Electromagnetic reactions from few to many-body systems Giuseppina Orlandini

Transcription

1 Electromagnetic reactions from few to many-body systems Giuseppina Orlandini ECT* Workshop on Recent advances and challenges in the description of nuclear reactions at the limit of stability, March 5-9, 2018

2 Physics of e.m. Interactions with Nuclei e' F > F > q > q = * e I > I > G. Orlandini Summer School - La Rabida, July 2009

3 The e.m. interaction is perturbative compared to the nuclear strong interaction H = H N + V em Therefore the reaction cross sections are proportional to ~ < F J I > 2 F > and I > are eigenstates of H N I > is a bound state (g.s.), F > can be a bound or a continuum (scattering) state J is the nuclear current G. Orlandini ECT* Workshop on Recent advances and G. challenges Orlandini Summer in the School description - La Rabida, of nuclear July 2009 reactions at the limit of stability, March 5-9, 2018

4 The ab-initio approach Start from neutrons and protons as building blocks (positions, spins, isospins) r2... r1 ra

5 The ab-initio approach Start from neutrons and protons as building blocks (positions, spins, isospins) Subtract the cm motion and remain with relative coordinates...

6 The ab-initio approach Start from neutrons and protons as building blocks (positions, spins, isospins) Subtract the cm motion and remain with relative coordinates... Solve the non-relativistic quantum mechanical problem of A-interacting nucleons

7 The ab-initio approach Start from neutrons and protons as building blocks (positions, spins, isospins) Subtract the cm motion and remain with relative coordinates... Solve the non-relativistic quantum mechanical problem of A-interacting nucleons Find numerical solutions with no approximations or controllable approximations (error bars)

8 The ab-initio approach Start from neutrons and protons as building blocks (positions, spins, isospins) Subtract the cm motion and remain with relative coordinates... Solve the non-relativistic quantum mechanical problem of A-interacting nucleons Find numerical solutions with no approximations or controllable approximations (error bars) Calculate low-energy observables and compare with experiment to test nuclear forces and provide predictions for future experiments and microscopic interpretations for older experiments

9 For the ab-initio program, as previously defined, the most interesting reactions are those at low-energy (up to ~ 50 - (100??) MeV ) I > is in general a ground state F > can be a bound or a continuum (scattering) state

10 I > F > F > F >...

11 Physics of e.m. Interactions with Nuclei e' F > F > q >?? q = * e I > I > G. Orlandini ECT* Workshop on Recent advances and G. challenges Orlandini Summer in the School description - La Rabida, of nuclear July 2009 reactions at the limit of stability, March 5-9, 2018

12 ~ < F J I > 2 where J = ( J ) J = ( J ) J = ( J T ) e' F > F > q > q = e * However I > I > G. Orlandini ECT* Workshop on Recent advances and G. challenges Orlandini Summer in the School description - La Rabida, of nuclear July 2009 reactions at the limit of stability, March 5-9, 2018

13 For q (and q small (q R -> 1) ~ = < F D I > < F J ~ E1 I > T e' F > F > q > q = * e I > I > G. Orlandini ECT* Workshop on Recent advances and G. challenges Orlandini Summer in the School description - La Rabida, of nuclear July 2009 reactions at the limit of stability, March 5-9, 2018

14 Experimental status Stable Nuclei We have data on ~180 stable nuclei Giant dipole resonances Unstable Nuclei Few data pigmy dipole resonances Leistenschneider et al. Photoabsorption experiments e - e - e - (p,p ) experiments Coulomb excitation experiments

15 Experimental status Stable Nuclei We have data on ~180 stable nuclei Giant dipole resonances Unstable Nuclei Few data pigmy dipole resonances Leistenschneider et al. Photoabsorption experiments e - e - e - (p,p ) experiments Coulomb excitation experiments

16 dipole strength collective interpretation

17 dipole strength collective interpretation Giant Dipole Resonance (GDR) E Pigmy dipole resonance (PDR) in neutron-rich nuclei core

18 Do we see the emergence of collective modes from first principle calculations?

19 Reactions to continuum Energies in the non-relativistic regime Non-Relativistic Quantum Mechanics (including Translation, Galileian, Rotational invariances) [ H, P cm ]=0 [ H, R cm ]=0 [ H, J ]=0 Degrees of freedom: total A nucleons ( microscopic model) H = T + V Framework: V = ij v ij + ( ijk v ijk +...)

