⚛️ phenomenology

The four-loop quark-to-quark splitting function in QCD

This paper presents the first fully analytic calculation of the four-loop pure-singlet contribution to the quark-to-quark splitting function in perturbative QCD, thereby completing the full splitting function at this order and providing precise numerical representations for parton evolution.

Original authors: Thomas Gehrmann, Andreas von Manteuffel, Vasily Sotnikov, Tong-Zhi Yang

Published 2026-09-30

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CC BY 4.0

Original authors: Thomas Gehrmann, Andreas von Manteuffel, Vasily Sotnikov, Tong-Zhi Yang

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 and neutron, the building blocks of ordinary matter, a chaotic storm of smaller particles swirls in a state of constant flux. These particles, known as quarks and gluons, do not sit still; they constantly split apart and recombine, exchanging energy and momentum in a dance governed by the strong nuclear force. To understand how these particles behave when they are smashed together in high-energy colliders, physicists must track how their distribution changes as the energy scale shifts. This tracking relies on a set of mathematical rules called splitting functions, which act as a map for how one particle transforms into another. For decades, scientists have been able to draw this map with increasing precision, but a specific, crucial section of the chart remained incomplete, forcing researchers to rely on approximations that left a shadow of uncertainty over their predictions.

The work presented in this paper fills that final gap. The researchers have calculated the four-loop contribution to the quark-to-quark splitting function, a term that describes how a quark emits a gluon and then reabsorbs it, or how it transforms into a different flavor of quark, all within the complex environment of the strong force. While previous calculations had managed to solve parts of this problem or estimate the answer using fixed points, this team has derived a complete, exact formula that works for every possible momentum fraction. By combining this new result with previously known data, they have now fully mapped the quark-to-quark splitting function at the fourth level of precision in quantum chromodynamics. This achievement removes the need for approximations in this specific area, allowing for a much sharper understanding of how particles evolve inside the proton.

To reach this level of detail, the team had to navigate a landscape of immense complexity. In the quantum world, calculating how particles interact involves summing up an infinite number of possible paths and configurations. At the fourth loop level, which represents a very high order of precision, the number of these configurations explodes into the millions. The researchers used powerful computer algorithms to generate the necessary diagrams, which represent the various ways quarks and gluons can interact, and then simplified the resulting mathematical expressions. They treated the problem as a puzzle of algebraic structures, breaking it down into manageable pieces and solving them step by step. This process required handling intricate patterns of numbers and symbols that describe the forces between particles, eventually reconstructing a single, unified expression that describes the entire behavior of the system.

The result is a fully analytic expression, meaning it is a precise mathematical formula valid for all conditions, rather than a collection of estimates. The team verified their work by checking it against known partial results and by examining how the function behaves when particles carry very little momentum. They found that their new formula aligns perfectly with previous calculations where they overlapped, confirming the accuracy of their method. Furthermore, they discovered that the formula behaves in a specific, predictable way at the extremes, with certain terms dominating the behavior when momentum fractions are very small. This detailed view allows physicists to see exactly how the distribution of particles changes, removing the guesswork that previously existed in this part of the evolution matrix.

One of the most significant outcomes of this work is the elimination of theoretical uncertainty in this specific entry of the evolution matrix. In the past, scientists had to use approximations based on fixed points or limited data, which introduced a residual error into predictions for particle collisions. With this new, exact formula, that source of error is gone. The researchers also provided a highly accurate numerical fit for the function, making it easy for other scientists to use in their own simulations of particle collisions. This precision is vital for the next generation of experiments, where the goal is to detect subtle signs of new physics hidden within the noise of standard particle interactions.

The team also compared their exact result with a previous approximation that was widely used. They found that while the approximation was generally good, it missed some subtle details, particularly in the way the function behaves at very small momentum fractions. The new calculation shows that the leading terms in the function are more complex than previously thought, and the approximation failed to capture the full strength of these terms. This discrepancy highlights the importance of having the exact formula, as relying on the approximation could lead to small but significant errors in high-precision predictions. The researchers noted that while the approximation worked well in many cases, the exact result reveals a deeper structure that was previously hidden.

In the broader context of particle physics, this calculation is a milestone in the effort to understand the strong force. It represents the completion of a major chapter in the study of how quarks and gluons evolve. The authors confirmed that their results agree with all available partial data and that the new formula is consistent with the known behavior of the system. However, they also pointed out a small discrepancy with a theoretical prediction based on high-energy resummation, a method used to estimate behavior at extreme energies. This mismatch, which appears in a specific color-dependent term, remains an open question that will require further theoretical investigation.

The paper concludes by making the new data widely available to the scientific community. The researchers have provided the full mathematical expression and a precise numerical fit, ensuring that other physicists can immediately incorporate these results into their models. This transparency accelerates the progress of the field, allowing for more accurate predictions of what will happen when particles collide at the highest energies. By removing the uncertainty in the quark-to-quark splitting function, this work sharpens the lens through which we view the fundamental building blocks of the universe, bringing us one step closer to a complete understanding of the strong force.

