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

P. Castro-Villarreal·R. Delgadillo-Blando·Badis Ydri
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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 .

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P. Castro-Villarreal, R. Delgadillo-Blando, Badis Ydri. 2004-09-25. A Gauge-Invariant UV-IR Mixing and The Corresponding Phase Transition For U(1) Fields on the Fuzzy Sphere. https://doi.org/10.1016/j.nuclphysb.2004.10.032

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