Light-Front Holography and AdS/QCD: A New Approximation to...
Transcript of 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
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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
Effective two-particle equation
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
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AdS/QCD and LF Holography Stan Brodsky SLAC
• 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
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AdS/QCD and LF Holography Stan Brodsky SLAC
∑ni xi = 1
∑ni
�k⊥i = �0⊥
Ψn(xi,�k⊥i, λi)
xiP+, xi
�P⊥ + �k⊥i
P+, �P⊥
x
Fixed τ = t + z/c
Invariant under boosts! Independent of Pμ
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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!
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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
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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
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Intrinsic heavy quarks s(x) �= s(x)
u(x) �= d(x)c(x), b(x) at high x
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AdS/QCD and LF Holography Stan Brodsky SLAC
zero for q+ = 0
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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
Ψn(xi,�k⊥i, λi)
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
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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
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HQCDLF |Ψh >= M2
h|Ψh >HQCDLC |Ψh〉 = M2
h |Ψh〉Heisenberg Matrix Formulation
Light-Front QCD H.C. Pauli & sjbDiscretized Light-Cone
Quantization
Eigenvalues and Eigensolutions give Hadron Spectrum and Light-Front wavefunctions
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
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AdS/QCD and LF Holography
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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
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AdS/QCD and LF Holography Stan Brodsky SLAC
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
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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
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AdS/QCD and LF Holography Stan Brodsky SLAC
• 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:
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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
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Stan Brodsky SLAC
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AdS/QCD and LF Holography
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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
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AdS/QCD and LF Holography Stan Brodsky SLAC 22
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
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AdS/QCD and LF Holography Stan Brodsky SLAC 23
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
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AdS/QCD and LF Holography Stan Brodsky SLAC
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
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ppolarization atSerber-Uehling
Π(Q2)→ α15π
Q2
m2 Q2 << 4m2
�+
�−
Shrock, sjb
Deur, Chen, Burkert, Korsch,
Υ → ggg
Stan Brodsky SLAC
SCGTDecember 8, 2009
AdS/QCD and LF Holography
• 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
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Maximal Wavelength of Confined Fields
Shrock, sjb
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AdS/QCD and LF Holography Stan Brodsky SLAC
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.
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invariant measure
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AdS/QCD and LF Holography Stan Brodsky SLAC 29
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AdS/QCD and LF Holography Stan Brodsky SLAC 30
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AdS/QCD and LF Holography Stan Brodsky SLAC 31
• 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).
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
φ(z)
ζ =√
x(1 − x)�b2⊥ z
x
(1 − x)
�b⊥
ψ(x,�b⊥)
LF(3+1) AdS5
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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)
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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)
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AdS/QCD and LF Holography Stan Brodsky SLAC 34
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography
ψ(z) ∼ zΔ at z → 0
ψ(z0) = 00 < z < z0 z0 = 1ΛQCD
x2μ → λ2x2
μ z → λz
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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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 36
• 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)
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AdS/QCD and LF Holography Stan Brodsky SLAC 37
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)
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 38
• 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
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AdS/QCD and LF Holography Stan Brodsky SLAC 39
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 41
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 43
Agrees with Klebanov and Maldacena for positive-sign exponent of
dilaton
ds2 = eκ2z2 R2
z2(dx2
0 − dx21 − dx2
3 − dx23 − dz2)
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 44
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 45
• 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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 46
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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)
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 51
-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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 52
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 53
-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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 54
• 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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
e+
e−γ∗
π+
π−
Dressed soft-wall current bring in higher Fock states and more vector meson poles
55
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 56
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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⊥
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 60
• 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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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)
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 62
• 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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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
Effective two-particle equation
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 65
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)
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 66
• 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
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 68
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.
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 69
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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)
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 71
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 73
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 74
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 75
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 76
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 77
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?
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 78
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 79
• 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.
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 80
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.
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 81
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).
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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)
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 83
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.
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 84
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
−
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 85
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.
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 86
• 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).
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 87
• 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).
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 88
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
HQCDLC |Ψh〉 = M2
h |Ψh〉Heisenberg Equation
Light-Front QCD
Use AdS/QCD basis functions
91
Stan Brodsky SLAC
SCGTDecember 8, 2009
AdS/QCD and LF Holography
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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
Deur, de Teramond, sjb, (preliminary)
-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
AdS/QCD with�s,g1 extrapolation
βAdS(Q2) =d
d log Q2αAdS
s (Q2) =πQ2
4κ2e−Q2/4κ2
Deur, de Teramond, sjb, (preliminary)
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
Deur, de Teramond, sjb, (preliminary)
Running Coupling for Static Potential from AdS/QCD
r (fm)
�s(
r)
Lattice QCD�s,g1:Jlab+pQCD+sum rules
AdS/QCD with�s,g1 extrapolation
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography
• 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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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!
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography
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)
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 106
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography
• 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
Ψn(xi,�k⊥i, λi)
xiP+, xi
�P⊥ + �k⊥i
P+ = P0 + Pz
P+, �P⊥
Stan Brodsky SLAC
SCGTDecember 8, 2009
AdS/QCD and LF Holography
• 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⊥
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 113
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 115
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography
116
δM2 = −2mq < ψψ > ×∫
dz φ2(z)z2
de Teramond, Shrock, sjb
Chiral Symmetry Breaking in AdS/QCD
Stan BrodskySCGTb
AdS/QCD and LF Holography
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 119
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 120
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.
e-Print: nucl-th/9708029
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.
e-Print: nucl-th/0301024
− 〈qq〉πζ = fπ〈0|qγ5q|π〉 .
“In-Meson Condensate”Valid even for mq → 0
fπ nonzero
SCGTDecember 8, 2009
AdS/QCD and LF Holography Stan Brodsky SLAC 121
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography
“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
SCGTDecember 8, 2009
AdS/QCD and LF Holography
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
SCGTDecember 8, 2009
AdS/QCD and LF Holography
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"
Ψn(xi,�k⊥i, λi)