Research Article | Open Access
Study on the Weak Decays
Motivated by the prospects of the potential particle at high-luminosity heavy-flavor experiments, we studied the weak decays, where = , , . The nonfactorizable contributions to hadronic matrix elements are taken into consideration with the QCDF approach. It is found that the CKM-favored decay has branching ratio of , which might be measured promisingly by the future experiments.
The first evidence for upsilons, the bound states of , was observed in collisions of protons with a stationary nuclear target at Fermilab in 1977 [1, 2]. From that moment on, bottomonia have been a subject of intensive experimental and theoretical research. Some of the salient features of upsilons are as follows . In the upsilons rest frame, the relative motion of the quark is sufficiently slow. Nonrelativistic Schrödinger equation can be used to describe the spectrum of bottomonia states and thus one can learn about the interquark binding forces. The particles, with the radial quantum number , and , decay primarily via the annihilation of the quark pairs into three gluons. Thus the properties of the invisible gluons and of the gluon-quark coupling can be gleaned through the study of the upsilons decay. Compared with the , , and light quarks, the relatively large mass of the heavy quark implies a nonnegligible coupling to the Higgs bosons, making upsilons one of the best hunting grounds for light Higgs particles. By now, our knowledge of the properties of bottomonia comes primarily from annihilation.
The particle is the ground state of the vector bottomonia with quantum number of . The mass of the particle, , is about three times heavier than the mass of the particle (the ground state of charmonia with the same quantum number of ). On one hand, compared with the decay, much richer decay channels could be accessed by the particle. On the other hand, the coupling constant for the decay is smaller than that for the decay due to the QCD nature of asymptotic freedom, which results in hadronic partial width , although the possible phase space in the decay is larger than that in the decay. In addition, the squared value of the quark charge, , is less than that of the quark charge, , which results in electromagnetic partial width . So the full decay width of the particle, , is less than that of the particle, . Furthermore, one of the outstanding properties of all upsilons below threshold is their narrow decay width of tens of keV .
The and particles share the similar decay mechanism. As is the case for the particle, strong decays of the particle are suppressed by the phenomenological OZI (Okubo-Zweig-Iizuka) rules [5–7], so electromagnetic interactions and radiative transitions become competitive. It is expected that, at the lowest order approximation, the decay modes of the particle could be subdivided into four types. The lion’s share of the decay width is the hadronic decay via the annihilation of the quark pairs into three gluons, that is, some via . The partial width of the electromagnetic decay via the annihilation of the quark pairs into a virtual photon could be written approximately as , where the value of is the ratio of inclusive production of hadrons to the pair production rate at the energy scale of and is the partial width of decay into dileptons. Branching ratio of the radiative decay is about . The up-to-date research for light Higgs bosons in the radiative decay has been performed by CLEO , Belle , and BaBar  collaborations. The magnetic dipole transition decay, , is very challenging to experimental physicist due to the very soft photon and pollution from other processes, such as . The experimental signal for has not been discovered until now. Besides, the particle could also decay via the weak interactions, although the branching ratio for a single or quark decay is tiny, about . In this paper, we will estimate the branching ratios for the flavor-changing nonleptonic weak decays with the QCD factorization (QCDF) approach [12, 13], where , and . The motivation is listed as follows.
From the experimental point of view, there is plenty of upsilons at the high-luminosity dedicated bottomonia factories, for example, over at Belle (see Table 1). Upsilons are also observed by the on-duty ALICE , ATLAS , CMS , and LHCb  experiments at LHC. It is hopefully expected that more upsilons could be accumulated with great precision at the running upgraded LHC and forthcoming SuperKEKB. The huge data samples will provide good opportunities to search for the weak decays which in some cases might be detectable. Theoretical studies on the weak decays are just necessary to offer a ready reference. For the two-body decay, final states with opposite charges have definite energies and momenta in the center-of-mass frame of the particle. Particularly, identification of a single charged meson in the final state, which is free from inefficiently double tagging of the bottomed hadron pairs occurring above the threshold, would provide an unambiguous signature of the weak decay. Of course, small branching ratios make the observation of the weak decays extremely difficult, and evidences of an abnormally large production rate of single mesons in the decay might be a hint of new physics beyond the standard model.
