\documentclass[12pt,reqno]{amsart}

\numberwithin{equation}{section}

\textheight 230mm
\textwidth 160mm
\hoffset -14mm
\voffset -26mm

\usepackage{amssymb}
%\usepackage{amsthm}
%\usepackage{amscd}
\usepackage{graphicx}
\usepackage{color}
\usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref}
%\usepackage{comment}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{fact}{Fact}[section]
\newtheorem{observation}{Observation}[section]
\newtheorem{claim}{Claim}[section]

\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{conjecture}{Conjecture}[section]
\newtheorem{open}{Open Problem}[section]
\newtheorem{problem}{Problem}[section]
\newtheorem{question}{Question}[section]

\theoremstyle{remark}
\newtheorem{remark}{Remark}[section]
\newtheorem{example}{Example}[section]
\newtheorem{note}{Note}[section]


\newcommand{\dd}{\mathrm{d}}
\newcommand{\prt}{\partial}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\def\ba#1\ea{\begin{align}#1\end{align}}
\def\bas#1\eas{\begin{align*}#1\end{align*}}
\newcommand{\bd}{\begin{displaymath}}
\newcommand{\ed}{\end{displaymath}}
\newcommand{\nn}{\nonumber\\}
\newcommand{\whr}{\ \textrm{, where} \ }
\newcommand{\mand}{\quad \textrm{and} \quad}
\newcommand{\mor}{\quad \textrm{or} \quad}
\newcommand{\mt}{\textrm}
\newcommand{\mb}{\mathbb}
\newcommand{\mc}{\mathcal}
\newcommand{\mf}{\mathfrak}
\newcommand{\id}{\mathrm{id}}
\newcommand{\tr}{\mathrm{tr}}
\newcommand{\Tr}{\mathrm{Tr}}
\newcommand{\ad}{\mathrm{ad}}
\newcommand{\diag}{\mathrm{diag}}
\newcommand{\sgn}{\mathrm{sgn}}
\newcommand{\st}{\ \textrm{s.t.}\ }
\newcommand{\1}{\mathbf{1}}
\newcommand{\ti}{\tilde}
\newcommand{\res}{\mathrm{Res}}

\definecolor{r}{rgb}{1,0,0}



\begin{document}


\title[Combinatorial Hopf algebra for tensor field
  theory]{Combinatorial Hopf algebra for\\the Ben Geloun--Rivasseau tensor field theory}

\newbox\AA
\setbox\AA\hbox{$^a$}
\newbox\BB
\setbox\BB\hbox{$^{a,b}$}


\author{Matti Raasakka\box\AA\ and Adrian Tanasa\box\BB}
\address{$^a$LIPN, Institut Galil\'ee, CNRS UMR 7030, 
Universit\'e Paris 13, Sorbonne Paris Cit\'e, 99
  av.\ Clement, 93430 Villetaneuse, France, EU\newline 
\indent
$^{b}$Horia Hulubei National Institute for Physics and
  Nuclear Engineering, 
P.O.B. MG-6, 077125 Magurele, Romania, EU
}

\thanks{\tt  matti.raasakka@lipn.univ-paris13.fr, 
adrian.tanasa@lipn.univ-paris13.fr}

\date{}


\begin{abstract}
The Ben Geloun--Rivasseau quantum field theoretical model is the first tensor model shown to be 
perturbatively renormalizable. 
We define here an appropriate Hopf algebra describing the 
combinatorics of this new tensorial renormalization.
The structure we propose is significantly different from the previously defined Connes--Kreimer 
combinatorial Hopf algebras due to the involved combinatorial and topological properties of the tensorial Feynman graphs. In particular, the 2- and 4-point function insertions must be defined to be 
non-trivial only if the superficial divergence degree of the associated Feynman integral 
is conserved.
\end{abstract}

\maketitle


\thispagestyle{myheadings}
\font\rms=cmr8 
\font\its=cmti8 
\font\bfs=cmbx8

\markright{\its S\'eminaire Lotharingien de
Combinatoire \bfs 70 \rms (2014), Article~B70d\hfill}
\def\thepage{}



\section{Introduction and Motivation}
The present article concerns the combinatorics of renormalization of tensor field theory (TFT) models. TFT models are a path integral formulation of quantum field theories (QFT) whose Feynman graphs correspond combinatorially to discrete geometries.
In the perturbative expansion of the partition function of a TFT model, with each such Feynman graph one associates a probability (called the Feynman amplitude in QFT), and accordingly we may understand the TFT model as a quantum model of discrete geometry. 
Recently, the first renormalizable rank four TFT model with Lie
group-valued field variables was formulated and studied by Ben Geloun
and Rivasseau \cite{BR,BR2}. Since then, the renormalizability of several
other TFT models has been established
\cite{BgL,BgS,COR1,COR2,SVT}. 

On the combinatorial level of Feynman graphs, the renormalizability property can be phrased in a purely 
algebraic way, as was shown by Connes and Kreimer \cite{CK}. This was followed by the formulation in terms of a Riemann--Hilbert correspondence for categories of
differential systems by Connes and Marcolli \cite{CM, cm2}.

The Connes--Kreimer combinatorial approach to renormalization was initially applied to local QFT models.
When one generalizes to QFT models on non-commutative spaces, such as the Moyal space, locality is lost due to the quantum uncertainty relations between space coordinates.
However, it was shown in \cite{TV,TK} that, by replacing the notion of locality with a 
new concept, the so-called `Moyality' property for interactions, the Hopf algebraic structure of Feynman graphs allowing for the renormalization of particular non-commutative models can be established.
The Connes--Kreimer  formalism of matrix field theory, whose two-stranded ribbon Feynman graphs are dual to two-dimensional triangulated surfaces, was also considered in \cite{M}. In addition, Hopf algebraic structures of quantum gravity spin foams, closely related to TFT models, have been considered in \cite{Tan}.
The Hopf algebra of tensorial Feynman graphs of group field 
theory and the associated Schwinger--Dyson equations have also been considered in \cite{Kra}.


The goal of the current paper is to continue the above direction of research, by formulating the combinatorial Connes--Kreimer Hopf algebraic structure of the Ben Geloun--Rivasseau (BGR) model. Let us emphasize that this constitutes the first application of the Connes--Kreimer combinatorial renormalization to a TFT model of rank-$D$ for $D>2$.


We note that a first important difference to the usual case of local QFT is the structure of external constraints; namely, the momentum conservation of local QFT is replaced by the so-called tensorial invariance. Moreover, the implementation of the Connes--Kreimer formalism requires particular attention at the level of the $n$-point function insertions. This follows from the fact that the divergences of the BGR $n$-point functions are not only dependent on the external topological data of the Feynman graph, as in ordinary local QFT models. Thus, only when the superficial divergence degree is conserved, a BGR tensor graph insertion is defined to be non-trivial. This is 
a crucial difference with respect to the previously defined Connes--Kreimer structures for local QFT.

%First page headline in AmS-LaTeX for S\'eminaire Lotharingien de Combinatoire
%--restoring the headers and pagenumbering
\pagenumbering{arabic}
\addtocounter{page}{1}
\markboth{\SMALL MATTI RAASAKKA AND ADRIAN TANASA}{\SMALL 
COMBINATORIAL HOPF ALGEBRA FOR TENSOR FIELD THEORY}

\section{Connes--Kreimer algebra for local QFT}\label{sec:CK}

\subsection{Feynman graphs of quantum field theory}
The basic feature of any QFT model is that it gives a prescription for
associating quantum probability amplitudes (taking values in $\mb{C}$)
with physical processes. In the path integral formulation of a QFT
model, these amplitudes can be extracted from the 
generating functional $\mc{W}$ of the model, that is typically expressed as an infinite dimensional functional integral of the type
$$
	\mc{W}[J] = \int \mc{D}\phi\ e^{-S[\phi] + \phi\cdot J}\ ,
$$
where $\phi$ and $J$ denote some fields (usually taken to be smooth sections of a fiber bundle), or collections of fields, on a fixed background manifold $X$. $S[\phi]$ is called the action functional defining the model, and the integral measure can typically be informally written as an infinite product of Lebesgue measures over the field values at each point in $X$, $\mc{D}\phi := \prod_{x\in X} \dd\phi(x)$. Then the correlation functions $\langle \cdots \rangle$ for the monomials in the field variables, which correspond to the basic physical observables of the model, can be written as functional derivatives of the generating functional,
\begin{equation}\label{eq:qftcorrelations}
	\langle \phi(x_1) \cdots \phi(x_n) \rangle \equiv \int
        \mc{D}\phi\ \phi(x_1) \cdots \phi(x_n) e^{-S[\phi]} =
        \left. \frac{\delta}{\delta J(x_1)} \cdots
        \frac{\delta}{\delta J(x_n)} \mc{W}[J] \right|_{J=0}\ . 
\end{equation}
The well-definedness of such an expression is, of course, highly questionable. Indeed, apart from rare exceptions, these expressions are highly divergent.

Nevertheless, one can extract finite quantities from these formal divergent expressions, which have been shown to match experimental results with remarkable accuracy. The main tool in this direction is perturbation theory. Typically the action functional is of the form $S[\phi] = S_{\mt{kin}}[\phi] + S_{\mt{int}}[\lambda;\phi]$, where $S_{\mt{kin}}[\phi]$ is quadratic in the field variables, and the higher order polynomial interaction part $S_{\mt{int}}[\lambda;\phi]$ is proportional to a collection of parameters $\lambda$, the coupling constants of the model. Thus, the kinetic part $S_{\mt{kin}}[\phi]$ by itself gives a simple (infinite dimensional) Gaussian measure, whose correlation functions can be computed explicitly. One can then expand the integrand in Equation~(\ref{eq:qftcorrelations}) in powers of the coupling constant(s) $\lambda$, and in each finite order one ends up with a sum over products of correlation functions of the Gaussian measure with given weights. It is these terms in the perturbative 
expansion, and their combinatorial structure in particular, for which Feynman graphs provide a very convenient book-keeping device. The contribution to the full correlation function associated with a particular Feynman graph is called the corresponding Feynman amplitude, and can be obtained from the graph by applying the set of Feynman rules of the particular model. Any finite order contribution is then obtained as a sum over the amplitudes of a finite number of Feynman graphs.

However, individual terms in this perturbative series are generically divergent. This particular problem is addressed by the framework of perturbative renormalization. 
The Connes--Kreimer combinatorial Hopf algebra of Feynman graphs is aimed at organizing the renormalization of the individual Feynman amplitudes in the perturbative expansion in a coherent unifying algebraic structure.

\bigskip

Let us give a standard example of the 
general points discussed above about the relationship of Feynman graphs to QFT models: the $\lambda\phi^4$ scalar field model on $\mb{R}^4$ (with Euclidean metric). This model is determined by the action functional $S[\phi] = S_{\mt{kin}}[\phi] + S_{\mt{int}}[\lambda;\phi]$, where
\begin{equation}
\label{act-x}
	S_{\mt{kin}}[\phi] = \frac{1}{2}\int_{\mb{R}^4} d^4x\  \phi (\Delta + m^2) \phi \mand S_{\mt{int}}[\lambda;\phi] = \frac{\lambda}{4!} \int_{\mb{R}^4} d^4x\ \phi^4 \ .
\end{equation}
Here, the single field $\phi$ takes values in $\mb{R}$, and the parameters of the model, $m,\lambda\in\mb{R}_+$, correspond to the mass of the particle species modelled and the coupling strength, respectively. 
$\Delta$ denotes the Laplace operator. 
The interaction is local in the sense that it couples the field values only at the same point in space. It is usual to consider the model in the momentum space, obtained through Fourier transform, instead of the direct space, since due to the translation symmetry of the model the total momentum is conserved. 
In momentum space, the action \eqref{act-x} reads explicitly
\begin{align}
	S_{\mt{kin}}[\ti{\phi}] &= \frac{1}{2}\int_{\mb{R}^4} \dd^4p\ \ti{\phi}({p}) (|{p}|^2 + m^2) \ti{\phi}({p}) \,, \\
	S_{\mt{int}}[\lambda;\ti{\phi}] &= \frac{\lambda}{4!} \int_{(\mb{R}^4)^{4}} \big[\prod_{i=1}^4 \dd^4p_i\big] \ti{\phi}({p}_1)\ti{\phi}({p}_2)\ti{\phi}({p}_3)\ti{\phi}({p}_4) \delta^4({p}_1+ {p}_2+ {p}_3+ {p}_4)\,,\label{eq:phi4int}
\end{align}
where $\ti{\phi}({p}) := \int_{\mb{R}^4}\dd^4x\ e^{-i{p}\cdot{x}} \phi({x})$ is the Fourier transform of the field $\phi$. In the interaction term, the Dirac delta distribution $\delta^4({p}_1+ {p}_2+ {p}_3+ {p}_4)$ imposes the 
so-called {\it momentum conservation}. The covariance of the kinetic part, i.e., the (free) \emph{propagator}, can be explicitly computed to yield
\begin{equation}\label{eq:freeprop}
	\mc{P}({p}_1,{p}_2):= \langle \ti{\phi}({p}_1) \ti{\phi}({p}_2) \rangle_{\mt{kin}} = \int \mc{D}\ti{\phi}\ \ti{\phi}({p}_1) \ti{\phi}({p}_2) e^{-S_{\mt{kin}}[\ti{\phi}]} = \frac{1}{|{p}_1|^2+m^2} \delta^4({p}_1-{p}_2) \,.
\end{equation}
Moreover, due to Wick's theorem, any correlation function of such Gaussian measure can be expressed as a sum of products of propagators.
For example, for the so-called four-point function, we have
\begin{align}
\langle \ti{\phi}({p}_1) \ti{\phi}({p}_2) \ti{\phi}({p}_3) &\ti{\phi}({p}_{4}) \rangle_{\mt{kin}}=
\langle \ti{\phi}({p}_{1}) \ti{\phi}({p}_{2}) \rangle_{\mt{kin}}
\langle \ti{\phi}({p}_{3}) \ti{\phi}({p}_{4}) \rangle_{\mt{kin}}
+\langle \ti{\phi}({p}_{1}) \ti{\phi}({p}_{3}) \rangle_{\mt{kin}}
\langle \ti{\phi}({p}_{2}) \ti{\phi}({p}_{4}) \rangle_{\mt{kin}}\nonumber\\
&\kern2cm
+\langle \ti{\phi}({p}_{1}) \ti{\phi}({p}_{4}) \rangle_{\mt{kin}}
\langle \ti{\phi}({p}_{2}) \ti{\phi}({p}_{3}) \rangle_{\mt{kin}}.
\end{align}
For a monomial of odd degree, the correlation function vanishes, since
the Gaussian measure is symmetric. The QFT perturbation expansion  is
based on this  fact. For example, the $k$-th order contribution to the
full correlation function of $n$ field values, the \emph{n-point
  function}, is obtained by expanding the interaction part of the
exponential integrand, and picking out the $k$-th order term 
$$
	\langle \ti{\phi}({p}_1) \ti{\phi}({p}_2) \cdots \ti{\phi}({p}_n) \rangle_k = \frac{(-1)^k}{k!} \int \mc{D}\ti{\phi}\ \ti{\phi}({p}_1) \ti{\phi}({p}_2)\cdots \ti{\phi}({p}_2) (S_{\mt{int}}[\lambda;\ti{\phi}])^k e^{-S_{\mt{kin}}[\ti{\phi}]} \,.
$$
By substituting the explicit expression for the interaction term \eqref{eq:phi4int}, and, writing all the resulting correlation functions of the free theory in terms of propagators,
 one can explicitly calculate this $k$-th order contribution to the $n$-point function of the theory. In QFT, the physical interpretation of this quantity is that it gives the (non-normalized) probability amplitude for a scattering process from an initial state with $m$ free particles having momenta $({p}_i)_{i=1,\ldots,m}$ to a final state with $n-m$ free particles having momenta $({p}_i)_{i=m+1,\ldots,n}$.

