---
title: Quadratically Coupled ULDM Phenomenology
url: https://www.emergentmind.com/topics/quadratically-coupled-ultralight-dark-matter
type: topic
---

# Quadratically Coupled ULDM Phenomenology

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 [2211.05174, 2507.12514].

## 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” \(d_i^{(2)}\) [2605.28248]. 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 \(\alpha_{\rm EM}\), \(\Lambda_{\rm QCD}\), and fermion masses [2211.05174].

These couplings imply
\[
\Lambda_3(\phi)=\Lambda_3\left(1+d_g^{(2)}\frac{(\kappa\phi)^2}{2}\right),\qquad
\alpha_{\rm EM}(\phi)=\alpha_{\rm EM}\left(1+d_e^{(2)}\frac{(\kappa\phi)^2}{2}\right),\qquad
m_f(\phi)=m_f\left(1+d_{m_f}^{(2)}\frac{(\kappa\phi)^2}{2}\right),
\]
so any observable with sensitivity to these parameters inherits a \(\phi^2\) dependence [2605.28248]. Because the halo field behaves approximately as \(\phi(t)\simeq \phi_0\cos(m_\phi t)\), quadratic couplings generate both a DC component and an oscillatory component at frequency \(2m_\phi\), rather than \(m_\phi\) as in linearly coupled models [2211.05174].

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 \(\partial_\mu\phi\,\partial_\nu\phi\) 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 [2404.17636]. 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
\[
S = \int d^4x \left[ \frac{1}{2}(\partial \phi)^2 -\frac{1}{2}m^2 \phi^2 -\frac{1}{2}\rho(\vec{x})\,\alpha\, \phi^2 \right],
\]
so that
\[
m_{\rm eff}^2(\vec{x}) = m^2 + \alpha \rho(\vec{x}) [2507.16526].
\]
In the composition-dependent treatment used for experimental analyses, this becomes
\[
m^2_{\rm eff}(\mathbf{x}) = m^2 + \rho_A(\mathbf{x})\alpha_A,
\]
with \(\alpha_A\) a linear combination of dilaton charges and quadratic couplings [2605.28248]. For \(\alpha_A>0\), the field is suppressed in dense matter; for sufficiently negative coupling one encounters tachyonic behavior and sourcing rather than screening [2507.12514].

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 \(1/r\) distortion of the ambient oscillating field, while inside the object the field is reduced by the induced mass barrier [2410.23350]. 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 \(|\alpha|\gtrsim 1/(\rho_c\Delta R^2)\). For \(\alpha>0\), the suppression inside a cavity is exponential in the wall thickness times the in-medium wave number, so signals proportional to \(\phi^2\) or \(\nabla(\phi^2)\) can be reduced by many orders of magnitude [2507.16526]. 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 \(t^{-1/2}\), after an onset time \(t_* \approx 0.22\,m r^2\) at radius \(r\) [2410.23350]. 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 [2410.23350].

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 \(m_0 \gtrsim 10^{-6}\,\mathrm{eV}\), 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 [2504.11522].

## 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 \(\phi^2 m_e \bar e e /M_{\rm Pl}^2\) still generates a radiative mass correction linear in the quadratic coupling, so ultralight masses require either tiny couplings or a very low cutoff [2507.12514]. 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 \(\Sigma\), with quadratic couplings to both the Standard Model and a twin copy of the Standard Model related by a mirror \(\mathbb{Z}_2\). The explicit-breaking operators are arranged as
\[
|\Sigma_{\rm SM}|^2 \mathcal{O}_{\rm SM} + |\Sigma_{\rm twin}|^2 \mathcal{O}_{\rm twin},
\]
with identical couplings in the two sectors [2507.12514]. Because the mirror symmetry makes the leading radiative correction proportional to \(|\Sigma_{\rm SM}|^2+|\Sigma_{\rm twin}|^2=|\Sigma|^2\), the linear-in-coupling contribution is \(G\)-invariant and does not generate a pNGB mass. The first non-vanishing correction appears at quadratic order:
\[
\delta m_\phi^2 = \frac{(d_{m_e}^{(2)})^2 m_e^2 f^2}{16\pi^2 M_{\rm Pl}^4}\,\Lambda^2,
\]
or, more generally, with the same \((d^{(2)})^2\) suppression and an extra factor \((f/M_{\rm Pl})^2\) [2507.12514]. For \(f\gtrsim 10\,\mathrm{TeV}\), \(\Lambda\sim 10\,\mathrm{TeV}\), and \(m_\phi\sim 10^{-20}-10^{-15}\,\mathrm{eV}\), this opens about \(25\) orders of magnitude in coupling space relative to the untwinned pNGB case [2507.12514].

