Few-Body SystDOI 10.1007/s00601-014-0871-x

C. Albertus

Semileptonic and Nonleptonic Decays of Bs Mesons

Received: 30 October 2013 / Accepted: 10 March 2014© Springer-Verlag Wien 2014

Abstract We investigate the semileptonic and nonleptonic decays of Bs mesons. We work within the contextof nonrelativistic constituent quark models. We calculate the different form factors that parameterize the hadronmatrix elements.

In this work we study the semileptonic decays of Bs mesons into Ds states and some of its nonleptonicdecays. We work in the nonrelativistic constituent quark model scheme. Since the discovery of the Bs and Bsmesons, their lifetimes and decay modes have been in the aim of both theoretical and experimental collaborationand, for instance, are currently being measured in the LHCb experiment at CERN [1–9].

The description of the different J P states involved is given in Ref. [10] and is omitted here for the shakeof brevity. The semileptonic decays are governed by the V-A b → c current. The matrix elements are parame-trized in terms of form factors as in Eq. (9) of Ref. [10] The calculation of the matrix elements and form factorscan be found in full detail in Ref. [10], as well.

The semileptonic double differential decay width with respect to q2 and the cosine, xl , of the angle betweenthe final meson momentum and the momentum of the final charged lepton, the latter measured in the lepton-neutrino center of mass frame (CMF), results to be, for a Bs at rest

d2Γ

dxldq2 = G2F

64m2Bs

|Vbc|28π3

λ1/2(q2,m2Bs,m2

cs)

2m Bs

q2 − m2l

q2 Hαβ(PBs, Pcs)Lαβ(pl , pν), (1)

where G F [11] is the Fermi constant, λ(a, b, c) = (a + b − c)2 − 4ab, ml is the mass of the charged lepton,H and L are the hadron and lepton tensors, and pl and pν are the lepton four momenta. PBs

and Pcs (withcs = Ds, D∗

s , D∗s2) are the meson four-momenta, m Bs

and mcs their masses respectively, P = PBs+ Pcs , and

q = PBs− Pcs . To calculate the contraction of the lepton and hadron tensors, that involve the form factors

from Eq. (9) of [10], we shall follow the helicity formalism of Ref. [12]. This calculation is given in full detailin Sec. III E of Ref. [10]. Integrating Eq. 1 we obtain the total decay widths. Numerical values can be foundin [10].

In this work we have also studied the nonleptonic Bs → cs MF two-meson decays, where MF is apseudoscalar or vector meson. These decays correspond to a b → c transition at the quark level as well. Thesetransitions are governed, neglecting penguin operators, by the effective Hamiltonian [13,14]

Heff = G F√2

(Vcb

[c1(μ)Q

cb1 + c2(μ)Q

cb2

]+ H.c.

), (2)

C. Albertus (B)Departamento de Física Atómica, Molecular y Nuclear, Facultad de Ciencias,Universidad de Granada, Avenida de Fuentenueva S/N, 37008 Granada, SpainE-mail: albertus@ugr.es

C. Albertus

Table 1 Branching fractions for the indicated decay channels, in percentage

BR in % BR in %Bs → D+

s e−νe 2.32 Bs → D+s π

− 0.53Bs → D∗+

s e−νe 6.26 Bs → D+s ρ

− 1.26Bs → D+

s τ−ντ 0.67 Bs → D+

s K − 0.04Bs → D∗+

s τ−ντ 1.53 Bs → D+s K ∗− 0.08

Bs → D∗+s0 μ

−νμ 0.39 Bs → D∗+s0 π

− 0.10Bs → D∗+

s1 (2,460)μ−νμ 0.60 Bs → D∗+s0 ρ

− 0.27Bs → D∗+

s1 (2,536)μ−νμ 0.19 Bs → D∗+s0 K − 0.009

Bs → D∗+s2 μ

−νμ 0.44 Bs → D∗+s0 K ∗− 0.16

where c1,2 are scale-dependent Wilson coefficients, and Q1,2 are local four-quark operators given by

Qcb1 =Ψc(0)γμ(I − γ5)Ψb(0)

[V ∗

ud Ψd(0)γμ(I − γ5)Ψu(0)+ V ∗

usΨs(0)γμ(I − γ5)Ψu(0)

+ V ∗cd Ψd(0)γ

μ(I − γ5)Ψc(0) +V ∗csΨs(0)γ

μ(I − γ5)Ψc(0)]

Qcb2 =Ψd(0)γμ(I − γ5)Ψb(0)

[V ∗

ud Ψc(0)γμ(I − γ5)Ψu(0)+ V ∗

cd Ψc(0)γμ(I − γ5)Ψc(0)

]

+ Ψs(0)γμ(I − γ5)Ψb(0)[V ∗

usΨc(0)γμ(I − γ5)Ψu(0)+ V ∗

csΨc(0)γμ(I − γ5)Ψc(0)

], (3)

where Vi j are CKM matrix elements. We follow the factorization approximation, i. e., the hadron matrixelements of the effective Hamiltonian are evaluated as a product of quark-current matrix elements. One ofthese is that of the Bs transition to one of the final mesons, while the other one corresponds to a transitionof the vacuum to the second final mesons, which is given by the corresponding meson decay constant. ForMF = π, ρ, K or K ∗ the decay width is given by

Γ = G2F

16πm2Bs

|Vbc|2|VF |2 λ1/2(m2

Bs,m2

cs,mMF )2

2m Bs

a21Hαβ(PBs , Pcs)Hαβ(PF ), (4)

where a1 ≈ 1.14 is a Wilson coefficient and VF is a CKM matrix elements which depends on the actual decayconsidered. As explained in Ref. [10], the contraction of the resulting tensors has been performed using thehelicity formalism.

Some results for the semileptonic and nonleptonic decay widths can be found in Table 1. More results andcomparison with previous calculations have been published in Ref. [10].

Acknowledgments The author thanks a contract granted by the CPAN Project and support from Universidad de Granada andJunta de Andalucía, through the Grant FQM-225.

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