Fourth-generation effects on the rare B → K*νν̄ decay

The rare B→K*νν̄ decay is analysed with a new up-like quark in a sequential fourth-generation model. Two possible solutions for the fourth-generation CKM (Cabibbo-Kabayashi-Maskawa) factor obtained as a function of the new -quark mass are used. The branching ratio (BR) and missing-energy spectrum of this decay in the two cases are estimated. In one case, it is shown that for GeV a significant enhancement to the BR and the missing-energy spectrum of this decay over the SM (standard model) is recorded, while the results are almost same in the other case. If a fourth generation should exist in nature and nature chooses this case, this B-meson decay mode could be a good probe for the existence of the fourth generation, or perhaps a signal for a new physics.


Introduction
The SM has been widely discussed in the literature, and serves as an explicit model for studying all low-energy experimental data. But there is no doubt that the SM is an incomplete theory. Among the unsolved problems within the SM is the CP violation, and the number of generations. In the SM there are three generations, and yet there is no theoretical argument to explain why there are three and only three generations of fermions in the SM. From the LEP result of the invisible partial decay width of the Z boson it follows that the mass of an extra-generation neutrino N should be larger than 45 GeV [1]. Having this experimental result in mind, and if we believe that the fourth-generation fermions really exist in nature, we should be able to find their mass spectrum, and take into account their physical effects in low-energy physics.
One of the promising areas in the experimental search for a fourth generation via its indirect loop effects is that of the rare B-meson decays. The experimental observation of the inclusive b → X s γ [2], and exclusive B → K * γ [3] decays, together with the recent Babar 25.2 Collaboration [4] upper limit on the exclusive decay BR(B → K * + − ) < 2.9 × 10 −6 , and the Belle Collaboration [5] result BR(B → K + − ) = (0.75 +025 −0.21 ± 0.9) × 10 −6 , at SLAC and KEK laboratories, respectively, will allow us to set up a complete programme to test the SM properties at the loop level and constrain various new physics scenarios.
On this basis, serious attempts to study the effects of the fourth-generation fermions on the rare B-meson have been made by many authors. For example, the effect of the fourth-generation fermions on ∆M B d,s in B 0 -B 0 mixing is discussed in [6]. A prediction of ∆M Bs , and constraints on a new fourth-generation CKM matrix factor V * ts Vt b are obtained, from ∆M B d . In [7] the possibility of a fourth generation in the minimal supersymmetric standard model (MSSM) is explored. It is shown that the new generation must have masses mν, mτ < 86 GeV, mt < 178 GeV, and mb < 156 GeV to ensure that the MSSM remains perturbative up to the unification scale M U of the Yukawa couplings. The implications of a fourth generation of quarks on the process b → s have been previously investigated in [8,9], and it is shown that in the fourth-generation model the b → sγ branching ratio (BR) is essentially within the range allowed by CLEO [9]. On the other hand, the B → X s + − , and the B → X s γ decays with the fourth-generation fermions are analysed in [10]. Recently, the fourth-generation effects on the rare decays B → K * + − [11], B → + − , B → + − γ [12], and B s → ννγ [13] were studied.
The exclusive decay B → K * νν provokes special interest in the SM and beyond [14]- [18]. In particular, the SM has been exploited to establish a bound on the BR of the abovementioned decay of the order ∼10 −5 , which is quite measurable at KEK and SLAC B-factories, where it is hoped that a definite answer on possible fourth-generation fermions at the KEK and SLAC B-factories will be found; this year the upgraded B-factories are providing us with first experimental data.
Therefore, in this work we will investigate the rare B → K * νν decay and the existence of a new up-like quarkt in a sequential fourth-generation model SM, which we shall call SM4 hereafter for the sake of simplicity. This model is considered as a natural extension of the SM, where the fourth-generation model is introduced in the same way as the three generations are introduced in the SM [19], so no new operators appear, and clearly the full operator set is exactly the same as in the SM. Hence, the fourth generation will change only the values of the Wilson coefficients via virtual exchange of the fourth-generation up-like quarkt.
