⚛️ high-energy theory

Finite path integral limits work in cases where the perturbative series is not Borel summable

This paper demonstrates that employing finite path integral limits yields absolutely convergent perturbative series that exactly reproduce analytical results for non-Borel summable systems, such as double-well potentials, even when expanding around a single minimum.

Original authors: Ariel Edery

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ariel Edery

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: The "Broken Calculator"

Imagine you are trying to calculate the total weight of a very complex object. You have a calculator (a mathematical method called perturbation theory) that works by breaking the object down into smaller, simpler pieces and adding them up.

Usually, this works great. But sometimes, the calculator starts acting weird. As you add more and more pieces, the numbers get bigger and bigger, and the sum never settles down. It's like trying to add up an infinite list of numbers that keeps growing forever. In physics, this is called an asymptotic series.

To fix this, scientists usually use a special trick called Borel resummation. Think of this as a "magic filter" that can take that broken, infinite list and squeeze it into a single, correct number.

However, there is a catch. Sometimes, the object you are measuring has a "split personality." In physics, this is called a non-trivial vacuum or a double-well potential. Imagine a landscape with two deep valleys separated by a hill. The system can sit in either valley.

When the system has this "two-valley" structure, the standard calculator breaks, and the "magic filter" (Borel resummation) fails completely. It's as if the filter is clogged with a blockage right in the middle of the path, making it impossible to get a result. For a long time, physicists thought: "If the filter is broken, we can't calculate the answer using this method."

The New Idea: Building a Fence

The author of this paper, Ariel Edery, asks a simple question: What if we don't try to measure the whole infinite world at once?

In the standard method, the math assumes the "world" (the path integral) stretches from negative infinity to positive infinity. This is like trying to measure the entire ocean at once.

Edery suggests a different approach: Put up a fence.
Instead of measuring from $-\infty$ to $+\infty$ , let's measure from $-L$ to $+L$ , where $L$ is just a very large, but finite, number. We put a fence around the area we are interested in.

The Experiment: Two Ways to Look at the Valley

The author tested this "fence" idea on a simple mathematical model that represents a double-well landscape (two valleys). He tried two different ways to start his calculation:

Starting from the Hilltop: He tried to expand the math starting from the top of the hill between the two valleys (the origin).
- Old way: Without the fence, the math explodes because the hilltop is unstable.
- New way: With the fence, the math stays calm. He got a series of numbers that added up perfectly to a finite answer.
Starting from the Valley: He tried to expand the math starting from the bottom of one of the two valleys.
- Old way: Without the fence, the math produced the broken, infinite series that the "magic filter" couldn't fix.
- New way: With the fence, the math produced a convergent series. This means the numbers added up nicely and stopped growing out of control.

The Surprise Result

Here is the most surprising part of the paper:

When the author took his "fenced-in" calculation (which was perfectly stable) and slowly moved the fence further and further away (letting $L$ go to infinity), the result didn't just give him the answer for one valley.

It gave him the exact answer for the entire system, including both valleys.

Even though he started his calculation standing in just one valley, the "fenced" method somehow captured the physics of the other valley too. It was as if he measured one side of a room, but the calculation magically told him the exact dimensions of the whole room, including the part he wasn't looking at.

Why This Matters

The Old View: If a system has two valleys (a non-trivial vacuum), standard math fails, and you can't fix it with the usual "magic filter."
The New View: If you simply change the boundaries of your calculation (put up a fence), the math works perfectly. You get a series that converges (adds up correctly) even when the coupling (the strength of the interaction) is very strong.
The Takeaway: The "broken" nature of the old math wasn't because the physics was impossible to solve; it was because the method was trying to handle an infinite range too aggressively. By limiting the range first, the math becomes stable and reveals the true, exact answer.

Summary in a Nutshell

Imagine trying to count all the grains of sand on Earth. If you try to do it all at once, your brain (the math) might break. But if you count the sand in a single bucket (finite limits), you get a perfect number. Then, if you realize that the number of buckets is infinite, you can use a pattern to figure out the total.

This paper shows that for systems with "two valleys" (which usually break the math), putting a "bucket" around the problem first allows you to get a perfect, convergent answer that captures the whole picture, even if you only started looking at one part of it.

