Existence of nuclear modifications on longitudinal-transverse structure-function ratio

Technical Summary: Existence of Nuclear Modifications of the Nucleon Longitudinal-Transverse Structure-Function Ratio

Problem Statement
In the analysis of lepton deep inelastic scattering (DIS), it has been a standard assumption that nuclear modifications do not exist for the ratio of the longitudinal to transverse structure functions, $R_N = F_L^N / (2xF_1^N)$. This assumption is widely employed to extract "nucleon" structure functions from nuclear data, particularly using the deuteron as a proxy for the neutron. While nuclear modifications of the structure function $F_2$ are well-established across the $x$ spectrum (shadowing at small $x$, binding effects at medium $x$, and Fermi motion at large $x$), no experimental evidence has confirmed modifications for $R_N$ to date. Consequently, theoretical models often treat $R_N$ as invariant in the nuclear medium. However, this paper argues that this assumption is theoretically flawed because nucleons within a nucleus possess Fermi motion in arbitrary directions, not strictly aligned with the virtual photon's momentum. This transverse motion necessitates a mixing of the nucleon's longitudinal and transverse structure functions within the nuclear medium, implying that $R_N$ must undergo nuclear modifications.

Methodology
The author employs a convolution formalism to describe nuclear structure functions, treating the nucleus as a collection of nucleons with a specific momentum distribution (spectral function).

• Formalism: The nuclear hadron tensor $W_\mu\nu^A$ is expressed as a convolution of the nucleon tensor $W_\mu\nu^N$ and the nucleon momentum distribution $S(p_N)$.
• Mixing Mechanism: By applying projection operators to the hadron tensor, the author derives expressions for the nuclear structure functions $F_1^A$ and $F_L^A$. Crucially, the formalism reveals that due to the transverse momentum ($\vec{p}_{N\perp}$) of the bound nucleons, the nuclear longitudinal structure function $F_L^A$ becomes a linear combination of the free nucleon's $F_1^N$ and $F_L^N$. The mixing coefficients are proportional to $\vec{p}_{N\perp}^2 / Q^2$ (or more precisely $\vec{p}_{N\perp}^2 / \bar{p}_N^2$).
• Calculation Setup: Numerical calculations were performed specifically for the deuteron ($D$).
  • Inputs: The nucleon structure function $F_2^N$ was calculated using MSTW08 parton distribution functions (PDFs) at leading order. The ratio $R_N$ was taken from the 1990 SLAC parametrization. The deuteron wave function $\phi$ was modeled using the Bonn potential.
  • Variables: Calculations were conducted at $Q^2 = 1, 5,$ and $100 \text{ GeV}^2$ across the Bjorken-$x$ range.
  • Comparisons: The author compared results with and without the mixing terms (setting $\vec{p}_{N\perp}^2 \to 0$) to isolate the effect of the transverse motion. Additionally, the impact of Target Mass Corrections (TMCs) was investigated using the $\xi$-scaling prescription.

Key Contributions and Results

1. Existence of Nuclear Modifications: The study explicitly demonstrates that nuclear modifications of $R_N$ exist in the deuteron. The ratio $R_D/R_N$ deviates from unity, contradicting the standard assumption of no modification.
2. Two Sources of Modification: The paper identifies two distinct mechanisms driving these modifications:
   • Longitudinal-Transverse Mixing: The primary source, driven by the $\vec{p}_{N\perp}^2/Q^2$ admixture of $F_1^N$ and $F_L^N$. This effect is most pronounced at lower $Q^2$ (e.g., $1 \text{ GeV}^2$) but remains non-negligible even at high $Q^2$.
   • Convolution Integrals: A secondary source arising from the convolution of the nucleon's light-cone momentum distributions ($f_{LL}$ and $f_{11}$) with the $x$-dependent functional forms of $F_1^N$ and $F_L^N$. Even if mixing terms are removed, modifications persist due to the different functional dependencies of the structure functions on the momentum fraction $y$.
3. Quantitative Findings:
   • At $Q^2 = 5 \text{ GeV}^2$, the nuclear modification of the longitudinal structure function $F_L^D/F_L^N$ differs significantly from that of the transverse function $F_1^D/F_1^N$, particularly in the medium-$x$ region ($x \sim 0.5$).
   • The ratio $R_D/R_N$ shows deviations of a few percent. At $Q^2 = 100 \text{ GeV}^2$, while the mixing-induced effects diminish, a residual modification remains due to the convolution integrals.
   • Target Mass Corrections (TMCs), specifically using $\xi$-scaling, were found to suppress the Fermi-motion-induced rise in modifications at large $x$, particularly at lower $Q^2$.
4. Implications for Polarized PDFs: The author notes that current global analyses of polarized parton distribution functions (PDFs) extract neutron data from polarized deuteron and $^3$He targets assuming $R_D = R_N$. The demonstrated existence of $R_D \neq R_N$ suggests that these extractions may contain systematic errors, which could become significant as polarized PDFs are determined with higher precision.

Significance and Claims
The paper claims that the assumption of no nuclear modification for $R_N$ is theoretically incorrect and must be abandoned for precise determinations of nucleon structure functions.

• Theoretical Necessity: The mixing of longitudinal and transverse components due to nucleon Fermi motion is an unavoidable consequence of the kinematics in a nuclear medium.
• Experimental Relevance: The predicted modifications are within the range of detectability for current and future facilities. The author highlights that upcoming experiments at the Thomas Jefferson National Accelerator Facility (JLab), specifically scheduled for 2026, and future Electron-Ion Colliders (EICs) are well-positioned to test these predictions.
• Broader Impact: Beyond standard nuclear binding and Fermi motion, the study suggests that investigating $R_N$ at small $x$ could provide new insights into gluon dynamics and saturation in nuclei, linking nuclear structure to the gluon distribution.
• Short-Range Correlations (SRCs): The paper argues that the neglect of $R_N$ modifications in SRC analyses (where $R_N$ is used to interpret electron-nucleus cross-sections) should be re-evaluated, as these effects could alter the interpretation of high-momentum nucleon pairs.

In conclusion, the work provides a rigorous theoretical framework and numerical evidence that $R_N$ is subject to nuclear modifications in the deuteron, urging a re-examination of how nuclear data is used to extract fundamental nucleon properties.