  Using $15.6$ $\rm fb^{-1}$ of $e^+e^-$ collision data collected at
twenty-four center-of-mass energies from $4.0$ to $4.6$ GeV with the BESIII
detector, the helicity amplitudes of the process $e^+e^-\to
\pi^{+}\pi^{-}\omega$ are analyzed for the first time. Born cross section
measurements of two-body intermediate resonance states with statistical
significance greater than 5$\sigma$ are presented, such as $f_{0}(500)$,
$f_{0}(980)$, $f_{2}(1270)$, $f_{0}(1370)$, $b_{1}(1235)^{\pm}$, and
$\rho(1450)^{\pm}$. In addition, evidence of a resonance state in $e^+e^-\to
\pi^+\pi^-\omega$ production is found. The mass of this state obtained by line
shape fitting is about 4.2 GeV/$c^2$, which is consistent with the production
of $\psi(4160)$ or $Y(4220)$.
