Non-perturbative tests for the Asymptotic Freedom in the $\mathcal{P}\mathcal{T}$-symmetric $(-{\varphi }^4{)}_{3+1}$ theory

Abstract

In the literature, the asymptotic freedom property of the $(-{\varphi }^4)$ 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 $(-{\varphi }^4{)}_{3+1}$ theory using 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 $\mathcal{P}\mathcal{T}$-symmetric $(-{\varphi }^4)$ theory. In this respect, we calculated three different amplitudes of different positive dimensions (in mass units) and find that all of them goes 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 $\Lambda$ has the value $\frac{(4\pi {)}^2}{6}=26.319$ compared to the result $\Lambda =26.3209$ from the literature which was obtained via numerical predictions. We assert that the non-blow up 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 $\mathcal{P}\mathcal{T}$-symmetric Higgs mechanism.

18 pages, 3 figures. In this version we obtained the asymptotic behaviour of the condensate, analytically, and corrected some formulas

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Cited by in corpus (9)

• Gauge invariance and the Englert-Brout-Higgs mechanism in non-Hermitian field theories
• Spontaneously Breaking Non-Abelian Gauge Symmetry in Non-Hermitian Field Theories
• Discrete spacetime symmetries and particle mixing in non-Hermitian scalar quantum field theories
• Extrapolating the precision of the Hypergeometric Resummation to Strong couplings with application to the $\mathcal{P}\mathcal{T}-$Symmetric $i{\varphi }^3$ Field Theory
• Poincaré symmetries and representations in pseudo-Hermitian quantum field theory
• Original study of the $g{\varphi }^2(i\varphi {)}^{\epsilon }$ theory. Analysis of all orders in $\epsilon$ and resummations
• Isomorphic Hilbert spaces associated with different Complex Contours of the $\mathcal{P}\mathcal{T}$-Symmetric $(-x^4)$ Theory
• Representation Dependence of Superficial Degree of Divergences in Quantum Field Theory
• Vacuum Stability of the wrong sign $(-{\varphi }^6)$ Scalar Field Theory