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GdT
Hadron Spectroscopy and Dynamics from Light-Front Holography
and Conformal Symmetry
Guy F. de Teramond
Universidad de Costa Rica
13th International Conference on Meson-Nucleon
Physics and the Structure of the Nucleon
Pontificia Universita della Santa Croce
Rome, 30 September - October 4, 2013
Image credit: ic-msquare
In collaboration with Stan Brodsky (SLAC) and Hans G. Dosch (Heidelberg)
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Contents
1 Dirac Forms of Relativistic Dynamics 5
2 Light-Front Dynamics 7
3 Semiclassical Approximation to QCD in the Light Front 8
4 Conformal Quantum Mechanics and Light Front Dynamics 11
5 Gravity in AdS and Light Front Holographic Mapping 17
6 Higher Integer-Spin Wave Equations in AdS Space 20
7 Meson Spectrum 23
8 Higher Half-Integer Spin Wave Equations in AdS Space 28
9 Baryon Spectrum 30
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• SLAC and DESY: elementary interactions of quarks and gluons at short distances remarkably well
described by QCD
• No understanding of large distance strong dynamics of QCD: how quarks and gluons are confined and
how hadrons emerge as asymptotic states
• Euclidean lattice important first-principles numerical simulation of nonperturbative QCD: excitation
spectrum of hadrons requires enormous computational efforts
• Important theoretical goal: find initial analytic approximation to QCD in its strongly coupled regime, like
Schrodinger EQ in atomic physics corrected for quantum fluctuations
• We give an overview of the holographic connection between a one-dim semiclassical approximation to
light-front (LF) dynamics (LFQM) with gravity in a higher dim AdS space, and the constraints imposed
by the invariance properties under the full conformal group in one dim: Conformal QM (CQM)
• Result is a relativistic LF QM wave equation which incorporates important nonperturbative features of
hadron physics
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4 – dim Light –
Front Dynamics
SO(2,1)
5 – dim
Classical Gravity
dAFF
1 – dim
Conformal Group
AdS5
AdS2LFQM
CQM
LFD
8-20138840A1
• Effective theory for QCD with underlying SO(2, 1)confining algebraic structure (spectrum generating algebra)
follows from LFQM, classical gravity in AdS space and CQM
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1 Dirac Forms of Relativistic Dynamics[Dirac (1949)]
• The Poincare group is the full symmetry group of any form of relativistic dynamics
[Pµ, P ν ] = 0,
[Mµν , P ρ] = i (gµρP ν − gνρPµ) ,
[Mµν ,Mρσ] = i (gµρMνσ − gµσMνρ + gνσMµρ − gνρMµσ)
• Poincare generators Pµ and Mµν separated into kinematical and dynamical
• Kinematical generators act along the initial hypersurface where initial conditions are imposed and
contain no interactions (leave invariant initial surface)
• Dynamical generators are responsible for evolution of the system and depend on the interactions
(map initial surface into another surface)
• Each front has its Hamiltonian and evolve with a different time, but results computed in any front should
be identical (different parameterizations of space-time)
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ct(a) (b) (c)
y
z
ct
y
z
ct
y
z
8-20138811A4
• Instant form: initial surface defined by x0 = 0
P 0, K dynamical, P, J kinematical
• Front form: initial surface tangent to the light cone x+ = x0 + x3 = 0 ( P± = P 0 ± P 3)
P−, Jx, Jy dynamical P⊥, J3, K kinematical
• Point form: initial surface is the hyperboloid x2 = κ2 > 0, x0 > 0
Pµ dynamical, Mµν kinematical
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2 Light-Front Dynamics
• Hadron with 4-momentum P = (P+, P−,P⊥), P± = P 0 ± P 3, mass-shell relation
P 2 = M2 leads to LF Hamiltonian equation
P−|ψ(P )〉 =M2 + P2
⊥P+
|ψ(P )〉
• Construct LF invariant Hamiltonian HLF = P 2 = P−P+ −P2⊥ (P+ and P⊥ kinematical)
HLF |ψ(P )〉 = M2|ψ(P )〉
• Longitudinal momentum P+ is kinematical: sum of single particle constituents p+i of bound state,
P+ =∑
i p+i , p+
i > 0
• Bound-state is off the LF energy shell, P− −∑n
i p−i < 0
• Vacuum is the state with P+ = 0 and contains no particles: all other states have P+ > 0
• Simple structure of vacuum allows definition of partonic content of hadron in terms of wavefunctions:
quantum-mechanical probabilistic interpretation of hadronic states
• Light-front quantization ideal framework to describe internal constituent structure of hadrons
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3 Semiclassical Approximation to QCD in the Light Front[GdT and S. J. Brodsky, PRL 102, 081601 (2009)]
• Reduce effectively LF multiparticle dynamics to a 1-dim QFT with no particle creation and absorption:
LF quantum mechanics !
