ab initio frameworks in nuclear physics: from basic

Kai Hebeler

Ab initio frameworks in nuclear physics:from basic concepts to current developments

Belfast, August 30-September 1, 2017

Lecturer information

Teaching schedule

The final schedule for the school is in page 3 of this document. You should be able to find yournames and lecture dates on it. We would normally ask you to arrive on the day before anddepart on the day after the teaching is scheduled.

We will be providing USB keys to students that include all teaching material. We thereforerequest that you send all your teaching material (slides, question sheets, codes, ...) by Friday

11

thAugust via email to one of us. We assume most of you will bring your own laptops and

we will have A/V equipment in place. If you have any other requirements in terms of teachingand lectures (e.g. blackboard, whiteboards, microphone, etc), it would be useful to know inadvance.

Daily schedule

1. Your lectures will take place on a lecture theatre in Riddel Hall in 1 hour slots every morn-ing (9 - 12h).

2. After the morning session, students (and lecturers!) get a few hours of leisure.

3. At 16h, students will reconvene for a tutorial session with Early Career Researchers tutors.The idea of these tutorials is to discuss the morning lectures and address any issues thatmight have arisen. While this is by no means compulsory, you might want to prepare afinal slide with potential discussion points that students could take over to the tutorials.

4. A Q&A session will follow at 17h, where students will be given the opportunity to asklecturers specific questions about the lectures.

5. After dinner, 2h have been reserved for student talks from 20h on. For some PhD students,these are often their very first talks to an audience beyond their own research group. Yourfeedback as internationally leading researchers will no doubt be appreciated..

Practicalities

1. Accommodation at Queen’s University Belfast will be booked on the basis of your lec-tures, and will be on a single ensuite room. If your travel plans differ from the standardschedule, please do let us know in advance. If you plan to travel with a partner, weare afraid we must ask you to arrange the accommodation separately. Beware that, be-cause of the Bank Holiday weekend, there are limited accommodation options availablein Belfast right now.

1

The plan…

1. Motivation: goals of nuclear structure theory

The plan…


2. Basic ideas: what does ab initio mean? (my definition…)

The plan…



3. Fundamental ingredients:

The plan…




A. Nuclear interactions and chiral effective field theory

The plan…





B. Renormalization Group methods

The plan…






C. Ab initio many-body frameworks

The plan…







4. Current status and frontiers in ab initio nuclear theory

The plan…







4. Current status and frontiers in ab initio nuclear theory

ask questions!

The plan…

ask questions!

ask questions!

20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170100

N

10

20

Z

30

40

50

60

70

80

90

100

110

The nuclear landscape:New frontiers from rare isotope facilities

Ca

O

Ni

Sn

Pb

stablebeta+beta-pnalphafission

Z

www-nds.iaea.org

0

CONTENTS

I. Supranuclear Density Matter 3A. Introduction 3B. The nature of matter: Major open questions 3C. Methodology: How neutron star mass and radius

specify the EOS 4D. Current observational constraints on the

cold dense EOS 6E. Future observational constraints on the cold dense EOS 8

II. Hard X-ray Timing Techniques that Deliver M and R 8A. Waveform modeling 9

1. Factors affecting the waveform 92. Spacetime of spinning neutron stars 103. Inversion: From waveform to M and R 104. Instrument requirements and observing strategy 11

B. Spin measurements 121. Rapid rotation 122. Spin distribution and evolution 14

C. Asteroseismology 14III. Summary 16Acknowledgments 16References 16

I. SUPRANUCLEAR DENSITY MATTER

A. Introduction

Neutron stars are the densest observable objects in theUniverse, attaining physical conditions of matter that cannotbe replicated on Earth. Inside neutron stars, the state of matterranges from ions (nuclei) embedded in a sea of electrons atlow densities in the outer crust, through increasingly neutron-rich ions in the inner crust and outer core, to the supranucleardensities reached in the center, where particles are squeezedtogether more tightly than in atomic nuclei, and theorypredicts a host of possible exotic states of matter (Fig. 1).The nature of matter at such densities is one of the greatunsolved problems in modern science, and this makes neutronstars unparalleled laboratories for nuclear physics and quan-tum chromodynamics (QCD) under extreme conditions.The most fundamental macroscopic diagnostic of dense

matter is the pressure-density-temperature relation of bulkmatter, the equation of state (EOS). The EOS can be used toinfer key aspects of the microphysics, such as the role ofmany-body interactions at nuclear densities or the presence ofdeconfined quarks at high densities (Sec. I.B). Measuring theEOS of supranuclear density matter is therefore of majorimportance to nuclear physics. However, it is also critical toastrophysics. The dense matter EOS is clearly central tounderstanding the powerful, violent, and enigmatic objectsthat are neutron stars. However, neutron star–neutron star andneutron star–black hole binary inspiral and merger, primesources of gravitational waves and the likely engines of shortgamma-ray bursts (Nakar, 2007) also depend sensitively onthe EOS (Shibata and Taniguchi, 2011; Bauswein et al., 2012;Faber and Rasio, 2012; Lackey et al., 2012; Takami, Rezzolla,and Baiotti, 2014). The EOS affects merger dynamics, blackhole formation time scales, the precise gravitational wave andneutrino signals, any associated mass loss and r-processnucleosynthesis, and the attendant gamma-ray bursts and

optical flashes (Metzger et al., 2010; Hotokezaka et al.,2011; Kumar and Zhang, 2015; Rosswog, 2015). The EOSof dense matter is also vital to understanding core collapsesupernova explosions and their associated gravitational waveand neutrino emission (Janka et al., 2007).1

B. The nature of matter: Major open questions

The properties of neutron stars, like those of atomic nuclei,depend crucially on the interactions between protons andneutrons (nucleons) governed by the strong force. This isevident from the seminal work of Oppenheimer and Volkoff(1939), which showed that the maximal mass of neutron starsconsisting of noninteracting neutrons is 0.7M⊙. To stabilizeheavier neutron stars, as realized in nature, requires repulsiveinteractions between nucleons, which set in with increasingdensity. At low energies, and thus low densities, the inter-actions between nucleons are attractive, as they have to be tobind neutrons and protons into nuclei. However, to preventnuclei from collapsing, repulsive two-nucleon and three-nucleon interactions set in at higher momenta and densities.Because neutron stars reach densities exceeding those inatomic nuclei, this makes them particularly sensitive tomany-body forces (Akmal, Pandharipande, and Ravenhall,1998), and recently it was shown that the dominant uncer-tainty at nuclear densities is due to three-nucleon forces(Hebeler et al., 2010; Gandolfi, Carlson, and Reddy, 2012).

FIG. 1. Schematic structure of a neutron star. The outer layer is asolid ionic crust supported by electron degeneracy pressure.Neutrons begin to leak out of ions (nuclei) at densities∼4 × 1011 g=cm3 (the neutron drip density, which separatesthe inner from the outer crust), where neutron degeneracy alsostarts to play a role. At densities ∼2 × 1014 g=cm3, the nucleidissolve completely. This marks the crust-core boundary. In thecore, densities reach several times the nuclear saturation densityρsat ¼ 2.8 × 1014 g=cm3 (see text).

1Note that while most neutron stars, even during the binaryinspiral phase, can be described by the cold EOS that is the focus ofthis Colloquium (see Sec. I.C), temperature corrections must beapplied when describing either newborn neutron stars in theimmediate aftermath of a supernova or the hot differentially rotatingremnants that may survive for a short period of time following acompact object merger. The cold and hot EOS must of course connectand be consistent with one another.

Anna L. Watts et al.: Colloquium: Measuring the neutron star …

Rev. Mod. Phys., Vol. 88, No. 2, April–June 2016 021001-3

http://www.nds.iaea.org

20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170100

N

10

20

Z

30

40

50

60

70

80

90

100

110


Ca

O

Ni

Sn

Pb


New frontiers from rare isotope beams (e.g., FRIB)

Asymptotic freedom ?

from B. Sherrill

DFT$FRIB$

current$

Pushing to the boundaries and exploiting isotope chainsRequires critical interplay of experiment and theory

,FAIR...

RIFs

adapted from Balantekin et al.,Mod. Phys. Lett. A 29,1430010 (2014)

Z

www-nds.iaea.org

0

CONTENTS









A. Introduction











20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170100

N

10

20

Z

30

40

50

60

70

80

90

100

110


Ca

O

Ni

Sn

Pb


Z

www-nds.iaea.org

0

CONTENTS









A. Introduction










application of modern optimization and statistical methods, togetherwith high-performance computing, has revolutionized nuclear DFTduring recent years.In our study, we use quasi-local Skyrme functionals15 in the

particle–hole channel augmented by the density-dependent, zero-range pairing term. The commonly used Skyrme EDFs reproduce totalbinding energies with a root mean square error of the order of1–4MeV (refs 15, 16), and the agreement with the data can be signifi-cantly improved by adding phenomenological correction terms17. TheSkyrme DFT approach has been successfully tested over the entirechart of nuclides on a broad range of phenomena, and it usually per-forms quite well when applied to energy differences (such as S2n), radiiand nuclear deformations. Other well-calibrated mass models include

the microscopic–macroscopic finite-range droplet model (FRDM)18,the Brussels–Montreal Skyrme–HFB models based on the Hartree–Fock–Bogoliubov (HFB) method17 and Gogny force models19,20.Figure 2 illustrates the difficulties with theoretical extrapolations

towards drip lines. Shown are the S2n values for the isotopic chain ofeven–even erbium isotopes predicted with different EDF, SLy421, SV-min13, UNEDF015, UNEDF122, and with the FRDM18 and HFB-2117

models. In the region for which experimental data are available, allmodels agree and well reproduce the data. However, the discrepancybetween various predictions steadily grows when moving away fromthe region of known nuclei, because the dependence of the effectiveforce on the neutron-to-proton asymmetry (neutron excess) is poorlydetermined. In the example considered, the neutron drip line is

0

4

8

12

16

20

24

S2n

(MeV

)

Er

Neutron number, N

80 100 120 140 160

ExperimentDrip line

0

2

4

140 148 156 164N

0

4

8

58 62 66Z

N = 76 162 154

S2n

(MeV

)

S2p

(MeV

)

FRDM

HFB-21

SLy4

UNEDF1

UNEDF0

SV-min

exp

Er

Figure 2 | Calculated and experimental two-neutron separation energies ofeven–even erbium isotopes. Calculations performed in this work using SLy4,SV-min, UNEDF0 andUNEDF1 functionals are compared to experiment2 andFRDM18 andHFB-2117 models. The differences betweenmodel predictions aresmall in the region where data exist (bracketed by vertical arrows) and grow

steadily when extrapolating towards the two-neutron drip line (S2n5 0). Thebars on the SV-min results indicate statistical errors due to uncertainty in thecoupling constants of the functional. Detailed predictions around S2n5 0 areillustrated in the right inset. The left inset depicts the calculated andexperimental two-proton separation energies at N5 76.

0 40 80 120 160 200 240 280

Neutron number, N

0

40

80

120P

roto

n nu

mbe

r, Z

Two-proton drip line

Two-neutron drip line

90

110

100

Z = 50

Z = 82

N = 50

N = 82

N = 126

N = 20

N = 184

SV-min

N = 28

Z = 28

230 244

N = 258Drip line

Known nuclei

Stable nuclei

S2n = 2 MeV

Z = 20

232 240 248 256

Figure 1 | Nuclear even–even landscape as of 2012. Mapof bound even–evennuclei as a function of Z and N. There are 767 even–even isotopes knownexperimentally,2,3 both stable (black squares) and radioactive (green squares).Mean drip lines and their uncertainties (red) were obtained by averaging theresults of different models. The two-neutron drip line of SV-min (blue) is

shown together with the statistical uncertainties at Z5 12, 68 and 120 (blueerror bars). The S2n5 2MeV line is also shown (brown) together with itssystematic uncertainty (orange). The inset shows the irregular behaviour of thetwo-neutron drip line around Z5 100.

RESEARCH LETTER

5 1 0 | N A T U R E | V O L 4 8 6 | 2 8 J U N E 2 0 1 2

Macmillan Publishers Limited. All rights reserved©2012

Erler et al., Nature 486, 509 (2012)


20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170100

N

10

20

Z

30

40

50

60

70

80

90

100

110


Ca

O

Ni

Sn

Pb


Z

www-nds.iaea.org

0

CONTENTS









A. Introduction










2 D. Habs, P.G. Thirolf, M. Gross, K. Allinger, J. Bin, A. Henig, D. Kiefer, W. Ma, J. Schreiber

Fig. 1 Chart of the nuclides indicating various pathways for astrophysical nucleosynthesis: thermonuclear fusion reactions instars (orange vector), s-process path (red vector) and the r-process generating heavy nuclei in the Universe (red pathway).The nuclei marked in black indicate stable nuclei. For the green nuclei some nuclear properties are known, while the yellow, yetunexplored regions extend to the neutron and proton drip lines. The blue line connects nuclei with the same neutron/protonratio as for (almost) stable actinide nuclei. On this line the maximum yield of nuclei produced via fission-fusion (withoutneutron evaporation) will be located. The elliptical conture lines correspond to the expected maximum fission-fusion crosssections decreased to 50% ,10% and 0.1%, respectively, for primary 232Th beams.

beam bunches of solid-state density. ii) The strongly re-duced stopping power of these dense bunches in a sec-ond thick Th target, where the decomposition into fis-sion fragments and the fusion of these fragments takesplace. After the laser flash we want to extract ratherlong-lived isotopes (> 100 ms) in flight, separate theme.g. in a (gas-filled) recoil separator and study them viadecay spectroscopy or lifetime and nuclear mass mea-surements.

In the following we outline the relevance of the projectfor nuclear astrophysics, describe the new laser acceler-ation scheme and in particular the new fission-fusion re-action method. Finally the planned ELI-Nuclear Physicsfacility will be briefly introduced, where the productionof these nuclei and the experiments to measure theirproperties will be realized.

2 The Relevance of the N=126 Waiting Pointfor Nuclear Astrophysics

Fig. 1 shows the nuclidic chart marked with differentnucleosynthesis pathways for the production of heavyelements in the Universe: the thermonuclear fusion pro-cesses in stars producing elements up to iron (orange ar-row), the slow neutron capture process (s-process) alongthe valley of stability leading to about half of the heaviernuclei (red arrow) and the rapid neutron capture pro-cess (r-process) proceeding along pathways with neu-tron separation energies Sn in the range of 2–3 MeV. Inthis scenario, rather neutron-rich nuclei are populatedin an intense neutron flux [9]. The r-process path ex-

hibits characteristic vertical regions for constant magicneutron numbers of 50, 82 and 126, where the r-processis slowed down due to low neutron capture cross sectionswhen going beyond the magic neutron numbers. Thesedecisive bottlenecks of the r-process flow are called wait-ing points [10].

The astrophysical site of the r-process nucleosynthe-sis is still under debate: it may be cataclysmic core col-lapse supernovae (II) explosions with neutrino winds [2,3,11,12] or mergers of neutron-star binaries [13,14,15].The r-process element abundances from galactic halostars tell us that the r-process site for lighter and heavierneutron capture processes may occur under different as-trophysical conditions [10]. For the heavier elements be-yond barium, the isotopic abundancies are always verysimilar (called universality) and the process seems to bevery robust. Perhaps also the recycling of fission frag-ments from the end of the r-process strengthens thisstability. Presently, it seems more likely that a mergerof neutron star binaries is the source for the heavier r-process branch, while core collapsing supernova explo-sions contribute to the lighter elements below barium.The modern nuclear equations of state, neutrino inter-actions and recent supernova explosion simulations [3]lead to detailed discussions of the waiting point N=126.Here measured nuclear properties along the N=126 wait-ing point may help to clarify the sites of the r-process.

Fig. 2 shows the measured solar elemental abundancesof the r-process nuclei together with a calculation, wheremasses from the Extended Thomas-Fermi plus Strutin-ski Integral (ETFSI) mass model [16] have been used to-

figure taken from Habs et al.,Appl. Phys. B 103, 471 (2011)


20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170100

N

10

20

Z

30

40

50

60

70

80

90

100

110

The theoretical nuclear landscape:Scope of ab initio methods for atomic nuclei

0

The 1980’s…

20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170100

N

10

20

Z

30

40

50

60

70

80

90

100

110

Ca

O

Ni

Sn


…2017

20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170100

N

10

20

Z

30

40

50

60

70

80

90

100

110

Ca

O

Ni

Sn


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

…2017

LQCD = 1

4F aµF

aµ + q(iµ@µ m)q + gqµTaqAaµ

AB INITIO:The theory of the strong interaction (QCD)

• theory perturbative at high energies

• highly non-perturbative at low energies

↵s g

4

nuclear structure and reaction observables

Quantum Chromodynamics

Ab initio nuclear structure and reaction theory

Lattice QCD

• requires extreme amounts of computational resources

• currently limited to 1- or 2-nucleon systems

• current accuracy insufficient for precision nuclear structure




Chiral effective field theorynuclear interactions and currents






ab initio many-body frameworksFaddeev, Quantum Monte Carlo, no-core shell model, coupled cluster ...






Renormalization Group methods

ab initio many-body frameworksFaddeev, Quantum Monte Carlo, no-core shell model, coupled cluster ...


Ab initio: my definition for this lecture

Methods for solving the quantum mechanical

many-body problem, which…


• …use neutrons and protons as fundamental degrees of freedom





• …require only nuclear interactions as microscopic input






• …treat all particles explicitly as active degrees of freedom









• in practice calculations are not exact due to finite computational resources








• most frameworks in addition employ approximations on conceptual level,

but these can be systematically improved (we still call them ‘ab initio’)





Do we expect perfect agreement with experiment when assuming infinite computational resources for ‘exact’ methods?




• most frameworks in addition employ approximations on conceptual level,

but these can be systematically improved (we still call them ‘ab initio’)

The many-body Schrödinger equation

H | i = T + V | i = E | i

| i

T

V = VNN + V3N + ...+ VAN

E

: intrinsic A-body wave function (center-of-mass motion decoupled)

: relative kinetic energy

: interparticle interactions

: energy eigenvalues (binding energy, nuclei masses, excitation spectra…)

: Hamiltonian in center-of-mass reference frameH

The many-body Schrödinger equation

H | i = T + V | i = E | i

| i

T

V = VNN + V3N + ...+ VAN

E

: intrinsic A-body wave function (center-of-mass motion decoupled)

: relative kinetic energy

: interparticle interactions

: energy eigenvalues (binding energy, nuclei masses, excitation spectra…)

: Hamiltonian in center-of-mass reference frameH

How are the interaction terms , ,… determined?VNN V3N

Overview RG Summary Extras Physics Resolution Forces Filter Coupling

Why is textbook nuclear physics so hard?

VL=0(k , k )

r2 dr j0(kr) V (r) j0(k r) = k |VL=0|k = Vkk matrix

Momentum units ( = c = 1): typical relative momentumin large nucleus 1 fm1 200 MeV but . . .

Repulsive core = large high-k ( 2 fm1) componentsDick Furnstahl RG in Nuclear Physics

• constructed to fit NN-scattering data (long-wavelength information)

• long-range part dominated by one pion exchange interaction

• short range part strongly model dependent!

• traditional NN interactions contain strongly repulsive core at small distance‣ strong coupling between low and high-momenta‣ many-body problem hard to solve!

“Traditional” NN interactions

k|V |k

V3N

k k

k k

V



VL=0(k , k )






VL=0(k , k )




• constructed to fit NN-scattering data (long-wavelength information)

• long-range part dominated by one pion exchange interaction

• short range part strongly model dependent!

• traditional NN interactions contain strongly repulsive core at small distance‣ strong coupling between low and high-momenta‣ many-body problem hard to solve!

“Traditional” NN interactions

k|V |k

V3N

k k

k k

V



VL=0(k , k )



Repulsive core = large high-k ( 2 fm1) componentsDick Furnstahl RG in Nuclear PhysicsIs it a problem that various potentials look quite different?

How do we estimate uncertainties for many-body observables?

Modern approach to nuclear interactions:Chiral effective field theory


• if a nucleus is probed at high energies, nucleon substructure is resolved

• at low energies, details are not resolved


• 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 in multipole expansion)

Basic ideas of effective theories:Multipole expansion

Figure 2: A localized charge distribution generates an electrostatic potential which can be describedin terms of the multipole expansion.

3 Chiral perturbation theory: An elementary introduction

E↵ective field theories have proved to be an important and very useful tool in nuclear and particlephysics. One understands under an e↵ective (field) theory an approximate theory whose scope is todescribe phenomena which occur at a chosen length (or energy) range. The main idea of this methodcan be illustrated with the following example from classical electrodynamics. Consider a localizedcharge distribution in space of a size a. The resulting electrostatic potential at any given position ~Rcan be calculated by integrating over the elementary charges and using the familiar expression for theCoulomb potential generated by a point charge:

V (~R) /Z

d3r(~r )

|~R ~r |(3.11)

Expanding 1/|~R ~r | for r R,

1

|~R ~r |=

1

R+X

i

riRi

R3+

1

2

X

ij

rirj3RiRj ijR2

R5+ . . . , (3.12)

with i, j denoting the Cartesian components allows to rewrite the integral as

Z

d3r(~r )

|~R ~r |=

q

R+

1

R3

X

i

RiPi +1

6R5

X

ij

(3RiRj ijR2)Qij + . . . (3.13)

8

Consider electrostatic potential generated

by a localized charge distribution :(r)

Figure taken from E. Epelbaum, arXiv:1001.3229

(r)





V (~R) /Z

d3r(~r )

|~R ~r |(3.11)


1

|~R ~r |=

1

R+X

i

riRi

R3+

1

2

X

ij

rirj3RiRj ijR2

R5+ . . . , (3.12)


Z

d3r(~r )

|~R ~r |=

q

R+

1

R3

X

i

RiPi +1

6R5

X

ij

(3RiRj ijR2)Qij + . . . (3.13)

8




(r)

exact

(R) = C

Zd3r

(r)

|R r|(set )C = 1





V (~R) /Z

d3r(~r )

|~R ~r |(3.11)


1

|~R ~r |=

1

R+X

i

riRi

R3+

1

2

X

ij

rirj3RiRj ijR2

R5+ . . . , (3.12)


Z

d3r(~r )

|~R ~r |=

q

R+

1

R3

X

i

RiPi +1

6R5

X

ij

(3RiRj ijR2)Qij + . . . (3.13)

8




(r)

Let's assume we don’t know the exact form of .

How can we approximately determine for ?

(r)

exact

(R) = C

Zd3r

(r)

|R r|

(R) |R| a

(set )C = 1





V (~R) /Z

d3r(~r )

|~R ~r |(3.11)


1

|~R ~r |=

1

R+X

i

riRi

R3+

1

2

X

ij

rirj3RiRj ijR2

R5+ . . . , (3.12)


Z

d3r(~r )

|~R ~r |=

q

R+

1

R3

X

i

RiPi +1

6R5

X

ij

(3RiRj ijR2)Qij + . . . (3.13)

8




(r)

Let's assume we don’t know the exact form of .

How can we approximately determine for ?

(r)

exact

(R) = C

Zd3r

(r)

|R r|

(R) |R| a

Expand : 1

|R r|

(R) =q

R+

1

R3

X

i

RiPi +1

6R5

X

i,j

(3RiRj ijR2)Qij + ...

with: q =

Zd3r(r), Pi =

Zd3r(r)ri, Qij =

Zd3r(r)(3rirj ijr

2)

(set )C = 1

Monopole Dipole Quadrupole


(r)

(r)

(R) =q

R+

1

R3

X

i

RiPi +1

6R5

X

i,j


q =

Zd3r(r), Pi =

Zd3r(r)ri, Qij =

Zd3r(r)(3rirj ijr

2)


Multipole expansion calculable if is known,

but also allows to make predictions if is unknown!

(r)

(r)


(r)

(r)

Strategy (assuming that we don’t even know Poisson’s law!):

1.Symmetries: write down most general expression in terms of vectors

and that is consistent with symmetries (which symmetry here?)

2.Naturalness: unknown multipole moments expected to be of natural size:

3.Fixing constants: measure at several locations ( ),

determine and make predictions for other locations

R

r

(R) =q

R+

1

R3

X

i

RiPi +1

6R5

X

i,j


q =

Zd3r(r), Pi =

Zd3r(r)ri, Qij =

Zd3r(r)(3rirj ijr

2)




(r)

(r)



and that is consistent with symmetries (which symmetry here?)