That improvement does not eliminate every theoretical tension. Matter couplings unavoidably induce scalar self-interactions through loops. For quadratic couplings to fermions,
\[
\Delta\lambda_{\rm quad}\propto N_c\,\frac{m_f^4}{M_{\rm Pl}^4}\,\left(d_{m_f}^{(2)}\right)^2,
\]
so cosmological bounds on the effective quartic self-coupling map into bounds on \(d_{m_f}^{(2)}\) [2605.03477]. 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 [2605.03477]. 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 [2605.03477].

## 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 \(\phi^2\), so the characteristic frequency is \(2m_\phi\), while force-based observables depend on gradients of the background-induced \(\phi^2\) profile [2211.05174].

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 \({}^{229}\mathrm{Th}\) nuclear clock, for which an assumed \(\delta\alpha/\alpha\sim 10^{-24}\) would probe deeply into the twin-natural region [2507.12514]. 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 [2507.12514].

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
\[
5\times10^{-20}\,\mathrm{eV} \lesssim m \lesssim 3\times10^{-11}\,\mathrm{eV}.
\]
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 \(d_g^{(2)}\) because they probe the absolute Earth–satellite force rather than only differential composition effects [2605.28248].

Quadratic couplings also motivate qualitatively different search strategies. One proposal uses stimulated annihilation \(\phi\phi\to\gamma\gamma\) in the presence of a background radio beam, producing a reflected electromagnetic wave at frequency
\[
\nu_\phi \simeq \frac{m_\phi}{2\pi},
\]
with fractional width \(\Delta\nu_\phi/\nu_\phi\sim 10^{-6}\). For a \(50\,\mathrm{MW}\) 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 \(8\) orders of magnitude in an Earth-halo scenario in the \(5\)–\(20\,\mathrm{MHz}\) band [2308.08477].

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 \(10^{-23}\,\mathrm{eV}\lesssim m_\phi \lesssim 10^{-5}\,\mathrm{eV}\) can extend orders below existing cold-DM limits [2402.06736].

## 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 \(\langle|\psi|^2\rangle\), and the resulting signal in a satellite EP experiment decomposes into a main band at \(\omega_{\rm EP}\) plus sidebands at
\[
\omega_{\rm EP}\pm n\omega_{\rm orb},
\]
with \(n\geq 1\) [2606.28481]. 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 [2606.28481]. In the high-mass regime where \(k_0R_\oplus\gtrsim 1\), 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 [2606.28481].

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 \(m_\phi\sim 10^{-11}\,\mathrm{eV}\) 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 [2606.28481]. 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 [2606.28481].

Pulsar timing arrays probe a different observable sector. Quadratically coupled ULDM produces both coherent PTA signals, associated with the narrow fast mode near \(2m_\phi\), and stochastic signals, associated with slow-mode fluctuations of \(\phi^2\). 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 [2510.13945]. 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 [2510.13945].

## 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 [2507.16526]. 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 [2606.28481].

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 \(\mathbb{Z}_2\) breaking, such as asymmetric reheating, the twin bath would conflict with \(N_{\rm eff}\). 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 [2507.12514]. 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 [2507.16526]. 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 [2606.28481, 2510.13945]. The third is the exploration of qualitatively distinct channels—stimulated annihilation in radio beams, bosenova bursts, and matter-effect scattering at \(m_0\gtrsim 10^{-6}\,\mathrm{eV}\)—that probe parameter space inaccessible to standard oscillating-constants experiments [2308.08477, 2504.11522].

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 \(\phi^2\) 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.

Source: https://www.emergentmind.com/topics/quadratically-coupled-ultralight-dark-matter