may encounter the mentioned problem.

After all, we indeed confront a serious problem on the calculation of superfluid density.

![\[fig:SFf2\] Normalized superfluid density $\rho_{s}(T)/\rho_{s}(0)$ versus normalized temperature T/T$_{c}$ for $d_{x^{2}-y^{2}}$ (a) and extended-s (b) wave pairing symmetry by using Eq. \[eq10\].](SF33.eps){width="1.0\columnwidth"}

A useful and correct superfluid response formula
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![\[fig:SFf3\] Normalized superfluid density $\rho_{s}(T)/\rho_{s}(0)$ versus normalized temperature T/T$_{c}$ for $d_{x^{2}-y^{2}}$ (a) and extended-s (b) wave pairing symmetry by using Eq. \[eq11\].](SF11.eps){width="1.0\columnwidth"}

In last section, we have seen that the standard process of calculating superfluid response in fact breaks down. However, there exists a hint to resolve this question by noting that when collective Kondo screening develops, the quasi-particle of the system is the heavy fermion quasi-particle $A_{k\sigma}$ and $B_{k\sigma}$. If all interesting physics occurs at energy-scale smaller than the width of quasi-particle band $E_{k}^{+}$ and $E_{k}^{-}$, the heavy fermion quasi-particle can be seen as the only fundamental or true particles in our problem, thus we may expect the low-energy thermodynamics and transport are dominated by those heavy quasi-particle alone. For heavy fermion superconductivity, it is commonly believed that the pairing results from heavy quasi-particle but not directly involving with original conduction electron.[@Poole; @Monthoux2001; @Coleman; @Pfleiderer2009] Therefore, by considering these two 