Advances in High Energy Physics
Volume 2016 (2016), Article ID 2352041, 10 pages
http://dx.doi.org/10.1155/2016/2352041

Research Article

Semileptonic Decays of to Light Tensor Mesons

Reza Khosravi and S. Sadeghi

Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran

Received 23 June 2016; Revised 21 August 2016; Accepted 6 September 2016

Academic Editor: Juan José Sanz-Cillero

Copyright © 2016 Reza Khosravi and S. Sadeghi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

Abstract.

The semileptonic , , transitions are investigated in the frame work of the three-point QCD sum rules. Considering the quark condensate contributions, the relevant form factors of these transitions are estimated. The branching ratios of these channel modes are also calculated at different values of the continuum thresholds of the tensor mesons and compared with the obtained data for other approaches.

1. Introduction

Investigation of the meson decays into tensor mesons is useful in several aspects such as CP asymmetries, isospin symmetries, and the longitudinal and transverse polarization fractions. A large isospin violation has already been experimentally detected in mode [1]. Also, the decay mode is mainly dominated by the longitudinal polarization [2, 3], in contrast with , where the transverse polarization is comparable with the longitudinal one [4]. Therefore, nonleptonic and semileptonic decays of meson can play an important role in the study of the particle physics.

In the flavor symmetry, the light -wave tensor mesons with containing isovector mesons , isodoublet states , and two isosinglet mesons and are building the ground state nonet which has been experimentally established [5, 6]. The quark content for the isovector and isodoublet tensor resonances is obvious. The isoscalar tensor states, and , have mixing wave functions where mixing angle should be small [7, 8]. Therefore, is primarily a state, while is dominantly [9].

As a nonperturbative method, the QCD sum rules is a well established technique in the hadron physics since it is based on the fundamental QCD Lagrangian [10]. The semileptonic decays of to the light mesons involving , , and have been studied via the three-point QCD sum rules (3PSR), for instance, [11], , [12–14],   [15], [16], and [17]. The determination of the form factor value relevant for the and [14, 18] decays allowed prediction of the ratio , which agrees with the experimental measurements [19–21]. The obtained results of the decay [11] and simulations on the lattice [22–24] are in a reasonable agreement.

In this work, we investigate decays within the 3PSR method. For analysis of these decays, the form factors and their branching ratio values are calculated. So far, the form factors of the semileptonic decays have been studied via different approaches such as the LCSR [25], the perturbative QCD (PQCD) [5], the large energy effective theory (LEET) [26–28], and the ISGW II model [29]. A comparison of our results for the form factor values in and branching ratio data with predictions obtained from other approaches, especially the LCSR, is also made.

The plan of the present paper is as follows: the 3PSR approach for calculation of the relevant form factors of decays is presented in Section 2. In the final section, the value of the form factors in and the branching ratio of the considered decays are reported. For a better analysis, the form factors and differential branching ratios related to these semileptonic decays are plotted with respect to the momentum transfer squared .

2. Theoretical Framework

In order to study decays, we focus on the exclusive decay via the 3PSR. The decay governed by the tree level transition (see Figure 1). In the framework of the 3PSR, the first step is appropriate definition of correlation function. In this work, the correlation function should be taken as

where and are four-momentum of the initial and final mesons, respectively. is the squared momentum transfer and is the time ordering operator. is the transition current. and are also the interpolating currents of and the tensor meson , respectively. With considering all quantum numbers, their interpolating currents can be written as follows [33]:

where is the four-derivative vector with respect to acting at the same time on the left and right. It is given as

where and are the Gell-Mann matrices and the external gluon fields, respectively. It should be noted that the second current in (2) interpolates a spin particle for massless quarks. In the general case, to describe a spin state one has to use a current such that the trace of vanishes.

Figure 1: Schematic picture of the spectator mechanism for the decay.

