NLO QCD method of the polarized semiinclusive DIS data analysis

Method of polarized semi-inclusive deep inelastic scattering (SIDIS) data analysis in the next to leading order (NLO) QCD is developed. Within the method one first directly extracts in NLO few first truncated (available to measurement) Mellin moments of the quark helicity distributions. Second, using these moments as an input to the proposed modification of the Jacobi polynomial expansion method (MJEM), one eventually reconstructs the local quark helicity distributions themselves. All numerical tests demonstrate that MJEM allows us to reproduce with the high precision the input local distributions even inside the narrow Bjorken $x$ region accessible for experiment. It is of importance that only four first input moments are sufficient to achieve a good quality of reconstruction. The application of the method to the simulated SIDIS data on the pion production is considered. The obtained results encourage one that the proposed NLO method can be successfully applied to the SIDIS data analysis. The analysis of HERMES data on pion production is performed. To this end the pion difference asymmetries are constructed from the measured by HERMES standard semi-inclusive spin asymmetries. The LO results of the valence distribution reconstruction are in a good accordance with the respective leading order SMC and HERMES results, while the NLO results are in agreement with the existing NLO parametrizations on these quantities.

Figure 1

Results of $\Delta u_V(x)$ (${\alpha }_{\mathrm{opt}}=8.189221$, ${\beta }_{\mathrm{opt}}=-0.99000$) and $\Delta d_V(x)$ (${\alpha }_{\mathrm{opt}}=-0.99000$, ${\beta }_{\mathrm{opt}}=-0.387196$) reconstruction with the usual JEM. Solid line corresponds to the input (reference) parametrization GRSV2000NLO (symmetric sea). Dotted line corresponds to the distribution reconstructed with JEM for $N_{\max }=12$.

Figure 2

Results of $\Delta u_V(x)$ (${\alpha }_{\mathrm{opt}}=-0.827885$, ${\beta }_{\mathrm{opt}}=-0.011505$) and $\Delta d_V(x)$ (${\alpha }_{\mathrm{opt}}=-0.989752$, ${\beta }_{\mathrm{opt}}=-0.012393$) reconstruction with MJEM. Solid line corresponds to the input (reference) parametrization GRSV2000NLO (symmetric sea). Dotted line corresponds to the distribution reconstructed with MJEM for $N_{\max }=12$.

Figure 3

Results of the valence PDFs reconstruction with JEM and MJEM in comparison. Top part corresponds to $\Delta u_V(x)$ (${\alpha }_{\mathrm{opt}}=-0.99$, ${\beta }_{\mathrm{opt}}=0.054010$) and $\Delta d_V(x)$ (${\alpha }_{\mathrm{opt}}=0.174096$, ${\beta }_{\mathrm{opt}}=0.162567$) reconstructed with the usual JEM. Bottom part corresponds to $\Delta u_V(x)$ (${\alpha }_{\mathrm{opt}}=-0.0025869$, ${\beta }_{\mathrm{opt}}=-0.071591$) and $\Delta d_V(x)$ (${\alpha }_{\mathrm{opt}}=0.110331$, ${\beta }_{\mathrm{opt}}=-0.049255$) reconstructed with MJEM. Solid lines correspond to input (reference) parametrization GRSV2000NLO (symmetric sea). Dotted lines correspond to the distributions reconstructed with JEM (top) and MJEM (bottom).

Figure 4

Results of $\Delta u_V$ and $\Delta d_V$ reconstruction for GRSV2000NLO parametrization for both symmetric (top) and broken sea (bottom) scenarios. Solid line corresponds to the reference curve (input parametrization). Dotted line is reconstructed with MJEM and criterion (8) inside the accessible for measurement region ([0.023,0.6] here). Optimal values of parameters for symmetric sea scenario for $\Delta u_V$ are ${\alpha }_{\mathrm{opt}}=-0.15555$, ${\beta }_{\mathrm{opt}}=-0.097951$ and for $\Delta d_V$ are ${\alpha }_{\mathrm{opt}}=-0.002750$, ${\beta }_{\mathrm{opt}}=-0.07190$. Optimal values of parameters for broken sea scenario for $\Delta u_V$ are ${\alpha }_{\mathrm{opt}}=-0.209346$, ${\beta }_{\mathrm{opt}}=0.153417$ and for $\Delta d_V$ are ${\alpha }_{\mathrm{opt}}=0.702699$, ${\beta }_{\mathrm{opt}}=-0.293231$.

Figure 5

Idealized LO testing of the integration procedures given by Eq. (14, 16). The input parametrizations GRSV2000LO with symmetric sea scenario (top) and broken sea scenario (bottom) are used for direct calculation of asymmetries with Eq. (18). Solid line corresponds to the input parametrization. Closed circles correspond to the values of input parametrization in the middle of each bin. Broken line shows the way of integral approximation corresponding to application of Eq. (14). Dashed line is obtained with MJEM and application of Eq. (16) for the moment calculation. Dot-dashed line is obtained with MJEM and application of Eq. (14) for the moment calculation. Parameters ${\nu }_1$ and ${\nu }_2$ are given by Eq. (5) and show the quality of reconstruction for the dashed and dot-dashed lines, respectively.

Figure 6

Results of the simulated difference asymmetry analysis in LO. GRSV2000LO parametrizations for symmetric sea (top) and broken sea (bottom) scenarios are used for simulations. Solid line corresponds to the input parametrization. Dashed line corresponds to the reconstructed with MJEM curve. Points with error bars correspond to direct extraction of the valence distributions with Eq. (18).

Figure 7

Results of the simulated difference asymmetry analysis in NLO. GRSV2000NLO parametrizations for symmetric sea (top) and broken sea (bottom) scenarios are used for simulations. Solid line corresponds to the input parametrization. Dashed line corresponds to the reconstructed with MJEM curve.

Figure 8

Combined results of LO and NLO analysis (top) of the simulated difference asymmetries in comparison with the respective LO and NLO versions of GRSV2000 (symmetric sea) parametrization (bottom). Dashed and solid lines correspond to LO and NLO curves, respectively.

Figure 9

Combined results of LO and NLO analysis (top) of the simulated difference asymmetries in comparison with the respective LO and NLO versions of GRSV2000 (broken sea) parametrization (bottom). Dashed and solid lines correspond to LO and NLO curves, respectively.

Figure 10

Difference asymmetries constructed with application of Eq. (20).

Figure 11

LO extraction of the valence PDFs from the difference asymmetries constructed with Eq. (20) (up-oriented triangles) in comparison with the respective published HERMES results (down-oriented triangles). The HERMES results are shifted to the right for better visibility.

Figure 12

Different LO procedures of the valence PDFs extraction in comparison. Solid line and up-oriented triangles correspond to LO extraction with the proposed method and direct extraction with Eq. (18), respectively. The difference asymmetries constructed with application of Eq. (20) are used. Down-oriented triangles correspond to LO results of HERMES obtained with application of the purity method to the measured by HERMES usual virtual photon spin asymmetries.

Figure 13

Results of both LO and NLO analysis of the difference asymmetries constructed with Eq. (20). Solid and dashed lines correspond to NLO and LO results, respectively.

Figure 14

NLO results obtained with application of MJEM to the moments extracted from the difference asymmetries corrected due to evolution. Dotted lines correspond to corrections estimated with the different parametrizations. Dashed line corresponds to the curve averaged over corrections. Solid line corresponds to reconstruction without corrections.