modified by replacing the bosonic field-dependant masses by their thermally corrected values [@DWG]. The thermal corrections to the bosonic masses are given in Appendix B.

The $B+L$ anomalous interactions [@B+L] that violate the baryon number do not have the same rate in the symmetric and broken phases ( i.e., at both sides of the bubble wall). In the symmetric phase, this rate behaves like $%
\sim T^{4}$ [@Symm], and is suppressed as $exp(-E_{Sp}/T)$ [@bro], in the broken phase, where $E_{Sp}$ is the system static energy within such field configuration called the sphaleron [@Sph]. Therefore any generated baryon number at the symmetric phase will get erased , unless these interactions are switched off in the broken phase, which translates to the condition (\[v/t\]). In reality, the $B+L$ anomalous interactions should be switched off at a temperature $T_{0}<T_{n}<T_{c}$, where $T_{n}$ is the temperature at which bubbles start to nucleate, and $T_{0}$ is the temperature at which the barrier between the two minima completely disappears. Then the condition in (\[v/t\]) should be fulfilled at $T_{n}$, but, in general, the two values are significantly close, and if $v(T_{c})/{%
T_{c}}>1$, it is necessarily satisfied at $T_{n}$. Indeed, in some earlier works (e.g. [@T0; @MSSMcheck]), the 