⚛️ phenomenology

Extension of the non-pole technique to the twist-3 gluon distribution contribution in $pp$ collisions

This paper resolves the challenge of applying the nonpole formalism to the Sivers effect in $pp$ collisions by developing a generalized method that successfully reproduces known results despite the complexities of initial- and final-state interactions.

Original authors: Longjie Chen, Shinsuke Yoshida

Published 2026-09-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Longjie Chen, Shinsuke Yoshida

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

Inside every proton, the building blocks of ordinary matter, a chaotic and dynamic world unfolds. These particles, known as protons, are not solid spheres but rather complex bags of smaller constituents called quarks and gluons. While early models pictured these components as simple points moving in straight lines, modern physics reveals a far richer reality: the quarks and gluons possess their own spin, or intrinsic rotation, and they move in intricate orbital paths around the center of the proton. Understanding how this internal motion relates to the proton's overall spin is one of the great challenges of particle physics. A key to unlocking this mystery is a phenomenon called the Sivers effect. This effect describes a subtle asymmetry: when a proton is spinning, its internal quarks and gluons do not move randomly. Instead, they tend to drift to one side, creating a preferred direction of motion that is linked to the proton's spin. Detecting this drift requires smashing protons together at incredibly high speeds and measuring how the resulting particles scatter.

For decades, physicists have used a specific mathematical tool to calculate these scattering events, a method that relies on identifying a particular type of mathematical "singularity" or pole in the equations. This approach, often called the pole technique, has successfully explained many experimental results, particularly in collisions between electrons and protons. However, when scientists tried to apply this same tool to collisions between two protons, the mathematics became tangled. In proton-proton collisions, the particles interact in two distinct ways: they can influence each other before the main collision occurs, known as an initial-state interaction, and they can influence each other after the collision, known as a final-state interaction. The old method struggled to handle the simultaneous presence of both interactions, leading to calculations that did not match known results. This created a gap in our ability to study the gluon Sivers effect, which is crucial for understanding the orbital motion of gluons, the particles that hold quarks together.

In a recent study, researchers Longjie Chen and Shinsuke Yoshida have bridged this gap by developing a new mathematical approach that avoids the problematic "pole" entirely. Instead of focusing on the singularities that caused the old method to stumble, they adopted a different perspective known as the nonpole technique. This method was originally designed to study a different phenomenon called the Collins effect, which involves how particles break apart after a collision. The researchers realized that this alternative framework could be adapted to the Sivers effect, but only if they could resolve the confusion caused by having both initial and final interactions present at the same time. Their work involved a meticulous re-examination of the fundamental rules governing the interactions, specifically a set of relationships known as Ward-Takahashi identities. These identities act as a consistency check, ensuring that the mathematical description of the collision remains valid regardless of how the observer views the event.

The core of the researchers' discovery lies in how they sorted the complex web of interactions. They demonstrated that in proton-proton collisions, the general rules can be broken down into smaller, distinct sets. One set of rules applies strictly to the interactions happening before the collision, while another set applies to those happening after. By separating these interactions, the researchers were able to determine the correct mathematical signs and directions for each part of the calculation, a step that was previously ambiguous. This separation allowed them to construct a complete and consistent formula for the scattering process. They then used this new formula to calculate the contribution of the three-gluon distribution to the single transverse-spin asymmetry, a specific measurement of the Sivers effect in proton-proton collisions.

The results of their calculation were striking. When they applied their new nonpole method to the problem, the final numbers they obtained matched perfectly with the results derived from the old pole technique, which had been the standard for years. This agreement is significant because it proves that the new method is not just a theoretical curiosity but a robust and reliable tool. Furthermore, the study revealed a unique feature that only appears when both initial and final interactions are present: a specific term involving a particular combination of color charges that had never been isolated in this way before. This term, which arises solely from the coexistence of the two types of interactions, was essential for reproducing the correct physical result. Without including this specific contribution, the calculation would have failed.

This work provides a comprehensive toolkit for future experiments. By establishing a method that works consistently for both the Sivers and Collins effects within the same framework, the researchers have simplified the path forward for theoretical calculations. This is particularly important as the next generation of particle colliders, such as the proposed Electron-Ion Collider, prepares to explore the three-dimensional structure of the proton in unprecedented detail. These machines will cover a wide range of collision energies and angles, requiring precise theoretical predictions to interpret the data. The ability to calculate the gluon Sivers effect accurately in proton-proton collisions is a major step toward understanding how the spin of the proton is generated by the orbital motion of its gluons. The study confirms that the nonpole approach is a viable and powerful alternative to traditional methods, offering a clearer path to deciphering the complex internal dynamics of the matter that makes up our universe.

