Nuclear Forces and Their Impact on Structure, Reactions and Astrophysics

Nuclear forces and their impact on structure, reactions and astrophysics

Dick Furnstahl Ohio State University July, 2013

Lectures for Week 2

M. Chiral EFT 1 (as); χ-symmetry in NN scattering, QCD 2 (rjf) T. Chiral EFT 2 (rjf); Three-nucleon forces 1 (as) W. Renormalization group 1 (rjf); Forces from LQCD (zd) Th. Renormalization group 2 (rjf); Three-nucleon forces 2 (as) F. Many-body overview (rjf); Nuclear forces and electroweak interactions (as) SRG Overview Methods Lattice 3NF Perturbativeness Numerical Outline

SRG leftovers

Overview of nuclear many-body problem

Many-body methods

Nuclear lattice simulations

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice 3NF Perturbativeness Numerical Tjon line revisited

30 Tjon line for NN-only potentials SRG NN-only λ −1 29 SRG NN+NNN ( >1.7 fm ) λ=1.8 λ=1.5 Expt. 28 λ=2.0 λ=2.5 27 He) [MeV] λ 4 =1.2 λ=3.0 ( b E 26 28.4 3 N LO 28.3 25 (500 MeV) 28.2 8.45 8.5 24 7.6 7.8 8 8.2 8.4 8.6 8.8 3 Eb( H) [MeV]

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice 3NF Perturbativeness Numerical Weinberg eigenvalue analysis of convergence 1 1 1 Born Series: T (E) = V + V V + V V V + E H E H E H ··· − 0 − 0 − 0 For ﬁxed E, ﬁnd (complex) eigenvalues ην(E) [Weinberg] 1 V Γ = η Γ = T (E) Γ = V Γ (1 + η + η2 + ) E H | ν i ν | ν i ⇒ | ν i | ν i ν ν ··· − 0 = T diverges if any η (E) 1 [nucl-th/0602060] ⇒ | ν | ≥ Im η Im η 1 3 3 S0 S1− D1 1 1

−3 −2 −1 1 −3 −2 −1 1 Re η Re η

−1 −1 AV18 CD-Bonn 3 AV18 N LO CD-Bonn −2 −2 3 N LO (500 MeV)

Dick Furnstahl TALENT: Nuclear forces −3 −3 Pauli blocking in nuclear matter increases it even more! at Fermi surface, pairing revealed by η > 1 | ν |

SRG Overview Methods Lattice 3NF Perturbativeness Numerical Lowering the cutoff increases “perturbativeness” Weinberg eigenvalue analysis (repulsive) [nucl-th/0602060]

Im η 1 0 S 1 0 1 S0 −0.2 −3 −2 −1 1 Re η −0.4 (E=0)

−1 -1 ν Λ = 10 fm η −0.6 -1 Λ = 7 fm Argonne v18 -1 Λ = 5 fm 2 -1 N LO-550/600 [19] −2 Λ = 4 fm 3 -1 −0.8 Λ = 3 fm N LO-550/600 [14] -1 3 Λ = 2 fm N LO [13] 3 N LO −1 −3 1.5 2 2.5 3 3.5 4 -1 Λ (fm )

Dick Furnstahl TALENT: Nuclear forces Pauli blocking in nuclear matter increases it even more! at Fermi surface, pairing revealed by η > 1 | ν |

SRG Overview Methods Lattice 3NF Perturbativeness Numerical Lowering the cutoff increases “perturbativeness” Weinberg eigenvalue analysis (repulsive) [nucl-th/0602060]

Im η 3 3 0 S − D 3 3 1 1 1 S1− D1

−3 −2 −1 1 −0.5 Re η

−1 (E=0) −1 ν

-1 η Λ = 10 fm Argonne v -1 18 Λ = 7 fm 2 -1 N LO-550/600 Λ = 5 fm −1.5 3 -1 −2 Λ = 4 fm N LO-550/600 -1 3 Λ = 3 fm N LO [Entem] -1 Λ = 2 fm 3 N LO −2 −3 1.5 2 2.5 3 3.5 4 -1 Λ (fm )

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice 3NF Perturbativeness Numerical Lowering the cutoff increases “perturbativeness” Weinberg eigenvalue analysis (η at 2.22 MeV vs. density) ν −

1 3 S1 with Pauli blocking

0.5 -1 Λ = 4.0 fm

) -1 d Λ = 3.0 fm

(B 0 -1 ν Λ = 2.0 fm η

-0.5

-1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 -1 kF [fm ] Pauli blocking in nuclear matter increases it even more! at Fermi surface, pairing revealed by η > 1 | ν | Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice 3NF Perturbativeness Numerical Comments on computational aspects Although momentum is continuous in principle, in practice represented as discrete (gaussian quadrature) grid:

= ⇒

Calculations become just matrix multiplications! E.g.,

k V k 0 k 0 V k 1 k V k + h | | ih | | i+ = Vii + Vij Vji + 2 2 ( 2 2) h | | i 0 (k k 0 )/m ··· ⇒ ki kj /m ··· Xk − Xj − 100 100 resolution is sufﬁcient for two-body potential × Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice 3NF Perturbativeness Numerical Discretization of integrals = matrices! Momentum-space ﬂow equations⇒ have integrals like:

I(p, q) dk k 2 V (p, k)V (k, q) ≡ Z Introduce gaussian nodes and weights k , w (n = 1, N) { n n} = dk f (k) w f (k ) ⇒ ≈ n n n Z X Then I(p, q) I , where p = k and q = k , and → ij i j I = k 2w V V V V where V = √w k V k w ij n n in nj → in nj ij i i ij j j n n X X e e e p Lets us solve SRG equations, integral equation for phase shift, Schrodinger¨ equation in momentum representation, . . . In practice, N=100 gauss points more than enough for accurate nucleon-nucleon partial waves

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice 3NF Perturbativeness Numerical MATLAB Code for SRG is a direct translation! d The ﬂow equation ds Vs = [[T , Hs], Hs] is solved by discretizing, so it is just matrix multiplication.

If the matrix Vs is converted to a vector by “reshaping”, it can be fed to a differential equation solver, with the right side:

% V_s is a vector of the current potential; convert to square matrix V_s_matrix = reshape(V_s, tot_pts, tot_pts); H_s_matrix = T_matrix + V_s_matrix; % form the Hamiltontian

% Matrix for the right side of the SRG differential equation if (strcmp(evolution,’T’)) rhs_matrix = my_commutator( my_commutator(T_matrix, H_s_matrix), ... H_s_matrix ); elseif (strcmp(evolution,’Wegner’)) rhs_matrix = my_commutator( my_commutator(diag(diag(H_s_matrix)), ... H_s_matrix), H_s_matrix ); [etc.]

% convert the right side matrix to a vector to be returned dVds = reshape(rhs_matrix, tot_pts*tot_pts, 1);

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice 3NF Perturbativeness Numerical Pseudocode for SRG evolution

1 Set up basis (e.g., momentum grid with gaussian quadrature or HO wave functions with Nmax)

2 Calculate (or input) the initial Hamiltonian and Gs matrix elements (including any weight factors)

3 Reshape the right side [[Gs, Hs], Hs] to a vector and pass it to a coupled differential equation solver 1/4 4 Integrate Vs to desired s (or λ = s− )

5 Diagonalize Hs with standard symmetric eigensolver = energies and eigenvectors ⇒ (i) (i) 6 Form U = ψ ψ from the eigenvectors i | s ih s=0| 7 Output or plotP or calculate observables

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice 3NF Perturbativeness Numerical Many versions of SRG codes are in use

Mathematica, MATLAB, Python, C++, Fortran-90 Instructive computational project for undergraduates! Once there are discretized matrices, the solver is the same with any size basis in any number of dimensions! Still the same solution code for a many-particle basis Any basis can be used For 3NF, harmonic oscillators, discretized partial-wave momentum, and hyperspherical harmonics are available An accurate 3NF evolution in HO basis takes 20 million ∼ matrix elements = that many differential equations ⇒

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice Outline

SRG leftovers

Overview of nuclear many-body problem

Many-body methods

Nuclear lattice simulations

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice Low-energy playground: Table of the nuclides

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice Overlapping theory methods cover all nuclei

dimension of the problem

Interfaces provide crucial clues

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice

What is “new” about theory methods? (examples) Really an explosion of new things! New methods for theoretical inputs (Hamiltonians and operators) Three-body (and higher) forces (N3LO chiral 3NF, RG methods) New extensions of established microscopic techniques e.g., IT-NCSM, MBPT, Berggren basis, LIT Spectroscopic factors, ANCs, . . . (e.g., with GFMC, CC) New microscopic many-body techniques e.g., Lattice EFT, IM-SRG, NCSM/RGM New analysis methods/philosophy (theory error bars!!) Correlation analysis of energy functionals Power counting, benchmarking, . . . New computational reach (e.g., from SciDAC projects) Better scaling: massively parallel codes, load balancing Improved algorithms: e.g., optimization (POUNDERS)

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice Why do we need so many different methods?

Each method has strengths and limitations dimension of the problem Need to cross-check results Exploit overlapping domains Trade-offs: Superior scaling vs. accuracy or more microscopic or . . . Interfaces provide crucial clues

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NCSM Outline

SRG leftovers

Overview of nuclear many-body problem

Many-body methods

Nuclear lattice simulations

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NCSM Fully microscopic (from input NN + NNN) Ab initio theory for light nuclei and uniform matter Ab initio: QMC, NCSM, CC,… (nuclei, neutron droplets, nuclear matter) Input choices: Ab initio input • Accurate forces based on phase shift analysis and few-body data • EFT-based nonlocal chiral NN and !"#$%&'($% NNN potentials )*+,'(% • RG-softened potentials evolved from NN+NNN interactions

! Quantum Monte Carlo -&.*/0"&1*.% (GFMC, lattice EFT) 12C • Direct comparison ! No-Core Shell Model with experiment 14F, 14C • Pseudo-data to ! Coupled-Cluster Techniques inform theory 17F, 48Ca

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NCSM

Size and sparsity of Hamiltonian matrices [from P. Maris]

DimensionsHamiltonian and sparsity matrices of grow matrices rapidly – with stable basis nuclei size (Nmax) and A = N + Z from combinatorics:

10 10 9 10 8 10 7 10 6 10 4He 5 6Li 10 8Be 4 10B 10 12C 3 16O 10 19F 2 10 23Na

M-scheme basis space dimension 1 27Al 10 0 10 0 2 4 6 8 10 12 14 Nmax

Dick Furnstahl TALENT: Nuclear forces

Many Fermion Dynamics – nuclear physics – p.7/16 Many-Body Hamiltonian matrix

Harmonic oscillator single-particle basis states SRG Overview Methods Lattice NCSM Many-Body basis states: Size and sparsitySlater of Determinants Hamiltonian matrices of single-particle[from P. Maris] states But fortunately there are many zero elements so storage is large but feasible.Construct How can we Many–Body take advantage of Hamiltonian sparsity? matrix

large sparse matrix Eigenvalues: bound state spectrum Eigenvectors: nuclear wavefunctions

Dick Furnstahl TALENT: Nuclear forces

Many Fermion Dynamics – nuclear physics – p.4/16 Dimensions and sparsity of matrices – light neutron-rich nuclei

8He (Z= 2, N= 6) 14Be (Z= 4, N= 10) 18 18 10 10

dimension dimension # nonzero’s NN # nonzero’s NN 15 # nonzero’s NNN 15 10 10 # nonzero’s NNN # nonzero’s NNNN # nonzero’s NNNN # nonzero’s NNNNN # nonzero’s NNNNN # matrix elements # matrix elements 12 12 10 10

9 9 10 10

6 6 10 10 SRG Overview Methods Lattice NCSM

3 3 10 10 Size and sparsity of Hamiltonian matrices [from P. Maris]

0 0 10 10 But0 fortunately there 4 are many zero 8 elementsso 12 storage is large but0246 8 10 basis space truncation N basis space truncation N feasible. How can we take advantagemax of sparsity? max 11Li (Z= 3, N= 8) 20C (Z= 6, N= 14) 18 18 10 10

dimension dimension # nonzero’s NN # nonzero’s NN 15 15 # nonzero’s NNN 10 # nonzero’s NNN 10 # nonzero’s NNNN # nonzero’s NNNN # nonzero’s NNNNN # nonzero’s NNNNN # matrix elements # matrix elements 12 12 10 10

9 9 10 10

6 6 10 10

3 3 10 10

0 0 10 10 0246 8 10 12 0246 8 basis space truncation Nmax basis space truncation Nmax

Dick Furnstahl TALENT: Nuclear forces

Many Fermion Dynamics – nuclear physics – p.8/16 SRG Overview Methods Lattice NCSM Lanczos method in short

Consider an arbitrary vector Ψ and its expansion in eigenstates | i of H, where H ψ = E ψ . Then Hm Ψ = C E m ψ | k i k | k i | i k k k | k i If m large enough, largest E will dominate the sum | k | P = project out the corresponding eigenvector ⇒ To get lowest eigenvalue, use (H σI)m with σ > 0 large − enough so that E σ > E σ | 0 − | | max − | More efﬁcient to diagonalize H in the basis spanned by H ψ , | k i H2 ψ ,..., Hm ψ | k i | k i Called the “Krylov space” Lanczos: orthogonalize basis states as you go, generating H in tri-diagonal form, which is efﬁciently diagonalized Re-orthonormalization for numerical stability Many computational advantages to treating sparse matrices with Lanczos [see J. Vary et al., arXiv:0907.0209]

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NCSM !"#$%&'()%*+,-'./01%234(%)'%2'.567%

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•*Z+6,+4)*'5*[9;<\*$'##",'4"&'(*6"-+*60$4')$'30$* ***(%$#+"4*)74%$7%4+*$"#$%#"&'()*7'*)'#>+*7/+*3%]]#+* •*[)+-*).)7+6"&$*$/04"#*^"60#7'(0"(*54'6*#'@2+(+41.* ***+_+$&>+*`+#-*7/+'4.*'5*a!<* •*b+.*5+"7%4+A*$'()0)7+(7* !"#$%&'(#)0(7+4"$&'()*

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Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NCSM ImportanceImportance Truncated Truncated NCSM NCSM [Roth et al.]

