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In Le Bellac's book, Quantum and Statistical Field Theory, the renormalization constant $Z_3$ is introduced with the equation $$ \Gamma^{(2)}_R(k^2, m^2, g) = Z_3 \Gamma^{(2)}(k^2, m_0^2, g_0; \Lambda) $$ where $m_0$ and $g_0$ are the bare mass and coupling and $m$ and $g$ are the renormalized mass and coupling. Shortly after, the author writes that, by dimensional analysis, $Z_3$ can be a function only of $g$ and $\Lambda/m$ ( $Z_3 = Z_3(g,\Lambda/m$) ). It's not immediate to me why is it so; can somebody explain it?

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$Z_3$ is dimensionless. The only independent dimensionless quantities are : $g, \frac{\Lambda}{m}$ and $\frac{\Lambda^2}{k^2}$ But the momenta $k$ are not renormalizable quantities, momenta are which they are, so $Z_3$ cannot depends on momenta. The renormalizable quantities are the coefficients of kinetic terms, mass terms, or coupling terms in the Lagrangian. So, finally, $Z_3 = Z_3(g, \frac{\Lambda}{m})$ – Trimok Aug 18 '13 at 10:59
Thank you very much for the quick and clear answer! – Alex A Aug 18 '13 at 11:15

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