\begin{figure}
\centering
\begin{minipage}{0.20\linewidth}\centering
\def\svgwidth{0.8\columnwidth}
\input{propphi4.eps_tex}
\end{minipage}\hspace{0.05\linewidth}
\begin{minipage}{0.20\linewidth}\centering
\def\svgwidth{0.8\columnwidth}
\input{intphi4.eps_tex}
\end{minipage}\hspace{0.1\linewidth}
\begin{minipage}{0.4\linewidth}\centering
\def\svgwidth{0.8\columnwidth}
\input{examplephi4-2.eps_tex}
\end{minipage}
\caption{\label{fig:phi4}On the right: the Feynman graph
  representation of the propagator and the 4-point-interaction for the
  $\lambda\phi^4$-model. On the left: an example of a second order
  Feynman graph with four external edges.} 
\end{figure}

\medskip
We now introduce Feynman graphs in QFT models formally.

\begin{definition}
A \emph{Feynman graph} $\Gamma$ of a QFT model is a graph that consists of a set of vertices $\Gamma^{[0]}$ and a set of edges $\Gamma^{[1]}$. The edges $\Gamma^{[1]}_{\mt{int}}\subseteq \Gamma^{[1]}$ that are connected to vertices (or to the same vertex) at both ends are called \emph{internal}, while the remaining edges $\Gamma^{[1]}_{\mt{ext}}\subseteq \Gamma^{[1]}$ 
are connected to one vertex at one end (the other end of the edge not being connected to any vertex) and they   
are called \emph{external}. The external edges $e^{ext}_i \in \Gamma^{[1]}_{\mt{ext}}$ are labelled by incoming external momenta $e^{ext}_i \mapsto \vec{p}_i$. 
The external momenta obey the momentum conservation $\sum_i
\vec{p}_i=0$. We denote the space of the external momenta of $\Gamma$
by $P_\Gamma$.
We denote the set of all Feynman graphs of a model by $\mc{G}$.
\end{definition}

\bigskip

Let us briefly describe the Feynman graphs and the associated Feynman rules of the $\lambda\phi^4$-model. Each Feynman graph corresponds to an individual summand in the perturbative expansion of the correlation functions with respect to $\lambda$. The Feynman graphs of the $\lambda\phi^4$-model consist of a single type of edges (the propagator) labelled by a momentum variable and four-valent vertices (the interaction), as shown in Figure~\ref{fig:phi4}. The number of vertices in a Feynman graph equals the order of the corresponding summand in the perturbative expansion. An arrow on an edge denotes the direction of the momentum flow $p$, and can be assigned arbitrarily; the opposite orientation corresponds to $-p$. In the following we will often just drop the momentum labels and arrows, since their specific assignment is immaterial --- they can always be reintroduced arbitrarily. The Feynman amplitude associated with a particular Feynman graph is constructed as follows:
\begin{enumerate}
	\item Let $p_l$ be the momenta labels on the edges $l\in\Gamma$ of the Feynman graph $\Gamma$. We form the product of free propagators
	$$
		\prod_{l\in\Gamma} \frac{1}{|p_l|^2 + m^2} \,.
	$$
	\item For each vertex $v\in\Gamma$, multiply by the coupling constant $\lambda$, and impose momentum conservation by multiplying the above product by
	$$
		\prod_{v\in\Gamma} \delta^4\left(\sum_{l\sim v} \sigma_{vl} p_l\right) \,,
	$$
	where $l\sim v$ symbolizes that the edge $l$ is connected to the vertex $v$, and $\sigma_{vl}=\pm 1$ according to whether $p_l$ is incoming or outgoing with respect to the vertex $v$.
	\item Integrate over the $p_l$'s which are associated with the internal edges of the graph.
\end{enumerate}
The quantity that results from applying these rules to a Feynman graph is the Feynman amplitude of that graph.

%Note that usually in physics one is only concerned with connected Feynman graphs, because these are the ones that describe scattering processes. 
For any Feynman graph, the amplitude factorizes into the product of
the amplitudes of its connected components, so one can just focus on
connected Feynman graphs. On the level of the generating functional,
the disconnected graphs can be removed by replacing $\mc{W}[J]$ with
the logarithm $\log \mc{W}[J]$ in
\eqref{eq:qftcorrelations}. Moreover, it is common to restrict oneself to consider the so-called one-particle-irreducible (1PI) Feynman graphs, or what are called bridgeless graphs in graph theory language.
\begin{definition}
A Feynman graph $\Gamma$ is called \emph{one-particle-irreducible}
(1PI), if $|\Gamma^{[1]}_{\mt{ext}}|>1$ and it is bridgeless, i.e., it
cannot be disconnected by removing a single edge. We also require
$|\Gamma^{[1]}_{\mt{int}}|\geq 1$. The set of all 1PI
Feynman graphs is denoted by $\mc{G}_{1\mt{PI}}$. 
\end{definition}
If there would exist a bridge, i.e., an edge connecting two separate components
of the graph, the momentum conservation would fix the momentum on
this edge. It would then be trivial to consider the amplitude 
as the product of the amplitudes of the two contributions coming from this two separate components. 

For example, by applying the above Feynman rules to the Feynman graph on the right-hand side of Figure~\ref{fig:phi4}, we obtain for the corresponding Feynman amplitude the expression
\begin{multline*}
 \lambda^2 \int_{\mb{R}^4}\dd^4p_5 \int_{\mb{R}^4}\dd^4p_6 \left[ \prod_{l=1}^{6} \frac{1}{|p_l|^2 + m^2} \right] \delta^4(p_1+p_2-p_5-p_6) \delta^4(p_5+p_6-p_3-p_4) \\
	= \left[ \prod_{l=1}^{4} \frac{1}{|p_l|^2 + m^2} \right] \delta^4(p_1+p_2-p_3-p_4) \times \lambda^2 \int_{\mb{R}^4}\dd^4p_5 \frac{1}{|p_5|^2 + m^2} \frac{1}{|p_1+p_2-p_5|^2 + m^2} \,,
\end{multline*}
where we integrated over $p_6$. The first part of the bottom line
contains the propagators and momentum conservation associated with the
external edges of the graph. The second part corresponds to the
internal structure, in particular, to the loop formed by edges $5$ and
$6$ that allows for one free momentum integration, unconstrained by
the momentum conservation at the vertices. 
This last integral
\begin{equation*} %\label{} 
	\lambda^2 \int \dd^4p_5 \frac{1}{|p_5|^2 + m^2} \frac{1}{|p_1+p_2-p_5|^2 + m^2} 
	\end{equation*}
	is divergent when one wants to integrate for the different values of $p_5$. 
If one integrates only over momenta $p_5$ with $|p_5|\le \Lambda$, where $\Lambda>0$ is a cutoff parameter, one obtains
	\begin{equation*}
%	\lim_{\Lambda\rightarrow\infty} 
	\lambda^2 \int_{|p_5|\le \Lambda} \frac{\dd |p_5|^3|p_5|}{(|p_5|^2+m^2)^2}
\end{equation*}
(up to some irrelevant constant coming from integrating over 
the angular variables of the $4$-dimensional volume element in spherical coordinates).
Finally, this last integral behaves like 
	%\begin{equation}	
	 $\log \Lambda$,
%\end{equation}
which diverges when taking the limit $\Lambda\rightarrow\infty$.
Therefore, the momentum integration related to the loop is logarithmically divergent for large momentum --- thus it exhibits the so-called ultraviolet divergences of QFT.

This is where renormalization comes in (when possible).
In order to establish renormalizability, of paramount importance is the so-called \emph{power counting theorem}. 
Denote by $N_{int}$ and $V$ the number of internal edges and the number of vertices, respectively. Then, the number of independent momentum integrations is $N_{int} - V + 1$, and thus the superficial divergence degree in this case reads $\omega := 4(N_{int} - V + 1) - 2N_{int} = 2N_{int} - 
4(V-1)$. On the other hand, we have $4V = 2N_{int} + N_{ext}$, where $N_{ext}$ is the number of external edges, so $\omega = 4 - N_{ext}$. This is the power counting theorem for the $\lambda\phi^4$-model. 
Due to this theorem, only graphs with two or four external edges are proven to be {\it superficially divergent}.

These considerations prompt us to give the following definitions concerning the analytic aspects of Feynman amplitudes associated with Feynman graphs.
\begin{definition}
The \emph{regularized Feynman rules} of a local QFT model is a map
$\Gamma \mapsto \phi_\Lambda(\Gamma) \in E_{\Gamma}$ for
$\Gamma\in\mc{G}$, where $E_{\Gamma}$ stands for the linear vector space of generalized functions on the space $P_\Gamma$ of the external parameters $\vec{p}_i$ associated with the external edges $e^{ext}_i \in \Gamma^{[1]}_{\mt{ext}}$ of $\Gamma$. We call $\phi_\Lambda(\Gamma)$ the \emph{regularized Feynman amplitude} of the graph $\Gamma\in\mc{G}$. The parameter $\Lambda\in\mb{R}_+$ is the \emph{regularization cut-off}, such that in the limit $\Lambda\rightarrow\infty$ we recover the unregularized Feynman rules $\Gamma \mapsto \phi(\Gamma)$.
\end{definition}
\begin{remark}
In the following, we will consider the propagators associated with the external edges of $\Gamma$ not to be included in $\phi(\Gamma)$. Otherwise, one can also simply factor these out of the Feynman amplitudes.
\end{remark}
\begin{definition}
A \emph{renormalization scheme} is specified by a linear operator $R$, which extracts the divergent part of a Feynman amplitude, so that $\lim_{\Lambda\rightarrow\infty} \rho_\Lambda(\Gamma) - R(\rho_\Lambda(\Gamma)) < \infty$.
%, and satisfies the Rota-Baxter relation
%\ba
%	R(ab) + R(a)R(b) = R(aR(b)) + R(R(a)b) \,.
%\ea
\end{definition}
\begin{remark}
There is inherent ambiguity in choosing the renormalization scheme, but all the schemes should lead to the same results, when only differences of amplitudes are considered. Also in practice, there are many different choices for implementing the regularization. In order to define the $R$-operator, one may consider the regularized Feynman amplitudes to lie in the space of formal Laurent series in a cut-off parameter $\epsilon \sim \Lambda^{-1}$, and then the $R$-operator projects onto the series with negative powers of $\epsilon$.
Nevertheless, most renormalization schemes used in QFT are not
specified by a Rota--Baxter operator 
(for example, this is the case of the celebrated BPHZ scheme). The
interested reader is referred to \cite{patras} for details.
\end{remark}
\begin{definition}
A \emph{superficial divergence degree} is a map
$\omega:\mc{G}\rightarrow \mb{Z}$ such that any Feynman graph $\Gamma$
with $\lim_{\Lambda\rightarrow \infty}\rho_\Lambda(\Gamma)<\infty$
satisfies $\omega(\gamma)< 0$ for all $\gamma\subset\Gamma$. Let us
call $\Gamma\in\mc{G}_{1\mt{PI}}$ with $\omega(\Gamma)\geq 0$
\emph{superficially divergent}, and write $\mc{G}_{sd}^{\omega} :=
\{\Gamma\in\mc{G}_{1\mt{PI}} : \omega(\Gamma) \geq 0\}$ for 
the set of superficially divergent 1PI Feynman graphs with respect to $\omega$.
\end{definition}
\begin{remark}
Such a superficial divergence degree is usually obtained via a \emph{power counting theorem}, which relates features of the combinatorial and topological structure of Feynman graphs to the divergence of the corresponding Feynman amplitudes. Notice that the superficial divergence degree only sets bounds on the divergence. That is, not all superficially divergent graphs necessarily have divergent Feynman amplitudes. Conversely, a superficially convergent graph may have divergent subgraphs.
\end{remark}


\subsection{Hopf algebra of Feynman graphs}
\label{subsec:HopfFeynman}

%We now describe the Connes--Kreimer algebra for local QFT. A Hopf
%algebra over $\mb{C}$ is a tuple $(\mc{H},u,m,\epsilon,\Delta,S)$,
%where 
%\begin{itemize}
%	\item $\mc{H}$ denotes the set of elements,
%	\item $u:\mb{C}\rightarrow \mc{H}$ denotes the unit,
%	\item $m: \mc{H} \otimes \mc{H} \rightarrow \mc{H}$ denotes the product,
%	\item $\epsilon: \mc{H} \mapsto \mb{C}$ denotes the counit,
%	\item $\Delta: \mc{H} \rightarrow \mc{H} \otimes \mc{H}$ denotes the coproduct, and
%	\item $S:\mc{H}\rightarrow \mc{H}$ denotes the antipode.
%\end{itemize}
%These objects are required to satisfy several compatibility conditions
%in order for\break 
%$(\mc{H},u,m,\epsilon,\Delta,S)$ to form a Hopf algebra. (See, e.g., \cite{Manchon} %for an exposition of Hopf algebras.)

