There is a result R and teams T and U.
Team T submitted into a journal result R, but not on Arxiv. Team U then submitted result R to Arxiv, without knowing about the submission of Team T. Team U also submitted to a journal after publishing on Arxiv but got rejected. Now the paper of Team T is already published, but later than the Arxiv submission of Team U.
What should or can the Team U do?
What should or can the Team U do?
Team U should improve their paper and try to publish it again. Be sure to cite your competition. It would be an ethical obligation, even if their paper had also been rejected, because the citation will make it easier to understand your paper.
The value of being first to publish is exaggerated. An ethical journal will publish replication studies without regard to claims that the replication was developed first. Most authors will cite papers because they are helpful, without regard to which paper was first.
Disputes about priority are petty and best avoided.
The fact that both teams got the same result is very good for science.
Team U can genuinely claim the following in a future version of their article:
"After making our result publicly available on arXiv, we became aware of a similar result obtained in \cite{teamTresultPublishedPaper}."
Team U can congratulate team T and ask the following:
"Dear Team T, congratulation on your result R in journal J. As you may be aware, we have been working independently on a similar result as your recently published paper; our article was available on arXiv on Sept 2019. We were planning on acknowledging your independently obtained result in the next revision on our article, and we were wondering if you would agree to do the same." As pointed out in the comment, this is potentially obnoxious, and might not have any benefits.
Team U can submit to paper to the same journal that accepted team T's paper. "Dear Editor, we have noticed a recently published article by team T in your journal on result R. The same result was obtained independently by our team and made public on Sept 2019, a few months before team T's result was publicly available. We are wondering about the journal policy regarding publication of two papers that prove the same result, when the public record shows that the results were obtained independently and contemporaneously."
Alternatively, instead of asking the Editor on the possibility of such submissions, Team U may directly submit the article to the same journal with a cover letter asking to be considered.
If members of team T were referees for the paper of Team U (the paper that was eventually rejected), this may constitute a conflict of interest. If a reviewer reviews a paper that proves the same result as an article written by the reviewer not yet public but currently under review, the reviewer should at least declare this as a conflict of interest during the review. Of course, reviewers are anonymous so there is no way for team U to know. Team U may contact the Editor that rejected the journal, saying "Dear Editor, thanks for your consideration for our submission XXX that was rejected on $RejectDate. After the submission of our article to your journal, the same result was published and made public by Team U in OtherJournal. We are wondering if you could double check that no conflict of interest among the reviewers hindered the review process." this is potentially obnoxious, and might not have any benefits.
Beyond these, team U should try to have their result published. Team U can certainly claim the result in their CV, research statement etc.
There are numerous cases of independently obtained results in hot fields and many papers actually include a sentence as in 1. above. When that happens, curious readers and historians can lookup arXiv or other public repositories (e.g. PhD thesis, etc) to check the actual timeline.
I'm guessing here that the team T finished first and submitted. The fact that you (assuming "you" = U) didn't know about it doesn't give you priority. The paper might have been in the works and under review for a long time.
Team U seems to have been later even though their result was visible generally earlier. I doubt that a journal would want to retract based on an arXiv version by someone else. Almost certainly the journal editor and, possibly, reviewers saw T's work before the arXiv version of U.
Congratulate team T. See if there are opportunities for collaboration on future projects since you are on the same trajectory.
But this is one of the dangers of working on hot problems. There is likely to be other work going on simultaneously.
Get your follow-up paper submitted STAT.
