Joint Lecture Groningen-Osaka
Spontaneous Breaking of Chiral Symmetry in Hadron Physics
30 Sep 09:00- CEST/16:00- JST Atsushi HOSAKA
07 Oct 09:00- CEST/16:00- JST
Nuclear Structure
21 Oct 09:00- CEST/16:00- JST Nasser KALANTAR-NAYESTANAKI
28 Oct 09:00- CET/17:00- JST
Low-energy tests of the Standard Model
25 Nov 09:00- CET/17:00- JST Rob TIMMERMANS
02 Dec 09:00- CET/17:00- JST
Relativistic chiral mean field model description of finite nuclei
09 Dec 09:00- CET/17:00- JST Hiroshi TOKI
16 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 and
What is broken symmetry
Symmetry
The key concept in the modern Physics
Example of translation
Symmetry
The key concept in the modern Physics
Example of translation
Symmetric
Translation causes
nothing
Uniform density
Symmetry
The key concept in the modern Physics
Example of translation
Symmetric
Translation causes
nothing
Uniform density
Less symmetric
Symmetry
The key concept in the modern Physics
Example of translation
Symmetric
Translation causes
nothing
Uniform density
Less symmetric
Translation changes
the location of the cluster
Localize
Clusterize
Symmetry
Example of rotation
Symmetric
Symmetry
Example of rotation
Symmetric
Rotation causes
nothing
Spherical
Symmetry
Example of rotation
Symmetric
Rotation causes
nothing
Spherical
Less symmetric
Symmetry
Example of rotation
Symmetric
Rotation causes
nothing
Spherical
Less symmetric
Rotation changes
the appearance
Deformed
Symmetry
Example of rotation
Symmetric
Rotation causes
nothing
Random
Less symmetric
Rotation changes
the appearacnce
Ordered
Spontaneous breaking
Symmetric
Simple
Disordered
Less symmetric
Complex
Ordered
Spontaneous breaking
Symmetric
Simple
Disordered
Symmetry is spontaneously broken
(Dynamical: due to interactions)
Phase transition
Reality in our
world
Less symmetric
Complex
Ordered
With Variety
Role of interaction
High temperature
Kinetic motion > Interaction
Random
Like gas
Role of interaction
High temperature
Kinetic motion > Interaction
Random
Like gas
Interaction breaks the symmetry
=> Spontaneously broken
Role of interaction
High temperature
Kinetic motion > Interaction
Random
Like gas
Interaction breaks the symmetry
=> Spontaneously broken
Low temperature
Kinetic motion < Interaction
Ordered
Like solid
Examples of interaction
(1) Translational invariance
r 2
r 2
p1
p2
r r
H

