Local chiral NN potentials and the structure of light nuclei Maria Piarulli @ELBA XIV WORKSHOP June 7-July 1 16, Marciana Marina, Isola d Elba PHYSICAL REVIEW C 91, 43(15) Minimally nonlocal nucleon-nucleon potentials with chiral two-pion exchange including resonances M. Piarulli, 1 L. Girlanda,,3 R. Schiavilla, 1,4 R. Navarro Pérez, 5 J. E. Amaro, 5 and E. Ruiz Arriola 5 1 Department of Physics, Old Dominion University, Norfolk, Virginia 359, USA https://arxiv.org/abs/166.6335 Local chiral potentials and the structure of light nuclei M. Piarulli a, L. Girlanda b,c, R. Schiavilla d,e, A. Kievsky f, A. Lovato a, L.E. Marcucci g,f, Steven C. Pieper a, M. Viviani f, and R.B. Wiringa a
Nuclear χeft Approach: S. Weinberg, Phys. Lett. B51, 88 (199); Nucl. Phys. B363, 3 (1991); Phys. Lett. B95, 114 (199) χeft uses the chiral-symmetry to constrain the interactions of π s among themselves or with baryons (N and -isobar) π s couple by powers of its momentum Q, and the Lagrangian (L eff ) can be expanded systematically in powers of Q/Λ; (Q Λ 1 GeV is the chiral-symmetry breaking scale and Q m π ) L eff = L () + L (1) + L () +... χeft allows for a perturbative treatment in terms of powers of Q The unknown coefficients of the perturbative expansion are called LEC s and are determined fitting the experimental data The χ-expansion gives rise to potentials and external currents can be naturally incorporated
The derivation of nuclear forces from χeft has been a topic of active interest for the past 5 years Previous work: S. Weinberg, Phys. Lett. B 51, 88 (199); Nucl. Phys. B 363, 3 (1991) C. Ordóñez, L. Ray, and U. van Kolck, Phys. Rev. Lett. 7, 198 (1994); Phys. Rev. C 53, 86 (1996) N. Kaiser, R. Brockmann, and W. Weise, Nucl. Phys. A 65, 758 (1997) N. Kaiser, S. Gerstendörfer, and W. Weise, Nucl. Phys. A 637, 395 (1998) N. Kaiser, Phys. Rev. C 61, 143 (1999) N. Kaiser, Phys. Rev. C 6, 41 () E. Epelbaum, W. Glöckle, and U.-G. Meißner, Nucl. Phys. A 637, 17 (1998); 671, 95 () N. Kaiser, Phys. Rev. C 63, 441 (1); 64, 571 (1); 65, 171 (1) D. R. Entem and R. Machleidt, Phys. Rev. C 66, 14 (); 68, 411 (3) E. Epelbaum, W. Glöckle, and U.-G. Meißner, Eur. Phys. J. A 19, 15 (4) E. Epelbaum, W. Glöckle, and U.-G. Meißner, Nucl. Phys. A 747, 36 (5) H. Krebs, E. Epelbaum, and Ulf.-G. Meißner, Eur. Phys. J. A 3, 17 (7) A. Ekström, G. Baardsen, C. Forssen, G. Hagen, M. Hjorth-Jensen, G. R. Jansen, R. Machleidt, W. Recent work: A. Ekström, G. Baardsen, C. Forssen, G. Hagen, M. Hjorth-Jensen, G. R. Jansen, R. Machleidt, W. Nazarewicz, T. Papenbrock, J. Sarich, and S. M. Wild, Phys. Rev. Lett. 11, 195 (13) A. Gezerlis, I. Tews, E. Epelbaum, S. Gandolfi, K. Hebeler, A. Nogga, and A. Schwenk, Phys. Rev. Lett. 111, 351 (13) A. Gezerlis, I. Tews, E. Epelbaum, M. Freunek, S. Gandolfi, K. Hebeler, A. Nogga, and A. Schwenk, Phys. Rev. C 9, 5433 (14) D. R. Entem, N. Kaiser, R. Machleidt, and Y. Nosyk, Phys. Rev. C 91, 14 (15); 9, 641 (15) N. Kaiser, Phys. Rev. C 9, 4 (15) E. Epelbaum, H. Krebs, and U.-G. Meißner Phys. Rev. Lett. 115, 131 (15)
