Determination of the quantum numbers of $\Sigma_b(6097)^{\pm}$ via their strong decays

The progresses in the experimental sector has been the harbinger of the observations of many new hadrons. Very recently, LHCb Collaboration announced the observation of two new $\Sigma_b(6097)^{\pm}$ states in the $\Lambda^0_b\pi^{\pm}$ invariant mass distribution, which are considered as the excited states of the ground state $\Sigma^{(*)}_b$ baryon. Though, almost all of the ground state baryons were observed, having a limited number of excited states observed so far makes them intriguing. Understanding the properties of the excited baryons improve our knowledge on the strong interaction as well as the nature and internal structures of these baryons. To specify the quantum numbers of the $\Sigma_b(6097)^{\pm}$ an analysis on their strong decays to $\Lambda_b^0$ and $\pi^{\pm}$ are performed within the light cone QCD sum rule formalism. To this end, they are considered as possible $1P$ or $2S$ excitation of either the ground state $\Sigma_b$ baryon with $J=\frac{1}{2}$ or $\Sigma_b^{*}$ baryon with $J=\frac{3}{2}$. The corresponding masses are also calculated considering the same scenarios for their quantum numbers. The results of the analyses indicate that the $\Sigma_b(6097)^{\pm}$ baryons are excited $1P$ baryons having quantum numbers $J^P=\frac{3}{2}^-$.


I. INTRODUCTION
In the quark model, the heavy baryons containing one heavy and two light quarks form multiplets using the symmetry of flavor, spin, and spatial wave functions [1]. These considerations lead to the results that they belong to the sextet and the antitriplet representations of SU (3). At present time, almost all the ground-state heavy baryons have been observed in experiments. According to the quark model predictions, in addition to the ground states, their excited states are also expected to be existed. Now, only a few excited baryons are observed in the bottom sector [2][3][4][5][6]. The detailed study of the experimentally discovered states and looking for new, yet to be observed, states can play a critical role for understanding their internal structures and give essential information about the dynamics of QCD at the non-perturbative domain.
The authors concluded that this state is a P -wave baryon with the quantum numbers J P = 3 2 − or J P = 5 2 − [9]. Another prediction for the quantum numbers of the observed Σ b (6097) − and Σ b (6097) + states was presented in Ref. [10] via the quark-pair creation model, which shows the possibility of their being again either J P = 3 2 − or In the present study, the properties of these baryons are studied in the framework of the QCD sum rule method [11]. In our calculations, the observed states are considered as 1P or 2S excitated states with J = 1 2 or J = 3 2 . We analyze the Σ ± b → Λ b π ± decays and compare the values of the obtained decay widths with the experimental results, which allows us to determine the quantum numbers of Σ b (6097) ± states. To calculate the decay widths the main ingredient is the coupling constants corresponding the considered transitions. For calculation of these coupling constants we use the light cone QCD sum rules (LCSR) method [12]. In this work, we also calculate the masses and the decay constants of the states under consideration by taking into account again all possibilities, i.e. assuming that these states are 1P or 2S excited states of the ground state Σ b and Σ * b baryons with J = 1 2 or J = 3 2 . The obtained masses and decay constants are used as inputs in the numerical computations of the strong coupling constants of the related decays. Similar coupling constants for the ground state baryons with single heavy quark having J = 1 2 and J = 3 2 have been calculated in Refs. [13][14][15][16].
The paper is organized as follows: In Section 2, the strong decays Σ b (6097) ± → Λ 0 b π ± are studied within the LCSR method [12] by taking into account the possible configurations assigned to the Σ b (6097) ± states. In this section, we also formulate the sum rules for the masses and decay constants of Σ b (6097) ± with J = 1 2 or J = 3 2 . The numerical results of the masses and decay constants are used as input parameters in the analyses of the strong coupling constants defining the above strong decay channels. The numerical results of the strong coupling constants are also used to obtain the numerical values of the decay widths of the transitions under consideration. The last section contains our concluding remarks.
