X hyperons in the nuclear medium described by - - PowerPoint PPT Presentation

◼ It is basic to obtain a better description of baryon-baryon interactions in

the strangeness sector to understand strangeness physics

◼ Various potentials since 1970s ➢ Boson exchange picture: Nijmegen group [NHC-D, NHC-F, NSC89,

NSC97, ESC04, ESC08, ESC16], Jülich YN(2005), ⋯

➢ Quark picture: SU6 quark model by Kyoto-Niigata group [fss2]

⚫ role of Pauli forbidden state: strong repulsion in the SN 3S1 T=3/2 channel ⚫ role of antisymmetric spin-orbit component: small Λ spin-orbit splitting

◼ Recent development: construction in chiral effective field theory

光学模光学模型ポテンシャルにおける

X hyperons in the nuclear medium described by the chiral NLO interactions

• M. Kohno, RCNP Osaka University

◼

B-B interactions in the strangeness sector by the Jülich-Bonn-München group

◼

𝑇 = −1

➢

LO [Polinder, Haidenbauer, and Meißner, Nucl. Phys. A779, 244 (2006)], NLO13, NLO19 [Haidenbauer, Meißner, and Nogga, arXiv:1906.11681]

◼

𝑇 = −2

➢ LO [Polinder, Haidenbauer, and Meißner, Phys. Lett. B653, 29 (2016)], NLO

[Petschauer, Kaiser, Haidenbauer, Meißner, and Weise, Phys. Rev. C 93, 014001 (2016)], Updated version [Haidenbauer and Meißner, Euro.Phys.J. A55, 23 (2019)] which is used in the present calculations.

NLO diagrams [ 𝜌, 𝐿, and 𝜃 exchanges in SU(3) ]

◼

Parameters are determined by SU(3) symmetry and (scarce) two-body data.

◼

It is important to examine the predictions of the interaction for X properties in the nuclear medium and compare them with (yet scarce) experimental data.

Construction of baryon-baryon interactions in ChEFT

XN phase shifts: NLO ChEFT and HAL-QCD

◼ It is interesting to see that ChEFT NLO potential and HAL-QCD potential

(by T. Inoue†) yield similar XN phase shifts in all s-waves.

†Takashi Inoue for HAL QCD Collaboration, AIP Conference Proceedings 2130, 020002

(2019) [ https://doi.org/10.1063/1.5118370 ]

◼

Experimental data is scarce.

➢ Potential parameters are not well controlled. ◼

Recent (K-, K+) experiments at KEK (1) Emulsion experiments

➢ Double L hypernuclei, most likely ΛΛ

11Be with ΔBΛΛ = 1.87 ± 0.37 MeV

[Ekawa et al., PTEP 2019, 021D02 (2019)]

ΔBΛΛ = − ΛΛ V ΛΛ 𝐵 − (positive rearrangement effect, ∼1 MeV)

➢ Deeply bound (4.38 ± 0.28 MeV) state of X-–14N (X- + 14N →

Λ 10Be + Λ 5He)

[Nakazawa et al., PTEP 2015, 033D02 (2015)]

(2) X production (K-, K+) inclusive spectrum on 12C (Cross sections are not available yet)

➢ Peak position and its dependence on the momentum transfer

(namely, K+ scattering angle) depend on the X optical potential.

➢ Cross sections around the threshold are sensitive to the X potential.

Recent experiments concerning the 𝑇 = −2 sector

◼

The cutoff scale ~550 MeV of ChEFT is not so soft to use the interactions in a perturbative scheme.

◼

The high-momentum components are treated by the Bethe-Goldstone equation (Lowest-order Brueckner theory) 𝐻XNｰXN = 𝑊

XN−XN + ෍

𝐶1𝐶2

𝑊

XN−𝐶1𝐶2

𝑅 𝜕 − 𝑢𝐶1 − 𝑢𝐶2 − 𝑉𝐶1 − 𝑉𝐶2 𝐻𝐶1𝐶2−XN

◼

Continuous prescription: potentials 𝑉𝑂 , 𝑉Λ , 𝑉Σ in intermediate propagators are prepared by ChEFT NN, ΛN and ΣN interactions including 3BF effects.

