Effective interactions in the delta-shell model

Transcription

1 Effective interactions in the delta-shell model Rodrigo Navarro Pérez Enrique Ruiz Arriola José Enrique Amaro Soriano University of Granada Atomic, Molecular and Nuclear Physics Department Few Body, Fukuoka, August Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August /

2 Motivation Delta Shell Potential Fitting np Phaseshifts Scattering Observables Skyrme parameters 5 Summary Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August /

3 Motivation S P P P S D ǫ D D P F ǫ F F D G ǫ G G F H ǫ 5 δ [deg.] 5 5 ELAB [MeV] No unique determination of the NN interaction Different phenomenological potentials Fitted to experimental scattering data High accuracy χ /d.o.f. Dispersion in Phaseshifts OPE as a long range interaction parameters for the short and intermediate range Repulsive core for most of them Short range correlations Nuclear structure calculations become complicated Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August /

4 Motivation Effective coarse graining Oscillator Shell Model Euclidean Lattice EFT V lowk interaction Characteristic distance.5. fm Nyquist Theorem Optimal sampling Finite Bandwidth r k () de Broglie wavelength of the most energetic particle Sampling resolution determined V [fm ] - AV8 uk (r) u(r) u5(r) r [fm] fm - u(r) Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August /

5 Motivation Effective coarse graining Oscillator Shell Model Euclidean Lattice EFT V lowk interaction Characteristic distance.5. fm Nyquist Theorem Optimal sampling Finite Bandwidth r k () de Broglie wavelength of the most energetic particle Sampling resolution determined V [fm ] - AV8 uk (r) u(r) u5(r) r [fm] fm - u(r) Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August /

6 Motivation Effective Interaction [Skyrme, Moshinsky] Useful simplifications in many body calculations Power expansion in CM momenta V (p, p) = d xe ix (p p) ˆV (x) = t ( + x P σ ) + t ( + x P σ )(p + p ) +t ( + x P σ )p p + it V S (p p) + t T + t U +O(p ) [ σ pσ p + σ p σ p ] σ σ (p + p ) [ σ pσ p + σ p σ p ] σ σ p p Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August 5 /

7 Delta Shell Potential A sum of delta functions V (r) = i λ i µ δ(r r i) () [Aviles Phys.Rev. C6 (97) 67] Optimal and minimal sampling of the nuclear interaction Pion production threshold k = fm Optimal sampling around.5 fm Analitic Potential Delta Shell r [fm] 5 k =. fm, u(r) r [fm] Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August 6 /.5.5 5

8 Coarse Graining the AV8 potential V [fm ] -.5 AV8 Delta Shell.5.5 r [fm].5 V [fm ] -.5 AV8 Delta Shell.5.5 r [fm].5 δ [deg] N = 8 N = N = N = 5 N = E LAB [MeV] 5 δ [deg] N = N = N = N = N = E LAB [MeV] 5 Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August 7 /

9 Delta Shell Potential Comparison with V lowk S S Vlowk [fm] Delta Shell AV8.5 k [fm ].5 Vlowk [fm] Delta Shell AV8.5 k [fm ].5 Nuclear structure calculations B [MeV] He UCOM GFMC Exp..6 rm [fm].8 B [MeV] O BHF...6 rm [fm] CC Exp.8 B [MeV] Ca Exp rm [fm] Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August 8 /

10 Delta Shell Potential well defined regions Innermost region r.5 fm Short range interaction No delta shell (No repulsive core) Intermediate region.5 r. fm Unknown interaction λ i parameters fitted to scattering data Outermost region r. fm Long range interaction Described by OPE Sampled with delta shells as well Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August 9 /

11 Delta Shell Potential well defined regions Innermost region r.5 fm Short range interaction No delta shell (No repulsive core) Intermediate region.5 r. fm Unknown interaction λ i parameters fitted to scattering data Outermost region r. fm Long range interaction Described by OPE Sampled with delta shells as well Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August 9 /

12 Delta Shell Potential well defined regions Innermost region r.5 fm Short range interaction No delta shell (No repulsive core) Intermediate region.5 r. fm Unknown interaction λ i parameters fitted to scattering data Outermost region r. fm Long range interaction Described by OPE Sampled with delta shells as well Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August 9 /

