Measurement of the Inclusive e{\pm}p Scattering Cross Section at High Inelasticity y and of the Structure Function FL

A measurement is presented of the inclusive neutral current e\pm p scattering cross section using data collected by the H1 experiment at HERA during the years 2003 to 2007 with proton beam energies Ep of 920, 575, and 460 GeV. The kinematic range of the measurement covers low absolute four-momentum transfers squared, 1.5 GeV2<Q2<120 GeV2, small values of Bjorken x, 2.9 \cdot 10-5<x<0.01, and extends to high inelasticity up to y = 0.85. The structure function FL is measured by combining the new results with previously published H1 data at Ep = 920 GeV and Ep = 820 GeV. The new measurements are used to test several phenomenological and QCD models applicable in this low Q2 and low x kinematic domain.


Introduction
The neutral current double differential ep scattering cross section at low absolute four momentum transfer squared Q 2 in a reduced form can be represented by two structure functions: where f (y) = y 2 /(1+(1−y) 2 ). Here x is the Bjorken scaling variable, and the inelasticity y is related with x, Q 2 and the centre-of-mass energy squared s as y = Q 2 /(sx).
Due to the kinematic factor f (y) and the relation 0 ≤ F L ≤ F 2 , the F L term contributes significantly to the cross section only for y > 0.5. In the quark-parton model, F 2 is given by the charge squared weighted sum of the quark densities while F L is equal zero. In QCD, the gluon emission gives rise to a non-vanishing structure function F L . Therefore, measuring F L in the region of deep inelastic scattering (DIS) provides information about the gluon density and tests perturbative QCD as well.
2 Measurement of σ r at high inelasticity y One of the main aimes of the measurements presented here is to reach as high y as possible to increase the sensitivity to the structure function F L . At low Q 2 it means to perform a measurement at low energies of the scattered electron E ′ e which is difficult because of the high background of hadrons from photoproduction. Therefore, to reduce the systematic uncertainty, a special background determination procedure was developed which relies on the charge of the scattered electron candidate in data [1]. The method allows a reliable background estimation.
The cross-section measurements with different proton beam energies E p are shown in Fig. 1. Both runs at reduced E p extend the H1 cross section measurements at high inelasticity y (up to y = 0.85) down to Q 2 = 1.5 GeV 2 . Compared to previous H1 measurements [2,3,4] the precision of the new data is signicantly better in the high y region. Moreover combining the H1 data at E p = 920 GeV leads to an improvement in precision at high y by a factor of two.
3 Measurement of the structure function F L To determine the two structure functions F 2 (x, Q 2 ) and F L (x, Q 2 ) from the reduced cross section shown in Eq. (1) it is necessary to perform measurements at the same values of x and Q 2 but at different y. This was achieved at HERA by reducing the proton beam energy to E p = 460 GeV and E p = 575 GeV. The run at E p = 460 GeV gives the highest sensitivity to F L while the run at E p = 575 GeV extends the kinematic range of the measurement and provides a cross check. Compared to the previous H1 publication on F L [5] we apply an improved F L determination procedure which takes into account correlations due to systematic uncertainties. The measured structure functions F 2 and F L are shown in Fig. 2 and compared to a recent result from the ZEUS [6] Collaboration. The measurement spans over two decades in x at low 0.00002 < x < 0.002 and it is well described by a NLO DGLAP fit in the ACOT scheme [7]. We also observe a good agreement between the H1 and ZEUS data in the regions of overlap.

H1 Collaboration
The values of F L (x; Q 2 ) resulting from averages over x at fixed Q 2 are shown in Fig. 3. The average is performed taking into account correlations. The measured structure function F L is compared with theoretical predictions based on HERAPDF1.0 [8], CT10 [9], NNPDF2.1 [10,11], MSTW08 [12], GJR08 [13,14] and ABKM09 [15] sets. Depending on the PDF set, the calculations are performed at NLO or NNLO in perturbative QCD. x Figure 2: The proton structure functions F 2 (x; Q 2 ) and F L (x; Q 2 ) measured by the H1 [1] and ZEUS [6] Collaborations. The inner error bars represent the statistical error, the full error bars include the statistical and systematic uncertainties added in quadrature. The curves represent predictions of the NLO DGLAP fit in the ACOT scheme [7].
Within the uncertainties all predictions describe the data reasonably well. At low x and Q 2 , where F L is measured for the first time, some difference between the predictions is observed. The measurement of the structure functions F 2 and F L can be used to determine the ratio R = F L /(F 2 − F L ). This ratio is shown in Fig. 4. Also results from other experiments are shown [6,[16][17][18][19]. Within the range of this measurement the ratio R is consistent with a constant behaviour with R = 0.26 ± 0.05.

