[PPT] - Chiral three-body forces: From neutron matter to neutron stars Kai PowerPoint Presentation

SLIDE 1

Kai Hebeler (OSU)

Darmstadt, April 20, 2012

In collaboration with:

E. Anderson(OSU), S. Bogner (MSU), R. Furnstahl (OSU), J. Lattimer (Stony Brook),
A. Nogga (Juelich), C. Pethick (Nordita), A. Schwenk (Darmstadt)

EMMI program The Extreme Matter Physics of Nuclei: From Universal Properties to Neutron-Rich Extremes

Chiral three-body forces: From neutron matter to neutron stars

SLIDE 2

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, leading order 3N forces

lead to theoretical uncertainties in many-body observables

SLIDE 3

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Low-momentum interactions: The (Similarity) Renormalization Group

goal: generate unitary transformation of “hard” Hamiltonian
basic idea: change resolution in small steps:

with the resolution parameter

SLIDE 4

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Low-momentum interactions: The (Similarity) Renormalization Group

goal: generate unitary transformation of “hard” Hamiltonian
basic idea: change resolution in small steps:

with the resolution parameter

SLIDE 5

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Low-momentum interactions: The (Similarity) Renormalization Group

goal: generate unitary transformation of “hard” Hamiltonian
basic idea: change resolution in small steps:

with the resolution parameter

SLIDE 6

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Low-momentum interactions: The (Similarity) Renormalization Group

goal: generate unitary transformation of “hard” Hamiltonian
basic idea: change resolution in small steps:

with the resolution parameter

SLIDE 7

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Low-momentum interactions: The (Similarity) Renormalization Group

goal: generate unitary transformation of “hard” Hamiltonian
basic idea: change resolution in small steps:

with the resolution parameter

SLIDE 8

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Low-momentum interactions: The (Similarity) Renormalization Group

goal: generate unitary transformation of “hard” Hamiltonian
basic idea: change resolution in small steps:

with the resolution parameter

SLIDE 9

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Low-momentum interactions: The (Similarity) Renormalization Group

goal: generate unitary transformation of “hard” Hamiltonian
basic idea: change resolution in small steps:

with the resolution parameter

SLIDE 10

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Low-momentum interactions: The (Similarity) Renormalization Group

goal: generate unitary transformation of “hard” Hamiltonian
basic idea: change resolution in small steps:

with the resolution parameter

SLIDE 11

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Low-momentum interactions: The (Similarity) Renormalization Group

goal: generate unitary transformation of “hard” Hamiltonian
basic idea: change resolution in small steps:

with the resolution parameter

SLIDE 12

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Low-momentum interactions: The (Similarity) Renormalization Group

goal: generate unitary transformation of “hard” Hamiltonian
basic idea: change resolution in small steps:

with the resolution parameter

SLIDE 13

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Low-momentum interactions: The (Similarity) Renormalization Group

goal: generate unitary transformation of “hard” Hamiltonian
basic idea: change resolution in small steps:

with the resolution parameter

SLIDE 14

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Low-momentum interactions: The (Similarity) Renormalization Group

goal: generate unitary transformation of “hard” Hamiltonian
basic idea: change resolution in small steps:

with the resolution parameter

SLIDE 15

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Low-momentum interactions: The (Similarity) Renormalization Group

goal: generate unitary transformation of “hard” Hamiltonian
basic idea: change resolution in small steps:

with the resolution parameter

SLIDE 16

Hλ = UλHU †

λ

dHλ dλ = [ηλ, Hλ]

Low-momentum interactions: The (Similarity) Renormalization Group

goal: generate unitary transformation of “hard” Hamiltonian
basic idea: change resolution in small steps:

with the resolution parameter

SLIDE 17

elimination of coupling between low- and high momentum components,

calculations much easier

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.

