Selling a house with a current one year lease

[Jeff D.]

A neighbor wants to buy our California rental property in the city of Watsonville. We just signed up a new family for a one-year lease.  The potential buyer says they will "assume" the lease for its term.   

If we do sell, does this require the tenant to sign a new lease?  

One concern is that if they don't sign a new lease (or refuse), and the new owner breaks some provision of the lease or a landlord tenant law, we might be liable. 

If it matters, the lease is between the tenants and our business name (a fictitious business name, not a registered corp or an LLC), but signed by us.

Thanks for any wisdom.