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We provide a unified interpretation of both paramagnon and plasmon modes inhigh-$T_c$ copper-oxides, and verify it quantitatively against availableresonant inelastic $x$-ray scattering (RIXS) data across the hole-doped phasediagram. Three-dimensional extended Hubbard model, with included long-rangeCoulomb interactions and doping-independent microscopic parameters for bothclasses of quantum fluctuations, is used. Collective modes are studied usingVWF+$1/\mathcal{N}_f$ approach which extends variational wave function (VWF)scheme by means of an expansion in inverse number of fermionic flavors($1/\mathcal{N}_f$). We show that intense paramagnons persist along theanti-nodal line from the underdoped to overdoped regime and undergo rapidoverdamping in the nodal direction. Plasmons exhibit a three-dimensionalcharacter, with minimal energy corresponding to anti-phase oscillations onneighboring $\mathrm{CuO_2}$ planes. The theoretical spin- and chargeexcitation energies reproduce semi-quantitatively RIXS data for $\mathrm{(Bi,Pb)_2 (Sr, La)_2 CuO_{6+\delta}}$. The present VWF+$1/\mathcal{N}_f$ analysisof dynamics and former VWF results for static quantities combine into aconsistent description of the principal properties of hole-doped high-$T_c$cuprates as strongly correlated systems.

A unified theory of spin and charge excitations in high-$T_c$ cuprates: Quantitative comparison with experiment and interpretation