Testing the Anthropic Principle with Lattice Simulations ... · PDF fileA SHORT HISTORY of the HOYLE STATE 7 ... Suppose you calculate not only the energy ... ,!no inﬂuence on the

1

Testing the Anthropic Principlewith Lattice Simulations

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

Testing the Anthropic Principle with Lattice Simulations – Ulf-G. Meißner – INT, Oct., 2012 · ◦ C < ∧ O > B •

2

•Nuclear Lattice Effective Field Theory collaboration

Evgeny Epelbaum (Bochum)

Hermann Krebs (Bochum)

Timo Lahde (Julich)

Dean Lee (NC State)

Ulf-G. Meißner (Bonn/Julich)


3

CONTENTS

• Introduction I: The Anthropic Principle & the Hoyle state

• Introduction II: Effective Field Theory for Nuclear Physics

• Nuclear lattice simulations: methods

• Nuclear lattice simulations: results

• How anthropic is the Hoyle state?

• Summary & outlook


4

Introduction I:The Anthropic Principle

& the Hoyle state


5THE 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)[505 cites]

• Ex. 2: “The anthropic string theory landscape” Susskind (2003) [681 cites]


6

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


7A 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 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


8THE TRIPLE-ALPHA PROCESS

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 !


9

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


10The NON-ANTHROPIC SCENARIO

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

g g+ ∆gg ∆g−

En

erg

ie−D

iffe

ren

z

4He+8BeHoyle

−84.80

−84.51

fundamentaler Parameter


11The ANTHROPIC SCENARIO

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

g g+ ∆gg ∆g−

En

erg

ie−D

iffe

ren

z

4He+8BeHoyle

−84.80

−84.51

fundamentaler Parameter


12EARLIER 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”


13

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]]


14CHIRAL 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]


15

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


16

Nuclear lattice simulations– Formalism –


17NUCLEAR 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)


18

CONFIGURATIONS

⇒ all possible configurations are sampled⇒ clustering emerges naturally


19

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


20

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


21

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


22

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

⇒


23

COMPUTATIONAL EQUIPMENT

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


24

Nuclear lattice simulations– Results –

nuclei neutron matter


25

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)


26

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


27PREDICTIONS: 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


28

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


29

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


30

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)




31

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


32SPECTRUM OF 12C & the HOYLE STATE

Epelbaum, Krebs, Lee, UGM, Phys. Rev. Lett. 106 (2011) 192501Viewpoint: Hjorth-Jensen, Physics 4 (2011) 38Epelbaum, Krebs, Lahde, Lee, UGM, arxiv:1208.1328 (numbers from this ref.)


33

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


34THE 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)


35A 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


36SPECTRUM 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


37

Testing theAnthropic Principle


38

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


39PION 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


40

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


41DETERMINATION 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


42RESULTS

• 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)


43INTERPRETATION

• (∂∆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?


44CONSTRAINTS 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


45CORRELATIONS

• 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!


46THE 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


47

SUMMARY & OUTLOOK

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

• Formulate continuum EFT on space-time lattice V = Ls × Ls × Ls × Lt

• New method to extract phase shifts & mixing angles

• Fix parameters in few-nucleon systems→ predictions

• Promising results for A = 2, 3, 4, 8, 12, 16 at NNLO

• 12C spectrum at NNLO→ Hoyle state & 2+ excitation

• First ever ab initio MC calculation of 16O

• Testing the anthropic principle→ strong correlations of α-cluster type

⇒ the Hoyle state does not appear anthropic (Coulomb to be done)

⇒ larger A and higher precision


48


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