Quantitative Comparison between Electronic Raman Scattering and Angle-Resolved Photoemission Spectra in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ Superconductors: Doping Dependence of Nodal and Antinodal Superconducting Gaps

Both electronic Raman scattering (ERS) and angle-resolved photoemission spectra (ARPES) revealed two energy scales for the gap in different momentum spaces in the cuprates. However, the interpretations were different, and the gap values were also different in two experiments. In order to clarify the origin of these discrepancies, we directly compared ERS and ARPES by calculating ERS from the experimental data of ARPES through the Kubo formula. The calculated ERS spectra were in good agreement with the experimental results except for the B$_{1g}$ peak energies. The doping-dependent B$_{2g}$ peak energy was well reproduced from a doping-independent d-wave gap deduced from ARPES, by assuming a particular spectral weight distribution along the Fermi surface. The B$_{1g}$ peak energies could not be reproduced by the ARPES data. The difference between B$_{1g}$ ERS and antinodal ARPES became larger with underdoping, which implies that the effect of the pseudogap is different in these two techniques.