%A tuple $(\mc{H},u,m,\epsilon)$ as above, but without the antipode, is called a bialgebra.
%\begin{definition}
%A (positively) \emph{graded bialgebra} is a bialgebra $\mc{B}$, which is graded as a linear space,
%\ba
%	\mc{B} = \bigoplus_{n=0}^\infty \mc{B}^{(n)},
%\ea
%so that the grading is compatible with the algebra and coalgebra structures:
%\ba\label{eq:grading}
%	m(\mc{B}^{(n)}\otimes\mc{B}^{(m)}) \subseteq \mc{B}^{(n+m)} \mand \Delta(B^{(n)}) \subseteq \bigoplus_{k=0}^n \mc{B}^k \otimes \mc{B}^{(n-k)}.
%\ea
%\end{definition}
%\begin{definition}
%	A \emph{connected bialgebra} is a graded bialgebra $\mc{B}$, for which $\mc{B}^{(0)} = u(\mb{C})$.
%\end{definition}
%\begin{remark}
%	The coproduct of a connected graded bialgebra is always of the form
%	\ba\label{eq:Deltagradedecomp}
%		\Delta(\Gamma) = \Gamma \otimes \1 + \1 \otimes \Gamma + \Delta'(\Gamma)\ \forall\ \Gamma\in\mc{B} \whr \Delta'(B^{(n)}) \subseteq \bigoplus_{k=1}^{n-1} \mc{B}^k \otimes \mc{B}^{(n-k)}.
%	\ea
%\end{remark}
Let us first recall a few 
definitions of basic algebraic structures.
\begin{definition}\label{def:alg}
A (unital associative) \emph{algebra} $\mc{A}$ over a field $\mb{K}$ is a $\mb{K}$-linear space endowed with the following two linear homomorphisms:
\begin{description}
	\item[a product] $m: \mc{A} \otimes \mc{A} \rightarrow \mc{A}$ such that (associativity)
	\ba
		m \circ (m \otimes \id)(\Gamma)= m \circ (\id \otimes m)(\Gamma) \quad \text{for all } \Gamma\in \mc{A}^{\otimes 3},
	\ea
	\item[a unit] $u:\mb{K}\rightarrow\mc{A}$ such that
	\ba
		m\circ (u\otimes\id)(1\otimes\Gamma) = \Gamma = m\circ (\id\otimes u)(\Gamma \otimes 1) \quad \text{for all } \Gamma\in\mc{A}.
	\ea
\end{description}
\end{definition}
\begin{definition}\label{def:coalg}
A (counital coassociative) \emph{coalgebra} $\mc{C}$ over a field $\mb{K}$ is a $\mb{K}$-linear space endowed with the following two linear homomorphisms:
\begin{description}
	\item[a coproduct] $\Delta: \mc{C} \rightarrow \mc{C} \otimes \mc{C}$ such that (coassociativity)
	\ba
		(\Delta \otimes \id)\circ\Delta(\Gamma) = (\id \otimes \Delta)\circ\Delta(\Gamma) \quad \text{for all } \Gamma\in \mc{A},
	\ea
	\item[a counit] $\epsilon:\mc{A}\rightarrow\mb{K}$ such that
	\ba
		(\epsilon\otimes\id)\circ\Delta(\Gamma) = \Gamma = (\id\otimes\epsilon)\circ\Delta(\Gamma) \quad \text{for all } \Gamma\in\mc{A}.
	\ea
\end{description}
\end{definition}
\begin{definition}
A \emph{bialgebra} $\mc{B}$ over a field $\mb{K}$ is a $\mb{K}$-linear space endowed with both an algebra and a coalgebra structure such that the coproduct and the counit are unital algebra homomorphisms (or, equivalently, the product and the unit are coalgebra homomorphisms), namely,
\begin{align*}
	\Delta\circ m_\mc{B} &= m_{\mc{B}\otimes\mc{B}} \circ (\Delta \otimes \Delta),\quad  \text{such that } \Delta(\1) = \1\otimes\1, \\
	\epsilon\circ m_\mc{B} &= m_\mb{K} \circ (\epsilon \otimes \epsilon), 
\quad \text{such that } \epsilon(\1) = 1,
\end{align*}
where we wrote $\1:=u(1)$ for the unit in $\mc{B}$.
\end{definition}
\begin{definition}
A (positively) \emph{graded bialgebra} is a bialgebra $\mc{B}$, which is graded as a linear space,
\ba
	\mc{B} = \bigoplus_{n=0}^\infty \mc{B}^{(n)},
\ea
so that the grading is compatible with the algebra and coalgebra structures,
that is,
\ba\label{eq:grading}
	\mc{B}^{(n)}\mc{B}^{(m)} \subseteq \mc{B}^{(n+m)} \mand \Delta(B^{(n)}) \subseteq \bigoplus_{k=0}^n \mc{B}^k \otimes \mc{B}^{(n-k)}.
\ea
\end{definition}

\begin{definition}
	A \emph{connected bialgebra} is a graded bialgebra $\mc{B}$, for which $\mc{B}^{(0)} = u(\mb{K})$.
\end{definition}
\begin{definition}
	A \emph{Hopf algebra} $\mc{H}$ over a field $\mb{K}$ is a bialgebra over $\mb{K}$, which is equipped with an antipode map $S:\mc{H}\rightarrow\mc{H}$ satisfying
\ba\label{eq:antipode}
	m\circ(S\otimes\id)\circ\Delta = u\circ\epsilon = m\circ(\id\otimes S)\circ\Delta .
\ea
\end{definition}

A fundamental result is the following.

\begin{lemma}[\sc \cite{Manchon}, Corollary II.3.2]
A connected graded bialgebra is a Hopf algebra. The antipode map $S:\mc{H}\rightarrow\mc{H}$ may be obtained through the recursive formula
\begin{equation} \label{eq:Srecurs}
	S(\Gamma) = -\Gamma - m\circ (\mathrm{id}\otimes
        S)\circ\Delta'(\Gamma), 
\end{equation}
or, alternatively,
\begin{equation}\label{eq:Srecurs2}
	S(\Gamma) = -\Gamma - m\circ (S \otimes
        \mathrm{id})\circ\Delta'(\Gamma) ,
\end{equation}
by setting $S(\1)=\1$.
\end{lemma}
%\begin{remark}
%Note that the recursion is facilitated by the property (\ref{eq:Deltagradedecomp}) that $\Delta'(\Gamma)$ 
%can be expressed 
%in terms of elements in $\mc{B}$ of grade smaller than that of $\Gamma$.
%\end{remark}
%\begin{definition}
%The \emph{convolution product} $\phi\star\phi':\mc{H}\rightarrow \mc{A}$ of maps $\phi,\phi':\mc{H}\rightarrow \mc{A}$, where $\mc{A}$ is an algebra, is given by $\phi \star \phi' := m_\mc{A} \circ (\phi \otimes \phi') \circ \Delta$, where $m_\mc{A}:\mc{A}\otimes\mc{A}\rightarrow \mc{A}$ is the product in $\mc{A}$. The convolution product is associative.
%\end{definition}

\bigskip

We now define the corresponding operations on a set of Feynman graphs
of a given QFT model, 
where the $\lambda\phi^4$ scalar field model has already been introduced in the
previous subsection. 
\begin{definition}\label{def:res}
The \emph{residue} $\res(\Gamma)$ of $\Gamma\in\mc{G}$ is the graph obtained by contracting all internal edges of $\Gamma$ to a point (or points, one for each connected component, if $\Gamma$ is disconnected). Clearly, $P_{\res(\Gamma)}=P_\Gamma$, since any Feynman graph of a local QFT model satisfies momentum conservation.
\end{definition}

\begin{definition}
	A graph $\gamma\in\mc{G}$ is a (proper) \emph{subgraph} of a graph $\Gamma\in\mc{G}$, denoted by $\gamma\subseteq\Gamma$ ($\gamma\subsetneq\Gamma$), if its internal edges are a (proper) subset of the internal edges of $\Gamma$. Two subgraphs are \emph{disjoint}, if they do not share any internal edges.
\end{definition}

Let us now introduce an algebraic structure on the set $\mc{H}$ of
disjoint unions of 1PI Feynman graphs of a local QFT model. Consider
the associative commutative product
$m:\mc{H}\otimes\mc{H}\rightarrow\mc{H}$ of graphs as given by the
disjoint union $m(\Gamma\otimes\Gamma')=\Gamma \dot\cup \Gamma'$. The
unit with respect to this product is given by $u: \mb{C} \rightarrow
\mc{H}$ such that $u(1)=\1$, where $\1\in\mc{H}$ denotes the empty
graph. Thus, $\mc{H} \equiv \mb{C}[\1,\mc{G}_{1\mt{PI}}]$, the
$\mb{C}$-module of polynomials generated by the elements of
$\mc{G}_{1\mt{PI}}$ and the empty graph $\1$, equipped with the above
operations linearly extended to $\mc{H}$,  
constitutes a unital associative algebra.

Let us next define a coalgebra structure on $\mc{H}$. First we need to define the operation of subgraph contraction.
\begin{definition}\label{def:subcontr}
Define the operation of a \emph{proper subgraph contraction} as follows: For 
$\gamma \subsetneq \Gamma$ such that $\res(\gamma)$ corresponds to an interaction vertex of the model, let $\Gamma/\gamma\in\mc{G}$ be the graph that is obtained from $\Gamma$ by replacing $\gamma\subset\Gamma$ by $\res(\gamma)$ inside $\Gamma$. This definition of proper subgraph contraction has an obvious extension to the case of $\gamma$ being a disjoint union $\gamma=\bigcup_{i=1}^n\gamma_i$ of disjoint proper subgraphs $\gamma_i\subset\Gamma$ ($|(\gamma_i)^{[1]}_{int} \cap (\gamma_j)^{[1]}_{int}| = 0$), such that $\Gamma/\gamma \equiv (\cdots ((\Gamma/\gamma_1)/\gamma_2)\cdots /\gamma_n)$ is obtained from $\Gamma$ by replacing all $\gamma_i$ by $\res(\gamma_i)$ inside $\Gamma$.
\end{definition}
\begin{figure}
\centering
\def\svgwidth{0.9\columnwidth}
\input{contrexamplephi4-3.eps_tex}
\caption{\label{fig:contrphi4}An example of subgraph contractions for
  the $\lambda\phi^4$-model.}
\end{figure}

We now introduce an operation $\Delta$, acting on the generators
$\Gamma\in\mc{G}$ of $\mc{H}$ by
$$
	\Delta(\Gamma) = \Gamma \otimes \1 + \1 \otimes \Gamma + \sum_{\substack{\gamma\in\bigcup\mc{G}_{sd}^\omega\\\gamma\subsetneq\Gamma}} \gamma \otimes \Gamma/\gamma \,.
$$
Notice that here the sum runs over all disjoint unions
$\gamma=\bigcup_i\gamma_i\in \bigcup\mc{G}_{sd}^\omega$ of
superficially divergent disjoint 1PI subgraphs $\gamma_i$ of
$\Gamma$. Since $\Delta$ is an algebra homomorphism, it may be
extended to $\mc{H}$. We have the following fact.

\begin{lemma}[\sc \cite{CM}, Theorem~1.27]
	$\Delta: \mc{H} \rightarrow \mc{H}\otimes\mc{H}$ is a coassociative coproduct.
\end{lemma}
The counit of $\mc{H}$ with respect to $\Delta$ is given by the linear
extension of $\epsilon: \mc{G} \rightarrow \mb{C}$, where
$\epsilon(\1)=1$ and $\epsilon(\Gamma)=0$ for all
$\Gamma\neq\1$. $\mc{H}$ equipped with the above structure thus forms
a bialgebra. 
\begin{figure}
\centering
\def\svgwidth{\columnwidth}
\input{coprodexamplephi4_thick.eps_tex}
\caption{\label{fig:coprodphi4}An example of the coproduct structure
  for the $\lambda\phi^4$-model.}
\end{figure}

Moreover, we have a natural grading for the elements of $\mc{H}$ given
by the number of internal edges, which is compatible with the
coproduct $\Delta$. Clearly, $\mc{H}^{(0)} = \mb{C}\1 =
u(\mb{C})$. Thus, $\mc{H}$ constitutes a graded connected
bialgebra. Accordingly, we have the following result.

\begin{theorem}[\sc \cite{CM}, Theorem 1.27]
	$(\mc{H},u,m,\epsilon,\Delta,S)$ is a Hopf algebra, where the
  antipode $S$ is given by the formula \eqref{eq:Srecurs} 
{\em(}or \eqref{eq:Srecurs2}{\em)}.
\end{theorem}

Consequently, the combinatorial algebraic properties of Feynman graphs and amplitudes join their forces in the following theorem by Connes and Kreimer \cite{CK}.
\begin{theorem}[\sc \cite{CK}]\label{thm:ren}
	For a local perturbatively renormalizable QFT model, the renormalized Feynman amplitudes are given by the formula
\begin{equation} 
	\phi_R(\Gamma) = S^\phi_R(\phi(\Gamma)) \star \phi(\Gamma) \,,
\end{equation}
where $S^\phi_R$ is given recursively by
\begin{equation} 
	S^\phi_R(\phi(\Gamma)) = - R[\phi(\Gamma)] -
        R\left[\sum_{\substack{\gamma\subsetneq\Gamma\\\gamma\in\mc{G}_{sd}^\omega}}
          S^\phi_R(\phi(\gamma)) \phi(\Gamma/\gamma) \right] .
\end{equation}
Here, the $R$-operator defines the corresponding renormalization scheme. Notice that we have $S^\phi_R(\phi(\Gamma)) = -R(\phi(\Gamma))$ for $\Gamma$ without superficially divergent subgraphs, which facilitates the recursion.
\end{theorem}


Finally, we describe \emph{subgraph insertion}, 
the operation which is dual to subgraph contraction.
\begin{definition}\label{def:insert}
Let $\Gamma\in\mc{G}$ and
$v\in\Gamma^{[0]}$. Then, for $\gamma\in\mc{G}^\omega_{sd}$ such that
$\res(\gamma)\sim v$ (which means that $\res(\gamma)$ is of the
type of $v$),
we define the subgraph insertion at vertex~$v$, denoted by
$\gamma \circ_{v} \Gamma$, as the graph obtained from
$\Gamma$ by replacing $v$ by $\gamma$. We also define 
\begin{equation}\label{eq:insum}
	\gamma \circ \Gamma = \sum_{\substack{v\in\Gamma^{[0]}\\v\sim\res(\gamma)}} \gamma \circ_{v} \Gamma \,.
\end{equation}
Notice that, in general, there may be several inequivalent ways of inserting $\gamma$ into a vertex $v$ of $\Gamma$. In this case we must provide additional gluing data and sum also over all different ways of insertion in (\ref{eq:insum}).
\end{definition}

In the $\lambda\phi^4$-model considered above, Feynman graphs consist originally of a single type of edge and a single type of 4-valent vertex. The external edges are labelled by the external 4-momentum variables $\vec{p}_i$, which for any connected component of a Feynman graph satisfy the momentum conservation constraint $\sum_i \vec{p}_i = 0$. It is important in view of subgraph insertions and contractions to note the almost trivial fact that any graph consisting of vertices, which satisfy the momentum conservation constraint, and propagators, which preserve momentum, satisfies itself the momentum conservation constraint.

