Tests of the Parton Model for Inclusive Semileptonic B Decays with the Heavy Quark Effective Theory

In order to study the end-point spectrum of inclusive semileptonic b → u decays, we investigate the parton model motivated by deep inelastic scattering process. The reliability of the model is carefully examined by comparing the decay rates and the lepton spectra for b → c decay with those of the heavy quark effective theory. We also compute the decay rates of the inclusive semileptonic decay into tau lepton in terms of the parton model, which give another test of the model independently. We conclude that one can make the parton model consistent with the heavy quark effective theory, and it would be phenomenologically useful in extracting |Vub|. Typeset using REVTEX [email protected] [email protected] 1 Semileptonic decays of B mesons play an important role for understanding the Standard Model (SM). They are essential for testing and determining the parameters in the SM, and also provide valuable informations on the bound state structure of the mesons. Recently, there has been much progress in the study of inclusive semileptonic decays of b hadrons with the help of heavy quark effective theory (HQET) [1–3]. One can get a QCD-based expansion in powers of 1/mQ by performing an operater product expansion on the hadronic tensor. It has been shown that at leading order the contributions to the decay rates in the expansion coincide with those of the free quark decay. The terms of order of 1/m2Q are the leading corrections of the nonperturbative effects because terms of order of 1/mQ vanish. These enable one to obtain model-independent calculations of the decay rates and lepton spectra for most region of phase space. However, a model study is still required. The study of the end-point region of the electron spectrum in b → u decay where B̄ → Xclν̄ decays are forbidden is of great interest in the decay of B̄ → Xulν̄, because it is potentially useful to determine the CabbiboKobayashi-Maskawa (CKM) angle |Vub|. Unfortunately, the HQET calculation of the lepton spectrum for the inclusive semileptonic decay gives singularities in the end-point region [2,3] and modelling the end-point is inevitable for the extraction of |Vub| at present. Thus the following strategy may be presented: one picks a model, fixes its shape parameters from fitting the spectrum in B̄ → Xclν̄ and applies it to the end-point region from where one extracts |Vub/Vcb|. Up to now the most popular model for the inclusive semileptonic decays of heavy mesons is the model presented by Altarelli et al. [4], to be refered as ACCMM. They improve the free quark decay by considering the Fermi motion of the spectater quark in the meson. The ACCMM model has been reanalysed in the framework of the HQET recently by several authors [5]. They showed that the ACCMM model can be made to be consistent with the HQET by properly redefining the model. In the recent publication [6], a parton model motivated by the deep inelastic scattering process (DIS) was developed and the results of the model was compared with experimental results. It is well known that inclusive semileptonic decays of heavy flavours are intimately 2 related to DIS via channel crossing. The model shows a good agreement with the experimental data and yields well-behaved lepton energy spectrum at the end-point region where it may serve as a useful model. The parton model seems to be one of the useful models that can describe this region. In this letter, we investigate the reliability of the parton model approach by explicitly comparing the results of the model with those of HQET. For the model to be phenomenologically reliable, we require that the decay rates and the lepton spectra (at least far from the end-point region) coincide with those of the HQET. This requirement would fix the parameter of the model and we can compare the lepton spectra of the parton model with those of the HQET. For the test of consistency of the model, we also study the lepton mass effect. Theoretically the lepton mass does not affect the picture of the parton model and we expect that the parameters which we have fixed in the case of the decay into an electron as a final state should be also valid in the decay into a massive lepton. The lepton mass effects are significant in the semileptonic decay involving a tau lepton. The tau channel in semileptonic decays of B mesons is one of the latest measured process. Recently a branching ratio Br(B̄ → Xτν̄) has been measured in LEP [8]. Br(B̄ → Xτν̄) = 2.4± 0.7 (stat.)± 0.8 (syst.) % This mode is of some theoretical interest by several reasons, since the decay kinematics and dynamics become complicated in the decay involving a tau lepton. There are more structure functions which does not contribute to the decays involving an electron. Since these changes are independent of the CKM angle |Vcb| and |Vub|, computation of the ratio of the decay rates for the tau lepton channel to those for the electron channel is useful in testing the model. Phenomenologically the study of B → Xτν̄ is useful to determine the CKM angle |Vcb| independently of the decay into an electron. This channel is also interesting as being the window to show the effects of physics beyond the SM. It has been used to probe the charged Higgs boson signal and gives some constraints on parameters of the minimal supersymmetric standard model [9]. In the following we briefly review the parton model for inclusive semileptonic decays of 3 B mesons [6,7] and describe its extension to the decay of B → Xτν̄. The main ideas of this model are followings: i) it pictures the mesonic decay as the decay of the partons. We consider that the parton in the B meson carries a fraction of the B meson momentum. ii) The distribution of a parton in the B meson is determined by a single function, which will be discussed later. The hadronic tensor is defined as