  The analytical estimations on the Friedberg-Lee-Sirlin typed Q-balls is
performed. The two-field Q-balls are also discussed under the one-loop
motivated effective potential subject to the temperature. We argue under the
analytical consideration that the parameters from the potential can be
regulated to lead the energy per unit charge of Q-balls to be lower to keep the
model stable. If the energy density is low enough, the Q-balls can become
candidates of dark matter. It is also shown rigorously that the two-field
Q-balls can generate in the first-order phase transition and survive while they
are affected by the expansion of the universe. The analytical evaluations show
that the Q-balls with one-loop motivated effective potential can exist with the
adjustment of coefficients of terms. We cancel the infinity in the energy to
obtain the necessary conditions consist with those imposed in the previous
work. According to the approximate expressions instead of curves versus the
model parameters with a series of fixed values, the lower temperature will
reduce the energy density, so there probably have been more and more stable
Friedberg-Lee-Sirlin typed Q-balls to become the dark matter in the expansion
of the universe.
