††institutetext: Institut des Hautes Études Scientifiques, 91440 Bures-sur-
Yvette, France

# Local/Short-range conformal field theories from long-range perturbation
theory

Junchen Rong<EMAIL_ADDRESS>
###### Abstract

We show that by imposing the conformal Wald identity, one can extract
conformal data of the corresponding short-range/local CFT from the long-range
perturbation theory. We first apply this to the O(N) vector model. We
demonstrate that by properly re-sum the perturbative series, one gets
reasonable estimations of the critical exponents of the local/short-range
CFTs. We then apply this method to study fermionic models with four-fermion
interactions. In 2+1 dimensions, the model has the Gross-Neveu coupling and
the Thirring coupling. We also consider a 4+1 dimensional theory with a
generalized Thirring coupling.

##  1 Introduction

Conformal field theory (CFT) plays an important role in the theory of phase
transitions. Nearly all second-order phase transitions—whether quantum or
statistical—are described by CFTs cardy1996scaling ; sachdev1999quantum . To
study a CFT, certain methods, such as Monte Carlo simulation landau2021guide
and conformal bootstrap rattazzi2008bounding ; poland2019conformal  have been
developed. In some cases kos2016precision ; chester2020carving ;
chester2021bootstrapping ; atanasov2022precision ; erramilli2023gross , the
bootstrap method was able to provide the highest precision results for
critical exponents. In some other cases, on the other hand, the precise
determination of the critical exponents remains unavailable, which include
various gauge theories in 2+1 dimensions Chester:2016wrc ; Albayrak:2021xtd ;
He:2021sto ; He:2021xvg . The more traditional way to study CFTs using
perturbation theories remains valuable. These include the $4-\epsilon$
expansion PhysRevLett.28.240  and the large N expansion Moshe:2003xn . Even
though the corresponding perturbative series are usually asymptotic, after
proper re-summation bender1969anharmonic ; brezin1977perturbation ;
brezin1978critical ; kazakov1979calculation ; zinn1981perturbation ;
le2012large , they give nice results for the critical exponents.

It is known that some short-range/local CFTs are in fact special points in a
conformal manifold given by long-range generalizations of the CFT. Let us take
the O(N) vector model as an example, consider the action

$S=\frac{\mathcal{N}^{-1}}{2}\int
d^{D}xd^{D}y\frac{\sum_{i}\phi_{i}(x)\phi_{i}(y)}{|x-y|^{D+s}}+\int
d^{D}x\frac{1}{8}\lambda\left(\sum_{i}\phi_{i}(x)^{2}\right)^{2}.$

Here $\mathcal{N}$ is a normalization factor which we will fix later. The
scaling dimension of the scalar field is $\Delta_{\phi}=\frac{D-s}{2}$. We set
$s=\frac{d+\delta}{2}$, so that $\Delta_{\phi}=\frac{D-\delta}{4}$. With small
and positive $\delta$, the interaction term is slightly relevant. This can be
viewed as a version of the analytic regularization 10.1007/BF02395016 ;
gel2014generalized ; bollini1964analytic ; guttinger1966generalized ;
caianiello1973combinatorics , that was introduced in the early days of quantum
field theory. It is well-known that the theory flows to an IR fixed point, and
one can use perturbation theory to study the corresponding long-range CFT
fisher1972critical . In addition to the early works in fisher1972critical ;
sak1977low ; PhysRevB.8.281 ; honkonen1989crossover , the above theory was
also recently studied in small $\delta$-expansion in Paulos:2015jfa ;
Behan:2019lyd ; Behan:2018hfx ; Giombi:2019enr ; Benedetti:2020rrq , in the
large N limit in Chai:2021arp , and using the conformal bootstrap method in
Behan:2019lyd ; Behan:2018hfx ; Behan:2023ile . Some Monte Carlo simulation
results are also available for small N models picco2012critical ;
angelini2014relations ; luijten2002boundary ; Zhao:2023inj .

The scaling dimension of operators will depend on the parameter $\delta$. In
the $\delta\rightarrow 0$ limit, one can calculate these anomalous dimensions,
such as for the mass operator $\phi^{2}$ and the leading spin-2 operator

$\Delta_{\phi^{2}}(\delta),\quad\text{ and
}\quad\Delta_{T_{\mu\nu}}(\delta)\text{. }$

The scalar operator $\phi$ will not be renormalized, so that

$\Delta_{\phi}=\frac{D-\delta}{4},$

is valid without higher loop corrections fisher1972critical ; sak1977low ;
honkonen1989crossover . The three loop results of $\Delta_{\phi^{2}}(\delta)$
and $\Delta_{T_{\mu\nu}}(\delta)$ were calculated in Benedetti:2020rrq  and in
the present paper. Their explicit forms are given in (LABEL:deltaT3dindelta)
and (5).

It was commonly believed that the above conformal manifold contains the short-
range O(N) vector model. However, a comprehensive understanding of the long-
range to short-range crossover was only achieved recently Behan:2017emf ;
Behan:2017dwr . The IR CFT at the crossover is in fact the short-range-model
accompanied by a generalized free field. This explained many long-standing
puzzles of the crossover.

In the present paper, we focus on one question, how can we extract data of the
short-range CFT from the long-range perturbation theory at small $\delta$?
Compared to the long-range CFTs in the conformal manifold, the short-range
point has a local conserved stress tensor operator, it also has a sub-sector
that satisfies the conformal Ward Identity. Reverse the logic, this means that
by imposing the corresponding conformal Ward Identity, we can extract
conformal data of the short-range models by imposing the conformal Ward
Identity. For the O(N) vector model, one can simply impose

$\Delta_{T_{\mu\nu}}(\delta)=D.$

This allows us to solve for $\delta$. Plug the result back in
$\Delta_{\phi}(\delta)$ and $\Delta_{\phi^{2}}(\delta)$ gives us the critical
exponents of the short range model.

The above procedure gives us a complementary method to study short-range/local
CFTs in addition to the $4-\epsilon$ expansion and the large N expansion. The
relation of the three methods is summarized in Fig. 1.

Figure 1: The relation between the $4-\epsilon$ expansion, the large N limit
and the long range perturbation limits. The shaded plane is where the short-
range models are located. The $d=3$, $N\sim\mathcal{O}(1)$ and $\delta\approx
1$ region is the strong coupling region.

In this paper, we first use this approach to study the O(N) vector model. In
Section 2.1, we first work in $D=4-\epsilon$ dimensions and show that the
above procedure allows us to recover the $\epsilon$-expansion results of the
short-range models. We then work directly in $D=3$, after re-sum the
perturbative series, we show that the above procedure allows us to obtain
reasonable results for the short-range critical exponents, the results are
summarized in Section 2.2.

We then initiate the study of short-range fermionic models using long-range
perturbation theory. We note here an important motivation to develop the long-
range perturbation theory approach, namely we do not need to worry about the
complication due to the famous $\gamma^{5}$ problem leibbrandt1975introduction
in dimensional regularization. Namely, when evaluating the Feynman diagram in
$D=4-\epsilon$, one encounters
$\operatorname{Tr}[\gamma^{\mu}\gamma^{v}\gamma^{\rho}\gamma^{\sigma}]\sim\epsilon^{\mu\nu\rho\sigma}\operatorname{Tr}[1]+\ldots$,
and the Levi-civita tensor can not be generalized to continuous dimensions.
When we try to dimensional continue to $D=3$, on the other hand, one has to
modify the Clifford algebra so that
$\operatorname{Tr}[\gamma^{\mu}\gamma^{v}\gamma^{\rho}]\sim\epsilon^{\mu\nu\rho}\operatorname{Tr}[1]+\ldots$.
At higher loops, one may need to add back certain diagrams to recover the full
symmetries of the $D=3$ theory111This prescription was introduced in
Zerf:2017zqi  and named “DREG3”. Using this prescription, one successfully
recovered the supersymmetry relations for the $N=1$ super-Ising model. See
also Bennett:1999he  for a treatment of Clifford algebra in $2+\epsilon$
expansion..

In Section 3.1, we study, in two-loop order, the $2+1$ dimensional long-range
models with four-fermion couplings including the Gross-Neveu coupling222The
long-range version of the Gross-Neveu model was first introduced in
Chai:2021wac . and the Thirring coupling. In Section 3.2, we study, in two-
loop order, a $4+1$ dimensional long-range fermionic model with a generalized
Thirring coupling.

To calculate the Feynman integrals, we use the Mellin-Barnes method (see for
example smirnov2013analytic ), which can be conveniently applied using the
packages AMBRE Gluza:2007rt  and MB.m Czakon:2005rk .

##  2 the (scalar) O(N) vector model

We consider the long-range O(N) vector model (1). The propagator in real space
is

$G_{ij}(x-y)=(H_{s}H_{-s})^{-1}\frac{\mathcal{N}}{|x-y|^{D-s}}\delta_{ij},\qquad\operatorname{with}\quad
H_{s}=\frac{2^{s}\pi^{\frac{d}{2}}\Gamma\left(\frac{s}{2}\right)}{\Gamma\left(\frac{d-s}{2}\right)},$

which satisfies the long range equation of motion,

$\frac{1}{\mathcal{N}}\int
d^{D}y\frac{\delta_{ij}}{|x-y|^{D+s}}G_{jk}(y-z)=\delta_{ik}\delta^{(D)}(x-z).$

We have used the equation,

$\int
d^{D}y\frac{1}{|x-y|^{D+s}}\frac{1}{|y-z|^{D-s}}=H_{s}H_{-s}\delta^{(D)}(x-z).$

Using the Fourier transformation, one can show that

$\int d^{D}x\frac{1}{r^{D-s}}e^{-ip\cdot
x}=H_{s}\frac{1}{|p|^{s}},\qquad\frac{1}{(2\pi)^{D}}\int
d^{D}p\frac{1}{|p|^{s}}e^{ip\cdot x}=(H_{s})^{-1}\frac{1}{r^{D-s}}.$

We take $\mathcal{N}=H_{-s}$, so that the propagator in the momentum space is
given by,

$G(p)_{ij}=\frac{\delta_{ij}}{(p^{2}+\mu^{2})^{s/2}}.$

We work in Euclidean signature, so that $p^{2}=\sum^{3}_{i=1}p_{i}^{2}$.
Notice that we have introduced the IR regulator $\mu$ for the propagator. It
is important to introduce this regulator to remove the IR singularity, so that
the divergence of the Feynman integral contains UV divergence only. The
Feynman rule for the four-point vertex is

$V_{ijkl}=\lambda(\delta_{ij}\delta_{kl}+\delta_{il}\delta_{jk}+\delta_{ik}\delta_{jl}).$

We take $s=\frac{D+\delta}{2}$, so that the scaling dimension of the scalar
operator is

$\Delta_{\phi}=\frac{D-\delta}{4}.$

In the $\delta\rightarrow 0_{+}$ limit, the $\lambda\phi^{4}$ coupling term is
slightly relevant. We therefore take $\delta$ to be the perturbation
parameter. The short-range models are located in the $\delta\approx 1$ region.

We collect here the perturbative calculation results of the theory, the beta
function was calculated in earlier papers, in our convention (which is the
same convention as in Benedetti:2020rrq ),

$\beta_{g}=-\delta g+\frac{1}{2}\delta g^{2}(N+8)D_{0}+\delta
g^{3}(5N+22)(D_{0}^{2}-2S_{0}),$

We use $g$ to denote the renormalized coupling constant, and $\lambda$ to
denote the bare coupling (see Benedetti:2020rrq ). The constants $D_{0}$ and
$S_{0}$ were calculated in Benedetti:2020rrq ,

$D_{0}=\frac{2^{-D}\pi^{-\frac{D}{2}}\Gamma\left(\frac{\delta}{2}\right)}{\Gamma\left(\frac{D+\delta}{2}\right)},\qquad
S_{0}=\frac{2^{1-2D}\pi^{-D}}{\delta^{2}\Gamma\left(\frac{D}{2}\right)^{2}}-\frac{(4\pi)^{-D}\left(\psi^{(0)}\left(\frac{D}{2}\right)+2\psi^{(0)}\left(\frac{D}{4}\right)+3\gamma\right)}{\delta\Gamma\left(\frac{D}{2}\right)^{2}}.$
(1)

The anomalous dimension of the leading spin-2 operator is given by

$\Delta_{T^{\mu\nu}}=2\Delta_{\phi}+2-3\delta
g^{2}(N+2)S_{2}+\frac{3}{4}\delta
g^{3}(N^{2}+10N+16)(-4D_{0}S_{2}+4I_{1}+I_{2}),$ (2)

with $S_{2}$ defined in (26), $I_{1}$ and $I_{2}$ defined in (27). The
anomalous dimension of the leading spin-1 operator (in the adjoint irrep of
O(N)) is

$\Delta_{J^{\mu}}=2\Delta_{\phi}+1-\delta g^{2}(N+2)S_{1}+\frac{1}{4}\delta
g^{3}(N+2)(-4D_{0}S_{1}+24I_{1}^{\prime}+3(N+4)I_{2}^{\prime}).$ (3)

with $S_{1}$ defined in (32), $I_{1}^{\prime}$ defined in (34) and
$I_{2}^{\prime}$ defined in (35).

