Gedenk-Kolloquium für Dr. Klaus Erkelenz und zur Gründung ... · 16:30 Vortrag. Professor Dr....

1'

&

$

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Die Mathematisch-Naturwissenschaftliche Fakultät und

das Helmholtz-Institut für Strahlen-und Kernphysik, Abteilung Theorie

laden ein zum

Gedenk-Kolloquium für Dr. Klaus Erkelenz und zur Gründung der Dr. Klaus Erkelenz Stiftung

Programm 16:15 Einführung Professor Dr. Ulf-G. Meißner Dekan der Mathematisch-Naturwissenschaftlichen Fakultät an der Universität Bonn 16:30 Vortrag Professor Dr. Ruprecht Machleidt University of Idaho, Moscow, USA Klaus Erkelenz and the Bonn Potential 17:15 Vortrag Professor Dr. Ulf-G. Meißner Direktor der Abteilung Theorie des Helmholtz-Instituts für Strahlen- und Kernphysik der Universität Bonn Nuclear Theory: A Modern Perspective 18:00 Professor e.m. Dr. Peter David Erinnerungen an Dr. Klaus Erkelenz

Prof. R. Machleidt, Univ. of Idaho, USA Klaus Erkelenz and the Bonn Potential Dr. Klaus Erkelenz was a research associate at the Institute for Theoretical Nuclear Physics of the Bonn University until November 1973, when he died untimely at the age of 42. Even though his career was cut short, he made substantial contributions to nuclear physics. His main focus was the meson theory of nuclear forces. In particular, he worked on a consistent relativistic derivation of the nucleon-nucleon interaction based upon meson field theory. His initiatives inspired a decade of further work on the subject at the Bonn Institute. The resulting nuclear force models have become known in the international community as the „Bonn Potentials“.

Prof. U.-G. Meißner, Univ. Bonn & FZ Jülich Nuclear Theory: A Modern Perspective Effective Field Theory is a modern tool in many branches of theoretical physics. In particular, the problem of the forces between two and three nucleons has gained renewed interest based on this framework. I discuss the essentials of this approach and show how it ties to the meson field theory potentials like the one set up by Klaus Erkelenz and others. In addition, the use of high performance computers allows for ab initio calculations of the properties of atomic nuclei and gives new insight into the fine-tuning underlying nucleosynthesis in the Big Bang and in stars.

Freitag, 15. November 2013, 16.00 Uhr c.t. im Hörsaal I des Physikalischen Instituts, Nussallee 12, 53115 Bonn

Nuclear Theory: A Modern Perspective – Ulf-G. Meißner – Bonn, November 2013 · C < ∧ O > B •

2

Nuclear Theory:A Modern Perspective

Ulf-G. Meißner, Univ. Bonn & FZ Julich

Supported by DFG, SFB/TR-16 and by DFG, SFB/TR-110 and by EU, I3HP EPOS and by BMBF 06BN9006 and by HGF VIQCD VH-VI-417

Nuclear Astrophysics Virtual Institute

NLEFT


3

CONTENTS

• Some basic facts

• Ab initio calculation of atomic nuclei

• The fate of carbon-based life

• Towards medium-mass nuclei

• Summary & outlook


4

Some basicfacts


5

STRUCTURE FORMATION in QCD

• The strong interactions are described by QCD

• Quarks and gluons are confined within hadrons

• Protons and neutrons form atomic nuclei

⇒ This requires the inclusion of electromagnetism

⇒ Atomic nuclei make up the visible matter in the Universe

• up and down quarks are very light, a few MeV

So how are these strongly interacting composites generated?

How sensitive are they to changes in the fundamental parametersof QCD+QED?

QCDO(α )

245 MeV

181 MeV

ΛMS(5) α (Μ )s Z

0.1210

0.1156

0.1

0.2

0.3

0.4

0.5

αs(Q)

1 10 100Q [GeV]

Heavy QuarkoniaHadron Collisionse+e- AnnihilationDeep Inelastic Scattering

NL

O

NN

LO

TheoryData L

attic

e

211 MeV 0.1183s4


6THE NUCLEAR LANDSCAPE: AIMS & METHODS

• Theoretical methods:

− Lattice QCD: A = 0,1,2, . . .

− NCSM, Faddeev-Yakubowsky, GFMC, ... :A = 3 - 16

− coupled cluster, . . .: A = 16 -100

− density functional theory, . . .: A ≥ 100

• Chiral EFT:

− provides accurate NN and 3N forces

− successfully applied in light nucleiwith A = 2, 3, 4

− combine with simulations to get to larger A

⇒ Nuclear Lattice Simulations


7

Ab initio calculationsof atomic nuclei


8Ingredients

• Nuclear binding is shallow: E/A ≤ 8 MeV

⇒ Nuclei can be calculated from the A-body Schrodinger equation: HΨA = EΨA

• Forces are of (dominant) two- and (subdominant) three-body nature:V = VNN + VNNN

⇒ can be calculated systematically and to high-precisionWeinberg, van Kolck, Epelbaum, M., Entem, Machleidt, . . .

⇒ fit all parameters in VNN + VNNN from 2- and 3-body data

⇒ exact calc’s of systems with A ≤ 4 using Faddeev-Yakubowsky machinerysee fig.

