= Cubic-scalar exchange corrections at zero momentum
In the <six-dimensional cubic scalar field theory>, use full external-propagator amputation and <tadpole subtraction>. The exchange part of the zero-momentum four-point amplitude is
$$
\mathcal A_{4,\mathrm{exchange}}=\frac{3g^2}{m^2}+\frac{6g^4}{m^2}I_3+\frac{3g^4}{2m^4}I_2+O(\text{two loops}).
$$
The triangle corrects either end of each of three exchanges, and the symmetry factor of an internal bubble is $1/2$. Equivalently insert $\Gamma^{(3)}=g+g^3I_3$ and $\Gamma^{(2)}=m^2-g^2I_2/2$ into $3[\Gamma^{(3)}]^2/\Gamma^{(2)}$. Without tadpole subtraction, the shift $v=-gI_1/(2m^2)$ adds $3g^4I_1/(2m^6)$. None of these is an additional local 1PI quartic coupling; zero-momentum bridges are also excluded from a strict high-mode Wilsonian shell. If bare rather than full external propagators are amputated, external-line decorations additionally contribute $6g^4I_2/m^4$, and $6g^4I_1/m^6$ without tadpole subtraction. Every displayed term has <mass dimension> minus two. In a renormalized expansion, the exchange diagrams also receive the <counterterm> insertions $6g\delta g/m^2-3g^2\delta m^2/m^4$. Without enforcing a zero one-point function, a linear <counterterm> $\delta h\phi$ contributes $3g^3\delta h/m^6$ through the background shift; choosing $\delta h=-gI_1/2$ cancels the attached tadpole. A kinetic <counterterm> is proportional to the exchanged momentum squared, so its insertion vanishes at this zero-momentum point.
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