![Page 1: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/1.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
![Page 2: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/2.jpg)
Structure of the Vacuum in Light-Front Dynamics
∆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
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 2
![Page 3: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/3.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 3
![Page 4: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/4.jpg)
Structure of the Vacuum in Light-Front Dynamics
10 Higher Half-Integer Spin Wave Equations in AdS Space 34
11 Baryon Spectrum 36
12 Conclusions 38
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 4
![Page 5: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/5.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 5
![Page 6: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/6.jpg)
Structure of the Vacuum in Light-Front Dynamics
• 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)
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 6
![Page 7: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/7.jpg)
Structure of the Vacuum in Light-Front Dynamics
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)
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 7
![Page 8: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/8.jpg)
Structure of the Vacuum in Light-Front Dynamics
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+, P⊥, J3, K kinematical
• Point-form: initial surface is the hyperboloid x2 = κ2 > 0, x0 > 0
Pµ dynamical, Mµν kinematical
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 8
![Page 9: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/9.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
i
p+i
, p+i > 0
• Longitudinal momentum P+ is kinematical: sum of single particle constituents p+i of bound state
P+ =∑i
p+i , p+
i > 0
• LF Hamiltonian P− is dynamical: bound-state is arbitrarly off the LF energy shell
P− −n∑i
p−i < 0
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 9
![Page 10: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/10.jpg)
Structure of the Vacuum in Light-Front Dynamics
• LF evolution given by relativistic Schrodinger-like equation (x+ = x0 + x3 LF time)
i∂
∂x+|ψ(P )〉 = P−|ψ(P )〉
where
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)
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 10
![Page 11: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/11.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 11
![Page 12: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/12.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 12
![Page 13: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/13.jpg)
Structure of the Vacuum in Light-Front Dynamics
4 Light-Front Quantization of QCD
• Start with SU(3)C QCD Lagrangian
LQCD = ψ (iγµDµ −m)ψ − 14G
aµνG
aµν
• 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
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 13
![Page 14: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/14.jpg)
Structure of the Vacuum in Light-Front Dynamics
• 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
]with
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
q+
)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
q+
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 14
![Page 15: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/15.jpg)
Structure of the Vacuum in Light-Front Dynamics
• 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
kµi
)2=∑i
k2⊥i +m2
i
xi
• M2−M2n is the measure of the off-energy shell: key variable which controls the bound state
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 15
![Page 16: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/16.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
]∑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
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 16
![Page 17: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/17.jpg)
Structure of the Vacuum in Light-Front Dynamics
• 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︸ ︷︷ ︸M2n=
Pi
k2⊥i+m
2i
xi
), 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⊥
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 17
![Page 18: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/18.jpg)
Structure of the Vacuum in Light-Front Dynamics
• 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 valueL = 0 corresponds to lowest possible stable solution, the ground state of the LF invariant
Hamiltonian P 2
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 18
![Page 19: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/19.jpg)
Structure of the Vacuum in Light-Front Dynamics
• 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
q
x(1− x)
obtain
U = V 2 + 2√
p2 +m2q V + 2V
√p2 +m2
q
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]
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 19
![Page 20: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/20.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
Q2
)where g is a dimensionless number (Casimir operator which depends on the representation)
• The equation of motion
Q− g
Q3= 0
and the generator of evolution in t, the Hamiltonian
H = 12
(Q2 +
g
Q2
)follow from the dAFF action
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 20
![Page 21: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/21.jpg)
Structure of the Vacuum in Light-Front Dynamics
• 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
• Applying Noether’s theorem one obtains 3 conserved operators
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,
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 21
![Page 22: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/22.jpg)
Structure of the Vacuum in Light-Front Dynamics
• 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 =
12
(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
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 22
![Page 23: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/23.jpg)
Structure of the Vacuum in Light-Front Dynamics
• 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) =
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
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 23
![Page 24: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/24.jpg)
Structure of the Vacuum in Light-Front Dynamics
• 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
G = 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 !
