strangeness in nuclei and neutron stars: many-body forces ... · strangeness in nuclei and neutron...

Strangeness in nuclei and neutron stars: many-body forces and the hyperon puzzle

HYP2018 - June 29, 2018

In collaboration with:

✓ S. Gandolfi, LANL

✓ J. A. Carlson, LANL

✓ A. Lovato, ANL & INFN

✓ F. Pederiva, UniTN

Diego LonardoniFRIB Theory Fellow

!2Outline

1. Neutron Star Matter & Strangeness

2. Nuclear-Hypernuclear Hamiltonians

3. Quantum Monte Carlo

4. Results: Hypernuclei

5. Results: Neutron Star Matter

6. Summary

!3

same underlying physics

neutron stars

R ⇠ 10 km ⇠ 104 m

M ⇠ 1.4M� ⇠ 1030 kg

(hyper)nuclei

R ⇠ fm ⇠ 10�15 m

M ⇠ 10�27 kg

Question: neutron star matter, why do we care?

observationsexpe

rimen

ts

theory

nuclear force

‣ energies

‣ radii, densities

‣ neutron skin

‣ …

‣ mass

‣ radii

‣ GW: tidal polarizability

‣ …

equation of state (EOS) neutron star matter

TOV equations:M(R), Mmax

Neutron Star Matter

nuclear potentials

!4

same underlying physics

R ⇠ 10 km ⇠ 104 m

M ⇠ 1.4M� ⇠ 1030 kg

R ⇠ fm ⇠ 10�15 m

M ⇠ 10�27 kg

Question: neutron star matter, why do we care?

observationsexpe

rimen

ts

theory

nuclear force

‣ energies

‣ radii, densities

‣ neutron skin

‣ …

‣ mass

‣ radii

‣ GW: tidal polarizability

‣ …theory

observationsexperiments

nuclear potentials

NS: excellent laboratory

neutron stars(hyper)nuclei

Neutron Star Matter

< 2M�

soft

µY = µN

⇢thY ⇠ 2� 3⇢0

!5

Question: neutron star matter & strangeness, why do we care?

F. Özel et al., Annu. Rev. Astro. 54, 401-440 (2016)

J0453+1559

J0453+1559c

J0737-3039A

J0737-3039B

B1534+12

B1534+12cJ1756-2251

J1756-2251c

J1906+0746J1906+0746c

B1913+16B1913+16cB2127+11CB2127+11Cc

J1518+4904

J1518+4904c

J1811-1736

J1811-1736c

J1829+2456J1829+2456cJ1930-1852

J1930-1852c

J0337+1715

J0348+0432

J0437-4715

J0621+1002

J0751+1807

J1012+5307

J1614-2230

J1713+0747

J1738+0333

J1802-2124

J1807-2500B

B1855+09J1903+0327

J1909-3744

J1910-5958A

J1946+3417

J2222-0137

J2234+0611J0024-7204H

J0514-4002A

B1516+02B

J1748-2021B

J1748-2446I

J1748-2446J

J1750-37A

B1802-07

J1824-2452C

EXO 1745-248

4U 1608-52

4U 1820-30

KS 1731-260

SAX J1748.9-20214U 1724-207

Cyg X-2

Vela X-1

SMC X-1

LMC X-4

Cen X-3SAX J1802.7-2017

XTE J1855-026

EXO 1722-363OAO 1657-415

Her X-14U1538-52

B2303+46J1141-6545

Mmax = 1.67(2)M�

PSR J1903+0327

P. C. C. Freire et al., MNRAS (2011) 412 (4): 2763

Mmax = 2.01(4)M�

J. Antoniadis et al., Science 340, 1233232 (2013)

PSR J0348+0432

Mmax = 1.93(2)M�

PSR J1614-2230

E. Fonseca et al., ApJ 832,167 (2016)

2008

2013

stiffE

⇢b

M

R

> 2M�

12 km

⇢0

obs:⇠ 2M�

?

hyperon puzzle

theory observations

Neutron Star Matter & Strangeness

M. Linares et al., ApJ 859, 54 (2018)

PSR J2215+5135Mmax = 2.27+0.17

�0.15M�<latexit sha1_base64="+GjRfy4d3WqbJ5n9kA+v/LkG3E0=">AAACl3icfVHbThsxEHW2F2h6C+Wp6gsiPFRqGu1GVKEPCCQkxEtVkBpAYtOV7R2IG1+W9SwQWf4NvoZX+If+TZ2Qlw1Sj2zN0Zw5o9EMK6SwGMd/G9Gz5y9eLi2/ar5+8/bd+9bKh2NrqpLDgBtpylNGLUihYYACJZwWJVDFJJyw8d5UP7mC0gqjf+GkgKGiF1qcC04xpLJWnM56OCYr8D8ylyp647d73V7/t/sSd5O+z9zXEL8FMTW5wazVjrvxDGtPSTInbTLHYbbS+JnmhlcKNHJJrT1L4gKHjpYouATfTCsLBeVjegFnFZ5vDZ3QRYWg+YJmQVMFduhmI9dER5W1E8U6ISqKo2nEkeqw8GbfyLxuYMaMkTLbUZVEUZrrhX75n8riGjM39bywBk1Rn9oVo4kV3PpmM9VwzY1SVOcuvYJLP10ojkqQLi0NUoTQ0X2PvdvY3vDeLxj0fx2h9nJqCidIFhf+lBz3ukk43NFme3drfoxl8omsk88kIX2ySw7IIRkQTm7JHbknD9HHaCfajw4eS6PG3LNKaoiO/gGootA/</latexit><latexit sha1_base64="+GjRfy4d3WqbJ5n9kA+v/LkG3E0=">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</latexit><latexit sha1_base64="+GjRfy4d3WqbJ5n9kA+v/LkG3E0=">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</latexit><latexit sha1_base64="+GjRfy4d3WqbJ5n9kA+v/LkG3E0=">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</latexit>

20182010

(to be confirmed)

!6Nuclear Hamiltonians

Model: non-relativistic nucleons interacting with an effective two-body (NN) and three- body (NNN) nucleon force

p, n

⇤

not many data, but new info!

not many data, but new info!what do we have? what can we use?

