Light-Front Holography

Guy F. de Teramond´

University of Costa Rica

In Collaboration with Stan Brodsky

2009 JLab Users Group Meeting Jefferson Lab, June 8-10, 2009

GdT and Brodsky, PRL 102, 081601 (2009)

2009 JLab Users Group Meeting, June 8, 2009 Page 1 Outline

1. Introduction

2. Light-Front Dynamics

Light-Front Fock Representation

3. Semiclassical Approximation to QCD

Hard-Wall Model

Holographic Mapping

4. Higher-Spin Bosonic Modes

Hard-Wall Model

Soft-Wall Model

5. Higher-Spin Fermionic Modes

Hard-Wall Model

Soft-Wall Model

6. Other Applications of Gauge/Gravity Duality

2009 JLab Users Group Meeting, June 8, 2009 Page 2 1 Introduction

• Most challenging problem of strong interaction dynamics: determine the composition of hadrons in terms of their fundamental QCD quark and gluon degrees of freedom

• Recent developments inspired by the AdS/CFT correspondence [Maldacena (1998)] between string states in AdS space and conformal ﬁeld theories in physical space-time have led to analytical insights into the conﬁning dynamics of QCD

• Description of strongly coupled gauge theory using a dual gravity description!

• Strings describe spin-J extended objects (no quarks). QCD degrees of freedom are pointlike particles and hadrons have orbital angular momentum: how can they be related?

• Light-front (LF) quantization is the ideal framework to describe hadronic structure in terms of quarks and gluons: simple vacuum structure allows unambiguous deﬁnition of the partonic content of a hadron, exact formulae for form factors, physics of angular momentum of constituents ...

µ 2 • Frame-independent LF Hamiltonian equation PµP |P i = M |P i similar structure of AdS EOM

• First semiclassical approximation to the bound-state LF Hamiltonian equation in QCD is equivalent to equations of motion in AdS and can be systematically improved

2009 JLab Users Group Meeting, June 8, 2009 Page 3 2 Light Front Dynamics

• Different possibilities to parametrize space-time [Dirac (1949)]

• Parametrizations differ by the hypersurface on which the initial conditions are speciﬁed. Each evolve with different ‘times’ and has its own Hamiltonian, but should give same physical results

• Instant form: hypersurface deﬁned by t = 0, the familiar one

• Front form: hypersurface is tangent to the light cone at τ = t + z/c = 0

x+ = x0 + x3 light-front time

x− = x0 − x3 longitudinal space variable

k+ = k0 + k3 longitudinal momentum (k+ > 0)

k− = k0 − k3 light-front energy

1 + − − + k · x = 2 (k x + k x ) − k⊥ · x⊥

2 2 2 2 − k⊥+m On shell relation k = m leads to dispersion relation k = k+

2009 JLab Users Group Meeting, June 8, 2009 Page 4 • QCD Lagrangian 1 L = − Tr (GµνG ) + iψD γµψ + mψψ QCD 4g2 µν µ + − • LF Momentum Generators P = (P ,P , P⊥) in terms of dynamical ﬁelds ψ, A⊥ 1 Z (i∇ )2 + m2 P − = dx−d2x ψ γ+ ⊥ ψ + interactions 2 ⊥ i∂+ Z + − 2 + + P = dx d x⊥ ψ γ i∂ ψ 1 Z P = dx−d2x ψ γ+i∇ ψ ⊥ 2 ⊥ ⊥

• LF energy P − generates LF time translations ∂ ψ(x),P − = i ψ(x) ∂x+ + and the generators P and P⊥ are kinematical

2009 JLab Users Group Meeting, June 8, 2009 Page 5 Light-Front Fock Representation

• Dirac ﬁeld ψ, expanded in terms of ladder operators on the initial surface x+ = x0 + x3 Z + 2 2 2 X dq d q⊥ q + m P − = ⊥ b† (q)b (q) + interactions (2π)3 q+ λ λ λ

