Effects of pair creation on the expansion of vacuum bubbles

The real-time behavior of true-vacuum bubbles nucleated in the false-vacuum background is studied in a λ?⁴ theory. Classically, the bubble expansion rate has been known to approach the velocity of light asymptotically. A quantum effect, creation of Higgs and fermion pairs, is studied by a semiclassical method, and is shown to lead to a slower expansion rate. Within the thin-wall approximation, several possible asymptotic behaviors are examined. A solution that behaves like the classical one but with different coefficients is shown to be stable. The first-order corrections to the coefficients are calculated approximately. The created Higgs pairs are found to remain inside the bubble and carry an energy of approximately 10^?3 λ of the energy released from the false vacuum. The fermions that are massless in the false vacuum go out of the bubble and have energy ～10^?3g², where g is the Yukawa coupling constant. It is shown that the resulting asymptotic state is Lorentz invariant.