If it doesn't leaf out in a few weeks, it's a goner.

The cause of many mesquite die-backs is a bacterial disease, and bad pruning practices gives it an entrance.

Any reason why part of the tree is still green and the other side dead?

Hopefully, it just went dormant and will come back.

You can do a scratch test, or cut off a small branch in the "dead" area. Any green under the bark? If yes, it is dormant.

If not, it is dead. You can keep progressively keep cutting back until you reach green wood to locate the source of the problem.

My guess would be the Y split allows water to get into the V crotch and rot the tree or borers. Or that side has split separating from the other half and is dangerously just hanging on. One side has been girdled and has died. The other will likely die too or look like hell the rest of its life. BUT, looking at your setup it is hard to identify a water source that would do this. Your yard looks mesquite friendly unless a sprinkler is periodically hitting the tree.

It does look like inappropriate pruning down below may have opened the upper trunk/main branches to too much sun and lion's tailing to stress it in the wind. Otherwise it is a fine looking tree.

Here is a link that might be useful: Mesquite and Palo Verde Trees for the Urban Landscape