Microscopic calculation of the. in the FMD approach. 3 He(α,γ) 7 Be capture reaction

Microscopic calculation of the 3 He(α,γ) 7 Be capture reaction in the FMD approach S-factor [kev b].7.6.5.4.3 Weizmann 4 LUNA 6 LUNA 7 Seattle 7 ERNA 9 FMD Thomas Neff Interfaces between structure and reactions INT, Seattle Aug 17, 211.2 1 2 E cm [MeV]

Overview Effective Nucleon-Nucleon interaction: Unitary Correlation Operator Method R. Roth, T. Neff, H. Feldmeier, Prog. Part. Nucl. Phys. 65 (21) 5 Short-range Correlations and Effective Interaction NCSM calculations Many-Body Method: Fermionic Molecular Dynamics Model 3 He(α,γ) 7 Be Radiative Capture Reaction T. Neff, Phys. Rev. Lett. 16, 4252 (211)

Unitary Correlation Operator Method Nuclear Force Argonne V18 (T=) spins aligned parallel or perpendicular to the relative distance vector 2 AV18 strong repulsive core: nucleons can not get closer than.5fm central correlations VNN [MeV] 1 r 12 strong dependence on the orientation of the spins due to the tensor force tensor correlations -1 r 12-2..5 1. 1.5 2. 2.5 3. r 12 [fm]

Universality of short-range correlations One-body densities coordinate space ρ (1) (r 1 ) [fm -3 ].35.3 α t.25 h.2 d.15 α*.1.5 1 2 3 4 r 1 [fm] momentum space 1 n (1) (k 1 )/A [fm 3 ] 1 2 1 1 1 1-1 1-2 1-3 1-4 1-5 α t h d α* 1 2 3 4 k 1 [fm -1 ] one-body densities calculated from exact wave functions for AV8 interaction coordinate space densities reflect different sizes and densities of 2 H, 3 H, 3 He, 4 He and the + 2 state in 4 He similar high-momentum tails in the momentum densities Feldmeier, Horiuchi, Neff, Suzuki, arxiv:117.4956

Universality of short-range correlations Two-body densities coordinate space S=1,M S =1, T= momentum space S=1, T= C N 1, ρrel 11, (r) [fm-3 ].5.4.3.2.1 z x α t h d α* C N 1, nrel 1, (k) [fm3 ] 1 4 1 3 1 2 1 1 1 1-1 1-2 1-3 α t h d α* 1 2 3 4 1-4 1 2 3 4 x/z [fm] k [fm -1 ] normalize two-body density in coordinate space at r=1. fm normalized two-body densities in coordinate are identical at short distances for all nuclei use the same normalization factor in momentum space high momentum tails agree for all nuclei Feldmeier, Horiuchi, Neff, Suzuki, arxiv:117.4956

UCOM Unitary Correlation Operator Method Correlation Operator induce short-range (two-body) central and tensor correlations into the many-body state C =C Ω C r=exp g Ω,j exp g r,j, C C =1 <j correlation operator should conserve the symmetries of the Hamiltonian and should be of finite-range, correlated interaction phase shift equivalent to bare interaction by construction Correlated Operators correlated operators will have contributions in higher cluster orders <j C O C =Ô [1] +Ô [2] +Ô [3] +... two-body approximation: correlation range should be small compared to mean particle distance Correlated Interaction C (T +V )C =T +V UCOM+V [3] UCOM +...

UCOM Central and Tensor Correlations C =C Ω C r p r = 1 2 r r r r p + p r r p=p r +p Ω r r, pω = 1 2r l r r r r l

UCOM Central and Tensor Correlations C =C Ω C r p r = 1 2 r r r r p + p r r p=p r +p Ω r r, pω = 1 2r l r r r r l Central Correlations c r =exp pr s(r)+s(r)p 2 r probability density shifted out of the repulsive core S=,T=1 2 AV18.35.3 ρ (2) (r) V [MeV] 1 AV18 [fm 3 ].25.2.15.1.5 ˆρ (2) (r) -1 v c (r) -2..5 1. 1.5 2. 2.5 3. r [fm]..5 1. 1.5 2. 2.5 3. r [fm]

