In chapter 10 of Peskin & Schroeder the renormalization of the $\phi^4$ theory is discussed. The renormalization conditions are imposed such that the infinities are absorbed into the counter term variables, if I understand correctly. My question is, why does one use the specific renormalization conditions? I find in a later post why do the renormalization conditions related to the exact propagator exist (Understanding renormalization conditions in the $\phi^4-$theory), but it is not discussed much, neither in P&S, nor in the aforementioned post why do we have to impose the amputated 4 point diagram should be equal to $-i\lambda$ at $s=4m^2$ and $t=u=0$?
Any help would be appreciated.
Best Answer
I think P&S explain their motivation in the book, if I'm not wrong (I haven't opened it in a long time!). I think the choice is motivated from an experimental POV. Suppose you were performing an $2\to2$ scattering experiment in the CM frame at energy $2E$. Then, $$s=4E^2, \qquad t=-4(E^2-m^2) \sin^2\frac{\theta}{2} , \qquad t=-4(E^2-m^2) \cos^2\frac{\theta}{2}. $$ To experimentally test out our theory, we first need to do a couple of "setup" experiments. Note that $\lambda$ and $m$ are parameters in our theory which must be fixed by experiment (i.e. the theory does not by itself predict these values). To fix these, we do one experiment at some energy scale $E_0$ which will fix $\lambda(E_0)$ and $m(E_0)$. Once this is known, the theory predicts the results of all other experiments at all energy scales. Every other experiment then can be used to test out the theory.
OK, now what's the best choice for $E_0$? Well, one natural choice is to measure it at the lowest possible energy scale, which in this case is $E_0=m$ (when $E<m$ there is no particle excitation so there is nothing to scatter). So, we perform the experiment at this scale and find $\lambda(m) = \lambda_0$ and $m(m) = m_0$. The renormalization condition chosen by P&S precisely defines the renormalized coupling and mass to these experimentally measured values.
Note that there are many other useful renormalization conditions. Sometimes, it is convenient to fix $\lambda$ in the unphysical regime of scattering (say when $t>0$). This choice makes the theoretical calculations easier, but then a few extra steps have to be taken to match anything to experiment.