R

r


(r)

(r)



and that is consistent with symmetries (here: rotational symmetry)




R

r

(R) =q

R+

1

R3

X

i

RiPi +1

6R5

X

i,j


q =

Zd3r(r), Pi =

Zd3r(r)ri, Qij =

Zd3r(r)(3rirj ijr

2)




(r)

(r)







R

r


(r)

(r)







R

r

q a, Pi a, Qij a

(R) =q

R+

1

R3

X

i

RiPi +1

6R5

X

i,j


q =

Zd3r(r), Pi =

Zd3r(r)ri, Qij =

Zd3r(r)(3rirj ijr

2)




(r)

(r)







R

r

q a, Pi a, Qij a

(R) =q

R+

1

R3

X

i

RiPi +1

6R5

X

i,j


q =

Zd3r(r), Pi =

Zd3r(r)ri, Qij =

Zd3r(r)(3rirj ijr

2)




(r)

(r)







R

r

q a0, Pi a1, Qij a2 ,terms in mult. expansion scale like a

R

n

(R) =q

R+

1

R3

X

i

RiPi +1

6R5

X

i,j


q =

Zd3r(r), Pi =

Zd3r(r)ri, Qij =

Zd3r(r)(3rirj ijr

2)




(r)

(r)







R

r

q a0, Pi a1, Qij a2 ,terms in mult. expansion scale like a

R

n

(R) |R| a

q, Pi, Qij

Chiral effective field theoryFundamental ingredient of effective theories is a separation of scales

(e.g.: ). How can this idea be applied to nuclear physics?|R| a



Energy scales in nuclear physics: mN 940 MeV

m 770 MeV

m 135 MeV

m mN 300 MeV

1/aS 8 35 MeV




m 770 MeV

m 135 MeV

m mN 300 MeV

1/aS 8 35 MeV

Mass gap!




m 770 MeV

m 135 MeV

m mN 300 MeV

1/aS 8 35 MeV

Mass gap!

pion-less EFT

chiral (pion-full) EFT

Chiral effective field theory

Basic idea: utilize separation of energy scales!

Q

typical momenta of nucleons

breakdown scale of EFT (~500 MeV)

(compare multipole expansion in ) a

RStrategy:

1.Symmetries: write down most general Lagrangian which respects symmetries

of QCD in terms of low-energy degrees of freedom (pions and nucleons)

2.Power counting: identify contributions (i.e. Feynman diagrams) that give

contributions at a given order in

3.Fixing of low-energy constants: perform calculations at a given order in

expansion, determine constants by matching to experimental data (e.g. NN

scattering observables) and make predictions for other observables,

in principle constants can be computed from QCD (cf. multipole expansion)

Chiral effective field theory

Basic idea: utilize separation of energy scales!

Q

typical momenta of nucleons

breakdown scale of EFT (~500 MeV)

(compare multipole expansion in ) a

RStrategy:

1.Symmetries: write down most general Lagrangian which respects symmetries

of QCD in terms of low-energy degrees of freedom (pions and nucleons)

2.Power counting: identify contributions (i.e. Feynman diagrams) that give

contributions at a given order in

3.Fixing of low-energy constants: perform calculations at a given order in

expansion, determine constants by matching to experimental data (e.g. NN

scattering observables) and make predictions for other observables,

in principle constants can be computed from QCD (cf. multipole expansion)

Q

Symmetries of QCD - chiral symmetry

Dµ = @µ igqµTaqAaµ

LQCD = 1

4F aµF

aµ + qf (iµ@µ mf )qf + gqµTaqA

aµ

1

4F aµF

aµ + qf (iµDµ mf )qf

• : flavour index, here we consider only light flavors ( )

• ‘covariant derivative’ (independent of flavor):

f f = u, d

Decompose quark fields (for each flavor) into left- and right-handed

chiral components via:

qL = PLq, qR = PRq, PL/R =1

2(1 5) , PL + PR = 1

• introduce mass matrix (in flavour space):

M = diag(mu,md)


LQCD = 1

4F aµF

aµ

+ iqL,f iµDµqL,f + iqR,f

µDµqR,f

qL,fMqR,f qR,fMqL,f


LQCD = 1

4F aµF

aµ


µDµqR,f

qL,fMqR,f qR,fMqL,f

• for is invariant under independent global rotations

in flavour space ( symmetry):

M = 0 LQCD

SU(2)L SU(2)R

qL =

qL,u

qL,d

! q0L = exp (iL · /2) qL

qR =

qR,u

qR,d

! q0R = exp (iR · /2) qR


LQCD = 1

4F aµF

aµ


µDµqR,f

qL,fMqR,f qR,fMqL,f



M = 0 LQCD

SU(2)L SU(2)R

qL =

qL,u

qL,d

! q0L = exp (iL · /2) qL

qR =

qR,u

qR,d

! q0R = exp (iR · /2) qR

• the total symmetry group in flavour space is

U(1)V U(1)A SU(2)L SU(2)R


LQCD = 1

4F aµF

aµ


µDµqR,f

qL,fMqR,f qR,fMqL,f



M = 0 LQCD

SU(2)L SU(2)R

qL =

qL,u

qL,d

! q0L = exp (iL · /2) qL

qR =

qR,u

qR,d

! q0R = exp (iR · /2) qR



quark number

conservation


LQCD = 1

4F aµF

aµ


µDµqR,f

qL,fMqR,f qR,fMqL,f



M = 0 LQCD

SU(2)L SU(2)R

qL =

qL,u

qL,d

! q0L = exp (iL · /2) qL

qR =

qR,u

qR,d

! q0R = exp (iR · /2) qR



quark number

conservation

broken by

anomaly


LQCD = 1

4F aµF

aµ


µDµqR,f

qL,fMqR,f qR,fMqL,f



M = 0 LQCD

SU(2)L SU(2)R

qL =

qL,u

qL,d

! q0L = exp (iL · /2) qL

qR =

qR,u

qR,d

! q0R = exp (iR · /2) qR



quark number

conservation

broken by

anomaly

spontaneously broken down to subgroup

explicitly broken by finite MSU(2)V

• choose relevant degrees of freedom: here nucleons and pions

• operators constrained by symmetries of QCD

• short-range physics captured in short-range couplings

• separation of scales: Q << Λb, breakdown scale Λb~500 MeV

• power-counting: expand in Q/Λb

• systematic, obtain error estimates

• many-body forces appear naturally

Chiral effective field theory for nuclear forces NN 3N 4N

2006

1994

2011

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

c1, c3, c4 terms cD term cE term

NN 3N 4N



Many-body forces in chiral EFT

2006

1994

2011

need to be fit to three-body and/or higher-body systems



NN 3N 4N



first incorporation in calculations of neutron and nuclear matterTews, Krüger, KH, Schwenk, PRL 110, 032504 (2013)Krüger, Tews, KH, Schwenk, PRC 88, 025802 (2013)


2006

1994

2011

all terms predicted(no new low-energy couplings)

NN 3N 4N



first incorporation in calculations of neutron and nuclear matterTews, Krüger, KH, Schwenk, PRL 110, 032504 (2013)Krüger, Tews, KH, Schwenk, PRC 88, 025802 (2013)

2006

1994

2011

first calculation of matrix elements for ab initio studies of matter and nucleiKH, Krebs, Epelbaum, Golak, Skibinski, PRC 91, 044001(2015)


Aren’t 3N forces unnatural? Do we really need them?

Why are there three-nucleon (3N) forces?

Nucleons are finite-mass composite particles,

can be excited to resonances

dominant contribution from !(1232 MeV)

+ shorter-range parts

tidal effects leads to 3-body forces in earth-sun-moon system

Why are there three-nucleon (3N) forces?

Nucleons are finite-mass composite particles,

can be excited to resonances

dominant contribution from !(1232 MeV)

+ shorter-range parts

tidal effects leads to 3-body forces in earth-sun-moon system

Consider classical analog: tidal effects in earth-sun-moon system

• force between earth and moon depends on the position of sun

• tidal deformations represent internal excitations

• describe system using point particles 3N forces inevitable!

• nucleons are composite particles, can also be excited

• change of resolution change excitations that can be described explicitly

‣ existence of three-nucleon forces natural

‣ crucial question: how important are their contributions?

Nuclear forces for different ab-initio many-body frameworks

Hyperspherical harmonics

coupled cluster method

no-core shell model

Faddeev,Faddeev-Yakubovski

Many-bodyperturbation theory

Self-consistentGreens function

In-medium SRG

!"#$%&!'%"(&)*+!

!"#$%&!'*,,-.&!!

/0*1"%-!2!

!"#$!

One-body Green’s function (or propagator) describes the motion of quasi- particles and holes: …this contains all the structure information probed by nucleon transfer !"#$%&'()*+,-%./-0:

Green’s functions in many-body theory

gαβ(E) =∑

n

⟨ΨA0 |cα|Ψ

A+1n ⟩⟨ΨA+1

n |c†β|ΨA0 ⟩

E − (EA+1n − EA

0 ) + iη+∑

k

⟨ΨA0 |c

†β|Ψ

A−1k ⟩⟨ΨA−1

k |cα|ΨA0 ⟩

E − (EA0 − EA−1

k )− iη

1234*5678/'&9:;$-"$-4*<="6*>$?6*2%&4*@ABCDC*!E@@F0G*234*<9="6*>$?6*H$I6*'()4*E@EJ@E*!E@@F0K*

Shab() =

1

Im gab()

2

(ph)

(ph)

!

(pp/hh)gII

!

FIG. 1: (Color online) Left. One of the diagrams included in the correlated self-energy, Σ(ω). Arrows up (down) refer to quasiparticle

(quasihole) states, the Π(ph) propagators include collective ph and charge-exchange resonances, and the gII include pairing between two

particles or two holes. The FRPA method sums analogous diagrams, with any numbers of phonons, to all orders [21, 25]. Right. Single-

particle spectral distribution for neutrons in 56Ni, obtained from FRPA. Energies above (below) EF are for transitions to excited states of57Ni (55Ni). The quasiparticle states close to the Fermi surface are clearly visible. Integrating over r [Eq. (4)] gives the SFs reported in Tab. I.

poles give the experimental energy transfer for nucleon pickup

(knockout) to the excited states of the systems with A+1 (A-1)

particles. The propagator (2) is obtained by solving the Dyson

equation [g(ω) = g(0)(ω) + g(0)(ω) Σ⋆(ω) g(ω)], where

g(0)(ω) propagates a free nucleon. The information on nuclear

structure is included in the irreducible self-energy, which was

split into two contributions:

Σ⋆(r, r′;ω) = ΣMF (r, r′;ω) + Σ(r, r′;ω) . (3)

The term ΣMF (ω) includes both the nuclear mean field (MF)

and diagrams describing two-particle scattering outside the

model space, generated using a G-matrix resummation [24].

As a consequence, it acquires an energy dependence which

is induced by SRC among nucleons [23]. The second term,

Σ(ω), includes the LRC. In the present work, Σ(ω) is calcu-

lated in the so-called Faddeev random phase approximation

(FRPA) of Refs. [21, 25]. This includes diagrams for particle-

vibration coupling at all orders and with all possible vibration

modes, see Fig. 1, as well as low-energy 2p1h/2h1p configu-

rations. Particle-vibration couplings play an important role in

compressing the single-particle spectrum at the Fermi energy

to its experimental density. However, a complete configura-

tion mixing of states around the Fermi surface is still missing

and would require SM calculations.

Each spectroscopic amplitude ψA±1(r) appearing in Eq. (2)

has to be normalized to its respective SF as

Zα =

∫

dr |ψA±1α (r)|2 =

1

1 −∂Σ⋆αα

(ω)

∂ω

∣

∣

∣

∣

∣

∣

∣

∣

ω=±(EA±1α −EA

0)

, (4)

where Σ⋆αα

(ω) ≡< ψα|Σ⋆(ω)|ψα > is the matrix element of

the self-energy calculated for the overlap function itself but

normalized to unity (∫

dr |ψα(r)|2 = 1). By inserting Eq. (3)

into (4), one distinguishes two contributions to the quenching

of SFs. For model spaces sufficiently large, all low-energy

physics is described by Σ(ω). Then, the derivative of ΣMF (ω)

accounts for the coupling to states outside the model space

and estimates the effects of SRC alone [33].

In general, the SC self-energy (3) is a functional of the one-

body propagator itself, Σ⋆ = Σ⋆[g]. Hence the FRPA equa-

tions for the self-energy and the Dyson equation have to be

solved iteratively. The mean-field part, ΣMF [g], was calcu-

lated exactly in terms of the fully fragmented propagator (2).

For the FRPA, this procedure was simplified by employing the

Σ[gIPM] obtained in terms of a MF-like propagator

gIPM(r, r′;ω) =∑

n /∈F

(φn(r))∗ φn(r′)

ω − εIMPn + iη

+∑

k∈F

φk(r) (φk(r′))∗

ω − εIMPk− iη

,

(5)

which is updated at each iteration to approximate Eq. (2) with

a limited number of poles. Eq. (5) defines a set of undressed

single-particle states that can be taken as a basis for SM ap-

plications. This feature will be used below to estimate the im-

portance of configuration mixing effects on the quenching of

spectroscopic factors. The present calculations employed the

N3LO interaction from chiral perturbation theory [26] with a

modification of the tensor monopoles to correct for missing

three-nucleon interactions [27].

Results.— The calculated single-particle spectral function

[S 56Ni(r,ω) = 1π|g(r = r

′;ω)|2] is shown in Fig. 1 for the case

of neutron transfer on 56Ni. This picture puts in evidence the

quasiparticle and quasihole states associated with valence or-

bits in the 0p1 f shell. The corresponding SFs are reported

in Tab. I, including both protons and neutrons. The first col-

umn is obtained by including only the derivative of ΣMF (ω)

when calculating Eq. (4). Since N3LO is rather soft com-

pared to other realistic interactions the effect of SRC is rela-

tively small. From other models one could expect a quenching

up to about 10% [16], as confirmed by recent electron scatter-

ing experiments [14, 15, 28]. This difference would not affect

sensibly the conclusions below. The complete FRPA result for

SFs is given in the second column. For the transition between

the 56Ni and 57Ni ground states, our result agrees with knock-

C. Barbieri, PRL 103,202520 (2009)

3BF beyond the EoS

Shear viscosity with CBF

Benhar & Valli, PRL 99, 232501 (2007)Benhar & Carbone, arxiv:0912.0129

PNS dynamical evolution with BHF

Burgio et al., arxiv:1106.2736

• Many-body modelers are aiming at complete descriptions!• Consistent description of transport coefficients• Response of nuclear & neutron matter• Transport coefficients & dynamical evolution of NS 27 / 30

!"#$%&!'%"(&)*+!

!"#$%&!'*,,-.&!!

/0*1"%-!2!

!"#$!

One-body Green’s function (or propagator) describes the motion of quasi- particles and holes: …this contains all the structure information probed by nucleon transfer !"#$%&'()*+,-%./-0:

Green’s functions in many-body theory

gαβ(E) =∑

n

⟨ΨA0 |cα|Ψ

A+1n ⟩⟨ΨA+1

n |c†β|ΨA0 ⟩

E − (EA+1n − EA

0 ) + iη+∑

k

⟨ΨA0 |c

†β|Ψ

A−1k ⟩⟨ΨA−1

k |cα|ΨA0 ⟩

E − (EA0 − EA−1

k )− iη

1234*5678/'&9:;$-"$-4*<="6*>$?6*2%&4*@ABCDC*!E@@F0G*234*<9="6*>$?6*H$I6*'()4*E@EJ@E*!E@@F0K*

Shab() =

1

Im gab()

C. Barbieri, PRL 103,202520 (2009)

Details!

Introduction

VNNV3N

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



Development of nuclear interactions

predictionsvalidation

optimizationpower counting

Basis function expansion

H | ii = T + V | ii = Ei | ii

Basis size and matrix dimensions

Overview RG Basics Resolution Forces Filter Coupling

Many short wavelengths =) Large matricesHarmonic oscillator basis with N

max

shells for excitationsGraphs show convergence for soft chiral EFT potential(although not at optimal ~ for 6Li)

2 4 6 8 10 12 14 16 18 20Matrix Size [N

max]

−29

−28

−27

−26

−25

−24

−23

−22

−21

−20

Gro

un

d S

tate

En

erg

y [

MeV

]

Original

expt.

VNN

= N3LO (500 MeV)

Helium-4

VNNN

= N2LO

ground-state energy

2 4 6 8 10 12 14 16 18

Matrix Size [Nmax

]

−36

−32

−28

−24

−20

−16

−12

−8

−4

0

4

8

12

16

Gro

un

d-S

tate

En

erg

y [

MeV

]

Lithium-6

expt.

Originalh- Ω = 20 MeV

ground-state energy

VNN

= N3LO (500 MeV)

VNNN

= N2LO

0 2 4 6 8 10 12 14N

max

101

102

103

104

105

106

107

108

109

Mat

rix d

imen

sion

4He

6Li

12C

16O

Factorial growth of basis with A =) limits calculationsToo much resolution from potential =) mismatch of scales

Dick Furnstahl Nuclei at Low Resolution

Basis size of convergence of NCSM calculations

• many short wavelengths large matrices

• harmonic oscillator basis with Nmax shells for excitations

• convergence for chiral potential (EM 500 MeV):

• factorial growth of basis with A, limits calculations severely




max


2 4 6 8 10 12 14 16 18 20Matrix Size [N

max]

−29

−28

−27

−26

−25

−24

−23

−22

−21

−20

Gro

un

d S

tate

En

erg

y [

MeV

]

Original

expt.

VNN

= N3LO (500 MeV)

Helium-4

VNNN

= N2LO

ground-state energy

2 4 6 8 10 12 14 16 18

Matrix Size [Nmax

]

−36

−32

−28

−24

−20

−16

−12

−8

−4

0

4

8

12

16

Gro

und-S

tate

Ener

gy [

MeV

]

Lithium-6

expt.


ground-state energy

VNN

= N3LO (500 MeV)

VNNN

= N2LO

0 2 4 6 8 10 12 14N

max

101

102

103

104

105

106

107

108

109

Mat

rix

dim

ensi

on

4He

6Li

12C

16O










max


2 4 6 8 10 12 14 16 18 20Matrix Size [N

max]

−29

−28

−27

−26

−25

−24

−23

−22

−21

−20

Gro

un

d S

tate

En

erg

y [

MeV

]

Original

expt.

VNN

= N3LO (500 MeV)

Helium-4

VNNN

= N2LO

ground-state energy

2 4 6 8 10 12 14 16 18

Matrix Size [Nmax

]

−36

−32

−28

−24

−20

−16

−12

−8

−4

0

4

8

12

16

Gro

un

d-S

tate

En

erg

y [

MeV

]

Lithium-6

expt.


ground-state energy

VNN

= N3LO (500 MeV)

VNNN

= N2LO

0 2 4 6 8 10 12 14N

max

101

102

103

104

105

106

107

108

109

Mat

rix d

imen

sion

4He

6Li

12C

16O








Petascale ~ 200Tbytes for matrix + index arrays





max


2 4 6 8 10 12 14 16 18 20Matrix Size [N

max]

−29

−28

−27

−26

−25

−24

−23

−22

−21

−20

Gro

un

d S

tate

En

erg

y [

MeV

]

Original

expt.

VNN

= N3LO (500 MeV)

Helium-4

VNNN

= N2LO

ground-state energy

2 4 6 8 10 12 14 16 18

Matrix Size [Nmax

]

−36

−32

−28

−24

−20

−16

−12

−8

−4

0

4

8

12

16

Gro

und-S

tate

Ener

gy [

MeV

]

Lithium-6

expt.


ground-state energy

VNN

= N3LO (500 MeV)

VNNN

= N2LO

0 2 4 6 8 10 12 14N

max

101

102

103

104

105

106

107

108

109

Mat

rix

dim

ensi

on

4He

6Li

12C

16O










max


2 4 6 8 10 12 14 16 18 20Matrix Size [N

max]

−29

−28

−27

−26

−25

−24

−23

−22

−21

−20

Gro

un

d S

tate

En

erg

y [

MeV

]

Original

expt.

VNN

= N3LO (500 MeV)

Helium-4

VNNN

= N2LO

ground-state energy

2 4 6 8 10 12 14 16 18

Matrix Size [Nmax

]

−36

−32

−28

−24

−20

−16

−12

−8

−4

0

4

8

12

16

Gro

un

d-S

tate

En

erg

y [

MeV

]

Lithium-6

expt.


ground-state energy

VNN

= N3LO (500 MeV)

VNNN

= N2LO

0 2 4 6 8 10 12 14N

max

101

102

103

104

105

106

107

108

109

Mat

rix d

imen

sion

4He

6Li

12C

16O








20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170100

N

10

20

Z

30

40

50

60

70

80

90

100

110


0

since 1980‘s

‘Exact’ methods: nore-core shell model, Greens function Monte Carlo

factorial scaling

20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170100

N

10

20

Z

30

40

50

60

70

80

90

100

110


0

since 1980‘s

‘Exact’ methods: nore-core shell model, Greens function Monte Carlo

factorial scaling

plus frameworks for few-body systems (A=3,4)

Faddeev, Faddeev-Yakubovski

20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170100

N

10

20

Z

30

40

50

60

70

80

90

100

110

Ca

O

Ni

0

since year ~2000

closed-shell nuclei: coupled cluster, in-medium SRG, self-consistent Greens function

polynomial scaling


20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170100

N

10

20

Z

30

40

50

60

70

80

90

100

110

0

Ca

O

Ni


since year ~2010

open-shell nuclei: multi-reference IMSRG, Gorkov Greens function, Bogoliubov coupled cluster

polynomial scaling

20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170100

N

10

20

Z

30

40

50

60

70

80

90

100

110

Ca

O

Ni

Sn


since year ~2014

ab initio valence shell model based on nonperturbative calculation of effective interactions

mixed scaling

20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170100

N

10

20

Z

30

40

50

60

70

80

90

100

110

Ca

O

Ni

Sn


since year ~2014

ab initio valence shell model based on nonperturbative calculation of effective interactions

mixed scaling

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

S.K. Bogner et al. / Progress in Particle and Nuclear Physics 65 (2010) 94–147 105

Fig. 13. Ground-state energy of 3H as a function of the maximum momentum kmax for three different values of [14]. The cutoff function isexp[(k2/k2max)

n] with n = 8. The initial potentials are (a) the N3LO NN potential of Ref. [20] and (b) the Argonne v18 potential [18].

Fig. 14. Momentum–spacematrix elements in the 3S1 channel for (a) the N3LO potential ( = 500MeV) of Ref. [20], (b) the sharp cutoff N3LOWpotential( = 2.0 fm1) of Ref. [72], and (c) for the N3LO potential ( = 500 MeV) evolved by a smooth Vlow k to = 2.0 fm1.

Fig. 15. Flow of Vlow k(k0 = 0, k = 0; ) compared to the corresponding momentum-independent contact interaction C0() at LO and NLO, where thiscoupling is determined entirely from RG invariance and fits to the scattering length as (at LO) plus effective range re (at NLO) [73].

We can gain further insights into the interplay of the RG and EFT by considering chiral EFT as providing a general operatorbasis that can be used to expand the RG evolution. At a given order (Q/b)

n, chiral EFT includes contributions from one-or multi-pion exchanges and from contact interactions, with short-range couplings that depend on the resolution or cutoffscale. As part of the RG evolution, short-range couplings included in the initial potential evolve. This is illustrated in Fig. 15by comparing the flow of Vlow k(k0 = 0, k = 0; ) with the corresponding momentum-independent contact interactionC0() in subsequent orders of pionless EFT. In addition, the RG generates higher-order short-range contact interactions sothat observables are exactly reproduced and the truncation error is unchanged. Consequently, the cutoff variation can be

AV18

(1S0 channel)

Low- and high momentum couplings in interactions

EM 500 MeV N3LO

(3S1 channel)

• strong couplings of low- and high-momenta in interactions

complicates convergence ab initio many-body calculations

• these couplings are reduced in chiral potentials compared to

‘traditional’ interactions, but still present!

Regularization schemes fornuclear interactions (here: NN)

V3NV

p01 p0

2

p1 p2

Separation of long- and short-range physics

p = (p1 p2)/2

p0 = (p01 p0

2)/2

q = (p1 p01)


nonlocalEpelbaum, Glöckle, Meissner, NPA 747, 362 (2005)Entem, Machleidt, PRC 68, 041001 (2003)

V3NV

p01 p0

2

p1 p2


VNN(p,p0) ! exp

h(p2 + p02)/2

niVNN(p,p

0)

p = (p1 p2)/2

p0 = (p01 p0

2)/2

q = (p1 p01)

VNN(q) ! exp

hq2/2

niVNN(q)


nonlocal

local (momentum space)

Epelbaum, Glöckle, Meissner, NPA 747, 362 (2005)Entem, Machleidt, PRC 68, 041001 (2003)

cf. Navratil, Few-body Systems 41, 117 (2007)

V3NV

p01 p0

2

p1 p2


VNN(p,p0) ! exp

h(p2 + p02)/2

niVNN(p,p

0)

p = (p1 p2)/2

p0 = (p01 p0

2)/2

q = (p1 p01)

VNN(q) ! exp

hq2/2

niVNN(q)

V NN(r) !