UCOM Central and Tensor Correlations r p r p p Ω C =C Ω C r p r = 1 2 r r r r p + p r r p=p r +p Ω r r, pω = 1 2r l r r r r l Central Correlations c r =exp pr s(r)+s(r)p 2 r probability density shifted out of the repulsive core Tensor Correlations c Ω =exp ϑ(r) 3 2 (σ 1 p Ω )(σ 2 r)+ 3 2 (σ 1 r)(σ 2 p Ω ) tensor force admixes other angular momenta S=,T=1 2 AV18 S=1,T= 2 AV18 [fm 3 ].35.3.25.2.15.1.5 ρ (2) (r) V [MeV] 1-1 AV18 v c (r) UCOM(variational): correlation functions s(r) und ϑ(r) are determined by variation of ˆρ (2) the (r) energy in the twobody system for each S,T channel r [fm]..5 1. 1.5 2. 2.5 3. -2..5 1. 1.5 2. 2.5 3. r [fm] [fm 3 ].35.3.25.2.15.1.5 V [MeV] 1-1 AV18 ρ (2) (r) v c (r) ˆρ (2) (r) v t (r) UCOM(SRG): correlation functions s(r) und ϑ(r) are determined ˆρ (2) from mapping wave L= functions obtained with (2) bare (r)ˆρ L=2 interaction (r) to wave functions obtained with SRG interaction r [fm]..5 1. 1.5 2. 2.5 3. -2..5 1. 1.5 2. 2.5 3. r [fm]

Unitary Correlation Operator Method Correlations and Energies 2 two-body densities z [fm] 1 1 C r ~ ~ C Ω 2 2 1 1 2 2 1 1 2 2 1 1 2 x [fm] x [fm] x [fm] ρ (2) S,T (r 1 r 2 ) S=1,M S =1,T= central correlator C r shifts density out of the repulsive core tensor correlator C Ω aligns density with spin orientation Neff and Feldmeier, Nucl. Phys. A713 (23) 311

Unitary Correlation Operator Method Two-body Densities coordinate space S=1,M S =1, T= momentum space S=1, T= ρ rel 11, (r) [fm-3 ].4.3.2.1 x z UCOM4 UCOM2 uncorr. CG n rel 1, (k) [fm3 ] 1 2 1 1 1 1-1 1-2 1-3 1-4 1-5 L=2 UCOM4 UCOM2 CG L= 1 2 3 4 1-6 1 2 3 4 x/z [fm] k [fm -1 ] two-body densities calculated from ħω 4 He and correlated density operators UCOM2 correlators derived from λ 1.5 fm 1 SRG interaction reproduce coordinate space two-body density and high-momentum components very well high-momentum components dominated by tensor correlations long-range correlations should fill up momentum space two-body density above the Fermi momentum Feldmeier, Horiuchi, Neff, Suzuki, arxiv:117.4956

Unitary Correlation Operator Method Correlated Interaction in Momentum Space 3 S 1 bare 3 S 1-3 D 1 bare bare interaction has strong off-diagonal matrix elements connecting to high momenta correlated interaction 3 S 1 correlated is more attractive at low momenta 3 S 1-3 D 1 correlated off-diagonal matrix elements connecting low- and high- momentum states are strongly reduced Roth, Hergert, Papakonstaninou, Neff, Feldmeier, Phys. Rev. C 72, 342 (25)

Unitary Correlation Operator Method Correlated Interaction in Momentum Space 3 S 1 bare 3 S 1-3 D 1 bare bare interaction has strong off-diagonal matrix elements connecting to high momenta correlated interaction 3 S 1 correlated is more attractive at low momenta 3 S 1-3 D 1 correlated off-diagonal matrix elements connecting low- and high- momentum states are strongly reduced similar to V low-k, SRG Roth, Hergert, Papakonstaninou, Neff, Feldmeier, Phys. Rev. C 72, 342 (25)

UCOM(SRG) No-Core Shell Model Calculations 4 HeUCOMSRG 4 HeUCOMSRG EMeV 5 1 15 2 α=.4 fm 4 2 EMeV 5 1 15 2 ħω 25 3 15 2 25 3 35 4 45 ħmev 4 25 3 converged.5.1.15.2 Αfm 4 convergence much improved compared to bare interaction effective interaction in two-body approximation converges to different energy then bare interaction transformed interaction can be tuned to obtain simultaneously (almost) exact 3 He and 4 He binding energies

