Probing Neutron Skins with KDAR Neutrinos: From Coherent to Diffractive Elastic Neutrino--Nucleus Scattering

This paper demonstrates that kaon-decay-at-rest (KDAR) neutrinos enable coherent elastic neutrino-nucleus scattering measurements to extend beyond the strict coherent limit into the diffractive regime, offering a competitive and complementary method to electron scattering for precisely probing neutron skin thickness in various nuclei.

The Gist

Imagine an atom as a tiny, dense solar system. In the center, you have a nucleus made of protons (positively charged) and neutrons (neutral). Usually, we think of protons and neutrons as being packed together in a neat, uniform ball. But in reality, the neutrons often spill out a little further than the protons, creating a fuzzy, extra layer around the edge. Physicists call this extra layer the "neutron skin."

Measuring how thick this skin is is crucial. It helps us understand how stars explode, how neutron stars behave, and even the fundamental laws of the universe. However, measuring it is incredibly hard because the skin is thinner than a single atom's width.

This paper proposes a clever new way to measure this "skin" using neutrinos—ghostly particles that pass through almost everything without stopping.

The Problem: The "Flashlight" Was Too Dim

Previously, scientists used neutrinos created by stopping pions (a type of particle) in a target. These neutrinos are like a dim, low-power flashlight. When they hit an atomic nucleus, they bounce off, but because the neutrinos are moving relatively slowly, they can't "see" the details of the nucleus. They only see the nucleus as a single, blurry blob. It's like trying to see the texture of a basketball by shining a dim light on it from a mile away; you just see a round shape, not the bumps and seams.

The Solution: The "High-Power Laser"

The authors suggest switching to neutrinos created by stopping kaons (another particle). These "KDAR" neutrinos are much more energetic—they are like a high-powered laser beam.

Because these neutrinos are moving so fast, they can penetrate deeper and interact with the nucleus in a more complex way. Instead of just seeing a blurry blob, the fast neutrinos start to "diffract" (bend and scatter) off the edges of the nucleus, much like light diffracting when it passes through a fine mesh or around a small obstacle.

The Analogy: The Orchestra and the Conductor

To understand how this works, imagine an orchestra (the nucleus) and a conductor (the neutrino).

1. The Old Way (Pion Neutrinos): The conductor moves very slowly. The musicians (protons and neutrons) all play in perfect unison because they can't hear the conductor's subtle cues. The sound you hear is just one giant, loud note. You can't tell if the violin section is slightly larger than the brass section. This is the "Coherent" regime.
2. The New Way (Kaon Neutrinos): The conductor moves very fast and gives sharp, distinct cues. Now, the musicians start to react individually. The violins might play a slightly different rhythm than the brass. The sound becomes complex, with echoes and patterns. This is the "Diffractive" regime.

By listening to these complex patterns (the "shape" of the sound), the conductor can figure out exactly how many violins are in the back row versus the front row. In physics terms, the fast neutrinos create a pattern of bounces that reveals the thickness of the neutron skin.

Why This Matters

The paper calculates that if we use these fast "Kaon" neutrinos at a facility like JSNS2 (a neutrino lab in Japan), we can measure the thickness of the neutron skin with incredible precision.

• For a medium-sized atom (Calcium-48): They could measure the skin thickness to within about 0.02 to 0.03 femtometers (a femtometer is one-quadrillionth of a meter).
• For a heavy atom (Lead-208): They could get a similar level of precision.

This is a big deal because it matches or even beats the precision of other methods, like shooting electrons at nuclei (which is expensive and complex). Neutrinos have a special advantage: they interact only with the "weak force," making them a very clean probe that isn't confused by the electromagnetic forces that complicate electron experiments.

The Bottom Line

This paper is a blueprint for a new kind of "neutron microscope." By switching from slow, dim neutrinos to fast, bright ones, scientists can stop seeing atoms as blurry balls and start seeing the detailed "skin" on their surface. This will help us solve mysteries about the heaviest elements in the universe and the extreme physics inside neutron stars.