Motivations: WHY? Many of the available versions of chiral potentials are strongly non-local Non-localities due to contact interactions Non-localities due to regulator functions p i relative momentum operator Non-local interactions hard (but not impossible; see A. Roggero et al. PRL 11, 113 (14)) handle in for example Quantum Monte Carlo (QMC) methods GOAL: Construct a local χeft NN potential with chiral TPE including -isobar: Minimize the number of non-localities due to contact interactions and remove those due to the regulator functions M. Piarulli et al. PRC 91, 43 (15) Minimally nonlocal nucleon-nucleon potentials with chiral two-pion exchange including s The LECs multiplying these non-localities are now absent M. Piarulli et al. arxiv/166.6335 Local chiral NN potentials and the structure of light nuclei
Local chiral NN Potentials: v EM 1 : EM interaction component v L 1 : long-range component LO : Q p k -p p -p v 1 = v EM 1 + v L 1 + v S 1 Leading Coulomb interaction Second order Coulomb interaction Darwin-Foldy interaction Vacuum polarization interaction Magnetic moment interaction NLO : Q k = p p Dependence on g A, F π and h A =3g A / NLO : Q 3 c 1, c, c 3, c 4 (L () πn ) b 3 + b 8 (L () πn ) taken from π-n scattering H. Krebs, E. Epelbaum, U.-G. Meissner (7)
v S 1 LO : Q p : short range component p () NLO : Q K = N3LO : Q 4 1 (p + p ) k = p p In the NLO and N3LO contact interactions terms proportional to K and K 4 have been removed by Fierz rearrangements: P exc f = f f O i = f P exc O i K m 1+τ 1 τ (7) P exc = 1+σ 1 σ 1+σ 1 σ 1+τ 1 τ (15) P space k m m with m= or 4 k K and K 1/ k Fierz rearrangement is effective in completely removing non-localities at NLO (see A. Gezerlis et al. PRC 9, 5433 (14), M. Piarulli et al. PRC 91, 43 (15)), but it cannot do so at N3LO. Of course mixed terms as k K or K k can not Fierz-transformed away k K 1+τ 1 τ 1+σ 1 σ K k
Coordinate-space v L 1: 6 v L 1 = l=1 v l L(r) O l 1 + v σt L O l=1,...,6 1 =[1, σ 1 σ,s 1 ] [1, τ 1 τ ] (r) O1 σt + vl tt (r) O1 tt O σt 1 = σ 1 σ T 1 O tt 1 = S 1 T 1 T 1 =3τ 1z τ z τ 1 τ CIB terms v l L(r) v σt L (r) v tt L (r) divergencies of type 1/r n, 1 n 6 C RL (r) =1 1 (r/r L ) 6 e (r R L)/a L +1 R L =(.8, 1., 1.)fm a L = R L / CRL (r) r [fm]
Coordinate-space v S 1: For the short-range terms the regularization is achieved by employing a local regulator C RS (k)=e R S k /4 C RS (r) = v S 1 = 1 16 l=1 O l=1,...,6 1 =[1, σ 1 σ,s 1 ] [1, τ 1 τ ] π 3/ R 3 S v l S(r) O l 1 O l=7,...,11 1 = L S, L S τ 1 τ, (L S), L, L σ 1 σ O l=1,...,16 1 = T 1, (τ z 1 + τ z ), σ 1 σ T 1,S 1 T 1, L S T 1 In this parametrization we removed { v p S (r)+vpσ S (r) σ 1 σ + v pt S (r) S 1 + v ptτ S (r) S 1 τ 1 τ, p } and also we considered CD terms e (r/r S) In combination with R L =(.8, 1., 1.)fm Λ S =/R S (7, 6, 5)MeV R S =(.6,.7,.8)fm
Fitting Procedure I: In this work the LECs are fixed by fitting the pp and np Granada database up to laboratory frame energies E lab = 15 MeV and E lab = MeV, the deuteron binding energy and the nn scattering length 3 σ-criterion to remove inconsistencies in the database [1] There are 493 exp data up to 15MeV (3476 data up to MeV) There are N sets each one corresponding to a different experiment Each data set contains measurements at fixed energy and different scattering angle (except total cross sections) We fit first phase shifts and then refine the fit by minimizing the χ obtained from a direct comparison with the database [1] R. Navarro Pérez, J.E. Amaro, and E. Ruiz Arriola, Phys. Rev. C 88, 64 (13) http://www.ugr.es/ amaro/nndatabase/
