[PPT] - Baryon-baryon interactions and the quest to constrain them by PowerPoint Presentation

SLIDE 1

Baryon-baryon interactions and the quest to constrain them by measuring correlation functions

Johann Haidenbauer

IAS & JCHP , Forschungszentrum Jülich, Germany

YITP Workshop, Kyoto, March 25-29, 2019

Johann Haidenbauer Baryon-baryon interactions

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Outline

1

Introduction

2

Baryon-baryon interaction in chiral effective field theory

3

Strangeness S=-1 sector

4

Strangeness S=-2 sector

5

Three- and four-body systems

6

Hyperons in nuclear matter

7

Final-state interaction

8

Summary

Johann Haidenbauer Baryon-baryon interactions

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Introduction

BB interaction in chiral effective field theory

ΛN and ΣN scattering

→ Role of SU(3) flavor symmetry

Few-body systems with hyperons: 3

ΛH, 4 ΛH, 4 ΛHe, ...

→ Role of three-body forces large charge symmetry breaking 4

ΛH ↔ 4 ΛHe

(Λ, Σ) hypernuclei and hyperons in nuclear matter

→ very small spin-orbit splitting: weak spin-orbit force existence of Ξ hypernuclei repulsive Σ nuclear potential

implications for astrophysics

→ hyperon stars stability/size of neutron stars

Johann Haidenbauer Baryon-baryon interactions

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BB interaction in chiral effective field theory

Baryon-baryon interaction in SU(3) χEFT à la Weinberg (1990) Advantages: Power counting systematic improvement by going to higher order Possibility to derive two- and three-baryon forces and external current operators in a consistent way

degrees of freedom: octet baryons (N, Λ, Σ, Ξ), pseudoscalar mesons (π, K, η)
pseudoscalar-meson exchanges
contact terms – represent unresolved short-distance dynamics

LO : NLO :

LO:

H. Polinder, J.H., U.-G. Meißner, NPA 779 (2006) 244

NLO: J.H., S. Petschauer, N. Kaiser, U.-G. Meißner, A. Nogga, W. Weise, NPA 915 (2013) 24 Johann Haidenbauer Baryon-baryon interactions

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Structure of the potential

V OBE

B1B2→B′

1B′ 2 = −fB1B′ 1PfB2B′ 2P

(

σ1· q)( σ2· q)

q 2+m2

P

fB1B′

1P ... coupling constants fixed by standard SU(3) relations utilizing fNNπ

mP ... mass of the exchanged pseudoscalar meson SU(3) symmetry breaking due to the mass splitting of the ps mesons

(mπ = 138.0 MeV, mK = 495.7 MeV, mη = 547.3 MeV)

taken into account already at LO!

(TBE ⇒ J.H. et al., NPA 915 (2013) 24)

V CT

B1B2→B′

1B′ 2 = ˜

Cα + Cα(p2 + p′2) (or Cβpp′) α = 1S0, 3S1, 3S1 −3 D1 β = 3P0, 1P1, 3P1, 3P2

Johann Haidenbauer Baryon-baryon interactions

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SU(3) structure of contact terms for BB

SU(3) structure for scattering of two octet baryons → 8 ⊗ 8 = 1 ⊕ 8a ⊕ 8s ⊕ 10∗ ⊕ 10 ⊕ 27 BB interaction can be given in terms of LECs corresponding to the SU(3)f irreducible representations: C1, C8a, C8s, C10∗, C10, C27 Channel I Vα Vβ Vβ→α S = NN → NN – C10∗

β

– NN → NN 1 C27

α

– – S = −1 ΛN → ΛN

1 2 1 10

9C27

α + C8s α

1

2

C8a

β + C10∗ β

−C8sa

ΛN → ΣN

1 2 3 10

−C27

α + C8s α

1

2

−C8a

β + C10∗ β

−3C8sa

C8sa ΣN → ΣN

1 2 1 10

C27

α + 9C8s α

1

2

C8a

β + C10∗ β

3C8sa

ΣN → ΣN

3 2

C27

α

C10

β

– α = 1S0,3 P0,3 P1,3 P2, β = 3S1,3 S1 −3 D1,1 P1

No. of contact terms (LECs): limited by SU(3) symmetry

LO : 6 [2 (NN, ΞΞ) + 3 (YN, ΞY) + 1 (YY)] NLO: 22 [7 (NN, ΞΞ) + 11 (YN, ΞY) + 4 (YY)] (No. of spin-isospin channels in NN+YN: 10 S = −2, −3, −4: 27)

Johann Haidenbauer Baryon-baryon interactions

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SU(3) symmetry + breaking

S. Petschauer, N. Kaiser, NPA 916 (2013) 1:

example: 1S0 partial wave for pure {27} states V I=1

NN

= ˜ C27 + C27(p2 + p′2) + 1 2 Cχ

1 (m2 K − m2 π)

V I=3/2

ΣN

= ˜ C27 + C27(p2 + p′2) + 1 4 Cχ

1 (m2 K − m2 π)

V I=2

ΣΣ

= ˜ C27 + C27(p2 + p′2) V I=3/2

ΞΣ

= ˜ C27 + C27(p2 + p′2) + 1 4 Cχ

2 (m2 K − m2 π)

V I=1

ΞΞ

= ˜ C27 + C27(p2 + p′2) + 1 2 Cχ

2 (m2 K − m2 π)

Cχ

1 , Cχ 2 , LECs that break SU(3) symmetry of LO contact terms

Our strategy: impose SU(3) symmetry only for BB systems with same strangeness S {NN} or {ΛN, ΣN} or {ΛΛ, ΣΣ, ΞN, ΛΣ} BB scattering for S = 0 to S = −4: 6 Cχ

i s for 1S0 and 6 Cχ i s for 3S1

→ cannot be determined from presently available data

Johann Haidenbauer Baryon-baryon interactions

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Coupled channels Lippmann-Schwinger Equation

T ν′ν,J

ρ′ρ

(p′, p) = V ν′ν,J

ρ′ρ

(p′, p) +

ρ′′,ν′′

∞ dp′′p′′2 (2π)3 V ν′ν′′,J

ρ′ρ′′

(p′, p′′) 2µρ′′ p2 − p′′2 + iη T ν′′ν,J

ρ′′ρ

(p′′, p) ρ′, ρ = ΛN, ΣN (ΛΛ, ΞN, ΛΣ, ΣΣ) LS equation is solved for particle channels (in momentum space) Coulomb interaction is included via the Vincent-Phatak method The potential in the LS equation is cut off with the regulator function: V ν′ν,J

ρ′ρ

(p′, p) → f Λ(p′)V ν′ν,J

ρ′ρ

(p′, p)f Λ(p); f Λ(p) = e−(p/Λ)4 consider values Λ = 550 - 700 MeV [LO] 500 - 650 MeV [NLO]

Johann Haidenbauer Baryon-baryon interactions

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YN integrated cross sections

100 200 300 400 500 600 700 800 900

plab (MeV/c)

100 200 300

σ (mb)

EFT LO Sechi-Zorn et al. Kadyk et al. Alexander et al. Jülich ’04

Λp -> Λp

500 600 700 800

plab (MeV/c)

10 20 30 40 50 60 70

σ (mb)

EFT LO Kadyk et al. Jülich ’04 Hauptman

Λp -> Λp Σ

+n ->

<- Σ

0p 100 120 140 160 180

plab (MeV/c)

50 100 150 200 250 300

σ (mb)

EFT LO NLO Jülich ’04 Engelmann et al.

