When performing loop integrals in theories containing Dirac fermions, one almost always confronts terms of the form


For instance, in $d$ dimensions, we could compute the simple trace $\text{Tr}\left[\gamma^{\mu}\gamma_{\mu}\right]$ as


Typically, at this point, textbooks set $\text{Tr}[\textbf{1}]$. However, it is well known that the size of the $\gamma$ matrices in various dimensions is not always $4$, but is given by

$$\textbf{dim}(\gamma)=2^{\lfloor d/2\rfloor}.$$

Since this is clearly not an analytic function, how does one deal with this in dimensional regularization? Do we need to define a smooth function that interpolates between all of the integer $d$ values of this equation?

I have a slight feeling that the answer is going to be something along the lines of "It doesn't matter, because the order $\epsilon$ term in the trace can only contribute a constant to the $\mathcal{O}(1)$ terms in the full answer (after being multiplied by a $1/\epsilon$ term), and such constants can be absorbed by the counter-term, or in the definition of $\mu$." And I feel like that might be right, but I'm having a hard time convincing myself that this works.


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