20 Reactions to continuum perturbative (electro-weak) First order perturbation theory (Fermi-Golden Rule) Linear Response theory + b ----> b* ( n 0 n n + 0 H n > = E n n >

21 Reactions to continuum PERTURBATIVE INCLUSIVE S ( n n 0 n + 0 S ( represents the crucial quantity Requires the solution of both the bound and continuum A-body problem

22

23 Integral transform (IT) Φ = dω K(ω,σ) S(ω ) One IS NOT able to calculate S( ) (the quantity of direct physical meaning) but IS able to calculate Φ In order to obtain S( ω) one needs to invert the transform Problem: Sometimes the inversion of Φ may be problematic

24 S Φ S( ) K(ω,σ) d 1) integrate in d using delta function

25 S Φ S( ) K(ω,σ) d 1) integrate in d using delta function Φ K( E -E n,σ) 0 n n 0 n 0

26 S Φ S( ) K(ω,σ) d 1) integrate in d using delta function Φ K( E -E n,σ) 0 n n 0 n 0 0 n K( H-E 0,σ) n n 0

27 S Φ S( ) K(ω,σ) d 1) integrate in d using delta function Φ K( E -E n,σ) 0 n n 0 n 0 2) Use 0 n K( H-E 0,σ) n n 0 n n n = I Φ 0 K( K(H-E 0, ) 0

28 Φ S( ) K(ω,σ) d 0 H-E, ) 0 0

29 The calculation of ANY transform seems to require, in principle, only the knowledge of the ground state! However, H-E 0, ) can be quite a complicate operator. Φ 0 H-E, ) 0 0

30 The calculation of ANY transform seems to require, in principle, only the knowledge of the ground state! However, H-E 0, ) can be quite a complicate operator. So, which kernel is suitable for calculation of this? Φ 0 H-E, ) 0 0

31 a good Kernel has to satisfy two requirements 1) one must be able to calculate the integral transform 2) one must be able to invert the transform minimizing uncertainties

32 Which is the best kernel?

33 The -function!

34 What would be the perfect Kernel? the delta-function! in fact Φ S S d

35 but what about a representation of the -function?

36 The Lorentzian kernel: complex! R +i K( ω, ) = C (ω - σ) (ω + σ* ) R It is a representation of the -Function Φ R C [ R S d

37 Illustration of requirement N.1: one can calculate the integral transform

38 Remember! Φ S( ) K(ω,σ) d 0 H-E, ) 0 0

39 K(ω,σ) Φ S( S( ) (ω - σ) (ω + σ* ) d 0 (H-E ) (H-E )

40 K(ω,σ) Φ S( S( ) (ω - σ) (ω + σ* ) d 0 (H-E ) (H-E )

41 main point of the LIT : Schrödinger-like equation with a source ~ ( H - E 0 - R - i 0>

42 main point of the LIT : Schrödinger-like equation with a source ~ ( H - E 0 - R - i 0> Theorem: The solution is unique and has bound state asymptotic conditions one can apply bound state methods

43 Illustration of requirement N.2: one can invert the integral transform minimizing uncertainties

44 Illustration of the problem of inversion: Suppose that H-E 0, )=e - H-E 0 ) S Φ Laplace transform

45 Illustration of the problem of inversion: Suppose that H-E 0, )=e - H-E 0 ) S Laplace transform Φ Numerical errors

46 Illustration of the problem of inversion: Suppose that H-E 0, )=e - H-E 0 ) S Laplace transform Φ Numerical errors???

47 How can one easily understand why the inversion is much less problematic if H-E 0, )= lorentzian

48 How can one easily understand why the inversion is much less problematic if H-E 0, )= lorentzian S Φ Lorentz transform blurred, but still distinguishable

49 How can one easily understand why the inversion is much less problematic if H-E 0, )= lorentzian S Φ Numerical errors Lorentz transform blurred, but still distinguishable also with errors!

50 How can one easily understand why the inversion is much less problematic if H-E 0, )= lorentzian Inversion: e.g. regularization method at fixed width S Φ Lorentz transform Numerical errors!!!

