Renormalization of a tensorial field theory on the homogeneous space SU(2)/U(1)

We consider a tensorial quantum field theory on d copies of the homogeneous space SU(2)/U(1), which is isomorphic to the two dimensional sphere ${{\mathcal{S}}_{2}}$. The phase space of the theory is the cotangent bundle ${{\left({{\mathcal{T}}^{\ast}}{{\mathcal{S}}_{2}}\right)}^{\times d}}\cong {{\left({{\mathcal{S}}_{2}}\times {{\mathbb{R}}^{2}}\right)}^{\times d}}$. The complex field $\psi \in {{L}_{2}}\left(\mathcal{S}_{2}^{\times d}\right)$, assumed to be square-integrable, is defined as

$$\begin{eqnarray}&&\begin{array}{c} \psi :{{\left[SU(2)/U(1)\right]}^{d}} \to \mathbb{C}, \\ \left({{x}_{1}},...,{{x}_{d}}\right)\in {{\left[SU(2)/U(1)\right]}^{d}} \to \psi \left({{x}_{1}},...,{{x}_{d}}\right), \\ {{{\int}^{}}_{\mathcal{S}_{2}^{\times d}}}\underset{i=1}{\overset{d}{\prod}}\,\text{d}{{x}_{i}}\bar{\psi}\left({{x}_{1}},...,{{x}_{d}}\right) \psi \left({{x}_{1}},...,{{x}_{d}}\right)<\infty. \end{array}\end{eqnarray}$$

The quantum dynamics is defined by the partition function

$$\begin{eqnarray}&&\mathcal{Z}={\int}^{}\text{d}{{\mu}_{{{C}_{0}}}}\left(\psi,\bar{\psi}\right){{\text{e}}^{-{{S}_{\text{int}}}\left(\psi,\bar{\psi}\right)}},\end{eqnarray} \tag{ 1 }$$

where the Gaussian measure with covariance C₀, $\text{d}{{\mu}_{{{C}_{0}}}}\left(\psi,\bar{\psi}\right)$ encode the kinetic part of the classical action, and define the free 2-point function, and the interaction part S_int of the action is constructed with all the trace invariant contractions

$$\begin{eqnarray}&&{{S}_{\text{int}}}=\underset{b}{\sum}\,{{\lambda}_{b}}\text{T}{{\text{r}}_{b}}\left(\psi,\bar{\psi}\right).\end{eqnarray} \tag{ 2 }$$

These traces are labeled by a d-colored bipartite regular graph (i.e. strictly d-valent, with links colored with d colors at each node), called bubbles, whose some examples are pictured on figure 1 below. Each black and white nodes correspond respectively to the fields ψ and $\bar{\psi}$, the d half-lines hooked to a black (resp. white) node picture the d variable of the corresponding field ψ (resp. $\bar{\psi}$), and the connectivity of the graph give the pattern of contraction between each fields. For instance, the bubble on figure 1(a) corresponds to the following interaction:

$$\begin{eqnarray}&&\text{T}{{\text{r}}_{{{b}_{\text{figure}\,1(a)}}}}\left(\psi,\bar{\psi}\right)={\int}^{}\underset{i=1}{\overset{3}{\prod}}\,\text{d}{{x}_{i}}\text{d}x_{i}^{\prime}\psi \left({{x}_{1}},{{x}_{2}},{{x}_{3}}\right)\bar{\psi}\left(x_{1}^{\prime},{{x}_{2}},{{x}_{3}}\right)\psi \left(x_{1}^{\prime},x_{2}^{\prime},x_{3}^{\prime}\right)\bar{\psi}\left({{x}_{1}},x_{2}^{\prime},x_{3}^{\prime}\right).\end{eqnarray} \tag{ 3 }$$

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This trace or tensorial invariant provides a well characterization of the theory space. Moreover, the tensorial structure of the interaction allows to organize a power-counting, and as we will see in section 6, it provides a well definition of locality, essential for the definitions of counter-terms and renormalization.

If the kinetic action is properly defined (i.e. if the propagator is invertible), the Gaussian measure can be defined by the choice of a kinetic action as follows:

$$\begin{eqnarray}&&\text{d}{{\mu}_{{{C}_{0}}}}\left(\psi,\bar{\psi}\right):={{\text{e}}^{-{{S}_{\text{kin}}}\left[\bar{\psi},\psi \right]}}\text{d}\psi \text{d}\bar{\psi},\end{eqnarray} \tag{ 4 }$$

with

$$\begin{eqnarray}&&{{S}_{\text{kin}}}\left[\bar{\psi},\psi \right]:={\int}_{\mathcal{S}_{2}^{\times d}}\underset{i=1}{\overset{d}{\prod}}\,\left[\text{d}{{x}_{i}}\sqrt{|g|}\right]\bar{\psi}\left(\vec{x}\right)\left(-\underset{i=1}{\overset{d}{\sum}}\,{{ \Delta }_{i}}+{{m}^{2}}\right)\psi \left(\vec{x}\right).\end{eqnarray} \tag{ 5 }$$

where ${{ \Delta }_{i}}$ is the Laplacian operator on the 2-sphere of unit radius, $|g|$ the determinant of the metric in coordinates $\left\{{{x}_{i}},i=1,2\right\}$, and m² be a real parameter playing the role of a mass term. As an integrable function on $\mathcal{S}_{2}^{\times d}$, the field ψ can be expanded on the spherical harmonics basis $\left\{{{Y}_{l,m}}(\theta,\phi )\right\}$, which is a complete basis of L²-functions on the 2-sphere. In this basis, the propagator (or covariance) ${{C}_{0}}\left(\left\{{{\theta}_{i}},{{\phi}_{i}},\theta _{i}^{\prime},\phi _{i}^{\prime}\right\}\right)$, defined by the kinetic action (5) writes as:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{C}_{0}}\left(\left\{{{\theta}_{i}},{{\phi}_{i}},\theta _{i}^{\prime},\phi _{i}^{\prime}\right\}\right)={\int}^{}\text{d}{{\mu}_{{{C}_{0}}}}\bar{\psi}\left(\left\{{{\theta}_{i}},{{\phi}_{i}}\right\}\right)\psi \left(\left\{\theta _{i}^{\prime},\phi _{i}^{\prime}\right\}\right)=\underset{\left\{{{l}_{i}},{{m}_{i}}\right\}}{\sum}\,{{C}_{0\,\left\{{{l}_{i}},{{m}_{i}}\right\}}}\underset{i=1}{\overset{d}{\prod}}\,Y_{{{l}_{i}},{{m}_{i}}}^{\ast}\left({{\theta}_{i}},{{\phi}_{i}}\right){{Y}_{{{l}_{i}},{{m}_{i}}}}\left(\theta _{i}^{\prime},\phi _{i}^{\prime}\right), \end{array}\end{eqnarray} \tag{ 6 }$$

where the coefficients

$$\begin{eqnarray}&&{{C}_{0\,\left\{{{l}_{i}},{{m}_{i}}\right\}}}:=\frac{1}{\underset{i}{\sum}\,{{l}_{i}}\left({{l}_{i}}+1\right)+{{m}^{2}}},\end{eqnarray} \tag{ 7 }$$

do not depend on the magnetic indices m_i. This definition of the theory is in fact highly formal, because some divergences can occur in the perturbative expansion. In order to circumvent this difficulty, we introduce an ultra-violet cut-off $\Lambda$, and define the regularized propagator using Schwinger regularization:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{C}_{0\, \Lambda }}\left(\left\{{{\theta}_{i}},{{\phi}_{i}},\theta _{i}^{\prime},\phi _{i}^{\prime}\right\}\right)={\int}_{1/{{ \Lambda }^{2}}}^{+\infty}\text{d}\alpha {{\text{e}}^{-\alpha {{m}^{2}}}}\times \underset{\left\{{{l}_{i}},{{m}_{i}}\right\}}{\sum}\,\underset{i=1}{\overset{d}{\prod}}\,{{\text{e}}^{-\alpha {{l}_{i}}\left({{l}_{i}}+1\right)}}Y_{{{l}_{i}},{{m}_{i}}}^{\ast}\left({{\theta}_{i}},{{\phi}_{i}}\right){{Y}_{{{l}_{i}},{{m}_{i}}}}\left(\theta _{i}^{\prime},\phi _{i}^{\prime}\right). \end{array}\end{eqnarray} \tag{ 8 }$$

Interestingly for the computation of Feynman amplitudes, this propagator involves the heat kernel

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{K}_{\alpha}}\left(\left\{\theta,\phi,{{\theta}^{\prime}},{{\phi}^{\prime}}\right\}\right)=\underset{\left\{l,m\right\}}{\sum}\,{{\text{e}}^{-\alpha l(l+1)}}Y_{l,m}^{\ast}(\theta,\phi ){{Y}_{l,m}}\left({{\theta}^{\prime}},{{\phi}^{\prime}}\right), \end{array}\end{eqnarray} \tag{ 9 }$$

which verifies the heat equation,

$$\begin{eqnarray}&&\frac{\partial}{\partial \alpha}{{K}_{\alpha}}= \Delta {{K}_{\alpha}},\end{eqnarray} \tag{ 10 }$$

and boundary conditions:

$$\begin{eqnarray}&&{{K}_{\alpha =0}}\left(\theta,\phi ;{{\theta}^{\prime}},{{\phi}^{\prime}}\right)=\delta \left(\cos \theta -\cos {{\theta}^{\prime}}\right)\delta \left(\phi -{{\phi}^{\prime}}\right).\end{eqnarray} \tag{ 11 }$$

Moreover, the heat kernel satisfies the composition law:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {\int}^{}\sin \theta \text{d}\theta \text{d}\phi & {{K}_{{{\alpha}_{1}}}}\left(\left\{{{\theta}^{\prime}},{{\phi}^{\prime}},\theta,\phi \right\}\right){{K}_{{{\alpha}_{2}}}}\left(\left\{\theta,\phi,{{\theta}^{\prime \prime}},{{\phi}^{\prime \prime}}\right\}\right)={{K}_{{{\alpha}_{1}}+{{\alpha}_{2}}}}\left(\left\{{{\theta}^{\prime}},{{\phi}^{\prime}},{{\theta}^{\prime \prime}},{{\phi}^{\prime \prime}}\right\}\right). \end{array}\end{eqnarray} \tag{ 12 }$$

This is in turn the key property to obtain the expression for the Feynman amplitudes entering in the perturbative expansion of the N-point correlation functions S_N. Such a function can be expanded as a sum indexed by Feynman graphs:

$$\begin{eqnarray}&&{{S}_{N}}=\underset{G}{\sum}\,\frac{1}{s(G)}\left(\underset{b\in G}{\prod}\,-{{\lambda}_{b}}\right){{\mathcal{A}}_{G}}.\end{eqnarray} \tag{ 13 }$$

where s(G), the symmetry factor. Using (12), the amplitude ${{\mathcal{A}}_{G}}$ for the graph $G$ can be written as (see [19]):

$$\begin{eqnarray}\begin{array}{*{35}{l}} {} & {{\mathcal{A}}_{G}}\,=\,\left[\underset{e\in L(G)}{\prod}\,{\int}_{1/{{ \Lambda }^{2}}}^{\infty}\text{d}{{\alpha}_{e}}{{\text{e}}^{-{{\alpha}_{e}}{{m}^{2}}}}\right]\times \underset{f\in F(G)}{\prod}\,\underset{{{l}_{f}}}{\sum}\,\left(2{{l}_{f}}+1\right){{\text{e}}^{-\alpha (\,f){{l}_{f}}\left({{l}_{f}}+1\right)}} \\ {} & \,\,\,\,\,\,\quad \,\,\,\,\times \,\,\left(\underset{f\in {{F}_{\text{ext}}}(G)}{\prod}\,\underset{{{l}_{f}},{{m}_{f}}}{\sum}\,{{\text{e}}^{-\alpha (\,f){{l}_{f}}\left({{l}_{f}}+1\right)}}Y_{{{l}_{f}},{{m}_{f}}}^{\ast}\left({{\theta}_{s(\,f)}},{{\phi}_{s(\,f)}}\right){{Y}_{{{l}_{f}},{{m}_{f}}}}\left({{\theta}_{t(\,f)}},{{\phi}_{t(\,f)}}\right)\right), \end{array}\end{eqnarray} \tag{ 14 }$$

where L(G),F(G), and ${{F}_{\text{ext}}}(G)$ denote respectively the sets of lines, faces, and external faces of the graph G, and s and t map open faces to their boundary variables, and $\alpha (\,f):={\sum}_{e\in \partial f}{{\alpha}_{e}}$, where $\partial f$ denote the set of boundary lines of f. Feynman graph can be depicted graphically as in figure 2, with the rule that bubble vertices are depicted as in figure 1, and Wick contractions with a dotted line between a black and a white vertex, both in the same bubble or not.

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For physical reasons, in which we will return in the appendix, we will impose an additional condition on our field, which we call closure constraint, and which make sense only for d  >  1. It can be understood as a gauge symmetry for the field, which reduces the manifold $\mathcal{S}_{2}^{\times d}$ as:

$$\begin{eqnarray}&&{{\left[SU(2)/U(1)\right]}^{d}}\to {{\left[SU(2)/U(1)\right]}^{d}}/SU(2),\end{eqnarray} \tag{ 15 }$$

identifying the field components up to a global SU(2) group action. More precisely, if we denote the action of the group element $g\in SU(2)$ on the field ψ as $\hat{\mathcal{R}}(g)\triangleright \psi$, where $\hat{\mathcal{R}}$ and $\triangleright$ are defined by the explicit group action of SU(2) (see below), the closure constraint identifies, for a given ψ, all the elements $\hat{\mathcal{R}}(g)\triangleright \psi \,\forall g\in SU(2)$.

Let us specify further the group action. We observe that the 2-sphere admits a natural embedding in ${{\mathbb{R}}^{3}}$, and, using this, into SO(3):

$$\begin{eqnarray}\begin{array}{*{35}{l}} \quad \quad \pi :{{\mathcal{S}}_{2}} & \to SO(3) \end{array}\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} (\theta,\phi ) & \to \pi (\theta,\phi )\in SO(3) \end{array}\end{eqnarray} \tag{ 17 }$$

with the following explicit expression in local coordinates $(\theta,\phi )$:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \pi (\theta,\phi )\left[\hat{z}\right]=\vec{n}(\theta,\phi ) \end{array}\end{eqnarray} \tag{ 18 }$$

where:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {} & \vec{n}:(\theta,\phi )\to \vec{n}(\theta,\phi ):=(\sin \theta \cos \phi,\sin \theta \sin \phi,\cos \theta )\in {{\mathbb{R}}^{3}} \end{array}\end{eqnarray} \tag{ 19 }$$

Hence, $\pi (\theta,\phi )$ is the rotation of SO(3) mapping the $\hat{z}$ axis in the direction $\vec{n}$⁴. Starting from our field on $\mathcal{S}_{2}^{\times d}$, this mapping enable us to define a new field on $SO{{(3)}^{\times d}}$, $\tilde{\psi}\in {{L}_{2}}\left(SO{{(3)}^{\times d}}\right)$ such as ${{\pi}^{\ast}}\tilde{\psi}\in {{L}_{2}}\left(\mathcal{S}_{2}^{\times d}\right):=\psi$, with the notation ${{\pi}^{\ast}}$ defined as: ${{\pi}^{\ast}}\tilde{\psi}\left({{x}_{1}},...,{{x}_{d}}\right):=\tilde{\psi}\left(\pi \left({{x}_{1}}\right),...,\pi \left({{x}_{d}}\right)\right)$.

For the field $\tilde{\psi}\in {{L}_{2}}\left(SO{{(3)}^{\times d}}\right)$, there are a natural right action of the group SO(3). Hence, we can define the gauge symmetry as the identification of all the fields up to a global right action of SO(3). More concretely, we introduce the symmetric rotation $\hat{R}$ on ${{L}_{2}}\left(SO{{(3)}^{\times d}}\right)$, such that $\hat{R}(g)\equiv \hat{R}(-g)\,\forall g\in SU(2)$ (it is more convenient to work with a compact simply connected group). Hence, the operator $\hat{R}$ can be understood as a function on $SO(3)\sim SU(2)/{{\mathbb{Z}}_{2}}$. Let $\mathcal{R}$ the map $\mathcal{R}:SU(2)\to SO(3)$, which identify g and  −g. We can therefore define the transformation law:

$$\begin{eqnarray}\begin{array}{*{35}{l}} \hat{R}(g):{{L}_{2}}\left(SO{{(3)}^{\times d}}\right) & \to {{L}_{2}}\left(SO{{(3)}^{\times d}}\right) \\ \left[\hat{R}(g)\tilde{\psi}\right]\left(\left\{\pi \left({{\theta}_{i}},{{\phi}_{i}}\right)\right\}\right) & :=\tilde{\psi}\left(\left\{\pi \left({{\theta}_{i}},{{\phi}_{i}}\right)\mathcal{R}(g)\right\}\right), \end{array}\end{eqnarray} \tag{ 20 }$$

Now, we can clarify the definition of the action $\triangleright$ introduced before. More precisely, $\hat{\mathcal{R}}(g)$ acts on ψ as:

$$\begin{eqnarray}\begin{array}{*{35}{l}} \hat{\mathcal{R}}(g)\triangleright \psi \left(\left\{{{\theta}_{i}},{{\phi}_{i}}\right\}\right) & :={{\pi}^{\ast}}\left[\hat{R}(g)\tilde{{\psi}}\,\right]\left(\left\{{{\theta}_{i}},{{\phi}_{i}}\right\}\right) \\ {} & \,=\psi \left({{\pi}^{-1}}\left[\left(\pi \left({{\theta}_{i}},{{\phi}_{i}}\right)\right)\mathcal{R}(g)\right]\right). \end{array}\end{eqnarray} \tag{ 21 }$$

Then, the closure constraint, or gauge invariance, is the requirement:

$$\begin{eqnarray}&&\hat{\mathcal{R}}(g)\triangleright \psi =\psi,\quad \quad \forall g\in SU(2).\end{eqnarray} \tag{ 22 }$$

Let us consider the projector into gauge invariant fields ${\int}^{}\text{d}g\hat{\mathcal{R}}(g)$. We can think to implement the closure constraint by projection of the fields involved in the definition of the kinetic action S_kin. But in this way, we can not define easily an explicit propagator, because the kinetic kernel is not invertible on the space of fields. However, the Wick theorem states that the Gaussian measure, and with it, the perturbative expansion of the quantum theory, is well defined as long as the 2-point function at ${{\lambda}_{b}}=0$ (the propagator ${{C}_{ \Lambda }}\left(\left\{{{\theta}_{i}},{{\phi}_{i}}\right\},\left\{\theta _{i}^{\prime},\phi _{i}^{\prime}\right\}\right)$) is properly defined. As a result, we choose⁵:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {} & {\int}^{}\text{d}{{\mu}_{C}}\left(\psi,\bar{\psi}\right)\psi \left(\left\{{{\theta}_{i}},{{\phi}_{i}}\right\}\right)\bar{\psi}\left(\left\{\theta _{i}^{\prime},\phi _{i}^{\prime}\right\}\right)={{{\int}^{}}_{SU(2)}}\text{d}g{\int}^{}\text{d}{{\mu}_{{{C}_{0}}}}\left(\psi,\bar{\psi}\right)\hat{\mathcal{R}}(g)\triangleright \psi \left(\left\{{{\theta}_{i}},{{\phi}_{i}}\right\}\right)\bar{\psi}\left(\left\{\theta _{i}^{\prime},\phi _{i}^{\prime}\right\}\right), \end{array}\end{eqnarray} \tag{ 23 }$$

or

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {\int}^{}\text{d}{{\mu}_{C}}\left(\psi,\bar{\psi}\right)\psi \left(\left\{{{\theta}_{i}},{{\phi}_{i}}\right\}\right)\bar{\psi}\left(\left\{\theta _{i}^{\prime},\phi _{i}^{\prime}\right\}\right)={\int}_{SU(2)}\text{d}g{\int}_{1/{{ \Lambda }^{2}}}^{\infty}\text{d}\alpha {{\text{e}}^{-\alpha {{m}^{2}}}}\underset{i=1}{\overset{d}{\prod}}\,{{K}_{\alpha}}\left(\left\{{{\pi}^{-1}}\left[\left(\pi \left({{\theta}_{i}},{{\phi}_{i}}\right)\right)\mathcal{R}(g)\right];\theta _{i}^{\prime},\phi _{i}^{\prime}\right\}\right). \end{array}\end{eqnarray} \tag{ 24 }$$

We can obtain an explicit expression for this constrained propagator. Using the expression for the heat kernel (9) and the decomposition,

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{Y}_{l,m}} & \left(\left(\pi \left({{\theta}_{i}},{{\phi}_{i}}\right)\right)\mathcal{R}(g)\hat{z}\right)=\underset{{{m}^{\prime}}=-l}{\overset{+l}{\sum}}\,D_{{{m}^{\prime}}m}^{(l)}\left[\mathcal{R}{{(g)}^{-1}}\pi {{\left({{\theta}_{i}},{{\phi}_{i}}\right)}^{-1}}\right]{{Y}_{l,{{m}^{\prime}}}}\left(\hat{z}\right), \end{array}\end{eqnarray}$$

where D^(l) is the well known Wigner matrix defined, in the usual Dirac notation for the canonical basis of angular momentum, as:

$$\begin{eqnarray}&&D_{m{{m}^{\prime}}}^{(l)}\left[\mathcal{R}(g)\right]:=\langle m,l|\hat{\mathcal{R}}(g)|l,{{m}^{\prime}}\rangle \quad l\in \mathbb{N},\end{eqnarray}$$

we obtain, using the fact that $Y_{l,m}^{\,\ast}={{\left[\frac{2l+1}{4\pi}\right]}^{1/2}}D_{m0}^{(l)}$, and that ${{Y}_{l,m}}(0,\phi )={{\left[\frac{2l+1}{4\pi}\right]}^{1/2}}{{\delta}_{m,0}}$:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {\int}^{}\text{d}{{\mu}_{C}}\left(\psi,\bar{\psi}\right)\bar{\psi}\left(\left\{{{\theta}_{i}},{{\phi}_{i}}\right\}\right)\psi \left(\left\{\theta _{i}^{\prime},\phi _{i}^{\prime}\right\}\right) & ={{{\int}^{}}_{SU(2)}}\text{d}g{\int}_{1/{{ \Lambda }^{2}}}^{\infty}\text{d}\alpha {{\text{e}}^{-\alpha {{m}^{2}}}} \\ {} & \,\,\,\,\times \,\,\underset{\left\{{{l}_{i}}\right\}}{\sum}\,\underset{i=1}{\overset{d}{\prod}}\,{{\text{e}}^{-\alpha {{l}_{i}}\left({{l}_{i}}+1\right)}}\frac{2l+1}{4\pi}D_{00}^{\left({{l}_{i}}\right)}\left[\mathcal{R}(g)\pi {{\left({{\theta}_{i}},{{\phi}_{i}}\right)}^{-1}}\pi \left(\theta _{i}^{\prime},\phi _{i}^{\prime}\right)\right]. \end{array}\end{eqnarray} \tag{ 25 }$$

