First Constraint on P-odd/T-odd Cross Section in Polarized Neutron Transmission through Transversely Polarized $^{139}$La

This paper presents the first constraint on time-reversal invariance violating effects in polarized neutron transmission through a transversely polarized $^{139}$La target by applying a density matrix formalism to existing data, yielding an upper limit of $|W_T|<15~\mathrm{eV}$ at 90% confidence level despite the data's lack of optimization for such observables.

The Gist

The Big Picture: Hunting for a "Time-Travel" Glitch

Imagine the universe has a set of strict rules, like the laws of physics in a video game. One of these rules is Time-Reversal Invariance. In simple terms, it means that if you recorded a movie of a particle interaction and played it backward, the backward movie would look just as physically possible as the forward one.

However, physicists suspect there might be a tiny "glitch" in the code—a violation of this rule. If they can find it, it would explain why our universe is made of matter instead of antimatter (a mystery known as CP violation).

This paper is about a team of scientists trying to find that glitch using neutrons (tiny particles inside atoms) and a special target made of Lanthanum-139 (a heavy metal).

The Experiment: The "Spin" Dance

To find this glitch, the scientists set up a very specific dance:

1. The Dancers: They shot a beam of neutrons at a target. But these weren't just any neutrons; they were polarized, meaning all their internal "spins" were pointing in the same direction, like a crowd of people all raising their right hands.
2. The Stage: The target (Lanthanum-139) was also polarized, but in a different way. Imagine the target atoms are spinning like tops, but the scientists forced them to spin sideways (transversely) using a massive magnet.
3. The Goal: They wanted to see if the neutrons passed through the target differently depending on the direction of their spin. If the laws of physics were perfectly symmetrical in time, the neutrons would behave the same way going forward or backward. If there's a glitch (TRIV), the neutrons would act slightly differently.

The Challenge: Finding a Needle in a Haystack

The problem is that this "glitch" is incredibly small. It's like trying to hear a pin drop in the middle of a rock concert.

• The Haystack: The experiment was originally designed to measure something else entirely (how neutrons interact with the target's spin). It wasn't built specifically to hunt for the time-reversal glitch.
• The Needle: The signal for the glitch is so faint that it gets drowned out by other, much stronger effects.

The scientists had to use a very sophisticated mathematical toolkit (called density matrix formalism) to filter out the noise. Think of it like using a high-end noise-canceling headphone app to isolate a single whisper from a loud crowd. They had to account for the shape of the target, the strength of the magnets, and even the tiny imperfections in the neutron beam.

The Results: No Glitch Found (Yet)

After crunching the numbers and analyzing the data:

• The Verdict: They did not find a statistically significant signal. The neutrons behaved exactly as the standard rules of physics predicted.
• The Silver Lining: Even though they didn't find the glitch, they proved their "noise-canceling" math works. They showed that their theoretical framework can handle real-world data.
• The New Limit: They set a new "speed limit" for how big this glitch could possibly be. They concluded that if the glitch exists, it must be smaller than 15 eV (a unit of energy).

The Analogy: The Broken Compass

Imagine you are trying to find a tiny, invisible magnetic field that pulls a compass needle slightly off-course.

1. You have a compass (the neutron) and you spin it around a giant magnet (the Lanthanum target).
2. You expect the compass to point North.
3. You look very closely to see if it points slightly East or West due to a mysterious force.
4. The wind is blowing hard (other physical effects), making the compass wobble.
5. You use a super-precise model to calculate exactly how much the wind should make it wobble.
6. You subtract the "wind wobble" from the actual movement.
7. Result: The compass is still pointing exactly North.
8. Conclusion: The mysterious force is either non-existent, or it is so weak that it's smaller than the smallest wobble your model can detect.

Why Does This Matter?

Even though they didn't find the "Time-Travel Glitch," this paper is a crucial step forward.

• Proof of Concept: It proves that the complex math they developed actually works on real data.
• Roadmap for the Future: Now that they know how to filter the noise, they can build better experiments. Future versions of this experiment will be much more sensitive, potentially able to detect a glitch that is 100,000 times smaller than what they could see here.

In short: They didn't find the treasure, but they successfully mapped the terrain and proved their metal detector works. This gives them the confidence to keep digging in deeper, more sensitive places.

Technical Summary

1. Problem Statement

The search for Time-Reversal Invariance Violating (TRIV) effects is a critical experimental avenue for discovering CP violation beyond the Standard Model, mediated by the CPT theorem. While Parity-Violating (PV) effects in neutron-nucleus interactions have been observed with significant enhancement (up to $10^{-2}$) in p-wave resonances, TRIV effects remain elusive.

The specific challenges addressed in this work include:

• Theoretical Complexity: Developing a rigorous theoretical framework to describe neutron spin propagation through polarized nuclear targets, specifically accounting for higher-rank tensor polarizations (up to rank 3) in the forward scattering amplitude.
• Experimental Limitations: Existing high-precision transmission data (specifically from Ref. [18] on $^{139}\text{La}$) were originally designed to measure spin-dependent cross sections (PV effects), not TRIV observables. Consequently, the sensitivity to TRIV parameters in these datasets is intrinsically limited by the experimental geometry (transverse polarization) and lack of optimization for P-odd/T-odd signals.
• Goal: To reanalyze existing transmission data using an extended density matrix formalism to derive the first quantitative constraints on TRIV effects in this specific experimental configuration.

