2009/01/31 by Abouzeid M. Shalaby, Abouzeid Shalaby, Suleiman S. Al-Thoyaib +1
Mathematics · Physics and Astronomy · #Asymptotic freedom #Combinatorics #Coupling (piping) #Energy (signal processing) #Gauge theory #Hamiltonian (control theory) #Lambda #Mathematical physics #Mathematics #Neutrino Physics Research #Physics #Quantum Mechanics and Non-Hermitian Physics #Quantum mechanics #Topological Materials and Phenomena #hep-th
paper · pdf · doi:10.1103/physrevd.82.085013
published as Phys.Rev.D82:085013,2010 · 18 pages, 3 figures. In this version we obtained the asymptotic behaviour of the condensate, analytically, and corrected some formulas
arxiv created 2010/05/25 · openalex publication_date 2010/10/12 · arxiv updated 2014/11/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
In the literature, the asymptotic freedom property of the (\ensuremath-\ensuremathφ4) theory is always concluded from real-line calculations while the theory is known to be a non-real-line one. In this article, we test the existence of the asymptotic freedom in the (\ensuremath-\ensuremathφ4)3+1 theory using the mean field approach. In this approach and contrary to the original Hamiltonian, the obtained effective Hamiltonian is rather a real-line one. Accordingly, this work resembles the first reasonable analysis for the existence of the asymptotic freedom property in the PT-symmetric (\ensuremath-\ensuremathφ4) theory. In this respect, we calculated three different amplitudes of different positive dimensions (in mass units) and find that all of them go to very small values at high energy scales (small coupling) in agreement with the spirit of the asymptotic freedom property of the theory. To test the validity of our calculations, we obtained the asymptotic behavior of the vacuum condensate in terms of the coupling, analytically, and found that the controlling factor \ensuremathΛ has the value \frac(4\ensuremathπ)26=26.319 compared to the result \ensuremathΛ=26.3209 from the literature, which was obtained via numerical predictions. We assert that the nonblowup of the massive quantities at high energy scales predicted in this work strongly suggests the possibility of the solution of the famous hierarchy puzzle in a standard model with the PT-symmetric Higgs mechanism.