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The theory of a complex scalar field $\chi$ is given by

$$\mathcal{L}=\partial_{\mu}\chi^{*}\partial^{\mu}\chi-m_{\chi}^{2}\chi^{*}\chi.$$


Why is it not common to include a factor of $\frac{1}{2}$ in front of the complex $\chi$ kinetic term?

What is the effect on the propagator of including a factor of $\frac{1}{2}$?

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The prefactor of the kinetic term is just a convention - you can rescale your field by a constant factor to make the prefactor anything you want without affecting the physics (doing so would of course change the other coefficients in the Lagrangian as well). The conventional prefactor is chosen to be 1/2 for a real scalar field and 1 for a complex scalar field so that the equations of motion you get from the Euler-Lagrange equations are normalized in a "simple" way without any overall prefactors. (For a real scalar field you have $\frac{1}{2} \partial_\mu \varphi\, \partial^\mu \varphi$, and differentiating the square cancels the factor of 1/2 in the equations of motion. For a complex scalar field you have $\partial_\mu \varphi^\dagger \partial^\mu \varphi$ and you differentiate w.r.t. $\partial_\mu \varphi$, so there's no square would need to be canceled by a factor of 1/2.)

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