Weak nuclear decays deep-underground as a probe of axion dark matter

This paper proposes using time modulation in weak nuclear decays as a probe for axion dark matter, deriving a theoretical framework to predict such variations, constraining axion parameters using existing Gran Sasso data on $^{40}$K and $^{137}$Cs, and suggesting a new electron capture measurement to extend sensitivity to higher axion masses.

The Gist

Imagine the universe is filled with an invisible, ghostly wind called axion dark matter. We can't see it, but physicists suspect it's everywhere, making up most of the universe's mass. This paper proposes a clever way to "feel" this wind by listening to the ticking of atomic clocks inside radioactive atoms.

Here is the breakdown of the paper's ideas, using simple analogies:

1. The Invisible Wind and the Atomic Clocks

Think of a radioactive atom (like a tiny, unstable clock) as a pendulum. Normally, it swings back and forth at a perfectly steady rate. This rate is called its "decay rate."

The paper suggests that if this invisible axion wind blows through the atom, it might slightly nudge the pendulum, making it speed up or slow down in a rhythmic pattern. Just as a strong wind might make a clock's pendulum wobble, the axion wind might cause the atom to decay (break down) slightly faster or slower depending on the time of day or year.

2. The "Deep Underground" Lab

To hear this tiny wobble, you need a very quiet room. On the surface of the Earth, there is a lot of "noise" from space (cosmic rays) that hits atoms and makes them act up, which would drown out the subtle signal from the axion wind.

The researchers used the Gran Sasso Laboratory in Italy. This lab is buried deep under a mountain. The rock above acts like a giant soundproof blanket, blocking out the cosmic noise. This allows them to listen to the atoms in near-perfect silence.

3. The Experiment: Listening to Two Different "Tones"

The team looked at two specific types of radioactive atoms to see if their "ticking" changed over time:

• Potassium-40 (The Electron Catcher): Imagine an atom that catches an electron and swallows it. The team looked at old data from a potassium experiment run between 2015 and 2017. They checked if the rate of this "swallowing" changed in a pattern over days, months, or years.
• Cesium-137 (The Beta Emitter): Imagine an atom that spits out a particle. They looked at data from a cesium experiment run between 2011 and 2012. They checked if the rate of this "spitting" changed over time.

4. The Results: The Wind is Quiet (For Now)

After analyzing the data, the researchers found no evidence that the axion wind was making these atoms wobble. The atoms kept ticking at a steady pace.

However, this "nothing happened" result is actually very useful. It's like saying, "We didn't find a ghost in the house, so we know the ghost can't be hiding in the corners we checked." By not finding a wobble, they were able to set strict rules on how heavy or light the axion wind could be. They ruled out a specific range of "axion weights" (masses) that scientists were curious about.

5. The Future: Building a Faster Microphone

The researchers realized that their current "microphones" (detectors) were too slow to catch very fast wobbles. The old experiments could only detect changes that happened over hours or days.

They propose building a new, super-fast experiment using Potassium-40 again. This new setup will be able to detect wobbles that happen in just one-millionth of a second (microseconds).

• Why do this? If the axion wind is very heavy, it would wiggle the atoms very fast. The old experiments were too slow to see this. The new experiment will act like a high-speed camera, allowing them to look for much heavier axions than before.

Summary

• The Goal: To find axion dark matter by seeing if it makes radioactive atoms change their decay speed.
• The Method: Use deep underground labs to block out noise and listen to the "ticking" of Potassium and Cesium atoms.
• The Finding: No wobble was found in the old data, which helps scientists rule out certain types of axions.
• The Next Step: Build a faster detector to listen for much quicker wobbles, potentially finding heavier axions that the old experiments missed.

The paper concludes that while they haven't found the axion wind yet, their method works, and with faster equipment, they can keep searching for it in new, faster timeframes.

Technical Summary

1. Problem Statement

The paper addresses the search for axion dark matter, specifically focusing on the QCD axion which solves the Strong CP problem and constitutes dark matter.

