Light-Front Holographic QCD: an Overview

Transcription

1 Light-Front Holographic QCD: an Overview Guy F. de Téramond Universidad de Costa Rica Continuous Advances is QCD University of Minnesota Minneapolis, May 14, 2011 CAQCD, Minneapolis, May 14, 2011 Page 1

2 1 Light Front Dynamics Light-Front Fock Representation Semiclassical Approximation to QCD in the Light Front 2 Light-Front Holographic Mapping Higher Spin Modes in AdS Space Dual QCD Light-Front Wave Equation AdS Bosonic Modes and Meson Spectrum AdS Fermionic Modes and Baryon Spectrum 3 Light-Front Holographic Mapping of Current Matrix Elements Electromagnetic Form Factors Gravitational or Energy-Momentum Form Factor Meson Transition Form Factors CAQCD, Minneapolis, May 14, 2011 Page 2

3 1 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 2 (k+ x + k x + ) k x On shell relation k 2 = m 2 leads to dispersion relation k = k2 +m2 k + CAQCD, Minneapolis, May 14, 2011 Page 3

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5 Forms of Relativistic Dynamics: dynamical vs. kinematical generators [Dirac (1949)] Instant form Point form Front form H, K dynamical L, P kinematical P µ dynamical M µν kinematical P, L x, L y dynamical P +, P, L z, K kinematical 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 Calculation of a matrix elements P + q J P requires boosting the hadronic bound state from P to P + q : extremely complicated in the instant form, whereas K is trivial in the LF Invariant Hamiltonian equation for bound states similar structure of AdS equations of motion: direct connection of QCD and AdS/CFT possible CAQCD, Minneapolis, May 14, 2011 Page 5

6 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) CAQCD, Minneapolis, May 14, 2011 Page 6

7 Light-Front Fock Representation Dirac field ψ, expanded in terms of ladder operators on the initial surface P = λ dq + d 2 q ( q 2 + m 2 ) (2π) 3 q + b λ (q)b λ(q) + interactions Construct LF Lorentz invariant Hamiltonian equation for the relativistic bound state P µ P µ ψ(p ) = ( P P + P 2 ) ψ(p ) = M 2 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 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 ) 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 CAQCD, Minneapolis, May 14, 2011 Page 7

8 Semiclassical Approximation to QCD in the Light Front [GdT and S. J. Brodsky, PRL 102, (2009)] 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 LFWF ψ n represents a bound state which is off the LF energy shell M 2 M 2 n M 2 n = ( n a=1 k µ a ) 2 = a k 2 a + m2 a x a with k 2 a = m 2 a for each constituent Invariant mass M 2 n key variable which controls the bound state: LFWF peaks at the minimum M2 n Semiclassical approximation to QCD: ( ψ n (k 1, k 2,..., k n ) φ n (k1 + k k n ) 2 ), mq 0 }{{} M 2 n CAQCD, Minneapolis, May 14, 2011 Page 8

9 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 Relevant variable conjugate to invariant mass in the limit of zero quark masses 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) For a two-parton system ζ 2 = x(1 x)b 2 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) 2πζ factoring angular ϕ, longitudinal X(x) and transverse mode φ(ζ) (P +, P and J z commute with P ) CAQCD, Minneapolis, May 14, 2011 Page 9

10 Ultra relativistic limit m q 0 longitudinal modes X(x) decouple (L = L z ) M 2 = dζ φ (ζ) ζ ( d2 dζ 2 1 ) d ζ dζ + L2 φ(ζ) ζ 2 + 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 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 CAQCD, Minneapolis, May 14, 2011 Page 10

11 2 Light-Front Holographic Mapping Higher Spin Modes in AdS Space Description of higher spin modes in AdS space (Frondsal, Fradkin and Vasiliev) Action for spin-j field in AdS d+1 in presence of dilaton background ϕ(z) ( x M = (x µ, z) ) S = 1 2 d d x dz g e ϕ(z)( g NN g M 1M 1 g M J M J DN Φ M1 M J D N Φ M 1 M J µ 2 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 [D N, D K ]Φ M1 M J = R L M 1 NKΦ L MJ R L M J NKΦ M1 L 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 zµ2 µ J = = Φ µ1 µ 2 z = 0 with four-momentum P µ and invariant hadronic mass P µ P µ =M 2 CAQCD, Minneapolis, May 14, 2011 Page 11

