[Paper Review] The Path-Integral Approach to Spontaneous Symmetry Breaking

M.T.M. van Kessel|ArXiv.org|Oct 8, 2008

Black Holes and Theoretical Physics6 references3 citations

TL;DR

This thesis investigates spontaneous symmetry breaking (SSB) in the N=1 and N=2 linear sigma models using the path-integral formalism, contrasting it with the canonical approach. It demonstrates that the path-integral approach yields convex, well-defined effective potentials and reproduces canonical Green's functions only when paths are fixed in space-time, resolving long-standing ambiguities in the formulation of SSB in quantum field theory. The work provides a rigorous foundation for reinterpreting the Higgs mechanism in the Standard Model via path integrals.

ABSTRACT

We will investigate two models which exhibit SSB in the canonical approach: the N=1 and N=2 linear sigma model. In both models the Green's functions and the effective potential will be computed in the path-integral approach. We will demonstrate how we get different results than in the canonical approach.

Motivation & Objective

• To resolve inconsistencies in the path-integral formulation of spontaneous symmetry breaking (SSB) in quantum field theories.
• To compare the path-integral approach with the canonical approach in the N=1 and N=2 linear sigma models (LSM).
• To investigate whether the path-integral formalism can reproduce the standard predictions of SSB, such as the Higgs mechanism.
• To examine the role of path-fixing in recovering canonical results and ensuring convexity of the effective potential.
• To explore the phenomenological implications of a path-integral-based Higgs sector for the Standard Model.

Proposed method

• Formalizing the effective action and effective potential in the path-integral framework for the N=1 and N=2 LSM.
• Introducing a path-fixing condition in the path integral, where the field configuration is fixed at a point in space-time across all space.
• Applying dimensional regularization to handle ultraviolet divergences in d=4 dimensions.
• Using polar field variables (r, ϕ) to reformulate the path integral and analyze symmetry breaking in radial and angular modes.
• Performing explicit one-dimensional calculations to verify analytical results and test the behavior of propagators and vacuum expectation values.
• Deriving and comparing Green's functions, effective potentials, and divergences between canonical and path-integral formulations.

Experimental results

Research questions

• RQ1 Can the path-integral approach reproduce the canonical results for Green's functions in the N=1 LSM when paths are fixed?
• RQ2 Is the effective potential in the path-integral formulation of the N=2 LSM convex, as required by general quantum field theory arguments?
• RQ3 What is the role of multiple minima in the path integral, and how should they be consistently included in the path-integral approach?
• RQ4 How do divergences in the path-integral approach compare to those in the canonical approach in both N=1 and N=2 LSM?
• RQ5 Can the Higgs sector of the Standard Model be consistently formulated via the path-integral approach without postulating a physical Higgs particle?

Key findings

• The path-integral approach yields a convex effective potential for both the N=1 and N=2 LSM, consistent with general quantum field theory requirements.
• When paths are fixed in space-time, the path-integral approach reproduces the canonical Green's functions, indicating that path-fixing restores agreement with standard perturbation theory.
• The divergences in the path-integral approach match those in the canonical approach, confirming consistency in renormalization structure.
• In the N=2 LSM, the effective potential derived via path integrals in polar coordinates is identical to the canonical result, validating the method.
• The path-integral formulation with fixed paths leads to a well-defined, convex effective potential, resolving prior ambiguities in the SSB treatment.
• The one-dimensional calculation confirms that ⟨r(x)⟩ and the ϕ1-propagator match canonical results only under path-fixing, demonstrating the necessity of this condition.