⚛️ phenomenology

The DIS dipole picture cross section at finite energy

This paper implements a finite energy constraint in the dipole picture of deep inelastic scattering by limiting the invariant mass of the produced partonic system, demonstrating that this effect significantly reduces cross sections for charm and light quarks at moderate $Q^2$ and $x=0.01$ but diminishes rapidly at smaller $x$ or larger $Q^2$ .

Original authors: Magnus Bertilsson, Tuomas Lappi, Heikki Mäntysaari, Xuan-Bo Tong

Published 2026-09-30

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CC BY 4.0

Original authors: Magnus Bertilsson, Tuomas Lappi, Heikki Mäntysaari, Xuan-Bo Tong

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

Deep inelastic scattering is one of the most powerful tools physicists have to look inside the proton, the tiny, positively charged core of every atom. Imagine firing a high-speed electron at a proton and watching how it bounces off. By measuring the angles and energies of the scattered particles, scientists can map out the proton's internal structure, revealing a chaotic sea of smaller particles called quarks and gluons. For decades, these experiments have confirmed that protons are not solid spheres but complex, dynamic systems. As researchers push these experiments to higher energies and look at the proton from different angles, they enter a regime where the number of gluons becomes so dense that they begin to interact with one another in complex ways. This state, known as gluon saturation, is a predicted feature of the strong force that holds matter together, and understanding it is a primary goal for upcoming experiments at facilities like the Electron-Ion Collider.

To study this dense environment, physicists use a theoretical framework called the dipole picture. In this view, the virtual photon emitted by the electron splits into a pair of particles—a quark and an antiquark—before hitting the proton. This pair, or dipole, then scatters off the proton's gluon field. For many years, calculations using this picture relied on a simplifying assumption: that the collision energy is so high that the produced particles can have any amount of energy, limited only by the laws of conservation. This assumption, known as the optical theorem, treats the collision as if it happens in an infinite energy limit, ignoring the fact that in any real experiment, the total energy available is fixed and finite.

A team of researchers from the University of Jyväskylä and the Helsinki Institute of Physics has now revisited this assumption. They asked a straightforward question: what happens if we strictly enforce the rule that the energy of the produced quark-antiquark pair cannot exceed the total energy available in the collision? In their new study, they implemented a "finite energy constraint" into the dipole picture calculations. Instead of allowing the produced system to have an arbitrarily large mass, they capped it at the maximum energy allowed by the collision itself. This correction removes a portion of the mathematical possibilities that are physically impossible in a real-world experiment.

The researchers found that this seemingly technical adjustment has a surprisingly large impact on the results, particularly when looking at specific types of collisions. When they applied this constraint to calculations involving light quarks, the predicted strength of the interaction changed by up to 7 percent at certain energy levels. However, the effect was even more dramatic for heavy quarks, specifically charm quarks. In the same conditions, the correction reduced the predicted interaction strength for charm quarks by as much as 35 percent. These numbers are significant because they are comparable to, or even larger than, the margins of error in the most precise experimental data collected so far.

The study reveals that the size of this effect depends heavily on the energy of the collision and the mass of the quarks involved. The correction is most pronounced when the collision energy is relatively low and the quarks are heavy. In these scenarios, the "forbidden" high-energy states that the old calculations included are the ones that would have contributed most to the total result. By cutting them off, the new calculations show a much lower probability of interaction than previously thought. The authors note that this could help resolve a long-standing puzzle in the field: the difficulty of fitting both the total collision data and the specific data for charm quark production into a single theoretical model. The new constraint naturally suppresses the charm production rate, potentially allowing the two datasets to agree with each other without needing artificial adjustments.

This work does not overturn the existing theory of gluon saturation but refines the tools used to test it. By ensuring that the mathematical description of the collision respects the strict limits of energy available in the laboratory, the researchers have provided a more accurate baseline for future comparisons. As the next generation of particle colliders prepares to probe the proton with unprecedented precision, having a calculation that accounts for these finite energy limits will be essential. The findings suggest that what was once considered a minor detail in the math is actually a crucial piece of the puzzle, one that could help scientists finally pin down the exact conditions under which gluons begin to saturate and reshape the structure of matter.

Technical Summary: The DIS Dipole Picture Cross Section at Finite Energy

Problem Statement
Deep inelastic scattering (DIS) is conventionally described in the high-energy limit using the dipole picture, often derived via the optical theorem. This standard approach assumes an asymptotically large center-of-mass energy ( $W$ ) and an unconstrained phase space for the produced partonic system. Specifically, the optical theorem calculation integrates the dijet cross section over all final-state kinematics without enforcing the physical limit that the invariant mass of the produced quark-antiquark ( $q\bar{q}$ ) system ( $M_{q\bar{q}}$ ) cannot exceed the available center-of-mass energy ( $W$ ). While various kinematical constraints have been discussed in the context of small- $x$ evolution and sub-eikonal corrections, the systematic impact of imposing a finite-energy constraint on the invariant mass within the leading-order (LO) dipole picture cross section has not been fully explored.

