AdS/QCD, Light-Front Holography, and Color Confinement

0.29 MB·English

by  Stanley J. Brodsky

AdS/QCD, Light-Front Holography, and Color
Confinement
Stanley J. Brodsky∗†
SLACNationalAcceleratorLaboratory
3 StanfordUniversity,Stanford,CA94309,USA
1
E-mail: [email protected]
0
2
Guy F. de Téramond

n
UniversidaddeCostaRica,SanJosé,CostaRica
a
J E-mail: [email protected]

2
1
AremarkableholographicfeatureofdynamicsinAdSspaceinfivedimensionsisthatitisdual

] toHamiltoniantheoryinphysicalspace-time,quantizedatfixedlight-fronttimeτ=t+z/c. This
h
p light-frontholographicprincipleprovidesapreciserelationbetweenthebound-stateamplitudes
- inAdSspaceandtheboost-invariantlight-frontwavefunctionsdescribingtheinternalstructureof
p
e hadronsinphysicalspace-time. Thefifthdimensioncoordinatezisdualtothelightfrontvariable
h
ζ describingtheinvariantseparationofthequarkconstituents. TheresultingvalenceFock-state
[

wavefunction eigensolutions of the light-front QCD Hamiltonian satisfy a single-variable rela-
1
v tivisticequationofmotion,analogoustothenonrelativisticradialSchrödingerequation.Thesoft-
3 wall dilaton profile eκ2z2 provides a model for the light-front potential which is color-confining
3
7 andreproduceswellthelinearReggebehaviorofthelight-quarkhadronspectruminbothLthe
2
orbital angular momentum and n the radial node number. The pion mass vanishes in the chiral
.
1
limit and other features of chiral symmetry are satisfied. The resulting running QCD coupling
0
3 displaysaninfraredfixedpoint. Theelasticandtransitionformfactorsofthepionandthenucle-
1
onsarealsofoundtobewelldescribedinthisframework. Thelight-frontAdS/QCDholographic
:
v
approachthusgivesaframe-independentanalyticfirstapproximationofthecolor-confiningdy-
i
X namics,spectroscopy,andexcitationspectraofrelativisticlight-quarkboundstatesinQCD.
r
a
XthQuarkConfinementandtheHadronSpectrum,
October8-12,2012
TUMCampusGarching,Munich,Germany
∗Speaker.
†ThisresearchwassupportedbytheDepartmentofEnergycontractDE–AC02–76SF00515.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Light-FrontHolography StanleyJ.Brodsky
1. Introduction
The AdS/QCD approach to hadron physics, combined with light-front holography, provides
a remarkable description of both hadron spectroscopy and dynamics. It is relativistic, frame-
independent, color-confining, and analytic. It also successfully predicts many phenomenological
featuresofQCD,suchasspace-likeandtime-likeformfactors,scalinglaws,Reggespectroscopy,
and has many properties associated with chiral symmetry. In this approach, baryons have three
colors, and constraints from the operator product at short distances are satisfied. Light-front (LF)
holographypredictsnotonlythehadronspectrum,butalsotheanalyticformofthelight-frontwave
functions (LFWFs) of hadrons, thus allowing many new phenomenological predictions and tests.
It also gives the form of the running QCD coupling in the nonperturbative domain and provides
newinsightintothefateofgluonsatsoftscales. Aswediscussbelow, theequationsofmotionin
AdS space have a remarkable holographic mapping to the equations of motion obtained in light-
front Hamiltonian theory in physical space-time. Thus, in principle, one can compute physical
observablesinastronglycoupledgaugetheoryintermsofaneffectiveclassicalgravitytheory.
The quantization of QCD at fixed light-front time [1] (Dirac’s Front Form) provides a first-
principles Hamiltonian method for solving nonperturbative QCD. It is rigorous, has no fermion-
doubling, is formulated in Minkowski space, and it is frame-independent. Given the light-front
wavefunctionsψ ,onecancomputealargerangeofhadronobservables,startingwithstructure
n/H
functions, generalized parton distributions, and form factors. It is also possible to compute jet
hadronizationattheamplitudelevelfromfirstprinciplesfromtheLFWFs.[2]Asimilarmethodhas
been used to predict the production of antihydrogen from the off-shell coalescence of relativistic
antiprotons and positrons. [3] The LFWFs of hadrons thus provide a direct connection between
observablesandtheQCDLagrangian.