1 exp

hr2/R2

niV NN(r)


nonlocal


local (coordinate space)



Gezerlis et. al, PRL, 111, 032501 (2013)

V3NV

p01 p0

2

p1 p2


VNN(p,p0) ! exp

h(p2 + p02)/2

niVNN(p,p

0)

p = (p1 p2)/2

p0 = (p01 p0

2)/2

q = (p1 p01)

(r) ! ↵n exp(r2/R2

)

n

VNN(q) ! exp

hq2/2

niVNN(q)

V NN(r) !

1 exp

hr2/R2

niV NN(r)


nonlocal


local (coordinate space)

semi-local



Gezerlis et. al, PRL, 111, 032501 (2013)

Epelbaum et. al, PRL, 115, 122301 (2015)

V3NV

p01 p0

2

p1 p2


VNN(p,p0) ! exp

h(p2 + p02)/2

niVNN(p,p

0)

(r) ! C ! exp

h(p2 + p02)/2

niC

p = (p1 p2)/2

p0 = (p01 p0

2)/2

q = (p1 p01)

V NN(r) !

1 exp

r2/R2

nV NN(r)

(r) ! ↵n exp(r2/R2

)

n

Resolution: the higher the better?in nuclear structure we are interested in low-energy observables

(long-wavelength information!)

Resolution: the higher the better?

• resolution of short-distance structures can obscure this information

• small details have nothing to do with long-wavelength information!

in nuclear structure we are interested in low-energy observables

(long-wavelength information!)

• long-wavelength information is preserved

• much less information necessary

Strategy: use a lower-resolution version

low-pass filter

• long-wavelength information is preserved

• much less information necessary

... however, it’s not that easy in nuclear physics.


low-pass filter

0 100 200 300E

lab (MeV)

−20

0

20

40

60

ph

ase

shif

t (d

egre

es)

1S

0

AV18 phase shifts

k = 2 fm−1


low-pass filter

low-pass filter

0 100 200 300E

lab (MeV)

−20

0

20

40

60

ph

ase

shif

t (d

egre

es)

1S

0

AV18 phase shifts

k = 2 fm−1

0 100 200 300E

lab (MeV)

−20

0

20

40

60

ph

ase

shif

t (d

egre

es)

1S

0

AV18 phase shifts

after low-pass filter

k = 2 fm−1


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!

dH

d= [, H]

One solution: the Similarity Renormalization Group• generate unitary transformation which decouples low- and high momenta:

• change resolution systematically in small steps:

with the resolution parameter

• generator can be chosen and tailored to different applications

• observables are preserved due to unitarity of transformation

H = UHU †

dH

d= [, H]





Resolution


H = UHU †

dH

d= [, H]





Resolution


H = UHU †

0 1 2 3 4r [fm]

−100

0

100

200

V(r

) [M

eV]

λ = 20 fm−1

1 2 3 4r [fm]

λ = 4 fm−1

1 2 3 4r [fm]

λ = 3 fm−1

1 2 3 4r [fm]

λ = 2 fm−1

1 2 3 4r [fm]

λ = 1.5 fm−1

AV18

N3LO

V (r) =

Zdr0r02V(r, r

0)

• 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 transformations also change three-body (and higher-body) interactions!

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

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

Implementation of 3NF SRG evolution in a momentum basis:

Alternative implementations in harmonic oscillator basis:Jurgenson, Navratil, Furnstahl, PRL 103, 082501 (2009)

Roth, Calci, Langhammer, Binder, PRC 90, 024325 (2014)

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






max


2 4 6 8 10 12 14 16 18 20Matrix Size [N

max]

−29

−28

−27

−26

−25

−24

−23

−22

−21

−20

Gro

un

d S

tate

En

erg

y [

MeV

]

Original

expt.

VNN

= N3LO (500 MeV)

Helium-4

VNNN

= N2LO

ground-state energy

2 4 6 8 10 12 14 16 18

Matrix Size [Nmax

]

−36

−32

−28

−24

−20

−16

−12

−8

−4

0

4

8

12

16

Gro

und-S

tate

Ener

gy [

MeV

]Lithium-6

expt.


ground-state energy

VNN

= N3LO (500 MeV)

VNNN

= N2LO

0 2 4 6 8 10 12 14N

max

101

102

103

104

105

106

107

108

109

Mat

rix

dim

ensi

on

4He

6Li

12C

16O








Figure 3: Top: The Argonne 18

potential in the 3S1

channel is evolved with SRG. Bottom: The localprojections of the

18

and a higher order contribution to an EFT potential are shown. The e↵ects ofSRG can be seen in both plots [1].

Figure 4: Left: The dimension of matrices that occur in NCSM calculations for selected nuclei depend-ing on the number of available shells. Matrices larger than 109 cannot be diagonalized with nowadayscomputational possibilities. Right: Calculations for 6Li using a SRG evolved potential converge andyield a ground state energy close to experimental results [1].

4

Ground state energies of nuclei based on RG-evolved chiral EFT interactions

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

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

• very promising results for light nuclei: excellent agreement with experiment small dependence on RG scale

• issues for heavier nuclei: significant dependence on RG scale

(induced 4N forces?) not possible to compare with exp.

evolution equations at three-body level was demonstratedonly recently [6,7]. In view of the application in theNCSM it is convenient to solve the flow equation for thethree-body system using a harmonic-oscillator (HO)Jacobi-coordinate basis [12]. The intermediate sums inthe 3N Jacobi basis are truncated at Nmax ¼ 40 for chan-nels with J " 5=2 and ramp down linearly to Nmax ¼ 24for J # 13=2. Based on this and the corresponding solutionof the flow equation in two-body space (using either apartial-wavemomentum or harmonic-oscillator representa-tion) we extract the irreducible two- and three-body termsof the Hamiltonian for the use in A-body calculations.

We have made major technical improvements regardingthe SRG transformation, reducing the computational effortby 3 orders of magnitude compared to Ref. [7], e.g., byusing a solver with adaptive step-size and optimized matrixoperations. Furthermore, we have developed a transforma-tion from 3N Jacobi matrix elements to a JT-coupledrepresentation with a highly efficient storage scheme,which allows us to handle 3N matrix-element sets ofunprecedented size. A detailed discussion of these aspectsis presented elsewhere.

Importance-truncated NCSM.—Based on the SRG-evolved Hamiltonian we treat the many-body problem inthe NCSM; i.e., we solve the large-scale eigenvalue prob-lem of the Hamiltonian, represented in a many-body basisof HO Slater determinants truncated with respect to themaximum HO excitation energy Nmax@!. In order to copewith the factorial growth of the basis dimension with Nmax

and particle number A, we use the importance-truncation(IT) scheme introduced in Refs. [13,14]. The IT-NCSMuses an importance measure !" for the individual basisstates j""i derived from many-body perturbation theoryand retains only states with j!"j above a threshold !min inthe model space. Through a variation of the threshold andan a posteriori extrapolation !min ! 0 the contribution ofdiscarded states is recovered. We use the sequential updatescheme discussed in Ref. [14], which connects to the fullNCSMmodel space and thus the exact NCSM results in thelimit of vanishing threshold. In the following we alwaysreport threshold-extrapolated results including an estimatefor the extrapolation uncertainties. For the present appli-cation we have extended the IT-NCSM to include full 3Ninteractions. Using the JT-coupled 3N matrix elements weare able to perform calculations up to Nmax ¼ 12 or 14 forall p-shell nuclei with moderate computational resources.

Ground-state energies.—We first focus on IT-NCSMcalculations for the ground states of 4He, 6Li, 12C, and16O using SRG-transformed chiral NN þ 3N interactions.Throughout this work we use the chiral NN interaction atN3LO of Entem and Machleidt [1] and the 3N interactionat N2LO [15] with low-energy constants determined fromthe triton binding energy and #-decay half-life [16]. Inorder to disentangle the effects of the initial and theSRG-induced 3N contributions, we consider three different

Hamiltonians. (i) NN only: starting from the chiral NNinteraction only the SRG-evolved NN contributions arekept. (ii) NN þ 3N-induced: starting from the chiral NNinteraction the SRG-evolvedNN and the induced 3N termsare kept. (iii) NN þ 3N-full: starting from the chiralNN þ 3N interaction all SRG-evolved NN and 3N termsare kept. For each Hamiltonian we assess the dependenceof the observables, here the ground-state energies, on theflow-parameter $. We use the five values $ ¼ 0:04, 0.05,0.0625, 0.08, and 0:16 fm4, which correspond to momen-tum scales #¼$%1=4¼2:24, 2.11, 2, 1.88, and 1:58 fm%1,respectively. For extrapolations to infinite model-space,Nmax ! 1, we use simple exponential fits based on thelast 3 or 4 data points. The extrapolated energy is given bythe average of the two extrapolations, the uncertainty bythe difference.The ground-state energies obtained in IT-NCSM calcu-

lations for 4He and 6Li with the three Hamiltonians aresummarized in Fig. 1. Analogous calculations in the fullNCSM for the same SRG-evolved initial Hamiltonian havebeen presented in Ref. [6] for 4He and in Ref. [7] for 6Li.We have cross-checked our results with Refs. [6,7] andfound excellent agreement.The first and foremost effect of the SRG transformation

is the acceleration of the convergence of NCSM calcula-tions with Nmax. With increasing $ the convergence issystematically improved for all three versions of the

-29

-28

-27

-26

-25

-24

-23

.

Egs

[MeV

]

(a)

NN only

4He20 MeV

(b)

NN 3N-induced

exp.

(c)

NN 3N-full

2 4 6 8 10 12 14Nmax

-34

-32

-30

-28

-26

-24

-22

.

Egs

[MeV

]

(d)6Li

20 MeV

2 4 6 8 10 12 14Nmax

(e)

exp.

2 4 6 8 10 12 14Nmax

(f)

FIG. 1 (color online). IT-NCSM ground-state energies for 4Heand 6Li as function of Nmax for the three types of Hamiltonians(see column headings) for a range of flow parameters: $ ¼ 0:04(blue,&), 0.05 (red,r), 0.0625 (green,m), 0.08 (violet,j), and0:16 fm4 (light blue,w). Error bars indicate the uncertainties ofthe threshold extrapolations. The bars at the right-hand sideof each panel indicate the results of exponential extrapolationsof the individual Nmax sequences (see text).

PRL 107, 072501 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending

12 AUGUST 2011

072501-2

Hamiltonian. With the initial Hamiltonian, i.e., ! ¼ 0,even our large model spaces are not sufficient to obtainconverged results.

For the NN-only Hamiltonian Fig. 1 shows a clear!-dependence of the extrapolated ground-state energiesfor 4He and 6Li, hinting at sizable SRG-induced 3N con-tributions. When including those induced 3N terms, i.e.,when using the NN þ 3N-induced Hamiltonian, the ex-trapolated ground-state energies are shifted significantlyand become ! independent within the uncertainties of theNmax-extrapolation. Thus, induced contributions beyondthe 3N level originating from the initial NN interactionare negligible in the ! range considered here, indicatingthat the NN þ 3N-induced Hamiltonian is unitarilyequivalent to the initial NN Hamiltonian. The extrapolatedground-state energies for a subset of !-values are summa-rized in Table I.

By including the initial chiral 3N interaction, i.e., byusing the NN þ 3N-full Hamiltonian, the ground-stateenergies are lowered and are in good agreement withexperiment for both, 4He and 6Li. There is no sizable !dependence in the range considered here. We conclude thatinduced 3N terms originating from the initial NN interac-tion are important, but that induced 4N (and higher) termsare not relevant for light p-shell nuclei, since the ground-state energies obtained with the NN þ 3N-induced and theNN þ 3N-full Hamiltonian are practically ! independent.

This picture changes if we consider nuclei in the upperp-shell. In Fig. 2 we present the first accurate ab initiocalculations for the ground states of 12C and 16O startingfrom chiral NN þ 3N interactions. By combining theIT-NCSM with the JT-coupled storage scheme for the3N matrix elements we are able to reach model spacesup to Nmax ¼ 12 for the upper p-shell at moderate compu-tational cost. Previously, even the most extensive NCSMcalculations including full 3N interactions were limited toNmax ¼ 8 in this regime [17]. As evident from the Nmax

dependence of the ground-state energies, this increase inNmax 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 severe! dependence indicating sizable induced 3N contributions.Their inclusion in the NN þ 3N-induced Hamiltonianleads to ground-state energies that are practically indepen-dent of !, confirming that induced 4N contributions areirrelevant when starting from the NN interaction only.Therefore, the NN þ 3N-induced results can be consid-ered equivalent to a solution for the initial NN interaction.The 16O binding energy per nucleon of 7.48(4) MeV is ingood agreement with a recent coupled-cluster !-CCSD(T)result of 7.56MeV for the ‘‘bare’’ chiralNN interaction [18].In contrast to light nuclei the ground-state energies of

12C and 16O obtained with the NN þ 3N-full Hamiltoniando show a significant ! dependence, as evident fromFig. 2(c) and 2(f) and, for a subset of ! values, fromTable I. The inclusion of the initial chiral 3N interactionleads to induced 4N contributions whose omission causesthe ! dependence.A direct comparison of the ! 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 about25 MeV (0.7 MeV) for 16O (4He) in the range from! ¼ 0:04 fm4 to 0:16 fm4. The inclusion of the induced3N terms eliminates this ! dependence. The inclusion ofthe initial 3N interaction again generates an ! dependenceof about 10 MeV for 16O. Note that the induced 4N (andhigher) contributions that are needed to compensate the! dependence for 16O reach about half the size of the total

TABLE I. Summary of Nmax-extrapolated IT-NCSM ground-state energies in MeV for a subset of ! values with @" ¼20 MeV (see text).

!½fm4$ 4He 6Li 12C 16O

NN 0.05 %28:08ð2Þ %31:5ð2Þ %99:1ð6Þ %161:0ð2Þonly 0.0625 %28:25ð1Þ %31:8ð1Þ %101:4ð3Þ %164:9ð6Þ

0.08 %28:38ð1Þ %32:2ð1Þ %103:7ð2Þ %170:2ð4ÞNNþ 0.05 %25:33ð1Þ %27:7ð2Þ %76:9ð2Þ %119:5ð3Þ3N-ind. 0.0625 %25:34ð1Þ %27:6ð2Þ %77:2ð1Þ %119:7ð6Þ

0.08 %25:34ð1Þ %27:6ð1Þ %77:4ð2Þ %119:5ð2ÞNNþ 0.05 %28:45ð3Þ %31:8ð2Þ %96:1ð4Þ %143:7ð2Þ3N-full 0.0625 %28:45ð1Þ %31:8ð1Þ %96:8ð3Þ %145:6ð2Þ

0.08 %28:46ð1Þ %31:8ð1Þ %97:6ð1Þ %147:8ð1Þexp. %28:30 %31:99 %92:16 %127:62

-100

-90

-80

-70

-60

.

Egs

[MeV

]

(a)

NN only

12C20 MeV

(b)

NN 3N-induced

exp.

(c)

NN 3N-full

2 4 6 8 10 12 14Nmax

-180

-160

-140

-120

-100

.

Egs

[MeV

]

(d)16O

20 MeV

2 4 6 8 10 12Nmax

(e)

exp.

2 4 6 8 10 12Nmax

(f)

FIG. 2 (color online). IT-NCSM ground-state energies for 12Cand 16O as function of Nmax for the three types of Hamiltoniansand a range of flow parameters (for details see Fig. 1).


12 AUGUST 2011

072501-3

Ground state energies of nuclei based on RG-evolved chiral EFT interactions

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

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

• very promising results for light nuclei: excellent agreement with experiment small dependence on RG scale

• issues for heavier nuclei: significant dependence on RG scale

(induced 4N forces?) not possible to compare with exp.

evolution equations at three-body level was demonstratedonly recently [6,7]. In view of the application in theNCSM it is convenient to solve the flow equation for thethree-body system using a harmonic-oscillator (HO)Jacobi-coordinate basis [12]. The intermediate sums inthe 3N Jacobi basis are truncated at Nmax ¼ 40 for chan-nels with J " 5=2 and ramp down linearly to Nmax ¼ 24for J # 13=2. Based on this and the corresponding solutionof the flow equation in two-body space (using either apartial-wavemomentum or harmonic-oscillator representa-tion) we extract the irreducible two- and three-body termsof the Hamiltonian for the use in A-body calculations.

We have made major technical improvements regardingthe SRG transformation, reducing the computational effortby 3 orders of magnitude compared to Ref. [7], e.g., byusing a solver with adaptive step-size and optimized matrixoperations. Furthermore, we have developed a transforma-tion from 3N Jacobi matrix elements to a JT-coupledrepresentation with a highly efficient storage scheme,which allows us to handle 3N matrix-element sets ofunprecedented size. A detailed discussion of these aspectsis presented elsewhere.

Importance-truncated NCSM.—Based on the SRG-evolved Hamiltonian we treat the many-body problem inthe NCSM; i.e., we solve the large-scale eigenvalue prob-lem of the Hamiltonian, represented in a many-body basisof HO Slater determinants truncated with respect to themaximum HO excitation energy Nmax@!. In order to copewith the factorial growth of the basis dimension with Nmax

and particle number A, we use the importance-truncation(IT) scheme introduced in Refs. [13,14]. The IT-NCSMuses an importance measure !" for the individual basisstates j""i derived from many-body perturbation theoryand retains only states with j!"j above a threshold !min inthe model space. Through a variation of the threshold andan a posteriori extrapolation !min ! 0 the contribution ofdiscarded states is recovered. We use the sequential updatescheme discussed in Ref. [14], which connects to the fullNCSMmodel space and thus the exact NCSM results in thelimit of vanishing threshold. In the following we alwaysreport threshold-extrapolated results including an estimatefor the extrapolation uncertainties. For the present appli-cation we have extended the IT-NCSM to include full 3Ninteractions. Using the JT-coupled 3N matrix elements weare able to perform calculations up to Nmax ¼ 12 or 14 forall p-shell nuclei with moderate computational resources.

Ground-state energies.—We first focus on IT-NCSMcalculations for the ground states of 4He, 6Li, 12C, and16O using SRG-transformed chiral NN þ 3N interactions.Throughout this work we use the chiral NN interaction atN3LO of Entem and Machleidt [1] and the 3N interactionat N2LO [15] with low-energy constants determined fromthe triton binding energy and #-decay half-life [16]. Inorder to disentangle the effects of the initial and theSRG-induced 3N contributions, we consider three different

Hamiltonians. (i) NN only: starting from the chiral NNinteraction only the SRG-evolved NN contributions arekept. (ii) NN þ 3N-induced: starting from the chiral NNinteraction the SRG-evolvedNN and the induced 3N termsare kept. (iii) NN þ 3N-full: starting from the chiralNN þ 3N interaction all SRG-evolved NN and 3N termsare kept. For each Hamiltonian we assess the dependenceof the observables, here the ground-state energies, on theflow-parameter $. We use the five values $ ¼ 0:04, 0.05,0.0625, 0.08, and 0:16 fm4, which correspond to momen-tum scales #¼$%1=4¼2:24, 2.11, 2, 1.88, and 1:58 fm%1,respectively. For extrapolations to infinite model-space,Nmax ! 1, we use simple exponential fits based on thelast 3 or 4 data points. The extrapolated energy is given bythe average of the two extrapolations, the uncertainty bythe difference.The ground-state energies obtained in IT-NCSM calcu-

lations for 4He and 6Li with the three Hamiltonians aresummarized in Fig. 1. Analogous calculations in the fullNCSM for the same SRG-evolved initial Hamiltonian havebeen presented in Ref. [6] for 4He and in Ref. [7] for 6Li.We have cross-checked our results with Refs. [6,7] andfound excellent agreement.The first and foremost effect of the SRG transformation

is the acceleration of the convergence of NCSM calcula-tions with Nmax. With increasing $ the convergence issystematically improved for all three versions of the

-29

-28

-27

-26

-25

-24

-23

.

Egs

[MeV

]

(a)

NN only

4He20 MeV

(b)

NN 3N-induced

exp.

(c)

NN 3N-full

2 4 6 8 10 12 14Nmax

-34

-32

-30

-28

-26

-24

-22

.

Egs

[MeV

]

(d)6Li

20 MeV

2 4 6 8 10 12 14Nmax

(e)

exp.

2 4 6 8 10 12 14Nmax

(f)

FIG. 1 (color online). IT-NCSM ground-state energies for 4Heand 6Li as function of Nmax for the three types of Hamiltonians(see column headings) for a range of flow parameters: $ ¼ 0:04(blue,&), 0.05 (red,r), 0.0625 (green,m), 0.08 (violet,j), and0:16 fm4 (light blue,w). Error bars indicate the uncertainties ofthe threshold extrapolations. The bars at the right-hand sideof each panel indicate the results of exponential extrapolationsof the individual Nmax sequences (see text).


12 AUGUST 2011

072501-2

Hamiltonian. With the initial Hamiltonian, i.e., ! ¼ 0,even our large model spaces are not sufficient to obtainconverged results.

For the NN-only Hamiltonian Fig. 1 shows a clear!-dependence of the extrapolated ground-state energiesfor 4He and 6Li, hinting at sizable SRG-induced 3N con-tributions. When including those induced 3N terms, i.e.,when using the NN þ 3N-induced Hamiltonian, the ex-trapolated ground-state energies are shifted significantlyand become ! independent within the uncertainties of theNmax-extrapolation. Thus, induced contributions beyondthe 3N level originating from the initial NN interactionare negligible in the ! range considered here, indicatingthat the NN þ 3N-induced Hamiltonian is unitarilyequivalent to the initial NN Hamiltonian. The extrapolatedground-state energies for a subset of !-values are summa-rized in Table I.

By including the initial chiral 3N interaction, i.e., byusing the NN þ 3N-full Hamiltonian, the ground-stateenergies are lowered and are in good agreement withexperiment for both, 4He and 6Li. There is no sizable !dependence in the range considered here. We conclude thatinduced 3N terms originating from the initial NN interac-tion are important, but that induced 4N (and higher) termsare not relevant for light p-shell nuclei, since the ground-state energies obtained with the NN þ 3N-induced and theNN þ 3N-full Hamiltonian are practically ! independent.

This picture changes if we consider nuclei in the upperp-shell. In Fig. 2 we present the first accurate ab initiocalculations for the ground states of 12C and 16O startingfrom chiral NN þ 3N interactions. By combining theIT-NCSM with the JT-coupled storage scheme for the3N matrix elements we are able to reach model spacesup to Nmax ¼ 12 for the upper p-shell at moderate compu-tational cost. Previously, even the most extensive NCSMcalculations including full 3N interactions were limited toNmax ¼ 8 in this regime [17]. As evident from the Nmax

dependence of the ground-state energies, this increase inNmax 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 severe! dependence indicating sizable induced 3N contributions.Their inclusion in the NN þ 3N-induced Hamiltonianleads to ground-state energies that are practically indepen-dent of !, confirming that induced 4N contributions areirrelevant when starting from the NN interaction only.Therefore, the NN þ 3N-induced results can be consid-ered equivalent to a solution for the initial NN interaction.The 16O binding energy per nucleon of 7.48(4) MeV is ingood agreement with a recent coupled-cluster !-CCSD(T)result of 7.56MeV for the ‘‘bare’’ chiralNN interaction [18].In contrast to light nuclei the ground-state energies of

12C and 16O obtained with the NN þ 3N-full Hamiltoniando show a significant ! dependence, as evident fromFig. 2(c) and 2(f) and, for a subset of ! values, fromTable I. The inclusion of the initial chiral 3N interactionleads to induced 4N contributions whose omission causesthe ! dependence.A direct comparison of the ! 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 about25 MeV (0.7 MeV) for 16O (4He) in the range from! ¼ 0:04 fm4 to 0:16 fm4. The inclusion of the induced3N terms eliminates this ! dependence. The inclusion ofthe initial 3N interaction again generates an ! dependenceof about 10 MeV for 16O. Note that the induced 4N (andhigher) contributions that are needed to compensate the! dependence for 16O reach about half the size of the total

TABLE I. Summary of Nmax-extrapolated IT-NCSM ground-state energies in MeV for a subset of ! values with @" ¼20 MeV (see text).

!½fm4$ 4He 6Li 12C 16O

NN 0.05 %28:08ð2Þ %31:5ð2Þ %99:1ð6Þ %161:0ð2Þonly 0.0625 %28:25ð1Þ %31:8ð1Þ %101:4ð3Þ %164:9ð6Þ

0.08 %28:38ð1Þ %32:2ð1Þ %103:7ð2Þ %170:2ð4ÞNNþ 0.05 %25:33ð1Þ %27:7ð2Þ %76:9ð2Þ %119:5ð3Þ3N-ind. 0.0625 %25:34ð1Þ %27:6ð2Þ %77:2ð1Þ %119:7ð6Þ

0.08 %25:34ð1Þ %27:6ð1Þ %77:4ð2Þ %119:5ð2ÞNNþ 0.05 %28:45ð3Þ %31:8ð2Þ %96:1ð4Þ %143:7ð2Þ3N-full 0.0625 %28:45ð1Þ %31:8ð1Þ %96:8ð3Þ %145:6ð2Þ

0.08 %28:46ð1Þ %31:8ð1Þ %97:6ð1Þ %147:8ð1Þexp. %28:30 %31:99 %92:16 %127:62

-100

-90

-80

-70

-60

.

