AdS/QCD and Light-Front Holography Stan Brodsky

AdS/QCD and Light-Front Holography 

P + = P 0 + P z 

Fixed ⌅ = t + z/c 

Guy de Teramond 

Hans Günter Dosch 

c 

c 

x i = k+ 

P + = k0 +k 3 

P 0 +P z 

¯c 

⇧(⇤, b ) 

⇥ = d s(Q 2 ) 

d ln Q 2 < 0 

Exploring QCD, Cambridge, August 20-24, 2007 

Page 

u 

Stan Brodsky 

International Conference on Nuclear Theory in the Supercomputing Era - 2013 

LC 2013 

May 21, 2013 

Valparaiso, Chile May 19-20, 2011 

Skiathos, Greece 

1

QCD Lagrangian 

L QCD = 1 4 Tr(Gµ⌫ G µ⌫ )+ 

n f 

X 

f=1 

i ¯ f D µ 

µ 

f + 

n f 

X 

f=1 

m f ¯ f f 

iD µ = i@ µ gA µ G µ⌫ = @ µ A µ @ ⌫ A µ g[A µ ,A ⌫ ] 

Chiral Lagrangian is Conformally Invariant 

Where does the QCD Mass Scale Λ QCD come from 

How does color confinement arise 

Scale 

• 

de Alfaro, Fubini, Furlan: 

can appear in Hamiltonian and EQM 

without affecting conformal invariance of action! 

Unique potential! 

May 21, 2013 LC2013 AdS/QCD & LF-Holography Stan Brodsky 

2

de Teramond, Dosch, sjb 

AdS/QCD 

Soft-Wall Model 

Exploring QCD, Cambridge, August 20-24, 2007 Page 9 

Light-Front Holography 

Semi-Classical Approximation to QCD 

Relativistic, frame-independent 

Unique color-confining potential 

Zero mass pion for massless quarks 

Regge trajectories with equal slopes in n and L 

Light-Front Wavefunctions 

Light-Front Schrödinger Equation 

Conformal 

Symmetry 

of the action 


3






⇥ 

⇥ 

d 2 

d 2 

+ d⇣ 2 1V 

( 4L2 ) = M 2 ⇥( 

4⇣ 2 + U(⇣) ⇤ ) 

(⇣) =M 2 (⇣) 

⇥ 

d 2 

d 2 + V ( ) = M 2 ⇥( ) 

U(⇣) =apple 4 ⇣ 2 +2apple 2 (L + S 1) 


2 = x(1 x)b 2 ⇥ . 

Confinement scale: apple ' 0.5 GeV 

J z = S z 1/apple ' 0.4 fm 

p = ⇤ n 

i=1 

Si z + ⇤ n 1 

Conformal Symmetry 

of the action 

i=1 ⌥z i = 1 2 

Unique 

Confinement Potential! 


4

m q =0 

Massless pion in Chiral Limit! 

Same slope in n and L! 

Mass ratio of the ρ and the a1 mesons: coincides with Weinberg sum rules 


5

⇤(z) 

• Light-Front Holography 

⇥ = (x(1 x)|b ⇤ | 

General remarks about orbita 

z 

(x, k ) 

xploring QCD, Cambridge, August 20-24, 2007 Page 9 

mentum 

0.8 0.60.40.2 

0.2 

z 

0.15 

z 0 = 1 

• Light Front Wavefunctions: 

⇥ QCD 

d ⇥ np 

n(x i , k i , i) 

n 

Schrödinger Wavefunctions 

of Hadron Physics 

i=1 (x iR + b i ) = R 


x R 

+ b 

0.1 

0.05 

0 

1.3 

(x, k )(GeV) 

1.4 

1.5

Prediction from AdS/QCD: Meson LFWF 

(x, k ) 

⇥ M (x, Q 0 ) ⇥ 0.8 x(1 0.60.40.2 x) 

0.2 

de Teramond, 

sjb 

⇤ M (x, k 2 ⇤ ) 

µ R 

Note coupling 

µ R = Q 

k 2 , x 

0.15 

0.1 

0.05 

0 

1.3 

1.4 

(x, k ) (GeV) 

1.5 

“Soft Wall” 

model 

massless quarks 

x 

1 x 

µ F = µ R 

⇤ M (x, k ⇥ ) = 

4⇥ 

x(1 x) e k 2 ⇥ 

2 2 x(1 x) 

Provides Q/2 < Connection µ R < 2Q of Confinement 

µ 

to TMDs 

R 

⇥ M (x, Q 0 ) ⇥ x(1 x) 


7

J. R. Forshaw, 

R. Sandapen 

L 

T 

⇤ p ! ⇢ 0 p 0 

9

0.6 

Q 2 F Π Q 2 

0.5 

0.4 

0.3 

0.2 

0.1 

Q 2 GeV 2 

0.0 

0 1 2 3 4 5 6 7 

10

Each element of 

flash photograph 

illuminated 

at same LF time 

= t + z/c 

Evolve in LF time 

P 

= i d 

d 

Eigenvalue 

P = M2 + ~ P 2 

P + 

H QCD 

LF | h >= M 2 h| h > 

11

Light-Front Quantization 

Causal, Frame-independent, Simple Vacuum, 

Current Matrix Elements are overlap of LFWFS 

12

Light-Front Wavefunctions: rigorous representation of 

composite systems in quantum 

P + = P 0 field 

+ P z 

theory 

x = k+ 

P + = k0 + k 3 

P 0 + P 3 


x i = k+ 

P + , ↵ P + 

P + = k0 +k 3 

P 0 +P z 

General remark 

mentum 

A(⇤, ⇤) = 

General 

2⇥ 1 d e2 i ⇤ M( , ⇤) 

remarksx i P + about 

, x i 

↵ 

orbi 

⇧(⇤, b ) P⇤ + ↵ k⌃R 

⇤i 

P + , P apple ⇤ 

mentum 

ẑ 

x iR 

⌃ + ⌃ b i 

x i P + , x iP⇤ apple + apple ⇥ = d s(Q 2 ) 

k 

d ln Q 2 < 0 

⇤i 

↵L = R ↵ ⇥ P ↵ 

u 

n 

ni ⌃ b i = ⌃0 

= Q2 

i=1 (x iP ⌃ + ⌃ k 

2p·q 

i 

↵L i = (x iR⇤ ↵ + ↵ b ⇤i ) ⇥ P ↵ 

n 

n(x 

i x i = 1 

ẑ 

i , k 

x iP 

⌃ + ⌃ 

i , i) 

k i 

↵ ⇧i = ↵ b ⇤i ⇥ ↵ k ⇤i 

appleL = R apple ⇥ P apple n 

i ⌃ k i = ⌃0 

↵ ⇧i = L ↵ i x iR⇤ ↵ ⇥ P ↵ = ↵ b ⇤i ⇥ P ↵ 

appleL i = (x iR⇤ apple + apple b ⇤i ) ⇥ P apple n 

i x i = 1 

Invariant under boosts! Independent of P 

n 

i=1 (x iR + b i ) = R 

Bethe-Salpeter WF integrated over k - 

13

Hadron Distribution Amplitudes 

M (x, Q) = 

Q 

d 2 k ⇥ q¯q (x, k ) 

i 

x i = 1 

P + = P 0 + P z 


x 

1 x 

k 2 < Q 2 

• Fundamental gauge invariant non-perturbative 

x i = 

input k+ 

Pto 0 +P hard z 

P + = k0 +k 3 

exclusive processes, heavy hadron decays. Defined for 

⇧(⇤, b ) 

Mesons, Baryons 

• Evolution Equations from PQCD, OPE 

• Conformal Expansions 

Lepage, sjb 

Efremov, Radyushkin 

⇥ = d s(Q 2 ) 

d ln Q 

Sachrajda, 2 < 0 

Frishman Lepage, sjb 

u 

Braun, Gardi 

• Compute from valence light-front wavefunction in light-cone 

gauge 

May 21, 2013 LC2013 AdS/QCD & LF-Holography Stan Brodsky

Light-Front QCD 

Physical gauge: A + = 0 

Exact frame-independent formulation of 

nonperturbative QCD! 338 

L QCD H QCD 

LF 

H QCD = [ m2 + k 2 

] 

LF i + H int 

x 

LF 

i 

HLF int: 

Matrix in Fock Space 


LF | h >= M 2 h| h > 

|p, J z >= X n=3 

n(x i , ~ k i , i)|n; x i , ~ k i , i > 

Eigenvalues and Eigensolutions give Hadronic 

Spectrum and Light-Front wavefunctions 

LFWFs: Off-shell in P- and invariant mass 

15 

Fig. 6. A few selected 

H int 

LF

illustrated in Fig. 2 in terms of the block matrix n : x , k , λ Hn : x, k , λ . The structure of th 

 

Light-Front matrix depends QCD of course on 

H QCD the way 

S.J. Brodsky et LC al. / Physics | one has arranged 

h⇧ 

Reports 

= M301 2 h | the FockDLCQ: space, see Solve Eq. QCD(1+1) (3.7). Note that for mo 

of the block matrix elements vanish due to the nature (1998) h⇧ of the 299—486 any light-cone quark interaction mass and asflavors 

defined 

Heisenberg Equation 

Hornbostel, Pauli, sjb 

ig. 6. A few selected matrix elements of the QCD front form Hamiltonian H"P 

in LB-convention. 

Minkowski space; frame-independent; no fermion doubling; no ghosts 

trivial vacuum 

Fig. 2. The Hamiltonian matrix for a SU(N)-meson. The matrix elements are represented by energy diagrams. With 

r the instantaneous each block they fermion are all of the lines sameuse type: eitherfactor vertex, fork ¼or seagull in Fig. diagrams. 5 or Zero Fig. matrices 6, or are thedenoted correspond by a dot ( 

The single gluon is absent since it cannot be color 16 neutral. 

les in Section 4. For the instantaneous boson lines use the factor ¼ .

LIGHT-FRONT MATRIX EQUATION 

the interaction part of HLC. Diagrammatically, V involves 

number Rigorous of coupled Method integral eigenvalue for Solving equations, Non-Perturbative QCD! 

interactions--i.e. diagrams having no internal propagators-c 

(Fig. 5). These equations determine the hadronic spe 

- 

0 

. 

where V is the 

xJ= 

interaction part of HLC. Diagrammatically, V involves completely 

irreducible interactions--i.e. diagrams having no internal propagators-coupling 

Fock states :(Fig. 3 II - IL 7 - 

A 

. . 

+ = 0 

0 l . f 

⇥ ggg 

5). These equations determine the hadronic spectrum and 

xJ= 

: 3 II 

0 

IL 7 

⇥ ggg 

. 

- -. 

I- . 

R = (⇥ 

l . . f 

(⇥ ¯p 

1 II 

Minkowski space; frame-independent; 0 

no fermion doubling; no ghosts 

. 

R = C 

. 

• Light-Front I- . 

1 II 

Vacuum = vacuum of free Hamiltonian! 

ū(x) ⇥= ¯d(x 


Figure 5. Coupled eigenvalue equations 17 for the light-cone wa.vefunctious of a 

re 5. Coupled eigenvalue equations for the light-cone wa.vefunctio

DLCQ: Solve QCD(1+1) for any quark mass and flavors 


18

Kinks in Discrete Light-Cone Quantization 

Chakrabarti, Harindranath, Martinovic, Vary 

3 

2 

1 

0 

(a) 

K = 24 

K = 32 

K = 41 

-1 

-2 

φ c 

(x - ) 

-3 

3 

2 

1 

0 

(b) 

DLCQ K = 41 

Constrained Variational 

= 41 

-1 

-2 

-3 

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 

x - 

19

Wavefunction at fixed LF time: Arbitrarily Off-Shell in Invariant Mass 

Eigenstate: all Fock states contribute 

c 

¯c 

Higher Fock States of the Proton 

Fixed LF time 


21

n=3 

|p,S z >= ∑ 

n=3 

The Light Front Fock State Wavefunctions 

Ψ n (x i ,~k i ,λ i ) 

Ψ n (x i ,~k i ,λ i )|n;~k i ,λ i > 

are boost invariant; they are independent of the hadron’s energy 

and momentum P µ . 

