term in Eq.(\[eq:3.1\]): \[eqs:3.2\] A\_(x)A\_(y)= \_[D]{}(x-y), \[eq:3.2a\] with (x-y) = -4(\_\^2/c\^2 + \^2)\^[-1]{}(x-y), \[eq:3.2b\] or, in Fourier space, (k) = 4/(\_n\^2/c\^2 + [k]{}\^2). \[eq:3.2c\] Here $k \equiv (i\Omega_n,{\bm k})$ comprises a bosonic Matsubara frequency $\Omega_n = 2\pi T n$ and a wave vector ${\bm k}$.

The result of integrating out the photons exactly is very complicated and involves interacting electronic modes in both the spin-singlet and spin-triplet channels, the particle-particle and particle-hole channels, and all angular-momentum channels. We will restrict ourselves to those terms that are most relevant for magnetism, i.e. interactions between spin-density fluctuations, or modes in the $s$-wave particle-hole spin-triplet channel.

### $O(1/c^0)$: Exchange interaction {#subsubsec:III.B.1}

We organize the various contributions to the effective electron-electron interaction in powers of $1/c$. To zeroth order only the term coupling $\varphi$ to the number density $n$ in Eq. (\[eq:3.1\]) contributes. Integrating out $\varphi$ leads to a Coulomb interaction \[eqs:3.3\] S\_ = - dxdy n(x)v\_([x]{}-[y]{}) (\_x - \_y) n(y), \[eq:3.3a\] with v\_([x]{}) = e\^2/. \[eq:3.3b\] Here we have neglected the dynamical nature of the photon propagator ${\cal D}$ and have replaced it by its value at $\Omega_n=0$. The reason for this approximation is that Fermi-liquid effects lead to a dynamical screening of 