in a covariant constituent quark model
with infrared confinement
We investigate the exclusive semileptonic decays , where , within the Standard Model. The relevant transition form factors are calculated in the framework of a relativistic constituent quark model with built-in infrared confinement. Our calculations predict the branching fractions to be of the order of for and for . Most of our numerical results are consistent with other theoretical studies. However, some branching fractions are larger than those calculated in QCD sum rules approaches but smaller than those obtained in the covariant light-front quark model by a factor of about .
Low lying states of quarkonia systems similar to usually decay through intermediate photons or gluons produced by the parent quark pair annihilation Sharma:1998gc (). As a result, strong and electromagnetic decays of have been largely investigated while weak decays of have been put aside for decades. However, in the last few years many improvements in instruments and experimental techniques, in particular, the luminosity of colliders, have led to observation of many rare processes including the extremely rare decays , announced lately by the CMS and LHCb collaborations CMS:2014xfa (). The branching fractions were measured to be and . This raises the hope that one may also explore the rare weak decays of charmonium and draws researchers’ attention back to these modes.
Recently, BESIII Collaboration reported on their search for semileptonic weak decays Ablikim:2014fpb (), where “” indicates that the signals were sum of these modes and the relevant charge conjugated ones. The results at confidence level were found to be and . Although these upper limits are far above the predicted values within the Standard Model (SM), which are of the order of SanchisLozano:1993ki (); Wang:2007ys (); Shen:2008zzb (), one should note that this was the first time an experimental constraint on the branching fraction was set, and moreover, the constraint on the branching fraction was times more stringent than the previous one Agashe:2014kda (). With a huge data sample of events accumulated each year, BESIII is expected to detect these decays, even at SM levels, in the near future.
From the theoretical point of view, these weak decays are of great importance since they may lead to better understanding of nonperturbative QCD effects taking place in transitions of heavy quarkonia. Moreover, the semileptonic modes , as three-body weak decays of a vector meson, supply plentiful information about the polarization observables that can be used to probe the hidden structure and dynamics of hadrons. Additionally, these decays may also provide some hints of new physics beyond the SM, such as TopColor models Hill:1994hp (), the Minimal Supersymmetric Standart Model (MSSM) with or without R-parity Martin:1997ns (), and the two-Higgs-doublet models (2HDMs) Hou:1992sy (); Baek:1999ch ().
The very first estimate of was made based on the (approximate) spin symmetry of heavy mesons, giving an inclusive branching fraction of , summed over , , , and both charge conjugate modes SanchisLozano:1993ki (). In this work the transition form factors were parametrized through a universal function, similar to the Isgur-Wise function in the heavy quark limit. However, the zero-recoil approximation adopted in calculating the hadronic matrix elements led to large uncertainties in the decay width evaluation. For that reason, author of SanchisLozano:1993ki () noted that these results should be viewed as an estimate suggesting experimental searching, rather than a definite prediction. Recently, by employing QCD sum rules (QCD SR) Wang:2007ys () or making use of the covariant light-front quark model (LFQM) Shen:2008zzb (), new theoretical studies found the branching fractions of to be of the order of . However, the results presented in Shen:2008zzb () were about times larger than those calculated in Wang:2007ys (). Besides, one can significantly reduce hadronic uncertainties and other physical constants like and by considering the ratio of branching fractions . This ratio had been predicted to be in SanchisLozano:1993ki () while the recent study Wang:2007ys () suggested . Clearly, more theoretical studies and cross-check are necessary.
In the present work we offer an alternative approach to the investigation of the exclusive decays , in which we employ the covariant constituent quark model with built-in infrared confinement [for short, confined covariant quark model (CCQM)] as dynamical input to calculate the nonperturbative transition matrix elements. Our paper is organized as follows: In Sec. II, we set up our framework by briefly introducing the CCQM. Sec. III contains the definitions and derivations of the form factors of the decays based on the effective Hamiltonian formalism. In this section we also describe in some detail how calculation of the form factors proceeds in our approach. Sec. IV is devoted to the numerical results for the form factors, including comparison with the available data. Sec. V contains our numerical results for the branching fractions. And finally, we make a brief summary of our main results in Sec. VI.
The CCQM has been developed in some of our earlier papers (see Ivanov:2011aa () and references therein). In the CCQM framework one starts with an effective Lagrangian describing the coupling of a meson to its constituent quarks and ,
where is the relevant Dirac matrix and is the coupling constant. The vertex function is related to the scalar part of the Bethe-Salpeter amplitude and characterizes the finite size of the meson. Transitions between mesons are evaluated by one-loop Feynman diagrams with free quark propagators.The high energy divergence of quark loops is tempered by nonlocal Gaussian-type vertex functions with a falloff behavior. We adopt the following form,
where . This form of is invariant under the translation , which is necessary for the Lorence invariance of the Lagrangian (1).
