The Three-Loop Splitting Functions in QCD: The Non-Singlet Case

TL;DR

The paper delivers the complete three-loop (NNLO) non-singlet splitting functions in QCD, enabling precise DGLAP evolution of quark densities. It employs a Mellin-$N$ space calculation with optical theorem and OPE, using harmonic sums and harmonic polylogarithms to produce both $N$-space anomalous dimensions and their $x$-space counterparts. A key finding is a new leading-log small-$x$ contribution arising from the color structure $d^{abc}d_{abc}$, which alters the naive small-$x$ expectations while maintaining agreement with fixed moments and resummation predictions. Numerically, NNLO corrections are typically small for moderate to large $x$ and renormalization-scale uncertainties are substantially reduced, with detailed parametrizations and Fortran routines provided to facilitate practical use in phenomenology.

Abstract

We compute the next-to-next-to-leading order (NNLO) contributions to the three splitting functions governing the evolution of unpolarized non-singlet combinations of quark densities in perturbative QCD. Our results agree with all partial results available in the literature. We find that the correct leading logarithmic (LL) predictions for small momentum fractions x do not provide a good estimate of the respective complete results. A new, unpredicted LL contribution is found for the colour factor d^{abc}d_{abc} entering at three loops for the first time. We investigate the size of the corrections and the stability of the NNLO evolution under variation of the renormalization scale. Except for very small x the corrections are found to be rather small even for large values of the strong coupling constant, in principle facilitating a perturbative evolution into the sub-GeV regime.

Figures (9)

• Figure 1: The perturbative expansion of the anomalous dimension $\gamma_{\,\rm ns}^{\, +}(N)$ for four flavours at $\alpha_{\rm s} = 0.2$. In the right part the leading $N$-dependence for large $N$ has been divided out, and the corresponding asymptotic limits are indicated as discussed in the text.
• Figure 2: The $n_{\,\rm f\,}^{}$-independent three-loop contribution $P_{+,0}^{\,(2)}(x)$ to the splitting function $P_{\rm ns}^{\, +}(x)$, multiplied by $(1-x)$ for display purposes. Also shown in the left part is the uncertainty band derived in Ref. vanNeerven:2000wp from the lowest six even-integer moments Larin:1994vuLarin:1997wdRetey:2000nq. In the right part our exact result is compared to the small-$x$ approximations defined in Eq. (\ref{['eq:Plnx']}) and the text below it.
• Figure 3: As Fig. 2, but for the splitting function $P_{\rm ns}^{\,-}(x)$. The first seven odd moments underlying the previous approximations vanNeerven:2000wp also shown in the left part have been computed in Ref. Retey:2000nq.
• Figure 4: The $n_{\,\rm f\,}^{\, 1}$ three-loop contributions $P_{\pm,1}^{\,(2)}(x)$ to the splitting functions $P_{\rm ns}^{\,\pm}(x)$, compared to the uncertainty bands of Ref. vanNeerven:2000wp based on the integer moments calculated in Refs. Larin:1994vuLarin:1997wdRetey:2000nq.
• Figure 5: The first non-vanishing contribution $P_{{\rm s},1}^{\,(2)} (x)$ to the splitting functions $P_{\rm ns}^{\: s}(x)$, compared to the approximations of Ref. vanNeerven:2000wp (where, assuming the completeness of the resummation Kirschner:1983diBlumlein:1996jp, the possibility of a $\ln^{\,4}x$ term was disregarded) and to the small-$x$ expansion in powers of $\ln x$.