A. V. Berezhnoy, I. N. Belov, A. K. Likhoded
In the framework of diquark model for production we discuss the yields and observation prospects for excited states of doubly heavy baryons at LHC kinematic conditions.
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The theoretical analysis of production, lifetime, and decays of doubly heavy baryons is presented. The lifetime of $Ξ_{cc}^{++}$ baryon recently measured by the LHCb Collaboration is used to estimate the lifetimes of other doubly heavy baryons. The production and the possibility of observation of $Ξ_{bc}$ baryon at LHC are discussed.
In this article, we study the ${1\over 2}^+$ doubly heavy baryon states $Ω_{QQ}$ and $Ξ_{QQ}$ by subtracting the contributions from the corresponding ${1\over 2}^-$ doubly heavy baryon states with the QCD sum rules, and make reasonable predictions for their masses. Those doubly heavy baryon states maybe observed at the Tevatron, the LHCb and the PANDA.
We speculate on a possible interpretation of events selected by the SELEX collaboration and stress an insufficient evidence for the observation of doubly charmed baryon.
We analytically calculate the total and differential cross sections for the pair production of doubly heavy baryons in the framework of diquark model. The processes of electron-positron and quark-antiquark annihilations are considered. The fractions of doubly heavy baryons in the yields of heavy quarks are evaluated numerically.
This paper studies the leading chiral corrections to heavy quark-diquark symmetry predictions for doubly heavy baryon semileptonic decay form factors. We derive the coupling between heavy diquarks and weak current in the limit of heavy quark-diquark symmetry, and construct the chiral Lagrangian for doubly heavy baryons coupled to weak current. We evaluate chiral corrections to doubly heavy baryon zero-recoil semileptonic decay for both unquenched and partially quenched QCD. This theory is used to derive chiral extrapolation expressions for measurements of form factors of doubly heavy baryon ze…
Recently, some singly bottom baryons have been established experimentally, but none of doubly or triply bottom baryons has been observed. Under the Regge phenomenology, the mass of a ground state unobserved doubly or triply bottom baryon is expressed as a function of masses of the well established light baryons and singly bottom baryons. Then, the values of Regge slopes and Regge intercepts for baryons containing one, two, or three bottom quarks are calculated. After that, masses of the orbitally excited singly, doubly, and triply bottom baryons are estimated. Our predictions may be useful for…
The mass spectra of doubly heavy baryons are systematically calculated in the framework of QCD sum rules. With a tentative heavy-diquark--light-quark configuration, the interpolating currents representing the doubly heavy baryons are proposed. Contributions of the operators up to dimension six are included in operator product expansion. The numerical results are compatible with other theoretical predictions, which may support the $(QQ)-(q)$ structure of doubly heavy baryons.
The masses and residues of the spin--3/2 doubly heavy baryons are calculated within the QCD sum rules method. A comparison of our predictions with those existing in the literature is also made.
In this paper we give a short review of the recently obtained results on inclusive and exclusive decays of doubly heavy baryons.
Using the most general form of the interpolating current for baryons, the strong electric and magnetic coupling constants of light vector mesons $ ρ$ and $ K^* $ with doubly heavy baryons are computed within the light-cone sum rules. We consider 2- and 3-particle distribution amplitudes of the aforementioned vector mesons. The obtained results can be useful in the analysis of experimental data on the properties of doubly heavy baryons conducted at LHC.
The study of the properties of doubly heavy baryons represents a promising area in particle physics. It can provide us with information about Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and the low energy dynamics of QCD. They have a very rich phenomenology. The investigation of weak, electromagnetic, and strong decays has vital importance for understanding the dynamics of doubly heavy baryons. The main ingredients of such studies are the spectroscopic parameters, the strong coupling constants, and the transition form factors. For calculations of these quantities, non-perturbative methods …
We study the inclusive production of doubly heavy baryon $Ξ_{cc}$ at polarized photon collider. Our results show that proper choice of the initial beam polarizations may increase the production rate of $Ξ_{cc}$ approximately 10%.
We perform a detailed investigation of total lifetimes for the doubly heavy baryons $Ξ_{QQ'}$, $Ω_{QQ'}$ in the framework of operator product expansion over the inverse heavy quark mass, whereas, to estimate matrix elements of operators obtained in OPE, approximations of nonrelativistic QCD are used.
