# One-loop diagram calculation: Wick rotation and gamma matrices

I have a question about a calculation which is performed in David Tong's lecture notes on Gauge Theory (page 400, chapter 8). At the bottom (see below for a screenshot), we want to calculate the one-loop diagram. He Wick rotated to Euclidean space. I don't understand how to obtain the second line from the first? When he rewrites the trace?

Also, given that $$\left\{ \gamma^{\mu}, \gamma^{\nu} \right\} = 2 \eta^{\mu \nu}$$ with $$\eta^{\mu \nu} = (+1, -1, -1)$$ (we are in $$(2+1)$$-dimensions), what does this become after Wick rotation? Does it become $$\left\{ \gamma^{\mu}, \gamma^{\nu} \right\} = 2 \delta^{\mu \nu}$$ with the standard Euclidean metric?

I was thinking of doing: $$(\gamma^{\mu} k_{\mu} + m) (\gamma^{\nu} k_{\nu} - m) = k^2 - m^2$$ (this is true in Lorentzian signature, not sure if after Wick rotation). From this it would follow that $$\frac{1}{ \gamma^{\mu} k_{\mu} + m} = \frac{ \gamma^{\nu} k_{\nu} - m}{ k^2 - m^2}.$$ But I think there is a sign mistake, and not sure how to rewrite the term with $$1/ ((\gamma^{\mu} p_{\mu} + \gamma^{\nu} k_{\nu}) + m)$$.

• The last equation seems to be correct, if we say that we already did a Wick rotation, i.e. $k = (ik^0, \vec k)$ (although I am not sure where the $k^0$ interal went?) and $k^2$ actually denotes the euclidean scalar product, i.e. $k^2 = (k^0)^2 + \vec k^2$. Does that help you? Sep 20, 2021 at 15:14
• Not really, my question is how to obtain the trace in the last equation of Tongs notes from the previous equation. Sep 21, 2021 at 7:25

The last equality on the page follows immediately from seeing that the inverse of $$\gamma_{\mu} k^{\mu} + m$$ is $$-\frac{\gamma_{\mu} k^{\mu} - m}{k^2 + m^2}$$. If we assume that all scalar products are euclidean scalar products and either the $$k^{\mu}$$ is numerator is considered Wick rotate, $$k^{\mu} = (i k^0 ,\vec{k})$$ (this would be somewhat incosistent) or the gamma matrices are considered Wick rotated $$\gamma^{\mu} \rightarrow (i \gamma^0, \vec \gamma)$$, which implies $$\{ \gamma_{\mu}, \gamma_{\nu} \} = -2 \delta_{\mu \nu}$$ (this might be what he means by "gamma matrices square to $$-1$$"). $$\frac{\gamma_{\mu} k^{\mu} - m}{k^2 + m^2} (\gamma_{\nu} k^{\nu} + m) = \frac{\gamma_{\mu} \gamma_{\nu} k^{\mu} k^{\nu} - m^2}{k^2 + m^2} = \frac{-(k^0)^2 - \vec k^2 - m^2}{k^2 + m^2} = - 1.$$ I agree that the notation would be somehow weird and since I am not familiar with Tong's lecture notes this may also be an error.
• Ok I understand this. But what about the other term? Containing the $p$ ? It will contain cross terms if you try to do the same trick? Sep 22, 2021 at 9:12