Precise Few-body calculations for kaonic nuclear and atomic systems Asia-Pacific Conference on Few-Body Problems in Physics (APFB2017) Ronghu Hotel, Guilin, China 2017.8.25-29 Wataru Horiuchi (Hokkaido Univ.) Collaborators: S. Ohnishi, T. Hoshino (Hokkaido β company) K. Miyahara (Kyoto), T. Hyodo (YITP, Kyoto) W. Weise (TUM, Germany; YITP, Kyoto)
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Precise Few-body calculations for kaonic nuclear and atomic systems
Asia-Pacific Conference on Few-Body Problems in Physics (APFB2017)Ronghu Hotel, Guilin, China
2017.8.25-29
Wataru Horiuchi (Hokkaido Univ.)
Collaborators: S. Ohnishi, T. Hoshino (Hokkaido β company)K. Miyahara (Kyoto), T. Hyodo (YITP, Kyoto)W. Weise (TUM, Germany; YITP, Kyoto)
Outline
β’ Precise few-body calculations for kaonic systemsβ Modern πΎπΎππ interaction K. Miyahara, T. Hyodo, PRC93 (2016)
β’ Kaonic nuclei: πΎπΎππππ to πΎπΎππππππππππππ (7-body)
β’ Kaonic deuterium: πΎπΎππππ three-body system
β Unified approach to atomic and nuclear kaonic systems β’ Nucleus ο½few fmβ’ Atom ο½several hundreds fm
Precise variational calculations with correlated Gaussian method
S. Ohnishi, WH, T. Hoshino, K. Miyahara, T. Hyodo, Phys. Rev. C95, 065202 (2017)
T. Hoshino, S. Ohnishi, WH, T. Hyodo, W. Weise, arXiv:1705.06857
Reproduce Ξ(1405) as πΎπΎππ quasi-bound state
for N-body
Choice of πΎπΎππ interaction
Y.Ikeda, T.Hyodo, W.Weise, NPA881 (2012) 98
A. Dote, T. Hyodo, W. Weise, NPA804, 197 (2008).
Akaishi, Yamazaki, PRC65, 04400(2002).
K.Miyahara, T.Hyodo, PRC 93, 015201 (2016)
βField pictureβ
βParticle pictureβ
Variational calculation for many-body quantum system
β’ Many-body wave function Ξ¨ has all information of the systemβ’ Solve many-body Schoedinger equation
β Eigenvalue problem with Hamiltonian matrixHΞ¨ = EΞ¨
β’ Variational principle <Ξ¨|H|Ξ¨> = E β§ E0 (βExactβ energyοΌοΌEqual holds if Ξ¨ is the βexactβ solutionοΌ
Many degrees of freedomβ Expand Ξ¨ with several sets of basis functions
Correlated Gaussian + Global vectors
Formulation for N-particle systemAnalytical expression for matrix elements
Explicitly correlated basis approach
x: any relative coordinates (cf. Jacobi)
Correlated Gaussian with two global vectors Y. Suzuki, W.H., M. Orabi, K. Arai, FBS42, 33-72 (2008)
x1
x4x3
x2
y1
y4
y3y2 x1 x3
x2y1 y2
y3
Shell and cluster structure Rearrangement channels
Functional form does not change under any coordinate transformation
See Review: J. Mitroy et al., Rev. Mod. Phys. 85, 693 (2013)
Possibility of the stochastic optimization1. increase the basis dimension one by one2. set up an optimal basis by trial and error procedures3. fine tune the chosen parameters until convergence
Y. Suzuki and K. Varga, Stochastic variational approach to quantum-mechanical few-body problems, LNP 54 (Springer, 1998).K. Varga and Y. Suzuki, Phys. Rev. C52, 2885 (1995).
