DoD of Stories that need to be merges into multiple prod branches

The Definition of Done should capture all of the integration, testing, documentation etc. that is required for an increment to be immediately usable. That's the answer. Once an increment is Done, the imperative is to put it to use, so empirical feedback is obtained and the validated learning loop closed.

Scrum is very good at exposing organizational issues and shortcomings, many of which can be systemic. For example, you are experiencing constraints regarding CICD cycle time and branching and merging, which can take up to five weeks. That's what needs to be resolved. The problem is not with the DoD for exposing these issues, but you would have a problem if you adjusted the DoD to cover these issues up.

As @Ian mentions it would be worth checking CICD cycle time (if production deployment is a part of your definition of done). Also, I don't think deployment to production is a mandatory thing to mark a story as done (however some teams have that as a criteria). You could probably have UAT Approved/Regression tested as the last step in DoD (ensure customers perform the acceptance test/regression test within the sprint). Make sure to draft a definition of done with the whole team based on your situation and ensure it is understood and agreed upon by all. I have worked in a team where we had a 4-week sprint which includes 2-week development, 1-week UAT and 1-week regression testing and the sprints ends with a production deployment (done on weekends with some downtime). Once deployed to production, we close all the stories (so production deployment was a part of the DoD).

Done would mean the story meets the desired quality standards and is good enough to be delivered i.e. deployed to production and ready for users to use.

Thanks Ian and Anand for the input . I agree with the ideal approach, however the current reality here is that the CICD -branching and merging are with different department and they have their own way to do the quality check and those 3-4 production branches  has different timings. (ie each branch has different time to get into that production line ). And average cycle time for each of this branches to production is around 4 weeks due to the quality-check and other procedure .  That department is trying to improve the cycle time , however that is not going to fit into single week for 4 production branches.  So if we consider to keep the ticket to be in open state and simply wait for the  entire 4 different branches to be in production  , we need to wait for average 4 weeks ( 2 sprints )   and sprint backlog pilesup in un managed way after few sprints.

or  we can put my problem statement in this way-  what we can do if we have the story DoD  that has items which are not in the control of team( merging to 4 different production branches )

what we can do if we have the story DoD  that has items which are not in the control of team( merging to 4 different production branches )

What does the team think about this? -  It looks like the team is not sufficiently cross functional to deliver a Done increment.

Scrum is not prescriptive about the DoD the guide mentions The Definition of Done is a formal description of the state of the Increment when it meets the quality measures required for the product.

In your scenario, you have mentioned quality checks which means it is essentially a part of the DoD by the description above. So may be cutting corners on the DoD is not an option here.

Some questions to ponder upon are.

Why are there items not in control of the team in the DoD? 

Is a sprint duration of 2 weeks right in your scenario? 

Would increasing the sprint duration to 1 month ensure that you do all activities mentioned as a part of DoD?

Also should the scope (amount of stories) be decreased to accommodate the delay in the downstream process?