text: ωc 
  W (ωc) =  Re σ(Ω) dΩ = WK + f (ωc) 
  
  0 
  Z  ′ 
  ′ 
  ∞ 
  f (ωc) = −  Re σ(Ω) dΩ  (4) 
  ωc 
  Z 
  
  The Kubo formula, Eq. (3) is obtained assuming that 
  the second part is negligible. This is not guaranteed, 
  however, as typical ωc ∼ 1 − 2eV are comparable to the 
  bandwidth. 
  The diﬀerential sum rule ∆W is also a sum of two 
  terms 
  
  ∆W (ωc) = ∆WK + ∆f (ωc)  (5) 
  
  where ∆WK is the variation of the r.h.s. of Eq. 3, 
  and ∆f (ωc) is the variation of the cutoﬀ term. Because 
  conductivity changes with T at all frequencies, ∆f (ωc) 
  also varies with temperature. It then becomes the issue 
  whether the experimentally observed ∆W (ωc) is predom- 
  inantly due to “intrinsic” ∆WK, or to ∆f (ωc). [A third 
  possibility is non-applicability of the Kubo formula be- 
  cause of the close proximity of other bands, but we will 
  not dwell on this.] 
  
  For the NS, previous works21,22 on particular models 
  for the cuprates indicated that the origin of the temper- 
  ature dependence of W (ωc) is likely the T dependence 
  of the cutoﬀ term f (ωc). Speciﬁcally, Norman et. al.22 
  approximated a fermionic DOS by a constant (in which 