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 24
![Page 25: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/25.jpg)
Structure of the Vacuum in Light-Front Dynamics
• Evolution in τ in terms of original field operator Q(t) at t = 0
G(Q, Q) =12u
(Q2 +
g
Q2
)− 1
4v(QQ+ QQ
)+
12wQ2
= 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
i∂
∂τψ(x, τ) = Hτ
(x,−i d
dx
)ψ(x, τ)
• dAFF Hamiltonian G
G =12u
(− d2
dx2+
g
x2
)+i
4v
(xd
dx+
d
dxx
)+
12wx2
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 25
![Page 26: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/26.jpg)
Structure of the Vacuum in Light-Front Dynamics
• Compare dAFF Hamiltonian
G =12u
(− d2
dx2+
g
x2
)+i
4v
(xd
dx+
d
dxx
)+
12wx2
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+
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 26
![Page 27: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/27.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 27
![Page 28: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/28.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 28
![Page 29: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/29.jpg)
Structure of the Vacuum in Light-Front Dynamics
• 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)]
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 29
![Page 30: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/30.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
)+(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
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 30
![Page 31: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/31.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
(n+
J + L
2
)• λ < 0 incompatible with LF constituent interpretation
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 31
![Page 32: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/32.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 32
![Page 33: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/33.jpg)
Structure of the Vacuum in Light-Front Dynamics
• 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
(m2qx
+m2q
1−x
)e−
12λ ζ
2
• 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+
J + L
2
)• Effective quark masses from reduction of higher Fock states as functionals of the valence state:
mu = md = 46 MeV, ms = 357 MeV
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 33
![Page 34: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/34.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
0
1
2
3
4
5
6
L
M2IGeV2M n 2 n 1 n 0
K*H892L
K*H1410L
K*H1680L
K2*H1430L
K3*H1780L
K4*H2045L
HbL
0 1 2 3 40
1
2
3
4
5
6
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
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 34
![Page 35: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/35.jpg)
Structure of the Vacuum in Light-Front Dynamics
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)]
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 35
![Page 36: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/36.jpg)
Structure of the Vacuum in Light-Front Dynamics
• 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 !
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 36
![Page 37: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/37.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
2
P = + ν = L ν = L+ 12
P = - ν = L+ 12 ν = L+ 1
describes the full light baryon orbital and radial excitation spectrum
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 37
![Page 38: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/38.jpg)
Structure of the Vacuum in Light-Front Dynamics
LL1-20148844A5
M2
(G
eV
2)
(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
3
5
7
20 40
2
4
6
M2
(G
eV
2)
(a) (b)
n=2n=3 n=1 n=0
N(1710)
N(1700)
4λ
N(1720)
N(1680)
N(1675)
N(2250)
N(2600)
N(2190)
N(2220)
ν=L
ν=L+1
N(1650)
N(940)
N(1440)
N(940)
N(1900) N(2220)
N(1720)
N(1680)
0 2 40
2
4
6
0 2 4 60
4
8
Baryon orbital and radial excitations for√λ = 0.49 GeV (nucleons) and 0.51 GeV (Deltas)
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 38
![Page 39: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/39.jpg)
Structure of the Vacuum in Light-Front Dynamics
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
dAFF
1 – dim
Quantum Field Theory
AdS5
AdS2LFQM
Conf(R1)
LFD
1-20148840A1
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 39
![Page 40: Structure of the Vacuum in Light-Front Dynamics and ... · Structure of the Vacuum in Light-Front Dynamics Structure of the Vacuum in Light-Front Dynamics and Emergent Holographic](https://reader034.vdocument.in/reader034/viewer/2022042102/5e7f60d08f4e4907c26325e4/html5/thumbnails/40.jpg)
Structure of the Vacuum in Light-Front Dynamics
Thank you !
Niccolo Cabeo 2014, Ferrara, May 20, 2012Page 40