Z factor in renormalization disappeared?

[geoduck]

Suppose you have λø⁴ theory and calculate the bare 4-point function:

$$
\Gamma_0(s,t,u)=\lambda_0+\lambda_0^2f(s,t,u)\\
=\left[\lambda_0+\lambda_0^2f(0,0,0)\right]+\lambda_0^2\left[f(s,t,u)-f(0,0,0)\right]
$$

We then take a measurement at (s,t,u)=(0,0,0) and call the result λ_R. Then

$$
\Gamma_0(s,t,u)=\lambda_R+\lambda_R^2\left[f(s,t,u)-f(0,0,0)\right]
$$

But isn't this also the finite, renormalized \Gamma_R(s,t,u)? We didn't use any Z factors to convert bare to renormalized, instead we just plugged in a measurement.

Shouldn't the relation be \Gamma_0(s,t,u)=\frac{1}{Z^2}\Gamma_R (s,t,u)

 

[geoduck]

Never mind. When you make a measurement at (s,t,u)=(0,0,0), you're really measuring Z^2 \Gamma_0(0,0,0), since the pole of each propagator contributes Z and each external line has \frac{1}{\sqrt{Z}}. So \lambda_R=Z^2 \Gamma_0(0,0,0), leading to

$$\Gamma_0(s,t,u)=\frac{1}{Z^2}\lambda_R+\lambda_0^2\left[f(s,t,u)-f(0,0,0)\right]\\
\frac{1}{Z^2}\lambda_R+\lambda_R^2\left[f(s,t,u)-f(0,0,0)\right]\\
=\frac{1}{Z^2}\Gamma_R(s,t,u)
$$

which is correct to order λ².