2017/07/31 by Stephen R. Sharpe
Mathematics · Physics and Astronomy · #Amplitude #Formalism (music) #Geometry #High-Energy Particle Collisions Research #Lambda #Mathematical physics #Mathematics #Particle physics theoretical and experimental studies #Perturbation theory (quantum mechanics) #Physics #Quantum Chromodynamics and Particle Interactions #Quantum mechanics #Scalar (mathematics) #Scattering #Scattering amplitude #Statistical physics #hep-lat
paper · pdf · doi:10.1103/physrevd.96.054515
published as Phys. Rev. D 96, 054515 (2017) · 33 pages, 8 figures (v2: Typos corrected, explanations improved, results unchanged---consistent with published version) (v3: Missing diagram in Fig.5 added, along with discussion of why final results unchanged---consistent with erratum.)
openalex created_date 2017/07/21 · openalex publication_date 2017/09/26 · arxiv created 2018/10/20 · arxiv updated 2018/10/23 · openalex updated_date 2026/08/05
A relativistic formalism for relating the energies of the states of three scalar particles in finite volume to infinite volume scattering amplitudes has recently been developed. This formalism has been used to predict the energy of the state closest to threshold in an expansion in powers of 1/L, with L the box length. This expansion has been tested previously by a perturbative calculation of the threshold energy in \ensuremathλ\ensuremathφ4 theory, working to third order in \ensuremathλ and up to O(1/L6) in the volume expansion. However, several aspects of the predicted threshold behavior do not enter until fourth (three-loop) order in perturbation theory. Here I extend the perturbative calculation to fourth order and find agreement with the general prediction. This check also requires a two-loop calculation of the infinite-volume off-shell two-particle scattering amplitude near threshold. As a spin-off, I check the threshold expansion for two particles to the same order, finding agreement with the result that follows from L"uscher's formalism.