Lesson from QED and Lamb Shift:
maximum wavelength of bound quarks and gluons
k>
1
ΛQCD
λ < ΛQCD
g
b̄
q
BMeson
Shrock, sjb
gluon and quark propagators cutoff in IR
because of color confinement
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
47
Stan Brodsky
SLAC
Maximal Wavelength of Confined Fields
•
Colored elds conned to nite domain
•
All perturbative calculations regulated in IR
•
High momentum calculations unaected
•
Boundstate DysonSchwinger Equation
•
Analogous to Bethes Lamb Shift Calculation
(x − y)2 < Λ−2
QCD
Quark and Gluon vacuum polarization insertions
decouple: IR fixed Point
J. D. Bjorken,
SLACPUB 1053
Cargese Lectures 1989
A strictly-perturbative space-time region can be defined as one which
has 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).
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
48
Stan Brodsky
SLAC
IR Conformal Window for QCD?
•
Dyson-Schwinger Analysis:
QCD gluon coupling has IR
Fixed Point
•
Evidence from Lattice Gauge Theory
•
Dene coupling from observable: indications of IR
xed point for QCD eective charges
Shrock, de Teramond, sjb
•
Conned gluons and quarks have maximum wavelength:
Decoupling of QCD vacuum polarization at small Q2
SerberUehling
•
+
Q2
α
2
Q2 << 4m2
Π(Q ) → 15π m2
Justies application of AdS/CFT in strongcoupling−
conformal window
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
49
Stan Brodsky
SLAC
The Holographic Correspondence
• In the semi-classical approximation to QCD with massless quarks and no quantum loops the β 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
R2
ds = 2 (ημν dxμ dxν − dz 2 ).
z
2
• Semi-classical correspondence as a first approximation to QCD (strongly coupled at all scales).
• xμ → λxμ , z → λz, 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 correspond 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 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 Regge dependence can be obtained (Soft-Wall Model) Karch, Katz, Son and Stephanov (2006).
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
50
Stan Brodsky
SLAC
AdS/CFT
•
Use mapping of conformal group SO4,2 to AdS5
•
Scale Transformations represented by wavefunction in
2 2
z → λz
x2
5th dimension
μ → λ xμ
•
Match solutions at small z to conformal twist dimension
of hadron wavefunction at short distances ψ(z) ∼ z Δ at z → 0
•
Hard wall model: Connement at large distances and
conformal symmetry in interior
•
Truncated space simulates bag boundary conditions
0 < z < z0
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
ψ(z0) = 0
AdS/QCD
Q
51
z0 = Λ 1
QCD
Stan Brodsky
SLAC
Bosonic Solutions: Hard Wall Model
• Conformal metric:
ds2
= gm
dx dxm .
x
=
(xμ , z),
gm →
R2 /z 2
ηm .
• Action for massive scalar modes on AdSd+1 :
√
1
√
dd+1 x g 12 g m ∂ Φ∂m Φ − μ2 Φ2 ,
S[Φ] =
g → (R/z)d+1 .
2
• Equation of motion
1 ∂ √ m ∂
2
g
g
Φ
+
μ
Φ = 0.
√
m
g ∂x
∂x
• Factor out dependence along xμ -coordinates , ΦP (x, z) = e−iP ·x Φ(z), Pμ P μ = M2 :
2 2
2
2
2
z ∂z − (d − 1)z ∂z + z M − (μR) Φ(z) = 0.
• Solution: Φ(z) → z Δ as z → 0,
Φ(z)
d
2
z) =JCz JΔ−
d (zM) ,
=Φ(x,
Cz d/2
(zM)
Δ=2+L
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
Δ−d/2
2
d=4
Δ=
1
2
d + d2 + 4μ2 R2 .
(μR)2 = L2 − 4
AdS/QCD
52
Stan Brodsky
SLAC
Let Φ(z) = z 3/2 φ(z)
AdS Schrodinger Equation for bound state
of two scalar constituents:
2
d
1 − 4L
2
φ(z)
=
M
− 2−
φ(z)
dz
4z 2
2
L: lightfront orbital angular
momentum
Derived from variation of Action in AdS5
Hard wall model: truncated space
φ(z = z0 = Λ1 ) = 0.
c
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
53
Stan Brodsky
SLAC
Match fao at sma z to conformal twistdimensio
at short distances
• Pseudoscalar mesons: O
O2+L
3+L = ψγ5 D{1 . . . Dm } ψ (Φμ = 0 gauge).
