Light-Front Holography and Gauge/Gravity Correspondence: Applications to Hadronic Physics

Transcription

1 Light-Front Holography and Gauge/Gravity Correspondence: Applications to Hadronic Physics Guy F. de Téramond University of Costa Rica In Collaboration with Stan Brodsky 4 th International Sakharov Conference on Physics Lebedev Physics Institute Moscow, May 18-3, 009 GdT and Brodsky, PRL 10, (009) Sakharov Conference, Moscow, May 19, 009 Page 1

2 Outline 1. Introduction. 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 Sakharov Conference, Moscow, May 19, 009 Page

3 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... Frame-independent LF Hamiltonian equation P µ P µ P = M P 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 Sakharov Conference, Moscow, May 19, 009 Page 3

4 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 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 (k+ x + k x + ) k x On shell relation k = m leads to dispersion relation k = k +m k + Sakharov Conference, Moscow, May 19, 009 Page 4

5 QCD Lagrangian L QCD = 1 4g 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 x ψ γ + (i ) + m i + ψ + interactions P + = dx d x ψ γ + i + ψ P = 1 dx d x ψ γ + i ψ LF energy P generates LF time translations [ ψ(x), P ] = i and the generators P + and P are kinematical x + ψ(x) Sakharov Conference, Moscow, May 19, 009 Page 5

6 Light-Front Fock Representation Dirac field ψ, expanded in terms of ladder operators on the initial surface x + = x 0 + x 3 P = dq + d q ( q + m ) (π) 3 q + b λ (q)b λ(q) + interactions λ Sum over free quanta q = q +m q + plus interactions (m = 0 for gluons) Construct light-front invariant Hamiltonian for the composite system: H LF = P µ P µ = P P + P H LC ψ H = M H ψ H State ψ H (P ) = ψ H (P +, P, J z ) is an expansion in multi-particle Fock eigenstates n of the free LF Hamiltonian: ψ H = ψ n/h n n Fock components ψ n/h (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 n x i = 1, k i = 0. i=1 i=1 Sakharov Conference, Moscow, May 19, 009 Page 6

7 Compute M from hadronic matrix element ψ H (P ) H LF ψ H (P ) =M H ψ H (P ) ψ H (P ) Find [dxi ][ d ] k i ( k l + m l ) ψ n/h (x i, k i ) + interactions M H = n l x q Phase space normalization of LFWFs n [dxi ] [ d ] k i ψ n/h (x i, k i ) = 1 In terms of n 1 independent transverse impact coordinates b j, j = 1,,..., n 1, M H = n n 1 j=1 dx j d b j ψ n/h (x i, b i ) l ( b l + m ) l ψ x n/h (x i, b i )+interactions q Normalization n n 1 j=1 dx j d b j ψ n (x j, b j ) = 1 Sakharov Conference, Moscow, May 19, 009 Page 7

8 3 Semiclassical Approximation to QCD Consider a two-parton hadronic bound state in transverse impact space in the limit m q 0 1 M dx = d b ψ (x, b ) ( ) b 1 x ψ(x, b ) + interactions 0 Separate angular, transverse and longitudinal modes in terms of boost invariant transverse variable: ζ = x(1 x)b (In k space key variable is the LF KE k x(1 x) ) ψ(x, ζ, ϕ) = φ(ζ) πζ e imϕ f(x) Find (L = M ) M = dζ φ (ζ) ζ ( d dζ 1 ζ ) d dζ + L φ(ζ) ζ + 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) Sakharov Conference, Moscow, May 19, 009 Page 8

9 Ultra relativistic limit m q 0 longitudinal modes decouple and LF eigenvalue equation H LF φ = M φ is a LF wave equation for φ ( d dζ 1 4L ) 4ζ }{{ + U(ζ) φ(ζ) = M 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 LF modes φ(ζ) = ζ φ are normalized by φ φ = dζ ζ φ = 1 Semiclassical approximation to light-front QCD does not account for particle creation and absorption but can be implemented in the LF Hamiltonian EOM Sakharov Conference, Moscow, May 19, 009 Page 9

10 Hard-Wall Model Consider the potential (hard wall) U(ζ) = 0 if ζ 1 Λ QCD if ζ > 1 Λ QCD If L 0 the Hamiltonian is positive definite φ H L LF φ 0 and thus M 0 If L < 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 = β Lk Λ QCD Sakharov Conference, Moscow, May 19, 009 Page 10

11 Excitation spectrum hard-wall model: M n,l L + n Light-meson orbital spectrum Λ QCD = 0.3 GeV Sakharov Conference, Moscow, May 19, 009 Page 11

