A positive definite formulation of vacuum decay with reduced symmetry

This paper presents a positive definite formulation for calculating vacuum decay tunneling actions in the presence of spherically symmetric impurities that reduce symmetry from $O(4)$ to $O(3)$, generalizing the tunneling potential method and providing analytic examples with arbitrary wall thickness.

The Gist

The Big Picture: Rolling Down a Hill with a Bump

Imagine the universe is a landscape made of hills and valleys. A "vacuum" is like a ball sitting in a valley. Sometimes, a ball is stuck in a shallow valley (a "false vacuum") when a deeper, more stable valley (a "true vacuum") exists nearby. To get to the deeper valley, the ball has to roll up and over a hill (a "potential barrier").

In the world of quantum physics, balls don't just roll; they can sometimes "tunnel" straight through the hill, appearing on the other side. This is called vacuum decay. When this happens, it can trigger a massive change in the universe, like a first-order phase transition.

Physicists want to calculate how fast this tunneling happens. The speed depends on a number called the "action." The lower the action, the faster the decay.

The Old Problem: The Perfect Sphere vs. The Messy Room

For decades, physicists used a method that assumed the universe was perfectly symmetrical, like a smooth, round ball. In this perfect world, the tunneling ball rolls out in a perfect sphere. This made the math relatively easy, but it was a bit like trying to calculate how a ball rolls through a messy room by pretending the room is empty.

In reality, the universe isn't empty. It has "impurities"—things like monopoles (magnetic defects), black holes, or other cosmic objects. These objects act like obstacles or bumps in the landscape.

• The Problem: When a ball tries to tunnel near a black hole, the symmetry breaks. The tunneling shape isn't a perfect sphere anymore; it gets squished or distorted.
• The Consequence: The old "perfect sphere" math doesn't work well anymore. The standard way to calculate the tunneling speed in these messy situations is very difficult and often requires guessing (finding a "saddle point"), which is computationally messy and prone to errors.

The New Solution: A "Positive Definite" Map

The authors of this paper (Espinosa, Jinno, et al.) have created a new mathematical map to calculate tunneling in these messy, asymmetrical situations.

Here is the core of their innovation, explained through an analogy:

1. Changing the Perspective (The "Time-Field Swap")
Imagine you are watching a movie of the ball rolling over the hill.

• Old Way: You track the ball's position ($x$) as time ($t$) passes. You have to figure out the exact path the ball takes.
• New Way: The authors flip the script. They treat the position as the "time" and the time as the "position." Instead of asking "Where is the ball at time $t$?", they ask "At what time does the ball reach position $x$?"

This seems like a weird trick, but it turns a complicated, wobbly problem into a much cleaner one.

2. The "Positive Definite" Advantage
In the old method, finding the tunneling path was like trying to find the lowest point in a mountain range that has both peaks and valleys (a "saddle point"). It's easy to get lost or find a fake low point.

The new method transforms the math so that the "action" (the number we want to minimize) is always positive.

• Analogy: Imagine you are looking for the deepest hole in a field.
  • Old Method: The ground is a mix of hills and holes. You have to find the specific spot where the slope is zero, which is tricky.
  • New Method: The authors reshape the landscape so that everywhere is a hill, and the "tunneling path" is simply the lowest valley in that field. You just look for the bottom. There are no confusing peaks or saddle points to trick you. This makes the calculation much more stable and easier to solve, especially for complex shapes.

3. Handling the "Impurities" (O(3) Symmetry)
The paper specifically focuses on situations where the impurity (like a black hole) is spherical, but the tunneling happens in a way that is only symmetric in 3 dimensions (like a sphere in space), not 4 dimensions (space + time).

• They developed a new formula (Equation 3.26 in the paper) that acts like a generalized ruler. It can measure the tunneling cost even when the shape is distorted by the impurity.
• They proved that if you remove the impurity, their new formula magically turns back into the old, trusted formula. This shows their new method is a true generalization, not a replacement.

