(The good, bad and ugly of)Optical potentials and nucleon scattering

from ab-initio Green functionAndrea Idini

C. Barbieri

Toward Predictive Theories of Nuclear Reactions

Across the Isotopic Chart

INT - Seattle, 21 MarAndrea Idini

Objective: an effective, consistent description of structure and reactions with a single formalism.(Hopefully) Predictive power of nuclear reactions measurementsover a range of exotic isotopes.

Method: Optical potential derived from Self Consistent Green Function and χEFT interactions.

1. reproduce nuclear bulk properties, i.e. binding energy and radii; NNLOsat

2. use the same description to consistently generate an optical potential reproducing elastic scattering data.

Optical Potentials

Andrea Idini

Σ∗= +

Particle hole‘polarization’

propagator (ph-RPA)

Particle-particle (pp-RPA) two-body correlation ‘ladder’ propagator

Faddeev RPAADC(3)

n p

Green Functions (Dyson Equation)

Andrea Idini

HFADC(1)

Central Part

Andrea Idini

More details in C. Barbieri, A. Carbone, Lect. Notes Phys. arXiv:1611.03923 [nucl-th]

Σ∗(𝜔)Dyson

Equation

VerticesSummation

(𝜔)

Vertices(ME)

𝐺(𝑟, 𝜔)

particlestructure

self-energy(optical potential)

Building Blocks

Constitues

Unknown

2p1h

2p2h

The irreducible self-energy is a nucleon-nucleus optical potential*

This provides consistent many-body and scattering wave functions

Σ𝛼𝛽∗ 𝑟, 𝑟′; 𝜔 = Σ𝛼𝛽

∞ +

𝑖

𝑚𝛼𝑖 𝑚𝛽

𝑖∗

𝜔 − 𝜖𝑖 ± 𝑖𝜂

Nucleon elastic scattering

Σ𝐴+1

E

*Mahaux & Sartor, Adv. Nucl. Phys. 20 (1991), Escher & Jennings PRC66:034313 (2002)

Σ𝐴−1

𝜖𝐹

correlated mean-field

resonances beyond mean-field

Σ∞

Andrea Idini

16O neutron propagator

Different colors to different 𝑙

𝑆(2𝑗+1)

𝑔𝛼𝛽 𝜔 =

𝑖

𝑋𝛼𝑖 𝑋𝛽

𝑖∗

𝜔 − 𝜖𝑖 ± 𝑖𝜂

Σ𝑛,𝑛′𝑙,𝑗∗

(𝐸)

Σ𝑙,𝑗∗

(𝑘, 𝑘′, 𝐸)

Σ∗= +

- Solve Dyson equation in HO Space, find

𝑘2

2𝑚𝜓𝑙,𝑗 𝑘 + ∫ 𝑑𝑘′𝑘′

2Σ𝑙,𝑗∗

𝑘, 𝑘′, 𝐸 𝜓𝑙,𝑗 𝑘′ = E 𝜓𝑙,𝑗(𝑘)

- diagonalize in full continuum momentum space

Σ𝑛,𝑛′𝑙,𝑗∗

(𝐸)

Andrea Idini

Nmax

RESULTS

Andrea Idini

arXiv:1612.01478 [nucl-th]

Navràtil, Roth, Quaglioni, PRC82, 034609 (2010)

Σ∞

SRG-N3LO, Λ = 2.66 fm−1

𝑛 + 16O 𝑔. 𝑠.

Andrea Idini

SRG-N3LO, Λ = 2.66 fm−1

Andrea Idini

Navràtil, Roth, Quaglioni, PRC82, 034609 (2010)

𝑛 + 16O

SRG-N3LO, Λ = 2.66 fm−1

Andrea Idini

Navràtil, Roth, Quaglioni, PRC82, 034609 (2010)

𝑛 + 16O

Volume integrals

Non local potential

Σ 𝜖𝐹 = 0

different Fermi energies and

particle-hole gap for differentinteractions

48Ca protons 𝐽𝑊

Andrea Idini S. Waldecker et al. PRC84, 034616(2011)

Ca isotopes

neutron and protonvolume integrals of self energies.

Andrea Idini

𝑑3/2 𝑠1/2

𝑠1/2

𝑑3/2

Σ∞ Σ∞

Σ∗

Σ∗

Nmax convergence

𝑏 = 22.36 MeV

«Imaginary» Parameter

Andrea Idini

Andrea Idini

experiment 2.699±0.005 fm

NNLOsat 2.734 fm

N3LO NN 2.354 fm

16𝑂 𝑟𝑝

Andrea Idini

Nmax=11

𝜎𝛾 𝐸 = 4𝜋2𝛼𝐸𝑅(𝐸)

𝛼𝐷 = 2𝛼∫ 𝑑𝐸 𝑅𝑍(𝐸)/𝐸

𝑅 𝐸 =

𝑖

𝜓𝑖𝐴 𝐷 𝜓0

𝐴 2𝛿 𝐸𝑖 − 𝐸

𝛼𝛽

𝛼 𝐷 𝛽 𝜓𝑖𝐴 𝑐𝛼

+𝑐𝛽 𝜓0𝐴

Andrea Idini

WithFrancesco Raimondi

RPAHO ME

Dipole Response and Polarizability

Hagen et al., Nature Physics 12, 186

Ψ𝑖 𝑟 = 𝐴∫ 𝑑𝑟1…𝑑𝑟𝐴Φ 𝐴−1+ (𝑟1, … , 𝑟𝐴−1)Φ 𝐴

+ (𝑟1, … , 𝑟𝐴)𝑟𝑖 𝑟𝑖 𝑟𝑖

Overlap function

Andrea Idini EM results from A. Cipollone PRC92, 014306 (2015)

Conclusions and Perspectives• We are developing an interesting tool to study nuclear

reactions effectively. We have defined a non-local generalized optical potentialcorresponding to nuclear self energy.

• This tool is useful to probe properties of nuclearinteractions.

• Radii, saturation and bulk properties are fundamental!(but not enough? Where do we want to go?)

Andrea Idini

Hellemans et al. PRC 88, 064323 (2013)

𝛼 = 1/6

Density dependence introduces instabilitiesin the mean field

Tarpanov et al. PRC89, 014307 (2014)Andrea Idini

Andrea IdiniJ.Phys. G44 (2017) no.4, 045106

Finite Range pseudopotential

K. Bennaceur

J. DobaczewskiM. Korteleinen

A.Idini, K. Bennaceur, J. Dobaczewski, arXiv:1612.00378

Landau parameters and error analysisError Analysis and Landau parameters guide us to know which channel

of the interaction are less constrained

Andrea Idini

𝐹 ⇒ 𝑆 = 0, 𝑇 = 0𝐺 ⇒ 𝑆 = 1, 𝑇 = 0

𝐹′ ⇒ 𝑆 = 0, 𝑇 = 1𝐺′ ⇒ 𝑆 = 1, 𝑇 = 1

Andrea Idini

Escher & Jennings PRC66 034313 (2002)

Σ∗= +

Σ corresponds to the Feshbach’s generalized optical potential

Dyson Equation

Equation of motion

Corresponding Hamiltonian

Why Green’s Functions?

Why optical potentials?

A.I. et al. Phys. Rev. C 92, 031304 (2015)Broglia et al. Phys. Scr. 91 06301* (2016)

- Optical potentials reduce many-body complexity decoupling structure contribution and reactions dynamics.

1 particle transfer

2 particle transfer

Andrea Idini

40𝐶𝑎 + 𝑛 @3.2 MeV

16O and 24O