SolvingnonperturbativeQCDisequivalenttosolvingthelight-frontHeisenbergmatrixeigen-
valueproblem. AngularmomentumJz isconservedateveryvertex. TheLFvacuumisdefinedas
thestateoflowestinvariantmassandistrivialuptozeromodes. Therearethusnoquarkorgluon
vacuumcondensatesintheLFvacuum–thecorrespondingphysicsiscontainedwithintheLFWFs
themselves,[4,5]thuseliminatingamajorcontributiontothecosmologicalconstant.
Thesimplicityofthefrontformcontrastswiththeusualinstant-formformalism. Currentma-
trixelementsdefinedatordinarytimet mustincludethecouplingofthephotonsandvectorbosons
fieldstoconnectedvacuumcurrents;otherwise,theresultisnotLorentz-invariant. Thustheknowl-
edgeofthehadroniceigensolutionsoftheinstant-formHamiltonianareinsufficientfordetermining
formfactorsorotherobservables. Inaddition,theboostofaninstantformwavefunctionfrom pto
p+qchangesparticlenumberandisanextraordinarilycomplicateddynamicalproblem.
TheLFHamiltonianmethodthusprovidesafirst-principlemethodforsolvingnonperturbative
QCD. It is rigorous, has no fermion-doubling, is formulated in Minkowski space, and it is frame-
independent. AdS/QCD provides a remarkable, analytic, first approximation to the LF approach
which is systematically improvable using a basis light-front quantization (BLFQ) [6] and other
methods.
It is remarkable fact that AdS/QCD, which was originally motivated by the AdS/CFT corre-
spondencebetweengravityonahigher-dimensionalanti–deSitter(AdS)spaceandconformalfield
theories (CFT) in physical space-time, [7] has a direct holographic mapping to light-front Hamil-
2
Light-FrontHolography StanleyJ.Brodsky
tonian theory. [8] For example, the equation of motion for mesons on the light-front has exactly
thesamesingle-variableformastheAdS/QCDequationofmotion;onecantheninterprettheAdS
fifthdimensionvariablezintermsofthephysicalvariableζ,representingtheinvariantseparation
oftheqandqatfixedlight-fronttime. TheAdSmassparameterµRmapstotheLForbitalangular
momentum. Theformulaeforelectromagnetic[9]andgravitational[10]formfactorsinAdSspace
maptotheexactDrell-Yan-Westformulaeinlight-frontQCD.[11,12,13]Thus,asweshallshow,
the light-front holographic approach provides an analytic frame-independent first approximation
tothecolor-confiningdynamics,spectroscopy,andexcitationspectraoftherelativisticlight-quark
boundstatesofQCD.
2. TheLight-FrontSchrödingerEquation: ASemiclassicalApproximationtoQCD
Unlike ordinary instant-time quantization, light-front Hamiltonian equations of motion are
frameindependent;remarkably,theyhaveastructurewhichmatchesexactlytheeigenmodeequa-
tions in AdS space. This makes a direct connection of QCD with AdS methods possible. In fact,
onecanderivethelight-frontholographicdualityofAdSbystartingfromthelight-frontHamilto-
nianequationsofmotionforarelativisticbound-statesysteminphysicalspace-time.[8]Toafirst
semiclassical approximation, where quantum loops and quark masses are not included, this leads
toaLFHamiltonianequationwhichdescribestheboundstatedynamicsoflighthadronsinterms
of an invariant impact variable ζ, which measures the separation of the partons within the hadron
atfixedlight-fronttimeτ =t+z/c.[1]ThisallowsonetoidentifythevariablezinAdSspacewith
theimpactvariableζ,[11,13,8]thusgivingtheholographicvariableaprecisedefinitionandvery
intuitivemeaninginlight-frontQCD.Inthecaseofatwo-partonstate,ζ2=x(1−x)b2.
⊥
It is advantageous to reduce the full multiparticle eigenvalue problem of the LF Hamilto-
nian to an effective light-front Schrödinger equation (LFSE) which acts on the valence sector LF
wavefunctionanddetermineseacheigensolutionseparately.[14]Incontrast,diagonalizingtheLF
Hamiltonian yields all eigensolutions simultaneously, a complex task. The central problem for
deriving the LFSE becomes the derivation of the effective interaction U which acts only on the
valence sector of the theory and has, by definition, the same eigenvalue spectrum as the initial
Hamiltonian problem. For carrying out this program one must systematically express the higher
Fock components as functionals of the lower ones. The method has the advantage that the Fock
spaceisnottruncatedandthesymmetriesoftheLagrangianarepreserved.[14]Onestartswiththe
QCDLagrangianandderivesthelight-frontHamiltonian.[15]OnecanthentransformtheLFequa-
tion of motion to a semiclassical QCD equation operating on the lowest Fock state of the hadron.