2 2 − q⊥+m 2 Sum over free quanta q = q+ plus interactions (m = 0 for gluons) µ − + 2 • Construct light-front invariant Hamiltonian for the composite system: HLF = PµP = P P −P⊥ 2 HLC | ψH i = MH | ψH i

+ • State |ψH (P )i = |ψH (P , P⊥,Jz)i is an expansion in multi-particle Fock eigenstates | ni of the free LF Hamiltonian: X |ψH i = ψn/H |ni n z + • Fock components ψn/H (xi, k⊥i, λi ) are independent of P and P⊥ and depend only on relative + + z partonic coordinates: momentum fraction xi = ki /P , transverse momentum k⊥i and spin λi n n X X xi = 1, k⊥i = 0. i=1 i=1

2009 JLab Users Group Meeting, June 8, 2009 Page 6 • Compute M2 from hadronic matrix element

0 2 0 hψH (P )|HLF |ψH (P )i=MH hψH (P )|ψH (P )i

• Find Z 2 2 2 X 2 X k⊥` + m` 2 MH = dxi d k⊥i ψn/H (xi, k⊥i) + interactions xq n `

• Phase space normalization of LFWFs Z X 2 2 dxi d k⊥i ψn/h(xi, k⊥i) = 1 n

• In terms of n−1 independent transverse impact coordinates b⊥j, j = 1, 2, . . . , n−1,

n−1 Z −∇2 + m2 2 X Y 2 ∗ X b⊥` ` MH = dxjd b⊥jψn/H (xi, b⊥i) ψn/H (xi, b⊥i)+interactions xq n j=1 `

• Normalization n−1 Z X Y 2 2 dxjd b⊥j |ψn(xj, b⊥j)| = 1 n j=1

2009 JLab Users Group Meeting, June 8, 2009 Page 7 3 Semiclassical Approximation to QCD

• Consider a two-parton hadronic bound state in transverse impact space in the limit mq → 0 Z 1 dx Z M2 = d2b ψ∗(x, b )−∇2 ψ(x, b ) + interactions ⊥ ⊥ b⊥ ⊥ 0 1 − x

• Separate angular, transverse and longitudinal modes in terms of boost invariant transverse variable: 2 2 2 k⊥ ζ = x(1 − x)b⊥ — In k⊥ space key variable is the LF KE x(1−x) — φ(ζ) ψ(x, ζ, ϕ) = √ eiMϕf(x) 2πζ • Find (L = |M|)

Z d2 1 d L2 φ(ζ) Z M2 = dζ φ∗(ζ)pζ − − + √ + dζ φ∗(ζ) U(ζ) φ(ζ) dζ2 ζ dζ ζ2 ζ

where the conﬁning forces from the interaction terms is summed up in the effective potential U(ζ)

2009 JLab Users Group Meeting, June 8, 2009 Page 8 • Ultra relativistic limit mq → 0 longitudinal modes decouple and LF eigenvalue equation 2 HLF |φi = M |φi is a LF wave equation for φ

2 2 d 1 − 4L 2 − 2 − 2 + U(ζ) φ(ζ) = M φ(ζ) dζ 4ζ | {z } | {z } confinement kinetic energy of partons

• Effective light-front Schrodinger¨ equation: relativistic, frame-independent and analytically tractable

• Eigenmodes φ(ζ) determine the hadronic mass spectrum and represent the probability amplitude to ﬁnd n-massless partons at transverse impact separation ζ within the hadron at equal light-front time

• LF modes φ(ζ) = hζ|φi are normalized by Z hφ|φi = dζ |hζ|φi|2 = 1

• Semiclassical approximation to light-front QCD does not account for particle creation and absorption but can be implemented in the LF Hamiltonian EOM

2009 JLab Users Group Meeting, June 8, 2009 Page 9 Hard-Wall Model

• Consider the potential (hard wall)   0 if ζ ≤ 1 U(ζ) = ΛQCD  ∞ if ζ > 1 ΛQCD

2 L 2 • If L ≥ 0 the Hamiltonian is positive deﬁnite hφ HLF φi ≥ 0 and thus M ≥ 0

• If L2 < 0 the Hamiltonian is not bounded from below ( “Fall-to-the-center” problem in Q.M.)

