Quadratically Coupled ULDM Phenomenology

• Quadratically coupled ultralight dark matter is defined by a scalar field with φ² interactions that oscillates coherently, inducing shifts in fundamental constants.
• Its quadratic couplings result in a density-dependent effective mass in matter, leading to screening effects that modify signals in both terrestrial and orbital experiments.
• Theoretical frameworks using twin protection mitigate radiative corrections, paving the way for experimental searches via atomic clocks, interferometers, and space-based tests.

Quadratically coupled ultralight dark matter denotes a class of models in which a real scalar or pseudo-Nambu–Goldstone boson with $m_\phi \ll \mathrm{eV}$ constitutes dark matter and couples to Standard Model operators through $\phi^2$, rather than through a term linear in $\phi$. In the local halo, such a field is well described as a coherently oscillating classical wave. Its quadratic interactions induce shifts in fundamental constants, composition-dependent forces sourced by the dark-matter background, and a density-dependent effective mass in matter. The topic now spans low-energy dilaton-like effective theories, shift-symmetric dimension-8 constructions, environmental screening analyses, and twin-protected pNGB models designed to make experimentally interesting couplings technically natural (Banerjee et al., 2022, Delaunay et al., 16 Jul 2025).

1. Effective description and operator structure

A standard low-energy parametrization writes the scalar as a real field $\phi$ with quadratic couplings to photons, gluons, and fermion mass operators. In the Damour–Donoghue notation used in force and equivalence-principle analyses, the interaction takes the form

$$\mathcal{L} \supseteq \frac{1}{2}(\partial\phi)^2 - \frac{1}{2} m^2 \phi^2 + \frac{(\kappa \phi)^2}{2} \Bigg[ \frac{d_e^{(2)}}{4} F_{\mu\nu} F^{\mu\nu} - \frac{d_g^{(2)} \beta_3}{2 g_3} F^A_{\mu\nu} F^{A\,\mu\nu} - \sum_{f=u,d,e} (d_{m_f}^{(2)}+\gamma_{m_f} d_g^{(2)})\, m_f \bar\psi_f \psi_f \Bigg] + \mathcal{L}_{\rm SM},$$

with $\kappa=\sqrt{4\pi G}$ and dimensionless quadratic “dilaton coefficients” $d_i^{(2)}$ (Burrage et al., 27 May 2026). Equivalent descriptions appear in the varying-constants literature, where the same operators are normalized by $M_{\rm Pl}^{-2}$ and interpreted as $\phi^2$-dependent shifts of $\alpha_{\rm EM}$, $\phi^2$0, and fermion masses (Banerjee et al., 2022).

These couplings imply

$\phi^2$1

so any observable with sensitivity to these parameters inherits a $\phi^2$2 dependence (Burrage et al., 27 May 2026). Because the halo field behaves approximately as $\phi^2$3, quadratic couplings generate both a DC component and an oscillatory component at frequency $\phi^2$4, rather than $\phi^2$5 as in linearly coupled models (Banerjee et al., 2022).

A technically distinct realization imposes an exact shift symmetry in the interaction sector. In that case, the leading CP-even operators are derivative and first appear at dimension 8, schematically through $\phi^2$6 contracted with fermion and photon bilinears. This produces both oscillations of fundamental constants and Lorentz-violating spin-2 backgrounds, but the direct terrestrial bounds on such operators are comparatively weak, corresponding to a UV cutoff scale of keV order (Jiang et al., 2024). This derivative, shift-symmetric branch is conceptually separate from the non-derivative dilaton-like quadratic couplings that dominate the present phenomenology.

2. Propagation in matter, screening, and time-dependent profiles

A defining feature of quadratically coupled scalar dark matter is that ordinary matter modifies the scalar’s effective mass. In the universal-density approximation one may write

$\phi^2$7

so that

$\phi^2$8

In the composition-dependent treatment used for experimental analyses, this becomes

$\phi^2$9

with $\phi$0 a linear combination of dilaton charges and quadratic couplings (Burrage et al., 27 May 2026). For $\phi$1, the field is suppressed in dense matter; for sufficiently negative coupling one encounters tachyonic behavior and sourcing rather than screening (Delaunay et al., 16 Jul 2025).

The resulting scalar profile around macroscopic bodies is not Yukawa-like in the usual sense, because one is distorting an already present coherent background rather than sourcing a new vacuum field. For a uniform sphere, one finds an exterior solution with a $\phi$2 distortion of the ambient oscillating field, while inside the object the field is reduced by the induced mass barrier (Burrage et al., 2024). In the strong-coupling regime, the suppression becomes surface-dominated: only a thin shell effectively participates in the exterior profile.