Σ

−p -> Λn

LO: H. Polinder, J.H., U.-G. Meißner, NPA 779 (2006) 244 NLO: J.H., S. Petschauer, et al., NPA 915 (2013) 24 Jülich ’04: J.H., U.-G. Meißner, PRC 72 (2005) 044005

Johann Haidenbauer Baryon-baryon interactions

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YN integrated cross sections

100 120 140 160 180

plab (MeV/c)

100 200 300 400 500

σ (mb)

EFT LO NLO Jülich ’04 Engelmann et al.

Σ

−p -> Σ 0n 100 120 140 160 180

plab (MeV/c)

50 100 150 200 250 300

σ (mb)

EFT LO NLO Jülich ’04 Eisele et al.

Σ

−p -> Σ −p 100 120 140 160 180

plab (MeV/c)

50 100 150 200 250

σ (mb)

EFT LO NLO Jülich ’04 Eisele et al.

Σ

+p -> Σ +p

LO: H. Polinder, J.H., U.-G. Meißner, NPA 779 (2006) 244 NLO: J.H., S. Petschauer, et al., NPA 915 (2013) 24 Jülich ’04: J.H., U.-G. Meißner, PRC 72 (2005) 044005

Johann Haidenbauer Baryon-baryon interactions

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Λp differential cross sections

1.0
0.5

0.0 0.5 1.0

cos θ

10 20 30 40 50

no. of events

Sechi-Zorn (1968) LO600 NLO600 J04 NSC97f

Λp -> Λp plab = 180-248 MeV/c

1.0
0.5

0.0 0.5 1.0

cos θ

10 20 30 40 50

no. of events

Sechi-Zorn (1968) LO600 NLO600 J04 NSC97f

Λp -> Λp plab = 248-330 MeV/c

100 200 300 400 500

plab (MeV/c)

1 2 3

dσ/dΩ forward/backward ratio

Alexander (1968) Sechi-Zorn (1968) Jülich 04 NSC97f LO NLO

Λp -> Λp

LO: H. Polinder, J.H., U.-G. Meißner, NPA 779 (2006) 244 NLO: J.H., S. Petschauer, et al., NPA 915 (2013) 24 Jülich ’04: J.H., U.-G. Meißner, PRC 72 (2005) 044005 Nijmegen NSC97f: T.A. Rijken et al., PRC 59 (1999) 21

Johann Haidenbauer Baryon-baryon interactions

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YN differential cross sections

1.0
0.5

0.0 0.5 1.0

cos θ

20 40 60 80 100 120 140

dσ/dcosθ (mb) Σ

−p -> Λn

plab = 135 MeV/c

1.0
0.5

0.0 0.5 1.0

cos θ

20 40 60 80 100 120 140

dσ/dcosθ (mb) Σ

−p -> Λn

plab = 160 MeV/c

1.0
0.5

0.0 0.5 1.0

cos θ

50 100 150

dσ/dcosθ (mb) Σ

−p -> Σ −p

plab = 160 MeV/c

Johann Haidenbauer Baryon-baryon interactions

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YN differential cross sections

1.0
0.5

0.0 0.5 1.0

cos θ

10 20 30 40 50 60

dσ/dcosθ (mb) Σ

−p -> Σ −p

plab = 550 MeV/c

1.0
0.5

0.0 0.5 1.0

cos θ

20 40 60 80 100

dσ/dcosθ (mb) Σ

+p -> Σ +p

plab = 170 MeV/c

1.0
0.5

0.0 0.5 1.0

cos θ

10 20 30 40 50 60 70

dσ/dcosθ (mb) Σ

+p -> Σ +p

plab = 450 MeV/c

Johann Haidenbauer Baryon-baryon interactions

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YN scattering lengths [fm]

EFT LO EFT NLO Jülich ’04 NSC97f experiment∗ Λ [MeV] 550 · · · 700 500 · · · 650 aΛp

s

−1.90 · · · −1.91 −2.90 · · · −2.91 −2.56 −2.51 −1.8+2.3

−4.2

aΛp

t

−1.22 · · · −1.23 −1.51 · · · −1.61 −1.66 −1.75 −1.6+1.1

−0.8

aΣ+p

s

−2.24 · · · −2.36 −3.46 · · · −3.60 −4.71 −4.35 aΣ+p

t

0.60 · · · 0.70 0.48 · · · 0.49 0.29 −0.25 χ2 ≈ 30 15.7 · · · 16.8 ≈ 22 16.7 (3

ΛH) EB

−2.34 · · · −2.36 −2.30 · · · −2.33 −2.27 −2.30 −2.354(50)

∗G. Alexander et al., PR 173 (1968) 1452

Note: (3

ΛH) EB is used as additional constraint in EFT and Jülich ’04

Λp data alone do not allow to disentangle 1S0 (s) and 3S1 (t) contributions

Johann Haidenbauer Baryon-baryon interactions

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Information on S = −2 systems

Constraints on the ΛΛ scattering length:

∆BΛΛ = BΛΛ( 6

ΛΛHe) − 2BΛ(5 ΛHe) = 1.01 ± 0.20+0.18 −0.11 MeV Nagara event (H. Takahashi et al., Phys. Rev. Lett. 87 (2001) 212502)

0.67 ± 0.17 MeV

(K. Nakazawa, Nucl. Phys. A 835 (2010) 207) 6 ΛΛHe calculations (Filikhin; Fujiwara; Rijken; ...)

