Infinite Heat Order in 3+1 Dimensions

Abstract

We investigate whether spontaneous symmetry breaking can persist up to arbitrarily high temperature in ultraviolet-complete quantum field theories in four spacetime dimensions. We focus on completely asymptotically free models with gauge group $\mathrm{SU}(N_{c1})\times \mathrm{SU}(N_{c2})$ and two complex scalar fields, each transforming in the fundamental representation of one gauge factor and singlet under the other. The scalar potential contains quartic self-interactions together with a negative portal coupling between the two sectors. In the Veneziano limit, this class of theories was previously shown to admit fixed-flow trajectories for which one scalar acquires a negative thermal mass at asymptotically large temperature, leading to symmetry non-restoration. Here we extend that analysis to finite numbers of colours and flavours. We derive the finite-$N$ fixed-flow equations, compute the leading $1/N$ corrections to the large-$N$ solutions, and solve the full finite-$N$ system numerically. We find explicit finite-$N$ benchmark theories for which the scalar potential remains bounded from below, the gauge sector is asymptotically free, and one scalar thermal mass stays negative at arbitrarily high temperature. This provides an explicit perturbative example of infinite heat order in a four-dimensional ultraviolet-complete quantum field theory with a finite field content.

Figures (4)

• Figure 1: The shadowed region shows where in the $\tilde{\alpha}_+-\tilde{\lambda}$ parameter space the squared mass of $\varphi_1$ is negative. Here we set $\tilde{\alpha}_+^{(1)}=\tilde{\lambda}^{(1)}=0$ (as allowed by the RG equations), $x=0.1$ and $N_{c2}=1000$.
• Figure 2: (a) The necessary condition for symmetry non-restoration to happen in the theory is that the ratio $N_{c2}/N_{c1}$ lies above the blue curve. The yellow line denotes the lowest possible bound for $N_{c2}/N_{c1}$ occurring for $N_{c1}\to\infty$. (b) The necessary condition for the solution to exist is that $N_{c2}$ is above the blue curve.
• Figure 3: (a) Region in the $\tilde{\alpha}_+$-$\tilde{\lambda}$ plane with negative thermal mass square for $(N_{c1},N_{c2})=(100,1000)$. This is in agreement with Fig. \ref{['fig:regionplot']}. (b) Mass square as a function of $\tilde{\lambda}$. Different colours represent different values of $\tilde{\alpha}_+$. For each $\tilde{\alpha}_+$ and $\tilde{\lambda}$ there are two solutions for the other parameters $\tilde{\alpha}_-$, $\tilde{\lambda}_{1,2}$.
• Figure 4: (a) Regions in the $N_{c1}-N_{c2}$ plane with (blue dots) and without (yellow dots) a solution with symmetry non-restoration. The boundary between them is well described by the constraint (\ref{['N2']}). Here $\tilde{\alpha}_{1,2}$ are arbitrary real numbers, not $1/(2b_{1,2})$. (b) Taking into account \ref{['eq:alpha12def']} and varying integers $N_{f1,2}$, we plot the possible solutions above the blue continuous curve in panel (a). We see that some blue points, which represented allowed solutions before, become yellow, i.e. they are not solutions. This happens near the lower boundary.