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We determine the missing finite-size corrections to the asymptotic one-loopdilatation operator of the real $\beta$-deformed $\mathcal{N}=4$ SYM theory forthe gauge groups $U(N)$ and $SU(N)$ in the 't Hooft limit. In the $SU(N)$ case,the absence of the $U(1)$ field components leads to a new kind of finite-sizeeffect, which we call prewrapping. We classify which states are potentiallyaffected by prewrapping at generic loop orders and comment on the necessity toinclude it into the integrability-based description. As a further result, weidentify classes of $n$-point correlation functions which at all loop orders inthe planar theory are given by the values of their undeformed counterparts.Finally, we determine the superconformal multiplet structure and one-loopanomalous dimensions of all single-trace states with classical scalingdimension $\Delta_0 \leq 4.5$.

The complete one-loop dilatation operator of planar real beta-deformed N=4 SYM theory