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We calculate the energy of the state closest to threshold for two and three identical, spinless particles confined to a cubic spatial volume with periodic boundary conditions and with zero total momentum in the finite-volume frame. The calculation is performed in relativistic quantum field theory with particles coupled via a λφ⁴ interaction, and we work through order λ³. The energy shifts begin at O(1/L³), and we keep subleading terms proportional to 1/L⁴, 1/L⁵ and 1/L⁶. These terms allow a nontrivial check of the results obtained from quantization conditions that hold for arbitrary interactions, namely that of L\"uscher for two particles and our recently developed formalism for three particles. We also compare to previously obtained results based on nonrelativistic quantum mechanics.
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Hansen et al. (2016) studied this question.
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