  When conformal field theories (CFTs) are perturbed by marginally relevant
deformations, renormalization group (RG) flows ensue that can be studied with
perturbative methods, at least as long as they remain close to the original
CFT. In this work we study such RG flows in the vicinity of six-dimensional
unitary CFTs. Neglecting effects of scalar operators of dimension two and four,
we use Weyl consistency conditions to prove the $a$-theorem in perturbation
theory, and establish that scale implies conformal invariance. We identify a
quantity that monotonically decreases in the flow to the infrared due to
unitarity, showing that it does not agree with the one studied recently in the
literature on the six-dimensional $\phi^3$ theory.
