[Paper Review] Review of Indirect Methods Used to Determine the $^1S_0$ Neutron-Neutron Scattering Length

C. R. Howell|ArXiv.org|May 8, 2008

Nuclear Physics and Applications1 references4 citations

TL;DR

This paper reviews indirect methods for determining the neutron-neutron $^1S_0$ scattering length $a_{nn}$, focusing on $\pi^{-}d$ capture and neutron-deuteron breakup reactions. It finds that recent $\pi^{-}d$ capture experiments yield a consistent value of $a_{nn} = -18.6 \pm 0.3$ fm (experimental) $\pm 0.3$ fm (theoretical), while kinematically incomplete nd breakup analyses show a significant downward shift in $a_{nn}$ when reanalyzed with modern theory, indicating unresolved theoretical or experimental systematics.

ABSTRACT

We have determined a value for the $^1S_0$ neutron-neutron scattering length ($a_{nn}$) from high-precision measurements of time-of-flight spectra of neutrons from the $^2H(π^-,n γ)n$ capture reaction. The measurements were done at the Los Alamos Meson Physics Facility by the E1286 collaboration. The high spatial resolution of our gamma-ray detector enabled us to make a detailed assessment of the systematic uncertainties in our techniques. The value obtained in the present work is $a_{nn} = -18$.63 $\pm $0.10 (statistical) $\pm$ 0.44 (systematic) $\pm$ 0.30 (theoretical) fm. This result is consistent with previous determinations of $a_{nn}$ from the $π^-d$ capture reaction. We found that the analysis of the data with calculations that use a relativistic phase-space factor gives a more negative value for $a_{nn}$ by 0.33 fm over the analysis done using a nonrelativistic phase-space factor. Combining the present result with the previous ones from $π^-d$ capture gives: $a_{nn} = - 18$.63 $\pm$ 0.27 (expt) $\pm$ 0.30 fm (theory). For the first time the combined statistical and systematic experimental uncertainty in $a_{nn}$ is smaller than the theoretical uncertainty and comparable to the uncertainty in the proton-proton $^1S_0$ scattering length ($a_{pp}$). This average value of $a_{nn}$ when corrected for the magnetic-moment interaction of the two neutrons becomes -18.9 $\pm$ 0.4 fm which is 1.6 $\pm$ 0.5 fm different from the recommended value of $a_{pp}$, thereby confirming charge symmetry breaking at the 1% confidence level.

Motivation & Objective

• To evaluate the reliability and consistency of indirect experimental methods for determining the neutron-neutron $^1S_0$ scattering length $a_{nn}$, which remains experimentally inaccessible via direct free-neutron scattering.
• To resolve the long-standing discrepancy between $a_{nn}$ values derived from $\pi^{-}d$ capture and nd breakup experiments, both of which are sensitive to final-state interactions.
• To identify and address theoretical and experimental systematics affecting $a_{nn}$ determinations, particularly the large shift observed in reanalysis of kinematically incomplete nd breakup data.
• To recommend future steps, including direct measurements of $a_{nn}$ using reactor-based neutron beams, as a path to resolving current inconsistencies.

Proposed method

• The study reviews experimental techniques and theoretical frameworks used in $\pi^{-}d$ capture and nd breakup reactions, focusing on kinematically complete (KC) and incomplete (KI) measurements.
• For $\pi^{-}d$ capture, $a_{nn}$ is extracted by fitting the $\gamma$-ray energy spectrum or neutron time-of-flight spectrum using modern three-nucleon Faddeev calculations.
• In nd breakup, $a_{nn}$ is determined by fitting the neutron-neutron final-state interaction (FSI) enhancement in the proton energy spectrum, particularly at high energies.
• The analysis includes reprocessing of older KI nd breakup data using updated theoretical models, revealing a systematic shift in $a_{nn}$ values by approximately 3 fm toward smaller magnitudes.
• Theoretical uncertainties are quantified by comparing results from different potential models and Faddeev calculations, with emphasis on sensitivity to the NN potential strength.
• A statistical average of recent high-precision $\pi^{-}d$ capture experiments is used to derive a recommended $a_{nn}$ value, accounting for both experimental and theoretical uncertainties.

Experimental results

Research questions

• RQ1 Why do $\pi^{-}d$ capture and nd breakup experiments yield significantly different values for the neutron-neutron $^1S_0$ scattering length $a_{nn}$?
• RQ2 What causes the observed 3 fm downward shift in $a_{nn}$ when kinematically incomplete nd breakup data are reanalyzed with modern theory?
• RQ3 How do experimental systematics—such as neutron detection efficiency and target attenuation—affect the determination of $a_{nn}$ in $\pi^{-}d$ capture and nd breakup experiments?
• RQ4 Why is the $\pi^{-}d$ capture method considered more reliable than nd breakup for $a_{nn}$ determination despite both being indirect?
• RQ5 What are the key theoretical and experimental challenges that must be overcome to achieve a direct measurement of $a_{nn}$?

Key findings

• The most recent high-precision $\pi^{-}d$ capture experiments yield consistent results, supporting a recommended value of $a_{nn} = -18.6 \pm 0.3$ fm (experimental) $\pm 0.3$ fm (theoretical).
• Reanalysis of kinematically incomplete nd breakup data using modern Faddeev theory results in a systematic shift of $a_{nn}$ by approximately 3 fm toward smaller values, averaging to about 16 fm instead of the original 19 fm.
• The discrepancy in $a_{nn}$ values between $\pi^{-}d$ capture and reanalyzed KI nd breakup data is attributed to a mismatch between measured proton energy spectra in the flat region and theoretical predictions, affecting normalization and thus $a_{nn}$ extraction.
• Kinematically complete nd breakup experiments show significant discrepancies among recent results, suggesting unresolved systematics in either experimental setup or theoretical modeling.
• The $^1S_0$ scattering length is highly sensitive to NN potential strength, with a 1% change in potential depth causing a 10–30% shift in $a_{nn}$, underscoring its importance in probing charge symmetry breaking.
• A direct measurement of $a_{nn}$ is recommended as the next critical step, with the DIANNA collaboration planning such an experiment using the high-flux YAGUAR reactor in Russia.