On the Coulomb and higher-order sum rules in the relativistic Fermi gas |
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| Institution: | 1. Dipartimento di Fisica Teorica dell''Università di Torino, Istituto Nazionale di Fisica Nucleare, Sezione di Torino, via P. Giuria 1, I-10125 Torino, Italy;2. Dipartimento di Fisica dell''Università di Genova, Istituto Nazionale di Fisica Nucleare, Sezione di Genova, via Dodecaneso, 33 - 16146 Genova, Italy;3. Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA;1. Metabolic Disease Research Center, Zhengzhou University Affiliated Zhengzhou Central Hospital, Zhengzhou, 450007, China;2. Research Center for Basic Medicine, Academy of Medical Sciences, Zhengzhou University, Zhengzhou, 450052, China;3. Key laboratory of Zhengzhou Obesity and Metabolic Diseases, Zhengzhou, Zhengzhou, 450007, China;1. College of Education, Health and Human Performance, Fairmont State University, Fairmont, WV, USA;2. College of Applied Human Science, West Virginia University, Morgantown, WV, USA;3. Programa de Pós-graduação em Ciências Aplicadas à Saúde, Universidade Federal de Jataí, Jataí, Brazil;4. College of Liberal Arts, Community Health Fairmont State University, Fairmont, WV, USA;5. Department of Health Promotion and Development, School of Nursing, University of Pittsburgh, Pittsburgh, PA, USA;1. ESIC Business & Marketing School, Barcelona, Spain;2. ESIC Business & Marketing School, ESIC University, Barcelona, Spain;1. Student Research Committee, Gerash University of Medical Sciences, Gerash, Iran;2. Department of Biostatistics, School of Health, Kermanshah University of Medical Sciences, Kermanshah, Iran;3. Department of Operating Room, Nahavand School of Allied Medical Sciences, Hamadan University of Medical Sciences, Hamadan, Iran;4. Cellular and Molecular Research Center, Gerash University of Medical Sciences, Gerash, Iran;5. Department of Biology, Faculty of Science, University Putra Malaysia, Serdang, Selangor, Malaysia |
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| Abstract: | Two different methods for establishing a space-like Coulomb sum rule for the relativistic Fermi gas are compared. Both of them divide the charge response by a normalizing factor such that the reduced response thus obtained fulfills the sum rule at large momentum transfer. To determine the factor, in the first approach one exploits the scaling property of the longitudinal response function, while in the second one enforces the completeness of the states in the space-like domain via the Foldy-Wouthuysen transformation. The energy-weighted and the squared-energy-weighted sum rules for the reduced responses are explored as well and the extension to momentum distributions that are more general than a step-function is also considered. The two methods yield reduced responses and Coulomb sum rules that saturate in the non-Pauli-blocked region, which can hardly be distinguished for Fermi momenta appropriate to atomic nuclei. Notably the sum rule obtained in the Foldy-Wouthuysen approach coincides with the well-known non-relativistic one. Only at quite large momentum transfers (say 1 GeV/c) does a modest softening of the Foldy-Wouthuysen reduced response with respect to that obtained in the scaling framework show up. The two responses have the same half-width to second order in the Fermi momentum expansion. However, when distributions extending to momenta larger than that at the Fermi surface are employed, then in both methods the Coulomb sum rule saturates only if the normalizing factors are appropriately modified to account for the high momentum components of the nucleons. |
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