Effective Potential on Fuzzy Sphere

TL;DR

This paper evaluates the effective potential of a scalar field on a fuzzy sphere at two-loop level, revealing symmetry restoration effects and the dominance of planar diagrams, with implications for quantum field behavior in noncommutative geometries.

Contribution

It provides a two-loop analysis of the effective potential on a fuzzy sphere, demonstrating symmetry restoration and the dominance of planar diagrams at high mass or temperature.

Findings

Planar diagrams dominate over nonplanar diagrams by a factor of N^2.

Symmetry breaking tends to be restored at high temperature or heavy mass.

Nonplanar contributions are suppressed when particle wavelength is below the noncommutativity scale.

Abstract

The effective potential of quantized scalar field on fuzzy sphere is evaluated to the two-loop level. We see that one-loop potential behaves like that in the commutative sphere and the Coleman-Weinberg mechanism of the radiatively symmetry breaking could be also shown in the fuzzy sphere system. In the two-loop level, we use the heavy-mass approximation and the high-temperature approximation to perform the evaluations. The results show that both of the planar and nonplanar Feynman diagrams have inclinations to restore the symmetry breaking in the tree level. However, the contributions from planar diagrams will dominate over those from nonplanar diagrams by a factor N^2. Thus, at heavy-mass limit or high-temperature system the quantum field on the fuzzy sphere will behave like those on the commutative sphere. We also see that there is a drastic reduction of the degrees of freedom in the nonplanar diagrams when the particle wavelength is smaller than the noncommutativity scale.