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The cumulant expansion of the shell integral keeps precisely connected Feynman diagrams built from shell-restricted scalar propagators. The external low fields are backgrounds, not propagators. Connected graphs with bridges can contribute, so the expansion is not limited to one-particle-irreducible Feynman diagrams.

Writing the interaction as $V$ and averaging over the fast modes, which have a Gaussian distribution, gives the cumulant expansion $F_{\mathrm{eff}}=F_0+⟨V⟩-\frac{1}{2}(⟨V^2⟩-⟨V{⟩}^2)+\cdots$. The second cumulant selects connected Feynman diagrams.

A Feynman diagram with more than one connected component. Products of component contributions exponentiate; taking the logarithm of a Gaussian perturbation expansion removes disconnected products and keeps connected Feynman diagrams.

With signature $(+---)$, the momentum-space Feynman rules for phi-fourth theory follow by expanding the interaction exponential and using free-field Wick contractions. Every internal scalar propagator supplies $i/(k^2-m_0^2+i0)$, every quartic Feynman vertex supplies $-i{\lambda }_0$, and each vertex conserves four-momentum. Integrate every independent loop momentum with $d^{4}k/(2\pi {)}^4$ and divide by the Feynman-diagram symmetry factor $S$. Equivalently, assign momenta to all lines and attach $(2\pi {)}^4{\delta }^4(\sum k)$ at each vertex before eliminating constrained integrations. A connected amputated graph therefore has the formwith the remaining overall momentum-conservation delta function understood. For an unamputated correlation function, retain the external propagators; scattering amplitudes instead use LSZ reduction. The factor $4!$ in the interaction cancels the number of ways to attach the four identical fields at a vertex, leaving the stated vertex factor.

For a connected Feynman diagram, $L=N_l-N_v+1$. The superficial degree of divergence is the power under simultaneous large rescaling of all independent loop momenta before considering cancellations or divergent subgraphs. Since each integration contributes four powers and each internal propagator removes two,Counting vertex half-edges gives $4N_v=2N_l+N_e$. Substitution yieldsThe connectedness assumption matters: a disconnected interaction graph with $C$ components instead has the corresponding count $D=4C-N_e$. Tree graphs have no loop integral even when this formal power count is nonnegative.

In four dimensions the possible primitively divergent proper vertices have $N_e=2$ or $4$, with $D=2$ or $0$; odd vertices vanish by the $\varphi \mapsto -\varphi$ symmetry. The divergent local structures are a mass term, a kinetic term and a quartic coupling term. If vacuum diagrams are retained, a vacuum-energy term is also required. Thus one can renormalize using a finite family of operators already allowed by the theory's symmetries: four-dimensional phi-fourth theory is perturbatively renormalizable. The superficial count alone is not an all-orders convergence proof. After divergent subgraphs are subtracted with these counterterms, higher-point graphs have no new overall ultraviolet divergence; a negative overall $D$ does not by itself exclude subdivergences.

The requested lowest-order graphs and the three channels needed for the coupling correction are shown here:

For the two-point function, the free propagator is order one, while its first interacting correction is the tadpole diagram, with $N_v=N_l=L=1$, $N_e=2$, $S=2$ and $D=2$. Its loop integral behaves as $\int d^{4}k/k^2$ and is quadratically divergent. For the connected four-point function, the lowest graph is the order-${\lambda }_0$ contact vertex, with $N_v=1,N_l=L=0$ and formal $D=0$ but no loop divergence. The first loop correction comprises the $s,t,u$ bubble diagrams. Each has $N_v=2,N_l=2,L=1$, $N_e=4$, $S=2$ and $D=0$, matching its logarithmic integral $\int d^{4}k/(k^2{)}^2$.

If the full four-point correlation function rather than its connected part is considered, there is also an order-one contribution: the sum of the three Wick contractions into two free propagators, with pairings $(12)(34),(13)(24),(14)(23)$. Explicitly, $G_0^{(4)}=G_{12}{G}_{34}+G_{13}{G}_{24}+G_{14}{G}_{23}$, with $G_{ij}$ the free scalar propagator between insertions $i,j$. These disconnected pairings are drawn in the bottom row of the plate. They contain no loop integral and are not contributions to the proper four-point vertex.

A six-point amplitude in phi-fourth theory has a one-loop triangle Feynman diagram with two external legs at each of three vertices. It has $N_v=3,N_l=3,L=1$, $N_e=6$, and $D=-2$. For a fixed external-leg partition its loop integral contains three scalar denominators, for exampleAfter Wick rotation, its ultraviolet radial behavior is ${\int }^{\infty }dk{k}^3/k^6={\int }^{\infty }dk/k^3$, which converges. The triangle has no proper loop subgraph, so no ultraviolet subdivergence is hidden in this example. It generates no new six-field counterterm. Nonzero mass avoids a separate infrared issue; with massless propagators exceptional external momenta can produce infrared singularities without changing this ultraviolet conclusion.