twis
Δ=2+L
• 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,k ΛQCD
• Normalizable AdS modes Φ(z)
5
4
4
Φ(z)
2
3
Φ(z)
2
zΔ
0
1
-2
0
-4
2-2006
8721A7
0
1
2
z
3
4
2-2006
8721A8
1
z0
0
z0 = Λ
QCD
1
2
z
4
3
S = Fig:
0 Meson orbital and radial AdS modes for ΛQCD = 0.32 GeV.
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
54
Stan Brodsky
SLAC
4
3
2
Φ(z)
Φ(z) 2
0
1
-2
-4
0
2-2007
8721A18
0
1
2
z
3
4
2-2007
8721A19
0
1
2
z
3
4
Fig: Orbital and radial AdS modes in the hard wall model for ΛQCD = 0.32 GeV .
S=0
(b)
S=1
4
π2 (1670)
a0 (1450)
a2 (1320)
f1 (1285)
b1 (1235)
2
f4 (2050)
a4 (2040)
ρ (1700)
ρ3 (1690)
2
(GeV )
(a)
ω3 (1670)
ω (1650)
ω (782)
ρ (770)
π (140)
f2 (1270)
a1 (1260)
0
0
1-2006
8694A16
2
4
0
L
2
L
Fig: Light meson and vector meson orbital spectrum ΛQCD
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
4
AdS/QCD
55
= 0.32 GeV
Stan Brodsky
SLAC
Higher Spin Bosonic Modes HW
• Each hadronic state of integer spin S ≤ 2 is dual to a normalizable string mode
Φ(x, z)μ1 μ2 ···μS = μ1 μ2 ···μS e−iP ·x ΦS (z).
with four-momentum Pμ and spin polarization indices along the 3+1 physical coordinates.
• Wave equation for spin S -mode W. S. l’Yi, Phys. Lett. B 448, 218 (1999)
2 2
2
2
2
z ∂z − (d+1−2S)z ∂z + z M −(μR) ΦS (z) = 0,
• Solution
S=
Φ(z)
z S
R
d
Φ(z)S = Ce−iP ·x z 2 JΔ− d (zM) (P )μ1 μ2 ···μS ,
2
• We can identify the conformal dimension:
1
2
2
2
Δ = d + (d − 2S) + 4μ R .
2
• Normalization:
Λ−1
QCD
Rd−2S−1
0
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
dz
z d−2S−1
AdS/QCD
56
Φ2S (z) = 1.
Stan Brodsky
SLAC
Soft-Wall Model
• Soft-wall model [Karch, Katz, Son and Stephanov (2006)] retain conformal AdS metrics but introduce
smooth cutoff wich depends on the profile of a dilaton background field ϕ(z) = ±κ2 z 2
√ ϕ(z)
d
S = d x dz g e
L,
1
2
μ2 Φ2
• Equation of motion for scalar field L =
−
2 2 2 2
2
2
2
z ∂z − d − 1 ∓ 2κ z z ∂z + z M − (μR) Φ(z) = 0
with (μR)2
g m ∂ Φ∂m Φ
≥ −4. See also [Metsaev (2002), Andreev (2006)]
+ sign: Fen Zuo
• LH holography requires ‘plus dilaton’ ϕ = +κ2 z 2 . Lowest possible state (μR)2 = −4
M2 = 4κ2 n,
Φn (z) ∼ z 2 e−κ
2 z2
Ln (κ2 z 2 )
Φ0 (z) a chiral symmetric bound state of two massless quarks with scaling dimension 2: the pion
Massless pion
Thursday, September 24, 2009
• Erlich, Karch, Katz, Son, Stephanov
• de Teramond, sjb
AdS Soft-Wall Schrodinger Equation for
bound state of two scalar constituents:
d
1 − 4L
2
− 2−
+ U (z) φ(z) = M φ(z)
2
dz
4z
2
2
U (z) = κ z + 2κ (L + S − 1)
4 2
2
Derived from variation of Action Φ(z)
+κ2 z 2
e
=e
Dilaton-Modified AdS5
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
58
Stan Brodsky
SLAC
2 2
ds2 = eκ
z
R2
2
2
2
2
2
(dx
−
dx
−
dx
−
dx
−
dz
)
3
3
0
1
2
z
/2
/2
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
59
Agrees with
Klebanov and
Maldacena for
positive-sign
exponent of dilaton
Stan Brodsky
SLAC
2
R
ds2 = eκ z 2 (dx20 − dx21 − dx23 − dx23 − dz 2 )
z
d a “warped” metric of the form
2 2
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;
z→∞
y = R/z
z→0
e
Figure 5
Klebanov and Maldacena
ds2 = eA(y) (−dx20 + dx21 + dx23 + dx23 ) + dy 2
Thursday, September 24, 2009
Quark separation
increases with L
6
5
4
Φ(z)
Φ(z)
0
2
-5
0
2-2007
8721A20
0
4
8
z
0
2-2007
8721A21
4
8
z
Soft Wall
Model
Fig: Orbital and radial AdS modes in the soft wall model for κ = 0.6 GeV .
