Light-Front Quantization Approach to the Gauge/Gravity Correspondence and Applications to the Light Hadron Spectrum

Transcription

1 Light-Front Quantization Approach to the Gauge/Gravity Correspondence and Applications to the Light Hadron Spectrum Guy F. de Téramond Universidad de Costa Rica Ferrara International School Niccolò Cabeo 1 Hadronic Spectroscopy Ferrara, May 1-6, 1 Image credit: ic-msquare Niccolò Cabeo 1, Ferrara, May 5, 1 Recent Review: GdT and S.J. Brodsky, arxiv:13.45 [hep-ph] Page 1

2 The objective of these lectures is to describe recent analytical insights into the nonperturbative nature of the strong coupled dynamics of light-hadron bound states. The holographic approach described in the lectures is at the confluence of phenomenology, light-front physics and the gauge/gravity correspondence. Important dynamical properties of hadrons such as their light-front wavefunctions, form factors and the systematics of their excitation spectrum are well described in this framework. Niccolò Cabeo 1, Ferrara, May 5, 1 Page

3 1 General Introduction Internal Structure of the Proton Quantum Chromodynamics Lattice QCD General Theory of Relativity to the Rescue? Gauge/Gravity Correspondence and QCD Light Front Dynamics Light-Front Quantization of QCD Light-Front Fock Representation Semiclassical Approximation to QCD in the Light Front Meson Spectrum in Hard-Wall Model 3 Light-Front Holographic Mapping of Wave Equations Higher Spin Wave Equations in AdS Space Dual QCD Light-Front Wave Equation Meson Spectrum in Soft-Wall Model Niccolò Cabeo 1, Ferrara, May 5, 1 Page 3

4 4 Fermionic Modes in AdS Space and Baryon Spectrum Higher Spin Wave Equations in AdS Space Baryon Spectrum in Soft-Wall Model 5 Light-Front Holographic Mapping of Current Matrix Elements Nucleon Elastic Form Factors Nucleon Transition Form Factors Flavor Decomposition of Elastic Nucleon Form Factors Pion Transition Form Factor Higher Fock Components in LF Holographic QCD Conclusions Niccolò Cabeo 1, Ferrara, May 5, 1 Page 4

5 1 Introduction Internal Structure of the Proton High Energy ( GeV) scattering at SLAC (1969) revealed the internal structure of the proton Deep inelastic scattering experiments ( ): Bjorken and Feynman partons identified with GellMann and Zweig quarks Quarks were not just hypothetical mathematical entities but the true building blocks of hadrons Niccolo Cabeo 1, Ferrara, May 5, 1 Page 5

6 Quantum Chromodynamics (QCD) Quarks should have an additional quantum number color ψ(x) i, i = R, G, B QCD Lagrangian follows from the gauge invariance of the theory Find QCD Lagrangian ψ(x) e iαa (x)t a ψ(x), [ T a, T b] = if abc T c (T a ) ij, i, j = 1,, 3, a, b = 1,, 8 L QCD = 1 4g 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: strongly coupled non-abelian gauge theory confinement! But how could the actual existence of gluons be demonstrated experimentally? Niccolò Cabeo 1, Ferrara, May 5, 1 Page 6

7 In the mid 197s QCD was referred as the candidate theory of the strong interactions Asymptotically free QFT which can describe scaling First hint of gluons from deep-inelastic scattering experiments: only half of a proton s momentum is carried by the quarks PETRA storage ring at DESY (1979): first direct experimental proof of the existence of the gluon QCD becomes fundamental theory of quarks and gluons: interactions of quarks and gluons at high energies is well described by QCD Most challenging problem of strong interaction dynamics: determine the composition of hadrons in terms of their fundamental QCD quark and gluon degrees of freedom Enormous complexity out of a very simple Lagrangian (H. Fritsch) Niccolò Cabeo 1, Ferrara, May 5, 1 Page 7

8 Lattice QCD Lattice numerical simulations at the petaflop/sec scale (resolution L/a) Sums over quark paths with billions of dimensions a L Dynamical properties in Minkowski space-time not amenable to Euclidean lattice computations Computational complexity of hadronic excitation spectrum beyond ground state configuration Niccolò Cabeo 1, Ferrara, May 5, 1 Page 8