The inclusive production of doubly heavy baryons $Ξ_{cc}$ and $Ξ_{bb}$ at $γγ$ collider is investigated. It is found that the contribution from the heavy quark pair $QQ$ in color triplet and color sextet are important.
Mass spectra and semileptonic decay rates of doubly heavy baryons are studied in the framework of the relativistic quark model in the quark-diquark approximation.
Since the LHCb collaboration announced the observation of the doubly charmed baryon $Ξ_{cc}^{++}$, a series of studies of doubly heavy baryons have been presented. In this work, I analyse the non-leptonic weak decays of doubly heavy baryons $Ξ_{bc}$ and $Ω_{bc}$ under the flavor $SU(3)$ symmetry. I mainly focus on the $W$-exchange diagrams, which will contribute to the decay channels with final states are light meson and light baryon. These channels would be helpful for searching for $Ξ_{bc}$ and $Ω_{bc}$ at LHC. And these channels and relations of corresponding decay widths could be examined …
The high production rate and $\langle x_{F} \rangle > 0.33$ of the doubly charmed baryons measured by the SELEX experiment is not amenable to perturbative QCD analysis. In this paper we calculate the production of the doubly heavy baryons with the double intrinsic charm Fock states whose existence is rigorously predicted by QCD. The production rate and the longitudinal momentum distribution are both reproduced. We also show that the production rates of the doubly charmed baryons and double $J/ψ$ production observed by NA3 collaboration are comparable. Recent experimental results are reviewed. …
In this article, we study the ${3\over 2}^+$ heavy and doubly heavy baryon states $Ξ^*_{cc}$, $Ω^*_{cc}$, $Ξ^*_{bb}$, $Ω^*_{bb}$, $Σ_c^*$, $Ξ_c^*$, $Ω_c^*$, $Σ_b^*$, $Ξ_b^*$ and $Ω_b^*$ by subtracting the contributions from the corresponding ${3\over 2}^-$ heavy and doubly heavy baryon states with the QCD sum rules, and make reasonable predictions for their masses.
The analysis of singly charmed hadrons has been extended to the case of doubly charmed baryons, $Ξ_{cc}^{++}$, $Ξ_{cc}^{+}$ and $Ω_{cc}^{+}$. Doubly charmed baryons are described as a system containing a heavy $cc$-diquark and a light quark, similarly as in a heavy-light meson. This leads to preasymptotic effects in semileptonic and nonleptonic decays which are essentially proportional to the meson wave function. Interplay between preasymptotic effects in semileptonic and/or nonleptonic decay rates leads to very clear predictions for semileptonic branching ratios and lifetimes of doubly charme…
We study the masses of the doubly bottom baryons and the charmed bottom baryons up to $\mathcal{O}\left(p^{3}\right)$ in heavy baryon chiral perturbation theory. For the low-energy constants, $g_{A}$ was determined in the quark model, and we fit to the lattice QCD data to determine $m_{0}$, $c_{1}$, and $c_{7}$. We show the numerical results up to $\mathcal{O}\left(p^{3}\right)$, and our results are compared with other theoretical results.
We extend the analysis of weak decays of heavy hadrons to the case of doubly charmed baryons, Ξ_{cc}^{++}, Ξ_{cc}^{+} and Ω_{cc}^{+}. Doubly charmed baryons are modeled as a heavy-light system containing a heavy cc-diquark and a light quark. Such a model leads to preasymptotic effects in semileptonic and nonleptonic decays which are essentially proportional to the meson wave function. Very clear predictions for semileptonic branching ratios and lifetimes of doubly charmed baryons are obtained.
In the framework of a non-relativistic quark model, the mass spectra of the ground and excited states of doubly heavy $Ξ$ and $Ω$ baryons are calculated. We estimate the mass difference between the $ Ω$ and corresponding $ Ξ$ baryons as $M_Ω-M_Ξ\simeq178 $ MeV for all the states containing $ cc,~bc $, or $ bb $ quarks. A simple form of the universal Isgur-Wise function, as the transition form factor between the doubly heavy baryons, is introduced. Working in the close-to-zero recoil limit, we investigate the $ b \rightarrow c $ semileptonic decay widths and branching fractions of the doubly he…
In the paper we discuss contribution of the doubly intrinsic heavy mechanism into the production of $B_c$ meson, the doubly heavy baryons and the all-charm tetraquark at a future fixed-target experiment at the LHC (AFTER@LHC). The production cross sections and the mean values of Feynman-x for the finale states are presented.
