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 a posteriori, just as the intensive parameter in the
micro-canonical 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
multi-critical point principle. As a concrete toy model for solving the Higgs
hierarchy problem, we study how the mass parameter is fixed in the Ï•4
theory at the one-loop level in the micro-canonical 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 Ï•4
theory and find that it can be fixed to a finite value.Comment: 29 pages, 3 figure