51 main point of the LIT : Schrödinger-like equation with a source ~ ( H - E - - i 0> 0 R Theorem: The solution is unique and has bound state asymptotic conditions one can apply bound state methods

52 bound state methods: ~ ( H - E - - i 0> 0 R ~ Represent H, 0> on a complete b.s. basis and invert the linear problem

53 A very efficient basis for few-body systems: Hyperspherical Harmonics (HH) [generalization to Spherical Harmonics Y lm to a 3(A-1) dimensional space]

54 Photodisintegration of 4 He Figure from Bacca and Pastore, Journal of Physics G.: Nucl. Part. Phys. 41, (2014)

55 Photodisintegration of 4 He Figure from Bacca and Pastore, Journal of Physics G.: Nucl. Part. Phys. 41, (2014)

56 What about many-body systems? LIT Lorentz Integral Transform A method that allows to circumvent the continuum problem by reducing it to the solution of a bound-state-like equation + CC Coupled-cluster theory Accurate many-body theory with mild polynomial scaling in mass number = LIT-CC An approach to many-body break-up induced reactions with a proper accounting of the continuum

57 Coupled-cluster theory Many-body method that can extend the frontiers of ab-initio calculations to heavier and neutron nuclei CC future aims CC theory now 0 0 cluster expansion CCSD CCSDT

58 Coupled-cluster theory formulation of LIT Phys. Rev. Lett. 111, (2013) Results with implementation at CCSD level

59 Benchmark Validation for 4 He Comparison of CCSD with exact hyperspherical harmonics with NN forces at N 3 LO HH HH The comparison with exact theory is very good

60 Photonuclear reactions Stable nuclei Dipole Response Functions with NN forces from chiral EFT (N 3 LO) PRC 90, (2014) Unstable nuclei NN(N 3 LO) core

61 Another I.T. with a different Kernel: The Stieltjes Kernel K( ω, σ ) = ( ) 1 real

62 It may be useful for a specific purpose:

63 In fact: given S Lim. Φ Lim. S( ) ) ( ( d S( ) d generalized polarizability e.g. electric polarizability, magnetic susceptibility, compressibility etc... depending on

64 Main point of the Stieltjes Transform : Schrödinger-like equation with a source ~ ( H - E + 0> 0 Theorem: The solution is unique and has bound state asymptotic conditions one can apply bound state methods

65 bound state methods: Represent ~ H, 0> on a complete b.s. basis and invert the linear problem

66 Recent results on with D (El. Dipole Polarizability)

67 Electric Dipole Polarizability as limit of the Stieltjes transform for ---> 0 Φ D 16 O. b.s. expansion: Coupled Cluster (non hermitian) Lanczos diagonalization

68 Electric dipole polarizability M. Miorelli et al., PRC (2016) D S( ) d D Interesting correlation with the proton charge radius NN only +3NF Role of 3b-force G. Hagen et al. Nature Phys NN only +3NF A. Ekström et al., Phys. Rev. C91, (2015) K. Hebeler et al., Phys. Rev. C83, (2011)

69 Electric dipole polarizability M. Miorelli et al., PRC (2016) D S( ) d D Interesting correlation with the proton charge radius +3NF Role of 3b-force +3NF Much better agreement with experimental data with 3NF Variation of Hamiltonian can be used to assess the theoretical error bar

70 48 Ca from first principles

71 48 Ca from first principles Theory provides predictions for future experiments International collaboration (USA/Canada/Europe/Israel) using coupled-cluster theory Hagen et al., Nature Physics 12, 186 (2016) Ab initio with three nucleon forces from chiral EFT Density Functional Theory Rskin will be measured at JLab and Mainz with Parity violation electron scattering (CREX and MREX)

72 48 Ca from first principles Theory provides predictions for future experiments International collaboration (USA/Canada/Europe/Israel) using coupled-cluster theory Hagen et al., Nature Physics 12, 186 (2016) Ab initio with three nucleon forces from chiral EFT Density Functional Theory Strong correlations with Rp allow to put narrow constraints Rskin will be to Rskin measured and at JLab and Mainz with Parity violation electron scattering Ab-initio predictions: (CREX and MREX)

73 48 Ca from first principles Theory provides predictions for future experiments International collaboration (USA/Canada/Europe/Israel) using coupled-cluster theory Hagen et al., Nature Physics 12, 186 (2016) Ab initio with three nucleon forces from chiral EFT Density Functional Theory Strong correlations with Rp allow to put narrow constraints Rskin will be to Rskin measured and at JLab and Mainz with Parity violation electron scattering Ab-initio predictions: (CREX Rskin will and be MREX) measured at JLab and Mainz with Parity violation electron scattering (CREX and MREX)

74 Summary: The electromagnetic probe is a clean mean to investigate nuclear dynamics (pertubation theory is valid) Ab initio methods help building the bridge between QCD and nuclear phenomena: (what( is the effective V???) They are moving from the traditional few-body (A=2-4) regime to larger systems Integral transform methods are alternative approaches to overcome the many-body scattering problem Giuseppina Orlandini, Colloquium, Universita' di Torino, June 10, 2016