• Compute M2 from hadronic matrix element 〈ψ(P ′)|PµPµ|ψ(P )〉=M2〈ψ(P ′)|ψ(P )〉
• Find
M2 =∑n
∫ [dxi][d2k⊥i
]∑q
(k2⊥q +m2
q
xq
)|ψn(xi,k⊥i)|2 + interactions
with phase space normalization∑n
∫ [dxi] [d2k⊥i
]|ψn(xi,k⊥i)|2 = 1
• Central problem is derivation of effective interaction which acts only on the valence sector:
express higher Fock states as functionals of the lower ones
• Advantage: Fock space not truncated and symmetries of the Lagrangian preserved[H. C. Pauli, EPJ, C7, 289 (1999)
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• In terms of n−1 independent transverse impact coordinates b⊥j , j = 1, 2, . . . , n−1,
M2 =∑n
n−1∏j=1
∫dxjd
2b⊥jψ∗n(xi,b⊥i)∑q
(−∇2
b⊥q+m2
q
xq
)ψn(xi,b⊥i) + interactions
with normalization ∑n
n−1∏j=1
∫dxjd
2b⊥j |ψn(xj ,b⊥j)|2 = 1.
• Semiclassical approximation
ψn(k1, k2, . . . , kn)→ φn(
(k1 + k2 + · · ·+ kn)2︸ ︷︷ ︸M2
n=P
i
k2⊥i
+m2i
xi(Invariant mass)
), mq → 0
• M2−M2n is the measure of the off-energy shell: key variable which controls the bound state
• For a two-parton bound-state in the mq → 0 M2n=2 = k2
⊥x(1−x)
• Conjugate invariant variable in transverse impact space is
ζ2 = x(1− x)b2⊥
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• To first approximation LF dynamics depend only on the invariant variable ζ , and dynamical properties
are encoded in the hadronic mode φ(ζ)
ψ(x, ζ, ϕ) = eiLϕX(x)φ(ζ)√2πζ
,
where we factor out the longitudinal X(x) and orbital kinematical dependence from LFWF ψ
• Ultra relativistic limit mq → 0 longitudinal modes X(x) decouple (L = Lz)
M2 =∫dζ φ∗(ζ)
√ζ
(− d2
dζ2− 1ζ
d
dζ+L2
ζ2
)φ(ζ)√ζ
+∫dζ φ∗(ζ)U(ζ)φ(ζ),
where effective potential U includes all interaction terms upon integration of the higher Fock states
• LF eigenvalue equation PµPµ|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)
• Relativistic and frame-independent LF Schrodinger equation: U is instantaneous in LF time
• Critical value L = 0 corresponds to lowest possible stable solution, the ground state of HLF
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4 Conformal Quantum Mechanics and Light Front Dynamics[S. J. Brodsky, GdT and H.G. Dosch, arXiv:1302.4105]
• Incorporate in a 1-dim QFT – as an effective theory, the fundamental conformal symmetry of the 4-dim
classical QCD Lagrangian in the limit of massless quarks
• Invariance properties of 1-dim field theory under the full conformal group from dAFF action
[V. de Alfaro, S. Fubini and G. Furlan, Nuovo Cim. A 34, 569 (1976)]
S = 12
∫dt(Q2 − g
Q2
)where g is a dimensionless number (Casimir operator which depends on the representation)
The translation operator in t, the Hamiltonian, is
Ht = 12
(Q2 +
g
Q2
)• The equations of motion for the state vector and field operator Q(t) are given by the usual quantum-
