Renormalization Group Equations in Gauge Theories with Multiple U(1)   Groups
  Two-loop renormalization group equations in gauge theories with multiple U(1) groups are presented. Instead of normalizing the abelian gauge fields in canonical forms, we retain kinetic-mixing terms and treat the mixing coefficients as free parameters. The beta- and gamma-functions are then obtained in a straightforward manner. 
Mingxing Luo and Yong Xiao