⇒ connection to boson-exchange models (as in Machleidt’s talk)?→ slide

But how about ab initio calculations for systems with A ≥ 5?


9

Examples

0 60 120 180

θ [deg]

12

16

20

dσ/d

Ω [

mb/

sr]

EGM N3LOEM N3LOCD Bonn 2000Gross, Stadler

Nijmegen PWA

60 120 180

θ [deg]

0

0,1

0,2

0,3

Ay

10

100

dσ/dΩ [mb/sr]

10 MeV

1

10

100

dσ/dΩ [mb/sr]

65 MeV

-0,1

0

T20

-0,4

-0,2

0

0,2T20

0 60 120θ [deg]

-0,08

-0,03

0,02

0,07 T21

0 60 120 180θ [deg]

-0,2

-0,1

0

T21

0 60 120 180-0.4

0.0

0.4

0.8K

X X’ (N

)

0 60 120 180

0.4

0.6

0.8

1.0

K Y Y

’ (N)

0 60 120 180CM [deg]

-0.6

-0.4

-0.2

0.0

K Z X

’ (N)


10CONNECTION to BOSON–EXCHANGE MODELSEpelbaum, M., Glockle, Elster, Phys. Rev. C65 (2002) 044001 [nucl-th/0106007]

• Basic idea:

Expand meson-exchange in powers of t/M2R

and map on 4N operators

...M

= + +

ε1

CD-BonnNijm-93

Bonn-B Nijm-INijm-IIAV-18

C C C C C C C C1S0 1S0 3S1 3S1 1P1 3P0 3P1 C3P2

0.5

1.0

1.5

-0.5

0.0

g2

t−M2R

= − g2

M2R− g2 t

M4R

+ . . .

with • ∝ t0 ∝ ~∇0

∝ t ∝ ~∇2

• works amazingly well

all LECs of natural size

(in units of 1/Fnπ Λmχ )

NLO

NNLO


11NUCLEAR LATTICE SIMULATIONSFrank, Brockmann (1992), Koonin, Muller, Seki, van Kolck (2000) , Lee, Schafer (2004), . . .

Borasoy, Krebs, Lee, UGM, Nucl. Phys. A768 (2006) 179; Borasoy, Epelbaum, Krebs, Lee, UGM, Eur. Phys. J. A31 (2007) 105

• new method to tackle the nuclear many-body problem

• discretize space-time V = Ls × Ls × Ls × Lt:nucleons are point-like fields on the sites

• discretized chiral potential w/ pion exchangesand contact interactions

• typical lattice parameters

Λ =π

a' 300 MeV [UV cutoff]

p

p

n

n a

~ 2 fm

• strong suppression of sign oscillations due to approximate Wigner SU(4) symmetryJ. W. Chen, D. Lee and T. Schafer, Phys. Rev. Lett. 93 (2004) 242302

• hybrid Monte Carlo & transfer matrix (similar to LQCD)


12

CONFIGURATIONS

⇒ all possible configurations are sampled⇒ clustering emerges naturally⇒ perform ab initio calculations using only VNN and VNNN as input⇒ grand challenge: the spectrum of 12C


13SPECTRUM OF 12C & the HOYLE STATE

Epelbaum, Krebs, Lee, UGM, Phys. Rev. Lett. 106 (2011) 192501Epelbaum, Krebs, Lahde, Lee, UGM, Phys. Rev. Lett. 109 (2012) 252501Viewpoint: Hjorth-Jensen, Physics 4 (2011) 38


14THE TRIPLE-ALPHA PROCESS→ MOVIE

c©ANU

• the 8Be nucleus is instable, long lifetime→ 3 alphas must meet

• the Hoyle state sits just above the continuum threshold→ most of the excited carbon nuclei decay

(about 4 out of 10000 decays produce stable carbon)

• carbon is further turned into oxygen but w/o a resonant condition

⇒a triple wonder !


15RESULTS

• some groundstate energies and differences

E [MeV] NLEFT Exp.3He -3H 0.78(5) 0.764He −28.3(6) −28.38Be −55(2) −56.512C −92(3) −92.2

0.5

0.6

0.7

0.8

0.9

1

5 10 15 20 25

E3H

e −

Etr

iton (

MeV

)

L (fm)

latticephysical (infinite volume)

• promising results

• excited states more difficult

⇒ new projection MC method


16The SPECTRUM of CARBON-12

• After 8 · 106 hrs JUGENE/JUQUEEN (and “some” human work)

2+

Exp Th

−92

Hoyle

2+

0−84

−86

−88

−90

−87.72

−84.51 −85(3)

−82

0 +

−92(3)

−88(2)

2+−82.6(1)

2 +

0 + +−83(3)

E [

MeV

]

0 +

−92.16

⇒ First ab initio calculationof the Hoyle state

√

Structure of the Hoyle state:


17

The fate of carbon-based lifeas a function of the fundamental

parameters of QCD+QED


18FINE-TUNING of FUNDAMENTAL PARAMETERS


19

EARLIER STUDIES of the ANTHROPIC PRINCIPLE

• rate of the 3α-process: r3α ∼(Nα

kT

)3

Γγ exp

(−

∆E

kT

)∆E = E?12 − 3Eα = 379.47(18) keV

• how much can ∆E be changedso that there is still enough12C and 16O?

