Dark QCD and Chiral Symmetry Breaking

• Dark QCD is a class of QCD-like gauge theories with hidden sectors where chiral symmetry breaking arises from strong, nonperturbative dynamics.
• Effective models such as the NJL framework and gap-equation approaches accurately capture the dynamics of χSB, influencing order parameters and cosmological signals.
• External dark electromagnetic fields modulate the chiral phase transition's order, impacting phenomena like gravitational wave production and dark matter relic abundance.

Dark QCD denotes a class of QCD-like gauge theories hypothesized as hidden sectors, typically based on $SU(N_d)$ groups with $N_{f_d}$ dark-quark flavors. Central to dark QCD is the mechanism of chiral symmetry breaking (χSB), echoing visible QCD, and a rich interplay with effective field theory, topological structures, and cosmology. The fate of the chiral phase transition—especially its order and critical properties—has implications ranging from dark matter phenomenology to gravitational wave production. Chiral symmetry breaking arises from strong dynamics, is encoded in various non-perturbative phenomena, and can be modified by extensions such as external (dark) fields or additional interactions.

1. Theoretical Foundations of Chiral Symmetry Breaking in Dark QCD

Chiral symmetry in QCD-like gauge theories emerges due to the near-masslessness of constituent fermions. For a generic $SU(N_d)$ dark sector gauge group with $N_{f_d}$ massless dark-quark flavors, the classical Lagrangian exhibits $$SU(N_{f_d})_L \times SU(N_{f_d})_R \times U(1)_V \times U(1)_A$$ symmetry. Quantum anomalies break $U(1)_A$, while the non-abelian chiral symmetry is believed to spontaneously break as

$$SU(N_{f_d})_L \times SU(N_{f_d})_R \rightarrow SU(N_{f_d})_V,$$

accompanied by a non-zero condensate $$\langle \bar\psi\psi \rangle \sim \Lambda_\text{dark}^3$$ (Dvali, 2017).

The topological and anomaly structure, as analyzed via the vacuum topological susceptibility $$\chi_\text{top}^{\text{dark}} \sim \Lambda_\text{dark}^4$$, determines the mass spectrum and χSB pattern. Non-vanishing $\chi_\text{top}$ implies dynamical breaking of chiral symmetry even without explicit reference to confinement, as demonstrated in three-form reformulations of the QCD vacuum (Dvali, 2017).

Gap-equation approaches, including Schwinger–Dyson analysis and effective four-fermion models, confirm that chiral symmetry breaking is intimately linked to the nonperturbative IR dynamics of the gauge theory (Doff et al., 2011). In all such constructions, the scale of χSB (and the associated pseudo-Nambu–Goldstone bosons, or dark pions) is set by $\Lambda_\text{dark}$, up to group-theoretic coefficients and the proximity to the conformal window.

2. Nambu–Jona-Lasinio Models and Order Parameters

The Nambu–Jona-Lasinio (NJL) model, extended to three or more flavors, provides an effective description of χSB in dark QCD, capturing both the spontaneous breaking driven by strong dynamics and the influence of explicit symmetry breaking terms (Wang et al., 2022, Aoki et al., 2021). The mean-field NJL Lagrangian for three dark-quark flavors with a $U(1)_D$ dark photon reads

$$\mathcal{L} = \bar\psi (i\gamma^\mu D_\mu - \mathbf{m})\psi + G\sum_{a=0}^8 \left[ (\bar\psi\lambda^a\psi)^2 + (\bar\psi i\gamma_5\lambda^a\psi)^2 \right] - K \left[\det\bar\psi(1+\gamma_5)\psi + \det\bar\psi(1-\gamma_5)\psi \right],$$

with

• $D_\mu = \partial_\mu - i e Q A_\mu$, $Q = \text{diag}(q_f)$,
• $G$ four-fermion coupling,
• $K$ ’t Hooft determinant coupling (accounts for axial anomaly).

The dynamical masses $M_f$ and condensates $\phi_f = \langle \bar q_f q_f \rangle$ are governed by gap equations and an effective thermodynamic potential $\Omega(\{\phi_f\}; T, eB)$. The critical order-parameter is the condensate or, equivalently, the constituent quark mass at zero momentum (Wang et al., 2022, Aoki et al., 2021).

Condensate-to-mass and pion-decay constant relations—for example, the Gell-Mann–Oakes–Renner relation for dark pions ($m_\pi^2 f_\pi^2 = -m_q\langle\bar\psi\psi\rangle$)—carry over directly, with all scales mapped to the $\Lambda_\text{dark}$ regime (Aoki et al., 2021).

3. Role of Confinement, Effective Propagators, and the Gap Equation

The interlinked phenomena of confinement and χSB are formalized in nonperturbative gap-equation frameworks. The Cornwall–Machado–Natale approach introduces an effective IR confining propagator

$$D_c^{\mu\nu}(k) = \delta^{\mu\nu} \frac{1}{(k^2 + m^2)^2},$$

which—at $m\to 0$—yields the linear rising potential $V(r)\sim r$. The zero-momentum dynamical mass $M(0)$ and condensate are derived from the gap equation

$M(p^2) = M_c(p^2) + M_{1g}(p^2),$

where $M_c$ encodes the confining contribution and $M_{1g}$ the one-gluon exchange with dynamical gauge boson mass (Doff et al., 2011). In the deep IR, the gap equation reduces to an effective four-fermion interaction with coupling $G_\text{eff} \simeq \frac{2K_F}{\pi m^2}$.