EMeV UCOM(SRG) NCSM 6 Li/ 7 Li ground state energy 5 1 15 2 25 3 6 LiUCOMSRG 16 MeV 2 MeV 24 MeV 35 2 4 6 8 1 12 14 16 18 2 N max EMeV 1 2 3 4 7 LiUCOMSRG 16 MeV 2 MeV 24 MeV 2 4 6 8 1 12 14 16 18 2 N max tuned interaction also works reasonably well for heavier nuclei

UCOM and SRG NCSM 7 Li spectrum 13 12 7 Li ħ 16MeV EMeV 11 1 9 8 7 6 5 4 12 72 32 52 52 72 72 12 32 52 52 72 3 2 1 12 32 UCOM var. UCOM SRG SRG ħ 2ħ 4ħ 6ħ 8ħ 1ħ 1ħ 1ħ Experiment 12 32

UCOM(SRG) NCSM 6 Li/ 7 Li radii Rpfm 2.4 2.2 2. 1.8 1.6 16 MeV 2 MeV 24 MeV 6 LiUCOMSRG 2 4 6 8 1 12 14 16 18 2 N max Rpfm 2.4 2.2 2. 1.8 16 MeV 2 MeV 24 MeV 7 LiUCOMSRG 1.6 2 4 6 8 1 12 14 16 18 2 N max radii converge worse than energies harmonic oscillator basis not well suited to describe tails of weakly bound nuclei

FMD Halos, Clusters,... Al-Khalili, Nunes, J. Phys. G 29, R89 (23)

FMD Fermionic Molecular Dynamics Fermionic Slater determinant Q = A q 1 q A antisymmetrized A-body state Feldmeier, Schnack, Rev. Mod. Phys. 72 (2) 655 Neff, Feldmeier, Nucl. Phys. A738 (24) 357

FMD Fermionic Molecular Dynamics Fermionic Slater determinant Q = A q 1 q A antisymmetrized A-body state Molecular single-particle states x q =exp (x b)2 2 χ,χ ξ Gaussian wave-packets in phase-space (complex parameter b encodes mean position and mean momentum), spin is free, isospin is fixed width is an independent variational parameter for each wave packet use one or two wave packets for each single particle state Feldmeier, Schnack, Rev. Mod. Phys. 72 (2) 655 Neff, Feldmeier, Nucl. Phys. A738 (24) 357

FMD Fermionic Molecular Dynamics Fermionic Slater determinant Q = A q 1 q A Antisymmetrization antisymmetrized A-body state Molecular single-particle states x q =exp (x b)2 2 χ,χ ξ Gaussian wave-packets in phase-space (complex parameter b encodes mean position and mean momentum), spin is free, isospin is fixed width is an independent variational parameter for each wave packet use one or two wave packets for each single particle state Feldmeier, Schnack, Rev. Mod. Phys. 72 (2) 655 Neff, Feldmeier, Nucl. Phys. A738 (24) 357

FMD Interaction Matrix Elements (One-body) Kinetic Energy qk T q = k b k T b χk χ ξk ξ k b T 1 3 k b = 2m k + (b k b ) 2 ( k + R ) 2 k (Two-body) Potential fit radial dependencies by (a sum of) Gaussians G(x 1 x 2 )=exp (x 1 x 2 ) 2 2κ Gaussian integrals α kmn = k m + n k + m + n ρ kmn = mb k + k b m k + m R km = k b k m b m nb + b + n k b k, b G m κ 3/2 ρ b m, n b n =Rkm R n exp 2 kmn α kmn +κ 2(α kmn +κ) analytical formulas for matrix elements