The light-cone momentum fraction 

are boost invariant. 

sum over states with n=3, 4, ...constituents 

n 

∑k i + = P + , 

i 

x i = k+ i 

p + = k0 i + kz i 

P 0 + P z 

n 

∑x i = 1, 

i 

Intrinsic heavy quarks 

n 

∑ 

i 

s(x), c(x), b(x) at high x ! 

Mueller: gluon Fock states 

~k i =~0 . 

ep ⇥ e + n 

P /p ⇤ 30% 

Violation of Gottfried sum rule 

¯s(x) ⇤= s(x) 

Fixed LF time 

ū(x) ⌅= ¯d(x) 

⇥ M (x, Q 0 ) ⇥ x(1 x) 

BFKL Pomeron 

Hidden Color

E866/NuSea (Drell-Yan) 

¯d(x) = ū(x) 

s(x) = ¯s(x) 

Intrinsic glue, sea, 

heavy quarks 


23

Proton Self Energy 

Intrinsic Heavy Quarks 

Fixed LF time 

x Q (m 2 Q + k 2 ) 1/2 

p 

Q 

p 

Q 

Probability (QED) 

1 

M 4 Probability (QCD) 

1 

M 2 Q 

Collins, Ellis, Gunion, Mueller, sjb 

M. Polyakov, et al. 

24

Fixed LF time 

Proton 5-quark Fock State : 

Intrinsic Heavy Quarks 

p 

Q 

QCD predicts 

Intrinsic Heavy 

p 

Quarks at high x! 

Q 

Minimal off-shellness 

x Q (m 2 Q + k 2 ) 1/2 

Probability (QED) 

1 

M 4 Probability (QCD) 

1 

M 2 Q 

Collins, Ellis, Gunion, Mueller, sjb 

Polyakov, et al. 

25

W. C. Chang and 

J.-C. Peng 

arXiv:1105.2381 

HERMES: Two components to s(x,Q 2 )! 

x(s+s − x(s+s 

) − ) 

0.3 

0.3 

0.2 

0.2 

0.1 

0.1 

0 

0 

BHPS (µ=0.5 GeV) 

HERMES BHPS (µ=0.3 GeV) 



10 -1 

10 -1 

HERMES 

Extrinsic (DGLAP) 

strangeness! 

Figure 2: Comparison of the HERMES x(s(x)+¯s(x)) data with the Figure 3: Comparison 

calculations based on the BHPS model. The solid and dashed curves calculations based on t 

Figure are obtained 2: Comparison by evolving of the the HERMES BHPS x(s(x)+¯s(x)) result to Q 2 =2.5 data with GeVthe 

2 usingFigure are 3: from Comparison the HERME of the 

calculations 

µ =0.5 GeVandµ 

based on the 

=0.3 

BHPS 

GeV,respectively. 

model. The solid and dashed curves calculations based on the BH 

are obtained by evolving the BHPS result to Q 2 Thenormalizationsof 

=2.5 GeV 2 are obtained from the 

the calculations are adjusted to fit the data at x>0.1 withstatistical 

using are from 

curves 

the 

are 

HERMES 

obtained 

expe 

by 

µ =0.5 GeVandµ =0.3 GeV,respectively. Thenormalizationsof are obtained from the PDF s 

errors only, denoted by solid circles. 

using µ =0.5 GeVand 

the calculations are adjusted to fit the data at x>0.1 withstatistical curves are obtained by evolv 

of the calculations are 

errors only, denoted by solid circles. 

using µ =0.5 GeVandµ =0 

of the calculations are adjust 

x 

x 

their measurement of charged kaon production in SIDIS reaction 

measurement [6]. The HERMES of charged data, kaon production shown in Fig. in SIDIS 2, exhibits their rex(d 

− +u − -s-s − ) 

x(d − +u − -s-s − ) 

BHPS: Hoyer, Sakai, 

Peterson, sjb 

0.3 

0.2 

0.3 

0.2 

0.1 

Intrinsic 

strangeness! 0.1 

0 

Consistent with 

0 

intrinsic charm 10 -2 

data 

s(x, Q 2 )=s(x, Q 2 ) extrinsic + s(x, Q 2 ) intrinsic 

10 -2 

QCD: 

1 

M 2 Q 

scaling 

the x( ¯d(x)+ū(x)) 26

0.024 

(7) 

.5 GeV) (6) 

It is reand 

the 

to0.029 

check 

.3 GeV) (7) 

k states, 

lities for 

cale W. µ. C. Chang It is re-and 

+¯s(x), J.-C. and Peng the 

allow ilityus for to check 

uark Fock states, 

%. 

probabilities 

It is 

for 

0.08 

0.06 

c − 

0.1 

0.08 

0.04 

0.06 

0.02 

0.04 

0.02 

0 


QCD (1/m Q 2 ) scaling: predict IC ! 

BHPS 



Hoyer, Peterson, Sakai, sjb 

ta in the 

% prob- 

d probability for 

e 

round 

[13], arXiv:1105.2381 

0 0.2 0.4 0.6 0.8 1 

40%. It is 

¯d−ū in 

data in the 

y. 

has 

A∼60% sign 

of state the[13], in 

probleon 

0 

x 

0 0.2 0.4 0.6 0.8 1 

he thiskaon- 

so worth 

study. A sigextraction 

of the Figure 4: Calculations of the ¯c(x) distributionsbasedontheBHPS 

x 

ted to the kaonl. 

〉 Itstates 

is also worth and the dashed and dotted curves are obtained by evolving the BHPS 

model. The solid curve corresponds to the calculation using Eq. 1 

Figure 4: Calculations of the ¯c(x) distributionsbasedontheBHPS 

|uudQ ¯Q〉 

model. The solid curve corresponds to the calculation using Eq. 1 

etic mo- states resultand tothe Qdashed 2 = 75 andGeV dotted 2 curves using are µ obtained = 3.0by GeV,andµ evolving the BHPS = 0.5 GeV, 

n’s magnetic moe 

Q and ¯Q in the 

result to Q 2 = 75 GeV 2 using µ = 3.0 GeV,andµ = 0.5 GeV, 

¯Q in the 

respectively. The normalization is set at P c¯c 

respectively. The normalization is set at P5 c¯c =0.01. 

Intrinsic Charm 

5 

Consistent with EMC 

=0.01. 27

Measurement of Charm Structure 

Function! 

J. J. Aubert et al. [European Muon Collaboration], “Production 

Of Charmed Particles In 250-Gev Mu+ - Iron Interactions,” 

Nucl. Phys. B 213, 31 (1983). 

First Evidence for Intrinsic Charm 

Hoyer, Peterson, Sakai, sjb 

factor of 30 ! 

gluon splitting 

(DGLAP) 

DGLAP / Photon-Gluon Fusion: factor of 30 too small 

Two Components (separate evolution): 

c(x, Q 2 )=c(x, Q 2 ) extrinsic + c(x, Q 2 ) intrinsic

Goldhaber, Kopeliovich, Schmidt, sjb 

Intrinsic Charm Mechanism for Inclusive 

High-X F Higgs Production 

p 

¯c 

c 

g 

H 

pp 

HX 

p 

Also: intrinsic bottom, top 

Higgs can have 80% of Proton Momentum! 

New production mechanism for Higgs 

29

Intrinsic Bottom Contribution to Inclusive 

Higgs Production 

⌅ = t + z/c 

d⇤ 

dx F 

(pp ⇥ HX)[fb] 

fb 

LHC : 

⇥q ⇥ 

s = 14TeV 

q 

⇥ 

Tevatron : 

s = 2TeV 

p 

Goldhaber, Kopeliovich, Schmidt, sjb 

30

=2p + F (q 2 ) 

P + = P 0 + P z 

⇤ 

q 2 = Q 2 = q 2 Fixed ⌅ = t + z/c 

Interaction 

picture 

q + =0 

~q 

x i = k+ 

⇧(⇤, b ) 

Form Factors are 

Overlaps of LFWFs 

P + = k0 +k 3 

P 0 +P z 

p 

x, ~ k x, ~ k + ~q 

⇥ = d s(Q 2 ) 

d ln Q 2 < 0 

u 

(x (x i , ~ ki) 

0 i , ~ k i ) 

p + q 

Drell &Yan, West 

Exact LF formula! 

struck 

spectators 

~ k 

0 

i = ~ k i +(1 x i )~q 

~ k 

0 

i = ~ k i x i ~q 


31

P + = P 0 + P z 

Light-Front Wavefunctions and Electron-Proton Collisions 


Hoyer 

CF 

p 

|I> 

x i = k+ 

P + = k0 +k 3 

P 0 +P z 

⇧(⇤, b ) 

⇥ = d s(Q 2 ) 

d ln Q 2 < 0 

CF 

p+q 

|F> 

(x i , ~ k i ) 

u 

q 

(x i , ~ k 0 i) 

e’ 

e 

q 2 = Q 2 = q 2 |F> = resonances, multiparticle states. 

q + =0 

~q 

All final states |F> in electroproduction produced 

Diagonal n to n overlap of LFWFs 

Confinement:Only Color Singlets! 

32

Calculation of Form Factors in Equal-Time Theory 

Instant Form 

Need vacuum-induced currents 

Calculation of Form Factors in Light-Front Theory 

Front Form 

Exact Answer! 

Absent zero for forq + q + = 00 zero !! 


33 

No vacuum graphs

Calculation of proton form factor in Instant Form 

 

p 

p + q p 

p + q 

• Need to boost proton wavefunction from p to p 

+q: Extremely complicated dynamical problem; 

particle number changes 

• Need to couple to all currents arising from 

vacuum!! Remains even after normal-ordering 

• Each time-ordered contribution is framedependent 

• Divide by disconnected vacuum diagrams 

• Instant form: Violates causality 


eas the Pauli and electric [dx] [d kdipole ⇧ ] ⇤ form factors are given by 16⇤ 1 x 

Exact LF 

i,c ,f i i=1 

2(2⇤) 3 i 

te between the struck and spectator constituents; namely, we 

i=1 

F 2 (q 2 ) 

A(⇤, 

2M 

⇧ ⌅) ⌥ = 1 

Formula 

2⇥ 

d 

[dx][d 2 k ⇧ ] ⇧ e2 i ⇤ for Pauli Form Factor 

1 

M( , ⌅) 

e j 

a 

j 

2 ⇥ 

Drell, sjb 

where n denotes the number of constituents in Fock state a 

k ⌅ ⇧j = k ⇧j + (1 x j )q ⇧ (14) 

P 

1 

+ , Ppossible {⇥ i }, {c i }, and {f i } 

q L ⌅⇥ a (x i , k ⌅ ⇧i, ⇥ i ) ⌅a(x ⇤ i , k ⇧i , ⇥ i ) + 1 

in state a. The arguments of the 

⌅ 

t j and wave function di erentiate between 

q R ⌅⇤ a (xthe i , k ⌅ struck 

⇧i, ⇥ i ) ⌅and a(x ⇥ = spectator i , 0k ⇧i , ⇥ i ) co , 

have [13, 15] 

xk i ⌅ P + 

⇧i = , kx k ⌅ ⇧i i P ⌅ x+ i qk ⇧ ⌅i 

⇧j = k ⇧j + (15) B(0) x j )q= ⇧ 0 Fock 

⌥ 

re i 

F 

⌅= 3 (q 

j. 2 ) 

