# Zee's explanation of expressing bare coupling by physical coupling

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 $$\lambda_P$$?

• It is supposed to be a perturbation series and $\lambda_P(\lambda)$. This is just a formal matter as you are managing infinities here as $\Lambda$ should go to infinity. Anyway, if properly done, it works very well. It is called renormalization. – Jon Apr 9 at 13:56

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.