ON THE COULOMB AND HIGHER-ORDER SUM-RULES IN THE RELATIVISTIC FERMI GAS

Two different methods for establishing a space-like Coulomb sum rule f or the relativistic Fermi gas are compared. Both of them divide the ch arge response by a normalizing factor such that the reduced response t hus obtained fulfills the sum rule at large momentum transfer. To dete rmine the factor, in the first approach one exploits the scaling prope rty of the longitudinal response function, while in the second one enf orces the completeness of the states in the space-like domain via the Foldy-Wouthuysen transformation. The energy-weighted and the squared-e nergy-weighted sum rules for the reduced responses are explored as wel l and the extension to momentum distributions that are more general th an a step-function is also considered. The two methods yield reduced r esponses and Coulomb sum rules that saturate in the non-Pauli-blocked region, which can hardly be distinguished for Fermi momenta appropriat e to atomic nuclei. Notably the sum rule obtained in the Foldy-Wouthuy sen approach coincides with the well-known non-relativistic one. Only at quite large momentum transfers (say 1 GeV/c) does a modest softenin g of the Foldy-Wouthuysen reduced response with respect to that obtain ed in the scaling framework show up. The two responses have the same h alf-width to second order in the Fermi momentum expansion. However, wh en distributions extending to momenta larger than that at the Fermi su rface are employed, then in both methods the Coulomb sum rule saturate s only if the normalizing factors are appropriately modified to accoun t for the high momentum components of the nucleons. (C) 1997 Elsevier Science B.V.