Binary stellar evolution has been studied as an important pathway to initiate various transient events like supernovae (SNe). Although the common envelope (CE) phase in a binary, outcomes of the CE phase, and conditions for SN explosion during the CE phase are uncertain, it has been suggested that SN explosions can be triggered during the CE phase. In this work, we explore the formation and evolution routes of carbon/oxygen (CO) white dwarf (WD) binaries in order to investigate mergers of CO WDs and cores of nondegenerate stars during the CE phase as possible origins for SNe under the core merger detonation (CMD) scenario by considering several physical binary models. The evolution of CO WD + intermediate-mass normal (nondegenerate and hydrogen-rich) star binaries lead to mergers during the CE phases and may still trigger Type Ia SNe (SNe Ia) interacting with circumstellar material in different models. Mergers between CO WDs and the cores of He (nondegenerate and helium-rich) stars within the common envelope (CE) phase are rare compared to other CE merger events. These two channels may produce peculiar SNe Ia such as overluminous/super-Chandrasekhar-mass SNe Ia as a certain fraction of them have combined core masses ≥2 M ⊙. In the channel of CO WD + massive star (≥8 M ⊙) CE events, we find that rates of mergers between CO WDs and He cores of massive stars are (0.85–12.18) , which may initiate Type II superluminous SNe like SN 2006gy, and delay times from this scenario are in the range of 34–120 Myr. Our results based on the CMD model are comparable with observational results of peculiar SNe.
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