Electroweak splitting functions in the Standard Model and beyond

We derive quasi-collinear factorization formulas in generic spontaneously broken gauge theories with scalars, fermions, and vector bosons. Specifically, we obtain polarized leading-order splitting functions for all possible final-state and initial-state 1->2 processes in the considered gauge theory. The main complication lies in the presence of mass-singular terms in longitudinal polarization vectors, prohibiting the direct application of the usual factorization procedure known from Quantum Electrodynamics and Quantum Chromodynamics. We overcome this issue with two different strategies, using gauge invariance and Ward identities as guiding principle. Our derivations do not use any explicit component-wise parametrizations of momenta and wave functions and bear no reference to a particular Lorentz frame. Furthermore, our results are valid for completely general definitions of the spin reference axes of the individual external particles. The various massless limits, the special case of the Electroweak Standard Model, the reproduction of existing literature results, and symmetry relations among our splitting functions are discussed in detail.

💡 Research Summary

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The paper presents a comprehensive derivation of quasi‑collinear factorisation formulas and the associated leading‑order (LO) splitting functions for generic spontaneously broken gauge theories that contain scalars, fermions and massive vector bosons. The authors work in full generality, without fixing a specific Lorentz frame or a particular choice of spin‑reference axes, and they formulate all results in a manifestly Lorentz‑covariant way using generic Sudakov parametrisations of the splitting momentum.

A central technical obstacle is the presence of mass‑singular terms ∝ pμ/MV in the longitudinal polarisation vectors εLμ(p) of massive gauge bosons. These terms would spoil the usual collinear factorisation if treated naively, because the gauge cancellations that normally remove them involve diagrams that are omitted in an approximate factorisation. The authors resolve this problem with two complementary strategies.

1. Ward‑identity / Goldstone‑boson approach – By exploiting Ward identities and the Goldstone‑boson equivalence theorem (GBET), the problematic pμ/MV pieces are systematically replaced by amplitudes involving the corresponding would‑be Goldstone bosons. In this way the mass‑singular contributions cancel already at the level of the squared matrix element, reproducing the correct gauge‑invariant result.

2. Explicit term‑by‑term cancellation – Without redefining the polarisation vectors, the authors identify the ill‑behaved pieces in each diagram and demonstrate how they cancel against contributions from other diagrams, following the logic of earlier works that performed a careful diagrammatic analysis. Both methods are shown to give identical splitting functions, providing a non‑trivial cross‑check.

The derivation proceeds by writing the splitting momentum k in terms of the parent momentum p, the daughter momentum q and a transverse component k⊥ (the generic Sudakov parametrisation). This eliminates any need for component‑wise parametrisations of momenta, spinors or polarisation vectors. The factorisation formula for the squared amplitude takes the form

|M(p→q + k)|² ≈ |M₀(p→q)|² · P_{a→bc}(z, k⊥², masses)

where P_{a→bc} is a tensorial splitting function that encodes all spin correlations. By contracting this tensor with the appropriate hard‑matrix element one can obtain the azimuthally‑averaged scalar splitting kernels needed for subtraction schemes.

The authors compute the full set of polarized splitting functions for every possible 1→2 process involving the three field types. This includes:

• Bosonic splittings (V*→VV, V*→ϕϕ, ϕ*→Vϕ, ϕ*→ϕϕ) with all possible polarisation combinations (transverse–transverse, transverse–longitudinal, longitudinal–longitudinal).
• Fermionic splittings (f*→f V, f*→f ϕ) with left‑ and right‑handed helicities, and also the corresponding anti‑fermion processes.

The results are presented both as full tensorial expressions and as azimuthally‑averaged scalar kernels. In the massless limit the kernels reduce to the well‑known Altarelli‑Parisi functions of QED and QCD, providing a stringent check. For massive particles new terms proportional to ratios such as m_f²/M_V² or M_V²/s appear, and they correctly reproduce the logarithmic enhancements ∝ log(s/M_V²) that dominate at multi‑TeV energies.

Symmetry relations are derived from the underlying gauge group structure: charge conjugation, CP, and SU(2)↔U(1) interchange lead to non‑trivial identities among the splitting functions, which can be used to reduce the number of independent kernels and to verify the consistency of the calculation.

A detailed comparison with the existing literature (Refs.