# Subleading Isgur-Wise Function of using QCD sum rules

###### Abstract

Subleading Isgur-Wise form factor at for weak transition is calculated by using the QCD sum rules in the framework of the heavy quark effective theory (HQET), where and are the orbitally excited charmed baryon doublet with . We consider the subleading contributions from the weak current matching in the HQET. The interpolating currents with transverse covariant derivative are adopted for and in the analysis. The slope parameter in linear approximation of is obtained to be and the interception to be GeV.

###### pacs:

## I Introduction

The ground state bottom baryon weak decays [1] provide a testing ground for the standard model (SM). They reveal some important features of the physics of bottom quark. The experimental data on these decays have been accumulated to wait for reliable theoretical calculations. With the discovery of the orbitally excited charmed baryons and [2], it would be of great interest for one to investigate the semileptonic decays into these baryons.

From the phenomenological point of view, these semileptonic transitions are interesting since in principle they may account for a sizeable fraction of the inclusive semileptonic rate of decay. In addition, the properties of excited baryons have attracted attention in recent years. Investigation on them will extend our ability in the application of QCD. It can also help us foresee any other excited heavy baryons that have not been discovered yet.

The heavy quark symmetry [3] is a useful tool to classify the hadronic spectroscopy containing a heavy quark . In the infinite mass limit, the spin and parity of the heavy quark and that of the light degrees of freedom are separately conserved. Coupling the spin of light degrees of freedom with the spin of heavy quark yields a doublet with total spin (or a singlet if ). This classification can be applied to the -type baryons. For the charmed baryons the ground state contains light degrees of freedom with spin-parity , being a singlet. The excited states with are spin symmetry doublet with (,). The lowest states of such excited charmed states, and , have been observed to be identified with and respectively [2].

However, the difficulties in the SM calculations are mainly due to the poor understanding of the nonperturbative aspects of the strong interaction (QCD). The heavy quark effective theory (HQET) based on the heavy quark symmetry provides a model-independent method for analyzing heavy hadrons containing a single heavy quark [3]. It allows us to expand the physical quantity in powers of systematically, where is the heavy quark mass. Within this framework, the classification of the exclusive weak decay form factors has been greatly simplified. The decays such as [4], [5], [6], [7] have been studied.

To obtain detailed predictions for the hadrons, at this point, some nonperturbative QCD methods are also required. We have adopted QCD sum rules [8] in this work. QCD sum rule is a powerful nonperturbative method based on QCD. It takes into account the nontrivial QCD vacuum which is parametrized by various vacuum condensates in order to describe the nonperturbative nature. In QCD sum rule, hadronic observables can be calculated by evaluating two- or three-point correlation functions. The hadronic currents for constructing the correlation functions are expressed by the interpolating fields. In describing the excited heavy baryons, transverse covariant derivative is included in the interpolating field. The static properties of and ( denotes the generic charmed state) have been studied with QCD sum rules in the HQET in Ref. [9] and Ref. [10, 11], respectively. Recently, the leading order Isgur-Wise (IW) function is also calculated in the HQET QCD sum rule in Ref. [12].

In decay, corrections are very important. At the heavy quark limit of , the transition matrix elements should vanish at zero recoil since the light degrees of freedom change their configurations. Nonvanishing contribution to, say, at zero recoil appears at order. Since both and are heavy enough, the behavior of the matrix elements near the zero recoil is very important. That explains why people pay attention to the next-to-leading order (NLO) contributions. The same situation occurs in heavy mesons. As for decay, leading and subleading Isgur-Wise (IW) functions have been computed using QCD sum rule in Ref. [13, 14, 15, 16, 17]. They showed that the branching ratio is enhanced considerably when the subleading contributions are included.

In HQET, corrections appear in a two-fold way. At the Lagrangian level, subleading terms are summarized in and . parametrizes the kinetic term of higher derivative, while represents the chromomagnetic interaction which explicitly breaks the heavy quark spin symmetry. At the current level, corrections come from the small portion of the heavy quark fields which correspond to the virtual motion of the heavy quark. In this work, the subleading IW function from the latter case, i.e., at the current level, is analyzed in the HQET QCD sum rules.

In Sec. II, the weak transition matrix elements are parametrized by the leading and subleading IW functions. By evaluating the three-point correlation function, we give the subleading IW function in Sec. III. We present, in Sec. IV, the numerical analysis and discussions. The summary is given in Sec. V.

## Ii Weak Transition Matrix Elements and the Subleading Isgur-Wise Functions

The weak transition matrix elements for are parametrized by the 14-form factors as

(1a) | |||||

(1b) | |||||

(1c) | |||||

(1d) |

where and are the four-velocity and spin of , respectively. And the form factors , , and are functions of . In the limit of , all the form factors are related to one independent universal form factor called Isgur-Wise (IW) function. A convenient way to evaluate hadronic matrix elements is by introducing interpolating fields in HQET developed in Ref. [18] to parametrize the matrix elements in Eqs. (1). With the aid of this method the matrix element can be written as [19]

(2) |

at leading order in and , where is any collection of -matrices. The ground state field, , destroys the baryon with four-velocity ; the spinor field is given by

(3) |

where is the ordinary Dirac spinor and is the spin 3/2 Rarita-Schwinger spinor, they destroy and baryons with four-velocity , respectively. To be explicit,

(4) | |||||

In general, the IW form factor is a decreasing function of the four velocity transfer . Since the kinematically allowed region of for heavy to heavy transition is very narrow around unity,

(5) |

and hence it is convenient to approximate the IW function linearly as

(6) |

where is the slope parameter which characterizes the shape of the leading IW function.

