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

Transcription

1 1 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 Freitag, 15. November 2013, Uhr c.t. im Hörsaal I des Physikalischen Instituts, Nussallee 12, Bonn 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.

2 2 Nuclear Theory: A Modern Perspective Ulf-G. Meißner, Univ. Bonn & FZ Jülich NLEFT 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

3 3 CONTENTS Some basic facts Ab initio calculation of atomic nuclei The fate of carbon-based life Towards medium-mass nuclei Summary & outlook

4 4 Some basic facts

5 STRUCTURE FORMATION in QCD 5 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 0.5 α s (Q) Data Theory Deep Inelastic Scattering e + e - Annihilation Hadron Collisions Heavy Quarkonia NLO NNLO Λ(5) MS α s(μ Z) 245 MeV QCD O(α MeV s) { 181 MeV Q [GeV] Lattice So how are these strongly interacting composites generated? How sensitive are they to changes in the fundamental parameters of QCD+QED?

6 THE NUCLEAR LANDSCAPE: AIMS & METHODS 6 Theoretical methods: Lattice QCD: A = 0,1,2,... NCSM, Faddeev-Yakubowsky, GFMC,... : A = 3-16 coupled cluster,...: A = density functional theory,...: A 100 Chiral EFT: provides accurate NN and 3N forces successfully applied in light nuclei with A = 2, 3, 4 combine with simulations to get to larger A Nuclear Lattice Simulations

7 7 Ab initio calculations of atomic nuclei

8 Ingredients 8 Nuclear binding is shallow: E/A 8 MeV Nuclei can be calculated from the A-body Schrödinger equation: HΨ A = EΨ A Forces are of (dominant) two- and (subdominant) three-body nature: V = V NN + V NNN can be calculated systematically and to high-precision Weinberg, van Kolck, Epelbaum, M., Entem, Machleidt,... fit all parameters in V NN + V NNN from 2- and 3-body data exact calc s of systems with A 4 using Faddeev-Yakubowsky machinery connection to boson-exchange models (as in Machleidt s talk)? slide see fig. But how about ab initio calculations for systems with A 5?

9 Examples dσ/dω [mb/sr] (N) K X X (N) EGM N3LO EM N3LO CD Bonn 2000 Gross, Stadler Nijmegen PWA K Y Y θ [deg] A y 0,3 0, , θ [deg] ,1 dσ/dω [mb/sr] T MeV T MeV dσ/dω [mb/sr] ,2 0-0,2-0,4 (N) K Z X ,07 T 21 0,02-0,03-0, θ [deg] T θ [deg] 0-0,1-0, CM [deg]

10 CONNECTION to BOSON EXCHANGE MODELS 10 Epelbaum, M., Glöckle, Elster, Phys. Rev. C65 (2002) [nucl-th/ ] Basic idea: Expand meson-exchange in powers of t/m 2 R M = and map on 4N operators 1.5 g 2 t M 2 R = g2 M 2 R g2 t M 4 R +... Bonn-B CD-Bonn Nijm-93 Nijm-I Nijm-II AV-18 with t 0 0 t 2 works amazingly well all LECs of natural size (in units of 1/F nλm χ ) NLO NNLO -0.5 C C C C C C C C 1S0 1S0 3S1 3S1 ε1 1P1 3P0 3P1 C 3P2

11 NUCLEAR LATTICE SIMULATIONS 11 Frank, Brockmann (1992), Koonin, Müller, Seki, van Kolck (2000), Lee, Schäfer (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 = L s L s L s L t : nucleons are point-like fields on the sites p n discretized chiral potential w/ pion exchanges and contact interactions typical lattice parameters n p a Λ = a 300 MeV [UV cutoff] ~ 2 fm strong suppression of sign oscillations due to approximate Wigner SU(4) symmetry hybrid Monte Carlo & transfer matrix (similar to LQCD) J. W. Chen, D. Lee and T. Schäfer, Phys. Rev. Lett. 93 (2004)

12 CONFIGURATIONS 12 all possible configurations are sampled clustering emerges naturally perform ab initio calculations using only V NN and V NNN as input grand challenge: the spectrum of 12 C

13 SPECTRUM OF 12 C & the HOYLE STATE 13 Epelbaum, Krebs, Lee, UGM, Phys. Rev. Lett. 106 (2011) Epelbaum, Krebs, La hde, Lee, UGM, Phys. Rev. Lett. 109 (2012) Viewpoint: Hjorth-Jensen, Physics 4 (2011) 38 Nuclear Theory: A Modern Perspective Ulf-G. Meißner Bonn, November 2013 C < O > B

14 THE TRIPLE-ALPHA PROCESS MOVIE 14 c ANU the 8 Be 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 decays produce stable carbon) carbon is further turned into oxygen but w/o a resonant condition a triple wonder!