S=0
4
Pion has
zero mass!
sss!
π (1300)
π (140)
0
8-2007
8694A19
2
4 0
2
4
n
Light meson orbital (a) and radial (b) spectrum for κ
Thursday, September 24, 2009
mq = 0
π (140)
L
QNP09 IHEP Beijing
September 25, 2009
Pion mass
automatically
zero!
π (1800)
b1 (1235)
2
0
S=0
(b)
π2 (1670)
2
(GeV )
(a)
AdS/QCD
61
= 0.6 GeV.
Stan Brodsky
SLAC
Quark separation increases with L
6
Φ(z)
(a)
L=3
Φ(z)
4
L=2
Φ(z)
((b)
b)
5
Φ(z)
n=2
n=0
0
L=1
2
n=1
--5
5
n=3
L=0
0
2-2007
8721A20
0
4
8
z
z
S = 1 ρ (1690)
2
4
0
4
z
8
z
u
f4 (2050)
a4 (2040)
(a)
(GeV )
2-2007
8721A21
S=1
3
a2 (1320)
2
ω3 (1670)
ω (782)
ρ (770)
f2 (1270)
0
0
5-2006
8694A20
2
L
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
4
AdS/QCD
62
Stan Brodsky
SLAC
Quark separation increases with L
6
Φ(z)
(a)
L=3
Φ(z)
4
L=2
Φ(z)
(b)
5
Φ(z)
n=2
n=0
0
L=1
2
n=1
-5
n=3
L=0
0
2-2007
8721A20
0
4
8
z
z
0
4
2
S=1
3
ω3 (1670)
ω (782)
ρ (770)
ρ (1700)
ρ (1450)
a2 (1320)
2
z
u
(b)
S = 1 ρ (1690)
4
8
z
f4 (2050)
a4 (2040)
(a)
(GeV )
2-2007
8721A21
ρ (770)
f2 (1270)
0
0
5-2006
8694A20
2
2
4
n
L
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
0
4
AdS/QCD
63
Stan Brodsky
SLAC
1−−
2++
3−−
4++
JPC
M2
L
Parent and daughter Regge trajectories for the I
and the I
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
= 1 ρ-meson family (red)
= 0 ω -meson family (black) for κ = 0.54 GeV
AdS/QCD
64
Stan Brodsky
SLAC
Higher Spin Bosonic Modes SW
Soft-wall model
LF Schrödinger wave equation
• Effective
2
2
2
2
d
1 − d4L
1 − 4L
4 2 4 2 2 2
2
ζ +(L+
2κ (L+
S −1) φ
M2M
φS (ζ)
− 2 − − dζ22 − +4ζκ2 z ++κ 2κ
S −1)
φSS(ζ)
(z)= =
φS (z)
dz
4z
Same slope in n and L
M2 = 2κ2 (2n + 2L + S).
with eigenvalues
• Compare with Nambu string result (rotating flux tube): Mn2 (L) = 2πσ (n + L + 1/2) .
S=1
(a)
(b)
ρ (1700)
ρ (1450)
a2 (1320)
2
S=1
ρ3 (1690)
2
(GeV )
4
f4 (2050)
a4 (2040)
ω3 (1670)
ω (782)
ρ (770)
ρ (770)
f2 (1270)
0
0
5-2006
8694A20
2
4
0
L
2
4
n
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).
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
65
Stan Brodsky
SLAC
Hadron Form Factors from AdS/CFT
Propagation of external perturbation suppressed inside AdS.
z) =
zQK
(zQ)
interaction
occurs
in 1the
large-r conformal region. Importa
At large enough Q ∼ r/R2 , theJ(Q,
contribution to the FF integral from the boundary near z ∼ 1/Q.
