The lattice Sommerfield model, describing a massive vector gauge field coupled to a light fermion in 2d, is an ideal candidate to verify perturbative conclusions. In contrast with continuum exact solutions, we prove that there is no infinite field renormalization, implying the reduction of the degree of the ultraviolet divergence, and that anomalies are non renormalized. Such features are the counterpart of analogue properties at the basis of the Standard model perturbative renormalizability. The results are non-perturbative, in the sense that the averages of gauge invariant observables are expressed in terms of convergent expansions uniformly in the lattice and volume.
Nonperturbative Chiral Anomaly Cancellation and Irrelevant Operators
Università degli Studi Roma Tre, Mathematics and Physics Department（罗马第三大学数学与物理系）；Università degli Studi di Roma “La Sapienza”, Physics Department（罗马第一大学物理系）；SISSA, Mathematics Area（的里雅斯特国际高等研究学校数学区）
We present a method of short-distance analysis in quantum field theory that does not require choosing a renormalization prescription a priori. We set out from a local net of algebras with associated pointlike quantum fields. The net has a naturally defined scaling limit in the sense of Buchholz and Verch; we investigate the effect of this limit on the pointlike fields. Both for the fields and their operator product expansions, a well-defined limit procedure can be established. This can always be interpreted in the usual sense of multiplicative renormalization, where the renormalization factors…
Within the realm of contact potentials, the key structures intrinsic of nonperturbative renormalization of $T$-matrices are unraveled using rigorous solutions and an inverse form of algebraic Lippmann-schwinger equation. The intrinsic mismatches between effective field theory power counting and nonperturbative divergence structures are shown for the first time to preclude the conventional counterterm algorithm from working in the renormalization of EFT for $NN$ scattering in nonperturbative regimes.
We investigate the influence of the momentum cutoff function on the field-dependent nonperturbative renormalization group flows for the three-dimensional Ising model, up to the second order of the derivative expansion. We show that, even when dealing with the full functional dependence of the renormalization functions, the accuracy of the critical exponents can be simply optimized, through the principle of minimal sensitivity, which yields $ν= 0.628$ and $η= 0.044$.
Within the covariant formulation of Light-Front Dynamics, in a scalar model with the interaction Hamiltonian $H=-gψ^{2}(x)ϕ(x)$, we calculate nonperturbatively the renormalized state vector of a scalar "nucleon" in a truncated Fock space containing the $N$, $Nπ$ and $Nππ$ sectors. The model gives a simple example of non-perturbative renormalization which is carried out numerically. Though the mass renormalization $δm^2$ diverges logarithmically with the cutoff $L$, the Fock components of the "physical" nucleon are stable when $L\to\infty$.
On the example of the three-dimensional Ising model, we show that nonperturbative renormalization group equations allow one to obtain very accurate critical exponents. Implementing the order $\partial^4$ of the derivative expansion leads to $ν=0.632$ and to an anomalous dimension $η=0.033$ which is significantly improved compared with lower orders calculations.
We present a universal form of the $T$-matrices renormalized in nonperturbative regime and the ensuing notions and properties that fail conventional wisdoms. A universal scale is identified and shown to be renormalization group invariant. The effective range parameters are derived in a nonperturbative scenario with some new predictions within the realm of contact potentials. Some controversies are shown to be due to the failure of conventional wisdoms.
We analyze the dissipative quantum tunneling in the Caldeira-Leggett model by the nonperturbative renormalization-group method. We classify the dissipation effects by introducing the notion of effective cutoffs. We calculate the localization susceptibility to evaluate the critical dissipation for the quantum-classical transition. Our results are consistent with previous semiclassical arguments, but give considerably larger critical dissipation.
In this work, we study the nonperturbative renormalization of the supercurrent operator in $\mathcal{N} = 1$ Supersymmetric Yang-Mills (SYM) theory, using a gauge-invariant renormalization scheme (GIRS). The proposed prescription addresses successfully the unwanted mixing of the supercurrent with other operators of equal or lower dimension, which respect the same global symmetries. This mixing is introduced by the unavoidable breaking of supersymmetry on the lattice. In GIRS all gauge-noninvariant operators, which mix with the supercurrent, are excluded from the renormalization procedure. The …
The behaviour of a d-dimensional vectorial N=3 model at a m-axial Lifshitz critical point is investigated by means of a nonperturbative renormalization group approach that is free of the huge technical difficulties that plague the perturbative approaches and limit their computations to the lowest orders. In particular being systematically improvable, our approach allows us to control the convergence of successive approximations and thus to get reliable physical quantities in d=3.
