TLDR: If you've downgraded to 4.2 to get a working touchscreen, try upgrading and see if it stays working.
I had a very similar experience today. I replaced my digitizer with an lcd/digitizer assembly I bought from ebay.
The touch controller chip is a "Synapdcs" [sic] 5500CB-5 with additional markings FCA6616 (I might have some of the numbers wrong, the silkscreening is very poor quality):
It seems to match the digitizer that Phillipe described, except with 6 chips instead of 5 (4 decoupling caps, 1 resistor, 1 diode).
My phone was running Cyanogenmod 12.1 nightly build (Android 5.1 based). When I powered it on with the new digitizer it didn't register any touches!
Disassembled and jostled connectors, reassembled, same problem. Eventually I found this thread here and also a post on the CM forums that sounds the same.
Based on the advice here I reflashed entirely back to a factory 4.2.2 image via fastboot, erasing absolutely everything, and the touchscreen worked. Yay!
(After the touchscreen came good I found out that the proximity sensor didn't work, so I had to disassemble the phone again to apply this fix.)
I didn't want to be stuck on 4.2 so I figured I'd try to fix the driver. So I installed CM 10.1.3 (Android 4.2 based), which according to the other forum post didn't work (but 10.1.2 worked). The touchscreen worked. Installed CM 10.2 (Android 4.3 based) and the touchscreen kept working.
Now I was confused. I installed the latest CM 12.1 nightly build 20150619 (ie Android 5.1 again). Touchscreen still works.
Hard power cycled the phone a few times, including unplugging from USB. Touchscreen still works. Issued a Factory Wipe/Erase User Data from Recovery mode, touchscreen still works.
So... I'm really confused!
I don't think I physically made the screen come good, although maybe I did.
The touchscreen driver includes firmware upgrade code, and it seems the firmware is stored internally on the driver chip rather than loaded from the kernel each time the phone boots. My theory is that downgrading to 4.2.2 and then upgrading through the various versions somehow created a working firmware upgrade path for the controller!
To test this theory - if anyone else downgraded to 4.2.2 to get a working touchscreen, can you try upgrading and see if it stays working? I don't know if you can upgrade directly to 5.1 or if the interim steps I went through somehow helped as well - maybe try 5.1 first and if it fails then roll back again and try step-by-step.
Finally, to help debug here's some debug output from the touchscreen kernel driver via 'adb shell'. That's under 5.1 with a working touchscreen. I didn't get output under any other versions. :(
I will make sure to post back if my touchscreen suddenly stops working again...