The properties of boson Regge trajectories $\alpha \mathrm{}\left(s\right)\mathrm{}$ with a left-hand cut for $s<\mathrm{}0\mathrm{}$ are studied, with the following assumptions: (a) $\mathrm{Re}\alpha \mathrm{}\left(s\right)\mathrm{}$ is linear in the physical regions of the $s$ and $t$ channels. (b) The imaginary part is asymptotically smaller than the real part. (c) At $s=0\mathrm{}$, $\alpha \mathrm{}\left(s\right)\mathrm{}$ has a branch point due to the existence of a Regge cut ${\alpha }_c\mathrm{}\left(s\right)\mathrm{}$ in the angular momentum plane, with ${\alpha }_c\mathrm{}\left(0\right)=\alpha \mathrm{}\left(0\right)\mathrm{}$. The branch point is attributed to the collision of the physical pole $\alpha \mathrm{}\left(s\right)\mathrm{}$ with a second-sheet pole $\tilde{\alpha }\mathrm{}\left(s\right)\mathrm{}$, at $s=0\mathrm{}$. (d) The function $\alpha \mathrm{}\left(s\right)\mathrm{}$ is analytic in the $s$ plane. Under these assumptions the imaginary part for each boson trajectory is obtained in terms of a parameter $A$ and a parameter-free universal function determined by solving an integral equation numerically. The parameter $A$ is determined by requiring that the first meson of each trajectory have the observed width. The model gives imaginary parts increasing almost linearly with $s$ for large $\mathrm{}\left|s\right|\mathrm{}$. The widths of the recurrences increase linearly with their mass. The results support exchange degeneracy $p-f$, $K^{*}-K_N$, $\omega -A_2$. The imaginary part, for $s<\mathrm{}0\mathrm{}$, is compared with the phenomenological results of other authors.

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