The France right-back, 31, made his first Gunners appearance since November 2015 in the 3-1 win over Bournemouth - but was forced off after 16 minutes.
"He is absolutely disappointed but he is a fighter and will fight to come back" said manager Arsene Wenger.
"We are in December and he will not be available until mid-January. He usually recovers quicker than expected."
The news comes a day after the club confirmed midfielder Santi Cazorla is set to have ankle surgery, ruling him out for three months.
Full-back Hector Bellerin, Cazorla's Spain team-mate, is also out with an ankle injury.
Fourth-placed Arsenal travel to London rivals West Ham for a league match on Saturday (kick-off 17:30 GMT).
Arsenal defender Mathieu Debuchy has been ruled out for six weeks with a hamstring injury.