Sigma model and applications

1. The linear sigma model (& NJL model)

2. Chiral perturbation

3. Applications

Summary • The law of the nature is simple.

• But spontaneous symmetry breaking must occur, which brings us varieties.

• For hadrons it also generates mass.

• SSB induces collective (Nambu-Goldstone) mode (Pion) which governs the low energy dynamics of the broken world. => Chiral dynamics

• The chiral dynamics can be extended to resonance physics

• This might predict new form of hadronic matter.

1. The linear sigma model

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Left Right

c

Spin

cVelocity

p̂ 1

p̂ 1

LagrangianThe simplest that contain all the essences.

L i g i5

1

2( )2 ( )2 V ,

Potential: V , 4

2 2 f 2 2 0

Nucleon (or quark)Scalar meson (sigma)Pseudoscalar meson (pion)

Chiral mesons

• The Lagrangian has chiral symmetry U(1)L x U(1)R

• Before SSB: the fermion is massless • After SSB: the fermion obtains a finite mass

the pion becomes massless

Structure of The vacuum

Left and Right

~1

r

rp

E M

M 0

1r p̂

5

0 1

1 0

L 1 5

2 ~ (1

r p̂)

~1 1

1 1

1

r p̂

1 r p̂

(1 r p̂)

R

15

2 ~ (1

r p̂)

5 L ~r p̂(1

r p̂) L

5 R ~r p̂(1

r p̂) R

are eigenstates of (chirality) L , R 5 ~r p̂

L R

U(1)LxU(1)R Chiral symmetry

L i g i5

1

2( )2 ( )2 V ,

Li L Ri

R

g L i5 R g R i5 L ...

U(1)LxU(1)R Chiral symmetry

L i g i5

1

2( )2 ( )2 V ,

Li L Ri

R

g L i5 R g R i5 L ...

Kinetic term is invariant under U(1)L x U(1)R

U(1)L : L eiL L , U(1)R : R eiR R

U(1)LxU(1)R Chiral symmetry

L i g i5

1

2( )2 ( )2 V ,

Li L Ri

R

g L i5 R g R i5 L ...

Kinetic term is invariant under U(1)L x U(1)R

U(1)L : L eiL L , U(1)R : R eiR R

The chiral mesons transform

U(1)L : i5 eiL i5 ,

cosL

sinL

sinL

cosL

U(1)R : i5 i5 e iR ,

cosR

sinR

sinR

cosR

The total Lagrangian is invariant under U(1)L x U(1)R

The ground state: For f 2 < 0The minimum energy configuration

The ground state: For f 2 < 0

L i g i5

1

2( )2 ( )2 V ,

H ir

r g i5

1

2(0 )2 (0 )2 1

2(r )2 (

r )2 V ,

StaticNo FermionUniform mesons V ,

4 2 2 f 2 2

The minimum energy configuration

The ground state: For f 2 < 0

V ,

L i g i5

1

2( )2 ( )2 V ,

H ir

r g i5

1

2(0 )2 (0 )2 1

2(r )2 (

r )2 V ,

StaticNo FermionUniform mesons V ,

4 2 2 f 2 2

The minimum energy configuration

Minimum energy (density) is given by , 0, 0

Invariance of the vacuumV ,

Minimum energy (density) is given by , vac

0, 0

This vacuum is invariant under the chiral transformation

cosL

sinL

sinL

cosL

, 0,0 0,0

This corresponds to

Translation causes nothing

Uniform density

The ground state: For f 2 > 0Minimum energy (density) is given by an any point on the circle

V ,

, vav f , 0

This vacuum is not invariant under the chiral transformation

cosL

sinL

sinL

cosL

, f ,0 f cosL , f sinL f ,0

LocalizeClusterize

Translation changes the location of the cluster

This corresponds to

Symmetry natureis determined by the parameter f

A microscopic model is needed to determine f => Nambu (NJL) model

Liniear sigma model <=> Ginzburg-Landau modelNJL model <=> BCS model

LNJL i g ( )2 ( i5 )2

Attractive interaction causes the instability of the ground stateCooperative phenomena of infinitely many-body systems