From the theoretical point of view, the bottom-changing upsilon weak decay could permit overconstraining parameters obtained from meson decay, but few studies are devoted to the nonleptonic upsilon weak decay in the past. For the decay, the amplitude is usually treated as the factorizable product of two independent factors: one describing the transition between heavy quarkonium and and the other depicting the production of the state from the vacuum. Previous works, such as  based on the spin symmetry and nonrecoil approximation,  based on the heavy quark effective theory, and  based on the Bauer-Stech-Wirbel (BSW) model , concentrated mainly upon the transition form factors which are related to the space integrals of the meson wave functions. As is well known, there exist hierarchical scales with nonrelativistic quantum chromodynamics (NRQCD) [25–27] which is an approach to deal with the heavy quarkonium; that is, , where is the mass of heavy quark with typical velocities . The physical contributions at scales of are absorbed into wave functions of the and particles; thus the transition should be dominated by the nonperturbative dynamics. The nonfactorizable contributions above scales of have not been taken seriously in previous works. About 2000, Beneke et al. proposed the QCDF approach [12, 13], where nonfactorizable contributions could be estimated systematically with the perturbation theory based on collinear factorization approximation and power countering rules in the heavy quark limit , and the QCDF approach has been widely applied to nonleptonic meson decays. So it should be very interesting to study the weak decays by considering nonfactorizable contributions with the attractive QCDF approach.
This paper is organized as follows. In Section 2, we will present the theoretical framework and the amplitudes for the nonleptonic two-body weak decays with the QCDF approach. Section 3 is devoted to numerical results and discussion. The last section is our summary.
2. Theoretical Framework
2.1. The Effective Hamiltonian
The low energy effective Hamiltonian responsible for the decays is where the Fermi coupling constant . Using the Wolfenstein parameterization, the Cabibbo-Kobayashi-Maskawa (CKM) factors can be expanded as a power series in the small parameter ,The Wilson coefficients summarize the physical contributions above scales of . The Wilson coefficients are calculable with the perturbation theory and have properly been evaluated to the next-to-leading order (NLO) with the renormalization group (RG) equation. The numerical values of the Wilson coefficients at scales of in naive dimensional regularization scheme are listed in Table 2. The local tree four-quark operators are defined as follows:where and are color indices and the sum over repeated indices is understood. To obtain the decay amplitudes, the remaining and the most intricate works are to calculate accurately hadronic matrix elements of local operators.
2.2. Hadronic Matrix Elements
Phenomenologically, the simplest treatment with hadronic matrix elements of four-quark operators is approximated by the product of two current matrix elements with color transparency ansatz  and naive factorization (NF) scheme [30, 31], and current matrix elements are further parameterized by decay constants and transition form factors. For example, previous studies on the decay [22, 23] were based on NF approach.
As is well known, NF’s defect is the disappearance of the renormalization scale dependence, the strong phases, and the nonfactorizable contributions from hadronic matrix elements, resulting in nonphysical amplitudes and no violating asymmetries. To remedy NF’s deficiencies, Beneke et al. proposed that hadronic matrix elements could be written as the convolution integrals of hard scattering kernels and light cone distribution amplitudes with the QCDF approach [12, 13].
For the decay, the spectator quark is the heavy bottom (anti)quark. According to the QCDF’s power counting rules , contributions from the spectator scattering are power suppressed. With the QCDF master formula, hadronic matrix elements could be written aswhere transition form factor and light cone distribution amplitudes of the emitted meson are nonperturbative input parameters and hard scattering kernels are computable order by order with the perturbation theory in principle.
The leading twist two-valence-particle distribution amplitudes of pseudoscalar and longitudinally polarized vector meson are defined in terms of Gegenbauer polynomials [32, 33]:where , is the Gegenbauer moment, and .
After calculation, the decay amplitudes could be written as
For the transversely polarized vector meson, the factor is zero beyond leading twist (twist-2) contributions. For the pseudoscalar and longitudinally polarized vector meson, with the modified minimal subtraction () scheme, the factor is written as whereand the relations are as follows:
The numerical values of coefficient at scales of are listed in Table 2.
2.3. Decay Constants and Form Factors
The matrix elements of current operators are defined as follows:where and are the decay constants of pseudoscalar and vector mesons, respectively, and and denote the mass and polarization of vector meson, respectively.
It is clearly seen that there are only three independent form factors, and , at the pole for the decays. The form factors at the pole could be written as the overlap integrals of wave functions of mesons ; that is,where is a Pauli matrix acting on the spin indices of the decaying bottom quark and and denote the fraction of the longitudinal momentum and the transverse momentum carried by the nonspectator quark, respectively.
Using the separation of momentum and spin variables, the wave functions of mesons can be written aswith the normalization condition,where denote the spin of valence quark in meson; ; and for the and particles, respectively.
For the ground states of heavy quarkonia and particles, we will take the solution of the Schödinger equation with nonrelativistic three-dimensional scalar harmonic oscillator potential,where the parameter determines the average transverse quark momentum; that is, . According to the power counting rules of NRQCD , the characteristic magnitude of the moment is order of and . So we will take in our calculation. Employing the substitution ansatz ,where and is the mass of valence quark. Setting , we can obtainwhere is a normalization factor.
