Assuming carrier user has equal or more attack upgrades as the HT user has armor/shield upgrades then a carrier will "1-shot" a HT in a single volley. It takes 1.5 seconds (with the interceptor release upgrade which anyone going mass carriers will have) to launch all the interceptors for the first volley, then they have a cooldown of 3 seconds, so averaging out first HT kill in 1.5 sec, then one every 3 seconds (theorycrafting based on Liquidpedia so if anyone had real measured numbers feel free to chime in)
It takes 6 full duration storms to kill a carrier, but realistically against a smart opponent you won't get full damage, especially if they're attacking from just in range, so it's more like 8-12 storms. Storm has range of 9, Carriers range out 8, however that's just 8 range to start attacking. They can move back up to 14 range after interceptors are out and still keep attacking.
Carriers are also pretty big, so unless they're intentionally clumping them together, you'd hit 3 max (guessing here) with a single storm. So 10 storms to kill 3 carriers, and presumably most of the HT aren't going to survive long enough to cast more than 1 storm (unless you have some other units like stalkers in the front line and you opponent doesn't specifically target the HTs)
That's 1500 gas in HT compared to 750 in carriers. Yes, they cost more minerals, however if you spent all your gas on HT's the only units you can build with just minerals are zealots which can't attack air, or cannons which can attack air but have shorter range than carriers.
You'll probably be able to kill off some interceptors to slow down the carriers, but I doubt it's enough to make a difference.
Even with the mothership's vortex if it's really mass carriers you're not gonna hit them all, again unless they're clumped intentionally for some reason. The cloak might buy enough of a delay to do something, maybe you can get off a 2nd storm, but you can only have 1 mothership, so it's going to survive maybe 5-10 seconds.