Environmental structures can therefore become part of the detector. Spherical and cylindrical cavity calculations show that a vacuum chamber, cavity wall, or satellite hull can strongly suppress both the scalar amplitude and its gradient in the interior once $\phi$3. For $\phi$4, the suppression inside a cavity is exponential in the wall thickness times the in-medium wave number, so signals proportional to $\phi$5 or $\phi$6 can be reduced by many orders of magnitude (Burrage et al., 22 Jul 2025). This materially changes the interpretation of strong-coupling limits from experiments performed inside dense enclosures.

Time dependence around macroscopic objects is also nontrivial. For a spherical source that appears, disappears, or changes radius, the field relaxes toward the stationary configuration with a late-time deviation that falls as $\phi$7, after an onset time $\phi$8 at radius $\phi$9 (Burrage et al., 2024). This analysis resolves the apparent divergences that arise when one compares different infinite-volume stationary solutions: the divergent energy differences correspond to taking an infinite-time limit, whereas energies, pressures, and forces in any finite region remain finite once causality is respected (Burrage et al., 2024).

At higher masses, matter effects remain relevant even when oscillation-based searches lose bandwidth. A scattering-theory treatment of the repulsive quadratic scalar-photon interaction shows that, for $\phi$0, the phenomenology is controlled by a matter-induced potential barrier, producing both a DM-wind scattering force and a background-induced force between bodies. In the non-perturbative region with large incident momentum, a descreening effect appears and alleviates decoherence suppression (Gan et al., 15 Apr 2025).

3. Naturalness, twin protection, and radiative self-interactions

Quadratic couplings are not automatically natural. In a pNGB model without additional structure, an operator such as $\phi$1 still generates a radiative mass correction linear in the quadratic coupling, so ultralight masses require either tiny couplings or a very low cutoff (Delaunay et al., 16 Jul 2025). This is the “Goldstone naturalness” problem emphasized in recent model-building work.

A proposed resolution is the “quadratic twin” mechanism. In that construction, the ULDM field is a pNGB contained in a complex scalar multiplet $\phi$2, with quadratic couplings to both the Standard Model and a twin copy of the Standard Model related by a mirror $\phi$3. The explicit-breaking operators are arranged as

$\phi$4

with identical couplings in the two sectors (Delaunay et al., 16 Jul 2025). Because the mirror symmetry makes the leading radiative correction proportional to $\phi$5, the linear-in-coupling contribution is $\phi$6-invariant and does not generate a pNGB mass. The first non-vanishing correction appears at quadratic order: $\phi$7 or, more generally, with the same $\phi$8 suppression and an extra factor $\phi$9 (Delaunay et al., 16 Jul 2025). For $$\mathcal{L} \supseteq \frac{1}{2}(\partial\phi)^2 - \frac{1}{2} m^2 \phi^2 + \frac{(\kappa \phi)^2}{2} \Bigg[ \frac{d_e^{(2)}}{4} F_{\mu\nu} F^{\mu\nu} - \frac{d_g^{(2)} \beta_3}{2 g_3} F^A_{\mu\nu} F^{A\,\mu\nu} - \sum_{f=u,d,e} (d_{m_f}^{(2)}+\gamma_{m_f} d_g^{(2)})\, m_f \bar\psi_f \psi_f \Bigg] + \mathcal{L}_{\rm SM},$$0, $$\mathcal{L} \supseteq \frac{1}{2}(\partial\phi)^2 - \frac{1}{2} m^2 \phi^2 + \frac{(\kappa \phi)^2}{2} \Bigg[ \frac{d_e^{(2)}}{4} F_{\mu\nu} F^{\mu\nu} - \frac{d_g^{(2)} \beta_3}{2 g_3} F^A_{\mu\nu} F^{A\,\mu\nu} - \sum_{f=u,d,e} (d_{m_f}^{(2)}+\gamma_{m_f} d_g^{(2)})\, m_f \bar\psi_f \psi_f \Bigg] + \mathcal{L}_{\rm SM},$$1, and $$\mathcal{L} \supseteq \frac{1}{2}(\partial\phi)^2 - \frac{1}{2} m^2 \phi^2 + \frac{(\kappa \phi)^2}{2} \Bigg[ \frac{d_e^{(2)}}{4} F_{\mu\nu} F^{\mu\nu} - \frac{d_g^{(2)} \beta_3}{2 g_3} F^A_{\mu\nu} F^{A\,\mu\nu} - \sum_{f=u,d,e} (d_{m_f}^{(2)}+\gamma_{m_f} d_g^{(2)})\, m_f \bar\psi_f \psi_f \Bigg] + \mathcal{L}_{\rm SM},$$2, this opens about $$\mathcal{L} \supseteq \frac{1}{2}(\partial\phi)^2 - \frac{1}{2} m^2 \phi^2 + \frac{(\kappa \phi)^2}{2} \Bigg[ \frac{d_e^{(2)}}{4} F_{\mu\nu} F^{\mu\nu} - \frac{d_g^{(2)} \beta_3}{2 g_3} F^A_{\mu\nu} F^{A\,\mu\nu} - \sum_{f=u,d,e} (d_{m_f}^{(2)}+\gamma_{m_f} d_g^{(2)})\, m_f \bar\psi_f \psi_f \Bigg] + \mathcal{L}_{\rm SM},$$3 orders of magnitude in coupling space relative to the untwinned pNGB case (Delaunay et al., 16 Jul 2025).

