The most common and most likely answer to this crossword clue is the 4 plus 8 letter word COLD SHOULDER
I've seen this crossword clue in The Times
C
O
L
D
S
H
O
U
L
D
E
R
More clues leading to the result COLD SHOULDER
Buddy explains!

The straight part
We know that "silent treatment" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case silent treatment = cold shoulder.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does COLD SHOULDER mean?
COLD SHOULDER verb- To disrespect someone, especially by ignoring them.