Synthetic antibodies would leave your body within a few days.
The sources I looked at seem to agree that natural IgG antibodies have a half-life of about 21 days, and monoclonal antibodies are similar (range of 10-30 days). This is antibodies in general, I didn't find something specific for Covid antibodies.
This means that immediately after an infusion of monoclonal antibodies, the antibody level in your blood will be at its highest. Three weeks after infusion, it'll be half that, and six weeks after infusion it'll be 25% of the original peak (that is, another half will be gone), etc.
It's similar after you get Covid - Of the antibodies produced at that time half will be gone after three weeks, 75% gone after 6 weeks, etc.
But the difference is that during the natural infection, the B cells have been stimulated to become antibody-producing factories, and so after the first antibody molecules are gone, the B cells can keep churning out more antibodies. One particular antibody molecule is gone, but an identical one takes its place, so the antibody level measured in the tests remains high for longer than three weeks. With the infusion of antibodies, the B cells haven't been activated, so you get just that one helpful surge of antibodies, and when it's gone, it's gone.
Now, the truth is, I'm not sure if this applies to someone like YitzyS who was probably infected for a few days before getting the infusion. Maybe some B cells
were activated, and maybe his current antibody level is actually a mixture of antibodies that he made and those that were infused? Could be. Or maybe the infused antibodies are so high that they send a signal to the B cells that "We've got this covered. Don't need you." and the B cells inactivate? I don't know, and I don't know if it's been studied, but I don't think it's important.
What's important is this: The reason they recommend waiting is because they're concerned that if the vaccine is given soon after the antibody infusion, the vaccine may not be as effective. What the mRNA vaccine does is stimulate your body to make Spike protein, which is similar to the Spike protein on the virus. When the B cells and T cells see that Spike protein, they're "fooled" into thinking it's the virus, and they react by producing many more B and T cells, churning out antibodies, and creating the memory B and T cells that are important for long-lasting immunity. But the infusion contained antibodies to that same Spike protein and they will glom onto Spike proteins and neutralize them before they have the chance to be seen by the B and T cells. This would be good, of course, if the Spike were on a virus, and the antibodies were neutralizing the virus. But here, the antibodies would be neutralizing the protein that's trying to activate your immunity.
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