chiral three-nucleon forces: from neutron matter to neutron...

Kai Hebeler (OSU)

Hirschegg, January 29, 2013

Hirschegg 2013:Astrophysics and Nuclear Structure

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

Resolution: The higher the better?in the nuclear physics here we are interested in low-energy observables

(long-wavelength information)

Resolution: The higher the better?

• resolution of very small (irrelevant) structures can obscure this information• small details have nothing to do with long-wavelength information

in the nuclear physics here we are interested in low-energy observables

(long-wavelength information)

Strategy: Use a low-resolution version

• long-wavelength information is preserved• distortion at small distance significantly reduced• much less information necessary

In nuclear physics: Use renormalization group (RG) to change resolution.

• elimination of coupling between low- and high momentum components,many-body problem more perturbative and tractable

• 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

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 inmany-body observables

use chiral interactions as input for RG evolution

RG evolution of 3N interactions


• So far (in momentum basis): intermediate (cD) and short-range (cE) 3NF couplings fitted to few-body systems at different resolution scales:

E3H = �8.482 MeV and

coupling constants of natural size

• in neutron matter contributions from , and terms vanishcD cE c4• long-range contributions assumed to be invariant under RG evolution 2�

r4He = 1.464 fm







r4He = 1.464 fm

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 [1

033 d

yne/

cm2 ]

00

NN only, EM NN only, EGM

KH and Schwenk PRC 82, 014314 (2010)

0.05 0.10 0.15 [fm-3]

0

5

10

15

20

Ener

gy/n

ucle

on [

MeV

]

0

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)







0.8 1.0 1.2 1.4 1.6kF [fm

1]

20

15

10

5

0

5

Ener

gy/n

ucle

on [M

eV] = 1.8 fm 1

= 2.0 fm 1

= 2.2 fm 1

= 2.8 fm 1

0.8 1.0 1.2 1.4 1.6kF [fm

1]0.8 1.0 1.2 1.4 1.6

kF [fm1]

Hartree-FockEmpiricalsaturationpoint 2nd order

Vlow k NN from N3LO (500 MeV)

3NF fit to E3H and r4He

3rd order pp+hh

2.0 < 3NF < 2.5 fm1

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

r4He = 1.464 fm

symmetric nuclear matter




Otsuka et al.,PRL 105, 032501 (2010)

4

−60

−40

−20

0

8 16 2014

Neutron Number (N) Neutron Number (N)Neutron Number (N)

En

erg

y (

MeV

)

(a) Energies calculated from phenomenological forces

SDPF-MUSD-B

Exp.

(b) Energies calculated from G-matrix NN + 3N (∆) forces

NN NN + 3N (∆)

8 16 20148 16 2014

Exp.

(c) Energies calculated from V NN + 3N (∆,N LO) forces

low k 2

NN

NN + 3N (N LO) NN + 3N (∆)

Exp.2 O core

16

(d) Schematic picture of two- valence-neutron interaction induced from 3N force

FIG. 4: Ground-state energies of oxygen isotopes measured from 16O, including experimental values of the bound 16−24O.Energies obtained from (a) phenomenological forces SDPF-M [14] and USD-B [15], (b) a G matrix and including FM 3N forcesdue to ∆ excitations, and (c) from low-momentum interactions Vlow k and including chiral EFT 3N interactions at N

2LO aswell as only due to ∆ excitations [26]. The changes due to 3N forces based on ∆ excitations are highlighted by the shadedareas. (d) Schematic illustration of a two-valence-neutron interaction generated by 3N forces with a nucleon in the 16O core.

tween valence neutrons. Using microscopic NN and 3Nforces as well as known SPE, our shell-model calculationsnaturally explain why 24O is the heaviest oxygen isotope.The changes due to 3N forces are amplified and testablein neutron-rich nuclei and are expected to play a crucialrole for matter at the extremes.We thank S. Bogner, R. Furnstahl, A. Nogga and T.

Nakamura for useful discussions. This work was sup-ported in part by grants-in-aid for Scientific Research (A)20244022 and (C) 18540290, by the JSPS Core-to-Coreprogram EFES, by EMMI and NSERC. TRIUMF re-ceives funding via a contribution through the NationalResearch Council Canada. Part of the numerical calcu-lations have been performed at the JSC, Jülich, Germany.

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Guillemaud-Mueller et al., Phys. Rev. C 41, 937 (1990);M. Fauerbach et al., Phys. Rev. C 53, 647 (1996).

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Hoffman et al., Phys. Rev. Lett. 100, 152502 (2008); R.Kanungo et al., Phys. Rev. Lett. 102, 152501 (2009).

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[8] E. Epelbaum, H.-W. Hammer and U.-G. Meißner, Rev.Mod. Phys. 81, 1773 (2009).

[9] S. C. Pieper and R. B. Wiringa, Annu. Rev. Nucl. Part.Sci. 51, 53 (2001); S. C. Pieper, Nucl. Phys. A 751, 516(2005).

