We consider few-body systems in which only a certain subset of the particle-particle interactions is resonant. We characterize each subset by a unitary graph in which the vertices represent distinguishable particles and the edges resonant two-body interactions. Few-body systems whose unitary graph is connected will collapse unless a repulsive three-body interaction is included. We find two categories of graphs, distinguished by the kind of three-body repulsion necessary to stabilize the associated system. Each category is characterized by whether the graph contains a loop or not: for tree-like graphs (graphs containing a loop) the three-body force renormalizing them is the same as in the three-body system with two (three) resonant interactions. We show numerically that this conjecture is correct for the four-body case as well as for a few five-body configurations. We explain this result in the four-body sector qualitatively by imposing Bethe-Peierls boundary conditions on the pertinent Faddeev-Yakubovsky decomposition of the wave function.

We obtain the s - and p -wave low-energy scattering parameters for n 3 H elastic scattering and the position of the 4 H J?=1? resonance using the pionless effective field theory at leading order. Results are extracted with three numerical techniques: confining the system in a harmonic oscillator trap, solving the Faddeev-Yakubovsky equations in configuration space, and using an effective two-body cluster approach. The renormalization of the theory for the relevant amplitudes is assessed in a cutoff-regulator range between 1fm?1 and 10fm?1. Most remarkably, we find a cutoff-stable/RG-invariant resonance in the 4 H J?=1? system. This p -wave resonance is a universal consequence of a shallow two-body state and the introduction of a three-body s -wave scale set by the triton binding energy. The stabilization of a resonant state in a few-fermion system through pure contact interactions has a significant consequence for the powercounting of the pionless theory. Specifically, it suggests the appearance of similar resonant states also in larger nuclei, like 16-oxygen, in which the theory's leading order does not predict stable states. Those resonances would provide a starting state to be moved to the correct physical position by the perturbative insertion of sub-leading orders, possibly resolving the discrepancy between data and contact EFT.

We investigate a method to extract response functions (dynamical polarisabilities) directly from a bound-state approach applied to calculations of perturbation-induced reactions. The use of a square-integrable basis leads to a response in the form of a sum of ? functions. We integrate this over energy and fit a smooth function to the resulting stepwise-continuous one. Its derivative gives the final approximation to the physical response function. We show that the method reproduces analytical results where known, and analyse the details for a variety of models. We apply it to some simple models, using the stochastic variational method as the numerical method. Albeit we find that this approach, and other numerical techniques, have some difficulties with the threshold behavior in coupled-channel problems with multiple thresholds, its stochastic nature allows us to extract robust results even for such cases.