[PPT] - Equation of state and neutron star properties constrained by PowerPoint Presentation

SLIDE 1

Kai Hebeler

Equation of state and neutron star properties constrained by nuclear physics and observation

Stockholm, August 17, 2015 MICRA 2015: Workshop on Microphysics in Computational Relativistic Astrophysics

SLIDE 2

SLIDE 3

LETTER

doi:10.1038/nature11188

The limits of the nuclear landscape

Jochen Erler1,2, Noah Birge1, Markus Kortelainen1,2,3, Witold Nazarewicz1,2,4, Erik Olsen1,2, Alexander M. Perhac1 & Mario Stoitsov1,2{

LETTER

doi:10.1038/nature12226

Masses of exotic calcium isotopes pin down nuclear forces

F. Wienholtz1, D. Beck2, K. Blaum3, Ch. Borgmann3, M. Breitenfeldt4, R. B. Cakirli3,5, S. George1, F. Herfurth2, J. D. Holt6,7,
M. Kowalska8, S. Kreim3,8, D. Lunney9, V. Manea9, J. Mene

´ndez6,7, D. Neidherr2, M. Rosenbusch1, L. Schweikhard1,

A. Schwenk7,6, J. Simonis6,7, J. Stanja10, R. N. Wolf1 & K. Zuber10

LETTER

doi:10.1038/nature09466

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 Massive Pulsar in a Compact Relativistic Binary

John Antoniadis,* Paulo C. C. Freire, Norbert Wex, Thomas M. Tauris, Ryan S. Lynch, Marten H. van Kerkwijk, Michael Kramer, Cees Bassa, Vik S. Dhillon, Thomas Driebe, Jason W. T. Hessels, Victoria M. Kaspi, Vladislav I. Kondratiev, Norbert Langer, Thomas R. Marsh, Maura A. McLaughlin, Timothy T. Pennucci, Scott M. Ransom, Ingrid H. Stairs, Joeri van Leeuwen, Joris P. W. Verbiest, David G. Whelan

RESEARCH ARTICLE SUMMARY

Exciting recent developments on many fronts...

F

O

PSR J0348+0432

LETTER

doi:10.1038/nature12522

Evidence for a new nuclear ‘magic number’ from the level structure of 54Ca

D. Steppenbeck1, S. Takeuchi2, N. Aoi3, P. Doornenbal2, M. Matsushita1, H. Wang2, H. Baba2, N. Fukuda2, S. Go1, M. Honma4,
J. Lee2, K. Matsui5, S. Michimasa1, T. Motobayashi2, D. Nishimura6, T. Otsuka1,5, H. Sakurai2,5, Y. Shiga7, P.-A. So

¨derstro ¨m2,

T. Sumikama8, H. Suzuki2, R. Taniuchi5, Y. Utsuno9, J. J. Valiente-Dobo

´n10 & K. Yoneda2

SLIDE 4

Asymptotic freedom ? from B. Sherrill

DFT$ FRIB$ current$

New frontiers from rare isotope facilities

Balantekin et al. , arXiv:1401.6435

FRIB- Facility for Rare Isotope Beams

SLIDE 5

LQCD = −1 4F a

µνF aµν + q(iγµ∂µ − m)q + gqγµTaqAa µ

Theory of the strong interaction: Quantum chromodynamics

theory perturbative at high energies
highly non-perturbative at low energies

αs ≡ g 4π

SLIDE 6

nuclear structure and reaction observables Quantum Chromodynamics

Ab initio nuclear structure theory

SLIDE 7

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 8

Chiral effective field theory

nuclear interactions and currents

nuclear structure and reaction observables Quantum Chromodynamics

Ab initio nuclear structure theory

SLIDE 9

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 10

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 11

Nuclear effective degrees of freedom

if a nucleus is probed at high energies,

nucleon substructure is resolved

at low energies, details are not resolved

SLIDE 12

Nuclear effective degrees of freedom

if a nucleus is probed at high energies,

nucleon substructure is resolved

at low energies, details are not resolved
replace fine structure by something

simpler (like multipole expansion), low-energy observables unchanged

Resolution

effective field theory

SLIDE 13

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

3NF at N3LO completely

predicted, no new couplings

Chiral effective field theory for nuclear forces

NN 3N 4N 2006 1994 2011

SLIDE 14

NN 3N 4N

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

c1, c3, c4 terms

cD term cE term

1.5

large uncertainties in coupling constants at present:

Chiral EFT for nuclear forces

2006 1994 2011

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)

SLIDE 15

NN 3N 4N

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

c1, c3, c4 terms

cD term cE term

1.5

large uncertainties in coupling constants at present: 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)

Chiral EFT for nuclear forces

2006 1994 2011

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

first partial wave decomposition,

pens the way to new ab initio studies

SLIDE 16

Chiral effective field theory

nuclear interactions and currents

nuclear structure and reaction observables

Ab initio nuclear structure theory predictions validation

ptimization

power counting?