What They Actually Did (and Didn't Do)

• What they did: They derived a new mathematical formula that is "positive definite" (always positive) for calculating vacuum decay around spherical impurities. They showed how to turn the old "Euclidean" math into this new "Tunneling Potential" math.
• What they did: They created specific, solvable examples using math to prove their formula works. They showed that you can have "thick" walls (slow tunneling) or "thin" walls (fast tunneling) in these new scenarios.
• What they did NOT do: They did not apply this to a specific real-world event (like predicting when our universe will decay). They did not solve the problem for all types of impurities (like spinning black holes or flat walls), though they mentioned that as a future step. They did not discuss clinical uses (since this is theoretical physics, not medicine).

The Takeaway

Think of this paper as inventing a new, more robust GPS for navigating a bumpy, obstacle-filled landscape.

• The old GPS only worked on perfectly flat, round roads.
• The new GPS works even when there are potholes and bumps (impurities).
• The best feature of the new GPS is that it always gives you a "distance" that is a positive number, making it much easier to find the shortest route without getting confused by false shortcuts.

This allows physicists to calculate how likely it is for the universe to change states in the presence of cosmic objects like black holes, with much greater precision and less computational headache.

Technical Summary

1. Problem Statement

Vacuum decay (tunneling from a metastable false vacuum to a stable true vacuum) is a fundamental process in quantum field theory and cosmology, relevant for early universe phase transitions and gravitational wave signatures.

• Standard Approach: The decay rate is calculated using the Euclidean path integral, where the exponent is the Euclidean action ($S_E$) of the "bounce" solution. In a vacuum, this bounce possesses $O(4)$ symmetry (rotational invariance in 4D Euclidean space).
• The Challenge: The presence of impurities (e.g., monopoles, primordial black holes, Q-balls) or defects breaks the $O(4)$ symmetry. If the impurity is spherically symmetric, the bounce retains $O(3)$ symmetry (rotational invariance in 3D space, but not in Euclidean time).
• Limitations of Current Methods:
  • The standard Euclidean method requires finding a saddle point of the action, which is numerically difficult, especially for multi-field potentials.
  • Existing "Tunneling Potential" methods (Refs. [6,7,11]) provide a positive-definite action (allowing for minimization rather than saddle-point search) but strictly assume $O(4)$ symmetry.
  • For lower symmetry cases (like $O(3)$), researchers often rely on the "thin-wall approximation," which is not guaranteed to be valid for arbitrary wall thickness or specific impurity configurations.
• Goal: Develop a formulation for calculating the tunneling action that is explicitly positive definite and applicable to $O(3)$-symmetric configurations without relying on the thin-wall approximation.

2. Methodology

The authors generalize the "Tunneling Potential" formalism using a canonical transformation approach.

A. Variable Transformation

Instead of treating the scalar field $\phi(\tau, r)$ as the dynamical variable in Euclidean time $\tau$ and radial distance $r$, the authors exploit the monotonicity of the bounce in the $\tau$ direction. They invert the relationship to treat the Euclidean time $\tau$ as a function of the field $\phi$ and radius $r$, i.e., $\tau(\phi, r)$.

• This changes the integration measure from $d\tau dr$ to $d\phi dr$.
• The Lagrangian density is rewritten in terms of $\tau(\phi, r)$ and its derivatives.

B. Canonical Transformation

The authors perform a canonical transformation to move from the $(\tau, p)$ phase space to a new set of variables $(Q, P)$.

1. Generating Function: They introduce a generating function $W[\tau, P, \phi] = \int dr \, \tau \dot{P}$ (where dot denotes $\partial/\partial \phi$).
2. New Variables: This yields new conjugate variables $Q = -\dot{\tau}$ and $P$.
3. Hamiltonian: The Hamiltonian is transformed, and $Q$ is eliminated by minimizing the action with respect to it.