9 General Theory of Relativity to the Rescue? Space curvature determined by the mass-energy present following Einstein s equations R µν 1 R g µν = κ T µν }{{}}{{} mater geometry R µν Ricci tensor, R space curvature g µν metric tensor ( ds = g µν dx µ dx ν ) T µν energy-momentum tensor Matter curves space and space determines how matter moves Perhaps, but in a holographic sense: duality between theories in different number of space-time dimensions! Annalen der Physik 49 (1916) p. 3 Niccolò Cabeo 1, Ferrara, May 5, 1 Page 9

10 Gauge/Gravity Correspondence and QCD Recent developments inspired by the AdS/CFT correspondence [Maldacena (1998)] between gravity in AdS space and conformal field theories in physical space-time provide physical insights into the non-perturbative dynamics of QCD Description of strongly coupled gauge theory using a dual gravity description in a higher dimensional space (holographic) Isomorphism of SO(4, ) 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+) Mapping of AdS gravity to QCD quantized at fixed light-front time gives a precise relation between wave functions in AdS space and the LF wavefunctions describing the internal structure of hadrons Niccolò Cabeo 1, Ferrara, May 5, 1 Page 1

11 Niccolò Cabeo 1, Ferrara, May 5, 1 Page 11

12 AdS 5 metric: ds }{{} L AdS = R ( ηµν z dx µ dx ν dz ) }{{} L Minkowski A distance L AdS shrinks by a warp factor z/r as observed in Minkowski space (dz = ): 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 maps to UV conformal AdS 5 boundary z 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 Use the isometries of AdS to map the local interpolating operators at the UV boundary of AdS into the modes propagating inside AdS Niccolò Cabeo 1, Ferrara, May 5, 1 Page 1

13 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 Forms of Relativistic Dynamics: dynamical vs. kinematical generators [Dirac (1949)] ct Instant form: hypersurface defined by t =, the familiar one z H, K dynamical, L, P kinematical y A1 Point form: hypersurface is an hyperboloid ct P µ dynamical, M µν kinematical Front form: hypersurface is tangent to the light cone at τ = t + z/c = y z A P, L x, L y dynamical, P +, P, L z, K kinematical ct P ± = P ± P 3 y z A3 Niccolò Cabeo 1, Ferrara, May 5, 1 Page 13

14 LF coordinates x + = x + x 3 x = x x 3 x = ( x 1, x ) light-front time longitudinal space variable transverse space variable P + = P + P 3 longitudinal momentum (P + > ) P = P P 3 P = ( P 1, P ) light-front energy transverse momentum Compute P x to identify Hamiltonian as conjugate to LF time x + P µ x µ = 1 ( P + x + P x +) P x On shell relation P µ P µ = P P + P = M leads to dispersion relation for LF Hamilnotian P P = P + M P + Niccolò Cabeo 1, Ferrara, May 5, 1 Page 14

15 LF quantization is the ideal framework to describe hadronic structure in terms of quarks and gluons: simple vacuum structure allows unambiguous definition of partonic content of a hadron LFWFs Calculation of matrix elements P + q J P requires boosting the relativistic hadronic bound state from P to P + q : extremely complicated in the instant form but boosts are trivial in the LF No coupling to vacuum-induced currents or off-diagonal contributions in the LF Form factors in LF expressed as convolution of frame-independent LFWFs (Drell-Yan-West formula) + Image credit: M. Vanderhaeghen Mapping to AdS transition amplitudes possible (Polchinski-Strassler formula: overlap of AdS WFs) Hamiltonian equation for bound states similar structure of AdS equations: direct connection of QCD and AdS/CFT possible Niccolò Cabeo 1, Ferrara, May 5, 1 Page 15

16 Light-Front Quantization of QCD Express the hadron four-momentum generator P = (P +, P, P ) in terms of dynamical fields 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 +, where the integrals are over the null plane τ = x + = x + x 3 LF Hamiltonian P generates LF time translations and the generators P + and P are kinematical Hamiltonian equation for the relativistic bound state P ψ(p ) = M + P P + ψ(p ) Construct LF Lorentz invariant Hamiltonian H LF P = P P + P H LF ψ(p ) = M ψ(p ) Niccolò Cabeo 1, Ferrara, May 5, 1 Page 16