FMD Operator Representation of V UCOM C (T +V )C = T one-body kinetic energy + ST + T + T ˆV ST c (r)+ 1 2 pr 2 ˆV ST p 2 (r)+ ˆV ST ˆV T s (r)l s+ ˆV T 2 s (r)l 2 l s p 2 (r)p r 2 + ˆV ST 2 (r)l 2 ˆV T t (r)s 12 (r,r)+ ˆV T trp Ω (r)p r S 12 (r,p Ω )+ ˆV T t (r)s 12 (l,l)+ ˆV T tp Ω p Ω (r)s 12(p Ω,p Ω )+ ˆV T 2 tp Ω p Ω (r)l 2 S 12(p Ω,p Ω ) central potentials spin-orbit potentials tensor potentials Nucl. Phys. A745 (24) 3 bulk of tensor force mapped onto central part of correlated interaction tensor correlations also change the spin-orbit part of the interaction

FMD Mean-Field Calculations Minimization minimize Hamiltonian expectation value with respect to all singleparticle parameters q k QH T cm Q min {q k } QQ this is a Hartree-Fock calculation in our particular single-particle basis the mean-field may break the symmetries of the Hamiltonian 16 O 4 C 2 Ne 27 Al spherical nuclei intrinsically deformed nuclei

FMD Projection and Multiconfiguration Mixing Projection Slater determinant may break symmetries of Hamiltonian restore symmetries by projection on parity, linear and angular momentum Creating Basis States full Variation after Angular Momentum and Parity Projection (VAP) for spins of lowest states constrain radius, dipole, quadrupole or octupole moments to generate additonal basis states For heavier nuclei (sd-shell) only Projection after Variation possible P π = 1 2 (1+π) J P MK = 2J+1 d 3 ΩD J MK (Ω)R (Ω) 8π 2 P P = 1 (2π) 3 d 3 X exp{ (P P) X}

FMD Projection and Multiconfiguration Mixing Projection Slater determinant may break symmetries of Hamiltonian restore symmetries by projection on parity, linear and angular momentum Creating Basis States full Variation after Angular Momentum and Parity Projection (VAP) for spins of lowest states constrain radius, dipole, quadrupole or octupole moments to generate additonal basis states For heavier nuclei (sd-shell) only Projection after Variation possible Multiconfiguration Mixing Calculations diagonalize Hamiltonian in set of projected intrinsic states Q (), =1,...,N P π = 1 2 (1+π) J P MK = 2J+1 d 3 ΩD J MK (Ω)R (Ω) 8π 2 P P = 1 (2π) 3 d 3 X exp{ (P P) X} Q () J H P π P= KK P Q (b) c α K b = K b E Jπ α K b Q () P J π KK P P= Q (b) c α K b

PAV and VAP 4 He- 4 He -2 4 He- 4 He- 4 He 1 12 C -3 intrinsic projected + z [fm] 5 E [MeV] -4-5 -5.1.1.1-1 -1-5 5 1 x [fm] -6 2 4 6 8 1 R [fm] E intr = -49.39 MeV E proj = -61.31 MeV angular momentum projection lowers kinetic energy by delocalizing clusters correlation energies can be very significant

3 He(α,γ) 7 Be radiative capture one of the key reactions in the solar pp-chains Effective Nucleon-Nucleon interaction: UCOM(SRG) α=.2 fm 4 λ 1.5 fm 1 Many-Body Approach: Fermionic Molecular Dynamics Internal region: VAP configurations with radius constraint External region: Brink-type cluster configurations Matching to Coulomb solutions: Microscopic R-matrix method Results: 7 Be bound and scattering states Astrophysical S-factor

3 He(α,γ) 7 Be Theoretical Approaches Potential models (Kim et al. 1982, Mohr 29,...) 4 He and 3 He are considered as point-like particles interacting via an effective nucleus-nucleus potential fitted to bound state properties and phase shifts ANCs calculated from ab initio wave functions (Nollett 21, Navratil et al. 27) Microscopic Cluster Model (Tang et al. 1981, Langanke 1986, Kajino 1986...) antisymmetrized wave function built with 4 He and 3 He clusters some attempts to include polarization effects by adding other channels like 6 Li plus proton interacting via an effective nucleon-nucleon potential, adjusted to describe bound state properties and phase shifts Our Aim fully microscopic wave functions with cluster configurations at large distances and additional polarized A-body configurations in the interaction region using a realistic effective interaction

