[PPT] - Atomic nuclei: from fundamental interactions to structure and stars PowerPoint Presentation

SLIDE 1

Kai Hebeler

Atomic nuclei: from fundamental interactions to structure and stars

Mainz, April 7, 2016 New Vistas in Low-Energy Precision Physics (LEPP)

SLIDE 2

The theoretical nuclear landscape several years ago...

SLIDE 3

nuclear structure and reaction observables Quantum Chromodynamics

Ab initio nuclear structure theory

SLIDE 4

Lattice QCD

requires extreme amounts
f computational resources
currently limited to 1- or 2-nucleon systems
current accuracy insufficient for

precision nuclear structure

nuclear structure and reaction observables Quantum Chromodynamics

Ab initio nuclear structure theory

SLIDE 5

Chiral effective field theory

nuclear interactions and currents

nuclear structure and reaction observables Quantum Chromodynamics

Ab initio nuclear structure theory

SLIDE 6

Chiral effective field theory

nuclear interactions and currents

nuclear structure and reaction observables ab initio many-body frameworks

Faddeev, Quantum Monte Carlo, no-core shell model, coupled cluster ...

Quantum Chromodynamics

Ab initio nuclear structure theory

SLIDE 7

Chiral effective field theory

nuclear interactions and currents

nuclear structure and reaction observables Quantum Chromodynamics Renormalization Group methods

Ab initio nuclear structure theory

ab initio many-body frameworks

Faddeev, Quantum Monte Carlo, no-core shell model, coupled cluster ...

SLIDE 8

choose relevant degrees of

freedom: here nucleons and pions

operators constrained by

symmetries of QCD

short-range physics captured in

few short-range couplings

separation of scales: Q << Λb,

breakdown scale Λb~500 MeV

power-counting:

expand in powers Q/Λb

systematic: work to desired

accuracy, obtain error estimates

Chiral effective field theory for nuclear forces

NN 3N 4N 2006 1994 2011

SLIDE 9

NN 3N 4N

long (2π) intermediate (π) short-range

c1, c3, c4 terms

cD term cE term

Many-body forces in chiral EFT

2006 1994 2011

SLIDE 10

NN 3N 4N

long (2π) intermediate (π) short-range

c1, c3, c4 terms

cD term cE term

first incorporation in calculations of neutron and nuclear matter

Tews, Krueger, KH, Schwenk, PRL 110, 032504 (2013) Krueger, Tews, KH, Schwenk, PRC 88, 025802 (2013)

Many-body forces in chiral EFT

2006 1994 2011

all terms predicted

SLIDE 11

NN 3N 4N

long (2π) intermediate (π) short-range

c1, c3, c4 terms

cD term cE term

first incorporation in calculations of neutron and nuclear matter

Tews, Krueger, KH, Schwenk, PRL 110, 032504 (2013) Krueger, Tews, KH, Schwenk, PRC 88, 025802 (2013)

2006 1994 2011

first calculation of matrix elements for ab initio studies of matter and nuclei

KH, Krebs, Epelbaum, Golak, Skibinski, PRC 91, 044001(2015)

Many-body forces in chiral EFT

SLIDE 12

Chiral effective field theory

nuclear interactions and currents

nuclear structure and reaction observables

Development of nuclear interactions predictions validation

ptimization

power counting

LENPIC

SLIDE 13

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

The Similarity Renormalization Group

generate unitary transformation which decouples low- and high momenta
basic idea: change resolution successively in small steps:

with the resolution parameter

observables are preserved due to unitarity of transformation
generator can be chosen and tailored to different applications

ηλ

SLIDE 14

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

The Similarity Renormalization Group

generate unitary transformation which decouples low- and high momenta
basic idea: change resolution successively in small steps:

with the resolution parameter

observables are preserved due to unitarity of transformation
generator can be chosen and tailored to different applications

ηλ

SLIDE 15

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

The Similarity Renormalization Group

generate unitary transformation which decouples low- and high momenta
basic idea: change resolution successively in small steps:

with the resolution parameter

observables are preserved due to unitarity of transformation
generator can be chosen and tailored to different applications

ηλ

SLIDE 16

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

The Similarity Renormalization Group

generate unitary transformation which decouples low- and high momenta
basic idea: change resolution successively in small steps:

with the resolution parameter

observables are preserved due to unitarity of transformation
generator can be chosen and tailored to different applications

