Determination of $\mathrm{SU}(4{)}_{\mathrm{TC}}$ technicolor gauge group from embedding in extended technicolor

In technicolor theories using a $\mathrm{SU}(N_{\mathrm{TC}})$ gauge group, the value of $N_{\mathrm{TC}}$ is not, a priori, determined and is typically chosen by phenomenological criteria. Here we present a novel way to determine $N_{\mathrm{TC}}$ from the embedding of a one-family technicolor model, with fermions in the fundamental represention of $\mathrm{SU}(N_{\mathrm{TC}})$, in an extended technicolor theory, and use it to deduce that $N_{\mathrm{TC}}=4$ in this framework.