Note that the integral over the group of a product of such representation matrices defines a resolution of the identity in the space of intertwiners of the group $SO{{(3)}^{\otimes d}}$:

$$\begin{eqnarray}&&{\int}^{}\text{d}g\underset{i}{\prod}\,D_{{{m}_{i}}m_{i}^{\prime}}^{\left({{l}_{i}}\right)}\left[\mathcal{R}(g)\right]\in \text{inv}\left(SO{{(3)}^{\otimes d}}\right).\end{eqnarray}$$

We wish to obtain now the expression of the N-point correlation functions in perturbative expansion. The argument involving (12) is still valid, and not affected by the closure constraint. Using the addition formula:

$$\begin{eqnarray}&&\underset{m=-l}{\overset{+l}{\sum}}\,Y_{l,m}^{\ast}(\theta,\phi ){{Y}_{l,m}}\left({{\theta}^{\prime}},{{\phi}^{\prime}}\right)=\frac{2l+1}{4\pi}{{P}_{l}}\left(\vec{u}\centerdot {{\vec{u}}^{\prime}}\right),\end{eqnarray} \tag{ 26 }$$

where $\vec{u}$ (resp ${{\vec{u}}^{\prime}}$) is the unit vector pointing on the 2-sphere of radius unity in the direction $(\theta,\phi )$ (resp $\left({{\theta}^{\prime}},{{\phi}^{\prime}}\right)$), we deduce, using the explicit expression (25), the expression for the Feynman amplitudes of the constrained theory:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {} & {{\mathcal{A}}_{G}}=\left[\underset{e\in L(G)}{\prod}\,{\int}_{1/{{ \Lambda }^{2}}}^{\infty}\text{d}{{\alpha}_{e}}{{\text{e}}^{-{{\alpha}_{e}}{{m}^{2}}}}{{{\int}^{}}_{{{\left[SU(2)\right]}^{|L(G)|}}}}\underset{e\in L(G)}{\prod}\,\text{d}{{h}_{e}}\right] \\ {} & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\times \,\,\left(\underset{f\in F(G)}{\prod}\,\underset{{{l}_{f}}}{\sum}\,\left(2{{l}_{f}}+1\right)D_{00}^{\left({{l}_{f}}\right)}\left[\mathcal{R}\left({{\overrightarrow{\underset{e\in \partial f}{\prod}\,}}_{{}}}{{h}^{{{\epsilon}_{ef}}}}\right)\right]{{\text{e}}^{-\alpha (\,f){{l}_{f}}\left({{l}_{f}}+1\right)}}\right) \\ {} & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\times \,\,\left(\underset{f\in {{F}_{\text{ext}}}(G)}{\prod}\,\underset{{{l}_{f}}}{\sum}\,{{\text{e}}^{-\alpha (\,f){{l}_{f}}\left({{l}_{f}}+1\right)}}\frac{2{{l}_{f}}+1}{4\pi}D_{00}^{\left({{l}_{f}}\right)}\left(\mathcal{R}\left({{\overline{\underset{e\in \partial f}{\prod}\,}}_{{}}}{{h}^{{{\epsilon}_{ef}}}}\right)\pi {{\left({{\theta}_{s(\,f)}},{{\phi}_{s(\,f)}}\right)}^{-1}}\pi \left({{\theta}_{t(\,f)}},{{\phi}_{t(\,f)}}\right)\right)\right), \end{array}\end{eqnarray} \tag{ 27 }$$

where the notation $|Q|$ means the cardinality of the sets Q, and ${{\epsilon}_{ef}}$ is the incidence matrix, which contains the information on whether a line belongs to the boundary of a face and their relative orientation: ${{\epsilon}_{ef}}=0$ if $e\notin \partial f$, +1 if $e\in \partial f$ and they have the same orientations, −1 otherwise.

We choose an element of the Lie algebra $\mathfrak{s}\mathfrak{u}(2)$, ${{\sigma}_{z}}$ for instance, and note that the set of group elements $g\in SU(2)$, such as $g{{\sigma}_{z}}{{g}^{-1}}={{\sigma}_{z}}$, the stabilizer group of ${{\sigma}_{z}}$, is isomorphic to the group U(1):

$$\begin{eqnarray}&&U{{(1)}_{{{\sigma}_{z}}}}:=\left\{g={{\text{e}}^{\text{i}\theta {{\sigma}_{z}}}}\,\,\forall \theta \in \left[0,2\pi \right)\,\right\}\sim U(1).\end{eqnarray} \tag{ 28 }$$

Now we can simply define a field theory for a new field $\Psi :SU{{(2)}^{\times d}}\to \mathbb{C}$ with the constraint:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \Psi \left({{g}_{1}},...,{{g}_{d}}\right)= \Psi \left({{g}_{1}}{{h}_{1}},...,{{g}_{d}}{{h}_{d}}\right)\quad \forall \left({{h}_{1}},...,{{h}_{d}}\right)\in U(1)_{{{\sigma}_{z}}}^{\times d}. \end{array}\end{eqnarray} \tag{ 29 }$$

For this new field we define the partition function:

$$\begin{eqnarray}&&\mathcal{Z}={\int}^{}\text{d}{{\mu}_{{{{\tilde{C}}}_{ \Lambda }}}}\left(\Psi,\rm{\bar \Psi }\right){{\text{e}}^{-{{S}_{\text{int}}}\left(\Psi,\rm{\bar \Psi }\right)}},\end{eqnarray} \tag{ 30 }$$

where, as in the previous construction, S_int is a sum of tensorial invariants. The only difference between the two formulations is that in the first one, the fields have 2d local coordinates, while the new field has 3d local coordinates, with a constraint which reduces the number of degrees of freedom from 3d to 3d  −  d  =  2d, so that we left in the end with the same degrees of freedom.

The covariance ${{\tilde{C}}_{ \Lambda }}$ for this model is defined as:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {} & {\int}^{}\text{d}{{\mu}_{{{\tilde{{C}}\,}_{ \Lambda }}}} \Psi \left(\left\{{{g}_{i}}\right\}\right)\rm{\bar \Psi }\left(\left\{g_{i}^{\prime}\right\}\right):={{{\int}^{}}_{U(1)_{{{\sigma}_{z}}}^{\times d}}}\underset{i=1}{\overset{d}{\prod}}\,\text{d}{{h}_{i}}{\int}_{1/{{ \Lambda }^{2}}}^{+\infty}\text{d}\alpha {{\text{e}}^{-\alpha {{m}^{2}}}}\underset{i=1}{\overset{d}{\prod}}\,{{K}_{\alpha}}\left({{g}_{i}}{{h}_{i}}{{g}^{\prime}}_{i}^{-1}\right). \end{array}\end{eqnarray} \tag{ 31 }$$

from which we deduce the Feynman expansion of a N-point function S_N, indexed by graphs G:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\mathcal{A}}_{G}} & =\,\left[\underset{e\in L(G)}{\prod}\,{\int}_{1/{{ \Lambda }^{2}}}^{\infty}\text{d}{{\alpha}_{e}}{{\text{e}}^{-{{\alpha}_{e}}{{m}^{2}}}}\underset{i=1}{\overset{d}{\prod}}\,\text{d}{{h}_{{{i}_{e}}}}\right] \\ {} & \,\,\,\,\,\,\,\times \,\,\left(\underset{f\in F(G)}{{{\prod}^{}}}\,{{K}_{{{\alpha}_{(\,f)}}}}\left({{\overrightarrow{\underset{e\in \partial f}{\prod}\,}}_{{}}}h_{i{{(\,f)}_{e}}}^{{{\epsilon}_{ef}}}\right)\right)\times \left(\underset{f\in {{F}_{\text{ext}}}(G)}{\prod}\,{{K}_{{{\alpha}_{(\,f)}}}}\left({{g}_{s(\,f)}}{{\overrightarrow{\underset{e\in \partial f}{\prod}\,}}_{{}}}h_{i{{(\,f)}_{e}}}^{{{\epsilon}_{ef}}}g_{t(\,f)}^{-1}\right)\right), \end{array}\end{eqnarray} \tag{ 32 }$$

where i(f) is the color of the face f and ${{K}_{\alpha}}$ is the solution of the heat equation on SU(2) (given by the same formula (10) with $\Delta$ replaced by the Laplace operator on SU(2)).

We now confirm briefly the equivalence of the two constructions at the dynamical level. This can be seen immediately noting that the spherical harmonics are just the Wigner representation matrices for SU(2) integrated over a one-dimensional subgroup isomorphic to U(1). Indeed, the heat kernel is a class function on SU(2) and, by virtue of the Peter–Weyl theorem, it can be expanded on the (class invariant) basis of characters as:

$$\begin{eqnarray}&&{{K}_{\alpha}}\left({{g}_{1}}g_{2}^{-1}\right):=\underset{j\in \mathbb{N}/2}{\sum}\,(2j+1){{\text{e}}^{-4\alpha j(\,j+1)}}{{\chi}^{j}}\left({{g}_{1}}g_{2}^{-1}\right),\end{eqnarray} \tag{ 33 }$$

where the characters ${{\chi}^{j}}:=\text{T}{{\text{r}}_{j}}{{D}^{(\,j)}}$ of the irreducible representation j, verify:

$$\begin{eqnarray}&&{{ \Delta }_{SU(2)}}{{\chi}^{j}}(g)=-4j(\,j+1){{\chi}^{j}}(g).\end{eqnarray} \tag{ 34 }$$

Now, in the Euler angles parametrization

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\chi}^{j}}\left(g{{\text{e}}^{\text{i}{{\sigma}_{z}}\theta}}\right) & =\underset{m}{\sum}\,D_{mm}^{(\,j)}\left(g{{\text{e}}^{\text{i}{{\sigma}_{z}}\theta}}\right)=\underset{m}{\sum}\,\langle m,j|{{\text{e}}^{\text{i}\gamma {{J}_{z}}}}{{\text{e}}^{\text{i}\beta {{J}_{y}}}}{{\text{e}}^{\text{i}(\alpha +\theta ){{J}_{z}}}}|\,j,m\rangle, \end{array}\end{eqnarray}$$

and:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {\int}_{0}^{2\pi}\frac{\text{d}\theta}{2\pi}{{\chi}^{j}}\left(g{{\text{e}}^{\text{i}{{\sigma}_{z}}\theta}}\right)=\langle 0,j|{{\text{e}}^{\text{i}\beta {{J}_{y}}}}|\,j,0\rangle =D_{00}^{(\,j)}(g). \end{array}\end{eqnarray}$$

Note that because m  =  0, j is necessarily an integer. When applying the previous result to ${\int}^{}\text{d}\theta \chi \left({{g}_{1}}{{\text{e}}^{\text{i}\theta {{\sigma}_{z}}}}g_{2}^{-1}\right)$, we find:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {\int}_{0}^{2\pi} & \frac{\text{d}\theta}{2\pi}{{\chi}^{j}}\left({{g}_{1}}{{\text{e}}^{\text{i}{{\sigma}_{z}}\theta}}g_{2}^{-1}\right)=D_{00}^{(\,j)}\left(g_{2}^{-1}{{g}_{1}}\right)=\underset{m}{\sum}\,D_{0m}^{(\,j)}\left(g_{2}^{-1}\right)D_{m0}^{(\,j)}\left({{g}_{1}}\right)=\underset{m}{\sum}\,D_{m0}^{(\,j)\ast}\left({{g}_{2}}\right)D_{m0}^{(\,j)}\left({{g}_{1}}\right). \end{array}\end{eqnarray}$$

Hence, because of the relation: $Y_{l,m}^{\,\ast}={{\left[\frac{2l+1}{4\pi}\right]}^{1/2}}D_{m0}^{(l)}$, the equivalence between the two representations (up to a change of normalization of α and m: $\alpha \to \alpha /4$, $m\to 2m$) follows easily.

We now move on to the imposition of the gauge invariance (closure) constraint in a covariant way. The aim is to combine the constraint (29) with a global constraint of the form $\psi \left({{g}_{1}},...{{g}_{d}}\right)=\psi \left({{g}_{1}}l,...,{{g}_{d}}l\right)\,\forall l\in SU(2)$. We first define the two transformations

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{{\hat{T}}}_{l}}: \Psi \left({{g}_{1}},...,{{g}_{d}}\right) & \mapsto \Psi \left({{g}_{1}}l,...,{{g}_{d}}l\right) \end{array}\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \hat{t}_{{{h}_{i}}}^{(i)}: \Psi \left({{g}_{1}},...,{{g}_{d}}\right) & \mapsto \Psi \left({{g}_{1}},...,{{g}_{i}}{{h}_{i}},...,{{g}_{d}}\right), \end{array}\end{eqnarray} \tag{ 36 }$$

satisfying:

$$\begin{eqnarray}&&{{\hat{T}}_{l}}\circ \hat{t}_{{{h}_{i}}}^{(i)}=\hat{t}_{{{l}^{-1}}{{h}_{i}}l}^{(i)}\circ {{\hat{T}}_{l}}\end{eqnarray} \tag{ 37 }$$

Hence, by defining

$$\begin{eqnarray}&&{{ \Psi }_{{{\sigma}_{z}}}}\left({{g}_{1}},...,{{g}_{d}}\right):={\int}_{U(1)_{{{\sigma}_{z}}}^{\times d}}\underset{i=1}{\overset{d}{\prod}}\,\text{d}{{h}_{i}}\hat{t}_{{{h}_{i}}}^{(i)}[\Psi ]\left({{g}_{1}},...,{{g}_{d}}\right),\end{eqnarray} \tag{ 38 }$$

we have:

$$\begin{eqnarray}&&{{\hat{T}}_{l}}\left[{{ \Psi }_{{{\sigma}_{z}}}}\right]\left({{g}_{1}},...,{{g}_{d}}\right)={{ \Psi }_{{{l}^{-1}}{{\sigma}_{z}}l}}\left({{g}_{1}}l,...,{{g}_{d}}l\right),\end{eqnarray} \tag{ 39 }$$

with, for any $k\in \mathfrak{s}\mathfrak{u}(2)$:

$$\begin{eqnarray}&&{{ \Psi }_{k}}\left({{g}_{1}},...,{{g}_{d}}\right):={\int}_{U(1)_{k}^{\times d}}\underset{i=1}{\overset{d}{\prod}}\,\text{d}{{h}_{i}}\hat{t}_{{{h}_{i}}}^{(i)}[\Psi ]\left({{g}_{1}},...,{{g}_{d}}\right).\end{eqnarray} \tag{ 40 }$$

As a result, to include the closure constraint, we recast the theory in terms of the fields ${{\phi}_{k}}\left(\left\{{{g}_{i}}\right\}\right),{{\bar{\phi}}_{k}}\left(\left\{{{g}_{i}}\right\}\right):{{\left[SU(2)\right]}^{d}}\times \mathfrak{s}\mathfrak{u}(2)\to \mathbb{C}$, with the gauge invariance condition:

$$\begin{eqnarray}&&{{\phi}_{k}}\left({{g}_{1}},...,{{g}_{d}}\right)={{\phi}_{{{h}^{-1}}kh}}\left({{g}_{1}}h,...,{{g}_{d}}h\right).\end{eqnarray} \tag{ 41 }$$

Note that, because of the invariance of the Haar measure, this definition implies that the field $\psi :={\int}^{}\text{d}k{{\phi}_{k}}$ satisfies the standard closure constraint: ${{\hat{T}}_{h}}[\psi ]\left({{g}_{1}},...,{{g}_{d}}\right)=\psi \left({{g}_{1}},...,{{g}_{d}}\right)\forall h\in SU(2)$. In terms of this new fields ${{\phi}_{k}}$ and ${{\bar{\phi}}_{k}}$, our covariant quantum field theory is defined by the partition function:

$$\begin{eqnarray}&&\mathcal{Z}={\int}^{}\text{d}{{\mu}_{C}}\left(\phi,\bar{\phi}\right){{\text{e}}^{-{{S}_{\text{int}}}\left(\phi,\bar{\phi}\right)}},\end{eqnarray} \tag{ 42 }$$

where, following [7] the interaction is chosen of the form:

$$\begin{eqnarray}&&{{S}_{\text{int}}}\left(\phi,\bar{\phi}\right)=\underset{b}{\sum}\,{{\lambda}_{b}}\text{T}{{\text{r}}_{b}}\left(\hat{P}[\phi ],\hat{P}\left[\bar{\phi}\right]\right),\end{eqnarray} \tag{ 43 }$$

where $\hat{P}$ denotes the projector into the subset of gauge invariant fields:

$$\begin{eqnarray}&&\hat{P}:={{{\int}^{}}_{{{\mathcal{S}}_{2}}}}\text{d}k{\int}_{SU(2)}\text{d}l{\int}_{U(1)_{k}^{\times d}}\underset{i=1}{\overset{d}{\prod}}\,\text{d}{{h}_{i}}\hat{t}_{{{h}_{i}}}^{(i)}\circ {{\hat{T}}_{l}}.\end{eqnarray} \tag{ 44 }$$

This choice allows to choose a Gaussian measure without gauge projection, and we adopt the following definition for $\text{d}{{\mu}_{C}}\left(\phi,\bar{\phi}\right)$:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {\int}^{}\text{d}{{\mu}_{C}} & \left(\phi,\bar{\phi}\right){{\phi}_{k}}\left(\left\{{{g}_{i}}\right\}\right){{{\bar{\phi}}}_{{{k}^{\prime}}}}\left(\left\{g_{i}^{\prime}\right\}\right):\,={{\delta}_{k,{{k}^{\prime}}}}{\int}_{1/{{ \Lambda }^{2}}}^{+\infty}\text{d}\alpha {{\text{e}}^{-\alpha {{m}^{2}}}}\underset{i=1}{\overset{d}{\prod}}\,{{K}_{\alpha}}\left({{g}_{i}}g_{i}^{'-1}\right). \end{array}\end{eqnarray} \tag{ 45 }$$

Our model is then completely defined, and divergence free due to the cut-off over α integration. Note that in the perturbative expansion, due to the Wick contractions, we can think in terms of the effective field $\left(\Psi,\rm{\bar \Psi }\right):=\left(\hat{P}[\phi ],\hat{P}\left[\bar{\phi}\right]\right)$. These effective fields satisfy the closure constraints, and are associated to the effective propagator $\bar{C}$:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {\int}^{}\text{d}{{\mu}_{{\bar{C}}}}\left(\Psi,\rm{\bar \Psi }\right) \Psi \left(\left\{{{g}_{i}}\right\}\right)\rm{\bar \Psi }\left(\left\{g_{i}^{\prime}\right\}\right) & ={{{\int}^{}}_{SU(2)}}\text{d}l{\int}^{}\text{d}k{{{\int}^{}}_{U(1)_{k}^{\times d}}}\underset{i=1}{\overset{d}{\prod}}\,\text{d}{{h}_{i}}{\int}_{1/{{ \Lambda }^{2}}}^{+\infty}\text{d}\alpha {{\text{e}}^{-\alpha {{m}^{2}}}}\underset{i=1}{\overset{d}{\prod}}\,{{K}_{\alpha}}\left({{g}_{i}}l{{h}_{i}}g_{i}^{'-1}\right). \end{array}\end{eqnarray}$$

From the Wick theorem, using this effective propagator, we find the Feynman amplitude ${{\mathcal{A}}_{G}}$, whose boundary variable are projected into the gauge invariant subspace $\ker \left[\hat{P}-\mathbb{I}\right]$:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\mathcal{A}}_{G}}= & \left[\underset{e\in L(G)}{\prod}\,{\int}^{}\text{d}{{\alpha}_{e}}{{\text{e}}^{-{{\alpha}_{e}}{{m}^{2}}}}{\int}^{}\text{d}{{l}_{e}}{\int}^{}\text{d}{{k}_{e}}\underset{i=1}{\overset{d}{\prod}}\,{{D}^{{{k}_{e}}}}{{h}_{i,e}}\right] \\ {} & \times \,\,\left(\underset{f\in F(G)}{{{\prod}^{}}}\,{{K}_{\alpha (\,f)}}\left(\underset{e\in \partial f}{\prod}\,{{\left({{l}_{e}}{{h}_{i(\,f),e}}\right)}^{{{\epsilon}_{ef}}}}\right)\right) \\ {} & \times \,\,\left(\underset{f\in {{F}_{\text{ext}}}(G)}{\prod}\,{{K}_{\alpha (\,f)}}\left(\underset{e\in \partial f}{\prod}\,{{g}_{s(\,f)}}{{\left({{l}_{e}}{{h}_{i(\,f),e}}\right)}^{{{\epsilon}_{ef}}}}g_{t(\,f)}^{-1}\right)\right) \end{array}\end{eqnarray} \tag{ 46 }$$

where s and t map open faces to their boundary variables, ε is the incidence matrix, i(f) is the 'color' of the face f, and

$$\begin{eqnarray}&&{{D}^{k}}{{h}_{i}}:=\text{d}{{h}_{i}}\delta \left(k-{{h}_{i}}k{{\left({{h}_{i}}\right)}^{-1}}\right)\end{eqnarray} \tag{ 47 }$$

which reduces the integration from $SU{{(2)}^{\times d}}$ to $U(1)_{k}^{\times d}$. Because of the integration over k_e, we deduce the following proposition:

Proposition 1. The amplitude ${{\mathcal{A}}_{G}}$ for a connected graph G has a $SU{{(2)}^{\times |V(G)|}}$ gauge symmetry, which allows to fix variables along a spanning tree $\mathcal{T}\subset G$, such as ${{l}_{e}}=\mathbb{I}\,\,\forall e\in L\left(\mathcal{T}\,\,\right)$.

