Abstract
We investigate the renormalization group flows of multicomponent scalar theories with $U(1)$ gauge symmetry using the functional renormalization group method. The scalar sector is built up from traces of matrix fields that belong to simple, compact Lie algebras. We find that in general these theories are non-renormalizable even at zero gauge coupling, but if we add a $U(1)$ factor to the Lie algebra structure, then they are consistent. In accordance with our earlier findings, fluctuations introduce anomalous, regulator dependent gauge contributions, which are only consistent with the flow equation for a given set of gauge fixing parameters. We establish connections between regularization procedures in the standard covariant and the $R_\xi$ gauges arguing that one is not tied by introducing regulators at the level of the functional integral, and it is allowed to switch between schemes at different levels of the calculations. We calculate $\beta$ functions, classify fixed points, and clarify compatibility of the flow equation and the Ward-Takahashi identity between the scalar wavefunction renormalization and the charge rescaling factor.
Highlights
Modern implementation of the idea of the Wilsonian renormalization group (RG), the functional or exact RG, has had great success in the past for field theories with global, linearly realized symmetries [1,2]
Satisfied at any scale, they are satisfied at all scales, given that the effective action obeys the flow equation. This statement is sometimes argued to be violated by approximate solutions [10], but in our earlier work, we found compatibility [11], and in this paper, we aim to provide further evidence that in the local potential approximation the flow equation and the modified Ward-Takahashi identities (mWTIs) lead to the same scale dependence of the couplings
We have investigated if the local potential approximation (LPA) is a one-loop closed truncation of the RG flows, for classes of theories, where two quartic couplings, belonging to operators ∼jTrðΦ†ΦÞj2 and ∼TrðΦ†ΦΦ†ΦÞ, are present
Summary
Modern implementation of the idea of the Wilsonian renormalization group (RG), the functional or exact RG, has had great success in the past for field theories with global, linearly realized symmetries [1,2]. We wish to perform our investigations in the usual covariant gauge, i.e., ∼ð∂iAiÞ2=2ξ (here Ai is the gauge field), rather than in the Rξ gauge, which we used in a previous study [11] Even though the latter was very convenient from several computational points of view, it did not respect the symmetry generated by the interchange between the real and imaginary parts of the scalars. We wish to rederive and extend our earlier results, but in the covariant gauge, show the aforementioned compatibility and draw some new conclusions regarding the interplay between regularization schemes and gauge fixing terms We believe that these contributions help facilitate a deeper understanding of the application of the FRG method in gauge theories and opens up new approximations for the future.
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