Is a period used after a stand alone phrase?

No, you don't need one. I agree with you that it looks better without one.

Consider UPS whose slogan is "synchronizing the world of commerce". They don't use a period:

The following image has quite a few corporate taglines:

Most don't have periods. The ones that do do so to be more assertive, which is not something your particular tagline has to convey. The only exception is Microsoft's which is two separate clauses.