The Nuclear Force through the Decades

Ch. Elster

Lecture 2

• Chadwick: Discovery of the neutron• Heisenberg: Postulates Iso-Spin

Early History of Nuclear Physics

… suggests the assumption that atomic nuclei are built from protons and neutrons without electrons ..

Concept of Iso-Spin (Isobaric Spin)

• The mass of the neutron and the proton are almost identical: they are nearly degenerate, and both are thus often called nucleon

• Although the proton has a positive charge, and the neutron is neutral, they are almost identical in all other respects.

• The strong interaction between any pair of nucleons is almost the same, independent of whether they are interacting as protons or as neutrons

• Iso-spin operator τ in iso-spin space• Proton: τz = ½ Neutron: τz = - ½

Iso-Spin:• SU(2) symmetry • Follows the same algebra as

spin• Can be considered as rotation

in iso-spin space• Neutron and proton

characterized by projections of τ

Palladio: Villa Rotonda

Early History of Nuclear Physics• Chadwick: Discovery of the neutron• Heisenberg: Postulates Iso-Spin

Yukawa: Meson Hypothesis

From:

H. Yukawa,

Proc. Phys.Math.Soc. Japan 17, 48 (1935)

Nuclear Force is mediated by the exchange of a particle with mass

Repulsive force mediated through exchange of a particle (here a wrench)

Attractive force: think of Boomerang exchanged

Yukawa prediction:

Mass ≈140 MeV

Birth of Particle Physics

Early History of Nuclear Physics• Chadwick: Discovery of the neutron• Heisenberg: Postulates Iso-Spin

Yukawa: Meson Hypothesis

Discovery of the pion in cosmic rays (1947)and in the Berkeley Cyclotron Lab (1948)

Yukawa: Nobelprize (1949)

One-Pion-Exchange (OPE) o.k. Taketani, Nakamura, Sasaki (1951) 3 ranges Multi-pion exchange theories – BUT - problems

Most General Form of the NN Force:Galilei Invariance

Let’s be explicit: m = nucleon mass

NN c.m. frame:Other single nucleon operators:

Properties of Pauli matrices:

Only option for operator linear in τ(i)and scalar is 1 and

Thus:

Thus the so the most general form of a nucleon-nucleon potential can only have the form:

Information about nuclear forceDeuteron

Deuteron is not spherical

Probability density for bothNucleons having spin down

Deuteron shapes from AV18

www.phy.anl.gov/theory/movie-run.html

Spin Orbit Force

Nijm93

Radial Schrödinger equation:

S = 0 (singlet)

S = 1 (triplett)

J = L+S is the conserved quantum number

Each channel has its own parameters

MeV

Repulsion

Attraction

AV18

nn-online.org/NN/

1960’s: Golden Era of Particle Physics

• Discovery of mesons and baryons

• Attempt: Mesons heavier than the pion responsible for the short range part of the nuclear force

• Field theory and Feynman diagrams

• Symmetry:• Lorentz invariance

Covariant bilinear forms:

Meson Exchange NN forces

Open Questions:Which mesons and nucleon resonances ?

Loop diagrams of strong interaction diverge => Cutoff’s at vertices as effective size of the nucleon.

Mesons in the NN Force

Progress in Three-Nucleon Physics asked for more accurate NN forces as

input

Nijm and CD-Bonn: Partial waves fitted

AV18: Operators with scalar functions

Gell-Mann:3 quarks

Interaction between quarks via gluon exchange:

Quantum-Chromo-Dynamics (QCD)

Fundamental “Strong Force” acts between quarks

NOT between nucleons

Residual forces are typically weak compared to the original force (order of magnitude)

Effective Field Theory (EFT) for the NN force:

• Degrees of freedom relevant to nuclear physics (atenergies where no particles are produced): – Pions and Nucleons (NOT quarks and gluons)

• Connection to QCD via characteristic symmetries– Chiral Symmetry

• EFT should allow for an order by order controlled expansion and improvement of predictions

• 1990: Weinberg suggests an EFT with pions and nucleons preserving chiral symmetry together with a power counting scheme

For m=0 chirality = helicity

Chiral EFT for Nuclear Forces• Framework

– Use ChPT to calculate irreducible contributions (=nuclear force)

– Solve Schrödinger equation to calculate observables

• Power Counting

• Vertices

Chiral Perturbation Theory

• Asymptotically observed states are effective degrees of freedom EFT

• Spontaneously broken approximate chiral symmetry of QCD plays important role

• Light (mπ) and heavy (mρ) mass scales are well separated

Chiral EFT for Nuclear Forces• Framework

– Use ChPT to calculate irreducible contributions (=nuclear force)

– Solve Schrödinger equation to calculate observables

• Power Counting

• Vertices

Hierarchy of nuclear forces

Chiral NN Forces

• Contact terms are fixed by observables

• In practice: different partial waves fixed different contact terms

• The higher the order, the better the description of the phase shift.

Eur.Phys.J. A54 (2018) no.5, 86

Take away: any NN potential contains an operator structure compatible with the Okubo-Marshak invariantsThere may be more operators, but then those are not linearly independentThe underlying theory from which the potential are derived changed over time.Today’s NN potentials take their guidance from QCD.

Pedagogical review:

Application in Many-Body System

• NN forces constructed to describe NN observables– cross sections, spin observables

• Three-body system already needs 3N forces to describe 3N observables

• Additional issue in many-body calculations:– Short range part too repulsive or too strong high momentum

components• The repulsive part of the NN (and 3N) force needs to be

renormalized– Apply similarity renormalization group to do so.– Leaves NN observables invariant