singularity at around $d$ $=$ $3.6$. Equally for $N_{t}$
$=$ $2$ the $[0,2]$ Padé of $1/\nu^{\mbox{\footnotesize{XY}}}$ had a maximum
near a similar value of $d$. We have omitted plots of both Padé’s for each
exponent.

What is evident in the plots of $\eta^{\mbox{\footnotesize{XY}}}$ for various
$N_{t}$ is that there is convergence of the $O(1/N^{3})$ values down to
$N_{t}$ $=$ $4$. For $N_{t}$ $=$ $2$ there is a difference between the
$O(1/N^{2})$ and $O(1/N^{3})$ curves but the latter is bounded by the leading
two orders. This is similar to higher values of $N_{t}$. Given this it might
not be unreasonable to give $0.083$ as a rough estimate of
$\eta^{\mbox{\footnotesize{XY}}}$ for $N_{t}$ $=$ $2$ being the average of the
two Padé estimates. This is not inconsistent with the $N_{t}$ $=$ $2$
estimates from other approaches which are summarized in Table $2$. In the case
of $\eta_{\phi}^{\mbox{\footnotesize{XY}}}$ it is clear from Figure 23 that
there is very little difference for the value of the exponent between
successive orders in $1/N$. Indeed comparing the values for $N_{t}$ $=$ $2$
with other estimates there is a degree of consistency. Finally the situation
for $1/\nu^{\mbox{\footnotesize{XY}}}$ does not appear to be as good
especially at leading order for low values of $N_{t}$ due to the kink that is
evident in Figure 24 in its behaviour across $d$-dimensions. Although it
appears that this feature washes out for larger values of $N_{t}$, it is not
clear whether this would be the case for $N_{t}$ $=$ $2$ if an $O(1/N^{3})$
expression were available. Again one could assume that the two large $N$ Padé
approximants for $1/\nu^{\mbox{\footnotesize{XY}}}$ were bounds of the true
result. In this instance one would arrive at an estimate of around $0.845$
which is not dissimilar to the perturbative Padé or functional renormalization
group estimates of [11, 15].

Figure 24: Padé approximants of $\frac{1}{\nu^{\mbox{\footnotesize{XY}}}}$ for
$N_{t}$ $=$ $2$, $4$, $6$ and $8$.

##  9 Discussion.

We have provided the large $N$ critical exponents for the Nambu-Jona-Lasinio
universality class with Lagrangian (2.4) for a general Lie group using the
large $N$ critical point formalism pioneered in [18, 19, 20]. This included
the large $N$ conformal bootstrap formalism that allowed us to extract
$\eta_{3}$. To achieve this we had to extend the formalism to the situation
where there are two independent $3$-point interactions. By independent we mean
the vertex anomalous dimensions of the separate vertices were inequivalent.
The derivation of the underlying bootstrap equations was straightforward but
was discussed at length. Indeed from that it is evident that they can be
readily generalized to the case of more independent $3$-point interactions.
Taking the more general approach with a general Lie group means that results
for exponents in various models can be extracted in certain limits. Previously
$\eta_{3}$ was calculated for a limited number of fermionic models including
the $SU(2)$ Nambu-Jona-Lasinio model, [35], as the anomalous dimensions of
each vertex were the same. To produce the general Lie group results we were
able to benefit from the Form encoded color.h package which was based on [42]
that allowed us to express our exponents in terms of the general group Casimir
invariants. This was particularly important for the light-by-light graphs that
contribute to several exponents at $O(1/N^{2})$ and $\eta_{3}$. One of the
main consequences was that the abelian limit could be taken smoothly to deduce
exponents for the chiral XY model. This circumvented the application of the
bootstrap formalism directly to (2.2) due to an ill-defined intermediate
quantity that formally obstructs the direct derivation in the abelian case.
The ultimate general expressions are analytic in the colour group invariants
allowing the chiral XY model exponents to emerge smoothly. Finally this more
general group approach for the large $N$ formalism offers a more compact way
of extracting critical exponents for other universality classes. One benefit
for instance is that it gives access to additional information on the terms in
the explicit perturbative series of the underlying renormalization group
functions.

Acknowledgements. This work was supported by a DFG Mercator Fellowship and in
part through the STFC Consolidated ST/T000988/1. The graphs were drawn with
the Axodraw package [49]. Computations were carried out using the symbolic
manipulation language Form, [40, 41].

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