Joint Lecture Groningen-Osaka Spontaneous Breaking of Chiral Symmetry in Hadron Physics30 Sep 09:00- CEST/16:00- JST Atsushi HOSAKA07 Oct 09:00- CEST/16:00- JST Nuclear Structure21 Oct 09:00- CEST/16:00- JST Nasser KALANTAR-NAYESTANAKI28 Oct 09:00- CET/17:00- JST Low-energy tests of the Standard Model25 Nov 09:00- CET/17:00- JST Rob TIMMERMANS02 Dec 09:00- CET/17:00- JST Relativistic chiral mean field model description of finite nuclei09 Dec 09:00- CET/17:00- JST Hiroshi TOKI16 Dec 09:00- CET/17:00- JST + WRAP-UP/DISCUSSION

Spontaneous Breaking of Chiral Symmetry

in Hadron Physics

• What does spontaneous mean? • What is the breaking of Symmetry? • What is chiral? • What is hadron? • . . . .

Contents

• General discussions Aspects of symmetry and of spontaneous breaking

• Concrete examples NJL model for hadron physics

What is symmetric andWhat is broken symmetry

SymmetryThe key concept in the modern Physics

Example of translation

SymmetryThe key concept in the modern Physics

SymmetricTranslation causes nothing

Example of translation

Uniform density

SymmetryThe key concept in the modern Physics

SymmetricTranslation causes nothing

Less symmetric

Example of translation

Uniform density

SymmetryThe key concept in the modern Physics

SymmetricTranslation causes nothing

Less symmetric

Example of translation

Uniform density

LocalizeClusterize

Translation changes the location of the cluster

Symmetry

Symmetric

Example of rotation

Symmetry

SymmetricRotation causes nothing

Example of rotation

Spherical

Symmetry

SymmetricRotation causes nothing

Less symmetric

Example of rotation

Spherical

Symmetry

SymmetricRotation causes nothing

Less symmetric Rotation changes the appearance

Example of rotation

Spherical

Deformed

SymmetricRotation causes nothing

Less symmetric Rotation changes the appearacnce

Symmetry

Example of rotation

Random

Ordered

Symmetric

Less symmetric

Spontaneous breaking

ComplexOrdered

SimpleDisordered

Symmetric

Less symmetric

ComplexOrdered

Reality in our world

Symmetry is spontaneously broken(Dynamical: due to interactions)

Phase transition

Spontaneous breaking

SimpleDisordered

With Variety

Role of interaction

Random

Kinetic motion > Interaction

High temperature

Like gas

Role of interaction

Random

Kinetic motion > Interaction

High temperature

Interaction breaks the symmetry=> Spontaneously broken

Like gas

Role of interaction

Random

Ordered

Kinetic motion > Interaction

Kinetic motion < Interaction

High temperature

Low temperature

Interaction breaks the symmetry=> Spontaneously broken

Like gas

Like solid

Examples of interaction(1) Translational invariance

H

rp1

2

2m1

rp2

2

2m2

v(rr1

rr2 )

r1

rr1

rR,

rr2

rr2

rRH is invariant under

This causes localization (clustering) of a two-particle system

(2) Rotational invariance

vT (

rr )

r1

rr

r2

rr

1

3

r1

r2 r2

r2 v(r)

This causes deformation of two-particle system (deuteron)

(3) Isospin invariance

N p

n

, , , 0 ~ (1, 2 , 3)

Iso-spinor Iso-vector

H gN †r N r

“Internal symmetry” Isospin (flavor), chiral, color, ….

Recover the broken symmetry

This does not mean the phase transition between them

There is a special way to recover the broken symmetry

Low T High T

Recover the broken symmetrySymmetry transformation

pTranslation Rotation

Recover the broken symmetrySymmetry transformation

This does not require energy => Zero energy mode

Classical mechanics: No need to move an object on a flat/smooth surface

Field theory: Appearance of a massless particle => pion

W = Fs = 0

m = 0

pTranslation Rotation

Quantum mechanicsUncertainty principle

Quantum mechanics

Starts to movepeipx

eimZeromode excitations

Uncertainty principleFlctuations

Uncertainty principle

Quantum mechanics

Starts to movepeipx

eim

For small moment of inertia => Easy to fluctuateSymmetric states are realized in the quantum world

For large moment of inertia => hard to moveSymmetry is left broken ~ Classical world

Zeromode excitations

Uncertainty principleFlctuations

Uncertainty principle

Collective vs single particle motion

Collective vs single particle motion

In these motions, the shape does not change. The objects move collectively (simultaneously)

Nambu-GoldstoneBoson =Pion

Collective vs single particle motion

In these motions, the shape does not change. The objects move collectively (simultaneously)

Change in the shape requires more energy.Parts move => Motion of fewer particles

Nambu-GoldstoneBoson =Pion

Massive Modes=Massgeneration

Hadrons

Molecule

Atom

Nucleus

NucleonsMesons

Quarks

Electromagnetic interaction

Strong interaction

Many-body dynamics of electrons around atomic nuclei and/or ions

Many-body dynamics of nucleons => Nuclear Physics mesonsMany-body-dynamics of quarks and gluson => Hadron physics

Subatomic physics

Where to study?

Molecule

Atom

Nucleus

NucleonsMesons

Quarks

Electromagnetic interaction

Strong interaction

Many-body dynamics of electrons around atomic nuclei and/or ions

Many-body dynamics of nucleons => Nuclear Physics mesonsMany-body-dynamics of quarks and gluons Hadron Physics

Subatomic physics

Where to study?

