The Migdal effect in Semiconductors for the Effective Field Theory of Dark Matter Direct Detection

This paper combines effective field theories for dark matter-nucleus interactions and the Migdal effect in semiconductors to calculate signals for all ten dimension-six operators, derive new experimental bounds using EDELWEISS germanium data, and demonstrate that the resulting parameter space is disfavored by constraints on heavy mediators in simple UV completions.

The Gist

The Big Picture: Hunting the Invisible Ghost

Imagine the universe is filled with a ghostly substance called Dark Matter. We know it's there because it has gravity (it holds galaxies together), but we've never seen it, touched it, or felt it. Scientists are trying to catch these "ghosts" using detectors deep underground.

Usually, scientists look for a Dark Matter particle to bump into an atomic nucleus (the center of an atom) like a bowling ball hitting a pin. But if the Dark Matter is very light (like a ping-pong ball), it might not knock the pin over hard enough for our detectors to see.

The Problem: Our detectors have a "noise floor." If the bump is too gentle, the detector doesn't register it.

The Solution (The Migdal Effect): What if, instead of just knocking the nucleus, the Dark Matter particle also accidentally rips an electron (a tiny, fast-moving particle orbiting the nucleus) off the atom? Electrons are much lighter and easier to detect than nuclei. This "side effect" is called the Migdal effect. It's like a bowling ball hitting a pin, but the impact is so sudden that a tiny marble flying off the pin (the electron) hits a sensor, alerting us that a collision happened.

The Setting: The Crystal Ballroom

Most previous studies treated atoms like lonely people standing in a field (free atoms). But this paper focuses on semiconductors (like Germanium crystals used in detectors like EDELWEISS).

Think of a semiconductor crystal not as a field, but as a crowded ballroom where everyone is holding hands in a rigid dance formation (the crystal lattice).

• The Nucleus: A dancer in the middle of the room.
• The Electrons: Tiny sparks of light dancing around the dancer.
• The Lattice: The floor and the other dancers holding hands.

When a Dark Matter ghost bumps into a dancer in this crowded room, the reaction is complicated. The dancer can't just fly backward; they have to jiggle the whole line of dancers (creating phonons, or vibrations) and maybe knock a spark of light loose (the ionization).

The New Tool: The "Universal Translator" (Effective Field Theory)

The authors of this paper are like master mechanics who built a Universal Translator for these collisions.

In physics, there are many different theories about how Dark Matter might interact with normal matter. Some theories say it hits like a hammer; others say it pushes like a magnet; some say it spins.

• The Old Way: Scientists had to build a new, specific calculator for every single theory.
• The New Way (This Paper): The authors created one master formula (an Effective Field Theory) that can describe all 10 possible ways Dark Matter might interact with a nucleus in a crystal.

They figured out that even though the crystal is complex, the math simplifies. The collision breaks down into three separate parts that multiply together:

1. The Nuclear Hit: How hard the Dark Matter hits the nucleus.
2. The Electron Rip: How likely it is to knock an electron loose (the Migdal effect).
3. The Floor Jiggle: How the crystal lattice vibrates.

Because they separated these parts, they could calculate the signal for any type of Dark Matter interaction without starting from scratch every time.

The Experiment: Listening to the Germanium Detector

The authors took data from a real experiment called EDELWEISS, which uses a Germanium crystal detector. They asked: "If Dark Matter is hitting this crystal and causing the Migdal effect, what would the signal look like for each of our 10 theories?"

They found that:

• Low Mass is King: These detectors are amazing at finding very light Dark Matter (millions of times lighter than a proton) because the energy needed to knock an electron loose is tiny (like the energy of a single photon of light).
• The Signal Shape: They predicted exactly what the "sound" of the collision would look like on the detector graph. It's a specific curve that depends on the mass of the Dark Matter and which "type" of interaction it is.

The Verdict: The Ghost is Hiding (or the Theory is Wrong)

After crunching the numbers, they compared their predictions to the actual data from EDELWEISS.