• Critical value of the potential corresponds to L = 0, the lowest possible stable state

• Solutions: p φL(ζ) = CL ζJL (ζM)

• Mode spectrum from boundary conditions 1 φ ζ = = 0 ΛQCD

Thus 2 M = βLkΛQCD

2009 JLab Users Group Meeting, June 8, 2009 Page 10 • Excitation spectrum hard-wall model: Mn,L ∼ L + 2n

Light-meson orbital spectrum ΛQCD = 0.32 GeV

2009 JLab Users Group Meeting, June 8, 2009 Page 11 Holographic Mapping

• Holographic mapping found originally by matching expressions of EM and gravitational form factors of hadrons in AdS and LF QCD [Brodsky and GdT (2006, 2008)]

• Substitute Φ(ζ) ∼ ζ3/2φ(ζ), ζ → z in the conformal LFWE d2 1 − 4L2 − − φ(ζ) = M2φ(ζ) dζ2 4ζ2

• Find: 2 2 2 2 2 z ∂z − 3z ∂z + z M − (µR) Φ(z) = 0 2 2 with (µR) = −4 + L , the wave equation of string mode in AdS5 !

• Isomorphism of SO(4, 2) group of conformal QCD with generators P µ,M µν,D,Kµ with the group µ µ of isometries of AdS5 space: x → λx , z → λz R2 ds2 = (η dxµdxν − dz2) z2 µν

• AdS Breitenlohner-Freedman bound (µR)2 ≥ −4 equivalent to LF QM stability condition L2 ≥ 0

• Conformal dimension ∆ of AdS mode Φ given in terms of 5-dim mass by (µR)2 = ∆(∆ − 4). Thus ∆ = 2 + L in agreement with the twist scaling dimension of a two parton object in QCD

2009 JLab Users Group Meeting, June 8, 2009 Page 12 • AdS5 metric: 2 2 R µ ν 2 ds = 2 (ηµνdx dx −dz ) |{z} z | {z } LAdS LMinkowski

• A distance LAdS shrinks by a warp factor as observed in Minkowski space (dz = 0): z L ∼ L Minkowski R AdS • Different values of z correspond to different scales at which the hadron is examined

µ µ µ • Since x → λx , z → λz, short distances xµx → 0 maps to UV conformal AdS5 boundary z → 0, which corresponds to the Q → ∞ UV zero separation limit

µ 2 • Large conﬁnement dimensions xµx ∼ 1/ΛQCD maps to large IR region of AdS5, z ∼ 1/ΛQCD, thus there is a maximum separation of quarks and a maximum value of z at the IR boundary

• Local operators like O and LQCD deﬁned in terms of quark and gluon ﬁelds at the AdS5 boundary

• Use the isometries of AdS to map the local interpolating operators at the UV boundary of AdS into the modes propagating inside AdS

2009 JLab Users Group Meeting, June 8, 2009 Page 13 4 Higher-Spin Bosonic Modes

Hard-Wall Model ` µ • AdSd+1 metric x = (x , z): R2 ds2 = g dx`dxm = (η dxµdxν − dz2) `m z2 µν

• Action for gravity coupled to scalar ﬁeld in AdSd+1 Z √ 1 1 S = dd+1x g (R − 2Λ) + g`m∂ Φ∂ Φ − µ2Φ2 κ2 2 ` m | {z } | {z } SG SM • Equations of motion for SM 1 µR2 z3∂ ∂ Φ − ∂ ∂ρΦ − Φ = 0 z z3 z ρ z −iP ·x • Physical AdS modes ΦP (x, z) ∼ e Φ(z) are plane waves along the Poincare´ coordinates µ µ 2 with four-momentum P and hadronic invariant mass states PµP = M µ • Factoring out dependence of string mode ΦP (x, z) along x -coordinates 2 2 2 2 2 z ∂z − (d − 1)z ∂z + z M − (µR) Φ(z) = 0

