Analysis of strong decays of SU(3) partners of Ω(2012) baryon

We estimate the coupling constants and decay widths of the partners of the hyperon, as discovered by the BELLE Collaboration, using the distribution amplitudes of the octet baryons within the light cone sum rules method. Our study includes a comparison of the obtained results for the relevant decay widths with those derived within the framework of the flavor analysis. We observe a good agreement between the predictions of both approaches. Moreover, our result on the decay width of is compatible with the existing experimental result within the uncertainties of the model predictions. These results can provide helpful insights for determining the nature of the partners of the baryon.


I. INTRODUCTION
In 2018, the BELLE Collaboration made an exciting announcement regarding the discovery of the hyperon.This discovery was based on the and decay channels, with a measured mass of and decay width of [1].However, knowing only the mass of the state is not sufficient to determine the quantum numbers of a state.For instance, within the QCD sum rule method, the mass of the baryon is estimated, assuming it to be either the 1P or 2S excitation state [2].Both assumptions yield the same mass value, although the estimated residues differ.Thus, additional physical quantities, such as the decay width, are necessary to identify the quantum numbers of newly discovered particles. Ω(2012)

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In a previous study [3], the transition was investigated, and its corresponding decay width was estimated by considering two possible scenarios for : either a or state.A comparison of the total decay widths obtained in this work led to the conclusion that is itself a state.Moreover, predictions from various theoretical models also converge on the likely quantum numbers for the observed state [4−15].Ω(2012) In this study, considering as the state, the strong couplings of the SU(3) partners of this state are investigated within the framework of light cone sum rules (LCSRs) using the distribution amplitudes (DAs) of the octet baryon.Note that this problem was also investigated in [16] using the flavor SU(3) symmetry approach.

+
The structure of this paper is as follows.Section II introduces the LCSRs for the strong couplings of the transition + pseudoscalar mesons.Section III provides a numerical analysis of the LCSRs, focusing on the relevant strong couplings.Within this section, we also present the computed values of the decay widths based on the obtained coupling constants.Additionally, we compare our results with those obtained using the flavor SU(3) symmetry method.Finally, our conclusions are summarized in Section IV.
where represents the interpolating current of the decuplet baryons, is the interpolating current of the pseudoscalar mesons, and represents the octet baryon state.
The interpolating current of the decuplet baryons can be written as where are the color indices, C is the charge conjugation operator, and A is the normalization factor.The quark content of the decuplet baryons and the normalization factor A are presented in Table 1.
To derive the LCSRs for the strong coupling constants, the correlation function is computed in two ways: in terms of hadrons and in terms of quark-gluon fields within the deep Euclidean domain.By applying the quark-hadron duality ansatz, the relevant sum rules can be derived.
Strong coupling constants appear in the double dispersion relation for the correlation function given in Eq. (1).Hence, to calculate these constants, the double dispersion relation for the correlation function must be calculated.The double dispersion relation is obtained via analytical continuation of the imaginary part of the corresponding invariant amplitudes with respect to the variables and in the spin-3/2 and pseudoscalar meson channels, respectively.
− Before delving into the details of the calculations, it is important to highlight the following aspect: the interpolating current for the decuplet baryons interacts not only with the ground positive parity states , but also with the negative parity states and even with the states .
To eliminate the contributions from unwanted states, and , a technique involving the linear contributions of different Lorentz structures is employed (for more details about this approach, refer to [17]).
Following the standard procedure, we insert the total set of baryons with into the correlation function along with the corresponding pseudoscalar mesons.Then, we obtain where summation is over positive and negative states, and is the mass of the corresponding pseudoscalar meson with momentum q.The matrix elements in the above equation are defined as where are the residues of the related baryons, represents the coupling constants of the baryons with the octet baryons and pseudoscalar mesons, is the decay constant of the pseudoscalar meson and q denotes its 4-momentum, and and are the Rarita-Schwinger and Dirac spinors, respectively.Performing summation over the spins of the Rarita-Schwinger spinors using the formula and using Eqs.(3) and (4), we can obtain an expression for the correlation function from the hadronic part.It should be reminded that the interpolating current interacts not only with spin states, but also with spin states.
Table 1.Quark content of the decuplet baryons and the normalization factor A.
Using the condition , it can easily be shown that It follows from this equation that any structure containing or is "contaminated" by the contributions of spin -states.Hence, to remove the contributions of spin -states, such structures are all discarded.
Another problem is all Dirac structures not being independent of each other.To overcome this issue, Dirac structures must be arranged in a specific order.In this study, we choose the ordering .Keeping this in mind, and using Eqs.( 3), (4), and ( 5), we obtain the correlation function from the phenomenological part as follows: where is the mass of the relevant octet baryon, and is the mass of the spin-positive (negative) parity baryon.Here, indicates the contributions of the excited states and continuum.

