The Bosonization of the Electroweak Penguin Operators

We give the complete O(p) bosonization of the electroweak Penguin operators Q7,8 and compare the result with that of the gluon Penguin operators Q5,6. We find that, in addition to the usual (constant and current-current) parts, there are three new terms not discussed previously in the literature. Two of these are present in the factorization approximation and should be included in the standard definition of the B7,8-factors. The impact of these corrections on the direct CP -violating parameter ε/ε is briefly discussed. SISSA 74/96/EP May 1996 Permanent address: Ain Shams University, Faculty of Science, Dept. of Physics, Cairo, Egypt. 1. Penguin operators play an important role in the physics of the standard model at low energies and, in particular, in the non-leptonic decays of kaons. The estimate of their matrix elements has been a controversial subject from the very beginning because of a subtle cancellation of the leading contribution in chiral perturbation theory. Most of the debate [1, 2], however, was centered around the gluon Penguin operators—at the time the only relevant ones—that we write as Q5 = (sd)V−A ∑ q (qq)V+A (1) Q6 = (sαdβ)V−A ∑ q (qβqα)V+A (2) by means of the by-now-standard notation in which α, β denote color indices (α, β = 1, . . . , Nc), the subscripts (V ± A) refer to the chiral projections γμ(1 ± γ5) and color indices for the color singlet operators are omitted. The other gluon Penguin operators (Q3,4) discussed in the literature are readily bosonized as the product of two left-handed currents and we will not discuss them. The electroweak Penguin operators [4] are defined as Q7 = 3 2 (sd)V−A ∑ q êq (qq)V+A (3) Q8 = 3 2 (sαdβ)V−A ∑ q êq(qβqα)V+A , (4) where êq are the quark charges (êd = ês = −1/3 and êu = 2/3). The appereance of these operators has not revived the discussion on their bosonization and they have been treated along the same lines as the gluon ones. Yet, new features emerge when considering this new class and the argument followed in the bosonization and the determination of the chiral coefficient of the gluon Penguin fail us here. Let us see why. 2. In order to proceed we first write the two electroweak operators as Q7 = 3 2 êdQ5 + 3 2 (êu − êd)∆Q7 Q8 = 3 2 êdQ6 + 3 2 (êu − êd)∆Q8 (5)