Quasi-two-body decays in the perturbative QCD approach
In this work, we calculate the -averaged branching ratios and direct -violating asymmetries of the quasi-two-body decays by employing the perturbative QCD (PQCD) factorization approach, where is a light pseudoscalar meson , and . The considered decay modes are studied in the quasi-two-body framework by parametrizing the two-pion distribution amplitude . The -wave timelike form factor in the resonant regions associated with the and is estimated based on available experimental data. The PQCD predictions for the -averaged branching ratios of the decays are in the order of . The branching ratios of the two-body decays are extracted from the corresponding quasi-two-body decay modes. The whole pattern of the squared pion form factor measured by BABAR Collaboration could also be understood based on our studies. The PQCD predictions in this work will be tested by the precise data from the LHCb and the future Belle II experiments.
In recent years, prompted by a large number of experimental measurements bfbook (); prd78-072006 (); prd79-072004 (); lhcb0 (); prl111-101801 (); jhep10-143 (); prd90-112004 (); prl112-011801 (); prd95-012006 (); hfag2016 (), three-body hadronic -meson decays have been studied by using different theoretical frameworks plb564-90 (); prd91-014029 (); CY16 (); Wang-2014a (); Wang-2015a (); Wang-2016 (); ly15 (); ly16 (); zhou17 (). For such three-body decays, both resonant and nonresonant contributions may appear, as well as the possible final state interactions prd89-094013 (); 1512-09284 (); 89-053015 (). The nonresonant contributions have been studied with the method of heavy meson chiral perturbation theory prd46-1148 (); prd45-2188 (); plb280-287 () in Ref. CY16 (). Meanwhile, the resonant contributions are usually described with the isobar model prd11-3165 () in terms of the Breit-Wigner formalism BW-model (). Based on the QCD-improved factorization prl83 (), such decays have been studied by many authors plb622-207 (); prd74-114009 (); B.E:2009th (); ST:15 (); CY01 (); prd76-094006 (); CY16 (); prd89-094007 (); prd87-076007 (). By employing the perturbative QCD (PQCD) approach, the decays have also been investigated in Refs. Wang-2014a (); Wang-2015a (); Wang-2016 (); ly15 (); ly16 (); zhou17 (); ma16 (); ma17 (); Chen:2002th (); chen:2004th ().
In the PQCD approach Chen:2002th (); chen:2004th () for the cases of a -meson decaying into three final states, we restrict ourselves to the specific kinematical configurations, in which two energetic mesons are almost collimating to each other. The contribution from the region, where there is at least one pair of light mesons having an invariant mass below , being the meson and quark mass difference, as discussed in Refs. Wang-2014a (); Wang-2015a (); Chen:2002th (); chen:2004th (), is assumed dominant. The final state interactions are expected to be suppressed in such conditions. As a result, the dynamics associated with the meson pair could be factorized into a two-meson distribution amplitude MP (); MT01 (); MT02 (); MT03 (); MN (); Grozin01 (); Grozin02 (). The typical PQCD factorization formula for the decay amplitude can be written in the form of Chen:2002th ()
The hard kernel describes the dynamics of the strong and electroweak interactions in the three-body hadronic decays in a similar way as the cases of the two-body decays, and and are the wave functions for the meson and the final state , which absorb the nonperturbative dynamics in the related processes.
In this work, we extend the previous studies Wang-2016 (); ly16 () to the decays and in the PQCD approach with the help of the two-pion distribution amplitudes , where the stands for the light pseudoscalar mesons, , or . For simplicity, in the following parts of this work, and will be adopted to take the place of and , respectively. The theoretical studies of the excited states will provide us with a deeper understanding of the internal structure of hadrons. For and , there are not many studies except Refs. prd60-094020 (); prc79-025201 (); prd77-116009 (); 1205-6793 () in the frameworks of the quark model, the large- limits, or the double-pole QCD sum rules. For the phenomenological study of the two-body decays and , we still lack the distribution amplitudes of the states and at present. Fortunately, we are allowed to single out the (and ) component according to the two-pion distribution amplitudes as has been done in Ref. Wang-2016 (). Following Ref. Wang-2016 (), we here make an attempt to study the and decays in the quasi-two-body framework based on the PQCD factorization approach. And the branching fractions for the two-body decays will be extracted from the quasi-two-body processes .
This paper is organized as follows. In Sec. II, we give a brief introduction for the theoretical framework. The numerical values, some discussions, and the conclusions will be given in the last two sections.
