Once clonal selection is done, B cells would start dividing and producing antibodies. So, after an antigen is eliminated, what stops the division of B cells and antibody production?
Upon activation plasma B cells upregulate death receptors as part of being activated. Presence of the antigen overcomes the death signal, thus the cell survives. When antigen is lost, the death signals overcome the survival signals as there is no antigen, so the B cell dies via apoptosis.
Memory B cells don't do this. They survive and continue to produce antibodies for years, although this slowly wanes. Antibodies have to be produced as antibody mediated immune mechanisms are extremely important for memory and the reason why a subsequent infection is cleared so rapidly. Furthermore for the example of HPV, antibodies induced by the vaccine prevent the virus even infecting cells (so called neutralising antibodies).
You are right to say that Plasma Cells are terminally differentiated, so they don't become the memory cells, or calm back down or anything. But all Plasma Cells are not the same, there are Short-Lived Plasma Cells (SLPCs) that are less specific, but arise rapidly after stimulation, and there are Long-Lived Plasma Cells (LLPCs) that come about later and have more specific antigen binding. SLPCs are rapidly activated, but also rapidly lost. These are the cells that Android was talking about when s/he said that apoptotic programs are initiated as soon as cells become activated and are eventually eliminated by these mechanisms. LLPCs are the cells that stay in your body for a much longer time (theoretically for life, but probably not practically unless you get hit by a bus in a couple of years). When you get a vaccination, these are the cells that produce antibody that can be detected in the blood for a number of years (what testing your antibody titer looks for). These cells are in pretty limited supply, so we know less about them. My guess is that they are not actively eliminated, but may become eliminated over time by other mechanisms (purposely undefined here).
My understanding of this question is: Antibodies are produced by plasma cells which are terminally-differentiated B cells. Maturation of B cells to plasma cells is stimulated by the presence of antigen (T cells, cytokines, etc. etc).
In the absence of antigen the existing plasma cells undergo apoptosis, and the maturation of the cognate B cells will not be stimulated, so no more plasma cells will be produced.
I have found it surprisingly difficult to find references in support of these statements, although there is a large literature on the suppression of apoptosis in memory B cells.