15 RESULTS 15 some groundstate energies and differences E [MeV] NLEFT Exp. 3 He - 3 H 0.78(5) He 28.3(6) Be 55(2) C 92(3) 92.2 E3 He E triton (MeV) lattice physical (infinite volume) promising results L (fm) excited states more difficult new projection MC method

16 The SPECTRUM of CARBON After hrs JUGENE/JUQUEEN (and some human work) E [MeV] (1) (3) (3) (2) First ab initio calculation of the Hoyle state Hoyle Structure of the Hoyle state: Exp (3) Th

17 17 The fate of carbon-based life as a function of the fundamental parameters of QCD+QED

18 FINE-TUNING of FUNDAMENTAL PARAMETERS 18

19 EARLIER STUDIES of the ANTHROPIC PRINCIPLE 19 rate of the 3α-process: r 3α ( Nα kt ) 3 ( Γ γ exp E ) kt E = E 12 3E α = (18) kev too few 16 O too few 12 C how much can E be changed so that there is still enough 12 C and 16 O? E 100 kev Oberhummer et al., Science 289 (2000) 88 Csoto et al., Nucl. Phys. A 688 (2001) 560 Schlattl et al., Astrophys. Space Sci. 291 (2004) 27 [Livio et al., Nature 340 (1989) 281]

20 FINE-TUNING: MONTE-CARLO ANALYSIS 20 Epelbaum, Krebs, Lähde, Lee, UGM, PRL 110 (2013) ; Eur.Phys.J. A49 (2013) 82 consider first QCD only calculate E/ M relevant quantities (energy differences) E h E 12 E 8 E 4, E b E 8 2E 4 energy differences depend on parameters of QCD (LO analysis) ) E i = E i (M OPE, m N (M ), g N (M ), C 0 (M ), C I (M ) g N g A 2F remember: M 2 ± (m u + m d ) Gell-Mann Oakes Renner (1968) quark mass dependence pion mass dependence

21 PION MASS VARIATIONS 21 consider pion mass changes as small perturbations with E i M M phys x 1 m N M = E i M OPE M phys M phys, x 2 g N M E + x i 1 m N + x 3 E i C 0 M phys m phys N C phys 0, x 3 C 0 M E + x i 2 g N + x 4 E i C I M phys g phys N C phys I, x 4 C I M M phys problem reduces to the calculation of the various derivatives using AFQMC and the determination of the x i x 1 and x 2 can be obtained from LQCD plus CHPT x 3 and x 4 can be obtained from two-body scattering and its M -dependence

22 VISUALIZATION of the PION MASS VARIATIONS 22 At LO, one-pion exchange and 4N contact terms pion nucleon coupling g(m ) pion propagator 1/(q 2 M ) 2 Ā s,t a 1 s,t M M phys four nucleon couplings C(M ) nucleon mass m N (M )

23 CORRELATIONS 23 vary the quark mass derivatives of a 1 s,t within 1,..., +1: E b = E( 8 Be) 2E( 4 He) E h = E( 12 C ) E( 8 Be) E( 4 He) E h+b = E( 12 C ) 3E( 4 He) K Eb K Eh K Eh+b O H M = K H O H M K E4 clear correlations: α-particle BE and the energies/energy differences anthropic or non-anthropic scenario depends on whether the 4 He BE moves!

