Lattice Simulations with Chiral Nuclear Forces

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1 Lattice Simulations with Chiral Nuclear Forces Hermann Krebs FZ Jülich & Universität Bonn July 23, 2008, XQCD 2008, NCSU In collaboration with B. Borasoy, E. Epelbaum, D. Lee, U. Meißner

2 Outline EFT and Lattice QCD Nuclear forces in EFT Lattice EFT (Formalism) Lattice EFT to LO Nucleon-Nucleon scattering within Lattice EFT up to NLO Dilute neutron matter up to NLO First results for NNLO Conclusion & Outlook

ChPT and low energy QCD Spontaneous + explicit (by small quark masses) breaking of chiral symmetry in QCD Existence of light weakly interacting Goldstone bosons Chiral

3 ChPT and low energy QCD Spontaneous + explicit (by small quark masses) breaking of chiral symmetry in QCD Existence of light weakly interacting Goldstone bosons Chiral Perturbation theory (ChPT) Expansion in small momenta and masses of Goldstone bosons Systematic description of QCD by ChPT in low energy sector ( low momenta q Â ' 1 GeV )

: : Computable by LQCD Nucleon in LQCD Nucleons as point particles

4 EFT and Lattice QCD Free parameters of QCD Free parameters of ChPT s ; quark masses F ¼ ; g ¼N ; c i : : : ; d i : : : ; C S ; C T : : : Computable by LQCD Nucleon in LQCD Nucleons as point particles on the lattice Difference in computational effort» 0:1fm» 1: : : 4fm

5 Weinberg s scheme for NN Weinberg, Nucl. Phys. B 363: 3 (1991) No perturbative description for bound states V V NN cuts violate power counting Construct effective potential perturbatively V Solve Lippmann-Schwinger equation nonperturbatively T V V T

6 Nucleon-nucleon force up to N 3 LO Ordonez et al. 94; Friar & Coon 94; Kaiser et al. 97; Epelbaum et al. 98, 03; Kaiser 99-01; Higa et al. 03; Chiral expansion for the 2N force: (0) (2) (3) (4) V 2N = V 2N +V 2N + V 2N + V 2N + LO: 2 LECs NLO: renormalization of 1π-exchange renormalization of contact terms 7 LECs leading 2π-exchange N 2 LO: renormalization of 1π-exchange subleading 2π-exchange N 3 LO: renormalization of 1π-exchange 15 LECs renormalization of contact terms sub-subleading 2π-exchange 3π-exchange (small) + 1/m and isospin-breaking corrections

7 Neutron-proton phase shifts up to N 3 LO N 3 LO np scattering at 50 MeV dσ/dω [mb/sr] N 2 LO NLO A y N 2 LO N 3 LO PWA Deuteron binding energy & asymptotic normalizations A s and η d Entem & Machleidt 03; Epelbaum, Glöckle & Meißner 05

8 Few-nucleon forces with the Delta Isospin-symmetric contributions Ordonez et al. 96, Kaiser et al. 98 H.K., Epelbaum & Meißner 07

9 3 F 3 partial waves up to NNLO with and without Δ Nijmegen Virginia Tech NNLO-Δ NNLO-Δ NLO-Δ NLO-Δ LO (OPE) (calculated in the first Born approximation)

too complicated use EFT on the Lattice Nucleons are represented as point-like Grassmann fields Point-like instantaneous pions to reproduce

10 Nuclear lattice simulations Borasoy, H.K., Lee, Meißner, Nucl.Phys. A768: 179 (2006) Low-energy constants fixed from experimental data with A 3 All nuclear spectra for A 4 pure prediction Explicit methods for A 4 too complicated use EFT on the Lattice Nucleons are represented as point-like Grassmann fields Point-like instantaneous pions to reproduce effective potential Typical lattice parameters = ¼ ' 300 MeV(a ' 2 fm) a L ' 20 fm Strong suppression of sign oscillations due to approximate Wigner SU(4) symmetry Method: hybrid Monte Carlo & transfer matrix (similar to LQCD) a