Roth, PRC 79, 064324 (2009); PRL 99, 092501 (2007)

I converged NCSM calcula- -110 16 tions essentially restricted O -120 to lower/mid p-shell ] NNonly -130 α 0.04 fm4 MeV I full Nmx 10 calculation [ hΩ 20 MeV = for 16Overydifﬁcult E -140 ̵ = (basis dimension= 1010) -150 . > 0 24681012 14 16 18 20 Importance -110 G Nmx IT-NCSM Truncation -120 full NCSM ] reduce model space -130 + MeV to the relevant basis [

E -140 states using an apriori importance measure -150 derived from MBPT . 0 24681012 14 16 18 20 Nmx Robert Roth – TU Darmstadt – 05/2013 13 Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NCSM

Coupled cluster method [from D. Dean, G. Hagen, T. Papenbrock] Coupled-cluster method (in CCSD approximation)

! >6$-,#(?,"%-@(A0*-@"*B7$-C(D7%2(7"6+,$#7"?( !"#$%&' ((( 0+*E-,B(#7&,(*4;F(8( ! G+;"6$.*"(7#(%2,(*"-@($00+*57B$.*"8( ! >7&,(,5%,"#7H,(A,++*+(#6$-,#(D7%2(!C( "((I*#%(,J67,"%(:*+(3*;E-@(B$?76(";6-,7(

)*++,-$.*"#($+,(!"#$%!%&'(!)(/01/2($"3(40142(,567%$.*"#8(9$+%(*:("01"2(,567%$.*"#( 7"6-;3,3<( ( ( (

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Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NCSM

Coupled cluster method [from D. Dean, G. Hagen, T. Papenbrock] Asymmetry dependence and spectroscopic factors

•!"#$%&'($'")$*+,$%'&(*,&#*-'%* '.(#&/,.0#(** •12#3*,&#*#4%&,$%#5*+&'6*,*$&'((* (#$7'-*.,(#5*'-*,*("#$)8$*(%&9$%9&#* ,-5*&#,$7'-*6'5#0** •!%&9$%9&#*,-5*&#,$7'-*6'5#0(*-##5(* %'*.#*$'-()(%#-%:**

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Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NCSM

Coupled cluster method [from D. Dean, G. Hagen, T. Papenbrock] Quenching of spectroscopic factors for proton removal in neutron rich oxygen isotopes

!"#$%&'($'")$*+,$%'&*)(*,*-(#+-.*%''.*%'*(%-/0* $'&&#.,1'2(*%'3,&/(*%4#*/&)".)2#5** * !6*+'&*"&'%'2*'8,.*)2*2#-%&'2*&)$4*9:;*(4'3* (%&'2<*=>-#2$4)2

!%&'2<*,(077#%&0*/#"#2/#2$#*'2*%4#*!6* +'&*"&'%'2*,2/*2#-%&'2*'8,.*)2* 2#-%&'2*&)$4*'Q0<#2*)('%'"#(5** * !6RI*+'&*2#-%&'2*'8,.*34).#*"&'%'2(* ,&#*(%&'2<.0*$'&&#.,%#/*!6*RJ5S@J5K*)2* 99L9:L9T;**

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NCSM

In-mediumDecoupling SRG decoupling in A-Body[slides Space from H. Hergert]

ConsiderDecoupling SRG with 0p –0inh A-Bodyreference stateSpace (instead of vacuum)

P / Q P / Q

K. Tsukiyama, S. K. Bogner, and A. Schwenk, PRL 106, 222502 (2011)

Dick Furnstahl TALENT: Nuclear forces H. Hergert - Ohio State University - ANL Theory Seminar, 03/22/12

H. Hergert - Ohio State University - ANL Theory Seminar, 03/22/12 SRG Overview Methods Lattice NCSM In-mediumDecoupling SRG decouplingin A-Body[H. Space Hergert]

Aim: decouple reference state (0p–0h) from excitations

0p-0h 1p-1h 2p-2h 3p-3h

1p-1h 0p-0h

= 2p-2h ⇒ 3p-3h

K. Tsukiyama,P / S.Q K. Bogner, and A. Schwenk, PRL 106, 222502 (2011) Dick Furnstahl TALENT: Nuclear forces

H. Hergert - Ohio State University - ANL Theory Seminar, 03/22/12 Results:SRG Oxygen Overview Methods Chain Lattice NCSM

[H. Hergert] Oxygen0 chain with multi-reference0 IM-SRG ÙÊ A A Û NN + 3N-ind. O ÛÙÊ NN + 3N-full (400) O -25 E3 Max=14 -25 E3 Max=14 l=1.9 fm-1 l=1.9 fm-1 Ê‡ -50 -50 Ê‡

D -75 D -75 ÛÙÊ‡ MeV MeV

@ @ Ù‡ -100 -100 ÛÊ E E

exp. Ù‡ -125 ÛÊ -125 Ê‡ ÛÙÊ‡ ‡ Ê IM-SRG Ê Ê IM-SRG Ê‡ ‡ ÙÊ ‡ -150 ‡ IT-NCSM Û ‡ -150 ‡ IT-NCSM Ê ÙÊ Ê‡ Ù CCSD Û Ù CCSD ÙÊ‡ Û ‡ Ê‡ Û L-CCSD T Û L-CCSD T ÙÊ -175 -175 Û 10 12 14 16 18 20 22 24 26 10 12 14 16 18 20 22 24 26 H L A H L A • ref. state: number-projected Hartree-Fock-Bogoliubov vacuum • results (mostly) insensitive to choice of generator for same Hod • consistency between diﬀerent many-body methods

Scott Bogner - Michigan State University - NUCLEI Collaboration Meeting, Indiana University Bloomington, 06/25/13

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT Outline

SRG leftovers

Overview of nuclear many-body problem

Many-body methods

Nuclear lattice simulations

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT

Lattice EFT collaboration [E. Epelbaum] NuclearEffective Lattice field theory Effective on the lattice: Field Theory Ab initio calculations of nuclei and many-body systems The Collaboration: E.E., Hermann Krebs (Bochum), Timo Lähde (Jülich), Dean Lee (NC State), Ulf-G. Meißner (Bonn/Jülich), Gautam Rupak (Mississippi State)

!"#$%&'()&**+#%(,-.(/0$$&10'&*+02 Borasoy, E.E., Krebs, Lee, Meißner, Eur. Phys. J. A31 (07) 105, n Eur. Phys. J. A34 (07) 185,Evgeny Epelbaum (Bochum) Eur. Phys. J. A35 (08) 343, Hermann Krebs (Bochum) Eur. Phys. J. A35 (08) 357, p

E.E., Krebs, Lee, Meißner, Eur. Phys. J A40 (09) 199, Timo Lähde (Jülich) 20 fm 1 Eur. Phys. J A41 (09) 125, Dean Lee (NC State) ...

Phys. Rev. Lett 104 (10)Ulf 142501,-G. Meißner (Bonn/Jülich) 10

Eur. Phys. J. 45 (10) 335, ⇠ Phys. Rev. Lett. 106 (11) 192501, TU Darmstadt L E.E., Krebs, Lähde, Lee, Meißner Phys. Rev. Lett. 109 (12) 252501, Phys. Rev. Lett. 110 (13) 112502, Institut für Kernphysik arXiv:1303.4856 May 22, 2012 a 1 ...2fm ⇠

Dick Furnstahl TALENT: Nuclear forces

1 SRG Overview Methods Lattice NM DFT

Lattice QCD versus lattice EFT [from Dean Lee]

Compare variables and lattice spacing a:

Lattice effective field theory Lattice quantum chromodynamics

u n d n

p n p u

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT NUCLEAR LATTICE SIMULATIONS 13 Lattice EFT basics [from U. Meißner] Frank, 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 = L L L L : p • s × s × s × t nucleons are point-like ﬁelds on the sites n discretized chiral potential w/ pion exchanges • and contact interactions p

typical lattice parameters n a • π Λ = 300 MeV [UV cutoff] a ~ 2 fm

strong suppression of sign oscillations due to approximate Wigner SU(4) symmetry • J. W. Chen, D. Lee and T. Schafer,¨ Phys. Rev. Lett. 93 (2004) 242302 hybrid Monte Carlo & transfer matrix (similar to LQCD) •

Nuclear Physics from LatticeDick Simulations Furnstahl – Ulf-G. MeißnerTALENT: – EMMI, MayNuclear 10, 2012 forces < > · ◦ ∧ • 14 CONFIGURATIONS SRG Overview Methods Lattice NM DFT

Lattice EFT basics [from U. Meißner]

all possible conﬁgurations are sampled ⇒ clustering emerges naturally ⇒

Nuclear Physics from LatticeDick FurnstahlSimulations – Ulf-G.TALENT: Meißner – Nuclear EMMI, May forces 10, 2012 < > · ◦ ∧ • 15 SRG Overview Methods Lattice NM DFT TRANSFER MATRIX METHOD Lattice EFT basics [from U. Meißner] Correlation–function for A nucleons: Z (t)= Ψ exp( tH) Ψ • A A| − | A with ΨA a Slater determinant for A free nucleons Euclidean time

Ground state energy from the time derivative of the correlator • d EA(t)= ln ZA(t) −dt

0 ground state ﬁltered out at large times: E = lim EA(t) A t → →∞

Expectation value of any normal–ordered operator • O

ZO = Ψ exp( tH/2) exp( tH/2) Ψ A A| − O − | A

ZAO(t) lim = ΨA ΨA t →∞ ZA(t) |O |

Nuclear Physics from Lattice Simulations – Ulf-G. Meißner – EMMI, May 10, 2012 < > Dick Furnstahl TALENT: Nuclear forces· ◦ ∧ • SRG Overview Methods Lattice NM DFT 16 TRANSFER MATRIX CALCULATION Lattice EFT basics [from U. Meißner] Expectation value of any normal–ordered operator • O

ΨA exp( tH/2) exp( tH/2) ΨA ΨA ΨA = lim | − O − | |O | t Ψ exp( tH) Ψ →∞ A| − | A Anatomy of the transfer matrix •

operator insertion for expectation value Ψ free SU π O SU π Ψ free Z,N (4) full LO full LO (4) Z,N

L + L L + L L + L L 2 to ti to ti to ti/2 to 0 { { inexpensive ﬁlter inexpensive ﬁlter

Dick Furnstahl TALENT: Nuclear forces

Nuclear Physics from Lattice Simulations – Ulf-G. Meißner – EMMI, May 10, 2012 < > · ◦ ∧ • SRG Overview Methods Lattice NM DFT 17

LatticeMONTE EFT CARLO basics with AUXILIARY[from U. Meißner] FILEDS

Contact interactions represented by auxiliary ﬁelds s, s • I + 2 ∞ 2 exp(ρ /2) ds exp( s /2 sρ) , ρ N †N ∝ − − ∼ −∞ Correlation function = path-integral over pions & auxiliary ﬁelds •

S ⇒

Nuclear Physics from LatticeDick Simulations Furnstahl – Ulf-G. MeißnerTALENT: – EMMI, Nuclear May 10, forces 2012 < > · ◦ ∧ • SRG Overview Methods Lattice NM DFT 8 12 Ground statesGround of statesBe and ofC 8Be[E. Epelbaum] and 12C E.E., Krebs, Lee, Meißner, PRL 106 (11) 192501 Ground state of Beryllium-8 Ground state of Carbon-12 Simulations for 8Be and 12C, L=11.8 fm

Epelbaum, Krebs, D.L, Meißner, PRL 106 (2011) 192501 Epelbaum, Krebs, D.L, Meißner, PRL 104 (2010) 142501; Ground state energies (L=11.8 fm) of 4EPJAHe, 45 8 (2010)Be, 335; 12C PRL & 106 16 (2011)O 192501 31 33 4He 8Be 12C 16O LO [Q0], in MeV 28.0(3) 57(2) 96(2) 144(4) NLO [Q2], in MeV 24.9(5) 47(2) 77(3) 116(6) NNLO [Q3], in MeV 28.3(6) 55(2) 92(3) 135(6) Experiment, in MeV 28.30 56.5 92.2 127.6

