# Some of semileptonic and nonleptonic decays of meson in a Bethe-Salpeter relativistic quark model

###### Abstract

The semileptonic decays and the nonleptonic decays , where denotes a pseudoscalar (vector) charmonium or ()-meson, and denotes a light meson, are studied in the framework of improved instantaneous Bethe-Salpeter (BS) equation and the Mandelstam formula. The numerical results (width and branching ratio of the decays) are presented in tables, and in order to compare conveniently, those obtained by other approaches are also put in the relevant tables. Based on the fact that the ratio estimated here is in good agreement with the observation by the LHCb , one may conclude that with respect to the decays the present framework works quite well.

###### pacs:

13.20.He, 13.20.Fc, 13.25.Ft, 13.25.Hw, 11.10.St^{†}

^{†}preprint: APS/123-QED

meson carries two heavy flavor quantum numbers explicitly, and it decays only via weak interactions, although the strong and electromagnetic interactions can affect the decays. As consequences, meson has a comparatively long lifetime and very rich weak decay channels with sizable branching ratios. Being an explicit double heavy flavor meson, its production cross section can be estimated by perturbative QCD quite reliably and one can conclude that only via strong interaction and at hadronic high energy collisions the meson can be produced so numerously that it can be observed experimentally 07 (); 06 (); D0 (). Therefore, the meson is specially interesting in studying its production and decays both.

The first successful observation of was achieved through the semileptonic decay channel by CDF collaboration in 1998 from Run-I at Tevatron. They obtained the mass of : GeV and the lifetime: ps CDF (). Later on CDF collaboration further gave a more precise mass (stat)5(syst) MeV/ obtained through the exclusive non-leptonic decay CDF2006 (), and upgraded their results CDF2008 (). In the meantime D0 collaboration at Tevatron also carried out the observations and confirmed CDF results D02008 (). Resently, LHCb reported several new observations on decays LHCb (). Thus we may reasonably expect that at LHCb in the near future the data will be largely enhanced and new results are issued in time.

In literatures, there are many works studying various decays
Chang (); Wise (); d2 (); d5 (); d6 (); Colangelo (); Abd (); Anisimov (); Ebert (); Ivanov (); Sanchis (); Nobes (); Yang (); Lusignoli (); Liu (); Du (); Hern (); caid (); Choi (); Xiaozj ()
under different approaches. Among the approaches in the market, the one used in Ref. Chang ()
is that when the components in the concerned meson(s) in initial and final states are heavy quarks, an instantaneous Bethe-Salpeter (BS) equation BE () (also called Salpeter equation
E ()) with an instantaneous QCD-inspired kernel (interaction)^{1}^{1}1With the equation, the spectrum and relevant wave function as an eigenvalue
problem derived from the BS equation can be computed. is used to depict the meson(s) and the Mandelstam formula mand () is adopted to compute hadron matrix elements relevant to the concerned decays. This
approach has a comparatively solid foundation because the relativistic ‘recoil effects’
in the decays^{2}^{2}2The difference between masses of the initial meson and the decay product e.g. charmonium is great, so the recoil in a decay must be relativistic, and the ”recoil effects” in the decay should be taken into account well. may be taken into account better than in potential model and else approaches. The reason is that the BS equation and the Mandelstam formula
both are established on relativistic quantum field theory, although the BS equation is deduced into an instantaneous one. Generally, when solving the instantaneous BS equation, the wave function needs to be formulated by a basis of angular momentum with the spin of its components according to the bound state quantum numbers, such as pseudoscalar or vector or else, whereas in Ref. Chang () to do the formulation the authors, followed Ref. E (), took an extra
approximation. Since now a way to solve the instantaneous BS equation
without the extra approximation is avaliable changwang (), and a way more properly to treat the relevant transition matrix elements in Mandelstam formulation has been explored for years changwang1 (), so we think that now it is the right time by using the new wave functions obtained by solving the instantaneous BS equation without the extra approximation and the improved formula for the transition matrix elements to estimate the decays theoretically and then to compare the results with the newly experimental data to see how well the new improved approach changwang (); changwang1 () works. Considering the progresses in experiments, especially those at LHCb,
in this paper we would like to restrict ourselves to focus lights on the Cabibbo-Kobayashi-Maskawa (CKM) favored decays: the semileptonic ones and the nonleptonic ones precisely,
where represents pseudoscalar (vector) charmonium or a bound state.

