Transcript Chiral Symmetry Breaking

Low Energy Dynamics of QCD
& Realistic AdS/QCD
Yue-Liang Wu
Kavli Institute for Theoretical Physics China
Key Laboratory of Frontiers in Theoretical Physics
Institute of Theoretical Physics, Chinese Acadeny of Sciences
2010.11.25
Outline

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Success of Quantum Field Theory
Why Loop Regularization Method
Dynamically Generated Spontaneous Chiral
Symmetry Breaking
Scalars as Composite Higgs and Mass
Spectra of Lowest Lying Mesons
Why AdS/QCD & Realistic Model
Consistent Prediction for the Mass Spectra of
Resonance Mesons (Y.Q.Sui, YLW, Z.F.Xie, Y.B.Yang)
Conclusions
Symmetry & Quantum Field Theory

Symmetry has played an important role in
physics

All known basic forces of nature:
electromagnetic, weak, strong & gravitational
forces, are governed by
U(1)_Y x SU(2)_L x SU(3)_c x SO(1,3)

Real world has been found to be successfully
described by quantum field theories (QFTs)
Why Quantum Field Theory
So Successful

Folk’s theorem by Weinberg:
Any quantum theory that at sufficiently low energy and
large distances looks Lorentz invariant and satisfies the
cluster decomposition principle will also at sufficiently
low energy look like a quantum field theory.

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Indication: existence in any case a characterizing energy
scale (CES) M_c
At sufficiently low energy then means:
E << M_c  QFTs
Why Quantum Field Theory
So Successful

Renormalization group by Wilson or Gell-Mann & Low
Allow to deal with physical phenomena at any
interesting energy scale by integrating out the physics
at higher energy scales.
To be able to define the renormalized theory at any
interesting renormalization scale .

Implication: Existence of sliding energy scale (SES) μ_s
which is not related to masses of particles.

The physical effects above the SES μ_s are integrated in
the renormalized couplings and fields.
How to Avoid Divergence


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QFTs cannot be defined by a straightforward
perturbative expansion due to the presence of
ultraviolet divergences.
Regularization: Modifying the behavior of field theory
at very large momentum Feynman diagrams
become well-defined finite quantities
String/superstring: Underlying theory might not be a
quantum theory of fields, it could be something else.
Regularization Methods
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Cut-off regularization
Keeping divergent behavior, spoiling gauge symmetry &
translational/rotational symmetries
Pauli-Villars regularization
Modifying propagators, destroying non-abelian gauge
symmetry
Dimensional regularization: analytic continuation in dimension
Gauge invariance, widely used for practical calculations
Gamma_5 problem, losing scaling behavior (incorrect gap eq.),
problem to chiral theory and super-symmetric theory
All the regularizations have their advantages and shortcomings
Criteria of Consistent Regularization
(i) The regularization is rigorous that it can maintain the
basic symmetry principles in the original theory, such
as: gauge invariance, Lorentz invariance and
translational invariance
(ii) The regularization is general that it can be applied to
both underlying renormalizable QFTs (such as QCD)
and effective QFTs (like the gauged Nambu-JonaLasinio model and chiral perturbationtheory).
Criteria of Consistent Regularization
(iii) The regularization is also essential in the sense
that it can lead to the well-defined Feynman
diagrams with maintaining the initial divergent
behavior of integrals, so that the regularized theory
only needs to make an infinity-free renormalization.
(iv) The regularization must be simple that it can
provide the practical calculations.
Symmetry-Preserving Loop Regularization
with String Mode Regulators

Yue-Liang Wu, SYMMETRY PRINCIPLE PRESERVING AND
INFINITY FREE REGULARIZATION AND RENORMALIZATION
OF QUANTUM FIELD THEORIES AND THE MASS GAP.
Int.J.Mod.Phys.A18:2003, 5363-5420.

Yue-Liang Wu, SYMMETRY PRESERVING LOOP
REGULARIZATION AND RENORMALIZATION OF QFTS.
Mod.Phys.Lett.A19:2004, 2191-2204.
Irreducible Loop Integrals (ILIs)
Loop Regularization
Simple Prescription:
in ILIs, make the following replacement
With the conditions
So that
Gauge Invariant Consistency Conditions
Checking Consistency Condition
Checking Consistency Condition
Vacuum Polarization
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Fermion-Loop Contributions
Gluonic Loop Contributions
Cut-Off & Dimensional Regularizations

Cut-off violates consistency conditions

DR satisfies consistency conditions

But quadratic behavior is suppressed in DR
Symmetry–preserving & Infinity-free
Loop Regularization
With String-mode Regulators