H =� ~22mN

X

i

r2i +

X

i<j

vij +X

i<j<k

vijk + . . .

+� ~22m⇤

X

�

r�2 +

X

�<µ

v�µ +X

�<µ<⌫

v�µ⌫ + . . .

+X

�i

v�i +X

�,i<j

v�ij +X

�<µ,i

v�µi + . . .

+� ~22m⌃

X

✏

r✏2 + . . .

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⌃�,⌃0,⌃+

vij fit to NN scattering data & deuteron

vijk fit to properties of nuclei

v�i fit to ⇤N scattering data

v�ij fit to properties of ⇤-hypernuclei

v�µ, v�µi fit to properties of ⇤⇤-hypernuclei

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!8Hypernuclear Hamiltonians

p, n

⇤

not many data. HYP2018: new info!

not many data. HYP2018: new info!

many data

what do we have?

H =� ~22mN

X

i

r2i +

X

i<j

vij +X

i<j<k

vijk + . . .

+� ~22m⇤

X

�

r�2 +

X

�<µ

v�µ +X

�<µ<⌫

v�µ⌫ + . . .

+X

�i

v�i +X

�,i<j

v�ij +X

�<µ,i

v�µi + . . .

+� ~22m⌃

X

✏

r✏2 + . . .

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⌃�,⌃0,⌃+

vij fit to NN scattering data & deuteron

vijk fit to properties of nuclei

v�i fit to ⇤N scattering data

v�ij fit to properties of ⇤-hypernuclei

v�µ, v�µi fit to properties of ⇤⇤-hypernuclei

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Model: non-relativistic baryons interacting with an effective two-body (BB) and three- body (BBB) baryon force

!9

Idea: revisit the problem of the hyperon-nucleon interaction using Quantum Monte Carlo

H =� ~22mN

X

i

r2i +

X

i<j

vij +X

i<j<k

vijk

+� ~22m⇤

X

�

r�2 +

X

�i

v�i +X

�,i<j

v�ijphenomenological

interactions

⇤-hypernuclei

Proceedings of

The IX International Conference on Hypernuclear and Strange

Particle Physics

HYP 2006

October 10-14 , 2006 Mainz, Germany

edited by

J. Pochodzalla and Th. Walcher

NZ

|S|

Hypernuclear Hamiltonians

⇡ 40 A⇤Z

< 10 A⇤⇤Z

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(not updated)

fit to ⇤p scattering data<latexit sha1_base64="9P9/Pl7AVnSiYrnezFyUy7LEcno=">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</latexit><latexit sha1_base64="9P9/Pl7AVnSiYrnezFyUy7LEcno=">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</latexit><latexit sha1_base64="9P9/Pl7AVnSiYrnezFyUy7LEcno=">AAAClXicfVFNaxsxEJW3SZu4Teu0hxxyWeocejBmtxSaHgKBltJDISnUSSBrjKQd26r1FWk2iRH6F/01vbZ/ov8ma8eXdSAPiXnMvDeMNMxK4THL/reSJxubT59tbbefv9h5+aqz+/rMm8pxGHAjjbtg1IMUGgYoUMKFdUAVk3DOZp8X9fNrcF4Y/RPnFoaKTrQYC06xTo06/QLhFsNYYIomjcX32lrS1N6nU1/rEJzQk7SkSOOo08362RLpQ5KvSJescDrabZ0UpeGVAo1cUu8v88ziMFCHgkuI7aLyYCmf0QlcVjg+HAahbYWg+VrNg6YK/DAsH90oBqq8nyvWq6OiOF1EnKoeq8/yGlk2DcyYGVLme6qSKJy5WetX/qo8pszcNvPCGzS2OXWw07kX3Md2u9Bww41SVJehuIarGIrFOA5kKJxBilB3DJ+yGA6ODmKMawb9qKPWXi1M9Qry9Q9/SM7e9/Osn//40D0+XC1ji+yTt+QdyclHcky+kVMyIJz8Jn/IX/Iv2UuOki/J13tp0lp53pAGkpM7kKLRSg==</latexit><latexit sha1_base64="9P9/Pl7AVnSiYrnezFyUy7LEcno=">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</latexit>

T. Schimmert et al., Nucl. Phys. A 343 (1980) 429-434

!10

Idea: revisit the problem of the hyperon-nucleon interaction using Quantum Monte Carlo

H =� ~22mN

X

i

r2i +

X

i<j

vij +X

i<j<k

vijk

+� ~22m⇤

X

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r�2 +

X

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v�i +X

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v�ijphenomenological

interactions

⇤-hypernuclei

i. low number of parameters ii. fit to available exp data

Proceedings of

The IX International Conference on Hypernuclear and Strange

Particle Physics

HYP 2006

October 10-14 , 2006 Mainz, Germany

edited by

J. Pochodzalla and Th. Walcher

NZ

|S|

Hypernuclear Hamiltonians

Note: try to use both light & medium-mass systemsA ⇠ 5, 16, 40

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B⇤ = E�A�1Z

�� E

�A⇤Z

�

core nucleus hypernucleus

⇡ 40 A⇤Z

< 10 A⇤⇤Z

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(not updated)

!11

phenomenological potentials

N

N

⇡

N

N

N

NN

N

N

N

�

⇡

⇡

⇡

⇡

N

NN

N

N

N

N

N

⇡

⇡N

N

N

�

Note: the energy contribution of a many-body force is scheme (and scale) dependent