mechanical evolution
Ht|ψ(t)〉 = id
dt|ψ(t)〉, i [Ht, Q(t)] =
dQ(t)dt
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• Absence of dimensional constants implies that the action
S = 12
∫dt(Q2 − g
Q2
)is invariant under a larger group of transformations, the general conformal group
t′ =αt+ β
γt+ δ, Q′(t′) =
Q(t)γt+ δ
with αδ − βγ = 1
I. Translations in t: H = 12
(Q2 + g
Q2
),
II. Dilatations: D = 12
(Q2 + g
Q2
)t− 1
4
(QQ+QQ
),
III. Special conformal transformations: K = 12
(Q2 + g
Q2
)t2 − 1
2
(QQ+QQ
)t+ 1
2Q2,
• Using canonical commutation relations [Q(t), Q(t)] = i find
[Ht, D] = iHt, [Ht,K] = 2iD, [K,D] = −iK,
the algebra of the generators of the conformal group
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• Introduce the linear combinations
J12 =12
(1aK + aHt
), J01 =
12
(1aK − aHt
), J03 = D,
where a has dimension t since Ht and K have different dimensions
• Generators J have commutation relations
[J12, J01] = iJ02, [J12, J02] = −iJ01, [J01, J02] = −iJ12,
the algebra of SO(2, 1)
• J0i, i = 1, 2, boost in space direction i and J12 rotation in the (1,2) plane
• J12 is compact and has thus discrete spectrum with normalizable eigenfunctions
• The relation between the generators of conformal group and generators of SO(2, 1) suggests that
the scale a may play a fundamental role
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• dAFF construct a generator as a superposition of the 3 constants of motion
G = uHt + vD + wK
and introduce new time variable τ and field operator q(τ)
dτ =dt
u+ vt+ wt2, q(τ) =
Q(t)
[u+ vt+ wt2]12
• Find usual quantum mechanical evolution for time τ
G|ψ(τ)〉 = id
dτ|ψ(τ)〉
i [G, q(τ)] =dq(τ)dτ
and usual equal-time quantization [q(t), q(t)] = i
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• In terms of τ and q(τ)
S = 12
∫dt(Q2 − g
Q2
)= 1
2
∫dτ(q2 − g
q2− 4uω − v2
4q2)
+ surface term
Action is conformal invariant invariant up to a surface term !
• The corresponding Hamiltonian
Hτ = 12
(q2 +
g
q2+
4uω − v2
4q2)
is a compact operator for4uω − v2
4> 0
• Scale appears in the Hamiltonian without affecting the conformal invariance of the action !
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• The Schrodingier picture follows from the representation of q and p = q
q → x, q → −i ddx
• Hτ dAFF Hamiltonian
Hτ = 12
(− d2
dx2+
g
x2+
4uω − v2
4x2)
• Identical with LF Schrodingier equation
HLF = − d2
dζ2− 1− 4L2
4ζ2+ U(ζ)
provided that:
– x is identified with the LF variable ζ : x = ζ/√
2
– Casimir g with the LF orbital angular momentum L: g = L2 − 14
– Effective LF confining interaction
U(ζ) ∼ λ2ζ2 with λ2 =4uω − v2
16
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5 Gravity in AdS and Light Front Holographic Mapping
RNKLM = −1
R2(gNLgKM − gNM gKL)
• Why is AdS space important?
