Constraint on $B-L$ cosmic string from leptogenesis with degenerate neutrinos

In the early Universe, as a consequence of $U(1)_{B-L}$ gauge symmetry-breaking, the so-called $B-L$ cosmic strings are expected to be produced at the breaking scale $\eta_{B-L}$ according to the Kibble mechanism. The decaying, collapsing closed loops of these strings can release the right-handed neutrinos, whose subsequent decay can contribute to the baryon asymmetry of the Universe (BAU), through the "slow death"(SD) process and/or the "quick death"(QD) process. In this paper, we assume that the decay of the lightest heavy Majorana neutrinos released from the $B-L$ cosmic string loops can produce a baryon asymmetry consistent with the cosmic microwave background (CMB) observations. Considering the fact that both the neutrinoless double beta decay experiment and the cosmological data show a preference for degenerate neutrinos, we give the lower limits for the breaking scale $\eta_{B-L}$ with the neutrino masses $0.06 eV \leq \bar{m}=(m_{1}^{2}+m_{2}^{2}+m_{3}^{2})^{1/2} \leq 1.0 eV$, where the full possible cases of degenerate neutrinos are included. We obtain $\eta_{B-L}\gtrsim 3.3 \times 10^{15} GeV$, $5.3 \times 10^{15} GeV$ and $9.5 \times 10^{15} GeV$ for $\bar{m}=0.2 eV$, $0.4 eV$ and $1.0 eV$ respectively in the SD process, and find the $B-L$ cosmic string has a very small contribution to the BAU in the QD process.