Dear Rabbi, Is it permissible for one to brush teeth on a minor fast day without toothpaste? Thank you.

ב"ה Shalom The Shulchan Aruch states that even on a minor fast day, it is improper for one to rinse his mouth .(שו"ע או"ח תקס"ז :ג) However, the Mishna Brura there says that if one feels discomfort not rinsing his mouth, one may rinse his mouth provided that he is careful not to swallow anything. If one feels great discomfort it is permissible even on tisha B'av, but not Yom kippur. In regard to your question, there is no problem at all since you are not putting any liquid in your mouth to swallow. According to the Mishna brura above you are allowed to use mouth wash to rinse your mouth on a fast day. All the best