Flows of multicomponent scalar models with $U\mathbf{(}1\mathbf{)}$ gauge symmetry

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 these theories the local potential approximation (LPA) is not a one-loop closed truncation in general, even at zero gauge coupling. If, however, we add a $U(1)$ factor to the Lie algebra structure, then the LPA always becomes one-loop closed. 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 wave function renormalization and the charge rescaling factor.

Figure 1

The diagram that is responsible for the flow of the scalar wave function renormalization factor $Z_k$. Solid lines refer to the scalar (and pseudoscalar) fields, while the wiggly one is the gauge propagator. No tensor structure is indicated explicitly. In the regularization proposed here, the gauge propagator is not regulated.

Figure 2

Diagrams that contribute to the flow of the gauge wave function renormalization factor, $Z_{A,k}$. Solid lines mean scalar (and pseudoscalar) propagators, while the wiggly ones refer to the gauge field.

Figure 3

Diagrams (with zero external momenta) that contribute to the flows of the coupling constants $g_{1,k}$ and $g_{2.k}$. Solid lines refer to the scalar (and pseudoscalar) fields, while the wiggly one is the gauge propagator. No tensor structure is indicated explicitly.

Figure 4

Diagrams that contribute to the charge rescaling factor, $Z_{e,k}$. Solid lines refer to the scalar (or pseudoscalar) fields, while the wiggly one corresponds to the gauge field. The external gauge momentum is set to zero for simplicity in the calculation.