I Introduction

Decay constants and form factors of s-wave and p-wave mesons in the covariant light-front quark model

11.30.Hv, 12.39.Ki, 12.39.Hg, 13.25.-k, 14.40.-n

March, 2011

Decay constants and form factors of -wave and -wave mesons

in the covariant light-front quark model

R. C. Verma

Institute of Physics, Academia Sinica, Taipei, Taiwan, 11529

Permanent Address: Department of Physics, Punjabi University, Patiala, India, 147002


We reanalyze the decay constants of -wave and -wave mesons and form factors, where represents a pseudoscalar meson, a vector meson, a scalar meson, or an axial vector meson within a covariant light-front quark model. The parameter for wave-functions of most of -wave mesons and of a few axial-vector mesons are fixed with latest experimental information, wherever available or using the lattice calculations. The treatment of masses and mixing angles for strange axial vector mesons is improved for the purpose. We extend our analysis to determine the form factors appearing in the transition of transitions, and to the isoscalar final state mesons. Numerical results of the form factors for transitions between a heavy pseudoscalar meson and an -wave or -wave light meson and their momentum dependence are presented in detail. Further, their sensitivity to uncertainties of parameters of the initial as well as the final mesons is investigated. Some experimental measurements of the charmed and bottom meson decays are employed to compare the decay constants and transition form factors obtained in this and other works.

I Introduction

In the previous work Cheng04 (), various form factors, where represents a heavy pseudoscalar meson or , and represents either -wave or low-lying -wave meson, were calculated within the framework of the covariant light-front (CLF) approach. This formalism preserves the Lorentz covariance in the light-front framework and has been applied successfully to describe various properties of pseudoscalar and vector mesons Jaus90 (); Jaus91 (); Jaus99 (). The analysis of the covariant light-front quark model to transitions of the charmed and bottom mesons was extended to even parity, -wave mesons Cheng04 (). Recently, the CLF approach has also been used to the studies of the quarkonia Shen08 (); Hwang07 (), the -wave meson emitting decays of the bottom mesons Cheng10 () and the system Wang09 () and so on. In the present work, we update our results for and meson form factors, and extend this analysis to determine the form factors appearing in the transitions, and to the flavor-diagonal final state mesons . Experimental measurements of the decays of the lepton, pseudoscalar and vector mesons are employed to determine the decay constants, which in turn fix the shape parameters, , of the respective mesons. For a few cases, the decay constants estimated by lattice calculations have been used for this purpose. We have now used the improved estimation of the and mixing angle, where and are the and states of , respectively, which are related to the physical and states.

We then study transitions of the heavy flavor pseudoscalar mesons to pseudoscalar mesons (), vector mesons (), scalar mesons () and axial vector mesons () within the CLF model. Numerical results of the form factors for these transitions and their momentum dependence are presented in detail. In particular, all the form factors for heavy-to-light and heavy-to-heavy transitions for charmed mesons and bottom mesons are calculated. Further, their sensitivity to uncertainties of parameters of the initial as well as of the final mesons is investigated separately. Theoretically, the Isgur-Scora-Grinstein-Wise (ISGW) quark model ISGW (); IW89 () has been the only model for a long time that could provide a systematical estimate of the transition of a ground-state -wave meson to a low-lying -wave meson. However, this model is based on the nonrelativistic constituent quark picture. We have earlier pointed out Cheng04 () that relativistic effects could manifest in heavy-to-light transitions at maximum recoil where the final-state meson can be highly relativistic. For example, the form factor is found to be 0.13 in the relativistic light-front model Cheng04 (), while it is as big as 1.01 in the ISGW model ISGW (). Hence there is no reason to expect that the nonrelativistic quark model is still applicable there, though in the improved version of the model (ISGW2) ISGW2 () a number of improvements, such as the constraints imposed by heavy quark symmetry and hyperfine distortions of wave functions have been incorporated. We believe that the CLF quark model can provide useful and reliable information on transitions particularly at maximum recoil.

