Light-Front Holography and AdS/QCD: A New Approximation to...

Stan Brodsky

Light-Front Holography and AdS/QCD: A New Approximation to QCD

December 9, 2009

SCGT09

Dec 2009 Nagoya Global COE Workshop "Strong Coupling Gauge Theories in LHC Era"

Ψn(xi,�k⊥i, λi)

SCGTDecember 8, 2009

AdS/QCD and LF Holography Stan Brodsky SLAC

Light-Front Holography and Non-Perturbative QCD

Goal: Use AdS/QCD duality to construct

a first approximation to QCD

Hadron Spectrum Light-Front Wavefunctions,

Form Factors, DVCS, etc

Ψn(xi,�k⊥i, λi)in collaboration with

Guy de Teramond

2

Central problem for strongly-coupled gauge theories


AdS/QCD and LF Holography Stan Brodsky SLAC 3

Need a First Approximation to QCD

Comparable in simplicity to Schrödinger Theory in Atomic Physics

Relativistic, Frame-Independent, Color-Confining

HQED

[− Δ2

2mred+ Veff(�S,�r)] ψ(�r) = E ψ(�r)

[− 12mred

d2

dr2+

12mred

�(� + 1)r2

+ Veff(r, S, �)] ψ(r) = E ψ(r)

(H0 + Hint) |Ψ >= E |Ψ > Coupled Fock states

Effective two-particle equation

Spherical Basis r, θ, φ

Coulomb potential

Includes Lamb Shift, quantum corrections

Bohr SpectrumVeff → VC(r) = −α

r

QED atoms: positronium and muonium

Semiclassical first approximation to QED

HQED

Coupled Fock states


Azimuthal Basis

Confining AdS/QCD potential

QCD Meson SpectrumHLFQCD

(H0LF + HI

LF )|Ψ >= M2|Ψ >

[�k2⊥ + m2

x(1 − x)+ V LF

eff ] ψLF (x,�k⊥) = M2 ψLF (x,�k⊥)

[− d2

dζ2+

−1 + 4L2

ζ2+ U(ζ, S, L)] ψLF (ζ) = M2 ψLF (ζ) ζ, φ

U(ζ, S, L) = κ2ζ2 + κ2(L + S − 1/2)

ζ2 = x(1 − x)b2⊥

Semiclassical first approximation to QCD

Dirac’s Amazing Idea: The Front Form

Instant Form Front Form

τ = t + z/cσ = ct − z

Evolve in light-front time!

Evolve in ordinary time

P.A.M Dirac, Rev. Mod. Phys. 21, 392 (1949)

Each element of flash photograph

illuminated at same Light Front

time

τ = t + z/c

Evolve in LF time

P− = id

dτ

DIS, Form Factors, DVCS, etc. measure proton WF at fixed

τ = t + z/c



• QCD Lagrangian

LQCD = − 14g2

Tr (GμνGμν) + iψDμγμψ + mψψ

• LF Momentum Generators P = (P+, P−,P⊥) in terms of dynamical fields ψ, A⊥

P− =12

∫dx−d2x⊥ψ γ+ (i∇⊥)2+ m2

i∂+ψ + interactions

P+ =∫

dx−d2x⊥ ψ γ+i∂+ψ

P⊥ =12

∫dx−d2x⊥ ψ γ+i∇⊥ψ

• LF Hamiltonian P− generates LF time translations

[ψ(x), P−]

= i∂

∂x+ψ(x)

and the generators P+ and P⊥ are kinematical

8



∑ni xi = 1

∑ni

�k⊥i = �0⊥


xiP+, xi

�P⊥ + �k⊥i

P+, �P⊥

x

Fixed τ = t + z/c

Invariant under boosts! Independent of Pμ

9

Light-Front Wavefunctions: rigorous representation of composite systems in quantum field theory

x =k+

P+=

k0 + k3

P 0 + P 3

Process Independent Direct Link to QCD Lagrangian!



J z =n∑

i=1

szi +

n−1∑

j=1

lzj .

Conserved LF Fock state by Fock State!

plzj = −i

(k1j

∂

∂k2j

− k2j

∂

∂k1j

)

i h bi l l

n-1 orbital angular momenta

Angular Momentum on the Light-Front

Nonzero Anomalous Moment --> Nonzero orbital angular momentum!

LF Spin Sum Rule



nn

|p,Sz >=∑n=3

Ψn(xi,�k⊥i,λi)|n;�k⊥i,λi >

The Light Front Fock State Wavefunctions

Ψn(xi,�k⊥i,λi)

are boost invariant; they are independent of the hadron’s energyand momentum Pμ.

The light-cone momentum fraction

xi =k+

i

p+ =k0

i + kzi

P0 +Pz

are boost invariant.

n

∑i

k+i = P+,

n

∑i

xi = 1,n

∑i

�k⊥i =�0⊥.

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

Fixed LF time

11

Intrinsic heavy quarks s(x) �= s(x)

u(x) �= d(x)c(x), b(x) at high x



zero for q+ = 0

12

Calculation of Form Factors in Equal-Time Theory

Instant Form

Calculation of Form Factors in Light-Front Theory

Front Form

Absent for q+ = 0 zero !!

Need vacuum-induced currentseddddddddddddddddddddddddddd

AComplete Answer

QCD and the LF Hadron Wavefunctions

DVCS, GPDs. TMDs

Baryon Decay

Distribution amplitudeERBL Evolution

Heavy Quark Fock StatesIntrinsic Charm

Gluonic propertiesDGLAP

Quark & Flavor Struct

Coordinate space representation

Quark & Flavor Structure

Baryon Excitations


Initial and Final State Rescattering

DDIS, DDIS, T-Odd

Non-Universal Antishadowing

Nuclear ModificationsBaryon Anomaly

Color Transparency

Distriiibbbbbbbbbbbbbbbubub

NuclBa

Col

s

Hard Exclusive AmplitudesForm Factors

Counting Rules

φp(x1, x2, Q2)

AdS/QCDLight-Front Holography

LF Schrodinger Eqnront hrororororororororororororororororororroorroror d.

LF Overlap, incl ERBL

J=0 Fixed Pole

Orbital Angular MomentumSpin, Chiral Properties

Crewther Relation

Hadronization at Amplitude Level

re



Heisenberg Matrix FormulationLight-Front QCD

Eigenvalues and Eigensolutions give Hadron Spectrum and Light-Front wavefunctions

HQCDLF |Ψh >= M2

h|Ψh >

HQCDLF =

∑i

[m2 + k2

⊥x

]i + HintLF

LQCD → HQCDLF

HintLF : Matrix in Fock Space

Physical gauge: A+ = 0

14

HQCDLF |Ψh >= M2

h|Ψh >HQCDLC |Ψh〉 = M2

h |Ψh〉Heisenberg Matrix Formulation

Light-Front QCD H.C. Pauli & sjbDiscretized Light-Cone

Quantization


DLCQ: Frame-independent, No fermion doubling; Minkowski Space

HQCDLF |Ψh >= M2

h|Ψh >

DLCQ: Periodic BC in x−. Discrete k+; frame-independent truncation

Stan Brodsky SLAC


AdS/QCD and LF Holography

16

Light-Front QCD Features and Phenomenology

• Hidden color, Intrinsic glue, sea, Color Transparency

• Physics of spin, orbital angular momentum

• Near Conformal Behavior of LFWFs at Short Distances; PQCD constraints

• Vanishing anomalous gravitomagnetic moment

• Relation between edm and anomalous magnetic moment

• Cluster Decomposition Theorem for relativistic systems

• OPE: DGLAP, ERBL evolution; invariant mass scheme



Applications of AdS/CFT to QCD

in collaboration with Guy de Teramond

Changes in physical

length scale mapped to

evolution in the 5th dimension z

17

Application of AdS/CFT to QCD

Changes in physical

length scale mapped to

evolution in the 5th dimension z

Bottom-Up Top-Down

String Theory



• Use AdS/CFT to provide an approximate, covariant, and analytic model of hadron structure with confinement at large distances, conformal behavior at short distances

• Analogous to the Schrödinger Theory for Atomic Physics

• AdS/QCD Light-Front Holography

• Hadronic Spectra and Light-Front Wavefunctions

Goal:

19



Mμν,Pμ,D,Kμ,

Conformal Theories are invariant under the Poincare and conformal transformations with

the generators of SO(4,2)

SO(4,2) has a mathematical representation on AdS5

20

Stan Brodsky SLAC



21

αs(Q2) � const at small Q2

Maldacena:

AdS/CFT: Anti-de Sitter Space / Conformal Field Theory

Map AdS5 X S5 to conformal N=4 SUSY

s2

• QCD is not conformal; however, it has manifestations of a scale-invariant theory: Bjorken scaling, dimensional counting for hard exclusive processes

• Conformal window:

• Use mathematical mapping of the conformal group SO(4,2) to AdS5 space



Conformal QCD Window in Exclusive Processes

• Does αs develop an IR fixed point? Dyson–Schwinger Equation Alkofer, Fischer, LLanes-Estrada,

Deur . . .

