Beta-functions of enhanced quartic tensor field theories

Joseph Ben Gelouna,c,† and Reiko Toriumib,‡

aLaboratoire d’Informatique de Paris Nord UMR CNRS 7030
Université Paris 13, 99, avenue J.-B. Clement, 93430 Villetaneuse, France

bOkinawa Institute of Science and Technology
1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa, Japan 904-0495

cInternational Chair in Mathematical Physics and Applications
(ICMPA-UNESCO Chair), University of Abomey-Calavi,
072B.P.50, Cotonou, Rep. of Benin

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
Enhanced tensor field theories (eTFT) have dominant graphs that do not
correspond to melonic diagrams or ordinary tensor field theories (TFT). They
therefore describe pertinent candidates to escape the so-called branched
polymer phase, the universality geometry found for tensor models. For generic
rank $d$ of the tensor field, we compute the perturbative $\beta$-functions at
one-loop of two just-renormalizable quartic eTFT coined by $+$ or $\times$,
depending on the combinatorics of their vertex weights. Both models, $+$ and
$\times$, have two quartic coupling constants $(\lambda,\lambda_{+})$ and
$(\lambda,\lambda_{\times})$, respectively, other 2-point couplings, (mass,
$Z_{a}$) for $+$ and (mass, $Z_{a}$, $Z_{2a}$) for $\times$. At all loops,
both models have a constant wave function renormalization: $Z=1$ and therefore
no anomalous dimension. This signals the fact that these quantum models could
behave like classical ones, however, we find that there are nontrivial
renormalizations to couplings in our models. The RG flow of the first model
coined by $+$ exhibits neither asymptotic freedom nor the ordinary Landau
ghost of $\phi^{4}_{4}$ model: $\lambda_{+}$ is a fixed point and $\lambda$
has linear behavior in time scale $t=\log(k/k_{0})$. The mass behaves likewise
(linear in $t$) in the UV, and diverge in the IR, whereas $Z_{a}$ decreases
exponentially in the UV and diverge in the IR, exactly like any relevant
operator. For the second model $\times$, both $\lambda$ and $\lambda_{\times}$
do not flow, whereas all remaining 2-point coupling constants are linear
function of the time scale.

Pacs numbers: 11.10.Gh, 04.60.-m, 02.10.Ox
Key words: Renormalization group, beta-function, tensor models, tensor field
theories, quantum gravity

###### Contents

  1. 1 Introduction
  2. 2 Review of the renormalizable enhanced quartic TFT
    1. 2.1 Enhanced TFT models
    2. 2.2 Amplitudes and renormalizability
    3. 2.3 An alternative enhanced TFT model
  3. 3  One-loop beta-functions of the model $+$
    1. 3.1 Effective action
    2. 3.2 4-point function and its beta-function
    3. 3.3 Computing $\Gamma_{2}$ and $Z_{a}$ renormalization
    4. 3.4 Self energy and mass renormalization
    5. 3.5 Integration at arbitrary loops
  4. 4 One-loop beta-functions of the model $\times$
    1. 4.1 Effective coupling equations
    2. 4.2 Self energy and mass renormalization
    3. 4.3 Computing $\Gamma_{2;a}$ and $Z_{a}$ renormalization
    4. 4.4 Computing of $\Gamma_{2;2a}$ and $Z_{2a}$ renormalization
    5. 4.5 Integration at arbitrary loops
  5. 5 Conclusion
  6. 6 Acknowledgements
  7. A Euler-Maclaurin formula and Feynman amplitude approximations
  8. B Second order in perturbation in Model $+$.
    1. B.1 4-point function in Model $+$.
    2. B.2 $\Gamma_{2}$ in Model $+$.
    3. B.3 Self energy and mass renormalization in Model $+$.
  9. C Second order in perturbation in Model $\times$.
    1. C.1 Self energy and mass renormalization in Model $\times$
    2. C.2 $\Gamma_{2;a}$ in Model $\times$.
    3. C.3 $\Gamma_{2;2a}$ in Model $\times$.

##  1 Introduction

In the search for a gravitational theory resulting from the random generation
of geometries, the most notable success to date is restricted in two
dimensions [1]. The models at the base of this success are those of the
celebrated random matrices. These statistical models generate simplicial
complexes of dimension 2 and, therefore, discretized random surfaces. An
impressive number of results exist on these models (KPZ equation, integrable
models, string theory, etc) [2] [3] [4] [5] [6]. More recently, it has been
proven [7] [8] that the continuous limit of such random surfaces provide a
spherical geometry (2D) endowed with a conformal field called Liouville (the
Brownian sphere). Building on the success of statistical matrix models, models
of random tensors [9] [10] generate discrete random geometries in higher
dimensions and aim at extending these results to higher dimensions [11] [12]
[13] [14]. However, their continuous limit is singular and of (Hausdorff)
dimension less than 2 [15]. The generated continuous space is called the
branched polymers. These cannot be used for the geometric description of our
4-dimensional space-time.

Let us take a closer look at these tensor models and their universality
classes. By now it is known that the continuous limit of tensor statistical
models is dominated by a class of graphs called melons. They topologically
define a subclass of spheres and form a tree structure that characterizes a
so-called branched polymer phase [16]. It is a striking feature of tensor
patterns that such melons dominate so universal and can be studied
analytically. Nevertheless, in the context of quantum gravity, this class of
universal branched polymers is undesirable, therefore, we aspire to escape
from this branched polymer phase.

In order to improve its critical behavior, an interesting idea then emerged:
one could adjust the scale $N$, or the size of the indices of the tensors,
considering constants of couplings of non-melonic interactions of particular
scales in such a way that a broader class of graphs, including non-melonic
ones, contributes to the critical behavior of tensor models [17]. Thus, making
the graphs that were previously suppressed contribute to the large scale limit
$N$ defines the working hypothesis now put forward. We call the models of such
a program enhanced tensor models, and a model exhibiting a phase transition to
a two-dimensional geometry was found [17]. Therefore, it is possible that
enhanced tensor models help to produce a different phase than branched
polymers. However, notably, the counterpart of these results in quantum field
theory (QFT) still remains a vast territory to explore.

QFT is one of the most successful languages of modern physics. It describes
with incredible precision complex systems with an infinity of degrees of
freedom via the renormalization group. The most illustrious prediction of the
QFT is that of the measurement of the fine structure constant of
electromagnetism. In a different perspective from statistical models whose
degrees of freedom are fixed, the QFT, characterized by a nontrivial
propagator, unfolds a flow of a model from microscopic (ultraviolet) scales to
macroscopic (infrared) scales. We consider this point of view to be relevant
for the generation of random geometries: if the degrees of freedom, or modes
of the tensor fields, are geometric (like spacetime atoms) and the presence of
the nontrivial kinetic term induces the flow from UV to IR moved by dynamics
in the ultraviolet, we consider that going towards the infrared, these degrees
could agglomerate and form new degrees of freedom such as during a phenomenon
of condensation [18] [19] [20] [21] [22] . The fact that some theories of
tensor fields, or QFT of tensor fields, are asymptotically free, and therefore
act like Quantum Chromodynamics, brings one more stone to this building [23]
[24] [25].

Indeed, enhanced tensor field theories are proven renormalizable [26] [27] and
non-melonic graphs contribute to the flow of the renormalization group.
Therefore, they are crucial candidates for generating a geometric phase
different from that of branched polymers and those of 2D geometries.

Studying tensor field theories requires adopting powerful renormalization
methods of QFT to extract their critical behavior. The renormalization of a
tensor field theories is complex because it is a nonlocal field theory [22]
[28] [29] [30] [31] [32]. Additionally, one needs to carefully treat the
dimensions of coupling constants in order to analyze their flows [33] [34]
[35].

In this work, we take further the analysis of [26] and compute
$\beta$-functions of two eTFT models. One model is called $+$ and the second
model is labeled by $\times$. They were shown to be perturbatively just-
renormalizable, at all orders, for given sets of parameters $(d,D,a,b)$, where
$d$ is the tensor field rank, $D$ is the background group dimension, $b$
parametrizes the kinetic term $\sim p^{2b}(\phi_{p})^{2}$, and $a$ determines
the vertex weight $\sim p^{2a}(\phi_{p})^{4}$. The $\beta$-function of model
$+$ is studied for generic rank $d\geq 3$ of the tensor field and
$(D=1,a=(d-2)/2,b=(d-3/2)/2)$. The second model $\times$ has fixed parameters
$(D=1,d=3,a=1/2,b=1)$. Both models show exotic features compared to the
ordinary TFT.

Our results are the following:

(1) In both models, the wave function renormalization is constant $Z=1$ and
therefore the anomalous dimension is vanishing. Although this hints at a
classical behavior, there is still a RG flow; the system of $\beta$-functions
is remarkably simple and explicitly integrable at one loop; we give the
solutions of all equations in terms of the so-called time scale ($\log k$).
Based on the power counting theorem of each model and given the type of
corrections that appear, we conjecture that the system of $\beta$-functions of
each model can be explicitly solved at all orders of perturbation theory. This
is a genuine feature of these quantum models.

(2) In the model $+$, one quartic coupling is still running and integrates as
a linear function of the time scale. There is therefore running of the
coupling and an evolution. The second coupling does not flow: this was
expected as there were no divergent terms associated with that coupling. For
generic non vanishing couplings, this behavior is neither comparable to
asymptotic freedom, asymptotic safety, nor Landau ghost. The model $+$ has a
mass and another 2-point coupling $Z_{a}$ for a quadratic term
$p^{2a}(\phi_{p})^{2}$ (which is not the wave function renormalization, and
was introduced to stabilize the renormalization). They behave just like
relevant operators and tend to blow up in the IR; however, at the contrary of
relevant operators of ordinary QFT, the mass grows linearly in the UV and
therefore is also relevant in that direction. $Z_{a}$ on the other hand, is
suppressed in the UV, so behaves just like a relevant operator in the UV as
well as IR;

(3) Concerning the model $\times$, the quartic couplings do not flow for the
same reason above: no divergences are associated with these couplings.
Inspecting the flow of the mass and the 2-point couplings $Z_{a}$ and $Z_{2a}$
introduced to make sense of a renormalization analysis, they all behave
linearly in terms of the time scale.

The paper is organized as follows. In section 2, we review the essense of [26]
and present the two models named the model $+$ and the model $\times$. In
section 3, we compute the $\beta$-functions of the model $+$ at 1-loop and
interpret the results. The next section 4 presents the same analysis for the
the model $\times$. The conclusion in section 5 provides a summary of our work
and present some perspectives. In appendix A, we detail the integral
approximation techniques that we will use to tackle the spectral sums which
appear in the calculation of the $\beta$-functions. In appendices B and C, we
illustrate the divergent graphs at second order in perturbation theory.

##  2 Review of the renormalizable enhanced quartic TFT

We review here the main results of [26] introducing a class of enhanced
quartic TFTs parametrized by
$(D,d,a,b)\in{\mathbbm{N}}\times{\mathbbm{N}}\times{\mathbbm{R}}_{+}\times{\mathbbm{R}}_{+}$.
In the following sections, we will specify the 4-tuple dealing only with
specific models.

###  2.1 Enhanced TFT models

Consider a field theory defined by a complex function $\phi:(U(1)^{D})^{\times
d}\to{\mathbbm{C}}$. The Fourier transform of this complex field yields a rank
$d$ complex tensor $\phi_{\bf P}$, with ${\bf P}=(p_{1},p_{2},\dots,p_{d})$ a
multi-index. $\bar{\phi}_{\bf P}$ denotes its complex conjugate. Note that in
the notation $\phi_{\bf P}$ the indices $p_{s}$ define themselves multi-
indices:

$p_{s}=(p_{s,1},p_{s,2},\dots,p_{s,D})\,,\;\,p_{s,i}\in{\mathbbm{Z}}\,.$ (1)

A tensor field theory action $S$ is built by a sum of convolutions of the
tensors $\phi_{\bf P}$ and $\bar{\phi}_{\bf P}$:

$\displaystyle S[\bar{\phi},\phi]={\rm Tr}_{2}(\bar{\phi}\cdot{\bf
K}\cdot\phi)+\mu\,{\rm Tr}_{2}(\phi^{2})+S^{{\rm{int\,}}}[\bar{\phi},\phi]\,,$
(2) (3) $\displaystyle{\rm Tr}_{2}(\bar{\phi}\cdot{\bf K}\cdot\phi)=\sum_{{\bf
P},\,{\bf P}^{\prime}}\bar{\phi}_{{\bf P}}\,{\bf K}({\bf P};{\bf
P}^{\prime})\,\phi_{{\bf P}^{\prime}}\,,\qquad{\rm
Tr}_{2}(\phi^{2})=\sum_{{\bf P}}\bar{\phi}_{{\bf P}}\phi_{{\bf P}}\,,$ (4)

and where $S^{{\rm{int\,}}}[\bar{\phi},\phi]$ is another convolution
involving, in the case we are focusing on, 4 tensors.

Hence, giving ${\bf K}$ and $S^{{\rm{int\,}}}[\bar{\phi},\phi]$ entirely
determines the models. For a real parameter $b\geq 0$, we introduce the class
of kernels ${\bf K}={\bf{K}}_{b}({\bf P};{\bf
P^{\prime}})={\boldsymbol{\delta}}_{{\bf P};{\bf P^{\prime}}}{\bf P}^{2b}$
where

$\displaystyle{\boldsymbol{\delta}}_{{\bf P};{\bf
P^{\prime}}}=\prod_{s=1}^{d}\prod_{i=1}^{D}\delta_{p_{s,i},p^{\prime}_{s,i}}\,,\qquad{\bf
P}^{2b}=\sum_{s=1}^{d}|p_{s}|^{2b}\,,\qquad|p_{s}|^{2b}=\sum_{i=1}^{D}|p_{s,i}|^{2b}\,,$
(5)

where $\delta_{p,q}$ denotes the usual Kronecker symbol on ${\mathbbm{Z}}$. It
is clear that ${\bf K}_{b}$ represents a power sum of eigenvalues of $d$
Laplacian operators over the $d$ copies of $U(1)^{D}$ ($b=1$ precisely
corresponds to Laplacian eigenvalues on the torus). Seeking renormalizable
theories [26], more freedom for $b$ values, allowing even values different
from integers, leads to interesting models. In ordinary quantum field theory
(QFT), the restriction $b\leq 1$ ensures the Osterwalder-Schrader (OS)
positivity axiom [36, 37]. In any case, $b$ will be set as a strictly positive
real parameter with no other restriction.

We concentrate on the interaction part. Given a parameter $a\geq 0$, we
distinguish the following quartic interactions:

$\displaystyle{\rm
Tr}_{4;1}(\phi^{4})=\sum_{p_{s},p^{\prime}_{s}\in{\mathbbm{Z}}^{D}}\phi_{12\dots
d}\,\bar{\phi}_{1^{\prime}23\dots
d}\,\phi_{1^{\prime}2^{\prime}3^{\prime}\dots
d^{\prime}}\,\bar{\phi}_{12^{\prime}3^{\prime}\dots d^{\prime}}\,,$ (6)
$\displaystyle{\rm Tr}_{4;1}([p^{2a}+p^{\prime
2a}]\,\phi^{4})=\sum_{p_{s},p^{\prime}_{s}\in{\mathbbm{Z}}^{D}}\Big{(}|p_{1}|^{2a}+|{p^{\prime}}_{1}|^{2a}\Big{)}\phi_{12\dots
d}\,\bar{\phi}_{1^{\prime}23\dots
d}\,\phi_{1^{\prime}2^{\prime}3^{\prime}\dots
d^{\prime}}\,\bar{\phi}_{12^{\prime}3^{\prime}\dots d^{\prime}}\,,$ (7) (8)
$\displaystyle=2\sum_{p_{s},p^{\prime}_{s}\in{\mathbbm{Z}}^{D}}|p_{1}|^{2a}\,\phi_{12\dots
d}\,\bar{\phi}_{1^{\prime}23\dots
d}\,\phi_{1^{\prime}2^{\prime}3^{\prime}\dots
d^{\prime}}\,\bar{\phi}_{12^{\prime}3^{\prime}\dots d^{\prime}}=2\,{\rm
Tr}_{4;1}(p^{2a}\,\phi^{4})\,,$ (9) $\displaystyle{\rm
Tr}_{4;1}([p^{2a}p^{\prime
2a}]\,\phi^{4})=\sum_{p_{s},p^{\prime}_{s}\in{\mathbbm{Z}}^{D}}\Big{(}|p_{1}|^{2a}|{p^{\prime}}_{1}|^{2a}\Big{)}\phi_{12\dots
d}\,\bar{\phi}_{1^{\prime}23\dots
d}\,\phi_{1^{\prime}2^{\prime}3^{\prime}\dots
d^{\prime}}\,\bar{\phi}_{12^{\prime}3^{\prime}\dots d^{\prime}}\,.$ (10)

where the notation $\phi_{12\dots d}$ stands for
$\phi_{p_{1},p_{2},\dots,p_{d}}=\phi_{\bf P}$. In the above equations (6), (9)
and (10), the color index 1 plays a special role. The sum over all possible
color configurations delivers a colored symmetric interactions:

$\displaystyle{\rm Tr}_{4}(\phi^{4}):={\rm Tr}_{4;1}(\phi^{4})+{\rm Sym}(1\to
2\to\dots\to d)\,,$ (11) $\displaystyle{\rm Tr}_{4}(p^{2a}\,\phi^{4}):={\rm
Tr}_{4;1}(p^{2a}\,\phi^{4})+{\rm Sym}(1\to 2\to\dots\to d)\,,$ (12)
$\displaystyle{\rm Tr}_{4}([p^{2a}p^{\prime 2a}]\,\phi^{4}):={\rm
Tr}_{4;1}([p^{2a}p^{\prime 2a}]\,\phi^{4})+{\rm Sym}(1\to 2\to\dots\to d)\,.$
(13)

One might consider the momentum weights in the above interactions as
derivative couplings for particular choices of $a$. Seeking just
renormalizable models in our theory space, we will constrain $a$ a-posteriori
to some particular values. It turns out that the two interactions (12) and
(13) generate two new 2-point diverging graphs that neither the mass nor the
wave function renormalization can absorb. They are of the form:

$\displaystyle{\rm Tr}_{2}(p^{2\xi}\phi^{2})={\rm Tr}_{2}(\bar{\phi}\cdot{\bf
K}_{\xi}\cdot\phi)\,,\qquad\xi=a,2a\,,$ (14)

In [26], the authors handled these by adding them in kinetic term, in addition
to the former term ${\rm Tr}_{2}(p^{2b}\phi^{2})$. The models that proves
renormalizable have the following kinetic terms and interactions:

$\displaystyle\text{model }+:$ $\displaystyle
S^{{\rm{int\,}}}_{+}[\bar{\phi},\phi]=\frac{\lambda}{2}\,{\rm
Tr}_{4}(\phi^{4})+\frac{\lambda_{+}}{2}\,{\rm
Tr}_{4}(p^{2a}\,\phi^{4})+CT_{2}[\bar{\phi},\phi]+\sum_{\xi=a,b}CT_{2;\xi}[\bar{\phi},\phi]$
(17) $\displaystyle
S^{{\rm{kin\,}}}_{+}[\bar{\phi},\phi]=\sum_{\xi=a,b}Z_{\xi}{\rm
Tr}_{2}(p^{2\xi}\phi^{2})+\mu{\rm Tr}_{2}(\phi^{2})\,,$
$\displaystyle\text{model }\times:$ $\displaystyle
S^{{\rm{int\,}}}_{\times}[\bar{\phi},\phi]=\frac{\lambda}{2}\,{\rm
Tr}_{4}(\phi^{4})+\frac{\lambda_{\times}}{2}\,{\rm Tr}_{4}([p^{2a}p^{\prime
2a}]\,\phi^{4})+CT_{2}[\bar{\phi},\phi]+\sum_{\xi=a,2a,b}CT_{2;\xi}[\bar{\phi},\phi]\,,$
(20) $\displaystyle
S^{{\rm{kin\,}}}_{\times}[\bar{\phi},\phi]=\sum_{\xi=a,2a,b}Z_{\xi}{\rm
Tr}_{2}(p^{2\xi}\phi^{2})+\mu{\rm Tr}_{2}(\phi^{2})$

where $\lambda$, $\lambda_{+}$, $\lambda_{\times}$ are coupling constants,
$CT_{\cdot}$ are counterterms, $\mu$ is the mass coupling, $Z_{a}$ and
$Z_{2a}$ are other kinetic term couplings, and $Z_{b}$ will be called the wave
function renormalization.

Note that we could be also interested in a model $\lambda=\lambda_{+}$, in
which case these couplings merge in a single one. We will address this after
the extraction of the $\beta$-function.

Another important issue is the following: the choice of modifying the
covariance of the theory using a kinetic term as $\sum_{\xi}Z_{\xi}{\rm
Tr}_{2}(p^{2\xi}\phi^{2})+\mu{\rm Tr}_{2}(\phi^{2})$ make computations
lengthier. Indeed, the analysis of the amplitude could be made easier by
keeping a kinetic term as $Z_{b}{\rm Tr}_{2}(p^{2b}\phi^{2})+\mu{\rm
Tr}_{2}(\phi^{2})$ and let the extra term $Z_{a}{\rm Tr}_{2}(p^{2a}\phi^{2})$
in the interaction part. This makes the covariance much simpler. We will show
that both model share the same power counting and the renormalization analysis
applies equally well on both. The $\beta$-function will be computed with the
second kind of models (with simpler covariance).

