Decays with Perturbative QCD Approach
Besides the traditional strong and electromagnetic decay modes, meson can also decay through the weak interactions within the standard model of elementary particle. With anticipation of copious data samples at the running LHC and coming SuperKEKB experiments, the two-body nonleptonic bottom-changing decays () are investigated with perturbative QCD approach firstly. The absolute branching ratios for and decays are estimated to reach up to about and , respectively, which might possibly be measured by the future experiments.
The upsilon meson is the spin-triplet -wave state of bottomonium (bound state consisting of bottom quark and antibottom quark ) with well-established quantum number of . The characteristic narrow decay widths of mesons for , 2, and 3 provide insight into the study of strong interactions (see Table 1, and note that, for simplicity, will denote , , and mesons in the following content if not specified definitely). The mass of meson is below meson pair threshold. meson decays into bottomed hadrons through strong and electromagnetic interactions are forbidden by the law of conservation of flavor number. The bottom-changing decays can occur only via the weak interactions within the standard model, although with tiny incidence probability. Both constituent quarks of upsilons can decay individually, which provide an alternative system for investigating the weak decay of heavy-flavored hadrons. In this paper, we will study the nonleptonic ( and ) weak decays with perturbative QCD (pQCD) approach [2–4].
Experimentally, over data samples have been accumulated at Belle and BaBar experiments . More and more upsilon data samples will be collected at the running hadron collider LHC and the forthcoming collider SuperKEKB (the SuperKEKB has started commissioning test run (http://www.kek.jp/en/NewsRoom/Release/)). There seems to exist a realistic possibility to explore weak decay at future experiments. Signals of decays should be easily distinguished with “charge tag” technique, due to the facts that the back-to-back final states with different electric charges have definite momentum and energy in the rest frame of meson. meson has not been observed experimentally by now. meson production via the strong interaction is suppressed due to the simultaneous presence of two heavy quarks with different flavors and higher order in QCD coupling constant . , decays provide a novel pattern to study meson production. The identification of a single explicitly flavored meson could be used as an effective selection criterion to detect upsilon weak decays. Moreover, the radiative decay of meson provides a useful extra signal and a powerful constraint (the investigation on the radiative decay of meson can be found in, e.g., , with QCD sum rules). Of course, any discernible evidences of an anomalous production rate of single bottomed meson from upsilon decays might be a hint of new physics.
Theoretically, many attractive QCD-inspired methods have been developed recently to describe the exclusive nonleptonic decay of heavy-flavored mesons, such as the pQCD approach [2–4], the QCD factorization approach [7–9], and soft and collinear effective theory [10–13], and have been applied widely to vindicate measurements on meson decays. The upsilon weak decay permits one to further constrain parameters obtained from meson decay, and cross comparisons provide an opportunity to test various phenomenological models. The upsilon weak decay possesses a unique structure due to the Cabibbo-Kobayashi-Maskawa (CKM) matrix properties which predicts that the channels with one meson are dominant. decay belongs to the favorable transition, which should, in principle, have relatively large branching ratio among upsilon weak decays. However, there is still no theoretical study devoted to decay for the moment. In this paper, we will present a phenomenological investigation on weak decay with the pQCD approach to supply a ready reference for the future experiments.
This paper is organized as follows. Section 2 focuses on theoretical framework and decay amplitudes for weak decays. Section 3 is devoted to numerical results and discussion. The last section is a summary.
2. Theoretical Framework
2.1. The Effective Hamiltonian
Theoretically, , weak decays are described by an effective bottom-changing Hamiltonian based on operator product expansion : where  is the Fermi coupling constant; the CKM factors and correspond to and decays, respectively; with the Wolfenstein parameterization, the CKM factors are expanded as a power series in a small Wolfenstein parameter : The local tree operators are defined as where and are color indices and the sum over repeated indices is understood.
The scale factorizes physics contributions into short- and long-distance dynamics. The Wilson coefficients summarize the physics contributions at scale higher than and are calculable with the renormalization group improved perturbation theory. The hadronic matrix elements (HME), where the local operators are inserted between initial and final hadron states, embrace the physics contributions below scale of . To obtain decay amplitudes, the remaining work is to calculate HME properly by separating from perturbative and nonperturbative contributions.
2.2. Hadronic Matrix Elements
Based on Lepage-Brodsky approach for exclusive processes , HME is commonly expressed as a convolution integral of hard scattering subamplitudes containing perturbative contributions with universal wave functions reflecting nonperturbative contributions. In order to effectively regulate endpoint singularities and provide a naturally dynamical cutoff on nonperturbative contributions, transverse momentum of valence quarks is retained and the Sudakov factor is introduced within the pQCD framework [2–4]. Phenomenologically, pQCD’s decay amplitude could be divided into three parts: the Wilson coefficients incorporating the hard contributions above typical scale of , process-dependent rescattering subamplitudes accounting for the heavy quark decay, and wave functions of all participating hadrons, which is expressed as where is the momentum of valence quarks and is the Sudakov factor.
2.3. Kinematic Variables
The light cone kinematic variables in rest frame are defined as follows: where and are the longitudinal momentum fraction and transverse momentum of valence quarks, respectively; and are the longitudinal and transverse polarization vectors, respectively, and satisfy relations and ; the subscript on variables , , , and corresponds to participating hadrons; namely, for meson, for the recoiled meson, and for the emitted pseudoscalar meson; and are positive and negative null vectors, respectively; , , and are the Lorentz-invariant variables; is the common momentum of final states. The notation of momentum is displayed in Figure 2(a).