[10] P. Navratil et al., Phys. Rev. Lett. 99, 042501 (2007).[11] M. Hjorth-Jensen, T. T. S. Kuo and E. Osnes, Phys.

Rept. 261, 125 (1995).[12] S. K. Bogner, T. T. S. Kuo and A. Schwenk, Phys. Rept.

386, 1 (2003); S. K. Bogner et al., Nucl. Phys. A 784,

79 (2007).[13] D. R. Entem and R. Machleidt, Phys. Rev. C 68,

041001(R) (2003).[14] Y. Utsuno et al., Phys. Rev. C 60, 054315 (1999); Phys.

Rev. C 70, 044307 (2004).[15] B. A. Brown and W. A. Richter, Phys. Rev. C 74, 034315

(2006).[16] R. K. Bansal and J. B. French, Phys. Lett. 11, 145

(1964).[17] E. Caurier et al., Rev. Mod. Phys. 77, 427 (2005).[18] T. Otsuka et al., Phys. Rev. Lett 104, 012501 (2010).[19] T. Otsuka et al., Phys. Rev. Lett 87, 082502 (2001).[20] T. Otsuka et al., Phys. Rev. Lett. 95, 232502 (2005).[21] A. P. Zuker, Phys. Rev. Lett. 90, 042502 (2003).[22] G. E. Brown and A. M. Green, Nucl. Phys. A 137, 1

(1969).[23] A. M. Green, Rep. Prog. Phys. 39, 1109 (1976).[24] U. van Kolck, Phys. Rev. C 49, 2932 (1994); E. Epel-

baum et al., Phys. Rev. C 66, 064001 (2002).[25] The one-∆ contribution also corresponds to the leading

(NLO) 3N force in an EFT with explicit ∆ fields [8].[26] S. K. Bogner et al., arXiv:0903.3366 [nucl-th]. We use

the 3N couplings fit to the 3H binding energy and the4He radius for the smooth-cutoff Vlow k with Λ = Λ3N =2.0 fm−1. The one-∆ excitation 3N force corresponds toparticular values for the two-pion-exchange part [8].

[27] M. Stanoiu et al., Phys. Rev. C 69, 034312 (2004).[28] G. Hagen et al., Phys. Rev. C 76, 034302 (2007).[29] The inclusion of multipole terms changes the oxygen

ground-state energies by less than 300 keV.[30] S. Fritsch, N. Kaiser and W. Weise, Nucl. Phys. A 750,

259 (2005); L. Tolos, B. Friman and A. Schwenk, Nucl.Phys. A 806, 105 (2008); K. Hebeler and A. Schwenk,arXiv:0911.0483 [nucl-th].

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[34] Y. Utsuno et al., Phys. Rev. C 64, 011301(R) (2001).[35] G. Hagen et al., Phys. Rev. C 80, 021306(R) (2009).

nuclear shell model




r4He = 1.464 fm




• Ideal case: evolve 3NF consistently with NN interactions within the SRG

• has been achieved in oscillator basis (Jurgenson, Roth)• promising results in very light nuclei • puzzling effects in heavier nuclei (higher-body forces?)• not immediately applicable to infinite systems• limitations on ~⌦




r4He = 1.464 fm

RG evolution of 3N interactions 3

-100

-90

-80

-70

-60

.

E gs[MeV]

(a)NN only

12C!Ω = 20MeV

(b)NN+3N-induced

exp.

(c)NN+3N-full

2 4 6 8 10 12 14 ∞Nmax

-180

-160

-140

-120

-100

.

E gs[MeV]

(d)16O

!Ω = 20MeV

2 4 6 8 10 12 ∞Nmax

(e)

exp.

2 4 6 8 10 12 ∞Nmax

(f)

FIG. 2: (color online) IT-NCSM ground-state energies for 12C and16O as function of Nmax for the three types of Hamiltonians and arange of flow parameters (for details see Fig. 1).

initial NN interaction are negligible in the α-range consideredhere, indicating that the NN+3N-induced Hamiltonian is uni-tarily equivalent to the initial NN Hamiltonian. The extrapo-lated ground-state energies for different α are summarized inTab. I.By including the initial chiral 3N interaction, i.e., by using

the NN+3N-full Hamiltonian, the ground-state energies arelowered and are in good agreement with experiment for both,4He and 6Li. As for the NN+3N-induced there is no sizableα-dependence in the range considered here. We conclude thatinduced 3N terms originating from the initial NN interactionare important, but that induced 4N (and higher) terms are notrelevant for light p-shell nuclei, since the ground-state ener-gies obtained with the NN+3N-induced and the NN+3N-fullHamiltonian are practically α-independent.This picture changes if we consider nuclei in the upper p-

shell. In Fig. 2 we show the first accurate ab initio calcula-

TABLE I: Summary of Nmax-extrapolated IT-NCSMground-state en-ergies in MeV for !Ω = 20MeV (see text).