This paper is organized as follows. In section 2, the relevant effective Hamiltonian for the decay B → K * νν and the existence of a new up-like quarkt in a sequential fourth-generation model (SM4) is presented; therein the dependence of the BR and the missing-energy spectrum on the fourth-generation model parameters for the decay of interest are investigated using the results of the light-cone QCD sum rules for estimating form factors. Section 3 is devoted to the numerical analysis and concluding remarks.

Effective Hamiltonian
In the SM, the process B → K * νν is described at quark level by the b → sνν transition, and receives contributions from Z-penguin and box diagrams, where dominant contributions come from intermediate top quarks. The effective Hamiltonian responsible for b → sνν decay is described by only one Wilson coefficient, namely C (SM ) 11 , and its explicit form is [20]

25.3
where G F is the Fermi coupling constant, α is the fine-structure constant (at the Z mass scale), and V * ts V tb are products of Cabibbo-Kabayashi-Maskawa matrix elements. In equation (1), the Wilson coefficient C SM 11 in the context of the SM has the following form including O(α s ) corrections [21]: with where x t = m 2 t /m 2 W , and Here From the theoretical point of view, the transition b → sνν is a very clean process, since it is practically free from scale dependence, and free from any long-distance effects. In addition, the presence of a single operator governing the inclusive b → sνν transition is an appealing property. Therefore, the theoretical uncertainty within the SM is just related to the value of the Wilson coefficient C SM 11 due to the uncertainty in the top-quark mass. In this work, we have considered possible new physics in b → sνν only through the value of the Wilson coefficient.
In this spirit, the transition b → sνν in equation (1) can only include extra contribution due to the fourth-generation fermion mt; hence, the fourth-generation fermion contribution modifies only the value of the Wilson coefficient C SM 11 , and it does not induce any new operators: where C (new) can be obtained from C SM 11 by making the substitution m t → mt. As a result, we obtain a modified effective Hamiltonian, which represents b → sνν decay in the presence of the fourth-generation fermion: However, in spite of such theoretical advantages, it would be a very difficult task to detect the inclusive b → sνν decay experimentally, because the final state contains two missing neutrinos and many hadrons. Therefore, only the exclusive channels, namely B → K * (ρ)νν, are well suited for consideration in the search for possible 'new physics' effects and constraints.
In order to compute B → K * νν decay, we need the matrix elements of the effective Hamiltonian, equation (6), connecting the final and initial meson states. This problem is related to the non-perturbative sector of QCD and can be solved only by using non-perturbative methods. The matrix element K * |H ef f |B has been investigated in a framework of different approaches, 25.4 such as chiral perturbation theory [22], three-point QCD sum rules [23], the relativistic quark model with the light front formalism [24], effective heavy-quark theory [25], and light-cone QCD sum rules [26,27]. To begin with, let us denote by P B , and P K * the four-momentum of the initial and final mesons, and define q = P B − P K * as the four-momentum of the νν pair, and x ≡ E miss /M B as the missing-energy fraction, which is related to the squared four-momentum transfer q 2 by with M B and M K * being the initial and final meson masses. Then, the hadronic matrix element for B → K * νν can be parametrized in terms of five form factors: where (h) is the polarization four-vector of the K * -meson. The form factor A 3 (q 2 ) can be written as a linear combination of the form factors A 1 and A 2 : After performing summation over K * -meson polarization and taking into account the number of light neutrinos N ν = 3 for the differential of the decay width, one can get [20] where λ(1, r K * , s) = 1 + r 2 K * + s 2 − 2r K * s − 2r K * − 2s is the usual triangle function, and s = q 2 /M 2 B . From equation (7), it is easy to derive the missing-energy distribution corresponding to the helicity h = 0, ±1 of the K * -meson: From equations (9) rules in [26]- [28]. However, in this work, in estimating the total decay width, we have used the results of [28], where these form factors were calculated by including one-loop radiative corrections to the leading twist-two contribution: and the relevant values of the form factors at q 2 = 0 are and V B→K * (q 2 = 0) = 0.46 ± 0.07, with a F = 1.55, and b F = 0.575.
