A New Low $Q^2$ Measurement of the Proton's $g_1$ Spin Structure Function from Longitudinal & Transverse Polarized Data

Technical Summary of "A New Low Q² Measurement of the Proton's g1 Spin Structure Function from Longitudinal & Transverse Polarized Data"

Problem and Motivation
The internal structure of the nucleon, which constitutes the majority of visible mass, remains incompletely understood, particularly in the low momentum transfer ($Q^2$) regime. While Quantum Chromodynamics (QCD) is well-established at high $Q^2$ due to asymptotic freedom, effective theories such as Chiral Perturbation Theory ($\chi$PT) face challenges in the low $Q^2$ domain, where predictions for various observables are often untested or in internal conflict. A critical method for testing these theories involves measuring spin structure function moments, specifically those derived from the proton's spin structure functions $g_1$ and $g_2$. Prior to this work, while low $Q^2$ measurements of $g_1$ existed, there were no published results utilizing data from both longitudinally and transversely polarized targets collected at the same kinematics. This limitation necessitated reliance on models to disentangle the contributions of $g_1$ and $g_2$.

Methodology
The paper presents results from the Jefferson Lab E08-027 experiment, conducted in Hall A in 2012. The experiment utilized a longitudinally polarized ammonia ($NH_3$) target, polarized via Dynamic Nuclear Polarization, and a spin-polarized electron beam with energies ranging from 1.1 to 3.3 GeV and a current of 85 nA.

The core methodology involved measuring double-spin asymmetries for both longitudinally ($A_\parallel$) and transversely ($A_\perp$) polarized targets. These asymmetries were combined with the Bosted-Christy unpolarized cross-section model to determine the polarized cross-section differences ($\Delta\sigma_\parallel$ and $\Delta\sigma_\perp$). A unique feature of this analysis is the simultaneous collection of longitudinal and transverse data at a specific kinematic setting (beam energy 2.2 GeV, average $Q^2 = 0.045$ GeV$^2$). This allowed for the direct extraction of both $g_1$ and $g_2$ without relying on a Spin-Structure-Function Model to separate the two, a significant departure from previous analyses that required such model inputs.

The extracted $g_1$ values were corrected to a constant $Q^2$ of 0.045 GeV$^2$ using the Hall B model, a procedure contributing less than 2% to the systematic uncertainty. The data covers the invariant mass range $W = 1.078$ to 1.8 GeV, extending down to the pion production threshold ($W = 1078.2$ MeV).

Key Results
The primary result is the first presentation of the $g_1$ proton structure function from the E08-027 experiment at very low $Q^2$.

• $g_1$ Behavior: The data reveals that $g_1$ remains negative across the measured resonance region, including just above the pion production threshold. This contrasts with phenomenological models (Hall B and MAID) and other experimental data (EG4, EG1b) which suggest $g_1$ crosses zero and becomes positive in this threshold region. While the E08-027 data is consistent with a positive result within uncertainties, the central value remains negative.
• Resonance Structure: The $\Delta(1232)$ resonance observed in the E08-027 data is less negative (smaller in magnitude) than the EG4 results, showing better agreement with EG1b.
• Moments: The paper calculates several integrated moments:
  • $\Gamma_1$ (First Moment): Calculated by integrating $g_1$ from $x=0$ to 1, with the low-$x$ region filled using the CLAS Hall B model.
  • $\gamma_0$ (Generalized Forward Spin Polarizability): This moment is highly sensitive to the pion production threshold due to kinematic weighting ($x^2$ and $x^4$ terms). The E08-027 result favors the Bernard et al. $\chi$PT calculation, whereas Hall B data (EG4, EG1b) favors the Alarcon et al. calculation. The discrepancy is attributed to the sign difference in $g_1$ near the threshold and the inclusion of E08-027's transverse data in the $g_2$ extraction, which carries a strong $x^4$ weighting in the $\gamma_0$ integral.
  • $I_A$ (Generalized GDH Integral): The E08-027 data for $I_A$ is in close agreement with Hall B results and $\chi$PT calculations, though the convergence to the Gerasimov-Drell-Hearn (GDH) sum rule limit at $Q^2 \to 0$ requires further lower $Q^2$ measurements.

Significance and Claims
The paper asserts that the E08-027 results provide a proton $g_1$ extraction measured with very high precision across the resonance region without the model dependence inherent in previous low $Q^2$ extractions. The significance of these findings lies in:

1. Model Independence: The ability to extract $g_1$ and $g_2$ simultaneously from longitudinal and transverse data eliminates the need for model assumptions to separate the structure functions.
2. Threshold Behavior: The data highlights a tension between the E08-027 results and other measurements/models near the pion production threshold. The authors note that the behavior of $g_1$ in this region is crucial for higher moments (like $\gamma_0$) and impacts $\chi$PT calculations of hydrogen hyperfine splitting and lattice QCD extrapolations.
3. Validation of Sum Rules: The results provide a high-precision cross-check for the magnitude of $\Gamma_1$ and $I_A$ at low $Q^2$, contributing to the validation of the GDH slope and sum rule.

The authors conclude that these results, combined with previously published transversely polarized data from the same experiment, offer a new piece of the nucleon structure puzzle, helping to illuminate the structure of the proton and test the limits of effective QCD theories. They acknowledge that the source of the tension near the pion threshold is not yet well understood and that additional measurements at even lower $Q^2$ are necessary to fully observe the convergence to the GDH sum rule.