From the beta function, we can solve the coupling constant at the fixed point
${g_{\star}}$ in terms of the perturbation parameter $\delta$. From which we
get the scaling dimension of the spin-2 operator at the O(N) long-range fixed
points at three-loop,

$\displaystyle\Delta_{T^{\mu\nu}}$ $\displaystyle=$
$\displaystyle\frac{D+4}{2}-\frac{\delta}{2}-\frac{12\delta^{2}(N+2)}{D(D+2)(N+8)^{2}}$
$\displaystyle+\frac{2\delta^{3}(N+2)}{D^{2}(D+2)(N+8)^{4}}\bigg{[}\frac{(D(3D-20)-64)(N+8)^{2}}{D+4}$
$\displaystyle+D(N-4)(7N+20)\left(\psi^{(0)}\left(\frac{D}{2}\right)+\gamma\right)-2D(N-4)(7N+20)\left(\psi^{(0)}\left(\frac{D}{4}\right)+\gamma\right)\bigg{]}.$

(The two loop result of $\Delta_{T^{\mu\nu}}$ was previously calculated in
Behan:2019lyd ; Behan:2018hfx ; Behan:2023ile ). Similarly using the
$\Delta_{J}$ calculated in (3), we get the scaling dimension of the leading
spin-1 operator,

$\displaystyle\Delta_{J^{\mu}}$ $\displaystyle=$
$\displaystyle\frac{D+2}{2}-\frac{\delta}{2}-\frac{2\delta^{2}(N+2)}{D(N+8)^{2}}+$
$\displaystyle+\frac{\delta^{3}(N+2)}{D^{2}(N+8)^{4}}\bigg{(}D((N-16)N-48)\left(\psi^{(0)}\left(\frac{D}{2}\right)+\gamma\right)$
$\displaystyle-2D((N-16)N-48)\left(\psi^{(0)}\left(\frac{D}{4}\right)+\gamma\bigg{)}-32(N+8)\right).$

For completeness, we also note down the scaling dimension of the $\phi^{2}$
operator calculated at three-loop calculated in Benedetti:2020rrq , which is

$\displaystyle\nu^{-1}$ $\displaystyle=$
$\displaystyle\frac{D}{2}+\delta\left(\frac{6}{N+8}-\frac{1}{2}\right)+\frac{\delta^{2}(N+2)(7N+20)\left(-\psi^{(0)}\left(\frac{D}{2}\right)+2\psi^{(0)}\left(\frac{D}{4}\right)+\gamma\right)}{(N+8)^{3}}$
(5)
$\displaystyle-\frac{\delta^{3}(N+2)}{8(N+8)^{5}\Gamma\left(\frac{D}{2}\right)}\bigg{(}\Gamma\big{(}\frac{D}{2}\big{)}\big{(}-2(N(N(19N-60)-432)-256)$
$\displaystyle\big{(}2\psi^{(0)}\big{(}\frac{D}{4}\big{)}-\psi^{(0)}\big{(}\frac{D}{2}\big{)}\big{)}\big{(}2\big{(}\psi^{(0)}\big{(}\frac{D}{4}\big{)}+\gamma\big{)}-\psi^{(0)}\big{(}\frac{D}{2}\big{)}\big{)}-6N((N-4)N-112)\psi^{(1)}$
$\displaystyle\big{(}\frac{D}{2}\big{)}+768\psi^{(1)}\big{(}\frac{D}{2}\big{)}+\pi^{2}(N+8)((N-12)N-16)+2\gamma^{2}(N(N(60-19N)+432)+256)\big{)}$
$\displaystyle+8\pi(N+8)(5N+22)\csc\big{(}\frac{\pi
D}{4}\big{)}\Gamma\big{(}\frac{D}{4}+1\big{)}^{2}\big{(}\pi^{2}-6\psi^{(1)}\big{(}\frac{D}{4}\big{)}\big{)}\bigg{)}.$

The critical exponent $\omega$, calculated in Benedetti:2020rrq , is

$\displaystyle\omega$
$\displaystyle=\delta+\frac{2\delta^{2}(5N+22)\left(-\psi^{(0)}\left(\frac{D}{2}\right)+2\psi^{(0)}\left(\frac{D}{4}\right)+\gamma\right)}{(N+8)^{2}}$
(6)
$\displaystyle+\frac{\delta^{3}}{2(N+8)^{4}}\left((N(N+36)+152)(N(N+76)+328)\left(\psi^{(0)}\bigg{(}\frac{D}{2}\right)+\gamma\right)^{2}$
$\displaystyle-32(5N+22)(N(N+46)+196)\left(\psi^{(0)}\left(\frac{D}{4}\right)+\gamma\right)\left(\psi^{(0)}\left(\frac{D}{2}\right)+\gamma\right)$
$\displaystyle+192(5N+22)^{2}\left(\psi^{(0)}\left(\frac{D}{4}\right)+\gamma\right)^{2}\bigg{)}.$

###  2.1 $4-\epsilon$ expansion, a consistency check

With the results in the previous section, we now show that one can recover the
short-range CFT data. We make an ansatz
$\delta=a_{1}\epsilon+a_{2}\epsilon^{2}+\ldots.$ and solve the following
equations

$\Delta_{T^{\mu\nu}}(\epsilon,\delta)=4-\epsilon,\quad\operatorname{or}\quad\Delta_{J^{\mu}}(\epsilon,\delta)=3-\epsilon,$

up to $\epsilon^{3}$ order. For both equations, we get the same solution,

$\delta=2\epsilon-\frac{4(N+2)\epsilon^{2}}{(N+8)^{2}}+\frac{2(N^{3}-54N^{2}-384N-544)\epsilon^{3}}{(N+8)^{4}}+\ldots,$
(7)

One recognizes that $\Delta_{\phi}=\frac{d-\delta}{4}$ agrees with the
calculation in $4-\epsilon$ perturbation theory kleinert2001critical . Plug in
(7), we again recover the result of short-range models in $4-\epsilon,$

$\displaystyle\nu^{-1}$ $\displaystyle=$ $\displaystyle
2+\frac{(-N-2)\epsilon}{N+8}+\frac{(-N-2)(13N+44)\epsilon^{2}}{2(N+8)^{3}}+\frac{(N+2)\epsilon^{3}}{8(N+8)^{5}}(3N^{3}-452N^{2}$
$\displaystyle+96(N+8)(5N+22)\zeta(3)-2672N-5312).$

Notice these results do not generalize to the critical exponents
$\omega=\frac{d\beta(g)}{dg}$. Namely, plug (7) into $\omega$ does not recover
the $\omega$ critical exponents in $4-\epsilon$ for the short-range O(N)
vector model. In the traditional short-range $4-\epsilon$ expansion, we have
three almost marginal operators,

$\partial_{\mu}\phi\partial_{\mu}\phi,\quad\phi\partial^{2}\phi,\quad\operatorname{and}\quad\phi^{4},$
(8)

One combination of the three operators is the level two descendant of
$\phi^{2}$. Another combination becomes a null operator and decouples from the
spectrum due to the equation of motion $\partial^{2}\phi\sim g\phi^{3}$. We
are then left with a single operator which is a conformal primary, whose
scaling dimension gives us the critical exponents $\omega$. We call this
operator $O_{1}$. When long-range coupling is allowed, the above operators are
allowed to mix with the non-local operators. The operator

$O_{2}(x)=\phi(x)\chi(x),\quad\operatorname{with}\quad\chi(x)=\int
d^{D}y\frac{\phi(y)}{|x-y|^{2D-2\Delta_{\phi}}}.$ (9)

will mix with $O_{1}$ in $4-\epsilon$ expansion. In other words, both
$\omega_{\operatorname{long}}=\frac{d\beta(g)}{dg}$ and
$\omega_{\operatorname{short}}$ correspond to linear combinations of $O_{1}$
and $O_{2}$. To get $\omega_{\operatorname{short}}$ we have to resolve such a
mixing, which we leave for future work.

###  2.2 re-summation the $D=3$ perturbative series

Even though our perturbative series is known only to three-loop order, it is,
however, interesting to think about how we can re-summation this series. For a
nice review of the re-summation theory in quantum field theory, see
kleinert2001critical . For an asymptotic series with zero radius of
convergence

$A^{\prime}(g)=\sum_{k=0}^{\infty}f_{k}\lambda^{k},$

one can define the Borel-Leroy series to be

$B^{\beta}(g)=\sum_{k=0}^{\infty}\frac{f_{k}}{\Gamma(k+\beta+1)}\lambda^{k}.$

The inverse transformation is defined as

$A(g)=\int dtt^{\beta}e^{-t}B^{\beta}(\lambda t).$

For a perturbative series, the Borel-Leroy procedure leaves the series
invariant. This may seem useless. However, in many cases, $B^{\beta}(\lambda)$
is a convergent series. If one can sum the $B^{\beta}(\lambda)$ series, the
inverse Borel-Leroy transformation will give us an analytic function $A(g).$

Suppose the singularity of $B^{\beta}(g)$ that is closest to the origin is
located at $\lambda=-1/\alpha$, with a branch cut given by

$B^{\beta}(\lambda)\sim\frac{\gamma}{(1+\alpha\lambda)^{\beta+1}},$

The inverse Borel-Leroy transformation tells us the $f_{k}$ asymptotes to

$f_{k}=\gamma k^{\beta}k!(-\alpha)^{k},\qquad k\rightarrow\infty.$

For a field theory, large $k$ behavior is usually controlled by the saddle
points of the path integral, in other words, soliton solutions of the
classical equation of motion. The constant $\alpha$ is the action of the
leading soliton solution.

If the function
$B^{L,\beta}(\lambda)=\sum_{k=0}^{L}\frac{f_{k}}{\Gamma(k+\beta+1)}\lambda^{k}$
is only known to a limited number of terms, one can perform a conformal
mapping

$w(g)=\frac{\sqrt{\alpha\lambda+1}-1}{\sqrt{\alpha\lambda+1}+1},$

to map the branch cut onto the unit circle. Re-expansion $B^{\beta}(\lambda)$
using the new variable $w$, the truncated series

$B^{{}^{\prime}(w)\beta}(\lambda)\approx\sum_{k=0}^{L}W_{k}w^{k}(\lambda)$

are expected to converge faster. Clearly the above approach is only possible
if we know the solition solutions.

In addition to the conformal mapping technique, another way to approximate the
Borel function is to use the Pade approximation. This is the so-called Pade-
Borel re-summation method. To do this, one imposes that

$\operatorname{PB}^{L,\beta}(\lambda)=\frac{\sum^{M}_{i}a_{i}\lambda^{i}}{\sum^{N}_{i}b_{i}\lambda^{i}}+\mathcal{O}(\lambda^{L+1})=\sum_{k=0}^{L}\frac{f_{k}}{\Gamma(k+\beta+1)}\lambda^{k},\quad\operatorname{with}\quad
M+N=L.$

In other words, we require
$\operatorname{PB}^{L,\beta}(\lambda)=B^{L,\beta}(\lambda)$ up to order
$\lambda^{L+1}$. The freedom in choosing $(M,N)$ is an ambiguity. This
ambiguity can be partially resolved by requiring the singularity of
$\operatorname{PB}^{L,\beta}(\lambda)$ to resemble the singularity structure
of $B^{\beta}(\lambda)$. In our case, this means that the poles of
$\operatorname{PB}^{L,\beta}(\lambda)$ should lie close to the negative
$\lambda$-axis.

We will use both the conformal mapping method and the Borel-Pade re-summation
method in our analysis. We now first discuss the soliton solutions. In the
$\delta\rightarrow 0$ limit, the solution at $g<0$ is given by the solution of
the following nonlinear equation with fractional Laplacian,

$\frac{1}{\mathcal{N}}\int
d^{D}y\frac{1}{|x-y|^{D+s}}\phi_{i}(y)=-\lambda\frac{1}{2}\phi_{i}(x)\sum_{j}\phi_{j}(x)^{2}.$

The equation of motion has a solution given by

$\overrightarrow{\phi_{c}}(x)=\hat{n}\times\sqrt{2}\left(-\frac{\pi^{-\frac{D}{2}}\lambda\text{N}\Gamma\left(\frac{D-s}{2}\right)}{\Gamma\left(-\frac{s}{2}\right)}\right)^{\frac{s-D}{2s}}\left(\frac{a}{a^{2}+|x-x_{0}|^{2}}\right)^{\frac{D-s}{2}},\qquad\operatorname{with}\quad
s=\frac{D}{2}.$

This is a simple generalization of the $N=1$ soliton solution which has been
extensively studied by mathematicians (such as in chen2006classification
)333We thank Kihyun Kim for pointing out to us the long-range solition in the
$N=1$ model.. The moduli space for the soliton solution is parametrized by the
vector $\hat{n}$ satisfying $\hat{n}\cdot\hat{n}=1$ and the parameter $a$,
which controls the size of the soliton. The action of the solution is

$S=-\frac{1}{8}\lambda\int
d^{D}x\left(\sum_{i}\phi_{c,i}(x)^{2}\right)^{2}=-\frac{1}{\lambda}\frac{1}{\text{N}^{2}}\frac{\pi^{d+\frac{1}{2}}\Gamma\left(D-\frac{1}{2}\right)\Gamma\left(-\frac{D}{4}\right)^{2}}{16\Gamma\left(\frac{D}{4}\right)^{2}\Gamma(D)}.$

When $D=3$,
$S=-\frac{\pi^{2}\Gamma\left(\frac{9}{4}\right)^{2}}{\lambda\Gamma\left(\frac{3}{4}\right)^{2}}=-\frac{1}{\alpha\lambda}$.

The constant $\gamma$ and $\beta$ in (2.2) depend on the (massive and
massless) fluctuation around the soliton solution. For the short-range model,
one needs to solve a Schrodinger problem with the potential given by the
soliton. This can be done analytically (see, for example, Zinn-Justin:1980oco
). It will be interesting to study the analogous problem for long-range
models, we leave this for future work. For simplicity, we will take

$\beta=0.$

One will see that this already gives us reasonable results.