Therearenoapproximationsatthispoint,andtheresultisrelativisticandframe-independent.
For example, we can write the qq LFSE for a meson as ψ(x,ζ,ϕ) = eiLϕX(x)√φ(ζ) , thus
2πζ
factoring the angular dependence ϕ and the longitudinal, X(x), and transverse mode φ(ζ). In
the limit of zero quark masses the longitudinal mode decouples and the LF eigenvalue equation
P Pµ|φ(cid:105)=M2|φ(cid:105)isthusalight-frontwaveequationforφ [8]
µ
(cid:20)− d2 −1−4L2 +U(cid:0)ζ2,J,M2(cid:1)(cid:21)φ (ζ2)=M2φ (ζ2), (2.1)
dζ2 4ζ2 J,L,n J,L,n
arelativisticsingle-variableLFSchrödingerequation.
3
Light-FrontHolography StanleyJ.Brodsky
The potential in the LFSE is determined from the two-particle irreducible (2PI) qq → qq
Greens’ function. In particular, the higher Fock states in intermediate states leads to an effective
interactionU(ζ2,J,M2)forthevalence|qq(cid:105)Fockstate.[14]ThepotentialU thusdependsonthe
hadroniceigenvalueM2viatheLFenergydenominatorsP− −P− +iε oftheintermediate
initial intermediate
states which connect different LF Fock states. Here P− =(M2+P2)/P+. The dependence of
initial ⊥
U onM2 isanalogoustotheretardationeffectinQEDinteractions,suchasthehyperfinesplitting
in muonium, which involves the exchange of a propagating photon. Accordingly, the eigenvalues
M2 must be determined self-consistently. [16] The M2 dependence of the effective potential thus
reflectsthecontributionsfromhigherFockstatesintheLFSE(2.1), sinceU(ζ2,J,M2)isalsothe
kernelforthescatteringamplitudeqq→qqats=M2.Ithasonly“proper”contributions;i.e.,ithas
no qq intermediate state. The potential can be constructed, in principle systematically, using LF
time-ordered perturbation theory. In fact, as we shall see below, the QCD theory has the identical
form as the AdS theory, but with the quantum field-theoretic corrections due to the higher Fock
states giving a general form for the potential. This provides a novel way to solve nonperturba-
tive QCD. The LFSE for QCD becomes increasingly accurate as one includes contributions from
very high particle number Fock states. In the case of QED, this provides a new approach to the
spectroscopyofpositroniumandotheratomicboundstates.
Theabovediscussionassumesmasslessquarks. Moregenerallywemustincludequarkmass
terms,[17,18]m2/x+m2/(1−x),inthekineticenergytermandallowthepotentialU(ζ2,x,J,M2)
a b
to have dependence on the LF momentum fraction x. The quark masses also appear inU due to
the presence in the LF denominators as well as the chirality-violating interactions connecting the
valence Fock state to the higher Fock states. In this case, however, the equation of motion cannot
bereducedtoasinglevariable.
TheLFSEapproachalsocanbeappliedtoatomicboundstatesinQEDandnuclei. Inprinciple,
one could compute the spectrum and dynamics of atoms, such as the Lamb shift and hyperfine
splitting of hydrogenic atoms to high precision by a systematic treatment of the potential. Unlike
the ordinary instant form, the resulting LFWFs are independent of the total momentum and can
thus describe “flying atoms” without the need for dynamical boosts, such as the “true muonium”
(µ+µ−) bound states which can be produced by Bethe-Heitler pair production γZ →(µ+µ−)Z
belowthreshold.[19]Arelatedapproachfordeterminingthevalencelight-frontwavefunctionand
studyingtheeffectsofhigherFockstateswithouttruncationhasbeengiveninRef.[20].