As we have already seen, the superficial degree of divergence for a Feynman graph $\Gamma$ of the $\lambda\phi^4$-model is given by the formula $\omega=4-N_{ext}$, where $N_{ext}:=|\Gamma^{[1]}_{ext}|$ denotes the cardinality of $\Gamma^{[1]}_{ext}$. Thus, only graphs with two or four external edges need to be renormalized.
Accordingly, we must consider in addition two types of $2$-valent vertices, corresponding to mass and wave function renormalization counter-terms, which arise as residues of superficially divergent graphs. 
We decorate them with the label ``2",  as an indication of the quadratic character of the corresponding divergence.
The notion of external structure in local QFT is introduced in order to distinguish the contributions of 2-point functions to mass and wave-function renormalization.
\begin{definition}
The \emph{external structure} of a Feynman graph $\Gamma$ is specified
by the action of a set of distributions $\{\sigma_v \in E_\Gamma^*\}$
labelled by the vertices of the model, such that $\langle
\sigma_v,\phi(\Gamma)\rangle = \rho_v(\Gamma) \langle
\sigma_v,\phi(\res(\Gamma))\rangle$, where the $\rho_v(\Gamma) \in E_\Gamma$ are characters of $\mc{H}$, and $\langle\sigma_v,\phi(\res(\Gamma))\rangle$ is the kernel of the interaction functional associated with $v$.
\end{definition}
However, we do not make a distinction here at the level of graph
drawing. For a graph with two external edges, its contributions to
wave function and mass renormalization are given by the external
structures for each corresponding vertex. In Figures~\ref{fig:contrphi4} 
and \ref{fig:coprodphi4} we have provided some
basic examples of the subgraph contraction operation and the
coproduct, respectively, for the $\lambda\phi^4$-model.

\bigskip

Before ending this section, let us emphasize the following fact of particular 
importance in the sequel.
The 2-valent vertices are associated with a quadratic divergence,
since the power-counting simply gives $\omega=4-N_{ext}=2$ for
them. Due to this simple fact, one may insert any graph with two external edges into a bare propagator of any Feynman graph without changing the divergence degree of the graph, because one simultaneously needs to add a propagator, which adds a counter-balancing $-2$ to the divergence degree. Similarly, all 4-valent graphs are associated with a logarithmic divergence, which allows one to insert any graph with four external edges into a vertex without affecting the divergence degree. We already anticipate that this simple behaviour does not hold for models with more complicated power-counting, such as the BGR model, in which case the notions of subgraph contraction insertion must be treated more carefully.




\section{The Ben Geloun--Rivasseau tensor field theory model}\label{sec:BGR}

The form of the BGR model derives from the combinatorics of
4-dimensional simplicial geometry. Consider a 4-simplex. Its boundary
is given by five 3-simplices, i.e., tetrahedra. Each pair of these
boundary tetrahedra shares one boundary triangle. This combinatorial
structure can be illustrated by a stranded graph, as in
Figure~\ref{fig:colorvertex}. Here, each individual strand represents
a triangle, while the five rectangular boxes, each connected to four
strands entering the graph, represent the five boundary
tetrahedra. The tetrahedra are considered to be coloured, and we allow
for two opposite orientations of the colouring.  
One considers coloured tetrahedra and only two orientations of the colouring in order to simplify the combinatorics and the topology of this type of models (see, for example,  the paper \cite{Gurau}, where this type of model was first introduced).

One can then build larger stranded graphs from these building blocks by identifying boundary tetrahedra with matching colours. Moreover, we allow only for identification of tetrahedra belonging to 4-simplices of opposite orientations, taken into account by alignment of arrows in the boxes representing tetrahedra. 
%This corresponds to the principle of tensorial invariance in TFT. 
The resulting stranded graph will then be combinatorially dual to a 4-dimensional simplicial pseudo-manifold. %The colouring imposes some restrictions on the 
%topology of the dual complexes, which is the reason it was originally introduced in \cite{Gurau} --- in particular, one cannot identify two boundary tetrahedra of the same 4-simplex, so there are no extended singularities.
\begin{figure}
\centering
\def\svgwidth{0.6\columnwidth}
\input{colorvertexcolor_v2.eps_tex}
\caption{\label{fig:colorvertex}The two orientations of the stranded representation of the combinatorics of a 4-simplex, where we consider the colours to be: blue, red, green, grey and yellow.}
\end{figure}

A (quantum) statistical description of such simplicial geometries can be formulated as a coloured tensor model.
 This is a QFT model, whose Feynman graphs have exactly such stranded structure. 
The model has fields of five different colours (each coloured block in Figure~\ref{fig:colorvertex} being a field),
and each such field is  a complex-valued function on $U(1)^{\times 4}$.
The two stranded graphs in Figure~\ref{fig:colorvertex} correspond to the two interaction terms of the model, while the propagator simply identifies the strands of the same colour associated with interaction terms with opposite orientations. 
Such models are sometimes called {\it combinatorially non-local} due to the combinatorial nature of the non-locality of their interaction terms.

The Feynman graphs of the BGR model are obtained from such stranded representations of simplicial geometry by removing four of the five fields (namely the blue, the red, the green, and the yellow fields in Figure~\ref{fig:colorvertex} above). Thus, one considers interaction terms
such as those illustrated in Figures~\ref{fig:BGRvertices6} and
\ref{fig:BGRvertex41}.  
These terms correspond to one-to-one relations between particular coloured graphs and 
new ``uncoloured'' vertices which, by construction, only have grey 
fields on the boundaries. 

These vertices are dual to four-dimensional polytopes, whose boundaries are composed of six or four tetrahedra. Again, the combinatorial connectivity of the strands represents the identifications of boundary triangles of the boundary tetrahedra. 



Note that these are not the only possible ``uncoloured'' vertices that one can construct. However,
these are the only type of vertices considered in the BGR model in order to ensure renormalizability (see again \cite{BR,BR2} for details).

In addition, a non-trivial kinetic term is added in order to induce dynamics, which allows for renormalization (see Figure~\ref{propa} for a graphical representation). Note that propagators cannot shuffle strands.

\begin{figure}
\centering
\def\svgwidth{0.25\columnwidth}
\input{BGRprop.eps_tex}
\caption{\label{propa}The kinetic term.}
\end{figure}




In Figure~\ref{fig:BGRgraph1} we illustrate a simple example of a Feynman graph of the BGR model, where the propagators are represented by transparent rectangles, while the external edges end in solid rectangles.
\begin{figure}
\centering
\def\svgwidth{0.6\columnwidth}
\input{BGRvertex61contr_v2.eps_tex}
\vspace{10pt}
\def\svgwidth{0.8\columnwidth}
\input{BGRvertex62contr_v2.eps_tex}
\caption{\label{fig:BGRvertices6}Stranded structures of the vertices $V_{6;1}$ (upper) and $V_{6;2}$ (lower) of the BGR model.} 
\end{figure}
\begin{figure}
\centering
\def\svgwidth{0.7\columnwidth}
\input{BGRvertex41contr_v3.eps_tex}
\caption{\label{fig:BGRvertex41}Stranded structure of the vertex $V_{4;1}$ of the BGR model.}
\end{figure}
\begin{figure}
\centering
\def\svgwidth{0.4\columnwidth}
\input{BGRgraphexample1_v2.eps_tex}
\caption{\label{fig:BGRgraph1}A simple example of a Feynman graph of the BGR model. There is a 
$V_{6;1}$ vertex on the left and a $V_{6;2}$ vertex on the right. $\vec{k},\vec{l},\vec{m},\vec{n},\vec{p},\vec{q}\in\mb{Z}^4$ label the external momenta, while $\vec{r},\vec{s},\vec{t}\in\mb{Z}^4$ label momenta running through internal propagators. The momentum components $k_i\in\mb{Z}$, $i=1,2,3,4$, are associated with the strands as indicated by the numbers (and similarly for the other momenta).}
\end{figure}

Let us describe in detail the action functional of the BGR model. The fields of the model live on a direct product of four copies of the group $U(1)$. The momentum representation considered in the following is obtained through harmonic analysis, which yields the momentum space $\mb{Z}^4$. In the momentum representation the kinetic term of the action functional reads explicitly
$$
	S_{kin}[\phi] = \sum_{\vec{p} \in \mb{Z}^4} \overline{\phi(\vec{p})}(|\vec{p}|^2 + m^2)\phi(\vec{p}) \,,
$$
where $|p|^2 = \sum_{i=1}^4 |p_i|^2$. This leads to the propagator
$$
	\mc{P}(\vec{p},\vec{q}) = \frac{1}{\sum_i |p_i|^2 + m^2} \prod_{i=1}^4 \delta_{p_iq_i}.
$$
In the direct (i.e., $U(1)$) representation, by Fourier transform, the kinetic term reads
$$
S_{kin}[\tilde \phi]=\int \overline{\tilde\phi}_{1,2,3,4}\left( - \sum_{s=1}^4 \Delta_s +m^2\right)
\tilde \phi_{1,2,3,4},
$$
where the integral is performed for the Haar measure over the respective group variables, 
$\Delta_s$ denotes the Laplace operator on $U(1)$ acting on the strand index $s$.
Note that $\tilde \phi$ denotes the Fourier transform of the function $\phi$.


The interaction terms are based on a completely different principle --- instead of locality, they are built on the notion of simplicial combinatorics, as described above. The interaction terms read
\begin{multline*}
	S_{int}^{6;1}[\phi] =
        \lambda_{6;1}\sum_{\vec{p},\vec{p}',\vec{p}'' \in \mb{Z}^4}
        \sum_{\sigma\in\Sigma_4} 
        \phi_{\sigma_1,\sigma_2,\sigma_3,\sigma_4}\,
        \overline{\phi_{\sigma_1',\sigma_2,\sigma_3,\sigma_4}}\,
        \phi_{\sigma_1',\sigma_2',\sigma_3',\sigma_4'}\,
        \overline{\phi_{\sigma_1'',\sigma_2',\sigma_3',\sigma_4'}}\,\nn 
	  \times \phi_{\sigma_1'',\sigma_2'',\sigma_3'',\sigma_4''}\, \overline{\phi_{\sigma_1,\sigma_2'',\sigma_3'',\sigma_4''}} \,,
\end{multline*}
and
\begin{multline*}
	S_{int}^{6;2}[\phi] =
        \lambda_{6;2}\sum_{\vec{p},\vec{p}',\vec{p}'' \in \mb{Z}^4}
        \sum_{\sigma\in\Sigma_4} 
        \phi_{\sigma_1,\sigma_2,\sigma_3,\sigma_4}\,
        \overline{\phi_{\sigma_1',\sigma_2',\sigma_3',\sigma_4}}\,
        \phi_{\sigma_1',\sigma_2',\sigma_3',\sigma_4'}\,
        \overline{\phi_{\sigma_1'',\sigma_2,\sigma_3,\sigma_4'}}\, \nn 
	 \times \phi_{\sigma_1'',\sigma_2'',\sigma_3'',\sigma_4''}\, \overline{\phi_{\sigma_1,\sigma_2'',\sigma_3'',\sigma_4''}} \,,
\end{multline*}
where we wrote 
$$\phi(p_1,p_2,p_3,p_4)=:\phi_{1,2,3,4},\
\phi(p_1',p_2,p_3,p_4)=:\phi_{1',2,3,4},\
\phi(p_1,p_2',p_3,p_4)=:\phi_{2',1,3,4}, \text{ etc.},
$$ 
for simplicity,
and we sum over all permutations $\sigma\in\Sigma_4$ of the four colour indices. These interactions lead exactly to the identifications of the field variables represented by the stranded Feynman graphs of the type in Figure~\ref{fig:BGRvertices6} modulo permutations of the ordering of the strands.

