content_image/410306.jpg content_image/410310.jpg content_image/410311.jpg content_image/410312.jpg content_image/410314.jpg content_image/410319.jpg content_image/410321.jpg content_image/410323.jpg content_image/410327.jpg content_image/410334.jpg content_image/410331.jpg content_image/410325.jpg content_image/410329.jpg
<img src='content_image/410306.jpg'>

## IV. REGAINING THE LIMIT OF ZERO TEMPERATURE 

Comparing Eq. (3.20) with the zero temperature result derived in Ref. [34], and previously known for the cylin- drical shell [8, 10], it is not obvious that our expression simplifies to the zero temperature result as τ → 0. In this section we show that upon careful examination the correct limit is in fact obtained. Let us write down the zero temperature result ˜ E 0 for general p in its regularized form suitable for comparison 1 (c.f. Ref. [34], Eq. (4.14)): 

<img src='content_image/410310.jpg'>

<img src='content_image/410311.jpg'>

<img src='content_image/410312.jpg'>

The finite temperature quantity ¯ E of Eq. (3.5) is ana- lytic as τ → 0 and inverse application of the transition (2.5) simply gives us 

What remains is essentially to determine the low τ be- havior of K ( τ ) and S ( τ, a ) to check that the last term of Eq. (4.1a) may be regained. 

<img src='content_image/410314.jpg'>

To study the behavior of S it is convenient to employ the symmetry relation S ( τ, p ) = S ( p, τ ) which gives 

<img src='content_image/410319.jpg'>

For large arguments K$_{0}$ ( x ) ∝ $^{exp(}$− x ), so the sum over l is exponentially small as τ → 0. This immediately gives the asymptotic behavior: 

<img src='content_image/410321.jpg'>

Next we turn to K ( τ ). Using the Euler-Maclaurin for- mula (e.g. Ref. [39], p.806) we have 

<img src='content_image/410323.jpg'>

with 

The integral has the solution 

<img src='content_image/410327.jpg'>

as τ → 0. Moreover, κ ( τ ) = − 1 /τ + 1 − 1 $_{2}$τ 2 + .. , and upon inspection we recognize that 

<img src='content_image/410334.jpg'>

To leading order in τ , thus, the sum in Eq. (4.5) reads 1 τ ∑ ∞ l =1 B$_{2}$$_{l}$/ 2 l . As is typically the case for series expan- sions close to non-analytical points, the series is formally divergent. It can, however, be regularized by means of Borel summation [41]. For a highly similar problem and details on how to approach it, see [42]. We show in Ap- pendix A that the Borel regularized sum evaluates to 

Thus we have found the low- τ expansion of K ( τ ): 

<img src='content_image/410331.jpg'>

Further terms cancel at least to order τ $^{2}$, the leading order correction being at least of order τ $^{4}$. 

<img src='content_image/410325.jpg'>

<img src='content_image/410329.jpg'>