The re-summations of $\Delta_{T}$ and $\Delta_{J}$ are given in Fig 2. The re-
summation of $\nu^{-1}=3-\Delta_{\phi^{2}}$ is given in Fig 3. The re-
summation of the critical exponents $\omega$ is given in Fig. 4. Notice
$\omega=0$ at the short-range limit, as explained near equation (8). Notice
that even though we are dealing with the perturbation series only in three-
loop order, the re-summed result is very good. For comparison with the three-
loop re-summation of the $4-\epsilon$ result, see, for example, Chapter 17.4
of kleinert2001critical .

Figure 2: Re-summation of $\Delta_{T}$ and $\Delta_{J}+1$ for $N=1,2,3$. The
solid line is the result of the re-summation of $\Delta_{T}$ using conformal
mapping technique. The dashed line is the result of the re-summation of
$\Delta_{J}+1$ using conformal mapping technique. The dash-dotted line is
$\Delta_{T}$ without re-summation. The dotted line is $\Delta_{J}+1$ without
re-summation. We also tried to use the Pade-Borel re-summation method. It
turns out the Pade approximations of the corresponding Borel functions have
poles with positive real parts, we therefore discard all cases. The dots
indicate the results from conformal bootstrap kos2016precision ;
chester2020carving ; chester2021bootstrapping .

Figure 3: Re-summation of $\nu^{-1}=3-\Delta_{\phi^{2}}$for $N=1,2,3$. The
solid line is the result of the re-summation using conformal mapping
technique. The dashed line is the result using Pade[1,2]-Borel method. The
dash-dotted line are the results using Pade[2,1]-Borel method. The dotted line
is the result without re-summation. For the Pade[0,3] result, the Pade
approximations of the corresponding Borel function has poles with positive
real parts, we discard this case. The dots indicate the results from conformal
bootstrap kos2016precision ; chester2020carving ; chester2021bootstrapping .

Figure 4: Re-summation of $\omega$ for $N=1,2,3$. The solid line is the result
of the re-summation using conformal mapping technique. The dashed line is the
result using Pade[1,2]-Borel method. The dotted line is the result without re-
summation. Sometimes, the Pade approximations of the corresponding Borel
functions have poles with positive real parts, we discard these cases. The
dots indicate the results from conformal bootstrap kos2016precision ;
chester2020carving ; chester2021bootstrapping .

For completeness, we note down our prediction for the critical exponents in
Table 1.

| $\Delta_{\phi}$ | $\nu^{-1}$ | $\omega_{\operatorname{long}}$
---|---|---|---
N=1 | $0.516(8)$ | 1.583(1) | 0.0(2)
N=2 | $0.517(8)$ | 1.495(1) | 0.0(2)
N=3 | 0.517(7) | 1.430(4) | 0.0(2)
Table 1: Critical exponents.

We now give some general remarks about the critical exponents
$\omega_{\operatorname{long}}$ for the long-range models in Table 1. As we
have explained in the previous section, the $\omega_{\operatorname{long}}$
corresponds to the operator $O_{2}$ defined in (9), which is exactly marginal
at the short-range fixed point. When we work in $D=3$, near the
$\delta\rightarrow 0$ limit, there is another operator

$O_{1}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-y\phi\partial^{2}\phi,\quad\operatorname{with}\quad
y=\frac{\Delta_{\phi}}{2\Delta_{\phi}-D+2}$

which will give us $\omega_{\operatorname{short}}$ when we approach the short-
range fixed point. Unlike in $D=4-\epsilon$, we do not need to worry about
their mixing since they have different scaling dimensions in the
$\delta\rightarrow 0$ limit.

The existence of such the marginal operator $O_{2}$, will lead to logarithmic
finite size effect in Monte Carlo simulation Behan:2017emf ; Behan:2017dwr ,
and therefore making the result converge slowly. This explains the poor
quality of the numerical results near the crossover Zhao:2023inj .

##  3 Models with fermions

###  3.1 Models with four-fermion interactions in $D=2+1$

We now turn to fermionic models in D=3. We formulate our models using Majorana
fermions,

$\mathcal{L}=\mathcal{}\frac{1}{2}\mathcal{N}^{-1}\bar{\psi}_{a}(-\partial)^{s-2}\gamma^{\mu}\partial_{\mu}\psi_{a}+\lambda_{Y}(\overline{\psi_{a}}\psi_{a})^{2}+\lambda_{T}(\overline{\psi_{a}}\gamma^{\mu}\psi_{b}\Omega_{ab})^{2}$
(10)

Here we use the notation such that
$\bar{\psi}_{\alpha}=\epsilon_{\alpha\beta}\psi^{\beta}$, which is the
Majorana conjugate available in $D=2+1$ dimensions. We take the signature to
be $\eta^{\mu\nu}=(-1,1,1).$ The gamma matrices are

$\gamma^{0}=i\sigma^{2},\quad\gamma^{1}=\sigma^{1},\quad\gamma^{2}=\sigma^{3},$

notice they are real matrices. The matrix $\epsilon.\gamma^{\mu}$ is symmetric
in our convention. The antisymmetric matrix

$\Omega=\mathbf{l}_{\frac{N}{2}\times\frac{N}{2}}\otimes\epsilon=\left(\begin{array}[]{ccccc}0&1&&&\\\
-1&0&&&\\\ &&\ldots.&&\\\ &&&0&1\\\ &&&-1&0\end{array}\right),$

is block diagonal. The kinetic term and the Gross-Neveu coupling term
$g_{Y}(\overline{\psi_{i}}\psi_{i})^{2}$ preserves the O(N) symmetry. The
Thirring coupling
$g_{T}(\overline{\psi_{i}}\gamma^{\mu}\psi_{j}\Omega_{ij})^{2}$ break the
symmetry to U(${\frac{N}{2}}$). One can re-write the action (10) in a more
familiar form using the Dirac fermions. The Lagrangian (10) is the most
general Lagrangian that preserves the U(${\frac{N}{2}}$) symmetry. When $N=1$,
we have only two grassmann variables, so that both the Gross-Neveu and the
Thirring coupling vanish, the theory is just the free theory. When $N=2$, we
have four grassmann variables, the Gross-Neveu and the Thirring coupling are
equivalent, and we have only one coupling constant, which is a long-range
generalization of the original Thirring model thirring1958soluble .

The short-range version of the above model was previously studied using the
functional renormalization group method in gies2010uv ; janssen2012critical ,
using $2+\epsilon$ expansion in Bennett:1999he ; gracey1991computation ;
gracey1990three ; Hikami:1976at . The lattice version of the Thrring model was
recently studied using Monte Carlo simulation in del1996monte ; del1999three ;
del1997three ; hands1999phase ; christofi2007critical ; hands2019critical ;
Hands:2020itv ; hands2023planar ; hands2023spectroscopy . The Thirring model
is renormalizable in the large N limit hands19951 . It has a hidden gauge
symmetry Itoh:1994cr , which we now explain. The theory has a fixed point at
which the Gross-Neveu coupling vanishes at the large N limit, we get

$\mathcal{L}=\mathcal{}\frac{1}{2}i\bar{\psi}_{a}\gamma^{\mu}\partial_{\mu}\psi_{a}+\frac{\lambda}{N}(\overline{\psi_{i}}\gamma^{\mu}\psi_{j}\Omega_{ij})^{2},$

introduce the Hubbard-Stratonovich field $A_{\mu}$, we can write the above
action as

$\mathcal{L}=i\bar{\psi}\gamma\cdot\partial\psi-\sqrt{\frac{1}{N}}(\overline{\psi_{a}}\gamma^{\mu}\psi_{a}\Omega_{ab})A^{\prime}_{\mu}+\frac{1}{2\lambda}{A^{\prime}}^{2}.$
(11)

One can view the field $A^{\prime}_{\mu}$ as the massive vector boson from
Stuckelberg formalism. Just like the Gross-Neveu-Yukawa model is the UV
completion of the four-fermion Gross-Neveu model, the Thirring model has a UV
completion given by fermions coupled with a “massive” gauge field.

The long-range version of the Gross-Neveu model was first introduced in
Chai:2021wac , and a large N study was performed in $1<D<4$ and compared with
the long-range Gross-Neveu-Yukawa model in $D=4-\epsilon$. We will instead
focus on the $s\rightarrow 2$ limit at $D=3$, aiming to explain how to recover
short-range data from long-range perturbation theory.

The Feynman rules of the long-range model are the following. For each vertex,
we have

$V_{\alpha\beta\gamma\delta;abcd}=i\lambda_{Y}V^{Y}_{\alpha\beta\gamma\delta;abcd}+i\lambda_{T}V^{T}_{\alpha\beta\gamma\delta;abcd}$

with

$V^{T}_{\alpha\beta\gamma\delta;abcd}=\Omega_{\operatorname{ab}}\Omega_{\operatorname{cd}}(\epsilon.\gamma^{\mu})_{\alpha\beta}(\epsilon.\gamma_{\mu})_{\gamma\delta}+\operatorname{permutations}$

and

$V^{Y}_{\alpha\beta\gamma\delta;abcd}=\delta_{\operatorname{ab}}\delta_{\operatorname{cd}}\epsilon_{\alpha\beta}\epsilon_{\gamma\delta}+\operatorname{permutations}$

For each internal momentum, we need to integrate with

$\int\frac{dp^{D}}{(2\pi)^{D}},$

For non-local theory, the propagator is given by

$G^{\alpha\beta}_{ab}(p)=\frac{-ip_{\mu}(\gamma^{\mu}\epsilon)^{\alpha\beta}+\mu\epsilon^{\alpha\beta}}{(p^{2}+\mu^{2})^{s/2}}\delta_{ab}.$

We have introduced the IR regulator $\mu$ to make the Feynman integrals IR
finite. The final results will not depend on the mass. We take
$s=\frac{D+2+\delta}{2}$, so that the scaling dimension of the scalar operator
is

$\Delta_{\psi}=\frac{D-\delta}{4}.$ (12)

The four-fermion interaction is slightly irrelevant in the $\delta\rightarrow
0_{-}$ in the limit. Notice in this limit, the theory a (strongly) relevant
operator

$\bar{\psi}_{a}\gamma^{\mu}\partial_{\mu}\psi_{a},$

which is the short-range kinetic term for the fermions. We assume that we have
tuned the coupling of this term to zero. In other words, the fixed points that
we will discuss later are tri-critical points in the $\delta\rightarrow 0_{-}$
limit. The short-range models are located in the $\delta\approx-1$ region. The
above operator will have anomalous dimension, and may become irrelevant.

After evaluating all the Feynman diagram (given in Fig. 11 and Fig 12) which
contribute to the beta function renormalization up to two loops, we get

$\displaystyle\beta_{Y}$ $\displaystyle=$ $\displaystyle-\delta
g_{Y}+\frac{1}{6\pi^{2}}(g_{Y}^{2}(4\delta-6\delta\log(2)+N(\delta(\log(8)-3)+3)-6)$
$\displaystyle-2(\delta(\log(64)-5)+6)g_{T}^{2}+6(\delta(\log(8)-3)+3)g_{T}g_{Y})$
$\displaystyle+\frac{1}{108\pi^{4}}\bigg{(}4g_{T}^{3}(-205\delta+165\delta\log(2)-33\delta\log(2)\log(64)+41\delta\log(64)$
$\displaystyle+N(-\delta(\log(8)-7)(\log(64)-5)+9\pi+177+36\log(2))+99\pi+489+396\log(2))$
$\displaystyle+3g_{T}^{2}g_{Y}(-250\delta-378\delta\log^{2}(2)+768\delta\log(2)\delta\log(4)-24\delta\log(64)+24\delta\log(4)\log(64)$
$\displaystyle+3N(\delta(-6+\log(4)\log(8)-\log(16)\log(64)+\log(16777216))+9\pi+6+90\log(2)$
$\displaystyle-12\log(4)-2\log(8)-4\log(64))+45\pi+354-396\log(2)+144\log(4)+48\log(64))$
$\displaystyle+54g_{T}g_{Y}^{2}\big{(}\delta\big{(}12N\log^{2}(2)-N\log(4)\log(64)+4-30\log^{2}(2)+6\log^{2}(4)+\log(4)\big{)}$
$\displaystyle+3\pi+4+\log(4096)\big{)}+g_{Y}^{3}(-142\delta+2\delta\log^{2}(8)+44\delta\log(8)$
$\displaystyle+9N(\delta(2-6\log^{2}(2)+\log(16))+3\pi+6+\log(4096))-9\pi+246-36\log(2))\bigg{)}$
$\displaystyle\beta_{T}$ $\displaystyle=$ $\displaystyle-\delta
g_{T}+\frac{g_{T}((N+2)(\delta(\log(8)-7)+3)g_{T}+6(\delta+\delta(-\log(8))-3)g_{Y})}{18\pi^{2}}$
(13)
$\displaystyle+\frac{g_{T}}{324\pi^{4}}(g_{T}^{2}(-1958\delta+1192\delta\log(8)-510\delta\log(2)\log(8)+2N(\delta(\log(8)-7)^{2}+519$
$\displaystyle-18\log(2))-9\pi
N+765\pi+4110+3060\log(2))+6g_{T}g_{Y}(-194\delta-78\delta\log(2)\log(8)$
$\displaystyle+148\delta\log(8)+2N(-\delta(\log(8)-7)(\log(64)-5)+9\pi+141+9\log(16))+117\pi+834$
$\displaystyle+468\log(2))+9g_{Y}^{2}(-118\delta-18\delta\log^{2}(2)+144\delta\log(2)+3N(\delta(-2-6\log^{2}(2)$
$\displaystyle+\log(256))+3\pi+10+\log(4096))+9\pi+270+6\log(64))).$