α [fm4] 4He 6Li 12C 16ONN 0.05 -28.08(2) -31.5(2) -99.1(6) -161.0(2)only 0.0625 -28.25(1) -31.8(1) -101.4(3) -164.9(6)

0.08 -28.38(1) -32.2(1) -103.7(2) -170.2(4)NN+ 0.05 -25.33(1) -27.7(2) -76.9(2) -119.5(3)3N-ind. 0.0625 -25.34(1) -27.6(2) -77.2(1) -119.7(6)

0.08 -25.34(1) -27.6(1) -77.4(2) -119.5(2)NN+ 0.05 -28.45(3) -31.8(2) -96.1(4) -143.7(2)3N-full 0.0625 -28.45(1) -31.8(1) -96.8(3) -145.6(2)

0.08 -28.46(1) -31.8(1) -97.6(1) -147.8(1)exp. -28.30 -31.99 -92.16 -127.62

0 0.05 0.1 0.15α [fm4]

-29

-28

-27

-26

-25

.

E gs,∞

[MeV]

(a)

4He!Ω = 20MeV

0 0.05 0.1 0.15α [fm4]

-180

-170

-160

-150

-140

-130

-120

-110

.

E gs,∞

[MeV]

(b)

16O

!Ω = 20MeV

FIG. 3: (color online) Nmax-extrapolated ground-state energies of 4Heand 16O as function of the flow parameter α for the NN-only (•), theNN+3N-induced ( !), and the NN+3N-full Hamiltonian (").

tions for the ground states of 12C and 16O starting from chiralNN+3N interactions. By combining the IT-NCSM with theJT -coupled storage scheme for the 3N matrix elements weare able to reach model spaces up to Nmax = 12 for the upperp-shell at moderate computational cost. Previously, even themost extensive NCSM calculations including full 3N interac-tions were limited to Nmax = 8 in this regime [16]. As evidentfrom the Nmax-dependence of the ground-state energies, thisincrease in Nmax is vital for obtaining precise extrapolations.The general pattern for 12C and 16O is similar to the light

p-shell nuclei: The NN-only Hamiltonian exhibits a severeα-dependence indicating sizable induced 3N contributions.Their inclusion in the NN+3N-induced Hamiltonian leads toground-state energies that are practically independent of α,confirming that induced 4N contributions are irrelevant whenstarting from the NN interaction only. Therefore, the NN+3N-induced results can be considered equivalent to a solution forthe initial NN interaction. The 16O binding energy per nucleonof 7.48(4)MeV is in good agreement with a recent coupled-cluster Λ-CCSD(T) result of 7.56MeV for the ‘bare’ chiralNN interaction [17].In contrast to light nuclei the ground-state energies of 12C

and 16O obtained with the NN+3N-full Hamiltonian do showa significant α-dependence, as evident from Fig. 2(c) and (f).The inclusion of the initial chiral 3N interaction does induce4N contributions whose omission leads to the α-dependence.A direct comparison of the α-dependence of the extrapo-

lated ground-state energies for 4He and 16O is presented inFig. 3. For both nuclei, the NN-only Hamiltonian exhibitsa sizable variation of the ground-state energies of about 25MeV (0.7 MeV) for 16O (4He) in the range from α = 0.04 fm4to 0.16 fm4. The inclusion of the induced 3N terms elimi-nates this α-dependence. The inclusion of the initial 3N in-teraction again generates an α-dependence of about 10 MeVfor 16O. Note that the induced 4N (and higher) contributionsthat are needed to compensate the α-dependence for 16O reachabout half the size of the total 3N contribution in the SRG-transformed Hamiltonian. This is evidence that the hierarchyof the many-body forces in chiral EFT may not be preservedby the SRG transformation.

Roth et al. PRL 107, 072501 (2011)

• Ideal case: evolve 3NF consistently with NN interactions within the SRG

• has been achieved in oscillator basis (Jurgenson, Roth)• promising results in very light nuclei • puzzling effects in heavier nuclei (higher-body forces?)• not immediately applicable to infinite systems• limitations on ~⌦

• application to infinite systems‣ equation of state‣ systematic study of induced many-body contributions, scaling behavior‣ include initial N3LO 3N interactions

Consistent 3NF evolution in momentum basis:Current/future developments


• application to infinite systems‣ equation of state (first results for neutron matter)‣ systematic study of induced many-body contributions, scaling behavior‣ include initial N3LO 3N interactions (see talk by Thomas Krueger)


• transformation of evolved interactions to oscillator basis‣ application to finite nuclei, complimentary to HO evolution (no core shell model, coupled cluster, shell model)


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


k2

k�2


• study of various generators


Overview RG Summary Extras Flow Results History Eqs. Problem

Two ways to decouple with RG equations“Vlow k ”

Λ0

Λ1

Λ2

k’

k

Lower a cutoff �i in k , k �,e.g., demanddT (k , k �; k2)/d� = 0

Similarity RG

λ0

λ1

λ2

k’

k

Drive the Hamiltonian towarddiagonal with “flow equation”[Wegner; Glazek/Wilson (1990’s)]