Changing the resolution: The (Similarity) Renormalization Group

SLIDE 18

RG evolution of 3N interactions

c1, c3, c4 terms

cD term cE term

So far:

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

E3H = −8.482 MeV

r4He = 1.95 − 1.96 fm

and

Ideal case: evolve 3NF consistently with NN to lower resolution using the RG
has been achieved in oscillator basis (Jurgenson, Roth)
promising results in very light nuclei
problems in heavier nuclei
not suitable for infinite systems

coupling constants of natural size

in neutron matter contributions from , and terms vanish

cD cE c4

long-range contributions assumed to be invariant under RG evolution

2π

SLIDE 19

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-resolution interactions much more perturbative
inclusion of 3N interaction contributions crucial
use chiral interactions as initial input for RG evolution

+

. . .

Hartree-Fock

VNN VNN

+ + +

V3N V3N V3N VNN VNN V3N

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

SLIDE 20

significantly reduced cutoff dependence at 2nd order perturbation theory
small resolution dependence indicates converged calculation
energy sensitive to uncertainties in 3N interaction
variation due to 3N input uncertainty much larger than resolution dependence

Equation of state of pure neutron matter

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)

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

Hartree-Fock 2nd-order

SLIDE 21

significantly reduced cutoff dependence at 2nd order perturbation theory
small resolution dependence indicates converged calculation
energy sensitive to uncertainties in 3N interaction
variation due to 3N input uncertainty much larger than resolution dependence

Equation of state of pure neutron matter

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)

0.05 0.10 0.15

[fm-3]

5 10 15 20

Energy/nucleon [MeV]

ENN+3N,eff+c3+c1 uncertainties Schwenk+Pethick (2005) Akmal et al. (1998) QMC s-wave GFMC v6 GFMC v8’

good agreement with other approaches (different NN interactions)

Hartree-Fock 2nd-order

SLIDE 22

Symmetry energy constraints

symmetry energy parameters consistent with other constraints Sv = ∂2E/N ∂2x

ρ=ρ0,x=1/2

L = 3 8 ∂3E/N ∂ρ∂2x

ρ=ρ0,x=1/2

KH, Lattimer, Pethick and Schwenk, in preparation

extend EOS to finite proton fractions

x

and extract symmetry energy parameters

SLIDE 23

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)

Credit: NASA/Dana Berry

Mmax = 1.65M⊙ → 1.97 ± 0.04 M⊙

Calculation of neutron star properties requires EOS up to high densities. Strategy: Use observations to constrain the high-density part of the nuclear EOS.

SLIDE 24

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!

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

SLIDE 25

Constraints on the nuclear equation of state

use the constraints:

vs(ρ) =

dP/dε < c

Mmax > 1.97 M⊙

causality recent NS observation

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]

add full band

significant reduction of possible equations of state

SLIDE 26

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]

Constraints on the nuclear equation of state

use the constraints:

vs(ρ) =

dP/dε < c

causality NS mass

Mmax > 2.4 M⊙

increased systematically leads to stronger constraints

Mmax

SLIDE 27

8 10 12 14 16

Radius [km]

0.5 1 1.5 2 2.5 3

Mass [Msun]

causality

radius constraint 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, in preparation see also KH, Lattimer, Pethick, Schwenk, PRL 105, 161102 (2010)

1.4 M⊙

9.8 − 13.4 km

SLIDE 28

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 arXiv:1204.1888

high-density part of nuclear EOS only loosely constrained
simulations of NS binary mergers show strong correlation between between
f the GW spectrum and the raduis 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 29

0.8 1.0 1.2 1.4 1.6

kF [fm

!1]

!30 !25 !20 !15 !10 !5

Energy/nucleon [MeV]

" = 1.8 fm

!1 NN only

" = 2.8 fm

!1 NN only

Vlow k NN from N

3LO (500 MeV)

3NF fit to E3H and r4He "3NF = 2.0 fm

!1

3rd order pp+hh

NN only

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) nS ∼ 0.16 fm−3 Ebinding/N ∼ −16 MeV

empirical nuclear saturation properties

¯ lS ∼ 1.8 fm

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

SLIDE 30

¯

lS

nuclear saturation delicate due to cancellations of large kinetic and

potential energy contributions

¯

lS

3N forces are essential! 3N interactions fitted to and properties

0.8 1.0 1.2 1.4 1.6

kF [fm

!1]