In this section, we analyze the strong transitions of the Σ b (6097) ± states to the Λ 0 b and π ± particles. As we have already noted, our primary goal is the determination of the quantum numbers of the recently observed Σ b (6097) ± baryons. To this end, we assume that these states are 1P or 2S excitations of the the corresponding ground-state baryons with J = 1 2 or J = 3 2 . We calculate the widths of these baryons under these assumptions and compare our results with that of the experimental data.
Each decay is characterized by its own strong coupling constant. Therefore, in the first step, we calculate the corresponding coupling constant defining the strong Σ b → Λ b π transition for each case within the LCSR. For the ground-state Σ b and Σ * b particles, these strong coupling constants are defined as For their corresponding 1P and 2S excitations, similar definitions as Eq. (1) with the following replacements are used: a) For the 1P excitations: In this section and in all the following discussions, the ground state and its 1P and 2S excitations are denoted by Σ b (Σ * b ), Σ b1 (Σ * b1 ) and Σ b2 (Σ * b2 ) for corresponding J = 1 2 ( 3 2 ) baryons, respectively. Here, u(q, s) and u µ (q, s) are spinors corresponding to the J = 1 2 and J = 3 2 states, respectively. For the determination of the aforementioned coupling constants from the LCSR, we introduce the following vacuum to the pseudo-scalar meson correlation function: where the on-shell π-meson state is represented by π(q)| with momentum q, η Σ ( * ) b (µ) is used to represent the interpolating current of Σ ± b (Σ * ± b ), and η Λ b is the interpolating current for the Λ b baryon having J = 1 2 . The interpolating fields for the J = 1 2 -particles are given as and For the states with J = 3 2 , we have: In above equations, q is the u(d) quark field for Σ ). The indices a, b, and c represent the colors, C is the charge conjugation operator and β is an arbitrary parameter.
To obtain the sum rules for the strong coupling constants we start with the standard procedures of the QCD sum rules derivations. To obtain the physical or phenomenological sides of the desired sum rules, we insert complete sets of the Σ b (Σ * b ) and Λ b baryons into the correlation function. As a result, we get where p is the momentum of the Λ b baryon and p ′ = p + q is the momentum of the considered Σ bi initial states, with i = 1 or 2 indicating the 1P or 2S excited state. The dots at the ends of equations are used to represent the contributions of the higher states and the continuum.
After using the matrix elements given in Eq. (1) together with the following matrix elements, defined in terms of the decay constants, λ ( * ) , λ for the J = 3 2 -states, inside Eqs. (6) and (7), and making the summations over spins using the results become Π Phys where we only keep the terms that we use in the analyses and the dots in all the final results represent the contributions coming from other structures as well as the higher states and continuum. By applying the double Borel transformation with respect to −p 2 and −p ′2 , we suppress the contributions of the higher states and the continuum, and after this process, the Eqs. (12)-(15) becomẽ where M 2 1 and M 2 2 are the corresponding Borel parameters to be fixed later. In the above equations, the notatioñ Π Phys (µ) (p, q) is used to show the Borel transformed form of Π Phys (µ) (p, q), and we use q 2 = m 2 π . Among the presented Lorentz structures, to get the sum rules for the coupling constants, we choose the / q / pγ 5 and / pγ 5 for J = 1 2 -scenarios. The structures considered for the J = 3 2 -scenarios are the / q / pγ µ and / qq µ . For J = 3 2 -scenarios, the selected structures are free from the undesired spin-1 2 pollution. Besides the physical sides of the calculations we need the theoretical or QCD sides of the desired sum rules obtained from the correlation function, Eq. (2), via the operator product expansion (OPE). To this end, the explicit forms of the interpolating currents are placed in the correlator and possible contractions are made between the quark fields using Wick's theorem. As a results of these contractions, we obtain the outcomes in terms of the heavy-and light-quark propagators. There also appear terms containing the matrix elements of the quark-gluon field operators between vacuum and π-meson states having the common form π(q)|q(x)ΓG µν q(y)|0 or π(q)|q(x)Γq(y)|0 . Their explicit expressions are given in terms of the π-meson distribution amplitudes (DAs) (see Refs. [17][18][19]). The Γ and G µν denote the full set of Dirac matrices and the gluon field strength tensor, respectively. Using these matrix elements, one gets the nonperturbative parts contributing to the results in coordinate space. We transfer the calculations to the momentum space and apply a double Borel transformation over the same variables as the physicsl sides. After applying the continuum subtraction procedure, the coefficients of same Lorentz structures as in the physical sides are considered, and the matching of these coefficients from both sides leads to the QCD sum rules for the strong coupling  , we can depict the mentioned matches as follows: OPE ) represent the Borel transformed coefficients of the / q / pγ 5 ( / q / pγ µ ) and / pγ 5 ( / qq µ ) structures for the J = 1 2 ( 3 2 ) cases. The expressions of these functions are very lengthy, hence, we do not present them here.
