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Splitting a scalar field into slow and fast components $\varphi =\phi +\eta$, the Wick contractions in $\lambda ⟨(\phi +\eta {)}^6⟩$ generate $15\lambda ⟨{\eta }^2⟩{\phi }^4$. A vanishing bare quartic coupling is therefore not generally preserved by renormalization-group flow.

With cosmic-time interaction coefficient $g$, constant $H,ϵ,c_s$, and modes whose positive-frequency part is $H(1+ik{c}_s\tau )e^{-ik{c}_s\tau }/\sqrt{4ϵc_s{k}^3}$ in units $M_p=1$, two Wick contractions per differentiated pair and the late-time scalar gradient vertex integral giveHere $⟨{\zeta }^3{⟩}_c=(2\pi {)}^3{\delta }^3(\sum {\mathbf{k}}_i)B$. The extra factor $a$ in $dt=ad\tau$ must be included. A positive $g$ yields a negative bispectrum in an equilateral bispectrum configuration; changing the sign of the interaction changes the sign of $B$.

In three dimensions, after tuning the quadratic and quartic relevant operators, the positive sextic coupling has a renormalization-group beta function $d\lambda /ds=-C{\lambda }^2+\cdots$ with $C>0$ for increasing length scale $s$. The Wick contractions connecting two sextic vertices with three propagators give the primitive logarithmic correction. The interaction is marginally irrelevant.

For $H_{\mathrm{int}}=-\int d^{3}x{a}^3ϵ\mathcal{F}{\dot{\zeta }}^3/H$, $a=-1/(H\tau )$ and stipulated modes $u_k=H(1+ik\tau )e^{-ik\tau }/\sqrt{4ϵk^3}$, six connected Wick contractions and ${\int }_{-\infty (1-i0)}^0{\tau }^2{e}^{iK\tau }d\tau =2i/K^3$ yield $B=3H^4\mathcal{F}/(8{ϵ}^2{k}_1{k}_2{k}_3{K}^3)$. Here $⟨{\zeta }^3{⟩}_c=(2\pi {)}^3\delta (\sum {\mathbf{k}}_i)B$. Using unstarred vertex modes with the external-before-vertex prefactor incorrectly reverses the sign. Non-unit sound speed also changes the quadratic modes, so this result is conditional on the specified unit-speed modes.

For Gaussian initial conditions and the Einstein-de Sitter second-order standard perturbation theory density kernel, the unsmoothed density contrast obeysThere are three choices of the second-order field and two connected Wick contractions; the angular average of $F_2$ is $5/7+(2/7)/3=17/21$. Thus the reduced skewness is $6(17/21)=34/7$. Smoothing changes this result, and the unsmoothed calculation requires a finite variance.

Three external fields can be joined to the three cubic vertex fields in six connected Wick contractions. There are three choices for the external field at the undifferentiated slot and two exchanges of the differentiated slots. Three cyclic terms therefore require an explicit factor of two. Ordered contractions with an earlier vertex use complex-conjugated mode functions.

For $m^2>0$, the zero-dimensional scalar field theory with Euclidean action $m^2{x}^2/2+\lambda x^4/4!$ has a Gaussian integral normalization $I(0)=\sqrt{2\pi /m^2}$. Its Feynman rules assign $1/m^2$ to each propagator, $-\lambda$ to each quartic vertex, and the reciprocal Feynman-diagram symmetry factor to each vacuum diagram. The integral itself includes disconnected vacuum diagrams; its logarithm selects connected ones. At second order the three possible numbers of edges joining the two vertices are zero, two and four, with Feynman-diagram symmetry factors $128,16,48$ respectively. Their weights add to $35{\lambda }^2/(384m^8)$.

The Wick contractions give the general coefficient $I(0)(-1{)}^v(4v-1)!!/[v!(4!{)}^v{m}^{4v}]$. Successive absolute coefficients have ratio $(4v+3)(4v+1)/[24(v+1)m^4]$, which tends to infinity. Thus this is an asymptotic expansion as $\lambda \downarrow 0$, rather than a convergent power series. For every fixed truncation the Taylor theorem remainder estimate for $e^{-u}$, $u\ge 0$, can be integrated against the Gaussian distribution to justify the corresponding remainder bound.

A scalar field with no spacetime coordinates reduces its path integral to an ordinary integral. It preserves Wick contractions and Feynman-diagram symmetry factors, making it a useful model of perturbative quantum field theory.