Constraints on RG Flow for Four Dimensional Quantum Field Theories

The response of four dimensional quantum field theories to a Weyl rescaling of the metric in the presence of local couplings and which involve $a$, the coefficient of the Euler density in the energy momentum tensor trace on curved space, is reconsidered. Previous consistency conditions for the anomalous terms, which implicitly define a metric $G$ on the space of couplings and give rise to gradient flow like equations for $a$, are derived taking into account the role of lower dimension operators. The results for infinitesimal Weyl rescaling are integrated to finite rescalings $e^{2σ}$ to a form which involves running couplings $g_σ$ and which interpolates between IR and UV fixed points. The results are also restricted to flat space where they give rise to broken conformal Ward identities. Expressions for the three loop Yukawa $β$-functions for a general scalar/fermion theory are obtained and the three loop contribution to the metric $G$ for this theory are also calculated. These results are used to check the gradient flow equations to higher order than previously. It is shown that these are only valid when $β\to B$, a modified $β$-function, and that the equations provide strong constraints on the detailed form of the three loop Yukawa $β$-function. ${\cal N}=1$ supersymmetric Wess-Zumino theories are also considered as a special case. It is shown that the metric for the complex couplings in such theories may be restricted to a hermitian form.

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