In addition to the above 6-valent interaction vertices, there are also two 4-valent vertices in this model. The first one is shown in Figure~\ref{fig:BGRvertex41}, while the second one is a disconnected combination of two propagators, which arises from the vertex $V_{6;2}$ through the insertion of a propagator, as in Figure~\ref{fig:BGRvertex42}.
The corresponding interaction terms of the action functional read
$$
	S_{int}^{4;1}[\phi] = \lambda_{4;1} \sum_{\vec{p},\vec{p}' \in
          \mb{Z}^4} \sum_{\sigma\in\Sigma_4}
        \phi_{\sigma_1,\sigma_2,\sigma_3,\sigma_4}\,
        \overline{\phi_{\sigma_1',\sigma_2,\sigma_3,\sigma_4}}\,
        \phi_{\sigma_1',\sigma_2',\sigma_3',\sigma_4'}\,
        \overline{\phi_{\sigma_1,\sigma_2',\sigma_3',\sigma_4'}} \,, 
$$
and
$$
	S_{int}^{4;2}[\phi] = \lambda_{4;2} \sum_{\vec{p},\vec{p}' \in \mb{Z}^4} \sum_{\sigma\in\Sigma_4} \phi_{\sigma_1,\sigma_2,\sigma_3,\sigma_4}\, \overline{\phi_{\sigma_1,\sigma_2,\sigma_3,\sigma_4}}\, \phi_{\sigma_1',\sigma_2',\sigma_3',\sigma_4'}\, \overline{\phi_{\sigma_1',\sigma_2',\sigma_3',\sigma_4'}} \,.
$$
\begin{figure}
\centering
\def\svgwidth{0.45\columnwidth}
\input{BGRvertex42contr_v2.eps_tex}
\caption{\label{fig:BGRvertex42}Vertex $V_{4;2}$ of the BGR model obtained through the contraction of a self-loop from the vertex $V_{6;2}$.}
\end{figure}
Notice the peculiar disconnected structure of the vertex $V_{4;2}$. This 
can be interpreted geometrically as a four-sphere with two holes of the form of four-simplices.


The Feynman amplitude associated with a particular Feynman graph is constructed as follows:
\begin{enumerate}
	\item Let $\vec{p}_e\in\mb{Z}^4$ be the momenta labels on the edges $e\in\Gamma^{[1]}$ of the Feynman graph $\Gamma$.  We 
	construct the product of free propagators
	$$
		\prod_{e\in\Gamma^{[1]}} \frac{1}{|\vec{p}_e|^2 + m^2} \,.
	$$
	\item For each vertex $v_t\in\Gamma^{[0]}$, multiply by the coupling constant $\lambda_t$, where $t$ labels the type of the vertex, and impose identifications of strands (momentum variables) by multiplying the above product by
	$$
		\prod_{v\in\Gamma^{[0]}} \prod_k \delta(c^{t(v)}_k(\vec{p}_l)) \,,
	$$
	where $\{c_{t(v)}^k(\vec{p}_l)=0\}$ is the set of constraints imposed on the incoming momenta by the vertex type $t(v)$ of vertex $v\in\Gamma^{[0]}$. Explicitly, these are given by the following
	\begin{itemize}
		\item $\prod_k \delta(c^{6;1}_k(\vec{p}_l)) = \delta_{p_1n_1}\delta_{p_2q_2}\delta_{p_3q_3}\delta_{p_4q_4}\delta_{q_1k_1}\delta_{k_2l_2}\delta_{k_3l_3}\delta_{k_4l_4}\delta_{l_1m_1}\delta_{m_2n_2}\delta_{m_3n_3}\delta_{m_4n_4}$,
		\item $\prod_k \delta(c^{6;2}_k(\vec{p}_l)) = \delta_{p_1n_1}\delta_{p_2l_2}\delta_{p_3l_3}\delta_{p_4q_4}\delta_{q_1k_1}\delta_{q_2k_2}\delta_{q_3k_3}\delta_{k_4l_4}\delta_{l_1m_1}\delta_{m_2n_2}\delta_{m_3n_3}\delta_{m_4n_4}$,
		\item $\prod_k \delta(c^{4;1}_k(\vec{p}_l)) = \delta_{p_1l_1}\delta_{p_2q_2}\delta_{p_3q_3}\delta_{p_4q_4}\delta_{q_1k_1}\delta_{k_2l_2}\delta_{k_3l_3}\delta_{k_4l_4}$,
		\item $\prod_k \delta(c^{4;2}_k(\vec{p}_l)) = \delta_{p_1q_1}\delta_{p_2q_2}\delta_{p_3q_3}\delta_{p_4q_4}\delta_{k_1l_1}\delta_{k_2l_2}\delta_{k_3l_3}\delta_{k_4l_4}$,
	\end{itemize}
	where $\vec{p},\vec{k},\vec{m}$ are incoming and $\vec{q},\vec{l},\vec{n}$ outgoing momenta. Notice that the vertex functions are not symmetric with respect to all permutations of the edges, so in addition to specifying the type of vertices, one must also specify the orientations. (This is naturally taken care of in the stranded representation.)
	\item Sum over $\vec{p}_e\in\mb{Z}^4$ associated with the internal edges $e \in \Gamma^{[1]}_{int}$ of the graph.
\end{enumerate}
The quantity that results from applying these rules to a Feynman graph is the Feynman amplitude of that graph.

Since the momenta are conserved by a simple identification of
variables in the propagator and the vertices, as mentioned above, the
momentum vector components may be associated with the individual
strands $s\in \Gamma^{s}$ of a Feynman graph in the stranded
representation. We emphasize that this combinatorial identification of
field variables is, in fact, what allows for the stranded
representation in the first place. As a result, the Feynman amplitude
of a stranded graph consists of a product over Kronecker deltas
identifying the external momentum vector components through the
strands, multiplied by the product over internal propagators of the
graph, which depend on the momenta associated with the strands that
pass through them, summed over the momenta associated with internal
(looped) strands. For example, for the Feynman graph in
Figure~\ref{fig:BGRgraph1}, the amplitude reads
\begin{multline}
	\phi(\Gamma_{\mt{Fig.\ \ref{fig:BGRgraph1}}}) :=
        \lambda_{6;1}\lambda_{6;2}\ \delta_{k_1 q_1} \delta_{k_2 l_2}
        \delta_{k_3 l_3} \delta_{k_4 l_4} \delta_{l_1 m_1} \delta_{m_2
          n_2} \delta_{m_3 n_3} \delta_{m_4 n_4} \delta_{n_1 p_1}
        \delta_{p_2 q_2} \delta_{p_3 q_3} \delta_{p_4 q_4} \\ 
	\times \sum_{\substack{r_2,r_3,\\r_4\in\mb{Z}}}
        \left(\tfrac{1}{k_1^2 + r_2^2 + r_3^2 + r_4^2 + m^2}\right)
        \sum_{s_4\in\mb{Z}} \left(\tfrac{1}{l_1^2 + p_2^2 + p_3^2 +
          s_4^2 + m^2}\right) \left(\tfrac{1}{k_1^2 + p_2^2 + p_3^2 +
          s_4^2 + m^2}\right), 
\end{multline}
where $\vec{k},\vec{l},\vec{m},\vec{n},\vec{p},\vec{q}\in{\mathbb Z}^4$ 
are the momenta associated with the external edges
%ordered in right-handed orientation starting from lower left corner respectively, 
and $r_i$ ($i=2,3,4$) and $s_4$ are the momentum vector components associated with the internal strands. 
%The components of a momentum vector associated with an edge are ordered from left to right when the arrow in the rectangle points upwards. 
We observe that, for large absolute values of the internal momenta
$r_2,r_3,r_4,s_4$, the sums behave asymptotically as 
$$
	\phi(\Gamma_{\mt{Fig.\ \ref{fig:BGRgraph1}}}) \propto \sum_{\substack{r_2,r_3,\\r_4\in\mb{Z}}} \frac{1}{r_2^2 + r_3^2 + r_4^2}\ \sum_{s_4\in\mb{Z}} \frac{1}{s_4^4} \approx \int_{\mb{R}^3} \frac{\dd^3r}{r^2} \int_{\mb{R}} \frac{\dd s}{s^4} \propto \Lambda \times  \Lambda^{-3} = \Lambda^{-2}\,,
$$
where $\Lambda$ is a cut-off on the momentum summation. Here, the first factor, leading to a linear divergence in the limit $\Lambda\rightarrow\infty$, corresponds to the three strand loops in the upper left part of the graph, and the second convergent factor corresponds to the single strand loop in the middle. Although the first factor leads to a subdivergence, the superficial divergence degree of the total graph $\Gamma_{\mt{Fig.\ \ref{fig:BGRgraph1}}}$ is $-2$, and therefore the graph is superficially convergent.

Similarly, the Feynman amplitude for the graph in Figure~\ref{fig:BGRnonstrandedgraph} reads
\begin{multline}
	\phi(\Gamma_{\mt{Fig.\ \ref{fig:BGRnonstrandedgraph}}}) :=  \lambda_{6;1}\lambda_{6;2}\ \delta_{k_1 l_1} \delta_{k_2 l_2} \delta_{k_3 l_3} \delta_{k_4 l_4} \delta_{m_1 n_1} \delta_{m_2 n_2} \delta_{m_3 n_3} \delta_{m_4 q_4} \delta_{n_4 p_4} \delta_{p_1 q_1} \delta_{p_2 q_2} \delta_{p_3 q_3} \\
	  \times \sum_{\substack{r_2,r_3,\\r_4\in\mb{Z}}} \left(\tfrac{1}{k_1^2 + r_2^2 + r_3^2 + r_4^2 + m^2}\right) \left(\tfrac{1}{l_1^2 + p_2^2 + p_3^2 + m_4^2 + m^2}\right) \left(\tfrac{1}{k_1^2 + p_2^2 + p_3^2 + m_4^2 + m^2}\right),
\end{multline}
with the same notation for the momenta as in
Figure~\ref{fig:BGRgraph1}. The difference to the previous graph 
is that now there are only three internal strands corresponding to
$r_i$, $i=2,3,4$. The superficial divergence degree of this graph is
$-3$ despite the subdivergence, since the propagators contribute
negatively to the power counting. In the following we state the power
counting more explicitly. 

The Feynman amplitude of any other BGR tensor graph may be computed in
an analogous way. We leave the calculation of amplitudes for the rest
of the graphs of this section as an exercise to the interested reader.

We may also draw the Feynman graphs as ordinary graphs, but then the combinatorics of the internal structure must be taken into account by additional labelling of vertices. We have illustrated the correspondence between the ordinary and the stranded representation of Feynman graphs of the BGR model in Figure~\ref{fig:BGRnonstrandedgraph}.
\begin{figure}
\centering
\def\svgwidth{0.8\columnwidth}
\input{BGRgraphnonstrandedexample2_v2.eps_tex}
\caption{\label{fig:BGRnonstrandedgraph}An example of the correspondence between stranded and non-stranded representation of a Feynman graph of the BGR model.}
\end{figure}
The vertices are labelled, for example, as $V_{6;1}^{1234}$, where the lower index indicates the vertex type and the upper indices indicate the permutation of field variables. Since the vertex is not symmetric under permutations of edges, the edges meeting at a vertex must be distinguished. This can be accomplished, for example, by placing the vertex label between the first and the last edge, where the edges are ordered in a right-handed orientation around the vertex with respect to the normal pointing upwards from the surface, on which the graph is drawn.


\begin{definition}
	A \emph{face} of a stranded Feynman graph is a strand that forms a loop.
\end{definition}

Note that a face of a stranded graph is dual to the corresponding triangle of the tetrahedra of the four-dimensional triangulation.

The integrations over face momenta may give rise to divergences, as we
have already observed. In order for a strand to form a loop, the
non-stranded representation must clearly have a loop as well. However,
a loop in the non-stranded representation does not imply the existence
of a face as can for instance be seen in Figure~\ref{fig:BGRgraphexample3}.
\begin{figure}
\centering
\def\svgwidth{0.8\columnwidth}
\input{BGRgraphexample3_v2.eps_tex}
\caption{\label{fig:BGRgraphexample3}An example of a graph without a face but with a loop in the non-stranded representation.}
\end{figure}
Moreover, a face may extend around several loops of the non-stranded graph, as in Figure~\ref{fig:BGRgraphexample4}.
\begin{figure}
\centering
\def\svgwidth{0.8\columnwidth}
\input{BGRgraphexample4_v2.eps_tex}
\caption{\label{fig:BGRgraphexample4}An example of a graph with faces, which extend around several loops of the non-stranded representation.}
\end{figure}
Therefore, the faces can be highly non-local objects on the graph. Fortunately, the power counting restricts the non-locality of divergent faces significantly, as we will see in the following. 
Notice also that faces cannot extend over bridges of the non-stranded Feynman graph, since due to the colouring a single strand can pass the same propagator only once. Therefore, 1PI Feynman graphs are still sufficient to resolve the divergence structure of Feynman amplitudes of the BGR model.