Egs

[MeV

]

(a)

NN only

12C20 MeV

(b)

NN 3N-induced

exp.

(c)

NN 3N-full

2 4 6 8 10 12 14Nmax

-180

-160

-140

-120

-100

.

Egs

[MeV

]

(d)16O

20 MeV

2 4 6 8 10 12Nmax

(e)

exp.

2 4 6 8 10 12Nmax

(f)

FIG. 2 (color online). IT-NCSM ground-state energies for 12Cand 16O as function of Nmax for the three types of Hamiltoniansand a range of flow parameters (for details see Fig. 1).


12 AUGUST 2011

072501-3

reduction of cutoff in 3NF allows to suppress induced 4N forces…

Ab initio calculations of heavier nuclei4

-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

16O24O

36Ca40Ca

48Ca52Ca

54Ca48Ni

56Ni60Ni

62Ni66Ni

68Ni78Ni

88Sr90Zr

100Sn106Sn

108Sn114Sn

116Sn118Sn

120Sn132Sn

-0.5

0.5 (d)

FIG. 5: (Color online) Ground-state energies from CR-CC(2,3) for (a) the NN+3N-induced Hamiltonian starting from the N3LO and N2LO-optimized NN interaction and (c) the NN+3N-full Hamiltonian with Λ3N = 400 MeV/c and Λ3N = 350 MeV/c. The boxes represent thespread of the results from α = 0.04 fm4 to α = 0.08 fm4, and the tip points into the direction of smaller values of α. Also shown are thecontributions of the CR-CC(2,3) triples correction to the (b) NN+3N-induced and (d) NN+3N-full results. All results employ !Ω = 24 MeVand 3N interactions with E3max = 18 in NO2B approximation and full inclusion of the 3N interaction in CCSD up to E3max = 12. Experimentalbinding energies [32] are shown as black bars.

ies have shown that for both cutoffs, the induced 4N inter-action are small up into the sd-shell [6, 9]. For heavier nuclei,Fig. 5(c) reveals that the α-dependence of the ground-stateenergies remains small for Λ3N = 400 MeV/c up to the heav-iest nuclei. Thus, the attractive induced 4N contributions thatoriginate from the initial NN interaction are canceled by ad-ditional repulsive 4N contributions originating from the ini-tial chiral 3N interaction. By reducing the initial 3N cutoffto Λ3N = 350 MeV/c, the repulsive 4N component resultingfor the initial 3N interaction is weakened [9] and the attrac-tive induced 4N from the initial NN prevails, leading to anincreased α-dependence indicating an attractive net 4N con-tribution. All of these effects are larger than the truncation un-certainties of the calculations, such as the cluster truncation,as is evident by the comparatively small triples contributionsshown in Fig. 5(b) and (d).

Taking advantage of the cancellation of induced 4N termsfor the NN+3N-full Hamiltonian with Λ3N = 400 MeV/c wecompare the energies to experiment. Throughout the differentisotopic chains starting from Ca, the experimental pattern ofthe binding energies is reproduced up to a constant shift ofthe order of 1 MeV per nucleon. The stability and qualitativeagreement of the these results over an unprecedented massrange is remarkable, given the fact that the Hamiltonian wasdetermined in the few-body sector alone.

When considering the quantitative deviations, one has toconsider consistent chiral 3N interaction at N3LO, and theinitial 4N interaction. In particular for heavier nuclei, the

contribution of the leading-order 4N interaction might be siz-able. Another important future aspect is the study of otherobservables, such as charge radii. In the present calcula-tions the charge radii of the HF reference states are sys-tematically smaller than experiment and the discrepancy in-creases with mass. For 16O, 40Ca, 88Sr, and 120Sn the cal-culated charge radii are 0.3 fm, 0.5 fm, 0.7 fm, and 1.0 fmtoo small [32]. These deviations are larger than the ex-pected effects of beyond-HF correlations and consistent SRG-evolutions of the radii. This discrepancy will remain a chal-lenge for future studies of medium-mass and heavy nucleiwith chiral Hamiltonians.

Conclusions. In this Letter we have presented the firstaccurate ab initio calculations for heavy nuclei using SRG-evolved chiral interactions. We have identified and eliminateda number of technical hurdles, e.g., regarding the SRG modelspace, that have inhibited state-of-the-art medium-mass ap-proaches to address heavy nuclei. As a result, many-bodycalculations up to 132Sn are now possible with controlled un-certainties on the order of 2%. The qualitative agreement ofground-state energies for nuclei ranging from 16O to 132Snobtained in a single theoretical framework demonstrates thepotential of ab initio approaches based on chiral Hamiltoni-ans. This is a first direct validation of chiral Hamiltonians inthe regime of heavy nuclei using ab initio techniques. Futurestudies will have to involve consistent chiral Hamiltonians atN3LO considering initial and SRG-induced 4N interactionsand provide an exploration of other observables.

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

coupled cluster (CC) framework

Ab initio calculations of heavier nuclei

• spectacular increase in range of applicability of ab initio many body frameworks

• significant discrepancies to experimental data for heavy nuclei for

(most of) presently used nuclear interactions

• need to quantify theoretical uncertainties

4

-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

16O24O

36Ca40Ca

48Ca52Ca

54Ca48Ni

56Ni60Ni

62Ni66Ni

68Ni78Ni

88Sr90Zr

100Sn106Sn

108Sn114Sn

116Sn118Sn

120Sn132Sn

-0.5

0.5 (d)

FIG. 5: (Color online) Ground-state energies from CR-CC(2,3) for (a) the NN+3N-induced Hamiltonian starting from the N3LO and N2LO-optimized NN interaction and (c) the NN+3N-full Hamiltonian with Λ3N = 400 MeV/c and Λ3N = 350 MeV/c. The boxes represent thespread of the results from α = 0.04 fm4 to α = 0.08 fm4, and the tip points into the direction of smaller values of α. Also shown are thecontributions of the CR-CC(2,3) triples correction to the (b) NN+3N-induced and (d) NN+3N-full results. All results employ !Ω = 24 MeVand 3N interactions with E3max = 18 in NO2B approximation and full inclusion of the 3N interaction in CCSD up to E3max = 12. Experimentalbinding energies [32] are shown as black bars.

ies have shown that for both cutoffs, the induced 4N inter-action are small up into the sd-shell [6, 9]. For heavier nuclei,Fig. 5(c) reveals that the α-dependence of the ground-stateenergies remains small for Λ3N = 400 MeV/c up to the heav-iest nuclei. Thus, the attractive induced 4N contributions thatoriginate from the initial NN interaction are canceled by ad-ditional repulsive 4N contributions originating from the ini-tial chiral 3N interaction. By reducing the initial 3N cutoffto Λ3N = 350 MeV/c, the repulsive 4N component resultingfor the initial 3N interaction is weakened [9] and the attrac-tive induced 4N from the initial NN prevails, leading to anincreased α-dependence indicating an attractive net 4N con-tribution. All of these effects are larger than the truncation un-certainties of the calculations, such as the cluster truncation,as is evident by the comparatively small triples contributionsshown in Fig. 5(b) and (d).

Taking advantage of the cancellation of induced 4N termsfor the NN+3N-full Hamiltonian with Λ3N = 400 MeV/c wecompare the energies to experiment. Throughout the differentisotopic chains starting from Ca, the experimental pattern ofthe binding energies is reproduced up to a constant shift ofthe order of 1 MeV per nucleon. The stability and qualitativeagreement of the these results over an unprecedented massrange is remarkable, given the fact that the Hamiltonian wasdetermined in the few-body sector alone.

When considering the quantitative deviations, one has toconsider consistent chiral 3N interaction at N3LO, and theinitial 4N interaction. In particular for heavier nuclei, the

contribution of the leading-order 4N interaction might be siz-able. Another important future aspect is the study of otherobservables, such as charge radii. In the present calcula-tions the charge radii of the HF reference states are sys-tematically smaller than experiment and the discrepancy in-creases with mass. For 16O, 40Ca, 88Sr, and 120Sn the cal-culated charge radii are 0.3 fm, 0.5 fm, 0.7 fm, and 1.0 fmtoo small [32]. These deviations are larger than the ex-pected effects of beyond-HF correlations and consistent SRG-evolutions of the radii. This discrepancy will remain a chal-lenge for future studies of medium-mass and heavy nucleiwith chiral Hamiltonians.

Conclusions. In this Letter we have presented the firstaccurate ab initio calculations for heavy nuclei using SRG-evolved chiral interactions. We have identified and eliminateda number of technical hurdles, e.g., regarding the SRG modelspace, that have inhibited state-of-the-art medium-mass ap-proaches to address heavy nuclei. As a result, many-bodycalculations up to 132Sn are now possible with controlled un-certainties on the order of 2%. The qualitative agreement ofground-state energies for nuclei ranging from 16O to 132Snobtained in a single theoretical framework demonstrates thepotential of ab initio approaches based on chiral Hamiltoni-ans. This is a first direct validation of chiral Hamiltonians inthe regime of heavy nuclei using ab initio techniques. Futurestudies will have to involve consistent chiral Hamiltonians atN3LO considering initial and SRG-induced 4N interactionsand provide an exploration of other observables.

Binder et al., Phys. Lett B 736, 119 (2014) 4He 16O

22O 24O36Ca 40Ca

48Ca 52Ca54Ca 60Ca

48Ni 56Ni68Ni 78Ni

-9

-8

-7

-6

E/A

(MeV

)

2.0/2.0 (PWA)2.2/2.0 (EM)2.0/2.0 (EM)1.8/2.0 (EM)exp.extrapol.

Simonis, Stroberg et al., PRC 96, 014303 (2017)

coupled cluster (CC) framework

in-medium SRG (IMSRG) framework

• remarkable agreement between different many-body frameworks• very good agreement between theory and experiment for masses of oxygen and calcium isotopes based on specific chiral interactions• contributions from 3N force play important role for drip line

Studies of neutron-rich nuclei:Neutron dripline and the oxygen anomaly

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

16 18 20 22 24 26 28Mass Number A

-180

-170

-160

-150

-140

-130

Ener

gy (M

eV)

MR-IM-SRGIT-NCSMSCGFCC AME 2012

oxygenisotopes

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

neutron dripline

• remarkable agreement between different many-body frameworks• very good agreement between theory and experiment for masses of oxygen and calcium isotopes based on specific chiral interactions• contributions from 3N force play important role for drip line

Studies of neutron-rich nuclei:Neutron dripline and the oxygen anomaly

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

16 18 20 22 24 26 28Mass Number A

-180

-170

-160

-150

-140

-130

Ener

gy (M

eV)

MR-IM-SRGIT-NCSMSCGFCC AME 2012

oxygenisotopes

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

neutron dripline

Need to quantify theoretical uncertainties!

The size of the atomic nucleus: challenges from novel high-precision measurements

ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS3529

0.15 0.18 0.21Rskin (fm)

3.2

3.3

3.4

3.5

R p (f

m)

a

3.4 3.5 3.6

Rn (fm)

b

2.0 2.4 2.8

c

D (fm3)α

Figure 2 | Predictions for observables related to the neutron distribution in 48Ca. Neutron skin Rskin (a), r.m.s. point-neutron radius Rn (b) and electricdipole polarizability ↵D (c) plotted versus the r.m.s. point-proton radius Rp. The ab initio predictions with NNLOsat (red circles) and chiral interactions ofref. 29 (squares) are compared to the DFT results with the energy density functionals SkM, SkP, SLy4, SV-min, UNEDF0 and UNEDF1 (ref. 20; diamonds).This is a representative subset of DFT results; for other DFT predictions, the reader is referred to ref. 20. The theoretical error bars estimate uncertaintiesfrom truncations of the employed method and model space (see Methods for details). The blue line represents a linear fit to the data. The blue bandencompasses all error bars and estimates systematic uncertainties. The horizontal green line marks the experimental value of Rp. Its intersection with theblue line and the blue band yields the vertical orange line and orange band, respectively, giving the predicted range for the ordinate.

0.15 0.20 0.25 0.30FW (qc)

3.25

3.35

3.45

3.55

3.65

R n (f

m)

qc = 0.778 fm−1

a

0.0 0.4 0.8 1.2 1.6q (fm−1)

0.0

0.2

0.4

0.6

0.8

1.0

F W (q

)

bNNLOsatDFT

0 1 2 3 4 5 6 7 8r (fm)

0.00

0.02

0.04

0.06

0.08

0.10

ρ− W

NNLOsat

c

(fm

−3)

ρ

chρ

Figure 3 | Weak-charge observables in 48Ca. a, Root mean square point-neutron radius Rn in 48Ca versus the weak-charge form factor FW(qc) at the CREXmomentum qc =0.778 fm1 obtained in ab initio calculations with NNLOsat (red circle) and chiral interactions of ref. 29 (squares). The theoretical errorbars estimate uncertainties from truncations of the employed method and model space (see Methods for details). The width of the horizontal orange bandshows the predicted range for Rn and is taken from Fig. 2b. The width of the vertical orange band is taken from Supplementary Fig. 2 and shows thepredicted range for FW(qc). b, Weak-charge form factor FW(q) as a function of momentum transfer q with NNLOsat (red line) and DFT with the energydensity functional SV-min21 (diamonds). The orange horizontal band shows FW(qc). c, Charge density (blue line) and (negative of) weak-charge density(red line). The weak-charge density extends well beyond ch as it is strongly weighted by the neutron distribution. The weak charge of 48Ca, obtained byintegrating the weak-charge density is QW =26.22 (for the weak charge of the proton and neutron see Methods).

is 0.12.Rskin . 0.15 fm. Figure 2a shows two remarkable features.First, the ab initio calculations yield neutron skins that are almostindependent of the employed interaction. This is due to the strongcorrelation between the Rn and Rp in this nucleus (Fig. 2b). Incontrast, DFT models exhibit practically no correlation betweenRskin and Rp. Second, the ab initio calculations predict a significantlysmaller neutron skin than the DFT models. The predicted rangeis also appreciably lower than the combined DFT estimate of0.176(18) fm (ref. 20) and is well below the relativistic DFT value ofRskin =0.22(2) fm (ref. 20). To shed light on the lower values of Rskinpredicted by ab initio theory, we computed the neutron separationenergy and the three-point binding energy dierence in 48Ca (bothbeing indicators of the N =28 shell gap). Our results are consistentwith experiment and indicate the pronounced magicity of 48Ca(Supplementary Table 2), whereas DFT results usually significantlyunderestimate the N =28 shell gap30. The shortcoming of DFT for48Ca is also reflected in Rp. Although many nuclear energy densityfunctionals are constrained to the Rp of 48Ca (refs 18,30), the resultsof DFT models shown in Fig. 2a overestimate this quantity.

For Rn (Fig. 2b) we find 3.47.Rn . 3.60 fm. Most of the DFTresults for Rn are outside this range, but fall within the blueband. Comparing Fig. 2a,b suggests that a measurement of asmall neutron skin in 48Ca would provide a critical test for abinitio models. For the electric dipole polarizability (Fig. 2c) ourprediction 2.19.↵D.2.60 fm3 is consistent with the DFT valueof 2.306(89) fm3 (ref. 20). Again, most of the DFT results fallwithin the ab initio uncertainty band. The result for ↵D will betested by anticipated experimental data from the Darmstadt–Osakacollaboration13,14. The excellent correlation between Rp, Rn and ↵Dseen in Fig. 2b,c demonstrates the usefulness of Rn and ↵D as probesof the neutron density.

The weak-charge radiusRW is another quantity that characterizesthe size of the nucleus. The CREX experiment will measure theparity-violating asymmetry Apv in electron scattering on 48Caat the momentum transfer qc = 0.778 fm1. This observable isproportional to the ratio of the weak-charge and electromagneticcharge form factors FW(qc)/Fch(qc) (ref. 12). Making someassumptions about the weak-charge form factor, one can deduce RW

188

© 2016 Macmillan Publishers Limited. All rights reserved

NATURE PHYSICS | VOL 12 | FEBRUARY 2016 | www.nature.com/naturephysics

Hagen. et al.. Nature Phys. 12, 186 (2015)

Horowitz

Rskin

48Ca

208Pb

A piece of the weak interaction violates parity (mirror symmetry) which allows to isolate it.

Negative longitudinal spin

Positive longitudinal spin

Pb 208

P

S (spin)

(momentum)

Incident electron

Target

Lead ( Pb) Radius Experiment : PREX

E = 1 GeV, electrons on lead

Elastic Scattering

Parity Violating Asymmetry PREXPb Radius ExperimentCREXCa Radius Experiment



15049Ca (I = 3/2)

51Ca (I = 3/2)

52Ca (I = 0)

100

50

20

10

Coun

ts p

er p

roto

n pu

lse

2

1

0 2,000Relative frequency (MHz)

3,000

Figure 2 | Examples of hyperfine structure spectra measured for the Caisotopes in the 393-nm 4s 2S1/2!4p 2P3/2 ionic transition. The solid linesshow the fit with a Voigt profile. Frequency values are relative to the centroidof 40Ca. The position of each hfs centroid is indicated by the dashed lines.

magnitude. It is now possible to routinely perform experiments withbeams of 104 ions s1 (ref. 23).

In this work, we have further optimized the photon detectionsensitivity and at the same time reduced further the photonbackground events8, now allowing the study of calcium isotopesproduced with a yield of only a few hundred ions per second. Whilepreserving the high resolution, this sensitivity surpasses the previouslimit by two orders of magnitude, achieved by an ultrasensitiveparticle detection technique employed on Ca isotopes18.

The short-lived Ca isotopes studied in this work were producedat the ISOLDE on-line isotope separator, located at the EuropeanCenter for Nuclear Research, CERN. High-energy proton pulseswith intensities of 3 1013 protons/pulse at 1.4GeV impingedevery 2.4 s on an uranium carbide target to create radioactivespecies of a wide range of chemical elements. The Ca isotopeswere selected from the reaction products by using a three-steplaser ionization scheme provided by the Resonance Ionization LaserIon Source (RILIS; ref. 24). A detailed sketch of the dierentexperimental processes from the ion beam production to thefluorescence detection is shown in Fig. 1.

After selective ionization, Ca ions (Ca+) were extracted fromthe ion source and accelerated up to 40 keV. The isotope ofinterest was mass-separated by using the High-Resolution MassSeparator (HRS). The selected isotopes were injected into a gas-filled radiofrequency trap (RFQ) to accumulate the incomingions. After a few milliseconds, bunches of ions of 5 µs temporalwidth were extracted and redirected into a dedicated beamline for collinear laser spectroscopy experiments (COLLAPS). AtCOLLAPS, the ion beamwas superimposed with a continuous wavelaser beam fixed at a wavelength of 393 nm (see Methods), closeto the 4s 2S1/2 !4p 2P3/2 transition in the Ca+. The laser frequencywas fixed to a constant value, while the ion velocity was variedinside the optical detection region. A change in the ion velocitycorresponds to a variation of laser frequency in the ion rest frame.This Doppler tuning of the laser frequency was used to scan thehyperfine structure (hfs) components of the 4s 2S1/2 ! 4p 2P3/2transition. At resonance frequencies, transitions between dierenthfs levels were excited, and subsequently the fluorescence photonswere detected by a light collection system consisting of four lensesand photomultiplier tubes (PMT) (see ref. 8 for details). The photonsignals were accepted only when the ion bunch passed in front ofthe light collection region, reducing the background counts fromscattered laser light and PMT dark counts by a factor of 104. Asample of the hfs spectra measured during the experiment is shownin Fig. 2. Isotopes with nuclear spin I =0 do not exhibit hyperfinestructure splitting. Consequently, only a single transition is observedfor 52Ca.

0.0

0.1

0.2

0.3

0.4

⟨r2 ⟩48

,52 (

fm2 )

δ

0.5

0.6NNLO

satSRG1SRG2

UNEDF0DF3

-a D1SD1S+co

rrHFB

-24DD-M

E2ZBM2+HF

Exp.

SRG2

SRG1

NNLO satUNEDF0

Mass number AR ch

(fm

)

Experiment (this work)

Ab initio(this work) DFT CI

ZBM2+HO

40 42 44 46 48 50 52 543.4

3.5

3.6

a

b

Figure 3 | Charge radii of Ca isotopes. a, Experimental charge radiicompared to ab initio calculations with chiral EFT interactions NNLOsat,SRG1, SRG2, as well as DFT calculations with the UNEDF0 functional.Experimental error bars are smaller than the symbols. The absolute valueswere obtained from the reference radius of 40Ca (Rch =3.478(2) fm;ref. 26). The values of 39Ca and 41,42Ca are taken from refs 45,46,respectively. A systematic theoretical uncertainty of 1% is estimated for theabsolute values due to the truncation level of the coupled-cluster methodand the finite basis space employed in the computation. b, Experimentalr.m.s. charge radius in 52Ca relative to that in 48Ca compared to the ab initioresults as well as those of representative density functional theory (DFT)and configuration interaction (CI) calculations. The systematicuncertainties in the theoretical predictions are largely cancelled when thedierences between r.m.s. charge radii are calculated (dotted horizontalblue lines). Experimental uncertainties are represented by the horizontalred lines (statistical) and the grey shaded band (systematic).

The isotope shifts were extracted from the fit of the hfsexperimental spectra, assuming multiple Voigt profiles in the 2-minimization (see Methods). The measured isotope shift relativeto the reference isotope 40Ca, and the corresponding change in themean-square charge radius are shown in Table 1. Statistical errors(parentheses) correspond to the uncertainty in the determinationof the peak positions in the hfs spectra. The systematic errors inthe isotope shift (square brackets) are mainly due to the uncertaintyin the beam energy, which is also the main contribution tothe uncertainty in the charge radius. Independent high-precisionmeasurements of isotope shifts on stable Ca isotopes25 were used foran accurate determination of the kinetic energy of each isotope. Thestability of the beam energy was controlled by measuring the stable40Ca, before and after the measurement of each isotope of interest.

Our experimental results (Table 1 and Fig. 3) show that the root-mean-square (r.m.s.) charge radius of 49Ca presents a considerableincrease with respect to 48Ca, hr 2i48,49 = 0.097(4) fm2, but muchsmaller than previously suggested17. The increase continues towardsN = 32, resulting in a very large charge radius for 52Ca, with anincrease relative to 48Ca of hr 2i48,52 =0.530(5) fm2.

596


NATURE PHYSICS | VOL 12 | JUNE 2016 | www.nature.com/naturephysics

Garcia Ruiz. et al.,Nature Phys. 12, 594 (2016)


15049Ca (I = 3/2)

51Ca (I = 3/2)

52Ca (I = 0)

100

50

20

10

Coun

ts p

er p

roto

n pu

lse

2

1


3,000






0.0

0.1

0.2

0.3

0.4

⟨r2 ⟩48

,52 (

fm2 )

δ

0.5

0.6

NNLOsat

SRG1SRG2

UNEDF0DF3

-a D1SD1S+co

rrHFB

-24DD-M

E2ZBM2+HF

Exp.

SRG2

SRG1

NNLO satUNEDF0

Mass number A

R ch (f

m)



ZBM2+HO

40 42 44 46 48 50 52 543.4

3.5

3.6

a

b




596




0.15 0.18 0.21Rskin (fm)

3.2

3.3

3.4

3.5

R p (f

m)

a

3.4 3.5 3.6

Rn (fm)

b

2.0 2.4 2.8

c

D (fm3)α


0.15 0.20 0.25 0.30FW (qc)

3.25

3.35

3.45

3.55

3.65

R n (f

m)

qc = 0.778 fm−1

a

0.0 0.4 0.8 1.2 1.6q (fm−1)

0.0

0.2

0.4

0.6

0.8

1.0

F W (q

)

bNNLOsatDFT

0 1 2 3 4 5 6 7 8r (fm)

0.00

0.02

0.04

0.06

0.08

0.10

ρ− W

NNLOsat

c

(fm

−3)

ρ

chρ





188




calciumisotopes

Horowitz

Rskin

48Ca

208Pb

A piece of the weak interaction violates parity (mirror symmetry) which allows to isolate it.

