The infrared limit of the SRG evolution and Levinson's theorem
Tipo
Artigo
Data de publicação
2014
Periódico
Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics
Citações (Scopus)
10
Autores
Arriola E.R.
Szpigel S.
Timoteo V.S.
Szpigel S.
Timoteo V.S.
Orientador
Título da Revista
ISSN da Revista
Título de Volume
Membros da banca
Programa
Resumo
On a finite momentum grid with N integration points p n and weights wn (n = 1, ..., N) the Similarity Renormalization Group (SRG) with a given generator G unitarily evolves an initial interaction with a cutoff λ on energy differences, steadily driving the starting Hamiltonian in momentum space Hn,m0=pn2δn,m+Vn,m to a diagonal form in the infrared limit (λ→0), Hn,mG,λ→0=Eπ(n)δn,m, where π(n) is a permutation of the eigenvalues E n which depends on G. Levinson's theorem establishes a relation between phase-shifts δ(p n) and the number of bound-states, n B, and reads δ(p1) - δ(p N) = n Bπ. We show that unitarily equivalent Hamiltonians on the grid generate reaction matrices which are compatible with Levinson's theorem but are phase-inequivalent along the SRG trajectory. An isospectral definition of the phase-shift in terms of an energy-shift is possible but requires in addition a proper ordering of states on a momentum grid such as to fulfill Levinson's theorem. We show how the SRG with different generators G induces different isospectral flows in the presence of bound-states, leading to distinct orderings in the infrared limit. While the Wilson generator induces an ascending ordering incompatible with Levinson's theorem, the Wegner generator provides a much better ordering, although not the optimal one. We illustrate the discussion with the nucleon-nucleon (NN) interaction in the S01 and S13 channels. © 2014.