NUCLEAR STRUCTURE AND REACTIONS FROM LATTICE QCD

Transcription

1 NUCLEAR STRUCTURE AND REACTIONS FROM LATTICE QCD U. van Kolck Institut de Physique Nucléaire d Orsay and University of Arizona Supported by CNRS and US DOE 1

2 Outline QCD at Low Energies and the Lattice Nuclear Effective Field Theories EFT for Lattice Nuclei Outlook and Conclusion

3 Two goals of nuclear physics Nucleus as a laboratory: properties of the Standard Model and beyond nuclear matrix elements for symmetry tests reaction rates for nucleosynthesis equation of state for stellar structure variation of parameters for cosmology ab initio methods, phenomenology, etc. Nucleus as the simplest complex system: quarks and gluons interacting strongly, yet exhibiting many regularities QCD at large distances an unsolved part of the SM tools for non-perturbative quantum (field) theories, e.g. cold atoms TODAY

4 QCD QCD Basic mass scales u a d.o.f.s quarks: q = gluons: G (photon: A ) d µ µ symmetries SO(3,1) global, SU (3) gauge (+ U (1) gauge) 1 ( ) Tr µν = q i / + gg s / q G G 2 µν MQCD m,, N m 4 f, 1 GeV ρ π π c em ( 1 ετ ) + m q q + 3 m π mm QCD 140 MeV PDG V QCD qq qiγ τq 5 fπ M QCD 4 π + ( m) 100 MeV f π M QCD π

5 Lattice QCD lattice model space quark gluon L l l path integral solved with Monte Carlo methods, typically for unrealistically large quark masses Λ 1 l λ 1 L 2 π 2 ml cotδ ( E) 2 π 2 ml L 4 n < L l 1 L = π 2 2 mel n ( 2π ) mel l n Lϋscher 91

6 nucleon l 1 MQCD 1 M 0.3 fm QCD 1 m π 1.4 fm L 1 m π

7 nucleus l 1 MQCD 1 M 0.3 fm QCD ( ) A A R ρ m f m π π π 13 L ρ( mπ fπ) A mπ 1.2 fm

8 uclear cattering

9 How? Lattice QCD + Effective Field Theory c mass scales M m Λ S ( ν ) Q λ UV regulator order normalization expansion parameter ν ( ) 1 ( ) Q Q, Q λ Q m M M, ; m = + Fν γ i, λm ν= ν M m Q min Λ Λ IR regulator N ν ν min LO (unfortunately not the usage by nuclear potential modelers) Most general S matrix with given symmetries ν + 1 ν + 1 ν Q Q λq 1 +,, ν + 1 ν ν + 1 M M Λ M CONTROLLED UNCERTAINTY ν non-analytic functions, from solution of dynamical equation low-energy constants renormalizationgroup invariance Λ S ( ν ) λ S ( ν ) ( ν ) ν + 1 S Q = ν Λ M Λ ( ν ) ν S Q λ = ν + 1 λ M MODEL INDEPENDENCE (insensitivity to high-mom details) Want large model space Λ > λ < M Q

10 two-step strategy ab initio primo m π II) I) solve fit LECs EFT for for A any a A 3, 4 Many 300, m π 400 MeV π l 1 MQCD 1 M 0.3 fm QCD Also for reactions L ρ ρ 1 m π ρ ( 13) a ( m f ) A m 13 ( M f ) a 13 M f M π π π π π π M π π π 1 M π

11 Experimental and LQCD data + Inoue et al. 12 (MeV) [10] Yamazaki et al. 15 [7] Yamazaki et al. 12 [8] Beane et al Berkowitz et al Beane et al. in progress Beane et al. 13

12 J. Kirscher

13 Scales ( MeV) m N 2 m ( m m ) N N m π ( ) 2m B A A= 2 4 N A Q ℵ mn B < m < M 2 π QCD

14 M QCD ~ m, m, 4π f, N ρ Mnuc ~ Q π ~1 GeV f π ~100 MeV pert pion eventually Chiral EFT non-pert pion real nuclei phys m π =140 MeV The Nuclear EFT Landscape pqcd?? lattice nuclei now Pionless EFT m π Halo/ Cluster EFT See Ji s talk Deformed EFT

15 Extrapolation in pion mass Pionful (Chiral) EFT Q mπ M QCD Another talk

16 Pionless EFT degrees of freedom: nucleons symmetries: Lorentz, P, T EFT = + C + N C N N N N + expansion in: Q ℵ m B m M N i 0 N N N N N N N N N N N 2mN N 3 8mN 4 Q M N Q m = N Q m, multipole π 2 π D non-relativistic omitting spin, isospin Universality: first orders apply also to neutral atoms m l r = + 6 r 4 vdw π 1 l vdw where V () 2mat Bedaque, Hammer + v.k Bedaque, Braaten + Hammer 01

