# Fixed Lines in Four Fermion Models in Two Dimensions

Sidan A, Tom Banks
Department of Physics and NHETC
Rutgers University, Piscataway, NJ 08854 USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Motivated by conjectures about near-horizon dynamics in quantum gravity, we
search for lines of perturbatively accessible fixed points emanating from
models of $N$ free fermions. Through two loops we find a new class of models,
apart from the well-known Abelian Thirring models. Further study is needed to
see whether these can lead to true conformal manifolds, or perhaps a new class
of large N fixed points.

##  1 Introduction

In a recent paper [1], one of us proposed a general strategy for studying the
local physics of models of quantum gravity in given classical space-time
backgrounds. Following Jacobson [2], the classical space-time is assumed to
reproduce the hydrodynamics of the underlying quantum theory, assigning a
local entropy to each causal diamond according to the BHJB [3] area law, which
is equivalent to the double null projection of Einstein’s equations with some
conserved stress tensor source. The ansatz of Carlip and Solodukhin [4, 5]
(C-S), generalized to arbitrary causal diamonds in [6], implies that the
modular Hamiltonian of each causal diamond is the Virasoro generator $L_{0}$
of a $1+1$ dimensional conformal field theory (CFT) living on the (past or
future)111The copies of the CFT on the past and future boundary of the diamond
are related by the unitary time evolution operator in the diamond. stretched
horizon. The CFT has a UV cutoff, related to the smallest size of causal
diamond for which the C-S analysis is valid, and a central charge proportional
to the area of the holographic screen of the diamond, in Planck units. Cardy’s
formula then matches the CFT entropy to the BHJB entropy, with the UV cutoff
on the Virasoro spectrum taken just at the Cardy matching point.

In [1] the CFT was taken to be built from free fermion fields $\psi_{i}$ where
the label $i$ runs over the spectrum of the Dirac operator on the holographic
screen of the diamond, with a transverse UV cutoff determined by the finite
central charge of the CFT. The motivation for this choice, had its roots in
the Holographic Space-time formalism [7], which was in turn based on A. Connes
[8] observation that Riemannian geometry is completely determined by the Dirac
operator. One can think of the fields $\psi_{i}(t,z)$ as the expansion
coefficients of the solutions of a fluctuating Dirac equation (which
represents a fluctuating geometry of the holographic screen) in eigenspinors
of the Dirac operator on the background geometry. The background geometry
represents the coarse-grained hydrodynamic average of the system.

The authors of [9] have argued that the dynamical system on the stretched
horizon of a black hole must be a fast scrambler of quantum information. Free
fields are not fast scramblers. In [1] a certain set of abelian Thirring
deformations of the free fermion system above was argued to preserve conformal
invariance and the central charge but act as a fast scrambler. The
preservation of the central charge is a property of lines of fixed points in
two dimensions. The purpose of the present paper is to explore whether there
can be other lines of fixed points emanating from the free fermion point. A
necessary, but not sufficient condition for this is the vanishing of the one
and two-loop beta functions. We will explore this condition for general four-
fermion interactions.

##  2 The Generalized Thirring model

Before defining the generalized Thirring model, we will introduce the original
Thirring model [10] in two dimensions. The original Thirring model of a
massless Dirac fermion with a four-fermion interaction of the current-current
type has the following Lagrangian,

$\begin{split}\mathcal{L}_{Th}&=i\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi+\frac{1}{2}gj^{\mu}j_{\mu}\hskip
2.84526pt,\end{split}$ (1)

with $g$ being the Thirring coupling and the current is defined as
$j^{\mu}=\overline{\psi}\gamma^{\mu}\psi$. This interaction is unique because
any four-fermi interactions can be rewritten in terms of this current-current
form using the Fierz transformation, a mathematical procedure used to deal
with expressions involving products of Dirac gamma matrices and spinors. A
derivation of the Fierz identity can be found in the appendix. A.
$\overline{\psi}=\psi^{\dagger}\gamma^{0}$ is the Dirac adjoint of a two-
component complex fermion $\psi=\begin{pmatrix}\psi_{1}\\\
\psi_{2}\end{pmatrix}$. The equations of motion for $\overline{\psi}$ and
$\psi$ are determined to be

$\begin{split}i\gamma^{\mu}\partial_{\mu}\psi-
gj^{\mu}\gamma_{\mu}\psi&=0\hskip 2.84526pt,\\\
i\partial_{\mu}\overline{\psi}\gamma^{\mu}+gj_{\mu}\overline{\psi}\gamma^{\mu}&=0\hskip
2.84526pt,\end{split}$ (2)

respectively. Then we find that the conservation law is given by
$\partial_{\mu}j^{\mu}=0$. The theory is renormalizable because the coupling
constant is dimensionless. In addition, the Lagrangian has $U(1)$ symmetry
$\psi\rightarrow e^{i\alpha}\psi$ and chiral symmetry $\psi\rightarrow
e^{i\alpha\gamma^{3}}\psi$ with parameter $\alpha$. In this paper we choose
the Weyl representation where $\gamma^{0}=\hskip
2.84526pt\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\hskip 2.84526pt$
$\gamma^{1}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$, and
$\gamma^{3}=\gamma^{0}\gamma^{1}=\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix}$
since it is more convenient for massless theories.

For the generalized Thirring model, we consider an arbitrary number $N$
massless fermions, denote different fermions with upper indices
$a,b,\dots=1,\dots,N$, then the Lagrangian of the theory can written as

$\begin{split}\mathcal{L}&=i\overline{\psi}^{a}\gamma^{\mu}\partial_{\mu}\psi^{a}+R_{abcd}\overline{\psi}^{a}\gamma^{\mu}\psi^{b}\overline{\psi}^{c}\gamma_{\mu}\psi^{d}\hskip
2.84526pt,\end{split}$ (3)

where $R_{abcd}$’s are coupling constants, and repeated indices are summed
over $1,\dots,N$ in the expression. In two dimensions, fermionic fields have
mass dimension $1/2$, and $R_{abcd}$ is dimensionless, thus the theory is
renormalizable. This model has minimum symmetry. Consider global
transformations $\psi^{a}\rightarrow e^{i\lambda_{a}}\psi^{a}$ and chiral
transformations $\psi^{a}\rightarrow e^{i\lambda_{a}\gamma^{3}}\psi^{a}$ in
each fields with parameter $\lambda_{a}$. These transformations in general
break both the global and chiral symmetry unless specific conditions are
imposed. As we can see, for any interaction
$R_{abcd}\overline{\psi}^{a}\gamma^{\mu}\psi^{b}\overline{\psi}^{c}\gamma_{\mu}\psi^{d}$
to be invariant under global and chiral transformations, we need the
parameters to satisfy $\lambda_{a}-\lambda_{b}+\lambda_{c}-\lambda_{d}=0$. A
special case of this model is when $a=b$ and $c=d$, this is known as the
Abelian Thirring model, which is invariant under the global and chiral
transformations.

Write down the current-current interaction explicitly in terms of spinor
components to obtain symmetry properties in the indices of the current-current
couplings. These properties come purely from the Grassmannian nature of
fermions, meaning that they are self-anticommuting.

$\begin{split}\overline{\psi}^{a}\gamma^{\mu}\psi^{b}\overline{\psi}^{c}\gamma_{\mu}\psi^{d}&=g_{\mu\nu}\overline{\psi}^{a}\gamma^{\mu}\psi^{b}\overline{\psi}^{c}\gamma^{\nu}\psi^{d}=2\psi^{a\dagger}_{1}\psi^{b}_{1}\psi^{c\dagger}_{2}\psi^{d}_{2}+2\psi^{a\dagger}_{2}\psi^{b}_{2}\psi^{c\dagger}_{1}\psi^{d}_{1}\end{split}$
(4)

where the metric $g_{\mu\nu}=diag(+1,-1)$ and the lower indices are spinor
indices of the two-dimensional fermionic fields.

From Eq.(4) we see that for Dirac fermions, couplings $R_{abcd}$ have the
following conditions in their indices:

$R_{abcd}=R_{cdab}\hskip 2.84526pt.$ (5)

On the other hand, Majorana fermions are real fields, i.e.
$\psi^{\dagger}=\psi$, which give additional symmetry properties in the
indices. The Majorana case couplings $R_{abcd}$ are also antisymmetric in
their first and the second pair of indices,

$R_{abcd}=-R_{bacd}=-R_{abdc}\hskip 2.84526pt.$ (6)

Now we will use the renormalization method to remove the UV divergence that
appeared in the Feynmann diagrams due to unknown small-scale physics, we
separate the fields and couplings into a renormalized part and a counterterm,
where the counterterm is intrinsically infinite and removes the UV divergence
in the diagrams. Let $\psi^{a}_{0}$ and $R_{abcd}^{0}$ be the bare quantities,
and let $\psi^{a}$ and $R_{abcd}$ be renormalized ones. Up to two-loop level,

$\begin{split}\psi^{a}_{0}=\sqrt{Z_{a}}\psi^{a}\hskip
8.53581pt&\text{with}\hskip
8.53581ptZ_{a}=1+\delta^{(1)}_{a}+\delta^{(2)}_{a},\\\
R^{0}_{abcd}=Z_{abcd}R_{abcd}\hskip 8.53581pt&\text{with}\hskip
8.53581ptZ_{abcd}=1+\delta^{(1)}_{abcd}+\delta^{(2)}_{abcd},\end{split}$ (7)

where $Z$’s are the renormalization factors of the fields and coupling
constants, $\delta^{(k)}_{a}$ and $\delta^{(k)}_{abcd}$ are the counterterms
at k-loop level. Note that the field strength renormalization is
$\psi^{a}_{0}=\sqrt{Z^{a}_{b}}\psi^{b}$ with $b$ summed over $1,\dots,N$, but
as we will show that at one and two-loop levels off-diagonal entries in
$Z^{a}_{b}$ vanish, therefore, we denote diagonal terms $Z_{a}\equiv
Z^{a}_{a}$ for convenience. We also define
$\tilde{Z}_{abcd}=(Z_{a}Z_{b}Z_{c}Z_{d})^{1/2}Z_{abcd}=1+\tilde{\delta}^{(1)}_{abcd}+\tilde{\delta}^{(2)}_{abcd}$
for convenience, note that here repeated indices are NOT summed over. Using
the above expressions to rewrite the Lagrangian that was originally written in
terms of bare quantities, we can separate it into the renormalized and the
counterterm parts,

$\begin{split}\mathcal{L}&=\sum_{a=1,\dots,N}i\overline{\psi}^{a}\gamma^{\mu}\partial_{\mu}\psi^{a}+\sum_{\begin{subarray}{c}a,b,c,d\\\
=1,\dots,N\end{subarray}}R_{abcd}\overline{\psi}^{a}\gamma^{\mu}\psi^{b}\overline{\psi}^{c}\gamma_{\mu}\psi^{d}\\\
&\hskip
42.67912pt+\sum_{a=1,\dots,N}i\delta^{(1)}_{a}\overline{\psi}^{a}\gamma^{\mu}\partial_{\mu}\psi^{a}+\sum_{\begin{subarray}{c}a,b,c,d\\\
=1,\dots,N\end{subarray}}\tilde{\delta}^{(1)}_{abcd}R_{abcd}\overline{\psi}^{a}\gamma^{\mu}\psi^{b}\overline{\psi}^{c}\gamma_{\mu}\psi^{d}\\\
&\hskip
85.35826pt+\sum_{a=1,\dots,N}i\delta^{(2)}_{a}\overline{\psi}^{a}\gamma^{\mu}\partial_{\mu}\psi^{a}+\sum_{\begin{subarray}{c}a,b,c,d\\\
=1,\dots,N\end{subarray}}\tilde{\delta}^{(2)}_{abcd}R_{abcd}\overline{\psi}^{a}\gamma^{\mu}\psi^{b}\overline{\psi}^{c}\gamma_{\mu}\psi^{d}.\end{split}$
(8)

The idea of renormalization is that, at $L$-loop level, the $L$-loop
counterterm absorbs the divergence from $L$-loop diagrams, thus making the
correlation function correction finite at this level. To compute the
$\beta$-function of $R_{abcd}$, we need to determine the counterterms. Treat
the last three terms in Lagrangian Eq.(8) as a perturbation and the
correlation function is given by

$\begin{split}\braket{\Omega}{\mathcal{T}\\{\overline{\psi}^{a}\gamma^{\mu}\psi^{b}\overline{\psi}^{c}\gamma_{\mu}\psi^{d}\\}}{\Omega}&=\lim_{T\rightarrow\infty(1-i\epsilon)}\frac{\braket{0}{\mathcal{T}\\{\overline{\psi}^{a}\gamma^{\mu}\psi^{b}\overline{\psi}^{c}\gamma_{\mu}\psi^{d}\exp[-i\int_{-T}^{T}dtH_{I}(t)]\\}}{0}}{\braket{0}{\mathcal{T}\\{\exp[-i\int_{-T}^{T}dtH_{I}(t)]\\}}{0}}\\\
&=\text{Sum of all connected and amputated diagrams}\\\ &\hskip
14.22636pt\text{(By Wick's theorem) }\end{split}$ (9)

Besides standard Feynman rules for fermions, we define a new rule in momentum
space for the interaction vertices
$\overline{\psi}^{a}\gamma^{\mu}\psi^{b}\overline{\psi}^{c}\gamma_{\mu}\psi^{d}$,
and the counterterms
$i\delta_{a}\overline{\psi}^{a}\gamma^{\mu}\partial_{\mu}\psi^{a},\tilde{\delta}_{abcd}R_{abcd}\overline{\psi}^{a}\gamma^{\mu}\psi^{b}\overline{\psi}^{c}\gamma_{\mu}\psi^{d}$
as shown in Figure.1.

bdca$=-iR_{abcd}(\gamma^{\mu}\dots\gamma_{\mu})$

ppba$=-i\not{p}\delta_{ab}\delta^{(k)}_{a}$

bdca$=-i\tilde{\delta}^{(k)}_{abcd}R_{abcd}(\gamma^{\mu}\dots\gamma_{\mu})$

Figure 1: Feynman diagrams in momentum space for the three interactions in
Lagrangian Eq.(8). Note that $\delta_{ab}$ is the Kronecker delta,
$\delta^{(k)}_{a}$ is the field strength counter term. In the current-current
vertex, if two fermions $\psi^{a}$ and $\psi^{b}$ form a current
$\overline{\psi}^{a}\gamma^{\mu}\psi^{b}$, then their legs join at the vertex.
If they do not join, e.g. $\psi^{a}$ and $\psi^{d}$, then they are not in the
same current.

