Scalar propagator ( $Δ_{E} (p)$ )

For a free massive scalar in a Euclidean path integral, inversion of the quadratic kernel gives

Δ_{E} (p) = 1/ (p^{2} + m^{2})

. The Minkowski version follows by Wick rotation.

Table of contents
- Smooth-cutoff scalar propagator Scalar propagator
- Shell-restricted scalar propagator Scalar propagator

Smooth-cutoff scalar propagator ( $C_{Λ} (p) = 1/ f_{Λ} ((p^{2} + m^{2}) / Λ^{2})$ )

 0  0

A smooth-cutoff scalar propagator is the inverse of a positive regulated quadratic kernel. It agrees with the unregulated scalar propagator at low momentum and decreases rapidly at high momentum. Smooth suppression is not identical to vanishing support. Its cutoff derivative is the line weight in an exact renormalization-group flow.

Shell-restricted scalar propagator ( $C_{>} (p) = 1_{Λ^{'} < ∣ p ∣ \leq Λ} / (p^{2} + m^{2})$ )

 0  0

The Gaussian contraction used while integrating out a momentum shell. Its Fourier support lies entirely in that shell. In particular, convolution with a purely low-momentum field is zero. This distinguishes shell Feynman diagrams from unrestricted loop diagrams.