Baryons containing two heavy quarks are treated in the Born-Oppenheimer approximation. Schrödinger equation for two center Coulomb plus harmonic oscillator potential is solved by the method of ethalon equation at large intercenter separations. Asymptotical expansions for energy term and wave function are obtained in the analytical form. Using those formulas, the energy spectra of doubly heavy baryons with various quark compositions are calculated analytically.
The existence of doubly heavy flavor baryons has not been well established experimentally so far. In this Letter we systematically investigate the weak decays of the doubly charmed baryons, $Ξ_{cc}^{++}$ and $Ξ_{cc}^{+}$, which would be helpful for experimental searches for these particles. The branching fractions are studied under the factorization hypothesis for the short-distance contributions and considering the rescattering effect for the long-distance contributions which are significantly enhanced. Comparing all the decay modes, we recommend the processes of $Ξ_{cc}^{++}\to Λ_c^+K^-π^+π^…
The strong coupling constants of doubly heavy baryons with heavy mesons
针对实验仅发现一种双粲重子、待探索更多双重味重子的背景，本研究采用光锥QCD求和规则计算双重味重子、单重味重子与重介子的强耦合常数，给出数值结果与误差分析，为双重味重子衰变研究及实验数据分析提供理论支撑。
The finite size of doubly heavy diquark gives a positive correction to the masses of baryons calculated in the local diquark approximation. We evaluate this correction for the basic states of doubly charmed baryons to give quite accurate predictions actual for current searches of those baryons at LHCb: $m[{Ξ_{cc}^{1/2}}^{+}]\approx m[{Ξ_{cc}^{1/2}}^{++}]=3615\pm 55$ MeV and $m[{Ξ_{cc}^{3/2}}^{+}]\approx m[{Ξ_{cc}^{3/2}}^{++}]=3747\pm 55$ MeV.
In this paper we present in greater detail previous work on the Born-Oppenheimer approximation to treat the hydrogen bond of QCD, and add a similar treatment of doubly heavy baryons. Doubly heavy exotic resonances X and Z can be described as color molecules of two-quark lumps, the analogue of the H_2 molecule, and doubly heavy baryons as the analog of the H_2^+ ion, except that the two heavy quarks attract each other. We compare our results with constituent quark model and lattice QCD calculations and find further evidence in support of this upgraded picture of compact tetraquarks and baryons.
Mass spectra of the ground-state baryons consisting of three or two heavy (b or c) and one light (u,d,s) quarks are calculated in the framework of the relativistic quark model and the hyperspherical expansion. The predictions of masses of the triply and doubly heavy baryons are obtained by employing the perturbation theory for the spin-independent and spin-dependent parts of the three-quark Hamiltonian.
The strong coupling constant of doubly heavy baryons with light pseudoscalar mesons $ π$ and $ K $ are computed within the light cone sum rules. We take into account two-particle and three-particle distribution amplitudes of the said pseudoscalar mesons. We compare our result with the one existing in the literature.
We present the results for the masses of the doubly heavy baryons $Ξ_{QQ'}$ and $Ω_{Q'Q}$ where $Q,Q'=b,c$ obtained in the framework of the simple approximation within the nonperturbative string approach.
The chiral effective meson-baryon Lagrangian for the description of interactions between the doubly charmed baryons and Goldstone bosons is constructed up to the order of $q^{4}$. The numbers of linearly independent invariant monomials of $\mathcal{O}(q^2)$, $\mathcal{O}(q^3)$ and $\mathcal{O}(q^4)$ are 8, 32 and 218, in order. The obtained Lagrangian can be used to study the chiral dynamics and relevant phenomenology of the doubly charmed baryons at complete one-loop level in future. For completeness, the non-relativistic reduction of the Lagrangian is also discussed.
The spectrum of doubly heavy baryons, hadrons made up of two heavy quarks and one light quark, is computed through a potential model with relativistic kinematics. The expression for the $Q\bar Q$ potential comes from the AdS/QCD correspondence.
We present some piloting calculations of the masses of the doubly heavy baryons in the framework of the simple approximation within the nonperturbative string approach. The simple analytical results for dynamical masses of heavy and light quarks and eigenvalues of the effective QCD Hamiltonian are presented.