###  2.1 Field Strength and Coupling Constant Corrections at the One-loop
Level

To determine the field strength counterterm $\delta^{(1)}_{a}$ for the fields,
we need to compute the two-point function. For any $a\in\\{1,\dots,N\\}$, the
two-point correlation functions have the following corrections at the one-loop
level:

ppq$G_{1}^{(2)}(x,y)$$=$$+$ppq$+$pp Figure 2: Feynman diagrams at one-loop
level for the two-point function.

Write down the momentum space correlation function at the one-loop level for
each diagram, the first diagram in Figure.2 contributes

$\begin{split}-iR_{efgh}\braket{\wick{\Omega}{\c{3}{\overline{\psi}^{a}_{x}}(\c{4}{\overline{\psi}^{e}_{z}}\gamma^{\mu}\c{2}\psi^{f}_{z}\c{2}{\overline{\psi}^{g}_{z}}\gamma_{\mu}\c{3}\psi^{h}_{z})\c{4}{\psi^{b}_{y}}}{\Omega}}{}{&}\Rightarrow\delta_{ah}\delta_{be}\delta_{fg}\frac{-iR_{efgh}}{2}\int\frac{d^{2}q}{(2\pi)^{2}}\frac{i\not{p}}{p^{2}}\gamma^{\mu}\frac{i\not{q}}{q^{2}+i\epsilon}\gamma_{\mu}\frac{i\not{p}}{p^{2}}\\\
&=-\frac{R_{beea}}{2}\frac{p_{\alpha}p_{\beta}}{p^{4}}\gamma^{\alpha}\gamma^{\mu}\gamma^{\nu}\gamma_{\mu}\gamma^{\beta}\int\frac{d^{2}q}{(2\pi)^{2}}\frac{q_{\nu}}{q^{2}+i\epsilon}=0,\end{split}$
(10)

where index $e$ is summed over $1,\dots,N$. The expression vanishes because
both $\gamma^{\mu}\gamma^{\nu}\gamma_{\mu}=0$ by gamma matrices identities in
Eq.(43) and the integrand is an odd function of $q$ and it is integrated over
$(-\infty,\infty)$ thus gives 0. This calculation indicates that any diagram
that has one-end-open loops has 0 amplitude since
$\gamma^{\mu}(\cdots)\gamma_{\mu}$ where $(\cdots)$ consists of an odd number
of Dirac matrices vanishes.

Similarly, the second diagram in Figure.2 contributes

$\begin{split}-iR_{efgh}\braket{\wick{\Omega}{\c{2}{\overline{\psi}^{a}_{x}}(\c{3}{\overline{\psi}^{e}_{z}}\gamma^{\mu}\c{2}\psi^{f}_{z}\c{2}{\overline{\psi}^{g}_{z}}\gamma_{\mu}\c{2}\psi^{h}_{z})\c{3}{\psi^{b}_{y}}}{\Omega}}{}{&}\Rightarrow\delta_{be}\delta_{af}\delta_{gh}\frac{iR_{efgh}}{2}\frac{i\not{p}}{p^{2}}\gamma^{\mu}\frac{i\not{p}}{p^{2}}\int\frac{d^{2}q}{(2\pi)^{2}}\operatorname{tr}\bigg{(}\frac{i\not{q}}{q^{2}+i\epsilon}\gamma_{\mu}\bigg{)}\\\
&=\frac{R_{baee}}{2}\frac{p_{\alpha}p_{\beta}}{p^{4}}\gamma^{\alpha}\gamma^{\mu}\gamma^{\beta}\int\frac{d^{2}q}{(2\pi)^{2}}\frac{q_{\mu}}{q^{2}+i\epsilon}=0,\end{split}$
(11)

with index $e$ summed over $1,\dots,N$. This diagram also vanishes since the
integrand is an odd function in $q$ integrated over $q\in(-\infty,\infty)$.
Since the sum of the three diagrams is zero, we conclude that, at the one-loop
level, the field-strength correction $\delta^{(1)}_{a}=0$.

Now let us compute the current-current correlation function to determine the
one-loop counterterm to the coupling $R_{abcd}$. Recall that for the
$\phi^{4}$ theory at the one-loop level, we have three diagrams, i.e. the s,
t, and u-channels, contributing corrections to the $\phi^{4}$ coupling
constant $\lambda$. In analogy to the scalar theory, we have diagrams in the
three channels, but the current-current vertex in our theory generates
multiple diagrams in each channel due to the broken permutation symmetry of
this vertex. However, the observation we made from the calculations above
tells us that, using $\gamma$-matrices identities in two dimensions and
features of integration over undetermined momenta, we can eliminate most of
the diagrams. The trick is presented in appendix.B.

At the one-loop level, the $288$ Wick contractions reduce to $10$ distinct
Feynman diagrams, using the trick explained in appendix.B, $8$ more vanish and
there are only $2$ diagrams left that contribute to the correlation function.
For any choice of $a,b,c,d\in\\{1,\dots,N\\}$, the correlation function of the
current-current interaction is given by

cdab$G_{1}^{(4)}(x,y)$$=$$+$cdab$+$cdab Figure 3: Feynman diagrams at one-loop
level for the four-point function.

The first diagram in Figure.3 contributes

$\begin{split}&(-iR_{e_{1}f_{1}g_{1}h_{1}})(-iR_{e_{2}f_{2}g_{2}h_{2}})\braket{\Omega}{\wick{\c{3}{\overline{\psi}^{a}_{x}}\gamma^{\mu}\c{2}\psi^{b}_{x}(\c{2}{\overline{\psi}^{e_{1}}_{z}}\gamma^{\nu}\c{2}\psi^{f_{1}}_{z}\c{5}{\overline{\psi}^{g_{1}}_{z}}\gamma_{\nu}\c{4}\psi^{h_{1}}_{z})(\c{2}{\overline{\psi}^{e_{2}}_{w}}\gamma^{\rho}\c{3}\psi^{f_{2}}_{w}\c{4}{\overline{\psi}^{g_{2}}_{w}}\gamma_{\rho}\c{4}\psi^{h_{2}}_{w})\c{4}{\overline{\psi}^{c}_{y}}\gamma_{\mu}\c{5}\psi^{d}_{y}}}{\Omega}\\\
\Rightarrow&-R_{eafc}R_{bedf}\int\frac{d^{2}q}{(2\pi)^{2}}\operatorname{tr}\bigg{(}\frac{i\not{p}_{b}}{p_{b}^{2}}\gamma^{\nu}\frac{i\not{q}}{q^{2}+i\epsilon}\gamma^{\rho}\frac{i\not{p}_{a}}{p_{a}^{2}}\gamma^{\mu}\bigg{)}\operatorname{tr}\bigg{(}\frac{i\not{p}_{d}}{p_{d}^{2}}\gamma_{\nu}\frac{i(\not{p}_{a}-\not{p}_{c}-\not{q})}{(p_{a}-p_{c}-q)^{2}+i\epsilon}\gamma_{\rho}\frac{i\not{p}_{c}}{p_{c}^{2}}\gamma_{\mu}\bigg{)}\\\
=&R_{eafc}R_{bedf}P\Gamma_{1}\int\frac{d^{2}q}{(2\pi)^{2}}\frac{q_{\alpha}(p_{a}-p_{c}-q)^{\beta}}{(q^{2}+i\epsilon)[(p_{a}-p_{c}-q)^{2}+i\epsilon]},\\\
&\text{where}\hskip
8.53581pt\Gamma_{1}=\operatorname{tr}(\gamma^{b}\gamma^{\nu}\gamma^{\alpha}\gamma^{\rho}\gamma^{a}\gamma^{\mu})\operatorname{tr}(\gamma_{d}\gamma_{\nu}\gamma_{\beta}\gamma_{\rho}\gamma_{c}\gamma_{\mu})\hskip
8.53581pt\text{and}\hskip
8.53581ptP=\frac{(p_{a})_{a}(p_{b})_{b}(p_{c})^{c}(p_{d})^{d}}{p_{a}^{2}p_{b}^{2}p_{c}^{2}p_{d}^{2}}.\end{split}$
(12)

The second diagram contributes

$\begin{split}&(-iR_{e_{1}f_{1}g_{1}h_{1}})(-iR_{e_{2}f_{2}g_{2}h_{2}})\braket{\Omega}{\wick{\c{3}{\overline{\psi}^{a}_{x}}\gamma^{\mu}\c{2}\psi^{b}_{x}(\c{2}{\overline{\psi}^{e_{1}}_{z}}\gamma^{\nu}\c{2}\psi^{f_{1}}_{z}\c{4}{\overline{\psi}^{g_{1}}_{z}}\gamma_{\nu}\c{5}\psi^{h_{1}}_{z})(\c{2}{\overline{\psi}^{e_{2}}_{w}}\gamma^{\rho}\c{3}\psi^{f_{2}}_{w}\c{3}{\overline{\psi}^{g_{2}}_{w}}\gamma_{\rho}\c{4}\psi^{h_{2}}_{w})\c{5}{\overline{\psi}^{c}_{y}}\gamma_{\mu}\c{3}\psi^{d}_{y}}}{\Omega}\\\
\Rightarrow&-R_{eadf}R_{befc}\int\frac{d^{2}q}{(2\pi)^{2}}\operatorname{tr}\bigg{(}\frac{i\not{p}_{b}}{p_{b}^{2}}\gamma^{\nu}\frac{i\not{q}}{q^{2}+i\epsilon}\gamma^{\rho}\frac{i\not{p}_{a}}{p_{a}^{2}}\gamma^{\mu}\bigg{)}\operatorname{tr}\bigg{(}\frac{i\not{p}_{d}}{p_{d}^{2}}\gamma_{\rho}\frac{i(\not{p}_{b}-\not{p}_{c}+\not{q})}{(p_{b}-p_{c}+q)^{2}+i\epsilon}\gamma_{\nu}\frac{i\not{p}_{c}}{p_{c}^{2}}\gamma_{\mu}\bigg{)}\\\
=&R_{eadf}R_{befc}P\Gamma_{2}\int\frac{d^{2}q}{(2\pi)^{2}}\frac{q_{\alpha}(p_{b}-p_{c}+q)^{\beta}}{(q^{2}+i\epsilon)[(p_{b}-p_{c}+q)^{2}+i\epsilon]},\\\
&\text{where}\hskip
8.53581pt\Gamma_{2}=\operatorname{tr}(\gamma^{b}\gamma^{\nu}\gamma^{\alpha}\gamma^{\rho}\gamma^{a}\gamma^{\mu})\operatorname{tr}(\gamma_{d}\gamma_{\rho}\gamma_{\beta}\gamma_{\nu}\gamma_{c}\gamma_{\mu})=\Gamma_{1}\hskip
8.53581pt\text{since}\hskip
8.53581pt\gamma_{\rho}\gamma_{\beta}\gamma_{\nu}=\gamma_{\nu}\gamma_{\beta}\gamma_{\rho}.\end{split}$
(13)

Note that in the above two diagrams, the indices $e,f$ are summed over
$1,\dots,N$.

The counterterm contributes in two ways

$\begin{split}&-i\tilde{\delta}^{(1)}_{efgh}R_{efgh}\braket{\Omega}{\wick{\c{3}{\overline{\psi}^{a}_{x}}\gamma^{\mu}\c{2}\psi^{b}_{x}(\c{2}{\overline{\psi}^{e}_{z}}\gamma^{\nu}\c{3}\psi^{f}_{z}\c{3}{\overline{\psi}^{g}_{z}}\gamma_{\nu}\c{2}\psi^{h}_{z})\c{2}{\overline{\psi}^{c}_{y}}\gamma_{\mu}\c{3}\psi^{d}_{y}}}{\Omega}\Rightarrow-i\tilde{\delta}^{(1)}_{badc}R_{badc}P\Gamma,\\\
&\text{where}\hskip
8.53581pt\Gamma=\operatorname{tr}(\gamma^{b}\gamma^{\nu}\gamma^{a}\gamma^{\mu})\operatorname{tr}(\gamma_{d}\gamma_{\nu}\gamma_{c}\gamma_{\mu}).\\\
&-i\tilde{\delta}^{(1)}_{efgh}R_{efgh}\braket{\Omega}{\wick{\c{3}{\overline{\psi}^{a}_{x}}\gamma^{\mu}\c{2}\psi^{b}_{x}(\c{2}{\overline{\psi}^{e}_{z}}\gamma^{\nu}\c{4}\psi^{f}_{z}\c{2}{\overline{\psi}^{g}_{z}}\gamma_{\nu}\c{3}\psi^{h}_{z})\c{4}{\overline{\psi}^{c}_{y}}\gamma_{\mu}\c{2}\psi^{d}_{y}}}{\Omega}\Rightarrow
i\tilde{\delta}^{(1)}_{dabc}R_{dabc}P\Gamma^{\prime},\\\ &\text{where}\hskip
8.53581pt\Gamma^{\prime}=\operatorname{tr}(\gamma^{b}\gamma^{\nu}\gamma_{c}\gamma_{\mu}\gamma_{d}\gamma_{\nu}\gamma^{a}\gamma^{\mu})=0\hskip
8.53581pt\text{thus this term does not contribute}.\end{split}$ (14)

To evaluate the integral over undetermined momentum, we apply Feynman
parametrization with $A=q^{2}+i\epsilon$ and
$B=(p_{a}-p_{c}-q)^{2}+i\epsilon$,

$\frac{1}{AB}=\int_{0}^{1}dx\frac{1}{[xA+(1-x)B]^{2}}=\int_{0}^{1}dx\frac{1}{(l^{2}-\Delta)^{2}},$
(15)

where $l\equiv q-(p_{a}-p_{c})(1-x)$ and
$\Delta\equiv(p_{a}-p_{c})^{2}(x^{2}-x)-i\epsilon$ and the integral over
undetermined momentum gives

$\begin{split}\int\frac{d^{2}q}{(2\pi)^{2}}\frac{q_{\alpha}(p_{a}-p_{c}-q)^{\beta}}{(q^{2}+i\epsilon)[(p_{a}-p_{c}-q)^{2}+i\epsilon]}&=\int_{0}^{1}dx\int\frac{d^{2}l}{(2\pi)^{2}}\frac{-l_{\alpha}l^{\beta}+B^{\beta}l_{\alpha}+C}{(l^{2}-\Delta)^{2}}.\end{split}$
(16)

We use $l_{\mu}l_{\nu}=\frac{1}{2}g_{\mu\nu}l^{2}$ which gives $A=-1/2$. $B,C$
are functions of $p_{a}-p_{c}$, and we can ignore $B$ since $l_{\alpha}$ is an
odd fcn which gives 0 when integrated over $l\in[-\infty,\infty]$.

