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Momentum Dependence of the Single-Particle Self-Energy and Fluctuation Spectrum of Slightly Underdoped Bi_2 Sr_2 CaCu_2 O_{8+δ} from High Resolution Laser ARPES

We deduce the normal state angle-resolved single-particle self-energy $Σ(θ, ω)$ and the Eliashberg function (i.e., the product of the fluctuation spectrum and its coupling to fermions) $α^2 F(θ,ω)$ for the high temperature superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ from the ultra high resolution laser angle-resolved photoemission spectroscopy (ARPES). The self-energy $Σ(θ, ω)$ at energy $ω$ along several cuts normal to the Fermi surface at the tilt angles $θ$ with respect to the nodal direction in a slightly underdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ were extracted by fitting the ARPES momentum distribution curves. Then, using the extracted self-energy as the experimental input, the $α^2 F(θ,ω)$ is deduced by inverting the Eliashberg equation employing the adaptive maximum entropy method. Our principal new result is that the Eliashberg function $α^2F(θ,ω)$ collapse for all $θ$ onto a single function of $ω$ up to the upper cut-off energy despite the $θ$ dependence of the self-energy. The in-plane momentum anisotropy is therefore predominantly due to the anisotropic band dispersion effects. The obtained Eliashberg function has a small peak at $ω\approx0.05$ eV and flattens out above 0.1 eV up to the angle-dependent cut-off. It takes the intrinsic cut-off of about 0.4 eV or the energy of the bottom of the band with respect to the Fermi energy in the direction $θ$, whichever is lower. The angle independence of the $α^2 F(θ,ω)$ is consistent only with the fluctuation spectra which have the short correlation length on the scale the lattice constant. This implies among others that the antiferromagnetic fluctuations may not be underlying physics of the deduced fluctuation spectrum.