2. Methodology

Theoretical Framework

The authors formulated the transmission asymmetry using the density matrix formalism, explicitly incorporating the forward scattering amplitude ($f$) for $^{139}\text{La}$ ($I=7/2$).

• Scattering Amplitude: The amplitude was expanded to include tensor polarization terms up to third-rank ($q \leq 3$). The amplitude $f$ includes spin-independent terms and three spin-dependent terms ($\beta$), where the component $\beta_{\hat{k}_n \times \hat{I}}$ is odd under both Parity (P) and Time-Reversal (T).
• Asymmetry Definition: The theoretical analyzing power $A_I(\vartheta_{kI})$ was derived as a function of the angle between neutron momentum ($\hat{k}_n$) and nuclear spin ($\hat{I}$). The expression relates the measured asymmetry to the interference between spin-independent and spin-dependent amplitudes ($A^*B$ and $C^*D$).
• TRIV Contribution: In the specific experimental geometry (transverse polarization), the scalar triple product $\sigma_n \cdot (\hat{k}_n \times \hat{I})$ vanishes. However, TRIV effects arise through the absorption associated with the imaginary part of the TRIV amplitude ($D'$). The observable contribution scales as $\text{Re}(D')\text{Im}(D')$, which is proportional to the square of the nuclear polarization ($P_1^2$) and inversely proportional to the external magnetic field.

Experimental Reanalysis

• Data Source: The study utilized existing transmission data from the RADEN beamline at J-PARC (Japan Proton Accelerator Research Complex). The experiment involved a pulsed neutron beam passing through a transversely polarized $^{139}\text{La}$ target at a temperature of $\sim 76$ mK.
• Configuration: Neutrons were polarized by a $^3\text{He}$ spin filter. The target was placed in a strong transverse magnetic field ($6.8$ T). The neutron spin was flipped periodically to measure the asymmetry between parallel and antiparallel spin configurations.
• Fitting Procedure:
  • The measured asymmetry was fitted over a Time-of-Flight (TOF) range of $0$ to $3.4$ ms.
  • A quadratic background was used to model s-wave resonances ($s_0$ and $s_1$), while regions around the $s_1$ resonance ($72.3$ eV) and an impurity resonance ($^{138}\text{La}$ at $2.99$ eV) were excluded.
  • The p-wave resonance ($0.75$ eV) was modeled using the derived theoretical expression involving the TRIV parameter $W_T$, the mixing angle $\phi$, and the resonance energy $E_p$.
  • A global $\chi^2$ minimization was performed over the parameter space ($W_T, \phi, E_p$) to extract confidence intervals.

3. Key Contributions

1. Extended Formalism: The paper provides a complete derivation of the transmission asymmetry for transversely polarized targets, explicitly including third-rank tensor polarization terms. This fills a gap in the theoretical description required for high-precision TRIV searches.
2. Validation on Real Data: It demonstrates the applicability of this complex theoretical framework to real, non-optimized experimental data, proving that constraints can be extracted even when the experiment was not originally designed for TRIV.
3. Methodology for Degeneracy Handling: The authors developed a robust statistical approach to handle the multimodal structure of the $\chi^2$ surface (where different combinations of resonance energy $E_p$ and mixing angle $\phi$ yield similar fits). This ensures that the derived limits on $W_T$ are conservative and robust against parameter degeneracies.

4. Results

• Statistical Significance: No statistically significant TRIV signal was observed. The best-fit value for the TRIV matrix element was $W_T = 0.02 \pm 12$ eV, consistent with zero.
• Upper Limit on $W_T$: By analyzing the global $\chi^2$ structure and profile distributions, the authors established a 90% confidence level upper limit:
  $$|W_T| < 15 \text{ eV}$$
• Upper Limit on Cross Section: Using the optical theorem and the resonance-averaged parameters, this translates to an upper limit on the TRIV cross section:
  $$|\Delta \sigma_{\not{T}\not{P}}| < 8.3 \times 10^2 \text{ b}$$
• Parameter Correlations: The analysis revealed that $W_T$ has weak correlations with other parameters ($\phi$ and $E_p$). However, the $\chi^2$ surface for $E_p$ exhibited a multimodal structure with local minima around $0.766$ eV and $0.792$ eV. Despite these variations, the upper limit on $W_T$ remained stable across all solutions.
• Sensitivity Estimate: The derived limit corresponds to a ratio $|W_T/W| \lesssim 10^4$ (assuming a representative PV matrix element $W \sim 1$ meV). This is orders of magnitude less sensitive than future dedicated experiments (which aim for $10^{-6}$), but serves as a crucial proof-of-concept.

5. Significance

• Proof of Concept: This work validates the theoretical density matrix formalism for P-odd/T-odd observables in polarized neutron transmission, confirming that the framework correctly describes real experimental data.
• Guidance for Future Experiments: The study highlights the specific challenges of transverse polarization geometry (suppression of TRIV signals by the external magnetic field) and the necessity of optimizing experimental parameters (e.g., maximizing tensor polarization $P_1$) for future dedicated searches.
• Benchmark for NOPTREX: As part of the J-PARC E99 program (NOPTREX collaboration), these results provide a baseline and methodological roadmap for the next generation of experiments aiming to probe CP violation at the $10^{-6}$ level.
• Constraint on New Physics: While the limit is not yet competitive with other TRIV searches, it provides a new, independent constraint on time-reversal violating interactions in the nucleon-nucleon sector using a different experimental observable (transmission asymmetry vs. absorption).