Mass relations in heavy hadrons from Jensen-like inequalities

TL;DR

This paper derives mass inequalities for heavy hadrons based on the concavity of binding energies, using empirical data to validate and predict hadron masses with Jensen-like inequalities.

Contribution

It introduces a model-independent approach linking binding energy concavity to mass inequalities and provides new mass predictions for unobserved heavy baryons.

Findings

Empirical binding energies exhibit concavity in reduced mass.

Jensen-like inequalities reproduce known mass relations.

Predicted masses for unobserved heavy baryons are provided.

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.