ηλ

SLIDE 17

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

The Similarity Renormalization Group

generate unitary transformation which decouples low- and high momenta
basic idea: change resolution successively in small steps:

with the resolution parameter

observables are preserved due to unitarity of transformation
generator can be chosen and tailored to different applications

ηλ

SLIDE 18

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

The Similarity Renormalization Group

generate unitary transformation which decouples low- and high momenta
basic idea: change resolution successively in small steps:

with the resolution parameter

observables are preserved due to unitarity of transformation
generator can be chosen and tailored to different applications

ηλ

SLIDE 19

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

The Similarity Renormalization Group

generate unitary transformation which decouples low- and high momenta
basic idea: change resolution successively in small steps:

with the resolution parameter

observables are preserved due to unitarity of transformation
generator can be chosen and tailored to different applications

ηλ

SLIDE 20

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

The Similarity Renormalization Group

generate unitary transformation which decouples low- and high momenta
basic idea: change resolution successively in small steps:

with the resolution parameter

observables are preserved due to unitarity of transformation
generator can be chosen and tailored to different applications

ηλ

SLIDE 21

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

The Similarity Renormalization Group

generate unitary transformation which decouples low- and high momenta
basic idea: change resolution successively in small steps:

with the resolution parameter

observables are preserved due to unitarity of transformation
generator can be chosen and tailored to different applications

ηλ

SLIDE 22

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

The Similarity Renormalization Group

generate unitary transformation which decouples low- and high momenta
basic idea: change resolution successively in small steps:

with the resolution parameter

observables are preserved due to unitarity of transformation
generator can be chosen and tailored to different applications

ηλ

SLIDE 23

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

The Similarity Renormalization Group

generate unitary transformation which decouples low- and high momenta
basic idea: change resolution successively in small steps:

with the resolution parameter

observables are preserved due to unitarity of transformation
generator can be chosen and tailored to different applications

ηλ

SLIDE 24

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

The Similarity Renormalization Group

generate unitary transformation which decouples low- and high momenta
basic idea: change resolution successively in small steps:

with the resolution parameter

observables are preserved due to unitarity of transformation
generator can be chosen and tailored to different applications

ηλ

SLIDE 25

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

The Similarity Renormalization Group

generate unitary transformation which decouples low- and high momenta
basic idea: change resolution successively in small steps:

with the resolution parameter

observables are preserved due to unitarity of transformation
generator can be chosen and tailored to different applications

ηλ

SLIDE 26

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

The Similarity Renormalization Group

generate unitary transformation which decouples low- and high momenta
basic idea: change resolution successively in small steps:

with the resolution parameter

observables are preserved due to unitarity of transformation
generator can be chosen and tailored to different applications

ηλ

SLIDE 27

1 2 3 4 r [fm] −100 100 200 V(r) [MeV] λ = 20 fm

−1

1 2 3 4 r [fm] λ = 4 fm

−1

1 2 3 4 r [fm] λ = 3 fm

−1

1 2 3 4 r [fm] λ = 2 fm

−1

1 2 3 4 r [fm] λ = 1.5 fm

−1

AV18 N

3LO

V λ(r) = Z dr0r02Vλ(r, r0)

Systematic decoupling of high-momentum physics: The Similarity Renormalization Group

SLIDE 28

elimination of coupling between low- and high momentum components,

simplified many-body calculations, smaller required model spaces

observables unaffected by resolution change (for exact calculations)
residual resolution dependences can be used as tool to test calculations

Not the full story: RG transformation also changes three-body (and higher-body) interactions.