Dick Furnstahl RG in Nuclear Physics

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

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


k2

k�2


• study of various generators


Overview RG Summary Extras Flow Results History Eqs. Problem

Two ways to decouple with RG equations“Vlow k ”

Λ0

Λ1

Λ2

k’

k

Lower a cutoff �i in k , k �,e.g., demanddT (k , k �; k2)/d� = 0

Similarity RG

λ0

λ1

λ2

k’

k

Drive the Hamiltonian towarddiagonal with “flow equation”[Wegner; Glazek/Wilson (1990’s)]

Dick Furnstahl RG in Nuclear Physics

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

• explicit calculation of unitary 3N transformation

‣ RG evolution of operators‣ study of correlations in nuclear systems factorization

RG evolution of 3N interactions in momentum space

|pq�⇥ i � |piqi; [(LS)J(lsi)j]JJz(Tti)T Tz⇥

pq

Three-body Faddeev basis:

p

qp

q

|pq��1 |pq��2 |pq��3

i�pq�|P |p�q��⇥i =i�pq�|p�q��⇥j

1

2

3

22

11

33

|�i� = G0�2tiP + (1 + tiG0)V i3N (1 + 2P )

⇥|�i�

Faddeev bound-state equation:

dVijds

= [[Tij , Vij ] , Tij + Vij ] ,

dV123ds

= [[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 momentum basis

�s = [Trel, Hs]dHsds

= [�s, 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 dV123ds

Hs

H = T + V12 + V13 + V23 + V123

SRG evolution in momentum space• evolve the antisymmetrized 3N interaction

• embed NN interaction in 3N basis:

V 123 =ihpq↵| (1 + P123 + P132)V (i)123(1 + P123 + P132) |p0q0↵0ii

V13 = P123V12P132, V23 = P132V12P123

with 3hpq↵|V12|p0q0↵0i3 = hp↵̃|VNN|p0↵̃0i �(q � q0)/q2

• use P123V 123 = P132V 123 = V 123

) dV 123/ds = C1(s, T, VNN, P )+ C2(s, T, VNN, V 123, P )

+ C3(s, T, V 123)

SRG evolution of 3N interactions in momentum space:Results for the Triton

Hebeler PRC(R) 85, 021002 (2012)

1.5 2 3 4 5 7 10 15 [fm 1]

8.5

8.4

8.3

8.2

8.1

E gs

[MeV

] NN-only

0.00010.0010.010.1s [fm4]

8.5

8.4

8.3

8.2

8.1

J12 max=5550/600 MeV

N =42

exp.


Hebeler PRC(R) 85, 021002 (2012)

It works:Invariance of within for consistent chiral interactions at E

3Hgs N

2LO

1.5 2 3 4 5 7 10 15 [fm 1]

8.5

8.4

8.3

8.2

8.1

E gs

[MeV

] NN-onlyNN + 3N-induced

0.00010.0010.010.1s [fm4]

8.5

8.4

8.3

8.2

8.1

J12 max=5550/600 MeV

N =42

1 eV

exp.


Hebeler PRC(R) 85, 021002 (2012)


3Hgs N

2LO

1.5 2 3 4 5 7 10 15 [fm 1]

8.5

8.4

8.3

8.2

8.1

E gs

[MeV

] NN-onlyNN + 3N-inducedNN + 3N-full

0.00010.0010.010.1s [fm4]

8.5

8.4

8.3

8.2

8.1

J12 max=5550/600 MeV

N =42

1 eV

exp.


Hebeler PRC(R) 85, 021002 (2012)


3Hgs N

2LO

1.5 2 3 4 5 7 10 15 [fm 1]

8.5

8.4

8.3

8.2

8.1

E gs

[MeV

] NN-onlyNN + 3N-inducedNN + 3N-full

0.00010.0010.010.1s [fm4]

8.5

8.4

8.3

8.2

8.1

450/500 MeV600/500 MeV450/700 MeV600/700 MeV

J12 max=5550/600 MeV

N =42

1 eV

exp.

Decoupling of matrix elements

450/500 MeV

�2 = p2 +34q2 tan � =

2 p�3 q

hyperradius: hyperangle:

�/�̃

550/600 MeV

same decoupling patterns like in NN interactions

� =⇥

12

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

T = J = 12

Universality in 3N interactions at low resolution

0 0.5 1 1.5 2

k [fm−1

]

−2

−1.5

−1

−0.5

0

0.5

1

1.5

VN

N(k

,k)

[fm

]

CD BonnArgonne v

18

Nijmegen I

Nijmegen II

0 0.5 1 1.5 2 2.5

k [fm−1

]

−2

−1.5

−1

−0.5

0

0.5

1

1.5

VN

N(0

,k)

[fm

]

Reid 93Bonn AParis

1S

0 V

NN initial

1S

0 V

NN initial

0 0.5 1 1.5 2

k [fm−1

]

−2

−1.5

−1

−0.5

0

0.5

1

1.5

Vlo

w k

(k,k

) [f

m]

CD BonnArgonne v

18

Nijmegen I

Nijmegen II

0 0.5 1 1.5 2 2.5

k [fm−1

]

−2

−1.5

−1

−0.5

0

0.5

1

1.5

Vlo

w k

(0,k

) [f

m]

Reid 93Bonn-AParis

1S

0 V

low k(k,k)

1S

0 V

low k(0,k)

Figure 17: Diagonal (left) and off-diagonal (right) momentum-space matrix elements for various phe-nomenological NN potentials initially (upper figures) and after RG evolution to low-momentum inter-actions Vlow k [5, 6] (lower figures) for a smooth regulator with Λ = 2.0 fm

−1 and nexp = 4.

