Cut moments approach in the analysis of DIS data

We review the main results on the generalization of the DGLAP evolution equations within the cut Mellin moments (CMM) approach, which allows one to overcome the problem of kinematic constraints in Bjorken $x$. CMM obtained by multiple integrations as well as multiple differentiations of the original parton distribution also satisfy the DGLAP equations with the simply transformed evolution kernel. The CMM approach provides novel tools to test QCD; here we present one of them. Using appropriate classes of CMM, we construct the generalized Bjorken sum rule that allows us to determine the Bjorken sum rule value from the experimental data in a restricted kinematic range of $x$. We apply our analysis to COMPASS data on the spin structure function $g_1$.

Figure 1

${\mathrm{\Gamma }}_{1;\omega }(x_0)$, Eq. (13), for different values of $A$ and the truncated BSR ${\mathrm{\Gamma }}_1(x_0)$, Eq. (8) (thick black curve) as a function of $x_0$. Input parametrization, Eq. (24), with $a=0$.

Figure 2

${\mathrm{\Gamma }}_{1;\omega }(x_0)$, Eq. (13), for different values of $A$ and the truncated BSR ${\mathrm{\Gamma }}_1(x_0)$, Eq. (8) (thick black curve) as a function of $x_0$. Input parametrization, Eq. (24), with $a=-0.4$.

Figure 3

The percent errors ${\epsilon }^{\mathrm{I}}(x_{\min },r)$, Eq. (30), for different $r:0.9,0.5,0.3$, together with ${\epsilon }^{\mathrm{IBSR}}(x_{\min })$, Eq. (31), as a function of $x_{\min }$. Small-$x$ behavior of $g_1$ with $a=-0.1$, Eq. (24).

Figure 4

The percent errors ${\epsilon }^{\mathrm{I}}(x_{\min },r)$, Eq. (30), for different $r:0.9,0.5,0.3$, together with ${\epsilon }^{\mathrm{IBSR}}(x_{\min })$, Eq. (31), as a function of $x_{\min }$. Small-$x$ behavior of $g_1$ with $a=-0.4$, Eq. (24).

Figure 5

${\mathrm{\Gamma }}_{1,\omega }(x,r,Q^2)$, Eq. (25), for $A(x_{\min }=0.0036,r)$, Eq. (28) for three values of $r:0.9,0.5,0.3$, together with the truncated Bjorken sum rule ${\mathrm{\Gamma }}_1(x,Q^2)$, Eq. (8), as a function of $x$. The results are based on our fit to the COMPASS data.

Figure 6

The percent errors ${\epsilon }^{\mathrm{I}}(r)$, Eq. (30), for $x_{\min }=0.0036$, as a function of $r$. The results are based on our fit to the COMPASS data.

Figure 7

${\mathrm{\Gamma }}_1(Q^2)$, Eq. (32), incorporating ${\mathrm{N}}^2\mathrm{LO}$ and HT corrections, ${\mu }_4^{p-n}/M^2=-0.047$, ${\mathrm{\Lambda }}_{\mathrm{qcd}}=311\mathrm{MeV}$, together with JLAB data [6]: black point, the single red point higher is the COMPASS result $0.181\pm 0.008\mathrm{stat}\pm 0.014\mathrm{syst}$ [18], $M$ is the nucleon mass.

Figure 8

Contour plot of $1\sigma$ error ellipse for ${\mathrm{\Lambda }}_{\mathrm{qcd}}^2$ and ${\mu }_4^{p-n}$, at the central point ${\chi }_{\mathrm{ndf}}^2=0.60$. The upper band (yellow strip) represents the Jlab result ${\mu }_4^{p-n}/M^2=-0.021\pm 0.016$ [6] and the lower band (blue strip) is a typical theoretical estimation, ${\mu }_4^{p-n}/M^2=-0.05\pm 0.02$ [25].