⚛️ phenomenology

On the status of the 4-loop splitting functions in QCD

This paper summarizes recent progress on four-loop QCD splitting functions, highlighting that while the three non-singlet cases are now fully determined with some unexpected small- $x$ double logarithms, the flavour-singlet sector reveals novel analytical structures, with overall numerical effects remaining below 1% for most parton distributions down to $x \approx 10^{-4}$ .

Original authors: S. Moch (Hamburg U., Inst. Theor. Phys. II), A. Vogt (Liverpool U., Dept. Math.,Hamburg U., Inst. Theor. Phys. II)

Published 2026-09-29

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CC BY 4.0

Original authors: S. Moch (Hamburg U., Inst. Theor. Phys. II), A. Vogt (Liverpool U., Dept. Math.,Hamburg U., Inst. Theor. Phys. II)

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 fundamental building blocks of the visible universe, there is a chaotic storm of smaller particles called quarks and gluons. These particles are not static; they constantly split apart and merge back together, a process that changes depending on how hard you probe the proton. To understand the results of high-energy collisions, such as those at the Large Hadron Collider, physicists must track how these internal distributions shift as the energy scale changes. This shifting is governed by mathematical rules known as splitting functions. For decades, scientists have calculated these rules with increasing precision, moving from simple approximations to highly complex calculations that account for multiple layers of interaction. The latest effort in this long-standing quest involves pushing these calculations to a new level of complexity, known as the four-loop order, to see if the established patterns hold up under the most extreme scrutiny.

Two researchers, Sven-Olaf Moch and Andreas Vogt, have recently completed a major milestone in this field by finalizing the calculations for the non-singlet splitting functions. These specific functions describe how quarks behave when they are not mixed with gluons, a scenario that is mathematically cleaner and serves as a crucial test bed for the theory. Their work confirms that the vast majority of the theoretical expectations for these interactions were correct. The new results align perfectly with previous predictions across almost the entire range of particle behavior. However, the team discovered a small but significant surprise in the behavior of particles carrying very low momentum. In this specific region, the calculations revealed a type of double logarithmic pattern that had never been seen before in lower-order calculations. This deviation suggests that the mathematical structure of these interactions is more intricate than previously thought, particularly when dealing with the smallest fractions of momentum.

The researchers also examined the "flavor-singlet" cases, which involve the more complicated interactions where quarks and gluons mix together. While the complete exact solution for these mixed interactions is still being finalized, the partial results available so far already show that they contain analytical structures never before observed in splitting functions. These new structures are distinct from anything seen in previous decades of research. Despite these complex mathematical underpinnings, the practical impact of these new four-loop corrections is surprisingly small for most applications. When the researchers applied these new rules to standard benchmark scenarios, the changes to the predicted evolution of the particle distributions amounted to about one percent or less, even for particles with momentum fractions as low as one ten-thousandth. The only notable exception was the total distribution of valence quarks, where the effects were slightly more pronounced at very low momentum.

The team provided detailed numerical approximations to make these complex results usable for other scientists. They found that for the non-singlet cases, the new corrections are so small that they represent a clear improvement in accuracy over previous methods, reducing uncertainties significantly. For the mixed singlet cases, the current approximations are sufficient for most practical needs, though the authors note that the uncertainties become relevant only at extremely low momentum values. The work confirms that the standard model of particle physics continues to hold up under the weight of these incredibly high-order calculations, while simultaneously revealing subtle new features in the mathematical fabric of the theory. The findings are now available for the broader community to use in refining predictions for future experiments, ensuring that the theoretical tools used to interpret the universe's most energetic events are as precise as possible.

Technical Summary: On the status of the 4-loop splitting functions in QCD

Problem and Context
Parton distribution functions (PDFs) are essential for analyzing hard-scattering observables in hadron collisions, yet their scale dependence (evolution) is calculable within perturbative QCD. This evolution is governed by integro-differential equations involving splitting functions, $P_{ik}(\alpha_s)$ , expanded in powers of the strong coupling constant. While 3-loop (N $^2$ LO) splitting functions have been standard for over two decades, achieving Next-to-Next-to-Next-to-Leading Order (N $^3$ LO or 4-loop) accuracy is increasingly required for high-precision determinations of PDFs and the strong coupling $\alpha_s$ , as well as for processes with large higher-order corrections like Higgs boson production. This paper addresses the current status of these 4-loop splitting functions, specifically focusing on the recently completed non-singlet sector and the approximate state of the flavor-singlet sector.

