Modified Anti-de-Sitter Metric, Light-Front Quantized QCD, and Conformal Quantum Mechanics

Transcription

1 GEOMETRY AND PHYSICS II Institut Henri Poincaré, Nov. 8 &9, 013 Modified Anti-de-Sitter Metric, Light-Front Quantized QCD, and Conformal Quantum Mechanics H.G.Dosch Institut für Theoretische Physik der Universität Heidelberg Stan Brodsky*, Guy de Téramond ** *Stanford, USA, **San José, CR arxiv: [hep-th]

2 GEOMETRY AND PHYSICS II Institut Henri Poincaré, Nov. 8 &9, 013 Modified Anti-de-Sitter Metric, Light-Front Quantized QCD, and Conformal Quantum Mechanics 1) Theory of Strong Interaction. The Problems ) Modified Anti-de-Sitter metric. A solution? 3) AdS metric, LF Quantized QCD and Conformal Q M. Une ménage à trois

3 Quantum Chromo Dynamics, theory of strongly interacting particles (hadrons: protons, neutrons, π-mesons ). Fundamental fields: Quarks: Spin 1 fields with colour quantum numbers, interacting through Gluons. Theory invariant under SU(3) gauge transformations. Cf. with Quantum Electrodynamics: electrons and photons, gauge group U(1). Confinement:To the quarks fields correspond no states in the Fock space of hadrons. Problem of all realistic Quantum Field Theories: The only analytically tractable treatment is perturbation theory around the free fields. Not adequate for the effects of strong interactions, like confining the quarks.

4 But even in QED: Need for a semiclassical treatment of bound state problems (Schrödinger, Dirac equation) Formidable task: Find semiclassical equations for QCD! Not completely unrealistic: Qualitative success of non-relativistic quark model (based on Schroedinger equation) Striking regularities in the hadronic spectra Notably Regge Trajectories «Øµ ¾ ¾ Ø Î ¾ µ 8 «Øµ Ô ½ ¾ Æ ½ ¾ ½½ ¾ Æ ¾ ¾ Æ ¾ Ø Î ¾ µ From Donnachie et al. QCD and Pomeron Physics

5 Light Front Quantization adequate framework in relativistic theories Dirac, Rev. Mod. Phys. 1, 39 (1949), Brodsky et al.... Instant-form: x 0 = 0 Light Front-form: x 0 + x 3 = 0 For the Fock state of two massless quarks: ( ) ζ + 4L 1 + U(ζ) ψ(ζ) = P ψ(ζ) 4ζ ζ: essentially the separation of the quarks in the transverse (1-) plane, L: longitud. angular momentum U(ζ) comprises all interactions, including those with higher Fock states Eigenvalues:Squared hadron masses

6 Strong boost: Maldacena Conjecture Adv. Theor. Math. Phys., 31 (1998): Gauge-QFT d = 4 Classical gravitational theory d = 5 Generating functional of the gauge QFT is given by the minimum of the classical action of the gravitational theory at the 4-dim border of the 5-dim space Graviation: AdS 5 metric, ds = R z ( 3 i=0 dx i dx i dz ) Poincaré coordinates. z = x 5 holographic coordinate, z = 0 border to 4-dim. space R(1, 3) QFT: greatly oversymmetrized: conformal supersymmetric gauge theory bottom-up approach: Start from realistic 4-dim Theory. Here: Light-Front Holography G. F. de Téramond and S. J. Brodsky Phys. Rev. Lett. 10, (009)

7 Consider scalar field in AdS 5 : Action: S = d 4 x dz g ( g MN M Φ(x, z) N Φ(x, z) µ Φ(x, z) ) With Φ(x, z) = e ipx Φ(z) e.o.m. can be brought into the form: ( z + 4(µR) z ) ϕ(z) = P ϕ(z) Compare ( with LF equation: ) ζ + 4L 1 + U(ζ) ψ(ζ) = P ψ(ζ) 4ζ same structure: z ζ (µr) + 4 L but U(ζ) = 0 No wonder: AdS 5 maximally symmetric, 15 isometries Conf(R 1,3 ),(10 Poincaré + 4 inversions + dilatation) Conformal symmetry: No scale no dicrete spectrum