Here $g_{Y}$ and $g_{T}$ are the renormalized couplings of the bare couplings
$\lambda_{Y}$ and $\lambda_{T}$, for there definition, see (37). The scaling
dimension of the stress tensor operator is given by

$\Delta_{T^{\mu\nu}}=2\Delta_{\psi}+1-\frac{\Gamma\left(\frac{1}{4}\right)(3Ng_{T}^{2}+Ng_{Y}^{2}+6g_{T}g_{Y}+3g_{T}^{2}-g_{Y}^{2})}{60\pi^{4}\Gamma\left(\frac{5}{4}\right)}.$
(14)

The scaling dimension of the fermion mass operator
$\overline{\psi_{i}}\psi_{i}$ is

$\displaystyle\Delta_{\bar{\psi}\psi}=$
$\displaystyle\frac{3-\delta}{2}+\frac{(\delta(\log(64)-6)+6)g_{T}}{4\pi^{2}}+\frac{g_{Y}((N-1)(\delta(\log(4)-2)+2))}{4\pi^{2}}$
(15)
$\displaystyle+\frac{g_{T}g_{Y}}{12\pi^{4}}(58\delta-2\delta\log(8)(4+\log(8))+16N(\delta(\log(8)-4)+3)+9\pi-114+36\log(2))$
$\displaystyle+\frac{(N-1)g_{Y}^{2}}{72\pi^{4}}(58\delta-2\delta\log(8)(4+\log(8))+16N(\delta(\log(8)-4)+3)+9\pi$
$\displaystyle-114+36\log(2))+\frac{g_{T}^{2}}{8\pi^{4}}(-2\delta(33+\log(2)(\log(8)-28))+N(-2\delta(\log(2)-1)(\log(8)-1)$
$\displaystyle+3\pi-22+\log(4096))+3\pi+26+\log(4096)).$

The Feynman integrals we encounter are given in Appendix B. One can check that
there is no wavefunction renormalization since the kinetic term is non-local.
The fixed points of the one-loop beta functions are

$\displaystyle F_{1}$ $\displaystyle:$ $\displaystyle g_{Y}=g_{T}=0,$
$\displaystyle F_{2}$ $\displaystyle:$ $\displaystyle
g_{Y}=\frac{2\pi^{2}}{N-2}\delta,\quad g_{T}=0,$ (16) $\displaystyle F_{3}$
$\displaystyle:$ $\displaystyle
g_{Y}=-\frac{\pi^{2}}{N^{3}+2N^{2}+32N-80}\left(-N^{2}+(N+2)\sqrt{N^{2}+112N+256}+14N-112\right)\delta,$
$\displaystyle
g_{T}=\frac{6\pi^{2}}{N^{3}+2N^{2}+32N-80}\left(N^{2}-\sqrt{N^{2}+112N+256}+N+16\right)\delta$
$\displaystyle F_{4}$ $\displaystyle:$ $\displaystyle
g_{Y}=\frac{\pi^{2}}{N^{3}+2N^{2}+32N-80}\left(N^{2}+(N+2)\sqrt{N^{2}+112N+256}-14N+112\right)\delta,$
$\displaystyle
g_{T}=\frac{6\pi^{2}}{N^{3}+2N^{2}+32N-80}\left(N^{2}+\sqrt{N^{2}+112N+256}+N+16\right)\delta.$

The free fixed point $F_{1}$ is the most stable fixed point without relevant
direction. The second fixed point $F_{2}$ is the long-range generalization of
the Gross-Neveu model, for large enough $N$, the stability matrix

$\frac{\partial\beta_{i}}{\partial g_{j}}$

has one positive eigenvalue, which means that the RG flow has one relevant
direction. The third fixed point $F_{3}$ is the long-range generalization of
the Thirring model, for large enough $N$, the fixed point has one relevant RG
directions. The fourth fixed point has two irrelevant direction, which is a
tri-critical point. The fixed point structure resembles those found in
gies2010uv ; janssen2012critical  for the short-range model.

Notice that the coupling $g_{Y}$ of the Gross-Neveu fixed point diverges at
$N=2$, this is analogous to wetzel1984two ; gracey1991computation ;
gracey1990three . This is because the Thirring coupling is equivalent to the
Gross-Neveu coupling, as we have mentioned. The theory at $N=2$ needs special
treatment, we can set $g_{T}=0$, since the two couplings are equivalent, the
beta function then becomes

$\displaystyle\beta_{Y}=$ $\displaystyle-\delta g_{Y}-\frac{\delta
g_{Y}^{2}}{3\pi^{2}}+\frac{g_{Y}^{3}}{108\pi^{4}}(2\delta(-53-54\log^{2}(2)+\log^{2}(8)+22\log(8)$
(17) $\displaystyle+9\log(16))+45\pi+354-36\log(2)+18\log(4096)).$

The critical point becomes

$g_{\star}=\pm 4.11877\sqrt{\delta}+\mathcal{O}(\delta^{1}).$

To solve the fixed point to the next order, we need the beta function at
$g_{Y}^{4}$ order (three-loop), which we leave for future work. The scaling
dimensions are

$\Delta_{T}=2.5-(0.5\pm 0.01161)\delta+\mathcal{O}(\delta^{3/2}),\quad{\rm
and}\quad{{\Delta_{\bar{\psi}\psi}}=\frac{3}{2}}\pm
0.208659\sqrt{\delta}+\mathcal{O}(\delta^{1}).$

At small $\delta$, these fixed points are clearly not smoothly connected to
the fixed point (16) as we vary $N$. For large $\delta$, it remains an open
question whether one can find a short-range CFT, which is smoothly connected
to the short-range CFTs with large $N$.

We now discuss the $N>2$ solutions. Now the question is whether the fixed
points will lead to unitary CFTs. It is too early to try to infer the final
result just from a naive two-loop calculation, we will however plot the result
just to get a feeling of it. For the Gross-Neveu fixed point, we have

$\displaystyle\text{Gross-Neveu}:$ $\displaystyle N=4$
$\displaystyle\Delta_{T}=2.5-0.5\delta-0.2\delta^{2},\hskip
30.00005pt{\Delta_{\bar{\psi}\psi}}=1.5+\delta-8.58085\delta^{2},$
$\displaystyle N=8$
$\displaystyle\Delta_{T}=2.5-0.5\delta-0.0518519\delta^{2},{\Delta_{\bar{\psi}\psi}}=1.5+0.666667\delta+1.33808\delta^{2}.$

Figure 5: $\Delta_{T_{\mu\nu}}$ and $\Delta_{\bar{\psi}\psi}$ of the Gross-
Neveu model. We have used Pade-Borel re-summation method. Sometimes, the Pade
approximations of the corresponding Borel functions have poles with positive
real parts, we discard these cases. The dots indicate the results from
conformal bootstrap erramilli2023gross .

The Gross-Neveu(-Yukawa) model has been solved by the conformal bootstrap
method. For the Thirring model, on the other hand, the long-range perturbation
theory may allow us to be ahead of the bootstrap method. For the Thirring
fixed point, we have,

$\displaystyle\operatorname{Thirring}:$ $\displaystyle N=8$
$\displaystyle\Delta_{T}=2.5-0.5\delta-1.7714\delta^{2},\hskip
20.00003pt{\Delta_{\bar{\psi}\psi}}=1.5+2.62541\delta-326.947\delta^{2},$
$\displaystyle N=100$
$\displaystyle\Delta_{T}=2.5-0.5\delta-0.0772493\delta^{2},\hskip
10.00002pt{\Delta_{\bar{\psi}\psi}}=1.5+0.734632\delta-4.81543\delta^{2}.$

The re-summed curve $\Delta_{T}(\delta)$ will intersect with the
$\Delta_{T}=3$ line when $N>12$. However, when $N$ is close to 12, the
intersection happens at $|\delta|\gg 1$ , we should not trust our perturbation
theory in that region. When $N>29$, the intersection happens at
$|\delta|\approx 1$ We therefore take $N_{c}\approx 29$ as our two-loop
prediction of the critical $N_{c}$ above which the short-range the Thirring
fixed point exists.

Figure 6: The Thirring fixed point from $\Delta_{T}=3$. We have used Pade-
Borel re-summation method. Sometimes, the Pade approximations of the
corresponding Borel functions has poles with positive real parts, we discard
these cases.

Notice that at the $\delta\rightarrow 0$ limit, $\Delta_{\psi}$ is below the
unitarity bound. It is important to check that when $\Delta_{T}=D$ happens,
the corresponding $\Delta_{\psi}$ is above the unitarity bound, so that the
short-range model is unitary. One can see in Fig. 5 and 6 that this is indeed
the case.

###  3.2 Generalized Thirring model in $D=4+1$

We also consider a fermionic model in D=4+1. We formulate our theory using
symplectic Majorana fermions (a Dirac fermion can be written as two symplectic
Majorana fermions). A nice review of the Clifford algebras in general
dimensions is given in freedman2012supergravity . We will study the following
model,

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathcal{N}^{-1}\psi_{i,\alpha}(C.\gamma^{\mu})^{\alpha\beta}(-\partial)^{s-2}\partial_{\mu}\psi_{j\beta}\Omega_{ij}$
$\displaystyle+\frac{1}{8}\lambda_{T_{2}}(\psi_{i,\alpha}(C.\gamma^{\mu\nu})^{\alpha\beta}\psi_{j,\beta}\Omega_{ij})(\psi_{i,\alpha}(C.\gamma_{\mu\nu})^{\alpha\beta}\psi_{j,\beta}\Omega_{ij}),\quad\operatorname{with}\quad
i=1\ldots N.$

Clearly, $N$ is an even number. We have also defined that
$\gamma^{\mu\nu}\equiv\gamma^{[\mu\nu]}=\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}$
. We take the convention of gamma matrix to be

$\displaystyle\gamma^{0}=i\sigma^{1}\otimes\mathbf{l}_{2\times
2},\quad\gamma^{1}=\sigma^{2}\otimes\mathbf{l}_{2\times 2},$
$\displaystyle\gamma^{2}=\sigma^{3}\otimes\sigma^{1},\quad\gamma^{3}=\sigma^{3}\otimes\sigma^{2},$
$\displaystyle\gamma^{4}=\sigma^{3}\otimes\sigma^{3},\quad
B=-\mathbf{l}_{2\times 2}\otimes\sigma^{2},$ $\displaystyle
C=B^{T}.\gamma^{0}.$

The symplectic Majorana fermions are defined as

$\psi_{i}=\Omega_{ij}B^{-1}(\psi_{j})^{\star}.$

In our convention, the matrix $C$ and $(C.\gamma^{\mu})$ are antisymmetric,
while $(C.\gamma^{\mu\nu})$ is symmetric. The gamma matrices satisfy

$B^{T}.(C)^{\star}.B=-C,\quad
B^{T}.(C.\gamma^{\mu})^{\star}.B=C.\gamma^{\mu},\quad
B^{T}.(C.\gamma^{\mu\nu})^{\star}.B=-C.\gamma^{\mu\nu},$

so that the action is Hermitian. We can consider other four-fermion couplings
such as Gross-Neveu coupling
$\frac{1}{8}\lambda_{Y}(\psi_{i,\alpha}(C)^{\alpha\beta}\psi_{i,\beta})^{2}$,
and
$\frac{1}{8}\lambda_{T}(\psi_{i,\alpha}(C.\gamma^{\mu})^{\alpha\beta}\psi_{i,\beta})(\psi_{i,\alpha}(C.\gamma_{\mu})^{\alpha\beta}\psi_{i,\beta})$.
These terms, however, will break the $\operatorname{Sp}(N)$ symmetry to the
$U\left(\frac{N}{2}\right)$ group. The above theory is the maximally symmetric
action in $D=4+1$.