Phenomenological analysis
The combined cross-section data for E p = 460, 575, 820 and 920 GeV are used for several phenomenological analyses. We have tested two QCD fits with different schemes for the treatment of heavy quarks and for the F L structure function -ACOT [  RT [20], [21]. The ACOT fit is found to agree better with the data, having a χ 2 /dof = 715/781 compared to the RT fit with χ 2 /dof = 765/781. Increasing the cut on the minimum Q 2 min of the data used in the fit from 1.5 to 7.5 GeV 2 improves the quality of the fits. Such a change in Q 2 min leads to an increase of the gluon distribution while the sea-quark distribution becomes smaller at low x, see Fig. 5.
The rise of the structure function F 2 towards low x has previously been described by a power law in x, F 2 = c(Q 2 )x −λ(Q 2 ) [22]. This simple parameterisation has been shown to model the ep data well for x < 0.01. This idea is extended to fit the reduced cross section σ r in different Q 2 bins in order to simultaneously extract λ(Q 2 ) and c(Q 2 ). The parameter λ exhibits an approximately linear increase as a function of ln Q 2 for Q 2 ≥ 2 GeV 2 . For lower Q 2 , the variation of λ deviates from that linear dependence. The normalisation coefficient c(Q 2 ) rises with increasing Q 2 for Q 2 < 2 GeV 2 and is consistent with a constant behaviour for higher Q 2 , as in [22]. The quality of the fit is poor with the offset method for systematic error estimation: χ 2 /dof = 538/350. Therefore the parameterisation of F 2 was extended by one parameter to allow for deviations from a simple power law: F 2 = c(Q 2 )x −λ(Q 2 )+λ ′ (Q 2 ) ln x . This fit returns a signicantly improved χ 2 /dof = 405/326. Taking into account the strong correlation between λ and λ ′ and also the fact that λ is consistent with a constant value of λ = 0.25, we performed a fit where λ is fixed to a value of 0.25 and two parameters are free: λ ′ (Q 2 ) and c(Q 2 ). The quality of this fit with a total χ 2 /dof = 464/350 is better compared to the original λ fit. The results are shown in Fig. 6.
At low x and low Q 2 , virtual photon-proton scattering can be described using the colour dipole model (CDM) [23]. In this model the initial photon splits into quarkantiquark pair (dipole) which interacts with the proton. The dipole approach is ap- plicable only for x < 0.01 where the gluon and the sea dominate since it neglects the valence contributions to the ep cross section. Hovewer these contributions can be sizable (up to 15%) also at low x. Therefore we tested three different dipole models (GBW [24], IIM [25] and B-SAT [26]) with and without a DGLAP based correction for the valence quark contributions. The fits were done in the range 3.5 ≤ Q 2 ≤ 150 GeV 2 and x < 0.01 where both CDM and DGLAP are expected to work. It was found that the addition of the valence contribution improves the description of the data at high x, but overall fit quality is not improving. From the models considered here the best description of the data is observed for the ACOT fit (χ 2 /dof = 248.3/249) which is closely followed by the pure dipole IIM fit (χ 2 /dof = 259.4/252). A comparison of the F L predictions from different models to the H1 data is illustrated in Fig. 7. The data are reasonably well described by all models, except the low Q 2 region where the RT fit falls below data. λ' λ' λ' λ' λ' λ' λ' λ' λ' λ' λ' λ' λ' λ' λ' λ' λ' λ' λ' λ' λ' λ' λ' λ b) Figure 6: Coefficients c and λ ′ determined from a fit to the F 2 data of the form F 2 = c(Q 2 )x −λ(Q 2 )+λ ′ (Q 2 ) ln x as a function of Q 2 with fixed λ = 0.25. The inner error bars represent the statistical uncertainties. The outer error bars contain the statistical and systematic uncertainties added in quadrature.