Four-quark Operators Relevant to B Meson Lifetimes from QCD Sum Rules

Content Provider	Semantic Scholar
Author	Baek, Min Sun Lee, Jungil Liu, Chun Song, Huesup
Copyright Year	1997
Abstract	At the order of 1/mb , the B meson lifetimes are controlled by the hadronic matrix elements of some four-quark operators. The nonfactorizable magnitudes of these four-quark operator matrix elements are analyzed by QCD sum rules in the framework of heavy quark effective theory. The vacuum saturation for color-singlet four-quark operators is justified at hadronic scale, and the nonfactorizable effect is at a few percent level. However for color-octet four-quark operators, the vacuum saturation is violated sizably that the nonfactorizable effect cannot be neglected for the B meson lifetimes. The implication to the extraction of some of the parameters from B decays is discussed. The B meson lifetime ratio is predicted as τ(B)/τ(B) = 1.09 ± 0.02. However, the experimental result of the lifetime ratio τ(Λb)/τ(B ) still cannot be explained. PACS: 12.38.Lg, 12.39.Hg, 13.25.Hw, 13.30.-a. Address after Dec. 1, 1997: Dept. of Phys., Ohio State Univ., Columbus OH43210, USA 1 Heavy hadron lifetimes provide us with testing ground to the Standard Model, especially to QCD in some aspects [1-3], because they can be systematically calculated within the framework of heavy quark expansion. Theoretically, if we do not assume the failure of the local duality assumption, the heavy hadron lifetime differences appear, at most, at the order of 1/mQ [4]. Recent experimental results on the lifetime ratio of Λb baryon and B meson [5] showed some deviation from the theoretical expectation. This has drawn a lot theoretical attentions [6-11]. The current experimental values for the lifetime ratios which we are interested in are [5] τ(B) τ(B0) = 1.06 ± 0.04 , τ(Λb) τ(B0) = 0.79 ± 0.06 . (1) This may imply that the O( 1 m b ) contribution is not enough for the explanation of above heavy baryon and heavy meson lifetime difference. To the order of 1/mb , the hadron lifetimes have been studied since mid-80s [11, 12, 4, 6, 7]. And the potential importance of the 1/mb corrections has been pointed out. The parameterization of the hadronic matrix elements of four-quark operators which appear in the hadron lifetimes at the order of 1/mb is generally expressed as [7] 〈B̄|b̄γμ(1 − γ5)qq̄γ (1 − γ5)b|B̄〉 ≡ B1F 2 Bm 2 B , 〈B̄|b̄(1 − γ5)qq̄(1 + γ5)b|B̄〉 ≡ B2F 2 Bm 2 B , 〈B̄|b̄γμ(1 − γ5)taqq̄γ (1 − γ5)tab|B̄〉 ≡ ǫ1F 2 Bm 2 B , 〈B̄|b̄(1 − γ5)taqq̄(1 + γ5)tab|B̄〉 ≡ ǫ2F 2 Bm 2 B , (2) and 1 2mΛb 〈Λb|b̄γμ(1 − γ5)qq̄γ (1 − γ5)b|Λb〉 ≡ − F 2 BmB 12 r , 1 2mΛb 〈Λb|b̄(1 − γ5)qq̄(1 + γ5)b|Λb〉 ≡ −B̃ F 2 BmB 24 r , (3) where the parameters Bi, ǫi (i = 1, 2), FB, r and B̃ should be calculated by some nonperturbative QCD method. In above equations, the renormalization scale is arbitrary, and the parameters depend on it. It can be taken naturally at the low hadronic scale to apply heavy quark expansion. On the other hand, if the scale is taken at mb, 2 parameter FB(mb) is just the well-defined measurable physical quantity − B meson decay constant fB. The QCD sum rule [13], which is regarded as a nonperturbative method rooted in QCD itself, has been used successfully to calculate the properties of various hadrons. In Ref. [8], in the framework of heavy quark effective theory (HQET), the baryonic parameters r and B̃ have been calculated by QCD sum rule, r ∼ 0.1 − 0.3, B̃ ≃ 1. For a complete analysis, the mesonic parameters Bi and ǫi should be also calculated from QCD sum rule. The four-quark operators, and hence Bi, ǫi, r and B̃, are scaledependent quantities when the QCD radiative corrections are included. It was proposed by Shifman and Voloshin [11] that at the low hadronic scale, the vacuum saturation approximation, namely Bi = 1 and ǫi = 0, makes sense. However in this case, the measured lifetime ratio τ(Λb)/τ(B ) cannot be explained [8]. There are some argument, on the other hand, that the vacuum saturation maybe a poor approximation [9]. Especially from a naive large Nc analysis, ǫi’s are about 1/Nc ∼ 0.3 [11, 7]. We will explore the violation of the vacuum saturation approximation in detail from QCD sum rules in the framework of HQET. Let us first consider the parameters Bi. We construct the following three-point Green’s function, Γ(ω, ω) = i ∫ dxdye ′〈0|T [q̄(x)γγ5h (b) v (x)]O(0)[q̄(y)γμγ5h (b) v (y)] |0〉 , (4) where ω = 2v · k, ω = 2v · k; h v is the b-quark field in the HQET with velocity v. And O denotes the color-singlet operators given in Eq. (2), O = b̄Γ1qq̄Γ2b , (5) with Γ1 = Γ2 = γ (1 − γ5) for B1 and Γ1 = 1 − γ5, Γ2 = 1 + γ5 for B2. In terms of the hadronic expression, the parameter Bi appears in the ground state contribution of Γ(ω, ω), Γ(ω, ω) = Bi F 4 Bm 2 B (2Λ̄ − ω)(2Λ̄ − ω′) + resonances , (6)
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