Renormalization footprints in the phase diagram of the Grosse-Wulkenhaar model

Dragan Prekrat

doi:10.1103/physrevd.104.114505

Abstract

We construct and analyze the phase diagram of a self-interacting matrix field in two dimensions coupled to the curvature of the non-commutative truncated Heisenberg space. In the infinite size limit, the model reduces to the renormalizable Grosse-Wulkenhaar's. The curvature term proves crucial for the diagram's structure: when turned off, the triple point collapses into the origin as matrices grow larger; when turned on, the triple point recedes from the origin proportionally to the coupling strength and the matrix size. The coupling attenuation that turns the Grosse-Wulkenhaar model into a renormalizable version of the $\phi^4_\star$-model cannot stop the triple point recession. As a result, the stripe phase escapes to infinity, removing the problems with UV/IR mixing.

Highlights

Noncommutativity (NC) of space-time coordinates was initially proposed in the 1940s in the hope of resolving the confusion about the infinities in the nascent quantum field theory [1]
We chose the minus sign of the mass term to enable positive c2 to parametrize the relevant portion of the phase diagram, while positive c4 ensures that SN is bounded from below
This paper aimed to see if the renormalizable GW redefinition of the φ4⋆ model is reflected in its corresponding phase diagram and if it affects the extent of the stripe phase connected to the UV/IR mixing

Summary

INTRODUCTION

Noncommutativity (NC) of space-time coordinates was initially proposed in the 1940s in the hope of resolving the confusion about the infinities in the nascent quantum field theory [1]. More details on the htr and Sh are provided later in the text Another possible source of the oscillator term was presented in [14], where it elegantly appears in the expansion of the kinetic term of the free scalar field situated in the Snyder-de Sitter space. Phase diagrams on NC spaces have been extensively studied in various matrix models, since they regularize corresponding continuum theories in a numerical simulation-friendly fashion [24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44] They generically feature three phases that meet at a triple point. We present the effects of the curvature coupling variation on the phase diagram, look at the coupled model as we turn the coupling off, and compare its limit with the uncoupled one

MATRIX MODEL

R-ON SHIFT AND RENORMALIZATION

CONCLUDING REMARKS