Systematic decoupling of high-momentum physics: The Similarity Renormalization Group

SLIDE 29

Recent advances in ab-initio many-body theory

10
9
8
7
6

NN+3N-induced

N3LO N2LOopt

(a) exp

0.5

0.5 (b)

10
9
8
7

. E/A [MeV] NN+3N-full

Λ3N = 400 MeV/c Λ3N = 350 MeV/c

(c) exp

16O 24O 36Ca 40Ca 48Ca 52Ca 54Ca 48Ni 56Ni 60Ni 62Ni 66Ni 68Ni 78Ni 88Sr 90Zr 100Sn 106Sn 108Sn 114Sn 116Sn 118Sn 120Sn 132Sn

0.5

0.5 (d)

Binder et al. , Phys. Lett B 736, 119 (2014)

SLIDE 30

Recent advances in ab-initio many-body theory

spectacular increase in range of applicability
f ab initio many body frameworks
significant overbinding in heavy nuclei

for presently used nuclear interactions

10
9
8
7
6

NN+3N-induced

N3LO N2LOopt

(a) exp

0.5

0.5 (b)

10
9
8
7

. E/A [MeV] NN+3N-full

Λ3N = 400 MeV/c Λ3N = 350 MeV/c

(c) exp

16O 24O 36Ca 40Ca 48Ca 52Ca 54Ca 48Ni 56Ni 60Ni 62Ni 66Ni 68Ni 78Ni 88Sr 90Zr 100Sn 106Sn 108Sn 114Sn 116Sn 118Sn 120Sn 132Sn

0.5

0.5 (d)

Binder et al. , Phys. Lett B 736, 119 (2014) Hagen et al., Nature Physics 12, 186 (2016)

SLIDE 31

0.8 1.0 1.2 1.4 1.6

kF [fm

−1]

−30 −25 −20 −15 −10 −5

Energy/nucleon [MeV]

Λ = 1.8 fm

−1 NN only

Λ = 2.8 fm

−1 NN only

Vlow k NN from N

3LO (500 MeV)

3NF fit to E3H and r4He Λ3NF = 2.0 fm

−1

3rd order pp+hh

NN only

¯

lS

“Very soft potentials must be excluded because they do not give saturation; they give too much binding and too high density. In particular, a substantial tensor force is required.” Hans Bethe (1971)

KH, Bogner, Furnstahl, Nogga, PRC(R) 83, 031301 (2011)

1 2 3 4 r [fm] −100 100 200 V(r) [MeV] λ = 20 fm

−1

4 1 2 3 4 r [fm] λ = 1.5 fm

−1

AV18 N

3LO

Fitting the 3NF LECs at low resolution scales

SLIDE 32

0.8 1.0 1.2 1.4 1.6

kF [fm

−1]

−30 −25 −20 −15 −10 −5

Energy/nucleon [MeV]

Λ = 1.8 fm

−1

Λ = 2.8 fm

−1

Λ = 1.8 fm

−1 NN only

Λ = 2.8 fm

−1 NN only

Vlow k NN from N

3LO (500 MeV)

3NF fit to E3H and r4He Λ3NF = 2.0 fm

−1

3rd order pp+hh

NN + 3N NN only

“Very soft potentials must be excluded because they do not give saturation; they give too much binding and too high density. In particular, a substantial tensor force is required.” Hans Bethe (1971)

KH, Bogner, Furnstahl, Nogga, PRC(R) 83, 031301 (2011)

¯

lS

1 2 3 4 r [fm] −100 100 200 V(r) [MeV] λ = 20 fm

−1

4 1 2 3 4 r [fm] λ = 1.5 fm

−1

AV18 N

3LO

Fitting the 3NF LECs at low resolution scales

intermediate (cD) and short-range (cE) 3NF couplings fitted to few-body systems at different resolution scales:

E3H = −8.482 MeV r4He = 1.464 fm

c1, c3, c4 terms

cD term cE term

SLIDE 33

0.8 1.0 1.2 1.4 1.6

kF [fm

−1]

−30 −25 −20 −15 −10 −5

Energy/nucleon [MeV]

Λ = 1.8 fm

−1

Λ = 2.8 fm

−1

Λ = 1.8 fm

−1 NN only

Λ = 2.8 fm

−1 NN only

Vlow k NN from N

3LO (500 MeV)

3NF fit to E3H and r4He Λ3NF = 2.0 fm

−1

3rd order pp+hh

NN + 3N NN only

“Very soft potentials must be excluded because they do not give saturation; they give too much binding and too high density. In particular, a substantial tensor force is required.” Hans Bethe (1971)

KH, Bogner, Furnstahl, Nogga, PRC(R) 83, 031301 (2011)