• The Mechanism: The axion field ($a$) promotes the QCD $\theta$-term to a dynamical field ($\theta \to a/f_a$). In the presence of axion dark matter, this $\theta$-term oscillates in time: $\theta(t) \approx \theta_0 \cos(m_a t)$.
• The Phenomenon: This oscillating $\theta$-term induces time-dependent variations in fundamental constants, specifically the pion mass ($M_\pi$) and the neutron-proton mass difference ($\Delta m_N$).
• The Gap: While previous studies have looked for time modulation in $\alpha$-decays and $\beta$-decays of light nuclei (like tritium), there was a lack of a comprehensive theoretical framework and experimental constraints for weak nuclear decays (specifically Electron Capture and $\beta^-$ decay) of heavier nuclei in the ultra-low mass axion range ($10^{-23}$ to $10^{-9}$ eV).
• The Challenge: Distinguishing potential axion-induced modulations from environmental noise (cosmic rays, temperature) requires deep underground facilities to suppress background fluxes.

2. Methodology

A. Theoretical Framework

The authors developed a formalism to calculate the $\theta$-dependence of weak nuclear decay rates ($\Gamma_{weak}$).

1. $\theta$-Dependence of Nuclear Parameters:
   • The pion mass depends on $\theta$ via $M_\pi^2(\theta) = M_\pi^2 \cos(\theta/2) \sqrt{1 + \epsilon^2 \tan^2(\theta/2)}$.
   • The neutron-proton mass difference is derived from chiral perturbation theory: $\Delta m_N(\theta) \approx -4c_5 \epsilon M_\pi^4 / M_\pi^2(\theta)$.
   • While nuclear binding energies (BE) also depend on $\theta$, the authors argue that for weak decays (where mass number $A$ is conserved), the leading BE variations cancel out in the energy difference between initial and final states. Thus, the dominant effect arises from the neutron-proton mass difference affecting the $Q$-value (kinetic energy release).
2. Multipole Expansion of Weak Decays:
   • The authors utilized the multipole expansion of the nuclear weak current to derive decay widths for Electron Capture (EC) and $\beta$-decay.
   • They defined spherical tensor operators ($M_{JM}, L_{JM}, T^{el}_{JM}, T^{mag}_{JM}$) and applied selection rules based on angular momentum and parity conservation.
   • Key Approximation: The $\theta$-dependence of the reduced nuclear matrix elements was neglected, as the sensitivity to the phase space (kinetic energy) variation is the leading-order effect.
3. Modulation Observable:
   • The observable is defined as the relative time variation of the decay rate: $I(t) = (\Gamma(\theta(t)) - \langle \Gamma \rangle) / \langle \Gamma \rangle$.
   • For small $\theta$, $\Gamma(\theta) \approx \Gamma(0) + \dot{\Gamma}(0)\theta^2$.
   • The resulting modulation is $I(t) \propto \theta_0^2 \cos(2m_a t)$, where the amplitude depends on the derivative of the decay width with respect to $\theta^2$.

B. Specific Nuclei Analyzed

1. $^{40}\text{K}$ (Electron Capture):
   • Decays to the first excited state of $^{40}\text{Ar}^*$ (10.3% branching ratio).
   • Transition: $J^\pi_i = 4^- \to J^\pi_f = 2^+$.
   • Dominant multipole: $J=2$.
   • Sensitivity: The decay width scales as $\Gamma_{EC} \propto E_\nu^4$. The variation in $E_\nu$ due to $\Delta m_N$ leads to a significant modulation factor: $\dot{\Gamma}/\Gamma \approx 18.8$.
2. $^{137}\text{Cs}$ ($\beta^-$ Decay):
   • Decays to the second excited state of $^{137}\text{Ba}^{**}$ (94.57% branching ratio).
   • Transition: $J^\pi_i = 7/2^- \to J^\pi_f = 11/2^+$.
   • Dominant multipole: $J=2$.
   • Sensitivity: Calculated via numerical integration of the differential decay width, yielding $\dot{\Gamma}/\Gamma \approx -2.13$.