◼

X single-particle potential

𝑉Ξ = σ𝑂 X𝑂 𝐻XNｰXN X𝑂 X potential in nuclear matter: the lowest order Brueckner theory

G = + G

X N N X X N N X X N X N X N X N B1 B2 𝑊 𝑊

X Potentials in nuclear matter

real part imaginary part

◼

The weak attractive contribution comes from the baryon channel (ΣΣ) coupling in the T=0 3S1 state.

◼

Imaginary potential (from the ΛΛ coupling) is small at low momenta.

◼

Without baryon channel coupling, k-dependence is weak （no meson-exchange XN→NX process）.

X Potentials in pure neutron matter

real part imaginary part

◼

Dose X- hyperon appear in high-density neutron star matter?

◼

X- potential with ChEFT is repulsive in pure neutron matter at higher densities.

◼ Use local-density approximation （LDA） to convert the potential in NM

into the potential in a finite nucleus.

➢ Density distribution of a nucleus 𝜍(𝑠)

local Fermi momentum 𝑙𝐺 𝑠 = 3𝜌2𝜍(𝑠)/2 1/3

➢ X potential 𝑉X

𝑂𝑁(𝐹, 𝑙𝐺) in NM

as a function of 𝐹 (or 𝑙) X potential in a finite nucleus 𝑉X(𝐹, 𝑙𝐺(𝑠))

➢ To correct the finite range effect, introduce a Gaussian form factor

with the range 𝛾 = 1 fm (improve LDA). 𝑉X 𝑠; 𝐹 = ( 𝜌𝛾)−3න 𝑒𝒔 𝑓− 𝒔−𝒔′ 2/𝛾2 𝑉X(𝐹, 𝑙𝐺(𝑠′))

X Potential in finite nuclei: local-density approximation

2 4 6 0.1 0.2 0.5 1 1.5

12C

(r) kF(r)

r [fm] density [fm−3] local Fermi momentum [fm

−1]

normal density

◼

The reliability of the improved LDA is checked by comparing the potential with a more sophisticated folding procedure.

𝑉X 𝑠; 𝐹 in 12C and 14N

◼

Weakly attractive at low energies, mainly due to the XN-SS coupling in T=1 3S1 state.

◼

The attractive potential lowers s and p Coulomb bound levels.

◼

The potential becomes repulsive with increasing the energy.

2 4 6 −10 10

E 100 MeV 150 MeV 100 MeV E 50 MeV 20 MeV 0 MeV

UX(r;E) [MeV] r [fm] improved LDA

14N

2 4 6 −10 10

E 200 MeV 150 MeV 100 MeV E 50 MeV 20 MeV 0 MeV

UX(r;E) [MeV] r [fm] improved LDA

12C

Bound states provided by 𝑉X 𝑠; 𝐹

Coulomb: 𝑊𝐷 𝑠 = −

𝑎𝑓2 2𝑆𝐷 3 2 − 𝑠2 2𝑆𝐷

2

for 𝑠 < 𝑆𝐷, −

𝑎𝑓2 𝑠 for 𝑠 > 𝑆𝐷 , 𝑆𝐷 = 1.15𝐵1/3

• Exp. Data: Aoki et al.,
• Phys. Lett. B355, 45 (1995)
• Exp. Data: Nakazawa et al.,

PTEP 2015, 033D02 (2015)

−6 −5 −4 −3 −2 −1

X X-

E=0 MeV E=0 MeV pure Coul.

in

12C

with UX(r,E) X bound states

0s 0s 0d 0s 0d 0p Exp. 0p

candidates depending on the residual nuclei

MeV

−6 −5 −4 −3 −2 −1

X X-

E=0 MeV E=0 MeV pure Coul.

in

14N

with UX(r,E) X bound states

0s 0s 0d 0p 0s 0d 0p Exp.