An elementary introduction to the non-perturbative renormalization group is presented mainly in the context of statistical mechanics. No prior knowledge of field theory is necessary. The aim is this article is not to give an extensive overview of the subject but rather to insist on conceptual aspects and to explain in detail the main technical steps. It should be taken as an introduction to more advanced readings.
The Kosterlitz-Thouless phase transition is described by the nonperturbative renormalization flow of the two dimensional $ϕ^4$-model. The observation of essential scaling demonstrates that the flow equation incorporates nonperturbative effects which have previously found an alternative description in terms of vortices. The duality between the linear and nonlinear $σ$-model gives a unified description of the long distance behaviour for O(N)-models in arbitrary dimension $d$. We compute critical exponents in first order in the derivative expansion.
As a unified theory of integer and fractional quantum Hall plateau transitions, a nonperturbative theory of the two-parameter scaling renormalization group function is presented. By imposing global symmetries known as ``the law of corresponding states'', we seek a possible form of renormalization group flows. Asking for consistency with the result from weak-localization perturbation theory, such restriction is so intense that we can analytically determine its concrete form. Accordingly, the critical exponent $ν$ and the irrelevant scaling index $y$ are obtained analytically and turn out to be …
We investigate the massive Schwinger model in $d=1+1$ dimensions using bosonization and the nonperturbative functional renormalization group. In agreement with previous studies we find that the phase transition, driven by a change of the ratio $m/e$ between the mass and the charge of the fermions, belongs to the two-dimensional Ising universality class. The temperature and vacuum angle dependence of various physical quantities (chiral density, electric field, entropy density) are also determined and agree with results obtained from density matrix renormalization group studies. Screening of fra…
Using the nonperturbative renormalization group, we study the existence of bound states in the symmetry-broken phase of the scalar $ϕ^4$ theory in all dimensions between two and four and as a function of the temperature. The accurate description of the momentum dependence of the two-point function, required to get the spectrum of the theory, is provided by means of the Blaizot--Méndez-Galain--Wschebor approximation scheme. We confirm the existence of a bound state in dimension three, with a mass within 1% of previous Monte-Carlo and numerical diagonalization values.
Using the nonperturbative functional renormalization group (FRG) within the Blaizot-Méndez-Galain-Wschebor approximation, we compute the operator product expansion (OPE) coefficient $c_{112}$ associated with the operators $\mathcal{O}_1\simφ$ and $\mathcal{O}_2\simφ^2$ in the three-dimensional $\mathrm{O}(N)$ universality class and in the Ising universality class ($N=1$) in dimensions $2 \leq d \leq 4$. When available, exact results and estimates from the conformal bootstrap and Monte-Carlo simulations compare extremely well to our results, while FRG is able to provide values across the whole …
We derive the necessary conditions for implementing a regulator that depends on both momentum and frequency in the nonperturbative renormalization group flow equations of out-of-equilibrium statistical systems. We consider model A as a benchmark and compute its dynamical critical exponent $z$. This allows us to show that frequency regulators compatible with causality and the fluctuation-dissipation theorem can be devised. We show that when the Principle of Minimal Sensitivity (PMS) is employed to optimize the critical exponents $η$, $ν$ and $z$, the use of frequency regulators becomes necessar…
Starting from the Schwinger--Dyson equation and the renormalization group equation for the massless Wess--Zumino model, we compute the dominant nonperturbative contributions to the anomalous dimension of the theory, which are related by alien calculus to singularities of the Borel transform on integer points. The sum of these dominant contributions has an analytic expression. When applied to the two-point function, this analysis gives a tame evolution in the deep euclidean domain at this approximation level, making doubtful the arguments on the triviality of the quantum field theory with posit…
