Adiabatic response in the Migdal Effect

This paper presents the first first-principles calculation of the Migdal effect in isolated atoms, establishing the conditions for adiabatic suppression and confirming that direct dark matter searches operate within the unsuppressed regime.

The Gist

Imagine a dark matter particle (let's call it a "ghost") zooming through space and bumping into an atom inside a detector. Usually, scientists think of this like a billiard ball hitting another: the heavy nucleus gets knocked backward, and the tiny electrons just sit there, waiting to be shaken loose later.

However, there's a famous theory called the Migdal effect. It suggests that when the nucleus gets hit, it doesn't just move; it "jiggles" the electrons so violently that they get knocked out of the atom immediately. This is crucial because it allows scientists to detect very light dark matter that wouldn't otherwise leave a trace.

For years, everyone assumed this "jiggle" happened instantly, like a sudden snap. But this new paper asks a vital question: What if the hit isn't a snap, but a slow push?

The Core Problem: The "Snap" vs. The "Slow Push"

The authors of this paper wanted to test the limits of the "instant snap" idea. They asked: If the dark matter particle hits the nucleus slowly enough, will the electrons still get knocked out, or will they just ride along with the nucleus like a passenger in a car?

According to a fundamental rule of physics called the Adiabatic Theorem, if you move something slowly enough, the things attached to it will adjust smoothly and stay attached. In our analogy:

• The Snap (Impulse Approximation): You yank the car door open suddenly. The passenger (the electron) is thrown out.
• The Slow Push (Adiabatic Regime): You gently accelerate the car. The passenger (the electron) stays in their seat, holding on tight. No one gets ejected.

What the Paper Did

Instead of guessing or making up rules about how "fast" is fast enough, the authors performed a rigorous, first-principles calculation. They built a mathematical model from the ground up to see exactly what happens to the electrons when a nucleus is hit, without assuming the hit is instantaneous.

They treated the system as a closed loop, calculating the exact forces involved. They found that there is indeed a "crossover point":

1. Fast Hits: If the momentum transfer is fast, the electrons fly off (the standard Migdal effect works).
2. Slow Hits: If the momentum transfer is slow, the electrons stay bound to the nucleus. The ionization (the ejection of the electron) is suppressed—it effectively disappears.

The Big Discovery: Good News for Dark Matter Hunters

You might think, "Oh no, if the effect gets suppressed, our detectors won't work!" But here is the twist:

The authors mapped out the entire landscape of possibilities and found that real-world dark matter experiments are safe.

• The "Safe Zone": The dark matter particles that current detectors are looking for (specifically those under 1 GeV in mass) hit the nuclei so fast that they are firmly in the "Snap" regime. The electrons do get knocked out.
• The "Suppressed Zone": The "Slow Push" regime where electrons stay attached only happens in conditions that terrestrial detectors are shielded from or simply don't encounter with dark matter.

The Takeaway

Think of this paper as a quality control check for a safety mechanism.

• Before: Scientists used a rule of thumb (the Impulse Approximation) that assumed the "snap" always happened.
• Now: They proved mathematically that the "snap" can fail if the hit is too slow.
• The Result: They confirmed that for the specific dark matter we are hunting for, the hit is never too slow. The "snap" always happens.

In short: The theory behind the Migdal effect is solid. The "slow push" scenario where the effect vanishes exists in the math, but it doesn't happen in the real experiments we are running today. The detectors are working exactly as the standard models predicted, and the search for light dark matter remains valid.

A Note on Neutrons

The paper also mentions that while dark matter is safe, neutrons (which are used to test these detectors in labs) might actually hit nuclei slowly enough to trigger this "suppression" effect. This means neutron experiments are actually the perfect place to test this new "slow push" physics in the future.

Technical Summary

Technical Summary: Adiabatic Response in the Migdal Effect

Problem Statement
The Migdal effect, wherein a sudden nuclear recoil induces prompt ionization or excitation of bound electrons, is a critical mechanism for extending the sensitivity of direct dark matter (DM) detectors to sub-GeV candidates. Current theoretical estimates of Migdal scattering rates rely entirely on the impulse approximation. This approximation assumes the nuclear recoil occurs as a sudden momentum transfer $q$ on a timescale significantly shorter than any characteristic electron timescale. While this framework has motivated extensive experimental and theoretical work, its validity beyond the impulse limit remains unresolved. Specifically, there is a lack of first-principles derivation regarding the regime where momentum transfer is sufficiently slow that the electron cloud remains bound to the recoiling nucleus, leading to an adiabatic suppression of ionization probability. Existing literature treats this suppression heuristically rather than deriving it from fundamental scattering principles.