11 Monte Carlo with auxiliary fields Contact interactions can be represented by auxiliary fields exp(½ 2 =2)» Z 1 1 ds exp( s 2 =2 s ½) Hubbard-Stratonovich field Correlation function as path-integral over pions and auxiliary fields Z A (t) / Z 1 1 Ds Y Ds I D¼ I exp( S ¼¼ S ss ) det M(¼ I ; s; s I ) I=1;2;3 Slater-determinant of single nucleon matrix elements Single-nucleon matrix elements L t -th temporal lattice step M i;j (¼ I ; s; s I ) = hã i;x jm (L t 1) X M (0) X jã j;xi

12 Free nucleons and pions Positive definite free action: S ss (s; s I ) = 1 2 S ¼¼ (¼ I ) = t 2 X s(~n) ~n X I=1;2;3 X I=1;2;3 X s I (~n) 2 X ¼ I (~n)( + M¼)¼ 2 I (~n) ~n O(a 4 )-improved free nucleon lattice Hamiltonian H free = 1 3X X m k=0 ~n s ;l s ;i;j t = a t =a ~n Auxiliary field contributions Free instantaneous pions f 0;1;2;3 = 49 2 ; 3 4 ; 3 40 ; f k h a y i;j (~n s) a i;j (~n s + k ^l s ) + a i;j (~n s k ^l s ) i Nucleon density operators with different spin-isospin polarizations ½ ay ;a (~n s ) = X a y i;j (~n s)a i;j (~n s ) ½ ay ;a I (~n s ) = X a y i;j (~n s)[ I ] j;j 0a i;j 0(~n s ) i;j i;j;j 0 ½ ay ;a S;I (~n s) = X a y i;j (~n s)[¾ S ] i;i 0[ I ] j;j 0a i0 ;j 0(~n s) i;i 0 ;j;j 0

13 Leading order interactions Transfer matrix from n t - th step in temporal direction C < 0; C I > 0 M (nt) = : exp H free t g A t 2F ¼ + p X C t [s(~n s ; n t )½ a ~n s Small sign oscillation from pion-nucleon vertex X r S ¼ I (~n s ; n t )½ ay ;a S;I (~n s) S;I y ;a (~n s ) + i p X X C I t s I (~n s ; n t )½ a y ;a I (~n s ) I ~n s : No sign oscillation for contact interactions if the number of protons and neutrons are equal Real determinant in the pion-less case 2 M 2 = M det M = det M Chen, Kaplan, Phys. Rev. Lett. 92: (2004), Lee, Phys. Rev. C70: (2004) Antisymmetry of 2 real eigenvalues of M are doubly degenerate det M 0

14 Approximate SU(4) symmetry Wigner spin-isospin SU(4) symmetry transformation: Wigner, Phys. Rev. 51: 106 (1937) ±N = ¹º ¾ ¹ º N with ¾ ¹ = (1;~¾); ¹ = (1;~ ) SU(4) invariance in the limit of infinite a( 1 S 0 ) and a( 3 S 1 ) scattering length SU(4)-breaking terms» 1=a( 3 S 1 ) 1=a( 1 S 0 ); q= Â Mehen, Stewart, Wise, Phys. Rev. Lett. 83: 931 (1999) Large NN scattering length approximate SU(4) symmetry M (n t) SU(4) = : exp H free t + p X C t [s(~n s ; n t )½ a y ;a (~n s ) : ~n s Operator insertion for expectation value Pionless SU(4)-symmetric simulations are cheaper Decrease computational effort by using SU(4)-filter

15 Hybrid Monte Carlo Introduce conjugate fields p ¼I ; p s ; p si H HMC = 1 2 X (p 2 ¼ I (~n ) + p 2 s(~n ) + p 2 s I (~n )) + V (¼ I ; s; s I ) I;~n V (¼ I ; s; s I ) = S ¼¼ + S ss logfj det Mjg Generate new configurations for p ¼I ; p s ; p si ; ¼ I ; s; s I molecular dynamics trajectories by Repeat the steps many times Apply Metropolis accept or reject step for the new configuration according to probability exp( H HMC )

16 Leading-order considerations Promising results for A = 2; 3; 4 Borasoy, Epelbaum, H.K., Lee, Meißner, Eur.Phys.J.A31:105 (2007) Simulation Experiment r d [fm] 1:989(1) 1:9671(6) Q d [fm 2 ] 0:278(1) 0:2859(3) E 3H [MeV] 8:9(2) 8:482 r 3H [fm] 2:27(7) 1:755(9) E 4He [MeV] 21:5(9) 28:296 r 4He [fm] 1:50(14) 1:673(1) CPU time appears to scale linear with A (A 10) Nucleon density correlation in in the x-y plane 3 H

extension to mixing angles available L Z 0;0 (s; q 2 ) = p 1=4¼ X n2z 3 1 exp(2 i ± 0 ) =