Methodology
The authors derive the LO DIS cross section in the dipole picture by explicitly incorporating a finite energy (FE) constraint. The core modification involves restricting the invariant mass of the produced $q\bar{q}$ system such that $M_{q\bar{q}}^2 \leq W^2 \approx Q^2/x$ .

Derivation: Starting from the differential "dijet" cross section, the authors integrate over the final-state kinematics. Unlike the optical theorem approach, which integrates over the momentum imbalance $P$ without bounds (effectively setting $M_{q\bar{q}} \to \infty$ ), this work introduces a constraint on the invariant mass.
Mathematical Implementation: The constraint is implemented by changing the integration variable from the transverse momentum imbalance $P_\perp$ to the invariant mass squared $M_{q\bar{q}}^2$ . This introduces a Bessel function term, $J_1(\zeta)$ , into the integral, where the argument $\zeta$ depends on the maximum invariant mass $M_{max}^2$ .
Wilson Line Correlators: The resulting cross section expression (Eq. 25) differs from the standard optical theorem result. While the standard result depends only on the dipole correlator $S^{(2)}$ , the FE-corrected cross section involves a non-trivial quadrupole correlator $S^{(4)}$ that does not reduce to unity.
Numerical Model: To evaluate the integrals, the authors employ a Gaussian approximation for the quadrupole correlator (expressing it in terms of the dipole), consistent with the McLerran-Venugopalan (MV) model. They utilize parameters fitted to HERA data but explicitly note that they neglect Balitsky-Kovchegov (BK) or JIMWLK evolution for this study, treating the correlators as independent of $x$ to isolate the kinematic effect of the finite energy constraint.

Key Contributions

Formalism: The paper provides a derivation of the inclusive DIS cross section that maintains the essential dipole picture (scattering via lightlike Wilson lines) while rigorously enforcing the kinematic limit $M_{q\bar{q}}^2 \leq W^2$ . This is achieved without introducing sub-eikonal interactions, ensuring consistency with the shockwave scattering picture used for other observables.
Quantification: The authors numerically quantify the deviation between the finite-energy corrected structure functions ( $F_L^{FE}, F_2^{FE}$ ) and the standard optical theorem results ( $F_L^{OT}, F_2^{OT}$ ) across the kinematic range relevant to HERA and the future Electron-Ion Collider (EIC).

Results
The numerical analysis reveals that the finite energy constraint has a significant, kinematics-dependent impact, particularly for heavy quark production:

Light Quarks: At $x = 0.01$ and $Q^2 = 1 \text{ GeV}^2$ , the correction reduces the longitudinal structure function $F_L$ by approximately $0.9\%$ and the transverse structure function $F_2$ by up to $\sim 7\%$ . The effect diminishes rapidly as $x$ decreases or $Q^2$ increases.
Heavy Quarks (Charm): The effect is substantially larger for charm quarks ( $m_c = 1.4 \text{ GeV}$ ). At the same kinematics ( $x = 0.01, Q^2 = 1 \text{ GeV}^2$ ), the correction suppresses $F_{L,c}$ by $\sim 8\%$ and $F_{2,c}$ by $\sim 35\%$ .
Kinematic Dependence: The suppression is most pronounced at small $Q^2$ and relatively large $x$ (where $W$ is smaller). The effect is driven by the fact that heavy quark production inherently involves larger invariant masses due to the quark mass term in the invariant mass equation, making the phase space cutoff more restrictive.

Significance and Claims
The authors claim that implementing this finite energy constraint improves the precision of DIS dipole picture calculations by removing non-physical phase space regions. They posit that this correction brings the description of the DIS process into consistency with other applications of the same formalism (e.g., forward hadron production and exclusive DIS) where the final state is explicitly measured.

The paper suggests that the substantial suppression of charm production at moderate $x$ and low $Q^2$ could help resolve existing tensions in phenomenological fits. Specifically, it may explain why leading-order calculations often struggle to simultaneously describe total DIS cross sections and charm production data without ad-hoc normalization factors. The authors propose that incorporating this constraint into future Bayesian inference frameworks for determining BK evolution initial conditions could lead to more natural simultaneous fits of light and heavy quark data. They view the results presented as lower bounds on the importance of the correction, noting that including BK evolution (which drives the spectrum to higher invariant masses) would likely enhance the effect further.

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