17 Light-Front Fock Representation Dirac field ψ, expanded in terms of ladder operators on the initial surface P = λ dq + d q ( q + m ) (π) 3 q + b λ (q)b λ(q) + interactions LF Lorentz invariant Hamiltonian equation for the relativistic bound state P µ P µ ψ(p ) = M ψ(p ) State ψ(p ) is expanded in multi-particle Fock states n of the free LF Hamiltonian ψ = n ψ n n, n = { uud, uudg, uudqq, } with k i = m 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 ) 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 =. i=1 Niccolò Cabeo 1, Ferrara, May 5, 1 Page 17

18 Semiclassical Approximation to QCD in the Light Front [GdT and S. J. Brodsky, PRL 1, 8161 (9)] Compute M from hadronic matrix element ψ(p ) P µ P µ ψ(p ) =M ψ(p ) ψ(p ) Find M = n [dxi ][ d ] k i q ( k q + m q x q ) ψ n (x i, k i ) + interactions LFWF ψ n represents a bound state which is off the LF energy shell M M n M n = ( n i=1 k µ i ) = i k i + m i x i with k a = m a for each constituent Invariant mass M n key variable which controls the bound state Semiclassical approximation to QCD: ( ψ n (k 1, k,..., k n ) φ n (k1 + k + + k n ) ), mq }{{} M n Niccolò Cabeo 1, Ferrara, May 5, 1 Page 18

19 In terms of n 1 independent transverse impact coordinates b j, j = 1,,..., n 1, M = n 1 dx j d b j ψn(x i, b i ) ( + ) m b q q ψ n (x i, b i ) + interactions x n q q j=1 Relevant variable conjugate to invariant mass M n x ζ = 1 x (Cluster decomposition) n 1 j=1 x j b j the x-weighted transverse impact coordinate of the spectator system (x active quark) For a two-parton system ζ = x(1 x)b To first approximation LF dynamics depend only on the invariant variable ζ, and hadronic properties are encoded in the hadronic mode φ(ζ) from ψ(x, ζ, ϕ) = e imϕ X(x) φ(ζ) πζ factoring angular ϕ, longitudinal X(x) and transverse mode φ(ζ) (P +, P and J z commute with P ) Niccolò Cabeo 1, Ferrara, May 5, 1 Page 19

20 Ultra relativistic limit m q longitudinal modes X(x) decouple (L = L z ) M = dζ φ (ζ) ζ ( d dζ 1 ) d ζ dζ + L φ(ζ) ζ + dζ φ (ζ) U(ζ) φ(ζ) ζ where the confining forces from the interaction terms are summed up in the effective potential U(ζ) LF eigenvalue equation P µ P µ 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 relativistic and frame-independent LF Schrödinger equation: U is instantaneous in LF time Eigenmodes φ(ζ) represent the probability amplitude to find n-massless partons at transverse impact separation ζ within the hadron at equal LF time The SO() Casimir L corresponds to group of rotations in transverse LF plane Casimir operator for SO(N) is L(L + N ) Semiclassical approximation to LF QCD does not account for particle creation and absorption Niccolò Cabeo 1, Ferrara, May 5, 1 Page

21 Meson Spectrum in Hard Wall Model [LF Hard wall model: GdT and S. J. Brodsky, PRL 94, 161 (5)] Conformal model up to the confinement scale 1/Λ QCD [Polchinski and Strassler ()] if ζ 1 Λ U(ζ) = QCD if ζ > 1 Λ QCD Confinement scale 1 Λ QCD 1 Fm, Λ QCD MeV Covariant version of MIT bag model: quarks permanently confined inside a finite region of space Normalized eigenfunctions φ φ = Λ 1 QCD dζ φ (z) = 1 φ L,k (ζ) = ΛQCD J 1+L (β L,k ) ζjl (ζβ L,k Λ QCD ) Eigenvalues M L,k = β L,k Λ QCD Niccolò Cabeo 1, Ferrara, May 5, 1 Page 1

22 Table 1: I = 1 mesons. For a qq state P = ( 1) L+1, C = ( 1) L+S L S n J P C I = 1 Meson + π(14) 1 + π(13) + π(18) 1 1 ρ(77) ρ(145) 1 1 ρ(17) b 1 (135) a (98) a (145) a 1 (16) a (13) + π (167) 1 + π (188) 1 3 ρ 3 (169) a 4 (4) Niccolò Cabeo 1, Ferrara, May 5, 1 Page