3. EffectiveConfinementInteractionfromtheGauge/gravityCorrespondence
Recentlywehavederivedholographicwaveequationsforhadronswitharbitraryspinstarting
fromaneffectiveactioninahigher-dimensionalspaceasymptotictoAdSspace.[21]Anessential
element is the mapping of the higher-dimensional equations of motion to the light-front Hamil-
tonian equation for relativistic bound-states found in Ref. [8]. This procedure allows a clear dis-
tinctionbetweenthekinematicalanddynamicalaspectsofthelight-frontholographicapproachto
hadronphysics. Accordingly,thenon-trivialgeometryofpureAdSspaceencodesthekinematics,
andtheadditionaldeformationsofAdSspaceencodethedynamics,includingconfinement.[21]
A spin-J field in AdS is represented by a rank J tensor field Φ , which is totally
d+1 M1···MJ
symmetric in all its indices. In presence of a dilaton background field ϕ(z) the effective action
4
Light-FrontHolography StanleyJ.Brodsky
is[21]
Seff =(cid:90) ddxdz(cid:112)|g|eϕ(z)gN1N1(cid:48)···gNJNJ(cid:48)(cid:16)gMM(cid:48)DMΦ∗N1...NJDM(cid:48)ΦN1(cid:48)...NJ(cid:48)
(cid:17)
−µ2 (z)Φ∗ Φ , (3.1)
eff N1...NJ N1(cid:48)...NJ(cid:48)
√
where the indices M,N =0,···,d, g=(R/z)d+1 and D is the covariant derivative which in-
M
cludesparalleltransport. ThecoordinatesofAdSaretheMinkowskicoordinatesxµ andtheholo-
graphic variable z, xM = (xµ,z). The effective mass µ (z), which encodes kinematical aspects
eff
of the problem, is an a priori unknown function, but the additional symmetry breaking due to
its z-dependence allows a clear separation of kinematical and dynamical effects. [21] The dilaton
backgroundfieldϕ(z)in(3.1)introducesanenergyscaleinthefive-dimensionalAdSaction,thus
breakingconformalinvariance. Itvanishesintheconformalultravioletlimitz→0.
Aphysicalhadronhasplane-wavesolutionsandpolarizationindicesalongthe3+1physical
coordinates Φ (x,z) =eiP·xΦ (z)ε (P), with four-momentum P and invariant hadronic
P ν1···νJ J ν1···νJ µ
massP Pµ=M2. Allothercomponentsvanishidentically. Thewaveequationsforhadronicmodes
µ
follow from the Euler-Lagrange equation for tensors orthogonal to the holographic coordinate z,
Φ =0. Termsintheactionwhicharelinearintensorfields,withoneormoreindicesalong
zN2···NJ
the holographic direction, Φ , give us the kinematical constraints required to eliminate the
zN2···NJ
lower-spinstates.[21]UponvariationwithrespecttoΦˆ∗ ,wefindtheequationofmotion[21]
ν1...νJ
(cid:34) (cid:32) (cid:33) (cid:35)
zd−1−2J eϕ(z) (mR)2
− ∂ ∂ + Φ =M2Φ , (3.2)
eϕ(z) z zd−1−2J z z2 J J
with (mR)2 =(µ (z)R)2−Jzϕ(cid:48)(z)+J(d−J+1), which is the result found in Refs. [8, 22] by
eff
rescaling the wave equation for a scalar field. Similar results were found in in Ref. [23]. Upon
variationwithrespecttoΦˆ∗ wefindthekinematicalconstraintswhicheliminatelowerspin
N1···z···NJ
statesfromthesymmetricfieldtensor[21]
ηµνP ε (P)=0, ηµνε (P)=0. (3.3)
µ νν2···νJ µνν3···νJ
UponthesubstitutionoftheholographicvariablezbytheLFinvariantvariableζ andreplacing
Φ (z) = (R/z)J−(d−1)/2e−ϕ(z)/2φ (z) in (3.2), we find for d = 4 the LFSE (2.1) with effective
J J
potential[24]
1 1 2J−3
U(ζ2,J)= ϕ(cid:48)(cid:48)(ζ2)+ ϕ(cid:48)(ζ2)2+ ϕ(cid:48)(ζ2), (3.4)
2 4 2ζ
provided that the AdS mass m in (3.2) is related to the internal orbital angular momentum L =
max|Lz|andthetotalangularmomentumJz =Lz+Sz accordingto(mR)2 =−(2−J)2+L2. The
critical value L = 0 corresponds to the lowest possible stable solution, the ground state of the
LFHamiltonian. ForJ =0thefivedimensionalmassmisrelatedtotheorbitalmomentumofthe
hadronicboundstateby(mR)2=−4+L2andthus(mR)2≥−4. Thequantummechanicalstability
conditionL2≥0isthusequivalenttotheBreitenlohner-FreedmanstabilityboundinAdS.[25]
ThecorrespondencebetweentheLFandAdSequationsthusdeterminestheeffectiveconfining
interactionU in terms of the infrared behavior of AdS space and gives the holographic variable z
a kinematical interpretation. The identification of the orbital angular momentum is also a key
elementofourdescriptionoftheinternalstructureofhadronsusingholographicprinciples.
