The Red Sox, acting swiftly after losing Andrew Miller to a season-ending foot injury, announced in the second inning here Friday night that they had acquired left-handed reliever Matt Thornton and cash for minor-league outfielder Brandon Jacobs. Thornton, 36, is a fastball-slider pitcher whose velocity still averages a tick over 94 miles an hour, though his strikeout ratio has dropped each year from 12.02 per 9 innings to 6.75 this season. He has made 40 appearances (28 innings) this season for the White Sox and has held left-handed hitters to a .173 average this season (9 for 52), though three of the four home runs he has given up this season have been to left-handed hitters. Right-handed hitters have knocked him around, with a slash line of .320/.414/.420. But the veteran Thornton, who has an 0-3 record and 3.86 ERA, his highest ERA since 2007, gives Boston a veteran arm to pair with Craig Breslow in the absence of Miller, who was enjoying the best season of his career until he tore a ligament in his left foot while backing up home plate last Saturday night in Anaheim.