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We present a novel framework to decompose three-nucleon forces in a momentumspace partial-wave basis. The new approach is computationally much moreefficient than previous methods and opens the way to ab initio studies offew-nucleon scattering processes, nuclei and nuclear matter based onhigher-order chiral 3N forces. We use the new framework to calculate matrixelements of chiral three-nucleon forces at N2LO and N3LO in large basis spacesand carry out benchmark calculations for neutron matter and symmetric nuclearmatter. We also study the size of the individual three-nucleon forcecontributions for $^3$H. For nonlocal regulators, we find that the sub-leadingterms, which have been neglected in most calculations so far, provide importantcontributions. All matrix elements are calculated and stored in a user-friendlyway, such that values of low-energy constants as well as the form of regulatorfunctions can be chosen freely.

Efficient calculation of chiral three-nucleon forces up to N3LO for ab initio studies