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

Within the framework of quenched lattice QCD and using $O(a)$ improved Wilson fermions and nonperturbative renormalization, a high statistics computation of low moments of the unpolarized nucleon structure functions is given. Particular attention is paid to the chiral and continuum extrapolations.

Figure 1

The one-, two- and three-loop results for $[\Delta Z_{v_2}^{\overline{\mathrm{MS}}}(\mu ){]}^{-1}$ and $[\Delta Z_{v_4}^{\overline{\mathrm{MS}}}(\mu ){]}^{-1}$ for quenched QCD versus $\mu /{\Lambda }^{\overline{\mathrm{MS}}}$.

Figure 2

The one- and two-loop results for $E_{F_2;v_n}^{\overline{\mathrm{MS}}}$, $n=2,4$ for quenched QCD versus $Q/{\Lambda }^{\overline{\mathrm{MS}}}$. The two-loop results at $2\mathrm{GeV}$ are 1.011(1), 1.130(3) for $n=2,4$ respectively where the error is a reflection of the error in ${\Lambda }^{\overline{\mathrm{MS}}}{r}_0$.

Figure 3

World experimental data for $F_2^{\gamma ;\mathrm{NS}}(x,Q_0^2)$ [33] at $Q_0^2=4{\mathrm{GeV}}^2$ 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 $R(17a,\tau ;\overrightarrow{0};{\mathcal{O}}_{v_{2b}})$, $R(17a,\tau ;{\overrightarrow{p}}_1;{\mathcal{O}}_{v_{2b}})$, Eq. (39) left and middle pictures, respectively, and $v_{2a}$ from $R(17a,\tau ;{\overrightarrow{p}}_1;{\mathcal{O}}_{v_{2a}})$, right picture for ${\mathcal{O}}^{(u)}$, ${\mathcal{O}}^{(d)}$ (open circles) and NS (i.e. ${\mathcal{O}}^{(u)}-{\mathcal{O}}^{(d)}$) (solid circles) for $\beta =6.2$ at $\kappa =0.1344$. The chosen fit intervals are denoted by vertical dotted lines.

Figure 6

The ratios $a{r}_{v_{2a}}^1$, $a{r}_{v_{2a}}^2$ plotted against $(r_0{m}_{ps}{)}^2$. The solid symbols at nonzero $(r_0{m}_{ps}{)}^2$ are for $a{r}_{v_{2a}}^1$, while the open symbols denote $a{r}_{v_{2a}}^2$. Also shown is a linear chiral extrapolation (dashed line and solid line for $a{r}_{v_{2a}}^1$, $a{r}_{v_{2a}}^2$ respectively). The result in the chiral limit $(r_0{m}_{ps}{)}^2\equiv 0$ is also indicated, again using solid, open symbols for $a{r}_{v_{2a}}^1$, $a{r}_{v_{2a}}^2$ respectively. The small horizontal lines around the chiral limit represent the perturbative estimate, see Sec. 5, with again a dashed line corresponding to $a{r}_{v_{2a}}^1$ and a solid line to $a{r}_{v_{2a}}^2$ respectively. The rough values of the (hypothetical) pseudoscalar mass composed from $u/d$ or $s$ quark masses, evaluated from $m_{\pi }$ and $m_K$ respectively are shown as dashed vertical lines.

Figure 7

$a{r}_{v_{2b}}^1$, $a{r}_{v_{2b}}^2$ versus $(r_0{m}_{ps}{)}^2$ for $\beta =6.0$, $6.2$ and $6.4$ with $\overrightarrow{p}=\overrightarrow{0}$. The notation is the same as for Fig. 6.

Figure 8

Mixing terms, $v_3^{m_1}$, $v_3^{m_2}$ for $\beta =6.0$ (solid circles). Also shown is $v_3$ (open circles) and linear chiral extrapolations. The same notation as for Fig. 6.

Figure 9

$v_4$ for $\beta =6.0$ together with $v_4^{m_1}$, $v_4^{m_2}$, $v_4^{m_3}/a$. The notation is the same as for Fig. 6.

Figure 10

$Z_{v_{2b}}^{\mathrm{RGI}}$ at $\beta =6.2$ versus $(r_0{\mu }_p{)}^2$. NP method I is denoted by circles, while method II is given by solid squares. The dashed line is the TRB-PT result, while the solid line is the NP estimate from method II.

