Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front...

Structure of the Vacuum in Light-Front Dynamics

and Emergent Holographic QCD

Guy F. de Teramond

Universidad de Costa Rica

Ferrara International School

Niccolo Cabeo 2014

Vacuum and Symmetry Breaking:

From the Quantum to the Cosmos

Ferrara, May 19-23, 2014

Image credit: ic-msquare

In collaboration with Stan Brodsky (SLAC) and Hans G. Dosch (Heidelberg)

Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 1

∆ηµoκριτoσ460 - 370 BC

νoµω ψυχρoν, νoµω θερµoν, ετεη δε ατoµα και κενoν

(By convention cold, by convention hot, in reality atoms and the void)

Democritus 460 - 370 BC

Contents

1 Introduction 4

2 Dirac Forms of Relativistic Dynamics 6

3 Light-Front Dynamics and Light-Front Vacuum 8

4 Light-Front Quantization of QCD 12

5 Semiclassical Approximation to QCD in the Light Front 15

6 Effective Confinement from Underlying Conformal Invariance 19

7 Gravity in AdS and Light Front Holographic Mapping 26

8 Higher Integer-Spin Wave Equations in AdS Space 27

9 Meson Spectrum 30

10 Higher Half-Integer Spin Wave Equations in AdS Space 34

11 Baryon Spectrum 36

12 Conclusions 38

1 Introduction

• SLAC and DESY 70’s: 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 complexity beyond ground-state configurations

• Only known analytically tractable treatment in relativistic QFTh is perturbation theory: description of

confinement from perturbation theory is not possible (Kinoshita-Lee-Nauenberg Theorem)

• Important theoretical goal: find initial analytic approximation to strongly coupled QCD, like Schrodinger

or Dirac Eqs in atomic physics, corrected for quantum fluctuations

• Convenient frame-independent Hamiltonian framework for treating bound-states in relativistic theories

is light-front (LF) quantization

• Definition of invariant LF vacuum within causal horizon P 2|0〉 = 0

• Simple structure of vacuum allows definition of partonic content of hadron in terms of wavefunctions:

quantum-mechanical probabilistic interpretation of hadronic states

• Calculation of matrix elements 〈P + q|J |P 〉 requires boosting the hadronic bound state from |P 〉 to

|P + q〉: no change in particle number from boost in LF (kinematical problem)

• To a first semiclassical approximation one can reduce the strongly correlated multi-parton LF dynamics

to an effective one-dim QFTh, which encodes the conformal symmetry of the classical QCD Lagrangian

• Uniqueness of the form of confining interaction from one dim conformal QFTh: extension of conformal

quantum mechanics to LF bound-state dynamics

• Invariant Hamiltonian equation for bound states similar structure of AdS equations of motion: direct

connection of QCD and AdS/CFT possible

• Description of strongly coupled gauge theory in physical space-time using a dual gravity description in

a higher dimensional space (holographic)

2 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 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 form has its Hamiltonian and evolve with different time, but results computed in any form should

be identical (different parameterizations of space-time)

ct(a) (b) (c)

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+, P⊥, J3, K kinematical

• Point-form: initial surface is the hyperboloid x2 = κ2 > 0, x0 > 0

Pµ dynamical, Mµν kinematical

3 Light-Front Dynamics and Light-Front Vacuum

• Hadron with 4-momentum P = (P+, P−,P⊥), P± = P 0 ± P 3, mass-shell relation

P 2 = M2 leads to dispersion relation for LF Hamiltonian P−

P− =P2⊥ +M2

P+, P+ > 0

• n-particle bound state with p2i = m2

i , pi =(p+i , p

−i ,p⊥i

), for each constituent i

p−i =p2⊥i +m2

, p+i > 0

• Longitudinal momentum P+ is kinematical: sum of single particle constituents p+i of bound state

P+ =∑i

p+i , p+

• LF Hamiltonian P− is dynamical: bound-state is arbitrarly off the LF energy shell

P− −n∑i

p−i < 0

• LF evolution given by relativistic Schrodinger-like equation (x+ = x0 + x3 LF time)

∂x+|ψ(P )〉 = P−|ψ(P )〉

P−|ψ(P )〉 =P2⊥ +M2

P+|ψ(P )〉

• Construct LF invariant Hamiltonian P 2 = PµPµ = P−P+ −P2

P 2|ψ(P )〉 = M2|ψ(P )〉

• Since P+ and P⊥ are kinematical, linear QM evolution in LF Hamiltonian P− is maintained

• State |ψ(P )〉 is expanded in multi-particle Fock states |n〉 of the free LF Hamiltonian

|ψ〉 =∑n

ψn|n〉, |n〉 = |uud〉, |uudg〉, |uudqq〉, · · ·

with p2i = m2

i , pi = (p+i , p

−i ,p⊥i), for each constituent i in state n

• Fock component ψn is the light-front wave function (LFWF)

Vaccum in the Front-Form of Dynamics

• P+ =∑

i p+i , p

+i > 0: LF vacuum is the state with P+ = 0 and contains no particles: all other

states have P+ > 0 (usual vacuum bubbles are kinematically forbidden in the front form !)

