Tree Level Semileptonic \Sigma_{b} to Nucleon Decay in Light Cone QCD Sum Rules
pacs:
11.55.Hx, 13.30.-a, 14.20.Mr, 12.39.Hg

Using the most general form of the interpolating current of the heavy spin 1/2, baryon and distribution amplitudes of the nucleon, the transition form factors of the semileptonic decay are calculated in the framework of light cone QCD sum rules. It is obtained that the form factors satisfy the heavy quark effective theory relations. The obtained results for the related form factors are used to estimate the decay rate of this transition.

I Introduction

The baryons containing a heavy quark have been at the focus of much theoretical attention, especially since the development of the heavy quark effective theory (HQET) and its application to the spectroscopy of these baryons. The heavy quark provides a window that permits us to see further under the skin of the non-perturbative QCD as compared the light baryons. These states are expected to be narrow, so that their isolation and detection are relatively easy. Recently, experimental studies on the spectroscopy of these baryons have been accelerated and new heavy baryons have been discovered Mattson (); Ocherashvili (); Acosta (); Chistov (); Aubert1 (); Abazov1 (); Aaltonen1 (); Solovieva (). The channels are expected to be very rich, so it will be possible to check its semileptoic decays like its decay to the nucleon at LHC in the near future. There are many works in literature which are devoted to the investigation of the mass and magnetic moments of the heavy baryons using different approaches. The masses of these baryons have been discussed within QCD sum rules in kazem1 (); Shuryak (); Kiselev1 (); Kiselev2 (); Bagan1 (); Bagan3 (); Duraes (); Wang1 (); Zhang1 (), in heavy quark effective theory (HQET) in Grozin (); Groote (); Dai (); Lee (); Huang (); Liu (); Wang2 () and using different quark models in Ebert1 (); Ebert3 (); Capstick (); Matrasulov (); Gershtein (); Kiselev3 (); Vijande (); Martynenko (); Hasenfratz (). The magnetic dipole moment of heavy spin 1/2 and 3/2 baryons as well as the transition magnetic dipole and electric quadrupole moments of heavy spin 3/2 to heavy spin 1/2 baryons have been calculated in the framework of different approaches ( see for example kazem1 (); kazem2 (); kazem3 () and references therein). However, the semileptonic and nonleptonic decays of the heavy baryons have not been extensively discussed in the literature comparing their mass and electromagnetic properties. Transition form factors of the and decays have been studied in three points QCD sum rules in yeni1 (), and then used in the study of the semileptonic decays. The transition has also been investigated using three point QCD sum rules within the framework of heavy quark effective theory (HQET) in yeni2 () and using SU(3) symmetry and HQET in Datta (). Hyperfine mixing and the semileptonic decays of double-heavy baryons in a quark model Roberts (), strong decays of heavy baryons in Bethe-Salpeter formalism Guo (), strong decays of charmed baryons in heavy hadron Chiral perturbation theory Cheng () and semileptonic decays of some heavy baryons containing single heavy quark in different quark models D.Ebert (); albertus (); pervin () are some other works related to the heavy baryon decays.

In the present work, we calculate the form factors related to the semileptonic decay of the transition in the framework of the light cone QCD sum rules using the nucleon distribution amplitudes. Here, N refers to two members of the octet baryons, namely neutron and proton. The parameters appearing in the nucleon distribution amplitudes have been calculated using various methods. In this work, for the values of these parameters, we use the results of QCD sum rules approach Lenz () and also the results which are recently obtained from lattice QCD Gockeler1 (); Gockeler2 (); QCDSF (). Analyzing of such transitions can give essential information about the internal structure of the baryon as well as accurate calculation of the nucleon wave functions. Since the spin of the heavy baryon carries information on the spin of the heavy quark, the study of such transitions might also lead us to study the spin effects in the heavy quark sector of the standard model.

The outline of the paper is as follows: in section II, using the nucleon distribution amplitudes and the most general form of the interpolating currents for the baryon, we calculate the form factors entering to the semileptonic decay of the heavy baryon to nucleon in the framework of the light cone QCD sum rules. The heavy quark limit of the form factors and the relations between the form factors in this limit is also discussed in this section. Section III encompasses numerical analysis of the form factors, our predictions for the decay rate obtained in two different ways: first, using the DA’s obtained from QCD sum rules and second, the DA’s calculated in lattice QCD , and discussion.

