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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?

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    $\begingroup$ 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) $\endgroup$ Jul 4, 2021 at 9:58

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