The paper is organized as follows. The basic features of the covariant light-front (CLF) model are recapitulated in Sec. II. In Sec. III, decay constants are presented in the CLF model. Available experimental measurements for various decays are used to determine decay constants, which in turn are used to fix parameters of the CLF model. Sometimes, lattice predictions for few decay constants are also used for this purpose. In Sec. IV, the analysis of form factors appearing for transitions from pseudoscalar mesons to -wave mesons (pseudoscalar or vector) and -wave mesons (scalar and axial vector) is given. In Sec. V, numerical results are presented for these form factors and their - dependence taking proper inclusions of uncertainties in the shape parameter, . Summary and conclusions are given in Sec. VI.

Ii Formalism of a covariant light-front model

     (a)                                                           (b)

Figure 1: Feynman diagrams for (a) meson decay and (b) meson transition amplitudes, where is the incoming (outgoing) meson momentum, is the quark momentum, is the anti-quark momentum and denotes the corresponding current vertex.

In the conventional light-front framework, the constituent quarks of the meson are required to be on their mass shells and various physical quantities are extracted from the plus component of the corresponding current matrix elements. However, this procedure will miss the zero-mode effects and render the matrix elements non-covariant. Jaus Jaus90 (); Jaus91 () has proposed a covariant light-front approach that permits a systematical way of dealing with the zero mode contributions. Physical quantities such as the decay constants and form factors can be calculated in terms of Feynman momentum loop integrals which are manifestly covariant. This of course means that the constituent quarks of the bound state are off-shell. In principle, this covariant approach will be useful if the vertex functions can be determined by solving the QCD bound state equation. In practice, we would have to be contended with the phenomenological vertex functions such as those employed in the conventional light-front model. Therefore, using the light-front decomposition of the Feynman loop momentum, say , and integrating out the minus component of the loop momentum , one goes from the covariant calculation to the light-front one. Moreover, the antiquark is forced to be on its mass shell after integration. Consequently, one can replace the covariant vertex functions by the phenomenological light-front ones.

To begin with, we consider decay and transition amplitudes given by one-loop diagrams as shown in Fig. 1 for the decay constants and form factors of ground-state -wave mesons and low-lying -wave mesons. We follow the approach of Jaus99 (); Cheng04 () and use the same notation. The incoming (outgoing) meson has the momentum , where and are the momenta of the off-shell quark and antiquark, respectively, with masses and . These momenta can be expressed in terms of the internal variables ,


with . Note that we use , where , so that .

In the covariant light-front approach, total four momentum is conserved at each vertex where quarks and antiquarks are off-shell. These differ from the conventional light-front approach (see, for example  Jaus91 (); Cheng97 ()) where the plus and transverse components of momentum are conserved, and quarks as well as antiquarks are on-shell.

It is useful to define some internal quantities for on-shell quarks:


Here can be interpreted as the kinetic invariant mass squared of the incoming system, and the energy of the quark .

pseudoscalar ()
vector ()
scalar ()
axial ()
axial ()
Table 1: Feynman rules for the vertices () of the incoming mesons-quark-antiquark, where and are the quark and antiquark momenta, respectively. Under the contour integrals to be discussed below, and are reduced to and , respectively, whose expressions are given by Eq. (12). Note that for outgoing mesons, we shall use for the corresponding vertices.

It has been shown in CM69 () that one can pass to the light-front approach by integrating out the component of the internal momentum in covariant Feynman momentum loop integrals. We need Feynman rules for the meson-quark-antiquark vertices to calculate the amplitudes shown in Fig. 1. These Feynman rules for vertices () of ground-state -wave mesons and low-lying -wave mesons are summarized in Table 1. Next, we shall find the decay constants in the covariant light-front approach.

Iii Decay constants

The decay constants for mesons are defined by the matrix elements


where the , , , and states of mesons are denoted by , , , and , respectively. It is useful to note that in the SU(N)-flavor limit () we should have vanishing and . The former can be seen by applying equations of motion to the matrix element of the scalar resonance in Eq. (3) to obtain


The latter is based on the argument that the light and states transfer under charge conjugation as


where the light axial-vector mesons are represented by a matrix. Since the weak axial-vector current transfers as under charge conjugation, it is clear that the decay constant of the meson vanishes in the SU(3) limit Suzuki (). This argument can be generalized to heavy axial-vector mesons. In fact, under similar charge conjugation argument [, ] one can also prove the vanishing of in the SU(N) limit.