Ii Light cone QCD sum rules for the form factors

This section is devoted to the calculation of form factors relevant for the and transitions using the light cone QCD sum rules approach. At quark level, these transitions are governed by the tree level transition. Considering the SU(2) symmetry, the form factors of these two transitions are the same, so we will use the notation N instead of neutron and proton. The quark level transition is described by the effective Hamiltonian given by

(1)

Hence, to study decay, one needs the matrix element . To calculate this matrix element, following the general philosophy of QCD sum rules, we start by considering the correlation function,

(2)

where, is interpolating currents of baryon, is transition current and presents the proton sate. denotes the proton momentum and is the transferred momentum. To calculate the form factors, the following three steps will be applied:

  • The correlation function is calculated by saturating it with a tower of hadrons having the same quantum number as the interpolating current, called the phenomenological or physical side.

  • The correlation function is calculated in QCD or theoretical side via operator product expansion (OPE), where the short and long distance quark-gluon interactions are separated. The former is calculated using QCD perturbation theory, whereas the latter are parameterized in terms of the light-cone distribution amplitudes of the nucleon.

  • The sum rules for form factors are calculated equating the two representation of the correlation function mentioned above and applying Borel transformation to suppress the contribution of the higher states and continuum.

To calculate the physical side, a complete set of hadronic state is inserted to the correlation function. After performing integral over x, we obtain

(3)

where, the … represents the contribution of the higher states and continuum. The matrix element in (3) can be written as:

(4)

where is residue of baryon. The transition matrix element, is parameterized in terms of the form factors and as

where , and , and , are the form factors and and are the spinors of nucleon and , respectively. Using Eqs. (3), (4) and ,(II) and summing over spins of the baryon using

(6)

we obtain the following expression

(7)

Using

(8)

in Eq. (7), the final expression for the physical side of the correlation function is obtained as

(9)

Among many structures appearing in Eq. (7), we chose the independent structures , , , , , and to evaluate the form factors , , , , and , respectively.

On QCD side, to calculate the correlation function in deep Euclidean region where , we need to know the explicit expression for the interpolating current of the baryon. It is chosen as

(10)

where are the color indices and is the charge conjugation operator and is an arbitrary parameter with corresponding to the Ioffe current. Using the transition current, and and contracting out all quark pairs applying the Wick’s theorem, we obtain

where, is the heavy quark propagator which is represented as Balitsky ():

(12)

where

and are the Bessel functions. The terms proportional to the gluon strength tensor can give contribution to four and five particle distribution functions but they are expected to be small 17 (); 18 (); Braun1b () and for this reason, we will neglect these amplitudes in further analysis.

For the calculation of in Eq. (II), the matrix element is required. The nucleon wave function is given as Lenz (); 17 (); 18 (); Braun1b (); 8 ():

(14)

where, the calligraphic functions, which are functions of the scalar product and the parameters , , can be expressed in terms of the nucleon distribution amplitudes (DA’s) with the increasing twist. The distribution amplitudes with different twist are given explicitly in Tables 1, 2, 3, 4 and 5:

Table 1: Relations between the calligraphic functions and proton scalar DA’s.
Table 2: Relations between the calligraphic functions and proton pseudo-scalar DA’s.
Table 3: Relations between the calligraphic functions and proton vector DA’s.
Table 4: Relations between the calligraphic functions and proton axial vector DA’s.
Table 5: Relations between the calligraphic functions and proton tensor DA’s.

One can expresses the distribution amplitudes = , , , , as:

(15)

Here with corresponds to the longitudinal momentum fractions carried by the quarks.

Using the expressions for the heavy quark propagator and nucleon distribution amplitudes and performing integral over the expression for the correlation function in QCD or theoretical side is obtained. Equating the corresponding structures from both representations of the correlation function and applying Borel transformation with respect to to suppress the contribution of the higher states and continuum, one can obtain sum rules for the form factors , , , , and . Finally, to subtract the contribution of the higher states and the continuum, quark-hadron duality is assumed.

In heavy quark effective theory (HQET), the heavy quark symmetry reduces the number of independent form factors to two namely, and Mannel (); alievozpineci (), i.e.,

where, is any Dirac structure and . Comparison between Eq. (II) with the general definition of the form factors in Eq. (II) leads to the following relations among the form factors in HQET limit Chen (); ozpineci ()

(17)

Our calculations show that the deviation from the relations and are negligible in the case of HQET limit. However, when we consider finite mass, the violation is for and turns out to be large for values. The explicit expressions for the form factors are very lengthy, so considering the above relations, we will present only the expressions for and in the Appendix–A. However, we will give the extrapolation of all form factors in finite mass in terms of in the numerical analysis section.

