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Two Body Non-Leptonic Λb Decays in Quark Model with Factorization Ansatz

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Two Body Non-Leptonic Λb Decays in Quark Model with Factorization Ansatz Two body non-leptonic Λb decays in quark model with factorization ansatz Fayyazuddin Department of Physics Quaid-i-Azam University Islamabad, Pakistan. Riazuddin Department of Physics King Fahd University of Petroleum and Minerals Dhahran 31261, Saudi Arabia. Abstract The two body non-leptonic Λb decays are analyzed in factorization approximation, using quark model, ξ = 1/Nc as a free parameter. It is shown that the experimental branching ratio for Λ ΛJ/ψ b −→ restricts ξ and this ratio can be understood for a value of ξ which lies in the range 0 ξ 0.5 suggested by two body B meson decays. The ≤ ≤ ∗ branching ratios for Λb ΛcDs (Ds) are predicted to be larger than arXiv:hep-ph/9802326v1 15 Feb 1998 −→ the previous estimates. Finally it is pointed that CKM-Wolfenstein parameter ρ2 + η2, where η is CP phase, can be determined from the ratio of widths of Λ ΛD¯ and Λ ΛJ/ψ or that of Λ p D b −→ b −→ b −→ s and Λ Λ D independent of the parameter ξ. b −→ c s PACS: 13.30. Eg, 11.30.Hv, 12.39.Jh 1 1 INTRODUCTION Two body non-leptonic decays of bottom baryons provide useful informa- tion for QCD effects in weak decays and indirect CP asymmetries which involve CKM-Wolfenstein parameters ρ and η. The standard frame work to study non-leptonic decays of bottom baryons is provided by an effec- tive Hamiltonian approach, which allows a separation between short- and long-distance contributions in these decays. The latter involves the matrix elements < MB′ O B > at a typical hadronic scale, where O is an operator | i| i in the effective Hamitonian. These matrix elements cannot be calculated at present from first principles. Thus one has to resort to some approximate schemes. Such schemes are often complicated by competing mechanisms, such as factorization, baryon pole terms and W-exchange terms, each of which has uncertainties of its own. The purpose of this paper is to study a class of two body bottom baryon non-leptonic decays in the framework of factorization scheme, where neglecting final state interactions, hadronic ma- trix elements are factorized into a product of two matrix elements of the form < B′ J B > and < 0 J ′ M > for which more information may be available. | µ| | µ| Following the phenomenological sucess of factorization in the heavy to heavy non-leptonic B-meson decays [1], this frame work has been extended to the domain of heavy to light transitions [2]. The factorization anstaz here introduces one free parameter, called ξ = 1/N ( N being number of colors), which is introduced to compensate for the neglect of color octet-octet contribution in evaluating the hadronic matrix elements in the heavy to light sectors. The range 0 ξ 0.5 has been found [2] to be consistent with ≤ ≤ data on a number of measured B meson decays. We apply the factorization to decays Λ ΛJ/ψ, Λ Λ D (D∗), Λ ΛD¯ and Λ p D . b −→ b −→ c s s b −→ b −→ s In addition, we use quark model to fix current coupling constants which appear in the matrix elements < B′ J B >. We show that the measured | µ| branching ratio for Λ ΛJ/ψ can be accounted for in this approach with b −→ the parameter ξ in the above mentioned range. Our estimates for branching ratios for Λ Λ D (D∗) are larger than their previous estimates [3, 4]. b −→ c s s The decays Λ ΛD¯ and Λ p D can give information on the CKM- b −→ b −→ s 2 Wolfenstein parameter (ρ2 + η2) [5] or V /V independent of ξ. | ub cb| We write the effective Hamiltonian[6]: GF ∗ c c Heff (∆B =1) = VcbVqs (C1O1 + C2O2) √2 "q=u,c X ∗ u u + VubVqs (C1O1 + C2O2 ) (1) q=u,c # X where Ci are Wilson coefficients evaluated at the renormalization scale µ; the current-current operators O1,2 are c α β O1 = (¯c bα) − s¯ qβ V A V −A c α β O2 = (¯c bβ) − s¯ qα (2) V A V −A u and Oi are obtained through replacing c by u. Here α and β are SU(3) color α α indices while (¯c bβ)V −A =¯c γµ(1+ γ5)bβ etc. The related Wilson coefficients at µ =2.5GeV in next-to-leading logarithmic (NLL) precision are [2] C1 = 1.117 C2 = 0.257 (3) − These are not very different from those at µ =5GeV in the leading logarith- mic approximation (LLA) [7]: C1(m )=1.11 and C2(m )= 0.26. b b − In the factorization scheme we encounter matrix elements of the form ′ ′ < B(p ) Jµ Bb(p) > =u ¯(p )Γµu(p) | | ′ =u ¯(p )i [(g (s) g (s)γ5) γ V − A µ +(fV (s)+ hA(s)γ5) σµν qν + i (h (s) f (s)γ5) q ] u(p) (4) V − A µ where Bb is a baryon, which contains b quark while B is any baryon not containing it. Here s = q2 = (p p′)2. In the heavy quark spin symmetry − − − limit [8], the vector and axial vector form factors are related [when Bb belongs to the triplet representation of flavor SU(3)] as follows: gV (s)= gA(s)= f1 (5) 3 1 fV = hV = hA = fA = f2 (6) − mBb For a decay of the type B (p) B(p′)+ X(p ), the matrix elements b −→ X are of the form ′ ′ G ′ ′ 1 mm T = i < 0 Jµ X(pX ) > u¯(p )Γµu(p) 3 ′ (7) √2 | | (2π) s popo In the rest frame of Bb, the decay rate of Bb and its polarization are given by ′2 G 1 ′ ρ σ Γ= 2 dsp (s) ρ(s)Γ (s)+ σ(s)Γ (s) (8) 2 4πm Z { } where q = p = p p′, s = q2 and X − − Γρ(s) = Q(s) g2 + g2 3mm′s g2 g2 V A − V − A n ′ ′ 2 +3s (m + m ) (m m ) s g f − − V V h ′ ′ 2 (m m ) (m + m ) s g f − − − A A ′′ 2 2 ′ 2i 2 +s Q (s) f + h 3mm s f h V A − V − A h ′ 2 ′2 i 2mp (s)n s (m m ) 2s g g − · − − A V +s ((m 3m′)hg h (m +3m′)g f ) − V A − A V +sf h s m2 5m′2 (9) V A − − io Γσ(s) = Q′(s) g2 + g2 s (m m′) (m + m′)2 s g f V A − − − V V n ′ ′h2 +(m + m ) (m m ) s g f − − A A 1 i + s2 (m + m′)2 s h2 + (m m′)2 s f 2 2 − V − − A h i 2mp′(s)n s (m2 m′2)g g − · − V A ′ h ′ 2 s ((m + m )g h +(m m )g g )+ s h f (10) − A V − V A V A io Here it is understood that form factors are functions of s and ρ = ρV , ρA or 0 − + − according as X is 1 , 1 or O , while correspondingly σ = 0, σA or σp·s and 1 2 1/2 p′(s) = m2 + m′2 s 4m2m′2 (11) 2m − − h i 4 1 2 Q(s) = m2 m′2 + s m2 + m′2 2s2 (12) 2 − − 1 2 Q′(s) = m2 m′2 s m2 + m′2 (13) 2 − − 1 2 Q′′(s) = 2 m2 m′2 s m2 + m′2 s2 (14) 2 − − − The form factors defined in Eq.(4) are calculated in quark model at s = q2 = m2 where X is a vector or pseudoscalar particle in the decay B − X −→ B′X, thereby taking into account recoil correction. This is in contrast to the use of the non-relativistic quark model for the evaluation of form factors at zero recoil q = 0 [9]. This latter approach also necessitates the extrapolation 2 2 2 of form factors from maximum q [ q = t =(m m ′ ) ] to the desired − m m B − B s = q2 = m2 . We may point out that since q 1.75GeV in Λ − X | | ≈ b −→ Λ + J/ψ, for example, the no recoil approximation does not seem to be justified; in fact q m in Λ, making the s quark in Λ relativistic. In | | ≫ s our approach no recoil approximation, nor any extrapolation of form factors at the physical point are needed. Our quark model results do satisfy the constraints imposed by heavy quark spin symmetry. The plan of the paper is as follows: Sec.11 summarizes the calculation of the baryonic form factors within the frame work of quark model at the desired value of s = q2, rather than at the zero recoil point, relegating the − details in the appendix. In section III we apply the results to some specific nonleptonic decay modes of Λb. Section IV summarizes our conclusions. 2 BARYONIC FORM FACTORS IN QUARK MODEL In order to calculate the form factors we first reduce the matrix elements in Eq.(4) from four component Dirac spinors to Pauli spinors without making any approximation and do the same for the quark level current jµ = iqγ¯ µ(1 + γ5)b (15) 5 We treat the b quark in Bb extremely non relativistically (pb/mb 0) and ′ 2 2 ≈ set p p = q = p , E = q + m , E = m = m3. Then as shown in b − q − q q b b the appendix: q h = f , f = h (16) A − A V V ′ ′ gV (s) = ξV Ia(E , E3) 1 f (s) = ξ Ib(E′, E′ ) V m V 3 ′ ′ gA(s) = ξAIa(E , E3) 1 f (s) = ξ Ib(E′, E′ ) (17) A −m A 3 where ′ ′ ′ ′ ′ m ′ ′ m ′ ′ 1 E (E + m ) 1 +(E3 + m3) 1+ a(E , E ) = − m m 3 ′ ′ ′ ′ ′ 2sE3 (E + m )(E3 + m3) ′ ′ ′ q ′ ′ ′ ′ 1 E (E + m ) (E + m ) b(E , E ) = − 3 3 (18) 3 ′ ′ ′ ′ ′ 2sE3 (E + m )(E3 + m3) q ′ ′ ′ ′2 ′2 ′ and E = po, E3 = Eq = p + m3 , and m3 = mq.
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