SLIDE 17

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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]

Systematic decoupling of high-momentum physics: 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]

Systematic decoupling of high-momentum physics: 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]

Systematic decoupling of high-momentum physics: 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]

Systematic decoupling of high-momentum physics: 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]

Systematic decoupling of high-momentum physics: 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]

Systematic decoupling of high-momentum physics: 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]

Systematic decoupling of high-momentum physics: 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]

Systematic decoupling of high-momentum physics: 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]

Systematic decoupling of high-momentum physics: 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

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 28

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 29

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 30

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 31

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 32

elimination of coupling between low- and high momentum components,

simplified many-body calculations

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 33

Ground state energies of nuclei based on consistently evolved 3NF interactions

very promising results for light nuclei, issues for heavier nuclei

NN (N3LO) + 3NF (N2LO, 500 MeV)

29
28
27
26
25
24
23

. Egs [MeV]

(a) NN only

4He 20 MeV

(b) NN 3N-induced

exp.

(c) NN 3N-full

2 4 6 8 10 12 14 Nmax

34
32
30
28
26
24
22

. Egs [MeV]

(d)

6Li 20 MeV 2 4 6 8 10 12 14 Nmax

(e)

exp. 2 4 6 8 10 12 14 Nmax

(f)

@

100
90
80
70
60

. Egs [MeV]

(a) NN only

12C 20 MeV

(b) NN 3N-induced

exp.

(c) NN 3N-full

2 4 6 8 10 12 14 Nmax

180
160
140
120
100

. Egs [MeV]

(d)

16O 20 MeV 2 4 6 8 10 12 Nmax

(e)

exp. 2 4 6 8 10 12 Nmax

(f)

Roth, Langhammer, Calci, Binder, Navratil, PRL 107, 072501 (2011)

remarkable agreement of different MB calculations for a given Hamiltonian
calculations are based on NN (N3LO) and 3NF (N2LO) forces
need to quantify theoretical uncertainties

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

xygen isotopes

SLIDE 34

28 29 30 31 32

Neutron Number N

1 2 3

∆n

(3) (MeV)

28 29 30 31 32 8 10 12 14 16 18

S2n (MeV)

AME2003 TITAN NN+3N (MBPT) NN+3N (emp) TITAN+ AME2003

Gallant et al. PRL 109, 032506 (2012)

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

Wienholtz et al. Nature 498, 346 (2013)

Calculations and measurements of neutron-rich nuclei

high precision mass measurements at TITAN showed that 52Ca is 1.74

MeV more bound compared to atomic mass evaluation

neutron separation energies agree well with MBPT calculations based
n NN+3NF chiral interactions
need to quantify theoretical uncertainties

SLIDE 35

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)

Ground state energies of medium-mass and heavy nuclei

significant overbinding of heavy nuclei
need to quantify and reduce theoretical uncertainties

Binder, Calci, Langhammer, Roth

Phys. Lett B736, 119 (2014)

SLIDE 36

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)

Ground state energies of medium-mass and heavy nuclei

significant overbinding of heavy nuclei
need to quantify and reduce theoretical uncertainties
EFT power counting?
missing NN and/or many-body contributions?
optimized fitting procedures?

Binder, Calci, Langhammer, Roth

Phys. Lett B736, 119 (2014)

SLIDE 37

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 38

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 39

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)

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

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 40

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

Reproduction of saturation point without readjusting parameters!

“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

SLIDE 41

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, in preparation

SLIDE 42

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 43

First application to isospin asymmetric nuclear matter

uncertainty bands determined

by set of 7 Hamitonians

Drischler, KH, Schwenk, in preparation

x = np np + nn

SLIDE 44

0.05 0.1 0.15

n [fm-3]

5 10 15 20

E/N [MeV]

EM 500 MeV + N

2LO 3N

EM 500 MeV + N

3LO 3N + 4N

EM 500 MeV, RG evolved + N

2LO 3N

SCGF (Carbone et al.) CC (Hagen et al.) MBPT (Corragio et al.)