We study the optimization of nonperturbative renormalization group equations truncated both in fields and derivatives. On the example of the Ising model in three dimensions, we show that the Principle of Minimal Sensitivity can be unambiguously implemented at order $\partial^2$ of the derivative expansion. This approach allows us to select optimized cut-off functions and to improve the accuracy of the critical exponents $ν$ and $η$. The convergence of the field expansion is also analyzed. We show in particular that its optimization does not coincide with optimization of the accuracy of the cri…
We study the quantum sine-Gordon model within a nonperturbative functional renormalization-group approach (FRG). This approach is benchmarked by comparing our findings for the soliton and lightest breather (soliton-antisoliton bound state) masses to exact results. We then examine the validity of the Lukyanov-Zamolodchikov conjecture for the expectation value $\langle e^{\frac{i}{2}nβφ}\rangle$ of the exponential fields in the massive phase ($n$ is integer and $2π/β$ denotes the periodicity of the potential in the sine-Gordon model). We find that the minimum of the relative and absolute disagre…
Field-theoretical calculations performed in an approximation scheme often present a spurious dependence of physical quantities on some unphysical parameters associated with the details of the calculation setup (such as, the renormalization scheme or, in perturbation theory, the resummation procedure). In the present article, we propose to reduce this dependence by invoking conformal invariance. Using as a benchmark the three-dimensional Ising model, we show that, within the derivative expansion at order 4, performed in the nonperturbative renormalization group formalism, the identity associate…
Pionful nuclear effective field theory (NEFT) in the two-nucleon sector is examined from the Wilsonian renormalization group point of view. The pion exchange is cut off at the floating cutoff scale, $Λ$, with the short-distance part being represented as contact interactions in accordance with the general principle of renormalization. We derive the nonperturbative renormalization group equations in the leading order of the nonrelativistic approximation in the operator space including up to $\mathcal{O}(p^2)$ and find the nontrivial fixed points in the $^1S_0$ and $^3S_1$--$^3D_1$ channels which…
We consider a functional relation between a given Wilsonian RG flow, which has to be related to a specific coarse-graining procedure, and an infinite family of (UV cutoff) scale dependent field redefinitions. Within this framework one can define a family of Wilsonian proper-time exact RG equations associated to an arbitrary regulator function. New applications of these RG flow schemes to the Ising Universality class in three dimensions in the derivative expansion are shortly illustrated.
We propose a continuous real space renormalization group transformation based on gradient flow, allowing for a numerical study of renormalization without the need for costly ensemble matching. We apply our technique in a pilot study of SU$(3)$ gauge theory with $N_f = 12$ fermions in the fundamental representation, finding the mass anomalous dimension to be $γ_m = 0.23(6)$, consistent with other perturbative and lattice estimates. We also present the first lattice calculation of the nucleon anomalous dimension in this theory, finding $γ_N = 0.05(5)$.
Anomalous quark triangles with one axial and two vector currents are studied in special kinematics when one of the vector currents carries a soft momentum. According to the Adler-Bardeen theorem the anomalous longitudinal part of the triangle is not renormalized in the chiral limit. We derive a new nonrenormalization theorem for the transversal part of the triangle. This nonrenormalization, in difference with the longitudinal part, holds on only perturbatively. At the nonperturbative level we use the operator product expansion and the pion dominance in the longitudinal part to determine the ma…
The three-dimensional real scalar model, in which the $Z_2$ symmetry spontaneously breaks, is renormalized in a nonperturbative manner based on the Tamm-Dancoff truncation of the Fock space. A critical line is calculated by diagonalizing the Hamiltonian regularized with basis functions. The marginal ($ϕ^6$) coupling dependence of the critical line is weak. In the broken phase the canonical Hamiltonian is tachyonic, so the field is shifted as $ϕ(x)\toφ(x)+v$. The shifted value $v$ is determined as a function of running mass and coupling so that the mass of the ground state vanishes.
