Nuclear Schiff moment of fluorine isotope $^{19}$F

This study presents the first *ab initio* calculation of the nuclear Schiff moment for the $^{19}$F isotope using the no-core shell model, which, when combined with quantum chemistry calculations and experimental data on HfF$^+$, establishes the first experimental bound on this moment and its associated pion-nucleon-nucleon coupling constants.

The Gist

Imagine the universe as a giant, perfectly symmetrical dance floor. For decades, physicists believed that if you played the music backward (Time-Reversal) or looked at the dance in a mirror (Parity-Inversion), the dancers would move exactly the same way. But deep down, they suspect the music has a tiny, hidden "glitch" that makes the dance slightly different in the mirror or when played backward. Finding this glitch is the holy grail of modern physics because it could explain why our universe is made of matter instead of disappearing into nothingness.

This paper is about hunting for that glitch using a very specific, tiny dancer: the Fluorine-19 atom (a common isotope of fluorine, the stuff in toothpaste).

Here is the story of how they did it, broken down into simple concepts:

1. The "Ghost" in the Machine (The Nuclear Schiff Moment)

Inside an atom's nucleus, protons and neutrons are dancing. If the laws of physics are perfectly symmetrical, the nucleus is a perfect sphere. But if there is a "glitch" (a violation of symmetry), the nucleus gets a tiny, lopsided charge distribution.

Think of the nucleus as a spinning top. If it's perfectly balanced, it spins smoothly. If it's slightly unbalanced (due to the symmetry glitch), it wobbles. This wobble creates a "Schiff Moment." It's a bit like a ghostly magnetic field that shouldn't exist if the universe were perfectly symmetrical.

The problem? This ghost is usually hidden. The cloud of electrons swirling around the nucleus acts like a noise-canceling headphone, screening out most of the wobble so we can't see it. However, in heavy atoms or specific molecules, this "headphone" isn't perfect, and a tiny bit of the wobble leaks through.

2. The Detective Work: Two Steps to the Truth

To catch this ghost, the scientists had to do two very hard things:

1. Predict the Ghost: Calculate exactly how much the Fluorine-19 nucleus should wobble if the symmetry glitch exists.
2. Measure the Leak: Calculate how much that wobble would shake a specific molecule (Hafnium Fluoride, or HfF+) so they could compare it to real-world experiments.

Step A: The Nuclear Calculation (The "Ab Initio" Approach)

Usually, physicists use "phenomenological models" to guess how nuclei behave. This is like trying to predict the weather by looking at a map of clouds and guessing. It works okay, but it's not perfect.

In this paper, the team did something revolutionary. They used "Ab Initio" methods.

• The Analogy: Instead of guessing, they built a digital Lego simulation of the Fluorine-19 nucleus from the ground up. They took every single proton (9 of them) and neutron (10 of them) and simulated how they interact using the fundamental laws of quantum mechanics.
• The Challenge: Simulating 19 particles interacting is like trying to solve a Rubik's cube where every piece is moving and changing color at the same time. It requires a supercomputer.
• The Result: They found that the Fluorine-19 nucleus is actually a "super-dancer." Because it has a special excited state (a partner dance move just above its normal state), any symmetry glitch gets amplified. The wobble is much bigger in Fluorine-19 than in many heavier, more complex nuclei.

Step B: The Molecular Calculation (The "Sensitivity" Check)

Now that they knew how big the wobble could be, they needed to know how easy it would be to see in a molecule. They looked at a molecule called HfF+ (Hafnium Monofluoride cation).

• The Analogy: Imagine the Fluorine nucleus is a tiny, wobbly magnet. The HfF+ molecule is a sensitive seismograph. The scientists used quantum chemistry to calculate exactly how much the "seismograph" would shake if the "magnet" wobbled.
• The Discovery: They found that HfF+ is incredibly sensitive to the Fluorine wobble, even though the wobble itself is tiny.

3. The Big Reveal: Setting a Limit

The team combined their supercomputer nuclear simulation with their quantum chemistry seismograph calculations and compared it to a recent, ultra-precise experiment measuring the HfF+ molecule.

• The Experiment: Scientists measured the energy levels of HfF+ with such precision that they could detect changes smaller than a single atom's width.
• The Conclusion: They didn't find the "glitch" (the symmetry violation). But that's a good thing!
  • Because they didn't find it, they can now say: "If the glitch exists, it must be smaller than X."
  • This sets a new, strict boundary (a limit) on how big the symmetry violation can be.

