Vector-Like Lepton Pair Production With Polarized Beams at Linear Colliders:Sensitivity Projections and Chirality Observables

We study pair production of a vector-like lepton doublet at polarized future linear colliders, focusing on the charged and neutral channels $e^+e^-\to\tau^\prime\bar{\tau}^\prime$ and $e^+e^-\to\nu^\prime\bar{\nu}^\prime$. The benchmark masses are $M_{\rm VLL}=1000,1200~{\rm GeV}$ at CLIC with $\sqrt{s}=3~{\rm TeV}$ and $M_{\rm VLL}=390,460~{\rm GeV}$ at ILC with $\sqrt{s}=1~{\rm TeV}$. Using the realistic polarization configurations LR$=(-0.8,+0.3)$, RL$=(+0.8,-0.3)$, LL$=(-0.8,-0.3)$, and RR$=(+0.8,+0.3)$, we evaluate tree-level production-level cross sections and construct observables designed to test the electroweak structure of the doublet. The charged channel is consistently larger than the neutral channel because it receives both photon and $Z$ exchange. In the LR configuration, the charged-channel rates reach $23.32$ and $20.28~{\rm fb}$ at CLIC, and $188.83$ and $129.04~{\rm fb}$ at ILC, for the two benchmark masses at each collider. We express rate reach through the projected visible-fraction requirement $f_{\rm vis}^{95}=3/(\mathcal{L}\sigma_{\rm prod})$, keeping the result independent of a specific decay selection. To quantify charged--neutral separation we use the absolute discriminator $D_\sigma$, which reaches about $0.546$ at CLIC and $0.542$ at ILC in the LR benchmark. We also find a stable observed asymmetry separation, $|\Delta A_{LR}^{\rm obs}|\simeq 0.45$--$0.46$, between the charged and neutral channels. The corresponding production-level statistical projection gives sizeable $Z_A$ values for the benchmark luminosities, scaling as $\sqrt{\mathcal{L}_{\rm tot}f_{\rm vis}}$ under an equal LR/RL luminosity split. These results demonstrate that the beam polarization can provide a representation-sensitive diagnostic of vector-like lepton doublets, beyond a simple rate enhancement.