Organize heavy-hadron masses by the light degrees of freedom

Write the heavy-quark momentum as its mass times a timelike unit velocity plus a residual momentum of hadronic size. Factoring out the rapid mass-dependent phase leaves a slowly varying effective field. At leading order the action depends on its velocity and color charge but not its spin orientation or heavy flavor. The corresponding spin-flavor symmetry is an emergent low-energy property, not an exact symmetry rotating unequal quark masses in the full QCD Lagrangian.

The light degrees of freedom have angular momentum j l independent of the decoupled heavy spin one half. For nonzero j l, coupling those angular momenta gives a doublet with total spins j l minus one half and j l plus one half. The ground-state heavy-light mesons have light angular momentum one half, yielding a pseudoscalar and a vector. Their degeneracy becomes exact only at infinite heavy mass; finite chromomagnetic interactions split them. A light cloud with zero angular momentum instead has no such pair of distinct total spins.

The heavy-meson mass is its heavy-quark mass plus a light-cloud energy, followed by inverse-mass corrections. A conventional kinetic matrix element lambda one and a chromomagnetic parameter lambda two distinguish the pseudoscalar and vector expressions. The latter appears with coefficients minus three and plus one. Weighting the vector mass by its three spin polarizations and the pseudoscalar mass by one cancels this chromomagnetic contribution. It leaves the light-cloud and kinetic terms, rather than eliminating all nonperturbative physics.

The hyperfine mass difference is twice lambda two divided by the heavy mass at this leading order, up to short-distance logarithmic matching. More carefully, the chromomagnetic Wilson coefficient and its matrix element have compensating scale dependence. Heavy-quark mass schemes and the definition of the light-cloud parameter must also be coordinated; an isolated pole-mass interpretation has a hadronic-size ambiguity. The simple numerical model below absorbs the chosen leading coefficient into lambda two and ignores its logarithmic change between heavy flavors.

To compare heavy flavors, the light cloud should have matched flavor and angular quantum numbers and the heavy masses should be large enough for the expansion. Bottom typically offers a better hierarchy than charm, but the size of corrections must be assessed for each observable. Heavy-flavor symmetry also constrains transition form factors at fixed velocity transfer. Their special zero-recoil normalizations and corrections require separate current matching; a mass-doublet argument alone does not determine every semileptonic decay rate.

Heavy quarkonium contains two heavy constituents and additional velocity-dependent scales, so it generally uses a nonrelativistic expansion rather than simply copying the single-heavy light-cloud formula. The top quark decays before ordinary hadronization and does not populate analogous stable top mesons. Lattice calculations can determine many matrix elements but need continuum, finite-volume and heavy-mass systematics. Large-N counting supplies a different expansion and does not eliminate these tasks at three colors. The wave lab illustrates cancellation of weighted terms, without being a hadron spectrum solver.