Now apply dimensional regularization to find $\int
d^{2}l\frac{l^{2}}{(l^{2}-\Delta)^{2}}$ and $\int
d^{2}l\frac{1}{(l^{2}-\Delta)^{2}}$. We have the following results in
Minkowski signature

$\int\frac{d^{d}l}{(2\pi)^{d}}\frac{1}{(l^{2}-\Delta)^{2}}=\frac{i}{(4\pi)^{d/2}}\frac{\Gamma(2-d/2)}{\Gamma(2)}\frac{1}{\Delta}^{2-d/2},$
(17)
$\int\frac{d^{d}l}{(2\pi)^{d}}\frac{l^{\mu}l^{\nu}}{(l^{2}-\Delta)^{2}}=\frac{-i}{(4\pi)^{d/2}}\frac{g^{\mu\nu}}{2}\frac{\Gamma(1-d/2)}{\Gamma(2)}\frac{1}{\Delta}^{1-d/2}$
(18)

. The integral in Eq.(17) converges thus compute directly with $d=2$ by wick-
rotating to Euclidean space with $l^{0}=il^{0}_{E}$ and $l^{2}=-l^{2}_{E}$.
The integral in Eq.(18) diverges logarithmically, we evaluate using
dimensional regularization,

$\begin{split}\int\frac{d^{2}l}{(2\pi)^{2}}\frac{1}{(l^{2}-\Delta)^{2}}=\int\frac{d^{2}l_{E}}{(2\pi)^{2}}\frac{i}{(l_{E}^{2}+\Delta)^{2}}&=\frac{i}{4\pi\Delta},\end{split}$
(19)
$\begin{split}\int\frac{d^{2-\epsilon}l}{(2\pi)^{2-\epsilon}}\frac{l_{\alpha}l^{\beta}}{(l^{2}-\Delta)^{2}}&=\frac{-i{g_{\alpha}}^{\beta}}{8\pi}\frac{\Gamma(\epsilon/2)}{\Gamma(2)}\bigg{(}\frac{4\pi}{\Delta}\bigg{)}^{\epsilon/2}\\\
&\rightarrow\frac{-i{g_{\alpha}}^{\beta}}{8\pi}\bigg{(}\frac{2}{\epsilon}-\\#+\mathcal{O}(\epsilon)\bigg{)}\bigg{(}1+\frac{\epsilon}{2}\ln\frac{4\pi}{\Delta}+\mathcal{O}(\epsilon^{2})\bigg{)}\sim-\frac{i{g_{\alpha}}^{\beta}}{4\pi\epsilon},\end{split}$
(20)

where $\Delta\sim\mu^{2}$ with $\mu$ as the energy scale. Let $d=2-\epsilon$
and take $\epsilon\rightarrow 0$, note that here $\epsilon$ is the small
parameter in dimensional regularization, which is different from the
$\epsilon$ in the propagator. Expand $\Gamma(x)$ near $x=0$ we have
$\Gamma(x)=1/x-\\#+\mathcal{O}(x)$ where $\\#$ is the Euler-Mascheroni
constant, and expand $(1/\Delta)^{1-d/2}=1-(1-d/2)\ln\Delta+\dots$ near $d=2$.
We see that the current-current correlation function has a divergence of
$1/\epsilon$.

Using $\gamma$ matrices identities to simplify
${g_{\alpha}}^{\beta}\Gamma_{1}$ where ${g_{\alpha}}^{\beta}$ comes from doing
Feynman parametrization, we get

$\begin{split}{g_{\alpha}}^{\beta}\Gamma_{1}&=\operatorname{tr}(\gamma^{b}\gamma^{\nu}\gamma^{\alpha}\gamma^{\rho}\gamma^{a}\gamma^{\mu})\operatorname{tr}(\gamma_{d}\gamma_{\nu}\gamma_{\alpha}\gamma_{\rho}\gamma_{c}\gamma_{\mu})\\\
&=\operatorname{tr}(\gamma^{b}(g^{\nu\alpha}\gamma^{\rho}+g^{\alpha\rho}\gamma^{\nu}-g^{\nu\rho}\gamma^{\alpha})\gamma^{a}\gamma^{\mu})\operatorname{tr}(\gamma_{d}\gamma_{\nu}\gamma_{\alpha}\gamma_{\rho}\gamma_{c}\gamma_{\mu})\\\
&=4\operatorname{tr}(\gamma^{b}\gamma^{\nu}\gamma^{a}\gamma^{\mu})\operatorname{tr}(\gamma_{d}\gamma_{\nu}\gamma_{c}\gamma_{\mu})=4\Gamma.\end{split}$
(21)

Include the symmetry factor in each diagram and the three diagrams in Figure.3
contributes

$\begin{split}\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&=\frac{iP\Gamma}{4\pi\epsilon}\sum_{e,f=1}^{N}R_{eafc}R_{bedf},\\\
\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&=-\frac{iP\Gamma}{4\pi\epsilon}\sum_{e,f=1}^{N}R_{eadf}R_{befc},\\\
\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&=-\frac{iP\Gamma}{2}\tilde{\delta}^{(1)}_{badc}R_{badc},\end{split}$
(22)

and sum up to 0. Up to one-loop level,
$\tilde{Z}_{abcd}=(Z_{a}Z_{b}Z_{c}Z_{d})^{1/2}\Big{(}1+\delta^{(1)}_{abcd}\Big{)}$
and $Z_{a}=1+\delta^{(1)}_{a}=1$ because $\delta^{(1)}_{a}$ is determined to
be 0. Then
$\tilde{Z}_{abcd}=1+\tilde{\delta}^{(1)}_{abcd}=1+\delta^{(1)}_{abcd}$ then we
have $\tilde{\delta}^{(1)}_{abcd}=\delta^{(1)}_{abcd}$. Since for Dirac
fermions, we do not have antisymmetry in the first and second pair of indices,
$R_{abcd}\neq 0$. Then we get

$\begin{split}\delta^{(1)}_{abcd}R_{abcd}&=\frac{1}{2\pi\epsilon}\sum_{e,f=1}^{N}(R_{aecf}R_{ebfd}-R_{aefd}R_{ebcf})\equiv\frac{\mathcal{R}^{(1)}_{abcd}}{2\pi\epsilon}.\end{split}$
(23)

###  2.2 At Two-loop Level

The correction to the $\beta$-function at the two-loop level is given by

$G_{2}^{(2)}(x,y)$$=$$p$$p$$q_{1}$$q_{2}$$q^{\prime}$$+$$p$$p$$q_{1}$$q_{2}$$q^{\prime}$$+$$p$$p$$q$$+$$p$$p$$q$$+$$p$$p$
Figure 4: Feynman diagrams at the two-loop level for the two-point function.

The diagrams in Figure.4 contribute the following,

$\begin{split}\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&:-\frac{R_{befg}R_{gaef}}{4}\int\frac{d^{2}q_{1}d^{2}q_{2}}{(2\pi)^{4}}\frac{i\not{p}}{p^{2}}\gamma^{\mu}\frac{i\not{q}_{2}}{q_{2}^{2}+i\epsilon}\gamma_{\nu}\frac{i\not{q}_{1}}{q_{1}^{2}+i\epsilon}\gamma_{\mu}\frac{i(\not{p}+\not{q}_{1}-\not{q}_{2})}{(p+q_{1}-q_{2})^{2}+i\epsilon}\gamma^{\nu}\frac{i\not{p}}{p^{2}}=0,\\\
&\text{since $\gamma^{\mu}(\cdots)\gamma_{\mu}=0$ where $(\cdots)$ consists of
an odd number of gamma matrices.}\\\ \leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&:\frac{R_{befg}R_{eagf}}{4}\int\frac{d^{2}q_{1}d^{2}q_{2}}{(2\pi)^{4}}\frac{i\not{p}}{p^{2}}\gamma^{\mu}\frac{i(\not{p}-\not{q}_{1}+\not{q}_{2})}{(p-q_{1}+q_{2})^{2}+i\epsilon}\gamma^{\nu}\frac{i\not{p}}{p^{2}}\operatorname{tr}\bigg{(}\frac{i\not{q}_{1}}{q_{1}^{2}+i\epsilon}\gamma_{\mu}\frac{i\not{q}_{2}}{q_{2}^{2}+i\epsilon}\gamma_{\nu}\bigg{)}=0,\\\
&\text{the numerator of the integrand
$(p-q_{1}+q_{2})^{\lambda}q_{1}^{\alpha}q_{2}^{\beta}$ is a sum of odd (after
changes}\\\ &\text{of variables) functions of $q_{1}$ and/or $q_{2}$, thus
integrating over $\\{-\infty,\infty\\}$ gives 0.}\\\ \leavevmode\hbox
to14.18pt{\vbox to14.18pt{\pgfpicture\makeatletter\hbox{\hskip
7.09111pt\lower-7.09111pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&:-\frac{i\delta^{(1)}_{beea}R_{beea}}{2}\int\frac{d^{2}q}{(2\pi)^{2}}\frac{i\not{p}}{p^{2}}\gamma^{\mu}\frac{i\not{q}}{q^{2}+i\epsilon}\gamma_{\mu}\frac{i\not{p}}{p^{2}}=0,\\\
&\text{since $\gamma^{\mu}(\cdots)\gamma_{\mu}=0$ where $\cdots$ are odd
number of gamma matrices.}\\\ \leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&:\frac{i\delta^{(1)}_{baee}R_{baee}}{2}\frac{i\not{p}}{p^{2}}\gamma^{\mu}\frac{i\not{p}}{p^{2}}\int\frac{d^{2}q}{(2\pi)^{2}}\operatorname{tr}\bigg{(}\frac{i\not{q}}{q^{2}+i\epsilon}\gamma_{\mu}\bigg{)}=0,\\\
&\text{since the integrand is an odd function of $q$.}\end{split}$ (24)

Indices $e,f,g$ are summed over $1,\dots,N$. Therefore, at the two-loop level
the correction $\delta^{(2)}_{a}=0$.

$G_{2}^{(4)}(x,y)$$=$cdab$+$cdab$+$cdab$+$cdab$+$cdab$+$cdab$+$cdab$+$cdab$+$cdab$+$cdab$+$cdab
Figure 5: Feynman diagrams at the two-loop level for the four-point function.

At the two-loop level, 21600 different Wick contractions are reduced to 96
distinct Feynman diagrams. This is still a huge number to read off correlation
functions from. Applying tricks presented in appendix.B, we can drop 90
vanishing diagrams and keep only 6 of them. Using results from the current-
current correlation function at the one-loop level and bringing in Eq.(23), we
easily obtain contributions from the diagrams in the second row of Figure.5,

$\begin{split}\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{2}&=\frac{iP\Gamma}{4\pi}\sum_{e,f=1}^{N}\delta^{(1)}_{eafc}R_{eafc}R_{bedf}=\frac{iP\Gamma}{8\pi^{2}\epsilon^{2}}\sum_{e,f,g,h=1}^{N}(R_{egfh}R_{gahc}-R_{eghc}R_{gafh})R_{bedf},\\\
\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{2}&=-\frac{iP\Gamma}{4\pi}\sum_{e,f=1}^{N}\delta^{(1)}_{befc}R_{eadf}R_{befc}=-\frac{iP\Gamma}{8\pi^{2}\epsilon^{2}}\sum_{e,f,g,h=1}^{N}(R_{bgfh}R_{gehc}-R_{bghc}R_{gefh})R_{eadf},\\\
\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{2}&=\frac{iP\Gamma}{4\pi}\sum_{e,f=1}^{N}\delta^{(1)}_{bedf}R_{eafc}R_{bedf}=\frac{iP\Gamma}{8\pi^{2}\epsilon^{2}}\sum_{e,f,g,h=1}^{N}(R_{bgdh}R_{gehf}-R_{bghf}R_{gedh})R_{eafc},\\\
\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{2}&=-\frac{iP\Gamma}{4\pi}\sum_{e,f=1}^{N}\delta^{(1)}_{eadf}R_{eadf}R_{befc}=-\frac{iP\Gamma}{8\pi^{2}\epsilon^{2}}\sum_{e,f,g,h=1}^{N}(R_{egdh}R_{gahf}-R_{eghf}R_{gadh})R_{befc},\\\
\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{5}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{2}&=-\frac{iP\Gamma}{2}\tilde{\delta}^{(2)}_{badc}R_{badc}.\end{split}$
(25)