That improvement does not eliminate every theoretical tension. Matter couplings unavoidably induce scalar self-interactions through loops. For quadratic couplings to fermions,

$$\mathcal{L} \supseteq \frac{1}{2}(\partial\phi)^2 - \frac{1}{2} m^2 \phi^2 + \frac{(\kappa \phi)^2}{2} \Bigg[ \frac{d_e^{(2)}}{4} F_{\mu\nu} F^{\mu\nu} - \frac{d_g^{(2)} \beta_3}{2 g_3} F^A_{\mu\nu} F^{A\,\mu\nu} - \sum_{f=u,d,e} (d_{m_f}^{(2)}+\gamma_{m_f} d_g^{(2)})\, m_f \bar\psi_f \psi_f \Bigg] + \mathcal{L}_{\rm SM},$$4

so cosmological bounds on the effective quartic self-coupling map into bounds on $$\mathcal{L} \supseteq \frac{1}{2}(\partial\phi)^2 - \frac{1}{2} m^2 \phi^2 + \frac{(\kappa \phi)^2}{2} \Bigg[ \frac{d_e^{(2)}}{4} F_{\mu\nu} F^{\mu\nu} - \frac{d_g^{(2)} \beta_3}{2 g_3} F^A_{\mu\nu} F^{A\,\mu\nu} - \sum_{f=u,d,e} (d_{m_f}^{(2)}+\gamma_{m_f} d_g^{(2)})\, m_f \bar\psi_f \psi_f \Bigg] + \mathcal{L}_{\rm SM},$$5 (Aghaie et al., 5 May 2026). Using CMB+LSS and projected CMB-HD limits on repulsive self-interactions, recent work finds that the resulting bounds on quadratic couplings to electrons and light quarks can be comparable to or stronger than BBN bounds and several orders of magnitude stronger than equivalence-principle constraints across most of the viable ULDM mass range (Aghaie et al., 5 May 2026). A common misconception is therefore that symmetry protection of the mass automatically guarantees phenomenological viability at order-one quadratic couplings; the radiatively induced quartic remains a separate and potentially dominant constraint (Aghaie et al., 5 May 2026).

4. Experimental phenomenology across laboratory, space, and radio searches

Quadratically coupled ULDM produces oscillations of fundamental constants, composition-dependent forces, and environment-modified backgrounds. The traditional laboratory program includes atomic clocks, molecular spectroscopy, optical cavities, unequal-arm interferometers, resonant bar detectors, and equivalence-principle tests. In the cold-halo regime, the oscillatory observables scale as $$\mathcal{L} \supseteq \frac{1}{2}(\partial\phi)^2 - \frac{1}{2} m^2 \phi^2 + \frac{(\kappa \phi)^2}{2} \Bigg[ \frac{d_e^{(2)}}{4} F_{\mu\nu} F^{\mu\nu} - \frac{d_g^{(2)} \beta_3}{2 g_3} F^A_{\mu\nu} F^{A\,\mu\nu} - \sum_{f=u,d,e} (d_{m_f}^{(2)}+\gamma_{m_f} d_g^{(2)})\, m_f \bar\psi_f \psi_f \Bigg] + \mathcal{L}_{\rm SM},$$6, so the characteristic frequency is $$\mathcal{L} \supseteq \frac{1}{2}(\partial\phi)^2 - \frac{1}{2} m^2 \phi^2 + \frac{(\kappa \phi)^2}{2} \Bigg[ \frac{d_e^{(2)}}{4} F_{\mu\nu} F^{\mu\nu} - \frac{d_g^{(2)} \beta_3}{2 g_3} F^A_{\mu\nu} F^{A\,\mu\nu} - \sum_{f=u,d,e} (d_{m_f}^{(2)}+\gamma_{m_f} d_g^{(2)})\, m_f \bar\psi_f \psi_f \Bigg] + \mathcal{L}_{\rm SM},$$7, while force-based observables depend on gradients of the background-induced $$\mathcal{L} \supseteq \frac{1}{2}(\partial\phi)^2 - \frac{1}{2} m^2 \phi^2 + \frac{(\kappa \phi)^2}{2} \Bigg[ \frac{d_e^{(2)}}{4} F_{\mu\nu} F^{\mu\nu} - \frac{d_g^{(2)} \beta_3}{2 g_3} F^A_{\mu\nu} F^{A\,\mu\nu} - \sum_{f=u,d,e} (d_{m_f}^{(2)}+\gamma_{m_f} d_g^{(2)})\, m_f \bar\psi_f \psi_f \Bigg] + \mathcal{L}_{\rm SM},$$8 profile (Banerjee et al., 2022).