 v(r1  r2 )
2m1 2m2
r r r
r r
H is invariant under r1  r1  R, r2  r2  R
This causes localization (clustering) of a two-particle system
(2) Rotational invariance
r r r r 1r r 2
1  r  2  r  1   2 r
r
3
vT (r ) 
v(r)
2
r
This causes deformation of two-particle system (deuteron)
(3) Isospin invariance
 p
N    ,   ,   ,  0 ~ ( 1,  2 ,  3 )
 n
Iso-spinor
Iso-vector
†r
r
H  gN  N  
“Internal symmetry”
Isospin (flavor), chiral, color, ….
Recover the broken symmetry
Low T
High T
This does not mean
the phase transition
between them
There is a special way to recover the broken symmetry
Recover the broken symmetry
Symmetry transformation
Translation
p
Rotation
Recover the broken symmetry
Symmetry transformation
Translation
p
Rotation
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
Quantum mechanics
Uncertainty principle
Quantum mechanics
Uncertainty principle
e
ipx
p
Starts to move
Uncertainty principle
Flctuations
Zeromode excitations
e
im
Quantum mechanics
Uncertainty principle
e
ipx
p
Starts to move
Uncertainty principle
Flctuations
Zeromode excitations
e
im
For small moment of inertia => Easy to fluctuate
Symmetric states are realized in the quantum world
For large moment of inertia => hard to move
Symmetry is left broken ~ Classical world
Collective vs single particle motion
Collective vs single particle motion
In these motions, the shape does not change.
The objects move collectively (simultaneously)
NambuGoldstone
Boson =
Pion
Collective vs single particle motion
In these motions, the shape does not change.
The objects move collectively (simultaneously)
NambuGoldstone
Boson =
Pion
Massive
Modes=
Mass
generation
Change in the shape requires more energy.
Parts move => Motion of fewer particles
Hadrons
Where to study?
Molecule
Electromagnetic interaction
Many-body dynamics of electrons
around atomic nuclei and/or ions
Atom
Nucleus
Nucleons
Mesons
Quarks
Strong interaction
Many-body dynamics of nucleons
=> Nuclear Physics
mesons
Many-body-dynamics
of quarks and gluson
=> Hadron physics
Where to study?
Molecule
Electromagnetic interaction
Many-body dynamics of electrons
around atomic nuclei and/or ions
Atom
Strong interaction
Nucleus
Nucleons
Mesons
Quarks
Many-body dynamics of nucleons
=> Nuclear Physics
mesons
Many-body-dynamics
of quarks and gluons
Hadron Physics
Atoms
Many-electron system => Periodic table
Ne = 1, 2, 3…. [One dimensional plot]
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Nuclei
Many-nucleon system (protons and neutrons)
Nuclei
Many-nucleon system (protons and neutrons)
Proton number
=> Nucleat chart
Np = 1, 2, 3….
Nn = 1, 2, 3…. => [Two-dimensional plot]
Neutron number
Hadrons
Many(?)-quark system (u, d, c, s, b, t)
Particle Data
Proton/neutron
Particle Data Table
Mesons
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Baryons
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Hadrons
Many(?)-quark system (u, d, c, s, b, t)
Particle Data
Proton/neutron
However
Only qq and qqq?
Mesons
Why?
Baryons
Problems of hadron physics
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Clay Mathematics Institute, Millennium Problems
http://www.claymath.org/millennium/
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.
１Birch and Swinnerton-Dyer Conjecture
２Hodge Conjecture
３Navier-Stokes Equations
４P vs NP
５Poincare Conjecture
６Riemann Hypothesis
７ 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 nontrivially 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
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Potential energy surface
of the vacuum
Yoichiro Nambu
Chiral order
parameter
Quarks & gluons
Confinement,
Mass generation
Hadrons & nuclei
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
In the microscopic world
A particle
Vacuum = Ground state
is not empty
Particles are interacting with the vacuum
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
Analogy with BCS
QED
Phonon exchange ee
Cooper pair
qq  0
Order parameter
Gauge (local) symmetry
Superconductivity
Analogy with BCS
QED
QCD
Phonon exchange ee
Cooper pair
qq  0
Strong interaction qq
Quark-antiquark pair
qq  0
Order parameter
Gauge (local) symmetry
Flavor (global) symmetry
Superconductivity
Nambu-Goldstone boson
Superconductivity
• Gap in energy spectrum
Hadrons
• Mass of particles
N*
E=0
D
Ground state
Majorana
mass
• Meissner effect
Super
M
E=0
N
Dirac
mass
Vacuum
• Exclusion of color electric field
Normal
Normal Super
Chiral symmetry
Hand
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Left
²Ç ý emiTkciuQ
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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
We can not pass the particle moving at the speed of light
c
c’ = c
Spin
S-frame
Right-handed
Chirality remains
unchanged
Spin
S’-frame
Right-handed
Right-left do not mixing
Right and left can be independent
Isospin (internal) symmetry can be introduced separately
Massive fermion
v
v’
Spin
Boost can change
from right to left
Right-handed
Spin
Left-handed
The word chiral (handedness) comes from this
For massive particle, right and left mix
=> Chiral symmetry is broken
Summary 1
Symmetry 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
Vacuum point
Only one
V
A
Infinitely many on
-> choose one
V
Pions[NG boson] appear
A
Where and how pions appear
LQCD
1 a 
      g    Aa  F Fa
4
Strong interaction dynamics

Quarks
and gluons

a
      G ( )2  ( i 5 )2   L


       m*   g   i 5 
Mass generation
Pions
Constituent (quasi) quarks
π
q
q
q
π
Quarks
and mesons
π
q
q