2 in a given band is compensated by an appropriate change of the spectral weight in other bands such that the total spectral weight, integrated over all bands, is conserved, as in Eq. (1). Still, non-conservation of the spectral weight within a given band is an interesting phenomenon as the degree of non-conservation is an indicator of rele- vant energy scales in the problem. Indeed, when relevant energy scales are much smaller than the Fermi energy, i.e., changes in the conductivity are confined to a near vicinity of a Fermi surface (FS), one can expand ε$_{k}$ near k$_{F}$ as ε$_{k}$ = v$_{F}$ ( k − k$_{F}$ ) + ( k − k$_{F}$ ) $^{2}$/ (2 m$_{B}$ ) + O ( k − k$_{F}$ ) 3 and obtain ∇ 2 ⃗ $_{k$_{x}$}$ε$_{⃗}$ k ≈ 1 /m$_{B}$ [this approximation is equiv- alent to approximating the density of states (DOS) by a constant]. Then W$_{K}$ becomes πne $^{2}$/ (2 m$_{B}$ ) which does not depend on temperature. The scale of the tempera- ture dependence of W$_{K}$ is then an indicator how far in energy the changes in conductivity extend when, e.g., a system evolves from a normal metal to a superconductor. Because relevant energy scales increase with the interac- tion strength, the temperature dependence of W$_{K}$ is also an indirect indicator of whether a system is in a weak, intermediate, or strong coupling regime. In a conventional BCS superconductor the only rele- vant scales are the superconducting gap Δ and the impu- rity scattering rate Γ. Both are generally much smaller than the Fermi energy, so the optical integral should be almost T -independent, i.e., the spectral weight lost in a superconducting state at low frequencies because of gap opening is completely recovered by the zero-frequency δ - function. In a clean limit, the weight which goes into a δ − function is recovered within frequencies up to 4Δ. This is the essence of FGT sum rule $^{2,3}$. In a dirty limit, this scale is larger, O (Γ), but still W$_{K}$ is T -independent and there was no “violation of sum rule”. The issue of sum rule attracted substantial interest in the studies of high T$_{c}$ cuprates 5–18,21–26 in which pairing is without doubts a strong coupling phenomenon. From a theoretical perspective, the interest in this issue was orig- inally triggered by a similarity between W$_{K}$ and the ki- netic energy K = 2 ∑ ε$_{⃗}$ $_{k}$n$_{⃗}$$_{k}$. 18–20 For a model with a sim- ple tight binding cosine dispersion ε$_{k}$ ∝ (cos k$_{x}$ + cos k$_{y}$ ), d $^{2}$ε$_{⃗}$ k d k 2 x ∼ − ε$_{⃗}$ k and W$_{K}$ = − K . For a more complex dis- persion there is no exact relation between W$_{K}$ and K , but several groups argued 17,27,28 that W$_{K}$ can still be regarded as a good monitor for the changes in the kinetic energy. Now, in a BCS superconductor, kinetic energy increases below T$_{c}$ because n$_{k}$ extends to higher frequen- cies (see Fig.2). At strong coupling, K not necessary increases because of opposite trend associated with the fermionic self-energy: fermions are more mobile in the SCS due to less space for scattering at low energies than they are in the NS. Model calculations show that above some coupling strength, the kinetic energy decreases be- low T$_{c}$ $^{29}$. While, as we said, there is no one-to-one cor- respondence between K and W$_{K}$ , it is still likely that, when K decreases, W$_{K}$ increases. A good amount of experimental effort has been put into where Δ W$_{K}$ is the variation of the r.h.s. of Eq. 3, and Δ f ( ω$_{c}$ ) is the variation of the cutoff term. Because conductivity changes with T at all frequencies, Δ f ( ω$_{c}$ ) also varies with temperature. It then becomes the issue whether the experimentally observed Δ W ( ω$_{c}$ ) is predom- inantly due to “intrinsic” Δ W$_{K}$ , or to Δ f ( ω$_{c}$ ). [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 works 21,22 on particular models for the cuprates indicated that the origin of the temper- ature dependence of W ( ω$_{c}$ ) is likely the T dependence of the cutoff term f ( ω$_{c}$ ). Specifically, Norman et. al. 22 approximated a fermionic DOS by a constant (in which The differential sum rule Δ W is also a sum of two terms The Kubo formula, Eq. (3) is obtained assuming that the second part is negligible. This is not guaranteed, however, as typical ω$_{c}$ ∼ 1 − 2 eV are comparable to the bandwidth. Theoretical analysis of these results 21,22,25,28,30 added one more degree of complexity to the issue. It is tempt- ing to analyze the temperature dependence of W$_{K}$ and relate it to the observed behavior of the optical integral, and some earlier works 25,28,30 followed this route. In the experiments, however, optical conductivity is integrated only up to a certain frequency ω$_{c}$ , and the quantity which is actually measured is The analysis of the optical integral showed that in over- doped cuprates it definitely decreases below T$_{c}$ , in con- sistency with the expectations at weak coupling $^{11}$. For underdoped cuprates, all experimental groups agree that a relative change of the optical integral below T$_{c}$ gets much smaller. There is no agreement yet about the sign of the change of the optical integral : Molegraaf et al. 8 and Santander-Syro et al. 9 argued that the optical inte- gral increases below T$_{c}$ , while Boris et al. 10 argued that it decreases. addressing the issue of the optical sum rule in the c − axis and in-plane conductivities 8–16 in overdoped, optimally doped, and underdoped cuprates. The experimental re- sults demonstrated, above all, outstanding achievements of experimental abilities as these groups managed to de- tect the value of the optical integral with the accuracy of a fraction of a percent. The analysis of the change of the optical integral between normal and SCS is even more complex because one has to (i) extend NS data to T < T$_{c}$ and (ii) measure superfluid density with the same accuracy as the optical integral itself. W ( ω$_{c}$ ) = ∫ ω$_{c}$ 0 Re σ (Ω) d Ω = W$_{K}$ + f ( ω$_{c}$ ) f ( ω$_{c}$ ) = − ∫ $^{′}$∞′ ω$_{c}$ Re σ (Ω) d Ω (4) Δ W ( ω$_{c}$ ) = Δ W$_{K}$ + Δ f ( ω$_{c}$ ) (5)