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The heavy quark-antiquark asymmetry in the variable flavor number scheme

A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, K. Schönwald

TL;DR

This work resolves the heavy-quark–antiquark asymmetry within the variable flavor number scheme by computing the massive single-mass operator matrix elements that generate a nonzero difference between heavy-quark and heavy-antiquark distributions at three-loop order. It treats both unpolarized and polarized cases, deriving the three-loop heavy-flavor distributions from massive pure-singlet OMEs in the Larin scheme for polarization, with scale dependence governed by $\gamma^{\rm NS,s,(2)}_{qq}$ and $\Delta\gamma^{\rm NS,s,(2)}_{qq}$; a key result is the explicit three-loop expression for $\Delta\gamma_{qq}^{\rm NS,s,(2)}$ and the corresponding OME constants. Numerically, the asymmetry is very small and oscillatory in $x$, with the polarized case offering only a small signal that would require large luminosities to detect, indicating negligible impact on the nucleon momentum and spin budgets. The work completes the set of massive three-loop single-mass OMEs, provides corrected polarized anomalous dimensions, and delivers numerical tools and ancillary data for phenomenology.

Abstract

The twist-2 heavy-quark and antiquark distributions, as defined in the variable flavor number scheme, turn out to be different due to QCD corrections from three-loop onward. This is caused by terms containing the color factor $d_{abc} d^{abc}$ in the heavy-flavor massive pure-singlet operator matrix elements (OMEs) $A^{\rm PS, s, (3)}_{Qq}$ for odd moments in the unpolarized case and for $ΔA^{\rm PS, s, (3)}_{Qq}$ for even moments in the polarized case. The dependence on the factorization scale of the OMEs is ruled by the anomalous dimensions $γ^{\rm NS, s, (2)}_{qq}$ and $Δγ^{\rm NS, s, (2)}_{qq}$. The polarized calculations are performed in the Larin scheme. We compute the corresponding three-loop heavy-flavor distributions $(Δ) f_Q(x,Q^2) - (Δ) f_{\overline{Q}}(x,Q^2)$. Compared to the sum of the heavy-quark and antiquark parton distributions, their difference is small, however, non-vanishing.

The heavy quark-antiquark asymmetry in the variable flavor number scheme

TL;DR

This work resolves the heavy-quark–antiquark asymmetry within the variable flavor number scheme by computing the massive single-mass operator matrix elements that generate a nonzero difference between heavy-quark and heavy-antiquark distributions at three-loop order. It treats both unpolarized and polarized cases, deriving the three-loop heavy-flavor distributions from massive pure-singlet OMEs in the Larin scheme for polarization, with scale dependence governed by and ; a key result is the explicit three-loop expression for and the corresponding OME constants. Numerically, the asymmetry is very small and oscillatory in , with the polarized case offering only a small signal that would require large luminosities to detect, indicating negligible impact on the nucleon momentum and spin budgets. The work completes the set of massive three-loop single-mass OMEs, provides corrected polarized anomalous dimensions, and delivers numerical tools and ancillary data for phenomenology.

Abstract

The twist-2 heavy-quark and antiquark distributions, as defined in the variable flavor number scheme, turn out to be different due to QCD corrections from three-loop onward. This is caused by terms containing the color factor in the heavy-flavor massive pure-singlet operator matrix elements (OMEs) for odd moments in the unpolarized case and for for even moments in the polarized case. The dependence on the factorization scale of the OMEs is ruled by the anomalous dimensions and . The polarized calculations are performed in the Larin scheme. We compute the corresponding three-loop heavy-flavor distributions . Compared to the sum of the heavy-quark and antiquark parton distributions, their difference is small, however, non-vanishing.

Paper Structure

This paper contains 7 sections, 27 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Basic Formalism
  3. The massive operator matrix elements
  4. The operator matrix element ${A}_{Qq}^{\rm PS, s, (3)}$
  5. The operator matrix element $\Delta {A}_{Qq}^{{\rm PS},s}$
  6. The heavy quark-antiquark asymmetry
  7. Conclusions

Figures (6)

  • Figure 1: The constant part of the unrenormalized massive OME $\hat{{A}}_{Qq}^{\rm PS, s, (3)}$, ${a}_{Qq}^{\rm PS, s, (3)}$, rescaled by $x(1-x)$. Dashed line: small-$x$ expansion up to the constant term. Dash-dotted line: large-$x$ approximation. Full line: complete result.
  • Figure 2: The constant part of the unrenormalized massive OME $\Delta \hat{{A}}_{Qq}^{\rm PS, s, (3)}$, $\Delta {a}_{Qq}^{\rm PS, s, (3)}$, rescaled by $x(1-x)$. Dashed line: small-$x$ expansion up to the constant term. Dash-dotted line: large-$x$ approximation. Full line: complete result.
  • Figure 3: The unpolarized distributions $x[c(x,Q^2)-\overline{c}(x,Q^2)]$ (left panel) and $x[c(x,Q^2)+\overline{c}(x,Q^2)]$ (right panel). Dotted lines: $Q^2 = 4~\mathrm{GeV}^2$. Dashed lines: $Q^2 = 30~\mathrm{GeV}^2$. Full lines: $Q^2 = 100~\mathrm{GeV}^2$.
  • Figure 4: The unpolarized distributions $x[b(x,Q^2)-\overline{b}(x,Q^2)]$ (left panel) and $x[b(x,Q^2)+\overline{b}(x,Q^2)]$ (right panel). Dotted lines: $Q^2 = m_b^2$. Dashed lines: $Q^2 = 30~\mathrm{GeV}^2$. Full lines: $Q^2 = 100~\mathrm{GeV}^2$.
  • Figure 5: The polarized distributions $[\Delta c(x,Q^2)-\Delta \overline{c}(x,Q^2)]$ (left panel) and $[\Delta c(x,Q^2)+ \Delta \overline{c}(x,Q^2)]$ (right panel). Dotted lines: $Q^2 = 4~\mathrm{GeV}^2$. Dashed lines: $Q^2 = 30~\mathrm{GeV}^2$. Full lines: $Q^2 = 100~\mathrm{GeV}^2$.
  • ...and 1 more figures