Diagrams in the first row of Figure.5 contribute

$\begin{split}\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{1}&=-\frac{iP\Gamma}{8\pi^{2}\epsilon^{2}}\sum_{e,f,g,h=1}^{N}R_{eafc}R_{bgdh}R_{gehf},\hskip
28.45274pt\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{1}=-\frac{iP\Gamma}{8\pi^{2}\epsilon^{2}}\sum_{e,f,g,h=1}^{N}R_{eadf}R_{bghc}R_{gefh},\\\
\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{1}&=\frac{iP\Gamma}{8\pi^{2}\epsilon^{2}}\sum_{e,f,g,h=1}^{N}\frac{1}{2}R_{eafc}R_{bghf}R_{gedh},\hskip
28.45274pt\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{1}=\frac{iP\Gamma}{8\pi^{2}\epsilon^{2}}\sum_{e,f,g,h=1}^{N}\frac{1}{2}R_{eahf}R_{bgfc}R_{gedh},\\\
\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{5}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{1}&=\frac{iP\Gamma}{8\pi^{2}\epsilon^{2}}\sum_{e,f,g,h=1}^{N}\frac{1}{2}R_{eafg}R_{bhdf}R_{hegc},\hskip
29.87538pt\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{6}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{1}=\frac{iP\Gamma}{8\pi^{2}\epsilon^{2}}\sum_{e,f,g,h=1}^{N}\frac{1}{2}R_{eadf}R_{bgfh}R_{gehc}.\end{split}$
(26)

We see that each integral over $q_{i}$ gives a divergence of $1/\epsilon$, so
each diagram contributes a second-order infinity to the correction. Up to two-
loop level,
$\tilde{Z}_{abcd}=(Z_{a}Z_{b}Z_{c}Z_{d})^{1/2}\big{(}1+\delta^{(1)}_{abcd}+\delta^{(2)}_{abcd}\big{)}=1+\tilde{\delta}^{(1)}_{abcd}+\tilde{\delta}^{(2)}_{abcd}$,
we have shown that $\delta^{(1)}_{a}=\delta^{(2)}_{a}=0$ and
$\tilde{\delta}^{(1)}_{abcd}=\delta^{(1)}_{abcd}$, it is trivial that
$\tilde{\delta}^{(2)}_{abcd}=\delta^{(2)}_{abcd}$. The correction to the beta
function at the 2-loop level is given by

$\begin{split}\delta^{(2)}_{abcd}R_{abcd}&=\frac{1}{4\pi^{2}\epsilon^{2}}\sum_{e,f,g,h=1}^{N}\bigg{(}R_{aecf}R_{gbhd}R_{egfh}+R_{aefd}R_{gbch}R_{eghf}-\frac{1}{2}R_{aecf}R_{gbfh}R_{eghd}\\\
&\hskip
28.45274pt-\frac{1}{2}R_{aefg}R_{hbcf}R_{ehgd}-\frac{1}{2}R_{aefd}R_{gbhf}R_{egch}-\frac{1}{2}R_{aefg}R_{hbgd}R_{ehcf}\bigg{)}\equiv\frac{\mathcal{R}^{(2)}_{abcd}}{4\pi^{2}\epsilon^{2}}.\end{split}$
(27)

##  3 The $\beta$-functions and Constraints on Coupling Constants

Recall the definition of the $\beta$-function,

$\beta_{abcd}(R_{abcd}(\mu))\equiv\mu\frac{\partial R_{abcd}}{\partial\mu},$
(28)

where $R_{abcd}(\mu)$ is the renormalized coupling constant which depends on
the renormalization scale $\mu$. $\mu$, with mass dimension $[\mu]=1$, is a
parameter introduced to carry out dimensional regularization, which is not a
physical parameter but an artifice of the renormalization procedure. To
regularize the divergence in the loop diagrams, we perform dimensional
regularization and take $d=2-\epsilon$ with limit $\epsilon\rightarrow 0$. Now
we count the mass dimension of each bare quantity. Since the action
$\mathcal{S}=\int d^{d}x\mathcal{L}$ is dimensionless, we have
$[\mathcal{L}]=d$, then

$\begin{split}[\psi^{a}_{0}]=\frac{d-1}{2}=\frac{1-\epsilon}{2},\hskip
8.53581pt\text{and}\hskip
8.53581pt[R^{0}_{abcd}]=d-4[\psi_{0}^{a}]=\epsilon.\end{split}$ (29)

The bare couplings ought to be independent of $\mu$, therefore we need to
replace $R^{0}_{abcd}$ with $\mu^{\epsilon}R^{0}_{abcd}$ in the Lagrangian to
make it a dimensionless the following equation

$\begin{split}\frac{\partial
R^{0}_{abcd}}{\partial\mu}=\frac{\partial}{\partial\mu}(\mu^{\epsilon}Z_{abcd}R_{abcd})=0.\end{split}$
(30)

Expand the right hand side and multiply $\mu^{1-\epsilon}Z_{abcd}^{-1}$ on
both sides, we get the expression of the $\beta$-function for $R_{abcd}$,

$\begin{split}\mu\frac{\partial R_{abcd}}{\partial\mu}=-\epsilon
R_{abcd}-\frac{R_{abcd}}{Z_{abcd}}\bigg{(}\mu\frac{\partial
Z_{abcd}}{\partial\mu}\bigg{)}.\end{split}$ (31)

$\begin{split}\mu\frac{\partial
Z_{abcd}}{\partial\mu}=\mu\frac{\partial\delta_{abcd}}{\partial\mu}=-\frac{\mathcal{R}_{abcd}}{2\pi\epsilon
R^{2}_{abcd}}\bigg{(}\mu\frac{\partial
R_{abcd}}{\partial\mu}\bigg{)}+\frac{1}{2\pi\epsilon
R_{abcd}}\bigg{(}\mu\frac{\partial\mathcal{R}_{abcd}}{\partial\mu}\bigg{)}.\end{split}$
(32)

Bring Eq.(32) into Eq.(31) we get

$\begin{split}\beta_{abcd}&=-\epsilon
R_{abcd}-\frac{\mathcal{R}_{abcd}}{2\pi}-\frac{1}{2\pi\epsilon}\bigg{(}\mu\frac{\partial\mathcal{R}_{abcd}}{\partial\mu}\bigg{)}+\mathcal{O}\big{(}R^{4}_{abcd}\big{)}\\\
&\equiv\beta^{(0)}_{abcd}+\beta^{(1)}_{abcd}+\beta^{(2)}_{abcd}+\mathcal{O}\big{(}R^{4}_{abcd}\big{)},\end{split}$
(33)

where
$\mathcal{R}_{abcd}\equiv\mathcal{R}^{(1)}_{abcd}+\frac{1}{2\pi}\mathcal{R}^{(2)}_{abcd}$
is a function of coupling constants $R_{abcd}$’s to the second order in the
first term and third order in the second term. $\mathcal{R}^{(k)}_{abcd}$ is
defined as $\delta^{(k)}_{abcd}R_{abcd}\sim\mathcal{R}^{(k)}_{abcd}$ up to a
coefficient that depends on $\epsilon$ and is determined by cancelling k-loop
level divergence in the current-current correlation function. The beta
functions are just combinations of the coefficients of the divergent
logarithms and they are given by

$\begin{split}\beta^{(0)}_{abcd}&=-\epsilon R_{abcd},\hskip
56.9055pt\beta^{(1)}_{abcd}=\frac{\mathcal{R}^{(1)}_{abcd}}{2\pi},\hskip
56.9055pt\beta^{(2)}_{abcd}=\frac{2\mathcal{R}^{(2)}_{abcd}}{(2\pi)^{2}}.\end{split}$
(34)

As we can see in the calculations in section 2.2, the two-loop beta function
is obtained by replacing one of the vertices with the one-loop counterterms,
therefore vanishing beta functions at the one-loop level automatically yield
vanishing beta function at the two-loop level. Setting the one-loop
corrections to the $\beta$-function to zero, we get the following constraints
that

$\begin{split}\mathcal{R}^{(1)}_{abcd}&\equiv\sum_{e,f=1}^{N}(R_{aecf}R_{ebfd}-R_{aefd}R_{ebcf})=0.\end{split}$
(35)

To find the most general constraints on the coupling constants $R_{abcd}$’s
such that the model is a CFT, we look for solutions to Eq.(35). An obvious
solution takes the form of

$R_{abcd}^{2}=R_{abab}R_{cdcd},$ (36)

this solution is sufficient but not necessary, and there are solutions beyond
this form. In either the Majorana or the Dirac cases, the degrees of freedom
(DOFs) $D$ in the couplings grow as $\mathcal{O}(N^{4})$. The beta function
constraints remove some DOFs, to find the remaining DOFs one needs to
analytically solve the system of Eq.(35).

In the Dirac cases, the DOFs are larger due to fewer symmetry properties in
the indices, which is given by $D=\frac{1}{2}N^{2}(N^{2}+1)$. One can verify
that Eq.(36) is indeed a root of the beta functions, but there are other roots
to Eq.(35) beyond this form for any $N\geq 2$, this is numerically verified.
The form in Eq.(36) requires the number of DOFs in couplings to be reduced to
$N^{2}$, which is not the case even for small $N$. For example, consider the
case of $N=2$ Dirac fermions, there are $10$ independent couplings
$R_{abcd}$’s and the vanishing beta function conditions has an analytic
solution with minimal zero couplings,

$\begin{split}R_{1122}&=\frac{1}{2R_{1212}}\Big{[}R_{1111}R_{1212}+R_{1212}R_{2222}+2R_{1112}R_{1222}-R_{1112}^{2}-R_{1222}^{2}\Big{]},\\\
R_{1112}&=\frac{1}{R_{1212}}\Big{[}R_{1222}R_{2121}+R_{1221}(R_{1121}-R_{2122})\Big{]},\\\
R_{1221}^{2}&=R_{1212}R_{2121},\end{split}$ (37)

such a solution removes only $3$ DOFs with $7$ DOFs remaining, and $7>2^{2}$.
For $N\geq 3$, analytic solutions are difficult to find, but we can solve the
system numerically. Numerical examples of a general solution to vanishing beta
functions for $N=2,3$, and $4$ are shown in Tables 3, 4, and 5,6 in appendix
D, respectively.

The number of couplings grows as $\mathcal{O}(N^{4})$, which quickly becomes a
large amount of data to analyze. To visualize the solutions, we create color
maps using the numerical results and we brief the coloring procedure in the
following. The coupling constants $R_{abcd}$’s can be mapped into a $N\times
N$ matrix $\mathbf{R}$ of $N\times N$ matrices $\mathbf{R}_{ab}$,

$\begin{split}\mathbf{R}=\big{(}\mathbf{R}_{ab}\big{)}\hskip
14.22636pt\text{where}\hskip
14.22636pt\mathbf{R}_{ab}=\big{(}R_{ab;cd}\big{)}\hskip
14.22636pt\text{with}\hskip 14.22636pt1\leq a,b,c,d\leq N,\end{split}$ (38)

and we define $R_{ab;cd}\equiv R_{abcd}$. More explicitly, we map the
couplings in the following way

$\mathbf{R}=\begin{pmatrix}\begin{pmatrix}R_{1111}&\cdots&R_{111N}\\\
\vdots&\ddots&\vdots\\\
R_{11N1}&\cdots&R_{11NN}\end{pmatrix}&\cdots&\begin{pmatrix}R_{1N11}&\cdots&R_{1N1N}\\\
\vdots&\ddots&\vdots\\\ R_{1NN1}&\cdots&R_{1NNN}\end{pmatrix}\\\
\vdots&\ddots&\vdots\\\ \begin{pmatrix}R_{N111}&\cdots&R_{N11N}\\\
\vdots&\ddots&\vdots\\\
R_{N1N1}&\cdots&R_{N1NN}\end{pmatrix}&\cdots&\begin{pmatrix}R_{NN11}&\cdots&R_{NN1N}\\\
\vdots&\ddots&\vdots\\\ R_{NNN1}&\cdots&R_{NNNN}\end{pmatrix}\end{pmatrix}.$
(39)

Then we impose the symmetry properties on the couplings to remove some DOFs.
Replacing the couplings by their corresponding numerical results, we get a
$N^{2}\times N^{2}$ table of data. Then we assign a color scale to values with
gray for numerical zero, red for positive, and blue for negative, the darker
the color, the larger the absolute value. Now we fill in the table with colors
corresponding to numerical results of $R_{abcd}$’s to obtain a color map of a
set of solutions to the couplings. Consider the Abelian Thirring model as a
simple example, where $R_{abcd}=g\delta_{ab}\delta_{cd}$, its color map is
shown in Figure.7.

Figure 6: Color map of couplings in the Abelian Thirring model with $N=4$
Dirac fermions, where $R_{abcd}=g\delta_{ab}\delta_{cd}$ with $g=1/2$.

Figure 7: An example solution to the beta functions up to the two-loop level
for N=4 Dirac fermions that has no obvious pattern.

Following this procedure, we generate color maps for $N=2$ to $N=7$ Dirac
fermions, shown in Figures.9-13. We see a pattern of solutions in these color
maps as $N$ gets larger. Diagonal submatrices $\mathbf{R}_{aa}$’s share a
similar pattern for $1\leq a\leq N$. Diagonal terms in off-diagonal
submatrices are a few orders stronger than the rest couplings, creating
diagonal strips in the color maps. These locations in the maps correspond to
couplings in $R_{aabb}$ form with $1\leq a,b\leq N$, and the ones in
$R_{abcc}$ form with $1\leq a,b,c\leq N$ and $a\neq b$. These couplings are
more intense (i.e. have larger numerical values) than the rest, as the rest
couplings are mapped to gray. Such a solution is more general than one in
Eq.(36) form. In addition, couplings in $R_{abcc}$ form with the same pair of
$a,b$, for example, $R_{1233}$ and $R_{1244}$, have similar magnitude, as we
see in the maps that diagonal terms in the same off-diagonal submatrices share
very similar color, but diagonal terms in different off-diagonal submatrices
have different colors. If solutions of the couplings completely concentrate on
these patterns, the diagonal terms $R_{aabb}$’s give $\frac{1}{2}N(N+1)$ DOFs,
and the $N$ diagonal terms $R_{abcc}$’s in any off-diagonal submatrices gives
DOFs to $N^{2}(N-1)$, then in total the DOFs for a theory of $N$ Dirac
fermions is given by

$D_{N}=N^{3}-\frac{1}{2}N^{2}+\frac{1}{2}N.$ (40)

Bringing $N=1$ and $N=2$ into this formula, we get $D_{1}=1$ for the original
Thrring model, and $D_{2}=7$ which matches with the DOFs given by the symbolic
solution in Eq.(37).