In short: We are upgrading from a blurry snapshot to a high-definition video of the atomic nucleus, and it turns out the "ghost" particles (neutrinos) are the perfect camera for the job.

Technical Summary

1. Problem Statement

Coherent Elastic Neutrino–Nucleus Scattering (CE$\nu$NS) is a neutral-current process where a neutrino scatters off an entire nucleus coherently. While this process provides a clean probe of neutron distributions, existing measurements using Pion Decay at Rest ($\pi$DAR) neutrinos (energy $E_\nu \approx 30$ MeV) are kinematically confined to the strictly coherent regime ($qR \lesssim 1$, where $q$ is momentum transfer and $R$ is nuclear radius).

• Limitation: In this regime, the weak form factor $F_W(q) \approx 1$, meaning the cross-section scales primarily with the square of the weak charge ($Q_W^2 \propto N^2$). Consequently, recoil spectra are only weakly sensitive to the detailed shape of the neutron distribution or the neutron skin thickness ($\Delta R_{np} = R_n - R_p$).
• Goal: The authors aim to demonstrate that Kaon Decay at Rest (KDAR) neutrinos ($E_\nu = 236$ MeV) provide sufficient momentum transfer to access the coherent-to-diffractive transition ($qR \gtrsim 1$). This regime allows recoil spectra to develop genuine shape sensitivity to the nuclear weak form factor, enabling precise extraction of neutron skin thicknesses independent of parity-violating electron scattering (PVES) experiments like PREX and CREX.

2. Methodology

A. Kinematic Framework

The study utilizes the dimensionless variable $qR$ to categorize scattering regimes:

• Strict Coherence ($qR \ll 1$): Amplitudes add in phase; cross-section $\propto N^2$.
• Transition/Diffractive ($qR \gtrsim 1$): Spatial phases de-align; $F_W(q)$ deviates from unity, exhibiting oscillatory structure (diffraction minima).
• Kinematics: The paper derives exact elastic kinematics relating recoil energy $T$, momentum transfer $q = \sqrt{2MT}$, and scattering angle $\theta$. KDAR neutrinos ($236$ MeV) allow access to $qR \sim 1$ for light/medium nuclei and approach the first diffractive minimum ($qR \approx 4.493$) for heavy nuclei, whereas $\pi$DAR neutrinos remain largely in the $qR < 1$ region.

B. Nuclear Models

To model the nuclear weak charge density $\rho_W(r)$, the authors employ two standard parametrizations:

1. Helm Model: A convolution of a hard sphere (diffraction radius $R_0$) and a Gaussian (surface thickness $s$).
2. Two-Parameter Fermi (2pF) / Woods-Saxon: Defined by half-density radius $c$ and surface diffuseness $a$.

• Weak Charge: The weak form factor is dominated by neutrons due to the suppression of the proton contribution by the factor $(1 - 4\sin^2\theta_W)$. Thus, $F_W(q) \approx F_n(q)$.
• Targets: Calculations are performed for representative nuclei: $^{12}\text{C}$, $^{40}\text{Ca}$, $^{48}\text{Ca}$, and $^{208}\text{Pb}$.

C. Sensitivity Analysis

• Perturbative Approach: The authors derive an analytic expansion of the weak form factor in terms of $\Delta R_{np}$ to establish a baseline sensitivity in the low-$q$ limit.
• Full Form Factor Calculation: For the KDAR regime, they move beyond analytic approximations, calculating the full recoil spectrum distortion $\Delta \sigma(T)$ induced by variations in $\Delta R_{np}$ using the full Helm and 2pF form factors.
• Statistical Projection: They perform a binned $\chi^2$ analysis to estimate the $1\sigma$ statistical sensitivity to $\Delta R_{np}$ at a facility similar to JSNS2 (J-PARC Sterile Neutrino Search).
  • Exposure: Assumed total exposure of $10 \text{ ton}\cdot\text{year}$.
  • Fluence Scenarios: Based on JSNS2 milestones (2021 data, mid-2024 cumulative, 3-year, and 5-year projections).
  • Detector Assumptions: Recoil threshold $T_{th} = 10$ keV, Gaussian energy smearing ($\sigma_T/T = 10\%/\sqrt{T/\text{MeV}}$).