Furthermore, in the heavy quark limit (), the heavy quark spin decouples from the other degrees of freedom so that and the total angular momentum of the light antiquark are separately good quantum numbers. Hence, it is more convenient to use the , , and basis. It is obvious that the first and the last of these states are and , respectively, while IW91 ()


Heavy quark symmetry (HQS) requires IW89 (); HQfrules ()


where we have denoted the and states by and , respectively. These relations in the above equation can be understood from the fact that , and form three doublets in the HQ limit and that the tensor meson cannot be induced from the current.

Following the procedure described in Jaus99 (); Cheng04 (), we now evaluate meson decay constants through the following formulas:




are the appropriate replacements of the vertex functions,


appearing in the matrix elements of annihilation of a meson state via weak currents, and and are the light-front momentum distribution amplitudes for -wave and -wave mesons, respectively. There are several popular phenomenological light-front wave functions that have been employed to describe various hadronic structures in the literature. In the present work, we shall use the Gaussian-type wave function Gauss ()


The parameter , which describes the momentum distribution, is expected to be of order .

Note that with the explicit form of shown in Eq. (12), the familiar expression of in the conventional light-front approach Jaus91 (); Cheng97 (), namely,


is reproduced. For decay constants of vector and axial-vector mesons, we consider the case with the transverse polarization given by


For , the meson wave function is symmetric with respect to and , and hence , as it should be. Similarly, it is clear that for . The SU(N)-flavor constraints on and are thus satisfied.1

Table 2: The input parameter (in units of GeV) in the Gaussian-type wave function (14) for mesons. Note that is used for the state.

To perform numerical computations of decay constants and form factors, we need to specify the input parameters in the covariant light front model. These are the constituent quark masses and the shape parameter appearing in the Gaussian-type wave function (14). For constituent quark masses, we use Jaus96 (); Cheng97 (); Hwang02 (); Jaus99 (); Cheng04 ()


Shown in Tables 2 and 3 are the input parameter and decay constants, respectively. In Table 3 the decay constants in parentheses are used to determine using the analytic expressions in the covariant light-front model as given above. For most of -wave mesons, and a few axial vector mesons, these are fixed from the latest decay rates given in the Particle Data Group  PDG10 (), or other analysis based on some experimental results. For decay constants of some heavy flavor mesons, we have used recent lattice results to fix . For the remaining -wave mesons, we use the parameters obtained in the ISGW2 model ISGW2 (), the improved version of the ISGW model, up to some simple scaling. In this paper, we have investigated the variation of the form factors and their slope parameters for dependence with the variation of values. Wherever the experimental information is available, we have used that to fix the errors for the corresponding values, otherwise arbitrarily introduced an uncertainty of in for some -wave and -wave mesons.

Table 3: Meson decay constants (in units of MeV) obtained by using Eqs. (8), (10), (9) and (11). Those in parentheses are taken as inputs to determine the corresponding ’s shown in Table 2. Decay constants of some -wave mesons are also used as inputs (see the text for details). Here denotes decay constant for the state.

Several remarks are in order:

(i) Decay constants of the charged pseudoscalar mesons, and (and their charge-conjugate partners) can be determined from their purely leptonic decay rates. These mesons formed from a quark and anti-quark can decay to a charged lepton pair when their constituents annihilate via a virtual boson. Now quite precise measurements are available for the branching fractions of decays PDG10 (). Following the analysis of Rosner and Stone Rosner10 () for the available branching fractions, we take (all in MeV) to fix the parameters of the respective mesons.

(ii) For fixing , we have taken the world average value MeV for given by the Heavy Flavor Averaging Group HFAG10 () based on the BaBar, Belle and CLEO measurements of and . This value can be compared well to the results from the two precise lattice QCD calculations MeV and MeV, respectively, from the HPQCD Collaboration HPQCD10 () and the Fermilab/MILC Collaboration MILC08 (). For the bottom sector, the Belle and BaBar collaborations have found evidence for decay in collisions at the energy, however, the errors are rather large in the measured branching fractions with the computed average value . Further a more accurate value of is required for the determination of . Considering the large uncertainties on and the branching fraction measurements for , and sensitivity of this decay to the new physics, we rely upon MeV, used in Rosner10 () as the average of the two lattice results MeV MILC08 () and MeV HPQCD09 (), to fix the input parameter . Likewise, for meson, we use the lattice prediction of MeV HPQCD09 () for determining .

