We study a just-renormalizable tensorial group field theory of
rank six with quartic melonic interactions and Abelian group

Article funded by SCOAP3

1 $ and~define \begin{eqnarray} C_{0}(\vec{g},\vec{g'}) &=&\int_{1}^{\infty} \mathrm{d}\alpha e^{-\alpha m^{2}} \int{dh \prod_{c=1}^{d} K_{\alpha}(g_{c}hg{'}_{c}^{-1})} \nonumber\\ C_{i}(\vec{g},\vec{g'})&=&\int_{M^{-2i}}^{M^{-2(i-1)}} \mathrm{d}\alpha e^{-\alpha m^{2}} \int{dh \prod_{c=1}^{d} K_{\alpha}(g_{c}hg{'}_{c}^{-1})}, i \neq 0. \end{eqnarray} We choose the UV-regulator $\Lambda$ so that $\Lambda=M^{-2\rho}$, and the complete propagator $C_{\Lambda}\equiv C^{\rho}$ is then given by: \begin{equation} C^{\rho}=\sum_{i=0}^\rho C_i. \end{equation} A corresponding sharp momentum cutoff $\chi_{\le\rho} (\vec p) $ is 1 if $\vert \vec p\vert^2 \le M^{2\rho}$ and zero otherwise. The theory with cutoff $\rho$ is defined by using the covariance \begin{equation} C^{\rho}(\vec p) =C(\vec p) \chi_{\le\rho} (\vec p) . \end{equation} Then we slice the theory according to \begin{equation} C^{\rho}(\vec p) =\sum_{i=1}^\rho C_i (\vec p),\; C_i (\vec p) = C(\vec p) \chi_i (\vert \vec p\vert^2) \label{cutoffsha} \end{equation} where $\chi_1$ is 1 if $ \vert \vec p\vert^2 \le M^{2}$ and zero otherwise and for $i\ge 2$ $\chi_i$ is 1 if $M^{2(i-1)} < \vert \vec p\vert^2 \le M^{2i}$ and zero otherwise. A subgraph $S \subset G$ in an initial Feynman graph is a certain subset of lines (propagators $C$) plus the vertices attached to them; the half-lines attached to the vertices of $S$ (whether external lines of $G$ or half-internal lines of $G$ which do not belong to $S$) form the external lines of $G$. Translating to the intermediate representation, we find that a subgraph should be a \emph{set of arcs} of the intermediate field representation, plus all the wavy edges attached to these arcs. The external lines are then the (half)-arcs attached to these wavy edges which do not belong to $S$. A \emph{vertex} of the initial representation is called external for $S$ if it is hooked to at least one external line for $S$. Similarly a wavy \emph{line} of the intermediate representation will be called external to $S$ if it hooks to at least one external \emph{arc}. Particularly interesting subgraphs in the intermediate field representation are those for which the set of arcs are exactly those of a set $S \subset \cL\cV$ of loop vertices (excluding any chain, so no arc belongs to any ciliated vertex). Let us call such subgraphs \emph{proper intermediate} or PI. Remark that any PI graph is automatically 1PI in the initial representation (since all arcs belong to at least one loop, the one of their loop vertex). Also any PI graph can be considered amputated, hence as a graph for a particular vertex function. The converse is not true and many graphs for vertex functions do not correspond to PI graphs in the intermediate representation. PI subgraphs can be represented as graphs of the pure intermediate theory, simply by omitting the two half-arcs at the end of each external wavy line. In our model their amplitude depends only of the single strand momentum entering the wavy line, not of the full momentum of the two half-arcs hooked at its end. We shall see that in our theory only very particular non-vacuum connected subgraphs are superficially divergent, namely PI graphs which are trees with at most two external~lines. ]]>

