Nuclear Energy Density Functional for Heavy Ion …sirius.unist.ac.kr/SRC-CHEA/seminar/2019+chea+ppt/...Nuclear Energy Density Functional for Heavy Ion Collisions and Neutron Stars

Nuclear Energy Density Functional for Heavy Ion Collisions and

Neutron Stars

Young-Min Kim (UNIST)

in collaboration with

S. Jeon (McGill Univ.), Y. Kim (IBS), M. K. Kim, Chang-Hwan Lee (Pusan Nat’l Univ.),

Kyujin Kwak (UNIST), Yeunhwan Lim (Texas A&M), Chang Ho Hyun (Daegu Univ.), H. Gil (KNU)

1

CHEA-20190116 @ 경주

Motivation - Neutron Star

2

Phenomena:GW, XRB, etc.

Compositions?Interactions?

[Credit: NSF/LIGO/Sonoma State University/A. Simonnet]

CHEA-20190116 @ 경주

Gandolfi and Steiner (2016)

Motivation - Nuclear Symmetry Energy

3

CHEA-20190116 @ 경주

Motivation - Nuclear Symmetry Energy

4taken from Myungkuk Kim’s KPS slides

중이온활용과제 연차 워크샵@ 2018.12.20

Symmetry Energy from GW observation

5PhysRevLett.119.161101

GW170817

[D] Y.Lim and J. Holt, PRL.121.062701 (arXiv:1803.02803v2)

(isospin-asymmetry energy slope parameter)

Abbott et al. (LSC and Virgo), PhysRevLett.121.161101

CHEA-20190116 @ 경주

Connections to RAON/LAMPS and LIGO

6

RAON/LAMPS

DJBUU: HIC simulation,pion production

Astrophysical Phenomena: GW (LIGO)

Nuclear Symmetry Energy

CHEA-20190116 @ 경주


Theoretical Connections

7

Skyrme force: KIDS

DJBUU: HIC simulation,

pion production, etc.

GW Obs. : NS tidal deformability

RMF: σ-ω-ρ model

CHEA-20190116 @ 경주



7

Skyrme force: KIDS




RMF: σ-ω-ρ model

CHEA-20190116 @ 경주



7

Skyrme force: KIDS




RMF: σ-ω-ρ model

CHEA-20190116 @ 경주

KIDS (Korea: IBS-Daegu-Sungkyunkwan)

8

at zero temperature

Phys. Rev. C 97, 014312 (2018)

ci(�) = ↵i + �i�2

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E(⇢, �) = E(⇢, �)

A= T (⇢, �) + ⌃N�1

i=0 ci(�)⇢1+i/3

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KIDS Energy density functional form

kF = (3⇡2⇢/2)1/3<latexit sha1_base64="ZirpBCPxNTMk+Q3RGBRvey3B8jY=">AAACBHicbVDLSsNAFJ34rPUVddnNYBHqpk1aQTdCURCXFewDmjRMptN26GQSZiZCCVm48VfcuFDErR/hzr9x2mahrQcuHM65l3vv8SNGpbKsb2NldW19YzO3ld/e2d3bNw8OWzKMBSZNHLJQdHwkCaOcNBVVjHQiQVDgM9L2x9dTv/1AhKQhv1eTiLgBGnI6oBgpLXlmYewlN+llqeZEtFeFjhiFleppL7ErtdQzi1bZmgEuEzsjRZCh4ZlfTj/EcUC4wgxJ2bWtSLkJEopiRtK8E0sSITxGQ9LVlKOASDeZPZHCE6304SAUuriCM/X3RIICKSeBrzsDpEZy0ZuK/3ndWA0u3ITyKFaE4/miQcygCuE0EdingmDFJpogLKi+FeIREggrnVteh2AvvrxMWtWybZXtu7Ni/SqLIwcK4BiUgA3OQR3cggZoAgwewTN4BW/Gk/FivBsf89YVI5s5An9gfP4A7vuWVg==</latexit><latexit sha1_base64="ZirpBCPxNTMk+Q3RGBRvey3B8jY=">AAACBHicbVDLSsNAFJ34rPUVddnNYBHqpk1aQTdCURCXFewDmjRMptN26GQSZiZCCVm48VfcuFDErR/hzr9x2mahrQcuHM65l3vv8SNGpbKsb2NldW19YzO3ld/e2d3bNw8OWzKMBSZNHLJQdHwkCaOcNBVVjHQiQVDgM9L2x9dTv/1AhKQhv1eTiLgBGnI6oBgpLXlmYewlN+llqeZEtFeFjhiFleppL7ErtdQzi1bZmgEuEzsjRZCh4ZlfTj/EcUC4wgxJ2bWtSLkJEopiRtK8E0sSITxGQ9LVlKOASDeZPZHCE6304SAUuriCM/X3RIICKSeBrzsDpEZy0ZuK/3ndWA0u3ITyKFaE4/miQcygCuE0EdingmDFJpogLKi+FeIREggrnVteh2AvvrxMWtWybZXtu7Ni/SqLIwcK4BiUgA3OQR3cggZoAgwewTN4BW/Gk/FivBsf89YVI5s5An9gfP4A7vuWVg==</latexit><latexit sha1_base64="ZirpBCPxNTMk+Q3RGBRvey3B8jY=">AAACBHicbVDLSsNAFJ34rPUVddnNYBHqpk1aQTdCURCXFewDmjRMptN26GQSZiZCCVm48VfcuFDErR/hzr9x2mahrQcuHM65l3vv8SNGpbKsb2NldW19YzO3ld/e2d3bNw8OWzKMBSZNHLJQdHwkCaOcNBVVjHQiQVDgM9L2x9dTv/1AhKQhv1eTiLgBGnI6oBgpLXlmYewlN+llqeZEtFeFjhiFleppL7ErtdQzi1bZmgEuEzsjRZCh4ZlfTj/EcUC4wgxJ2bWtSLkJEopiRtK8E0sSITxGQ9LVlKOASDeZPZHCE6304SAUuriCM/X3RIICKSeBrzsDpEZy0ZuK/3ndWA0u3ITyKFaE4/miQcygCuE0EdingmDFJpogLKi+FeIREggrnVteh2AvvrxMWtWybZXtu7Ni/SqLIwcK4BiUgA3OQR3cggZoAgwewTN4BW/Gk/FivBsf89YVI5s5An9gfP4A7vuWVg==</latexit><latexit sha1_base64="ZirpBCPxNTMk+Q3RGBRvey3B8jY=">AAACBHicbVDLSsNAFJ34rPUVddnNYBHqpk1aQTdCURCXFewDmjRMptN26GQSZiZCCVm48VfcuFDErR/hzr9x2mahrQcuHM65l3vv8SNGpbKsb2NldW19YzO3ld/e2d3bNw8OWzKMBSZNHLJQdHwkCaOcNBVVjHQiQVDgM9L2x9dTv/1AhKQhv1eTiLgBGnI6oBgpLXlmYewlN+llqeZEtFeFjhiFleppL7ErtdQzi1bZmgEuEzsjRZCh4ZlfTj/EcUC4wgxJ2bWtSLkJEopiRtK8E0sSITxGQ9LVlKOASDeZPZHCE6304SAUuriCM/X3RIICKSeBrzsDpEZy0ZuK/3ndWA0u3ITyKFaE4/miQcygCuE0EdingmDFJpogLKi+FeIREggrnVteh2AvvrxMWtWybZXtu7Ni/SqLIwcK4BiUgA3OQR3cggZoAgwewTN4BW/Gk/FivBsf89YVI5s5An9gfP4A7vuWVg==</latexit>

