# On the nature of

###### Abstract

The single charmed excited state was discovered many years ago by BaBar collaboration and recently confirmed by Belle experiment. However, both of these experiments, unfortunately, could not fix the quantum numbers of this particle and its nature is under debates. In the present study we try to clarify its quantum numbers by calculating its mass and width of its dominant decay to . To this end we consider state once as orbitally excited and then radially excited single charmed baryon in channel. The comparison of the results obtained with the experimental data on its mass and width allows us to interpret the state as the negative parity P-wave ground state baryon and assign to it the quantum numbers: .

## I Introduction

With impressive developments of experimental techniques many new conventional and exotic states have been discovered PDG . These discoveries have opened a new direction in hadron physics. Indeed, the heavy baryon spectroscopy receives special attention as heavy baryons represent very suitable place for testing the ground of heavy quark symmetry and provide us with deep understanding of the details of the strong interaction.

While many new excited charmed baryons are discovered in different experiments and many theoretical works are devoted to establish their quantum numbers, still their nature is not clear and many open questions remain on their internal structure and quark organization. For instance, there have been made many suggestions on the structures of the newly observed five narrow resonances by LHCb Collaboration in the invariant mass spectrum Aaij:2017nav : some authors have treated them as usual three-quark resonances Agaev:2017jyt ; Agaev:2017lip ; Aliev:2017led ; Karliner:2017kfm ; Wang:2017vnc , while some others have interpreted them as new penta-quark states Yang:2017rpg ; Huang:2017dwn . At present time there is unfortunately no any phenomenological model, which can successfully describe the properties of such complicated systems Crede:2013sze ; Cheng . For this, more experimental and theoretical attempts are needed to understand dynamics of these new systems.

Some new excited states at , and channels have also been discovered that are of great importance and deserve investigations with the aim of clarification of their nature and internal structure. The observation of the charmed-strange baryon , which is the subject of the present study, has a long history in the experiment. This state was firstly observed by BaBar Collaboration in 2008 with mass and width of as an intermediate resonance in the decay Aubert . Note that the Belle Collaboration had before measured the branching ratios of the decays and in 2006 Gabyshev but could not find any intermediate charmed resonances. After observation of by BaBar, the state was investigated in the framework of different theoretical models like constituent quark model Wang:2017kfr ; Chen:2016iyi , chiral quark model Liu , QCD sum rules Chen , etc.

Very recently, Belle Collaboration performed an updated measurement on decay and observed the state in the invariant mass with a significance of Li:2017uvv . The measured mass and width are:

(1) |

respectively. As is seen, though the measured mass is close but the measured width by Belle collaboration differ considerably with that of the BaBar measurements. This automatically suggests more experimental and theoretical efforts on the properties of this resonance.

We aim to calculate some parameters of state in the present study to clarify its nature and fix its quantum numbers. To this end we assume it once as orbitally excited (1P) negative parity () and the second as radially excited (2S) positive parity () spin-1/2 baryon at channel. We evaluate the widths of the strong decays and . For this, firstly we compute the mass and residue of the ground state, first orbitally and radially excited baryons as well as the couplings of the strong and vertices allowing us to find the required decay widths. For calculation of the masses and residues we employ QCD two-point sum rule, whereas in the case of the strong couplings we apply the technique of QCD light-cone sum rule (LCSR).

The article is organized in the following way. In section II, the mass sum rules for baryons including its first orbital and radial excitations are calculated and the values of the masses and residues are found. Section III is devoted to calculation of the strong coupling constants defining the and vertices. We estimate the widths of the decay channels under consideration and compare the results obtained on the masses and widths with the experimental data with the aim of fixing the quantum numbers of the resonance. Last section is reserved for summary and concluding remarks.

## Ii Masses and Pole Residues of the first orbitally and radially excited states

As we noted, the has been seen as a peak in the invariant mass distribution. But unfortunately its quantum numbers have not established yet. In present work we consider two possible scenarios for it: a) The is considered as radial excitation of the ground state . In other words it carries the same quantum numbers as , i.e. . b) The is treated as the first orbital excitation of the , that is negative parity baryon with .

