In-medium K & mesons Mesic Nuclei, JU Krakow, Sept. 2013 - - PowerPoint PPT Presentation

In-medium ¯ K & η mesons

Mesic Nuclei, JU Krakow, Sept. 2013 Hadrons in Nuclei, YITP Kyoto, Oct. 2013 Avraham Gal Racah Institute of Physics, Hebrew University, Jerusalem

• ¯

KN − πY chiral dynamics and its consequences

• ¯

K nuclear few-body systems

• ¯

K-nucleus potentials from K− atoms A.Gal in HYP2012 Proc., NPA 914 (2013) 270

• Quest for η nuclear quasibound states

E.Friedman, A.Gal, J.Mareˇ s, PLB 725 (2013) 334

1

¯ KN − πY Chiral Dynamics

2

K−p scattering amplitude from NLO chiral SU(3) dynamics

• Y. Ikeda, T. Hyodo, W. Weise (IHW), PLB 706 (2011) 63; NPA 881 (2012) 98

Strong subthreshold K−p attraction; Λ(1405) physics Consequences for kaonic atoms and K− nuclear quasibound states K− absorption might be governed by out-of-model K−NN → Y N

3

K−p subthreshold ambiguity

Two NLO chiral-model fits by Guo-Oller, PRC 87 (2013) 035202

• Fit I: one value of meson weak-decay constant f = 125.7 ± 1.1 MeV.
• Fit II: separate fixed values for fπ, fK, fη.

Fit II will create problems when confronted with kaonic-atom data.

4

K−p → π±Σ∓ reaction data fitted by LEC of NLO scheme for ¯ KN − πY coupled channels (Y = Λ, Σ)

• Y. Ikeda, T. Hyodo, W. Weise, NPA 881 (2012) 98

Large difference in cross sections ⇒ Strong isospin dependence

5

4 2

• 2

FKN [fm] 1440 1400 1360 1320 s1/2 [MeV] Re F, 2ch Re F, full KN(I=0) 1.5 1.0 0.5 0.0

• 0.5
• 1.0

FπΣ [fm] 1440 1400 1360 1320 s1/2 [MeV] πΣ(I=0) Im F, 2ch Im F, full

• T. Hyodo, W. Weise, PRC 77 (2008) 035204

I = 0 coupled-channel amplitudes Location of ‘resonances’: ¯ KN ≈ 1420 MeV, πΣ ≈ 1405 MeV Are there two distinct ‘Λ(1405)’ resonances?

6

K nuclear few-body systems

7

Energy dependence in ¯ K nuclear few-body systems

• Λ(1405) induces strong energy dependence of the

scattering amplitudes f ¯

KN(√s) and the underlying

effective single-channel input potentials v ¯

KN(√s).

• s = (√sth − BK − BN)2 − (

pK + pN)2 ≤ sth

• Expanding nonrelativistically near √sth ≡ mK + mN:

δ√s = −B

A − A−1 A BK − ξN A−1 A TN:N − ξK A−1 A 2 TK,

δ√s ≡ √s − √sth, BK = −EK, ξN(K) ≡

mN(K) (mN+mK).

• Self-consistency: output √s from solving the

Schroedinger equation identical with input √s.

8

3– & 4–body B & Γ calculated self-consistently

−35 −30 −25 −20 −15 −10 −5 5 10 15 20 25 30 35 40 Eg.s. [MeV] Kmax ¯ KNN I = 1

2

¯ K ¯ KNN I = 0 ¯ KNNN I = 0 ¯ KNNN I = 1 20 40 60 80 100 120 140 160

• 80
• 70
• 60
• 50
• 40
• 30
• 20
• 10

Γ [MeV] δ√s [MeV] ¯ KNN I = 1

2

¯ K ¯ KNN I = 0 ¯ KNNN I = 0 ¯ KNNN I = 1

• N. Barnea, A. Gal, E.Z. Liverts, PLB 712 (2012)
• Variational calculation in hyperspherical basis controlled by Kmax
• ¯

KN energy dependence [Hyodo–Weise, PRC 77 (2008) 035204] restrains B & Γ by treating δ√s ¯

KN self-consistently

• B(4-body) small w.r.t. non-chiral estimates of over 100 MeV

9

• ¯

KNN: is there an excited I = 1/2 quasibound state ( ¯ Kd, dominantly INN = 0) on top of “K−pp” g.s. ?

