The Evaluation of a Quartic Integral via Schwinger, Schur and Bessel

TL;DR

This paper introduces three novel methods to evaluate a specific quartic integral, utilizing Schwinger, Schur, and Bessel function techniques, providing explicit polynomial expressions for the integral.

Contribution

The paper presents new approaches for evaluating a quartic integral, connecting it with Schwinger, Schur functions, and Bessel functions, and deriving explicit polynomial forms.

Findings

Derived explicit polynomial expressions for the integral

Connected integral evaluation with Schwinger, Schur, and Bessel methods

Provided three distinct proofs for the integral evaluation

Abstract

We provide additional methods for the evaluation of the integral \begin{eqnarray} N_{0,4}(a;m) & := & \int_{0}^{\infty} \frac{dx} {\left( x^{4} + 2ax^{2} + 1 \right)^{m+1}} \end{eqnarray} where $m \in {\mathbb{N}}$ and $a \in (-1, \infty)$ in the form \begin{eqnarray} N_{0,4}(a;m) & = & \frac{\pi}{2^{m+3/2} (a+1)^{m+1/2} } P_{m}(a) \end{eqnarray} where $P_{m}(a)$ is a polynomial in $a$. The first one is based on a method of Schwinger to evaluate integrals appearing in Feynman diagrams, the second one is a byproduct of an expression for a rational integral in terms of Schur functions. Finally, the third proof, is obtained from an integral representation involving modified Bessel functions.