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

Abstract

We compare three definitions of ${\Xi }_c$--${\Xi }_c^{\prime }$ mixing within a constituent-quark model: 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 two operator-based definitions give $2.48$--$2.58^{\circ }$; a second parameter set gives $2.38$--$2.49^{\circ }$. The operator-based angles are closer to the lattice determinations $1.2(1{)}^{\circ }$ and $0.810(50)(200{)}^{\circ }$, but remain larger. We then couple conventional $qsQ$ states to virtual $qsQg$ configurations, where $q=u,d$ and $Q=c,b$. An orbital-$S$-wave 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 normalized to reference short-range mass shifts. The hybrid probability stays below $1.7\%$, and the corresponding bottom angles are $0.42$--$0.48^{\circ }$. Eliminating the hybrid sector gives a rank-one mass correction precisely when all weighted transitions select the same retained-state combination. The mode-resolved calculation separates the flavor-breaking response that changes the mixing angle from the response that generates a second transition direction. The accompanying effective-current relation connects this mass analysis to radiative and semileptonic amplitudes through their current matrix elements.

arXiv comment16 pages, 3 figures; under revision for Phys. Rev. D