23 4 4 ρ 3 (169) a 4 (4) M (GeV ) π (167) M (GeV ) a (13) a 1 (16) a (98) b 1 (135) ρ(77) -1 88A19 π(14) L -1 88A3 L Orbital and radial excitations for the π and the ρ I=1meson families (Λ QCD =.3 GeV) Pion is not chiral M n + L in contrast to usual Regge dependence M n + L Important J L splitting (different J for same L) in mesons not described by hard-wall model Radial modes not well described in hard-wall model Niccolò Cabeo 1, Ferrara, May 5, 1 Page 3

24 3 Light-Front Holographic Mapping Higher Spin Wave Equations in AdS Space Description of higher spin modes in AdS space (Frondsal, Fradkin and Vasiliev) Spin-J in AdS represented by totally symmetric rank J tensor field Φ M1 M J Action for spin-j field in AdS d+1 in presence of dilaton background ϕ(z) ( x M = (x µ, z) ) S = 1 d d x dz g e ϕ(z)( g MN g M 1M 1 g M J M J DM Φ M1 M J D N Φ M 1 M J µ g M 1M 1 g M J M J ΦM1 M J Φ M 1 M J + ) where D M is the covariant derivative which includes parallel transport (affine connection) D M Φ M1 M J = M Φ M1 M J Γ K MM 1 Φ K MJ Γ K MM J Φ M1 K Physical hadron has plane-wave and polarization indices along 3+1 physical coordinates Φ P (x, z) µ1 µ J = e ip x Φ(z) µ1 µ J, Φ zµ µ J = = Φ µ1 µ z = with four-momentum P µ and invariant hadronic mass P µ P µ =M Niccolò Cabeo 1, Ferrara, May 5, 1 Page 4

25 Construct effective action in terms of spin-j modes Φ J with only physical degrees of freedom [H. G. Dosch, S. J. Brodsky and GdT] Introduce fields with Lorentz (tangent) indices g MN (x) = η AB e A M (x)eb N (x) ˆΦ A1 A A J = e M 1 A 1 e M A e M J A J Φ M1 M M J = ( z R) JΦA1 A A J where M, N = 1,, d + 1 curved space indices, A, B = 1,, d + 1 tangent indices Find effective action for the Lorentz spin-j mode ˆΦ J = ˆΦ µ1 µ J S = 1 d d x dz g e ϕ(z)( ) g NN N ˆΦJ N ˆΦJ µ ˆΦJ ˆΦJ upon µ-rescaling Variation of the action gives AdS wave equation for spin-j mode Φ J = Φ µ1 µ J [ ( zd 1 J e ϕ ) ( ) ] (z) µr e ϕ(z) z z d 1 J z + Φ J (z) = M Φ J (z) z [See also: T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega, Phys. Rev. D 85, 763 (1)] Niccolò Cabeo 1, Ferrara, May 5, 1 Page 5

26 Dual QCD Light-Front Wave Equation z ζ, Φ P (z) ψ(p ) [GdT and S. J. Brodsky, PRL 1, 8161 (9)] Upon substitution z ζ and φ J (ζ) ζ 3/+J e ϕ(z)/ Φ J (ζ) in AdS WE [ ( ) zd 1 J e ϕ(z) ( ) ] µr e ϕ(z) z z d 1 J z + Φ J (z) = M Φ J (z) z find LFWE (d = 4) ( d dζ 1 4L 4ζ ) + U(ζ) φ J (ζ) = M φ J (ζ) with U(ζ) = 1 ϕ (z) ϕ (z) + J 3 ϕ (z) z and (µr) = ( J) + L LF Schrödinger equation from AdS 5 mapping to physical 3+1 Minkowski space at fixed LF time x + AdS Breitenlohner-Freedman bound (µr) 4 equivalent to LF QM stability condition L Scaling dimension τ of AdS mode ˆΦ J is τ = + L in agreement with twist scaling dimension of a two parton bound state in QCD and determined by QM stability condition Niccolò Cabeo 1, Ferrara, May 5, 1 Page 6

