ar X iv : h ep - p h / 95 05 36 9 v 1 2 3 M ay 1 99 5 OITS - 576 A Method for Determining CP Violating Phase

A new way of determining the phases of weak amplitudes in charged B decays based on SU(3) symmetry is proposed. The CP violating phase γ can now be determined without the previous difficulty associated with electroweak penguins. Typeset using REVTEX Work supported in part by the Department of Energy Grant No. DE-FG06-85ER40224. 1 Detection of CP violation and verification of the unitarity triangle of the CKM matrix is a major goal of B factories [1]. Decisive information about the origin of CP violation will be obtained if the three phases α = arg(−VtdV ∗ tb/V ∗ ubVud), β = arg(−VcdV ∗ cb/V ∗ tbVtd) and γ = arg(−VudV ∗ ub/V ∗ cbVcd) can be independently measured experimentally [2]. The sum of these three phases must be equal to 180 if the Standard Model with three generations is the model for CP violation. There have been many studies to measure these phases. The phase β can be determined unambiguously by measuring time variation asymmetry in B̄(B) → ψKS decay rates [2]. The phase α can be measured in B → ππ and B → ρπ decays [3,4]. In these decays, there are contributions from the tree and the penguin (both strong and electroweak) amplitudes. The methods proposed in Refs. [3,4] are valid even if the strong penguin contributions are included. The inclusion of the electroweak penguin contributions may contaminate the result. However, because the electroweak penguin effects are small in this case, the error in α determination are small. Several methods using ∆S = 1 B decays to measure the phase γ had been proposed [5,6]. Most had assumed that the effects from electroweak penguin could be neglected. It has been recently shown by us [7] that this assumption is badly violated for top quark mass of order 170 GeV. For ∆S = 0 hadronic B decays, the strong penguin effects are much smaller than the leading tree contributions, and thus electroweak penguin effects which are even smaller can be safely ignored. In ∆S = 1 decays, because of the large enhancement factor |VtbV ∗ ts/VubV ∗ us| ≈ 55, the strong penguins dominate and the electroweak penguin effects are comparable to the tree contributions. This invalidates methods proposed in Refs. [5,6]. In this letter we give further consideration to measuring γ using ∆S = 1 decays, although other methods have been discussed in the literature [2,8]. An interesting method has recently been proposed by Hernadez, Gronau, London and Rosner [9] using SU(3) relations between the decay amplitudes for B → πK, πK̄, πK̄, πK, B → ππ, and Bs → ηπ. This method requires the reconstruction of the quadrangle from B → πK, πK̄, πK̄, πK decays. In order to do so, one not only needs to measure all the four B → πK decay amplitude but also needs to measure the 2 rare decay amplitude Bs → ηπ. It has been shown that this last decay is a pure ∆I = 1 transition, with the dominant contribution from the electroweak penguin. However, the branching ratio is extremely small O(10) [10]. In this letter we propose a new method to measure γ using ∆S = 1 decays B → πK̄, πK, ηK, and the ∆S = 0 decay B → ππ, which relies on SU(3) symmetry. This method is free from the electroweak penguin contamination problem, and further, all decays involved have relatively large (O(10)) branching ratios. More importantly these measurements can in principle be carried out at present facilities like CLEO or CDF/D0. In the SM the most general effective Hamiltonian for hadronic B decyas can be written as follows: H eff = GF √ 2 [VubV ∗ uq(c1O q 1 + c2O q 2)− 10 ∑ i=3 VtbV ∗ tqciO q i ] +H.C. , (1) where the O i are defined as O 1 = q̄αγμ(1− γ5)uβūβγ(1− γ5)bα , O 2 = q̄γμ(1− γ5)uūγ(1− γ5)b , O 3,5 = q̄γμ(1− γ5)bq̄γμ(1∓ γ5)q′ , O 4,6 = q̄αγμ(1− γ5)bβ q̄′ βγμ(1∓ γ5)q′ α , (2) O 7,9 = 3 2 q̄γμ(1− γ5)beq′ q̄γ(1± γ5)q′ , O 8,10 = 3 2 q̄αγμ(1− γ5)bβeq′ q̄′ βγμ(1± γ5)q′ α , Here q is summed over u, d, and s. For ∆S = 0 processes, q = d, and for ∆S = 1 processes, q = s. O2, O1 are the tree level and QCD corrected operators. O3−6 are the strong gluon induced penguin operators, and operators O7−10 are due to γ and Z exchange, and the “box” diagrams at one loop level (electroweak penguin). The Wilson coefficients ci are defined at the scale of μ ≈ mb which have been evaluated to the next-to-leading order in QCD [11]. In the above we have neglected small contributions from u and c quark loop contributions proportional to VubV ∗ uq. We can parametrise the decay amplitude of B as A =< final state|H eff |B >= VubV ∗ uqT (q) + VtbV ∗ tqP (q) , (3) where T (q) contains the tree contribution, while P (q) contains penguin contributions. SU(3) symmetry will lead to specific relations among B decay amplitudes. 3 SU(3) relations for B decays have been studied by several authors [12,13]. We will follow the notation used in Ref. [13]. The operators Q1,2, O3−6, and O7−10 transform under SU(3) symmetry as 3̄a + 3̄b +6+ 15, 3̄, and 3̄a + 3̄b + 6+ 15, respectively. In general, we can write the SU(3) invariant amplitude for B to two octet pseudoscalar mesons in the following form T = AT(3̄)BiH(3̄) (M l M l k) + C T (3̄)BiM i kM k j H(3̄) j + AT(6)BiH(6) ij kM l jM k l + C T (6)BiM i jH(6) jk l M l k + AT(15)BiH(15) ij kM l jM k l + C T (15)BiM i jH(15) jk l M l k , (4) where Bi = (B , B̄, B̄ s ) is a SU(3) triplet, M j i is the SU(3) pseudoscalar octet, and the matrices H represent the transformation properties of the operatorsO1−10. H(6) is a traceless tensor that is antisymmetric on its upper indices, and H(15) is also a traceless tensor but is symmetric on its upper indices. For q = d, the non-zero entries of the H matrices are given by H(3̄) = 1 , H(6) 1 = H(6) 23 3 = 1 , H(6) 21 1 = H(6) 32 3 = −1 , H(15) 1 = H(15) 21 1 = 3 , H(15) 22 2 = −2 , H(15) 3 = H(15) 3 = −1 . (5) For q = s, the non-zero entries are H(3̄) = 1 , H(6) 1 = H(6) 32 2 = 1 , H(6) 31 1 = H(6) 23 2 = −1 , H(15) 1 = H(15) 31 1 = 3 , H(15) 33 3 = −2 , H(15) 2 = H(15) 2 = −1 . (6) In terms of the SU(3) invariant amplitudes, the decay amplitudes T (ππ), T (πK) for B̄ → ππ, B̄ → πK are given by T (πK̄) = C (3̄) + A T (6) − C (6) + 3AT(15) − C (15) , T (πK) = 1 √ 2 (C (3̄) + A T (6) − C (6) + 3AT(15) + 7C (15)) , T (η8K −) = 1 √ 6 (−C (3̄) − AT(6) + C (6) − 3AT(15) + 9C (15)) , T (ππ) = 8 √ 2 C (15) , (7) 4 We also have similar relations for the amplitude P (q). The corresponding SU(3) invariant amplitudes will be denoted by Ai and C P i . It is easy to obtain the following triangle relation from above: √ 2A(πK)− 2A(πK̄) = √ 6A(η8K −) . (8) For the moment if we ingnore η−η′ mixing, it is clear that we can construct this triangle from the experimentally measured rates for the various B decays. A similar triangle can also be constructed for the modes of B decay: √ 2Ā(πK)− 2Ā(πK) = √ 6Ā(η8K ) . (9) We shall now use a hypothesis that the tree contribution to the mode B → πK̄ is negligible [14]. This is varified in factorization approximation and had been assumed by Ref. [6,9]. Further, if we work in Wolfenstein parametrization of the CKM matrix, the amplitude A(πK̄) contains no weak phase. Then A(πK̄) = Ā(πK) . (10) We can now obtain the magnitude and relative phases of A(π(η8)K ) and Ā(π(η8)K ) subject to two fold ambiguities related to whether the triangles for the B and B decays are on the same side (solution a) or opposite side (solution b) of A(πK̄) as shown in Figure 1. Now we construct two complex quantities (shown in Figure 1) B = √ 2A(πK)−A(πK̄) = 8(|VubV ∗ us|eC 15 + |VtbV ∗ ts|C 15) . (11)