Similar to the $D=2+1$ dimensional case, we take $s=\frac{D+2+\delta}{2}$, so
that the scaling dimension of the scalar operator is

$\Delta_{\psi}=\frac{D-\delta}{4}.$

The four fermion interaction is slightly irrelevant in the $\delta\rightarrow
0_{-}$ in the limit. We assume that we have tuned the coupling of the short-
range kinetic term

$\psi_{i,\alpha}(C.\gamma^{\mu})^{\alpha\beta}\partial_{\mu}\psi_{j\beta}\Omega_{ij}$

to zero. The Feynman rule for the propagators are

$G_{\alpha\beta,ij}(p)=-i\frac{p_{\mu}(\gamma^{\mu}C)_{\alpha\beta}\Omega_{ij}-\mu
C_{\alpha\beta}\delta_{ij}}{(p^{2}+\mu^{2})^{s/2}}.$

We have introduced the mass term to make the Feynman integrals IR finite,
which corresponds to add $-i\frac{1}{2}\mu\psi_{i}C\psi_{i}\delta_{ij}$ to the
action. The final results will not depend on the mass. The four-fermion
vertice is

$\displaystyle V^{T_{2}}_{\alpha\beta\gamma\delta;abcd}$ $\displaystyle=$
$\displaystyle
i\lambda_{T_{2}}(\Omega_{ad}\Omega_{bc}(C.\gamma^{\mu\nu})_{\alpha\delta}(C.\gamma_{\mu\nu})_{\beta\gamma}+\Omega_{ab}\Omega_{cd}(C.\gamma^{\mu\nu})_{\alpha\beta}(C.\gamma_{\mu\nu})_{\gamma\delta}+\Omega_{ac}\Omega_{bd}(C.\gamma^{\mu\nu})_{\alpha\gamma}(C.\gamma_{\mu\nu})_{\beta\delta}),$

The beta function at 2-loop order looks like,

$\displaystyle\beta_{T_{2}}$ $\displaystyle=$ $\displaystyle-\delta
g_{T_{2}}+\frac{g_{T_{2}}^{2}\left(\frac{4}{15}\delta(4N-337)-\delta(N-28)\log(4)-2N+56\right)}{15\pi^{3}}$
$\displaystyle+\frac{4g_{T_{2}}^{3}}{50625\pi^{6}}(64296\delta-21952\delta
N-450\delta(59N+3)\log^{2}(2)$
$\displaystyle+60\log(2)(-816\delta+(922\delta+885)N+45)+23160N-13275\pi
N-675\pi-2280).$

Here $g_{T_{2}}$ is the renormalized coupling of $\lambda_{T_{2}}$. The
anomalous dimension of the stress tensor operator $T_{\mu\nu}$ is

$\Delta_{T_{\mu\nu}}=\frac{7}{2}-\frac{\delta}{2}-\frac{4(N+1)\Gamma\left(\frac{3}{4}\right)g_{T_{2}}^{2}}{21\pi^{6}\Gamma\left(\frac{7}{4}\right)}.$

The anomalous dimension of the mass operator
$\psi_{i,\alpha}(C)^{\alpha\beta}\psi_{i,\beta}$ is

$\displaystyle\Delta_{\bar{\psi}\psi}=$
$\displaystyle\frac{5}{2}-\frac{\delta}{2}+\frac{5(\delta(\log(64)-8)+6)g_{T_{2}}}{9\pi^{3}}+\frac{2g_{T_{2}}^{2}}{405\pi^{6}}(-4128\delta-64\delta
N-90\delta(N-3)\log^{2}(2)$ (20)
$\displaystyle+12\log(2)(183\delta+(14\delta+15)N-45)-72N-45\pi
N+135\pi+2916).$

When $N\neq 28$, at the fixed point, we have

$\Delta_{T_{\mu\nu}}=\frac{7}{2}-\frac{\delta}{2}-\frac{75\delta^{2}(N+1)\Gamma\left(\frac{3}{4}\right)}{7(N-28)^{2}\Gamma\left(\frac{7}{4}\right)}.$
$\displaystyle\Delta_{\bar{\psi}\psi}=$
$\displaystyle\frac{5}{2}+\frac{\delta(N+22)}{56-2N}+\frac{\delta^{2}}{6(N-28)^{3}}(80(268N-2119)$
(21) $\displaystyle-75\pi(N(N+87)+90)+300(N(N+87)+90)\log(2)).$

When $N=28$, the solution is at

$g_{\star}=\pm 4.89026\sqrt{\delta}+\mathcal{O}(\delta^{1}).$

The scaling dimensions are

$\Delta_{T}=3.5-(0.5\pm
0.183207)\delta+\mathcal{O}(\delta^{3/2}),\quad{\Delta_{\bar{\psi}\psi}}=\frac{5}{2}\pm
0.525728\sqrt{\delta}+\mathcal{O}(\delta).$

Just like the 2+1 dimensional Thirring model can be though of as the fermions
coupled Stuckelberg vector field Itoh:1994cr , the 4+1 dimension model
(LABEL:5dThirring) can be though as fermions coupled with “Higgsed” two-form
fields. In other words, one can introduce the auxiliary field $B_{\mu\nu}$

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\psi_{i,\alpha}(C.\gamma^{\mu})^{\alpha\beta}\partial_{\mu}\psi_{j\beta}\Omega_{ij}-\sqrt{\frac{1}{N}}(\psi_{i,\alpha}(C.\gamma^{\mu\nu})^{\alpha\beta}\psi_{j,\beta}\Omega_{ij})B_{\mu\nu}$
(22)
$\displaystyle+\frac{4}{\lambda_{T_{2}}}(\psi_{i,\alpha}(C.\gamma^{\mu\nu})^{\alpha\beta}\psi_{j,\beta}\Omega_{ij})B_{\mu\nu}\quad\operatorname{with}\quad
i=1\ldots N.$

and think about $B_{\mu\nu}$ as a dynamical “massive” two form field. It will
be interesting to study the above model in the large $N$ limit.

##  4 Discussion

The main point of our paper is that by imposing the conformal Ward identity,
we can extract conformal data of local/short-range CFTs from long-range
perturbation theory. We have applied this idea to the O(N) vector models. It
is interesting to think about how to apply this idea to other quantum field
theories.

1\. the multi-scalar theories, irreducible fixed points. In the O(N) vector
model, all the scalar fields transform in a single irreducible representation
of the O(N) group. It would be interesting to consider smaller symmetry groups
so that the scalars transform in more than one irreducible representations. In
this case, scalars can have different scaling dimensions, to take this into
account, we consider the following theory

$\mathcal{L}=\int
d^{D}xd^{D}y\sum_{a=1}^{N}\frac{\phi^{a}(x)\phi^{a}(y)}{|x-y|^{D+s_{a}}}+\int
d^{D}x\frac{\lambda_{abcd}}{4!}\phi(x)^{a}\phi(x)^{b}\phi(x)^{c}\phi(x)^{d},$

we can take

$s_{a}=\frac{D+\gamma_{a}\delta}{2}.$

We normalize $\gamma_{1}=1$. For local CFTs, from the conformal Wald identity,
we can derive the following OPE formula (see for example Penedones:2016voo ),

$C_{OOT_{\mu\nu}}=-\frac{D\Delta_{O}}{D-1}\frac{1}{S_{D}},$

with $S_{D}=\frac{2\pi^{d/2}}{\Gamma(d/2)}$. Here $O$ can be any scalar
operator. Apply this to the $\phi_{a}$’s, we have the OPE ratios

$C_{\phi_{a}\phi_{a}T_{\mu\nu}}/C_{\phi_{b}\phi_{b}T_{\mu\nu}}=\frac{\Delta_{a}}{\Delta_{b}}.$
(23)

These relations, together with $\Delta_{T_{\mu\nu}}=D$, give us $N$
constraints, and allow us to solve for $\delta$ together with $\gamma_{a}$’s.
One can choose to impose these constraints directly at the $\delta\rightarrow
0$ limit, treating $\gamma_{a}=1+\mathcal{O}(\delta)$ as a perturbative
series. Recently, an efficient method to evaluate the OPE in the long-range
perturbation limit has been developed Behan:2023ile , which will be useful in
the above project444We thank Connor Behan for explaining this..

2\. the Gross-Neveu-Yukawa theory. Similar to their short-range cousins, the
long-range Gross-Neveu models also have UV completions given by the long-range
Gross-Neveu-Yukawa models (see Chai:2021wac  for the test of such a UV
completion). The short-range Gross-Neveu-Yukawa fixed point lives in a long-
range conformal manifold parametrized $\Delta_{\psi}$ and $\Delta_{\phi}$, the
conformal manifold contains various perturbative limits, one therefore has the
freedom to choose where to perform the perturbative calculation. We can
consider the couplings

$\lambda_{1}\overline{\psi_{i}}\psi_{i}\phi+\lambda_{2}\phi^{4},$

and introduce long-range kinetic terms for both the fermions and bosons. Take
$\Delta_{\psi}=\frac{3D}{8}+\gamma_{\psi}\delta$ and
$\Delta_{\phi}=\frac{D}{4}+\gamma_{\phi}\delta$, and treat $\delta$ as the
perturbation parameter. Similar to the bosonic theory, we can normalize
$\gamma_{\psi}=1$. The condition $\Delta_{T}=D$ and the OPE relations similar
to (23) allow us to fix $\delta$ and $\gamma_{\phi}$, and recover the data for
the local/short-range Gross-Neveu-Yukawa model. Alternatively, one can keep
only the Yukawa coupling (assume we have tune $\lambda_{2}=0$)

$\lambda_{1}\overline{\psi_{i}}\psi_{i}\phi,$

and take $\Delta_{\psi}=\frac{1}{6}(2D+1)+\gamma_{\psi}\delta$ and
$\Delta_{\phi}=\frac{1}{3}(D-1)+\gamma_{\phi}\delta$. In this limit, the two
candidate spin-2 operators,
$T^{(f)\mu\nu}=\bar{\psi}\partial_{\nu}\gamma_{\mu}\psi-\frac{1}{d}\delta_{\mu\nu}\bar{\psi}\partial\cdot\gamma\psi$
and
$T^{(b)\mu\nu}=(\partial^{\mu}\phi\partial^{\nu}\phi-y\phi\partial^{\mu}\partial^{\nu}\phi-\operatorname{trace})$
are degenerate. They will mix in the perturbation theory. Compared to the
previous setup, the OPE ratios can now be imposed directly at the
$\delta\rightarrow 0$ limit. It is however important to check that the
$\phi^{4}$ coupling is irrelevant when $\Delta_{T_{\mu\nu}}(\delta)=D$. The
Gross-Neveu-Ising model describes the phase transition from Dirac-semi metal
phase to charge density wave phase on graphene. Other models such as the
Gross-Neveu-Yukawa-Heinsberg and the Gross-Neveu-Yukawa-O(2) model can be
studied in a similar manner. We leave this for future work.

3\. the quantum O(N) models. In addition to the long-range models we studied
here, it is also interesting to consider the quantum long-range models. For
example, the local O(N) vector model can be embedded into a critical manifold,
which can be reached as the IR fixed point of the following action,

$\displaystyle S=$ $\displaystyle\int
dtd^{D-1}x\frac{1}{2}\sum_{i}(\partial_{t}\phi^{i})^{2}+\int
dtd^{D-1}xd^{D-1}y\frac{1}{2}\frac{\sum_{i}\phi^{i}(t,x)\phi^{i}(t,y)}{|x-y|^{D-1+\sigma}}$
(24) $\displaystyle+\lambda\int
dtd^{D-1}x\left(\sum_{i}\phi^{i}(t,x)^{2}\right)^{2}.$

Some of these models have recently been studied using quantum Monte Carlo
simulation Song:2023omz  and perturbatively in Benedetti:2024oif . It will be
interesting to use the above theory as the perturbation theory to study the
short-range models by imposing the conformal Ward identity.

It will also be interesting to generalize our work to the long-range
generalization of the non-Abelian Thirring model frishman1993bosonization ;
Bennett:1999he ; banks1976bosonization ; ludwig20034 , the long-range
generalization of QED as studied in di20193d ; heydeman2020renormalization ,
and the long-range non-linear sigma model Brezin:1975sq ; hikami1981three ;
friedan1985nonlinear ; sak1977low ; bernreuther1986four .

###### Acknowledgements.

We thank Rajeev Erramilli, Connor Behan, Jiaxin Qiao, Slava Rychkov, Edoardo
Lauria, Philine van Vliet, Bilal Hawashin, Michael Scherer, Tom Steudtner and
Emmanuel Stamou for stimulating discussions. The work of JR is supported by
the Simons programme général (2022-2031) in Institut des Hautes Études
Scientifiques.

##  Appendix A spin-2 and spin-1 operator of the scalar theory

Our calculation is performed using AMBRE and MB.m, which use mostly minus
signature, the change of convention is simply given by

$I^{\operatorname{Lorentz},\operatorname{most}\operatorname{plus}}=\frac{1}{i^{L}}\frac{1}{(-1)^{\nu_{1}+\nu_{2}+\cdots}}\frac{1}{(2^{D}\pi^{D/2})^{L}}I^{\operatorname{AMBRE}}.$

We follow Benedetti:2020rrq  and use the zero momentum BPHZ (Bogoliubov-
Parasiuk-Hepp-Zimmermann) subtraction scheme sibold2010bogoliubov . The tree-
level diagram is given in Fig. 7, the corresponding Feynman rule for the
stress-tensor vertex is

$i\lambda_{\rm
ST}((p-q)_{\mu}((p-q)_{\nu}-yp_{\nu})+p_{\mu}(p_{\nu}-y(p-q)_{\nu})-\operatorname{trace})\delta_{ij},$

with $y=-\frac{\Delta_{\phi}+1}{\Delta_{\phi}}$. The Feynman rule for
conserved current vertex is

$i\lambda_{J}(p_{\mu}+(p-q)_{\mu})\Omega_{ij}.$

Figure 7: for calculating $\Delta_{T}$ and $\Delta_{J}$.

In the leading order, the diagram which contributes to the renormalization of
the stress tensor is given in Fig. 8.

Figure 8: One loop diagram for calculating $\Delta_{T}$ and $\Delta_{J}$.

After introducing the IR regulator for our propagator, we can perform the
regularization at any kinematic point. For simplicity, we take

$q^{\mu}=0.$

In this special kinematic point, the diagram must vanish. Since there is no
external momentum that one can use to build an expression that transforms in
the spin-2 representation of the Euclidean group. We therefore conclude that
this diagram must be regular. This trick was introduced in Behan:2019lyd .

The two-loop diagrams are given in Fig. 9.

Figure 9: Two loop diagram for calculating $\Delta_{T}$ and $\Delta_{J}$..