• Frame independent definition of the vacuum within the causal horizon

P 2|0〉 = 0

(LF vacuum also has zero quantum numbers and P+ = 0)

• LF vacuum is defined at fixed LF time x+ = x0 + x3

over all x− = x0 − x3 and x⊥, the expanse of space

that can be observed within the speed of light

• Causality is maintained since LF vacuum only

requires information within the causal horizon

• The front form is a natural basis for cosmology:

universe observed along the front of a light wave

Vacuum in the Instant-Form of Dynamics

• Unsolved problems concerning the instant vacuum structure of QCD, confinement and chiral symmetry

breaking

• In Lattice QCD the structure of the instant-form vacuum is sampled in the Euclidean region

• If monopoles or vortices are removed from the vacuum confinement and chiral symmetry breaking

disappear

• Instantons provide a mechanism for symmetry breaking through the Banks-Casher relation

〈0|ψψ|0〉 = −πρ(0),

where ρ(0) is the density or Dirac-zero modes /Dψλ = λψλ, but do not lead to confinement

• No convincing proof that continuum QCD is confining in the infrared

• Confinement in continuum QCD is a fundamental unsolved problem in theoretical physics

4 Light-Front Quantization of QCD

• Start with SU(3)C QCD Lagrangian

LQCD = ψ (iγµDµ −m)ψ − 14G

aµνG

• Express the hadron four-momentum generator P = (P+, P−,P⊥) in terms of dynamical fields

ψ+ = Λ±ψ and A⊥(Λ± = γ0γ±

)quantized in null plane x+ = x0 + x3 = 0

P− = 12

∫dx−d2x⊥ψ+ γ

+ (i∇⊥)2 +m2

i∂+ψ+ + interactions

P+ =∫dx−d2x⊥ψ+γ

+i∂+ψ+

P⊥ = 12

∫dx−d2x⊥ψ+γ

+i∇⊥ψ+

where the integrals are over initial surface x+ = 0, where commutation relation for fields are defined

• Dirac field ψ+ expanded in terms of ladder operators b(q)|0〉 = d(q)|0〉 = 0

ψ+(x−,x⊥)α =∑λ

∫q+>0

dq+√2q+

d2q⊥(2π)3

[bλ(q)uα(q, λ)e−iq·x + dλ(q)†vα(q, λ)eiq·x

b(q), b†(q′)

=d(q), d†(q′)

= (2π)3 δ(q+ − q′+) δ(2)

(q⊥ − q′⊥

P− =∑λ

∫dq+d2q⊥

(2π)3(q2⊥ +m2

)b†λ(q)bλ(q) + interactions

P+ =∑λ

∫dq+d2q⊥

(2π)3q+ b†λ(q)bλ(q)

P⊥ =∑λ

∫dq+d2q⊥

(2π)3q⊥ b

†λ(q)bλ(q)

where q− = q2⊥+m2

• Relative partonic coordinates: k+i = xiP

+, p⊥i = xiP⊥i + k⊥in∑i=1

xi = 1,n∑i=1

k⊥i = 0

• Fock component ψn(xi,k⊥i, λzi ) depends only on relative coordinates: longitudinal momentum

fraction xi = k+i /P

+, transverse momentum k⊥i and spin λzi

• LFWF ψn is frame independent

• Momentum conservation: P+ =∑n

i=1 k+i , k+

i > 0, but LFWF represents a bound-state which is

off the LF energy shell, P− −∑n

i k−i < 0

• Express off-shell dependence of bound-state in terms of invariant variable

(invariant mass of constituents)

M2n = (k1 + k2 + · · ·+ kn)2 =

( n∑i=1

)2=∑i

k2⊥i +m2

• M2−M2n is the measure of the off-energy shell: key variable which controls the bound state

5 Semiclassical Approximation to QCD in the Light Front[GdT and S. J. Brodsky, PRL 102, 081601 (2009)]

• Effective reduction of strongly coupled LF multiparticle dynamics to a 1-dim QFT with no particle cre-

ation and absorption: LF quantum mechanics !