0 0.5 1 1.5 2 2.5

k [fm−1

]

−2

−1.5

−1

−0.5

0

0.5

1

1.5

VN

N(k

,k)

[fm

]

EGM 450/500EGM 550/600EGM 600/600EGM 450/700

0 0.5 1 1.5 2 2.5 3

k [fm−1

]

−2

−1.5

−1

−0.5

0

0.5

1

1.5

VN

N(0

,k)

[fm

]

EGM 600/700EM 500EM 600

1S

0 N

3LO initial

1S

0 N

3LO initial

0 0.5 1 1.5 2 2.5

k [fm−1

]

−2

−1.5

−1

−0.5

0

0.5

1

1.5

Vlo

w k

(k,k

) [f

m]

EGM 450/500EGM 550/600EGM 600/600EGM 450/700

0 0.5 1 1.5 2 2.5 3

k [fm−1

]

−2

−1.5

−1

−0.5

0

0.5

1

1.5

Vlo

w k

(0,k

) [f

m]

EGM 600/700EM 500EM 600

1S

0 V

low k(k,k)

1S

0 V

low k(0,k)

Figure 18: Diagonal (left) and off-diagonal (right) momentum-space matrix elements of different N3LONN interactions (EM [20] and EGM [44]) initially (upper figures) and after RG evolution to low-momentum interactions Vlow k [5,6] (lower figures) for a smooth regulator with Λ = 2.0 fm

−1 and nexp = 4.

18

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

Universality in 3N interactions at low resolution

0 1 2 3p [fm −1]

-0.02

0

0.02

0.04

0.06

0.08

0.1

[fm

4 ]450/500 MeV600/500 MeV550/600 MeV450/700 MeV600/700 MeV

0 1 2 3 4q [fm−1]

q = 1.5 fm−1 p = 0.75 fm−1

• remarkably reduced scheme 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)

VNN V3N

V3N

V3N

First application to neutron matter:Equation of state

E =

+ +

+ +

• evolve consistently NN + 3NF in the isospin channel• calculate EOS by taking all blue-boxed contributions into account• in this approximation NN and 3NF contributions factorize

+ . . .

Hartree-Fock

VNN

VNN

++ +V3N

V3N

V3N

VNN

VNN

V3N

2nd-order/NN-ladder

Hartree-Fock

kinetic energy

3rd-order and beyond

T = 3/2

�

First results for neutron matter

• all partial waves included up to in SRG evolution and EOS calculation• consistent 3NF with and

0 0.05 0.1 0.15 0.2

n [fm−3

]

0

5

10

15

20

Ener

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3N-full

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NN-only

EM 500 MeV

J = 9/2c1 = �0.81 GeV�1 c3 = �3.2 GeV�1

KH and Furnstahl, arXiv: 1301.xxxx


0 0.05 0.1 0.15 0.2

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Resolution-scale dependence at saturation density

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

λ [fm−1

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• solid lines: NN resummed, dashed lines: 2nd order• variations: NN-only 3.6 MeV, 3N-induced: 390 keV, 3N-full: 650 keV• indications for 4N forces at small ?

2nd order NN-3N contributions at � = 1

�

KH and Furnstahl, arXiv: 1301/02.xxxx


1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

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1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

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Partial-wave convergence of 3NF contributions


Partial-wave convergence of 3NF contributions

• E3N agrees within 0.4 % with the exact result at saturation density• E3N converged in partial waves at both scales, and � = 1 � = 2.0 fm�1


✓ =⇡

20

�2 = p2 +34q2 tan � =

2 p�3 q

show dominant channel for and positive total parity:J = 1/2

✓ =⇡

15

Matrix elements of evolved 3-neutron interactions

• strong renormalization effects of long-range two-pion exchange• moderate effects in range to � = 1 � = 2.0 fm�1


✓ =⇡

20

�2 = p2 +34q2 tan � =

2 p�3 q


✓ =⇡

10



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20

�2 = p2 +34q2 tan � =

2 p�3 q


✓ =⇡

8



✓ =⇡

20

�2 = p2 +34q2 tan � =

2 p�3 q


✓ =⇡

6



✓ =⇡

20

�2 = p2 +34q2 tan � =

2 p�3 q


✓ =⇡

4



✓ =⇡

20

�2 = p2 +34q2 tan � =

2 p�3 q


✓ =⇡

3



✓ =⇡

20

�2 = p2 +34q2 tan � =

2 p�3 q


✓ =⇡

2.5


Scaling of three-body contributions

�

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

λ [fm−1

]