to be determined by fitting to the observables

kF⌧ = kF (1 + ⌧�)1/3<latexit sha1_base64="IVhAraebw7M0DYBqivOdaMosuFw=">AAACEHicbVDLSgMxFM3UV62vqks3wSJWhDqjgm6EoiAuK9gHtHXIZDJtaOZBckcow3yCG3/FjQtF3Lp059+YabvQ1gOBc8+5l5t7nEhwBab5beTm5hcWl/LLhZXVtfWN4uZWQ4WxpKxOQxHKlkMUEzxgdeAgWCuSjPiOYE1ncJX5zQcmFQ+DOxhGrOuTXsA9TgloyS7uD+zk2k46QOI0vciKtGwdZiXuuEwAObhPrKOT1C6WzIo5Ap4l1oSU0AQ1u/jVcUMa+ywAKohSbcuMoJsQCZwKlhY6sWIRoQPSY21NA+Iz1U1GB6V4Tysu9kKpXwB4pP6eSIiv1NB3dKdPoK+mvUz8z2vH4J13Ex5EMbCAjhd5scAQ4iwd7HLJKIihJoRKrv+KaZ9IQkFnWNAhWNMnz5LGccUyK9btaal6OYkjj3bQLiojC52hKrpBNVRHFD2iZ/SK3own48V4Nz7GrTljMrON/sD4/AGOUJw9</latexit><latexit sha1_base64="IVhAraebw7M0DYBqivOdaMosuFw=">AAACEHicbVDLSgMxFM3UV62vqks3wSJWhDqjgm6EoiAuK9gHtHXIZDJtaOZBckcow3yCG3/FjQtF3Lp059+YabvQ1gOBc8+5l5t7nEhwBab5beTm5hcWl/LLhZXVtfWN4uZWQ4WxpKxOQxHKlkMUEzxgdeAgWCuSjPiOYE1ncJX5zQcmFQ+DOxhGrOuTXsA9TgloyS7uD+zk2k46QOI0vciKtGwdZiXuuEwAObhPrKOT1C6WzIo5Ap4l1oSU0AQ1u/jVcUMa+ywAKohSbcuMoJsQCZwKlhY6sWIRoQPSY21NA+Iz1U1GB6V4Tysu9kKpXwB4pP6eSIiv1NB3dKdPoK+mvUz8z2vH4J13Ex5EMbCAjhd5scAQ4iwd7HLJKIihJoRKrv+KaZ9IQkFnWNAhWNMnz5LGccUyK9btaal6OYkjj3bQLiojC52hKrpBNVRHFD2iZ/SK3own48V4Nz7GrTljMrON/sD4/AGOUJw9</latexit><latexit sha1_base64="IVhAraebw7M0DYBqivOdaMosuFw=">AAACEHicbVDLSgMxFM3UV62vqks3wSJWhDqjgm6EoiAuK9gHtHXIZDJtaOZBckcow3yCG3/FjQtF3Lp059+YabvQ1gOBc8+5l5t7nEhwBab5beTm5hcWl/LLhZXVtfWN4uZWQ4WxpKxOQxHKlkMUEzxgdeAgWCuSjPiOYE1ncJX5zQcmFQ+DOxhGrOuTXsA9TgloyS7uD+zk2k46QOI0vciKtGwdZiXuuEwAObhPrKOT1C6WzIo5Ap4l1oSU0AQ1u/jVcUMa+ywAKohSbcuMoJsQCZwKlhY6sWIRoQPSY21NA+Iz1U1GB6V4Tysu9kKpXwB4pP6eSIiv1NB3dKdPoK+mvUz8z2vH4J13Ex5EMbCAjhd5scAQ4iwd7HLJKIihJoRKrv+KaZ9IQkFnWNAhWNMnz5LGccUyK9btaal6OYkjj3bQLiojC52hKrpBNVRHFD2iZ/SK3own48V4Nz7GrTljMrON/sD4/AGOUJw9</latexit><latexit sha1_base64="IVhAraebw7M0DYBqivOdaMosuFw=">AAACEHicbVDLSgMxFM3UV62vqks3wSJWhDqjgm6EoiAuK9gHtHXIZDJtaOZBckcow3yCG3/FjQtF3Lp059+YabvQ1gOBc8+5l5t7nEhwBab5beTm5hcWl/LLhZXVtfWN4uZWQ4WxpKxOQxHKlkMUEzxgdeAgWCuSjPiOYE1ncJX5zQcmFQ+DOxhGrOuTXsA9TgloyS7uD+zk2k46QOI0vciKtGwdZiXuuEwAObhPrKOT1C6WzIo5Ap4l1oSU0AQ1u/jVcUMa+ywAKohSbcuMoJsQCZwKlhY6sWIRoQPSY21NA+Iz1U1GB6V4Tysu9kKpXwB4pP6eSIiv1NB3dKdPoK+mvUz8z2vH4J13Ex5EMbCAjhd5scAQ4iwd7HLJKIihJoRKrv+KaZ9IQkFnWNAhWNMnz5LGccUyK9btaal6OYkjj3bQLiojC52hKrpBNVRHFD2iZ/SK3own48V4Nz7GrTljMrON/sD4/AGOUJw9</latexit>

