Real modes and null memory contributions in effective-one-body models

We introduce a novel approach to describe real-valued $m=0$ modes from inspiral to merger and ringdown in effective-one-body models, including both oscillatory and null memory contributions. A crucial aspect of the modelization of the oscillatory part is the complexification of the real modes via a Hilbert transform. This procedure allows for an accurate description of the merger-ringdown waveform by applying standard approaches employed for the complex $m>0$ modes, which include source-driven effects. The physical signal is then recovered by solely considering the real part. We apply this method in the extreme-mass-ratio regime, considering particle-driven linear gravitational perturbations in Schwarzschild and Kerr spacetimes. We then extend our description to spin-aligned, quasi-circular, comparable-mass binaries providing hierarchical fits incorporating the test-mass limit. The post-merger waveform is then matched with an inspiral effective-one-body waveform. By adopting TEOBResumS-GIOTTO as our baseline, we also include the displacement memory in the (2,0) mode through Bondi-Metzner-Sachs balance laws, thus providing a complete effective-one-body model incorporating both oscillatory and null memory effects. The accuracy of this model is validated against the hybrid numerical relativity surrogate NRHybSur3dq8_CCE, finding, for the quadrupole of the equal mass nonspinning case, a LIGO noise-weighted mismatch of $\bar{{\cal F}} = 6\cdot 10^{-4}$ at $50 M_\odot$ for the inclination that maximizes the contribution of the (2,0) mode.