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Real and complex scalar fields mass terms in a lagrangian density

Physics Asked by Adrián Terrones on December 24, 2020

Consider a lagrangian density of the form:
$$ cal{L}=partial_muphi_1partial^muoverline{phi}_1+partial_muphi_2partial^muphi_2-lambda_1phi_1overline{phi}_1-lambda_2phi_2phi_2 $$
Where $phi_1$ is a complex scalar field and $phi_2$ is a real scalar field. In every example I have seen, the mass of the complex $phi_1$ field is taken as $m_1^2=lambda_1$, but the mass of the real $phi_2$ field is taken as $m_2^2=frac{1}{2}lambda_2$. Where does that $frac{1}{2}$ factor come from?

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