Light-Front Holography in QCD and Hadronic Physics

Guy F. de Teramond´

Universidad de Costa Rica

Rencontres de Moriond

QCD and High Energy Interactions

La Thuile, Valle d’Aosta, Italy

March 22-29, 2014

Image credit: N. Evans

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

Page 1 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

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

• 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 (including well deﬁned massless quark limit) is light-front (LF) quantization

• 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

Page 2 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

• Effective LF theory for QCD endowed with SO(2, 1) algebraic structure follows from one-dim semiclassical 4 – dim Light – 5 – dim Front Dynamics Classical Gravity approximation to LF dynamics in physical space-time, higher dimensional gravity in AdS space and 5 LFD AdS5 one-dimensional conformal quantum mechanics

• Result is a relativistic LF wave equation with a conﬁning LFQM AdS2 SO(2,1) structure which incorporates essential spectroscopic and dynamical features of hadron physics Conf(R1) • Remarkable connection follows from isomorphism of 1 one-dim conformal group Conf R with SO(2, 1), dAFF

which is also the isometry group of AdS2 1 – dim 1-2014 • One of the generators of SO(2, 1) is compact and 8840A1 Quantum Field Theory has therefore a discrete spectrum !

Conformal Quantum Mechanics: de Alfaro, Fubini and Furlan (dAFF)

Page 3 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

Dirac Forms of Relativistic Dynamics [Dirac (1949)]

• Poincare´ generators P µ and M µν separated into kinematical and dynamical

• Kinematical generators act along initial hypersurface and contain no interactions

• Dynamical generators are responsible for evolution of the system and depend on the interactions

• Each front has its Hamiltonian and evolve (a)ct (b)ct (c) ct with a different time, but results computed z z z in any front should be identical y y y (different parameterizations of space-time) 8-2013 8811A4

• 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)

− x y 3 P ,J ,J dynamical P⊥,J , K kinematical

• Point form: initial surface is the hyperboloid x2 = κ2 > 0, x0 > 0: P µ dynamical,M µν kinematical

Page 4 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

Light-Front Dynamics

+ − ± 0 3 • Hadron with 4-momentum P = (P ,P , P⊥), P = P ± P , mass-shell relation µ 2 PµP = M leads to LF Hamiltonian equation M 2 + P2 P −|ψ(P )i = ⊥ |ψ(P )i P +

µ − + 2 + • Construct LF invariant Hamiltonian PµP = P P − P⊥ (P and P⊥ kinematical)

µ 2 PµP |ψ(P )i = M |ψ(P )i

+ + • Longitudinal momentum P is kinematical: sum of single particle constituents pi of bound state, + P + + P = i pi , pi > 0 − Pn − • Bound-state is arbitrarly off the LF energy shell, P − i pi < 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

Page 5 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

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 !

• Central problem is derivation of effective interaction which acts only on the valence sector: express higher Fock states as functionals of the lower ones

2 0 µ 2 0 • Compute M from hadronic matrix element hψ(P )|PµP |ψ(P )i=M hψ(P )|ψ(P )i

• Semiclassical approximation

2 ψn(k1, k2, . . . , kn) → φn (k1 + k2 + ··· + kn) , mq → 0 | {z } k2 +m2 2 P ⊥i i Mn= (Invariant mass) i xi

2 2 • M − Mn is the measure of the off-energy shell: key variable which controls the bound state 2 2 k⊥ • For a two-parton bound-state in the mq → 0 Mqq = x(1−x)

Page 6 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

• Conjugate invariant variable in transverse impact space is

2 2 ζ = x(1 − x)b⊥

• To ﬁrst approximation LF dynamics depend only on the invariant variable ζ, and dynamical properties are encoded in the hadronic mode φ(ζ) φ(ζ) ψ(x, ζ, ϕ) = eiLϕX(x)√ , 2πζ

where we factor out the longitudinal X(x) and orbital kinematical dependence from LFWF ψ

z • Ultra relativistic limit mq → 0 longitudinal modes X(x) decouple (L = L ) 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 effective potential U includes all interaction terms upon integration of the higher Fock states

Page 7 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

µ 2 • LF eigenvalue equation PµP |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

• Critical value L = 0 corresponds to lowest possible stable solution, the ground state of the LF Hamil- tonian

• Relativistic and frame-independent LF Schrodinger¨ equation: U is instantaneous in LF time

• A linear potential Veff in the instant form implies a quadratic potential Ueff in the front form at large qq separation (thus linear Regge trajectories for small quark masses!) q q 2 2 2 2 2 Ueff = Veff + 2 p + mq Veff + 2 Veff p + mq

• Result follows from comparison of invariant mass in the instant form in the CMS, P = 0, with invariant

mass in front form in the constituent rest frame (CRF): pq + pq = 0 [A. P. Trawinski,´ S. D. Glazek, S. J. Brodsky, GdT, H. G. Dosch, arXiv: 1403.5651]