2.4. Wave Functions
Considering mass relations of and , it might assume that the motion of heavy valence quarks in and mesons is nearly nonrelativistic. The wave functions of and mesons could be approximately described with nonrelativistic quantum chromodynamics (NRQCD) [22–24] and time-independent Schrödinger equation. For an isotropic harmonic oscillator potential, the eigenfunctions of stationary state with quantum numbers are written as  where parameter determines the average transverse momentum; that is, . Employing the substitution ansatz , where and are the longitudinal momentum fraction and mass of valence quark, respectively; then integrating out and combining with their asymptotic forms, the distribution amplitudes (DAs) for and mesons can be written as  where ; . According to NRQCD power counting rules , with and QCD coupling constant . The exponential function represents distribution. Parameters of , , , , , , , and are normalization coefficients satisfying the conditions
The shape lines of normalized DAs for and mesons are shown in Figure 1. It is clearly seen that DAs for and mesons fall quickly down to zero at endpoint , due to suppression from exponential functions; DAs for meson are symmetric under the interchange of momentum fractions , and DAs for meson are basically consistent with the feature that valence quarks share momentum fractions according to their masses.
Our study shows that only the leading twist (twist-2) DAs of the emitted light pseudoscalar meson are involved in decay amplitudes (see Appendix). The twist-2 DAs have the expansion ,and are normalized as where are Gegenbauer polynomials: and each term corresponds to a nonperturbative Gegenbauer moment ; note that due to the normalization condition equation (12); -parity invariance of the pion DAs requires Gegenbauer moment for .
2.5. Decay Amplitudes
The Feynman diagrams for weak decay are shown in Figure 2. There are two types. One is factorizable emission topology where gluon attaches to quarks in the same meson, and the other is nonfactorizable emission topology where gluon connects to quarks between different mesons.
With the pQCD master formula equation (4), the amplitude for decay can be expressed as  which is conventionally written as the helicity amplitudes : where and the color number ; the subscript on corresponds to three different helicity amplitudes; that is, ; the superscript on denotes indices of Figure 2. The explicit expressions of building blocks are collected in Appendix.
3. Numerical Results and Discussion
In the center-of-mass of meson, branching ratio for decay is defined as
The input parameters are listed in Tables 1 and 2. If not specified explicitly, we will take their central values as the default inputs. Our numerical results are collected in Table 3, where the first uncertainty comes from scale and the expression of is given in (A.4) and (A.5); the second uncertainty is from mass of and ; the third uncertainty is from hadronic parameters including decay constants and Gegenbauer moments; the fourth uncertainty is from CKM parameters. The followings are some comments:(1)Branching ratio for decay is about with pQCD approach, which is well within the measurement potential of LHC and SuperKEKB. For example, experimental studies have showed that production cross sections for meson in p-p and p-Pb collisions are a few at the LHCb [27, 28] and ALICE [29, 30] detectors. Consequently, there will be more than data samples per data collected by the LHCb and ALICE, corresponding to a few hundreds of events. Branching ratio for decay, , is generally less than that for decay by one order of magnitude due to the CKM suppression, .(2)As it is well known, due to the large mass of , the momentum transition in decay may be not large enough. One might naturally wonder whether the pQCD approach is applicable and whether the perturbative calculation is reliable. Therefore, it is necessary to check what percentage of the contributions comes from the perturbative region. The contributions to branching ratio for decay from different region are shown in Figure 3. It can be clearly seen that more than 93% (97%) contributions come from (0.3) region, implying that decay is computable with the pQCD approach. As the discussion in [2–4], there are many factors for this, for example, the choice of the typical scale, retaining the quark transverse moment and introducing the Sudakov factor to suppress the nonperturbative contributions, which deserve much attention and further investigation.(3)Because of the relations among masses resulting in the fact that phase space increases with the radial quantum number in addition to the relations among decay widths , in principle, there should be relations among branching ratios for the same pseudoscalar meson . But the numerical results in Table 3 are beyond such expectation. Why? The reason is that the factor of in (16) has almost the same value for , so branching ratio is proportional to factor with the maximal value for . Besides, contributions from regions decrease with (see Figure 3), which enhance the decay amplitudes.(4)Besides the uncertainties listed in Table 3, other factors, such as the models of wave functions, contributions of higher order corrections to HME, and relativistic effects, deserve the dedicated study. Our results just provide an order of magnitude estimation.
decay via the weak interaction, as a complementary to strong and electromagnetic decay mechanism, is allowable within the standard model. Based on the potential prospects of physics at high-luminosity collider experiment, decay into and final states is investigated with the pQCD approach firstly. It is found that the dominant contributions come from perturbative regions , which might imply that the pQCD calculation is practicable and workable; there is a promiseful possibility of searching for () decay with branching ratio about () at the future experiments.
Building Blocks for Decays
The building blocks , where the superscript corresponds to indices of Figure 2 and the subscript relates to different helicity amplitudes, are expressed as follows: where ; variable is the longitudinal momentum fraction of the valence quark; is the conjugate variable of the transverse momentum ; and is the QCD coupling at the scale of ; .
The function is defined as follows : where and ( and ) are the (modified) Bessel function of the first and second kind, respectively; () is the gluon virtuality of the emission (annihilation) topological diagrams; the subscript of the quark virtuality corresponds to the indices of Figure 2. The definition of the particle virtuality is listed as follows :
The typical scale and the Sudakov factor are defined as follows, where the subscript corresponds to the indices of Figure 2: where is the quark anomalous dimension; the explicit expression of can be found in the appendix of .
The authors declare that there are no competing interests related to this paper.
The work is supported by the National Natural Science Foundation of China (Grant nos. 11547014, 11475055, U1332103, and 11275057).
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