• Recent lattice simulations: evidence that αs becomes constant and is not small in the infrared

Furui and Nakajima, hep-lat/0612009 (Green dashed curve: DSE).

-0.4-0.2 0 0.2 0.4 0.6 0.8 1Log_10�q�GeV��

0.5

1

1.5

2

2.5

3

3.5

4

Αs�q�

2222222222222

Conformal Behavior of QCD in Infrared



Deur, Korsch, et al.

�s/�

pQCD evol. eq.

�s,g1/� JLab

Cornwall

Fit

GDH limit

Godfrey-Isgur

Bloch et al.

Burkert-Ioffe

Fischer et al.

Bhagwat et al.Maris-Tandy

Q (GeV)

Lattice QCD

10-1

1

10-1

1

10-1

1 10-1

1

DSE gluoncouplings



IR Conformal Window for QCD

SCGT Stan Brod

• Dyson-Schwinger Analysis: QCD gluon coupling has IR Fixed Point

• Evidence from Lattice Gauge Theory

• Stability of

• Define coupling from observable: indications of IR fixed point for QCD effective charges

• Confined gluons and quarks have maximum wavelength: Decoupling of QCD vacuum polarization at small Q2

• Justifies application of AdS/CFT in strong-coupling conformal window

24

ppolarization atSerber-Uehling

Π(Q2)→ α15π

Q2

m2 Q2 << 4m2

�+

�−

Shrock, sjb

Deur, Chen, Burkert, Korsch,

Υ → ggg

Stan Brodsky SLAC



• Colored fields confined to finite domain

• All perturbative calculations regulated in IR

• High momentum calculations unaffected

• Bound-state Dyson-Schwinger Equation

• Analogous to Bethe’s Lamb Shift Calculation

A strictly-perturbative space-time region can be defined as one whichhas the property that any straight-line segment lying entirely within the region has an invariant length small compared to the confinement scale (whether or not the segment is spacelike or timelike).

J. D. Bjorken, SLAC-PUB 1053

Cargese Lectures 1989

(x − y)2 < Λ−2QCD

Quark and Gluon vacuum polarization insertions decouple: IR fixed Point

25

Maximal Wavelength of Confined Fields

Shrock, sjb



Scale Transformations

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

SO(1, 5)

ds2 =R2

z2(ημνdxμdxν − dz2),

xμ → λxμ, z → λz, maps scale transformations into the holographic coordinate z.

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

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

x2 → λ2x2, z → λz.

x2 = xμxμ: invariant separation between quarks

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

26

invariant measure



• Truncated AdS/CFT (Hard-Wall) model: cut-off at z0 = 1/ΛQCD 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 ϕ(z) – usual linear Regge dependence can

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



φ(z)

ζ =√

x(1 − x)�b2⊥ z

x

(1 − x)

�b⊥

ψ(x,�b⊥)

LF(3+1) AdS5

32

Holography: Unique mapping derived from equality of LF and AdS formula for current matrix elements

ψ(x, ζ) =√

x(1 − x)ζ−1/2φ(ζ)



soft wallconfining potential

Light-Front Holography: Map AdS/CFT to 3+1 LF Theory

ζ2 = x(1 − x)b2⊥.

Relativistic LF radial equation

G. de Teramond, sjb

x

(1 − x)

�b⊥

Frame Independent

[ − d2

dζ2+

1 − 4L2

4ζ2+ U(ζ)

]φ(ζ) = M2φ(ζ)

33

U(z) = κ4z2 + 2κ2(L + S − 1)



Light-Front Quantization of QCD and AdS/CFT

• Light-front (LF) quantization is the ideal framework to describe hadronic structure in terms of quarks and gluons: simple vacuum structure gives an unambiguous definition of the partonic content of a hadron, exact formulae for form factors, physics of angular momentum of constituents ...

• Frame-independent LF Hamiltonian equation: similar structure as AdS EOM

• First semiclassical approximation to the bound-state LF Hamiltonian equation in QCD is equivalent to equations of motion in AdS and can be systematically improved

GdT and Sjb PRL 102, 081601 (2009)

PμPμ|P >= (P−P+ − �P 2⊥)|P >= M2|P >

Stan Brodsky SLAC



ψ(z) ∼ zΔ at z → 0

ψ(z0) = 00 < z < z0 z0 = 1ΛQCD

x2μ → λ2x2

μ z → λz

35

AdS/CFT

ψ(z) ∼ zΔ at z →

x2μ → λ2x2

μ z → λz

• Use mapping of conformal group SO(4,2) to AdS5

• Scale Transformations represented by wavefunction in 5th dimension

• Match solutions at small z to conformal twist dimension of hadron wavefunction at short distances

• Hard wall model: Confinement at large distances and conformal symmetry in interior

• Truncated space simulates “bag” boundary conditions



• Conformal metric: ds2 = g�mdx�dxm. x� = (xμ, z), g�m → (R2/z2

)η�m .

• Action for massive scalar modes on AdSd+1:

S[Φ] =12

∫dd+1x

√g 1

2

[g�m∂�Φ∂mΦ − μ2Φ2

],

√g → (R/z)d+1.

• Equation of motion1√g

∂

∂x�

(√g g�m ∂

∂xmΦ

)+ μ2Φ = 0.

• Factor out dependence along xμ-coordinates , ΦP (x, z) = e−iP ·x Φ(z), PμPμ = M2 :[z2∂2

z − (d − 1)z ∂z + z2M2 − (μR)2]Φ(z) = 0.

• Solution: Φ(z) → zΔ as z → 0,

Φ(x, z) = Czd2 JΔ− d

2(zM) , Δ = 1

2

(d +

√d2 + 4μ2R2

).

Bosonic Solutions: Hard Wall Model

Δ = 2 + L (μR)2 = L2 − 4d = 4

Φ(z) = Czd/2JΔ−d/2(zM)



AdS Schrodinger Equation for bound state of two scalar constituents:

Derived from variation of Action in AdS5

φ(z = z0 = 1Λc

) = 0.

Hard wall model: truncated space

Let Φ(z) = z3/2φ(z)

L: light-front orbital angular momentum

[ − d2

dz2− 1 − 4L2

4z2

]φ(z) = M2φ(z)



• Pseudoscalar mesons: O3+L = ψγ5D{�1 . . . D�m}ψ (Φμ = 0 gauge).

• 4-d mass spectrum from boundary conditions on the normalizable string modes at z = z0,

Φ(x, zo) = 0, given by the zeros of Bessel functions βα,k: Mα,k = βα,kΛQCD

• Normalizable AdS modes Φ(z)

10 2 3 4

1

2

0

3

4

5

z

Φ(z)

2-20068721A7

10 2 3 4

-2

-4

0

2

4

z

Φ(z)

2-20068721A8

Fig: Meson orbital and radial AdS modes for ΛQCD = 0.32 GeV.

zΔ

333

z0

z0 = 1ΛQCD

Match fall-off at small z to conformal twist-dimensionat short distances

Δ = 2 + L2 L

twist

S = 0

O2+L



10 2 3 4

1

2

0

3

z

Φ(z)

2-20078721A18

-2

-4

0

2

4

Φ(z)

10 2 3 4z

2-20078721A19

Fig: Orbital and radial AdS modes in the hard wall model for ΛQCD = 0.32 GeV .

0

2

4

(GeV

2 )

(b)(a)

0 2 40 2 41-20068694A16

ω (782)ρ (770)π (140)

b1 (1235)

π2 (1670)a0 (1450)a2 (1320)f1 (1285)

f2 (1270)a1 (1260)

ρ (1700)ρ3 (1690)

ω3 (1670)ω (1650)

f4 (2050)a4 (2040)

L L

Fig: Light meson and vector meson orbital spectrum ΛQCD = 0.32 GeV

S = 0 S = 1

• Equation of motion for scalar field L = 12

(g�m∂�Φ∂mΦ − μ2Φ2

)[z2∂2

z − (3∓ 2κ2z2

)z ∂z + z2M2 − (μR)2

]Φ(z) = 0

with (μR)2 ≥ −4.

