I suggest that you first of all charge the phone for a couple of hours using the wall charger. You could also try to charge the phone using another charger.
If the phone still doesn't start you could try to update/repair the phone software using PC Companion.
http://www.sonymobile.com/update
In PC Companion, press start on Support Zone > Start on Update the phones software > "Repair phone"
If the problem still persists you might need to have the phone examined by a service center. Please contact your local support team for more information:
http://www.sonymobile.com/global-en/support/contact-us/
