From what I can work out no, not from losing the arm, keep in mind Phantom Limb Syndrome where people can still feel amputated limbs, Essence provides an excellent explanation for this as the 'soul' part of the limb is still there, but the flesh is missing.
AFAIK this character faces three options:
1: If possibly recover the arm and have a street doc reattach it. Relatively cheep and straight forward, just be careful to check for glitches from the doc during surgery for some hilarity.
2: Have a new arm cloned and then attached to the player. This will take time and be quite expensive, but as it's basically their own arm grown in a vat there's no essence loss involved.
3: Attach a cybernetic replacement, this is the point at witch Essence loss is incurred as the 'soul' part of the arm is destroyed by the mechanical replacement.
