The renormalization of the effective gauge Lagrangian with spontaneous symmetry breaking: the U(1) case

We study the renormalization of the nonlinear effective U(1) Lagrangian up to $O(p^4)$ with spontaneous symmetry breaking. The problems of the quartic divergences and of the truncation of infinite divergence tower are addressed. The renormalization group equations of the effective Lagrangian are derived, which make it possible to get the leading logarithm corrections of the heavy degree of freedom, even if the effective and full theories are matched at tree level. The method we use in the U(1) case can easily be extended to study the non-Abelian effective theories and the electroweak chiral Lagrangian.