Systematics of the Hadron Spectrum from Conformal Quantum Mechanics and Holographic QCD

Transcription

1 Systematics of the Hadron Spectrum from Conformal Quantum Mechanics and Holographic QCD Guy F. de Téramond Universidad de Costa Rica Twelfth Workshop on Non-Perturbative Quantum Chromodynamics Institut d Astrophysique de Paris June 0-3, 03 In collaboration with Stan Brodsky SLAC) and Hans G. Dosch Heidelberg) Image credit: ic-msquare IAP, June 3, 03 Page

2 Effective Confinement from Underlying Conformal Invariance [S. J. Brodsky, GdT and H.G. Dosch, arxiv:30.405] Incorporate in an effective theory the fundamental dilatation symmetry of the 4-dim QCD Lagrangian in the chiral limit of massless quarks Invariance properties of one dimensional field theory under the full conformal group from daff action [V. de Alfaro, S. Fubini and G. Furlan daff) [Nuovo Cim. A 34, )] S = dt Q g ) Q where g is a dimensionless number Casimir operator which depends on the representation) The equation of motion Q g Q 3 = 0 and the generator of evolution in t, the Hamiltonian H t = Q + g Q ) follow from the daff action IAP, June 3, 03 Page

3 Absence of dimensional constants implies that the action S = dt Q g ) Q is invariant under a larger group of transformations, the general conformal group with αδ βγ = t = αt + β γt + δ, Q t ) = Qt) γt + δ Applying Noether s theorem obtain conserved operators i) Translations: H t = Q + g ) Q ii) Dilatations: D = th t Q Q iii) Special conformal transformations: K = t H t tq Q + Q Any combination of the generators H t, D and K G = uh t + vd + ωk is also a constant of motion IAP, June 3, 03 Page 3

4 TIme evolution for state vector and field operator for daff generator G from canonical quantization [Qt), Qt)] = i G ψt) = ift) d dt ψt) i [G, Qt)] = ft) dqt) dt where ft) = u + vt + wt dft) dt Qt) daff introduce new time variable τ and field operator qτ) dτ = dt u + vt + wt, qτ) = Qt) [u + vt + wt ] Find usual quantum mechanical evolution for time τ and usual equal-time quantization G ψτ) = i d dτ ψτ) i [G, qτ)] = dqτ) dτ [qt), qt)] = i IAP, June 3, 03 Page 4

5 In terms of τ and qτ) S = = dt Q g ) Q dτ q g 4uω v q ) + surface term q 4 Action is conformal invariant invariant up to a surface term! The corresponding Hamiltonian is a compact operator for H τ = q + g 4uω v + q ) q 4 4uω v Scale appears in the Hamiltonian without affecting the conformal invariance of the action! 4 > 0 IAP, June 3, 03 Page 5

6 Conformal Quantum Mechanics The Schrödingier picture follows from the representation of q and p = q daff) q x, q i d dx Schrödinger wave equation determines evolution of bound states in terms of the variable τ i τ ψx, τ) = H τ x, i d ) ψx, τ) dx daff Hamiltonian H τ = d dx + g 4uω v + x ) x 4 IAP, June 3, 03 Page 6

7 Light Front Dynamics On shell relation P µ P µ = P P + P = M leads to dispersion relation for LF Hamiltonian P P = P + M P +, P + > 0, P ± = P 0 ± P 3 Hamiltonian equation for the relativistic bound state x + = x 0 + x 3 light-front time) i x + ψp ) = P ψp ) = M + P P + ψp ) Construct LF Lorentz invariant Hamiltonian P = P P + P P µ P µ ψp ) = M ψp ) LF quantization allows unambiguous definition of partonic content of hadrons wave function) LF Hamiltonian equation for bound states has similar structure of AdS and daff equations: direct connection of QCD with AdS/CFT and conformal QM daff) possible! IAP, June 3, 03 Page 7