AdS5 is a space of maximal symmetry, negative curvature and a four-dim boundary: Minkowski space
• Isomorphism of SO(4, 2) group of conformal transformations with generators Pµ,Mµν,Kµ, D with
the group of isometries of AdS5 Dim isometry group of AdSd+1: (d+1)(d+2)2
• AdS5 metric xM = (xµ, z):
ds2 =R2
z2(ηµνdxµdxν − dz2)
• Since the AdS metric is invariant under a dilatation of all coordinates xµ → λxµ, z → λz, the
variable z acts like a scaling variable in Minkowski space
• Short distances xµxµ → 0 map to UV conformal AdS5 boundary z → 0
• Large confinement dimensions xµxµ ∼ 1/Λ2
QCD map to IR region of AdS5, z ∼ 1/ΛQCD, thus
AdS geometry has to be modified at large z to include the scale of strong interactions
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Light Front Holographic Mapping
[S. J. Brodsky and GdT, PRL 96, 201601 (2006)] Mapping of EM currents
[S. J. Brodsky and GdT, PRD 78, 025032 (2008)] Mapping of energy-momentum tensor
• EM transition matrix element in QCD: local coupling to pointlike constituents
〈P ′|Jµ|P 〉 = (P + P ′)µF (Q2)
where Q = P ′ − P and Jµ =∑
q eqqγµq
• EM hadronic amplitude in AdS from coupling of external EM field propagating in AdS with extended
mode Φ(x, z)∫d4x dz
√g AM (x, z)Φ∗P ′(x, z)
←→∂ MΦP (x, z)
∼ (2π)4δ4(P ′− P
)εµ(P + P ′
)µF (Q2)
• Recover hard pointlike scattering counting rules at large Q out of soft collision of extended objects:
cut z-space to introduce hadronic scale [Polchinski and Strassler (2002)]
• Could the QCD and AdS expressions for the EM form factor (and energy-momentum tensor) become
identical in some approximation?
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• Write the DYW expression for the EM form factor in impact space as a sum of overlap of LFWFs of the
j = 1, 2, · · · , n− 1 spectator constituents [D. E. Soper (1977)]
F (q2) =∑n
n−1∏j=1
∫dxjd
2b⊥j exp(iq⊥ ·
n−1∑j=1
xjb⊥j)|ψn(xj ,b⊥j)|2
• Compare AdS EM FF with EM FF in LF QCD for arbitrary Q:
expressions can be matched only if LFWF is factorized
• For n = 2
ψ(x, ζ, ϕ) = eiMϕX(x)φ(ζ)√2πζ
we find
X(x) =√x(1− x), φ(ζ) =
(ζ
R
)−3/2
Φ(ζ), z → ζ =√x(1− x) |b⊥|
• For arbitrary n we find the LF cluster decomposition (active quark vs spectators)
ζ =√
x
1− x
∣∣∣ n−1∑j=1
xjb⊥j∣∣∣
x = xn is the longitudinal momentum fraction of the active quark
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6 Higher Integer-Spin Wave Equations in AdS Space
[GdT, H.G. Dosch and S. J. Brodsky, PRD 87, 075004 (2013)]
• Description of higher spin modes in AdS space (Frondsal, Fradkin and Vasiliev)
• Integer spin-J fields in AdS conveniently described by tensor field ΦN1···NJwith effective action
Seff =∫ddx dz
√|g| eϕ(z) gN1N ′1 · · · gNJN
′J
(gMM ′DMΦ∗N1...NJ
DM ′ΦN ′1...N′J
− µ2eff (z) Φ∗N1...NJ
ΦN ′1...N′J
)DM is the covariant derivative which includes affine connection and dilaton ϕ(z) breaks conformality
• Effective mass µeff (z) is determined by precise mapping to light-front physics
• Non-trivial geometry of pure AdS encodes the kinematics and the additional deformations of AdS
encode the dynamics, including confinement
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• Physical hadron has plane-wave and polarization indices along 3+1 physical coordinates and a profile
wavefunction Φ(z) along holographic variable z
ΦP (x, z)µ1···µJ = eiP ·xΦ(z)µ1···µJ , Φzµ2···µJ = · · · = Φµ1µ2···z = 0
with four-momentum Pµ and invariant hadronic mass PµPµ=M2
• Further simplification by using a local Lorentz frame with tangent indices
• Variation of the action gives AdS wave equation for spin-J field Φ(z)ν1···νJ = ΦJ(z)εν1···νJ[−z
d−1−2J
eϕ(z)∂z
(eϕ(z)zd−1−2J
∂z
)+(mR
z
)2]
ΦJ = M2ΦJ
with
(mR)2 = (µeff (z)R)2 − Jz ϕ′(z) + J(d− J + 1)
and the kinematical constraints
ηµνPµ ενν2···νJ = 0, ηµν εµνν3···νJ = 0.