Using the aforementioned convention, we getwhere the uncertainties come from variation of valence quark mass . In addition, according to the NRQCD argument, the relativistic corrections and higher-twist effects might give uncertainties of , about 10%~30%. Values at could, in principle, be extrapolated by assuming the form factors dominated by a proper pole which is unknown or calculated with other methods, such as the approach using the Bethe-Salpeter wave functions with the help of the nonrelativistic instantaneous approximation and the potential model based on the Mandelstam formalism . Here, we will follow the common practice for nonleptonic decays with the QCDF approach. Values of form factors at are taken to offer an order of magnitude estimation, because both and are heavy quarkonium and the recoil effects might be not so significant.
2.4. Decay Amplitudes
With the aforementioned definition of hadronic matrix elements, the decay amplitudes of decays can be written as
The definition of helicity amplitudes iswhere invariant amplitudes , , and and variable areThe scalar amplitudes , , and describe the , , and wave contributions, respectively. Clearly, compared with the wave amplitude, the and wave amplitudes are suppressed by a factor .
3. Numerical Results and Discussion
In the rest frame of particle, branching ratio for nonleptonic weak decays can be written aswhere the decay width ; the momentum of final states is
The input parameters, including the CKM Wolfenstein parameters, masses of and quarks, decay constants, and Gegenbauer moments of distribution amplitudes in (5), are collected in Table 3. If not specified explicitly, we will take their central values as the default inputs. Our numerical results on the -averaged branching ratios for the decays are displayed in Table 4, where theoretical uncertainties of the QCDF results come from the CKM parameters, the renormalization scale , masses of and quarks, and hadronic parameters (decay constants and Gegenbauer moments), respectively. For the sake of comparison, previous results of [22, 23] are reevaluated with coefficient , where the scenario of the flavor dependent parameter in  is taken. The following are some comments.(1)The QCDF’s results fall in between those of [22, 23], because the form factors in our calculation fall in between those of [22, 23].(2)There is a clear hierarchical relationship, . These are two dynamical reasons. One is that the CKM factor responsible for the decay is suppressed by a factor of relative to the CKM factor responsible for the , decays. The other is that the orbital angular momentum .(3)The CKM-favored dominated decay has the largest branching ratio, ~10−10, which should be sought for with high priority and firstly observed at the running LHC and forthcoming SuperKEKB. For example, the production cross section in -Pb collision can reach up to a few with the ALICE detector at LHC . Therefore, per 100 fb−1 data collected at ALICE, over particles are in principle available, corresponding to tens of events if with about 10% reconstruction efficiency.(4)There are many uncertainties on the QCDF’s results. The first uncertainty, about 7~8%, from the CKM factors could be lessened with the improvement on the precision of the Wolfenstein parameter . The second uncertainty from the renormalization scale should, in principle, be reduced by inclusion of higher order corrections to hadronic matrix elements. The third uncertainty is due to the fact that masses of and quark affect the shape lines of wave functions and hence the magnitude of form factors and branching ratios. The fourth uncertainty from hadronic parameters is expected to be reduced with the relative ratio of branching ratios.(5)Other factors, such as the contributions of higher order corrections to hadronic matrix elements, relativistic effects, and dependence of form factors, which are not considered in this paper, deserve the dedicated study. Our results just provide an order of magnitude estimation.
With the sharp increase of the data sample at high-luminosity dedicated heavy-flavor factories, the bottom-changing weak decays are interesting in exploring the underlying mechanism responsible for transition between heavy quarknoia, investigating perturbative and nonperturbative effects and overconstraining parameters from decays. The weak decays are allowable within the standard model, though their branching ratios are expected to be tiny in comparison to the conventional strong and electromagnetic decays. In this paper, we studied the nonleptonic weak decays, which are -dominated based on the low energy effective theory and hence should have large branching ratios among weak decay modes. Considering the nonfactorizable contributions to hadronic matrix elements with the QCDF approach, we estimated the branching ratios of the weak decays, where transition form factors are obtained by the integrals of wave functions with the nonrelativistic isotropic harmonic oscillator potential. The prediction on branching ratios for the decays is the same order as previous works [22, 23]. The CKM-favored decay has relatively large branching ratio, ~10−10, and might be detectable in future experiments.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the referees for their helpful comments. The work is supported by the National Natural Science Foundation of China (Grant nos. 11475055, 11275057, U1232101, and U1332103). Dr. Wang is thankful for the support from CCNU-QLPL Innovation Fund (QLPL201411).
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Copyright © 2015 Junfeng Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.