Make the thing that tunnels as large as possible ── the universe itself may not have finished falling
The finale of the main series. Episode 1 wrote that "the wall is a valley in imaginary time." Now we apply that line to the largest thing there is ── the vacuum. A vacuum is a state in which a field sits at the bottom of a valley; but what if there are two valleys? The one we are in need not be the deepest. If it is not, the vacuum tunnels to the deeper one. In 1977 Coleman computed the probability ── \(\Gamma/V=Ae^{-S_E/\hbar}\), exactly the same form as everything before. And remarkably, this is not an armchair exercise. Put in the Higgs mass measured in 2012 and the top quark mass, and the Standard Model vacuum comes out neither "stable" nor "unstable" but metastable ── it does decay, but takes vastly longer than the age of the universe, a delicately balanced position indeed. We live on an exponent.
"Vacuum" does not mean empty space; it means a field sitting still at the bottom of an energy valley (see the sister series "Fields That Click"). The Higgs field is one, currently settled at a value of 246 GeV.
The trouble is that there need not be only one valley. If there is another, deeper valley further out, then ours is a false vacuum.
Classically it cannot: there is a hill in between. But as Episode 1 showed, quantum mechanics can tunnel.
In field theory, though, the tunneling has a distinctive shape. Space does not shift all at once; instead a "bubble" of true vacuum nucleates at some point. The bubble has two competing budgets ── the energy its interior gains (proportional to volume) and the cost of building its wall (proportional to area).
Small bubbles are beaten by surface tension and collapse; large ones are won by volume and expand at nearly the speed of light. The dividing line is \(R_c=3\sigma/\varepsilon\).
"The probability of nucleating a critical bubble" is the tunneling probability, and with Coleman's bounce solution
Once more, \(e^{-S_E/\hbar}\). A particle in Episode 1, a proton in Episode 2, a phase in Episode 4, and now the universe itself. One equation, working at wildly different scales.
Now the real subject. In the Standard Model, the shape of the Higgs valley changes with energy scale (running couplings, from the sister series "Renormalization That Clicks"). We can compute how \(\lambda\), the steepness of the valley, evolves toward high energy.
The one-loop flow is dominated by these two terms:
$$16\pi^2\frac{d\lambda}{d\ln\mu}=\underbrace{24\lambda^2}_{\text{itself: pushes up}}\underbrace{-\,6y_t^4}_{\textbf{the top quark: pushes down}}+\cdots$$The top quark threatens the stability of the vacuum. It is the heaviest fermion (Yukawa coupling \(y_t\approx0.94\)) and enters at the fourth power. So \(\lambda\) is pushed down hard at high energy and crosses zero somewhere. \(\lambda<0\) means the valley opens downward ── i.e. there is a lower vacuum out there.
The figure runs the Standard Model Higgs self-coupling \(\lambda\) at one loop. The horizontal axis is energy scale (GeV, log, from \(10^2\) up to the Planck scale \(10^{19}\)).
Move the top-quark mass slider. A change of just 1 GeV moves the place where \(\lambda\) crosses zero by orders of magnitude. And with the measured values (\(m_t\approx172.5\) GeV, \(m_H\approx125.25\) GeV) it lands ── just inside the boundary between "metastable" and "stable." This uncanny positioning is called near-criticality and is one of the Standard Model's great puzzles.
| Category | Meaning | Are we here? |
|---|---|---|
| stable | \(\lambda\) stays positive up to the Planck scale. No lower vacuum exists | just barely not |
| metastable | a lower vacuum exists, but falling into it takes vastly longer than the age of the universe | here |
| unstable | it would decay within the age of the universe. We could not exist | clearly not |
Lifetime estimates vary with the calculation, but they run far beyond \(10^{100}\) years, often above \(10^{300}\). The universe is \(1.4\times10^{10}\) years old, so the comparison is meaningless.
One more large thing. In 2000 Parikh and Wilczek derived Hawking radiation as a tunneling process.
For a particle just inside a black hole, the horizon is an "impassable wall." Compute the action in the manner of Episode 1 and the escape probability is
$$\Gamma\propto e^{-2\,\mathrm{Im}S}=e^{-8\pi GME/\hbar c^3}=e^{-E/k_BT_{\rm H}}$$── exactly the Boltzmann factor at temperature \(T_{\rm H}=\hbar c^3/8\pi GMk_B\). The Hawking temperature from the finale of "Temperature That Clicks" emerges in the language of tunneling.