Note 

2M 

⇧ for the struck 

[dx][d 2 constituent 

k ⇧ ] ⇧ i 

e j 

a 

j 

2 ⇥ j and 

= Q2 that because of the frame choice q + = 0, onlyq R,L = q x ± iq y 

2p·q 

s of the light-front Fock states appear [14]. 

k ⌅ ⇧i = k ⇧i x 

1 

q L ⌅⇥ a (x i , k ⌅ ⇧i, ⇥ i ) ⌅a(x ⇤ 1 

⇤(x, i q ⇧ b ⌅ ) 

ˆx, ŷ plane 

i , k ⇧i , ⇥ i ) 

q R ⌅⇤ a (x i , k ⌅ ⇧i, ⇥ i ) ⌅a(x ⇥ i , k ⇧i , ⇥ i ) . 

for6each spectator i, where i ⌅= j. Note that because x of the fra 

diagonal (n ⌅ = n) overlaps of the light-front Fock states appea 

ummations M 2 are (L) over ⇤ all L- 

contributing Fock states a and struck constituent cha 

b 

Here, as earlier, we refrain from including the constituents’ ⌅ (GeV) 

color and 1 

fl 

6 

ndence in the arguments of the light-front wave functions. The phase-s 

Must have ↵ 

ration is 

z = ±1 to have nonzero F 2 (q 2 ) 

x] [d 2 k ⇧ ] ⇤ ⇧ i,c i ,f i 

⇤ n ⌃ 

Nonzero Proton Anomalous Moment --> 

⌥ ⌥ 

⇥⌅ 

Nonzero orbital quark angular momentum 

dxi d 2 k ⇧i 

2(2⇤) 3 

16⇤ 3 1 

May 21, 2013 LC2013 i=1 AdS/QCD & LF-Holography Stan i=1 Brodsky 

35 

n⇧ 

x i 

⇥ 

Identify z ⇤ ⇥ = 

(2) 

⇥ 

n⇧ 

k ⇧i 

i=1 

,

Vanishing Anomalous gravitomagnetic moment B(0) 

Terayev, Okun, et al: B(0) Must vanish because of 

Equivalence Theorem 

graviton 

sum over constituents 

- 

Hwang, Schmidt, sjb; 

Holstein et al 

B(0) = 0 

Each Fock State 


36

In general the light-cone wavefunctions satisfy cons 

y conservation of the z projection of 

Angular Momentum on the Light-Front 

ngular momentum: 

J z = 

n∑ 

i=1 

si z ∑n−1 

+ 

j=1 

l z j . 

Conserved 

LF Fock-State by Fock-State 

Every (62) Vertex 

he sum over si 

z represents the contribution of the intr 

onstituents. The sum over orbital angular momenta lj z = 

the intrinsic spins of the n Fock state 

ta l z j = −i( k 1 ∂ − k 2 ∂ ) 

derives from 

j j 

e n−1 relative momenta. This excludes the contribution 

∂k 2 j 

∂k 1 j 

ibution to the orbital angular momentum 

ue to the motion of the center of mass, which is not an in 

Parke-Taylor Amplitudes 

Stasto 

ot an intrinsic property of the hadron. 

We can see how the angular momentum sum rule 

avefunctions Eqs. (20) and (23) of the QED model sys 

m rule Eq. (62) is satisfied for the 

n-1 orbital angular 

momenta 

Nonzero Anomalous Moment Nonzero orbital angular momentum 

Drell, sjb 


37

Advantages of the Dirac’s Front Form for Hadron Physics 

• Measurements are made at fixed τ 

• Causality is automatic 

• Structure Functions are squares of LFWFs 

• Form Factors are overlap of LFWFs 

• LFWFs are frame-independent -- no boosts 

• No dependence on observer’s frame 

• Dual to AdS/QCD 

• LF Vacuum trivial -- no condensates 

• Implications for Cosmological Constant 


Light-Front Wave Function Overlap Representation 

Diehl, Hwang, sjb, NPB596, 2001 

See also: Diehl, Feldmann, Jakob, Kroll 

DVCS/GPD 

DGLAP 

region 

N 

N 

N+1 

N-1 

ERBL 

region 

N 

N 

DGLAP 

region 

Bakker & JI 

Lorce 


39

General remarks about orbital angular momentum 

• Light Front Wavefunctions: 

n(x i , k i , i) 

GTMDs 

Momentum space 

Position space 

Transverse density in 

n 

momentum space 

i=1 (x iR + b i ) = R 

Transverse density in position 

space 

x i R 

+ 

TMDs 

b i 

TMFFs 

GPDs 

Transverse 

n 

i b i = 0 

Lorce, 

Pasquini 

TMSDs 

n 

i x i = 1 

PDFs 

FFs 

Longitudinal 

Sivers, T-odd from lensing 

Charges 

40

Leading-Twist Contribution to Real Part of DVCS 

LF Instantaneous interaction 

⇤ 

⇤ 

T = 

2 X q 

e 2 q 

x q 

~✏ · ~✏ 0 

T / s 0 F C=+ (t =0) 

Origin of ‘D-Term’ 

in QCD 

s-independent 

‘J=0 fixed pole’ 

p 

Damashek, Gilman 

Close, Gunion, sjb 

Szczepaniak, 

Llanes Estrada, sjb 

p 


41

Static 

Dynamic 

• Square of Target LFWFs 

• No Wilson Line 

• Probability Distributions 

• Process-Independent 

• T-even Observables 

• No Shadowing, Anti-Shadowing 

• Sum Rules: Momentum and J z 

Modified by Rescattering: ISI & FSI 

Contains Wilson Line, Phases 

No Probabilistic Interpretation 

Process-Dependent - From Collision 

T-Odd (Sivers, Boer-Mulders, etc.) 

Shadowing, Anti-Shadowing, Saturation 

Sum Rules Not Proven 

Hwang, 

Schmidt, sjb, 

Mulders, Boer 

Qiu, Sterman 

Collins, Qiu 

Pasquini, Xiao, 

Yuan, sjb 

• DGLAP Evolution; mod. at large x DGLAP Evolution 

• No Diffractive DIS 

Hard Pomeron and Odderon Diffractive DIS 

General remarks about orbital e angular momentum 

current 

2 

– quark jet 

!* 

e – 

n(x i , ⇥ k i , i) 

S 

proton 

quark 

final state 

interaction 

spectator 

system 

11-2001 

8624A06 

n 

May 21, 2013 LC2013 

i=1 (x iR ⇥ + ⇥ b 

AdS/QCD i ) = R ⇥ & LF-Holography Stan Brodsky 

42

• LF wavefunctions play the role of Schrödinger wavefunctions 

in Atomic Physics 

• LFWFs=Hadron Eigensolutions: Direct Connection to QCD 

Lagrangian 

• Relativistic, frame-independent: no boosts, no disc 

contraction, Melosh built into LF spinors 

• Hadronic observables computed from LFWFs: Form factors, 

Structure Functions, Distribution Amplitudes, GPDs, TMDs, 

Weak Decays, .... modulo `lensing’ from ISIs, FSIs 

• Cannot compute current matrix elements using instant form 

from eigensolutions alone -- need to include vacuum currents! 

n 

• Hadron Physics without LFWFs is like Biology without DNA! 

General remark 

mentum 

n(x i , k i , i) 

n 

i=1 (x iR + b 

x i R 

n 

+ b i 

i b i = 0 

i x i = 1 


•Hadron Physics without LFWFs is like Biology without DNA! 

General remarks about orbital angular 

mentum 

n(x i , k i , i) 

n 

i=1 (x iR + b i ) = R 

x i R 

+ b i 


44

Light-Front Wavefunctions 

Dirac’s Front Form: Fixed τ = t + z/c 

(x i , ~ k i , i) 

x i = k+ i 

P + 

Invariant under boosts. Independent of P μ 


LF |ψ >= M2 |ψ > 

0 < x i < 1 

Direct connection to QCD Lagrangian 

Remarkable new insights from AdS/CFT, 

the duality between conformal field theory 

and Anti-de Sitter Space 

n 

i=1 

x i = 1 


45

[ 

[ 

H QED 

QED atoms: positronium and 

muonium 

2 

Effective two-particle equation 

Includes Lamb Shift, quantum corrections 

Spherical Basis 

V 

Coulomb potential 

eff ⇥ V C (r) = r 

(H 0 + H int ) | >= E | > Coupled Fock states 

2m red 

+ V e (S, r)] (r) = E (r) 

1 d 2 

2m red dr 2 + 1 ⌃(⌃ + 1) 

2m red r 2 + V e (r, S, ⌃)] (r) = E (r) 

r, , ⇥ 

Semiclassical first approximation to QED --> Bohr Spectrum 

46

Light-Front QCD 

H LF 

QED 

QCD 

(H 0 LF + H I LF )| >= M 2 | > 

Coupled Fock states 

[ k2 + m 2 

x(1 x) + V LF 

e ] LF (x, k ) = M 2 LF (x, k ) 

4 

Effective two-particle equation 

2 = x(1 x)b 2 

Azimuthal Basis 

, ⇥ 


Confining AdS/QCD 

potential! 

Semiclassical first approximation to QCD 47

⇥ 

Light-Front Schrödinger d 2 

+ V ( ) Equation 

= M 2 ⇥( ) 

d 2 

Relativistic LF single-variable radial 

equation for QCD & QED 

4 

⇥ 

d 2 

d 2 + V ( ) 

2 = x(1 x)b 2 ⇥ . 

G. de Teramond, sjb 

Frame Independent! 

= M 2 ⇥( ) 

AdS/QCD: 

J z = x S(1 

p z = x) ⇤ b n 

i=1 

Si z + ⇤ 

⇥ 

n 1 

⇤(x, b ⇥ ) = ⇤( ) 

each Fock State 

⇥(z) 

= x(1 x)b 2 ⇥ 


x (1 

x) b ⇥ 

J z p = S z q + S z g + L z q + L z g = 1 2 

= x(1 x)b 2 ⇥ 

z 

May 21, 2013 LC2013 AdS/QCD & LF-Holography Stan Brodsky z 

48 

z 

i=1 ⌥z i = 1 2 

⇤(x, b ⇥ ) = ⇤( ) 

x (1 x) b ⇥ 

⇥(z) 

⇤(x, b ⇥ ) = ⇤( ) 

⇥(z) 

z 

= x(1 x)b 2 ⇥

m q =0 

Same slope in n and L! 

Massless pion in Chiral Limit! 

Mass ratio of the ρ and the a1 mesons: coincides with Weinberg sum rules 


49

⇥ 



= M 2 ⇥( ) 

d 2 



4 

⇥ 

d 2 

d 2 + V ( ) 

2 = x(1 x)b 2 ⇥ . 



= M 2 ⇥( ) 


J z = x S(1 

p z = x) ⇤ b n 

i=1 

Si z + ⇤ 

⇥ 

n 1 

⇤(x, b ⇥ ) = ⇤( ) 

U is the exact QCD potential 

Conjecture: ‘H’-diagrams generate 


⇥(z) 

= x(1 x)b 2 ⇥ 


x (1 

x) b ⇥ 


= x(1 x)b 2 ⇥ 

z 


50 

z 

i=1 ⌥z i = 1 2 

⇤(x, b ⇥ ) = ⇤( ) 

x (1 x) b ⇥ 

⇥(z) 

⇤(x, b ⇥ ) = ⇤( ) 

⇥(z) 

z 

= x(1 x)b 2 ⇥

Remarkable Features of 


• Relativistic, frame-independent 

• QCD scale appears spontaneously - unique LF potential 

• Reproduces spectroscopy and dynamics of light-quark hadrons with 

one parameter 

• Zero-mass pion for zero mass quarks! 

