The Hoyle state and the fate of carbon-based life -...

1

The Hoyle state andthe fate of carbon-based life

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 05P12PDFTE and by HGF VIQCD VH-VI-417

Nuclear Astrophysics Virtual Institute

NLEFT

– Ulf-G. Meißner, The Hoyle state and the fate of carbon-based life – NC State, August 2013 · C < ∧ O > B •

2

CONTENTS

• Intro I: Element generation in stars

• Intro II: The anthropic principle

• Ab initio calculation of atomic nuclei

• Nuclear lattice simulations: results

• The fate of carbon-based life as a function of fundamental parameters

• Summary & outlook


3

Element generationin stars


4FIRST STEPS: PROTON-PROTON FUSION etc

• Elements are generated in the Big Bang & starsthrough the fusion of protons & nuclei

• All is simple until 4He

• Simply adding further protons does not work

• So how are the life-essential elementslike 12C and 16O generated?

• BTW: nuclei make up the visible matter in the Universe


5A SHORT HISTORY of the HOYLE STATE

• Heavy element generation in massive stars: triple-α processBethe 1938, Opik 1952, Salpeter 1952, Hoyle 1954, . . .

4He + 4He 8Be8Be + 4He 12C∗ →12C + γ12C + 4He 16O + γ

• Hoyle’s contribution: calculation of relative abundances of 4He, 12C and 160⇒ need a resonance close to the 8Be + 4He threshold at ER = 0.35 MeV⇒ this corresponds to a JP = 0+ excited state 7.7 MeV above the g.s.

• a corresponding state was experimentally confirmed at Caltech atE − E(g.s.) = 7.653± 0.008 MeV Dunbar et al. 1953, Cook et al. 1957

• still on-going experimental activity, e.g. EM transitions at SDALINACM. Chernykh et al., Phys. Rev. Lett. 98 (2007) 032501

• and how about theory ?→ this talk

• side remark: NOT driven by anthropic considerationsH. Kragh, Arch. Hist. Exact Sci. 64 (2010) 721


6THE 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 !


7

AN ENIGMA for NUCLEAR THEORY

• Ab initio calculation in the no-core shell model: ≈ 107 CPU hrs on JAGUARP. Navratil et al., Phys. Rev. Lett. 99 (2007) 042501; R. Roth et al., Phys. Rev. Lett. 107 (2011) 072501

0

4

8

12

16

NN+NNN Exp NN

0 +

0 +

2 +

2 +

1 +

1 +

4 +

4 +

1 +; 1

1 +; 1

2 +; 1

2 +; 1

0 +; 1

0 +; 1

0

4

8

12

16

NN+NNN Exp NN

1/2- 1/2

-

3/2-

3/2-

5/2-

5/2-

1/2-

1/2-

3/2-

3/2-

7/2-

7/2-

3/2-; 3/2

3/2-; 3/2

12C 13

C

0+??

⇒ excellent description, but no trace of the Hoyle state


8

The anthropicprinciple


9THE ANTHROPIC PRINCIPLE

• The anthropic principle:

“The observed values of all physical and cosmological quantities are notequally probable but they take on values restricted by the requirement thatthere exist sites where carbon-based life can evolve and by the requirementsthat the Universe be old enough for it to have already done so.”

Carter 1974, Barrow & Tippler 1988, . . .

⇒ can this be tested? / have physical consequences?

• Ex. 1: “Anthropic bound on the cosmological constant” Weinberg (1987)[550 cites]

• Ex. 2: “The anthropic string theory landscape” Susskind (2003) [724 cites]


10

A PRIME EXAMPLE for the ANTHROPIC PRINCIPLE

• Hoyle (1953):Prediction of an excited level in carbon-12 to allow for a sufficient production of heavyelements (12C, 16O,...) in stars

• was later heralded as a prime example for the AP:

“As far as we know, this is the only genuine anthropic principle prediction”Carr & Rees 1989

“In 1953 Hoyle made an anthropic prediction on an excited state – ‘level of life’ –for carbon production in stars” Linde 2007

“A prototype example of this kind of anthropic reasoning was provided byFred Hoyle’s observation of the triple alpha process...” Carter 2006


11

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 move 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


12The 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


13The 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


14

EARLIER STUDIES of the AP

• rate of the 3α-process: r3α ∼(Nα

kBT

)3

Γγ exp

(−

∆E

kBT

)∆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


15

Ab initio calculationsof atomic nuclei


16EMERGENCE of STRUCTURE in QCD

• The strong interactions are described by QuantumChromoDynamics:

LQCD = −1

4g2GµνG

µν +∑

f=u,d,s,c,b,t

qf (iγµDµ −mf) qf

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

• Quarks and gluons are confined within hadrons

• Protons and neutrons form atomic nuclei

⇒ This requires the inclusion of electromagnetismdescribed by QED with αEM ' 1/137

⇒ Atomic nuclei make up the visible matter in the Universe

So how are these strongly interacting composites generated?