###  2.2 Amplitudes and renormalizability

The models $+$ and $\times$ associated with the actions given by (17) and
(20), respectively, give the quantum models determined by the partition
function

$Z_{\bullet}=\int
d\nu_{C_{\bullet}}(\bar{\phi},\phi)\;e^{-S^{{\rm{int\,}}}_{\bullet}[\bar{\phi},\phi]}\,,$
(21)

where $\bullet=+,\times$, and $d\nu_{C_{\bullet}}(\bar{\phi},\phi)$ is a field
Gaussian measure with covariance $C_{\bullet}$ given by the inverse of the
kinetic term:

$C_{\bullet}({\bf P};{\bf P^{\prime}})=\tilde{C}_{\bullet}({\bf
P})\,{\boldsymbol{\delta}}_{{\bf P},{\bf
P}^{\prime}}\,,\qquad\tilde{C}_{\bullet}({\bf P})\,=\frac{1}{\sum_{\xi}{\bf
P}^{2\xi}+\mu}\,.$ (22)

where, if $\bullet=+$, $\xi=a,b$ and if $\bullet=\times$, $\xi=a,2a,b$.

Feynman graphs in TFTs. Feynman graphs of TFTs and enhanced TFTs have two
equivalent representations. One is called “stranded graph” representation and
incorporates more details of the structure of the Feynman graph. The other
representation of a Feynman graph in this theory is a bipartite colored graph
[38, 39].

The propagator is drawn as a set of $d$ non-intersecting segments called
strands, or as a dotted line (see Figure 1).

$p_{1}$$p_{2}$$\vdots$$p_{d}$ Figure 1: Two ways of representing the
propagator of the theory.

In Figure 2, an interaction is pictured as a stranded vertex (pictures above)
or by a $d$-regular colored bipartite graph (pictures below). We list therein
all possible quadratic and quartic vertices and a bold line represents a
momentum weight of the term.

Figure 2:  Rank $d$ vertices of the mass, $\phi^{2}$\- and $\phi^{4}$-terms.

In Figure 3, we give some examples of two 4-pt graphs.

Figure 3:  Rank $d=3$ Feynman graphs.

At the perturbative level, we compute the amplitude of a Feynman graph
${\mathcal{G}}({\mathcal{V}},{\mathcal{L}})$ with set of vertices
${\mathcal{V}}$ and set of propagator line ${\mathcal{L}}$, in the standard
way:

$\displaystyle A_{{\mathcal{G}}}(\\{p_{{\rm{ext\,}}}\\})=\sum_{{\bf
P}_{v}}\prod_{l\in{\mathcal{L}}}C_{\bullet,l}({\bf P}_{v},{\bf
P}^{\prime}_{v^{\prime}})\prod_{v\in{\mathcal{V}}}(-{\bf V}_{v}({\bf P}_{v}))$
(23)

where $C_{\bullet,l}$ is a propagator with line index $l$, ${\bf V}_{v}({\bf
P}_{v})$ is a given vertex weight that contains a coupling constant but also a
momentum weight if the vertex $v$ is enhanced. The sum is performed over
internal momenta and will follow the ordinary momentum routine whereas the set
$\\{p_{{\rm{ext\,}}}\\}$ defines external momenta that are not summed over.

Some notation is needed to distinguish the different vertices and their weight
that we shall deal with:

\- the set ${\mathcal{V}}_{4;s}$ of vertices ${\rm Tr}_{4;s}(\phi^{4})$ with
color $s$ and with kernel ${\bf V}_{4;s}$,
${\mathcal{V}}_{4}=\sqcup_{s=1}^{d}{\mathcal{V}}_{4;s}$ (disjoint union
notation);

\- the set ${\mathcal{V}}_{+;4;s}$ of vertices ${\rm
Tr}_{4;s}(p^{2a}\,\phi^{4})$ with color $s$ and with vertex kernel ${\bf
V}_{+;4;s}$, ${\mathcal{V}}_{+;4}=\sqcup_{s=1}^{d}{\mathcal{V}}_{+;4;s}$;

\- the set ${\mathcal{V}}_{\times;4;s}$ of vertices ${\rm
Tr}_{4;s}(([p^{2a}p^{\prime 2a}]\,\phi^{4})$ with vertex kernel ${\bf
V}_{+;4;s}$,
${\mathcal{V}}_{\times;4}=\sqcup_{s=1}^{d}{\mathcal{V}}_{\times;4;s}$;

\- the set ${\mathcal{V}}_{2}$ of mass vertices with kernel ${\bf V}_{2}$, the
set ${\mathcal{V}}_{2;\xi;s}$ of vertices $Z_{\xi}{\rm
Tr}_{2}(p^{2\xi}\phi^{2})$ each kind corresponding to a kernel ${\bf
V}_{2;\xi;s}$, ${\mathcal{V}}_{2;s}=\cup_{\xi}{\mathcal{V}}_{2;\xi;s}$.

We denote the cardinalities $|{\mathcal{V}}_{4;s}|=V_{4;s}$,
$|{\mathcal{V}}_{4}|=V_{4}$, $|{\mathcal{V}}_{\bullet;4;s}|=V_{\bullet;4;s}$,
$|{\mathcal{V}}_{\bullet;4}|=V_{\bullet;4}$, $\bullet=+,\times$;
$|{\mathcal{V}}_{2;\xi;s}|=V_{2;\xi;s}$, $V_{2;\xi}=\sum_{s}V_{2;\xi;s}$.
Then,
${\mathcal{V}}=\sqcup_{s=1}^{d}({\mathcal{V}}_{4;s}\cup{\mathcal{V}}_{\bullet;4;s}\cup{\mathcal{V}}_{2;s})$,
$|{\mathcal{V}}|=V$.

Power counting theorems and list of divergent graphs. We restrict now to
primitively divergent graphs. Given $a\leq b$, the degree of divergence of a
graph amplitude of the model $+$ is given by

$\displaystyle\omega_{{\rm d};+}({\mathcal{G}})=$ (24)
$\displaystyle-{2D\over(d-1)!}(\omega({\mathcal{G}}_{\rm
color})-\omega(\partial{\mathcal{G}}))-D(C_{\partial{\mathcal{G}}}-1)-{1\over
2}\left[(D\,(d-1)-2b)N_{{\rm{ext\,}}}-2D\,(d-1)\right]$ (25)
$\displaystyle+{1\over 2}\left[-2D\,(d-1)+(D\,(d-1)-2b)n\right]\cdot
V+2a\rho_{+}+2a\rho_{2;a}+2b\rho_{2;b}\,.$ (26)

where $\omega({\mathcal{G}}_{\rm color})$ is the Gurau degree [40, 41] of the
extended colored graph ${\mathcal{G}}_{\rm color}$ of ${\mathcal{G}}$,
$\partial{\mathcal{G}}$ is the boundary graph of ${\mathcal{G}}$ [22],
$C_{\partial{\mathcal{G}}}$ is the number of connected component of
$\partial{\mathcal{G}}$, $N_{{\rm{ext\,}}}$ is the number of external legs of
the diagram ${\mathcal{G}}$, $n\cdot V=\sum_{k}kV_{k}$ where $k$ is the
valence of the vertex $V_{k}$, $\rho_{+}$ is the number of times that a
momentum is enhanced passing through a strand of an enhanced vertex
${\mathcal{V}}_{\bullet;4}$; likewise $\rho_{2;\xi}$, $\xi=a,b$ is the number
of times that a momentum gets an enhancement passing through a vertex
${\mathcal{V}}_{2;\xi;s}$.

Concerning the model $\times$, one obtains the degree of divergence in this
model valid for $3a\leq 2b$,

$\displaystyle\omega_{{\rm d};\times}({\mathcal{G}})$ $\displaystyle=$
$\displaystyle-{2D\over(d^{-})!}(\omega({\mathcal{G}}_{\rm
color})-\omega(\partial{\mathcal{G}}))-D(C_{\partial{\mathcal{G}}}-1)-{1\over
2}\left[(D\,d^{-}-2b)N_{{\rm{ext\,}}}-2D\,d^{-}\right]$ (28)
$\displaystyle+{1\over 2}\left[-2D\,d^{-}+(D\,d^{-}-2b)n\right]\cdot
V+2a\rho_{\times}+\sum_{\xi=a,2a,b}2\xi\rho_{2;\xi}\,.$

The following statement has been proved in [26].

######  Proposition 1 (List of primitively divergent graphs for the model
$+$).

The $p^{2a}\phi^{4}$-model $+$ with parameters
$a=D(d^{-}-1)/2,b=D(d^{-}-\frac{1}{2})/2$ for two integers $d>2$ and $D>0$,
has primitively divergent graphs with
$(\Omega({\mathcal{G}})=\omega(\mathcal{G}_{\text{color}})-\omega({\partial\mathcal{G}}))$:

${\mathcal{G}}$ |  | $N_{{\rm{ext\,}}}$ |  | $V_{2}$ |  | $V_{2;a}$ |  | $V_{4}$ |  | $\rho_{+}$ |  | $C_{{\partial\mathcal{G}}}-1$ |  | $\Omega({\mathcal{G}})$ |  | $\omega_{{\rm d};+}({\mathcal{G}})$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | 4 |  | 0 |  | 0 |  | 0 |  | $V_{+;4}$ |  | 0 |  | 1 |  | 0
I |  | 2 |  | 0 |  | 0 |  | 0 |  | $V_{+;4}$ |  | 0 |  | 1 |  | ${D\over 2}$
II |  | 2 |  | 0 |  | 0 |  | 0 |  | $V_{+;4}-1$ |  | 0 |  | 0 |  | ${D\over 2}$
III |  | 2 |  | 0 |  | 0 |  | 1 |  | $V_{+;4}$ |  | 0 |  | 0 |  | ${D\over 2}$
IV |  | 2 |  | 0 |  | 1 |  | 0 |  | $V_{+;4}$ |  | 0 |  | 1 |  | $0$
V |  | 2 |  | 0 |  | 1 |  | 0 |  | $V_{+;4}-1$ |  | 0 |  | 0 |  | $0$
VI |  | 2 |  | 0 |  | 1 |  | 1 |  | $V_{+;4}$ |  | 0 |  | 0 |  | $0$
Table 1: List of primitively divergent graphs of the $p^{2a}\phi^{4}$-model
$+$.

Having a look at Table 1 and the row labeled by $\Omega({\mathcal{G}})$, we
see that the dominant graphs are not those labeled by
$\Omega({\mathcal{G}})=0$ which are the melonic diagrams, but are those that
are $\Omega({\mathcal{G}})=1$ and thus non-melonic graphs. This shows that the
model $+$ delivers the expected output.

######  Theorem 1.

The $p^{2a}\phi^{4}$ model $+$ with parameters
$a=D(d^{-}-1)/2,b=D(d^{-}-\frac{1}{2})/2$ for arbitrary rank $d\geq 3$ and
dimension $D>0$ with action defined by (2) is just-renormalizable at all
orders of perturbation theory.

######  Proposition 2 (List of primitively divergent graphs for the model
$\times$).

The $p^{2a}\phi^{4}$-model $\times$ with parameters $D=1,d=3,a={1\over
2},b=1$, has the following primitively divergent graphs which obey
$(\Omega({\mathcal{G}})=\omega(\mathcal{G}_{\text{color}})-\omega({\partial\mathcal{G}}))$

${\mathcal{G}}$ |  | $N_{{\rm{ext\,}}}$ |  | $V_{2}$ |  | $V_{2;a}$ |  | $V_{4}$ |  | $\rho_{\times}$ |  | $C_{{\partial\mathcal{G}}}-1$ |  | $\Omega({\mathcal{G}})$ |  | $\omega_{{\rm d};\times}({\mathcal{G}})$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
I |  | 2 |  | 0 |  | 0 |  | 0 |  | $2V_{\times;4}-1$ |  | 0 |  | 1 |  | ${0}$
II |  | 2 |  | 0 |  | 0 |  | 0 |  | $2V_{\times;4}-2$ |  | 0 |  | 0 |  | ${0}$
III |  | 2 |  | 0 |  | 0 |  | 1 |  | $2V_{\times;4}$ |  | 0 |  | 0 |  | ${0}$
Table 2: List of primitively divergent graphs of the $p^{2a}\phi^{4}$-model
$\times$.

######  Theorem 2.

The $p^{2a}\phi^{4}$ model $\times$ with parameters $D=1,d=3,a={1\over
2},b=1$, with the action defined by (2) is renormalizable at all orders of
perturbation.

This model $\times$ has an unexpected behavior: it does not have divergent
4pt-graphs, only log-divergent 2pt-graphs. One may ask if this is a super-
renormalizable model with a finite number of divergent graphs. The answer is
no because the model has infinite terms participating in the mass flow. The
property can be regarded as a particularity entailed by both the nonlocality
and the presence of enhanced vertices which make the mass behave like a
marginal coupling.

###  2.3 An alternative enhanced TFT model

As stated previously, the presence of the enhanced vertices generate new 2pt-
terms with external data following the pattern of ${\rm
Tr}_{2}(p^{2a}\phi^{2})$ for the model $+$, and the patterns of ${\rm
Tr}_{2}(p^{2a}\phi^{2})$ and ${\rm Tr}_{2}(p^{4a}\phi^{2})$ for the model
$\times$. Consequently, in [26], the authors have modified the covariance and
performed the renormalization analysis. We present here an alternative way of
dealing with these terms that make the whole simpler. We simply demand that
the new terms are interactions and therefore propose the following models: (in
the following, we use the same notation as in the previous sections as no
confusion may arise)

$\displaystyle\text{model }+:$ $\displaystyle
S^{{\rm{int\,}}}_{+}[\bar{\phi},\phi]=\frac{\lambda}{2}\,{\rm
Tr}_{4}(\phi^{4})+\frac{\lambda_{+}}{2}\,{\rm
Tr}_{4}(p^{2a}\,\phi^{4})+Z_{a}{\rm Tr}_{2}(p^{2}\phi^{2})$ (32)
$\displaystyle+CT_{2}[\bar{\phi},\phi]+CT_{2;b}[\bar{\phi},\phi]$
$\displaystyle S^{{\rm{kin\,}}}_{+}[\bar{\phi},\phi]=Z_{b}{\rm
Tr}_{2}(p^{2b}\phi^{2})+\mu{\rm Tr}_{2}(\phi^{2})\,,$
$\displaystyle\text{model }\times:$ $\displaystyle
S^{{\rm{int\,}}}_{\times}[\bar{\phi},\phi]=\frac{\lambda}{2}\,{\rm
Tr}_{4}(\phi^{4})+\frac{\lambda_{\times}}{2}\,{\rm Tr}_{4}([p^{2a}p^{\prime
2a}]\,\phi^{4})+\sum_{\xi=a,2a}Z_{\xi}{\rm Tr}_{2}(p^{2\xi}\phi^{2})$ (37)
$\displaystyle+CT_{2}[\bar{\phi},\phi]+CT_{2;b}[\bar{\phi},\phi]\,,$
$\displaystyle S^{{\rm{kin\,}}}_{\times}[\bar{\phi},\phi]=Z_{b}{\rm
Tr}_{2}(p^{2b}\phi^{2})+\mu{\rm Tr}_{2}(\phi^{2})$

Considering this proposal, the covariance of these models is unique and given
by

$C({\bf P};{\bf P^{\prime}})=\tilde{C}({\bf P})\,{\boldsymbol{\delta}}_{{\bf
P},{\bf P}^{\prime}}\,,\qquad\tilde{C}({\bf P})\,=\frac{1}{{\bf
P}^{2b}+\mu}\,.$ (38)

The difference between this propagator (38) and the former (22) is that, for a
given strand, the momentum which was previously $\sum_{\xi}p_{s}^{2\xi}$
becomes $p_{s}^{2b}$. The multiscale analysis of [26] with appropriate
parameter $M^{2b}$ ($M>1$) can be mimicked with no difficulty to obtain the
sliced propagator as

$\displaystyle C_{0}\leq K\,,\qquad\quad C_{i}({\bf P})\leq KM^{2bi}e^{\delta
M^{-bi}({\bf P}^{b}+\mu)}$ (39)

for $i>0$ a high slice index, and for some constant $K$ and $\delta$.

To write the multiscale amplitude where each propagator lives in an arbitrary
slice is identical to the previous analysis [26] for both the model $+$ and
the model $\times$. We need a convenient bound on that quantity. One should
arrive at the expression of a sliced amplitude according to a momentum
attribution $\boldsymbol{\mu}=(i_{1},\dots i_{|{\mathcal{L}}|})$:

$\displaystyle|A_{{\mathcal{G}};\boldsymbol{\mu}}|\leq\tilde{K}\prod_{l\in{\mathcal{L}}}M^{-2bi_{l}}\sum_{p_{f_{s}}}\prod_{f_{s}\in\
F_{{\rm{int\,}}}}e^{-\delta(\sum_{l\in
f_{s}}M^{-bi_{l}})|p_{f_{s}}^{b}|}\prod_{i=1}^{d}\prod_{v_{s}\in{\mathcal{V}}_{\bullet;4;s}}{\rm
weight}(v_{s})$ (40)

where $\tilde{K}$ is a constant, and weight$(v_{s})$ is self-explanatory and
is identical to the former case.

The rest of the analysis consists in performing the spectral sums
$\sum_{p_{f_{s}}}(\cdot)$. A close inspection of the method introduced in [26]
shows that one condition is imposed and gets rid of $a$ at leading order. The
parameter $a$ does not contribute to the sum. We have at $a\geq 0$:

$\displaystyle\sum_{p_{1};\dots
p_{D}=1}^{\infty}(\sum_{l=1}^{D}p_{l}^{c})^{n}e^{-B(p^{b}+p^{a})}=kB^{-\frac{cn+D}{b}}e^{-B^{1-\frac{a}{b}}}(1+{\mathcal{O}}(B^{\frac{1}{b}}))$
(41)

Requesting $a\leq b$ for the model $+$ and $3a\leq 2b$ for the model $\times$
removes the parameter $a$ from the expansion at leading order in $B$. Then,
the remaining part of the momentum integration exactly performs in the same
way. We therefore reach the same power counting, the same list of divergent
graphs, Propositions 1 and 2 are valid for the models $+$ and $\times$,
respectively. We perform the same subtraction procedure making the models (32)
and (37) renormalizable at all orders of perturbation theory. Theorems 1 and 2
hold for the models $+$ and $\times$, respectively.

From this point onwards, we consider the models (32) and (37). To simplify the
determination of the graph combinatorial factors, we will distinguish all
couplings by providing them with colors. For instance, for the model $+$, we
write:

$\displaystyle
S^{\prime{\rm{int\,}}}_{+}[\bar{\phi},\phi]:=\sum_{c=1}^{d}\frac{\lambda^{(c)}}{2}\;{\rm
Tr}_{4;c}(\phi^{4})+\sum_{c=1}^{d}\frac{\lambda^{(c)}_{+}}{2}\;{\rm
Tr}_{4;c}(p^{2a}\,\phi^{4})+\sum_{c=1}^{d}Z_{a}^{(c)}{\rm
Tr}_{2;c}(p^{2a}\phi^{2})\,.$ (42)

In the end, the $\beta$-functions of $\lambda$, $\lambda_{+}$, and $Z_{a}$
will be directly inferred by letting $\lambda^{(c)}\to\lambda$,
$\lambda^{(c)}_{+}\to\lambda_{+}$ and $Z_{a}^{(c)}\to Z_{a}$. The same will be
done for the model $\times$.

##  3  One-loop beta-functions of the model $+$

This section now addresses a first set of new results in this contribution: we
determine the RG flow at 1-loop of the model $+$. We compute the
$\beta$-function of the coupling $\lambda$, the coupling $Z_{a}$ and the mass.
Since, the action contains a vertex weight and an extra quadratic coupling, we
carefully carry out the formalism of finding the effective action and write
the resulting RG flow equations.

