aa r X i v : . [ nu c l - t h ] S e p He in cluster effective field theory Shung-Ichi Ando ∗ and Yongseok Oh
2, 3, † Department of Information Display, Sunmoon University, Asan, Chungnam 336-708, Korea Department of Physics, Kyungpook National University, Daegu 702-701, Korea Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Korea (Dated:)The hypernucleus He is studied as a three-body (ΛΛ α ) cluster system in cluster effective fieldtheory at leading order. We find that the three-body contact interaction exhibits the limit cycle whenthe cutoff in the integral equations is sent to the asymptotic limit and thus it should be promotedto leading order. We also derive a determination equation of the limit cycle which reproduces thenumerically obtained limit cycle. We then study the correlations between the double Λ separationenergy B ΛΛ of He and the scattering length a ΛΛ of the S -wave ΛΛ scattering. The role of thescale in this approach is also discussed. PACS numbers: 21.80.+a, 11.10.Hi, 21.45.-v
Although the first observation of He was reported in1960s [1], there have been only a few reports on this lighthypernucleus [2, 3]. Among them, a track of He wasclearly caught in an emulsion experiment of the KEK-E373 Collaboration [3], now known as the “NAGARA”event, and the two-Λ separation energy B ΛΛ of He isestimated as B ΛΛ = 6 . ± .
16 MeV after being averagedwith that from the “MIKAGE” event [4, 5]. This wouldbe an essential information to study the ΛΛ interaction.On the other hand, theoretical studies for double Λhypernuclei mainly aim at extracting information onbaryon-baryon interactions in the strangeness sector andsearching for new exotic systems for which the value of B ΛΛ of He plays an important role [6, 7]. Theoret-ical studies on He have been reported with variousissues [8–12], primarily employing the three-body (ΛΛ α )cluster model. One of those issues is the role of the mix-ing of the Ξ N channel in the ΛΛ interaction which istriggered by the small mass difference, about 23 MeV,between Ξ N and ΛΛ [11].Effective field theories at very low energies are ex-pected to provide a model-independent and systematicperturbative method where one introduces a high mo-mentum separation scale Λ H between relevant degreesof freedom in low energy and irrelevant degrees of free-dom in high energy for the system in question. Thenone constructs an effective Lagrangian expanded in termsof the number of derivatives order by order. Couplingconstants appearing in the effective Lagrangian shouldbe determined from available experimental or empiricaldata. For a review, see, e.g., Refs. [13, 14] and referencestherein. In the previous publication [15], we studied H ,a bound state of a light double Λ hypernucleus, and the S -wave scattering of Λ and H below the hypertritonbreakup threshold by treating H as a three-body (Λ-Λ-deuteron) system in cluster effective field theory (EFT)at leading order (LO). ∗ [email protected] † [email protected] In this work, we apply this approach to study thestructure of He as a three-body (ΛΛ α ) cluster sys-tem. For this purpose, we treat the α particle field as anelementary field. The binding energy of the α particle is B ≃ . J π = 0 + , I = 0) and the excitation energyof E ≃ . - p (19.8 MeV) from the ground state energy and thatof He - n (20.6 MeV). Thus the large momentum scale ofthe α -cluster theory is Λ H ≃ √ µE ∼
170 MeV where µ is the reduced mass of the ( H , p ) system or the ( He , n )system so that µ ≃ m N with m N being the nucleonmass. Therefore, the mixing of the Ξ N channel in theΛΛ interaction becomes irrelevant because the mass dif-ference ∼
23 MeV of the two channels is larger than E ,the large energy scale of this approach. On the otherhand, we choose the binding momentum of He as thetypical momentum scale Q of the theory. The Λ sepa-ration energy of He is B Λ ≃ .
