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I am trying to follow this paper concerning decay rates in QFT. In equations (E.5), (E.6), (E.7), a functional trace is calculated using Feynman diagrams. However, I am struggling to see why

$$Tr[(-\partial^2)^{-1}W(x)] = \tilde{W}(0)\int\frac{d^dp}{(2\pi)^d}\frac{1}{p^2}$$ and $$Tr[(-\partial^2)^{-1}W(x)(-\partial^2)^{-1}W(x)] = \int\frac{d^dq}{(2\pi)^d}\frac{d^dk}{(2\pi)^d}\frac{\tilde{W}(q)\tilde{W}(-q)}{k^2(k+q)^2}.$$

How should I think about these functional traces and their Feynman diagrams?

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1 Answer 1

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First write the expression in $x$ space, and then substitute the Fourier expression for $(-\partial^{2})^{-1}$ and do the $x$-space integrals

For example $$ {\rm Tr}\left\{ W (-\partial^{2})^{-1}W (-\partial^{2})^{-1}\right\}\\ \equiv \int d^nx d^n x' W(x) G(x,x') W(x') G(x',x). $$ where $$ G(x,x')= (-\partial^{2})^{-1}_{xx'}= \int \frac{d^np}{(2\pi)^n} \frac{e^{ip(x-x')}}{p^2}. $$ Doing the $x$, $x'$ integrals gives your second equation.

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