• LH holography requires ‘plus dilaton’ ϕ = +κ2z2. Lowest possible state (μR)2 = −4

M2 = 0, Φ(z) ∼ z2e−κ2z2, 〈r2〉 ∼ 1

κ2

A chiral symmetric bound state of two massless quarks with scaling dimension 2: the pion

Soft-Wall Model

S =∫

d4x dz√

g eϕ(z)L,

Retain conformal AdS metrics but introduce smooth cutoff which depends on the profile of a dilaton background field

Karch, Katz, Son and Stephanov (2006)]

Massless pion

ϕ(z) = ±κ2z2



AdS Soft-Wall Schrodinger Equation for bound state of two scalar constituents:

Derived from variation of Action Dilaton-Modified AdS5

[ − d2

dz2− 1 − 4L2

4z2+ U(z)

]φ(z) = M2φ(z)

U(z) = κ4z2 + 2κ2(L + S − 1)

• Erlich, Karch, Katz, Son, Stephanov • de Teramond, sjb

eΦ(z) = e+κ2z2

Positive-sign dilaton

Figure 5

d a “warped” metric of the form

oportional to eA(y). The function

found in the AdS metric, but its

minimum, eA(0), at y = 0. Thus

etain a nonzero energy

from the effects of confinement;

e

ds2 = eA(y)(−dx20 + dx2

1 + dx23 + dx2

3) + dy2

ds2 = eκ2z2 R2

z2(dx2

0 − dx21 − dx2

3 − dx23 − dz2)

y = R/zz → 0z → ∞

Klebanov and Maldacena



Agrees with Klebanov and Maldacena for positive-sign exponent of

dilaton

ds2 = eκ2z2 R2

z2(dx2

0 − dx21 − dx2

3 − dx23 − dz2)



00 4 8

2

4

6

Φ(z)

2-20078721A20 z

-5

0

5

0 4 8z

Φ(z)

2-20078721A21

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

0

2

4

(GeV

2 )

(a)

0 2 48-20078694A19

π (140)

b1 (1235)

π2 (1670)

L

(b)

0 2 4

π (140)

π (1300)

π (1800)

n

Light meson orbital (a) and radial (b) spectrum for κ = 0.6 GeV.

S = 0 S = 0

Soft Wall Model

Pion mass automatically

zero!

mq = 0

6Quark separation increases with L

Pion has zero mass!

0000000000000000000000000000000000000000000



• Obtain spin-J mode Φμ1···μJ with all indices along 3+1 coordinates from Φ by shifting dimensions

ΦJ(z) =( z

R

)−JΦ(z)

• Substituting in the AdS scalar wave equation for Φ[z2∂2

z − (3−2J − 2κ2z2

)z ∂z + z2M2− (μR)2

]ΦJ = 0

• Upon substitution z→ζ

φJ(ζ)∼ζ−3/2+Jeκ2ζ2/2 ΦJ(ζ)

we find the LF wave equation

(− d2

dζ2− 1 − 4L2

4ζ2+ κ4ζ2 + 2κ2(L + S − 1)

)φμ1···μJ = M2φμ1···μJ

with (μR)2 = −(2 − J)2 + L2

Higher-Spin Hadrons



Higher Spin Bosonic Modes SW

• Effective LF Schrodinger wave equation[− d2

dζ2− 1 − 4L2

4ζ2+ κ4ζ2 + 2κ2(L+ S−1)

]φS(ζ) = M2φS(ζ)

with eigenvalues M2 = 2κ2(2n + 2L + S).

• Compare with Nambu string result (rotating flux tube): M2n(L) = 2πσ (n + L + 1/2) .

0

2

(a) (b)

4

(GeV

2 )

0 2 45-20068694A20

ω (782)ρ (770)

a2 (1320)

f2 (1270)

ρ3 (1690)

ω3 (1670)

f4 (2050)a4 (2040)

L

0 2 4

n

ρ (770)

ρ (1450)

ρ (1700)

Vector mesons orbital (a) and radial (b) spectrum for κ = 0.54 GeV.

• Glueballs in the bottom-up approach: (HW) Boschi-Filho, Braga and Carrion (2005); (SW) Colangelo,

De Facio, Jugeau and Nicotri( 2007).

[− d2

dz2− 1 − 4L2

4z2+ κ4z2 + 2κ2(L+ S−1)

]φS(z) = M2φS(z)

S = 1S = 1

Soft-wall model

Same slope in n and L

Regge Trajectories



00 4 8

2

4

6

Φ(z)

2-20078721A20 z

-5

0

5

0 4 8z

Φ(z)

2-20078721A21 u

L = 0

L = 1

L = 2

L = 3n = 0

n = 3

n = 2

n = 1

(a) (b)

Φ(z)Φ(z)

z z

0

2

(a)

4

(GeV

2 )

0 2 45-20068694A20

ω (782)ρ (770)

a2 (1320)

f2 (1270)

ρ3 (1690)

ω3 (1670)

f4 (2050)a4 (2040)

L

(b)

0 2 4

n

ρ (770)

ρ (1450)

ρ (1700)

Quark separation increases with L

47

S = 1 S = 1

0

2

(a) (b)

4

(GeV

2 )

0 2 45-20068694A20

ω (782)ρ (770)

a2 (1320)

f2 (1270)

ρ3 (1690)

ω3 (1670)

f4 (2050)a4 (2040)

L

0 2 4

n

ρ (770)

ρ (1450)

ρ (1700)

M2 = 2κ2(2n + 2L + S).

S = 1



1−− 2++ 3−− 4++ JPC

M2

L

Parent and daughter Regge trajectories for the I = 1 ρ-meson family (red)

and the I = 0 ω-meson family (black) for κ = 0.54 GeV

49



Propagation of external perturbation suppressed inside AdS.

At large enoughQ ∼ r/R2, the interaction occurs in the large-r conformal region. Importa

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

J(Q, z), Φ(z)

1 2 3 4 5

0.2

0.4

0.6

0.8

1

z

Consider a specific AdS mode Φ(n) dual to an n partonic Fock state |n〉. At small z, Φscales as Φ(n) ∼ zΔn . Thus:

F (Q2) →[

1Q2

]τ−1

,

where τ = Δn − σn, σn =∑n

i=1 σi. The twist is equal to the number of partons, τ = n.

Dimensional Quark Counting Rules:General result from

AdS/CFT and Conformal Invariance

50

l t b ti d i id AdS

Hadron Form Factors from AdS/CFT

Polchinski, Strasslerde Teramond, sjb

F (Q2)I→F =∫ dz

z3ΦF (z)J(Q, z)ΦI(z)

J(Q, z) = zQK1(zQ)

High Q2from

small z ~ 1/Q

J(Q, z) Φ(z)



-10 -8 -6 -4 -2 00

0.2

0.4

0.6

0.8

1

q2(GeV 2)

Spacelike pion form factor from AdS/CFT

Fπ(q2)

Hard Wall: Truncated Space Confinement

Soft Wall: Harmonic Oscillator Confinement

One parameter - set by pion decay constant

Data CompilationBaldini, Kloe and Volmer

de Teramond, sjbSee also: Radyushkin



Current Matrix Elements in AdS Space (SW)

• Propagation of external current inside AdS space described by the AdS wave equation[z2∂2

z − z(1 + 2κ2z2

)∂z − Q2z2

]Jκ(Q, z) = 0.

• Solution bulk-to-boundary propagator

Jκ(Q, z) = Γ(

1 +Q2

4κ2

)U

(Q2

4κ2, 0, κ2z2

),

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

Γ(a)U(a, b, z) =∫ ∞

0e−ztta−1(1 + t)b−a−1dt.

• Form factor in presence of the dilaton background ϕ = κ2z2

F (Q2) = R3

∫dz

z3e−κ2z2

Φ(z)Jκ(Q, z)Φ(z).

• For large Q2 4κ2

Jκ(Q, z) → zQK1(zQ) = J(Q, z),

the external current decouples from the dilaton field.

sjb and GdT Grigoryan and Radyushkin

Soft Wall Model



-10 -8 -6 -4 -2 00

0.2

0.4

0.6

0.8

1

q2(GeV 2)

Spacelike pion form factor from AdS/CFT

Fπ(q2)

Hard Wall: Truncated Space Confinement

Soft Wall: Harmonic Oscillator Confinement

One parameter - set by pion decay constant

Data CompilationBaldini, Kloe and Volmer

Stan Brodsky

de Teramond, sjbSee also: Radyushkin



• Analytical continuation to time-like region q2 → −q2 (Mρ = 4κ2 = 750 MeV)

• Strongly coupled semiclassical gauge/gravity limit hadrons have zero widths (stable).

-10 -5 0 5 10

-3

-2

-1

0

1

2

7-20078755A4q2 (GeV2)

log

IFπ

(q2 )

I

Space and time-like pion form factor for κ = 0.375 GeV in the SW model.

• Vector Mesons: Hong, Yoon and Strassler (2004); Grigoryan and Radyushkin (2007).

