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 โ company)K. Miyahara (Kyoto), T. Hyodo (YITP, Kyoto)W. Weise (TUM, Germany; YITP, Kyoto)
Outline
โข Precise few-body calculations for kaonic systemsโ Modern ๐พ๐พ๐๐ interaction K. Miyahara, T. Hyodo, PRC93 (2016)
โข Kaonic nuclei: ๐พ๐พ๐๐๐๐ to ๐พ๐พ๐๐๐๐๐๐๐๐๐๐๐๐ (7-body)
โข Kaonic deuterium: ๐พ๐พ๐๐๐๐ three-body system
โ Unified approach to atomic and nuclear kaonic systems โข Nucleus ๏ฝfew fmโข Atom ๏ฝseveral 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)
โ strongly attractive ๐พ๐พ๐๐ interaction
Does Kaonic nucleus really exist? E15 exp. Can such a high density system be produced in laboratory? โ ๐พ๐พ๐๐ interaction is essential!
Dalitz, Wong, Tajasekaran, PR 153, 1617 (1967)
โข ฮ(1405); Jฯ=1/2-, S= -1โ uds constituent quark model
โข Energy is too high
โ ๐พ๐พ๐๐ 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)
โข Hamiltonianโข Correlated Gaussian basis
โ Many parameters ๏ฝ(N-1)(N-2)/2ร(# of basis)โ Stochastic variational method
โข Choice of NN potential (AV4โ, ATS3, MN)
All NN interaction models reproduce the binding energy of s-shell nuclei
K. Varga and Y. Suzuki, PRC52, 2885 (1995).
Kyoto ๐พ๐พ๐๐ potential Energy-dependent ๐พ๐พ๐๐ single-channel potential Chiral SU(3) dynamics at NLO Pole energy: 1424 - 26i and 1381 โ 81i MeV
๐พ๐พ๐๐ two-body energy in an N-body system are determined as:
Akaishi-Yamazaki (AY) potential Energy-independent
Reproduce ฮ(1405) as ๐พ๐พ๐๐ quasi-bound state
for N-body
Choice of ๐พ๐พ๐๐ 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)
โField pictureโ
โParticle pictureโ
Variational calculation for many-body quantum system
โข Many-body wave function ฮจ has all information of the systemโข Solve many-body Schoedinger equation
โ Eigenvalue problem with Hamiltonian matrixHฮจ = Eฮจ
โข Variational principle <ฮจ|H|ฮจ> = E โง E0 (โExactโ energy๏ผ๏ผEqual holds if ฮจ is the โexactโ solution๏ผ
Many degrees of freedomโ Expand ฮจ 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 โ 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 curvesโข Optimization only with a real
part of the ๐พ๐พ๐๐ pot.โข Two-body ๐พ๐พ๐๐ energy is self-
consistently determinedโข AV4โ NN pot. is employed
Full energy curves
Validity of this approach is confirmedin the three-body (K-pp) system
Properties of K-pp
Kyoto ๐ฒ๐ฒ๐ต๐ต pot.Similar binding energies with Types I and II B๏ฝ27-28 MeV
ฮ๏ฝ30-60 MeV
AY pot.Deeper binding energy ๏ฝ49MeVโ Smaller rms radii
Nucleon Density distributions
(deuteron is J=1)
Central nucleon density ฯ(0) is enhanced by kaonฯ(0)~0.7fm-3 at maximum, ๏ฝ2 times higher than that without ๐พ๐พ
(๏ฝ4 times higher than saturation dens.)
Interaction dependence
Not sensitive to the NN interaction models
AV4โ 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 ๐พ๐พ๐๐๐๐๐๐๐๐๐๐๐๐ with Jฯ=0- and 1-
๐พ๐พ๐๐ 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 ๐พ๐พ๐๐
1-
1-
0-
0-
๐ต๐ต๐ต๐ต ๐ฒ๐ฒ๐ต๐ต๐ต๐ตJ=0 unbound Bound
J=1 bound (d) unbound
A=2 A=6
Kaonic hydrogen (atomic system)
โข Bound mainly with Coulomb int. (like e- in H)
โข ๐พ๐พ๐๐ interaction induces โlevel shiftโโข Precise measurement
