$B \to K \etap$ decays in the SM with fourth generation fermions

By employing the perturbative QCD (pQCD) factorization approach, we calculate the new physics contributions to the four $B \to K \etap$ decays in the Standard Model (SM) with a fourth generation of fermions (SM4), induced by the loop diagrams involving $\tp$ quark. Within the considered parameter space of the SM4 we find that (a) the next-to-leading order (NLO) pQCD predictions for the branching ratios and CP-violating asymmetries in both the SM and SM4 generally agree with the data within one standard deviation; (b) for $Br(B \to K \eta)$, the inclusion of the fourth generation contributions can improve the agreement between the theoretical predictions and the data effectively; (c) for $Br(B \to K \etar)$, however, the decrease due to $\tp$ loops is disfavored by the data; and (d) the new physics corrections to the CP-violating asymmetries of the considered decays are about 10% only.


I. INTRODUCTION
As a simple extension of the standard model(SM), the standard model with the fourth generation fermion (SM4) was rather popular in 1980s [1][2][3][4]. But unfortunately, the direct searches at the LHC experiments [5][6][7] have not found any signs of the heavy fourth generation t ′ and b ′ quarks so far. The phenomenological studies of the electroweak precision observables (EWPOs) [8][9][10] and some B meson rare decays [11][12][13][14][15] also resulted in some constraints on the parameter space of the SM4. The observation of the SM Higgs boson at a mass of 126GeV as reported by the CMS and ATLAS Collaboration [16,17] leads to very strong limits on the SM4: it was claimed [18,19] that the SM4 was ruled out at 5.3σ by the Higgs data. But the authors of Ref. [20] also point out recently that a SM4 with two-Higgs-doublets (4G-2HDM) can explain current 126GeV Higgs signals. The loop diagrams (box or penguins) involving the fourth generation fermions t ′ and b ′ , as is well-known, can provide new physics(NP) corrections to the branching ratios and CP violating asymmetries of B meson decays, such as the B → Kη (′) decays. At present, it is still interesting to study the possible new physics effects to those well measured B meson rare decays and to draw additional constraints on the SM4 from the relevant phenomenological analysis. Such constraints are complimentary to those obtained from the EWPOs and/or the Higgs data.
The B → Kη (′) decays are penguin dominated decays, and have been studied intensively by many authors for example in Refs. [21][22][23][24][25] in the framework of the SM or various new physical models. These four decays are studied very recently [26] by employing the perturbative QCD (pQCD) factorization approach with the inclusion of all known next-to-leading order (NLO) contributions from different sources, and the NLO pQCD predictions for both the branching ratios and the CP violating asymmetries agree well with the precision experimental measurements [27,28].
In this paper, we will study the possible loop contributions induced by the heavy t ′ quark appeared in the SM4. We will focus on the following points: 1. Besides all the known NLO contributions already considered in Ref. [26], we here will consider the effects of the t ′ contributions to the relevant Wilson coefficients as presented in Refs. [11,13,14] on the B → Kη (′) decays in the conventional Feldmann-Kroll-Stech (FKS) η − η ′ mixing scheme [29].
2. We will check the SM4 parameter-dependence of the pQCD predictions for the branching ratios and CP-violating asymmetries, such as those The paper is organized as follows. In Sec. II, we give a brief review for the pQCD factorization approach and the SM4 model. In Sec. III, we will make numerical calculations and present the numerical results. A short summary will be given in the final section.