The bare mass is a coefficient in the regulated action, whereas the physical mass is the pole position of the full propagator. Write the amputated two-point insertion as $-i\Sigma (p^2)$. Summing its insertions givesAt order ${\lambda }_0$, only the tadpole contributes. Its factor one half follows directly from Wick pairing: of the four fields at the vertex, $4\cdot 3$ choices attach the two labeled external lines and the remaining pair closes the loop, so $12/4!=1/2$. ThusRotate $k^0=i{k}_E^0$ through the Feynman contour. The $i$ from the measure and the $i$ in the propagator, together with the denominator sign, give a positive Euclidean integral. With spherical cutoff $|k_E|<\Lambda$,This is the Wick-rotated cutoff tadpole mass shift, and it is independent of external momentum. Hence to first order,For a positive stable quartic coupling the first-order shift is positive. Keeping the measured pole fixed while changing the cutoff requires adjusting the bare mass. At $m_0=0$ the bracket is understood by its finite limit ${\Lambda }^2$. There is no momentum-dependent wave-function correction from this one-loop tadpole; such corrections begin at higher order.

Let the shell-restricted scalar propagator beWrite $\Delta S=S_0[\hat{\varphi }]+V[\varphi ,\hat{\varphi }]$ and average with the normalized Gaussian shell measure. The cumulant expansion givesThe 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.

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 areInterchanging the two vertices identifies the same topology. Enumerating these conditions gives exactly the following eight possibilities:

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 $\varphi$. 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 ${\varphi }^3(x)C_{>}(x-y){\varphi }^3(y)$ 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 ${\Lambda }^{\prime }$, D vanishes too; in particular it does not produce a zero-momentum local ${\varphi }^6$ 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.

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 $\varphi \mapsto -\varphi$ excludes odd-field vertices. The original quartic form is therefore not closed under exact Wilsonian integration.

For a connected Feynman diagram with cubic vertices with four external legs and one loop, $3V=2I+4$ and $1=I-V+1$ give $V=I=4$. In a one-particle-irreducible Feynman diagram, every internal edge lies on the single cycle. Each of the four cubic vertices therefore has two internal and one external leg, giving a box Feynman diagram. With labelled external momenta there are three inequivalent cyclic orderings, modulo rotation and reversal, represented by $1234$, $1243$ and $1324$:

Take all momenta incoming with ${\sum }_i{p}_i=0$. The Feynman rules for the Euclidean action give a propagator $(q^2+m^2{)}^{-1}$ and a cubic insertion $-g$. For order $1234$, the amputated connected insertion iswhere $\chi (q)=1_{\{\Lambda <|q|<{\Lambda }_0\}}$ is the high-mode projector for a strict Wilsonian effective action. Each labelled box has Feynman-diagram symmetry factor one. A common fixed-loop-cutoff convention instead restricts only the chosen $q$ integration to the shell; both prescriptions give the same zero-external-momentum value. The effective-action vertex has the opposite sign to this connected insertion. Thus $\boxed{\text{three labelled boxes, with one cyclic topology}.}$

Absorb $k_{B}T$ into the free energy, so the statistical weight is $e^{-F}$. Under the Gaussian fixed point rescaling, invariance of the gradient energy gives the engineering dimensionThe engineering dimensions of ${\varphi }^2,{\varphi }^4,{\varphi }^6$ are respectively $1,2,3$. Their coupling eigenvalues are $d-[{\varphi }^{2m}]$, hence $2,1,0$. The quadratic and quartic interactions are relevant operators; the sextic interaction is a marginal operator at this order.

The three steps of the momentum-shell renormalization group are:

Here $b=e^s>1$, so increasing $s$ means flowing toward longer distances. At leading order, the fast field has a Gaussian distribution with covarianceThis perturbative integration requires $k^2+{\mu }_0^2>0$ on the shell. By the Wick theorem, odd fast-field moments vanish and $⟨{\eta }^4⟩=3I_b^2$, $⟨{\eta }^6⟩=15I_b^3$. Therefore the first term of the cumulant expansion of a coarse-grained free energy isThis is the quartic interaction generated by a sextic interaction. Including the rescaling, the requested flow isIn spherical coordinates, $I_b=(2{\pi }^2{)}^{-1}{\int }_{\Lambda /b}^{\Lambda }{k}^2/(k^2+{\mu }_0^2)dk$. In particular, for ${\mu }_0^2=0$,The same integration shifts the quadratic coefficient by $90{\lambda }_0{I}_b^2$ before rescaling and adds an irrelevant constant to the free energy. Thus setting the bare quartic coupling to zero does not place the theory on the tricritical critical surface.

To calculate the sextic renormalization-group beta function to second order, let $V={\lambda }_0\int d^{3}x(\phi +\eta {)}^6$. The required cumulant expansion isThe subtraction removes disconnected contributions. The term $20{\lambda }_0{\phi }^3{\eta }^3$ at each vertex supplies the primitive sextic correction. The Wick contractions giveThe first term contains three propagators connecting the two vertices; its factor $6$ is the number of bijections between the three fields at each vertex. The second contains a local tadpole diagram at each vertex and one connecting propagator. The primitive contribution to the free energy is thereforeExpanding the slow fields about a common point gives a negative local sextic correction proportional towhere ${\chi }_b$ restricts a momentum to the shell. This connected Feynman diagram has two independent loop momenta. Other Wick contractions also generate lower interactions and tadpole diagrams; they must be included or subtracted consistently when fixing the tricritical critical surface.