Methodology
The authors utilize a combination of analytical and numerical techniques:

Reciprocity Respecting (RR) Kernels: The non-singlet anomalous dimensions are analyzed using the conformal symmetry of QCD in $D=4-2\epsilon$ dimensions. This allows the expression of evolution kernels in terms of reciprocity-respecting quantities, utilizing harmonic sums and binomial sums to ensure specific structural properties (e.g., invariance under $N \to -N-1$ ).
Small- $x$ Expansions: The authors perform small- $x$ expansions of the splitting functions to identify double-logarithmic (DL) and single-logarithmic terms, comparing them against previous theoretical expectations derived from lower-order results.
Numerical Evaluation: Exact results for non-singlet moments were computed using the Forcer package for specific values of the Mellin moment $N$ (up to $N=24$ for certain terms). These exact moments, combined with endpoint information (large- $x$ and small- $x$ limits), were used to construct accurate parametrizations in $x$ -space.
Approximation Schemes: For the singlet sector, where exact all- $N$ results are not yet fully available, the authors rely on approximations constructed from even- $N$ moments (up to $N=20$ ) supplemented by small- $x$ and large- $x$ constraints.

Key Contributions and Results

Completion of Non-Singlet Sectors:
The paper confirms the completion of the three non-singlet splitting functions ( $P^{(3)}_{\pm, ns}$ and $P^{(3)}_{s, ns}$ ) as presented in reference [32].
- Small- $x$ Anomalies: A significant finding is the deviation from previous expectations regarding small- $x$ behavior. Specifically, the 4-loop results exhibit non-vanishing next-to-next-to-leading log (NNLL) double-logarithmic coefficients ( $\zeta_2$ terms) for even powers of $x$ in the $P^+_{ns}$ case. These terms were previously predicted to vanish or be fixed by lower-order information but are found to be non-zero, indicating new analytical structures not present at lower orders.
- Reciprocity Structure: The results confirm that the universal evolution kernels respect reciprocity, implying that harmonic sums with index -1 do not appear in the massless perturbative QCD splitting functions.
- Parametrizations: The authors provide accurate numerical parametrizations (Fortran-ready forms) for $P^{(3)}_{\pm, ns}(x)$ and $P^{(3)}_{s, ns}(x)$ , valid across the full range of momentum fractions $x$ .
Status of Singlet Sectors:
The flavor-singlet sector ( $P_{qq}, P_{qg}, P_{gq}, P_{gg}$ ) remains partially known.
- Analytical Structures: The singlet anomalous dimensions display novel analytical structures not seen in lower orders, including $\zeta$ -function terms with $N$ in the numerator (e.g., $(N-1)$ , $(N+2)$ , $N(N+1)$ ) and specific combinations of harmonic sums like $f(N)$ and $\tilde{f}(N)$ . A notable feature is the appearance of a $\delta_{N,2}$ term in the analytical expressions, linked to the function $x^{-2}H_{-1,0}(x)$ .
- Approximations: Current knowledge is limited to specific color factors (e.g., $n_f^3$ , $n_f^2$ ) and approximate $x$ -space expressions constructed from moments up to $N=20$ . These approximations suffer from larger uncertainties at very small $x$ ( $x < 10^{-3}$ ) compared to the non-singlet case.
Numerical Impact:
- Non-Singlet: The N $^3$ LO corrections to the scale derivatives of non-singlet quark distributions are generally small, amounting to about 1% or less down to $x \approx 10^{-4}$ at a benchmark scale where $\alpha_s = 0.2$ . The exception is the total valence quark distribution at small $x$ , where contributions from $P^{(n)}_{s, ns}$ (for $n=2,3$ ) are more significant.
- Singlet: For singlet quark and gluon distributions, the N $^3$ LO corrections are also small (reaching $(2 \pm 1)\%$ at $x=10^{-5}$ ), with uncertainties becoming relevant only below $x \approx 10^{-4}$ due to the convolution with the input distributions.

Significance and Claims
The paper asserts that the completion of the non-singlet 4-loop splitting functions represents a major milestone, providing the necessary ingredients for fully consistent N $^3$ LO analyses of deep-inelastic scattering and other hard processes. The discovery of unexpected small- $x$ double logarithms challenges previous theoretical assumptions and suggests that the analytical structure of splitting functions is more complex than anticipated.

Regarding the singlet sector, the authors claim that while the current approximate results are likely sufficient for most practical applications, the determination of the full $x^{-1} \ln x$ terms (which are currently unknown for some components) would be a "game changer." They note that the state of the art for the small- $x$ behavior of the pure-singlet and gluon-to-quark splitting functions has not significantly advanced since 1994. The authors express confidence that the singlet functions will be computed exactly in the near future, rendering incremental improvements to current approximations unnecessary.

The work is supported by the availability of exact expansion coefficients and parametrization files, intended to facilitate immediate use in global PDF fits and phenomenological studies.

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