MeV

Summary in the first half

◼ X potentials are calculated in nuclear matter, using the NLO baryon-baryon

interactions in chiral effective field theory, in the framework of lowest-order Brueckner theory

➢ N, L, and S potentials in the propagators of the G-matrix equation are

prepared by the ChEFT interactions with including the effects of 3-body forces.

◼ The X potential in NM is converted to the potential in finite nuclei by

improved LDA method.

➢ X bound states in 12C and 14N are evaluated. ➢ Shallow bound states are reasonable in view of the recent emulsion

data.

◼ It is straightforward to study X bound states in heavier nuclei. ◼ Further experimental data should help to lessen the uncertainties in the

parametrization of the interaction in 𝑇 = −2.

• M. Kohno, “X hyperons in the nuclear medium described by chiral NLO interactions“,
• Phys. Rev. C100, 024313 (2019)

◼ K+ inclusive spectra of (K-, K+) X production reaction on nuclei provide

information on X optical potential as well as X bound states.

➢ Experimental data of new 12C(K-, K+)

measurements at KEK is being analyzed.

◼ 9Be(K-, K+) and 12C(K-, K+) spectra are calculated,

using a semiclassical distorted wave method.

◼ Data of previous experiment at BNL

[Khaustov et al., Phys. Rev. C61, 054603 (2000)] was used to deduce the attractive X potential with the strength of 14 MeV in a W-S shape.

◼ 𝑊 = −14 MeV has been a canonical value,

but not conclusive.

X production (K-, K+) inclusive spectrum

𝑒2𝜏 𝑒𝑋𝑒Ω = 𝜕𝑗,𝑠𝑓𝑒𝜕𝑔,𝑠𝑓𝑒 2𝜌 2 𝑞𝑗 𝑞𝑔 ෍

Ξ,ℎ

1 4𝜕𝑗𝜕𝑔 𝜓𝐿+

− 𝜚Ξ − 𝑤𝑔,𝑞,𝑗,ℎ 𝜓𝐿− + 𝜚ℎ 2

𝜀(𝑋 − 𝜗Ξ + 𝜗ℎ)

◼ 𝜓𝐿−

+ ,𝜓𝐿+ − ： distorted waves are evaluated by OMPｓ in a 𝑢𝜍 approximation

◼ 𝑤𝑔,𝑞,𝑗,ℎ ： elementary amplitude ◼ Calculations by the semi-classical distorted wave (SCDW) method.

➢ Proton Fermi motion and angle dependence of the elementary amplitude are

taken into account.

𝐿− 𝐿+ 𝑤𝑔,𝑞,𝑗,ℎ 𝜀(𝑋 − 𝜗Ξ + 𝜗ℎ) Ξ ℎ

DWBA description of X production (K-, K+) inclusive process

Double differential cross section in the SCDW method

𝑒2𝜏 𝑒𝑋𝑒Ω = 𝜕𝑗,𝑠𝑓𝑒𝜕𝑔,𝑠𝑓𝑒 2𝜌 2 𝑞𝑗 𝑞𝑔 ෍

𝑞,ℎ

1 4𝜕𝑗𝜕𝑔 𝜓𝐿+

− 𝜚Ξ − 𝑤𝑔,𝑞,𝑗,ℎ 𝜓𝐿− + 𝜚ℎ 2

𝜀(𝑋 − 𝜗Ξ + 𝜗ℎ)

𝜓+ 𝒔 𝜓+∗ 𝒔′ = 𝜓+ 𝑺 + 1 2 𝒕 𝜓+∗ 𝑺 − 1 2 𝒕 ≅ 𝜓+ 𝑺

𝟑𝑓𝑗𝒍 𝑺 ∙𝒕

⚫

𝒍 𝑺 =

Re{𝜓+∗ 𝑺 (−𝑗)∇𝜓+ 𝑺 } 𝜓+ 𝑺

𝟑

, 𝑛𝐿

2 + 𝒍2 𝑺 + 2𝜕 𝑉𝐿 𝑆 + 𝑊 𝐷 𝑆

− 𝑊

𝐷 2 𝑆 = 𝜕2 ⚫ Φℎ : Wigner transformation of the density matrix of the nucleon hole state ⚫