We again renormalize our diagram at the special kinematic point

$q^{\mu}=0.$

By the same argument as before, the first diagram should vanish at $q^{\mu}=0$
limit. The second diagram leads us to the integral

$\displaystyle\mu^{2\delta}\mathcal{I}_{0}^{\mu\nu}=$
$\displaystyle\frac{1}{((2\pi)^{D})^{2}}\int
dk_{1}^{D}dk_{2}^{D}\frac{k_{2}^{\mu}k_{2}^{\nu}-\operatorname{trace}}{(k_{2}^{2}+\mu^{2})^{s/2}((p-k_{1}+k_{2})^{2}+\mu^{2})^{s/2}(k_{1}^{2}+\mu^{2})^{s/2}(k_{1}^{2}+\mu^{2})^{s/2}}$
$\displaystyle=$ $\displaystyle(p^{\mu}p^{\nu}-\operatorname{trace})S_{2},$
(25)

Using AMBRE and then MB.m, we get

$\displaystyle S_{2}=$
$\displaystyle\frac{(4\pi)^{-D}}{\delta\Gamma\left(\frac{D}{2}+2\right)\Gamma\left(\frac{D}{2}\right)}+\frac{(4\pi)^{-D}}{\Gamma\left(\frac{D}{2}+2\right)\Gamma\left(\frac{D}{4}\right)^{2}\Gamma\left(\frac{D}{2}\right)}J_{2}(D)$
(26)
$\displaystyle-\frac{2^{-2D-1}\pi^{-D}}{\Gamma\left(\frac{D}{2}+2\right)\Gamma\left(\frac{D}{2}\right)}\left(\psi^{(0)}\left(\frac{D}{2}\right)+2\psi^{(0)}\left(\frac{D}{4}\right)+3\gamma-1\right).$

Here we introduce a constant $J_{2}(D)$, whose analytical expression are not
easy to calculate, we instead give the corresponding Mellin-Barnes integral,

$J_{2}(D)=\int_{0^{{}^{-}}}dz\frac{\Gamma(2-z)\Gamma(-z)\Gamma(z)\Gamma\left(\frac{D}{4}+z\right)^{2}\Gamma\left(\frac{D}{2}+z\right)}{\Gamma\left(\frac{D}{2}+2z\right)}.$

The integral is regular at $D=4$. The best way to evaluate is to numerically
integrate along the contour (after shifting the contour away from poles),
which can be easily done in MB.m. We also give an approximate result of the
integral in $4-\epsilon$,

$J_{2}(4-\epsilon)=-2.229271492460209-1.077724442678504\epsilon-0.5458130816580664\epsilon^{2}+O(\epsilon^{3}).$

Also, notice

$J_{2}(D=3)\approx-4.1944.$

In our calculation of the O(N) model, this constant will drop out in the final
expression, it is, however, important to consider it for other multi-component
scalar models.

At three-loop level, the diagrams that survive after taking $q^{\mu}=0$ are
given in Fig. 10.

Figure 10: The three loop diagram for $\Delta_{T}$ and $\Delta_{J}$.

The corresponding integrals are

$\displaystyle\mu^{2\delta}\mathcal{I}_{1}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\mu^{3\delta}}{((2\pi)^{D})^{3}}\int
d^{D}k_{1}d^{D}k_{2}d^{D}k_{3}$
$\displaystyle\times\frac{k_{3}^{\mu}k_{3}^{\nu}-\operatorname{trace}}{(k_{3}^{2}+\mu^{2})^{s}((p+k_{2})^{2}+\mu^{2})^{s/2}((k_{1}+k_{2})^{2}+\mu^{2})^{s/2}((k_{2}+k_{3})^{2}+\mu^{2})^{s/2}(k_{1}^{2}+\mu^{2})^{s/2}}$
$\displaystyle=$ $\displaystyle(p^{\mu}p^{\nu}-\operatorname{trace})I_{1}$
$\displaystyle\mu^{2\delta}\mathcal{I}_{2}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\mu^{3\delta}}{((2\pi)^{D})^{3}}\int
d^{D}k_{1}d^{D}k_{2}d^{D}k_{3}$
$\displaystyle\times\frac{k_{3}^{\mu}k_{3}^{\nu}-\operatorname{trace}}{(k_{3}^{2}+\mu^{2})^{s/2}((p+k_{1}-k_{3})^{2}+\mu^{2})^{s/2}((k_{2})^{2}+\mu^{2})^{s/2}((p+k_{2}-k_{3})^{2}+\mu^{2})^{s/2}(k_{3}^{2}+\mu^{2})^{s}}$
$\displaystyle=$ $\displaystyle(p^{\mu}p^{\nu}-\operatorname{trace})I_{2}$

with

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{2^{2-3D}\pi^{-\frac{3D}{2}}}{3\delta^{2}\Gamma\left(\frac{D}{2}+2\right)\Gamma\left(\frac{D}{2}\right)^{2}}$
$\displaystyle+\frac{2^{1-3D}\pi^{-\frac{3D}{2}}\left(3J_{0}(D)-\Gamma\left(\frac{D}{4}\right)^{2}\left(5\psi^{(0)}\left(\frac{D}{4}\right)+\psi^{(0)}\left(\frac{D}{4}+2\right)+6\gamma-2\right)\right)}{3\delta\Gamma\left(\frac{D}{2}+2\right)\Gamma\left(\frac{D}{4}\right)^{2}\Gamma\left(\frac{D}{2}\right)^{2}}$
$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\frac{8^{1-D}\pi^{-\frac{3D}{2}}}{3\delta^{2}\Gamma\left(\frac{D}{2}+2\right)\Gamma\left(\frac{D}{2}\right)^{2}}+\frac{2^{2-3D}\pi^{-\frac{3D}{2}}}{3\delta\Gamma\left(\frac{D}{2}+2\right)\Gamma\left(\frac{D}{4}\right)^{2}\Gamma\left(\frac{D}{2}\right)^{2}}\bigg{(}3J_{0}(D)$
(27)
$\displaystyle-\Gamma\left(\frac{D}{4}\right)^{2}\left(\psi^{(0)}\left(\frac{D}{2}\right)+4\psi^{(0)}\left(\frac{D}{4}\right)+5\gamma-2\right)\bigg{)}.$

Collect the results, we get the renormalization function of the stress-tensor
vertex (at $q^{\mu}=0$)

$\Gamma^{\mu\nu}=2(p^{\mu}p^{\nu}-\operatorname{trace})\Gamma_{\rm ST},$

with

$\Gamma_{\rm ST}=\lambda_{\rm
ST}\left(1+\frac{3}{2}\lambda^{2}(N+2)S_{2}\mu^{-2\delta}-I_{1}\lambda^{3}(N^{2}+10N+16)\mu^{-3\delta}-\frac{1}{4}I_{2}\lambda^{3}(N^{2}+10N+16)\mu^{-3\delta}\right).$

We define our renormalized coupling to be

$g_{\rm ST}=\mu^{-(D-2\Delta_{\phi}-2)}\Gamma_{\rm ST}.$ (28)

Invert the above relation, we get

$\displaystyle\lambda_{\rm ST}=$
$\displaystyle\mu^{(D-2\Delta_{\phi}-2)}g_{\rm
ST}\bigg{(}1-\frac{3}{2}\lambda^{2}(N+2)S_{2}\mu^{-2\delta}$ (29)
$\displaystyle+\frac{1}{4}\lambda^{3}\mu^{-3\delta}(4I_{2}(N^{2}+10N+16)+I_{2}(N^{2}+10N+16))+1\bigg{)}$

The beta function for the coupling $g_{\rm ST}$ is defined as

$\beta_{\rm ST}=\mu\frac{\partial}{d\mu}g_{\rm ST}.$

Notice the bare couplings $\lambda_{\rm ST}$ and $\lambda$ do not depend on
$\mu$, so that the above definition together with $\eqref{renormalizeLT}$ and

$\lambda=\mu^{\delta}\left(g+\frac{1}{2}D_{0}g^{2}(N+8)+\frac{1}{4}D_{0}^{2}g^{3}(N^{2}+6N+20)+3g^{3}\left(\frac{5N}{3}+\frac{22}{3}\right)\left(\text{$D_{0}$}^{2}-S_{0}\right)\right),$
(30)

gives us the scaling dimension of the (to-be) stress tensor, which is (2). The
constants $D_{0}$ and $S_{0}$ are defined in (1). The above relation comes
from the renormalization of the coupling constant, which was calculated in
Benedetti:2020rrq .

One can similarly calculate the scaling dimension of the current operator (3).
The diagrams that we need to calculate are the same as the ones for the stress
tensor. We can also use the special kinematics $q^{\mu}=0$, which means that
the one-loop diagram in Fig. 8 and the first two-loop diagram in Fig. 9 do not
contribute to the final result. The two-loop integral corresponding to the
second diagram in Fig. 9 is

$\displaystyle\mu^{2\delta}{\mathcal{I}^{\prime}_{0}}^{\mu\nu}=$
$\displaystyle\frac{1}{((2\pi)^{D})^{2}}\int
d^{D}k_{1}d^{D}k_{2}\frac{k_{1}^{\mu}}{(k_{2}^{2}+\mu^{2})^{s/2}((p-k_{1}+k_{2})^{2}+\mu^{2})^{s/2}(k_{1}^{2}+\mu^{2})^{s/2}(k_{1}^{2}+\mu^{2})^{s/2}}$
(31) $\displaystyle=p^{\mu}S_{1},$

with

$\displaystyle S_{1}=$
$\displaystyle\frac{(4\pi)^{-D}}{\delta\Gamma\left(\frac{D}{2}+1\right)\Gamma\left(\frac{D}{2}\right)}+\frac{(4\pi)^{-D}}{\Gamma\left(\frac{D}{2}+1\right)\Gamma\left(\frac{D}{4}\right)^{2}\Gamma\left(\frac{D}{2}\right)}J_{1}(D)$
(32)
$\displaystyle-\frac{2^{-2D-1}\pi^{-D}}{\Gamma\left(\frac{D}{2}+1\right)\Gamma\left(\frac{D}{2}\right)}\left(\psi^{(0)}\left(\frac{D}{2}\right)+2\psi^{(0)}\left(\frac{D}{4}\right)+3\gamma\right),$

and

$J_{1}(D)=\int_{0^{{}^{-}}}dz\frac{\Gamma(1-z)\Gamma(-z)\Gamma(z)\Gamma\left(\frac{D}{4}+z\right)^{2}\Gamma\left(\frac{D}{2}+z\right)}{\Gamma\left(\frac{D}{2}+2z\right)}.$

After numerical evaluation, we have $J_{1}(3)\approx-3.08557$. The integral is
regular at $D=4$. We can give an expression in $4-\epsilon$,

$\displaystyle J_{1}(4-\epsilon)\approx$
$\displaystyle-1.075405220902442\epsilon^{4}-1.401988434885228\epsilon^{3}-1.697234636355991\epsilon^{2}$
(33) $\displaystyle-1.640421373262027\epsilon-1.562604825793316.$

In our calculation of the O(N) model, this constant again drops out in the
final expression, it is however, important to consider it for other multi-
component scalar models. The three-loop integrals corresponding to the
diagrams in Fig. 10 are

$\displaystyle\mu^{2\delta}{\mathcal{I}^{\prime}_{1}}^{\mu\nu}$
$\displaystyle=$ $\displaystyle\frac{\mu^{3\delta}}{((2\pi)^{D})^{3}}\int
d^{D}k_{1}d^{D}k_{2}d^{D}k_{3}$
$\displaystyle\times\frac{k_{3}^{\mu}}{(k_{3}^{2}+\mu^{2})^{s}((p+k_{2})^{2}+\mu^{2})^{s/2}((k_{1}+k_{2})^{2}+\mu^{2})^{s/2}((k_{2}+k_{3})^{2}+\mu^{2})^{s/2}(k_{1}^{2}+\mu^{2})^{s/2}}$
$\displaystyle=$ $\displaystyle p^{\mu}I_{1}^{\prime}$
$\displaystyle\mu^{2\delta}{\mathcal{I}^{\prime}_{2}}^{\mu\nu}$
$\displaystyle=$ $\displaystyle\frac{\mu^{3\delta}}{((2\pi)^{D})^{3}}\int
d^{D}k_{1}d^{D}k_{2}d^{D}k_{3}$
$\displaystyle\times\frac{k_{3}^{\mu}}{(k_{3}^{2}+\mu^{2})^{s/2}((p+k_{1}-k_{3})^{2}+\mu^{2})^{s/2}((k_{2})^{2}+\mu^{2})^{s/2}((p+k_{2}-k_{3})^{2}+\mu^{2})^{s/2}(k_{3}^{2}+\mu^{2})^{s}}$
$\displaystyle=$ $\displaystyle p^{\mu}I_{4}$

The constants are

$\displaystyle I_{1}^{\prime}=$
$\displaystyle\frac{2^{2-3D}\pi^{-\frac{3D}{2}}}{3\delta^{2}\Gamma\left(\frac{D}{2}+1\right)\Gamma\left(\frac{D}{2}\right)^{2}}+\frac{2^{1-3D}\pi^{-\frac{3D}{2}}}{3\delta\Gamma\left(\frac{D}{2}+1\right)\Gamma\left(\frac{D}{4}\right)^{2}\Gamma\left(\frac{D}{2}\right)^{2}}\bigg{[}3J_{1}(D)$
(34)
$\displaystyle-\Gamma\left(\frac{D}{4}\right)^{2}\left(5\psi^{(0)}\left(\frac{D}{4}\right)+\psi^{(0)}\left(\frac{D}{4}+1\right)+6\gamma\right)\bigg{]}.$

and

$\displaystyle I_{2}^{\prime}=$
$\displaystyle\frac{8^{1-D}\pi^{-\frac{3D}{2}}}{3\delta^{2}\Gamma\left(\frac{D}{2}+1\right)\Gamma\left(\frac{D}{2}\right)^{2}}+\frac{2^{2-3D}\pi^{-\frac{3D}{2}}}{3\delta\Gamma\left(\frac{D}{2}+1\right)\Gamma\left(\frac{D}{4}\right)^{2}\Gamma\left(\frac{D}{2}\right)^{2}}\bigg{[}3J_{1}(D)$
(35)
$\displaystyle-\Gamma\left(\frac{D}{4}\right)^{2}\left(\psi^{(0)}\left(\frac{D}{2}\right)+4\psi^{(0)}\left(\frac{D}{4}\right)+5\gamma\right)\bigg{]}.$

With the above integral, we can calculate the renormalization of the conserved
current vertex, which gives $\Gamma^{\mu}=2p^{\mu}\Gamma_{J}$, with

$\Gamma_{J}=\lambda_{J}\left(1++\frac{1}{2}\lambda^{2}(N+2)S_{2}\mu^{-2\delta}-2I_{1}^{\prime}\lambda^{3}(N+2)\mu^{-3\delta}-\frac{1}{4}I_{2}^{\prime}\lambda^{3}(N^{2}+6N+8)\mu^{-3\delta}\right).$

We define our renormalized coupling to be

$g_{J}=\mu^{-(D-2\Delta_{\phi}-1)}\Gamma_{J}.$

Invert the above relation, we get

$\lambda_{J}=\mu^{(D-2\Delta_{\phi}-1)}g_{J}\left(1--\frac{1}{2}\lambda^{2}(N+2)S_{2}^{\prime}\mu^{-2\delta}+\frac{1}{4}\lambda^{3}\mu^{-3\delta}(8I_{1}^{\prime}(N+2)+I_{2}^{\prime}(N^{2}+6N+8))\right)$
(36)

Together with (30) and the definition
$\beta_{J}=\mu\frac{\partial}{d\mu}g_{J}$, the above results give us the
scaling dimension of the (to-be) conserved current, which is (3).

