[Paper Review] Dark Matter Detection With Bound Nuclear Targets: The Poisson Phonon Tail

Yonatan Kahn, Gordan Krnjaic+1 more|arXiv (Cornell University)|Nov 18, 2020

Dark Matter and Cosmic PhenomenaPhysics and Astronomy53 references33 citations

TL;DR

This paper proposes that bound nuclear targets in solid-state detectors can enhance sensitivity to sub-GeV dark matter by exploiting the Poisson distribution of multi-phonon excitations, which creates a high-energy tail in the phonon spectrum. Unlike elastic scattering, where energy deposition is sharply defined, the phonon occupation number follows a Poisson distribution with variance ∆n = q/q₀, leading to a broader energy spread ∆E = q√(ω₀/(2mₙ)) that allows low-threshold calorimeters to detect low-momentum dark matter scattering even when the average energy is below the single-phonon threshold.

ABSTRACT

Dark matter (DM) scattering with nuclei in solid-state systems may produce elastic nuclear recoil at high energies and single-phonon excitation at low energies. When the dark matter momentum is comparable to the momentum spread of nuclei bound in a lattice, $q_0 = \sqrt{2 m_N \omega_0}$ where $m_N$ is the mass of the nucleus and $\omega_0$ is the optical phonon energy, an intermediate scattering regime characterized by multi-phonon excitations emerges. We study a greatly simplified model of a single nucleus in a harmonic potential and show that, while the mean energy deposited for a given momentum transfer $q$ is equal to the elastic value $q^2/(2m_N)$, the phonon occupation number follows a Poisson distribution and thus the energy spread is $\Delta E = q\sqrt{\omega_0/(2m_N)}$. This observation suggests that low-threshold calorimetric detectors may have significantly increased sensitivity to sub-GeV DM compared to the expectation from elastic scattering, even when the energy threshold is above the single-phonon energy, by exploiting the tail of the Poisson distribution for phonons above the elastic energy. We use a simple model of electronic excitations to argue that this multi-phonon signal will also accompany ionization signals induced from DM-electron scattering or the Migdal effect. In well-motivated models where DM couples to a heavy, kinetically-mixed dark photon, we show that these signals can probe experimental milestones for cosmological DM production via thermal freeze-out, including the thermal target for Majorana fermion DM.

Motivation & Objective

• To investigate the scattering of sub-GeV dark matter off nuclei bound in a harmonic lattice potential, where standard elastic scattering assumptions break down.
• To understand the transition between single-phonon and elastic scattering regimes in solid-state detectors by modeling the nucleus as a quantum harmonic oscillator.
• To quantify how the Poisson distribution of phonon excitations extends the detectable energy range for low-mass dark matter.
• To assess the implications of this multi-phonon signal for next-generation calorimetric and ionization-based dark matter detectors.
• To explore the connection between nuclear recoil-induced ionization (Migdal effect) and phonon excitation, showing that phonon signals are an irreducible component of ionization signals.

Proposed method

• Model a single nucleus in a 3D isotropic harmonic oscillator potential with frequency ω₀, representing the lattice binding potential.
• Use non-relativistic quantum mechanics to compute the matrix element ⟨n|e^{iq·r̂_N}|0⟩ for momentum transfer q, showing that the phonon number distribution is exactly Poissonian.
• Derive the differential scattering rate as a function of momentum transfer q, incorporating the Poisson distribution of phonon number n = (q/q₀)².
• Calculate the energy spread due to phonon fluctuations: ∆E = q√(ω₀/(2mₙ)), which exceeds the elastic energy deposit for q > q₀.
• Generalize the model to include electronic excitations via the Migdal effect, showing that ionization probability factorizes from the nuclear recoil spectrum under certain conditions.
• Use a contact interaction model with a heavy dark photon mediator to compute the full multi-phonon spectrum and compare it to thermal relic targets.

Experimental results

Research questions

• RQ1 How does the energy deposition distribution change when a nucleus in a lattice is excited by dark matter scattering, compared to free-nucleus elastic scattering?
• RQ2 What is the role of the Poisson distribution of phonon excitations in extending the sensitivity of calorimetric detectors to sub-GeV dark matter?
• RQ3 Can the multi-phonon tail in the energy spectrum be exploited to detect dark matter with momentum transfer q ≫ 1/a but recoil energy ER ≲ Ed?
• RQ4 How does the Migdal effect—ionization from nuclear recoil—interact with phonon excitation in bound nuclei, and can it be used to distinguish particle signals from backgrounds?
• RQ5 To what extent can the full multi-phonon spectrum probe thermal relic dark matter models, including Majorana fermions?

Key findings

• The mean energy deposited for a given momentum transfer q is equal to the elastic recoil energy q²/(2mₙ), but the phonon occupation number follows a Poisson distribution with mean n = (q/q₀)².
• The energy spread due to phonon fluctuations is ∆E = q√(ω₀/(2mₙ)), which is significantly larger than the elastic energy deposit for q > q₀, enabling detection above the single-phonon threshold.
• The single-phonon rate is recovered as an upward Poisson fluctuation when q ≪ q₀, and the elastic limit is approached when q ≫ q₀, showing a smooth transition between regimes.
• For a silicon target with ω₀ ≈ 60 meV and mₙ ≈ 28 GeV, the characteristic momentum spread is q₀ ≈ 56 keV, and the Poisson tail extends to higher energies than the elastic peak.
• The Migdal effect produces ionization energy Eₑ that is independent of the nuclear recoil spectrum except at the highest kinematic limits, and the resulting electron spectrum factorizes from the nuclear recoil spectrum under spherical symmetry.
• In models with a heavy dark photon mediator, the full multi-phonon spectrum can probe the thermal relic target for Majorana fermion dark matter, even when the elastic cross section is velocity-suppressed.