I am recently trying to understand dimensional regularization in the context of quantum field theory. So to solve an integral $$\int_{\mathbb R^d} \frac{\text d ^d p}{(2 \pi)^d} \frac{1}{(p^2 + m^2)^2}$$ which is divergent for $$d = 4$$ it is often said that if one assumes that $$d \in \mathbb C$$ with sufficiently small $$\text{Re}(d)$$ it is possible to do some kind of analytic continuation. But in examples it is often shown that one uses $$d = 4 - \varepsilon$$ and then transforms to hyper spherical coordinates to get an expression like: $$\int_{\mathbb R^+}\frac{\text d p}{(2\pi)^{4-\varepsilon}} \frac{2\pi^{(4-\varepsilon)/2}}{\Gamma\left(\frac{4 - \varepsilon}{2}\right)} \frac{p^{3-\varepsilon}}{(p^2 + m^2)^2}$$ which has a solution. We then expand for small values of $$\varepsilon$$.

Question:

Why is it necessary for $$d$$ to be complex?$$\quad\rightarrow$$ is there a possibility to stay real ?

Why is it legal to transform in hyper spherical coordinates when complex dimension is assumed ?

Do i get right that we expand the solution to the second integral in a laurent series for small $$\varepsilon$$ to extract terms which do not diverge at $$\varepsilon=0$$ and then when adding amplitudes the divergent terms cancel out ( dependent on the process of course ) ?

• FWIW, analytic continuation into the complex plane comes for free. Do you want to stay on the real line? May 2 '20 at 14:00
• @Qmechanic i reread my question and commit that it was kind of unclear what i am asking. So i edited it. I know it is not welcome to ask multiple questions in one thread but since these are kind of related to the same problem i hope it is excused in this case. May 2 '20 at 14:40

3) Well the expansion in $$\epsilon$$ is right, you expand in a laurent series in $$\epsilon$$ (This uniquely defines your analytic continuation btw). You don't just add amplitudes that you got from your previous lagrangian. There are several ways to formulate the process of Renormalization but what I find the most convenient is via counterterms. You include additional, divergent terms in your lagrangian by redefinition of your masses, fields and coupling constants. These new terms induce new Feynman diagrams which you have to include in your amplitude calculation and the divergencies from your previous loop integrals cancel with these new feynman diagrams (at least for uv divergencies as far as I know, I'm still learning so I haven't gotten to infrared divergencies yet)