• Kinematical constrains in the LF imply that m must be a constant
[See also: T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega, Phys. Rev. D 85, 076003 (2012)]
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Light-Front Mapping
[GdT and S. J. Brodsky, PRL 102, 081601 (2009)]
• Upon substitution ΦJ(z) ∼ z(d−1)/2−Je−ϕ(z)/2 φJ(z) and z→ζ in AdS WE[−z
d−1−2J
eϕ(z)∂z
(eϕ(z)
zd−1−2J∂z
)+(mR
z
)2]
ΦJ(z) = M2ΦJ(z)
we find LFWE (d = 4)(− d2
dζ2− 1− 4L2
4ζ2+ U(ζ)
)φJ(ζ) = M2φJ(ζ)
withU(ζ) = 1
2ϕ′′(ζ) +
14ϕ′(ζ)2 +
2J − 32z
ϕ′(ζ)
and (mR)2 = −(2− J)2 + L2
• Unmodified AdS equations correspond to the kinetic energy terms of the partons inside a hadron
• Interaction terms in the QCD Lagrangian build the effective confining potential U(ζ) and correspond
to the truncation of AdS space in an effective dual gravity approximation
• AdS Breitenlohner-Freedman bound (mR)2 ≥ −4 equivalent to LF QM stability condition L2 ≥ 0
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7 Meson Spectrum
• Dilaton profile in the dual gravity model determined from conformal QM (dAFF) !
ϕ(z) = λz2, λ2 =4uω − v2
16
• Effective potential: U = λ2ζ2 + 2λ(J − 1)
• LFWE (− d2
dζ2− 1− 4L2
4ζ2+ λ2ζ2 + 2λ(J − 1)
)φJ(ζ) = M2φJ(ζ)
• Normalized eigenfunctions 〈φ|φ〉 =∫dζ φ2(z) = 1
φn,L(ζ) = |λ|(1+L)/2
√2n!
