Can negative bare couplings make sense? The $\vecφ^4$ theory at large $N$

Abstract

Scalar $λφ^4$ theory in 3+1D, for a positive coupling constant $λ>0$, is known to have no interacting continuum limit, which is referred to as quantum triviality. However, it has been recently argued that the theory in 3+1D with an $N$-component scalar $\vecφ$ and a $(\vecφ\cdot\vecφ)^{\,2}=\vecφ^{\,4}$ interaction term does have an interacting continuum limit at large $N$. It has been suggested that this continuum limit has a negative (bare) coupling constant and exhibits asymptotic freedom, similar to the $\mathcal{P}\mathcal{T}$-symmetric $-gφ^4$ field theory. In this paper I study the $\vecφ^{\,4}$ theory in 3+1D at large $N$ with a negative coupling constant $-g<0$, and with the scalar field taking values in a $\mathcal{P}\mathcal{T}$-symmetric complex domain. The theory is non-trivial, has asymptotic freedom, and has a Landau pole in the IR, and I demonstrate that the thermal partition function matches that of the positive-coupling $λ>0$ theory when the Landau poles of the two theories (in the $λ>0$ case a pole in the UV) are identified with one another. The spirit of renormalization is that observables do not depend on the renormalization scale. Here we see even if the coupling is taken negative above the scale of the Landau pole, thermodynamic observables are unaffected. Thus the $\vecφ^{\,4}$ theory at large $N$ appears to have a negative bare coupling constant; the coupling only becomes positive in the IR, which in the context of other $\mathcal{P}\mathcal{T}$-symmetric and large-$N$ quantum field theories I argue is perfectly acceptable.

Figures (5)

• Figure 1: A plot of the renormalized running coupling $\lambda_{\textsc{r}}(\bar{\mu})$ from equation \ref{['eq:running-coupling']} in the $\lambda\vec{\phi}^4/N$ theory as a function of $\overline{\textrm{MS}}$ scale $\bar{\mu}$. One sees that at the scale $\bar{\mu}=\Lambda_{\textsc{lp}}$ of the Landau pole, the coupling diverges, and above that scale, the coupling becomes negative.
• Figure 2: A visualization of the $\mathcal{P}\mathcal{T}$-symmetric domains of path integration $\mathcal{C}_{\mathcal{P}\mathcal{T}}$ in \ref{['eq:domain-PT']} and $\mathcal{C}_-$ in \ref{['eq:domain-neg-coup']}. The axes are expressed in some unit of mass (it does not matter which). For the case of $\mathcal{C}_{\mathcal{P}\mathcal{T}}$, $\vec{\phi}(x)\cdot\hat{e}$ lives on either the solid red line or the solid blue line, and every component of $\vec{\phi}(x)-(\vec{\phi}(x)\cdot\hat{e})\hat{e}$ lives on the corresponding dashed line of the same color (red or blue). For the case of $\mathcal{C}_-$, the zero mode $\vec{\phi}_0\cdot\hat{e}$ lives on either the solid red or the solid blue line, while the non-zero modes $\vec{\phi}'(x)\cdot\hat{e}$ and every component of $\vec{\phi}(x)-(\vec{\phi}(x)\cdot\hat{e})\hat{e}$ live on the corresponding dashed line of the same color.
• Figure 3: The imaginary part of the pressure $p(\beta,\zeta_0)$ in \ref{['eq:pressure-renormed-neg-coup']} at a temperature $T=1/\beta=0.65\,\Lambda_{\textsc{lp}}$, showing that the function is multi-sheeted. The horizontal axes are expressed in units of $\Lambda_{\textsc{lp}}^2$ and the vertical axis in units of $\Lambda_{\textsc{lp}}^4$. There is a branch point at $\zeta_0=0$ and a branch cut can be made along the negative half of the real axis. In order to avoid the branch cut, one has to integrate $e^{N\beta V p(\beta,\zeta_0)}$ along a contour $\zeta_0+\textrm{i}0^+\in\mathbb{R}$ slightly shifted in the imaginary direction, which is shown in red.
• Figure 4: The downward flow $-(\partial p(\beta,\zeta_0)/\partial\zeta_0)^*$ of the pressure $p(\beta,\zeta_0)$ at a temperature $T=1/\beta=0.65\,\Lambda_{\textsc{lp}}$. Both axes are expressed in units of $\Lambda_{\textsc{lp}}^2$. The line of integration of $e^{N\beta V p(\beta,\zeta_0)}$ over $\zeta_0+\textrm{i}0^+\in\mathbb{R}$ is indicated in red. Note that it picks up a contribution only from the lower saddle indicated in red. The line of integration $\zeta_0-\textrm{i}0^+\in\mathbb{R}$ in blue corresponds to the integral of $e^{N\beta V p(\beta,-\zeta_0)}$, and it picks up a contribution only from the upper saddle indicated in blue. The total partition function is a sum of both integrals, so it gets a contribution from both complex conjugate saddles for $T>T_{\textsc{c}}$.
• Figure 5: A plot of the pressure per component $p(\beta)$ as a function of temperature $T=1/\beta$ for the $-g\vec{\phi}^4/N$ theory. The vacuum pressure is subtracted and $p(\beta)$ is scaled by $T^4$. This agrees with the result of Romatschke in Romatschke:2022jqg. The dashed line indicates the Stefan--Boltzmann limit $p(\beta)=\pi^2 T^4/90$ for a free scalar boson. One can see that the pressure approaches the free theory limit at high temperatures, which is a consequence of asymptotic freedom.