Semileptonic decays of doubly heavy baryons are studied in the framework of the relativistic quark model. The doubly heavy baryons are treated in the quark-diquark approximation. The transition amplitudes of heavy diquarks bb and bc going respectively to bc and cc are explicitly expressed through the overlap integrals of the diquark wave functions in the whole accessible kinematic range. The relativistic baryon wave functions of the quark-diquark bound system are used for the calculation of the transition matrix elements, the Isgur-Wise function and decay rates in the heavy quark limit.
In this paper, we analyze the doubly heavy spin-3/2--spin-1/2 baryons and light meson vertices within the method of light-cone QCD sum rules. These vertices are parametrized in terms of one (three) coupling constant(s) for the pseudoscalar (vector) mesons. The said coupling constants are calculated for all possible transitions. The results presented here can serve to be useful information in experimental as well as theoretical studies of the properties of doubly heavy baryons.
In the framework of two-point sum rules of NRQCD, the masses and couplings of doubly heavy baryons to the corresponding quark currents are evaluated with account of coulomb-like corrections in the system of doubly heavy diquark as well as the contribution of nonperturbative terms determined by the quark, gluon, mixed condensates and the product of gluon and quark condensates. The introduction of nonzero light quark mass destroys the factorization of baryon and diquark correlators even at the perturbative level and provides the better convergency of sum rules. We estimate the difference M_Omega…
In this work, we calculate the mass spectrum of doubly heavy baryons with the diquark model in terms of the QCD sum rules. The interpolating currents are composed of a heavy diquark field and a light quark field. Contributions of the operators up to dimension six are taken into account in the operator product expansion. Within a reasonable error tolerance, our numerical results are compatible with other theoretical predictions. This indicates that the diquark picture reflects the reality and is applicable to the study of doubly heavy baryons.
In this article, we study the doubly heavy baryon states and pentaquark states with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension $7$ and $13$ respectively in a consistent way. In calculations, we separate the contributions of the negative parity and positive parity hadron states unambiguously, and study the masses and pole residues of the doubly heavy baryon states and pentaquark states in details. The present predictions can be confronted to the experimental data in the future.
In the framework of quark-hadron duality, the cross section values of charmonium and doubly charmed baryon production in $e^+e^-$-annihilations have been estimated at interaction energy $\sqrt{s}=10.6$ GeV. It has been shown that the approach under discussion allows to resolve some contradictions between the experimental data and pQCD predictions.
The mass and residues of the first radial and orbital excitations of $J=\frac{1}{2}$ baryons containing two heavy quarks (b or c) are calculated within the QCD sum rules method. In calculations, the general forms of the interpolating currents with symmetric and anti-symmetric forms with respect to the exchange of heavy quarks have been used. Our results on the spectroscopic parameters of doubly heavy baryons are compared with the predictions of other theoretical approaches.
In this article, we study the ${(1/2)^-}$ and ${(3/2)^-}$ heavy and doubly heavy baryon states $Σ_Q({(1/2)^-})$, $Ξ'_Q({(1/2)^-})$, $Ω_Q({(1/2)^-})$, $Ξ_{QQ}({(1/2)^-})$, $Ω_{QQ}({(1/2)^-})$, $Σ_Q^*({(3/2)^-})$, $Ξ_Q^*({(3/2)^-})$, $Ω_Q^*({(3/2)^-})$, $Ξ^*_{QQ}({(3/2)^-})$ and $Ω^*_{QQ}({(3/2)^-})$ by subtracting the contributions from the corresponding ${1\over 2}^+$ and ${3\over 2}^+$ heavy and doubly heavy baryon states with the QCD sum rules in a systematic way, and make reasonable predictions for their masses.
