Clarifying the coupling between electrons and bosonic excitations (phonons or magnetic fluctuations) that mediate the formation of Cooper pairs is pivotal to understand superconductivity. Such coupling effects are contained in the electron self-energy, which is experimentally accessible via angle-resolved photoemission spectroscopy (ARPES). However, in unconventional superconductors, identifying the nature of the electron-boson coupling remains elusive partly because of the significant band renormalization due to electron correlation. Until now, to quantify the electron-boson coupling, the self-energy is most often determined by assuming a phenomenological 'bare' band. Here, we demonstrate that the conventional procedure underestimates the electron-boson coupling depending on the electron-electron coupling, even if the self-energy appears to be self-consistent via the Kramers-Kronig relation. Our refined method explains well the electron-boson and electron-electron coupling strength in ruthenate superconductor Sr 2 RuO 4 , calling for a critical revision of the bosonic coupling strength from ARPES self-energy in strongly correlated electron systems.T he many-body effects, the coupling between electrons and various excitations, are contained in the electron self-energywhich is accessible via the spectral function A(k, v) measured by angle-resolved photoemission spectroscopy (ARPES) aswhere e 0 (k) is the non-interacting (bare) energy-band dispersion and the real and imaginary parts of the selfenergy, . However, the derivation of the self-energy depends on the unfounded assumption of a bare band, which cannot be directly observed. To overcome this difficulty, the self-consistency of the self-energy via Kramers-Kronig relation is the most widely used criteria to extract the self-energy 2,[5][6][7][8] . Moreover, iterative fitting algorithms for the determination of the bare band have also been developed based on the maximum entropy method 9-11 or the Kramers-Kronig relation [12][13][14] . In contrast, many previous ARPES experiments have regarded the bare band as a renormalized band due to the electron-electron coupling, which has enabled evaluation of the 'effective' bosonic self-energy and coupling strength [2][3][4][5][6][7][8] . However, the term 'effective' is often abbreviated or overlooked in many cases, although it has been proposed that the effective electron-boson coupling strength is smaller than the true coupling strength because of the electron-electron coupling 15,16 . There is a need, therefore, to reconsider and/or refine the most commonly used method for extracting the bosonic self-energy from ARPES data.
ResultsWe assume two main scattering channels, the electron-boson and electron-electron couplings. The energy-scales associated with these two are quite different: the electron-boson coupling dominates at low energy-scales (at most ,100 meV), whereas the electron-electron coupling over a wide energy-scale (eV-order). The two are thus nearly independent in the scattering process and hence the Matth...