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Power-counting theorem for non-local matrix models and renormalisation

Harald Grosse, Raimar Wulkenhaar

TL;DR

This work establishes a rigorous power-counting theorem for non-local matrix models by solving the Wilson–Polchinski exact renormalisation group equation in perturbation theory. The central result expresses the divergence degree of ribbon-graph amplitudes in terms of two scaling exponents $\delta_0,\delta_1$ of the cut-off propagator and topological data (genus $g$, boundary components $B$, segmentation index $\iota$) with an additional exponent $\delta_2$ encoding dimensional factors. The authors show that renormalisability requires a finite set of base couplings, achievable only when suitable locality properties are present, and they illustrate this with the noncommutative $\mathbb{R}^D$ matrix models, where a harmonic oscillator potential can render the theory renormalisable (e.g., $\delta_0=\delta_1=2$). The theorem provides a practical criterion to assess renormalisability of general non-local matrix models and clarifies how topology suppresses non-planar divergences. Overall, the framework clarifies how UV/IR-like issues arise from propagator non-locality and guides the construction of renormalisable noncommutative field theories and related matrix-model realizations.

Abstract

Solving the exact renormalisation group equation a la Wilson-Polchinski perturbatively, we derive a power-counting theorem for general matrix models with arbitrarily non-local propagators. The power-counting degree is determined by two scaling dimensions of the cut-off propagator and various topological data of ribbon graphs. As a necessary condition for the renormalisability of a model, the two scaling dimensions have to be large enough relative to the dimension of the underlying space. In order to have a renormalisable model one needs additional locality properties--typically arising from orthogonal polynomials--which relate the relevant and marginal interaction coefficients to a finite number of base couplings. The main application of our power-counting theorem is the renormalisation of field theories on noncommutative R^D in matrix formulation.

Power-counting theorem for non-local matrix models and renormalisation

TL;DR

This work establishes a rigorous power-counting theorem for non-local matrix models by solving the Wilson–Polchinski exact renormalisation group equation in perturbation theory. The central result expresses the divergence degree of ribbon-graph amplitudes in terms of two scaling exponents of the cut-off propagator and topological data (genus , boundary components , segmentation index ) with an additional exponent encoding dimensional factors. The authors show that renormalisability requires a finite set of base couplings, achievable only when suitable locality properties are present, and they illustrate this with the noncommutative matrix models, where a harmonic oscillator potential can render the theory renormalisable (e.g., ). The theorem provides a practical criterion to assess renormalisability of general non-local matrix models and clarifies how topology suppresses non-planar divergences. Overall, the framework clarifies how UV/IR-like issues arise from propagator non-locality and guides the construction of renormalisable noncommutative field theories and related matrix-model realizations.

Abstract

Solving the exact renormalisation group equation a la Wilson-Polchinski perturbatively, we derive a power-counting theorem for general matrix models with arbitrarily non-local propagators. The power-counting degree is determined by two scaling dimensions of the cut-off propagator and various topological data of ribbon graphs. As a necessary condition for the renormalisability of a model, the two scaling dimensions have to be large enough relative to the dimension of the underlying space. In order to have a renormalisable model one needs additional locality properties--typically arising from orthogonal polynomials--which relate the relevant and marginal interaction coefficients to a finite number of base couplings. The main application of our power-counting theorem is the renormalisation of field theories on noncommutative R^D in matrix formulation.

Paper Structure

This paper contains 9 sections, 3 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. The exact renormalisation group equation
  3. Ribbon graphs and their topologies
  4. Formulation of the power-counting theorem
  5. Proof of the power-counting theorem
  6. Tree-contractions of two subgraphs
  7. Loop-contractions at the same vertex
  8. Loop-contractions at different vertices
  9. Discussion

Key Result

Theorem 6

The homogeneous parts $A^{(V,V^e,B,g,\iota)}_{ m_1n_1;\dots;m_Nn_N}[\Lambda]$ of the coefficients of the effective action describing a $\phi^4$-matrix model with initial interaction (L4start) and cut-off propagator characterised by the three exponents $\delta_0,\delta_1,\delta_2$ are for $2 \leq N\l provided that for all $V'{<}V,\;2\leq N'\leq 2V'{+}2$ and $V'{=}V,\;N{+}2\leq N'\leq 2V{+}2$ the in

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Definition 4
  • Definition 5
  • Theorem 6
  • Corollary 7
  • Proposition 8