[Paper Review] Heavy Flavor Wilson Coefficients in Deep-Inelastic Scattering: Recent Results

Jakob Ablinger, Arnd Behring+4 more|arXiv (Cornell University)|Jan 1, 2017

Quantum Chromodynamics and Particle Interactions3 references33 citations

TL;DR

This paper presents advanced analytic calculations of 3-loop heavy flavor Wilson coefficients in deep-inelastic scattering, focusing on the massive operator matrix element $A^{(3)}_{Qg}$. Using the method of arbitrarily large moments and symbolic computation tools like Sigma and HarmonicSums, the authors compute 18 out of 28 color-ζ terms at $\mathcal{O}(\varepsilon^0)$, achieving full analytic results for most contributions and identifying remaining terms involving iterative non-iterative integrals as the final frontier in this calculation.

ABSTRACT

We present recent analytic results for the 3-loop corrections to the massive operator matrix element $A_{Qg}^{(3)}$for further color factors. These results have been obtained using the method of arbitrarily large moments. We also give an overview on the results which were obtained solving all difference and differential equations for the corresponding master integrals that factorize at first order.

Motivation & Objective

• To compute the 3-loop massive operator matrix element $A^{(3)}_{Qg}$ in deep-inelastic scattering with high precision for global QCD fits.
• To extend analytic results for massive OMEs beyond the 2-loop level, particularly for the $A^{(3)}_{Qg}$ channel.
• To resolve the remaining 10 out of 28 color-ζ terms involving iterative non-iterative integrals, which are not factorizable in first-order difference equations.
• To validate the 3-loop anomalous dimension $\gamma^{(2)}_{qg}(N)$ independently using the method of arbitrarily large moments.

Proposed method

• The method of arbitrarily large moments is used to compute high Mellin moments of master integrals, enabling the reconstruction of difference equations.
• Integration-by-parts (IBP) relations reduce the 1358 Feynman diagrams to 340 master integrals.
• Differential and difference equations for master integrals are solved using symbolic computation packages: Sigma, EvaluateMultiSums, SumProduction, and HarmonicSums.
• The method of guessing is applied to derive difference equations from high-moment sequences, particularly for terms with pure color factors and $\zeta_3$-factors.
• Iterative non-iterative integrals are identified as the source of non-factorizing difference equations in the remaining 10 color-ζ terms.
• The unrenormalized OME $\hat{A}^{(3)}_{Qg}$ is reconstructed from moment data using rational function reconstruction and special function algebra.

Experimental results

Research questions

• RQ1 What is the analytic structure of the 3-loop massive operator matrix element $A^{(3)}_{Qg}$ in deep-inelastic scattering, particularly for the $\mathcal{O}(\varepsilon^0)$ term?
• RQ2 Which color-ζ factors in $A^{(3)}_{Qg}$ can be fully resolved using first-order factorizing difference equations?
• RQ3 How do iterative non-iterative integrals emerge in the remaining 10 color-ζ terms, and what is their role in the non-factorizing structure of the difference equations?
• RQ4 Can the 3-loop anomalous dimension $\gamma^{(2)}_{qg}(N)$ be independently recalculated from first principles using the method of arbitrarily large moments?
• RQ5 What is the computational limit of the current toolbox in resolving all 28 color-ζ terms of $A^{(3)}_{Qg}$?

Key findings

• The 3-loop anomalous dimension $\gamma^{(2)}_{qg}(N)$ has been independently recalculated and confirmed using the method of arbitrarily large moments.
• 18 out of 28 color-ζ terms in the $\mathcal{O}(\varepsilon^0)$ contribution to $A^{(3)}_{Qg}$ have been computed analytically, including terms with $\zeta_3$, $\zeta_4$, and $B_4$.
• The remaining 10 terms involve non-factorizing difference equations of order 45 and degree ~1500, indicating the presence of iterative non-iterative integrals and elliptic structures.
• The constant part of the unrenormalized OME $\hat{A}^{(3)}_{Qg}$ is fully expressed in terms of nested harmonic sums, zeta values, polynomials, and $B_4$, with explicit rational functions for all but 10 terms.
• The method of arbitrarily large moments successfully computes the 3-loop anomalous dimension from first principles in the massive case, demonstrating its robustness.
• 1122 out of 1358 Feynman diagrams contributing to $A^{(3)}_{Qg}$ have been computed, with the remaining 236 diagrams involving new functions requiring further analysis.