For the complex scalar field theory

$L = -\partial_{\mu}\phi^{*}\partial_{\mu}\phi - m^{2}\phi^{*}\phi + J\phi^{*}+J^{*}\phi$

1. Why is there no factor of 1/2 in the lagrangian like in the real scalar field?

2. Can we say Y = 0 (normalization) because we know the two-point function $<0|T\phi(x)\phi(x')|0> = 0$ and so $<0|\phi(x)|0> = 0$ is satisfied?

For the complex scalar field theory

$$L = -\partial_{\mu}\phi^{*}\partial_{\mu}\phi - m^{2}\phi^{*}\phi + J\phi^{*}+J^{*}\phi,$$

1. Why is there no factor of 1/2 in the lagrangian like in the real scalar field?

2. Can we say $ Y = 0$ (renormalization) because we know the two-point function $<0|T\phi(x)\phi(x')|0> = 0$ and so $<0|\phi(x)|0> = 0$ is satisfied?