Contact terms in Dyson-Schwinger equation can be ignored?

Question

According to this text here

http://www.physics.indiana.edu/~dermisek/QFT_09/qft-II-4-4p.pdf

contact terms do not affect the scattering amplitude. But These contact Terms are there; the question is: When contact Terms are relevant for scattering Amplitude computation?

My idea:

By starting with the connected Partition function $G[J]:=\log Z[J]$ where $Z[J]$ is ordinary Partition function corresponding to the Action

$$S = S_\mathrm{theory} + \int d^4x\ J\phi$$

for some fields $\phi$ and the source $J$ one can derive cumulants belonging to $S_\mathrm{theory}$ by multiple Derivation of $G[J]$ by $J$ and Setting $J=0$. Only the equation for quadratic cumulants $\langle0|\phi(x) \phi(y)|0\rangle$ will contain an equation with the contact term $\delta(x-y)$. More precisely

$$\mathcal{H} \langle0|\phi(x) \phi(y)|0\rangle = f(\text{others})+\delta(x-y)$$

for an Operator $\mathcal{H}$ that I assume to be linear and nonlinear corrections $f(\text{others})$.

Neglecting nonlinearities I see that $\langle0|\phi(x) \phi(y)|0\rangle$ is exactly the Green function generated by $\mathcal{H}$. This Green function $\Delta(x-y)$ vanishes if the Observation time $t$ is set to $\infty$. And infinitely Observation times are assumed in the LSZ formula for scattering amplitudes.

Will contact Terms be relevant for finite Observation times? Why on scattering Amplitude/ cross section computation infinitely Long Observation times are assumed?

No real process has infinitely Long Observation times. But maybe uncertainty in energy is cancelled if $\Delta t \mapsto \infty$ is assumed.

Help would be greatly appreciated.

Answer 1

The contact terms for a correlator of $n$ fields are made of the correlators of $n-1$ fields and can therefore not have $n$ poles in the momenta of the fields.

The LSZ formalism generally shows that a correlator contributing to a scattering amplitude, which is by definition a "matrix element" of the connected part of the S-matrix (i.e. the amplitude corresponding to connected Feynman diagrams), has precisely $n$ poles. Therefore, whatever the contact terms do, they cannot appear in scattering amplitudes. This is a subtlety in the LSZ reduction formula - when we write (schematically) $$\langle p \vert S \vert q \rangle\vert_\text{connected} = (-p^2 + m^2)(-q^2+m^2)\langle \phi(q)\phi(p)\rangle,$$ we really only mean the part of $\langle \phi(q)\phi(p)\rangle$ with the correct pole structure. The parts with fewer poles - the contact terms - contribute to disconnected scattering processes, which we generally do not consider when talking about "scattering amplitudes".

This observation is crucial e.g. in deriving the Ward identities for scattering amplitudes from the Schwinger-Dyson equations.