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Sorry to hear this but you would need to either return your device to the place of purchase or contact Xperia care http://www.sonymobile.com/global-en/support/contact-us/
A small crack could cause this issue or no issues at all, this is true to other brands, I'm certain you've seen phones with cracked screens but they still work, I'm really sorry about your situation but you have 2 options
1. Get it repaired with your local Xperia care
or
2. buy the screen and replace it yourself, by doing so, it will void your warranty but should be cheaper.
Follow Thommo's link and instead of sending emails why don't you call them and/or stop by in person.
Keep in mind that this forum, is for us, users, to help each other.
It's hard to tell, that's why you have to contact them directly.