Hadron Spectroscopy and Dynamics from Light …menu2013.roma2.infn.it/.../3-DeTeramond_menu2013.pdfGdT Hadron Spectroscopy and Dynamics from Light-Front Holography and Conformal Symmetry

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)

MENU 2013, PUSC, October 3, 2013Page 1

GdT

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


GdT

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−


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• 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


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)]


GdT

Many thanks !


Documents

Hadron Spectroscopy and Dynamics from Light …menu2013.roma2.infn.it/.../3-DeTeramond_menu2013.pdfGdT Hadron Spectroscopy and Dynamics from Light-Front Holography and Conformal Symmetry