The paper is organized as follows. In Sec. I we outline the useful formulas. In Sec. II we present numerical results for the semileptonic and nonleptonic decays and compare the results with those obtained by other approaches. Sec. III is contributed to discussions. We put the relativistic BS equation with covariant instantaneous approximation, the forms of relativistic wave functions for pseudoscalar and vector mesons, the formulations of the form factors, and the parameters used to solve the BS equation into Appendices.

## I Formulations for semileptonic and nonleptonic decays

For the semileptonic decays shown in Fig. 1, the -matrix element can be written as hadronic component and leptonic component:

(1) |

where is the CKM matrix element, is the charged weak current responsible for the decays, , are the momenta of the initial state and the final state respectively, while is the polarization vector when is a vector particle. The square of the matrix element, summed and averaged over the spin (unpolarized), is:

(2) |

where the leptonic tensor:

(3) |

is easy to compute, and the hadronic tensor is defined by:

(4) |

where . The general form of based on Lorentz-covariance analysis can be written as:

(5) | |||||

By a straightforward calculation, the differential decay rate is obtained:

(6) | |||||

where and , is the mass of meson, is the mass of the final state . The coefficient functions , , , , and relate to the form factors of weak currents directly (see below).

To evaluate the exclusive semileptonic differential decay rates of meson, one needs to calculate the hadron matrix element of the weak current sandwiched by the meson state as the initial state and a single-hadron state of the concerned final state, i.e., with being a given suitable meson. In fact, the hadron matrix elements of weak currents can be generally expressed in terms of the momenta and of the mesons in initial state and final state respectivelly, as well as their coefficients. The coefficients, being functions of the momentum transfer , are Lorentz-invariant and are called as form factors usually. As emphasised in Refs. Chang (); d6 (), with the help of the Mandelstam formalism mand (), no matter how great the recoil happens the weak current hadron matrix element can be well calculated, thus here we adopt the method used in Refs. Chang (); d6 () but with improvements changwang (); changwang1 (), i.e., the used wave functions are obtained by solving the relevant instantaneous BS equation with the improved approach changwang (); changwang1 (). Although the improved approach is used to calculate the hadron matrix elements of weak currents, the form factors are still written as overlap integrals of the relevant wave functions for the bound states (mesons). To show the general feature of the improved approach, we put its outline in Appendices. Moreover, here we temperately constrain ourselves to consider the cases that is a -wave meson only.

According to the Mandelstam formalism and with wave functions of instantaneous BS equation(s), in the leading order, the matrix element can be written as changwang1 ():

(7) | |||||

here for the last equal sign we have chosen the center of mass system of initial mason ; is the propagator of the second component (“spectator”); is the three dimensional momentum of finial hadron state and ; is the component of BS wave function projected onto the “positive energy” for the relevant mesons, and may be obtained by solving the BS equation. Its definition can be found in Appendix A. Since the initial and finial states in the transition are both heavy mesons, as adopted in Eq. (7), it is a good approximation that only positive energy projected BS wave functions are included (the contributions from the component of the wave functions projected onto the “negative energy” are much smaller than that from the positive energy one).

The form factors can be generally related to the weak current matrix element as follows:

1. If is a state, of the weak current the axial vector matrix element vanishes, and the vector current matrix element can be written as:

(8) |

2. If is a state, of the weak current the axial vector matrix element can be written as:

(9) | |||||

and the vector current matrix element as:

(10) |

where is the polarization vector of the final hadron .

With the relation between the matrix element and the form factors above and using Eq. (7), the form factors can be calculated out. Explicitly expressions for the form factors as overlap integrals of meson wave functions are given in Appendix B.3. Correspondingly, the coefficient functions , and in Eq. (6) can be expressed in terms of the form factors. For example, for the decay ( is a pseudoscalar meson) we have:

(11) |

For the decay ( is a vector meson) we have:

(12) |

Putting the above form factors into the formula for differential decay rates Eq. (6), the concerned semileptonic decay rates can be calculated.

