Dimension-eight operators in the weak OPE

We argue that there is a potential flaw in the standard treatment of weak decay amplitudes, including that of ′/ . We show that (contrary to conventional wisdom) dimension-eight operators do contribute to weak amplitudes, at order GF αs and without 1/M2 W suppression. We demonstrate the existence of these operators through the use of a simple weak hamiltonian. Their contribution appears in different places depending on which scheme is adopted in performing the OPE. If one performs a complete separation of short and long distance physics within a cutoff scheme, dimension-eight operators occur in the weak hamiltonian at order GF αs/μ, μ being the separating scale. However, in an MS renormalization scheme for the OPE the dimension-eight operators do not appear explicitly in the hamiltonian at order GF αs. In this case, matrix elements must include physics above the scale μ, and it is here that dimension eight effects enter. The use of a cutoff scheme (especially quark model methods) for the calculation of the matrix elements of dimension-six operators is inconsistent with MS unless there is careful matching including dimension-eight operators. The contribution of dimension-eight operators can be minimized by working at large enough values of the scale μ. We find from sum rule methods that the contribution of dimension-eight operators to the dimension-six operator Q 7 is at the 100% level for μ = 1.5 GeV. This suggests that presently available values of μ are too low to justify the neglect of these effects. Finally, we display the dimension-eight operators which appear within the Standard Model at one loop. Typeset using REVTEX