⇒ |∆E| . 100 keV

Oberhummer et al., Science 289 (2000) 88

Csoto et al., Nucl. Phys. A 688 (2001) 560Schlattl et al., Astrophys. Space Sci. 291 (2004) 27[Livio et al., Nature 340 (1989) 281]

too few 16O too few 12C


20FINE-TUNING: MONTE-CARLO ANALYSISEpelbaum, Krebs, Lahde, Lee, UGM, PRL 110 (2013) 112502; Eur.Phys.J. A49 (2013) 82

• consider first QCD only→ calculate ∂∆E/∂Mπ

• relevant quantities (energy differences)

∆Eh ≡ E∗12 − E8 − E4, ∆Eb ≡ E8 − 2E4

• energy differences depend on parameters of QCD (LO analysis)

Ei = Ei

(MOPEπ ,mN(Mπ), gπN(Mπ), C0(Mπ), CI(Mπ)

)gπN ≡

gA2Fπ

• remember: M2π± ∼ (mu +md) Gell-Mann–Oakes–Renner (1968)

⇒ quark mass dependence ≡ pion mass dependence


21PION MASS VARIATIONS

• consider pion mass changes as small perturbations

∂Ei∂Mπ

∣∣∣∣Mphysπ

=∂Ei

∂MOPEπ

∣∣∣∣Mphysπ

+ x1

∂Ei∂mN

∣∣∣∣mphysN

+ x2

∂Ei∂gπN

∣∣∣∣gphysπN

+ x3

∂Ei∂C0

∣∣∣∣Cphys

0

+ x4

∂Ei∂CI

∣∣∣∣CphysI

with

x1 ≡∂mN

∂Mπ

∣∣∣∣Mphysπ

, x2 ≡∂gπN∂Mπ

∣∣∣∣Mphysπ

, x3 ≡∂C0

∂Mπ

∣∣∣∣Mphysπ

, x4 ≡∂CI∂Mπ

∣∣∣∣Mphysπ

⇒ problem reduces to the calculation of the various derivativesusing AFQMC and the determination of the xi

• x1 and x2 can be obtained from LQCD plus CHPT

• x3 and x4 can be obtained from two-body scattering and its Mπ-dependence


22

VISUALIZATION of the PION MASS VARIATIONS

• At LO, one-pion exchange and 4N contact terms

N (M )πnucleon mass m

four−nucleoncouplings C(M )π

pion propagator2

pion−nucleon

coupling g(M )π

π1/(q − M )2As,t ≡

∂a−1s,t

∂Mπ

∣∣∣∣M

physπ

m


23CORRELATIONS

• vary the quark mass derivatives of a−1s,t within −1, . . . ,+1:

-6 -4 -2 0 2 4 6K

E4

π

-600

-400

-200

0

200

400

600

K∆Ebπ

K∆Ehπ

K∆Eh+bπ

• clear correlations: α-particle BE and the energies/energy differences

⇒ anthropic or non-anthropic scenario depends on whether the 4He BE moves!

∆Eb = E(8Be)− 2E(4He)

∆Eh = E(12C∗)− E(8Be)− E(4He)

∆Eh+b = E(12C∗)− 3E(4He)

∂OH

∂Mπ

= KπH

OH

Mπ


24THE END-OF-THE-WORLD PLOT• |δ(∆Eh+b)| < 100 keV

→∣∣∣∣(0.571(14)As + 0.934(11)At − 0.069(6)

)δmq

mq

∣∣∣∣ < 0.0015

As,t ≡∂a

−1s,t

∂Mπ

∣∣∣∣M

physπ

The light quark massis fine-tuned to' 2−3 %

Similarly:αEM is fine-tunedto ' 2.5%

Berengut et al.,Phys. Rev. D 87 (2013) 085018


25

Towards medium-mass nuclei


26GOING up the ALPHA CHAIN

• Consider the α ladder 12C, 16O, 20Ne, 24Mg, 28Si as tCPU ∼ A2

• Improved “multi-state” technique to extract g.s. energies

⇒ higher A, better accuracy

⇒ overbinding at LO beyond A = 12 persists up to NNLO

-155

-150

-145

-140

-135

-130

-125

-120

-115

0 2 4 6 8 10 12 14

E16

(LO

) [M

eV]

Nt

-8.0e-5-6.5e-5-5.5e-5-4.5e-5

-210

-200

-190

-180

-170

-160

-150

0 2 4 6 8 10 12 14

E20

(LO

) [M

eV]

Nt

-8.0e-5-6.5e-5-5.5e-5-4.5e-5

-270

-260

-250

-240

-230

-220

-210

-200

0 2 4 6 8 10 12 14

E24

(LO

) [M

eV]

Nt

-8.0e-5-6.5e-5-5.5e-5-4.5e-5

-360

-340

-320

-300

-280

-260

0 2 4 6 8 10 12 14

E28

(LO

) [M

eV]

Nt

-5.5e-5-4.5e-5-3.5e-5

E = −131.3(5) E = −165.9(9) E = −232(2) E = −308(3)

[−127.62] [−160.64] [−198.26] [−236.54]

16O 20Ne 24Mg 28Si


27REMOVING the OVERBINDINGLahde et al., arXiv:1311.0477 [nucl-th]

• Overbinding is due to four α clusters in close proximity

⇒ remove this by an effective 4N operator [long term: N3LO]

