[Paper Review] Mellin moments of heavy flavor contributions to F 2 (x, Q 2 ) at NNLO

Sebastian Klein|arXiv (Cornell University)|Nov 16, 2009

Particle physics theoretical and experimental studiesPhysics and Astronomy56 references21 citations

TL;DR

This paper computes fixed moments of the O(α_s^3) heavy flavor contributions to the Wilson coefficients of the proton structure function F_2(x,Q^2) at next-to-next-to-leading order (NNLO), using massive operator matrix elements (OMEs) and known light flavor Wilson coefficients. It provides explicit results for OMEs A_{Qg}^(3), A_{qg,Q}^(3), A_{gg,Q}^(3) from N=2 to 10, A_{Qq}^{(3),PS} up to N=12, and A_{qq,Q}^{(3),NS}, A_{qq,Q}^{(3),PS}, A_{gq,Q}^(3) up to N=14, along with odd moments for flavor non-singlet combinations, all in the asymptotic limit Q^2 >> m^2.

ABSTRACT

This thesis is concerned with the calculation of fixed moments of the O(a_s^3) heavy flavor contributions to the Wilson coefficients of the structure function F_2(x,Q^2) in the limit Q^2 >> m^2, neglecting power corrections. The massive Wilson coefficients in the asymptotic region are given as convolutions of massive operator matrix elements (OMEs) and the known light flavor Wilson coefficients. The former derive from the twist--2 operators emerging in the light--cone--expansion and are calculated at the 3--loop level for fixed moments. We also compute the massive OMEs which are needed to evaluate heavy flavor parton distributions in the variable flavor number scheme to the same order. All contributions to the Wilson coefficients and OMEs but the genuine constant terms at O(a_s^3) of the OMEs are derived in terms of quantities, which are known for general values in the Mellin variable N. For the OMEs A_{Qg}^(3), A_{qg,Q}^(3) and A_{gg,Q}^(3) the moments N = 2 to 10, for A_{Qq}^{(3), PS} to N = 12, and for A_{qq,Q}^{(3), NS}, A_{qq,Q}^{(3), PS}, A_{gq,Q}^(3)}to N=14 are computed. These terms contribute to the light flavor '+'-combinations. For the flavor non-singlet terms, we calculate as well the odd moments N=1 to 13, corresponding to the light flavor '-'-combinations. We also obtain moments of the terms ~ T_F of the 3-loop anomalous dimensions in an independent calculation, which agree with results given in the literature. The mathematical structure of the occurring momentum integrals and of the final results in terms of harmonic sums is discussed. We study applications of the same techniques to the polarized and transversity case at the NLO and NNLO level as well.

Motivation & Objective

• To compute fixed moments of the O(α_s^3) heavy flavor contributions to the Wilson coefficients of F_2(x,Q^2) in the asymptotic region Q^2 >> m^2.
• To derive massive operator matrix elements (OMEs) at the 3-loop level for use in the variable flavor number scheme.
• To provide explicit results for both singlet and non-singlet combinations of OMEs up to N=14, covering both even and odd moments.
• To validate the results by reproducing known 3-loop anomalous dimension terms, confirming consistency with existing literature.
• To extend the methodology to polarized and transversity distributions at NLO and NNLO for future applications.

Proposed method

• The massive Wilson coefficients are constructed as convolutions of massive OMEs and known light flavor Wilson coefficients.
• The OMEs are computed at the 3-loop level using the light-cone expansion of twist-2 operators.
• Fixed moments of the OMEs are evaluated for N=2 to 14, with specific treatment for both singlet and non-singlet combinations.
• The mathematical structure of momentum integrals is analyzed in terms of harmonic sums, enabling analytical computation.
• The results are cross-checked by independently computing T_F terms of the 3-loop anomalous dimensions, confirming agreement with literature.
• Techniques are extended to the polarized and transversity cases at NLO and NNLO, demonstrating methodological consistency.

Experimental results

Research questions

• RQ1 What are the fixed moments of the O(α_s^3) heavy flavor contributions to the Wilson coefficients of F_2(x,Q^2) in the asymptotic limit?
• RQ2 How can massive operator matrix elements (OMEs) be computed at the 3-loop level for use in the variable flavor number scheme?
• RQ3 What is the mathematical structure of the momentum integrals and final results in terms of harmonic sums for these OMEs?
• RQ4 How do the computed OMEs compare with known results for the 3-loop anomalous dimensions, particularly the T_F terms?
• RQ5 Can the same computational framework be applied to polarized and transversity distributions at NLO and NNLO?

Key findings

• The massive OMEs A_{Qg}^(3), A_{qg,Q}^(3), and A_{gg,Q}^(3) are computed for moments N=2 to 10.
• The OMEs A_{Qq}^{(3),PS} are computed up to N=12, and A_{qq,Q}^{(3),NS}, A_{qq,Q}^{(3),PS}, A_{gq,Q}^(3) up to N=14.
• Odd moments N=1 to 13 are computed for flavor non-singlet combinations, corresponding to the light flavor '-'-combinations.
• The T_F terms of the 3-loop anomalous dimensions are reproduced independently and agree with published results.
• The mathematical structure of the results is fully expressed in terms of harmonic sums, enabling further analytical and numerical applications.
• The methodology is extended to the polarized and transversity cases, demonstrating its broader applicability at NLO and NNLO.