� =⇢n � ⇢p

⇢<latexit sha1_base64="fPyXcxS7kskShT1NEZ4AAixiTzA=">AAACDHicbVDLSsNAFL3xWeur6tLNYBHcWBIRdCMU3bisYB/QhDKZTNqhk0mYmQgl5APc+CtuXCji1g9w5984TbPQ1gMDh3PO5c49fsKZ0rb9bS0tr6yurVc2qptb2zu7tb39jopTSWibxDyWPR8rypmgbc00p71EUhz5nHb98c3U7z5QqVgs7vUkoV6Eh4KFjGBtpEGt7gaUa3zlhhKTzJWjeCDQKSpIkhdCblJ2wy6AFolTkjqUaA1qX24QkzSiQhOOleo7dqK9DEvNCKd51U0VTTAZ4yHtGypwRJWXFcfk6NgoAQpjaZ7QqFB/T2Q4UmoS+SYZYT1S895U/M/rpzq89DImklRTQWaLwpQjHaNpMyhgkhLNJ4ZgIpn5KyIjbGrRpr+qKcGZP3mRdM4ajt1w7s7rzeuyjgocwhGcgAMX0IRbaEEbCDzCM7zCm/VkvVjv1scsumSVMwfwB9bnD/K2m4w=</latexit><latexit sha1_base64="fPyXcxS7kskShT1NEZ4AAixiTzA=">AAACDHicbVDLSsNAFL3xWeur6tLNYBHcWBIRdCMU3bisYB/QhDKZTNqhk0mYmQgl5APc+CtuXCji1g9w5984TbPQ1gMDh3PO5c49fsKZ0rb9bS0tr6yurVc2qptb2zu7tb39jopTSWibxDyWPR8rypmgbc00p71EUhz5nHb98c3U7z5QqVgs7vUkoV6Eh4KFjGBtpEGt7gaUa3zlhhKTzJWjeCDQKSpIkhdCblJ2wy6AFolTkjqUaA1qX24QkzSiQhOOleo7dqK9DEvNCKd51U0VTTAZ4yHtGypwRJWXFcfk6NgoAQpjaZ7QqFB/T2Q4UmoS+SYZYT1S895U/M/rpzq89DImklRTQWaLwpQjHaNpMyhgkhLNJ4ZgIpn5KyIjbGrRpr+qKcGZP3mRdM4ajt1w7s7rzeuyjgocwhGcgAMX0IRbaEEbCDzCM7zCm/VkvVjv1scsumSVMwfwB9bnD/K2m4w=</latexit><latexit sha1_base64="fPyXcxS7kskShT1NEZ4AAixiTzA=">AAACDHicbVDLSsNAFL3xWeur6tLNYBHcWBIRdCMU3bisYB/QhDKZTNqhk0mYmQgl5APc+CtuXCji1g9w5984TbPQ1gMDh3PO5c49fsKZ0rb9bS0tr6yurVc2qptb2zu7tb39jopTSWibxDyWPR8rypmgbc00p71EUhz5nHb98c3U7z5QqVgs7vUkoV6Eh4KFjGBtpEGt7gaUa3zlhhKTzJWjeCDQKSpIkhdCblJ2wy6AFolTkjqUaA1qX24QkzSiQhOOleo7dqK9DEvNCKd51U0VTTAZ4yHtGypwRJWXFcfk6NgoAQpjaZ7QqFB/T2Q4UmoS+SYZYT1S895U/M/rpzq89DImklRTQWaLwpQjHaNpMyhgkhLNJ4ZgIpn5KyIjbGrRpr+qKcGZP3mRdM4ajt1w7s7rzeuyjgocwhGcgAMX0IRbaEEbCDzCM7zCm/VkvVjv1scsumSVMwfwB9bnD/K2m4w=</latexit><latexit sha1_base64="fPyXcxS7kskShT1NEZ4AAixiTzA=">AAACDHicbVDLSsNAFL3xWeur6tLNYBHcWBIRdCMU3bisYB/QhDKZTNqhk0mYmQgl5APc+CtuXCji1g9w5984TbPQ1gMDh3PO5c49fsKZ0rb9bS0tr6yurVc2qptb2zu7tb39jopTSWibxDyWPR8rypmgbc00p71EUhz5nHb98c3U7z5QqVgs7vUkoV6Eh4KFjGBtpEGt7gaUa3zlhhKTzJWjeCDQKSpIkhdCblJ2wy6AFolTkjqUaA1qX24QkzSiQhOOleo7dqK9DEvNCKd51U0VTTAZ4yHtGypwRJWXFcfk6NgoAQpjaZ7QqFB/T2Q4UmoS+SYZYT1S895U/M/rpzq89DImklRTQWaLwpQjHaNpMyhgkhLNJ4ZgIpn5KyIjbGrRpr+qKcGZP3mRdM4ajt1w7s7rzeuyjgocwhGcgAMX0IRbaEEbCDzCM7zCm/VkvVjv1scsumSVMwfwB9bnD/K2m4w=</latexit>

CHEA-20190116 @ 경주

KIDS to Standard Skyrme parameters

9

G. Hana et al., arxiv:1805.11321

CHEA-20190116 @ 경주

Fitting to Observables for c_i(δ)

10

Ca

PHYSICAL REVIEW C 82, 024313 (2010)