Dick Furnstahl TALENT: Nuclear forces

⇤⇥ ⇤q ⇤⇥ ⇤q V = 1 · 1 3 · 3 ⌧ ⌧ (q )+⌧ ⌧ ⌧ ⇤q ⇤q ⇤⇥ (q ) 2 [q2 + M 2][q2 + M 2] 1 · 3 A 2 1 ⇥ 3 · 2 1 ⇥ 3 · 2 B 2 1 3 ⇣ ⌘

g2 g2 c (3)(q )= A (2c 4c )M 2 + c q2 , (3)(q )= A 4 , A 2 8F 4 3 1 3 2 B 2 8F 4 ⇣ ⌘

g4 (4)(q )= A A(q ) 2M 4 +5M 2q2 +2q4 + 4g2 +1 M 3 +2 g2 +1 M q2 , A 2 256F 6 2 2 2 A A 2 h ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ i g4 (4)(q )= A A(q ) 4M 2 + q2 +(2g2 +1)M B 2 256F 6 2 2 A h ⇣ ⌘ i

g (5)(q )= A M 2q2(F 2 23042g (4e¯ +2e¯ e¯ e¯ ) 23042d¯ c A 2 46082F 6 2 A 14 19 22 36 18 3 h ⇣ ⌘ + g (144c 53c 90c )) + M 4 F 2 46082d¯ (2c c )+46082g (2e¯ +2e¯ e¯ 4e¯ ) A 1 2 3 18 1 3 A 14 19 36 38 ⇣ ⇣ ⌘ + g 72 642¯l +1 c 24c 36c + q4 23042e¯ F 2g 2g (5c +18c ) A 3 1 2 3 2 14 A A 2 3 ⇣ 2 ⇣ ⌘ ⌘⌘ ⇣ ⌘i gA 2 2 2 2 2 6 L(q2) M +2q2 4M (6c1 c2 3c3)+q2( c2 6c3) , 768 F g ⇣ ⌘⇣ ⌘ (5)(q )= A M 2 F 2 11522d¯ c 11522g (2e¯ +2e¯ e¯ ) +108g3 c +24g c B 2 23042F 6 18 4 A 17 21 37 A 4 A 4 h ⇣ ⇣ ⌘ ⌘ g2 c + q2 5g c 11522e¯ F 2g + A 4 L(q ) 4M 2 + q2 2 A 4 17 A 3842F 6 2 2 ⇣ ⌘i ⇣ ⌘

p⇤1 p⇤10 p⇤2 p⇤20 p⇤3 p⇤30

22 V loc. = F (r ,r ,r )+5perm.. 3N Gi i 12 23 31 Xi=1 =1, = ⌧ ⌧ , = ⇤⇥ ⇤⇥ ,... G1 G2 1 · 2 G3 1 · 3

1 222 ticeticetice spacing spacing spacingaaa=1=1=1.97..9797 fm fm fm and and and total total total length length lengthLLL=== 12 12 12 fm. fm. fm. In In In SRG Overview Methods Lattice NM DFT TABLETABLETABLE I: I: I: Lattice Lattice Lattice results results results and and and experimental experimental experimental values values values for for for the the the thethe time time direction, direction, we we use use lattice lattice time time step stepaat =1=1..3232 fm fm 44 88 the time direction, we use lattice time step att=1.32 fm groundgroundground state state state energies energies energies of of of4HeHeHe and and and8Be.Be.Be. All All All energies energies energies are are are in in in and vary the propagation time L to extrapolate to the Hoyle State [E. Epelbaum] andand vary vary the the propagation propagation time timeLLttttoto extrapolate extrapolate to to the the unitsunitsunits of of of MeV. MeV. MeV. The Hoyle state EE, Krebs,Nuclear Lähde, Physics Lee, withMeißner, Chiral PRL Effective 106 (2011) Field 192501; Theory PRL 109 (2012) 252501 Evgeny Epelbaum limitlimitlimitLLLt ... The The The nucleons nucleons nucleons are are are treated treated treated as as as point-like point-like point-like 444 888 tt ⇥ HeHeHe BeBeBe ⇥⇥ 12 particlesparticlesparticles on on on lattice lattice lattice sites, sites, sites, and and and interactions interactions interactions due due due to to to the the the ex- ex- ex- Lattice0 results for low-lying even-parity states of C LOLOLO [O [ [OO((Q(QQ00)])])] 282828.0(3)..0(3)0(3) 57(2)57(2)57(2) changechangechange of of of pions pions pions and and and multi-nucleon multi-nucleon multi-nucleon operators operators operators are are are gener- gener- gener- + + + + + + 1 222 01 21 (E ) 02 22 (E ) atedatedated using using using auxiliary auxiliary auxiliary fields. fields. fields. Lattice Lattice Lattice e e⌅ e⌅⌅ectiveectiveective field field field theory theory theory NLONLONLO [O [ [OO((Q(QQ)])])] 242424.9(5)..9(5)9(5) 47(2)47(2)47(2) LO 3 96(2) 94(2) 89(2) 88(2) was originally used to calculate the many-body proper- NNLONNLONNLO [O [ [OO((Q(QQ33)])])] 282828.3(6)..3(6)3(6) 55(2)55(2)55(2) 0.5 waswas originally originally used used to to calculate calculate the the many-body many-body proper- proper- NLO 77(3) 74(3) 72(3) 70(3) ties of homogeneous nuclear and neutron matter [19, 20]. Experiment 28.30 56.50 tiesties of of homogeneous homogeneous nuclear nuclear and and neutron neutron matter matter [19, [19, 20]. 20]. ExperimentExperimentNNLO 289228(.330.)30 568956.(50.350) 85(3) 83(3) t