Methodology
The authors present a first-principles ($ab$ $initio$) calculation of the Migdal scattering rate for isolated atoms that does not rely on the impulse approximation.

• Formalism: The system is treated as a closed quantum system using a Hamiltonian formulation without integrating out degrees of freedom. The authors utilize Jacobi coordinates to separate the relative motion of the incident particle $\chi$ (DM or neutron) and the nucleus from the internal electronic coordinates.
• Interaction: The $\chi$-nucleus interaction potential $V_{\chi N}$ is expanded to leading order in the mass ratio $\beta = m_e/M_A$. The term coupling the electron coordinates to the $\chi$-nucleus relative coordinate induces the Migdal effect.
• Scattering Amplitude: The ionization amplitude $M_{fi}$ is expressed in terms of an exact force matrix element $\tilde{F}_{fi}$, which is the expectation value of the gradient of the potential between exact scattering eigenstates $|\psi^\pm_{p}\rangle$ (solutions to the Lippmann-Schwinger equation), rather than plane waves.
• Model Potential: To evaluate the matrix element beyond the Born approximation, the authors employ a Yukawa potential $V_{\chi N}(\rho) = (\alpha'/\rho)e^{-m_\phi \rho}$, characterizing the interaction by dimensionless parameters: the Sommerfeld parameter $\eta_i$ (interaction strength relative to velocity) and the mediator mass parameter $\lambda_i$ (light vs. heavy mediator regimes).

Key Contributions and Results

1. Identification of the Adiabatic Crossover: The study establishes the transition between the impulse regime and the adiabatic cutoff regime. It demonstrates that the suppression of ionization is fully encoded in the scattering amplitude of the $\chi$-nucleus system, requiring no ad hoc assumptions about timescales.
2. Adiabatic Suppression Factor ($F_{ad}$): The authors derive a suppression factor $F_{ad}$ such that the inelastic matrix element is $|\tilde{F}_{fi}| = F_{ad} \cdot q |A_{el}|$, where $A_{el}$ is the exact elastic scattering amplitude.
   • $F_{ad}$ approaches 1 in the limit of zero electronic energy transfer ($\omega \to 0$), recovering the standard result.
   • For finite $\omega$, $F_{ad}$ acts as a super-exponential suppression factor, $|F_{ad}|^2 \approx \exp[-\omega \tau_{\chi N}]$, where $\tau_{\chi N}$ is the timescale of the $\chi$-nucleus scattering process.
3. Semiclassical Interpretation: In the regime of large Sommerfeld parameters ($\eta_i \gg 1$), corresponding to long-range interactions and classical trajectories, the suppression is shown to be of truly adiabatic origin. The scattering timescale $\tau_{\chi N}$ asymptotes to the classical scattering time $\tau_{cl}$, confirming that "accelerating the nucleus slowly" fails to induce ionization.
4. Parameter Space Mapping: Numerical results for Yukawa interactions map the parameter space $(\lambda_i, \eta_i)$. Significant deviations from the impulse approximation (Born limit) occur in the semiclassical regime ($\eta_i > 1$) and for long-range interactions ($\lambda_i < 1$).

Significance and Claims
The paper claims that the validity of the standard Migdal treatment for direct dark matter searches is robust.

• Dark Matter Regime: The authors find that the parameter space relevant for sub-GeV dark matter direct detection lies firmly within the unsuppressed regime. Consequently, the limits derived using standard impulse approximation calculations are theoretically sound and do not require correction for adiabatic effects.
• Theoretical Resolution: The work removes a major theoretical assumption behind previous Migdal searches by establishing the validity of the impulse approximation from first principles for the relevant DM parameter space.
• Experimental Implications: While DM searches are unaffected, the authors note that neutron scattering experiments, which can probe dynamics beyond the Born limit, serve as a critical setting to investigate the identified adiabatic suppression effects. Additionally, neutrino- and photon-induced Migdal scattering remain unsuppressed.
• Future Work: The current formulation applies to isolated atoms. The authors identify the translation of this framework to solid-state detectors (semiconductors and crystals) as a necessary next step, which will be addressed in separate future work.