Pseudo-Nambu-Goldstone Bosons (pNGBs)

Updated 11 December 2025

Pseudo-Nambu-Goldstone bosons (pNGBs) are light scalar excitations emerging from the spontaneous breaking of an approximate global symmetry with a small explicit breaking that gives them mass.
They exhibit universal relations in effective field theories, where low-energy couplings and derivative interactions are determined by the underlying coset geometry independent of UV details.
pNGBs play pivotal roles in composite Higgs models, dark matter, dark energy, and cosmological phase transitions, with experimental probes spanning collider, astrophysical, and laboratory searches.

A pseudo-Nambu-Goldstone boson (pNGB) is a light scalar excitation resulting from the spontaneous breaking of an approximate global symmetry, with a small explicit breaking term that gives it a nonzero mass. pNGBs are structurally distinguished from exact Nambu-Goldstone bosons by their explicit mass and their role as IR remnants of broken symmetries, often subject to universal low-energy relations determined by their coset geometry and invariance properties, rather than UV model details. They play central roles in theories of physics beyond the Standard Model, including composite Higgs scenarios, dark matter, dark energy (quintessence), flavor physics, and cosmological phase transitions.

1. Symmetry Breaking, Coset Structure, and pNGB Dynamics

The essential framework for pNGBs originates from the spontaneous breaking of a global continuous symmetry group $G$ down to a subgroup $H$ at a high scale $f$ , such that the vacuum manifold is the coset $G/H$ . The resulting fields $\pi^a$ parameterize this coset nonlinearly:

$U(x) = \exp\left( \frac{i}{f}\pi^a(x) X^a \right),$

where $X^a$ are the broken generators. Under $G$ , $U(x)$ transforms with a compensating $h \in H$ ensuring the physical fields’ nonlinear transformation properties.

The low-energy EFT at leading order is the two-derivative chiral Lagrangian,

$\mathcal{L}^{(2)} = \frac{f^2}{4} \operatorname{Tr} \left[ (D_\mu U)^\dagger (D^\mu U) \right]$

with the appropriate gauging in place (e.g., for the electroweak subgroup in the composite Higgs case).

A small explicit breaking -- e.g., via spurion terms, incomplete representations, or explicit mass parameters -- lifts the pNGBs above zero mass. The resulting mass and potential terms are controlled both by the structure of the explicit breaking and the underlying UV completion (Liu et al., 2018, Koutroulis et al., 2023, Gangopadhyay et al., 2022).

2. Universal Relations and Effective Couplings

pNGBs exhibit universal relationships between couplings and self-interactions at low energies, tightly dictated by their origin in the coset structure. Once the decay constant $f$ (or equivalently, the IR misalignment parameter $\xi = v^2 / f^2$ ) is fixed by a measured coupling, all shift-symmetric (gauge sector) interactions are determined, independent of the full UV coset $G/H$ -- this is the principle of "universal nonlinearity."

For the minimal SO(4)→SO(3) composite Higgs, this universality leads to explicit predictions for on-shell gauge and triple gauge couplings: $\mathcal{L}^{(2)} \supset \frac{1}{2} \partial_\mu h \partial^\mu h + \frac{g^2 f^2}{4} \sin^2\left(\theta + \frac{h}{f}\right) W^+_\mu W^{-\mu} + \frac{g^2 f^2}{8\cos^2\theta_W} \sin^2\left(\theta + \frac{h}{f}\right) Z_\mu Z^\mu$ with measured couplings expressed as (Liu et al., 2018): $g_{hWW} = g m_W \sqrt{1-\xi}, \quad g_{hhWW} = \frac{g^2}{2}(1-2\xi)$ and related (model-independent) structure for triple gauge couplings and higher-dimension operators at ${\cal O}(p^4)$ .

Universal relations also control the ratios of higher-order form factors (for $hVV$ , $hhVV$ operators, etc.). For example,

$\frac{C_3^{2h}}{C_3^h} = \frac{1}{2} \cos\theta,$

where $C_i^h$ and $C_i^{2h}$ are the coefficients of dimension-6 and -8 $hVV$ and $hhVV$ operators, respectively.