12 MeV and thus thebinding momentum of He as the (Λ α ) cluster system is γ Λ α = p µ Λ α B Λ , where µ Λ α is the reduced mass of theΛ- α system. This leads to γ Λ α ≃ . Q/ Λ H ∼ γ Λ α / Λ H ≃ . S -wave neutron-deuteron( nd ) scattering for spin doublet channel in the pionlessEFT [16]. The limit cycle in a renormalization groupanalysis was suggested by Wilson [17] and it is alsoknown that the limit cycle is associated with the Efi-mov states [18] in the unitary limit, where the scatteringlength in the N N interaction becomes infinity. Further-more, a “determination equation” of the limit cycle, asan expression of the homogeneous part of the integralequation in the asymptotic limit, was obtained earlier byDanilov [19].In this work, we investigate the bound state of the(ΛΛ α ) cluster system in cluster EFT at LO in order todescribe He . We find that the three-body contact in-teraction exhibits the limit cycle behavior when the cou-pled integral equations with a sharp cutoff are numeri-cally solved. Thus the contact interaction should be pro-moted to LO. We also derive a determination equationof the limit cycle for the ΛΛ α system and find that thesolution of the equation reproduces remarkably well thenumerically obtained limit cycle. In addition, we inves-tigate the correlation between B ΛΛ and the scatteringlength a ΛΛ of the S -wave ΛΛ interaction including thethree-body contact interaction with different cutoff val-ues. The case without the three-body contact interactionwill be studied as well.The LO effective Lagrangian relevant to our studyreads L = L Λ + L α + L s + L t + L Λ t . (1)Here, L Λ and L α are one-body Lagrangians for spin-1/2 Λ and spin-0 α -cluster field in heavy-baryon formal-ism [20, 21], respectively, L Λ = B † Λ (cid:20) iv · D + ( v · D ) − D m Λ (cid:21) B Λ + · · · , (2) L α = φ † α (cid:20) iv · D + ( v · D ) − D m α (cid:21) φ α + · · · , (3)where v µ is the velocity vector v µ = (1 , ) and m Λ and m α are the Λ and α mass, respectively. The dots denotehigher order terms. The Lagrangian of the auxiliary fields s and t are given by L s and L t , respectively, where s isthe dibaryon field of two Λ particles in S channel and t is the composite field of the (Λ α ) system in the He ( S = 1 /
2) channel [15, 22, 23], L s = σ s s † (cid:20) iv · ∂ + ( v · ∂ ) − ∂ m Λ + ∆ s (cid:21) s − y s h s † (cid:16) B T Λ P ( S ) B Λ (cid:17) + H.c. i + · · · , (4) L t = σ t t † (cid:20) iv · ∂ + ( v · ∂ ) − ∂ m Λ + m α ) + ∆ t (cid:21) t − y t (cid:2) t † B Λ φ α + H.c. (cid:3) + · · · , (5)where σ s,t are sign factors. The mass differences betweenΛΛ and the s dibaryon state and between Λ α and the composite t state ( He ) are represented by ∆ s,t , respec-tively. P ( S ) = − i σ is the spin projection operator tothe S state. The dibaryon s state is coupled to two Λin S state and the composite t state to the S -wave Λ α state with the coupling constants y s,t , respectively. TheLagrangian for the contact interaction of Λ and t reads L Λ t = − m α y t g (Λ c )Λ c (cid:16) B T Λ P ( S ) t (cid:17) † (cid:16) B T Λ P ( S ) t (cid:17) + · · · , (6)where the coupling g (Λ c ) is a function of the cutoff Λ c ,which is defined in the coupled integral equations below.In the present work we consider two composite statesin the two-body part, namely, the s field and t field. Thedibaryon s state was investigated in our previous publi-cation [15], where the Feynman diagrams for the dresseddibaryon propagator can be found. The renormalizeddressed dibaryon propagator is obtained as D s ( p ) = 4 πy s m Λ a ΛΛ − q − m Λ p + p − iǫ , (7)where y s = − m Λ q πr ΛΛ . The scattering length and theeffective range of S -wave ΛΛ scattering are representedby a ΛΛ and r ΛΛ , respectively. We note that the expres-sion of the dressed dibaryon propagator in Eq. (7) is forthe large value of a ΛΛ . In the case of a small value of a ΛΛ one can expand it in terms of the kinetic square rootterm [24]. The diagrams for the dressed t (Λ α ) propa-gator can be found, e.g., in Ref. [15], which lead to therenormalized dressed t (Λ α ) propagator as D t ( p ) = 2 πy t µ Λ α γ Λ α − r − µ Λ α (cid:16) p − m α + m Λ ) p + iǫ (cid:17) , (8)where y t = − µ Λ α q πr Λ α . We also note that the depen-dence of y s,t on the effective ranges r ΛΛ and r Λ α disap-pears in the final expression of the three-body coupledintegral equations at LO [15].The amplitude for S -wave elastic Λ- He scattering inCM frame can be described by the coupled integral equa-tions at LO as a ( p, k ; E ) = K ( a ) ( p, k ; E ) − m α y t g (Λ c )Λ c − π Z Λ c dl l (cid:20) K ( a ) ( p, l ; E ) − m α y t g (Λ c )Λ c (cid:21) D t (cid:18) E − l m Λ , l (cid:19) a ( l, k ; E ) − π Z Λ c dl l K ( b ( p, l ; E ) D s (cid:18) E − l m α , l (cid:19) b ( l, k ; E ) ,b ( p, k ; E ) = K ( b ( p, k ; E ) − π Z Λ c dl l K ( b ( p, l ; E ) D t (cid:18) E − l m Λ , l (cid:19) a ( l, k ; E ) , (9)where the amplitudes a ( p, k ; E ) and b ( p, k ; E ) are half- off shell amplitudes for the elastic Λ t channel and the -10-5 0 5 10 100 1000 10000 100000 g ( Λ c ) Λ c (MeV)a ΛΛ = -1.8 fm = -1.2 fm = -0.6 fm FIG. 1. (Color Online) The coupling g (Λ c ) as a function ofΛ c for a ΛΛ = − . − . − . g (Λ c )are fitted by B ΛΛ = 6 .