8 Semiclassical Approximation to QCD in the Light Front [GdT and S. J. Brodsky, PRL 0, )] Compute M from hadronic matrix element ψp ) P µ P µ ψp ) =M ψp ) ψp ) To first approximation LF dynamics depends only on the invariant variable ζ = x x)b Factor angular ϕ, longitudinal Xx) and transverse mode φζ) ψx, ζ, ϕ) = e ilϕ Xx) φζ) πζ Ultra relativistic limit m q 0 longitudinal modes Xx) decouple L = L z ) M = dζ φ ζ) ζ d dζ ) d ζ dζ + L φζ) ζ + dζ φ ζ) Uζ) φζ) ζ where the confining forces from the interaction terms are summed up in the effective potential Uζ) IAP, June 3, 03 Page 8

9 LF eigenvalue equation P µ P µ φ = M φ is a LF wave equation for φ d dζ 4L ) 4ζ }{{ + Uζ) φζ) = M φζ) }{{}} conf inement kinetic energy of partons Effective relativistic and frame-independent LF Schrödinger equation, U is instantaneous in LF time The SO) Casimir L corresponds to group of rotations in transverse LF plane Semiclassical approximation to LF QCD does not account for particle creation and absorption Compare with daff Hamiltonian H τ = d dx + g 4uω v + x ) x 4 Identical with LF Hamiltonian provided x is identified with the LF variable ζ: x = ζ/, g with the LF orbital angular momentum L: g = L /4 with effective LF confining interaction U λ ζ IAP, June 3, 03 Page 9

10 3 Higher Integer-Spin Wave Equations in AdS Space and Light Front Holographic Mapping Why is AdS space important? R NKLM = R g NLg KM g NM g KL ) AdS 5 is a space of maximal symmetry, negative curvature and a four-dim boundary: Minkowski space Isomorphism of SO4, ) group of conformal transformations with generators P µ, M µν, K µ, D with the group of isometries of AdS 5 Dim isometry group of AdS d+ : d+)d+) AdS 5 metric x M = x µ, z): ds = g MN dx M dx N = R z η µνdx µ dx ν dz ) 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 /Λ QCD map to large IR region of AdS 5, z /Λ QCD, thus there is a maximum separation of quarks and a maximum value of z IAP, June 3, 03 Page 0

11 Higher Spin Wave Equations in AdS Space [GdT, H.G. Dosch and S. J. Brodsky, PRD 87, )] Description of higher spin modes in AdS space Frondsal, Fradkin and Vasiliev) Integer spin-j fields in AdS conveniently described by tensor field Φ N N J with effective action S eff = d d x dz g e ϕz) g N N g N J N J g MM D M Φ N...N J D M Φ N...N J ) µ eff z) Φ N...N J Φ N...N J where D M is the covariant derivative which includes affine connection The z-dependent effective AdS mass µ eff z) can absorb the contribution from different contractions in the action and is a priori unknown Effective mass µ eff z) allows a separation of kinematical and dynamical effects and is determined by precise mapping to light-front physics Non-trivial geometry of pure AdS encodes the kinematics and the additional deformations of AdS encode the dynamics, including confinement IAP, June 3, 03 Page

12 Physical hadron has plane-wave and polarization indices along 3+ physical coordinates Φ P x, z) µ µ J = e ip x Φz) µ µ J, Φ zµ µ J = = Φ µ µ z = 0 with four-momentum P µ and invariant hadronic mass P µ P µ =M Further simplification by using a local Lorentz frame with tangent indices Variation of the action gives AdS wave equation for spin-j field Φz) ν ν J = Φ J z)ɛ ν ν J [ zd J e ϕ ) ) ] z) mr e ϕz) z z d J z + Φ J = M Φ J z with m R) = µ eff z)r) Jz ϕ z) + Jd J + ) and the kinematical constraints η µν P µ ɛ νν ν J = 0, η µν ɛ µνν3 ν J = 0. Kinematical constrains in the LF imply that m must be a constant [See also: T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega, Phys. Rev. D 85, )] IAP, June 3, 03 Page