Mρ = 2κ = 750 MeV



e+

e−γ∗

π+

π−

Dressed soft-wall current bring in higher Fock states and more vector meson poles

55



Note: Analytical Form of Hadronic Form Factor for Arbitrary Twist

• Form factor for a string mode with scaling dimension τ , Φτ in the SW model

F (Q2) = Γ(τ)Γ

(1+ Q2

4κ2

)

Γ(τ + Q2

4κ2

) .

• For τ = N , Γ(N + z) = (N − 1 + z)(N − 2 + z) . . . (1 + z)Γ(1 + z).

• Form factor expressed as N − 1 product of poles

F (Q2) =1

1 + Q2

4κ2

, N = 2,

F (Q2) =2(

1 + Q2

4κ2

)(2 + Q2

4κ2

) , N = 3,

· · ·F (Q2) =

(N − 1)!(1 + Q2

4κ2

)(2 + Q2

4κ2

)· · ·

(N−1+ Q2

4κ2

) , N.

• For large Q2:

F (Q2) → (N − 1)![4κ2

Q2

](N−1)

.

Spacelike and timelike pion form factor

κ = 0.534 GeV

Pqqqq = 15%

|π >= ψqq|qq > + ψqqqq|qqqq >

Q2 (GeV2)

Q2 (GeV2)

Q2 (GeV2)

Γρ = 120 MeV, Γ′ρ = 300 MeV

Preliminary



Holographic Model for QCD Light-Front WavefunctionsSJB and GdT in preparation

• Drell-Yan-West form factor in the light-cone (two-parton state)

F (q2) =∑

q

eq

∫ 1

0dx

∫d2�k⊥16π3

ψ∗P ′(x,�k⊥ − x�q⊥) ψP (x,�k⊥).

• Fourrier transform to impact parameter space�b⊥

ψ(x,�k⊥) =√

4π

∫d2�b⊥ ei�b⊥·�k⊥ψ(x,�b⊥)

• Find (b = |�b⊥|) :

F (q2) =∫ 1

0dx

∫d2�b⊥ eix�b⊥·�q⊥∣∣ψ(x, b)

∣∣2

= 2π

∫ 1

0dx

∫ ∞

0b db J0 (bqx)

∣∣ψ(x, b)∣∣2,

Soper

58

Light-Front Representation of Two-Body Meson Form Factor

�q2⊥ = Q2 = −q2

Holographic Mapping of AdS Modes to QCD LFWFs

• Integrate Soper formula over angles:

F (q2) = 2π∫ 1

0dx

(1 − x)x

∫ζdζJ0

(ζq

√1 − x

x

)ρ(x, ζ),

with ρ(x, ζ) QCD effective transverse charge density.

• Transversality variable

ζ =√

x

1 − x

∣∣∣n−1∑j=1

xjb⊥j

∣∣∣.

• Compare AdS and QCD expressions of FFs for arbitrary Q using identity:

∫ 1

0dxJ0

(ζQ

√1 − x

x

)= ζQK1(ζQ),

the solution for J(Q, ζ) = ζQK1(ζQ) !

ζ =√

x(1 − x)�b2⊥



• Electromagnetic form-factor in AdS space:

Fπ+(Q2) = R3

∫dz

z3J(Q2, z) |Φπ+(z)|2 ,

where J(Q2, z) = zQK1(zQ).

• Use integral representation for J(Q2, z)

J(Q2, z) =∫ 1

0dx J0

(ζQ

√1 − x

x

)

• Write the AdS electromagnetic form-factor as

Fπ+(Q2) = R3

∫ 1

0dx

∫dz

z3J0

(zQ

√1 − x

x

)|Φπ+(z)|2

• Compare with electromagnetic form-factor in light-front QCD for arbitrary Q

∣∣∣ψqq/π(x, ζ)∣∣∣2 =

R3

2πx(1 − x)

|Φπ(ζ)|2ζ4

with ζ = z, 0 ≤ ζ ≤ ΛQCD



x

(1 − x)

�b⊥

φ(z)

ζ =√

x(1 − x)�b2⊥ z

ψ(x,�b⊥)

LF(3+1) AdS5

61

Light-Front Holography: Unique mapping derived from equality of LF and AdS formula for current matrix elements

ψ(x,�b⊥)

ψ(x,�b⊥) =

√x(1 − x)

2πζφ(ζ)



• Hadronic gravitational form-factor in AdS space

Aπ(Q2) = R3

∫dz

z3H(Q2, z) |Φπ(z)|2 ,

where H(Q2, z) = 12Q2z2K2(zQ)

• Use integral representation for H(Q2, z)

H(Q2, z) = 2∫ 1

0x dxJ0

(zQ

√1 − x

x

)

• Write the AdS gravitational form-factor as

Aπ(Q2) = 2R3

∫ 1

0x dx

∫dz

z3J0

(zQ

√1 − x

x

)|Φπ(z)|2

• Compare with gravitational form-factor in light-front QCD for arbitrary Q

∣∣∣ψqq/π(x, ζ)∣∣∣2 =

R3

2πx(1 − x)

|Φπ(ζ)|2ζ4

,

which is identical to the result obtained from the EM form-factor

Abidin & Carlson

Gravitational Form Factor in AdS space

Identical to LF Holography obtained from electromagnetic current



soft wallconfining potential:

Light-Front Holography: Map AdS/CFT to 3+1 LF Theory

ζ2 = x(1 − x)b2⊥.

Relativistic LF radial equation!

G. de Teramond, sjb

x

(1 − x)

�b⊥

Frame Independent

[− d2

dζ2+

1 − 4L2

4ζ2+ U(ζ)

]φ(ζ) = M2φ(ζ)

63

U(ζ) = κ4ζ2 + 2κ2(L + S − 1)

HQED

Coupled Fock states


Azimuthal Basis

Confining AdS/QCD potential

QCD Meson SpectrumHLFQCD

(H0LF + HI

LF )|Ψ >= M2|Ψ >

[�k2⊥ + m2

x(1 − x)+ V LF

eff ] ψLF (x,�k⊥) = M2 ψLF (x,�k⊥)

[− d2

dζ2+

−1 + 4L2

ζ2+ U(ζ, S, L)] ψLF (ζ) = M2 ψLF (ζ) ζ, φ

U(ζ, S, L) = κ2ζ2 + κ2(L + S − 1/2)

ζ2 = x(1 − x)b2⊥

Semiclassical first approximation to QCD



Derivation of the Light-Front Radial Schrodinger Equation directly from LF QCD

M2 =∫ 1

0

dx

∫d2�k⊥16π3

�k2⊥

x(1 − x)

∣∣∣ψ(x,�k⊥)∣∣∣2 + interactions

=∫ 1

0

dx

x(1 − x)

∫d2�b⊥ ψ∗(x,�b⊥)

(−�∇2

�b⊥�

)ψ(x,�b⊥) + interactions.

(�ζ, ϕ), �ζ =√

x(1 − x)�b⊥:Change variables ∇2 =

1ζ

d

dζ

(ζ

d

dζ

)+

1ζ2

∂2

∂ϕ2

M2 =∫

dζ φ∗(ζ)√

ζ

(− d2

dζ2− 1

ζ

d

dζ+

L2

ζ2

)φ(ζ)√

ζ

+∫

dζ φ∗(ζ)U(ζ)φ(ζ)

=∫

dζ φ∗(ζ)(− d2

dζ2− 1 − 4L2

4ζ2+ U(ζ)

)φ(ζ)



• Find (L = |M |)

M2 =∫

dζ φ∗(ζ)√

ζ

(− d2

dζ2− 1

ζ

d

dζ+

L2

ζ2

)φ(ζ)√

ζ+∫

dζ φ∗(ζ) U(ζ)φ(ζ)

where the confining forces from the interaction terms is summed up in the effective potential U(ζ)

• Ultra relativistic limit mq → 0 longitudinal modes X(x) decouple and LF eigenvalue equation

HLF |φ〉 = M2|φ〉 is a LF wave equation for φ

(− d2

dζ2− 1 − 4L2

4ζ2︸︷︷︸kinetic energy of partons

+ U(ζ)︸︷︷︸confinement

)φ(ζ) = M2φ(ζ)

• Effective light-front Schrodinger equation: relativistic, frame-independent and analytically tractable

• Eigenmodes φ(ζ) determine the hadronic mass spectrum and represent the probability amplitude to

find n-massless partons at transverse impact separation ζ within the hadron at equal light-front time

• Semiclassical approximation to light-front QCD does not account for particle creation and absorption

but can be implemented in the LF Hamiltonian EOM or by applying the L-S formalism

HQED

[− Δ2

2mred+ Veff(�S,�r)] ψ(�r) = E ψ(�r)

[− 12mred

d2

dr2+

12mred

( + 1)r2

+ Veff(r, S, )] ψ(r) = E ψ(r)

(H0 + Hint) |Ψ >= E |Ψ > Coupled Fock states


Spherical Basis r, θ, φ

Coulomb potential

Includes Lamb Shift, quantum corrections

Bohr SpectrumVeff → VC(r) = −α

r

QED atoms: positronium and muonium

Semiclassical first approximation to QED



Example: Pion LFWF

• Two parton LFWF bound state:

ψHWqq/π(x,b⊥) =

ΛQCD

√x(1 − x)√

πJ1+L(βL,k)JL

(√x(1 − x) |b⊥|βL,kΛQCD

)θ

(b2⊥ ≤ Λ−2

QCD

x(1 − x)

),

ψSWqq/π(x,b⊥) = κL+1

√2n!

