Light-Front Quantization Approach to the Gauge/Gravity Correspondence and Strongly Coupled Dynamics

Transcription

1 Light-Front Quantization Approach to the Gauge/Gravity Correspondence and Strongly Coupled Dynamics Guy F. de Téramond University of Costa Rica Instituto Tecnológico de Aeronáutica São José dos Campos, Brazil, March 31, 2010 ITA, March 31, 2010 Page 1

2 1. General Introduction Quantum Electrodynamics Internal Structure of the Proton Quantum Chromodynamics Lattice QCD General Theory of Relativity Gauge/Gravity Correspondence and QCD Light-Front Quantization and AdS/CFT 2. Light-Front Quantization of QCD Light-Front Partonic Representation Light-Front Bound State Hamiltonian Equation Semiclassical Approximation to QCD Hard-Wall Model Light-Front Holographic Mapping 3. Gravity and QCD in a Soft-Wall Model Bosonic Modes Fermionic Modes ITA, March 31, 2010 Page 2

3 1 General Introduction Quantum Electrodynamics (QED) QED fundamental theory of the interaction of electrons and photons QED Lagrangian follows from the gauge invariance of the theory ψ(x) e iα(x) ψ(x) L QED = 1 4 F µν F µν + iψd µ γ µ ψ + mψψ where D µ = µ + iea µ and F µν = µ A ν ν A µ QED describes electrodinamics, atomic physics and basic properties of the electron with extraodinary precision. Ej. factor the g factor of the electron: g exp = (15) g QED = ITA, March 31, 2010 Page 3

4 Internal Structure of the Proton High Energy (20 GeV) scattering at SLAC (1969) revealed the internal structure of the proton Deep inelastic scattering experiments ( ) : Bjorken and Feynman partons identified with Gell-Mann and Zweig quarks Interactions of quarks and gluons are well described by Quantum Chromodynamics (QCD) a remarkable generalization of QED Most challenging problem of strong interaction dynamics: determine the composition of hadrons in terms of their fundamental QCD quark and gluon degrees of freedom ITA, March 31, 2010 Page 4

5 Quantum Chromodynamics (QCD) QCD fundamental theory of quarks and gluons QCD Lagrangian follows from the gauge invariance of the theory ψ(x) e iαa (x)t a ψ(x), Non-abelian thery with 3 color charges Find QCD Lagrangian [ T a, T b] = if abc T c L QCD = 1 4g 2 Tr (Gµν G µν ) + iψd µ γ µ ψ + mψψ where D µ = µ igt a A a µ, Ga µν = µ A a ν ν A a µ + f abc A b µa c ν Quarks and gluons interactions from color charge, but... gluons also interact with each other color confinement ITA, March 31, 2010 Page 5

6 Lattice QCD Lattice numerical simulations at the teraflop/sec scale (resolution L/a) LQCD (2009) > 1 petaflop/sec a L ITA, March 31, 2010 Page 6

7 General Theory of Relativity Space curvature determined by the mass-energy present following Einstein s equations R µν 1 2 R g µν = κ T µν }{{}}{{} mater geometry R µν Ricci tensor, R space curvature g µν metric tensor ( ds 2 = g µν dx µ dx ν ) (10 gravitational potentials: gravity is tensorial!) T µν energy-momentum tensor κ = 8πG/c 4, G Newton s constant Matter curves space and space determines how matter moves Annalen der Physik 49 (1916) p. 30 ITA, March 31, 2010 Page 7

8 Gauge/Gravity Correspondence and QCD Recent developments inspired by the AdS/CFT correspondence [Maldacena (1998)] between gravity theories 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? Isomorphism of SO(4, 2) group of conformal transformations with generators P µ, M µν, K µ, D, with the group of isometries of AdS 5, a space of maximal symmetry, negative curvature and a four-dim boundary: Minkowski space Isometry group: most general group of transformations which leave invariant the distance between two points Ej: S N O(N +1) Dimension of isometry group of AdS d+1 is (d+1)(d+2) 2 ITA, March 31, 2010 Page 8