Why Does This Matter?

1. New Physics: If the "glitch" is bigger than their limit, our current understanding of the universe (The Standard Model) is wrong, and we need new physics. Since they set a very tight limit, they are forcing new theories to be even more creative.
2. A New Tool: This is the first time anyone has calculated this "Schiff Moment" using a pure, first-principles simulation (Ab Initio) rather than a guess-work model. It's like moving from drawing a map of a city based on rumors to using a satellite to see every street.
3. The Future: Fluorine-19 is light, so it's easier to simulate. The scientists hope this success will teach them how to simulate heavier, more complex nuclei (like Radium or Thorium) in the future, which are even better at amplifying these symmetry glitches.

The Bottom Line

This paper is a masterclass in precision. The team built a digital twin of a Fluorine atom, figured out how it would wobble if the universe's laws were slightly broken, and then checked a real molecule to see if that wobble was there. They didn't find the break, but they proved that if it exists, it's hiding in a very small corner. This tightens the noose on new physics theories and proves that we can now simulate these tiny nuclear dances with incredible accuracy.

Technical Summary

1. Problem Statement

The search for physics beyond the Standard Model (BSM) often focuses on detecting violations of Parity (P) and Time-reversal (T) symmetries. A primary experimental signature of such violations is the Electric Dipole Moment (EDM) of atoms and molecules.

• The Challenge: Interpreting molecular EDM measurements requires a precise theoretical link between the observed energy shift and the underlying P,T-violating fundamental interactions (specifically, pion-nucleon-nucleon coupling constants, $\bar{g}_{0,1,2}$).
• The Bottleneck: This link requires two components:
  1. Molecular Sensitivity: How the molecule amplifies the nuclear effect (calculable with high precision via quantum chemistry).
  2. Nuclear Structure: How the nuclear P,T-violating forces generate a Nuclear Schiff Moment (NSM). Historically, NSM calculations have relied on phenomenological nuclear models (e.g., shell models with tuned parameters), which suffer from large uncertainties (often varying by an order of magnitude or even changing sign) and lack systematic error quantification.
• The Gap: There was no ab initio (first-principles) calculation of an NSM to provide a reliable, systematic benchmark for heavier nuclei or to constrain BSM physics with controlled uncertainties.

2. Methodology

The authors employed a dual approach combining Nuclear Ab Initio Theory and Relativistic Quantum Chemistry.

A. Nuclear Structure Calculations (The NSM of $^{19}$F)

• Framework: The No-Core Shell Model (NCSM) was used. This treats all 9 protons and 10 neutrons explicitly, avoiding the need for an inert core approximation.
• Interactions:
  • Parity-Conserving (PC): Chiral Effective Field Theory ($\chi$EFT) interactions were used, specifically NN-N4LO + 3N$^*_{lnl}$ and NN-N3LO + 3N$_{lnl}$. These were softened using the Similarity Renormalization Group (SRG) technique.
  • P,T-Violating (PTV): The leading-order (LO) P,T-violating nucleon-nucleon interaction arising from one-pion exchange was included.
• Computational Technique:
  • The Lanczos strength method was employed to solve the generalized Schrödinger equation. This allows for the exact calculation of the nuclear many-body Green's function, effectively summing over all intermediate states connected by the P,T-violating interaction and the Schiff operator.
  • Basis Size: Calculations were performed up to $N_{max}=6$ (dimension $\sim 1.35$ billion) for positive parity states and $N_{max}=5$ for negative parity states. For $N_{max}=7$, Importance-Truncated NCSM (IT-NCSM) was used to handle the computational cost.
  • Energy Shift: Calculated ground-state energies were shifted to match experimental excitation energies of the lowest opposite-parity state ($1/2^-$) to ensure accuracy in the energy denominators.

B. Molecular Sensitivity Calculations

• Target: The Hafnium monofluoride cation (HfF$^+$), specifically the $a\,^3\Delta_1$ state, which was used in recent high-precision EDM experiments.
• Method: Relativistic Exact Two-Component Coupled-Cluster Singles and Doubles (X2C-CCSD) with analytic gradient techniques.
• Basis Sets: Extended basis sets were constructed to accurately describe the electron density gradient near the $^{19}$F nucleus (critical for NSM sensitivity).
  • Fluorine: ETB0 (30s30p4d3f2g).
  • Hafnium: ANO-RCC (24s21p15d11f4g2h).
• Error Analysis: Systematic studies of basis set convergence, electron correlation (up to CCSD(T)), and relativistic effects were performed to estimate uncertainties.

