baxb - this is Bacopa and is grown in the UK as tender summer bedding. If you want to keep it over the winter it needs to be in a light frost free place such as a greenhouse or porch.

Are you certain it has no roots? If that is really the case it is just behaving like a cut flower and is surviving on the moisture in the soil. All you can do is leave it in the conditions I suggest and see what happens. These are usually discarded in the Autumn and bought new the following year.

I can not believe these cuttings do not have roots, look closer. Soft wood or tip cuttings either root quickly or die as they have very little energy stored in their soft wood. I do have hardwood cuttings taken last January still alive and gradually using the stored energy stored in the wood. They have gradually died over time directly in proportion to the diameter of the wood and amount of stored energy. Although they all calloused none grew roots. Al