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New Heavy Flavor Contributions to the DIS Structure Function $F_2(x,Q^2)$ at $O(α_s^3)
J. Ablinger, J. Blümlein, A. Hasselhuhn, S. Klein, C. Schneider, F. Wißbrock
TL;DR
The paper addresses the need for O(α_s^3) heavy-flavor Wilson coefficients for precise F_2 predictions at Q^2/m^2 ≳ 10. It advances massive-OME calculations across several diagram classes and identifies cyclotomic harmonic structures, while noting scheme and matching-scale limitations.
Problem
Precise small-x F_2 data require accurate three-loop heavy-flavor Wilson coefficients because charm contributions are large and measured to 1% accuracy.
Source: §1 · p. 2Method
The paper calculates selected three-loop massive OMEs for unequal-mass diagrams, ladder graphs, and gluonic contributions using moment and general-N methods.
Source: §1 · p. 2, §2 · pp. 2–3, §4 · p. 6Results
The work reports progress on new massive-Wilson-coefficient contributions and intermediate cyclotomic harmonic sums and polylogarithms.
Source: §1 · p. 2Takeaways & Limitations
The results extend the ingredients available for heavy-flavor F_2 calculations and variable-flavor-scheme construction in the stated asymptotic region.
Source: §1 · p. 2, §4 · p. 6Unequal-mass contributions are uniquely defined in the fixed-flavor scheme but cannot be assigned separately to charm or bottom distributions in the variable-flavor scheme.
Source: §2 · p. 3
Abstract
from arXiv · showhide
We report on recent results obtained for the massive Wilson coefficients which contribute to the structure function $F_2(x,Q^2)$ at $O(α_s^3)$ in the region $Q^2/m^2 \gsim 10$. In the calculation new species of harmonic sums and harmonic polylogarithms generated by cyclotomic polynomials arise in intermediary results which are briefly discussed.
1. Introduction
The paper targets missing O(α_s^3) heavy-flavor corrections to F_2 in the high-Q^2/m^2 region, where precision data require accurate Wilson coefficients. It reports progress on massive OMEs, including new cyclotomic harmonic sums and polylogarithms.
Motivation
- 3-loop corrections are needed because charm contributions are large at small x, where F_2 is measured to 1% accuracy.These corrections support precise strong-coupling and parton-density extractions and affect precision measurements at the Tevatron and LHC.Source: §1 · p. 2
Asymptotic method
- For Q^2/m^2 ≳ 10, massive Wilson coefficients can be obtained by convolving massive OMEs with massless Wilson coefficients.Known two-loop OMEs and massless three-loop coefficients determine logarithmic O(α_s^3) terms, but unknown constants are not subdominant in the HERA region.Source: §1 · p. 2
Related results
- Earlier work computed selected O(α_s^3) coefficients and related heavy-flavor corrections, while the present calculation extends the set of contributions under study.The cited results include complete general-N coefficients for two Wilson coefficients and asymptotic F_L corrections.Source: §1 · p. 2
Computational framework
- Massive Wilson-coefficient contributions with one massless and one massive internal fermion line reduce to the known weight-4 harmonic-sum set after algebraic relations.General-N calculations use tform and Sigma.Source: §1 · p. 2
Contributions
- The calculation advances three-loop massive-OME results for charm and bottom contributions, gluonic OMEs, ladder graphs, and cyclotomic harmonic structures.The reported intermediate expressions include harmonic sums and polylogarithms generated by cyclotomic polynomials.Source: §1 · p. 2
2. Two massive quarks of unequal mass
This section treats diagrams containing unequal charm and bottom masses by expanding in their small mass ratio and extending moment calculations to N = 6. It also identifies scheme and matching-scale limitations.
Mass expansion
- For charm and bottom, the mass ratio b ≃ 1/10 permits an expansion of the corresponding unequal-mass diagrams.Earlier N = 2 and N = 4 moments were extended using qexp and projector methods.Source: §2 · p. 2
Moment calculation
- N = 6 is presented for the gluonic operator matrix element A_Qg after calculations became increasingly time-consuming at higher moments.The computation required several months of run time.Source: §2 · p. 3
Explicit result
- The cited passages list three explicit polynomial terms: 14586075000 x + 1511074426112, 180093375 x2 + 2287164970759, and 2552563125 x + 1048702178522.Source: §2 · p. 3
Scheme limitation
- The unequal-mass contributions are uniquely calculable in the fixed-flavor scheme with three massless initial-state quarks.They cannot be assigned separately to charm or bottom distributions in a variable-flavor scheme, which ends at two loops for this representation.Source: §2 · p. 3
Matching-scale caveat
- The matching scale is process dependent and may differ significantly from the heavy-quark mass m_Q, although this dependence is often ignored.The cited comparisons to complete calculations establish this as a practical qualification on matching choices.Source: §2 · p. 3
3. Contributions due to 3-Loop Ladder Graphs
The paper begins calculating a topology class of three-loop ladder graphs with up to six massive lines and local operator insertions. It gives an example scalar-diagram result through polynomial expressions in N.
Ladder topology
- Three-loop ladder graphs with up to six massive lines and local operator insertions form a topology class whose calculation is underway.The paper presents a scalar diagram as an example.Source: §3 · p. 4
Example result
- The scalar-diagram example is expressed through polynomial factors P23, P24, P26, and related terms in the Mellin moment N.The supplied passages list explicit coefficients but do not state a numerical benchmark or qualitative comparison.Source: §3 · pp. 5–6
4. Contributions to Gluonic Matrix Elements
The paper obtains first general-N results for gluonic OMEs with color structure O(N_f T_F^2 C_A,F). These results help define the variable-flavor scheme for single-heavy-mass contributions.
Gluonic OMEs
- First general-N results are obtained for gluonic OMEs with color structure O(N_f T_F^2 C_A,F).The paper shows a renormalized example for A^(3)_FCA,F.Source: §4 · p. 6
Scheme relevance
- These gluonic OME results are important for defining the variable-flavor scheme for single-heavy-mass contributions.The section connects the new results to earlier single-mass OME work.Source: §4 · p. 6
5. Cyclotomic Harmonic Sums and Harmonic Polylogarithms
The calculation introduces cyclotomic generalizations of harmonic polylogarithms and sums to represent structures arising in massive operator matrix elements. Their mathematical properties and implementation are connected to iterated integrals, roots of unity, multiple zeta values, and HarmonicSums.
5. Cyclotomic Harmonic Sums and Harmonic Polylogarithms
- Cyclotomic harmonic polylogarithms generalize the usual harmonic-polylogarithm alphabet for structures arising in massive operator matrix elements.The usual alphabet contains 1/x, 1/(1−x), and 1/(1+x).Source: §5 · p. 6
- Cyclotomic harmonic polynomials are formed as iterated integrals of letters built from cyclotomic polynomials and Euler’s totient function.Source: §5 · p. 7
- The resulting sums relate to usual harmonic sums with numerator weights that are nth roots of unity and to generalized multiple zeta values.Their properties and interrelations are documented in [31].Source: §5 · p. 7
- These cyclotomic harmonic structures are implemented in the HarmonicSums package.Source: §5 · p. 7