We note that for $N=4$ Dirac fermions some numerical solutions are more random
than the one shown in Figure.11 and do not have an obvious pattern as for
other values of $N$, an example is shown in Figure.7. A theory with $N=4$
Majorana fermions also behaves oddly where multiple patterns appear in the
color map. We include these color maps and a discussion in appendix C.

Figure 8: A solution for N=2 Dirac fermions.

Figure 9: A solution for N=3 Dirac fermions.

In Majorana cases, $D=\frac{1}{8}N^{2}(N-1)^{2}+\frac{1}{4}N(N-1)$. When $N=1$
it is a free theory as no interaction is allowed by the Grassmannian nature of
fermions. When $N=2$ the theory has only one nonzero coupling $R_{1212}$ and
the beta function vanishes automatically. $N=3$ theory has $6$ DOFs by
symmetry properties in the indices, and the beta functions have an analytic
solution with minimal zero couplings, which is given by

$R_{1213}^{2}=R_{1212}R_{1313},\hskip
28.45274ptR_{1223}^{2}=R_{1212}R_{2323},\hskip
28.45274ptR_{1323}^{2}=R_{1313}R_{2323},$ (41)

where $3$ DOFs are removed and only $3$ remain. One can also numerically check
that for $N=1,2,3$, there are no other solutions to the one-loop beta function
other than the one given in Eq.(36). However, for $N\geq 4$ analytically
solving the system becomes difficult. Numerical results show that there are
indeed additional solutions that are not in the above form. The DOFs of the
most general solution grow at least as $\mathcal{O}(N^{2})$. This bound is
given by the solution form in Eq.(36), it reduces the DOFs to $N$ choose $2$
in the Majorana case, i.e. $C_{N}^{2}=\frac{1}{2}N(N-1)$ for $\forall N\geq
2$. See appendix C for color map samples of numerical solutions. However, for
Majorana fermions color maps are not as helpful in DOF reduction as in the
Dirac cases. As one can see in the figures in the appendix, all couplings that
survive after symmetry properties are imposed are turned on, and no obvious
relation between the couplings manifests. A theory with $N=4$ Majorana
fermions is odd compared to other numbers of fermions because multiple
patterns appear in the color maps, see appendix C for more details.

Even though exact symbolic solutions have not yet been determined, the
calculation successfully reproduces some of the known interacting fermionic
CFTs. The original Thirring model of $N=1$ Dirac fermion is a CFT that is
equivalent to a free bosonic CFT compactified on a circle [21]. The Thirring
coupling $g$ relates to the compact free boson coupling which is the radius
$R$ of the circle in the following way

$\frac{R^{2}}{4}=\frac{\pi}{\pi+g}\hskip 2.84526pt,$ (42)

When the Thirring coupling vanishes, i.e. $g=0$, the Thirring model becomes
simply the free Dirac fermion, and the radius of the equivalent compact free
boson corresponding to $g=0$ is $R=2$, which, through the T-duality, is dual
to the $R=1$ case (recall that we have chosen $\alpha^{\prime}=2$). Therefore,
the above relation reduces the duality we discussed in the previous section.
The $N=1$ Majorana fermion theory is a free fermionic theory since the
interaction $(\overline{\psi}\gamma^{\mu}\psi)^{2}$ does not exist in this
case due to the Grassmannian nature of fermions, and a free fermionic QFT is
conformally invariant. When the theory has $N=2$ Majorana fermions, one can
form a complex fermion. In general, this requires the Majorana fermions to
have the same mass, in the massless case the requirement is satisfied. This
model is a CFT, equivalent to a free boson compactified on a $\mathbb{Z}_{2}$
orbifold through bosonization [22]. In addition, letting
$R_{abcd}=g\delta_{ab}\delta_{cd}$ in the generalized Thirring model yields
Abelian Thirring model, which is also a CFT of an arbitrary number of $N$
Dirac fermions with an interaction of the abelian currents. The above theories
have vanishing $\beta$-function at the one and two-loop level because they are
CFTs, and this is verified using Mathematica.

Figure 10: A solution for N=4 Dirac fermions.

Figure 11: A solution for N=5 Dirac fermions.

Figure 12: A solution for N=6 Dirac fermions.

Figure 13: A solution for N=7 Dirac fermions.

##  4 Conclusions

Motivated by the quantum gravity considerations outlined in [1] we have
revisited the question of lines of fixed points emanating from models of free
fermions in $1+1$ dimensions. Through two loop order we have found a new
family of possibilities, in addition to the well known Abelian Thirring
models. In addition there are some numerical indications of additional
solutions. Of course, vanishing of the beta functions through two loop order
means that the three loop contributions are scheme independent and might well
be non-zero, so our work is only the first step in establishing the existence
of true fixed lines. We reserve the difficult task of higher loop calculations
to future work.

We note however that for potential applications to quantum gravity we are
primarily interested in models with a very large number of fermion fields. In
this case it is possible that various kinds of large $N$ re-scalings of the
couplings $R_{abcd}$ might lead to models where approximate fixed points could
be found using only the two loop beta functions. Since the requirement of
conformal invariance only appears in the limit of semi-classical gravity,
corresponding to large $N$, this might be sufficient. Again, we will have to
leave the explorations of such speculations to future research.

Acknowledgments

S. A thanks D. Rogerson, R. Singh, and F. Stubbs for discussions. The authors
thank S. Lukyanov and D. Diaconescu for helpful comments. The work of T. B.
and S. A is supported in part by the DOE under grant DE-SC0010008.

##  Appendix A Gamma Matrix Identities and Fierz Transformations in 2D

In the correlation function calculations, we used the following gamma matrices
identities in two dimensions, where the metric $g^{\mu\nu}=diag(+,-)$:

$\begin{split}\\{\gamma^{\mu},\gamma^{\nu}\\}&=2g^{\mu\nu}\mathbb{I}_{2}\\\
\gamma^{\mu}\gamma_{\mu}&=2\mathbb{I}_{2}\\\
\gamma^{\mu}\gamma^{\nu}\gamma_{\mu}&=0\\\
\gamma^{\mu}\gamma^{3}\gamma_{\mu}&=-2\gamma^{3}\\\
\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma_{\mu}&=2\gamma^{\rho}\gamma^{\nu}\\\
\gamma^{\mu}(\gamma^{\nu_{1}}\dots\gamma^{\nu_{2k+1}})\gamma_{\mu}&=0\\\
\operatorname{tr}(\gamma^{\mu}\gamma^{\nu})&=2g^{\mu\nu}\\\
\operatorname{tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma})&=2(g^{\mu\nu}g^{\rho\sigma}-g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho})\\\
\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}&=g^{\mu\nu}\gamma^{\rho}+g^{\nu\rho}\gamma^{\mu}-g^{\mu\rho}\gamma^{\nu}\\\
&=\gamma^{\rho}\gamma^{\nu}\gamma^{\mu}\\\
\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\lambda}&=(g^{\mu\nu}g^{\rho\sigma}+g^{\nu\rho}g^{\mu\sigma}-g^{\mu\rho}g^{\nu\sigma})\gamma^{\lambda}\\\
&\hskip
14.22636pt-(g^{\mu\nu}g^{\rho\lambda}+g^{\nu\rho}g^{\mu\lambda}-g^{\mu\rho}g^{\nu\lambda})\gamma^{\sigma}\\\
&\hskip
28.45274pt+g^{\mu\nu}g^{\sigma\lambda}\gamma^{\rho}+g^{\nu\rho}g^{\sigma\lambda}\gamma^{\mu}-g^{\mu\rho}g^{\sigma\lambda}\gamma^{\nu}\\\
&=\gamma^{\lambda}\gamma^{\sigma}\gamma^{\rho}\gamma^{\nu}\gamma^{\mu}\end{split}$
(43)

The current-current interaction is unique up to Fierz transformations, that is
to say any four-fermi interaction can be rearranged to the current-current
type using the Fierz transformation. The Fierz identity can be written as

$\begin{split}(\overline{u}_{1}\Gamma^{A}u_{2})(\overline{u}_{3}\Gamma^{B}u_{4})=\sum_{C,D}{C^{AB}}_{CD}(\overline{u}_{1}\Gamma^{D}u_{4})(\overline{u}_{3}\Gamma^{D}u_{2}).\end{split}$
(44)

Where $\Gamma^{A}\in\\{\mathbb{I}_{2},\gamma^{0},i\gamma^{1},\gamma^{3}\\}$ is
any combinations of the normalized elements (with the convention
$\operatorname{tr}(\Gamma^{A}\Gamma^{B})=2\delta^{AB}$). The set in general
dimensions is
$\\{\mathbb{I},\gamma^{\mu},\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}],\gamma^{3}\gamma^{\mu},\gamma^{3}\\}$.
Left multiply
$(\overline{u}_{2}\Gamma^{F}u_{3})(\overline{u}_{4}\Gamma^{E}u_{1})$, then

$\begin{split}LHS&=\overline{u}_{4}\Gamma^{E}\Gamma^{A}\Gamma^{F}\Gamma^{B}u_{4}=\operatorname{tr}(\Gamma^{E}\Gamma^{A}\Gamma^{F}\Gamma^{B})\\\
RHS&=\sum_{C,D}{C^{AB}}_{CD}\operatorname{tr}(\Gamma^{E}\Gamma^{C})\operatorname{tr}(\Gamma^{F}\Gamma^{D})=4\sum_{C,D}{C^{AB}}_{CD}\delta^{EC}\delta^{FD}=4{C^{AB}}_{EF}\\\
\Longrightarrow{C^{AB}}_{CD}&=\frac{1}{4}\operatorname{tr}(\Gamma^{C}\Gamma^{A}\Gamma^{D}\Gamma^{B})\end{split}$
(45)
$\begin{split}(\overline{u}_{1}\gamma^{\mu}u_{2})(\overline{u}_{3}\gamma_{\mu}u_{4})&=\sum_{C,D}\frac{1}{4}\operatorname{tr}(\Gamma^{C}\gamma^{\mu}\Gamma^{D}\gamma_{\mu})(\overline{u}_{1}\Gamma^{C}u_{4})(\overline{u}_{3}\Gamma^{D}u_{2})\\\
&=\sum_{C}\frac{1}{4}\operatorname{tr}(\Gamma^{C}\gamma^{\mu}\Gamma^{C}\gamma_{\mu})(\overline{u}_{1}\Gamma^{C}u_{4})(\overline{u}_{3}\Gamma^{C}u_{2})\\\
&=\frac{1}{4}\operatorname{tr}(\gamma^{\mu}\gamma_{\mu})(\overline{u}_{1}u_{4})(\overline{u}_{3}u_{2})+\frac{1}{4}\operatorname{tr}(\gamma^{3}\gamma^{\mu}\gamma^{3}\gamma_{\mu})(\overline{u}_{1}\gamma^{3}u_{4})(\overline{u}_{3}\gamma^{3}u_{2})\\\
&=(\overline{u}_{1}u_{4})(\overline{u}_{3}u_{2})-(\overline{u}_{1}\gamma^{3}u_{4})(\overline{u}_{3}\gamma^{3}u_{2})\end{split}$
(46)

$\begin{split}(\overline{u}_{1}u_{2})(\overline{u}_{3}u_{4})&=\sum_{C}\frac{1}{4}\operatorname{tr}(\Gamma^{C}\Gamma^{C})(\overline{u}_{1}\Gamma^{C}u_{4})(\overline{u}_{3}\Gamma^{C}u_{2})\\\
&=\frac{1}{2}(\overline{u}_{1}u_{4})(\overline{u}_{3}u_{2})+\frac{1}{2}(\overline{u}_{1}\gamma^{0}u_{4})(\overline{u}_{3}\gamma^{0}u_{2})\\\
&\hskip
28.45274pt-\frac{1}{2}(\overline{u}_{1}\gamma^{1}u_{4})(\overline{u}_{3}\gamma^{1}u_{2})+\frac{1}{2}(\overline{u}_{1}\gamma^{3}u_{4})(\overline{u}_{3}\gamma^{3}u_{2})\\\
&=\frac{1}{2}(\overline{u}_{1}u_{4})(\overline{u}_{3}u_{2})+\frac{1}{2}(\overline{u}_{1}\gamma^{3}u_{4})(\overline{u}_{3}\gamma^{3}u_{2})+\frac{1}{2}(\overline{u}_{1}\gamma^{\mu}u_{4})(\overline{u}_{3}\gamma_{\mu}u_{2})\end{split}$
(47)
$\begin{split}(\overline{u}_{1}\gamma^{3}u_{2})(\overline{u}_{3}\gamma^{3}u_{4})&=\sum_{C}\frac{1}{4}\operatorname{tr}(\Gamma^{C}\gamma^{3}\Gamma^{C}\gamma^{3})(\overline{u}_{1}\Gamma^{C}u_{4})(\overline{u}_{3}\Gamma^{C}u_{2})\\\
&=\frac{1}{4}\operatorname{tr}(\gamma^{3}\gamma^{3})(\overline{u}_{1}u_{4})(\overline{u}_{3}u_{2})+\frac{1}{4}\operatorname{tr}((\gamma^{0}\gamma^{3})^{2})(\overline{u}_{1}\gamma^{0}u_{4})(\overline{u}_{3}\gamma^{0}u_{2})\\\
&\hskip
28.45274pt-\frac{1}{4}\operatorname{tr}(-(\gamma^{1}\gamma^{3})^{2})(\overline{u}_{1}\gamma^{1}u_{4})(\overline{u}_{3}\gamma^{1}u_{2})-\frac{1}{4}\operatorname{tr}(-(\gamma^{3})^{4})(\overline{u}_{1}\gamma^{3}u_{4})(\overline{u}_{3}\gamma^{3}u_{2})\\\
&=\frac{1}{2}(\overline{u}_{1}u_{4})(\overline{u}_{3}u_{2})-\frac{1}{2}(\overline{u}_{1}\gamma^{0}u_{4})(\overline{u}_{3}\gamma^{0}u_{2})\\\
&\hskip
28.45274pt+\frac{1}{2}(\overline{u}_{1}\gamma^{1}u_{4})(\overline{u}_{3}\gamma^{1}u_{2})+\frac{1}{2}(\overline{u}_{1}\gamma^{3}u_{4})(\overline{u}_{3}\gamma^{3}u_{2})\\\
&=\frac{1}{2}(\overline{u}_{1}u_{4})(\overline{u}_{3}u_{2})-\frac{1}{2}(\overline{u}_{1}\gamma^{\mu}u_{4})(\overline{u}_{3}\gamma_{\mu}u_{2})+\frac{1}{2}(\overline{u}_{1}\gamma^{3}u_{4})(\overline{u}_{3}\gamma^{3}u_{2})\end{split}$
(48)

##  Appendix B Elimating Vanishing Diagrams in the Current-Current Correlation
Function

| 1 |  $\exists$ $n_{i}>6$ | $\gamma^{\mu}(\cdots)\gamma_{\mu}=0$
---|---|---|---
✓ | 2 | 6+6 | $\operatorname{tr}(\cdots)\operatorname{tr}(\cdots)$
| 3 | 4+4+4 |  | ext(2+2+0)
---
int(0+0+2)
$\overset{F.P.}{\Longrightarrow}$ 0
| 4 |  $\exists$ $n_{i}$ = 2 |  $\braket{\overline{\psi}\psi}$ correction $\Longrightarrow$ 0
Table 1: At one-loop level, 12 gamma matrices can be grouped to form
different-sized loops, there are 4 different cases listed on the left. On the
right, the reason why each case contributes or not is shown. Only the second
case contributes to the current-current correlation function.