Fitting Procedure II: To minimize the total χ, we use the Practical Optimization Using No Derivatives (for Squares), POUNDerS M. Kortelainen, PRC 8, 4313 (1) Model a :(R L,R S )=(1.,.8) Model b :(R L,R S )=(1.,.7) Model c :(R L,R S )=(.8,.6) (with the help of J. Sarich and S. Wild) model order R L (fm) R S (fm) E LAB (MeV) χ /datum Model b LO 1..7 15 59.88 Model b NLO 1..7 15.18 Model b NLO 1..7 15.3 Model b N3LO 1..7 15 1.7 Model a N3LO 1..8 15 1.5 Model c N3LO.8.6 15 1.11 Model a N3LO 1..8 1.37 Model b N3LO 1..7 1.37 Model c N3LO.8.6 1.4
Phase shifts model b up to 15 MeV (order by order): Phase Shift [deg] Phase Shift [deg] 6 4-5 -1 3 P1-15 5 5 75 1 15 Lab. Energy [MeV] np T=1 6 4 1 S 1 3 D 3 P 4 1 5 3 P 5 5 75 1 15 Lab. Energy [MeV] 1-1 - -3 LO NLO NLO N3LO Granada Nijm! Mixing Angle [deg] -4 5 5 75 1 15 Lab. Energy [MeV] Gross 1 LO NLO NLO N3LO 3
Phase shifts model b up to 15 MeV (order by order): Phase Shift [deg] Phase Shift [deg] -5-1 -15 15 1 5 Gross 1 P1 3 1-1! 3 3 1 S1 D1 3 D 3 1 4 3 D3 - LO - 5 5 75 1 15 5 5 75 1 15 NLO 5 5 75 1 15 Lab. Energy [MeV] Lab. Energy [MeV] NLO Lab. Energy [MeV] N3LO Granada LO NLO Nijm NLO Gross N3LO np T= 1 Mixing Angle [deg] 3
Phase shifts: np T=1
Phase shifts: pp T=1
Nuclear Many-Body Problem: Few- and many-body systems provide a laboratory to study nuclear forces with a variety of numerical and computational techniques H Ψ(R; s 1,..,s A ; t 1,..,t A ) = E Ψ(R; s 1,..,s A ; t 1,..,t A ) H = m i + v ij + i i<j i<j<k V ijk R =(r 1, r,...,r A ) configuration space; s i are nucleon spins: ± 1 ; t i are nucleon isospins (proton or neutron): ± 1 HH method (A. Kievsky et al., NPA577, 511(1994);A.Kievskyet al., FBS, 1 (1997); M. Viviani et al., PRC71, 46(5);A.Kievskyet al., JPG:NPP35, 6311 (8)) is used to calculate the ground-state energies of 3 H and 4 He: provide a benchmark for the corresponding QMC calculations QMC methods (J. Carlson et al., RMP87, 167(15)) are then applied to compute BE and rms radii of the 3 He ground state, of the 6 Li and 6 He ground and excited states
Minimize the expectation value of H: E T = Ψ T H Ψ T E Ψ T Ψ T Trial wave function (involves variational parameters): S i<j represents a symmetrized product pair correlation operators U ij = u p (r ij ) O p ij Ψ J = Variational Monte Carlo (VMC): R.B. Wiringa, PRC 43, 1585 (1991) Ψ T = S p=,6 pair correlation obtained by solving the (two-body) Euler-Lagrange equations (in spin S and isospin T channels) no three-body correlations induced by three-body force i<j f c(r ij ) Φ(JMTT z ) (s-shell nuclei): Jastrow wave function, fully antisymmetric A (1 + U ij ) i<j Ψ J The search in parameter space is made using COBYLA (Constrained Optimization BY Linear Approximations) algorithm available in the NLopt library http://ab-initio.mit.edu/wiki/index.php/nlopt
