Register automaton (RA) is an extension of finite automaton by adding registers storing data values. RA has good properties such as the decidability of the membership and emptiness problems. Linear temporal logic with the freeze quantifier (LTL^↓) proposed by Demri and Lazić is a counterpart of RA. However, the expressive power of LTL^↓ is too high to be applied to automatic verification. In this paper, we propose a subclass of modal µ-calculus with the freeze quantifier, which has the same expressive power as RA. Since a conjunction ψ₁ ∧ ψ₂ in a general LTL^↓ formula cannot be simulated by RA, the proposed subclass prohibits at least one of ψ₁ and ψ₂ from containing the freeze quantifier or a temporal operator other than X (next). Since the obtained subclass of LTL^↓ does not have the ability to represent a cycle in RA, we adopt µ-calculus over the subclass of LTL^↓, which allows recursive definition of temporal formulas. We provide equivalent translations from the proposed subclass of µ-calculus to RA and vice versa and prove their correctness.