2009 JLab Users Group Meeting, June 8, 2009 Page 14 • Spin J-ﬁeld on AdS represented by rank-J totally symmetric tensor ﬁeld Φ(x, z)`1···`J [Fronsdal; Fradkin and Vasiliev]

• Action in AdSd+1 for spin-J ﬁeld 1 Z √ S = dd+1x g ∂ Φ ∂`Φ`1···`J − µ2Φ Φ`1···`J + ... M 2 ` `1···`J `1···`J

• Each hadronic state of total spin J is dual to a normalizable string mode

−iP ·x ΦP (x, z)µ1···µJ = e Φ(z)µ1···µJ

with four-momentum Pµ, spin polarization indices along the 3+1 physical coordinates and hadronic µ 2 invariant mass PµP = M

• For string modes with all indices along Poincare´ coordinates, Φzµ2···µJ = Φµ1z···µJ = ··· = 0 and appropriate subsidiary conditions system of coupled differential equations from SM reduce to a

homogeneous wave equation for Φ(z)µ1···µJ

2009 JLab Users Group Meeting, June 8, 2009 Page 15 • Obtain spin-J mode Φµ1···µJ with all indices along 3+1 coordinates from Φ by shifting dimensions z −J Φ (z) = Φ(z) J R

• Normalization [Hong, Yoon and Strassler (2006)]

Z zmax d−2J−1 dz 2 R d−2J−1 ΦJ (z) = 1 0 z

• Substituting in the AdS scalar wave equation for Φ

2 2 2 2 2 z ∂z − (d−1−2J)z ∂z + z M − (µR) ΦJ = 0

upon ﬁfth-dimensional mass rescaling (µR)2 → (µR)2 − J(d − J)

• Conformal dimension of J-mode 1 p ∆ = d+ (d − 2J)2 + 4µ2R2 2

and thus (µR)2 = (∆ − J)(∆ − d + J)

2009 JLab Users Group Meeting, June 8, 2009 Page 16 • Upon substitution z →ζ and −3/2+J φJ (ζ)∼ζ ΦJ (ζ)

we recover the QCD LF wave equation (d = 4)

d2 1 − 4L2 − − φ = M2φ dζ2 4ζ2 µ1···µJ µ1···µJ

with (µR)2 = −(2 − J)2 + L2

• J-decoupling in the HW model

2 • For L ≥ 0 the LF Hamiltonian is positive deﬁnite hφJ |HLF |φJ i ≥ 0 and we ﬁnd the stability bound (µR)2 ≥ −(2 − J)2

• The scaling dimensions are ∆ = 2 + L independent of J in agreement with the twist scaling dimension of a two parton bound state in QCD

2009 JLab Users Group Meeting, June 8, 2009 Page 17 Soft-Wall Model

• Soft-wall model [Karch, Katz, Son and Stephanov (2006)] retain conformal AdS metrics but introduce smooth cutoff wich depends on the proﬁle of a dilaton background ﬁeld ϕ(z) = ±κ2z2 Z √ S = ddx dz g eϕ(z)L,

1 `m 2 2 • Equation of motion for scalar ﬁeld L = 2 g ∂`Φ∂mΦ − µ Φ 2 2 2 2 2 2 2 z ∂z − d − 1 ∓ 2κ z z ∂z + z M − (µR) Φ(z) = 0

with (µR)2 ≥ −4 (KKSS AdS/QCD model ‘minus’ dilaton ϕ = −κ2z2)

• LF holographic model ‘plus dilaton’ ϕ =+ κ2z2. Lowest possible state (µR)2 = −4

