  The diagrammatic formalism and transport equation are conventionally
considered as separate but complementary techniques to tackle the impurity
scattering effect. To compare with the previous studies from the
gauge-invariant kinetic equation approach [F. Yang and M. W. Wu, Phys. Rev. B
98, 094507 (2018); 102, 144508 (2020)], we analytically perform a diagrammatic
formulation of the impurity scattering in superconductors, with both transport
and collective Higgs mode studied, in order to fill the long missing
calculation of the Kubo current-current correlation in superconductors with
impurity scattering and resolve the controversy (whether the impurity
scattering can lead to the damping of Higgs mode) between gauge-invariant
kinetic equation and Eilenberger equation. For transport behavior, through a
special unitary transformation that is equivalent to the Wilson-line technique
for the diamagnetic response, we derive the Meissner-supercurrent vertex. Then,
by formulating the supercurrent-supercurrent correlation with Born and vertex
corrections from impurity scattering, we recover the previously revealed
microscopic momentum-relaxation rate of superfluid by gauge-invariant kinetic
equation. This rate is finite only when the superconducting velocity is larger
than a threshold, at which the normal fluid emerges and causes the friction
with the superfluid current, similar to the Landau's superfluid theory of
liquid helium. This derivation also provides a physical understanding of the
relaxation-time approximation in the previous diagrammatic formulation in the
literature, which leads to the friction resistance of the Meissner
supercurrent. For the collective Higgs mode, we calculate the
amplitude-amplitude correlation with Born and vertex corrections from impurity
scattering. The vertex correction, which only emerges at nonequilibrium case,
leads to a Higgs-mode damping, ......