###  3.1 Effective action

The presence of several coupling constants in our theory urges us to handle
the effective action with care. We will compute the renormalization of
coupling constants via multiscale analysis. We start with performing a slice
decomposition of the covariance. We let $M>1$ a positive real number and we
define the (sharp) cutoff functions as

$\chi^{0}(\alpha)=\begin{cases}1&\text{if $1\leq\alpha$}\\\ 0&\text{if
$1>\alpha$}\end{cases}\,,$ (43)

$\forall i>0$,

$\chi^{i}(\alpha)=\begin{cases}0&\text{if $\alpha\leq M^{-2bi}$}\\\ 1&\text{if
$M^{-2bi}<\alpha\leq M^{-2b(i-1)}$}\\\ 0&\text{if
$M^{-2b(i-1)}<\alpha$}\end{cases}$ (44)

The presence of $2b$ in the cutoff function is related to the propagator
momentum power and has a normalization effect for the degree of divergence
$\omega_{{\rm d};+}({\mathcal{G}})$. Then we write the covariance in (22) as
expressed in Schwinger parametrization,

$C({\bf P};{\bf P^{\prime}})=\tilde{C}({\bf P})\,{\boldsymbol{\delta}}_{{\bf
P},{\bf P}^{\prime}}\,,\qquad\tilde{C}({\bf P})\,=\frac{1}{{\bf
P}^{2b}+\mu}=\sum_{i=0}^{\infty}\tilde{C}_{i}({\bf P})\,,$ (45)

with

$\tilde{C}_{i}({\bf P})=\int_{0}^{\infty}d\alpha\,e^{-\alpha({\bf
P}^{2b}+\mu)}\chi^{i}(\alpha)\,.$ (46)

We integrate out the fields at high scales greater than $i$ and include their
effects in the effective action $W^{i}$. As written below,

$Z=\int d\nu_{C_{\leq i}}(\bar{\phi}_{\leq i},{{\phi}}_{\leq
i})\,e^{-W^{i}(\bar{\phi}_{\leq i},\,{{\phi}}_{\leq i})}\,,$ (47)

where

$C_{\leq i}({\bf P};{\bf P}^{\prime})={\boldsymbol{\delta}}_{{\bf P},{\bf
P}^{\prime}}\sum_{j\leq i}{\tilde{C}}_{j}({\bf P})\,.$ (48)

Following Wilsonian renormalization group idea, after integrating up to the
scale $i$, we continue integrating the slice $i$ in order to obtain an
effective action at scale $i-1$. We can do this by using the property of
Gaussian measure; we can readily decompose $C_{\leq i}=C_{i}+C_{\leq i-1}$ and
the corresponding fields $\phi_{\leq i}=\psi_{i}+\phi_{\leq i-1}$
(${\bar{\phi}}_{\leq i}={\bar{\psi}}_{i}+{\bar{\phi}}_{\leq i-1}$). Then the
partition function becomes

$Z=\int d\nu_{C_{\leq i-1}}(\bar{\phi}_{\leq i-1},{\phi}_{\leq
i-1})e^{-W^{i-1}(\bar{\phi}_{\leq i-1},\,{\phi}_{\leq i-1})}\,,$ (49)

where the effective action at scale $i-1$ is given by

$e^{-W^{i-1}(\bar{\phi}_{\leq i-1},\,{\phi}_{\leq i-1})}=\int
d\nu_{C_{i}}(\bar{\psi}_{i},{\psi}_{i})e^{-W^{i}(\bar{\psi}_{i}+\bar{\phi}_{\leq
i-1},\,{\psi}_{i}+{\phi}_{\leq i-1})}\,,$ (50)

Note that here, in the case that the theory is renormalizable, one can assert
the effective action at any scale $i$ takes the same form as the interaction
action,

$W^{i}(\bar{\phi}_{\leq i},\phi_{\leq i})=S^{{\rm{int\,}}}(\bar{\phi}_{\leq
i},\phi_{\leq i})\,,$ (51)

We can, then, write formally:

$\displaystyle-W^{i-1}(\bar{\phi}_{\leq i-1},{\phi}_{\leq i-1})={\rm
Tr}_{2}(\bar{\phi}_{\leq i-1}\cdot\Sigma\cdot\phi_{\leq i-1})+\frac{1}{2}{\rm
Tr}_{4}({\phi^{4}_{\leq i-1}}\cdot\Gamma_{4})+R(\phi_{\leq i-1})\,,$ (52)

where $\Sigma(\\{p\\})$ is the sum over all amputated 1PI 2-pt graphs,
$\Gamma_{4}(\\{p\\})$ is the sum of 1PI 4-pt graphs following the pattern of
${\rm Tr}_{4}(\phi^{4})$, and $R(\phi_{\leq i-1})$ is the rest of the terms
containing 1PR graphs (they do not contribute to the iteration process) and
the finite terms. In the above equation, $\Sigma$ and $\Gamma_{4}$ are kernels
that are convoluted with the tensors fields.

We separate the graph amplitudes into local and nonlocal parts, therefore,

$\Sigma(\\{p\\})=\Sigma(\\{0\\})+\sum_{c}|p_{c}|^{2b}\partial_{|p_{c}|^{2b}}\Sigma\big{|}_{\\{p\\}=0}+\sum_{c}|p_{c}|^{2a}\partial_{|p_{c}|^{2a}}\Sigma\big{|}_{\\{p\\}=0}+\cdots$
(53)

As a result of [26], the renormalization analysis of the model dictated by the
rows of I, III, IV, and VI in Table 1 proved that
$\partial_{|p_{c}|^{2b}}\Sigma\big{|}_{\\{p\\}=0}=0$ and the rest of the terms
in $\cdots$ are finite, whereas the mass renormalization $\Sigma(\\{0\\})$ and
$\partial_{|p_{c}|^{2a}}\Sigma|_{\\{p\\}=0}\equiv\Gamma^{(c)}_{2}(\\{0\\})$
are divergent. $|p_{c}|^{2a}\Gamma^{(c)}_{2}(\\{p\\})$ is the sum of all
amputated 1PI (one-particle-irreducible) 2pt-functions following the pattern
of ${\rm Tr}_{2}(p_{c}^{2a}\phi^{2})$ on their boundary graphs as dictated by
II and V rows of Table 1. The boundary of the graphs contributing to this
function are all melonic without the $|p|^{2a}$-enhancement (see Proposition 1
and lines with $\Omega({\mathcal{G}})=0$).

Similarly, one can expand the contribution coming from $4$-point function

$\Gamma_{4}(\\{p\\})=\sum_{c}\Big{\\{}\Gamma_{4}^{(c)}(\\{0\\})+|p_{c}|^{2a}\partial_{|p_{c}|^{2a}}\Gamma_{4}^{(c)}\Big{|}_{\\{p\\}=0}+|p_{c}|^{2b}\partial_{|p_{c}|^{2b}}\Gamma_{4}^{(c)}\Big{|}_{\\{p\\}=0}\Big{\\}}+\cdots\,,$
(54)

where $\sum_{c}\Gamma_{4}^{(c)}(\\{0\\})\equiv\Gamma_{4}(\\{0\\})$ is the sum
of all amputated 1PI 4pt-functions following the pattern of ${\rm
Tr}_{4}(\phi^{4})$ on their boundary graphs as dictated by the first row of
Table 1. We define
$\partial_{|p_{c}|^{2a}}\Gamma^{(c)}_{4}\big{|}_{\\{p\\}=0}\equiv\Gamma_{4;+}^{(c)}(\\{0\\})$
which are all amputated 1PI 4pt-functions following the pattern of ${\rm
Tr}_{4;c}(p^{2a}\phi^{4})$, with characteristics given by the first row of
Table 1 and having a boundary with external $|p|^{2a}$-enhancement (see
Proposition 1). In fact, from Proposition 1, there is only the leading order
${\mathcal{O}}(\lambda_{+})$ contribution in $\Gamma_{4;+}^{(c)}(\\{0\\})$ and
there are no contributions from higher orders in perturbation theory in
$\lambda_{+}$. Also, from the Proposition 1, the rest denoted by $\cdots$ and
$\partial_{|p_{c}|^{2b}}\Gamma_{4}^{(c)}\big{|}_{\\{p\\}=0}$ are finite.

Now, reorganizing and absorbing all the finite parts into
${\tilde{R}}({\phi_{\leq i-1}})$, however intentionally leaving the term with
$\partial_{|p_{c}|^{2b}}\Sigma\big{|}_{\\{p\\}=0}$ even though it is zero,

$\displaystyle-W^{i-1}(\phi_{\leq i-1},{\bar{\phi}}_{\leq i-1})$
$\displaystyle=$ $\displaystyle\Sigma_{i-1}(\\{0\\}){\rm
Tr}_{2}(\phi^{2}_{\leq
i-1})+(\partial_{|{p_{c}|}^{2b}}\Sigma\big{|}_{\\{p\\}=0})_{i-1}{\rm
Tr}_{2}(p^{2b}\phi^{2}_{\leq i-1})$ (55)
$\displaystyle+\sum_{c}\Gamma^{(c)}_{2,\;i-1}(\\{0\\}){\rm
Tr}_{2;c}(p^{2a}\phi^{2}_{\leq
i-1})+\sum_{c}\frac{\Gamma^{(c)}_{4,\;i-1}(\\{0\\})}{2}{\rm
Tr}_{4;c}({\phi^{4}_{\leq i-1}})$
$\displaystyle+\sum_{c}\frac{\Gamma^{(c)}_{4;+,\;i-1}(\\{0\\})}{2}{\rm
Tr}_{4;c}(p^{2a}\phi^{4}_{\leq i-1})+{\tilde{R}}(\phi_{\leq i-1})\,.$

The effective theory is then defined by a new measure given by

$\displaystyle d\nu_{\tilde{C}^{i-1}(\phi_{\leq
i-1})}\exp\Big{[}{\Sigma_{i-1}(\\{0\\}){\rm Tr}_{2}(\phi_{\leq
i-1}^{2})+\sum_{c}(\partial_{|p_{c}|^{2b}}\Sigma|_{\\{p\\}=0})_{i-1}{\rm
Tr}_{2}(p_{c}^{2b}\phi_{\leq i-1}^{2})}\Big{]}\,,$ (56)

which is still a Gaussian measure. Let us compute the new covariance for the
above Gaussian measure,

$\displaystyle\frac{1}{Z_{b,\,i-1}}\int_{0}^{\infty}d\alpha\,e^{-\alpha(|p|^{2b}+\mu_{{\rm
ren},i-1})}\chi^{i-1}(\alpha)=\frac{1}{Z_{b,\,i-1}}{\tilde{C}}^{i-1}(p)\,,$
(57)

where we defined the renormalized mass $\mu_{{\rm ren},i-1}$ to be

$\mu_{{\rm ren},i-1}=\frac{\mu_{i-1}-\Sigma_{i-1}(\\{0\\})}{Z_{b,\,i-1}}\,,$
(58)

and the wave function renormalization to be

$Z_{b,\,i-1}\equiv
1+(\partial_{|p_{c}|^{2b}}\Sigma\big{|}_{\\{p\\}=0})_{i-1}\,.$ (59)

Note that the color dependence on
$(\partial_{|p_{c}|^{2b}}\Sigma\big{|}_{\\{p\\}=0})$ should be actually
absent. Then, the effective theory for $\phi_{\leq i-1}$ can be written as

$\displaystyle d\nu_{\frac{1}{Z_{b,\,i-1}}{\tilde{C}^{i-1}}}(\phi_{\leq i-1})$
$\displaystyle\exp$
$\displaystyle\Big{[}\sum_{c}\Gamma^{(c)}_{2,\,i-1}(\\{0\\}){\rm
Tr}_{2}(p_{c}^{2a}\phi_{\leq
i-1}^{2})+\sum_{c}\frac{\Gamma^{(c)}_{4,\,i-1}(\\{0\\})}{2}{\rm
Tr}_{4}(\phi_{\leq i-1}^{4})$ (61)
$\displaystyle+\sum_{c}\frac{\Gamma^{(c)}_{4;+,\,i-1}(\\{0\\})}{2}{\rm
Tr}_{4}(p_{c}^{2a}\phi_{\leq i-1}^{4})+{\tilde{R}}(\phi_{\leq i-1})\Big{]}\,.$

With a field rescaling $\phi_{\leq i-1}\rightarrow\sqrt{Z_{b,\,i-1}}\phi_{\leq
i-1}$ (which in our specific case, there is no actual rescaling because
$Z_{b,\,i-1}=1$ and trivial), the effective theory for $\phi_{\leq i-1}$ can
be recast:

$\displaystyle d\nu_{{\tilde{C}}_{i-1}}(\phi_{\leq i-1})$ $\displaystyle{\rm
exp}$
$\displaystyle\Big{[}\sum_{c}\frac{\Gamma^{(c)}_{2,i-1}(\\{0\\})}{Z_{b,i-1}}{\rm
Tr}_{2;c}(p^{2a}\phi_{\leq
i-1}^{2})+\sum_{c}\frac{\Gamma^{(c)}_{4,i-1}(\\{0\\})}{{2}Z_{b,i-1}^{2}}{\rm
Tr}_{4;c}(\phi_{\leq i-1}^{4})$ (62)
$\displaystyle+\sum_{c}\frac{\Gamma^{(c)}_{4;+,i-1}(\\{0\\})}{{2}Z_{b,i-1}^{2}}{\rm
Tr}_{4;c}(p^{2a}\phi_{\leq i-1}^{4})+{\tilde{R}}(\sqrt{Z_{b,\,i-1}}\phi_{\leq
i-1})\Big{]}\,.$

Now we can identify the effective couplings at scale $i-1$,

$\displaystyle Z_{a,i-1}$ $\displaystyle=$
$\displaystyle-\frac{\Gamma^{(c)}_{2,i-1}(\\{0\\})}{Z_{b,i-1}}\,,$
$\displaystyle\lambda^{(c)}_{i-1}$ $\displaystyle=$
$\displaystyle-\frac{\Gamma^{(c)}_{4,i-1}(\\{0\\})}{Z_{b,i-1}^{2}}\,,$
$\displaystyle\lambda^{(c)}_{+;\,i-1}$ $\displaystyle=$
$\displaystyle-\frac{\Gamma^{(c)}_{4;+,i-1}(\\{0\\})}{Z_{b,i-1}^{2}}\,.$ (63)

Note that in our case,
$(\partial_{|p_{c}|^{2b}}\Sigma\big{|}_{\\{p\\}=0})_{i-1}=0$ therefore,
throughout, we actually had

$\displaystyle Z_{b,\,i-1}$ $\displaystyle=$ $\displaystyle 1\,,$ (64)
$\displaystyle\mu_{{\rm ren},i-1}$ $\displaystyle=$
$\displaystyle\mu_{i-1}-\Sigma_{i-1}(\\{0\\})\,,$ (65) $\displaystyle
Z_{a,i-1}$ $\displaystyle=$ $\displaystyle-\Gamma^{(c)}_{2,i-1}(\\{0\\})\,,$
(66) $\displaystyle\lambda^{(c)}_{i-1}$ $\displaystyle=$
$\displaystyle-\Gamma^{(c)}_{4,i-1}(\\{0\\})\,,$ (67)
$\displaystyle\lambda^{(c)}_{+;\,i-1}$ $\displaystyle=$
$\displaystyle-\Gamma^{(c)}_{4;+,i-1}(\\{0\\})\,.$ (68)

In the following sections, we will compute these perturbative renormalization
group flow equations restricting to 1-loop corrections. For the model $+$, the
$\beta$-functions can be computed for generic parameters $a=(d-2)/2$, and
$b=(d-3/2)/2$ and $d>2$ but with fixed group dimension $D=1$. For numerics, we
will concentrate on a certain model $+$, specified by $d=3$, $a=1$, and
$b=3/4$. More general equations for $D>1$ can be also computed with a bit more
work 1.

###  3.2 4-point function and its beta-function

We will restrict the analysis by only considering up to one-loop corrections
to the $\beta$-function in the perturbation theory. Note that, from
Proposition 1, $\Gamma^{(c)}_{4;+}(\\{p\\})$ only contains the leading order
zero-loop graph with one $\lambda_{+}$ coupling with four external legs. Next,
we focus on $\Gamma^{(c)}_{4}(\\{p\\})$ and write

$\displaystyle\Gamma^{(c)}_{4}(\\{p\\})=\sum_{{\mathcal{G}}_{4,\iota}^{(c)}}K_{{\mathcal{G}}_{4,\iota}^{(c)}}\;S_{{\mathcal{G}}_{4,\iota}^{(c)}}(\\{p\\})\,,$
(69)

where the sum over ${\mathcal{G}}_{4,\iota}^{(c)}$ runs over a list of 4pt-
graphs obeying the first row of Table 1, $K_{{\mathcal{G}}_{4,\iota}^{(c)}}$
is a combinatorial factor and $S_{{\mathcal{G}}_{4,\iota}^{(c)}}(\\{p\\})$ is
a formal amplitude sum. At zero-loop, the leading order graph is made of one
$\lambda$ interaction bubble with four external legs. At one-loop a single
graph that we call $n_{4}^{(c)}$ shown in Fig. 4 contributes. If one is
further interested in two-loop contribution, see Appendix B.1.

$c$

$c$

$c$

$c$

Figure 4:  The only one-loop primitively divergent graph, $n_{4}^{(c)}$ that
contributes to the renormalization of $4$-pt function in
$\Gamma_{4}^{(c)}(\\{p\\})$. Here the rank $d=3$.

Explicitly, at one-loop,

$\displaystyle K_{n_{4}^{(c)}}=2\,,$ (70) $\displaystyle
S_{n_{4}^{(c)}}(\\{\mathbf{p},\mathbf{p}^{\prime}\\})=\frac{1}{2!}\Big{(}\frac{-\lambda^{(c)}_{+}}{2}\Big{)}^{2}\sum_{q_{c}}\frac{|q_{c}|^{2a}}{(|\mathbf{p}_{\check{c}}|^{2b}+|q_{c}|^{2b}+\mu)}\frac{|q_{c}|^{2a}}{(|\mathbf{p}^{\prime}_{\check{c}}|^{2b}+|q_{c}|^{2b}+\mu)}\,,$
(71)

where $\mathbf{p}_{\check{c}}=(p_{1},\dots,p_{c-1},p_{c+1},\dots,p_{d})$, and
$|\mathbf{p}_{\check{c}}|^{2b}=\sum_{l=1|l\neq c}^{d}p_{l}^{2b}$.

We compile these equations and deliver the $\beta$-functions of the couplings
up to one-loop as

$\displaystyle\lambda^{(c)}_{\rm
ren}=\lambda^{(c)}-\frac{1}{4}(\lambda^{(c)}_{+})^{2}S_{0}\,,\qquad\quad
S_{0}=\sum_{q}\frac{|q|^{4a}}{(|q|^{2b}+\mu_{i})^{2}}\,,$ (72)
$\displaystyle\lambda^{(c)}_{\rm+;ren}=\lambda^{(c)}_{+}\,,$ (73)

where $\lambda_{\rm ren}$ and $\lambda_{+,{\rm ren}}$ are the obvious
corresponding renormalized coupling constant. We set all couplings to
$\lambda^{(c)}=\lambda$, and $\lambda^{(c)}_{+}=\lambda_{+}$ to simplify these

$\displaystyle\lambda_{\rm ren}=\lambda-\frac{1}{4}(\lambda_{+})^{2}S_{0}\,,$
(74) $\displaystyle\lambda_{+,{\rm ren}}=\lambda_{+}\;.$ (75)

These RG flow equations carry already a lot of information. The second
equation displays the fact that the coupling $\lambda_{+}$ does not run and
defines a fixed point at all orders of perturbation. A second fact is that the
two couplings $\lambda$ and $\lambda_{+}$ never coincide and could not be set
at equal value. At the perturbative level, the model $\lambda=\lambda_{+}$ is
ill-defined (nonrenormalizable). Another piece of information conveyed by the
first equation is that $\lambda_{\rm ren}<\lambda$ ($S_{0}$ may be formal but
is positive) which means that $\lambda$ increases in the UV. One may ask if
this leads to the ordinary Landau ghost (like the $\phi^{4}_{4}$ theory for
instance), the answer is no. Indeed, note that the second term is not a
correction in $\lambda^{2}$, but rather a correction in $\lambda_{+}^{2}$.
This is a new effect that we must investigate better.

We want to understand the qualitative feature of RG flow given by the above
coupled system. In the multiscale analysis with discrete scale $i$, the system
can be written as

$\displaystyle\lambda_{i-1}=\lambda_{i}-\frac{1}{4}\lambda_{+,i}^{2}S_{0,i}\,,$
(76) $\displaystyle\lambda_{+,i-1}=\lambda_{+,i}\,,$ (77)

where $S_{0,i}$ stands for the cut-off amplitude of the formal sum $S_{0}$ and
formulates as

$\displaystyle S_{0,i}$ $\displaystyle=$
$\displaystyle\sum_{q}|q|^{4a}\int_{0}^{\infty}d\alpha\chi^{i}(\alpha)e^{-\alpha(|q|^{2b}+\mu_{i})}\int_{0}^{\infty}d\alpha^{\prime}\chi^{i}(\alpha^{\prime})e^{-\alpha^{\prime}(|q|^{2b}+\mu_{i})}$
(78) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\alpha\,\chi^{i}(\alpha)\int_{0}^{\infty}d\alpha^{\prime}\chi^{i}(\alpha^{\prime})e^{-(\alpha+\alpha^{\prime})\mu_{i}}\sum_{q}|q|^{4a}e^{-(\alpha+\alpha^{\prime})|q|^{2b}}\,,$
(79)

where the sharp cutoff function $\chi^{i}(\alpha)$ was defined in (44). The
sum over $q$ will be handled using an integral via Euler-Maclaurin
approximation (Appendix A details the following result, and we have
$S_{0,i}=\widetilde{S}_{0,i}$ and therefore,
$\lambda_{i}=\widetilde{\lambda}_{i}$ and
$\lambda_{+,i}=\widetilde{\lambda}_{+,i}$, where $\;\widetilde{}\;$ indicates
a dimensionless quantity):

$\displaystyle
S_{0,i}=\frac{1}{b}\log\frac{(M^{2b}+1)^{2}}{4M^{2b}}+{\mathcal{O}}(M^{-2bi}\log(M^{-2bi}))\,.$
(80)

We write the $\beta$-function for a given coupling $g$ as
$\beta_{g}(k)=k\partial_{k}g(k)$, where $k$ is a momentum scale. In our
present setting dealing with discrete slices (multiscale analysis), we have
finite difference equations that we will turn into differential equation. The
momentum scale must be compared to the slice range as $k/k_{0}\sim M^{i}$,
given in terms of a given momentum unit $k_{0}$. Using the so-called time
scale $t=\log(k/k_{0})\sim i\log M$, given that the coupling
$\lambda_{+}=\lambda_{+,i}$ does not run and that, at leading order
$S_{0,i}=c\log M$, where $c>0$, the difference $\lambda_{i-1}-\lambda_{i}$
takes the form:

$\displaystyle-(\lambda_{i-1}-\lambda_{i})=\frac{\partial\lambda_{i}}{\partial
i}$ $\displaystyle=$ $\displaystyle\frac{1}{4}\lambda_{+,i}^{2}\,S_{0,i}\,,$
(81)

which can be translated into the following first order ODE

$\displaystyle\frac{\partial\lambda_{i}}{\partial((\log
M)i)}=\partial_{t}\lambda(t)$ $\displaystyle=$
$\displaystyle-\beta_{\lambda}\lambda_{+}^{2}$ (82)
$\displaystyle\beta_{\lambda}$ $\displaystyle=$
$\displaystyle-\frac{1}{4b}\frac{\log\frac{(M^{2b}+1)^{2}}{4M^{2b}}}{\log(M)}<0\,.$
(83)

One may wonder why $M$, the propagator slice parameter, appears in the
perturbative expansion and if this the present equation does not depend on the
slicing scheme. The multi-scale analysis justifies this entirely: we are
computing a flow between two scales $\sim M^{-2bi}$, therefore $M$ becomes an
input of our equation.

