Post on 22-Dec-2015
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
Low Energy Dynamics of QCD & Realistic AdS/QCD
Outline
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.
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
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
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-Jona-Lasinio 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.
Loop Regularization
Simple Prescription: in ILIs, make the following replacement
With the conditions
So that
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
Renormalization Constants of Non- Abelian gauge Theory and β Function of QCD in Loop Regularization
Lagrangian of gauge theory
Possible counter-terms
Jian-Wei Cui, Yue-Liang Wu, Int.J.Mod.Phys.A23:2861-2913,2008
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
Supersymmetry
Supersymmetry is a full symmetry of quantum theory
Supersymmetry should be Regularization-independent
Supersymmetry-preserving regularization
J.W. Cui, Y.Tang,Y.L. Wu Phys.Rev.D79:125008,2009
Check of Ward Identity
Gamma matrix algebra in 4-dimension and translational invariance of integral momentumLoop regularization satisfies these conditions
Check of Ward Identity
Gamma matrix algebra in 4-dimension and translational invariance of integral momentumLoop regularization satisfies these conditions
Triangle Anomaly Amplitudes
Using the definition of gamma_5
The trace of gamma matrices gets the most general and unique structure with symmetric Lorentz indices
Anomaly of Axial Current
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
Chiral limit: Taking vanishing quark masses mq→ 0.
QCD Lagrangian
s
d
u
q
GgD
GGqiDqqiDqL
LR
s
RRLLoQCD
)1(2
1
2/
4
1
5,
)(
has maximum global Chiral symmetry :
)1()1()3()3( BARL UUSUSU
QCD Lagrangian and SymmetryQCD Lagrangian and Symmetry
Effective Lagrangian based on Loop Regularization
Y.B. Dai and Y-L. Wu, Euro. Phys. J. C 39 s1 (2004)
L o w E n e r g y D y n a m i c s O f Q C DQ C D 低 能 强 相 互 作 用
g
q q
q q
L Rq q
Dynamically Generated Higgs Potential For Scalar Mesons
QCD低能动力学量子效应生成的标量介子Higgs势
V() = - 2 2 + 4
Spontaneous Symmetry Breaki ng
膺标介子作为Goldstone粒子
标量介子作为Higgs粒子
自发对称破缺
标量介子
夸克
赝标介子
Composite Higgs FieldsComposite Higgs Fields
Scalars as Partner of Pseudoscalars & Lightest Composite Higgs Bosons
Scalar mesons:
Pseudoscalar mesons :
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 Chiral Symmetry Breaking
Field Theory Gravity theory=
Gauge Theories
QCD
Quantum Gravity
String theory
Use the field theory to learn about gravity
Use the gravity description to learn about the field theory
Theories of Field and Gravity
(J.M.)
Most SUSY QCD SU(N)
String theory on AdS x S 5
5=
Radius of curvature
(J.M.)
sYMAdSSlNgRR
4/12
55
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.
Particle Theory Gravity Theory
N colors N = magnetic flux through S5
AdS/CFT Dictionary
N=4 SYM
U(Nc)
= g N
SO(2,4) superconformal group
SO(6) flavour (R) symmetry
RG scale
Sources and operators
Glueballs
Type IIB strings in AdS5xS5
Only gauge invariant operators
R’ = 4 π g N
SO(2,4) metric isometries
SO(6) S5 isometries
Radial coordinate
Constants of integration in SUGRA field solutions
Regular linearized fluctuations of dilaton
s2 2
AdS/CFT
• Qualitative Similarities to QCD
• Real QCD, full string construction?
X
AdS5
S5
Bulk Space
N D-branes
SU(N) Yang-Mills Symmetry
More D-branes
For flavor
• Deviations from AdS for finite N
• Top down string-model approach appears to be far away
• Difficult to find reasonable supergravity background & brane configurations
Top Down
Quantum ChromoDynamics QCD
colors (charges)
They interact exchanging gluons
Electrodynamics QED Chromodynamics (QCD)
electron
photon
gluong gg
g
Gauge group
U(1) SU(3)
3 x 3 matrices
Gluons carry color charge, sothey interact amongthemselves
Bottom up
QCD Strings & GravityGluon: color and anti-color
Closed strings glueballs
Open strings mesons
At distances larger than the typical size of the string
Gravity theory
Radius of curvature >> string length gravity is a good approximation
ls
R
Gauge Theory + Large N_c String Theory Gravity Theory
Large N_c
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
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
4D CFT QCD5D AdS
operators 5D bulk fields
global symmetries local gauge symmetries
correlation functions correlation functions
Resonance hadrons KK mode states
4D generating functional 5D (classical) effective action
Chiral symmetry breaking 5D bulk VEV
Linear Confinement IR boundary condition of Dilaton
IR brane, QCD confinementBulk fields
J
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
Other insights into QCD:
Chiral symmetry breaking
Linear confinement
mass
a1
f0
’
KK modes mass spectra
IR Brane
σ
5D bulk
It has constant negative curvature, with a radius of curvature given by R.
ds2 = R2 (dx23+1 + dz2)
z2
Boundary
R4
AdS5
z = 0z
z = infinity
Gravitational potential w(z)
z
Anti-de-Sitter space
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
5D Gauge fields AL and AR
4D Operators 5D Bulk fields Xij
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
Hard-Wall AdS/QCD Lagrangian
SU(3)LXSU(3)R gauge symmetry in AdS5
4D Operators
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 QCDNonvanishing breaks chiral symmetry to diagonal subgroup
J J
Goldstones
Hard-Wall AdS/QCD with/without Back-Reacted Effects
Quark massesExplicit chiral breaking
Relevant in the UV
Spontaneous chiral breaking
Relevant in the IR
Quark condensatejust solve equations of motion!
=
Soft-Wall AdS/QCD
Solving equations of motion for vector field
Linear trajectory for mass spectra of vector mesons
Dilaton field
Achievements & Challenges
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
Minimal condition for the bulk vacuum
UV & IR boundary conditions of the bulk vacuum
Solutions for the dilaton field at the UV & IR boundary
PRD
arXiv:0909.3887
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:
Solutions via Solving Equations of Motion
Scalar Sector
Equation of Motion
IR & UV Boundary Condition
Solutions via Solving Equations of Motion
Vector Sector
Equation of Motion
IR & UV Boundary Condition
Solutions via Solving Equations of Motion
Axial-vector Sector
Equation of Motion
IR & UV Boundary Condition
The PrimEx Experimental Project @ The PrimEx Experimental Project @ JLabJLab
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 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