[Paper Review] A Non-perturbative Solution of the Zero-Dimensional lambda phi^4 Field Theory

A. P. C. Malbouisson, Renato Portugal+1 more|arXiv (Cornell University)|Sep 23, 1999

Black Holes and Theoretical Physics2 references4 citations

TL;DR

This paper presents an exact, non-perturbative solution of the zero-dimensional λϕ⁴ scalar field theory by expressing the partition function in terms of Macdonald's function for Re(λ) > 0. It extends this result via analytic continuation to the entire complex λ-plane, excluding a branch cut along the negative real axis, and confirms consistency with perturbative Borel summation in the common domain of validity.

ABSTRACT

We have done a study of the zero-dimensional $λϕ^{4}$ model. Firstly, we exhibit the partition function as a simple exact expression in terms of the Macdonald's function for $Re(λ)>0$. Secondly, an analytic continuation of the partition function for $Re(λ)<0$ is performed, and we obtain an expression defined in the complex coupling constant plane $λ$, for $|arg λ|0$, it coincides precisely with the exact expression.

Motivation & Objective

• To provide an exact, non-perturbative expression for the partition function of the zero-dimensional λϕ⁴ field theory.
• To extend the domain of the partition function to complex coupling constants λ through analytic continuation.
• To resolve the issue of convergence and non-perturbative behavior in the λϕ⁴ model in the zero-dimensional limit.
• To verify consistency between the exact solution and perturbative results obtained via Borel summation.

Proposed method

• Derive the partition function exactly using the integral representation of the Macdonald function for Re(λ) > 0.
• Perform analytic continuation of the partition function into the region Re(λ) < 0 using complex analysis techniques.
• Define the partition function on the complex λ-plane with a branch cut along the negative real axis, valid for |arg λ| < π.
• Apply the Borel summation technique to the perturbative series of the partition function in the region Re(λ) > 0.
• Compare the Borel-summable perturbative result with the exact expression to confirm agreement in the overlapping domain.
• Use properties of special functions and contour integration to ensure analyticity and convergence of the solution.

Experimental results

Research questions

• RQ1 Can the partition function of the zero-dimensional λϕ⁴ theory be expressed in closed form for Re(λ) > 0?
• RQ2 How can the partition function be analytically continued to Re(λ) < 0 and complex λ values?
• RQ3 What is the structure of the singularity in the complex λ-plane, and how does it affect the non-perturbative behavior?
• RQ4 Does the non-perturbative solution agree with the Borel-summable perturbative series in the common domain?
• RQ5 What is the role of the Macdonald function in representing the exact partition function?

Key findings

• The partition function is exactly expressed in terms of the Macdonald function K_{1/2}(z) for Re(λ) > 0.
• The partition function is analytically continued to the complex λ-plane with a branch cut along the negative real axis, valid for |arg λ| < π.
• The exact solution agrees precisely with the Borel-summable perturbative result in the domain Re(λ) > 0.
• The model exhibits a non-perturbative structure that is fully captured by the analytic continuation of the partition function.
• The solution demonstrates the consistency between non-perturbative and perturbative approaches in a solvable field theory model.
• The absence of renormalization in the zero-dimensional case allows for a clean derivation of the exact partition function using special functions.