So something I am struggling to fully grasp is how P-values, and their corresponding z scores differ between 1 and two tailed tests.

I think this is best shown through example.

First lets say I am conducting a 1 tailed test, I calculate my test statistic (lets call this X0), the p value (prob of getting a more extreme value) is the integral from X0 to inf of my H0 sampling distribution. Then I frequently wish to turn this into a z score, which is the point on my standard normal that corresponds to this p value and gives us some more intuition into how likly this event is (coming from particle physics we typically want a z score greater than 5 to say anything conclusive). And I can find this by simply putting the compliment of my p value into the inverse of the standard normal.

Now for the 2 tailed case. which is the point i start losing understanding.

So like in the first case I have some H0 and I then measure some X0. My upper tail probability (lets call this a) is the same integral I calculated in the one tailed case. But now my p value is 2a to account for the fact I am dealing with 2 tails. So in this 2 tailed case each tail accounts for half of the total p value. (this still makes sense to me). My issue is now converting this p value into a z score which is the part that confuses me. My lecture notes say i may take the inverse of the standard normal of (1-p/2). But this means I will end up the inverse of the standard nromal of (1-a) which is the same z-score as i had in the one tailed case. This feels strange and incorrect to me.

Can anyone offer any advice on the source of my confusion? Most resources I find online refer to the difference in 1 and 2 tailed tests in the perspective of calculating critical regions (which I understand) but i cant find much on the finding of p values and the corresponding z scores.

Any help is hugely appreciated.