Proof. The expression (46) is invariant under the transformation:

$$\begin{eqnarray}&&{{l}_{e}}\to {{g}_{t(e)}}{{l}_{e}}g_{s(e)}^{-1},\quad \quad {{k}_{e}}\to {{g}_{s(e)}}{{k}_{e}}g_{s(e)}^{-1},\end{eqnarray} \tag{ 48 }$$

where t(e) and s(e) are the target and source vertex of an oriented edge e, and with the additional rule that one of the two group elements is the identity for open lines. Because of this invariance, $|V(G)|$ gauge variables can be freely fixed. Because there are only $|V(G)|\,-\,1$ lines in a spanning tree, the proposition is proved. □

Moreover note that the gauge invariance at each (black or white) node allows to choose all the k_e in the same direction, say Oz, up to a global translation for boundary variables (i.e. the variables attached to the external lines). Then, is this gauge, the integration over k_e drops out of the integral. Following [7], we call time gauge this gauge fixing.

This formulation is more convenient for the study of the renormalizability of the model, and it also lends itself more easily to generalisation to other homogeneous spaces $SO(D)/SO(D-1)\simeq {{\mathcal{S}}_{D-1}}$, making clearer the role of the group manifold dimension parametrized by D.

In the appendix, we give some details on the geometrical interpretation of this construction using group Fourier transform, which also motivates its interest from a quantum gravity perspective.

In this paper we focus on the field theory on $\mathcal{S}_{2}^{d}$, and on its renormalisation. However, most of our construction as well as part of the renormalizability analysis, can easily be extended to the homogeneous space ${{\left[SO(D)/SO(D-1)\right]}^{\times d}}$, using the projector formulation introduced above. In this section, we reframe the essential results obtained in the previous section for the homogeneous space ${{\left[SO(D)/SO(D-1)\right]}^{\times d}}$. The extension is straightforward, therefore we give only the essential steps, without too many details. Note that the motivation to extend the analysis to this case from the quantum gravity perspective, is that this is the basis for model building of d-dimensional euclidean quantum gravity models in the spin foam and discrete gravity context, via a generalised Barrett–Crane construction [24, 25, 27].

Let $\left\{{{\left({{L}_{\mu \nu}}\right)}_{\rho \sigma}}\right\}$, a basis of anti-symmetric $D\times D$ matrices of the Lie Algebra $\mathfrak{s}\mathfrak{o}(D)$, and $k=\left\{{{k}_{\mu}}\right\}$ an unit vector of ${{\mathbb{R}}^{D}}$ (the Greek indices run over 1,...,D and label the Euclidean coordinates on ${{\mathbb{R}}^{D}}$). Any element $g\in SO(D)$ can be written as (we use the Einstein convention for sums over Greek indices):

$$\begin{eqnarray}&&g={{\text{e}}^{{{ \Omega }_{\mu \nu}}{{L}_{\mu \nu}}}},\end{eqnarray} \tag{ 49 }$$

and any element h of the stabilizer group of k, isomorphic to SO(D  −  1) (denoted SO_k(D  −  1)), can be written as:

$$\begin{eqnarray}&&h={{\text{e}}^{{{ \Omega }_{\mu \nu}}\mathcal{P}_{\mu {{\mu}^{\prime}}}^{k}\mathcal{P}_{\nu {{\nu}^{\prime}}}^{k}{{L}_{{{\mu}^{\prime}}{{\nu}^{\prime}}}}}}\in S{{O}_{k}}(D-1)\end{eqnarray} \tag{ 50 }$$

where ${{\mathcal{P}}^{k}}=\mathbb{I}-k\otimes k$, ${{\left({{\mathcal{P}}^{k}}\right)}^{2}}={{\mathcal{P}}^{k}}$ is the projector onto the subspace orthogonal to k. As for the SU(2) case, we define a field theory on $SO{{(D)}^{\times d}}$ as a map $\Psi :SO{{(D)}^{\times d}}\to \mathbb{C}$, and we reduce the manifold to the homogeneous space ${{\left[SO(D)/SO(D-1)\right]}^{\times d}}$ imposing the constraint

$$\begin{eqnarray} &&\Psi \left({{g}_{1}},...,{{g}_{d}}\right)= \Psi \left({{g}_{1}}{{h}_{1}},...,{{g}_{\text{d}}}{{h}_{d}}\right)\quad \forall \left({{h}_{1}},...,{{h}_{d}}\right)\in {{\left[S{{O}_{k}}(D-1)\right]}^{\times d}}.\end{eqnarray} \tag{ 51 }$$

The corresponding quantum theory is defined by the choice of a partition function, or in other worlds, by the choice of an action S_kin and of a (UV regularized) Gaussian measure $\text{d}{{\mu}_{{{{\bar{C}}}_{ \Lambda }}}}$. As before, the action is a sum of tensorial invariants, built again as in correspondence with colored bipartite graphs (bubbles). And the closure constraint have to be implemented in a covariant way. To this end, we define the operators ${{\hat{T}}_{l}}$ and $\hat{t}_{{{h}_{i}}}^{(i)}$

$$\begin{eqnarray}&&{{\hat{T}}_{l}}: \Psi \left({{g}_{1}},...,{{g}_{d}}\right)\to \Psi \left({{g}_{1}}l,...,{{g}_{d}}l\right)\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}&&\hat{t}_{{{h}_{i}}}^{(i)}: \Psi \left({{g}_{1}},...,{{g}_{d}}\right)\to \Psi \left({{g}_{1}},...,{{g}_{d}}\right),\end{eqnarray} \tag{ 53 }$$

satisfying again: ${{\hat{T}}_{l}}\circ \hat{t}_{{{h}_{i}}}^{(i)}=\hat{t}_{{{l}^{-1}}{{h}_{i}}l}^{(i)}\circ {{\hat{T}}_{l}}$, and implying that the field

$$\begin{eqnarray}&&{{ \Psi }_{k}}\left({{g}_{1}},...,{{g}_{d}}\right)={{{\int}^{}}_{S{{O}_{k}}{{(D-1)}^{d}}}}\underset{i=1}{\overset{d}{\prod}}\,\text{d}{{h}_{i}}\hat{t}_{{{h}_{i}}}^{(i)}[\Psi ]\left({{g}_{1}},...,{{g}_{d}}\right),\end{eqnarray} \tag{ 54 }$$

verifies:

$$\begin{eqnarray}&&{{\hat{T}}_{l}}\left[{{ \Psi }_{k}}\right]\left({{g}_{1}},...,{{g}_{d}}\right)={{ \Psi }_{\mathcal{R}_{l}^{-1}[k]}}\left({{g}_{1}},...,{{g}_{d}}\right),\end{eqnarray} \tag{ 55 }$$

where $\mathcal{R}_{l}^{-1}[k]$ is the vector k rotated by $l\in SO(D)$. At this stage, all the definitions following (42) can be applied formally without change. We define the partition function as

$$\begin{eqnarray}&&\mathcal{Z}={\int}^{}\text{d}{{\mu}_{C}}\left(\phi,\bar{\phi}\right){{\text{e}}^{-{{S}_{\text{int}}}\left(\phi,\bar{\phi}\right)}},\end{eqnarray} \tag{ 56 }$$

with the action

$$\begin{eqnarray}&&{{S}_{\text{int}}}\left(\phi,\bar{\phi}\right)=\underset{b}{\sum}\,{{\lambda}_{b}}\text{T}{{\text{r}}_{b}}\left(\hat{P}[\phi ],\hat{P}\left[\bar{\phi}\right]\right),\end{eqnarray} \tag{ 57 }$$

and the gauge invariant propagator for the effective field $\Psi :=\hat{P}[\phi ]$:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {\int}^{}\text{d}{{\mu}_{C}}\left(\Psi,\rm{\bar \Psi }\right) \Psi \left(\left\{{{g}_{i}}\right\}\right)\rm{\bar \Psi }\left(\left\{g_{i}^{\prime}\right\}\right) & ={{{\int}^{}}_{SO(D)}}\text{d}l{\int}^{}\text{d}k{{{\int}^{}}_{S{{O}_{k}}{{(D-1)}^{\times d}}}}\underset{i=1}{\overset{d}{\prod}}\,\text{d}{{h}_{i}}{\int}_{1/{{ \Lambda }^{2}}}^{+\infty}\text{d}\alpha {{\text{e}}^{-\alpha {{m}^{2}}}}\underset{i=1}{\overset{d}{\prod}}\,{{K}_{\alpha}}\left({{g}_{i}}l{{h}_{i}}g_{i}^{'-1}\right). \end{array}\end{eqnarray}$$

The Feynman amplitudes for the corresponding field theory take then the form (46). Note that for the action (57), we have adopted the definition of [7].

We move on to a systematic analysis provided by the multi-scale expansion [19]. It attributes a scale to each line $e\in \mathcal{L}(G)$ of any amplitude of any Feynman graph $G$, and allows to deduce power-counting in a more systematic and rigorous way. Moreover, it renormalizes any graph in a sequence of successive steps, providing a concrete implementation of Wilson's ideas directly at the graphical level. Note that, for the rest of this section, we set $\mathbf{G}=SU(2)$.

For convenience, we choose the UV-regulator $\Lambda$ so that $\Lambda ={{M}^{-2\rho}}$, and the complete effective propagator⁶ ${{C}_{ \Lambda }}\equiv {{C}^{\rho}}$ is sliced according to

$$\begin{eqnarray}&&{{C}_{ \Lambda }}=\underset{i<\rho}{\sum}\,{{C}_{i}},\end{eqnarray} \tag{ 58 }$$

where the cut-off $\Lambda$ is chosen of the form $\Lambda ={{M}^{\rho}}$, M  >  1, and the effective propagator 'in the slice i' C_i is

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{C}_{i}} & ={\int}^{}\text{d}k{\int}_{SU(2)}\text{d}h{\int}_{{{\left[SU(2)\right]}^{d}}}\underset{j=1}{\overset{d}{\prod}}\,\text{d}{{l}_{j}}\delta \left(k-{{l}_{j}}k{{\left({{l}_{j}}\right)}^{-1}}\right){\int}_{{{M}^{-i}}}^{{{M}^{-(i-1)}}}\text{d}\alpha {{\text{e}}^{-\alpha {{m}^{2}}}}\underset{i=1}{\overset{d}{\prod}}\,{{K}_{\alpha}}\left({{g}_{i}}h{{l}_{i}}{{\left(g_{i}^{\prime}\right)}^{-1}}\right). \end{array}\end{eqnarray} \tag{ 59 }$$

Let us start by establishing general power counting via a multi-scale analysis, following the notations and general strategy of [37].

The amplitude of a graph $\mathcal{G}$, $\mathcal{A}\left(\mathcal{G}\right)$, with fixed external momenta, is thus divided into the sum of all the scale attributions $\mu =\left\{{{i}_{e}},e\in \mathcal{L}\left(\mathcal{G}\right)\right\}$, where i_e is the scale of the momentum p of line e:

$$\begin{eqnarray}&&{{\mathcal{A}}_{G}}=\underset{\mu}{\sum}\,{{\mathcal{A}}_{G,\mu}}.\end{eqnarray} \tag{ 60 }$$

We will prove the following key theorem, which gives the power counting of the theory and a divergence criterion for a graph amplitude, and is the first step of the perturbative renormalizability analysis at all orders.

Theorem 1. Let a Feynman graph G with sliced amplitude ${{\mathcal{A}}_{G,\mu}}$ and scale assignment $\mu =\left\{{{i}_{{{l}_{1}}}},...,{{i}_{{{l}_{|L(G)|}}}}\right\}\,{{l}_{i}}\in L(G)$. This amplitude satisfies the uniform bound:

$$\begin{eqnarray}&&|{{\mathcal{A}}_{G,\mu}}|\leqslant {{K}^{|L(G)|}}\underset{i}{\prod}\,\underset{k=1}{\overset{\rho (i)}{\prod}}\,{{M}^{\omega \left(G_{i}^{\rho}\right)}},\end{eqnarray} \tag{ 61 }$$

where $G_{i}^{\rho}$ is the ρth connected component of the sub-graph ${{G}_{i}}\subset G$, which contains only the lines of the graph G with a slice ${{i}_{l}}\geqslant i$, and where the divergence degree $\omega \left(G_{i}^{\rho}\right)$ is given by:

$$\begin{eqnarray}&&\omega \left(G_{i}^{\rho}\right)=-2|L\left(G_{i}^{\rho}\right)|\,+\,2\left(|F\left(G_{i}^{\rho}\right)|\,-\,R\left(G_{i}^{\rho}\right)\right).\end{eqnarray} \tag{ 62 }$$

Proof. The first step is to bound the heat kernel. The heat kernel ${{K}_{\alpha}}(g)$ on SU(2) has a complicated expression (see for example [33]). However, it can be approximated in the UV regime, i.e. for large representation labels, by the following uniform bound:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{C}_{i}}\left(\left\{{{g}_{j}}\right\},\left\{g_{j}^{\prime}\right\}\right)\leqslant K{{M}^{(3d-2)i}}{\int}^{}\text{d}k{{{\int}^{}}_{SU(2)}}\text{d}h{{{\int}^{}}_{{{\left[U{{(1)}_{k}}\right]}^{d}}}}\underset{j=1}{\overset{d}{\prod}}\,\text{d}{{l}_{j}}{{\text{e}}^{-\delta {{M}^{i}}\underset{{}}{\overset{{}}{{{\sum}_{j=1}^{d}}}}\,|{{g}_{j}}h{{l}_{j}}g_{j}^{\prime \,-1}|}}. \end{array}\end{eqnarray} \tag{ 63 }$$

where here $|{{g}_{1}}g_{2}^{-1}|$ indicates the geodesic distance (using the standard metric on $SU(2)\simeq {{\mathcal{S}}_{3}}$) between the two group elements g₁ and g₂, and δ, K are two positive constants which can be precisely computed (the values of these constants do not affect the proof).

This result allows us to bound the (multi-)scale decomposition ${{\mathcal{A}}_{G,\mu}}$ of the amplitude. The first step is to rewrite in a suitable manner the term ${\prod}_{l\in L(G)}{{M}^{(3d-2){{i}_{l}}}}$. To this end, note that, trivially: ${{M}^{i}}={\prod}_{i}M$. This allows to rewrite the product over the lines of the graph so that ${\prod}_{l\in L(G)}{{M}^{(3d-2){{i}_{l}}}}={\prod}_{l\in L(G)}{\prod}_{i=1}^{{{i}_{l}}}{{M}^{(3d-2)}}$. Now, we wish to invert the order of the double product. Selecting a scale-assignment i, and a subset of lines in G so that, for each of these lines, the scale assignment is higher than or equal to i, we define the subgraph G_i of G. It follows that

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \underset{l\in L(G)}{\prod}\,{{M}^{(3d-2){{i}_{l}}}}=\underset{l\in L(G)}{\prod}\,\underset{i=1}{\overset{{{i}_{l}}}{\prod}}\,{{M}^{(3d-2)}}=\underset{i}{\prod}\,\underset{l\in L\left({{G}_{i}}\right)}{\prod}\,{{M}^{(3d-2)}}. \end{array}\end{eqnarray}$$

Because the graph G_i is not necessarily connected, we introduce the notation $G_{i}^{\rho}$ for its connected components, so that ${{G}_{i}}=\underset{\rho =1}{\overset{\rho (i)}{\cup}}\,G_{i}^{\rho}$. It follows that the previous decomposition becomes

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \underset{l\in L(G)}{\prod}\,{{M}^{(3d-2){{i}_{l}}}}=\underset{i}{\prod}\,\underset{l\in L\left({\cup}_{\rho =1}^{k(i)}G_{i}^{\rho}\right)}{\prod}\,{{M}^{(3d-2)}}=\underset{i}{\prod}\,\underset{\rho =1}{\overset{\rho (i)}{\prod}}\,\underset{l\in L\left(G_{i}^{\rho}\right)}{\prod}\,{{M}^{(3d-2)}}=\underset{i}{\prod}\,\underset{\rho =1}{\overset{\rho (i)}{\prod}}\,{{M}^{(3d-2)L\left(G_{i}^{\rho}\right)}}. \end{array}\end{eqnarray}$$

The second step is to bound the contributions of the internal faces. Using the same trick to reorganize the products and the compactness of the group U(1), we obtain the following contribution for the internal faces:

$$\begin{eqnarray}&&\underset{i}{\prod}\,\underset{\rho =1}{\overset{\rho (i)}{\prod}}\,{{M}^{-3d|L\left(G_{i}^{\rho}\right)|+3|F\left(G_{i}^{\rho}\right)|}}.\end{eqnarray} \tag{ 64 }$$

Note that, to obtain this formula, we have chosen an optimal tree in each face on which we perform the integrations over the angle variables. This result, combined with the first one provided by the factors M^(3d−2)i gives

$$\begin{eqnarray}&&\underset{i}{\prod}\,\underset{\rho =1}{\overset{\rho (i)}{\prod}}\,{{M}^{-2|L\left(G_{i}^{\rho}\right)|+3|F\left(G_{i}^{\rho}\right)|}}.\end{eqnarray} \tag{ 65 }$$

The third and last contribution comes from the remaining integrals

$$\begin{eqnarray}&&{\int}^{}\underset{e}{\prod}\,\text{d}{{h}_{e}}\text{d}{{k}_{e}}\underset{f}{\prod}\,\text{d}l_{{{k}_{e}}}^{c(\,f)}{{\text{e}}^{-\delta {{M}^{i(\,f)}}|\underset{e\in \partial f}{\prod}\,{{\left({{h}_{e}}l_{{{k}_{e}}}^{c(\,f)}\right)}^{{{\epsilon}_{ef}}}}|}},\end{eqnarray} \tag{ 66 }$$

where $i(\,f):=\inf \left\{{{i}_{l}},\,l\in \partial f\right\}$ and c(f) is the color of the face f. It is at this point that the Abelian approximation intervenes. As shown in [19] for a non-Abelian TGFT on SU(2), the exact power counting is uniformly bounded by its Abelian version, which corresponds to the linearized version of the exact one around identity for all the group elements (i.e. the non-commutativity of the group variables improves the convergence of a graph amplitude compared to the Abelian version, see section 4.3).

The Abelian version (66) is

$$\begin{eqnarray}&&{\int}^{}\underset{e\in L(G)}{\prod}\,\text{d}{{\vec{\lambda}}_{e}}\text{d}{{k}_{e}}{\int}^{}\underset{f}{\prod}\,\text{d}\theta _{{{k}_{e}}}^{c(\,f)}{{\text{e}}^{-\delta {{M}^{i(\,f)}}|\underset{e}{\sum}\,{{\epsilon}_{ef}}\left({{{\vec{\lambda}}}_{e}}+\theta _{{{k}_{e}}}^{c(\,f)}{{{\vec{e}}}_{{{k}_{e}}}}\right)|}}.\end{eqnarray} \tag{ 67 }$$

Where ${{\vec{e}}_{k}}$ is the unit 3d vector associated to the unit Lie algebra element k and $|\vec{q}|\,:=\sqrt{{\sum}_{j=1}^{d}q_{j}^{2}}$ is the ${{\mathbb{R}}^{3}}$ norm (we use here explicitly the trivial isomorphism between the elements of the Lie algebra $\mathfrak{s}\mathfrak{u}(2)$, and the vectors of ${{\mathbb{R}}^{3}}$).

Integrating over a selected tree ${{\mathcal{T}}_{2}}$ of faces, such that the number of faces in this set equals the rank of the incidence matrix, and in an optimal way, in the sense that the faces of this set proceed recursively from the leaves to the root of the Gallavotti–Nicoló tree, the integral (67) over the $\vec{\lambda}$ and θ variables gives the power counting contribution

$$\begin{eqnarray}&&\underset{i}{\prod}\,\underset{\rho =1}{\overset{\rho (i)}{\prod}}\,{{M}^{-3R\left(G_{i}^{\rho}\right)}},\end{eqnarray} \tag{ 68 }$$

and the remaining integration

$$\begin{eqnarray}\begin{array}{*{35}{l}} {} & {\int}^{}\underset{e\in L(G)}{\prod}\,\text{d}{{k}_{e}}{\int}^{}\underset{f\in F/{{\mathcal{T}}_{2}}}{\prod}\,\text{d}\theta _{{{k}_{e}}}^{c(\,f)}{{\text{e}}^{-\delta {{M}^{i(\,f)}}|\underset{e\in \partial f}{\sum}\,{{\epsilon}_{ef}}\theta _{{{k}_{e}}}^{c(\,f)}|}}={\int}^{}\underset{f\in F/{{\mathcal{T}}_{2}}}{\prod}\,{\int}^{}\underset{e\in \partial f}{\prod}\,\text{d}\theta _{{{k}_{e}}}^{c(\,f)}\text{d}{{k}_{e}}{{\text{e}}^{-\delta {{M}^{i(\,f)}}|\underset{e\in \partial f}{\sum}\,{{\epsilon}_{ef}}\theta _{{{k}_{e}}}^{c(\,f)}|}}, \end{array}\end{eqnarray} \tag{ 69 }$$

gives, up to a positive constant,

$$\begin{eqnarray}\begin{array}{*{35}{l}} \underset{f\in F/{{\mathcal{T}}_{2}}}{\prod}\,{{M}^{-i(\,f)}} & =\underset{i}{\prod}\,\underset{\rho}{\prod}\,\underset{f\in F/{{\mathcal{T}}_{2}}\left(G_{i}^{\rho}\right)}{\prod}\,{{M}^{-1}}=\underset{i,\rho}{\prod}\,{{M}^{-|F\left(G_{i}^{\rho}\right)|+R\left(G_{i}^{\rho}\right)}}, \end{array}\end{eqnarray} \tag{ 70 }$$

from which we deduce the bound on the amplitude ${{A}_{G,\mu}}$:

$$\begin{eqnarray}\begin{array}{*{35}{l}} |{{\mathcal{A}}_{G,\mu}}| & \leqslant {{K}^{|L(G)|}}\underset{i}{\prod}\,\underset{\rho =1}{\overset{\rho (i)}{\prod}}\,{{M}^{-2|L\left(G_{i}^{\rho}\right)|+2|F\left(G_{i}^{\rho}\right)|-2R\left(G_{i}^{\rho}\right)}}={{K}^{|L(G)|}}\underset{i}{\prod}\,\underset{\rho =1}{\overset{\rho (i)}{\prod}}\,{{M}^{\omega \left(G_{i}^{\rho}\right)}}. \end{array}\end{eqnarray} \tag{ 71 }$$

□

Note that for the same model without simplicity constraint, the power counting involves a 3 and not a 2 in front of the contribution F  −  R (see [19]). It seems that it is just the dimension of the manifolds, 3 for SU(2) and 2 for SU(2)/U(1) which differs at this point. The next section provides a confirmation of this intuition.