3. Key Contributions

1. First Ab Initio NSM Calculation: This is the first-ever calculation of a Nuclear Schiff Moment using a fully ab initio nuclear many-body framework (NCSM).
2. Benchmarking Heavy Nuclei: By calculating the NSM for the light nucleus $^{19}$F, the authors established a rigorous benchmark for future ab initio efforts targeting heavier, octupole-deformed nuclei (like $^{225}$Ra or $^{227}$Ac) where phenomenological models currently dominate.
3. Structural Insight: The study revealed that the NSM of $^{19}$F is dominated by the contribution from the lowest opposite-parity excited state ($1/2^-$), whereas the nuclear EDM is negligible for this transition. This is attributed to the distinct cluster structures: the ground state ($1/2^+$) is shell-model-like ($^{18}$O+p), while the excited state ($1/2^-$) exhibits strong $\alpha$-clustering ($^{15}$N+$\alpha$).
4. Experimental Constraint: The work provides the first theoretical framework to interpret recent HfF$^+$ EDM measurements specifically in terms of the $^{19}$F NSM, rather than just the electron EDM.

4. Results

Nuclear Schiff Moment ($^{19}$F)

The calculated NSM is expressed in terms of the P,T-violating $\pi$NN coupling constants ($\bar{g}_0, \bar{g}_1, \bar{g}_2$):
$$S(^{19}\text{F}) = (-2.9 \bar{g}_0 - 2.4 \bar{g}_1 - 2.0 \bar{g}_2) \times 10^{-2} \, e \, \text{fm}^3$$

• Uncertainty: Estimated at approximately 50%, arising from basis size extrapolation, HO frequency variation, and interaction model dependence.
• Magnitude: The magnitude is comparable to that of the heavy nucleus $^{129}$Xe, despite $^{19}$F being much lighter, due to the low-lying opposite-parity state and large matrix elements.

Molecular Sensitivity ($W_S$)

The molecular sensitivity factor for HfF$^+$ was calculated as:
$$W_S = 115 \, \left( \frac{e}{4\pi\epsilon_0 a_0^4} \right) \approx 5100 \, \text{Hz}/(e \, \text{fm}^3)$$

• Precision: Theoretical uncertainty is estimated to be < 10%.

Experimental Bounds

Combining the theoretical results with the recent HfF$^+$ EDM measurement (Roussy et al., Science 2023):

• NSM Bound: $|S(^{19}\text{F})| < 9.0 \times 10^{-9} \, e \, \text{fm}^3$ (90% confidence level).
• Coupling Constants:
  • $|\bar{g}_0| < 2.3 \times 10^{-8}$
  • $|\bar{g}_1| < 2.8 \times 10^{-8}$
  • $|\bar{g}_2| < 3.3 \times 10^{-8}$
• QCD $\theta$-term: Assuming standard relations, the bound on the strong CP-violating parameter is $|\bar{\theta}| < 1.6 \times 10^{-6}$.

Note: While these bounds are not yet the most stringent compared to those derived from heavy nuclei (like $^{199}$Hg or $^{225}$Ra), they are the first derived using a systematic ab initio nuclear framework.

5. Significance

• Validation of Theory: The work demonstrates that ab initio methods can successfully tackle P,T-violating observables, providing a path to reduce the theoretical uncertainties that currently plague the interpretation of EDM experiments.
• Future Roadmap: It sets a critical benchmark for extending ab initio calculations to heavier, deformed nuclei. Once the many-body challenges for heavy systems are overcome, the precision of NSM predictions will improve dramatically, potentially surpassing current phenomenological limits.
• Complementary Constraints: The study highlights that molecules containing $^{19}$F (like HfF$^+$) can provide competitive constraints on P,T-violating physics, especially as experimental precision improves and theoretical uncertainties are reduced.
• Structural Physics: The discovery that the NSM is dominated by a specific low-lying state with a distinct cluster structure offers new insights into how nuclear structure amplifies symmetry-violating effects, distinguishing it from the behavior of nuclear EDMs.

In summary, this paper bridges the gap between fundamental particle physics and nuclear structure theory, establishing a new standard for interpreting EDM experiments through rigorous, first-principles nuclear calculations.