Atomic parity violation in highly charged $^{40,48}$Ca and $^{208}$Pb ions

This paper calculates parity-violation-induced E1 amplitudes for specific transitions in H- and Li-like calcium and lead ions, concluding that while neutron-skin corrections can be neglected for calcium isotope comparisons in searches for new $Z'$ boson physics, both neutron-skin effects and sensitivity to such interactions are significant for $^{208}$Pb.

The Gist

Imagine the atom as a tiny solar system. In the center, you have the nucleus (the sun), made of protons and neutrons. Orbiting around it are electrons (the planets).

For decades, physicists have been trying to understand the "rules of the game" that govern how these particles interact. The current rulebook is called the Standard Model. It's a great book, but physicists suspect there might be missing chapters—clues to a "New Physics" that explains things like dark matter or gravity in a new way.

One way to find these missing chapters is to look for Atomic Parity Violation (APV).

The "Handedness" of the Universe

Most laws of physics are "symmetrical." If you look at a mirror image of a ball bouncing, it looks just as real as the original. But the Weak Force (one of the four fundamental forces) is different. It has a "handedness." It prefers left over right.

In a normal atom, electrons don't really care about this preference. But if you look very closely, the Weak Force causes a tiny, almost invisible "wobble" in how electrons move. This wobble is the Parity Violation.

The Problem: Too Much Noise

Usually, scientists study this in neutral atoms (like Cesium or Ytterbium). But these atoms are messy. They have many electrons buzzing around, making it hard to calculate exactly what should happen according to the Standard Model. If the experiment doesn't match the theory, is it because of a new force, or just because our math for the messy atom was slightly off?

The Solution: The authors of this paper suggest using Highly Charged Ions.
Think of this as stripping an atom down to its bare essentials.

• Hydrogen-like ions: Just one electron orbiting a heavy nucleus.
• Lithium-like ions: Just three electrons.

It's like comparing a chaotic mosh pit (neutral atoms) to a quiet conversation between two people (highly charged ions). In the quiet conversation, it's much easier to hear if someone whispers a secret (a new force).

The Cast of Characters

The paper focuses on three specific "actors":

1. Calcium-40 ($^{40}\text{Ca}$) and Calcium-48 ($^{48}\text{Ca}$): Two versions of the same element. They have the same number of protons (20), but different numbers of neutrons (20 vs. 28).
2. Lead-208 ($^{208}\text{Pb}$): A very heavy nucleus with a lot of neutrons.

The "Neutron Skin" Mystery

Inside the nucleus, protons and neutrons aren't perfectly mixed. Sometimes, the neutrons form a fuzzy "skin" around the core of protons. This is called the Neutron Skin.

• Why it matters: The Weak Force talks to neutrons much more loudly than it talks to protons. If the neutron skin is thick, the "wobble" (APV) gets bigger.
• The Confusion: If we see a big wobble, is it because a new force is there? Or is it just because the neutron skin is thicker than we thought?

The Great Detective Story

The authors ran complex calculations to see how these ions would behave under two scenarios:

1. The Standard Model: Only the known Weak Force (mediated by the $Z^0$ boson).
2. New Physics: A hypothetical new force mediated by a new, heavy particle called the $Z'$ boson.

Here is what they found, using some analogies:

1. The Calcium Twins ($^{40}\text{Ca}$ and $^{48}\text{Ca}$)

Imagine you have two identical twins. They wear the same shirt (same protons) but one is slightly heavier (more neutrons).

• The Surprise: The authors found that for these Calcium ions, the "fuzzy skin" of neutrons doesn't mess up the signal much.
• The Advantage: Because the "noise" from the neutron skin is so low, if you measure a difference between Calcium-40 and Calcium-48, you can be almost certain it's not just a nuclear shape issue. It's likely a sign of that new $Z'$ boson.
• Verdict: These are the perfect "clean rooms" for hunting new physics.