⇒ −1.32 < aΛΛ < −0.73 fm [based on 2001 value!] aΛΛ = -1.2 ± 0.6 fm

(A. Gasparyan et al., Phys. Rev. C 85 (2012) 0152047)

deduced from ΛΛ invariant mass spectrum of the reaction

12C(K −, K +ΛΛX) (C.-J. Yoon et al., PRC 75 (2007) 022201)

−1.92 < aΛΛ < −0.50 fm

(A. Ohnishi et al., NPA 954 (2016) 294)

deduced from analyzing ΛΛ correlations in relativistic heavy-ion collisions (STAR Collaboration, PRL 114 (2015) 022301)

Johann Haidenbauer Baryon-baryon interactions

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Information on S = −2 systems

Constraints on the ΞN interaction:

data/limits for the range 200 < p Ξ < 800 MeV/c σΞ−p→ΛΛ < 12 mb, at 90% confidence level σΞ−p→Ξ−p < 24 mb, at 90% confidence level σΞ−p→ΛΛ = 4.3+6.3

−2.7 mb, at p Ξ = 500 MeV/c (J.K. Ahn et al., PLB 633 (2006) 214)

in-medium cross sections: inelastic σΞ−N = 12.7+3.5

−3.1 mb (400 < p Ξ < 600 MeV/c) (S. Aoki et al., NPA 644 (1998) 365)

lder data at higher energies (Charlton, 1970; Muller, 1972):

σΞ−p→Ξ−p = 13 ± 6 mb (p Ξ = 1 − 4 GeV/c) σΞ0p→Ξ0p = 19 ± 10 mb (p Ξ = 1 − 4 GeV/c) σΞ0p→Ξ0p = 8 mb σΞ0p→Σ+Λ = 24 mb (p Ξ ≈ 2 GeV/c) ⇒ ΞN cross sections are small → ΞN interaction cannot be very strong !

Johann Haidenbauer Baryon-baryon interactions

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Information on S = −2 systems: ΞN

Ξ-nuclear potential: attractive or repulsive?

Measurements of the reaction 12C(K −, K +)X

T. Fukuda et al., PRC 58 (1998) 1306

P . Khaustov et al., PRC 61 (2000) 054603 (possible evidence for 12

Ξ Be)

⇒ UΞ ≈ −15 MeV Deeply bound state of Ξ− 14N (KISO event)

K. Nakazawa et al., PTEP (2015) 033D02

binding energy: 4.38 ± 0.25 MeV or 1.11 ± 0.25 MeV Ξ bound state in 12C(K −, K +)X

T. Nagae et al. (E05 Collaboration), PoS (INPC2016) (2016) 038; HYP2018

binding energy: ≈ 9 MeV (and 2.1 MeV)

Johann Haidenbauer Baryon-baryon interactions

SLIDE 18

Information on S = −2 systems: ΞN

12C(K −, K +) inclusive spectrum at Ξ− production threshold region

(P . Khaustov et al., PRC 61 (2000) 054603)

100 200 100 200 300 −BΞ [MeV] d2σ/dEdΩlab [nb/(sr!&MeV)] pK−=1.8 GeV/c θK+=5.5o UΞ

0 [MeV]

−20 −10 +10 fss2 −20 20 40 50 100 150 −BΞ [MeV] d2σ/dEdΩlab [nb/(sr!&MeV)] pK−=1.8 GeV/c θK+=5.5o UΞ

0 [MeV]

−20 −10 +10 fss2

M. Kohno & S. Hashimoto, PoTP 123 (2010) 157 ⇒ UΞ ≈ 0 MeV is preferable

Johann Haidenbauer Baryon-baryon interactions

SLIDE 19

Information on S = −2 systems: ΞN

608

D. P. GOYAL, J. N. MISRA,

AND

A. V. SODHI

21

100

80- 60-

Vl L0

LU

40-

O

ALL

EVENTS

SEF N SPECTATORS

40- 80- 60-

LLJ

() 40-

20-

2.24 2.39 2.54,

M(:

— p) (GeV)

2.69

p.p 0.03

0.06

( MISSING

MASS) (GeV

)

I

l

0.0

P.02 0.04

FIG. l.

(Missing mass) distribution

for K d -= K p (seen K decay).

events,

which are used in the present work.

This sample includes events where the proton in re- actions (I) to (4) is either seen or unseen.

In the

latter case, the proton is inserted

and then ki-

nematical fits are obtained. "

FIG, 3. - p invariant-mass distribution

for all events.

4(a). This shows a clear enhancement

in the

2480-MeV mass region and the shoulder

around

2360 MeV is still indicated. The peak is -3 standard deviation

above an estimated background level of 22-25 events. Our reaction contains :" E in the final state and is therefore expected to show an effect due to the well established

Z (2030) in the

Z' invariant-mass

distribution. To check the interference

f Z(2030) on the " p
III. RESULTS

The ™~p invariant-mass distribution

for 557

events is shown

in Fig. 3. An enhancement in the

2480-MeV mass region along with a shoulder at 2360 MeV are visible here. Since the events

with

unseen protons are poorly constrained

and may have ambiguities,

in the present work we have given reliance to those events

nly where the pro-

ton is seen. The " p invariant-mass distribution

for 265 events with seen protons is shown

in Fig,

20

20-

L

10-

)

O

z

(c)

(b) 120-

ALL

EVENTS

30-

Vl

I-

80-

hl

O O

40- 80- 40-

SEEN SPECTATORS 20

.

10-

224 2.39 2.54

M(=- p)(GeV) 2.69

0.0

I

0.15 0.25 0 35

I

0.5 00

I

0.15

0 25 0.35

0.5 (MISSING

MASS ) ( GeV

)

2 2

FIG. 2. (Missing mass)

distribution

for K d- - K p

(unseen Kp decay).

FIG. 4.

"p invariant-mass

distribution

for events

with seen protons (a) all events, (b) excluding events in the region 2060 MeV &M(" K }& 2000 MeV, (c) in- cluding

nly those events

lying in the region 2060 MeV

&M(= K') &2000 MeV.

10

G. d 'Agostini et al. / Search for strange dibaryons

tial cross section for X÷ backward production [17]. This efficiency factor turns out to be about 0.35, with a 25% systematic uncertainty mainly due to the large errors in the published data.

4. Results

The data presented in this paper have been collected during a running period of about 14 months. The total incident flux is 4.1 × 101° K-. The missing-mass spectrum of our total sample (8767 events) is shown in fig. 8. A dibaryonic state should appear in the missing-mass spectrum as an enhancement above the physical background, which is mainly due to the interaction of the incident K- with a single nucleon of the deuteron. The primary contribution comes from the reaction K-+ d-+ K++ ~-+ n. (3) The solid line is the result of a fit in which the background coming from reaction (3), as evaluated by Monte Carlo calculation, is considered, together with the "out of target" contribution (- 5%). A constant background (- 4%) is included in order to account for residual contaminations due to beam decay in the spectrometer or to

ther physical processes, such as K ÷ production through double charge exchange

800

%

600 ~n

.

