Additional results on the four-loop flavour-singlet splitting functions in QCD

Abstract: We have extended our previous computations, performed analytically for a general gauge group, of the even-$N$ moments $γ{\rm ik}^{\,(3)}(N)$ of the four-loop flavour-singlet splitting functions $P{\rm ik}^{\,(3)}(x)$ to $N = 22$. The numerical QCD results perfectly agree with all predictions resulting from the approximations for $P_{\rm ik}^{\,(3)}(x)$ that we obtained before from the moments $N \leq 20 $ and endpoint constraints, confirming their reliability for collider-physics applications. Due to the additional analytical constraints provided by our new $N = 22$ results, we are now closing in on determining the all-$N$ forms of all non-rational ($ζ$-function) contributions to $γ{\rm ik}^{\,(3)}(N)$: only the $n_f^{0} ζ_3$ parts of the quark-to-gluon (gq) and the $n_f^{1} ζ_3$ parts of the gluon-to-gluon (qg) cases still need to be completed. Finally we extend our approximations for all $P{\rm ik}^{\,(3)}(x)$ to $n_f^{} = 6$ light flavours and update those for $P_{\rm gq}^{\,(3)}(x)$ at lower $n_f^{}$.

• The paper computes new even-N Mellin moments up to N=22, significantly extending four-loop flavour-singlet calculations in QCD.
• It derives all-N analytic expressions, detailing zeta function contributions and harmonic sum structures critical for precision DGLAP evolution.
• The study validates updated x-space approximations, confirming their reliability in reducing uncertainties in small-x QCD phenomenology.

Four-Loop Flavour-Singlet Splitting Functions in QCD: New Even-$N$ Moments and All-$N$ Structures

Introduction and Objectives

The paper "Additional results on the four-loop flavour-singlet splitting functions in QCD" (2512.10783) advances the analytic computation of flavour-singlet splitting functions $P_{\rm ik}^{(3)}(x)$ in massless perturbative QCD at the four-loop order (N$^3$LO), focusing on the determination of higher even Mellin moments up to $N=22$. These splitting functions govern the renormalization-group evolution (DGLAP) of parton distribution functions (PDFs) and are essential for precision phenomenology at high-energy colliders, notably for processes such as Higgs boson production at the LHC operating at its current accuracy frontier.

Emphasis is placed on:

• Calculating new even-$N$ moments $\gamma_{\rm ik}^{(3)}(N)$ for $N=22$, extending previous computations to $N\leq 20$.
• Providing further analytic constraints for the all-$N$ dependence, especially for contributions involving zeta functions ($\zeta_3$, $\zeta_4$, $\zeta_5$).
• Validating the $x$-space approximations used in phenomenological studies and examining their uncertainty bands.
• Progressing towards closed-form expressions for all-$N$ Mellin moments, particularly for non-rational terms, to better understand the small-$x$ regime and facilitate resummation studies.

Expansion to $N=22$ Moments and Validation of $x$-space Approximations

Using an optimized version of the FORCER program within FORM, the authors compute exact analytic expressions for the four-loop flavour-singlet anomalous dimensions $\gamma_{\rm ik}^{(3)}(N)$ at $N=22$ for general simple gauge groups and, in particular, for QCD relevant values of $n_f = 3,4,5,6$. The numerical results for physically relevant cases ($n_f=3,4,5$) are compared to predictions from previous $x$-space fits based on $N\leq 20$ moments and DGLAP endpoint constraints. The agreement is precise to all quoted digits, demonstrating that no systematic underestimation of uncertainties at low $x$ exists for these state-of-the-art approximations.

This comparison provides robust evidence for the reliability of phenomenological studies at N$^3$LO and the stability of error bands in the small-$x$ region ($x\sim10^{-3}$), which had remained a delicate point given their enhanced contribution from logarithmic terms.

Analytic Structure at All-$N$: Zeta Function Contributions and New Features

The new $N=22$ results provide additional constraints enabling the derivation of all-$N$ analytic expressions for the contributions proportional to zeta values (notably $\zeta_3$). For the pure-singlet and selected off-diagonal cases (quark-to-gluon $qg$ and gluon-to-quark $gq$), the paper presents explicit, all-$N$ decompositions for the $\zeta_3$ terms.

Key advancements include:

• Identification of analytic structures not present at three loops, in particular, terms independent of $N$ and specific weight-3 harmonic sums combinations that vanish as $1/N^2$, as well as contributions related to $\delta(N-2)$.
• Explicit determination of most non-rational $\zeta$ function contributions, with the remaining uncertainty limited to quadratic Casimir terms in specific off-diagonal entries.
• Verification that the $\zeta_5$ and $\zeta_4$ parts are now fully determined for all $N$, building on previous work.

These analytic results also allow for precise predictions and cross-checks in Mellin space, underpinning further mathematical understanding of splitting functions and their resummation properties.

Updated Approximations and Extensions for Larger $n_f$

Responding to requests from the community, the authors provide extensions of $x$-space approximations for $P_{\rm ik}^{(3)}(x)$ to $n_f=6$—relevant for PDFs at ultra-high energy scales—and update previous fits for the $gq$ channel at $n_f=3,4,5$ by incorporating the improved knowledge of the coefficient of $x^{-1}\ln x$. These updates are important for precision fits of PDFs in future collider scenarios and provide bounds on the theoretical uncertainties entering precision observables.

Implications and Further Developments

The extension to higher moments and the improved analytic control over non-rational terms in the singlet evolution kernels have several implications:

• PDF Evolution: The results solidify the foundation for DGLAP evolution at N$^3$LO in phenomenological analyses, reducing uncertainties in QCD backgrounds for precision collider measurements.
• Small-$x$ Physics: By tightening the constraints on small-$x$ behaviour, the findings enable more reliable extrapolations into regimes where higher-order logarithms and possibly BFKL-type corrections become relevant.
• Operator Product Expansion: The techniques and closed-form results for Mellin moments contribute to improved understanding of twist-two operator anomalous dimensions at four loops.
• Resummation Studies: The explicit all-$N$ analytic structures support investigations into threshold and high-energy resummation schemes, especially regarding non-power-law $\zeta$ terms.

Future work will address the remaining Casimir contributions for all-$N$ expressions and further refine the singlet and non-singlet splitting functions at this order. Improved results on the non-singlet sector, including large-$n_c$ suppressed terms, are anticipated as computational and algorithmic techniques advance.

Conclusion

This work completes a significant step towards the full analytic understanding and phenomenological control of four-loop, flavour-singlet splitting functions in QCD. By pushing the computation to $N=22$ and extracting all-$N$ formulae for many contributing terms, it substantiates the validity and stability of $x$-space splitting function approximations for N$^3$LO DGLAP evolution, particularly at small $x$. The new analytic results on the Mellin moment structure, especially regarding zeta function contributions, will facilitate ongoing efforts to fully resolve the four-loop singlet evolution and underpin high-precision studies in QCD and collider phenomenology (2512.10783).