We obtain analytic, nonperturbative, approximate solutions of Yukawa theory in the one-fermion sector using light-front quantization. The theory is regulated in the ultraviolet by the introduction of heavy Pauli-Villars scalar and fermion fields, each with negative norm. In order to obtain a directly soluble problem, fermion-pair creation and annihilation are neglected, and the number of bosonic constituents is limited to one of either type. We discuss some of the features of the wave function of the eigensolution, including its endpoint behavior and spin and orbital angular momentum content. …
We investigate scalar field theories in de Sitter space by means of nonperturbative renormalization group techniques. We compute the functional flow equation for the effective potential of O(N) theories in the local potential approximation and we study the onset of curvature-induced effects as quantum fluctuations are progressively integrated out from subhorizon to superhorizon scales. This results in a dimensional reduction of the original action to an effective zero-dimensional Euclidean theory. We show that the latter is equivalent both to the late-time equilibrium state of the stochastic a…
Non-perturbative renormalizability, or non-triviality, of the gauged Nambu-Jona-Lasinio (NJL) model in four dimensions is examined by using non-perturbative (exact) renormalization group in large $N_c$ limit. When running of the gauge coupling is asymptotically free and slow enough, the two dimensional renormalized trajectory (subspace) spanned by the four fermi coupling and the gauge coupling is found to exist, which implies renormalizability of the gauged NJL model. In the case of fixed gauge coupling, renormalizability of the model turns out to be guaranteed by the line of the UV fixed poin…
We study the $O(2)$ model with $\mathbb{Z}_4$-symmetric perturbations within the framework of nonperturbative renormalization group (RG) for spatial dimensionality $d=2$ and $d=3$. In a unified framework we resolve the relatively complex crossover behavior emergent due to the presence of multiple RG fixed points. In $d=3$ the system is controlled by the $XY$, Ising, and low-$T$ fixed points in presence of a dangerously irrelevant anisotropy coupling $λ$. In $d=2$ the anisotropy coupling is marginal and the physical picture is governed by the interplay between two distinct lines of RG fixed poi…
The renormalization group plays an essential role in many areas of physics, both conceptually and as a practical tool to determine the long-distance low-energy properties of many systems on the one hand and on the other hand search for viable ultraviolet completions in fundamental physics. It provides us with a natural framework to study theoretical models where degrees of freedom are correlated over long distances and that may exhibit very distinct behavior on different energy scales. The nonperturbative functional renormalization-group (FRG) approach is a modern implementation of Wilson's RG…
The renormalization of EFT for nucleon-nucleon scattering in nonperturbative regimes is investigated in a compact parametrization of the $T$-matrix. The key difference between perturbative and nonperturbative renormalization is clarified. The underlying theory perspective and the 'fixing' of the prescriptions for the $T$-matrix from physical boundary conditions are stressed.
We reexamine the two-dimensional linear O(2) model ($φ^4$ theory) in the framework of the nonperturbative renormalization-group. From the flow equations obtained in the derivative expansion to second order and with optimization of the infrared regulator, we find a transition between a high-temperature (disordered) phase and a low-temperature phase displaying a line of fixed points and algebraic order. We obtain a picture in agreement with the standard theory of the Kosterlitz-Thouless (KT) transition and reproduce the universal features of the transition. In particular, we find the anomalous d…
We present an approximation scheme of the nonperturbative renormalization group that preserves the momentum dependence of correlation functions. This approximation scheme can be seen as a simple improvement of the local potential approximation (LPA) where the derivative terms in the effective action are promoted to arbitrary momentum-dependent functions. As in the LPA the only field dependence comes from the effective potential, which allows us to solve the renormalization-group equations at a relatively modest numerical cost (as compared, e.g., to the Blaizot--Mendéz-Galain--Wschebor approxim…
We compute the critical exponents $ν$, $η$ and $ω$ of $O(N)$ models for various values of $N$ by implementing the derivative expansion of the nonperturbative renormalization group up to next-to-next-to-leading order [usually denoted $\mathcal{O}(\partial^4)$]. We analyze the behavior of this approximation scheme at successive orders and observe an apparent convergence with a small parameter -- typically between $1/9$ and $1/4$ -- compatible with previous studies in the Ising case. This allows us to give well-grounded error bars. We obtain a determination of critical exponents with a precision …
We set up a nonperturbative gravitational coarse graining flow and the corresponding functional renormalization group equation on the as to yet unexplored "tetrad only" theory space. It comprises action functionals which depend on the tetrad field (along with the related background and ghost fields) and are invariant under the semi-direct product of spacetime diffeomorphisms and local Lorentz transformations. This theory space differs from that of Quantum Einstein Gravity (QEG) in that the tetrad rather than the metric constitutes the fundamental variable and because of the additional symmetry…
The Light-Front Tamm-Dancoff method of finding the nonperturbative solutions in field theory is based on the Fock decomposition of the state vector, complemented with the sector-dependent nonperturbative renormalization scheme. We show in detail how to implement the renormalization procedure and to solve the simplest nontrivial example of the scalar Yukawa model in the two- and three-body Fock space truncations incorporating scalar "nucleon" and one or two scalar "pions".