400 w

,,>,

I [ I I I 200 2.15 K-+d ~K++MM t 1.4 GeV/c (AAx-)

(~-ns)

i{r-r o)

. . . . . . . i ....... I [ ~ I ~4J~-~t&~- 2.20 2.25 2.30 2.35 2.40 2.45 MISSING MASS (GeV/c 2)

Fig. 8. Missing-mass spectrum for the reaction K-d ~ K++ MM at 1.4 GeV/c. In this sample the

Cerenkov C 2 was required in the trigger. The solid curve is the result of a fit (see text). The thresholds for the different possible channels are also shown. Only (-~-n) and (X-A) channels have been taken into account in the fit.

K −d → K 0 + X K −d → K + + X

Left: D.P . Goyal et al., PRD 21 (1980) 607 Right: G. d’Agostini et al., NPB 209 (1982) 1

⇒ no visible enhancement around the ΞN threshold

Johann Haidenbauer Baryon-baryon interactions

SLIDE 20

Results for S = −2

200 400 600 800 plab (MeV/c) 50 100 150 200 250 300 350 400 σ (mb)

LO NLO

(a) ΛΛ -> ΛΛ

200 400 600 800 plab (MeV) 10 20 30 40 50 60 70 σ (mb)

Ahn (2006)

(c) Ξ

−p -> ΛΛ

-->
-->

ΛΛ effective range parameters

NLO LO Λ 500 550 600 650 550 600 650 700 a1S0 −0.62 −0.61 −0.66 −0.70 −1.52 −1.52 −1.54 −1.67 r1S0 7.00 6.06 5.05 4.56 0.82 0.59 0.31 0.34 empirical: aΛΛ = -1.2 ± 0.6 fm −1.92 < aΛΛ < −0.50 fm J.H., U.-G. Meißner, S. Petschauer, NPA 954 (2016) 273 Johann Haidenbauer Baryon-baryon interactions

SLIDE 21

Results for S = −2

200 400 600 800 plab (MeV/c) 10 20 30 40 50 60 70 80 90 100 σ (mb)

Ahn (2006)

Ξ

−p -> Ξ −p

-->
-->

200 400 600 800 plab (MeV/c) 10 20 30 40 50 60 70 80 90 100 σ (mb)

Aoki (1998)

Ξ

−p -> ΛΛ,Ξ 0n,Σ 0Λ

hatched band: NLO interaction from NPA 954 (2016) filled band: JH, U.-G. Meißner, EPJA 55 (2019) 23 NLO LO Λ 500 550 600 650 550 600 650 700 Ξ0n a3S1 −0.70 −0.73 −0.84 −0.85 −0.34 −0.25 −0.20 −0.15 (2016) −0.25 −0.20 −0.26 −0.34 Ξ0p a1S0 0.37 0.39 0.34 0.31 0.21 0.19 0.17 0.13 a3S1 −1.17 −1.15 −1.13 −0.90 0.02 0.00 0.02 0.03 (2016) −0.20 −0.04 0.02 0.04 Σ+Σ+ a1S0 −2.19 −1.94 −1.83 −1.82 −6.23 −7.76 −9.42 −9.27 Johann Haidenbauer Baryon-baryon interactions

SLIDE 22

ΛΛ 1S0 phase shift

isospin basis particle basis

100 200 300 400 500 600 plab (MeV/c)

20
10

10 20 30 40 50 60 70 δ (degrees) ΛΛ

1S0

100 200 300 400 500 600 plab (MeV/c) 0.0 0.2 0.4 0.6 0.8 1.0 η ΛΛ

1S0

100 200 300 400 500 600 plab (MeV/c)

20
10

10 20 30 40 50 60 70 δ (degrees) ΛΛ

1S0

H-dibaryon appears as a virtual state close to the ΞN threshold ⇒ large cusp for physical masses thresholds are splitted up: ΞN (25.8 MeV) → Ξ0n (23.2) − Ξ−p (28.3) ⇒ effect is much smaller

Johann Haidenbauer Baryon-baryon interactions

SLIDE 23

ΞN: Comparison with HAL QCD results

10 20 30 40 50 60 70 80 Ecm (MeV)

10

10 20 30 40 50 60 70 80 90 100 δ (degrees)

t=12 t=11 t=10

ΞN

1S0 (I=0) 10 20 30 40 50 60 70 80

Ecm (MeV)

10

10 20 30 δ (degrees)

t=13 t=12 t=11

ΞN

3S1 (I=0)

HAL QCD Collaboration, from K. Sasaki’s talk at Lattice2017, Granada, Spain results are for different sink-source time-separations t

Johann Haidenbauer Baryon-baryon interactions

SLIDE 24

Results for S = −2

200 400 600 800 plab (MeV/c) 10 20 30 40 50 60 70 80 90 100 σ (mb)

Ahn (2006)

Ξ

−p -> Ξ 0n 200 400 600 800

plab (MeV/c) 10 20 30 40 50 60 70 80 90 100 110 120 130 σ (mb)

Ξ

0p -> Ξ 0p

hatched band: NLO interaction from NPA 954 (2016) filled band: JH, U.-G. Meißner, EPJA 55 (2019) 23 Johann Haidenbauer Baryon-baryon interactions

SLIDE 25

Hypertriton (Faddeev calculation by A. Nogga)

[MeV] Λ

4 5 5 5 5 6 6 5 7

[MeV]

Λ

E

. . 5 . 1 . 1 5 . 2 . 2 5

separation energy Λ H

Λ 3

X

L O N L O

I

d a h

5

N L O

I

d a h

6

E x p .

separation energy Λ H

Λ 3

X

[ M e V ] Λ

4 5 5 5 5 6 6 5 7 2

χ

1 1 2 1 4 1 6 1 8 2 2 2 2 4 2 6 2 8 3 2

χ d e p e n d e n c e

f

Λ Y N

L O N L O 2

χ d e p e n d e n c e

f

Λ Y N

EΛ = E(core) − E(hypernucleus)

separation energies:

Λp 1S0 / 3S1 scattering lengths are chosen so that 3

ΛH is bound

however, effect of three-body forces needs to be explicitly estimated
cutoff variation:

* NNN → is lower bound for magnitude of higher order contributions * ΛNN - correlation with χ2 of YN interaction?