Hypernuclear Hamiltonians: many-body forces

⇤

⇤

⌃

N

N

⇡

⇡

⇤

⇤

⌃

N

N

⇡

⇡N

⇤

⇤

⌃

⇡

⇡

N

NN

N

⇤

⇤

⇡

⇡

N

NN

N

2b

3b

nucleons & pion exchange AV18 + UIX

nucleons/lambda & pion exchange Bodmer/Usmani

1⇡ � exchange<latexit sha1_base64="ej3gxk/22Fq1ZFg0Gg7QzcaKmjc=">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</latexit><latexit sha1_base64="ej3gxk/22Fq1ZFg0Gg7QzcaKmjc=">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</latexit><latexit sha1_base64="ej3gxk/22Fq1ZFg0Gg7QzcaKmjc=">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</latexit><latexit sha1_base64="ej3gxk/22Fq1ZFg0Gg7QzcaKmjc=">AAACe3icfVFNTxsxEHW2hUIoEFpuXCLCoUIh2q1aNT1UisSFW1OpCUhsFNnOJDHxF+vZkNTa/9Ir/CN+DFI3IZdNpD7Zmqd580ZjD7NSOAzD51Lw5u3W9rud3fLe+/2Dw8rRh64zacKhw400yQ2jDqTQ0EGBEm5sAlQxCddscrnQr6eQOGH0b5xb6Ck60mIoOMU81a8cR7EVFzHCDD3M+JjqEWT9Si1shEtUN0m0IjWyQrt/VPoZDwxPFWjkkjp3G4UWe54mKLiErBynDizlEzqC2xSHzZ4X2qYImq9pDjRV4Hp++bSC6Klybq5YPY+K4ngRcazqLD/La+SgaGDGTJAyV1epRJGYh7V+g7vUYZWZWTEvnEFji1N7O547wV1WLscaHrhRiuqBj6dwn/l4MU4C0seJQYqQd/Tfw8yf/TjLsmzNoP/ryGvvF6Z8BdH6h2+S7udGFDaiX19qreZqGTvkhJySTyQi30iLXJE26RBO/pC/5JE8lV6CWnAe1F9Lg9LK85EUEHz9B7tXx2E=</latexit>

2⇡ � exchange<latexit sha1_base64="ZaSlVA0GX0VSXPa9XpaMcJ73bgo=">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</latexit><latexit sha1_base64="ZaSlVA0GX0VSXPa9XpaMcJ73bgo=">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</latexit><latexit sha1_base64="ZaSlVA0GX0VSXPa9XpaMcJ73bgo=">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</latexit><latexit sha1_base64="ZaSlVA0GX0VSXPa9XpaMcJ73bgo=">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</latexit>

1⇡ � exchange<latexit sha1_base64="ej3gxk/22Fq1ZFg0Gg7QzcaKmjc=">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</latexit><latexit sha1_base64="ej3gxk/22Fq1ZFg0Gg7QzcaKmjc=">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</latexit><latexit sha1_base64="ej3gxk/22Fq1ZFg0Gg7QzcaKmjc=">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</latexit><latexit sha1_base64="ej3gxk/22Fq1ZFg0Gg7QzcaKmjc=">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</latexit>

2⇡ � exchange<latexit sha1_base64="ZaSlVA0GX0VSXPa9XpaMcJ73bgo=">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</latexit><latexit sha1_base64="ZaSlVA0GX0VSXPa9XpaMcJ73bgo=">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</latexit><latexit sha1_base64="ZaSlVA0GX0VSXPa9XpaMcJ73bgo=">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</latexit><latexit sha1_base64="ZaSlVA0GX0VSXPa9XpaMcJ73bgo=">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</latexit>

this scheme: S = �1 : hV2bi & hV3bi<latexit sha1_base64="R4SLsYo2B96IS+UmCAGaRxd+FT4=">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</latexit><latexit sha1_base64="R4SLsYo2B96IS+UmCAGaRxd+FT4=">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</latexit><latexit sha1_base64="R4SLsYo2B96IS+UmCAGaRxd+FT4=">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</latexit><latexit sha1_base64="R4SLsYo2B96IS+UmCAGaRxd+FT4=">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</latexit>

S = 0 : hV2bi � hV3bi<latexit sha1_base64="AglG5M7g6/lxT4kXjYi/B7TFOIs=">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</latexit><latexit sha1_base64="AglG5M7g6/lxT4kXjYi/B7TFOIs=">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</latexit><latexit sha1_base64="AglG5M7g6/lxT4kXjYi/B7TFOIs=">AAAC1HicfVFLbxMxEHa2PEp4pXDkEpEeOESRt9tHCqpUiUtvFEHSSnEU2c50Y+LH1va2RGZPiCt/jlN/Cjd2lwgpicRorPk0832fRh6WSeE8xneNaOve/QcPtx81Hz95+ux5a+fF0JncchhwI429ZNSBFBoGXngJl5kFqpiECzZ/X80vbsA6YfRnv8hgrGiqxZXg1JetSUt9OsFvyTsiqU4ltIeTQGrTkU3ZOOAe7h8cHCVd3EsO+8nxYQkw3o+TpCBWtfdYWWohSdMNi2BhWtOSf7RJq1MZVNHeBPESdNAyzic7jQ9kaniuQHsuqXOjGGd+HKj1gksomiR3kFE+pymMcn/VHwehs9yD5mszB5oqcONQ77YyDFQ5t1CsW1ZF/ayqfqa6rMz6GTldFTBj5p4y11W59MKa2zW/6Zfc+TYzX1f7whlvstWtQzZbOMFd0WwSDbfcKEX1NJAbuC4CqdaxIAOxxlMPpWM4xkXYPdktimJNoP+rKLnXlag8Qbz+4ZtguNeLcS/+uN857S+PsY1eodfoDYrRETpFZ+gcDRBHv9DvRtTYiobRt+h79OMvNWosNS/RSkQ//wA7ceSb</latexit><latexit sha1_base64="AglG5M7g6/lxT4kXjYi/B7TFOIs=">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</latexit>

