[Paper Review] Some methods for the evaluation of complicated Feynman integrals

A. V. Kotikov|ArXiv.org|Dec 28, 2001

Advanced Mathematical Identities3 references3 citations

TL;DR

This paper presents advanced analytical techniques—Gegenbauer Polynomial and Differential Equation Methods—for evaluating complex two-loop Feynman integrals in quantum field theory, particularly for massless and massive propagator diagrams. It achieves high-precision results for critical exponents in $φ^4$-theory at $\mathcal{O}(1/N^3)$, including explicit Laurent series in $\varepsilon$-dimensional regularization with zeta functions, polylogarithms, and inverse sine integrals, validated numerically via PSLQ and Padé approximants.

ABSTRACT

We discuss a progress in calculations of Feynman integrals based on the Gegenbauer Polynomial Technique and the Differential Equation Method. We demonstrate the results for a class of two-point two-loop diagrams and the evaluation of most complicated part of O(1/N^3) contributions to critical exponents of ϕ^4-theory. An illustration of the results obtained with help of above methods is considered.

Motivation & Objective

• To develop and apply analytical methods for evaluating highly nontrivial two-loop Feynman integrals in quantum field theory.
• To compute the most complicated part of $\mathcal{O}(1/N^3)$ corrections to critical exponents in $\phi^4$-theory.
• To provide precise Laurent expansions in $\varepsilon$-dimensional regularization for two-point and three-point diagrams with massless and massive propagators.
• To validate results numerically using PSLQ and Padé approximants against Taylor expansions and master integrals.
• To establish connections between results and transcendental constants arising in three-loop diagrams, particularly at sixth roots of unity.

Proposed method

• Employs the Gegenbauer Polynomial (GP) technique to expand propagators in $D$-dimensional momentum space using orthogonal polynomials in hyperspherical coordinates.
• Uses the traceless product formalism to express tensor structures in terms of Gegenbauer polynomials and $x^2$-dependent factors.
• Applies the Differential Equation Method to derive coupled ODEs for master integrals, solved via series expansions in $\varepsilon$.
• Utilizes Mellin-Barnes representations and analytic continuation to extract coefficients in Laurent series in $\varepsilon$.
• Employs numerical checks via Taylor series expansions and Padé approximants to validate analytical results.
• Applies the PSLQ algorithm to identify combinations of transcendental constants (zeta values, polylogarithms, inverse sine integrals) in numerical results.

Experimental results

Research questions

• RQ1 How can the Gegenbauer Polynomial method be systematically applied to two-loop two-point diagrams with massless and massive propagators?
• RQ2 What is the structure of the $\mathcal{O}(1/N^3)$ contribution to critical exponents in $\phi^4$-theory, and how can it be evaluated analytically?
• RQ3 How do the results from the Gegenbauer and Differential Equation methods compare in terms of accuracy and convergence for complex diagrams?
• RQ4 Which transcendental constants (e.g., $\zeta(2), \zeta(3), \text{Ls}_3(2\pi/3)$) emerge in the Laurent expansion of two-loop integrals?
• RQ5 Can numerical evaluations of the integrals confirm the analytical structure involving polylogarithms at sixth roots of unity?

Key findings

• The master integral $\mathbf{V}\{\mathcal{IJKL}\}(1,1,1,1,m)$ is evaluated as $\frac{1}{2\varepsilon^2} + \frac{1}{\varepsilon}\left(\frac{5}{2} - \frac{\pi}{\sqrt{3}}\right) + \frac{19}{2} + \frac{b_1}{2}\zeta(2) - 4\frac{\pi}{\sqrt{3}} - \frac{63}{4}S_2 + \cdots + \mathcal{O}(\varepsilon^2)$, with $b_1 = -1$.
• The integral $\mathbf{J}_{111}(1,1,1,m)$ is found to be $-m^2\left(\frac{3}{2\varepsilon^2} + \frac{17}{4\varepsilon} + \frac{59}{8} + \cdots\right) + \mathcal{O}(\varepsilon^3)$, with coefficients $b_2 = -6$, $b_3 = 9/2$, $b_4 = 4$, $b_5 = 9$.
• The integral $\mathbf{J}_{011}(1,1,2,m)$ yields $\frac{1-4\varepsilon}{2(1-2\varepsilon)(1-3\varepsilon)}\left(\frac{1}{\varepsilon^2} + 2\frac{\pi}{\sqrt{3}} - \frac{2}{3}\zeta(2) + \cdots\right) + \mathcal{O}(\varepsilon^2)$.
• The integral $\mathbf{J}_{011}(1,1,1,m)$ is expressed as $-\frac{m^2}{2}\frac{4-15\varepsilon}{(1-2\varepsilon)(1-3\varepsilon)(2-3\varepsilon)}\left(\frac{1}{\varepsilon^2} + \frac{3}{2}\frac{\pi}{\sqrt{3}} + \cdots\right) + \mathcal{O}(\varepsilon^3)$, with $b_1 = 3$, $b_2 = 8$, $b_3 = -3/2$, $b_4 = 0$, $b_5 = 21$.
• The integral $\mathbf{ONS11}(1,1,m)$ is computed as $\frac{1}{1-2\varepsilon}\left(\frac{1}{\varepsilon} - \frac{\pi}{\sqrt{3}} + \varepsilon\left(\frac{\pi}{\sqrt{3}}\ln 3 - 9S_2\right) + \cdots\right) + \mathcal{O}(\varepsilon^3)$.
• Numerical checks confirm the analytical results, with PSLQ identifying combinations of $\zeta(2), \zeta(3), \text{Ls}_3(2\pi/3)$, and $\text{Li}_3(e^{2\pi i/3})$-type constants, consistent with three-loop transcendental structures.