Nuclear forces and their impact on structure, reactions and astrophysics

Transcription

1 Nuclear forces and their impact on structure, reactions and astrophysics Lectures for Week 2 Dick Furnstahl Ohio State University July, 213 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, hyperon-nucleon (as) Th. Renormalization group 2 (rjf); Nuclear forces and electroweak interactions (as) F. Many-body overview (rjf); Three-nucleon forces 2 (as)

2 Outline PWA and χ-symmetry LQCD HALQCD High partial waves and chiral symmetry Lattice QCD NN from LQCD (slides from recent talk by S. Aoki ) Dick Furnstahl TALENT: Nuclear forces

3 PWA and χ-symmetry LQCD HALQCD Figures from Kaiser, Brockmann, Weise, Nucl. Phys. A 625, 758 (1997) Key: dashed = OPE; full = OPE iterated + irred. 2π; others = NN PSA I6 3 I6 MIXING ANGLE [deg] ε I5 3 I H H4 3 H5 3.4 H6 MIXING ANGLE [deg] ε [What is a mixing angle?] What do you conclude about pion exchange? Dick Furnstahl TALENT: Nuclear forces

4 PWA and χ-symmetry LQCD HALQCD Figures from Kaiser, Brockmann, Weise, Nucl. Phys. A 625, 758 (1997) Key: dashed = OPE; full = OPE iterated + irred. 2π; others = NN PSA G4 3 G4 MIXING ANGLE [deg] ε 4 3 G3 3 G H H4 3 H5 3.4 H6 MIXING ANGLE [deg] ε [What is a mixing angle?] What do you conclude about pion exchange? Dick Furnstahl TALENT: Nuclear forces

5 PWA and χ-symmetry LQCD HALQCD Figures from Kaiser, Brockmann, Weise, Nucl. Phys. A 625, 758 (1997) Key: dashed = OPE; full = OPE iterated + irred. 2π; others = NN PSA F3 3 F F2 3 F D2 3 D2 2 3 D D1 MIXING ANGLE [deg] ε 3 MIXING ANGLE [deg] 2 ε Why does OPE work better at low energy and high L? Dick Furnstahl TALENT: Nuclear forces

6 Outline PWA and χ-symmetry LQCD HALQCD High partial waves and chiral symmetry Lattice QCD NN from LQCD (slides from recent talk by S. Aoki ) Dick Furnstahl TALENT: Nuclear forces

7 PWA and χ-symmetry LQCD HALQCD Visualization of the lattice [from T. Hatsuda] a quarks q on the sites gluons U μ =exp(iaa μ ) on the links What ca hadr hot L 4-dim. Euclidean Lattice What is cold phen Dick Furnstahl TALENT: Nuclear forces

8 PWA and χ-symmetry LQCD HALQCD QCDSF LQCD calculations of f π and g A [arxiv: ] FIG. 4: The renormalized axial charge ga R in the infinite volume plotted against with the experimental value g A =1.27 ( ). The shaded area shows the fit of eq. Two flavors of nonperturbatively O(a) improved Wilson fermions and Wilson plaquette action 3 lattice spacings: a =.76,.71, and.6 fm Renormalized fπ R = 89.7 ± 1.5 ± 1.8 MeV at m π = 13 MeV Solid line is fit to chiral ansatz = also determines l 4 from ChPT Very encouraging but are all the systematic errors under control? FIG. 5: The renormalized pion decay constant fπ R in the infinite volume plotted together with the experimental value f π =92.2MeV ( ). The curve shows a fit data. Dick Furnstahl TALENT: Nuclear forces

9 PWA and χ-symmetry LQCD HALQCD QCDSF LQCD calculations of f π and g A [arxiv: ] g A is known experimentally from neutron β decay Renormalized ga R = 1.24 ±.4 at m π = 13 MeV Shaded band is fit to ChEFT g R A (m π) = note rapid rise Very encouraging but are all the systematic errors under control? FIG. 4: The renormalized axial charge ga R in the infinite volume plotted against with the experimental value g A =1.27 ( ). The shaded area shows the fit of eq Dick Furnstahl TALENT: Nuclear forces

10 Outline PWA and χ-symmetry LQCD HALQCD High partial waves and chiral symmetry Lattice QCD NN from LQCD (slides from recent talk by S. Aoki ) Dick Furnstahl TALENT: Nuclear forces

11 Extensions of the HAL QCD approach to inelastic and multi-particle scatterings in lattice QCD Sinya Aoki University of Tsukuba HAL QCD Collaboration INT Workshop INT-15-53W Nuclear Reactions from Lattice QCD Institute for Nuclear Theory, University of Washington, Seattle, USA, March 11-12, 213

12 1. Introduction HAL QCD approach to Nuclear Force

13 Potentials in QCD? What are potentials (quantum mechanical objects) in quantum field theories such as QCD? Potentials themselves can NOT be directly measured. cf. running coupling in QCD scheme dependent, ambiguities in inelastic region experimental data of scattering phase shifts potentials, but not unique Potentials are still useful tools to extract observables such as scattering phase shift. One may adopt a convenient definition of potentials as long as they reproduce correct physics of QCD.

