The β function of the two-dimensional non-linear σ model

Abstract

Improved MCRG methods are used to determine the ~ function of the two-dimensional 0(3) standard action o modeL We argue that the previously observed anomalous behavior of the mass gap can be understood as a general scaling behavior according to the full # function.

Four-loop free energy for the 2D O(n) non-linear σ-model with 0-loop and 1-loop Symanzik improved actions

“…of this equation can produce terms like 11) which are finite after the removal of the IR-regulator. These terms are exclusive of the finite size L regularization: notice for example that they cannot be expressed as usual integrals, neither after the L → ∞ limit.…”

“…For instance, the calculation of the mass gap in this model with O(3) symmetry has yielded results [7][8][9][10][11][12][13][14][15] in conflict with the exact analytical value m HMN of Ref. [16].…”

Continuum limit of the hierarchicalO(N) non-linear ?-model

“…The simplest O(N) version of the a-model exhibits many properties expected for QCD and serves as a good playground for the Monte-Carlo (renormalization group) simulations of non-linear theories with symmetries, see e.g. [4]. Its S-matrix is known [5] as well as its spectrum for N--3 [6].…”

Asymptotic safety on the lattice: The nonlinear O(N) sigma model

“…Thus, a large number of results are available, both from numerical high-precision Monte Carlo methods as well as analytical calculations using the high-temperature expansion or renormalization group method. Even experimental data from condensed matter physics are available, see for example [11,15,16,20,26,[29][30][31][32]. We are particularly interested in the flow diagram of the threedimensional model that is conjectured to show a non-trivial UV fixed point, a necessary requirement for the asymptotic safety scenario to be at work.…”

“…In an alternative recent approach one tries to directly integrate out momentum shells on the lattice by using Fourier Monte Carlo simulation [9]. In the present work we make use of the well-known Monte Carlo Renormalization Group method (MCRG) [10][11][12][13]. It is based on the idea of blockspin transformations and can be applied to theories with fermionic or gauge fields [14].…”