A Gauge-Invariant UV-IR Mixing and The Corresponding Phase Transition For U(1) Fields on the Fuzzy Sphere

TL;DR

This paper investigates gauge-invariant UV-IR mixing and phase transitions in U(1) gauge theory on the fuzzy sphere, revealing a noncommutative anomaly, a first-order phase transition, and the effects of scalar mass terms.

Contribution

It provides a detailed analysis of UV-IR mixing, phase transitions, and the impact of scalar mass terms in U(1) gauge theory on the fuzzy sphere, including explicit calculations and predictions.

Findings

Gauge-invariant UV-IR mixing persists in the continuum limit.

A first-order phase transition is predicted at one-loop level.

Adding a large scalar mass term removes UV-IR mixing from the gauge sector.

Abstract

From a string theory point of view the most natural gauge action on the fuzzy sphere {\bf S}^2_L is the Alekseev-Recknagel-Schomerus action which is a particular combination of the Yang-Mills action and the Chern-Simons term . Since the differential calculus on the fuzzy sphere is 3-dimensional the field content of this model consists naturally of a 2-dimensional gauge field together with a scalar fluctuation normal to the sphere . For U(1) gauge theory we compute the quadratic effective action and shows explicitly that the tadpole diagrams and the vacuum polarization tensor contain a gauge-invariant UV-IR mixing in the continuum limit L{\longrightarrow}{\infty} where L is the matrix size of the fuzzy sphere. In other words the quantum U(1) effective action does not vanish in the commutative limit and a noncommutative anomaly survives . We compute the scalar effective potential and prove the gauge-fixing-independence of the limiting model L={\infty} and then show explicitly that the one-loop result predicts a first order phase transition which was observed recently in simulation . The one-loop result for the U(1) theory is exact in this limit . It is also argued that if we add a large mass term for the scalar mode the UV-IR mixing will be completely removed from the gauge sector . It is found in this case to be confined to the scalar sector only. This is in accordance with the large L analysis of the model . Finally we show that the phase transition becomes harder to reach starting from small couplings when we increase M .