Technical Summary: Finite Path Integral Limits and Non-Borel Summable Series

Problem Statement
In quantum field theory (QFT) and quantum mechanics, the perturbative expansion of observables in powers of the coupling constant typically yields an asymptotic series. While Borel resummation can often extract finite, physical values from these asymptotic series, there are well-known cases where the series is not Borel summable. This non-summability usually arises in systems with a non-trivial vacuum structure, such as the double-well potential in quantum mechanics or infrared renormalons in Quantum Chromodynamics (QCD). In these scenarios, the Borel transform possesses singularities on the positive real axis, obstructing the standard resummation path.

Recent work has demonstrated that evaluating perturbative series under finite path integral limits (integrating from $-L$ to $L$ rather than $-\infty$ to $\infty$ ) can yield absolutely convergent series, even at strong coupling, for systems that are Borel summable (e.g., the anharmonic oscillator). However, it remained an open question whether this method is effective for systems where the standard perturbative series is not Borel summable due to a degenerate vacuum.

Methodology
To address this, the author analyzes a basic integral with an integrand of the form $e^{-f(x)}$ , where $f(x) = \lambda x^4 - \frac{1}{2}ax^2$ (with $a, \lambda > 0$ ). This function exhibits a double-well shape with two degenerate minima at $x = \pm\sqrt{a/4\lambda}$ and a local maximum at the origin $x=0$ . The integral possesses an exact analytical solution in terms of modified Bessel functions, providing a rigorous benchmark.

The study proceeds in two stages:

Standard Perturbative Analysis: The author first derives the standard perturbative series by expanding about one of the minima ( $x_+$ ). This reproduces the known result: an asymptotic series with positive coefficients that grows factorially. The Borel transform is shown to have a logarithmic singularity at $b = a^2/16$ on the positive real axis, confirming the series is not Borel summable.
Finite Limit Perturbative Analysis: The author then re-evaluates the problem by imposing finite integration limits, $L$ $L$ , such that the integral is defined as $\lim_{L \to \infty} \int_{-L}^{L}$ $lim_{L \to \infty} \int_{- L}^{L}$ . Two distinct perturbative expansions are developed under these finite limits:
- Expansion about the local maximum ( $x=0$ ): Unlike the infinite case where the quadratic term diverges, the finite limits allow for a valid expansion of $e^{-\lambda x^4}$ about $x=0$ .
- Expansion about one of the minima ( $x_+$ ): The expansion is performed around the positive minimum, but the integration over the fluctuation variable $y$ is restricted to $[-L, L]$ .

In both cases, the resulting series are expressed in terms of incomplete gamma functions (arising from the finite limits) and generalized hypergeometric functions.

Key Results

Absolute Convergence: Both perturbative series derived under finite limits are proven to be absolutely convergent for all values of the coupling $\lambda$ (weak and strong), regardless of the expansion point. This is established via the ratio test, where the limit of the ratio of successive terms vanishes.
Recovery of Exact Solution: By summing the convergent series analytically and subsequently taking the limit $L \to \infty$ , the author demonstrates that the series converges to the exact analytical expression of the original integral (involving modified Bessel functions).
Non-Perturbative Capture: Crucially, the expansion performed about only one of the two minima successfully reproduces the full non-perturbative result, which inherently includes the effects of both minima.

Significance and Claims
The paper claims that finite path integral limits provide a mechanism to obtain correct, finite results for systems with non-trivial vacuum structures where standard perturbation theory fails (i.e., yields non-Borel summable asymptotic series).

The primary significance lies in the demonstration that a perturbative expansion about a single vacuum state, when conducted under finite integration limits, can capture the full physics of the degenerate vacuum system. This challenges the conventional view that perturbation theory about one minimum cannot account for the "other side" of a double-well potential without non-perturbative input (such as instantons). The author posits that the finiteness of the integration limits prevents the divergence that leads to non-summability, thereby allowing the series to converge to the exact result.

Future Directions
The author identifies this work as a first step. Future applications of this method are proposed for more complex non-Borel summable cases, specifically:

The energy series of the double-well potential in quantum mechanics (typically studied via multi-instanton methods).
Infrared (IR) renormalons in QCD, which are associated with non-trivial vacuum states and Borel ambiguities.

The paper does not claim to have solved these complex systems yet but suggests that the finite limit approach offers a promising avenue for addressing Borel ambiguities arising from non-trivial vacua in realistic physical systems.