(2011, SIDDHARTA experiment , DAฮฆNE ) Bazzi et al., NPA881 (2012)
ฯต1s = 283 ยฑ 36(stat) ยฑ 6(syst)eVฮ1s = 541 ยฑ 89(stat) ยฑ 22(syst)eV
p
Kโ
X-rayCoulomb + ๏ฟฝ๐พ๐พ๐๐
(experiment)2p
1s
Coulomb (analytic)
1S level shift
Constraint for ๐พ๐พ๐๐ interaction: Kaonic deuterium
โข Isospin dependence of ๐พ๐พ๐๐ interactionsโ I=0: well determined by ฮ(1405) propertiesโ I=1: constraint is weak only with kaonic hydrogen
โข Precise kaonic deuterium data (Exp. and theor.) are highly desired
๐พ๐พโ๐๐ = โโ= |๐ผ๐ผ = 0โฉ + |๐ผ๐ผ = 1โฉ
๐พ๐พโ๐๐ = โโ= ๐ผ๐ผ = 1
I=0:I=1=1:3
p
Kโ
Kaonic deuterium
n
Three-body calculation for kaonicdeuterium
Hamiltonian
๐ป๐ป = ๐๐ + ๐๐ = ๏ฟฝ๐๐=1
3๐๐๐๐2
2๐๐๐๐โ ๐๐๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐ ๏ฟฝ๐พ๐พ๐๐ + ๐๐๐๐๐๐๐๐๐๐
๐๐๐๐๐๐ (Minnesota potential) D. R. Thompson, M. Lemere and Y. C. Tang, NPA286 (1977)
๐๐ ๏ฟฝ๐พ๐พ๐๐(Kyoto ๐พ๐พ๐๐ potential) K.Miyahara, T.Hyodo, PRC93 (2016)
โข The Kyoto ๐พ๐พ๐๐ 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 ๏ฟฝ๐พ๐พ๐๐ two-body scattering3. Branching ratio of the ๏ฟฝ๐พ๐พ๐๐ decay
Three-body calculationVariational Method Wave function
Correlated Gaussian basis
๐ด๐ด: 2 ร 2 positive definite symmetric matrix๐๐ = ๐๐1,๐๐2 , ๏ฟฝ๐๐๐๐ = ๐ข๐ข1๐๐1 + ๐ข๐ข2๐๐2, ๏ฟฝ๐๐๐๐ = ๐ฃ๐ฃ1๐๐1 + ๐ฃ๐ฃ2๐๐2
๐ฆ๐ฆ๐๐๐๐ ๏ฟฝ๐๐๐๐ = ๏ฟฝ๐๐๐๐ ๐๐๐๐๐๐๐๐(๏ฟฝ๏ฟฝ๐๐๐๐)๐๐๐ฝ๐ฝ๐ฝ๐ฝ: spin function, ๐๐๐๐๐ฝ๐ฝ๐ก๐ก: isospin function
ฮจ = ๏ฟฝ๐๐=1
๐๐
๐๐๐๐๐๐๐๐ , ๐๐ = ๐ด๐ด๐๐๐๐ {๐๐โ12๏ฟฝ๐๐๐ด๐ด๐๐[[๐ฆ๐ฆ๐ฟ๐ฟ1 ๏ฟฝ๐๐๐๐ ๐ฆ๐ฆ๐ฟ๐ฟ2 ๏ฟฝ๐๐๐๐ ]๐ณ๐ณ๐๐๐๐]๐ฝ๐ฝ๐ฝ๐ฝ๐๐๐๐๐ฝ๐ฝ๐ก๐ก} N๏ฟฝ๐พ๐พ
N
๐๐1
๐๐2
Configuration 1
N๏ฟฝ๐พ๐พ
N๐๐1
๐๐2
Configuration 2
โข Geometric progression for Gaussian fall-off parameters โ Cover 0.1-500 fmโ ๐ฟ๐ฟ1 + ๐ฟ๐ฟ2 โค 4โ About 8000 basis states
โข Channel coupling, physical masses
Precise calculation with huge number of nonorthogonal bases
New orthonormal set
Generalized eigenvalue problem
Cutoff parameter
โ
Energy levels of 1S, 2P, 2S states
โข No level shift of 2P stateโข Transition energy can directly be used for the
extraction of the 1S level shiftCoulomb + ๏ฟฝ๐พ๐พ๐๐+etc.
(experiment) 2p
1s
Point charge(analytic)
1S level shift
Level shift of kaonic deuterium
Sensitivity of I=1 component ๐ ๐ ๐๐ ๐๐ ๏ฟฝ๐พ๐พ๐๐ = ๐ ๐ ๐๐ ๐๐๐ผ๐ผ=0 + ฮฒ ร ๐ ๐ ๐๐ ๐๐๐ผ๐ผ=1 Varying the strength of the real part of the KN potential within the
SIDDHARTA constraint for the level shift of kaonic hydrogen ฮE = 283 ยฑ 36(stat) ยฑ 6(syst)eV ฮ = 541 ยฑ 89(stat) ยฑ 22(syst)eV
Energy shifts of kaonic hydrogen and deuterium
๏ฝ25% uncertaintyPossible constraint for I=1
Conclusions
Unified description of kaonic atom and nuclear systemsโ Kaonic nucleus (3- to 7-body)
โข Central density is increased by ๏ฝ2 times higher (4 times than ฯ0)โ Soft NN interaction induces too high central densities
โข Inverted spin-parity in the g.s, of 6Li ๐พ๐พ is predictedโ Isospin dependence of ๐พ๐พ๐๐ interaction is essential
โ Kaonic atom (3-body)โข Prediction of the energy shift of the kaonic deuterium
โข I=1 component can be constrained if measurement is performed within 25% uncertainty
Planned exp. accuracy ๏ฝ5-10%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)