_ A 0 SinceSinceSince then then then the the the properties properties properties of of of several several several atomic atomic atomic nuclei nuclei nuclei have have have Exp 92.16 87.72 84.51 82(1) 4 beenbeenbeen investigated investigated investigated [21, [21, [21, 22]. 22]. 22]. A A A recent recent recent review review review of of of the the the liter- liter- liter- δ Figure 7: Left panel: Lattice results of Ref. [83] for the ener- -0.5 | mq /mq| = 0.01 δ ature can be found in Ref. [23]. 12 | mq /mq| = 0.05 tial financial support from the DFG, NSFC (CRC 110), atureature can can be be found found in in Ref. Ref. [23]. [23]. Probing (α(I)(I)-cluster)(I)gies of low-lying structure even-parity of the states0(II)1+(II)(II), 0 of2+ statesCTABLE compared III: to Lattice exper- results at leading order and experimen- RMS radii and quadrupole HGF (VH-VI-417), BMBF (06BN7008) USDOE (DE-FG02- Euclidean time propagation is used to project on to -50 imental values (in-50 units of MeV). Right panel:tal values “Survivability for the root-mean-square charge radius and the EuclideanEuclidean time time propagation propagation is is used used to to project project on on to to -50-50 -50-50 moments12 -1 03ER41260), EU HadronPhysics3 project “Study of strongly LOLOLO [A] [A] [A] LOLOLO [C] [C] [C] quadrupole moments for C. -1 -0.5 _0 0.5 1 low-energy states of our interacting system. Let H be LObands” [B] of carbon-oxigen based lifeLO [D] obtained from lattice sim- interacting matter”, and ERC project 259218 NUCLE- low-energylow-energy states states of of our our interacting interacting system. system. Let LetHH bebe -60 LOLO [B] [B] -60 LOLO [D] [D] LO Experiment A -60-60 LOLOLO [ulations ![ [!]!]] of Ref.-60-60 [84] as explainedLOLOLO [ in"[ ["]" the]] text. 3 s AREFT. Computational resources were provided by the the Hamiltonian. For any initial quantum state ⇤,the + thethe Hamiltonian. Hamiltonian. For For any any initial initial quantum quantum state state⇤⇤,the,the r(01 ) [fm] 2.2(2) 2.47(2) [26] Jülich Supercomputing Centre at the Forschungszentrum -70 -70 -70-70 -70-70 r(2+) [fm] 2.2(2) projectionprojectionprojection amplitude amplitude amplitude is is is defined defined defined as as as the the the expectation expectation expectation value value value 1 Jülich. The crucial quantity that controls the production rate+ is the energy of the Hoyle state relative to Ht Q(2 )[e fm2] 6(2) 6(3) [27] e HtHt . For large Euclidean time t, the exponential -80 -80 1 ee .. For For large large Euclidean Euclidean time timett,, the the exponential exponential -80-80 -80-80 + ) (MeV) the triple-alpha threshold which is experimentally known to be = 397.47(18) keV. Changing t ) (MeV) ) (MeV) r(02 ) [fm] 2.4(2) t ( t ( Ht ( HtHt E 3 + 12 16 E operator e enhances the signal of low-energy states. E by an amount of 100 keV results in a strong reductionr(22 ) [fm] of the2.4(2) formation of C and O in the operatoroperatoree enhancesenhances the the signal signal of of low-energy low-energy states. states. -90-90-90 -90-90-90 ⌦⌦⌦ ↵↵↵ FIG. 2: This shows initial state , a wavefunction consisting± FIG. 3: This shows initial state ⇥, a wavefunction+ consisting2 Q(22 )[e fm ] 7(2) ¨ EnergiesEnergiesEnergies can can can be be be determined determined determined from from from the the the exponential exponential exponential decay decay decay of three alpha clusters formeduniverse by Gaussian making packets centered the emergenceof three alpha of clusters carbon-based formed by Gaussian life packets impossible. centered It is, therefore, very interesting[1] E. J. Opik, Proc. Roy. Irish Acad. A54,49(1951). -100-100on the vertices of a compact triangle. There are-100 a-100 total of 12 on the vertices of a bent-arm or obtuse triangular configura- [2] E. E. Salpeter, Astrophys. J. 115,326(1952). of these projection amplitudes. The first few time steps -100equivalent orientations ofto this configuration. investigate-100 how thistion. seemingly There are a fine-tuned total of 24 equivalent quantity orientations depends of this on the fundamental constants of ofof these these projection projection amplitudes. amplitudes. The The first first few few time time steps steps configuration. [3] E. E. Salpeter, Ann. Rev. Nucl. Sci. 2,41(1953). TABLE IV: Lattice results at leading order and experimen- [4] F. Hoyle, Astrophys. J. Suppl. 1,121(1954). and last few time steps are evaluated using a simpler -110-110-110 nature such as-110-110-110mq. We have studied the sensitivity of to variations of mq within nuclear lattice andand last last few few time time steps steps are are evaluated evaluated using using a a simpler simpler FIG. 2: This showsenergy, initial state96(2), a MeV. wavefunction For consistinginitial state A, we start with tal values for electromagnetic transitions involving the even- 0 0.02 0.04 0.06 0.08 0.1FIG. 0.12 3: This shows 0 initial 0.02 state 0.04⇥, a wavefunction 0.06 0.08 consisting 0.1 0.12 12 [5] C. Cook, W. A. Fowler, C. C. Lauritsen, and T. Laurit- of three alpha clustersfour 0 0 nucleons formed 0.02 0.02 by 0.04 Gaussian each0.04 0.06 at0.06 packets zero 0.08 0.08 centered momentum, 0.1 0.1 0.12of 0.12 three apply alpha creation 0 clusters 0 0.02 0.02 formed 0.04TABLE 0.04 by Gaussian 0.06 0.06 II: Lattice 0.08 packets 0.08 results centered0.1 0.1parity for0.12 0.12 the states low-lying of even-parityC. states Hamiltonian H based upon Wigner’s SU(4) symme- simulations-1 in Ref. [84].12 Fig. 7 shows-1 the survivability bands of carbon-based life under 1% andsen, Phys. Rev. 107,508(1957). HamiltonianHamiltonianHHSU(4)SU(4)SU(4)basedbased upon upon Wigner’s Wigner’s SU(4) SU(4) symme- symme- on the vertices ofoperators a compact triangle. aftert the Therett (MeV first are-1 a time-1 total) ofstep 12 toon inject the vertices four of more a bent-armof ort obtusetCt (MeV compared triangular-1-1 ) with configura- the experimental results in units of equivalent orientations of this configuration. (MeV (MeV) ) tion. There are a total of 24MeV. equivalent (MeV¯ (MeV orientations) ) of1 this LO Experiment [6] H. Morinaga, Phys. Rev. 101,254(1956). nucleons at rest, and then5% inject changes fourconfiguration. more of nucleonsmq. atHere, As,t (⇤a+ + /⇤M+⇥ ) phys+ denote the slope of the inverse NN S-wave try for protons and neutrons [24]. This Hamiltonian is 01S0,23S(1E+) 0 M⇥ 2 (E+) trytry for for protons protons and and neutrons neutrons [24]. [24]. This This Hamiltonian Hamiltonian is is rest after the second time step. The reverse process ⇥ 1 1 2 + 2 + 2 4 [7] A. Tohsaki, H. Horiuchi, P. Schuck, and G. Ropke, Phys. energy, 96(2) MeV. For initial state A, we start with Dick FurnstahlLO [O(Q0)]TALENT:96(2) Nuclear94(2) B89(2)(E forces2, 21 88(2)01 )[e fm ] 5(2) 7.6(4) [29] is used to extract nucleonsscattering for final state lengths A. The same as functions of the pion mass. These! quantities can, in principle, be computedRev. Lett. 87,192501(2001). computationally inexpensive and is used as a low-energy four nucleons each at zero momentum, apply creation TABLE II: Lattice results for the low-lying2 even-parity states + + 2 4 computationallycomputationally inexpensive inexpensive and and is is used used as as a a low-energy low-energy scheme is used for initial state B, though12 the12 interactions NLO [O(Q )] 77(3) 74(3) B72(3)(E2, 2 70(3)0 )[e fm ] 1.5(7) 2.6(4) [29] operators after the first time step to inject four more of1212C compared with the experimental results in units of 1 2 2 [8] M. Freer et al., Phys. Rev. C 80,041303(2009). FIG. 1: Lattice resultsin lattice-QCD.for theMeV. C The spectrum data point at in3 leading the right order. panel of Fig.! 7 corresponds to the recent N LO results FIG.FIG.nucleons 1: 1: at Latticerest, Latticein andHSU(4) thenused inject results results are four not more as nucleons stronglyfor for the at the attractiveC asC spectrumthose spectrum for NNLO at at [O( leadingQ leading)] 92(3) order.89(3) order.85(3) + 83(3)+ 2 4 filter before starting the main calculation. This tech- 0+ 2+(E+) 0+ 2+(E+) B (E2, 22 01 )[e fm ] 2(1) [9] S. Hyldegaard et al., Phys. Rev. C 81,024303(2010). filterfilter before before starting starting the the main main calculation. calculation. This This tech- tech- rest after the secondA. time step. The reverse process 1 1 2 2 1 82.6(1)! [8, 10] InIn Panel Panel I Iwe we show show results resultsof Ref. from from [28LO three] [ threeO for(Q0)] chiral di di96(2)⇤⇤erent extrapolationserent94(2) 89(2) initial initial88(2) states, of states,a A, A,shown + in+ Fig.2 34. These findings suggest that[10] the W. R. Zimmerman, N. E. Destefano, M. Freer, M. Gai, Inis Panel used to extract IForwe nucleons initial show for state final state results, we A. use The a samefrom wavefunction three consisting di⇤erentExperiment initial 92. states,16 871.S720,B3S84 A,(1E.512, 2822 .32(6)0 [11]2 )[e fm ] 6(2) nique is described in Ref. [23]. NLO [O(Q2)] 77(3) 74(3) 72(3) 70(3) niquenique is is described described in in Ref. Ref. [23]. [23]. scheme is used forof threeinitial state alpha B,clusters though the as interactions shown in Fig. 2. The alpha + ! + 2 and F. D. Smit, Phys. Rev. C 84,027304(2011). B, and , each approaching the ground3 state energy. In Panel 81.1(3) [9] B,B,in and andHSU(4) used, are, each eachnot as strongly approaching approaching attractiveformation as those for the theNNLO of ground ground carbon [O(Q )] 92(3) and state state89(3) oxygen energy.85(3) energy. in83(3) our In In universe Panel Panelm(E0 would, 02 0 survive1 )[e fm a] 3(1)2% change5.5(1) [17] in the light quark clusters are formed by Gaussian packets centered on the ! [11] M. Itoh et al., Phys. Rev. C 84,054308(2011). In Table I we present lattice results for the ground A. 82.6(1) [8, 10] ⇤ InIn Table Table I I we we present present lattice lattice results results for for the the ground ground II we showvertices results of a compact starting triangle. In from order to threeconstruct other initial states, C, IIII we weFor initial show show state results, results we use a wavefunction starting startingmass. consisting from fromExperiment three three92. other16 other87.72 84 initial. initial51 82.32(6) states, [11] states, C, C, [12] F. Smit, F. Nemulodi, Z. Buthelezi, J. Carter, R. Fearick, 4 8 eigenstates of total momentum and lattice cubic rota- 44 88 of three alpha clusters as shown in Fig. 2. The alpha figuration as81 shown.1(3) [9] in Fig. 3. There are a total of 24 et al. (2012), 1206.4217. statestate energies energies of of HeHe and and BeBe up up to to NNLO. NNLO. The The method method D,clusters and are formed⇥tions,. by These we Gaussian consider packets trace all centeredpossible out on translations the an intermediate and rotations plateau at energy state energies of He and Be up to NNLO. The method D,D, and and⇥⇥.. These These trace trace out out an an intermediate intermediateequivalent plateau plateau orientations at atmoments of energy thisenergy configuration. of the two We spin-2 do not states reflects the oblate shape [13] E. Epelbaum, H. Krebs, D. Lee, and U.-G. Meißner, vertices of a compactof the triangle.initial state. In order There to construct are a total of 12 equivalent find the same plateau starting from+ other configurations + abouteigenstates7 of total MeV momentum above and lattice the cubic rota-ground state. ofofof calculation calculation calculation is is is nearly nearly nearly the the the same same same as as as that that that described described described in in in aboutabout 7 7 MeV MeVorientations above above of this configuration. the the ground ground Wefiguration do state. not state. as find shown fast in Fig.of 3. alpha There clusters. are a total We ofof 24 conclude the 21 thatstate the configurations and prolate shape of the 22 state. Phys. Rev. Lett. 106,192501(2011). tions, we consider all possible translations andReferences rotations equivalent orientations of this configuration. We do not + of the initial state.convergence There are to a the total ground of 12 equivalent state whenfind starting the same from plateau any startingin Fig. from 3 other have configurations the strongestThe overlap leading with order the 02 resultsHoyle for the electromagnetic tran- [14] R. Roth, J. Langhammer, A. Calci, S. Binder, Ref. [13, 22, 25]. The higher-order corrections are com- 12 Ref.Ref. [13, [13, 22, 22, 25]. 25]. The The higher-order higher-order corrections corrections are are com- com- orientations ofother this configuration. configuration We of do alpha not find clusters. fast of From alpha this clusters. we con- We concludestate ofthat theC. configurationssitions among the even-parity states of 12C are shown in and P. Navratil, Phys.Rev.Lett. 107,072501(2011), convergence toclude the ground that state the alpha when starting cluster from configurations any in Fig. in 3 Fig. have 2 the have strongest overlap with the 0+ Hoyle [1] W. Detmold, these proceedings.We use the same2 multi-channel method developed in 1105.3173. puted using perturbation theory. The coe⇧cients of other configuration of alpha clusters. From this we con-+ state of 12C. 12 Table IV. The definitions for these quantities can be putedputed using using perturbation perturbation theory. theory. The The coe coe⇧⇧cientscients of of the strongest overlap with the 01 ground state of C. Ref. [13] to find a spin-2 excitation above the ground state clude that the alpha cluster configurations in Fig. 2 have We use the same multi-channel method developed in [15] C. Forssen, R. Roth, and P. Navratil (2011), 1110.0634. The fact that it+ is an isosceles right12 triangle rather than as well as a spin-2 excitationfound above in Ref. the Hoyle [28]. state. The In agreement with available ex- timethe strongest step overlap to with inject the 01 ground four state[2] of moreS.C. Weinberg,Ref. [13] nucleons to find Phys. a spin-2 excitation Lett. at zeroB above251 the, groundmomentum. 288 state (1990); Nucl. Phys. B 363, 3 (1991). the nucleon-nucleon interactions are set by fitting to low- timetime step stepan to equilateralto inject inject triangle four four is just more more an artifact nucleons nucleons of the lattice at atboth zero zero cases we momentum. aremomentum. taking the E+ representation of the cu- [16] M. Chernykh, H. Feldmeier, T. Ne, P. von Neumann- thethe nucleon-nucleon nucleon-nucleon interactions interactions are are set set by by fitting fitting to to low- low- The fact that it is an isosceles right triangle rather than as well as a spin-2 excitation above the Hoyle state.perimental In values is reasonable. The lattice results at Thean equilateral analogousspacing. triangle is just an artifactprocess of the lattice isboth done cases we toare taking extract thebicE+ rotationrepresentation fourgroup of on the the nucleons cu- lattice. We show the results for Cosel, and A. Richter, Phys. Rev. Lett. 98,032501 energyenergyenergy scattering scattering scattering data. data. data. In In In our our our calculations calculations calculations the the the NNLO NNLO NNLO TheThespacing. analogous analogousIn Panel II process of process Fig. 1 we[3] show isH.-W. is leading-order done donebicHammer rotation to groupenergies to extractonand extract the lattice.L.the Platter, binding We showfour four energies Ann.the results nucleons of nucleonsRev.leading thefor low-lying Nucl. order even-parity Part. have Sci. states a60 tendency, of 207 (2010). to be somewhat smaller (2007). In Panel II offor Fig. three 1 we di show⇤erent leading-order initial states, energies C, D,the and binding⇥, each energies ap- of the12 low-lyingC in Table even-parity II. states We find of that the binding energies at beforefor three di⇤erent the initial last states, C, step. D, and ⇥, each This ap- 12C injection in Table II. We find and that theextraction binding energiesthan at experimental pro- values. This presumably reflects the [17] M. Chernykh, H. Feldmeier, T. Ne, P. von Neumann- corrections correspond with three-nucleon forces. A de- beforebefore the theproaching last last an step. intermediate step.This plateau This at injection injection89(2) MeV. If and andNNLO extraction areextraction in agreement with pro- experimentalpro- values. correctionscorrections correspond correspond with with three-nucleon three-nucleon forces. forces. A A de- de- proaching an intermediate plateau at 89(2)[4] MeV.V. If Bernard,NNLO are in Prog. agreement Part. with experimentalNucl. Phys. values.60, 82 (2008). Euclidean time is taken to infinity, these curves eventu- In Table III we presentgreater results binding at leading energies order for the and smaller radii of the nuclei at Cosel, and A. Richter, Phys. Rev. Lett. 105,022501 tailed description of the interactions at each order can cessEuclidean of time nucleonsis taken to infinity, these at curves zero eventu- momentumIn Table III we present results helps at leadingto order for eliminate the (2010), 1004.3877. tailedtailed description description of of the the interactions interactions at at each each order order can can cesscessally approach of of nucleons thenucleonsally ground approach state energythe at ground at like thezero zero state curves energy in momentum momentumroot-mean-square like the curves charge in radiushelpsroot-mean-square helps and quadrupole to toeliminatecharge moment eliminateleading radius and order. quadrupole We moment also predict electromagnetic decays [5] E. Epelbaum and U.-G.12 Meißner, Ann. Rev.12 Nucl. Part. Sci. 62, 159 (2012). Panel I. HoweverPanel it is I. clear However that a di it⇤erent is clear state that is first a di⇤oferent the even-parity state is first states ofof theC. even-parityWe also show states experi-involving of C.We the also 2+ showstate experi- that may be measured experimen- [18] E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Rev. be found in Ref. [25]. We have used the triton binding directionaldirectionalbeing formed whichbeing is biasesformednot biases the ground which state. causedis notcaused We the identify ground by state.bymental initial values Weinitial identify where available. and andmental Infinal this values final study where we state compute state available. mo- Inmo- this study2 we compute bebe found found in in Ref. Ref. [25]. [25]. We We have have used used the the triton triton binding binding directional+ biases caused+ by initial and final state mo- Mod. Phys. 81,1773(2009). the 0 state inthe this 0plateau+ state region in this as the plateau 02 Hoyle region state. as theelectromagnetic 0+ Hoyle state. moments onlyelectromagnetic at leading order. moments We notetally only in at the leading near order. future. We note The common thread connecting each of the initial[6] statesE. Epelbaum,that2 moments suchH. as-W. the Hammercharge radius for and resonances U.-G. Meißner, Rev. Mod. Phys. 81, 1773 (2009). [19] H. M. Müller, S. E. Koonin, R. Seki, and U. van Kolck, energy and the weak axial vector current to fix the low- menta.menta. The common thread connecting each of the initial states that moments such as the charge radius for resonances energyenergy and and the the weak weak axial axial vector vector current current to to fix fix the the low- low- menta.C, D, and ⇥, is that each produces a state which has an above threshold are dependent on boundary conditionsIn summary we have presented ab initio lattice cal- Phys. Rev. C61,044320(2000). extended or prolateC, D, geometry. and ⇥, is This that is in each contrast[7] produces toR. the aMachleidt stateused to which regulate has and the an continuum-state D.above R. Entem, threshold asymptotics Phys. are of dependentculations the Rept. 503 on whichboundary, 1 (2011). show conditions the structure of the Hoyle state energy constants c and c entering the three-nucleon oblateWe triangular makeextended configuration oruse prolate in Fig. of 2.geometry. many Thisdi iswavefunction. in⌅ contrasterent to We the avoidinitial this problem and because final all of the states [20] D. Lee, B. Borasoy, and T. Schäfer, Phys. Rev. C70, energyenergy constants constants ccDDDandandccEEEenteringentering the the three-nucleon three-nucleon WeWe make make use use of of many many di di⌅⌅erenterent initial initialused toand and regulate final final the continuum-state states states asymptotics of the For initial stateoblate C, we triangular take four nucleons configuration at rest, fourin Fig.low-lying 2.1212 states are boundwavefunction. at leading order. One We expects avoidand this find12 problem12 evidence because all for of thea low-lying spin-2 rotational ex- 014007 (2004). interaction. towith probe momenta (2For the/L, initial2/L, structure2 state/L), andC, we four take[8] with fourof mo-C. nucleons theOrdonez,that as12 at theC rest, higher-order L. states. four Ray corrections andlow-lying U.In pushvan states the all Kolck, binding areof bound en- the Phys. at leading12 Rev.C order. Lett. One72 expects, 1982 (1994). interaction.interaction. totomenta probe probe ( 2/L, the the2/L, structure2 structure/L). For initial stateof of D,the theergies closerCC to states. thestates. triple alpha threshold, In In all theall correspond- of ofcitation. the the CC For the ground state and first spin-2 state, [21] E. Epelbaum, H. Krebs, D. Lee, and U. G. Meißner, Eur. we use a similarwith configuration momenta with (2/L, four2 at/L, rest,2 four/L), anding radii four will with increase mo- accordingly.that as the A detailed higher-order studywe of corrections find mostly push the a binding compact en- triangular configuration of al- Phys. J. A41,125(2009). In comparison with the calculations in Ref. [13], some stateswith momenta investigatedmenta (2/L, 2 (/L,20),/L, and2 four/L, with here2 momenta/L). we Forthese measure initial resonances state as D, a function four-nucleonergies of closer finite volume to the sizetriple will correla- alpha threshold, the correspond- InIn comparison comparison with with the the calculations calculations in in Ref. Ref. [13], [13], some some statesstates investigated investigated here here[9] weD. we R. measure measure Entem and four-nucleonR. four-nucleon Machleidt, Phys. correla- correla-Rev. C 68, 041001 (2003). [22] E. Epelbaum, H. Krebs, D. Lee, and U.-G. Meißner, ( 2/L, 2/L,we0). use For a initialsimilar state configuration⇥,weuseaset withbe four investigated at rest, in four future work.ing radii We find will good increase agreement4 pha accordingly. clusters. A detailed For study the of Hoyle state and second spin-2 three alpha clusters formed by Gaussian packets centered with the experimental value for the 2+ quadrupole44 mo- Phys. Rev. Lett. 104,142501(2010). improvementsimprovements have have been been made made using using higher-derivative higher-derivative tionstionstions by by bywith calculating calculating calculating momenta (2/L, 2 the/L, the the0), expectation and expectation expectation four with momenta value value valuethese resonances of1 of of as,where,where a,wherestate, function we of finite find volume a bent-arm size will or obtuse triangular config- improvements have been made using higher-derivative on the vertices( of2 a bent-arm/L, 2/L, or obtuse0). triangular For[10] initial con-E. state Epelbaum,ment.⇥,weuseaset The di⇤erence W. Glöckle in signsbe forinvestigated the and electric U.-G. inquadrupole future Meißner, work. We Nucl. find good Phys. agreement A 747, 362 (2005). [23] D. Lee, Prog. Part. Nucl. Phys. 63,117(2009). lattice operators which eliminate the overbinding of the is the totalthree alpha nucleon clusters formed density. by Gaussian packets We centered find strongwith the experimental four-nucleonuration value for theof 2alpha+ quadrupole clusters. mo- We have calculated charge latticelattice operators operators which which eliminate eliminate the the overbinding overbinding of of the the isis the the total total nucleon nucleon density. density. We We find find strong strong four-nucleon four-nucleon1 [24] E. Wigner, Phys. Rev. 51,106(1937). on the vertices of a bent-arm[11] orV. obtuse G. J. triangular Stoks, R.con- A. M.ment. Kompl, The di⇤erence M. C. inradii, signsM. Rentmeester forquadrupole the electric quadrupole moments,and J. J. de and Swart, electromagnetic Phys. Rev. C transi-48, 792 (1993).[25] E. Epelbaum, H. Krebs, D. Lee, and U.-G. Meißner, Eur. leading order action when calculating larger nuclei such correlations consistent with the formation of alpha clus- 12 leadingleading order order action action when when calculating calculating larger larger nuclei nuclei such such correlationscorrelations consistent consistent with with the the formation formation of of alpha alphations clus- clus- among the low-lying even-parity states of C at Phys. J. A45,335(2010). 16 [12] N. A. Arndt et al., SAID online program, http://gwdac.phys.gwu.edu. asas1616O.O. The The details details of of this this improved improved action action will will be be dis- dis- ters.ters.ters. In In In Fig. Fig. Fig. 1 1 1 we we we present present present lattice lattice lattice results results results for for for the the theleading energy energy energy order. All of the results are in reasonable agree- [26] L. A. Schaller, L. Schellenberg, T. Q. Phan, G. Piller, as O. The details of this improved action will be dis- ment with experimental values. More work is still needed A. Ruetschi, and H. Schneuwly, Nucl. Phys. A379,523 of121212C at leading order[13] E. versus Epelbaum Euclidean and J. Gegelia, projection Phys. Lett. B time716, 338 (2012). cussedcussedcussed in in in a a a forthcoming forthcoming forthcoming publication. publication. publication. The The The error error error bars bars bars in in in ofof CC at at leading leading order order versus versus Euclidean Euclidean projection projectionsuch time as time calculations using smaller lattice spacings. How- (1982). t. For each of the initial states A, B, C, andever D, these we results provide a deeper understanding of the [27] W. J. Vermeer et al., Phys. Lett. B122,23(1983). TableTableTable I I I are are are one one one standard standard standard deviation deviation deviation estimates estimates estimates which which which inin- in- tt.. For For each each of of the the initial initial states states A, A, B, B, C, C, and and D, D, we we [28] A. Bohr and B. R. Mottelson, Nuclear Structure. Volume clude both Monte Carlo statistical errors and uncertain- startstart with with delocalized delocalized nucleon nucleon standing standing waves waves and andstructure use use a and13 a rotations of the Hoyle state starting from I: Single-Particle Motion (W. A. Benjamin, New York, cludeclude both both Monte Monte Carlo Carlo statistical statistical errors errors and and uncertain- uncertain- start with delocalized nucleon standing waves andfirst useprinciples. a 1969). ties due to extrapolation at large Euclidean time. We strong attractive interaction in HSU(4) to allow the nu- [29] F. Ajzenberg-Selove, Nucl. Phys. B506,1(1990). tiesties due due to to extrapolation extrapolation at at large large Euclidean Euclidean time. time. We We strongstrong attractive attractive interaction interaction in inHHSU(4)SU(4)toto allow allowAcknowledgements the the nu- nu- We thank T. Ne, G. Rupak, H. 4 8 seeseesee that that that the the the binding binding binding energy energy energy results results results for for for44HeHeHe and and and88BeBeBe at at at cleonscleonscleons to to to self-organize self-organize self-organize into into into a a a nucleus. nucleus. nucleus. For For For initial initial initialWeller, states states states and W. Zimmerman for useful discussions. Par- NNLONNLONNLO are are are in in in agreement agreement agreement with with with experimental experimental experimental values. values. values. andandand⇥⇥⇥,,,wewewe use use use alpha alpha alpha cluster cluster cluster wavefunctions wavefunctions wavefunctions to to to recover recover recover InInIn our our our projection projection projection Monte Monte Monte Carlo Carlo Carlo calculations calculations calculations we we we use use use a a a thethethe same same same states states states found found found using using using initial initial initial states states states A, A, A, B, B, B, C, C, C, and and and D. D. D. larger class of initial and final states than considered in For these calculations, the interaction in HSU(4) is not as largerlarger class class of of initial initial and and final final states states than than considered considered in in ForFor these these calculations, calculations, the the interaction interaction in inHHSU(4)SU(4)isis not not as as 4 previouspreviousprevious work. work. work. For For For the the the calculation calculation calculation of of of44HeHeHe we we we use use use an an an strongstrongstrong and and and the the the projected projected projected states states states retain retain retain their their their original original original alpha alpha alpha initialinitialinitial state state state with with with four four four nucleons, nucleons, nucleons, each each each at at at zero zero zero momentum. momentum. momentum. clusterclustercluster character. character. character. 8 ForForFor the the the calculation calculation calculation of of of88BeBeBe we we we use use use the the the same same same initial initial initial state state state InInIn Panel Panel Panel I, I, I, we we we show show show results results results from from from three three three di di di⌅⌅⌅erenterenterent initial initial initial 4 asasas44He,He,He, but but but then then then apply apply apply creation creation creation operators operators operators after after after the the the first first first states,states,states, A, A, A, B, B, B, and and and,,, each each each approaching approaching approaching the the the ground ground ground state state state 8 THETHE TRIPLE-ALPHA TRIPLE-ALPHA PROCESS PROCESS 8

