Deep-Inelastic Inclusive ep Scattering at Low x and a Determination of α_s

A precise measurement of the inclusive deep-inelastic e p scattering cross section is reported in the kinematic range 1:5 Q x 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 . The double differential cross section, from which the proton structure function F 2 (x; Q ) and the longitudinal structure function FL ) are extracted, is measured with typically 1% statistical and 3% systematic uncertainties. The measured derivative (@F 2 (x; Q )=@ ln Q ) x is observed to rise continuously towards small x for fixed . The cross section data are combined with published H1 measurements at high Q for a next-to-leading order DGLAP QCD analysis. The H1 data determine the gluon momentum distribution in the range 3 10 x 0:1 to within an experimental accuracy of about 3% for Q . A fit of the H1 measurements and the p data of the BCDMS collaboration allows the strong coupling constant s and the gluon distribution to be simultaneously determined. A value of s (M ) = 0:1150 0:0017(exp) +0:0009 0:0005 (model) is obtained in NLO, with an additional theoretical uncertainty of about 0:005, mainly due to the uncertainty of the renormalisation scale