A 1996 Analysis of the CP Violating Ratio ε ′ / ε †

We update our 1993 analysis of the CP violating ratio ε/ε in view of the changes in several input parameters, in particular the improved value of the top quark mass. We also investigate the strange quark mass ms dependence of ε/ε in view of rather low values found in the most recent lattice calculations. A simple scanning of the input parameters within one standard deviation gives the ranges: −1.2 · 10−4 ≤ ε/ε ≤ 16.0 · 10−4 and 0 ≤ ε/ε ≤ 43.0 · 10−4 for ms(mc) = 150 ± 20 MeV and ms(mc) = 100 ± 20 MeV respectively. If the experimentally measured numbers and the theoretical input parameters are used with Gaussian errors, we find ε/ε = (3.6 ± 3.4) · 10−4 and ε/ε = (10.4 ± 8.3) · 10−4 respectively. We also give results for ImV ∗ tsVtd. Analyzing the dependence of ε /ε on various parameters we find that only for ms(mc) ≤ 100 MeV and a conspiration of other input parameters, values for ε/ε as large as (2 − 4) · 10−3 and consistent with the NA31 result can be obtained. Supported by the German Bundesministerium für Bildung und Forschung under contract 06 TM 743 and DFG Project Li 519/2-1. The measurement of ε/ε at the 10 level remains as one of the important targets of contemporary particle physics. A non-vanishing value of this ratio would give the first signal for the direct CP violation ruling out the superweak models. The experimental situation on Re(ε/ε) is unclear at present. While the result of NA31 collaboration at CERN with Re(ε/ε) = (23 ± 7) · 10 [1] clearly indicates direct CP violation, the value of E731 at Fermilab, Re(ε/ε) = (7.4± 5.9) · 10 [2], is compatible with superweak theories [3] in which ε/ε = 0. Hopefully, in about two years the experimental situation concerning ε/ε will be clarified through the improved measurements by the two collaborations at the 10 level and by the KLOE experiment at DAΦNE. There is no question about that the direct CP violation is present in the standard model. Yet, accidentally it could turn out that it will be difficult to see it in K → ππ decays. Indeed in the standard model ε/ε is governed by QCD penguins and electroweak (EW) penguins. In spite of being suppressed by α/αs relative to QCD penguin contributions, the electroweak penguin contributions have to be included because of the additional enhancement factor ReA0/ReA2 = 22 relative to QCD penguins. With increasing mt the EW penguins become increasingly important [4, 5], and entering ε/ε with the opposite sign to QCD penguins suppress this ratio for largemt. Formt ≈ 200 GeV the ratio can even be zero [5]. Because of this strong cancelation between two dominant contributions and due to uncertainties related to hadronic matrix elements of the relevant local operators, a precise prediction of ε/ε is not possible at present. In spite of all these difficulties, a considerable progress has been made in this decade to calculate ε/ε. First of all the complete next-to-leading order (NLO) effective hamiltonians for ∆S = 1 [6–8], ∆S = 2 [9–11] and ∆B = 2 [9] are now available so that a complete NLO analysis of ε/ε including constraints from the observed indirect CP violation (εK) and the B 0 d − B̄ d mixing (∆mBd) is possible. The improved determination of the Vub and Vcb elements of the CKM matrix [12,13], and in particular the determination of the top quark mass mt [14] had of course also an important impact on ε/ε. The main remaining theoretical uncertainties in this ratio are then the poorly known hadronic matrix elements of the relevant QCD penguin and electroweak penguin operators, the values of the non-perturbative parameters BK and √ BBFB and as stressed in [7] the values of ms and ΛMS. In 1993 we have presented a detailed NLO analysis of ε/ε [7] using all information available at that time. A similar NLO analysis has been made by the Rome group [15]. In 1995 the latter group has updated their analysis to predict a very small ratio ε/ε = (3.1 ± 2.5) · 10 [16], essentially consistent with the superweak scenario. An analysis of ε/ε with different treatments of hadronic matrix elements can be found in [17–19] and will be briefly discussed below. The purpose of the present letter is to update our 1993 analysis and to confront our new result with the one of the Rome group [16] and with [17–19]. Let us list the main new ingredients of our present analysis compared to the