24 THE END-OF-THE-WORLD PLOT 24 δ( E h+b ) < 100 kev ( ) δmq 0.571(14)Ā s (11)Ā t 0.069(6) m q < Ā s,t a 1 s,t M M phys The light quark mass is fine-tuned to 2 3 % Similarly: α EM is fine-tuned to 2.5% Berengut et al., Phys. Rev. D 87 (2013)

25 25 Towards medium-mass nuclei

26 GOING up the ALPHA CHAIN 26 Consider the α ladder 12 C, 16 O, 20 Ne, 24 Mg, 28 Si as t CPU A 2 Improved multi-state technique to extract g.s. energies higher A, better accuracy overbinding at LO beyond A = 12 persists up to NNLO E 16 (LO) [MeV] e-5-6.5e-5-5.5e-5-4.5e-5 E 20 (LO) [MeV] e-5-6.5e-5-5.5e-5-4.5e O 20 Ne 24 Mg 28 Si E 24 (LO) [MeV] e-5-6.5e-5-5.5e-5-4.5e-5 E 28 (LO) [MeV] e-5-4.5e-5-3.5e N t N t N t N t E = 131.3(5) E = 165.9(9) E = 232(2) E = 308(3) [ ] [ ] [ ] [ ]

27 REMOVING the OVERBINDING 27 Overbinding is due to four α clusters in close proximity Lähde et al., arxiv: [nucl-th] remove this by an effective 4N operator [long term: N3LO] V (4N eff ) = D (4N eff ) ρ( n 1 )ρ( n 2 )ρ( n 3 )ρ( n 4 ) 1 ( n i n j ) 2 2 fix the coefficient D (4N eff ) from the BE of 24 Mg excellent description of the g.s. energies A Th 90.3(2) 131.3(5) 165.9(9) 198(2) 233(3) Exp

28 GROUND STATE ENERGIES 28 Experiment NNLO NNLO + 4N eff 4 He 8 Be 12 C 16 O 20 Ne 24 Mg 28 Si E (MeV)

29 SUMMARY & OUTLOOK 29 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) 12 C spectrum at NNLO Hoyle state and its structure Fine-tuning of m quark and α EM viability of life changes in m quark of about 2% and in α EM of about 2.5% are allowed First ab initio results for medium mass nuclei the strong interactions remain a challenge

30 30 SPARES

31 The RELEVANT QUESTION 31 Date: Sat, 25 Dec :03: From: Steven Weinberg To: Ulf-G. Meissner Subject: Re: Hoyle state in 12C E (MeV) J P = J P = J z = 0, J P = J z = 2, J P = Dear Professor Meissner, -100 Thanks for the colorful graph. It makes a nice Christmas -110 card. But I have a detailed question. Suppose you calculate not only the energy LO NLO EM+IB NNLO Exper. of the Hoyle state in C12, but also of the ground states of He4 and Be8. How sensitive is the result that the energy of the Hoyle state is near the sum of the rest energies of He4 and Be8 to the parameters of the theory? I ask because I suspect that for a pretty broad range of parameters, the Hoyle state can be well represented 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!

32 The NON-ANTHROPIC SCENARIO 32 Weinberg s assumption: The Hoyle state stays close to the 4He+8Be threshold energy difference Hoyle 4He+8Be g g g g+ g fundamental parameter

33 The ANTHROPIC SCENARIO 33 The AP strikes back: The Hoyle state moves away from the 4He+8Be threshold energy difference Hoyle 4He+8Be g g g g+ g fundamental parameter

34 EARLIER STUDIES of the AP 34 By hand modification of the energy diff. & network calcs in massive stars 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 Livio et al., Nature 340 (1989) 281 Changing NN and em interactions in a microscopic model & network calcs Oberhummer 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 [α QED ] would destroy all carbon (oxygen) in stars should be of interest to AP considerations

35 35 Introduction II: Effective Field Theory for Nuclear Physics only a brief reminder details in E. Epelbaum, H.-W. Hammer, UGM, Rev. Mod. Phys. 81 (2009) 1773 [arxiv: [nucl-th]]

36 CHIRAL EFT FOR FEW-NUCLEON SYSTEMS 36 Gasser, Leutwyler, Weinberg, van Kolck, Epelbaum, Bernard, Kaiser, UGM,... Scales in nuclear physics: Natural: λ = 1/M 1.5 fm (Yukawa 1935) Vc (r) [MeV] 100 repulsive core 0 Unnatural: a np ( 1 S 0 ) = 23.8 fm, a np ( 3 S 1 ) = 5.4 fm 1/M -100 CD Bonn Reid93 AV this can be analyzed in a suitable EFT based on L QCD L EFF = L + L N + L NN +... pion and pion-nucleon sectors are perturbative in Q/Λ χ chiral perturbation th y L NN collects short-distance contact terms, to be fitted NN interaction requires non-perturbative resummation chirally expand V NN(N), use in regularized LS/FY equation