3. Key Contributions

1. Kinematic Lever Arm: The paper quantitatively establishes that KDAR neutrinos uniquely bridge the gap between the coherent and diffractive regimes. Unlike $\pi$DAR, KDAR spectra for medium and heavy nuclei extend into the region where $F_W(q)$ exhibits significant shape dependence on nuclear structure.
2. Model Dependence Quantification: The authors demonstrate that while Helm and 2pF models agree in the deep coherent limit, they diverge in the transition region ($qR \sim 1-2$). This divergence translates to $\sim 5-15\%$ variations in the recoil spectrum shoulder for medium-mass nuclei, providing a handle to constrain nuclear surface models.
3. Perturbative vs. Full Form Factor: The study highlights that low-$q$ expansions fail to capture the enhanced sensitivity near diffractive minima. They provide a rigorous framework using the full form factor to interpret spectral distortions.
4. Complementarity to PVES: The work positions KDAR-CE$\nu$NS as an independent electroweak probe. Unlike PVES (PREX/CREX), which measures at a single averaged $q$, KDAR provides a continuous recoil spectrum, offering a complementary check on systematic uncertainties.

4. Key Results

A. Spectral Sensitivity

• Light Nuclei ($^{12}\text{C}$): Remain largely in the near-coherent regime even with KDAR; sensitivity to $\Delta R_{np}$ is weak (dominated by normalization).
• Medium Nuclei ($^{40,48}\text{Ca}$): KDAR accesses the transition region ($qR \sim 1-2$). The recoil spectra show distinct shape sensitivity to the neutron skin.
• Heavy Nuclei ($^{208}\text{Pb}$): KDAR accesses the diffractive regime ($qR \sim 2-4$). While relative spectral distortions are large near diffraction minima, the absolute event rate is suppressed, balancing the statistical gain.

B. Projected Statistical Sensitivities ($1\sigma$)

For a total exposure of $10 \text{ ton}\cdot\text{year}$ and increasing KDAR fluence, the projected precision on neutron skin thickness ($\Delta R_{np}$) is:

Nucleus	2021 Fluence ($6.82 \times 10^{11} \text{ cm}^{-2}$)	5-Year Fluence ($9.75 \times 10^{12} \text{ cm}^{-2}$)
$^{48}\text{Ca}$	$\sim 0.093$ fm	$\sim 0.024$ fm
$^{208}\text{Pb}$	$\sim 0.068$ fm	$\sim 0.018$ fm
$^{40}\text{Ca}$	$\sim 0.12$ fm	$\sim 0.032$ fm
$^{12}\text{C}$	$\sim 0.13$ fm	$\sim 0.034$ fm

• Comparison to Current Data: These projected sensitivities ($\sim 0.02$ fm) are competitive with or superior to current PVES results (CREX: $\pm 0.026$ fm; PREX-II: $\pm 0.071$ fm).
• Optimal Target: $^{48}\text{Ca}$ is identified as the optimal target, offering the best balance between event statistics and shape sensitivity.

5. Significance

• New Physics Avenue: This work establishes KDAR-based CE$\nu$NS as a robust, complementary method for measuring neutron skins, distinct from hadronic probes and PVES. It relies on a clean neutral-current interaction with different systematic uncertainties.
• Nuclear Equation of State: Precise measurements of $\Delta R_{np}$ directly constrain the density dependence of the nuclear symmetry energy and the pressure of neutron-rich matter, which are critical for understanding neutron star structure and mergers.
• Experimental Roadmap: The results provide a quantitative justification for future CE$\nu$NS experiments (like JSNS2 or future KDAR sources) to prioritize medium-mass targets ($^{48}\text{Ca}$) and aim for high-statistics runs to resolve the neutron skin thickness with $\sim 0.02$ fm precision.
• Theoretical Validation: By validating the transition from coherence to diffraction in neutrino scattering, the paper confirms that neutrino beams can serve as high-resolution probes of nuclear surface structure, extending the utility of CE$\nu$NS beyond simple cross-section measurements.