Phase Diagram and Finite Temperature Properties of Negative Coupling Scalar Field Theory

Abstract

In this work, I consider scalar field theory with negative quartic self-interaction, corresponding to an upside-down classical potential. Despite not possessing a classically stable ground state, such potentials are known to behave properly when treated quantum mechanically, leading to stable and unitary time evolution. Using two different saddle-point expansions for the same theory, I discuss the phase diagram in terms of bare parameters in Euclidean dimensions one to four, as well as the generalization to finite temperature. Comparing to other methods where available, I find that negative coupling field theory is a promising candidate for an interacting scalar field theory in the continuum. In particular, in four dimensions it exploits a loophole in mathematical proofs of quantum triviality, suggesting that negative coupling scalar field theory could offer a UV-complete and interacting description of the Higgs.

Figures (5)

• Figure 1: Left: Comparison of lowest lying eigenvalues $E_0,E_1$ of the Hamiltonian (\ref{['hami']}) to minus the pressure $p(M,T=0),\tilde{p}(\tilde{M},T=0)$ from the symmetric and broken saddles obtained in the R1-level resummation. Right: Phase Diagram at finite temperature, indicating which saddle is thermodynamically preferred. Note that there is no actual phase transition here, just a transition from one saddle to another saddle. See text for details.
• Figure 2: Left: Free Energy (negative pressure) from the symmetric and broken phase saddle expansions, as a function of the coupling. Right: Phase diagram at finite temperature, indicating which saddle is thermodynamically preferred. Line labeled 'high T approx' is Eq. (\ref{['d2hightap']}). See text for details.
• Figure 3: Pressure for the dominant phase as a function of temperature for $g_B=5 \Lambda_{\overline{\rm MS}}^2$. Left part of the plot shows $\frac{p(M,T)}{\Lambda_{\overline{\rm MS}}^2}$ whereas right part shows $\frac{p(M,T)}{T^2}$ to indicate approach to the Stefan-Boltzmann limit $p_{SB}(T)=\frac{\pi T^2}{6}$ (grey line).
• Figure 4: Left: Phase diagram at finite temperature, indicating which saddle is thermodynamically preferred. Right: pressure as a function of temperature for $m_B^2=0.5 g_B^2$. Left part of the plot shows $\frac{p(M,T)}{g_B^3}$ whereas right part shows $\frac{p(M,T)}{T^3}$ to indicate approach to the Stefan-Boltzmann limit $p_{SB}(T)=\frac{\zeta(3) T^3}{2\pi}$ (grey line). For $m_B^2=0.5 g_B^2$, the broken saddle pressure exceeds the symmetric saddle branch above $T_c\simeq 0.71 g_B$. See text for details.
• Figure 5: Pressure as a function of temperature for $m_B^2=0$. Left part of the plot shows $\frac{p(M,T)}{\Lambda_{\overline{\rm MS}}^4}$ whereas right part shows $\frac{p(M,T)}{T^4}$ to indicate approach to the Stefan-Boltzmann limit $p_{SB}(T)=\frac{\pi^2 T^4}{90}$ (grey line). See text for details.