Study of HERA $ep$ data at low $Q^2$ and low $x_{\mathrm{Bj}}$ and the need for higher-twist corrections to standard perturbative QCD fits

A detailed comparison of HERA data at low Bjorken-$x$ and low four-momentum-transfer squared, $Q^2$, with predictions based on $\ln Q^2$ evolution (DGLAP) in perturbative quantum chromodynamics suggests inadequacies of this framework. The standard DGLAP evolution was augmented by including an additional higher-twist term in the description of the longitudinal structure function, $F_{\mathrm{L}}$. This additional term, $F_{\mathrm{L}}{A}_{\mathrm{L}}^{\mathrm{HT}}/Q^2$, improves the description of the reduced cross sections significantly. The resulting predictions for $F_{\mathrm{L}}$ suggest that further corrections are required for $Q^2$ less than about $2{\mathrm{GeV}}^2$.

Figure 1

The HHT parton distribution functions, $x{u}_v$, $x{d}_v$, $xS$ and $xg$, at the scale ${\mu }_{\mathrm{f}}^2=10{\mathrm{GeV}}^2$ compared to the PDFs from HERAPDF2.0 at NNLO (top) and NLO (bottom). The gluon and sea distributions are scaled down by a factor 20. The bands represent the experimental, i.e. fit, uncertainties.

Figure 2

The ${\chi }^2/\mathrm{ndof}$ versus $Q_{\min }^2$ for HHT and HERAPDF2.0 fits at NNLO and NLO.

Figure 3

The ${\chi }^2/\mathrm{ndof}$ versus $Q_{\min }^2$ for HHT and HERAPDF2.0 fits at NNLO and NLO using the FONLL heavy-flavor scheme instead of the default RTOPT scheme.

Figure 4

The predictions of HHT NNLO (top) and HERAPDF2.0 NNLO (bottom), both with $Q_{\min }^2=3.5{\mathrm{GeV}}^2$, compared to the HERA measurements of ${\sigma }_r$. The bands represent the experimental, i.e. fit, uncertainties. Extrapolations are indicated as dotted lines.

Figure 5

The predictions of HHT NLO (top) and HERAPDF2.0 NLO (bottom), both with $Q_{\min }^2=3.5{\mathrm{GeV}}^2$, compared to the HERA measurements of ${\sigma }_r$. The bands represent the experimental, i.e. fit, uncertainties. Extrapolations are indicated as dotted lines.

Figure 6

The predictions of HHT NNLO (top) and HHT NLO (bottom) with $Q_{\min }^2=2.0{\mathrm{GeV}}^2$ compared to the HERA measurements of ${\sigma }_r$. The bands represent the experimental, i.e. fit, uncertainties.

Figure 7

The extrapolated predictions of HHT NNLO with $Q_{\min }^2=3.5{\mathrm{GeV}}^2$ (top) and with $Q_{\min }^2=2.0{\mathrm{GeV}}^2$ (bottom) compared to the HERA NC $e^{+}p$ measurements of ${\sigma }_r$ at $Q^2$ of 1.2 and $1.5{\mathrm{GeV}}^2$.

Figure 8

The predictions of HHT NNLO (top) and HERAPDF2.0 NNLO (bottom), both with $Q_{\min }^2=3.5{\mathrm{GeV}}^2$, compared to extracted values $F_2^{\text{extracted}}$. The bands represent the experimental, i.e. fit, uncertainties. Extrapolations are indicated as dotted lines.

Figure 9

The predictions of HHT NLO (top) and HERAPDF2.0 NLO (bottom) both with $Q_{\min }^2=3.5{\mathrm{GeV}}^2$, compared to extracted values $F_2^{\text{extracted}}$. The bands represent the experimental, i.e. fit, uncertainties. Extrapolations are indicated as dotted lines.

Figure 10

The predictions of HHT NNLO (top) and HHT NLO (bottom) with $Q_{\min }^2=2.0{\mathrm{GeV}}^2$ compared to extracted values $F_2^{\text{extracted}}$. The bands represent the experimental, i.e. fit, uncertainties.

Figure 11

The predictions for $F_{\mathrm{L}}$ from HHT and HERAPDF2.0, both with $Q_{\min }^2=3.5{\mathrm{GeV}}^2$, compared to the separate direct measurements published by the H1 and ZEUS collaborations. The bands represent the experimental, i.e. fit, uncertainties on the predictions. Hatched bands represent extrapolations.

Figure 12

The $F_2^{\text{extracted}}$ and $F_{\mathrm{L}}^{\text{extracted}}$ values as extracted from HHT NNLO and HERAPDF2.0 NNLO together with the corresponding predictions from HHT NNLO and HERAPDF2.0 NNLO for three selected values of $W$. Also shown are predictions from the GBW model.

Figure 13

The $F_2^{\text{extracted}}$ and $F_{\mathrm{L}}^{\text{extracted}}$ values as extracted from HHT NLO and HERAPDF2.0 NLO together with the corresponding predictions from HHT NLO and HERAPDF2.0 NLO for three selected values of $W$. Also shown are predictions from the GBW model.

Figure 14

The predictions from HHT and HERAPDF2.0 at NNLO and NLO for $W=232\mathrm{GeV}$, together with direct measurements of $F_{\mathrm{L}}$ published by the H1 Collaboration for $W$ between 230 and 234 GeV. Also shown is the prediction from the GBW model.