Screening strongly affects which experiments dominate. For quadratically coupled electrons, the “twin naturalness” region identified in the quadratic-twin construction overlaps with ongoing and future tabletop searches; the paper highlights optical clocks, molecular spectroscopy, interferometers, resonant detectors, MICROSCOPE, and especially a future $$\mathcal{L} \supseteq \frac{1}{2}(\partial\phi)^2 - \frac{1}{2} m^2 \phi^2 + \frac{(\kappa \phi)^2}{2} \Bigg[ \frac{d_e^{(2)}}{4} F_{\mu\nu} F^{\mu\nu} - \frac{d_g^{(2)} \beta_3}{2 g_3} F^A_{\mu\nu} F^{A\,\mu\nu} - \sum_{f=u,d,e} (d_{m_f}^{(2)}+\gamma_{m_f} d_g^{(2)})\, m_f \bar\psi_f \psi_f \Bigg] + \mathcal{L}_{\rm SM},$$9 nuclear clock, for which an assumed $\kappa=\sqrt{4\pi G}$0 would probe deeply into the twin-natural region (Delaunay et al., 16 Jul 2025). Environmental effects can suppress amplitude-sensitive signals while enhancing gradient-sensitive ones, so the relative importance of clocks and EP tests depends on the sign of the coupling and the local density profile (Delaunay et al., 16 Jul 2025).

Free-space orbital dynamics provide a complementary regime. Using the measured pericentre precession of LAGEOS II, one can constrain quadratic couplings in the approximate range

$\kappa=\sqrt{4\pi G}$1

This method is especially valuable at strong coupling, where laboratory and enclosed-satellite experiments become ineffective because surrounding material screens the field. LAGEOS II, by contrast, is in a relatively clean environment and remains sensitive to the saturated but non-vanishing scalar fifth force; the bounds are particularly strong for the gluon coupling $\kappa=\sqrt{4\pi G}$2 because they probe the absolute Earth–satellite force rather than only differential composition effects (Burrage et al., 27 May 2026).

Quadratic couplings also motivate qualitatively different search strategies. One proposal uses stimulated annihilation $\kappa=\sqrt{4\pi G}$3 in the presence of a background radio beam, producing a reflected electromagnetic wave at frequency

$\kappa=\sqrt{4\pi G}$4

with fractional width $\kappa=\sqrt{4\pi G}$5. For a $\kappa=\sqrt{4\pi G}$6 emitter and low-frequency arrays such as LOFAR, UTR-2, and ngLOBO, the forecast reach depends strongly on the assumed local halo model: it is modest in an isothermal halo, stronger in a caustic ring, and can exceed BBN constraints by up to $\kappa=\sqrt{4\pi G}$7 orders of magnitude in an Earth-halo scenario in the $\kappa=\sqrt{4\pi G}$8–$\kappa=\sqrt{4\pi G}$9 band (Gong et al., 2023).

Transient searches are another branch. If ULDM forms boson stars that undergo bosenova collapse, quantum sensors can search for relativistic scalar bursts rather than only the cold Galactic field. For quadratic couplings, Earth screening again makes space-based experiments especially attractive, and the projected reach in the mass range $d_i^{(2)}$0 can extend orders below existing cold-DM limits (Arakawa et al., 2024).