• 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)

• Compute M2 from hadronic matrix element 〈ψ(P ′)|PµPµ|ψ(P )〉=M2〈ψ(P ′)|ψ(P )〉

• Find

M2 =∑n

∫ [dxi][d2k⊥i

(k2⊥q +m2

)|ψn(xi,k⊥i)|2 + interactions

with phase space normalization∑n

∫ [dxi] [d2k⊥i

]|ψn(xi,k⊥i)|2 = 1

• 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

)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︸︷︷︸M2n=

k2⊥i+m

), mq → 0

• For a two-parton system in the mq → 0 M2qq = k2

⊥x(1−x)

• Conjugate invariant variable in transverse impact space is

ζ2 = x(1− x)b2⊥

• 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)

(− d2

dζ2− 1ζ

dζ+L2

)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 valueL = 0 corresponds to lowest possible stable solution, the ground state of the LF invariant

Hamiltonian P 2

• Compare invariant mass in the instant-form in the hadron center-of-mass system P = 0,

M2qq = 4m2

q + 4p2

with the invariant mass in the front-form in the constituent rest frame, kq + kq = 0

M2qq =

k2⊥ +m2

x(1− x)

obtain

U = V 2 + 2√

p2 +m2q V + 2V

√p2 +m2

where p2⊥ = k2

⊥4x(1−x) , p3 = mq(x−1/2)√

x(1−x), and V is the effective potential in the instant-form

• For small quark masses a linear instant-form potential V implies a harmonic front-form potential U

and thus linear Regge trajectories

[A. P. Trawinski, S. D. Glazek, S. J. Brodsky, GdT, H. G. Dosch, arXiv: 1403.5651]

6 Effective Confinement from Underlying Conformal Invariance[S. J. Brodsky, GdT and H.G. Dosch, PLB 729, 3 (2014)]

• 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

)where g is a dimensionless number (Casimir operator which depends on the representation)

• The equation of motion

Q− g

and the generator of evolution in t, the Hamiltonian

H = 12

)follow from the dAFF action

• Absence of dimensional constants implies that the action

S = 12

∫dt(Q2 − g

)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

• Applying Noether’s theorem one obtains 3 conserved operators

I. Translations in t: H = 12

(Q2 + g

II. Dilatations: D = 12

(Q2 + g

)t− 1

(QQ+QQ

III. Special conformal transformations: K = 12

(Q2 + g

)t2 − 1

(QQ+QQ

• Using canonical commutation relations [Q(t), Q(t)] = i find

[H,D] = iH, [H,K] = 2iD, [K,D] = −iK,

the algebra of the generators of the conformal group Conf(R1)

• Introduce the linear combinations

J12 =12

(1aK + aH

), J01 =

(1aK − aH

), J02 = D,

where a has dimension t since H 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 and a ground state

• The relation between the generators of conformal group and generators of SO(2, 1) suggests that

the scale a may play a fundamental role

• dAFF construct a new generator as a superposition of the 3 constants of motion

G = uH + vD + wK

and introduce new time variable τ and field operator q(τ)

dτ =dt

u+ vt+ wt2, q(τ) =

[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

• In terms of τ and q(τ)

S = 12

∫dt(Q2 − g

∫dτ(q2 − g

q2− 4uω − v2

+ surface term

Action is conformal invariant invariant up to a surface term !

• The corresponding Hamiltonian

G = 12

4uω − v2

is a compact operator for4uω − v2

• Scale appears in the Hamiltonian without affecting the conformal invariance of the action !

• Evolution in τ in terms of original field operator Q(t) at t = 0

G(Q, Q) =12u

)− 1

4v(QQ+ QQ

= uH + vD + wK,

• The Schrodinger picture follows from the representation of Q and P = Q

Q→ x, Q→ −i ddx

• dAFF Conformal Quantum Mechanics !

• Schrodinger wave equation determines evolution of bound states in terms of the variable τ

∂τψ(x, τ) = Hτ

(x,−i d

)ψ(x, τ)

• dAFF Hamiltonian G

G =12u

(− d2

• Compare dAFF Hamiltonian

G =12u

(− d2

with LF Hamiltonian for u = 2, v = 0,

P 2 = − d2

dζ2− 1− 4L2

4ζ2+ U(ζ)

• x is identified with the LF variable ζ : x = ζ

• Casimir g with the LF orbital angular momentum L: g = L2 − 14

• w = 2λ2 ∼ 1a2 fixes effective LF confining interaction to quadratic λ2 ζ2 dependence

U ∼ λ2ζ2

• Also, dAFF evolution variable τ is identified with LF time τ = x+/P+

7 Gravity in AdS and Light Front Holographic Mapping

RNKLM = −1

R2(gNLgKM − gNMgKL)

• Why is AdS space important?

