The victim was walking on Queen Street in the early hours of December 20 when the suspect started making advances to his wife.
When he intervened, a fight ensued and the suspect bit him on the nose.
Two other men, who are believed to have known the suspect, approached the group and tried to calm the situation.
The suspect and the other two men were last seen heading down Queen Street Gardens East.
The victim was rushed to Royal Infirmary of Edinburgh and faces months of treatment.
In a renewed appeal for information, officers have released a shocking photo of the nose injury, which shows that the man’s septum is missing.
The suspect is described as white, between 20 and 30-years-old 5ft 11ins tall with a stocky build and a local accent.He had short fair hair and stubble and was possibly wearing a grey-coloured top with thin black stripes.
One of the men who intervened is described as white, in his early thirties, 6ft 2ins tall with a slight build. He had short ginger hair, a clean-shaven face and was wearing a white t-shirt and dark trousers.
The second man is described as white, in his early thirties 5ft 8ins tall with a slight build, dark hair and stubble on his face. He was wearing a grey-hooded jacket and dark jeans.
Detective Constable Alison Wyllie said: “This was a violent attack which has left the victim requiring ongoing treatment, and we’re conducting extensive enquiries in order to trace the suspect.
“We have released the image of the victim to show the extent of his injuries and to perhaps convince whoever was responsible to come forward.
“I would also ask that the two men who intervened contact police immediately and assist us with our ongoing investigation.
“Similarly, anyone else who witnessed this assault, or who has any further information relevant to this inquiry, should also get in touch.”
Anyone with information is asked to contact Police Scotland on 101 or contact Crimestoppers anonymously on 0800 555 111.