II. THEORETICAL FRAMEWORK
For the charmless B → Kη (′) decays, the corresponding weak effective Hamiltonian can be written as [30]: where q = d, s, G F = 1.16639 × 10 −5 GeV −2 is the Fermi constant, O i (i = 1, ..., 10) are the local four-quark operators [30]. The Wilson coefficients C i in Eq. (1) and the corresponding renormalization group evolution matrix are known currently at LO and NLO level [30].
In the B-rest frame, we assume that the light final state meson M 2 and M 3 ( here M i refers to K or η (′) ) is moving along the direction of n = (1, 0, 0 T ) and v = (0, 1, 0 T ), respectively. Using the light-cone coordinates, the B meson momentum P B and the two final state mesons' momenta P 2 and P 3 (for M 2 and M 3 respectively) can be written as while the anti-quark momenta are chosen as where r i = m i /M B with m i is the mass of meson M i , and x i refers to the momentum fraction of the anti-quark in each meson. After making the same integrations over the small components k − 1 , k − 2 , and k + 3 as in Ref. [26] we obtain the decay amplitude conceptually where b i is the conjugate space coordinate of k iT . In above equation, The function S t (x i ) and e −S(t) are the threshold and K T Sudakov factors which suppresses the soft dynamics effectively [31].
In pQCD approach, the B meson is treated as a very good heavy-light system. Following Ref. [32], we can write the wave function of B meson as the form of Here we adopted the widely used B-meson distribution amplitude where the normalization factor N B depends on the value of ω b and f B and defined through the normalization relation . We here also take the shape parameter ω b = 0.4 ± 0.04 GeV. For the final state kaon and η (′) mesons, we use the same wave functions and distribution functions as those used in Ref. [26].
In the SM4 model, the classic 3 × 3 Cabibbo-Kobayashi-Maskawa (CKM) matrix is extended into a 4 × 4 CKM-like mixing matrix [33] with the t ′ and b ′ denote the fourth generation up-and down-type quark.
In the SM4, the t ′ quark play a similar role as the top quark in the loop diagrams and will provide new physics terms, such as [34]. When the new physics contributions are taken into account, the ordinary SM Wilson coefficients C i (M W ) will be changed accordingly.
In the SM4, one can generally write the Wilson coefficients as the combination of the SM part and the additional fourth generation contribution [33] As mentioned in previous sections, the three new physics input parameters in the SM4 include λ t ′ , φ t ′ and m t ′ .
For the mixing scheme of η −η ′ , we here use conventional FKS scheme [29] in the quark-flavor basis: η q = (uū + dd)/ √ 2 and η s = ss; where φ is the mixing angle, and the mixing parameters are defined as The relation between the decay constants (f q η , f s η , f q η ′ , f s η ′ ) and (f q , f s , ) can be found in Ref. [25]. The chiral enhancement m q 0 and m s 0 have been defined in Ref. [35] by assuming the exact isospin symmetry m q = m u = m d . The three input parameters f q , f s and φ in the FKS mixing scheme have been extracted from the data of the relevant exclusive processes [29]: with f π = 0.13 GeV.

A. NLO contributions in the pQCD approach
In Ref. [26], the authors studied the four B → Kη (′) decays with the inclusion of all known NLO contributions by using the pQCD factorization approach. For the SM part of the relevant decay amplitudes we use the formulaes as presented in Ref. [26], where the authors confirmed numerically that the still unknown NLO contributions from the relevant spectator and annihilation diagrams are indeed small in size and can be neglected safely.
In this paper, we will take all known NLO contributions as considered in Ref. [26] into account. For the sake of the reader, we list these NLO contributions as follows: (1) The NLO Wilson coefficients C i (m W ) and the NLO renormalization group evolution matrix U(t, m, α) as defined in Ref. [30], and the strong coupling constant α s (t) at two-loop level.
For the explicit expressions of the decay amplitudes for the four B → Kη (′) decays and the relevant functions, one can see Ref. [26]. We here focus on the NP contributions from the heavy t ′ quark.
We use the following input parameters [27,28] For the CKM quark-mixing matrix in the SM, we adopt the Wolfenstein parametrization as given in Ref. [27,28] and take A = 0.832, λ = 0.2246,ρ = 0.130 ± 0.018,η = 0.350 ± 0.013. For the three NP parameters, we choose similar values as in Ref. [13]: We here firstly calculate the branching ratios of the considered decay modes in both the SM and SM4 by employing the pQCD factorization approach. In the B-rest frame, the branching ratio of a general B → M 2 M 3 decay can be written as where τ B is the lifetime of the B meson, χ ≈ 1 is the phase space factor and equals to unit when the masses of final state light mesons are neglected. When all currently known NLO contributions are included, we find the pQCD predictions for Br(B → Kη (′) ) in the SM4 (in unit of 10 −6 ): In Table I, we list the NLO pQCD predictions in the framework of the SM (column two) or the SM4 (column three). In column four we show the NLO SM predictions based on the QCD factorization (QCDF) approach as given in Ref. [36]. And finally, the world averaged values of experimental measurements [27] are given in the last column. The SM predictions in the column two of Table I agree perfectly with those as given in Ref. [26] when the ordinary FKS η − η ′ mixing scheme was employed. The theoretical errors labeled with "SM" or "NP" denote the quadrature combination of the theoretical errors from the uncertainties of two sets of input parameters (ω b , m s , f B , a η 2 ) and (|λ t ′ |, φ t ′ , m t ′ ), respectively. From the numerical results as shown in Eq. (15) and Table I, we find the following points:  TABLE I. The NLO pQCD predictions for the branching ratios (in unit of 10 −6 ) in the framework of the SM (column two) and SM4 ( column three). As a comparison, the QCDF predictions [36] and the measured values [27] are also listed in the last two columns. 1. The pQCD predictions for Br(B → Kη (′) ) become smaller than the SM ones after the inclusion of the new physics contributions due to the destructive interference between the SM and NP contributions, but they still agree with the measure values within one standard deviation Since the theoretical errors are still large.