F (Q2)I→F =
J(Q, z), Φ(z)
1
Q2
High
from
small z ~ 1/Q
dz
Φ (z)J(Q, z)ΦI (z)
z3 F
J(Q, z)
Φ(z)
0.8
Polchinski, Strassler
de Teramond, sjb
0.6
0.4
0.2
1
2
3
Consider a specific AdS mode Φ(n) dual to an
scales as Φ(n) ∼ z Δn . Thus:
where τ = Δn − σn , σn =
1
F (Q ) →
Q2
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
i=1 σi .
4
5
n partonic Fock state |n. At small z , Φ
2
n
z
τ −1
,
Dimensional Quark Counting Rules:
General result from
AdS/CFT and Conformal Invariance
The twist is equal to the number of partons, τ = n.
AdS/QCD
66
Stan Brodsky
SLAC
Spacelike pion form factor from AdS/CFT
1
Fπ (q 2) 0.8
0.6
Data Compilation
Baldini, Kloe and Volmer
0.4
0.2
0
-10
-8
-6
-4
-2
0
q 2(GeV 2)
Soft Wall: Harmonic Oscillator Connement
Hard Wall: Truncated Space Connement
One parameter set by pion decay constan
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
67
de Teramond, sjb
See also: Radyushkin
Stan Brodsky
SLAC
sjb and GdT
Grigoryan and Radyushkin
Current Matrix Elements in AdS Space (SW)
• Propagation of external current inside AdS space described by the AdS wave equation
2 2
2 2
2 2
z ∂z − z 1 + 2κ z ∂z − Q z Jκ (Q, z) = 0.
• Solution bulk-to-boundary propagator
2
Q
2 2
Jκ (Q, z) = Γ 1 + 2 U
,
0,
κ
z ,
4κ
4κ2
Q2
where U (a, b, c) is the confluent hypergeometric function
Γ(a)U (a, b, z) =
0
∞
Soft Wall
Model
e−zt ta−1 (1 + t)b−a−1 dt.
• Form factor in presence of the dilaton background ϕ = κ2 z 2
dz −κ2 z 2
F (Q2 ) = R3
e
Φ(z)Jκ (Q, z)Φ(z).
z3
• For large Q2 4κ2
Jκ (Q, z) → zQK1 (zQ) = J(Q, z),
the external current decouples from the dilaton field.
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
68
Stan Brodsky
SLAC
Spacelike pion form factor from AdS/CFT
1
Fπ (q 2) 0.8
0.6
Data Compilation
Baldini, Kloe and Volmer
0.4
0.2
0
-10
-8
-6
-4
-2
0
q 2(GeV 2)
Soft Wall: Harmonic Oscillator Connement
Hard Wall: Truncated Space Connement
One parameter set by pion decay constan
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
69
de Teramond, sjb
See also: Radyushkin
Stan Brodsky
SLAC
2
• Analytical continuation to time-like region q 2 → −q 2 (Mρ =
750=MeV)
M4κ
750 MeV
ρ ==2κ
• Strongly coupled semiclassical gauge/gravity limit hadrons have zero widths (stable).
2
log IFπ (q2)I
1
0
-1
-2
-3
-10
-5
0
q2 (GeV2)
Space and time-like pion form factor for κ
5
10
7-2007
8755A4
= 0.375 GeV in the SW model.
• Vector Mesons: Hong, Yoon and Strassler (2004); Grigoryan and Radyushkin (2007).
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
70
Stan Brodsky
SLAC
Spacelike and timelike pion form factor
κ = 0.534 GeV
Q2 (GeV2 )
GdT and SJB
preliminary
Q2 (GeV2 )
|π >= ψqq̄ |q q̄ > + ψqq̄qq̄ |q q̄q q̄ >
Γρ = 120 MeV, Γρ = 300 MeV
Pqq̄qq̄ = 15%
Q2 (GeV2 )
Thursday, September 24, 2009
Dressed soft-wall current bring in higher
Fock states and more vector meson poles
π+
e+
e−
γ∗
π−
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
72
Stan Brodsky
SLAC
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
Q2
Γ 1+ 4κ2
2
.
F (Q ) = Γ(τ ) Q2
Γ τ + 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+
F (Q2 ) = 1
Q2
4κ2
, N = 2,
2
2
Q
+ 4κ
2
···
F (Q2 ) = 2
Q
1 + 4κ
2
2
Q2
+ 4κ
2
(N − 1)!