13 Light-Front Mapping [GdT and S. J. Brodsky, PRL 0, )] Upon substitution Φ J z) z d )/ J e ϕz)/ φ J z) and z ζ in AdS WE [ ) zd J e ϕz) ) ] mr e ϕz) z z d J z + Φ J z) = M Φ J z) z we find LFWE d = 4) d dζ 4L 4ζ ) + Uζ) φ J ζ) = M φ J ζ) with Uζ) = ϕ ζ) + 4 ϕ ζ) + J 3 ϕ ζ) z and mr) = J) + L Unmodified AdS equations correspond to the kinetic energy terms of the partons inside a hadron Interaction terms in the QCD Lagrangian build the effective confining potential Uζ) and correspond to the truncation of AdS space in an effective dual gravity approximation AdS Breitenlohner-Freedman bound mr) 4 equivalent to LF QM stability condition L 0 IAP, June 3, 03 Page 3

14 Meson Spectrum Dilaton profile in the dual gravity model is also determined from conformal QM daff)! ϕz) = λz, Effective potential: U = λ ζ + λj ) λ = 4uω v 6 LFWE d dζ ) 4L 4ζ + λ ζ + λj ) φ J ζ) = M φ J ζ) Normalized eigenfunctions φ φ = dζ φ z) = φ n,l ζ) = λ +L)/ n! n+l)! ζ/+l e λ ζ / L L n λ ζ ) Eigenvalues for λ > 0 For λ < 0, of hadronic states M n,j,l = 4λ n + J + L ) M = 4λ n + + L J)/), incompatible with the LF constituent interpretation IAP, June 3, 03 Page 4

15 φζ) 0.4 φζ) A ζ A ζ LFWFs φ n,l ζ) in physical space-time: L) orbital modes and R) radial modes IAP, June 3, 03 Page 5

16 Table : I = mesons. For a qq state P = ) L+, C = ) L+S L S n J P C I = Meson π40) π300) π800) 0 0 ρ770) 0 ρ450) 0 ρ700) b 35) a 0 980) 0 ++ a 0 450) 0 ++ a 60) 0 ++ a 30) π 670) 0 + π 880) 0 3 ρ 3 690) a 4 040) IAP, June 3, 03 Page 6

17 J = L + S, I = meson families M n,l,s = 4λ n + L + S/) 4 n= n= n=0 4 n= n= n=0 π 880) ρ700) a 4 040) M GeV ) π800) π 670) M GeV ) ρ 3 690) π300) b 35) ρ450) a 30) A0 π40) 0 L A4 ρ770) 0 L Orbital and radial excitations for the π λ = 0.59 GeV) and the ρ I=meson families λ = 0.54 GeV) IAP, June 3, 03 Page 7

18 5 4 M GeV a M n,j,l = 4λ n + J + L ) 3 Ρ a 30 Triplet splitting for vector meson a-states L =, J = 0,, ) Ρ 770 a 60 a L M a 30) > M a 60) > M a0 980) 0 3 Systematics of I = light meson spectra orbital and radial excitations as well as important J L splitting, well described by light-front harmonic confinement model Linear Regge trajectories, a massless pion and relation between the ρ and a mass M a /M ρ = usually obtained from Weinberg sum rules [Weinberg 967)] IAP, June 3, 03 Page 8

19 4 Higher Half-Integer Spin Wave Equations in AdS Space and Light-Front Holographic Mapping [J. Polchinski and M. J. Strassler, JHEP 0305, 0 003)] [GdT and S. J. Brodsky, PRL 94, )] [GdT, H.G. Dosch and S. J. Brodsky, PRD 87, )] Image credit: N. Evans 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 Extension of holographic ideas to spin- and higher half-integral J) hadrons by considering wave equations for Rarita-Schwinger spinor fields in AdS space and their mapping to light-front physics ζ = x x n j= x j b j where x = x n is the longitudinal momentum fraction of the active quark LF clustering decomposition: Same multiplicity of states for mesons and baryons! IAP, June 3, 03 Page 9