5
Light-FrontHolography StanleyJ.Brodsky
n=2 n=1 n=0 n=2 n=1 n=0
4 4
π2(1880) ρ(1700) a4(2040)
2) 2)
V V
2(Ge 2 π(1800) π2(1670) 2(Ge 2 ρ3(1690)
M M
ρ(1450)
π(1300) a2(1320)
b1(1235)
π(140)
ρ(770)
0 0
0 2 0 2
28-822001A220 L 28-822001A224 L
Figure 1: I =1 parent and daughter Regge trajectories for the π-meson family (left) with κ =0.59 GeV;
andtheρ-mesonfamily(right)withκ =0.54GeV.
4. ASoft-WallHolographicModel
The soft-wall model ϕ(z) = ±κ2z2 leads to linear Regge trajectories [26] and avoids the
ambiguities in the choice of boundary conditions at the infrared wall. However, the solution
ϕ(z)=−κ2z2 is incompatible with the light-front constituent interpretation of hadronic states as
shown in Ref. [21] . In fact, it can be shown that if one starts with a dilaton of the general form
ϕ(z,s)=κzs, for arbitrary values of s, the constraints imposed by chiral symmetry in the limit of
massless quarks determine uniquely the value s=2. [27] This is a remarkable result, since this
valuecorrespondspreciselytothedilatonprofilerequiredtoreproducethelinearReggebehavior.
The confining solution ϕ = exp(cid:0)κ2z2(cid:1) leads to the effective potential U(ζ2,J) = κ4ζ2+
2κ2(J−1)andEq. (2.1)haseigenvaluesM2 =4κ2(cid:0)n+J+L(cid:1), withastringReggeformM2 ∼
n,J,L 2
n+L. A discussion of the light meson and baryon spectrum, as well as the elastic and transition
form factors of the light hadrons using LF holographic methods, is given in Ref. [22]. As an ex-
ample, the spectral predictions for the J =L+S light pseudoscalar and vector meson states are
compared with experimental data in Fig. 1 for the positive sign dilaton model. The data is from
PDG.[28]TheresultsforQ4Fp(Q2)andQ4Fn(Q2)inthevalenceapproximationareshowninFig.
1 1
2. WealsoshowinFig. 2theresultsforFp(Q2)andFn(Q2)forthesamevalueofκ normalizedto
2 2
thestaticquantitiesχ andχ . Tocomparewithphysicaldatawehaveshiftedthepolesintheform
p n
factortotheirphysicalvalueslocatedatM2=4κ2(n+1/2)followingthediscussioninRef.[22].
Thevalueκ =0.545GeVisdeterminedfromtheρ mass. ThedatacompilationisfromRef.[29].
HolographicQCDalsoprovidepowerfulnonperturbativeanalyticaltoolstoextendthemulti-
component light-front Fock structure of amplitudes to the time-like region. Consider for example
the elastic pion form factor F . The pion is a superposition of an infinite number of Fock com-
π
ponents |N(cid:105), |π(cid:105)=∑NψN|N(cid:105), and thus the pion form factor is given by [22] Fπ(s)=∑τPτFτ(s),
whereP istheprobabilityfortwistτ (thenumberofcomponentsN inagivenFock-state)andF
τ τ
M2 M2 ···M2
F (s)= n=0 n=1 n=τ−2 , (4.1)
τ (cid:0)M2 −s(cid:1)(cid:0)M2 −s(cid:1)···(cid:0)M2 −s(cid:1)
n=0 n=1 n=τ−2
iswrittenasaτ−1productofpoles.[12]NormalizationatQ2=0,F (0)=1,impliesthat∑ P =
π τ τ
1ifallpossiblestatesareincluded.