27 Meson Spectrum in Soft Wall Model Linear Regge trajectories [Karch, Katz, Son and Stephanov (6)] Dilaton profile ϕ(z) = +κ z Effective potential: U(z) = κ 4 ζ + κ (J 1) LF WE ( d dζ 1 ) 4L 4ζ + κ 4 ζ + κ (J 1) φ J (ζ) = M φ J (ζ) Normalized eigenfunctions φ φ = dζ φ (z) = 1 φ n,l (ζ) = κ 1+L n! (n+l)! ζ1/+l e κ ζ / L L n(κ ζ ) Eigenvalues ( M n,j,l = 4κ n + J + L ) Niccolò Cabeo 1, Ferrara, May 5, 1 Page 7

28 .8.5 φ(ζ).4 φ(ζ) A9 4 8 ζ -1 88A1 4 8 ζ LFWFs φ n,l (ζ) in physical space-time: (L) orbital modes and (R) radial modes Niccolò Cabeo 1, Ferrara, May 5, 1 Page 8

29 J = L + S, I = 1 meson families M n,l,s = 4κ (n + L + S/) 4κ for n = 1 4κ for L = 1 κ for S = 1 4 n= n=1 n= 4 n= n=1 n= π (188) ρ(17) a 4 (4) M (GeV ) π(18) π (167) M (GeV ) ρ 3 (169) π(13) b 1 (135) ρ(145) a (13) -1 88A π(14) L -1 88A4 ρ(77) L Orbital and radial excitations for the π (κ =.59 GeV) and the ρ I=1meson families (κ =.54 GeV) Triplet splitting for the L = 1, J =, 1,, I = 1 vector meson a-states M a (13) > M a1 (16) > M a (98) J L splitting in mesons and radial excitations are well described in soft-wall model Light scalar mesons P. Colangelo, F. De Fazio et al., Phys. Rev. D 78, 559 (8)] Niccolò Cabeo 1, Ferrara, May 5, 1 Page 9

30 Fermionic Modes in AdS Space and Baryon Spectrum [GdT and S. J. Brodsky, PRL 94, 161 (5)] Image credit: N. Evans Lattice calculations of the ground state hadron masses agree very with experimental values However, excitation spectrum of nucleon represents important challenge to LQCD due to enormous computational complexity beyond ground state configuration and multi-hadron thresholds Large basis of interpolating operators required in LQCD since excited nucleon states are classified according to irreducible representations of the lattice, not the angular momentum 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 and nucleon form factors Extension of holographic ideas to spin- 1 (and higher half-integral J) hadrons by considering propagation of RS spinor field Ψ αm1 M J 1/ in AdS space Niccolò Cabeo 1, Ferrara, May 5, 1 Page 3

31 Higher Spin Wave Equations in AdS Space For fermion fields in AdS one cannot break conformality with introduction of dilaton background since it can be scaled away leaving the action conformally invariant [I. Kirsch (6)] Introduce an effective confining potential V (z) in the action for a Dirac field in AdS d+1 S F = d d x dz gg M ( 1M 1 g M T M T (Ψ M1 M T ie M A Γ A D M µ V (z) ) Ψ M 1 M T + ) where D M is the covariant derivative of the spinor field Ψ αm1 M T, T = J 1 D M Ψ M1 M T = M Ψ M1 M T i ωab M Σ AB Ψ M1 M T Γ K MM 1 Ψ K MT Γ K MM T Ψ M1 K M, N = 1,, d + 1 curved space indices, A, B = 1,, d + 1 tangent indices e M A is the vielbein, wab M spin connection, Σ AB generators of the Lorentz group, Σ AB = i 4 [Γ A, Γ B ] Γ A { tangent space Dirac matrices Γ A, Γ B} = η AB For d even we choose Γ A = (Γ µ, Γ z ) with Γ z = Γ z = Γ Γ 1 Γ d 1 For d = 4: Γ A = (γ µ, iγ 5 ) Niccolò Cabeo 1, Ferrara, May 5, 1 Page 31