(n + L)![x(1 − x)]

12+L|b⊥|Le−

12 κ2x(1−x)b2

⊥LLn

(κ2x(1 − x)b2

⊥).

(a) (b)b b

xx

ψ(x

,b)

7-20078755A1

1.0

00

0 10 200 10 20

0.05

0.10

0.5

1.0

0

0.5

0

0.1

0.2

Fig: Ground state pion LFWF in impact space. (a) HW model ΛQCD = 0.32 GeV, (b) SW model κ = 0.375 GeV.



Consider the AdS5 metric:

ds2 = R2

z2 (ημνdxμdxν − dz2).

ds2 invariant if xμ → λxμ, z → λz,

Maps scale transformations to scale changes of the the holographic coordinate z.

We define light-front coordinates x± = x0 ± x3.

Then ημνdxμdxν = dx02 − dx3

2 − dx⊥2 = dx+dx− − dx⊥2

and

ds2 = −R2

z2 (dx⊥2 + dz2) for x+ = 0.

• ds2 is invariant if dx⊥2 → λ2dx⊥2, and z → λz, at equal LF time.

• Maps scale transformations in transverse LF space to scale changes of the holographic coordinate z.

• Holographic connection of AdS5 to the light-front.

• Casimir for the rotation group SO(2).• The effective wave equation in the two-dim transverse LF plane has the Casimir representation L2

corresponding to the SO(2) rotation group [The Casimir for SO(N) ∼ SN−1 is L(L + N − 2) ].

Light-Front/AdS5 Duality



Prediction from AdS/CFT: Meson LFWF

x0.20.40.60.8

1.3

1.4

1.5

0

0.05

0.1

0.15

0.2

0

5

“Soft Wall” model

k⊥ (GeV)

de Teramond, sjb

70

φM(x, Q0) ∝√

x(1 − x)

ψM(x, k2⊥)

κ = 0.375 GeV

massless quarksNote coupling

k2⊥, x

Connection of Confinement to TMDs

ψM (x, k⊥) =4π

κ√

x(1 − x)e− k2

⊥2κ2x(1−x)



Example: Evaluation of QCD Matrix Elements

• Pion decay constant fπ defined by the matrix element of EW current J+W :

⟨0∣∣ψuγ+ 1

2(1 − γ5)ψd

∣∣π−⟩ = iP+fπ√

2with

∣∣π−⟩ = |du〉 =1√NC

1√2

NC∑c=1

(b†c d↓d

†c u↑ − b†c d↑d

†c u↓) ∣∣0⟩.

• Find light-front expression (Lepage and Brodsky ’80):

fπ = 2√

NC

∫ 1

0dx

∫d2�k⊥16π3

ψqq/π(x, k⊥).

• Using relation between AdS modes and QCD LFWF in the ζ → 0 limit

fπ =18

√32

R3/2 limζ→0

Φ(ζ)ζ2

.

• Holographic result (ΛQCD =0.22 GeV and κ=0.375 GeV from pion FF data): Exp: fπ =92.4 MeV

fHWπ =

√3

8J1(β0,k)ΛQCD = 91.7 MeV, fSW

π =√

38

κ = 81.2 MeV,

Stan Brodsky SLAC



72

φH(xi, Q)

φM (x,Q) =∫ Q

d2�k ψqq(x,�k⊥)

Fixed τ = t + z/c

x

1 − x

k2⊥ < Q2

∑i

xi = 1

Braun, Gardi

Lepage, sjbEfremov, Radyushkin

Sachrajda, Frishman Lepage, sjb

Lepage, sjb

Hadron Distribution Amplitudes

Q2

Braun, Gard

Lepage, sjbEEfrfrEE emov, RadyR dfrfrfr

Sachrajda, Frishman Lep

• Fundamental gauge invariant non-perturbative input to hard exclusive processes, heavy hadron decays. Defined for Mesons, Baryons

• Evolution Equations from PQCD, OPE, Conformal Invariance

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



Second Moment of Pion Distribution Amplitude

< ξ2 >=∫ 1

−1

dξ ξ2φ(ξ)

ξ = 1 − 2x

φasympt ∝ x(1 − x)

φAdS/QCD ∝√

x(1 − x)

Braun et al.

Donnellan et al.

< ξ2 >π= 1/5 = 0.20

< ξ2 >π= 1/4 = 0.25

Lattice (I) < ξ2 >π= 0.28 ± 0.03

Lattice (II) < ξ2 >π= 0.269 ± 0.039



0 100 200 300 400 5000.0

0.1

0.2

0.3

0.4

500 x

Φ�x,

Q2 �

�f Π

ERBL evolution at Q2 �2,10,100 GeV2

0 0.5 1.0

√x(1 − x)

x(1 − x)

F. Cao, GdT, sjb (preliminary)

ERBL Evolution of Pion Distribution Amplitude

Q2 = 100 GeV2

Q2 = 2 GeV2



Baryons in Ads/CFT

• Baryons Spectrum in ”bottom-up” holographic QCD

GdT and Brodsky: hep-th/0409074, hep-th/0501022.

• Action for massive fermionic modes on AdSd+1:

S[Ψ,Ψ] =∫

dd+1x√

g Ψ(x, z)(iΓ�D� − μ

)Ψ(x, z).

• Equation of motion:(iΓ�D� − μ

)Ψ(x, z) = 0

[i

(zη�mΓ�∂m +

d

2Γz

)+ μR

]Ψ(x�) = 0.

GdT and Sjb



4 Fermionic Modes

Hard-Wall Model

From Nick Evans• Action for massive fermionic modes on AdS5:

S[Ψ,Ψ] =∫

d4x dz√

g Ψ(x, z)(iΓ�D� − μ

)Ψ(x, z)

• Equation of motion:(iΓ�D� − μ

)Ψ(x, z) = 0

[i

(zη�mΓ�∂m +

d

2Γz

)+ μR

]Ψ(x�) = 0

• Solution (μR = ν + 1/2)

Ψ(z) = Cz5/2 [Jν(zM)u+ + Jν+1(zM)u−]

• Hadronic mass spectrum determined from IR boundary conditions ψ± (z = 1/ΛQCD) = 0

M+ = βν,k ΛQCD, M− = βν+1,k ΛQCD

with scale independent mass ratio

• Obtain spin-J mode Φμ1···μJ−1/2, J > 1

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

• Baryons Spectrum in ”bottom-up” holographic QCD

GdT and Brodsky: hep-th/0409074, hep-th/0501022.

Baryons in Ads/CFT



Baryons

SCGT AdS/QCD and LF Holography Stan Brodsky

Holographic Light-Front Integrable Form and Spectrum

• In the conformal limit fermionic spin-12 modes ψ(ζ) and spin-3

2 modes ψμ(ζ)are two-component spinor solutions of the Dirac light-front equation

αΠ(ζ)ψ(ζ) = Mψ(ζ),

where HLF = αΠ and the operator

ΠL(ζ) = −i

(d

dζ− L + 1

2

ζγ5

),

and its adjoint Π†L(ζ) satisfy the commutation relations

[ΠL(ζ),Π†

L(ζ)]

=2L + 1

ζ2γ5.

• Supersymmetric QM between bosonic and fermionic modes in AdS?



Soft-Wall Model

• Equivalent to Dirac equation in presence of a holographic linear confining potential[i

(zη�mΓ�∂m +

d

2Γz

)+ μR + κ2z

]Ψ(x�) = 0.

• Solution (μR = ν + 1/2, d = 4)

Ψ+(z) ∼ z52+νe−κ2z2/2Lν

n(κ2z2)

Ψ−(z) ∼ z72+νe−κ2z2/2Lν+1

n (κ2z2)

• Eigenvalues

M2 = 4κ2(n + ν + 1)

• Obtain spin-J mode Φμ1···μJ−1/2, J > 1

2 , with all indices along 3+1 from Ψ by shifting di

by shifting dimensions



• Note: in the Weyl representation (iα = γ5β)

iα =

⎛⎝ 0 I

−I 0

⎞⎠ , β =

⎛⎝0 I

I 0

⎞⎠ , γ5 =

⎛⎝I 0

0 −I

⎞⎠ .