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As a final step, we must eliminate the contributions of states.For this purpose, the linear combinations of the invariant functions corresponding to different Lorentz structures are considered.
We now turn our attention to the calculation of the correlation function using operator product expansion (OPE) in the deep Euclidean region for the variables and .To calculate OPE, the explicit forms of the interpolating current are placed in the correlator, and possible contractions are performed between quark fields using Wick's theorem.As an example, for the correlation function of , we obtain where is the strange quark propagator.From this expression, it follows that the OPE results are obtained via convolution of the quark propagator to the sum of the nucleon DAs, obtained from the matrix element.A diagrammatic description of Eq. ( 8) is given in Fig. 1.Once we use the explicit expressions of the quark propagators and the definition of the DAs of octet baryons, the following master integral appears in the coefficients of different Lorentz structures: For the calculation of the double spectral densities, it is sufficient to find the double spectral representations of the master integrals.The details of the spectral density calculations for the case are presented in Appendix A. The cases of and are calculated in a similar manner.
The invariant amplitudes are related to the spectral densities via the double dispersion relation as follows: The spectral density can be obtained from by applying two subsequent double Borel transformations (for more details of the calculation, see Appendix A.) Matching the OPE results with the double dispersion relations for the relevant Lorentz structures of the hadrons, applying the quark-hadron duality ansatz, and performing double Borel transformation with respect to the variables and , we obtain the LCSRs for the relevant coupling constants whose explicit form can be written as ) where and are the invariant functions of the Lorentz structures and , respectively, and where m is the corresponding mass of light quarks.Here, is the continuum threshold in the pseudoscalar meson channel.The continuum threshold is chosen as a mass square of the first radial excitation of the corresponding pseudoscalar meson.Finally, note that in the limit, the applied method must be modified (for more details, see [18,19]).

III. NUMERICAL ANALYSIS
This section is devoted to the numerical analysis of the coupling constants derived in the previous section within the LCSRs.The main nonperturbative input of the considered LCSRs is the DAs of the octet baryons, namely, N, Σ, and Ξ.The explicit expressions of the relevant DAs are obtained in [20−23].The DAs contain the normalization constants f, , and , which are determined from the analysis of mass sum rules as well as lattice QCD [24,25].The normalization constant of the leading twist f, (for N, Σ, and Ξ baryons) is defined via the matrix element of the local current (all quark fields are at the same point).
Moreover, the DAs of higher twist contributions involve two additional normalization constants, and , which are defined as the matrix elements of local three quark twist-four operators, where n is the light-like vector, and is the Dirac bispinor.The normalization constants f, , and for the Λ baryon can be obtained from Eqs. ( 11) and ( 12) via the following replacements: SU( 3) In our analysis, we use the parameter values obtained from lattice QCD that are presented in Table 2 for completeness.The masses of the partners of are obtained in [16] and presented below.
These mass values are used in our numerical analysis.For the masses of the ground state baryons, we adapt values from the PDG [26].In addition, the value of the quark condensate is taken as [17] and [27].
The residues of the negative parity baryons are related to the residues of the radial excitations of the decuplet baryons as follows: The residues of the radial excitations of the decuplet baryons are calculated in [2].Using these results, we can easily determine the residues of the baryons.
The working regions of the Borel mass parameters and continuum thresholds, and , used in the numerical analysis are presented in Table 3. Determination of the working regions of the Borel parameters is based on the criteria that both power corrections and continuum butions should be suppressed.Moreover, the continuum threshold is obtained under the condition that the mass  Having the values of all input parameters at hand, we can perform the numerical analysis of the relevant coupling constants.As an example, in Fig. 2, we present the dependency of the coupling constant on at fixed values of the continuum thresholds , , and for the transition, because this transition has already been discovered.From this figure, we observe that there is good stability of the coupling constant when varies in its working region (Table 3).The obtained coupling constants are presented in Table 4.The errors in the results for the coupling constants can be attributed to the uncertainties in the input parameters as well as errors to the Borel mass parameters and and continuum threshold and .Having determined the coupling constants, we can calculate the decay widths of the corresponding transitions.Using the matrix elements for the considered transitions, the decay width can be written as where is the momentum of octet baryon, and and are the mass of the octet baryon and pseudoscalar meson, respectively.Using the values of the coupling constants obtained within this study, we estimate the decay widths of the relevant transitions summarized in Table 4.For comparison, we also present the results of the decay widths obtained from the flavor analysis [16].We would like to make the following remark at this point.From the expression of the decay width, it is evidently sensitive to the mass splitting among the partners of the and ground state baryons.Thus, for a fair comparison, we use the same mass values as in [16].