Page 8 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

Conformal Quantum Mechanics and Light Front Dynamics [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)] Z g S = 1 dt Q˙ 2 − , g dimensionless 2 Q2

• Absence of dimensional constants implies action is invariant under a large group of transformations, the general conformal group αt + β Q(t) t0 = ,Q0(t0) = , αδ − βγ = 1 γt + δ γt + δ 1 ˙ 2 g I. Translations in t: H = 2 Q + Q2 , 1 ˙ 2 g 1 ˙ ˙ II. Dilatations: D = 2 Q + Q2 t − 4 QQ + QQ , 1 ˙ 2 g 2 1 ˙ ˙ 1 2 III. Special conformal transformations: K = 2 Q + Q2 t − 2 QQ + QQ t + 2 Q ,

Page 9 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

• 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

11 11 J 12 = K + aH ,J 01 = K − aH ,J 02 = D, 2 a 2 a where a has dimension t since H and K have different dimensions

• Generators J have commutation relations

[J 12,J 01] = iJ 02, [J 12,J 02] = −iJ 01, [J 01,J 02] = −iJ 12,

the algebra of SO(2, 1)

• J 0i, i = 1, 2, boost in space direction i and J 12 rotation in the (1,2) plane

• J 12 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

Page 10 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

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

G = uH + vD + wK

and introduce new time variable τ dt dτ = u + vt + wt2 • Find usual quantum mechanical evolution for time τ d G|ψ(τ)i = i |ψ(τ)i dτ with the new Hamiltonian G 1 g 1 1 G = u Q˙ 2 + − v QQ˙ + QQ˙ + wQ2 2 Q2 4 2

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

• Operator G is compact for 4uw − v2 > 0 4 Bound states !

Page 11 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

Connection to Light-Front Dynamics

• The Schrodinger¨ picture follows from the representation of Q and P = Q˙ d Q → x, Q˙ → −i dx 1 d2 g i d d 1 G = u − + + v x + x + wx2. 2 dx2 x2 4 dx dx 2

• Compare dAFF Hamiltonian G with LF Hamiltonian P 2 d2 1 − 4L2 P 2 = − − + U(ζ) dζ2 4ζ2

and identify dAFF variable x with LF invariant variable ζ

2 1 • u = 2, v = 0, g = L − 4 from kinematical constraints 2 1 2 2 • w = 2λ ∼ a2 ﬁxes the LF potential to quadratic λ ζ dependence U ∼ λ2ζ2

Page 12 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

Gravity in AdS and Light Front Holographic Mapping 1 RNKLM = − (gNLgKM − gNM gKL) R2

• Description of strongly coupled gauge theory using a dual gravity description in a higher dimensional space (holographic)

• Why is AdS space important? AdS5 is a 5-dim space of maximal symmetry, negative curvature and a four-dim boundary: Minkowski space

• Isomorphism of SO(4, 2) group of conformal transformations with generators P µ,M µν,Kµ,D with (d+1)(d+2) the group of isometries of AdS5 Dim isometry group of AdSd+1: 2 • AdS metric xM = (xµ, z): 5 R2 ds2 = (η dxµdxν − dz2) z2 µν • 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 µ • Short distances xµx → 0 map to UV conformal AdS5 boundary z → 0 µ 2 • Large conﬁnement dimensions xµx ∼ 1/ΛQCD 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

Page 13 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

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

Z 0 0 0 d p ϕ(z) N1N NJ N MM ∗ S = d x dz |g| e g 1 ··· g J g DM Φ D 0 Φ 0 0 eﬀ N1...NJ M N1...NJ

2 ∗ − µ (z)Φ Φ 0 0 eﬀ N1...NJ N1...NJ

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

Page 14 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

• Physical hadron has plane-wave and polarization indices along 3+1 physical coordinates and a proﬁle wavefunction Φ(z) along holographic variable z

iP ·x ΦP (x, z)µ1···µJ = e Φ(z)µ1···µJ , Φzµ2···µJ = ··· = Φµ1µ2···z = 0 µ 2 with four-momentum Pµ and invariant hadronic mass PµP =M

• 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)ν1···νJ " # zd−1−2J eϕ(z) mR2 − ∂ ∂ + Φ = M 2Φ eϕ(z) z zd−1−2J z z J J

with 2 2 0 (m R) = (µeﬀ (z)R) − Jz ϕ (z) + J(d − J + 1)

and the kinematical constraints

µν µν η Pµ νν2···νJ = 0, η µνν3···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)]

Page 15 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

Light-Front Mapping [GdT and S. J. Brodsky, PRL 102, 081601 (2009)]