V (4Neff ) = D(4Neff )∑

1≤(~ni−~nj)2≤2

ρ(~n1)ρ(~n2)ρ(~n3)ρ(~n4)

• fix the coefficient D(4Neff ) from the BE of 24Mg

⇒ excellent description of the g.s. energies

A 12 16 20 24 28Th −90.3(2) −131.3(5) −165.9(9) −198(2) −233(3)

Exp −92.16 −127.62 −160.64 −198.26 −236.54


28GROUND STATE ENERGIES

28Si

24Mg

20Ne

16O

12C

8Be

4He

-400 -350 -300 -250 -200 -150 -100 -50 0

E (MeV)

ExperimentNNLO

NNLO + 4Neff


29

SUMMARY & OUTLOOK

• Nuclear forces can be calculated accurately and systematically in chiral EFT

• Connection to boson-exchange models can be made

• Nuclear lattice simulations as a new quantum many-body approach

• Fix parameters in few-nucleon systems→ predictions(ab initio calculations)

• 12C spectrum at NNLO→ Hoyle state and its structure

• Fine-tuning of mquark and αEM→ viability of life

⇒ changes in mquark of about 2% and in αEM of about 2.5% are allowed

• First ab initio results for medium mass nuclei

⇒ the strong interactions remain a challenge


30

SPARES


31

The RELEVANT QUESTIONDate: Sat, 25 Dec 2010 20:03:42 -0600From: Steven Weinberg 〈[email protected]〉To: Ulf-G. Meissner 〈[email protected]〉Subject: Re: Hoyle state in 12C

Dear Professor Meissner,Thanks for the colorful graph. It makes a nice Christmas

card. But I have a detailed question. Suppose you calculate not only the energyof the Hoyle state in C12, but also of the ground states of He4 and Be8. Howsensitive is the result that the energy of the Hoyle state is near the sum of therest energies of He4 and Be8 to the parameters of the theory? I ask because Isuspect that for a pretty broad range of parameters, the Hoyle state can be wellrepresented as a nearly bound state of Be8 and He4.

All best,Steve Weinberg

• How does the Hoyle state relative to the 4He+8Be threshold,if we change the fundamental parameters of QCD+QED?

• not possible in nature, but on a high-performance computer!

-110

-100

-90

-80

-70

-60

LO NLO EM+IB NNLO Exper.

E (

MeV

)

JP = 0

+1

JP = 0

+2

Jz = 0, J

P = 2

+1

Jz = 2, J

P = 2

+1


32The NON-ANTHROPIC SCENARIO

•Weinberg’s assumption: The Hoyle state stays close to the 4He+8Be threshold

g g+ ∆gg ∆g−

4He+8BeHoyle

−84.80

−84.51

fundamental parameter

en

erg

y d

iffe

ren

ce


33The ANTHROPIC SCENARIO

•The AP strikes back: The Hoyle state moves away from the 4He+8Be threshold

g g+ ∆gg ∆g−

4He+8BeHoyle

−84.80

−84.51

en

erg

y d

iffe

ren

ce

fundamental parameter


34EARLIER STUDIES of the AP

• By hand modification of the energy diff. & network calcs in massive starsLivio et al., Nature 340 (1989) 281

→ a 60 keV increase does not significantly alter carbon production

→ a 60 keV decrease roughly doubles the carbon production rate

→ a ±277 keV change leaves essentially no carbon (just oxygen)

→ weak conclusion: the strong AP might be in trouble

• Changing NN and em interactions in a microscopic model & network calcsOberhummer et al., Science 289 (2000) 88

→ modified NN strength & fine structure constant in [0.996, 1.004]

→ no influence on the width but on the relative position of the Hoyle state

→ use up-to-date stellar evolution model

→ more than 0.5[4]% in the strong coupling [αQED] would destroyall carbon (oxygen) in stars

→ “should be of interest to AP considerations”


35

Introduction II:Effective Field Theoryfor Nuclear Physics

only a brief reminder→ details in

E. Epelbaum, H.-W. Hammer, UGM, Rev. Mod. Phys. 81 (2009) 1773[arXiv:0811.1338 [nucl-th]]


36CHIRAL EFT FOR FEW-NUCLEON SYSTEMSGasser, Leutwyler, Weinberg, van Kolck, Epelbaum, Bernard, Kaiser, UGM, . . .

• Scales in nuclear physics:

Natural: λπ = 1/Mπ ' 1.5 fm (Yukawa 1935)

Unnatural: |anp(1S0)| = 23.8 fm , anp(3S1) = 5.4 fm 1/Mπ

• this can be analyzed in a suitable EFT based on

LQCD → LEFF = Lππ + LπN + LNN + . . .

• pion and pion-nucleon sectors are perturbative in Q/Λχ → chiral perturbation th’y

• LNN collects short-distance contact terms, to be fitted

• NN interaction requires non-perturbative resummation

→ chirally expand VNN(N), use in regularized LS/FY equation

repulsive

core

CD Bonn

Reid93

AV18

0 0.5 1 1.5 2 2.5

300

200

100

0

-100

Vc

(r)

[ Me

V]


37

CHIRAL POTENTIAL and NUCLEAR FORCES

• explains naturally the observed hierarchy of nuclear forces

• MANY successfull tests in few-nucleon systems (continuum calc’s)

O((Q/Λχ)0)

O((Q/Λχ)2)

O((Q/Λχ)3)

O((Q/Λχ)4)

2 LECs

7 LECs

15 LECs

2 LECs


38

Nuclear lattice simulations– Formalism –


39NUCLEAR LATTICE SIMULATIONSFrank, Brockmann (1992), Koonin, Muller, Seki, van Kolck (2000) , Lee, Schafer (2004), . . .