!30 !25 !20 !15 !10 !5

Energy/nucleon [MeV]

" = 1.8 fm

!1

" = 2.8 fm

!1

" = 1.8 fm

!1 NN only

" = 2.8 fm

!1 NN only

Vlow k NN from N

3LO (500 MeV)

3NF fit to E3H and r4He "3NF = 2.0 fm

!1

3rd order pp+hh

NN + 3N NN only

3H

4He

Equation of state of symmetric nuclear matter, Nuclear saturation

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

SLIDE 31

saturation point consistent with experiment, without free parameters
cutoff dependence at 2nd order significantly reduced
3rd order contributions small
cutoff dependence consistent with expected size of 4N force contributions

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

0.8 1.0 1.2 1.4 1.6

kF [fm

!1]

!20 !15 !10 !5 5

Energy/nucleon [MeV]

" = 1.8 fm

!1

" = 2.0 fm

!1

" = 2.2 fm

!1

" = 2.8 fm

!1

0.8 1.0 1.2 1.4 1.6

kF [fm

!1]

0.8 1.0 1.2 1.4 1.6

kF [fm

!1]

Hartree-Fock

Empirical saturation point

2nd order

Vlow k NN from N

3LO (500 MeV)

3NF fit to E3H and r4He

3rd order pp+hh

2.0 < "3NF < 2.5 fm

!1

Equation of state of symmetric nuclear matter, Nuclear saturation

SLIDE 32

Hierarchy of many-body contributions

0.05 0.1 0.15 0.2 0.25 0.3 ! [fm

3]
60
40
20

20 40 Energy/nucleon [MeV] Ekinetic 0.05 0.1 0.15 ! [fm

3]
40
20

20 40 Energy/nucleon [MeV] Ekinetic

binding energy results from cancellations of much larger kinetic and potential

energy contributions

chiral hierarchy of many-body terms preserved for considered density range
cutoff dependence of natural size, consistent with chiral exp. parameter ∼ 1/3

neutron matter nuclear matter

SLIDE 33

Hierarchy of many-body contributions

0.05 0.1 0.15 0.2 0.25 0.3 ! [fm

3]
60
40
20

20 40 Energy/nucleon [MeV] Ekinetic ENN Ekinetic + ENN 0.05 0.1 0.15 ! [fm

3]
40
20

20 40 Energy/nucleon [MeV] Ekinetic ENN Ekinetic + ENN

binding energy results from cancellations of much larger kinetic and potential

energy contributions

chiral hierarchy of many-body terms preserved for considered density range
cutoff dependence of natural size, consistent with chiral exp. parameter ∼ 1/3

neutron matter nuclear matter

SLIDE 34

Hierarchy of many-body contributions

0.05 0.1 0.15 0.2 0.25 0.3 ! [fm

3]
60
40
20

20 40 Energy/nucleon [MeV] Ekinetic ENN E3N + 3N-NN Etotal 0.05 0.1 0.15 ! [fm

3]
40
20

20 40 Energy/nucleon [MeV] Ekinetic ENN E3N + 3N-NN Etotal

binding energy results from cancellations of much larger kinetic and potential

energy contributions

chiral hierarchy of many-body terms preserved for considered density range
cutoff dependence of natural size, consistent with chiral exp. parameter ∼ 1/3

neutron matter nuclear matter

SLIDE 35

RG evolution of 3N interactions in momentum space

|pqαi ≡ |piqi; [(LS)J(lsi)j] J Jz(Tti)T Tz

p

q

Three-body Faddeev basis:

p

q

p

q

|pqα1 |pqα2 |pqα3 |ψi = G0

2tiP + (1 + tiG0) V i

3N(1 + 2P)

|ψi

Faddeev bound state equations:

ipqα|P|p′q′α′i =ipqα|p′q′α′j

SLIDE 36

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]