We adopt a Gaussian form for the vertex function:
The parameter characterizes the size of the meson. The calculations of the Feynman diagrams proceed in the Euclidean region where and therefore the vertex function has the appropriate falloff behavior to provide for the ultraviolet convergence of the loop integral.
The normalization of particle-quark vertices is provided by the compositeness condition Z=0 ()
where is the wave function renormalization constant of the meson and is the derivative of the meson mass function. To better understand the physical meaning of the compositeness condition we want to remind the reader that the constant can be view as the matrix element between the physical particle state and the corresponding bare state. The compositeness condition implies that the physical bound state does not contain the bare state. The constituents are virtual and they are introduced to realize the interaction described by the Lagrangian (1). As a result of the interaction, the physical particle becomes dressed and its mass and wave function are renormalized. Technically, the compositeness condition allows one to evaluate the coupling constant . The meson mass function in (4) is defined by the Feynman diagram shown in Fig. 1. It has the explicit form
for a pseudoscalar meson and a vector meson, respectively. Note that we use the free quark propagator
where is the constituent quark mass.
The confinement of quarks is embedded in an effective way: first, by introducing a scale intergration in the space of -parameters; and second, by truncating this scale intergration on the upper limit that corresponds to an infrared cutoff. By doing this one removes all possible thresholds in the quark diagram. The cutoff parameter is taken to be universal. Other model parameters are adjusted by fitting to available experimental data. Once these parameters are fixed, one can employ the CCQM as a frame-independent tool for hadronic calculation. One of the advantages of the CCQM is that in this framework the full physical range of momentum transfer is available, making calculation of hadronic quantities straightforward without any extrapolation.
Iii hadronic matrix elements
The effective Hamiltonian describing the semileptonic decays is given by
where , and is the weak Dirac matrix with left chirality.
In the CCQM the hadronic matrix elements of the semileptonic meson decays are defined by the diagram in Fig. 2 and are given by
We use the on-shell conditions , , and . Because there are three quark types involved in the transition, we have introduced a two-subscript notation such that .
where is the loop momentum and is the external momentum. As described later on, the use of the Fock-Schwinger representation allows one to do tensor loop integrals in a very efficient way since one can convert loop momenta into derivatives of the exponent function.
All loop integrations are performed in Euclidean space. The transition from Minkowski space to Euclidean space is performed by using the Wick rotation
so that Simultaneously one has to rotate all external momenta, i.e. so that . Then the quadratic form in Eq. (11) becomes positive definite,
and the integral over is absolutely convergent. We will keep the Minkowski notation to avoid excessive relabeling. We simply imply that and .
Collecting the representations for the vertex functions and quark propagators given by Eqs. (3) and (11), respectively, one can perform the Gaussian integration in the expressions for the matrix elements in Eqs. (III) and (10). The exponent has the form , where . Using the following properties,
one can replace by which allows one to exchange the tensor integrations for a differentiation of the Gaussian exponent which appears after integration over loop momentum. The -dependent Gaussian exponent can be moved to the left through the differential operator by using the following properties,
Finally, one has to move the derivatives to the right by using the commutation relation
The last step has been done by using a form code which works for any numbers of loops and propagators. In the remaining integrals over the Fock-Schwinger parameters we introduce an additional integration which converts the set of Fock-Schwinger parameters into a simplex. We use the transformation
The integral over is well defined and convergent below the threshold . The convergence of the integral above threshold is guaranteed by the addition of a small imaginary to the quark mass, i.e. in the quark propagator. It allows one to rotate the integration variable to the imaginary axis . As a result the integral becomes convergent but obtains an imaginary part corresponding to quark pair production.
However, by cutting the scale integration at the upper limit corresponding to the introduction of an infrared cutoff
one can remove all possible thresholds present in the initial quark diagram Branz:2009cd (). Thus the infrared cutoff parameter effectively guarantees the confinement of quarks within hadrons. This method is quite general and can be used for diagrams with an arbitrary number of loops and propagators. In the CCQM the infrared cutoff parameter is taken to be universal for all physical processes Ivanov:2015tru ().
Finally, the matrix elements in Eqs. (III) and (10) are written down as linear combinations of the Lorentz structures multiplied by the scalar functions–form factors which depend on the momentum transfer squared. For the transition one has
where , , , .
For comparison of results we relate our form factors to those defined, e.g., in Khodjamirian:2006st (), which are denoted by a superscript . The relations read
We note in addition that the form factors satisfy the constraints
to avoid the singularity at .
In the case of transition we follow the authors in Wang:2007ys () and define the form factors as follows:
The form factors in our model are represented by the threefold integrals which are calculated by using fortran codes in the full kinematical momentum transfer region.
Iv Form factors
Before listing our numerical results we need to specify parameters of the CCQM that cannot be evaluated from first principles. They are the size parameter of hadrons , the universal infrared cutoff parameter and the constituent quark masses . These parameters are determined by a least-squares fit of calculated meson leptonic decay constants and several fundamental electromagnetic decays to experimental data and/or lattice simulations within a root-mean-square deviation of Ivanov:2000aj (). This value can provide a reasonable estimate of our theoretical error since the calculations in our work are, in principle, not different from those used in the fit. For example, based on a widespread application in a previous paper Ivanov:2007cw (), we suggested that a reasonable estimate of our theoretical error is .
