Nuclear effects on longitudinal-transverse structure function ratio in the deuteron

This paper theoretically demonstrates and numerically calculates that nuclear modifications to the longitudinal-transverse structure function ratio $R_N$ in the deuteron, arising from transverse Fermi motion, are of the order of a few percent and should be accounted for in high-energy nuclear data analysis and future experimental investigations.

The Gist

Imagine you are trying to take a perfect photograph of a single, stationary dancer (a nucleon) to understand their moves. You have two cameras: one capturing their "longitudinal" moves (forward and backward) and another capturing their "transverse" moves (side-to-side).

For decades, scientists believed that if you took a photo of a whole dance troupe (an atomic nucleus) made of many dancers, the ratio of forward moves to side-to-side moves would look exactly the same as it did for the single dancer standing still. They assumed the troupe was just a collection of individual, perfect copies.

The Big Surprise
This paper argues that this assumption is wrong. The author, S. Kumano, explains that inside a nucleus, the dancers aren't standing still. They are jittering, spinning, and moving in all directions (a phenomenon called "Fermi motion").

Because the dancers are moving sideways while you try to photograph them, your "forward" camera accidentally captures some of their "side" moves, and vice versa. It's like trying to measure the speed of a car driving straight down a highway, but the car is also swerving slightly left and right. If you don't account for the swerving, your measurement of "straight speed" will be slightly off.

The "Mixing" Effect
The paper uses a mathematical recipe (called a "convolution model") to show how this happens.

• The Recipe: Imagine you have a smoothie made of fruit (the nucleon's structure functions). Usually, you just blend the fruit.
• The Twist: In a nucleus, the blender is shaking the cup sideways while it spins. This causes the "forward" fruit juice and "sideways" fruit juice to mix together in a way that depends on how fast the cup is shaking (the transverse momentum) compared to how hard you are blending (the energy of the experiment).

What the Numbers Say
The author ran the numbers for the simplest nucleus, the deuteron (which is just a pair of dancers holding hands).

• The Result: The "mixing" changes the ratio of forward-to-sideways moves by a few percent.
• The Scale: While a few percent might sound small, in the world of subatomic physics, it's a significant error if you are trying to measure something with high precision.
• The Future: The paper notes that this effect gets much bigger for heavier nuclei (larger dance troupes with more jittery dancers).

Why This Matters Now
For a long time, scientists ignored this effect because they thought it didn't exist. However, new experiments are being prepared at the Jefferson Lab (JLab) to measure this ratio specifically for the deuteron.

The author's main message is: Don't ignore the jitter. If scientists want to get precise measurements from these new experiments, they must account for the fact that nucleons inside a nucleus are moving sideways, which mixes up the data. If they don't, their "photographs" of the subatomic world will be slightly blurry and inaccurate.

In a Nutshell
Just as a spinning dancer looks different depending on the angle of the camera, a nucleon inside a moving nucleus looks different than a stationary one. This paper proves that this "motion blur" changes the fundamental ratio of how these particles behave, and scientists need to fix their math to see the true picture.

Technical Summary

Technical Summary: Nuclear Effects on Longitudinal-Transverse Structure Function Ratio in the Deuteron

Problem Statement
While nuclear modifications of the nucleon's structure function $F_2^N$ (the EMC effect) have been extensively studied since 1983, it has been a standard assumption that the longitudinal-transverse structure function ratio, defined as $R_N = F_L^N / (2xF_1^N)$, remains unmodified in nuclear environments. Previous experimental analyses of charged-lepton scattering, neutrino deep inelastic scattering (DIS), and Jefferson Lab (JLab) data generally found no evidence for such nuclear effects. However, theoretical considerations suggest this assumption is flawed. In a nuclear target, nucleons possess transverse Fermi motion relative to the virtual-photon momentum direction (defined as the $z$-axis). This motion causes a mixing between the longitudinal and transverse structure functions of the bound nucleon. The magnitude of this mixing is proportional to the square of the nucleon's transverse momentum divided by the momentum transfer squared ($\vec{p}_T^2/Q^2$). Despite theoretical proposals suggesting these effects exist, they have not been quantitatively demonstrated for the deuteron in the context of standard convolution models, creating a gap between theoretical expectations and the assumptions used in analyzing nuclear data.