9 AdS 5 metric: ds 2 }{{} L AdS = R 2 ( ηµν z 2 dx µ dx ν 2 dz ) }{{} L Minkowski A distance L AdS shrinks by a warp factor z/r as observed in Minkowski space (dz = 0): L Minkowski z R L AdS Since the AdS metric is invariant under a dilatation of all coordinates x µ λx µ, z λz, the variable z acts like a scaling variable in Minkowski space Short distances x µ x µ 0 maps to UV conformal AdS 5 boundary z 0 Large confinement dimensions x µ x µ 1/Λ 2 QCD maps to large IR region of AdS 5, z 1/Λ QCD, thus there is a maximum separation of quarks and a maximum value of z Local operators like O and L QCD defined in terms of quark and gluon fields at the AdS 5 boundary Use the isometries of AdS to map the local interpolating operators at the UV boundary of AdS into the modes propagating inside AdS ITA, March 31, 2010 Page 9

10 V (z) e +κ2 z 2 Nonconformal metric dual to a confining gauge theory ds 2 = R2 z 2 e2a(z) ( η µν dx µ dx ν dz 2) where A(z) 0 at small z for geometries which are asymptotically AdS 5 x e κ2 z 2 Gravitational potential energy for object of mass m V = mc 2 g 00 = mc 2 R ea(z) z Consider warp factor exp(±κ 2 z 2 ) Plus solution: V (z) increases exponentially confining any object in modified AdS metrics to distances z 1/κ z ITA, March 31, 2010 Page 10

11 Light-Front Quantization and AdS/CFT 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... Frame-independent LF Hamiltonian equation similar structure of AdS EOM P µ P µ ψ(p ) = ( P P + P 2 ) ψ(p ) = M 2 ψ(p ) ( ) ( ) 1 µr 2 z 3 z z 3 zφ µ µ Φ Φ = 0 }{{} z M 2 First semiclassical approximation to the bound-state LF Hamiltonian equation in QCD is equivalent to EOM in AdS and can be systematically improved [GdT and S. J. Brodsky, PRL 102, (2009)] Mapping at fixed light-front time ψ(p ) dual to Φ(z) ITA, March 31, 2010 Page 11

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13 2 Light-Front Quantization of QCD 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 the 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 + = x 0 + x 3 x = x 0 x 3 light-front time longitudinal space variable k + = k 0 + k 3 longitudinal momentum (k + > 0) k = k 0 k 3 light-front energy k x = 1 2 (k+ x + k x + ) k x On shell relation k 2 = m 2 leads to dispersion relation k = k2 +m2 k + ITA, March 31, 2010 Page 13

14 QCD Lagrangian L QCD = 1 4g 2 Tr (Gµν G µν ) + iψd µ γ µ ψ + mψψ LF Momentum Generators P = (P +, P, P ) in terms of dynamical fields ψ, A P = 1 dx d 2 x ψ γ + (i ) 2 + m 2 2 i + ψ + interactions P + = dx d 2 x ψ γ + i + ψ P = 1 dx d 2 x ψ γ + i ψ 2 LF Hamiltonian P generates LF time translations [ ψ(x), P ] = i and the generators P + and P are kinematical x + ψ(x) ITA, March 31, 2010 Page 14

15 Light-Front Partonic Representation Dirac field ψ, expanded in terms of ladder operators on the initial surface x + = x 0 + x 3 ψ(x, x ) α = λ q + >0 dq + 2q + d 2 q [ (2π) 3 b λ (q)u α (q, λ)e iq x + d λ (q) v α (q, λ)e iq x] LF Generators P = (P +, P, P ) in terms of constituents with momentum q = (q +, q, q ) P = λ dq + d 2 q ( q 2 + m 2 ) (2π) 3 q + b λ (q)b λ(q) + interactions P + = λ dq + d 2 q (2π) 3 q + b λ (q)b λ(q) P = λ dq + d 2 q (2π) 3 q b λ (q)b λ(q) ITA, March 31, 2010 Page 15

16 Light-Front Bound State Hamiltonian Equation Construct LF Lorentz invariant Hamiltonian equation for the relativistic bound state system P µ P µ ψ(p ) = ( P P + P 2 ) ψ(p ) = M 2 ψ(p ) State ψ(p +, P, J z ) is expanded in multi-particle Fock states n of the free LF Hamiltonian ψ = uud ψ n n, n = uudg n uudqq with k 2 i = m2 i, k i = (k + i, k i, k i), for each constituent i in state n Fock components ψ n (x i, k i, λ z i ) are independent of P + and P and depend only on relative partonic coordinates: momentum fraction x i = k + i /P +, transverse momentum k i and spin λ z i n x i = 1, i=1 n k i = 0. i=1 ITA, March 31, 2010 Page 16