It is not hard to extend the previous analysis to the homogeneous space ${{\left[SO(D)/SO(D-1)\right]}^{d}}\simeq {{\mathcal{S}}_{D-1}}$, allowing to obtain a preliminary classification of potentially just-renormalizable models, for various choices of D and d. Of course, such classification is valid only to the extent in which the Abelian power counting captures in fact the exact power counting of these non-Abelian models. This is however not straightforward, and we have actually reasons not to believe it, as we are going to discuss in the following.

We obtain the following result:

Theorem 2. The Abelian superficial divergence degree of any Feynman graph G associated to a field theory on ${{\left[SO(D)/SO(D-1)\right]}^{\times \,d}}$ with closure constraint is given by

$$\begin{eqnarray}&&\omega (G)=-2|L(G)|\,+\,(D-1)\left[|F(G)|\,-\,R(G)\right].\end{eqnarray} \tag{ 72 }$$

The proof is the exact generalization of the previous one, and we will only give the main steps. The previous bound (63) for the propagator becomes, on SO(D):

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{C}_{i}}\left(\left\{{{g}_{j}}\right\},\left\{g_{j}^{\prime}\right\}\right)\leqslant K{{M}^{\left(d\frac{D(D-1)}{2}-2\right)i}}{\int}^{}\text{d}k{\int}_{SU(2)}\text{d}h{{{\int}^{}}_{{{\left[S{{O}_{k}}(D-1)\right]}^{d}}}}\underset{j=1}{\overset{d}{\prod}}\,\text{d}{{l}_{j}}{{\text{e}}^{-\delta {{M}^{i}}\underset{j=1}{\overset{d}{\sum}}\,|{{g}_{j}}h{{l}_{j}}g_{j}^{\prime \,-1}|}}. \end{array}\end{eqnarray} \tag{ 73 }$$

After integration over group variables g_i, the product (65) becomes

$$\begin{eqnarray}&&\underset{i}{\prod}\,\underset{\rho =1}{\overset{\rho (i)}{\prod}}\,{{M}^{-2|L\left(G_{i}^{\rho}\right)|+\frac{D(D-1)}{2}|F\left(G_{i}^{\rho}\right)|}}\end{eqnarray} \tag{ 74 }$$

and the remaining integration (67), in the Abelian approximation, which corresponds to the linearized version of (66), becomes

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {\int}^{}\underset{e\in L(G)}{\prod}\,\text{d}{{\lambda}_{e}}\text{d}{{k}_{e}}\times {\int}^{}\underset{f}{\prod}\,\text{d}\theta _{{{k}_{e}}}^{c(\,f)}{{\text{e}}^{-\delta {{M}^{i(\,f)}}|\underset{e}{\sum}\,{{\epsilon}_{ef}}\left({{\lambda}_{e,\mu \nu}}+\theta _{{{k}_{e}},{{\mu}^{\prime}}{{\nu}^{\prime}}}^{c(\,f)}\mathcal{P}_{{{\mu}^{\prime}}\mu}^{{{k}_{e}}}\mathcal{P}_{{{\nu}^{\prime}}\nu}^{{{k}_{e}}}\right){{L}_{\mu \nu}}|}}. \end{array}\end{eqnarray} \tag{ 75 }$$

The integration over the ${{\lambda}_{e}}$ variables replaces (68) by:

$$\begin{eqnarray}&&\underset{i}{\prod}\,\underset{\rho =1}{\overset{\rho (i)}{\prod}}\,{{M}^{-\frac{D(D-1)}{2}R\left(G_{i}^{\rho}\right)}}\end{eqnarray} \tag{ 76 }$$

and the remaining integration over the $\theta _{{{k}_{e}}}^{c(\,f)}$ gives, instead of (70),

$$\begin{eqnarray}&&\underset{i,\rho}{\prod}\,{{M}^{-\frac{(D-1)(D-2)}{2}\left[|F\left(G_{i}^{\rho}\right)|-R\left(G_{i}^{\rho}\right)\right]}}\quad.\end{eqnarray} \tag{ 77 }$$

Combining the results (74), (75) and (77), we obtain the divergence degree (72).

In this section we want to examine further the validity of the Abelian power counting for our non-Abelian model. To this end, we will study the behaviour of the integral (66)⁷. In order to simplify the reasoning, we choose the orientations of faces and lines such that ${{\epsilon}_{ef}}\geqslant 0$. A moment of reflection shows that it is always possible to do so: one just has to exploit the bipartite structure of the Feynman graphs, and choose the orientation of lines from black to white vertices, for instance. Hence, we will study the behaviour in $\Lambda$ of the simpler integral:

$$\begin{eqnarray}&&{{\mathcal{I}}_{ \Lambda }}={\int}^{}\underset{e}{\prod}\,\text{d}{{h}_{e}}\text{d}{{k}_{e}}\underset{f}{\prod}\,\text{d}l_{{{k}_{e}}}^{c(\,f)}{{\text{e}}^{-{{ \Lambda }^{2}}|\underset{{}}{{{{{\prod}^{}}_{e\in \partial f}}}}\,{{h}_{e}}l_{{{k}_{e}}}^{c(\,f)}{{|}^{2}}}},\end{eqnarray} \tag{ 78 }$$

in the large $\Lambda$ limit. Because of the normalization of each integration measure, $\mathcal{I}$ goes to zero when $\Lambda \to \infty$, and we expect a behavior of the type ${{ \Lambda }^{- \Omega (G)}}$. The aim is therefore to find $\Omega$, or, at least, an optimal bound for it. In addition, note that the integral is absolutely convergent, and trivially bounded by a constant.

The large $\Lambda$ limit enforces the relations:

$$\begin{eqnarray}&&\underset{e\in \partial f}{\prod}\,{{h}_{e}}l_{{{k}_{e}}}^{c(\,f)}=\mathbb{I},\end{eqnarray} \tag{ 79 }$$

and the strategy is to expand the exponent in the vicinity of these solutions, and integrating around them, by the Laplace method. Let $x=\left\{{{\bar{h}}_{e}},\bar{l}_{\text{e}}^{c(\,f)}\right\}$ a point in the space of solutions of (79), expected to be a manifold with a priori many connected parts, eventually of null dimension (a single point). We define $\mathcal{A}=\left\{{{h}_{e}}\right\}$ the set of group variables attached to each line and H_f the map from $SO{{(D)}^{\times |L|}}$ to $SO{{(D)}^{\times |F|}}$ defined by:

$$\begin{eqnarray}&&{{H}_{f}}\left(\mathcal{A}\right)=\underset{e\in \partial f}{\prod}\,{{h}_{e}}l_{{{k}_{e}}}^{c(\,f)}\end{eqnarray} \tag{ 80 }$$

whose differential around x is:

$$\begin{eqnarray}&&\text{d}{{H}_{f}}(x)=\underset{e\in \partial f}{\sum}\,\text{A}{{\text{d}}_{\left\{\underset{{}}{{{{{\prod}^{}}_{{{\text{e}}^{\prime}}\in \partial f|{{\text{e}}^{\prime}}<e}}_{{}}}}\,{{{\bar{h}}}_{{{\text{e}}^{\prime}}}}\bar{l}_{{{\text{e}}^{\prime}}}^{c(\,f)}\right\}}}\left[{{\hat{\delta}}_{e}}+{{\bar{h}}_{e}}\hat{\delta}_{e}^{f}\bar{h}_{e}^{-1}\right]=:\underset{e\in \partial f}{\sum}\,{{L}_{fe}}\left[{{\hat{\delta}}_{e}}+{{\bar{h}}_{e}}\hat{\delta}_{e}^{f}\bar{h}_{e}^{-1}\right],\end{eqnarray} \tag{ 81 }$$

where ${{\hat{\delta}}_{e}}$ and $\hat{\delta}_{\text{e}}^{f}$, living in the Lie Algebra $\mathfrak{s}\mathfrak{u}(2)$, are the right variations of h_e and $l_{\text{e}}^{c(\,f)}$ respectively. Note that, we fix the gauge by setting ${{h}_{e}}=\mathbb{I}$ along a spanning tree before linearizing, otherwise, the gauge orbit becomes non-compact and leads to spurious divergences. Defining $\hat{\delta}_{\text{e}}^{f\,\sharp}={{\bar{h}}_{e}}\hat{\delta}_{e}^{f}\bar{h}_{e}^{-1}$, we obtain, around x:

$$\begin{eqnarray}&&{{\mathcal{I}}_{ \Lambda }}(x)={\int}^{}\underset{e,f}{\prod}\,\text{d}{{\hat{\delta}}_{e}}\text{d}\hat{\delta}_{\text{e}}^{f}\underset{f}{\prod}\,{{\text{e}}^{-{{ \Lambda }^{2}}|\underset{e\in \partial f}{\sum}\,{{L}_{fe}}\left({{{\hat{\delta}}}_{e}}+\hat{\delta}_{\text{e}}^{f\,\sharp}\right){{|}^{2}}}},\end{eqnarray} \tag{ 82 }$$

where only the lines in the complementary of the gauge tree are selected. In order to integrate it, we introduce the quantities ${{\hat{\delta}}_{f}}$ and $\hat{\delta}_{f}^{\sharp}$ as:

$$\begin{eqnarray}&&{{\hat{\delta}}_{f}}=\underset{e\in \partial f}{\sum}\,{{L}_{fe}}{{\hat{\delta}}_{e}}\quad \quad \quad \quad \hat{\delta}_{f}^{\sharp}=\underset{e\in \partial f}{\sum}\,{{L}_{fe}}\delta _{e}^{f\,\sharp},\end{eqnarray} \tag{ 83 }$$

and the notations ${{\hat{\delta}}_{f\,\parallel}}$ and ${{\hat{\delta}}_{f\,\bot}}$, designating respectively the components parallel and orthogonal to $\hat{\delta}_{f}^{\sharp}$. Inserting this in (82), we find

$$\begin{eqnarray}&&{{\mathcal{I}}_{ \Lambda }}(x)={\int}^{}\underset{e,f}{\prod}\,\text{d}{{\hat{\delta}}_{e}}{{\text{e}}^{-{{ \Lambda }^{2}}|{{{\hat{\delta}}}_{f\,\bot}}{{|}^{2}}}}{\int}^{}\text{d}\hat{\delta}_{\text{e}}^{f}\underset{f}{\prod}\,{{\text{e}}^{-{{ \Lambda }^{2}}|{{{\hat{\delta}}}_{f\,\parallel}}+\hat{\delta}_{f}^{\sharp}{{|}^{2}}}},\end{eqnarray} \tag{ 84 }$$

which behaves as

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\mathcal{I}}_{ \Lambda }}(x) & \lesssim \,\,|\det {{[L]}_{\ker {{(L)}_{\bot}}}}{{|}^{-1}}{{ \Lambda }^{-\left\{\dim\left[SO(D)\right]-\dim\left[SO(D-1)\right]\right\}\text{rk}[L]}}{{ \Lambda }^{-\dim\left[SO(D-1)\right]|F|}} \\ {} & =\,\,|\det {{[L]}_{\ker {{(L)}_{\bot}}}}{{|}^{-1}}{{ \Lambda }^{-(D-1)\text{rk}[L]}}{{ \Lambda }^{-\frac{(D-1)(D-2)}{2}|F|}}, \end{array}\end{eqnarray} \tag{ 85 }$$

where $\text{rk}[L]$ is the rank of L, and the notation $\det {{[L]}_{\ker {{(L)}_{\bot}}}}$ indicates the determinant over the complementary space $\ker {{(L)}_{\bot}}$ of $\ker (L)$. Because the rank $\text{rk}[L]$ is at least equal to the rank of ${{\epsilon}_{ef}}$, the previous bound in always bounded by its Abelian version. The sum over x, however can eventually spoil this result. As explained before, the support of this sum splits into continuous and discrete components, and the integral over the continuous component can be ill-defined. However, these singularities occur when the determinant vanishes, and because the integral is absolutely convergent, it is a snag of the Laplace method. Moreover, for these points, the co-dimension of the kernel of L becomes bigger than the co-dimension of the kernel on the other points. Hence, presumably these singularities do not affect the conclusion.

This result is important for the rest of this paper, because it allows to find some just-renormalizable models only from the Abelian divergent degree. However, it seems to indicate that the abelian power counting is a pessimistic one, and as a result, that the list of just-renormalizable models obtained using the Abelian divergent degree is certainly incomplete. In addition, the flatness condition (79) is different of the one obtained in standard TGFT, which is ${\prod}_{e\in \partial f}{{h}_{e}}=\mathbb{I}$. We will return on this subtlety in the following.

This section give some definitions and properties of colored graphs. Most of these properties are well-known in tensor model literature, so we simply adopt them and refer to, say, [10] for their proof. From [19], we adopt the following definitions.

Definition 1 (contraction operation). Let G be a Feynman graph and ${{L}_{0}}=\left\{{{l}_{i}}\right\}\subset L(G)$ an ordered subset of dotted (i.e. propagation) lines in G (including tadpole lines). The graph G/L₀ is obtain from G by the following steps:

considering the dotted line ${{l}_{i}}\in {{L}_{0}}$:

• deleting the line l_i and its two (black and white) end vertices and all the colored lines joining these two vertice;
• identifying the colored line linked to the deleted black vertex with the corresponding line linked to the white vertex;
• repeating the same steps for l_i+1, and so on.

Definition 2. For a connected graph G with $|L|$ lines, $|V|$ vertices and a spanning tree $\mathcal{T}\subset G$, we call tensorial rosette or simply rosette, the contracted graph $G/\mathcal{T}$ with one vertex and $|L|\,-\,|V|\,+\,1$ lines.

The figure 3 below illustrates the definition 1 in a simple example.

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From this definition, we have the following lemma:

Lemma 1. Let a connected graph G, with $|F|$ and R respectively its number of faces and the rank of the incidence matrix ${{\epsilon}_{ef}}$. Under contraction of a spanning tree $\mathcal{T}$, $|F|$ and R do not change.

Proof. Because $\mathcal{T}$ is a spanning tree, its lines bound faces with a number of boundary lines bigger or equal to two, so their number does not change under contraction. In the same way, because of the gauge invariance of the Feynman amplitude, allowing to fix the gauge such that ${{h}_{e}}=\mathbb{I}$ along an arbitrary spanning tree, we deduce that the rank does not change. □

Definition 3. Consider a Feynman graph G. The colored extension G_c of this graph is the bipartite regular graph for which:

• the vertices are partitioned in the form $\mathcal{V}\left({{G}_{c}}\right)=V{\cup}^{}\bar{V}$, where V (respectively $\bar{V}$) is the set of black (respectively white) vertices;
• the set of lines $\mathcal{E}\left({{G}_{c}}\right)$ is formed by all the lines (colored plus dotted) joining any pair $\left\{v,\bar{v}\right\}\in V\times \bar{V}$; by definition, the dotted lines have color 0;
• the set of faces is of the form $\mathcal{F}\left({{G}_{c}}\right)=F{\cup}^{}F_{c}^{\ne 0}$, where F is the set of faces in G, i.e. the set of faces of the form f_0i with boundary lines of color 0 and i ($i\ne 0$), and $F_{c}^{\ne 0}$ is the set of faces of the form f_ij with boundary lines of color i and j($i\ne j;i,j\ne 0$);

Definition 4. Consider a colored extension G_c. A k-dipole d_k is a set of k colored lines necessarily including the color 0 and linking two vertices v and $\bar{v}$. An example is depicted on the figure 4 below.

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In addition, we recall the following three definitions about colored graphs:

Definition 5 (jacket). Consider a colored extension G_c in dimension d. A jacket $\mathcal{J}$ is a 2-subcomplex of G_c, labeled by a (d  +  1)-cycle τ, such that $\mathcal{J}$ has the same number of lines and vertices as G_c, but only a subset of its faces: ${{\mathcal{F}}_{\mathcal{J}}}=\left\{\,f\in {{\mathcal{F}}_{{{G}_{c}}}}|\,f=\left({{\tau}^{q}}(0),{{\tau}^{q+1}}(0)\right),q\in {{\mathbb{Z}}_{D+1}}\right\}$.

A jacket is a ribbon graph, corresponding to a sub-manifold of dimension 2 and of Euler–Poincaré characteristic given by $\chi \left(\mathcal{J}\,\right)=\,|{{\mathcal{F}}_{\mathcal{J}}}|\,-\,|{{\mathcal{E}}_{\mathcal{J}}}|\,+\,|{{\mathcal{V}}_{\mathcal{J}}}|=2\,-\,2{{g}_{\mathcal{J}}}$, where ${{g}_{\mathcal{J}}}$ is the genus of the surface.

Definition 6 (degree). The degree $\varpi \left({{G}_{c}}\right)$ of a colored extension G_c is the sum over all the genus of its jackets:

$$\begin{eqnarray}&&\varpi \left({{G}_{c}}\right)=\underset{\mathcal{J}}{\sum}\,{{g}_{\mathcal{J}}}\quad \Rightarrow \quad \varpi \left({{G}_{c}}\right)\geqslant 0\end{eqnarray}$$

Definition 7. The graphs whose degree is equal to zero are called melonic graphs.

In addition to these definitions, we have the three following lemmas:

Lemma 2. The melonic graphs are dual to a d-dimensional sphere.

Lemma 3. In dimension d, the degree $\varpi \left({{G}_{c}}\right)$ is related to the number of bi-colored faces and to the number of black (or white) vertices p, by the following two relations:

$$\begin{eqnarray}&&|\mathcal{F}|\,=\frac{\text{d}(d-1)}{2}p+d-\frac{2}{(d-1)!}\varpi \left({{G}_{c}}\right)\end{eqnarray}$$

$$\begin{eqnarray}&&\varpi \left({{G}_{c}}\right)=\frac{(d-1)!}{2}\left(\,p+d-{{\mathcal{B}}^{[d]}}\right)+\underset{i;\rho}{\sum}\,\varpi \left(\mathcal{B}_{(\rho )}^{{\hat{i}}}\right).\end{eqnarray}$$

In addition, we can show that $p+d-{{\mathcal{B}}^{[d]}}\geqslant 0$.

Note that, in this lemma, the sum over i in the second relation includes the color 0. In addition, $\mathcal{B}_{(\rho )}^{{\hat{i}}}$ is the connected component ρ of the sub-graph obtained from G_c by deleting all the lines with color i (including the color 0). ${{\mathcal{B}}^{[d]}}$ is the number of these connected sub-graphs. These sub-graphs are the so-called d-bubbles. From this lemma, we easily deduce the following proposition:

Proposition 2. Under any 1-dipole contraction, the degree of a graph is unchanged.

With this material at hand, we now move on to the renormalizability analysis, which is the object of the next section.

We have seen that, for these models, the divergence degree of a graph grows with the number of faces. The first question is: which are the graphs that have a maximum number of faces? To answer this question, consider a vacuum graph G and its colored extension G_c. We can choose a tree $\mathcal{T}$ in G and build the rosette $G/\mathcal{T}=\hat{G}$, ${{\hat{G}}_{c}}$ being its colored extension. Then, from the lemma 3, we have:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} |F\left(\hat{G}\right)|=(d-1)|L\left(\hat{G}\right)|\,+\,1- \Delta \left(\hat{G}\right), \end{array}\end{eqnarray} \tag{ 86 }$$

where we have used the fact that $L\left(\hat{G}\right)=p$ in the lemma 3, and where

$$\begin{eqnarray} &&\Delta \left(\hat{G}\right):=\frac{2}{(d-2)!}\left[\frac{1}{d-1}\varpi \left(\hat{G}\right)-\varpi \left({{\hat{G}}^{0}}\right)\right],\end{eqnarray}$$

where ${{\hat{G}}^{0}}$ is the d-bubble of color 0 obtained from $\hat{G}$ by deleting all the lines of color 0. Note that because the rosette $\hat{G}$ have only one vertex, ${{\hat{G}}^{0}}$ have only one connected component. Because one can prove that $\varpi \left(\hat{G}\right)\geqslant d\varpi \left({{\hat{G}}^{0}}\right)$, we easily deduce that $|F|$ is bounded by:

$$\begin{eqnarray}&&(d-1)p+1-\frac{2}{(d-1)!}\varpi \left(\hat{G}\right)\leqslant |F|\leqslant (d-1)p+1-\frac{2}{(d-1)!}\varpi \left({{\hat{G}}^{0}}\right).\end{eqnarray}$$

The number of faces is then maximal when $\varpi \left(\hat{G}\right)=0$, implying $\varpi \left({{\hat{G}}^{0}}\right)=0$ from the lemma 3. Hence we deduce that the number of faces is maximal for the melonic (colored extension) graphs. This result is actually a key one for all the TGFT models that have been studied to date. In addition to the formal definition 7, the melonic graphs have an iterative definition. From the simplest melon with p  =  1, the so-called supermelon, pictured in figure 5 below, we obtain the refined melons of order p by replacing an edge by a d-dipole as in the figure 6 below.