At the one-loop level, the $6!=720$ Wick contractions reduce to 10 distinct
Feynman diagrams, which is still relatively a large number to read off and
calculate the correlation functions. Luckily, observations made in the field
strength correction calculation in Subsection 2.1 become useful in determining
which diagrams contribute and which vanish.

The current-current correlation function at the one-loop level corresponds to
the second-order expansion of the exponential in Eq.(9) which consists of 6
currents, i.e. $\overline{\psi}\gamma^{\mu}\psi$. The 6 currents contain 6
$\gamma$ matrices and 6 fermion-antifermion pairs in the function. Through
Wick contraction, the 6 fermion-antifermion pairs give 6 fermionic propagators

$\begin{split}D_{F}(x-y)\propto\int\frac{d^{2}p}{(2\pi)^{2}}\frac{i\not{p}}{p^{2}-m^{2}},\end{split}$
(49)

where the numerator $\not{p}=p_{\mu}\gamma^{\mu}$, so the 6 propagators
contribute 6 more $\gamma$’s and in total there are 12 in the expression. The
$\gamma$’s are traced over in several traces and the number of traces depends
on how the fermionic fields are contracted. Any trace must contain an even
number of $\gamma$’s because any propagator is connected to 2 $\gamma$’s and
for a fermion loop consisting of $P\geq 1$ propagators, it is a trace over
$P+2P/2=2P$ Dirac matrices, where the first $P$ comes from $\gamma$’s in the
propagators and there are $2P$ $\gamma$’s connected to each propagator,
divided by 2 due to double counting. Another way to see this is that any
$\gamma$ provided by a propagator (propagator gamma) must be accompanied by a
$\gamma$ from the current $\overline{\psi}\gamma^{\mu}\psi$ (current gamma),
therefore any trace contains an even number of gamma matrices. Then
determining which diagrams contribute becomes a simple math problem. Break the
12 $\gamma$’s into several sets,

$12=\sum_{i=1}^{k}n_{i}\hskip 8.53581pt\text{where}\hskip 8.53581pt2\leq
n_{i}\leq 12\hskip 8.53581pt\text{and}\hskip 8.53581ptn_{i}\in 2\mathbb{Z},$
(50)

where $k$ is the number of fermion loops (number of traces) in the diagram,
and $n_{i}$ is the number of $\gamma$’s in a trace. Then we have the following
cases:

1\. $\exists$ $i$ such that $n_{i}>6$: this case does NOT contribute because
of $\gamma$-matrices identities in two dimensions. By the pigeonhole
principle, any loop containing more than 3 current gammas must have
$\gamma^{\mu}(\cdots)\gamma_{\mu}$ where $(\cdots)$ are an odd number of gamma
matrices, thus it vanishes for the same reason shown in Eq.(10).

2\. $k=2$ and $n_{i}=6$ for $\forall$ $i$: this case gives diagrams consisting
of two equal loops of the same size, which contribute to the current-current
correlation function.

3\. $k=3$ and $n_{i}=4$ for $\forall$ $i$: this case does NOT contribute. One
of the three loops contains both incoming momentum flows; one contains both
outgoing momentum flows; one contains 2 internal propagators with momentum
flows $p,p^{\prime}$. We denote such configurations as

ext(2+2+0)
---
int(0+0+2)

where ”ext” and ”int” stand for ”external” and ”internal” momentum flow, and
each column represents a loop containing a number of external and internal
momentum flows. We can easily see that in the third loop, $p^{\prime}$
linearly depends on the undetermined momentum $p$, i.e.
$p^{\prime}=p^{\prime}(p)\propto p$, and when performing Feynman
parametrization, the third loop contributes a term that is

$\begin{split}\sim\operatorname{tr}\big{(}\not{p}\gamma^{\mu}\not{p}^{\prime}\gamma^{\nu}\big{)}=p_{\alpha}p^{\prime}_{\beta}\operatorname{tr}\big{(}\gamma^{\alpha}\gamma^{\mu}\gamma^{\beta}\gamma^{\nu}\big{)}\sim
g_{\alpha\beta}p^{2}\operatorname{tr}\big{(}\gamma^{\alpha}\gamma^{\mu}\gamma^{\beta}\gamma^{\nu}\big{)}=p^{2}\operatorname{tr}\big{(}\gamma_{\beta}\gamma^{\mu}\gamma^{\beta}\gamma^{\nu}\big{)}=0,\end{split}$
(51)

as shown in Eq.(16).

4\. $\exists$ $i$ such that $n_{i}=2$: this case does NOT contribute because
such diagrams either are disconnected or vanish for the same reason shown in
Eq.(11) due to integration of an odd function.

Table.1 is a summary of the 4 cases and their reasons for whether or not they
contribute.

At the two-loop level, the $8!=40320$ Wick contractions reduce to $96$
distinct Feynman diagrams. Applying the same reasoning above, $90$ of which
vanishes and only $6$ diagrams remain. Group $16$ gamma matrices into loops
and we have $6$ cases. Table.2 gives a summary of the 6 cases. Cases $1,2,5$,
and $6$ follow the same reasons as cases $1,2,3$, and $4$, respectively, at
the one-loop level. Cases $3$ and $4$ are more complicated which we will
explain in the following.

| 1 |  $\exists$ $n_{i}>8$ | $\gamma^{\mu}(\cdots)\gamma_{\mu}=0$
---|---|---|---
✓ | 2 | 8+8 | $\operatorname{tr}(\cdots)\operatorname{tr}(\cdots)$
| 3 | 8+4+4 |  | ext(0+2+2) | ext(2+2+0) | ext(4+0+0)
---|---|---
int(4+0+0) | int(2+0+2) | int(0+2+2)
$\overset{F.P.}{\Longrightarrow}$ 0
| 4 | 6+6+4 |  | ext(0+2+2) | ext(2+2+0)
---|---
int(3+1+0) | int(1+1+2)
$\overset{F.P.}{\Longrightarrow}$ 0
| 5 | 4+4+4+4 |  | ext(2+2+0+0)
---
int(0+0+2+2)
$\overset{F.P.}{\Longrightarrow}$ 0
| 6 |  $\exists$ $n_{i}$ = 2 |  $\braket{\overline{\psi}\psi}$ correction $\Longrightarrow$ 0
Table 2: A summary of whether or not a configuration contributes at the two-
loop level, there are $6$ ways to group the $16$ gamma matrices and only the
second case contributes. Cases $3$ and $4$ have $3$ and $2$ subcases,
respectively.

For case $3$, there are $3$ ways to form $3$ loops consisting $8,4$, and $4$
gamma matrices, shown in Table.2 case 3:

3.1) One loop contains $4$ internal flows and two loops contain $2$ external
flows each. The loop with $4$ internal flows $p,p^{\prime},q,q^{\prime}$
yields 0 by Feynman parametrization, as $p^{\prime}=p^{\prime}(p)$ and
$q^{\prime}=q^{\prime}(q)$ similar to the case in Eq.(51).

3.2) One loop contains $2$ external and $2$ internal flows, one contains $2$
external flows, and one contains $2$ internal flow. The loop with only $2$
external flows forces one of the other two loops to contain a pair of linearly
dependent internal flows, and such diagrams vanish similar to the case in
Eq.(51).

3.3) One loop contains $4$ external flows, and two loops each contain $2$
internal flows. The loop with $4$ external flows is in
$\operatorname{tr}[\gamma^{\mu}(\cdots)\gamma_{\mu}\cdots]$ form, and it
vanishes by two-dimensional gamma matrices identities.

For case 4, $2$ ways to form $3$ loops both vanish:

4.1) One loop contains $3$ internal flows, one contains $2$ external flows,
and one contains $2$ external and $1$ internal flows. Similar to case 3.1),
the loop with only $2$ external flows forces one of the other two loops to
contain a pair of linearly dependent internal flows, thus this case vanishes.

4.2) Two loops contain $2$ external and $1$ internal flows each, and one loop
contains $2$ internal flows. In this case, one can verify that $3$ out of the
$4$ internal flows depend on one of the two undetermined momenta, thus the
integrands are odd functions of undetermined momenta. Such diagrams also
vanish.

Therefore, the only cases left at the two-loop level are diagrams consisting
of $2$ equal-sized loops of $8$ gamma matrices.

##  Appendix C Color maps of solutions for Majorana fermions and odd behaviour
for $N=4$

We present color maps for Majorana fermions in this appendix and explain
behaviours of $N=4$ for Majorana cases.

The coloring procedure for Majorana is the same as the one for Dirac, we first
arrange all couplings into a matrix of matrices shown in Eq.(38). The
antisymmetry properties in the first pairs of indices lead to diagonal
symmetry in the big matrix and the antisymmetry properties in the second pairs
of indices lead to diagonal symmetry in the submatrices. The antisymmetry in
indices gives vanishing diagonal terms. All independent couplings are covered
by these upper triangle parts, with $R_{abab}$’s appearing only once and all
the rest nonvanishing couplings appearing twice. For example, $N=3$ has $6$
DOFs, where $R_{1213},R_{1223},R_{1323}$ repeat themselves, thus the $9$ red
blocks at the top right corner. As one can see in Figures.16-19, all allowed
couplings are turned on in a solution. In Majorana cases, no obvious pattern
is observed other than the antisymmetry and symmetry properties in the
indices, unlike in Dirac cases where one similar pattern of solution emerges
for all $N$.

Figure 14: N=2 Majorana.

Figure 15: N=3 Majorana.

Figure 16: N=5 Majorana.

Figure 17: N=6 Majorana.

Figure 18: N=7 Majorana.

Figure 19: N=8 Majorana.

However, color maps show different patterns for $N=4$ Majorana fermions shown
in Figures.20-22 for different numerical solutions that generate different
patterns. Figure.20 gives an example of how couplings concentrate more and
more and eventually settle at $3$ DOFs $R_{1313},R_{2424}$, and cross coupling
$R_{1324}$. The cross coupling of $R_{abab}$ and $R_{cdcd}$ is defined as
$R_{abcd}$. Similarly, couplings can also settle at $R_{1414},R_{2323}$, and
$R_{1423}$ as shown in the middle color map in Figure.21, or
$R_{1212},R_{3434}$, and $R_{1234}$ as shown in the color map on the right in
Figure.21. These are the only three patterns with $3$ DOFs, and the form of
Eq.(36) is satisfied by these solutions. A theory with this structure is
equivalent to a theory of two bosons with an interacting term in the
bosonization picture where a free boson is equivalent to two real fermions
with a four-fermion interaction.

Other middle stages of coupling concentration are shown in Figure.22. These
are the only patterns with $10$ DOFs. Once couplings completely concentrate
following these patterns, these stages are solutions in the form of Eq.(36).
In the left most color map in Figure.22, couplings concentrate on
$R_{1313},R_{2424},R_{1414},R_{2323}$ and their cross couplings
$R_{1324},R_{1314},R_{1323},R_{1423},R_{1424},R_{2324}$. The middle one in
Figure.22 is an example of couplings concentrating on
$R_{1212},R_{3434},R_{1313},R_{2424}$ and their 6 cross couplings. The color
map on the right in Figure.22 shows couplings concentrate on
$R_{1212},R_{3434},R_{1414},R_{2323}$ and their 6 cross couplings. A further
step is to explore whether the fermionic theory with such solutions has a
bosonic equivalence, if this is the case, is there a way to map the four-
fermion couplings to couplings of this bosonic theory?

For $N>4$ Majorana fermions, patterns of solutions become more and more
difficult to observe as $N$ gets larger, therefore, further work is needed to
determine the most general symbolic solutions to the vanishing beta functions
up to the two-loop level.

Figure 20: The color map on the left is an example of a general solution to
the vanishing beta functions for $N=4$ Majorana fermions. The middle color map
is a solution where couplings become weaker except for $R_{1313},R_{2424}$,
and $R_{1324}$. The concentration continues in the color map on the right.

Figure 21: Three different ways of coupling concentration. The couplings
concentrate on $R_{1313},R_{2424}$, and $R_{1324}$ in the diagram on the left.
The couplings concentrate on $R_{1414},R_{2323}$, and $R_{1423}$ in the middle
diagram. The couplings concentrate on $R_{1212},R_{3434}$, and $R_{1234}$ in
the diagram on the right.

Figure 22: Three patterns emerge when $10$ DOFs survive the coupling
concentration. These solutions follow the form in Eq.(36). Left:
$R_{1212},R_{3434},R_{1313},R_{2424}$ and their cross couplings. Middle:
$R_{1313},R_{2424},R_{1414},R_{2323}$ and their cross couplings. Right:
$R_{1212},R_{3434},R_{1414},R_{2323}$ and their cross couplings.

