Gell-Mann–Okubo Mass Relations

• Gell-Mann–Okubo mass relations are parameter-independent sum rules derived from SU(3) flavor symmetry breaking in QCD that accurately relate baryon and meson masses.
• They feature linear and equal-spacing formulas for different multiplets and can incorporate higher-order, electromagnetic, and isospin corrections to refine mass predictions.
• Data-driven techniques like lattice QCD, QCD sum rules, and symbolic regression confirm GMO predictions and support extensions such as q-deformations and parity doubling.

The Gell-Mann–Okubo (GMO) mass relations are parameter-independent sum rules for baryon and meson multiplets arising from the breaking of SU(3) flavor symmetry in quantum chromodynamics (QCD). Originally formulated for the baryon octet and decuplet, these relations connect the masses of different members in a multiplet with remarkable precision and serve as a powerful diagnostic of the underlying symmetry structure and symmetry-breaking mechanisms in strong interaction physics.

1. Foundational Principles and Derivation

The classical GMO relation for the baryon octet arises from first-order SU(3) flavor symmetry breaking, where the quark mass differences ($m_s \neq m_u = m_d$) act as the dominant perturbation. The baryons are organized into SU(3) multiplets, and the most general mass operator at this order is constructed from flavor singlet and octet operators, yielding the formula

$$M = M_0 + a\,Y + b\Bigl[I(I+1) - \tfrac{1}{4}Y^2\Bigr]$$

where $M_0$ is an SU(3)-symmetric mass, $Y$ is the hypercharge, and $I$ is the isospin (Wagner, 2021). Eliminating the parameters, the empirical relation between octet masses reads:

$2M_N + 2M_\Xi = 3M_\Lambda + M_\Sigma$

or

$$\frac{1}{2}(M_N+M_\Xi) = \frac{3}{4}M_\Lambda + \frac{1}{4}M_\Sigma$$

(Morpurgo, 2010, Kryshen, 2011).

For the decuplet, where all states are symmetric, only a single flavor-breaking parameter appears, leading to the equal-spacing rule:

$$M_{\Sigma^*} - M_\Delta = M_{\Xi^*} - M_{\Sigma^*} = M_\Omega - M_{\Xi^*}$$

(Fernando et al., 2014, Gresnigt, 2016).

2. Extensions: Higher-Order Breaking and Generalization

The first-order GMO formula can be extended to include second-order SU(3)-breaking, electromagnetic corrections, and isospin breaking. Morpurgo’s general parameterization method yields the most general octet mass operator up to second order (Morpurgo, 2010):

$$M_B = M_0 + A\,T_3(B) + B\,Y(B) + C[T_3^2(B) - \tfrac{1}{4}Y^2(B)] + D\,Y^2(B) + E\,T_3(B)\,Y(B)$$

The non-vanishing second-order coefficients ($C,D,E$) generate small empirical deviations ($T \sim 5.7$ MeV) from the exact classical GMO sum rule. The corrected mass relation for the octet becomes:

$$M_\Lambda + \frac{1}{2}(M_\Sigma + M_N) = \frac{1}{2}(M_\Xi + M_N) + T$$

with $$T = (M_{\Xi^*} - M_\Xi) + (M_{\Sigma^*} - M_\Sigma)$$ (Morpurgo, 2010).

Generalizations also incorporate heavy-quark multiplets and any three-flavor baryon system. The mass operator for a generic ijk-flavor octet is (Beaudoin et al., 2013):

$$M(I^{ij}, I_z^{ij}, n_k) = a_0^{ijk} + a_1^{ijk} n_k + a_2^{ijk} [I^{ij}(I^{ij}+1) - \tfrac{1}{4} n_k^2] - a_3^{ijk} I_z^{ij}$$

This formalism quantifies symmetry quality and accurately predicts masses of $\Lambda$-like and $\Sigma$-like baryons for any flavor combination.

3. Lattice QCD, QCD Sum Rules, and Data-Driven Discovery

Ab initio lattice QCD calculations implement the GMO relations for baryon masses extracted from correlators, using static mass operator bases classified under SU(6)×O(3) irreps (Fernando et al., 2014). The fit to lattice spectra at pion masses up to 702 MeV verifies that the GMO and equal-spacing relations are satisfied to within a few percent, with deviations suppressed as $\mathcal{O}(\epsilon^2, \epsilon/N_c)$, where $\epsilon = m_s - \hat{m}$.

QCD sum rules confirm that both vacuum and in-medium baryon masses satisfy the GMO formulas at linear order in SU(3)-breaking condensates, with higher-order corrections suppressed by the truncation of the operator product expansion and density (Kryshen, 2011).

