content_image/417636.jpg content_image/417639.jpg content_image/417640.jpg content_image/417650.jpg content_image/417632.jpg content_image/417633.jpg content_image/417634.jpg content_image/417643.jpg content_image/417644.jpg content_image/417645.jpg content_image/417649.jpg content_image/417648.jpg content_image/417635.jpg content_image/417637.jpg content_image/417638.jpg content_image/417642.jpg content_image/417646.jpg content_image/417651.jpg content_image/417652.jpg content_image/417653.jpg content_image/417641.jpg
factor 

where 

<img src='content_image/417636.jpg'>

<img src='content_image/417639.jpg'>

<img src='content_image/417640.jpg'>

Next we expand these functions in powers of small pa- rameter y to order O ( y $^{2}$) 

where the coefficients of the expansion (A9) can be writ- ten using the shorthand expressions D ≡ ω 2 − 4 ϵ 2 p and ζ ≡ ϵ$_{p}$/ 2 T as 

<img src='content_image/417650.jpg'>

<img src='content_image/417632.jpg'>

<img src='content_image/417633.jpg'>

<img src='content_image/417634.jpg'>

<img src='content_image/417643.jpg'>

<img src='content_image/417644.jpg'>

<img src='content_image/417645.jpg'>

<img src='content_image/417649.jpg'>

<img src='content_image/417648.jpg'>

where sech $^{2}$( x ) ≡ 1 − tanh ( x ) $^{2}$. On substituting Eqs. (A7) and (A9) in Eq. (A1) we obtain 

<img src='content_image/417635.jpg'>

The terms that are odd in x drop out on integration in symmetrical limits. Similarly, 

<img src='content_image/417637.jpg'>

<img src='content_image/417638.jpg'>

We further use the relation 

to write 

<img src='content_image/417642.jpg'>

Finally, 

<img src='content_image/417646.jpg'>

and A $_{0}$/ C 0 = − 4Δ $^{2}$/ω$^{2}$, and B $_{0}$/ C 0 = − 2Δ /ω . 

## Appendix B: Evaluating phase-space integrals 

Here we evaluate the integrals (39)–(41), which are given as 

<img src='content_image/417651.jpg'>

<img src='content_image/417652.jpg'>

<img src='content_image/417653.jpg'>

<img src='content_image/417641.jpg'>

