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Callan–Symanzik–Lifshitz approach to generic competing systems

Authors
Journal
Annals of Physics
0003-4916
Publisher
Elsevier
Publication Date
Volume
325
Issue
1
Identifiers
DOI: 10.1016/j.aop.2009.07.006
Keywords
  • Renormalization Group With Independent Mass Scales
  • Generic Competing Systems
  • Lifshitz Points Of Arbitrary Higher Character

Abstract

Abstract We present the massive method to approaching the critical behaviors of systems with arbitrary competing interactions. Every distinct competition subspace in the anisotropic cases define an independent set of renormalized vertex parts via normalization conditions with nonvanishing distinct masses at zero external momenta. Otherwise, only one mass scale is required in the isotropic behaviors. At the critical dimension, we prove: (i) the existence of the Callan–Symanzik–Lifshitz equations and (ii) the multiplicative renormalizability of the vertex functions using the inductive method. Away from the critical dimension, we utilize the orthogonal approximation to compute higher loop Feynman integrals, anisotropic as well as isotropic, necessary to get the exponents η n and ν n at least up to two-loop level. Moreover, we calculate the latter exactly for isotropic behaviors at the same perturbative order. Similarly to the computation in the massless formalism, the orthogonal approximation is found to be exact at one-loop order. The outcome for all critical exponents matches exactly with those computed using the zero mass field-theoretic description renormalized at nonvanishing external momenta.

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