Study on the <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.9939pt" id="M1" height="15.2545pt" version="1.1" viewBox="-0.0657574 -12.2606 39.7303 15.2545" width="39.7303pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-149" d="M648 625C638 644 619 665 597 665C518 665 445 585 404 489C377 436 359 379 345 309H340C335 367 324 448 296 515C263 599 214 665 123 665C83 665 33 649 1 632L6 606C19 608 32 609 44 609C124 609 186 567 230 473C270 390 283 308 283 200V122C283 42 274 33 184 27V0H468V27C376 33 368 42 368 122V194C368 293 402 401 461 492C512 573 576 611 629 611C634 611 636 611 642 610L648 625Z"/></g><g transform="matrix(.017,0,0,-0.017,11.245,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,17.182,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g><g transform="matrix(.017,0,0,-0.017,25.419,0)"><path id="g113-84" d="M449 634C442 637 425 643 405 650C376 660 341 666 307 666C181 666 98 590 98 485C98 400 170 343 215 310L246 288C307 243 343 204 343 147C343 67 291 18 219 18C104 18 61 124 51 202L23 199C28 124 27 71 27 47C47 22 122 -16 204 -16C324 -16 428 60 428 174C428 256 379 309 307 360L276 382C223 419 179 455 179 516C179 576 221 632 293 632C379 632 410 564 418 487L448 490C446 536 446 592 449 634Z"/></g><g transform="matrix(.017,0,0,-0.017,33.519,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg> <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.3443pt" id="M2" height="15.7437pt" version="1.1" viewBox="-0.0657574 -11.3994 59.9551 15.7437" width="59.9551pt"><g transform="matrix(.017,0,0,-0.017,2.876,0)"><path id="g117-148" d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H796L694 83L716 68C790 149 861 220 901 255Z"/></g><g transform="matrix(.017,0,0,-0.017,26.917,0)"><path id="g113-67" d="M578 512C578 619 494 650 387 650H140L134 622C219 615 223 609 210 542L127 119C112 40 104 34 23 28L17 0H235C317 0 387 7 444 33C515 65 564 122 564 195C564 289 495 335 412 352V354C505 373 578 422 578 512ZM486 510C486 422 423 367 314 367H257L294 565C299 591 305 602 315 608C324 614 339 617 362 617C421 617 486 595 486 510ZM466 200C466 100 393 35 296 35C222 35 198 51 212 127L250 333H303C388 333 466 303 466 200Z"/></g><g transform="matrix(.012,0,0,-0.012,37.145,4.134)"><path id="g50-100" d="M387 400C387 425 348 451 303 451C247 451 176 414 132 376C69 322 24 228 24 148C24 43 74 -12 147 -12C211 -12 301 33 363 103L346 128C319 99 249 51 193 51C148 51 112 84 112 165C112 230 130 287 154 330C170 359 199 400 243 400C277 400 304 383 326 354C333 345 343 343 354 348C378 360 387 382 387 400Z"/></g><g transform="matrix(.017,0,0,-0.017,42.757,0)"><path id="g113-78" d="M962 650H795L470 145L347 650H176L170 622C268 613 275 606 244 503L190 322C150 188 132 126 118 91C102 50 80 33 18 28L12 0H245L251 28C175 35 170 48 174 93C177 128 191 180 220 284L292 542H294C331 392 383 150 409 4H432L774 555H776L714 137C700 40 694 34 612 28L606 0H868L874 28C793 34 784 37 797 137L849 533C859 612 863 616 956 622L962 650Z"/></g></svg> Weak Decays

Sun, Junfeng; Chen, Lili; Wang, Na; Chang, Qin; Huang, Jinshu; Yang, Yueling

doi:https://doi.org/10.1155/2015/691261

Advances in High Energy Physics

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Research Article | Open Access

Volume 2015 | Article ID 691261 | https://doi.org/10.1155/2015/691261

Study on the Weak Decays

Junfeng Sun,¹Lili Chen,¹Na Wang,^1,2Qin Chang,^1,2Jinshu Huang,^1,3and Yueling Yang¹

Academic Editor: Enrico Lunghi

Received07 Jun 2015

Revised08 Jul 2015

Accepted27 Jul 2015

Published10 Aug 2015

Abstract

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.

1. Introduction

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 [3]. 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 [4]. The mass of the particle, [4], 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, [4]. Furthermore, one of the outstanding properties of all upsilons below threshold is their narrow decay width of tens of keV [4].

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 [4]. 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 [4]. The up-to-date research for light Higgs bosons in the radiative decay has been performed by CLEO [8], Belle [9], and BaBar [10] 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 [11]. 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 [4]. 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 [17], ATLAS [18], CMS [19], and LHCb [20] 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 [21] based on the spin symmetry and nonrecoil approximation, [22] based on the heavy quark effective theory, and [23] based on the Bauer-Stech-Wirbel (BSW) model [24], 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 [13], 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 [28]where the Fermi coupling constant [4]. Using the Wolfenstein parameterization, the Cabibbo-Kobayashi-Maskawa (CKM) factors can be expanded as a power series in the small parameter [4],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 [29] 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 [13], 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

The coefficient in (6), including nonfactorizable contributions from QCD radiative vertex corrections, is written as [34]

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 [34]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.

The transition form factors are defined as follows [22–24]:where and is required compulsorily to cancel singularities at the pole . There is a relation among these form factors:

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 [24]; 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 [25], the characteristic magnitude of the moment is order of and . So we will take in our calculation. Employing the substitution ansatz [35],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 [36]. 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

For the decays, the hadronic matrix elements in (6) can also be expressed as [37]

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 [4]; 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 [23] 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 [38]. 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.

4. Summary

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.

Acknowledgments

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

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.

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