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The O(α_s^3) Massive Operator Matrix Elements of O(n_f) for the Structure Function F_2(x,Q^2) and Transversity

J. Ablinger, J. Blümlein, S. Klein, C. Schneider, F. Wißbrock

arXiv:1008.3347v1hep-phhep-exhep-thmath-ph

TL;DR

The paper computes nf-dependent O(α_s^3) massive operator matrix elements for F2 and transversity in the Q^2 ≫ m^2 limit at general Mellin N. It completes two OMEs and independently recalculates selected three-loop anomalous dimensions using nested-sum methods. The results are expressed with nested harmonic sums and agree with the literature for the recalculated anomalous dimensions.

  • Problem

    Percent-level deep-inelastic data require NNLO heavy-flavor corrections, whose general-N three-loop massive OMEs were not yet fully available.

    Source: §1 · p. 2

  • Method

    The calculation evaluates Feynman-parameter integrals as nested sums over generalized hypergeometric functions and harmonic sums, then applies modern summation technologies.

    Source: §1 · p. 2, §5 · p. 15

  • Results

    Two three-loop OMEs, Aqq,Q^PS(N) and Aqg,Q(N), are completed, and selected three-loop anomalous dimensions are independently recalculated and confirmed against the literature.

    Source: §5 · p. 16

  • Takeaways & Limitations

    The completed OMEs and independently confirmed anomalous dimensions extend general-N heavy-flavor results for F2 and transversity in the asymptotic region.

    Source: §5 · pp. 15–16

    The formulation is presented in the MS scheme with factorization and renormalization scales identified for simplicity.

    Source: §2 · p. 4

Abstract

from arXiv · show

The contributions $\propto n_f$ to the $O(α_s^3)$ massive operator matrix elements describing the heavy flavor Wilson coefficients in the limit $Q^2 \gg m^2$ are computed for the structure function $F_2(x,Q^2)$ and transversity for general values of the Mellin variable $N$. Here, for two matrix elements, $A_{qq,Q}^{\sf PS}(N)$ and $A_{qg,Q}(N)$, the complete result is obtained. A first independent computation of the contributions to the 3--loop anomalous dimensions $γ_{qg}(N)$, $γ_{qq}^{\sf PS}(N$ and $γ_{qq}^{\sf NS,(TR)}(N)$ is given. In the computation advanced summation technologies for nested sums over products of hypergeometric terms with harmonic sums have been used. For intermediary results generalized harmonic sums occur, while the final results can be expressed by nested harmonic sums only.

1 Introduction

The paper addresses heavy-flavor corrections needed for percent-level deep-inelastic precision in the asymptotic region, computing nf-dependent three-loop massive OMEs for F2 and transversity. It completes two OMEs and derives corresponding anomalous-dimension contributions using nested-sum summation methods.

  • Heavy-flavor corrections are especially large at low Bjorken x, while approximately 1% data precision requires NNLO corrections for precision determinations of αs and parton distributions.Source: §1 · p. 2
  • For Q^2 ≫ m^2, power corrections become small and logarithmic plus constant terms describe F2 at the 1%-level for Q^2/m^2 ≳ 10.Source: §1 · p. 2
  • The work computes nf-dependent O(α_s^3) massive OMEs for F2 and transversity at general Mellin N, completely determining Aqq,Q^PS(N) and Aqg,Q(N).Source: §1 · p. 2
  • Feynman-parameter integrals are converted into nested sums over hypergeometric terms and harmonic sums, with generalized harmonic sums appearing intermediately and cancelling from final nested-harmonic-sum expressions.Source: §1 · pp. 2–3
  • The single-pole terms yield nf parts of three-loop anomalous dimensions, including γqg(N) and γqq^PS(N), in an independent calculation compared with earlier results.Source: §1 · p. 3

2 The heavy flavor Wilson coefficients in the asymptotic region

This section sets up heavy-flavor Wilson coefficients by distinguishing photon couplings and partonic channels, then relates their asymptotic forms to renormalized massive OMEs and massless Wilson coefficients. The formulation uses the MS scheme with identified factorization and renormalization scales.

  • Wilson coefficients are labeled Li and Hi according to whether the photon couples to a light or heavy quark, with NS, PS, and S flavor structures.Source: §2 · p. 3
  • For Q^2 ≫ m^2, massive Wilson coefficients are expressed through renormalized massive OMEs and massless Wilson coefficients through O(a_s^3).Source: §2 · p. 3
  • The asymptotic mass dependence comes from unrenormalized massive OMEs after mass, coupling, operator, and factorization renormalization.Source: §2 · p. 4
  • The calculation uses the MS scheme and identifies the factorization and renormalization scales for simplicity.Source: §2 · p. 4

3 The Massive Operator Matrix Elements

The paper computes O(n_f) three-loop massive operator matrix elements for F2 and transversity at general Mellin N, obtaining complete results for selected OMEs and expressing them through nested harmonic sums. It also analyzes their mathematical structure and small- and large-x behavior.