c ANU c ANU 8 thetheBe8Be nucleus nucleus is instable, is instable, long long lifetime lifetime3 alphas3 alphas must mustmeet meet • • the Hoyle state sits just above the continuum threshold • the Hoyle state sits just above the continuum threshold • most of the excited carbon nuclei decay most of the excited carbon nuclei decay The triple alpha reaction(about 4 out of 10000 rate decays produce stable carbon) (about 4 out of 10000 decays produce stable carbon) carbon is further turned into oxygen but w/o a resonant condition • carbon is further turned into oxygen but w/o a resonant condition q • as a functionK of the quark (1)massa triple wonder ! ⌅ ⇥ a triple wonder ! ⇥

Testing the Anthropic Principle with Lattice Simulations – Ulf-G. Meißner – INT, Oct., 2012 < > · ⇥ ⇥ ⇤ ⇤ • Testing the Anthropic Principle with Lattice Simulations – Ulf-G. Meißner – INT, Oct., 2012 < > · ⇥ ⇥ ⇤ ⇤ • 12 ⇥E E 2E , (2) Production of C in stars dependsb ⇥ 8sensitively 4 on the energy differences: ⇥Eb E8 2E4 , (1) ⇧ ⇥ ⌅ ⇥Eh E12 E8 E4 (3) ⇥E E E E (2) ⇥ h ⇥ 12 8 4

2 3 2 3 3 2⇤h¯ ⌃ 3 3 ⇥ 2 3 3 2⇥h¯ ⇥ ⇧2 3 Reaction rate for the triple2 alpha process: r3 3 N exp 3 2⇥h¯ , ⇥ (1)⇧ r3 3 N exp ⇤ ,M k T ⇥r h¯(4)3 2 Nk T ⇥ exp , (3) Oberhummer, Csoto, Schlattl,⌅ Science 289M (2000)k T 88⇥ h¯ k T ⇥ B 3 B ⇥Eb E8 2E4 , (1) B B ⌅ MkBT ⇥ h¯ kBT ⇥ ⇥ ⌅ 1 ⇥Eh E12 E8 E4 (2) ⇧ ELO(t)= at log[Z(t + t)/Z(t)] ⌅ (2) ⇥ where ⇧ ⇥Eb + ⇥Eh = E12 3E4 =379.47(18) keV. - crucial⇧ ⇥ controlE(5)+ ⇥ Eparameter= E 3E =397, 47(18) keV . (4) ⇥Eb E8 2E4 , ⇥ (1) ⇥ b h 12 4 ⇥ ⌅ ⇥Eh 2 E3 12 E8 E4 (2) 1 E(Nt)=12E( )+16 cE exp( Nt/⇧), (3) 3 3 2⇥h¯ ⇥ ⇥Changing ⇧ by ~100ELO keV(t)= destroysa log[ productionZ(t + t)/Z of( teither)] C or O Livio(6) et al.ʼ89; Oberhummer,1 et al.ʼ00 2 t ⌅ ELO(t)= at log[Z(t + t)/Z(t)] (5) r3 3 N exp , (3) ⌅ M k T ⇥ h¯ k T ⇥ B 2 3 B 3 3 2⇥h¯ How⇥ robust is ⇧ with respect to variations of fundamental constantsLt (QCD+QED)? (4) 2 ⌅ E(N )=E( )+c exp( N /⌅), (7) r3 3 N exp , t (3) E t ⇥⌅ E(Nt)=E( )+cE exp( Nt/⌅), (6) ⇧ ⇥Eb +⌅⇥Eh = E12 3E4 = 397, 47(18) keV . (4) ⇧ ⇥ MkBT ⇥ h¯ kBT ⇥ ⇧ ij i j ⌅ ZA (t)= ⇥A exp( tH) ⇥A (5) ⇧ ⇥Eb + ⇥Eh =1 E 3E4 = 397, 47(18) keV . L(4)t ⇧ | (8) | ⌃ L (7) ELO(t)= a log[12Z(t + t)/Z (t)] (5) ⇤⇧ t ⇥ t ⇤⇧ ij j d(ln ZA) 1 i E = lim ij i j(6) E (t)= a log[Z(t + )/Z (t)] ZA (t)= ⇤(5)A exp( tH) ⇤A A (9) E(N LO)=E( )+t c exp( N /t⌅ ), (6)⌃ | | ⌥ t dt ZA (t)= ⇤A exp( tH) ⇤A (8) t ⇧ E t !1 ⌃ | | ⌥

d(ln ZA) EA = lim d(ln ZA)/dt d(ln Z ) (7) E(N )=E( )+c exp( N /⌅ ), E = (6)lim t (10) A t Lt E t A(7) EA = lim (9) ⇧ t dt !1 t ⇤⇧ ⇥⇤ ⇥⇤ dt ij i j Z (t)= ⇥ exp( tHLO) ⇥ (8) ij i Lt j E = (7)lim d(ln Z )/dt LO A (11) A Z (t)= ⇤ exp( tH) ⇤ A (8) A ⇧ | E| A =⌃ lim d(ln ZA)/dt (10) A A ⇤⇧ A t t ⌃ | | ⌥ ⇥⇤ ⇥⇤ O ij i j ij i Z (t)=j ⇥A exp( tH/2) O exp( tH/2) ⇥A (9) Z (t)= ⇤ dexp((ln ZA)tH) ⇤ Z (t)= ⇤(8)exp( tH A) ⇤ (12) ij i j A A A LO A LO A ⇧ | Z (t)= ⇤| exp(⌃ tHLO) ⇤ (11) EA = lim⌃ | | ⌥ (9)⌃ | | ⌥ LO A A t ⌃ | | ⌥ ⇥⇤ dt d(ln Z ) O A Z (t)= ⇤ exp( tH/2) O exp( tH/2) ⇤ O (13) 2 EA = lim A A (1 (9)⌥⌅ i ⌥⌅ j) (3 + i j) A (3Z +(t⌥⌅ )=i ⌥⌅ j⇤) A(1exp( i tH/ j)2) Ogexp(A ⌥⌅ i tH/⌥q ⌥⌅ j2)⌥q⇤A (12) EA = lim dt(ln ZA)/dt V = C ⌃ f(|q(10)) · · +| C⌥ f(q) A · · · · t ⇥⇤ dt LO3 0,1 1,0 ⌃ | 2 2 2| i⌥ j ⇥⇤ 4 4 4 4 4F⇤ q + M⇤ · (10) ij EA =i lim d(ln ZA)/dtj (10) 2 Z (t)= ⇤ exp(t (1tHLO⌃⇤ i) ⇤⌃⇤ j) (3 + i j) (11) (3 + ⌃⇤ i ⌃⇤ j) (1 i j) g ⌃⇤ i ⌃q ⌃⇤ j ⌃q 2 LO A A (1 ⌃⇤ i ⌃⇤ j) (3A + i j) (3 + ⌃⇤ i ⌃⇤ j) (1 i j) gA ⌃⇤ i ⌃q ⌃⇤ j ⌃q VLO3⌃= C|0,1 ⇥⇤f(q) |· ⌥ · + C1,0 f(q) · 22· · · i j VLO3 = C0,1 f(q) · 2 2 · 2+ C1,0 f(q) · · · · i j 4 4 4 4 4 4F⇤ q4 + M⇤ · 4 4 4F 2 q2 + M 2 · ij i j V3N = i Fi(r12,r23,r31)+5perm. (11) ⇤ ⇤ O (14) Z (t)= ⇤ZLOexp((t)=tH/⇤A2)exp(O exp(tHLOtH/) ⇤2)A ⇤ (12)(11) i=1 G (13) A ⌃ A| ⌃ | | |⌥ A⌥ ⇤ 22 22 O 1, 2, 3, ⌥⌅ 1, ⌥⌅ 2, ⌥⌅ 3, ⌥r12, ⌥r23, ⌥r31 (12) V3N = i Fi(r12,r23,r31)+5perm. (15) ZA (t)= ⇤A exp( tH/2) O exp( tH/2) ⇤A (12) V3N = i Fi(r12,r23,r31)+5perm. (14) ⌃ | | ⌥i=1 G 2 G (1 ⌃⇤ i ⌃⇤ j) (3 + ⌅ i ⌅ j) (3 + ⌃⇤ i ⌃⇤ j) (1 ⌅ i ⇤⌅ j) g ⌃⇤ i q⌃ ⌃⇤ j q⌃ i=1 V = C f(q) · · + C f(q) · · A · · ⌅ ⌅2 ⇤ LO3 0,1 1,0 2 2 (0) 2 i gAj ⌥⌅ 1 ⌥q ⌥⌅ 2 ⌥q 4 4 4 4 4F⇤ q +V2NM⇤= · 1 2 · · + CS + CT ⌥⌅ 1 ⌥⌅ 2, (13) 2 4F 2 · ⌥q 2 + M 2 1, 2, 3, ⌃⇤ 1, ·⌃⇤ 2, ⌃⇤ 3, ⌃r12, ⌃r23, ⌃r31 (15) (1 ⌃⇤ i ⌃⇤ j) (3 + ⌅ i ⌅ j) (3 + ⌃⇤ i ⌃⇤ j) (1 ⌅ i (13)⌅ j) g ⌃⇤ i q⌃ ⌃⇤ j q⌃ ⇤ ⇤ V = C f(q) · · + C f(q) · · 1 A · · ⌅ ⌅ LO3 0,1 1,0 2 2 2 i· j 4 22 4 4 4 4F⇤ q + M⇤ (13) d3k m V3N = i Fi(r12,r23,r31)+5 perm. (14) (0) (0) ⌥ N ⌥ T (p⌥ 0, p⌥)=V2N (p⌥ 0, p⌥)+ V2N (p⌥ 0, k) T (k, p⌥) (14) i=1 G (2 ⇤)3 p2 k2 + i ⇥ 1 ⇤ 22 ⌅ V3N = i Fi(r12,r23,r31)+5 perm. (14) ⌅ 1, ⌅ 2, ⌅ 3, ⌃⇤ G1, ⌃⇤ 2, ⌃⇤ 3, ⌃r12, ⌃r23, ⌃r31 (15) i⇤=1 ⌅ 1, ⌅ 2, ⌅ 3, ⌃⇤ 1, ⌃⇤ 2, ⌃⇤ 3, ⌃r12, ⌃r23, ⌃r31 (15) 1 1