• Bayar & Oset [NPA 881 (2012) 127]: YES, bound by

about 9 MeV, from a peak in |T ¯

KNN|2 calculated in a

fixed-scatterer approximation to Faddeev equations.

• Shevchenko [NPA 890-1 (2012) 50]: UNLIKELY,

judging from the K−d scattering length and effective range deduced from a ¯ KNN Faddeev calculation.

• Barnea, Gal & Liverts do not find such a bound state

below the Λ∗N threshold at B = 11.4 MeV.

10

K−pp calculated binding energies & widths (in MeV)

chiral, energy dependent non-chiral, static calculations

• var. [1]
• var. [2]
• Fad. [3]
• var. [4]
• Fad. [5]
• Fad. [6]
• var. [7]

B 16 17–23 9–16 48 50–70 60–95 40–80 Γ 41 40–70 34–46 61 90–110 45–80 40–85

• 1. N. Barnea, A. Gal, E.Z. Liverts, PLB 712 (2012)
• 2. A. Dot´

e, T. Hyodo, W. Weise, NPA 804 (2008) 197, PRC 79 (2009) 014003

• 3. Y. Ikeda, H. Kamano, T. Sato, PTP 124 (2010) 533
• 4. T. Yamazaki, Y. Akaishi, PLB 535 (2002) 70
• 5. N.V. Shevchenko, A. Gal, J. Mareˇ

s, PRL 98 (2007) 082301

• 6. Y. Ikeda, T. Sato, PRC 76 (2007) 035203, PRC 79 (2009) 035201
• 7. S. Wycech, A.M. Green, PRC 79 (2009) 014001 (including p waves)

Robust binding & large widths; chiral models give weak binding

11

Yamazaki et al. PRL 104 (2010) 132502, DISTO data reanalysis at 2.85 GeV

Broad K−pp structure in pp → ΛpK+ at πNΣ threshold

Forthcoming experiments: pp → (K−pp) + K+ at GSI K−3He → (K−pp) + n (E15) & π+d → (K−pp) + K+ (E27) at J-PARC

12

RMF quasibound spectra calculated self-consistently (NLO30 ‘+ SE’)

10 20 30 40 A2/3 20 40 60 80 100 BK (MeV)

1s 1d 2s 1f 2p 2d 3s 1h 1i 3p 2f 1g

O Ca Zr Pb

1p

C Li He 10 20 30 40 A2/3 10 20 30 40 ΓK (MeV)

1s 1p 2s 1d 2p 1f 1g 2d 1h 2f 3p 1i

O Ca Zr Pb

3s

C Li He

• D. Gazda, J. Mareˇ

s, NPA 881 (2012) 159

• NLO30 is a chirally motivated coupled channel separable model

with in-medium versions [A. Ciepl´

y, J. Smejkal, NPA 881 (2012) 115]

• ΓK due only to K−N → πY (no K−NN → Y N) decay modes
• Self consistency: deep K− levels are narrower than shallow ones

13

What do K− atoms tell us?

14

10 20 30 40 50 60 70 80 90 100 Z 10

−5

10

−4

10

−3

10

−2

10

−1

10 10

1

width (keV)

kaonic atoms F model n=2 3 4 5 6 7 n=8

K−

atom widths across the periodic table in model F (deep pot.)

Lowest χ2 phenom. model, χ2 = 84 per 65 data points,

• J. Mareˇ

s, E. Friedman, A. Gal, NPA 770 (2006) 84.

15

1 2 3 4 5 6 7 8 9 10

r (fm)

0.1 0.2 0.3 0.4

fm

−3 or fm −4

10

6|R| 2ρm DD model

ρm Ni +4f K

− overlap

10

6|R| 2ρm ‘tρ’ model

6 8 10 12 14 16 18 mass number 0.0 0.5 1.0 1.5 2.0 R(1sΛ) x10

3

normalized formation rates SH (shallow) DD (deep)

Left: K−-Ni 4f atomic wavefunction overlap with nuclear density for deep potential, revealing a nuclear ℓ = 3 quasibound state. Right: FINUDA 1sΛ formation rates in K−

stop capture in nuclei

[Ciepl´ y-Friedman-Gal-Krejˇ ciˇ r´ ık, PLB 698 (2011) 226]. Deep K− nuclear potential is favored.