From the explicit expressions of the form factors, it is clear that we need to know the expression for the residue of the baryon. The residue is determined from sum rule and its expression is given in Ozpineci1 () as:

(18)

with

(20)

where, is continuum threshold, is the Borel mass parameter and are some dimensionless functions.

Iii Numerical results

This section is devoted to the numerical analysis for the form factors and total decay rate for transition. Some input parameters used in the analysis of the sum rules for the form factors are , , , , and Belyaev (). The nucleon DA’s are the main input parameters, whose explicit expressions can be found in Lenz (). These DA’s contain 8 independent parameters and . These parameters have been calculated also in Lenz () within the light cone QCD sum rules. Recently, most of these parameters have been calculated in the framework of the lattice QCD Gockeler1 (); Gockeler2 (); QCDSF (). We will use these two sets of data from QCD sum rules and lattice QCD and for each parameter which have not been calculated in lattice, we will use the values from QCD sum rules prediction. These parameters are given in Table 6.

QCD sum rules Lenz () Lattice QCD Gockeler1 (); Gockeler2 (); QCDSF ()
Table 6: The values of independent parameters entering to the nucleon DA’s. The first errors in lattice values are statistical and the second errors represent the uncertainty due to the chiral extrapolation and renormalization.

The sum rules for form factors also contain 3 auxiliary parameters namely, continuum threshold , Borel mass parameter and general parameter entering to the general current of the baryon. These are not physical quantities, hence the form factors should be independent of them. Therefore, we look for working regions such that in these regions our results are practically independent of these mathematical objects. The continuum threshold, is not completely arbitrary and it is related to the energy of the exited states. Our numerical analysis for form factors show that the results are weakly depend on in the interval, . In order to obtain the working region for , we plot the form factors with respect to in the interval which is corresponds to , where and look for a region at which the dependency is weak. The common working region for is obtained to be . The Ioffe current which corresponds to is out of this region. The similar results have been obtained in kazem3 (). The lower limit on Borel mass squared, is determined from condition that the contribution of higher states and continuum to the correlation function should be enough small, i.e., the contribution of the highest term with power is less than, say, 20–25% of the highest power of . The upper limit of this parameter is acquired from the condition that series of the light cone expansion with increasing twist should be convergent. Generally, this means that the higher states, higher twists and continuum contributions to the correlation function should be less than 40–50% of the total value. Our numerical analysis show that both conditions are satisfied in the region , which we will use in numerical analysis. Considering the above requirements, we obtained that the form factors obey the following extrapolations in terms of :

(21)

The values of the parameters and are given in Tables 7 and 8 related to the QCD sum rules and lattice QCD input parameters, respectively. These parameterizations show that increasing in the value of leads to increasing in the absolute value of the form factors and they have no pole inside the physical region. The values of presents the pole outside the allowed region of and related to this and accordance to mesons, one can calculate the coupling constant , where, can be considered as the exited state of baryon. For detailed analysis in this respect see damir1 (); damir2 (); damir3 (). Note that, as we work near the light cone, , from the considered correlation function it is clear that our predictions at low are not reliable and we need the above parameterization to extend the results to full physical region. As an example, to show how the actual sum rules results, and the parameterization fit to each other, we present the dependency of (both actual sum rule result and fit parameterization) on for QCD sum rules input parameters and at fixed values of auxiliary parameters in Fig. 1.

Figure 1: The dependency of (both actual sum rule result and fit parameterization) on for QCD sum rules input parameters at , and .
0.13 0.005 4.92
0.03 -0.10 5.40
-0.09 -0.02 4.92
0.20 -0.05 5.56
-0.02 0.015 5.96
-0.02 -0.009 5.65
Table 7: Parameters appearing in the fit function for QCD sum rules set of data.
0.19 0.004 4.88
0.038 - 0.067 5.38
-0.06 -0.015 4.93
0.25 -0.064 4.97
-0.03 -0.002 5.97
-0.028 -0.009 5.95
Table 8: Parameters appearing in the fit function for lattice QCD set of data.

The values of form factors at is also obtained as presented in Table 9.

For QCD sum rules input parameters For lattice QCD input parameters
Table 9: The value of the form factors at

.

Our next task is to calculate the total decay rate of transition in the whole physical region, i.e., . The decay width for such transition is given by the following expression Faessler (); Pietschmann:1974ap ()

(22)

where

(23)

Where , , , , , ,