And the derivation has a sequel. Emitting a particle makes the black hole lighter, so imposing exact energy conservation modifies the exponent to \(e^{-8\pi GE(M-E/2)/\hbar c^3}\). Then the spectrum departs from exactly thermal and correlations appear between emitted quanta ── which feeds the argument that information might come out after all (see the information paradox in the sister series "Black Holes That Click").
| Episode | What tunnelled | What rode on the exponent |
|---|---|---|
| 1 | a particle (a wall) | \(2S_E/\hbar\) |
| 2 | a proton (the Coulomb barrier) | \(\sqrt{E_G/E}\) |
| 3 | the divide between heat and tunneling | \(2\pi E_b/\hbar\omega_b\) |
| 4 | the phase difference (one macroscopic variable) | \(\Delta U/k_BT\) and \(2S_E/\hbar\) |
| 5 | (not tunneling) | \(1/N(0)V\) |
| 6 | the outside of perturbation theory itself | \(S/g\) |
| 7 | the vacuum | \(S_E/\hbar\) |
The "sizes" of the world are usually written on an exponent.
The Sun's lifetime, the superconducting transition temperature, the proton's mass, the vacuum's lifetime ── all of them settle in the form \(e^{-(\text{something})}\), where a change of a few tens on the exponent moves the answer by a few tens of orders. That is precisely how phenomena spanning wildly different scales can coexist in one universe. If everything were linear, the Sun would burn out instantly, the vacuum would have collapsed first, and we would not be here.
And the quantities riding on that exponent share a common origin ── classical motion in imaginary time. The wall is a valley in imaginary time (Episode 1), heat is a circle in imaginary time (Episode 3), vacuum decay is a bubble in imaginary time (this episode). What looks "impossible" in real time is usually "ordinary" in imaginary time.
On the collection's map, this series walked the non-perturbative side of the \(\hbar\) axis ── and at the end touched \(G\) (vacuum decay, Hawking) and \(k_B\) (Episode 3's \(T_0\)) as well. Somewhere near the diagonal of the Physics Cube.
Established: the theory of false-vacuum decay (Coleman 1977, Callan–Coleman 1977) and \(\Gamma/V=Ae^{-S_E/\hbar}\), the thin-wall action \(S_E=27\pi^2\sigma^4/2\varepsilon^3\) and critical radius \(R_c=3\sigma/\varepsilon\); that the Higgs self-coupling \(\lambda\) can turn negative at high energy in the Standard Model, driven mainly by the \(-6y_t^4\) term from the top Yukawa coupling; that with the measured values the Standard Model vacuum is assessed as metastable with a lifetime exceeding the age of the universe by many orders (Degrassi et al. 2012, Buttazzo et al. 2013 and others); the cosmic-ray safety argument (Hut–Rees 1983); and the tunneling picture of Hawking radiation (Parikh–Wilczek 2000) reproducing the Hawking temperature. All standard results.
Assumptions and caveats: (1) The metastability conclusion rests on the strong assumption that the Standard Model holds unchanged all the way to the Planck scale. A single new particle changes the running and could change the conclusion. (2) The result is extremely sensitive to \(m_t\), and the systematic uncertainty in which top-mass definition is meant (pole mass, MS-bar, their relation to the Monte-Carlo mass) is the dominant uncertainty. (3) The figure runs at one loop and is simplified. Real analyses use two or three loops and place the zero-crossing of λ at around \(10^{10\text{–}11}\) GeV. Read the figure's values as order-of-magnitude guides. (4) The metastable/unstable boundary is likewise an approximate formula. (5) Bubble nucleation in curved spacetime (an expanding universe, near a black hole) is not simply the flat-space calculation, and remains debated. (6) Parikh–Wilczek is one derivation of Hawking radiation, and whether information can genuinely be extracted is unresolved. (7) Vacuum decay is not something that can be triggered artificially (the cosmic-ray argument).
We made the thing that tunnels as large as possible ── the vacuum. With two valleys, the one we occupy may be false, and a bubble nucleates and tunnels to the true one. The probability is \(\Gamma/V=Ae^{-S_E/\hbar}\) (Coleman 1977) ── exactly the form from Episode 1.
And this is not hypothetical. The Higgs self-coupling \(\lambda\) is pushed down at high energy by the top quark's \(-6y_t^4\) and turns negative; with the measured values the Standard Model vacuum comes out metastable ── it decays, but on a timescale of order \(10^{300}\) years. And it sits just inside the boundary with "stable," an uncannily delicate place (near-criticality). Though the result is acutely sensitive to \(m_t\) and rests on assuming no new physics up to the Planck scale.
Even Hawking radiation reads as "tunneling through the horizon," producing \(e^{-E/k_BT_{\rm H}}\). The conclusion: the "sizes" of the world are usually written on an exponent. The Sun's lifetime, the proton's mass, the vacuum's lifetime. And the quantities on that exponent share one origin ── classical motion in imaginary time. What looks impossible in real time is usually ordinary in imaginary time.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen, moving the top-quark mass by 1 GeV shifts the place where λ crosses zero by orders of magnitude. Check that the measured values sit just inside the boundary between "stable" and "metastable." "See the answer" opens each solution.