[Paper Review] New Results on Massive 3-Loop Wilson Coefficients in Deep-Inelastic Scattering

Jakob Ablinger, Arnd Behring+8 more|arXiv (Cornell University)|Jan 1, 2016

Particle physics theoretical and experimental studiesPhysics and Astronomy98 references5 citations

TL;DR

This paper presents new analytic calculations of 3-loop massive Wilson coefficients in deep-inelastic scattering (DIS), focusing on charm and bottom quark contributions. Using advanced symbolic computation and integration techniques, the authors compute 2864 Feynman diagrams, reduce them to 687 master integrals (116 of which involve non-iterative elliptic structures), and derive results in Mellin and x-space using harmonic and generalized polylogarithms, enabling precision QCD analysis at NNLO.

ABSTRACT

We present recent results on newly calculated 2- and 3-loop contributions to the heavy quark parts of the structure functions in deep-inelastic scattering due to charm and bottom.

Motivation & Objective

• To compute 3-loop massive Wilson coefficients for deep-inelastic scattering involving charm and bottom quarks.
• To achieve analytic results in the asymptotic region Q² ≫ m² for precision determination of parton distribution functions, αs, and heavy quark masses.
• To extend the existing framework of 2-loop results to 3-loop order, completing the missing link for next-to-next-to-leading order (NNLO) global fits of DIS data.
• To develop and apply advanced symbolic computation tools to solve complex 3-loop Feynman integrals, including those with non-trivial elliptic structures.
• To provide results in both Mellin and x-space representations for use in global PDF fits and precision tests of the Standard Model at the LHC.

Proposed method

• Feynman diagrams (2864) are generated using QGRAF and their Dirac and color structures processed via FORM and Color.
• Integration-by-parts (IBP) reduction via Reduze 2 maps the problem to 687 master integrals, with 571 solvable using iterative sum-structures and special functions.
• Master integrals are evaluated using symbolic tools including Sigma, HarmonicSums, MultiIntegrate, and EvaluateMultiSums, leveraging difference and differential equations.
• Solutions are expressed in terms of harmonic polylogarithms, generalized harmonic polylogarithms, cyclotomic harmonic polylogarithms, and root-valued iterated integrals.
• The remaining 116 master integrals involve second-order differential equations, expected to yield complete elliptic integrals and their iterated structures.
• Results are transformed from Mellin space to x-space using inverse Mellin transforms, with applications to structure functions F2, F_L, and g1.

Experimental results

Research questions

• RQ1 What are the analytic 3-loop massive Wilson coefficients for deep-inelastic scattering in the asymptotic limit Q² ≫ m²?
• RQ2 How can the 3-loop heavy flavor contributions to F2(x,Q²) and F_L(x,Q²) be computed and expressed in terms of special functions?
• RQ3 What is the structure of the 116 master integrals that do not factorize into first-order equations, and how do they relate to elliptic functions?
• RQ4 How do the new 3-loop results improve the precision of global fits for parton distribution functions, αs, and heavy quark masses?
• RQ5 What symbolic and algorithmic techniques are required to compute and simplify 3-loop Feynman integrals with multiple mass scales?

Key findings

• The authors compute 2864 Feynman diagrams and reduce them to 687 master integrals, with 571 solvable using known special function techniques.
• Fifty-seven-one master integrals are solved using iterative sum-structures and special functions such as harmonic polylogarithms and generalized harmonic polylogarithms.
• The remaining 116 master integrals are expected to involve complete elliptic integrals and their iterated structures due to second-order differential equations.
• The 3-loop heavy flavor Wilson coefficients for F2(x,Q²) are computed at O(α³s) and validated against known 2-loop results and Mellin moments.
• The results are expressed in both Mellin and x-space, enabling direct use in global PDF fits and precision tests of QCD at the LHC.
• The computation relies on a suite of advanced symbolic tools (Sigma, HarmonicSums, MultiIntegrate) to handle recurrence relations and special function identities.