The Unresolved Behaviour of Polarized Scattering Matrix Elements at NNLO in QCD

Spin asymmetries in collisions of spin-polarized hadrons probe polarized parton distributions, which encode the spin structure of the colliding hadrons. To perform precision physics studies with spin asymmetries, higher order QCD corrections to the underlying polarized cross sections are required. Their numerical implementation relies on the use of an infrared subtraction scheme, which extracts the infrared singular pieces from the real and virtual subprocesses. We derive the universal behaviour of longitudinally polarized real radiation matrix elements in their infrared-singular limits up to next-to-next-to-leading order (NNLO) in QCD, thereby enabling the construction of infrared subtraction schemes for generic polarized cross sections at NNLO.

💡 Research Summary

The paper addresses a critical gap in the theoretical description of spin‑dependent processes at next‑to‑next‑to‑leading order (NNLO) in quantum chromodynamics (QCD). While unpolarized NNLO calculations have become routine, the corresponding polarized calculations are hampered by the need to handle γ₅ and the Levi‑Civita tensor in dimensional regularisation. The authors adopt the Larin scheme, which extends these objects to d = 4 − 2ε dimensions at the cost of breaking axial‑current Ward identities that must be restored by finite renormalisation.

The central goal is to derive the universal unresolved (infrared‑singular) behaviour of matrix elements involving longitudinally polarized partons, up to NNLO. This is achieved by extracting the factorised collinear limits of colour‑ordered amplitudes and expressing them in terms of polarized splitting functions (also called antenna or collinear functions). The work proceeds in several logical steps:

1. Kinematics of collinear limits – Using a Sudakov parametrisation, the authors define momentum fractions (z) and transverse momenta (k_T) for single‑ and triple‑collinear configurations. They show how initial‑state splittings can be obtained from final‑state ones by simple substitutions, thereby unifying the treatment of all collinear limits.

2. Angular averaging – In the collinear limit, terms proportional to (k_T·p_a)(k_T·p_b)/k_T² generate azimuthal correlations. For gluon‑initiated splittings these correlations are essential to obtain a locally subtracted cross‑section. The authors demonstrate that, after averaging over the transverse‑momentum direction, only symmetric tensor structures survive, while antisymmetric pieces involving ε^{μν}p_νn_μ appear only in triple‑collinear limits and require careful handling in the Larin scheme.

3. Colour ordering – By working in a colour‑ordered basis, each parton is colour‑adjacent only to its neighbours, which restricts the set of possible unresolved limits. This simplifies the identification of the relevant colour factors for single‑ and double‑collinear emissions and is crucial for constructing subtraction terms that are colour‑coherent.

4. Derivation of universal collinear behaviour – The authors extract polarized splitting functions from the simplest current‑decay amplitudes that contain the required parton content. For single‑collinear limits they use three‑parton decay matrix elements (both at tree level and one loop). For triple‑collinear limits they employ four‑parton decay amplitudes of colour‑neutral currents (photon, neutralino, graviton). The polarized projectors for quarks and gluons are given explicitly in the Larin scheme:

   • P_quark = −i/(2 p·q) ε^{μ₁μ₂μ₃μ₄} p_{μ₃} q_{μ₄} γ^{μ₁}γ^{μ₂}/p₁,
   • P_gluon^{μ₁μ₂} = i ε^{μ₁μ₂μ₃μ₄} p_{μ₃} n_{μ₄}/(p·n). These projectors isolate the helicity‑difference component of the squared amplitude.

5. Single‑collinear splitting functions up to one loop – The paper presents the polarized tree‑level splitting kernels P_{a→bc}^{(0)}(z) and their one‑loop corrections P_{a→bc}^{(1)}(z, ε). The results reproduce known unpolarized kernels when summed over helicities, but contain additional Δ‑terms that encode the spin asymmetry. The authors discuss the renormalisation of the axial current and the scheme‑dependent finite pieces required to restore Ward identities.

6. Triple‑collinear splitting functions at tree level – By analysing four‑parton decay amplitudes, the authors obtain the full set of polarized triple‑collinear kernels P_{a→bcd}^{(0)}(z_i, s_{ij}). They treat all possible flavour combinations, including quark‑gluon‑gluon and gluon‑gluon‑gluon configurations. The kernels exhibit a richer dependence on the azimuthal angle and on the Levi‑Civita tensor, reflecting the fact that only one of the three partons can be polarized. The results are cross‑checked against unpolarized triple‑collinear functions and against a separate calculation using graviton decay.

7. Implications for NNLO subtraction schemes – The derived polarized splitting functions constitute the “unresolved factors” needed in any NNLO subtraction method (e.g. antenna, sector‑improved residue subtraction, q_T‑subtraction) when applied to polarized observables. The authors outline how these factors can be inserted into existing unpolarized frameworks, emphasizing that the only new ingredient is the spin‑dependent part of the splitting kernels.

The paper concludes with an outlook toward the upcoming Electron‑Ion Collider (EIC), where a wealth of polarized data (jets, heavy‑flavour, di‑hadron correlations) will demand NNLO accuracy. The universal polarized splitting functions presented here provide the essential building blocks for constructing subtraction terms that render such calculations finite and numerically stable.

In summary, this work delivers a comprehensive, scheme‑consistent set of polarized collinear splitting functions up to NNLO, establishes the methodology for their extraction from colour‑ordered decay amplitudes, and paves the way for the implementation of infrared‑subtraction schemes for polarized processes at the precision frontier.