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I want to know how equation 2.11 (page 9) follows from 2.10 (page 8) in this paper.


The two references mentioned just before 2.11 also seem to skip this crucial step. Unless I am missing something obvious everywhere!


It would be great if someone can help with this!

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up vote 1 down vote accepted

A taste of answer, skipping numerical coefficients.

The integrand, in the integral$(2.10)$, is $ I(y)= (y^2-1)^{\large \frac{d-3}{2}}$

We are interested at the behaviour as $y \rightarrow +\infty$

So, the integrand is : $I(y)=y^{d-3}(1 - \frac{1}{y^2})^{\large \frac{d-3}{2}}$

So, you get terms, in the integrand, as :

$I(y)=a_{d-3}y^{d-3} + a_{d-5}y^{d-5} + .....a_{d-2n}y^{d-2n}....\tag{1}$

Now, there is a difference between odd and even $d$, that is :

$I_{even}(y) = a_{d-3}y^{d-3} + a_{d-5}y^{d-5} + ..a_1 y + a_{-1}y^{-1} + O(y^{-2})\tag{2}$

$I_{odd}(y) = a_{d-3}y^{d-3} + a_{d-5}y^{d-5} + ..a_0 +O(y^{-2})\tag{3}$

Now, we have to integrate the integrand between $1$ and $\frac{R}{\delta}$, with $\delta \rightarrow 0$: $J = \int_1^{\frac{R}{\delta}} dy~I(y) $

For even dimensions, you integrate the term $a_{-1}y^{-1}$ between $1$ and $\frac{R}{\delta}$, adn this gives a term $ a_{-1} \ln (\frac{R}{\delta})$. The complete integral is ((the constant $K,K'$ above ,comes frome the integration at the point $1$)):

$$J_{even} = b_{d-2}(\frac{R}{\delta})^{d-2} + b_{d-4}(\frac{R}{\delta})^{d-4} + ...+b_{2}(\frac{R}{\delta})^{2} + a_{-1} \ln (\frac{R}{\delta}) + 0(\frac{\delta}{R})\tag{4}+ K$$

$$J_{odd} = b_{d-2}(\frac{R}{\delta})^{d-2} + b_{d-4}(\frac{R}{\delta})^{d-4} + ...+b_{1}(\frac{R}{\delta}) + 0(\frac{\delta}{R})\tag{5}+ K'$$

The paper said that "Of course, the power-law divergences in the R´enyi entropies are not universal, e.g., see [5, 6], however, a universal contribution can be extracted from the subleading terms."

So, we have, finally (and noting that $K$ is neglectible relatively to $ln (\frac{R}{\delta}$)):

$J_{even} \sim a_{-1} \ln (\frac{R}{\delta}), J_{odd} \sim K'$

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Thanks! I think I got confused in understanding what is meant by "the subleading terms" - so you basically wrote down "all" the terms in 2.10 (of which they wrote just the first 2) to find the required terms. But physically its not obvious as to why one should be interested in these terms and not the rest of the series. I guess I have to go to these references 5 and 6 to understand that! (..unless you have a short answer to that!...) –  user6818 Aug 14 '13 at 14:48
    
May be you can help answer my other 2 entanglement questions also? :) - physics.stackexchange.com/questions/73909/… and physics.stackexchange.com/questions/73612/… –  user6818 Aug 14 '13 at 14:50
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