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Light-Front Holography 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 jP i = M jP 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 ) + i D γµ + m QCD 4g2 µν µ + − • LF Momentum Generators P = (P ;P ; P?) in terms of dynamical fields , A? 1 Z (ir )2 + m2 P − = dx−d2x γ+ ? + interactions 2 ? i@+ Z + − 2 + + P = dx d x? γ i@ 1 Z P = dx−d2x γ+ir ? 2 ? ? • LF energy P − generates LF time translations @ (x);P − = i (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 , expanded in terms of ladder operators on the initial surface x+ = x0 + x3 Z + 2 2 2 X dq d q? q + m P − = ? by (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 j H i = MH j H i + • State j H (P )i = j H (P ; P?;Jz)i is an expansion in multi-particle Fock eigenstates j ni of the free LF Hamiltonian: X j H i = n=H jni n z + • Fock components n=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 h H (P )jHLF j H (P )i=MH h H (P )j H (P )i • Find Z 2 2 2 X 2 X k?` + m` 2 MH = dxi d k?i n=H (xi; k?i) + interactions xq n ` • Phase space normalization of LFWFs Z X 2 2 dxi d k?i n=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?j n=H (xi; b?i) n=H (xi; b?i)+interactions xq n j=1 ` • Normalization n−1 Z X Y 2 2 dxjd b?j j n(xj; b?j)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 ∗(x; b )−∇2 (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) (x; ζ; ') = p eiM'f(x) 2πζ • Find (L = jMj) Z d2 1 d L2 φ(ζ) Z M2 = dζ φ∗(ζ)pζ − − + p + 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 jφi = M jφi 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) = hζjφi are normalized by Z hφjφi = dζ jhζjφij2 = 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) 8 < 0 if ζ ≤ 1 U(ζ) = ΛQCD : 1 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 ! 1 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 p 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 p 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
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