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14/29/20151 Symposium on Astro-Particle and Nuclear Physics In Honour of 70th Birthday of Prof. Q.N. Usmani 4/29/2015

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Professor M. Z. Rahman Khan 4/29/20153

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4 4 Energies of multi-strange α-cluster hypernuclei using variational Monte Carlo Method MOHAMMAD SHOEB Department of Physics, Aligarh Muslim University, Aligarh-202 002, India 4/29/2015

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5 5 Outline 1.Introduction 2. Hamiltonian in α-cluster model 3. Potential models 4.Variational wavefunctions 5. Results and discussion 6. Summary 4/29/2015

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6 6 1. Introduction: Aim of nuclear physics Complete knowledge of the interaction among octet of baryons in a unified way 4/29/2015

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7 Motivation of studying Strange and multi- strange hypernuclei to extract interaction between hyperon-N and hyperon-hyperon Existence of hypernuclei represent a new state of matter may exhibit new symmetries, selection rules, etc. 4/29/2015

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8 Presence of hyperon(s) may modify the properties of the core moment inertia of deformed nucleus rotational and vibrational states structure of nucleus Hypernucleus provides us a opportunity to investigate properties of hyperon(s) in nuclear medium Hyperon(s) inside nuclei may be used as probe to study the nuclear structure 4/29/2015

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9 9 It is believed that hyperons matter forming the inner core of neutron stars would have significant effect on their properties. Schaffner-Bielich [NP 804(2008)309 and ref. their in ] has discussed that hypernuclear potential depths, two-body hyperon- nucleon and hyperon three-body forces as well as hyperon-hyperon interaction would 4/29/2015

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10 Therefore, determination of hyperon- nucleon and hyperon-hyperon interaction becomes very important for investigating the properties of neutron stars. have a impact on the maximum mass, mass- radius relation, and cooling properties of neutron stars. 4/29/2015

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11 Hypernuclear physics is likely to play key role in the study: Properties of neutron stars Equation state of nuclear matter Figure in the next slide shows interdisciplinary nature of hypernuclei linking particle, nuclear, many-body, astrophysics etc. [ref. Erni et al arXiv: hep-ex/0903.3905] 4/29/2015

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134/29/201513 Segr table 4/29/2015

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14 Extension of the nuclear chart in a new dimension, strangeness S 4/29/2015

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15 -Hypernuclear events, ( ground and excited states), or (Hida event ) and 4/29/2015

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16 Hypernuclear experiments planned or operative at various (nine) laboratories all over the World 4/29/2015

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17 Experimental facilities for hypernuclear physics program List of a few leading laboratories where Hypernuclear physics program to produce and identify hypernuclei with strangeness S= -1 to -3 is being pursued TJNAF(Thomas Jefferson National Accelerator Facility) at Newport news in USA Electro-production FINUDA(FIsica NUclearea DA NE): A special accelerator, DA NE (Double Annular ring For Nice Experiment), designed at INFN (Instituto Nazionale di Fisica Nucleare) 4/29/2015

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184/29/201518 510 MeV ( M=1020 MeV, 20s) Head on collision decaysCopious production A beam of of extremely high intensity and precise low energy is expected to insert “strangeness” inside nucleus to produce hypernuclei. 4/29/2015

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19 J-PARC(Japan Proton Accelerator Research Complex) at KEK: Already a rich data related to both the spectroscopy and decay of hypernuclei at KEK have been measured. Program for production and unambiguous identification of hypernuclei and excited states using reaction ( ) Excited states of double-Λ hypernuclei 4/29/2015194/29/2015

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20 ANDA ( ANnihilation at DArmstadt ): A beam hits primary target to produce ; Stopping and absorption of in the secondary target produce hypernucei. Program to produce S= -3, -hypernucle i 4/29/2015204/29/2015

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21 In primary target In secondary target, e.g. Li, Be, B, Schematic picture describing production of double Λ hypernuclei at PANDA 4/29/2015

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22 Multi-strange hypernuclei Schaffner et al [Ann. Phys.(NY)235(1994) 35 ] observed that a would become particle stable against the strong decay if a sufficient number of bound ’s Pauli blocked this decay mode. Thus is the lightest system suggested to study. At present production of multi- strange hypernuclei seems to be impossible. 4/29/2015224/29/2015

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23 However, it will interesting to theoretically study the stability of multi-strange systems. Such a study is likely to have implication on future experimental efforts in producing, identifying and measuring the properties of multi-strange hypernuclei. Therefore, we have included in our study multi-strange hypernuclei apart from strange ones. 4/29/2015

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244/29/201524 2. Hamiltonian in α-cluster model Hypernuclei studied in the α-cluster model using VMC s-shell: p-shell: Systems within rectangular boxes are the ones whose stability predictions are to be made. 4/29/2015

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25 Hamiltonian for the five-body system in Ξαα model with α treated as rigid : 4 5 α α Ξ 3 2 1 Λ Λ 4/29/2015

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264/29/201526 …..(1) K. E. operator,potential energy for the phenomenological dispersive three-body potential with Yukawa form factors. the particle pair 4/29/2015