We have the following auxiliary result.
\begin{lemma}[\sc \cite{BR}, Lemma 2]
	A superficial divergence degree for the BGR model is given by $\omega(\Gamma) = F_\Gamma - 2 P_\Gamma$, where $F_\Gamma$ and $P_\Gamma$ are the numbers of faces and propagators in $\Gamma$, respectively.
\end{lemma}
Thus, we see that not all faces lead to divergences, but there is a balance between the number of propagators and the number of strands. It was further proved that the superficial divergence degree $\omega$ for connected graphs, where $V_{4;2}$ is considered disconnected, can be written in terms of the combinatorial and topological structure of the stranded Feynman graphs as follows.
\begin{definition}[\sc\cite{BR}, Definition 1]
	Let $\Gamma$ be a Feynman graph of the BGR model. Then
	\begin{itemize}
		\item the \emph{coloured extension} $\Gamma_{col}$ of
                  $\Gamma$ is the corresponding unique stranded graph
                  obtained by replacing the vertices of $\Gamma$ by
                  the corresponding graphs of the simplicial coloured
                  model from Figures~\ref{fig:BGRvertices6} and
                  \ref{fig:BGRvertex41}.
		\item a \emph{jacket} $J$ of $\Gamma_{col}$ is a ribbon graph obtained from $\Gamma_{col}$ by removing a subset of the strands. Specifically, a jacket $J$ is determined by a permutation $(0abcd)$ of colours (up to cyclic permutations), and the strands that are included in $J$ are the ones that connect the tetrahedra of colours $(0a),(ab),(bc),(cd),(d0)$, where $a,b,c,d \in \{R,G,B,Y\}$ are all different. Here, we have chosen 0 to represent the grey colour, while the other colours are labelled by their initials.
		\item the \emph{pinched jacket} $\tilde{J}$ is the vacuum ribbon graph obtained from the jacket $J$ by closing all external ribbon edges of $J$. (See Figure~\ref{fig:jacket} for an example.)
\begin{figure}
\centering
\def\svgwidth{0.95\columnwidth}
\input{BGRgraphjacketexample_v2.eps_tex}
\caption{\label{fig:jacket}An example of a pinched jacket.}
\end{figure}
		\item the \emph{boundary graph} $\prt\Gamma$ of $\Gamma$ is the 3-dimensional tensor graph, which corresponds to the boundary of $\Gamma$. It is obtained by first removing all internal faces of $\Gamma$, closing the external edges by connecting their strands to a vertex per external edge, replacing the strands by bundles of three strands, and finally replacing the vertices of external edges by tensorial simplicial vertices. (See Figure~\ref{fig:bound} for an example.)
\begin{figure}
\centering
\def\svgwidth{0.7\columnwidth}
\input{BGRgraphboundexample_v2.eps_tex}
\caption{\label{fig:bound}An example of a boundary graph.}
\end{figure}
		\item the \emph{boundary jacket} $J_\prt$ is a jacket of the boundary graph $\prt\Gamma$.
	\end{itemize}
\end{definition}
\noindent
(See \cite{BR,BR2} and references therein for further details.) Ben Geloun and Rivasseau proved the following result.
\begin{theorem}
The superficial divergence degree $\omega_{BGR}$ of a connected Feynman graph $\Gamma$ can be written as
$$
	\omega_{BGR}(\Gamma) = -\frac{1}{3}\left[ \sum_J g_{\tilde{J}} - \sum_{J_\prt} g_{J_\prt} \right] - (C_\prt -1) - V_4 -2V''_2 - \frac{1}{2}(N_{ext} - 6) \,,
$$
where $g_{J}$ denotes the genus of the ribbon graph $J$, $C_\prt$ is the number of disconnected components of $\prt\Gamma$, $V_4$ is the number of vertices of type $V_{4;1}$, 
%$V_2$ is the number of mass counter-terms, 
$V''_2/2$ is the number of vertices of type $V_{4;2}$, and $N_{ext}$ is the number of external edges of $\Gamma$.
\end{theorem}
Moreover, they classified all superficially divergent Feynman graphs in Table~\ref{tab:supdivclass}, which is the departure point for our subsequent analysis.
%\begin{table}
%\begin{center}
%\begin{tabular}{c|cc|cc|c||c}
%	$N_{ext}$ & $V_2 + V''_2$ & $V_4$ & $\sum_{J_\prt} g_{J_\prt}$ & $C_{\prt} -1$ & $\sum_{\tilde{J}} g_{\tilde{J}}$ & $\omega_{BGR}$ \\ \hline\hline
%	6 & 0 & 0 & 0 & 0 & 0 & 0 \\\hline
%	4 & 0 & 0 & 0 & 0 & 0 & 1 \\
%	4 & 0 & 1 & 0 & 0 & 0 & 0 \\
%	4 & 0 & 0 & 0 & 1 & 0 & 0 \\\hline
%	2 & 0 & 0 & 0 & 0 & 0 & 2 \\
%	2 & 0 & 1 & 0 & 0 & 0 & 1 \\
%	2 & 0 & 2 & 0 & 0 & 0 & 0 \\
%	2 & 0 & 0 & 0 & 0 & 6 & 0 \\
%	2 & 1 & 0 & 0 & 0 & 0 & 0 \\\hline\hline
%\end{tabular}
%\caption{\label{tab:bgr}The classification of superficially divergent Feynman graphs of the BGR model \cite{BR,BR2}.}
%\end{center}
%\end{table}
\begin{table}[ht!]
\begin{center}
\begin{tabular}{c||c|c|c|c|c||c|c}
	class & $N_{ext}$ & $V_4$ & $\sum_{J_\prt} g_{J_\prt}$ & $C_{\prt} -1$ & $\sum_{\tilde{J}} g_{\tilde{J}}$ & $\omega_{BGR}$ & $\res$\\ \hline\hline
	$6_1$ & 6 & 0 & 0 & 0 & 0 & 0 & $V_{6;i}$ \\\hline
	$4_1$ & 4 & 0 & 0 & 0 & 0 & 1 & $V_{4;1}^*$ \\
	$4_2$ & 4 & 1 & 0 & 0 & 0 & 0 & $V_{4;1}$ \\
	$4_3$ & 4 & 0 & 0 & 1 & 0 & 0 & $V_{4;2}$ \\\hline
	$2_1$ & 2 & 0 & 0 & 0 & 0 & 2 & $V_{2;1}$ \\
	$2_2$ & 2 & 1 & 0 & 0 & 0 & 1 & $V_{2;2}$ \\
	$2_3$ & 2 & 2 & 0 & 0 & 0 & 0 & $V_{2;3}$ \\
	$2_4$ & 2 & 0 & 0 & 0 & 6 & 0 & $V_{2;3}$ \\\hline\hline
\end{tabular}
\vskip10pt
\end{center}
\caption{\label{tab:supdivclass}Classification and residues of superficially divergent Feynman graphs of the BGR model.}
\end{table}

Notice that all superficially divergent graphs have a combinatorial structure that corresponds to one of the vertices of the original model, which facilitates perturbative renormalizability. Ben Geloun and Rivasseau \cite{BR,BR2} showed that the BGR model is perturbatively renormalizable. 


\section{Hopf algebraic description of the combinatorics of the renormalizability 
of the BGR model}
\label{sec:CKforBGR}
The formal definitions for local QFT models of Subsection~\ref{subsec:HopfFeynman} also apply to the case of the BGR model, with four important exceptions:

\medskip
(i) The constraints on the external parameters of a Feynman graph $\Gamma$ are more envolved. One does not need to impose only momentum conservation. Instead, one needs to identify the variables represented by the stranded structure of
the Feynman diagrams.
 
\begin{definition}
	A Feynman graph $\Gamma$ of a QFT model is equipped with a set
        of \emph{external constraints} $\{c^\Gamma_k(\vec{p}_i) = 0\}$
        on the external momenta. We denote the space of the external
        momenta of $\Gamma$ by $P_\Gamma$. 
The Feynman amplitudes are then generalized functions on $P_\Gamma$, as before.
\end{definition}
\begin{remark}
Notice that this generalizes the case of local QFT, where the external
constraints impose the momentum conservation. In contrast, in tensor
field theories the interactions are combinatorially non-local in the
sense that the boundary variables are identified according to
tensorial invariance. Thus, the notion of tensorial invariance in TFT
constitutes the substitute for 
the notion of locality in local QFT. A gluing of tensor
invariant graphs always yields another tensor invariant graph. 
\end{remark}
Accordingly, we also need to generalize the notion of residue to tensorial Feynman graphs.
\begin{definition}
A residue $\res(\Gamma)$ of a tensorial Feynman graph
$\Gamma\in\mc{G}_{1\mt{PI}}$ of the BGR model is the graph obtained,
in the non-stranded representation, by shrinking all internal edges of
$\Gamma$ to a point, and retaining the external constraints and
structure of the graph. In the stranded representation, 
this corresponds to the removal of all internal faces and internal
propagators of $\Gamma$. 
\end{definition}

(ii) There are vertices of valence four and vertices of valence six. By inserting a
self-loop into a 6-valent vertex $V_{6;1}$, we obtain a self-loop
Feynman graph with four external edges and divergence degree 1. This 
leads us to introduce a new 4-valent counter-term vertex $V_{4;1}^*$ that has the same structure as $V_{4;1}$ but a linear counter-term attached. This may be illustrated as in Figure~\ref{fig:BGRgraph4pointexample}.

\medskip
(iii) Similarly, 
the contraction of superficially divergent subgraphs with two external edges leads us to introduce 2-valent counter-term vertices $V_{2;1}$, $V_{2;2}$, and $V_{2;3}$, with a quadratic, linear, and logarithmic counter-term attached, respectively. This is very different, for example, from the $\phi^4$-model, for which we only have quadratically divergent 2-point functions. See Figure~\ref{fig:BGRgraph2pointexample} for illustration. Of the 2-valent counter-term vertices, only the one with a quadratic divergence can be replaced by the original propagator, since only in this case the divergence degree of the graph is preserved.
\begin{figure}
\centering
\def\svgwidth{0.35\columnwidth}
\input{BGRgraph4pointexample_v2.eps_tex}
\caption{\label{fig:BGRgraph4pointexample}Illustration of the linear 4-valent counter-term vertex.}
\vspace{40pt}
\def\svgwidth{0.7\columnwidth}
\input{BGRgraph2pointexample_v3.eps_tex}
\caption{\label{fig:BGRgraph2pointexample}Some examples of the 2-valent counter-term as residues of 2-point functions from the classes $2_1$, $2_2$, $2_3$, and $2_4$, in the respective order. Only the quadratic counter-term vertex may be inserted into a propagator, since this conserves the divergence degree.}
\end{figure}

\medskip
Let us make here a few more remarks on the renormalizability of the BGR model. 
When one does the Taylor expansion of the two-point function (see Lemmas~11 and 12 
in \cite{BR,BR2} for the respective analytic details), the subleading divergence of the first graph in 
Figure~\ref{fig:BGRgraph2pointexample} is logarithmic and renormalizes the wave function.
Note that there is no linear subleading divergence. 
Analogously to the ``usual'' $\lambda \phi^4$ model (see Section~\ref{sec:CK}), there is no distinction between these two counterterms (quadratic, mass and logarithmic, wave function) at the level of the graph
drawing, but this is given by the external structure.
Moreover, there are no subleading divergences for the rest of the graphs of Figure~\ref{fig:BGRgraph2pointexample}.

\begin{remark}
Note that a given two-point graph cannot have several counterterms of type $0$, $1$ and $2$ (see 
again Figure~\ref{fig:BGRgraph2pointexample}), since one can identify to which line of Table~\ref{tab:supdivclass}
 the respective tensor graph belongs (due to the numbers $V_4$, $V''_2$, and to the graph topology). If the graph belongs to line $2_1$, then it has a counterterm of type $2$ (which means a quadratic counterterm of mass type and a logarithmic counterterm of wave function-type, see above), if it belongs to line $2_2$, then it has a counterterm of type $1$, and so on.
\end{remark}

(iv) The superficial divergence degree does not only depend on the
external data of a Feynman graph, such as the number of external
edges, but also some internal combinatorial and topological
information, in particular, the number of 4-valent vertices and the
sum over the genera of pinched jackets. Out of these, the first one is
easy enough to understand, but to have a control over the second
requires a bit more work. To this aim, we prove the following
auxiliary result.

\begin{lemma}
Let $\Gamma\in\mc{G}$, $\gamma\in\mc{G}_{sd}$ be Feynman graphs of the
BGR model such that $\gamma\subseteq\Gamma$. For the sum over the
genera of pinched jackets, we have the relation 
$$
	\sum_J g_{\tilde{J}}(\Gamma) = \sum_J g_{\tilde{J}}(\gamma) + \sum_J g_{\tilde{J}}(\Gamma/\gamma) \,.
$$
\end{lemma}
\begin{proof}
First note that the genus for a single pinched jacket may be expressed
via the Euler characteristic formula as $g_{\tilde{J}} =
-\frac{1}{2}(F_{\tilde{J}}-L_{\tilde{J}}+V_{\tilde{J}}
-2C_{\tilde{J}})$, where $F_{\tilde{J}}$, $L_{\tilde{J}}$,
$V_{\tilde{J}}$, and $C_{\tilde{J}}$ are the numbers of faces, edges,
vertices, and connected components of $\tilde{J}$. (For a disconnected
jacket, we define the genus to be given by the sum over the genera of
its connected components.) One may further write $F_{\tilde{J}} =
F_{\tilde{J},ext} + F_{\tilde{J},int}$, where $F_{\tilde{J},ext}$ is
the number of faces formed by pinching the external edges of $J$, and
$F_{\tilde{J},int}$ is the number of (internal) faces of $J$. Now, one
may easily verify that each of $F_{\tilde{J},int}$, $L_{\tilde{J}}$,
and $V_{\tilde{J}}$ separately satisfies the relation 
$$
	Q(\Gamma) - Q(\gamma) = Q(\Gamma/\gamma) - Q(\res(\gamma))
$$
with respect to the contraction of subgraphs $\gamma\in\mc{G}_{sd}$, as defined for Feynman graphs of the BGR model. (Notice, in particular, that, in general, $F_{\tilde{J},int}(\res(\gamma))\neq 0$, $L_{\tilde{J}}(\res(\gamma))\neq 0$, and $V_{\tilde{J}}(\res(\gamma))\neq 0$, since the vertices of the BGR model correspond to extended subgraphs in the coloured extension of the Feynman graphs, from which the jackets are derived.) For example, it is clear that both sides of
$$
	Q(\Gamma) - Q(\Gamma/\gamma) = Q(\gamma) - Q(\res(\gamma))
$$
represent the number of internal faces, edges, or vertices, which are removed from $\tilde{J}$ by contracting $\gamma\subseteq\Gamma$, since by definition the subgraph contraction operation replaces $\gamma$ by $\res(\gamma)$ inside $\Gamma$, which does not affect the numbers of internal faces, edges, or vertices that do not belong to $\gamma$. (Notice that some of the internal faces of $\Gamma$ not internal to $\gamma$ may still pass through $\gamma$, but are not affected by the contraction of $\gamma$.) Accordingly, we have
$$
	Q(\Gamma) - Q(\res(\gamma)) = Q(\gamma) - Q(\res(\gamma)) + Q(\Gamma/\gamma) - Q(\res(\gamma)) \,,
$$
so $G(\Gamma):= Q(\Gamma) - Q(\res(\gamma))$ gives a grading of jackets with respect to the contraction of $\gamma$, in the sense that $G(\Gamma)=G(\gamma)+G(\Gamma/\gamma)$. Moreover, if we have a set of gradings $G_i$, then it is easy to verify that $\sum_{i} (-1)^{n_i} G_i$ is also a grading for any choice of $n_i\in\{0,1\}$.