• Regge slope same for n and L -- not usual HO 

• Splitting in L persists to high mass -- contradicts conventional 

wisdom based on breakdown of chiral symmetry 

• Phenomenology: LFWFs, Form factors, electroproduction 

• Extension to heavy quarks 








⇥ d 2 

d⇣ 2 + 1 

4L2 

4⇣ 2 + U(⇣) ⇤ (⇣) =M 2 (⇣) 



Confinement scale: 

apple ' 0.5 GeV 

1/apple ' 0.4 fm 


of the action 


52

The Holographic Correspondence 

In the “ semi-classical” approximation to QCD with massless quarks and no quantum loops the 

function is zero, and the approximate theory is scale and conformal invariant. 

Isomorphism of SO(4, 2) of conformal QCD with the group of isometries of AdS space 

ds 2 = R2 

z 2 (⇥ µ⇥dx µ dx ⇥ dz 2 ). 

Semi-classical correspondence as a first approximation to QCD (strongly coupled at all scales). 

x µ ⇤ ⇤x µ , z ⇤ ⇤z, maps scale transformations into the holographic coordinate z. 

Different values of z correspond to different scales at which the hadron is examined: AdS boundary at 

z ⇤ 0 corresponds to the Q ⇤ ⌅, UV zero separation limit. 

There is a maximum separation of quarks and a maximum value of z at the IR boundary 

Truncated AdS/CFT (Hard-Wall) model: cut-off at z 0 = 1/ QCD breaks conformal invariance and 

allows the introduction of the QCD scale (Hard-Wall Model) Polchinski and Strassler (2001). 

Smooth cutoff: introduction of a background dilaton field ⌅(z) – usual linear Regge dependence can 

be obtained (Soft-Wall Model) Karch, Katz, Son and Stephanov (2006). 

Changes in 

physical 

length scale 

mapped to 

evolution in the 

5th dimension z 


Scale Transformations 

• Isomorphism of SO(4, 2) of conformal QCD with the group of isometries of AdS space 

SO(1, 5) 

AdS/CFT 

ds 2 = R2 

z 2 ( µ⇥dx µ dx ⇥ dz 2 ), 

x µ ⇤ ⇥x µ , z ⇤ ⇥z, maps scale transformations into the holographic coordinate z. 

• AdS mode in z is the extension of the hadron wf into the fifth dimension. 

invariant measure 

• Different values of z correspond to different scales at which the hadron is examined. 

x 2 ⇤ ⇥ 2 x 2 , 

z ⇤ ⇥z. 

x 2 = x µ x µ : invariant separation between quarks 

• The AdS boundary at z ⇤ 0 correspond to the Q ⇤ ⌅, UV zero separation limit. 


54 

altech High Energy Seminar, Feb 6, 2006 Page 11

2 Bosonic Modes 

Bosonic Solutions: Hard Wall Model 

• Conformal metric: ds 2 = g ⌅m dx ⌅ dx m . x ⌅ = (x µ , z), g ⌅m ⇤ R 2 /z 2⇥ ⌅m . 

• Action for massive scalar modes on AdS d+1 : 

S[⇥] = 1 ⌥ 

d d+1 x ⇧ g 1 

2 

2 g⌅m ⌃ ⌅ ⇥⌃ m ⇥ µ 2 ⇥ 2 , 

⇧ g ⇤ (R/z) d+1 . 

• Equation of motion 

1 ⌃ ⇧ 

⇧ g g 

⌅m ⌃ 

g ⌃x ⌅ ⌃x m ⇥⇥ + µ 2 ⇥ = 0. 

• Factor out dependence along x µ -coordinates , ⇥ P (x, z) = e iP ·x ⇥(z), P µ P µ = M 2 : 

⇤ 

z 2 ⌃ 2 z (d 1)z ⌃ z + z 2 M 2 (µR) 2⌅ ⇥(z) = 0. 

• Solution: ⇥(z) ⇤ z as z ⇤ 0, 

• Normalization 

⇥(x, z) = Cz d 2 J d 

2 

(z) = Cz d/2 J 

d/2(zM) 

R d 1 ⌥ ⇥ 

1 

(zM) , = 1 2 

QCD 

dz 

z d 1 ⇥2 S=0(z) = 1. 

⇧d + ⌦ d 2 + 4µ 2 R 2 ⌃ 

= 2 + L d = 4 (µR) 2 = L 2 4 

May 21, 2013 LC2013 AdS/QCD & 0LF-Holography 


55 

.

e '(z) = e +apple2 z 2 

Positive-sign dilaton 

• de Teramond, sjb 

⇥ d 2 

AdS Soft-Wall Schrodinger Equation for 

bound state of two scalar constituents: 

⌅(x, b ⇤ ) = ⌅(⇥) 

1 4L 2 ⇤ 

dz 2 4z 2 + U(z) (z) =M 2 (z) 

⇤(z) 

U(z) = 4 z 2 + 2 2 ⌅(x, (L b+ S 1) 

⇤ ) = ⌅(⇥) 

⇥ = x(1 x)b 2 ⇤ 

Derived from variation of Action for Dilaton-Modified AdS 5 

⇤(z) 

Identical to Light-Front Bound State Equation! 

z 

⇥ = x(1 x)b 2 ⇤ 

z 

56

LF(3+1) AdS 5 

57 

⌅(x, b ⇤ ) = ⌅(⇥) 

⇤(z) 

x (1 x) b ⇥ 

⇤(x, b ⇥ ) = ⇤( ) 

⇥(z) 

⇥ = x(1 x)b 2 ⇤ 

⇤(z) 

⌅(x, b ⇤ ) = ⌅(⇥) 

⇥ = x(1 

⇤(z) 

⇥ = (x(1 x) 

z 

de Teramond, sjb 

z 

z 

z 

z 

z 0 = 1 


= x(1 x)b 2 ⇥ 

x (1 

⇤(x, ) = x(1 x) 

z 0 = 1 

z 

x) b ⇥ 

z 

z 

⇤(x, b ⇥ ) = ⇤( ) 

⇥(z) 

= x(1 x)b 2 ⇥ 

1/2 ⇥( ) 

x (1 

z 0 = 1 

z 0 = 1 

x) b ⇥ 

⇤(x, b ⇥ ) = ⇤( ) 

x (1 x) b ⇥ 

⇥(z) 

⇤(x, b ⇥ ) = ⇤( ) 

⇥(z) 

= x(1 x)b 2 ⇥ 


z 

= x(1 x)b 2 ⇥ 

Light d Front ⇥ np 

Holography: ⇥ Unique mapping derived from equality of LF 

and AdS formula for EM QCD 

and gravitational current matrix 

z 

⇥ 

elements 

and identical equations of motion QCD 

z

G. de Teramond and sjb, PRL 102 081601 (2009) 

Dual QCD Light-Front Wave Equation z ⌃ , P (z) ⌃ |⇧(P ) 

[ GdT and S. J. Brodsky, PRL 102, 081601 (2009)] 

• Upon substitution z ⇧ and ⌅ J ( ) ⌅ 3/2+J e (z)/2 J( ) in A dS W E 

⇤ 

z d 1 2J 

⇥ z 

z d 1 2J z + µR ⇥ ⌅ 2 

e '(z) 

J(z) =M 2 

z 

e (z) 

J(z) 

fi nd L F W E (d =4) 

d 2 1 4L 2 ⇥ 

d 2 4 2 + U( ) ⌅ J ( )=M 2 ⌅ J ( ) 

w ith 

A F e e q e L F Q c 

e A m e m e w e 

U( )= 1 2 ⌃⇥⇥ (z)+ 1 4 ⌃⇥ (z) 2 + 2J 3 ⌃ ⇥ (z) 

2z 

and (µR) 2 = (2 J) 2 + L 2 

• dS Bre ite nl oh ne r- re dm an bound (µR) 2 ⇤ 4 uiv al nt to Mstabil ity ondition L 2 ⇤ 0 

• Sc al ing dim nsion ⇤ of dS ode ˆ J is ⇤ =2+L in ag re nt ith tw ist sc al ing dim nsion of a 

tw o parton bound state in Q C D and de te rm ine d by Q Mstabil ity c ondition 


58 

L C 2011 2011, D al l as, May 23 , 2011 Pag e 10

H. G. Dosch, G. de Teramond, sjb PRD 87 (2013) 


59

General-Spin Hadrons 


• Obtain spin-J mode µ 1···µ J 

with all indices along 3+1 coordinates from by shifting dimensions 

J(z) = 

⇧ z 

R 

⌃ J 

(z) 

e '(z) = e +apple2 z 2 

• Substituting in the AdS scalar wave equation for 

⇤ 

z 2 ⇧ 2 z 3 2J 2⇥ 2 z 2⇥ z ⇧ z + z 2 M 2 (µR) 2⌅ J =0 

• Upon substitution z ⌅ 

⌅ J ( )⇤ 3/2+J e ⇥2 2 /2 

J( ) 

we find the LF wave equation 

⌥ d 

2 

d 2 1 4L 2 

4 2 + ⇥ 4 2 +2⇥ 2 (L + S 1) ⌅ µ1···µ J 

= M 2 ⌅ µ1···µ J 

with (µR) 2 = (2 J) 2 + L 2 


60

⇥ 



= M 2 ⇥( ) 

d 2 



⇥ 

d 2 

d 2 + V ( ) 



= M 2 ⇥( ) 

2 = x(1 x)b 2 ⇥ . 

x (1 

x) b ⇥ 

x (1 x) b ⇥ 

⇤(x, b ⇥ ) = ⇤( ) 

x (1 x) b ⇥ 

⇤(x, b ⇥ ) = ⇤( ) 

⇥(z) 


each Fock State⇤(x, b ⇥ ) = ⇤( ) 

⇥(z) 


= x(1 x)b 2 ⇥ 

⇥(z) 

Jp z = 

= Sq z x(1 

+ Sg z x)b 2 ⇥ 

Semiclassical first approximation to QCD 

+ L z z 

q + L 

potential z g = 1 = x(1 x)b 2 ⇥ 

z 


61 

z 

J z = S z p = ⇤ n 

i=1 

S z i + ⇤ n 1 

i=1 ⌥z i = 1 2 

Confining AdS/QCD

z ! ⇣ 

U(⇣ 2 )=apple 4 ⇣ 2 +2apple 2 (J 1) 


62

May 21, 2013 LC2013 Stan Brodsky 

AdS/QCD & LF-Holography 

63

Quark separation 

increases with L 

6 

4 

5 

Φ(z) 

Φ(z) 

0 

2 

-5 

2-2007 

8721A20 

0 

0 4 8 

z 

2-2007 

8721A21 

0 4 8 

z 

Fig: Orbital and radial AdS modes in the soft wall model for = 0.6 GeV . 

(a) 

S = 0 (b) S = 0 

Soft Wall 

Model 

Pion has 

zero mass! 

(GeV 2 ) 

4 

2 

π (140) 

b 1 

(1235) 

π 2 

(1670) 

π (140) 

π (1300) π (1800) 

Pion mass 

automatically zero! 

m q = 0 

0 

8-2007 

8694A19 

0 2 4 

L 

0 2 4 

n 

Light meson orbital (a) and radial (b) spectrum for 

= 0.6 GeV. 