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


17Ingredients

• Nuclear binding is shallow: E/A ≤ 8 MeV mN = 940 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, UGM, Entem, Machleidt, . . .

see slides

⇒ 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.

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


18CHIRAL 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 Schrodinger 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]


19

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


20Examples

• np scattering • nd scattering

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

• pol. transfer in pd scattering

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)

Epelbaum, Hammer, UGM,Rev. Mod. Phys. 81 (2009) 1773


21NUCLEAR LATTICE SIMULATIONSFrank, Brockmann (1992), Koonin, Muller, Seki, van Kolck (2000) , Lee, Borasoy, Schafer, Phys.Rev. C70 (2004) 014007, . . .

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)


22

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


23

COMPUTATIONAL EQUIPMENT

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

6 Pflops

6×1000000000000000 flops


24SPECTRUM 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


25RESULTS• fix parameters from 2N scattering and two 3N observables [NNLO: 9+2]

• 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) −92.216O −141(1) −127.6

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 [3NFs very important]

• excited states more difficult

⇒ new projection MC method [large class of initial wfs]

preliminary


26The 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:


27SPECTRUM of 12C• Summarizing the results for carbon-12 at NNLO:

0+1 2+

1 0+2 2+

2

2N −77 MeV −74 MeV −72 MeV −70 MeV3N −15 MeV −15 MeV −13 MeV −13 MeV

2N+3N −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] Zimmermann et al., arXiv:1303.4326 [nucl-ex]

• importance of consistent 2N & 3N forces

• good agreement w/ experiment, can be improved


28

The fate of carbon-based lifeas a function of the quark mass


29FINE-TUNING of FUNDAMENTAL PARAMETERSFig. courtesy Dean Lee


30FINE-TUNING: MONTE-CARLO ANALYSISEpelbaum, Krebs, Lahde, Lee, UGM, PRL 110 (2013) 112502

• consider first 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 ≡ gA/(2Fπ)

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

⇒ quark mass dependence ≡ pion mass dependence


31PION 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


32

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


33

AFQMC RESUTS for the DERIVATIVES

• 4He • 12C(0+1 , 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


34DETERMINATION 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


35RESULTS

• 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)


36INTERPRETATION

• (∂∆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?


37CONSTRAINTS 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 of Mπ: −∂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: Berengut et al., Phys. Rev. D 87 (2013) 085018

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

• leads also to constraints from Big Bang Nucleosynthesis!


38CORRELATIONS

• 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

⇒ the anthropic view of the Universe depends on whether the 4He BE moves!


39THE END-OF-THE-WORLD PLOT• |δ(∆Eh+b)| < 100 keV Schlattl et al. (2004)

→∣∣∣∣(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


40

SUMMARY & OUTLOOK

• Chiral nuclear EFT: best approach to nuclear forces and few-body systems

• Many successes in the description of few-nucleon dynamics (A = 2, 3, 4)

• 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 2-3 % and in αEM of about 2.5% are allowed

• Results for 16O, 20Ne, 24Mg, 28Si and 32S forthcoming → fig.

• must improve and extend these calculations, e.g. α+12C→16O+γ Lee et al.

⇒ the strong interactions remain a challenge


41ALPHA-CLUSTER NUCLEI

• Preliminary results for A = 16, 20, 24, 28

28Si

24Mg

20Ne

16O

12C

8Be

4He

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

E (MeV)

ExperimentNNLO [O(Q3)]

+ smeared 4N correction (estimated)

tCPU = const×A2

PRELIMINARY


42

SPARES


43

Nuclear lattice simulations– Formalism –


44

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


45

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


46

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


47

PROJECTION MONTE CARLO TECHNIQUE

• General wave function:

ψj(~n) , j = 1, . . . , A

• States with well-defined momentum:

L−3/2∑~m

ψj(~n+ ~m) exp(i ~P · ~m) , j = 1, . . . , A

• 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

shell-model type cluster typeψj(~n) = exp[−c~n2] ψj(~n) = exp[−c(~n− ~m)2]

ψ′j(~n) = nx exp[−c~n2] ψ′j(~n) = exp[−c(~n− ~m′)2]

ψ′′j (~n) = ny exp[−c~n2] ψ′′j (~n) = exp[−c(~n− ~m′′)2]

ψ′′′j (~n) = nz exp[−c~n2] ψ′′′j (~n) = exp[−c(~n− ~m′′′)2]

• shell-model w.f.s do not have enough 4N correlations ∼ 〈(N†N)2〉


48

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

⇒


49

Nuclear lattice simulations– Results –

nuclei neutron matter


50

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)


51

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


52PREDICTIONS: 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


53

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


54

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


55

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)




56

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


57

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


58THE 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)


59A 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


60EARLIER 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 change consistent with carbon-oxygen based life

→ 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

→ a ±100 keV change of ε = EHoyle − 3Eα = 380 keV can be tolerated

→ “should be of interest to AP considerations”


61


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