, N.
Q2
Q2
2 + 4κ2 · · · N −1+ 4κ2
• For large Q2 :
F (Q2 ) → (N − 1)!
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
, N = 3,
4κ
Q2
AdS/QCD
73
2 (N −1)
.
Stan Brodsky
SLAC
HolographicLight-Front
Model for QCDRepresentation
Light-Front Wavefunctions
SJB and GdT in preparation
of Two-Body
Meson Form Factor
2
• Drell-Yan-West form factor in the light-cone (two-parton state)
q⊥
= Q2 = −q 2
1 d2k⊥
∗
k⊥ ).
F (q 2 ) =
eq
dx
ψ
q
)
ψ
(x,
(x, k⊥ − x
P
⊥
P
16π 3
0
q
• Fourrier transform to impact parameter space b⊥
√ 2
b⊥ )
ψ(x, k⊥ ) = 4π d b⊥ eib⊥ ·k⊥ ψ(x,
• Find (b = |b⊥ |) :
2
F (q ) =
dx
0
= 2π
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
1
2
d b⊥ e
1
∞
dx
0
ixb⊥ ·
q⊥ 0
Soper
2
b db J0 (bqx) ψ(x, b) ,
AdS/QCD
74
2
ψ(x, b)
Stan Brodsky
SLAC
Holographic Mapping of AdS Modes to QCD LFWFs
• Integrate Soper formula over angles:
1
1−x
(1 − x)
ρ̃(x, ζ),
F (q 2 ) = 2π
dx
ζdζJ0 ζq
x
x
0
with ρ
(x, ζ) QCD effective transverse charge density.
• Transversality variable
ζ
n−1
x 2 =
ζ = x(1 −
x)xbj b
⊥j .
⊥
1−x
j=1
• Compare AdS and QCD expressions of FFs for arbitrary Q using identity:
1
1−x
= ζQK1 (ζQ),
dxJ0 ζQ
x
0
the solution for J(Q, ζ)
Thursday, September 24, 2009
= ζQK1 (ζQ) !
• Electromagnetic form-factor in AdS space:
Fπ+ (Q2 ) = R3
where J(Q2 , z)
dz
2
2
J(Q
,
z)
|Φ
+ (z)| ,
π
z3
= zQK1 (zQ).
• Use integral representation for J(Q2 , z)
2
J(Q , z) =
0
1
dx J0 ζQ
1−x
x
• Write the AdS electromagnetic form-factor as
1 1−x
dz
2
|Φ
zQ
Fπ+ (Q2 ) = R3 dx
J
+ (z)|
0
π
z3
x
0
• Compare with electromagnetic form-factor in light-front QCD for arbitrary Q
2 R 3
|Φπ (ζ)|2
x(1 − x)
ψ̃qq/π (x, ζ) =
2π
ζ4
with ζ
= z , 0 ≤ ζ ≤ ΛQCD
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
76
Stan Brodsky
SLAC
AdS5
LF(3+1)
φ(z)
ψ(x, b⊥)
ζ=
z
x(1 − x)b2
⊥
x
ψ(x, b⊥)
b⊥
(1 − x)
ψ(x, b⊥ ) =
x(1 − x)
φ(ζ)
2πζ
Light-Front Holography: Unique mapping derived from equality of
LF and AdS formula for current matrix elements
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
77
Stan Brodsky
SLAC
Gravitational Form Factor in AdS space
• Hadronic gravitational form-factor in AdS space
dz
2
2
Aπ (Q2 ) = R3
H(Q
,
z)
|Φ
(z)|
,
π
z3
where H(Q2 , z)
Abidin & Carlson
= 12 Q2 z 2 K2 (zQ)
• Use integral representation for H(Q2 , z)
1
1−x
H(Q2 , z) = 2 x dx J0 zQ
x
0
• Write the AdS gravitational form-factor as
1
1−x
dz
2
zQ
|Φ
Aπ (Q2 ) = 2R3 x dx
J
(z)|
0
π
z3
x
0
• Compare with gravitational form-factor in light-front QCD for arbitrary Q
2 R 3
|Φπ (ζ)|2
x(1 − x)
,
ψ̃qq/π (x, ζ) =
2π
ζ4
which is identical
to the
result obtained from
the EM form-factor
Identical
to LF
Holography
obtained
from electromagnetic current
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
78
Stan Brodsky
SLAC
Light-Front Holography:
Map AdS/CFT to 3+1 LF Theory
Frame Independent
Relativistic LF radial equation!