Choosing the regulator masses to have the
string-mode Reggie trajectory behavior

Coefficients are completely determined

from the conditions
Explicit One Loop Feynman Integrals
With
Two intrinsic mass scales
and
play the roles
of UV- and IR-cut off as well as CES and SES
Interesting Mathematical Identities
which lead the functions to the following simple forms
Renormalization Constants of Non- Abelian gauge
Theory and β Function of QCD in Loop Regularization
Jian-Wei Cui, Yue-Liang Wu, Int.J.Mod.Phys.A23:2861-2913,2008

Lagrangian of gauge theory

Possible counter-terms
Ward-Takahaski-Slavnov-Taylor Identities
Gauge Invariance
Two-point Diagrams
Three-point Diagrams
Four-point Diagrams
Ward-Takahaski-Slavnov-Taylor Identities

Renormalization Constants

All satisfy Ward-Takahaski-Slavnov-Taylor identities
Renormalization β Function

Gauge Coupling Renormalization
which reproduces the well-known QCD β function (GWP)
Supersymmetry in Loop Regularization
J.W. Cui, Y.Tang,Y.L. Wu Phys.Rev.D79:125008,2009
Supersymmetry
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Supersymmetry is a full symmetry of
quantum theory
Supersymmetry should be Regularizationindependent
Supersymmetry-preserving regularization
Massless Wess-Zumino Model
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Lagrangian
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Ward identity

In momentum space
Check of Ward Identity
Gamma matrix algebra in 4-dimension and
translational invariance of integral momentum
Loop regularization satisfies these conditions
Massive Wess-Zumino Model
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Lagrangian

Ward identity
Check of Ward Identity
Gamma matrix algebra in 4-dimension and
translational invariance of integral momentum
Loop regularization satisfies these conditions
Triangle Anomaly
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Amplitudes

Using the definition of gamma_5
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The trace of gamma matrices gets the most general and
unique structure with symmetric Lorentz indices
Anomaly of Axial Current

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Explicit calculation based on Loop Regularization with the most
general and symmetric Lorentz structure
Restore the original theory in the limit
which shows that vector currents are automatically conserved, only
the axial-vector Ward identity is violated by quantum corrections
Chiral Anomaly Based on Loop Regularization
Including the cross diagram, the final result is
Which leads to the well-known anomaly form
Anomaly Based on Various Regularizations


Using the most general and symmetric trace formula for
gamma matrices with gamma_5.
In unit
Dynamically Generated
Spontaneous Chiral Symmetry Breaking
In Chiral Effective Field Theory
QCD Lagrangian and Symmetry
Chiral limit: Taking vanishing quark masses mq→ 0.
QCD Lagrangian
(o)
QCD
L
1  
 qL  iD qL  qR  iD qR  G G
4

D     g s  / 2G
u 
 
q  d 
s 
 
qR , L
1
 (1   5 )q
2
has maximum global Chiral symmetry :
SU L (3)  SU R (3) U A (1) U B (1)
QCD Lagrangian and Symmetry

QCD Lagrangian with massive light quarks
Effective Lagrangian based on Loop Regularization
Y.B. Dai and Y-L. Wu, Euro. Phys. J. C 39 s1 (2004)
Dynamically Generated
Spontaneous Symmetry Breaking
Dynamically Generated
Spontaneous Symmetry Breaking
Composite Higgs Fields
Spont aneous
Symmet r y Br eaki ng
膺标介子作为
Gol dst one粒子
自发对称破缺
标量介子作为
Hi ggs粒子
Scalars as Partner of Pseudoscalars &
Lightest Composite Higgs Bosons
Scalar mesons:
Pseudoscalar mesons :
Mass Formula
Pseudoscalar mesons :
Mass Formula
Predictions for Mass Spectra & Mixings
Predictions
Chiral Symmetry Breaking
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Chiral SUL(3)XSUR(3)
spontaneously broken
Goldstone mesons
π0, η8
Chiral UL(1)XUR(1) breaking
Instanton Effect of anomaly
Mass of η0
Flavor SU(3) breaking
The mixing of
π0, η and η‫׳‬
Chiral Symmetry Breaking & QCD
Confinement in AdS/QCD Models
Theories of Field and Gravity
Field Theory
Gauge Theories
QCD
=
Gravity theory
Quantum Gravity
String theory
Use the field theory to learn about gravity
Use the gravity description to learn about the field theory
(J.M.)
T
o
p
D
o
w
n
Particle Theory
Most SUSY QCD
SU(N)
N colors
Radius of curvature
Gravity Theory
=
String theory on
AdS5 x S 5
N = magnetic flux through S5