This easily integrates to give a solution:

$\displaystyle\lambda(t)={(-\beta_{\lambda})}\,\lambda_{+}^{2}\,t+C\,,\qquad
C=cste$ (84) (85)
$\displaystyle\lambda(t)={(-\beta_{\lambda})}\,\lambda_{+}^{2}\,(t-t_{0})+\lambda(t_{0})\,,$
(86)

where at some IR scale $e^{t_{0}}\ll\Lambda/k_{0}=e^{t}$. As opposed to the
usual $\phi^{4}_{4}$ model, there is no pole in the solution at first order.
This proves that this is not a Landau ghost and therefore this model is not
similar to the ordinary $\phi^{4}_{4}$ model. Furthermore, at large $t\geq
t_{0}$, and for non vanishing $\lambda_{+}\neq 0$, since
$\lambda>\lambda(t_{0})$, the bare coupling is suppposedly not vanishing. We
conclude that the model is not asymptotically free. This hints at an
asymptotically safe model that only non perturbative calculation can make
rigorous. If $\lambda_{+}=0$, then $\lambda(t)=C$ and we have a fixed point.
However, the enhancement disappears, both couplings $\lambda$ and
$\lambda_{+}$ do not flow, all amplitudes become finite. Note that the
resulting model is not the usual $T^{4}$-TFT model with only $\lambda$
coupling and a different class of dominant graphs (melonic ones).

###  3.3 Computing $\Gamma_{2}$ and $Z_{a}$ renormalization

Now we compute the renormalization of the $2$-point coupling,
$\Gamma_{2}^{(c)}(\\{0\\})$, at fixed color $c$:

$\displaystyle|p_{c}|^{2a}\Gamma_{2}^{(c)}(\\{p\\})$ $\displaystyle=$
$\displaystyle\sum_{{\cal G}_{2;a;\iota}^{(c)}}K_{{\cal
G}_{2;a;\iota}^{(c)}}S_{{\cal G}_{2;a;\iota}^{(c)}}(\\{p\\})\,,$ (87)

where the sum is over all amputated $1$PI $2$-pt graphs at $1$-loop whose
boundaries to be in the form of ${\rm Tr}_{2;(c)}({p}^{2a}\phi^{2})$.

Up to the first order in perturbation theory, we have ${\cal
G}_{2;a;\iota}^{(c)}\in\\{z_{a}^{(c)},m_{e}^{(c)}\\}$, where $z_{a}^{(c)}$ is
the leading order (zero-loop) graph with one $Z_{a}^{(c)}$ interaction with
two external legs. (Appendix B shows additional graphs up to the second order
in perturbation theory.) In (53), we have identified
$\partial_{|p_{c}|^{2a}}\Sigma|_{\\{p\\}=0}\equiv\Gamma^{(c)}_{2}(\\{0\\})$ as
divergent. The other contributing graphs, namely $m_{e}^{(c)}$, $c=1,2,3$, are
divergent at first order in perturbation theory (row II of Table 1) with
$\omega_{m_{e}^{(c)}}=\frac{D}{2}$.

$c=1$

$q_{2}$

$q_{3}$

$p_{c=1}$

$p_{2}$

$p_{3}$

Figure 5:  $m_{e}^{(c=1)}$ for $d=3$ is illustrated in colored (left) and
stranded (right) representations. $\\{q_{\check{c}}\\}=\\{q_{2},\;q_{3}\\}$,
${|{\bf q}_{\check{c}}|}^{2b}={|q_{2}|}^{2b}+{|q_{3}|}^{2b}$.

$\displaystyle K_{m_{e}^{(c)}}S_{m_{e}^{(c)}}(\\{p\\})$ $\displaystyle=$
$\displaystyle
K_{m_{e}^{(c)}}\Big{(}-\frac{{\lambda_{+}}^{(c)}}{2}\Big{)}\sum_{\\{q_{\check{c}}\\}}\frac{|p_{c}|^{2a}}{({|{\bf
q}_{\check{c}}|}^{2b}+|p_{c}|^{2b}+\mu)}\,,$ (88)

where $K_{m_{e}^{(c)}}=2$. Putting all together, up to first order in
perturbation theory,

$\displaystyle|p_{c}|^{2a}\Gamma_{2}^{(c)}(\\{p\\})=-|p_{c}|^{2a}Z_{a}^{(c)}-\lambda_{+}^{(c)}|p_{c}|^{2a}\sum_{\\{q_{\check{c}}\\}}\frac{1}{({|{\bf
q}_{\check{c}}|}^{2b}+{|p_{c}|}^{2b}+\mu)}\,.$ (89)

$\Gamma_{2}^{(c)}(\\{p\\})$ is therefore identified easily. For general $d$,
recalling the renormalization group equations (68) and setting all the
external momenta $\\{p\\}=0$, we obtain:

$\displaystyle Z^{(c)}_{a,{\rm ren}}$ $\displaystyle=$
$\displaystyle-\Gamma^{(c)}_{2}(\\{0\\})=Z^{(c)}_{a}+\lambda_{+}^{(c)}\sum_{\\{q_{\check{c}}\\}}\frac{1}{({|{\bf
q}_{\check{c}}|}^{2b}+\mu)}\,.$ (90)

Note that we only keep the first order in Taylor expansion in couplings.
Setting the couplings independent of colors,

$\displaystyle Z_{a,{\rm ren}}$ $\displaystyle=$ $\displaystyle
Z_{a}+\lambda_{+}\sum_{\\{q_{1},\dots,q_{d-1}\\}}\frac{1}{({|{\bf
q}|}^{2b}+\mu)}\,.$ (91)

In the multiscale analysis language,

$\displaystyle Z_{a,i-1}=Z_{a,i}+\lambda_{+,i}S_{1,i}\,,$ (92)

where $S_{1,i}$ is given in (110) with explicit expression given in (112),
where we fixed $D=1$. We recall that $\lambda_{+}=\lambda_{+,i}$ does not run.
Making explicit the dimensions, we obtain the renormalization group equation
for $Z_{a}$ as

$\displaystyle Z_{a,i-1}=Z_{a,i}+k^{1/2}\lambda_{+,i}\,\widetilde{S}_{1,i}\,,$
(93)

which, using $\widetilde{S}_{1,i}$ given in (A.50), gives the $\beta$-function
for $Z_{a}$:

$\displaystyle-(Z_{a,i-1}-Z_{a,i})$ $\displaystyle=$
$\displaystyle\frac{\partial Z_{a,i}}{\partial
i}=-k^{1/2}\lambda_{+,i}\,\widetilde{S}_{1,i}\,,$ (94)
$\displaystyle\frac{\partial Z_{a,i}}{\partial((\log M)i)}$ $\displaystyle=$
$\displaystyle\partial_{t}Z_{a}(t)=-k^{1/2}\beta_{Z_{a}}\,\lambda_{+}\,,$ (95)
$\displaystyle\beta_{Z_{a}}$ $\displaystyle=$
$\displaystyle\frac{\widetilde{S}_{1,i}}{\log(M)}>0\,,$ (96)

Introducing dimensionless quantities, $Z_{a}(t)=k^{1/2}\widetilde{Z}_{a}(t)$,
the dimensionless RG equation can be written

$\displaystyle\partial_{t}\widetilde{Z}_{a}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\widetilde{Z}_{a}(t)+k^{-1/2}\partial_{t}Z_{a}(t)$
(97) $\displaystyle=$
$\displaystyle-\frac{1}{2}\widetilde{Z}_{a}(t)-\beta_{Z_{a}}\,\lambda_{+}\,.$
(98)

This integrates easily with respect to $t$ and gives

$\displaystyle\widetilde{Z}_{a}(t)=c_{1}\,e^{-t/2}-2\beta_{Z_{a}}\,\lambda_{+}\,,$
(99)

where $c_{1}$ is again an integration constant. This equation just expresses
the fact that $\widetilde{Z}_{a}(t)$ is a relevant coupling and decreases
exponentially in the UV ($t\rightarrow\infty$) and suppressed, whereas in the
IR ($t\rightarrow-\infty$), it blows up.

###  3.4 Self energy and mass renormalization

Our following task is to compute the so-called self energy, which we denote by
$\Sigma_{b}(\\{p\\})$:

$\displaystyle\Sigma_{b}(\\{p\\})=\sum_{c=1}^{d}\sum_{{\cal
G}^{(c)}_{2,\iota}}{K_{{\cal G}^{(c)}_{2,\iota}}}{S_{{\cal
G}^{(c)}_{2,\iota}}}(\\{p\\})\,,$ (100)

where the first sum is broken by color $c$ and the second sum is performed
over all amputated 1PI 2pt-graphs ${\cal G}^{(c)}_{2,\iota}$ (with color label
$c$) at 1-loop with boundary to be in the form of ${\rm
Tr}_{2}(p^{2b}\phi^{2})$ (hence the index $b$ in subscript in
$\Sigma_{b}(\\{p\\})$).

It is noteworthy that $\Sigma_{b}(\\{p\\})$ corresponds to the part
$\Sigma(\\{0\\})+\sum_{c}|p_{c}|^{2b}\partial_{|p_{c}|^{2b}}\Sigma\big{|}_{\\{p\\}=0}$
of total self-energy function $\Sigma(\\{p\\})$ in (53). Noting that the
second term $\partial_{|p_{c}|^{2b}}\Sigma\big{|}_{\\{p\\}=0}=0$, we only
focus on the contribution $\Sigma(\\{0\\})$, namely the contribution to the
mass renormalization.

The graphs we are interested in are denoted ${{\cal
G}^{(c)}_{2,\iota}}\in\\{m^{(c)},n^{(c)}\\}_{c=1,2,\dots,d}$, see Figures 6
and 7.

  * •

For the graph $m^{(c)}$, $c=1,\dots,d$, the degree of divergence
$\omega_{d;+}(m^{(c)})$ reaches $\frac{D}{2}$, as shown in the class III of
Table 1. In Figure 6, we display $m^{(c)}$, when the vertical line is color
$c$.

$c$

$c$

$p_{c}$

$p_{1}$

$p_{2}$

$q_{2}$

$q_{1}$

Figure 6:  The graph $m^{(c)}$ in the case $d=3$ in colored (left) and
stranded (right) representations.

The contribution to the amplitude brought by the graphs $m^{(c)}$ is

$\displaystyle\sum_{c=1}^{d}K_{m^{(c)}}S_{m^{(c)}}(\\{p\\})$ $\displaystyle=$
$\displaystyle\sum_{c=1}^{d}2\big{(}-\frac{\lambda^{(c)}}{2}\big{)}\sum_{\\{q_{\check{c}}\\}}\frac{1}{({|{\bf
q}_{\check{c}}|}^{2b}+|p_{c}|^{2b}+\mu)}$ (101) $\displaystyle=$
$\displaystyle\sum_{c=1}^{d}\sum_{\\{q_{\check{c}}\\}}(-\lambda^{(c)})\frac{1}{({|{\bf
q}_{\check{c}}|}^{2b}+|p_{c}|^{2b}+\mu)}\,,$ (102)

where $K_{m^{(c)}}=2$, for any $c$.

  * •

In the second class of graphs denoted each $n^{(c)}$, the amplitude has
divergence degree $\omega_{d;+}(n^{(c)})=\frac{D}{2}$. This graph belongs to
the class I in Table 1. Figure 7 shows $n^{(c)}$ at rank $d=3$ with horizontal
line colored by index $c$.

$c$

$q_{c}$

$p_{3}$

$p_{2}$

$p_{1}$

Figure 7:  For $d=3$, $n^{(c)}$ is shown above in colored (left) and stranded
(right) representations.

The sum of Feynman amplitude associated with the graph $n{(c)}$ yields

$\displaystyle\sum_{c=1}^{d}K_{n^{(c)}}S_{n^{(c)}}(\\{p\\})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{c=1}^{d}(-\lambda_{+}^{(c)})\sum_{q_{c}}\frac{|q_{c}|^{2a}}{(|q_{c}|^{2b}+{|{\bf
p}_{\check{c}}|}^{2b}+\mu)}\,,$ (103)

where $K_{n^{(c)}}=1$ for any $c$.

Up to the first order in perturbation theory and evaluating at $0$ external
momenta, we obtain

$\displaystyle\Sigma_{b}(\\{0\\})=\sum_{c=1}^{d}\Big{(}K_{m^{(c)}}S_{m^{(c)}}(\\{p\\})+K_{n^{(c)}}S_{n^{(c)}}(\\{p\\})\Big{)}\Big{|}_{p=0}$
(104)
$\displaystyle=\sum_{c=1}^{d}\Big{[}(-\lambda^{(c)})\sum_{\\{q_{\check{c}}\\}}\frac{1}{({|{\bf
q}_{\check{c}}|}^{2b}+\mu)}+\frac{1}{2}(-{\lambda_{+}}^{(c)})\sum_{q_{c}}\frac{|q_{c}|^{2a}}{(|q_{c}|^{2b}+\mu)}\Big{]}$
(105)

Therefore, for general $d$, the renormalized mass equation up to the first
order in perturbation is given by:

$\displaystyle{\mu_{\rm
ren}}=\mu+\sum_{c=1}^{d}\Bigg{\\{}\lambda^{(c)}\sum_{\\{q_{\check{c}}\\}}\frac{1}{({|{\bf
q}_{\check{c}}|}^{2b}+\mu)}+\frac{1}{2}{\lambda_{+}}^{(c)}\sum_{q_{c}}\frac{|q_{c}|^{2a}}{(|q_{c}|^{2b}+\mu)}\Bigg{\\}}\,.$
(107)

Now assert color independence, namely $\lambda^{(c)}=\lambda$,
${\lambda_{+}}^{(c)}=\lambda_{+}$, one gets

$\displaystyle\mu_{\rm ren}=\mu+d\Big{(}\lambda
S_{1}\>+\;\frac{1}{2}{\lambda_{+}}S_{2}\Big{)}$ (108) $\displaystyle
S_{1}=\sum_{\\{q_{1},\dots,q_{d-1}\\}}\frac{1}{({|{\bf q}|}^{2b}+\mu)}\qquad
S_{2}=\sum_{q}\frac{|q|^{2a}}{(|q|^{2b}+\mu)}\,,$ (109)

where we restrict ourselves to $D=1$.

To compute the perturbative flow, we must reach dimensionless coupling
equations and therefore must set up the scaling dimension of the mass coupling
(note that $\lambda$ and $\lambda_{+}$ were marginal and therefore
dimensionless). Following [Geloun:2016qyb], the mass scaling dimension is
determined by the maximal degree of divergence of the 2pt amplitudes.

We cut off the propagators, switch to dimensionful quantities, and write the
sums

$\displaystyle S_{1,i}=\sum_{{\bf q}\in
k{\mathbbm{Z}}^{d-1}}\int_{0}^{\infty}d\alpha\;\chi^{i}(\alpha)\,e^{-\alpha(|{\bf
q}|^{2b}+\mu_{i})}\,,$ (110) $\displaystyle S_{2,i}=\sum_{q\in
k{\mathbbm{Z}}}\int_{0}^{\infty}d\alpha\;\chi^{i}(\alpha)\,|q|^{2a}e^{-\alpha(|q|^{2b}+\mu_{i})}\,.$
(111)

In writing explicitly the dimensions of $q$ and $\alpha$ in terms of momentum
scale $k$, (A.58) in Appendix A proves that we can approximate those as

$\displaystyle
S_{1,i}=k^{1/2}\Bigg{(}\Big{(}\frac{1}{b}\Gamma\left(\frac{1}{2b}\right)\Big{)}^{d-1}(2d-3)\Big{(}1-M^{-1/2}\Big{)}M^{i/2}+{\mathcal{O}}(M^{-i/2})\Bigg{)}\,,$
(112) $\displaystyle
S_{2,i}=k^{1/2}\Bigg{(}4\,\Gamma\Big{(}\frac{2(d-1)}{2d-3}\Big{)}(1-M^{-1/2})M^{i/2}+{\mathcal{O}}\big{(}M^{-i(d-2)}\big{)}\Bigg{)}\,,$
(113)

where we have set $D=1$ at any rank $d$ and $b=\frac{1}{2}(d^{-}-\frac{1}{2})$
given in Proposition 1. Compiling this, we obtain the following equation at
leading order,

$\displaystyle\mu_{i-1}$ $\displaystyle=$
$\displaystyle\mu_{i}+k^{1/2}d\Big{(}\lambda_{i}\widetilde{S}_{1,i}+\frac{1}{2}\lambda_{+,i}\widetilde{S}_{2,i}\Big{)}\,,$
(114) $\displaystyle\widetilde{S}_{1,i}$ $\displaystyle=$ $\displaystyle
k^{-1/2}S_{1,i}\,,$ (115) $\displaystyle\widetilde{S}_{2,i}$ $\displaystyle=$
$\displaystyle k^{-1/2}S_{2,i}\,,$ (116)

where we recall that $\lambda$ and $\lambda_{+}$ are marginal, therefore
dimensionless in scales, and where $\widetilde{S}_{1,i}$ and
$\widetilde{S}_{2,i}$ are given in (A.50) and (A.40).

We obtain the $\beta$-function for the mass

$\displaystyle-(\mu_{i-1}-\mu_{i})$ $\displaystyle=$
$\displaystyle\frac{\partial\mu_{i}}{\partial
i}=-k^{1/2}d\Big{(}\widetilde{S}_{1,i}\,\lambda_{i}+\frac{1}{2}\widetilde{S}_{2,i}\,\lambda_{+,i}\Big{)}\,,$
(117) $\displaystyle\frac{\partial\mu_{i}}{\partial((\log M)i)}$
$\displaystyle=$
$\displaystyle\partial_{t}\mu=-k^{1/2}(\beta_{\mu,1}\,\lambda+\beta_{\mu,2}\,\lambda_{+})\,,$
(118) $\displaystyle\beta_{\mu,1}$ $\displaystyle=$
$\displaystyle\frac{d}{\log M}\,\widetilde{S}_{1,i}>0\,,$ (119)
$\displaystyle\beta_{\mu,2}$ $\displaystyle=$ $\displaystyle\frac{d}{2\,\log
M}\,\widetilde{S}_{2,i}>0\,.$ (120)

We switch to dimensionless quantities, fixing $\mu=k^{1/2}\widetilde{\mu}$,
where $\widetilde{\mu}$ is dimensionless in scale, therefore,
$\partial_{t}\mu=k\partial_{k}\mu=k^{1/2}(\frac{1}{2}\widetilde{\mu}+\partial_{t}\widetilde{\mu})$.
The previous equation becomes

$\displaystyle\partial_{t}\widetilde{\mu}(t)=-\frac{1}{2}\widetilde{\mu}(t)+k^{-1/2}\partial_{t}\mu(t)\,.$
(121)

Given that the coupling $\lambda_{+}=\lambda_{+,i}$ does not run and that,
$\lambda$ runs according to (86), we make the $t$ dependence explicit and

$\displaystyle\partial_{t}\widetilde{\mu}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\widetilde{\mu}(t)-\beta_{\mu,1}\,\lambda(t)-\beta_{\mu,2}\,\lambda_{+}$
(122) $\displaystyle=$
$\displaystyle-\frac{1}{2}\widetilde{\mu}(t)+\beta_{\mu,1}\,\beta_{\lambda}\,\lambda_{+}^{2}\,t-\Big{(}\beta_{\mu,2}\,\lambda_{+}+\beta_{\mu,1}\,\beta_{\lambda}\,\lambda_{+}^{2}\,t_{0}+\beta_{\mu,1}\,\lambda(t_{0})\Big{)}\,,$
(123)

where we recall $\beta_{\lambda}<0$ . We can solve this differential equation
and obtain

$\displaystyle\widetilde{\mu}(t)$ $\displaystyle=$ $\displaystyle
c_{1}e^{-t/2}+4\beta\,t-4\beta+2\gamma\,,$ (124) $\displaystyle\beta$
$\displaystyle=$
$\displaystyle\beta_{\mu,1}\,\beta_{\lambda}\,\lambda_{+}^{2}<0\,,$ (125)
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle-\Big{(}\beta_{\mu,2}\,\lambda_{+}+\beta_{\mu,1}\,\beta_{\lambda}\,\lambda_{+}^{2}\,t_{0}+\beta_{\mu,1}\,\lambda(t_{0})\Big{)}\,.$
(126)

where $c_{1}$ is an integration constant that should be set by an initial
condition of the flow.