17 Compute M 2 from hadronic matrix element ψ(p ) P µ P µ ψ(p ) =M 2 ψ(p ) ψ(p ) Find M 2 = n [dxi ][ d 2 ] k i l ( k 2 l + m 2 l x q ) ψ n (x i, k i ) 2 + interactions Phase space normalization of LFWFs n [dxi ] [ d 2 ] k i ψn (x i, k i ) 2 = 1 In terms of n 1 independent transverse impact coordinates b j, j = 1, 2,..., n 1, M 2 = n 1 dx j d 2 b j ψn(x i, b i ) ( 2 + ) m2 b l l ψ n (x i, b i ) + interactions x n j=1 q l Normalization n n 1 j=1 dx j d 2 b j ψ n (x j, b j ) 2 = 1 ITA, March 31, 2010 Page 17

18 Semiclassical Approximation to QCD [GdT and S. J. Brodsky, PRL 102, (2009)] Semiclassical approximation to QCD: ( ψ n (k 1, k 2,..., k n ) φ n (k1 + k k n ) 2 ) }{{} M 2 n with k 2 i = m2 i for each constituent Functional dependence of Fock state n given by invariant mass M 2 n = ( n a=1 k µ a ) 2 = a k 2 a + m2 a x a Key variable controlling bound state: off-energy shell E = M 2 M 2 n In impact space the relevant variable conjugate to the invariant mass is ζ = x 1 x n 1 j=1 x j b j the x-weighted transverse impact coordinate of the spectator system (x active quark) ITA, March 31, 2010 Page 18

19 Consider a two-parton hadronic bound state in transverse impact space in the limit m q 0 1 M 2 dx = d 2 b ψ (x, b ) ( 2 ) b 1 x ψ(x, b ) + interactions 0 For a two-parton system ζ 2 = x(1 x)b 2 To first approximation LF dynamics depend only on the invariant variable M n or ζ, and hadronic properties are encoded in the hadronic mode φ(ζ) from ψ(x, ζ, ϕ) = e imϕ X(x) φ(ζ) 2πζ factoring out angular ϕ, longitudinal X(x) and transverse mode φ(ζ) Find (L = M ) M 2 = dζ φ (ζ) ζ ( d2 dζ 2 1 ζ ) d dζ + L2 φ(ζ) ζ 2 + ζ dζ φ (ζ) U(ζ) φ(ζ) where the confining forces from the interaction terms is summed up in the effective potential U(ζ) ITA, March 31, 2010 Page 19

20 Ultra relativistic limit m q 0 longitudinal modes X(x) decouple and LF eigenvalue equation P µ P µ φ = M 2 φ is a LF wave equation for φ ( d2 dζ 2 1 4L2 ) 4ζ }{{ 2 + U(ζ) φ(ζ) = M 2 φ(ζ) }{{}} conf inement kinetic energy of partons Effective light-front Schrödinger 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 Semiclassical approximation to light-front QCD does not account for particle creation and absorption but can be implemented in the LF Hamiltonian EOM or by applying the L-S formalism ITA, March 31, 2010 Page 20

21 Hard-Wall Model Consider the potential (hard wall) U(ζ) = 0 if ζ 1 Λ QCD if ζ > 1 Λ QCD If L 2 0 the Hamiltonian is positive definite φ P µ P µ φ 0 and thus M 2 0 If L 2 < 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: φ L (ζ) = C L ζjl (ζm) Mode spectrum from boundary conditions φ(ζ = 1/Λ QCD ) = 0 Thus M 2 = β Lk Λ QCD ITA, March 31, 2010 Page 21

22 Excitation spectrum hard-wall model: M n,l L + 2n Light-meson orbital spectrum Λ QCD = 0.32 GeV ITA, March 31, 2010 Page 22

23 Light-Front Holographic Mapping LF 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) ζ 3/2 φ(ζ), ζ z in the conformal LFWE ( d2 dζ 2 1 ) 4L2 4ζ 2 φ(ζ) = M 2 φ(ζ) Find: [ z 2 2 z 3z z + z 2 M 2 (µr) 2] Φ(z) = 0 with (µr) 2 = 4 + L 2, the wave equation of string mode in AdS 5! ds 2 = R2 z 2 (η µνdx µ dx ν dz 2 ) AdS Breitenlohner-Freedman bound (µr) 2 4 equivalent to LF QM stability condition L 2 0 Scaling dimension τ of AdS mode Φ given in terms of 5-dim mass by (µr) 2 = τ(τ 4). Thus τ = 2 + L in agreement with the twist scaling dimension of a two parton object in QCD ITA, March 31, 2010 Page 23

