Study of asymptotic free scalar field theories from adaptive perturbation method

We focus on the behavior of (2+1)d $\lambda {\varphi }^4$ and (5+1)d $\lambda {\varphi }^3$ or $\lambda |\varphi {|}^3$ theories in different regimes and compare the results obtained from the adaptive perturbation method with those obtained from lattice simulation. These theories are simple models that exhibit asymptotic freedom, which is a property that is also observed in more complex theories such as QCD, which describes the strong interaction between quarks and gluons. Asymptotic freedom is an important feature of these theories because it allows for a perturbative treatment of interactions at high energies. However, the standard perturbation scheme breaks down in the presence of strong interactions, and the adaptive perturbation method, which involves resuming the Feynman diagrams, is more suitable for studying these interactions. Our research involves comparing the perturbation result to lattice simulation. In the case of the ${\varphi }^3$ theory, there is no stable vacuum, so we explore evidence from the $|\varphi {|}^3$ theory instead. Our results appear to show that resummation improves the strong coupling result for both the $\lambda {\varphi }^4$ and $\lambda |\varphi {|}^3$ theories. Additionally, we improve the resummation method for the three-point coupling vertex and study the RG flow to analyze the resummation contribution and theoretical properties.