0 , \end{equation} and the divergence degree $\omega(\mathcal{H})$ of a connected subgraph $\mathcal{H}$ is given by: \begin{equation} \omega(\mathcal{H})=-2L(\mathcal{H})+F(\mathcal{H})-R(\mathcal{H}), \end{equation} where $L(\mathcal{H})$ and $F(\mathcal{H})$ are respectively the number of lines and internal faces of the subgraph $\mathcal{H}$, and $R(\mathcal{H})$ is the rank of the adjacency matrix $\epsilon_{\ell f}$ for the lines and faces of $\mathcal{H}$. \end{theorem} \emph{Proofs.} Obviously we have (for $K=M^2$) \begin{equation} \vert C_i(\vec{p}) \vert \leq K \delta (\sum_c p_c) M^{-2i} \chi_{\le i } ( \vec p). \end{equation} Fixing the external momenta of all external faces the Feynman amplitude (in this momentum representation) is bounded by \begin{align} |A_{\mu}(\mathcal{G})|\leq & \left[ \prod_{\ell\in\mathcal{L}(\mathcal{G})}KM^{-2i_\ell}\right] \prod_{f\in F_{\rm int}(\mathcal{G})} \sum_{p_f\in {\mathbb Z}} \prod_{\ell \in \partial f} \chi_{\le i_\ell } ( \vec p) \prod_{\ell\in\mathcal{L}(\mathcal{G})} \delta (\sum_c p^\ell_c ) . \end{align} The key to multiscale power counting is to attribute the powers of $M$ to the $\mathcal{G}_i^{(k)}$ connected components. For this, we note that, trivially: $M^i=M^{-1}\prod_{j=0}^i M$, a trivial but useful identity which allows e.g.\ to rewrite $\prod_{\ell\in L(G)}M^{-2i_l}=M^2\prod_{\ell\in L(G)}\prod_{i=0}^{i_\ell}M^{-2}$. Then, inverting the order of the double product leads to \begin{equation} \prod_{\ell\in L(\mathcal{G})}M^{-2i_\ell}=\prod_i\prod_{\ell\in \mathcal{L}(\cup_{k=1}^{k(i)}\mathcal{G}_i^k)}M^{-2}=\prod_i\prod_{k=1}^{k(i)}\prod_{l\in \mathcal{L}(\mathcal{G}_i^k)}M^{-2}=\prod_i\prod_{k=1}^{k(i)}M^{-2L(\mathcal{G}_i^k)}. \end{equation} The goal is now to optimize the cost of the sum over the momenta $p_f$ of the internal faces. Summing over $p_f$ with a factor $\chi_{\le i } ( \vec p)$ leads to a factor $KM^i$, hence we should sum with the smallest values $i(f)$ of slices $i$ for the lines $\ell \in \partial f$ along the face $f$. This is exactly the value at which, starting form $i$ large and going down towards $i=0$ the face becomes first internal for some $\mathcal{G}_i^k$. Hence in this way we could bound the sums $\prod_{f\in F_{\rm int}(\mathcal{G})} \sum_{p_f\in {\mathbb Z}} $ by \begin{equation} \prod_i\prod_{k=1}^{k(i)}M^{F(\mathcal{G}_i^k)}. \end{equation} However this can be still improved, because we have not yet taken into account the gauge factor $\prod_{\ell\in\mathcal{L}(\mathcal{G})} \delta (\sum_c p _c ) $. It clearly tells us that some sums over $p_f$ do not occur at all. How many obviously depends of the \emph{rank} $R$ of the incidence matrix $\epsilon_{\ell f}$. Indeed rewriting the delta functions in terms of the $p_{f(\ell, c)}$ we have \begin{equation} \prod_{\ell\in\mathcal{L}(\mathcal{G})} \delta (\sum_c p^\ell_c )= \prod_{\ell\in\mathcal{L}(\mathcal{G})} \delta (\sum _f \epsilon_{\ell f} p_f) . \end{equation} Hence writing the linear system of $L$ equations $\sum _f \epsilon_{\ell f} p_f =0$ corresponding to the delta functions we can solve for $R$ momenta $p_f$ in terms of $L-R$ others. It means that in the previous argument we should pay only for $F-R$ sums over internal face momenta instead~of~$F$.\footnote{Remark that the remaining product unused or redundant $\delta$ functions are simply bounded by 1 because the $p_f$ variables are discrete, hence the $\delta$ function are simply Kronecker symbols, all bounded by 1; of course this would not be true for continuous variables as a product of redundant $\delta$ distributions in the continuum is ill-defined.