Let the logarithm of the random varible $T_i$, associated with the lifetime of the $i$th individual in a survival study, follow the disitribution
$$log(T_i) = \mu + X_i\beta + \sigma\epsilon_i$$
with $\mu$ being a constant that is equal for all individuals, $X_i$ a vector of covariate values with its coeffcient vector $\beta$, $\epsilon_i$ a random error term for individual $i$ that follows a particular probability distribution, and $\sigma$ a so called scale parameter. Since the effect of covariates is proportional with regard to the survival time, this model is called an accelerated failure time model.
I am about confused that this scale paramter $\sigma$ is included. Why not simply leave it out, so that the log-linear formulation of this accelerated failure time model looks the formulation of a linear regression model? Is this so that $T_i$ follows a known probability distribution?