In the light-cone coordinates, the meson momentum , the total momentum of the pion pair, , the momentum of the final state meson , the momentum of the spectator quark in the meson, the momentum for the resonant state , and for the final state are chosen as
where is the mass of meson, and the variable is defined as with the invariant mass squared . The parameter denotes the momentum fraction of the positive quark in each meson and runs from zero to unity. , and denote the transverse momentum of the positive quark, respectively. If we choose as one of the pion pair’s momentum fractions, the two pions momenta can be written as
The two-pion distribution amplitudes can be described in the same way as in Ref. Wang-2016 () ,
where the Legendre polynomial .
where is the two-pion invariant mass square, and () is the decay width (mass) for the relevant resonance . The mass and width for these excited mesons, and the values of the complex parameters and in Eq. (8) can be found in Ref. prd86-032013 (). The explicit expressions of the resonant state functions , and can be found, for example, in Ref. ly16 (). In this paper, we only consider the contributions from and . Following Ref. Wang-2016 (), we also assume that
for the two-pion distribution amplitudes as used in Ref. ly16 ().
The normalization factor depends on the value of and , which is defined through the normalization relation . We set GeV and GeV li2003 (); Xiao:2011tx () in the numerical calculations. The wave function of the final state pseudoscalar meson (, or ) is of the form
where is the chiral mass. The expressions of the relevant distribution amplitudes can also be found, for example, in Refs. ly16 (); ball99 (); ball9901 (); ball05 (); ball06 (); prd76-074018 (); ly14 (); prd90-114028 ().
Iii Numerical results
In numerical calculations, we use the following input parameters (in units of GeV except ) pdg2016 ():
The values of the Wolfenstein parameters are the same as those given in Ref. pdg2016 (): , , .
For the decay , the differential branching ratio is written as pdg2016 ()
with the mean lifetime of meson, and the invariant mass squared. The kinematic variables and denote one of the pion pair’s and ’s momentum in the center-of-mass frame of the pion pair,
|Decay modes||Quasi-two-body (in )||Two-body (in )|
|Decay modes||Quasi-two-body (in )||Two-body (in )|
By using the differential branching fraction, Eq. (16), and the decay amplitudes in the Appendix of Ref. ly16 (), we calculate the averaged branching ratios () and direct -violating asymmetries () for the decays and list the results in Table 1. Meanwhile, and for the decays are shown in Table 2. The four errors of these PQCD predictions as listed in Tables 1 and 2 come from the uncertainties of , , , and , respectively.
For and the other three decay modes, the PQCD predictions for their branching ratios as listed in Table I are a little different from those as given previously in Table I of Ref. . The reason is very simple: the Gegenbauer moments used here have been modified slightly from those in Ref.  as discussed in Ref. .
Taking the quasi-two-body decay as an example, the PQCD prediction for its branching ratio and -violating asymmetry are the following:
Here the individual errors as listed in Table 1 have been added in quadrature. Such a PQCD prediction for its branching ratio agrees well with the measured value from Collaboration within errors prd79-072006 (). Furthermore, the PQCD prediction for this decay mode is also consistent with the measured value from prd79-072006 ().
The width for the process was found to be MeV in Ref. ijmpa13-5443 (), which is consistent with the value MeV estimated from the annihilation experiments zpc62-455 (). The branching fraction could be induced with the GeV zpc62-455 (). The branching fraction, on the other hand, could be estimated from the relation EPJC2-269 ()
where the coupling is fetched from the component of the timelike form factor in Eq. (8) according to at . The decay constant GeV resulting from the data keV zpc62-455 () is adopted in this work, which agrees with GeV from the double-pole QCD sum rules 1205-6793 (), GeV from the perturbative analysis in the large- limit prd77-116009 (), or GeV from the relativistic constituent quark model prd60-094020 (). Utilizing Eq. (20), we find . From the definition of the decay rates between the quasi-two-body and the corresponding two-body decay modes
we then can find the PQCD predictions for , as listed in the third column of Table 1, where the individual errors have been added in quadrature.
For the cases of the considered quasi-two-body and two-body decays involving instead of , in principle, one can obtain the PQCD predictions for the branching ratios and -violating asymmetries in a similar way as the case for . But, there is not much reliable information about the properties of the meson except its mass and width ( GeV and GeV) pdg2016 (). What we can do here is to make some rough estimations of the branching ratios and violating asymmetries for the considered decays, and list the PQCD predictions in Table 2. For given keV zpc62-455 (), we find the longitudinal decay constant GeV. And then can be obtained by using the same methods as for the decays involving . The errors of the PQCD predictions listed in the third column of Table 2 have been added in quadrature.
In Fig. 1(a), we show the dependence of the differential decay rate after the inclusion of the contributions from the resonant state , , and . One can see that there exists a clear dip near in Fig. 1(a). The position of this dip and the pattern of the whole curve do agree well with Fig. 45 of Ref. prd86-032013 (), where the pion form factorsquared measured by are illustrated as a function of [i.e., ] in the region from to GeV. In fact, the differential decay rate does depend on the values of . In Fig. 1(a), we find the prominent peak, a shoulder around the and a clear dip followed by an enhancement (second a little lower and wide peak) in the region. The clear dip at is caused by the destructive interference between the resonant state and . We calculated numerically the interference terms between and amplitudes and found the large negative contribution to the branching ratios.