37 CHIRAL POTENTIAL and NUCLEAR FORCES 37 2 LECs 7 LECs 15 LECs 2 LECs O((Q/Λ χ ) 0 ) O((Q/Λ χ ) 2 ) O((Q/Λ χ ) 3 ) O((Q/Λ χ ) 4 ) explains naturally the observed hierarchy of nuclear forces MANY successfull tests in few-nucleon systems (continuum calc s)

38 38 Nuclear lattice simulations Formalism

39 NUCLEAR LATTICE SIMULATIONS 39 Frank, Brockmann (1992), Koonin, Müller, Seki, van Kolck (2000), Lee, Schäfer (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 = L s L s L s L t : nucleons are point-like fields on the sites p n discretized chiral potential w/ pion exchanges and contact interactions typical lattice parameters n p a Λ = a 300 MeV [UV cutoff] ~ 2 fm strong suppression of sign oscillations due to approximate Wigner SU(4) symmetry hybrid Monte Carlo & transfer matrix (similar to LQCD) J. W. Chen, D. Lee and T. Schäfer, Phys. Rev. Lett. 93 (2004)

40 CONFIGURATIONS 40 all possible configurations are sampled clustering emerges naturally

41 TRANSFER MATRIX METHOD 41 Correlation function for A nucleons: Z A (t) = Ψ A exp( th) Ψ A with Ψ A a Slater determinant for A free nucleons Euclidean time Ground state energy from the time derivative of the correlator E A (t) = d dt ln Z A(t) ground state filtered out at large times: E 0 A = lim t E A(t) Expectation value of any normal ordered operator O Z O A = Ψ A exp( th/2) O exp( th/2) Ψ A lim t Z O A (t) Z A (t) = Ψ A O Ψ A

42 TRANSFER MATRIX CALCULATION 42 Expectation value of any normal ordered operator O Ψ A O Ψ A = lim t Ψ A exp( th/2) O exp( th/2) Ψ A Ψ A exp( th) Ψ A Anatomy of the transfer matrix operator insertion for expectation value Ψ free Z,N SU(4) full LO O full LO SU(4) Ψ free Z,N 2L to + L ti L to + L ti L to + L ti /2 L to 0 { inexpensive filter { inexpensive filter

43 PROJECTION MONTE CARLO TECHNIQUE 43 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 12 C

44 MONTE CARLO with AUXILIARY FILEDS 44 Contact interactions represented by auxiliary fields s, s I exp(ρ 2 /2) + ds exp( s 2 /2 sρ), ρ N N Correlation function = path-integral over pions & auxiliary fields p p

45 COMPUTATIONAL EQUIPMENT 45 Past = JUGENE (BlueGene/P) Present = JUQUEEN (BlueGene/Q)

46 46 Nuclear lattice simulations Results nuclei neutron matter

47 FIXING PARAMETERS & FIRST PREDICTIONS 47 work at NNLO including strong and em isospin breaking 1 S0 3 S1 9 NN LECs from np scattering and Q d 2 LECs for isospin-breaking (np, pp, nn) 2 LECs D, E related to the leading 3NF make predictions δ( 1 S 0 ) (degrees) LO 3 NLO 3 PWA93 (np) p CM (MeV) δ( 3 S 1 ) (degrees) 180 LO 3 NLO PWA93 (np) p CM (MeV) 1 S0 pp vs np scattering nd spin-3/2 quartet channel... δ( 1 S 0 ) (degrees) LO NLO + IB + EM PWA93 (pp) p cot δ (fm -1 ) p-d (exp.) n-d (exp.) LO NLO NNLO p CM (MeV) p 2 (fm -2 )

48 48 Ground states Epelbaum, Krebs, Lähde, Lee, UGM, arxiv:

49 PREDICTIONS: TRITON & HELIUM-3 49 Epelbaum, Krebs, Lee, UGM, Phys. Rev. Lett. 104 (2010) ; Eur. Phys. J. A 45 (2010) 335 binding energies of 3N systems: E(L) = B.E. a L exp( bl) predict the energy difference E( 3 He) E( 3 H) see also Hammer, Kreuzer (2011) lattice physical (infinite volume) E3 He E triton (MeV) (5) MeV 0.76 MeV [exp.] L (fm)