32 Physical hadron has plane-wave, spinors, and polarization along 3+1 physical coordinates Ψ P (x, z) µ1 µ T = e ip x Ψ(z) µ1 µ T, Ψ zµ µ T = = Ψ µ1 µ z = with four-momentum P µ and invariant hadronic mass P µ P µ =M Construct effective action in terms of spin-j modes Ψ J with only physical degrees of freedom [H. G. Dosch, S. J. Brodsky and GdT] Introduce fields with Lorentz indices ˆΨ A1 A A T = e M 1 A 1 e M A ( ) T = J 1 e M T A T Ψ M1 M M T = ( z R) T ΨA1 A A T Find effective action for the Lorentz spin-j mode ˆΨ J = ˆΦ µ1 µ J 1/ S F = d d x dz ( ( g ˆΨ J izη MN Γ M N + i d ) Γ z µr RV (z) )ˆΨJ Variation of the action gives AdS wave equation for spin-j mode Φ J = Φ µ1 µ J 1/ [ ( i zη MN Γ M N + d ) ] Γ z µr RV (z) Ψ J = upon µ-rescaling [See also: T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega, Phys. Rev. D 85, 763 (1)] Niccolò Cabeo 1, Ferrara, May 5, 1 Page 3

33 Baryon Spectrum in Soft-Wall Model Upon substitution z ζ and Ψ J (x, z) = e ip x z ψ J (z)u(p ), find LFWE for d = 4 d dζ ψj + + ν + 1 ψ+ J + U(ζ)ψ+ J = Mψ J ζ, d dζ ψj + ν + 1 ζ ψ J + U(ζ)ψ J = Mψ J +, where U(ζ) = R ζ V (ζ) Choose linear potential U = κ ζ Eigenfunctions ψ J +(ζ) ζ 1 +ν e κ ζ / L ν n(κ ζ ), ψ J (ζ) ζ 3 +ν e κ ζ / L ν+1 n (κ ζ ) Eigenvalues M = 4κ (n + ν + 1), ν = L + 1 (τ = 3) Full J L degeneracy (different J for same L) for baryons along given trajectory! Niccolò Cabeo 1, Ferrara, May 5, 1 Page 33

34 L S n Baryon State N 1 + (94) 1 N 1 + (144) N 1 + (171) 3 + (13) (16) N 1 (1535) N 3 (15) (165) N 3 (17) N 5 (1675) N 1 1 (16) 3 (17) N 3 + (17) N 5 + (168) 1 N 5 + (19) 1 + (191) 3 + (19) 5 + (195) 7 N 5 N 3 N 5 5 N N 9 N 7 N 9 N 7 N 7 + (195) (19) N 9 (5) 7 N 9 + () 9 + N 11 N (4) (6) N 13 Niccolò Cabeo 1, Ferrara, May 5, 1 Page 34

35 Gap scale 4κ determines trajectory slope and spectrum gap between plus-parity spin- 1 and minusparity spin- 3 nucleon families! No J L splitting! 8 N(6) 6 N(5) N(19) M (GeV ) 4 4κ N(17) N(1675) N(165) N() N(17) N(168) A6 N(94) 4 6 L Plus-minus nucleon spectrum gap for κ =.49 GeV Niccolò Cabeo 1, Ferrara, May 5, 1 Page 35

36 Fix the energy scale to the proton mass for the lowest state n =, L = Subtraction to mass scale may be understood as displacement required to describe nucleons with N C =3 as composite system with twist 3 + L instead of a quark-squark bound state with twist + L Phenomenological rules for increase in mass M to construct full baryon spectrum from proton state 4κ for n = 1 4κ for L = 1 κ for S = 1 κ for P = ± Eigenvalues M (+) n,l,s = 4κ (n + L + S/ + 3/4) M ( ) n,l,s = 4κ (n + L + S/ + 5/4) Niccolò Cabeo 1, Ferrara, May 5, 1 Page 36

37 6 n=3 n= n=1 n= 6 n=3 n= n=1 n= Δ(4) M (GeV ) 4 N(171) N(144) N(19) N(17) N(168) N() M (GeV ) 4 Δ(16) Δ(195) Δ(19) Δ(191) Δ(195) N(94) Δ(13) -1 88A1 4 L -1 88A3 4 L Orbital and radial excitations for positive parity N and baryon families (κ = GeV) [See also: H. Forkel, M. Beyer and T. Frederico, JHEP 77, 77 (7)] Niccolò Cabeo 1, Ferrara, May 5, 1 Page 37

38 8 N(6) 8 6 N(5) N(19) 6 M (GeV ) A13 N(94) N(17) N(1675) N(165) N(17) N(168) N(1535) N(15) N() 4 6 L M (GeV ) A4 Δ(13) Δ(17) Δ(16) Δ(195) Δ(19) Δ(191) Δ(195) Δ(4) 4 6 L Baryon orbital trajectories for n = and κ = GeV (193) quantum number assignment (E. Klempt and J. M. Richard (1): S = 3/, L = 1, n = 1 Find M (193) = 4κ GeV compared with experimental value 1.96 GeV Niccolò Cabeo 1, Ferrara, May 5, 1 Page 38

