Zee's explanation of expressing bare coupling by physical coupling

Question

In terms of bare parameter $\lambda$, the $\phi\phi\to\phi\phi$ scattering amplitude is $\lambda\phi^4$ theory is given by $$\mathcal{M}=-i\lambda+iC\lambda^2\Big[\ln\Big(\frac{\Lambda^2}{s}\Big)+\ln\Big(\frac{\Lambda^2}{t}\Big)+\ln\Big(\frac{\Lambda^2}{u}\Big)\Big]+\mathcal{O}(\lambda^3)\tag{1}$$ where $C$ is a computable numerical constant independent of $\lambda$ and $\Lambda$. In expression $(1)$, the parameter $\lambda$ is bare parameter and the cut-off $\Lambda$ is unknown.

According to Anthony Zee's book of QFT (See Eq.4, page 149 of Quantum Field Theory in a Nutshell) the whole point of renormalization is to measure the physical coupling $\lambda_P$ at a reference value of $s_0,u_0,t_0$ and express the bare parameter $\lambda$ in terms of the physical coupling $\lambda_P$ measured experimentally. To this end, he writes $$-i\lambda_P=-i\lambda+iC\lambda^2\Big[\ln\Big(\frac{\Lambda^2}{s_0}\Big)+\ln\Big(\frac{\Lambda^2}{t_0}\Big)+\ln\Big(\frac{\Lambda^2}{u_0}\Big)\Big]+\mathcal{O}(\lambda^3)\tag{2}.$$

Why does he write $-i\lambda_P$ in the LHS? More concretely, how did he know the LHS is linear in the physical coupling $\lambda_P$?

Answer 1

Eq. (2) is the definition of the renormalized coupling $\lambda_P$. The definition of $\lambda_P$ is that it is the value such that the resummed 4-vertex has scattering amplitude $-\mathrm{i}\lambda_P$. That's what we mean by "physical coupling": the value the coupling would have to have so that a single tree-level vertex would yield the same amplitude as the fully resummed vertex of the actual theory.