We investigate the regime of fully developed homogeneous and isotropic turbulence of the Navier-Stokes (NS) equation in the presence of a stochastic forcing, using the nonperturbative (functional) renormalization group (NPRG). Within a simple approximation based on symmetries, we obtain the fixed point solution of the NPRG flow equations that corresponds to fully developed turbulence both in $d=2$ and $d=3$ dimensions. Deviations to the dimensional scalings (Kolmogorov in $d=3$ or Kraichnan-Batchelor in $d=2$) are found for the two-point functions. To further analyze these deviations, we deriv…
Bosonization allows one to describe the low-energy physics of one-dimensional quantum fluids within a bosonic effective field theory formulated in terms of two fields: the "density" field $φ$ and its conjugate partner, the phase $\vartheta$ of the superfluid order parameter. We discuss the implementation of the nonperturbative functional renormalization group in this formalism, considering a Luttinger liquid in a periodic potential as an example. We show that in order for $\vartheta$ and $φ$ to remain conjugate variables at all energy scales, one must dynamically redefine the field $\vartheta$…
We study the "Higgs" amplitude mode in the relativistic quantum O($N$) model in two space dimensions. Using the nonperturbative renormalization group and the Blaizot--Méndez-Galain--Wschebor approximation (which we generalize to compute 4-point correlation functions), we compute the O($N$) invariant scalar susceptibility at zero temperature in the vicinity of the quantum critical point. In the ordered phase, we find a well-defined Higgs resonance for $N=2$ and $N=3$ and determine its universal properties. No resonance is found for $N\geq 4$. In the disordered phase, the spectral function exhib…
Nonperturbative renormalization group techniques have recently proven a powerful tool to tackle the nontrivial infrared dynamics of light scalar fields in de Sitter space. In the present article, we develop the formalism beyond the local potential approximation employed in earlier works. In particular, we consider the derivative expansion, a systematic expansion in powers of field derivatives, appropriate for long wavelength modes, that we generalize to the relevant case of a curved metric with Lorentzian signature. The method is illustrated with a detailed discussion of the so-called local po…
Nonperturbative approximation schemes based on two-particle irreducible (2PI) effective actions provide an important means for our current understanding of (non-)equilibrium quantum field theory. A remarkable property is their renormalizability, since these approximations involve selective summations to infinite perturbative orders. In this paper we show how to renormalize all $n$-point functions of the theory, which are given by derivatives of the 2PI-resummed effective action $Γ[ϕ]$ for scalar fields $ϕ$. This provides a complete description in terms of the generating functional for renormal…
We consider the O(N)-symmetric phi4 theory in two and three dimensions and determine the nonperturbative mass renormalization needed to obtain the phi4 continuum theory. The required nonperturbative information is obtained by resumming high-order perturbative series in the massive renormalization scheme, taking into account their Borel summability and the known large-order behavior of the coefficients. The results are in good agreement with those obtained in lattice calculations.
We investigate the scaling regimes of the Kardar-Parisi-Zhang equation in the presence of spatially correlated noise with power law decay $D(p) \sim p^{-2ρ}$ in Fourier space, using a nonperturbative renormalization group approach. We determine the full phase diagram of the system as a function of $ρ$ and the dimension $d$. In addition to the weak-coupling part of the diagram, which agrees with the results from Refs. [Europhys. Lett. 47, 14 (1999), Eur. Phys. J. B 9, 491 (1999)], we find the two fixed points describing the short-range (SR) and long-range (LR) dominated strong-coupling phases. …
Massachusetts Institute of Technology（麻省理工学院）；NSF AI Institute for Artificial Intelligence and Fundamental Interactions（美国国家科学基金会人工智能与基础相互作用研究所）；Université Paris Saclay（巴黎萨克雷大学）；CEA（法国原子能和替代能源委员会）；LIST（法国原子能和替代能源委员会实验室集成系统与技术研究所）；Université d’Abomey-Calavi（阿波美-卡拉维大学）
In a recent work arXiv:2008.08601, Halverson, Maiti and Stoner proposed a description of neural networks in terms of a Wilsonian effective field theory. The infinite-width limit is mapped to a free field theory, while finite $N$ corrections are taken into account by interactions (non-Gaussian terms in the action). In this paper, we study two related aspects of this correspondence. First, we comment on the concepts of locality and power-counting in this context. Indeed, these usual space-time notions may not hold for neural networks (since inputs can be arbitrary), however, the renormalization …
