[Paper Review] Definite integrals by the method of brackets. Part 1

Iván González, Victor H. Moll|ArXiv.org|Dec 17, 2008

Algebraic and Geometric Analysis47 references11 citations

TL;DR

This paper introduces the method of brackets, a heuristic technique for evaluating definite integrals over [0, ∞) by reducing them to solving linear systems of equations. It leverages bracket notation for divergent integrals and applies it to hypergeometric and Feynman diagram integrals, yielding closed-form results in terms of special functions like the Appell F4 function with rigorous validation underway.

ABSTRACT

A new heuristic method for the evaluation of definite integrals is presented. This method of brackets has its origin in methods developed for theevaluation of Feynman diagrams. We describe the operational rules and illustrate the method with several examples. The method of brackets reduces the evaluation of a large class of definite integrals to the solution of a linear system of equations.

Motivation & Objective

• To develop a systematic, heuristic method for evaluating definite integrals over [0, ∞) that bypasses ad-hoc techniques.
• To apply the method to integrals arising in quantum field theory, particularly Feynman diagrams, using bracket series formalism.
• To provide a practical tool for verifying and deriving entries in standard integral tables such as Gradshteyn and Ryzhik.
• To demonstrate the method’s effectiveness through examples involving hypergeometric and Appell functions.
• To lay the groundwork for a rigorous foundation of the bracket rules, currently under development.

Proposed method

• The method assigns a bracket ⟨a⟩ to each parameter a, representing the divergent integral ∫₀^∞ x^{a−1} dx.
• A function f(x) = Σₙ aₙ x^{αn + β − 1} is associated with a bracket series ∫₀^∞ f(x) dx ⋍ Σₙ aₙ ⟨αn + β⟩.
• The evaluation reduces to solving a system of linear equations derived from setting the bracket indices to zero (n*).
• The solution is expressed in terms of special functions such as the Appell F₄ function, particularly in the context of Feynman diagrams.
• The method handles convergence formally, ignoring convergence issues, and focuses on integrals over [0, ∞).
• Indices are replaced by n* values, which may be complex, to evaluate the formal sum.

Experimental results

Research questions

• RQ1 Can a unified heuristic method be developed to evaluate a broad class of definite integrals over [0, ∞)?
• RQ2 How can the method of brackets be systematically applied to Feynman diagrams in quantum field theory?
• RQ3 To what extent can the method reproduce or verify entries in standard integral tables like Gradshteyn and Ryzhik?
• RQ4 What is the formal structure of the bracket series, and how can it be rigorously justified?
• RQ5 How do the results from the method compare with known special function representations, such as Appell F₄?

Key findings

• The method reduces the evaluation of definite integrals to solving a system of linear equations derived from bracket conditions.
• The integral G from a Feynman diagram is expressed as a sum of three terms depending on the mass ordering: G₁₂ + G₁₄ + G₂₅, G₁₃ + G₃₅, or G₂₃ + G₃₄.
• Each component Gi,j is expressed in terms of the Appell F₄ function with specific parameters involving gamma functions and powers of P², m₁², and m₂².
• For example, G₁₂ = (−1)^{−D/2} Γ(a₁+a₂−D/2)Γ(D/2−a₁)Γ(D/2−a₂)/[Γ(a₁)Γ(a₂)Γ(D−a₁−a₂)] × (P²)^{D/2−a₁−a₂} × F₄(1+a₁+a₂−D, a₁+a₂−D/2; 1+a₁−D/2, 1+a₂−D/2 | m₁²/P², m₂²/P²).
• The method successfully evaluates integrals that are otherwise intractable, such as the one in entry 3.248.5 of Gradshteyn and Ryzhik, which was later found to be incorrect.
• The method provides a consistent framework for deriving closed-form results, even when standard tables contain errors, and enables systematic verification of integral entries.