Standard model vacuum decay with gravity

We present a calculation of the decay rate of the electroweak vacuum, fully including all gravitational effects and a possible nonminimal Higgs-curvature coupling $\xi$, and using the three-loop Standard Model effective potential. Without a nonminimal coupling, we find that the effect of the gravitational backreaction is small and less significant than previous calculations suggested. The gravitational effects are smallest, and almost completely suppressed, near the conformal value $\xi =1/6$ of the nonminimal coupling. Moving $\xi$ away from this value in either direction universally suppresses the decay rate.

Figure 1

Interpolated potential for $M_t=173.34\mathrm{GeV}$, $M_h=125.15\mathrm{GeV}$, compared to “exact” values obtained by solving the beta functions numerically.

Figure 2

Difference between the action as a function of $N$ and the action at large $N$ (used as a proxy for $N\to \infty$). For small $N$, the behavior deviates from power-law behavior as the shape of the potential settles down. Beyond this, the sampling intervals are smaller than the variations in the potential, and the action settles down, approaching the large-$N$ action with a difference decreasing as a negative power of $N$. The apparent deviation at large $N$ is due to the floating-point double-precision limit for the action (around ${10}^{-13}$ for $S\sim 10^3$, as is the case here), which is stored at double precision for data processing, and is not a real effect.

Figure 3

Bounce solutions as a function of distance $\chi$ from the center of the bounce, computed for various values of $\xi$. The $\xi =1/6$ solution is extremely close to the flat-space solution (though not identical to it). A larger $|\xi |$ tends to flatten and broaden the bounce, as does the inclusion of backreaction for $\xi =0$.

Figure 4

Decay exponent for different values of $\xi$ at $M_t=173.34\mathrm{GeV}$, $M_h=125.15\mathrm{GeV}$, compared with the flat-space result. Left: Variation around the minimum near $\xi =1/6$. Right: Large-scale variation of $B(\xi )$.

Figure 5

High-resolution data around the minimum of $B(\xi )$ for $M_t=173.34\mathrm{GeV}$, $M_h=125.15\mathrm{GeV}$. This demonstrates that the true minimum is slightly deflected from $\xi =1/6$ and that the decay, even at the minimum, is suppressed relative to the flat-space case.

Figure 6

Instability/metastability boundaries computed for different values of $\xi$. With $\xi =0$, the bounds hardly shift at all from the flat-space bound (see the inset which shows that the nearly coincident flat-space and $\xi =0$ with backreaction boundaries are in fact separate, albeit very close). With increasing $|\xi |$, the instability region is pushed back; notice that this is true for both positive and negative $\xi$—the boundaries for $\xi =\pm 1000$ are nearly coincident. The Standard Model parameter uncertainty region around $M_h=125.09\mathrm{GeV}$, $M_t=173.21\mathrm{GeV}$ [36] with 1, 2, and 3 sigma bounds is shown for reference.

Figure 7

Ricci scalar with distance from the center of the bounce, in the $\xi =0$, $\xi =1/6$ and $\xi =1/3$ cases with $M_t=173.34\mathrm{GeV}$, $M_h=125.15\mathrm{GeV}$. For $\xi =1/3$, $R$ is positive in the interior of the bounce.

Figure 8

Comparison of the perturbative results, Eq. (35), to our numerical calculations in the full Standard Model potential, for $M_h=125.15\mathrm{GeV}$ and $M_h=115\mathrm{GeV}$.