The properties of boson Regge trajectories $\ensuremath{\alpha}(s)$ with a left-hand cut for $s<0$ are studied, with the following assumptions: (a) $\mathrm{Re}\ensuremath{\alpha}(s)$ 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 $s=0$, $\ensuremath{\alpha}(s)$ has a branch point due to the existence of a Regge cut ${\ensuremath{\alpha}}_{c}(s)$ in the angular momentum plane, with ${\ensuremath{\alpha}}_{c}(0)=\ensuremath{\alpha}(0)$. The branch point is attributed to the collision of the physical pole $\ensuremath{\alpha}(s)$ with a second-sheet pole $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\alpha}}(s)$, at $s=0$. (d) The function $\ensuremath{\alpha}(s)$ 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 $|s|$. The widths of the recurrences increase linearly with their mass. The results support exchange degeneracy $p\ensuremath{-}f$, ${K}^{*}\ensuremath{-}{K}_{N}$, $\ensuremath{\omega}\ensuremath{-}{A}_{2}$. The imaginary part, for $s<0$, is compared with the phenomenological results of other authors.