SRG flow equations of NN and 3N forces in Faddeev basis

ηs = [Trel, Hs] dHs ds = [ηs, Hs]

spectators correspond to delta functions, matrix representation of ill-defined
solution: explicit separation of NN and 3N flow equations

see Bogner, Furnstahl, Perry PRC 75, 061001(R) (2007)

only connected terms remain in , ‘dangerous’ delta functions cancel

dV123 ds

Hs

H = T + V12 + V13 + V23 + V123

SLIDE 37

RG evolution of 3N interactions in momentum space

1.5 2 3 4 5 7 10 15 λ [fm

−1]

−8.8 −8.7 −8.6 −8.5 −8.4 −8.3 Egs [MeV] NN-only NN + 3N-induced NN + 3N-full 0.0001 0.001 0.01 0.1 s [fm

4]

−8.8 −8.7 −8.6 −8.5 −8.4 −8.3 450/500 MeV 600/500 MeV 450/700 MeV 600/700 MeV Nα=26 550/600 MeV np-only Nα=42

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

First implementation: Invariance of within for consistent chiral interactions at

E

3

H gs

16 keV

N2LO

SLIDE 38

Decoupling of 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)

SLIDE 39

Universality in 3N interactions at low resolution

0.5 1 1.5 2 2.5

k [fm

1]

2 1.5 1 0.5 0.5 1 1.5

VNN(k,k) [fm] EGM 450/500 EGM 550/600 EGM 600/600 EGM 450/700

0.5 1 1.5 2 2.5 3

k [fm

1]

2 1.5 1 0.5 0.5 1 1.5

VNN(0,k) [fm] EGM 600/700 EM 500 EM 600

1S0 N 3LO initial 1S0 N 3LO initial

0.5 1 1.5 2 2.5

k [fm

1]

2 1.5 1 0.5 0.5 1 1.5

Vlow k(k,k) [fm] EGM 450/500 EGM 550/600 EGM 600/600 EGM 450/700

0.5 1 1.5 2 2.5 3

k [fm

1]

2 1.5 1 0.5 0.5 1 1.5

Vlow k(0,k) [fm] EGM 600/700 EM 500 EM 600

1S0 Vlow k(k,k) 1S0 Vlow k(0,k)

phase-shift equivalence common long- range physics (approximate) universality of low-resolution NN interactions

To what extent are 3N interactions constrained at low resolution?

only two low-energy constants
3N interactions give only subleading contributions to observables

cD and cE

SLIDE 40

Universality in 3N interactions at low resolution

1 2 3 p [fm

−1]

0.02

0.02 0.04 0.06 0.08 0.1 <p q α | V123 | p q α > [fm

4]

450/500 MeV 600/500 MeV 550/600 MeV 450/700 MeV 600/700 MeV 1 2 3 4 q [fm

−1]

q = 1.5 fm

−1

p = 0.75 fm

−1

remarkably reduced model dependence for typical momenta ,

matrix elements with significant phase space well constrained at low resolution

new momentum structures induced at low resolution
study based on chiral interactions, improved universality at ?

∼ 1 fm−1

N2LO N3LO

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

SLIDE 41

different decoupling patterns (e.g. Vlow k)
improved efficiency of evolution
suppression of many-body forces

Current/future directions

k2 k′2

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

(no core shell model, coupled cluster)

study of alternative generators
application to infinite systems
equation of state
systematic study of induced many-body contributions
include initial N3LO 3N interactions (see also next talk!)

low k

Λ0 Λ1 Λ2 k’ k

Anderson et al. , PRC 77, 037001 (2008)

explicit calculation of unitary 3N transformation
RG evolution of operators
study of correlations in nuclear systems factorization

SLIDE 42

Summary

low-resolution interactions allow simpler calculations for nuclear systems
observables invariant under resolution changes, interpretation of results can change!
chiral EFT provides systematic framework for constructing nuclear Hamiltonians
3N interactions are essential at low resolution
nucleonic matter equation of state based on low-resolution interactions consistent

with empirical constraints

constraints for the nuclear equation of state and structure of neutron stars

Outlook

RG evolution of three-nucleon interactions: microscopic study of light nuclei and

nucleonic matter using chiral nuclear interactions at low resolution

RG evolution of operators: nuclear scaling and correlations in nuclear systems