2. The Heavy Lead ($^{208}\text{Pb}$)

Lead is like a giant, crowded city. It has a very thick neutron skin.

• The Result: In Lead, the neutron skin creates a huge signal.
• The Use: This isn't great for finding new forces (because the "skin noise" is too loud), but it's perfect for measuring the skin itself. By measuring the APV in Lead, we can learn exactly how thick the neutron skin is, which helps astrophysicists understand neutron stars.

The "Z'" Boson (The New Force)

The paper also looked at how heavy this new hypothetical particle ($Z'$) might be.

• If the $Z'$ is very light, it acts like a long-range whisper.
• If it's heavy, it's a short-range shout.
• The calculations show that for the Calcium ions, the "whisper" of the new force is easy to hear because the background noise (neutron skin) is quiet.

Summary: Why Should You Care?

This paper is a roadmap for future experiments (like those planned at CERN's "Gamma Factory").

• For New Physics: Use Calcium. It's the cleanest test tube. If we see a deviation there, it's a smoking gun for new physics, not just a quirk of nuclear shapes.
• For Nuclear Physics: Use Lead. It's the best tool to map out the "neutron skin," helping us understand the densest matter in the universe.

In short, the authors have figured out which "microscopes" (ions) to use to see the smallest, most hidden secrets of the universe, separating the signal of new laws of physics from the noise of old nuclear shapes.

Technical Summary

1. Problem Statement

Atomic Parity Violation (APV) is a critical low-energy probe for testing the Standard Model (SM) and searching for new physics, such as light $Z'$ bosons. While APV has been extensively studied in neutral atoms, highly charged ions (HCIs) offer distinct advantages: simpler electronic structures (facilitating precise theoretical calculations) and enhanced wavefunction overlap with the nucleus (amplifying the APV effect).

However, APV measurements are sensitive to nuclear structure, specifically the neutron skin (the difference between neutron and proton root-mean-square radii). Uncertainties in neutron distributions can mimic or obscure signals of new physics. The paper addresses two specific challenges:

1. How do neutron skin effects and nuclear uncertainties impact APV amplitudes in H-like and Li-like ions of Calcium ($^{40,48}\text{Ca}$) and Lead ($^{208}\text{Pb}$)?
2. Can specific isotope pairs (like $^{40}\text{Ca}$ and $^{48}\text{Ca}$) be used to disentangle standard model neutron effects from hypothetical new physics couplings?

2. Methodology

The authors performed high-precision relativistic calculations for transitions in H-like ($1s \to 2s$) and Li-like ($1s^2 2s \to 1s^2 3s$) ions.

• Theoretical Framework:

  • Standard Model (SM): Calculated the parity-violating E1 amplitude induced by the $Z^0$ boson using an effective four-fermion Hamiltonian. The interaction includes spin-independent couplings to protons ($C_1^p$) and neutrons ($C_1^n$), with radiative corrections included.
  • New Physics (NP): Modeled a hypothetical light $Z'$ boson with mass $m_{Z'}$ using a parity-violating Yukawa potential. The interaction strength is parameterized by coupling constants $g_{eN}$ for protons and neutrons.
  • Matrix Elements: The APV amplitude ($E_{PV}$) was calculated using second-order perturbation theory, summing over intermediate states (dominated by $np_{1/2}$ states). The matrix elements were decomposed into proton and neutron contributions to isolate specific effects.

• Nuclear Modeling:

  • Charge Densities: Used normalized Fermi distributions for proton densities, with parameters ($c, z$) tuned to reproduce experimental Barrett moments and moment ratios from electron scattering and muonic atom spectroscopy. Two different parametrizations were used for each isotope to estimate nuclear model uncertainties.
  • Neutron Distributions: Derived from experimental neutron skin data ($\Delta r_{rms} = r_{n,rms} - r_{p,rms}$).
    • $^{40}\text{Ca}$: Assumed a small negative skin ($\approx -0.01$ fm).
    • $^{48}\text{Ca}$: Used CREX collaboration data ($\approx 0.121$ fm).
    • $^{208}\text{Pb}$: Used PREX-II data ($\approx 0.283$ fm).
  • Energy Levels: Calculated using ab initio QED methods (for $n=1,2$) and Dirac-Fock/CI-QEDMOD approaches (for $n \ge 3$) to ensure high precision in energy denominators.