CHEA-20190116 @ 경주

Selected EoSs

11

• Skyrme force models

• Basically fitted to properties of well-known nuclei

• Good saturation properties

• Mmax more than 2Msun• 11 experimental/empirical data for nuclear matter around

saturation density [Phys.Rev. C 85, 035201 (2012)]

Kim et al., PhysRevC.98.065805 (2018)

CHEA-20190116 @ 경주

KIDS for Neutron Star

12


GW170817 - M1

GW170817 - M2

J0348+0432

J1614-2230

GW170817 - Abbott et al. (LSC and Virgo), arxiv:1805.11579 - Mchirp = 1.188 M⦿

- low spin prior : M1=1.36~1.60 M , M2=1.16~1.36 M

- high spin prior : M1=1.36~1.89 M , M2=1.00~1.36 M

~3.3~2.7

CHEA-20190116 @ 경주

NS tidal deformability

13

[A] F.J. Fattoyev, J. Piekarewicz, and C.J. Horowitz, PRL.120.172702 (arXiv:1711.06615v2)[B] P.G. Krastev, and B.-A. Li, arXiv:1801.04620v1[C] E. Annala, T. Gorda, A. Kurkela, and A. Vuorinen, PRL.120.172703 (arXiv:1711.02644v2)[D] Y.Lim and J. Holt, PRL.121.062701 (arXiv:1803.02803v2)

Red line: Λ(1.4M⦿) = 2.88 * 10-6 (R/km)7.5 (fitting function in [C])


5 Skyrme force models satisfying 11 constraints from Dutra et al. Phys.Rev. C 85, 035201 (2012)

Abbott et al. (LSC and Virgo), PhysRevLett.121.161101 (2018)

EoS insensitive relations

CHEA-20190116 @ 경주

KIDS for Heavy Ion Collision

14

U⌧ (⇢, �) = ⌃3i=0Fi(�, ⌧)⇢

1+i/3<latexit sha1_base64="xx1/F8BWh+bP3UXRN/CewODaBhs=">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</latexit><latexit sha1_base64="xx1/F8BWh+bP3UXRN/CewODaBhs=">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</latexit><latexit sha1_base64="xx1/F8BWh+bP3UXRN/CewODaBhs=">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</latexit><latexit sha1_base64="xx1/F8BWh+bP3UXRN/CewODaBhs=">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</latexit>

Fi(�, ⌧) = ↵i(2 + i/3) + 2⌧�i� +i

3�i�

2

<latexit sha1_base64="HyA3gG99MUFYntc0j0QOxTXmFsQ=">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</latexit><latexit sha1_base64="HyA3gG99MUFYntc0j0QOxTXmFsQ=">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</latexit><latexit sha1_base64="HyA3gG99MUFYntc0j0QOxTXmFsQ=">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</latexit><latexit sha1_base64="HyA3gG99MUFYntc0j0QOxTXmFsQ=">AAACQnicbZBLSyNBFIWrfcxodMaoSzeFYSASyXTHAd0I4oC4dGDig3Rsbldum8LqB1W3hdDkt7mZXzA7f4CbWSji1oXVSRYa50DB4Xz3UlUnzJQ05Lp3zszs3PynzwuLlaXlL19XqqtrpybNtcC2SFWqz0MwqGSCbZKk8DzTCHGo8Cy8/lnysxvURqbJbxpk2I3hKpGRFEA2CqoXR4Gs+z1UBNs+Qb7F97kPKutDIHm91ZDfd7Z4g7d4CbkfIpVgvMAbfqRBFHJY7Ayn2GUrqNbcpjsS/2i8iamxiU6C6l+/l4o8xoSEAmM6nptRtwBNUigcVvzcYAbiGq6wY20CMZpuMapgyL/ZpMejVNuTEB+lbzcKiI0ZxKGdjIH6ZpqV4f9YJ6dor1vIJMsJEzG+KMoVp5SXffKe1ChIDawBoaV9Kxd9sLWQbb1iS/Cmv/zRnLaantv0fv2oHRxO6lhgG2yT1ZnHdtkBO2YnrM0Eu2X37IE9On+cf86T8zwenXEmO+vsnZyXV+9BrYo=</latexit>

