Aye Aye Min, Khin Swe Myint, J. Esmaili & Yoshinori AKAISHI August 23, 2011 By Theoretical Investigation for Production of Double-  Hypernuclei from Stopped Hyperon on APFB2011

Abstract Investigation of the formation ratio of to for various absorptions from 2S, 2P and 3D orbitals of atom by assuming a d-  cluster model for Two kinds of d-  relative wave function namely 1s d-  relative wave function with phenomenological One Range Gaussian (ORG) potential and that with Orthogonality Condition Model (OCM ) were used in our calculations. We have also investigated differential cross section for single-   hypernuclei, and.

(K. Nakazawa, Nucl. Phys. A 835 (2010)) It is worthwhile to measure the masses of double-  hypernuclei for several nuclear species to determine  -  interaction without ambiguities. t p Emulsion Experiment  B  :  interaction energy  B  = B  ( A  Z) - 2B  ( A-1  Z) Weakly attractive  Interaction ! (T. Fukuda et. al., Phys. Rev. Lett. 87 (2001)) BNL

S = 0 sector S = -1 sector S = -2 sector NN NN NN NN  NN ~ 300 MeV ~ 80 MeV ~ 28 MeV Although coupling effect is not significant in non-strangeness sector, coupling effect plays an important role in strangeness sector. (K.S. Myint, S. Shinmura and Y. Akaishi, Nucl. Phys. A 721 (2003) 21)  N coupling effect in

d-  cluster structure Production of Double-  Hypernuclei In order to produce and, the reaction is Target ( ) P t d P n  P t d  P n  

two  hyperons and ordinary nucleus       H-dibaryon and ordinary nucleus S P D       Elementary process for the reaction 28.33MeV Single-  hypernucleus and  -hyperon    Two single-  hypernuclei         Double-  hypernucleus Absorption of  - in atom and Production of hypernuclei

No.ReactionsQ-value (MeV) Table 1. Possible reactions for the stopped hyperon on

d   t  P t d Transition matrix, Transition matrix in terms of relevant momenta, internal wave function of sub-systems relative wave functions Triton(t), deuteron(d) proton-triton (p-t) deuteron-alpha (d-  ) n n p t p n d p d   t wave function for target Formation from stopped on

Interaction for elementary process, is described by separable potential. where Interaction for elementary process By assumption the interaction is zero range, p  

We will discuss later! Decay width to and deutron Decay width ( ) is

 P n     n Formation p n n n p  p 1. GBWF (one range phenomenonlogiacl Gaussian potential) 2. GBWF (OCM model)    n

Construction of relative d-  wave function by using one range phenomenological Gaussian potential Gaussian basis radial wave function for d-  cluster is Gaussian one range potential b j = range parameter and c j = the expasion coefficient we adjusted the potential strength( MeV) to give energy eigen value of 1s state(-1.48 MeV ) and eigen function corresponding to this 1s bound state. By applying Fourier transform, 2.0 fm

The Gaussian potential between  and x particle Where, For our system, case, x is deuteron and. The potential strengths and range parameters for  -d system Construction of relative d-  wave function with OCM (E. Hiyama et.al., arXiv:nucl-th 24 (2002) ) = the spin of x = relative angular momentum between  and x

The Pauli principle between nucleons belonging to  and x (x = n, p, d, t ) clusters is taken into account by the Pauli projection operator or OCM projection operator The forbidden states for d-  cluster are 0s and 0p states. Monte Carlo integration Method

Models of single  -hypernuclei

d-  density distribution of In coordinate space Results and Discussions B.E ( ) = 5.0 MeV Table 2. Formation ratio of to from stopped hyperon on d-  wave function types Atomic absorption (arbitary unit) GBWF (1s) (one range pot.) 2S P D OCM 2S P D K.S. Myint, S. Shinmura and Y. Akaishi, Eur. Phys. J. A 16 (2003) 21.

effect of low and high momenta component of d-  relative motion ??? effect of low and high momenta component of d-  relative motion ??? to clarify this argument more profoundly! q d  (MeV/c) d-  density distribution of (in momentum space) This wave function ( 0s′ ) is obtained by reducing the strength of one range Gaussian potential ( MeV) to give the ground state energy, E = MeV.

