I have changed my default browser to firefox, google chrome, and internet explorer. I have had the same results using each browser. Each time I try a new browser, I delete the LR app within my fb settings. When I try authorize my fb account through LR it takes me to the login page, I login, next screen says I am logging into Adobe Photoshop Lightroom and a button to Login with facebook, I login with facebook, next screen: Adobe Photoshop would like permission to..., I click ALLOW, the screen changes to my fb page, so I click DONE in LR, and LR continues to show AUTHORIZING until I quit out.
The "failed to authorize" message only came up when I was running LR 4.2. That was because after clicking "allow" on fb, it would direct me to an error page. I would copy and paste the URL from that page and after entering the URL into LR, I would receive a message saying something like "error, missing name_link". I'm not sure of the exact wording because I don't get that "failed to authorize" message anymore in 4.3 only a continuing authorizing message.