It was 40 years ago today, to paraphrase 'Sgt Pepper's', that John Lennon was fatally gunned down outside the Dakota building in New York. His killer was Mark Chapman, eerily photographed with the ex-Beatle earlier that day as Lennon signed a copy of the Double Fantasy album for him. But what was the real reason behind this senseless slaying?

Was it all JD Salinger's fault for writing Catcher In The Rye, a novel Chapman was obsessed with, and which featured an anti-hero railing against life's 'phonies'? Did the CIA get to Chapman with their MK-Ultra mind-control programme, given how fearful the establishment was of this anti-war campaigner's potential influence on America's youth? Or was Chapman simply a delusional, deranged loner whose narcissism led him to snuffing out the most famous person he knew he could get close to simply in order to grab his slice of fame?

Jealous Guy: The Assassination Of John Lennon never really settles on a specific theory or motive over the course of its 80 minutes' airtime, this despite having terrific access to Chapman himself via his biographer Jack Jones. We hear Chapman's voice (calm, composed and rational) as the pair chat on the phone, mainly discussing the possibility of him getting parole at the 11th attempt. Jones also has the ear of Chapman's wife Gloria, whom we also hear speaking to her husband moments after his arrest in 1980. The testimony of interviewees (including Beatles' biographers, Lennon's friends, and on-the-scene police officers), archive footage and doom-laden soundtrack all contribute to a sense of overwhelming sadness and a continued confusion which probably still won't clear when we reach the crime's half-century anniversary.

For this documentary, the lack of closure may be impeded by too many narratives thrown into the pot. Not only do we hear about Chapman's origins story (his parents seemed to love him, though he harboured murderous thoughts towards his father) and Lennon's upbringing (the loss of his mother had a profound effect), but Jones' own sad biography is inserted for no obvious reason. The story of John Lennon and his killer Mark Chapman was a shock to the collective psyche for anyone who heard the news that day, and reverberates still. Despite being sensitively put-together, this documentary is just another hollow contribution to a maddening and inconclusive canon.

Sky Documentaries, Tuesday 8 December, 9pm. Available on NOW TV after transmission.