¯

lS

1 2 3 4 r [fm] −100 100 200 V(r) [MeV] λ = 20 fm

−1

4 1 2 3 4 r [fm] λ = 1.5 fm

−1

AV18 N

3LO

Fitting the 3NF LECs at low resolution scales

Drischler, KH, Schwenk, arXiv:1510.06728

SLIDE 34

2 8 20 28 Z N 2 8

s t a b i l i t y l i n e

O 1970 F 1999 N 1985 C 1986 B 1984 Be 1973 Li 1966 H 1934 He 1961 Ne 2002 Na 2002 Mg2007 Al 2007 Si 2007

unstable oxygen isotopes unstable fluorine isotopes stable isotopes unstable isotopes neutron halo nuclei

remarkable agreement between different many-body frameworks
excellent agreement between theory and experiment for masses of
xygen and calcium isotopes based on specific chiral interactions
need to quantify theoretical uncertainties

Gallant et al. PRL 109, 032506 (2012) Wienholtz et al. Nature 498, 346 (2013)

Studies of neutron-rich nuclei

S2n (MeV) Neutron number, N 2 6 10 14 18 22

[S2n(theo) – S2n(exp)] (MeV) 2.0 1.0 0.0 –1.0 –2.0 30 31 32 33 34

a

28 30 32 34 36 38

Neutron number, N

ISOLTRAP Experiment NN+3N (MBPT) CC (ref. 5) KB3G GXPF1A

Ca isotopes

16 18 20 22 24 26 28

Mass Number A

180
170
160
150
140
130

Energy (MeV)

MR-IM-SRG IT-NCSM SCGF Lattice EFT CC

btained in large many-body spaces

AME 2012

O isotopes

KH et al. , Ann. Rev. Nucl. Part. Sci.165, 457 (2015)

SLIDE 35

10

10 20 30 40

S2n (MeV)

10

10 20 30 40 10 20 30 40 50 10 12 14 16 18 20 10 12 14 16 18 20 10 20 30 40 50 10 12 14 16 18 20 10 20 30 40 50 10 12 14 16 18 20 10 12 14 16 18 20

Neutron Number N

10 20 30 40 50

2nd order 3rd order AME 2012 8O 9F 11Na 13Al 10Ne 14Si 12Mg 15P 19K 18Ar 16S 17Cl 20Ca

Towards theoretical uncertainty quantification

calculations based on NN+3N interactions fitted to NN, 3N and 4N systems
reasonable reproduction of experimental trends
uncertainties dominated by differences in nuclear Hamiltonians

Simonis, KH, Holt, Menendez, Schwenk,

Phys. Rev. C93, 011302(R) (2016)

SLIDE 36

10

10 20 30 40

S2n (MeV)

10

10 20 30 40 10 20 30 40 50 10 12 14 16 18 20 10 12 14 16 18 20 10 20 30 40 50 10 12 14 16 18 20 10 20 30 40 50 10 12 14 16 18 20 10 12 14 16 18 20

Neutron Number N

10 20 30 40 50

2nd order 3rd order AME 2012 8O 9F 11Na 13Al 10Ne 14Si 12Mg 15P 19K 18Ar 16S 17Cl 20Ca

Towards theoretical uncertainty quantification

calculations based on NN+3N interactions fitted to NN, 3N and 4N systems
reasonable reproduction of experimental trends
uncertainties dominated by differences in nuclear Hamiltonians

Simonis, KH, Holt, Menendez, Schwenk,

Phys. Rev. C93, 011302(R) (2016)

2 4 6

2

+ Energy (MeV)

8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 8 10 12 14 16 18 20

Neutron Number N

2 4 6

2nd order 3rd order exp.

8O 10Ne 12Mg 14Si 16S 18Ar 20Ca

SLIDE 37

Results for the neutron matter equation of state

c1, c3, c4 terms

cD term cE term

nly long-range 3NF

contribute in leading order neutron matter is a unique system for chiral EFT: pure neutron matter

0.2 0.4 0.6 0.8 1.0

[

0]

0.5 1.0 1.5 2.0 2.5

P [1033dyne / cm2]

NN only, EM NN only, EGM

KH and Schwenk PRC 82, 014314 (2010)

0.05 0.10 0.15

[fm-3]

5 10 15 20

Energy/nucleon [MeV]

ENN+3N,eff+c3+c1 uncertainties ENN+3N,eff+c3 uncertainty ENN

(1) + ENN (2)

