Structure of the $\Xi_b(6227)^-$ Resonance

We explore the recently observed $\Xi_b(6227)^{-}$ resonance to fix its quantum numbers. To this end, we consider various possible scenarios: It can be considered as either 1P/2S excitations of the $\Xi_b^-$ and $\Xi_b'(5935)^{-}$ ground state baryons with spin-$\frac{1}{2}$ or 1P/2S excitations of the ground state $\Xi_b(5955)^{-}$ with spin-$\frac{3}{2}$. We calculate the masses of the possible angular-orbital $1P$ and $2S$ excited states corresponding to each channel employing the QCD sum rule technique. It is seen that all the obtained masses are in agreement with the experimentally observed value, implying that the mass calculations are not enough to distinguish the quantum numbers of the state under question. Therefore, we extend the analysis to investigate the possible decays of the same excited states into $\Lambda_b^0 K^-$ and $\Xi_b^-\pi$. Using the light cone QCD sum rule method, we calculate the corresponding strong coupling constants, which are used to extract the decay widths of the modes under consideration. Our results on decay widths indicate that the $\Xi_b(6227)^{-}$ is $1P$ angular-orbital excited state of the $\Xi_b(5955)^{-}$ baryon with quantum numbers $J^P=\frac{3}{2}^{-}$.

The outline of the article is as follows: In Section II, we derive the QCD sum rules for the masses and decay constants of the Ξ b (6227) − with the possible quantum numbers J P = 1 2 ± and 3 2 ± . This section also contains the numerical values obtained for masses and decay constants of considered states which will be used as inputs in the following section. In Section III, within light cone QCD sum rule method, we obtain the coupling constants for the considered transitions with possible configurations assigned to the Ξ b (6227) − and present the results obtained from the analysis. This section includes also the decay width calculations for the considered transitions. Section IV is devoted to the summary and discussion of the results.

II. SPECTROSCOPIC PARAMETERS OF THE Ξ b STATES
In this section, the details of the calculations for spectroscopic properties, i.e. masses and decay constants, of 1P and 2S excited Ξ b states are presented for three different total angular momentum J possibilities. The calculations for all three considerations are performed using the QCD sum rule formalism which starts from the following correlation function The correlation function is written in terms of the interpolating current of the considered state, i.e. the current J B(µ) corresponding to the considered J = 1 2 ( 3 2 ) state, which is formed using quark fields and considering the quantum numbers of the state. The sub-index B represents one of the states, Ξ b (J = 1 2 ), Ξ ′ b (J = 1 2 ) or Ξ b (J = 3 2 ). We will use the following interpolating currents in the calculations: a) For J = 1 2 particles: b) For J = 3 2 particles: The indices a, b, and c in the current expressions are used to represent the color indices, C is the charge conjugation operator, and β present in the J = 1 2 currents is an arbitrary parameter. In QCD sum rule calculations, we calculate the correlator in two ways. In the first step, it is calculated in terms of hadronic degrees of freedom, considering the interpolating fields as operators annihilating or creating those hadrons. This side is expressed in terms of hadronic degrees of freedom and denoted as physical or phenomenological side. For the calculation of this side, complete sets of hadronic states with the same quantum numbers of the considered hadrons are inserted in the correlation function. As a result we have Π Phys (µν) (q) = 0|J B(µ) |B(q, s) B(q, s)|J B(ν) |0 and Π Phys (µν) (q) = 0|J B(µ) |B(q, s) B(q, s)|J B(ν) |0 m 2 − q 2 + 0|J B(µ) |B ′ (q, s) B ′ (q, s)|J B(ν) |0 m ′2 − q 2 + . . . , when the 1P and the 2S excitations are considered, respectively. Here m, m and m ′ are the mass of the ground, 1P and 2S excited states of the Ξ b baryons, correspondingly. J B(µ) represents either the current J B of J = 1 2 or that J Bµ of J = 3 2 baryon. The contributions of higher states and the continuum are represented by the dots. The matrix elements between the vacuum and one-particle states are defined as 0|J B |B(q, s) = λu(q, s), Parameters Values for the coefficients of qg µν and g µν attained in QCD side. To perform the numerical analysis, various input