Noether Gauge Symmetry of Modified Teleparallel Gravity Minimally Coupled with a Canonical Scalar Field

This paper is devoted to the study of Noether gauge symmetries of $f(T)$ gravity minimally coupled with a canonical scalar field. We explicitly determine the unknown functions of the theory $f(T),V(\varphi ),W(\varphi )$. We have shown that there are two invariants for this model, one of which defines the Hamiltonian $H$ under time invariance (energy conservation) and the other is related to scaling invariance. We show that the equation of state parameter in the present model can cross the cosmological constant boundary. The behavior of Hubble parameter in our model closely matches to that of $\mathrm{\Lambda }$CDM model, thus our model is an alternative to the later.