3N neutron star matter

KH, Lattimer, Pethick, Schwenk, PRL 105, 161102 (2010)

ENN+3N,eff

(1)

ENN+3N,eff 2.0 <

3N < 2.5 fm-1

0. 05
0. 10

0.15

[fm-3]

5 10 15 20

Energy/nucleon [MeV]

0. 05
0. 10
0. 15

[fm-3]

= 1.8 fm-1 = 2.0 fm-1 = 2.4 fm-1 = 2.8 fm-1

KH and Schwenk PRC 82, 014314 (2010)

SLIDE 38

Symmetry energy and neutron skin constraints

neutron matter give tightest constraints
in agreement with all other constraints

Sv = ∂2E/N ∂2x

ρ=ρ0,x=1/2

L = 3 8 ∂3E/N ∂ρ∂2x

ρ=ρ0,x=1/2

1303.4662

KH, Lattimer, Pethick, Schwenk, ApJ 773,11 (2013)

rskin[208Pb] = 0.14 − 0.2 fm

KH, Lattimer, Pethick, Schwenk, PRL 105, 161102 (2010)

neutron skin constraint from neutron matter results:

S(208Pb) (fm) slope of neutron EOS 0.0 0.1 0.2 0.3

50

50 100 150 200

!"#$ "%&!

Brown, PRL 85, 5296 (2000) Piekarewicz, PRC 85, 041302 (2012)

SLIDE 39

Hagen et al., Nature Physics 12, 186 (2016)

Predictions for the neutron skin of 48Ca

microscopic coupled cluster results based on a set of different

nuclear NN+3N interactions (see also Phys. Rev. C91, 051301 (2015))

correlations between different observables and the precisely measured Rp
prediction of significantly smaller neutron skin compared to EDF results:

0.12 . Rskin . 0.15 fm

SLIDE 40

Charge radii of calcium isotopes

Garcia Ruiz at al., Nature Physics (advanced online, 2016)

novel precise measurements of Ca isotope shifts
unexpectedly large radii of neutron-rich isotopes
reasonable theoretical reproduction of radius trends

in coupled cluster calculations based on chiral EFT interactions

radius increase quantitatively underestimated in all theoretical studies

SLIDE 41

Constraints on the nuclear equation of state (EOS)

A two-solar-mass neutron star measured using Shapiro delay

P. B. Demorest1, T. Pennucci2, S. M. Ransom1, M. S. E. Roberts3 & J. W. T. Hessels4,5

a b

–40 –30 –20 –10 10 20 30 –40 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Orbital phase (turns)

Timing residual (μs)

Demorest et al., Nature 467, 1081 (2010)

Mmax = 1.65M → 1.97 ± 0.04 M

Calculation of neutron star properties require EOS up to high densities. Strategy: Use observations to constrain the high-density part of the nuclear EOS. New constraints from recent observations:

A Massive Pulsar in a Compact Relativistic Binary

Antoniadis et al., Science 340, 448 (2013)

→ 2.01 ± 0.04 M

SLIDE 42

Neutron star radius constraints

incorporation of beta-equilibrium: neutron matter neutron star matter parametrize piecewise high-density extensions of EOS:

use polytropic ansatz
range of parameters

p ∼ ρΓ

13.0 13.5 14.0

log 10 [g / cm3]

31 32 33 34 35 36 37

log 10 P [dyne / cm2]

1 2 3

with ci uncertainties

crust

crust EOS (BPS) neutron star matter

12 23 1

Γ1, ρ12, Γ2, ρ23, Γ3 limited by physics

KH, Lattimer, Pethick, Schwenk, ApJ 773, 11 (2013) KH, Lattimer, Pethick, Schwenk, PRL 105, 161102 (2010)

SLIDE 43

Constraints on the nuclear equation of state

use the constraints:

vs(ρ) =

dP/dε < c

Mmax > 1.97 M

causality recent NS observations constraints lead to significant reduction of EOS uncertainty band

KH, Lattimer, Pethick, Schwenk, ApJ 773,11 (2013)

SLIDE 44

vs(ρ) =

dP/dε < c

causality fictitious NS mass

Mmax > 2.4 M

increased systematically reduces width of band

Mmax

use the constraints:

Constraints on the nuclear equation of state

KH, Lattimer, Pethick, Schwenk, ApJ 773,11 (2013)