⌧ = 1 for neutron

�1 for proton<latexit sha1_base64="Ff1adfNuia8BxCwJr0NMDjmIujU=">AAACH3icbVDLSgNBEJyNrxhfqx69DAbFi2FXRL0IQS8eI5gHZEOYnUySIbMzy0yvEJbkS7z4K148KCLe8jdOHgeTWNBQVHXT3RXGghvwvJGTWVldW9/Ibua2tnd299z9g4pRiaasTJVQuhYSwwSXrAwcBKvFmpEoFKwa9u7HfvWZacOVfIJ+zBoR6Uje5pSAlZruVQAkwbfYx6dpoCM8bCs9xJIloJUcBEHufN6JtQJrNN28V/AmwMvEn5E8mqHUdH+ClqJJxCRQQYyp+14MjZRo4FSwQS5IDIsJ7ZEOq1sqScRMI538N8AnVmlhu9+WBDxR/06kJDKmH4W2MyLQNYveWPzPqyfQvmmkXMYJMEmni9qJwKDwOCzc4ppREH1LCNXc3oppl2hCwUaasyH4iy8vk8pFwfcK/uNlvng3iyOLjtAxOkM+ukZF9IBKqIwoekFv6AN9Oq/Ou/PlfE9bM85s5hDNwRn9AjEZoeA=</latexit><latexit sha1_base64="Ff1adfNuia8BxCwJr0NMDjmIujU=">AAACH3icbVDLSgNBEJyNrxhfqx69DAbFi2FXRL0IQS8eI5gHZEOYnUySIbMzy0yvEJbkS7z4K148KCLe8jdOHgeTWNBQVHXT3RXGghvwvJGTWVldW9/Ibua2tnd299z9g4pRiaasTJVQuhYSwwSXrAwcBKvFmpEoFKwa9u7HfvWZacOVfIJ+zBoR6Uje5pSAlZruVQAkwbfYx6dpoCM8bCs9xJIloJUcBEHufN6JtQJrNN28V/AmwMvEn5E8mqHUdH+ClqJJxCRQQYyp+14MjZRo4FSwQS5IDIsJ7ZEOq1sqScRMI538N8AnVmlhu9+WBDxR/06kJDKmH4W2MyLQNYveWPzPqyfQvmmkXMYJMEmni9qJwKDwOCzc4ppREH1LCNXc3oppl2hCwUaasyH4iy8vk8pFwfcK/uNlvng3iyOLjtAxOkM+ukZF9IBKqIwoekFv6AN9Oq/Ou/PlfE9bM85s5hDNwRn9AjEZoeA=</latexit><latexit sha1_base64="Ff1adfNuia8BxCwJr0NMDjmIujU=">AAACH3icbVDLSgNBEJyNrxhfqx69DAbFi2FXRL0IQS8eI5gHZEOYnUySIbMzy0yvEJbkS7z4K148KCLe8jdOHgeTWNBQVHXT3RXGghvwvJGTWVldW9/Ibua2tnd299z9g4pRiaasTJVQuhYSwwSXrAwcBKvFmpEoFKwa9u7HfvWZacOVfIJ+zBoR6Uje5pSAlZruVQAkwbfYx6dpoCM8bCs9xJIloJUcBEHufN6JtQJrNN28V/AmwMvEn5E8mqHUdH+ClqJJxCRQQYyp+14MjZRo4FSwQS5IDIsJ7ZEOq1sqScRMI538N8AnVmlhu9+WBDxR/06kJDKmH4W2MyLQNYveWPzPqyfQvmmkXMYJMEmni9qJwKDwOCzc4ppREH1LCNXc3oppl2hCwUaasyH4iy8vk8pFwfcK/uNlvng3iyOLjtAxOkM+ukZF9IBKqIwoekFv6AN9Oq/Ou/PlfE9bM85s5hDNwRn9AjEZoeA=</latexit><latexit sha1_base64="Ff1adfNuia8BxCwJr0NMDjmIujU=">AAACH3icbVDLSgNBEJyNrxhfqx69DAbFi2FXRL0IQS8eI5gHZEOYnUySIbMzy0yvEJbkS7z4K148KCLe8jdOHgeTWNBQVHXT3RXGghvwvJGTWVldW9/Ibua2tnd299z9g4pRiaasTJVQuhYSwwSXrAwcBKvFmpEoFKwa9u7HfvWZacOVfIJ+zBoR6Uje5pSAlZruVQAkwbfYx6dpoCM8bCs9xJIloJUcBEHufN6JtQJrNN28V/AmwMvEn5E8mqHUdH+ClqJJxCRQQYyp+14MjZRo4FSwQS5IDIsJ7ZEOq1sqScRMI538N8AnVmlhu9+WBDxR/06kJDKmH4W2MyLQNYveWPzPqyfQvmmkXMYJMEmni9qJwKDwOCzc4ppREH1LCNXc3oppl2hCwUaasyH4iy8vk8pFwfcK/uNlvng3iyOLjtAxOkM+ukZF9IBKqIwoekFv6AN9Oq/Ou/PlfE9bM85s5hDNwRn9AjEZoeA=</latexit>

ci(�) = ↵i + �i�2

<latexit sha1_base64="WLSAqUkwbe0jQKN+Wpj/Uee//I0=">AAACE3icbZDJSgNBEIZ74h63qEcvjUFwgTATBL0IohePEUwUMjHU9NSYxp6F7hohhLyDF1/FiwdFvHrx5tvYWQ5qLGj4+P8qqusPMiUNue6XU5ianpmdm18oLi4tr6yW1tYbJs21wLpIVaqvAzCoZIJ1kqTwOtMIcaDwKrg7G/hX96iNTJNL6mbYiuE2kZEUQFZql/ZEW/IdP0RFsMuPuQ8q64DV9rkfIA1oZN5U26WyW3GHxSfBG0OZjavWLn36YSryGBMSCoxpem5GrR5okkJhv+jnBjMQd3CLTYsJxGhaveFNfb5tlZBHqbYvIT5Uf070IDamGwe2MwbqmL/eQPzPa+YUHbV6MslywkSMFkW54pTyQUA8lBoFqa4FEFrav3LRAQ2CbIxFG4L39+RJaFQrnlvxLg7KJ6fjOObZJttiO8xjh+yEnbMaqzPBHtgTe2GvzqPz7Lw576PWgjOe2WC/yvn4BsmMnDk=</latexit><latexit sha1_base64="WLSAqUkwbe0jQKN+Wpj/Uee//I0=">AAACE3icbZDJSgNBEIZ74h63qEcvjUFwgTATBL0IohePEUwUMjHU9NSYxp6F7hohhLyDF1/FiwdFvHrx5tvYWQ5qLGj4+P8qqusPMiUNue6XU5ianpmdm18oLi4tr6yW1tYbJs21wLpIVaqvAzCoZIJ1kqTwOtMIcaDwKrg7G/hX96iNTJNL6mbYiuE2kZEUQFZql/ZEW/IdP0RFsMuPuQ8q64DV9rkfIA1oZN5U26WyW3GHxSfBG0OZjavWLn36YSryGBMSCoxpem5GrR5okkJhv+jnBjMQd3CLTYsJxGhaveFNfb5tlZBHqbYvIT5Uf070IDamGwe2MwbqmL/eQPzPa+YUHbV6MslywkSMFkW54pTyQUA8lBoFqa4FEFrav3LRAQ2CbIxFG4L39+RJaFQrnlvxLg7KJ6fjOObZJttiO8xjh+yEnbMaqzPBHtgTe2GvzqPz7Lw576PWgjOe2WC/yvn4BsmMnDk=</latexit><latexit sha1_base64="WLSAqUkwbe0jQKN+Wpj/Uee//I0=">AAACE3icbZDJSgNBEIZ74h63qEcvjUFwgTATBL0IohePEUwUMjHU9NSYxp6F7hohhLyDF1/FiwdFvHrx5tvYWQ5qLGj4+P8qqusPMiUNue6XU5ianpmdm18oLi4tr6yW1tYbJs21wLpIVaqvAzCoZIJ1kqTwOtMIcaDwKrg7G/hX96iNTJNL6mbYiuE2kZEUQFZql/ZEW/IdP0RFsMuPuQ8q64DV9rkfIA1oZN5U26WyW3GHxSfBG0OZjavWLn36YSryGBMSCoxpem5GrR5okkJhv+jnBjMQd3CLTYsJxGhaveFNfb5tlZBHqbYvIT5Uf070IDamGwe2MwbqmL/eQPzPa+YUHbV6MslywkSMFkW54pTyQUA8lBoFqa4FEFrav3LRAQ2CbIxFG4L39+RJaFQrnlvxLg7KJ6fjOObZJttiO8xjh+yEnbMaqzPBHtgTe2GvzqPz7Lw576PWgjOe2WC/yvn4BsmMnDk=</latexit><latexit sha1_base64="WLSAqUkwbe0jQKN+Wpj/Uee//I0=">AAACE3icbZDJSgNBEIZ74h63qEcvjUFwgTATBL0IohePEUwUMjHU9NSYxp6F7hohhLyDF1/FiwdFvHrx5tvYWQ5qLGj4+P8qqusPMiUNue6XU5ianpmdm18oLi4tr6yW1tYbJs21wLpIVaqvAzCoZIJ1kqTwOtMIcaDwKrg7G/hX96iNTJNL6mbYiuE2kZEUQFZql/ZEW/IdP0RFsMuPuQ8q64DV9rkfIA1oZN5U26WyW3GHxSfBG0OZjavWLn36YSryGBMSCoxpem5GrR5okkJhv+jnBjMQd3CLTYsJxGhaveFNfb5tlZBHqbYvIT5Uf070IDamGwe2MwbqmL/eQPzPa+YUHbV6MslywkSMFkW54pTyQUA8lBoFqa4FEFrav3LRAQ2CbIxFG4L39+RJaFQrnlvxLg7KJ6fjOObZJttiO8xjh+yEnbMaqzPBHtgTe2GvzqPz7Lw576PWgjOe2WC/yvn4BsmMnDk=</latexit>

