Progress on double-logarithmic large-x and small-x resummations for (semi-)inclusive hard processes

TL;DR

The paper addresses the challenge of double-logarithmic resummations at both large-$x$ and small-$x$ for (semi-)inclusive hard processes, focusing on DIS and SIA within massless perturbative QCD. It develops and applies two complementary strategies to extract all-order highest logarithms from NNLO inputs: physical evolution kernels that reveal single-log enhancement across endpoint expansions, and unfactorized quantities that enable systematic end-point resummations via epsilon-expansions and dimensional decompositions. The authors report closed-form, all-order expressions (often in terms of Bernoulli functions) for key off-diagonal and timelike quantities, identify intriguing patterns such as vanishing leading logs in certain four-loop singlet terms, and discuss the limitations and channels where these resummations remain challenging. These results sharpen theoretical control over endpoint regions, guiding high-precision analyses and pointing to future work needed to extend coverage to remaining processes and moments.

Abstract

Over the past few years considerable progress has been made on the resummation of double-logarithmically enhanced threshold (large-x) and high-energy (small-x) higher-order contributions to the splitting functions for parton and fragmentation distributions and to the coefficient functions for inclusive deep-inelastic scattering and semi-inclusive e^+e^- annihilation. We present an overview of the methods which allow, in many cases, to derive the coefficients of the highest three logarithms at all orders in the strong coupling from next-to-next-to-leading order results in massless perturbative QCD. Some representative analytical and numerical results are shown, and the present limitations of these resummations are discussed.

Figures (2)

• Figure 1: nameref-fig1 fith LAB: fig1 Left: the third-order coefficient function for $F_{2,\rm ns}$MVV6 compared to large-$N$ approximations by only the soft$\,$+$\,$virtual $N^{\,0\!}$ terms and by those plus the ($\,r=0\,$) $N^{\,-1\!}$ contributions. Right: the LO, NLO and NNLO P2Tgg approximations to the timelike splitting function $P_{\rm gg}^{\,T}$ for five flavours at a typical scale $Q^{\,2} \simeq M_{Z\!}^{\:2\!}$.
• Figure 2: nameref-fig2 fith LAB: fig2 Left: the fourth-order coefficient function for the structure function $F_{2,\rm ns}$. Shown are the large $N$ estimates by the known seven (of eight) $\ln^n N$ soft-gluon contributions and by adding the highest four (of seven) $N^{\,-1}\ln^n N$ corrections. Right: the timelike gluon-gluon splitting functions (multiplied by $x$) for a very wide range of the momentum fraction $x$ at a value of $\alpha_{\,\sf s}$ corresponding to $Q^{\,2} \simeq M_{Z\!}^{\:2\!}$. The all-$x$ ($N\!=\!1$ finite) LO$\,+\,$LL and NLO$\,+\,$NNLL results are compared to the LO and NLO approximations valid only at large $x$.