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Phenomenological Estimate for the Longitudinal Unpolarized Structure Function in Semi-inclusive Deep Inelastic Scattering

This paper presents a phenomenological study combining low- and high-transverse-momentum contributions to estimate the longitudinal unpolarized structure function in SIDIS, revealing that its sizable ratio to the transverse structure function ( $R_{\text{SIDIS}} \simeq 0.3$ ) necessitates its inclusion for accurate interpretation of experimental data and motivates dedicated measurements to further constrain this longitudinal contribution.

Original authors: Alessandro Bacchetta, Matteo Cerutti, Leonard Gamberg, Richard Whitehill

Published 2026-10-06

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CC BY 4.0

Original authors: Alessandro Bacchetta, Matteo Cerutti, Leonard Gamberg, Richard Whitehill

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For decades, physicists have used high-energy collisions to look inside protons and neutrons, the tiny building blocks of all visible matter. By firing electrons at these particles and watching how they scatter, scientists can map out the internal landscape of the atom, revealing how quarks and gluons move and carry energy. This process, known as deep inelastic scattering, has long been understood as a game of two main players: the transverse force, which pushes sideways, and the longitudinal force, which pushes forward along the path of the collision. For a long time, the forward push was considered a minor player, a subtle effect that could be safely ignored in most calculations because it was thought to be too weak to matter.

However, the story changes when scientists look at a more complex version of this experiment, called semi-inclusive deep inelastic scattering. In this setup, researchers do not just watch the electron bounce off; they also track a new particle, usually a proton or a pion, that flies out of the collision. This extra step allows scientists to see not just how much energy the particles carry, but also how they move sideways. When the sideways motion is very small, the rules of the game shift. The forward push, which was previously dismissed as negligible, suddenly becomes a significant factor. Understanding exactly how strong this force is and how it behaves across different speeds is crucial for getting an accurate picture of the proton's internal structure. Without it, the maps scientists build of the subatomic world could be slightly distorted.

A team of researchers has now taken a fresh look at this overlooked force, specifically the longitudinal structure function in semi-inclusive deep inelastic scattering. Their goal was to create a complete picture of how this force behaves from the slowest sideways movements to the fastest, bridging a gap that previous theories had left open. They found that the forward push is much stronger than many had assumed, particularly when the sideways motion of the particles is low. In these specific conditions, the ratio of the forward push to the sideways push can reach about thirty percent. This is a substantial amount, suggesting that ignoring this force is no longer an option if scientists want to interpret their experimental data correctly.

To reach this conclusion, the researchers combined two different ways of describing the collision. At low speeds, where the particles move slowly sideways, they used a model based on the internal, non-perturbative dynamics of the proton. At high speeds, where the particles are kicked hard by the collision, they used standard calculations based on the exchange of energy between quarks. The challenge was to connect these two descriptions smoothly so that the theory works across the entire range of speeds. The team developed a method to stitch these two regimes together, creating a continuous curve that shows how the longitudinal force changes as the collision energy increases. They tested this model against existing data and found that it produces a consistent and realistic description of the physical world.

The results indicate that this longitudinal contribution is not just a tiny correction but a meaningful part of the total picture. When the researchers added this force into their calculations for the total number of particles produced in these collisions, they saw a change of about ten percent in the final numbers. This might sound small, but in the precise world of particle physics, a ten percent shift is significant. It means that previous estimates of how quarks and gluons are distributed inside the proton might need to be adjusted. If this force is left out, the resulting maps of the proton's interior could be slightly off, leading to misunderstandings about how matter is built.

The study also highlighted that the strength of this force depends heavily on the energy of the collision. At lower energies, the effect is most pronounced, reaching those thirty percent levels. As the energy increases, the effect becomes smaller, but it never disappears completely. The researchers noted that while their model provides a solid estimate, there are still uncertainties. The exact shape of the curve and the precise size of the effect depend on how the two different theoretical descriptions are matched together. These uncertainties are currently the largest source of error in their predictions, suggesting that more precise measurements are needed to pin down the exact behavior of this force.

Despite these uncertainties, the work provides a clear roadmap for future experiments. The researchers argue that dedicated measurements of this specific ratio are now essential. By measuring the longitudinal force directly across different speeds, experimentalists can test the predictions of this new model and refine the theoretical tools used to describe the proton. This will help resolve tensions between different sets of experimental data and lead to a more unified understanding of the subatomic world. The paper concludes that while the longitudinal force was once considered a minor detail, it is now clear that it plays a vital role in the dynamics of high-energy collisions, and accounting for it is necessary for the next generation of discoveries in particle physics.

Technical Summary: Phenomenological Estimate for the Longitudinal Unpolarized Structure Function in SIDIS

Problem Statement
In semi-inclusive deep inelastic scattering (SIDIS), the unpolarized cross section is traditionally dominated by the transverse structure function, $F_{UU,T}$ . The longitudinal structure function, $F_{UU,L}$ , is often neglected in phenomenological analyses, particularly in the low-transverse-momentum ( $q_T \ll Q$ ) region where it is power-suppressed relative to $F_{UU,T}$ by a factor of $q_T^2/Q^2$ (Next-to-Next-to-Leading Power, or NNLP). However, recent theoretical developments suggest that kinematic power corrections can generate sizable NNLP contributions to $F_{UU,L}$ from twist-two Transverse Momentum Dependent (TMD) distributions. Despite this, a consistent description of $F_{UU,L}$ across the full kinematic range—connecting the non-perturbative TMD region to the perturbative collinear region—has not been established. This gap is critical because global TMD analyses and the interpretation of hadron multiplicities (ratios of semi-inclusive to inclusive cross sections) currently neglect the longitudinal contribution in the numerator, potentially introducing mismatches with the denominator where the longitudinal structure function is included.