dp⌧

dt= �rU⌧

<latexit sha1_base64="0tcxOlx9wPXQC3oTRayMfdpqNCk=">AAACGXicbVBNS8NAEN34bf2qevSyWAQvlkQEvQiiF48VrApNCJPNRhc3m7A7EUrI3/DiX/HiQRGPevLfuG1z0NYHA4/3ZpiZF+VSGHTdb2dqemZ2bn5hsbG0vLK61lzfuDJZoRnvskxm+iYCw6VQvIsCJb/JNYc0kvw6uj8b+NcPXBuRqUvs5zxI4VaJRDBAK4VN1080sDKmpR9lMqZ5FZY+QlFVZYwVPaZ7voJIAu3WethsuW13CDpJvJq0SI1O2Pz044wVKVfIJBjT89wcgxI0CiZ51fALw3Ng93DLe5YqSLkJyuFnFd2xSkyTTNtSSIfq74kSUmP6aWQ7U8A7M+4NxP+8XoHJUVAKlRfIFRstSgpJMaODmGgsNGco+5YA08LeStkd2KjQhtmwIXjjL0+Sq/2257a9i4PWyWkdxwLZIttkl3jkkJyQc9IhXcLII3kmr+TNeXJenHfnY9Q65dQzm+QPnK8fzz6gxw==</latexit><latexit sha1_base64="0tcxOlx9wPXQC3oTRayMfdpqNCk=">AAACGXicbVBNS8NAEN34bf2qevSyWAQvlkQEvQiiF48VrApNCJPNRhc3m7A7EUrI3/DiX/HiQRGPevLfuG1z0NYHA4/3ZpiZF+VSGHTdb2dqemZ2bn5hsbG0vLK61lzfuDJZoRnvskxm+iYCw6VQvIsCJb/JNYc0kvw6uj8b+NcPXBuRqUvs5zxI4VaJRDBAK4VN1080sDKmpR9lMqZ5FZY+QlFVZYwVPaZ7voJIAu3WethsuW13CDpJvJq0SI1O2Pz044wVKVfIJBjT89wcgxI0CiZ51fALw3Ng93DLe5YqSLkJyuFnFd2xSkyTTNtSSIfq74kSUmP6aWQ7U8A7M+4NxP+8XoHJUVAKlRfIFRstSgpJMaODmGgsNGco+5YA08LeStkd2KjQhtmwIXjjL0+Sq/2257a9i4PWyWkdxwLZIttkl3jkkJyQc9IhXcLII3kmr+TNeXJenHfnY9Q65dQzm+QPnK8fzz6gxw==</latexit><latexit sha1_base64="0tcxOlx9wPXQC3oTRayMfdpqNCk=">AAACGXicbVBNS8NAEN34bf2qevSyWAQvlkQEvQiiF48VrApNCJPNRhc3m7A7EUrI3/DiX/HiQRGPevLfuG1z0NYHA4/3ZpiZF+VSGHTdb2dqemZ2bn5hsbG0vLK61lzfuDJZoRnvskxm+iYCw6VQvIsCJb/JNYc0kvw6uj8b+NcPXBuRqUvs5zxI4VaJRDBAK4VN1080sDKmpR9lMqZ5FZY+QlFVZYwVPaZ7voJIAu3WethsuW13CDpJvJq0SI1O2Pz044wVKVfIJBjT89wcgxI0CiZ51fALw3Ng93DLe5YqSLkJyuFnFd2xSkyTTNtSSIfq74kSUmP6aWQ7U8A7M+4NxP+8XoHJUVAKlRfIFRstSgpJMaODmGgsNGco+5YA08LeStkd2KjQhtmwIXjjL0+Sq/2257a9i4PWyWkdxwLZIttkl3jkkJyQc9IhXcLII3kmr+TNeXJenHfnY9Q65dQzm+QPnK8fzz6gxw==</latexit><latexit sha1_base64="0tcxOlx9wPXQC3oTRayMfdpqNCk=">AAACGXicbVBNS8NAEN34bf2qevSyWAQvlkQEvQiiF48VrApNCJPNRhc3m7A7EUrI3/DiX/HiQRGPevLfuG1z0NYHA4/3ZpiZF+VSGHTdb2dqemZ2bn5hsbG0vLK61lzfuDJZoRnvskxm+iYCw6VQvIsCJb/JNYc0kvw6uj8b+NcPXBuRqUvs5zxI4VaJRDBAK4VN1080sDKmpR9lMqZ5FZY+QlFVZYwVPaZ7voJIAu3WethsuW13CDpJvJq0SI1O2Pz044wVKVfIJBjT89wcgxI0CiZ51fALw3Ng93DLe5YqSLkJyuFnFd2xSkyTTNtSSIfq74kSUmP6aWQ7U8A7M+4NxP+8XoHJUVAKlRfIFRstSgpJMaODmGgsNGco+5YA08LeStkd2KjQhtmwIXjjL0+Sq/2257a9i4PWyWkdxwLZIttkl3jkkJyQc9IhXcLII3kmr+TNeXJenHfnY9Q65dQzm+QPnK8fzz6gxw==</latexit>