0

0.05

0.1

0.15

0.2

0.25

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0.35

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|E3N

|/|E

NN

|

n=ns, no initial 3NF

n=ns/2, no initial 3NF

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

λ [fm−1

]

0

0.05

0.1

0.15

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|E3N

|/|E

NN

|

n=ns, with initial 3NF

n=ns/2, with initial 3NF

n=ns, no initial 3NF

n=ns/2, no initial 3NF

• relative size of 3N contribution grows systematically towards smaller• no obvious trend with density (may be obscured by cancellations among contributions)

Constraints on the nuclear equation of state (EOS)LETTER

doi:10.1038/nature09466

A two-solar-mass neutron star measured usingShapiro delayP. B. Demorest1, T. Pennucci2, S. M. Ransom1, M. S. E. Roberts3 & J. W. T. Hessels4,5

Neutron stars are composed of the densest form of matter knownto exist in our Universe, the composition and properties of whichare still theoretically uncertain. Measurements of the masses orradii of these objects can strongly constrain the neutron starmatterequation of state and rule out theoretical models of their composi-tion1,2. The observed range of neutron star masses, however, hashitherto been too narrow to rule out many predictions of ‘exotic’non-nucleonic components3–6. The Shapiro delay is a general-relat-ivistic increase in light travel time through the curved space-timenear a massive body7. For highly inclined (nearly edge-on) binarymillisecond radio pulsar systems, this effect allows us to infer themasses of both the neutron star and its binary companion to highprecision8,9. Here we present radio timing observations of the binarymillisecond pulsar J1614-223010,11 that show a strong Shapiro delaysignature.We calculate the pulsarmass to be (1.976 0.04)M[, whichrules out almost all currently proposed2–5 hyperon or boson con-densate equations of state (M[, solar mass). Quark matter can sup-port a star thismassive only if the quarks are strongly interacting andare therefore not ‘free’ quarks12.In March 2010, we performed a dense set of observations of J1614-

2230 with the National Radio Astronomy Observatory Green BankTelescope (GBT), timed to follow the system through one complete8.7-d orbit with special attention paid to the orbital conjunction, wheretheShapirodelay signal is strongest.Thesedatawere takenwith thenewlybuilt Green Bank Ultimate Pulsar Processing Instrument (GUPPI).GUPPI coherently removes interstellar dispersive smearing from thepulsar signal and integrates the data modulo the current apparent pulseperiod, producing a set of average pulse profiles, or flux-versus-rota-tional-phase light curves. From these, we determined pulse times ofarrival using standard procedures, with a typical uncertainty of,1ms.We used themeasured arrival times to determine key physical para-

meters of the neutron star and its binary system by fitting them to acomprehensive timing model that accounts for every rotation of theneutron star over the time spanned by the fit. The model predicts atwhat times pulses should arrive at Earth, taking into account pulsarrotation and spin-down, astrometric terms (sky position and propermotion), binary orbital parameters, time-variable interstellar disper-sion and general-relativistic effects such as the Shapiro delay (Table 1).We compared the observed arrival times with the model predictions,and obtained best-fit parameters by x2 minimization, using theTEMPO2 software package13. We also obtained consistent resultsusing the original TEMPO package. The post-fit residuals, that is,the differences between the observed and the model-predicted pulsearrival times, effectively measure how well the timing model describesthe data, and are shown in Fig. 1. We included both a previouslyrecorded long-term data set and our new GUPPI data in a single fit.The long-term data determine model parameters (for example spin-down rate and astrometry) with characteristic timescales longer thana few weeks, whereas the new data best constrain parameters ontimescales of the orbital period or less. Additional discussion of the

long-termdata set, parameter covariance and dispersionmeasure vari-ation can be found in Supplementary Information.As shown in Fig. 1, the Shapiro delay was detected in our data with

extremely high significance, and must be included to model the arrivaltimes of the radio pulses correctly.However, estimating parameter valuesand uncertainties can be difficult owing to the high covariance betweenmany orbital timing model terms14. Furthermore, the x2 surfaces for theShapiro-derived companionmass (M2) and inclination angle (i) are oftensignificantly curved or otherwise non-Gaussian15. To obtain robust errorestimates, we used a Markov chain Monte Carlo (MCMC) approach toexplore the post-fitx2 space andderive posterior probability distributionsfor all timing model parameters (Fig. 2). Our final results for the model

1National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, Virginia 22093, USA. 2Astronomy Department, University of Virginia, Charlottesville, Virginia 22094-4325, USA. 3EurekaScientific, Inc., Oakland, California 94602, USA. 4Netherlands Institute for Radio Astronomy (ASTRON), Postbus 2, 7990 AA Dwingeloo, The Netherlands. 5Astronomical Institute ‘‘Anton Pannekoek’’,University of Amsterdam, 1098 SJ Amsterdam, The Netherlands.