AdS5 is 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

8 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···NJ with 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

• 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

ΦJ = M2ΦJ

(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)]

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 =√x(1− x)b in AdS WE[

−zd−1−2J

eϕ(z)∂z

(eϕ(z)

zd−1−2J∂z

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

9 Meson Spectrum

• Dilaton profile in the dual gravity model determined from one-dim QFTh (dAFF)

ϕ(z) = λz2, λ2 = w/2

• 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

)• λ < 0 incompatible with LF constituent interpretation

φ(ζ)

0 4 80

2-2012

8820A9 ζ

φ(ζ)

2-2012

8820A10

LFWFs φn,L(ζ) in physical space-time: (L) orbital modes and (R) radial modes

• Results easily extended to light quarks masses (Ex: K-mesons)

[GdT, S. J. Brodsky and H. G.Dosch, arXiv:1405.2451 [hep-ph]]

• First order perturbation in the quark masses

∆M2 = 〈ψ|∑a

m2a/xa|ψ〉

• Holographic LFWF with quark masses

[S. J. Brodsky and GdT, arXiv:0802.0514 [hep-ph]

ψ(x, ζ) ∼√x(1− x) e−

12λ ζ

• Ex: Description of diffractive vector meson production at HERA

[J. R. Forshaw and R. Sandapen, PRL 109, 081601 (2012)]

• For the K∗

M2n,L,S = M2

K± + 4λ(n+

)• Effective quark masses from reduction of higher Fock states as functionals of the valence state:

mu = md = 46 MeV, ms = 357 MeV

M2IGeV2MHaL n 3 n 2 n 1 n 0

ΡH1700L

ΡH1450L

ΡH770L

a2H1320L

Ρ3H1690L

a4H2040L

f2H1270L

f4H2050L

f2H2300L

f2H1950L

ΩH782L

ΩH1650L

ΩH1420L

Ω3H1670L

L0 1 2 3 4

M2IGeV2M n 2 n 1 n 0

K*H892L

K*H1410L

K*H1680L

K2*H1430L

K3*H1780L

K4*H2045L

0 1 2 3 40

Orbital and radial excitations for vector mesons for√λ = 0.54 GeV

• Linear Regge trajectories, a massless pion and relation between the ρ and a1 massMa1/Mρ =√

2usually obtained from Weinberg sum rules described by LF harmonic confinement model

10 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

ζ =√

x1−x

∣∣∣∑n−1j=1 xjb⊥j

∣∣∣ (spectators vs active quark)

• 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)]

• 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

dζψ− −

ν + 12

ζψ− − V (ζ)ψ− = Mψ+

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 !

11 Baryon Spectrum

• Choose linear potential V = λ ζ , λ > 0 to satisfy dAFF

• Eigenfunctions

ψ+(ζ) ∼ ζ12+νe−λζ

2/2Lνn(λζ2), ψ−(ζ) ∼ ζ32+νe−λζ

2/2Lν+1n (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

• ν depends on internal spin and parity

The assignment

S = 12 S = 3

P = + ν = L ν = L+ 12

P = - ν = L+ 12 ν = L+ 1

describes the full light baryon orbital and radial excitation spectrum

LL1-20148844A5

(c) (d)

n=2 n=1 n=0

n=2n=3 n=1 n=0

Δ(1950)

Δ(1920)

Δ(1700)

Δ(1620)

Δ(1910)

Δ(1905)

Δ(1232)

Δ(1600)

Δ(2420)

N(1875)

N(1535)

N(1520)

0 2 41

(a) (b)

n=2n=3 n=1 n=0

N(1710)

N(1700)

N(1720)

N(1680)

N(1675)

N(2250)

N(2600)

N(2190)

N(2220)

ν=L+1

N(1650)

N(940)

N(1440)

N(940)

N(1900) N(2220)

N(1720)

N(1680)

0 2 40

0 2 4 60

Baryon orbital and radial excitations for√λ = 0.49 GeV (nucleons) and 0.51 GeV (Deltas)

12 Conclusions

We have described a threefold connection between

a one-dimensional QFT semiclassical approximation

to light-front QCD with gravity in a higher dimensional

AdS space, and the constraints imposed by the invariance

properties under the full conformal group in one dimension.

This provides a new insight into the physics underlying the

QCD vacuum, confinement, chiral invariance and the QCD

mass scale

4 – dim Light –

Front Dynamics

SO(2,1)

5 – dim

Classical Gravity

1 – dim

Quantum Field Theory

AdS2LFQM

Conf(R1)

1-20148840A1

Thank you !

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