RS 5  RAdS5  g
2
YM

1/ 4
N
ls
Duality:
g2 N is small  perturbation theory is easy – gravity is bad
g2 N is large  gravity is good – perturbation theory is hard
Strings made with gluons become fundamental strings.
(J.M.)
AdS/CFT Dictionary
N=4 SYM
Type IIB strings in AdS5xS5
U(Nc)
2
=g N
Only gauge invariant operators
2
R/’ = 4 π gs N
SO(2,4) superconformal group
SO(2,4) metric isometries
SO(6) flavour (R) symmetry
SO(6) S5 isometries
RG scale
Radial coordinate
Sources and operators
Constants of integration in
SUGRA field solutions
Glueballs
Regular linearized fluctuations
of dilaton
AdS/CFT
More D-branes
For flavor
• Qualitative Similarities to QCD
• Real QCD, full string construction?
N D-branes
• Deviations from AdS for finite N
SU(N) Yang-Mills Symmetry
AdS5
Bulk Space
X
Top Down
S5
• Top down string-model approach appears to be far away
• Difficult to find reasonable supergravity background &
brane configurations
Quantum ChromoDynamics QCD
B
o
t
t
o
m
u
p
colors (charges)
They interact exchanging gluons
Electrodynamics QED
Chromodynamics (QCD)
photon
g
g
g
gluon
g
3 x 3 matrices
electron
Gauge group
U(1)
SU(3)
Gluons carry
color charge, so
they interact among
themselves
QCD Strings & Gravity
Gluon: color and anti-color
Large N_c
Closed strings  glueballs
Open strings 
mesons
At distances larger than the typical size of the string
ls
Gravity theory
R
Radius of curvature >> string length  gravity is a good approximation
Gauge Theory + Large N_c  String Theory  Gravity Theory
Dual Theory of QCD

In the UV regime: highly nonlocal,
corresponding to asymptotic freedom.

In the IR regime: local, corresponding
to the strongly correlated QCD.

QCD Strings to the gravitational dual
as a local theory.
Holographic QCD

Sum over all geometries that have an AdS boundary.

Large N  typically one geometry  dominant contribution

Determined by the boundary conditions  Holographic QCD

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Holographic QCD is a gravitational theory of gauge
invariant fields in 5 dimensions.
The 5th dimension play the role of the energy scale.
+
+ ….
AdS/CFT(QCD)


A scale invariant (conformal) field theory in 1+3 dimensions
has symmetry group SO(2,4)
Classical AdS has an SO(2,4) symmetry group.
(Such a symmetry is analogous to Lorentz symmetry, in the infinity
limit of the curvature radius, it becomes the Poincare group )
 It is the same as symmetries of 1+4
dimensional Anti-de-Sitter space = the
simplest and most symmetric negatively
curved spacetime
 Quantum gravity in AdS is the same as
a conformal field theory on the boundary
This symmetry is preserved by the
quantization, in the sense that the dual field
theory has the full conformal symmetry.
AdS/CFT(QCD) Dictionary
Bulk fields
IR brane, QCD confinement
Jm
4D CFT QCD
5D AdS
4D generating functional
5D (classical) effective action
operators
5D bulk fields
global symmetries
local gauge symmetries
correlation functions
correlation functions
Resonance hadrons
KK mode states
Chiral symmetry breaking
Linear Confinement
5D bulk VEV
IR boundary condition of Dilaton
AdS/CFT(QCD) Bottom Up
• Bottom up approach is directly related to QCD data & fit to QCD
• Works around the conformal limit, check consistency as an effective field theory
• Carries out calculations for non-perturbative quantites
• Predicts mass spectra, form factors, hadronic matrix elements
• Insights into chiral dynamics, vector meson dominance, quark models, instantons
• Understands chiral symmetry breaking & linear confinement
5D bulk
mass
h’
σ
IR Brane
f0
a1
r
p
Other insights into QCD:
KK modes mass spectra
Chiral symmetry breaking
Linear confinement
AdS/QCD Model
AdS/CFT Correspondence
Klebanov and Witten
1999
Anti-de-Sitter space
R4
AdS5
Boundary
z
z=0
z = infinity
It has constant negative curvature, with a radius of curvature given by R.
w(z)
ds2
=
R2
(dx2
3+1
z2
+
Gravitational potential
dz2)
z
Solution of Einstein’s equations with negative cosmological constant
Hard-Wall AdS/QCD Model
Global SU(3)L x SU(3)R symmetry in QCD
SU(3)LXSU(3)R gauge symmetry in AdS5
4D Operators
5D Gauge fields AL and AR
4D Operators
5D Bulk fields Xij
Hard-Wall AdS/QCD Lagrangian
AL, AR, Xij
Mass term is determined by the scaling dimension
Xij has dimensionΔ= 3 and form p=0, AL & AR have dimension Δ= 3 and form p=1
Gauge coupling determined from correlation functions
In QCD, correlation function of vector current is
Leading order of quark loop
In AdS, correlation function of source fields on UV brane
Bulk to boundary propagator solution to the equation of motion
with V(0)=1
Chiral Symmetry Breaking in QCD
Nonvanishing
J
breaks chiral symmetry to diagonal subgroup
J
Goldstones
Hard-Wall AdS/QCD
with/without Back-Reacted Effects
=
Quark masses
Quark condensate
Explicit chiral breaking
Relevant in the UV
Spontaneous chiral breaking
Relevant in the IR
just solve equations of motion!
Results from hard-wall AdS/QCD
J.P. Shock, F.Wu,YLW, Z.F. Xie, JHEP 0703:064,2007
Soft-Wall AdS/QCD
Dilaton field
Solving equations of motion for vector field
Linear trajectory for mass spectra of vector mesons
Achievements & Challenges

