  Within effective field theory we explore the properties of the $d$=3+1
quantum XY model at low temperatures and in weak magnetic or staggered fields.
For this parameter regime only few results appear to be known, and furthermore
are restricted to one-loop order. In the present study we systematically
analyze the thermodynamics of the $d$=3+1 quantum XY model up to three-loop
order. In the low-temperature expansion of the free energy density, the free
Bose gas term of order $T^4$ receives corrections of order $T^6$ and $T^8$. The
discussion also includes the pressure, (staggered) magnetization and
susceptibility. In particular, we show how these quantities are influenced by
the spin-wave interaction. We then compare our findings with those for the
quantum XY model in $d$=2+1.