(iii) The decay constants of the diagonal pseudoscalar mesons and , in principle, could be obtained from branching fractions. In the case of , the value of MeV Suzuki03 () has been extracted from the measured decay width, which is compatible with , as is expected from isospin symmetry. However, decay constants of the system cannot be extracted from two-photon decay rates alone and get more complicated due to the mixing, the chiral anomaly and gluonium mixing Feldmann00 (); ChengLi09 (). For describing the mixing between and , it is more convenient to employ the flavor states , and labeled by the and , respectively. We then write


where follows from the analysis of Feldmann et al. Feldmann00 () to fit the experimental data. This analysis also gives and , which are used in the present work. For , the decay width is poorly known with PDG PDG10 () estimate given as keV giving GeV. Alternatively, one may extract from decay using the factorization approximation, for which CLEO Edwards01 () obtained MeV. In the literature, is expected to be quite close to on the basis of quark model considerations Gourdin95 (). Recently, the HPQCD collaboration HPQCD10 () has reported a more precise result for to be MeV consistent with other estimates, and is in fact very close to the experimental result MeV obtained from the leptonic decay width of PDG10 (). So we use the lattice prediction to fix . In the absence of any experimental estimate for , we shall assume to fix following the heavy-quark spin symmetry.

(iv) For vector mesons, we extract the decay constants for diagonal states from the experimental values of their respective branching fractions of leptonic decays decays PDG10 (). Thus we obtain and (all in MeV) for ideal mixing, and use them to fix the parameters of the respective mesons.

(v) The decay constant determines not only the coupling of the neutral vector mesons to a photon, but also the coupling of charged vector mesons, like and , to the weak vector bosons . There are no data available for the leptonic decay of these charged vector mesons, but the couplings can be extracted indirectly from the decays and . With the experimental values for the branching fractions of these decays and , the decay width formula


where is the appropriate CKM- factor corresponding to the vector meson , yield MeV and MeV, respectively. It is worth noting that the difference in and seems consistent with the expected size of isospin breaking, and we take the average of the two values, i.e., MeV, the error chosen so as to satisfy the two cases in extreme limits.

(vi) Contrary to the non-strange charmed meson case, where has a slightly larger decay constant than , the recent measurements of PDG10 (); BaBarDs () indicate that the decay constants of and are relatively similar. As for the decay constant of , a recent lattice calculation yields Bernard (). Explicitly, for naked charmed and bottom states , and , we have used the lattice predictions, , and (all in MeV) Becirevic99 () to fix the central value of the respective parameters , and allow variation in each case, giving decay constant ratios as , and , to leave the scope for matching with other results.

(vii) For axial vector mesons, there are two different nonets of in the quark model as the orbital excitation of the system. In terms of the spectroscopic notation , there are two types of -wave axial vector mesons, namely, and , which have distinctive C quantum numbers, and , respectively. Experimentally, the nonet consists of , and , while the nonet has , and .

(viii) It is generally argued that should have a similar decay constant as the meson. Presumably, can be extracted from the decay . Though this decay is not shown in the Particle Data Group PDG10 (), an experimental value of MeV is nevertheless quoted in Bloch ().2 The decay constant MeV obtained using the QCD sum rule method Yang07 () is slightly higher than this value as well as MeV. In Table 3 we have employed MeV as input following our sign convention.

(ix) The nonstrange axial-vector mesons, for example, and cannot have mixing because of the opposite -parities. On the contrary, physical strange axial-vector mesons are the mixture of and states, while the heavy axial-vector resonances are generally taken as the mixture of and . For example, the physical mass eigenstates and are a mixture of and states owing to the mass difference of the strange and nonstrange light quarks:


Using the experimental results and