X iv : h ep - p h / 94 08 40 4 v 2 8 S ep 1 99 4 OITS - 553 Isospin Structure of Penguins And Their Consequences in B Physics ∗

Isospin structure of gluon mediated or strong penguin is significantly altered when the full electroweak corrections are included. This has the consequence that some previous analyses which relied on a simple isospin structure in charmless B decays become inapplicable. We present the general Hamiltonian in next-to-leading order QCD, and illustrate our conclusion quantitatively for B → ππ and B → Kπ decays in the factorization approximation. Some remarks on CP asymmetries in B decays are also made. 12.15.Ji, 12.38.Bx, 13.20.He, 13.25.Hw Typeset using REVTEX Work supported in part by the Department of Energy Grant No. DE-FG06-85ER40224. 1 Flavor changing one loop processes mediated by a gluon play an important role in understanding the Standard Model (SM) [1–4]. The CP violating parameter ǫ/ǫ in the K system is estimated by evaluating the flavor changing process s → dqq̄, where q = u or d. It is by now well known that substantial corrections arise in this estimate from inclusion of the full electroweak effects [5,6] (the γ, Z penguins and the “box” contributions). These additional electroweak corrections add ∆I = 3/2 contribuitons to the ∆I = 1/2 pure gluon exchange, and substantially reduce the estimate for ǫ/ǫ. A complete next-to-leading order (NLO) calculation now exists for the full electroweak contributions for such flavor changing process [6]. In the K system the enhancement of ∆I = 1/2 amplitude over ∆I = 3/2 by a factor of about 20 in K → ππ decays further amplifies the electroweak contribution to ǫ/ǫ. One might think that in the B system electroweak corrections to gluon exchange may be negligible. This in fact is incorrect. The large mass of the top increases the relative contributions of the electroweak corrections and has the effect of causing significant reduction in the estimates for gluon mediated decays. Our recent estimates for b → sφ process suggests a reduction in the rate of 20%∼ 30% [3,4]. The gluon mediated process b → sqq̄ where q = u , d , s is pure ∆I = 0. Inclusion of the full electroweak corrections will introduce a significant ∆I = 1 admixture. Any analysis that relies on penguin contributions being pure ∆I = 0 is therefore suspect. We will comment on an interesting analysis assuming the above property which was recently published in Physical Review Letters [7]. Similarly, one might worry about the wrong isospin admixture in b → dqq̄ process where the gluon exchange is pure ∆I = 1/2. Extraction of penguin effects from the tree effects are important in B → ππ process [8,9] because of the need to remove penguin contamination to make a measurement of the angle α in the unitarity triangle of the KM matrix. We shall first present the full Hamiltonian describing flavor changing processes in the NLO approximation. The estimate of matrix elements to quantify our results can only be done in the context of a model in our present calculational limitation. We shall use factorization approximation and some models for form factors when necessary. The effective Hamiltonian responsible for charmless B decays can be parametrized as 2 H∆B=1 = GF √ 2 [VubV ∗ uq(c1O u 1 + c2O u 2 )− VtbV ∗ tq 10