),,,( 21 mkkk EEE 2. Get the eigenvalues
4. k β k+1
),,,( 21 mkkk AAA 1. Generate randomly
3. Select nkEn
kA corresponding to the lowest and Include it in a basis set
Basis optimization: Stochastic Variational Method
Energy curvesβ’ Optimization only with a real
part of the πΎπΎππ pot.β’ Two-body πΎπΎππ energy is self-
consistently determinedβ’ AV4β NN pot. is employed
Full energy curves
Validity of this approach is confirmedin the three-body (K-pp) system
Properties of K-pp
Kyoto π²π²π΅π΅ pot.Similar binding energies with Types I and II Bο½27-28 MeV
Ξο½30-60 MeV
AY pot.Deeper binding energy ο½49MeVβ Smaller rms radii
Nucleon Density distributions
(deuteron is J=1)
Central nucleon density Ο(0) is enhanced by kaonΟ(0)~0.7fm-3 at maximum, ο½2 times higher than that without πΎπΎ
(ο½4 times higher than saturation dens.)
Interaction dependence
Not sensitive to the NN interaction models
AV4β and ATS3 potential: strong short-range repulsionMN: weak short-range repulsion
Nucleon density distributions
Binding energy and decay width with different NN potential models
Structure of πΎπΎππππππππππππ with JΟ=0- and 1-
πΎπΎππ interaction in I=0 is more attractive than in I=1, and J=0 state containing more I=0 component than J=1
Energy gain in J=0 is larger than J=1 channelAY potential in I=0 is strongly attractive
Constraint for πΎπΎππ interaction: Kaonic deuterium
β’ Isospin dependence of πΎπΎππ interactionsβ I=0: well determined by Ξ(1405) propertiesβ I=1: constraint is weak only with kaonic hydrogen
β’ Precise kaonic deuterium data (Exp. and theor.) are highly desired
β’ Geometric progression for Gaussian fall-off parameters β Cover 0.1-500 fmβ πΏπΏ1 + πΏπΏ2 β€ 4β About 8000 basis states
β’ Channel coupling, physical masses
Precise calculation with huge number of nonorthogonal bases
New orthonormal set
Generalized eigenvalue problem
Cutoff parameter
β
Energy levels of 1S, 2P, 2S states
β’ No level shift of 2P stateβ’ Transition energy can directly be used for the
extraction of the 1S level shiftCoulomb + οΏ½πΎπΎππ+etc.
(experiment) 2p
1s
Point charge(analytic)
1S level shift
Level shift of kaonic deuterium
Sensitivity of I=1 component π π ππ ππ οΏ½πΎπΎππ = π π ππ πππΌπΌ=0 + Ξ² Γ π π ππ πππΌπΌ=1 Varying the strength of the real part of the KN potential within the
SIDDHARTA constraint for the level shift of kaonic hydrogen ΞE = 283 Β± 36(stat) Β± 6(syst)eV Ξ = 541 Β± 89(stat) Β± 22(syst)eV
Energy shifts of kaonic hydrogen and deuterium
ο½25% uncertaintyPossible constraint for I=1
Conclusions
Unified description of kaonic atom and nuclear systemsβ Kaonic nucleus (3- to 7-body)
β’ Central density is increased by ο½2 times higher (4 times than Ο0)β Soft NN interaction induces too high central densities
β’ Inverted spin-parity in the g.s, of 6Li πΎπΎ is predictedβ Isospin dependence of πΎπΎππ interaction is essential
β Kaonic atom (3-body)β’ Prediction of the energy shift of the kaonic deuterium
β’ I=1 component can be constrained if measurement is performed within 25% uncertainty
Planned exp. accuracy ο½5-10%C. Curceanu et al., Nucl. Phys. A914 251 (2013).M. Iliescu et al., J. Phys. Conf. Ser. 770, no. 1, 012034(2016).J. Zmeskal et al., Acta Phys. Polon. B 46, no. 1, 101 (2015).
T. Hoshino, S. Ohnishi, WH, T. Hyodo, W. Weise, arXiv:1705.06857
S. Ohnishi, WH, T. Hoshino, K. Miyahara, T. Hyodo, Phys. Rev. C95, 065202 (2017)