(n+L)!ζ1/2+Le−|λ|ζ
2/2LLn(|λ|ζ2)
• Eigenvalues for λ > 0 M2n,J,L = 4λ
(n+
J + L
2
)• λ < 0 incompatible with LF constituent interpretation
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ζ
φ(ζ)
0 4 80
0.4
0.8
2-2012
8820A9 ζ
φ(ζ)
0 4 8
0
0.5
-0.5
2-2012
8820A10
LFWFs φn,L(ζ) in physical space-time: (L) orbital modes and (R) radial modes
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Table 1: I = 1 mesons. For a qq state P = (−1)L+1, C = (−1)L+S
L S n JPC I = 1 Meson
0 0 0 0−+ π(140)0 0 1 0−+ π(1300)0 0 2 0−+ π(1800)0 1 0 1−− ρ(770)0 1 1 1−− ρ(1450)0 1 2 1−− ρ(1700)
1 0 0 1+− b1(1235)1 1 0 0++ a0(980)1 1 1 0++ a0(1450)1 1 0 1++ a1(1260)1 1 0 2++ a2(1320)
2 0 0 2−+ π2(1670)2 0 1 2−+ π2(1880)2 1 0 3−− ρ3(1690)
3 1 0 4++ a4(2040)
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• J = L+ S, I = 1 meson families M2n,L,S = 4λ (n+ L+ S/2)
0
2
4
0 2
L
M2
(G
eV
2)
2-20128820A20
π(140)
π(1300)
π(1800)
b1(1235)
n=2 n=1 n=0
π2(1670)
π2(1880)
0
2
4
0 2
M2
(G
eV
2)
2-20128820A24 L
ρ(770)
ρ(1450)
ρ(1700)
a2(1320)
a4(2040)
ρ3(1690)
n=2 n=1 n=0
Orbital and radial excitations for the π (√λ = 0.59 GeV) and the ρ I=1meson families (
√λ = 0.54 GeV)
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M2IGeV2M
L
ΡH770L
a2H1320L
Ρ3H1690L
a4H2040L
a1H1260L
a0H980L
0 1 2 3
0
1
2
3
4
5
M2n,J,L = 4λ
(n+
J + L
2
)
• Triplet splitting for vector meson a-states
(L = 1, J = 0, 1, 2)
Ma2(1320) >Ma1(1260) >Ma0(980)
• Systematics of I = 1 light meson spectra – orbital and radial excitations as well as important J − Lsplitting, well described by light-front harmonic confinement model
• Linear Regge trajectories, a massless pion and relation between the ρ and a1 massMa1/Mρ =√
2usually obtained from Weinberg sum rules [Weinberg (1967)]
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8 Higher Half-Integer Spin Wave Equations in AdS Space
[J. Polchinski and M. J. Strassler, JHEP 0305, 012 (2003)]
[GdT and S. J. Brodsky, PRL 94, 201601 (2005)]
[GdT, H.G. Dosch and S. J. Brodsky, PRD 87, 075004 (2013)]
Image credit: N. Evans
• The gauge/gravity duality can give important insights into the strongly coupled dynamics of nucleons
using simple analytical methods: analytical exploration of systematics of light-baryon resonances
• Extension of holographic ideas to spin-12 (and higher half-integral J ) hadrons by considering wave
equations for Rarita-Schwinger spinor fields in AdS space and their mapping to light-front physics
• LF clustering decomposition of invariant variable ζ : same multiplicity of states for mesons and baryons
• But in contrast with mesons there is important degeneracy of states along a given Regge trajectory for
a given L: no spin-orbit coupling
[See also: T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega, Phys. Rev. D 85, 076003 (2012)]
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• Half-integer spin J = T + 12 conveniently represented by RS spinor [ΨN1···NT
]α with effective AdS
action
Seff = 12
∫ddx dz
√|g| gN1N ′1 · · · gNT N
′T[
ΨN1···NT
(iΓA eMA DM − µ− U(z)
)ΨN ′1···N ′T + h.c.
]where the covariant derivative DM includes the affine connection and the spin connection
• eAM is the vielbein and ΓA tangent space Dirac matrices{
ΓA,ΓB}
= ηAB
• LF mapping z → ζ find coupled LFWE
− d
dζψ− −
ν + 12
ζψ− − V (ζ)ψ− = Mψ+
d
dζψ+ −
ν + 12
ζψ+ − V (ζ)ψ+ = Mψ−
provided that |µR| = ν + 12 and
V (ζ) =R
ζU(ζ)
a J -independent potential – No spin-orbit coupling along a given trajectory !
