Chiral Symmetry Breaking Minima

• Chiral symmetry breaking minima are distinct vacuum states in effective potentials where chiral symmetry is spontaneously broken, determining the system's phase behavior.
• Analyzed using stationary conditions, FRG, and Coleman–Weinberg methods, these minima reveal key insights into degeneracy, stability, and first-order phase transitions.
• Insights extend to applications in gauge theories, supersymmetric models, and condensed matter systems, elucidating mechanisms behind massless modes and anomaly-driven transitions.

Chiral symmetry breaking minima refer to the distinct vacuum configurations of effective potentials in field-theoretic models with chiral (handedness-based) symmetry, where spontaneous symmetry breaking (SSB) occurs. In strongly interacting field theories, condensed matter systems, and effective models (e.g., linear sigma models, Nambu–Jona-Lasinio (NJL), and supersymmetric gauge theories), the classification, stability, and global structure of these minima directly dictate the dynamical realization of chiral symmetry breaking and the nature of associated phase transitions. A detailed analysis of these minima reveals insights into the ground state structure, massless Nambu–Goldstone spectrum, metastability, and the underlying mechanisms of SSB for various classes of symmetries and interactions.

1. Chiral Symmetry and Effective Potentials

Chiral symmetry in field theories, typically of the form $U_L(n)\times U_R(n)$, $SU(N_f)_L\times SU(N_f)_R$, or their variants, constrains the form of the scalar sector and dictates invariance properties of the effective potential. The minimization of the vacuum effective potential $V(\phi)$ (where $\phi$ denotes the relevant multiplet) yields isolated or continuous sets of minima, each characterized by its unbroken symmetry subgroup. The nature and multiplicity of these minima depend crucially on the structure of the potential, the representations, and any explicit symmetry breaking or anomaly-induced terms present in the Lagrangian.

In the $U_L(n)\times U_R(n)$ meson model, the basic field $M(x)$ is an $n\times n$ complex matrix built from real scalar and pseudoscalar multiplets, and the potential is constructed from invariants such as $I_1 = \mathrm{Tr}(MM^\dagger)$ and $I_2 = \mathrm{Tr}(MM^\dagger MM^\dagger)$ (Fejos, 2012). For $SU(3)_L\times SU(3)_R$, the potential may also feature trilinear (determinant) terms, leading to additional cubic invariants and, for $n=3$, to a richer structure of extrema (Bai et al., 2017).

2. Structure and Classification of Minima

The classification of chiral symmetry breaking minima is determined by the stationary conditions for the effective potential, leading to a set of coupled polynomial equations for the vacuum expectation values (VEVs) of the condensates along the invariant directions. In the $U_L(n)\times U_R(n)$ model, homogeneous condensate ansatzes such as $\langle M\rangle = v_0 T^0 + v_8 T^8$ yield cubic stationarity equations that have multiple solutions, corresponding to physically inequivalent vacua (Fejos, 2012).

A summary classification for the $U_L(n)\times U_R(n)$ model includes:

Minima label	$(v_0, v_8)$ Structure	Symmetry breaking pattern	Stability
[I]	$(0, 0)$	No SSB (trivial maximum)	Maximum
[II]	$(\sqrt{-m^2 n^2/(g_1+g_2)},0)$	$U_L(n)\times U_R(n)\to U_V(n)$	Minimum
[III]	$$((n-2)\sqrt{-m^2/(g_1+g_2)},\ 2\sqrt{-(n-1)m^2/(g_1+g_2)})$$	$U_L(n)\times U_R(n)\to U_V(n)$	Minimum
[IV]/[V]	other analytic roots	saddle points	Saddle

Key features include exact degeneracy between minima [II] and [III] due to discrete axial symmetries, as well as the property that all minima related by $U_A(n)$ transformations commute with $MM^\dagger$, ensuring identical vacuum energy and excitation spectra (Fejos, 2012).

For $SU(3)_L\times SU(3)_R$-symmetric models with trilinear terms, only three types of minima arise at tree level: (i) the trivial vacuum ($\langle\Phi\rangle=0$), (ii) the diagonal vacuum preserving $SU(3)_{\text{diag}}$, and (iii) an $SU(2)\times SU(2)\times U(1)$-preserving vacuum. The latter are separated by first-order phase boundaries, and the global structure depends on quartic and cubic couplings (Bai et al., 2017).

3. Discrete Symmetries and Degeneracy Structure

Discrete axial symmetry transformations can relate apparently different minima. In the $U_L(n)\times U_R(n)$ model, acting with elements $A\in U_A(n)$ on the standard vacuum generates new minima via $$\langle M\rangle \to A^\dagger \langle M\rangle A^\dagger$$. The requirement that the transformed vacuum remain diagonal and unitary constrains the group of possible vacua to $2^n$ discrete minima, all sharing the same residual vector symmetry and vacuum energy (Fejos, 2012). These degeneracies are not lifted, even under inclusion of quantum fluctuations, as proven using functional renormalization group methods—since the spectrum depends only on the invariants of $MM^\dagger$, all such vacua are physically equivalent in energy and spectrum.