For the nonleptonic decays concerned here, we follow Ref. Chang () to take the CKM-favored effective Hamiltonian with QCD leading logarithm correction to be responsible for them:

(13) |

where and are the Wilson coefficients, and the four-fermion operators and are defined:

and denote ’down’ and ’strange’ weak eigenstates^{3}^{3}3Since we restrict ourselves to consider
the decays here, so we list the main operators the and only
which relate and greatly contribute to the decays.. Based on the QCD Renormalization Group (RG) calculation,
and in terms of the combination operators which have diagonal anomalous dimensions,
the corresponding Wilson coefficients read as follows Chang (); Wise ():

(14) |

Then to use ”naive factorization” as done in Ref. Chang (), the -matrix element can be written as:

(15) |

where are the relevant CKM matrix elements to and accordingly, , are the momenta of , and respectively, and are the polarization vectors for or and when is a vector meson. The parameter

(16) |

in Eq. (15) is attributed to the contribution from the operators and that from the Fierz-reordered with a suppressed factor to the concerned decays Chang ().

For the two-body decays concerned here, having the matrix element Eq. (15), it is straightforward to calculate the decay widths.

## Ii Numerical Results

The components of the meson are and quarks, and it happens that the contributions from each of them to the total decay rate are comparable in magnitude. Thus the semileptonic decay modes of meson can be classified into two: -quark decays with the quark inside the meson as a spectator, and -quark decays with the quark as a spectator. The former causes decays into charmonium or -meson pair, while the latter causes decays into or mesons. In this paper, we restrict ourselves to compute decays to charmonium or meson only because the approach adopted here is good for double heavy mesons.

When calculating the decays under the adopted approach, we need to fix several parameters. In fact, the parameters are fixed by fitting well-measured experimental data and the established potential model. The parameters appearing in the potential (the kernel of Salpeter equation) used in this work are fixed by the spectra of heavy quarkonia as done in Ref. changwang1 () and outlined in Appendix B.4. The masses of the ground states are used as inputs, while the masses of excited states are considered as predictions. According to the fits we obtain GeV and GeV, and to compare with experimental data GeV and GeV, one may see the fits are quite good.

Mode | Ours | Chang () | d2 () | Abd () | Anisimov () | Ebert () | Ivanov () | Sanchis () | Yang () | Lusignoli () | Liu () | Du () |
---|---|---|---|---|---|---|---|---|---|---|---|---|

14.2 | 11 | 11.1 | 13.05 | 5.9 | 14 | 10 | 4.3 | 10.6 | 8.31 | 6.5 | ||

26.6 | 59 | 14.3 | 22.0 | 12 | 29 | 18 | 11.75 | 16.4 | 26.8 | 11.1 | ||

34.4 | 28 | 30.2 | 26.6 | 17.7 | 33 | 42 | 16.8 | 38.5 | 20.3 | 21.8 | ||

44.0 | 65 | 50.4 | 51.2 | 25 | 37 | 43 | 32.56 | 40.9 | 34.6 | 43.7 | ||

0.727 | 0.28 | 0.46 | 0.605 | |||||||||

1.45 | 1.36 | 0.44 | 0.186 |

The values of the CKM matrix elements adopted in this paper are , , and . The properties of relevant light mesons appearing in the concerned nonleptonic decays are served as phenomenological inputs, namely we take

where the masses and the decay constants are taken from PDG PDG (), except and , which are quoted from Ref. Ball ().

The numerical results of semileptonic decays are presented in Table I, and here the uncertainties for our results are obtained by varying the model parameters , , , and by . For comparison precisely, the results from other typical approaches are also listed in the tables. To see the feature of the decays, we plot the lepton spectrum for the decays in Fig. 2 and Fig. 3 respectively.

The concerned nonleptonic decay modes (some for -decays and as spectator and some for -decays and as a spectator) are computed with uncertainties precisely too. The results, as well as some from other approaches for comparisons, are presented in Table II and Table III, respectively.