𝜊 =

𝐵 𝐵−1

𝑒2𝜏 𝑒𝑋𝑒Ω = 𝜕𝑗,𝑠𝑓𝑒𝜕𝑔,𝑠𝑓𝑒 (2𝜌)2 𝑞𝑗 𝑞𝑔 𝜊6 ඵ 𝑒𝑺𝑒𝑳 ෍

𝑞

1 4𝜕𝑗𝜕𝑔 𝜓𝐿+

− (𝑺) 2

𝜓𝐿−

+ (𝑺) 2

𝜚𝑞

(−)(𝜊𝑺) 2

𝑤𝑔,𝑞,𝑗,ℎ

2 (2𝜌)3

𝜊3 ෍

ℎ

Φℎ 𝜊𝑺, 1 𝜊 𝑳 𝜀 𝑳 + 𝑙𝐿− 𝑺 − 𝑙𝐿+ 𝑺 − 𝑙𝑞 𝑺 𝜀(𝑋 − 𝜗𝑞 − 𝜗ℎ)

◼ X absorptive effect is taken care of by Gaussian folding (half width Γ/2).

SCDW calculations of (K-, K+) X production reaction: 9Be and 12C

➢ Experimental data : PhD thesis by

Tamagawa, Univ. of Tokyo, 2000

1000 1200 1400 0.1 0.2 0.3 0.4 0.5 2= MeV pK

+ [MeV/c]

d2/(dLdpK) [b/sr/(MeV c−1)] threshold

9Be(K−,K+)

pK−=1.8 GeV/c K+=1.5−8.5 deg. ChEFT K+ [deg] 3 5 7 9

◼ Absolute values of the cross

section are reproduced.

100 200

0.1 0.2 0.3 0.4 0.5 −B.E. [MeV] d2/dEdL [b/(MeV sr)]

12C(K−,K+)

pK−=1.8 GeV/c SCDW calculations with ChEFT potential 2= MeV K+ 3 deg. 5 deg. 7 deg. 9 deg.

◼ Peak position shifts according to

the momentum transfer (K+-angle).

➢ Cross section data of the new

experiment at KEK is waited for.

X potential in 12C by improved LDA

◼ The X potential described by the

NLO ChEFT interactions is weakly attractive at low energies and becomes repulsive with increasing the energy.

◼ X−Bound states of 𝑉X 𝑠; 𝐹 = 0

in 12C.

𝑽X(𝒔, 𝑭 = 𝟏)

W-S

(𝑋 = −14 MeV)

state 𝑓X (MeV) 𝑠 (fm) 𝑓X (MeV) 𝑠(fm)

0s

• 4.07

3.3

• 5.83

2.8 0p

• 0.35

16.0

• 0.43

12.4 0d

• 0.13

41.8

• 0.13

42.6

12C (K-, K+) X spectra with ChEFT and W-S with 3 choices of the strength

◼

Peak position naturally depends

• n the potential strength.

◼

The spectrum of ChEFT is similar to that of zero potential.

◼

Cross sections around the threshold are sensitive to the potential.

Summary in the second half

◼ (K-, K+) X

− production inclusive spectra are calculated, using a SCDW method.

➢ Distorted waves of K-and K+ are described by optical potentials in 𝑢

approximation

➢ Proton Fermi motion and energy- and angle-dependence of the elementary

amplitude are taken care of.

◼ Results of 9Be(K-, K+) are consistent with the experimental data in

magnitude.

◼ Dependence of the cross sections of 12C(K-, K+) on K+ angle is shown. ➢ Differential cross sections of the new experiments on 12C at KEK are awaited. ◼ Is it possible to explain simultaneously both X bound states (emulsion data)

and (K-, K+) spectra?