17 Scattering from finite volume Rotation group SO(3) SO(3; Z) Every irreducible repr. includes definite J mod4 quantum numbers Scattering from finite volume L Large Note: No extension to mixing angles available L Z 0;0 (s; q 2 ) = p 1=4¼ X n2z 3 1 exp(2 i ± 0 ) = Z 0;0(1; q 2 ) + i ¼ 3=2 q Z 0;0 (1; q 2 ) i ¼ 3=2 q Lüscher formula for phase-shifts (n 2 q 2 ) s

18 Spherical wall method Borasoy, Epelbaum, H.K.,Lee, Meißner, Eur.Phys.J.A34:185 (2007) Spherical wall imposed in the center-of-mass frame ª(~r ) = [cos ± L j L (kr) sin± L y L (kr)]y L;m (µ; Á) ª( ~ R Wall ) = 0 tan ± L = j L(kR Wall ) y L (kr Wall ) Similar for Spin-triplet case Spherical wall removes copies of interactions due to periodic boundaries Energy spectrum by solving Schrödinger Eq. on the lattice (Lanczos method) Illustration for a toy-model: C = 2 MeV; R 0 = MeV 1 ¾ µ V (~r) = C ½1 + r2 [3(^r ~¾ 1 )(^r ~¾ 2 ) ~¾ 1 ~¾ 2 ] exp 1 r 2 2 R 2 0 Very shallow bound state in the R S(D) 1 channel with energy 0:155MeV

19 Illustration for the toy model Borasoy, Epelbaum, H.K.,Lee, Meißner, Eur.Phys.J.A34:185 (2007) Free particle spectrum for R Wall = 10 + ² Interacting spectrum for S = 0 L z mod 4 L z mod 4 L z mod 4 L z mod S 0 energy = 0:9280MeV 1 1 S 0 energy = 0:6445MeV k free = 29:52 MeV; j 0 (k free R Wall ) = 0 R Wall = E = k 2 free=m ¼ k free = 0:1064 MeV 1 k = 24:60MeV ±( 1 S 0 ) = tan 1 j0 (kr Wall ) = 30:0 o y 0 (kr Wall )

20 Chiral EFT at NLO: Scattering results Borasoy, Epelbaum, H.K., Lee, Meißner, Eur. Phys. J. A35: 343 (2008) 9 NLO LECs fitted to the various S(D) and P(F)-wave phase-shifts + Quadrupole moment of the deuteron NLO results with different actions estimate errors Fairly accurate description for momenta M ¼ Deviations appear consistent with higher-order effects Small systematic errors at NLO

Dilute neutron matter Neutron-Neutron scattering matrix: f 0 (k) = 1 k cot(± 0 ) i k Effective range expansion k cot(± 0 ) = a 1 scatt + 1 2 k2 r e + : : : Unitary limit: a scatt! 1; r e! 0 f 0 (k)!

21 Dilute neutron matter Neutron-Neutron scattering matrix: f 0 (k) = 1 k cot(± 0 ) i k Effective range expansion k cot(± 0 ) = a 1 scatt k2 r e + : : : Unitary limit: a scatt! 1; r e! 0 f 0 (k)! i k Free fermion ground states Scattering amplitude is as strong as possible Energy per particle free fermion case: E free 0 =A = 3 5 E F; E F = k2 F 2m In the unitary limit: E 0 =A =»E free 0 =A =» 3 5 E F» :dimensionless measurable number

22 Dilute neutron matter Neutron matter at k F» 80 MeV is close to the unitary limit Scattering length at lower density Effective range and other effects at higher density Experiment: ultracold» = 0: » = 0: » = 0:51(4) Important corrections 6 Li and 40 K using a magnetic-field Feshbach resonance Stewart et al., Phys. Rev. Lett. 97, (2006) Kinast et al., Science 307, 1296 (2005) Bartenstein et al., Phys. Rev. Lett. 92, (2004) Larger values in earlier experiments Further work to be needed Numerous calculations of» :»» 0:2 0:6 For review see: Furnstahl, Rupak, Schäfer, arxiv: [nucl-th]

