Clebsch-Gordan construction of lattice interpolated fields for excited baryons

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.

Figure 1

Young tableaux for three-quark $SU(2{)}_I$ irreducible representations. The first tableau is antisymmetric in labels of particles 1 and 2 (denoted MA for mixed-antisymmetric), while the second tableau is symmetric in the labels of particles 1 and 2 (denoted MS for mixed-symmetric). The third tableau is fully symmetric (denoted S).

Figure 2

Young tableaux for three Dirac indices.

Figure 3

Illustration of two-link displacements.