Lifetime of a particle at the maxima of an unstable potential

Question

Consider an unstable potential of the form $$V(\phi) = - \lambda \phi^n $$ with $\lambda > 0$ (hence unstable). Now due to quantum fluctuations, a particle at the top is going to roll down. So what is the lifetime of the particle at the maxima of the potential?

Nima in his 2019 TASI lecture on EFT gives a hand-wavey argument via dimensional analysis in 4D. Particularly, the lifetime $\tau \sim \frac{1}{M}$ where $M$ is the mass dimension of the proportionality constant in the potential. While it works for $n=2,3$, at $n=4$ he derives the lifetime from some other argument to be $\tau \sim e^{1/\lambda}$ . For $n>4$, again this procedure doesn't give the right answer. e.g. for $n=6$, $\tau \sim \sqrt{\lambda}$ which is incorrect/opposite to what you expect.

So is there a way to derive the lifetime in a more concrete way and which works for any power of potential (in any number of spacetime dimensions)?

Answer 1

In this answer we generalize the discussion of the unstable potential $$\begin{align} {\cal L}~\sim~&(\partial \phi)^2 - {\cal V}(\phi), \cr {\cal V}(\phi)~=~& -\lambda\phi^n,\cr \lambda~>~&0,\end{align}\tag{1} $$ in the 1st 2019 TASI lecture on EFT by Nima Arkani-Hamed to arbitrary spacetime dimension $D>0$ and interaction power $n\geq 2$.

The system is more formally described by quantum tunnelling through a kinetic energy barrier, cf. the semiclassical theory of false vacuum decay by Coleman & de Luccia.

We'll find that we are (the system is) always eventually dead (delocalized), but the tunnelling may take longer than the age of the universe.

• 24:30 Dimensional analysis of mass dimension $[\cdot]$ in absolute units $\hbar=1=c$: $$\begin{align} [\phi]~\stackrel{(1)}{=}~&\frac{D-2}{2}, \cr [\lambda]~\stackrel{(1)}{=}~&D-n[\phi] ~=~(1-\frac{n}{2})D+n .\end{align}\tag{2}$$

  1. Relevant interaction: $[\lambda]>0\quad\Leftrightarrow\quad\frac{1}{n}+\frac{1}{D}>\frac{1}{2}$.

  2. Marginal interaction: $[\lambda]=0\quad\Leftrightarrow\quad\frac{1}{n}+\frac{1}{D}=\frac{1}{2}$.

  3. Irrelevant interaction: $[\lambda]<0\quad\Leftrightarrow\quad\frac{1}{n}+\frac{1}{D}<\frac{1}{2}$.

• 32:10 Groundstate wavefunction $$ \begin{align}\text{wavefunction}\quad \Psi ~\sim~& \prod_{\bf k} \exp\left( -\omega_{{\bf k}}|\tilde{\phi}({\bf k},t)|^2\right)\cr ~\sim~& \exp\left( -\int d^d{\bf k}~\omega_{{\bf k}}|\tilde{\phi}({\bf k},t)|^2\right), \cr [\tilde{\phi}]~=~&-\frac{D}{2},\qquad D~=~d+1. \end{align}$$

• 33:30 Gaussian wavefunction for bubble of height $\phi_0$ and radius $R$ using dimensional analysis: $$ \begin{align}\text{wavefunction}\quad \Psi ~\sim~& \exp( -A), \cr \text{with}\quad A~\sim~&\phi_0^2 R^{D-2}. \end{align}\tag{3}$$

• 35:15 Using eq. (3) fluctuations grows $R\to\infty$ (dies $R\to0$) for $D<2$ ($D>2$), respectively.

• 37:00 Decay rate for non-marginal interaction $[\lambda]\neq 0$ from dimensional analysis: $$\begin{align} [\Gamma]~=~&-[\tau]~=~1\cr \Rightarrow\qquad \tau^{-1}~\equiv~&\Gamma~\sim~\lambda^{1/[\lambda]}. \end{align}\tag{4} $$

  Note that a mass-term $n=2$ is a relevant coupling in any spacetime dimension $D$. From now on we assume that $n>2$.

• 43:00 The radius $R$ of the bubble should be big enough that the negative potential term overcome the positive kinetic term/surface energy term $$\lambda\phi_0^n~\sim~\phi_0^2/R^2\tag{5}$$
  $$\Rightarrow \qquad \phi_0~\stackrel{(5)}{\sim}~ (\lambda R^2)^{-1/(n-2)}\tag{6}$$ $$\Rightarrow \qquad A~\stackrel{(3)+(6)}{\sim}~ R^{D-2}(\lambda R^2)^{-2/(n-2)}. \tag{7}$$

  1. Relevant interaction: $A$ is a negative power of $R$. $$\Rightarrow \qquad A ~\stackrel{(7)}{\to}~ 0\quad\text{for}\quad R~\to~\infty.\tag{8}$$

  2. Marginal interaction: $A$ is independent of $R$: $$ A~\stackrel{(7)}{\sim}~ \lambda^{-2/(n-2)} \tag{9}$$ Non-perturbative in the coupling constant $\lambda$. $$\Rightarrow \qquad \Gamma~\stackrel{(3)+(9)}{\sim}~ \exp\left(-\lambda^{-2/(n-2)}\right). \tag{10}$$

  3. Irrelevant interaction: $A$ is a positive power of $R$. $$\Rightarrow \qquad A ~\stackrel{(7)}{\to}~ 0\quad\text{for}\quad R~\to~0.\tag{11}$$
     (Note that it is technically unnatural if all interactions are irrelevant.)