Mode | Ours | Chang () | d2 () | Abd () | Anisimov () | Ebert () | Lusignoli () | Liu () |
---|---|---|---|---|---|---|---|---|

1.97 | 1.43 | 1.22 | 0.82 | 0.67 | 1.79 | 1.01 | ||

0.152 | 0.12 | 0.090 | 0.079 | 0.052 | 0.130 | 0.0764 | ||

5.95 | 4.37 | 3.48 | 2.32 | 1.8 | 5.07 | 3.25 | ||

0.324 | 0.25 | 0.197 | 0.18 | 0.11 | 0.263 | 0.174 | ||

0.251 | 0.12 | 0.0708 | ||||||

0.018 | 0.009 | 0.00499 | ||||||

0.710 | 0.20 | 0.183 | ||||||

0.038 | 0.011 | 0.00909 | ||||||

2.07 | 1.8 | 1.59 | 1.47 | 0.93 | 1.71 | 1.49 | ||

0.161 | 0.15 | 0.119 | 0.15 | 0.073 | 0.127 | 0.115 | ||

5.48 | 4.5 | 3.74 | 3.35 | 2.3 | 4.04 | 3.93 | ||

0.286 | 0.22 | 0.200 | 0.24 | 0.12 | 0.203 | 0.198 | ||

0.268 | 0.19 | 0.248 | ||||||

0.020 | 0.014 | 0.0184 | ||||||

0.622 | 0.40 | 0.587 | ||||||

0.031 | 0.021 | 0.0283 |

Mode | Ours | Chang () | d2 () | Abd () | Anisimov () | Ebert () | Lusignoli () | Liu () |
---|---|---|---|---|---|---|---|---|

58.4 | 167 | 15.8 | 34.8 | 25 | 44.0 | 65.1 | ||

4.20 | 10.7 | 1.70 | 2.1 | 3.28 | 4.69 | |||

44.8 | 72.5 | 39.2 | 23.6 | 14 | 20.2 | 42.7 | ||

1.06 | 0.03 | 0.292 | ||||||

51.6 | 66.3 | 12.5 | 19.8 | 16 | 34.7 | 25.3 | ||

2.96 | 3.8 | 1.34 | 1.1 | 2.52 | 1.34 | |||

150 | 204 | 171 | 123 | 110 | 152.1 | 139.6 |

## Iii Discussion and Conclusion

If comparing the semileptonic and nonleptonic decays estimated by various approaches via Tables I-III, one may find that the deviations among the theoretical predictions by the various approaches are quite wide. Specifically, the results with new solutions of the Salpeter equation and new formulation are quite different from those in Ref. Chang () too.

When calculating the decay branching ratio of semileptonic and nonleptonic decays, here the lifetime of meson is needed as input. For this purpose, we take the experimental lifetime from PDG PDG (). For the nonleptonic decays considered here, the parameter for nonleptonic decays appearing in Eq. (15), additionally, need to evaluate precisely too. Note that for quark (denoted as ) decays should be different from for quark (denoted as ) decays, and we take and as in Refs. d2 (); Ebert (); Ivanov (); Hern (); Choi (). Having the lifetime and the parameter fixed, the branching ratio of the concerned decay modes are straightforwardly calculated and we put the results in Table IV and Table V respectively.

Recently, LHCb has reported an observation of decays and i.e. the related ratio LHCb ()

(17) |

We would like to point out that, in contrary to the others observables, the above measured ratio, in which the production of meson is canceled totally, is a very essential test of the decays thus here we precisely give the corresponding ratio given by the approach adopted hare:

(18) |

and one may see that it is in good agreement with the observation. Here we should further note that the parameter which appears in Eq. (15) and the theoretical uncertainties caused by naive factorization for the nonleptonic decays would be canceled a lot in calculating the ratio. Namely the related ratio is mostly determined by hadron transition, so this agreement between the experimental value and the theoretical estimate on the ratio indicates a vary strong support of the present approach.

Mode | BR (%) |
---|---|

Mode | BR (%) | Mode | BR(%) |
---|---|---|---|

In summary, we have calculated the decay width and branching ratio of the exclusive semileptonic decays of meson to a charmonium or a meson plus leptons and nonleptonic decays to a charmonium or a meson plus a light meson under the improved instantaneous BS equation and Mandelstam approach. Under this approach, the full Salpeter equations for , and etc systems are solved with the respective full relativistic wave functions for and