The Two Loop Long Range Effect on the Proton Decay Effective Lagrangian
  We calculate the two loop long range effect on the proton decay effective Lagrangian. Numerical calculation for suppression factor gives $ A_L($2-loop$) = 0.321$ for the value of the strong coupling constant $ \alpha_s(m_Z) = 0.116 $. Two loop effect to more general effective Lagrangian is also given. 
Takeshi Nihei and Jiro Arafune