Energy Correlators Resolving Proton Spin

We investigate the partonic origin of the proton longitudinal spin using spin-dependent energy correlators measured in lepton-hadron collisions with longitudinally polarized proton beams. These observables encode angular correlations in energy flow and are sensitive to the spin-momentum structure of confined partons. Using soft-collinear effective theory, we analyze the correlation patterns in both nearly back-to-back and forward limits, which establishes a direct correspondence with longitudinally polarized transverse momentum-dependent distributions (TMDs) and nucleon energy correlators (NECs). The TMDs and NECs allow consistent matching onto hard radiation regions and provide a comprehensive description of the transition from perturbative parton branching to nonperturbative confinement. Using renormalization group evolution, we obtain joint next-to-next-to-next-to-leading and next-to-next-to-leading logarithmic quantitative predictions for spin-dependent energy correlation patterns in the current and target fragmentation regions. The framework provides new theoretical insight into how the internal motion and spin of partons contribute to the formation of the proton longitudinal spin and offers an experimental paradigm for probing the interplay between color confinement and spin dynamics at the forthcoming Electron-Ion Collider.

💡 Research Summary

The paper proposes a novel way to probe the origin of the proton’s longitudinal spin by introducing spin‑dependent energy‑correlators (SDECs) measured in lepton‑hadron collisions with a longitudinally polarized proton beam. Unlike traditional structure‑function observables, SDECs weight the energy deposited in the detector by an angular factor θ, thereby encoding detailed information about the flow of energy as a function of direction. Because they are defined as collinear‑safe operators, they can be consistently used across the full kinematic range—from the hard radiation region to the current‑fragmentation region (CFR) and the target‑fragmentation region (TFR).

The authors formulate SDECs in the Breit frame and decompose the associated hadronic tensor into unpolarized, longitudinal, and spin‑dependent pieces. They then focus on two complementary angular limits. In the back‑to‑back limit (θ→π), which corresponds to the CFR where the struck quark fragments into a jet, they derive a factorization theorem using soft‑collinear effective theory (SCET). The hard matching coefficient C(Q,μ), the soft function Sₙₙ̄(b⊥,μ,ν), and the transverse‑momentum‑dependent (TMD) beam and fragmentation functions appear in a product that is further dressed by a jet function J_f(b⊥,E,μ,ν). Zero‑bin subtraction and a rapidity regulator ν are employed to avoid double counting between collinear and soft sectors. The renormalization‑group (RG) evolution of the hard, beam, and soft pieces is governed by the cusp anomalous dimension γ_cusp and non‑cusp pieces γ_H, while the Collins‑Soper (CS) kernel K(b⊥,μ) controls rapidity evolution. By solving these RG equations the authors achieve next‑to‑next‑to‑next‑to‑leading order (N³LO) fixed‑order matching combined with next‑to‑next‑to‑leading logarithmic (NNLL) resummation, delivering high‑precision predictions for the θ‑distribution of the spin‑dependent energy flow in the CFR.

In the collinear limit (θ→0), which probes the TFR where the proton remnants dominate, the paper introduces nucleon energy correlators (NECs). These are non‑perturbative objects that directly involve the energy‑weighting operator and are matched onto the usual polarized parton distribution functions (Δq, Δg). The authors connect NECs to helicity evolution equations that resum small‑x double logarithms (the Kovchegov‑Sievert‑Pitonak framework) and discuss the Regge‑BFKL dynamics that may become relevant at very low x. By performing an operator product expansion (OPE) of the NECs, they express them in terms of the polarized PDFs evaluated at appropriate Collins‑Soper scales ξ_n and ξ̄_n, ensuring that rapidity logarithms are minimized.

A key conceptual contribution is the unified treatment of SDECs across both regions, allowing a smooth matching between the TMD‑based description in the CFR and the NEC‑based description in the TFR. This provides a complete picture of how partonic spin and transverse motion translate into observable energy‑flow asymmetries.

Finally, the authors present phenomenological predictions for the upcoming Electron‑Ion Collider (EIC). They provide numerical results for the spin‑dependent cross section Δσ(θ) and the unpolarized counterpart σ_U(θ) as functions of θ, Q², and Bjorken‑x, highlighting the sensitivity to both quark and gluon helicity distributions as well as to orbital angular momentum encoded in TMD pretzelosity‑type functions. They discuss experimental requirements such as angular resolution, energy‑weighting triggers, and the need for precise control of beam polarization.

In summary, the work establishes spin‑dependent energy correlators as a powerful, operator‑based observable that bridges perturbative and non‑perturbative QCD dynamics. By leveraging SCET factorization, high‑order RG evolution, and a careful treatment of rapidity divergences, the authors deliver state‑of‑the‑art theoretical predictions that can be directly tested at the EIC, opening a new avenue for unraveling the proton spin puzzle.