In this paper, the mass spectra are obtained for doubly heavy $Ξ$ baryons, namely, $Ξ_{cc}^{+}$, $Ξ_{cc}^{++}$, $Ξ_{bb}^{-}$, $Ξ_{bb}^{0}$, $Ξ_{bc}^{0}$ and $Ξ_{bc}^{+}$. These baryons are consist of two heavy quarks($cc$, $bb$ and $bc$) with a light($d$ or $u$) quark. The ground, radial and orbital states are calculated in framework of Hypercentral constituent quark model with coul- omb plus linear potential. Our outcomes are also compared with other predictions, thus, the average possible range of excited states masses of these $Ξ$ baryons can be determined. The study of the Regge trajectori…
A doubly heavy baryon can be regarded as composed of a heavy diquark and a light quark. In this picture, we study the masses of the doubly heavy diquarkes in the Bethe-Salpeter (BS) formalism first, which are then used as one of the inputs in studying the masses of the doubly heavy baryons in the quark-diquark model. We establish the BS equations for both the heavy diquarks and the heavy baryons with and without taking the heavy quark limit, respectively. These equations are solved numerically with the kernel containing the scalar confinement and one-gluon-exchange terms. The mass of the doubl…
The present work has been done on the baryons containing one, two, or three bottom quarks. In this article, the ground-state masses of $Ω_{b}^{-}$, $Ξ_{bb}^{0}$, $Ξ_{bb}^{-}$, $Ω_{bb}^{-}$, and $Ω_{bbb}^{*-}$ baryons are calculated within the framework of Regge phenomenology. Further, the values of Regge slopes and Regge intercepts for singly, doubly, and triply bottom baryons are estimated in both the ($J,M^{2}$) and ($n,M^{2}$) planes to calculate the excited-state masses of these baryons. Here our attempt is to assign a possible spin-parity to recently observed some singly bottom baryons an…
We construct a chiral Lagrangian for doubly heavy baryons and heavy mesons that is invariant under heavy quark-diquark symmetry at leading order and includes the leading O(1/m_Q) symmetry violating operators. The theory is used to predict the electromagnetic decay width of the J=3/2 member of the ground state doubly heavy baryon doublet. Numerical estimates are provided for doubly charm baryons. We also calculate chiral corrections to doubly heavy baryon masses and strong decay widths of low lying excited doubly heavy baryons.
The masses of baryons containing two heavy quarks and their couplings to the corresponding quark currents are evaluated in the framework of NRQCD sum rules. The coulomb-like corrections in the system of doubly heavy diquark are taken into account, and the contribution of nonperturbative terms coming from the quark, gluon and mixed condensates as well as the product of quark and gluon condensates, is analyzed. The higher condensates destroy the factorization of baryon and diquark correlators and provide the convergency of sum rule method. As a result the accuracy of estimates is improved.
We calculate the mixing angles between the spin--1/2, $Ξ_{bc}$--$Ξ^\prime_{bc}$ and $Ω_{bc}$--$Ω^\prime_{bc}$ states of doubly heavy baryons within the QCD sum rules method. It is found that the mixing angles are large and have the values $φ_{Ξ_{bc}} = 16^0 \pm 5^0$ and $φ_{Ω_{bc}} = 18^0 \pm 6^0$, respectively. The mixing angles are slightly smaller compared to the predictions of the non--relativistic quark model, $φ_{Ξ_{bc}} = 25.5^0$ and $φ_{Ω_{bc}} = 25.9^0$.
We give heavy quark-diquark symmetry predictions for doubly heavy baryons and tetraquarks in light of the recent discovery of the $Ξ_{cc}^{++}$ by LHCb. For five excited doubly charm baryons that are predicted to lie below the $Λ_c D$ threshold, we give predictions for their electromagnetic and strong decays using a previously developed chiral Lagrangian with heavy quark-diquark symmetry. Based on the mass of the $Ξ_{cc}^{++}$, the existence of a doubly heavy bottom $I=0$ tetraquark that is stable to the strong and electromagnetic decays has been predicted. If the mass of this state is below 1…
The LHCb Collaboration announced the observation of doubly charmed baryon through $Ξ_{c c}^{++} \rightarrow Λ_{c}^{+} K^{-} π^{+} π^{+}$ in 2017. Since then, a series of studies of doubly heavy baryons have been presented. $Ξ_{cc}^{++}$ was discovered through nonleptonic four-body decay mode, and experimental data has indicated that the decay modes of $Ξ_{c c}^{++}$ are not saturated by two and three-body intermediate states. In this work, we analyze the four-body weak decays of doubly heavy baryons $Ξ_{cc}^{++}, Ξ_{cc}^+$, and $Ω_{cc}^+$. Decay amplitudes for various channels are parametrized…
A comprehensive study of $b\to c$ weak decays of doubly heavy baryons is presented in this paper. The transition form factors as well as the pole residues of the initial and final states are respectively obtained by investigating the three-point and two-point correlation functions in QCD sum rules. Contributions from up to dimension-6 operators are respectively considered for the two-point and three-point correlation functions. The obtained form factors are then applied to a phenomenological analysis of semi-leptonic decays.
