The dimensional regularization and renormalization of the vulcanized \(\Phi^{*4}\) model on the Moyal space (Gross-Wulkenhaar scalar \(\Phi^{*4}\) model modified the kinetic part of the action in order to satisfy the Langmann-Szabo duality; H. Grosse and R. Wulkenhaar [Commun. Math. Phys. 254, No. 1, 91–127 (2005; Zbl 1079.81049), Lett. Math. Phys. 71, No. 1, 13–26 (2005; Zbl 1115.81055)], E. Langmann and R. J. Szabo [Phys. Lett., B 533, No. 1–2, 168–177 (2002; Zbl 0994.81116)]) are implemented. The action of a Grosse-Wilkenhaar model is
\[ {\mathcal S}= \int d^4 x\Biggl({1\over 2} \partial_\mu\widetilde\phi* \partial^\mu\phi+ {\Omega^2\over 2}(\widetilde x_\mu\overline\phi)* (\widetilde x^\mu\phi)+ \overline\phi* \phi*\overline\phi* \phi\Biggr). \]
Here \(x^{\mu}\), \(\mu= 1,\dots,D\) are noncommutative variables;
\[ [x^\mu, x^\nu]= i\Theta^{\mu}\nu,\quad \Theta= \left( \begin{matrix} 0 & \theta\\ -\theta & 0 \end{matrix} \right) \oplus\cdots\oplus \left( \begin{matrix} 0 & \theta\\ -\theta & 0\end{matrix} \right) , \]
\(f\star g\) is the (deformed) Moyal product
\[ (f\star g)(x)= \frac{1}{ \pi^D\text{det}\Theta} \int d^D yd^D zf(x+ y) g(x+ z) e^{2iy\Theta^{-1}z}, \]
\(\widetilde{x}_{\mu} = 2(\Theta^{-1})_{\mu\nu} x^\nu\) and the propagator of this model is the inverse of \(-\Delta+ \Omega^2\widetilde{x}^2\). The kernel of the propagator is
\[ C(x,y) = \int^\infty_0 \frac{\widetilde{\Omega} d\alpha}{ [2\pi\sinh(\alpha)]^{D/2}}e^{-\frac{\widetilde{\Omega}}{4} (\coth(\frac{\alpha}{2})(x- y)^2+ \tanh(\frac{\alpha}{2}(x+ y)^2)}, \]
where \(\widetilde{\Omega}= 2\Omega/\theta\).
To study this model, first the Filk move of a graph [cf. T. Filk, Divergence in a field theory on quantum space, Phys. Lett., B 376, 53–58 (1996)] is explained. Iterating this operation for the \(n- 1\) tree lines, a rosette; a single final vertex with all the loop lines hooked to it, is obtained. This process corresponds to the shrinking of a subgraph to a point (§2.1). This process is applied to the graph \(G\) representing \({\mathcal S}\). (§2.2). The leading terms (terms with the smallest global degree in the \(t\) variables) are precisely computed in §3. In §4. Under the rescaling \(t_\alpha\to \rho^2 t_\alpha\) of the parameter-corresponding to a divergent subgraph \(S\) of any Feynman graph \(G\), the factorization is worked out (Th. 4.1)and used to analyse the two point function (§4, (4.28)).
Then, following the approach of M. C. Bergère and F. David [Integral representation for the dimensionally regularized massive Feynman amplitude, J. Math. Phys. 30, 1244 (1979)], a function \({\mathcal A}_{G,\overline{v}}\) of \(D\) originally given on \(2\leq \operatorname{Re}D\leq 4\), \({\mathcal A}_{G,\overline V}\) is shown to be meromorphic on \(D^\sigma= \{D\mid 0< \operatorname{Re}D< 4+ \varepsilon_G\},\) where \(\varepsilon_G> 0\) depends on the graph \(G\) (§5.1). Renormalizes amplitude is obtained and shown to be analytic \(D1{\sigma}\) (Th. 5.1) in §5.2.
In §6, the last section, the authors say the factorization results (§4) are the starting point for the implementation of a Hopf algebra structure for NCQFT (the authors refer progressing work of A. Tanas, F. Vignes-Tourneret, Hopf algebra for non-commutative quantum field theory). Then claim NCQFT is a strong candiate for new physics beyond the standrd model, because of the absence of Landau ghost, and the Langmann-Szabo symmetry responsible for supressing the ghost could play a role similar to supersymmetry in taming UV divergence.