parameters entering to the sum rules are needed. Some of these input parameters are given in Table I. Besides these parameters, the sum rules contain three auxiliary parameters. These are the Borel parameter M 2 , threshold parameter s 0 , and an arbitrary parameter β existing in the calculations of J = 1 2 states. To fix their working intervals, we follow the standard criteria of the QCD sum rules formalism. To begin with, in the determination of threshold parameter s 0 , we need to emphasize that it is not completely arbitrary and has a relation with the energy of first excited state having the same quantum numbers with the considered state. However, since we have very limited knowledge on the energy of excited states, we fix its interval looking at the pole dominance condition. We demand that the pole contributions for each case are dominant and comprise the highest part of the total value. The Borel parameter region is determined looking at the convergence of the OPE. This requires also the dominance of the perturbative terms over the nonperturbative ones in the calculations. Claiming these, the lower limit of the Borel parameter is fixed. For the upper limit of this parameter, the pole dominance is required. Finally, for the calculations including the parameter β, its working interval is obtained from the analysis of the results, requiring least possible dependence on this parameter. For this purpose, one examines the variance of the results as a function of cos θ, where β = tan θ, and determines the regions where the results have relatively weak dependence on cos θ. Actually, the relatively weak dependence on the auxiliary parameters is another requirement in the QCD sum rules calculation to gain reliable results for the physical parameters under consideration. With all these requirements, the intervals for auxiliary parameters, for all the considered states, are attained as and Using the working regions for the auxiliary parameters, together with the parameters given in Table I, we obtain the final results for the masses and decay constants under consideration. Note that the ground state mass values of Ξ b baryons, i.e. the masses of Ξ − b , Ξ ′ b (5935) − and Ξ b (5955) − are taken as inputs in the equations. The values of the the masses corresponding to 1P and 2S excitations are obtained as presented in Table II. The errors in the results are due to the errors of the input parameters as well as those coming from the variations of the results with respect to the variations of the auxiliary parameters in their working intervals. As an example, in Fig. 1, we present the dependence of the mass of Ξ b ( 3 2 − )(1P ) state on M 2 and s 0 . From this figure, one can see that the requirement of the relatively weak dependence on these parameters is satisfied.    The masses of the other possibilities and decay constants for all the states under consideration are determined similarly and presented in Table II. From this table, it follows that the masses of the considered states with J P = 1 2 ± and J P = 3 2 ± are all very close to each other and therefore the mass determination is not enough to identify Ξ b (6227) − .
Hence, in the next section, we extend the analysis and obtain the widths of the considered excited states decaying to Λ 0 b K − and Ξ 0 b π − , which can provide us with the possibility to assign quantum numbers of the Ξ b (6227) − state.
This section is devoted to the analysis of the transitions Ξ b (6227) − → Λ 0 b K − and Ξ b (6227) − → Ξ 0 b π − by considering the Ξ b (6227) − as 1P or 2S excitation state of one of the ground state Ξ − b , Ξ ′ b (5935) − , or Ξ b (5955) − baryon. In calculations, the LCSR method is employed. The starting point of this method is consideration of the correlation function given as In this correlation function, J B(µ) represents one of the currents given in Eqs. (2) and (3) and the calculation of this correlation function will be carried on for each current given there, considering again both possibilities of being 1P or 2S states separately. The index (µ) is used only for transition of J = 3 2 state. K(π)(q)| is the on-shell K(π)-meson state with momentum q. J Λ 0 b and J Ξ 0 b are the interpolating currents of the J = 1 2 Λ 0 b and Ξ 0 b baryons. In this part of the study, again the correlation function is firstly calculated in terms of the hadronic parameters.
