B737-800 CREW SHUT DOWN #1 ENG AFTER THE FUEL IMBAL; ANNUNCIATOR ILLUMINATED.

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WHILE IN CRUISE AT FL370 WE GOT A FUEL IMBAL INDICATION ON THE L SIDE. THE FO WAS FLYING AND I TOOK CTL OF THE ACFT TO LET HER RUN THE APPROPRIATE CHKLIST AND CALLED THE COMPANY TO ASK FOR GUIDANCE. AFTER RUNNING THE CHKLIST WE BELIEVED WE HAD A LEAK AND THE NEXT STEP IS TO SHUT DOWN THE EFFECTED ENG. WE DECALED AN EMER; SHUT DOWN THE L ENG AND PROCEEDED TO LAND AT ZZZ. AFTER LNDG I CALLED OUR MAINT CTL AND; WHILE TALKING TO THEM; LEARNED THAT I MIGHT HAVE MISMANAGED THE XFEED SYS ON THE ACFT. OUR AIRLINE HAS HAD SOME FUEL LEAK PROBS WITH THE B737 OF WHICH BOTH THE FO AND I WERE AWARE. I BELIEVE THAT THIS KNOWLEDGE LET US DEVELOP TUNNEL VISION TO THE PERCEIVED INFLT ANOMALY. WHILE DISCUSSION OF THESE EVENTS IN TRAINING AND ON THE LINE IS VERY USEFUL; PERHAPS MORE EMPHASIS SHOULD BE PUT ON LOOKING PAST THE APPARENT AND DELVING INTO THE PROB. IT IS VERY DIFFICULT; IN A 2-PERSON CREW; TO FLY; COMMUNICATE WITH ATC; COMPANY; FLT ATTENDANTS; AND PAX WHILE FACING A TIME CRITICAL EMER. SUPPLEMENTAL INFO FROM ACN 627014: DURING CRUISE FLT; APPROX 30 MINS INTO THE FLT; THE FUEL IMBAL ANNUNCIATOR ILLUMINATED. AFTER COMPLETING THE CHKLIST; A DETERMINATION WAS MADE THAT THERE WAS A POSSIBLE L TANK FUEL LEAK. THE L TANK QUANTITY INDICATOR CONTINUED TO DECREASE. AT ABOUT 5 MINS AFTER THE DISCOVERY OF FUEL IMBAL; THE IMBAL HAD INCREASED TO 2000 LBS. CONSIDERING THIS SIT; THE THOUGHTS OF LOSING MORE FUEL AND BECOMING INCREASINGLY OUT OF BAL; AS WELL AS POSSIBLE FIRE; WERE IN MY MIND. WE DECIDED TO TAKE THE SAFEST COURSE OF ACTION BY SHUTTING DOWN AND SECURING THE L ENG AND WE DIVERTED TO THE NEAREST SUITABLE ARPT WHICH WAS ZZZ ARPT APPROX 50 MI AWAY. INITIALLY; I WAS THE PF. THE CAPT DECIDED TO XFER CTLS BEFORE WE SECURED THE ENG SO HE WAS THE PF AND I WAS THE PNF FOR THE REMAINDER OF THE FLT. ATC GAVE US RADAR VECTORS TO ZZZ AND WE LANDED WITHOUT FURTHER INCIDENT. AFTER WE LANDED ON THE GND; TAXIED CLR OF THE RWY; AND SHUT DOWN THE R ENG; THE FIRE AND RESCUE TEAM VISUALLY INSPECTED THE ACFT AND FOUND NO EVIDENCE OF LEAKING FUEL. GIVEN THIS OUTCOME; I DON'T KNOW IF THE PROB WAS CAUSED BY INTERNAL FUEL SYS MALFUNCTIONS; A FAULTY FUEL INDICATOR; PLT ERROR; OR PLT JUDGEMENT ERROR.