23 Dilute neutron matter Recent results Put N = 8; 12; 16 neutrons in a box with L = 10; 12; 14 fm; k F = 1 L (3¼2 N) 1=3 2 8 % of normal nuclear matter density Energy per particle at NLO NLO lattice data in comparíson to earlier results Close to unitary limit E 0 E free 0 =»» 1 k F a scatt + 0:16k F r e (0:51fm 3 )k 3 F Results from the fit» ' 0:31» 1 ' 0:81 Predictions similar to earlier results

24 NNLO three-body forces Three-body forces at NNLO Fit D and E to neutron-deuteron scattering data + binding energy 3 H Spin-1/2 doublet channel 3 H binding energy p cot ±(fm 1 ) Fitting point E3 H (MeV) physical p 2 (fm 2 ) L(fm)

NNLO first applications Spin-3/2 quartet channel in neutron-deuteron scattering (Lanczos method) p cot ±(fm 1 ) Predictions for quartet phase-shifts are between p-d and n-d exp.

25 NNLO first applications Spin-3/2 quartet channel in neutron-deuteron scattering (Lanczos method) p cot ±(fm 1 ) Predictions for quartet phase-shifts are between p-d and n-d exp. data Small effects from NNLO corrections p 2 (fm 2 ) 4 He binding energy versus Euclidean time Monte-Carlo simulation with L = 16 fm he4 Hei = hª 4j exp( th=2)h exp( th=2)jª 4 i hª 4 j exp( th)jª 4 i he4 He i (MeV) physical 5% overestimation of physical binding energy with subtracted Coulomb-effects t(mev 1 )

NNLO simulations for light nuclei Conclusions Lattice EFT is a promising tool for a quantitative description of light nuclei At LO promising results for light nuclei up to 4 He At NLO 9 LECs are

26 NNLO simulations for light nuclei Conclusions Lattice EFT is a promising tool for a quantitative description of light nuclei At LO promising results for light nuclei up to 4 He At NLO 9 LECs are fitted to the phase-shifts by spherical wall method NLO Lattice EFT simulation of dilute neutron matter close to unitary limit with N=8,12,16 neutrons in a box NNLO three-body forces fitted to neutron-deuteron scattering data and triton binding energy Satisfactory NNLO description of quartet channel in neutron-deuteron scattering and 4 He binding energy Outlook NNLO simulations of neutron matter with larger number of neutrons in a box

27 Computational scaling Borasoy, H.K., Lee, Meißner, Nucl.Phys. A768: 179 (2006) Average phase versus the number of nucleons Almost linear behavior in CPU time for A<10 e i µ = det M j det Mj

28 Dilute neutron matter Borasoy, Epelbaum, H.K., Lee, Meißner, Eur. Phys. J. A35: 357 (2008) Put N = 8; 12 neutrons in a box with L = 10; 12; 14 fm; k F = 1 L (3¼2 N) 1=3 Energy per particle at NLO 2 8 % of normal nuclear matter density NLO lattice data in comparíson to earlier results Slope? Close to unitary limit E 0 E free 0 =»» 1 k F a scatt + 0:15k F r e Results from the fit» ' 0:25» 1 ' 1:0

29 Systematic error estimation Check for sensitivity on the cut off scale Deviations for cut offs 1 & 2 should be higher order corr. Fixed cut off in present work: check of cut off sensitivity in future Error estimation with fixed cut off At LO different actions V LO1 = V (0) 1 + V OPE + V Q2 = 2 1 V LO2 = V (0) 2 + V OPE + V Q2 = 2 2 Quasi-local operators with at least two powers of momenta Physical observables should agree up to NLO corrections At NLO different actions V NLO1 = V LO1 + V1 0 + V (2) 1 + V Q4 = 4 1 V NLO2 = V LO2 + V2 0 + V (2) 2 + V Q4 = 4 2 Expected agreement up to NNLO corr.