3. Key Contributions

• Decoupling Neutron Skin and New Physics in Calcium: The paper demonstrates that for the $^{40,48}\text{Ca}$ isotope pair, the sensitivity of the APV amplitude to the neutron skin is negligible when searching for light $Z'$ bosons. This is because the proton radii are nearly identical, and the neutron skin contribution to the difference in matrix elements is suppressed.
• Isotopic Ratios as a Diagnostic Tool: The authors derived analytical expressions for the difference and ratio of APV matrix elements between isotopes. They showed that the ratio is primarily sensitive to new proton couplings, while the difference is sensitive to neutron couplings and the neutron skin.
• Lead as a Neutron Skin Probe: Conversely, for $^{208}\text{Pb}$, the large neutron skin creates a significant effect in the SM APV amplitude, making it a viable candidate for measuring neutron distributions via APV, complementary to electron scattering experiments.

4. Key Results

• Standard Model Amplitudes:
  • Calculated total SM APV amplitudes for $1s \to 2s$ in H-like ions and $1s^2 2s \to 1s^2 3s$ in Li-like ions.
  • In $^{208}\text{Pb}$, the difference between proton and neutron contributions to the amplitude is $\approx 0.8\%$, driven by the thick neutron skin and high $Z$.
  • In $^{40,48}\text{Ca}$, this difference is negligible ($\approx 0.03\%$).
• New Physics Sensitivity ($Z'$ Boson):
  • For light $Z'$ bosons ($m_{Z'} < 10$ MeV), the APV amplitude is insensitive to the neutron skin. The matrix elements for protons and neutrons are nearly identical regardless of the skin thickness.
  • Neutron skin effects only become relevant for heavy $Z'$ bosons ($m_{Z'} > 0.1$ GeV), where the difference between proton and neutron contributions remains $< 0.9\%$ even for $^{208}\text{Pb}$.
• The $^{40,48}\text{Ca}$ Advantage:
  • Because the proton radii ($r_p$) and neutron radii ($r_n$) are effectively equal in both isotopes, and the neutron skin is small, the nuclear distribution uncertainty cancels out in isotopic comparisons.
  • The large difference in neutron number ($\Delta N = 8$) between the isotopes amplifies the SM signal, while the cancellation of nuclear uncertainties allows for a clean separation of hypothetical $Z'$ couplings to protons vs. neutrons.
• Uncertainty Analysis:
  • For light $Z'$ searches, nuclear distribution errors can be disregarded in Ca isotopes, significantly reducing the theoretical uncertainty compared to neutral atom experiments.

5. Significance

• New Physics Search: The $^{40,48}\text{Ca}$ isotope pair in highly charged ion form is identified as a "unique testing ground" for new parity-violating physics. The ability to neglect neutron skin uncertainties allows for tighter constraints on the mass and coupling strength of hypothetical $Z'$ bosons.
• Nuclear Structure: The study validates that highly charged $^{208}\text{Pb}$ ions can serve as a complementary tool to PREX-II for measuring neutron skins, leveraging the $Z^3$ scaling of APV effects in heavy ions.
• Experimental Feasibility: The results support the Gamma Factory project at CERN, suggesting that photoexcitation of HCIs offers a pathway to high-precision APV measurements that are theoretically cleaner than those in neutral atoms.

In conclusion, the paper provides a rigorous theoretical foundation suggesting that while $^{208}\text{Pb}$ is ideal for probing nuclear structure, the $^{40,48}\text{Ca}$ pair is superior for isolating and detecting new fundamental interactions beyond the Standard Model due to the cancellation of nuclear structure uncertainties.