d2
1 − 4L2
2
− 2+
+
U
(ζ)
φ(ζ)
=
M
φ(ζ)
dζ
4ζ 2
ζ 2 = x(1 − x)b2
⊥.
x
b⊥
(1 − x)
U (ζ) = κ ζ + 2κ (L + S − 1)
4 2
2
soft wall
confining potential:
G. de Teramond, sjb
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
79
Stan Brodsky
SLAC
Derivation of the Light-Front Radial Schrodinger Equation
directly from LF QCD
M2
2
2
k 2
d k⊥
⊥
ψ(x,
k
=
dx
)
⊥ + interactions
3
16π x(1 − x)
0
1
dx
2
=
d2b⊥ ψ ∗ (x, b⊥ ) −∇
b⊥ ψ(x, b⊥ ) + interactions.
0 x(1 − x)
Change
variables
1
ϕ), ζ =
(ζ,
x(1 − x)b⊥ :
1 d
∇ =
ζ dζ
2
d
ζ
dζ
1 ∂2
+ 2
ζ ∂ϕ2
2
2
d
1 d
L
φ(ζ)
∗
√
=
dζ φ (ζ) ζ − 2 −
+ 2
dζ
ζ dζ
ζ
ζ
+ dζ φ∗ (ζ)U (ζ)φ(ζ)
2
2
d
1 − 4L
∗
+ U (ζ) φ(ζ)
=
dζ φ (ζ) − 2 −
2
dζ
4ζ
M2
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
80
Stan Brodsky
SLAC
HQED
QED atoms: positronium and
muonium
(H0 + Hint ) |Ψ >= E |Ψ >
Δ2
r)] ψ(r) = E ψ(r)
[−
+ Veff (S,
2mred
Coupled Fock states
Effective two-particle equation
Includes Lamb Shift, quantum corrections
d2
1
1 ( + 1)
[−
+
+ Veff (r, S, )] ψ(r) = E ψ(r)
2mred dr2
2mred r2
Vef f
α
→ VC (r) = −
r
Semiclassical first approximation to QED
Thursday, September 24, 2009
Spherical Basis
r, θ, φ
Coulomb potential
Bohr Spectrum
LF
HQED
QCD
H
0
I
(HLF
+ HLF
)|Ψ >= M 2 |Ψ >
k 2 + m2
LF
[ ⊥
] ψLF (x, k⊥ ) = M 2 ψLF (x, k⊥ )
+ Veff
x(1 − x)
QCD Meson Spectrum
Coupled Fock states
Effective two-particle equation
ζ 2 = x(1 − x)b2⊥
d2
−1 + 4L2
2
[− 2 +
+
U
(ζ,
S,
L)]
ψ
(ζ)
=
M
ψLF (ζ) Azimuthal Basis
LF
dζ
ζ2
U (ζ, S, L) = κ2 ζ 2 + κ2 (L + S − 1/2)
Semiclassical first approximation to QCD
Thursday, September 24, 2009
ζ, φ
Confining AdS/QCD
potential
Prediction from AdS/CFT: Meson LFWF
x
de Teramond, sjb
0.40.2
0.8 0.6
0.2
0.15
2)
ψM (x, k⊥
Soft Wall
model
0.1
0.05
5
0
κ = 0.375 GeV
1.3
Note coupling
1.4
k⊥ GeV
2
k⊥ , x
4π
massless quarks
1.5
2
k⊥
− 2κ2 x(1−x)
e
ψM (x, k⊥ ) = κ x(1 − x
φM (x, Q0) ∝
x(1 − x)
Connection of Confinement to TMDs
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
83
Stan Brodsky
SLAC
Hadron Distribution Amplitudes
Lepage, sjb
x
φH (xi , Q)
2
k⊥
< Q2
1−x
xi = 1
Fixed τ = t + z/c
i
•
•
Fundamental gauge invariant nonperturbative input to
hard exclusive processes, heavy hadron decays. Dened
for Mesons, Baryons
Evolution Equations from
PQCD, OPE, Conformal Invariance
Lepage
Lepage,
a e, sjb
Eemov,
E
E
emov
v, Ra
Radyushki
dy
Sachrajda,
Sachrajda
r a, F
Frishman
rishman Lepage,
Lep
sjb
Braun,
Bra
r un, Gardi
G rd
Ga
r
•
Compute from valence lightfront wavefunction in light
Q
cone gauge
φM (x, Q) =
d2k ψqq̄ (x, k⊥ )
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
84
Stan Brodsky
SLAC
Lepage, sjb
C. Ji, A. Pang, D. Robertson, sjb
Choi, Ji
F ␲共 Q 兲 ⫽
2
冕
1
0
dx ␾ ␲ 共 x 兲
冕
1
0
16␲ C F ␣ V 共 Q V 兲
dy ␾ ␲ 共 y 兲
共 1⫺x 兲共 1⫺y 兲 Q 2
φ(x, Q0) ∝
x(1 − x)
φasymptotic ∝ x(1 − x)
