Reassessing the gallium anomaly using self-consistent electron wave functions

This paper reassesses the long-standing gallium anomaly by calculating neutrino capture cross sections using self-consistent electron wave functions derived from the Dirac-Coulomb equation, thereby updating the global significance of the discrepancy and its interpretation regarding sterile neutrinos.

The Gist

Imagine you are trying to count how many raindrops hit a specific bucket. You have a very precise mathematical formula that predicts exactly how many drops should hit the bucket based on the size of the rain and the size of the bucket. However, every time you actually count the drops in real life, you find there are fewer than your formula predicted. This missing rain is what physicists call the "Gallium Anomaly."

For over 30 years, experiments using Gallium-71 (a type of metal) as a "bucket" to catch neutrinos (tiny, ghostly particles from the sun or radioactive sources) have consistently found fewer neutrinos than expected. The gap between the prediction and reality has grown so large that it's now considered a major mystery in physics.

This paper by Cadeddu and colleagues is like a team of master mechanics deciding to rebuild the engine of that prediction formula from the ground up to see if they made a calculation error.

The Old Way vs. The New Way

The Old Engine (The Approximation):
Previously, scientists calculated how neutrinos interact with the Gallium atoms using a "rough draft" version of the math. They treated the electrons (the tiny particles orbiting the atom's nucleus) as if they were simple, smooth waves that didn't change much inside the atom. It was like estimating the shape of a bumpy road by just looking at a flat map. They assumed the electron wave was the same everywhere inside the tiny nucleus.

The New Engine (The Exact Solution):
In this new study, the authors decided to stop using the flat map. Instead, they used a high-definition GPS to solve the exact equations (called the Dirac-Coulomb equation) that describe how electrons actually behave.

• The Analogy: Imagine the nucleus is a crowded dance floor. The old method assumed everyone on the dance floor was standing still in a perfect circle. The new method actually counts every dancer, accounting for how they bump into each other and move around in the crowded space. They solved the math for both the electrons stuck to the atom and the ones flying out, using a specialized computer program to get the exact shape of the electron's "wave."

The "Averaging" Trick

Another key change in this paper is how they handle the size of the nucleus.

• The Old Way: They picked one single point in the center of the nucleus (like measuring the temperature of a room by sticking a thermometer in the exact middle) and assumed that represented the whole thing.
• The New Way: They realized the nucleus has a size, so they "averaged" the electron's behavior across the entire volume of the nucleus. It's like taking the temperature at every spot in the room and finding the true average, rather than just guessing based on the center.

What Did They Find?

When they ran their new, more precise calculations:

1. The Prediction Changed: Their new, more accurate formula predicted that fewer neutrinos should be caught than the old formula did.
2. The Gap Widened: Because their new prediction is lower, the difference between what they expect to see and what the experiments actually saw became even bigger.
3. The Result: The "missing neutrinos" are now a 5.5-sigma problem. In the world of science, "sigma" is a measure of confidence. A 5-sigma result is the gold standard for a discovery, meaning there is less than a 1-in-3.5-million chance that this discrepancy is just a random fluke.

The "Sterile Neutrino" Hypothesis

Physicists have a favorite theory to explain this missing rain: Sterile Neutrinos.

• The Metaphor: Imagine the neutrinos are like a flock of birds flying toward the bucket. The theory suggests that some of these birds are "invisible" (sterile). They don't interact with the bucket at all; they just fly right through it. If these invisible birds exist, they would explain why the bucket is emptier than expected.

The authors updated the math to see if this "invisible bird" theory still fits.

• The Good News: The math still allows for these invisible birds to exist. The Gallium data still points strongly toward their presence.
• The Bad News: Other experiments (looking at reactor neutrinos, solar neutrinos, and mass measurements) have set up "fences" that say these invisible birds shouldn't be able to fly in the way the Gallium data suggests. The Gallium data wants the birds to be very active, but the other fences say they must be very shy.

The Conclusion

The authors didn't solve the mystery; they actually made it more mysterious. By using better math and more precise electron models, they confirmed that the "missing neutrinos" are a real, robust problem, not a calculation error.

They conclude that while the "Sterile Neutrino" idea is still the leading suspect, it is currently in a standoff with other experimental evidence. The mystery remains unsolved, and the authors suggest that a new experiment with a different type of detector might be needed to finally catch the culprit.