The correlation function is a complex function of which the imaginary part comprises the computations of the phenomenology and real part comprises the computations of the theoretical part (QCD). By linking these two parts via the dispersion relation, the physical quantities are calculated. In the phenomenological part of the QCD sum rules approach, the correlation function in (1) is calculated by inserting two complete sets of intermediate states with the same quantum numbers as and . After performing four integrals over and , it will be

In (4), the vacuum to initial and final meson state matrix elements is defined as

where and are the leptonic decay constants of and mesons, respectively. is polarization tensor of . The transition current gives a contribution to these matrix elements and it can be parametrized in terms of some form factors using the Lorentz invariance and parity conservation. The correspondence between a vector meson and a tensor meson allows us to get these parametrizations in a comparative way (for more information see [5]). The parametrization of form factors is analogous to the case except that is replaced by , as follows:

where , , and . The factor accounts for the flavor content of particles: for , and for [34]. Inserting (5) and (6) in (4) and performing summation over the polarization tensor as

where , the final representation of the physical side is obtained as

For simplicity in calculations, the following redefinitions have been used in (9):

Now, the QCD part of the correlation function is calculated by expanding it in terms of the OPE at large negative value of as follows:

where are the Wilson coefficients, is the unit operator, is the local fermion field operator, and is the gluon strength tensor. In (11), the first term is contribution of the perturbative and the other terms are contribution of the nonperturbative part.

To compute the portion of the perturbative part (Figure 1), using the Feynman rules for the bare loop, we obtain

taking the partial derivative with respect to of the quark free propagators and performing the Fourier transformation and using the Cutkosky rules, that is, , imaginary part of is calculated as

where is four-momentum of the spectator quark . To solve the integral in (13), we will have to deal with the integrals such as , , , and with respect to . For example, can be as

where and . , , , and can be taken as an appropriate tensor structure as follows:

The quantities , , , and are indicated in Appendix. Using the relations in (15), can be calculated for each structure corresponding to (9) as follows:

where the spectral densities    are found as

Using the dispersion relation, the perturbative part contribution of the correlation function can be calculated as follows:

For calculation of the nonperturbative contributions (condensate terms), we consider the condensate terms of dimensions 3, 4, and 5 related to the contributions of the quark-quark, gluon-gluon, and quark-gluon condensate, respectively. They are more important than the other terms in the OPE. In the 3PSR, when the light quark is a spectator, the gluon-gluon condensate contributions can be easily ignored [35]. On the other hand, the quark condensate contributions of the light quark, which is a nonspectator, are zero after applying the double Borel transformation with respect to both variables and , because only one variable appears in the denominator. Therefore, only two important diagrams of dimensions 3, 4, and 5 remain from the nonperturbative part contributions. The diagrams of these contributions corresponding to and are depicted in Figure 2. After some calculations, the nonperturbative part of the correlation function is obtained as follows:

where , [35], and , ; that is, we choose the value of the condensates at a fixed renormalization scale of about 1 GeV [36, 37].

(a)

(b)

Figure 2: The diagrams of the effective contributions of the condensate terms.

The next step is to apply the Borel transformations with respect to and on the phenomenological as well as the perturbative and nonperturbative parts of the correlation functions and equate these two representations of the correlations. The following sum rules for the form factors are derived:

where and and are the continuum thresholds in the initial and final channels, respectively. The lower limit in the integration over is . Also, transformation is defined as follows:

where and are Borel mass parameters.

In (20), to subtract the contributions of the higher states and the continuum, the quark-hadron duality assumption is also used; that is, it is assumed that

We would like to provide the same results for and decays. With a little bit of change in the above expressions such as and , we can easily find similar results in (20) for the form factors of the new transitions.

3. Numerical Analysis

In this section, we numerically analyze the sum rules for the form factors , , , and as well as branching ratio values of the transitions , where can be one of the tensor mesons , or . The values of the meson masses and leptonic decay constants are chosen as presented in Table 1. Also, = 4.820 GeV, = 0.150 GeV [38], = 1.776 GeV, and = 0.105 GeV [30].

Table 1: The values of the meson masses [30] and decay constants [31, 32] in GeV.