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274/29/201527 Hamiltonian forin ΛΛααα model 3 4 5 1 2 ….(2) α α α Λ Λ phenomenological repulsive three-body potential with Gaussian form factors 4/29/2015

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284/29/201528 3. Potential Models 3.1 Two-body potentials Three-range Gaussian BB(=ΛΛ, Ξ) potentials in spin state (=s,t) ….(3) (7.26 MeV) 4/29/2015

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294/29/201529 Potentials For = 4/29/2015

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304/29/201530 Potentials 4/29/2015

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314/29/201531 …(4) (4) 4/29/2015

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324/29/201532 potentials for l th partial waves that fit scattering phase shifts. potential of Chien and Brown has been used for only as the energy is not very sensitive to the choice of the potential. Ali-Bodmer potentials 4/29/2015

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334/29/201533 Two-range Gaussian potentials Isle fits of and its weak decay modes MSA is obtained from Brueckner-Hartee-Fock Theory and slightly modified to fit of [Euro. Phys. J. 16(2003)21] 4/29/2015

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344/29/201534 …..(5) 4/29/2015

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354/29/201535 (4). In the previous slide its graph is shown by black color line. WS24 with = 24.0, as suggested by Dover and Gal [Ann.Phys. 146 (1983 )309], has been obtained from a analysis of old and ambiguous data energy = -2.09 MeV for WS24 and Isle potential = -0.06 MeV for WS14 4/29/2015

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364/29/201536 3.2 Phenomenological Three-body potential among and clusters Microscopic calculations of Bodmer and Usmani for shows that contribution of dispersive three-body NN force for the triad where one nucleon from each is participating 4/29/2015

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374/29/201537 is quite significant, neglecting it among cluster overbinds and. In cluster model calculation we [Pramana68 (2007)943] have proposed to simulate phenomenologically the dispersive energy in the triad through a simple form (6) 4/29/2015

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384/29/201538 and. Phenomenological three-body potential gives good fit to the binding energy and rms radius of in the cluster model for AB [ NP 83(1966)66 & phys Lett B 389(1996)631] potential. ( 7) 4/29/2015

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394/29/201539 4.Variational wavefunctions Construction of good trial wavefunction Physics necessary to describe the ground and excited state Reasonably efficient to compute Wavefunctions are product of two-body correlation functions and the appropriate spin functions 4.1 Wavefunctions (i) and model g.s., degenerate doublet and,. Replacing by gives w.f. for. 4/29/2015

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404/29/201540 (ii) and : model g.s., excited state, ( iii) and : model g.s., degenerate doublet, 4/29/2015

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414/29/201541 Replacing by gives w.f. of. (iv) Wavefunctions for, and (a) wavefunction for in model: = 4/29/2015

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424/29/201542 (b) Wavefunction for : suppress a and a indices in the wavefunction of in (a) above (c) Wavefunction for : suppress a index in the wavefunction of in (a) above (d) Similarly wavefunction for can be obtained. 4/29/2015

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434/29/201543 4.2 Calculation of correlation function A procedure developed by Urbana group. Solution of the following Schroedinger type equation etc. pair. Potential between particles. 4/29/2015

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454/29/201545 4/29/2015

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464/29/201546 5.Procedure for energy calculation ( 8 ) For local operator H the energy can be written in a form suitable for Monte Carlo calculation. Defining local energy 4/29/2015

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474/29/201547 and a multivariate probability distribution ( 9) The variational energy is written as (10 ) 4/29/2015

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484/29/201548 (11) General procedure for calculation of energy in VMC method: (12) 4/29/2015

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504/29/201550 The energy is evaluated using (i) model of : and (ii) model of : and (iii) model of : and (iv) model of : and Similarly for other hypernuclei 4/29/2015

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514/29/201551 To explore the structure of and The quadrupole moment ( ) in the cluster model is calculated using where runs over coordinate of two s, treated as point particles and distances are being measured from the cm of two alphas. 4/29/2015

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524/29/201552 5. Result and Discussion VMC energy: s-Shell Hypernuclei and NSC97e, :ESC00, ND, NSC97b 4/29/2015

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534/29/201553 : Isle and WS = 2.09 MeV, =3.45 (Isle) and 3.36 (WS) fm ( ) fm for WS24 Isle WS (12.6, 2.93) (4.8, 2.2) potentials 4/29/2015

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544/29/201554 Filikhin et al Faddeev Faddeev-Yakubovsky predicted for unbound for Isle soft replusive potential bound for WS24 potential Prediction: binding energy capable of discriminating between radial shape of central potentials, very unlikely, as this violates shape independence of low energy data 4/29/2015

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554/29/201555 potential dependent configuration for stronger potential a configuration speculated i.e. screened by. configuration for weaker potential Detailed VMC calculation :ESC00, ND, NAGSIM, NSC97e, NSC97b :NSC97e : Isle, WS24, WS14 4/29/2015

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564/29/201556 F and FY [Filikhin et al JPG35(2008)] bold face within round bracket 4/29/2015