##  Appendix B Fermionic models in $D=2+1$ and $D=4+1$

The one and two-loop diagrams which contribute to the beta function
renormalization are given in Fig. 11 and Fig 12.

Figure 11: The one-loop diagram for the beta function of the fermionic models.

Figure 12: The two-loop diagrams for the beta function of the fermionic
models.

The diagrams contributing to the renormalization of $T_{\mu\nu}$ have the same
topology as the bosonic case. The Feynman rule for the stress-tensor vertex is

$i\lambda_{\rm
ST}((\epsilon.\gamma^{\mu})_{\alpha\beta}(p^{\nu}+(p-q)^{\nu})+(\epsilon.\gamma^{\nu})_{\alpha\beta}(p^{\mu}+(p-q)^{\mu}-{\rm
trace})\delta_{ab}$

for the $D=2+1$ model (10). For the $D=4+1$ dimensional model
(LABEL:5dThirring), on the other hand, the Feynman rule is

$i\lambda_{\rm
ST}((C.\gamma^{\mu})_{\alpha\beta}(p^{\nu}+(p-q)^{\nu})+(C.\gamma^{\nu})_{\alpha\beta}(p^{\mu}+(p-q)^{\mu}-{\rm
trace})\Omega_{ij}.$

We will first summarize the integrals that we will encounter in our
calculation. After evaluating the $\gamma$-matrix algebra, we are left with
the following integrals, which we again evaluate using the Mellin-Barnes
method.

At one loop, we will encounter two integral. The first integral is

$\mu^{\delta}\times I_{0}^{\mu\nu}=\frac{1}{(2\pi)^{D}}\int
d^{D}k\frac{k^{\mu}k^{\nu}}{\left(k^{2}+\mu^{2}\right)^{s/2}\left(k^{2}+\mu^{2}\right)^{s/2}}=f_{1}\eta^{\mu\nu}=-\frac{2^{-D-1}\pi^{-\frac{D}{2}}\cdot\Gamma\left(\frac{\delta}{2}\right)}{\Gamma\left(\frac{1}{2}\cdot(D+\delta+2)\right)}\eta^{\mu\nu}.$

The second integral is

$\mu^{\delta}\times I_{1}^{\mu\nu}=\frac{1}{(2\pi)^{D}}\int
d^{D}k\frac{1}{\left(k^{2}+\mu^{2}\right)^{s/2}\left(k^{2}+\mu^{2}\right)^{s/2}}=\frac{1}{{\mu}^{2}}f_{2}=\frac{1}{{\mu}^{2}}\frac{2^{-D}\cdot\pi^{-\frac{D}{2}}\cdot\Gamma\left(\frac{\delta}{2}+1\right)}{\Gamma\left(\frac{1}{2}\cdot(D+\delta+2)\right)}.$

At two loop, we need to evaluate

$\displaystyle\mu^{2\delta}I_{3}^{\mu\nu}=\frac{1}{\left((2\pi)^{D}\right)^{2}}\int
d^{D}k_{1}d^{D}k_{2}\frac{k_{2}^{\mu}k_{2}^{\nu}}{\left(k_{2}^{2}+\mu^{2}\right)^{s/2}\left(\left(p-k_{1}+k_{2}\right)^{2}+\mu^{2}\right)^{s/2}\left(k_{1}^{2}+\mu^{2}\right)^{s/2}\left(k_{1}^{2}+\mu^{2}\right)^{s/2}}=$
$\displaystyle\frac{1}{\delta}\eta^{\mu\nu}c_{3}+\mathcal{O}\left(\delta^{0}\right)$

Here the constant is
$c_{3}=-\frac{(4\cdot\pi)^{-D}}{\mu^{2}\cdot\Gamma\cdot\left(\frac{D}{2}+1\right)^{2}}$.
The next integral is

$\displaystyle\mu^{2\delta}I_{5}^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}$
$\displaystyle=\frac{1}{\left((2\pi)^{D}\right)^{2}}\int
d^{D}k_{1}d^{D}k_{2}\frac{k_{1}^{\mu_{1}}k_{1}^{\mu_{2}}k_{1}^{\mu_{3}}k_{2}^{\mu_{4}}}{\left(k_{2}^{2}+\mu^{2}\right)^{s/2}\left(\left(p-k_{1}+k_{2}\right)^{2}+\mu^{2}\right)^{s/2}\left(k_{1}^{2}+\mu^{2}\right)^{s/2}\left(k_{1}^{2}+\mu^{2}\right)^{s/2}}$
$\displaystyle=\frac{1}{\delta}\left(\eta^{\mu_{1}\mu_{2}}\eta^{\mu_{3}\mu_{4}}+\eta^{\mu_{1}\mu_{3}}\eta^{\mu_{2}\mu_{4}}+\eta^{\mu_{1}\mu_{4}}\eta^{\mu_{2}\mu_{3}}\right)c_{5}+\mathcal{O}\left(\delta^{0}\right)$

with the constant is
$c_{5}=\frac{4^{-D-1}\cdot\pi^{-D}\cdot\Gamma\left(\frac{D-2}{4}\right)}{\Gamma\cdot\left(\frac{D}{2}+2\right)\cdot\Gamma\left(\frac{D}{2}\right)\cdot\Gamma\left(\frac{D+2}{4}\right)}$.
The next integral is

$\displaystyle\mu^{2\delta}I_{6}^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}$
$\displaystyle=\frac{1}{\left((2\pi)^{D}\right)^{2}}\int
d^{D}k_{1}d^{D}k_{2}\frac{k_{1}^{\mu_{1}}k_{1}^{\mu_{2}}k_{2}^{\mu_{3}}k_{2}^{\mu_{4}}}{\left(k_{2}^{2}+\mu^{2}\right)^{s/2}\left(\left(p-k_{1}+k_{2}\right)^{2}+\mu^{2}\right)^{s/2}\left(k_{1}^{2}+\mu^{2}\right)^{s/2}\left(k_{1}^{2}+\mu^{2}\right)^{s/2}}$
$\displaystyle=\frac{1}{\delta^{2}}\eta^{\mu_{1}\mu_{2}}\eta^{\mu_{3}\mu_{4}}c_{6,0}+\frac{1}{\delta}\eta^{\mu_{1}\mu_{2}}\eta^{\mu_{3}\mu_{4}}c_{6,1}+\frac{1}{\delta}\left(\eta^{\mu_{1}\mu_{3}}\eta^{\mu_{2}\mu_{4}}+\eta^{\mu_{1}\mu_{4}}\eta^{\mu_{2}\mu_{3}}\right)c_{6,2}+\mathcal{O}\left(\delta^{0}\right)$

with the constants

$\displaystyle
c_{6,0}=\frac{2^{-2D-1}\cdot\pi^{-D}}{\Gamma\left(\frac{D}{2}+1\right)^{2}}$
$\displaystyle c_{6,1}=\frac{(4\cdot\pi)^{-D}\cdot\left((2-D)\cdot{\rm
Ha}\left(\frac{D}{2}\right)-2\cdot(D-2)\cdot{\rm
Ha}\left(\frac{D-2}{4}\right)+2\right)}{(D-2)\cdot
D^{2}\cdot\Gamma\left(\frac{D}{2}\right)^{2}}$ $\displaystyle
c_{6,2}=\frac{2^{-2\cdot
D-1}\cdot\pi^{-D}}{(D-2)\cdot\Gamma\left(\frac{D}{2}+1\right)^{2}}.$

Here $\mathrm{Ha}(x)$ is the Harmonic number. Next, we have

$\displaystyle\mu^{2\delta}I_{7}^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}$
$\displaystyle=\frac{1}{\left((2\pi)^{D}\right)^{2}}\int
d^{D}k_{1}d^{D}k_{2}\frac{k_{1}^{\mu_{1}}k_{1}^{\mu_{2}}k_{1}^{\mu_{3}}k_{1}^{\mu_{4}}}{\left(k_{2}{}^{2}+\mu^{2}\right)^{s/2}\left(\left(p-k_{1}+k_{2}\right)^{2}+\mu^{2}\right)^{s/2}\left(k_{1}{}^{2}+\mu^{2}\right)^{s/2}\left(k_{1}{}^{2}+\mu^{2}\right)^{s/2}}$
$\displaystyle=\frac{1}{\delta}\left(\eta^{\mu_{1}\mu_{2}}\eta^{\mu_{3}\mu_{4}}+\eta^{\mu_{1}\mu_{3}}\eta^{\mu_{2}\mu_{4}}+\eta^{\mu_{1}\mu_{4}}\eta^{\mu_{2}\mu_{3}}\right)c_{7}+\mathcal{O}\left(\delta^{0}\right)$

with the constants
$c_{7}=\frac{2^{-2D-1}\cdot\pi^{-D}}{(D-2)\cdot\Gamma\left(\frac{D}{2}+1\right)^{2}}$.
We also encounter three regular integrals that appears in the calculation,

$\displaystyle\mu^{2\delta}I_{8}$
$\displaystyle=\frac{1}{\left((2\pi)^{D}\right)^{2}}\int
d^{D}k_{1}d^{D}k_{2}\frac{1}{\left(k_{2}^{2}+\mu^{2}\right)^{s/2}\left(\left(p-k_{1}+k_{2}\right)^{2}+\mu^{2}\right)^{s/2}\left(k_{1}^{2}+\mu^{2}\right)^{s/2}\left(k_{1}^{2}+\mu^{2}\right)^{s/2}},$
$\displaystyle\mu^{2\delta}I_{2}^{\mu\nu}$
$\displaystyle=\frac{1}{\left((2\pi)^{D}\right)^{2}}\int
d^{D}k_{1}d^{D}k_{2}\frac{k_{1}^{\mu}k_{1}^{\nu}}{\left(k_{2}^{2}+\mu^{2}\right)^{s/2}\left(\left(p-k_{1}+k_{2}\right)^{2}+\mu^{2}\right)^{s/2}\left(k_{1}^{2}+\mu^{2}\right)^{s/2}\left(k_{1}^{2}+\mu^{2}\right)^{s/2}},$
$\displaystyle\mu^{2\delta}I_{4}^{\mu\nu}$
$\displaystyle=\frac{1}{\left((2\pi)^{D}\right)^{2}}\int
d^{D}k_{1}d^{D}k_{2}\frac{k_{1}^{\mu}k_{2}^{\nu}}{\left(k_{2}^{2}+\mu^{2}\right)^{s/2}\left(\left(p-k_{1}+k_{2}\right)^{2}+\mu^{2}\right)^{s/2}\left(k_{1}^{2}+\mu^{2}\right)^{s/2}\left(k_{1}^{2}+\mu^{2}\right)^{s/2}}.$

For the $2+1\mathrm{D}$ fermionic model, we can calculate the renormalization
functions (the one-particle irreducible two/three/four-point functions),

$\Gamma_{\lambda_{Y}},\quad\Gamma_{\lambda_{T}},\quad\Gamma_{\mathrm{ST}}^{\mu\nu}\text{,
and }\quad\Gamma_{\text{Mass. }}$