Green s Function Monte Carlo (GFMC): J. Carlson, PRC 36, 6 (1987); J. Carlson, PRC, 1879 (1988) Projects out lowest energy state from the best variational Ψ T : Ψ lim Ψ(τ) = lim τ τ e (H E ) τ Ψ T Ψ(τ = ) = Ψ T the imaginary-time evolution operator is computed for small time steps τ (τ = n τ) Propagator does not contain p, L,(L S) : it is carried out with a simplified version H of the full Hamiltonian H; H contains a charge independent eight-operator projection: Fermion sign problem limits maximum τ: [1, σ 1 σ,s 1, L S] [1, τ 1 τ ] Constrained path approximation: R.B. Wiringa PRC 6, 141 () limits the initial propagation to regions where the propagated Ψ(τ) and trial Ψ T wave functions have a positive overlap and discards those configurations that instead have a small or vanishing overlap small number of unconstrained time steps n uc are used when evaluating the expectation values
Results for binding energies: HH vs QMC The 3 H ground-state energies E (MeV) and rms proton radii r p (fm) Model a Model a Model b Model b Method E rp E rp E rp E rp VMC 7.59(6) 1.65 7.691(6) 1.6 7.317(7) 1.68 7.643(5) 1.63 GFMC 7.818(8) 1.6 7.917(1) 1.6 7.67(17) 1.65 7.863(8) 1.57 HH 7.818 7.949 7.599 7.866 The 4 He ground-state energies E (MeV) and rms proton radii r p (fm) Model a Model a Model b Model b Method E rp E rp E rp E rp VMC 4.38(1) 1.51 5.3(1) 1.49.89() 1.54 4.46() 1.49 GFMC 5.13(5) 1.49 5.71(3) 1.5 3.88(5) 1.53 5.1(4) 1.45 HH 5.15 5.8 3.96 5.8
Results for binding energies: The 3 H, 3 He, 4 He, 6 He, and 6 Li ground- and excited-state energies in MeV and proton rms radii r p in fm with model b compared with the corresponding GFMC results obtained with the AV18. VMC GFMC GFMC(AV18) A Z(J π ; T ) E rp E rp E rp 3 H( 1 + ; 1 ) 7.643(5) 1.63 7.863(8) 1.57 7.61(5) 1.66 He( 1 + ; 1 ) 6.97(5) 1.84 7.115(9) 1.84 6.88(5) 1.85 He( + ; ) 4.46() 1.49 5.1(4) 1.45 4.14(1) 1.49 6 He( + ; 1).58(3).5 4.53(6).7(1) 3.76(9).6(1) 6 He( + ; 1).94().6.87(6).18() 1.85(9).11(1) 6 Li(1 + ; ) 5.86(3).58 7.71(8).6(1) 6.87(9).58(1) 6 Li(3 + ; ).73(3).59 4.56(8).59(1) 4.11(7).87(1) 6 Li( + ; ) 1.4(3).61 4.4(9).79().75(11).63(1) 6 Li(1 + ;, 3 D[]).4(3).58 3.9(11).89() 1.99(1).85(3)
Conclusions: We constructed a family of local NN potential with chiral TPE including -isobar up to NLO (Q 3 ) and contact interactions up to N3LO (Q 4 ) Three versions of this chiral potential for three different cutoffs have been developed with fits to np and pp data up to E lab = 15 MeV and MeV, deuteron binding energy and nn scattering length A subset of the potentials a, a, b, and b have been used in HH, VMC, and GFMC calculations of binding energies and proton rms radii of nuclei with A = 6 Plans: The next stage in the program of studies of light nuclei structure with chiral interactions will be the inclusion of a 3N potential A chiral version of it at leading order, including -isobar intermediate states, has been developed, and is currently being constrained by reproducing observables in the A =3 systems.
Phase Shifts: 15 and MeV
The S-wave and D-wave components of the deuteron wave function corresponding to models a (dashed lines), b (dotted-dashed lines) and c (dotted-dasheddotted lines) are compared with those corresponding to the AV18 (solid lines) Deuteron wave functions [fm -1/ ].5.4.3. AV18 Model a Model b Model c.1 1 3 4 5 r [fm]
Phase Shift [deg] Phase Shift [deg] 6 4-5 -1-15 6 4 1 D 4 3 3 P 1 S 1 15-1 1-3 P1 5 3-3 P! Mixing Angle [deg] - 5 5 75 1 15 Lab. Energy [MeV] 5 5 75 1 15 Lab. Energy [MeV] -4 5 5 75 1 15 Lab. Energy [MeV]