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9 Baryon Spectrum
• Choose linear potential V = λ ζ , λ > 0 to satisfy dAFF
• Eigenfunctions
ψ+(ζ) ∼ ζ12
+νe−λζ2/2Lνn(λζ2)
ψ−(ζ) ∼ ζ32
+νe−λζ2/2Lν+1
n (λζ2)
• Eigenvalues
M2 = 4λ(n+ ν + 1)
• Lowest possible state n = 0 and ν = 0
• Orbital excitations ν = 0, 1, 2 · · · = L
• L is the relative LF angular momentum
between the active quark and spectator cluster
• For λ < 0 no solution possible
L
n � 0n = 1n � 2n � 3
NH940L
NH1440L
NH1710L NH1720LNH1680L
NH1900L
NH2220L
Λ � 0.49 GeV
M2IGeV2M
0 1 2 3 40
1
2
3
4
5
6
7
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SU(6) S L n Baryon State
56 12 0 0 N 1
2+
(940)
32 0 0 ∆ 3
2+
(1232)
56 12 0 1 N 1
2+
(1440)
32 0 1 ∆ 3
2+
(1600)
70 12 1 0 N 1
2−
(1535) N 32−
(1520)
32 1 0 N 1
2−
(1650) N 32−
(1700) N 52−
(1675)
12 1 0 ∆ 1
2−
(1620) ∆ 32−
(1700)
56 12 0 2 N 1
2+
(1710)
12 2 0 N 3
2+
(1720) N 52+
(1680)
32 2 0 ∆ 1
2+
(1910) ∆ 32+
(1920) ∆ 52+
(1905) ∆ 72+
(1950)
70 32 1 1 N 1
2−
N 32−
(1875) N 52−
32 1 1 ∆ 5
2−
(1930)
56 12 2 1 N 3
2+
(1900) N 52+
70 12 3 0 N 5
2−
N 72−
32 3 0 N 3
2−
N 52−
N 72−
(2190) N 92−
(2250)
12 3 0 ∆ 5
2−
∆ 72−
56 12 4 0 N 7
2+
N 92+
(2220)
32 4 0 ∆ 5
2+
∆ 72+
∆ 92+
∆ 112
+(2420)
70 12 5 0 N 9
2−
N 112−
32 5 0 N 7
2−
N 92−
N 112−
(2600) N 132−
MENU 2013, PUSC, October 3, 2013Page 31
GdT
• Eigenvalues
M2 = 4λ(n+ ν + 1)
ν is related to the 5-dim AdS mass: ν = |µR| − 12
• Gap scale 4λ determines trajectory slope and
spectrum gap between plus-parity spin-12 and
minus-parity spin-32 nucleon families !
• ν depends on internal spin and parity
• For nucleons ν+1/2 = L, ν−3/2 = L+ 1
• The assignment
ν+1/2 = L, ν+
3/2 = L+ 1/2
ν−1/2 = L+ 1/2, ν−3/2 = L+ 1
describes the full light baryon orbital and radial excitation spectrum
MENU 2013, PUSC, October 3, 2013Page 32
GdT
• ν = L+ 1/2 baryons (√λ ' 0.5 GeV)
M2
NH1520L
NH1535L
NH1875L
n � 2 n � 1 n � 0
0 1 2 3 41
2
3
4
5
6
7
DH1950L
DH1920L
DH1910L
DH1905L
DH2420L
DH1232L
DH1600L
M2 n � 0n � 1n � 2n � 3
DH1700L
DH1620L
0 1 2 3 40
1
2
3
4
5
6
7
• ∆(1930) non SU(6) assignment (Klempt and Richard (2010): S = 3/2, L = 1, n = 1
• FindM∆(1930) = 4κ ' 2 GeV compared with experimental value 1.96 GeV
[See also: de Paula, Frederico, Forkel and Beyer, Phys. Rev. D 79, 075019 (2009)
and Forkel and Klempt, Phys. Lett. B 679, 77 (2009)]
MENU 2013, PUSC, October 3, 2013Page 33
GdT
Many thanks !
MENU 2013, PUSC, October 3, 2013Page 34