Note that all errors which come out are due to the uncertainties of the b-quark mass; the Borel parameter variation, wavefunctions, and radiative corrections are added in quadrature. Finally, to obtain quantitative results one needs the values of the fourth-generation CKM matrix elements V * ts Vt b . Following [6], the values of the fourth-generation CKM factor V * ts V ± tb due to the masses oft are listed in table 1.
A few comments about the numerical values of (V * ts Vt b ) ± are in order. From the unitarity condition for the CKM matrix we have  When the values of the CKM matrix elements in the SM are used [29], the sum of the first three terms in equation (16) is about 7.6 × 10 −2 . Substituting in the values of (V * ts Vt b ) + from table 1, it is observed that the sum of the four terms on the left-hand side of equation (16) in this case is much better, and much closer to zero than that in the SM, because the value of (V * ts Vt b ) + is very close to the sum of the first three terms, but has the opposite sign. This holds true, and is clear when mt ≥ 200 GeV, and this may be the most direct lower bound on the mass oft. If one considers (V * ts Vt b ) − , whose value is about 10 −3 , which is one order of magnitude smaller compared to the previous case, it is easy to see then that the values of (V * ts Vt b ) − also satisfy the unitarity condition of CKM, and the error in the sum of the four terms in equation (16) 25.8 is within the SM error range. Therefore, the (V * ts Vt b ) − contribution to the physical quantities should be practically indistinguishable from SM results, and our numerical analysis will confirm this expectation. On the other hand, there are no available upper bounds on the mass of thet, but one generally expects to have mt ≤ 1 TeV in order for the perturbation theory to remain valid. The large mt-values will lead on to a detailed discussion of the ρ 0 -parameter [30].

Numerical analysis
In studying the influence of the fourth-generation model parameters on the BR BR(B → K * νν), the missing-energy spectra, and BRs of rare B → K * L νν and B → K * T νν decays (where K * L , and K * T stand for longitudinally and transversely polarized K * -mesons, respectively), the following values have been used as input parameters: G F = 1.17 × 10 −5 GeV −2 , α = 1/137, µ b = m b = 5.0 GeV, M B = 5.28 GeV, |V * ts V tb | = 0.045, M K * = 0.892 GeV, and the lifetime is taken as τ (B d ) = 1.56 × 10 −12 s [30]; also we have run calculations of equations (9)-(11) adopting the two sets of values of (V * ts Vt b ) ± from table 1. The numerical results for the BR and the missing-energy spectra are presented in series of graphs. The BR for B → K * νν decay as a function of mt, with the different values of (V * ts Vt b ) ± , is shown in figure 1. It can be seen that when V * ts Vt b takes positive values, i.e. for (V * ts Vt b ) − , the BR almost matches that of the SM. That is, the results in SM4 are the same as those in the SM, except a peak in the curve when mt takes values mt ≥ 210 GeV. The reason for this is not that there is new predicted deviation from SM,  Figure 6. The dependence of the differential of the BR of the decay B → K * νν on the missing-energy fraction x at a fixed value of mt when K * is polarized transversely.