SLIDE 45

current radius prediction for typical neutron star:
low-density part of EOS sets scale for allowed high-density extensions

14.2 14.4 14.6 14.8 15.0 15.2 15.4

log 10 [g / cm3]

33 34 35 36

log 10 P [dyne / cm2]

WFF1 WFF2 WFF3 AP4 AP3 MS1 MS3 GM3 ENG PAL GS1 GS2

14.2 14.4 14.6 14.8 15.0 15.2 15.4 33 34 35 36

PCL2 SQM1 SQM2 SQM3 PS

Constraints on neutron star radii

KH, Lattimer, Pethick, Schwenk, ApJ 773, 11 (2013) see also KH, Lattimer, Pethick, Schwenk, PRL 105, 161102 (2010)

1.4 M

8 10 12 14 16

Radius [km]

0.5 1 1.5 2 2.5 3

Mass [Msun]

8 10 12 14 16

Radius [km]

0.5 1 1.5 2 2.5 3

Mass [Msun]

causality

9.7 − 13.9 km

SLIDE 46

current radius prediction for typical neutron star:
low-density part of EOS sets scale for allowed high-density extensions

14.2 14.4 14.6 14.8 15.0 15.2 15.4

log 10 [g / cm3]

33 34 35 36

log 10 P [dyne / cm2]

WFF1 WFF2 WFF3 AP4 AP3 MS1 MS3 GM3 ENG PAL GS1 GS2

14.2 14.4 14.6 14.8 15.0 15.2 15.4 33 34 35 36

PCL2 SQM1 SQM2 SQM3 PS

Constraints on neutron star radii

KH, Lattimer, Pethick, Schwenk, ApJ 773, 11 (2013) see also KH, Lattimer, Pethick, Schwenk, PRL 105, 161102 (2010)

1.4 M

8 10 12 14 16

Radius [km]

0.5 1 1.5 2 2.5 3

Mass [Msun]

8 10 12 14 16

Radius [km]

0.5 1 1.5 2 2.5 3

Mass [Msun]

causality

radius measurements could significantly improve constraints

9.7 − 13.9 km

Large

bservatory

for X-ray timing

SLIDE 47

Summary

recent advances allow ab initio studies of medium-mass nuclei
remarkable agreement between different methods for given interaction,

uncertainties dominated by differences in nuclear interactions

results presented for properties of neutron-rich nuclei and matter based
n sets of current chiral EFT NN+3N interactions

LENPIC

Future directions

derivation of systematic uncertainty estimates for many-body observables,
rder-by-order convergence studies
exploration of different fitting strategies, include bayesian analysis for

statistical interpretation of uncertainties?

role of regulators, clean separation of short- and long-range physics,

naturalness of coupling constants, power counting schemes, inclusion of delta excitations...

SLIDE 48

In collaboration with: computing support:

JUROPA

C. Pethick
C. Drischler, T. Krüger, R. Roth,
A. Schwenk
J. Lattimer
S. Bogner
R. Furnstahl, S. More
A. Nogga
E. Epelbaum, H. Krebs
J. Golak, R. Skibinski
A. Gezerlis

international collaborator in

LENPIC

G. Hagen, T. Papenbrock

SLIDE 49

Thank you!

SLIDE 50

Backup slides

SLIDE 51

VNN V3N V3N V3N

Equation of state: Many-body perturbation theory

E = + + + +

central quantity of interest: energy per particle E/N

“hard” interactions require non-perturbative summation of diagrams
with low-momentum interactions much more perturbative
inclusion of 3N interaction contributions crucial!

+

. . .

Hartree-Fock

VNN VNN

+ + +

V3N V3N V3N VNN VNN V3N

2nd-order kinetic energy 3rd-order and beyond H(λ) = T + VNN(λ) + V3N(λ) + ...