SU(3)
Ω → ΞK Finally, we compare our results with the values obtained within the framework of the flavor method [16].In this analysis, the coupling constant for is taken as the input parameter, and all the remaining    4. Decay widths of the baryons.3)

Decay channels
couplings are expressed in terms of this coupling using symmetry relations.Using the experimental value of the decay width , we can determine the coupling constant of this transition via Eq.( 16), and hence all the other coupling constants can be determined.When we compare our results on the coupling constants and decay widths of the considered decays with those obtained within the flavor analysis, we find that they are compatible within the uncertainties of the the model predictions.Small deviations in the results can be attributed to the violation effects and uncertainties of the input parameters of the theory.Furthermore, our prediction of the decay width for is compatible with those observed by the BELLE Collaboration within the uncertainties [1].Moreover, note that the coupling constant, and hence the decay width of , within LCSRs method was calculated using the DAs of pseudoscalar mesons in [3].However, in this study, we recalculate these quantities within the same framework using the DAs of the Ξ baryon.In this method, the calculations of the theoretical part of the sum rules can be achieved using only one quark propagator; however, in [3], two quark propagators were required, making the calculations difficult because each quark propagator contains many terms.Another advantage of the present method lies in dealing with the contributions of baryons with different parities, especially when mass splittings are small.In this method, no pollution arise due to negative parity baryons.However, with the methods used in [3], the problem of the separation of the contributions of positive baryons remains unsolved.Another difference between the two methods is that in this study, we consider both Borel mass parameters and but [3], was considered.The uncertainties of the parameters entering the DAs of baryons are larger than those of meson DAs.Once the errors are minimized in the determination of these parameters, more precise results can be obtained.When we compare our results on the coupling constant for , we find that our result is consistent with that in [3] within the uncertainties.

IV. CONCLUSION SU(3)
Ω(2012) In conclusion, we employ the LCSR method to compute the strong coupling constants and decay widths for the partners of the baryon in transitions.The "contamination" caused by the baryons are eliminated by considering the linear combinations of the sum rules obtained from different Lorentz structures.By comparing our decay width results with the findings of [16], we ascertain the compatibility of our decay width predictions with the outcomes of the flavor symmetry analysis.The small discrepancy between the predictions of the two methods may be attributed to the violation effects.Moreover, our estimated decay width for the transition is also compatible with the measurement of the BELLE Collaboration within the uncertainty limits.In addition, our result on the coupling constant for calculated using the DAs of Ξ is consistent with the prediction in [3], where the DAs of pseudoscalar mesons are used.

SU(3) Ω(2012)
Our results on the branching ratios can provide useful hints about the nature of the partners of the baryon.

ACKNOWLEDGMENTS
We are grateful to Y. M. Wang for discussions on numerical analysis.We also thank A. Ozpineci for useful remarks.

DENSITY
Here, we provide the detailed derivation of the spectral density (see also [28]).

−p ′2
−q 2 After applying the double Borel transformation over the variables and to Eq. ( 9), we obtain Before implementing the second double Borel transformation, we introduce the new variables .The second Borel transformation can be performed over the new Borel parameter using the relation As a result, we have Hence, the double spectral density can be obtained as follows: Let us now focus on the double spectral density for the case.Using where , can be written as where m is the corresponding quark mass.Using the Schwinger representation for the denominator and performing the first double Borel transformation over the variables and , we obtain where .To perform the second double Borel transformation, we use the relation ò .
Then, we obtain After performing the second Borel transformation, we obtain the the spectral density corresponding to : As a result of the above summarized calculations, the spectral density can take the following form: To evaluate the x integral, we introduce a new variable through the relation so that the spectral density can be written as The double spectral densities for and can be calculated using the following relations: (see also [17] for the calculation of the spectral densities and ).

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To calculate the strong couplings of partners, denoted as states in the following discussions, we introduce the vacuum-to-octet baryon correlation function: Chinese Physics C Vol. 48, No. 8 (2024) 083101 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Article funded by SCOAP 3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd 083101-1

Fig. 1 .
Fig. 1.Diagrammatic representation of the correlation function.The wavy lines denote the external currents, solid lines correspond to the quark fields, and shaded regions correspond to the DAs of the nucleon.

Fig. 2 .
Fig. 2. (color online) Dependency of the coupling constant of the transition on the Borel mass parameter at several fixed values of the Borel parameter and the continuum threshold .

Table 3 .
Working regions of the Borel mass parameters and continuum threshold .
2Using two Dirac delta functions, we can easily perform integrals over t and :