Johann Haidenbauer Baryon-baryon interactions

SLIDE 26

4 ΛH results (Faddeev-Yakubovsky – by A. Nogga)

[MeV] Λ

4 5 5 5 5 6 6 5 7

[MeV]

Λ

E

. . 5 1 . 1 . 5 2 . 2 . 5

separation energy Λ )

+

H (J=0

Λ 4

X

LO NLO-Idaho 500 Exp.

separation energy Λ )

+

H (J=0

Λ 4

X

[MeV] Λ

450 500 550 600 650 700

[MeV]

Λ

E

0.0 0.5 1.0 1.5 2.0 2.5

separation energy Λ )

+

H (J=1

Λ 4

X

LO NLO-Idaho 500 Exp.

separation energy Λ )

+

H (J=1

Λ 4

X

LO: unexpected small cutoff dependence in 0+ result
NLO: underbinding → comparable to what is observed in calculations with

phenomenological potentials (Jülich ’04, NSC97f)

possible effects of long ranged three-body forces?
open problem: charge symmetry breaking

4 ΛH ↔ 4 ΛHe Johann Haidenbauer Baryon-baryon interactions

SLIDE 27

Three-body forces

three-body force (nominally at N2LO): S. Petschauer et al., PRC 93 (2016) 014001 arising 3BF LECs can be constrained by resonance saturation (via decuplet baryons):

S. Petschauer et al., NPA 957 (2017) 347 (for ΛNN, based on the Σ∗(1385) excitation)

⇒ inclusion in bound state calculation in progress

Johann Haidenbauer Baryon-baryon interactions

SLIDE 28

Three-body forces

N Λ N N Λ N

✉ ② ✉

N Λ N N Λ N

✉ ②

N Λ N N Λ N

②

(a) (b) (c) N Λ N N Λ N Σ∗

✉ ✉ ✉ ✉

N Λ N N Λ N Σ

✉ ✉ ✉ ✉

(d) (e)

(a) - (c) appear at N2LO (d) appears at NLO – in EFT that includes decuplet baryons (e) is already included by solving coupled-channel Faddeev equations

Johann Haidenbauer Baryon-baryon interactions

SLIDE 29

Brueckner reaction-matrix formalism

conventional non-relativistic lowest order Brueckner theory: YN|GYN(ζ)|YN = YN|V|YN +

Y ′N

YN|V|Y ′N Y ′N| Q ζ − H0 |Y ′N Y ′N|GYN(ζ)|YN Q ... Pauli projection operator ζ = EY(pY) + EN(pN) Eα(pα) = Mα + p2

α

2Mα + Uα(pα), α = Y, N UY(pY) =

pN≤kF

d3pN YN|GYN(ζ(UY))|YN BY(∞) = −UY(pY = 0) - evaluated at saturation point of nuclear matter

⇒ JH, U.-G. Meißner, NPA 936 (2015) 29

S. Petschauer, et al., EPJA 52 (2016) 15

JH, U.-G. Meißner, EPJA 55 (2019) 23 Johann Haidenbauer Baryon-baryon interactions

SLIDE 30

Nuclear matter properties

UY(pY = 0) [in MeV] at saturation density, kF = 1.35 fm−1 (ρ0 = 0.166 fm−3) EFT LO EFT NLO Jülich ’04 Jülich ’94 Λ [MeV] 550 · · · 700 500 · · · 650 UΛ(0) −38.0 · · · −34.4 −28.2 · · · −22.4 −51.2 −29.8 UΣ(0) 28.0 · · · 11.1 17.3 · · · 11.9 −22.2 −71.4 “Empirical” value for the Λ binding energy in nuclear matter: ≈ 27 − 30 MeV ΣN (I=3/2): 3S1–3D1: decisive for Σ properties in nuclear matter A description of YN data is possible with an attractive as well as a repulsive 3S1–3D1 interaction adopt the repulsive solution in accordance with evidence from

level shifts and widths of Σ− atoms
(π−, K +) inclusive spectra related to Σ− formation in heavy nuclei

Lattice QCD calculations support also a repulsive 3S1–3D1!

S. Beane et al., PRL 109 (2012) 172001; H. Nemura, EPJ Web Conf. 175 (2018) 05030

Jülich ’94: A. Reuber, K. Holinde, J. Speth, NPA 570 (1994) 543 Johann Haidenbauer Baryon-baryon interactions

SLIDE 31

kF dependence of s.p. potentials

0.6 0.8 1.0 1.2 1.4 1.6 kF (fm

1)
60
50
40
30
20
10

UΛ (pΛ=0) (MeV)

EFT LO EFT NLO Jülich ’04

0.6 0.8 1.0 1.2 1.4 1.6 kF (fm

1)
10

10 20 30 40 50 UΣ (pΣ=0) (MeV)

Johann Haidenbauer Baryon-baryon interactions

SLIDE 32

density dependent effective YN interaction

three-body force (nominally at N2LO): density dependent effective YN interaction: ... close two baryon lines by sum over occupied states within the Fermi sea arising 3BF LECs can be constrained by resonance saturation (via decuplet baryons)

J.W. Holt, N. Kaiser, W. Weise, PRC 81 (2010) 064009 (for NNN)

S. Petschauer et al., NPA 957 (2017) 347 (for ΛNN)

Johann Haidenbauer Baryon-baryon interactions

SLIDE 33

Results for UΛ at larger density ρ

symmetric nuclear matter neutron matter

0.5 1.0 1.5 2.0 ρ / ρ0

40
20

20 UΛ (MeV)

(a)

0.5 1.0 1.5 2.0 ρ / ρ0

40
20

20 UΛ (MeV)

(b)

——— χEFT at NLO − · − χEFT at NLO + density-dependent ΛN interaction derived from chiral ΛNN 3-body force (S. Petschauer et al., NPA 957 (2017) 347) − − Jülich ’04; · · · Nijmegen NSC97f

⇒ χEFT: less attractive or even repulsive for ρ > ρ0 neutron stars: hyperons appear at higher density impact on the so-called hyperon puzzle

Johann Haidenbauer Baryon-baryon interactions

SLIDE 34

Explanation of in-medium properties

200 400 600 800 plab (MeV/c)

80
70
60
50
40
30
20
10

10 20 δ (degrees) EFT NLO Jülich ’04 NSC97f Λp

3S1

200 400 600 800 plab (MeV/c)

30

30 60 90 120 150 180 δ (degrees) Λp

3S1

ΛN–ΣN coupling switched off full result

⇒ consequences for in-medium properties: ΛN–ΣN coupling is suppressed for increasing no. of nucleons (dispersive effects; Pauli blocking effects)

V eff

ΛN(E) = VΛN + VΛN→ΣN(E − H0)−1VΣN→ΛN

N Λ N N Λ N Σ

✉ ✉ ✉ ✉

N Λ N N Λ N Σ

✉ ✉ ✉ ✉

→ ΛN interaction from EFT (NLO) is less attractive in the medium than the Nijmegen NSC97f and, specifically, the Jülich ’04 meson-exchange potentials

