Diffractive deep inelastic scattering in the dipole picture: the $q\bar{q}g$ contribution in exact kinematics

This paper computes the exact kinematic $q\bar{q}g$ contribution to diffractive deep inelastic scattering structure functions, demonstrating that previous high-energy approximations are insufficient and revealing an equally important soft quark contribution alongside the soft gluon term at high $Q^2$.

The Gist

Imagine you are trying to understand the inside of a proton (a tiny particle inside an atom) by smashing a high-speed electron into it. This is called "Deep Inelastic Scattering." Usually, when you smash things, they break apart into a chaotic mess. But sometimes, the proton stays intact, and only a specific, organized group of particles flies out. This is called "Diffractive Scattering." It's like throwing a ball at a wall, and instead of the wall crumbling, the ball bounces off, and a perfectly formed bouquet of flowers flies out the other side, leaving the wall untouched.

Physicists use a tool called the "Color Glass Condensate" (CGC) to predict what happens in these collisions. Think of the proton not as a solid ball, but as a dense fog of tiny particles called "partons" (quarks and gluons).

The Problem: The "Three-Person" Dance

In the simplest version of this theory, the electron hits the proton, and the proton splits into just two particles: a quark and an antiquark (a pair). This is like a dance with two partners. Scientists have been very good at calculating this "two-person dance."

However, reality is messier. Sometimes, a third dancer joins the party: a gluon. Now you have a trio (a quark, an antiquark, and a gluon). This is the $q\bar{q}g$ contribution.

For a long time, physicists tried to calculate this trio dance using shortcuts. They assumed that one of the dancers was "lazy" or "soft"—moving very slowly compared to the others. They also assumed the dance happened in a very specific, extreme way (like only looking at the dance when the music is extremely fast). These shortcuts are called "approximate kinematics."

The New Discovery: The Full Dance Floor

This paper, by Kaushik, Mäntysaari, and Penttala, says: "Stop using the shortcuts. Let's calculate the whole dance exactly."

They performed a massive, complex calculation (a "numerical implementation") that tracks all three particles moving around without making any "lazy dancer" assumptions. They looked at the exact rules of the game, including all the tricky angles and speeds.

Here is what they found, using simple analogies:

1. The "Lazy Dancer" Myth
Previous studies assumed that the "soft gluon" (the lazy third dancer) was the most important part of the trio. They thought if you just calculated the soft gluon, you'd get a good answer.

• The Paper's Finding: This is wrong. The soft gluon is important, but it's only about one-third of the story. If you only count the soft gluon, you are missing a huge chunk of the action.

2. The Surprise Guest: The Soft Quark
The paper discovered that there is another "lazy dancer" that is just as important as the soft gluon: a soft quark.

• The Analogy: Imagine you thought the party was only about the slow-moving DJ (the gluon). But you just realized there's also a slow-moving singer (the quark) who is just as crucial to the vibe. If you ignore the singer, your description of the party is incomplete.
• The Result: At high energies, the "soft quark" contribution is just as big as the "soft gluon" contribution. You need both to get the right answer.

3. The "Approximation" Gap
The authors compared their "exact" calculation with the old "shortcut" calculations.

• The Finding: The old shortcuts are not very accurate. In the conditions expected for the future Electron-Ion Collider (EIC)—a giant new particle accelerator—the old methods underestimate the result by a factor of three.
• Why it matters: The EIC is designed to measure things with extreme precision (like measuring a hair's width from a mile away). If you use a method that is off by 300%, you can't trust your measurements. The old shortcuts are too rough for the new, high-precision experiments.

4. The "Munier-Shoshi" Limit
There is another extreme case where the third particle is extremely soft and the energy is huge. The paper checked this too. They found that while this extreme limit is interesting, it doesn't match the "exact" calculation well in the middle ground where real experiments happen.

The Bottom Line

This paper is a "reality check" for physicists. It says:

• We used to think we could get away with simple math (approximations) for these particle collisions.
• We were wrong. The math is much more complex.
• To understand the proton at the high precision of the future Electron-Ion Collider, we must include the full, exact calculation of the three-particle (quark-antiquark-gluon) interaction.
• Specifically, we cannot ignore the "soft quark" just because we used to focus on the "soft gluon."

The authors have built a new, precise mathematical engine (a computer code) that can handle this complexity. This engine is now ready to be used to interpret data from the next generation of particle colliders, ensuring that when we look at the proton's "fingerprint," we aren't looking at a blurry, distorted image.

Technical Summary

Technical Summary: Diffractive Deep Inelastic Scattering in the Dipole Picture: The $q\bar{q}g$ Contribution in Exact Kinematics

Problem Statement
Diffractive deep inelastic scattering (DIS) is a critical probe for understanding gluon saturation and the non-linear dynamics of Quantum Chromodynamics (QCD) at small momentum fractions ($x$). While the Color Glass Condensate (CGC) framework successfully describes inclusive and diffractive observables using the dipole picture, achieving Next-to-Leading Order (NLO) precision is essential to match the accuracy of existing HERA data and future Electron-Ion Collider (EIC) measurements.