• The Result: They didn't see the Dark Matter ghosts.
• The Constraint: This means they can now rule out a huge range of possibilities. If Dark Matter interacts in these specific ways, it must be even weaker than we thought, or the Dark Matter particles must be even lighter than we can currently detect.

The "UV Completion" Warning:
The paper also adds a "reality check." It says, "Hey, these theories are mathematically possible, but if you try to build a real-world model of what creates these forces (the 'UV completion'), the universe seems to forbid it."

• Analogy: It's like designing a car engine that runs on water. The math says it could work, but the laws of thermodynamics (the "UV constraints") say it's impossible. The authors found that for many of these Dark Matter theories, the "laws of the universe" (constraints from other experiments) make them very unlikely to exist.

Summary in a Nutshell

1. The Goal: Catch light Dark Matter particles that are too weak to knock a nucleus over.
2. The Trick: Look for the "Migdal effect"—when the collision knocks an electron loose instead.
3. The Innovation: Created a single mathematical tool to predict this signal for all 10 major theories of Dark Matter interactions inside a crystal.
4. The Finding: Used real data to say, "If Dark Matter interacts this way, we would have seen it. Since we didn't, these specific theories are likely wrong or the interaction is incredibly weak."

This paper is a roadmap for future experiments. It tells scientists exactly what to look for in their data to either catch the ghost or prove that the ghost doesn't exist in the way we thought.

Technical Summary

1. Problem Statement

Direct detection of Dark Matter (DM) has traditionally focused on elastic scattering between DM and atomic nuclei, which is sensitive to DM masses in the GeV–TeV range. However, sensitivity drops significantly for sub-GeV DM due to detector energy thresholds. To probe lighter DM (MeV–GeV), researchers utilize:

• DM-electron scattering: Direct ionization of electrons.
• The Migdal effect: An inelastic process where a DM-nucleus scattering event imparts enough recoil energy to the nucleus to cause the sudden ionization (shake-off) of bound atomic electrons.

While the Migdal effect has been studied in noble liquids (where nuclei are treated as free atoms), its calculation in semiconductors is significantly more complex due to the presence of a crystal lattice, complex electronic band structures, and collective vibrational modes (phonons). Existing calculations often relied on the "free-atom" approximation, which fails at low momentum transfers relevant for light DM. Furthermore, previous studies of the Migdal effect in semiconductors were largely limited to spin-independent (SI) interactions, lacking a comprehensive framework for general non-relativistic DM-nucleus operators.

2. Methodology

The authors develop a unified theoretical framework combining two Effective Field Theory (EFT) approaches:

1. Non-Relativistic EFT for DM-Nucleus Interactions: They utilize the complete set of ten dimension-six Galilean-invariant operators ($O_1$ through $O_{11}$) that describe general DM-nucleon interactions, including spin-dependent, velocity-dependent, and momentum-dependent couplings.
2. Semiconductor EFT for the Migdal Effect: They generalize the recent EFT framework (originally developed for SI interactions) to handle general operators. This framework exploits two separation of scales:
   • Momentum Scale: $q \ll \Lambda$ (where $\Lambda$ is the UV cutoff).
   • Energy Scale: The energy of the ionized electron ($\omega \sim$ eV) is much larger than the energy of the mediating lattice mode ($E \lesssim 100$ meV).

Key Theoretical Steps:

• Effective Hamiltonian Derivation: By integrating out the intermediate lattice modes, the authors derive a velocity-dependent effective Hamiltonian ($H_{eff}$) that describes the interaction between the DM, the lattice, and the electrons.
• Factorization: They demonstrate that the differential scattering rate factorizes into three distinct components:
  1. Nuclear Response: Described by nuclear form factors and response functions ($M, \Sigma', \Sigma'', \Delta, \Phi''$) dependent on the specific operator.
  2. Electronic Response: Encoded in the electron loss function (derived from the material's dielectric function, $\epsilon^{-1}$).
  3. Vibrational Response: Encoded in the dynamic structure factor ($S(q, E)$), which accounts for phonon excitations.
• Rate Calculation: Using Fermi's Golden Rule, they derive the differential rate for both ionization (electron excitation) and phonon (lattice vibration) signals for all ten operators.