12 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 (006, 008)] Substitute Φ(ζ) ζ 3/ φ(ζ), ζ z in the conformal LFWE ( d dζ 1 ) 4L 4ζ φ(ζ) = M φ(ζ) Find: [ z z 3z z + z M (µr) ] Φ(z) = 0 with (µr) = 4 + L, the wave equation of string mode in AdS 5! Isomorphism of SO(4, ) group of conformal QCD with generators P µ, M µν, D, K µ with the group of isometries of AdS 5 space: x µ λx µ, z λz ds = R z (η µνdx µ dx ν dz ) AdS Breitenlohner-Freedman bound (µr) 4 equivalent to LF QM stability condition L 0 Conformal dimension of AdS mode Φ given in terms of 5-dim mass by (µr) = ( 4). Thus = + L in agreement with the twist scaling dimension of a two parton object in QCD Sakharov Conference, Moscow, May 19, 009 Page 1

13 AdS 5 metric: ds }{{} L AdS = R z (η µνdx µ dx ν dz ) }{{} L Minkowski A distance L AdS shrinks by a warp factor as observed in Minkowski space (dz = 0): L Minkowski z R L AdS Different values of z correspond to different scales at which the hadron is examined Since x µ λx µ, z λz, short distances x µ x µ 0 maps to UV conformal AdS 5 boundary z 0, which corresponds to the Q UV zero separation limit Large confinement dimensions x µ x µ 1/Λ 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 at the IR boundary 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 Sakharov Conference, Moscow, May 19, 009 Page 13

14 4 Higher-Spin Bosonic Modes Hard-Wall Model AdS d+1 metric x l = (x µ, z): Action for gravity coupled to scalar field in AdS d+1 S = ds = g lm dx l dx m = R z (η µνdx µ dx ν dz ) d d+1 x ( 1 g (R Λ) κ }{{} S G + 1 Equations of motion for S M ( 1 ) ( µr z 3 z z 3 zφ ρ ρ Φ z ( g lm l Φ m Φ µ Φ ) ) } {{ } S M ) Φ = 0 Physical AdS modes Φ P (x, z) e ip x Φ(z) are plane waves along the Poincaré coordinates with four-momentum P µ and hadronic invariant mass states P µ P µ = M Factoring out dependence of string mode Φ P (x, z) along x µ -coordinates [ z z (d 1)z z + z M (µr) ] Φ(z) = 0 Sakharov Conference, Moscow, May 19, 009 Page 14

15 Spin J-field on AdS represented by rank-j totally symmetric tensor field Φ(x, z) l1 l J Fradkin and Vasiliev] [Fronsdal; Action in AdS d+1 for spin-j field S M = 1 d d+1 x g ( ) l Φ l1 l J l Φ l 1 l J µ Φ l1 l J Φ l 1 l J +... Each hadronic state of total spin J is dual to a normalizable string mode Φ P (x, z) µ1 µ J = e ip x Φ(z) µ1 µ J with four-momentum P µ, spin polarization indices along the 3+1 physical coordinates and hadronic invariant mass P µ P µ = M For string modes with all indices along Poincaré coordinates, Φ zµ µ J = Φ µ1 z µ J = = 0 and appropriate subsidiary conditions system of coupled differential equations from S M reduce to a homogeneous wave equation for Φ(z) µ1 µ J Sakharov Conference, Moscow, May 19, 009 Page 15

16 Obtain spin-j mode Φ µ1 µ J with all indices along 3+1 coordinates from Φ by shifting dimensions ( z ) J Φ J (z) = Φ(z) R Normalization [Hong, Yoon and Strassler (006)] R d J 1 zmax 0 dz z d J 1 Φ J(z) = 1 Substituting in the AdS scalar wave equation for Φ [ z z (d 1 J)z z + z M (µr) ] Φ J = 0 upon fifth-dimensional mass rescaling (µr) (µr) J(d J) Conformal dimension of J-mode = 1 (d+ (d J) + 4µ R ) and thus (µr) = ( J)( d + J) Sakharov Conference, Moscow, May 19, 009 Page 16

17 Upon substitution z ζ and we recover the QCD LF wave equation (d = 4) φ J (ζ) ζ 3/+J Φ J (ζ) ( d dζ 1 ) 4L 4ζ φ µ1 µ J = M φ µ1 µ J with (µr) = ( J) + L J-decoupling in the HW model For L 0 the LF Hamiltonian is positive definite φ J H LF φ J 0 and we find the stability bound (µr) ( J) The scaling dimensions are = + L independent of J in agreement with the twist scaling dimension of a two parton bound state in QCD Sakharov Conference, Moscow, May 19, 009 Page 17