In order to calculate the mass and residue of baryon, we start with the following two point correlation function:

(2) |

where is the interpolating current for state with spin-parity and indicates the time ordering operator. The general form of the interpolating current for the heavy spin-, baryon belonging to antitriplet representations of can be written as:

(3) | |||||

where are the color indices and is an arbitrary parameter with corresponding to the Ioffe current. and are and quarks for baryon, respectively.

To derive the mass sum rules for the baryon we calculate this correlation function using two languages: hadronic, in terms of the masses and residues called the physical side and QCD, in terms of the fundamental QCD degrees of freedom called the QCD or theoretical side. By equating these two representations, one can get the QCD sum rules for the physical quantities of the baryons under consideration. Firstly we consider the case when is considered as a negative parity baryon. The physical side of the correlation function is obtained by inserting the complete sets of intermediate states with both parities:

(4) | |||||

where , and , are the masses and spins of the ground and first orbitally excited baryons, respectively. The dots denote contributions of higher resonances and continuum states. In Eq. (4) the summations over the spins are are implied.

The matrix elements in Eq. (4) are determined as

(5) |

Here and are the residues of the ground and first orbitally excited baryons, respectively. Using Eqs. (4) and (5) and carrying out summations over the spins of corresponding baryons, we obtain

(6) |

Performing Borel transformation of this expression we have

(7) | |||||

The QCD side of the aforementioned correlation function is calculated in terms of the QCD degrees of freedom in deep Euclidean region. After inserting the explicit form of the interpolating current given by Eq. (3) into the correlation function in Eq. (2) and performing contractions via the Wick’s theorem, we get the QCD side in terms of the light and heavy quarks propagators. By using light and heavy quark propagators in the coordinate space and performing the Fourier and Borel transformations, as well as applying the continuum subtraction, after lengthy calculations for the correlation function we obtain

(8) |

The expressions for and are presented in Appendix.

Having calculated both the hadronic and QCD sides of the correlation function, we match the coefficients of the structures and from these two sides and obtain the following sum rules, which are used to extract the masses and residues of the ground and first orbitally excited states:

(9) |

Using two equations given in Eq. (9) it is easy to show that

(10) |

where .

For obtaining the expressions for the mass and residue for radially excitation state it is enough to make replacement and redefine the residue as in expressions of Eq. (10).

To perform analysis of the sum rules for the masses and residues of the orbitally and radially excited state of the baryon as well as the residue of the ground state we need some inputs which are presented in Table 1. The mass of the ground state is also taken as an input parameter. Besides the input parameters, QCD sum rules contains three auxiliary parameters namely the continuum threshold , Borel parameter and an arbitrary mixing parameter . The working windows of these parameters are determined demanding that the physical quantities under consideration be roughly independent of these parameters. To determine the working interval of the Borel parameter one needs to consider two criteria: convergence of the series of operator product expansion (OPE) and adequate suppression of the higher states and continuum. Consideration of these criteria in the analysis leads to the following working interval of :

(11) |

To determine the working region of the continuum threshold, we impose the conditions of the pole dominance and OPE convergence. This leads to the interval

(12) |

In order to explore the sensitivity of the obtained results on the Borel parameter and continuum threshold , as examples, in Figs. 1-4 we depict the mass of the baryon and residues of the ground state , and baryons as functions of these parameters at fixed value of . From these figures we see very weak dependence of the quantities under consideration on and , satisfying the requirements of the method used.

To find the working region of , as examples, in Fig. 5 we present the dependence of the ’s mass and residue on at average values of and . From this figure we see that the results are practically insensitive to the variations of when it varies in the region

(13) |

We depict the numerical results of the masses and residues of the first orbitally and radially excited baryons as well as the residue of the ground state particle obtained using the above-presented working intervals for the auxiliary parameters in table 2. Note that we obtain the same mass for the first orbitally and radially excited baryons. The errors in the presented results are due to the uncertainties in determination of the working regions for the auxiliary parameters as well as the errors of other input parameters. The values presented in table 2 will be used as inputs in next section.