17 1 V = C r r + C r r + + { ( ) ( ) } 2 0( ) δ 2( ) δ ij s i j s i j s= 0 a, r a2 LO 2 2 NLO α + + r r i j NNLO and higher v.k. 97 Kaplan, Savage + Wise 98 König, Grießhammer, Hammer + v.k. 15 NLO NNLO and higher V = D δ r r r r + ( ) δ ( ) ijk 0 i j j k a 3 V = E δ r r δ r r δ r r + ( ) ( ) ( ) ijkl 0 i j j k k l etc. a 4 LO NNLO and higher not LO Bedaque, Hammer + v.k Hammer + Mehen 00 Platter, Hammer + Meißner Hammer + Platter 07

18 Regularization δ ( ri rj) δ Λ ( ri rj) 3 (arbitrary) Λ 2 2 e.g. local Gaussian regulator δλ ( r) = exp 32 ( Λ r 4) 8π Solution LO NLO H E Ψ = E Ψ (0) (0) (0) (0) A A A A = Ψ H Ψ (1) (0) (1) (0) A A A A Renormalization (essential) etc. C ( Λ) 0(0,1) fitted to two two-body data, e.g. a NN (0,1) D ( Λ) 0 etc. a nd fitted to one three-body datum, e.g. (1/2) A = 2 equivalent to v.k. 99 effective-range expansion pseudopotential boundary condition at origin Bethe 49 Fermi 37 Bethe + Peierls 35 A 3 not just the effective-range expansion: includes many-body forces!

19 NLO Three-Body Force at LO Bedaque, Hammer + v.k. 99, 00 Bedaque, Grießhammer, Hammer + Rupak 02 Hammer, Meißner + Platter 05 LO varying Λ* expt potential models single SU(4) W symmetric parameter Λ * Phillips line varying Λ* LO Tjon line vs. potential models expt

20 Pionless EFT reproduces effective range expansion explains Thomas collapse from improper renormalization explains correlations (Phillips, Tjon lines) from proper renormalization generates Efimov states and its descendants as consequence of (approximate) discrete scale invariance gives approximate Wigner SU(4) invariance but o applies only at momenta below pion mass o has unknown reach in terms of nucleon number

21 Extrapolation in nucleon number m π M QCD Pionful EFT Pionless EFT m π M QCD + any exact ab initio method That is, 1) truncate EFT expansion at desired order 2) solve Schrödinger equation for low A at fixed cutoff (exactly for LO, sublos in perturbation theory) 3) fit LECs to selected lattice input 4) solve Schrödinger equation for larger A 5) repeat steps 2-4 at other cutoffs 6) obtain observables at large cutoffs

22 Experimental and LQCD data LO pionless fit: m C N, C 0(0), D 0(1) 0 Stetcu, Barrett + v.k. 06, * * * * [10] Yamazaki et al. 15 [7] Yamazaki et al. 12 [8] Beane et al. 12

23 A > 4 As A grows, given computational power limits number of accessible one-nucleon states Lattice Box Λ 1 l λ 1 L Λ Nmax b λ 1 Nmax b IR cutoff Harmonic Oscillator No-Core Shell Model 2 π mn L L 2 2 π ml N 2 N max 1 mb N 2 b nuclear matter Mueller et al. 99 finite nuclei Stetcu et al. 06 few nucleons Lee et al mn Eb Γ 4 n < L l 1 L cotδ ( E cotδ ( E) = π ) = m ( 2 ) mn Eb 1 mn Eb N EL n π mn EL l n Γ 4 2 Lϋscher 91 Busch et al. 99

24 Pionless EFT: LO N max 16 Stetcu, Barrett + v.k. 06 (parameters fitted to d, t, a ground-state binding energies) ω [ MeV] Λ[ MeV] within ~10% works within ~30% Nmax 8 Bonus: N max < 30 Stetcu, Barrett, Vary + v.k. 08 gs.. J π 3 spin-1/2 particles

25 Experimental and LQCD data Barnea, Contessi, Gazit, Pederiva + v.k. 13 Kirscher, Barnea, Gazit, Pederiva + v.k. 15 Contessi, Lovato, Pederiva, Roggero, Kirscher + v.k. 17 * * * * LO pionless fit: m C N, C 0(0), D 0(1) 0, [23] Stetcu et al. 06 [10] Yamazaki et al. 15 [7] Yamazaki et al. 12 [8] Beane et al. 12 Beane et al. 13

26 Ab initio methods employed Effective-Interaction Hyperspherical Harmonics (EIHH) hyperspherical coordinates: hyperradius + 3A-4 hyperangles model space: hyperangular momentum K K max wavefunction: expanded in antisymmetrized spin/isospin states effective interaction: Lee-Suzuki projection to subspace in medium extrapolation: K max Barnea et al Refined Resonating Group Method (RRGM) Hoffmann 86 wavefunction: expanded in overcomplete basis of Gaussians in all cluster channels Kohn-Hulthen variational approach minimizing reactance matrix convergence (heavier channels, higher partial waves, Gaussian set) tested Auxiliary-Field Diffusion Monte Carlo (AFDMC) Schmidt + Fantoni 99 integral equation for evolution of wavefunction in imaginary time τ in terms of Green s function (diffusion) two- and more-body operators linearized by auxiliary fields (Hubbard- Stratonovich transformation) trial wavefunction probed stochastically with weight given by the Green s function lowest-energy state with symmetries projected onto as τ -->