6
Light-FrontHolography StanleyJ.Brodsky
1.2 0
4p24QF(Q) (GeV) 1 00..48 4n24QF(Q) (GeV) 1 -0.2
0
0 10 20 30 0 10 20 30
2-2012 Q2 (GeV2) 2-2012 Q2 (GeV2)
8820A18 8820A17
2 0
2) 2)
Q Q
np (2 1 n (2-1
F F
0 -2
0 2 4 6 0 2 4 6
2-2012 Q2 (GeV2) 2-2012 Q2 (GeV2)
8820A8 8820A7
Figure 2: Predictions for Q4Fp(Q2) (upper left) and Q4Fn(Q2) (upper right) in the soft wall model. The
1 1
results for Fp(Q2) (lower left) and Fn(Q2) (lower right) are normalized to static quantities. The nucleon
2 2
spin-flavorstructurecorrespondstotheSU(6)limit.
Inthestronglycoupledsemiclassicalgauge/gravitylimit,hadronshavezerowidthsandaresta-
ble,asintheN →∞limitofQCD.Onecannonethelessmodify(4.1)byintroducingfinitewidths
C
√
usingthesubstitutions→s+i sΓ . Theresultingexpressionhasaseriesofresonancepolesinthe
n
√ √ (cid:16) (cid:112) (cid:17)
non-physicallower-halfcomplex s-plane(Im s<0)locatedatαn = 1 −iΓ ± 4M2−Γ2
√ ± 2 n n n
with n=0,1,···,τ−2. In the upper half plane (Im s>0) –which maps to the physical sheet,
the pion form factor is an analytic function and thus can be mapped to the space-like region by a
√
π/2rotationinthecomplex s-planesuchastoavoidcuttingtherealaxis. Thissimpleprocedure
allows us to analytically continue the holographic results to the space-like region where the pion
formfactorisalsomodifiedbythefinitewidthsofthevectormesonresonancesfrominteractions
withthecontinuum.[30]
To illustrate the relevance of higher Fock states in the analytic structure of the pion form
factor we consider a simple phenomenological model where we include the first three compo-
nents in a Fock expansion of the pion state |π(cid:105)=ψL=0|qq;L=0(cid:105) +ψL=0|qqqq;L=0(cid:105) +
qq/π τ=2 qqqq τ=4
ψL=1|qqqq;L=1(cid:105) +···, and no constituent dynamical gluons. [31] The JPC =0−+ twist-2,
qqqq τ=5
twist-4andtwist-5statesarecreatedbytheinterpolatingoperatorsO =qγ+γ q, O =qγ+γ qqq
2 5 4 5
and O =qγ+γ Dµqqγ q. The results for the space-like and time-like form factor are shown in
5 5 µ
Fig. 3. We choose the values Γ =149 MeV, Γ =300 MeV, Γ =250 MeV and Γ =200
ρ ρ(cid:48) ρ(cid:48)(cid:48) ρ(cid:48)(cid:48)(cid:48)
MeV. The chosen width of the ρ(cid:48) is slightly smaller than the PDG value listed in Ref. [28]. The
7
Light-FrontHolography StanleyJ.Brodsky
2
|
)
s 0
(
π
F
|

g -2
o
l
-4
-10 -5 0 5 10 15
2 11-2012
s (GeV )
8829A1
Figure3: Structureofthespace-like(s<0)andtime-like(s>4m2)pionformfactorinlight-frontholog-
π
raphy for a truncation of the pion wave function up to twist five. The space-like data are the compilation
fromBaldinietal.[32](black)andJLABdata[33](greenandred). Thetime-likedataarefromtherecent
precisemeasurementsfromBABAR[34](black)andCLEO[35](red).