## Iii and Transitions To

In this section we calculate the strong coupling constants and , which are necessary to calculate widths of the decays and . For this aim we introduce the correlation function

(14) |

where is the interpolating current for the baryon which can be obtained from Eq. (3) with and .

Firstly let consider the transition. Before calculations we note that the interpolating current for interact with both positive and negative parity baryons. Taking into consideration this fact, inserting complete sets of hadrons with the same quantum numbers as the interpolating currents and isolating the ground states, we obtain

(15) | |||||

where and are the momenta of the , baryons and meson, respectively. and are the positive and negative parity of the spin- baryon. In this expression is the mass of the baryon. The dots in Eq. (15) stand for contributions of the higher resonances and continuum states.

The matrix elements in Eq. (15) are parameterized as

where are the strong coupling constants for corresponding transitions.

Using the matrix elements given in Eq.(LABEL:eq14) and performing summation over spins of and baryons and applying the double Borel transformations with respect and for physical part of the correlation function we get

(17) |

where is the mass of the meson, and and are the Borel parameters.

From Eq. (17) follows that we have different structures which can be used to derive the sum rules for the strong coupling constants for channel. We have four couplings (see Eq.15), and in order to determine the coupling we need four equations. Therefore we select the structures , , and . Solving four algebraic equations for , we obtain

(18) | |||||

where , , and are the invariant amplitudes corresponding to the structures , , and , respectively.

The general expressions obtained above contain two Borel parameters and . In our analysis we choose

(19) |

which is traditionally justified by the fact that masses of the involved heavy baryons and are close to each other. The sum rules corresponding to the coupling constant defining the transition can be easily obtained from Eq.(18), by replacing and .

The QCD side of the correlation function for can be obtained by contracting out the quark fields using Wick’s theorem and inserting into the obtained expression the relevant quark propagators. For obtaining nonperturbative contributions in light cone QCD sum rules, which are described in terms of the -meson distribution amplitudes, one can use the Fierz rearrangement formula

where is the full set of Dirac matrices. Sandwiched between the K-meson and vacuum states, these terms as well as the ones generated by insertion of the gluon field strength tensor from quark propagators, give these distribution amplitudes (DAs) of various quark-gluon contents in terms of wave functions with definite twists. The DAs are main nonperturbative inputs of light cone QCD sum rules. For -meson they are derived in Ball:2006wn ; Belyaev:1994zk ; Ball:2004ye , which will be used in our numerical analysis. All these steps summarized above result in lengthy expression for the QCD side of correlation function. In order not to overwhelm the study with overlong mathematical expressions we prefer not to present them here. Apart from parameters in the distribution amplitudes, the sum rules for the couplings depend also on numerical values of the baryon’s mass and pole residue. In numerical calculations we utilize

(20) |

where the value for the residue has been extracted from the corresponding mass sum rules in the present study and we use the mass of state from PDG PDG . The working regions of the Borel mass , threshold and parameters for calculations of the relevant strong couplings are chosen the same as the mass sum rules analyses.

Using the couplings and we can easily calculate the width of and decays. After some computations we obtain:

(21) | |||||

and

(22) | |||||

In expressions above the function is given as:

Numerical values obtained from our analyses for coupling constants and decay widths are presented in table 3. The obtained value for the decay width of the case is in nice consistency with the experimental value given in Eq. (1) Li:2017uvv . However our prediction for the width of is considerably small compared to the experimental value presented in Eq. (1).

## Iv Summary and Concluding remarks

We performed a QCD sum rule based analysis on the mass and width of the considering it as first orbitally/ radially excited charmed-strange baryon in channel. We obtained the same mass for both the orbitally and radially excited states, and in nice agreement with the experimental value by Belle Collaboration, preventing us assigning one of these possibilities for the structure of this state. In next step, we considered the dominant decay of to in both scenarios. The obtained result for width is nicely consistent with the experimental value of Belle Collaboration when we consider state as the orbitally excited charmed-strange baryon. From these results we conclude
that the state has the spin-parity
, i.e. it represents a negative parity baryon in channel.

*

## Appendix A The QCD side of the correlation function

In this appendix we present the explicit expressions of the functions and used in mass sum rules:

(A.23) | |||||

(A.24) | |||||