For
Br(B 0 → K 0 η) ( Br(B + → K + η)), the NP decrease of the central value of the pQCD prediction is about 40% (10%). The agreement between the theoretical predictions for Br(B → Kη) is improved effectively after the inclusion of NP contributions.

For Br
, however, the NP decrease is about 23% (12%), but such changes are disfavored by the data.
Although the four B → Kη (′) decays are generally penguin-dominated decays, the relative strength of the penguin part against the tree and/or other parts can be rather different for different decay modes. The explicit numerical calculations tell us that the penguin contribution play a more important rule in B 0 → K 0 η decay than in other three decay modes in consideration, the t ′penguins consequently provide a much larger modification to Br(B 0 → K 0 η) ( a decrease about 40% ) than to other decays ( a decrease from 10% to 23% in magnitude).

C. CP-violating asymmetries in SM4
Now we turn to the CP-violating asymmetries of B → Kη (′) decays in pQCD approach. For B ± → K ± η decays, there is a large direct CP asymmetry (A dir CP ), due to the destructive interference between the penguin amplitude and the tree amplitude. The NLO pQCD predictions for the direct CP asymmetries (in units of 10 −2 ) A dir CP (B ± → K ± η) and A dir CP (B ± → K ± η ′ ) in the SM (column two) and the SM4 (column three)are listed in Table II, the QCDF predictions and the data as given in Refs. [27,36] are also given in last two columns as a comparison.
As to the CP-violating asymmetries for the neutral decays B 0 → K 0 η (′) , the effects of B 0 − B 0 mixing should be considered. The explicit formulae for the CP-violating asymmetries of B 0 (B 0 ) → K 0 η (′) decays can be found easily, for example, in Ref. [26], we here make the numerical calculations and then show the NLO pQCD predictions for the direct and mixing-induced CP asymmetries in Table III. The theoretical errors labeled with "SM" or "NP" have specified previously.
From the numerical results as listed in Table II and III, one can see that the new physics effects on the pQCD predictions for the CP-violating asymmetries of the considered four decays are generally much smaller than the theoretical errors.

IV. SUMMARY
In this paper, we calculated the new physics contributions to the four B → Kη (′) decays in the SM4. From our numerical calculations and phenomenological analysis, we find the following points 1. In both the SM and SM4, the pQCD predictions for the branching ratios and CP-violating asymmetries agree with the data within one standard deviation, of course, partially due to the still large theoretical errors.
2. For Br(B 0 → K 0 η) and Br(B + → K + η), the NP decrease is about 40% and 10% respectively, the agreement between the theoretical predictions and the data is improved effectively after the inclusion of NP contributions.
3. For Br(B 0 → K 0 η ′ ) and Br(B + → K + η ′ ), however, the NP decrease is about 23% and 12% respectively, but such changes are disfavored by the data. 4. The new physics corrections on the CP-violating asymmetries of the considered decays are about 10% only.