39 4 Light-Front Holographic Mapping of Current Matrix Elements [S. J. Brodsky and GdT, PRL 96, 161 (6) [S. J. Brodsky and GdT, PRD 78, 53 (8)] Mapping of EM currents Mapping of energy-momentum tensor EM transition matrix element in QCD: local coupling to pointlike constituents P J µ P = (P + P ) µ F (Q ) where Q = P P and J µ = e q qγ µ q EM hadronic matrix element in AdS space from coupling of external EM field propagating in AdS with extended mode Φ(x, z) d 4 x dz g A M (x, z)φ P (x, z) M Φ P (x, z) (π) 4 δ 4 ( P P ) ( ɛ µ P + P ) µ F (Q ) How to recover hard pointlike scattering at large Q out of soft collision of extended objects? [Polchinski and Strassler ()] Mapping of J + elements at fixed light-front time: Φ P (z) ψ(p ) Niccolò Cabeo 1, Ferrara, May 5, 1 Page 39

40 Compare with electromagnetic FF in LF QCD for arbitrary Q. Expressions can be matched only if LFWF is factorized ψ(x, ζ, ϕ) = e imϕ X(x) φ(ζ) πζ Find X(x) = x(1 x), φ(ζ) = ( ) ζ 3/ Φ(ζ), z ζ R Form factor in soft-wall model expressed as τ 1 product of poles along vector radial trajectory (twist τ = N + L) [Brodsky and GdT, Phys.Rev. D77 (8) 567] F τ (Q ) = 1 ( )( ) ( ) 1 + Q 1 + Q 1 + Q M ρ M ρ M ρ τ Analytical form F (Q ) incorporates correct scaling from constituents and mass gap from confinement M ρ n 4κ (n + 1/) since VM is twist- qq and not twist 3 squark-squark with L = 1 Niccolò Cabeo 1, Ferrara, May 5, 1 Page 4

41 Nucleon Elastic Form Factors [Brodsky and GdT, arxiv:13.45 [hep-ph]] Nucleon EM form factor P J µ () P = u(p ) [γ µ F 1 (q ) + iσµν q ν ] M F (q ) u(p ) EM hadronic matrix element in AdS space from non-local coupling of external EM field in AdS with fermionic mode Ψ P (x, z) d 4 x dz g Ψ P (x, z) e A MΓ A A M (x, z)ψ P (x, z) Effective AdS/QCD model: additional term in the 5-dim action [Abidin and Carlson, Phys. Rev. D79, 1153 (9) ] d 4 x dz g Ψ e A Me B N [Γ A, Γ B ] F MN Ψ Generalized Parton Distributions in AdS/QCD (π) 4 δ 4 ( P P q ) ɛ µ u(p )γ µ F 1 (q )u(p ) (π) 4 δ 4 ( P P q ) ɛ µ u(p ) iσµν q ν [Vega, Schmidt, Gutsche and Lyubovitskij, Phys.Rev. D83 (11) 361] M F (q )u(p ) Niccolò Cabeo 1, Ferrara, May 5, 1 Page 41

42 Using SU(6) flavor symmetry and normalization to static quantities 1. Q 4 F p 1 (Q ) (GeV 4 ).8.4 Q 4 F n 1 (Q ) (GeV 4 ) A Q (GeV ) -1 88A Q (GeV ) F np (Q ) 1 F n (Q ) A8 4 6 Q (GeV ) A7 4 6 Q (GeV ) Niccolò Cabeo 1, Ferrara, May 5, 1 Page 4

43 Nucleon Transition Form Factors F p 1 N N (Q ) = 3 Q M ρ ( )( )( ). 1 + Q 1 + Q 1 + Q M ρ M ρ M ρ F p 1N N* (Q ) A16 4 Q (GeV ) Proton transition form factor to the first radial excited state. Data from JLab Niccolò Cabeo 1, Ferrara, May 5, 1 Page 43

