Precise Few-body calculations for kaonic nuclear and...
Transcript of Precise Few-body calculations for kaonic nuclear and...
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
Kaonic nuclei (Nucleus with antikaon)
ud
s
Isgur, Karl, PRD 18, 4187(1978)
β strongly attractive πΎπΎππ interaction
Does Kaonic nucleus really exist? E15 exp. Can such a high density system be produced in laboratory? β πΎπΎππ interaction is essential!
Dalitz, Wong, Tajasekaran, PR 153, 1617 (1967)
β’ Ξ(1405); JΟ=1/2-, S= -1β uds constituent quark model
β’ Energy is too high
β πΎπΎππ quasi-bound state
Y. Akaishi, T. Yamazaki, PRC 65, 044005 (2002).
Dote, et. al., PLB590, 51(2004).
Y. Sada et al., Prog. Theor. Exp. Phys. 2016, 051D01 (2016).
Kaonic nuclear systems (3 to 7-body)
β’ Hamiltonianβ’ Correlated Gaussian basis
β Many parameters ο½(N-1)(N-2)/2Γ(# of basis)β Stochastic variational method
β’ Choice of NN potential (AV4β, ATS3, MN)
All NN interaction models reproduce the binding energy of s-shell nuclei
K. Varga and Y. Suzuki, PRC52, 2885 (1995).
Kyoto πΎπΎππ potential Energy-dependent πΎπΎππ single-channel potential Chiral SU(3) dynamics at NLO Pole energy: 1424 - 26i and 1381 β 81i MeV
πΎπΎππ two-body energy in an N-body system are determined as:
Akaishi-Yamazaki (AY) potential Energy-independent
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
J=0 ground state
6Li
6Li K-
1+0+
AYKyoto πΎπΎππ
1-
1-
0-
0-
π΅π΅π΅π΅ π²π²π΅π΅π΅π΅J=0 unbound Bound
J=1 bound (d) unbound
A=2 A=6
Kaonic hydrogen (atomic system)
β’ Bound mainly with Coulomb int. (like e- in H)
β’ πΎπΎππ interaction induces βlevel shiftββ’ Precise measurement
(2011, SIDDHARTA experiment , DAΦNE ) Bazzi et al., NPA881 (2012)
Ο΅1s = 283 Β± 36(stat) Β± 6(syst)eVΞ1s = 541 Β± 89(stat) Β± 22(syst)eV
p
Kβ
X-rayCoulomb + οΏ½πΎπΎππ
(experiment)2p
1s
Coulomb (analytic)
1S level shift
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
πΎπΎβππ = ββ= |πΌπΌ = 0β© + |πΌπΌ = 1β©
πΎπΎβππ = ββ= πΌπΌ = 1
I=0:I=1=1:3
p
Kβ
Kaonic deuterium
n
Three-body calculation for kaonicdeuterium
Hamiltonian
π»π» = ππ + ππ = οΏ½ππ=1
3ππππ2
2ππππβ ππππππ + ππππππ + ππ οΏ½πΎπΎππ + ππππππππππ
ππππππ (Minnesota potential) D. R. Thompson, M. Lemere and Y. C. Tang, NPA286 (1977)
ππ οΏ½πΎπΎππ(Kyoto πΎπΎππ potential) K.Miyahara, T.Hyodo, PRC93 (2016)
β’ The Kyoto πΎπΎππ pot. simulates scattering amplitude calculated from NLO chiral SU(3) dynamics Y.Ikeda, T.Hyodo, W.Weise, NPA881 (2012)
1. Scattering length extracted from the energy shift measured in SIDDHARTA experiment.
2. Cross section of a οΏ½πΎπΎππ two-body scattering3. Branching ratio of the οΏ½πΎπΎππ decay
Three-body calculationVariational Method Wave function
Correlated Gaussian basis
π΄π΄: 2 Γ 2 positive definite symmetric matrixππ = ππ1,ππ2 , οΏ½ππππ = π’π’1ππ1 + π’π’2ππ2, οΏ½ππππ = π£π£1ππ1 + π£π£2ππ2
π¦π¦ππππ οΏ½ππππ = οΏ½ππππ ππππππππ(οΏ½οΏ½ππππ)πππ½π½π½π½: spin function, πππππ½π½π‘π‘: isospin function
Ξ¨ = οΏ½ππ=1
ππ
ππππππππ , ππ = π΄π΄ππππ {ππβ12οΏ½πππ΄π΄ππ[[π¦π¦πΏπΏ1 οΏ½ππππ π¦π¦πΏπΏ2 οΏ½ππππ ]π³π³ππππ]π½π½π½π½πππππ½π½π‘π‘} NοΏ½πΎπΎ
N
ππ1
ππ2
Configuration 1
NοΏ½πΎπΎ
Nππ1
ππ2
Configuration 2
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)