Quantum phase transition and absence of quadratic divergence in generalized quantum field theories

Abstract

In ordinary thermodynamics, around first-order phase transitions, the intensive parameters such as temperature and pressure are automatically fixed to the phase transition point when one controls the extensive parameters such as total volume and total energy. From the microscopic point of view, the extensive parameters are more fundamental than the intensive parameters. Analogously, in conventional quantum field theory (QFT), coupling constants (including masses) in the path integral correspond to intensive parameters in the partition function of the canonical formulation. Therefore, it is natural to expect that, in a more fundamental formulation of QFT, coupling constants are dynamically fixed , just as the intensive parameter in the microcanonical formulation. Here, we demonstrate that the automatic tuning of the coupling constants is realized at a quantum phase transition point at zero temperature, even when the transition is of higher order, due to the Lorentzian nature of the path integral. This naturally provides a basic foundation for the multicritical point principle. As a concrete toy model for solving the Higgs hierarchy problem, we study how the mass parameter is fixed in the ϕ4 theory at the one-loop level in the microcanonical or further generalized formulation of QFT. We find that there are two critical points for the renormalized mass: zero and of the order of ultraviolet cutoff. In the former, the Higgs mass is automatically tuned to be zero and thus its fine-tuning problem is solved. We also show that the quadratic divergence is absent in a more realistic two-scalar model that realizes the dimensional transmutation. Additionally, we explore the possibility of fixing quartic coupling in ϕ4 theory and find that it can be fixed to a finite value. Published by the American Physical Society 2024