Connected generating functional ( $W [J] = - i lo g Z [J]$ )

The logarithm of a generating functional generates connected correlation functions because its cumulant expansion removes products of disconnected vacuum components.

Table of contents
- Centered four-point cumulant decomposition Connected generating functional
- Source renormalization of a scalar generating functional Connected generating functional

Centered four-point cumulant decomposition ( $G_{4} = C_{4} + \sum C_{2} C_{2}$ )

For a normalized Minkowski source functional

Z = e^{iW}

with vanishing one-point function,

C_{n} = i^{1 - n} δ^{n} W / δ J^{n}

and

G_{1234} = C_{1234} + C_{12} C_{34} + C_{13} C_{24} + C_{14} C_{23} .

Differentiating

e^{iW}

assigns the four labels to connected blocks. Blocks of size one vanish, leaving one block of four or one of the three pair partitions. Thus

G_{4} = i W_{4} - \sum W_{2} W_{2}

in this source convention.

// Target: perturbative-quantum-field-theory.bigb

Source renormalization of a scalar generating functional ( $h_{0} = μ^{- ϵ /2} Z^{- 1/2} h$ )

For

d = 4 - ϵ

, write the bare field as

ϕ_{0} = Z^{1/2} ϕ

and the renormalized source coupling as

μ^{- ϵ /2} h ϕ

. In bare variables this is

h_{0} ϕ_{0}

, with the displayed source. At fixed bare parameters, the source running is

μ d lo g h / d μ = ϵ /2 + \frac{1}{2} β \partial_{λ} lo g Z

. Normalizing the generating functional by its zero-source value cancels source-independent vacuum diagrams and the constant measure factor from field rescaling. It does not remove source-dependent ultraviolet divergences or infrared divergences; the former require the ordinary field, mass and interaction counterterms.

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