18 Note: p-forms In tensor notation EOM for a p-form in AdS d+1 are p + 1 coupled differential equations [l Yi (1998)] [ z z (d + 1 p)z z z M (µr) + d + 1 p ] Φ zα α p = 0 [ z z (d 1 p)z z z M (µr) ] Φ α1 α α p = z ( α1 Φ zα α p + α Φ α1 z α p + ) For modes with all indices along the Poincaré coordinates Φ zα α p = Φ α1 z α p = = 0 [ z z (d 1 p)z z + z M (µr) ] Φ α1 α p = 0 with (µr) = ( p)( d + p) Identical with spin-j solution from shifting dimensions Sakharov Conference, Moscow, May 19, 009 Page 18

19 Note: Algebraic Construction If L > 0 the LF Hamiltonian, H LF, is written as a bilinear form [Bargmann (1949)] H L LF (ζ) = Π L (ζ)π L(ζ) in terms of the operator Π L (ζ) = i ( d dζ L + 1 ζ ) and its adjoint ( Π L (ζ) = i d dζ + L + 1 ζ ) with commutation relations [Π L (ζ), Π L (ζ) ] = L + 1 ζ If L 0 the LF Hamiltonian is positive definite φ HLF L φ = dζ Π L φ(z) 0 Higher-spin fields in AdS [Metsaev (1998) and (1999)] Sakharov Conference, Moscow, May 19, 009 Page 19

20 Soft-Wall Model Soft-wall model [Karch, Katz, Son and Stephanov (006)] retain conformal AdS metrics but introduce smooth cutoff wich depends on the profile of a dilaton background field ϕ(z) = ±κ z S = d d x dz g e ϕ(z) L, Equation of motion for scalar field L = 1 ( g lm l Φ m Φ µ Φ ) [ z z ( d 1 κ z ) z z + z M (µr) ] Φ(z) = 0 with (µr) 4. See also [Metsaev (00), Andreev (006)] LH holography requires plus dilaton ϕ = +κ z. Lowest possible state (µr) = 4 M = 4κ n, Φ n (z) z e κ z L n (κ z ) Φ 0 (z) a chiral symmetric bound state of two massless quarks with scaling dimension : the pion Sakharov Conference, Moscow, May 19, 009 Page 0

21 Action in AdS d+1 for spin J-field S M = 1 d d x dz ( ) g e κ z l Φ l1 l J l Φ l 1 l J µ Φ l1 l J Φ l 1 l J +... Obtain spin-j mode Φ µ1 µ J with all indices along 3+1 coordinates from Φ by shifting dimensions ( z ) J Φ J (z) = Φ(z) R Normalization R d J 1 0 dz z d J 1 eκ z Φ J(z) = 1. Substituting in the AdS scalar wave equation for Φ [ z z ( d 1 J κ z ) z z + z M (µr) ] Φ J = 0 upon mass rescaling (µr) (µr) J(d J) and M M Jκ Sakharov Conference, Moscow, May 19, 009 Page 1

22 Upon substitution z ζ (J z = L z + S z ) we find for d = 4 φ J (ζ) ζ 3/+J e κ ζ / Φ J (ζ), (µr) = ( J) + L ( d dζ 1 ) 4L 4ζ + κ 4 ζ + κ (L + S 1) φ µ1 µ J = M φ µ1 µ J Eigenfunctions φ nl (ζ) = κ 1+L n! (n+l)! ζ1/+l e κ ζ / L L n(κ ζ ) Eigenvalues ) 6 5 M n,l,s = 4κ ( n + L + S 4 4κ for n = 1 4κ for L = 1 Φ(z) Φ(z) 0-5 κ for S = A0 z A z Orbital and radial states: ζ increase with L and n Sakharov Conference, Moscow, May 19, 009 Page

23 J P C M 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 Sakharov Conference, Moscow, May 19, 009 Page 3

24 Note: Algebraic Construction Write the LF Hamiltonian, H LF H L LF (ζ) = Π L (ζ)π L(ζ) + C in terms of the operator and its adjoint Π L (ζ) = i ( d dζ L + 1 ζ ( Π L (ζ) = i d dζ + L + 1 ζ ) κ ζ ) + κ ζ with commutation relations [Π L (ζ), Π L (ζ) ] = L + 1 ζ κ The LF Hamiltonian is positive definite, φ H L LF φ 0, for L 0 and C 4κ Sakharov Conference, Moscow, May 19, 009 Page 4

