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Relativistic Covariance of Light-Front Few-Body Systems in Hadron Physics
^{†}^{†}thanks: Presented at the 20th International IUPAP Conference on Few-Body Problems in Physics, 20 - 25 August, 2012, Fukuoka, Japan

###### Abstract

We study the light-front covariance of a vector-meson decay constant using a manifestly covariant fermion field theory model in dimensions. The light-front zero-mode issues are analyzed in terms of polarization vectors and method of identifying the zero-mode operator and of obtaining the light-front covariant decay constant is discussed.

###### Keywords:

Weak Decay Decay Constant Light-Front Zero-Mode^{†}

^{†}journal: Few-Body Systems (FB20)

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## 1 Introduction

Mesonic weak transition form factors and decay constants are two of the most important ingredients in studying weak decays of mesons, which enter in various decay rates. Many theoretical efforts were undertaken to calculate these observables. The light-front quark model (LFQM) based on the LF dynamics (LFD) has been quite successful in describing various exclusive decays of mesons Jaus99 ; Cheng04 ; BCJ_spin1 ; BCJ_PV ; CJ_Bc ; CJ_PV ; CJ_tensor . However, one should also realize that the success of LFD in hadron physics cannot be realized unless the treacherous points in LFD such as the zero-mode contributions in the hadron form factors Jaus99 ; Cheng04 ; BCJ_spin1 ; BCJ_PV ; CJ_Bc ; CJ_PV ; CJ_tensor are well taken care of with proper methods.

In this paper, we study the LF covariance of the vector-meson decay constant using a manifestly covariant fermion field theory model in dimensions. Although the LF covariant issue for the vector meson decay constant has been raised by Jaus Jaus99 some time ago, systematic analyses of the zero-modes depending on different polarization vectors have not yet been explored much. Here, we attempt to systematically investigate the LF zero-mode issues in terms of polarization vectors of a vector meson and show a method of identifying the zero-mode operator to obtain the LF covariant decay constant even in the case that there exists a zero-mode contribution.

## 2 Manifestly Covariant Calculation

The decay constant of a vector meson of mass and bound state of a quark of mass , and an antiquark of mass , is defined by the matrix element of the vector current

(1) |

where the polarization vector of a vector meson satisfies the Lorentz condition .

The matrix element is given in the one-loop approximation as a momentum integral

(2) |

where and with and denotes the number of colors. For simplicity in regularizing the covariant loop, we take in a multipole ansatz for the bound-state vertex function of a vector meson, where , and and are constant parameters. The vector meson vertex operator in the trace term

(3) |

where .

## 3 Light-Front Calculation

Performing the LF calculation in parallel with the manifestly covariant calculation, we use two different approaches, i.e. (1) plus component () of the currents with the longitudinal polarization and (2) perpendicular components () of the currents with the transverse polarization , to obtain the decay constant. The explicit form of polarization vectors of a vector meson is given in CJ_PV .

By the integration over in Eq. (2) and closing the contour in the lower half of the complex plane, one picks up the residue at (on shell value of ) in the region (or ). Thus, the Cauchy integration formula for the integral in Eq. (2) gives

(4) |

where is the result of the trace when and with .

Firstly, using with the longitudinal polarization vector in Eq. (4), the decay constant is obtained from the relation

(5) |

For the purpose of analyzing zero-mode contribution to the decay constant, we denote the decay constant as when the matrix element is obtained for in the region of . Comparing with the manifestly covariant result , we find that is exactly the same as when (or ) is used. The same observation has also been made in BCJ_PV . However, is different from when is used. The difference between the two results, i.e. , corresponds to the zero-mode contribution to the full solution .

As in the case of zero-mode contribution to the weak transition form factors for semileptonic and decays CJ_Bc ; CJ_PV , the zero-mode contribution to comes (if exists) from the singular (or equivalently ) term in in the limit of when . For the case of , we find the following singular term in as follows

(6) |

As we presented in the weak transition form factor calculation CJ_Bc ; CJ_PV , we identify the zero-mode operator by replacing with the operator Jaus99 ; CJ_Bc ; CJ_PV : i.e. , where . The zero-mode contribution to the matrix element is given by

(7) |

and the corresponding zero-mode contribution to the decay constant is obtained as . Finally, we obtain the full result of the decay constant for the longitudinal polarization as

(8) | |||||

It can be checked that Eq. (8) is identical to the manifestly covariant result of Eq. (3).

Secondly, using with the transverse polarization vector , the decay constant is obtained from the relation

(9) |

In this case, the decay constant receives the zero mode from the simple vertex term but not from the term including the factor. Following the same procedure as for the case of , we find the following singular term in as

(10) |

and thus the corresponding zero-mode operator is given by . Finally, we obtain the full result of the decay constant as follows

(11) | |||||

From Eq. (11) one can check that that is the same as [Eq. (8)], which confirm the result obtained by Jaus Jaus99 .

## 4 Conclusion

In this work, we investigate the LF zero-mode issue for the vector meson decay constant using two different polarization vectors of a vector meson. We find that the decay constant obtained from transverse polarization vectors cannot avoid the zero-mode even at the level of model-independent simple vector meson vertex, i.e. . Although the decay constant obtained from longitudinal polarization vector may receive a zero-mode depending on a model-dependent form of factor, it is immune to the zero-mode at the level of simple vertex. The independence of the decay constant on the polarization vectors is also explicitly shown. Our results do not depend on the value of in the multipole ansatz and may give an important guidance on a more realistic model building.

###### Acknowledgements.

This work was supported in part by the Korea Research Foundation Grant(KRF-2010-0009019) and in part by Kyungpook National University Research Fund, 2012## References

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