14 HAL QCD strategy Full details: Aoki, Hatsuda & Ishii, PTP123(21)89. Step 1 define (Equal-time) Nambu-Bethe-Salpeter (NBS) Wave function Spin model: Balog et al., 1999/21 k(r) = N(x + r, )N(x, ) NN, W k W k = 2 k 2 + m 2 N energy N(x) =" abc q a (x)q b (x)q c (x): local operator

15 Key Property 1 Lin et al., 21; CP-PACS, 24/25 k(r) l,m C l sin(kr l /2 + l (k)) Y ml ( r ) kr r = r!1 l(k) scattering phase shift (phase of the S-matrix by unitarity) in QCD! How can we extract it? cf. Luescher s finite volume method

16 Step 2 define non-local but energy-independent potential as [ k H ] k(x) = d 3 y U(x, y) k (y) µ = m N /2 reduced mass ɛ k = k2 2µ H = 2 2µ Key Property 2 non-local potential A non-local but energy-independent potential can be constructed as W k,w k U(x, y) = k,k W th [ k H ] k(x) 1 k,k k (y) inner product 1 k,k : inverse of k,k = ( k, k ) k is linearly independent. For W p < W th = 2m N + m (threshold energy) d 3 y U(x, y) p (y) = k,k [ k H ] k(x) 1 k,k k,p = [ p H ] p(x) Note 1: Potential satisfying this is not unique. Note2: Non-relativistic approximation is NOT used. We just take the equal-time frame.

17 Step 3 expand the non-local potential in terms of derivative as U(x, y) =V (x, r) 3 (x y) V (x, ) =V (r)+v σ (r)(σ 1 σ 2 )+V T (r)s 12 + V LS (r)l S + O( 2 ) NLO NNLO LO LO LO spins tensor operator S 12 = 3 r 2 (σ 1 x)(σ 2 x) (σ 1 σ 2 ) V A (x) local and energy independent coefficient function (cf. Low Energy Constants(LOC) in Chiral Perturbation Theory)

18 Step 4 Step 5 extract the local potential. At LO, for example, we simply have V LO (x) = [ k H ] k(x) k(x) solve the Schroedinger Eq. in the infinite volume with this potential. phase shifts and binding energy below inelastic threshold (We can calculate the phase shift at all angular momentum.) L(k) L(p = k) exact by construction approximated one by the derivative expansion expansion parameter W p W k W th 2m N m E p We can check a size of errors at LO of the expansion. We can improve results by extracting higher order terms in the expansion.

19 2. Results from lattice QCD Ishii et al. (HALQCD), PLB712(212) 437.

20 Extraction of NBS wave function NBS wave function Potential k(r) = N(x + r, )N(x, ) NN, W k [ k H ] k(x) = d 3 y U(x, y) k (y) 4-pt Correlation function source for NN F (r,t t )= T{N(x + r,t)n(x,t)}j (t ) complete set for NN F (r,t t ) = T {N(x + r,t)n(x,t)} 2N,W n,s 1,s 2 2N,W n,s 1,s 2 J (t ) + = n,s1,s2 ground state saturation at large t n,s1,s2 A n,s1,s2ϕ Wn (r)e Wn(t t), A n,s1,s2 = 2N,W n,s 1,s 2 J (). lim F (r,t t )=A ϕ W (r)e W(t t) + O(e W n (t t ) ) (t t ) NBS wave function This is a standard method in lattice QCD and was employed for our first calculation.

21 Improved method normalized 4-pt Correlation function R(r,t) F (r,t)/(e mn t ) 2 = n Ishii et al. (HALQCD), PLB712(212) 437 A n ϕ Wn (r)e Wnt W n = W n 2m N = k2 n m N ( W n) 2 4m N t R(r,t)= { H + U 1 4m N 2 t } R(r,t) Leading Order potential energy-independent { H t } 4m N t 2 R(r,t)= d 3 r U(r, r )R(r,t)=V C (r)r(r,t)+ total 1st 2nd 3rd 3rd term(relativistic correction) is negligible. V C (r) [MeV] total -3 1st term 2nd term 3rd term r [fm] Ground state saturation is no more required. (advantage over finite volume method.)