These predictions are robust under integrating out heavy resonances; only shift-symmetry-violating effects -- from the Higgs potential, Yukawa couplings, or other explicit breakings -- can modify them, and only in non-gauge sectors (Liu et al., 2018).

3. Phenomenology and Experimental Probes

pNGBs arising in various sectors yield diverse signatures and constraints:

Composite Higgs: Experimental tests focus on precise measurements of single- and double-Higgs couplings ( $g_{hWW}$ , $g_{hZZ}$ , $g_{hhWW}$ , $g_{hhZZ}$ ), triple gauge couplings ( $\delta g_1^Z$ , $\delta \kappa_\gamma$ , $\delta \kappa_Z$ ), and momentum-dependent form factors in vector-boson scattering and diboson production. Deviation from universal relations would falsify the simplest pNGB Higgs scenarios (Liu et al., 2018).
Dark Matter: pNGBs as dark matter candidates arise in hidden sectors (technicolor, non-Abelian gauge extensions, composite sectors) with relic density set either by thermal freeze-out, out-of-equilibrium production (freeze-in), or asymmetric mechanisms. Direct-detection cross sections are typically suppressed due to derivative couplings, leading to vanishing non-relativistic scattering at tree level (protected by the original shift symmetry) (Belyaev et al., 2010, Otsuka et al., 2022, Abe et al., 2021, Sheikh et al., 28 Apr 2025, Abe et al., 2020, Haisch et al., 2021). Indirect detection limits, collider signatures (missing energy channels, production in association with SM particles), and relic density matching constrain viable parameter space.
Dark Energy (Quintessence): Canonical pNGB models with cosine-type potentials require $f \gtrsim M_\mathrm{P}$ to achieve slow roll and $w\approx -1$ , but introducing Galileon-type kinetic terms or structurally richer CW potentials enables sub-Planckian $f$ and generic initial misalignment angle, maintaining observational viability (Adak et al., 2014, Gangopadhyay et al., 2022).
Cosmological Phase Transitions: pNGBs with nontrivial vacuum structure in SO(N+1)→SO(N) theories experience additional sub- $f$ vacuum transitions (thermal or supercooled). While thermal transitions are typically very weak (order parameter jump, transition strength parameter $\alpha \lesssim 10^{-3}$ ), supercooled symmetry-restoring transitions can be strongly first-order, producing gravitational wave backgrounds observable by future interferometers if explicit symmetry-breaking $\epsilon$ and $f$ are in suitable ranges (Koutroulis et al., 2023).
Flavor Sector: In non-Abelian flavor models, pNGBs with unsuppressed flavor-violating derivative couplings result in rare flavor-changing decays. Constraints from processes such as $K\to\pi\pi'$ , $\mu\to e\pi'$ or $K-\bar K$ mixing often require symmetry breaking scales $v_\phi\gtrsim 10^{11}$ – $10^{12}$ GeV for light pNGBs, surpassing direct collider and astrophysical bounds (Calibbi et al., 13 Nov 2025).
Laboratory Searches for Light pNGBs: Stimulated photon-photon scattering in high-intensity laser experiments provides a direct probe for pNGBs with photon couplings $g_{a\gamma\gamma}$ down to $10^{-15}\;\mathrm{GeV}^{-1}$ in the sub-keV mass range, competitive with and complementary to astrophysical and cosmological limits (Homma et al., 2017).

4. Cosmological and Astrophysical Implications

Thermal relic pNGBs, especially those with two-photon or weak couplings (axion-like), are subject to stringent cosmological and astrophysical constraints. Key phenomena and observables:

Thermal Production and Decay: pNGBs in thermal equilibrium efficiently decouple via freeze-out once Primakoff or similar processes become inefficient. Their late decay to photons modifies the effective number of neutrino species $N_\mathrm{eff}$ , induces entropy injection, distorts the CMB ( $\mu$ , $y$ distortions), affects Big Bang Nucleosynthesis yields, and contributes to the diffuse photon background.
Parameter Space Exclusion: The comprehensive exclusion regions in the ( $m_a,g_{a\gamma\gamma}$ ) plane are set by N_eff, BBN (deuterium, $^4$ He), CMB distortions, late-time photon fluxes, and overclosure. For example, late decaying pNGBs after neutrino decoupling are excluded for $N_\mathrm{eff}<2.11$ (99% C.L.), BBN and CMB observations exclude much of the parameter space for lifetimes and masses corresponding to $\tau \gtrsim 10^2$ – $10^{13}$ s (Cadamuro et al., 2011).
Non-Thermal and Freeze-In Production: If pNGB-SM couplings are extremely small (freeze-in regime), pNGBs never thermalize, and their abundance can result from out-of-equilibrium decays, scatterings, or directly from inflaton decay. In unified models, the radial symmetry-breaking scalar can simultaneously be the inflaton; the pNGB abundance is then tied to both freeze-in and inflaton decay yields, with compatibility with observed $\Omega_\mathrm{DM}$ imposing $m_\chi$ below a few GeV (freeze-in) or $<\mathcal{O}(100)$ GeV (inflaton decay) in the allowed parameter space (Abe et al., 2020, Kaneta et al., 13 Jun 2024).
Dark Radiation and $\Delta N_\mathrm{eff}$ : Direct non-thermal production of light pNGBs during reheating predicts a nonzero contribution to dark radiation ( $\Delta N_\mathrm{eff}$ ), testable by future CMB experiments. The yield is calculable in terms of the inflaton-pNGB coupling, decay constant, and reheating temperature (Kaneta et al., 13 Jun 2024).

5. Model Realizations and Parameter Dependence

Various structural choices for the symmetry breaking and explicit breaking determine the detailed mass spectrum, stability, and couplings of the pNGBs:

Soft breaking terms: The mass of the pNGB is typically directly proportional to the strength of the explicit breaking parameter (e.g., $m_\chi^2\propto \mu_c$ in models with cubic scalar terms or $m_{12}^2$ in $\mathbb{Z}_3$ -stabilized dark sectors) (Abe et al., 2020, Sheikh et al., 28 Apr 2025).
Residual discrete symmetries: Exact stability at renormalizable level may be enforced by residual symmetries (e.g., $\mathbb{Z}_3$ , $\mathbb{Z}_2$ ), often surviving the spontaneous and explicit symmetry breaking pattern, as in triplet complex-scalar models (Sheikh et al., 28 Apr 2025).
Mixing and portal couplings: pNGB couplings to SM Higgs and gauge bosons can arise via mixing with singlet scalars, Higgs portal terms, or derivative interactions. Their structure is highly constrained by the requirement to preserve the shift symmetry in the massless limit, leading to amplitude suppression in the non-relativistic limit.
Embedding in Grand Unified Theories: Complete UV frameworks, e.g., SO(10) unification, fix the gauge couplings and symmetry breaking scales, yielding narrow windows of viable parameter space consistent with relic, direct/indirect detection, and stability bounds (Abe et al., 2021).

6. Theoretical Consistency, EFT Bounds, and Experimental Outlook

For trustable EFTs describing pNGBs, the explicit breaking parameter $\epsilon$ must be sufficiently small for the effective potential and its radiative corrections to remain under control. One-loop analyses establish concrete upper bounds on $|\epsilon|$ in terms of the decay constant and the UV cutoff (e.g., $|\epsilon| \ll \epsilon_{\max}$ proportional to $f^2 / M^2$ ) (Koutroulis et al., 2023).

Future directions for pNGB searches and theory include:

Collider measurements: Next-generation LHC/e $^+$ e $^-$ machines will further constrain universal relations of composite Higgs sectors, rare decays, and missing energy channels for pNGB DM (Liu et al., 2018, Haisch et al., 2021).
Laboratory and laser searches: Experiments exploiting stimulated photon-photon scattering extend the reach to ultra-weakly coupled pNGBs in the sub-keV mass range, probing regions inaccessible to astrophysical searches (Homma et al., 2017).
CMB and gravitational wave observatories: Measurements of $N_\mathrm{eff}$ and searches for stochastic gravitational wave backgrounds from strong supercooled pNGB phase transitions will probe currently open parameter space in cosmology (Koutroulis et al., 2023, Kaneta et al., 13 Jun 2024).
Flavor and rare decay experiments: Low-energy flavor tests (rare $K$ , $B$ decays, $\mu\to e$ processes) are expected to constrain the flavor-breaking scales associated with pNGBs to well above the direct collider and astrophysical reach (Calibbi et al., 13 Nov 2025).