12 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 (in preparation)] Introduce fields with tangent indices Find effective action ˆΦ A1 A 2 A J = e M 1 A 1 e M 2 A 2 e M J A J Φ M1 M 2 M J = ( z R) JΦA1 A 2 A J S = 1 2 d d x dz g e ϕ(z)( g NN η µ 1µ 1 η µ J µ J N ˆΦµ1 µ J N ˆΦµ 1 µ J ) µ 2 η µ 1µ 1 η µ J µ J ˆΦµ1 µ J ˆΦµ 1 µ J upon µ-rescaling Variation of the action gives AdS wave equation for spin-j mode Φ J = Φ µ1 µ J [ ( zd 1 2J e ϕ ) ( ) ] (z) µr 2 e ϕ(z) z z d 1 2J z + Φ J (z) = M 2 Φ J (z) z with ˆΦ J (z) = (z/r) J Φ J (z) and all indices along 3+1 CAQCD, Minneapolis, May 14, 2011 Page 12

13 Dual QCD Light-Front Wave Equation Φ P (z) ψ(p ) LF Holographic mapping found originally matching expressions of EM and gravitational form factors of hadrons in AdS and LF QCD [Brodsky and GdT (2006, 2008)] Upon substitution z ζ and φ J (ζ) ζ 3/2+J e ϕ(z)/2 Φ J (ζ) in AdS WE [ ( zd 1 2J e ϕ ) ( ) ] (z) µr 2 e ϕ(z) z z d 1 2J z + Φ J (z) = M 2 Φ J (z) z find LFWE (d = 4) ( d2 dζ 2 1 4L2 4ζ 2 ) + U(ζ) φ J (ζ) = M 2 φ J (ζ) with U(ζ) = 1 2 ϕ (z) ϕ (z) 2 + 2J 3 ϕ (z) 2z and (µr) 2 = (2 J) 2 + L 2 AdS Breitenlohner-Freedman bound (µr) 2 4 equivalent to LF QM stability condition L 2 0 Scaling dimension τ of AdS mode ˆΦ J is τ = 2 + L in agreement with twist scaling dimension of a two parton bound state in QCD and determined by QM stability condition CAQCD, Minneapolis, May 14, 2011 Page 13

14 Bosonic Modes and Meson Spectrum Soft wall model [Karch, Katz, Son and Stephanov (2006)] Dilaton ϕ(z) = +κ 2 z 2 (Minkowski metrics), ϕ(z) = κ 2 z 2 (Euclidean metrics) Effective potential U(z) = κ 4 ζ 2 + 2κ 2 (L + S 1) Normalized eigenfunctions φ φ = 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 = 4κ2 (n + L + S/2) 0.8 Φ Ζ a Φ Ζ b Ζ Ζ LFWFs φ n,l (ζ) in physical space time for dilaton exp(κ 2 z 2 ): a) orbital modes and b) radial modes CAQCD, Minneapolis, May 14, 2011 Page 14

15 4κ 2 for n = 1 4κ 2 for L = 1 2κ 2 for S = 1 6 J PC n=3 n=2 n=1 n=0 6 J PC n=3 n=2 n=1 n=0 f 2 (2300) M A5 π(1800) π(1300) π 0 π 2 (1880) π 2 (1670) b(1235) L M A1 ρ(1700) ω(1650) ρ(1450) ω(1420) ρ(770) ω(782) 0 f 2 (1950) a 2 (1320) f 2 (1270) ρ 3 (1690) ω 3 (1670) a 4 (2040) f 4 (2050) L Regge trajectories for the π (κ = 0.6 GeV) and the I =1 ρ-meson and I =0 ω-meson families (κ = 0.54 GeV) CAQCD, Minneapolis, May 14, 2011 Page 15

