Hadronic decays of in the perturbative QCD approach
Abstract
We calculate the branching ratios and polarization fractions of the decays in the perturbative QCD(pQCD) approach at leading order, where () stands for the axialvector state. By combining the phenomenological analyses with the perturbative calculations, we find the following results: (a) the large decay rates around to of the decays dominated by the longitudinal polarization(except for the mode) are predicted and basically consistent with those in the QCD factorization(QCDF) within errors, which are expected to be tested by the Large Hadron Collider and BelleII experiments. The large branching ratio could provide hints to help explore the mechanism of the colorsuppressed decays. (b) the rather different QCD behaviors between the and mesons result in the destructive(constructive) contributions in the nonfactorizable spectator diagrams with emission. Therefore, an interesting pattern of the branching ratios appears for the colorsuppressed and modes in the pQCD approach, , which is different from in the QCDF and would be verified at future experiments. (c) the large naive factorization breaking effects are observed in these decays. Specifically, the large nonfactorizable spectator(weak annihilation) amplitudes contribute to the mode(s), which demand confirmations via the precise measurements. Furthermore, the different phenomenologies shown among , , and decays are also expected to be tested stringently, which could shed light on the typical QCD dynamics involved in these modes, even further distinguish those two popular pQCD and QCDF approaches.
pacs:
13.25.Hw, 12.38.Bx, 14.40.NdIt is well known that the nonleptonic meson decays can provide highly important information to understand the physics within and/or beyond the standard model(SM). Specifically, they can help us to study the perturbative and nonperturbative quantum chromodynamics(QCD), search for the chargeparity(CP) violation to further find out its origin, determine the CabibboKobayashiMaskawa(CKM) phases , and in the unitary triangle, even identify the possible new physics hidden in the higher energy scale, etc. Moreover, one can also indirectly conjecture the inner structure of the hadrons involved in the final states through the precise measurements experimentally. The great efforts have been extensively contributed to the exclusive and decays at both theoretical and experimental aspects in the past decades, for example, see Refs. Wirbel:1985ji (); Bauer:1986bm (); Ali:1997nh (); Ali:1998eb (); Du:2001hr (); Beneke:2003zv (); Beneke:2006hg (); Li:2006jv (); Ali:2007ff (); Wang:2008rk (); Cheng:2009cn (); Zou:2015iwa (); Zhou:2015jba (); Zhou:2016jkv (); Wang:2017hxe (); Wang:2017rmh (); Olive:2016xmw (); Amhis:2016xyh (), where and denote the wave pseudoscalar and vector states, respectively. However, the known ”puzzles”, for example, the large observed , , and decay rates, the experimental inequality of the direct CP asymmetries between and modes, the unknown mechanism of the polarization in the penguindominated processes etc., are still not elegantly resolved Cheng:2009xz (); Olive:2016xmw (); Amhis:2016xyh (). Therefore, a large variety of relevant meson decay modes should be opened to help us get deep understanding complementarily.
Fortunately, two successful factory experiments, i.e., BABAR at SLAC and Belle at KEK, have measured many nonleptonic meson decays into the final states containing wave light hadrons in the last decade Olive:2016xmw (); Amhis:2016xyh (). Then the Large Hadron Colliderbeauty(LHCb) experiments at CERN almost became the only apparatus to explore the physics of quark in recent years. A large number of data related to nonleptonic decays have been reported Olive:2016xmw (); Amhis:2016xyh (). The forthcoming start of the upgraded BelleII experiment will further improve the measurements. The Future Circular Collider and Circular ElectronPositron Collider are expected to give further chance for the studies on meson decays CEPC (). Therefore, it is believed that the great supports coming from these current running and forthcoming experiments could dramatically promote our understanding of the nature.
In this work, we will study the nonleptonic charmless decays of in the SM. For the sake of simplicity, the abbreviation and will be used in the following content to denote the and mesons, respectively, unless otherwise stated. As we know, the considered processes contain the same components as the modes at the quark level. The latter decays have contributed to the determination and constraints on the CKM angle Olive:2016xmw (). Certainly, the , and decays can also provide useful information to the angle complementarily Lombardo:2009kt (); Aubert:2009ab (); Cheng:2007mx (); Wang:2008hu (); Cheng:2008gxa (); Zhang:2012ew (); Liu:2012jb (). Particularly, because and behave differently from each other, these considered decays could provide opportunities for us to explore the interesting QCD dynamics. Furthermore, the decays with emission could provide more evidence for probing the naive factorization breaking effects Diehl:2001xe () because the decay constant vanishes owing to the charge conjugation invariance for the neutral state or the even Gparity validity in the isospin limit for the charged states Yang:2005gk (); Yang:2007zt (); Cheng:2007mx ().
As stated in the naive factorization hypothesis Bauer:1986bm (), the hadronic matrix element of a meson decay amplitude can be expressed by the factorizable emission amplitudes as a production of the decay constants and the transition form factors. Then, for example, the mode with emission almost receives no factorizable contributions due to the vanishing decay constant and the branching ratio would approach to zero in the naive factorization. While, it is worth emphasizing that the corresponding decay rate predicted in the QCD factorization(QCDF) Beneke:1999br (); Du:2001hr () by including the nonfactorizable spectator and annihilation contributions can reach Cheng:2008gxa (), which is detectable at the current experiments. It means that these important contributions violate the naive factorization if this large decay rate would be confirmed by the related experiments. However, because of the unavoidable endpoint singularities, the nonfactorizable spectator amplitudes, as well as the annihilation ones, have to be determined by data fitting accompanied with large uncertainties in the framework of QCDF Beneke:1999br (); Beneke:2003zv (). Luckily, the perturbative QCD(pQCD) approach Keum:2000ph (); Lu:2000em (), which bases on the framework of factorization theorem, is appropriate to calculate the decay amplitudes with the nonfactorizable spectator and annihilation topologies. Since it keeps the transverse momentum of the valence quark in the hadrons, then the resultant Sudakov factor[] and threshold factor[], which smear the endpoint singularities, make the pQCD approach more selfconsistent. More details about this pQCD approach can be found in the review paper Li:2003yj ().
We will therefore study the branching ratios and polarization fractions of the considered decays in the pQCD approach, with which the nonfactorizable spectator and annihilation Feynman diagrams can be calculated perturbatively. It is worth stressing that the observations of the pure annihilation and decays performed by the CDF Ruffini:2013jea () and LHCb Aaij:2012as () collaborations have confirmed the pQCD calculations Li:2004ep (); Ali:2007ff (); Xiao:2011tx () of the annihilation type diagrams ^{2}^{2}2Certainly, the softcollinear effective theory(SCET) Bauer:2000yr () has a different point of view on the calculations of the annihilation diagrams Arnesen:2006vb () . We believe that this discrepancy between the pQCD approach and SCET could be finally resolved through the precise measurements experimentally. Therefore, this conversation will be put aside in the present work.. Moreover, both of and are axialvector() states but with different quantum numbers and correspondingly. It is believed that the decays could provide more information on the helicity structure of the decay mechanism because, like decays, they also contain three polarization states Cheng:2008gxa (), which would be helpful to understand the famous ”polarization puzzle” in a different way.
For the considered decays with transition, the related weak effective Hamiltonian Buchalla:1995vs () can be written as
(1) 
with the Fermi constant , CKM matrix elements , and Wilson coefficients at the renormalization scale . The local fourquark operators are written as

