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.

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

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

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

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

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

Massless Wess-Zumino Model

Lagrangian

Ward identity

In momentum space

Check of Ward Identity

Gamma matrix algebra in 4-dimension and translational invariance of integral momentumLoop regularization satisfies these conditions

Massive Wess-Zumino Model

Lagrangian

Ward identity

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

Dynamically Generated Spontaneous Chiral Symmetry Breaking

In Chiral Effective Field Theory

Chiral limit: Taking vanishing quark masses mq→ 0.

QCD Lagrangian

qq

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

QCD Lagrangian and QCD Lagrangian and SymmetrySymmetry 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

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 :

Mass Formula Pseudoscalar mesons :

Mass Formula

Predictions for Mass Spectra & Mixings

Predictions

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

Chiral Symmetry Breaking & QCD Confinement in AdS/QCD Models

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.)

Top

Down

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

AdS/CFT CorrespondenceAdS/CFT Correspondence

AdS/QCD Model

Klebanov and Witten 1999

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!

=

Results from hard-wall AdS/QCD

J.P. Shock, F.Wu,YLW, Z.F. Xie, JHEP 0703:064,2007

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:

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 @ 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

THANKSTHANKS