2 2 1 M2 = 0, Φ(z) ∼ z2e−κ z , hr2i ∼ κ2 Chiral symmetric bound state of two massless quarks with scaling dimension 2: the pion

2009 JLab Users Group Meeting, June 8, 2009 Page 18 • Action in AdSd+1 for spin J-ﬁeld

1 Z √ 2 2 S = ddx dz g eκ z ∂ Φ ∂`Φ`1···`J − µ2Φ Φ`1···`J + ... M 2 ` `1···`J `1···`J

• Obtain spin-J mode Φµ1···µJ with all indices along 3+1 coordinates from Φ by shifting dimensions z −J Φ (z) = Φ(z) J R

• Normalization Z ∞ d−2J−1 dz κ2z2 2 R d−2J−1 e ΦJ (z) = 1. 0 z • Substituting in the AdS scalar wave equation for Φ

2 2 2 2 2 2 2 z ∂z − d−1−2J − 2κ z z ∂z + z M − (µR) ΦJ = 0

upon mass rescaling (µR)2 → (µR)2 − J(d − J) and M2 → M2 − 2Jκ2

2009 JLab Users Group Meeting, June 8, 2009 Page 19 • Upon substitution z →ζ (Jz = Lz + Sz) we ﬁnd for d = 4

−3/2+J κ2ζ2/2 2 2 2 φJ (ζ)∼ζ e ΦJ (ζ), (µR) = −(2 − J) + L

d2 1 − 4L2 − − + κ4ζ2 + 2κ2(L + S − 1) φ = M2φ dζ2 4ζ2 µ1···µJ µ1···µJ

• Eigenfunctions s 2n! 2 2 φ (ζ) = κ1+L ζ1/2+Le−κ ζ /2LL(κ2ζ2) nL (n+L)! n

• Eigenvalues 6 S 5 M2 = 4κ2 n + L + n,L,S 2 4 Φ 0 (z) Φ(z) 2 4κ for ∆n = 1 2 -5 4κ2 for ∆L = 1

2 0 0 4 8 2κ for ∆S = 1 2-2007 0 4 8 2-2007 8721A20 z 8721A21 z Orbital and radial states: hζi increase with L and n

2009 JLab Users Group Meeting, June 8, 2009 Page 20 1−− 2++ 3−− 4++ J PC M2

Parent and daughter Regge trajectories for the I = 1 ρ-meson family (red) and the I = 0 ω-meson family (black) for κ = 0.54 GeV

2009 JLab Users Group Meeting, June 8, 2009 Page 21 5 Higher-Spin Fermionic Modes

Hard-Wall Model

• Action for massive fermionic modes on AdSd+1: From Nick Evans Z d √ ` S[Ψ, Ψ] = d x dz g Ψ(x, z) iΓ D` − µ Ψ(x, z)

` • Equation of motion: iΓ D` − µ Ψ(x, z) = 0 d i zη`mΓ ∂ + Γ + µR Ψ(x`) = 0 ` m 2 z

• Solution (µR = ν + 1/2, d = 4)

5/2 Ψ(z) = Cz [Jν(zM)u+ + Jν+1(zM)u−]

• Hadronic mass spectrum determined from IR boundary conditions ψ± (z = 1/ΛQCD) = 0

+ − M = βν,k ΛQCD, M = βν+1,k ΛQCD

with scale independent mass ratio

Obtain spin- mode , 1 , with all indices along 3+1 from by shifting dimensions • J Φµ1···µJ−1/2 J > 2 Ψ

