AdS/QCD and Its Holographic Light-Front Partonic Representation Page: 3 of 5
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where all the complexity of the interaction terms in the QCD Lagrangian is summed up
in the effective potential U ((). The light-front eigenvalue equation HLF 0) = 2 0) is
thus a light-front wave equation for 0
Cd 2 14 L + U( )) (() = 42(),(3)
an effective single-variable light-front Schrtdinger equation which is relativistic, covari-
ant and analytically tractable. One can readily generalize the equations to allow for the
kinetic energy of massive quarks .
As the simplest example we consider a bag-like model where the partons are free
inside the hadron and the interaction terms will effectively build confinement. The
effective potential is a hard wall: U() = 0 if ( < 1 and U( ) = if > A .
If L2 > 0 the LF Hamiltonian is positive definite (4)HLF 4)) > 0 and thus _W2 > 0. If
L2 < 0 the LF Hamiltonian is unbounded from below and the particle "falls towards the
center". The critical value corresponds to L = 0. The mode spectrum follows from the
boundary conditions 4(( = 1/AQCD) = 0, and is given in terms of the roots of Bessel
functions: AL k = /L,kAQCD. Since in the conformal limit U() -- 0, the hard-wall
LF model discussed here is equivalent to the AdS/CFT hard wall model of Ref. .
Likewise a two-dimensional transverse oscillator with effective potential U() ~ (2 is
equivalent to the soft-wall model of Ref.  which reproduce the usual linear Regge
trajectories. Upon the substitution ( ->z and <Dj(z) - (z/R)3/2-JO(z) in (3) we find the
equation of motion
z2d2 - (d-1-2J)zdz +z22- (gR)2]<DJ = 0, (4)
describing the propagation of a spin-J mode in AdSd+1space. For d = 4 the fifth dimen-
sional mass (yR)2 = -(2 - J)2 +L2. The scaling dimensions are A = 2+L independent
of J in agreement with the twist scaling dimension of a two parton bound state in QCD.
TRANSITION MATRIX ELEMENTS
Light-Front Holography can be derived by observing the correspondence between ma-
trix elements obtained in AdS/CFT with the corresponding formula using the LF rep-
resentation  . The light-front electromagnetic form factor in impact space [3, 4] can
be written as a sum of overlap of light-front wave functions of the j = 1,2, - - - , n - 1
F(q2) = f fdxjd2bij Leq exp (iqr- Lxjbij) yn(xj,bij) 2. (5)
n j=1 q j=1
The formula is exact if the sum is over all Fock states n. For definiteness we shall
consider a two-quark )r+ valence Fock state Iud) with charges en = 2 and eg = 3. For
n = 2, there are two terms which contribute to the q-sum in (5). Exchanging x <-> 1 - x
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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/m1/3/: accessed April 26, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.