What is the notation for bounding running time in worst case with concrete example resulting in that worst case running time I know that Big O is used to bound worst case running time. So an algorithm with running time $O(n^5)$ means its running time in worse case is less than $n^5$ asymptotically.

Similarly, one can say that for example merge sort's running time is $O(n^2)$  which is correct. But we know that there is a better bound for it: $O(n\log n)$. Technically speaking, one can say that every polytime algorithm has running time $O(2^n)$. This is correct, but not useful.

So my question is: what is the notation used for the case of worst case running time such that there exists an input in which the worst case running time happens.

In the merge sort example, one cannot construct an input example so that merge sort would take $n^2$ comparisons, but one can construct an example that requires the number of comparisons being about $n\log n$.
In words, such a bound is called a tight bound. In the specific case of Landau notation, $\Theta$ can be used to indicate a tight bound.
You have to be somewhat careful about your wording, though:


"The runtime of quicksort is in $O(n^2)$," is a correct statement (but somewhat imprecise).
"The runtime of quicksort is in $O(n^2)$, and this bound is tight," is correct and expresses, what you want to express.
"The worst-case runtime of quicksort is in $\Theta(n^2)$," is equivalent to the previous statement.
But "The runtime of quicksort is in $\Theta(n^2)$," is not correct, since this statement implies that quicksort takes quadratic time on every input.