C. Definition of the Generalized Tunneling Potential

A new dynamical variable, the generalized tunneling potential $V_t(\phi, r)$, is defined in terms of the canonical momentum $P$:
$$V_t(\phi, r) = \frac{3}{4\pi r^3} P$$
This variable resides in the $(\phi, r)$ space, reflecting the reduced $O(3)$ symmetry (dependence on both field value and radial coordinate).

3. Key Contributions and Results

A. The Positive-Definite Action

The primary result is the derivation of a new action functional $S[V_t]$ that is explicitly positive definite. The action is given by:
$$S[V_t] = \int d\phi \int dr \sqrt{128\pi^2 r^4 \left[ V(\phi, r) - \left( V_t + \frac{r}{3}\dot{V}_t + \frac{r^2}{18}V_t'^2 \right) \right]}$$

• Positivity: The integrand is a square root of a positive quantity (assuming physical conditions), ensuring the action is positive.
• Advantage: This allows the bounce solution to be found by minimizing the action (a global minimum search) rather than finding a saddle point, significantly improving numerical stability and tractability for complex potentials.

B. Equation of Motion (EoM)

The authors derive the Euler-Lagrange equation for $V_t(\phi, r)$, which is a partial differential equation (PDE):
$$6(V - V_t)V_t'' + (4V_t' - 3V')V_t' + 3\ddot{V}_t + 2r(V_t' \dot{V}_t' - V_t'' \dot{V}_t) + \frac{3}{r}(4\dot{V}_t - 3\dot{V}) = 0$$

• Here, primes denote $\partial/\partial \phi$ and dots denote $\partial/\partial r$.
• This PDE replaces the ordinary differential equation (ODE) used in the $O(4)$ case.

C. Consistency Checks

• $O(4)$ Limit: The authors prove that if $V_t$ is independent of $r$ (restoring $O(4)$ symmetry), the action and EoM reduce exactly to the known $O(4)$ tunneling potential formulas (Refs. [6, 11]).
• Boundary Terms: They demonstrate that for finite-action bounces, the boundary terms arising from the canonical transformation vanish at spatial infinity ($r \to \infty$) and the origin ($r=0$), provided the potential behaves regularly.

D. Analytic Examples

The paper constructs analytic solutions for $O(3)$ symmetric bounces by deforming known $O(4)$ solutions (specifically the Fubini bounce and a thin-wall example).

• Deformation: They introduce a deformation function $a(r)$ such that the Euclidean radius becomes $r_E^2 = a^2(r)r^2 + \tau^2$.
• Results: They derive the specific position-dependent potentials $V(\phi, r)$ and tunneling potentials $V_t(\phi, r)$ required to support these deformed bounces.
• Numerical Verification: They show that for a fixed impurity profile (fixed $a(r)$), the action is minimized when the bounce profile matches the impurity, validating the variational principle.

4. Significance

• Beyond Thin-Wall Approximation: This formulation allows for the calculation of tunneling rates around impurities with arbitrary wall thickness, removing the need for the often-inaccurate thin-wall approximation in non-symmetric scenarios.
• Numerical Efficiency: By converting the problem into a positive-definite minimization problem in field space, it opens the door to efficient numerical algorithms (e.g., gradient flow) for multi-field models with impurities, which are notoriously difficult to solve using standard Euclidean shooting methods.
• Foundation for Further Extensions: The framework is a stepping stone toward:
  • Configurations with even less symmetry (e.g., $O(2)$ for rotating black holes, or no symmetry for cosmic strings/domain walls).
  • Including gravitational effects (Coleman-De Luccia and Hawking-Moss bounces) in the presence of impurities.
  • Calculating one-loop determinants (fluctuations) around the bounce in lower-symmetry backgrounds.

In summary, the paper successfully extends the powerful "Tunneling Potential" method to $O(3)$ symmetric systems, providing a robust, positive-definite framework for studying vacuum decay catalyzed by spherical impurities like black holes and monopoles.