Charmless hadronic decays in the perturbative QCD approach
Abstract
We study 20 charmless hadronic decays in the perturbative QCD(pQCD) formalism with denoting , , and mesons; standing for the light pseudoscalar mesons; and representing axialvector mesons and that result from a mixing of quarkflavor and states with the angle . The estimations of CPaveraged branching ratios and CPasymmetries of the considered decays, in which the modes are investigated for the first time, are presented in the pQCD approach with from recently measured decays. It is found that (a) the tree(penguin) dominant decays with large branching ratios[] and large direct CPviolations(around in magnitude) simultaneously are believed to be clearly measurable at the LHCb and Belle II experiments; (b) the and decays with nearly pure penguin contributions and safely negligible tree pollution also have large decay rates in the order of , which can be confronted with the experimental measurements in the near future; (c) as the alternative channels, the and decays have the supplementary power in providing more effective constraints on the CabibboKobayashiMaskawa weak phases , , and , correspondingly, which are explicitly analyzed through the large decay rates and the direct and mixinginduced CPasymmetries in the pQCD approach and are expected to be stringently examined by the measurements with high precision; (d) the weak annihilation amplitudes play important roles in the , , decays, and so on, which would offer more evidence, once they are confirmed by the experiments, to identify the softcollinear effective theory and the pQCD approach on the evaluations of annihilation diagrams and to help further understand the annihilation mechanism in the heavy meson decays; (e) combined with the future precise tests, the considered decays can provide more information to further understand the mixing angle and the nature of the mesons in depth after the confirmations on the reliability of the pQCD calculations in the present work.
pacs:
13.25.Hw, 12.38.Bx, 14.40.NdI Introduction
It is well known that nonleptonic weak decays of heavy (specifically, , , , and ) mesons can not only provide the important information to search for CPviolation and further constrain the CabibboKobayashiMaskawa(CKM) parameters in the standard model(SM), but also reveal the deviations from the SM, i.e., the signals of exotic new physics beyond the SM. Furthermore, comparison of theoretical predictions and experimental data for the physical observables may also help us understand the hadronic structure of the involved bound states deeply Feldmann:2014iha (). In contrast to the traditional , and decays, the alternative channels such as decays ( is the axialvector meson) to be largely detected in experiments in the near future may give additional and complementary information on exclusive nonleptonic weak decays of heavy mesons Calderon:2007nw (); e.g., due to , the penguindominated decays have the same CKM phase as the tree level decays Agashe:2014kda (). Therefore, the mediated decays such as , etc. can provide measurements ( is the CKM weak phase) in the SM complementarily.
Very recently, the Large Hadron Collider beauty(LHCb) Collaboration reported the first measurements of decays Aaij:2013rja (), where the final state was observed for the first time in heavy meson decays. In the conventional two quark structure, and its partner Close:1997nm (); Li:2000dy () [hereafter, for the sake of simplicity, we will use to denote both and unless otherwise stated] are considered as the orbital excitation of the system, specifically, the light wave axialvector flavorless mesons. In terms of the spectroscopic notation with , both mesons belong to nonet carrying the quantum number Agashe:2014kda (). Similar to the mixing in the pseudoscalar sector Agashe:2014kda (), these two mesons are believed to be a mixture resulting from the mixing between nonstrange and strange states in the popular quarkflavor basis with a single mixing angle . And for the mixing angle , there are several explorations that have been performed from theory and experiment sides. However, the value of is still in controversy presently (see Ref. Liu:2014doa () and references therein). It is necessary to point out that the mixing angle has important roles in investigating the properties of mesons themselves, but also of strange axialvector mesons, i.e., and , by constraining the mixing angle between two distinct types of axialvector and states. The underlying reason is that when the mixing angle is determined from the mass relations related with the masses of and , it eventually depends on the mixing angle Cheng:2011pb (). With the successful running of LHC and the forthcoming Belle II experiments, it is therefore expected that these first observations of decays will motivate the people to explore the mixing angle and the properties of both mesons in more relevant meson decay processes at both experimental and theoretical aspects. Of course, in view of some of the axialvector mesons such as and that have been seen in twobody hadronic meson decays Agashe:2014kda (), it is also believed that the information on both mesons could be obtained from heavy quark decays in the near future.
In this work, we will therefore study 20 charmless hadronic ^{5}^{5}5In the literature Liu:2010da (), two of us(X. Liu and Z.J. Xiao) have studied the decays occurring only via the pure annihilation diagrams in the SM within the framework of the perturbative QCD(pQCD) factorization approach Keum:2000ph (); Lu:2000em (); Li:2003yj (). decays, in which stands for , , and , respectively, and denotes the light pseudoscalar pion, kaon, and and mesons. From the experimental point of view, up to now, only two penguindominated decays have been measured by the BABAR Collaboration in 2007 Burke:2007zz (). The preliminary upper limits on the decay rates have been placed at the 90% confidence level as Agashe:2014kda ()
(1) 
for decay and
(2)  
(3) 
for decay, respectively. However, due to the lack of the information on the decay rates of and decays, the upper limits of cannot be extracted effectively. But, this status will be greatly improved in present and future experiments, notably at running LHCb and forthcoming Belle II. Also, other decays are expected to be detected with good precision at the relevant experiments in the near future.