Chiral symmetry breaking thus occurs when the confining kernel dominates at small $k^2$, and a critical value of $m$ exists above which no non-trivial solution for $M(p^2)$ appears. For typical QCD parameters, $m\sim 200$ MeV constitutes the upper threshold. Translating to dark QCD involves replacing the scale parameters with those of the dark sector, with the result that $\langle \bar\psi\psi \rangle_\text{dark}$, $f_{\pi,\text{dark}}$, and $M(0)_\text{dark}$ all scale with $\Lambda_\text{dark}$. No significant separation between confinement and χSB scales is expected unless $N_{f_d}$ approaches the conformal window (Doff et al., 2011).

4. Topological Susceptibility, Anomalies, and the Spectrum

The topological structure of the pure-gauge dark QCD vacuum, quantified by the topological susceptibility $\chi_\text{top}^{\text{dark}}$, underpins χSB and the mass spectrum, independent of explicit confinement dynamics (Dvali, 2017). Using a three-form gauge theory formulation, introducing $N_{f_d}$ massless fermions causes a chiral condensate and the associated breaking:

$$\langle \bar\psi_{R,j}\psi_{L,i}\rangle = v \delta_{ij}, \qquad \langle \det(\psi_L\psi_R) \rangle \propto e^{i\eta/f} \neq 0,$$

where $\eta$ is the would-be $U(1)_A$ Goldstone boson. The anomaly structure (axial and mixed gravitational) dictates that only the maximal anomaly-free subgroup remains unbroken, and the $\eta'$-like pseudoscalar receives a mass:

$$m_{\eta'}^2 f_{\eta'}^2 = 2N_{f_d} \chi_\text{top}^{\text{dark}}.$$

Spectator field methods confirm that all massless composite fermions are eliminated in the IR—mirroring the familiar Witten–Veneziano construction for visible QCD (Dvali, 2017). These results generalize to gravitational analogues and establish a group-theoretic and topological origin for χSB in any dark QCD scenario.

5. Influence of External “Magnetic” Fields on the Chiral Phase Transition

When dark quarks couple to an external $U(1)_D$ background—effectively a “dark-photon” magnetic field—the order of the chiral phase transition is modified (Wang et al., 2022). Within the NJL framework, two central effects operate:

• Magnetic catalysis: The presence of $|eB| > 0$ enhances the chiral condensate, deepening the broken-phase minimum and delaying chiral restoration.
• Scale anomaly-induced “tadpole”: At one loop, a new term proportional to $-(eB)\sigma$ appears in the effective potential due to the electromagnetic scale anomaly, favoring nonzero condensates even above the expected critical temperature.

In massless three-flavor dark QCD, the ’t Hooft determinant induces a $-\sigma^3$ term (driving a robust first-order transition). As $eB$ increases,

• The discontinuity in the order parameter $M_u(T)$ at $T_c$ (the critical temperature) diminishes.
• When $eB \gtrsim 2f_\pi^2$, the first-order barrier vanishes and the transition becomes a smooth crossover.

The phase boundary in the $(m_s/f_\pi, eB/f_\pi^2)$ “extended Columbia plot” confirms the first-order domain shrinks continuously and is destroyed above $eB/f_\pi^2 \approx 2$ for $m_s \rightarrow 0$ (Wang et al., 2022).

Effect	Mechanism	Impact on Transition
Magnetic catalysis	Quadratic deepening via $\Delta \sigma^2$	Restoration delayed
Tadpole (anomaly)	Linear $-(eB)\,\sigma$ term	Suppresses second minimum
’t Hooft determinant	$-\sigma^3$ cubic term	Favors first order

6. Cosmological and Phenomenological Implications

A first-order chiral transition in dark QCD drives out-of-equilibrium phenomena during cosmological history:

• Gravitational wave generation: Bubble nucleation and latent heat release in a first-order transition can source detectable gravitational waves. Suppressing the first-order nature reduces the latent heat $L/T_c^4$ and thus weakens the gravitational wave amplitude; for $eB/f_\pi^2 = 1.8$, $L/T_c^4$ is reduced by $\sim 15\%$ (Wang et al., 2022).
• Baryogenesis: Efficacy of out-of-equilibrium baryogenesis mediated by the dark sector’s condensate dynamics is tied to the strength of the transition and is curtailed as the transition softens.
• Constraints on magnetogenesis: If early-universe magnetogenesis produces a dark-photon background with $eB > 2f_\pi^2$, it can prevent a strong first-order transition, constraining scenarios that require sizable primordial dark magnetic fields.

In theories where the dark sector couples to cosmology via a real scalar (as inflaton), χSB in the hidden QCD sector can generate the Planck scale, the right-handed neutrino scale (and thus accommodate radiative neutrino mass generation), and provide ultra-heavy dark pions as dark matter candidates (Aoki et al., 2021). The relic abundance and stability of such dark pions result from the symmetry and production mechanisms detailed in NJL effective models.

7. Summary and Outlook

Dark QCD and its chiral symmetry breaking phenomena are governed by robust theoretical structures: chiral anomalies, topological susceptibility, and strong coupling gap dynamics. The order of the chiral phase transition—and hence the cosmological and astrophysical signatures—can be dramatically modified by interaction with dark electromagnetic backgrounds. Spectral properties and dynamical condensate formation closely parallel visible QCD, with all scales set by $\Lambda_\text{dark}$. NJL and gap-equation approaches, as well as anomaly-based topological methods, converge on a consistent picture: chiral symmetry breaking in dark QCD is a generic and calculable phenomenon, intertwined with early-universe dynamics and observable signatures such as dark matter relic abundance and gravitational waves (Doff et al., 2011, Dvali, 2017, Aoki et al., 2021, Wang et al., 2022).