Exploring the damping of Alfv\'en waves along a long off-limb coronal loop, up to 1.4 R$_\odot$

The Alfv\'en wave energy flux in the corona can be explored using the electron density and velocity amplitude of the waves. The velocity amplitude of Alfv\'en waves can be obtained from the non-thermal velocity of the spectral line profiles. Previous calculations of the Alfv\'en wave energy flux with height in active regions and polar coronal holes have provided evidence for the damping of Alfv\'en waves with height. We present off-limb Hinode EUV imaging spectrometer (EIS) observations of a long coronal loop up to 1.4~R$_\odot$. We have obtained the electron density along the loop and found the loop to be almost in hydrostatic equilibrium. We obtained the temperature using the EM-loci method and found the loop to be isothermal across, as well as along, the loop with a temperature of about 1.37 MK. We significantly improve the estimate of non-thermal velocities over previous studies by using the estimated ion (equal to electron) temperature. Estimates of electron densities are improved using the significant updates of the CHIANTI v.8 atomic data. More accurate measurements of propagating Alfv\'en wave energy along the coronal loop and its damping are presented up to distances of 1.4 R$_\odot$, further than have been previously explored. The Alfv\'en wave energy flux obtained could contribute to a significant part of the coronal losses due to radiation along the loop.