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Parton distributions in the presence of target mass corrections

F. M. Steffens, M. D. Brown, W. Melnitchouk, S. Sanches

TL;DR

This work scrutinizes how target mass corrections (TMCs) are incorporated into parton distributions within the operator product expansion at low $Q^2$. It argues that the conventional Nachtmann-variable-based inversion introduces energy–momentum conservation issues and undermines PDF universality at finite $Q^2$. To address this, the authors develop a series expansion of inverted moments in $oldsymbol{ extmu}=M^2/Q^2$ that avoids $\xi$, expressing all unpolarized structure functions as derivative operators acting on PDFs, and they demonstrate convergence properties for representative kinematics. The approach preserves moment consistency, improves the reliability of TMCs down to $W oughly 1.3$ GeV for moderate $Q^2$, and offers a practical path for including low-$W$ data in global PDF analyses, with caveats near the elastic limit and potential extension to spin structure functions in future work.

Abstract

We study the consistency of parton distribution functions in the presence of target mass corrections (TMCs) at low Q^2. We review the standard operator product expansion derivation of TMCs in both x and moment space, and present the results in closed form for all unpolarized structure functions and their moments. To avoid the unphysical region at x > 1 in the standard TMC analysis, we propose an expansion of the target mass corrected structure functions order by order in M^2/Q^2, and assess the convergence properties of the resulting forms numerically.

Parton distributions in the presence of target mass corrections

TL;DR

This work scrutinizes how target mass corrections (TMCs) are incorporated into parton distributions within the operator product expansion at low . It argues that the conventional Nachtmann-variable-based inversion introduces energy–momentum conservation issues and undermines PDF universality at finite . To address this, the authors develop a series expansion of inverted moments in that avoids , expressing all unpolarized structure functions as derivative operators acting on PDFs, and they demonstrate convergence properties for representative kinematics. The approach preserves moment consistency, improves the reliability of TMCs down to GeV for moderate , and offers a practical path for including low- data in global PDF analyses, with caveats near the elastic limit and potential extension to spin structure functions in future work.

Abstract

We study the consistency of parton distribution functions in the presence of target mass corrections (TMCs) at low Q^2. We review the standard operator product expansion derivation of TMCs in both x and moment space, and present the results in closed form for all unpolarized structure functions and their moments. To avoid the unphysical region at x > 1 in the standard TMC analysis, we propose an expansion of the target mass corrected structure functions order by order in M^2/Q^2, and assess the convergence properties of the resulting forms numerically.

Paper Structure

This paper contains 9 sections, 49 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Target mass corrections in the OPE
  3. Moments of structure functions
  4. Parton distributions with TMCs
  5. Consistency of the standard TMCs?
  6. Series expansion of inverted moments
  7. Conclusion
  8. Derivation of structure function moments
  9. Structure function inversion

Figures (4)

  • Figure 1: $n=2$ moments of the $F_2$ structure function, illustrating the convergence of the series in Eq. (\ref{['eq:M2']}) for $j=0$ (dotted), $j<2$ (dot-dash-dashed), $j<3$ (dot-dashed) and $j<4$ (dashed), compared with the standard TMC result from GP GP76 using Eq. (\ref{['eq:F2']}) with the upper limit of integration $\xi_{\rm max}=\xi_0$ (\ref{['eq:M2GP']}) (dot-dot-dashed) and $\xi_{\rm max}=1$ (solid).
  • Figure 2: Target mass corrected $F_2$ structure function at $Q^2=1$ GeV$^2$ from Eq. (\ref{['eq:F2ser']}), showing the convergence with increasing $j$, and compared with the standard TMC result from GP GP76 using Eq. (\ref{['eq:F2']}), shown as a function of (a) Bjorken-$x$, and (b) the hadronic final state mass $W$. The arrows in (a) indicate the locations of the resonance region ($W=2$ GeV) and the $\Delta$ resonance, while the arrow in (b) denotes the elastic limit, $W=M$.
  • Figure 3: Ratio of target mass corrected $F_2$ structure function at $Q^2=5$ GeV$^2$ from Eq. (\ref{['eq:F2ser']}) for various $j$ ($j=0$ up to $j<5$) to the standard TMC result from GP GP76 using Eq. (\ref{['eq:F2']}), shown as a function of (a) Bjorken-$x$, and (b) the hadronic final state mass $W$. The arrows in (a) indicate the locations of the resonance region ($W=2$ GeV) and the $\Delta$ resonance.
  • Figure 4: Convergence of the series expansion for the target mass corrected $F_2$ structure function for large values of $j$ ($j<101, 102, 201, 202, 301$ and 302), at $Q^2 = 2$ GeV$^2$. The arrows indicate the values of $x$ at which the final state mass corresponds to the $\Delta$ resonance and to $W = 1.25$ GeV. Note the limited $x$ range on the ordinate.