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Hard-wall AdS/QCD models contain the chiral symmetry
breaking, the resulting mass spectra for the excited mesons are
contrary to the experimental data
Soft-wall AdS/QCD models describe the linear confinement and
desired mass spectra for the excited vector mesons, while the
chiral symmetry breaking can't consistently be realized.
A quartic interaction in the bulk scalar potential was introduced
to incorporate linear confinement and chiral symmetry
breaking. While it causes an instability of the scalar potential
and a negative mass for the lowest lying scalar meson state.
How to naturally incorporate two important features into a
single AdS/QCD model and obtain the consistent mass spectra.
Modified Soft-Wall AdS/QCD
by Y.Q.Sui, YLW, Z.F.Xie, Y.B.Yang
Deformed 5D Metric in IR Region & Quartic Interaction
PRD arXiv:0909.3887
Minimal condition for the bulk vacuum
UV & IR boundary conditions of the bulk vacuum
Solutions for the dilaton field at the UV & IR boundary
Various Modified Soft-wall AdS/QCD Models
Some Exact Forms of bulk VEV in Models: I, II, III
Two IR boundary conditions of the bulk VEV
Ia, IIa, IIIa:
Ib, IIb, IIIb:
Behaviors of
VEV & Dilaton
Determination of Model Parameters
Two Energy Scales as Input Parameters
Fitted Parameters
Without Quartic Interaction
Effective IR Cut-off Scale in Soft-Wall AdS/QCD
Fitted Parameters
With Quartic Interaction of bulk scalar
Solutions via Solving Equations of Motion
Pseudoscalar Sector
Equation of Motion
Mass Spectra of Pseudoscalar Mesons
Mass Spectra of Pseudoscalar Mesons
Resonance
States of
Pseudoscalars
Solutions via Solving Equations of Motion
Scalar Sector
Equation of Motion
IR & UV Boundary Condition
Mass Spectra of Scalar Mesons
Mass Spectra of Scalar Mesons
Resonance
States of
Scalars
Wave Functions of Resonance Scalars
Solutions via Solving Equations of Motion
Vector Sector
Equation of Motion
IR & UV Boundary Condition
Mass Spectra of Vector Mesons
Mass Spectra of Vector Mesons
Resonance
States of
Vectors
Wave Functions of Resonance Vectors
Solutions via Solving Equations of Motion

Axial-vector Sector
Equation of Motion
IR & UV Boundary Condition
Mass Spectra of Axial-vector Mesons
Mass Spectra of Axial-vector Mesons
Resonance
States of
Axial-vectors
Vector Coupling & Pion Form Factor
Structure of Pion Form Factor
The PrimEx Experimental Project @ JLab
Experimental program
Precision measurements of:


Two-Photon Decay Widths:
Γ(0→), Γ(→), Γ(’→)
Transition Form Factors
at low Q2 (0.001-0.5 GeV2/c2):
F(*→ 0), F(* →),
F(* →)
Test of Chiral Symmetry and Anomalies via the Primakoff Effect
Conclusions
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Why such a simply modified soft-wall AdS/QCD model
works so well
How to understand dynamical origin of the metric
induced conformal symmetry breaking in the IR
region.
What is dynamics of the dilaton and gravity beyond as
the background
The important role of the dilaton field and the effect
from the back-reacted geometry.
The possible higher order interaction terms and their
effects on the mass spectra and form factors.
Extend to the three flavor case and consider the SU(3)
breaking and instanton effects.
Can lessons from AdS/QCD be applied to other gauge
theories and symmetry breaking systems
THANKS