First complete calculations of neutron matter at N3LO

bands include uncertainties from many-body calculations and NN, 3NF and 4NF
good agreement with other methods
significant contributions from 3NF at N3LO

Tews, Krueger, KH, Schwenk, PRL 110, 032504 (2013) KH, Holt, Menendez, Schwenk, in press

SLIDE 45

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 46

Symmetry energy and neutron skin constraints

0.15 0.18 0.21

Rskin (fmD

3.2 3.3 3.4 3.5

Rp (fmD A

3.4 3.5 3.6

Rn (fmD B

2.0 2.4 2.8

αD (fmn D C

Hagen et al.,

ab intio coupled cluster calculations of neutron skin and dipole polarizability of 48Ca

SLIDE 47

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 48

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 49

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 50

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 51

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 52

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

new observatories could significantly improve constraints

9.7 − 13.9 km

Large

bservatory

for X-ray timing

SLIDE 53

constructed 3 representative EOS compatible with uncertainty bands for

astrophysical applications: soft, intermediate and stiff

allows to probe impact of current theoretical EOS uncertainties on

astrophysical observables

Representative set of EOS

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

100 1000

[MeV / fm3]

1 10 100 1000

P [MeV / fm3]

8 10 12 14 16

Radius [km]

0.5 1 1.5 2 2.5 3

Mass [Msun]

causality

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

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

develop the most advanced chiral Hamiltonians to enable

controlled microscopic calculations of matter and light as well as medium-mass nuclei

improve EOS constraints at high densities (LOFT, GW waves, ?),

explore limits of chiral EFT interactions

extend EOS calculations to finite temperature
calculate response functions and neutrino interactions in matter
benchmarks between different many-body frameworks based on

a set of Hamiltonians

derivation of systematic uncertainty estimates by

performing order-by-order calculations in chiral expansion

Future directions, open problems

SLIDE 56

In collaboration with: computing support:

JUROPA

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

international collaborator in

SLIDE 57

Thank you!

SLIDE 58

backup slides

SLIDE 59

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 60

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 61

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 62

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

SLIDE 63

Goal Calculate matrix elements of 3NF in a momentum partial-wave decomposed form, which is suitable for all these few- and many-body frameworks.

Chiral 3N forces at subleading order (N3LO)

SLIDE 64

Goal Calculate matrix elements of 3NF in a momentum partial-wave decomposed form, which is suitable for all these few- and many-body frameworks. Challenge Due to the large number of matrix elements, the traditional way of computing matrix elements requires extreme amounts of computer resources.

Chiral 3N forces at subleading order (N3LO)

SLIDE 65

Goal Calculate matrix elements of 3NF in a momentum partial-wave decomposed form, which is suitable for all these few- and many-body frameworks. Challenge Due to the large number of matrix elements, the traditional way of computing matrix elements requires extreme amounts of computer resources. Strategy Development of a general framework, which allows to decompose efficiently arbitrary local 3N interactions.

Chiral 3N forces at subleading order (N3LO)

o o oo
o
o o o o
o

2 4 6 8 q @fm-1D

0.007
0.006
0.005
0.004
0.003

V123@fm4D

o o o o
o oo
o o o o
o o o o
o

2 4 6 8 q @fm-1D

0.005
0.004
0.003
0.002
0.001

V123@fm4D

o o
o oo
o
o o
o

2 4 6 8 q @fm-1D

0.002
0.001

0.001 0.002 V123@fm4D

o o
o oo
o
o o
o

2 4 6 8 q @fm-1D

0.0010
0.0005

0.0005 0.0010 V123@fm4D

o o o o o
o o o o
o o o o o
2

4 6 8 q @fm-1D

0.0005
0.0004
0.0003
0.0002
0.0001

V123@fm4D

o
2

4 6 8 q @fm-1D

0.0005

0.0005 0.0010 V123@fm4D

perfect agreement with results based on traditional approach
speedup factors of >1000
very general, can also be applied to pion-full EFT, N4LO terms, currents...

SLIDE 66

Incorporation in different many-body frameworks

valence shell model Hyperspherical harmonics coupled cluster method no-core shell model Faddeev, Faddeev-Yakubovski

Holt (TRIUMF), Menendez, Simonis, Schwenk (TU Darmstadt) Binder, Hagen, Papenbrock (Oak Ridge) Roth (TU Darmstadt), Navratil (TRIUMF), Vary (Iowa) Bacca (TRIUMF), Barnea (Hebrew U.) Nogga (Juelich), Witala (Kracow)

Many-body perturbation theory Self-consistent Greens function In-medium SRG

Bogner (MSU), Hergert (OSU), Holt (TRIUMF)

!"#$! VNN V3N V3N V3N VNN VNN V3N V3N VNN V3N

Required inputs:

