Stability of axion-dilaton wormholes

We study the perturbative stability of Euclidean axion-dilaton wormholes that asymptotically approach flat space, both with a massless and a massive dilaton, and focus on homogeneous perturbations. We find massless wormholes to always be perturbatively stable. The phenomenologically more relevant case of a massive dilaton presents us with a wide variety of wormhole solutions, depending on the dilaton coupling and mass, and on the axion charge. We find that the solutions with the smallest dilaton potential are perturbatively stable and dominant, even in cases where the wormhole solutions are not continuously connected to the massless case by decreasing the mass. For branches of solutions emanating from a bifurcation point, one side of the branch always contains a negative mode in its spectrum, rendering such solutions unstable. The existence of classes of perturbatively stable wormhole solutions with massive dilaton sharpens the puzzles associated with Euclidean wormholes.

Figure 1

Summary of the branch structure and stability properties for axion-dilaton wormhole solutions for dilaton coupling $\beta =1.579$ (just below hierarchy inversion, left panel) and $\beta =1.58$ (just above hierarchy inversion, right panel). The solutions are indicated by their dilaton field value ${\varphi }_0$ at the neck and as a function of $m^{2}N$, where $m$ is the dilaton mass and $N$ the axion charge. At sufficiently large $m^{2}N$ new branches of solutions appear, starting from a bifurcation point. The bifurcation point admits an extra zero mode, which turns into a negative mode along one branch of solutions (shown in blue). For dilaton couplings below a critical value ${\beta }_i$ the lowest ${\varphi }_0$ solutions are stable and dominant, and moreover, they are continuously connected to the massless solution. Above this dilaton coupling ${\beta }_i$, the branch that connects to the massless case does not remain dominant at large $m^{2}N$, where a new branch of solutions takes over. More details are provided in Sec. 4.

Figure 2

Existence of axion-dilaton wormhole solutions, as found in [3] for four typical values of $\beta$. The solutions are displayed via their initial dilaton value ${\varphi }_0$, as a function of their mass and charge combination $m^{2}N$. Below the critical value ${\beta }_c=2\sqrt{\frac{2}{3}}\approx 1.63$ solutions exist which are continuously connected to the massless case. In addition, for all dilaton couplings new bifurcating branches appear at sufficiently large values of $m^{2}N$. Full details can be found in [3].

Figure 3

An example of a Euclidean wormhole solution displaying oscillations in the fields. The evolution of the scale factor is shown on the left (linear growth at late times indicates flat space in polar coordinates), and that of the scalar field on the right. We are only plotting the evolution from the neck of the wormhole to infinity, as would be appropriate for describing the nucleation of a baby universe. The full wormhole connecting two asymptotically flat regions would then be obtained by adding a reflection across $t_E=0$. Here $\beta =1.2$, $m^{2}N=3$, ${\varphi }_0=8.1964321797$.

Figure 4

Massless dilatonic wormhole with $\beta =1$. In the upper row we show the evolution of the fields, and in the lower row one can see that the kinetic function ${\dot{\varphi }}^2/Q$ as well as the fluctuation potential $U$ are positive definite. These solutions are thus perturbatively stable.

Figure 5

Summary plots for various values of the dilaton coupling $\beta$: a) small dilaton coupling; b) just below the inversion point; c) just above the inversion point; d) above the critical value. Dots indicate the existence of a wormhole solution, with the initial dilaton value ${\varphi }_0(t_{\mathrm{E}}=0)$ given as a function of the dilaton mass $m$ and axion charge $N$. The color coding is as follows: Green dots indicate stable solutions with $Q>0$, $U>0$ everywhere. Cyan dots indicate stable solutions with $Q>0$ everywhere, but $U<0$ in some regions. Blue dots indicate unstable solutions with one negative mode ($Q>0$ everywhere, $U<0$ in some regions). Red dots indicate solutions for which the kinetic function $Q$ contains negative regions, so that we cannot meaningfully study their stability in the present setup.

Figure 6

Massive dilatonic wormhole with $\beta =1.579$ and $m^{2}N=2.2$. This wormhole starts out at ${\varphi }_0\approx 2.1705146359$. In the upper row we show the evolution of the fields and in the lower row the kinetic function ${\dot{\varphi }}^2/Q$, as well as the fluctuation potential $U$. Both are positive definite, and thus this is an example of a perturbatively stable massive axion-dilaton wormhole.

Figure 7

Massive dilatonic wormhole with $\beta =1.579$ and $m^{2}N=2.2$. This wormhole starts out at ${\varphi }_0\approx 3.4770669337$. In the upper row we show the evolution of the fields and in the lower row the kinetic function ${\dot{\varphi }}^2/Q$, as well as the fluctuation potential $U$, which starts out negative. This solution turns out to admit a negative mode, shown in the left panel of Fig. 8, and is thus unstable.

Figure 8

Left panel: the negative mode associated with the wormhole solution at $\beta =1.579$, $m^{2}N=2.2$ shown in Fig. 7. The eigenvalue is $\lambda \approx -10.3602531840$. One can see that the eigenmode is concentrated near the neck of the wormhole, in the region where the fluctuation potential is negative. Right panel: negative eigenvalues as a function of $m^{2}N$, for $\beta =1.579$ [for the blue dot solutions near the bifurcation point shown in Fig. 5]. The eigenvalues diverge near $m^{2}N\approx 2.22$.

Figure 9

A closer look at the intermediate-${\varphi }_0$ solutions for $\beta =1.579$ shows that the kinetic function $Q$ evolves from starting out positive to starting out negative as the dilaton mass and axion charge are increased past $m^{2}N\approx 2.22$. This is the reason that the eigenvalues of the negative modes of these solutions [blue dots in Fig. 5] diverge here, as shown in the right panel of Fig. 8. When $Q$ starts out negative and then crosses zero, this also brings about a singularity in the fluctuation potential $U$, as shown in the right panel above.