Effects of Universal Extra Dimensions on $B^0 - {\bar B^0}$ Mixing

We study contributions coming to $\Delta m_B$ from one or more universal extra dimensions (UED) in which all the Standard Model fields can propagate. In the model with a single UED, the box diagrams for mixing are convergent and therefore insensitive to the cutoff scale of the theory. In the case of two UEDs, the result is not very sensitive to the cut-off scale due to GIM mechanism. Within the present range of the parameters at $1 \sigma$ level, the lower bound on the compactification scale 1/R has been estimated to be 165 GeV for one UED and 280 GeV for two UEDs. The bound increases drastically if one can have a better determination of the $B$ meson decay constant $f_B$ and the QCD correction parameter $B_B$. For example, it rises to 740 GeV (for one UED) if the error (at $1\sigma$) in the determination of $f_B\sqrt{B_B}$ from quenched lattice calculation is reduced to one-third from its present value. The UED contributions to the $K$ system are strongly suppressed.