widths of the ρ and ρ(cid:48)(cid:48) are the PDG values and the width of the ρ(cid:48)(cid:48)(cid:48) is unknown. The results cor-
respond to PL=0 = 2.5 % and PL=1 = 4.5 % the admixture of the |qqqq;L=0(cid:105) and |qqqq;L=1(cid:105)
qqqq qqqq
states. The value of PL=0 and PL=1 (and the widths) are input in the model. The main features of
qqqq qqqq
the space-like and time-like regions are well described by the same formula with a minimal num-
ber of parameters. In contrast, models based on hadronic degrees of freedom involve sums over a
large number of intermediate states and thus require, for large s, a very large number of hadronic
parameters [36, 37, 38]. A detailed data comparison requires the introduction of thresholds and
s-dependent widths from multiparticle states coupled to the ρ-resonances. This will be described
elsewhere.[39]
5. Conclusions
DespitesomelimitationsofAdS/QCD,theLFholographicapproachtothegauge/gravitydu-
ality, has given significant physical insight into the strongly-coupled nature and internal structure
of hadrons. In particular, the light-front AdS/QCD soft-wall model provides an elegant analytic
frameworkfordescribingnonperturbativehadrondynamics,thesystematicsoftheexcitationspec-
trum of hadrons, including their empirical multiplicities and degeneracies. The pion is massless
in the chiral limit and the light-hadron spectrum displays linear Regge trajectories with the same
slopeinthenodenumbernandorbitalangularmomentumL. Italsoprovidespowerfulnewanalyt-
ical tools for computing hadronic transition amplitudes incorporating conformal scaling behavior
atshortdistancesandthetransitionfromthehard-scatteringperturbativedomain,wherequarkand
gluonsaretherelevantdegreesoffreedom,tothelong-rangeconfininghadronicregion.
The confining light-front potential for hadronic bound-state equations for particles with ar-
bitrary spin can be derived from an effective invariant action in the higher dimensional classical
gravitational theory. The fact that we can map the equations of motion for arbitrary spin from the
gravitationaltheorytoaHamiltonianequationofmotioninlight-frontquantizedQCDisaremark-
8
Light-FrontHolography StanleyJ.Brodsky
able result. The undisturbed AdS geometry reproduces the kinematical aspects of the light-front
Hamiltonian, notably the emergence of a LF angular momentum which is holographically identi-
fiedwiththemassinthegravitationaltheory. ThebreakingofthemaximalsymmetryofAdSthen
allowstheintroductionoftheconfinementdynamicsofthetheoryinphysicalspace-time. Inorder
tofullypreserveallthekinematicalaspects,aconsistentmappingtoLFquantizedQCDrequiresa
clearseparationbetweenthekinematicalanddynamicaleffects.[21]
TheeffectiveinteractionU(ζ2)=κ4ζ2+2κ2(J−1)thatisderivedfromtheAdS/QCDmodel
is instantaneous in LF time and acts on the lowest state of the LF Hamiltonian. This equation
describesthespectrumofmesonsasafunctionofn,thenumberofnodesinζ2andthetotalangular
momentum J = Jz, with Jz = Lz+Sz the sum of the internal orbital angular momentum of the
constituents. Itistherelativistic,frame-independentfront-formanalogofthenon-relativisticradial
Schrödingerequationformuoniumandotherhydrogenicatomsinthepresenceofaninstantaneous
Coulomb potential. The holographic harmonic oscillator potential could in fact emerge from the
exact QCD formulation when one includes contributions from the LFSE potential U which are
due to the exchange of two connected gluons; i.e., “H” diagrams. [40] We notice thatU becomes
complex for an excited state since a denominator can vanish; this gives a complex eigenvalue and
thedecaywidth.
Acknowledgments
Section 3 is based on a collaboration with Hans Guenter Dosch, and we thank him for help-
ful comments. This research was supported by the Department of Energy contract DE–AC02–
76SF00515. SLAC-PUB-15350.
References
[1] P.A.M.Dirac,Rev.Mod.Phys.21,392(1949).
[2] S.J.Brodsky,G.deTeramondandR.Shrock,AIPConf.Proc.1056,3(2008)[arXiv:0807.2484
[hep-ph]].
[3] C.T.Munger,S.J.BrodskyandI.Schmidt,Phys.Rev.D49,3228(1994).
[4] S.J.BrodskyandR.Shrock,Proc.Nat.Acad.Sci.108,45(2011)[arXiv:0905.1151[hep-th]].
[5] S.J.Brodsky,C.D.Roberts,R.Shrock,P.Tandy,Phys.Rev.C82,022201(2010)[arXiv:1005.4610
[nucl-th]].
[6] J.P.Vary,H.Honkanen,J.Li,P.Maris,S.J.Brodsky,A.Harindranath,G.F.deTeramondand
P.Sternbergetal.,Phys.Rev.C81,035205(2010)[arXiv:0905.1411[nucl-th]].
[7] J.M.Maldacena,Adv.Theor.Math.Phys.2,231(1998)[arXiv:hep-th/9711200].
[8] G.F.deTeramondandS.J.Brodsky,Phys.Rev.Lett.102,081601(2009)[arXiv:0809.4899
[hep-ph]].