This manuscript aims at giving our new advance on the functional renormalization group applied to tensorial group field theory. It is based on a series of our three papers [arXiv:1803.09902], [arXiv:1809.00247] and [arXiv:1809.06081]. We consider the polynomial Abelian $U(1)^d$ models without closure constraint, especially we discuss the case of quartic melonic interaction. We present a new approach to solve the exact Wetterich flow equation called the effective vertex expansion method, and investigate the resulting flow equations, especially about the existence of non-Gaussian fixed points fo…
We present a nonperturbative renormalization-group approach to the Bose-Hubbard model. By taking as initial condition of the renormalization-group flow the (local) limit of decoupled sites, we take into account both local and long-distance fluctuations in a nontrivial way. This approach yields a phase diagram in very good quantitative agreement with quantum Monte Carlo simulations, and reproduces the two universality classes of the superfluid--Mott-insulator transition. The critical behavior near the multicritical points, where the transition takes place at constant density, agrees with the or…
We investigate the strong-coupling regime of the stationary Kardar-Parisi-Zhang equation for interfaces growing on a substrate of dimension d=1, 2, and 3 using a nonperturbative renormalization group (NPRG) approach. We compute critical exponents, correlation and response functions, extract the related scaling functions and calculate universal amplitude ratios. We work with a simplified implementation of the second-order (in the response field) approximation proposed in a previous work [PRE 84, 061128 (2011) and Erratum 86, 019904 (2012)], which greatly simplifies the frequency sector of the N…
This paper, in addition to our recent works, intends to explore the behavior of the Wetterich flow equations in the portion of the theory space spanned by non-branching melons constrained with Ward-identities. We focus on a rank-5 just-renormalizable tensorial group field theory and consider a non-trivial extension of the local potential approximation namely effective vertex expansion for just-renormalizable quartic melonic interactions, disregarding effects coming from disconnected interactions. Investigating the dynamically constrained flow, we show explicitly that results weakly rely on the…
In this paper, we study three dimensional NL$σ$Ms within two kind of nonperturbative methods; WRG and large-N expansion. First, we investigate the renormalizability of some NL$σ$Ms using WRG equation. We find that some models have a nontrivial UV fixed point and are renormalizable within nonperturbative method. Second, we study the phase structure of $CP^{N-1}$ and $Q^{N-2}$ models using large-N expansion. These two models have two and three phases respectively. At last, we construct the conformal field theories at the fixed point of the nonperturbative WRG $β$ function. This is the review of …
This article is devoted to the study of the critical properties of classical XY and Heisenberg frustrated magnets in three dimensions. We first analyze the experimental and numerical situations. We show that the unusual behaviors encountered in these systems, typically nonuniversal scaling, are hardly compatible with the hypothesis of a second order phase transition. We then review the various perturbative and early nonperturbative approaches used to investigate these systems. We argue that none of them provides a completely satisfactory description of the three-dimensional critical behavior. …
We discuss questions related to renormalization group and to nonperturbative aspects of non-Abelian gauge theories with N=2 and/or N=1 supersymmetry. Results on perturbative and nonperturbative $β$ functions of these theories are reviewed, and new mechanisms of confinement and dynamical symmetry breaking recently found in a class of $SU(n_c)$, $USp(2n_c)$ and $SO(n_c)$ theories are discussed.
We project the Wilson/Polchinski renormalization group equation onto its uniform external field dependent effective free energy and connected Green's functions. The result is a hierarchy of equations which admits a choice of "natural" truncation and closure schemes for nonperturbative approximate solution. In this way approximation schemes can be generated which avoid power series expansions in either fields or momenta. When following one closure scheme the lowest order equation is the mean field approximation, while another closure scheme gives the "local potential approximation." Extension o…
We study renormalization in a scalar field theory on the fuzzy sphere. The theory is realized by a matrix model, where the matrix size plays the role of a UV cutoff. We define correlation functions by using the Berezin symbol identified with a field and calculate them nonperturbatively by Monte Carlo simulation. We find that the 2-point and 4-point functions are made independent of the matrix size by tuning a parameter and performing a wave function renormalization. The results strongly suggest that the theory is nonperturbatively renormalizable in the ordinary sense.