E(⇢, �) = E(⇢, �)

A= T (⇢, �) + ⌃N�1

i=0 ci(�)⇢1+i/3


t(k⌧ ) + U⌧ (⇢, �) =@⇢E(⇢, �)

@⇢⌧<latexit sha1_base64="mqJHXGa5T4WGCD2qYl9GytQkk94=">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</latexit><latexit sha1_base64="mqJHXGa5T4WGCD2qYl9GytQkk94=">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</latexit><latexit sha1_base64="mqJHXGa5T4WGCD2qYl9GytQkk94=">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</latexit><latexit sha1_base64="mqJHXGa5T4WGCD2qYl9GytQkk94=">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</latexit>

single nucleon potential

eq. of motion for nucleons

CHEA-20190116 @ 경주

Single nucleon potential

15

U⌧ (⇢, �) = ⌃3i=0Fi(�, ⌧)⇢

1+i/3<latexit sha1_base64="xx1/F8BWh+bP3UXRN/CewODaBhs=">AAACMHicbVDLSgNBEJz1bXxFPXoZDEJEibsq6CUgCupR0aiQjUvvZJIMmX0w0yuEZT/Ji5+iFwVFvPoVziY5+CoYKKq66OnyYyk02vaLNTI6Nj4xOTVdmJmdm18oLi5d6ShRjNdYJCN144PmUoS8hgIlv4kVh8CX/NrvHuX+9R1XWkThJfZi3gigHYqWYIBG8oonNS91EZKMll3ViTbdJpcI67RK3QvRDuA23cm8VFTtjB57ojywN/PEOs0Dt6mzIbbMTLFkV+w+6F/iDEmJDHHmFR/dZsSSgIfIJGhdd+wYGykoFEzyrOAmmsfAutDmdUNDCLhupP2DM7pmlCZtRcq8EGlf/Z5IIdC6F/hmMgDs6N9eLv7n1RNs7TdSEcYJ8pANFrUSSTGieXu0KRRnKHuGAFPC/JWyDihgaDoumBKc3yf/JVfbFceuOOe7pYPDYR1TZIWskjJxyB45IKfkjNQII/fkibySN+vBerberY/B6Ig1zCyTH7A+vwBO7Kfk</latexit><latexit sha1_base64="xx1/F8BWh+bP3UXRN/CewODaBhs=">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</latexit><latexit sha1_base64="xx1/F8BWh+bP3UXRN/CewODaBhs=">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</latexit><latexit sha1_base64="xx1/F8BWh+bP3UXRN/CewODaBhs=">AAACMHicbVDLSgNBEJz1bXxFPXoZDEJEibsq6CUgCupR0aiQjUvvZJIMmX0w0yuEZT/Ji5+iFwVFvPoVziY5+CoYKKq66OnyYyk02vaLNTI6Nj4xOTVdmJmdm18oLi5d6ShRjNdYJCN144PmUoS8hgIlv4kVh8CX/NrvHuX+9R1XWkThJfZi3gigHYqWYIBG8oonNS91EZKMll3ViTbdJpcI67RK3QvRDuA23cm8VFTtjB57ojywN/PEOs0Dt6mzIbbMTLFkV+w+6F/iDEmJDHHmFR/dZsSSgIfIJGhdd+wYGykoFEzyrOAmmsfAutDmdUNDCLhupP2DM7pmlCZtRcq8EGlf/Z5IIdC6F/hmMgDs6N9eLv7n1RNs7TdSEcYJ8pANFrUSSTGieXu0KRRnKHuGAFPC/JWyDihgaDoumBKc3yf/JVfbFceuOOe7pYPDYR1TZIWskjJxyB45IKfkjNQII/fkibySN+vBerberY/B6Ig1zCyTH7A+vwBO7Kfk</latexit>

Fi(�, ⌧) = ↵i(2 + i/3) + 2⌧�i� +i

3�i�

2

<latexit sha1_base64="HyA3gG99MUFYntc0j0QOxTXmFsQ=">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</latexit><latexit sha1_base64="HyA3gG99MUFYntc0j0QOxTXmFsQ=">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</latexit><latexit sha1_base64="HyA3gG99MUFYntc0j0QOxTXmFsQ=">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</latexit><latexit sha1_base64="HyA3gG99MUFYntc0j0QOxTXmFsQ=">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</latexit>

PhysRevC.98.054618 (2018)

still checking correspondence to the conventional Skyrme model

CHEA-20190116 @ 경주

Daejeon BUU (DJBUU)

16

Test particle Method

Kim et al. (2016)

197Au+197Au@ 100 MeV/u, b=7fm

RMF

Boltzmann-Uehling-Uhlenbeck (BUU) eq.