Atoms

Many-electron system

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Many-electron system => Periodic tableNe = 1, 2, 3….　 [One dimensional plot]

NucleiMany-nucleon system (protons and neutrons)

NucleiMany-nucleon system (protons and neutrons) => Nucleat chartNp = 1, 2, 3…. 　Nn = 1, 2, 3…. => [Two-dimensional plot]

Neutron number

Pro

ton

nu

mb

er

Hadrons

Many(?)-quark system (u, d, c, s, b, t)

Particle DataProton/neutron

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MesonsBaryons

Particle Data Table

Hadrons

Many(?)-quark system (u, d, c, s, b, t)

Only qq and qqq?

Mesons Baryons

However

Particle DataProton/neutron

Why?

Problems of hadron physics

Clay Mathematics Institute, Millennium Problems

Millennium Problems In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven Prize Problems. The Scientific Advisory Board of CMI selected these problems, focusing on important classic questions that have resisted solution over the years. The Board of Directors of CMI designated a $7 million prize fund for the solution to these problems, with $1 million allocated to each. During the Millennium Meeting held on May 24, 2000 at the Collège de France, Timothy Gowers presented a lecture entitled The Importance of Mathematics, aimed for the general public, while John Tate and Michael Atiyah spoke on the problems. The CMI invited specialists to formulate each problem.

http://www.claymath.org/millennium/

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１Birch and Swinnerton-Dyer Conjecture２Hodge Conjecture３Navier-Stokes Equations４P vs NP５Poincare Conjecture６Riemann Hypothesis７ Yang-Mills Theory => QCD

A. Jaffe and E. Witten

• It must have a “mass gap,” that is, there must be some strictly positive constant such that every excitation of the vacuum has energy at least . ∆ ∆

• It must have “quark confinement,” that is, even though the theory is described in terms of elementary fields, such as the quarks, that transform non-trivially under S U (3), the physical particle states – such as the proton, neutron, and pion – are S U (3)-invariant.

• It must have “chiral symmetry breaking,” which means that the vacuum is potentially invari- ant (in the limit that the quark bare masses vanish) only under a certain subgroup of the full symmetry group that acts on the quark fields.

Where qqqq, qqqqq and more ?

Tetraquark Pentaquark

Exotic hadrons

Spontaneous breaking of chiral () symmetry

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Yoichiro Nambu

Spontaneous breaking of chiral () symmetry

Quarks & gluons

Hadrons & nucleiConfinement, Mass generation

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Potential energy surface of the vacuum

Chiral order parameter

Yoichiro Nambu

Dynamics of Spontaneous symmetry breaking

in the strongly interacting system

Tasks of Physics

• Find the ultimate law of everything• Reconstruct phenomena from the law

They are not independent due to the presence of interactions

We are on the vacuum.Particles are the excitations of the vacuum.

Complicated system

Physics is to find the properties of the vacuum and its excitations in the presence of interactions

Vacuum = Ground state is not empty

A simply looking system can be more complicated due to the interaction and change its properties drastically.

E.G. from quarks to Hadrons with mass generation

Particles are interacting with the vacuum

A particle

In the microscopic world

Analogy with BCS

Phonon exchange ee

QED

Cooper pairqq 0

Gauge (local) symmetry Superconductivity

Order parameter

Analogy with BCS

Phonon exchange ee

QED

Strong interaction qq

QCD

Cooper pairqq 0qq 0

Quark-antiquark pair

Gauge (local) symmetry Superconductivity

Flavor (global) symmetry Nambu-Goldstone boson

Order parameter

Superconductivity Hadrons

• Gap in energy spectrum • Mass of particles

E = 0Ground state

E = 0Vacuum

N

N*

• Meissner effect • Exclusion of color electric field

NormalSuper

Normal Super

Majorana mass

Dirac mass

Chiral symmetryHand

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Left Right

Chiral symmetry => Left-hand world has a symmetry (law) Right-hand world has a symmetry (law)If they mix, we say that chiral symmetry is broken

Massless fermion

c

Spin

c’ = c

Spin

S-frame S’-frame

Right-handed Right-handed

Right-left do not mixingRight and left can be independentIsospin (internal) symmetry can be introduced separately

We can not pass the particle moving at the speed of light

Chirality remains unchanged

Massive fermionv

Spin

v’

Spin

Right-handed Left-handed

For massive particle, right and left mix => Chiral symmetry is broken

The word chiral (handedness) comes from this

Boost can changefrom right to left

Summary 1Symmetry can be spontaneously broken by interactions.

Symmetry and broken phase can change each other.(Temperature, density, …)

In the broken phase, symmetry is recovered by the presence the Nambu-Goldstone mode. Zero energy mode ~ pion

Collective, and single particle modes are distinguished.

The zero mode (pions) governs the dynamics at low energy.

Summary 2

Hadrons are made of quarks and gluons Baryons qqq, mesons qq*, others (exotics)??

Quark properties changes drastically by the strong interaction (nearly massless -> massive)

Chiral symmetry is broken spontaneously

Quark masses are dynamically generated (by interaction)

Pions become massless (Nambu-Goldstone mode)

Dynamics of L and R <=> V and A

V = R + L, A = R - L

Potential

V A

Vacuum pointOnly one Infinitely many on

-> choose one

V APions[NG boson] appear

Where and how pions appear

LQCD g

a Aa

1

4F

a Fa

G ( )2 ( i5 )2 L

m* g i5

Strong interaction dynamics

Mass generationConstituent (quasi) quarks

Pions

q q q q qππ

π

Quarks and gluons

Quarks and mesons