but just the second term of equation (5). In this case, it does not show the new effects of mt. Also, we cannot establish the existence of the fourth generation from the BR for B → K * νν, although we cannot exclude the possibility of existence either. This is because, as seen from table 1, the values of (V * ts Vt b ) − are positive. But they are of order 10 −3 , i.e. very small. The values of V * ts V tb are about ten times larger than V * ts = 0.045, V tb = 0.9995 (see [30]). But in the second case, when the values of V * ts Vt b are negative, i.e. for (V * ts Vt b ) + , the curve of BR for B → K * νν is quite different from that of the SM. This can be clearly seen from figure 1. The BR increases rapidly with increase of mt. In this case, the fourth-generation effects are shown clearly. The reason is that (V * ts Vt b ) + is 2-3 times larger than V * ts V tb , so the last term in equation (5) becomes important, and it depends on thet mass strongly. Thus the effect of the fourth generation is significant. In figure 2, we show the differential of the BR, dBR(B → K * νν)/ds, as functions of s (0 ≤ s ≤ (1 + √ r K * ) 2 ) when mt = 300 GeV. The curve of the differential decay width is quite different from that of the SM when one considers (V * ts Vt b ) + . This can be clearly seen from figure 2. The differential decay width increases rapidly, and the energy spectrum of the K * -meson is almost symmetrical. In figure 3, the ratio R = BR SM 4 (B → K * νν)/BR SM (B → K * νν) is depicted as a function of (V * ts Vt b ) ± for various values of mt. Figure 3 shows that for all values of mt ≥ 210 GeV, the value of R becomes >1, meaning that the value of R = 1 is shifted. In other words, by defining the position for which R = 1, information can be obtained about mt, the mass of the fourth-generation fermion. For completeness, we also consider the ratio R1 = BR SM 4 (B → K * νν)/BR SM (B →X s νν). This ratio is plotted as a function of (V * ts Vt b ) ± for various values of mt in figure 4. It is well known that the inclusive decay width in the SM corresponding to B → X s νν is given as (see [20])

25.10
where the theoretical uncertainties related to the b-quark mass dependence disappear. In equation (17) the factor 3 corresponds to the number of light neutrinos. The phase space factor f (m c /m b ) 0.44, the QCD correction factors η 0 0.87,η = 1 + (2α s (m b )/3π)( 25 4 − π 2 ) 0.83 [21], and the experimental measured value BR(B → X c lν) = 10.14%. In this figure, the SM4 prediction on R1 for all the values of mt is greater than the SM prediction. This means that the SM4 contributes constructively to the decay width. In figures 5 and 6, we show the missingenergy distribution for the decays dBR(B → K * L νν)/dx and dBR(B → K * T νν)/dx as functions of x; (1 − r K * )/2 ≤ x ≤ 1 − √ r K * for mt = 250 GeV. It can be seen there that, when V * ts Vt b takes positive values, i.e. for (V * ts Vt b ) − , the missing-energy spectrum almost matches that of the SM. That is, the results in SM4 are the same as those in the SM. But in the second case, when the values of V * ts Vt b are negative, i.e. for (V * ts Vt b ) + , the curve of the missing-energy spectrum is quite different from that of the SM. This can be clearly seen from figures 5 and 6. The enhancement of the missing-energy spectrum is rapid, and the missing-energy spectrum of the K * -meson is almost symmetrical. In figures 7 and 8, the BRs BR(B → K * L νν) and BR(B → K * T νν) are depicted as a function of mt. mt. In this case, the fourth-generation effects are shown clearly, whereas in our approach, the predictions for the ratio (B → K * L νν)/(B → K * T νν), as well as the transverse asymmetry A T , are model independent.
In conclusion, the BR and the missing-energy spectra of the rare exclusive semileptonic B → K * νν decay have been investigated in a sequential fourth-generation model. The effects of a possible fourth-generation fermiont-quark mass have been considered, and sensitivities of the BR and the missing-energy spectra to thet-quark mass are observed.
Finally, note that the results for B → ρνν decay can be easily obtained from B → K * νν if the following replacement is made in all equations: V tb V * ts is replaced by V tb V * td and M K * is replaced by M ρ . In viewing these results, one must keep in mind that the values of the form factors for the B → ρ transition generally differ from those for the B → K * transition. However, these differences must be in the range of SU(3) violation, namely of the order 15-20%.