2009 JLab Users Group Meeting, June 8, 2009 Page 22 SU(6) SL Baryon State

1 1 + 56 2 0 N 2 (939) 3 3 + 2 0 ∆ 2 (1232) 1 1 − 3 − 70 2 1 N 2 (1535) N 2 (1520) 3 1 − 3 − 5 − 2 1 N 2 (1650) N 2 (1700) N 2 (1675) 1 1 − 3 − 2 1 ∆ 2 (1620) ∆ 2 (1700) 1 3 + 5 + 56 2 2 N 2 (1720) N 2 (1680) 3 1 + 3 + 5 + 7 + 2 2 ∆ 2 (1910) ∆ 2 (1920) ∆ 2 (1905) ∆ 2 (1950) 1 5 − 7 − 70 2 3 N 2 N 2 3 3 − 5 − 7 − 9 − 2 3 N 2 N 2 N 2 (2190) N 2 (2250) 1 5 − 7 − 2 3 ∆ 2 (1930) ∆ 2 1 7 + 9 + 56 2 4 N 2 N 2 (2220) 3 5 + 7 + 9 + 11 + 2 4 ∆ 2 ∆ 2 ∆ 2 ∆ 2 (2420) 1 9 − 11 − 70 2 5 N 2 N 2 (2600) 3 7 − 9 − 11 − 13 − 2 5 N 2 N 2 N 2 N 2

2009 JLab Users Group Meeting, June 8, 2009 Page 23 • Excitation spectrum for baryons in the hard-wall model: M ∼ L + 2n

Light baryon orbital spectrum for ΛQCD = 0.25 GeV in the HW model. The 56 trajectory corresponds to L even P = + states, and the 70 to L odd P = − states: (a) I = 1/2 and (b) I = 3/2

2009 JLab Users Group Meeting, June 8, 2009 Page 24 Soft-Wall Model

• Equivalent to Dirac equation in presence of a holographic linear conﬁning potential d i zη`mΓ ∂ + Γ + µR + κ2z Ψ(x`) = 0. ` m 2 z

• Solution (µR = ν + 1/2, d = 4)

5 +ν −κ2z2/2 ν 2 2 Ψ+(z) ∼ z 2 e Ln(κ z ) 7 +ν −κ2z2/2 ν+1 2 2 Ψ−(z) ∼ z 2 e Ln (κ z )

• Eigenvalues M2 = 4κ2(n + ν + 1)

Obtain spin- mode , 1 , with all indices along 3+1 from by shifting dimensions • J Φµ1···µJ−1/2 J > 2 Ψ

2009 JLab Users Group Meeting, June 8, 2009 Page 25 2 2 Glazek and Schaden [Phys. Lett. B 198, 42 (1987)]: (ωB/ωM ) = 5/8 4κ for ∆n = 1 4κ2 for ∆L = 1 2κ2 for ∆S = 1 M2

L Parent and daughter 56 Regge trajectories for the N and ∆ baryon families for κ = 0.5 GeV

2009 JLab Users Group Meeting, June 8, 2009 Page 26 6 Other Applications of Gauge/Gravity Duality

• Chiral symmetry breaking [Erlich, Katz, Son and Stephanov, Da Rold and Pomarol . . . ]

• Electromagnetic, gravitational and transition form-factors of composite hadrons [Abidin and Carlson, Grigoryan and Radyushkin, Kwee and Lebed, Brodsky and GdT ...]

• Quark and gluon matter at extreme conditions in heavy ion physics (RHIC, LHC) [Policastro, Son, Starinets, Kovtun, Gubser, Kim, Sin, Zahed, Caceres,´ Guijosa,¨ Edelstein, . . . ]

• Condensed matter physics [Herzog, Kovtun, Son . . . ]

Future Applications of Light-Front Holography

• Introduction of massive quarks (heavy and heavy-light quark systems)

• Systematic improvement (QCD Coulomb forces, higher Fock states (HFS) . . . )

2009 JLab Users Group Meeting, June 8, 2009 Page 27 Example: Space- and Time Like Pion Form-Factor (HFS) PRELIMINARY

|πi = ψqq/π|qqi + ψqqqq/π|qqqqi 2 2 Mρ → 4κ (n + 1/2)

κ = 0.54 GeV

Γρ = 130, Γρ0 = 400, Γρ00 = 300 MeV

Pqqqq = 13 %

2009 JLab Users Group Meeting, June 8, 2009 Page 28