factors, we propose that the heavy fermion quasi-particle is directly responsible for superfluid electromagnetic response, thus the corresponding paramagnetic current is $$j_{i}^{p}(q,\tau)=e\sum_{k\sigma}(v_{A}^{i}A_{k+q\sigma}^{\dag}(\tau)A_{k\sigma}(\tau)+v_{B}^{i}B_{k+q\sigma}^{\dag}(\tau)B_{k\sigma}(\tau))\nonumber$$ where the effective velocity for heavy fermion quasi-particle $A_{\sigma}$ and $B_{\sigma}$ are defined by $v_{A}^{i}=\frac{\partial E_{k}^{+}}{\partial k_{i}}$ and $v_{B}^{i}=\frac{\partial E_{k}^{-}}{\partial k_{i}}$, respectively. At the same time, the diamagnetic current is found to be $$\begin{aligned}
j_{i}^{d}(q,\tau)&&=e^{2}A_{j}\sum_{k\sigma}((\frac{1}{m_{A}})_{ij}A_{k+q\sigma}^{\dag}(\tau)A_{k\sigma}(\tau)\nonumber\\
&&+(\frac{1}{m_{B}})_{ij}B_{k+q\sigma}^{\dag}(\tau)B_{k\sigma}(\tau)).\nonumber\end{aligned}$$

Then, with the same treatment in last subsection, one can use these two heavy quasi-particle currents to derive Eq. \[eq11\]. $$\begin{aligned}
\frac{\rho_{s}(T)}{m}&&=\sum_{k}[-\frac{\partial^{2}E_{k}^{+}}{\partial k_{x}^{2}}\frac{E_{k}^{+}}{E_{k}^{A}}\tanh\frac{E_{k}^{A}}{2T}+2(\frac{\partial E_{k}^{+}}{\partial k_{x}})^{2}\frac{\partial f_ {F}(E_{k}^{A})}{\partial E_{k}^{A}}\nonumber\\
&&-\frac{\partial^{2}E_{k}^{-}}{\partial k_{x}^{2}}\frac{E_{k}^{-}}{E_{k}^{B}}\tanh\frac{E_{k}^{B}}{2T}
+2(\frac{\partial E_{k}^{-}}{\partial k_{x}})^{2}\frac{\partial f_ {F}(E_{k}^{B})}{\partial E_{k}^{B}}], \label{eq11}\end{aligned}$$ which is a useful and physically correct superfluid density formula and is the most important finding of the present paper. As shown in Fig. \[fig:SFf3\], the superfluid density is correctly vanished when transition temperature $T_{c}$ is approached from heavy fermion superconducting phase. Meanwhile, the global behavior of temperature-dependence in $\rho_{s}(T)$ is also consistent with general physical expectation, e.g. extended-s wave should gives rise to a flat $T$-dependence while the nodal $d_{x^{2}-y^{2}}$-wave pairing usually leads to a power-law behavior in low-temperature regime.[@Xiang; @Poole]

We should emphasize that our formula Eq. \[eq11\] is not only valid in superconducting mean-field model Eq.\[eq8\] but can be used in other 