In the UV, the first exponential term vanishes and the second linear term
dominates. As $\beta=\beta_{\mu,1}\,\beta_{\lambda}\,\lambda_{+}^{2}<0$, the
mass becomes negative and grows linearly. This is not the ordinary behavior of
scalar field theory where relevant operators in the UV are suppressed. In the
IR, the exponential term dominates and this is the common behavior of relevant
operators in ordinary $\phi^{4}$-theory.

Summary – We list our dimensionless 1-loop RG flow equations for the model $+$
and their solutions below.

$\partial_{t}\,\lambda(t)=-\beta_{\lambda}\lambda_{+}^{2}$ , $\beta_{\lambda}<0$ | $\lambda(t)=-\beta_{\lambda}\,\lambda_{+}^{2}\,(t-t_{0})+\lambda(t_{0})$
---|---
$\partial_{t}\,\lambda_{+}=0$ | $\lambda_{+}=cste$
$\partial_{t}\,\widetilde{\mu}(t)=-\frac{1}{2}\widetilde{\mu}(t)-\beta_{\mu,1}\,\lambda(t)-\beta_{\mu,2}\,\lambda_{+}$ | $\widetilde{\mu}(t)=c_{1}e^{-t/2}+4\beta\,t-4\beta+2\gamma$
$\beta_{\mu,1}>0\,,\;\,\beta_{\mu,2}>0$ |  $\beta=\beta_{\mu,1}\,\beta_{\lambda}\,\lambda_{+}^{2}<0$ ,
|
$\gamma=-\Big{(}\beta_{\mu,2}\,\lambda_{+}+\beta_{\mu,1}\,\beta_{\lambda}\,\lambda_{+}^{2}\,t_{0}+\beta_{\mu,1}\,\lambda(t_{0})\Big{)}$
$\partial_{t}\,\widetilde{Z}_{a}(t)=-\frac{1}{2}\widetilde{Z}_{a}(t)-\beta_{Z_{a}}\,\lambda_{+}$ , $\beta_{Z_{a}}>0$ | $\widetilde{Z}_{a}(t)=c_{2}\,e^{-t/2}-2\beta_{Z_{a}}\,\lambda_{+}$

###  3.5 Integration at arbitrary loops

The power counting of the model $+$, provided by Proposition 1, gives us a lot
of information about the $\beta$-functions of the couplings even in an
arbitrarily high order in perturbation theory. In this section, we investigate
general forms of $\beta$-functions of the couplings in the model $+$ at all
orders in perturbation theory.

4-pt couplings $\lambda$ and $\lambda_{+}$ RG equation – Proposition 1
dictates that there are no diverging amplitudes contributing to the
renormalization of $\lambda_{+}$ at all orders in perturbation, therefore,
$\lambda_{+}$ is constant at all orders. Furthermore, from the first row of
Table 1 of Proposition 1, which governs the renormalization of the coupling
$\lambda$, we know that in order for the amplitude to be divergent, Feynman
graph must only contain $\lambda_{+}$ couplings, and no other couplings. Then,
we can readily conclude that, the subleading corrections to $\lambda$ at
$n$-loops, $n$ being arbitrary, can be written as a polynomial
$P_{n}(\lambda_{+})$ in the variable $\lambda_{+}$ that is fixed to a
constant. At arbitrary $n$-th order in perturbation, the $\beta$-function of
the coupling $\lambda$ assumes the form

$\displaystyle\partial_{t}\lambda(t)$ $\displaystyle=$ $\displaystyle
P_{n}(\lambda_{+})\,,$ (127) $\displaystyle\lambda(t)$ $\displaystyle=$
$\displaystyle P_{n}(\lambda_{+})(t-t_{0})+\lambda(t_{0})\,.$ (128)

The particular form of
$P_{n}(\lambda_{+})=-\beta_{\lambda}\lambda_{+}^{2}+\dots$ is left for future
investigation. Note that we may find an asymptotic safe theory we must solve
$P_{n}(\lambda_{+})=0$ which may have several solutions $\lambda_{+}^{*}$.

2-pt coupling $Z_{a}$ RG equation – The classes II and V of Proposition 1
contribute to the renormalization of the $2$-pt coupling $Z_{a}$. All class II
amplitudes involve only the coupling $\lambda_{+}$ yielding a similar result
as to the one-loop computation above. The class V contains contributions with
exactly one $Z_{a}$ and several $\lambda_{+}$ (an example is given in Fig. 12
Appendix B). Therefore, at an arbitrary $n$-th order in perturbation theory,
we expect

$\displaystyle\partial_{t}Z_{a}(t)$ $\displaystyle=$ $\displaystyle
k^{1/2}Q_{1;n}(\lambda_{+})+\log(k/k_{0})\,Z_{a}(t)Q_{2;n}(\lambda_{+})\,,$
(129)

where $Q_{i;n}(\lambda_{+})$, $i=1,2$ are polynomials in $\lambda_{+}$. Then,
in terms of dimensionless quantities, we obtain

$\displaystyle\partial_{t}\widetilde{Z}_{a}(t)=\Big{(}t\,Q_{2;n}(\lambda_{+})-\frac{1}{2}\Big{)}\widetilde{Z}_{a}(t)+Q_{1;n}(\lambda_{+})\,.$
(130)

which will integrate to yield

$\displaystyle\widetilde{Z}_{a}(t)=e^{\frac{Q_{2;n}t^{2}}{2}-\frac{t}{2}}\Big{[}c_{1}+\sqrt{2}\,\frac{Q_{1;n}}{\sqrt{Q_{2;n}}}\,e^{\frac{1}{8Q_{2;n}}}\,\text{Erf'}\left(\frac{2Q_{2;n}t-1}{2\sqrt{2}\sqrt{Q_{2;n}}}\right)\Big{]}$
(131)

where Erf’ is the non normalized incomplete error function
Erf’$(z)=\int_{0}^{z}e^{-s^{2}}ds$, and $c_{1}$ an integration constant.
Understanding the flow of $\widetilde{Z}_{a}(t)$ rests upon the the
understanding of the polynomials $Q_{1;n}$ and $Q_{2;n}$. Focusing on the
asymptotic of the Erf function, Erf’$(z)\sim e^{-z^{2}}/z^{2}$, we see that
$\widetilde{Z}_{a}(t)\sim e^{\frac{Q_{2;n}t^{2}}{2}-\frac{t}{2}}$. Then,
$\widetilde{Z}_{a}(t)$ behaves the same in the UV ($t\rightarrow\infty$) and
the IR ($t\rightarrow-\infty$), either can be suppressed or blows up depending
on the sign of $Q_{2;n}$.

Mass RG equation – Looking at Proposition 1, the classes that contribute to
the renormalization of the mass are I, III, IV, and VI. The class I only
contains $\lambda_{+}$, which is held constant. The class III contains only
exactly one $\lambda$ and the rests are all $\lambda_{+}$. The class IV
contains only exactly one $Z_{a}$ coupling and the rests are all
$\lambda_{+}$. Finally, the class VI contains exactly only one $Z_{a}$, and
exactly only one $\lambda$, and the rests are $\lambda_{+}$. Noting that
$\lambda_{+}$ does not run at all orders, the most complicated one could have
is the class VI where $\lambda$ and $Z_{a}$ are coupled (whose example is
given in Fig.14). Therefore, we expect

$\displaystyle\partial_{t}\widetilde{\mu}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\widetilde{\mu}(t)+R_{1;n}(\lambda_{+})\,\lambda(t)+R_{2;n}(\lambda_{+})$
(133)
$\displaystyle+R_{3;n}(\lambda_{+})\,t\,\widetilde{Z}_{a}(t)+R_{4;n}(\lambda_{+})\,t\,\widetilde{Z}_{a}(t)\,\lambda(t)\,,$

to an arbitrary $n$-th order in perturbation theory, and $R_{i;n}$, with
$i=1,2,3,4$ are polynomials in $\lambda_{+}$ and some constants.

In summary, we expect that the coupled system of RG equations of the model $+$
to an arbitrary $n$-th order is given by

$\displaystyle\partial_{t}\lambda_{+}$ $\displaystyle=$ $\displaystyle 0\,,$
(134) $\displaystyle\partial_{t}\lambda(t)$ $\displaystyle=$ $\displaystyle
P_{n}(\lambda_{+})\,,$ (135) $\displaystyle\partial_{t}\widetilde{Z}_{a}(t)$
$\displaystyle=$
$\displaystyle\Big{(}t\,Q_{2;n}(\lambda_{+})-\frac{1}{2}\Big{)}\widetilde{Z}_{a}(t)+Q_{1;n}(\lambda_{+})$
(136) $\displaystyle\partial_{t}\widetilde{\mu}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\widetilde{\mu}(t)+R_{1;n}(\lambda_{+})\,\lambda(t)+R_{2;n}(\lambda_{+})$
(138)
$\displaystyle+t\,\widetilde{Z}_{a}(t)\,R_{3;n}(\lambda_{+})+t\,\widetilde{Z}_{a}(t)\,\lambda(t)\,R_{4;n}(\lambda_{+})\,.$

##  4 One-loop beta-functions of the model $\times$

Just as performed for the model $+$, we will compute the 1-loop RG flow
equations of coupling constants via multiscale analysis for the model
$\times$. Because the scheme and proofs are nearly identical, the derivations
and explanations are given in a streamlined analysis. We stress the following
fact: although the notation and expressions are similar to previous section,
they actually refer to different quantities. As the model is different, there
is no confusion and no need to introduce new notation.

###  4.1 Effective coupling equations

The integration of high modes yields a formal effective action of the form
(52) where notation keeps its meaning but adapts to the present model.
Therein, the 2-point amplitudes expand in local and nonlocal parts, obtaining,
a self-energy of the same form as given in (53) in addition with the following
term:

$\displaystyle\sum_{c}|p_{c}|^{4a}\partial_{|p_{c}|^{4a}}\Sigma\big{|}_{\\{p\\}=0}\,.$
(139)

According to [26], Table 2 dictates the renormalization analysis of the model.
One shows that

$\displaystyle\partial_{|p_{c}|^{2b}}\Sigma\big{|}_{\\{p\\}=0}=0$
$\displaystyle\qquad\text{implies }\;Z_{b}=1\quad\text{wave function
renormalization}$ (140) $\displaystyle\Sigma(\\{0\\})\sim\log-{\rm divergent}$
(row III) mass renormalization (141)
$\displaystyle\partial_{|p_{c}|^{2a}}\Sigma|_{\\{p\\}=0}\equiv\Gamma^{(c)}_{2;a}(\\{0\\})\sim\log-{\rm
divergent}$ $\displaystyle\qquad\text{(row I) \quad}Z_{a}\text{\;
renormalization}$ (142)
$\displaystyle\partial_{|p_{c}|^{4a}}\Sigma|_{\\{p\\}=0}\equiv\Gamma^{(c)}_{2;2a}(\\{0\\})\sim\log-{\rm
divergent}$ $\displaystyle\qquad\text{(row II) \quad}Z_{2a}\text{\;
renormalization}$ (143)

where $|p_{c}|^{2a}\Gamma^{(c)}_{2;a}(\\{p\\})$ and
$|p_{c}|^{4a}\Gamma^{(c)}_{2;2a}(\\{p\\})$ are the sum of all amputated 1PI
2pt-functions following the patterns of ${\rm Tr}_{2;c}({p}^{2a}\phi^{2})$ and
${\rm Tr}_{2;c}({p}^{4a}\phi^{2})$, respectively, on their boundary graphs.
Because $4$-point functions converge we do not need to report them.

We reorganize the effective action as

$\displaystyle-W^{i-1}(\phi_{\leq i-1},{\bar{\phi}}_{\leq i-1})$
$\displaystyle=$ $\displaystyle\Sigma_{i-1}(\\{0\\}){\rm
Tr}_{2}(\phi^{2}_{\leq i-1})+\sum_{c}\Gamma^{(c)}_{2;a,\;i-1}(\\{0\\}){\rm
Tr}_{2;c}({p}^{2a}\phi^{2}_{\leq i-1})$ (145)
$\displaystyle+\sum_{c}\Gamma^{(c)}_{2;2a,\;i-1}(\\{0\\}){\rm
Tr}_{2;c}({p}^{4a}\phi^{2}_{\leq i-1})+{\tilde{R}}(\phi_{\leq i-1})\,,$

where ${\tilde{R}}(\phi_{\leq i-1})$ contains all finite contributions.

Following step by step, the previous analysis, we deliver the the effective
couplings at scale $i-1$:

$\displaystyle Z_{b,\,i-1}$ $\displaystyle=$ $\displaystyle 1\,,$ (146)
$\displaystyle\mu_{{\rm ren},i-1}$ $\displaystyle=$
$\displaystyle\mu_{i-1}-\Sigma_{i-1}(\\{0\\})\,,$ (147) $\displaystyle
Z_{a,i-1}$ $\displaystyle=$ $\displaystyle-\Gamma^{(c)}_{2;a,i-1}(\\{0\\})\,,$
(148) $\displaystyle Z_{2a,i-1}$ $\displaystyle=$
$\displaystyle-\Gamma^{(c)}_{2;2a,i-1}(\\{0\\})\,,$ (149)

Following Proposition 2, we work with the set of parameters $d=3$, $D=1$,
$a=\frac{1}{2}$, and $b=1$ so that the model is just-renormalizable.

###  4.2 Self energy and mass renormalization

For the model $\times$, we calculate the self energy which we denote again by
$\Sigma_{b}(\\{p\\})$ and which has an expression similar to (100), where the
graphs ${\cal G}^{(c)}_{2,\iota}$ should be choosen among all amputated 1PI
2pt-graphs with boundary of the form ${\rm Tr}_{2;c}(p^{2b}\phi^{2})$.

Up to the first order in perturbation theory, ${\cal
G}^{(c)}_{2,\iota}\in\\{m^{(1)},m^{(2)},m^{(3)}\\}$. (At second order in
perturbation theory, the interested reader may look at Fig. 18 in Appendix C.)

Recall that $\Sigma_{b}(\\{p\\})$ corresponds to
$\Sigma(\\{0\\})+\sum_{c}|p_{c}|^{2b}\partial_{|p_{c}|^{2b}}\Sigma\big{|}_{\\{p\\}=0}$.
Considering that the second term vanishes, i.e.
$\partial_{|p_{c}|^{2b}}\Sigma\big{|}_{\\{p\\}=0}=0$, only the contribution
$\Sigma(\\{0\\})$, namely the mass renormalization, survives.

For $m^{(c)}$, $c=1,2,3$, $\omega_{d;\times}({m^{(c)}})=0$. This graph belongs
to the class III in Table 2, also, it appears at first order.

$c$

$c$

$p_{c}$

$p_{1}$

$p_{2}$

$q_{2}$

$q_{1}$

Figure 8: The graph $m^{(c)}$. In the case $d=3$. ${|{\bf
q}_{\check{c}}|}^{2b}=|q_{1}|^{2b}+|q_{2}|^{2b}$. Here the parameters should
follow that of Proposition 2.

We sum the contributions of the graphs $m^{(c)}$ and write:

$\displaystyle\Sigma_{b}(\\{p\\})=\sum_{c=1}^{d}K_{m^{(c)}}S_{m^{(c)}}(\\{p\\})$
$\displaystyle=$
$\displaystyle\sum_{c=1}^{d}2\big{(}-\frac{\lambda^{(c)}}{2}\big{)}\sum_{\\{q_{\check{c}}\\}}\frac{1}{({|{\bf
q}_{\check{c}}|}^{2b}+p^{2b}_{c}+\mu)}\,,$ (150)

where $K_{m^{(c)}}=2$.

The renormalized mass (147) finds the expression:

${\mu_{\rm
ren}}=\mu-\Sigma_{b}(\\{p\\})\Big{|}_{\\{p\\}=\\{0\\}}=\mu+\sum_{c=1}^{d}\lambda^{(c)}\sum_{\\{q_{\check{c}}\\}}\frac{1}{({|{\bf
q}_{\check{c}}|}^{2b}+\mu)}\,.$ (151)

Now assert color independence, namely $\lambda^{(c)}=\lambda$, and
${\lambda_{\times}}^{(c)}=\lambda_{\times}$, the renormalized mass can be
expressed as

$\displaystyle\mu_{\rm ren}$ $\displaystyle=$
$\displaystyle\mu+d\lambda\sum_{\\{q_{\check{c}}\\}}\frac{1}{({|{\bf
q}_{\check{c}}|}^{2b}+\mu)}=\mu+d\,\lambda\,S_{1}\,,$ (152)

where $S_{1}$ is defined earlier (109). We perform a similar analysis as done
in section 3.4, use (A.56) for $\widetilde{S}_{1,i}$ and (A.58) to obtain the
running of the mass at leading order,

$\displaystyle\mu_{i-1}=\mu_{i}+d\,\widetilde{S}_{1,i}\,\lambda_{i}\,.$ (153)

Note that the mass does not have a scaling dimension, and that $\lambda$ does
not run in this model (see (149)), then we write the RG flow equation for
$\mu$:

$\displaystyle-(\mu_{i-1}-\mu_{i})$ $\displaystyle=$
$\displaystyle\frac{\partial\mu_{i}}{\partial
i}=-d\,\widetilde{S}_{1,i}\,\lambda_{i}\,,$ (154)
$\displaystyle\frac{\partial\mu_{i}}{\partial((\log M)i)}$ $\displaystyle=$
$\displaystyle\partial_{t}\mu(t)=-\beta_{\mu,1}\,\lambda\,,$ (155)
$\displaystyle\beta_{\mu,1}$ $\displaystyle=$ $\displaystyle\frac{d}{\log
M}\,\widetilde{S}_{1,i}=2d\pi>0\,.$ (156)

Fixing an initial condition at $t_{0}$, this integrates to give

$\displaystyle\mu(t)=-(t-t_{0})\beta_{\mu,1}\,\lambda+\mu(t_{0})\,.$ (157)

Therefore, the mass in this theory, the model $\times$ grows linearly in $t$
in its magnitude.

###  4.3 Computing $\Gamma_{2;a}$ and $Z_{a}$ renormalization

We address here the flow of the 2-point coupling $Z_{a}$. The 2-pt diagram sum
$\Gamma^{(c)}_{2;a}(\\{p\\})$ follows again an equation similar to (87) of the
previous section 3.3. The sum performs over all amputated 1PI 2-pt graphs at
1-loop of the form ${\rm Tr}_{2;c}({p}^{2a}\phi^{2})$.

At first order, the diagrams $n_{e}^{(c)}$, $c=1,2,3$, see Fig. 9, contribute
to the flow. (The next order in perturbation theory will have the additional
graphs of Appendix C.) In (53), we have defined
$\partial_{|p_{c}|^{2a}}\Sigma|_{\\{p\\}=0}\equiv\Gamma^{(c)}_{2;a}(\\{0\\})$
which is divergent.

The graphs $n_{e}^{(c)}$ satisfy $\omega_{d;\times}(n_{e}^{(c)})=0$ and belong
to the class I in Table 2.

$c=1$

$p_{c=1}$

$p_{2}$

$p_{3}$

$q_{c=1}$

$c=1$

Figure 9:  For $d=3$, $n_{e}^{(c=1)}$ is shown above in colored (left) and
stranded (right) representations. $\\{p\\}=\\{p_{1},p_{2},p_{3}\\}$ and
${|{\bf p}_{\check{c}}|}^{2b}={|p|}^{2b}+{|p_{2}|}^{2b}$.

The Feynman amplitude associated with the graph $n_{e}^{(c)}$ shown in Fig. 9
is

$\displaystyle K_{n_{e}^{(c)}}S_{n_{e}^{(c)}}(\\{p\\})$ $\displaystyle=$
$\displaystyle
2\Big{(}-\frac{{\lambda_{\times}}^{(c)}}{2}\Big{)}|p_{c}|^{2a}\sum_{q_{c}}\frac{|q_{c}|^{2a}}{(|q_{c}|^{2b}+{|{\bf
p}_{\check{c}}|}^{2b}+\mu)}\,,$ (158)

where $K_{n_{e}^{(c)}}=2$.

Let us now compute the renormalization of $Z_{a}$, up to first order in
perturbation theory

$\displaystyle|p_{c}|^{2a}\Gamma_{2;a}(\\{p\\})$ $\displaystyle=$
$\displaystyle-|p_{c}|^{2a}Z_{a}^{(c)}+K_{n_{e}^{(c)}}S_{n_{e}^{(c)}}(\\{p\\})$
(159) $\displaystyle=$
$\displaystyle-|p_{c}|^{2a}Z_{a}^{(c)}-\lambda_{\times}^{(c)}|p_{c}|^{2a}\sum_{q_{c}}\frac{|q_{c}|^{2a}}{(|q_{c}|^{2b}+{|{\bf
p}_{\check{c}}|}^{2b}+\mu)}\,.$

Therefore, to the first order in perturbation,

$\displaystyle Z_{a,{\rm ren}}^{(c)}$ $\displaystyle=$
$\displaystyle-\Gamma_{2;a}^{(c)}(\\{p\\})\big{|}_{\\{p\\}=0}=Z_{a}^{(c)}+\lambda_{\times}^{(c)}\sum_{q}\frac{|q|^{2a}}{|q|^{2b}+\mu}\,.$
(160)

By imposing the color independence, we write

$\displaystyle Z_{a,{\rm ren}}=Z_{a}+\lambda_{\times}\,S_{2}\,,$ (161)

where $S_{2}$ is given in (109). In multiscale analysis, we write

$\displaystyle Z_{a,i-1}=Z_{a,i}+\lambda_{\times,i}\,S_{2,i}\,,$ (162)

where $S_{2,i}$ is given in (111).