27 H 1 = (0) 2 i 2mN i Λ rij C10( Λ) + C01( Λ) + ( C10( Λ) C01( Λ) ) σi σ j e + D1 ( Λ) i j e 4 τ τ i< j i< j< k cyc Barnea, Contessi, Gazit, Pederiva + v.k. 13 Kirscher, Barnea, Gazit, Pederiva + v.k. 15 ( rij rjk ) Λ + 4 m Λ ( s) 2 ( a0 Λ ) N 1 C 2 SI ( Λ) = θ0 + + O ( ) ( s) a0 Λ θ θ 0 = S 1 ( ) ( ) a = ± fm cutoff variation 2 to 14 fm -1

28 Neutron-deuteron scattering: quartet for all masses 4 a nd = ( 1 kpn ) k = 4m B 3 pn N D deuteron break-up threshold Kirscher, Barnea, Gazit, Pederiva + v.k. 15

29 H 1 = (0) 2 i 2mN i Λ rij C10( Λ) + C01( Λ) + ( C10( Λ) C01( Λ) ) σi σ j e + D1 ( Λ) i j e 4 τ τ i< j i< j< k cyc Barnea, Contessi, Gazit, Pederiva + v.k. 13 Kirscher, Barnea, Gazit, Pederiva + v.k. 15 ( rij rjk ) Λ + 4 m π = 510 MeV m N D 2 1 ( Λ ) = F ΛΛ Λ ( )

30 H 1 = (0) 2 i 2mN i Λ rij C10( Λ) + C01( Λ) + ( C10( Λ) C01( Λ) ) σi σ j e + D1 ( Λ) i j e 4 τ τ i< j i< j< k cyc Barnea, Contessi, Gazit, Pederiva + v.k. 13 Kirscher, Barnea, Gazit, Pederiva + v.k. 15 ( rij rjk ) Λ + 4

31 Kirscher, Barnea, Gazit, Pederiva + v.k. 15 Neutron-deuteron scattering: doublet Phillips correlations at various pion masses zero of T close to threshold 4 a nd = ( 1 kpn )

32 Kirscher, Barnea, Gazit, Pederiva + v.k. 15 Tjon correlations at various pion masses Alpha Particle

33 Barnea, Contessi, Gazit, Pederiva + v.k. 13 no excited states for A =2,3,4 no 3 n droplet * [23] Stetcu et al. 06 [10] Yamazaki et al. 15 [7] Yamazaki et al. 12 [8] Beane et al. 12 [4] Barnea et al. 13 predictions

34 Barnea, Contessi, Gazit, Pederiva + v.k. 13 Z. Davoudi consistency between EFT and LQCD, including plateau id B B 5 4 B B A=5 gap persists!? nuclear saturation survives!? Maybe pions play less of a role than we are used to think?

35 A Scene from Program INT Effective field theorist: (nonchalant) Pionless EFT works pretty well for light nuclei. --- Esteemed nuclear theorist: (doubtful) But has it been applied to a real nucleus? --- EFTst: (stunned) What the hell is a real nucleus? When we did deuteron we were told triton was more like a real nucleus. When we did triton we were told it s the alpha particle that s a real nucleus. Now you tell me the alpha particle is not a real nucleus? --- ENTst: (didactic) No, you need to show saturation for a real nucleus like 16 O. --- EFTst: (outraged) But we have shown saturation for triton and alpha particle! It s due to the three-body force, so 16 O will saturate. --- ENTst: (jaded) I believe it when I see it.

36 H 1 = (0) 2 i 2mN i + + ( rij rjk ) Λ r 4 Λ + ij C1( Λ) C2( Λ) σi σ j e + D0( Λ) e i< j i< j< k cyc 4 Contessi, Lovato, Pederiva, Roggero, Kirscher + v.k He within previous error bar in good agreement with previous calculations experiment 16 O indistinguishable from four-alpha threshold; if anything, too much saturation

37 Alpha-particle clusterization Contessi, Lovato, Pederiva, Roggero, Kirscher + v.k. 17 Single-nucleon point densities 4 He 16 O m π =140 MeV m π = 510 MeV m π = 805 MeV

38 16 O m π =140 MeV imaginary-time diffusion of a single walker Contessi, Lovato, Pederiva, Roggero, Kirscher + v.k. 17 Λ = 2 fm 1 Λ = 8 fm 1

39 What next? NLO Bazak et al., in progress Other A hypernuclei Chiral EFT at lower m π (when available)

40 Conclusion EFT is constrained only by symmetries and thus can be matched onto lattice QCD EFT allows controlled extrapolations of lattice results in nucleon number (and pion mass), including reactions! First, proof-of-principle calculations carried out at m π 500, 800 MeV with Pionless EFT World at large pion mass might be just a denser version of ours