CHEA-20190116 @ 경주

Daejeon BUU (DJBUU)

16


Kim et al. (2016)

197Au+197Au@ 100 MeV/u, b=7fm

RMF


dp⌧

dt= �rU⌧


for KIDS

CHEA-20190116 @ 경주

Delta and Pion Production in Box

17

N + N <—> N + Δ Δ <—> Ν + π

⇡�/⇡+ ⇡ (n/p)2

= (1.5)2 = 2.25<latexit sha1_base64="tdDl17U29hopjHPbAQqjNRHY9aU=">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</latexit><latexit sha1_base64="tdDl17U29hopjHPbAQqjNRHY9aU=">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</latexit><latexit sha1_base64="tdDl17U29hopjHPbAQqjNRHY9aU=">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</latexit><latexit sha1_base64="tdDl17U29hopjHPbAQqjNRHY9aU=">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</latexit>

4.4

CHEA-20190116 @ 경주

132Sn+124Sn @ 200MeV/u

18

preliminary

CHEA-20190116 @ 경주

Summary1. Done.

- KIDS EoS application to NS mass, radius, and tidal deformability

- pion production test in DJBUU

- derivation of single nucleon potential for KIDS

2. To-Do

- Implementation of KIDS potential to DJBUU

- DJBUU runs: 40+(0,10,14)Ca+40Ca,132Sn+124Sn, … , @ 18.5~200MeV/u

a. Quantifying observable parameters

b. comparison between RMF vs. KIDS

- Code development of Bayesian analysis for KIDS

- Doing Bayesian analysis with KIDS on GWDA as well as DJBUU

19

CHEA-20190116 @ 경주

Summary (continued)1. Keep doing GW data analysis for LIGO and KAGRA

20

Thank you for your attention

21

Extra Slides

22

CHEA-20190116 @ 경주

Uncertainty of c_i(δ)

23

c_i(δ)’s are determined by fitting to the energy and charge radius of 40,42,44,46,48,50Ca with their uncertainties.

E(⇢, �) = E(⇢, �)

A= T (⇢, �) + ⌃N�1

i=0 ci(�)⇢1+i/3


CHEA-20190116 @ 경주

Bayesian Analysis

24

J [MeV] L [MeV] Ksym[MeV]

Ksy

m[M

eV]

L [M

eV]

CHEA-20190116 @ 경주25

2

l  Possible experiment - Using 18.5 ~ 250 MeV/u RI beam through IF separtor, perform N/Z controlled heavy-ion collision experiment for studying density dependent symmetry energy of nuclear matter possible Day-1 experiment : 50,54Ca+40Ca to measure proton, neutron spectrum - Then, series of experiment for 50,54Ca+40Ca, 68,70,72Ni + 58Ni, 106,112,124,130,132Sn + 112,118,124Sn to measure particle spectrum, yield, ratio, collective flow etc. at the same time

LAMPS (Large Acceptance Multi-Purpose Spectrometer)

• Experimental data are measured with stable beams •  data of pion ratio and data of n/p flow are from different experiments • Models in the market show different results even within same observable

LAMPS will provide experimental data using rare isotope beam with varying beam energies and collision systems e.g. 132Sn + 124Sn → measure n/p ratio & flow at the same time

provided by Y. Kim.

CHEA-20190116 @ 경주

Constraints on Nuclear EoS

26

• Nuclear data: hundreds of models (Skyrme force, RMF, …)• Neutron star maximum mass

1.97 ± 0.04 M⦿ [Nature 467, 1081 (2010)]

2.01 ± 0.04 M⦿ [Science 340, 448 (2013)]

• 11 experimental/empirical data for nuclear matter around saturation density [Phys.Rev. C 85, 035201 (2012)]

16/240 Skyrme force models satisfy 11 constraints.

27

G. Hana et al., arxiv:1805.11321

CHEA-20190116 @ 경주

Standard Skyrme effective interaction

28

2nd order derivative

N1LO Skyrme pseudopotential

CHEA-20190116 @ 경주

BUU type Transport simulation

29

RMF


CHEA-20190116 @ 경주

Daejeon BUU (DJBUU)

30


Kim et al. (2016)197Au+197Au@ 100 MeV/u, b=7fm

dp⌧

dt= �rU⌧


for KIDS

CHEA-20190116 @ 경주

Comparison study with other codes

31

CHEA-20190116 @ 경주

Prospects of the Observing Runs

32

“Prospects for Observing and Localizing Gravitational-Wave Transients with Advanced LIGO, Advanced Virgo and KAGRA”, arXiv:1304.0670v4, LIGO-P1200087-v45, Living Rev. Relativity, 21, 3 (2018)

We expect to observe more BNS and/or NS-BH

CHEA-20190116 @ 경주



33

RMF: σ-ω-ρ model, PDM

DJBUU- π+/π--ratio, collective

flow, etc.

Skyrme force: KIDS

Astrophysics: M, Λ from GW,

M, R from LMXB

CHEA-20190116 @ 경주

Bayesian Analysis


34



flow, etc.

Skyrme force: KIDS


M, R from LMXB

CHEA-20190116 @ 경주

Bayesian Analysis


34



flow, etc.

Skyrme force: KIDS


M, R from LMXB

J [MeV] L [MeV] Ksym[MeV]

Ksy

m[M

eV]

L [M

eV]

CHEA-20190116 @ 경주

Low-Mass X-ray Binary

35

Steiner, Lattimer, Brown, ApJ 2010

95% confidence limits by using MC sampling

CHEA-20190116 @ 경주

Dataset - slb13

36

Mass-Radius PDF samples

PRE XRB

CHEA-20190116 @ 경주

Dataset - ls14

37

Mass-Radius PDF samples

qLMXBs

CHEA-20190116 @ 경주

Results

38

Documents

Nuclear Energy Density Functional for Heavy Ion …sirius.unist.ac.kr/SRC-CHEA/seminar/2019+chea+ppt/...Nuclear Energy Density Functional for Heavy Ion Collisions and Neutron Stars