Table 1 | Physical parameters for PSR J1614-2230Parameter Value

Ecliptic longitude (l) 245.78827556(5)uEcliptic latitude (b) 21.256744(2)uProper motion in l 9.79(7)mas yr21

Proper motion in b 230(3)mas yr21

Parallax 0.5(6)masPulsar spin period 3.1508076534271(6)msPeriod derivative 9.6216(9) 310221 s s21

Reference epoch (MJD) 53,600Dispersion measure* 34.4865pc cm23

Orbital period 8.6866194196(2) dProjected semimajor axis 11.2911975(2) light sFirst Laplace parameter (esinv) 1.1(3) 31027

Second Laplace parameter (ecosv) 21.29(3) 31026

Companion mass 0.500(6)M[Sine of inclination angle 0.999894(5)Epoch of ascending node (MJD) 52,331.1701098(3)Span of timing data (MJD) 52,469–55,330Number of TOAs{ 2,206 (454, 1,752)Root mean squared TOA residual 1.1 ms

Right ascension (J2000) 16h 14min 36.5051(5) sDeclination (J2000) 222u 309 31.081(7)99Orbital eccentricity (e) 1.30(4) 31026

Inclination angle 89.17(2)uPulsar mass 1.97(4)M[Dispersion-derived distance{ 1.2 kpcParallax distance .0.9 kpcSurface magnetic field 1.8 3108GCharacteristic age 5.2GyrSpin-down luminosity 1.2 31034 erg s21

Average flux density* at 1.4GHz 1.2mJySpectral index, 1.1–1.9GHz 21.9(1)Rotation measure 228.0(3) radm22

Timingmodel parameters (top), quantities derived from timingmodel parameter values (middle) andradio spectral and interstellar medium properties (bottom). Values in parentheses represent the 1suncertainty in the final digit, asdeterminedbyMCMCerror analysis. The fit includedboth ‘long-term’ dataspanning seven years and new GBT–GUPPI data spanning three months. The new data were observedusing an800-MHz-wide band centred at a radio frequency of 1.5GHz. The rawprofileswere polarization-and flux-calibrated and averaged into 100-MHz, 7.5-min intervals using the PSRCHIVE softwarepackage25, from which pulse times of arrival (TOAs) were determined. MJD, modified Julian date.*These quantities vary stochastically on>1-d timescales. Values presented here are the averages forour GUPPI data set.{Shown in parentheses are separate values for the long-term (first) and new (second) data sets.{Calculated using the NE2001 pulsar distance model26.

2 8 O C T O B E R 2 0 1 0 | V O L 4 6 7 | N A T U R E | 1 0 8 1

Macmillan Publishers Limited. All rights reserved©2010

parameters, withMCMC error estimates, are given in Table 1. Owing tothe high significance of this detection, our MCMC procedure and astandard x2 fit produce similar uncertainties.From the detected Shapiro delay, we measure a companion mass of

(0.50060.006)M[, which implies that the companion is a helium–carbon–oxygenwhite dwarf16. The Shapiro delay also shows the binary

system to be remarkably edge-on, with an inclination of 89.17u6 0.02u.This is the most inclined pulsar binary system known at present. Theamplitude and sharpness of the Shapiro delay increase rapidly withincreasing binary inclination and the overall scaling of the signal islinearly proportional to the mass of the companion star. Thus, theunique combination of the high orbital inclination and massive whitedwarf companion in J1614-2230 cause a Shapiro delay amplitudeorders of magnitude larger than for most other millisecond pulsars.In addition, the excellent timing precision achievable from the pulsarwith the GBT and GUPPI provide a very high signal-to-noise ratiomeasurement of both Shapiro delay parameters within a single orbit.The standardKeplerian orbital parameters, combinedwith the known

companionmass and orbital inclination, fully describe the dynamics of a‘clean’ binary system—one comprising two stable compact objects—under general relativity and therefore also determine the pulsar’s mass.Wemeasure a pulsar mass of (1.976 0.04)M[, which is by far the high-est preciselymeasured neutron star mass determined to date. In contrastwith X-ray-based mass/radius measurements17, the Shapiro delay pro-videsno informationabout theneutron star’s radius.However, unlike theX-ray methods, our result is nearly model independent, as it dependsonly on general relativity being an adequate description of gravity.In addition, unlike statistical pulsar mass determinations based onmeasurement of the advance of periastron18–20, pure Shapiro delay massmeasurements involve no assumptions about classical contributions toperiastron advance or the distribution of orbital inclinations.The mass measurement alone of a 1.97M[ neutron star signifi-

cantly constrains the nuclear matter equation of state (EOS), as shownin Fig. 3. Any proposed EOS whose mass–radius track does not inter-sect the J1614-2230 mass line is ruled out by this measurement. TheEOSs that produce the lowestmaximummasses tend to be thosewhichpredict significant softening past a certain central density. This is a

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Orbital phase (turns)

