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.
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