Normalized to fπ
AdS/CFT :
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
Increases PQCD leading twist prediction for
Fπ (Q2) by factor 16/9
AdS/QCD
85
Stan Brodsky
SLAC
Second Moment of Pion Distribution Amplitude
< ξ 2 >=
1
dξ ξ 2 φ(ξ)
−1
ξ = 1 − 2x
< ξ 2 >π = 1/5 = 0.20
< ξ 2 >π = 1/4 = 0.25
φasympt ∝ x(1 − x)
φAdS/QCD
Lattice (I) < ξ 2 >π = 0.28 ± 0.03
Lattice (II) < ξ 2 >π = 0.269 ± 0.039
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
86
∝ x(1 − x)
Donnellan et al.
Braun et al.
Stan Brodsky
SLAC
Diffractive Dissociation of Pion
into Quark Jets
E791 Ashery et al.
2
∂
M ∝ ∂ 2k ψπ (x, k⊥)
⊥
Measure Light-Front Wavefunction of Pion
Minimal momentum transfer to nucleus
Nucleus left Intact!
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
87
Stan Brodsky
SLAC
E791 FNAL Diffractive DiJet
Gunion, Frankfurt, Mueller, Strikman, sjb
Frankfurt, Miller, Strikman
Two-gluon exchange measures the second derivative of the pion
light-front wavefunction
q̄
2
∂
M ∝ ∂ 2k ψπ (x, k⊥)
⊥
π
q
N
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
N
AdS/QCD
88
Stan Brodsky
SLAC
Key Ingredients in E791 Experiment
Brodsky Mueller
Frankfurt Miller Strikman
Small color-dipole moment pion not absorbed;
interacts with each nucleon coherently
QCD COLOR Transparency
MA = A MN
q̄
π
dσ (πA → qq̄A ) = A2 dσ (πN → qq̄N ) F 2 (t)
A
dt
dt
q
N
A
A
Target left intact
Diraction, Rapidity gap
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
89
Stan Brodsky
SLAC
E791 Diractive DiJet transverse momentum distribution
dσ
dkT
Two Components
kT−6.5
Gaussian
High Transverse momentum
dependence consistent with k−6.5
T
PQCD, ERBL Evolution
Gaussian component similar
to AdS/CFT HO LFWF
kT (GeV)
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
90
Stan Brodsky
SLAC
asympt
mp
pt
CZ
Z
x
x
Ashery E791
Narrowing of x distribution at higher jet transverse momentum
Fig. 22. Thexu distribution
d
of diffractive dijets from the platinum target for 1.25 ≤ kt ≤ 1.5 GeV/c (left) and for
1.5 ≤ kt ≤ 2
2.5
5 GeV/c
G
(right). The solid line is a fit to a combination of the asymptotic and CZ distribution amplitudes.
The dashed line shows the contribution from the asymptotic function and the dotted line that of the CZ function.
Possibly two components: Nonperturbative
AdS/CFT and Perturbative ERBL φ(x) ∝ x(1 − x)
Evolution to asymptotic distribution
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
91
Stan Brodsky
SLAC
asympt
CZ
Ashery
E791
xx
x
x
x
x
Possibly two components:
Perturbative ERBL + Nonperturbative AdS/CFT
φ(x) =
2
Apert (k⊥
)x(1
2
− x) +Bnonpert (k⊥
) x(1 − x)
Narrowing of x distribution at high jet transverse momentum
QNP09 IHEP Beijing
September 25, 2009
Thursday, September 24, 2009
AdS/QCD
92
Stan Brodsky
SLAC