25 5 Higher-Spin Fermionic Modes Hard-Wall Model Action for massive fermionic modes on AdS d+1 : S[Ψ, Ψ] = d d x dz g Ψ(x, z) ( ) iγ l D l µ Ψ(x, z) From Nick Evans Equation of motion: ( iγ l D l µ ) Ψ(x, z) = 0 [ ( i zη lm Γ l m + d ) ] Γ z + µr Ψ(x l ) = 0 Solution (µr = ν + 1/, d = 4) Ψ(z) = Cz 5/ [J ν (zm)u + + J ν+1 (zm)u ] Hadronic mass spectrum determined from IR boundary conditions ψ ± (z = 1/Λ QCD ) = 0 M + = β ν,k Λ QCD, M = β ν+1,k Λ QCD with scale independent mass ratio Obtain spin-j mode Φ µ1 µ J 1/, J > 1, with all indices along 3+1 from Ψ by shifting dimensions Sakharov Conference, Moscow, May 19, 009 Page 5

26 SU(6) S L Baryon State N 1 + (939) (13) 1 N 1 1 N N 3 1 (1535) N 3 (150) (1650) N 3 (1700) N 5 (1675) + (1910) 3 3 N 5 3 N 3 N N (160) 3 (1700) + (170) N 5 + (1680) + (190) 5 + (1905) 7 + (1950) N 9 5 N 7 N 7 N 7 (190) N 9 (50) (1930) 7 N 9 N 9 + (0) 9 + N 11 N (600) + (40) N 13 Sakharov Conference, Moscow, May 19, 009 Page 6

27 Excitation spectrum for baryons in the hard-wall model: M L + n Light baryon orbital spectrum for Λ QCD = 0.5 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/ and (b) I = 3/ Sakharov Conference, Moscow, May 19, 009 Page 7

28 Note: Algebraic Construction Fermionic modes are solutions of the light-front Dirac equation for ψ(ζ) ζ Ψ(ζ) απ(ζ)ψ(ζ) = Mψ(ζ) where α = α, α = 1, {α, γ 5 } = 0 The operator and its adjoint satisfy the commutation relations Π ν (ζ) = i Π ν(ζ) = i ( ( [ ] Π ν (ζ), Π ν(ζ) d dζ ν + ) 1 γ 5 ζ d dζ + ν + ) 1 γ 5 ζ = ν + 1 ζ γ 5 Sakharov Conference, Moscow, May 19, 009 Page 8

29 Soft-Wall Model Equivalent to Dirac equation in presence of a holographic linear confining potential [ i ( zη lm Γ l m + d Γ z ) ] + µr + κ z Ψ(x l ) = 0. Solution (µr = ν + 1/, d = 4) Ψ + (z) z 5 +ν e κ z / L ν n(κ z ) Ψ (z) z 7 +ν e κ z / L ν+1 n (κ z ) Eigenvalues M = 4κ (n + ν + 1) Obtain spin-j mode Φ µ1 µ J 1/, J > 1, with all indices along 3+1 from Ψ by shifting dimensions Sakharov Conference, Moscow, May 19, 009 Page 9

30 Glazek and Schaden [Phys. Lett. B 198, 4 (1987)]: (ω B /ω M ) = 5/8 4κ for n = 1 4κ for L = 1 M κ for S = 1 L Parent and daughter 56 Regge trajectories for the N and baryon families for κ = 0.5 GeV Sakharov Conference, Moscow, May 19, 009 Page 30

31 Note: Algebraic Construction Fermionic modes are solutions of the light-front Dirac equation for ψ(ζ) ζ Ψ(ζ) where α = α, α = 1, {α, γ 5 } = 0 απ(ζ)ψ(ζ) = Mψ(ζ) The operator and its adjoint Π ν (ζ) = i Π ν(ζ) = i ( ( d dζ ν + ) 1 γ 5 κ ζγ 5 ζ d dζ + ν + ) 1 γ 5 κ ζγ 5 ζ satisfy the commutation relations [ ] Π ν (ζ), Π ν(ζ) = ( ) ν + 1 ζ κ γ 5 Supersymmetric QM at equal LF time? Sakharov Conference, Moscow, May 19, 009 Page 31

32 Other Applications of Light-Front Holography Nucleon form-factors: space-like region Pion form-factors: space and time-like regions Gravitational form-factors of composite hadrons Future Applications Q 4 F p 1 (Q ) (GeV 4 ) Pauli Form Factor Introduction of massive quarks Systematic improvement (QCD Coulomb forces... ) A Q (GeV ) log IF π (q )I q (GeV ) A4 SJB and GdT, PLB 58, 11 (004) GdT and SJB, PRL 94, (005) SJB and GdT, PRL 96, (006) SJB and GdT, PRD 77, (008) SJB and GdT, PRD 78, 0503 (008) GdT and SJB, PRL 10, (009) GdT and SJB, arxiv: Sakharov Conference, Moscow, May 19, 009 Page 3