Technical Summary: Extension of the Non-Pole Technique to the Twist-3 Gluon Distribution in $pp$ Collisions

Problem Statement
The Sivers effect, a twist-3 phenomenon describing the correlation between partonic orbital motion and nucleon spin, is traditionally calculated within the collinear factorization framework using a "pole" approach. This method relies on isolating the imaginary delta-function part of internal propagators ( $1/(p^2 + i\epsilon) \to -i\pi\delta(p^2)$ ) to generate the necessary complex phase for single transverse-spin asymmetries (SSAs). While this approach has been successfully applied to various processes in electron-proton ($ep$) and proton-proton ($pp$) collisions, a parallel "non-pole" formalism was developed for the Collins effect (arising from twist-3 fragmentation functions). The non-pole method, which avoids explicit pole decomposition and instead utilizes Ward-Takahashi identities (WTIs) to relate intrinsic, kinematical, and dynamical functions, offers advantages in gauge invariance and computational simplicity, particularly at next-to-leading order (NLO).

Recent attempts to apply the non-pole formalism to the Sivers effect have succeeded in semi-inclusive deep inelastic scattering (SIDIS), where only final-state interactions (FSI) are present. However, extending this method to $pp$ collisions has remained an open problem. In $pp$ collisions, the simultaneous presence of initial-state interactions (ISI) and final-state interactions (FSI) introduces an ambiguity in the sign of the $i\epsilon$ prescription within the WTIs. This ambiguity prevents the unique determination of pole structures required to derive the twist-3 cross-section formula, and consequently, known results for the gluon Sivers effect in $pp$ collisions had not been reproduced using the non-pole technique.

Methodology
The authors address this challenge by re-examining the WTIs in the context of $pp$ collisions, specifically focusing on the $gq \to qg$ hard parton scattering channel contributing to light hadron production. The core of their methodology involves:

Decomposition of WTIs: The authors demonstrate that the general WTIs, which are process-independent, can be decomposed into "minimal" subsets corresponding to distinct ISI and FSI contributions. By categorizing Feynman diagrams into subsets based on whether the extra gluon line connects to initial or final external lines (or internal propagators with specific color structures), they isolate contributions that contain only a single sign of the $i\epsilon$ prescription.
Resolution of $i\epsilon$ Ambiguity: Within these minimal subsets, the sign of the $i\epsilon$ prescription is uniquely determined. This allows for the derivation of specific identities for the hard parts $S_{L}^{\mu\nu\lambda}$ and $S_{R}^{\mu\nu\lambda}$ (representing left and right cut diagrams) that are free from the ambiguity present in the general WTI formulation.
Construction of Matrix Elements: Using these resolved identities, the authors perform a collinear expansion of the hadronic tensor. They systematically combine contributions from intrinsic, kinematical, and dynamical parts. A crucial step involves handling the color factors associated with the $d_{abc}$ and $f_{abc}$ structure constants, showing how they combine to form gauge-invariant matrix elements.
Verification: The derived cross-section formula is tested by calculating the hard cross sections in the Feynman gauge. The authors verify that the $n$ -dependence (where $n$ is an arbitrary lightlike vector) cancels out when combining kinematical and dynamical contributions, ensuring frame independence.

Key Contributions and Results
The primary result of the paper is the successful derivation of the twist-3 gluon distribution contribution to the SSA in $pp$ collisions using the non-pole formalism. Key findings include:

Unique Determination of Signs: The paper establishes that by decomposing the WTIs into minimal ISI and FSI subsets, the sign of the $i\epsilon$ prescription is uniquely fixed, resolving the theoretical barrier that previously hindered the application of the non-pole method to $pp$ collisions.
Reproduction of Known Results: The authors calculate the hard cross sections and demonstrate that the non-pole approach yields results consistent with those obtained via the traditional pole technique for the $gq \to qg$ channel.
Emergence of Mixed ISI-FSI Terms: A significant finding is the identification of a specific kinematical term proportional to $d_{abc}\Phi_{F}^{abc}(x,x)$ . This term arises exclusively from the coexistence of ISI and FSI and is associated with the coefficient $C_{Lkl}^{d(-)[+]}$ . The authors show that this term is non-zero and essential for reproducing the correct physical result.
Structure of the Cross Section: The final formula expresses the cross section in terms of dynamical functions ( $N(x,x)$ , $N(x,0)$ , $O(x,x)$ , $O(x,0)$ ), with the intrinsic contribution vanishing and kinematical functions being replaced by dynamical ones, mirroring the structure found in the Collins effect but adapted for the Sivers mechanism.

Significance
The paper claims that this work provides a comprehensive solution to the problem of applying the non-pole formalism to the Sivers effect in processes involving both ISI and FSI. By successfully reproducing known results for the gluon Sivers function in $pp$ collisions, the authors validate the non-pole technique as a robust alternative to the pole method.

The significance of this development is twofold:

Theoretical Unification: It enables the calculation of both Sivers-type and Collins-type contributions within the same theoretical framework, enhancing the consistency of twist-3 phenomenology.
Practical Utility: The authors note that the non-pole approach, which naturally incorporates kinematical functions and rewrites them in terms of dynamical functions, has been shown in recent studies to simplify NLO calculations. This work thus provides the necessary theoretical foundation and calculational techniques to extend these simplifications to the gluon Sivers effect in $pp$ collisions, which is a critical goal for future experiments aiming to understand the orbital motion of gluons.

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