Mass and decay angles in the ${\Xi }_c$--${\Xi }_c^{\prime }$ mixing tension: Virtual hybrids and a single-direction criterion

Abstract

We study the relation between mixing angles obtained from mass calculations and those inferred from decay analyses of the ${\Xi }_c$--${\Xi }_c^{\prime }$ system. Within a constituent-quark model, we compare three definitions: a spin-component probability angle, a common-source overlap angle, and an SU(3)-breaking mass-operator angle. The probability angle is $3.11^{\circ }$, while the latter two definitions give $2.48$--$2.58^{\circ }$; a second parameter set gives $2.38$--$2.49^{\circ }$. The change of definition reduces, but does not remove, the difference from the lattice values $1.2(1{)}^{\circ }$ and $0.810(50)(200{)}^{\circ }$. We then include virtual $qsQg$ states coupled to the conventional $qsQ$ states, where $q=u,d$ and $Q=c,b$. In the tested orbital-$S$-wave model, a chromomagnetic transition containing both $j_{Qg}=1/2$ and $3/2$ raises the source and mass-operator charm angles to $2.67$--$3.16^{\circ }$ when its strength is normalized to reference short-range mass shifts. The hybrid probability remains below $1.7\%$, and the corresponding bottom angles are $0.42$--$0.48^{\circ }$. We also give a single-direction condition for the induced mass correction to be proportional to a single rank-one projector. This condition concerns the mass operator, not a universal angle for decay amplitudes. Eliminating the hybrid sector changes the current operator as well as the Hamiltonian; we derive this relation but do not calculate process-dependent decay angles. The results include projected hyperfine coefficients and energy-cut sums for comparisons in specified continuum and lattice matching schemes.

Comments: 16 pages, 3 figures; under revision for Phys. Rev. D