##  Appendix D Samples of Numerical Results to Vanishing Beta Functions

$\begin{matrix}R_{1111}&R_{1112}\\\ R_{1121}&R_{1122}\end{matrix}$ | $\begin{matrix}R_{1112}&R_{1212}\\\ R_{1221}&R_{1222}\end{matrix}$
---|---
$\begin{matrix}R_{1121}&R_{1221}\\\ R_{2121}&R_{2122}\end{matrix}$ | $\begin{matrix}R_{1122}&R_{1222}\\\ R_{2122}&R_{2222}\end{matrix}$

-4.3860748e-2 | -2.4213040e-1 | -2.4213040e-1 | 3.1556370e-2
---|---|---|---
3.0402219e-1 | 2.4040258e-1 | 6.2101346e-3 | -2.4448646e-1
3.0402219e-1 | 6.2101346e-3 | 2.4040258e-1 | -2.4448646e-1
1.2221232e-3 | 3.0355853e-1 | 3.0355853e-1 | 5.2484182e-1

Table 3: A solution to the vanishing beta functions for the theory with N=2
Dirac fermions that does not follow the form of Eq.(36).

$\begin{matrix}R_{1111}&R_{1112}&R_{1113}\\\ R_{1121}&R_{1122}&R_{1123}\\\
R_{1131}&R_{1132}&R_{1133}\end{matrix}$
---
$\begin{matrix}R_{1112}&R_{1212}&R_{1213}\\\ R_{1221}&R_{1222}&R_{1223}\\\
R_{1231}&R_{1232}&R_{1233}\end{matrix}$
$\begin{matrix}R_{1113}&R_{1213}&R_{1313}\\\ R_{1321}&R_{1322}&R_{1323}\\\
R_{1331}&R_{1332}&R_{1333}\end{matrix}$
$\begin{matrix}R_{1121}&R_{1221}&R_{1321}\\\ R_{2121}&R_{2122}&R_{2123}\\\
R_{2131}&R_{2132}&R_{2133}\end{matrix}$
$\begin{matrix}R_{1122}&R_{1222}&R_{1322}\\\ R_{2122}&R_{2222}&R_{2223}\\\
R_{2231}&R_{2232}&R_{2233}\end{matrix}$
$\begin{matrix}R_{1123}&R_{1223}&R_{1323}\\\ R_{2123}&R_{2223}&R_{2323}\\\
R_{2331}&R_{2332}&R_{2333}\end{matrix}$
$\begin{matrix}R_{1131}&R_{1231}&R_{1331}\\\ R_{2131}&R_{2231}&R_{2331}\\\
R_{3131}&R_{3132}&R_{3133}\end{matrix}$
$\begin{matrix}R_{1132}&R_{1232}&R_{1332}\\\ R_{2132}&R_{2232}&R_{2332}\\\
R_{3132}&R_{3232}&R_{3233}\end{matrix}$
$\begin{matrix}R_{1133}&R_{1233}&R_{1333}\\\ R_{2133}&R_{2233}&R_{2333}\\\
R_{3133}&R_{3233}&R_{3333}\end{matrix}$

5.87648556482932e-1 | 4.15490588387255e-1 | 8.09728843069292e-2
---|---|---
-1.17589376850528e-1 | 6.35939476084822e-1 | 9.66442479189108e-2
-1.09541107265992e-1 | -2.52992302722233e-1 | 8.71030751509361e-2
4.15490588387255e-1 | 5.59563619288908e-4 | 3.28299313172747e-4
5.37601466194129e-4 | 4.1563647157943e-1 | 3.72746485831559e-4
4.94452190544431e-4 | 4.14961112540219e-4 | 4.15297389252339e-1
8.09728843069292e-2 | 3.28299313172747e-4 | 2.08068902716359e-4
3.52770559599846e-4 | 8.10293435524669e-2 | 2.32760156206622e-4
2.74020612749278e-4 | 2.8670364435102e-4 | 8.08365134052374e-2
-1.17589376850528e-1 | 5.37601466194129e-4 | 3.52770559599846e-4
6.28702609173722e-4 | -1.1754154676859e-1 | 3.95675644307164e-4
4.52106341843372e-4 | 4.91213947244407e-4 | -1.17842968195166e-1
6.35939476084822e-1 | 4.1563647157943e-1 | 8.10293435524669e-2
-1.1754154676859e-1 | 6.84345214578727e-1 | 9.67115256538758e-2
-1.09396141048504e-1 | -2.52955300347104e-1 | 1.35399551371414e-1
9.66442479189108e-2 | 3.72746485831559e-4 | 2.32760156206622e-4
3.95675644307164e-4 | 9.67115256538758e-2 | 2.61029933264969e-4
3.10949258252192e-4 | 3.20139640173062e-4 | 9.64918473705521e-2
-1.09541107265992e-1 | 4.94452190544431e-4 | 2.74020612749278e-4
4.52106341843372e-4 | -1.09396141048504e-1 | 3.10949258252192e-4
3.09068165743291e-4 | 4.59802986817494e-4 | -1.09686644813478e-1
-2.52992302722233e-1 | 4.14961112540219e-4 | 2.8670364435102e-4
4.91213947244407e-4 | -2.52955300347104e-1 | 3.20139640173062e-4
4.59802986817494e-4 | 2.9698568041399e-4 | -2.53201916372555e-1
8.71030751509361e-2 | 4.15297389252339e-1 | 8.08365134052374e-2
-1.17842968195166e-1 | 1.35399551371414e-1 | 9.64918473705521e-2
-1.09686644813478e-1 | -2.53201916372555e-1 | -4.13338584898267e-1

Table 4: A solution for N=3 Dirac fermions.

$\begin{matrix}R_{1111}&R_{1112}&R_{1113}&R_{1114}\\\
R_{1121}&R_{1122}&R_{1123}&R_{1124}\\\ R_{1131}&R_{1132}&R_{1133}&R_{1134}\\\
R_{1141}&R_{1142}&R_{1143}&R_{1144}\end{matrix}$
---
$\begin{matrix}R_{1112}&R_{1212}&R_{1213}&R_{1214}\\\
R_{1221}&R_{1222}&R_{1223}&R_{1224}\\\ R_{1231}&R_{1232}&R_{1233}&R_{1234}\\\
R_{1241}&R_{1242}&R_{1243}&R_{1244}\end{matrix}$
$\begin{matrix}R_{1113}&R_{1213}&R_{1313}&R_{1314}\\\
R_{1321}&R_{1322}&R_{1323}&R_{1324}\\\ R_{1331}&R_{1332}&R_{1333}&R_{1334}\\\
R_{1341}&R_{1342}&R_{1343}&R_{1344}\end{matrix}$
$\begin{matrix}R_{1114}&R_{1214}&R_{1314}&R_{1414}\\\
R_{1421}&R_{1422}&R_{1423}&R_{1424}\\\ R_{1431}&R_{1432}&R_{1433}&R_{1434}\\\
R_{1441}&R_{1442}&R_{1443}&R_{1444}\end{matrix}$
$\begin{matrix}R_{1121}&R_{1221}&R_{1321}&R_{1421}\\\
R_{2121}&R_{2122}&R_{2123}&R_{2124}\\\ R_{2131}&R_{2132}&R_{2133}&R_{2134}\\\
R_{2141}&R_{2142}&R_{2143}&R_{2144}\end{matrix}$
$\begin{matrix}R_{1122}&R_{1222}&R_{1322}&R_{1422}\\\
R_{2122}&R_{2222}&R_{2223}&R_{2224}\\\ R_{2231}&R_{2232}&R_{2233}&R_{2234}\\\
R_{2241}&R_{2242}&R_{2243}&R_{2244}\end{matrix}$
$\begin{matrix}R_{1123}&R_{1223}&R_{1323}&R_{1423}\\\
R_{2123}&R_{2223}&R_{2323}&R_{2324}\\\ R_{2331}&R_{2332}&R_{2333}&R_{2334}\\\
R_{2341}&R_{2342}&R_{2343}&R_{2344}\end{matrix}$
$\begin{matrix}R_{1124}&R_{1224}&R_{1324}&R_{1424}\\\
R_{2124}&R_{2224}&R_{2324}&R_{2424}\\\ R_{2431}&R_{2432}&R_{2433}&R_{2434}\\\
R_{2441}&R_{2442}&R_{2443}&R_{2444}\end{matrix}$

8.93891334289414e-2 | -9.08200765529855e-2 | 4.87993833917034e-2 | 2.23581578500429e-1
---|---|---|---
1.73381012395297e-1 | -4.17010032954388e-2 | 4.6646543694454e-1 | 9.93087222968853e-2
1.58558911566139e-1 | 4.52465020090685e-1 | 2.8389663637554e-2 | -1.63924330125004e-1
4.75721783983157e-3 | -3.58994878445356e-1 | 1.12327371281389e-1 | 4.44100160950618e-1
-9.08200765529855e-2 | 1.71052767932729e-3 | 1.93060133709441e-3 | 1.89843302179749e-3
2.96189119224102e-3 | -8.92895768322059e-2 | 1.97390000514186e-3 | 2.61074796896732e-3
2.40165860870197e-3 | 2.56284197024156e-3 | -9.03723841098154e-2 | 2.11490288448325e-3
2.35821991735503e-3 | 2.58807745278666e-3 | 1.8085886792796e-3 | -9.0800486359443e-2
4.87993833917034e-2 | 1.93060133709441e-3 | 1.6936758398197e-3 | 2.03107883583002e-3
1.54367500346283e-3 | 4.84899996107517e-2 | 1.86979604656575e-3 | 1.79080549133709e-3
1.78430153581487e-3 | 1.92339007103261e-3 | 4.85049584020254e-2 | 1.43254060620137e-3
1.25052863228629e-3 | 1.32845068929825e-3 | 1.07201615064092e-3 | 4.81159771724384e-2
2.23581578500429e-1 | 1.89843302179749e-3 | 2.03107883583002e-3 | 1.47610977973788e-3
2.0059157954446e-3 | 2.23846997738574e-1 | 2.18802237204685e-3 | 2.07659959472313e-3
1.84087370176185e-3 | 1.90974014151239e-3 | 2.23370066864299e-1 | 2.08605758348406e-3
1.7917763064753e-3 | 1.96278188235845e-3 | 1.30812316112946e-3 | 2.23151197796952e-1
1.73381012395297e-1 | 2.96189119224102e-3 | 1.54367500346283e-3 | 2.0059157954446e-3
2.82846144695018e-3 | 1.73671827323392e-1 | 2.98753320525608e-3 | 2.63791972200659e-3
2.52104369314041e-3 | 2.63789857454227e-3 | 1.733136299339e-1 | 2.38810631084216e-3
2.07794831327891e-3 | 2.12047093908491e-3 | 2.04441432655092e-3 | 1.73175284635749e-1
-4.17010032954388e-2 | -8.92895768322059e-2 | 4.84899996107517e-2 | 2.23846997738574e-1
1.73671827323392e-1 | -1.73818121724867e-1 | 4.67814084997426e-1 | 9.97096618974108e-2
1.58987547899646e-1 | 4.52863700838672e-1 | -1.03119353024888e-1 | -1.63372737242689e-1
4.78464959400363e-3 | -3.59139073909566e-1 | 1.12837363644479e-1 | 3.12850716672893e-1
4.6646543694454e-1 | 1.97390000514186e-3 | 1.86979604656575e-3 | 2.18802237204685e-3
2.98753320525608e-3 | 4.67814084997426e-1 | 1.92526733017194e-3 | 2.47974227969789e-3
2.44946384671077e-3 | 2.60411172815349e-3 | 4.66717323041201e-1 | 2.23656869548097e-3
2.28777290938034e-3 | 2.4482832432676e-3 | 2.3553065012906e-3 | 4.66282825250818e-1
9.93087222968853e-2 | 2.61074796896732e-3 | 1.79080549133709e-3 | 2.07659959472313e-3
2.63791972200659e-3 | 9.97096618974108e-2 | 2.47974227969789e-3 | 2.24391953895858e-3
2.39419476780728e-3 | 2.5393522972734e-3 | 9.92881322122194e-2 | 2.05063520505214e-3
1.88218275451014e-3 | 1.92510275257336e-3 | 1.63169569765752e-3 | 9.92424706998542e-2

Table 5: First half of a solution for N=4 Dirac fermions.

