Notes on Vacuum Decay

This is the content I presented at the previous group meeting, and I save it here for the record. Please note that the content in this page is not fixed. The main part of this survey is based on V. Mukhanov's Physical Foundations of Cosmology.

Introduction

True Vacuum, False Vacuum, Phase Transition

Vacuum Structure
Vacuum Structure

A vacuum is a minimum on a potential–state diagram. If the minimum is global, the vacuum is called a true vacuum(stable vacuum), else it is called a false vacuum(meta-stable vacuum).

A phase transition is a process that the system change from one vacuum state to another. If the state is discrete, the phase transition is called first order, else it is called continuous phase transition. First order phase transition have latent heat.

Non-Trivial Vacuum Structure

  1. If there exist both a true vacuum and a false vacuum in the potential–state diagram, the vacuum structure is called non-trivial.
  2. Not all model of electroweak interaction have the feature of non-trivial vacuum structure. Especially, the Standard Model do not have non-trivial vacuum structure. So the probe of vacuum structure is in a sense a probe of possible physics beyond SM. For instance

    V_\text{tree}(h)=\frac{1}{2}\mu^2 h^2+\frac{1}{4}\lambda h^4+\frac{\kappa}{8\Lambda^2}h^6

  3. In Quantum Field Theory considering temperature, the model’s coupling with temperature can also lead to the emergence of non-trivial vacuum structure. For instance

    V_\text{eff}(h,T)=V_\text{tree}(h)+V_1^{T=0}(h)+\Delta V_1^{T>0}(h,T)

Importance for Cosmology

  1. In the process of the Hot Big Bang Theory, the temperature keeps dropping after the Big Bang. This will lead to the emergence of non-trivial vacuum structure, therefore lead to the spontaneous symmetry breaking.

    CriticalTemperature
    Effective potential and Critical Temperature
  2. If the phase transition is first order, it will be quite strong and altering the energy-momentum tensor. This serves as a possible source for cosmological GW background.

What to Calculate about Vacuum Decay

  1. One of the most important parameter in many phase transitional gravitational wave model is the nucleation rate β, which represent the tunneling probability Γ from the false vacuum to the true vacuum.
  2. The main method we use to calculate the tunneling probability is Saddle Point Approximation or in a sense WKB approximation.
  3. The decay is obviously classically forbidden. So the nature of the tunneling is quantum fluctuation (instanton) & thermal fluctuation (sphaleron), so we will begin with a Quantum Mechanics situation and use analogy to go into QFT situation.

Quantum Mechanics Analogy

Potential of a 1D Particle
Potential of a 1D Particle

Decay via Instanton

The tunneling probability can be written using the propagator of quantum mechanics

P = |\langle t|f\rangle|^2 = \left|\int \mathcal{D}[x(t)]\exp(iS[x(t)]/\hbar)\right|^2

where S[x(t)] is the action of the particle defined by

S[x(t)]=\int_f^t\left(\frac{1}{2}m \left(\frac{\mathrm{d} x}{\mathrm{d} t}\right)^2 - V(x)\right) \mathrm{d} t

Doing the Wick Rotation  t \to \tau=it , we have

S[x(t)]=\int_f^t\left(-\frac{1}{2}m \left(\frac{\mathrm{d} x}{\mathrm{d} \tau}\right)^2 - V(x)\right) \mathrm{d} (i\tau)=iS_E[x(\tau)]

Where the S_E[x(\tau)] is the Euclidean action

S_E[x(\tau)]=\int_f^t\left(\frac{1}{2}m \left(\frac{\mathrm{d} x}{\mathrm{d} \tau}\right)^2 - [-V(x)]\right) \mathrm{d} \tau

while the tunneling probability can be written as

P = |\langle t|f\rangle|^2 = \left|\int \mathcal{D}[x(\tau)]\exp(-S_E[x(\tau)]/\hbar)\right|^2

Now introduce the saddle point approximation. Since \hbar is small, the main contribution of the integration comes from the extreme value of the Euclidean action \delta S_E[x(\tau)]=0 , this lead to the classical motion generated by S_E[x(\tau)] , i.e. particle moving x_\text{cl}(\tau) in a inverted potential -V(x) . This motion is called instanton, the corresponding action is called instanton action.