Figure 1 | Shapiro delay measurement for PSRJ1614-2230. Timing residual—the excess delaynot accounted for by the timing model—as afunction of the pulsar’s orbital phase. a, Fullmagnitude of the Shapiro delay when all othermodel parameters are fixed at their best-fit values.The solid line shows the functional form of theShapiro delay, and the red points are the 1,752timingmeasurements in ourGBT–GUPPI data set.The diagrams inset in this panel show top-downschematics of the binary system at orbital phases of0.25, 0.5 and 0.75 turns (from left to right). Theneutron star is shown in red, the white dwarfcompanion in blue and the emitted radio beam,pointing towards Earth, in yellow. At orbital phaseof 0.25 turns, the Earth–pulsar line of sight passesnearest to the companion (,240,000 km),producing the sharp peak in pulse delay.We foundno evidence for any kind of pulse intensityvariations, as from an eclipse, near conjunction.b, Best-fit residuals obtained using an orbitalmodelthat does not account for general-relativistic effects.In this case, some of the Shapiro delay signal isabsorbed by covariant non-relativistic modelparameters. That these residuals deviatesignificantly from a random, Gaussian distributionof zero mean shows that the Shapiro delay must beincluded to model the pulse arrival times properly,especially at conjunction. In addition to the redGBT–GUPPI points, the 454 grey points show theprevious ‘long-term’ data set. The drasticimprovement in data quality is apparent. c, Post-fitresiduals for the fully relativistic timing model(including Shapiro delay), which have a root meansquared residual of 1.1ms and a reduced x2 value of1.4 with 2,165 degrees of freedom. Error bars, 1s.

89.1

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Pro

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Pulsar mass (M()

Figure 2 | Results of theMCMCerror analysis. a, Grey-scale image shows thetwo-dimensional posterior probability density function (PDF) in theM2–iplane, computed from a histogram ofMCMC trial values. The ellipses show 1sand 3s contours based on a Gaussian approximation to the MCMC results.b, PDF for pulsar mass derived from the MCMC trials. The vertical lines showthe 1s and 3s limits on the pulsar mass. In both cases, the results are very welldescribed by normal distributions owing to the extremely high signal-to-noiseratio of our Shapiro delay detection. Unlike secular orbital effects (for exampleprecession of periastron), the Shapiro delay does not accumulate over time, sothe measurement uncertainty scales simply as T21/2, where T is the totalobserving time. Therefore, we are unlikely to see a significant improvement onthese results with currently available telescopes and instrumentation.

RESEARCH LETTER

1 0 8 2 | N A T U R E | V O L 4 6 7 | 2 8 O C T O B E R 2 0 1 0







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RESEARCH LETTER








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RESEARCH LETTER



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.

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 � ��

13.0 13.5 14.0log 10 [g / cm

3]

31

32

33

34

35

36

37

log 1

0P

[dyn

e/cm

2 ]

1

2

3

with ci uncertainties

crust

crust EOS (BPS)neutron star matter

12 231

�1, �12,�2, �23,�3 limited by physics!

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

0.2 0.4 0.6 0.8 1.0 [ 0]

0.5

1.0

1.5

2.0

2.5

P [1

033 d

yne/

cm2 ]

00

NN only, EM NN only, EGM

Constraints on the nuclear equation of state

use the constraints:

vs(�) =�

dP/d⇥ < c

Mmax > 1.97 M�

causality

recent NS observation

significant reduction of uncertainty band

KH, Lattimer, Pethick, Schwenk, in preparation

Constraints on the nuclear equation of state

vs(�) =�

dP/d⇥ < c

causality

NS mass

Mmax > 2.4 M�

increased systematically reduces width of bandMmax

KH, Lattimer, Pethick, Schwenk, in preparation

use the constraints:

8 10 12 14 16

Radius [km]

0

0.5

1

1.5

2

2.5

3

Mas

s [M

sun]

caus

ality

• 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 15.4log 10 [g / cm

3]

33

34

35

36

log 1

0P

[dyn

e/cm

2 ]

WFF1WFF2WFF3AP4AP3MS1MS3GM3ENGPALGS1GS2

14.2 14.4 14.6 14.8 15.0 15.2 15.4

33

34

35

36

PCL2SQM1SQM2SQM3PS

Constraints on neutron star radii

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

1.4 M� 9.8� 13.4 km

Summary• demonstrated the feasibility of SRG evolution of NN+3NF in momentum space• first results of neutron matter based on consistently evolved NN+3NF interactions• strong renormalization effects of chiral two-pion exch. interaction in neutron matter• no indications of significant contributions from 4N forces down toin neutron matter

Outlook• inclusion of 3NF N3LO contributions in RG evolution• extend RG evolution to channels, application to nuclear matter• transformation to HO basis, application to finite nuclei (CC, NCSM)• RG evolution of operators: nuclear scaling and correlations in nuclear systems

T = 1/2

� = 1.2 fm�1

chiral three-nucleon forces: from neutron matter to neutron...

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