SLIDE 52

First application to isospin asymmetric nuclear matter

uncertainty bands determined

by set of 7 Hamitonians

Drischler, KH, Schwenk, in preparation

x = np np + nn

many-body framework allows

treatment of any decomposed 3N interaction

SLIDE 53

Equation of state of symmetric nuclear matter, nuclear saturation

¯

lS

“Very soft potentials must be excluded because they do not give saturation; they give too much binding and too high density. In particular, a substantial tensor force is required.” Hans Bethe (1971)

SLIDE 54

1 2 3 4 5 −25 −24 −23 −22 −21

f [kHz] log(h+(f)f1/2/Hz1/2)

5 10 15 20 −2 2 x 10

−22

h+ at 50 Mpc t [ms]

eosUU fpeak Shen

Bauswein and Janka, PRL 108, 011101 (2012), Bauswein, Janka, KH, Schwenk, PRD 86, 063001 (2012)

simulations of NS binary mergers show strong correlation between between
f the GW spectrum and the radius of a NS
measuring is key step for constraining EOS systematically at large

fpeak fpeak

ρ

10 11 12 13 14 15 16 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 fpeak [kHz] R1.6 [km]

Gravitational wave signals from neutron star binary mergers

SLIDE 55

Representation of 3N interactions in momentum space

|pqα⇥i |piqi; [(LS)J(lsi)j] J Jz(Tti)T Tz⇥

p

q

p

q

p

q

|pqα1 |pqα2 |pqα3

1 2 3 2 2 1 1 3 3

Due to the large number of matrix elements, the traditional way of computing matrix elements requires extreme amounts of computer resources.

Np ' Nq ' 15

Nα ' 30 180

dim[hpqα|V123|p0q0α0i] ' 107 1010

Number of matrix elements was so far not sufficient for studies of systems.

A ≥ 4

SLIDE 56

First Quantum Monte Carlo based on local chiral EFT interactions

0.05 0.1 0.15

n [fm-3]

5 10 15

E/N [MeV]

QMC (2010) AFDMC N2LO 0.8 fm (2nd order) 0.8 fm (3rd order) 1.2 fm (2nd order) 1.2 fm (3rd order)

perfect agreement for soft interactions, first direct validation

f perturbative calculations

Gezerlis, Tews, Epelbaum, Gandolfi, KH, Nogga, Schwenk PRL 111, 032501 (2013)

SLIDE 57

0.05 0.1 0.15

n [fm-3]

4
2

2 4

E/N [MeV] Two-pion-exchange 3N

0.05 0.1 0.15

n [fm-3] Two-pion−one-pion-exchange 3N

0.05 0.1 0.15

n [fm-3] Pion-ring 3N

0.05 0.1 0.15 0.2

n [fm-3]

4
2

2 4 EM 500 MeV EGM 450/700 MeV EGM 450/500 MeV

Two-pion-exchange−contact 3N

Contributions of many-body forces at N3LO in neutron matter

Tews, Krüger, KH, Schwenk PRL 110, 032504 (2013)

NN 3N 4N

first calculations of N3LO 3NF and 4NF

contributions to EOS of neutron matter

found large contributions in Hartree Fock appr.,

comparable to size of N2LO contributions

SLIDE 58

0.05 0.1 0.15

n [fm-3]

4
2

2 4

E/N [MeV] Two-pion-exchange 3N

0.05 0.1 0.15

n [fm-3] Two-pion−one-pion-exchange 3N

0.05 0.1 0.15

n [fm-3] Pion-ring 3N

0.05 0.1 0.15 0.2

n [fm-3]

4
2

2 4 EM 500 MeV EGM 450/700 MeV EGM 450/500 MeV

Two-pion-exchange−contact 3N

Tews, Krüger, KH, Schwenk PRL 110, 032504 (2013)

NN 3N 4N

first calculations of N3LO 3NF and 4NF

contributions to EOS of neutron matter

found large contributions in Hartree Fock appr.,

comparable to size of N2LO contributions

4NF contributions small

0.05 0.1 0.15

n [fm-3]

0.4
0.3
0.2
0.1

0.1 0.2 0.3 0.4

E/N [MeV] Three-pion-exchange 4N V

a

0.05 0.1 0.15

n [fm-3] Three-pion-exchange 4N V

e

0.05 0.1 0.15

n [fm-3] Pion-pion-interaction 4N V

f

0.05 0.1 0.15 0.2

n [fm-3]

0.4
0.3
0.2
0.1

0.1 0.2 0.3 0.4 EGM 450/500 MeV EGM 450/700 MeV EM 500 MeV

pRelativistic-corrections 3Np

Contributions of many-body forces at N3LO in neutron matter

SLIDE 59

N3LO contributions in nuclear matter (Hartree Fock)