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Note that for a rosette, this recursion procedure excludes the line of color 0 because the d-bubble ${{\hat{G}}^{0}}$ has just one connected component.

Now we can turn to the analysis of the rank, the other main contribution to the divergence degree. It is obvious from the previous recursion that for a melonic rosette of order p, the rank is just equal to p, the number of lines $L\left(\hat{G}\right)$ in $\hat{G}$. Hence, the rank is maximal for the same melonic graphs, and each insertion of a d-dipole increases $|F|-R$ by d  −  2. It follows that, for a melonic rosette graph:

$$\begin{eqnarray}&&|F\left(\hat{G}\right)|-\,R\left(\hat{G}\right)=(d-2)|L\left(\hat{G}\right)|\,+\,1.\end{eqnarray} \tag{ 87 }$$

It is also the optimal bound for $|F|\,-\,R$ for arbitrary graphs. A statement that we can prove easily by recursion. Starting from the order p  =  1, the unique connected vacuum graph ${{\mathcal{M}}_{1}}$ is the supermelon in figure 5. For the next order p  =  2, we wish to add one black vertex and one white vertex, or, in other words, a new dotted line. Each line carries at least d faces ${{f}^{0i}}\,i\ne 0$ of length one, and can increase the rank at least of  +1. Because of the connectivity constraint, it seems that one colored line must be sacrificed, and bound a common face for the two dotted lines. Hence, the maximal number of faces is 2d  −  1  =  2(d  −  1)  +  1, in accordance with the formula (87). Concerning the rank, if we wish to minimize this variation in the step $p=1\to p=2$, the only possibility is to exclude the creation of a k-dipole for k  >  1, and so to create a new face f  ⁰ⁱ. Hence, we lose d  −  1 faces, and, if d  >  2, this possibility does not correspond to the leading order. Privileging the graphs with the maximum number of faces is then more advantageous, and the connectivity constraint implies that the only possibility is a melonic graph ${{\mathcal{M}}_{2}}$ as depicted in the figure 7.

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The same argument survives at order p. Starting with a melonic graph of order p, ${{\mathcal{M}}_{p}}$, we move from the order p to the order p  +  1 by adding a dotted line. This dotted line can carry at least d faces, but one is necessarily common with another dotted line, ensuring the graph connectivity. Hence, adding a new line increases at least by d  −  1 the number of faces, and the optimal graph corresponds to the melon ${{\mathcal{M}}_{p+1}}$. As to the rank, it can at least increase by 1. It is clear that the optimal graphs for the rank and the faces are incompatible, because to minimize the rank variation, so $\Delta R=0$, it is necessary that no k-dipole (for k  >  1) and no face is created by the new line. Then, we lose d  −  1 lines compared to the melonic graphs, and this solution does not correspond to a leading order graph.

Hence, the melonic rosettes correspond to the most divergent graphs, i.e. the leading order of the perturbative expansion. Because $L\left(\hat{G}\right)=\,|L(G)|\,-\,|V(G)|\,+\,1$, we deduce the following result:

Proposition 3. Let G be a vacuum Feynman graph. Its divergence degree is bounded by

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \omega (G)\leqslant {{\omega}_{\text{melo}}}(G) \end{array}\end{eqnarray} \tag{ 88 }$$

with

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\omega}_{\text{melo}}}(G):\,=[& (D-1)(d-2)-2]|L(G)|\,-\,(D-1)(d-2)(|V(G)|\,-\,1)+1, \end{array}\end{eqnarray} \tag{ 89 }$$

which correspond to the divergence degree of a melonic rosette graph.

As a result, it follows that the leading order graphs are also melonic. Indeed, it follows from another elementary result concerning the degree $\varpi \left({{G}_{c}}\right)$, i.e. that it is invariant under 1-dipole contraction [10, 19]. Yet, it is obvious that the tree contraction given by the rosette in the previous proposition is a succession of 1-dipole contractions, then:

Corollary 1. The leading order graphs are melonic: their degree ϖ vanish.

Corollary 2. Only the melonic interactions contribute to the leading order graphs.

It remains to consider the non-vacuum graphs. And we have the following proposition:

Proposition 4. Let G_N be a non-vacuum graph with N external lines. Its divergence degree is bounded as:

$$\begin{eqnarray}\begin{array}{*{35}{l}} \omega ({{G}_{N}})\leqslant [& (D-1)(d-2)-2]|L(G)|\,-\,(D-1)(d-2)(|V(G)|\,-\,1) \end{array}\end{eqnarray} \tag{ 90 }$$

with equality for melonic graphs. In addition, an external mono-color face connects all the external black and white vertices.

Proof. Let $\hat{G}$ be a vacuum graph containing G_N in the following sense. $G\subset \hat{G}$ means that $\hat{G}$ contains all the vertices of G_N and has the same connectivity, and that any face of $\hat{G}$ is either an internal face of G_N, either splitting in some external faces of this graph. In addition, this inclusion supposes that G_N can be obtained from $\hat{G}$ by cutting some internal lines. Because of proposition 3, it follows that $\hat{G}$ is a leading order graph if and only if it is melonic. Starting from this graph, we wish to build the most divergent graph with the same number of external lines as G_N. Suppose that G_N has 2 external lines. From $\hat{G}$, we begin by selecting a spanning tree $\mathcal{T}$ and we construct the rosette $\overline{{\hat{G}}}$. Now, the simplest way to obtain a non-vacuum graph from the rosette is to cut an internal dotted line. An internal line is necessarily a dipole line, and carries d faces. In addition, the rank has to be decreased by 1, and $|F|-\,R$ decreases by d  −  1  =  (d  −  2)  +  1. Another (more complicated) way to build a 2-points graph is to cut n internal dotted lines, to select two half dotted lines, and to reconnect the 2(n  −  1) remaining half dotted lines in an optimal way. But observe that cutting these n lines decreases $|F|-\,R$ at most by (d  −  2)n  +  1. Hence, we must at least reconstruct (d  −  1)(n  −  1) faces and increase the rank by (n  −  1), obtaining in a more involved way exactly the same result. It is then obvious, from the recursive definition of the melonic graphs and their inherited connectivity, that any reconnecting procedure, which does not correspond to the cutting of a singular dipole, increases the length of the d externals faces, and necessarily reduces the number of internal faces. Hence, cutting a single dipole is the more face-economic way to obtaine a 2-point graph which respects the melonicity condition, and thus the maximization of the divergence degree. Therefore, from the rosette $\overline{{\hat{G}}}$, we obtain the leading order 2-points graphs $\bar{G}_{2}^{\text{melo}}$ by cutting one dipole, and the divergence degree of G₂ is bounded by:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{\omega}_{\text{melo}}}\left({{G}_{2}}\right)\leqslant -2|L\left({{G}_{2}}\right)|\,+\,(D-1)(d-2)\left[|L(\overline{{\hat{G}}})|\,-\,1\right]=-2|L\left({{G}_{2}}\right)|\,+\,(D-1)(d-2)|L({{{\bar{G}}}_{2}})| \end{array}\end{eqnarray}$$

which bounds also the divergence degree of any 2-point graph.

Now, from the 2-point graph, we would like to build the 4-point graph. As previously, we start from a rosette graph $\bar{G}_{2}^{\text{melo}}$. The same argument as before shows that the leading order graphs are obtained by cutting a new dipole. But a new subtlety appears. Indeed, because of the special connectivity of the melonic graphs, two given lines in $\bar{G}_{2}^{\text{melo}}$ can be the boundary of one and only one internal face. If two lines do not have common faces, they are said to be face disconnected⁸, and if we cut two 'face-disconnected' lines, we lose 2d faces against 2(d  −  1)  +  1 if they are face-connected. Hence, it follows that the leading order graphs with four external lines are melonic with an external face of length upper than 2 in the colored extension graph. The same argument can be applied with N external lines. Then, from the complete graph $\overline{{\hat{G}}}$, $|F|\,-\,R$ decreases by (N/2)(d  −  1)  +  1 when we cut face-connected lines. Hence,

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\omega}_{\text{melo}}}\left({{G}_{N}}\right)\leqslant & -2|L\left({{G}_{N}}\right)|\,+\,(D-1)(d-2)\left[|L\left(\overline{{\hat{G}}}\right)|-\,N/2\right] \\ {} & =-2|L\left({{G}_{N}}\right)|\,+\,(D-1)(d-2)|L\left({{{\bar{G}}}_{N}}\right)|, \end{array}\end{eqnarray}$$

and an external mono-color face connects all the external black and white vertices. □

From proposition 4, we can easily deduce a criterion for just-renormalizability. Remember that a field theory is said to be just-renormalizable if its divergence degree does not increase with the number of vertices. Because of the following topological relationship:

$$\begin{eqnarray}&&|L(G)|=\underset{k=1}{\overset{{{k}_{\text{max}}}}{\sum}}\,k{{n}_{k}}(G)-{{N}_{\text{ext}}}/2\quad \quad |V(G)|\,=\underset{k=1}{\overset{{{k}_{\text{max}}}}{\sum}}\,{{n}_{k}}(G),\end{eqnarray} \tag{ 91 }$$

where n_k is the number of vertices of degree k in G (with k black (or white) vertices in their corresponding bubble interaction vertex), we deduce from the previous theorem 3:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\omega}_{\text{melo}}} & (G):=(D-1)(d-2)-\left[(D-1)(d-2)-2\right]\frac{{{N}_{\text{ext}}}}{2} \\ {} & +\underset{k=1}{\overset{{{k}_{\text{max}}}}{\sum}}\,\left(\left[(D-1)(d-2)-2\right]k-(D-1)(d-2)\right){{n}_{k}}(G). \end{array}\end{eqnarray}$$

Hence, renormalizability is ensured if, and only if, the maximal value k_max for the degree of the interactions does not exceed

$$\begin{eqnarray}&&{{k}_{R}}=\frac{(D-1)(d-2)}{(D-1)(d-2)-2}.\end{eqnarray} \tag{ 92 }$$

This result allows to classify the just- and super-renormalizable TGFT models, on the basis of the Abelian power counting.

The super-renormalizable models are those for which ${{k}_{\text{max}}}<{{k}_{R}}$, such that the divergence degree decreases with the number of vertices, implying that only a finite number of graphs needs to be renormalized. It is only when the divergence degree does not depend on the order of the pertubative expansion, i.e. when the higher degree k_max equals k_R, that the theory is said to be just-renormalizable, and that the divergences can be taken care of by a renormalization procedure, implying the definition of a finite number of counter-terms. The table 1 below lists some power-counting Abelian just-renormalizable TGFT models⁹, in the class we have been considering and on the basis of the Abelian power counting only.

Note that some promising models for quantum gravity are absent of this table. This is the case, for example, of models on SO(4)/SO(3) in dimension 4, the TGFT counterpart of the simplicial ones studied in [23, 24, 26] and which have been a source of inspiration for this paper. Their absence is due to the rather 'pessimistic' Abelian divergence degree, which, as discussed in section 4.3, is always higher than the exact divergence degree. Hence, if all the models in the table are certainly just-renormalizable, this classification certainly does not exhaust the class of just-renormalizable models. To emphasize this point, we adopt for the following definition:

Definition 8. Any model which is power-counting just-renormalizable on the basis of the Abelian power counting only is said to be Abelian just-renormalizable. Moreover, any power counting Abelian just-renormalizable model is also power counting just-renormalizable.

Table 1. Table of power-counting Abelian-just-renormalizable theories.

Type	d	D	kR	${{\omega}_{\text{melo}}}$
A	3	4	3	$3-N/2-2{{n}_{1}}-{{n}_{2}}$
B	4	3	2	4  −  N  −  2n1
C	5	2	3	$3-N/2-2{{n}_{1}}-{{n}_{2}}$
D	6	2	2	4  −  N  −  2n1

Note that models that are not Abelian renormalizable can be understood to be actually renormalizable by using a different scaling of the coupling constants and working with a different set of interactions. See [36], in preparation.

In the rest of this paper, however, we address the issue of perturbative renormalization for a localizable rank 4 just-renormalizable quartic melonic model over SU(2). Note that localizability, which means that one can define a contraction procedure and a set of counter-terms is an essential ingredient of the renormalization procedure, not obvious in a theory with non-local interactions.

The previous analysis shows that the special model in dimension 4 with group SU(2) and quartic melonic interaction is just-renormalizable. The interactions of this model are of the form depicted in the figure 8 below. There are exactly four interactions of this type, one for each choice of the color of the intermediate lines between the two 3-dipoles. In the following, each interaction bubble b_i will be labeled by the color of these intermediate lines.

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Normally, each bubble b_i can appear in the interaction part of the action with its own coupling but in the following we will limit our attention to the simplest case where all the interaction bubbles have the same coupling. Hence:

$$\begin{eqnarray}&&{{S}_{\text{int}}}=\lambda \underset{i=1}{\overset{4}{\sum}}\,\text{T}{{\text{r}}_{{{b}_{i}}}}\left(\bar{\psi},\psi \right).\end{eqnarray} \tag{ 93 }$$

As for the Gaussian measure, it is given by the formula (31) (or (25)) in the first formulation), and allows to write the regularized generating functional as:

$$\begin{eqnarray}&&{{\mathcal{Z}}_{ \Lambda }}\left[J,\bar{J}\right]:={\int}^{}\text{d}{{\mu}_{{{C}_{ \Lambda }}}}{{\text{e}}^{-{{S}_{\text{int}}}\left(\bar{\psi},\psi \right)+\langle \bar{J},\psi \rangle +\langle \bar{\psi},J\rangle}},\end{eqnarray} \tag{ 94 }$$

where:

$$\begin{eqnarray}&&\langle \bar{J},\psi \rangle :={\int}^{}{{\left[\text{d}g\right]}^{4}}\bar{J}\left({{g}_{1}},{{g}_{2}},{{g}_{3}},{{g}_{4}}\right)\psi \left({{g}_{1}},{{g}_{2}},{{g}_{3}},{{g}_{4}}\right).\end{eqnarray} \tag{ 95 }$$

In its minimal prescription, the aim of the renormalization procedure is to give a sense to the limit $\Lambda \to \infty$ perturbatively. This issue will be considered in the following two sections.

From the previous section, a question remains: if it is now clear that the most divergent graphs are melons, it is not obvious that all the divergent graphs are melonic. In other words, we have not proved that the melonic graphs contain all the divergences occurring in the graph expansion of correlation functions, and we dedicate the end of this section to the answer to this question.

The form of the interaction bubbles in figure 8 allows to use a very useful representation of the theory, the intermediate field representation, from which the problem can be translated into a simple recursion. The building rules of the intermediate field representation are the following. From the basic properties of the Gaussian integration, the generating functional (94) can be formally rewritten as:

$$\begin{eqnarray}&&{{\mathcal{Z}}_{ \Lambda }}\left[J,\bar{J}\right]={\int}^{}\text{d}{{\mu}_{{{C}_{ \Lambda }}}}\underset{i=1}{\overset{4}{\prod}}\,\text{d}{{\mu}_{1}}\left({{\sigma}_{i}}\right){{\text{e}}^{\text{i}\sqrt{2\lambda}\langle \bar{\psi}, \Sigma \psi \rangle +\langle \bar{J},\psi \rangle +\langle \bar{\psi},J\rangle}},\end{eqnarray} \tag{ 96 }$$

where $\text{d}{{\mu}_{1}}\left({{\sigma}_{i}}\right):={{\text{e}}^{-\text{tr}\left({{\sigma}^{2}}\right)}}\text{d}\sigma$ is the Gaussian measure for Hermitian matrices and:

$$\begin{eqnarray} &&\Sigma =\underset{i=1}{\overset{4}{\sum}}\,{{\mathbb{I}}^{\otimes (i-1)}}\otimes {{\sigma}_{i}}\otimes {{\mathbb{I}}^{\otimes (4-i)}},\end{eqnarray} \tag{ 97 }$$

where $\mathbb{I}$ is the identity operator for a single variable function: ${\int}^{}\text{d}g\mathbb{I}\left(g,{{g}^{\prime}}\right)\phi \left({{g}^{\prime}}\right)=\phi (g)$. Now, the Gaussian integration over ψ and $\bar{\psi}$ can be performed, leading to the following effective multi-matrix model:

$$\begin{eqnarray}&&{{\mathcal{Z}}_{ \Lambda }}\left[J,\bar{J}\right]=\underset{i=1}{\overset{4}{\prod}}\,\text{d}{{\mu}_{1}}\left({{\sigma}_{i}}\right){{\text{e}}^{-\text{Tr}\ln \left(1-\text{i}\sqrt{2\lambda}C \Sigma \right)+\langle \bar{J},\Re J\rangle}},\end{eqnarray} \tag{ 98 }$$

where: $\Re :={{\left(1-\text{i}\sqrt{2\lambda}C \Sigma \right)}^{-1}}C$. The Feynman rules are the following: expanding the logarithm, we generate interactions with one, two,... n-external points, and a typical Feynman graph is composed of several of these vertices, connected to each other by matrix lines.

These matrix lines, of color 1 to 4, are depicted by a wavy line, and the vertices, to which they are hooked, by a grey disk, as in figure 9. In this figure, one of the grey disks has a dotted arrow line (a cilium). This grey disc does not correspond to an interaction generated by the logarithm expansion, but comes from the expansion of the R operator defined before (98), and a ciliated disk with n external wavy lines correspond to the term of degree n in the expansion in powers of $\Sigma$.

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This representation has been studied in detail in several recent papers, e.g. [13]. An important result about this representation is that the leading order graphs, the melons of the original representation, appear as trees in the intermediate field representation. More precisely:

Proposition 5. In the intermediate field representation, the melonic graphs of the divergent sector are trees with one or two external wavy lines, all of the same color, and connected to the same external face.

This can be easily proven by recursion. Now, we will use this property to see if all the divergent graphs are melons. Starting from a leading order graph (a tree) with l wavy lines (to each wavy line corresponds a vertex of the original representation, and the number of these wavy lines is equivalent to the power of λ associated to the graph), we will investigate the different ways to build a graph with l  +  1 wavy lines, and the possibility that one of these gives a non-melonic divergent graph. Two of these ways are depicted in figure 10. They correspond to the addition of a tadpole graph over a grey disk, or to the replacement of a wavy line by a 2-points grey disk. But neither of them affects the tree structure of the starting graph, which remains a melon.

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The two operations depicted in figure 11 are more promising. They are both a deviation from the melonicity, because both affect the tree structure of the starting graph (because of the connectivity of the starting graph, the second operation necessarily builds a loop). As a result, only these two operations can give us a non-melonic divergent graph, and we will examine these two possibilities.

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In the two cases, we increase the number of field lines by two (the field lines are the dotted lines of the original formulation, which are hidden inside the disks in the intermediate field representation). In the worst case, the number of faces increases only by one, but it is exactly compensated by the variation of the rank. Therefore, the total variation of the divergence degree is:

$$\begin{eqnarray}&&\delta \omega \leqslant \delta {{\omega}_{\text{Abelian}}}=-2\delta |L|\,+\,2\delta (|F|\,-\,R)<-2,\end{eqnarray} \tag{ 99 }$$

where ${{\omega}_{\text{Abelian}}}$ is the Abelian divergence degree computed previously. Because of the Abelian divergence degree is 4  −  N for a graph with N external lines, any graph with $N\geqslant 2$ becomes superficially convergent. However, the previous result seems to show that a vacuum divergent subgraph can support one intermediate field loop. And we have proven the following result:

Proposition 6. All the divergent non-vacuum graphs of Abelian the melonic ${{\phi}^{4}}$ model are melonic.

Our conclusion about the Abelian just-renormalizable model ${{\phi}^{4}}$ in d  =  4 is fundamental for the following reason. The recent literature on TGFT renormalization has shown that melonic graphs are interesting for two (closely related) essential reasons. The first one is that the Abelian power counting become exact for these graphs, and the second one is that they are said to be tracial, meaning that any connected graph remains connected under contraction of a melonic subgraph—a property which is essential for renormalization. The renormalization procedure can be defined in a worst-case-scenario, in which all the Abelian divergent graphs are regarded as dangerous.

In this section we compute the divergent parts of the one particle irreducible (1PI) melonic 2- and 4-point functions, providing a simple illustration of the renormalization procedure that we will generalize at all order in the next section. As an interesting consequence for our models, we will deduce the so called beta function of the running coupling constant ${{\lambda}_{\text{eff}}}$, and deduce that the theory is asymptotically free in the vicinity of the Gaussian fixed point.

6.2.1. The 2-points function.

At one-loop order, the divergences are due to the melonic tadpole diagrams, an example of which is pictured in figure 12, and corresponds to the following Feynman amplitude, written in the time gauge:

$$\begin{eqnarray}\begin{array}{*{35}{l}} \mathcal{A}_{\mathcal{M}}^{(4)} & \left({{\mathbf{g}}_{t}},{{\mathbf{g}}_{s}}\right)={\int}_{1/{{ \Lambda }^{2}}}^{+\infty}\text{d}{{[\alpha ]}^{3}}{{\text{e}}^{-\left(\alpha +{{\alpha}_{1}}+{{\alpha}_{2}}\right){{m}^{2}}}}{\int}^{}\text{d}l\text{d}{{l}_{1}}\text{d}{{l}_{2}}{\int}^{}\text{d}k{{{\int}^{}}_{{{U}_{k}}{{(1)}^{\times 12}}}}{{\left[\text{d}{{h}_{1}}\right]}^{4}}{{\left[\text{d}{{h}_{2}}\right]}^{4}}{{\left[\text{d}h\right]}^{4}} \\ {} & \underset{i=1}{\overset{3}{\prod}}\,{{K}_{\alpha}}\left(l{{h}_{i}}\right){{K}_{{{\alpha}_{1}}+{{\alpha}_{2}}}}\left({{g}_{ti}}{{l}_{1}}{{h}_{1\,i}}{{l}_{2}}{{h}_{2\,i}}g_{si}^{-1}\right){{K}_{{{\alpha}_{1}}+{{\alpha}_{2}}+\alpha}}\left({{g}_{t4}}{{l}_{1}}h_{1}^{(4)}l{{h}_{4}}{{l}_{2}}h_{2}^{(4)}g_{s4}^{-1}\right), \end{array}\end{eqnarray} \tag{ 100 }$$

where the subscripts t and s meaning 'target' and 'source' label the boundary group variable.

