We have theoretically obtained an expression for the current density in a terahertz field due to hot-electron injection in carbon nanotubes. The injection modifies the stationary distribution function and leads to a qualitative change in the behavior of the current-voltage characteristics and causes absolute negative conductivity. We compared the current-voltage characteristic behavior at different injection rates and observed a drastic change in the current density and absolute negative conductivity values. We propose that carbon nanotubes with hot-electron injection may be useful for high-frequency applications.