Johann Haidenbauer Baryon-baryon interactions

SLIDE 35

Nuclear matter properties

UΞ(pΞ = 0) [in MeV] at saturation density, kF = 1.35 fm−1 (ρ0 = 0.166 fm−3) EFT NLO (2019) EFT NLO (2016) ESC08c fss2 Λ [MeV] 500 · · · 650 500 · · · 650 UΞ(0) −5.5 · · · −3.8 22.4 · · · 27.7 −7.0 −1.5 “Canonical” value for the depth of the Ξ single-particle potential: ≈ −15 MeV

Nijmegen ESC08c: M.M Nagels, T.A. Rijken, Y. Yamamoto, arXiv:1504:02634 Quark model fss2: Y. Fujiwara, Y. Suzuki, C. Nakamoto, Prog. Part. Nucl. Phys. 58 (2007) 439 (UΞ results from M. Kohno, S. Hashimoto, Prog. Theor. Phys. 123 (2010) 157) Johann Haidenbauer Baryon-baryon interactions

SLIDE 36

Baryon-baryon interaction in the final state

Johann Haidenbauer Baryon-baryon interactions

SLIDE 37

Extract information from final-state interaction

invariant mass spectrum in “high-momentum” transfer reaction pp → K +Λp (COSY - Jülich) γd → K +Λn (SPring-8, CLAS, ELPH) K −d → K 0ΛΛ, 12C(K −, K +ΛΛX), K −d → K +Ξ−n, ... Migdal-Watson approach Jost function approach Dispersion integral (A. Gasparyan et al., PRC 69 (2004) 034006) ⇒ extract effective-range parameters from final-state interaction Study two-particle momentum correlations in relativistic heavy-ion collisions (STAR, RHIC) high energetic pp collisions (ALICE, CERN) (S.E. Koonin, S. Pratt, R. Lednicky-V.L. Lyuboshitz, ...) (T. Hyodo, K. Morita, A. Ohnishi, ...) ⇒ extract effective-range parameters from correlations

Johann Haidenbauer Baryon-baryon interactions

SLIDE 38

Dispersion relations

P

K +

P

p

P

Λ

P

1

P

2

P

2

P

1− )2

s=(

+

A =

NEGLECT KΛ AND KN FSI

− )2 t=( − )2

P

K P 1

P

p P Λ

m =(

2 Assume point-like production operator (large momentum transfer)

is a constant with respect to variations in m2

Dispersion relation technique: A(s, t, m2) = exp

1

π ∞

m2

δ(m′2) m′2 − m2 − i0dm′2

×Φ(s, t, m2)

Johann Haidenbauer Baryon-baryon interactions

SLIDE 39

Jost function

DWBA: A = A0 + A0GΛpTΛp for a short-ranged production mechanism A0 is only weakly momentum (energy) dependent A ≈ A0[1 + GΛpTΛp] = A0ψ(−)∗

k

(0) = A0F−1(−k) (Jost-function approach) ERE: k cot δ(k) = − 1

a + 1 2r0k2

S = exp(2iδ(k)) = k cot δ(k) + ik k cot δ(k) − ik = (k + iα)(k + iβ) (k − iα)(k − iβ) where r0(α + β) = 2, a αβ = (α + β) F−1(−k) = (k + iβ) (k − iα) = (k2 + β2)r0/2 − 1

a + 1 2r0k2 − ik

Johann Haidenbauer Baryon-baryon interactions

SLIDE 40

Migdal-Watson approach

Migdal-Watson approximation: β ≫ k (|a| ≫ r0) F−1(−k) ≈ 1 − 1

a + 1 2r0k2 − ik

A ≈ N · A0 · TΛp works indeed reliably for interactions with a rather large scattering length, e.g. 1S0 np → a = -23.75 fm However: aΛN ≈ −2 fm aΛΛ ≈ −0.7 fm

Johann Haidenbauer Baryon-baryon interactions

SLIDE 41

Details of the method

A. Gasparyan et al., PRC 69 (2004) 034006

aS = lim

m2→m2

1 2π MΛ + MN √MΛMN

P

m2

max

m2

dm′ 2

m2

max − m2

m2

max − m′ 2

× 1

m′ 2 − m2

0 (m′ 2 − m2)

log 1 q′ d2σS dm′ 2dt

σS ... spin-dependent cross sections

⇒ required for separating the 1S0 and 3S1 states d2σS dm2dt ∝ k|AS(s, t, m2)|2 mmax − m0 = Qmax ≈ 20 − 50 MeV

in practice: parameterize d2σS/dm′ 2dt → perform integral analytically theoretical uncertainty: ±0.3 fm

(N.I. Muskhelishvili, “Singular Integral Equations”, 1953; R. Omnes, Nuovo Cim. 8, 316 (1958); W.R. Frazer and J.R. Fulco, PRL 2, 365 (1959); B.V. Geshkenbein, Yad.Fiz.9, 1232 (1969), PRD 61) Johann Haidenbauer Baryon-baryon interactions

SLIDE 42

Separating singlet and triplet for p p → pK +Λ

Special kinematical conditions (kΛN is small) + parity conservation + Pauli principle (for initial protons)

spin triplet contributions only:

A0yσ0(θ = 90◦) (1 − Axx)σ0(θ = 90◦ or φ = 90◦) spin singlet contributions only: (1 + Axx + Ayy − Azz)σ0(θ = 90◦)

Johann Haidenbauer Baryon-baryon interactions

SLIDE 43

pp → K +Λp at plab = 2.95 GeV/c (ǫ = 204 MeV)

Λp invariant mass spectrum |M|2 ∝

1 kΛp dσ dmΛp

2060 2080 2100 2120 2140 2160 2180 2200 2220 2240 2260

)

2

N/A/(2MeV/c 500 1000 1500 2000 2500 3000 3500 ]

2

[MeV/c

Λ p

m 2060 2080 2100 2120 2140 2160 2180 2200 2220 2240 2260 A 0.1 0.2

]

2

[MeV/c

Λ p

m 2050 2060 2070 2080 2090 2100 2110 [arbitrary units]

2

|M| 1 2 3 4 5 6 7 8 9 10

all data upper region +1.5 units lower region +1.5 units

triangles down: (2.590 ≤ m2

KΛ ≤ 3.176) GeV2/c4

triangles up: (3.176 ≤ m2

KΛ ≤ 3.287) GeV2/c4

→ distortion of FSI signal from N∗ resonances in πN → KΛ (COSY, Jülich) M. Röder et al., EPJA 49 (2013) 157

Johann Haidenbauer Baryon-baryon interactions

SLIDE 44

pp → K +Λp at plab = 2.70 GeV/c (ǫ = 121 MeV)

Results of a new measurement with 5 x higher statistics: allows one to isolate/remove distortions from N∗ resonances

Λp invariant mass spectrum A0y(cos θK , mΛp) ≈ α sin θK + β cos θK sin θK

)