3 He(α,γ) 7 Be FMD model space Frozen configurations Frozen antisymmetrized wave function built with 4 He and 3 He FMD clusters up to channel radius =12 fm R Polarized configurations FMD wave functions obtained by VAP on 1/2, 3/2, 5/2, 7/2 and 1/2 +, 3/2 + and 5/2 + combined with radius constraint in the interaction region Polarized 3/2 7/2 1/2 + Boundary conditions Match relative motion of clusters at channel radius to Whittaker/Coulomb functions with the microscopic R- matrix method of the Brussels group D. Baye, P.-H. Heenen, P. Descouvemont

Slater determinants and RGM wave functions Divide model space into internal and external region at channel radius In internal region wave function is described microscopically with FMD Slater determinants In external region wave function is considered as a system of two point-like clusters (Microscopic) cluster wave function Slater determinant Q b (R) = 1 Q A cb ( m b R) Q b ( m +m b m R) m +m b Projection on total linear momentum decouples intrinsic motion, relative motion of clusters and total center-of-masss Q b (R);P= = d 3 r (r R) b (r) Pcm = using RGM basis states ρ,ξ,ξ b b (r) = 1 cb A δ(ρ r) (ξ ) b (ξ b ) RGM norm kernel n b (r,r )= b (r) b (r )

Slater determinants and RGM wave functions Relative motion in Slater determinant described by Gaussian (r R)= βrel π 2 rel 3/4 exp (r R)2 2 rel with rel = A b + b A A A b, β rel = b A b + b A Overlap of full wave function with RGM cluster basis ψ(r)= d 3 r n 1/2 (r,r ) (r ) Ψ Match asymptotics to Whittaker or Coulomb functions ψ b (r)=a 1 r W η,l+1/2(2κr) ψ sctt (r)= 1 r L (η,kr) e 2δ O L (η,kr) with κ= 2μE b, k= 2μE, η=μ Z Z b e 2 k

3 He(α,γ) 7 Be p-wave Bound and Scattering States Bound states Experiment FMD 7 Be E 3/2-1.59 MeV -1.49 MeV 18 Phase shift analysis: Spiger and Tombrello, PR 163, 964 (1967) E 1/2-1.15 MeV -1.31 MeV r ch 2.647(17) fm 2.67 fm Q -6.83 e fm 2 7 Li E 3/2-2.467 MeV -2.39 MeV E 1/2-1.989 MeV -2.17 MeV r ch 2.444(43) fm 2.46 fm Q -4.(3) e fm 2-3.91 e fm 2 δ(e) [deg] 15 12 9 6 3 3/2-1/2 - centroid of bound state energies well described if polarized configurations included tail of wave functions tested by charge radii and quadrupole moments 2 4 6 8 1 E [MeV] dashed lines frozen configurations only solid lines polarized configurations in interaction region included Scattering phase shifts well described, polarization effects important

3 He(α,γ) 7 Be s-, d- and ƒ-wave Scattering States 21 18-2 15 δ(e) [deg] -4 δ(e) [deg] 12 9-6 1/2 + 3/2 + 5/2 + 6 3 7/2-5/2 - -8 1 2 3 4 5 6 E [MeV] 2 4 6 8 1 E [MeV] dashed lines frozen configurations only solid lines FMD configurations in interaction region included polarization effects important s- and d-wave scattering phase shifts well described 7/2 resonance too high, 5/2 resonance roughly right, consistent with no-core shell model calculations

3 He(α,γ) 7 Be S-Factor S-factor [kev b].7.6.5.4.3 Weizmann 4 LUNA 6 LUNA 7 Seattle 7 ERNA 9 FMD S-factor: S(E) = σ(e)e exp{2πη} η= μz 1Z 2 e 2 k Nara Singh et al., PRL 93, 26253 (24) Bemmerer et al., PRL 97, 12252 (26) Confortola et al., PRC 75, 6583 (27) Brown et al., PRC 76, 5581 (27) Di Leva et al., PRL 12, 23252 (29).2 1 2 E cm [MeV] dipole transitions from 1/2 +, 3/2 +, 5/2 + scattering states into 3/2, 1/2 bound states FMD is the only model that describes well the energy dependence and normalization of new high quality data fully microscopic calculation, bound and scattering states are described consistently