(1) currentcurrent (tree) operators
(2) 
(2) QCD penguin operators
(3) 
(3) electroweak penguin operators
(4)
with the color indices (not to be confused with the CKM angles) and the notations . The index in the summation of the above operators runs through , , and . We will use the leading order Wilson coefficients to keep the consistency since the calculations in this work are at leading order[] of the pQCD approach. For the renormalization group evolution of the Wilson coefficients from higher scale to lower scale, we use the formulas as given in Ref. Keum:2000ph () directly.
Similar to the vector meson, the axialvector one also has three kinds of polarizations, i.e., longitudinal (), normal (), and transverse (), respectively. Therefore, analogous to the decays, the decay amplitudes will be characterized by the polarization states of these axialvector mesons. In terms of helicities, the decay amplitudes for decays can be generally described by
(5)  
where the superscript denotes the helicity states of two mesons with standing for the longitudinal (transverse) component and the definitions of the amplitudes in terms of the Lorentzinvariant amplitudes , and are
(6)  
with and denoting the polarization vector and momentum of the state correspondingly. Here, with and , the masses of the light and heavy mesons, respectively. We will therefore analyze the helicity amplitudes based on the pQCD approach. According to the helicity amplitudes (6), the transversity ones can be defined as
(7) 
for the longitudinal, parallel, and perpendicular polarizations, respectively, where the ratio and the normalization factor with the decay width and the momentum of either of the outgoing axialvector mesons . These amplitudes satisfy the following relation,
(8) 
As illustrated in Fig. 1, analogous to the and decays Liu:2012jb (), there are 8 types of diagrams contributing to the decays at the lowest order in the pQCD approach. Because the amplitudes for the Feynman diagrams of the decays have been analyzed explicitly in Ref. Liu:2012jb (), then the decay amplitudes can be easily obtained from the Eqs. (25)(60) by appropriate replacements correspondingly:

(1) When the state flies(recoils) along with the direction in the meson rest frame, the above mentioned Eqs. (25)(60) Liu:2012jb () will describe the decays with transition, in which the related form factor can be factored out. The Feynman decay amplitudes will be expressed with and ;

(2) When the state flies(recoils) along with the direction in the meson rest frame, the above mentioned Eqs. (25)(60) Liu:2012jb () will describe the decays with transition, in which the related form factor can also be extracted out. The Feynman decay amplitudes will be presented with and .
Hence, for simplicity, we will not present the factorization formulas for these modes again in this work. The interested readers can refer to Ref. Liu:2012jb () for details. By combining various contributions from the relevant Feynman diagrams together, the decay amplitudes of the decays can then be collected straightforwardly with three polarizations as follows:
(9)  
(10)  
(11)  
(12)  
(13)  
where and stand for the products of CKM matrix elements and , respectively. The standard combinations of Wilson coefficients are defined as follows,
(14) 
where and the upper(lower) sign applies when is odd(even).
Now, we will turn to the numerical evaluations of the branching ratios and polarization fractions of the considered decays in the pQCD approach. The essential comments on the input parameters are given as follows:

For the heavy emsons and light axialvector and states, the same hadron wave functions and distribution amplitudes including Gegenbauer moments are adopted as those in Ref. Liu:2012jb (). And the same QCD scale, masses of hadrons, and decay constants are also utilized. The meson lifetime is updated as ps Olive:2016xmw ().

As for the CKM matrix elements, we adopt the Wolfenstein parametrization at leading order and the newly updated parameters , , , and Olive:2016xmw ().
The theoretical predictions for decays evaluated in the pQCD approach, together with the results in the QCDF approach, have been grouped in the Tables 13, in which the first error is induced by the uncertainties of the shape parameter GeV in the meson wave function, the second error arises from the combination of the uncertainties of Gegenbauer moments , and in the distribution amplitudes of and mesons, and the last error is also the combined uncertainty from the CKM matrix elements: and Olive:2016xmw (). It is easily seen that the theoretical predictions suffer from large uncertainties that mainly induced by the parameters describing the nonperturbative hadron dynamics. It is therefore expected that the predictions given in the pQCD approach could be improved greatly with the wellconstrained inputs based on the nonperturbative QCD, e.g., Lattice QCD, calculations with high precision and/or the future precise measurements experimentally.
Decay Channels  

Parameter  Definition  This work  QCDF  This work  QCDF  This work  QCDF 
BR()  
Decay Channels  

Parameter 
Definition  This work  QCDF  This work  QCDF 
BR()  
Branching ratios
We first analyze the branching ratios of the decays according to the numerical results obtained in the pQCD approach. And furthermore, since these considered modes have been studied in another popular QCDF approach, we also quote the related predictions to make an essential comparison and discussion, which could be helpful to further discriminate these two rather different tools through the future precise measurements.
As presented in Tables 12, the pQCD predictions for the branching ratios of the classified five modes ^{3}^{3}3It should be stressed that the final states in the former , , and modes are the CP eigenstates, while those in the latter and ones are not, which therefore result in the branching ratios with and without the CPaveraged final states as presented in Tables 1 and 2, respectively. are from to , explicitly,
(15) 
Decay Channels  

Parameter  Definition  This work  This work  This work 
BR()  