Why is autocorrelation used without normalization in signal processing field?
Autocorrelation (sometimes also termed aucovariance) is often used in the context of stationnary time signals.  As the $\sigma_t$ and $\sigma_s$ are "equal". Since the autocorrelation is symmetric and peaks at $\tau=0$, $R_x(\tau)$ (as noted by @robert bristow-johnson) is often regarded as the natural reference. In a lot of applications, one mostly look at its scale-variant properties: as a measure on how signal chunks look similar, or modeling its decay, or by fitting a linear model through it. 

Sometimes, to distinguish the normalized version, it is called an autocorrelation coefficient function.

When dealing with non-stationnary signals, it would be unnatural to stationnarize it artificially. For instance, the Wigner-Ville distribution is based on the Fourier transform of the unnormalized non-stationary autocorrelation function.