1 p +0.03 0 = [1.55 0.15(stat+syst) 0.03(model) ±

p +0.03 A¯s, A¯t 0 = [1.55 0.15(stat+syst) 0.03(model) ±

1 1 a [fm ] t A¯s, A¯t

1 1 ⇥mq/maq) <[fm0.05] | t |

q +1.9 q +0.17 Kas =2.3 1⇥.m8,Kq/maqt)=0< .032.050.18 | |

q q +1.9q q +0.17 K K=5as =25.3,K1.8,K=1at.1=0.032.9 0.18 as ± at ±

q q q q K =2Ka.s4=53.0,K5,Ka=3t =1.0.1 3.05.9 as ± ± at ±± V 3N = 3N q V = q 2N Kas =2.42N 3.0,Kat =3.0 ⇥m3q.5 V = 0.771(14) A¯s +0.934(11)V ±A¯=t 0.069(6) ± < 0.0015 mq V 3N = ⇥ V 3N = ⇤ q mq X q mq X K 2N 2N ⇥mq X V = 0.771(14) A¯s +0.934(11)V A¯=t 0.069(6) KX < 0.0015 Quark⇥ X m q mass phys dependence of the triple-mq ⇥ Xα mreactionq phys rate mq ⇥ m ⇤ mq q EE, Krebs, Lähde, Lee, Meißner,mq XPRL 110 (2013) 112502; arXiv:1303.4856 q (tom qappearX in EPJA) q KX KX ⇥ X mq phys ⇥ X mq phys m¯q mq mq Input: M -dependence of the long-range A forcet known, short-range M -dependence parametrized3N in terms of ⇥Eb E8 2E4 , V = (1) 1 1 ⇥ a A¯t a ⌅ 2N 1 p s +03.03 ¯ t ⇥E E E E spin-singletV = ( S(2)0): A¯s spin-triplet ( S1): At h 12 8 4 F 0 = [1.55 0.15(stat+syst) 0.03(model)F ⇥ 3N ⇥ M phys± ⇥ M phys V = M M 2N V = +0.029 2 3 K =1.652(25) A¯ +3.401(21) A¯ 0.183(7) ⇥ 2 E4 s ¯ ¯t 2 0.039 3 2⇥h¯ ⇥ 1) Is ⇧ a fine-tuned quantity (with respect to A s , A t )? ⇥Yes! 2 3 i 2N 3N i 2N 3N r3 3 N ⌅exp+ Vij + Vijk, ⇤ = E(3)⇤ F¯ ¯ ⌅ + +0V.ij034 + VFijk ⇤ = E ⇤ KE =2.593(19) As +4.940(13) At 0.287(8) 0.043+0.029 ⌅ M k T ⇥⇥2h¯mN k T ⇥ ⇤ | ⇤ 12 | ⇤ ¯ ¯⇥2 mN nearly⇤ | ⇤ the same| ⇤ ratio B ForB example: KE4 =1.652(25) As +3.401(21) At 0.183(7) 0.039 ⌅ ⌅ ¯ ¯1 1 +3.7 for all energies K⇥ = 213(7) As ¯ 348(6) At +24(2)¯ 3.4 +0.027 ⌅ KE 2 =1.940(80) As +3.a870(60)t [fm A]t 0.217(21)2 0.041 ⇧ ⇥E + ⇥E = E 3E = 397, 47(18) keV . ⇥8(4) ⇥ b h 12 4 i 2N 3N i 2N 3N ⇥ ⌅ + Vij + Vijk¯ ⇤ = E ⇤¯ ⌅ ++0V.034ij + Vijk ⇤ = E ⇤ 8 4 K =2.593(19) As +4.940(13) At 0.287(8) 8 4 ⇥Eh EHoyle E Be E He⇥=278keV2 Em12N ⇤ | ⇤ ⇥| E⇤ h ⇥2 mENHoyle 0.043E Be E ⇤He| =278keV⇤ | ⇤ ⇥ ⌅ ⌅⇥ 1 ⇥mq/mq) < 0.05 +3.7+0.026 ELO(t)= a log[⇥EZb(t + Et8)Be/Z (t2)]E4He =92keVbut: KK (5)=2= .27(4)213(7)A¯A¯ +4.348(6)32(3) A¯A¯ ⇥+24(2)E0b.239(11)E8Be 2E4He =92keV t E12⇥ s s | t|t 3.4 0.034 ⇥ ⇥ ⇥Eh+b ⇥Eh + ⇥Eb =370 ⇥E +3⇥.7E + ⇥E =370 K = 213(7) A¯s 348(6) A¯t +24(2)h+b h b ⇥ ⇥Eh ⇥ EHoyle E8Be EvgenyEEvgeny4He Epelbaum Epelbaum=278keV et al.: et The al.: Thedependence⇥ dependenceE⇥h of the3 of. triple4 the tripleE alphaHoyle alpha process process on theE on8 fundamental theBe fundamentalE constants4He constants=278keV of nature of nature 11 11 ⇥ Kq =21.3+1.9,Kq =0.32+0⇥.17 E(Nt)=E( )+cE exp( Nt/⌅ ), (6) as 1.8 at 0.18 ⇧ ⇥Eb E8Be 2E4He6=92keV6 ⇥Eb E6008Be600 2E4He =92keV 2) Strong correlations⇥ between 1 ⇥ ⇥E h + bdifferent⇥Eh +energies:⇥Ehb+b ⇥Eh + ⇥Eb =3704 4 ⇥Eh+b ⇥Eh⇥+b400Eh400+⇥E⇥hE+b =370⇥Eb ⇥ q q ⇥ L ⇥(7) π π ⇥ t K 2 =52 KE*K *5,K =1.1 0.9 as 12 E12 at 200200 ⇤⇧ π ±π ± KE KE 0 0 0 0 12 12 0 p K-factors nfor all energies and π π p 0 0 0n E = 1.16 , E =+2.13 KE KE ij i 0+ j 0+ ⇥Eh+b ⇥Eh + ⇥E8 b 8 E0+ = ⇥E1.h16+b, E⇥0+πEπh =+2+ ⇥E.b13 Z (t)= ⇤ exp( tH) ⇤ energy differences are(8) expres- q -2 -2 π π q ⇥ K∆EK∆E A A A ⇥ K∆EK∆E -200-200 b b K =2.4 c 3c .0,K =3.0 3.5 ⌃ | | ⌥ 4 as at π π sible in terms of that of He -4 -4 K∆EK∆E ± ± h h 0 -400-400 0p 0n p Kπ Kπ 0n d(ln ZA) E = 1.16 , E-6 -6 =+2.13 E0+ = 1.16 , ε Eε =+2.13 0+ 0+ -600-600 0+ EA = lim str (9) -6 -6 -4-4 -2 -2 0 0 2 2str 4 4 6 ⇥m6 q-6 -6 -4-4 -2 -2 0 0 2 2 4 4 6 6 t =118 2MeV ¯ (1) ¯ =118 2MeV (1) ⇥⇤ dt 0.771(14) As +0.934(11) AKt π Kπ 0.069(6) < 0.0015 π π ± E E ± KEKE em str em 4 4 em strmq em 4 4 /( + )=0.55 ...0.65%⇥ (2) /(⇤ + )=0.55 ...0.65% (2) Fig.Fig. 5. Correlation 5. Correlation of the of⇥ the-particle⇥-particle energy energyE4 withE4 with the ener- the ener-Fig.Fig. 6. Correlation 6. Correlation of the of⇥ the-particle⇥-particle energy energyE4 withE4 with the energy the energy EA = lim d(ln ZA)/dt (10)str gies Egies8, EE128,, EE1212, andE12 theand Hoyle the Hoyle state state excitation excitation energy energystrEcEdic erencesdierences pertinent pertinent to the to triple-alpha the triple-alpha process process under under variation variation underunder variation variation of A¯ of Ain¯ thein range the range1 ...1 .... 1 . of A¯of Ain¯ thein range the range1 ...1 .... 1 . t =118 2MeV s,t s,t { { } } =118s,t(1)s,t 2MeV{ { } } (1) ⇥⇤ ± ± em str em mq em str em /( + di)=0erences,dierences,.55 are strongly are... strongly0. correlated.65% correlated. In order In order to quantify to quantify/(wherewhere the(2)+ restriction the restriction )=0⇤(⌅)⇤<(⌅)100.55< keV100... keVleads, leads,0 using.65% using Eq. (50), Eq. (50), (2) ij i j to the tolerance ⇤⇥| ||/⇥ | 2.5% for compatibility with str the correlations,the correlations, we plot we plot in Figs. in Figs. 5 the 5 changes the changes in the inK the- Kto- the tolerance ⇤⇥em/⇥emem em2.5% for compatibility with Z (t)= ⇤ exp( tHLO) ⇤ (11) ⇤ ⇤ the existencethestr existence of| carbon-oxygen of| carbon-oxygen| ⌃ | based⌃ based life. life. This This is consis- is consis- LO A A =118 2MeV factorsfactors for the(3) for energiesthe energiesE8, E812, ,EE1212, Eand12 and for the for Hoylethe Hoyle =118 2MeV (3) ⌃ | | ⌥ tenttent with with the the4% bound4% bound reported reported in Ref. in Ref. [11]. [11]. statestate excitation excitation energy energyEcversusEc versus the changesthe changes in K inE4 KE4 ± ¯ ¯ ¯ ¯ For shiftsFor shifts in⌃ the in⌃ lightthe light quark± quark masses, masses, we find we find inducedinduced by the by independentthe independent¯ variations variations of A ofs andAs andAt inAt in em str em the rangethe range1 ...11.... Correlation1 . CorrelationAt of the ofem⇥ the-particle⇥-particlestr energy energy em /( + )=0.55 ...0.65 {(4){ } } /( + )=0.55 ...0.65 ⇤m ⇤mq (4) O E withE with the energy the energy dierences dierencesE , EEb, andEh and⌅ pertinent⌅ pertinent ¯ ¯ q 4 4 b h 0.572(19)0.572(19)A¯ +0As.933(15)+0.933(15)A¯ A0t.064(6)0.064(6) Z (t)= ⇤ exp( tH/2) O exp( tH/2) ⇤ (12) str to the triple-alpha process under the same variation of str s t ⇥ ⇥m m A A A to the triple-alpha process under the same variation of =118 2MeV⇥ ⇥q ⇤q ⇤ (3) 3N =118A¯ A¯are visualizedare visualized2MeV in Fig. in Fig. 6. Such 6. Such behavior behavior can canreadily readily ⌅ (3)⌅ ⇧ ⇧ ⌃ | | ⌥ s,t s,t < 0.15%< 0..15%. (64) (64) V = be explainedbe explained in± terms in terms of the of⇥ the-cluster⇥-cluster structure structure of 8Be, of 8Be, ± 12 12 em str em ¯ 2Nem str em C andC and the Hoylethe Hoyle state. state. Notice Notice that that such such correlations correlationsTheThe obtaibed obtaibed constraints constraints on the on valuesthe values of A¯ ofs,t Acompati-s,t compati- V =/( + )=0relatedrelated to.55 carbon to carbon... production0 production.65 have have been been speculated speculated /( upon upon ble+ withble(4) with the conditionthe)=0 condition⇤(⌅)⇤.(<55⌅)100<...100 keV keV are0.65 visualized are visualized in in (4) | | str earlierearlier [10,38]. [10,38]. Fig.Fig. 7. The 7. The light light (dark) (dark)| shaded| shaded bands bands cover cover the allowedthe allowed 2 str ¯ ¯ (1 ⌃⇤ i ⌃⇤ j) (3 + ⌅ i ⌅ j) (3 +⌃⇤i ⌃⇤=118j) (1 2MeV⌅ i ⌅ j,) g ⌃⇤ i q⌃ ⌃⇤ j q(5)⌃ valuesvalues=118 for the for quantities the quantitiesA2MeVs,t Aifs,t theif light the, light quark quark mass massmq ismq is (5) A variedvaried by 1% by (5%). 1% (5%). In the In most the most generic generic scenario, scenario, assuming assuming VLO3 = C0,1 f(q) · · + C1,0 f(q) em · ± · · · ⌅ i ⌅ j em ± ¯ ¯ ¯ ¯ 2 2 2 thatthat both both dimensionless dimensionless quantities quantitiesAs andAs andAt areAt ofare or- of or- 8 Reaction8 Reaction rate rate of· the of the triple triple alpha alpha process process +0.029 ¯ ¯ 4 4 4 4 4F q + M ¯ ¯ der oneder and,one and, therefore, therefore, 0.572(19) 0.572(19)A¯ +0As.933(15)+0.933(15)A¯ At (1), (1), K ⇤=1.652(25)⇤ A +3.401(21) A 0.183(7) s t ⇤ O =0.55 ...0.65 4He (6) s t our resultsour=0 results imply. imply55 that0.039... that a change a0 change.65 in the in light the light quark quark⇤ massesO masses (6) str em str We now turn to the reaction rate in thestr triple alphaem pro- str + (13) We now turn to the reaction rate in the triple alpha+ pro- of onlyof only0.15%0.15% would would su⇥ce su⇥ toce render to render the existencethe existence of of =118cesscess which which is2MeV approximately is approximately, given given by the by expressionthe expression in in(5)=118⌃ 2MeV, (5) ¯ ¯ carbon-oxygencarbon-oxygen⌃ based+0 based.034 life unlikely. life unlikely. Stated Stated dierently, dierently, the the K em =2.593(19)Eq. (3)Eq.± and (3) anddetermineA determine+4 the rangethe. range940(13) of variations of variations ofAm ofq andmq emand0.287(8) ± Hoyle s t “survivability“survivability band” band”0. corresponding043 corresponding to ⇤ tomq/m⇤mq/m