Significance of d-  relative momentum contribution Wave function types atomic absorption GBWF( 1s )3D GBWF( )3D formation is enhanced and formation is dropped off significantly! It is important to understand the structure of a target to propose a feasible reaction to populate double-  hypernuclei from hyperon captured at rest.

For single-  hypernuclei case, and are at rest! 150 MeV/c MeV/c

It may be deduced the significance of  -  coupling effect from this experiment. Formation of is more dominant than that of for all absorption orbitals; 2S, 2P and 3D states from this reaction ( 1.1 for ORG and 2.0 for OCM for the major 3D absorption case). Concluding remarks Binding energy of can be measured without ambiguities. Thus, we have proposed a feasible reaction which can produce,, and with comparable branching ratios. Low momentum component of d-  relative wave function favors the formation.

Thank you for your kind attention!

0.0 MeV MeV MeV  +  + t 8.0 MeV    +     + n  t  + p + t    + p + t Pauli Suppression effect  N coupling effect in Coupling effect enhancement

Strength ( ) BE of d-  cluster (MeV) remark Unphysical forbidden state Unphysical forbidden state Unphysical forbidden state Unphysical forbidden state Allowed state Allowed state Allowed state Binding Energy of d-  cluster by changing the strength of value

E =-1.48 MeV

 P t d  P n  P t d  P n   Proton speration ~ MeV energy B.E(d ) =2.224 MeV n formation t d   n     (  6 He) 2S absorption  (  6 He) 2P absorption  (  6 He) 3D absorption BE(2.224 MeV) BE(3.5MeV)

P d = MeV/c KE d =9 MeV KE(   H)= 3.04 MeV Q=12.04 MeV P n = MeV KE n =28 MeV KE(   He)= 3.88 MeV Q=31.88 MeV

The required data are;

 P t d  P n  P t d  P n   Proton speration ~ MeV energy B.E(d ) =2.224 MeV n formation t d   n     (  6 He) 2S absorption  (  6 He) 2P absorption  (  6 He) 3D absorption BE(2.224 MeV) BE(3.5MeV)

Abundant of Lithium 7% 93%  P t d 6 Li MeV  P t t 7 Li -2.5 MeV

d-  wave function types Atomic absorption (arbitary unit) GBWF (1s) (one range pot.) 2S P D OCM 2S P D (arbitary unit) Old data from Nagara_paper (BE(LLHe6)) New data from Nagazawa Sensei (BE(LLHe6))

Wave function types Probabilities of low momentum component Probabilities of high momentum component GBWF(1s) OCM Table 3. Probabilities of momentum components of d-  relative wave unction of

Introduction  hyperon can stay in the nucleus deeply without obeying Pauli exclusion principle  hypernucleus probes a deep interior of the nucleus and investigates the nuclear structure gives a new dimension to the traditional world of nuclei provides the rich information on the baryon dynamic involving the strange particles

Strangeness-exchange process Combination of strangeness exchange and associated production of strangeness process Associated production of strange-hadrons process Possible production of hypernuclei etc.

participant Spectator -projectile fragment Spectator -target fragment coalescence of hyperons to projectile fragnent  theoretical model (Wakai, Bando, Sano) High energy heavy-ion collisions From Professor Dr T. Fukuda’s Presentation

High energy heavy-ion collisions Coalescence of strange particles with a nuclear fragment produced in projectile nuclear fragmentation Coalescence of strange particles and nucleons both produced in the participant part Secondary process by  and K mesons produced in the primary nuclear collisions p n     K F F Conversion of   hypernucleus into single and double-  hypernucleus ( at 2.1 GeV/nucleon ) ( at 3.7 GeV/nucleon ) ( at 2.1 GeV/nucleon ) ( at 14.5 GeV/nucleon ) etc.

