Jets constructed via clustering algorithms (e.g., anti-$k_T$, soft-drop) have been proposed for many precision measurements, such as the strong coupling $alpha_s$ and the nucleon intrinsic dynamics. However, the theoretical accuracy is affected by mi
ssing QCD corrections at higher orders for the jet functions in the associated factorization theorems. Their calculation is complicated by the jet clustering procedure. In this work, we propose a method to evaluate jet functions at higher orders in QCD. The calculation involves the phase space sector decomposition with suitable soft subtractions. As a concrete example, we present the quark-jet function using the anti-$k_T$ algorithm with E-scheme recombination at next-to-next-to-leading order.
We study the threshold corrections for inclusive deep-inelastic scattering (DIS) and their all-order resummation. Using recent results for the QCD form factor, related anomalous dimensions and Mellin moments of DIS structure functions at four loops w
e derive the complete soft and collinear contributions to the DIS Wilson coefficients at four loops. For a general $SU(n_c)$ gauge group the results are exact in the large-$n_c$ approximation and for QCD with $n_c=3$ we present precise approximations. We extend the threshold resummation exponent $G^N$ in Mellin-$N$ space to the fifth logarithmic (N$^4$LL) order collecting the terms $alpha_{rm s}^{,3} (alpha_{rm s} ln N)^n$ to all orders in the strong coupling constant $alpha_{rm s}$. We study the numerical effect of the N$^4$LL corrections using both the fully exponentiated form and the expansion of the coefficient function in towers of logarithms. As a byproduct, we derive a numerical result for the complete pole structure of the QCD form factor in the parameter of dimensional regularization $varepsilon$ at four loops.
