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The main goal of the paper is to investigate theoretically the experimentallyobserved fragmentation of the isovector $M1$ resonance in $^{208}$Pb within aself-consistent model based on an energy-density functional (EDF) of the Skyrmetype. This fragmentation (spread of the $M1$ strength) is not reproduced in aconventional one-particle--one-hole ($1p1h$) random-phase approximation (RPA)and thus has to be investigated in the framework of more complicated models.However, previously applied models of this type were not self-consistent. Inthe present work, we use a recently developed renormalized version of theself-consistent time blocking approximation (RenTBA) in which the$1p1h\otimes$phonon configurations are included on top of the RPA $1p1h$configurations. We have determined several sets of the parameters of themodified Skyrme EDF fitted within the RenTBA and RPA and have found thenecessary condition of producing the fragmentation of the $M1$ resonance in$^{208}$Pb in our model. We present also the results of the RenTBA and RPAcalculations for the first excited states of the natural parity modes in$^{208}$Pb obtained with these modified parametrizations.

M1 resonance in $^{208}$Pb within the self-consistent phonon-coupling model