Borasoy, Krebs, Lee, UGM, Nucl. Phys. A768 (2006) 179; Borasoy, Epelbaum, Krebs, Lee, UGM, Eur. Phys. J. A31 (2007) 105

• new method to tackle the nuclear many-body problem

• discretize space-time V = Ls × Ls × Ls × Lt:nucleons are point-like fields on the sites

• discretized chiral potential w/ pion exchangesand contact interactions

• typical lattice parameters

Λ =π

a' 300 MeV [UV cutoff]

p

p

n

n a

~ 2 fm

• strong suppression of sign oscillations due to approximate Wigner SU(4) symmetryJ. W. Chen, D. Lee and T. Schafer, Phys. Rev. Lett. 93 (2004) 242302

• hybrid Monte Carlo & transfer matrix (similar to LQCD)


40

CONFIGURATIONS

⇒ all possible configurations are sampled⇒ clustering emerges naturally


41

TRANSFER MATRIX METHOD

• Correlation–function for A nucleons: ZA(t) = 〈ΨA| exp(−tH)|ΨA〉

with ΨA a Slater determinant for A free nucleons

• Ground state energy from the time derivative of the correlator

EA(t) = −d

dtlnZA(t)

→ ground state filtered out at large times: E0A = lim

t→∞EA(t)

• Expectation value of any normal–ordered operator O

ZOA = 〈ΨA| exp(−tH/2)O exp(−tH/2) |ΨA〉

limt→∞

ZOA (t)

ZA(t)= 〈ΨA|O |ΨA〉

Euclidean time


42

TRANSFER MATRIX CALCULATION

• Expectation value of any normal–ordered operator O

〈ΨA|O |ΨA〉 = limt→∞

〈ΨA| exp(−tH/2)O exp(−tH/2) |ΨA〉〈ΨA| exp(−tH)|ΨA〉

• Anatomy of the transfer matrix

OΨfree

Ψfree

02Lto

+ Lti

SU(4) π

Lto

+ Lti/2 L

toL

to+ L

ti

full LO full LO SU(4) π

operator insertion forexpectation value

Z,N Z,N

inexpensive filter inexpensive filter


43

PROJECTION MONTE CARLO TECHNIQUE

• Insert clusters of nucleons at initial/final states (spread over some time interval)→ allows for all type of wave functions (shell model, clusters, . . .)→ removes directional bias

• Example: two basic configurations in the spectrum of 12C


44

MONTE CARLO with AUXILIARY FILEDS

• Contact interactions represented by auxiliary fields s, sI

exp(ρ2/2) ∝∫ +∞

−∞ds exp(−s2/2− sρ) , ρ ∼ N†N

• Correlation function = path-integral over pions & auxiliary fields

p

p

⇒


45

COMPUTATIONAL EQUIPMENT

• Past = JUGENE (BlueGene/P)• Present = JUQUEEN (BlueGene/Q)


46

Nuclear lattice simulations– Results –

nuclei neutron matter


47

FIXING PARAMETERS & FIRST PREDICTIONS

• work at NNLO including strong and em isospin breaking

• 9 NN LECs from np scattering and Qd

• 2 LECs for isospin-breaking (np, pp, nn)

• 2 LECs D,E related to the leading 3NF

⇒ make predictions 0

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140

pCM (MeV)

1S0

δ(1S

0)

(deg

rees

)

LO3NLO3

PWA93 (np)

40

60

80

100

120

140

160

180

0 20 40 60 80 100 120 140

pCM (MeV)

3S1

δ(3S

1)

(deg

rees

)

LO3NLO3

PWA93 (np)

40

60

80

100

120

140

160

180

0 20 40 60 80 100 120 140

pCM (MeV)

3S1

δ(3S

1)

(deg

rees

)

LO3NLO3

PWA93 (np)

• pp vs np scattering

• nd spin-3/2quartet channel

• . . .

0 0.10 0.20 0.30 0.40

p2 (fm-2)

0

-0.05

0.05

0.10

-0.10

-0.15

p c

ot

δ (

fm-1

)

0.15

n-d (exp.)

NNLO

NLO

LO

p-d (exp.)

