Optical and Radio Variability of the Blazar S4 0954+658

We present an optical-to-radio study of the BL Lac object S4 0954+658 observations during 1998–2023. The measurements were obtained with the SAO RAS Zeiss-1000 and AS-500/2 0.5-m telescopes in 2003–2023, with the RATAN-600 radio telescope at 1.25 (0.96, 1.1), 2.3, 4.7 (3.7, 3.9), 8.2 (7.7), 11.2, 22.3 (21.7) GHz in 1998–2023, with the IAA RAS RT-32 Zelenchukskaya and Badary telescopes at 5.05 and 8.63 GHz in 2020–2023, and with the RT-22 single-dish telescope of CrAO RAS at 36.8 GHz in 2009–2023. In this period the blazar was showing extremely high broadband activity with the variability amplitude of the flux densities up to 70–100\(\%\) both in the optical and radio domains. During the period of 2014–2023 the blazar displayed extremely high activity in the radio wavelengths, and we detected multiple radio flares of varying amplitude and duration. The large flares last on average from 0.3 to 1 year at 22–36.8 GHz and slightly longer at 5–11.2 GHz. The optical flares are shorter and last 7–50 days. The characteristic time scale \(\tau\) of variation at 5–22 GHz is about 100 days in the most active epoch of 2014–2023 and about 1000 days for the state with lower activity in 2009–2014. We found a general correlation between the optical, radio, and \(\gamma\)-ray flux variations, which suggesting that we observe the same photon population from different emission regions. We estimated the linear size of this region as 0.5–2 pc for different conditions. A broadband radio spectrum with two components of the S4 0954+658 jet was modeled using both electrons and protons as emitting particles. The results suggest that the synchrotron radio waves in this AGN may be produced by relativistic protons.