44 Flavor Decomposition of Elastic Nucleon Form Factors G. D. Cates et al. Phys. Rev. Lett. 16, 53 (11) Proton SU(6) WF: F p u,1 = 5 3 G G, F p d,1 = 1 3 G G Neutron SU(6) WF: F n u,1 = 1 3 G G, F n d,1 = 5 3 G G G + (Q 1 ) = ( )( and 1+ Q M ρ ) 1+ Q M ρ G (Q 1 ) = ( )( 1+ Q M ρ )( 1+ Q M ρ ) 1+ Q M ρ G Q G Q Q PRELIMINARY Niccolò Cabeo 1, Ferrara, May 5, 1 Page 44

45 e Pion Transition Form-Factor [S. J. Brodsky, F.-G. Cao and GdT, arxiv:15.39xx] Definition of π γ TFF from γ π γ vertex in the amplitude eπ eγ Γ µ = ie F πγ (q )ɛ µνρσ (p π ) ν ɛ ρ (k)q σ, k = g g* Q q x q 1- x p e Asymptotic value of pion TFF is determined by first principles in QCD: Q F πγ (Q ) = f π [Lepage and Brodsky (198)] Pion TFF from 5-dim Chern-Simons structure [Hill and Zachos (5), Grigoryan and Radyushkin (8)] d 4 x dz ɛ LMNP Q A L M A N P A Q Find for A z Φ π (z)/z (π) 4 δ (4) (p π + q k) F πγ (q )ɛ µνρσ ɛ µ (q)(p π ) ν ɛ ρ (k)q σ F πγ (Q ) = 1 π with normalization fixed by asymptotic QCD prediction V (Q, z) bulk-to-boundary propagator of γ dz z Φ π(z)v ( Q, z ) Niccolò Cabeo 1, Ferrara, May 5, 1 Page 45

46 1 8 Q 6 dsêdq 6 4 Niccolò Cabeo 1, Ferrara, May 5, Q Page 46

47 Higher Fock Components in LF Holographic QCD Effective interaction leads to qq qq, qq qq but also to q qqq and q qqq Higher Fock states can have any number of extra qq pairs, but surprisingly no dynamical gluons Example of relevance of higher Fock states and the absence of dynamical gluons at the hadronic scale π = ψ qq/π qq τ= + ψ qqqq qqqq τ=4 + Modify form factor formula introducing finite width: q q + imγ (P qqqq = 13 %).6 Q F π (Q ) (GeV ).4. Log I F π (q )I A1 4 6 Q (GeV ) -1 88A 4 q (GeV ) Niccolò Cabeo 1, Ferrara, May 5, 1 Page 47

48 5 Conclusions The gauge/gravity duality leads to a simple analytical frame-independent nonperturbative semiclassical approximation to the light-front Hamiltonian problem for QCD: Light-Front Holography Unlike usual instant-time quantization the Hamiltonian equation in the light-front is frame independent and has a structure similar to eigenmode equations in AdS AdS transition matrix elements (overlap of AdS wave functions) map to current matrix elements in LF QCD (convolution of frame-independent light-front wave functions) Mapping of AdS gravity to boundary QFT quantized at fixed light-front time gives a precise relation between holographic wave functions in AdS and LFWFs describing the internal structure of hadrons No constituent gluons Improve the semiclassical approximation: introduce nonzero quark masses and short-range Coulomblike gluonic corrections (heavy and heavy-light quark systems) Apply Lippmann-Schwinger methods to systematically improve the light-front Hamiltonian of the semiclassical holographic approximation Niccolò Cabeo 1, Ferrara, May 5, 1 Page 48

49 Despite some limitations of AdS/QCD, the light-front holographic approach to the gauge/gravity duality has provided so far significant physical insight into the strongly-coupled nature and internal structure of hadrons. The resulting model provides a simple framework for describing nonperturbative hadron dynamics: the systematics of the excitation spectrum of hadrons: the mass spectrum, observed multiplicities and degeneracies. It also provides powerful new analytical tools for computing hadronic transition amplitudes, incorporating the scaling behavior and the transition from the hard-scattering perturbative domain, where quark and gluons are the relevant degrees of freedom, to the long range confining hadronic region. The holographic mapping provides the basis for a profound connection between physical QCD quantized in the light-front and the physics of hadronic modes in a higher dimensional AdS space Niccolò Cabeo 1, Ferrara, May 5, 1 Page 49