!12Quantum Monte Carlo

Method: why Quantum Monte Carlo?

i. solve the many-body problem for strongly correlated systems in a non-perturbative fashion

ii. accurate description of ground-state properties iii. statistical uncertainties

GFMCVMC AFDMC

12

CVMC

40

A

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AFDMC nuclear matter

CVMC light to medium- AFDMC mass nuclei

(C)VMC GFMC AFDMC

light systems A 12<latexit sha1_base64="Qo5L+ljOCeRZDFdv/+KiKQlgg6k=">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</latexit><latexit sha1_base64="Qo5L+ljOCeRZDFdv/+KiKQlgg6k=">AAACbHicfVHbSgMxEE3Xe7216psIYov4UMpuEdQHQfHFNxWsFdwiSTq1sblsN7NqCfsRvuqX+RN+g9val63gIWEOc+YMkwyLpLDo+18Fb2Z2bn5hcam4vLK6tl4qb9xZk8QcmtxIE98zakEKDU0UKOE+ioEqJqHF+hcjvfUCsRVG3+IwgraiT1p0BaeYpVrnoYRB0HgsVfy6P8buXxJMSIVMcP1YLlyFHcMTBRq5pNY+BH6EbUdjFFxCWgwTCxHlffoEDwl2j9tO6ChB0HxKs6CpAtt247fkREeVtUPFallUFHujiD1VY9kZXyM7eQMzpo+U2ZpKJIrYvE716zwnFneZecvnhTVoovzULuoNreA2LRZDDa/cKEV1x4UvMEhdOBonBunC2CBFyDq6Ez911dNqmqZTBv2vI6sdjEzZCoLpD/9L7hr1wK8HN4eVs+PJMhbJNtkjByQgR+SMXJJr0iSc9Mk7+SCfhW9vy9v2dn5LvcLEs0ly8PZ/AKZ3wQE=</latexit><latexit sha1_base64="Qo5L+ljOCeRZDFdv/+KiKQlgg6k=">AAACbHicfVHbSgMxEE3Xe7216psIYov4UMpuEdQHQfHFNxWsFdwiSTq1sblsN7NqCfsRvuqX+RN+g9val63gIWEOc+YMkwyLpLDo+18Fb2Z2bn5hcam4vLK6tl4qb9xZk8QcmtxIE98zakEKDU0UKOE+ioEqJqHF+hcjvfUCsRVG3+IwgraiT1p0BaeYpVrnoYRB0HgsVfy6P8buXxJMSIVMcP1YLlyFHcMTBRq5pNY+BH6EbUdjFFxCWgwTCxHlffoEDwl2j9tO6ChB0HxKs6CpAtt247fkREeVtUPFallUFHujiD1VY9kZXyM7eQMzpo+U2ZpKJIrYvE716zwnFneZecvnhTVoovzULuoNreA2LRZDDa/cKEV1x4UvMEhdOBonBunC2CBFyDq6Ez911dNqmqZTBv2vI6sdjEzZCoLpD/9L7hr1wK8HN4eVs+PJMhbJNtkjByQgR+SMXJJr0iSc9Mk7+SCfhW9vy9v2dn5LvcLEs0ly8PZ/AKZ3wQE=</latexit><latexit sha1_base64="Qo5L+ljOCeRZDFdv/+KiKQlgg6k=">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</latexit>

A ⇠ 50<latexit sha1_base64="KP0+Sy6yrsmOVtB+GqLvsXUnj+0=">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</latexit><latexit sha1_base64="KP0+Sy6yrsmOVtB+GqLvsXUnj+0=">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</latexit><latexit sha1_base64="KP0+Sy6yrsmOVtB+GqLvsXUnj+0=">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</latexit><latexit sha1_base64="KP0+Sy6yrsmOVtB+GqLvsXUnj+0=">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</latexit>

A = 1<latexit sha1_base64="adWfxjdjDXYf3FWW9q7fmYXh1xQ=">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</latexit><latexit sha1_base64="adWfxjdjDXYf3FWW9q7fmYXh1xQ=">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</latexit><latexit sha1_base64="adWfxjdjDXYf3FWW9q7fmYXh1xQ=">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</latexit><latexit sha1_base64="adWfxjdjDXYf3FWW9q7fmYXh1xQ=">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</latexit>

S

| (⌧)i = e�(H�E0)⌧ | (0)i ⌧!1�! | 0i<latexit sha1_base64="mGIqIClNtm6eprcNs+VRsHpj+4s=">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</latexit><latexit sha1_base64="mGIqIClNtm6eprcNs+VRsHpj+4s=">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</latexit><latexit sha1_base64="mGIqIClNtm6eprcNs+VRsHpj+4s=">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</latexit><latexit sha1_base64="mGIqIClNtm6eprcNs+VRsHpj+4s=">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</latexit>