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As the divergences occur in the vicinity of $\alpha =0$, we can make use of the corresponding approximation for the heat kernels appearing in (100)

$$\begin{eqnarray}&&{{K}_{\alpha}}\left(g={{\text{e}}^{\text{i}X}}\right)\underset{\alpha \to 0}{{\longrightarrow }}\,{{(4\pi \alpha )}^{-3/2}}{{\text{e}}^{-\frac{\langle X,X\rangle}{4\alpha}}},\end{eqnarray} \tag{ 101 }$$

where $\langle.,.\rangle :\text{su}(2)\to \mathbb{R}$ is the (normalized) Killing form on the Lie algebra. The vicinity of $\alpha =0$ forces us, in the saddle point approximation, to evaluate only the fluctuations around the identity for each group arguments:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{\text{e}}^{{{X}_{l}}}}{{\text{e}}^{{{Y}_{{{h}_{i}}}}}}\approx \mathbb{I}\quad \text{where}\quad l=:{{\text{e}}^{{{X}_{l}}}}\,,{{h}_{i}}=:{{\text{e}}^{{{Y}_{{{h}_{i}}}}}}. \end{array}\end{eqnarray} \tag{ 102 }$$

Hence, the integral in (100) behaves as

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\left(\frac{1}{4\pi \alpha}\right)}^{9/2}}{{{\int}^{}}_{{{\mathbb{R}}^{2}}}}{{\text{d}}^{2}}x{\int}_{\mathbb{R}}\underset{i=1}{\overset{3}{\prod}}\,\text{d}{{y}_{i}} & {{\text{e}}^{-\frac{3x_{1}^{2}+3x_{2}^{2}+{{\left({{x}_{3}}-{{y}_{1}}\right)}^{2}}+{{\left({{x}_{3}}-{{y}_{2}}\right)}^{2}}+{{\left({{x}_{3}}-{{y}_{3}}\right)}^{2}}}{4\alpha}}}\sim {{\alpha}^{-2}}. \end{array}\end{eqnarray}$$

Note that the power of α coincides with the divergence degree computed previously. Hence, we confirm its validity in this simple example.

In order to extract the divergences of the expression (100), we begin by fixing k along the Oz axis. Then, defining:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{\mathcal{I}}_{k}}\left(l,g_{4}^{-1}g_{4}^{\prime}l\right):={{{\int}^{}}_{U(1)_{k}^{\times 4}}}{{\left[\text{d}h\right]}^{4}}\times {\int}^{}\text{d}{{g}_{4}}\text{d}g_{4}^{\prime}{{K}_{\alpha}}\left(l{{h}_{1}}\right){{K}_{\alpha}}\left(l{{h}_{2}}\right){{K}_{\alpha}}\left(l{{h}_{3}}\right){{K}_{\alpha}}\left(g_{4}^{-1}g_{4}^{\prime}l{{h}_{4}}\right), \end{array}\end{eqnarray}$$

and integrating over the h_i gives:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{\mathcal{I}}_{k}}\left(l,g_{4}^{-1}g_{4}^{\prime}l\right):=\underset{\left\{{{l}_{i}}\right\}\in {{\mathbb{N}}^{4}}}{\sum}\,\underset{{{m}_{1}}=-{{l}_{1}}}{\overset{{{l}_{1}}}{\sum}}\,\underset{i=1}{\overset{3}{\prod}}\,\left(2{{l}_{i}}+1\right){{\text{e}}^{-4\alpha {{l}_{i}}\left({{l}_{i}}+1\right)}}D_{00}^{\left({{l}_{i}}\right)}(l)\left(2{{l}_{4}}+1\right){{\text{e}}^{-4\alpha {{l}_{4}}\left({{l}_{4}}+1\right)}}D_{{{m}_{4}}0}^{\left({{l}_{4}}\right)}(l)D_{0{{m}_{4}}}^{\left({{l}_{4}}\right)}\left(g_{4}^{-1}g_{4}^{\prime}\right). \end{array}\end{eqnarray}$$

The first product is nothing that ${{\left[4\pi K_{\alpha}^{{{\mathcal{S}}_{2}}}(|x|)\right]}^{3}}$, where $x:=\pi (l)$ is the path over ${{\mathcal{S}}_{2}}$ corresponding to $l\in SU(2)$. With our definitions, using the decomposition: $l={{\text{e}}^{\text{i}\varphi {{\sigma}_{z}}/2}}{{\text{e}}^{\text{i}\theta {{\sigma}_{y}}/2}}{{\text{e}}^{\text{i}\gamma {{\sigma}_{z}}/2}}$, one find explicitly: $|x|\,=\theta$ i.e. the geodesic length from the north pole to $\vec{n}=(\theta,\phi )$. As explained before, the divergences come from the negative (or null) powers of α, allowing to make an expansion around the identity. When $\alpha \to 0$, $K_{\alpha}^{{{\mathcal{S}}_{2}}}(x)\approx \delta (x)$, and using the expansion of the heat kernel over the 2-sphere in the vicinity of $\alpha =0$:

$$\begin{eqnarray}&&K_{\alpha}^{{{\mathcal{S}}_{2}}}(x)\approx \frac{{{\text{e}}^{-|x{{|}^{2}}/4\alpha}}}{4\pi \alpha},\end{eqnarray} \tag{ 103 }$$

where the discarded terms involve higher power of α and generate sub-divergent corrections for the mass parameter, which are not relevant for our discussion. Then, because of the fact that, for any C² function $f:{{\mathcal{S}}_{2}}\to \mathcal{C}$ which is regularized at the origin:

$$\begin{eqnarray}&&{\int}^{}{{\text{d}}^{2}}x{{\text{e}}^{-3|x{{|}^{2}}/4\alpha}}f(x)=\frac{4}{3}\pi \alpha f(0)+\left(\frac{4}{3}\pi \alpha \right)\frac{\alpha}{3}{{ \Delta }_{{{\mathcal{S}}_{2}}}}\,f(0)+\cdots,\end{eqnarray} \tag{ 104 }$$

where ${{ \Delta }_{{{\mathcal{S}}_{2}}}}$ denote the Laplacian over ${{\mathcal{S}}_{2}}$. And because ${{ \Delta }_{{{\mathcal{S}}_{2}}}}D_{{{m}_{4}}0}^{\left({{l}_{4}}\right)}=-{{l}_{4}}\left({{l}_{4}}+1\right)D_{{{m}_{4}}0}^{\left({{l}_{4}}\right)}$, we deduce (up to sub-divergent term for the first correction without Laplacian):

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {\int}^{}\text{d}l{{\mathcal{I}}_{k}}\left(l,g_{4}^{-1}g_{4}^{\prime}l\right)={{{\int}^{}}_{U{{(1)}_{k}}}}\text{d}{{h}_{k}}\delta \left(g_{4}^{-1}g_{4}^{\prime}{{h}_{k}}\right)\left\{\frac{1}{6}\frac{1}{{{\alpha}^{2}}}-\frac{13}{18}\frac{1}{\alpha}{{l}_{4}}\left({{l}_{4}}+1\right)+\cdots \right\}. \end{array}\end{eqnarray} \tag{ 105 }$$

where we have taking into account a normalization factor $1/8\pi$ coming from the normalized Haar measure $\text{d}l$. Then, integrating over the $\left\{h_{\ell}^{(i)}\right\},\ell =1,2$ variables in (100), we find the leading divergent par of the 2-points function:

$$\begin{eqnarray}\begin{array}{*{35}{l}} \mathcal{A}_{\mathcal{M}}^{(4)\,\infty}\left({{\mathbf{g}}_{t}},{{\mathbf{g}}_{s}}\right)= & \underset{\left\{{{l}_{i}}\right\}\in {{\mathbb{N}}^{4}}}{\sum}\,{\int}_{1/{{ \Lambda }^{2}}}^{+\infty}\text{d}{{[\alpha ]}^{2}}{{\text{e}}^{-\left({{\alpha}_{1}}+{{\alpha}_{2}}\right)\left({{m}^{2}}+4\underset{i=1}{\overset{4}{\sum}}\,{{l}_{i}}\left({{l}_{i}}+1\right)\right)}}{\int}^{}\text{d}l \\ {} & \times \,\,\left\{\frac{1}{6}{{\mathcal{I}}_{1}}-\frac{13}{18}{{\mathcal{I}}_{2}}{{l}_{4}}\left({{l}_{4}}+1\right)+\cdots \right\}\underset{i=1}{\overset{4}{\prod}}\,\left(2{{l}_{i}}+1\right)D_{00}^{{{l}_{i}}}\left(g_{si}^{-1}{{g}_{ti}}l\right) \end{array}\end{eqnarray} \tag{ 106 }$$

where we have defined:

$$\begin{eqnarray}&&{{\mathcal{I}}_{1}}:={\int}_{1/{{ \Lambda }^{2}}}^{+\infty}\text{d}\alpha \frac{{{\text{e}}^{-\alpha {{m}^{2}}}}}{{{\alpha}^{2}}},\end{eqnarray} \tag{ 107 }$$

$$\begin{eqnarray}&&{{\mathcal{I}}_{2}}:={\int}_{1/{{ \Lambda }^{2}}}^{+\infty}\text{d}\alpha \frac{{{\text{e}}^{-\alpha {{m}^{2}}}}}{\alpha}.\end{eqnarray} \tag{ 108 }$$

and where the subscript $\infty$ means that we retain only the divergent part. The first term of (106) correspond to the mass renormalization and the second one to the wave function. Similar expressions can be obtained for each interaction bubble, and the divergent part of the complete 1PI 2-points function $\Gamma _{\infty}^{(2)}\left(\left\{{{g}_{{{t}_{i}}}}\right\},\left\{{{g}_{{{s}_{i}}}}\right\}\right)$ writes as (a global factor 2 comes from the Wick-theorem):

$$\begin{eqnarray}\begin{array}{*{35}{l}} \Gamma _{\infty}^{(2)} & =-\frac{4\lambda}{3}\left[{{\mathcal{I}}_{1}}+\frac{13}{24}{{\mathcal{I}}_{2}}\,\underset{i=1}{\overset{4}{\sum}}\,{{ \Delta }_{{{\mathcal{S}}_{2}},i}}\right]. \end{array}\end{eqnarray} \tag{ 109 }$$

6.2.2. The 4-point function.

At one-loop order, a typical melonic contribution involves one loop between two vertices, as an example is pictured in figure 13 below.

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The amputated Feynman amplitude, pictured on figure 13 can be easily obtained from the Feynman rules and from the previous calculation. From the Feynman rules—and taking into account symmetry factors:

$$\begin{eqnarray} &&\Gamma _{{{\mathcal{M}}_{4}}}^{(4)}\left({{g}_{1}},{{\bar{g}}_{1}},g_{1}^{\prime},\bar{g}_{1}^{\prime}\right)=8{{\lambda}^{2}}\underset{\left\{{{l}_{i}}\right\}\in {{\mathbb{N}}^{5}}}{\sum}\,{\int}_{1/{{ \Lambda }^{2}}}^{\infty}\text{d}{{[\alpha ]}^{2}}{{\text{e}}^{-\left({{\alpha}_{1}}+{{\alpha}_{2}}\right){{m}^{2}}}}{\int}_{SU{{(2)}^{2}}}\text{d}{{h}_{1}}\text{d}{{h}_{2}}\mathcal{I}_{{{\mathcal{M}}_{4}},{{\alpha}_{1}}+{{\alpha}_{2}},\left\{{{l}_{i}}\right\}}^{(4)}\left({{g}_{1}},{{\bar{g}}_{1}},g_{1}^{\prime},\bar{g}_{1}^{\prime},{{h}_{1}},{{h}_{2}}\right),\end{eqnarray} \tag{ 110 }$$

with

$$\begin{eqnarray}\begin{array}{*{35}{l}} \mathcal{I}_{{{\mathcal{M}}_{4}},{{\alpha}_{1}}+{{\alpha}_{2}},\left\{{{l}_{i}}\right\}}^{(4)} & \left({{g}_{1}},{{{\bar{g}}}_{1}},g_{1}^{\prime},\bar{g}_{1}^{\prime},{{h}_{1}},{{h}_{2}}\right)=\underset{i=1}{\overset{3}{\prod}}\,\left(2{{l}_{i}}+1\right){{\text{e}}^{-4\left({{\alpha}_{1}}+{{\alpha}_{2}}\right){{l}_{i}}\left({{l}_{i}}+1\right)}}D_{00}^{\left({{l}_{i}}\right)}\left({{h}_{1}}{{h}_{2}}\right) \\ {} & \times \,\left(2{{l}_{4}}+1\right)\left(2{{l}_{5}}+1\right){{\text{e}}^{-4{{\alpha}_{1}}{{l}_{4}}\left({{l}_{4}}+1\right)-4{{\alpha}_{2}}{{l}_{5}}\left({{l}_{5}}+1\right)}}D_{00}^{\left({{l}_{4}}\right)}\left({{g}^{\prime}}_{1}^{\,-1}{{g}_{1}}{{h}_{1}}\right)D_{00}^{\left({{l}_{5}}\right)}\left(\bar{g}_{1}^{-1}{{{\bar{g}}}_{1}}{{h}_{2}}\right). \end{array}\end{eqnarray} \tag{ 111 }$$

Note that the amplitude is invariant under the transformation of the gauge variables: ${{h}_{1}}\to {{G}_{1}}{{h}_{1}}\bar{G}_{1}^{-1}$, ${{h}_{2}}\to {{\bar{G}}_{1}}{{h}_{2}}G_{1}^{-1}$ if ${{G}_{1}}\in U{{(1)}_{{{\sigma}_{z}}}}$, up to an irrelevant global translation of the boundary variable: ${{g}_{1}},{{\bar{g}}_{1}}\to {{g}_{1}}G_{1}^{-1},{{\bar{g}}_{1}}\bar{G}_{1}^{-1}$—the 4-points function being assumed to be a gauge invariant function i.e. $\Gamma _{{{\mathcal{M}}_{4}}}^{(4)}\in {{\mathbb{G}}^{4}}$, where $\mathbb{G}=\ker \left[\hat{P}-\mathbb{I}\right]$ has been defined in section 3.1. Fixing the gauge such that ${{\bar{G}}_{1}}{{h}_{2}}=\mathbb{I}$, the amplitude (112) writes as:

$$\begin{eqnarray} &&\Gamma _{{{\mathcal{M}}_{4}}}^{(4)}\left({{g}_{1}},{{\bar{g}}_{1}},g_{1}^{\prime},\bar{g}_{1}^{\prime}\right)=8{{\lambda}^{2}}\underset{\left\{{{l}_{i}}\right\}\in {{\mathbb{N}}^{5}}}{\sum}\,{\int}_{1/{{ \Lambda }^{2}}}^{\infty}\text{d}{{[\alpha ]}^{2}}{{\text{e}}^{-\left({{\alpha}_{1}}+{{\alpha}_{2}}\right){{m}^{2}}}}{\int}_{SU(2)}\text{d}h\mathcal{I}_{{{\mathcal{M}}_{4}},{{\alpha}_{1}}+{{\alpha}_{2}},\left\{{{l}_{i}}\right\}}^{(4)}\left({{g}_{1}},{{\bar{g}}_{1}},g_{1}^{\prime},\bar{g}_{1}^{\prime},h,\mathbb{I}\right),\end{eqnarray} \tag{ 112 }$$

and the computation of the divergent part of ${{\mathcal{I}}_{{{\mathcal{M}}_{4}},{{\alpha}_{1}}+{{\alpha}_{2}},\left\{{{l}_{i}}\right\}}}$ follows the same strategy as for the computation of the mass correction in the previous section. As a matter of fact, because the diagram on figure 13 scale as $\ln (\Lambda )$, only the first term in the local expansion is relevant for the extraction of the divergences. Let us consider separately the Hepp sectors ${{\alpha}_{1}}\geqslant {{\alpha}_{2}}$ and ${{\alpha}_{2}}\geqslant {{\alpha}_{1}}$, and the change of variables:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\alpha}_{1}} & =\alpha \end{array}\end{eqnarray} \tag{ 113 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\alpha}_{2}}-\frac{1}{{{ \Lambda }^{2}}} & =\beta \left({{\alpha}_{1}}-\frac{1}{{{ \Lambda }^{2}}}\right), \end{array}\end{eqnarray} \tag{ 114 }$$

in the first one of these sectors. We then obtain truly the same integral as for the computation of the mass correction. Because of the melonic structure of the graph, the local approximation at order ${{\lambda}^{2}}$ writes as (note that because the 4-points function scale as $\ln (\Lambda )$, the first term of the local expansion contain all the divergences):

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \mathcal{A}_{{{\mathcal{M}}_{4}}}^{(4)}\left({{\mathbf{g}}_{1}},{{\mathbf{g}}_{2}},{{\mathbf{g}}_{3}},{{\mathbf{g}}_{4}}\right)=\left(-4\lambda +{{ \Gamma }^{(4)}}\right)\frac{1}{2}\left[\mathcal{W}_{{{\mathbf{g}}_{1}},{{\mathbf{g}}_{2}},{{\mathbf{g}}_{3}},{{\mathbf{g}}_{4}}}^{(4)}+{{\mathbf{g}}_{1}}\longleftrightarrow {{\mathbf{g}}_{3}}\right], \end{array}\end{eqnarray}$$

and one finds:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {} & {{ \Gamma }^{(4)\,\infty}}=\frac{4{{\lambda}^{2}}}{3}{{\mathcal{I}}_{2}}. \end{array}\end{eqnarray} \tag{ 115 }$$

6.2.3. Running coupling and asymptotic freedom.

In order to extract the dangerous part of the one-loop amplitudes, i.e. the terms involving a positive or null power of $\Lambda$, we study the behavior of the integrals ${{\mathcal{I}}_{1}}$ and ${{\mathcal{I}}_{2}}$. Firstly, observe that an integration by parts gives:

$$\begin{eqnarray}&&{{\mathcal{I}}_{1}}={{ \Lambda }^{2}}{{\text{e}}^{-{{m}^{2}}/{{ \Lambda }^{2}}}}-{{m}^{2}}{{\mathcal{I}}_{2}}.\end{eqnarray}$$

Secondly, observe that the divergence of ${{\mathcal{I}}_{2}}$ is at most logarithmic. Hence,

$$\begin{eqnarray}&&{{\mathcal{I}}_{2}}=A\ln (\Lambda )+\mathcal{O}(1/ \Lambda ).\end{eqnarray}$$

By differentiating the two members of this equality, we obtain A  =  2, and finally:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\mathcal{I}}_{1}} & \sim {{ \Lambda }^{2}}-2{{m}^{2}}\ln (\Lambda ) \end{array}\end{eqnarray} \tag{ 116 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\mathcal{I}}_{2}} & \sim 2\ln (\Lambda ). \end{array}\end{eqnarray} \tag{ 117 }$$

Hence, for the 1PI 2-point function, the dangerous part $\Gamma _{\text{div}}^{2}$ is equal to

$$\begin{eqnarray}\begin{array}{*{35}{l}} {} & \Gamma _{\infty}^{(2)}=-\frac{4\lambda}{3}\left[{{ \Lambda }^{2}}-2{{m}^{2}}\ln (\Lambda )+\frac{26}{24}\ln (\Lambda )\,\underset{i=1}{\overset{4}{\sum}}\,{{ \Delta }_{{{\mathcal{S}}_{2}},i}}\right], \end{array}\end{eqnarray} \tag{ 118 }$$

which fixes the divergent parts of the mass and wave function counter-terms as:

$$\begin{eqnarray}\begin{array}{*{35}{l}} \delta m_{\text{div}}^{2} & :=-\frac{4\lambda}{3}\left[{{ \Lambda }^{2}}-2{{m}^{2}}\ln (\Lambda )\right] \end{array}\end{eqnarray} \tag{ 119 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \delta {{Z}_{\text{div}}} & :=\frac{13\lambda}{9}\ln (\Lambda ). \end{array}\end{eqnarray} \tag{ 120 }$$

Similarly, from the expression (115), we deduce that the divergence is exactly compensated by an interaction of the initial form with an intermediate line of color 4, if this interaction is proportional to the counter-term $\delta \lambda$, with:

$$\begin{eqnarray}&&\delta \lambda =\frac{{{\lambda}^{2}}}{3}\ln (\Lambda ).\end{eqnarray} \tag{ 121 }$$

This result allows to obtain the dependence of the effective coupling at scale $\Lambda$. Indeed, the effective coupling includes the effect of the wave-function renormalization. Hence:

$$\begin{eqnarray}&&{{\lambda}_{\text{eff}}}(\Lambda ):=\frac{\lambda +\delta \lambda}{{{Z}^{2}}}.\end{eqnarray} \tag{ 122 }$$

By differentiating the two terms, and using the relations (118) and (121), we find:

$$\begin{eqnarray} &&\Lambda \frac{\text{d}{{\lambda}_{\text{eff}}}}{\text{d} \Lambda }=-\frac{23}{9}\lambda _{\text{eff}}^{2},\end{eqnarray} \tag{ 123 }$$

where the minus sign means that the model is perturbatively asymptotically free in the deep UV. As a result, we expect that no-Landau pole occurs in the high energy limit, and that the theory can be properly defined beyond the perturbative regime using constructive methods.

The table 1 shows that in the present case, the divergence degree of a Feynman graph G with N external lines is bounded by 4  −  N. Hence, a priori, only the 2 and 4-points functions are potentially dangerous. Then, we adopt the following definitions:

Definition 9. Consider a Feynman graph G and let $h\subseteq G$ be a melonic subgraph of G with n_h external lines. The subgraph h is said to be:

• superficially convergent if ${{\omega}_{\text{Abelian}}}(h)<0$.
• superficially divergent or dangerous if ${{\omega}_{\text{Abelian}}}(h)\geqslant 0\Rightarrow N(h)\leqslant 4$.