Technical Summary

Technical Summary: Reassessing the Gallium Anomaly Using Self-Consistent Electron Wave Functions

Problem Statement
The paper addresses the "gallium anomaly," a persistent discrepancy observed in neutrino capture experiments using $^{71}\text{Ga}$ as a target. Experiments employing $^{51}\text{Cr}$ and $^{37}\text{Ar}$ radioactive sources (GALLEX, SAGE, and recently BEST) have consistently measured neutrino-induced inverse beta decay (IBD) rates that are systematically lower than theoretical predictions. This deficit, now exceeding $4\sigma$ significance, challenges the robustness of theoretical cross-section calculations and has fueled hypotheses regarding physics beyond the Standard Model, such as sterile neutrinos. The authors identify that previous theoretical evaluations relied on leading-order (LO) approximations for electronic wave functions and inconsistent treatments of nuclear and leptonic matrix elements.

Methodology
The authors undertake a comprehensive revision of the theoretical neutrino capture cross section on $^{71}\text{Ga}$ by abandoning conventional LO approximations in favor of a self-consistent numerical approach. Key methodological components include:

1. Exact Wave Function Solutions: Instead of using approximate wave functions, the authors numerically solve the Dirac-Coulomb equation for both bound (electron capture, EC) and continuum (IBD) electron states. This is achieved using a custom software package based on the RADIAL library, employing the Dirac-Hartree-Fock-Slater (DHFS) equations.
2. Self-Consistent Potentials: The calculations utilize a DHFS potential that includes nuclear, electronic, and exchange potentials. The nuclear potential is modeled using a two-parameter Fermi distribution, incorporating the first direct measurement of the $^{71}\text{Ge}$ nuclear charge radius ($R_{ch} = 4.032 \pm 0.002$ fm). The exchange potential parameter ($C_{ex}$) is averaged to $1.25$ to account for model variations between Slater ($3/2$) and Kohn-Sham ($1$) approximations.
3. Averaging Procedure: Moving beyond the convention of evaluating wave functions at a single point (e.g., $r_0 = 0$ or $r_0 = R_{box}$), the authors propose averaging the electron wave functions and the Fermi function over the nuclear charge density ($\rho_{ch}(r)$). This is applied to both the IBD cross-section calculation and the electron capture (EC) rate used to extract the nuclear matrix element via the detailed balance principle.
4. Updated Inputs: The analysis incorporates recent experimental inputs, including:
   • A weighted average of four independent measurements of the $^{71}\text{Ge}$ EC half-life ($t_{1/2} = 11.465 \pm 0.003$ d).
   • An updated Q-value for the EC process ($232.47 \pm 0.09$ keV).
   • Refined electron-capture probability ratios ($P_L/P_K$ and $P_M/P_K$) and binding energies.
   • Gamow-Teller strengths for excited states derived from $(p,n)$ and $(^3\text{He}, ^3\text{H})$ scattering data.

Key Contributions and Results

• Revised Cross Sections: The new calculation yields a ground-state cross section that is approximately 2.8% lower than previous estimates due to the averaging procedure. When including excited-state contributions, the total cross sections for $^{51}\text{Cr}$ and $^{37}\text{Ar}$ sources are found to be approximately 2% smaller than those reported in the most recent literature (Ref. [27]).
• Significance of the Anomaly: By comparing the updated theoretical predictions with experimental data from GALLEX, SAGE, and BEST, the authors re-evaluate the anomaly's significance:
  • Using $(p,n)$ scattering data for excited-state contributions, the discrepancy reaches $5.5\sigma$ ($R = 0.835^{+0.030}_{-0.048}$).
  • Using $(^3\text{He}, ^3\text{H})$ scattering data, the discrepancy is $4.7\sigma$ ($R = 0.811^{+0.037}_{-0.035}$).
  • A conservative estimate using only the ground-state cross section yields a $3.8\sigma$ anomaly.
• Self-Consistency: The authors emphasize that by using the same software and underlying assumptions for both the Fermi function (IBD) and the electron density at the nucleus (EC), systematic shifts are canceled out in the ratio, rendering the result more robust.

Significance and Interpretation
The paper claims that the gallium anomaly is now more significant than previously reported, reaching up to $5.5\sigma$. The authors re-examine the interpretation of this anomaly in terms of short-baseline active-sterile neutrino oscillations (3+1 model). Their results confirm that the gallium data, including the precise BEST measurements, require a large active-sterile mixing angle ($0.16 \lesssim \sin^2 2\vartheta_{ee} \lesssim 0.54$ at $2\sigma$ for $\Delta m^2_{41} \gtrsim 0.98 \text{ eV}^2$).

However, the authors note that this required large mixing is in tension with constraints from reactor antineutrino data, solar neutrino bounds, and the KATRIN neutrino mass measurement. Consequently, the paper concludes that while the statistical significance of the anomaly has increased, the sterile neutrino explanation remains in tension with other experimental data. The authors state that the solution to the gallium anomaly remains an open problem, potentially requiring a new source experiment with a different detector to resolve the discrepancy.