⌧ =it

~ : � @

@⌧| (⌧)i = (H � E0)| (⌧)i

<latexit sha1_base64="qbjFQhTzn67PPIyV3xJ5xV+jh6I=">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</latexit><latexit sha1_base64="qbjFQhTzn67PPIyV3xJ5xV+jh6I=">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</latexit><latexit sha1_base64="qbjFQhTzn67PPIyV3xJ5xV+jh6I=">AAACyXicfVFba1NBEN4cbzXeUn30pZg+pJCGc0SwKoGCCAUfrGDaQjeE3c0k2WYvJ7tz2sZ1n/xz/gX/hK/66DlpEJKAwy7zMd98M8MMz5X0mKY/a8mt23fu3tu6X3/w8NHjJ43tpyfeFk5AT1hl3RlnHpQ00EOJCs5yB0xzBad8+r7iTy/BeWnNF5zn0NdsbORICoZlaNDgFFnRpSPHRJAYA51w5uJb+m7/JkZz5lAyFf+hShC/0dzLVgX3qGNmrKDbOtr/MEj3NplBo5l20oXtbIJsCZpkaceD7donOrSi0GBQKOb9eZbm2A9Vf6Eg1mnhIWdiysZwXuDooB+kyQsEI9Y4D4Zp8P2wWNQKGZj2fq55u/Sa4aTyONFtXr7Ft2q4KuDWTpFx39aFQuns1Vq94UXhcYfb69W49BZtvjp1yCdzL4WP9To1cCWs1swMA72EWbnoahwHKlBnkSGUFcObNIbd7m6McU1g/qsoc2eVqDxBtr7wTXDyspOlnezzq+bhwfIYW+Q5eUFaJCOvySE5IsekRwT5QX6R3+RP8jGZJdfJ15vUpLbUPCMrlnz/C3En5vU=</latexit><latexit sha1_base64="qbjFQhTzn67PPIyV3xJ5xV+jh6I=">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</latexit>

!13Results: Hypernuclei

AFDMC: ⇤-hypernuclei

D. L. and F. Pederiva, arXiv:1711.07521

0

5

10

15

20

25

30

0.0 0.1 0.2 0.3 0.4 0.5

208

89

40

28

1612

9

76

54

3

spdfg

BΛ

(M

eV)

A−2/3

emulsion

(K−,π−

)

(π+,K

+)

(e,e’K+)

0

5

10

15

20

25

30

0.0 0.1 0.2 0.3 0.4 0.5

updated data:

+0.54(5)MeV<latexit sha1_base64="0jJG1AcTcN31OWMvK/zG/LNqBlQ=">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</latexit><latexit sha1_base64="0jJG1AcTcN31OWMvK/zG/LNqBlQ=">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</latexit><latexit sha1_base64="0jJG1AcTcN31OWMvK/zG/LNqBlQ=">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</latexit><latexit sha1_base64="0jJG1AcTcN31OWMvK/zG/LNqBlQ=">AAACpXicfVFLaxsxEJa3r3T7ctpjL6YONKXG7JaEpoWWQC+9hKYQO4HIGK12EqvWK9JsEiP0b/Jrck0u/Tdduz50HeiHNDPMzPcxzBRWCo9Z9ruV3Lv/4OGjtcfpk6fPnr9or78celM5DgNupHFHBfMghYYBCpRwZB0wVUg4LKbf5vXDc3BeGH2AMwsjxU61OBGcYZ0at7/ShUZwUEaKcImhsiVDKDu1ZZ8jpek/He+z/vbW5vY72qNOdfZgOG53s362QOdukC+DLllif7ze+kFLwysFGrlk3h/nmcVRYA4FlxBTWnmwjE/ZKRxXeLIzCkLbCkHzlZoHzRT4UViM1ygGpryfqaJXe8VwMvc4Ub2ifotvZNkkFMZMkRW+pyqJwpmLFb3yV+WxU5jLZl54g8Y2pw52MvOC+5imVMMFN0oxXQZ6Dmcx0Pk4DmSgzmC95loxfMpi2PiyEWNcIej/MureszmpPkG+uvC7wfBDP8/6+c+t7u7O8hhr5DV5QzZJTj6SXfKd7JMB4eSKXJMbcpu8TfaSg2T4tzVpLTmvSAPJ+A+RttaM</latexit>

T. Gogami et al., Phys. Rev. C 93, 034314 (2016)

!14


0

5

10

15

20

25

30

0.0 0.1 0.2 0.3 0.4 0.5

208

89

40

28

1612

9

76

54

3

spdfg

BΛ

(M

eV)

A−2/3

emulsion

(K−,π−

)

(π+,K

+)

(e,e’K+)

AFDMC

0

5

10

15

20

25

30

0.0 0.1 0.2 0.3 0.4 0.5

Results: Hypernuclei


!14


0

5

10

15

20

25

30

0.0 0.1 0.2 0.3 0.4 0.5

208

89

40

28

1612

9

76

54

3

spdfg

BΛ

(M

eV)

A−2/3

emulsion

(K−,π−

)

(π+,K

+)

(e,e’K+)

AFDMC

0

5

10

15

20

25

30

0.0 0.1 0.2 0.3 0.4 0.5


AFDMC: ⇤-hypernucleiA⇤Z (J⇡, T ) B⇤ Exp

3⇤H

⇣12

+, 0⌘

0.23(9) 0.13(5)

4⇤H

�0+, 1

2

�1.95(9) 2.16(8)

4⇤He

�0+, 1

2

�2.37(9) 2.39(3)

5⇤He

⇣12

+, 0⌘

2.60(6) 3.12(2)<latexit sha1_base64="+uoQ6KwUAkYPl1nrzbFMRODT2OQ=">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</latexit><latexit sha1_base64="+uoQ6KwUAkYPl1nrzbFMRODT2OQ=">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</latexit><latexit sha1_base64="+uoQ6KwUAkYPl1nrzbFMRODT2OQ=">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</latexit><latexit sha1_base64="+uoQ6KwUAkYPl1nrzbFMRODT2OQ=">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</latexit>

old

spin-independent

CSB V⇤N<latexit sha1_base64="/CFddgzwSxoORcTyyFjUeXLrc8k=">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</latexit><latexit sha1_base64="/CFddgzwSxoORcTyyFjUeXLrc8k=">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</latexit><latexit sha1_base64="/CFddgzwSxoORcTyyFjUeXLrc8k=">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</latexit><latexit sha1_base64="/CFddgzwSxoORcTyyFjUeXLrc8k=">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</latexit>

