$n_{c}<1.06$, ($n_{c}=1.06$ corresponds to the observed critical doping $x_{c}=0.2$.) $d_{x^{2}-y^{2}}$-wave pairing dominates over extended-s wave, but if $n_{c}>1.06$, the situation is inverted and the effectively nodeless extended-s wave is the leading pairing symmetry. Therefore, this simple calculation indeed indicates there may be a transition at $n_{c}=1.06$ ($x_{c}=0.2$) from nodal d-wave to nodeless s-wave symmetry. When checking the doping evolution of effective mass, it can be seen that there is no clear signal of any singular behavior in both $\alpha$ ($E_{k}^{+}$) and $\beta$ ($E_{k}^{-}$) heavy quasi-particle bands, thus we expect the changes of electronic band or Lifshitz transition observed in ARPES and dHvA measurements may not be an active factor for the pairing symmetry transition, but it is an issue by itself and needs further studies.[@Dudy2013; @Polyakov2012]

Furthermore, Fig. \[fig:30\] shows the temperature-dependent superfluid density in both nodal d-wave and nodeless s-wave state for $n_{c}=1.0$ ($x=0.125<x_{c}=0.2$) and $n_{c}=1.10$ ($x=0.25>x_{c}=0.2$), respectively.

![\[fig:30\] The superfluid density of nodal-d wave (a) and nodeless-s wave (b) for $n_{c}=1.0$ ($x=0.125<x_{c}=0.2$) and $n_{c}=1.10$ ($x=0.25>x_{c}=0.2$), respectively.](ds1.eps){width="1.0\columnwidth"}

Importantly, the behaviors of calculated results are consistent with the experimental data in Ref. , where low-temperature behavior of London penetration depth is changed from power-law to exponential when doping is over 