Energy recovery twin linear $e^+e^-$, $e^-e^-$ colliders (ERLC ) with high luminosities and accelerating gradients

A recently proposed superconducting linear collider with energy recovery (ERLC) and multiple beam reuse employs twin RF structures to eliminate parasitic collisions in the linacs. Such a collider can operate in either pulsed or continuous-wave (CW) mode, achieving a luminosity of ${\cal O}(10^{36})$ cm$^{-2}$s$^{-1}$ at $2E_0$ = 250--500 GeV. This paper demonstrates that in pulsed mode, the ERLC luminosity is independent of the accelerating gradient for a fixed total power, enabling operation at the highest available gradients. A similar independence holds for the CW mode when the available power significantly exceeds the operational threshold. The luminosity scales with the cavity quality factor as $L\propto Q_0^{1/2}$. We also present, for the first time, a study of a twin $e^-e^-$ ERLC and estimate its performance. This configuration is simpler than the $e^+e^-$ version as it eliminates the need for beam recirculation; electrons can be generated anew for each cycle. In this case, the luminosity scales as $L\propto Q_0^{1/4}$. Furthermore, the use of traveling-wave (TW) RF structures allows for higher gradients and reduced thermal loading. We show that an ERLC with $G$ = 40 MeV/m can operate in CW mode, reaching luminosities of $L_{e^+e^-}$= (1-2.5)$\times 10^{36}$ and $L_{e^-e^-}$= (3-7)$\times 10^{36}$ cm$^{-2}$s$^{-1}$ at $2E_0$ = 250 and 500 GeV, respectively, with a total power consumption of 150-300 MW. These results position the ERLC as a highly promising candidate for a future Higgs factory.