0

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140 160

pCM (MeV)

1S0

δ(1S

0)

(deg

rees

)

LONLO + IB + EM

PWA93 (pp)


48

Ground statesEpelbaum, Krebs, Lahde, Lee, UGM, arxiv:1208.1328


49PREDICTIONS: TRITON & HELIUM-3Epelbaum, Krebs, Lee, UGM, Phys. Rev. Lett. 104 (2010) 142501; Eur. Phys. J. A 45 (2010) 335

• binding energies of 3N systems: E(L) = B.E.−a

Lexp(−bL)

see also Hammer, Kreuzer (2011)

⇒ predict the energy difference E(3He)− E(3H)

0.5

0.6

0.7

0.8

0.9

1

5 10 15 20 25

E3H

e −

Etr

iton (

MeV

)

L (fm)

latticephysical (infinite volume)

0.76 MeV [exp.]0.78(5) MeV


50

Ground state of 4He

L = 11.8 fm

-30

-28

-26

-24

-22

-20

0 0.02 0.04 0.06 0.08 0.1 0.12

E(t

) (M

eV)

t (MeV-1)

(A)

LO

-6

-4

-2

0

2

4

6

0 0.02 0.04 0.06 0.08 0.1 0.12

t (MeV-1)

(B)

∆NLO-IS∆IB + ∆EM

∆NNLO

LO (O(Q0)) −28.0(3) MeVNLO (O(Q2)) −24.9(5) MeV

NNLO (O(Q3)) −28.3(6) MeVExp. −28.3 MeV


51

Ground state of 8Be

L = 11.8 fm

-65

-60

-55

-50

-45

-40

0 0.02 0.04 0.06 0.08 0.1 0.12

E(t

) (M

eV)

t (MeV-1)

(A)

LO

-10

-5

0

5

10

15

0 0.02 0.04 0.06 0.08 0.1 0.12

t (MeV-1)

(B)


∆NNLO

LO (O(Q0)) −57(2) MeVNLO (O(Q2)) −47(2) MeV

NNLO (O(Q3)) −55(2) MeVExp. −56.5 MeV


52

Ground state of 12C

L = 11.8 fm

-120

-110

-100

-90

-80

-70

-60

0 0.02 0.04 0.06 0.08 0.1 0.12

E(t

) (M

eV)

t (MeV-1)

(A)

LO (1)LO (2)

-20

-10

0

10

20

30

40

0 0.02 0.04 0.06 0.08 0.1 0.12

t (MeV-1)

(B)

∆NLO-IS (1)∆NLO-IS (2)

∆IB + ∆EM (1)∆IB + ∆EM (2)

∆NNLO (1)∆NNLO (2)




53

Ground state of 16O

L = 11.8 fm

-170

-160

-150

-140

-130

-120

-110

0 0.02 0.04 0.06 0.08 0.1

E(t

) (M

eV)

t (MeV-1)

(A)

LO (1)LO (2)

-30

-20

-10

0

10

20

30

40

50

60

0 0.02 0.04 0.06 0.08 0.1

t (MeV-1)

(B)

∆NLO-IS (1)∆NLO-IS (2)

∆IB + ∆EM (1)∆IB + ∆EM (2)

∆NNLO (1)∆NNLO (2)



to be published


54

EXCITED STATES of 12C

• Lowest excited state is 2+1 (as in nature)

E(2+1 ) = −89(3) MeV

[−87.7 MeV]

-4

-3

-2

-1

0

1

0 0.02 0.04 0.06 0.08 0.1

log

[Z(t

)/Z

0+ 1(t)]

t (MeV-1)

(A)

LO

-20

-10

0

10

20

30

40

0 0.02 0.04 0.06 0.08 0.1E

(t)

(MeV

)t (MeV-1)

(B)


∆NNLO


55THE HOYLE STATE (0+2 )

• energy: E(0+2 ) = −85(3) MeV

• close to E(4He) + E(8Be) = −83.3(2.0) MeV

• structure: “bent” alpha-chain like (not “BEC”)

-110

-100

-90

-80

-70

-60

-50

0 0.02 0.04 0.06 0.08 0.1 0.12

E(t

) (M

eV)

t (MeV-1)

(A)

LO (1)LO (2)

-20

-10

0

10

20

30

40

0 0.02 0.04 0.06 0.08 0.1 0.12

t (MeV-1)

(B)

∆NLO-IS (1)∆IB + ∆EM (1)

∆NNLO (1)


56A HOYLE STATE EXCITATION (2+2 )

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.02 0.04 0.06 0.08 0.1

log

[Z(t

)/Z

0+ 2(t)]

t (MeV-1)

(A)

LO

-20

-10

0

10

20

30

40

0 0.02 0.04 0.06 0.08 0.1

E(t

) (M

eV)

t (MeV-1)

(B)


∆NNLO• a 2+ state 2 MeV above the Hoyle state

• interpretation:a rotational band of the Hoyle stategenerated from excitations of the alpha-chain

• what’s in the data ?a 2+ state 3.51 MeV above the Hoyle state seen in 11B(d, n)12Cnot included in the level scheme! Ajzenberg-Selove, Nucl. Phys. A506 (1990) 1

a 2+ state 3.8(4) MeV above the Hoyle state seen in 12C(α,α)12CBency John et al., Phys. Rev. C 68 (2003) 014305

• and much more, see next slide and:→ talk by Henry Weller

⇒ ab initio prediction requires experimental confirmation


57SPECTRUM OF 12C• Summarizing the results for carbon-12:

0+1 2+

1 0+2 2+

2

LO −96(2) MeV −94(2) MeV −89(2) MeV −88(2) MeVNLO −77(3) MeV −74(3) MeV −72(3) MeV −70(3) MeV

NNLO −92(3) MeV −89(3) MeV −85(3) MeV −83(3) MeV−82.6(1) MeV [1,2]

Exp. −92.16 MeV −87.72 MeV −84.51 MeV −82.32(6) MeV [3]−81.1(3) MeV [4]−82.13(11) MeV [5]

