Nonperturbative tests for asymptotic freedom in the $\mathcal{P}\mathcal{T}$-symmetric $(-{\varphi }^4{)}_{3+1}$ theory

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 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 $\mathcal{P}\mathcal{T}$-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 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 $\Lambda$ 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 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 $\mathcal{P}\mathcal{T}$-symmetric Higgs mechanism.

Figure 1

The vacuum energy (has mass dimensions of 4) of the ($-{\varphi }^4$) scalar field theory. For either $G$ small (high energy scales) or large (IR energy scales) the vacuum energy is finite.

Figure 2

The mass squared $t$ ($t=\frac{M^2}{m^2}$) of the $(-{\varphi }^4{)}_{3+1}$ scalar field theory, which has mass dimensions of 2. In this case also for either $G$ small (high energy scales) or large (IR energy scales) $t$ is finite.

Figure 3

The absolute value of the vacuum condensate (has mass dimensions of 1) $|b|$ of the ($-{\varphi }^4$) scalar field theory. For either $G$ small (high energy scales) or large (IR energy scales) the vacuum condensate is finite too.