SU(3) Symmetry-Breaking Parameters

• SU(3) symmetry-breaking parameters quantify the deviations in hadronic observables caused by the nondegeneracy of light quark masses.
• Effective field theories and global fits use spurions and Clebsch–Gordan coefficients to parameterize corrections, typically yielding 20–30% modifications.
• Precise characterization of these parameters refines extraction of quark distribution functions and guides the development of symmetry-breaking potentials in BSM models.

Flavor SU(3) symmetry-breaking parameters are central to the theoretical and phenomenological analysis of hadronic and electroweak processes involving the lightest three quark flavors (up, down, strange). Precise characterization and quantification of SU(3) violation is essential in the extraction of fundamental quantities, modeling of baryon and meson dynamics, and the construction of realistic symmetry-breaking potentials in extended gauge and family symmetry scenarios.

1. Conceptual Framework for SU(3) Symmetry Breaking

Flavor SU(3) symmetry is an approximate symmetry of QCD under the interchange of u, d, and s quarks. Its breaking arises primarily due to the nondegeneracy of the light quark masses, with $m_s \gg m_{u,d}$. This explicit breaking modifies hadron spectra, coupling constants, and matrix elements, requiring careful parameterization in both effective field theory and data-driven analyses.

The canonical parameterization introduces SU(3)-breaking spurions proportional to the quark mass matrix, leading linear corrections in $m_s-m_{u,d}$ for most hadronic observables. In collective or soliton models, the breaking Hamiltonian $H_{\rm sb}$ is expressed as

$$H_{\rm sb} = (m_s-\bar m) [\alpha D^{(8)}_{88} + \beta Y + \frac{\gamma}{\sqrt{3}} D^{(8)}_{8i} J_i] + (\text{isospin breaking terms}),$$

where $D^{(8)}_{ab}$ are octet Wigner D-functions and $J_i$ are collective spin operators. The parameters $\alpha, \beta, \gamma$ are model-dependent but can be fit to baryon masses, magnetic moments, or decay observables (Yang et al., 2015).

In flavor-changing processes, SU(3) breaking enters both in operator matrix elements and in wavefunction mixings between different flavor multiplets. Importantly, certain quantities, such as the vector current form factors at zero momentum transfer, are protected against first-order breaking by the Ademollo–Gatto theorem.

2. Quantitative Parameterizations in Phenomenological Fits

In global fits to polarized parton distribution functions (PDFs), SU(3) symmetry underpins the axial-vector current sum rules relating nonsinglet first moments $a_3$ and $a_8$ to the $F$ and $m_s-m_{u,d}$0 couplings:

$m_s-m_{u,d}$1

SU(3) breaking is parameterized by allowing $m_s-m_{u,d}$2 to deviate from its symmetry-limit value (e.g., $m_s-m_{u,d}$3 from hyperon decays). In (Khorramian et al., 2020), three scenarios are considered: (A) exact SU(3) ($m_s-m_{u,d}$4), (B) moderate breaking ($m_s-m_{u,d}$5), and (C) large explicit breaking with $m_s-m_{u,d}$6 fit freely. The degree of breaking (expressed as a fractional reduction $m_s-m_{u,d}$7) controls the extraction of the polarized strange quark density, with changes in $m_s-m_{u,d}$8 shifting the first moment $m_s-m_{u,d}$9 by factors up to $H_{\rm sb}$0 (Khorramian et al., 2020).

3. Dynamical SU(3)-Breaking Parameters in Effective Models

In chiral soliton and collective quantization models, symmetry breaking is encapsulated by operator expansions in $H_{\rm sb}$1 and a set of dynamical parameters $H_{\rm sb}$2:

• $H_{\rm sb}$3: SU(3)-symmetric leading and $H_{\rm sb}$4 corrections.
• $H_{\rm sb}$5: linear $H_{\rm sb}$6 contributions parameterizing explicit SU(3) breaking.

