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Repair the phone with PCC and report back
Bridge (for Mac)
Alternatives on How to backup Xperias
http://talk.sonymobile.com/thread/36355
Settings > power management > battery > check and take screenshots of
the list of apps
the graph
tap on the top 4 items on the list and take screenshots of each
post them all here or upload them on box or dropbox and post the link here.
keep us updated
The problem is how often you sync the Email app, as you can see your phone is awake most of the time.