Negative longitudinal spin

Positive longitudinal spin

Pb 208

P

S (spin)

(momentum)

Incident electron

Target

Lead ( Pb) Radius Experiment : PREX

E = 1 GeV, electrons on lead

Elastic Scattering

Parity Violating Asymmetry PREXPb Radius ExperimentCREXCa Radius Experiment



15049Ca (I = 3/2)

51Ca (I = 3/2)

52Ca (I = 0)

100

50

20

10

Coun

ts p

er p

roto

n pu

lse

2

1


3,000






0.0

0.1

0.2

0.3

0.4

⟨r2 ⟩48

,52 (

fm2 )

δ

0.5

0.6NNLO

satSRG1SRG2

UNEDF0DF3

-a D1SD1S+co

rrHFB

-24DD-M

E2ZBM2+HF

Exp.

SRG2

SRG1

NNLO satUNEDF0

Mass number AR ch

(fm

)



ZBM2+HO

40 42 44 46 48 50 52 543.4

3.5

3.6

a

b




596



Garcia Ruiz. et al.,Nature Phys. 12, 594 (2016)


15049Ca (I = 3/2)

51Ca (I = 3/2)

52Ca (I = 0)

100

50

20

10

Coun

ts p

er p

roto

n pu

lse

2

1


3,000






0.0

0.1

0.2

0.3

0.4

⟨r2 ⟩48

,52 (

fm2 )

δ

0.5

0.6

NNLOsat

SRG1SRG2

UNEDF0DF3

-a D1SD1S+co

rrHFB

-24DD-M

E2ZBM2+HF

Exp.

SRG2

SRG1

NNLO satUNEDF0

Mass number A

R ch (f

m)



ZBM2+HO

40 42 44 46 48 50 52 543.4

3.5

3.6

a

b




596




0.15 0.18 0.21Rskin (fm)

3.2

3.3

3.4

3.5

R p (f

m)

a

3.4 3.5 3.6

Rn (fm)

b

2.0 2.4 2.8

c

D (fm3)α


0.15 0.20 0.25 0.30FW (qc)

3.25

3.35

3.45

3.55

3.65

R n (f

m)

qc = 0.778 fm−1

a

0.0 0.4 0.8 1.2 1.6q (fm−1)

0.0

0.2

0.4

0.6

0.8

1.0

F W (q

)

bNNLOsatDFT

0 1 2 3 4 5 6 7 8r (fm)

0.00

0.02

0.04

0.06

0.08

0.10

ρ− W

NNLOsat

c

(fm

−3)

ρ

chρ





188




calciumisotopes

Horowitz

Rskin

VOLUME 85, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 18 DECEMBER 2000

S(2

08P

b) (

fm)

slope of neutron EOS

0.0

0.1

0.2

0.3

-50 0 50 100 150 200

FIG. 3. The derivative of the neutron EOS at rn !0.10 neutron!fm3 (in units of MeV fm3!neutron) vs the S valuein 208Pb for 18 Skyrme parameter sets. The cross is SkX.

provided by Wiringa, Fiks, and Fabrocini [17] and Akmaland Pandharipande [18]. Generally the agreement with FPis good up to about rn ! 0.10 neutron!fm3. At higherdensity the differences in the various NN potentials [17]and the very uncertain NNN potential become important.Thus, although the FP neutron EOS serves as a reasonablestarting point, we do not have a truly fundamental theoryfor neutron EOS. Any constraints coming from the prop-erties of nuclei such as the neutron radii are extremelyimportant.

Given the difficulty of the JLAB measurement, it isimportant to know to what extent a measurement of Sin one nucleus such as 208Pb will be applicable to othernuclei. There are two points to investigate: the dependenceof S on mass and the dependence of S on the asymmetryin the Fermi energy for protons and neutrons. For the firstcase, I compare in Fig. 4 the S values for two nuclei nearthe valley of stability (where the Fermi energies for protonsand neutrons are about equal to each other), those for 208Pband 138Ba. One observes a nearly linear relationship whichstarts at S ! 0. For the second case, I compare in thesame figure the S value in 208Pb to the S value for 132Snwhere the neutrons at the Fermi surface are bound about8 MeV less than the protons (see Figs. 4 and 5 in Ref. [6]).Again there is a tight correlation, but the asymmetry inthe Fermi energy produces a systematic increase in theneutron skin for all of the 18 SHF parameter sets. Thusthere are two clear mechanisms for producing a neutronskin. One which is related to the asymmetry in the Fermienergy is well determined within SHF, and another whichdepends on the neutron EOS is undetermined unless oneadds a constraint to the neutron EOS. It is the Fermi-energy asymmetry effect which dominates the increase inthe matter radii of neutron-rich light nuclei such as in the

S fo

r 20

8Pb

(fm

)

S for 132Sn and 138Ba (fm)

0.0

0.1

0.2

0.3

0.0 0.1 0.2 0.3

FIG. 4. The S value for 208Pb vs the S values for 132Sn (filledcircles) and 138Ba (plusses) for 18 Skyrme parameter sets. Thehorizontal line is the SkX value for 208Pb.

Na isotopes [11]. Thus it is most important to accuratelydetermine the neutron rms radius in a stable nucleus suchas 208Pb. The neutron rms radius of 208Pb will providean important new constraint on the neutron EOS modelswhich are used to calculate the properties of neutron stars[17]. The results discussed here are based upon a widevariety of parametrizations for the Skyrme Hartree-Fockmodel for finite nuclei and nucleon matter. It will beimportant to explore the generality of these conclusionswithin the Skyrme model as well as in other mean-fieldmodels.

This work was stimulated by discussions with ChuckHorowitz and Dick Furnstahl during the ECT workshopon “Parity Violation in Atomic, Nuclear and Hadronic Sys-tems” which was held in Trento, Italy, June 5–16 (2000).Support for this work was provided by the U.S. NationalScience Foundation Grant No. PHY-0070911.

[1] G. Fricke et al., At. Data Nucl. Data Tables 60, 177 (1995).[2] L. Ray, Phys. Rep. 212, 223 (1992).[3] Jefferson Laboratory experiment E-00-003, spokespersons

R. Michaels, P. A. Souder, and G. M. Urciuoli.[4] C. J. Horowitz, S. J. Pollock, P. A. Souder, and R. Michaels,

nucl-th/9912039 [Phys. Rev. C (to be published)].[5] D. Vautherin and D. M. Brink, Phys. Rev. C 5, 626 (1972).[6] B. A. Brown, Phys. Rev. C 58, 220 (1998).[7] B. A. Brown, W. A. Richter, and R. Lindsay, Phys. Lett. B

483, 49 (2000).[8] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Scha-

effer, Nucl. Phys. A627, 710 (1997).[9] S. J. Pollack and M. C. Welliver, Phys. Lett. B 464, 177

(1999).[10] B. Friedman and V. R. Pandharipande, Nucl. Phys. A361,

502 (1981).

5298

Brown, PRL 85, 5296 (2000)

48Ca

direct connections to astrophysics!

208Pb

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() = T + VNN() + V3N() + ...

0 0.5 1 1.5 2 2.5

kF (fm-1)

-25

0

25

50

75

100

125

150

175

E/A

-M (M

eV)

QHD (547 MeV)NLC (224 MeV)NL3 (271 MeV)Z271 (271 MeV)

Symmetric Nuclear Matter

Krishna S. Kumar NSKIN2016: The PREX-I Result

Relativistic Electron Scattering

4

e

e

γ

Differential Cross Section

Heavy, spinlessnucleus

d

d

Mott=

4Z22E2

q4

d

d=

d

d

Mott

F (q)2

Neglecting recoil, form factor F(q) is the Fourier transform of charge

distribution

As Q increases, nuclear size modifies formula

and nuclear size

Q2: -(4-momentum)2 of the virtual photon

€

q2 = −4 E # E sin2 θ2

4-momentum transfer

Uniform interior is a clearmanifestation of nuclear saturation, namely the

existence of an equilibrium density

Nuclear Saturation A Hallmark of the Nuclear DynamicsOverview RG Summary Extras Physics Resolution Forces Filter Coupling


VL=0(k , k )




lS

Equation of state of symmetric nuclear matter:nuclear saturation

Batty et. al, Karlsruhe (1987)

0.8 1.0 1.2 1.4 1.6kF [fm−1]

−30

−25

−20

−15

−10

−5

0

Ener

gy/n

ucle

on [M

eV]

Λ = 1.8 fm−1 NN onlyΛ = 2.8 fm−1 NN only

Vlow k NN from N3LO (500 MeV)

3NF fit to E3H and r4He Λ3NF = 2.0 fm−1

3rd order pp+hhNN only



VL=0(k , k )




lS

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

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

0 1 2 3 4r [fm]

−100

0

100

200

V(r

) [M

eV]

λ = 20 fm−1

1 2 3 4r [fm]

λ = 4 fm−1

1 2 3 4r [fm]

λ = 3 fm−1

1 2 3 4r [fm]

λ = 2 fm−1

1 2 3 4r [fm]

λ = 1.5 fm−1

AV18

N3LO

0 1 2 3 4r [fm]

−100

0

100

200

V(r

) [M

eV]

λ = 20 fm−1

1 2 3 4r [fm]

λ = 4 fm−1

1 2 3 4r [fm]

λ = 3 fm−1

1 2 3 4r [fm]

λ = 2 fm−1

1 2 3 4r [fm]

λ = 1.5 fm−1

AV18

N3LO

Fitting the 3NF LECs at low resolution scales

0.8 1.0 1.2 1.4 1.6kF [fm−1]

−30

−25

−20

−15

−10

−5

0

Ener

gy/n

ucle

on [M

eV]




3rd order pp+hhNN only



VL=0(k , k )




lS


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



0 1 2 3 4r [fm]

−100

0

100

200

V(r

) [M

eV]

λ = 20 fm−1

1 2 3 4r [fm]

λ = 4 fm−1

1 2 3 4r [fm]

λ = 3 fm−1

1 2 3 4r [fm]

λ = 2 fm−1

1 2 3 4r [fm]

λ = 1.5 fm−1

AV18

N3LO

0 1 2 3 4r [fm]

−100

0

100

200

V(r

) [M

eV]

λ = 20 fm−1

1 2 3 4r [fm]

λ = 4 fm−1

1 2 3 4r [fm]

λ = 3 fm−1

1 2 3 4r [fm]

λ = 2 fm−1

1 2 3 4r [fm]

λ = 1.5 fm−1

AV18

N3LO


0.8 1.0 1.2 1.4 1.6kF [fm−1]

−30

−25

−20

−15

−10

−5

0

Ener

gy/n

ucle

on [M

eV]

Λ = 1.8 fm−1

Λ = 2.8 fm−1




3rd order pp+hh

NN + 3N

NN only

Reproduction of saturation point without readjusting parameters!





VL=0(k , k )




lS

0 1 2 3 4r [fm]

−100

0

100

200

V(r

) [M

eV]

λ = 20 fm−1

1 2 3 4r [fm]

λ = 4 fm−1

1 2 3 4r [fm]

λ = 3 fm−1

1 2 3 4r [fm]

λ = 2 fm−1

1 2 3 4r [fm]

λ = 1.5 fm−1

AV18

N3LO

0 1 2 3 4r [fm]

−100

0

100

200

V(r

) [M

eV]

λ = 20 fm−1

1 2 3 4r [fm]

λ = 4 fm−1

1 2 3 4r [fm]

λ = 3 fm−1

1 2 3 4r [fm]

λ = 2 fm−1

1 2 3 4r [fm]

λ = 1.5 fm−1

AV18

N3LO


0.8 1.0 1.2 1.4 1.6kF [fm−1]

−30

−25

−20

−15

−10

−5

0

Ener

gy/n

ucle

on [M

eV]

Λ = 1.8 fm−1

Λ = 2.8 fm−1




3rd order pp+hh

NN + 3N

NN only





VL=0(k , k )




lS

0 1 2 3 4r [fm]

−100

0

100

200

V(r

) [M

eV]

λ = 20 fm−1

1 2 3 4r [fm]

λ = 4 fm−1

1 2 3 4r [fm]

λ = 3 fm−1

1 2 3 4r [fm]

λ = 2 fm−1

1 2 3 4r [fm]

λ = 1.5 fm−1

AV18

N3LO

0 1 2 3 4r [fm]

−100

0

100

200

V(r

) [M

eV]

λ = 20 fm−1

1 2 3 4r [fm]

λ = 4 fm−1

1 2 3 4r [fm]

λ = 3 fm−1

1 2 3 4r [fm]

λ = 2 fm−1

1 2 3 4r [fm]

λ = 1.5 fm−1

AV18

N3LO


Drischler, KH, Schwenk, PRC93, 054314 (2016)

0 0.05 0.1 0.15n [fm-3]

-4

-2

0

2

4

E/N

[MeV

]

Two-pion-exchange 3N

0 0.05 0.1 0.15n [fm-3]

Two-pion−one-pion-exchange 3N

0 0.05 0.1 0.15n [fm-3]

Pion-ring 3N

0 0.05 0.1 0.15 0.2n [fm-3]

-4

-2

0

2

4

EM 500 MeVEGM 450/700 MeVEGM 450/500 MeV

Two-pion-exchange−contact 3N

Contributions of many-body forces at N3LO in neutron matter

Tews, Krüger, KH, SchwenkPRL 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,

(power counting?)

0 0.05 0.1 0.15n [fm-3]

-4

-2

0

2

4

E/N

[MeV

]


0 0.05 0.1 0.15n [fm-3]


0 0.05 0.1 0.15n [fm-3]

Pion-ring 3N

0 0.05 0.1 0.15 0.2n [fm-3]

-4

-2

0

2

4



Tews, Krüger, KH, SchwenkPRL 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,

(power counting?)

• 4NF contributions small

3

0 0.05 0.1 0.15n [fm-3]

-4

-2

0

2

4

E/N

[MeV

]


0 0.05 0.1 0.15n [fm-3]


0 0.05 0.1 0.15n [fm-3]

Pion-ring 3N

0 0.05 0.1 0.15 0.2n [fm-3]

-4

-2

0

2

4



0 0.05 0.1 0.15n [fm-3]

-0.4-0.3-0.2-0.1

00.10.20.30.4

E/N

[MeV

]

Three-pion-exchange 4N Va

0 0.05 0.1 0.15n [fm-3]

Three-pion-exchange 4N Ve

0 0.05 0.1 0.15n [fm-3]

Pion-pion-interaction 4N Vf

0 0.05 0.1 0.15 0.2n [fm-3]

-0.4-0.3-0.2-0.100.10.20.30.4EGM 450/500 MeV

EGM 450/700 MeVEM 500 MeV

pRelativistic-corrections 3Np

FIG. 2. (Color online) Energy per particle versus density for all individual N3LO 3N and 4N force contributions to neutronmatter at the Hartree-Fock level. The bands are obtained by varying the 3N/4N cutoff Λ = 2 − 2.5 fm−1. For the two-pion-exchange–contact and the relativistic-corrections 3N forces, the different bands correspond to the different NN contacts, CT

and CS , determined consistently for the N3LO EM/EGM potentials. The inset diagram illustrates the 3N/4N force topology.

ity of the energy to the single-particle spectrum used.We find that the energy changes from second to thirdorder, employing a free or Hartree-Fock spectrum, by0.8, 0.4, 1.3MeV (1.4, 0.9, 2.7MeV) per particle at n0/2(n0) for the EGM 450/500, 450/700, EM 500 N3LO po-tentials, respectively. The results, which include all theseuncertainties, are displayed by the bands in Fig. 1. Un-derstanding the cutoff dependence and developing im-proved power counting schemes remain important openproblems in chiral EFT [21]. For the neutron matter en-ergy at n0, our first complete N3LO calculation yields14.1 − 21.0MeV per particle. If we were to omit theresults based on the EM 500 N3LO potential, as it con-verges slowest at n0, the range would be 14.1−18.4MeV.

As we find relatively large contributions from N3LO3N forces, it is important to study the EFT convergencefrom N2LO to N3LO. This is shown in Fig. 3 for theEGM potentials (N2LO is not available for EM), wherethe N3LO results are found to overlap with the N2LOband across a ±1.5MeV range around 17MeV at satura-tion density. As expected from the net-attractive N3LO3N contributions in Fig. 2, the N3LO band yields lowerenergies. For the N2LO band, we have estimated the the-oretical uncertainties in the same way, and the neutronmatter energy ranges from 15.5 − 21.4MeV per particle

at n0. The theoretical uncertainty is reduced from N2LOto N3LO to 14.1 − 18.4MeV, but not by a factor ∼ 1/3based on the power counting estimate. This reflects the

0 0.05 0.1 0.15

n [fm-3]

0

5

10

15

20

E/N

[MeV

]

N2LON3LO (only EGM)

FIG. 3. (Color online) Neutron matter energy per particle asa function of density at N2LO (upper/blue band that extendsto the dashed line) and N3LO (lower/red band). The bandsare based on the EGM NN potentials and include uncertaintyestimates as in Fig. 1.

Contributions of many-body forces at N3LO in neutron matter

CONTENTS









A. Introduction










09/03/2015 | SFB 1245 | Nuclei: From Fundamental Interactions to Structure and Stars | Projects B05 and B06 | 72

B05 and B06: Physics introduction From fundamental interactions to supernovae

Core-collapse supernova: end of massive stars, birth of neutron stars All forces of nature are involved Major contribution to chemical history of the universe

•  Supernova simulations

•  Matter properties, equation of state

•  Neutrino-matter interactions

•  Reactions on nuclei

Nucleosynthesis

B04

B01

B02

The equation of state of high-density matter:constraints for neutron stars from nuclear physics

CONTENTS









A. Introduction

















Nucleosynthesis

B04

B01

B02


figure taken from Krüger,doctoral thesis (2016) www.stellarcollapse.org/nsmasses

http://www.stellarcoolapse.org/nsmasses

CONTENTS









A. Introduction

















Nucleosynthesis

B04

B01

B02


companionwith awell-determinedmass of 0.20M(15) that appears to be hot (10), suggesting that itsenvelope is thick. For this reason, we base theWD mass estimate on cooling tracks with thickhydrogen atmospheres for masses up to 0.2M,which we constructed by using the MESA stellarevolution code (8, 16). Initial models were builtfor masses identical to the ones in (11), for whichprevious comparisons have yielded good agree-ment with observations (14), with the additionof tracks with 0.175 and 0.185 M for finercoverage (Fig. 2). For masses up to 0.169M, ourmodels show excellent agreement with (11);however, our 0.196 M model is quite different,because it has a thick envelope instead of a thinone. Being closer to the constraints for the WDcompanion to PSR J0348+0432, it yields a moreconservative mass constraint, MWD = 0.165 to0.185 at 99.73% confidence (Fig. 3 and Table 1),which we adopt. The corresponding radius isRWD = 0.046 to 0.092 R at 99.73% confidence.Our models yield a cooling age of tcool ∼ 2 Gy.

Pulsar MassThe derived WD mass and the observed massratio q imply a NSmass in the range from 1.97 to2.05M at 68.27% or 1.90 to 2.18M at 99.73%confidence. Hence, PSR J0348+0432 is only thesecond NS with a precisely determined massaround 2M, after PSR J1614−2230 (2). It has a3-s lower mass limit 0.05M higher than the latterand therefore provides a verification, using a dif-ferent method, of the constraints on the EOS ofsuperdense matter present in NS interiors (2, 17).For these masses and the known orbital period,GR predicts that the orbital period should decrease

at the rate of P:GRb ¼ ð−2:58þ0:07

−0:11 Þ % 10−13 s s−1

(68.27%confidence) because of energy loss throughGW emission.

Radio ObservationsSince April 2011, we have been observing PSRJ0348+0432 with the 1.4-GHz receiver of the305-m radio telescope at the Arecibo Observatoryby using its four wide-band pulsar processors (18).In order to verify the Arecibo data, we have beenindependently timing PSR J0348+0432 at 1.4 GHzby using the 100-m radio telescope in Effelsberg,Germany. The two timing data sets produce con-sistent rotational models, providing added con-fidence in both. Combining the Arecibo andEffelsberg data with the initial GBTobservations(7), we derived the timing solution presented inTable 1. To match the arrival times, the solutionrequires a significant measurement of orbital de-cay, P

:b ¼ −2:73 % 10−13 T 0:45% 10−13 s s−1

(68.27% confidence).The total proper motion and distance estimate

(Table 1) allowed us to calculate the kinematiccorrections to P

:b from its motion in the Galaxy,

plus any contribution from possible variations ofG: dP

:b ¼ 0:016% 10−13 T 0:003% 10−13 s s−1.

This is negligible compared to the measurementuncertainty. Similarly, the small rate of rotationalenergy loss of the pulsar (Table 1) excludes anysubstantial contamination resulting frommass lossfrom the system; furthermore, we can excludesubstantial contributions to P

:b from tidal effects

[see (8) for details]. Therefore, the observedP:b is

caused by GW emission, and its magnitude isentirely consistent with the one predicted by GR:P:b=P

:GRb ¼ 1:05 T 0:18 (Fig. 3).

If we assume that GR is the correct theory ofgravity, we can then derive the component massesfrom the intersection of the regions allowed byq and P

:b (Fig. 3): MWD ¼ 0:177þ0:017

−0:018 M andMPSR ¼ 2:07þ0:20

−0:21 M (68.27% confidence). Thesevalues are not too constraining yet. However, theuncertainty of the measurement of P

:b decreases

with T baseline−5/2 (where Tbaseline is the timing base-

line); therefore, this method will yield very precisemass measurements within a couple of years.

Discussion

PSR J0348+0432 as a Testbed for GravityThere are strong arguments for GR not to be validbeyond a (yet unknown) critical point, like itsincompatibility with quantum theory and its pre-diction of the formation of spacetime singularities.Therefore, it remains an open question whetherGR is the final description of macroscopic gravity.This strongly motivates testing gravity regimesthat have not been tested before, in particularregimes where gravity is strong and highly non-linear. Presently, binary pulsars provide the besthigh-precision experiments to probe strong-fielddeviations from GR and the best tests of theradiative properties of gravity (19–23). The orbitalperiod of PSR J0348+0432 is only 15 s longerthan that of the double pulsar system PSR J0737–3039, but it has ∼two times more fractional grav-itational binding energy than each of the double-pulsar NSs. This places it far outside the presentlytested binding energy range (Fig. 4A) (8). Be-cause the magnitude of strong-field effects gener-ally depends nonlinearly on the binding energy,the measurement of orbital decay transforms the

Fig. 3. System masses andorbital-inclination constraints.Constraints on system masses andorbital inclination from radio andoptical measurements of PSRJ0348+0432 and its WD compan-ion. Each triplet of curves corre-sponds to the most likely valueand standard deviations (68.27%confidence) of the respective pa-rameters. Of these, two (q and MWD)are independent of specific gravitytheories (in black). The contourscontain the 68.27 and 95.45% ofthe two-dimensional probabilitydistribution. The constraints fromthe measured intrinsic orbital decay(P:bint, in orange) are calculated as-

suming that GR is the correct theoryof gravity. All curves intersect inthe same region, meaning thatGR passes this radiative test (8).(Bottom left) cosi-MWD plane. Thegray region is excluded by the con-dition MPSR > 0. (Bottom right)MPSR-MWD plane. The gray regionis excluded by the condition sini ≤ 1. The lateral graphs depict the one-dimensional probability-distribution function for the WD mass (right), pulsar mass(top right), and inclination (top left) based on the mass function, MWD, and q.

www.sciencemag.org SCIENCE VOL 340 26 APRIL 2013 1233232-3

RESEARCH ARTICLE

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

a

b

c

–40

–30

–20

–10

0

10

20

30

–40

–30

–20

–10

0

10

20

30

Tim

ing

resi

dual

(μs)

–40

–30

–20

–10

0

10

20

30

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)

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

89.12

89.14

89.16

89.18

89.2

89.22

89.24a b

0.48 0.49 0.5 0.51 0.52

Incl

inat

ion

angl

e, i

(°)

Companion mass, M2 (M()1.8 1.85 1.9 1.95 2 2.05 2.1 2.15

Pro

babi

lity

dens

ity

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







a

b

c

–40

–30

–20

–10

0

10

20

30

–40

–30

–20

–10

0

10

20

30

Tim

ing

resi

dual

(μs)

–40

–30

–20

–10

0

10

20

30

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0



89.1

89.12

89.14

89.16

89.18

89.2

89.22

89.24a b

0.48 0.49 0.5 0.51 0.52

Incl

inat

ion

angl

e, i

(°)


Pro

babi

lity

dens

ity

Pulsar mass (M()


RESEARCH LETTER








a

b

c

–40

–30

–20

–10

0

10

20

30

–40

–30

–20

–10

0

10

20

30

Tim

ing

resi

dual

(μs)

–40

–30

–20

–10

0

10

20

30

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0



89.1

89.12

89.14

89.16

89.18

89.2

89.22

89.24a b

0.48 0.49 0.5 0.51 0.52

Incl

inat

ion

angl

e, i

(°)


Pro

babi

lity

dens

ity

Pulsar mass (M()


RESEARCH LETTER



Demorest et al.,Nature 467, 1081 (2010) Antoniadis et al.,Science 340, 448 (2013)

The list of author affi liations is available in the full article online.*Corresponding author. E-mail: [email protected]

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

Introduction: Neutron stars with masses above 1.8 solar masses (M), possess extreme gravitational fi elds, which may give rise to phenomena outside general relativity. Hitherto, these strong-fi eld devia-tions have not been probed by experiment, because they become observable only in tight binaries containing a high-mass pulsar and where orbital decay resulting from emission of gravitational waves can be tested. Understanding the origin of such a system would also help to answer fundamental ques-tions of close-binary evolution.