1. consistent NN and 3N forces at N3LO in partial-wave-decomposed form
2. softened forces for judging approximations and pushing to heavier nuclei

Barbieri (Surrey), Duguet, Soma (CEA)

SLIDE 67

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 68

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 69

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 70

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 71

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 72

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 73

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 74

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 75

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 76

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 77

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 78

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 79

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 80

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Systematic decoupling of high-momentum physics: 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 81

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 82

elimination of coupling between low- and high momentum components,

simplified many-body calculations

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 83

Applications of chiral 3N forces at N3LO

Problem: Basis size for converged results of ab initio calculations including 3N forces grows rapidly with the number of particles. Calculations limited to light nuclei. Strategy: Use SRG transformations to decouple low- and high momentum states. Required basis size decreases drastically.

Hebeler PRC(R) 85, 021002 (2012)

First implementation of consistent SRG evolution of 3NF in a momentum basis:

SLIDE 84

Problem: Basis size for converged results of ab initio calculations including 3N forces grows rapidly with the number of particles. Calculations limited to light nuclei. Strategy: Use SRG transformations to decouple low- and high momentum states. Required basis size decreases drastically.

Hebeler PRC(R) 85, 021002 (2012)

First implementation of consistent SRG evolution of 3NF in a momentum basis:

2 4 6 8 10 12 14 16 18 20 22 Nmax

9
8.5
8
7.5
7
6.5
6

Egs [MeV] λ=infty λ=3.0 fm

1

λ=2.0 fm

1

λ=1.8 fm

1

λ=1.6 fm

1

λ=1.4 fm

1

3H (N2LO 450/500 MeV)

Transformation to HO basis:

Applications of chiral 3N forces at N3LO

SLIDE 85

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

λ [fm

−1]

6 8 10 12 14 16 18 20

Energy per neutron [MeV]

3N-full 3N-induced NN-only

n=ns EM 500 MeV

First results for neutron matter equation of state based on consistently evolved 3N (N2LO) forces

0.05 0.1 0.15 0.2

n [fm

−3]

5 10 15 20

Energy per neutron [MeV]

λ=2.8 fm

1

λ=2.4 fm

1

λ=2.0 fm

1

λ=1.8 fm

1

3N-full 3N-induced NN-only

EM 500 MeV

3NF contributions treated in Hartree-Fock approximation
no indications for unnaturally large 4N force contributions

KH and Furnstahl, PRC 87, 031302(R) (2013)

SLIDE 86

different decoupling patterns (e.g.

Vlow k)

improved efficiency of evolution
suppression of many-body forces

k2 k2

transformation of evolved interactions to oscillator basis
application to nuclei, complimentary to HO evolution

(already implemented and tested)

study of various generators
application to infinite systems
equation of state (first applications to neutron matter)
systematic study of induced many-body contributions

low k

Λ0 Λ1 Λ2 k’ k

evolution of arbitrary operators
needed for all observables
study of correlations in nuclear systems factorization

3NF evolution in momentum basis: Current developments and applications

170
160
150
140
130
120
110
100

. E [MeV] NN-only exp. NN+3N-ind. 16O Ω = 20 MeV NN+3N-full 2 4 6 8 10 12 14

240
220
200
180
160
140
120

. E [MeV] exp. 2 4 6 8 10 12 14 24O Ω = 20 MeV 2 4 6 8 10 12 14

600
550
500
450
400
350
300
250

. E [MeV] NN-only exp. NN+3N-ind. 40Ca Ω = 20 MeV NN+3N-full 2 4 6 8 10 12 14 emax

800
700
600
500
400
300

. E [MeV] exp. 2 4 6 8 10 12 14 emax 48Ca Ω = 20 MeV 2 4 6 8 10 12 14 emax

SLIDE 87

1.5 2 3 4 5 7 10 15 λ [fm

−1]

−8.5 −8.4 −8.3 −8.2 −8.1 Egs [MeV] NN-only 0.0001 0.001 0.01 0.1 s [fm

4]

−8.5 −8.4 −8.3 −8.2 −8.1 J12 max=5 550/600 MeV Nα=42 1.5 2 3 4 5 7 10 15 λ [fm

−1]

−8.5 −8.4 −8.3 −8.2 −8.1 Egs [MeV] NN-only NN + 3N-induced 0.0001 0.001 0.01 0.1 s [fm

4]

−8.5 −8.4 −8.3 −8.2 −8.1 J12 max=5 550/600 MeV Nα=42 1.5 2 3 4 5 7 10 15 λ [fm

−1]

