# Log-interaction term calculation

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I have a question regarding calculating the following integral with cutoff. $$\int_{-\infty}^{\infty} \frac{d\omega}{|\omega|} \cos(\omega(\tau_i-\tau_j)-1)$$

How should I set up the correct cutoff so that the result of the integral becomes: $$-2\ln \frac{\tau_i-\tau_j}{\tau_c}$$ where $\tau_c$ is the short time cutoff~$E_F^{-1}$.

This calculation is from certain resonant tunneling problem of Luttinger liquid, for which you can obtain renormalization group flow equation.

This post imported from StackExchange Physics at 2014-04-13 11:24 (UCT), posted by SE-user huyichen
retagged Apr 13, 2014

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rewrite the cosine in terms of exponenials, and add a tiny real part to the exponents.

In more detail, rewrite the integral as twice the integral from 0 to $\infty$, abbreviate $\tau_i-\tau_j$ by $t$, differentiate with respect to $t$ to get rid of the denominator, use $cos x = (e^{ix}+e^{-ix})/2$, and change $e^{iz}$ to $e^{iz-\epsilon\omega}$ to be able to perform the integration. Then set $\epsilon=0$ and integrate the result from $t=\tau_c \approx 0$ to $t=\tau_i$-$\tau_j$.

answered Apr 13, 2014 by (15,787 points)
edited Apr 15, 2014

Could you explain a little bit more in detail?

In the final integration w.r. to $t$, one needs the cutoff mentioned.
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