  We derive the shape of the high-energy features due to a weakly coupled boson
in cuprate superconductors, as seen experimentally in Bi_2 Sr_2 Ca_1 Cu_2 O_8+x
(BSCCO) by Lee et al. [Nature (London) 442, 546 (2006)]. A simplified model is
used of d-wave Bogoliubov quasiparticles coupled to Einstein oscillators with a
momentum-independent electron-boson coupling and an analytic fitting form is
derived, which allows us (a) to extract the boson mode's frequency and (b) to
estimate the electron-boson coupling strength. We further calculate the maximum
possible superconducting gap due to an Einstein oscillator with the extracted
electron-boson coupling strength, which is found to be less than 0.2 times of
the observed gap indicating at the observed boson's non-dominant role in the
superconductivity's mechanism. The extracted momentum-independent
electron-boson coupling parameter (that we show a posteriori to indeed be in
the weak-coupling regime) is then to be interpreted as an (band-structure
detail dependent weighted) average over the Brillouin zone of the actual
momentum-dependent electron-boson coupling in BSCCO.
