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

TL;DR

This paper investigates the asymptotic freedom of the $ ext{PT}$-symmetric $(-^{4})_{3+1}$ theory using a mean field approach, finding results consistent with asymptotic freedom and suggesting implications for the hierarchy problem.

Contribution

It provides the first non-perturbative analysis of asymptotic freedom in the $ ext{PT}$-symmetric $(-^{4})$ theory using an effective real-line Hamiltonian.

Findings

All calculated amplitudes decrease at high energy scales.

The vacuum condensate's asymptotic behavior matches numerical predictions.

The theory's quantities do not blow up at high energies.

Abstract

In the literature, the asymptotic freedom property of the $(-\phi^{4})$ 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 $(-\phi^{4})_{3+1}$ 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 $\mathcal{PT}$-symmetric $(-\phi^{4})$ 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 $\mathcal{PT}$-symmetric Higgs mechanism.