A lattice determination of moments of unpolarised nucleon structure functions using improved Wilson fermions

TL;DR

This study performs a high-statistics lattice QCD analysis of low moments $v_n$ ($n=2,3,4$) of unpolarised nucleon structure functions in the quenched approximation using $O(a)$-improved Wilson fermions and non-perturbative renormalisation. By employing carefully chosen lattice operators, addressing operator mixing and $O(a)$ improvement, and comparing perturbative, TRB, and non-perturbative renormalisation constants, the work delivers continuum-limit results for the moments in the $\overline{MS}$ scheme at $\mu=2$ GeV, including RG-invariant forms. The lattice results show good agreement with phenomenology for the lowest moment in the RGI frame but significant discrepancies remain for $v_2^{\overline{MS}}$ and especially for $v_4$, highlighting potential effects from quenching and heavier-than-physical quark masses. The study demonstrates the methodological viability of combining lattice QCD with RG-improved, non-perturbative renormalisation to access nucleon structure, while underscoring the need for unquenched simulations and lighter quark masses to reconcile with experimental data and global fits.

Abstract

Within the framework of quenched lattice QCD and using O(a) improved Wilson fermions and non-perturbative renormalisation, a high statistics computation of low moments of the unpolarised nucleon structure functions is given. Particular attention is paid to the chiral and continuum extrapolations.

Figures (22)

• Figure 1: The one, two and three loop results for $[\Delta Z^{\hbox{\tiny $\overline{MS}$}}_{v_2}(\mu)]^{-1}$ and $[\Delta Z^{\hbox{\tiny $\overline{MS}$}}_{v_4}(\mu)]^{-1}$ for quenched QCD versus $\mu/\Lambda^{\hbox{\tiny $\overline{MS}$}}$.
• Figure 2: The one and two loop results for $E_{F_2;v_n}^{\hbox{\tiny $\overline{MS}$}}$, $n=2$, $4$ for quenched QCD versus $Q/\Lambda^{\hbox{\tiny $\overline{MS}$}}$. The two loop results at $2\,\hbox{GeV}$ are 1.011(1), 1.130(3) for $n = 2$, $4$ respectively where the error is a reflection of the error in $\Lambda^{\hbox{\tiny $\overline{MS}$}} r_0$.
• Figure 3: World experimental data for $F_2^{\gamma;\hbox{\tiny $N\! S$}}(x,Q_0^2)$, bruell95a, at $Q_0^2=4\,\hbox{GeV}^2$ in the form of bins, plotted against $x$ using a linear scale. Errors in the bins are also shown. The dot-dashed line is a rough estimate, obtained by a linear extrapolation of the last bin to nought (at $x=1$).
• Figure 4: The $3$-point quark correlation function for a baryon.
• Figure 5: $v_{2b}$ versus $\tau/a$ from the ratios $R(17a,\tau;\vec{0};{\cal O}_{v_{2b}})$, $R(17a,\tau;\vec{p}_1;{\cal O}_{v_{2b}})$, eq. (\ref{['R_def_practical']}) left and middle pictures respectively and $v_{2a}$ from $R(17a,\tau;\vec{p}_1;{\cal O}_{v_{2a}})$, right picture for ${\cal O}^{(u)}$, ${\cal O}^{(d)}$ (empty circles) and NS (ie ${\cal O}^{(u)}-{\cal O}^{(d)}$) (filled circles) for $\beta = 6.2$ at $\kappa = 0.1344$. The chosen fit intervals are denoted by vertical dotted lines.