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Isospin splitting of nucleon effective mass from giant resonances in $^{208}$Pb

Based on mean field calculations with Skyrme interactions, we extract a constraint on the isovector effective mass in nuclear matter at saturation density $ρ_0$, i.e., $m_{v}^{\ast}(ρ_0)=(0.77\pm0.03) m$ by combining the experimental data of the centroid energy of the isovector giant dipole resonance (IVGDR) and the electric dipole polarizability $α_{\mathrm{D}}$ in $^{208}$Pb. Meanwhile, the isoscalar effective mass at $ρ_0$ is determined to be $m_{s}^{\ast}(ρ_0)=(0.91\pm0.05) m$ by analyzing the measured excitation energy of the isoscalar giant quadrupole resonance (ISGQR) in $^{208}$Pb. From the constrained $m_{s}^{\ast}(ρ_0)$ and $m_{v}^{\ast}(ρ_0)$, we obtain the isospin splitting of nucleon effective mass in asymmetric nuclear matter of isospin asymmetry $δ$ at $ρ_0$ as $[m_n^{\ast}(ρ_0,δ)-m_p^{\ast}(ρ_0,δ)]/m = Δm^*_1(ρ_0) δ+ O(δ^3)$ with the linear isospin splitting coefficient $Δm^*_1(ρ_0) = 0.33\pm0.16$. We notice that using the recently corrected data on the $α_{\mathrm{D}}$ in $^{208}$Pb with the contribution of the quasideuteron effect subtracted slightly enhances the isovector effective mass to $m_{v}^{\ast}(ρ_0)=(0.80\pm0.03) m$ and reduces the linear isospin splitting coefficient to $Δm^*_1(ρ_0) = 0.27\pm0.15$. Furthermore, the constraints on $m_{v}^{\ast}(ρ)$, $m_{s}^{\ast}(ρ)$ and $Δm^*_1(ρ)$ at other densities are obtained from the similar analyses and we find that the $Δm^*_1(ρ)$ increases with the density.