• Baryon: twist-dimension 3 + L (ν = L + 1)

O3+L = ψD{�1 . . . D�qψD�q+1 . . . D�m}ψ, L =m∑

i=1

i.

• Solution to Dirac eigenvalue equation with UV matching boundary conditions

ψ(ζ) = C√

ζ [JL+1(ζM)u+ + JL+2(ζM)u−] .

Baryonic modes propagating in AdS space have two components: orbital L and L + 1.

• Hadronic mass spectrum determined from IR boundary conditions

ψ± (ζ = 1/ΛQCD) = 0,

given by

M+ν,k = βν,kΛQCD, M−

ν,k = βν+1,kΛQCD,

with a scale independent mass ratio.



I = 1/2 I = 3/2

0 2L

4 60 2L

4 6

2

0

4

6

8

N (939)

N (1520)

N (2220)N (1535)N (1650)N (1675)N (1700)

N (1680)N (1720)

N (2190)N (2250)

N (2600)

� (1232)

� (1620)

� (1905)

� (2420)

� (1700)

� (1910)� (1920)� (1950)

(b)(a)

(GeV

2 )

� (1930)

56567070

1-20068694A14

Fig: Light baryon orbital spectrum for ΛQCD = 0.25 GeV in the HW model. The 56 trajectory corresponds to L

even P = + states, and the 70 to L odd P = − states.



Non-Conformal Extension of Algebraic Structure (Soft Wall Model)

• We write the Dirac equation

(αΠ(ζ) −M)ψ(ζ) = 0,

in terms of the matrix-valued operator Π

Πν(ζ) = −i

(d

dζ− ν + 1

2

ζγ5 − κ2ζγ5

),

and its adjoint Π†, with commutation relations

[Πν(ζ),Π†

ν(ζ)]

=(

2ν + 1ζ2

− 2κ2

)γ5.

• Solutions to the Dirac equation

ψ+(ζ) ∼ z12+νe−κ2ζ2/2Lν

n(κ2ζ2),

ψ−(ζ) ∼ z32+νe−κ2ζ2/2Lν+1

n (κ2ζ2).

• Eigenvalues

M2 = 4κ2(n + ν + 1).



Glazek and Schaden [Phys. Lett. B 198, 42 (1987)]: (ωB/ωM )2 = 5/8 4κ2 for Δn = 14κ2 for ΔL = 1

2κ2 for ΔS = 1

M2

L

Parent and daughter 56 Regge trajectories for the N and Δ baryon families for κ = 0.5 GeV

82

8• Δ spectrum identical to Forkel and Klempt, Phys. Lett. B 679, 77 (2009)



L+N

M2 (GeV2)

N=1

N=0

0 1 2 3 4 5 60

2

4

6

8

�3/2+(1232)

�3/2+(1600)

�1/2+(1750)

�3/2�(1700)

�1/2�(1620)

�5/2�(1930)

�3/2�(1940)

�1/2�(1900)

�7/2+(1950)

�5/2+(1905)

�3/2+(1920)

�1/2+(1910)

7/2+

�5/2+(2200)3/2

+1/2

+

�7/2�(2200)

�5/2�(2223)

�9/2�(2400)7/2

�

�5/2�(2350)3/2

�

�11/2+(2420)

�9/2+(2300)

�7/2+(2390)5/2

+

11/2+

9/2+

7/2+

5/2+

11/2�

9/2�

�13/2�(2750)

11/2�

9/2�

7/2�

�15/2+(2950)

13/2+

11/2+

9/2+

E. Klempt et al.: Δ∗ resonances, quark models, chiral symmetry and AdS/QCDy, y ( )

H. Forkel, M. Beyer and T. Frederico, JHEP 0707 (2007)077.H. Forkel, M. Beyer and T. Frederico, Int. J. Mod. Phys.E 16 (2007) 2794.



SU(6) S L Baryon State

56 12 0 N 1

2

+(939)32 0 Δ 3

2

+(1232)

70 12 1 N 1

2

−(1535) N 32

−(1520)32 1 N 1

2

−(1650) N 32

−(1700) N 52

−(1675)12 1 Δ 1

2

−(1620) Δ 32

−(1700)

56 12 2 N 3

2

+(1720) N 52

+(1680)32 2 Δ 1

2

+(1910) Δ 32

+(1920) Δ 52

+(1905) Δ 72

+(1950)

70 12 3 N 5

2

−N 7

2

−

32 3 N 3

2

−N 5

2

−N 7

2

−(2190) N 92

−(2250)12 3 Δ 5

2

−(1930) Δ 72

−

56 12 4 N 7

2

+N 9

2

+(2220)32 4 Δ 5

2

+ Δ 72

+ Δ 92

+ Δ 112

+(2420)

70 12 5 N 9

2

−N 11

2

−(2600)32 5 N 7

2

−N 9

2

−N 11

2

−N 13

2

−



Space-Like Dirac Proton Form Factor

• Consider the spin non-flip form factors

F+(Q2) = g+

∫dζ J(Q, ζ)|ψ+(ζ)|2,

F−(Q2) = g−∫

dζ J(Q, ζ)|ψ−(ζ)|2,

where the effective charges g+ and g− are determined from the spin-flavor structure of the theory.

• Choose the struck quark to have Sz = +1/2. The two AdS solutions ψ+(ζ) and ψ−(ζ) correspond

to nucleons with Jz = +1/2 and −1/2.

• For SU(6) spin-flavor symmetry

F p1 (Q2) =

∫dζ J(Q, ζ)|ψ+(ζ)|2,

Fn1 (Q2) = −1

3

∫dζ J(Q, ζ)

[|ψ+(ζ)|2 − |ψ−(ζ)|2] ,where F p

1 (0) = 1, Fn1 (0) = 0.



• Scaling behavior for large Q2: Q4F p1 (Q2) → constant Proton τ = 3

0

0.4

0.8

1.2

0 10 20 30Q2 (GeV2)

Q4 F

p 1 (Q

2 ) (

GeV

4 )

9-20078757A2

SW model predictions for κ = 0.424 GeV. Data analysis from: M. Diehl et al. Eur. Phys. J. C 39, 1 (2005).



• Scaling behavior for large Q2: Q4Fn1 (Q2) → constant Neutron τ = 3

0

-0.1

-0.2

-0.3

-0.40 10 20 30

Q2 (GeV2)

Q4 F

n 1 (Q

2 ) (

GeV

4 )

9-20078757A1

SW model predictions for κ = 0.424 GeV. Data analysis from M. Diehl et al. Eur. Phys. J. C 39, 1 (2005).



0 1 2 3 4 5 60

0.5

1

1.5

2

Spacelike Pauli Form Factor

Q2(GeV2)

Harmonic Oscillator Confinement

Normalized to anomalous moment

F p2 (Q2)

κ = 0.49 GeV

G. de Teramond, sjb

PreliminaryFrom overlap of L = 1 and L = 0 LFWFs

AdS/QCD No chiral

divergence!

F2(Q2) = 1 + O Q2

mπmp

in chiral perturbation theory



String Theory

AdS/CFTSSS///////////////////////////////////////////

Semi-Classical QCD / Wave EquationsSemi-Classical QCD / Wave Equations////////////////////////////

Mapping of Poincare’ and Conformal SO(4,2) symmetries of 3+1

space to AdS5 space

Integrable!

Boost Invariant 3+1 Light-Front Wave Equations

/

Boost Invariant 3+1 Light-Front Wave Equationsh

Hadron Spectra, Wavefunctions, Dynamics

ggggggggggggggggggggggggggggggggggggggggh

ve

/

AdS/QCDConformal behavior at short

distances+ Confinement at large distance

Counting rules for Hard Exclusive Scattering

Regge Trajectories

Holography

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

Goal: First Approximant to QCD

QCD at the Amplitude Level

89

Heisenberg Matrix FormulationLight-Front QCD


HQCDLF |Ψh >= M2

h|Ψh >

HQCDLF =

∑i

[m2 + k2

⊥x

]i + HintLF

LQCD → HQCDLF

HintLF : Matrix in Fock Space

Physical gauge: A+ = 0



HQCDLC |Ψh〉 = M2

h |Ψh〉Heisenberg Equation

Light-Front QCD

Use AdS/QCD basis functions

91

Stan Brodsky SLAC



Vary, Harinandrath, Maris, sjb

92

Pauli, Hornbostel, Hiller, McCartor, sjb

Use AdS/CFT orthonormal LFWFs as a basis for diagonalizing

the QCD LF Hamiltonian

Vary, Harinandrath, M

Pauli, Hornbostel, Hiller, McCart

• Good initial approximant

• Better than plane wave basis

• DLCQ discretization -- highly successful 1+1

• Use independent HO LFWFs, remove CM motion

• Similar to Shell Model calculations



New Perspectives for QCD from AdS/CFT

• LFWFs: Fundamental frame-independent description of hadrons at amplitude level

• Holographic Model from AdS/CFT : Confinement at large distances and conformal behavior at short distances

• Model for LFWFs, meson and baryon spectra: many applications!