p +0.03 (Eh+b) < 100 keV 0 = [1.55 0.15(stat+syst) 0.03(model) 12 16 3) How± much change in m q can be accommodated| to still| have enough C, O production? p +0.03 0 = [1.55 0.15(stat+syst) 0.03(model) ±mq ⇥⇤ < 100 keV 0.771(14) A¯s +0.934(11) A¯t 0.069(6) < 0.0015 | | m A¯ ⇥ ⇤ q t ⇥⇤ < 100 keV 1 1 | | 3N „Survivabilityat [fm bands“] for carbon-oxygen based life V = mq due to 0.5%, 1%, 5% variation of mq V 2N = 1 1 1 at [fm ] ⇥mq/m1 q) < 0.05 | | A¯t +0.029 up-to-date⇥m /m chiral) < EFT0.05 ¯ 3N ¯ q q K4Heq =1+1.652(25).9 q As +3+0.17.401(21) At 0.183(7) 0.039 | | K =2.3 ,K =0.32V = 2 as 0.50.51.8 at 0.18 calculation (N LO): ¯ V 2N = ¯ +0.034 KHoyle =2.593(19) As +4.940(13) At 0.287(8) 0.043 q +1.9 q +0.17 Kas =2.3 1.8,Kat =0.32 0.18 q q +3.7 t

t ¯ ¯ K Ka =5= 5213(7),Ka =1As .1 348(6)0.9 At +24(2) Eh+b s t 3.4 ¯ Berengut¯ et al., PRD+0 87.029 (2013) _ _ A A ±0 0 ± K4He =1.652(25) As +3.401(21) At 0.183(7) 0.039 q q ¯ Ka¯s =5 5,Ka+0t =1.034 .1 0.9 q q KHoyle =2.593(19) As +4.940(13) At 0±.287(8) 0.043 ± Kas =2.4 3.0,Kat =3.0 3.5 ± |mq /m±q| = 0.005K = 213(7) A¯ 348(6) A¯ +24(2)+3.7 -0.5-0.5 m XEh+b s t 3.4 q q q q KX =|mq /mq| = 0.01 Kas =2.4 3.0,Kat =3.0 3.5 X ⇥mmq phys ± ± q mq 0.771(14) A¯s +0.934(11) A¯t 0|.069(6)mq /mq| = 0.05 < 0.0015 m ⇥ -1 ⇤ q m ⇥X -1 -1 -0.5 0 0.5 q 1 q ¯ ¯ ⇥mq -1 -0.5 0_ 0.5 KX =01.771(14) As +0.934(11) At 0.069(6) < 0.0015 X ⇥mq phys mq 1 _ ⇥ mq ⇤ as A mq ¯ A s As s ⇥ M phys M ⇥a 1 mq ¯ s A¯t As ⇥ ⇥M phys M 3N ¯ V = F At V 2N = V 3N = F ⇥ 2 V 2N = ¯ ⌅i + V 2N¯ + V 3N ⇥+0.029= E ⇥ KE4 =1.652(25) As +3.401(21)ijAt 0.183(7)ijk 0.039 ⇤ 2 ⇥2 mN ⇤ | ⇤ | i ⇤ 2N 3N ⌅¯ ¯ +0.027 ⌅ + Vij + Vijk ⇥ = E ⇥ KE =1.940(80) As +3.870(60) At 0.217(21) 0.041⌅2 mN ⇧ | ⇤ ¯ | ⇤ ¯ +0.029 8 KE4 =1.652(25) As +3.401(21) At 0.183(7) 0.039 ⌃ +0.034 +0.027 K =2.593(19) A¯s +4.940(13) A¯t 0.287(8) ¯ ¯ E12 0.043 KE =1.940(80) As +3.870(60) At 0.217(21) 0.041 8 Eh EHoyle E8Be E4He =278keV ¯ ¯ +0.026 Eh EHoyle E8Be E4He =278keV +0.034 K =2.27(4) A +4.⇥32(3) A 0.239(11) ¯ ¯ E12 s t 0.034 ⇥ KE =2.593(19) As +4.940(13) At 0.287(8) 0.043 8 4 12 Eb E Be 2E He =92keVEb E8Be 2E4He =92keV K = 213(7) A¯ 348(6)⇥ A¯ +24(2) +3.7 ⇥ ¯ ¯ +0.026 ⇥ s t 3.4 KE =2.27(4) As +4.32(3) At 0.239(11) 0.034 Eh+b Eh + Eb=370Eh+b Eh + 12Eb =370 ⇥ ⇥ ¯ ¯ +3.7 1 K⇥ = 213(7) As 348(6) At +24(2)3.4 1 E E + E 1 h+b ⇥ h b

0 E p = 1.16 , E0n =+2.13 0+ 0+ 1 SRG Overview Methods Lattice NM DFT The shell model revisited Configuration interaction techniques • light and heavy nuclei • detailed spectroscopy • quantum correlations (lab-system description)

9&:-2;*.%&<$-,#)%&*7:#.3*/*=%,.37* 132>%0*

!"#$%&#'"(#)%&* +,-&.#)%&/0"#$%&#'"(#)%&** 1%&23*4#,'%*

5673,8#6'37* • Direct comparison with experiment • Pseudo-data to inform reaction theory and DFT

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT

Confronting theory and experiment to both driplines 0 Precision mass measurements test N=8 -2 Exciting advances for neutron-richimpact of nuclei chiral 3NF 24 -4 3N forces key to explain O as heaviest oxygenProton isotope rich [Holt et al., arXiv:1207.1509] Otsuka, Suzuki, Holt, Schwenk, Akaishi, Phys. Rev. Lett. 105, 032501 (2010). -6 Neutron rich [Gallant et al., arXiv:1204.1987]

-8 Many new tests possible! NN predicted increased binding for neutron-rich calcium NN+3N -10 NN+3N (sdf7/2p3/2) Ground-State Energy (MeV) AME2011 confirmed in precision Penning trap exp. -12 IMME 51,52 16 17 18 19 20 21 22 23 24 5! and 3! deviation in Ca from AME Mass Number A TITAN collaboration + Holt, Menendez, Schwenk, submitted. Shell model description using chiral potential evolved to Impact on global predictions? Vlow k plus 3NF ﬁt to A = 3, 4 Excitations outside valence space included in 3rd order MBPT

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT

Confronting theory and experiment to both driplines 0 Precision mass measurements test N=20 Exciting advances for neutron-richimpact of nuclei chiral 3NF -2 24 3N forces key to explain O as heaviest oxygenProton isotope rich [Holt et al., arXiv:1207.1509] -4 Otsuka, Suzuki, Holt, Schwenk, Akaishi, Phys. Rev. Lett. 105, 032501 (2010). Neutron rich [Gallant et al., arXiv:1204.1987] -6

Many new tests possible! -8 NN predicted increased binding for neutron-rich calcium NN+3N

NN+3N (pfg9/2)

Ground-State Energy (MeV) -10 Exp and AME2011 extrapolation IMME confirmed in precision Penning trap exp. -12 51,52 40 41 42 43 44 45 46 47 48 5! and 3! deviation in Ca from AME Mass Number A TITAN collaboration + Holt, Menendez, Schwenk, submitted. Shell model description using chiral potential evolved to Impact on global predictions? Vlow k plus 3NF ﬁt to A = 3, 4 Excitations outside valence space included in 3rd order MBPT

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT

Non-empirical shell model [from J. Holt] Solving the Nuclear Many-Body Problem Nuclei understood as many-body system starting from closed shell, add nucleons Interaction and energies of valence space orbitals from original Vlow k This alone does not reproduce experimental data

0h,1f,2p

0g,1d,2s

0f,1p

-“ sd”-valence space Active nucleons occupy 0p valence space

0s Assume filled core

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT

Hjorth-Jensen, Kuo, Osnes (1995)

0h,1f,2p

0g,1d,2s Strong interactions with core generate effective interaction 0f,1p between valence nucleons

-“ sd”-valence space Active nucleons occupy 0p valence space

0s Assume filled core

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT

Single-particle energies (SPEs) Hjorth-Jensen, Kuo, Osnes (1995)

0h,1f,2p

0g,1d,2s Strong interactions with core generate effective interaction 0f,1p between valence nucleons

-“ sd”-valence space Active nucleons occupy 0p valence space

0s Assume filled core

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT

Chiral 3NFs meet the shell model [from J. Holt] Drip Lines and Magic Numbers: The Evolving Nuclear Landscape Important in light nuclei, nuclear matter… What are the limits of nuclear existence? How do magic numbers form and evolve?

82! Heaviest oxygen isotope

126!

50!

82! protons 28! Otsuka, Suzuki, Holt, Schwenk, Akaishi, PRL (2010) 20! 50!

8! 28! 2! 20! 2! 8! neutrons 3N forces essential for medium mass nuclei Dick Furnstahl TALENT: Nuclear forces 4 4 2 2 (a) G-matrix NN + 3N (∆) forces (b) Vlow k NN + 3N (∆,N(a) LO G-matrix) forces NN + 3N (∆) forces (b) Vlow k NN + 3N (∆,N LO) forces

0 0 d3/2 d3/2 d3/2 d3/2

-4 -4 s 1/2 s 1/2 SRG Overview Methodss 1/2 Lattice NM DFT s 1/2 d5/2 d5/2 d5/2 d5/2

-8 -82 2 Chiral 3NFs meetNN + 3N (∆) the shell NN model + 3N (N LO) NN[from + 3N (∆) J. Holt] NN + 3N (N LO) ∆) ∆) Single-Particle Energy (MeV) NN NN + 3NSingle-Particle Energy (MeV) ( NN NN + 3N ( 3N Forces for Valence-ShellNN Theories NN 814 16 20 814 168 20 14 16 20 814 16 20 Normal-orderedNeutron Number 3N: contribution(N) toNeutron valence Number neutron(N) Neutron Number interactions (N) Neutron Number (N)

Effective(c) 3-body interaction two-body (d) 3-body interactionsEffective with(c) 3-bodyone interaction one-body (d) 3-body interactions with one more neutron added to (c) more neutron added to (c)

16O core 16O core

ab V a'b' ab V a'b' 1 3N,eff = # " 3N " a V3N,eff a' = $ "#a V3N "#a' " =core 2 "# =core

Combine with microscopic NN: eliminate empirical adjustments ! ! Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT

Chiral 3NFs meet the shell model [from J. Holt] (a) Phenomenological Forces (b) NN-only Theory Drip Lines and Magic Numbers: 3N Forces in Medium-Mass Nuclei

Important in light 1 nuclei, nuclear matter… What are the limits of nuclear existence? How do magic numbers form and evolve? N=28 magic number in calcium 82! (c) NN + 3N (pf-shell) (d) NN + 3N (pfg shell) Holt, Otsuka, Schwek, 6 9/2 Suzuki, arXiv:1009.5984 5 4 126! 3 Energy (MeV) 1 + 2 2 1 V (NN) low50 k ! 0 2 Vlow k + 3N(N LO) 2 Vlow k + 3N(N LO) [MBPT] 8242! 44 46 48 50 52 54 56 58 60 62 64 66 68 Mass Number A protons 28! 20! 50!

8! 28! 2! 20! 3N forces essential for medium mass nuclei 2! 8! neutrons Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT Overlapping theory methods cover all nuclei

dimension of the problem

Interfaces provide crucial clues

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT What do (ordinary) nuclei look like?

Charge densities of magic nuclei (mostly) shown Proton density has to be “unfolded” from ρcharge(r), which comes from elastic electron scattering Roughly constant interior density with R (1.1–1.2 fm) A1/3 ≈ · Roughly constant surface thickness

= Like a liquid drop! ⇒

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT What do (ordinary) nuclei look like?

= Like a liquid drop! ⇒

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT

Semi-empirical mass formula (A = N + Z) Z 2 (N Z)2 E (N, Z ) = a A a A2/3 a a − + ∆ B v − s − C A1/3 − sym A

Many predictions! Rough numbers: a 16 MeV, v ≈ a 18 MeV, a 0.7 MeV, s ≈ C ≈ a 28 MeV sym ≈ Pairing ∆ 12/√A MeV ≈ ± (even-even/odd-odd) or 0 [or 43/A3/4 MeV or . . . ] Surface symmetry energy: a (N Z)2/A4/3 surf sym − Much more sophisticated mass formulas include shell effects, etc.

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT Semi-empirical mass formula per nucleon 2 2 EB(N, Z) 1/3 Z (N Z ) = a a A− a a − A v − s − C A4/3 − sym A2

Divide terms by A = N + Z Rough numbers: a 16 MeV, a 18 MeV, v ≈ s ≈ a 0.7 MeV, a 28 MeV C ≈ sym ≈ Surface symmetry energy: a (N Z)2/A7/3 surf sym − Now take A with → ∞ Coulomb 0 and ﬁxed → N/A, Z/A Surface terms negligible

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT Nuclear and neutron matter energy vs. density

[Akmal et al. calculations shown] 40 Uniform with Coulomb turned off APR NRAPR Density n (or often ρ) RAPR 2 3 Fermi momentum n = (ν/6π )kF 20 Neutron matter (Z = 0) has Neutron matter positive pressure E/A (MeV) 0 Symmetric nuclear matter (N = Z = A/2) saturates Nuclear matter Empirical saturation at about -20 E/A 16 MeV and 0 0.05 0.1 0.15 0.2 0.25 -3 ≈ − 3 n (fm ) n 0.17 0.03 fm− ≈ ±

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT Low resolution calculations of nuclear matter Evolve NN by RG to low momentum, ﬁt NNN to A = 3, 4 Predict nuclear matter in MBPT [Hebeler et al. (2011)] 5 3 −1 V NN from N LO (500 MeV) Λ = 1.8 fm low k −1 Λ = 2.0 fm 0 3NF fit to E and r −1 3H 4He Λ = 2.2 fm −1 Λ = 2.8 fm −5 Λ −1 2.0 < 3NF < 2.5 fm

−10

−15 Empirical saturation Energy/nucleon [MeV] Hartree-Fock point 2nd order 3rd order pp+hh −20 0.8 1.0 1.2 1.4 1.6 0.8 1.0 1.2 1.4 1.6 0.8 1.0 1.2 1.4 1.6 −1 −1 −1 kF [fm ] kF [fm ] kF [fm ] Cutoff dependence at 2nd order signiﬁcantly reduced 3rd order contributions are small Remaining cutoff dependence: many-body corrections, 4NF?