Figure 11

$Z_{v_3}^{\mathrm{RGI}}$ at $\beta =6.2$ versus $(r_0{\mu }_p{)}^2$. The same notation as for Fig. 10.

Figure 12

$Z_{v_4}^{\mathrm{RGI}}$ at $\beta =6.2$ versus $(r_0{\mu }_p{)}^2$. The same notation as for Fig. 10.

Figure 13

$Z_{v_{2b}}^{\mathrm{RGI}}$ versus $(r_0{\mu }_p{)}^2$ for $\beta =6.0$ (circles), $6.2$ (solid squares), $6.4$ (diamonds) using method II. The corresponding dashed lines are the TRB-PT results, while the solid lines are the NP estimates.

Figure 14

$v_{2b}^{\mathrm{RGI}}$ versus $(r_0{m}_{ps}{)}^2$ for $\beta =6.0$ (circles), $6.2$ (squares) and $6.4$ (diamonds). The solid symbols are obtained when using the TI value for $c_0$ while the open symbols have $c_0=0$. Also shown are linear extrapolations to the chiral limit (dashed lines). Other notation as in Fig. 6.

Figure 15

$v_{2a}^{\mathrm{RGI}}$ versus $(r_0{m}_{ps}{)}^2$ for $\beta =6.0$ (circles), $6.2$ (squares) and $6.4$ (diamonds). Also shown are linear extrapolations to the chiral limit (dashed lines). Other notation as in Fig. 6.

Figure 16

$v_3^{\mathrm{RGI}}$ versus $(r_0{m}_{ps}{)}^2$ for $\beta =6.0$ (circles), $6.2$ (squares) and $6.4$ (diamonds). Also shown are linear extrapolations to the chiral limit (dashed lines). Other notation as in Fig. 6.

Figure 17

$v_4^{\mathrm{RGI}}$ versus $(r_0{m}_{ps}{)}^2$ for $\beta =6.0$ (circles), $6.2$ (squares) and $6.4$ (diamonds). Also shown are linear extrapolations to the chiral limit (dashed lines). Other notation as in Fig. 6.

Figure 18

$v_{2a}^{\mathrm{RGI}}$ and $v_{2b}^{\mathrm{RGI}}$ (both for $\overrightarrow{p}={\overrightarrow{p}}_1$ and $\overrightarrow{0}$) versus $(a/r_0{)}^2$, using the results at $\beta =6.0$, $6.2$, $6.4$ (solid circles). A linear continuum extrapolation in $a^2$ is also given (dashed line and open circle). The star is the MRST value given in Table 1.

Figure 19

$v_3^{\mathrm{RGI}}$ and $v_4^{\mathrm{RGI}}$ versus $(a/r_0{)}^2$, using the results at $\beta =6.0$, $6.2$, $6.4$. A continuum extrapolation is also given. The same notation as for Fig. 18.

Figure 20

$v_{2b}^{\mathrm{RGI}}-c_2(a/r_0{)}^2-d_{2}a{r}_0{m}_{ps}^2\equiv F_{\chi }^{(2)}=a_2(r_0{m}_{ps}{)}^2+b_2$ [i.e. using the chiral function of Eq. (76)] versus $(r_0{m}_{ps}{)}^2$, dashed line. Filled circles, squares and diamonds represent $\beta =6.0$, $6.2$ and $6.4$ respectively. The open square represents the chirally extrapolated value. Again the MRST phenomenological value of $v_2^{\mathrm{RGI}}$ is represented by a star.

Figure 21

$v_{2b}^{\mathrm{RGI}}-c_2(a/r_0{)}^2-d_{2}a{r}_0{m}_{ps}^2\equiv F_{\chi }^{(2)}$ [using the chiral function of Eq. (80)] versus $(r_0{m}_{ps}{)}^2$, dashed line. The chiral limit was fixed to the MRST phenomenological value. Other notation is the same as in Fig. 20.

Figure 22

Relativistic (open circles) and nonrelativistic (solid circles) two-point operators, Eq. (c4) with $\overrightarrow{p}=\overrightarrow{0}$, for $\beta =6.2$ at $\kappa =0.1344$.