13 Fitting np Phase shifts Several high quality potentials produce different np phase shifts PWA Nijm I Nijm II Reid9 Argonne V8 CD-Bonn Covariant Spectator Fit for every partial wave with j 5 Strength coefficients λ i as fit parameters Fixed and equidistant concentration radii r =.6 fm Fit at different energy ranges Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August /

14 Fitting np Phase shifts Several high quality potentials produce different np phase shifts PWA Nijm I Nijm II Reid9 Argonne V8 CD-Bonn Covariant Spectator Fit for every partial wave with j 5 Strength coefficients λ i as fit parameters Fixed and equidistant concentration radii r =.6 fm Fit at different energy ranges Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August /

15 Fitting np Phaseshifts χ /d.o.f. = S P H5 5 D F G P H5 5 D F G F 5 D G5 5 5 ELAB [MeV] Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August / S P ǫ ǫ ǫ ǫ5 5 5 ǫ P F H I5 5 D G 5 5 5

16 Scattering Observables Comparing with High Quality Potentials and Experimental data I 5.8 MeV D. MeV Rt 5. MeV I 5. MeV I. MeV At 5. MeV Dt 5. MeV P 5. MeV Compilation Experimental OPE rc =. fm P 5. MeV θ cm [deg] Good agreement with data and other potentials Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August /

17 Skyrme parameters Skyrme parameters in terms of paratial waves Partial Wave potential in momentum space V JS l l (p, p) = (π) M drr j l (p r)j l (pr)vl JS l (r) Using the Bessel function expansion x l [ x ] j l (x) = (l + )!! (l + ) + Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August /

18 Skyrme parameters Comparing similar terms (t, x t ) = d x [ V S (r) ± V S (r) ] (t, x t ) = d xr [ V S (r) ± V S (r) ] (t, x t ) = d xr [ 5 V P (r) + V P (r) + 5V P (r) ± 9V P (r) ] t V = d xr [ 7 V P (r) + V P (r) 5V P (r) ] t U = d xr [ 6 V P (r) + V P (r) V P (r) ] t T = 5 d xr V ɛ (r) Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August /

19 Skyrme parameters Straightforward for δ-shell potential t λi r n i Integrable for OPE starting at r c t f πnn m π Γ(n, m π r c ) Where fπnn /(π).8 Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August 5 /

20 Skyrme parameters Skyrme parameters fitting at different energy ranges DS Coupled S-Waves DS Uncoupled S-Waves AV k CM [MeV] t [MeV fm ] t [MeV fm 5 ] DS Coupled S-Waves DS Uncoupled S-Waves AV k CM [MeV] x x.5.5 DS Coupled S-Waves DS Uncoupled S-Waves AV k CM [MeV]..5.5 DS Coupled S-Waves DS Uncoupled S-Waves AV k CM [MeV] Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August 6 /

21 Skyrme parameters Skyrme parameters fitting at different energy ranges t [MeV fm 5 ] 78 DS 775 AV kcm [MeV] x DS Coupled S-Waves AV kcm [MeV] tv [MeV fm 5 ] DS 5 AV kcm [MeV] tu [MeV fm 5 ] 8 7 DS AV kcm [MeV] tt [MeV fm 5 ] DS AV kcm [MeV] Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August 7 /

22 Skyrme Parameters Fermi type shape density ρ(x) = ρ + e (r R)/a R = r A /, r =.fm and a =.7fm Error band for stable nuclei binding energy 8 -B(A)/A MeV 6 B A = 8A t d x ρ(x) A Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August 8 /

23 Skyrme Parameters Nuclear and Neutron matter Error grows linearly with the density B n.m. A B n A = 8 t ρ.75ρ = [t ( x )]ρ n.5ρ n B/N MeV n fm - B/A MeV n.m. fm - Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August 9 /

24 Summary Sampling of the NN interaction by a delta shell potential / m π M r /m π well defined regions Fit to the np phase shifts given by 7 High Quality Potentials Every partial wave with j 5 Systematic error propagation Good description of scattering observables Compatible with experimental data Effective interaction parametrized trough low momentum expansion δ-shell representation allows straightforward calculations Comparison between δ-shell and a V lowk representation of AV8 Something missing about coupled and uncoupled partial waves Error estimates for stable nuclei, nuclear and neutron matter Navarro-Pérez R. (UGR) Effective interactions in the δ-shell FB, August /