We approximate $S_{2,i}$ in (A.53). Now, we use the fact that $Z_{a}$ has no
scaling dimension (A.58), and that $\lambda_{\times}$ does not run as
presented in (149), and express the $\beta$-function of the coupling $Z_{a}$
as:

$\displaystyle-(Z_{a,i-1}-Z_{a,i})$ $\displaystyle=$
$\displaystyle\frac{\partial Z_{a,i}}{\partial
i}=-\,\widetilde{S}_{2,i}\,\lambda_{\times,i}\,,$ (163)
$\displaystyle\frac{\partial Z_{a,i}}{\partial((\log M)i)}$ $\displaystyle=$
$\displaystyle\partial_{t}Z_{a}(t)=-\beta_{Z_{a}}\,\lambda_{\times}\,,$ (164)
$\displaystyle\beta_{Z_{a}}$ $\displaystyle=$
$\displaystyle\frac{\widetilde{S}_{2,i}}{\log M}=2>0\,,$ (165)

where $\widetilde{S}_{2,i}$ is given in (A.53). The solution is
straightforward, after fixing an initial condition at some $t_{0}$

$\displaystyle
Z_{a}(t)=-(t-t_{0})\,\beta_{Z_{a}}\,\lambda_{\times}+Z_{a}(t_{0})\,.$ (166)

Therefore, in exactly the same manner as the mass in this theory, the $2$-pt
coupling $Z_{a}$ in the model $\times$, grows linearly in $t$ in its
magnitude.

###  4.4 Computing of $\Gamma_{2;2a}$ and $Z_{2a}$ renormalization

We focus on the RG flow for $Z_{2a}$. We denote by
$|p_{c}|^{4a}\Gamma_{2;2a}^{(c)}(\\{p\\})$ the sum is over all amputated 1PI
2-pt graphs at 1-loop whose boundaries follow the pattern of ${\rm
Tr}_{2;c}({p}^{4a}\phi^{2})$.

The diagrams that will contribute at 1-loop are denoted $m_{ee}^{(c)}$ and
depicted in Fig. 10. (Higher order diagrams are listed in Appendix C) For
$m_{ee}^{(c)}$, $\omega_{d;\times}(m_{ee}^{(c)})=0$ and belongs to the class
II in Table 2.

$c=1$

$c=1$

$p_{c=1}$

$p_{2}$

$p_{3}$

$q_{3}$

$q_{2}$

Figure 10: $m_{ee}^{(c=1)}$ for $d=3$ is illustrated in colored (left) and
stranded (right) representations. $\\{q_{\check{c}}\\}=\\{q_{2},\;q_{3}\\}$,
${|{\bf q}_{\check{c}}|}^{2b}={|q_{2}|}^{2b}+{|q_{3}|}^{2b}$.

$\displaystyle K_{m_{ee}^{(c)}}S_{m_{ee}^{(c)}}(\\{p\\})$ $\displaystyle=$
$\displaystyle
2\Big{(}-\frac{{\lambda_{\times}}^{(c)}}{2}\Big{)}|p_{c}|^{4a}\sum_{\\{q_{\check{c}}\\}}\frac{1}{({|{\bf
q}_{\check{c}}|}^{2b}+|p_{c}|^{2b}+\mu)}\,,$ (167)

where $K_{m_{ee}^{(c)}}=2$. Given this, we compute the renormalization of
$Z_{2a}$: where

$\displaystyle|p_{c}|^{4a}\Gamma_{2;2a}^{(c)}\big{(}\\{p\\})$ $\displaystyle=$
$\displaystyle-|p_{c}|^{4a}Z_{2a}^{(c)}+K_{m_{ee}^{(c)}}S_{m_{ee}^{(c)}}(\\{p\\})$
(168) $\displaystyle=$
$\displaystyle-|p_{c}|^{4a}Z_{2a}^{(c)}-{\lambda_{\times}}^{(c)}|p_{c}|^{4a}\sum_{\\{q_{\check{c}}\\}}\frac{1}{({|{\bf
q}_{\check{c}}|}^{2b}+|p_{c}|^{2b}+\mu)}\,.$

After a small manipulation, we obtain

$\displaystyle Z_{2a,{\rm ren}}^{(c)}$ $\displaystyle=$ $\displaystyle
Z_{2a}^{(c)}+{\lambda_{\times}}^{(c)}\sum_{\\{q_{\check{c}}\\}}\frac{1}{({|{\bf
q}_{\check{c}}|}^{2b}+\mu)}\,.$ (169)

The equation for $Z_{2a}$ is reached after requiring color independence:

$\displaystyle Z_{2a,{\rm ren}}=Z_{2a}+\lambda_{\times}\,S_{1}\,,$ (170)

where we recognise $S_{1}$ (109). Passing through the same steps via
multiscale regularization, the coupling equation can be written as:

$\displaystyle Z_{2a,i-1}$ $\displaystyle=$ $\displaystyle
Z_{2a,i}+\lambda_{\times,i}\,S_{1,i}\,,$ (171)
$\displaystyle-(Z_{2a,i-1}-Z_{2a,i})$ $\displaystyle=$
$\displaystyle\frac{\partial Z_{2a,i}}{\partial
i}=-\,\widetilde{S}_{1,i}\,\lambda_{\times,i}\,,$ (172)
$\displaystyle\frac{\partial Z_{2a,i}}{\partial((\log M)i)}$ $\displaystyle=$
$\displaystyle\partial_{t}Z_{2a}(t)=-\beta_{Z_{2a}}\,\lambda_{\times}\,,$
(173) $\displaystyle\beta_{Z_{2a}}$ $\displaystyle=$
$\displaystyle\frac{\widetilde{S}_{1,i}}{\log M}=2\pi>0\,,$ (174)

where $\widetilde{S}_{1,i}$ is computed in (A.56). Then, at this order of
perturbation, $Z_{2a}$ yields also a linear function in the time scale $t$.

$\displaystyle Z_{2a}(t)$ $\displaystyle=$
$\displaystyle-(t-t_{0})\,\beta_{Z_{2a}}\,\lambda_{\times}+Z_{a}(t_{0})\,.$
(175)

The argument goes the same as the mass and the other $2$-point coupling
$Z_{a}$, i.e., the 2-pt coupling $Z_{2a}$ in the model $\times$ grows linearly
in $t$ in its magnitude.

Summary – We give a summary of the 1-loop RG flow equations for the model
$\times$ and their solutions:

$\partial_{t}\,\lambda(t)=0$ | $\lambda(t)=cste$
---|---
$\partial_{t}\,\lambda_{\times}(t)=0$ | $\lambda_{\times}(t)=cste$
$\partial_{t}\,\mu(t)=-\beta_{\mu,1}\lambda$ , $\beta_{\mu,1}=2d\pi>0$ | $\mu(t)=-2d\pi\lambda(t-t_{0})+\mu(t_{0})$
$\partial_{t}\,Z_{a}(t)=-\beta_{Z_{a}}\,\lambda_{\times}$ , $\beta_{Z_{a}}=2>0$ | $Z_{2a}(t)=-2\lambda_{\times}\,(t-t_{0})+Z_{a}(t_{0})$
$\partial_{t}\,Z_{a}(t)=-\beta_{Z_{2a}}\,\lambda_{\times}$ , $\beta_{Z_{a}}=2\pi>0$ | $Z_{2a}(t)=-2\pi\lambda_{\times}\,(t-t_{0})\,+Z_{a}(t_{0})$

###  4.5 Integration at arbitrary loops

The power counting of the model $\times$, provided by Proposition 2, gives us
a remarkable amount of information on the $\beta$-functions of the couplings
in an arbitrarily high order in perturbation theory. The information is
stringent enough to let us present explicit general forms of $\beta$-functions
of the couplings in the model $\times$ at all orders in perturbation theory.

4-pt couplings $\lambda$ and $\lambda_{\times}$ RG equation – The power
counting theorem of the model $\times$ with Proposition 2 determines that at
all orders in pertubation theory, there are no amplitudes which are divergent
contributing to the renormalization of 4-pt couplings $\lambda$ and
$\lambda_{\times}$. Hence, $\lambda$ and $\lambda_{\times}$ of the model
$\times$ are constant and do not flow with scales, therefore

$\displaystyle\partial_{t}\lambda$ $\displaystyle=$ $\displaystyle 0\,,$ (176)
$\displaystyle\partial_{t}\lambda_{\times}$ $\displaystyle=$ $\displaystyle
0\,,$ (177)

which trivially yield

$\displaystyle\lambda(t)$ $\displaystyle=$ $\displaystyle cste\,,$ (178)
$\displaystyle\lambda_{\times}(t)$ $\displaystyle=$ $\displaystyle cste\,.$
(179)

Mass, 2-pt couplings $Z_{a}$ and $Z_{2a}$ RG equations – Observation of
Proposition 2 tells us that the mass renormalization is decided by the class
III, where only exactly one $\lambda$ and a number of $\lambda_{\times}$
contribute.

The $Z_{a}$ renormalization is decided by the class I, where only
$\lambda_{\times}$ contributes.

We also notice that only $\lambda_{\times}$ contributes to the renormalization
of $Z_{2a}$, as class II dictates.

With (179), one immediately conclude that the coupled differential RG
equations for the couplings of the model $\times$, which generalize trivially
the equations for the first order RG equations (156), (165), and (174), to
arbitrary $n$-th orders, by introducing polynomials $P_{n}(\lambda_{\times})$,
$Q_{n}(\lambda_{\times})$, and $R_{n}(\lambda_{\times})$, are

$\displaystyle\partial_{t}\mu(t)$ $\displaystyle=$
$\displaystyle\lambda\,P_{n}(\lambda_{\times})\,,$ (180)
$\displaystyle\partial_{t}Z_{a}(t)$ $\displaystyle=$ $\displaystyle
Q_{n}(\lambda_{\times})\,,$ (181) $\displaystyle\partial_{t}Z_{2a}(t)$
$\displaystyle=$ $\displaystyle R_{n}(\lambda_{\times})\,.$ (182)

One can immediately solve such system of equations and obtain too all orders
in perturbation theory,

$\displaystyle\mu(t)$ $\displaystyle=$
$\displaystyle(t-t_{0})\lambda\,P_{n}(\lambda_{\times})+\mu(t_{0})\,,$ (183)
$\displaystyle Z_{a}(t)$ $\displaystyle=$
$\displaystyle(t-t_{0})Q_{n}(\lambda_{\times})+Z_{a}(t_{0})\,,$ (184)
$\displaystyle Z_{2a}(t)$ $\displaystyle=$
$\displaystyle(t-t_{0})R_{n}(\lambda_{+})+Z_{2a}(t_{0})\,,$ (185)

where we notice that all couplings above grow linearly in $t$ in their
respective magnitudes.

##  5 Conclusion

We have computed explicitly the one-loop $\beta$-functions of the couplings of
enhanced tensor field theories, the model $+$ and the model $\times$ in the
first order in perturbation theory, summarized in Table 3.4 and Table 4.4. The
system RG flow equations can be explicitly computed.

Both models $+$ and $\times$ do have a constant wavefunction renormalization
($Z_{b}=1$). Nevertheless, we have obtained some nontrivial renormalization
group flows in some of the couplings.

For the model $+$, the enhanced $4$-pt coupling $\lambda_{+}$ does not run and
therefore is constant, whereas the ordinary $4$-pt coupling $\lambda$ grows
linearly in $t=\log k/k_{0}$, where $k$ is momentum scale, in its magnitude.
This statement is true for all orders in perturbation theory. Once we include
all orders of perturbation theory, we may find asymptotic safety for the
coupling $\lambda$. The $2$-pt coupling $Z_{a}$ has an exponential behavior in
$t$. To the first order in perturbation theory, we found that the mass behaves
different from relevant operators and grows linearly (but negative) in $t$ in
the UV. In the IR, the mass behaves like a relevant operator and blows up
exponentially. Additionally, the renormalization equation for the mass to all
orders in perturbation theory is also predicted in (133).

For the model $\times$, the $4$-pt couplings $\lambda$ and $\lambda_{\times}$
do not run and therefore constant at all orders. On the other hand, all the
$2$-pt couplings (mass, $Z_{a}$ and $Z_{2a}$) grow in their magnitude linearly
in the scale $t=\log k/k_{0}$.

Furthermore, with the explicit forms of the running of the couplings at first
order, together with the observation from the power counting theorems for each
model that we obtained earlier in [26], we are able to deduce the behaviors of
$\beta$-functions of couplings to all orders in perturbation theory for both
the models $+$ and $\times$, see Section 3.5 and Section 4.5. We will leave it
in the future, to compute precisely the coefficients of all-loop order. The
$\beta$-functions of these models seem simple enough that we should be able to
resum at all orders in perturbation theory.

Another interesting study in the future is to interpret the propagator
differently and compute the renormalization group flows in perturbation theory
based on such a covariance measure. In our current study, we chose the kinetic
term to be ${\rm Tr}(p^{2b}\phi^{2})+\mu\phi^{2}$. However, one could have
chosen the kinetic term for the model $+$ to be composed of several terms
$Z_{b}\,{\rm Tr}_{2}(p^{2b}\phi^{2})+Z_{a}\,{\rm
Tr}_{2}(p^{2a}\phi^{2})+\mu\,{\rm Tr}_{2}(\phi^{2})$ . This will entail that
we redefine the field as:

$\phi\to Z_{b}^{1/4}Z_{a}^{1/4}\phi\,,$ (186)

and lead us to an action in the form of

$\displaystyle{\rm
Tr}_{2}\Bigg{(}\big{(}Z_{b}^{1/4}Z_{a}^{1/4}{\bar{\phi}}\big{)}\cdot\Big{(}\frac{\sqrt{Z_{b}}}{\sqrt{Z_{a}}}|p|^{2b}+\frac{\sqrt{Z_{a}}}{\sqrt{Z_{b}}}|p|^{2a}+\frac{\mu}{\sqrt{Z_{a}Z_{b}}}\Big{)}\cdot\big{(}Z_{b}^{1/4}Z_{a}^{1/4}\phi\big{)}\Bigg{)}$
(187) $\displaystyle+$ $\displaystyle\frac{\lambda_{+}}{2\,Z_{a}Z_{b}}{\rm
Tr}_{4}\big{(}p^{2a}\phi^{4}\big{)}+\frac{\lambda}{2\,Z_{a}\,Z_{b}}{\rm
Tr}_{4}(\phi^{4})\,.$ (188)

In the case that $Z_{b}$ is still a constant and $Z_{a}$ divergent, this will
send ${\sqrt{Z_{b}}}/{\sqrt{Z_{a}}}\rightarrow 0$. Therefore, one may
interpret that even though we started out with a theory with a kinetic term
${\rm Tr}_{2}(p^{2b}\phi^{2})+{\rm Tr}_{2}(p^{2a}\phi^{2})+\mu{\rm
Tr}_{2}(\phi^{2})$, we may flow toward and end up with a theory with a kinetic
term with only ${\rm Tr}_{2}(p^{2a}\phi^{2})+\mu{\rm Tr}_{2}(\phi^{2})$ .

##  6 Acknowledgements

We would like to thank Dario Benedetti, Sylvain Carrozza, Riccardo Martini,
and Fabien Vignes-Tourneret for the insightful discussions. The authors would
also like to thank the thematic program "Quantum Gravity, Random Geometry, and
Holography" 9 January - 17 February 2023 at Institut Henri Poincaré, Paris,
France for the platform for discussions and the collaboration and letting us
progress further on this project. The authors acknowledge support of the
Institut Henri Poincaré (UAR 839 CNRS-Sorbonne Université) and LabEx CARMIN
(ANR-10-LABX-59-01).

##  Appendix

##  Appendix A Euler-Maclaurin formula and Feynman amplitude approximations

We briefly review the approximation of a discrete spectral sum by an integral
using Euler-Maclaurin formula [24]. In perturbation theory, this approximation
is well controlled and sufficient to achieve our calculations.

Consider $h_{n}(x)=x^{n}e^{-Ax^{a}}$, with $x\geq 0$, $n\in{\mathbbm{N}}$, and
the sum $\sum_{p=0}^{\infty}h_{n}(p)$. We use Euler-Maclaurin formula to
obtain for a finite integer $q\leq 1$,

$\sum_{p=1}^{q}h_{n}(p)=\int_{1}^{q}h_{n}(p)dp+R(q)\,,$ (A.1)

where

$R(q)=-B_{1}\big{(}h_{n}(1)+h_{n}(q)\big{)}+\sum_{k=1}^{\infty}\frac{B_{2k}}{2k!}\big{(}h_{n}^{(2k-1)}(q)-h_{n}^{(2k-)}(1)\big{)}\,,$
(A.2)

where $B_{k}$ are Bernoulli numbers and $h^{(2k-1)}$ denotes $2k-1^{\rm th}$
derivative of $h_{n}(p)$ with respect to $p$.

One can show that

${\rm
lim}_{q\rightarrow\infty}R(q)=-B_{1}-\sum_{k=1}^{\infty}\frac{B_{2k}}{2k}\binom{n}{2k-1}+{\mathcal{O}}(A)={\mathcal{O}}(1)+{\mathcal{O}}(A)\,.$
(A.3)

For any $A$, the integral below is exact,

$\displaystyle{\rm
lim}_{q\rightarrow\infty}\int_{1}^{q}h_{n}(p)dp=\frac{1}{a}A^{-\frac{1+n}{a}}\Gamma\Big{[}\frac{1+n}{a},A\Big{]}=\frac{1}{a}A^{-\frac{1+n}{a}}\Gamma\Big{[}\frac{1+n}{a}\Big{]}-\frac{1}{1+n}+{\mathcal{O}}(A)\,,$
(A.4)

with the incomplete Gamma function $\Gamma[\cdot,\cdot]$, and Euler Gamma
function $\Gamma[\cdot]$. This then gives us

$\displaystyle\sum_{p=1}^{\infty}h_{n}(p)={\rm
lim}_{q\rightarrow\infty}\sum_{p=1}^{q}h_{n}(p)=\frac{1}{a}A^{-\frac{1+n}{a}}\Gamma\Big{[}\frac{1+n}{a}\Big{]}-\frac{1}{1+n}+{\mathcal{O}}(1)+{\mathcal{O}}(A)\,.$
(A.5)

Expanding $\widetilde{S}_{0,i}$. The goal is to approximate $S_{0,i}$ (79)
using the above developments. Use Euler-Maclaurin expansion, we compute the
sum

$\displaystyle\tilde{\tau}$ $\displaystyle=$
$\displaystyle\sum_{q\in{\mathbbm{Z}}}|q|^{4a}e^{-(\alpha+\alpha^{\prime})|q|^{2b}}=2\sum_{q=1}^{\infty}q^{4a}e^{-(\alpha+\alpha^{\prime})|q|^{2b}}$
(A.8)
$\displaystyle=2\int_{1}^{\infty}dq\,q^{4a}e^{-(\alpha+\alpha^{\prime})|q|^{2b}}+R=2\times\frac{1}{2b(\alpha+\alpha^{\prime})^{\frac{4a+1}{2b}}}\Gamma\left(\frac{4a+1}{2b},\alpha+\alpha^{\prime}\right)+R$
$\displaystyle=\frac{(\alpha+\alpha^{\prime})^{\frac{-(4a+1)}{2b}}}{b}\Gamma\left(\frac{4a+1}{2b},\alpha+\alpha^{\prime}\right)+R\,,$

where $\Gamma(s,x)=\int_{x}^{\infty}t^{s-1}e^{-t}dt$ denotes the upper
incomplete Gamma function. $R$ is the Euler-Maclaurin remainder which behaves,
according to (A.5), as
$R={\mathcal{O}}(1)+{\mathcal{O}}(\alpha+\alpha^{\prime})$. $\Gamma(s,x)$
admits a known asymptotic expansion when $x\to 0^{+}$ given by

$\displaystyle\Gamma(s,x)=\Gamma(s)-\frac{x^{s}}{s}+{\mathcal{O}}(x^{s+1})$
(A.9)

with the obstruction $s\notin\\{0,-1,-2,\dots\\}$. Now, for simplicity, we
constrain our model and use $D=1$, but keep $a$ and $b$ accordingly as given
by Theorem 1. We use the above expansion (A.9) in (A.8) as
$\alpha+\alpha^{\prime}$ is a small parameter (recalling that $\alpha$ and
$\alpha^{\prime}$ are small in the UV)

