Mass relations in heavy hadrons from Jensen-like inequalities

Abstract

We demonstrate that mass inequalities for hadrons with one or more heavy quarks arise primarily from the concavity of binding energies in the quark model, reflecting short-range Coulombic interactions and long-range confinement. Empirical two-body bindings $B_{i\bar{j}}$ are extracted from spin-averaged meson masses, ensuring model independence and direct use of experimental data. Fitting these as functions of reduced mass $\mu_{ij}$ reveals a critical confinement scale of 1.34~fm where bindings turn positive. The concave $B(1/\mu)$ justifies Jensen-like inequalities under flavor permutation, reproducing relations like $m_{x\bar{y}} > \frac{1}{2}(m_{x\bar{x}} + m_{y\bar{y}})$ and baryon analogs, including $m_{xyz} > \frac{1}{3}(m_{xxx} + m_{yyy} + m_{zzz})$. Hadron mass decomposition validates $\Delta M_{\textrm{EXP}} \approx \Delta B + \Delta C$ with $\sigma \sim 2.07$~MeV for mesons and baryons. Promoting inequalities to equalities, we predict masses for unobserved heavy baryons (e.g., $M(\Omega_{b}^{\ast})=6076.6\,$MeV, $M(\Xi_{cc}^{\ast})=3703.6\,$MeV and $M(\Omega_{cc}^{\ast})=3802.4\,$MeV) and identify favored quark-exchange scattering channels.