From the theoretical point of view, to our best knowledge, G. Caldern et al. have carried out the calculations of decays in the framework of naive factorization with the form factors of obtained in the improved Isgur ScoraGrinsteinWise quark model Calderon:2007nw (), while Cheng and Yang have studied the decay rates and direct CPasymmetries of modes within the framework of QCD factorization (QCDF) with the form factors evaluated in the QCD sum rule Cheng:2007mx (). Note that the decays have never been studied yet in any methods or approaches up to this date. And, it should be stressed that the predictions of the branching ratios for decays in naive factorization are so crude that we cannot make effective comparison for relevant modes. For decays for example, on one hand, the authors did not specify and Calderon:2007nw (), which then could not provide effectively the useful information on the mixing angle from these considered decays; on the other hand, as discussed in Ref. Cheng:2007mx (), the meson behaves analogously to the vector meson, it is then naively expected that and if and are significantly dominated by the and components, respectively. Furthermore, in principle, in view of the mixing, the branching ratios of are generally a bit smaller than those of ones correspondingly. However, the branching ratio of predicted in the naive factorization is around , which is much larger than that of the corresponding modes, i.e., and Agashe:2014kda (). As for the investigations of decays in the QCDF approach Cheng:2007mx (), the authors specified and and considered their mixing with two different sets of angles, and , in the flavor singletoctet basis. And the decay rates are barely consistent with the preliminary upper limits within very large errors. However, the pattern exhibited from the numerical results with is more favored by the available upper limits. As aforementioned, because of the similar behavior between the vector meson and axialvector meson, the relation is expected to be highly preferred, as it should be.
In order to collect more information on the nature of both mesons and further understand the heavy flavor physics, we will study the physical observables such as CPaveraged branching ratios and CPviolating asymmetries of 20 charmless hadronic decays by employing the low energy effective Hamiltonian Buchalla:1995vs () and the pQCD approach Keum:2000ph (); Lu:2000em (); Li:2003yj () based on the factorization theorem. Though some efforts have been made on the nexttoleading order pQCD formalism Li:2010nn (); Cheng:2014gba (), we here will still consider the perturbative evaluations at leading order, which are believed to be the dominant contributions perturbatively. As is well known, the pQCD approach is free of endpoint singularity and the Sudakov formalism makes it more selfconsistent by keeping the transverse momentum of the quarks. More importantly, as the wellknown advantage of the pQCD approach, we can explicitly calculate the weak annihilation types of diagrams without any parametrizations, apart from the traditional factorizable and nonfactorizable emission ones, though a different viewpoint on the evaluations and the magnitudes^{6}^{6}6As a matter of fact, the recent works Zhu:2011mm (); Chang:2014rla () in the framework of QCDF confirmed that there should exist complex annihilation contributions with large imaginary parts in decays by fits to the experimental data, which support the concept on the calculations of the annihilation diagrams in the pQCD approach Chay:2007ep () to some extent. of weak annihilation contributions has been proposed in the softcollinear effective theory Arnesen:2006vb (). It is worth stressing that the pQCD predictions on the annihilation contributions in the heavy meson decays have been tested at various aspects, e.g., see Refs. Lu:2002iv (); Li:2004ep (); Ali:2007ff (); Xiao:2011tx (); Keum:2000ph (); Lu:2000em (); Hong:2005wj (). Typically, for example, the evaluations of the pure annihilation and decays in the pQCD approach Li:2004ep (); Ali:2007ff (); Xiao:2011tx () are in good consistency with the recent measurements by both CDF and LHCb Collaborations Aaltonen:2011jv (); Ruffini:2013jea (); Aaij:2012as (). Therefore, the weak annihilation contributions to the considered decays will be explicitly analyzed in this work, which are expected to be helpful to understand the annihilation mechanism in the heavy meson decays.
The paper is organized as follows. In Sec. II, we present the formalism, hadron wave functions and perturbative calculations of the considered 20 decays in the pQCD approach. The numerical results and the corresponding phenomenological analyses are addressed in Sec. III. Finally, Sec. IV contains the main conclusions and a short summary.