[1] Freer et al., Phys. Rev. C 80 (2009) 041303[2] Zimmermann et al., Phys. Rev. C 84 (2011) 027304[3] Hyldegaard et al., Phys. Rev. C 81 (2010) 024303[4] Itoh et al., Phys. Rev. C 84 (2011) 054308[5] Weller et al., in preparation

• importance of consistent 2N & 3N forces

• good agreement w/ experiment, can be improved


58

Testing theAnthropic Principle


59

MC ANALYSIS of the AP

• consider QCD only→ calculate ∂∆E/∂Mπ

• relevant quantities (energy differences)

∆Eh ≡ E∗12 − E8 − E4, ∆Eb ≡ E8 − 2E4 ∆Ec ≡ E∗12 − E12

• energy differences depend on parameters of QCD (LO analysis)

Ei = Ei

(MOPEπ ,mN(Mπ), gπN(Mπ), C0(Mπ), CI(Mπ)

)gπN ≡

gA2Fπ

• remember: M2π± ∼ (mu +md)

⇒ quark mass dependence ≡ pion mass dependence


60PION MASS VARIATIONS

• consider pion mass changes as small perturbations

∂Ei∂Mπ

∣∣∣∣Mphysπ

=∂Ei

∂MOPEπ

∣∣∣∣Mphysπ

+ x1

∂Ei∂mN

∣∣∣∣mphysN

+ x2

∂Ei∂gπN

∣∣∣∣gphysπN

+ x3

∂Ei∂C0

∣∣∣∣Cphys

0

+ x4

∂Ei∂CI

∣∣∣∣CphysI

with

x1 ≡∂mN

∂Mπ

∣∣∣∣Mphysπ

, x2 ≡∂gπN∂Mπ

∣∣∣∣Mphysπ

, x3 ≡∂C0

∂Mπ

∣∣∣∣Mphysπ

, x4 ≡∂CI∂Mπ

∣∣∣∣Mphysπ

⇒ problem reduces to the calculation of the various derivativesusing AFQMC and the determination of the xi

• x1 and x2 can be obtained from LQCD plus CHPT

• x3 and x4 can be obtained from two-body scattering and its Mπ-dependence


61

AFQMC RESUTS for the DERIVATIVES

• 4He • 12C(0+2 )

1.1

1.15

1.2

1.25

1.3

1.35

1.4

2 4 6 8 10 12 14 16

fitdE/dc11 [l.u.]not fitted

-1.7

-1.65

-1.6

-1.55

-1.5

-1.45

-1.4

-1.35

2 4 6 8 10 12 14 16

fitdE/dcii [l.u.]

not fitted

-0.046

-0.045

-0.044

-0.043

-0.042

-0.041

-0.04

-0.039

2 4 6 8 10 12 14 16

fitdE/dm

π (OPE)

not fitted

-0.21

-0.205

-0.2

-0.195

-0.19

-0.185

-0.18

-0.175

2 4 6 8 10 12 14 16

fitdE pion dM [MeV]

not fitted

-0.14

-0.138

-0.136

-0.134

-0.132

-0.13

-0.128

-0.126

-0.124

-0.122

-0.12

-0.118

2 4 6 8 10 12 14 16

fitdE pion IB [MeV]

not fitted

-0.068

-0.066

-0.064

-0.062

-0.06

-0.058

-0.056

-0.054

-0.052

2 4 6 8 10 12 14 16

fitdE/dmN

not fitted

0.255

0.26

0.265

0.27

0.275

0.28

0.285

0.29

0.295

0.3

0.305

2 4 6 8 10 12 14 16

fitdE/dg

πιN [l.u.]not fitted

0.33

0.34

0.35

0.36

0.37

0.38

0.39

0.4

2 4 6 8 10 12 14 16

fitdE dCpp [MeV]not fitted

0.54

0.545

0.55

0.555

0.56

0.565

0.57

0.575

0.58

0.585

2 4 6 8 10 12 14 16

fitdE dCoul [MeV]not fitted

-29

-28.5

-28

-27.5

-27

-26.5

-26

-25.5

-25

2 4 6 8 10 12 14 16

fitold dataE [MeV]

6.5

6.6

6.7

6.8

6.9

7

7.1

7.2

2 4 6 8 10 12 14 16

fitdE/dc11 [l.u.]

gnd state

-8.2

-8.1

-8

-7.9

-7.8

-7.7

-7.6

-7.5

2 4 6 8 10 12 14 16

fitdE/dcii [l.u.]

gnd state

-0.18

-0.178

-0.176

-0.174

-0.172

-0.17

-0.168

-0.166

2 4 6 8 10 12 14 16

fitdE/dm

π(OPE)

gnd state

-0.82

-0.815

-0.81

-0.805

-0.8

-0.795

-0.79

-0.785

-0.78

-0.775

-0.77

-0.765

2 4 6 8 10 12 14 16

fitdE pion dM [MeV]gnd state

-0.545

-0.54

-0.535

-0.53

-0.525

-0.52

-0.515

-0.51

-0.505

2 4 6 8 10 12 14 16

fitdE pion IB [MeV]gnd state

-0.44

-0.435

-0.43

-0.425

-0.42

-0.415

-0.41

-0.405

-0.4

-0.395

2 4 6 8 10 12 14 16

fitdE/dmN

gnd state

1.27

1.28

1.29

1.3

1.31

1.32

1.33

1.34

1.35

1.36

1.37

1.38

2 4 6 8 10 12 14 16

fitdE/dg

πιN [l.u.]gnd state

1.96

1.98

2

2.02

2.04

2.06

2.08

2.1

2.12

2.14

2 4 6 8 10 12 14 16

fitdE dCpp [MeV]

