Information retrieval axioms are formalized constraints that retrieval systems should ideally satisfy (e.g., to rank documents higher that contain the query terms more often). In this paper, we propose new axioms that focus on the scenario of argument retrieval: retrieval for queries that need arguments in the results. Our underlying axiomatic idea is that in such scenarios, documents should be prioritized with argumentative units that are similar to the query. We test our new axioms in re-ranking experiments on the data of the Touché ~2020 and~2021 shared task on argument retrieval for controversial questions, and show that the new axioms can improve the effectiveness of Touché's strong DirichletLM baseline model and even of the top-performing system from Touché ~2021, a system already specifically optimized for argument retrieval. Finally, we also propose a new method for visualizing the relationships between axioms based on their effects in re-ranking settings.