Methodology
The authors present a phenomenological study to estimate $F_{UU,L}$ and its ratio to the transverse function, $R_{\text{SIDIS}} = F_{UU,L}/F_{UU,T}$ , across the entire transverse-momentum spectrum. The approach involves three main components:

Low- $q_T$ Modeling (W-term): In the region $q_T \ll Q$ , the authors model $F_{UU,L}$ using TMD factorization. They adopt a weight function $\omega(p_T^2)$ acting on the convolution of unpolarized TMD parton distribution functions ( $f_1$ ) and fragmentation functions ( $D_1$ ). While previous literature suggested a quadratic weight $\omega \propto p_T^2$ , the authors propose a modified parameterization, $\omega(p_T^2) = \frac{4p_T^2/Q^2}{1 + 4p_T^2/Q^2}$ . This modification retains the expected $q_T^2/Q^2$ suppression at small $p_T$ but prevents excessive broadening of the $q_T$ distribution, facilitating a smoother connection to the high- $q_T$ regime. The calculation utilizes TMD grids from the MAP22 extraction at NNLL accuracy.
High- $q_T$ Calculation (FO-term): In the region $q_T \sim Q$ , the structure functions are computed using fixed-order collinear factorization at $\mathcal{O}(\alpha_s)$ . The authors use NLO collinear PDFs (MMHT2014) and FFs (DSS14). In this regime, $F_{UU,L}$ is a leading-power quantity.
Phenomenological Matching: To bridge the two regimes, the authors employ a transition function $w(q_T)$ to interpolate between the TMD (W) and fixed-order (FO) contributions: $F = w(q_T)F_W + [1-w(q_T)]F_{FO}$ . The transition function is defined by parameters controlling the onset and width of the interpolation region, chosen to ensure the W-term dominates at low $q_T$ and the FO-term at high $q_T$ , while maintaining a smooth transition. This approach circumvents the lack of a complete NNLP TMD factorization framework that would rigorously match the two regimes.

Key Contributions and Results

First Systematic Estimate: This work provides the first systematic phenomenological estimate of the transverse-momentum dependence of $F_{UU,L}$ and $R_{\text{SIDIS}}$ across the full $q_T$ spectrum.
Magnitude of $R_{\text{SIDIS}}$ : The study finds that $R_{\text{SIDIS}}$ can be sizable. At low transverse momentum and hard scales relevant to fixed-target experiments (e.g., $Q \sim 2$ GeV), the ratio reaches values of approximately $0.3 $(30%). As$ Q$ increases, the ratio decreases due to the $q_T^2/Q^2$ suppression, dropping to $\sim 5\%$ at higher $Q$ bins.
Impact on Integrated Observables: When integrated over transverse momentum, the inclusion of the $F_{UU,L}$ contribution modifies the longitudinal structure function $F_L$ by approximately 5–10% at $Q \sim 2$ GeV. This effect diminishes at higher $Q$ ( $Q \gtrsim 10$ GeV).
Uncertainty Analysis: The authors quantify uncertainties arising from the matching procedure, statistical errors in TMD and collinear sets, and missing higher-order corrections (scale variations). They find that matching uncertainty dominates at intermediate $q_T$ , while scale variations are significant at small $q_T$ for the ratio.
Comparison with Literature: The results at low $q_T$ show qualitative agreement with independent theoretical calculations based on resummation (Madrid) and rapidity factorization (BP) approaches, despite different underlying frameworks.
NNLP Corrections to $F_{UU,T}$ : The study investigates the impact of NNLP corrections to the transverse structure function on the ratio. It concludes that these corrections have a limited effect on $R_{\text{SIDIS}}$ and do not significantly alter the main conclusions.

Significance and Claims
The paper asserts that accounting for $F_{UU,L}$ is likely important for the precise interpretation of SIDIS multiplicities, particularly in the kinematic regimes of fixed-target experiments (e.g., COMPASS, Jefferson Lab) where $Q$ is relatively low. Neglecting this term in the numerator of multiplicity ratios while including it in the denominator may lead to normalization mismatches and affect the extraction of unpolarized TMD PDFs and FFs.

The authors emphasize that their results are exploratory. They explicitly state that their matching scheme is phenomenological and does not resolve the theoretical difficulties associated with rigorously matching power-suppressed TMD terms to fixed-order collinear results. Consequently, the estimates remain subject to substantial theoretical uncertainties. The work serves as a motivation for dedicated experimental measurements of $R_{\text{SIDIS}}$ differential in transverse momentum to better constrain the longitudinal contribution and for further theoretical progress toward a complete NNLP factorization and resummation framework.

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