Particle propertiesFluctuations around the vacuum

For f 2 < 0

L i g i5

1

2( )2 ( )2 f 2

2 2 2

4 2 2 2 f 4

4

Masses: : 0,

, : f 2 Degenerate between P=+,- particles

For f 2 < 0

L i g ( f ) i5

1

2( )2 ( )2

4( f )2 2 f 2 2

i gf g i5

1

2( )2 ( )2

44 f 2 2 4 f ( 2 2 ) ( 2 2 )2

Masses: : gf ,

: 2 f 2 , : 0

Expansion

, f ,

Fermion acquires a mass, and the pion becomes masslessNambu-Goldstone theorem

The Goldberger-Treiman relation

Two modes in the broken phaseV ,

V ,

They correspond to

Massive mode Massless mode

2. Chiral perturbationL i

g i5 1

2( )2 ( )2

4

2 2 f 2 2

At low energy, massless modes dominates =>

L i g i5

1

2( )2 ( )2

2 2 f 2

The constraint 2 2 f 2implies the use of an angle variable

i5

fU5 ei5 / f ,

if

U ei / f

Nonlinear modelL i

g i5 1

2( )2 ( )2

i gfU5

f 2

4UU

†

Also introduce U51/2 5 , 5 N

L NiN gfNN

f 2

4UU

† iN 5 15

1 N

5 15

1 i

f5

NiN MNN

f 2

4UU†

1

fN

5N

Pion interactions

L NiN MNN

f 2

4UU†

1

fN

5N

All pion terms contan derivatives ~ momentum

Small momentum can be a small expansion parameter=> Chiral perturbation theory

• NN vertex

L

1

fN

5N 1

f

r

rq

• interaction (needs isospin) in f 2

4UU†

rq

3. Application

Force

photon,

weak boson, W, Z

gluon

graviton

u

d

c

s

b

t

e

e

Quarks

Leptons

Matter

Octet mesons and baryons

(d u ) (ud )

K0(ds ) K (us )

K(su) K 0(sd )

0 (uu dd)

(uu dd )

p(uud)n(udd)

(uds)

0 (uds) (dds) (uus)

0(ssu) (ssd)

Their interactions are dictated by chiral symmetry and may reproduce resonances=>New type of hadrons ~ hadron mokecules

Key question: What multiquark configurations are possible?

Diquark

Observation of exotic hadron resonances

Triquark Meson-baryonmolecule

Colored correlation Colorless correlation

Θ+, N*(1670), Λ(1405), …, X(3872), Z+(4430), etcPentaquarks Hadronic molecule Tetraquarks

(1405)

• Quark model ~ uds, one of them is in p-state • But this state is the lightest among the family of 1/2–

• Also small LS splitting with (1520)

Spin, parity; 1/2–

l = 0 l = 1

… …

It could be KN (hadron-hadron) molecule, a new form of matter

Solving the LS equation

SU(3) (flavor) extension of this Lagrangian

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L NiN gfNN

f 2

4UU

† iN 5 15

1 N

Large attraction

Coefficients of the WT interaction

T-matrix

T ( s ) V VG( s )T ( s )

V VGV VGVGV ...

=T V V VG

V V VG G

+

+ + …

m

M

s

Two ingredientsV and G

V: Chiral interaction (Weinberg-Tomozawa)G: 1/(E – H0)

Poles on the complex energy plane

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• Two states near that energy?• Molecular like structure; the new form of hadrons

Summary • The law of the nature is simple.

• But spontaneous symmetry breaking must occur, which brings us varieties.

• For hadrons it also generates mass.

• SSB induces collective (Nambu-Goldstone) mode (Pion) which governs the low energy dynamics of the broken world. => Chiral dynamics

• The chiral dynamics can be extended to resonance physics

• This might predict new form of hadronic matter.