{<latexit sha1_base64="EZryA9Zd7LjI+PQTjHOwlj0hmg8=">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</latexit><latexit sha1_base64="EZryA9Zd7LjI+PQTjHOwlj0hmg8=">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</latexit><latexit sha1_base64="EZryA9Zd7LjI+PQTjHOwlj0hmg8=">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</latexit><latexit sha1_base64="EZryA9Zd7LjI+PQTjHOwlj0hmg8=">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</latexit>

2.04(4)<latexit sha1_base64="b3mzmMkVl8mRZBLx95cQtc1NO1A=">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</latexit><latexit sha1_base64="b3mzmMkVl8mRZBLx95cQtc1NO1A=">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</latexit><latexit sha1_base64="b3mzmMkVl8mRZBLx95cQtc1NO1A=">AAACeXicfVFNTxsxEHWWloa0hUCPXFLCIa2iaDdEKhwqReLCjVRqAImNIts7EBN/LPYsEFn7V3otf4nfwoVNyGWDxGiseZqZ9/TkYakUDsPwqRKsffi4/qm6Ufv85evmVn1758yZzHIYciONvWDUgRQahihQwkVqgSom4ZxNj+fz8zuwThj9F2cpjBS91uJKcIpFa1zfiRca3kKSdzthr9X7Ma43w064iMZbEC1BkyxjMN6unMaJ4ZkCjVxS5y6jMMWRpxYFl5DX4sxBSvmUXsNlhleHIy90miFovjJzoKkCN/ILU6Whp8q5mWLtoiqKk3nFiWqzIhfPyKRMYMZMkTLXVplEYc39il5ykzlsMPNQ7gtn0KRl1z6dzJzgLq/VYg333ChFdeLjO7jNfTy3Y0H62BqkCIWiPwpzv/97P8/zFYJ+l1Hs3s5JxQmi1Q9/C866nSjsRH96zf7h8hhVskv2SItE5BfpkxMyIEPCyQP5R/6Tx8pz8D1oBT9fV4PKkvONlCI4eAFwJsVV</latexit><latexit sha1_base64="b3mzmMkVl8mRZBLx95cQtc1NO1A=">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</latexit>

!16

0

5

10

15

20

25

30

0.0 0.1 0.2 0.3 0.4 0.5

208

89

40

28

1612

9

76

54

3

spdfg

BΛ

(M

eV)

A−2/3

emulsion

(K−,π−

)

(π+,K

+)

(e,e’K+)

AFDMC

0

5

10

15

20

25

30

0.0 0.1 0.2 0.3 0.4 0.5

fit to s-orbit separation energy only




!17

AFDMC

Exp



A⇤Z (J⇡, T ) s-orbit p-orbit d-orbit

40⇤K

�2+, 1

2

�18.6(3)

�3�, 1

2

�11.0(3)

�4+, 1

2

�3.9(3)

41⇤Ca

⇣12

+, 0⌘18.3(4)

⇣32

�, 0⌘11.5(4)

⇣52

+, 0⌘4.3(4)

45⇤Ca

⇣12

+, 2⌘19.0(8)

⇣32

�, 2⌘12.1(8)

⇣52

+, 2⌘4.6(8)

48⇤K

�1+, 9

2

�20.4(4)

�2�, 9

2

�12.9(4)

�3+, 9

2

�5.8(4)

49⇤Ca

⇣12

+, 4⌘21.0(5)

⇣32

�, 4⌘12.9(9)

⇣52

+, 4⌘6.5(7)

40⇤Ca

�2+, 1

2

�19.3(1.1)

�3�, 1

2

�11.0(5)

�4+, 1

2

�1.0(5)

51⇤V (?, 2) 21.5(6) (?, 2) 13.4(6) (?, 2) 5.1(6)

52⇤V

�3�, 5

2

�21.9(7) — —

�4�, 5

2

�21.9(7) — —

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!17

AFDMC

Exp




40⇤K

�2+, 1

2

�18.6(3)

�3�, 1

2

�11.0(3)

�4+, 1

2

�3.9(3)

41⇤Ca

⇣12

+, 0⌘18.3(4)

⇣32

�, 0⌘11.5(4)

⇣52

+, 0⌘4.3(4)

45⇤Ca

⇣12

+, 2⌘19.0(8)

⇣32

�, 2⌘12.1(8)

⇣52

+, 2⌘4.6(8)

48⇤K

�1+, 9

2

�20.4(4)

�2�, 9

2

�12.9(4)

�3+, 9

2

�5.8(4)

49⇤Ca

⇣12

+, 4⌘21.0(5)

⇣32

�, 4⌘12.9(9)

⇣52

+, 4⌘6.5(7)

40⇤Ca

�2+, 1

2

�19.3(1.1)

�3�, 1

2

�11.0(5)

�4+, 1

2

�1.0(5)

51⇤V (?, 2) 21.5(6) (?, 2) 13.4(6) (?, 2) 5.1(6)

52⇤V

�3�, 5

2

�21.9(7) — —

�4�, 5

2

�21.9(7) — —



P. H. Pile et al., Phys. Rev. Lett 66, 2585 (1991)

VOLUME 66, NUMBER 20 PHYSICAL REVIEW LETTERS 20 MAY 1991

plets is small. Because the A hyperon is uninhibited by the Pauli principle, it is free to occupy any shell-model orbital,and it is therefore an unsurpassed probe of nuclear single-particie states.The experiment described here was performed using the low-energy separated beam (LESB1) at the Brookhaven Na-

1 I I I I I I I 5 I I I 8 I I I I I I I I I I I I I I I

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12A

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0-BE (MeV)

I 1 I

10

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G

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0 10

I" IG. 1. The least-squares-htted spectra obtained in this experiment for targets of Be, C, 6, Si, Ca, and V. The cross sectionsshown refer to the laboratory frame and are plotted against the negative of the A-particle binding energy.