Definition 10. Consider a Feynman graph $G$. A Zimmermann forest $\mathbb{F}$ is a forest of connected divergent subgraphs $\left\{h\subseteq \mathcal{G}|\omega (h)\geqslant 0\right\}$. Here the word forest should be understood in the sense of inclusion relations. It simply means that taking two elements ${{h}_{1}},{{h}_{2}}\in \mathbb{F}$, they are either line and vertex disjoint or included one into the other. The set of all Zimmermann forests of $G$ is noted $D(G)$

The definition 9 is motivated by the following theorem, which states that the Feynman amplitude is finite if it does not contain any subdivergent graph in the sense of the definition 9:

Theorem 3 (Weinberg uniform). Consider a completely convergent graph G, i.e. a graph with no subdivergences. Its corresponding Feynman amplitude ${{\mathcal{A}}_{G}}$ has the following bounds:

$$\begin{eqnarray}&&|{{\mathcal{A}}_{G}}|\leqslant {{K}^{|V(G)|}},\,\,K\in {{\mathbb{R}}^{+}}.\end{eqnarray} \tag{ 124 }$$

Proof. The proof is standard in renormalization theory, and we will only give the main steps. The first step is to note that, when N  >  4, one has $4-N\leqslant -N/3$ (the graph with five external lines does not exist). Hence, for a given scale attribution μ, the graph amplitude ${{\mathcal{A}}_{G}}$ verifies the following trivial bounds:

$$\begin{eqnarray}&&|{{\mathcal{A}}_{G\,\mu}}|\leqslant {{K}^{l(G)}}\underset{i,\rho}{\prod}\,{{M}^{-N\left(G_{i}^{\rho}\right)/3}}.\end{eqnarray}$$

Definition 11. 

$$\begin{eqnarray}&&{{i}_{b}}(\mu )=\underset{l\in {{L}_{b}}(G)}{{\sup}}\,{{i}_{l}}(\mu )\quad \quad {{e}_{b}}(\mu )=\underset{l\in {{L}_{b}}(G)}{{\inf}}\,{{i}_{l}}(\mu ),\end{eqnarray}$$

where b stands for a vertex bubble $b\in G$, and L_b(G) is the set of its external lines. Note that b touches a connected subgraph $G_{i}^{\rho}$ if and only if $i\leqslant {{i}_{b}}(\mu )$, and is an external vertex if ${{e}_{b}}<i\leqslant {{i}_{b}}$. Therefore, because each vertex touches at most 4 subgraphs:

$$\begin{eqnarray}&&\underset{i,\rho}{\prod}\,{{M}^{-N\left(G_{i}^{\rho}\right)/3}}\leqslant \underset{i,\rho}{\prod}\,\underset{\begin{array}{c} b\in G_{i}^{\rho} \\ {{e}_{b}}<i\leqslant {{i}_{b}} \end{array}}{\prod}\,{{M}^{1/12}},\end{eqnarray}$$

and

$$\begin{eqnarray}&&|{{\mathcal{A}}_{G,\mu}}|\leqslant {{K}^{l(G)}}\underset{b}{\prod}\,{{M}^{-\frac{|{{i}_{b}}(\mu )-{{e}_{b}}(\mu )|}{12}}}.\end{eqnarray}$$

Using the fact that there are at most 4 half-lines, and thus $6=4\times 3/2$ pairs of half-lines hooked to a given vertex, and that, for two lines l and ${{l}^{\prime}}$ of a bubble b, $|{{e}_{b}}-{{i}_{b}}|\geqslant |{{i}_{l}}-{{i}_{{{l}^{\prime}}}}|$, we obtain:

$$\begin{eqnarray}&&|{{\mathcal{A}}_{G,\mu}}|\leqslant {{K}^{l(G)}}\underset{b}{\prod}\,\underset{\left(l,{{l}^{\prime}}\right)\in {{L}_{b}}\times {{L}_{b}}}{\prod}\,{{M}^{-\frac{|{{i}_{l}}-{{i}_{{{l}^{\prime}}}}|}{72}}}.\end{eqnarray}$$

This expression implies directly the finiteness of A_G. To understand why, observe that we can choose a total ordering of the lines $L(G)=\left\{{{l}_{1}},...,{{l}_{|L(G)|}}\right\}$ such that l₁ is hooked to an external vertex b₀ and that each subset $\left\{{{l}_{1}},...,{{l}_{m}}\right\},\,m\leqslant |L(G)|$ is connected. Therefor, for any line l_j, we can choose l_p(j) (with p(j)  <  j) a line sharing a vertex with l_j, from which we deduce:

$$\begin{eqnarray}&&\underset{b}{\prod}\,\underset{\left(l,{{l}^{\prime}}\right)\in {{L}_{b}}\times {{L}_{b}}}{\prod}\,{{M}^{-\frac{|{{i}_{l}}(\mu )-{{i}_{{{l}^{\prime}}}}(\mu )|}{78}}}\leqslant \underset{j=1}{\overset{|L(G)|}{\prod}}\,{{M}^{-\frac{|{{i}_{{{l}_{j}}}}-{{i}_{{{l}_{p(\,j)}}}}|}{72}}}.\end{eqnarray}$$

And because

$$\begin{eqnarray}&&\underset{{{i}_{{{l}_{j}}}}}{\sum}\,{{M}^{-\frac{|{{i}_{{{l}_{j}}}}-{{i}_{{{l}_{p(\,j)}}}}|}{72}}}\leqslant \underset{{{i}_{{{l}_{j}}}}\geqslant {{i}_{{{l}_{p(\,j)}}}}}{\sum}\,{{M}^{-\frac{|{{i}_{{{l}_{j}}}}-{{i}_{{{l}_{p(\,j)}}}}|}{72}}}=\frac{1}{1-{{M}^{-1/72}}},\end{eqnarray}$$

we have finally:

$$\begin{eqnarray}&&|{{\mathcal{A}}_{G\,\mu}}|\leqslant {{K}^{l(G)}}\underset{\mu =\left\{{{i}_{1}},...,{{i}_{l(G)}}\right\}}{\sum}\,\underset{j=1}{\overset{|L(G)|}{\prod}}\,{{M}^{-\frac{|{{i}_{{{l}_{j}}}}-{{i}_{{{l}_{p(\,j)}}}}|}{72}}}\leqslant {{K}^{\prime l(G)}}\quad.\end{eqnarray}$$

□

When the graphs contain some dangerous subgraphs, the proof given above breaks down, and the finiteness of the sum over scale attribution is not guaranteed. Therefore, the case of the presence of these subgraphs must be considered in details.

•$\boldsymbol{N}=2$. Let us consider the case of a subdivergent graph with two external lines. The situation is depicted in figure 14 below. The structure of the amplitude ${{\mathcal{A}}_{G,\mu}}$ for the scale attribution μ is

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{\mathcal{A}}_{G,\mu}}={\int}^{}\underset{l}{\prod}\,\text{d}{{{\bar{g}}}_{1l}}\text{d}{{{\bar{g}}}_{2l}}\text{d}{{g}_{1l}}\text{d}{{g}_{2l}}{{{\bar{\mathcal{A}}}}_{G,\mu}}\left(\left\{{{{\bar{g}}}_{1l}}\right\},\left\{{{{\bar{g}}}_{2l}}\right\}\right){{C}_{{{i}_{1}}}}\left(\left\{{{g}_{1l}}\right\},\left\{{{{\bar{g}}}_{1l}}\right\}\right){{C}_{{{i}_{2}}}}\left(\left\{{{g}_{2l}}\right\},\left\{{{{\bar{g}}}_{2l}}\right\}\right){{\mathcal{M}}_{j}}\left({{g}_{11}},{{g}_{21}}\right), \end{array}\end{eqnarray} \tag{ 125 }$$

where ${{\mathcal{M}}_{j}}$ is the 2-points subgraph pictured in figure 14, j its scale, i.e. the scale of its highest line, and ${{\bar{\mathcal{A}}}_{G,\mu}}$ is a completely convergent amplitude. The scale attribution is chosen such as $j>{{i}_{1}},{{i}_{2}}$, and the subgraph ${{\mathcal{M}}_{j}}$ is said to be high. This is typically the region in which this graph is potentially divergent. Obviously from the propagator structure one has ${{\mathcal{M}}_{j}}\left(g,{{g}^{\prime}}\right)={{\mathcal{M}}_{j}}\left(g{{g}^{\prime \,-1}}\right)$.

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The one-loop computations show that we have to replace the standard local expansion by an equivalence up to a right multiplication by an element of U(1)_k. In order to make this identification more conveniently in our analysis, we first consider the following definitions. Using the Euler parametrization, each $g\in SU(2)$ writes as: $g={{\text{e}}^{\text{i}\phi {{\sigma}_{z}}/2}}{{\text{e}}^{\text{i}\theta {{\sigma}_{y}}/2}}{{\text{e}}^{\text{i}\gamma {{\sigma}_{z}}/2}}$, where $0\leqslant \phi <2\pi$, $0\leqslant \theta \leqslant \pi$, $0\leqslant \gamma <4\pi$. $(\theta,\phi,\gamma )$ are coordinates over the group manifold, with metric: $\text{d}{{s}^{2}}=\left(\text{d}{{\phi}^{2}}+2\cos (\gamma )\text{d}\phi \text{d}\gamma +\text{d}{{\gamma}^{2}}+\text{d}{{\theta}^{2}}\right)/4$. Let the map $\pi :SU(2)\to {{\mathcal{S}}_{2}}$ such that $\pi (g)=\vec{n}(\theta,\phi )\in {{\mathcal{S}}_{2}},\,g\in SU(2)$. It provides a basic fibre bundle, say P with fibre U(1), acting on the right to each element $u\in P\sim SU(2)$: $u\to {{u}^{\prime}}=uh,\forall h\equiv {{\text{e}}^{\text{i}\gamma {{\sigma}_{z}}/2}}\in U(1)$, getting from one horizontal space to another. Moreover, we introduce the global section $s:{{\mathcal{S}}_{2}}\to SU(2)$, defined as $s(x)={{\text{e}}^{\text{i}\phi {{\sigma}_{z}}/2}}{{\text{e}}^{\text{i}\theta {{\sigma}_{y}}/2}}\in SU(2),\,$ $\,x\equiv \vec{n}(\theta,\phi )$, inducing a metric over S₂: ${{g}_{{{\mathcal{S}}_{2}}}}(x,x)={{g}_{SU(2)}}\left(s(x),s(x)\right)={{s}^{\ast}}{{g}_{SU(2)}}(x,x)$. Finally, for any $g\equiv (\theta,\phi,\gamma )\in P$, let ${{T}_{g}}(P)\sim \mathfrak{s}\mathfrak{u}(2)$ be the tangent space at g. We say that ${{X}_{g}}\in {{T}_{g}}(P)$ is horizontal, if and only if $\text{d}\gamma \left({{X}_{g}}\right)=0\Leftrightarrow \exists (\phi,\theta )|{{\text{e}}^{{{X}_{g}}}}={{\text{e}}^{\text{i}\phi {{\sigma}_{z}}/2}}{{\text{e}}^{\text{i}\theta {{\sigma}_{y}}/2}}$.

Now, we define the real parameter $t\in [0,1]$ in order to interpolate between g₂₁ and g₁₁. More precisely, in the 'time gauge', choosing k along Oz for each lines of ${{\mathcal{M}}_{j}}$ and of its external lines, we have the parametrization: $G:={{g}_{21}}g_{11}^{-1}={{\text{e}}^{\text{i}\phi {{\sigma}_{z}}/2}}{{\text{e}}^{\text{i}\theta {{\sigma}_{y}}/2}}{{\text{e}}^{\text{i}\gamma {{\sigma}_{z}}/2}}$, and as explained in the section 6.2, $|\theta |$ correspond to the geodesic distance between the north pole and the point $\vec{n}(\theta,\phi )$. Let X be the horizontal vector such that ${{\text{e}}^{X}}={{\text{e}}^{\text{i}\phi {{\sigma}_{z}}/2}}{{\text{e}}^{\text{i}\theta {{\sigma}_{y}}/2}}$. We define the smooth trajectory $G(t)={{\text{e}}^{tX}}{{\text{e}}^{\text{i}\gamma {{\sigma}_{z}}/2}}$, such that $\pi \left(G(1)\right)=\vec{n}(\theta,\phi )$ and $\pi \left(G(0)\right)={{\vec{e}}_{z}}$. Therefore, we have the parametrized amplitude:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{\mathcal{A}}_{G,\mu}}(t)={\int}^{}\underset{l}{\prod}\,\text{d}{{{\bar{g}}}_{1l}}\text{d}{{{\bar{g}}}_{2l}}\text{d}{{g}_{1l}}\text{d}{{g}_{2l}}{{{\bar{\mathcal{A}}}}_{G\mu}}\left(\left\{{{{\bar{g}}}_{1l}}\right\},\left\{{{{\bar{g}}}_{2l}}\right\}\right){{C}_{{{i}_{1}}}}\left(\left\{{{g}_{1l}}\right\},\left\{{{{\bar{g}}}_{1l}}\right\}\right){{C}_{{{i}_{2}}}}\left(\left\{{{g}_{12}}(t),{{g}_{2l\,l\ne 1}}\right\},\left\{{{{\bar{g}}}_{2l}}\right\}\right){{\mathcal{M}}_{j}}\left({{g}_{11}},{{g}_{21}}\right), \end{array}\end{eqnarray} \tag{ 126 }$$

such that ${{\mathcal{A}}_{G,\mu}}={{\mathcal{A}}_{G,\mu}}(t=1)$, and where ${{g}_{12}}(t):=G(t){{g}_{11}}$. We then introduce the $\ast$ application ${{\tau}_{\mathcal{M}}}$ as:

$$\begin{eqnarray}&&\tau _{\mathcal{M}}^{\ast}{{\mathcal{A}}_{G,\mu}}(t):=\underset{n=0}{\overset{\omega \left(\mathcal{M}\right)}{\sum}}\,\frac{1}{n!}\frac{{{\text{d}}^{n}}{{\mathcal{A}}_{G,\mu}}}{\text{d}{{t}^{n}}}(t=0),\end{eqnarray} \tag{ 127 }$$

and the goal, motivated by the usual quantum field theory, is to prove that

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \mathcal{A}_{G,\mu}^{R}=\left(1-\tau _{\mathcal{M}}^{\ast}\right){{\mathcal{A}}_{G,\mu}}{{|}_{t=1}}={\int}_{0}^{1}\text{d}t\frac{{{(1-t)}^{\omega \left({{\mathcal{M}}_{j}}\right)}}}{\omega \left({{\mathcal{M}}_{j}}\right)!}\frac{{{\text{d}}^{\omega \left({{\mathcal{M}}_{j}}\right)+1}}{{\mathcal{A}}_{G,\mu}}(t)}{\text{d}{{t}^{\omega \left({{\mathcal{M}}_{j}}\right)+1}}} \end{array}\end{eqnarray} \tag{ 128 }$$

is finite. The first key result is the following obvious bound of the derivative of the propagator in the slice i:

$$\begin{eqnarray}&&|C_{i}^{(k)}\left(\left\{{{g}_{i}}\right\},\left\{g_{i}^{\prime}\right\}\right)|\leqslant |X{{|}^{k}}K{{M}^{(3d-2+k)i}},\end{eqnarray} \tag{ 129 }$$

Because of this bounds, the derivative appearing in (128) increases the bound of the propagator by a factor ${{M}^{\left(\omega \left({{\mathcal{M}}_{j}}\right)+1\right){{i}_{2}}}}$. Furthermore, because the length $|X|$ scales as M^−j with M, the power $|X{{|}^{\omega \left({{\mathcal{M}}_{j}}\right)+1}}$ has a bound of the form $K{{M}^{-\left(\omega \left({{\mathcal{M}}_{j}}\right)+1\right)j}}$. Taking into account all the contributions, the total exponential decay is

$$\begin{eqnarray}&&\omega \left({{\mathcal{M}}_{j}}\right)+\left(\omega \left({{\mathcal{M}}_{j}}\right)+1\right)\left({{i}_{2}}-j\right)<-1,\end{eqnarray} \tag{ 130 }$$

meaning that the subgraph becomes superficially convergent in the sense of the definition 9.

•$\boldsymbol{N}=4$. A typical sub-divergence of this type is depicted in figure 15 below, in which the graphs $\bar{\mathcal{A}}_{G,\mu}^{(i)}$ are free of sub-divergences. As in the previous case, we begin by writing the amplitude in terms of the three blocks defined in figure 15. We define, with the same notations as in the case N  =  2:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {} & {{\mathcal{A}}_{G,\mu}}(t)={\int}^{}\underset{l=1}{\overset{4}{\prod}}\,\underset{k=1}{\overset{4}{\prod}}\,\text{d}{{g}_{lk}}\text{d}{{{\bar{g}}}_{lk}}\mathcal{A}_{G,\mu}^{(1)}\left(\left\{{{{\bar{g}}}_{1l}}\right\},\left\{{{{\bar{g}}}_{2l}}\right\}\right) \\ {} & \quad \times \underset{k=1}{\overset{2}{\prod}}\,{{C}_{{{i}_{k}}}}\left(\left\{{{{\bar{g}}}_{kl}}\right\};{{g}_{k1}}(t),\left\{{{g}_{kl\,l\ne 0}}\right\}\right)\underset{k=3}{\overset{4}{\prod}}\,{{C}_{{{i}_{k}}}}\left({{g}_{k1}}(t),\left\{{{g}_{kl\,l\ne 0}}\right\};\left\{{{{\bar{g}}}_{kl}}\right\}\right) \\ {} & \quad \times \,\mathcal{A}_{G,\mu}^{(2)}\left(\left\{{{{\bar{g}}}_{3l}}\right\},\left\{{{{\bar{g}}}_{4l}}\right\}\right)\mathcal{M}_{j}^{(4)}\left({{g}_{11}},{{g}_{12}};{{g}_{13}},{{g}_{14}}\right). \end{array}\end{eqnarray} \tag{ 131 }$$

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As in the case of the 2-point function, we introduce the $\ast$ operator ${{\tau}_{{{\mathcal{M}}^{(4)}}}}$ whose action is defined by the equation (128). The same argument as in the previous section can be applied to this case (there are two terms instead of one after differentiation with respect to t), and the conclusion is unchanged: in the domain $j>{{i}_{k}},\forall k$, the subgraph ${{\mathcal{M}}^{(4)}}$ becomes superficially convergent, in the sense of the definition 9.

The previous analysis motivates the following definition:

Definition 12. The renormalized amplitude $\mathcal{A}_{G}^{R}$ associated with the graph $G$ is deduced from the bare amplitude ${{\mathcal{A}}_{G}}$ though the Zimmermann formula (or forest formula):

$$\begin{eqnarray}&&\mathcal{A}_{G}^{R}:=\underset{\mathbb{F}\subset D(G)}{\sum}\,\underset{\gamma \in \mathbb{F}}{\prod}\,\left(-\tau _{\gamma}^{\ast}\right){{\mathcal{A}}_{G}},\end{eqnarray} \tag{ 132 }$$

where $D(G)$ is the set of Zimmerman forests.

The explicit form of the counter-term $\tau _{\mathcal{M}}^{\ast}{{\mathcal{A}}_{G,\mu}}$ is of interest for the next section. As in the previous paragraph, we start with the case N  =  2. From the definition (128), we have three terms, corresponding to the zeroth, first and second derivative with respect to t.

• Up to a change of variable, the zero derivative writes as:
  $$\begin{eqnarray}&&\tau _{\mathcal{M}}^{1\ast}{{\mathcal{A}}_{G,\mu}}=\left\{{\int}_{SU(2)}\text{d}g{{\mathcal{M}}_{j}}(g)\right\}{{\mathcal{A}}_{G/\mathcal{M},\mu}},\end{eqnarray} \tag{ 133 }$$
  where $G/\mathcal{M},\mu$ is the graph obtained from G by cutting the two lines of color 1 linked to the melon $\mathcal{M}$, and joining to one another the two half lines of color 1 linked to ${{\bar{\mathcal{A}}}_{G,\mu}}$. In the usual terminology, the term in square brackets corresponds to the mass renormalization. Note that this term includes presumably some sub-melonic contributions, as mentioned at the end of section 6.1.
• Because in the UV ${{\mathcal{M}}_{j}}$ is a symmetric function of θ (that can be proved recursively in the melonic sector from the one-loop case) $\mathcal{A}_{G,\mu}^{\prime}(0)=0$.
• Finally, the last case involve two derivative terms with respect to t. Note that:
  $$\begin{eqnarray}&&\frac{\text{d}}{\text{d}t}D_{0m}^{(l)}\left(G{{g}_{1}}\right){{|}_{t=0}}={{\mathcal{L}}_{X}}D_{0m}^{(l)}\left({{g}_{1}}\right)=\underset{i=1}{\overset{3}{\sum}}\,{{X}^{i}}{{\mathcal{L}}_{{{\tau}_{i}}}}D_{0m}^{(l)}\left({{g}_{1}}\right)\end{eqnarray} \tag{ 134 }$$
  where ${{\tau}_{i}}:=\text{i}{{\sigma}_{i}}/2$ and ${{\mathcal{L}}_{X}}$ denote the Lie derivative with respect to X. In the same way, the second derivative gives, for any test function f of ${\sum}_{i}{{\left({{X}^{i}}\right)}^{2}}$:
  $$\begin{eqnarray}&&{\int}^{}{{\text{d}}^{3}}Xf(X)\underset{i,j}{\sum}\,{{X}^{i}}{{X}^{j}}{{\mathcal{L}}_{{{\tau}_{i}}}}{{\mathcal{L}}_{{{\tau}_{j}}}}D_{0m}^{(l)}\left({{g}_{1}}\right)\propto \frac{1}{3}{\int}^{}{{\text{d}}^{3}}Xf(X){{g}_{SU(2)}}(X,X){{ \Delta }_{SU(2)}}D_{0m}^{(l)}\left({{g}_{1}}\right),\end{eqnarray} \tag{ 135 }$$
  where we have used to the fact that ${\sum}_{i}{{\left({{\mathcal{L}}_{{{\tau}_{i}}}}\right)}^{2}}={{ \Delta }_{SU(2)}}$. Then, taking into account that X is horizontal, ${{g}_{SU(2)}}(X,X)={{s}^{\ast}}{{g}_{SU(2)}}(x,x)$, and with: ${{ \Delta }_{SU(2)}}D_{0m}^{(l)}\left({{g}_{1}}\right)=4l(l+1)D_{0m}^{(l)}\left({{g}_{1}}\right)$, we deduce:
  $$\begin{eqnarray}&&\begin{array}{*{35}{l}} \tau _{\mathcal{M}}^{2\ast} {{\mathcal{A}}_{G,\mu}}\propto \left\{{\int}^{}{{\text{d}}^{2}}x{{\mathcal{M}}_{j}}(|x|)|x{{|}^{2}}\right\}\times {\int}^{}\underset{l}{\prod}\,\text{d}{{{\bar{g}}}_{1l}}\text{d}{{{\bar{g}}}_{2l}}\text{d}{{g}_{1l}}\text{d}{{g}_{2l}}{{{\bar{\mathcal{A}}}}_{G,\mu}}\left(\left\{{{{\bar{g}}}_{1l}}\right\},\left\{{{{\bar{g}}}_{2l}}\right\}\right) \\ \quad \quad \quad \quad \times \,{{C}_{{{i}_{1}}}}\left(\left\{{{g}_{1l}}\right\},\left\{{{{\bar{g}}}_{1l}}\right\}\right){{{\int}^{}}_{SU(2)}}\text{d}h\underset{\left\{{{l}_{i}},{{m}_{i}}\right\}}{\sum}\,\left(-{{l}_{1}}\left({{l}_{1}}+1\right)\right)\underset{i=1}{\overset{4}{\prod}}\,\left(2{{l}_{i}}+1\right){{\text{e}}^{-4{{M}^{-{{i}_{2}}}}{{l}_{i}}\left({{l}_{i}}+1\right)}}D_{00}^{\left({{l}_{i}}\right)}\left(g_{2i}^{-1}{{{\bar{g}}}_{2i}}h\right), \end{array}\end{eqnarray} \tag{ 136 }$$
  where the term in square brackets corresponds to the so-called wave function renormalization term, and gives the 'first deviation from locality', in the sense that the combination with the Laplacian operator does not correspond exactly to an invariant trace.