0.05 0.1 0.15

n [fm-3]

10
8
6
4
2

2 4

E/N [MeV] Two-pion-exchange 3N

0.05 0.1 0.15

n [fm-3] Two-pion-

one-pion-exchange 3N

0.05 0.1 0.15

n [fm-3] Pion-ring 3N

0.05 0.1 0.15

n [fm-3]

10
8
6
4
2

2 4 EM 500 MeV1 1 EGM 450/700 MeV EGM 450/500 MeV1 1

Two-pion-exchange-

contact 3N

0.05 0.1 0.15

n [fm-3]

0.6
0.4
0.2

0.2 0.4

E/N [MeV]

EGM 450/500 MeV1 1 EM 500 MeV EGM 450/700 MeV1 1

Relativistic-corrections 3N

0.05 0.1 0.15

n [fm-3] Three-pion-exchange 4N Va

0.05 0.1 0.15

n [fm-3] Three-pion-exchange 4N Vc

0.05 0.1 0.15 0.2

n [fm-3]

0.6
0.4
0.2

0.2 0.4

Three-pion-exchange 4N Ve

0.05 0.1 0.15

n [fm-3]

0.2
0.1

0.1 0.2 0.3 0.4 0.5

E/N [MeV] Pion-pion-interaction 4N Vf

0.05 0.1 0.15

n [fm-3]

EM 500 MeV11 EGM 450/700 MeV EGM 450/500 MeV1 1

Two-pion-exchange-contact 4N Vk

0.05 0.1 0.15

n [fm-3]

EGM 500 MeV11 EGM 450/700 MeV EGM 450/500 MeV1 1

Two-pion-exchange-

contact 4N Vl

0.05 0.1 0.15 0.2

n [fm-3]

0.2
0.1

0.1 0.2 0.3 0.4 0.5 EM 500 MeV11 EGM 450/700 MeV EGM 450/500 MeV1 1

Pion-exchange-

two-contact 4N Vn

Krüger, Tews, KH, Schwenk PRC88, 025802 (2013)

SLIDE 60

N3LO contributions in nuclear matter (Hartree Fock)

0.05 0.1 0.15

n [fm-3]

10
8
6
4
2

2 4

E/N [MeV] Two-pion-exchange 3N

0.05 0.1 0.15

n [fm-3] Two-pion-

one-pion-exchange 3N

0.05 0.1 0.15

n [fm-3] Pion-ring 3N

0.05 0.1 0.15

n [fm-3]

10
8
6
4
2

2 4 EM 500 MeV1 1 EGM 450/700 MeV EGM 450/500 MeV1 1

Two-pion-exchange-

contact 3N

0.05 0.1 0.15

n [fm-3]

0.6
0.4
0.2

0.2 0.4

E/N [MeV]

EGM 450/500 MeV1 1 EM 500 MeV EGM 450/700 MeV1 1

Relativistic-corrections 3N

0.05 0.1 0.15

n [fm-3] Three-pion-exchange 4N Va

0.05 0.1 0.15

n [fm-3] Three-pion-exchange 4N Vc

0.05 0.1 0.15 0.2

n [fm-3]

0.6
0.4
0.2

0.2 0.4

Three-pion-exchange 4N Ve

0.05 0.1 0.15

n [fm-3]

0.2
0.1

0.1 0.2 0.3 0.4 0.5

E/N [MeV] Pion-pion-interaction 4N Vf

0.05 0.1 0.15

n [fm-3]

EM 500 MeV11 EGM 450/700 MeV EGM 450/500 MeV1 1

Two-pion-exchange-contact 4N Vk

0.05 0.1 0.15

n [fm-3]

EGM 500 MeV11 EGM 450/700 MeV EGM 450/500 MeV1 1

Two-pion-exchange-

contact 4N Vl

0.05 0.1 0.15 0.2

n [fm-3]

0.2
0.1

0.1 0.2 0.3 0.4 0.5 EM 500 MeV11 EGM 450/700 MeV EGM 450/500 MeV1 1

Pion-exchange-

two-contact 4N Vn

Krüger, Tews, KH, Schwenk PRC88, 025802 (2013)

Conclusions/Indications:

N3LO 3N contributions significant
N3LO 4N contributions small