3 He(α,γ) 7 Be Overlap Functions and Dipole Matrixelements r Ψ 1/2+ (r) [a.u.] r Ψ 3/2- (r) [a.u.] 1/2 + scattering state E cm = 5 kev 3/2 - bound state 5 1 15 r [fm] r 2 Ψ 3/2- (r) r Ψ 1/2+ (r) [a.u.] 5 1 15 2 25 3 r [fm] Overlap functions from projection on RGM-cluster states Coulomb and Whittaker functions matched at channel radius =12 fm Dipole matrix elements calculated from overlap functions reproduce full calculation within 2% cross section depends significantly on internal part of wave function, description as an external capture is too simplified

3 He(α,γ) 7 Be Energy dependence of the S-Factor S-factor [kev b].6.5.4.3.2 FMD Kajino MHN s-wave d-wave.1 1 2 E cm [MeV] low-energy S-factor dominated by s-wave capture at 2.5 MeV equal contributions of s- and d-wave capture FMD results differ from Kajino results mainly with respect to s-wave capture related to short-range part of wave functions?

3 H(α,γ) 7 Li S-Factor S-factor [kev b].15.1.5 Caltech 94 FMD S-factor: S(E) = σ(e)e exp{2πη} η= μz 1Z 2 e 2 k Brune et al., PRC 5, 225 (1994).5 1 E cm [MeV] isospin mirror reaction of 3 He(α,γ) 7 Be 7 Li bound state properties and phase shifts well described FMD calculation describes energy dependence of Brune et al. data but cross section is larger by about 15%

3 He(α,γ) 7 Be and 3 H(α,γ) 7 Li S-Factors consistent? 3 He(α,γ) 7 Be 3 H(α,γ) 7 Li S-factor [kev b].7.6.5.4 Weizmann 4 LUNA 6 LUNA 7 Seattle 7 ERNA 9 Kajino MHN FMD FMD x.95 S-factor [kev b].15.1 Caltech 94 Kajino MHN FMD FMD x.85.3.5.2 1 2 E cm [MeV].5 1 E cm [MeV] FMD calculation agrees with normalization and energy dependence of 3 He(α,γ) 7 Be data FMD calculation agrees with energy dependence but not normalization of 3 H(α,γ) 7 Li data similar inconsistency observed in other models

4 He-n scattering Phase shift analysis: Bond and Kirk, Nucl. Phys. A287, 317 (1977) 18 18 15 15 3/2-1/2 - δ(e) [deg] 12 9 6 3 1/2 + δ(e) [deg] 12 9 6 3 2 4 6 8 1 E [MeV] dashed lines frozen configurations only, solid lines polarized configurations included 2 4 6 8 1 E [MeV] UCOM(SRG) interaction FMD VAP (1/2 +, 3/2, 1/2 ) plus radius constraint configurations in interaction region polarization effects very small in S-wave scattering splitting between 3/2 and 1/2 states too small consistent with GFMC (two-body interaction only) and NCSM results

4 He- 4 He scattering 2 15 + 2 + 4 + δ(e) [deg] 1 5 2 4 6 8 1 E [MeV] UCOM(SRG) interaction FMD VAP ( +, 2 +, 4 + ) plus radius constraint configurations in interaction region polarization effects shift S- and D-wave resonances by about 1 MeV

Summary Unitary Correlation Operator Method Explicit description of short-range central and tensor correlations Realistic low-momentum interaction V UCOM NCSM calculations with UCOM Fermionic Molecular Dynamics Microscopic many-body approach using Gaussian wave-packets Projection and multiconfiguration mixing 3 He(α,γ) 7 Be Radiative Capture Bound states, resonance and scattering wave functions S-Factor: energy dependence and normalization Analyzed in terms of overlap functions Inconsistency of 3 He(α,γ) 7 Be and 3 H(α,γ) 7 Li data? Thanks to my collaborators: Hans Feldmeier(GSI), Wataru Horiuchi (RIKEN), Karlheinz Langanke (GSI), Robert Roth (TUD), Yasuyuki Suzuki (Niigata), Dennis Weber (GSI)