Coupling of Lambda to theAtomic NucleusJAN POKORNÝ, WEJČF 2018, BÍLÝ POTOK U FRÝDLANTU

Motivation – nuclear physics

Lack of the general theory of nuclear structure –one has to rely on models and approximations

Nuclei in the high-energy regime – statisticalmodels

Nuclei in the low-energy regime – quantummechanics

Main challenges of the nuclear structure theory– solution of the quantum many-body problemand the formulation of the N-N potential

Many-body problem – nuclear models (liquiddrop, shell, mean-field, …)

N-N potential – cannot be determined fromQCD – models + Chiral Perturbation Theory

~ 10-10 [m] ~ 10-15 [m]

Atom Nucleus Nucleons

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Motivation – hypernuclear physics

Discovery – 1952 – Pniewski, Danysz

Proton from cosmic radiation interacted with anucleus in the nuclear emulsion – formedheavy fragment – first thought nucleus + pion –too small cross section for such capture –correct hypothesis with Lambda

Production mechanisms

Strangeness exchange – kaon + nucleus

Associated production – pion + nucleus

Electroproduction – electron scattering on anucleus – virtual photon

Study of hypernuclei – Lambda-N interaction,neutron stars, nuclear structure – deeper binding

CERN, BNL, FINUDA, JLab, JPARC, GSI, MAMI-C

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Nuclear chart with strangeness

Taken from [Hypernuclear Physics at JPARC, link:

http://slideplayer.com/slide/9056034/]

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Chiral Perturbation Theory

Effective field theory

Chiral symmetry of the QCD –SU(2) x SU(2)

Nucleons and pions degrees offreedom

Coupling constants fitted to phaseshifts and scattering data

Valid for low momenta (« 1 GeV)

Pertubative expansion – 2-body, 3-body, etc. interactions

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Mean-Field model of atomic nuclei

Equivalent of the shell model of atomic nuclei(magic numbers in nuclear physics2,8,20,28,50,86,128)

Each nucleon treated as a point-like particleplaced in a potential well

Inner structure of nucleons is not taken intoaccount – protons and neutrons are spin ½particles with mass 𝑀≈938 MeV

Based on the Hartree-Fock (HF) method –system of interacting nucleons is changed tosystem of non-interacting nucleons bound in apotential well

Starting basis – spherical harmonic oscillator –doubly magic nuclei

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Spherical Harmonic Oscillator

Quantum numbers N, l, j, m

Principal, orbital angular momentum,total angular momentum (spin-orbitcoupling), projection of the total angularmomentum

Energy

Wave functions

Truncation – Nmax

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Mathematical formalism

Second quantization – creation and annihilation operators

Hamiltonian with three-body N-N-Nand Lambda-N-N interactions

HF – variational method – minimalizing the Hamiltonian with respect to a unitary transformation between HO and self-consistent basis

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Hartree-Fock Equations

Three equations (one for each type ofparticles)

Self-consistent solution

Single-particle energies – spectra ofprotons, neutrons and Lambda

Mean-field operator

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Results – nuclear densities

Radial density distribution

Two-body and three-body interactions

Compared to RMF model – a nuclearmodel that describes well the bulkproperties of doubly magic nuclei

Nuclei without the 3B interactions aretoo compressed

3B interactions are crucial to obtainrealistic nuclear density distribution

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Results – neutron single-particle spectra

Spectra of neutrons – protons yieldanalogic results

The gaps between 0s and 0p shellshrink when 3B forces are included

40Ca – too small configuration space –not converged – we expect betterresults in larger basis

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Results – Lambda single-particle spectra

Preliminary calculations of hypernuclei

Spectra of Lambda

3B forces shrink the gaps between the0s and 0p shell

The implemented Lambda-N interactionhas a strong dependence on cut-off –change in cut-off leads to a shift inenergy

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Collective motion – excitations

Ground state – all shells filled up to theFermi level

Excitation – particle occupies the levelsabove Fermi and creates a hole in thelevels beyond Fermi

Particle-hole (ph) formalism – analogywith semiconductors

General excitations – sum of all possibleone ph, two ph, three ph, etc.combinations

Unfeasible: Tamm-Dancoff Approximation– phonon operators

Fermi level

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Tamm-Dancoff Approximation

Normal ordered Hamiltonian – Wick‘stheorem

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TDA Equation

Eigenvalue problem

Result is the coefficients of linearcombination

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Nucleon-Lambda TDA

Two types – proton-Lambda TDA andneutron-Lambda TDA

Motivation – study of hypernuclei witheven-odd or odd-even core

Alternative of the NTDA which was usedto study odd-even or even-odd nuclei,e.g. in [G. De Gregorio et al., Phys. Rev. C95, 034327 (2017); Phys. Rev. C 94,061301(R) (2016)].

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Nucleon-Lambda TDA

Results not available yet

Calculations are completed

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Conclusions and future plans

Nuclear physics

N-N potentials, nuclear models

ChPT – realistic interactions

mean-field model based on realisticinteractions

Three-body N-N-N forces – vital fordescription of nuclei

Hypernuclear physics

LO Lambda-N potential

Cut-off dependent

Implementation of better Lambda-Npotential

SRG, three-body L-N-N interactions

Lambda – Sigma mixing

Introducing three Sigma channels intoour model

NY TDA calculations

Even-odd and odd-even hypernuclei

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