Why we choose d=4 in renormalization? Why we choose $d=4$ in renormalization?
In the QFT book by Ryder in chapter 9, page 308 after doing some dimension analysis for $D$, superficial degree of divergence, he wrote "in the case $d=4$, we have $D=4-E$ which gives the correct result for diagrams." and in 312 there is a table for comparison Canonical dimensions of some quantities in $d=4$ and $d$ in general.
But why exactly $d=4$ works? Could you please some help?
 A: By "choose $d = 4$", do you mean "expand about $d = 4$ at the beginning" or "set $d = 4$ at the end"? If it's the latter, the answer is just that we live in $d = 4$ dimensions, so that's the most interest case to consider in the real world.
If you mean the former, the answer is more interesting. In condensed matter and statistical physics, we often work with $\varphi^4$ theory in two or three dimensions, but still expand around $d = 4$ dimensions and at the end set the $\epsilon$ in the actual dimension $d - \epsilon$ to $\epsilon = 1$ or $2$ rather than zero. This is because $d = 4$ is the upper critical dimension for $\varphi^4$ theory (and all the other interactions of the Standard Model). Above the upper critical dimension, the interaction becomes irrelevant under RG flow and therefore negligible in the IR limit - in terms of the path integral, the saddle-point approximation becomes exact. Since the Feynman expansion can be thought of as corrections to the saddle-point approximation, the upper critical dimension is the lowest one over which we have analytic control, making it the natural one about which to expand. Put another way, $d = 4$ is the dimension for which the coupling constant $g$ is dimensionless, so we can expand directly in $g$ itself rather than in some ratio of $g$ to some characteristic energy scale. Put yet another way, the infinities that give us trouble start to kick in right below the upper critical dimension, so they are the "smallest" (without being zero) near that dimension, making it the natural base dimension for a perturbative expansion.
