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In the literature, the asymptotic freedom property of the $(-\phi^{4})$theory is always concluded from real-line calculations while the theory isknown to be a non-real-line one. In this article, we test the existence of theasymptotic freedom in the $(-\phi^{4})_{3+1}$ theory using mean field approach.In this approach and contrary to the original Hamiltonian, the obtainedeffective Hamiltonian is rather a real-line one. Accordingly, this workresembles the first reasonable analysis for the existence of the asymptoticfreedom property in the $\mathcal{PT}$-symmetric $(-\phi^{4})$ theory. In thisrespect, we calculated three different amplitudes of different positivedimensions (in mass units) and find that all of them goes to very small valuesat high energy scales (small coupling) in agreement with the spirit of theasymptotic freedom property of the theory. To test the validity of ourcalculations, we obtained the asymptotic behavior of the vacuum condensate interms 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 numericalpredictions. We assert that the non-blow up of the massive quantities at highenergy scales predicted in this work strongly suggests the possibility of thesolution of the famous hierarchy puzzle in a standard model with$\mathcal{PT}$-symmetric Higgs mechanism.

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