[9] J.PolchinskiandM.J.Strassler,JHEP0305,012(2003)[arXiv:hep-th/0209211].
[10] Z.AbidinandC.E.Carlson,Phys.Rev.D77,095007(2008)[arXiv:0801.3839[hep-ph]].
[11] S.J.BrodskyandG.F.deTeramond,Phys.Rev.Lett.96,201601(2006)[arXiv:hep-ph/0602252];
9
Light-FrontHolography StanleyJ.Brodsky
[12] S.J.BrodskyandG.F.deTeramond,Phys.Rev.D77,056007(2008)[arXiv:0707.3859[hep-ph]].
[13] S.J.BrodskyandG.F.deTeramond,Phys.Rev.D78,025032(2008)
[14] H.C.Pauli,Eur.Phys.J.C7,289(1999)[arXiv:hep-th/9809005].
[15] S.J.Brodsky,H.C.PauliandS.S.Pinsky,Phys.Rept.301,299(1998)[arXiv:hep-ph/9705477].
[16] G.F.deTeramondandS.J.Brodsky,arXiv:1206.4365[hep-ph].
[17] S.J.BrodskyandG.F.deTeramond,WorldScientificSubnuclearSeries,45,139(2007)
[arXiv:0802.0514[hep-ph]].
[18] T.Gutsche,V.E.Lyubovitskij,I.SchmidtandA.Vega,Prog.Part.Nucl.Phys.67,206(2012)
[arXiv:1111.5169[hep-ph]].
[19] S.J.BrodskyandR.F.Lebed,Phys.Rev.Lett.102,213401(2009)[arXiv:0904.2225[hep-ph]].
[20] S.S.ChabyshevaandJ.R.Hiller,Phys.Lett.B711,417(2012)[arXiv:1103.0037[hep-ph]].
[21] G.F.deTeramond,H.G.DoschandS.J.Brodsky,arXiv:1301.1651[hep-ph].
[22] G.F.deTeramondandS.J.Brodsky,arXiv:1203.4025[hep-ph].
[23] T.Gutsche,V.E.Lyubovitskij,I.Schmidt,A.Vega,Phys.Rev.D85,076003(2012)
[arXiv:1108.0346[hep-ph]].
[24] G.F.deTeramondandS.J.Brodsky,AIPConf.Proc.1296,128(2010)[arXiv:1006.2431[hep-ph]].
[25] P.BreitenlohnerandD.Z.Freedman,AnnalsPhys.144,249(1982).
[26] A.Karch,E.Katz,D.T.SonandM.A.Stephanov,Phys.Rev.D74,015005(2006)
[arXiv:hep-ph/0602229].
[27] S.J.Brodsky,G.F.deTeramondandH.G.Dosch,“ConfinementandChiralPropertiesin
HolographicQCD,”arXiv:1301.XXXX[hep-ph].
[28] J.Beringeretal.[ParticleDataGroup],Phys.Rev.D86,010001(2012).
[29] M.Diehl,Nucl.Phys.Proc.Suppl.161,49(2006)[arXiv:hep-ph/0510221].
[30] E.R.Arriola,W.BroniowskiandP.Masjuan,arXiv:1210.7153[hep-ph].
[31] S.J.BrodskyandG.F.deTeramond,PoSQCD-TNT-II,008(2011)[arXiv:1112.4212[hep-th]].
[32] R.Baldinietal.,Eur.Phys.J.C11,709(1999).
[33] V.Tadevosyanetal.,Phys.Rev.C75,055205(2007)[arXiv:nucl-ex/0607007];T.Hornetal.,Phys.
Rev.Lett.97,192001(2006)[arXiv:nucl-ex/0607005].
[34] B.Aubertetal.Phys.Rev.Lett.103,231801(2009)[arXiv:0908.3589[hep-ex]].
[35] K.K.Seth,S.Dobbs,etal.,arXiv:1210.1596[hep-ex].
[36] J.H.KuhnandA.Santamaria,Z.Phys.C48,445(1990).
[37] C.Bruch,A.KhodjamirianandJ.H.Kuhn,Eur.Phys.J.C39,41(2005)[hep-ph/0409080].
[38] C.Hanhart,Phys.Lett.B715,170(2012)[arXiv:1203.6839[hep-ph]].
[39] G.F.deTeramondandS.J.Brodsky(Inpreparation).
[40] T.Appelquist,M.DineandI.J.Muzinich,Phys.Lett.B69,231(1977).
10