50 50 Ground state of 4 He L = 11.8 fm E(t) (MeV) (A) 6 LO t (MeV -1 ) (B) NLO-IS IB + EM NNLO t (MeV -1 ) LO (O(Q 0 )) NLO (O(Q 2 )) NNLO (O(Q 3 )) Exp. 28.0(3) MeV 24.9(5) MeV 28.3(6) MeV 28.3 MeV

51 51 Ground state of 8 Be L = 11.8 fm E(t) (MeV) (A) LO (B) NLO-IS IB + EM NNLO t (MeV -1 ) t (MeV -1 ) LO (O(Q 0 )) NLO (O(Q 2 )) NNLO (O(Q 3 )) Exp. 57(2) MeV 47(2) MeV 55(2) MeV 56.5 MeV

52 52 Ground state of 12 C L = 11.8 fm E(t) (MeV) (A) LO (1) LO (2) (B) NLO-IS (1) NLO-IS (2) IB + EM (1) IB + EM (2) NNLO (1) NNLO (2) t (MeV -1 ) t (MeV -1 ) LO (O(Q 0 )) NLO (O(Q 2 )) NNLO (O(Q 3 )) Exp. 96(2) MeV 77(3) MeV 92(3) MeV 92.2 MeV

53 53 Ground state of 16 O L = 11.8 fm E(t) (MeV) (A) LO (1) LO (2) (B) NLO-IS (1) NLO-IS (2) IB + EM (1) IB + EM (2) NNLO (1) NNLO (2) to be published t (MeV -1 ) t (MeV -1 ) LO (O(Q 0 )) NLO (O(Q 2 )) NNLO (O(Q 3 )) Exp. 144(4) MeV 116(6) MeV 135(6) MeV MeV

54 EXCITED STATES of 12 C 54 Lowest excited state is (as in nature) E(2 + 1 ) = 89(3) MeV [ 87.7 MeV] (A) (B) 1 0 LO NLO-IS IB + EM NNLO log [Z(t)/Z (t)] E(t) (MeV) t (MeV -1 ) t (MeV -1 )

55 THE HOYLE STATE (0 + 2 ) 55 energy: E(0 + 2 ) = 85(3) MeV close to E( 4 He) + E( 8 Be) = 83.3(2.0) MeV structure: bent alpha-chain like (not BEC ) (A) LO (1) LO (2) (B) NLO-IS (1) IB + EM (1) NNLO (1) E(t) (MeV) t (MeV -1 ) t (MeV -1 )

56 A HOYLE STATE EXCITATION (2 + 2 ) 56 a 2 + state 2 MeV above the Hoyle state interpretation: a rotational band of the Hoyle state generated from excitations of the alpha-chain what s in the data? log [Z(t)/Z (t)] (A) LO t (MeV -1 ) E(t) (MeV) (B) NLO-IS IB + EM NNLO a 2 + state 3.51 MeV above the Hoyle state seen in 11 B(d, n) 12 C not included in the level scheme! Ajzenberg-Selove, Nucl. Phys. A506 (1990) 1 t (MeV -1 ) a 2 + state 3.8(4) MeV above the Hoyle state seen in 12 C(α, α) 12 C and much more, see next slide and: talk by Henry Weller Bency John et al., Phys. Rev. C 68 (2003) ab initio prediction requires experimental confirmation

57 SPECTRUM OF 12 C 57 Summarizing the results for carbon-12: LO 96(2) MeV 94(2) MeV 89(2) MeV 88(2) MeV NLO 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 MeV MeV 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) [2] Zimmermann et al., Phys. Rev. C 84 (2011) [3] Hyldegaard et al., Phys. Rev. C 81 (2010) [4] Itoh et al., Phys. Rev. C 84 (2011) [5] Weller et al., in preparation importance of consistent 2N & 3N forces good agreement w/ experiment, can be improved

58 58 Testing the Anthropic Principle

59 MC ANALYSIS of the AP 59 consider QCD only calculate E/ M relevant quantities (energy differences) E h E 12 E 8 E 4, E b E 8 2E 4 E c E 12 E 12 energy differences depend on parameters of QCD (LO analysis) ) E i = E i (M OPE, m N (M ), g N (M ), C 0 (M ), C I (M ) remember: M 2 ± (m u + m d ) g N g A 2F quark mass dependence pion mass dependence