$\begin{matrix}R_{1131}&R_{1231}&R_{1331}&R_{1431}\\\
R_{2131}&R_{2231}&R_{2331}&R_{2431}\\\ R_{3131}&R_{3132}&R_{3133}&R_{3134}\\\
R_{3141}&R_{3142}&R_{3143}&R_{3144}\end{matrix}$
---
$\begin{matrix}R_{1132}&R_{1232}&R_{1332}&R_{1432}\\\
R_{2132}&R_{2232}&R_{2332}&R_{2432}\\\ R_{3132}&R_{3232}&R_{3233}&R_{3234}\\\
R_{3241}&R_{3242}&R_{3243}&R_{3244}\end{matrix}$
$\begin{matrix}R_{1133}&R_{1233}&R_{1333}&R_{1433}\\\
R_{2133}&R_{2233}&R_{2333}&R_{2433}\\\ R_{3133}&R_{3233}&R_{3333}&R_{3334}\\\
R_{3341}&R_{3342}&R_{3343}&R_{3344}\end{matrix}$
$\begin{matrix}R_{1134}&R_{1234}&R_{1334}&R_{1434}\\\
R_{2134}&R_{2234}&R_{2334}&R_{2434}\\\ R_{3134}&R_{3234}&R_{3334}&R_{3434}\\\
R_{3441}&R_{3442}&R_{3443}&R_{3444}\end{matrix}$
$\begin{matrix}R_{1141}&R_{1241}&R_{1341}&R_{1441}\\\
R_{2141}&R_{2241}&R_{2341}&R_{2441}\\\ R_{3141}&R_{3241}&R_{3341}&R_{3441}\\\
R_{4141}&R_{4142}&R_{4143}&R_{4144}\end{matrix}$
$\begin{matrix}R_{1142}&R_{1242}&R_{1342}&R_{1442}\\\
R_{2142}&R_{2242}&R_{2342}&R_{2442}\\\ R_{3142}&R_{3242}&R_{3342}&R_{3442}\\\
R_{4142}&R_{4242}&R_{4243}&R_{4244}\end{matrix}$
$\begin{matrix}R_{1143}&R_{1243}&R_{1343}&R_{1443}\\\
R_{2143}&R_{2243}&R_{2343}&R_{2443}\\\ R_{3143}&R_{3243}&R_{3343}&R_{3443}\\\
R_{4143}&R_{4243}&R_{4343}&R_{4344}\end{matrix}$
$\begin{matrix}R_{1144}&R_{1244}&R_{1344}&R_{1444}\\\
R_{2144}&R_{2244}&R_{2344}&R_{2444}\\\ R_{3144}&R_{3244}&R_{3344}&R_{3444}\\\
R_{4144}&R_{4244}&R_{4344}&R_{4444}\end{matrix}$

1.58558911566139e-1 | 2.40165860870197e-3 | 1.78430153581487e-3 | 1.84087370176185e-3
---|---|---|---
2.52104369314041e-3 | 1.58987547899646e-1 | 2.44946384671077e-3 | 2.39419476780728e-3
2.27817944586346e-3 | 2.39833248770982e-3 | 1.5852468439324e-1 | 2.15416233905997e-3
1.93903547033657e-3 | 2.03385597278184e-3 | 1.72377197400074e-3 | 1.58324230374422e-1
4.52465020090685e-1 | 2.56284197024156e-3 | 1.92339007103261e-3 | 1.90974014151239e-3
2.63789857454227e-3 | 4.52863700838672e-1 | 2.60411172815349e-3 | 2.5393522972734e-3
2.39833248770982e-3 | 2.52237722419909e-3 | 4.52406482033699e-1 | 2.28354266360319e-3
2.02819472846199e-3 | 2.1239967070906e-3 | 1.79094217772047e-3 | 4.52211803714519e-1
2.8389663637554e-2 | -9.03723841098154e-2 | 4.85049584020254e-2 | 2.23370066864299e-1
1.733136299339e-1 | -1.03119353024888e-1 | 4.66717323041201e-1 | 9.92881322122194e-2
1.5852468439324e-1 | 4.52406482033699e-1 | -3.30102173992401e-2 | -1.63742580345448e-1
4.65639058481899e-3 | -3.59168172715776e-1 | 1.12426473698445e-1 | 3.83087886565564e-1
-1.63924330125004e-1 | 2.11490288448325e-3 | 1.43254060620137e-3 | 2.08605758348406e-3
2.38810631084216e-3 | -1.63372737242689e-1 | 2.23656869548097e-3 | 2.05063520505214e-3
2.15416233905997e-3 | 2.28354266360319e-3 | -1.63742580345448e-1 | 1.83875902890197e-3
1.81220258635743e-3 | 1.90480511326621e-3 | 1.75603803032357e-3 | -1.64191775784397e-1
4.75721783983157e-3 | 2.35821991735503e-3 | 1.25052863228629e-3 | 1.7917763064753e-3
2.07794831327891e-3 | 4.78464959400363e-3 | 2.28777290938034e-3 | 1.88218275451014e-3
1.93903547033657e-3 | 2.02819472846199e-3 | 4.65639058481899e-3 | 1.81220258635743e-3
1.6593530291522e-3 | 1.7197160649665e-3 | 1.47111490679553e-3 | 4.47958869966015e-3
-3.58994878445356e-1 | 2.58807745278666e-3 | 1.32845068929825e-3 | 1.96278188235845e-3
2.12047093908491e-3 | -3.59139073909566e-1 | 2.4482832432676e-3 | 1.92510275257336e-3
2.03385597278184e-3 | 2.1239967070906e-3 | -3.59168172715776e-1 | 1.90480511326621e-3
1.7197160649665e-3 | 1.77567805960787e-3 | 1.52879724701502e-3 | -3.59360882202121e-1
1.12327371281389e-1 | 1.8085886792796e-3 | 1.07201615064092e-3 | 1.30812316112946e-3
2.04441432655092e-3 | 1.12837363644479e-1 | 2.3553065012906e-3 | 1.63169569765752e-3
1.72377197400074e-3 | 179094217772047e-3 | 1.12426473698445e-1 | 1.75603803032357e-3
1.47111490679553e-3 | 1.52879724701502e-3 | 1.28467778185848e-3 | 1.12317122847217e-1
4.44100160950618e-1 | -9.0800486359443e-2 | 4.81159771724384e-2 | 2.23151197796952e-1
1.73175284635749e-1 | 3.12850716672893e-1 | 4.66282825250818e-1 | 9.92424706998542e-2
1.58324230374422e-1 | 4.52211803714519e-1 | 3.83087886565564e-1 | -1.64191775784397e-1
4.47958869966015e-3 | -3.59360882202121e-1 | 1.12317122847217e-1 | 7.9903188041306e-1

Table 6: Second half of a solution for N=4 Dirac fermions.

## References

  * [1] T. Banks,“Hilbert Bundles and Holographic Space-time Models,” [arXiv:2306.07038 [hep-th]].
  * [2] T. Jacobson,“Thermodynamics of space-time: The Einstein equation of state,” Phys. Rev. Lett. 75, 1260-1263 (1995) doi:10.1103/PhysRevLett.75.1260 [arXiv:gr-qc/9504004 [gr-qc]].
  * [3] J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D 7 (1973), 2333-2346 doi:10.1103/PhysRevD.7.2333 S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. 43 (1975), 199-220 [erratum: Commun. Math. Phys. 46 (1976), 206] doi:10.1007/BF02345020 T. Jacobson, “Thermodynamics of space-time: The Einstein equation of state,” Phys. Rev. Lett. 75 (1995), 1260-1263 doi:10.1103/PhysRevLett.75.1260 [arXiv:gr-qc/9504004 [gr-qc]]. R. Bousso, “The Holographic principle,” Rev. Mod. Phys. 74 (2002), 825-874 doi:10.1103/RevModPhys.74.825 [arXiv:hep-th/0203101 [hep-th]].
  * [4] S. Carlip, “Black hole entropy from horizon conformal field theory,” Nucl. Phys. B Proc. Suppl. 88, 10-16 (2000) doi:10.1016/S0920-5632(00)00748-9 [arXiv:gr-qc/9912118 [gr-qc]]; S. Carlip, Phys. Rev. D 51, 632 (1995), arXiv:gr-qc/9409052’S. Carlip, Phys. Rev. Lett. 82, 2828 (1999), arXiv:hep-th/9812013, S. Carlip, Class. Quant. Grav. 15, 3609 (1998), arXiv:hep-th/9806026, S. Carlip, AIP Conf. Proc. 1483, 54 (2012), arXiv:1207.1488 [gr-qc].
  * [5] S. N. Solodukhin,“Conformal description of horizon’s states,” Phys. Lett. B 454, 213-222 (1999) doi:10.1016/S0370-2693(99)00398-6 [arXiv:hep-th/9812056 [hep-th]], and references therein.
  * [6] T. Banks and K. M. Zurek, “Conformal description of near-horizon vacuum states,” Phys. Rev. D 104, no.12, 126026 (2021) doi:10.1103/PhysRevD.104.126026 [arXiv:2108.04806 [hep-th]].
  * [7] T. Banks and W. Fischler, “Discretely Charged Dark Matter in Inflation Models Based on Holographic Space-Time,” Universe 8 (2022) no.11, 600 doi:10.3390/universe8110600 [arXiv:2209.08361 [hep-th]]. T. Banks and W. Fischler, “The holographic spacetime model of cosmology,” Int. J. Mod. Phys. D 27 (2018) no.14, 1846005 doi:10.1142/S0218271818460057 [arXiv:1806.01749 [hep-th]]. T. Banks and W. Fischler, “Holographic Inflation Revised,” doi:10.1017/9781316535783.013 [arXiv:1501.01686 [hep-th]]. T. Banks and W. Fischler, “Holographic Theory of Accelerated Observers, the S-matrix, and the Emergence of Effective Field Theory,” [arXiv:1301.5924 [hep-th]]. T. Banks, “Holographic Space-time Models in $1+1$ Dimensions,” [arXiv:1506.05777 [hep-th]]. T. Banks and W. Fischler, “Holographic Space-time, Newton‘s Law, and the Dynamics of Horizons,” [arXiv:2003.03637 [hep-th]].
  * [8] A. Connes, Non-commutative Geometry, Academic Press, New York 1994, ISBN0-12-185860-X, page 539 et. seq., and references therein.
  * [9] L. Susskind and J. Lindesay, “An introduction to black holes, information and the string theory revolution: The holographic universe,” P. Hayden and J. Preskill, “Black holes as mirrors: Quantum information in random subsystems,” JHEP 09, 120 (2007) doi:10.1088/1126-6708/2007/09/120 [arXiv:0708.4025 [hep-th]]. Y. Sekino and L. Susskind, “Fast Scramblers,” JHEP 10, 065 (2008) doi:10.1088/1126-6708/2008/10/065 [arXiv:0808.2096 [hep-th]].
  * [10] W. E. Thirring, Annals Phys. 3 (1958), 91-112 doi:10.1016/0003-4916(58)90015-0
  * [11] D. N. Page, “Average entropy of a subsystem,” Phys. Rev. Lett. 71, 1291-1294 (1993) doi:10.1103/PhysRevLett.71.1291 [arXiv:gr-qc/9305007 [gr-qc]].
  * [12] E. Verlinde and K. M. Zurek, “Modular fluctuations from shockwave geometries,” Phys. Rev. D 106, no.10, 106011 (2022) doi:10.1103/PhysRevD.106.106011 [arXiv:2208.01059 [hep-th]].
  * [13] E. P. Verlinde and K. M. Zurek, “Observational signatures of quantum gravity in interferometers,” Phys. Lett. B 822, 136663 (2021) doi:10.1016/j.physletb.2021.136663 [arXiv:1902.08207 [gr-qc]].
  * [14] G. ’t Hooft,“Quantum mechanics at the black hole horizon,” G. ’t Hooft,“More on the black hole S matrix,” G. ’t Hooft, “The black hole interpretation of string theory,” Nucl. Phys. B 335, 138-154 (1990) doi:10.1016/0550-3213(90)90174-C; G. ’t Hooft,“The Black hole horizon as a quantum surface,” Phys. Scripta T 36, 247-252 (1991) doi:10.1088/0031-8949/1991/T36/026.
  * [15] E. P. Verlinde and H. L. Verlinde,“High-energy scattering in quantum gravity,” Class. Quant. Grav. 10, S175-S184 (1993) doi:10.1088/0264-9381/10/S/018 H. L. Verlinde and E. P. Verlinde,“Scattering at Planckian energies,” Nucl. Phys. B 371, 246-268 (1992) doi:10.1016/0550-3213(92)90236-5 [arXiv:hep-th/9110017 [hep-th]].
  * [16] E. Verlinde and K. M. Zurek,“Spacetime Fluctuations in AdS/CFT,” JHEP 04, 209 (2020) doi:10.1007/JHEP04(2020)209 [arXiv:1911.02018 [hep-th]].
  * [17] E. Perlmutter, “A universal feature of CFT Rényi entropy,” JHEP 03, 117 (2014) doi:10.1007/JHEP03(2014)117 [arXiv:1308.1083 [hep-th]].
  * [18] T. Banks, W. Fischler and L. Mannelli, “Microscopic quantum mechanics of the p = rho universe,” Phys. Rev. D 71, 123514 (2005) doi:10.1103/PhysRevD.71.123514 [arXiv:hep-th/0408076 [hep-th]]; T. Banks and W. Fischler, Int. J. Mod. Phys. D 27, no.14, 1846005 (2018) doi:10.1142/S0218271818460057 [arXiv:1806.01749 [hep-th]].
  * [19] T. Banks,“On the Impossibility of Precise Verification of Models of Quantum Gravity,” [arXiv:2309.07203 [gr-qc]].
  * [20] E. Verlinde and K. M. Zurek, “Spacetime Fluctuations in AdS/CFT,” JHEP 04, 209 (2020) doi:10.1007/JHEP04(2020)209 [arXiv:1911.02018 [hep-th]]; Y. Zhang and K. M. Zurek, “Stochastic description of near-horizon fluctuations in Rindler-AdS,” Phys. Rev. D 108, no.6, 066002 (2023) doi:10.1103/PhysRevD.108.066002 [arXiv:2304.12349 [hep-th]]; D. Li, V. S. H. Lee, Y. Chen and K. M. Zurek, “Interferometer response to geontropic fluctuations,” Phys. Rev. D 107, no.2, 024002 (2023) doi:10.1103/PhysRevD.107.024002 [arXiv:2209.07543 [gr-qc]]; K. M. Zurek, “On vacuum fluctuations in quantum gravity and interferometer arm fluctuations,” Phys. Lett. B 826, 136910 (2022) doi:10.1016/j.physletb.2022.136910 [arXiv:2012.05870 [hep-th]]; Y. Zhang and K. M. Zurek, “Stochastic description of near-horizon fluctuations in Rindler-AdS,” Phys. Rev. D 108, no.6, 066002 (2023) doi:10.1103/PhysRevD.108.066002 [arXiv:2304.12349 [hep-th]]; T. He, A. M. Raclariu and K. M. Zurek, “From Shockwaves to the Gravitational Memory Effect,” [arXiv:2305.14411 [hep-th]]; K. M. Zurek, “Snowmass 2021 White Paper: Observational Signatures of Quantum Gravity,” [arXiv:2205.01799 [gr-qc]].
  * [21] E. Abdalla, M. C. B. Abdalla and K. D. Rothe, “Nonperturbative methods in two-dimensional quantum field theory,”
  * [22] P. Di Francesco, P. Mathieu and D. Senechal, “Conformal Field Theory,” Springer-Verlag, 1997, ISBN 978-0-387-94785-3, 978-1-4612-7475-9 doi:10.1007/978-1-4612-2256-9