93 MeV of He . inelastic Λ t to αs channel, respectively. Here, p = | p | and k = | k | where p ( k ) is the off-shell final (on-shellinitial) relative momentum in the CM frame. Thus thetotal energy E is determined as E = µ Λ(Λ α ) k − B Λ where µ Λ(Λ α ) = m Λ ( m Λ + m α ) / (2 m Λ + m α ). A sharpcutoff Λ c is introduced in the loop integrals of Eq. (9).The one- α and one-Λ exchange interactions are given by K ( a ) ( l, k ; E ) and K ( b ,b ( l, k ; E ), respectively, where K ( a ) ( p, l ; E ) = m α y t pl ln " m α µ Λ α ( p + l ) + pl − m α E m α µ Λ α ( p + l ) − pl − m α E ,K ( b ( p, l ; E ) = √ m Λ y s y t pl ln " p + m Λ µ Λ α l + pl − m Λ Ep + m Λ µ Λ α l − pl − m Λ E ,K ( b ( p, l ; E ) = √ m Λ y s y t pl ln " m Λ µ Λ α p + l + pl − m Λ E m Λ µ Λ α p + l − pl − m Λ E . (10)In Fig. 1, we plot curves of g (Λ c ) as a function of Λ c with a ΛΛ = − . − . − . B ΛΛ = 6 .
93 MeV. One cansee that the curves exhibit the limit cycle and the firstdivergence appears at Λ c ∼ | a ΛΛ | behaves as giving a larger attractive forceand shifts the curves of g (Λ c ) to the left in Fig. 1.As pointed out in Ref. [25], one can check if the systemexhibits the limit cycle behavior by studying the homo-geneous part of the integral equation in the asymptoticlimit. From Eq. (9), assuming the form of the amplitudein the asymptotic limit p ≫ k as a ( p, k ) ∼ p − − s , wehave 1 = C I ( s ) + C I ( s ) I ( s ) , (11) B ΛΛ ( M e V ) Λ c (MeV)a ΛΛ = - 1.8 fm = - 1.2 fm = - 0.6 fm FIG. 2. (Color Online) The two-Λ separation energy B ΛΛ as a function of the cutoff Λ c for a ΛΛ = − . , − . , − . B ΛΛ = 6 .
93 MeV is included as a reference line. where C = 12 π m α µ Λ α s µ Λ(Λ α ) µ Λ α , C = q m Λ µ Λ(Λ α ) µ α (ΛΛ) π µ / α , (12)and µ α (ΛΛ) = 2 m Λ m α / (2 m Λ + m α ). The functions I , , ( s ) are obtained by the Mellin transformation [26]and their explicit expressions are given in Appendix. Theimaginary solution s = ± is indicates the limit cycle so-lution and we have s = 1 . · · · . (13)On the other hand, the value of s can be obtained fromthe curves of the limit cycle of g (Λ c ) in Fig. 1. The( n + 1)-th values of Λ n at which g (Λ c ) vanishes can beparameterized as Λ n = Λ exp( nπ/s ). By using the sec-ond and third vales of Λ n for the three values of a ΛΛ , wehave s = π/ ln(Λ / Λ ) ≃ .
05, which is in a very goodagreement with the value of Eq. (13). Furthermore, thevalue of s may be checked by using Fig. 52 in Ref. [14]which is a plot of exp ( π/s ) versus m /m for the mass-imbalanced system where m = m = m . In our case, m /m = m Λ /m α ≃ .
3, which leads to s ≃ .