The QCD sum rules for the coupling constants are obtained from the numerical solutions of the equation pairs given in the Eqs. (20) and (21) for the J = 1 2 scenarios and the Eqs. (22) and (23) for the J = 3 2 scenarios. The calculations for the coupling constants require some input parameters presented in Table I. Since the masses of the considered baryons are close to each other, we choose As is seen from the equations, Eqs. (20)-(23), for the analyses of the considered coupling constants we also need the mass values of the considered baryons, and their decay constants. To obtain the masses and the decay constants we consider the following correlation function: where the current η Σ ( * ) b (µ) corresponds to the considered J = 1 2 ( 3 2 ) state, composed of the quark fields regarding the related quantum numbers. The sub-index Σ b is used to represent one of the states, Σ ± b having spin 1 2 and Σ * ± b having J = 3 2 . To determine the masses of the Σ ( * ) b states, we again consider two assumptions for each of the above-mentioned baryons, and four different QCD sum rules are obtained. For this purpose, the interpolating currents given in Eqs. (4) and (5) are used.
In the two-point QCD sum rule method for mass, one again follows two ways in the calculation of the corresponding correlator. The first way includes the calculation of the correlator in terms of the hadronic degrees of freedom and therefore it is called as the physical or the phenomenological side. For this purpose, the interpolating fields are treated as the operators creating or annihilating the states under consideration. Insertion of complete sets of hadronic states having the same quantum numbers of the hadrons under question results in and where Eq. b2 , respectively. The dots represent contributions of the higher states and the continuum. As seen from the last equations, these calculations also require the matrix elements given in the Eqs. (8) and (9). In these calculations again, the ground state and its 1P and 2S excitations are notated by Σ b (Σ * b ), Σ b1 (Σ * b1 ), and Σ b2 (Σ * b2 ) for corresponding J = 1 2 ( 3 2 ) baryon, respectively, and λ(λ * ), λ 1 (λ * 1 ) and λ 2 (λ * 2 ) are their corresponding decay constants. After the usage of expressions for the matrix elements and using the summation relations for spinors u(q, s) and u µ (q, s) given in Eqs (10) and (11), the physical sides for the J = 1 2 cases are obtained as and The similar steps give the results for the J = 3 2 cases as and As already mentioned, we need to follow a second way to calculate the same correlation function, Eq. (25), which proceeds in terms of the quark and gluon degrees of freedom. For this side of the calculation, we exploit the explicit expressions of the interpolating currents and OPE. After making the possible contractions between the quark fields, the results turn into expressions containing heavy-and light-quark propagators. To attain the final results, the expressions of these quark propagators are used and Fourier transformation from coordinate space to momentum space is applied to obtain the final form of the QCD sides. The results of this side are very lengthy; therefore, we will not give them here explicitly.
The calculations of the physical and the QCD sides are followed by the application of a Borel transformation to both sides, which suppresses the contributions coming from the higher states and continuum. Finally, the QCD sum rules are attained by matching the coefficients of the same Lorentz structures from both sides. In the present work, the mentioned structures are q and I for the J = 1 2 cases and qg µν and g µν for the J = 3 2 cases. While choosing the structures for the J = 3 2 states, among the present various ones, the structures qg µν and g µν are considered since the others contain the undesired contributions from the J = 1 2 states as well. After the application of the continuum subtraction, the obtained equation pairs are solved numerically for each state under consideration. These equations are given as (32) In the second term of the second equation, we use the − and + signs to represent the results for 1P excitation, Σ b1 , and 2S excitation, Σ b2 , respectively. To represent the expressions obtained in the QCD side of the calculations, we useT OPE i with i = 1, 2, which are the coefficient of the structures q and I for J = 1 2 cases. To obtain the results corresponding to the J = 3 2 cases, it suffices to make the changes i is used to represent the coefficients obtained from qg µν and g µν in the QCD side.