gnd state

5.55

5.6

5.65

5.7

5.75

5.8

5.85

2 4 6 8 10 12 14 16

fitdE dCoul [MeV]

gnd state

-95

-90

-85

-80

-75

-70

-65

-60

2 4 6 8 10 12 14 16

fitold data, fitted

E [MeV], Hoyle state

-100

-95

-90

-85

-80

-75

2 4 6 8 10 12 14 16

fitE [MeV], gnd state

E(Nt) = E(∞) + const exp(−Nt/τ )

Nt Nt


62DETERMINATION of the xi

• x1 from the quark mass expansion of the nucleon mass: x1 ' 0.8± 0.2

• x2 from the quark mass expansion of the pion decay constantand the nucleon axial-vector constant: x2 ' −0.056 . . . 0.008

• x3 and x4 can be obtained from a two-nucleon scattering analysis& can be deduced from:

−∂a−1

∂Mπ

≡A

aMπ

=1

πLS′(η)

∂η

∂Mπ

, η ≡ mNE

(L

2π

)2

⇒ while this can straightforwardly be computed, we prefer to use a representationthat substitutes x3 and x4 by:

∂a−1s

∂Mπ

∣∣∣∣∣Mphysπ

,∂a−1

t

∂Mπ


⇒ we are ready to study the pertinent energy differences


63RESULTS

• putting pieces together:

∂∆Eh∂Mπ

∣∣∣∣Mphysπ

= −0.455(35)∂a−1

s

∂Mπ


− 0.744(24)∂a−1

t

∂Mπ


+ 0.056(10)

∂∆Eb∂Mπ

∣∣∣∣Mphysπ

= −0.117(34)∂a−1

s

∂Mπ


− 0.189(24)∂a−1

t

∂Mπ


+ 0.012(9)

∂∆Ec∂Mπ

∣∣∣∣Mphysπ

= − 0.07(3)∂a−1

s

∂Mπ


− 0.14(2)∂a−1

t

∂Mπ


+ 0.017(9)

• x1 and x2 only affect the small constant terms

• also calculated the shifts of the individual energies (not shown here)


64INTERPRETATION

• (∂∆Eh/∂Mπ)/(∂∆Eb/∂Mπ) ' 4

⇒ ∆Eh and ∆Eb cannot be independently fine-tuned

•Within error bars, ∂∆Eh/∂Mπ & ∂∆Eb/∂Mπ appear unaffectedby the choice of x1 and x2→ indication for α-clustering

• For ∆Eh & ∆Eb, the dependence on Mπ is small when

∂a−1s /∂Mπ ' −1.6× ∂a−1

t /∂Mπ

• the triple alpha process is controlled by :∆Eh+b ≡ ∆Eh + ∆Eb = E?12 − 3E4

∂∆Eh+b

∂Mπ

∣∣∣∣Mphysπ

= −0.571(14)∂a−1

s

∂Mπ


− 0.934(11)∂a−1

t

∂Mπ


+ 0.069(6)

⇒ so what can we say about the quark mass dependence of the scattering lengths?


65CONSTRAINTS on the SCATTERING LENGTHS

• Quark mass dependence of hadron properties:δOH

δmf

≡ KfH

OH

mf

, f = u, d, s

• NN scattering lengths as a function ofMπ: −∂a−1

s,t

∂Mπ

≡As,t

as,tMπ

, As,t ≡Kqas,t

Kqπ

• earlier determinations from chiral EFT at NLOBeane, Savage (2003), Epelbaum, Glockle, UGM (2003)

• new determination at NNLO: Epelbaum et al. (2012)

Kqas

= 2.3+1.9−1.8 , K

qat

= 0.32+0.17−0.18 →

∂a−1t

∂Mπ

= −0.18+0.10−0.10 ,

∂a−1s

∂Mπ

= 0.29+0.25−0.23

• note the magical central value:∂a−1

s /∂Mπ

∂a−1t /∂Mπ

' −1.6+1.0−1.7


66CORRELATIONS

• vary the quark mass derivatives of a−1s,t within −1, . . . ,+1:

-6 -4 -2 0 2 4 6K

E4

π

-6

-4

-2

0

2

4

6

KE*12π

KE12π

KE8π

K∆Ecπ

-6 -4 -2 0 2 4 6K

E4

π

-600

-400

-200

0

200

400

600

K∆Ebπ

K∆Ehπ

K∆Eh+bπ

• clear correlations: α-particle BE and the energies/energy differences

⇒ anthropic or non-anthropic scenario depends on whether the 4He BE moves!


67THE END-OF-THE-WORLD PLOT• |δ(∆Eh+b)| < 100 keV

→∣∣∣∣(0.571(14)As + 0.934(11)At − 0.069(6)

)δmq

mq

∣∣∣∣ < 0.0015

As,t ≡∂a

−1s,t

∂Mπ

∣∣∣∣M

physπ


68


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