4. Quantum Corrections and Functional/Loop Methods

The stability and degeneracy of the chiral symmetry breaking minima under quantum corrections have been extensively analyzed using loop and nonperturbative functional methods. The functional renormalization group (FRG) analysis demonstrates that the one-loop effective action, when evaluated on the $U_V(n)$-invariant (and their discrete copies) vacua, retains all classical degeneracies, as the effective potential and spectrum are functions only of $MM^\dagger\propto \mathbf{1}$ on each minimum (Fejos, 2012).

In models such as the orientifold Seiberg duality scenario, the one-loop Coleman–Weinberg potential for meson fields $M$ admits a unique chiral-breaking minimum with a VEV proportional to the identity, breaking $SU(N_f)_L \times SU(N_f)_R \to SU(N_f)_V$ and generating $N_f^2$ Nambu–Goldstone bosons (Armoni, 2013). The stability of this minimum is ensured by positivity of the Hessian in all directions, and quantum corrections only provide small shifts.

Supersymmetric QCD (SQCD) and $Sp(N_c)$ gauge theories perturbed by small anomaly-mediated SUSY breaking (AMSB) exhibit rich structures of chiral symmetry breaking minima. For $N_f < N_c$ (SQCD) and all calculable regimes of $Sp(N_c)$, minimization of the combined SUSY and AMSB potentials selects unique, stable chiral-symmetry-breaking vacua breaking $SU(N_f)_L\times SU(N_f)_R \to SU(N_f)_V$ and $SU(2N_f)\to Sp(2N_f)$, respectively (Luzio et al., 2022, Varier et al., 3 Dec 2025).

5. Anomaly-Driven and Multicomponent Minima

In chiral effective models such as the linear sigma model and the NJL model, the inclusion of a $U_A(1)$-breaking term (e.g., a cubic 't Hooft determinant interaction) can independently drive chiral symmetry breaking, even if the quadratic mass term is positive. The tree-level potential takes the form

$$V(\sigma_0) = \frac12\mu^2 \sigma_0^2 + \frac14\left(\frac{\lambda}{3}+\lambda'\right)\sigma_0^4 -\frac23 B\sigma_0^3,$$

with the cubic anomaly term inducing a pair of nontrivial minima. When the anomaly is sufficiently strong ($B^2 > \mu^2(\lambda/3+\lambda')$), the “anomaly-driven” minimum becomes the global minimum, resulting in spontaneous breaking of $SU(3)_L\times SU(3)_R$ with a characteristic mass relation $m_{\sigma_0}^2 < \tfrac13 m_{\eta_0}^2$ (Kono et al., 2019). A key phenomenological consequence is that, in the anomaly-driven scenario, the lightest scalar ($\sigma_0$) remains sub-800 MeV.

This mechanism gives rise to a double-minimum structure, with the potential supporting both symmetric ($\sigma_0=0$) and broken ($\sigma_0\neq0$) vacua, indicating a strong first-order transition at finite temperature or chemical potential (Kono et al., 2019).

6. Alternative and Runaway Minima in Extended Models

Analysis of vacua in extended frameworks such as s-confining and baryonic branches of SQCD, as well as global-symmetry bifundamental scalar models, reveals the possible existence of metastable or runaway minima. For $N_f\geq N_c$ in SQCD with tree-level AMSB, minimization can lead to vacua breaking baryon number or with unbounded “runaway” directions, where the vacuum energy is lower than in the QCD-like baryon-preserving minimum (Luzio et al., 2022). In $Sp(N_c)$ gauge theories, the absence of a baryonic branch ensures the uniqueness and stability of the mesonic chiral-breaking minimum (Varier et al., 3 Dec 2025).

Models with $(SU(3)\times SU(3))$ symmetry and a bifundamental scalar present a phase diagram with minima preserving either $SU(3)_\mathrm{diag}$ or $SU(2)\times SU(2)\times U(1)$, depending on the parameters of quartic and trilinear couplings. Both symmetry breaking patterns can coexist as local minima, separated by first-order transitions, with existence dictated by analytic criteria in coupling space (Bai et al., 2017).

7. Chiral Minima in Condensed Matter and Frustrated Magnet Systems

In certain condensed matter settings, such as fully polarized frustrated magnets, the free energy can develop multiple minima associated with different magnon valley occupations due to frustration-created band degeneracies. Thermal fluctuations, combined with strong inter-valley magnon-magnon repulsion, can spontaneously break the discrete “chiral” symmetry by favoring occupation of one momentum valley, selecting a chiral minimum and generating a nonzero expectation value for vector chirality operators such as $S_m\times S_n^z$ (Ueda, 2014). The selection of chiral minima in such systems is strongly temperature and field dependent, resulting in a thermally induced Ising-like chiral phase.

• "Chiral symmetry breaking patterns in the $U_L(n)\times U_R(n)$ meson model" (Fejos, 2012)
• "Minimal $SU(3)\times SU(3)$ symmetry breaking patterns" (Bai et al., 2017)
• "A Note on Seiberg Duality and Chiral Symmetry Breaking" (Armoni, 2013)
• "On the Derivation of Chiral Symmetry Breaking in QCD-like Theories and S-confining Theories" (Luzio et al., 2022)
• "AMSB in $Sp(N_c)$ Gauge Theories" (Varier et al., 3 Dec 2025)
• "Role of U$_{A}$(1) breaking term in dynamical chiral symmetry breaking of chiral effective theories" (Kono et al., 2019)
• "Spontaneous chiral symmetry breaking on a fully polarized frustrated magnet at finite temperature" (Ueda, 2014)