Thus we have

P \propto |\exp(-S_E[x_\text{cl}(\tau)]/\hbar)|^2=\exp(-S_E[x_\text{cl}^\text{cyc}(\tau)]/\hbar)

 Note that we absorb the power 2 into the instanton action and change the instanton motion into a cyclic loop, denoted by x_\text{cl}^\text{cyc}(\tau) .

Decay via Sphaleron

If the system is in a hot bath with temperature T , there is another way to decay from false vacuum to true vacuum beside quantum tunneling, that is thermal fluctuation. This way to decay adds another Boltzmann factor to the total tunnelling probability

P \propto \int \exp(-\frac{E}{kT}-S_E[x_{\text{cl},E}^\text{cyc}(\tau)]/\hbar)\ \omega(E)\mathrm{d} E

\omega(E) is the energy distribution in thermal equilibrium. The real interest of us, however, lies in the situation where the temperature $T$ is extremely high. In such case, we use saddle point approximation to calculate the dominant contribution.

The saddle point of the exponent gives

-\frac{1}{kT}-\frac{\partial}{\partial E}S_E[x_{\text{cl},E}^\text{cyc}(\tau)]/\hbar=0

 It can be easily shown (leave for Ex.) that

-\frac{\partial}{\partial E}S_E[x_{\text{cl},E}^\text{cyc}(\tau)]=\text{period of instanton with energy E}

 Thus the saddle point function gives

\frac{\hbar}{kT}=\text{period of instanton with energy E}

For extreme high temperature, the period of the instanton is approximately zero. That is the instanton motion localized around the minimum of -V(x) , i.e. the maximum of \max V(x)=V(x_m) . the system's motion x_\text{cl}(\tau)=x_m is called a sphaleron (Greek name for "ready to fall").

Thus the tunneling probability is given by substituting the action in the exponent part

P\propto \exp(-V(x_m)/T)

Summary

  • For instanton: The tunneling probability of a particle with energy $E$ is calculated by

    P \propto \exp(-S_E[x_{\text{cl},E}^\text{cyc}(\tau)]/\hbar)

     where x_{\text{cl},E}^\text{cyc}(\tau) is the periodic instanton motion with Euclidean energy -E generated by inverted potential -V(x) , which changes the potential barrier to a potential well.
  • For Sphaleron: At temperature T the tunneling probability is given by

    P \propto \int \exp(-\frac{E}{kT}-S_E[x_{\text{cl},E}^\text{cyc}(\tau)]/\hbar)\ \omega(E)\mathrm{d} E

     The dominant instanton energy E contribution to the tunneling probability at temperature T is given by

    \frac{1}{kT}=\text{period of instanton with energy E}

     At ultra-high temperature, the tunneling probability is given by

    P\propto \exp(-V(x_m)/T)

QFT Correspondence

QFT can be treated as Quantum Mechanics with infinite DOF, so the correspondence is

Screen Shot 2016-05-10 at 19.41.39

Thus we can get the corresponding EoM of Instanton and Sphaleron:

Screen Shot 2016-05-10 at 19.41.51

Equation of Motion: Bubble Solution

From the instanton and sphaleron EoM in variation form, one can obtain the EoM in its normal PDE form. If Spherical Symmetric solution is assumed (i.e. bubble solution), then h=h(r)

\begin{cases}\displaystyle\frac{\partial^2 h}{\partial r^2}-\frac{\alpha}{r}\frac{\partial h}{\partial r}=\frac{\partial V}{\partial h}\\h'(0)=0,\ h(\infty)=0\end{cases}

and

  • for instanton: \alpha=3 , r is the distance in Euclidean space \mathbb{E}^4
  • for instanton: \alpha=3 , r is the distance in Euclidean space \mathbb{E}^4

For different potential, one can obtain the numerical solution of the bubble profile and use the previous formula to calculate the tunneling probability. Note that the boundary condition for the ODE is strange: one condition at each side of the solution interval. This kind of ODE should be tackled by using overshoot and undershoot method.

One Numerical Solution Using scipy.optimize.odeint

One Numerical Solution Using h^6 Potential
One Numerical Solution Using h^6 Potential

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