Now, note that we may write
\begin{multline*}
g_{\tilde{J}}(\Gamma) + \frac{1}{2} \big(F_{\tilde{J},ext}(\Gamma) +
F_{\tilde{J}}(\res(\gamma)) - L_{\tilde{J}}(\res(\gamma)) +
V_{\tilde{J}}(\res(\gamma)) - 2C_{\tilde{J}}(\res(\gamma))\big) \nn 
	= -\frac{1}{2}\big((F_{\tilde{J},int}(\Gamma) -
        F_{\tilde{J},int}(\res(\gamma))) -(L_{\tilde{J}}(\Gamma) -
        L_{\tilde{J}}(\res(\gamma))) + (V_{\tilde{J}}(\Gamma) -
        V_{\tilde{J}}(\res(\gamma)))\big)  \,. 
\end{multline*}
Thus, we find that $g_{\tilde{J}}(\Gamma) + \frac{1}{2}F_{\tilde{J},ext}(\Gamma) + X_{\tilde{J}}(\res(\gamma))$ is a grading with respect to the contraction of $\gamma$, where we wrote
$$
	X_{\tilde{J}}(\res(\gamma)) := \frac{1}{2} \big(F_{\tilde{J},int}(\res(\gamma)) - L_{\tilde{J}}(\res(\gamma)) + V_{\tilde{J}}(\res(\gamma)) - 2C_{\tilde{J}}(\res(\gamma))\big) \,.
$$
Accordingly, we find that, as a sum of gradings,
$$
	\sum_{J} g_{\tilde{J}}(\Gamma) + \frac{1}{2}\sum_J F_{\tilde{J},ext}(\Gamma) + \sum_J X_{\tilde{J}}(\res(\gamma))
$$
is also a grading with respect to the contraction of $\gamma$. Let us
write $X(\res(\gamma)) := \sum_{J}
X_{\tilde{J}}(\res(\gamma))$. Moreover, $\sum_J
F_{\tilde{J},ext}(\Gamma) = 2F_{\prt\Gamma}$, the number of faces of
the boundary $\prt\Gamma$ of $\Gamma$. Then, due to the grading
property, we have
$$
	\sum_J g_{\tilde{J}}(\Gamma) + F_{\prt\Gamma} = \sum_J g_{\tilde{J}}(\gamma) + F_{\prt\gamma} + \sum_J g_{\tilde{J}}(\Gamma/\gamma) + F_{\prt(\Gamma/\gamma)} + X(\res(\gamma)) \,.
$$
We may further simplify by noting that
$F_{\prt\Gamma}=F_{\prt(\Gamma/\gamma)}$, since a subgraph contraction
does not affect the boundary. Hence, we may write
$$
	\sum_J g_{\tilde{J}}(\Gamma) = \sum_J g_{\tilde{J}}(\gamma) + \sum_J g_{\tilde{J}}(\Gamma/\gamma) + F_{\prt\gamma}(\res(\gamma)) + X(\res(\gamma))\,.
$$
But now
\begin{align*}
 F_{\prt\gamma}(\res(\gamma)) &+ X(\res(\gamma)) \\
&= \frac{1}{2} \sum_{J} \big(F_{\tilde{J}}(\res(\gamma)) - L_{\tilde{J}}(\res(\gamma)) + V_{\tilde{J}}(\res(\gamma)) - 2C_{\tilde{J}}(\res(\gamma))\big) \nn
 &= -\sum_{J} g_{\tilde{J}}(\res(\gamma)) \,,
\end{align*}
so we finally obtain
$$
	\sum_J g_{\tilde{J}}(\Gamma) = \sum_J g_{\tilde{J}}(\gamma) + \sum_J g_{\tilde{J}}(\Gamma/\gamma) - \sum_J g_{\tilde{J}}(\res(\gamma)) \,.
$$

One may then explicitly check that all the genera of pinched jackets
vanish for different choices of $\res(\gamma)$ for
$\gamma\in\mc{G}_{sd}$, by using the Euler character formula for $g_{\tilde{J}}$. For example, for $\res(\gamma) = V_{6;1}$, we have $V_{\tilde{J}}(V_{6;1})=6$, $L_{\tilde{J}}(V_{6;1})=12$, and $C_{\tilde{J}}(V_{6;1})=1$ for all $J$, so $g_{\tilde{J}}=-\frac{1}{2}(F_{\tilde{J}}(V_{6;1})-8)$. We must then count the number of faces for each jacket, which gives $F_{\tilde{J}}(V_{6;1}) = 8$ for all $J$, so $g_{\tilde{J}}(V_{6;1}) = 0$ for all $J$. Similar calculations may be done for the other possible residues of superficially divergent graphs, which results in the above statement.
\end{proof}

%In our following analysis of the superficially divergent graphs, we do not need to consider in the power-counting the two-valent vertices, because these only arise as counter-terms of the original model, so we simplify the classification of the Feynman graphs of the original model. Table~\ref{tab:supdivclass} introduces the power-counting that we use to define the Connes--Kreimer combinatorial Hopf algebra of superficially divergent graphs, where we also list the possible residues for each class.



With the above
generalizations of the usual Connes--Kreimer framework, and the power-counting for the model given in Table~\ref{tab:supdivclass}, we are then ready to define the Hopf algebra of BGR model Feynman graphs in the same fashion as above for local QFT. In particular, we define the coproduct for the 1PI Feynman graphs $\Gamma\in\mc{G}_{\mt{1PI}}$ by
\begin{equation}\label{eq:coprod2}
	\Delta(\Gamma) = \Gamma \otimes \1 + \1 \otimes \Gamma + \sum_{\substack{\gamma\in\bigcup\mc{G}_{sd}^\omega\\\gamma\subsetneq\Gamma}} \gamma \otimes \Gamma/\gamma \,,
\end{equation}
where the subgraph contraction is defined as in Definition~\ref{def:subcontr}. Recall that the summation runs over the set $\bigcup\mc{G}_{sd}^\omega$ of all disconnected unions of superficially divergent 1PI subgraphs. (See Figure~\ref{fig:BGRgraphcoprodexample} for an example.)
\begin{figure}
\centering
\def\svgwidth{0.9\columnwidth}
\input{BGRgraphcoprodexample1_v2.eps_tex}
\caption{\label{fig:BGRgraphcoprodexample}An example of the coproduct $\Delta(\Gamma) \equiv \Gamma \otimes \1 + \1 \otimes \Gamma + \Delta'(\Gamma)$.}
\end{figure}

We have the following auxiliary result.

\begin{lemma} 
	The set of superficially divergent Feynman diagrams of the BGR model is closed under the operations of subgraph contraction and insertion.
\end{lemma}
\begin{proof}
	Let us first consider the subgraph contraction on a
        class-by-class basis. Notice that a subgraph contraction does
        not affect the boundary of a graph, and therefore we do not
        need to consider the boundary properties. Accordingly, the
        contractions of subgraphs lead to the following, depending on
        the class to which they belong:
\begin{itemize}
	\item[$6_1$:] Contracting a subgraph in $6_1,4_1,4_3,2_1$ gives a graph in $6_1$. Other classes are not possible, since $6_1$ does not contain 4-valent vertices, and the genera of its pinched jackets are zero.
	\item[$4_1$:] Contracting a subgraph in $6_1,4_1,4_3,2_1$ gives a graph in $4_1$. Other classes are not possible for the same reasons as above.
	\item[$4_2$:] Contracting a subgraph in $6_1,4_1,4_2,4_3,2_1$ gives a graph in $4_2$. Contracting a subgraph in $2_2$ gives a graph in $4_1$. Other subgraph classes are not possible.
	\item[$4_3$:] Contracting a subgraph in $6_1,4_1,4_3,2_1$ gives a graph in $4_3$. Other subgraph classes are not possible.
	\item[$2_1$:] Contracting a subgraph in $6_1,4_1,4_3,2_1$ gives a graph in $2_1$. Other subgraph classes are not possible.
	\item[$2_2$:] Contracting a subgraph in $6_1,4_1,4_2,4_3,2_1$ gives a graph in $2_2$. Contracting a subgraph in $2_2$ gives a graph in $2_1$. Other subgraph classes are not possible.
	\item[$2_3$:] Contracting a subgraph in $6_1,4_1,4_2,4_3,2_1$ gives a graph in $2_3$. Contracting a subgraph in $2_2$ gives a graph in $2_2$. Contracting a subgraph in $2_3$ gives a graph in $2_1$. Other subgraph classes are not possible.
	\item[$2_4$:] Contracting a subgraph in $6_1,4_1,4_3,2_1$ gives a graph in $2_4$. Contracting a subgraph in $2_4$ gives a graph in $2_1$. Other subgraph classes are not possible.
\end{itemize}
We may condense the above analysis into Table~\ref{tab3}.
\begin{table}
\begin{center}
\begin{tabular}{c||c|ccc|cccc}
	row/col & $6_1$ & $4_1$ & $4_2$ & $4_3$ & $2_1$ & $2_2$ & $2_3$ & $2_4$ \\\hline\hline
	$6_1$ & $6_1$ & $6_1$ & 0 & $6_1$ & $6_1$ & 0 & 0 & 0 \\\hline
	$4_1$ & $4_1$ & $4_1$ & 0 & $4_1$ & $4_1$ & 0 & 0 & 0 \\
	$4_2$ & $4_2$ & $4_2$ & $4_2$ & $4_2$ & $4_2$ & $4_1$ & 0 & 0 \\
	$4_3$ & $4_3$ & $4_3$ & 0 & $4_3$ & $4_3$ & 0 & 0 & 0 \\\hline
	$2_1$ & $2_1$ & $2_1$ & 0 & $2_1$ & $2_1$ & 0 & 0 & 0 \\
	$2_2$ & $2_2$ & $2_2$ & $2_2$ & $2_2$ & $2_2$ & $2_1$ & 0 & 0 \\
	$2_3$ & $2_3$ & $2_3$ & $2_3$ & $2_3$ & $2_3$ & $2_2$ & $2_1$ & 0 \\
	$2_4$ & $2_4$ & $2_4$ & 0 & $2_4$ & $2_4$ & 0 & 0 & $2_1$ \\\hline\hline
\end{tabular}
\vskip10pt
\end{center}
\caption{\label{tab3}Various possibilities of contractions of superficially divergent Feynman graphs of the BGR model.}
\end{table}
We have marked a vanishing entry in the table if a graph from the column class cannot be a subgraph of a graph from the row class.

\bigskip

Similarly, we have to consider the insertions of superficially
divergent graphs into other superficially divergent graphs, as defined
in Definition~\ref{def:insert}. If we consider insertions into the
original Feynman graphs of the model, which do not contain
counter-term vertices, we must note the following: first, we may only
insert graphs into a vertex, whose residue is equal to that
vertex. Moreover, into bare propagators we may only insert
superficially divergent graphs with two external edges from class
$2_1$, since these have divergence degree 2, which is needed in order
to conserve the divergence degree of the graph under insertions. The
other superficially divergent graphs with $N_{ext}=2$ can only be
inserted 
into those counter-term vertices which have the same divergence degree. 
%(and which are not a part of the original bare model). 
With these restrictions in mind, we provide the following analysis of the insertions into the  graphs of the model:
\begin{itemize}
	\item[$6_1$:] Inserting a graph from class $6_1,4_3,2_1$ yields a graph in class $6_1$. Insertion of a graph in $4_2$ is not possible, since $6_1$ does not contain $V_{4;1}$ vertices.
	\item[$4_1$:] Inserting a graph from class $6_1,4_3,2_1$ yields a graph in class $4_1$. Insertion of a graph in $4_2$ is not possible, since $4_1$ does not contain $V_{4;1}$ vertices.
	\item[$4_2$:] Inserting a graph from class $6_1,4_2,4_3,2_1$ yields a graph in class $4_2$.
	\item[$4_3$:] Inserting a graph from class $6_1,4_3,2_1$ yields a graph in class $4_3$. Insertion of a graph in $4_2$ is not possible, since $4_3$ does not contain $V_{4;1}$ vertices.
	\item[$2_1$:] Inserting a graph from class $6_1,4_3,2_1$ yields a graph in class $2_1$. Insertion of a graph in $4_2$ is not possible, since $2_1$ does not contain $V_{4;1}$ vertices.
	\item[$2_2$:] Inserting a graph from class $6_1,4_2,4_3,2_1$ yields a graph in class $2_2$.
	\item[$2_3$:] Inserting a graph from class $6_1,4_2,4_3,2_1$ yields a graph in class $2_3$.
	\item[$2_4$:] Inserting a graph from class $6_1,4_3,2_1$ yields a graph in class $2_4$. Insertion of a graph in $4_2$ is not possible, since $2_4$ does not contain $V_{4;1}$ vertices.
\end{itemize}
Accordingly, all the allowed insertions lead to superficially divergent graphs, and the claim follows.
\end{proof}

