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Register a new userClebsch-Gordan Construction of Lattice Interpolating Fields for Excited Baryons
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Large sets of baryon interpolating field operators are developed for use in lattice QCD studies of baryons with zero momentum. Operators are classified according to the double-valued irreducible representations of the octahedral group. At first, three-quark smeared, local operators are constructed for each isospin and strangeness and they are classified according to their symmetry with respect to exchange of Dirac indices. Nonlocal baryon operators are formulated in a second step as direct products of the spinor structures of smeared, local operators together with gauge-covariant lattice displacements of one or more of the smeared quark fields. Linear combinations of direct products of spinorial and spatial irreducible representations are then formed with appropriate Clebsch-Gordan coefficients of the octahedral group. The construction attempts to maintain maximal overlap with the continuum SU(2) group in order to provide a physically interpretable basis. Nonlocal operators provide direct couplings to states that have nonzero orbital angular momentum.
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We express each Clebsch-Gordan (CG) coefficient of a discrete group as a product of a CG coefficient of its subgroup and a factor, which we call an embedding factor. With an appropriate definition, such factors are fixed up to phase ambiguities. Particularly, they are invariant under basis transformations of irreducible representations of both the group and its subgroup. We then impose on the embedding factors constraints, which relate them to their counterparts under complex conjugate and therefore restrict the phases of embedding factors. In some cases, the phase ambiguities are reduced to sign ambiguities. We describe the procedure of obtaining embedding factors and then calculate CG coefficients of the group mathcal{PSL}_{2}left(7right) in terms of embedding factors of its subgroups S_{4} and mathcal{T}_{7}.
We report in this article three- and four-term recursion relations for Clebsch-Gordan coefficients of the quantum algebras $U_q(su_2)$ and $U_q(su_{1,1})$. These relations were obtained by exploiting the complementarity of three quantum algebras in a $q$-deformation of $sp(8, gr)$.
We present the ground and excited state spectra of singly, doubly and triply-charmed baryons by using dynamical lattice QCD. A large set of baryonic operators that respect the symmetries of the lattice and are obtained after subduction from their continuum analogues are utilized. These operators transform as irreducible representations of SU(3)$_F$ symmetry for flavour, SU(4) symmetry for Dirac spins of quarks and O(3) symmetry for orbital angular momenta. Using novel computational techniques correlation functions of these operators are generated and the variational method is exploited to extract excited states. The lattice spectra that we obtain have baryonic states with well-defined total spins up to 7/2 and the low lying states remarkably resemble the expectations of quantum numbers from SU(6)$otimes$O(3) symmetry.
The Clebsch-Gordan coefficients of the group SU(2) are shown to satisfy new inequalities. They are obtained using the properties of Shannon and Tsallis entropies. The inequalities associated with the Wigner 3-j symbols are obtained using the relation of Clebsch-Gordan coefficients with probability distributions interpreted either as distributions for composite systems or distributions for noncomposite systems. The new inequalities were found for Hahn polynomials and hypergeometric functions
We investigate the $D_{s0}^ast(2317)$ meson using lattice QCD and considering correlation functions of several $bar{c} s$ two-quark and $bar{c} s (bar{u} u + bar{d} d)$ four-quark interpolating fields. These interpolating fields generate different structures in color, spin and position space including quark-antiquark pairs, tetraquarks and two-meson scattering states. For our computation we use an ensemble simulated with pion mass $m_pi approx 0.296 , textrm{GeV}$ and spatial volume of extent $2.90 , textrm{fm}$. We find in addition to the expected spectrum of two-meson scattering states another state around $60 , textrm{MeV}$ below the $D K$ threshold, which we interpret as the $D_{s0}^ast(2317)$ meson. This state couples predominantly to a quark-antiquark interpolating field and only weakly to a $D K$ two-meson interpolating field. The coupling to the tetraquark interpolating fields is essentially zero, rendering a tetraquark interpretation of the $D_{s0}^ast(2317)$ meson rather unlikely. Moreover, we perform a scattering analysis using Luschers method and the effective range approximation to determine the $D_{s0}^ast(2317)$ mass for infinite spatial volume. We find this mass $51 , textrm{MeV}$ below the $D K$ threshold, rather close to both our finite volume result and the experimentally observed value.
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