22 2+1 flavor QCD, spin-singlet potential m 7 MeV a=.9fm, L=2.9fm V C (r) [MeV] NN potential S r [fm] [deg] phase shift 1 S exp lattice a exp ( 1 S ) = 23.7 fm a ( 1 S )=1.6(1.1) fm 3 35 E lab [MeV] Qualitative features of NN potential are reproduced. (1)attractions at medium and long distances (2)repulsion at short distance(repulsive core) It has a reasonable shape. The strength is weaker due to the heavier quark mass. Need calculations at physical quark mass.

23 Ref. [3].! deuterondef1ghijk9lm9n Potentials (a) 1 V(r) [MeV] ! PQ9KLMNRST6!"#$%9A 3 (b) 1 particle spectrum(uvwxyz[$)\] 1 VC(r) [3S] VC(r) [ S1] c=d]^\'ehfg<\hije5 VT(r) VC(r) [13S] VC(r) [ S1] VT(r) 3 25 quenched QCD full QCD -1. V(r) [MeV] flavor QCD result VT.5 mπ=71 MeV !2centrifugal barrier9lf6qmnsop u6v;<\uw=xykc (z(9{ =}k5j~d ÄÅH) Quenched QCD result 1-5. mπ=731 MeV r [fm] r [fm] L! 2.9 fm a!.91 a!.137 and fm tensor L! 4.4 fm Figure 9: (Left) 2+1fmflavor QCD results for the central potential potentials at mπ " 7 MeV. (Right) Quenched results for the same potentials at mπ " 731 MeV. Taken from Ref. [33]. 4.4 no repulsive core in the tensor potential. the tensor is enhanced in full QCD QCDpotential results Full from R.Machleidt, Adv.Nucl.Phys.19 Needless to say, it is important to repeat calculations of N N potentials in full QCD on larger volum t lighter pion masses. The PACS-CS collaboration is performing flavor QD simulations, whi over the physical pion mass[31, 32]. Gauge configurations are generated with the Iwasaki gauge acti nd non-perturbatively O(a)-improved Wilson quark action on a lattice. The lattice spacing s determined from mπ, mk and mω as a ".91 fm, leading to L " 2.9 fm. Three ensembles of gau Nconf=1 onfigurations are used to calculate N N potentials at!(m π, mn ) "(71 MeV, 1583 MeV), (57 Me

24 Quark mass dependence (full QCD) the tensor potential increases as the pion mass decreases. manifestation of one-pion-exchange? both repulsive core and attractive pocket are also grow as the pion mass decreases.

25 NN Potentials for the negative parity sector NN (I V ) NN ( r, ) (I = V ) (I (r) + V ) σ (r) ( σ 1 σ (I 2 ) + V ) (I T (r) S 12 + V ) LS (r) L S + O( 2 ) LO LO V C (r) V (r) + V σ (r) ( σ 1 σ 2 ) V (r) 3V σ (r) for S= = V (r) + V σ (r) for S=1 NLO S=,P=+)(I=1) S=1,P=+)(I=) S=,P=-)(I=) S=1,P=-)(I=1) V C (r) V C (r),v T (r),v LS (r) V C (r) V C (r),v T (r),v LS (r) 2S+1 L J S=1 channel: Central & tensor forces in LO Spin-orbit force in NLO

26 Preliminary Results (full QCD) Murano et al. (HAL QCD), lat212 2-flavor QCD, a=.16 fm m 1.1 GeV LO LO NLO MeV V C t=7 1 t= from 3 P from 3 1 P 2 MeV -2 V T MeV V LS t=7 from 3 P F fm fm fm Fig. 7. Potentials in odd parity sector obtained Very from weak 3 P!, 3 P 1 and 3 P F 2 NBS wave functions calculated at m π = 1136 MeV. Left, center, and right figure shows central, tensor and spin-orbit force in parity odd sector respectively. Hereafter potential in this report means the local potential at AV18 the leading order, unless otherwise stated Nuclear force in odd parity sectors and the spin-orbit force In this section, we consider the potentials in odd parity sectors. Together with the nuclear forces in even parity sectors, the information on odd parity sectors is nec-

27 PWA and χ-symmetry LQCD HALQCD HAL QCD Spin-orbit force from Lattice QCD K. Murano et al, arxiv: V(r) [MeV] Dick Furnstahl r [fm] V(r; 1 P 1 ) V(r; 3 P ) V(r, 3 P 1 ) V(r; 3 P 2 ) TALENT: Nuclear forces