CP violation, single lepton polarization asymmetry, and polarized CP asymmetry in B ! K  ‘ þ ‘  decay in the four-generation standard model V. Bashiry, 1, * N. Shirkhanghah, 2,† and K. Zeynali 3,‡ 1 Engineering Faculty, Cyprus International University, Via Mersin 10, Turkey 2 Islamic Azad University, Khalkhal Branch, Valiasr Street, Khalkhal, Iran 3 Faculty of Physics, University of Tabriz, Tabriz 51664, Iran (Received 23 April 2009; published 28 July 2009) In this paper we present a study of CP asymmetry, single lepton polarization asymmetry and polarized CP asymmetry in B ! K  ‘ þ ‘  decay within the four-generation standard model. Taking jV  t 0 s V t 0 b j¼ 0:01, 0.02, 0.03 with phase f60  –120  g, which is consistent with the b ! s‘ þ ‘  rate and the Bs mixing parameter m Bs , we find that CP asymmetry, single lepton polarization asymmetry and polarized CP asymmetry are sensitive to the existence of the fourth generation. This can serve as an indirect method to search for new physics effects, in particular, to search for the fourth-generation quarksðt 0 ;b 0 Þ via their indirect manifestations in loop diagrams. DOI: 10.1103/PhysRevD.80.015016 PACS numbers: 12.60.i, 13.30.a, 14.20.Mr I. INTRODUCTION The simple replication of chiral matter is a straightfor- ward extension of the standard model. This fourth genera- tion is often considered and ruled out or disfavored by many researchers. This extension has recently gained more attention (for the most recent studies see [1]) after mea- surement of direct CP-asymmetry of B ! K decays [2] and the nonvanished CP phase measured in b ! s transi- tion by CDF [3] and D0 [4]. These issues, in particular, the nonvanished CP phase measured in b ! s transition can- not be explained by 3  3 the Cabibbo-Kobayashi- Maskawa (CKM) matrix. A consequential extension of the three-generation standard model (SM3) to the four- generation standard model (SM4) has better solutions to some of the theoretical and experimental problems. Such extension can include extra weak phases in the quark mixing matrix. This might introduce a better solution to the problem related to baryogensis [5]. SM4 can also explain the direct CP-asymmetry of B ! K decays [6]. The reasons for discarding the fourth family are based on the interpretation of electroweak precision data and LEP II experiments. Kribs et al. indicated that the electroweak precision tests are not in conflict with the existence of the fourth family if the quark mass of the fourth generation satisfy the following relation [7]: m t 0  m b 0   1 þ 1 5 ln m H 115   55: (1) One of the consequences of their study is extension of the bounds on the Higgs mass [7]. Furthermore, a 4  4 CKM matrix includes more weak phases than the 3  3 CKM which leads to sizable increases in the calculation of CP asymmetry [8,9]. The 4  4 CKM matrix clearly af- fects flavor physics [10–14]. Considering many aspects of the fourth family, some of which are discussed above, the search for the fourth family must be included in the LHC. In this study, we enlarge the standard model with a complete sequential fourth family of the chiral quark. The fourth quark can contribute in the flavor-changing neutral processes (which are loop induced in the SM) like the other three generations. Thus, the mass and mixing parameters of 4  4 CKM can change the rate of the physical observables, with respect to the three-generation standard model (SM3) calculations. b ! s‘ þ ‘  is a loop- induced transition in the SM. Like u, c, t quarks, the fourth family t 0 quark can contribute in this transition. B ! K  ‘ þ ‘  decay in the quark level is described by b ! s‘ þ ‘  transition. B ! K  ‘ þ ‘  decay has been widely studied within SM and beyond [15–23]. The various ob- servables of this decay are going to be measured at the LHCb. Here, we study the CP asymmetry, single lepton polarization asymmetry and polarized CP asymmetry in B ! K  ‘ þ ‘  decay in the SM4. Note that single and double lepton polarization asymmetries have been studied in the SM3 and in the model independent approach by Refs. [24,25]. The outline of the paper is as follows: In Sec. II, we calculate the decay amplitude and single lepton polariza- tion CP asymmetry of the B ! K  ‘ þ ‘  decay within SM4. In Sec. III we present the calculation of unpolarized and polarized CP asymmetry. Section IV is devoted to the numerical analysis and discussion of the considered tran- sition as well as our conclusions. II. STRATEGY In this section, we present the theoretical expressions for the decay widths within SM4. As we mention above, t 0 can contribute to the b ! s transition as u, c and t quarks do. * bashiry@ciu.edu.tr † nasershirkhanghah@yahoo.com ‡ k.zeinali@arums.ac.ir PHYSICAL REVIEW D 80, 015016 (2009) 1550-7998= 2009=80(1)=015016(8) 015016-1 Ó 2009 The American Physical Society