16

Self-consistency requirement imposed in recent K− atom calculations

[Ciepl´ y-Friedman-Gal-Gazda-Mareˇ s, PLB 702 (2011) 402]:

√sK−N → Eth − BN − BK − ξN p2

N

2mN − ξK p2

K

2mK ξN(K) = mN(K) (mN + mK) p2

K

2mK ∼ −VK− ≈ 100 MeV

0.2 0.4 0.6 0.8 1

ρ/ρ0

−90 −70 −50 −30 −10

E−Eth (MeV)

E vs. ρ Ni Pb

K− is not at rest!

Friedman-Gal, NPA 899 (2013) 60

K−N subthreshold energy vs nuclear density in K− atoms. A dominant in-medium effect

17

1340 1360 1380 1400 1420 1440 s1/2 (MeV) −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 average f (fm) IHW average amplitude Real Imag.

0.2 0.4 0.6 0.8 1 ρ/ρ0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 F

eff (fm)

K

−Ni F eff

Real Imag. 1N only 1N only

Left: IHW free-space input fK−N Right: atomic-fit output Feff

tot

• Subthreshold energy shift is applied self consistently to in-medium

1N amplitude plus (2+...)N phenomenological amplitude.

• Multiple-scattering inclusion of in-medium correlations.
• K−-atom best-fit: χ2/Ndata = 118/65

[Friedman-Gal, NPA 899 (2013) 60].

18

1 2 3 4 5 6 7

r (fm)

−100 −60 −20

Im VK

− (MeV)

−200 −160 −120 −80 −40

Re VK

− (MeV)

K

− Ni potentials

IHW NLO30 IHW NLO30

1 2 3 4 5 6 7

r (fm)

−100 −60 −20

Im VK

− (MeV)

−200 −160 −120 −80 −40

Re VK

− (MeV)

K

− Ni potentials

IHW IHW 1N 1N mN mN

Kaonic-atom best-fit VK− for Ni & its non-additive breakdown into in-medium 1N and phenomenological m(any)N contributions.

NLO30: A. Cieply, J. Smejkal, NPA 881 (2012) 115 (in-medium). IHW: Y. Ikeda, T. Hyodo, W. Weise, NPA 881 (2012) 98. Figures taken from Friedman-Gal, NPA 899 (2013) 60.

19

• 140
• 120
• 100
• 80
• 60
• 40
• 20

0.05 0.1 0.15 0.2

ImV one [MeV] ρN [fm–3]

Mesonic pK = 0 MeV/c pK = 50 MeV/c pK = 100 MeV/c pK = 150 MeV/c pK = 200 MeV/c pK = 250 MeV/c A bound state

• 70
• 60
• 50
• 40
• 30
• 20
• 10

0.05 0.1 0.15 0.2

ImV two [MeV] ρN [fm–3]

Nonmesonic pK = 0 MeV/c pK = 50 MeV/c pK = 100 MeV/c pK = 150 MeV/c pK = 200 MeV/c pK = 250 MeV/c A bound state

K− nuclear 1N (left) and 2N (right) absorptive potentials, both calculated in a chiral unitary approach [PRC 86 (2012) 065205] by Sekihara, Yamagata-Sekihara, Jido, Kanada-En’yo. Note: empirical 25% 2N:1N BR is reached at too high density.

20

η nuclear quasibound states

21

fηN(√s) from K-matrix & N ∗(1535) chiral models

1300 1400 1500 1600 1700 1800

Ecm (MeV)

• 0.2

0.0 0.2 0.4 0.6 0.8 1.0

fηN (fm) Re Im

aηN model dependence

a(fm) M1 M2 GW GR CS Re 0.22 0.38 0.96 0.26 0.67 Im 0.24 0.20 0.26 0.24 0.20 Mai et al. PRD 86 (2012) 094033 Green-Wycech PRC 71 (2005) 014001 Garcia-Recio et al. PLB 550 (2002) 47 Cieply-Smejkal arXiv:1308.4300, NPA

• Re a varies between 0.2 to 1.0 fm;

Im a stable 0.2–0.3 fm.

• M1, M2, GW free-space models; GR, CS in-medium.
• In-medium: energy dependence, Pauli blocking, self-energies.

