# Loop integral in $d$ dimensions

I am studying large $$N$$ Quantum Field Theory and I am having a hard time calculating the expansion of the simple loop integral of eq.(13.123) of Peskin and Schroeder. $$\int\frac{d^dk}{(2\pi)^d}\frac{1}{k^2+m^2}=\begin{cases}C_1\Lambda^{d-2}-C_2m^{d-2}+...&\mbox{for d<4,}\\C_1\Lambda^{d-2}-\tilde C_2m^2\Lambda^{d-4}+...&\mbox{for d>4}\end{cases}\tag{13.123}$$

When I expand the integrand around $$m^2=0$$ I always get the form for $$d>0$$. I don't understand why should the form change under 4 dimensions and how the term of $$m^{d-2}$$ appears. There is a way to yield this term through the change $$k=mp$$, but shouldn't the two methods be equivalent?

• For d<4 if you substract 1/k^2 (the leading divergence) from the integrand, you obtain convergent integral (no need for cutoff anymore). Verify that such integral yields precisely what you need. Naive epansion around m=0 is not desirable, since integrals in the expansion quickly become infrared divergent (for small k) Jul 4, 2021 at 9:58