= Solution
Let the <shell-restricted scalar propagator> be
$$
C_>(x-y)=\int_{\Lambda'<|p|\leq\Lambda}\frac{d^4p}{(2\pi)^4}\frac{e^{ip\cdot(x-y)}}{p^2+m^2}.
$$
Write $\Delta S=S_0[\widehat\phi]+V[\phi,\widehat\phi]$ and average with the normalized Gaussian shell measure. The <cumulant expansion> gives
$$
S_{\Lambda'}^{\mathrm{eff}}=S_\Lambda^{\mathrm{eff}}-\log Z_>^0+\langle V\rangle_0-\frac12\left(\langle V^2\rangle_0-\langle V\rangle_0^2\right)+O(g^3).
$$
The subtraction removes <disconnected Feynman diagrams>; the logarithm retains <connected Feynman diagrams> made from shell contractions. This is the <connected shell-contraction expansion>. An external line denotes the background low field, not an additional low-momentum propagator.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-46-wilsonian-diagrams.png]
{title=Connected quartic shell diagrams through second order, including vacuum terms and the two momentum-support zeros}
{height=1260}
At order $g$, panel A is the existing four-field <interaction vertex>. The new shell contractions are B, a <tadpole diagram> contributing a two-field vertex, and C, a <Vacuum Feynman diagram> contributing a constant. The Gaussian <functional determinant> in panel L is another field-independent term, of order $g^0$; the original quadratic <kinetic term> and mass term are retained as well.
For completeness, all two-vertex topologies from <Wick contractions> at order $g^2$ are shown. Let $r$ be the number of shell lines joining the vertices and $t_1,t_2$ their numbers of self-contractions. The external-field counts are
$$
e_i=4-r-2t_i,\qquad r\geq1,\qquad t_i\geq0,\qquad e_i\geq0.
$$
Interchanging the two vertices identifies the same topology. Enumerating these conditions gives exactly the following eight possibilities:
* D: $(r,t_1,t_2)=(1,0,0)$, with $(e_1,e_2)=(3,3)$; a six-field vertex kernel.
* E: $(1,0,1)$, with $(3,1)$; a formal four-field kernel, which vanishes for the sharp shell split.
* F: $(1,1,1)$, with $(1,1)$; a formal two-field kernel, which also vanishes for the sharp shell split.
* G: $(2,0,0)$, with $(2,2)$; a four-field vertex kernel.
* H: $(2,0,1)$, with $(2,0)$; a two-field vertex kernel, containing a <tadpole diagram>.
* I: $(2,1,1)$, with $(0,0)$; a connected <Vacuum Feynman diagram>.
* J: $(3,0,0)$, with $(1,1)$; the two-field <sunset diagram> kernel.
* K: $(4,0,0)$, with $(0,0)$; a connected <Vacuum Feynman diagram> with four joining lines.
The support qualification is important. In E or F, a vertex with one external low field and one bridge also has a tadpole whose two momenta cancel. Conservation forces the bridge momentum to equal that single low momentum, outside the shell. Equivalently, convolution by $C_>$ annihilates $\phi$. This is <momentum-support exclusion for Wilsonian bridge diagrams>. D is different: its bridge carries the sum of three low momenta, which can lie in the shell. Its kernel is proportional to $\phi^3(x)C_>(x-y)\phi^3(y)$ and need not vanish. Thus <one-particle-reducible Feynman diagrams> must not be excluded merely because the object being computed is an effective action.
These kernels need not be local before a <derivative expansion>. Their momentum dependence generates derivative interactions where such an expansion is valid. If all external momenta are sufficiently far below $\Lambda'$, D vanishes too; in particular it does not produce a zero-momentum local $\phi^6$ coupling at order $g^2$. A local six-field term is allowed and is generated at higher orders, for example by a three-vertex shell triangle.
\b[To all orders, expect every scalar interaction allowed by the original symmetries: arbitrary even powers of the field and their allowed derivative couplings, together with vacuum terms.] The <Z2 symmetry> $\phi\mapsto-\phi$ excludes odd-field vertices. The original quartic form is therefore not closed under exact Wilsonian integration.
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