64 

Exploring QCD, Cambridge, August 20-24, 2007 Page 26

= apple 2 

J=2 

J=1 

J=0 


65

Prediction from AdS/CFT: Meson LFWF 

(x, k ) 

⇥ M (x, Q 0 ) ⇥ 0.8 x(1 0.60.40.2 x) 

0.2 

de Teramond, 

sjb 

⇤ M (x, k 2 ⇤ ) 

0.15 

0.1 

0.05 

“Soft Wall” 

model 

µ R 

Note coupling 

µ R = Q 

k 2 , x 

0 

1.3 

1.4 

(x, k ) (GeV) 

1.5 

= 0.375 GeV 

massless quarks 

µ F = µ R 

⇤ M (x, k ⇥ ) = 

4⇥ 

x(1 x) e k 2 ⇥ 

2 2 x(1 x) 

⇥ M (x, Q 0 ) ⇥ x(1 x) 

Provides Q/2 < Connection µ R < 2Q of Confinement 

µ 

to TMDs 

R 


66

0.6 

Q 2 F Π Q 2 

0.5 

0.4 

0.3 

0.2 

0.1 

Q 2 GeV 2 

0.0 

0 1 2 3 4 5 6 7 

67

J. R. Forshaw, 

R. Sandapen 

L 

T 

⇤ p ! ⇢ 0 p 0 

69





⇥ d 2 

d⇣ 2 + 1 

4L2 

4⇣ 2 + U(⇣) ⇤ (⇣) =M 2 (⇣) 






Unique 

Confinement Potential! 


of the action 


70

G| (⌧) >= i @ 

@⌧ | (⌧) > 

G = uH + vD + wK 

• 

de Alfaro, Fubini, Furlan 

G = H ⌧ = 1 2 

d 2 

dx 2 + g x 2 + 4uw 

4 

v2 

x 2 

Retains conformal invariance of action despite mass scale! 

4uw 

Identical to LF Hamiltonian with unique potential and dilaton! 

⇥ d 2 

d⇣ 2 + 1 

v 2 = apple 4 =[M] 4 • 

4L2 

4⇣ 2 + U(⇣) ⇤ (⇣) =M 2 (⇣) 

Dosch, de Teramond, sjb 



71

Uniqueness 


• ζ2 confinement potential and dilaton profile unique! 

• 

Linear Regge trajectories in n and L: same slope! 

• 

Massless pion in chiral limit! No vacuum condensate! 

• 

Derive from conformal invariance: conformally invariant action 

for massless quarks despite mass scale 

• Same principle, equation of motion as de Alfaro, Fubini, 

Furlan . 

• Conformal Invariance in Quantum Mechanics Nuovo Cim. 

A34 (1976) 569 

72

What determines the QCD mass scale QCD 

• Mass scale does not appear in the QCD Lagrangian 

(massless quarks) 

• Dimensional Transmutation Requires external constraint 

such as ↵ s (M Z ) 

• dAFF: Confinement Scale κ appears spontaneously via the 

Hamiltonian: 

G = uH + vD + wK 4uw v 2 = apple 4 =[M] 4 

• The confinement scale regulates infrared divergences, 

connects QCD to the confinement scale κ 

• Only dimensionless mass ratios (and M times R ) predicted 

• Mass and time units [GeV] and [sec] from physics external 

to QCD 

• New feature: bounded frame-independent relative time 

between constituents 


Quigg and Rosner (1979: 

Excitation energies of quarkonia appear to be flavor-independent 

E GeV 

1.5 

1 

0.5 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b¯b 

c¯c 

 

 

 

 

logarithmic 

potential 

0 

 

 

0 1 0 1 2 2 


74

Heavy-Quark Systems and Conformal Invariance 

H.G. Dosch, G. de Teramond, sjb 

q ! p mr, ˙q ! 1 p m 

i d dr 

[q(t), ˙q(t)] = i 


75

H.G. Dosch, G. de Teramond, sjb (in progress) 

! = ⇤ =0.28 GeV 


76

for the excitation energies, 

state 

that is the mass di↵erence of 

onic oscillator in nonrelativistic quantum mechanics 

and the J✓ PC =1 

1 

FIG. 1: Excitation energies 1 @ of c¯c (red) and b¯b (blue) mesons in GeV: M(n, J, P, C) M(0, 1, , ). 

H = 

@ ground state ◆are plotted 

l(l +1) 

For c¯c and b¯b 2m only states r 2 with @r r2 well defined @r quantumr 2 + 1 for all quarkon 

numbers below open charm and beauty numbers below2 m thresholds. 

⇤2 r 2 , 

open charm and open b 

threshold are shown. The data is from Ref. [6]. 

ntified gFrom withthe theHamiltonian Casimir operator (14) weof obtain the rotation the spectrum: group, g 

n 3 

✓ ◆ 

2 

1.5 

n 2 

)/4 with 

 

eigenvalues 

1 

given by H ` = E` `. The spectrum o 

 

E n` = 2 n + ` + 3 ⇤, 

b¯b 

 

f H U 2 

 

. 

1 

n 1 

 

0.5 

 

and the wave functions: 

 

⇤ =0.28 GeV 

t feature 

 

c¯c 

 

0.5 

of H is the 

 

independence of the spectrum from the m 

 

E GeV 

FIG. 2: 

E GeV 

0 

H.G. Dosch, G. de Teramond, sjb (in progress) 

seen directly from (14) by substituting n` = N p n` r` mr L`+1/2 

= y. This m⇤r n is 2 inde e m 

0 

 

 

n 0 

0 1 2 

0 1 2 

L 

Flavor independent 

Universal confinement time! 

egree, as can be seen from Fig. 1. In this figure4the experime 

L 

Excitation energies of c¯c (red boxes) and b¯b (blue diamonds) with di↵erent values of 

energies, that is the mass di↵erence of the radial and orbital 

angular momentum `. Only well established states below open flavour threshold are shown. The 

linear trajectories are the theoretical predictions from the Hamiltonian (14) with⇤ =0.28 GeV. 

ground state are plotted for all quarkonia with well establishe 


where the quantum numbers ` and n represent77 

the orbital angular and the radial excitation

Propagation of external perturbation suppressed inside AdS. 

At large enough Q ⇤ r/RX 2 , = thecū J(Q, interaction ¯dū z) = occurs zQK 1 in(zQ) 

the large-r conformal region. Importa 

contribution to the FF integral from the boundary near z ⇤ 1/Q. 

F (Q 2 ) I⇤F s(Q 2 = ) dz 

z 3 F (z)J(Q, z) I (z) 

J(Q, z), (z) 

⇥(Q 2 1 J(Q, ) = z) d s(Q 2 ) 

High Q 2 

from 

small z ~ 1/Q 

high Q 2 

0.8 

0.6 

0.4 

0.2 

1 2 3 4 5 

Consider a specific AdS mode ⇥ (n) dual to an n partonic Fock state |n⇧. At small z, ⇥ ( 

scales as ⇥ (n) ⇤ z n . Thus: 

Hadron z Form Factors from AdS/QCD 

i ⌅ m ⇧i = m 2 i + k2 ⇧ 


78 

⇥ 

(Q 2 ) 

15⇤ 

Q 2

Holographic Mapping of AdS Modes to QCD LFWFs 

• Integrate Soper formula over angles: 

F (q 2 ) = 2⇥ 

⇧ 1 

with ⌃⇤(x, ) QCD effective transverse charge density. 

• Transversality variable 

0 

Drell-Yan-West: Form Factors are 

Convolution of LFWFs 

⇧ ⇥ ⌥ ⇤ 

(1 x) 

1 x 

dx d J 0 q ˜⇤(x, ), 

x 

x 

⌥ n x ⌅1 

⇥ = = x(1 x)b x 2 j b ⇥j . 

1 x ⇤ 

• Compare AdS and QCD expressions of FFs for arbitrary Q 

z 

using identity: 

⇧ ⇥ 

1 

⌥ ⇤ 

1 x 

dxJ 0 Q = QK 1 ( Q), 

x 

the solution for J(Q, ) = QK 1 ( Q) ! 

0 

⌅(x, b ⇤ ) = ⌅(⇥) 

⇤(z) 

z 

z 0 = 1 


Identical to Polchinski-Strassler Convolution of AdS Amplitudes 

j=1 


79

Current Matrix Elements in AdS Space (SW) 

sjb and GdT 

Grigoryan and Radyushkin 

• Propagation of external current inside AdS space described by the AdS wave equation 

⇤ 

z 2 ⇧ 2 z z 1 + 2 2 z 2⇥ ⇧ z Q 2 z 2⌅ J (Q, z) = 0. 

• Solution bulk-to-boundary propagator 

J (Q, z) = 

⌃ ⇧1 + Q2 

4 2 U 

where U(a, b, c) is the confluent hypergeometric function 

(a)U(a, b, z) = 

⌥ ⇥ 

0 

⇧ Q 

2 

4 2 , 0, 2 z 2 ⌃ 

, 

e zt t a 1 (1 + t) b a 1 dt. 

Dressed 

Current 

in Soft-Wall 

Model 

• Form factor in presence of the dilaton background ⇥ = 2 z 2 

• For large Q 2 ⇤ 4 2 

F (Q 2 ) = R 3 ⌥ dz 

z 3 e 

2 z 2 ⇥(z)J (Q, z)⇥(z). 

J (Q, z) ⌅ zQK 1 (zQ) = J(Q, z), 

the external current decouples from the dilaton field. 


80 


0.6 

Q 2 F Π Q 2 

0.5 

0.4 

0.3 

0.2 

0.1 

Q 2 GeV 2 

0.0 

0 1 2 3 4 5 6 7 

81

Fermionic Modes and Baryon Spectrum 

[GdT and GdT S. J. and Brodsky, sjb, PRL PRL 94, 94, 201601 201601 (2005) (2005)] 

Yukawa interaction 

in 5 dimensions 

From Nick Evans 

• Action for Dirac field in AdS d+1 in presence of dilaton background ⇧(z) [Abidin and Carlson (2009)] 

⇧ 

S = d d+1⌃ ge ⌅ (z) i⌅e M A A D M ⌅ + h.c + ⇧(z)⌅⌅ µ⌅⌅ ⇥ 

• Factor out plane waves along 3+1: 

⌃ 

i 

e '(z) 

• Solution (⌅ = µR 1 2 , ⌅ = L + 1) 

⌅ P (x µ ,z)=e iP ·x ⌅(z) 

⌅ 

⌥ 

⇤z ⌦m ⌦ m +2 z + µR + ⇥ 2 z ⌅(x ⌦ )=0. 

⌅ + (z) ⇤ z 5 2 +⇤ e 

2 z 2 /2 L ⇤ n(⇥ 2 z 2 ), ⌅ (z) ⇤ z 7 2 +⇤ e 

2 z 2 /2 L ⇤+1 

n (⇥ 2 z 2 ) 

• Eigenvalues (how to fix the overall energy scale, see arXiv:1001.5193) 

M 2 =4⇥ 2 (n + L + 1) 

positive parity 

• Obtain spin-J mode ⇤ µ1···µ J 

1/2 

, J> 1 2 

, with all indices along 3+1 from ⌅ by shifting dimensions 

• Large N C : M 2 =4⇥ 2 (N C + n + L 2) =⌅ M ⇤ ⌃ N C ⇥ QCD 

82



83

Fermionic Modes and Baryon Spectrum 

[Hard wall model: GdT and S. J. Brodsky, PRL 94, 201601 (2005)] 

[Soft wall model: GdT and S. J. Brodsky, (2005), arXiv:1001.5193] 

From Nick Evans 

• Nucleon LF modes 

⇤ + ( ) n,L = ⇥ 2+L ⌅ 

2n! 

(n + L)! 

⌅ 

⇤ ( ) n,L = ⇥ 3+L 1 

⇤ 

n + L +2 

3/2+L e ⇥2 2 /2 L L+1 

n ⇥ 2 2⇥ 

2n! 

(n + L)! 

5/2+L e ⇥2 2 /2 L L+2 

n ⇥ 2 2⇥ 

• Normalization 

⇤ 

d ⇤ 2 +( )= 

⇤ 

d ⇤ 2 ( )=1 

• Eigenvalues 

• “Chiral partners” 

M 2 n,L,S=1/2 =4⇥2 (n + L + 1) 

M N(1535) 

M N(940) 

= ⇤ 2 

LC 2011 2011, Dallas, May 23, 2011 Page 13 

84

Table 1: SU(6) classification of confirmed baryons listed by the PDG. The labels S, L 

and n refer to the internal spin, orbital angular momentum and radial quantum number 

5 

respectively. The 

2 

(1930) does not fit the SU(6) classification since its mass is too low 

compared to other members 70-multiplet for n = 0, L = 3. 