The axial-vector current operator is decomposed as

$H_{\rm sb}$7

All $H_{\rm sb}$8 are directly fitted to data (hyperon decay constants and singlet axial charge) via linear algebra with the matrix of SU(3) Clebsch–Gordan coefficients. The breaking coefficients typically yield 20–30% corrections relative to the symmetric pieces, though in some channels (notably $H_{\rm sb}$9 couplings and $$H_{\rm sb} = (m_s-\bar m) [\alpha D^{(8)}_{88} + \beta Y + \frac{\gamma}{\sqrt{3}} D^{(8)}_{8i} J_i] + (\text{isospin breaking terms}),$$0 baryons) the breaking can be comparable to or larger than the SU(3)-symmetric part (Yang et al., 2018, Yang et al., 2015).

Explicit values (dimensionless) from (Yang et al., 2018): | $$H_{\rm sb} = (m_s-\bar m) [\alpha D^{(8)}_{88} + \beta Y + \frac{\gamma}{\sqrt{3}} D^{(8)}_{8i} J_i] + (\text{isospin breaking terms}),$$1 | $$H_{\rm sb} = (m_s-\bar m) [\alpha D^{(8)}_{88} + \beta Y + \frac{\gamma}{\sqrt{3}} D^{(8)}_{8i} J_i] + (\text{isospin breaking terms}),$$2 | $$H_{\rm sb} = (m_s-\bar m) [\alpha D^{(8)}_{88} + \beta Y + \frac{\gamma}{\sqrt{3}} D^{(8)}_{8i} J_i] + (\text{isospin breaking terms}),$$3 | $$H_{\rm sb} = (m_s-\bar m) [\alpha D^{(8)}_{88} + \beta Y + \frac{\gamma}{\sqrt{3}} D^{(8)}_{8i} J_i] + (\text{isospin breaking terms}),$$4 | $$H_{\rm sb} = (m_s-\bar m) [\alpha D^{(8)}_{88} + \beta Y + \frac{\gamma}{\sqrt{3}} D^{(8)}_{8i} J_i] + (\text{isospin breaking terms}),$$5 | $$H_{\rm sb} = (m_s-\bar m) [\alpha D^{(8)}_{88} + \beta Y + \frac{\gamma}{\sqrt{3}} D^{(8)}_{8i} J_i] + (\text{isospin breaking terms}),$$6 | |--------|--------|--------|--------|--------|--------| | -3.509 | +3.437 | +0.604 | -1.213 | +0.479 | -0.735 |

4. SU(3) Breaking in Hadronic Observables

The quantitative impact of SU(3)-breaking parameters is observable in baryon decay constants, form factors, and coupling constants. For instance, in hyperon semileptonic decays, the ratios $$H_{\rm sb} = (m_s-\bar m) [\alpha D^{(8)}_{88} + \beta Y + \frac{\gamma}{\sqrt{3}} D^{(8)}_{8i} J_i] + (\text{isospin breaking terms}),$$7 obey Gell-Mann–Okubo type relations in the SU(3) limit. Empirically, deviations in the range 5–11% are extracted by comparing measured form factor ratios to their symmetric predictions, with breaking parameters $$H_{\rm sb} = (m_s-\bar m) [\alpha D^{(8)}_{88} + \beta Y + \frac{\gamma}{\sqrt{3}} D^{(8)}_{8i} J_i] + (\text{isospin breaking terms}),$$8 defined as

$$H_{\rm sb} = (m_s-\bar m) [\alpha D^{(8)}_{88} + \beta Y + \frac{\gamma}{\sqrt{3}} D^{(8)}_{8i} J_i] + (\text{isospin breaking terms}),$$9

Precise fits yield, for example, $D^{(8)}_{ab}$0 in the range $D^{(8)}_{ab}$1 to $D^{(8)}_{ab}$2, and $D^{(8)}_{ab}$3 from $D^{(8)}_{ab}$4 to $D^{(8)}_{ab}$5 (Pham, 2012, Pham, 2014). A nontrivial feature is the Ademollo–Gatto protection of the vector current $D^{(8)}_{ab}$6 against first-order breaking, rendering the axial sector more sensitive to the detailed parameterization.