• New basis for diagonalizing Light-Front Hamiltonian

• Physics similar to MIT bag model, but covariant. No problem with support 0 < x < 1.

• Quark Interchange dominant force at short distances

93

• Consider five-dim gauge fields propagating in AdS5 space in dilaton background ϕ(z) = κ2z2

S = −14

∫d4x dz

√g eϕ(z) 1

g25

G2

• Flow equation1

g25(z)

= eϕ(z) 1g25(0)

or g25(z) = e−κ2z2

g25(0)

where the coupling g5(z) incorporates the non-conformal dynamics of confinement

• YM coupling αs(ζ) = g2Y M (ζ)/4π is the five dim coupling up to a factor: g5(z) → gY M (ζ)

• Coupling measured at momentum scale Q

αAdSs (Q) ∼

∫ ∞

0ζdζJ0(ζQ)αAdS

s (ζ)

• Solution

αAdSs (Q2) = αAdS

s (0) e−Q2/4κ2.

where the coupling αAdSs incorporates the non-conformal dynamics of confinement

Running Coupling from Modified AdS/QCDDeur, de Teramond, sjb

Q (GeV)

�s(

Q)/�

�s,g1/� (pQCD)�s,g1/� world data

�s,�/� OPAL

AdS/QCD with�s,g1 extrapolation

AdS/QCDLF Holography

Lattice QCDHall A/CLAS PLB 650 4 244JLab CLAS PLB 665 249

�s,F3/�GDH limit

0

0.2

0.4

0.6

0.8

1

10-1

1 10

Running Coupling from Light-Front Holography and AdS/QCD

αAdSs (Q)/π = e−Q2/4κ2

αs(Q)π

gnormalization

Deur, de Teramond, sjb, (preliminary)

κ = 0.54 GeV

�s/�

pQCD evol. eq.

�s,g1/� Hall A/CLAS JLab �s,g1/� CLAS JLab

Cornwall

Fit

GDH limit

Godfrey-Isgur

Bloch et al.

Burkert-Ioffe

Fischer et al.

Bhagwat et al.Maris-Tandy

Q (GeV)

Lattice QCDAdS/CFT(norm. to �, �=0.54),

10-1

1

10-1

1

10-1

1 1010-1

1 10


-2.25

-2

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

10-1

1 10Q

d�sef

f /dlo

g(Q

2 )

Bjorken sum ruleconstraint on �s,g1

GDH sum ruleconstraint on �s,g1

Lattice QCD

Hall A/CLAS PLB 650 4 244JLab CLAS PLB665 249

�s,F3/�

AdS/QCD LFHolography


βAdS(Q2) =d

d log Q2αAdS

s (Q2) =πQ2

4κ2e−Q2/4κ2


Conjectured behavior of the full β-function of QCD

β(Q → 0) = β(Q → ∞) = 0, (1)

β(Q) < 0, for Q > 0, (2)

dβdQ

∣∣Q=Q0

= 0, (3)

dβdQ < 0, for Q < Q0,

dβdQ > 0, for Q > Q0. (4)

1. QCD is conformal in the far UV and deep IR

2. Anti-screening behavior of QCD which leads to asymptotic freedom

3. Hadronic-partonic transition: the minimum is an absolute minimum

4. Since there is only one transition (4) follows from the above




Running Coupling for Static Potential from AdS/QCD

r (fm)

�s(

r)

Lattice QCD�s,g1:Jlab+pQCD+sum rules


AdS/QCD LF Holography

0

0.5

1

1.5

2

2.5

3

3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

99

Stan Brodsky SLAC



• Sivers Effect in SIDIS, Drell-Yan

• Double Boer-Mulders Effect in DY

• Diffractive DIS

• Heavy Quark Production at Threshold

100

Applications of Nonperturbative Running Coupling from AdS/QCD

All involve gluon exchange at small momentum transfer



T-OddPseudo-

11-2001 8624A06

S

current quark jet

final state interaction

spectator system

proton

e–

�*

e–

quark

Single-spin asymmetries

Leading Twist Sivers Effect

�Sp ·�q×�pq

Hwang, Schmidt, sjb

Light-Front Wavefunction S and P- Waves

QCD S- and P-Coulomb Phases

--Wilson Line

101

i

Collins, BurkardtJi, Yuan

Leading-Twist Rescattering Violates pQCD Factorization!



Final-State Interactions Produce Pseudo T-Odd (Sivers Effect)

• Leading-Twist Bjorken Scaling!

• Requires nonzero orbital angular momentum of quark

• Arises from the interference of Final-State QCD

Coulomb phases in S- and P- waves;

• Wilson line effect -- gauge independent

• Relate to the quark contribution to the target proton

anomalous magnetic moment and final-state QCD phases

• QCD phase at soft scale! Nonperturbative QCD

• New window to QCD coupling and running gluon mass in the IR

• QED S- and P-wave Coulomb phases infinite -- difference of phases

finite!

�S ·�p jet ×�qi

11-2001 8624A06

S

current quark jet

final state interaction

spectator system

proton

e–

�*

e–

quark

102

Stan Brodsky SLAC



103

Features of Soft-Wall AdS/QCD

• Single-variable frame-independent radial Schrodinger equation

• Massless pion (mq =0)

• Regge Trajectories: universal slope in n and L

• Valid for all integer J & S. Spectrum is independent of S

• Dimensional Counting Rules for Hard Exclusive Processes

• Phenomenology: Space-like and Time-like Form Factors

• LF Holography: LFWFs; broad distribution amplitude

• No large Nc limit

• Add quark masses to LF kinetic energy

• Systematically improvable -- diagonalize HLF on AdS basis

Stan Brodsky SLAC



104

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 3 quarks -- Large Nc limit not required

• Break Conformal symmetry with dilaton

• Dilaton introduces confinement -- positive exponent

• Origin of Linear and HO potentials: Stochastic arguments (Glazek); General ‘classical’ potential for Dirac Equation (Hoyer)

• Effective Charge from AdS/QCD

• Conformal Dimensional Counting Rules for Hard Exclusive Processes

• Use CRF (LF Constituent Rest Frame) to reconstruct 3D Image of Hadrons (Glazek, de Teramond, sjb)



H(pe+)p

e+

e−

Z

Formation of Relativistic Anti-Hydrogen

Munger, Schmidt, sjb

Measured at CERN-LEAR and FermiLab

γ∗

Coulomb field

ZCoalescence of off-shell co-moving positron and antiproton

“Hadronization” at the Amplitude Level

Wavefunction maximal at small impact separation and equal rapidity

b⊥ ≤ 1mredα

yp � ye+

ffff ff

105

eeeeeeeeeee

CCCCCCCCCCCCCCCCCCCCCCCCCCCoooooooooo



Hadronization at the Amplitude Level

e+

e−

γ∗g

q

q

γ∗qqqqqq

gggggggg

Construct helicity amplitude using Light-Front Perturbation theory; coalesce quarks via LFWFs

ψ(x,�k⊥, λi)

pH

x,�k⊥

1 − x,−�k⊥

τ = x+

Event amplitude generator

Hadronization at the Amplitude Level

e+

e−

γ∗g

q

q

γ∗qqqqqq

gggggggg

ψ(x,�k⊥, λi)

pH

x,�k⊥

1 − x,−�k⊥

τ = x+

Event amplitude generator

AdS/QCD Hard Wall

Confinement:

Capture if ζ2 = x(1 − x)b2⊥ > 1

Λ2QCD

i.e.,M2 = k2

⊥x(1−x) < Λ2

QCD

Stan Brodsky SLAC



• Coalesce color-singlet cluster to hadronic state if

• The coalescence probability amplitude is the LF wavefunction

• No IR divergences: Maximal gluon and quark wavelength from confinement

108

Features of LF T-Matrix Formalism“Event Amplitude Generator”

M2n =

n∑i=1

k2⊥i + m2

i

xi< Λ2

QCD


xiP+, xi

�P⊥ + �k⊥i

P+ = P0 + Pz

P+, �P⊥

Stan Brodsky SLAC



• 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 Lz =0

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

109

Features of LF T-Matrix Formalism“Event Amplitude Generator”

xiP+, xi

�P⊥ + �k⊥i

P+ = P0 + Pz

P+, �P⊥



p

u u

d

Baryon can be made directly within hard subprocess

nactive = 6

g g

φp(x1, x2, x3) ∝ Λ2QCD

110

Collision can produce 3 collinear quarks

Coalescence within hard subprocess

BjorkenBlankenbecler, Gunion, sjb

Berger, sjb Hoyer, et al: Semi-Exclusive

neff = 8

neff = 2nactive - 4qq → Bq

uu → pd

Small color-singletColor Transparent

Minimal same-side energyd

Sickles; sjb



p

u u

d

Baryon made directly within hard subprocess

nactive = 6

neff = 8

neff = 2nactive - 4uu → pd

Small color-singletColor Transparent

Minimal same-side energy

g gd

b⊥ � 1/pT

QGP

b⊥ � 1 fmFormation Time

proportional to Energy

111

Sickles; sjb



Proton production dominated by color-transparent direct high neff subprocesseseff