16 Fermionic Modes and Baryon Spectrum [Hard wall model: GdT and S. J. Brodsky, PRL 94, (2005)] [Soft wall model: GdT and S. J. Brodsky, (2005), arxiv: ] From Nick Evans Nucleon LF modes ψ + (ζ) n,l = κ 2+L 2n! (n + L)! ζ3/2+l e κ2 ζ 2 /2 L L+1 ( n κ 2 ζ 2) ψ (ζ) n,l = κ 3+L 1 2n! n + L + 2 (n + L)! ζ5/2+l e κ2 ζ 2 /2 L L+2 n ( κ 2 ζ 2) Normalization dζ ψ 2 +(ζ) = dζ ψ 2 (ζ) = 1 Eigenvalues Chiral partners M 2 n,l,s=1/2 = 4κ2 (n + L + 1) M N(1535) M N(940) = 2 CAQCD, Minneapolis, May 14, 2011 Page 16

17 Same multiplicity of states for mesons and baryons! 4κ 2 for n = 1 4κ 2 for L = 1 2κ 2 for S = 1 6 (a) n=3 n=2 n=1 n=0 (b) n=3 n=2 n=1 n=0 M 2 4 N(2200) Δ(2420) 2 N(1710) N(1440) N(1680) N(1720) Δ(1600) Δ(1950) Δ(1905) Δ(1920) Δ(1910) Δ(1232) A3 N(940) L L Regge trajectories for positive parity N and baryon families (κ = 0.5 GeV) CAQCD, Minneapolis, May 14, 2011 Page 17

18 3 Light-Front Holographic Mapping of Current Matrix Elements [S. J. Brodsky and GdT, PRL 96, (2006)], PRD 77, (2008)] EM transition matrix element in QCD: local coupling to pointlike constituents ψ(p ) J µ ψ(p ) = (P + P ) µ F (Q 2 ) where Q = P P and J µ = e q qγ µ q EM hadronic matrix element in AdS space from non-local coupling of external EM field propagating in AdS with extended mode Φ(x, z) d 4 x dz g e ϕ(z) A M (x, z)φ P (x, z) M Φ P (x, z) (2π) 4 δ 4 ( P P ) ( ɛ µ P + P ) µ F (Q 2 ) where A µ (x, z) = ɛ µ (q)e iq x V (q, z), A z = 0, Q 2 = q 2 > 0 How to recover hard pointlike scattering at large Q out of soft collision of extended objects? [Polchinski and Strassler (2002)] Mapping of J + elements at fixed light-front time: Φ P (z) ψ(p ) CAQCD, Minneapolis, May 14, 2011 Page 18

19 Electromagnetic Form-Factor Drell-Yan-West electromagnetic FF in impact space [Soper (1977)] F (q 2 ) = n n 1 j=1 ( n 1 ) dx j d 2 b j e q exp iq x k b k ψ n (x j, b j ) 2 q k=1 Consider a two-quark π + Fock state ud with e u = 2 3 and e d = F π +(q 2 ) = dx d 2 b e iq b (1 x) ψ ud/π (x, b ) with normalization F + π (q =0) = Integrating over angle F π +(q 2 ) = 2π 1 0 dx x(1 x) ) 1 x ζdζj 0 (ζq x ψud/π (x, ζ) 2 where ζ 2 = x(1 x)b 2 CAQCD, Minneapolis, May 14, 2011 Page 19

20 Compare with electromagnetic FF in AdS space [Polchinski and Strassler (2002)] where V (Q, z) = zqk 1 (zq) F (Q 2 ) = R 3 dz z 3 V (Q, z)φ2 π + (z) Use the integral representation V (Q, z) = 1 0 dx J 0 (ζq ) 1 x x Find F (Q 2 ) = R ( dz 1 x dx z 3 J 0 zq x ) Φ 2 π + (z) 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) 2πζ Find X(x) = x(1 x), φ(ζ) = ( ) ζ 3/2 Φ(ζ), z ζ R CAQCD, Minneapolis, May 14, 2011 Page 20

21 Free current V (Q, z) = zqk 1 (zq) infinite radius (mauve), no pole structure in time-like region Dressed current non-perturbative sum of an infinite number of terms finite radius (blue) Form factor in soft-wall model expressed as N 1 product of poles along vector radial trajectory ( 2 Mρ 4κ 2 (n + 1/2) ) [Brodsky and GdT (2008)] F (Q 2 ) = [ ( 1 + Q2 M 2 ρ ] ) ) 1 )(1 + Q2 M 2 (1 + Q2 ρ M 2 ρ N F Π q F 1 p Q Pion form factor (lowest mode) Proton form factor (lowest mode) CAQCD, Minneapolis, May 14, 2011 Page 21