SU(6) S L n Baryon State 

1 

56 

2 

0 0 N 1 + 

2 (940) 

1 

2 

0 1 N 1 + 

2 (1440) 

1 

2 

0 2 N 1 + 

2 (1710) 

3 

3 + 

2 

0 0 

2 (1232) 

3 

3 + 

2 

0 1 

2 (1600) 

70 

1 

2 

1 0 N 1 2 (1535) N 3 2 (1520) 

3 

2 

1 0 N 1 2 (1650) N 3 2 (1700) N 5 2 (1675) 

3 

2 

1 1 N 1 2 

N 3 2 (1875) N 5 2 

1 

1 

2 

1 0 

2 (1620) 3 

2 (1700) 

1 

56 

2 

2 0 N 3 + 

2 (1720) N 

5 + 

2 (1680) 

1 

2 

2 1 N 3 + 

2 (1900) N 

5 + 

2 

3 

1 + 

2 

2 0 

2 (1910) 

3 + 

2 (1920) 

5 + 

2 (1905) 

7 + 

2 (1950) 

70 

1 

2 

3 0 N 5 2 

N 7 2 

3 

2 

3 0 N 3 2 

N 5 2 

N 7 2 (2190) N 9 2 (2250) 

1 

5 

7 

2 

3 0 

2 

2 

1 

56 

2 

4 0 N 7 2 

3 

5 

2 

4 0 

2 

+ 7 

2 

+ 

+ 9 

2 

1 

70 

2 

5 0 N 9 2 

N 11 

2 

N 9 2 

+ (2220) 

+ 11 + 

2 (2420) 

3 

2 

5 0 N 7 2 

N 9 2 

N 11 

2 

(2600) N 13 

2 

PDG 2012 


85



86

Identify L with ν 


87

Space-Like Dirac Proton Form Factor 

• Consider the spin non-flip form factors 

⇤ 

F + (Q 2 ) = g + 

⇤ 

F (Q 2 ) = g 

d J(Q, )|⇥ + ( )| 2 , 

d J(Q, )|⇥ ( )| 2 , 

where the effective charges g + and g 

are determined from the spin-flavor structure of the theory. 

• Choose the struck quark to have S z = +1/2. The two AdS solutions ⇥ + ( ) and ⇥ ( ) correspond 

to nucleons with J z = +1/2 and 1/2. 

• For SU(6) spin-flavor symmetry 

F p 1 (Q2 ) = 

F n 1 (Q 2 ) = 

⇤ 

d J(Q, )|⇥ + ( )| 2 , 

⇤ 

1 

d J(Q, ) |⇥ + ( )| 2 |⇥ ( )| 2⇥ , 

3 

where F p 1 (0) = 1, F n 1 

(0) = 0. 


88 


• Compute Dirac proton form factor using SU(6) flavor symmetry 

⇧ 

F p dz 

1 (Q2 )=R 4 V (Q, z) 

2 

z4 +(z) 

• Nucleon AdS wave function 

+(z) = 

2+L 

R 2 

⌃ 

2n! 

(n + L)! z7/2+L L L+1 

n 

2 z 2⇥ e 

2 z 2 /2 

• Normalization (F 1 p (0) = 1, V(Q =0,z) = 1) 

R 4 ⇧ dz 

z 4 2 

+ (z) =1 

• Bulk-to-boundary propagator [Grigoryan and Radyushkin (2007)] 

V (Q, z) = 2 z 2 ⇧ 1 

0 

dx 

(1 x) 2 x Q2 

4apple 2 e 

2 z 2 x/(1 x) 

• Find 

F p 1 (Q2 )= 

⇤ 

1+ Q2 

M 2 ⇢ 

1 

⌅⇤ ⌅ 

1+ Q2 

M 2 ⇢ 0 

with M ⇥ 

2 

n ⇤ 4 2 (n +1/2) 


89 

LC 2011 2011, Dallas, May 23, 2011 Page 20

Using SU(6) flavor symmetry and normalization to static quantities 

1.2 

0 

Q 4 F p 1 (Q 2 ) (GeV 4 ) 

0.8 

0.4 

Q 4 F n 1 (Q 2 ) (GeV 4 ) 

-0.2 

0 

0 

10 20 30 

0 

10 20 30 

2-2012 

8820A18 

Q 2 (GeV 2 ) 

2-2012 

8820A17 

Q 2 (GeV 2 ) 

2 

0 

F np 2 (Q2 ) 

1 

F n 2 (Q2 ) 

-1 

0 

2-2012 

8820A8 

0 2 4 6 

Q 2 (GeV 2 ) 

-2 

2-2012 

8820A7 

0 2 4 6 

Q 2 (GeV 2 ) 


Niccolò Cabeo 2012, Ferrara, May 25, 2011 

90 

Page 42

Nucleon Transition Form Factors 

• Compute spin non-flip EM transition N(940) ⇥ N (1440): 

n=0,L=0 

+ ⇥ n=1,L=0 

+ 

• Transition form factor 

F 1 

p 

N⇥N (Q2 )=R 4 ⇧ dz 

z 4 

n=1,L=0 

+ (z)V (Q, z) n=0,L=0 

+ (z) 

• Orthonormality of Laguerre functions 

R 4 ⇧ dz 

F 1 

p 

N⇥N 

n ⇥ ,L 

z 4 + 

(0) = 0, 

(z) 

n,L 

+ (z) = n,n ⇥ 

V(Q =0,z)=1⇥ 

• Find 

F 1 

p 

N⇥N (Q2 )= 2⌅ 2 

3 

Q 2 

M 2 P 

⇤ 

1+ Q2 

M 2 ⌅⇤ 

1+ Q2 

M 2 ⇥ 

⌅⇤ 

1+ Q2 

M 2 ⇥⇥ 

⌅ 

with M 2 n ⇥ 4⇥2 (n +1/2) 

Consistent with counting rule, twist 3 



91

Nucleon Transition Form Factors 

F p 1 N!N ⇤ (Q 2 )= 

p 

2 

3 

Q 2 

M 2 ⇢ 

⇣ ⌘⇣ ⌘⇣ ⌘. 

1+ Q2 

1+ Q2 

1+ Q2 

M 2 ⇢ M 2 ⇢ 0 M 2 ⇢ 00 

F p 1N N* (Q 2 ) 

0.1 

2-2012 

8820A16 

0 

0 

2 4 

Q 2 (GeV 2 ) 

Proton transition form factor to the first radial excited state. Data from JLab 

May Niccolò 21, Cabeo 2013 2012, Ferrara, LC2013 May 25, 2011 AdS/QCD & LF-Holography Stan Brodsky 

92 

Page 43

Dressed soft-wall current brings in higher 

Fock states and more vector meson poles 

+ 

e + 

e 


93

Higher Fock Components in LF Holographic QCD 

• Effective interaction leads to qq ! qq, qq ! qq but also to q ! qqq and q ! qqq 

• Higher Fock states can have any number of extra qq pairs, but surprisingly no dynamical gluons 

• Example of relevance of higher Fock states and the absence of dynamical gluons at the hadronic scale 

|⇡ i = qq/ ⇡ |qqi ⌧ =2 + qqqq|qqqqi ⌧ =4 + ··· 

• Modify form factor formula introducing finite width: q 2 ! q 2 + p 2iM (P qqqq = 13 %) 

0.6 

2 

Q 2 F π (Q 2 ) (GeV 2 ) 

0.4 

0.2 

Log I Fπ (q 2 )I 

0 

-2 

2-2012 

8820A21 

0 

0 

2 4 6 

Q 2 (GeV 2 ) 

2-2012 

8820A22 

0 

2 4 

q 2 (GeV 2 ) 

Niccolò Cabeo 2012, Ferrara, May 25, 2011 


94 

Page 47

Timelike Pion Form Factor from AdS/QCD 

and Light-Front Holography 

log F Π q 2 

2 

F ⇡ (s) =(1 ) 

1 

(1 

s 

M 2 ⇢ 

) + 1 

s 

s 

(1 )(1 

M 2 ⇢ 

M 2 ⇢ 0 )(1 

s 

M 2 ⇢ 00 ) 

log |F ⇡ (s)| 

1 

M 2 ⇢ n 

=4apple 2 (1/2+n) 

=0.17 

0 

Twist 2 

Twist 2+4 

Prescription for 

Timelike poles : 

s 

1 

M 2 + i p s 

1 

Frascati data 

14% four-quark 

probability 

2 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

s(GeV 2 ) 

G. de Teramond & sjb 

95

2 

Pion Form Factor from AdS/QCD and Light-Front Holography 

1 

log |F ⇡ (s)| 

G deTeramond, sjb 

Preliminary 

0 

spacelike 

timelike 

1 

Frascati 

2 

3 

JLab 

BaBar ISR 

4 

P twist 2 =91%,P twist 4 =3%,P twist 5 =6% 

apple determined by the ⇢ mass, PDG widths. ⇢ 000 = ⇢ 00. 

10 5 0 5 

q 2 (GeV 2 ) 

96

log |F ⇡ (s)| 

q 2 (GeV 2 ) 

Timelike 

Spacelike 

|q 2 F ⇡ (q 2 )| ! (1 )m 2 ⇢ 

97

Meson Transition Form-Factors 

[S. J. Brodsky, Fu-Guang Cao and GdT, arXiv:1005.39XX] 

• Pion TFF from 5-dim Chern-Simons structure [Hill and Zachos (2005), Grigoryan and Radyushkin (2008)] 

⇤ ⇤ 

d 4 x dz ⇥ LMNP Q A L M A N P A Q 

⇤ (2⌅) 4 (4) (p ⇧ + q 

k) F ⇧ (q 2 )⇥ µ⌅⌃⌥ ⇥ µ (q)(p ⇧ ) ⌅ ⇥ ⌃ (k)q ⌥ 

• Take A z ⇧ ⇧ (z)/z, ⇧(z) = ⌃ 2P qq ⇤ z 2 e ⇥2 z 2 /2 , ⌥ ⇧ | ⇧ = P qq 

• Find ⇧(x) = ⌦ 3f ⇧ x(1 x), f ⇧ = ⌃ P qq ⇤/ ⌦ 2⌅ ⇥ 

Q 2 F ⇧ (Q 2 )= 4 ⌦ 

3 

⇤ 1 

0 

dx ⇧(x) 

1 x 

⌅1 e P qqQ 2 (1 x)/4⇧ 2 f 2 x ⇧ 

normalized to the asymptotic DA [P qq =1⌅ Musatov and Radyushkin (1997)] 

• Large Q 2 TFF is identical to first principles asymptotic QCD result 

Q 2 F ⇧ (Q 2 ⌅⌃)=2f ⇧ 

• The CS form is local in AdS space and projects out only the asymptotic form of the pion DA 


98

Q 2 F π γ (Q 2 )(GeV) 

0.30 

0.25 

0.20 

0.15 

0.10 

0.05 

0.00 

0 (1 + x) 3 

Photon-to-pion transition form factor 

nd term in Eq. (20) vanishes at the limit Q 2 ⇥⇤, one recovers 

ptotic prediction for Lepage, the pion sjbTFF: 

Q 2 F ⇤ (Q 2 ⇥⇤)=2f ⇤ .[11] 

ted with (19) forP q¯q =0.5 are shown as dashed curves in Figs. 1 

t the calculations with the dressed current are larger as compared 

uted with the free current and the experimental data at low- and 

2 < 10 GeV 2 ). The new results again disagree with BABAR’s data 

11 

F.-G. Cao, 

G. de Teramond, 

sjb 

Belle 

BaBar 

CLEO 

CELLO 

Free current; Twist 2 

Dressed current; Twist 2 

Dressed current; Twist 2+4 

0 10 20 30 40 

Q 2 (GeV 2 ) 

99

Non-Perturbative Running QCD Coupling from FromModified LF Holography AdS/QCD 

With A. Deur and S. J. Brodsky 

Deur, de Teramond, sjb 

• Consider five-dim gauge fields propagating in AdS 5 space in dilaton background ⇧(z) =⇤ 2 z 2 

S = 1 4 

d 4 xdz ⇧ ge ⇥(z) 1 g 2 5 

G 2 

• Flow equation 

1 

g5 2(z) = 1 

e⇥(z) 

g5 2(0) or g5(z) 2 =e 

2 z 2 g 2 5(0) 

where the coupling g 5 (z) incorporates the non-conformal dynamics of confinement 

• YM coupling 

s (⇥) =g 2 YM (⇥)/4⌅ is the five dim coupling up to a factor: g 5(z) ⌅ g YM (⇥) 

• Coupling measured at momentum scale Q 

AdS 

s (Q) ⇤ 

0 

⇥ 

⇥d⇥J 0 (⇥Q) 


s (⇥) 

• Solution 


s (Q 2 )= s AdS (0) e Q2 /4 2 . 

where the coupling 


s 

incorporates the non-conformal dynamics of confinement 

100

Deur, Korsch, et al. 