24 3 Gravity and QCD in a Soft-Wall Model Bosonic Modes Soft-wall model [Karch, Katz, Son and Stephanov (2006)] retain conformal AdS metrics but introduce smooth cutoff which depends on dilaton background field ϕ(z) 0, z 0 S = d 4 x dz g e ϕ(z) L Introduce energy scale in the five-dim action to break conformal invariance Lagrangian for scalar field x l = (x µ, z) L = 1 ( g lm l Φ m Φ µ 2 Φ 2) 2 Equation of motion ( ) 1 z 3 l z 3 eϕ(z) η lm m Φ + ( ) µr 2 e ϕ(z) Φ = 0 z ITA, March 31, 2010 Page 24

25 Hadronic state has plane waves along Minkowski coordinates and a profile function along the holographic variable z Upon substitution Φ P (x µ, z) = e ip x Φ(z) [ z 2 z 2 ( 3 2κ 2 z 2) z z + z 2 M 2 (µr) 2] Φ(z) = 0 with ϕ(z) = ± κ 2 z 2 and P µ P µ = M 2 Lowest possible state (µr) 2 = 4 (BF bound (µr) 2 4) for confining solution ϕ = +κ 2 z 2 M 2 = 0, Φ(z) z 2 e κ2 z 2, r 2 1 κ 2 A chiral symmetric bound state of two massless quarks with scaling dimension 2: the pion ITA, March 31, 2010 Page 25

26 Obtain spin-j mode Φ µ1 µ J with all indices along 3+1 coordinates from Φ by shifting dimensions Φ J (z) = ( z R ) J Φ(z) Substituting in the AdS scalar wave equation for Φ [ z 2 2 z ( 3 2J 2κ 2 z 2) z z + z 2 M 2 (µr) 2] Φ J = 0 Upon substitution z ζ φ J (ζ) ζ 3/2+J e κ2 ζ 2 /2 Φ J (ζ) we find the LF wave equation ( d2 dζ 2 1 4L2 4ζ 2 ) + U(z) φ µ1 µ J = M 2 φ µ1 µ J with and (µr) 2 = (2 J) 2 + L 2 U(z) = κ 4 ζ 2 + 2κ 2 (L + S 1) ITA, March 31, 2010 Page 26

27 Normalized eigenfunctions dζ φ(z) 2 = 1 φ nl (ζ) = κ 1+L 2n! (n+l)! ζ1/2+l e κ2 ζ 2 /2 L L n(κ 2 ζ 2 ) Eigenvalues ) M 2 n,l,s = 4κ 2( n + L + S 2 4κ 2 for n = 1 4κ 2 for L = 1 2κ 2 for S = Φ z Φ z z Orbital and radial states: ζ increase with L and n z ITA, March 31, 2010 Page 27

28 J P C M 2 L 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 ITA, March 31, 2010 Page 28

29 Fermionic Modes From Nick Evans Soft wall model equivalent to Dirac equation in presence of a holographic linear confining potential [ i Solution (µr = ν + 1/2, ν = L + 1) ) ] (zη lm Γ l m + 2Γ z + µr + κ 2 z Ψ(x l ) = 0. Ψ + (z) z 5 2 +ν e κ2 z 2 /2 L ν n(κ 2 z 2 ) Ψ (z) z 7 2 +ν e κ2 z 2 /2 L ν+1 n (κ 2 z 2 ) Eigenvalues (how to fix the overall energy scale, see arxiv: ) M 2 = 4κ 2 (n + L + 1) Obtain spin-j mode Φ µ1 µ J 1/2, J > 1 2, with all indices along 3+1 from Ψ by shifting dimensions ITA, March 31, 2010 Page 29

30 M 2 4κ 2 for n = 1 4κ 2 for L = 1 2κ 2 for S = 1 L Parent and daughter 56 Regge trajectories for the N and baryon families for κ = 0.5 GeV spectrum identical to de Paula, Frederico, Forkel and M.Beyer, Phys. Rev. D 79, (2009) and Forkel and Klempt, Phys. Lett. B 679, 77 (2009) ITA, March 31, 2010 Page 30