60 PION MASS VARIATIONS 60 consider pion mass changes as small perturbations with E i M M phys x 1 m N M = E i M OPE M phys M phys, x 2 g N M E + x i 1 m N + x 3 E i C 0 M phys m phys N C phys 0, x 3 C 0 M E + x i 2 g N + x 4 E i C I M phys g phys N C phys I, x 4 C I M M phys problem reduces to the calculation of the various derivatives using AFQMC and the determination of the x i x 1 and x 2 can be obtained from LQCD plus CHPT x 3 and x 4 can be obtained from two-body scattering and its M -dependence

61 AFQMC RESUTS for the DERIVATIVES 61 4 He 12 C(0 + 2 ) E(N t ) = E( ) + const exp( N t /τ ) fit old data E [MeV] fit de dc pp [MeV] not fitted fit old data, fitted E [MeV], Hoyle state fit E [MeV], gnd state fit de dc pp [MeV] gnd state fit de/dc ii [l.u.] not fitted fit de/dg ιn [l.u.] not fitted fit de dcoul [MeV] not fitted fit -8.1 de/dc ii [l.u.] gnd state fit de/dg ιn [l.u.] gnd state fit de dcoul [MeV] gnd state fit de/dc 11 [l.u.] not fitted fit de/dm N not fitted fit de pion IB [MeV] not fitted fit de/dc 11 [l.u.] gnd state fit de/dm N gnd state fit de pion IB [MeV] gnd state fit de/dm (OPE) not fitted fit de pion dm [MeV] not fitted fit de/dm (OPE) gnd state fit de pion dm [MeV] gnd state N t N t

62 DETERMINATION of the x i 62 x 1 from the quark mass expansion of the nucleon mass: x ± 0.2 x 2 from the quark mass expansion of the pion decay constant and the nucleon axial-vector constant: x x 3 and x 4 can be obtained from a two-nucleon scattering analysis & can be deduced from: a 1 M A am = 1 L S (η) η M, η m N E ( L while this can straightforwardly be computed, we prefer to use a representation that substitutes x 3 and x 4 by: a 1 s a 1 t, M M we are ready to study the pertinent energy differences M phys 2 ) 2 M phys

63 RESULTS 63 putting pieces together: E h M M phys = 0.455(35) a 1 s M M phys 0.744(24) a 1 t M M phys (10) E b M M phys = 0.117(34) a 1 s M M phys 0.189(24) a 1 t M M phys (9) E c M M phys = 0.07(3) a 1 s M M phys 0.14(2) a 1 t M M phys (9) x 1 and x 2 only affect the small constant terms also calculated the shifts of the individual energies (not shown here)

64 INTERPRETATION 64 ( E h / M )/( E b / M ) 4 E h and E b cannot be independently fine-tuned Within error bars, E h / M & E b / M appear unaffected by the choice of x 1 and x 2 indication for α-clustering For E h & E b, the dependence on M is small when a 1 s / M 1.6 a 1 t / M the triple alpha process is controlled by : E h+b E h + E b = E 12 3E 4 E h+b M M phys = 0.571(14) a 1 s M M phys 0.934(11) a 1 t M M phys (6) so what can we say about the quark mass dependence of the scattering lengths?

65 CONSTRAINTS on the SCATTERING LENGTHS 65 Quark mass dependence of hadron properties: δo H δm f K f H O H m f, f = u, d, s NN scattering lengths as a function of M : a 1 s,t A s,t, M a s,t M A s,t K q a s,t K q earlier determinations from chiral EFT at NLO Beane, Savage (2003), Epelbaum, Glöckle, UGM (2003) new determination at NNLO: Epelbaum et al. (2012) K q a s = , Kq a t = a 1 t M = , a 1 s = M note the magical central value: a 1 s / M a t / M

66 CORRELATIONS 66 vary the quark mass derivatives of a 1 s,t within 1,..., +1: K E*12 K E12 K E8 K Ec K E K Eb K Eh K Eh+b K E4 clear correlations: α-particle BE and the energies/energy differences anthropic or non-anthropic scenario depends on whether the 4 He BE moves!

67 THE END-OF-THE-WORLD PLOT 67 δ( E h+b ) < 100 kev ( ) δmq 0.571(14)Ā s (11)Ā t 0.069(6) m q < Ā s,t a 1 s,t M M phys

68