  We introduce the first family of parton showers that achieve next-to-leading
logarithmic (NLL) accuracy for processes involving a $t$-channel exchange of a
colour-singlet, and embed them in the PanScales framework. These showers are
applicable to processes such as deep inelastic scattering (DIS), vector boson
fusion (VBF), and vector boson scattering (VBS). We extensively test and verify
the NLL accuracy of the new showers at both fixed order and all orders across a
wide range of observables. We also introduce a generalisation of the
Cambridge-Aachen jet algorithm and formulate new DIS observables that exhibit a
simple resummation structure. The NLL showers are compared to a standard
transverse-momentum ordered dipole shower, serving as a proxy for the current
state-of-the-art leading-logarithmic showers available in public codes.
Depending on the observable, we find discrepancies at NLL of the order of
$15\%$. We also present some exploratory phenomenological results for Higgs
production in VBF. This work enables, for the first time, to resum
simultaneously global and non-global observables for the VBF process at NLL
accuracy.