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT Overlapping theory methods cover all nuclei

dimension of the problem

Interfaces provide crucial clues

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT

DFT for nuclei [UNEDF and NUCLEI projects] Nuclear Density Functional Theory and Extensions !"#$%& Technology to calculate observables Global properties Spectroscopy DFT Solvers Functional form Functional optimization Estimation of theoretical errors '"()*+&,("-.%+& /$"012"34&

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Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT Skyrme EDF and beyond τ 3 1 1 1 = + t ρ2 + t ρ2+α + (3t + 5t )ρτ+ (9t 5t ) ρ 2 + ESkyrme 2M 8 0 16 3 16 1 2 64 1 − 2 |∇ | ··· Kohn−Sham Potentials

Kohn-Sham density functional theory Skyrme = iterate to energy HFB ⇒ self-consistency functional solver t0, t1, t2, ... Pairing is critical Improve functional with same iteration scheme Orbitals and Occupation #’s Schematic equations to solve self-consistently: δE [ρ] 2 V (r) = int [ ∇ +V (x)]ψ = ε ψ = ρ(x) = n ψ (x) 2 KS δρ( ) ⇐⇒ −2m KS α α α ⇒ α| α | r α X Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT Skyrme EDF and beyond τ 3 1 1 1 = + t ρ2 + t ρ2+α + (3t + 5t )ρτ+ (9t 5t ) ρ 2 + ESkyrme 2M 8 0 16 3 16 1 2 64 1 − 2 |∇ | ··· Kohn−Sham Potentials

Kohn-Sham density functional theory DME = iterate to energy HFB ⇒ self-consistency functional solver A[ρ ], B[ρ ], ... Pairing is critical Improve functional with same iteration scheme Orbitals and Occupation #’s Schematic equations to solve self-consistently: δE [ρ] 2 V (r) = int [ ∇ +V (x)]ψ = ε ψ = ρ(x) = n ψ (x) 2 KS δρ( ) ⇐⇒ −2m KS α α α ⇒ α| α | r α X Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT RESEARCH LETTER“The limits of the nuclear landscape” J. Erler et al., Nature 486, 509 (2012) 120 Stable nuclei Known nuclei Drip line Two-proton drip line N = 258 S2n = 2 MeV Z = 82 Z 80 SV-min Two-neutron drip line N = 184 110 Z = 50

Proton number, number, Proton 40 N = 126 100 Z = 28 N = 82 Z = 20 90 230 244 232 240 248 256 N = 28 N = 50 N = 20 0 0 40 80 120 160 200 240 280 Neutron number, N Figure 1 | Nuclear even–evenProton landscape as and of 2012. neutronMap of bound driplines even–even predictedshown together by with Skyrme the statistical EDFs uncertainties at Z 5 12, 68 and 120 (blue nuclei as a function of Z and N. There are 767 even–even isotopes known error bars). The S2n 5 2 MeV line is also shown (brown) together with its 2,3 experimentally, both stable (blackTotal: squares) and6900 radioactive500 (green nuclei squares). withsystematicZ 120 uncertainty ( 3000 (orange). known) The inset shows the irregular behaviour of the Mean drip lines and their uncertainties (red) were obtained± by averaging the two-neutron≤ drip line≈ around Z 5 100. results of different models. The two-neutronEstimate drip line systematic of SV-min (blue) errors is by comparing models application of modern optimization and statistical methods, together the microscopic–macroscopic finite-range droplet model (FRDM)18, with high-performance computing, has revolutionizedDick nuclear Furnstahl DFT theTALENT: Brussels–Montreal Nuclear forces Skyrme–HFB models based on the Hartree– during recent years. Fock–Bogoliubov (HFB) method17 and Gogny force models19,20. In our study, we use quasi-local Skyrme functionals15 in the Figure 2 illustrates the difficulties with theoretical extrapolations particle–hole channel augmented by the density-dependent, zero- towards drip lines. Shown are the S2n values for the isotopic chain of range pairing term. The commonly used Skyrme EDFs reproduce total even–even erbium isotopes predicted with different EDF, SLy421, SV- binding energies with a root mean square error of the order of min13,UNEDF015, UNEDF122, and with the FRDM18 and HFB-2117 1–4 MeV (refs 15, 16), and the agreement with the data can be signifi- models. In the region for which experimental data are available, all cantly improved by adding phenomenological correction terms17. The models agree and well reproduce the data. However, the discrepancy Skyrme DFT approach has been successfully tested over the entire between various predictions steadily grows when moving away from chart of nuclides on a broad range of phenomena, and it usually per- the region of known nuclei, because the dependence of the effective forms quite well when applied to energy differences (such as S2n), radii force on the neutron-to-proton asymmetry (neutron excess) is poorly and nuclear deformations. Other well-calibrated mass models include determined. In the example considered, the neutron drip line is

4 N = 76 8 154 162 2 (MeV) (MeV)

4 2n 24 2p S S Er 0 0 20 58 62 66 140 148 156 164 Z N 16

FRDM (MeV)

2n 12

S HFB-21 SLy4 8 UNEDF1 Er UNEDF0 4 SV-min exp 0 Drip line Experiment 80 100 120 140 160 Neutron number, N

Figure 2 | Calculated and experimental two-neutron separation energies of steadily when extrapolating towards the two-neutron drip line (S2n 5 0). The even–even erbium isotopes. Calculations performed in this work using SLy4, bars on the SV-min results indicate statistical errors due to uncertainty in the 2 SV-min, UNEDF0 and UNEDF1 functionals are compared to experiment and coupling constants of the functional. Detailed predictions around S2n 5 0 are FRDM18 and HFB-2117 models. The differences between model predictions are illustrated in the right inset. The left inset depicts the calculated and small in the region where data exist (bracketed by vertical arrows) and grow experimental two-proton separation energies at N 5 76.

120 Stable nuclei Known nuclei Drip line Two-proton drip line N = 258 S2n = 2 MeV Z = 82 Z 80 SV-min Two-neutron drip line N = 184 110 Z = 50

Proton number, number, Proton 40 N = 126 100 Z = 28 N = 82 Z = 20 90 230 244 232 240 248 256 N = 28 N = 50 N = 20 0 0 40 80 120 160 200 240 280 Neutron number, N Figure 1 | Nuclear even–even landscape as of 2012. Map of bound even–even shown together with the statistical uncertainties at Z 5 12, 68 and 120 (blue nuclei as a function of Z and N. There are 767 even–even isotopes known error bars). The S2n 5 2 MeV line is also shown (brown) together with its experimentally,2,3 both stable (black squares) and radioactive (green squares). systematic uncertainty (orange). The inset shows the irregular behaviour of the Mean drip lines and their uncertainties (red) were obtained by averaging the two-neutron drip line around Z 5 100. results of different models. The two-neutron drip line of SV-min (blue) is application of modern optimization and statistical methods, together the microscopic–macroscopic finite-range droplet model (FRDM)18, with high-performance computing, has revolutionized nuclear DFT the Brussels–Montreal Skyrme–HFB models based on the Hartree– during recent years. Fock–Bogoliubov (HFB) method17 and Gogny force models19,20. In our study, we use quasi-local Skyrme functionals15 in the Figure 2 illustrates the difficulties with theoretical extrapolations particle–hole channel augmented by the density-dependent, zero- towards drip lines. Shown are the S2n values for the isotopic chain of range pairing term. The commonly used Skyrme EDFs reproduce total even–even erbium isotopes predicted with different EDF, SLy421, SV- binding energies with a root mean square error of the order of min13,UNEDF015, UNEDF122, and with the FRDM18 and HFB-2117 1–4 MeV (refs 15, 16), and the agreement with the data can be signifi- models. In the region for which experimental data are available, all cantly improved by adding phenomenological correction terms17. The models agree and well reproduce the data. However, the discrepancy Skyrme DFT approach has been successfully tested over the entire between various predictions steadily grows when moving away from chart of nuclides on a broad range of phenomena,SRG Overview and it usuallyMethods per- Latticethe regionNM of knownDFT nuclei, because the dependence of the effective forms quite well when applied to energy differences (such as S2n), radii force on the neutron-to-proton asymmetry (neutron excess) is poorly and nuclear deformations. Other well-calibrated mass models include determined. In the example considered, the neutron drip line is “The limits of the nuclear landscape”

4 N = 76 8 154 162 2 (MeV) (MeV)

4 2n 24 2p S S Er 0 0 20 58 62 66 140 148 156 164 Z N 16

FRDM (MeV)

2n 12

S HFB-21 SLy4 8 UNEDF1 Er UNEDF0 4 SV-min exp 0 Drip line Experiment 80 100 120 140 160 Neutron number, N

Figure 2 | Calculated and experimentalTwo-neutron two-neutron separation separation energies energies of steadily ofwhen even-even extrapolating towards erbium the two-neutron isotopes drip line (S2n 5 0). The even–even erbium isotopes. Calculations performed in this work using SLy4, bars on the SV-min results indicate statistical errors due to uncertainty in the 2 SV-min, UNEDF0 and UNEDF1 functionals are compared to experiment and coupling constants of the functional. Detailed predictions around S2n 5 0 are FRDM18 and HFB-2117 models. TheCompare differences between different model predictions functionals, are illustrated with in the right uncertainties inset. The left inset depictsof ﬁts the calculated and small in the region where data existDependence (bracketed by vertical arrows) on neutron and grow excessexperimental poorly two-proton determined separation energies at (cf.N 5 76. driplines) 510 | NATURE | VOL 486 | 28 JUNE 2012 Dick Furnstahl TALENT: Nuclear forces ©2012 Macmillan Publishers Limited. All rights reserved SRG Overview Methods Lattice NM DFT UNEDF Project: Use ab initio pseudo-data Neutron Matter: Neutrons in a Trap What are the properties of neutron-rich matter?

Uext

! Protons and neutrons Can bind neutrons by form self-bound system applying an external trap

Neutron Drops (mini neutron stars) calculated with Coupled-Cluster theory Use external harmonic oscillator potential, varying ! " ext

Put neutrons in a harmonic oscillator trap with ~ω (cf. cold atoms!) ! Calculate exact result with AFDMC [S. Gandolﬁ, J. Carlson, and S.C. Pieper, Phys. Rev. Lett. 106, 012501 (2011)] (or with other methods) UNEDF0 and UNEDF1 functionals improve over Skyrme SLy4!

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT Interaction with applied math experts Optimization Algorithms for Calibrating Extreme Scale Simulations

Typical Challenges New Algorithm POUNDERS

! Computational expense of simulation only allows ! Exploits mathematical structure for evaluating a few sets of parameter values in calibration problems ! Derivatives with respect to parameters can be ! Benefits from expert knowledge unavailable or intractable to compute/approximate " data, weights, uncertainties, etc. ! Experimental data incomplete or inaccurate ! Obtains good fits in minimal ! Sensitivity analysis/confidence regions desired number of simulations

POUNDERS obtains better solutions faster Energy density functionals (EDFs) for UNEDF ! Enables fitting of complex, state-of-the-art EDFs • Optimization previously avoided because too many evaluations required to obtain desirable features ! Substantial computational savings over alternatives ! Using resulting EDF parameterizations, the entire nuclear mass table was computed and is now distributed at www.massexplorer.org " Nuclear Energy Density Optimization. Kortelainen et al., Physical Review C 82, 024313, 2010 " Three joint physics & optimization publications @ SciDAC11!

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT

Ab Initio Configuration Interaction full space truncated space

Many SciDAC-2 UNEDF project body method

UniversalNuclearEnergy Diagonalizaon global properes Truncaon+diagonalizaon Observables spectroscopy Monte Carlo DensityFunctional Expt scaering Observables Expt spectroscopic Collaboration of physicists, informaon applied mathematicians, and

Nuclear Density computer scientists Functional Theory and Extensions Nuclear Energy Density US funding but international Funconal Nuclear DFT HFB (self-consistency) collaborators also Symmetry breaking

Symmetry restoraon Observables See unedf.org for highlights! Mul-reference DFT (GCM) Time dependent DFT (TDHFB) Expt global QRPA properes excited states Observables decays Expt ﬁssion

New SciDAC-3 NUCLEI project: Compound Nucleus and Direct Reaction Theory Coupled Channels NUclearComputational DWBA

Low-EnergyInitiative Opcal potenals Observables Expt cross secons

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT

Ab Initio Configuration Interaction ;"$$(.G&#%( 0'"*#&0%B(.G&#%(

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4G2#&$(G30%*2&$.( 45.%'6&5$%.( !"#$% #'3..(.%#23*.( SRG Overview Methods Lattice NM DFT Interaction with computer science experts “Load Balancing at Extreme Scale” – Ewing Lusk, Argonne National Laboratory

ASCR- SciDAC UNEDF Computer Science Highlight ()*+%,-+.'' !"#$%&' ! Enable Green’s Function Monte Carlo calculations ! Demonstrate capabilities of simple programming for 12C on full BG/P as part of UNEDF project models at petascale and beyond ! Show path forward with hybrid programming ! Simplify programming model models in library implementation ! Scale to leadership class machines

2'#0%3"4*56,7"(,&*8,%'#9)"*-'":/;$$*-'"<*/7)=*'%0"*(%4">* /0120+..' ! Initial load balancing was of CPU cycles !"#$%&'"()*%+*!")",-%(*.")/%01* ! Next it became necessary to balance memory utilization as well ! Finally ADLB acquired the capability to balance message flow ! “More Scalability, Less Pain” by E. Lusk, S.C. Pieper and R. Butler published in SciDAC Review 17, 30 (2010)

Dick Furnstahl TALENT: Nuclear forces SRG Overview Methods Lattice NM DFT SciDAC-3 NUCLEI Project

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Dick Furnstahl TALENT: Nuclear forces