Symbolic regression via Kolmogorov-Arnold Networks (KANs) has been applied to baryon mass data, autonomously rediscovers the classical GMO relations and extracts SU(3)-breaking parameters directly (He et al., 4 Jan 2026). The extracted coefficients, e.g., for the octet $a=-188.26$ MeV (hypercharge slope), $b=+39.14$ MeV (isospin), match conventional fits and indicate that data-driven approaches can reproduce analytic symmetry-based mass formulas with MeV-level precision.

4. Quantum Group Deformations, Parity Doubling, and Further Phenomenology

Quantum group deformations substitute the classical SU(3) flavor symmetry with its $q$-deformed counterpart SU$_q$(3), producing new mass formulas with up to 20-fold improved empirical accuracy (Gresnigt, 2016). For the octet and decuplet, explicit $q$-deformed relations incorporate electromagnetic corrections and are matched to specific roots of unity ($q = e^{i\pi/7}$ for octet, $q = e^{i\pi/21}$ for decuplet). A direct connection between deformation parameter $q$, baryon spin parity $J^P$, and the Cabibbo angle is found:

$\theta_C = -i J^P \ln q$

suggesting that quantum-group deformation captures physics associated with flavor mixing.

Parity-doubling models with SU(3)×SU(3) chiral symmetry yield linear mass relations for positive and negative parity states distinctly, preserving the GMO and equal-spacing rules for each parity sector as the order parameters ($\sigma_q, \sigma_s$) evolve, e.g., under temperature (Sasaki, 2017).

5. Linear vs. Quadratic GMO Relations: Application and Controversies

Classically, GMO relations for baryons are linear in masses, while meson multiplets prefer quadratic relations. Wagner (Wagner, 2021) demonstrates, however, that both linear and quadratic relations follow from first-order SU(3) breaking and apply to baryons and mesons alike, refuting the conventional spin-based distinction. Exceptions, e.g., for the pseudoscalar meson octet or charm/bottom multiplets, arise from large symmetry breaking or heavy-quark dynamics. Both variants satisfy multiplet masses to the expected accuracy, with experimental fits supporting the assignment of resonance states and prediction of missing baryons.

The incorporation of electromagnetic and isospin-breaking corrections into GMO relations is essential as precision reaches sub-percent levels. Charge-specific deformed relations constructed to cancel EM shifts further enhance accuracy (Gresnigt, 2016).

6. Practical Impact and Computational Summary

Precision baryon mass relations derived from symmetry-based arguments, confirmed by lattice QCD, QCD sum rules, and symbolic regression, establish the organizational principle for the hadron spectrum. The accuracy of these relations—few MeV for octet and decuplet, down to 0.02% with $q$-deformation—makes them indispensable for hadron classification, mass prediction of yet unseen states, and constraining phenomenological models (Fernando et al., 2014, Gresnigt, 2016, He et al., 4 Jan 2026, Beaudoin et al., 2013, Morpurgo, 2010).

The universality and generalizability of GMO-type relations, including generalized isospin and quantum-group extensions, enable systematic mass predictions in multiplets ranging from light-flavor hadrons through charm and bottom baryons, with quantified symmetry violation measures ($Q_\mathrm{rel}<0.22$ for octets) (Beaudoin et al., 2013).

Continued development of data-driven symbolic regression, lattice QCD computations, and group-theoretical generalization promises further refinement of mass formulas and deeper insight into the structure of QCD symmetry breaking.

7. Table: Standard GMO Mass Relations

Multiplet	Relation	Precision
Octet	$2M_N + 2M_\Xi = 3M_\Lambda + M_\Sigma$	$\lesssim$ few MeV
Decuplet	$M_{\Sigma^*} - M_\Delta = M_{\Xi^*} - M_{\Sigma^*}$	$\lesssim$ few MeV
Generalized	$M_{ijk}(I^{ij},...)$ formalism	3–50 MeV
$q$-Deformed	Explicit $q$-parameterized formulas	0.02–0.08%

The table summarizes key relations; the precision reflects empirical fit quality from group-theoretical, quantum-group, and generalized approaches.

• Group-theoretical derivations, lattice and sum rule tests: (Fernando et al., 2014, Kryshen, 2011, Beaudoin et al., 2013).
• Quantum group deformation and charge-specific refinement: (Gresnigt, 2016).
• Symbolic regression and algebraic extraction: (He et al., 4 Jan 2026).
• Parity-doubling chiral symmetry models: (Sasaki, 2017).
• Higher-order corrections and general parameterization: (Morpurgo, 2010).
• Linear and quadratic mass relations, heavy-quark multiplets: (Wagner, 2021).