05 by theresult of Ref. [14]. This is in a very good agreement withwhat we find in Eq. (13).One may also reproduce the experimental value of B ΛΛ by adjusting the value of Λ c without introducing thethree-body contact interaction. In this case, the boundstate of He with B ΛΛ = 6 .
93 MeV is found to appearonly when the cutoff parameter Λ c is larger than the crit-ical value Λ cr ≈
300 MeV, which is even larger than Λ H of the theory. We found that Λ c ≈
300 MeV leads to a ΛΛ ≈ − . × fm. When we use a ΛΛ = − . ∼− . ( K − , K + ΛΛ X ) data B ΛΛ ( M e V ) ΛΛ (fm -1 ) Λ c = 430 MeV = 300 MeV = 170 MeVPotential models FIG. 3. (Color Online) The two-Λ separation energy B ΛΛ as a function of 1 /a ΛΛ for Λ c = 170, 300, 430 MeV, where g (Λ c ) is renormalized at the point of B ΛΛ = 6 .
93 MeV and1 /a ΛΛ = − . − that is marked by a filled square. Opensquares are the results from the potential models in Table 5of Ref. [12]. in Ref. [27], we should have Λ c = 570 ∼
408 MeV. InFig. 2, we plot B ΛΛ as a function of the cutoff Λ c for a ΛΛ = − . − . − . B ΛΛ isquite sensitive to both Λ c and a ΛΛ and it becomes largeras Λ c or | a ΛΛ | increases.Figure 3 shows the two-Λ separation energy B ΛΛ asa function of 1 /a ΛΛ while g (Λ c ) is renormalized at thepoint marked by a filled square, i.e., B ΛΛ = 6 .
93 MeVand 1 /a ΛΛ = − . − . This leads to g (Λ c ) ≃ − . − . − .
254 for Λ c = 170, 300, 430 MeV, respec-tively. Open squares are the estimated values from thepotential models given in Table 5 of Ref. [12]. We findthat the curves are sensitive to the cutoff value and theresults from the potential models are remarkably well re-produced by the curve with Λ c = 300 MeV.In summary, we have studied the hypernucleus He as a three-body (ΛΛ α ) system in cluster EFT at LO. Wefound that the three-body contact interaction exhibitsthe limit-cycle and it is needed to be promoted to LO tomake the result independent of the cutoff. The determi-nation equation of the limit cycle for the bound state ofHe is derived and its solutions remarkably well repro-duce the numerically obtained results for the limit cycle.We here note that the determination equation dependson the masses and the spin-isospin quantum numbers ofthe state but not on the details of dynamics and that theimaginary solution of the determination equation impliesthe Efimov states in the unitary limit [16]. Even thoughthe system is not close to the unitary limit, the imaginarysolution could imply the presence of a bound state as seenin this study. Therefore, the determination equation inthree-body cluster systems may be useful to search foran exotic state.We also found that B ΛΛ of He can be reproduced even without introducing the three-body contact inter-action, which, however, requires Λ c = 570 ∼
410 MeVfor a ΛΛ = − . ∼ − . c may be converted to the length scale r c = Λ − c = 0 . ∼ .
48 fm, which overlaps the range of a hard core poten-tial in the early calculations of Ref. [9]. However, the a ΛΛ dependence is significant and it is unlikely to narrowthe range of a ΛΛ . More precise and diverse experimentaldata are thus required.Finally, the correlation between B ΛΛ and a ΛΛ was in-vestigated by introducing the three-body contact inter-action and changing the value of Λ c . We find that theresults of the potential models can be reasonably repro-duced by choosing Λ c = 300 MeV, which may be un-derstood to show the role of the two pion exchange asthe long range mechanism of the ΛΛ interaction. Mean-while, choosing Λ c > Λ H (Λ H ≃
170 MeV) is inconsis-tent within our cluster theory because such a large cutoffprobes the short range (or high momentum) degrees offreedom such as the first excitation state of α , which isbeyond the scope of the present calculation. ACKNOWLEDGMENTS
We are grateful to E. Hiyama for suggesting the presentwork. We also thank J.K. Ahn for useful discussionsand C. Ji for providing his Ph.D. dissertation. Thework of S.-I.A. was supported by the Basic Science Re-search Program through the National Research Foun-dation of Korea funded by the Ministry of Educationunder Grant No. NRF-2012R1A1A2009430. Y.O. wassupported in part by the National Research Founda-tion of Korea funded by the Korean Government (GrantNo. NRF-2011-220-C00011) and in part by the Min-istry of Science, ICT, and Future Planning (MSIP) andthe National Research Foundation of Korea under GrantNo. NRF-2013K1A3A7A06056592 (Center for Korean J-PARC Users).