Meson

Mass

Decay constant

From the 3PSR, it is clear that the form factors also contain the continuum thresholds and and the Borel parameters and as the main input. These are not physical quantities; hence the form factors should be independent of these parameters. The continuum thresholds, and , are not completely arbitrary, but these are in correlation with the energy of the first exiting state with the same quantum numbers as the considered interpolating currents. The value of the continuum threshold [39] is calculated from the 3PSR. The values of the continuum threshold for the tensor mesons , , and are taken to be , , and , respectively [9]. In this work, the variations of are considered to be . In these regions, the dependence of the form factors on the continuum threshold values is very small. For instance, we have shown the variations of the form factor for different values of in Figure 3. As can be seen, these plots are very close to each other.

Figure 3: The form factor of on for different values of .

We search for the intervals of the Borel parameters so that our results are almost insensitive to their variations. One more condition for the intervals of these parameters is the fact that the aforementioned intervals must suppress the higher states, continuum, and contributions of the highest-order operators. In other words, the sum rules for the form factors must converge. As a result, we get and . To show how the form factors depend on the Borel mass parameters, as examples, we depict the variations of the form factors , , , and for at with respect to the variations of the and parameters in their working regions in Figure 4. From these figures, it is revealed that the form factors weakly depend on these parameters in their working regions.

Figure 4: The form factor of on and .

In the Borel transform scheme, the ratio of the nonperturbative to perturbative part of the form factor is about . This value confirms that the higher order corrections are small, constituting a few percent, and can easily be neglected. Our calculation shows that the same suppression is observed for all other form factors.

The sum rules for the form factors are truncated at about . The dependence of the form factors , , , and on for transitions is shown in Figure 5. However, it is necessary to obtain the behavior of the form factors with respect to in the full physical region, , in order to calculate the decay width of the transitions. So, to extend our results, we look for a parametrization of the form factors in such a way that in the region , this parametrization coincides with the sum rules predictions. Our numerical calculations show that the sufficient parametrization of the form factors with respect to is as follows:

The values of the parameters , , and for the transition form factors of are given in Table 2.

Table 2: Parameter values appearing in the fit functions of the decays.

Form factor

0.13
2.19
0.83

0.10
1.36
0.09

0.13
2.10
0.75

0.11
1.45
0.23

0.12
2.01
0.60

0.10
1.40
0.16

0.23
3.77
4.21

0.05
0.21
−2.99

0.26
3.71
4.03

0.09
0.63
0.46

0.24
3.70
4.02

0.09
0.46
0.29

Figure 5: The SR predictions for the form factors of the transitions on .

In Table 3, our results for the form factors of decays in are compared with those of other approaches such as the LCSR, the PQCD, the LEET, and the ISGW II model. Our results are in good agreement with those of the LCSR, PQCD, and LEET in all cases.

Table 3: Comparison of the form factor values of decays in in different approaches.

Form factor
This work
LCSR [25]
PQCD [5]
LEET [26–28]
ISGW II [29]

—
—

—
—

—
—

—
—

At the end of this section, we would like to present the differential decay widths of the process under consideration. Using the parametrization of these transitions in terms of the form factors, the differential decay width for transition is obtained as

where represents the mess of the charged lepton. The other parameters are defined as

Integrating (24) over in the whole physical region and using [30], the branching ratios of the are obtained. The differential branching ratios of the decays on are shown in Figure 6. The branching ratio values of these decays are also obtained as presented in Table 4. Furthermore, this table contains the results estimated via the PQCD. Considering the uncertainties, our estimations for the branching ratio values of the decays are in consistent agreement with those of the PQCD.

Table 4: Comparison of the branching ratio values of decays with those of the PQCD (in units of ).

This work
PQCD [5]

Figure 6: The differential branching ratios of the semileptonic decays on .

It should be noted that the uncertainties in the branching ratio values come from the form factors, the CKM parameter, and the meson and lepton masses which are about 30% of the central values.

In summary, we considered channels and computed the relevant form factors considering the contribution of the quark condensate corrections. Our results are in good agreement with those of the LCSR, PQCD, and LEET in all cases. We also evaluated the total decays widths and the branching ratios of these decays. Our branching ratio values of these decays are in consistent agreement with those of the PQCD.

Appendix

In this appendix, the explicit expressions of the coefficients , , , and are given.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

Partial support of the Isfahan University of Technology Research Council is appreciated.

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