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Register a new userUV/IR Mixing and Anomalies in Noncommutative Gauge Theories
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Tadahito Nakajima
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Using path integral method (Fujikawas method) we calculate anomalies in noncommutative gauge theories with fermions in the bi-fundamental and adjoint representations. We find that axial and chiral gauge anomalies coming from non-planar contributions are derived in the low noncommutative momentum limit $widetilde{p}^{mu}(equiv theta^{mu u}p_{ u}) to 0$. The adjoint chiral fermion carries no anomaly in the non-planar sector in $D=4k (k=1,2,...,)$ dimensions. It is naturally shown from the path integral method that anomalies in non-planar sector originate in UV/IR mixing.
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In the absence of gauge fields, quantum field theories on the Groenewold-Moyal (GM) plane are invariant under a twisted action of the Poincare group if they are formulated following [1, 2, 3, 4, 5, 6]. In that formulation, such theories also have no UV-IR mixing [7]. Here we investigate UV-IR mixing in gauge theories with matter following the approach of [3, 4]. We prove that there is UV-IR mixing in the one-loop diagram of the S-matrix involving a coupling between gauge and matter fields on the GM plane, the gauge field being nonabelian. There is no UV-IR mixing if it is abelian.
We calculate conformal anomalies in noncommutative gauge theories by using the path integral method (Fujikawas method). Along with the axial anomalies and chiral gauge anomalies, conformal anomalies take the form of the straightforward Moyal deformation in the corresponding conformal anomalies in ordinary gauge theories. However, the Moyal star product leads to the difference in the coefficient of the conformal anomalies between noncommutative gauge theories and ordinary gauge theories. The $beta$ (Callan-Symanzik) functions which are evaluated from the coefficient of the conformal anomalies coincide with the result of perturbative analysis.
In this paper we study the phenomenon of UV/IR mixing in noncommutative field theories from the point of view of world-sheet open-closed duality in string theory. New infrared divergences in noncommutative field theories arise as a result of integrating over high momentum modes in the loops. These are believed to come from integrating out additional bulk closed string modes. We analyse this issue in detail for the bosonic theory and further for the supersymmetric theory on the $C^2/Z_2$ orbifold. We elucidate on the exact role played by the constant background $B$-field in this correspondence.
I review results from recent investigations of anomalies in fermion--Yang Mills systems in which basic notions from noncommutative geometry (NCG) where found to naturally appear. The general theme is that derivations of anomalies from quantum field theory lead to objects which have a natural interpretation as generalization of de Rham forms to NCG, and that this allows a geometric interpretation of anomaly derivations which is useful e.g. for making these calculations efficient. This paper is intended as selfcontained introduction to this line of ideas, including a review of some basic facts about anomalies. I first explain the notions from NCG needed and then discuss several different anomaly calculations: Schwinger terms in 1+1 and 3+1 dimensional current algebras, Chern--Simons terms from effective fermion actions in arbitrary odd dimensions. I also discuss the descent equations which summarize much of the geometric structure of anomalies, and I describe that these have a natural generalization to NCG which summarize the corresponding structures on the level of quantum field theory. Contribution to Proceedings of workshop `New Ideas in the Theory of Fundamental Interactions, Szczyrk, Poland 1995; to appear in Acta Physica Polonica B.
UV/IR mixing is one of the most important features of noncommutative field theories. As a consequence of this coupling of the UV and IR sectors, the configuration of fields at the zero momentum limit in these theories is a very singular configuration. We show that the renormalization conditions set at a particular momentum configuration with a fixed number of zero momenta, renormalizes the Greens functions for any general momenta only when this configuration has same set of zero momenta. Therefore only when renormalization conditions are set at a point where all the external momenta are nonzero, the quantum theory is renormalizable for all values of nonzero momentum. This arises as a result of different scaling behaviors of Greens functions with respect to the UV cutoff ($Lambda$) for configurations containing different set of zero momenta. We study this in the noncommutative $phi^4$ theory and analyse similar results for the Gross-Neveu model at one loop level. We next show this general feature using Wilsonian RG of Polchinski in the globally O(N) symmetric scalar theory and prove the renormalizability of the theory to all orders with an infrared cutoff. In the context of spontaneous symmetry breaking (SSB) in noncommutative scalar theory, it is essential to note the different scaling behaviors of Greens functions with respect to $Lambda$ for different set of zero momenta configurations. We show that in the broken phase of the theory the Ward identities are satisfied to all orders only when one keeps an infrared regulator by shifting to a nonconstant vacuum.
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