8 Way out: Distort maximal symmetry! S = d 4 x dz g ( g MN M Φ(x, z) N Φ(x, z) µ Φ(x, z) ) S = d 4 x dz g e φ(z) ( g MN M Φ(x, z) N Φ(x, z) µ Φ(x, z) ) U(z) = 1 4 (φ (z)) 3 z φ (z) + 1 φ (z) Soft wall model Karch, Katz, Son and Stephanov, Phys. Rev. D 74, (006), G. F. de Téramond and S. J. Brodsky, Nucl. Phys. Proc. Suppl. 199, 89 (010) ; arxiv: ,... G. F. de Téramond, H. G. D. and S. J. Brodsky, Phys. Rev. D 87, (013) φ(z) = λ z, U(ζ) = λ ζ λ

9 4 n= n=1 n=0 4 n= n=1 n=0 π (1880) ρ(1700) a 4 (040) M (GeV ) π(1800) π (1670) M (GeV ) ρ 3 (1690) π(1300) b 1 (135) ρ(1450) a (130) A0 π(140) 0 L S = 0 mesons A4 ρ(770) 0 L J = L + 1 mesons K star-meson M GeV Ν L 1 N 600 Msq L K mesons N 50 4Λ N 190 N 1700 N 1675 N 0 N 1650 Ν L N 170 N 1680 N 940 Λ 0.49 GeV L Nucleons from de Téramond and Brodsky, ArXiv: ; Brodsky, de Téramond, HGD, Josh Erlich, Phys. Rep. to appear. Choice of φ(z) = λ z only motivated by phenomenology. Some deeper reason?

10 Conformal symmetry. Classical QCD-Lagrangian with massless quarks conformally invariant. Need for renormalization introduces a scale Λ QCD Scale dependence of QCD coupling in perturbative regime (small distance = large momentum scales). Q dα s(q ) = β i dq i=0 4π α+i s +... α s (Q ) = 4π β 0 1 log(q /Λ QCD ) +... from PDG Indications: At large distances, small Q, α s becomes constant again. Restauration of conformal symmetry?

11 Fresh look at conformal symmetry Our goal: Find a semiclassical approximarion, i.e. approximate QFT by Quantum Mechanics Conformal QM = conformal QFT in 1 dimension. V. de Alfaro, S. Fubini and G. Furlan Nuovo Cim. A 34, 569 (1976) ( ) ( ) S conf = 1 dt Q(t) g H = 1 Q() Q + g Q P = δs δ Q = Q [Q, Q] = i Schrödinger Q(0) = r, Q(0) = i r ( ) Hψ(r) = 1 r + g r ψ(r) again back to free equation. ( AdS: z + 4(µR) LF : Big( ζ + 4L 1 4ζ 4z ) ϕ(z) = P ϕ(z) + U(ζ) }{{} =0 ) ψ(ζ) = P ψ(ζ) But, as stressed by de Alfaro, Fubini and Furlan:

12 There are besides H, the generator of translations t two more constants of motion, namely the two other Noether currents of the conformal action S conf D for dilatations, t t(1 + ϵ) and K for special conformal transformation t t 1 ϵt Allows to construct a generalized Hamiltonian: G = H + w K + v D translation in τ with dτ = Schrödinger G ψ(r) = 1 distorted Ads: ( z + 4(µR) z ( ζ + 4L 1 + U(ζ) 4ζ LF : dt 1+vt+wt ( r + g r + w r + i v (r r + r r) ) ψ(r) ) (φ (z)) 3 z φ (z) + 1 φ (z) ) ψ(ζ) = P ψ(ζ) ϕ(z) = P ϕ(z) The conformal Lagrangian S conf implies unambiguously : φ(z) = λ z U(ζ) = λ ζ

13 Back to geometry; The conformal group Conf (R 1 ) is isomorphic to the Lorentz group S0(, 1) and therefore to the isometries of AdS. Best seen by embedding AdS into a three-dimensional space: X0 X1 X 1 + X 0 X 1 = R Poincaré coordinates: z = R X 1 X 1 x 0 = X 0(X 1 X 1 ) R X-1 Generators of S0(, 1) Rotation J 10, boosts J 01 J 1,1 are the isometries of AdS

14 Isomorphism SO(, 1) Conf (R 1 ): a H = J 10 J 01, 1 a K = J 10 + J 01, D = J 1,1 The confining G G = H + wk : 1+θ a G = J 10 θ J 01 Schrödinger: ( G = ) 1 r + g θ r a 1+θ r