2

(MeV/c

Λ p

m 2060 2080 2100 2120 2140 2160 )

2

Entries / AC per (1 MeV/c 2000 4000 6000 8000 10000 12000 )

2

(MeV/c

Λ p

m 2060 2080 2100 2120 2140 2160 Legendre polynomial coefficients

0.2
0.1

0.1 0.2

(COSY, Jülich) F . Hauenstein et al., PRC 95 (2017) 034001

Johann Haidenbauer Baryon-baryon interactions

SLIDE 45

pp → K +Λp at plab = 2.70 GeV/c

Spin Triplet Scattering Length at

Hauenstein et al., PRC 95, 034001 (2017)

)

2

(MeV/c

Λ p

m 2060 2080 2100 2120 2140 2160 (arb. units)

2

| A ~ | 2 4 6

/ NDF = 1.25

2

χ = -1.39e-01 + 3.06e-02 - 3.25e-02 C = 1.21e+05 + 7.99e+03 - 7.26e+03

1

C = 4.16e+06 + 3.32e+03 - 3.58e+03

2

C

)

2

(MeV/c

Λ p

m 2060 2080 2100 2120 2140 2160 | (arb. units)

2

| A ~ | × )

Λ p

(m α | = |

1

|b 5 10

/ NDF = 2.17

2

χ = 5.99e-01 + 1.71e-01 - 2.33e-01 C = 4.55e+04 + 3.53e+04 - 1.96e+04

1

C = 4.20e+06 + 9.87e+03 - 1.71e+04

2

C

fit limit Rebin ×|αK |

Fit limit and parametrization as for effective scattering length Value and statistical errors determined with bootstrapping at = (−2.55+0.72

−1.39stat. ± 0.6syst. ± 0.3theo.) fm

First direct model-independent determination of at

Johann Haidenbauer Baryon-baryon interactions

SLIDE 46

Two-particle correlation function

Koonin-Pratt formalism

C(p1, p2) =

d4x1d4x2S1(x1, p1)S2(x2, p2)
Ψ(−)(r, k)
2
d4x1d4x2S1(x1, p1)S2(x2, p2)

≃

drS12(r)
Ψ(−)(r, k)
2

Si(xi, pi) ... single particle source function with momentum pi Ψ(−)(r, k) ... wave function in the outgoing state evaluate everything in center-of-mass frame: k = (m2p1 − m1p2)/(m1 + m2) assume a static and spherical Gaussian source with radius R: S12(r) = exp(−r2/4R2)/(2√πR)3

Johann Haidenbauer Baryon-baryon interactions

SLIDE 47

Two-particle correlation function

Koonin-Pratt formalism

Correlation function for identical particles (ΛΛ, Σ+Σ+, ...) C(k) ≃ 1 − 1 2 exp(−4k2R2) + 1 2 ∞ 4πr2 dr S12(r)

|ψ(k, r)|2 − |j0(kr)|2

Correlation function for non-identical particles (Λp, Ξ−p, K −p, ...) C(k) ≃ 1 + ∞ 4πr 2 dr S12(r)

|ψ(k, r)|2 − |j0(kr)|2

Boundary condition for wave function: ψ(k, r) → e−iδ kr sin(kr + δ) = 1 2ikr

eikr − e−2iδe−ikr

(r → ∞) (I consider only correlations in S-waves!!)

Johann Haidenbauer Baryon-baryon interactions

SLIDE 48

Two-particle correlation function

Lednicky-Lyuboshitz model

replace full wave function by its asymptotic form! ∞ 4πr 2dr S12(r)

|ψ(k, r)|2 − |j0(kr)|2

≈ |f(k)|2 2R2 F(r0) + 2Ref(k) √πR F1(x) − Imf(k) R F2(x) f(k) = (S(k) − 1)/2ik ... scattering amplitude (S ... S-matrix) ⇒ replace by effective-range expansion: f(k) ≈ 1/(− 1

a + r0k2/2 − ik)

F(r0) = 1 − r0/(2√πR) ... correction to wave function F1(x) = x

0 dt et2−x2/x,

F2(x) = (1 − e−x2)/x, x = 2kR

usually spin average is considered: f → f S; perform

S ρS

→ possible dependence on spin remains unresolved

valid only for elastic channels (pp, Λp, ΛΛ, ...)

but not for Ξ−p, K −p, ...

Johann Haidenbauer Baryon-baryon interactions

SLIDE 49

Extension to multi-channel problem

R. Lednicky, V.V. Lyuboshitz, V.L. Lyuboshitz, Phys. At. Nucl. 61 (1998) 2161

|ψ(k, r)|2 →

β

ωβ|ψβα(kα, r)|2 Cα(kα) ≃ 1 +

β

ωβ ∞ 4πr 2 dr Sβ(r)

|ψβα(kα, r)|2 − δβα |j0(kαr)|2
β ... over all two-body intermediate states that couple to α

ωβ ... weights of the various components (I put all to 1) Example: Λp (3S1) ψΛp−Λp(3S1 → 3S1), ψΛp−Λp(3D1 → 3S1) ψΣ+n−Λp(3S1 → 3S1), ψΣ+n−Λp(3D1 → 3S1) ψΣ0p−Λp(3S1 → 3S1), ψΣ0p−Λp(3D1 → 3S1)

Johann Haidenbauer Baryon-baryon interactions

SLIDE 50

Experimental determination of the correlation functions

Corrections due to misidentified particles and feed-down particles from strong and weak decays of resonances (pair purity parameter λ) residual source effects (ares, rres) Corrections of nonfemtoscopic correlation for larger k (a, b) (baseline correctons) Ohnishi, Morita, Miyahara, Hyodo, NPA 954 (2016) 294: Ccorr (k) = N

1 + λ(Ctheory (k) − 1) + arese−4r2

resk2

S. Acharya et al. (ALICE), PRC 99 (2019) 024001:

Cmodel (k) = 1 + λgenuine

Cgenuine(k) − 1
+
ij

λij

Cij (k) − 1
Ctot (k)

= N · (a + b k) · Cmodel (k) differences in the experiment:

Au+Au at √s = 200 GeV (STAR Collaboration)
pp at √s = 7, 13 TeV (ALICE Collaboration)

⇒ different λ, R, .... ⇒ differences in the resulting correlation function C(k) Johann Haidenbauer Baryon-baryon interactions

SLIDE 51

Correlation function for Λp

singlet S-wave triplet S-wave

50 100 150 200 250 300 k (MeV/c) 1.0 2.0 3.0 4.0 5.0 C(k)

Λp

1S0

50 100 150 200 250 300 k (MeV/c) 1.0 1.5 2.0 2.5 3.0 C(k) STAR ’06 ALICE ’19 R=3.09 R=1.2 LL 1.2 R=1.2, λ=0.47