22

In-medium ηN amplitudes

Friedman-Gal-Mareˇ s, PLB 725 (2013) 334 Ciepl´ y-Friedman-Gal-Mareˇ s, in preparation

• KG equation and self-energies:

[ ∇2 + ˜ ω2

η − m2 η − Πη(ωη, ρ) ] ψ = 0

˜ ωη = ωη − iΓη/2, ωη = mη − Bη Πη(ωη, ρ) ≡ 2ωηVη = −4π

√s mN fηN(√s, ρ)ρ

• Pauli blocking (Waas-Rho-Weise NPA 617 (1997) 449):

f WRW

ηN

(√s, ρ) =

fηN(√s) 1+ξ(ρ)(√s/mN)fηN(√s)ρ,

ξ(ρ) =

9π 4p2

F

• N ∗(1535) ⇒ energy dependent fηN(√s).

In medium ⇒ go subthreshold: δ√s = √s − √sth δ√s≈−BN

ρ ¯ ρ − ξNBη ρ ρ0 − ξNTN( ρ ρ0)2/3 + ξηRe Vη(√s, ρ)

Self-consistency relationship between δ√s & ρ

23

Self-consistency relationship

0.2 0.4 0.6 0.8 1 ρ/ρ0

• 50
• 40
• 30
• 20
• 10

E - Eth (MeV)

GW M2 M1

0.2 0.4 0.6 0.8 1 ρ/ρ0

• 60
• 50
• 40
• 30
• 20
• 10

E - Eth (MeV)

CS, +SE CS, no SE GW

δ√s vs. ρ for 1sη bound state in Ca using in-medium fηN

• 40–60 MeV subthreshold energy shifts at nuclear matter density ρ0,

larger than shifting down by Bη (GR) or by 30 MeV (Haider-Liu)

• Larger Re aηN ⇒ larger δ√s = E − Eth

24

Model dependence I

10 20 30 40 A

2/3

5 10 15 20 25 30 Bη (MeV) C Mg Ca Zr GW Pb M2 M1 10 20 30 40 A

2/3

5 10 15 20 25 Γη (MeV) C Mg Ca Zr GW Pb M2 M1

Binding energy and width of 1sη bound states across the periodic table using WRW Pauli-blocked fηN

• Larger Re aηN ⇒ larger Bη
• Widths are unrelated to Im aηN

25

Model dependence II

10 20 30 40 A

2/3

5 10 15 20 25 30 Bη (MeV) C Mg Ca Zr CS(δ√s) Pb GR(-Bη) GR(δ√s) 10 20 30 40 A

2/3

5 10 15 20 25 30 35 40 Γη (MeV) C Mg Ca Zr CS(δ√s) Pb GR(-Bη) GR(δ√s)

Sensitivity of calculated B1sη and Γ1sη to version of self-consistency

• δ√s recipe reduces both B1sη and Γ1sη w.r.t. −B1sη recipe
• GR’s widths are too large to resolve η bound states

Why Γη(GR) ≫ Γη(CS) for similar Im aηN?

26

Energy dependence of free-space & in-medium amplitudes

1400 1420 1440 1460 1480 1500 s

1/2 (MeV)

0.2 0.4 0.6 0.8 Re ηN amplitude (fm)

CS free CS ρ0 GR free GR ρ0

1400 1420 1440 1460 1480 1500 s

1/2 (MeV)

0.2 0.4 0.6 Im ηN amplitudes (fm)

CS free CS ρ0 GR free GR ρ0

• Subthreshold Re fηN similar in both in-medium models

in spite of large free-space difference at threshold

• Subthreshold Im fηN differ widely, which explains

the huge difference between Γη(GR) and Γη(CS)

27

Model predictions for small widths

10 20 30 40 A

2/3

5 10 15 20 25 30 Bη (MeV) C Mg Ca Zr 1s Pb 1p 1d 2s 1f 2p Li

GW model

10 20 30 40 A

2/3

5 10 15 20 25 30 Bη (MeV) C Mg Ca Zr 1s Pb 1p 1d 2s

CS model

• more theoretical work is needed to figure out what makes

subthreshold values of Im fηN sufficiently small to generate small widths.

28

Summary

• Large widths, ΓK > 50 MeV, expected for single-K

quasibound nuclear states. Focus on light systems. Searches for K−pp are underway in GSI and J-PARC.

• Major issues: (i) how deep is K nuclear spectrum?

(ii) how big is Γ(KNN → Y N) w.r.t. Γ(KN → πY )?

• Subthreshold behavior of fηN is crucial in

studies of η-nuclear bound states to decide whether (i) such states exist, (ii) can they be resolved (widths?), and (iii) which nuclear targets and reactions to try? Thanks to my collaborators N. Barnea, A. Ciepl´ y,

• E. Friedman, D. Gazda, J. Mareˇ

s

29