20 Half-integer spin J = T + conveniently represented by RS spinor [Ψ N N T ] α with effective AdS action S eff = d d x dz g g N N g N T N T ] [Ψ N N T i Γ A e M A D M µ ρz) )Ψ N N T + h.c. where the covariant derivative D M includes the affine connection and the spin connection e A M is the vielbein and ΓA tangent space Dirac matrices { Γ A, Γ B} = η AB For fermions one cannot introduce confinement with dilaton since it can be scaled away [I. Kirsch 006)] Introduce effective interaction ρz) constrained by the condition that the square of the Dirac equation leads to harmonic confinement daff) Linear covariant derivatives in the action prevents mixing between dynamical and kinematical effects In contrast with effective action for integer spin, the AdS mass µ is constant: systematics of meson and baryon spectrum different! IAP, June 3, 03 Page 0

21 Physical baryons have spinors, and T polarization indices along 3+ physical coordinates Ψ µ µ T, Ψ zµ µ T = = Ψ µ µ z = 0 Further simplification by using a local Lorentz frame with tangent indices Variation of the action gives AdS wave equation for spin-j field [ i zη MN Γ M N + d T Ψ ν ν J ) ] Γ z µr R ρz) Ψ ν...ν T = 0 and the Rarita-Schwinger condition γ ν Ψ νν... ν T = 0 [See also: T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega, Phys. Rev. D 85, )] IAP, June 3, 03 Page

22 Light-Front Mapping A physical baryon satisfies the Rarita-Schwinger equation for spinors in physical space-time iγ µ µ M) u ν ν T P ) = 0, γ ν u νν ν T P ) = 0 Upon substitution in AdS wave equation for spin J Ψ ± ν ν T x, z) = e ip x R z u ± chiral spinors) ) T d/ ψ ± T z) u± ν ν T P ) and z ζ find LFWE d dζ ψ ν + ζ ψ V ζ)ψ = Mψ + d dζ ψ + ν + ψ + V ζ)ψ + = Mψ ζ provided that µr = ν + and ψ± T = ψ ± with effective LF potential V ζ) = R ζ ρζ) a J-independent potential No spin-orbit coupling along a given trajectory! IAP, June 3, 03 Page

23 Baryon Spectrum Choose linear potential V = λ ζ, λ > 0 from underlying conformality of the theory daff) Eigenfunctions Eigenvalues ψ + ζ) ζ +ν e λζ / L ν nλζ ), ψ ζ) ζ 3 +ν e λζ / L ν+ n λζ ) M = 4λn + ν + ) Gap scale 4λ determines trajectory slope and spectrum gap between plus-parity spin- and minus-parity spin- 3 nucleon families! For nucleons ν + / = L, ν 3/ = L +, where L is the relative LF angular momentum between the active quark and spectator cluster For λ < 0 no solution possible IAP, June 3, 03 Page 3

24 L S n Baryon State N + 940) N + 440) N + 70) ) ) 0 N 535) N 3 50) 650) N 3 700) N 5 675) 0 N 0 60) 3 700) 0 N ) N ) N ) ) ) ) 7 0 N 5 0 N 3 N N N 9 0 N 7 N 9 N 7 N ) 90) N 9 50) 7 N 9 + 0) 9 + N N + 40) 600) N 3 IAP, June 3, 03 Page 4

25 Phenomenological identification to describe the full baryon spectrum: plus and negative sectors have internal spin S = and S = 3 ν + / = L, ν+ 3/ = L + / ν / = L + /, ν 3/ = L + 6 n=3 n= n= n=0 6 n=3 n= n= n=0 Δ40) M GeV ) 4 N70) N440) N900) N70) N680) N0) M GeV ) 4 Δ600) Δ950) Δ90) Δ90) Δ905) N940) Δ3) A 0 4 L A3 0 4 L Example: Orbital and radial excitations for positive parity N and baryon families λ 0.5 GeV) IAP, June 3, 03 Page 5

26 Many thanks! IAP, June 3, 03 Page 6