Λp

3S1

STAR Collaboration (J. Adams et al., PRC 74 (2006) 064906): Au+Au at √s = 200 GeV R = 3.09 ± 0.30 fm ALICE Collaboration (S. Acharya al., PRC 99 (2019) 024001): pp at √s = 7 TeV R = 1.125 ± 0.018 fm, λ = 0.4713 YN interaction: NLO (600) → as = −2.91 fm, at = −1.54 fm [only Λp component] LL ... Lednicky-Lyuboshitz model ⇒ strong sensitivity to λ, R [source function] Johann Haidenbauer Baryon-baryon interactions

SLIDE 52

Correlation function for Λp

50 100 150 200 250 300 k (MeV/c) 1.0 1.5 2.0 2.5 3.0 C(k) STAR ’06 ALICE ’19 R=1.2, λ=1 R=3.09, λ=1 R=1.125, λ=0.47

Λp

STAR Collaboration (J. Adams et al., PRC 74 (2006) 064906): Au+Au at √s = 200 GeV R = 3.09 ± 0.30 fm ALICE Collaboration (S. Acharya al., PRC 99 (2019) 024001): pp at √s = 7 TeV R = 1.125 ± 0.018 fm, λ = 0.4713 spin average: |ψ(k, r)|2 = 1

4 |ψ(1S0)(k, r)|2 + 3 4 |ψ(3S1)(k, r)|2

Johann Haidenbauer Baryon-baryon interactions

SLIDE 53

Effects of coupling Λp-ΣN

triplet S-wave

50 100 150 200 250 300 k (MeV/c) 1.0 1.5 2.0 2.5 3.0 C(k) STAR ’06 ALICE ’19 Λp Λp, ΣN-Λp λ=0.47 λ=0.47

Λp

3S1

250 260 270 280 290 300 k (MeV/c) 0.8 1.0 1.2 1.4 1.6 C(k) STAR ’06 ALICE ’19 Σ

+n-Σ 0p

Λp

3S1

YN interaction: NLO (600) R = 1.2 fm Λp (alone) + ΣN-Λp + Σ+n-Σ0p splitting ALICE Collaboration (L. Fabbietti) → see a structure at ΣN threshold in experiment Johann Haidenbauer Baryon-baryon interactions

SLIDE 54

Correlation function for ΛΛ

particle basis isospin basis

50 100 150 200 k (MeV/c) 0.7 0.8 0.9 1.0 1.1 1.2 C(k) ALICE STAR

ΛΛ

50 100 150 200 k (MeV/c) 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 C(k) STAR ALICE

ΛΛ STAR Collaboration (L. Adamczyk et al., PRL 114 (2015) 022301): Au+Au at √s = 200 GeV ALICE Collaboration (S. Acharya al., PRC 99 (2019) 024001): pp at √s = 7 TeV

·-· ΛΛ only

— ΛΛ + ΞN-ΛΛ · · · Lednicky-Lyuboshitz model black lines ... H-dibaryon at 20 MeV above ΛΛ threshold → not supported by present data JH, NPA 981 (2019) 1 (λ = 1, R = 1.2 fm) Johann Haidenbauer Baryon-baryon interactions

SLIDE 55

Correlation function for Ξ−p

no bound state ΞN bound state in I = 0 3S1-3D1 p.w.

50 100 150 200 k (MeV/c) 1.0 1.5 2.0 C(k) ALICE ’18

Ξ

−p

50 100 150 200 k (MeV/c) 1.0 1.5 2.0 C(k)

Ξ

−p

ALICE Collaboration (QNP2018, preliminary!!): p+Pb at √s = 5 TeV

·-· Ξ−p only
- - Ξ−p + ΛΛ-Ξ−p

— + Ξ0n-Ξ−p · · · + Coulomb (approximative) JH, NPA 981 (2019) 1 (λ = 1, R = 1.2 fm) Johann Haidenbauer Baryon-baryon interactions

SLIDE 56

Correlation function for K −p

riginal Jülich ¯

KN model (1990) + up-to-date potentials

50 100 150 200 k (MeV/c) 0.5 1.0 1.5 C(k)

K

p

50 100 150 200 k (MeV/c) 0.5 1.0 1.5 C(k)

K

p

· · · K −p averaged masses

·-· K −p physical masses
- - + ¯

K 0n-K −p

··- + π0Λ-K −p

— + πΣ-K −p — refitted Jülich ¯ KN model (2018) (¯ KN, πΛ, πΣ, ¯ K∆, ¯ K ∗N, ¯ K ∗∆) — chirally motivated ¯ KN model (Cieply & Smejkal, 2012) (¯ KN, πΛ, πΣ, ηΛ, ηΣ, KΞ) (both are in line with level shift and width of kaonic hydrogen (SIDDHARTA) ) Johann Haidenbauer Baryon-baryon interactions

SLIDE 57

Summary

Baryon-baryon interaction constructed within chiral EFT

Approach is based on a modified Weinberg power counting, analogous to the NN case The potential (contact terms, pseudoscalar-meson exchanges) is derived imposing SU(3)f constraints YN: Excellent results at next-to-leading order (NLO) low-energy data are reproduced with a quality comparable to phenomenological models S = −2: ΛΛ, ΞN results are in agreement with empirical constraints SU(3) symmetry breaking when going from NN to YN to YY! SU(3) symmetry provides a useful guiding line (fulfilled within 10 to 30 %) however, one should not follow SU(3) symmetry too strictly

3 ΛH, 4 ΛH, 4 ΛHe ... effects of three-body forces needs to be quantified Johann Haidenbauer Baryon-baryon interactions

SLIDE 58

Summary

Final-state interaction – momentum correlation functions

excellent source for information on the baryon-baryon interaction allows access to small momenta ⇒ infer effective range parameters (a, r0) works very well for elastic channels with simple spin-structure (ΛΛ: 1S0) (thanks to the Pauli principle) more delicate for baryon-baryon scattering in general: 1S0, 3S1, 5S2, 7S3 disentangling spin states experimentally is rather challening difficult for systems with open (inelastic) channels (ΞN, ¯ KN) dispersion integral method, Jost function approach, ... no longer work! effective range expansion ... no longer works! due to splitting of thresholds (Ξ0n - Ξ−p or K −p - ¯ K 0n) isospin symmetry is strongly broken for small momenta ⇒ proliferation of independent amplitudes not possible to separate those by measuring a single correlation function personal expectation: promising to look into unexplored territory: ΛcN, ΛcΛc, ΩΩ, ... challenge to add further constraints in cases like Λp, K −p, ...

Johann Haidenbauer Baryon-baryon interactions