I'm just a beginner learning renormalization. I have a short question: In interacting QFT, why the previous Lorentz invariant scaler mass $m$ will be replace by physical mass $m_\lambda$ (or say renormalized mass) once we introduce interaction into theory? Or equally: Why interaction will bring a correction to mass term?
And also, electric charge and field operator (wave function) will do corresponding corrections, are the physical nature of these corrections are same?
In QFT one notion of mass is the coefficient of the mass term quadratic in the fields. Coefficients in front of each term in the action are coupling constants, which potentially run. The elementary electric charge $e$, which is the coefficient in front of the $\bar{\psi}\gamma^{\mu}A_{\mu}\psi$ action term, is another running coupling constant.
Perhaps the running is easiest to see in the Wilsonian effective action (WEA) when integrating out heavy modes: Each action term in the WEA gets modified by amputated Feynman diagrams, made from heavy modes and with the same number of external fields as the action term in question, cf. e.g. my Phys.SE answer here.