Technical Summary: The Four-Loop Quark-to-Quark Splitting Function in QCD

Problem and Motivation
The scale evolution of parton distribution functions (PDFs) is governed by the DGLAP equations, which rely on QCD splitting functions. For consistent evolution at next-to-next-to-next-to leading order (N3LO) in perturbative QCD, four-loop splitting functions are required. While the non-singlet quark-to-quark splitting function at four loops was previously determined analytically, the pure-singlet contribution ( $P_{ps}$ ) remained incomplete. Prior to this work, the pure-singlet sector relied on approximations based on fixed Mellin moments, partial analytic results, and resummation behaviors at small and large momentum fractions. These approximations introduced residual theoretical uncertainties, particularly at small momentum fractions ( $x$ ), limiting the precision of PDF evolution predictions.

Methodology
The authors compute the four-loop pure-singlet contribution to the quark-to-quark splitting function, $P^{(3)}_{ps}(x)$ , within the $\overline{\text{MS}}$ scheme. The calculation proceeds through the following steps:

Operator Formalism: The splitting function is related via a Mellin transform to the anomalous dimension $\gamma_{qq}(n)$ of the flavor-singlet twist-two quark operator. The renormalization of this operator involves mixing with physical gluon operators and gauge-variant operators. The authors extract the relevant renormalization constant $Z_{qq}$ from the ultraviolet poles of off-shell two-parton operator matrix elements (OMEs).
Diagram Generation and Reduction: Using Qgraf, the authors generate contributing Feynman diagrams, including both planar and non-planar topologies. The Lorentz and Dirac algebra are handled with Form, and color algebra with Color.h.
Integral Evaluation: To evaluate the OMEs, the authors employ a strategy converting symbolic powers of the operator into linear propagators using an auxiliary parameter $t$ . The resulting scalar integrals are reduced to master integrals using integration-by-parts (IBP) identities via Reduze 2 and Finred, utilizing finite-field sampling and optimized reduction equations.
Differential Equations: Differential equations in the parameter $t$ are derived for the basis integrals. These are solved as Laurent series in the dimensional regularization parameter $\epsilon$ , with coefficients expanded as Taylor series in $t$ . Boundary conditions are fixed using standard four-loop self-energy integrals at $t=0$ .
Reconstruction: The authors extract the anomalous dimension $\gamma^{(3)}_{qq}$ from the $1/\epsilon$ poles. By subtracting the known non-singlet contribution, they isolate the pure-singlet term $\gamma^{(3)}_{ps}$ . They then reconstruct the full $n$ -dependence (where $n$ is the Mellin moment) using an ansatz based on harmonic sums up to weight six, including rational prefactors involving $n, n \pm 1, n \pm 2$ . Finally, an inverse Mellin transformation is performed to obtain the analytic splitting function $P^{(3)}_{ps}(x)$ .

Key Results

Analytic Expression: The paper provides the first fully analytic expression for the four-loop pure-singlet quark-to-quark splitting function, valid for all momentum fractions $x$ .
Small- $x$ Behavior: The authors derive the behavior of the splitting function in the small- $x$ limit, explicitly determining the coefficients of the leading and subleading logarithmic terms. This includes previously unknown coefficients for subleading logarithms at leading power, expressed in terms of color factors ( $C_A, C_F, n_f$ ) and Riemann zeta values ( $\zeta_2, \zeta_3, \zeta_5$ ).
Verification: The results confirm previous partial calculations, including fixed-moment results up to $n=22$ and specific coefficients of $\zeta$ -terms. The analytic result agrees with the known non-singlet contribution and the quartic-color contributions.
Numerical Implementation: To facilitate practical use in parton evolution, the authors provide a precise numerical fit for the remainder of the splitting function after subtracting the small- $x$ expansion. This fit achieves absolute and relative deviations of at most $10^{-5}$ and $10^{-8}$ , respectively.

Significance and Claims
The primary significance of this work is the completion of the four-loop quark-to-quark splitting function. By providing the exact analytic pure-singlet contribution, the authors eliminate the approximation uncertainty that previously existed in this entry of the evolution matrix.

The authors note that their exact result agrees with existing approximations within uncertainties across the range $x \in [10^{-5}, 1]$ . They attribute this agreement to the fact that the leading divergence in the pure-singlet sector is of higher logarithmic order ( $\log^2(x)/x$ ) and was already known exactly, rendering discrepancies in subleading terms subdominant.

However, the authors identify a specific discrepancy with the small- $x$ resummation prediction from Ref. [18] regarding the subleading-color contribution proportional to $\zeta_2$ at the next-to-next-to-leading logarithmic level. They state that the structural origin of this discrepancy remains an open question. Additionally, the exact result for $\gamma^{(3)}_{qq}$ provides the quark contribution to the singlet trace, which may assist in reconstructing the gluon-gluon splitting function $\gamma^{(3)}_{gg}$ from fixed Mellin moments via reciprocity relations.

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