Tx0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

)T

n(x

2

3

4

5

6

7

8

9

10

0�) for Tn(x

0-10%60-80%

Tx0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

2

- + h+h) for Tn(x

0-10%60-80%

Power-law exponent n(xT ) for �0 and h spectra in central and peripheral Au+Au collisions at√sNN = 130 and 200 GeV

S. S. Adler, et al., PHENIX Collaboration, Phys. Rev. C 69, 034910 (2004) [nucl-ex/0308006].

Peripheral

Central h+ includes protons

112



2

4

6

8

10

12

10-2

10-1

nef

f

�s=38.8/31.6 GeV E706�s=62.4/22.4 GeV PHENIX/FNAL�s=62.8/52.7 GeV R806�s=52.7/30.6 GeV R806�s=200/62.4 GeV PHENIX�s=500/200 GeV UA1�s=900/200 GeV UA1�s=1800/630 GeV CDF

�s=1800/630 GeV CDF � CDF jets�s=1800/630 GeV D0 � D0 jets

Leading-Twist PQCD

Edσ

d3p(pp → HX) =

F (xT , θCM = π/2)pneff

T

xT = 2pT /√

s

γ, jetsπ

Arleo, AurencheHwang, Sickles, sjb



partN0 50 100 150 200 250 300 350

yiel

d/t

rig

ger

0

0.02

0.04

0.06

0.08

0.1

< 4.0 GeV/cT

trigger: 2.5 < p

< 2.5 GeV/cT

associated: 1.8 < p

meson-meson, near sidebaryon-meson, near sidemeson-meson, away sidebaryon-meson, away side

proton trigger:# same-side

particles decreases with

centrality

partNProton production more dominated by

color-transparent direct high-neff subprocesses

Anne Sickles

114



Chiral Symmetry Breaking in AdS/QCD

A ∝ mq B ∝< ψψ >

We consider the action of the X field which encodes the effects of CSB inAdS/QCD:

SX =∫

d4xdz√

g(g�m∂�X∂mX − μ2

XX2), (1)

with equations of motion

z3∂z

(1z3

∂zX

)− ∂ρ∂

ρX −(

μXR

z

)2

X = 0. (2)

The zero mode has no variation along Minkowski coordinates

∂μX(x, z) = 0,

thus the equation of motion reduces to[z2∂2

z − 3z ∂z + 3]X(z) = 0. (3)

for (μXR)2 = −3, which corresponds to scaling dimension ΔX = 3. The solutionis

X(z) = 〈X〉 = Az + Bz3, (4)

where A and B are determined by the boundary conditions.

(2)

Erlich, Katz, Son, StephanovBabington, Erdmenger, Evans,

Kirsch, Guralnik, Thelfall

Expectation value taken inside hadron

de Teramond, Shrock, sjb

Stan Brodsky SLAC



116

δM2 = −2mq < ψψ > ×∫

dz φ2(z)z2

de Teramond, Shrock, sjb

Chiral Symmetry Breaking in AdS/QCD

Stan BrodskySCGTb


2q

2 2

• Chiral symmetry breaking effect in AdS/QCD depends on weighted z2 distribution, not constant condensate

• z2 weighting consistent with higher Fock states at periphery of hadron wavefunction

• AdS/QCD: confined condensate

• Suggests “In-Hadron” Condensates

•

Erlich et al.

ΩΛ = 0.76(expt)

(ΩΛ)EW ∼ 1056

(ΩΛ)QCD ∼ 1045

DARK ENERGY ANDTHE COSMOLOGICAL CONSTANT PARADOX

A. ZEE

Department of Physics, University of California, Santa Barbara, CA 93106, USAKavil Institute for Theoretical Physics, University of California,

Santa Barbara, CA 93106, [email protected]

“One of the gravest puzzles of theoretical physics”

QCD Problem Solved if Quark and Gluon condensates reside

within hadrons, not LF vacuumShrock, sjb

Light-Front (Front Form)

Formalism



Use Dyson-Schwinger Equation for bound-state quark propagator: find confined condensate

g

qb

k >1

ΛQCDλ < ΛQCD

Maximum wavelength of bound quarks and gluons

B-Meson

< b|qq|b > not < 0|qq|0 >

Shrock, sjb



Pion mass and decay constant.

Pieter Maris, Craig D. Roberts (Argonne, PHY) , Peter C. Tandy (Kent State U.) . ANL-PHY-8753-TH-97,

KSUCNR-103-97, Jul 1997. 12pp.

Published in Phys.Lett.B420:267-273,1998.

e-Print: nucl-th/9707003

Pi- and K meson Bethe-Salpeter amplitudes.

Pieter Maris, Craig D. Roberts (Argonne, PHY) . ANL-PHY-8788-TH-97, Aug 1997. 34pp.

Published in Phys.Rev.C56:3369-3383,1997.


Concerning the quark condensate.

K. Langfeld (Tubingen U.) , H. Markum (Vienna, Tech. U.) , R. Pullirsch (Regensburg U.) , C.D. Roberts

(Argonne, PHY & Rostock U.) , S.M. Schmidt (Tubingen U. & HGF, Bonn) . ANL-PHY-10460-TH-2002,

MPG-VT-UR-239-02, Jan 2003. 7pp.

Published in Phys.Rev.C67:065206,2003.


− 〈qq〉πζ = fπ〈0|qγ5q|π〉 .

“In-Meson Condensate”Valid even for mq → 0

fπ nonzero



In presence of quark masses the Holographic LF wave equation is (ζ = z)[− d2

dζ2+ V (ζ) +

X2(ζ)ζ2

]φ(ζ) = M2φ(ζ), (1)

and thus

δM2 =⟨

X2

ζ2

⟩. (2)

The parameter a is determined by the Weisberger term

a =2√x

.

Thus

X(z) =m√x

z −√x〈ψψ〉z3, (3)

and

δM2 =∑

i

⟨m2

i

xi

⟩− 2

∑i

mi〈ψψ〉〈z2〉 + 〈ψψ〉2〈z4〉, (4)

where we have used the sum over fractional longitudinal momentum∑

i xi = 1.

Mass shift from dynamics inside hadronic boundary

de Teramond, Sjb

Stan Brodsky SLAC



“Confined QCD Condensates”

122

Shrock, sjb

Casher Susskind

Maris, Roberts, Tandy

Quark and Gluon condensates reside within

hadrons, not LF vacuum

Shroc

CasSuss

Maris, RTan

M i• Bound-State Dyson-Schwinger Equations

• Spontaneous Chiral Symmetry Breaking within infinite-component LFWFs

• Finite size phase transition - infinite # Fock constituents

• AdS/QCD Description -- CSB is in-hadron Effect

• Analogous to finite-size superconductor!

• Phase change observed at RHIC within a single-nucleus-nucleus collisions-- quark gluon plasma!

• Implications for cosmological constant -- reduction by 45 orders of magnitude!

Stan Brodsky SLAC



x−αR

123

−αR

• Casher & Susskind model shows that spontaneous chiral symmetry breaking can occur in the finite domain of a hadronic LFWF

• Infinite number of partons required, but this is a feature of QCD LFWFs --

• Regge behavior of DIS due to behavior of structure functions (LFWFs squared )

• A.H. Mueller: BFKL Pomeron derived from the multi-gluon Fock States of the quarkonium LFWF

• F. Antonuccio, S. Dalley, sjb: Construct soft-gluon LFWF via ladder operators

• LF Vacuum Trivial up to zero modes

Stan Brodsky SLAC



124

Shrock and sjb

• Color Confinement: Maximum Wavelength of Quark and Gluons

• Conformal symmetry of QCD coupling in IR

• Conformal Template (BLM, CSR, ...)

• Motivation for AdS/QCD

• QCD Condensates inside of hadronic LFWFs

• Technicolor: confined condensates inside of technihadrons -- alternative to Higgs

• Simple physical solution to cosmological constant conflict with Standard Model

Stan Brodsky

Light-Front Holography and AdS/QCD: A New Approximation to QCD

December 9, 2009

SCGT09

Dec 2009 Nagoya Global COE Workshop "Strong Coupling Gauge Theories in LHC Era"


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