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We report on measurements of the oxygen isotope ($^{16}$O/$^{18}$O) effect(OIE) on the transition temperature $T_{\rm c}$ and the zero-temperaturein-plane magnetic penetration depth $\lambda_{\rm ab}(0)$ in the stoichiometriccuprate superconductor YBa$_2$Cu$_4$O$_8$ by means of muon-spinrotation/relaxation. An analysis of the temperature evolution of$\lambda^{-2}_{\rm ab}$ in terms of coexisting $s+d-$wave order parametersreveals that the OIE on the superfluid density $\rho_{\rms}(0)\propto\lambda^{-2}_{\rm ab}(0)$ stems predominantly from the $d-$wavecomponent while the contribution of the $s-$wave one is almost zero. The OIE onthe transition temperature $T_{\rm c}$ is found to be rather small: $\deltaT_{\rm c}/T_{\rm c}= -0.32(7)$%, compared to the total OIE on the superfluiddensity $\rho_{\rm s}(0)$: $\delta \rho_{\rm s}(0)/\rho_{\rm s}(0)=-2.8(1.0)$%. The partial OIE's on the corresponding $d-$wave and $s-$wavecomponents of $\rho_{\rm s}(0)$ are $\delta \rho_{\rm s,d}(0)/\rho_{\rm s}(0)=-3.0(1.2)$%, and $\delta \rho_{\rm s,s}(0)/\rho_{\rm s}(0)= 0.2(1.2)$%,respectively. Our results demonstrate that polaron formation in the CuO$_2$planes is the origin of the observed OIE in the $d-$wave channel. In the muchweaker $s-$wave channel, fermionic quasiparticles are present, which do notcontribute to the OIE on $\rho_{\rm s}(0)$. Our results support the originalidea of K. Alex M\"{u}ller on the polaronic nature of the supercarries inhigh-temperature cuprate superconductors.

Oxygen isotope effect on the superfluid density within the $d-$wave and $s-$wave pairing channels of YBa$_2$Cu$_4$O$_8$