Methods: We report on radio-timing observations of the pulsar J0348+0432 and phase-resolved optical spectroscopy of its white-dwarf companion, which is in a 2.46-hour orbit. We used these to derive the component masses and orbital parameters, infer the system’s motion, and constrain its age.

Results: We fi nd that the white dwarf has a mass of 0.172 ± 0.003 M, which, combined with orbital velocity measurements, yields a pulsar mass of 2.01 ± 0.04 M. Additionally, over a span of 2 years, we observed a signifi cant decrease in the orbital period, Pb

obs = –8.6 ± 1.4 µs year1 in our radio-timing data.

Discussion: Pulsar J0348+0432 is only the second neutron star with a precisely determined mass of 2 M and independently confi rms the existence of such massive neutron stars in nature. For these

masses and orbital period, general relativity predicts a significant orbital decay, which matches the observed value, Pb

obs/ PbGR = 1.05

± 0.18.The pulsar has a gravitational binding

energy 60% higher than other known neu-tron stars in binaries where gravitational-wave damping has been detected. Because the magnitude of strong-field deviations generally depends nonlinearly on the bind-ing energy, the measurement of orbital decay transforms the system into a gravita-tional laboratory for an as-yet untested grav-ity regime. The consistency of the observed orbital decay with general relativity therefore supports its validity, even for such extreme gravity-matter couplings, and rules out strong-fi eld phenomena predicted by physi-cally well-motivated alternatives. Moreover, our result supports the use of general rela-tivity–based templates for the detection of gravitational waves from merger events with advanced ground-based detectors.

Lastly, the system provides insight into pulsar-spin evolution after mass accretion. Because of its short merging time scale of 400 megayears, the system is a direct chan-nel for the formation of an ultracompact x-ray binary, possibly leading to a pulsar-planet system or the formation of a black hole.

Artist’s impression of the PSR J0348+0432 system. The compact pulsar (with beams of radio emission) produces a strong distortion of spacetime (illustrated by the green mesh). Conversely, spacetime around its white dwarf com-panion (in light blue) is substantially less curved. According to relativistic theories of gravity, the binary system is subject to energy loss by gravitational waves.

26 APRIL 2013 VOL 340 SCIENCE www.sciencemag.org

RESEARCH ARTICLE SUMMARY

448

FIGURES AND TABLE IN THE FULL ARTICLE

Fig. 1. Radial velocities and spectrum of the white dwarf companion to PSR J0348+0432

Fig. 2. Mass measurement of the white dwarf companion to PSR J0348+0432

Fig. 3. System masses and orbital-inclination constraints

Fig. 4. Probing strong fi eld gravity with PSR J0348+0432

Fig. 5. Constraints on the phase offset in gravitational wave cycles in the LIGO/VIRGO bands

Fig. 6. Past and future orbital evolution of PSR J0348+0432

Fig. 7. Possible formation channels and fi nal fate of PSR J0348+0432

Table 1. Observed and derived parameters for the PSR J0348+0432 system

SUPPLEMENTARY MATERIALS

Supplementary TextFigs. S1 to S11Tables S1 to S3References

ADDITIONAL RESOURCES

Fundamental Physics in Radio Astronomy group at the Max-Planck-Institut für Radioastronomie, www3.mpifr-bonn.mpg.de/div/fundamental/

The European Southern Observatory, www.eso.org/public/

D. R. Lorimer, Binary and millisecond pulsars. Living Rev. Relativ. 11, 8 (2008). http://dx.doi.org/10.12942/lrr-2008-8

T. Damour, “Binary systems as test-beds of gravity theories,” in Physics of Relativistic Objects in

Compact Binaries: From Birth to Coalescence, M. Colpi, P. Casella, V. Gorini, U. Moschella, A. Possenti, Eds. (Astrophysics and Space Science Library, Springer, Dordrecht, Netherlands, 2009), vol. 359, pp. 1–41.

READ THE FULL ARTICLE ONLINEhttp://dx.doi.org/10.1126/science.1233232

Cite this article as J. Antoniadis et al., Science 340, 1233232 (2013). DOI: 10.1126/science.1233232

Published by AAAS

DOI: 10.1126/science.1233232, (2013);340 Science

et al.John AntoniadisA Massive Pulsar in a Compact Relativistic Binary

This copy is for your personal, non-commercial use only.

clicking here.colleagues, clients, or customers by , you can order high-quality copies for yourIf you wish to distribute this article to others

here.following the guidelines

can be obtained byPermission to republish or repurpose articles or portions of articles

): September 30, 2013 www.sciencemag.org (this information is current as of

The following resources related to this article are available online at

http://www.sciencemag.org/content/340/6138/1288.3.full.htmlA correction has been published for this article at:

http://www.sciencemag.org/content/340/6131/1233232.full.htmlversion of this article at:

including high-resolution figures, can be found in the onlineUpdated information and services,

http://www.sciencemag.org/content/suppl/2013/04/25/340.6131.1233232.DC2.html http://www.sciencemag.org/content/suppl/2013/04/24/340.6131.1233232.DC1.html

can be found at: Supporting Online Material

http://www.sciencemag.org/content/340/6131/1233232.full.html#relatedfound at:

can berelated to this article A list of selected additional articles on the Science Web sites

http://www.sciencemag.org/content/340/6131/1233232.full.html#ref-list-1, 19 of which can be accessed free:cites 89 articlesThis article

http://www.sciencemag.org/content/340/6131/1233232.full.html#related-urls3 articles hosted by HighWire Press; see:cited by This article has been

http://www.sciencemag.org/cgi/collection/astronomyAstronomy

subject collections:This article appears in the following

registered trademark of AAAS. is aScience2013 by the American Association for the Advancement of Science; all rights reserved. The title

CopyrightAmerican Association for the Advancement of Science, 1200 New York Avenue NW, Washington, DC 20005. (print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last week in December, by theScience

on

Sept

embe

r 30,

201

3ww

w.sc

ienc

emag

.org

Down

load

ed fr

om

on

Sept

embe

r 30,

201

3ww

w.sc

ienc

emag

.org

Down

load

ed fr

om

on

Sept

embe

r 30,

201

3ww

w.sc

ienc

emag

.org

Down

load

ed fr

om

on

Sept

embe

r 30,

201

3ww

w.sc

ienc

emag

.org

Down

load

ed fr

om

on

Sept

embe

r 30,

201

3ww

w.sc

ienc

emag

.org

Down

load

ed fr

om

on

Sept

embe

r 30,

201

3ww

w.sc

ienc

emag

.org

Down

load

ed fr

om

on

Sept

embe

r 30,

201

3ww

w.sc

ienc

emag

.org

Down

load

ed fr

om

on

Sept

embe

r 30,

201

3ww

w.sc

ienc

emag

.org

Down

load

ed fr

om

on

Sept

embe

r 30,

201

3ww

w.sc

ienc

emag

.org

Down

load

ed fr

om

on

Sept

embe

r 30,

201

3ww

w.sc

ienc

emag

.org

Down

load

ed fr

om

on

Sept

embe

r 30,

201

3ww

w.sc

ienc

emag

.org

Down

load

ed fr

om

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

Mmax

= 2.0± 0.04M

R 10 km


• consider forces on a mass element:

rgravity:

Fp = A (pout

pin

) = A dppressuredifference:

Fg = GM(r)(r)A dr

r2

M(r)

A



rgravity:

Fp = A (pout

pin


Fg = GM(r)(r)A dr

r2

• hydrostatic equilibrium condition:

Fg = Fp ) dp

dr= GM(r)(r)

r2

M(r)

A

dp

dr= GM(r)"(r)

r2

1 +

p(r)

"(r)c2

1 +

4r3p(r)

M(r)c2

1 2GM(r)

c2r



rgravity:

Fp = A (pout

pin


Fg = GM(r)(r)A dr

r2


Fg = Fp ) dp

dr= GM(r)(r)

r2

M(r)

A

• include general-relativistic corrections:

‘Tolman-Oppenheimer-Volkov’ equation

dp

dr= GM(r)"(r)

r2

1 +

p(r)

"(r)c2

1 +

4r3p(r)

M(r)c2

1 2GM(r)

c2r



rgravity:

Fp = A (pout

pin


Fg = GM(r)(r)A dr

r2


Fg = Fp ) dp

dr= GM(r)(r)

r2

M(r)

A

• include general-relativistic corrections:

‘Tolman-Oppenheimer-Volkov’ equation

common feature of models that include the appearance of ‘exotic’hadronic matter such as hyperons4,5 or kaon condensates3 at densitiesof a few times the nuclear saturation density (ns), for example modelsGS1 and GM3 in Fig. 3. Almost all such EOSs are ruled out by ourresults. Our mass measurement does not rule out condensed quarkmatter as a component of the neutron star interior6,21, but it stronglyconstrains quark matter model parameters12. For the range of allowedEOS lines presented in Fig. 3, typical values for the physical parametersof J1614-2230 are a central baryondensity of between 2ns and 5ns and aradius of between 11 and 15 km, which is only 2–3 times theSchwarzschild radius for a 1.97M[ star. It has been proposed thatthe Tolman VII EOS-independent analytic solution of Einstein’sequations marks an upper limit on the ultimate density of observablecold matter22. If this argument is correct, it follows that our mass mea-surement sets an upper limit on this maximum density of(3.746 0.15)3 1015 g cm23, or ,10ns.Evolutionary models resulting in companion masses.0.4M[ gen-

erally predict that the neutron star accretes only a few hundredths of asolar mass of material, and result in a mildly recycled pulsar23, that isone with a spin period.8ms. A few models resulting in orbital para-meters similar to those of J1614-223023,24 predict that the neutron starcould accrete up to 0.2M[, which is still significantly less than the>0.6M[ needed to bring a neutron star formed at 1.4M[ up to theobserved mass of J1614-2230. A possible explanation is that someneutron stars are formed massive (,1.9M[). Alternatively, the trans-fer of mass from the companion may be more efficient than currentmodels predict. This suggests that systems with shorter initial orbitalperiods and lower companion masses—those that produce the vastmajority of the fully recycled millisecond pulsar population23—mayexperience even greater amounts of mass transfer. In either case, ourmass measurement for J1614-2230 suggests that many other milli-second pulsars may also have masses much greater than 1.4M[.

Received 7 July; accepted 1 September 2010.

1. Lattimer, J. M. & Prakash, M. The physics of neutron stars. Science 304, 536–542(2004).

2. Lattimer, J. M. & Prakash, M. Neutron star observations: prognosis for equation ofstate constraints. Phys. Rep. 442, 109–165 (2007).

3. Glendenning, N. K. & Schaffner-Bielich, J. Kaon condensation and dynamicalnucleons in neutron stars. Phys. Rev. Lett. 81, 4564–4567 (1998).

4. Lackey, B. D., Nayyar, M. & Owen, B. J. Observational constraints on hyperons inneutron stars. Phys. Rev. D 73, 024021 (2006).

5. Schulze, H., Polls, A., Ramos, A. & Vidana, I. Maximummass of neutron stars. Phys.Rev. C 73, 058801 (2006).

6. Kurkela, A., Romatschke, P. & Vuorinen, A. Cold quark matter. Phys. Rev. D 81,105021 (2010).

7. Shapiro, I. I. Fourth test of general relativity. Phys. Rev. Lett. 13, 789–791 (1964).8. Jacoby, B.A., Hotan, A., Bailes,M., Ord, S. &Kulkarni, S.R. Themassof amillisecond

pulsar. Astrophys. J. 629, L113–L116 (2005).9. Verbiest, J. P. W. et al. Precision timing of PSR J0437–4715: an accurate pulsar

distance, a high pulsarmass, and a limit on the variation of Newton’s gravitationalconstant. Astrophys. J. 679, 675–680 (2008).

10. Hessels, J.et al. inBinaryRadio Pulsars (edsRasio, F. A.&Stairs, I. H.)395 (ASPConf.Ser. 328, Astronomical Society of the Pacific, 2005).

11. Crawford, F. et al. A survey of 56midlatitude EGRET error boxes for radio pulsars.Astrophys. J. 652, 1499–1507 (2006).

12. Ozel, F., Psaltis, D., Ransom, S., Demorest, P. & Alford, M. The massive pulsar PSRJ161422230: linking quantum chromodynamics, gamma-ray bursts, andgravitational wave astronomy. Astrophys. J. (in the press).

13. Hobbs, G. B., Edwards, R. T. & Manchester, R. N. TEMPO2, a new pulsar-timingpackage - I. An overview.Mon. Not. R. Astron. Soc. 369, 655–672 (2006).

14. Damour, T. & Deruelle, N. General relativistic celestial mechanics of binarysystems. II. The post-Newtonian timing formula. Ann. Inst. Henri Poincare Phys.Theor. 44, 263–292 (1986).

15. Freire, P.C.C.&Wex,N.Theorthometricparameterisationof theShapirodelay andan improved test of general relativity with binary pulsars.Mon. Not. R. Astron. Soc.(in the press).

16. Iben, I. Jr & Tutukov, A. V. On the evolution of close binaries with components ofinitial mass between 3 solar masses and 12 solar masses. Astrophys. J Suppl. Ser.58, 661–710 (1985).

17. Ozel, F. Soft equations of state for neutron-star matter ruled out by EXO 0748 -676. Nature 441, 1115–1117 (2006).

18. Ransom, S. M. et al. Twenty-one millisecond pulsars in Terzan 5 using the GreenBank Telescope. Science 307, 892–896 (2005).

19. Freire, P. C. C. et al. Eight new millisecond pulsars in NGC 6440 and NGC 6441.Astrophys. J. 675, 670–682 (2008).

20. Freire, P. C. C.,Wolszczan, A., vandenBerg,M.&Hessels, J.W. T. Amassiveneutronstar in the globular cluster M5. Astrophys. J. 679, 1433–1442 (2008).

21. Alford,M.etal.Astrophysics:quarkmatterincompactstars?Nature445,E7–E8(2007).22. Lattimer, J. M. & Prakash, M. Ultimate energy density of observable cold baryonic

matter. Phys. Rev. Lett. 94, 111101 (2005).23. Podsiadlowski, P., Rappaport, S. & Pfahl, E. D. Evolutionary sequences for low- and

intermediate-mass X-ray binaries. Astrophys. J. 565, 1107–1133 (2002).24. Podsiadlowski, P. & Rappaport, S. CygnusX-2: the descendant of an intermediate-

mass X-Ray binary. Astrophys. J. 529, 946–951 (2000).25. Hotan, A.W., van Straten,W. &Manchester, R. N. PSRCHIVE andPSRFITS: an open

approach to radio pulsar data storage and analysis. Publ. Astron. Soc. Aust. 21,302–309 (2004).

26. Cordes, J.M. & Lazio, T. J.W.NE2001.I. A newmodel for theGalactic distribution offree electrons and its fluctuations. Preprint at Æhttp://arxiv.org/abs/astro-ph/0207156æ (2002).

27. Lattimer, J. M. & Prakash, M. Neutron star structure and the equation of state.Astrophys. J. 550, 426–442 (2001).

28. Champion, D. J. et al. An eccentric binary millisecond pulsar in the Galactic plane.Science 320, 1309–1312 (2008).

29. Berti, E., White, F., Maniopoulou, A. & Bruni, M. Rotating neutron stars: an invariantcomparison of approximate and numerical space-time models.Mon. Not. R.Astron. Soc. 358, 923–938 (2005).

Supplementary Information is linked to the online version of the paper atwww.nature.com/nature.

Acknowledgements P.B.D. is a Jansky Fellow of the National Radio AstronomyObservatory. J.W.T.H. is a Veni Fellow of The Netherlands Organisation for ScientificResearch.We thankJ. Lattimer for providing theEOSdataplotted inFig. 3, andP. Freire,F. Ozel and D. Psaltis for discussions. The National Radio Astronomy Observatory is afacility of the USNational Science Foundation, operated under cooperative agreementby Associated Universities, Inc.

Author Contributions All authors contributed to collecting data, discussed the resultsand edited themanuscript. In addition, P.B.D. developed theMCMCcode, reduced andanalysed data, and wrote the manuscript. T.P. wrote the observing proposal andcreated Fig. 3. J.W.T.H. originally discovered thepulsar.M.S.E.R. initiated the survey thatfound the pulsar. S.M.R. initiated the high-precision timing proposal.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Readers are welcome to comment on the online version of this article atwww.nature.com/nature. Correspondence and requests for materials should beaddressed to P.B.D. ([email protected]).

0.07 8 9 10 11

Radius (km)12 13 14 15

0.5

1.0

1.5

2.0

AP4

J1903+0327

J1909-3744

systemsDouble neutron sDouble neutron star sysy

J1614-2230

AP3

ENG

MPA1

GM3

GS1

PAL6

FSUSQM3

SQM1

PAL1

MS0

MS2

MS1

2.5 GR

Causality

Rotation

P < ∞

Mas

s (M

()

Figure 3 | Neutron star mass–radius diagram. The plot shows non-rotatingmass versus physical radius for several typical EOSs27: blue, nucleons; pink,nucleons plus exoticmatter; green, strange quarkmatter. The horizontal bandsshow the observational constraint from our J1614-2230 mass measurement of(1.976 0.04)M[, similar measurements for two other millisecond pulsars8,28

and the range of observed masses for double neutron star binaries2. Any EOSline that does not intersect the J1614-2230 band is ruled out by thismeasurement. In particular, most EOS curves involving exotic matter, such askaon condensates or hyperons, tend to predict maximum masses well below2.0M[ and are therefore ruled out. Including the effect of neutron star rotationincreases themaximum possiblemass for each EOS. For a 3.15-ms spin period,this is a=2% correction29 and does not significantly alter our conclusions. Thegrey regions show parameter space that is ruled out by other theoretical orobservational constraints2. GR, general relativity; P, spin period.

LETTER RESEARCH

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 3


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

Neutron star radius constraints

incorporation of beta-equilibrium: neutron matter neutron star matter

parametrize our ignorance via piecewise high-density extensions of EOS:

• use polytropic ansatz (results insensitive to particular form)

• range of parameters

p

13.0 13.5 14.0log 10 [g / cm3]

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

Constraints on the nuclear equation of state

use the constraints:

vs() =

dP/d < 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)

• 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 15.4log 10 [g / cm3]

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

1.4 M

8 10 12 14 16Radius [km]

0

0.5

1

1.5

2

2.5

3

Mas

s [M

sun]

8 10 12 14 16Radius [km]

0

0.5

1

1.5

2

2.5

3

Mas

s [M

sun] causality

9.7 13.9 km

Mmax

= 1.97M

Mmax

= 2.4M

• 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 15.4log 10 [g / cm3]

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

1.4 M

8 10 12 14 16Radius [km]

0

0.5

1

1.5

2

2.5

3

Mas

s [M

sun]

8 10 12 14 16Radius [km]

0

0.5

1

1.5

2

2.5

3

Mas

s [M

sun] causality

• proposed missions (LOFT,NICER...) could significantly improve constraints9.7 13.9 km

3

0 1 2 3 4Ecm (MeV)

0

20

40

60

80

100

120

140

(deg.)

32

12

(b) NLO

N2LO (D2, E )

N2LO (D2, EP)

Rmatrix

FIG. 1. (a) Couplings cE vs cD obtained by fitting the 4He binding energy for di↵erent 3N-operator forms. Triangles areobtained by using VD1 and VE , while the other symbols are obtained for VD2 and three di↵erent VE-operator structures. Theblue and green lines (lower and upper) correspond to R0 = 1.0 fm, while the red lines (central) correspond to R0 = 1.2 fm. TheGFMC statistical errors are smaller than the symbols. The stars correspond to the values of cD and cE which simultaneouslyfit the n-↵ P -wave phase shifts (see Table I and the right panel). No fit to both observables can be obtained for the case withR0 = 1.2 fm and VD1. (b) P -wave n-↵ elastic scattering phase shifts compared with an R-matrix analysis of experimental data.Colors and symbols correspond to the left panel. We also include phase shifts calculated at NLO which clearly indicate thenecessity of 3N interactions to fit the P -wave splitting.

TABLE I. Fit values for the couplings cD and cE for di↵erentchoices of 3N forces and cuto↵s.

V3N R0 (fm) cE cD

N2LO (D1, E)1.0 0.63 0.0

1.2

N2LO (D2, E)1.0 0.63 0.0

1.2 0.09 3.5

N2LO (D2, E ) 1.0 0.62 0.5

N2LO (D2, EP) 1.0 0.59 0.0

results in n-↵ P -wave scattering show a substantial sen-sitivity: VD1

appears to have a smaller e↵ect than VD2

.

In Fig. 2, we show ground-state energies and point pro-ton radii for A = 3, 4 nuclei at NLO and N2LO using VD2

and VE for R0

= 1.0 fm and R0

= 1.2 fm, in compar-ison with experiment. The ground-state energies of theA = 3 systems compare well with experimental values.The ground-state energy of 4He is used in fitting cD andcE , and so it is forced to match the experimental value towithin 0.03 MeV. The point proton radii also comparewell with values extracted from experiment. The theo-retical uncertainty at each order is estimated through theexpected size of higher-order contributions; see Ref. [32]for details. We include results from LO, NLO, and N2LO

in the analysis using the Fermi momentum and the pionmass as the small scales for neutron matter (discussedbelow) and nuclei, respectively. The error bars presentedhere are comparable to those shown in Ref. [33], althoughit is worth emphasizing that our calculations represent acomplete estimate of the uncertainty at N2LO since weinclude 3N interactions. Other choices for 3N structuresgive similar results.

It is noteworthy that NN and 3N interactions derivedfrom chiral EFT up to N2LO have sucient freedom suchthat n-↵ scattering phase shifts in Fig. 1(b) and proper-ties of light nuclei in Fig. 2 can be simultaneously de-scribed. The failures of the Urbana IX model in under-binding nuclei and underpredicting the spin-orbit split-ting in neutron-rich systems, including the n-↵, systemwere among the factors motivating the addition of thethree-pion exchange diagrams in the Illinois 3N mod-els [7]. Our results show that chiral 3N forces at N2LO,including the shorter-range parts in the pion exchanges,allow the simultaneous fit. These interactions should betested further in light p-shell nuclei.

Finally, we study the full chiral N2LO forces, includ-ing all 3N contributions, in neutron matter to extend theresults from Ref. [24]. More specifically, we examine thee↵ects of di↵erent VD and VE structures on the equationof state of neutron matter. Although these terms vanish

Gezerlis et al.,PRL 111, 032501 (2013)

3

0 50 100 150 200 250

Lab. Energy [MeV]

0

10

20

30

40

50

60

70

Phas

e Sh

ift [d

eg]

LONLON2LOPWA

0 50 100 150 200 250

Lab. Energy [MeV]

0

10

20

30

40

50

Phas

e Sh

ift [d

eg]

LONLON2LOPWA

0 50 100 150 200 250

Lab. Energy [MeV]

-25

-20

-15

-10

-5

0

LONLON2LOPWA

0 50 100 150 200 250

Lab. Energy [MeV]

0

10

20

30LONLON2LOPWA

1S0

3P0

3P1

3P2

FIG. 1. (Color online) Neutron-proton phase shifts as a function of laboratory energy Elab = 2p2/m in the 1S0, 3P0, 3P1, and3P2 partial waves (from left to right) in comparison to the Nijmegen partial-wave analysis (PWA) [43]. The LO, NLO, and

N2LO bands are obtained by varying R0 between 0.8 − 1.2 fm (with a spectral-function cutoff Λ = 800MeV).

and provide a measure of the theoretical uncertainty. Forthe R0 = 1.2 fm N2LO NN potential, we list the low-energy couplings at LO, NLO, and N2LO in Table I. AtN2LO, an isospin-symmetry-breaking contact interaction(Cnn for neutrons) is added in the spin S = 0 channel (toCS−3CT ), which is fit to a scattering length of −18.8 fm.As shown in Fig. 1, the comparison with NN phase shiftsis very good for Elab ! 150MeV. This is similar forhigher partial waves and isospin T = 0 channels, whichwill be reported in a later paper that will also study im-proved fits. In cases where there are deviations for higherenergies (such as in the 3P2 channel of Fig. 1), the widthof the band signals significant theoretical uncertaintiesdue to the chiral EFT truncation at N2LO. The NLOand N2LO bands nicely overlap (as shown for the casesin Fig. 1), or are very close, but it is also apparent thatthe N2LO bands are of a similar size as at NLO. This isbecause the width of the bands at both NLO and N2LOshows effects of the neglected order-Q4 contact interac-tions.

