### abstract

- Weyl semimetal (WSM) feature tilted Dirac cones and can be type I or II depending on the magnitude of the tilt parameter ($C$). The boundary between the two types is at $C=1$ where the cones are tipped and there is a Lifshitz transition. The topological charge of a WSM is one. In multi-Weyl it can be two or more depending on the value of the winding number $J$. We calculate the absorptive part of the AC optical conductivity both along the tilt direction ($\sigma_{zz}$) and perpendicular to it ($\sigma_{xx}$) as a function of the tilt ($C$) and chemical potential ($\mu$). For zero tilt there is a discontinuous rise in both $\sigma_{xx}$ and $\sigma_{zz}$ at photon energy $\Omega=2\mu$ followed by the usual linear in $\Omega$ law for $\sigma_{xx}$ at $J=1,2$ and $\sigma_{zz}$ at $J=1$. For $J=2$ and $\sigma_{zz}$ the interband background is constant rather than linear in $\Omega$. For type I there is a readjustment of optical spectral weight as the tilt is increased. The absorption starts from zero at $2\mu/(1+C)$ and then rises in a quasilinear fashion till it merge with the usual undoped untilted interband background at $2\mu/(1-C)$. The discontinuous rise at twice the chemical potential of the untilted case is lost. For type II the interband background of the undoped untilted case is never recovered. For noncentrosymmetric materials the energies of a pair of opposite chirality Weyl nodes become shifted by $\pm Q_{0}$ and this leads to two separate absorption edges corresponding to the effective chemical potential of each of the two nodes at $2(\mu+\chi Q_{0})$ depending on chirality $\chi=\pm$. We provide analytic expressions for the conductivity in this case which depend only on the ratio $Q_{0}/\mu$ and tilt when plotted against $\Omega/\mu$. The signature of finite energy shift $Q_{0}$ is more pronounced for $\sigma_{zz}$ and $J=2$ than for the other cases.