This paper is organized as follows. In Sec. II, we briefly review the HTC model Hamiltonian; in Sec. III, we dis-cuss the polaron decoupling effect and its influences on the polariton linewidth, as well as the scaling relations of polariton linewidth with N under the fast and slow limits; in Sec. IV, we provide analytic linewidth expressions for LP and UP under the fast limit and the general detuned cases; in Sec. V, we present numerical exact simulations using HEOM for the polariton LA spectra and compare their linewidths with the theoretical predictions; in Sec. VI, we briefly conclude the paper. In addition, we denote the full width at half maximum (FWHM) of the peaks in LA spectra as the linewidth throughout this paper unless specified.