} This argument can be made more precise and rigorous and distributed over all scales starting from the leaves of the Gallavotti-Nicol\`o tree (the smallest subgraphs $\mathcal{G}_i^k$) and progressing towards the root we can select faces such that the restricted sub-matrix $\epsilon_{\ell f} $ still has maximal rank $R(\mathcal{G}_i^k)$ in each $\mathcal{G}_i^k$. We discard the other faces decay factor. Then we can select lines so as to find a restricted \emph{square} submatrix $\epsilon_{\ell f} $ with maximal rank $R(\mathcal{G}_i^k)$ in each $\mathcal{G}_i^k$. This leads to \begin{equation} |A_{\mu}(\mathcal{G})| \leq K^{\mathcal{L}(\mathcal{G})}\prod_i\prod_{k=1}^{k(i)}M^{-2L(\mathcal{G}_i^k)+F(G_i^k)-R(\mathcal{G}_i^k)} = K^{|L(G)|}\prod_i\prod_{k=1}^{k(i)}M^{\omega(\mathcal{G}_i^k)}. \label{powerbound} \end{equation} This equation completes the proof, and the exponent $\omega(\mathcal{G}_i^k)=-2L(\mathcal{G}_i^k)+F(\mathcal{G}_i^k)-R(\mathcal{G}_i^k)$ identifies the divergence degree. \qed ]]>

2$, each new external line takes $L$ into $L+1$, can either keep $F$ unchanged (if it hits an already open face), in which case $R$ is also unchanged, or takes $F$ to $F-1$, in which case either $R$ is unchanged or goes to $R-1$; hence $\omega$ decreases at least by 1. This proves \begin{equation} \omega(\cG) \le - (N-2) \;\;{\rm if} \;\; N >2. \label{extlegs} \end{equation} \end{itemize} Consider next the case of a PI vacuum subgraph with $N$ external wavy lines and $q$ wavy loops. We can first pick a tree of wavy lines then add the wavy loops one by one. Each added loop creates two new arcs and changes the number of faces by -1, 0 or 1. It can change the rank at most by 1, and when it creates a face, then the rank cannot decrease (the matrix $\epsilon$ becoming bigger). Hence \begin{equation} \omega(\cG) \le - (N-2) - 3q \;\;{\rm if} \;\; N >2. \label{extlegsloops} \end{equation} In particular if $N=1$ and $q\ge 1$ we have $\omega(\cG) \le -1$ and the graph is convergent. Finally it remains to study the case of non-vacuum, non-PI graph. Since they add at least one new arc to a PI graph, it is easy to check they have $\omega <0$, except in two particular cases corresponding both to one-particle reducible graphs: \begin{itemize} \item a chain of arcs joining PI two-point trees, with one of them at both ends. Such subgraphs are one-particle reducible two point subgraphs of the initial theory with $\omega =2$. \item a chain of arcs joining PI two-point trees, with one of them at a single of its two ends. Such subgraphs are one-particle reducible four point subgraphs of the initial theory, with $\omega =0$. \end{itemize} These cases are not interesting since such subgraphs cannot occur as $\mathcal{G}_i^k$s and, as is well known, renormalization can be restricted to IPI subgraphs. These results in particular show that the degree of divergence $\omega$ does not depend on the number of vertices, but only on the number of external lines. This is typical of a just renormalizable field theory. Trees in the intermediate representation correspond to melonic subgraphs in the ordinary representation~\cite{Gurau-ml-2013pca}. Hence we have proved, in agreement with the other renormalizable TGFT's: \begin{theorem} The only superficially divergent PI subgraphs are melonic in the ordinary representation, with two or four external ordinary lines. In the intermediate representation, amputating the trivial external arcs, they are PI trees with a single external wavy line, or with two external wavy lines of the same color carrying the same strand momentum. \label{theordivequalmelo} \end{theorem} Melonic graphs are graphs with zero \emph{degree},\footnote{The degree in question is the ``degree of the colored graph", which characterizes the dominant order of the large-N limit of tensor models. It should not be confused with the degree of divergence, and we denote it by $\varpi(\mathcal{G}_c)$.