Consequences of symmetries in renormalizing collinear effective theory

We consider effects of symmetries on renormalization properties of the collinear effective theory. We investigate which types of operators are possible in the effective theory satisfying gauge invariance, reparameterization invariance and residual energy invariance. Each symmetry puts a constraint on the possible structure of the theory, and there can appear only specific combinations of operators in the effective Lagrangian satisfying all the symmetry requirements. And the final effective Lagrangian is not renormalized to all orders in alpha_s as long as no other nonlocal operators are induced at higher order. We explicitly prove this at one loop by renormalizing one-gluon vertices and discuss their features.