In order to produce a hypernucleus, where, q = momentum transfer to the hyperon The hyperon emerging from the reaction must remain in the nucleus. Formation probability of the hypernucleus Momentum transfer to the hyperon Sticking probability, n, = principal quantum number and orbital angular momentum for nucleon and hyperon state = bessel function with the orbital angular momentum transfer ( initial and final states are Harmonic Oscillator wave functions )

Direct Process Via  atom KEK-E 176 P H K-K- K+K+    K-K- 00 K+K+    K+K+ K-K- -- K-K- K+K+   - or H (?) K-K- K+K+   KEK-E 176 -E 224 BNL-E 813 -E 836 -E 885 KEK-E 176 -E 224 BNL-E 885 KEK-E 224  - atom  K+K+ K-K- -- A A     or H Prowse (?), Danysz et al. KEK- E 176, E373 BNL- E906 KEK- E 176 E373 KEK- E 176 E 224 BNL- E 885

H. Takahashi, “PhD Thesis”, Kyoto University (2003) Possible Candidates of double-  hypernuclei in emulsion experiments

KEK-PS E176 or  interaction energy attractive or repulsive ??? Double hyper event from E-176 experiment

Double hyper event from E-373 experiment t p  B  :  interaction energy  B  = B  ( A  Z) - 2B  ( A-1  Z) Weakly attractive  Interaction !

Nakazawa Sensei, 2003 Presentation (at J-Lab)

Nakazawa Sensei, 2003 Presentation (at J-Lab)

KEK-PS E176 or (Possibility of excited state was not considered!)

KEK-PS E373

Construction of relative d-  wave function by using one range phenomenological Gaussian potential Gaussian basis radial wave function for d-  cluster is Gaussian one range potential b j = range parameter and c j = the expasion coefficient Hamiltonian operator is we adjusted the potential strength( MeV) to give energy eigen value of 1s state(-1.48 MeV ) and eigen function corresponding to this 1s bound. By applying Fourier transform, 2.0 fm

approximate value of an integral Pick n randomly distributed points x 1, x 2, x 3,…, x n in the interval [ a,b ]. Average value of the function Approximate value of an integral Estimation for the error Monte Carlo Integration Method

Binding energy of Atomic absorption 3.59 MeV 2S P D MeV 2S P D MeV 2S P D 2.01 Binding energy of Atomic absorption 3.59 MeV 2S P D MeV 2S P D MeV 2S P D 3.17 GBWF (1s) OCM (1s) Binding energy effect of Ms. Hla Hla win (Ph D thesis, private communication)Binding energy of  6 He ( NAGARA event data ) K.S. Myint et.al., Eur. Phys. J. A 16 (2003) 21

Wave function types Probabilities of low momentum component Probabilities of high momentum component GBWF(1s) OCM(1s) GBWF(0s’) Table 2. Probabilities of momentum components of d-  relative wave function of

3Li6 no: of proton 3 no: of neutron 3 P  0s(2-1/2),0p(1-3/2) n  0s(2-1/2),0p(1-3/2) J=J(p)+J(n) =3/2+3/2 =3,2,1,0 ( 2 is impossible)  =(-1)**(l_p+l_n) =(-1)**(1+1) =+ J_  =3+,1+,0+ Iso_spin Transition matrix,

effect of low and high momenta component of d-  relative motion ??? effect of low and high momenta component of d-  relative motion ??? to clarify this argument more profoundly! This wave function (0s ’ ) is obtained by reducing the strength of one range Gaussian potential ( MeV) to give the ground state energy E = MeV. d-  density distribution of (in momentum space) q d  (MeV/c)

Our University will be held the International Conference on February, 2011.