text: In partial wave expansion, the thermal averaged cross section is given by 
  
  dw 
  ds (cid:19) 
  
  T 
  mN (cid:21)(cid:12) 
  s=4m2 
  (cid:12) 
  N 
  (cid:12) 
  (cid:12) 
  
  1 
  m2 
  N (cid:20) 
  dw 
  ds (cid:12) 
  s=4m2 
  (cid:12) 
  N 
  (cid:12) 
  (cid:12) 
  
  3 
  2 (cid:18) 
  4m2 
  N 
  σv  w(s)  2w(s)  = 
  h  i  −  − 
  
  T 
  mN 
  ,  = 6 
  
  with 
  
  4m2 
  s 
  
  s  d cos θ 
  
  1 
  8π r 
  2 =  2, 
  ﬁnal 
  4w(s)  dLIPS  − 
  
  2 X |M| 
  ≡ Z 
  X |M| 
  Z 
  
  where mﬁnal is the mass of ﬁnal state particle. 