Deciding to go after the Worldline Zero Catalyst is a bit tricky in Destiny 2. Swords in general are fun weapons to use but lack the options and power that come with ranged weapons. They have their place and can be useful, but generally speaking, a gun is almost always better.
The 
Worldline Zero Catalyst takes a fun weapon and gives it teeth. Whether or not it's enough to justify using is up to the player. For those still on the fence with this weapon here’s everything you need to know about what the Worldline Zero Catalyst does and how to obtain it.
What It Does
The Worldline Zero Catalyst gives the sword one perk:
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Another Dimension – Reduces the time it takes to activate
Tesseract.
Tesseract is the intrinsic perk of the Worldline Zero sword. When the player performs a heavy attack while sprinting it triggers Tesseract. Tesseract causes the player to blink forward, partially bypass elemental shielding, and strike the target with a heavy attack.
The Worldline Zero Catalyst reduces how long that process takes and allows the player to move faster and get the drop on their enemy sooner. It seems like a small thing on paper, but in practice, it’s great for taking enemies off guard and getting a powerful hit against opponents.
How To Get It
The Worldline Zero Catalyst is one of the easier Catalysts to obtain in the game. The player simply needs to kill a non-Guardian enemy using a sword of any kind. The player can kill literally any NPC in the game with a sword and it has a chance to drop the Catalyst.
Granted, it's still a rare drop. Players can get it very quickly or go weeks without obtaining it. But if the player is routinely using swords in PVE combat it should drop eventually without too much difficulty.
Upgrading
Upgrading the Worldline Zero Catalyst is also fairly simple. The player simply needs to kill 300 enemies using the Worldline Zero. It doesn’t matter what enemy or how the enemy is encountered. It could be a Gambit match or randomly strolling through story content.
The kills are also retroactive. The player will likely wrack up the majority of these kills trying to get the Worldline Zero Catalyst to drop. In terms of Catalyst grinds, this one is one of the most forgiving in the game.
Tips
Considering any sword can be used to obtain the Catalys,t players may consider using one of the top 10 swords in the game. Using something like the 
Black Talon or Falling Guillotine can speed up the process of slicing and dicing enemies.
The only problem with this strategy is that using the Worldline Zero will rack up retroactive kills that will apply to the Worldline Zero Catalyst upgrade, thereby cutting down on overall grind time. One method could be to soften enemies with a different weapon before dealing the finishing blow with the Worldline Zero.
The last thing to consider is that any NPC anywhere in the game can be killed for this Catalyst and subsequent upgrade. Players can and should tackle other quests, compete in Gambit, track down other Catalysts, or grind resources while pursuing the Worldline Zero Catalyst.