AdS/QCD and Its Holographic Light-Front Partonic Representation Page: 2 of 5
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pling a,(Q2) has an infrared fixed point at low Q2. As we have shown recently, there is
a remarkable mapping between the description of hadronic modes in AdS space and the
Hamiltonian formulation of QCD in physical space-time quantized on the light-front .
This procedure allows string modes <D(z) in the AdS holographic variable z to be pre-
cisely mapped to the LFWFs of hadrons in physical space-time in terms of a specific LF
variable which measures the separation of the quark and gluonic constituents within
the hadron. This mapping was originally obtained by matching the expression for elec-
tromagnetic current matrix elements in AdS space with the corresponding expression for
the current matrix element using LF theory in physical space time [3, 4]. More recently
we have shown that one obtains a consistent holographic mapping using the matrix ele-
ments of the energy-momentum tensor , thus providing an important verification of
holographic mapping from AdS to physical observables defined on the light front.
A SINGLE-VARIABLE LIGHT-FRONT EQUATION FOR QCD
To a first approximation light-front QCD is formally equivalent to an effective gravity
theory on AdS5. To prove this, we show that the LF Hamiltonian equation of motion of
QCD leads to an effective LF wave equation for physical modes 0 ( ) which encode the
hadronic properties. We compute the hadron mass _W2 from the hadronic matrix ele-
ment (/H ((P') HLF y/H(P)) = ((/H( l/iH(P)), where HLF is the Lorentz invariant
Hamiltonian HLF = PP1 = PP+- P2 and the state 1yH) is an expansion in multi-
particle Fock states In) of the free LF Hamiltonian: IYH) = n yn/H n). To simplify the
discussion we will consider a two-parton hadronic bound state in the limit of massless
constituents. We find 
1 fd2k k2
_2= odx 1d7k k( Iy(x, k ) 2+ interactions
= I x ) Jd2bify*(x, b_) (-v ) y(x, b) +interactions. (1)
- x(1 - x) .db(Vb
The functional dependence for a given Fock state is given in terms of the invariant mass
2 n k-1ka")2 -' x( ,the measure of the off-mass shell energy n2_
Similarly in impact space the relevant variable is (2 = x(1-x)b2 for a two-parton
state. Thus, to first approximation LF dynamics depend only on the boost invariant
variable Y4n or ( and hadronic properties are encoded in the hadronic mode 0( ):
y(x,k1) -0 4((). We choose the normalization of the LF mode 0(() = (( 0) with
fd I ( (10) 2 = 1. Comparing with the LFWF normalization, we find the functional
relation: = 2" yi(x, b)2.
We write the Laplacian operator in circular cylindrical coordinates ((, p), and factor
out the angular dependence of the modes in terms of the SO(2) Casimir representation
L2 of orbital angular momentum in the transverse plane: ( p) - e"'P((). We find
2= Jd(*(() (d2 14L2) ( )+ d( *(()U(((), (2)
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de Teramond, Guy F. & Brodsky, Stanley J. AdS/QCD and Its Holographic Light-Front Partonic Representation, article, November 12, 2008; United States. (https://digital.library.unt.edu/ark:/67531/metadc899730/m1/2/: accessed April 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.