Previous phenomenological applications of NLO corrections to the diffractive cross section have relied on approximate kinematics. Specifically, the contribution from the three-parton Fock state ($|q\bar{q}g\rangle$) has been evaluated only in limiting cases: the high-$Q^2$ or high-$M_X^2$ limits, often assuming a "soft" gluon (the Wüsthoff/GBW result) or a soft quark. These approximations neglect the full kinematical constraints of the exact eikonal interaction. The precise magnitude of the error introduced by these approximations in realistic kinematics, particularly those relevant for the EIC, has remained unquantified. Furthermore, the relative importance of the soft quark contribution compared to the soft gluon contribution in the exact kinematic regime has not been numerically established.

Methodology
The authors perform the first numerical implementation of the $|q\bar{q}g\rangle$ contribution to the diffractive cross section evaluated in exact eikonal kinematics. This contribution, referred to as the "NLO-trip" term, represents a finite subset of the full NLO corrections derived in Ref. [25].

1. Exact Kinematics: The calculation utilizes the full phase space integral for the $q\bar{q}g$ system interacting with the target shockwave. Unlike previous works, no kinematical approximations (such as $z \ll 1$ or $M_X^2 \gg Q^2$) are applied to the partonic splitting or the interaction with the target. The calculation involves a high-dimensional (9D) integral over transverse coordinates and longitudinal momentum fractions, constrained by the fixed invariant mass of the final state.
2. Numerical Implementation: The authors developed a Julia-based code utilizing the VEGAS algorithm (via the MCIntegration package) to handle the complex 9D integration. They employ a simple impact parameter factorization model (hard sphere target) and the Golec-Biernat–Wüsthoff (GBW) model for the dipole-target scattering amplitude to isolate the kinematic effects of the NLO-trip term.
3. Analytical Derivations: To benchmark the exact results, the authors analytically derive the soft quark contribution to the diffractive structure function in the high-$Q^2$ limit. This derivation adapts the TMD factorization formalism of Ref. [31] but explicitly excludes the divergent "emission-after-shockwave" contribution, retaining only the finite "emission-before-shockwave" term that constitutes the NLO-trip result. They also review the soft gluon (Wüsthoff) and large-$M_X$ (Munier–Shoshi) limits.

Key Contributions

• First Numerical Evaluation: The paper presents the first numerical evaluation of the finite NLO-trip contribution to diffractive structure functions in exact kinematics.
• Soft Quark Derivation: The authors derive the analytical expression for the soft quark contribution (Eq. 63) to the diffractive structure function, demonstrating that it is an equally important component to the soft gluon contribution at high $Q^2$.
• Quantification of Approximations: By comparing the exact numerical results with the soft gluon, soft quark, and combined soft parton approximations, the authors quantify the validity of existing phenomenological models.

Results

1. Inadequacy of Soft Gluon Approximation: The study demonstrates that the widely used "Wüsthoff" (soft gluon) result significantly underestimates the full NLO-trip contribution. In realistic EIC kinematics, the soft gluon approximation underestimates the exact result by a factor of approximately 3.
2. Importance of Soft Quarks: The soft quark contribution is found to be comparable to, and in some kinematic regions larger than, the soft gluon contribution. This is attributed to destructive interference between gluon emissions from the quark and antiquark in the soft gluon channel, which suppresses that specific contribution.
3. Combined Approximation Limits: While the sum of the soft quark and soft gluon contributions provides a reasonable estimate for the full result in the region $\beta \sim 0.3 - 0.6$, the deviation remains significant (better than $\sim 10\%$ only at very high $Q^2$). At small and large $\beta$, the aligned jet approximations break down completely.
4. Longitudinal Contribution: The authors calculate the longitudinal photon contribution to the $q\bar{q}g$ production, which is absent in leading-log approximations. While the transverse component dominates at high $Q^2$, the longitudinal component is non-negligible in EIC kinematics and dominates the longitudinal structure function $F_L^D$ below $\beta \lesssim 0.5$.
5. Improved Approximation: The authors show in Appendix A that including a subset of subleading corrections (specifically retaining the $\beta$-dependence inside the logarithm, $\log(Q^2/\beta k^2)$ rather than $\log(Q^2/k^2)$) significantly improves the accuracy of the aligned jet approximation, reducing the relative difference to under 9% over a broad kinematic range.

Significance and Claims
The paper claims that the existing phenomenological estimates for the $q\bar{q}g$ contribution, based on approximate kinematics, do not provide a good approximation for the full NLO contribution in the kinematic domain relevant for the EIC. The authors emphasize that to achieve the percent-level precision expected at the EIC, and to properly disentangle linear and non-linear dynamics in nuclear diffraction, complete NLO calculations in exact kinematics are necessary.

The work serves as a crucial step toward confronting NLO CGC calculations with experimental data from HERA and the future EIC. It establishes that the soft quark contribution is a necessary component of the NLO correction and that the "soft gluon only" approach is insufficient for precision physics. The authors state their intention to use these results to compute diffractive structure functions for protons and nuclei at full NLO accuracy in future work, incorporating the remaining NLO corrections and NLO-evolved dipole amplitudes.