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In the free $\Box^k$ scalar conformal field theory, there exist conserved andpartially-conserved higher-spin currents. We study their anomalous dimensionsassociated with $\phi^{2n}$ interaction in the $\epsilon$ expansion. We derivegeneral formulae for the leading corrections from the conformal multipletrecombination, and verify their consistency with crossing symmetry using theLorentzian inversion formula. The results are further extended to the O($N$)models.

Anomalous dimensions of partially-conserved higher-spin currents from conformal field theory: bosonic $\phi^{2n}$ theories