Post on 30-Dec-2015
transcript
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
ChiralityQuickTimeý Dz
TIFFÅiîÒàèkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉǾå©ÇÈǞǽDžÇÕïKóvÇ ÇÅB
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
QuickTimeý DzTIFFÅiîÒàèkÅj êLí£ÉvÉçÉOÉâÉÄ
ǙDZÇÃÉsÉNÉ`ÉÉǾå©ÇÈǞǽDžÇÕïKóvÇ ÇÅB
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
QuickTimeý DzTIFFÅiîÒàèkÅj êLí£ÉvÉçÉOÉâÉÄ
ǙDZÇÃÉsÉNÉ`ÉÉǾå©ÇÈǞǽDžÇÕïKóvÇ ÇÅB
• 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.