! s 

/" 

1 

! s,g1 

/" JLab 

Fit 

GDH limit 

pQCD evol. eq. 

Burkert-Ioffe 

Bloch et al. 

10 -1 

Cornwall 

Godfrey-Isgur 

Bhagwat et al. 

Maris-Tandy 

Lattice QCD 

1 

Fischer et al. 

10 -1 

DSE gluon 

couplings 

10 -1 1 10 -1 1 

Q (GeV) 

Fig: Infrared conformal window ( from Deur et al., arXiv:0803.4119 ) 


101

Running Coupling from Light-Front Holography and AdS/QCD 

Analytic, defined at all scales, IR Fixed Point 

s 

(Q)/ 

1 

s(Q) 

⇥ 

0.8 

0.6 

0.4 

0.2 

0 

Modified AdS 


g1 

/ (pQCD) 

g1 

/ world data 

GDH limit F3 

/ 

 

/ OPAL 

g1 

/ JLab CLAS 

g1 

/ Hall A/CLAS 

Lattice QCD (2004) (2007) 


s (Q)/⇥ = e Q2 /4 2 

apple =0.54 GeV 

10 -1 1 10 

Q (GeV) 

AdS/QCD dilaton captures the higher twist corrections to effective charges for Q < 1 GeV 

Sublimated gluons below 1 GeV 

e ' = e +apple2 z 2 

Deur, de Teramond, sjb 

102

Chiral Features of Soft-Wall 

AdS/QCD Model 

• Boost Invariant 

• Trivial LF vacuum! No condensate, but consistent with GMOR 

• Massless Pion 

• Hadron Eigenstates have LF Fock components of different L z 

• Proton: equal probability 

S z =+1/2,L z = 0; S z = 1/2,L z =+1 

J z =+1/2 :=1/2, 

• Self-Dual Massive Eigenstates: Proton is its own chiral partner. 

• Label State by minimum L as in Atomic Physics 

• Minimum L dominates at short distances 

• AdS/QCD Dictionary: Match to Interpolating Operator Twist at z=0. 

103

Gell-Mann Oakes Renner Formula in QCD 

m 2 ⇡ = (m u + m d ) 

f 2 ⇡ 

< 0|¯qq|0 > 

current algebra: 

effective pion field 

m 2 ⇡ = (m u + m d ) 

f ⇡ 

< 0|i¯q 5 q|⇡ > 

QCD: composite pion 

Bethe-Salpeter Eq. 

vacuum condensate actually is normal pion decay matrix element 

⇡ 

< 0|¯q 5 q|⇡ > 

Maris, Roberts, Tandy 

104

AdS/QCD and Light-Front Holography 

• AdS/QCD: Incorporates scale transformations 

characteristic of QCD with a single scale -- RGE 

• Light-Front Holography; unique connection of 

AdS5 to Front-Form 

• Profound connection between gravity in 5th 

dimension and physical 3+1 space time at fixed LF 

time τ 

• Gives unique interpretation of z in AdS to 

physical variable ζ in 3+1 space-time 


illustrated in Fig. 2 in terms of the block matrix n : x , k , λ Hn : x, k , λ . The structure of th 

 

matrix depends of course on the way one has arranged the Fock space, see Eq. (3.7). Note that mo 

Light-Front of the block QCD matrix elements 

H QCD vanish 

LC | due to the 

h⇧ = M 2 nature 

h | of the light-cone interaction as defined i 

h⇧ 

Heisenberg Equation 

S.J. Brodsky et al. / Physics Reports 301 (1998) 299—486 

BLFQ: Use AdS/QCD basis functions! 

Fig. 6. A few selected matrix elements of the QCD front form Hamiltonian H"P 

in LB-convention. 

Fig. 2. The Hamiltonian matrix for a SU(N)-meson. The matrix elements are represented by energy diagrams. With 

each block they are all of the same type: either vertex, fork or seagull diagrams. Zero matrices are denoted by a dot ( 

The single gluon is absent since it cannot be color neutral. 


106

Basis Light-Front Quantization Approach 

to Quantum Field Theory 

Use AdS/QCD orthonormal Light Front Wavefunctions 

as a basis for diagonalizing the QCD LF Hamiltonian 

• Good initial approximation 

BLFQ 

• Better than plane wave basis 

• DLCQ discretization -- highly successful 1+1 

• Use independent HO LFWFs, remove CM motion 

• Similar to Shell Model calculations 

• Xingbo Zhao 

• Anton Ilderton, 

• Heli Honkanen 

• Pieter Maris, 

• James Vary 

• Stan Brodsky 

• Hamiltonian light-front field theory within an AdS/QCD basis. 


Set of transverse 2D HO modes for n =1 

J.P. Vary, H. Honkanen, Jun Li, P. Maris, S.J. Brodsky, A. Harindranath, G.F. de Teramond, P. Sternberg, E.G. Ng, C. Yang, PRC 

m=0 m=1 m=2 

0.4 

0.2 

0 

−0.2 

0.2 

0 

−0.2 

0.2 

0 

−0.2 

−0.4 

5 

0 

−5 

−5 

0 

5 

−0.4 

5 

0 

−5 

−5 

0 

5 

−0.4 

5 

m=3 m=4 

0 

−5 

−5 

0 

5 

0.2 

0.2 

0 

0 

−0.2 

−0.2 

−0.4 

5 

0 

−5 

−5 

0 

5 

−0.4 

5 

0 

−5 

−5 

0 

5 

108

Hadronization 1 x, ↵ k⇥ at the Amplitude Level 

e + 

e 

⇥e + 

Event amplitude 

⇥ 

generator 

⇥ = x + 

e + 

e 

e 

g 

¯q q 

¯q 

g 

q 

e + 

e 

g 

q 

⇥ 

¯q 

pp ⇤ p + J/⇥ + p 

e 

¯q e 

⇥ 

pp ⇤ p + J/⇥ + p 

May 21, 2013 LC2013 q 

AdS/QCD & LF-Holography Stan Brodsky 

109 

e 

e + 

⇥ 

e + 

⇤ H (x, ↵ k ⇤ , ⇥ i ) 

⇥ 

g 

⇤ H (x, 

pp ⇤g 

p + J/⇥ ↵ k ⇤ , ⇥ i ) 

+ p 

¯q 

e 

g 

q 

⇥ 

¯q 

q 

pp ⇤ p + J/⇥ + p 

e 

pp ⇤ep + J/⇥ + p 

⇤(x, ↵ k ⇤ , ⇥ i ) 

Construct helicity amplitude using Light-Front Perturbation 

theory; coalesce quarks via LFWFs e + 

p H 

x, ↵ k ⇤ 

p H 

x, ↵ k ⇤ 

1 x, ↵ k⇤ 

1 x, 

e + 

↵ k⇤ 

e + 

⇥ 

e 

⇤ H (x, ↵ k ⇤ , ⇥ i ) 

p H 

x, ↵ k ⇤ 

1 x, ↵ k⇤ 

e +

Off -Shell T-Matrix 

• Quarks and Gluons Off-Shell 

• LFPth: Minimal Time-Ordering Diagrams-Only positive e + 

k+ 

• Jz Conservation at every vertex 

• Frame-Independent 

• Cluster Decomposition 

• “History”-Numerator structure universal 

• Renormalization- alternate denominators 

• LFWF takes Off-shell to On-shell 

• Tested in QED: g-2 to three loops 

Event amplitude generator 

Chueng Ji, sjb 


e + 

e 

e ⇥ 

+ 

e 

g 

¯q 

g 

q 

⇥ 

¯q 

pp ⇤ p + J/⇥ + p 

q 

e + 

e 

g 

¯q 

q 

⇥ 

pp ⇤ p + J/⇥ + p 

e 

g 

¯q 

q 

⇥ 

e + 

e 

g 

¯q 

q 

⇥ 

pp ⇤ p + J/⇥ + p 

e 

pp ⇤ep + J/⇥ + p 

⇥ 

pp ⇤ p + J/⇥ + p 

⇤ H (x, ↵ k ⇤ , ⇥ i ) 

p H 

⇤ H (x, ↵ k ⇤ , ⇥ i ) 

p H 

x, ↵ k ⇤ 

x, ↵ k ⇤ 

1 x, ↵ k⇤ 

1 x, 

e + 

↵ k⇤ 

e + 

⇤(x, ↵ k ⇤ , ⇥ i ) 

e + 

Roskies, Suaya, sjb 

⇥ 

e 

g 

⇥ 

⇤ H (x 

p H 

x, ↵ k ⇤ 

1 x 

e + 

e 

⇥

Features of LF T-Matrix Formalism 

“Event Amplitude Generator” 

• Same principle as antihydrogen production: off-shell coalescence 

• coalescence to hadron favored at equal rapidity, small transverse 

momenta 

• leading heavy hadron production: D and B mesons produced at 

large z 

• hadron helicity conservation if hadron LFWF has L z =0 

• Baryon AdS/QCD LFWF has aligned and anti-aligned quark spin 

P + , ↵ P + 

A(⇤, ⇤) = 1 

2⇥ 

P + , apple P ⇤ 

d e i 2 ⇤ 

x i P + , x iP⇤ ↵ + ↵ k ⇤i 

ẑ 


↵L = R ↵ ⇥ P ↵ x i = k+ 

x 

P + i P + , x i 

= P 0 P⇤ apple + 

+ P z apple k ⇤i 

= Q2 

2p·q 

P + = k0 +k 3 

P 0 +P z

p 


e + 

e 

g 

¯q 

q 

⇥ = x + 

e + 

e 

⇥ 

⇤(x, e ↵ k ⇤ , ⇥ i ) 

e + 

pp ⇤ p + J/⇥ + pp p ⇤ p + J/⇥ + p 

⇥ 

e + 

e 

⇤ H (x, ↵ k ⇤ , ⇥ i ) 

p H 

x, ↵ k ⇤ 

1 x, ↵ k⇤ 

e + 

⇥ 

e 

⇤ H (x, ↵ k ⇤ , ⇥ i ) 

⇥ 

p H 

g 

x, ↵ k ⇤ 

¯q 

1 x, ↵ k⇤ 

e + 

e 

q 

e + 

e 

g 

¯q 

q 

⇥ 

yp + ⇡ ,yp ⇡ + ` 

pp ⇤ p + J/⇥ + p 

Construct helicity amplitude using Light-Front Perturbation theory; 

coalesce quarks via LFWFs 

e + 

e 

g 

¯q 

q 

⇥ 

pp ⇤ p + J/⇥ + 

p ⇡ 

⇡(y, `, i) 