The most recent fit results for those parameters involved in this paper are given in (22) (all in GeV):
Model-independent parameters and other physical constants like the Cabibbo-Kobayashi-Maskawa matrix elements, mass and decay width of the particles are taken from Agashe:2014kda (). For clarity we note that we use the values and .
|415.0||4189||LAT and QCD SR Becirevic:2013bsa ()|
|206.1||204.65.0||PDG Agashe:2014kda ()|
|244.3||LAT Becirevic:1998ua ()|
|LAT Becirevic:2012ti ()|
|QCD SR Lucha:2014xla ()|
|257.5||257.54.6||PDG Agashe:2014kda ()|
|272.0||LAT Becirevic:1998ua ()|
|3119||LAT Becirevic:2012ti ()|
|QCD SR Lucha:2014xla ()|
|1.249||1.2580.038||PDG Agashe:2014kda ()|
|1.113||1.160.020.06||LAT Becirevic:2012ti ()|
We present our results for the leptonic decay constants of the and mesons in Table 1. We also list the values of these constants obtained from experiments or other theoretical studies for comparison. One can see that our calculated values are consistent (within ) with results of other studies.
In Fig. 3-5 we present the dependence of calculated form factors of the transitions in the full range of momentum transfer . We found that the form factors and defined in (21) are very similar to each other. As mentioned earlier, the CCQM allows one to evaluate form factors in the full kinematical range including the near-zero recoil region. This feature is one of those that distinguish the CCQM from other frameworks like QCD SR and some other approaches. For example, the physical region of for is . However, within the QCD SR approach, the authors of Wang:2007ys () had to restrict their calculations in the range of to avoid additional singularities and then use an extrapolation to obtain the form factors in large region. As a result, the extrapolation type becomes more sensitive.
The results of our numerical calculation are well represented by a double-pole parametrization
where . The double-pole approximation is quite accurate. The relative error relative to the exact results is less than over the entire range, as demonstrated in Fig. 6.
In Tables 4 and 5 we compare the values of our form factors at (maximum recoil) with those obtained within QCD SR Wang:2007ys () and LFQM Shen:2008zzb (). Our results are more consistent with those in Wang:2007ys (). For example, our predictions for the form factors at differ from results of Wang:2007ys () within while the discrepancy can come to a factor of comparing with the results of Shen:2008zzb ().
|QCD SR Wang:2007ys ()||1.09||0.34||0.81||1.71||0.35||1.07|
|LFQM Shen:2008zzb ()||2.75||0.18||1.6||3.05||0.13||1.8|
V Numerical results
The invariant matrix element for the decay is written down as
The unpolarized lepton tensor for the process is given by Gutsche:2015mxa ()
where the upper/lower sign refers to the two configurations. The sign change can be seen to result from the parity violating part of the lepton tensors. In our case we have to use the lower sign in Eq. (25). Summing up the vector polarizations, one finds the decay rate
Here , , and . The upper and lower bounds of are given by
where is the Källén function.
The hadron tensor reads
|Mode||Unit||This work||QCD SR Wang:2007ys ()||LFQM Shen:2008zzb ()|
We present our results for the branching fractions in Table 6 together with results of other theoretical studies based on QCD SR and LFQM for comparison. It is worth mentioning that all values for are fully consistent with those in Wang:2007ys (). Regarding , our results are larger than those in Wang:2007ys () by a factor of . We think this discrepancy is mainly due to the values of the meson leptonic decay constants and used in Wang:2007ys (), which are much smaller than and used in our present paper. In contrast, the constants and used in Wang:2007ys () are very close to our values of and , resulting in a full agreement in between the two studies. Comparing with another study, our results for are smaller than those in Shen:2008zzb () by a factor of .
It is interesting to consider the ratio , where a large part of theoretical and experimental uncertainties cancels. We list in (29) all available predictions for up till now:
Wang’s result for is about two times greater than our prediction because their branching fraction is about two times smaller than ours (mainly due to the leptonic decay constants). Therefore, we propose that the value is a reliable prediction.
Moreover, we also consider the ratios
which should be equal to under the flavor symmetry limit. These ratios are and in Wang:2007ys (). In this work we have the following values, and , which suggest a relative small symmetry breaking effect.
Vi Summary and conclusions
Let us summarize the main results of our paper. We have calculated the hadronic form factors relevant to the semileptonic decay in the framework of the confined covariant quark model. By using the calculated form factors and Standard Model parameters we have evaluated the decay rates and branching fractions. We have compared our results with those obtained in other approaches.
Acknowledgements.M.A.I. acknowledges Mainz Institute for Theoretical Physics (MITP) and the Heisenberg-Landau Grant for the support.
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