A positive definite formulation of vacuum decay with reduced symmetry

Key Points

• The aim is to present a positive definite formulation of the tunneling decay action considering impurities that reduce symmetry.
• Developed a formulation for calculating the tunneling decay action with explicit positivity.
• Explored the effects of spherically symmetric impurities that reduce $O(4)$ symmetry to $O(3)$.
• Provided analytic examples illustrating the action under varying wall thickness.
• The proposed action reduces to the tunneling potential for $O(4)$-symmetric cases.
• Successfully incorporates the effects of impurities on the decay action while maintaining positivity.
• Demonstrated analytic examples with $O(3)$ symmetry that were consistent with the expected behavior.