$\displaystyle\sum_{q\in{\mathbbm{Z}}}|q|^{4a}e^{-(\alpha+\alpha^{\prime})|q|^{2b}}$
(A.10)
$\displaystyle=\frac{(\alpha+\alpha^{\prime})^{\frac{-(4a+1)}{2b}}}{b}\left(\Gamma\left(\frac{4a+1}{2b}\right)-\frac{(\alpha+\alpha^{\prime})^{\frac{4a+1}{2b}}}{\frac{4a+1}{2b}}+{\mathcal{O}}\big{(}(\alpha+\alpha^{\prime})^{\frac{4a+1}{2b}+1}\big{)}\right)+R$
(A.11)
$\displaystyle=\frac{1}{b}\Gamma\left(\frac{4a+1}{2b}\right)(\alpha+\alpha^{\prime})^{\frac{-(4a+1)}{2b}}+{\mathcal{O}}(1)+{\mathcal{O}}(\alpha+\alpha^{\prime})\,.$
(A.12)

Having a look at (79), using
$e^{-(\alpha+\alpha^{\prime})\mu}=1+{\mathcal{O}}(\alpha+\alpha^{\prime})$, we
integrate this expression above over $\alpha$ and $\alpha^{\prime}$, with
$\frac{1}{b}\Gamma\left(\frac{4a+1}{2b}\right)=\frac{1}{b}\Gamma(2)=\frac{1}{b}$,
and write:

$\displaystyle\widetilde{S}_{0,i}$ $\displaystyle=$
$\displaystyle\int_{M^{-2bi}}^{M^{-2b(i-1)}}d\alpha\int_{M^{-2bi}}^{M^{-2b(i-1)}}d\alpha^{\prime}\Big{[}\frac{1}{b}(\alpha+\alpha^{\prime})^{-2}+{\mathcal{O}}\big{(}(\alpha+\alpha^{\prime})^{-1}\big{)}\Big{]}$
(A.13) $\displaystyle=$
$\displaystyle\int_{M^{-2bi}}^{M^{-2b(i-1)}}d\alpha\Big{[}-\frac{1}{b}(\alpha+\alpha^{\prime})^{-1}+{\mathcal{O}}\big{(}\log(\alpha+\alpha^{\prime})\big{)}\Big{]}_{\alpha^{\prime}=M^{-2bi}}^{\alpha^{\prime}=M^{-2b(i-1)}}$
(A.14) $\displaystyle=$
$\displaystyle\frac{1}{b}\Big{[}\log(\alpha+M^{-2bi})-\log(\alpha+M^{-2b(i-1)})$
(A.16)
$\displaystyle+\Big{(}\;{\mathcal{O}}\big{(}\alpha)+{\mathcal{O}}\big{(}\alpha\log(\alpha+M^{-2bi})\big{)}+{\mathcal{O}}\big{(}M^{-2bi}\log(\alpha+M^{-2bi})\big{)}\Big{)}\Big{]}_{\alpha=M^{-2bi}}^{\alpha=M^{-2b(i-1)}}$
$\displaystyle=$
$\displaystyle\frac{1}{b}\Big{[}-\log(2M^{-2bi})+2\log(M^{-2b(i-1)}+M^{-2bi})-\log(2M^{-2b(i-1)})$
(A.18)
$\displaystyle+\;{\mathcal{O}}\big{(}M^{-2bi})+{\mathcal{O}}\big{(}M^{-2bi}\log(M^{-2bi})\big{)}\Big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{b}\Big{[}2\log[M^{-2bi}(M^{2b}+1)]-\log(2M^{-2bi})-\log(2M^{-2bi+2b})\Big{]}$
(A.20)
$\displaystyle+{\mathcal{O}}(M^{-2bi})+{\mathcal{O}}\big{(}M^{-2bi}\log(M^{-2bi})\big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{b}\Big{[}2\log(M^{2b}+1)-\log(4)-\log(M^{2b})\Big{]}+{\mathcal{O}}(M^{-2bi})+{\mathcal{O}}\big{(}M^{-2bi}\log(M^{-2bi})\big{)}$
(A.22) $\displaystyle=$
$\displaystyle\frac{1}{b}\Big{[}\log\frac{(M^{2b}+1)^{2}}{4M^{2b}}\Big{]}+{\mathcal{O}}(M^{-2bi})+{\mathcal{O}}\big{(}M^{-2bi}\log(M^{-2bi})\big{)}$
(A.23)

Thus $\widetilde{S}_{0,i}$ is approximated by

$\displaystyle\widetilde{S}_{0,i}=\frac{1}{b}\log\frac{(M^{2b}+1)^{2}}{4M^{2b}}+{\mathcal{O}}(M^{-2bi}\log(M^{-2bi}))$
(A.25)

As $M\geq 1$, $\frac{(M^{2b}+1)^{2}}{(4M^{2b})}>1$, then the leading
coefficient in the above expanding is positive.

We can put the above in a standard $\log M$ approximation by expanding at
large enough $M\gg 1$:

$\displaystyle\frac{1}{b}\log\frac{(M^{2b}+1)^{2}}{(4M^{2b})}=\frac{1}{b}\Big{[}\log(M^{2b})+2\log(1+1/M^{2b})-\log(4)\Big{]}$
(A.26) $\displaystyle=\frac{1}{b}\log
M^{2b}+\frac{2}{b}\left(\frac{1}{M^{2b}}-\log(2)\right)+{\mathcal{O}}(M^{-4b})$
(A.27)

At large $M$, the first term will correspond to the standard $\log\Lambda$
divergence for marginal coupling.

Dimensionful computations for $S_{0,i}$. We address now the crucial question
of the dimension in our computation. The previous calculations of were
performed without taking care of that aspect that we now restore. The
couplings and fields have scaling dimensions. Expressing the propagator in
Schwinger parameterization as in (46), we notice that $\alpha$ should have a
dimension of $-2b$ in units of momentum scale $k$. In other words, we can
write $\alpha=k^{-2b}\,{\tilde{\alpha}}$ where ${\tilde{\alpha}}$ is
dimensionless. There is a dimensionful quantity $\tau$ associated with
$\tilde{\tau}$ (A.8), in which $q$ acquires dimension of $1$ in the units of
the momentum scale. The integral approximates the discrete sum over $q$ as in
the second line of (A.8), but gains one more dimension in the computation of
$\tau$. A similar fact concerning discrete sums having scaling dimensions was
advocated and used in [35]. This also can be understood by the power counting
theorem: in order to make the couplings marginal, so $\log$-divergent in the
cut-off, discrete sums must carry scale dimension. We perform the following
change of variables to let the dimensions be explicit in terms of a momentum
scale $k$:

$\displaystyle q=k\tilde{q}\,,\qquad\tilde{q}\in{\mathbbm{Z}}$ (A.28)
$\displaystyle\alpha=k^{-2b}\,{\tilde{\alpha}}$ (A.29)

We obtain in terms of dimensionless $\tilde{\tau}$ and $\widetilde{S}_{0,i}$:

$\displaystyle\tau=k^{4a+1}\tilde{\tau}$ (A.30) $\displaystyle
S_{0,i}=k^{4a+1}k^{-4b}\widetilde{S}_{0,i}=\widetilde{S}_{0,i}\,,$ (A.31)

where in the last equality, we notice $4a+1-4b=0$, and therefore we get
$S_{0,i}$ (80).

Expanding $\widetilde{S}_{1,i}$ and $\widetilde{S}_{2,i}$ for model $+$. We
use a similar technique to provide an approximation of $S_{1,i}$ (110) and
$S_{2,i}$ (111).

Express the propagators in Schwinger representation and carefully converting
the above sums into integrals and get

$\displaystyle\widetilde{S}_{2,i}=\int_{0}^{\infty}d\alpha\chi^{i}(\alpha)e^{-\alpha\mu_{i}}\sum_{q\in{\mathbbm{Z}}}|q|^{2a}e^{-\alpha|q|^{2b}}\,.$
(A.32)

This integral is similar to (A.8), changing only $\alpha+\alpha^{\prime}$ to
$\alpha$ and for a particular choice of $2a$. We perform an approximation in
an analogous way as before and get:

$\displaystyle\sum_{q\in{\mathbbm{Z}}}|q|^{2a}e^{-\alpha|q|^{2b}}=\frac{\alpha^{\frac{-(2a+1)}{2b}}}{b}\Gamma\left(\frac{2a+1}{2b},\alpha\right)+R=\frac{1}{b}\Gamma\left(\frac{2a+1}{2b}\right)\alpha^{\frac{-(2a+1)}{2b}}+{\mathcal{O}}(1)$
(A.33)

We insert this expression in (A.32) and obtain

$\displaystyle\widetilde{S}_{2,i}$ $\displaystyle=$
$\displaystyle\int_{M^{-2bi}}^{M^{-2b(i-1)}}d\alpha
e^{-\alpha\mu_{i}}\Big{[}c_{1;a,b}\,\alpha^{\frac{-(2a+1)}{2b}}+{\mathcal{O}}(1)\Big{]}$
(A.34) $\displaystyle=$
$\displaystyle\int_{M^{-2bi}}^{M^{-2b(i-1)}}d\alpha(1+{\mathcal{O}}(\alpha))\Big{[}c_{1;a,b}\,\alpha^{\frac{-(2a+1)}{2b}}+{\mathcal{O}}(1)\Big{]}$
(A.35) $\displaystyle=$
$\displaystyle\int_{M^{-2bi}}^{M^{-2b(i-1)}}d\alpha\Big{[}c_{1;a,b}\,\alpha^{\frac{-(2a+1)}{2b}}+{\mathcal{O}}\big{(}\alpha^{1-\frac{(2a+1)}{2b}}\big{)}\Big{]}$
(A.36) $\displaystyle=$
$\displaystyle\Big{[}\frac{1}{1-\frac{(2a+1)}{2b}}\,c_{1;a,b}\,\alpha^{1-\frac{(2a+1)}{2b}}+{\mathcal{O}}\big{(}\alpha^{2-\frac{(2a+1)}{2b}}\big{)}\Big{]}_{M^{-2bi}}^{M^{-2b(i-1)}}\,,$
(A.37)

where $c_{1;a,b}=\frac{1}{b}\Gamma\left(\frac{2a+1}{2b}\right)$.

For the model $+$, we use $a=\frac{1}{2}D(d^{-}-1)$ and
$b=\frac{1}{2}D(d^{-}-\frac{1}{2})$ for just-renormalizability (see
Proposition 1) with $D=1$, then,
$\frac{2a+1}{2b}=\frac{2(d-1)}{2d-3}=1+\frac{1}{2d-3}$. Thus

$\displaystyle\widetilde{S}_{2,i}$ $\displaystyle=$
$\displaystyle\Big{[}-\frac{(2d-3)}{(2d-3)/4}\,\Gamma\left(\frac{2(d-1)}{2d-3}\right)\alpha^{-1/(2d-3)}+{\mathcal{O}}\big{(}\alpha^{1-1/(2d-3)}\big{)}\Big{]}_{M^{-2bi}}^{M^{-2b(i-1)}}$
(A.38) $\displaystyle=$ $\displaystyle
4\,\Gamma\Big{(}\frac{2(d-1)}{2d-3}\Big{)}(M^{-2bi(-1/4b)}-M^{-2b(i-1)(-1/4b)})+{\mathcal{O}}\big{(}M^{-i(d-2)}\big{)}$
(A.39) $\displaystyle=$ $\displaystyle
4\,\Gamma\Big{(}\frac{2(d-1)}{2d-3}\Big{)}M^{i/2}(1-M^{-1/2})+{\mathcal{O}}\big{(}M^{-i(d-2)}\big{)}\,.$
(A.40)

Now we compute $\widetilde{S}_{1,i}$ namely,

$\displaystyle\widetilde{S}_{1,i}=\sum_{{\bf
q}\in{\mathbbm{Z}}^{d-1}}\int_{0}^{\infty}d\alpha\;\chi^{i}(\alpha)\,e^{-\alpha(|{\bf
q}|^{2b}+\mu_{i})}=\int_{0}^{\infty}d\alpha\chi^{i}(\alpha)e^{-\alpha\mu_{i}}\left(\sum_{q\in{\mathbbm{Z}}}e^{-\alpha|q|^{2b}}\right)^{d-1}\,.$
(A.41)

Let us focus on a single sum over $q$:

$\displaystyle\sum_{q\in{\mathbbm{Z}}}e^{-\alpha|q|^{2b}}=2\sum_{q=1}^{\infty}e^{-\alpha
q^{2b}}+1=2\int_{1}^{\infty}dqe^{-\alpha q^{2b}}+R+1$ (A.42)
$\displaystyle=2\Bigg{(}\frac{1}{2b}\alpha^{-\frac{1}{2b}}\Gamma\left(\frac{1}{2b},\alpha\right)\Bigg{)}+R+1\,,$
(A.43)

where $R={\mathcal{O}}(1)+{\mathcal{O}}(\alpha)$. We use (A.9) in (A.43) as
$\alpha$ is a small parameter in the UV,

$\sum_{q\in{\mathbbm{Z}}}e^{-\alpha|q|^{2b}}=\frac{1}{b}\alpha^{-\frac{1}{2b}}\Big{[}\Gamma\left(\frac{1}{2b}\right)-2b\alpha^{\frac{1}{2b}}+{\mathcal{O}}(\alpha^{\frac{1}{2b}+1})\Big{]}+R+1=\frac{1}{b}\Gamma\left(\frac{1}{2b}\right)\alpha^{-\frac{1}{2b}}+{\mathcal{O}}(1)\,.$
(A.44)

We insert this expression in (A.41),

$\displaystyle\widetilde{S}_{1,i}=\int_{0}^{\infty}d\alpha\chi^{i}(\alpha)e^{-\alpha\mu_{i}}\left(\frac{1}{b}\Gamma\left(\frac{1}{2b}\right)\alpha^{-\frac{1}{2b}}+{\mathcal{O}}(1)\right)^{d-1}$
(A.45) $\displaystyle=$
$\displaystyle\int_{M^{-2bi}}^{M^{-2b(i-1)}}d\alpha\,e^{-\alpha\mu_{i}}\Big{[}\Bigg{(}\frac{1}{b}\Gamma\left(\frac{1}{2b}\right)\Bigg{)}^{d-1}\alpha^{-\frac{d-1}{2b}}+{\mathcal{O}}(\alpha^{-\frac{d-2}{2b}})\Big{]}$
(A.46) $\displaystyle=$
$\displaystyle\int_{M^{-2bi}}^{M^{-2b(i-1)}}d\alpha(1+{\mathcal{O}}(\alpha))\Big{[}\Bigg{(}\frac{1}{b}\Gamma\left(\frac{1}{2b}\right)\Bigg{)}^{d-1}\alpha^{-\frac{d-1}{2b}}+{\mathcal{O}}(\alpha^{-\frac{d-2}{2b}})\Big{]}$
(A.47) $\displaystyle=$
$\displaystyle\Big{[}\Bigg{(}\frac{1}{b}\Gamma\left(\frac{1}{2b}\right)\Bigg{)}^{d-1}\frac{1}{1-\frac{d-1}{2b}}\alpha^{1-\frac{d-1}{2b}}+{\mathcal{O}}(\alpha^{1-\frac{d-2}{2b}})\Big{]}_{M^{-2bi}}^{M^{-2b(i-1)}}$
(A.48) $\displaystyle=$
$\displaystyle\Bigg{(}\frac{1}{b}\Gamma\left(\frac{1}{2b}\right)\Bigg{)}^{d-1}\frac{2b}{2b-d+1}M^{-i(2b-d+1)}\Big{(}M^{(2b-d+1)}-1\Big{)}+{\mathcal{O}}(M^{-i(2b-d+2)})\,,\qquad$
(A.49)

For the model $+$, we use $b=\frac{1}{2}D(d^{-}-\frac{1}{2})$ for just-
renormalizability (see Proposition 1) with $D=1$, then:

$\widetilde{S}_{1,i}=\Bigg{(}\frac{1}{b}\Gamma\left(\frac{1}{2b}\right)\Bigg{)}^{d-1}(2d-3)M^{i/2}\Big{(}1-M^{-1/2}\Big{)}+{\mathcal{O}}(M^{-i/2})\,.$
(A.50)

$\widetilde{S}_{1,i}$ and $\widetilde{S}_{2,i}$ for the model $\times$. For
the model $\times$, from the Proposition 2, we have a specific set of
parameters, i.e., $D=1$, $d=3$, $a=\frac{1}{2}$, $b=1$, and
$\frac{2a+1}{2b}=1$. We then specialize the previous computation. Starting
from (A.35), we have

$\displaystyle\widetilde{S}_{2,i}$ $\displaystyle=$
$\displaystyle\int_{M^{-2i}}^{M^{-2(i-1)}}d\alpha(1+{\mathcal{O}}(\alpha))\Big{[}\alpha^{-1}+{\mathcal{O}}(1)\Big{]}$
(A.51) $\displaystyle=$
$\displaystyle\Big{[}\log\alpha+{{\mathcal{O}}(\alpha)}\Big{]}_{M^{-2i}}^{M^{-2(i-1)}}$
(A.52) $\displaystyle=$ $\displaystyle 2\log M+{\mathcal{O}}(M^{-2(i-1)})\,.$
(A.53)

For the model $\times$, from the Proposition 2, we have a specific set of
parameters, i.e., $D=1$, $d=3$, $a=\frac{1}{2}$, $b=1$, and
$\frac{d-1}{2b}=1$. So recomputing (A.49), we get:

$\displaystyle\widetilde{S}_{1,i}$ $\displaystyle=$
$\displaystyle\int_{M^{-2i}}^{M^{-2(i-1)}}d\alpha(1+{\mathcal{O}}(\alpha))\Big{[}\Big{(}\Gamma\left(\frac{1}{2}\right)\Big{)}^{2}\alpha^{-1}+{\mathcal{O}}(\alpha^{-\frac{1}{2}})\Big{]}$
(A.54) $\displaystyle=$
$\displaystyle\Big{[}\pi\log\alpha+{\mathcal{O}}(\alpha^{\frac{1}{2}})\Big{]}^{M^{-2(i-1)}}_{M^{-2i}}$
(A.55) $\displaystyle=$ $\displaystyle 2\pi\log
M+{\mathcal{O}}\big{(}{M^{-i}}\big{)}\,,$ (A.56)

where we substitute $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$.

Dimensionful computations of $S_{1,i}$ (110) and $S_{2,i}$ (111). Following
the similar argument as for $S_{0,i}$, we perform the changes of variables
given in (A.29) so that the dimensions are explicit in computations. Then, we
obtain in terms of dimensionless $\widetilde{S}_{1,i}$ and
$\widetilde{S}_{2,i}$:

$\displaystyle S_{1,i}=k^{-2b}k^{d-1}\widetilde{S}_{1,i}$ (A.57)
$\displaystyle S_{2,i}=k^{-2b}k^{2a}k\widetilde{S}_{2,i}\,,$ (A.58)

where in the last equalities, we note from Propositions 1 and 2 that

$\displaystyle d-1-2b=2a-2b+1=\begin{cases}\frac{1}{2},&\text{for the model
}+\\\ 0,&\text{for the model }\times\end{cases}$ (A.59)

##  Appendix B Second order in perturbation in Model $+$.

###  B.1 4-point function in Model $+$.

  * •

The graph $n_{42}^{(c)}$ has degree of divergence $\omega_{n_{42}^{(c)}}=0$,
following the first row of Table 1, and its amplitude appears in the second
order in perturbation theory.

$c$

$c$

$c$

$c$

$c$

$c$

$q_{1}$

$q_{2}$

$q_{c}$

Figure 11:  The two-loop divergent graph, $n_{42}^{(c)}$ that contributes to
the renormalization of $4$-pt function in $\Gamma_{4}^{(c)}(\\{p\\})$. Here
the rank $d=3$.

Explicitly, at two-loop,

$\displaystyle K_{n_{42}^{(c)}}=12\,,$ (B.60) $\displaystyle
S_{n_{42}^{(c)}}(\\{\mathbf{p},\mathbf{p}^{\prime}\\})=\frac{1}{3!}\Big{(}\frac{-\lambda^{(c)}_{+}}{2}\Big{)}^{3}\sum_{\\{q\\}}\frac{\big{(}|q_{c}|^{2a}\big{)}^{3}}{(|\mathbf{p}_{\check{c}}|^{2b}+|q_{c}|^{2b}+\mu)(|\mathbf{p}^{\prime}_{\check{c}}|^{2b}+|q_{c}|^{2b}+\mu)^{2}(|\mathbf{q}|^{2b}+\mu)^{2}}\,.$

The contribution to the Feynman amplitude is given by

$\sum_{c}K_{n_{42}^{(c)}}\,S_{n_{42}^{(c)}}\,.$ (B.61)

###  B.2 $\Gamma_{2}$ in Model $+$.

  * •

For $m_{2e}^{(c\,c^{\prime})}$, $\omega_{m_{2e}^{(c\,c^{\prime})}}=0$. This
graph belongs to the class V in Table 1, and appears in the second order in
perturbation theory.

$q_{3}$

$q_{{c^{\prime}=2}}$

$p_{c=1}$

$p_{2}$

$p_{3}$

$c^{\prime}=2$

$c=1$

$c=1$

Figure 12: The graph $m_{2e}^{(c=1\,c^{\prime}=2)}$. For $d=3$ is illustrated.
$\\{q_{\check{c}}\\}=\\{q_{c^{\prime}},\;q_{3}\\}$, and ${|{\bf
q}_{\check{c}}|}^{2b}={|q_{c^{\prime}}|}^{2b}+{|q_{3}|}^{2b}$.