!18


40⇤K

�2+, 1

2

�18.6(3)

�3�, 1

2

�11.0(3)

�4+, 1

2

�3.9(3)

41⇤Ca

⇣12

+, 0⌘18.3(4)

⇣32

�, 0⌘11.5(4)

⇣52

+, 0⌘4.3(4)

45⇤Ca

⇣12

+, 2⌘19.0(8)

⇣32

�, 2⌘12.1(8)

⇣52

+, 2⌘4.6(8)

48⇤K

�1+, 9

2

�20.4(4)

�2�, 9

2

�12.9(4)

�3+, 9

2

�5.8(4)

49⇤Ca

⇣12

+, 4⌘21.0(5)

⇣32

�, 4⌘12.9(9)

⇣52

+, 4⌘6.5(7)

40⇤Ca

�2+, 1

2

�19.3(1.1)

�3�, 1

2

�11.0(5)

�4+, 1

2

�1.0(5)

51⇤V (?, 2) 21.5(6) (?, 2) 13.4(6) (?, 2) 5.1(6)

52⇤V

�3�, 5

2

�21.9(7) — —

�4�, 5

2

�21.9(7) — —



P. Bydžovský et al. / Nuclear Physics A 881 (2012) 199–217 213

Table 5Partial list of differential cross sections (in µb/sr) utilized at the 40Ca target.

[p-hole] sΛ1/2 pΛ

3/2 pΛ1/2 dΛ

5/2 dΛ3/2 1sΛ

1/2 f Λ7/2 1pΛ

Sum f Λ5/2

(0d−13/2) J = 0 – 0.0 – – 0.0 – – 0.1 –

J = 1 13.7 8.1 2.5 6.6 19.7 0.9 – 43.0 2.7J = 2 41.0 15.3 31.4 30.2 0.1 2.7 2.7 7.3 24.4J = 3 – 90.8 – 27.2 47.1 – 50.6 2.5 2.5J = 4 – – – 108.1 – – 36.9 – 42.0J = 5 – – – – – – 86.3 – –

(1s−11/2) J = 0 0.2 – 0.0 – – 0.0 – 0.0 –

J = 1 63.0 11.2 22.2 – 0.9 16.7 – 3.0 –J = 2 – 33.3 – 1.7 2.7 – – 2.9 5.3J = 3 – – – 3.5 – – 9.0 – 11.7J = 4 – – – – – – 14.9 – –

Fig. 6. Excitation function for the 40Ca(γ ,K+)40ΛK reaction calculated in DWIA at Eγ = 1.3 GeV and θL

K = 3◦ using theSaclay–Lyon A model. For simplicity to draw pronounced doublet peaks, the artificial spin–orbit splitting is introducedas 0.17(2l+ 1) in MeV. In actual case such multiplet may be seen as a degenerate one. The hypernuclear energy ismeasured from the 39K(g.s.) + Λ threshold, so it is expressed in terms of the hyperon energy EΛ .

stand that the series of major peaks is based on the conversion of 0f7/2 protons into a Λ particlesitting in the s-, p-, d-, and f -orbits. In fact, the dominant peaks are due to the unnatural parity[f −1

7/2jΛ> ]J=Jmax states with Jmax = j> + jΛ

> = lp + lΛ +1 = Lmax +1 where jΛ> = sΛ

1/2(J = 4−),pΛ

3/2(J = 5+), dΛ5/2(J = 6−), and f Λ

7/2(J = 7+), respectively. On the other hand, the Λ spin–

orbit partner states [f −17/2j

Λ< ]J=Lmax (J = 3−,4+,5−,6+) have about 60% of the production rate

of the corresponding biggest peak within each multiplet. The spin–orbit splitting is expected tobe of the order of 0.8 MeV for dΛ

5/2,3/2 based on the experimental data from 13ΛC. Therefore,

if the energy resolution is good enough, there will be a chance to get direct information on thespin–orbit splitting in this medium-mass region. It is interesting to note that another series of wellseparated peaks are obtained based on the proton d3/2 hole. It should be also mentioned that, ifwe improve the description of the 51V nuclear excited states, we will get very weakly excitedside peaks among the strong peaks shown here.

P. Bydžovský et al., Nucl. Phys. A 881 (2012) 199-217

Bs⇤ ' 18.0MeV Bp

⇤ ' 10.7MeV Bd⇤ ' 3.3MeV

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?<latexit sha1_base64="c+vX7aN3Xg+b6kN8PEm3/vQU088=">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</latexit><latexit sha1_base64="c+vX7aN3Xg+b6kN8PEm3/vQU088=">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</latexit><latexit sha1_base64="c+vX7aN3Xg+b6kN8PEm3/vQU088=">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</latexit><latexit sha1_base64="c+vX7aN3Xg+b6kN8PEm3/vQU088=">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</latexit>

!19


40⇤K

�2+, 1

2

�18.6(3)

�3�, 1

2

�11.0(3)

�4+, 1

2

�3.9(3)

41⇤Ca

⇣12

+, 0⌘18.3(4)

⇣32

�, 0⌘11.5(4)

⇣52

+, 0⌘4.3(4)

45⇤Ca

⇣12

+, 2⌘19.0(8)

⇣32

�, 2⌘12.1(8)

⇣52

+, 2⌘4.6(8)

48⇤K

�1+, 9

2

�20.4(4)

�2�, 9

2

�12.9(4)

�3+, 9

2

�5.8(4)

49⇤Ca

⇣12

+, 4⌘21.0(5)

⇣32

�, 4⌘12.9(9)

⇣52

+, 4⌘6.5(7)

40⇤Ca

�2+, 1

2

�19.3(1.1)

�3�, 1

2

�11.0(5)

�4+, 1

2

�1.0(5)