Given the above lemma, we may now follow along the lines of the coassociativity proof for the coproduct in \cite{TV}.
\begin{lemma}
	The coproduct defined in \eqref{eq:coprod2} is coassociative, if the set of superficially divergent Feynman diagrams is closed under the operations of subgraph contraction and insertion.
\end{lemma}
\begin{proof}
	First, by a simple calculation, we find that
$$
	(\Delta \otimes \id_\mc{H})\circ \Delta - (\id_\mc{H} \otimes \Delta)\circ \Delta = (\Delta' \otimes \id_\mc{H})\circ \Delta' - (\id_\mc{H} \otimes \Delta')\circ \Delta'
$$
for $\Delta(\Gamma) \equiv \Gamma \otimes \1 + \1 \otimes \Gamma + \Delta'(\Gamma)$, and therefore
$$
	(\Delta \otimes \id_\mc{H})\circ \Delta = (\id_\mc{H} \otimes
\Delta)\circ \Delta \quad \text{if and only if}\quad (\Delta' \otimes \id_\mc{H})\circ \Delta' = (\id_\mc{H} \otimes \Delta')\circ \Delta' \,.
$$
Moreover, it is enough to show the coassociativity property for 1PI Feynman graphs, which are the generators of $\mc{H}$, since $\Delta'$ is an algebra homomorphism, i.e., $\Delta(\Gamma\cup\Gamma')=\Delta(\Gamma)\cup\Delta(\Gamma')$, which is also easy to verify by a direct calculation. Now, on the one hand, we have
	$$
		(\Delta' \otimes \id_\mc{H})\circ \Delta' \Gamma = \sum_{\substack{\gamma\in\bigcup\mc{G}_{sd}^\omega\\\gamma\subsetneq\Gamma}} \Delta'\gamma \otimes \Gamma/\gamma = \sum_{\substack{\gamma\in\bigcup\mc{G}_{sd}^\omega\\\gamma\subsetneq\Gamma}} \sum_{\substack{\gamma'\in\bigcup\mc{G}_{sd}^\omega\\\gamma'\subsetneq\gamma}} \gamma'\otimes \gamma/\gamma' \otimes \Gamma/\gamma \,.
	$$
	On the other hand, we have
	$$
		(\id_\mc{H} \otimes \Delta')\circ \Delta' \Gamma = \sum_{\substack{\gamma'\in\bigcup\mc{G}_{sd}^\omega\\\gamma'\subsetneq\Gamma}} \gamma' \otimes \Delta'(\Gamma/\gamma') = \sum_{\substack{\gamma'\in\bigcup\mc{G}_{sd}^\omega\\\gamma'\subsetneq\Gamma}} \sum_{\substack{\gamma''\in\bigcup\mc{G}_{sd}^\omega\\\gamma''\subsetneq\Gamma/\gamma'}} \gamma' \otimes \gamma'' \otimes (\Gamma/\gamma')/\gamma'' \,.
	$$
	For the coassociativity property, we must show that these are equal for all algebra generators $\Gamma\in\mc{G}_{\mt{1PI}}$. Notice first that in the first sum we may interchange the order of the two summations,
	$$
		(\Delta' \otimes \id_\mc{H})\circ \Delta' \Gamma = \sum_{\substack{\gamma'\in\bigcup\mc{G}_{sd}^\omega\\\gamma'\subsetneq\Gamma}} \sum_{\substack{\gamma\in\bigcup\mc{G}_{sd}^\omega\\\gamma'\subsetneq\gamma\subseteq\Gamma}} \gamma'\otimes \gamma/\gamma' \otimes \Gamma/\gamma \,.
	$$
	It is therefore enough to show that, for a fixed $\gamma'$, we have
	$$
		\sum_{\substack{\gamma\in\bigcup\mc{G}_{sd}^\omega\\\gamma'\subsetneq\gamma\subsetneq\Gamma}} \gamma/\gamma' \otimes \Gamma/\gamma = \sum_{\substack{\gamma''\in\bigcup\mc{G}_{sd}^\omega\\\gamma''\subsetneq\Gamma/\gamma'}} \gamma'' \otimes (\Gamma/\gamma')/\gamma'' \,.
	$$
	These two expressions coincide, if we can set
        $\gamma''=\gamma/\gamma'$ in the summation for all
        $\gamma,\gamma',\gamma''\in\bigcup\mc{G}_{sd}^\omega$ such
        that $\gamma'\subsetneq\gamma\subsetneq\Gamma$ and
        $\gamma''\subsetneq\Gamma/\gamma'$, since then we have 
$$(\Gamma/\gamma')/\gamma'' =
        (\Gamma/\gamma')/(\gamma/\gamma')=\Gamma/\gamma. 
$$ 
This amounts to two requirements:
	\begin{enumerate}
		\item Closure of $\mc{G}_{sd}^\omega$ under contraction: $\gamma/\gamma'\in\mc{G}_{sd}^\omega$ for all $\gamma,\gamma'\in\mc{G}_{sd}^\omega$ such that $\gamma'\subsetneq\gamma$, so that we can find $\gamma'' = \gamma/\gamma' \subsetneq\Gamma/\gamma'$ in $\bigcup\mc{G}_{sd}^\omega$.
		\item Closure of $\mc{G}_{sd}^\omega$ under insertion: for all $\gamma',\gamma''\in\mc{G}_{sd}^\omega$ we have $\gamma'\circ_v \gamma''\in\mc{G}_{sd}^\omega$ for any allowed insertion, so that we can find $\gamma = \gamma'\circ_v \gamma'' \subsetneq\Gamma$ in $\bigcup\mc{G}_{sd}^\omega$.
	\end{enumerate}
	As we have assumed these properties to hold, this finishes the
proof of the lemma.
\end{proof}

We are now in the position to establish our main result.

\begin{theorem}
	Let $\mc{H}_{\mt{BGR}}$ be the unital associative algebra freely generated by the 1PI Feynman graphs of the Ben Geloun--Rivasseau TGFT model. Then $(\mc{H}_{\mt{BGR}},u,m,\epsilon,\Delta,S)$ is a Hopf algebra, where
	\begin{itemize}
		\item the unit $u:\mb{C}\rightarrow\mc{H}_{\mt{BGR}}$ is given by $u(1)=\1$, where $\1$ is the empty graph.
		\item the product $m:\mc{H}_{\mt{BGR}}\otimes\mc{H}_{\mt{BGR}}\rightarrow\mc{H}_{\mt{BGR}}$ is given by the the disjoint union of graphs.
		\item the counit $\epsilon:\mc{H}_{\mt{BGR}}\rightarrow\mb{C}$ is given by  $\epsilon(\1)=1$ and $\epsilon(\Gamma)=0$ for $\Gamma\neq\1$.
		\item the coproduct $\Delta:\mc{H}_{\mt{BGR}}\rightarrow\mc{H}_{\mt{BGR}}\otimes\mc{H}_{\mt{BGR}}$ is given by
			$$
				\Delta(\Gamma) = \Gamma \otimes \1 + \1 \otimes \Gamma + \sum_{\substack{\gamma\in\bigcup\mc{G}_{sd}^\omega\\\gamma\subsetneq\Gamma}} \gamma \otimes \Gamma/\gamma \,,
			$$
			which is extended to $\mc{H}_{\mt{BGR}}$ as an algebra homomorphism.
		\item the antipode $S:\mc{H}_{\mt{BGR}}\rightarrow\mc{H}_{\mt{BGR}}$ is given by the recursive formula
			$$
				S(\Gamma) = -\Gamma - \sum_{\substack{\gamma\in\mc{G}_{sd}^\omega\\\gamma\subsetneq\Gamma}} S(\gamma) \Gamma/\gamma,
			$$
			with $S(\1)=\1$.
	\end{itemize}
\end{theorem}
\begin{proof}
	We have shown above that the coproduct is coassociative. All the other Hopf algebra properties follow easily. In particular, we note that $(\mc{H}_{\mt{BGR}},u,m,\epsilon,\Delta)$ is a graded connected bialgebra, graded by the number of internal edges. Therefore, the antipode follows from Formula~(\ref{eq:Srecurs}).
\end{proof}

The 6-point function Feynman amplitude (without two- or four-point sub-divergences)
of the Ben Geloun--Rivasseau model can then by formally identified
with the general formula given in Theorem~\ref{thm:ren}, 
$$
	\phi_R(\Gamma) = S^\phi_R(\phi(\Gamma)) \star \phi(\Gamma) \,,
$$
where $S^\phi_R$ is given recursively by
$$
	S^\phi_R(\phi(\Gamma)) = - R[\phi(\Gamma)] - R\left[\sum_{\substack{\gamma\subsetneq\Gamma\\\gamma\in\mc{G}_{sd}^\omega}} S^\phi_R(\phi(\gamma)) \phi(\Gamma/\gamma) \right] \,.
$$
If one uses a dimensional regularization scheme, then the formula above can be identified with the renormalized Feynman amplitude of any graph (recall however that in the 
original paper \cite{BR,BR2} the position space multi-scale analysis Taylor expansion was used to prove renormalizability).


\section*{Acknowledgements}
The authors thank Joseph Ben Geloun, R\u azvan Gur\u au and Vincent Rivasseau for
various discussions and suggestions.
A.~Tanasa is partially supported by 
the grant\break PN~09~37~01~02 and he further thanks the Erwin Schr\"odinger Institute for the opportunity to carry out research work during the "Combinatorics, Geometry and Physics" programme.
M.~Raasakka gratefully acknowledges financial support from the Emil
Aaltonen Foundation. 


\begin{thebibliography}{99}

	\bibitem{BgL} J. Ben Geloun and E. Livine, \emph{Some classes of
          renormalizable tensor models}, 
          J. Math.\ Phys.\ {\bf 54} (2013), 082303, 25~pp;
          \href{http://arxiv.org/abs/1207.0416}{arXiv:1207.0416}. 

	\bibitem{BgS}J. Ben Geloun and D. O. Samary, \emph{$3D$
          tensor field theory: renormalization and one-loop
          $\beta$-functions},  Ann.\ Henri Poincar\'e {\bf 14} (2013), 1599--1642;
          \href{http://arxiv.org/abs/1201.0176}{arXiv:1201.0176}. 

	\bibitem{BR}J. Ben Geloun and V. Rivasseau, \emph{A
          renormalizable $4$-dimensional tensor field theory},
          Commun.\ Math.\ Phys.\ {\bf 318} (2013), 69--109;
          \href{http://arxiv.org/abs/1111.4997}{arXiv:1111.4997}.

\bibitem{BR2} J. Ben Geloun and V. Rivasseau, \emph{ Addendum to ``A
          renormalizable $4$-dimensional tensor field theory''},
      Comm.\ Math.\ Phys.\ {\bf 322} (2013), 957--965;
        \href{http://arxiv.org/abs/1209.4606}{arXiv:1209.4606}. 

	\bibitem{COR1} S. Carrozza, D. Oriti and V. Rivasseau,
          \emph{Renormalization of tensorial group field theories:
            Abelian $U(1)$ models in four dimensions}, 
    Comm.\ Math.\ Phys.\ {\bf 327} (2014), 603--641;
          \href{http://arxiv.org/abs/1207.6734}{arXiv:1207.6734}.

	\bibitem{COR2}S. Carrozza, D. Oriti and V. Rivasseau,
          \emph{Renormalization of 
          an $SU(2)$ tensorial group field theory in three
          dimensions}, preprint;
        \href{http://arxiv.org/abs/1303.6772}{arXiv:1303.6772}. 

	\bibitem{CK} A. Connes and D. Kreimer, \emph{Renormalization in
          quantum field theory and the Riemann--Hilbert problem~I: the
          Hopf algebra structure of graphs and the main theorem},
          Commun.\ Math.\ Phys.\ {\bf 210} (2000), 249--273;
          \href{http://arxiv.org/abs/hep-th/9912092}{arXiv:hep-th/9912092}. 

	\bibitem{CM}A. Connes and M. Marcolli, \emph{Noncommutative
          Geometry, Quantum Fields and Motives}, 
World Scientific Press,  Amer.\ Math.\ Soc.\ Coll.\ Publ., 
vol.~55, Amer.\ Math.\ Soc., Providence, R.I.; 
Hindustan Book Agency, New Delhi, 2008. 

\bibitem{cm2} A. Connes and M. Marcolli, \emph{Renormalization and motivic
  Galois theory}, Int.\ Math.\ Res.\ Not.\ {\bf 76} (2004), 4073--4091;
  \href{http://arxiv.org/abs/math/0409306}{arXiv:math/0409306}. 

\bibitem{patras} 
K. Ebrahimi-Fard and F. Patras,
{\it Exponential renormalisation, II. 
Bogoliubov's $R$-operation and momentum subtraction schemes}, 
J. Math.\ Phys.\ {\bf 53} (2012), 083505, 15~pp.

	\bibitem{Gurau} R. Gur\u au, \emph{Colored group field theory},
          Commun.\ Math.\ Phys.\  {\bf 304} (2011), 69--93;
          \href{http://arxiv.org/abs/0907.2582}{arXiv:0907.2582}. 

		\bibitem{Kra}T. Krajewski, \emph{Schwinger--Dyson
                  equations in group field theories of quantum
                  gravity}, preprint;
                  \href{http://arxiv.org/abs/1211.1244}{arXiv:1211.1244}. 

	\bibitem{M} D. V. Malyshev, \emph{Hopf algebra of ribbon graphs
          and renormalization}, J. High Energy Phys.\ {\bf 0205}
          (2002), 013, 28~pp.;
          \href{http://arxiv.org/abs/hep-th/0112146}{arXiv:hep-th/0112146}. 

	\bibitem{Manchon} D. Manchon, \emph{Hopf algebras, from basics
          to applications to renormalization}, Comptes Rendus des
          Rencontres Mathematiques de Glanon (2001), 2003;
          \href{http://arxiv.org/abs/math/0408405}{arXiv:math/0408405}. 

	\bibitem{SVT} D. O. Samary and F. Vignes-Tourneret,
          \emph{Just renormalizable TGFT's on $U(1)^{d}$ with gauge
            invariance}, preprint;
          \href{http://arxiv.org/abs/1211.2618}{arXiv:1211.2618}. 

	\bibitem{TV} A. Tanasa and F. Vignes-Tourneret, \emph{Hopf
          algebra of non-commutative field theory},
          J.\ Noncommut.\ Geom.\ {\bf 2} (2008), 125--139;
          \href{http://arxiv.org/abs/0707.4143}{arXiv:0707.4143}. 

	\bibitem{TK}A. Tanasa and D. Kreimer, \emph{Combinatorial
          Dyson--Schwinger equations in noncommutative field theory},
          J.\ Noncommut.\ Geom.\ {\bf 7} (2013), 255--289;
          \href{http://arxiv.org/abs/0907.2182}{arXiv:0907.2182}. 

	\bibitem{Tan} A. Tanasa, \emph{Algebraic structures in quantum
          gravity}, Class.\ Quant.\ Grav.\ {\bf 27} (2010), 095008, 16~pp;
          \href{http://arxiv.org/abs/0909.5631}{arXiv:0909.5631}. 

\end{thebibliography}


\end{document}