C. Experimental Data Re-analysis

The authors re-analyzed existing datasets from the Gran Sasso National Laboratory (LNGS):

• $^{40}\text{K}$ Experiment (2015–2017): Used a NaI(Tl) crystal to detect 1461 keV $\gamma$-rays. Data covered periods from 6 hours to 800 days.
• $^{137}\text{Cs}$ Experiment (2011–2012): Used a High Purity Germanium (HPGe) detector. Data covered periods from 6 hours to 1 year.
• Analysis Method: Fourier transform of residuals and $\chi^2$ minimization with cosine functions to search for periodic modulations. No significant modulation was found.

3. Key Contributions

1. Theoretical Extension: Extended the theoretical framework for axion-induced time modulation from $\alpha$-decays and light nuclei to weak decays of heavy nuclei ($A \sim 40-137$).
2. Quantitative Sensitivity Calculation: Derived the specific sensitivity factors ($\dot{\Gamma}/\Gamma$) for $^{40}\text{K}$ and $^{137}\text{Cs}$, demonstrating that $^{40}\text{K}$ EC is roughly an order of magnitude more sensitive to axion-induced $Q$-value shifts than $^{137}\text{Cs}$ $\beta$-decay.
3. Constraint Derivation: Translated the null results from the Gran Sasso experiments into exclusion limits on the axion decay constant ($f_a$) as a function of axion mass ($m_a$).
4. Proposal for Future Experiment: Designed a new, high-precision $^{40}\text{K}$ experiment with improved shielding and event-by-event data acquisition (160 ns time resolution) to probe much shorter timescales (down to 1 $\mu$s).

4. Results

• Exclusion Limits:
  • Using the $^{40}\text{K}$ and $^{137}\text{Cs}$ data, the authors set 2$\sigma$ exclusion limits on the axion decay constant $f_a$.
  • Mass Range: $3.0 \times 10^{-23} \text{ eV} < m_a < 9.6 \times 10^{-20} \text{ eV}$.
  • Constraint: Excluded $f_a$ values roughly between $10^{10}$ GeV and $10^{14}$ GeV in this mass range.
  • These limits are competitive with other laboratory bounds (e.g., EDM, atomic transitions) in this specific ultra-light mass window.
• Future Sensitivity:
  • The proposed new $^{40}\text{K}$ experiment, capable of resolving $\mu$s timescales, could extend the sensitivity to axion masses up to $10^{-9}$ eV.
  • Expected sensitivity amplitude: $4 \times 10^{-6}$ (2$\sigma$).
• Comparison with Other Nuclei:
  • The paper notes that nuclei with even lower $Q$-values (e.g., $^{187}\text{Re}$, $^{163}\text{Ho}$) could offer enhanced sensitivity ($\dot{\Gamma}/\Gamma \propto 1/Q$), potentially improving limits by an order of magnitude.

5. Significance

• Novel Probe: This work establishes weak nuclear decays in deep underground laboratories as a viable and competitive probe for ultra-light axion dark matter, complementing searches based on $\alpha$-decays and atomic clocks.
• Deep Underground Advantage: It highlights the critical role of the Gran Sasso Laboratory in suppressing cosmic ray backgrounds, which is essential for detecting the extremely small (per mille or sub-per mille) modulations predicted by axion models.
• Bridging Mass Gaps: The proposed future experiment aims to bridge the gap between current low-mass constraints and higher-mass searches, covering the "ultra-light" axion window ($10^{-23}$ to $10^{-9}$ eV) where other detection methods struggle.
• Model Independence: The approach relies on the generic coupling of the axion to gluons (via the $\theta$-term), making the constraints largely model-independent regarding the specific axion production mechanism, provided the axion constitutes the local dark matter density.

In conclusion, the paper successfully demonstrates that re-analyzing historical nuclear decay data can yield meaningful constraints on axion dark matter and proposes a concrete path forward to significantly expand the search parameter space using advanced underground detection techniques.