  We renormalize massless scalar effective field theories (EFTs) to higher loop
orders and higher orders in the EFT expansion. To facilitate EFT calculations
with the R* renormalization method, we construct suitable operator bases using
Hilbert series and related ideas in commutative algebra and conformal
representation theory, including their novel application to off-shell
correlation functions. We obtain new results ranging from full one loop at mass
dimension twelve to five loops at mass dimension six. We explore the structure
of the anomalous dimension matrix with an emphasis on its zeros, and
investigate the effects of conformal and orthonormal operators. For the real
scalar, the zeros can be explained by a `non-renormalization' rule recently
derived by Bern et al. For the complex scalar we find two new selection rules
for mixing $n$- and $(n-2)$-field operators, with $n$ the maximal number of
fields at a fixed mass dimension. The first appears only when the $(n-2)$-field
operator is conformal primary, and is valid at one loop. The second appears in
more generic bases, and is valid at three loops. Finally, we comment on how the
Hilbert series we construct may be used to provide a systematic enumeration of
a class of evanescent operators that appear at a particular mass dimension in
the scalar EFT.

High Energy Physics - Phenomenology;High Energy Physics - Theory