Methodology
The paper employs a standard convolution formalism to calculate nuclear structure functions for the deuteron. The approach involves the following steps:

1. Formalism: The lepton-hadron DIS cross-section is described via the hadron tensor $W_{\mu\nu}$. The transverse ($W_T$) and longitudinal ($W_L$) structure functions are derived from the helicity components of this tensor.
2. Convolution Model: The nuclear structure functions ($F_2^A, F_1^A, F_L^A$) are expressed as convolutions of the nucleon structure functions ($F_2^N, F_1^N, F_L^N$) with a spectral function $S(p_N)$. The spectral function is constructed using the momentum-space wave function $\phi_i$ of the nucleon within the nucleus.
3. Mixing Mechanism: The core of the calculation involves projection operators that reveal how the longitudinal and transverse components mix. The mixing coefficients ($f_{L1}, f_{1L}$) are explicitly proportional to the transverse momentum squared ($\vec{p}_{N\perp}^2$) relative to the virtual photon's momentum scale.
4. Inputs:
   • Deuteron Wave Function: The Bonn wave function is utilized.
   • Nucleon Structure Functions: $F_2^N$ is calculated using MSTW08 unpolarized Parton Distribution Functions (PDFs) at leading order. The ratio $R_N$ for the free nucleon is taken from the 1990 SLAC parametrization.
   • Kinematics: Calculations are performed for specific values of $Q^2$ (1, 5, 10, and 100 GeV$^2$) across the medium to large Bjorken-$x$ region.

Key Contributions
The primary contribution of this work is the numerical demonstration that nuclear modifications to $R_N$ do exist, even in the deuteron, contradicting the common assumption of their absence. The paper identifies two distinct mechanisms driving these modifications within the convolution framework:

1. Longitudinal-Transverse Admixture: The transverse Fermi motion of the nucleon causes a direct mixing of the longitudinal and transverse structure functions. This effect scales with $\vec{p}_T^2/Q^2$.
2. Functional Form Differences: Even in the absence of mixing (the scaling limit where $Q^2 \to \infty$), the convolution integral yields different results for $R_N$ because the $x$-dependent functional forms of the longitudinal and transverse structure functions differ. The convolution of these distinct functions with the nucleon momentum distribution inherently alters the ratio.

Results
Numerical results for the deuteron at $Q^2 = 5$ GeV$^2$ show that while the modifications to $F_2^A$ and $F_1^A$ follow expected patterns (negative at medium $x$ due to binding, positive at large $x$ due to Fermi motion), the modifications to $F_L^A$ differ significantly.

• Magnitude of $R_N$ Modifications: The ratio $R_D/R_N$ deviates from unity. The deviation is most pronounced at low $Q^2$ (e.g., 1 GeV$^2$) and decreases as $Q^2$ increases, consistent with the $\vec{p}_T^2/Q^2$ scaling. However, even at $Q^2 = 100$ GeV$^2$, a non-negligible deviation persists due to the second mechanism (functional form differences).
• Scale of Effects: In the deuteron, the magnitude of the modification is of the order of a few percent. The paper notes that while small in the deuteron, these effects are expected to be significantly larger in heavier nuclei.
• Comparison with "No Mixing" Scenario: The paper compares calculations with full mixing terms against those where mixing is artificially suppressed (setting $\vec{p}_T^2 \to 0$). The difference highlights the specific contribution of the transverse motion, while the residual difference in the "no mixing" case confirms the impact of the functional form disparity.

Significance and Claims
The paper asserts that the assumption of no nuclear modification for $R_N$ is theoretically incorrect and practically problematic for precision physics.

• Theoretical Validity: The existence of these modifications is a direct consequence of the kinematics of bound nucleons moving in arbitrary directions, not just the longitudinal direction.
• Experimental Relevance: With longitudinal-transverse structure function ratio measurements and tensor-polarized experiments currently under preparation for the deuteron at JLab, these theoretical effects must be accounted for to ensure the precise determination of physical quantities.
• Implications for Heavy Nuclei: While the effect is modest in the deuteron (a few percent), the paper emphasizes that for heavy nuclei, where nuclear effects are generally larger, neglecting these modifications could lead to significant systematic errors in the analysis of nuclear cross-sections.
• Future Outlook: The author expresses hope that upcoming experimental data will confirm these predicted nuclear modifications, validating the theoretical formalism presented.

The work concludes that for a precise analysis of nuclear data, particularly in the medium-to-large $x$ region, the nuclear modifications of $R_N$ arising from transverse Fermi motion and convolution kinematics must be included.