Subthreshold $K^+$ production in proton-nucleus collisions

Double differential K$^+$ cross sections have been measured in p+C collisions at 1.2, 1.5 and 2.5 GeV beam energy and in p+Pb collisions at 1.2 and 1.5 GeV. The K$^+$ spectrum taken at 2.5 GeV can be reproduced quantitatively by a model calculation which takes into account first chance proton-nucleon collisions and internal momentum with energy distribution of nucleons according to the spectral function. At 1.2 and 1.5 GeV beam energy the K$^+$ data excess significantly the model predictions for first chance collisions. When taking secondary processes into account the results of the calculations are in much better agreement with the data.