} hence for which all \emph{jackets} are planar. We include for completeness brief definitions of these two notions, referring to~\cite{review} for more~\mbox{details}. \begin{definition}[Jackets] A \emph{jacket} $\mathcal{J}$ of a regular $d+1$ colored graph $\mathcal{G}_{c}$ is the canonical ribbon graph associated to $\mathcal{G}_{c}$ and to a (D+1)-cycle $\xi$ up to orientation. It has the same number of lines and vertices than $\mathcal{G}_{c}$, but contains only a subset of the faces, those with consecutive colors in the cycle $\mathcal{F}_{\mathcal{J}}=\left\lbrace f\in \mathcal{F}_{\mathcal{G}_{c}}|f=({\xi}^q(0),{\xi}^{q+1}(0)), q\in \mathbb{Z}_{D+1}\right\rbrace $. \end{definition} Hence there are $d!/2$ jackets at rank $d$ and to each jacket is associated a Riemann surface of genus $g_{\mathcal{J}}$. \begin{definition}[Degree] The degree $\varpi(\mathcal{G}_{c})$ is by definition the sum over the \emph{genus} of all the jackets: \begin{equation} \varpi(\mathcal{G}_{c})=\sum_{\mathcal{J}}g_{\mathcal{J}} \Rightarrow \varpi(\mathcal{G}_{c})\geq 0 . \end{equation} \end{definition} The degree governs the $1/N$ tensorial expansion since the number of faces is a monotonically decreasing function of the degree. Melonic graphs have maximal number of faces at a given perturbation order. More precisely \begin{lemma} The number of faces $F_c$ of $\mathcal{G}_{c}$ is related to the number of black vertices $p$ and to the dimension $d$ by: \begin{equation} F_c=\dfrac{d(d-1)}{2}p+d-\dfrac{2}{(d-1)!}\varpi(\mathcal{G}_{c}) . \label{numberfaces} \end{equation} \end{lemma} A tensorial graph $\cG$ having a unique colored extension $\cG_c$, we can extend the notion of degree to tensorial graph. Since the colored extensions of type 1 vertices of our theory all have the same number of inner faces (faces without color 0), the degree of $\cG_c$ again governs the number of faces of $\cG$, which are the bicolored faces of $\cG_c$ which includes color 0. In our case the vertices of $\cG$ all have 25 inner faces and $p=2$ black vertices, so that~\eqref{numberfaces} tells~us \begin{equation} F(\cG) =5V +6-\dfrac{1}{60}\varpi(\mathcal{G}_{c}) . \label{numberfaces1} \end{equation} Returning to Theorem~\ref{theordivequalmelo} we can precise the divergent part of the theory in the language of the previous section. $\Gamma_2^{\rm melo}$ and $\Gamma_4^{\rm melo}$ are naturally defined as the melonic approximations to $\Gamma_2$ and $\Gamma_4$ and Theorem~\ref{theordivequalmelo} indeed proves that $\Gamma_2- \Gamma^{\rm melo}_2$ and $\Gamma_{4,\rm mono}- \Gamma^{\rm melo}_{4,\rm mono}$ are superficially convergent. Moreover they express simply as tree approximations of the pure $\tau$ intermediate field theory: we have \begin{equation} \Gamma_2^{\rm melo} (\vec p) = \sqrt{2 \lambda} \sum_{c=1}^6 W^{\rm tree}_1 (p_c),\; \Gamma_4^{\rm melo} (p_c,p_c') = \sqrt{2 \lambda} \delta(p_c, p'_c) W^{=,tree}_{2} (p_c, p_c) \label{goodeq1} \end{equation} where $W^{\rm tree}_1 $ and $W^{=,\rm tree}_{2}$ are respectively the tree approximation