−8.5 −8.4 −8.3 −8.2 −8.1 Egs [MeV] NN-only NN + 3N-induced NN + 3N-full 0.0001 0.001 0.01 0.1 s [fm

4]

−8.5 −8.4 −8.3 −8.2 −8.1 J12 max=5 550/600 MeV Nα=42 1.5 2 3 4 5 7 10 15 λ [fm

−1]

−8.5 −8.4 −8.3 −8.2 −8.1 Egs [MeV] NN-only NN + 3N-induced NN + 3N-full 0.0001 0.001 0.01 0.1 s [fm

4]

−8.5 −8.4 −8.3 −8.2 −8.1 450/500 MeV 600/500 MeV 450/700 MeV 600/700 MeV J12 max=5 550/600 MeV Nα=42

exp.

RG evolution of 3N interactions in momentum space

p

q

p

q

p

q

|pqα1 |pqα2 |pqα3

1

2 3 2 2

1 1 3 3

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

dVij ds = [[Tij, Vij] , Tij + Vij] , dV123 ds = [[T12, V12] , V13 + V23 + V123] + [[T13, V13] , V12 + V23 + V123] + [[T23, V23] , V12 + V13 + V123] + [[Trel, V123] , Hs]

represent interaction in basis
explicit equations for NN and 3N flow equations

Bogner, Furnstahl, Perry PRC 75, 061001(R) (2007) Hebeler PRC(R) 85, 021002 (2012)

SLIDE 88

100 200 300

Elab (MeV)

−20 20 40 60

phase shift (degrees)

1S0

AV18 phase shifts

k = 2 fm

−1

Strategy: Use a lower-resolution version

low-pass filter low-pass filter

SLIDE 89

100 200 300

Elab (MeV)

−20 20 40 60

phase shift (degrees)

1S0

AV18 phase shifts

k = 2 fm

−1

100 200 300

Elab (MeV)

−20 20 40 60

phase shift (degrees)

1S0

AV18 phase shifts after low-pass filter

k = 2 fm

−1

Strategy: Use a lower-resolution version

low-pass filter low-pass filter

truncated interaction fails completely to reproduce original phase shifts
problem: low- and high momentum states are coupled by interaction!

SLIDE 90

First Quantum Monte Carlo based on chiral EFT interactions

Problem: Current QMC frameworks can only applied to local Hamiltonians. Conventional interactions derived within chiral EFT are nonlocal.

regulate in coordinate space in relative distance: f(r) = 1 − e−(r/R0)4
use isospin dependent terms instead of non-local operators at NLO

Strategy: Use freedom in the choice of operators and the type

f regulator to construct local Hamiltonians up to N2LO:

50 100 150 200 250

Lab. Energy [MeV]

10 20 30 40 50 60 70

Phase Shift [deg]

LO NLO N2LO PWA 50 100 150 200 250

Lab. Energy [MeV]

10 20 30 40 50

Phase Shift [deg]

LO NLO N2LO PWA 50 100 150 200 250

Lab. Energy [MeV]
25
20
15
10
5

LO NLO N2LO PWA 50 100 150 200 250

Lab. Energy [MeV]

10 20 30 LO NLO N2LO PWA

1S0 3P0 3P1 3P2

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

SLIDE 91

First Quantum Monte Carlo based on 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)

Greens Function Monte Carlo calculations for light nuclei based on chiral interactions currently in progress

SLIDE 92

Decoupling in 3NF matrix elements

450/500 MeV

ξ2 = p2 + 3 4q2 tan θ = 2 p √ 3 q

hyperradius: hyperangle:

Λ/˜ Λ

550/600 MeV

same decoupling patterns like in NN interactions

θ = π 12

KH, PRC(R) 85, 021002 (2012)

T = J = 1 2

see also KH, Furnstahl, PRC(R) 87, 031302 (2013)

SLIDE 93

n n

9Li

Λpionless

Λchiral Λ Λchiral Q mπ

Q mπ

Resolution dependence of nuclear forces

quarks+gluons/partons: typical momenta in nuclei: Q ∼ mπ QCD Effective theory for NN, 3N, many-N interactions: chiral EFT: nucleons interacting via pion exchanges and short-range contact interactions pionless EFT: unitary regime, non-universal corrections large scattering length physics:

SLIDE 94

constructed to fit scattering data (long-wavelength information!)
“hard” NN interactions contain repulsive core at small relative distance
strong coupling between low and high-momentum components, hard to solve!

Problem: Traditional “hard” NN interactions

Claim: Problems due to high resolution from interaction.

k|V |k⇥

V3N

k −k

k

−k

V

SLIDE 95