Finally, we emphasize that the newly introduced localchiral EFT potentials include the same physics as themomentum-space versions. This is especially clear whenantisymmetrizing. Besides the new idea of removing thek2 terms, there are no conceptual differences between thetwo ways of regularizing (see also the early work [44]).

We then apply the developed local LO, NLO, andN2LO chiral EFT interactions in systematic QMC cal-culations for the first time. Since nuclear forces con-tain quadratic spin, isospin, and tensor operators (of theform σ

αi Aαβ

ij σβj ), the many-body wave function cannot

be expressed as a product of single-particle spin-isospinstates. All possible spin-isospin nucleon-pair states needto be explicitly accounted for, leading to an exponentialincrease in the number of possible states. As a result,Green’s Function Monte Carlo (GFMC) calculations arepresently limited to 12 nucleons and 16 neutrons [30]. Inthis Letter, we would like to simulate O(100) neutrons

0 0.05 0.1 0.15

n [fm-3]

0

5

10

15

20

E/N

[MeV

]

AFDMC LOAFDMC NLOAFDMC N2LO

R 0=0

.8 fm

R 0=1

.2 fm

FIG. 2. (Color online) Neutron matter energy per particleE/N as a function of density n calculated using AFDMCwith chiral EFT NN interactions at LO, NLO, and N2LO.The statistical errors are smaller than the points shown. Thelines give the range of the energy band obtained by varyingR0 between 0.8 − 1.2 fm (as for the phase shifts in Fig. 1),which provides an estimate of the theoretical uncertainty ateach order. The N2LO band is comparable to the one at NLOdue to the large ci couplings in the N2LO two-pion exchange.

to access the thermodynamic limit. We therefore turnto the auxiliary-field diffusion Monte Carlo (AFDMC)method [45], which is capable of efficiently handling spin-dependent Hamiltonians.Schematically, AFDMC rewrites the Green’s function

by applying a Hubbard-Stratonovich transformation us-ing auxiliary fields to change the quadratic spin-isospinoperator dependences to linear. As a result, when appliedto a wave function that is a product of single-particle

4

in the limit of infinite cuto↵, they contribute for finitecuto↵s. In Fig. 3 we show results for the neutron mat-ter energy per particle as a function of the density calcu-lated with the AFDMC method described in Refs. [3, 34].We show the energies for R

0

= 1.0 fm for the NN andfull 3N interactions. We use VD2

and the three di↵erentVE structures: VE (blue band), VE (red band), andVEP (green band). The error bands are determined as inthe light nuclei case. The VEP interaction fits A = 4, 5with a vanishing cD; hence, this choice of VE leads toan equation of state identical to the equation of statewith NN+ VC as in Ref. [24] (the projector P is zero forpure neutron systems), and qualitatively similar to pre-vious results using chiral interactions at N2LO [35] andnext-to-next-to-next-to-leading order [36].

As discussed, the contributions of VD and VE are onlyregulator e↵ects for neutrons. However, they are sizableand result in a larger error band. At saturation den-sity n

0

0.16 fm3, the di↵erence of the central valueof the energy per neutron after inclusion of the 3N con-tacts VE or VE is 2 MeV, leading to a total errorband with a range of 6.5 MeV when considering di↵er-ent VE structures. This relatively large uncertainty canbe qualitatively explained when considering the followinge↵ects. Because the expectation value h

Pi<j i · ji has

a sign opposite to that of the expectation value h i in4He, cE will also have opposite signs in the two cases to

9

8

7

6

5

E(M

eV)

3H 3He 4He31

25

19

13

3H 3He 4He0.8

1.0

1.2

1.4

1.6

1.8

2.0

hr

2pt i(fm

)

NLO

N2LO (D2, E )Exp.

FIG. 2. Ground-state energies and point proton radii for A =3, 4 nuclei calculated at NLO and N2LO (with VD2 and VE )compared with experiment. Blue (red) symbols correspondto R0 = 1.0 fm (R0 = 1.2 fm). The errors are obtained asdescribed in the text and also include the GFMC statisticaluncertainties.

FIG. 3. The energy per particle in neutron matter as afunction of density for the NN and full 3N interactions atN2LO with R0 = 1.0 fm. We use VD2 and di↵erent 3N contactstructures: The blue band corresponds to VE , the red bandto VE , and the green band to VEP . The green band coincideswith the NN+ 2-exchange-only result because both VD andVE vanish in this case. The bands are calculated as describedin the text.

fit the binding energy. However, in neutron matter bothoperators are the same, spreading the uncertainty band.A similar argument was made in Ref. [37].

With the regulators used here, the Fierz-rearrangement invariance valid at infinite cuto↵ isonly approximate at finite cuto↵, and hence the di↵erentchoices of VD and VE can lead to di↵erent results.The di↵erent local structures can lead to finite relativeP -wave contributions. These can be eliminated bychoosing VEP , which has a projection onto even-paritywaves (predominantly S waves). The usual nonlocalregulator in momentum space does not couple S and Pwaves.

In conclusion, we find for the first time that chiral in-teractions can simultaneously fit light nuclei and low-energy P -wave n-↵ scattering and provide reasonable es-timates for the neutron matter equation of state. Othercommonly used phenomenological 3N models do not pro-vide this capability. These chiral forces should be testedin light p-shell nuclei, medium-mass nuclei, and isospin-symmetric nuclear matter to gauge their ability to de-scribe global properties of nuclear systems.

We also find that the ambiguities associated withcontact-operator choices can be significant when mov-ing from light nuclei to neutron matter and possibly tomedium-mass nuclei, where the T = 3

2

triples play a

Recent and current developments of novel nuclear interactions

Lynn et al.,PRL 116, 062501 (2016)

1. local EFT interactions, suitable for Quantum Monte Carlo calculations status: NN plus 3N up to N2LO, calculations of few-body systems and neutron matter

Gezerlis et al.,PRC 90, 054323 (2014)

3

0 1 2 3 4Ecm (MeV)

0

20

40

60

80

100

120

140

(deg.)

32

12

(b) NLO

N2LO (D2, E )

N2LO (D2, EP)

Rmatrix



V3N R0 (fm) cE cD

N2LO (D1, E)1.0 0.63 0.0

1.2

N2LO (D2, E)1.0 0.63 0.0

1.2 0.09 3.5

N2LO (D2, E ) 1.0 0.62 0.5

N2LO (D2, EP) 1.0 0.59 0.0



.


and VE for R0

= 1.0 fm and R0






3

0 50 100 150 200 250

Lab. Energy [MeV]

0

10

20

30

40

50

60

70

Phas

e Sh

ift [d

eg]

LONLON2LOPWA

0 50 100 150 200 250

Lab. Energy [MeV]

0

10

20

30

40

50

Phas

e Sh

ift [d

eg]

LONLON2LOPWA

0 50 100 150 200 250

Lab. Energy [MeV]

-25

-20

-15

-10

-5

0

LONLON2LOPWA

0 50 100 150 200 250

Lab. Energy [MeV]

0

10

20

30LONLON2LOPWA

1S0

3P0

3P1

3P2






αi Aαβ



0 0.05 0.1 0.15

n [fm-3]

0

5

10

15

20

E/N

[MeV

]


R 0=0

.8 fm

R 0=1

.2 fm




4


0




0


Pi<j i · ji has


9

8

7

6

5

E(M

eV)

3H 3He 4He31

25

19

13

3H 3He 4He0.8

1.0

1.2

1.4

1.6

1.8

2.0

hr

2pt i(fm

)

NLO

N2LO (D2, E )Exp.







2

triples play a


Lynn et al.,PRL 116, 062501 (2016)



0 0.05 0.1 0.15

n [fm-3]

0

5

10

15

E/N

[MeV

]

QMC (2010)AFDMC N2LO0.8 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 of calculations within many-bodyperturbation theory

Gezerlis et. al,PRL 111, 032501 (2013)

first Quantum Monte Carlo of neutron matter based on chiral EFT interactions

3

0 1 2 3 4Ecm (MeV)

0

20

40

60

80

100

120

140

(deg.)

32

12

(b) NLO

N2LO (D2, E )

N2LO (D2, EP)

Rmatrix



V3N R0 (fm) cE cD

N2LO (D1, E)1.0 0.63 0.0

1.2

N2LO (D2, E)1.0 0.63 0.0

1.2 0.09 3.5

N2LO (D2, E ) 1.0 0.62 0.5

N2LO (D2, EP) 1.0 0.59 0.0



.


and VE for R0

= 1.0 fm and R0






3

0 50 100 150 200 250

Lab. Energy [MeV]

0

10

20

30

40

50

60

70

Phas

e Sh

ift [d

eg]

LONLON2LOPWA

0 50 100 150 200 250

Lab. Energy [MeV]

0

10

20

30

40

50

Phas

e Sh

ift [d

eg]

LONLON2LOPWA

0 50 100 150 200 250

Lab. Energy [MeV]

-25

-20

-15

-10

-5

0

LONLON2LOPWA

0 50 100 150 200 250

Lab. Energy [MeV]

0

10

20

30LONLON2LOPWA

1S0

3P0

3P1

3P2






αi Aαβ



0 0.05 0.1 0.15

n [fm-3]

0

5

10

15

20

E/N

[MeV

]


R 0=0

.8 fm

R 0=1

.2 fm




4


0




0


Pi<j i · ji has


9

8

7

6

5

E(M

eV)

3H 3He 4He31

25

19

13

3H 3He 4He0.8

1.0

1.2

1.4

1.6

1.8

2.0

hr

2pt i(fm

)

NLO

N2LO (D2, E )Exp.







2

triples play a


Lynn et al.,PRL 116, 062501 (2016)


good description of all A ≤ 4 data. Some of the πN LECsdisplay large variations, but the χ2=Ndof (without modelerror) for the πN data is within 2.28(4) for all of thesepotentials. The subleading πN LECs become more positivewhen NN scattering data at higher energies are included,and c1, in particular, carries a larger (relative) statisticaluncertainty than the others. It is noteworthy that for a givenTmaxLab , and up to 1σ precision, the πN LECs exhibit Λ

independence. The NNN LECs, cD and cE, tend to dependless on Tmax

Lab at larger values of Λ. However, they alwaysremain natural. It is also interesting to note that the tensorcontact CE1

is insensitive to Λ variations but stronglydependent on the Tmax

Lab cut. It was shown in Fig. 6 that CE1

and c4 correlate strongly. This effect can already beexpected from the structure of the underlying expressionfor the NNLO interaction.To gauge the magnitude of model variations in heavier

nuclei, we computed the binding energies of 4He and 16Oby using the previously mentioned family of 42 NNLOpotentials. The resulting binding energies for 4He and 16O,computed in the NCSM and CC, respectively, are shown inFig. 11. The NCSM calculations were carried out in a HOmodel space with Nmax ¼ 20 and ℏω ¼ 36 MeV. The CCcalculations were carried out in the so-called Λ−CCSD(T)approximation [7] in 15 major oscillator shells withℏω ¼ 22 MeV. The largest energy difference when goingfrom 13 to 15 oscillator shells was 3.6 MeV (observedfor Λ ¼ 600 MeV). From the observed convergence of thecorrelation energy we estimate the uncertainty of excludedhigher rank excitation clusters to "5 MeV. For ourpurposes, this provides well-enough converged results.The NNN force was truncated at the normal-orderedtwo-body level in the Hartree-Fock basis.

The Eð4HeÞ predictions vary within about a 2-MeVrange. For Eð16OÞ, this variation increases dramatically toabout 35 MeV. Irrespective of the discrepancy with themeasured value, the spread of the central values indicatesthe presence of a surprisingly large systematic error whenextrapolating to heavier systems.The statistical uncertainties remain small: tens of keV for

4He and a few hundred keV for 16O. These uncertainties areobtained from the quadratic approximation with the com-puted Jacobian and Hessian for 4He, while a brute-forceMonte Carlo simulation with 2.5 × 104 CC calculationswas performed for 16O. This massive set of CC calculationsemployed the singles and doubles approximation (CCSD)in nine major oscillator shells. We conclude that thestatistical uncertainties of the predictions for Eð4HeÞ andEð16OÞ at NNLO are much smaller than the variations dueto changing Λ or Tmax

Lab . However, this is only true forsimultaneously optimized potentials. For the separatelyoptimized NNLO potential (NNLOsep), the statisticaluncertainty of the Eð4HeÞ prediction is five times largerthan the observed variations due to changing Λ and Tmax

Lab .

V. OUTLOOK

The extended analysis of systematic uncertainties pre-sented above suggests that large fluctuations are induced inheavier nuclei (see Fig. 11). Furthermore, while predictionsfor 4He are accurate over a rather wide range of regulatorparameters, the binding energy for 16O turns out to beunderestimated for the entire range used in this study. Infact, there is no overlap between the theoretical predictionsand the experimental results, even though the former oneshave large error bars.Based on our findings, we recommend that continued

efforts towards an ab initio framework based on χEFTshould involve additional work in, at least, three differentdirections:(1) Explore the alternative strategy of informing the

model about low-energy many-body observables.(2) Diversify and extend the statistical analysis and

perform a sensitivity analysis of input data.(3) Continue efforts towards higher orders of the chiral

expansion, and possibly revisit the power counting.Let us comment briefly on these research directions. Thepoor many-body scaling observed in Fig. 11 was prag-matically accounted for in the construction of the so-calledNNLOsat potential presented in Ref. [35], where heaviernuclei were also included in the fit. The accuracy of many-body predictions was shown to be much improved, but theuncertainty analysis is much more difficult within such astrategy.Second, to get a handle on possible bias in the statistical

analysis due to the choice of statistical technique, it isimportant to apply different types of optimization anduncertainty quantification methods. Various choices exist,

FIG. 11. Binding-energy predictions for (a) 4He and (b) 16Owith the different reoptimizations of NNLOsim. On the x axisis the employed cutoff Λ. Vertically aligned red markerscorrespond to different Tmax

Lab for the NN scattering data usedin the optimization. The experimental binding energies areEð4HeÞ ≈ −28.30 MeV, represented by a gray band in panel(a), and Eð16OÞ≈−127.6MeV [98]. Statistical error bars on thetheoretical results are smaller than the marker size on thisenergy scale.

UNCERTAINTY ANALYSIS AND ORDER-BY-ORDER … PHYS. REV. X 6, 011019 (2016)

011019-19

Carlsson et al.,PRX 6, 011019 (2016)

potentials, there were no signs of convergence in thedescription of, e.g., np scattering data.If the experimental database of πN scattering cross

sectionswas complete, then itwould be possible to separatelyconstrain,with zerovariances, the correspondingLECs.Onlythis scenario would render it unnecessary to include the πNscattering data in the simultaneous objective function.Implicitly, this scenario also assumes a perfect theory, i.e.,that the employed χEFT can account for the dynamics ofpionic interactions. Of course, reality lies somewhere inbetween, and a simultaneous optimization approach ispreferable in the present situation. There exists ongoingefforts where the πN sector of χEFT is extrapolated and fittedseparately in the unphysical kinematical region, where itexhibits a stronger curvature with respect to the data [96].Overall, the importance of applying simultaneous

optimization is most prominent at higher chiral orderssince the subleading πN LECs enter first at NNLO. Infact, the separately optimized NNLOsep potential containsa large systematic uncertainty by construction. We findthat the scaling factor for the NN scattering model error,CNN , decreases from 1.6 to 1.0 mb1=2 when going fromNNLOsep to the simultaneously optimized NNLOsim.This implies that the separate, or sequential, optimizationprotocol introduces additional artificial systematic errorsnot due to the chiral expansion but due to incorrectly fittedLECs. This scenario is avoided in a simultaneous opti-mization. The scaling factor for the πN scattering modelerror, CπN, remains at 3.6 mb1=2 for both NNLOsep andNNLOsim.

The size of the model error is determined such that theoverall scattering χ2=Ndof is unity, which means that itdepends on the observables entering the optimization. Wecan explore the stability of our approach by reoptimizingNNLOsim with respect to different truncations of theinput NN scattering data. To this end, we adjust the allowedTmaxlab between 125 and 290 MeV in six steps. It turns out

that our procedure for extracting the model error is verystable. The resulting normalization constants CNN varybetween 1.0 mb1=2 and 1.3 mb1=2 as shown in Fig. 10(a).

FIG. 9. Comparison between selected NN and πN experimental data sets and theoretical calculations for chiral interactions at LO,NLO, and NNLO. The bands indicate the total errors (statistical plus model errors). (a) np total cross section for the sequentiallyoptimized interactions with no clear signature of convergence with increasing chiral order. All other results are for the simultaneouslyoptimized interactions: LOsim, NLOsim, and NNLOsim. (b) np total cross section; (c) np differential cross section; (d) πN charge-exchange, differential cross section; (e) πN elastic, differential cross section.

FIG. 10. Predictions for the different reoptimizations of NNLO-sim. On the x axis is the maximum T lab for the NN scatteringdata used in the optimization. (a) Model error amplitude (20)reoptimized so that χ2=Ndof ¼ 1 for the respective data subset.(b) Model prediction for the np total cross section at T lab ¼300 MeV with error bars representing statistical and modelerrors for the different reoptimizations.


011019-17

potentials, there were no signs of convergence in thedescription of, e.g., np scattering data.If the experimental database of πN scattering cross

sectionswas complete, then itwould be possible to separatelyconstrain,with zerovariances, the correspondingLECs.Onlythis scenario would render it unnecessary to include the πNscattering data in the simultaneous objective function.Implicitly, this scenario also assumes a perfect theory, i.e.,that the employed χEFT can account for the dynamics ofpionic interactions. Of course, reality lies somewhere inbetween, and a simultaneous optimization approach ispreferable in the present situation. There exists ongoingefforts where the πN sector of χEFT is extrapolated and fittedseparately in the unphysical kinematical region, where itexhibits a stronger curvature with respect to the data [96].Overall, the importance of applying simultaneous

optimization is most prominent at higher chiral orderssince the subleading πN LECs enter first at NNLO. Infact, the separately optimized NNLOsep potential containsa large systematic uncertainty by construction. We findthat the scaling factor for the NN scattering model error,CNN , decreases from 1.6 to 1.0 mb1=2 when going fromNNLOsep to the simultaneously optimized NNLOsim.This implies that the separate, or sequential, optimizationprotocol introduces additional artificial systematic errorsnot due to the chiral expansion but due to incorrectly fittedLECs. This scenario is avoided in a simultaneous opti-mization. The scaling factor for the πN scattering modelerror, CπN, remains at 3.6 mb1=2 for both NNLOsep andNNLOsim.

The size of the model error is determined such that theoverall scattering χ2=Ndof is unity, which means that itdepends on the observables entering the optimization. Wecan explore the stability of our approach by reoptimizingNNLOsim with respect to different truncations of theinput NN scattering data. To this end, we adjust the allowedTmaxlab between 125 and 290 MeV in six steps. It turns out

that our procedure for extracting the model error is verystable. The resulting normalization constants CNN varybetween 1.0 mb1=2 and 1.3 mb1=2 as shown in Fig. 10(a).

FIG. 9. Comparison between selected NN and πN experimental data sets and theoretical calculations for chiral interactions at LO,NLO, and NNLO. The bands indicate the total errors (statistical plus model errors). (a) np total cross section for the sequentiallyoptimized interactions with no clear signature of convergence with increasing chiral order. All other results are for the simultaneouslyoptimized interactions: LOsim, NLOsim, and NNLOsim. (b) np total cross section; (c) np differential cross section; (d) πN charge-exchange, differential cross section; (e) πN elastic, differential cross section.

FIG. 10. Predictions for the different reoptimizations of NNLO-sim. On the x axis is the maximum T lab for the NN scatteringdata used in the optimization. (a) Model error amplitude (20)reoptimized so that χ2=Ndof ¼ 1 for the respective data subset.(b) Model prediction for the np total cross section at T lab ¼300 MeV with error bars representing statistical and modelerrors for the different reoptimizations.


011019-17

2. simultaneous fit of NN and 3N forces to two- and few-body observables status: NN plus 3N up to N2LO, N3LO currently in development


Recent and current developments of novel nuclear interactions3. fits of NN plus 3N forces to two-, few- and many-body observables status: NN plus 3N up to N2LO, NN phase shifts only fitted up to Tlab~35 MeV

RAPID COMMUNICATIONS

A. EKSTROM et al. PHYSICAL REVIEW C 91, 051301(R) (2015)

FIG. 1. (Color online) Ground-state energy (negative of bindingenergy) per nucleon (top), and residuals (differences between com-puted and experimental values) of charge radii (bottom) for selectednuclei computed with chiral interactions. In most cases, theorypredicts too-small radii and too-large binding energies. References:a [40,41], b [24], c [23], d [22], e [42], f [43], g [44], h [45], i [46].The red diamonds are NNLOsat results obtained in this work.

to low-energy observables (as opposed to the traditionaladjustment of two-nucleon forces to NN scattering data athigher energies). Third, the impact of many-body effectsentering at higher orders (e.g., higher-rank forces) might bereduced if heavier systems, in which those effects are stronger,are included in the optimization.

Besides these theoretical arguments, there is also onepractical reason for a paradigm shift: predictive power andlarge extrapolations do not go together. In traditional ap-proaches, where interactions are optimized for A = 2,3,4,small uncertainties in few-body systems (e.g., by forcing arather precise reproduction of the A = 2,3,4 sectors at arather low order in the chiral power counting) get magnifiedtremendously in heavy nuclei; see, for example, Ref. [24].Consequently, when aiming at reliable predictions for heavynuclei, it is advisable to use a model that performs well forlight- and medium-mass systems. In our approach, light nucleiare reached by interpolation while medium-mass nuclei by amodest extrapolation. In this context, it is worth noting that themost accurate calculations for light nuclei with A ! 12 [59]employ NNN forces adjusted to 17 states in nuclei withA ! 8 [60]. Finally, we point out that nuclear saturation canbe viewed as an emergent phenomenon. Indeed, little in thechiral EFT of nuclear forces suggest that nuclei are self-boundsystems with a central density (or Fermi momentum) that ispractically independent of mass number. This viewpoint makesit prudent to include the emergent momentum scale into theoptimization, which is done in our case by the inclusion ofcharge radii for 3H, 3,4He, 14C, and 16O. This is similar in spiritto nuclear mean-field calculations [61] and nuclear densityfunctional theory [62,63] where masses and radii provide keyconstraints on the parameters of the employed models.

Optimization protocol and model details. We seek tominimize an objective function to determine the optimal setof coupling constants of the chiral NN + NNN interactionat NNLO. Our dataset of fit-observables includes the bindingenergies and charge radii of 3H, 3,4He, 14C, and 16O, as well

TABLE I. Binding energies (in MeV) and charge radii (in fm)for 3H, 3,4He, 14C, and 16,22,23,24,25O employed in the optimization ofNNLOsat.

Eg.s. Expt. [69] rch Expt. [65,66]

3H 8.52 8.482 1.78 1.7591(363)3He 7.76 7.718 1.99 1.9661(30)4He 28.43 28.296 1.70 1.6755(28)14C 103.6 105.285 2.48 2.5025(87)16O 124.4 127.619 2.71 2.6991(52)22O 160.8 162.028(57)24O 168.1 168.96(12)25O 167.4 168.18(10)

as binding energies of 22,24,25O as summarized in Table I.To obtain charge radii rch from computed point-proton radiirpp we use the standard expression [64]: ⟨r2

ch⟩ = ⟨r2pp⟩ +

⟨R2p⟩ + N

Z⟨R2

n⟩ + 3!2

4m2pc2 , where 3!2

4m2pc2 = 0.033 fm2 (Darwin–

Foldy correction), R2n = −0.1149(27) fm2 [65], and Rp =

0.8775(51) fm [66]. In this work we ignore the spin-orbitcontribution to charge radii [67]. From the NN sector, theobjective function includes proton-proton and neutron-protonscattering observables from the SM99 database [68] up to35 MeV scattering energy in the laboratory system as wellas effective range parameters, and deuteron properties (seeTable II). The maximum scattering energy was chosen suchthat an acceptable fit to both NN scattering data and many-body observables could be achieved.

In the present optimization protocol, the NNLO chiralforce is tuned to low-energy observables. The comparisonwith the high-precision chiral NN interaction N3 LOEM [49]and experimental data presented in Table II demonstrates thequality of NNLOsat at low energies.