We extend the chromomagnetic model by further considering the effect of color interaction. The effective mass parameters between quark pairs ($m_{qq}$ or $m_{q\bar{q}}$) are introduced to account both the effective quark masses and the color interaction between the two quarks. Using the experimental masses of hadrons, the quark pair parameters are determined between the light quark pairs and the light-heavy quark pairs. Then the parameters of heavy quark pairs ($cc$, $cb$, $bb$) are estimated based on simple quark model assumption. We calculate all masses of doubly and triply heavy-quark baryo…
We calculate the mass and residue of the heavy spin--1/2 baryons containing two heavy b or c quarks in the framework of QCD sum rules. We use the most general form of the interpolating current in its symmetric and anti-symmetric forms with respect to the exchange of heavy quarks, to calculate the two-point correlation functions describing the baryons under consideration. A comparison of the obtained results with existing predictions from various approaches is also made.
Using the quark-diquark approximation in the framework of Buchm\" uller-Tye potential model, we investigate the spectroscopy of doubly charmed baryons: $Ξ_{cc}^{++}$ and $Ξ_{cc}^{+}$. Our results include the masses, parameters of radial wave functions of states with the different excitations of both diquark and light quark-diquark system. We calculate the values of fine and hyperfine splittings of these levels and discuss some new features, connected to the identity of heavy quarks, in the dynamics of hadronic and radiative transitions between the states of these baryons.
The spectrum of baryons with two heavy quarks is predicted, assuming a configuration of a light quark and a heavy diquark. The masses are computed within a semirelativistic quark model, using a potential obtained in the framework of the AdS/QCD correspondence. All the parameters defining the model are determined fitting the meson spectrum. The obtained mass of Ξ_{cc} is in agreement with the measurements.
The doubly heavy baryon $Ξ_{QQ^{\prime}}$ ($Q^{(\prime)}$ = $b$ or $c$) is different from the ordinary baryons. The production of the doubly heavy baryon can provide valuable insight on how the colored $(QQ^{\prime})$-diquark can be transformed into the color-singlet baryon. As a sequential work of Ref.[2], we make a further study on the doubly heavy baryon production through the $e^+ e^-$ annihilation channel, $e^{+} + e^{-}\rightarrowγ/Z^0 \rightarrow Ξ_{QQ^{\prime}} +\bar{Q} +\bar{Q^{\prime}}$, within the nonrelativistic QCD framework. In addition to the total cross sections, we present the…
Quark Model Study of Doubly Heavy $Ξ$ and $Ω$ Baryons via Deep Neural Network and Hybrid Optimization
本研究采用超中心夸克模型，结合深度学习与粒子群优化求解六维薛定谔方程，预测了双重重子的质量谱与半轻子衰变宽度，为实验寻找未发现的重子态提供了重要理论信息。
Spectra of masses are calculated for the families of doubly heavy baryons in the framework of nonrelativistic quark model with the QCD potential by Buchmuller--Tye. We suppose the quark-diquark structure for the wave functions and take into account the spin-dependent splittings. The physical reasons causing the existence of quazi-stable excited states in the subsystem of heavy diquark are considered for the heavy quarks of identical flavors.
In present work we discuss the production of heavy quarkonia and diquarks in $pp$-interactions. The aim is to take into account the production of bound states of quarks originating from independent parton scatterings. This production is regulated by constraints on the invariant mass of constituents. Such an approach leads to larger value of the diquark production cross section than traditional consideration in single parton scattering. This production mechanism of doubly heavy baryons can sooner be verified by modern experiments (e.g. LHCb). We also discuss related contributions to double and …
We analyze the effects of hyperfine mixing in $b \to c$ semileptonic decays of doubly heavy baryons. We qualitatively confirm the results by W. Roberts and M. Pervin in Int. J. Mod. Phys. A, 2009, {\bf 24}: 2401-2413, finding that mixing has a great impact on those transitions. However, predictions without mixing differ by a factor of 2 and this discrepancy translates to the mixed case where large differences in decay widths are observed between the two calculations.
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