The Idea:
The title says it all. This ability is the posterchild ability which shows how much stamina damaging abilities are underperforming. Most stamina based DoTs do a wildly underwhelming amount of damage and poison arrow is the perfect example.

To prove as such, I'll compare the ability to a class ability with similar ability: The DK ability Searing Strike

The Comparison:
To test the difference I made a fresh DK wearing no gear and using no skill points, and took a look at the ability at it's base and can determine how the damage scales via skill calculators online. Here's what i came up with:

A fresh (level 1) DK in no gear but the started sword they give Searing Strike's tooltip at rank one reads 11 damage initially and another 15 over 8.5 seconds. Poison shot's tooltip at rank one in the same gear is 11 damage initially and 5 damage over 10 seconds. So even at rank one the abilities do the same initial damage but searing strike does 3x the amount that poison arrow in the DoT portion. That's just at rank one. It gets worse.

When they rank up, searing strikes initial damage and dot portion both scale up. The DoT portion scales up just over 40% higher than its rank one damage, the initial is around 20% higher. The morphs of this ability don't raise the damage scaling any further.

When poison arrow ranks up, the initial damage increases by 25% total but the DoT portion doesn't scale at all. Why? Then, with the morphs, Venom Arrow increases its initial damage with it's rank by another total of 25% from rank one to rank four. Poison Injection doesn't scale anything from rank one to four as far as we can tell, but from what I've seen it increases the damage of the DoT by 3x. Maybe that's what scales.

The Refined Comparison A La Napkin Math:
Using the % modifiers observed and mentioned above we can roughly show what the base damage of these abilities on the control character will be at their max ranks to compare them.

Searing strike at rank one does 11 damage initially, increased by 20% gives us around 13. 15 DoT increased by 40% gives us 21 damage over 8.5 seconds. So the total base damage output of this ability with initial and DoT added together is 35.

Poison arrow at rank one does 11 damage initially and 5 damage over 10 seconds. The initial damage raises 25% pre-morph and another 25% post morph with venom arrow. That leaves us with around 14 damage rounding up for the first 25%, then around 17 damage for hte next 25%. Add that with the 5 damage from the DoT and we get 22 damage total.

So Venom Arrow gives us a total of 22 damage (Poison injection is around 26 if counted during execute phase but only 16 outside of execute phase) while Searing Strike gives us a total of 35 damage. So Searing Strike does ~37% more total damage over less time than Venom Arrow, 54% more than Poison Injection pre-execute and 26% more even during execute. Searing strike does between 26% and 54% more damage than poison arrow and it's morphs, over LESS time. Granted, it's about a second and a half less, but it's the principle of the matter. What's more in this math comparison I'm completely ignoring the fact that the damage-oriented morph of Searing Strike increases its damage further the longer it's on the enemy and can be chained when cast multiple times.

And to top it all off, Searing Strike costs less resources. Without any cost reduction passives or enchantments, Searing Strike costs 18 magicka, Venom Arrow/Poison Injection costs 32 stamina. For an ability that draws from a resource pool that is being used for alternate purposes, you'd think it'd have a lower cost than its resource counterpart, but it costs 44% MORE than the one of the resource pool that has no alternate purpose.

Conclusion:
Where's the logic here ZoS? The only single target DoT that the bow tree has available to it is at best 25% (and that's a damn execute ability during the execute phase) worse than a similar class ability that DragonKnights have available, and costs 44% more. 25-54% less damage, 44% more cost out of a resource pool that needs to be conserved for defensive purposes like dodging and blocking, now I must ask, Where is the incentive in this or any of the bow abilities to actually use a bow when the damage efficiency is so monumentally less?

Suggestions:
Poison Arrow can still be saved which is the good news. Merely cause the DoT to scale up along with the initial damage pre-morph, then post morph keep venom arrow as it is so that it only scales up in initial damage since it will have a now higher DoT damage from the pre-morph scaling which is enough of a buff in itself.

Then just change Poison Injection to scale the DoT portion but not the initial portion. Poison Injection can do less damage in it's DoT portion than other similar DoTs to account for the execute damage bonus, but not so much as it is now where the DoT is basically non existent. As it is now, poison injections dot even in execute range does less damage per tick than the class based DoT I use. That in itself needs being looked at, an execute ability shouldn't do less damage than abilities that don't need execute range to be effective.

Parting Words:
Sorry for the wall of text guys! Just had to get my Bow-Blues out in the open, there needs to be some changes to the Bow tree and stamina damage builds in general, the latter of which ZoS has announced they're working on which is good, but I still think some player-details could help. Cheers!