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minish to zero at the high energy. This obvious weakness is a natural consequence of the neglect of the imaginary part of the self-energy (then the scattering rate). To obtain the full charge carrier Green function (6) within the kinetic energy driven SC mechanism, we 17,25 have made a self-consistent calculation in the static limit ap- proximation for the real part of the charge carrier self- energy ReΣ ( h ) 1 ( ω ) induced by the interaction between the charge carriers and spins, where its antisymmetric part has been obtained as Z − 1 hF = 1 − ReΣ ( h ) 1 o ( k , ω = 0) |$_{k}$$_{=[}$$_{π,}$$_{0]}$ and therefore is closely related to the charge carrier quasiparticle coherent weight, while its symmetric part ReΣ ( h ) 1 e ( k , ω = 0) is a constant around the Fermi surface, and it just renormalizes the chemical potential. How- ever, the imaginary part of the charge carrier self-energy ImΣ ( h ) 1 ( k , ω ) has been dropped $^{17,25}$. This leads to a fact that the imaginary part of the renormalized electron self- energy in the dressed electron Green’s function (10) due to the presence of the impurity scattering potential (9) 

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has been neglected. In this case, for the present discus- sions of the QSI extinction at the high energy, we need to introduce an imaginary part of the electron self-energy (then the scattering rate) in the electron Green function as, 

for a compensation due to the neglect of the imaginary part of the renormalized self-energy, where Γ and α are constant. It has been argued 34 that this type of the en- ergy dependent inelastic scattering rate (19) seems to be an intrinsic property of the electronic structure of cuprate superconductors, since it has been used to provide a ro- bust fit for the spatially inhomogeneous differential con- ductances of cuprate superconductor based on the phe- nomenological d-wave BCS formalism. By considering this scattering rate (19) in the electron Green function, we therefore find that the dispersing QSI always disap- pears at the high energy. To show this point clearly, we have made a series of calculations for δZ ( q , ω ) with different momenta and energies, and the result of the amplitudes of the lowest energy peaks in | δZ ( q , ω ) | as a function of energy along the [0 , 0] → [ π, π ] direction for temperature T = 0 . 002 J at the doping concentra- tion δ = 0 . 15 for parameters Γ = 0 . 005 J and α = 0 . 2 in the presence of single point-like potential scatterer of strength V = 0 . 1 J is plotted in Fig. 5 in comparison with the corresponding STM experimental result 14 of cuprate superconductors (inset). In comparison with the result in Fig. 4b, although the amplitude of the peak is severely suppressed due to the presence of the scattering rate (19), it diminishes to zero (then the QSI extinction) at the high energy. Moreover, during the calculations, we find that the constant scattering rate Γ in Eq. (19) only plays a subsidiary role, while the QSI modulation is further weakened by the effective scattering rate αω . The similar conclusion has been obtained in Ref. 34 where they also find that the effective scattering rate αω plays a key role. Using a reasonably estimative value of J ≈ 120 ∼ 150 meV, the peak disappears around the energy 6 meV ∼ 7.5 meV. Apparently, there is still a substantial difference be- tween theory and experiment, namely, the energy value of the QSI extinction calculated theoretically is smaller than the corresponding energy value measured in the ex- periment. However, upon a closer examination one can see immediately that the main difference is due to fact that the calculated peak energy decreases rapidly with energy at high energy. This energy value of the QSI ex- tinction is also smaller than the corresponding value of the charge carrier pairing gap $^{25}$. The simple t - J model can not be regarded as a complete model for the quantita- tive comparison with cuprate superconductors, however, as for a qualitative discussion in this paper, the overall shape seen in the theoretical result is qualitatively con- sistent with that observed in the STM experiment $^{14}$. 

A nature question is why QSI below the high energy in cuprate superconductors and its extinction at the high energy can be described qualitatively in the framework of the kinetic energy driven SC mechanism in the presence of a single point-like impurity. This may be understood from the kinetic energy driven SC mechanism itself $^{17}$. 

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