to $W_1 $ and $W^{=}_{2}$. \pagebreak But Theorem~\ref{theordivequalmelo} contains still an additional information on the divergent part of $\Gamma_4^{\rm melo}$. Defining $W^{=,\rm tree}_{2,\rm mono}$ as the part of $W^{=,\rm tree}_{2}$ in which all wavy lines along the unique path joining the two external lines must be of the same color $c$ than these two external lines, it states that the difference $W^{=,\rm tree}_{2}- W^{=,\rm tree}_{2,\rm mono}$ is also ultraviolet finite, hence can be neglected in the following section on renormalization. Since~\eqref{goodeq1} is nothing but~\eqref{goodeq} with $f= -i \sqrt{2/ \lambda} W^{{\rm tree},c}_1$ and $g= W^{\rm tree}_{2,\rm mono}$, this completes the proof of Theorem~\ref{goodth}. ]]>

4$ external arcs \begin{equation} \omega (\cH) \le - N(\cH)/3 = - 2N(\cH)/6. \end{equation} This is also true if $\cH$ is convergent with $N = 1$ or 2, since we saw that in this case $\omega \le -1 \le - N(\cH)/3$. For a $\phi^4$ graph of order $V=n$ with $2N$ external legs, we have $2L = 4V +2N$. Therefore~\eqref{powerbound} implies that for another constant $K$ \begin{equation} \cA(\mathcal{G}) \le K^n \sum_\mu \prod_i\prod_{k=1}^{k(i)}M^{-2N(\mathcal{G}_i^k)/6 }. \end{equation} Let us now define \begin{equation} i_v(\mu)=\sup_{\ell \in L_v(\mathcal{G})}i_\ell (\mu) , e_v(\mu)= \inf_{ \ell \in L_b(\mathcal{G})}i_\ell(\mu), \end{equation} where $v$ denotes a vertex $v\in\mathcal{G}$, and $L_v(\mathcal{G})$ the set of its external (half)-lines. $v$ is external to a high subgraph $\mathcal{G}_i^k$ if and only if $e_b < i \leq i_b$, and then it is hooked to at least one of the $2N(\mathcal{G}_i^k)$ external half-lines of $\mathcal{G}_i^k$. Therefore \begin{equation} \prod_{i,k}M^{-2N(\mathcal{G}^{(k)}_i)/6} \leq \prod_{i,k}\prod_{v\in \mathcal{G}^{(k)}_i | e_v

i$ and $\lambda_{i+1}$ for all vertices with highest scale $j\le i$; \item effective propagators $C_j$ for lines with indices $j >i$ and $C_{i+1}$ for all lines with indices $j\le i$, \item effective amplitudes ${\cal A}^{eff,i+1}(\mathcal{G}) $ with subtractions $\prod_{\gamma \in D^{i+1}_{\mu}(\mathcal{G})}(1-\tau^{*}_{\gamma})$, where $D^{i+1}_{\mu}$ is the forest of all divergent $\cG_{j}^k$ with $j > i$. \end{itemize} We define the next coupling $\lambda_{i}$ and propagator $C_i$ by considering in $\mu$ the scale number $i$. Adding and subtracting the counter-terms in $D^{i}_{\mu}\setminus D^{i+1}_{\mu}=\{\cH \in D_\mu(\mathcal{G}) | \inf i_\ell = i\} , \ell \in \cH$, we write \begin{equation} \cA^{eff,i+1}_{\mu}(\mathcal{G}):=\prod_{\cH\in D^{i}_{\mu}\setminus D^{i+1}_{\mu}} [(1-\tau^{*}_{\cH})+\tau^{*}_{\cH}]\prod_{\gamma \in D_{\mu}^{i+1}}(1-\tau^{*}_{\gamma})\cA_{\mu}^{eff,i+1}(\mathcal{G}), \end{equation} and we expand the product over $\cH\in D^{i}_{\mu}\setminus D^{i+1}_{\mu}$. The operators $(1-\tau^{*}_{\cH})$ will generate the next layer of subtraction in the formula to change the subtraction operations of $\cA^{eff,i+1}_{\mu}(\mathcal{G})$ into those of $A^{eff,i}_{\mu}(\mathcal{G})$. The counterterms $+\tau^{*}_{\cH}$ are then associated to collapsed graphs $\cG/\cH$ in which $\cH$ has been collapsed to a vertex (if $N(\cH) =2$) or to a mass or a wave function insertion (if $N(\cH)=1$). Collecting these pieces and rearranging them according to the collapsed graph rather than to the initial graph defines an (infinite series) redefinition of the couplings hooked to vertices with highest line of slice $j< i$ and of the propagators with scale $j