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As far as I know, the easiest way to do the integration by hand is for integer dimensions (i.e. Mathematica gives you the integral in terms of hypergeometric functions, but that's not really helpful). You can derive the integral (with a hard cut-off $\Lambda$) with respect to $m^2$, perform the integral, expand in $m/\Lambda$, and then integrate back. It ...


You can simply compute the integral using your preferred regularization method (cut-off, dimensional, Pauli-Villars...), and if all goes well (which is not guaranteed), the divergences will not depend on your parameter and they will eventually disappear when you compute physical stuff. If this does not happen, maybe your theory is simply ill-defined. And as ...

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