In the numerical analysis of the obtained results, we need some input parameters, which are presented in the Table I. The other ingredients of the sum rules are the three auxiliary parameters present in the results, namely the Borel parameter M 2 , threshold parameter s 0 , and an arbitrary parameter β. Note that the parameter β belongs to the currents of the states with J = 1 2 . Their working regions are fixed via following some criteria of the QCD sum rule formalism. To decide on the relevant region for the Borel parameter, the convergence of the OPE calculation is considered. To satisfy this requirement, we demand a dominant perturbative contribution compared to the nonperturbative ones which helps us determine the lower limit of the Borel parameter. As for its upper limit, the criterion is the pole dominance. From our analyses, we get this working interval as On the other hand, the threshold parameter, s 0 , is related to the energy of the first excited state of the considered state. Due to the lack of information about these excited states, this parameter is also determined considering pole dominance condition as 43 GeV 2 ≤ s 0 ≤ 47 GeV 2 .
The parameter β is determined from the analysis of the results searching for the region giving the least possible variation as a function of this parameter. This region is acquired via a parametric plot depicting the dependency of the result on cos θ, where β = tan θ. After this analysis, the region for this parameter is obtained as With the usage of these parameters and the ones given in the Table I, the obtained masses and the decay constants are presented in Table II. For the extraction of the masses for the considered excited states, the masses of corresponding ground-state baryons are used as inputs. This table also contains the errors in the results coming from uncertainties existing in the input parameters and the uncertainties arising in the determination of the working windows for auxiliary parameters.
As seen from the table, although the mass results are consistent with that of the experimental observation given as m Σ b (6097) − = 6098.0 ± 1.7 ± 0.5 MeV and m Σ b (6097) + = 6095.8 ± 1.7 ± 0.4 MeV [7], their central values are too close to indicate a deterministic information about the quantum numbers of the observed Σ b (6097) states. Therefore, for this purpose it would be much more helpful to resort to the results obtained for the decay widths. These decay widths are obtained from the usage of the results of strong coupling constant calculations with the application of the obtained mass and decay constant values.   After getting the masses and decay constants, we turn our attention again to the strong coupling constant calculations in which the results of above spectroscopic parameters are used as inputs. In the strong coupling constant analyses we adopt the auxiliary parameters used in the calculations of masses and decay constants with one exception. The Borel parameter M 2 in these calculations is revisited, and, considering the OPE series convergence and the pole dominance conditions, its interval for the strong coupling constants is determined as 15 GeV 2 ≤ M 2 ≤ 25 GeV 2 . (36) The coupling constants attained from the QCD sum rule analyses are used to get the related decay widths for the 1P and the 2S excitations of the considered states. To this end, we use the decay width formulas for the J = 1 2 cases given as: for 1P excitations and for the 2S excitations, respectively. For the J = 3 2 cases the respective decay-width equations are and The function f (x, y, z) present in the decay width equations is f (x, y, z) = 1 2x x 4 + y 4 + z 4 − 2x 2 y 2 − 2x 2 z 2 − 2y 2 z 2 . Table III presents the numerical results of the calculations for the coupling constants and decay widths. It can be seen from the table that our width results obtained for the scenarios considering Σ b (6097) ± as the 1P excitations of the ground state Σ * ± b with J P = 3 2 − are comparable to that of the experimental findings given as Γ Σ b (6097) − = 28.9 ± 4.2 ± 0.9 MeV and Γ Σ b (6097) + = 31.0 ± 5.5 ± 0.7 MeV [7].

III. CONCLUSSION
To investigate the properties of the recently observed Σ b (6097) ± , the light cone QCD sum rule calculations were performed and the strong coupling constants for their transitions to Λ 0 b π ± states were obtained. For the analyses,