Phantom energy supported wormhole model in $f(R,\,T)$ gravity assuming conformal motion

In this article, we have discussed Morris and Thorne (MT) wormhole solutions in a modified theory of gravity that admits conformal motion. Here we explore the wormhole solutions in $f(R,\,T)$ gravity, which is a function of the Ricci scalar ($R$) and the trace of the stress-energy tensor ($T$). To study wormhole geometries, we make assumption of spherical symmetric static spacetime and the existence of conformal Killing symmetry to get more acceptable astrophysical outcomes. To do this, we choose the expression of $f(R,\,T)$ as $f(R,T)= R+2 \gamma T$. Here we employ the phantom energy EoS relating to radial pressure and density given by $p_r=\omega \rho$ with $\omega<-1$ to constrain our model. Following a discussion of wormhole geometry and behavior of shape function, the study moves on to the computation of proper radial distance, active mass function, the nature of total gravitational energy and a discussion on the violation of energy conditions. We have shown that the wormhole solutions exist for positive as well as negative values of the coupling constant $\gamma$. From our analysis we see that no wormhole solution exists for $\gamma =-4\pi,\,-\pi(3+\omega)$. All the physical parameters have been drawn by employing the values of $\gamma$ as $\gamma=-0.3,\,-0.2,\,-0.1,\,0,\,0.1$ and $0.2$, where $\gamma=0$ corresponds to general relativity (GR) case. It is found that for our proposed model, a realistic wormhole solutions satisfying all the properties can be obtained.