The hypernucleus 6 He is studied as a three-body ( α) cluster system in cluster effective field theory at leading order. We find that the three-body contact interaction exhibits the limit cycle when the cutoff in the integral equations is sent to the asymptotic limit and thus it should be promoted to leading order. We also derive a determination equation of the limit cycle which reproduces the numerically obtained limit cycle. We then study the correlations between the double separation energy B of 6 He and the scattering length a of the S-wave scattering. The role of the scale in this approach is also discussed.Although the first observation of 6 He was reported in 1960s [1], there have been only a few reports on this light hypernucleus [2,3]. Among them, a track of 6 He was clearly caught in an emulsion experiment of the KEK-E373 Collaboration [3], now known as the "NAGARA" event, and the two-separation energy B of 6 He is estimated as B = 6.93 ± 0.16 MeV after being averaged with that from the "MIKAGE" event [4,5]. This would be essential information to study the interaction. On the other hand, theoretical studies for double hypernuclei mainly aim at extracting information on baryon-baryon interactions in the strangeness sector and searching for new exotic systems for which the value of B of 6 He plays an important role [6,7]. Theoretical studies on 6 He have been reported with various issues [8][9][10][11][12], primarily employing the three-body ( α) cluster model. One of those issues is the role of the mixing of the N channel in the interaction which is triggered by the small mass difference, about 23 MeV, between N and[11]. Effective field theories at very low energies are expected to provide a model-independent and systematic perturbative method where one introduces a high momentum separation scale H between relevant degrees of freedom in low energy and irrelevant degrees of freedom in high energy for the system in question. Then one constructs an effective Lagrangian expanded in terms of the number of derivatives order by order. Coupling constants appearing in the effective Lagrangian should be determined from available experimental or empirical data. For a review, see, e.g., Refs. [13,14] and references therein. In the previous publication [15], we studied 4 H, a bound state of a light double hypernucleus, and the S-wave scattering of and 3 H below the hypertriton breakup threshold by treating 4 H as a three-body ( --deuteron) system in cluster effective field theory (EFT) at leading order (LO).In this work, we apply this approach to study the structure of 6 He as a three-body ( α) cluster system. For this * sando@sunmoon.ac.kr † yohphy@knu.ac.kr purpose, we treat the α particle field as an elementary field. The binding energy of the α particle is B 4 28.3 MeV and its first excited state has the quantum numbers (J π = 0 + ,I = 0) and the excitation energy of E 1 20.0 MeV, which is between the energy gap of 3 H-p (19.8 MeV) from the ground state energy and that of 3 He-n (20.6 MeV). Thus the large momentu...