Ii Formalism and Perturbative Calculations
For the considered decays, the related weak effective Hamiltonian Buchalla:1995vs () can be written as
(4) 
with the light downtype quark or , the Fermi constant , CKM matrix elements , and Wilson coefficients at the renormalization scale . The local fourquark operators are written as

(1) currentcurrent(tree) operators
(5) 
(2) QCD penguin operators
(6) 
(3) electroweak penguin operators
(7)
with the color indices (not to be confused with the CKM weak phases and ) and the notations . The index in the summation of the above operators runs through , , and . The standard combinations of the Wilson coefficients are defined as follows:
(8) 
where is the largest one among all Wilson coefficients and the upper (lower) sign applies, when is odd (even). It is noted that, though the nexttoleading order Wilson coefficients have already been available Buchalla:1995vs (), we will still adopt the leading order ones to keep the consistency, since the short distance contributions of the considered decays are calculated at leading order[] in the pQCD approach. This is also a consistent way to cancel the explicit dependence in the theoretical formulas. For the renormalization group evolution of the Wilson coefficients from higher scale to lower scale, the expressions are directly taken from Ref. Lu:2000em ().
Nowadays, the pQCD approach has been one of the popular factorization methods based on the QCD theory to evaluate the hadronic matrix elements in the heavy meson decays. The unique point of the pQCD approach is that it keeps the transverse momentum , which will act as an infrared regulator and smear the endpoint singularity when the quark momentum fraction approaches 0, of the valence quarks in the calculation of the hadronic matrix elements. Then, all the meson transition form factors, nonfactorizable spectator and annihilation contributions are calculable in the framework of the factorization. The decay amplitude of decays in the pQCD approach can be conceptually written as
(9) 
where ’s are the momenta of (light) quarks in the initial and final states, represents the trace over Dirac and color indices, and is the Wilson coefficient which results from the radiative corrections at short distance. In the above convolution, includes the harder dynamics at larger scale than scale and describes the evolution of local Fermi operators from (the boson mass) down to scale, where is the hadronic scale. The stands for the wave function describing hadronization of the quark and antiquark to the meson, which is independent of the specific processes and usually determined by employing nonperturbative QCD techniques such as lattice QCD(LQCD) or other wellmeasured processes. The function describes the fourquark operator and the spectator quark connected by a hard gluon with the hard intermediate scale . Therefore, this hard part can be calculated perturbatively.
Since the quark is rather heavy, we thus work in the frame with the meson at rest for simplicity, i.e., with the meson momentum in the lightcone coordinate. For the considered decays, it is assumed that the and mesons move in the plus and minus direction carrying the momentum and , respectively. Then the momenta of the two final states can be written as
(10) 
respectively, where and the masses of light pseudoscalar pion, kaon, and and have been neglected. For the axialvector meson , its longitudinal polarization vector . By choosing the quark momenta in , and mesons as , , and , respectively, and defining
(11) 
then, the integration over , , and in Eq. (9) will give the more explicit form of decay amplitude in the pQCD approach,
(12)  
where is the conjugate space coordinate of , and is the largest running scale in the hard kernel . The large logarithms are included in the Wilson coefficients . Note that and are the two important elements in the perturbative calculations with the pQCD approach. The former is a jet function from threshold resummation, which can strongly suppress the behavior in the small region Li:2001ay (); Li:2002mi (); while the latter is a Sudakov factor from resummation, which can effectively suppress the soft dynamics in the small region Botts:1989kf (); Li:1992nu (). These resummation effects therefore guarantee the removal of the endpoint singularities. Thus it makes the perturbative calculation of the hard part applicable at intermediate scale. We will calculate analytically the function for the decays at LO in the expansion and give the convoluted amplitudes in the next section.
The heavy meson is usually treated as a heavylight system and its lightcone wave function can generally be defined as Keum:2000ph (); Lu:2000em (); Lu:2002ny ()
(13) 
in which are the color indices; is the momentum(mass) of the meson; is the color factor; and is the intrinsic transverse momentum of the light quark in the meson.
In Eq. (13), is the meson distribution amplitude, which satisfies the following normalization condition,
(14) 
where is the conjugate space coordinate of transverse momentum and is the decay constant of the meson.
For the pseudoscalar meson, the lightcone wave function can generally be defined as Chernyak:1983ej (); Ball:1998tj ()
(15) 
where and are the twist2 and twist3 distribution amplitudes, and is the chiral enhancement factor of the meson, while denotes the momentum fraction carried by quark in the meson and and are the dimensionless lightlike unit vectors.
The lightcone wave function of the axialvector mesons has been given in the QCD sum rule as Yang:2007zt (); Li:2009tx ()
(16) 
for longitudinal polarization with the polarization vector , satisfying , where (not to be confused with the angle in the mixing of mesons) and are the twist2 and twist3 distribution amplitudes, respectively. All the explicit forms of the aforementioned hadronic distribution amplitudes in the considered decays can be seen in the Appendix.
Now we come to the analytically perturbative calculations of the factorization formulas for the decays in the pQCD approach. From the effective Hamiltonian (4), there are eight types of diagrams contributing to the decays as illustrated in Fig. 1. For the factorizable emission() diagrams, with Eq. (12), the analytic expressions of the decay amplitudes from different operators are given as follows:

operators:
(17) 
operators:
(18) 
operators:
(19)
where and is a color factor. The convolution functions , the running hard scales , and the hard functions can be referred to in Ref. Liu:2005mm ().