The results for 3H and 3,4He (and 6Li) were computedwith the no-core shell model (NCSM) [6,10] accompaniedby infrared extrapolations [75]. The NNN force of NNLOsatyields about 2 MeV of binding energy for 4He. Heavier nuclei

TABLE II. Low-energy NN data included in the optimization.The scattering lengths a and effective ranges r are in units of fm. Theproton-proton observables with superscript C include the Coulombforce. The deuteron binding energy (ED , in MeV), structure radius(rD , in fm), and quadrupole moment (QD , in fm2) are calculatedwithout meson-exchange currents or relativistic corrections. Thecomputed d-state probability of the deuteron is 3.46%.

NNLOsat N3 LOEM [49] Expt. Ref.

aCpp −7.8258 −7.8188 −7.8196(26) [70]

rCpp 2.855 2.795 2.790(14) [70]

ann −18.929 −18.900 −18.9(4) [71]rnn 2.911 2.838 2.75(11) [72]anp −23.728 −23.732 −23.740(20) [73]rnp 2.798 2.725 2.77(5) [73]ED 2.22457 2.22458 2.224566 [69]rD 1.978 1.975 1.97535(85) [74]QD 0.270 0.275 0.2859(3) [73]

051301-2



0 5 10 15 20 25 30 35

TLab (MeV)

0

20

40

60

80

100

120

140

160

180

δ(3 S

1,1

S0)

(deg

)3S1

1S0

−10

−5

0

5

10

δ(3 D

1,3

P0)

(deg

)

3D1

3P0

FIG. 2. (Color online) Selected neutron-proton scattering phase-shifts as a function of the laboratory scattering energy TLab. (Top)NNLOsat prediction (solid lines) compared to the Nijmegen phaseshift analysis [95] (symbols) at low energies TLab < 35 MeV. Notethe two vertical scales. (Bottom) Neutron-proton scattering phaseshifts from NNLOsat (red diamonds) compared to the Nijmegenphase shift analysis (black squares) and the NNLO potentials (green)from Ref. [77].

dominated by about 90% of 1p-1h(p1/2 → d5/2) excitations,at 6.34 MeV. The energy of the 3−

1 state is strongly correlatedwith the charge radius of 16O, with smaller charge radiileading to higher excitation energies. For 1p-1h excited states,the excitation energy depends on the particle-hole gap andtherefore on one-nucleon separation energies of the A = 16and A = 17 systems. The charge radius depends also on theproton separation energy Sp. For 16O we find Sp = 10.69 MeVand the neutron separation energy Sn(17O) = 4.0 MeV, in anacceptable agreement with the experimental values of 12.12and 4.14 MeV, respectively. For 17F we find Sp = 0.5 MeV, tobe compared with the experimental threshold at 0.6 MeV.

The inset of Fig. 4 shows that the 2−1 state in 16O also comes

out well, suggesting a 1p-1h nature. However, the 1−1 state is

about 1.5 MeV too high compared with experiment. This stateis dominated by 1p-1h excitations from the occupied p1/2 tothe unoccupied s1/2 orbitals. In 17O the 1/2+ state is computedat an excitation energy of 2.2 MeV, which is about 1.4 MeV

FIG. 3. (Color online) Energies (in MeV) of selected excitedstates for various nuclei using NNLOsat. For 6Li we also includespectra from the NCSM (dotted lines), and isospin quantum numbersare also given. The NCSM results were obtained with Nmax = 10 and!! = 16 MeV. Parenthesis denote tentative spins assignments forexperimental levels. Data are from Refs. [100–103].

too high. This probably explains the discrepancy observed forthe 1− state in 16O.

Figure 4 shows that the experimental charge-density of 16Ois well reproduced with NNLOsat, and our charge form factoris, for momenta up to the second diffraction maximum, similarin quality to what Mihaila and Heisenberg [11] achieved withthe Av18 + UIX potential. For the heavier isotopes 22,24O and22,24F Fig. 3 shows good agreement between theory and experi-ment for excited states. For 22F our computed spin assignmentsagree with results from shell-model Hamiltonians [106] andwith recent ab initio results [89]. The binding energies for14N, 22,24F are 103.7, 163, and 175.1 MeV, respectively, ingood agreement with data (104.7, 167.7, and 179.1 MeV). Wealso computed the intrinsic charge (matter) radii of 22,24O andobtained 2.72 fm (2.80 fm) and 2.76 fm (2.95 fm), respectively.The matter radius of 22O agrees with the experimental resultfrom Ref. [91]. We note that the computed spectra in 18O is too

FIG. 4. (Color online) Charge density in 16O computed as inRef. [110] compared to the experimental charge density [111].The inset compares computed low-lying negative-parity states withexperiment.

051301-4

Ekström et al., PRC91, 051301 (2015)


ACCURATE NUCLEAR RADII AND BINDING ENERGIES . . . PHYSICAL REVIEW C 91, 051301(R) (2015)

FIG. 5. (Color online) Equation of state for symmetric nuclearmatter from chiral interactions. Solid red line is the prediction ofNNLOsat. Blue dashed-dotted and black dashed lines: Ref. [56].Symbols (red diamond, blue circle, black square) mark the corre-sponding saturation points. Triangles are saturation points from othermodels (upward triangles [33], rightward triangles [112], downwardtriangles [36]). The corresponding incompressibilities (in MeV) areindicated by numbers. Green box shows empirical saturation point.

compressed compared to experiment (theory yields 0.7 MeVcompared to 1.9 MeV for the first excited 2+ state), possiblydue to the too-high 1/2+ excited state in 17O. In general,the quality of our spectra for sd-shell nuclei is comparableto those of recent state-of-the-art calculations with chiralHamiltonians [44,107–109], while radii are much improved.

For 40Ca the computed binding energy E = 326 MeV,charge radius rch = 3.48 fm, and E(3−

1 ) = 3.81 MeV all agreewell with the experimental values of 342 MeV, 3.4776(19)fm [65], and 3.736 MeV respectively. We checked that ourenergies for the 3−

1 states in 16O and 40Ca are practicallyfree from spurious center-of-mass effects. The results for 40Caillustrate the predictive power of NNLOsat when extrapolatingto medium-mass nuclei.

Finally, we present predictions for infinite nuclear mat-ter. The accurate reproduction of the saturation point andincompressibility of symmetric nuclear matter has been achallenge for ab initio approaches, with representative resultsfrom chiral interactions shown in Fig. 5. The solid line showsthe equation of state for NNLOsat. Its saturation point is closeto the empirical point, and its incompressibility K = 253lies within the accepted empirical range [21]. At saturationdensity, coupled-cluster with doubles yields about 6 MeV perparticle in correlation energy, while triples corrections (andresidual NNN forces beyond the normal-ordered two-bodyapproximation) yield another 1.5 MeV.

Let us briefly discuss the saturation mechanism. Similarto Vlow k potentials [5], the NN interaction of NNLOsatis soft and yields nuclei with too-large binding energiesand too-small radii. The NNN interactions of NNLOsat areessential to arrive at physical nuclei, similarly to the roleof NNN forces in the saturation of nuclear matter withlow-momentum potentials [33]. This situation is reminiscentof the role the three-body terms play in nuclear densityfunctional theory [113].

Summary. We have developed a consistently optimizedinteraction from chiral EFT at NNLO that can be appliedto nuclei and infinite nuclear matter. Our guideline was thesimultaneous optimization of NN and NNN forces to experi-mental data, including two-body and few-body data, as well asproperties of selected light nuclei such as carbon and oxygenisotopes. The optimization is based on low-energy observablesincluding binding energies and radii. The predictions madewith the new interaction NNLOsat include accurate charge radiiand binding energies. Spectra for 40Ca and selected isotopesof lithium, nitrogen, oxygen and fluorine isotopes are wellreproduced, as well as the energies of 3−

1 excitations in 16Oand 40Ca. To our knowledge, NNLOsat is currently the onlymicroscopically founded interaction that allows for a gooddescription of nuclei (including their masses and radii) in awide mass range from few-body systems to medium mass.

Acknowledgments. We thank K. Hebeler and E. Epelbaumfor providing the matrix elements of the nonlocal three-bodyinteraction. This material is based upon work supportedby the U.S. Department of Energy, Office of Science, Of-fice of Nuclear Physics under Award Numbers DEFG02-96ER40963 (University of Tennessee), DE-SC0008499 andDE-SC0008511 (NUCLEI SciDAC collaboration), the FieldWork Proposal ERKBP57 at Oak Ridge National Laboratoryand the National Science Foundation with award number1404159. It was also supported by the Swedish Foundation forInternational Cooperation in Research and Higher Education(STINT, IG2012-5158), by the European Research Council(ERC-StG-240603), by the Research Council of Norwayunder contract ISP-Fysikk/216699, and by NSERC Grant No.401945-2011. TRIUMF receives funding via a contributionthrough the National Research Council Canada. Computertime was provided by the Innovative and Novel ComputationalImpact on Theory and Experiment (INCITE) program. This re-search used resources of the Oak Ridge Leadership ComputingFacility located in the Oak Ridge National Laboratory, which issupported by the Office of Science of the Department of Energyunder Contract No. DE-AC05-00OR22725 and used compu-tational resources of the National Center for ComputationalSciences, the National Institute for Computational Sciences,the Swedish National Infrastructure for Computing (SNIC),and the Notur project in Norway.

[1] P. F. Bedaque and U. van Kolck, Annu. Rev. Nucl. Part. Sci.52, 339 (2002).

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

[3] R. Machleidt and D. Entem, Phys. Rep. 503, 1(2011).

[4] H.-W. Hammer, A. Nogga, and A. Schwenk, Rev. Mod. Phys.85, 197 (2013).

[5] S. K. Bogner, T. T. S. Kuo, and A. Schwenk, Phys. Rep. 386,1 (2003).

[6] P. Navratil, S. Quaglioni, I. Stetcu, and B. R. Barrett, J. Phys.G: Nucl. Part. Phys. 36, 083101 (2009).

051301-5


4. semilocal NN forces, development of improved method to estimate uncertainties status: NN up to N4LO, 3N interactions in development (almost finished :-))









3H 8.52 8.482 1.78 1.7591(363)3He 7.76 7.718 1.99 1.9661(30)4He 28.43 28.296 1.70 1.6755(28)14C 103.6 105.285 2.48 2.5025(87)16O 124.4 127.619 2.71 2.6991(52)22O 160.8 162.028(57)24O 168.1 168.96(12)25O 167.4 168.18(10)


ch⟩ = ⟨r2pp⟩ +

⟨R2p⟩ + N

Z⟨R2

n⟩ + 3!2

4m2pc2 , where 3!2








aCpp −7.8258 −7.8188 −7.8196(26) [70]

rCpp 2.855 2.795 2.790(14) [70]


051301-2



0 5 10 15 20 25 30 35

TLab (MeV)

0

20

40

60

80

100

120

140

160

180

δ(3 S

1,1

S0)

(deg

)3S1

1S0

−10

−5

0

5

10

δ(3 D

1,3

P0)

(deg

)

3D1

3P0











051301-4

Ekström et al., PRC91, 051301 (2015)

S. BINDER et al. PHYSICAL REVIEW C 93, 044002 (2016)

FIG. 1. Chiral expansion of the 3H Eg.s. based on the NN potentials of Refs. [15,16] for the regulator R = 1.0 fm and using Q = Mπ/"b.Panel (a) shows incomplete results based on NN forces only, with uncertainties being estimated via Eqs. (5) and (6). Panel (b) shows incompleteresults based on NN forces only, with uncertainties being estimated via Eqs. (5) and (6) for chiral order i = 0,2 and via Eqs. (7) and (8) fori ! 3. Panel (c) shows the projected results assuming that the LECs in the N2LO 3NF are tuned to reproduce the 3H Eg.s. and using Eqs. (5)and (6) to specify the uncertainty.

long-range part of the NN potential is regularized in positionspace by multiplying with the function

f

(r

R

)=

[1 − exp

(− r2

R2

)]6

, (1)

with the cutoff R chosen in the range 0.8–1.2 fm.In this paper we, for the first time, apply these novel chiral

NN forces beyond the two-nucleon system and demonstratetheir suitability for modern ab initio few- and many-bodymethods. By applying the new method for error analysis, wepresent unambiguous evidence for missing 3NF effects anddemonstrate that the size of the required 3NF contributionsagrees well with expectations based on Weinberg’s powercounting. We also estimate the theoretical accuracy for variousobservables achievable at N4LO and identify the energy regionin elastic Nd scattering that is best suited for testing the chiral3NF.

II. UNCERTAINTY QUANTIFICATION

We first describe our procedure for estimating the the-oretical uncertainty. Let X(p) be some observable with preferring to the corresponding momentum scale and X(i)(p),i = 0,2,3, . . ., a prediction at order Qi in the chiral expansion.

We further define the order-Qi corrections to X(p) via

#X(2) ≡ X(2) − X(0), #X(i) ≡ X(i) − X(i−1), i ! 3, (2)

so that the chiral expansion for X takes the form

X(i) = X(0) + #X(2) + · · · + #X(i). (3)

Generally, the size of the corrections is expected to be

#X(i) = O(QiX(0)). (4)

In Ref. [16], the validity of this estimate was confirmedfor the total neutron-proton cross section. In Refs. [15,16],quantitative estimates of the theoretical uncertainty δX(i) of thechiral EFT prediction X(i) were made by using the expectedand actual sizes of higher-order contributions. Specifically, thefollowing procedure was employed:

δX(0) = Q2|X(0)|,δX(i) = max

2"j"i(Qi+1|X(0)|, Qi+1−j |#X(j )|), (5)

where i ! 2 and Q = max(p/"b, Mπ/"b) with "b =600, 500, and 400 MeV for the regulator choices of R =0.8–1.0, 1.1, and 1.2 fm, respectively. The sizes of actualhigher-order calculations provide additional information on

TABLE I. Ground-state energies Eg.s. of 3H and 4He (in MeV) and the point-proton radius rp of 4He (in fm) calculated by using theimproved NN chiral potentials of Refs. [15,16] up to N4LO for the cutoff R = 1.0 fm in comparison with empirical information. The quoteduncertainties for the theoretical predictions are estimated via Eqs. (5) and (6) for chiral order i = 0,2 and via Eqs. (7) and (8) for i ! 3.

LO NLO N2LO N3LO N4LO Empirical

Eg.s. (3H) −11.3(3.7) −8.36(83) −8.26(20) −7.53(5) −7.63(1) −8.48Eg.s. (4He) −45.5(21.7) −28.6(4.8) −28.1(1.2) −23.75(28) −24.27(6) −28.30rp (4He) 1.064(499) 1.389(174) 1.405(41) 1.563(9) 1.547(2) 1.462(6)

044002-2

S. BINDER et al. PHYSICAL REVIEW C 93, 044002 (2016)

FIG. 3. Predictions for the differential cross section and nucleonAy in elastic Nd scattering based on the NN potentials of Refs. [15,16]for R = 1.0 fm without including the 3NF. Theoretical uncertaintiesare estimated via Eqs. (5) and (6) for chiral order i = 2 and viaEqs. (7) and (8) for i ! 3. The bands of increasing width show theestimated theoretical uncertainty at N4LO (red), N3LO (blue), N2LO(green), and NLO (yellow). The dotted (dashed) lines show the resultsbased on the CD Bonn NN potential [20] (CD Bonn NN potential incombination with the Tucson–Melbourne 3NF [21]). For referencesto proton-deuteron data (symbols), see Ref. [5].

only, while using Eqs. (5) and (6) amounts to overestimatingthe actual error. The N3LO (N4LO) results for the 3H Eg.s. areexpected to be accurate at the level of ∼50 keV (∼10 keV)for the regulator choices of R = 0.8, 0.9, and 1.0 fm. Notethat the size of the inferred 3NF contribution agrees wellwith the uncertainty at NLO, which reflects the estimatedimpact of the N2LO contributions to the Hamiltonian. Thisis fully in line with expectations based on the Weinbergpower counting [1,2]. We further emphasize that the sizableunderbinding of the triton with the NN potentials at N3LOand N4LO is not limited to the employed regulator choice ofR = 1.0 fm. We find Eg.s. = −7.47 . . . − 7.56 MeV (Eg.s. =−7.48 . . . − 7.63 MeV) for the variation of the regulator in therange R = 0.8 . . . 1.2 fm at N3LO (N4LO).

We now turn to Nd scattering observables, which arecalculated by solving the Faddeev equation in the partial-wave

FIG. 4. Predictions for the tensor analyzing powers Ayy and Axx

in elastic Nd scattering based on the NN potentials of Refs. [15,16]for R = 1.0 fm without including the 3NF. For notations see Fig. 3.

basis. We take into account all partial waves up to thetotal angular momentum jmax = 5 in two-nucleon subsystems.Isospin-breaking effects are taken into account in the standardway as described in Ref. [18]. Our predictions for the Ndtotal cross section are visualized in Fig. 2, see also Table II.Similar to the 3H Eg.s., one observes a significant discrepancybetween the theoretical predictions based on the NN forcesonly and data, which provides clear evidence for missing 3NFcontributions. The size of the discrepancy agrees within 1.5times the estimated size of N2LO corrections shown by theNLO error bars. Interestingly, the discrepancy at the lowestenergy of 10 MeV is much smaller than the estimated size ofN2LO contributions. Given that the cross section at low energyis governed by the S-wave spin-doublet and spin-quartet Ndscattering lengths, this observation can be naturally explained.Indeed, the spin-quartet scattering length is almost an order ofmagnitude larger than that of the spin-doublet and much lesssensitive to the 3NF as a consequence of the Pauli principle.

Our predictions for Nd differential cross section andanalyzing powers Ay(N),Ayy , and Axx are shown in Figs. 3 and4. At the lowest energy of 10 MeV, there is little apparent needfor 3NF effects except for Ay . Interestingly, the fine-tuningnature of this observable is clearly reflected in large theoretical

044002-4

Binder et al.,PRC 93, 044002 (2016)

Epelbaum, Krebs, Meißner,PRL 115, 122301 (2015)

LENPIC



















051301-5


4. semilocal NN forces, development of improved method to estimate uncertainties status: NN up to N4LO, 3N interactions in development (almost finished :-))S. BINDER et al. PHYSICAL REVIEW C 93, 044002 (2016)

FIG. 1. Chiral expansion of the 3H Eg.s. based on the NN potentials of Refs. [15,16] for the regulator R = 1.0 fm and using Q = Mπ/"b.Panel (a) shows incomplete results based on NN forces only, with uncertainties being estimated via Eqs. (5) and (6). Panel (b) shows incompleteresults based on NN forces only, with uncertainties being estimated via Eqs. (5) and (6) for chiral order i = 0,2 and via Eqs. (7) and (8) fori ! 3. Panel (c) shows the projected results assuming that the LECs in the N2LO 3NF are tuned to reproduce the 3H Eg.s. and using Eqs. (5)and (6) to specify the uncertainty.

long-range part of the NN potential is regularized in positionspace by multiplying with the function

f

(r

R

)=

[1 − exp

(− r2

R2

)]6

, (1)

with the cutoff R chosen in the range 0.8–1.2 fm.In this paper we, for the first time, apply these novel chiral

NN forces beyond the two-nucleon system and demonstratetheir suitability for modern ab initio few- and many-bodymethods. By applying the new method for error analysis, wepresent unambiguous evidence for missing 3NF effects anddemonstrate that the size of the required 3NF contributionsagrees well with expectations based on Weinberg’s powercounting. We also estimate the theoretical accuracy for variousobservables achievable at N4LO and identify the energy regionin elastic Nd scattering that is best suited for testing the chiral3NF.

II. UNCERTAINTY QUANTIFICATION

We first describe our procedure for estimating the the-oretical uncertainty. Let X(p) be some observable with preferring to the corresponding momentum scale and X(i)(p),i = 0,2,3, . . ., a prediction at order Qi in the chiral expansion.

We further define the order-Qi corrections to X(p) via

#X(2) ≡ X(2) − X(0), #X(i) ≡ X(i) − X(i−1), i ! 3, (2)

so that the chiral expansion for X takes the form

X(i) = X(0) + #X(2) + · · · + #X(i). (3)

Generally, the size of the corrections is expected to be

#X(i) = O(QiX(0)). (4)

In Ref. [16], the validity of this estimate was confirmedfor the total neutron-proton cross section. In Refs. [15,16],quantitative estimates of the theoretical uncertainty δX(i) of thechiral EFT prediction X(i) were made by using the expectedand actual sizes of higher-order contributions. Specifically, thefollowing procedure was employed:

δX(0) = Q2|X(0)|,δX(i) = max

2"j"i(Qi+1|X(0)|, Qi+1−j |#X(j )|), (5)

where i ! 2 and Q = max(p/"b, Mπ/"b) with "b =600, 500, and 400 MeV for the regulator choices of R =0.8–1.0, 1.1, and 1.2 fm, respectively. The sizes of actualhigher-order calculations provide additional information on

TABLE I. Ground-state energies Eg.s. of 3H and 4He (in MeV) and the point-proton radius rp of 4He (in fm) calculated by using theimproved NN chiral potentials of Refs. [15,16] up to N4LO for the cutoff R = 1.0 fm in comparison with empirical information. The quoteduncertainties for the theoretical predictions are estimated via Eqs. (5) and (6) for chiral order i = 0,2 and via Eqs. (7) and (8) for i ! 3.

LO NLO N2LO N3LO N4LO Empirical

Eg.s. (3H) −11.3(3.7) −8.36(83) −8.26(20) −7.53(5) −7.63(1) −8.48Eg.s. (4He) −45.5(21.7) −28.6(4.8) −28.1(1.2) −23.75(28) −24.27(6) −28.30rp (4He) 1.064(499) 1.389(174) 1.405(41) 1.563(9) 1.547(2) 1.462(6)

044002-2

Binder et al.,PRC 93, 044002 (2016)

Epelbaum, Krebs, Meißner,PRL 115, 122301 (2015)

-3

-2

-1

0

1

2

3

<V

> [

MeV

]

cE=1.0,c

D=2.52

cE=0.5, c

D=-0.97

cE=0.0,c

D=-3.77

-4

-2

0

2

4

<V

> [

MeV

]

cE=1.0,c

D=-4.17

cE=1.5, c

D=-2.48

cE=2.0,c

D=-0.78

c1

c3

c4

cD

cE 2π 2π-1π rings 2π-cont. 1/m 1/m 1/m

-3

-2

-1

0

1

2

3

<V

> [

MeV

]

cE=-1.0,c

D=-4.31

cE=-0.5, c

D=-1.20

cE=0.0,c

D=3.80

EM 500 MeV

EGM 450/500 MeV

(CS) (C

T)(no c

i) (no C

i)

EGM 550/600 MeV

KH et al.,PRC 91, 044001 (2015)

c1 c3 c4 cD cE 2π 2π−1π rings 2π−cont. 1/m 1/m 1/m-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

<V>

[MeV

]

cD=0.0, cE=1.657cD=4.0, cE=0.344cD=10.0, cE=-2.32

(CS) (CT)(no ci) (no Ci)

3H, N3LO, R=0.9 fm

NN-only wave functionNN+3N wave function

c1 c3 c4 cD cE 2π 2π−1π rings 2π−cont. 1/m 1/m 1/m-1

-0.8

-0.6

-0.4

-0.2

0

0.2

<V>

[MeV

]

cD=0.0, cE=2.248cD=4.0, cE=1.300cD=10.0, cE=-0.282

(CS) (CT)(no ci) (no Ci)

3H, N3LO, R=1.0 fm

NN-only wave functionNN+3N wave function

nonlocal semilocalN2LO N3LO N2LO N3LO









3H 8.52 8.482 1.78 1.7591(363)3He 7.76 7.718 1.99 1.9661(30)4He 28.43 28.296 1.70 1.6755(28)14C 103.6 105.285 2.48 2.5025(87)16O 124.4 127.619 2.71 2.6991(52)22O 160.8 162.028(57)24O 168.1 168.96(12)25O 167.4 168.18(10)


ch⟩ = ⟨r2pp⟩ +

⟨R2p⟩ + N

Z⟨R2

n⟩ + 3!2

4m2pc2 , where 3!2








aCpp −7.8258 −7.8188 −7.8196(26) [70]

rCpp 2.855 2.795 2.790(14) [70]


051301-2



0 5 10 15 20 25 30 35

TLab (MeV)

0

20

40

60

80

100

120

140

160

180

δ(3 S

1,1

S0)

(deg

)3S1

1S0

−10

−5

0

5

10

δ(3 D

1,3

P0)

(deg

)

3D1

3P0











051301-4

Ekström et al.,PRC91, 051301 (2015)



















051301-5

discrepancies to experiment dominated by deficiencies of present nuclear interactions

remarkable agreement between different ab intio many-body methods

significant increase in scope of ab initio many-body frameworks

systematic estimates oftheoretical uncertainties

unified study of atomic nuclei, nuclear matter and reactions based on novel interactions

presently active efforts to develop improved nucleon interactions

(fits of LECs, power counting, regularization...)

Status and achievements

Current developments and open questions

Key goals

ab initio frameworks in nuclear physics: from basic

Documents