Hemisphere mass up to four-loops with generalised $k_t$ algorithms

We compute the fixed-order distribution of the non-global hemisphere mass observable in $e^+ e^-$ annihilation up to four loops for various sequential recombination jet algorithms. In particular, we focus on the $k_t$ and Cambridge/Aachen algorithms. Using eikonal theory and strong-energy ordering of the final-state partons, we determine the complete structure of both abelian (clustering) and non-abelian non-global logarithms through four loops in perturbation theory. We compare the resulting resummed expressions for both jet algorithms with the standard Sudakov form factor and demonstrate that neglecting these logarithms leads to unreliable phenomenological predictions for the observable's distribution.