51⇤V (?, 2) 21.5(6) (?, 2) 13.4(6) (?, 2) 5.1(6)

52⇤V

�3�, 5

2

�21.9(7) — —

�4�, 5

2

�21.9(7) — —



E12-15-008

48Ca�e, e0K+

�48⇤K

� = 0.188

40Ca�e, e0K+

�40⇤K

� = 0.025

� =N � Z

A<latexit sha1_base64="q3CAyxIUyq1i6JjRqoBwAJfsbHY=">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</latexit><latexit sha1_base64="q3CAyxIUyq1i6JjRqoBwAJfsbHY=">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</latexit><latexit sha1_base64="q3CAyxIUyq1i6JjRqoBwAJfsbHY=">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</latexit><latexit sha1_base64="q3CAyxIUyq1i6JjRqoBwAJfsbHY=">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</latexit>

nucleon-isospin asymmetry:

AFDMC



!20


12

15

18

21

24

17ΛO

41ΛCa

49ΛCa

BΛ

(M

eV)

CT = −2.0

CT = −1.0

CT = 0.0

CT = 0.5

12

15

18

21

24

17ΛO

41ΛCa

49ΛCa

n

p

⇤n

n

⇤p

p

⇤

TN = 0, 1 TN = 1

n

p

⇤n

n

⇤p

p

⇤

TN = 0, 1 TN = 1

control parameter CT<latexit sha1_base64="ZdD8urQ2yjYUQ9F6zkwzxWbQGkk=">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</latexit><latexit sha1_base64="ZdD8urQ2yjYUQ9F6zkwzxWbQGkk=">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</latexit><latexit sha1_base64="ZdD8urQ2yjYUQ9F6zkwzxWbQGkk=">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</latexit><latexit sha1_base64="ZdD8urQ2yjYUQ9F6zkwzxWbQGkk=">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</latexit>


is AFDMC sensitive to a

nucleon-isospin dependence

of the ⇤NN force?<latexit sha1_base64="PrlSoCAgIWbtmp591Md6Tm/yWEI=">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</latexit><latexit sha1_base64="PrlSoCAgIWbtmp591Md6Tm/yWEI=">AAAC1XicfVFNb9NAEF27BYr5SuHIZUUqxCFEdoVEOSCKihAHKEUiTaU6itbrSbPEu+t6x2mj1d4QV/4cF/4KJzapDziVGM1qRjPvjd7OZGUhDMbx7yDc2Lxx89bW7ejO3Xv3H3S2Hx4bXVccBlwXujrJmIFCKBigwAJOygqYzAoYZrODZX84h8oIrb7iooSRZGdKTARn6EvjjnqaIlyiFYa+ff/u0wE1oIxAMQeKmjKXplGDUDUvQKvnwmhTCkVzKEHloDj8A9ITilOgLv3oJeSMHh5e1elEe71v3LjTjfvxyuj1JGmSLmnsaLwdfE5zzWsJCnnBjDlN4hJHllUovB4XpbWBkvEZO4PTGid7IytUWaPXtdbzH2MSzMiultZqWiaNWcis56NkOF1GnMpe5n31dJG3CZnWM2SZ6cm6QFHpi7V5+bfaIM30Zbvud4e6bKu25XRhBDcuilIFF1xLyVRu0zmcO5su5VRQ2LTSyBD8RPsqdnbn9Y5zbo2g/svw2PMlyZ8gWV/49eR4t5/E/eTLi+7+XnOMLfKYPCHPSEJekn3ygRyRAeHkF/kTbASb4TB04ffwxxU0DBrOI9Ky8OdfiMnolg==</latexit><latexit sha1_base64="PrlSoCAgIWbtmp591Md6Tm/yWEI=">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</latexit><latexit sha1_base64="PrlSoCAgIWbtmp591Md6Tm/yWEI=">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</latexit>

E12-15-008


!21

AFDMC fit


D. L. et al., Phys. Rev. C 89, 014314 (2014) D. L. and F. Pederiva, arXiv:1711.07521

original parametrization

0

5

10

15

20

25

30

0.0 0.1 0.2 0.3 0.4 0.5

BΛ

(M

eV)

A−2/3

emulsion

(K−,π−

)

(π+,K

+)

(e,e’K+)

AFDMC ΛNN (I)

AFDMC ΛNN (II)

0

5

10

15

20

25

30

0.0 0.1 0.2 0.3 0.4 0.5


!22

2.45(1)M�

1.36(5)M�

2.09(1)M�

D. L. et al., Phys. Rev. Lett. 114, 092301 (2015)

AFDMC: ⇤-hypermatter

⇢th⇤ = 0.34(1) fm�3

⇢th⇤ > 0.56 fm�3

Results: Neutron Star Matter

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

11 12 13 14 15

PNM

ΛN + ΛNN (I)

ΛN + ΛNN (II)

M (

M0)

R (km)

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

11 12 13 14 15

PSR J1614−2230

PSR J0348+0432

!23Summary

‣ Quantum Monte Carlo calculations are feasible for medium-mass hypernuclei, and the accuracy of numerical predictions exceeds that of currently available experimental data.

‣ Phenomenological potentials developed in a pion exchange model with nucleons and lambdas as explicit degrees of freedom can describe the ground-state physics of hypernuclei in a wide mass range and for different hyperon orbits.

‣ A sensitivity study indicates the possibility to investigate the nucleon-isospin dependence of the three-body hyperon-nucleon-nucleon force in the medium-mass region of the hypernuclear chart using QMC methods.

‣ Future hypernuclear experiments on medium-mass targets will provide an opportunity to develop more accurate many-body hypernuclear interactions, crucial for the prediction of neutron star properties and for the solution of the hyperon puzzle.

Thank you!!

strangeness in nuclei and neutron stars: many-body forces ... · strangeness in nuclei and neutron...

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