Why are the renormalisation constants the same in LSZ and renormalisation?

To keep it simple I'll phrase everything in terms of scalar fields. We seem to have three constants called $$Z$$:

1. When we do LSZ reduction we say, as $$t\rightarrow-\infty$$ then $$\phi\rightarrow \sqrt{Z}\phi_{free}$$.

2. Later we renormalised the theory and rescaled the bare fields $$\phi_0$$ as $$\phi_0=\sqrt{Z}\phi_R$$.

3. We also have a factor $$Z$$ that is the residue of the two-point correlation function $$\frac{Z}{p^2-m^2_R}$$.

Why are these all the same $$Z$$? (ok, i sort of see that for 2 we can choose to rescale the field however we want, but that doesn't explain why 1 and 3 are the same)

• Apr 19 '20 at 13:00

The number $$Z$$ is defined as being overlap between the desired one particle state $$|k\rangle$$ with $$k^2=m_R^2$$ and the actual (more complicated) state created by $$\phi(x)$$. In other words
$$\langle k|\phi(x)|{\rm vac}\rangle= \sqrt Z e^{-ikx}.$$ This gives both the residue of the pole in the Lehman-Kallen expansion, and the need for $$Z$$ in LSZ. Note that $$Z\le 1$$, with equality if it's a free theory.