According to the Witch Bolt spell description:
Make a ranged spell attack against that creature. On a hit, the
target takes 1d12 lightning damage, and on each of your turns for the
duration, you can use your action to deal 1d12 lightning damage to the
target automatically.
The spell description says nothing about what happens when you miss. The intuitive answer would be that the spell fails. However, seeing as how the spell is a concentration spell, and the specific ways to end the spell are:
The spell ends if you use your action to do anything else. The spell
also ends if the target is ever outside the spell’s range or if it has
total cover from you.
What happens when Witch Bolt misses?
Best Answer
On a miss, nothing happens
The witch bolt spell states (emphasis mine):
From this, we can conclude that if you do not hit then you do not deal 1d12 lightning damage and you cannot use your action to deal 1d12 damage automatically later on. Further evidence of this can be found by comparison to the ice knife spell which explicitly calls out the effects on a miss:
Nothing happens if you miss with the spell, but...
You can continue to concentrate on the spell anyway
Even if you miss, the spell does not actually end because the spell does not say that it ends. Related to this is the following question:
The answer there states that you can continue to concentrate on a spell even if all of its targets succeed on their saving throws. Similarly, I see nothing stating you cannot continue to concentrate on witch bolt even if you miss with the attack roll. So even on a miss, you can continue to concentrate on the spell.
That said, there is seldom a reason (though not never a reason) to ever continue concentrating on a spell that does nothing on its own, but this is especially the case with witch bolt as the spell automatically ends if you use your action to do anything.