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Two-loop renormalization of multiflavor $ϕ^3$ theory in six dimensions and the trace anomaly

We use the background-field method and the heat kernel to obtain all counterterms to two-loop order of conformally-coupled multiflavor $ϕ^3$ theory in six spacetime dimensions, defined in curved spacetime and with spacetime-dependent couplings. We also include spacetime-dependent mass terms for completeness. We use these results to write a general expression for the trace anomaly. With the use of Weyl consistency conditions we are able to show that the strong $a$-theorem for a certain natural candidate quantity $\tilde{a}$ is violated in this theory, and obtain a three-loop expression for the coefficient $a$ of the Euler term in the anomaly.