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Choose four of the six fields at one sextic vertex to be slow and contract the remaining two fast fields into a tadpole diagram. There are $\left(\genfrac{}{}{0ex}{}{6}{4}\right)=15$ choices. Ifthen the first term of the cumulant expansion contributesAfter the canonical coordinate and field rescaling, its contribution to the quartic coupling is

In the displayed connected Feynman diagram, each sextic vertex carries two slow external legs and the four remaining legs at each vertex are paired across the vertices. The second cumulant expansion has a factor $-1/2$, the slow legs can be selected in ${\left(\genfrac{}{}{0ex}{}{6}{2}\right)}^2$ ways, and the four cross-contractions can be paired in $4!$ ways. The coefficient is thereforeWriting ${\mathcal{S}}_{\zeta }=\{q:\Lambda /\zeta <|q|<\Lambda \}$, the local zero-external-momentum contribution iswith the integral restricted further so that $q_1+q_2+q_3\in {\mathcal{S}}_{\zeta }$. The three independent internal momenta agree with the diagram's loop order $L=4-2+1=3$.

Classify the connected contractions by the numbers $(r,4-r)$ of external slow legs on the two sextic vertices and by the number $c$ of fast propagators joining them. Besides the displayed $(2,2;c=4)$ graph, the distinct topologies are:

Exchanging the two vertices gives no new topology. All four are connected Feynman diagrams selected by the logarithm in the cumulant expansion. With an ideal sharp momentum shell and a projection at exactly zero external momentum, the single joining line in the last topology cannot carry shell momentum, so that topology gives zero to the local quartic coupling; it is still the remaining formal connected contraction.

Split $\varphi ={\varphi }_{<}+{\varphi }_{>}$ and write $I_n={\int }_{\Lambda /b<|k|<\Lambda }\frac{d^{d}k}{(2\pi {)}^d}{G}_0(k{)}^n$. The first cumulant of $g_0{\varphi }^4$ contains $6g_0{\varphi }_{<}^2⟨{\varphi }_{>}^2⟩=6g_0{I}_1{\varphi }_{<}^2$. The connected second cumulant of the two terms $3{\lambda }_0{\varphi }_{<}{\varphi }_{>}^2$ uses Wick theorem to give $⟨{\varphi }_{>}^2(x){\varphi }_{>}^2(y){⟩}_c=2G_{>}(x-y{)}^2$. Keeping its leading local term, the quadratic coefficient after integrating out the shell is thereforeThe quartic tadpole raises the mass parameter, while the pair of cubic vertices gives the negative bubble contribution. A cubic interaction also generates a linear tadpole $3{\lambda }_0{I}_1{\varphi }_{<}$, which may be removed by shifting the background but is not part of ${\mu }^{\prime 2}$.

The phase screen adds the phase accumulated across its thickness, so the reduced field just after the screen isFor a zero-mean random variable $w$ with a normal distribution and variance ${\sigma }^2$, its characteristic function givesThe same expression is approximately valid for a non-Gaussian weak fluctuation: the cumulant expansion begins with $-k^2{\xi }^2{\sigma }^2/2$, while higher cumulants give higher-order corrections.

For $x>0$, every realization obeys the parabolic wave equationLinearity permits ensemble averaging, soThe initial mean is independent of $z$, hence diffraction does not change it and $⟨E(x,z)⟩=\exp (-k^2{\xi }^2{\sigma }^2/2)$ for every $x\ge 0$.