Deep inelastic inclusive e p scattering at low x and a determination of alpha(s)

A precise measurement of the inclusive deep-inelastic \(e^+ p\) scattering cross section is reported in the kinematic range \(1.5 \le Q^2 \le 150\) GeV\(^2\) and \( 3 \cdot 10^{-5} \le x \le 0.2\). The data were recorded with the H1 detector at HERA in 1996 and 1997, and correspond to an integrated luminosity of 20 pb\(^{-1}\). The double differential cross section, from which the proton structure function \(\) and the longitudinal structure function \(F_L(x,Q^2)\) are extracted, is measured with typically 1% statistical and 3% systematic uncertainties. The measured derivative \((\partial F_2(x,Q^2) / \partial \ln Q^2)_x \) is observed to rise continuously towards small x for fixed \(Q^2\). The cross section data are combined with published H1 measurements at high \(Q^2\) for a next-to-leading order DGLAP QCD analysis. The H1 data determine the gluon momentum distribution in the range \(3 \cdot 10^{-4} \le x \le 0.1\) to within an experimental accuracy of about 3% for \(Q^2 =20\) GeV\(^2\). A fit of the H1 measurements and the \(\mu p \) data of the BCDMS collaboration allows the strong coupling constant \(\alpha_s\) and the gluon distribution to be simultaneously determined. A value of \(\alpha_s (M_Z^2) =0.1150 \pm 0.0017 (exp)^{+0.0009}_{-0.0005} (model)\) is obtained in NLO, with an additional theoretical uncertainty of about \(\pm 0.005\), mainly due to the uncertainty of the renormalisation scale.