Appendix
The functions I , , ( s ) in Eq. (11) are obtained by theMellin transformation [26] as I ( s ) = Z ∞ dx ln (cid:18) x + ax + 1 x − ax + 1 (cid:19) x s − = 2 πs sin[ s sin − (cid:0) a (cid:1) ]cos (cid:0) π s (cid:1) , (A.1) I ( s ) = Z ∞ dx ln (cid:18) bx + x + 1 bx − x + 1 (cid:19) x s − = 2 πs b s/ sin[ s cot − (cid:0) √ b − (cid:1) ]cos (cid:0) π s (cid:1) , (A.2) I ( s ) = Z ∞ dx ln (cid:18) x + x + bx − x + b (cid:19) x s − = 2 πs b s/ sin[ s cot − (cid:0) √ b − (cid:1) ]cos (cid:0) π s (cid:1) , (A.3)where a = 2 µ Λ α /m α and b = m Λ / (2 µ Λ α ). When a = b = 1, they reproduce I ( s ) = Z ∞ dx ln (cid:18) x + x + 1 x − x + 1 (cid:19) x s − = 2 πs sin (cid:0) π s (cid:1) cos (cid:0) π s (cid:1) . (A.4) [1] D. J. Prowse, Phys. Rev. Lett. , 782 (1966).[2] R. H. Dalitz, D. H. Davis, P. H. Fowler, A. Montwill,J. Pniewski, and J. A. Zakrzewski, Proc. Roy. Soc. Lond.A , 1 (1989).[3] H. Takahashi et al. , Phys. Rev. Lett. , 212502 (2001).[4] K. Nakazawa for KEK-E176, E373, and J-PARC E07Collaborators, Nucl. Phys. A , 207 (2010).[5] E373 (KEK-PS) Collaboration, J. K. Ahn et al. , Phys.Rev. C , 014003 (2013).[6] E. Hiyama, T. Motoba, T. A. Rijken, and Y. Yamamoto,Prog. Theor. Phys. Suppl. , 1 (2010).[7] A. Gal, Prog. Theor. Phys. Suppl. , 270 (2010).[8] Y.C. Tang, R.C. Herndon, Phys. Rev. Lett. , 991(1965).[9] S. Ali and A. R. Bodmer, Nuovo Cim. A , 511 (1967).[10] H. Bando, K. Ikeda, and T. Motoba, Prog. Theor. Phys. , 508 (1982).[11] S.B. Carr, I.R. Afnan, B.F. Gibson, Nucl. Phys. A
143 (1997).[12] I. Filikhin and A. Gal, Nucl. Phys. A , 491 (2002).[13] P. F. Bedaque and U. van Kolck, Ann. Rev. Nucl. Part.Sci. , 339 (2002).[14] E. Braaten and H.-W. Hammer, Phys. Rep. , 259(2006). [15] S.-I. Ando, G.-S. Yang, and Y. Oh, Phys. Rev. C ,014318 (2014).[16] P. F. Bedaque, H. W. Hammer, and U. van Kolck, Phys.Rev. Lett. , 463 (1999); Nucl. Phys. A , 357 (2000).[17] K. G. Wilson, Phys. Rev. D , 1818 (1971).[18] V. N. Efimov, Yad. Fiz. , 1080 (1970), [Sov. J. Nucl.Phys. , 589–595 (1971)].[19] G. S. Danilov, Zh. Eksp. Teor. Fiz. , 498 (1961), [Sov.Phys. JETP , 349–355 (1961)].[20] V. Bernard, N. Kaiser, and U.-G. Meißner, Int. J. Mod.Phys. E , 193 (1995).[21] S.-I. Ando and D.-P. Min, Phys. Lett. B , 177 (1998).[22] S. R. Beane and M. J. Savage, Nucl. Phys. A , 511(2001).[23] S.-I. Ando and C. H. Hyun, Phys. Rev. C , 014008(2005).[24] U. van Kolck, Nucl. Phys. A , 273 (1999).[25] H. W. Griesshammer, Nucl. Phys. A , 110 (2005).[26] C. Ji, Universality and beyond: Effective field theory forthree-body physics in cold atoms and halo nuclei , PhDthesis, Ohio University, 2012.[27] A. Gasparyan, J. Haidenbauer, and C. Hanhart, Phys.Rev. C85