112

p 


e + 

e 

g 

¯q 

q 

⇥ = x + 

e + 

e 

⇥ 

⇤(x, e ↵ k ⇤ , ⇥ i ) 

e + 

pp ⇤ p + J/⇥ + pp p ⇤ p + J/⇥ + p 

⇥ 

e + 

e 

⇤ H (x, ↵ k ⇤ , ⇥ i ) 

p H 

x, ↵ k ⇤ 

1 x, ↵ k⇤ 

e + 

⇥ 

e 

⇤ H (x, ↵ k ⇤ , ⇥ i ) 

⇥ 

p H 

g 

x, ↵ k ⇤ 

¯q 

1 x, ↵ k⇤ 

e + 

e 

q 

e + 

e 

g 

¯q 

q 

⇥ 

yp + ⇡ ,yp ⇡ + ` 

pp ⇤ p + J/⇥ + p 

Construct helicity amplitude using Light-Front Perturbation theory; 

coalesce quarks via LFWFs 

Only Hadrons can Appear! 

e + 

e 

⇥ 

No gluons 

g 


potential 

¯q 

q 

pp ⇤ p + J/⇥ + 

p ⇡ 

⇡(y, `, i) 

113

Principle of Maximum Conformality 

QCD Observables 

O = C(↵ s (µ 2 0)) + B( 

log Q2 

µ 2 0 

)+D( m2 q 

Q 2 )+E(⇤2 QCD 

Q 2 

)+F ( ⇤2 QCD 

m 2 Q 

)+G( m2 q 

m 2 Q 

) 

Scale-Free 

Conformal Series 

Running Coupling 

Effects 

Higher Twist from 

Hadron Dynamics 

Intrinsic Heavy 

Quarks 

BLM/PMC: Absorb β-terms into running coupling 

Light by Light 

Loops 

O = C(↵ s (Q ⇤2 )) + D( m2 q 

Q 2 )+E(⇤2 QCD 

Q 2 

)+F ( ⇤2 QCD 

m 2 Q 

)+G( m2 q 

m 2 Q 

) 


114

Set multiple renormalization scales -- 

Lensing, DGLAP, ERBL Evolution ... 

Choose renormalization scheme; e.g. α R s (µinit R ) 

Result is independent of µ init 

R and scheme at fixed order No renormalization scale ambiguity! 

Choose µ init 

R ; arbitrary initial renormalization scale 

Result is independent of 

Renormalization scheme 

Identify {βi R} − terms using n f − terms 

and initial scale! 

through the P MC − BLM correspondence principle Same as QED Scale Setting 

Apply to Evolution kernels, 

Shift scale of α s to µ PMC 

R to eliminate {βi R } − terms 

hard subprocesses 

Eliminates unnecessary systematic uncertainty 

Conformal Series 

-scheme 

PMC/BLM 

automatically identifies 

QCD function terms 

Principle of Maximum Conformality 

Xing-Gang Wu, Matin Mojaza 

Leonardo di Giustino, SJB 


115

Contributes to the ¯pp ! ¯ttX asymmetry at the Tevatron 

t 

g 

¯t 

Interferes with Born term. 

Small value of renormalization scale increases asymmetry 

Xing-Gang Wu, sjb 


116

Eliminating the Renormalization Scale Ambiguity for Top-Pair Production 

Using the ‘Principle of Maximum Conformality’ (PMC) 

Xing-Gang Wu 

SJB 

Experimental asymmetry 

PMC Prediction 

Conventional: guess for 

renormalization scale and range 

t¯t asymmetry predicted by pQCD NNLO within 

1 of CDF/D0 measurements using PMC/BLM scale setting 

117

An analytic first approximation to QCD 

AdS/QCD + Light-Front Holography 

• As Simple as Schrödinger Theory in Atomic Physics 

• LF radial variable ζ conjugate to invariant mass squared 

• Relativistic, Frame-Independent, Color-Confining 

• Unique confining potential! 

• QCD Coupling at all scales: Essential for Gauge Link 

phenomena 

• Hadron Spectroscopy and Dynamics from one parameter 

• Wave Functions, Form Factors, Hadronic Observables, 

Constituent Counting Rules 

• Insight into QCD Condensates: Zero cosmological constant! 

• Systematically improvable with DLCQ-BLFQ Methods 






⇥ d 2 

d⇣ 2 + 1 

4L2 

4⇣ 2 + U(⇣) ⇤ (⇣) =M 2 (⇣) 







of the action 


119

⌅(x, b ⇤ ) = ⌅(⇥) 

⇤(z) 

⇥ = x(1 x)b 2 ⇤ 

z 

LF(3+1) AdS 5 

x (1 Light-Front x) b ⇥ Holography 

⇤(x, b ⇥ ) = ⇤( ) 

⇥(z) 

= x(1 x)b 2 ⇥ 

⌅(x, b ⇤ ) = ⌅(⇥) 

⇤(z) 

⇥ = (x(1 x) 

x (1 x) b 

z 

z 

⇥ 

⇤(x, b ⇥ ) = ⇤( ) 

x (1 x) b 

z 

⇥ 

⇤(x, b ⇥ ) = ⇤( ) 

⇥(z) 

z 

z 

z 0 = 

⇥ 1 ⇤(x, ) = x(1 x) 

1/2 ⇥( 

⇤(x, 

) 

b ⇥ ) = ⇤( 

z QCD 

0 = 

= x(1 1 

) 

⇥(z) 

x)b 2 ⇥ 

⇥(z) 

⇥ Light Front Holography: 

= x(1 x)b QCD 

z 2 0 = 1 Unique mapping derived from equality 

of LF and AdS formulae for bound-states and form ⇥ factors z 

d ⇥ np ⇥ QCD 

z 0 = = x(1 1 x)b 2 

May 21, 2013 LC2013 AdS/QCD & LF-Holography Stan Brodsky ⇥ 

⇥ 

z 

120 

z QCD 

z 

⇤(z) 

⇥ = x(1 

z 


x (1 x) b ⇥

String Theory 

Goal: First Approximant to QCD 

Counting rules for Hard Exclusive 

Scattering 

Regge Trajectories 

QCD at the Amplitude Level 

AdS/CFT 


Mapping of Poincare’ and Conformal 

SO(4,2) symmetries of 3+1 space 

to AdS5 space 

Conformal behavior at short distances 

+ Confinement at large distance 

Semi-Classical QCD / Wave Equations 

Holography 

Boost Invariant 3+1 Light-Front Wave Equations 

J =0,1,1/2,3/2 plus L 

Integrable! 

Hadron Spectra, Wavefunctions, Dynamics 


121

⇥ 



= M 2 ⇥( ) 

d 2 



⇥ 

d 2 

d 2 + V ( ) 



= M 2 ⇥( ) 

2 = x(1 x)b 2 ⇥ . 

J z = x S(1 

p z = x) ⇤ b n 

i=1 

Si z + ⇤ 

⇥ 

n 1 

⇤(x, b ⇥ ) = ⇤( ) 

U is the exact QCD potential 


Conjecture: ‘H’-diagrams ⇥(z) generate 

U( , S, L) = ⇥ 2 2 + ⇥ 2 (L + S 1/2) 

= x(1 x)b 2 ⇥ 

x (1 

x) b ⇥ 


= x(1 x)b 2 ⇥ 

z 


122 

z 

i=1 ⌥z i = 1 2 

⇤(x, b ⇥ ) = ⇤( ) 

x (1 x) b ⇥ 

⇥(z) 

⇤(x, b ⇥ ) = ⇤( ) 

⇥(z) 

z 

= x(1 x)b 2 ⇥





⇥ d 2 

d⇣ 2 + 1 

4L2 

4⇣ 2 + U(⇣) ⇤ (⇣) =M 2 (⇣) 







of the action 


123

Predict Hadron Properties from First Principles! 

Conformal 

Invariance 

QCD Lagrangian 

AdS/QCD! 

Light-Front 

Holography 

Lattice Gauge Theory 

Light-Front Hamiltonian 

DLCQ/ BLFQ 

Effective Field Theory 

Methods 

SCET, ChPT, ... 

PQCD 

Evolution Equations 

Counting Rules 

Hadron Masses and Observables 

Bound-State 

Dynamics! 

Confinement!

Basis Light-Front Quantization Approach 

to Quantum Field Theory 

Use AdS/QCD orthonormal Light Front Wavefunctions 

as a basis for diagonalizing the QCD LF Hamiltonian 

• Good initial approximation 

BLFQ 

• Better than plane wave basis 

• DLCQ discretization -- highly successful 1+1 

• Use independent HO LFWFs, remove CM motion 

• Similar to Shell Model calculations 

• Xingbo Zhao 

• Anton Ilderton, 

• Heli Honkanen 

• Pieter Maris, 

• James Vary 

• Stan Brodsky 

• Hamiltonian light-front field theory within an AdS/QCD basis. 


Solving nonperturbative QCD using the Front Form 

• Heisenberg: Diagonalize the QCD LF Hamiltonian 


• DLCQ: Complete solutions QCD(1+1): any number of colors, flavors, quark masses 

• AdS/QCD and Light-Front Holography: Soft-Wall Model predicts light-quark 

spectrum and dynamics de Teramond, sjb 

• BFLQ: Use AdS/QCD orthonormal basis functions 

Vary, Maris. et al 

• RGPEP: Systematically reduce off-diagonal elements; RG equations which evolve 

LFQCD in scale Glazek 

• Reduce QCD to equation for LF valence state with effective potential 

Pauli 

• Reduce QCD to one dimensional LF Schrödinger Equation in radial coordinate 

conjugate to the invariant mass. de Teramond, sjb 

• Lippmann-Schwinger expansion in ΔU = U QCD -U AdS Hiller-sjb 

• Cluster expansion methods 

Hiller-Chabysheva 


Features of AdS/QCD LF Holography 

• Based on Conformal Scaling of Infrared QCD Fixed Point 

• Conformal template: Use isometries of AdS5 

• Interpolating operator of hadrons based on twist, superfield 

dimensions 

• Finite Nc = 3: Baryons built on q +(qq) -- Large Nc limit not 

required 

• Break Conformal symmetry with dilaton 

• Dilaton introduces confinement -- positive exponent 

• Effective Charge from AdS/QCD at all scales 

• Conformal Dimensional Counting Rules for Hard Exclusive 

Processes 


New Directions 

• Hadronization at the Amplitude Level 

• LF Confinement potential and LFWFs predicted 

• Eliminate Factorization Scale: Fracture function determines off-shellness 

• Eliminate Renormalization Scale Ambiguity: Principle of Maximal 

Conformality (PMC) 

• Exclusive Channels: PQCD Gluon exchange versus Soft Interactions 

• Different mechanisms at x \to 1 and high k_\perp 

• Massive quark spectroscopy 

• Sublimated Gluons: Gluons appear at high virtuality 

• Hidden Color of Nuclear Wavefunctions 

• Duality: connection to DIS at high x 

128

AdS/QCD and 

P + = P 0 + P z 



c 

c 

x i = k+ 

P + = k0 +k 3 

P 0 +P z 

¯c 

⇧(⇤, b ) 

⇥ = d s(Q 2 ) 

d ln Q 2 < 0 

Exploring QCD, Cambridge, August 20-24, 2007 

Page 

u 


International Conference on Nuclear Theory in the Supercomputing Era - 2013 

LC 2013 

May 21, 2013 

Valparaiso, Chile May 19-20, 2011 

Skiathos, Greece 

129

AdS/QCD and Light-Front Holography Stan Brodsky

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