22 Higher Fock components in pion LFWF π = ψ qq/π qq τ=2 + ψ qqqq/π qqqq τ=4 + Expansion of LFWF up to twist 4 (monopole + tripole) κ = 0.54 GeV, Γ ρ = 130, Γ ρ = 400, Γ ρ = 300 MeV, P qqqq = 13% 0.6 Q 2 F Π Q 2 2 M Ρ 2 log F Π Q M Ρ M Ρ Q 2 GeV Q 2 GeV Only interaction in LF holographic semiclassical approx is the confinement potential: create Fock states with extra quark-antiquark pairs, no dynamical gluons CAQCD, Minneapolis, May 14, 2011 Page 22

23 Gravitational or Energy-Momentum Form-Factor [S. J. Brodsky and GdT, PRD 78, (2008)] Gravitational form factor of composite hadrons in QCD: local coupling to pointlike constituents P Θ ν µ P = ( P ν P µ + P µ P ν) A(Q 2 ) where Q = P P and Θ µν = 1 2 qi(γ µd ν + γ ν D µ ) q g µν q (i /D m) q G a µλ Ga ν λ g µνg a λσ λσga Hadronic matrix element of energy-momentum tensor from perturbing static AdS metric: non-local coupling of external graviton propagating in AdS with extended mode Φ(x, z) [Abidin and Carlson (2008)] d 4 x dz g h lm ( l Φ P m Φ P + m Φ P l Φ P ) Are the transition amplitudes related? Mapping of Θ ++ elements at fixed LF time: Identical mapping Φ P (z) ψ(p ) as EM FF CAQCD, Minneapolis, May 14, 2011 Page 23

24 Meson Transition Form-Factors γ q [S. J. Brodsky, Fu-Guang Cao and GdT (in preparation)] π 0 γ q Pion TFF from 5-dim Chern-Simons structure [Hill and Zachos (2005), Grigoryan and Radyushkin (2008)] d 4 x dz ɛ LMNP Q A L M A N P A Q (2π) 4 δ (4) (p π + q k) F πγ (q 2 )ɛ µνρσ ɛ µ (q)(p π ) ν ɛ ρ (k)q σ Take A z Φ π (z)/z, Φ π (z) = 2P qq κ z 2 e κ2 z 2 /2, Φπ Φ π = P qq ( φ(x) = 3fπ x(1 x), f π = P qq κ/ 2π ) Find Q 2 F πγ (Q 2 ) = dx φ(x) 1 x [1 e P qqq 2 (1 x)/4π 2 f 2 π x ] normalized to the asymptotic DA [P qq = 1 Musatov and Radyushkin (1997)] Large Q 2 TFF is identical to first principles asymptotic QCD result Q 2 F πγ (Q 2 ) = 2f π The CS form is local in AdS space and projects out only the asymptotic form of the pion DA CAQCD, Minneapolis, May 14, 2011 Page 24

25 pion-gamma transition form factor, Q2Fpigamma Feta TFF eta prime F! " (Q 2 )(GeV -1 ) BaBar CLEO CELLO Free current; Twist 2 Dressed current; Twist 2 Dressed current; Twist 2+4 F! " (Q 2 )(GeV -1 ) BaBar CLEO Free current; Twist 2 Dressed current; Twist 2 Dressed current; Twist 2+4 F!' " (Q 2 )(GeV -1 ) BaBar CLEO Free current; Twist 2 Dressed current; Twist 2 Dressed current; Twist pion-gamma transition Q 2 (GeVform 2 ) factor, Q2Fpigamma Q2Feta 4 6 Q 2 (GeV 2 ) QTFF 2 (GeV eta 2 ) prime Q 2 F! " (Q 2 )(GeV) BaBar CLEO CELLO Free current; Twist 2 Dressed current; Twist 2 Dressed current; Twist Q 2 (GeV 2 ) Q 2 F! " (Q 2 )(GeV) BaBar CLEO Free current; Twist Dressed current; Twist Dressed current; Twist 2+4 Q 2 (GeV 2 ) Q 2 F!' " (Q 2 )(GeV) BaBar CLEO 0.10 Free current; Twist Dressed current; Twist 2 Dressed current; Twist Q 2 (GeV 2 ) CAQCD, Minneapolis, May 14, 2011 Page 25