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574/29/201557 Our VMC calculation demonstrates and have negative energies and is bound for WS24 and Isle potential ( while is unbound for Isle potential in F-Y method). Gross property such as energy, not good discriminator of the shapes of the two-body potential. is unbound for WS14 potential. Due to strong conversion process,we will comment on stability of in the last. 4/29/2015

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584/29/201558 4/29/2015

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594/29/201559 Pyramid on triangular isosceles base Isosceles plane Two planes ESC00, NAGSIM, NSC97b and Arms in general increase with decrease in strength of potential but no change in the configuration 4/29/2015

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604/29/201560 being screened by two lambdas our calculation support not as opposed to speculation of Filikhin et al configuration whether interaction is weak or strong.[JPG36(2009)045104] 4/29/2015

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614/29/2015 61 p-Shell hypernuclei: (i),,, and Experimental =6.71 MeV =17.6 0.4 MeV ( 14.5 0.4 MeV assuming, a ray of about 3.0 MeV must have escaped the identification of decay product from the emulsion ). Excited states: = 3.66 MeV Demchi-Yanagi event = MeV 4/29/2015

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62 Revised = 11.90 0.13 MeV =11.69 MeV 4/29/2015

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634/29/201563 and of three-body potential are adjusted to fit ground state energy Ali-Bodmer [Chien-Brown]potential 4/29/2015

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644/29/201564 No free potential parameters 4/29/2015

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654/29/201565 No free potential parameters Isle+AB MSA+AB 4/29/2015

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664/29/201566 No free potential parameters Demchi-Yanagi event B =12.33 MeV 4/29/2015

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674/29/201567 -Ve quadrupole moment for : Oblate shape : Experimental Q =5.3 N replaced by Prolate Oblate Not much difference between the calculated energy using AB and CB alpha-alpha, Isle and MSA lambad-alpha interactions. Therefore, we consider only AB alpha-alpha and Isle lambda-alpha interactions. 4/29/2015

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684/29/201568 = 11.69 MeV First we analysed sub system Ground state energy = -7.26 MeV Excited state energy = -2.84 MeV Energies of ( and ) calculated variationally in cluster model. i) interaction(AB) very repulsive, gives energy -0.7 MeV. ii) + parameters and adjusted to fit the ground state. sitting at the vertices of equilateral triangle. Experimental 4.17 fm 2.4fm c.m. 4/29/2015

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694/29/201569 Filikhin et al JPG35 (2008) FY calculation with out 4/29/2015

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704/29/201570 and in < for (Shrinkage of core ) > in, wanders at the periphery of core 4/29/2015

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714/29/201571 Variational e nergy VMC energy (present work) = - 4.99 MeV in alpha cluster model using Jacobian-coordinate MeV Gaussian-basis function method ( Hiyama et al [Prog.Theo.Phys.97(1997)881] ). -7.17 -4.29 Exp. -2.84MeV Exp. -7.26MeV 4/29/2015

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724/29/201572 Ground and degenerate doublet Interaction (AB)+ (Isle) + + (no free parameter) Exp. -7.26+(-11.69) MeV -18.81 -14.98 Prediction 4/29/2015

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734/29/201573 : VMC prediction for G.S. energy in model fits energy(-7.26 MeV) + Isle + (AB) VMC Energy =-31.29 MeV; Predicted binding of = -7.16- (-31.29)= 24.12MeV 4/29/2015

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744/29/201574 in model :VMC Prediction (Isle & WS) + + Filikhin et al JPG35(2008) 4/29/2015

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754/29/201575 (ii) Prediction for the energies of and Calculated energy of the above systems in and models using combinations of (Isle, WS24, Ws14), (NSC97e) and (NSC97( b,e), NAGSIM) potentials along with dispersive three-body force. 4/29/2015

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764/29/201576 4/29/2015

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774/29/201577 4/29/2015

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784/29/201578 VMC energy for < Faddeev method Due to interplay between Calculation limited to s-wave two-body correlations, contribution from partial waves higher than s-wave is simulated. energy for Isle and WS24 differ by 3% for twice of Separation energy in nearly independent of strength. 4/29/2015

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79 Do the negative energies of and implies stabilty ? No Due to strong conversion depending on the depth of WS potential, these systems can decay as: ; 4/29/2015

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804/29/201580 Calculated level scheme for two depths of potential. It seems very unlikely that will ever observed in future experi- mental effort. 4/29/2015

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81 Note: All the calculations where appears were performed for hyperon. To obtain binding of hypernucleus a coulomb correction 1.5 to 2.0 MeV per alpha particle is to be added to the binding of hypernucleus containing. 4/29/2015814/29/2015

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82 6.Summary VMC for binding energy of three-, four-, and five-body alpha cluster s- and p-shell hypernuclei First cluster model VMC calculation for predicting the energy of multistrange hypernuclei VMC energy for is insensitive to the shape of potential as opposed to Faddeev- Yakubovsky method 4/29/2015824/29/2015

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83 is unbound and stability depends on potential depth Predicted the energies for and for ground state, and are predicted to be stable for WS14 Demachi-Yanagi event is interpretted as excited and exited state energy of degenerate doublet of explained. 4/29/2015

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84 Thank you 4/29/2015

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