(d−1)/2−J −ϕ(z)/2 • Upon substitution ΦJ (z) ∼ z e φJ (z) and z →ζ in AdS WE " ! # zd−1−2J eϕ(z) mR2 − ∂ ∂ + Φ (z) = M 2Φ (z) eϕ(z) z zd−1−2J z z J J

we ﬁnd LFWE (d = 4) d2 1 − 4L2 − − + U(ζ) φ (ζ) = M 2φ (ζ) dζ2 4ζ2 J J

with 1 2J − 3 U(ζ) = 1 ϕ00(ζ) + ϕ0(ζ)2 + ϕ0(ζ) 2 4 2z 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

Page 16 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

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 1 − 4L2 − − + λ2ζ2 + 2λ(J − 1) φ (ζ) = M 2φ (ζ) dζ2 4ζ2 J J

• Normalized eigenfunctions hφ|φi = R dζ φ2(z) = 1 s 2n! 2 φ (ζ) = |λ|(1+L)/2 ζ1/2+Le−|λ|ζ /2LL(|λ|ζ2) n,L (n+L)! n

• Eigenvalues for λ > 0 J + L M2 = 4λ n + n,J,L 2 • Resuts are easily extended to light quarks

• λ < 0 incompatible with LF constituent interpretation

Page 17 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

5 5 2 2 2 2 M IGeV M M IGeV M n 2 n 1 n 0 n 2 n 1 n 0 4 HaL 4 HbL K2H1820L Π2H1880L 3 3 ΠH1800L K2H1770L Π 2H1670L K1H1400L 2 2

Π K1H1270L 1 H1300L 1 b1H1235L

0 0 K 494 L ΠH140L L H L 0 1 2 3 0 1 2 3

6 2 2 6 M IGeV M M2 GeV2 n 2 n 1 n 0 n 2 n 1 n 0 I M a n 3 5 H L 5 HbL f2H2300L

4 4 K* 2045 a4H2040L 4H L ΡH1700L f2H1950L 3 f4H2050L 3 Ω 1650 * K* 1780 H L Ρ3H1690L K H1680L 3H L 2 Ω3H1670L 2 Ρ 1450 * H L K 1410 * a2 1320 H L 1 ΩH1420L H L K2H1430L f2H1270L 1 ΡH770L * 0 K 892 ΩH782L L H L L 0 0 1 2 3√ 4 0 1 2 3 4 Orbital and radial excitations for λ = 0.59 GeV (pseudoscalar) and 0.54 GeV (vector mesons)

Page 18 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

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

1 • Extension of holographic ideas to spin- 2 (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 (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)]

Page 19 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

1 • Half-integer spin J = T + 2 conveniently represented by RS spinor [ΨN1···NT ]α with effective AdS action

Z 0 0 1 d p N1 N1 NT NT Seﬀ = 2 d x dz |g| g ··· g

h A M i ΨN ···N i Γ e DM − µ − U(z) Ψ 0 0 + h.c. 1 T A N1···NT

where the covariant derivative DM includes the afﬁne connection and the spin connection A A A B AB • eM is the vielbein and Γ tangent space Dirac matrices Γ , Γ = η

• LF mapping z → ζ ﬁnd coupled LFWE d ν + 1 − ψ − 2 ψ − V (ζ)ψ = Mψ dζ − ζ − − + d ν + 1 ψ − 2 ψ − V (ζ)ψ = Mψ dζ + ζ + + −

1 provided that |µR| = ν + 2 and R V (ζ) = U(ζ) ζ a J-independent potential – No spin-orbit coupling along a given trajectory !

Page 20 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

Baryon Spectrum

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

• Eigenfunctions

1 +ν −λζ2/2 ν 2 3 +ν −λζ2/2 ν+1 2 ψ+(ζ) ∼ ζ 2 e Ln(λζ ), ψ−(ζ) ∼ ζ 2 e Ln (λζ )

• Eigenvalues: M 2 = 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 1 3 S = 2 S = 2

1 P = + ν = L ν = L + 2 1 P = - ν = L + 2 ν = L + 1

describes the full light baryon orbital and radial excitation spectrum

Page 21 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

(a) 8 (b) N(2600) 6 ν=L+1 n=3n=2 n=1 n=0 N(2250) N(2190) 4λ )

2 N(1900) N(2220) 4 N(1700) N(2220) (GeV N(1675)

2 N(1650) M N(1710) ν=L N(1720) 2 N(1680) N(1720) N(1440) N(1680)

N(940) N(940) 0 0 0 2 4 0 2 4 6 7 (c) (d) n=3n=2 n=1 n=0 6 n=2 n=1 n=0 (2420) 5 ) 2 4 (GeV

(1950) 2 (1920) M N(1875) 3 (1910) 2 (1600) (1700) (1620) (1905) N(1535) (1232) N(1520)

1 0 0 2 4 0 2 4 1-2014 8844A5 L √ L Baryon orbital and radial excitations for λ = 0.49 GeV (nucleons) and 0.51 GeV (Deltas)

Page 22 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013 Light-Front Holography in QCD and Hadronic Physics

Thanks !

Page 23 Moriond 2014, QCD and High Energy Interactions, La Thuile, March 25, 2013