3 The Massive Operator Matrix Elements

  • The calculation obtains O(n_f) three-loop massive OMEs for F2 and transversity, with complete results for A_qq,Q^PS(N) and A_qg,Q(N).The results concern general Mellin N and include unrenormalized constant contributions, with renormalized expressions also presented where applicable.Source: §3 · p. 4

3.1 Operator Matrix Elements contributing to F2(x, Q2)

  • The F2 calculation supplies constant contributions for pure-singlet, flavor non-singlet, and quark-gluon OMEs, and checks general-N expressions against fixed moments.The fixed-moment comparison agrees with results from Ref..Source: §3.1 · pp. 7–9

3.2 The Operator Matrix Elements for Transversity

  • The transversity O(n_f) contribution agrees with previously calculated moments and has the same harmonic-sum structures as the corresponding vector operator for this color factor.Transversity is accessed through polarized semi-inclusive deep-inelastic scattering and polarized Drell-Yan processes.Source: §3.2 · p. 11

3.3 The Mathematical Structure of the Operator Matrix Elements

  • The three-loop results contain nested harmonic sums up to weight 4; generalized sums arise during calculation, while algebraic and structural relations reduce the final representation to a smaller basis.The sums originate from expansions of generalized hypergeometric structures and are organized with rational functions of N and zeta values.Source: §3.3 · pp. 11–12

3.4 The OMEs in the Small and Large x Region

  • At small x, singlet and pure-singlet terms behave as 1/x while non-singlet terms are logarithmic; at large x, the O(n_f) terms are less singular than their massless counterparts.The two pure-singlet small-x asymptotics coincide, while the large-x pure-singlet limits differ by factors of -8 and -1/8 and the non-singlet contributions coincide.Source: §3.4 · p. 13

4 The Contributions to the Anomalous Dimensions

The paper derives three-loop anomalous-dimension contributions from single-pole terms of unrenormalized massive OMEs and compares them with prior results. Vector and tensor operators provide independent checks, including a two-calculation confirmation for the non-singlet case.

4 The Contributions to the Anomalous Dimensions

  • Single-pole terms of the unrenormalized OMEs determine the nf contributions to the three-loop anomalous dimensions.All other contributions to the 1/ε terms are known, allowing the new terms to be isolated by comparison.Source: §4 · p. 14

4.1 Vector Operators

  • The pure-singlet nf contribution is derived from the pure-singlet massive OME and expressed using harmonic-sum structures including S1^2 and S2.Source: §4.1 · p. 14
  • The flavor non-singlet anomalous dimension has been obtained through two independent new calculations.Source: §4.1 · p. 15
  • The anomalous dimensions agree with the moments and general results in Refs., while algebraic compactification yields fewer harmonic sums than Ref. and agrees with [45].For the flavor non-singlet case, the anomalous dimension was predicted in [18].Source: §4.1 · p. 15

4.2 Tensor Operator

  • The transversity anomalous-dimension contribution proportional to n_f^2 is obtained from the single-pole term of the tensor OME.Source: §4.2 · p. 15
  • The anomalous-dimension results provide a first independent check of Refs. [11], with vector and tensor operators yielding the same harmonic-sum structures.Source: §4.2 · p. 15

5 Conclusions

The study computes the nf contributions to three-loop massive OMEs for F2 and transversity at general Mellin N in the asymptotic MS scheme. It completes two OMEs, reduces the final expressions to harmonic sums, and confirms corresponding anomalous dimensions independently.

  • The calculation covers O(n_f) three-loop massive OMEs for F2 and transversity in the asymptotic MS scheme at general Mellin N.Source: §5 · p. 15
  • The complete results for two three-loop OMEs, Aqq,Q^PS and Aqg,Q, are now available.Source: §5 · p. 15
  • Nested sums over generalized hypergeometric functions are solved with modern difference- and product-field summation technologies.Intermediate generalized harmonic sums cancel, leaving rational functions of N and harmonic sums of maximal weight four.Source: §5 · pp. 15–16
  • The completed Aqg,Q OME contains no constant ζ2 after renormalization and agrees with fixed moments from Ref..Source: §5 · p. 16
  • Single-pole terms yield three-loop anomalous dimensions as a first independent recalculation, confirming results in the literature for deep-inelastic scattering and transversity.Source: §5 · p. 16

A Examples for sums occurring in the calculation

The appendix illustrates the generalized hypergeometric-type sums arising during the calculation and explains their reduction to nested harmonic sums. These sums can have broader arguments and higher intermediate weight than the final expressions.

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