The corrections come in two ways. One is from the subleading Lagrangian of the HQET while the other comes from the small portion of the heavy quark field to modify the effective currents. We only consider the latter case here.

Including and , the weak current is given by

(7) |

Keeping the Lorentz structure, the subleading terms are expanded in general as

(8) |

where are the subleading IW functions to be evaluated.

The matrix elements of these currents modify Eq. (4) as

(9) |

where . It is quite convenient to define

(10a) | |||||

(10b) |

Possible contractions of are listed in the Appendix. From the Eqs. (3) and (8), Eq. (1) can be reexpressed in terms of and :

(11) | |||||

A similar expression can be obtained for the spin-3/2 final states

(13) | |||||

## Iii QCD sum rule evaluation

As a starting point of QCD sum rule calculation, let us consider the interpolating field of heavy baryons. The heavy baryon current is generally expressed as

(14) |

where are the color indices, is the charge conjugation matrix, and is the isospin matrix while is a light quark field. and are some gamma matrices which describe the structure of the baryon with spin-parity . Usually and with least number of derivatives are used in the QCD sum rule method. The sum rules then have better convergence in the high energy region and often have better stability. For the ground state heavy baryon, we use , . In the previous work [10], two kinds of interpolating fields are introduced to represent the excited heavy baryon. In this work, we find that only the interpolating field of transverse derivative is adequate for the analysis. Nonderivative interpolating field results in a vanishing perturbative contribution. The choice of and with derivatives for the and is then

(15) |

where a transverse vector is defined to be , and in Eq. (15) is some hadronic mass scale. , are arbitrary numbers between 0 and 1.

The baryonic decay constants in the HQET are defined as follows,

(16a) | |||||

(16b) | |||||

(16c) |

where and are equivalent since and belong to the same doublet with . The QCD sum rule calculations give [9]

(17) |

and [10]

(18) | |||||

In the above equations, are the Borel parameters and are the continuum thresholds, and is the color number. In the heavy quark limit, the mass parameters and are defined as

(19) |

The main point in QCD sum rules for the IW function is to study the analytic properties of the 3-point correlators,

(20a) | |||||

(20b) | |||||

The variables , denote residual “off-shell” momenta which are related to the momenta of the heavy quark in the initial state and in the final state by , , respectively.

The coefficient in Eq. (20) is an analytic function in the “off-shell energies” and with discontinuities for positive values of these variables. It furthermore depends on the velocity transfer , which is fixed at its physical region for the process under consideration. By saturating with physical intermediate states in HQET, one finds the hadronic representation of the correlators as following

(21) |

In obtaining the above expression the Dirac and Rartia-Schwinger spinor sums

(22) |

have been used, where .

In the quark-gluon language, 20) is written as in Eq. (

(23) |

where the perturbative spectral density function and the condensate contribution are related to the calculation of the Feynman diagrams depicted in Fig. 1 . In Eq. (23), the -structures of spin-1/2 and 3/2 are the same as those in Eq. (20), respectively. Subleading IW functions, , obtained from spin-1/2 and 3/2 are therefore identical.

The six () are not independent. From the fact that

(24) |

Eq. (8) implies

(25) |

The above expression relates with as

(26a) | |||||

(26b) | |||||

(26c) |

Other relations are obtained from the equation of motion of the heavy quark, :

(27a) | |||||

(27b) |

From the above 5 equations in Eq. (26), (27), all the six subleading IW functions are reduced to only one independent form factor. We just pick up , then others are

(28a) | |||||

(28b) | |||||

(28c) | |||||

(28d) | |||||

(28e) |

Now that all the subleading IW functions are related to , we have only to extract the coefficient of (or for spin 3/2) in Eqs. (20) and (23).

The QCD sum rule is obtained by equating the phenomenological and theoretical expressions for . In doing this the quark-hadron duality needs to be assumed to model the contributions of higher resonance part of Eq. (21). Generally speaking, the duality is to simulate the resonance contribution by the perturbative part above some thresholds and , that is

(29) |

In the QCD sum rule analysis for semileptonic decays into ground state mesons, it was argued by Neubert in [20], and Blok and Shifman in [21] that the perturbative and the hadronic spectral densities can not be locally dual to each other, and therefore the necessary way to restore duality is to integrate the spectral densities over the “off-diagonal” variable , keeping the “diagonal” variable fixed. It is in that the quark-hadron duality is assumed for the integrated spectral densities. The same prescription shall be adopted in the following analysis. On the other hand, in order to suppress the contributions of higher resonance states a double Borel transformation in and is performed to both sides of the sum rule, which introduces two Borel parameters and .