  We propose various properties of renormalization group beta functions for
vector operators in relativistic quantum field theories. We argue that they
must satisfy compensated gauge invariance, orthogonality with respect to scalar
beta functions, Higgs-like relation among anomalous dimensions and a gradient
property. We further conjecture that non-renormalization holds if and only if
the vector operator is conserved. The local renormalization group analysis
guarantees the first three within power counting renormalization. We verify all
the conjectures in conformal perturbation theories and holography in the weakly
coupled gravity regime.