The case of the 4-point function follows the same pattern, but is simpler because only one term appears in the Taylor expansion: the zeroth derivative term. It follows that the divergent term can be written as

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \tau _{\mathcal{M}}^{1\ast}{{\mathcal{A}}_{G,\mu}}=\left\{{\int}^{}\text{d}g\text{d}{{g}^{\prime}}\mathcal{M}_{j}^{(4)}\left(g,{{g}^{\prime}}\right)\right\}\times {{\mathcal{A}}_{G/{{\mathcal{M}}^{(4)}},\mu}}. \end{array}\end{eqnarray} \tag{ 137 }$$

where $G/{{\mathcal{M}}^{(4)}}$ is the (connected) contracted graph obtained from G in the procedure detailed previously, and depicted in figure 16 below. This counter-term gives the coupling constant renormalization.

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The finiteness of the renormalized amplitude can be proved rigorously. In fact, we can prove that, when the graph contains some subdivergences, the renormalized amplitude $\mathcal{A}_{G}^{R}$ is finite, but increases dramatically as the factorial of the number of divergent forest. Proving this theorem requires to define precisely the dangerous and safe divergent forests:

Definition 13 (Dangerous and safe forests). Consider a graph G, ${{\mathcal{A}}_{G,\mu}}$ the corresponding amplitude for the scale attribution μ, and D(G) the set of divergent forests. Consider then $H\subset D(G)$. We define i_H and e_H as:

$$\begin{eqnarray}&&{{e}_{H}}=\sup \left\{{{i}_{l}}|l\in H/{{\mathbb{A}}_{D(G)}}(H)\right\}\quad {{i}_{H}}=\inf \left\{{{i}_{l}}|l\in {{L}_{H}}{\cap}^{}{{\mathbb{B}}_{{{D}_{G}}}}(H)\right\}\end{eqnarray}$$

where L_H is the set of external lines of H, ${{\mathbb{B}}_{D(G)}}$ is the ancestor of H in $D(G){\cup}^{}H$ and ${{\mathbb{A}}_{D(G)}}(H)$ the descendant, such as $\mathbb{A}_{D(G)}(H)=\cup_{h;H\supset h\in D(G)}h$. Moreover, H is said to be compatible with D(G), in the sense that, for any ${{H}^{\prime}}\subset D(G)$, ${{H}^{\prime}}{\cup}^{}H$ is still a forest.

The safe forest ${{\mathbb{F}}_{\mu}}$ is then the complementary in D(G) of the set ${{D}_{\mu}}\left({{\mathbb{F}}_{\mu}}\right)$ of dangerous or high subgraphs in G with respect to the scale assignment μ, defined as: ${{D}_{\mu}}=\left\{H\in D(G)|{{e}_{H}}>{{i}_{H}}\right\}$.

This definition allows to rewrite the renormalized amplitude as:

$$\begin{eqnarray}&&\mathcal{A}_{G}^{R}=\underset{f\in D(G)}{\sum}\,A_{G,f}^{R},\end{eqnarray} \tag{ 138 }$$

with:

$$\begin{eqnarray}&&A_{G,f}^{R}:=\underset{\mu |\,f\in {{\mathbb{F}}_{\mu}}}{\sum}\,\underset{g\in f}{\prod}\,\left(-\tau _{g}^{\ast}\right)\underset{h\in {{D}_{\mu}}(\,f)}{\prod}\,\left(1-\tau _{h}^{\ast}\right){{\mathcal{A}}_{G,\mu}},\end{eqnarray} \tag{ 139 }$$

or

$$\begin{eqnarray}&&A_{G,f}^{R}:=\underset{\mu |\,f\in {{\mathbb{F}}_{\mu}}}{\sum}\,\underset{g\in f}{\prod}\,\left(-\tau _{g}^{\ast}\right)\underset{g\in f{\cup}^{}\{G\}}{\prod}\,\underset{\begin{array}{c} h\in {{D}_{\mu}}(\,f) \\ {{\mathbb{B}}_{f}}(h)=g \end{array}}{\prod}\,\left(1-\tau _{h}^{\ast}\right){{\mathcal{A}}_{G,\mu}}.\end{eqnarray} \tag{ 140 }$$

Beginning with the contractions over the safe forest f, we obtain, after appropriate organization of the successive contractions:

$$\begin{eqnarray}&&\underset{g\in f}{\prod}\,\left(-\tau _{g}^{\ast}\right){{\mathcal{A}}_{G,\mu}}=\underset{g\in f{\cup}^{}\{G\}}{\prod}\,{{\nu}_{\mu}}\left(g/{{\mathbb{A}}_{f}}(g)\right),\end{eqnarray} \tag{ 141 }$$

where ${{\nu}_{\mu}}(g)$ is the discarded part of the amplitude. Note that all these terms are not exactly disconnected, because the contraction of the 2-point graph reveal a non-local operator, which acts on another contracted component. From the multiscale analysis, it follows that:

$$\begin{eqnarray}&&|\underset{g\in f{\cup}^{}\{G\}}{\prod}\,{{\nu}_{\mu}}\left(g/{{\mathbb{A}}_{f}}(g)\right)|\leqslant \underset{g\in f{\cup}^{}\{G\}}{\prod}\,\underset{i,\rho}{\prod}\,{{M}^{\omega \left[\left(g/{{\mathbb{A}}_{f}}(g)\right)_{i}^{\rho}\right]}}.\end{eqnarray} \tag{ 142 }$$

Now, observe that the contraction over the high divergent graphs only affects the components $g/{{\mathbb{A}}_{f}}(g)$. It follows then, from the analysis of the previous paragraph, that the decay of a renormalized graph g is at most ${{M}^{-|{{e}_{g}}-{{i}_{g}}|}}$. Hence, the renormalized amplitude is bounded by:

$$\begin{eqnarray}&&|A_{G,f}^{R}|\leqslant \underset{\mu |\,f\in {{\mathbb{F}}_{\mu}}}{\sum}\,\underset{g\in f{\cup}^{}\{G\}}{\prod}\,\underset{i,\rho}{\prod}\,{{M}^{{{\omega}^{\prime}}\left[\left(g/{{\mathbb{A}}_{f}}(g)\right)_{i}^{\rho}\right]}},\end{eqnarray} \tag{ 143 }$$

where:

$$\begin{eqnarray}&&{{\omega}^{\prime}}\left[\left(g/{{\mathbb{A}}_{f}}(g)\right)_{i}^{\rho}\right]:=\inf \left(-1,\omega \left[\left(g/{{\mathbb{A}}_{f}}(g)\right)_{i}^{\rho}\right]\right),\end{eqnarray}$$

except when $\left(g/{{\mathbb{A}}_{f}}(g)\right)_{i}^{\rho}=g/{{\mathbb{A}}_{f}}(g)$, in which case

$$\begin{eqnarray}&&{{\omega}^{\prime}}\left[\left(g/{{\mathbb{A}}_{f}}(g)\right)_{i}^{\rho}\right]=0.\end{eqnarray}$$

From the decay factor of equation (143), we can extract the factor ${{M}^{-\delta {{i}_{\text{max}}}(\mu )}}$, where ${{i}_{\text{max}}}(\mu ):=\sup (\mu )$. With the rest of the decay, we can sum over each component $g/{{\mathbb{A}}_{f}}(g)\,\,g\in f$, as in the proof of the Weinberg theorem. Because of the following bound:

$$\begin{eqnarray}&&\underset{g\in f{\cup}^{}\{G\}}{\prod}\,{{K}^{|V\left(g/{{\mathcal{A}}_{f}}(g)\right)|}}\leqslant {{K}^{\prime \,|V(G)|}},\end{eqnarray} \tag{ 144 }$$

the sum over internal scale assignments in each $g/{{\mathbb{A}}_{f}}(g)$ is bounded by ${{K}^{\prime \,|V(g)|}}$. The remaining sum over i_max is bounded by:

$$\begin{eqnarray}&&\underset{{{i}_{\text{max}}}}{\sum}\,{{\left({{i}_{\text{max}}}\right)}^{|\,f|}}{{M}^{-\delta {{i}_{\text{max}}}}}\leqslant |\,f|!{{K}^{|\,f|}},\end{eqnarray} \tag{ 145 }$$

where $|\,f|$ is the cardinality of the set f. Because the number of sub-forests in a graph G can be bounded by ${{2}^{|{{D}_{G}}|}}$, we finally deduce the following theorem:

Theorem 4 (BPH uniform). Consider a Feynman graph G of order $|V(G)|$. The renormalized amplitude $\mathcal{A}_{G}^{R}$ has the following bound:

$$\begin{eqnarray}&&|\mathcal{A}_{G}^{R}|\leqslant {{K}^{|V(G)|}}|D(G)|!,\,\,K\in {{\mathbb{R}}^{+}},\end{eqnarray} \tag{ 146 }$$

where $|D(G)|$ is the cardinality of the divergent forest set in G.

As announced, the amplitude is finite but arbitrarily large, increasing dramatically with the size of the divergent forest. This is the known problem of renormalons, which implies that the convergence of the renormalized series (132) is not guaranteed in a perturbative approach. To prove its convergence we would need the help of the constructive theory and of Borel summability technology, which is not the focus of this paper. To solve this technical difficulty, we use the effective series, defined in the next section, which is renormalons-free. This result, added to the asymptotic freedom, proved in section 6.2.3 at the one loop order, confirms the convergence of the effective series i.e. the perturbative series expressed in terms of the effective amplitudes and effective coupling.

The effective series is a more physical approach of renormalization, closely related to the Wilson approach. It is a way to solve the renormalons problem and to ensure the convergence of the perturbative series in many cases, as we will see below. The basic idea is the following. Consider a graph G and its bare amplitude ${{\mathcal{A}}_{G,\mu}}$ at scale attribution μ, as defined above. As we have seen before, in this graph, there are some divergent graphs, which form the set D(G). But in fact, only a subset of these subgraphs is potentially dangerous, the subset noted ${{D}_{\mu}}(G)$ in the previous section. The argument is that only this subset needs to be renormalized, and the effective amplitude $A_{G,\mu}^{\text{eff}}$ is defined by

$$\begin{eqnarray}&&A_{G,\mu}^{\text{eff}}:=\underset{\gamma \in {{D}_{\mu}}}{\prod}\,\left(1-\tau _{\gamma}^{\ast}\right){{\mathcal{A}}_{G,\mu}},\end{eqnarray} \tag{ 147 }$$

about which we have the following theorem [19, 35]:

Theorem 5 (Existence of the effective expansion). Consider the formal (bare) power series defined by:

$$\begin{eqnarray}&&S_{N}^{ \Lambda }=\underset{G,\mu}{\sum}\,\frac{1}{s(G)}\left(\underset{b\in \mathcal{V}(G)}{\prod}\,\left(-\lambda _{b}^{(\Lambda )}\right)\right){{A}_{G,\mu}},\end{eqnarray} \tag{ 148 }$$

where $\mathcal{V}(G)$ is the set of vertices in G including all the interactions compatible with the just-renormalizability criterion and $\lambda _{b}^{(\Lambda )}$ their coupling constants. This series can be rewritten in a more convenient form in terms of the effective amplitudes:

$$\begin{eqnarray}&&S_{N}^{ \Lambda }=\underset{G,\mu}{\sum}\,\frac{1}{s(G)}\left(\underset{b\in \mathcal{V}(G)}{\prod}\,\left(-\lambda _{b,{{e}_{b}}(G,\mu )}^{(\Lambda )}\right)\right)A_{G,\mu}^{\text{eff}},\end{eqnarray} \tag{ 149 }$$

where the $\lambda _{b,{{e}_{b}}(G,\mu )}^{(\Lambda )}$ are the effective couplings, generated by the local part of the high divergent subgraphs. They obey the following inductive relation

$$\begin{eqnarray}\begin{array}{*{35}{l}} - & \lambda _{b,i}^{(\Lambda )}=-\lambda _{b,i+1}^{(\Lambda )}+\underset{\begin{array}{c} \left(\mathcal{H},\mu,\hat{S}\right)\hat{S}\ne \varnothing \\ {{\phi}_{i}}\left(\mathcal{H},\mu,\hat{S}\right)=(b,\mu,\varnothing ) \end{array}}{\sum}\,\frac{1}{s\left(\mathcal{H}\right)}\left(\underset{{{b}^{\prime}}\in \mathcal{V}\left(\mathcal{H}\right)}{\prod}\,\left(-\lambda _{{{b}^{\prime}},i_{b}^{\prime}\left(\mathcal{H},\mu \right)}^{(\Lambda )}\right)\right)\,\times \,\left(\underset{m\in D_{\mu}^{i+1}\backslash \hat{S}}{\prod}\,\left(1-\tau _{m}^{\ast}\right)\right)\underset{M\in \hat{S}}{\prod}\,\tau _{M}^{\ast}{{A}_{\mathcal{H},\mu}}, \end{array}\end{eqnarray} \tag{ 150 }$$

with ${{e}_{b}}=\text{sup}\left\{{{\mu}_{l}},l~\text{hooked}\;\text{to}~b\right\}$.

The notation introduced above will be defined precisely in the proof, for which we give only the main steps, referring to [19, 35] for details.

Proof (Sketched). The basic idea is to introduce an intermediate step between the bare and the effective series as follows. We consider a slice i and define:

$$\begin{eqnarray}&&S_{N}^{ \Lambda }=\underset{G,\mu}{\sum}\,\frac{1}{s(G)}\left(\underset{b\in \mathcal{V}(G)}{\prod}\,\left(-\lambda _{b,\sup \left(i,{{i}_{b}}(G,\mu )\right)}^{(\Lambda )}\right)\right)A_{G,\mu}^{\text{eff},i},\end{eqnarray} \tag{ 151 }$$

where

$$\begin{eqnarray}&&A_{G,\mu}^{\text{eff},i}:=\underset{\gamma \in D_{\mu}^{i}}{\prod}\,\left(1-\tau _{\gamma}^{\ast}\right){{A}_{G,\mu}},\end{eqnarray} \tag{ 152 }$$

and

$$\begin{eqnarray}&&D_{\mu}^{i+1}(G)=\left\{m\in D(G)|{{i}_{m}}>i\right\}\quad {{i}_{m}}:=\inf \left\{{{\mu}_{l}},l~\text{hooked}~\text{to}~b\right\}.\end{eqnarray}$$

It is obvious that, if $i=\rho$, where $\Lambda ={{M}^{\rho}}$, the effective series reduces to the bare one. Assuming this is true at scale i  +  1, we can prove it at scale i by induction, by multiplying the effective amplitude at scale i  +  1 by a suitable form of the identity, adding and subtracting the counter-terms in $D_{\mu}^{i}(G)\backslash D_{\mu}^{i+1}(G)=\left\{m\in D(G)|{{i}_{m}}=i+1\right\}$, which changes $A_{G,\mu}^{eff,i+1}$ into $A_{G,\mu}^{\text{eff},i}$,

$$\begin{eqnarray}&&A_{\mu}^{\text{eff},i}(G):=\underset{\begin{array}{c} S\subseteq D_{\mu}^{i}\backslash D_{\mu}^{i+1} \\ S\ne \varnothing \end{array}}{\prod}\,\underset{M\in S}{\prod}\,\left(1-\tau _{M}^{\ast}+\tau _{M}^{\ast}\right)\underset{\gamma \in D_{\mu}^{i}}{\prod}\,\left(1-\tau _{\gamma}^{\ast}\right){{A}_{\mu}}(G).\end{eqnarray}$$

The completely subtracted piece changes $A_{G,\mu}^{eff,i+1}$ into $A_{G,\mu}^{\text{eff},i}$, and the second one is developed as a sum over S as follows:

$$\begin{eqnarray}&&S_{N}^{ \Lambda }=\underset{\begin{array}{c} (G,\mu,S) \\ S\subseteq D_{\mu}^{i}\backslash D_{\mu}^{i+1} \end{array}}{\sum}\,\frac{1}{s(G)}\left(\underset{b\in \mathcal{V}(G)}{\prod}\,\left(-\lambda _{b,\sup \left(i+1,{{i}_{b}}(G,\mu )\right)}^{(\Lambda )}\right)\right)A_{G,\mu,S}^{\text{eff},i},\end{eqnarray}$$

with

$$\begin{eqnarray}&&A_{\mu,S}^{\text{eff},i}:=\underset{M\in S}{\prod}\,\left(-\tau _{M}^{\ast}\right)\underset{m\in {{D}^{i}}\backslash S}{\prod}\,\left(1-\tau _{m}^{\ast}\right){{A}_{G,\mu}},\end{eqnarray}$$

and in particular $A_{\mu,\varnothing}^{\text{eff},i}=A_{\mu}^{\text{eff},i}$. A subtlety appears in this case because the 2-point divergent graphs (with degree $\omega =2$) introduce two counter-terms, one for the mass and one for the wave-function. For this reason we modify the previous definition of S, and introduce the new definition:

$$\begin{eqnarray}&&\hat{S}=\left\{\left(M,{{k}_{M}}\right)|M\in S,{{k}_{M}}\in 0,2,{{k}_{M}}\leqslant \omega (M)\right\}.\end{eqnarray}$$

Secondly, we introduce the collapse ${{\phi}_{i}}$ which sends the triplets $\left(G,\mu,\hat{S}\right)$ to its contracted version $\left({{\mathcal{G}}^{\prime}},{{\mu}^{\prime}},\varnothing \right)$, such that the previous sum can be rewritten as a sum on ${{\mathcal{G}}^{\prime}}$

$$\begin{eqnarray}&&S_{N}^{ \Lambda }=\underset{{{\mathcal{G}}^{\prime}},{{\mu}^{\prime}}}{\sum}\,\underset{\begin{array}{c} \left\{(G,\mu,S)\right\}= \\ \phi _{i}^{-1}\left({{\mathcal{G}}^{\prime}},{{\mu}^{\prime}},\varnothing \right) \end{array}}{\sum}\,\frac{A_{G,\mu,S}^{\text{eff},i}}{s(G)}\left(\underset{b\in \mathcal{V}(G)}{\prod}\,\left(-\lambda _{b,\sup \left(i+1,{{i}_{b}}(G,\mu )\right)}^{(\Lambda )}\right)\right).\end{eqnarray} \tag{ 153 }$$

Decomposing

$$\begin{eqnarray}&&\underset{M\in \hat{S}}{\prod}\,\left(-\tau _{M}^{\ast}\right)=\underset{{{b}^{\prime}}\in \mathcal{V}(G)}{\prod}\,\left(\underset{M\in \hat{S},\,M\subset \phi _{i}^{-1}\left({{b}^{\prime}}\right)}{\prod}\,\left(-\tau _{M}^{\ast}\right)\right)\end{eqnarray}$$

in the sum (153), we find that it gives exactly the effective sum at scale i given by (151), if the coupling satisfies the recursive relation of the theorem.

The coupling recursion defines a discrete flow, for which the initial data are, as usual in standard quantum field theory, imposed by the 1PI functions at zero momenta.

The main interest of the effective series is that all these amplitudes are bounded in the form [19]

$$\begin{eqnarray}&&|A_{G}^{\text{eff}}|\leqslant {{K}^{V(G)}},\end{eqnarray} \tag{ 154 }$$

a result which can be directly deduced from the theorem 4 proved in the previous section, in the special case where the set of inoffensive forest is empty. Remarkably, in the previous bound, renormalons do not appear.

Another important fact about the effective series and effective coupling constants is their relationship with the renormalized series. In fact, if we define the renormalized coupling by ${{\lambda}_{r}}:={{\lambda}_{-1}}$, and if we reframe the effective series in terms of the renormalized coupling, we find exactly the renormalized series.