Let us start with the familiar example of electron-phonon interactions when the irre- ducible interactions are specified by b$_{q}$$_{ν}$, b + q ν are the phonon creation and annihilation operators and g $^{ν}$( k , k + q ) = $^{√}$$_{ℏ}$/ (2 Mω$_{q}$$_{ν}$ ) I ν k , q is the electron-phonon vertex, M is the ion-mass and ω$_{q}$$_{ν}$ are the frequen- cies of phonons of momentum q and polarization ν . A specific form for the electron-phonon vertex will be introduced later. Optical phonons are to a first approximation dispersion free. Their interactions with elec- trons in a metal are effectively δ -function in real space, which only gives s -wave scattering. The interaction with acoustic phonons must vanish in the long wavelength limit. Moreover, the partial density of states of phonons is peaked near the zone-boundary. The effective interaction is therefore short-range, of the order of the lattice constant. As will be discussed later, this is true for pseudo-potential metals in which the e-ph interactions are weak. For transition metals and compounds and other co-valent metals with larger interactions and larger T$_{c}$ , the e-ph interactions are even shorter-range. Dominant attraction is therefore always in the s − wave channel. Electrons also repel. The effective dimensionless electron-electron interaction (normalized to the density of states at the chemical potential), and assumed to be cut-off in energy only at the Fermi-energy, where ϵ ( q ) is the dielectric function and the average is over initial and final states at the chemical potential in the same manner as in the definition of the λ in Eq. (24). It is interesting to consider the relationship of µ to λ with both calculated for jellium. The dimensionless electron-phonon coupling constant λ for jellium, defined in analogy with (23) 14 µ = N (0) < 4 πe $^{2}$/ ( q $^{2}$ϵ ( q )) > (5) 2. Effective electron-electron repulsion in the s-wave channel in Jellium H$_{e}$$_{−}$$_{ph}$ = ∑ k , q ,ν $^{√}$$_{ℏ}$/ (2 Mω$_{q}$$_{ν}$ ) I ν k , $_{q}$c + k + q $_{σ}$c$_{k}$$_{σ}$ ( b$_{q}$$_{ν}$ + b + − q $_{ν}$) , (4) 1. Pairing Symmetry due to e-ph Interactions A. Considerations on Pairing Symmetry