The renormalization function for four-fermion interaction is

$\displaystyle\Gamma_{\alpha\beta\gamma\delta,abcd}$ $\displaystyle=$
$\displaystyle\Gamma_{\lambda_{Y}}(\delta_{ad}\epsilon_{\alpha\delta}\delta_{bc}\epsilon_{\beta\gamma}+\delta_{ac}\epsilon_{\alpha\gamma}\delta_{bd}\epsilon_{\beta\delta}+\delta_{ab}\epsilon_{\alpha\beta}\delta_{cd}\epsilon_{\gamma\delta})$
$\displaystyle+\Gamma_{\lambda_{T}}(\Omega_{ad}\Omega_{bc}(\epsilon.\gamma^{\mu})_{\beta\gamma}(\epsilon.\gamma_{\mu})_{\alpha\delta}+\Omega_{ac}\Omega_{bd}(\epsilon.\gamma^{\mu})_{\alpha\gamma}(\epsilon.\gamma_{\mu})_{\beta\delta}+\Omega_{ab}\Omega_{cd}(\epsilon.\gamma^{\mu})_{\gamma\delta}(\epsilon.\gamma_{\mu})_{\alpha\beta})$

with

$\displaystyle\Gamma_{\lambda_{Y}}$ $\displaystyle=$
$\displaystyle\mu^{-2\delta}\lambda_{Y}^{3}\big{(}(6-18N)c_{6,0}-18Nc_{6,1}-12Nc_{6,2}+6c_{6,1}-36c_{6,2}+18c_{3}\mu^{2}N$
$\displaystyle-30c_{3}\mu^{2}-30c_{5}(N+1)+9f_{1}^{2}N^{2}+f_{2}^{2}N^{2}-6f_{1}f_{2}N^{2}$
$\displaystyle-27f_{1}^{2}N-3f_{2}^{2}N+18f_{1}f_{2}N+33f_{1}^{2}+5f_{2}^{2}-18f_{1}f_{2}\big{)}$
$\displaystyle-6\mu^{-2\delta}\lambda_{T}^{2}(2\lambda_{T}(2(N+11)c_{6,0}+2Nc_{6,1}+8Nc_{6,2}+22c_{6,1}$
$\displaystyle+28c_{6,2}+c_{3}\mu^{2}(-N)-8c_{3}\mu^{2}+10c_{5}(N+5))$
$\displaystyle+2f_{1}\mu^{\delta}-f_{2}\mu^{\delta})+\mu^{-2\delta}\lambda_{Y}^{2}(-36\lambda_{T}(3c_{6,0}+3c_{6,1}+2c_{6,2}-3c_{3}\mu^{2}+5c_{5})$
$\displaystyle+f_{2}(N-4)(-\mu^{\delta})+3f_{1}(N-2)(\mu^{\delta}-18f_{2}\lambda_{T})+81f_{1}^{2}(N-2)\lambda_{T}+9f_{2}^{2}(N-2)\lambda_{T})$
$\displaystyle+\mu^{-2\delta}\lambda_{Y}\bigg{(}-9\lambda_{T}^{2}\big{(}6Nc_{6,0}+6Nc_{6,1}+4Nc_{6,2}+10c_{6,0}+10c_{6,1}+20c_{6,2}+10c_{5}(N+3)$
$\displaystyle-41f_{1}^{2}+26f_{2}f_{1}-5f_{2}^{2}\big{)}+18c_{3}\mu^{2}(N-3)\lambda_{T}^{2}+6(3f_{1}-f_{2})\mu^{\delta}\lambda_{T}+\mu^{2\delta}\bigg{)},$
$\displaystyle\Gamma_{\lambda_{T}}$ $\displaystyle=$
$\displaystyle-3\mu^{-2\delta}\lambda_{T}\lambda_{Y}^{2}(6(N+1)c_{6,0}+6Nc_{6,1}+4Nc_{6,2}+6c_{6,1}+44c_{6,2}-6c_{3}\mu^{2}N+18c_{3}\mu^{2}$
$\displaystyle+10c_{5}(N+5)-15f_{1}^{2}-3f_{2}^{2}+6f_{1}f_{2})+\mu^{-2\delta}\lambda_{T}\big{(}\lambda_{T}^{2}\big{(}2Nc_{6,1}-52Nc_{6,2}$
$\displaystyle+2(N-85)c_{6,0}-170c_{6,1}-260c_{6,2}-10c_{5}(5N+43)+f_{1}^{2}N^{2}+f_{2}^{2}N^{2}$
$\displaystyle+2f_{1}f_{2}N^{2}+3f_{1}^{2}N+3f_{2}^{2}N+6f_{1}f_{2}N+125f_{1}^{2}+17f_{2}^{2}-74f_{1}f_{2}\big{)}$
$\displaystyle+2c_{3}\mu^{2}(N+5)\lambda_{T}^{2}+(f_{1}+f_{2})(N+2)\mu^{\delta}\lambda_{T}+\mu^{2\delta}\big{)}$
$\displaystyle+3\mu^{-2\delta}\lambda_{T}\lambda_{Y}\big{(}-4\lambda_{T}\big{(}2Nc_{6,0}+2Nc_{6,1}+8Nc_{6,2}+13c_{6,0}+13c_{6,1}$
$\displaystyle+22c_{6,2}+c_{3}\mu^{2}N-7c_{3}\mu^{2}+5c_{5}(2N+7)\big{)}$
$\displaystyle+2f_{2}\mu^{\delta}+f_{1}(2f_{2}(N+2)\lambda_{T}-2\mu^{\delta})+f_{1}^{2}(N+2)\lambda_{T}+f_{2}^{2}(N+2)\lambda_{T}\big{)},$

The renormalization function with one stress tensor and two fermions is

$(\Gamma^{\mu\nu})_{\alpha\beta,ab}=2i\Gamma_{\operatorname{ST}}\left(p^{\nu}(\epsilon.\gamma^{\mu})_{\alpha\beta}+p^{\nu}(\epsilon.\gamma^{\mu})_{\alpha\beta}-\frac{2}{3}\eta^{\mu\nu}p^{\sigma}(\epsilon.\gamma^{\sigma})_{\alpha\beta}\right)\delta_{ab},$

with

$\begin{array}[]{lll}\Gamma_{\operatorname{ST}}&=&\lambda_{\operatorname{ST}}+6c_{5}(N+1)\mu^{-2\delta}\lambda_{\operatorname{ST}}\lambda_{T}^{2}+2c_{5}(N-1)\mu^{-2\delta}\lambda_{\operatorname{ST}}\lambda_{Y}^{2}+12c_{5}\mu^{-2\delta}\lambda_{\operatorname{ST}}\lambda_{T}\lambda_{Y}.\end{array}$

The renormalization function with one mass operator and two fermions is,

$\Gamma_{\alpha\beta,ab}=\Gamma_{\operatorname{Mass}}\epsilon_{\alpha\beta}\delta_{ab},$

with

$\displaystyle\Gamma_{\operatorname{Mass}}$ $\displaystyle=$
$\displaystyle\lambda_{\operatorname{Mass}}\bigg{(}9\mu^{-2\delta}\lambda_{T}^{2}\big{(}-3(N+1)c_{6,1}-2Nc_{6,2}-2c_{6,2}+c_{3}\mu^{2}N+c_{3}\mu^{2}$
$\displaystyle+5c_{5}(N+1)+9f_{1}^{2}+f_{2}^{2}+6f_{1}f_{2}\big{)}+6\mu^{-2\delta}\lambda_{T}\lambda_{Y}\big{(}3(-3c_{6,1}-2c_{6,2}+c_{3}\mu^{2}+5c_{5})$
$\displaystyle+\big{(}3f_{1}+f_{2}\big{)}^{2}(N-1)\big{)}+(N-1)\mu^{-2\delta}\lambda_{Y}^{2}\big{(}3(-3c_{6,1}-2c_{6,2}+c_{3}\mu^{2}+5c_{5})$
$\displaystyle+\big{(}3f_{1}+f_{2}\big{)}^{2}(N-1)\big{)}+(3f_{1}-f_{2})(N-1)\mu^{-\delta}\lambda_{Y}+3(3f_{1}-f_{2})\mu^{-\delta}\lambda_{T}+1\bigg{)}.$

We define the renormalized coupling constants to be

$g_{Y}=\mu^{-\delta}\Gamma_{\lambda_{Y}},\quad
g_{T}=\mu^{-\delta}\Gamma_{\lambda_{T}},\quad
g_{\operatorname{ST}}=\mu^{-(D-2\Delta_{\psi}-1)}\Gamma_{\operatorname{ST}},\quad\operatorname{and}\quad
g_{\operatorname{Mass}}=\mu^{-(D-2\Delta_{\psi})}\Gamma_{\operatorname{Mass}}.\quad$
(37)

By inverting the above relations, we get the beta functions

$\beta_{Y}=\mu\frac{\partial}{\partial\mu}g_{Y},\quad\beta_{T}=\mu\frac{\partial}{\partial\mu}g_{T},\quad\beta_{\operatorname{ST}}=\mu\frac{\partial}{\partial\mu}g_{\operatorname{ST}},\quad\beta_{\operatorname{Mass}}=\mu\frac{\partial}{\partial\mu}g_{\operatorname{Mass}}$

for $g_{Y}$, $g_{T}$, $g_{\operatorname{ST}}$ and $g_{\operatorname{Mass}}$,
which then gives us (13), (14) and (15).

When calculating $\Gamma_{\mathrm{ST}}$, we can again take the special kinetic
point $q^{\mu}=0$ as the bosonic case. An argument similar to the scalar case
tells us that one can not build an invariant tensor that transforms in the
spin-2 representation of the Euclidean group (we can not use the external
momentum since $q^{\mu}=0$.) This means only the second diagram in Fig. 10
contributes to the renormalization to $T_{\mu\nu}$.

For the $4+1$ Dermionic model, we can calculate the corresponding
renormalization functions. The renormalization function for the four-fermion
coupling is

$\displaystyle\Gamma_{\alpha\beta\gamma\delta,abcd}=$
$\displaystyle\Gamma_{\lambda_{T_{2}}}\bigg{(}\Omega_{ad}\Omega_{bc}\left(C\cdot\gamma^{\mu\nu}\right)_{\alpha\delta}\left(C\cdot\gamma_{\mu\nu}\right)_{\beta\gamma}+\Omega_{ab}\Omega_{cd}\left(C\cdot\gamma^{\mu\nu}\right)_{\alpha\beta}\left(C\cdot\gamma_{\mu\nu}\right)_{\gamma\delta}$
(38)
$\displaystyle+\Omega_{ac}\Omega_{bd}\left(C\cdot\gamma^{\mu\nu}\right)_{\alpha\gamma}\left(C\cdot\gamma_{\mu\nu}\right)_{\beta\delta}\bigg{)},$

with

$\displaystyle\Gamma_{\lambda_{T_{2}}}$ $\displaystyle=$
$\displaystyle\lambda_{T_{2}}+(8f_{2}(N+2)-8f_{1}(N-28))\mu^{-\delta}\lambda_{T_{2}}^{2}$
$\displaystyle+64\mu^{-2\delta}\lambda_{T_{2}}^{3}\big{(}-2(59N+3)c_{6,0}-118Nc_{6,1}+20Nc_{6,2}-6c_{6,1}+20c_{6,2}+58c_{3}\mu^{2}N$
$\displaystyle+118c_{3}\mu^{2}+c_{5}(14-98N)+f_{1}^{2}N^{2}+f_{2}^{2}N^{2}-2f_{1}f_{2}N^{2}+3f_{1}^{2}N+3f_{2}^{2}N$
$\displaystyle-6f_{1}f_{2}N+787f_{1}^{2}+35f_{2}^{2}-6f_{1}f_{2}\big{)}.$

The renormalization function of the stress tensor-fermion-fermion vertex (at
$q^{\mu}=0$ ) is with

$\begin{gathered}\left(\Gamma^{\mu\nu}\right)_{\alpha\beta,ab}=2\cdot
i\cdot\Gamma_{\mathrm{ST}}\left(p^{\nu}\cdot\left(C\cdot\gamma^{\mu}\right)_{\alpha\beta}+p^{\nu}\cdot\left(C\cdot\gamma^{\mu}\right)_{\alpha\beta}-\frac{2}{5}\cdot\eta^{\mu\nu}p^{\sigma}\cdot\left(C\cdot\gamma^{\sigma}\right)_{\alpha\beta}\right)\Omega_{ab},\\\
\Gamma_{\mathrm{ST}}=\lambda_{\mathrm{ST}}+1920\cdot
c_{5}\cdot(N+1)\cdot\mu^{-2\delta}\cdot\lambda_{\mathrm{ST}}\cdot\lambda_{T_{2}}^{2}\end{gathered}$

The renormalization function of mass-fermion-fermion vertex is,

$\Gamma_{\alpha\beta,ab}=\Gamma_{\operatorname{Mass}}C_{\alpha\beta}\delta_{ab},$

with

$\displaystyle\Gamma_{{\rm Mass}}$ $\displaystyle=$
$\displaystyle\lambda_{{\rm
Mass}}(1+40(5f_{1}-f_{2})\mu^{-\delta}\lambda_{T_{2}}$
$\displaystyle+320(N-3)\mu^{-2\delta}\lambda_{T_{2}}^{2}(-5c_{6,1}-2c_{6,2}+7c_{5}+c_{3}\mu^{2})+1600(5f_{1}+f_{2})^{2}\mu^{-2\delta}\lambda_{T_{2}}^{2}).$

We define the renormalized coupling constants to be

$g_{T_{2}}=\mu^{-\delta}\Gamma_{T_{2}},\quad\quad
g_{\operatorname{ST}}=\mu^{-(D-2\Delta_{\psi}-1)}\Gamma_{\operatorname{ST}},\quad\operatorname{and}\quad
g_{\operatorname{Mass}}=\mu^{-(D-2\Delta_{\psi})}\Gamma_{\operatorname{Mass}}.\quad$

By inverting the above relations, and use we can get the beta functions

$\quad\beta_{T_{2}}=\mu\frac{\partial}{\partial\mu}g_{T_{2}},\quad\beta_{\operatorname{ST}}=\mu\frac{\partial}{\partial\mu}g_{\operatorname{ST}},\quad\beta_{\operatorname{Mass}}=\mu\frac{\partial}{\partial\mu}g_{\operatorname{Mass}}$

for $g_{T_{2}}$, $g_{\operatorname{ST}}$ and $g_{\operatorname{Mass}}$, which
then gives us (LABEL:5Dfermionbeta), (3.2) and (20).

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