Scheme invariants in ${\varphi }^4$ theory in four dimensions

We provide an analysis of the structure of renormalization scheme invariants for the case of ${\varphi }^4$ theory, relevant in four dimensions. We give a complete discussion of the invariants, up to four loops, and include some partial results at five loops, showing that there are considerably more invariants than one might naively have expected. We also show that one-vertex reducible contributions may consistently be omitted in a well-defined class of schemes, which of course includes $\overline{\mathrm{MS}}$.