To extract the logarithmic tricritical sextic beta function, use normal ordering to remove local tadpoles and tune the quadratic and quartic relevant operators. The massless three-dimensional propagator is $C(z)=1/(4\pi |z|)$. The same three-line contraction over short separations $a<|z|<ba$ givesIts positive coefficient and the negative sign of the second cumulant expansion term establish the sign of the flow. Matching this logarithm to a renormalized coupling, or performing consistent iterated shell integration, givesThe precise normalization of $C$ is not needed. In a sharp momentum cutoff scheme, simply forcing all three internal momenta into one infinitesimal shell does not extract this two-loop logarithm: the shell restrictions remove its linear-in-$ds$ phase space. The finite-shell cumulant and the logarithmic renormalized calculation above are distinct stages of the computation. Field normalization corrections at order ${\lambda }^2$ enter this renormalization-group beta function only at order ${\lambda }^3$.

A positive, stabilizing sextic coupling is marginally irrelevant in three dimensions on the tricritical critical surface. Reversing the convention for $s$ reverses the sign of the renormalization-group beta function. A negative sextic coupling without higher stabilizing powers makes the potential unbounded; moreover, the untuned bare choice $g=0$ generally flows away through its generated relevant operator.

The perturbative expansion of $Z[J]$ contains arbitrary Feynman diagrams, including disconnected products. The linked-cluster theorem givesbecause the factorials from repeated connected components reproduce the exponential series. Therefore $W[J]=-\hslash \log Z[J]$ is the sum of connected Feynman diagrams.

The Legendre transform removes diagrams that disconnect upon cutting one internal line. Equivalently, every connected diagram is a tree whose vertices are exact one-particle-irreducible vertices and whose edges are exact propagators. Thusand its functional derivatives are the one-particle-irreducible correlation functions. Algebraically, differentiating the Legendre relations givesso an exact quantum field theory propagator joining two proper vertices is precisely the inverse Hessian matrix needed to reconstruct connected diagrams.

With Euclidean source convention $+\int d^{d}x{J}_a{\varphi }_a$, the generating functional isHere $J_a(x)$ is a nondynamical source. Its functional derivatives insert fields:This is why $Z$ “generates” correlation functions.

With the notation of the question, $W[J]=-\log Z[J]$ is the Euclidean connected generating functional; it is distinct from a Wilsonian effective action. Define the classical fieldThe quantum effective action is the Legendre transformwhere $J$ is eliminated in favor of $\Phi$. Its first derivative isso at zero source the quantum expectation value is a stationary point of $\Gamma$. Moreover,where $G_{ab}=⟨{\varphi }_a{\varphi }_b{⟩}_{J,\mathrm{conn}}$ is the exact connected correlation function. Thus the second derivative of $\Gamma$ is the inverse exact propagator.

Perturbatively, $Z[J]$ sums all Feynman diagrams, including disconnected ones. The exponential formula for combinatorial structures says that its logarithm selects connected Feynman diagrams, so $W$ sums connected diagrams with the sign dictated by the convention above. The Legendre transform removes diagrams that disconnect when one internal line is cut; consequently $\Gamma$ sums one-particle-irreducible Feynman diagrams. Equivalently, every connected diagram is a tree assembled from one-particle-irreducible vertices and full propagators, and the Legendre transform inverts that tree construction.

Now let $J_a^{\prime }=J_b{U}_{ba}$. ThenChanging variables to ${\varphi }_b^{\prime }=U_{ba}{\varphi }_a$ and using invariance of both the action and functional measure gives $Z[J^{\prime }]=Z[J]$, hence $W[J^{\prime }]=W[J]$. In the Legendre transform, the pairing obeysChanging the source variable and using the invariance of $W$ therefore givesThe symmetry of the classical action and measure is inherited by the full quantum effective action when it has no quantum anomaly.

A connected Feynman diagram that is a cubic tree with five external legs has three vertices and two internal lines. Its unique unlabeled topology is a chain: two external legs meet at each end vertex and one external leg attaches to the middle vertex. The five-point tree amplitude in phi cubed theory therefore has fifteen labeled tree-level Feynman diagrams: choose the middle external leg in five ways, then partition the remaining four legs into two unordered pairs in three ways.

For $S=a\mathrm{tr}(M^2)/2+b\mathrm{tr}(M^4)/4$ with $a>0$, the Wick theorem givesThere are eight adjacent external-attachment contractions with one free matrix-index sum and four opposite attachments with no free sum. The second tensor structure is needed at finite $N$; omitting it is an additional large-$N$ approximation, not merely restricting to connected Feynman diagrams.