5. Background-induced forces, orbital sidebands, and pulsar timing arrays

The most detailed recent treatment of force phenomenology goes beyond the spherically symmetric approximation for the Earth-induced scalar profile. Using a full partial-wave calculation of dark-matter scattering by the Earth, the background-induced force can be expanded in multipoles of the ensemble-averaged density $d_i^{(2)}$1, and the resulting signal in a satellite EP experiment decomposes into a main band at $d_i^{(2)}$2 plus sidebands at

$d_i^{(2)}$3

with $d_i^{(2)}$4 (Bouley et al., 26 Jun 2026). Earth screening is responsible for this frequency-band structure, and the relative sideband amplitudes vary annually because the angle between the orbital plane and the DM-wind direction changes over the year (Bouley et al., 26 Jun 2026). In the high-mass regime where $d_i^{(2)}$5, the dominant contribution shifts from the monopole derivative to higher multipoles, so the first sidebands can become comparable to or larger than the main line (Bouley et al., 26 Jun 2026).

This matters directly for MICROSCOPE and its successors. Re-evaluating MICROSCOPE segment by segment with the full multipole template modifies the inferred bounds on quadratic couplings: near $d_i^{(2)}$6 the spherically symmetric estimate can be inaccurate by more than an order of magnitude, while at higher masses the EP-band sensitivity survives because higher multipoles remain active (Bouley et al., 26 Jun 2026). For next-generation space EP missions such as Galileo Galilei and STE-QUEST, a full-band analysis that includes the sidebands can improve sensitivity by about an order of magnitude in the high-mass regime relative to using only the main EP band (Bouley et al., 26 Jun 2026).

Pulsar timing arrays probe a different observable sector. Quadratically coupled ULDM produces both coherent PTA signals, associated with the narrow fast mode near $d_i^{(2)}$7, and stochastic signals, associated with slow-mode fluctuations of $d_i^{(2)}$8. The timing residuals arise through three channels: a Doppler signal from ULDM-induced accelerations of the Sun and pulsars, a clock signal from modulation of Terrestrial Time, and a pulsar-spin signal from changes in pulsar inertia (Gan et al., 15 Oct 2025). Recent analysis finds that the sensitivity of current PTA observations to the coherent signal competes with and sometimes exceeds that of other probes, including equivalence-principle tests and atomic clocks, whereas the stochastic PTA sensitivities underperform equivalence-principle constraints for both existing and upcoming data sets (Gan et al., 15 Oct 2025).

6. Conceptual tensions, common misconceptions, and current directions

Several recurrent misconceptions have been corrected by recent work. One is that stronger quadratic coupling always implies a stronger laboratory signal. In practice, once the in-medium mass becomes large, cavity walls, vacuum chambers, satellite housings, the atmosphere, or the Earth itself can suppress the field and its gradient at the detector, so naïve extrapolation of weak-coupling formulas overstates the excluded region (Burrage et al., 22 Jul 2025). Another is that the Earth-induced background force is effectively central at all relevant masses; beyond the spherical regime, the signal develops a directional wake, non-central components, and orbital sidebands (Bouley et al., 26 Jun 2026).

A second conceptual tension concerns cosmology. The quadratic-twin framework solves the scalar-mass naturalness problem by introducing a full twin Standard Model, but this brings the familiar cosmological challenge of extra relativistic degrees of freedom: without early-universe $d_i^{(2)}$9 breaking, such as asymmetric reheating, the twin bath would conflict with $M_{\rm Pl}^{-2}$0. Moreover, once thermal effects are included, the pNGB thermal mass is not protected by the mirror symmetry because the twin bath is not populated, suggesting non-standard cosmological histories that remain to be worked out (Delaunay et al., 16 Jul 2025). This suggests that naturalness, cosmology, and laboratory reach cannot be treated independently.

Current directions therefore combine three threads. The first is improved environmental modeling, including cavities, housings, and non-spherical bodies, so that strong-coupling constraints can be interpreted consistently (Burrage et al., 22 Jul 2025). The second is the construction of complete signal templates for force-based searches, especially the sideband structure in space EP tests and the coherent/stochastic separation in PTAs (Bouley et al., 26 Jun 2026, Gan et al., 15 Oct 2025). The third is the exploration of qualitatively distinct channels—stimulated annihilation in radio beams, bosenova bursts, and matter-effect scattering at $M_{\rm Pl}^{-2}$1—that probe parameter space inaccessible to standard oscillating-constants experiments (Gong et al., 2023, Gan et al., 15 Apr 2025).

Taken together, these developments have transformed quadratically coupled ultralight dark matter from a simple variation-on-dilaton theme into a technically distinct subject. Its defining structures are the $M_{\rm Pl}^{-2}$2 dependence of observables, the density-dependent effective mass in matter, the consequent screening and descreening phenomena, and the need to analyze both model-building consistency and experimental signatures in the presence of extended environments rather than in vacuum alone.