QCD splitting functions beyond kinematical limits

Technical Summary: QCD Splitting Functions Beyond Kinematical Limits

Problem Statement
In perturbative Quantum Chromodynamics (QCD), the precise description of particle production at high energies relies on handling infrared singularities arising from soft gluon radiation and the collinear decay of massless partons. Current Next-to-Leading Order (NLO) and Next-to-Next-to-Leading Order (NNLO) subtraction schemes rely on factorizing higher-order matrix elements into lower-order elements and universal splitting functions or soft insertion operators. However, a fundamental issue persists: the soft and collinear limits do not commute. Sub-leading power contributions in one limit often become leading power contributions in the other. Existing subtraction schemes, which typically rely on kinematical approximations (such as the double-soft limit or leading-power soft scaling), obscure the overlap between these singular regions. This leads to remainder functions that may contain residual sub-leading singularities or fail to capture the complete azimuthal angle dependence of splitting processes, complicating the construction of generic, process-independent NNLO subtraction terms and parton showers.

Methodology
The authors propose a systematic decomposition of QCD splitting functions that avoids taking kinematical limits (neither soft nor collinear) during the derivation. The core methodology involves:

1. Scalar QCD Decomposition: The authors isolate the universal component of splitting functions by making the scalar part of the theory manifest. They utilize scalar currents derived from the minimal coupling of a vector potential to a classical, accelerated charge. These scalar currents capture the singularity structure physically rather than purely mathematically.
2. Multipole Radiators: They derive multipole radiator functions based on scalar QCD. These radiators share essential features with known double-soft and one-loop soft gluon currents but are not based on kinematical approximations. They are formulated differentially in color space to maximize utility for practical calculations.
3. Spin Decomposition: The full QCD amplitudes are decomposed into scalar dipole radiators and "pure splitting remainders." The scalar components account for the leading and sub-leading soft singular terms, while the remainders contain spin-dependent terms and sub-leading singularities specific to the splitting configuration.
4. Gauge Choices:
   • Tree-Level: Calculations are performed in an axial gauge (specifically light-like axial gauge) to ensure the polarization tensor encodes only physical degrees of freedom and to mimic coherent gluon emission effects via color charge conservation.
   • One-Loop: The background field method is employed to compute scalar dipole radiator functions. This allows for the construction of individually renormalizable $n$-point Green's functions that satisfy naive Ward identities, ensuring gauge invariance and physical meaning. The results are matched to known one-loop splitting amplitudes computed in axial gauge.

Key Contributions and Results

• Tree-Level Decomposition (NNLO Double-Real and Real-Virtual):

  • The paper presents a complete decomposition of tree-level splitting functions for $1 \to 2$ and $1 \to 3$ parton splittings (including $q \to qg$, $g \to q\bar{q}$, $g \to gg$, $q \to qg\bar{q}$, $q \to qgg$, $g \to gq\bar{q}$, and $g \to ggg$).
  • The splitting functions are expressed as a sum of scalar dipole radiator functions (derived from scalar QCD) and spin-dependent remainder functions.
  • Crucially, the scalar radiators are derived without kinematical limits, allowing them to capture the full azimuthal angle dependence and sub-leading power terms that are often lost in standard soft approximations.
  • The authors demonstrate that the "pure" remainder functions (those not captured by the scalar radiators or lower-order factorized expressions) are sub-leading in both single-soft and double-collinear limits. This ensures that the singular structure is fully captured by the universal scalar components.
  • Explicit formulae for these decompositions are provided in Sections IV B through IV E, with scaling behaviors summarized in Table I.

• One-Loop Decomposition:

  • The authors extend the decomposition to one-loop order ($1 \to 2$ splittings).
  • Using the background field method, they compute scalar dipole radiator functions at one-loop level and match them to known one-loop splitting amplitudes.
  • They generalize a result from Ref. [22], proving that the cancellation of certain diagram classes in the soft limit is a consequence of Ward identities satisfied by $n$-point Green's functions in a physical scheme, rather than a coincidence of the soft approximation.
  • The one-loop splitting functions are decomposed into scalar components (which reproduce the pole structure of the full amplitude) and spin-dependent remainders. The scalar components are shown to match the collinear limits of the multipole radiators.
  • Results for one-loop scalar and spin-dependent remainders are detailed in Section V C, with scaling behaviors summarized in Table II.

Significance and Claims
The paper claims a twofold significance for its results:

1. Theoretical Insight: It demonstrates that neither the soft nor the collinear kinematical limit is strictly necessary to derive universally applicable subtraction terms for higher-order calculations. This is possible because a semi-classical limit of the theory (scalar QCD) exists and can be used to capture the universal singular structure.
2. Practical Application: The authors anticipate that these results will facilitate the development of subtraction or resummation schemes that do not suffer from overlaps between soft and collinear sectors, or between single- and double-unresolved limits. Specifically, they suggest this approach could aid in constructing:
   • A fully differential parton shower at Next-to-Next-to-Leading Logarithmic (NNLL) precision.
   • A generic NNLO subtraction scheme with improved convergence compared to existing methods based on leading-power approximations in soft regions.

The work provides the necessary building blocks (scalar radiators and remainder functions) and the explicit formulae to assemble splitting functions without relying on the ambiguous overlap regions that plague current subtraction techniques.