In QCD sum rule analyses of $D^{(8)}_{ab}$7 couplings, leading SU(3) breaking from $D^{(8)}_{ab}$8 mass, $D^{(8)}_{ab}$9-quark loop effects, and $J_i$0 contributes a 21% decrease:

$J_i$1

demonstrating that even for strongly-coupled matrix elements, SU(3)-breaking corrections are substantial (Singh et al., 2010).

5. SU(3) Symmetry-Breaking in Beyond Standard Model Potentials

Extension to greater symmetry structures (e.g., $J_i$2 or trinification) introduces additional symmetry-breaking parameters controlling vacuum alignments and phase transitions. For instance, in $J_i$3 supersymmetric trinification, adjoint and fundamental scalars acquire VEVs breaking the gauge and family symmetries at hierarchically separated scales. The relevant breaking parameters include mass terms ($J_i$4), cubic and quartic couplings ($J_i$5, A-terms), as well as soft SUSY-breaking terms. Vacuum expectation values such as $J_i$6 and $J_i$7 determine the stepwise symmetry breaking chain (Camargo-Molina et al., 2017).

Dynamical misalignment between Yukawa spurions in quark-flavor symmetry models is similarly controlled by explicit mass terms, determinant couplings, and non-hermitian trilinear terms, producing the observed quark mass and mixing hierarchies without introducing large ad hoc hierarchies in the symmetry-breaking parameters (Fong et al., 2013, Nardi, 2015).

6. Symmetry-Breaking Potentials and Vacuum Structure

For minimal $J_i$8 or discrete subgroup breaking, the complete scalar potential is constructed from all allowed invariant operators. In the bifundamental scalar context, the potential

$J_i$9

leads to phase diagrams delineated by critical values of $\alpha, \beta, \gamma$0 that select between fully symmetric, diagonally-broken, or $\alpha, \beta, \gamma$1-preserving vacua (Bai et al., 2017). The values for these parameters define phase boundaries and vacuum expectation values where SU(3) is spontaneously broken to residual subgroups or discrete non-Abelian symmetries (e.g., $\alpha, \beta, \gamma$2, $\alpha, \beta, \gamma$3, $\alpha, \beta, \gamma$4) as studied in (Luhn, 2011).

7. Summary Table: Representative SU(3)-Breaking Parameters

Sector/Process	Parameterization	Typical Size/Range	Reference
Octet axial charges	$\alpha, \beta, \gamma$5	$\alpha, \beta, \gamma$6 (SU(3) limit), $\alpha, \beta, \gamma$7 (20% breaking), fit: $\alpha, \beta, \gamma$8 (45% breaking)	(Khorramian et al., 2020)
Chiral soliton axial couplings	$\alpha, \beta, \gamma$9 in $a_3$0	$a_3$1: $a_3$2 20–30% of $a_3$3	(Yang et al., 2018, Yang et al., 2015)
Hyperon $a_3$4 ratios	$a_3$5 in $a_3$6	$a_3$7–$a_3$8\%	(Pham, 2012, Pham, 2014)
$a_3$9 coupling	$a_8$0	$a_8$1	(Singh et al., 2010)
$a_8$2 symmetry-broken vevs	$a_8$3, misalignment couplings	CP phase, mixings, mass ratios: $a_8$4	(Fong et al., 2013, Nardi, 2015)
Trinification (adjoint vevs)	$a_8$5 etc.	$a_8$6 GeV; $a_8$7 similar	(Camargo-Molina et al., 2017)

These parameters, and their precise determination or bounding from experiment and lattice calculations, are essential for accurate phenomenological predictions and for the discrimination among models of flavor and symmetry breaking in QCD and beyond.