30 L LO 1;2 C» X ~q f 1 (~q ) = 1 LO energy levels Two different LO actions in momentum space f 1;2 (~q ) CN y (~q )N(~q )N y ( ~q )N( ~q ) + C I N y (~q )~ N(~q ) N y ( ~q )~ N( ~q ) original repr. f 2 (~q )» exp( b X l=1;2;3 (1 cos q l )) Gaussian smearing LO 1 : 10-15% too low for 1 S 0 LO 2 : correct to within few % for 1 S 0 Higher partial waves within 1% LO 1 : correct within 1% for 1 P 1 LO 2 : 5% too low for 1 P 1 Deviations consistent with NLO corr.

31 Chiral EFT at NLO: Mixing angles Borasoy, Epelbaum, H.K., Lee, Meißner, arxiv: [nucl-th] 9 NLO LECs fitted to the various S(D) and P(F)-wave phase-shifts + Quadrupole moment of the deuteron NLO results with different actions estimate errors Two different LO contact interactions in momentum space L LO 1;2 C» X ~q f 1 (~q ) = 1 f 1;2 (~q ) CN y (~q )N(~q )N y ( ~q )N( ~q ) + C I N y (~q )~ N(~q ) N y ( ~q )~ N( ~q ) original repr. f 2 (~q )» exp( b X l=1;2;3 (1 cos q l )) Gaussian smearing Fairly accurate description for momenta M ¼ Mixing angle changes sign by LO NLO Deviations appear consistent with higher-order effects

32 S-wave phase shifts Two different LO contact interactions in momentum space L LO 1;2 C» X ~q f 1 (~q ) = 1 f 1;2 (~q ) CN y (~q )N(~q )N y ( ~q )N( ~q ) + C I N y (~q )~ N(~q ) N y ( ~q )~ N( ~q ) original repr. f 2 (~q )» exp( b X S-wave phase shifts with different actions l=1;2;3 (1 cos q l )) Gaussian smearing Accurate NLO-description in both cases for momenta 80 MeV Better convergence with smeared action

33 P-wave phase shifts P-wave phase shifts with different actions Better convergence with non-smeared action

34 Improving convergence Project the smearing out of the P-waves µ 1 V LO3 = C1 S 0 f(~q ) 4 1 µ 3 4 ~¾ 1 ~¾ ~ 1 ~ 2 +C3 S 1 f(~q ) µ µ 1 4 ~¾ 1 ~¾ ~ 1 ~ 2 +V OPE Spin 0/Isospin 1 projector Spin 1/Isospin 0 projector Fast convergence in all partial waves NLO can be treated perturbatively Small systematic errors at NLO

35 Dilute neutron matter Convergence of E 0 =E free 0 is sensitive to the P-waves O(k 3 F )-corrections are suppressed»» 1 k F a 0 + 0:16k F r e LO 2 NLO 2 LO 3 NLO 3»» 1 k F a 0»» 1 k F a 0 + 0:16k F r e (0:51fm 3 )k 3 F

36 Higher partial waves Accurate NLO description of higher partial waves Non of the D-wave data was used in the fitting of LECs

Instantaneous pions LO effective potential in continuum V C = C + C I ~ 1 ~ 2 V OPE = µ ga 2 ~ 1 ~ 2 ~¾ 1 ~q ~¾ 2 ~q 2F ¼ ~q 2 + M¼ 2 Instantaneous pion to reproduce iteration of V OPE " 1 3X # @¹ ~¼

37 Instantaneous pions LO effective potential in continuum V C = C + C I ~ 1 ~ 2 V OPE = µ ga 2 ~ 1 ~ 2 ~¾ 1 ~q ~¾ 2 ~q 2F ¼ ~q 2 + M¼ 2 Instantaneous pion to reproduce iteration of V OPE " 1 3X ¹ ~¼ M 2 2 ¼~¼ 2 i i ~¼ + M 2 2 ¼~¼ 2 i=1 No time-derivative Two-pion exchange potential V TPEP» L(q) = 1 2q p 4M 2 ¼ + q 2 ln p 4M 2 ¼ + q 2 + q p 4M 2 ¼ + q 2 q = 1 + q2 12M 2 ¼ + : : : Approximation by series of contact interactions. Valid for q<<l

Nuclear Physics from Lattice Effective Field Theory

Nuclear Physics from Lattice Effective Field Theory Dean Lee (NCSU/Bonn) work done in collaboration with Evgeny Epelbaum (Bochum) Hermann Krebs (Bochum) Ulf-G. Meißner (Bonn/Jülich) Buḡra Borasoy (now