The contribution to the Feynman amplitude is then,

$\displaystyle\sum_{c^{\prime}}K_{m_{2e}^{(c\,c^{\prime})}}S_{m_{2e}^{(c\,c^{\prime})}}(\\{p\\})$
$\displaystyle=$ $\displaystyle\sum_{c^{\prime}\neq c}^{(d-1)\;{\rm
terms}}2\Big{[}\Big{(}-\frac{{\lambda_{+}}^{(c)}}{2}\Big{)}(-Z_{a}^{(c^{\prime})})\Big{]}\sum_{\\{q_{\hat{c}}\\}}\frac{{|q_{c^{\prime}}|}^{2a}{|p_{c}|}^{2a}}{({|{\bf
q}_{{(\check{c})}}|}^{2b}+|{p_{c}|}^{2b}+\mu)^{2}}$ (B.62) $\displaystyle=$
$\displaystyle\lambda_{+}^{(c)}{|p_{c}|}^{2a}\sum_{c^{\prime}\neq
c}^{(d-1)\;{\rm
terms}}Z_{a}^{(c^{\prime})}\sum_{\\{q_{\check{c}}\\}}\frac{{|q_{c^{\prime}}|}^{2a}}{({|{\bf
q}_{{(\check{c})}}|}^{2b}+{|p_{c}|}^{2b}+\mu)^{2}}$ $\displaystyle=$
$\displaystyle\lambda_{+}^{(c)}{|p_{c}|}^{2a}\sum_{\\{q_{\check{c}}\\}}\frac{\sum_{c^{\prime}\neq
c}Z_{a}^{(c^{\prime})}{|q_{c^{\prime}}|}^{2a}}{({|{\bf
q}_{{(\check{c})}}|}^{2b}+{|p_{c}|}^{2b}+\mu)^{2}}\,,$ (B.63)

where $K_{m_{2e}^{(c\,c^{\prime})}}=2$.

  * •

For $meme_{2}^{(c\,c^{\prime})}$,
$\omega_{meme_{2}^{(c\,c^{\prime})}}=\frac{D}{2}$. This graph belongs to the
class II in Table 1, and appears in the second order in perturbation theory.

$c^{\prime}=2$

$q_{3}$

$q_{1}$

$q_{2}$

$p_{2}$

$p_{3}$

$p_{1}$

$c^{\prime}=2$

$c=1$

$c=1$

$q^{\prime}_{3}$

Figure 13:  For $d=3$, $meme_{2}^{(c=1\,c^{\prime}=2)}$ is illustrated.
$\\{q^{\prime}_{({\check{c}}{\check{c^{\prime}}})}\\}={q^{\prime}}_{3}$,
$\\{q\\}=\\{q_{1},q_{2},q_{3}\\}$, ${|{\bf
q}|}^{2b}={|q_{1}|}^{2b}+{|q_{2}|}^{2b}+{|q_{3}|}^{2b}$, ${|{\bf
q}_{({\check{c}}{\check{c^{\prime}}})}|}^{2b}={|q_{3}|}^{2b}$.

$\displaystyle\sum_{c^{\prime}\neq
c}K_{meme_{2}^{(c,c^{\prime})}}S_{meme_{2}^{(c,c^{\prime})}}(\\{p\\})$ (B.65)
$\displaystyle=$ $\displaystyle\sum_{c^{\prime}\neq c}^{(d-1)\;{\rm
terms}}K_{meme_{2}^{(c,c^{\prime})}}\Big{[}\Big{(}-\frac{\lambda^{(c)}}{2}\Big{)}\Big{(}-\frac{\lambda_{+}^{(c^{\prime})}}{2}\Big{)}\Big{]}{|p_{c}|}^{2a}$
$\displaystyle\qquad\quad\sum_{\\{q^{\prime}_{({\check{c}}{\check{c^{\prime}}})}\\},\\{q\\}}\frac{{|q_{c^{\prime}}|}^{2a}}{({|{\bf
q}|}^{2b}+\mu)({|{\bf
q^{\prime}}_{({\check{c}}{\check{c^{\prime}}})}|}^{2b}+{|q_{c^{\prime}}|}^{2b}+{|p_{c}|}^{2b}+\mu)^{2}}$
$\displaystyle=$
$\displaystyle\lambda^{(c)}{|p_{c}|}^{2a}\sum_{\\{q^{\prime}_{({\check{c}})}\\},\\{q\\}}\frac{\sum_{c^{\prime}\neq
c}\lambda_{+}^{(c^{\prime})}{|q_{c^{\prime}}|}^{2a}}{({|{\bf
q}|}^{2b}+\mu)({|{\bf
q^{\prime}}_{({\check{c}}{\check{c^{\prime}}})}|}^{2b}+{|q_{c^{\prime}}|}^{2b}+{|p_{c}|}^{2b}+\mu)^{2}}\,,\quad$
(B.66)

where $K_{meme_{2}^{(c,c^{\prime})}}=4$.

###  B.3 Self energy and mass renormalization in Model $+$.

  * •

For the graph $m_{2}^{(c,c^{\prime})}$, the degree of divergence is given by
$\omega_{d;+}(m_{2}^{(c,c^{\prime})})=0$. This graph belongs to the class VI
in Table 1, and also, appears in the second order in Taylor expansion of the
Feynman amplitude.

$c^{\prime}$

$c$

$c$

$p_{c}$

$p_{2}$

$p_{1}$

$q_{1}$

$q_{{c^{\prime}}}$

Figure 14: $m_{2}^{(c,c^{\prime})}$ in colored and stranded representations.
Here, we consider $d=3$. ${|{\bf
q}_{\check{c}}|}^{2b}={|q_{c^{\prime}}|}^{2b}+{|q_{1}|}^{2b}$ and
$\\{q_{\check{c}}\\}=\\{q_{c^{\prime}},\;q_{1}\\}$.

The contribution of the graph $m_{2}^{(c,c^{\prime})}$ shown in Fig. 14 in the
amplitude is

$\displaystyle\sum_{c,c^{\prime}\neq
c}K_{m_{2}^{(c,c^{\prime})}}S_{m_{2}^{(c,c^{\prime})}}(\\{p\\})$
$\displaystyle=$ $\displaystyle\sum_{c,c^{\prime}\neq c}^{d\times(d-1)\;{\rm
terms}}2\Big{[}\Big{(}-\frac{\lambda^{(c)}}{2}\Big{)}\Big{(}-{Z_{a}}^{(c^{\prime})}\Big{)}\Big{]}\sum_{\\{q_{\check{c}}\\}}\frac{|q_{c^{\prime}}|^{2a}}{({|{\bf
q}_{\check{c}}|}^{2b}+|p_{c}|^{2b}+\mu)^{2}}$ (B.67) $\displaystyle=$
$\displaystyle\sum_{c=1}^{d}\lambda^{(c)}\sum_{\\{q_{\check{c}}\\}}\frac{\sum_{c^{\prime}\neq
c}{Z_{a}}^{(c^{\prime})}|q_{c^{\prime}}|^{2a}}{({|{\bf
q}_{\check{c}}|}^{2b}+|p_{c}|^{2b}+\mu)^{2}}\,,$

where $K_{m_{2}^{(c,c^{\prime}\neq c)}}=2$.

  * •

For the graph $mme_{2}^{(c,c^{\prime})}$, the superfical degree of divergence
is $\omega_{d;+}(mme_{2}^{(c,c^{\prime})})=\frac{D}{2}$. This graph belongs to
the class III in Table 1, and appears in the second order in Taylor expansion.

$c^{\prime}=2$

$q_{3}$

$q_{1}$

$q_{2}$

$p_{2}$

$p_{3}$

$p_{1}$

$c^{\prime}=2$

$c=1$

$c=1$

$q^{\prime}_{3}$

Figure 15: $mme_{2}^{(c=1,c^{\prime}=2)}$. For $d=3$ is shown above in both
colored and stranded representations. $\\{p\\}=\\{p_{1},p_{2},p_{3}\\}$ and
$\\{q^{\prime}_{({\check{c}}{\check{c^{\prime}}})}\\}={q^{\prime}}_{3}$,
$\\{q\\}=\\{q_{1},q_{2},q_{3}\\}$, ${|{\bf
q}|}^{2b}={|q_{1}|}^{2b}+{|q_{2}|}^{2b}+{|q_{3}|}^{2b}$, ${|{\bf
q}_{({\check{c}}{\check{c^{\prime}}})}|}^{2b}={|q_{3}|}^{2b}$.

The amplitude associated with the graph $mme_{2}^{(c,c^{\prime})}$ shown in
Fig. 15 is given by

$\displaystyle\sum_{c,c^{\prime}\neq
c}K_{mme_{2}^{(c,c^{\prime})}}S_{mme_{2}^{(c,c^{\prime})}}(\\{p\\})$ (B.68)
$\displaystyle=$ $\displaystyle\sum_{c,c^{\prime}\neq c}^{d\times(d-1)\;{\rm
terms}}4\Big{[}\Big{(}-\frac{\lambda^{(c)}}{2}\Big{)}\Big{(}-\frac{\lambda_{+}^{(c^{\prime})}}{2}\Big{)}\Big{]}$
(B.70)
$\displaystyle\qquad\qquad\sum_{\\{q^{\prime}_{({\check{c}}{\check{c^{\prime}}})}\\},\\{q\\}}\frac{{|q_{c^{\prime}}|}^{2a}}{({|{\bf
q}|}^{2b}+\mu)({|{\bf
q^{\prime}}_{({\check{c}}{\check{c^{\prime}}})}|}^{2b}+{|q_{c^{\prime}}|}^{2b}+{|p_{c}|}^{2b}+\mu)^{2}}$
$\displaystyle=$
$\displaystyle\sum_{c=1}^{d}\lambda^{(c)}\sum_{\\{q^{\prime}_{({\check{c}}{\check{c^{\prime}}})}\\},\\{q\\}}\frac{\sum_{c^{\prime}\neq
c}\lambda_{+}^{(c^{\prime})}{|q_{c^{\prime}}|}^{2a}}{({|{\bf
q}|}^{2b}+\mu)({|{\bf
q^{\prime}}_{({\check{c}}{\check{c^{\prime}}})}|}^{2b}+{|q_{c^{\prime}}|}^{2b}+{|p_{c}|}^{2b}+\mu)^{2}}\,,\quad$
(B.71)

where $K_{mme_{2}^{(c,c^{\prime})}}=4$.

  * •

For $n_{2}^{(c)}$, $\omega_{d;+}(n_{2}^{(c)})=0$. This graph belongs to the
class IV in Table 1 and its amplitude appears in the second order in Taylor
expansion.

$c^{\prime}=c$

$c$

$q_{(c)}=q_{(c^{\prime})}$

$p_{(3)}$

$p_{(2)}$

$p_{(1)}$

Figure 16:  The illustration of $n_{2}^{(c)}$ in $d=3$. ${|{\bf
p}_{\check{c}}|}^{2b}={|p_{1}|}^{2b}+{|p_{2}|}^{2b}+{|p_{3}|}^{2b}$, and
$\\{p\\}=\\{p_{1},\;p_{2},\;p_{3}\\}$

Now the amplitude of $n_{2}^{(c)}$ is

$\displaystyle\sum_{c}K_{n_{2}^{(c)}}S_{n_{2}^{(c)}}(\\{p\\})$
$\displaystyle=$
$\displaystyle\sum_{c=1}^{d}\Big{[}\Big{(}-\frac{{\lambda_{+}}^{(c)}}{2}\Big{)}(-Z_{a}^{(c)})\Big{]}\sum_{q_{c}}\frac{(|q_{c}|^{2a})^{2}}{(|q_{c}|^{2b}+{|{\bf
p}_{\check{c}}|}^{2b}+\mu)^{2}}\,,\qquad\quad$ (B.72)

where $K_{n_{2}^{(c)}}=1$.

  * •

For $nme_{2}^{(c)}$, $\omega_{d;+}(nme_{2}^{(c)})=\frac{D}{2}$. This graph
belongs to the class I in Table 1 and the second order in Taylor expansion for
the Feynman amplitude.

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1

1

2

3

2

3

$2$

$1$

$1$

$3$

$2$

$1$

$1$

$2$

$3$

$3$

$1$

$1$

$1$

$1$

Figure 17:  For $d=3$, $nme_{2}^{(c=1)}$ is shown above.
$\\{p\\}=\\{p_{1},p_{2},p_{3}\\}$ and ${|{\bf
p}_{\check{c}}|}^{2b}={|p_{2}|}^{2b}+{|p_{3}|}^{2b}$, ${|{\bf
q}|}^{2b}={|q_{1}|}^{2b}+{|q_{2}|}^{2b}+{|q_{3}|}^{2b}$.

$\displaystyle\sum_{c}K_{nme_{2}^{(c)}}S_{nme_{2}^{(c)}}(\\{p\\})$ (B.73)
$\displaystyle=$
$\displaystyle\sum_{c=1}^{d}K_{nme_{2}^{(c)}}\frac{1}{2!}\Big{[}\Big{(}-\frac{{\lambda_{+}}^{(c)}}{2}\Big{)}^{2}\Big{]}\sum_{\\{q\\}}\frac{({|q_{c}|}^{2a})^{2}}{({|{\bf
q}|}^{2b}+\mu)({|q_{c}|}^{2b}+{|{\bf p}_{\check{c}}|}^{2b}+\mu)^{2}}\,,\qquad$
(B.74)

where we set $K_{nme_{2}^{(c)}}=4$.

##  Appendix C Second order in perturbation in Model $\times$.

###  C.1 Self energy and mass renormalization in Model $\times$

  * •

For $mm_{ee}^{(c,c^{\prime})}$, the superfical degree of divergence is
$\omega_{d;\times}(mm_{ee}^{(c,c^{\prime})})=0$. This graph belongs to the
class III in Table 2, and appears in the second order
${\mathcal{O}}(\lambda\,\lambda_{\times})$ in Taylor expansion.

$c^{\prime}=2$

$q_{3}$

$q_{1}$

$q_{2}$

$p_{2}$

$p_{3}$

$p_{1}$

$c^{\prime}=2$

$c=1$

$c=1$

$q^{\prime}_{3}$

Figure 18: $mm_{ee}^{(c=1,c^{\prime}=2)}$. For $d=3$ is shown above in both
colored and stranded representations. $\\{p\\}=\\{p_{1},p_{2},p_{3}\\}$ and
$\\{q^{\prime}_{({\check{c}}{\check{c^{\prime}}})}\\}={q^{\prime}}_{3}$,
$\\{q\\}=\\{q_{1},q_{2},q_{3}\\}$, ${|{\bf
q}|}^{2b}={|q_{1}|}^{2b}+{|q_{2}|}^{2b}+{|q_{3}|}^{2b}$, ${|{\bf
q}_{({\check{c}}{\check{c^{\prime}}})}|}^{2b}={|q_{3}|}^{2b}$.

The amplitude associated with the graph $mm_{ee}^{(c,c^{\prime})}$ shown in
Fig. 18 is given by

$\displaystyle\sum_{c,c^{\prime}\neq
c}K_{mm_{ee}^{(c,c^{\prime})}}S_{mm_{ee}^{(c,c^{\prime})}}(\\{p\\})$ (C.75)
$\displaystyle=$ $\displaystyle\sum_{c,c^{\prime}\neq c}^{d\times(d-1)\;{\rm
terms}}4\Big{[}\Big{(}-\frac{\lambda^{(c)}}{2}\Big{)}\Big{(}-\frac{\lambda_{\times}^{(c^{\prime})}}{2}\Big{)}\Big{]}$
(C.77)
$\displaystyle\qquad\qquad\sum_{\\{q^{\prime}_{({\check{c}}{\check{c^{\prime}}})}\\},\\{q\\}}\frac{{|q_{c^{\prime}}|}^{4a}}{({|{\bf
q}|}^{2b}+\mu)({|{\bf
q^{\prime}}_{({\check{c}}{\check{c^{\prime}}})}|}^{2b}+{|q_{c^{\prime}}|}^{2b}+{|p_{c}|}^{2b}+\mu)^{2}}$
$\displaystyle=$
$\displaystyle\sum_{c=1}^{d}\lambda^{(c)}\sum_{\\{q^{\prime}_{({\check{c}}{\check{c^{\prime}}})}\\},\\{q\\}}\frac{\sum_{c^{\prime}\neq
c}\lambda_{\times}^{(c^{\prime})}{|q_{c^{\prime}}|}^{4a}}{({|{\bf
q}|}^{2b}+\mu)({|{\bf
q^{\prime}}_{({\check{c}}{\check{c^{\prime}}})}|}^{2b}+{|q_{c^{\prime}}|}^{2b}+{|p_{c}|}^{2b}+\mu)^{2}}\,,$
(C.78)

where $K_{mm_{ee}^{(c,c^{\prime})}}=4$.

###  C.2 $\Gamma_{2;a}$ in Model $\times$.

  * •

For $n_{ee}m_{ee}^{(c)}$, $\omega_{d;\times}(n_{ee}m_{ee}^{(c)})=0$. This
graph belongs to the class I in Table 2 and the second order in Taylor
expansion for the Feynman amplitude.

$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$1123112112332311211233231123321121123323

1

1

2

3

2

3

$2$

$1$

$1$

$3$

$2$

$1$

$1$

$2$

$3$

$3$

$1$

$1$

$1$

$1$

Figure 19:  For $d=3$, $n_{ee}m_{ee}^{(c=1)}$ is shown above.
$\\{p\\}=\\{p_{1},p_{2},p_{3}\\}$ and ${|{\bf
p}_{\check{c}}|}^{2b}={|p_{2}|}^{2b}+{|p_{3}|}^{2b}$, ${|{\bf
q}|}^{2b}={|q_{1}|}^{2b}+{|q_{2}|}^{2b}+{|q_{3}|}^{2b}$.

$\displaystyle K_{n_{ee}m_{ee}^{(c)}}S_{n_{ee}m_{ee}^{(c)}}(\\{p\\})$ (C.79)
$\displaystyle=$ $\displaystyle
8\,\frac{1}{2!}\Big{(}-\frac{{\lambda_{\times}}^{(c)}}{2}\Big{)}^{2}|p_{c}|^{2a}\sum_{\\{q\\}}\frac{({|q_{c}|}^{2a})^{3}}{({|{\bf
q}|}^{2b}+\mu)({|q_{c}|}^{2b}+{|{\bf p}_{(\check{c})}|}^{2b}+\mu)^{2}}\,,$

where we set $K_{n_{ee}m_{ee}^{(c)}}=8$.

###  C.3 $\Gamma_{2;2a}$ in Model $\times$.

  * •

For $m_{ee}m_{ee}^{(c,c^{\prime})}$,
$\omega_{d;\times}(m_{ee}m_{ee}^{(c,c^{\prime})})=0$. This graph belongs to
the class II in Table 2, and appears in the second order in perturbation
theory.

$c^{\prime}=2$

$q_{3}$

$q_{1}$

$q_{2}$

$p_{2}$

$p_{3}$

$p_{1}$

$c^{\prime}=2$

$c=1$

$c=1$

$q^{\prime}_{3}$

Figure 20:  For $d=3$, $m_{ee}m_{ee}^{(c=1,c^{\prime}=2)}$ is illustrated.
$\\{q^{\prime}_{({\check{c}}{\check{c^{\prime}}})}\\}={q^{\prime}}_{3}$,
$\\{q\\}=\\{q_{1},q_{2},q_{3}\\}$, ${|{\bf
q}|}^{2b}={|q_{1}|}^{2b}+{|q_{2}|}^{2b}+{|q_{3}|}^{2b}$, ${|{\bf
q}_{({\check{c}}{\check{c^{\prime}}})}|}^{2b}={|q_{3}|}^{2b}$.

$\displaystyle\sum_{c^{\prime}\neq
c}K_{m_{ee}m_{ee}^{(c,c^{\prime})}}S_{m_{ee}m_{ee}^{(c,c^{\prime})}}(\\{p\\})$
(C.81) $\displaystyle=$ $\displaystyle\sum_{c^{\prime}\neq c}^{(d-1)\;{\rm
terms}}K_{meme_{2}}^{(c,c^{\prime})}\Big{[}\Big{(}-\frac{\lambda_{\times}^{(c)}}{2}\Big{)}\Big{(}-\frac{\lambda_{\times}^{(c^{\prime})}}{2}\Big{)}\Big{]}{|p_{c}|}^{4a}$
$\displaystyle\qquad\quad\sum_{\\{q^{\prime}_{({\check{c}}{\check{c^{\prime}}})}\\},\\{q\\}}\frac{{|q_{c^{\prime}}|}^{4a}}{({|{\bf
q}|}^{2b}+\mu)({|{\bf
q^{\prime}}_{({\check{c}}{\check{c^{\prime}}})}|}^{2b}+{|q_{c^{\prime}}|}^{2b}+{|p_{c}|}^{2b}+\mu)^{2}}$
$\displaystyle=$
$\displaystyle\lambda_{\times}^{(c)}{|p_{c}|}^{4a}\sum_{\\{q^{\prime}_{({\check{c}})}\\},\\{q\\}}\frac{\sum_{c^{\prime}\neq
c}\lambda_{\times}^{(c^{\prime})}{|q_{c^{\prime}}|}^{4a}}{({|{\bf
q}|}^{2b}+\mu)({|{\bf
q^{\prime}}_{({\check{c}}{\check{c^{\prime}}})}|}^{2b}+{|q_{c^{\prime}}|}^{2b}+{|p_{c}|}^{2b}+\mu)^{2}}\,,\quad$
(C.82)

where $K_{m_{ee}m{ee}^{(c,c^{\prime})}}=4$.

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