and for J = 3 2 we obtain Π Phys Again note that, in these equations, B(B * ), B( B * ), and B ′ (B * ′ ) represent the ground, 1P and 2S excitated states corresponding to each considered ground state baryon. Here, p ′ = p + q and p are the momenta of these baryons and Λ b (Ξ b ) baryon, respectively. The contributions of higher states and continuum are represented by dots. Now, we have some additional matrix elements in Eqs. (19)- (22). For J = 1 2 baryons, these matrix elements are determined as and, for J = 3 2 baryons, they are parametrized as where, in Eqs. (23) and (24) g's with various indices denote the strong coupling constants of the corresponding baryons with pseudoscalar mesons. Inserting these matrix elements into Eqs. (19)- (22) and applying summations over spins given in Eqs. (8) and (9), for physical sides of the correlation function, we get The function T αµ is given as To suppress the contributions coming from the higher states and continuum, a double Borel transformation with respect to −p 2 and −p ′2 is performed. As a result, we get BΠ Phys where, M 2 1 and M 2 2 are Borel parameters in initial and final channels, respectively. In these equations BΠ Phys (µ) (p, q) stands for the Borel transformed form of the Π Phys (µ) (p, q) function. These results contain different Lorentz structures from which we can get the sum rules to obtain the strong coupling constants under question. In the J = 1 2 scenario, we use / q / pγ 5 and / pγ 5 structures to obtain the coupling constants for each possibility that the state Ξ b (6227) may become. For obtaining the relevant coupling constants for J = 3 2 baryon, the Lorentz structures / q / pγ µ and / qq µ are considered.
For both J P = 1 2 ± and J P = 3 2 ± scenarios, we also need to calculate the theoretical sides of the correlation function, Eq. (18), with usage of the related interpolating currents, explicitly. Similar to the previous case after the possible contractions made using Wick's theorem between the quark fields of the interpolating currents, the results are expressed in terms of light and heavy quark propagators. Besides the propagators, we need matrix elements of remaining quark field operators between the K(π) meson and the vacuum. These matrix elements, whose common forms can be written as K(π)(q)|q(x)Γq(y)|0 or K(π)(q)|q(x)ΓG µν q(y)|0 with Γ and G µν being full set of Dirac matrices and gluon field strength tensor respectively, are parameterized in terms of K(π)-meson distribution amplitudes (DAs). Nonperturbative contributions are attained by exploiting these matrix elements as inputs in the calculations. The explicit form of these matrix elements are present in Refs. [48][49][50][51]. Again, considering the same structures in the physical and the theoretical sides and matching the coefficients of the same structures in both sides, performing the Borel transformation and continuum subtraction using quark hadron duality assumption, we obtain the QCD sum rules for the relevant coupling constants as where BΠ OPE entering the calculations, which leads to Moreover, for all of the auxiliary parameters, we adopt the values obtained in the mass and decay constant calculations only with one exception. Using the OPE series convergence and pole dominance conditions for working region of M 2 , in this part, we obtain Again to illustrate the sensitivity of the results to the auxiliary parameters, we pick out the coupling constant g Ξ b Λ b K for the transition of 1P excitation of Ξ b (5955) − baryon to Λ b and K final states and present the dependence of the corresponding coupling constant on M 2 and s 0 in Fig. 2. Similarly, we perform the analyses for all the coupling constants under consideration. Our final results for the relevant coupling constants are presented in Table III. The obtained coupling constants are used to extract the corresponding decay widths. The decay width formulas for 1P and 2S excitations for the J = 1 2 cases are  The state B(J P ) The similar decay width expressions for the J = 3 2 case are and The function f (x, y, z) appearing in the decay width equations is defined as The numerical results for the decay widths are also presented in Table III. In this table, we also present the total width values as a sum of the considered transitions in each case. Comparing our results given in Table III to that of the experimental width, which is Γ Ξ b (6227) − = 18.1 ± 5.4 ± 1.8 MeV [35], it can be seen that the state with J P = 3 2 − scenario nicely reproduces the experimental width value.
At the end of this section, we would like to have a comment on the heavy quark symmetry partners. In heavy quark effective theory (HQET), the heavy quark decouple with light quarks in the leading inverse heavy quark mass expansion. Therefore, properties of baryons with one heavy quark are determined with the properties of light quarks (so called diquark). The properties of P -wave baryons and their interpolating currents were systematically studied in [52,53]. Using the same notations given in these studies, the heavy baryons belonging to the baryon multiplet are characterized by the set of the quantum numbers [F, j l , s l , ρ/λ], where F means antitriplet or sextet representation of SU (3) F , j l and s l are the total angular and spin momenta of the diquark, and ρ and λ denote the ρ type (l ρ = 1, l λ = 0) and λ type (l ρ = 0, l λ = 1), respectively. Here, l ρ is the orbital angular momentum between two light quarks and l λ is the angular momentum between heavy quark and diquark. As a result, for the antitriplet negative parity baryons for instance, the following heavy quark symmetry partners are obtained: The strong decay widths of these states are calculated in [52].