3. Key Contributions

• Generalization to All Operators: This is the first comprehensive calculation of the Migdal effect in semiconductors for all ten dimension-six non-relativistic DM-nucleon operators, including those involving DM spin, nuclear spin, and angular momentum.
• Rigorous Semiconductor Treatment: Unlike previous works using the free-atom approximation, this paper rigorously incorporates the crystal lattice potential and collective modes via the dynamic structure factor, which is crucial for low-mass DM ($m_\chi \lesssim 50$ MeV).
• Phonon Signal Prediction: The authors calculate the accompanying phonon spectrum for Migdal events. They show that the phonon spectrum shape depends on the specific operator and DM mass, offering a potential discriminant against background events.
• Experimental Recasting: They recast existing data from the EDELWEISS experiment (using a germanium detector) to derive new experimental bounds on these operators.

4. Results

• Rate Spectra:
  • Ionization: The ionization spectrum peaks near the pair-creation energy of germanium ($\sim 2.9$ eV) and falls off as $\sim 1/\omega^4$. While the shape is largely dictated by the electron loss function, subtle differences exist between operators with different momentum/velocity dependencies.
  • Phonons: The phonon spectrum exhibits distinct shapes for different groups of operators. For example, operators with similar momentum dependencies (e.g., $O_1$ and $O_4$) share characteristic spectral shapes. At low DM masses ($10$ MeV), the phonon spectra for Migdal events differ slightly from non-Migdal nuclear recoil events, but the distinction becomes more pronounced at higher masses ($100$ MeV).
• Experimental Constraints:
  • Using EDELWEISS data (33.4 g detector, 58 hours), the authors derive new limits on the DM-nucleon cross-section ($\sigma_n$) for all ten operators.
  • Competitive Reach: For operators $O_7, O_8, O_{10},$ and $O_{11}$, the EDELWEISS constraints are the most competitive at low DM masses ($\lesssim 10-100$ MeV), surpassing constraints from larger noble liquid experiments (like XENON1T) which suffer from higher energy thresholds.
  • For $O_3$ and $O_5$, this work provides the only current direct detection limits.
• UV Completeness and Earth Shielding:
  • UV Bounds: The authors compare their derived limits against constraints from UV-complete models (e.g., heavy mediators). They find that the parameter space accessible to direct detection for operators $O_4, O_6,$ and $O_{10}$ is largely disfavored by astrophysical constraints (e.g., Bullet Cluster, meson decays) on heavy mediators.
  • Earth Shielding: They estimate the "Earth-shielding" effect (DM scattering in the Earth's crust before reaching the detector). For cross-sections above a certain threshold (dependent on the operator), DM particles would lose significant energy or be stopped, invalidating the standard halo model assumptions. This is particularly relevant for $O_7, O_8,$ and $O_{11}$.

5. Significance

• Bridging Theory and Experiment: This work provides the necessary theoretical tools to interpret low-threshold semiconductor data (like EDELWEISS, CDEX, and SuperCDMS) in the context of general DM interactions, moving beyond the simplified spin-independent assumption.
• New Discovery Channels: It highlights the unique sensitivity of semiconductor detectors to sub-GeV DM via the Migdal effect, particularly for operators that are velocity or momentum suppressed, which are often missed by traditional nuclear recoil searches.
• Multi-Channel Discrimination: By predicting both the ionization and phonon signals, the paper suggests that future experiments capable of measuring both channels can better distinguish DM signals from backgrounds and potentially identify the specific nature of the DM interaction (i.e., which operator is dominant).
• UV Sensitivity: The analysis underscores that while direct detection can probe specific parameter spaces, these regions are often tightly constrained by other physics (astrophysics and collider limits), guiding future model-building efforts toward light mediators or specific coupling structures.

In summary, this paper establishes a robust, model-independent framework for calculating the Migdal effect in semiconductors for a wide class of Dark Matter interactions, reinterprets existing data to set new world-leading limits on specific operators, and identifies the critical role of phonon signals in future low-mass DM searches.