Nucleon Resonance Spectrum and Form Factors from Superconformal Quantum Mechanics in Holographic QCD

Transcription

1 Nucleon Resonance Spectrum and Form Factors from Superconformal Quantum Mechanics in Holographic QCD Guy F. de Téramond Universidad de Costa Rica Nucleon Resonances: From Photoproduction to High Photon Virtualities ECT*, Trento, - 6 October 05 In collaboration with Stan Brodsky and Hans G. Dosch Page

2 Quest for a semiclassical approximation to describe bound states in QCD (Convenient starting point in QCD) I. Semiclassical approximation to QCD in the light-front: Reduction of QCD LF Hamiltonian leads to a relativistic LF wave equation, where complexities from strong interactions are incorporated in effective potential U II. Construction of LF potential U: Since the LF semiclassical approach leads to a one-dim QFT, it is natural to extend conformal and superconformal QM to the light front since it gives important insights into the confinement mechanism, the emergence of a mass scale and baryon-meson SUSY III. Correspondence between equations of motion for arbitrary spin in AdS space and relativistic LF boundstate equations in physical space-time: Embedding of LF wave equations in AdS leads to extension of LF potential U to arbitrary spin from conformal symmetry breaking in the AdS 5 action Page

3 Outline of this talk Semiclassical approximation to QCD in the light front Conformal quantum mechanics and light-front dynamics: Mesons 3 Embedding integer-spin wave equations in AdS space 4 Superconformal quantum mechanics and light-front dynamics: Baryons 5 Superconformal baryon-meson symmetry 6 Light-front holographic cluster decomposition and form factors Page 3

4 () Semiclassical approximation to QCD in the light front Start with SU(3) C QCD Lagrangian L QCD = ψ (iγ µ D µ m) ψ 4 Ga µνg a µν Express the hadron four-momentum generator P = (P +, P, P ) in terms of dynamical fields ( ψ + = Λ ± ψ and A Λ± = γ 0 γ ±) quantized in null plane x + = x 0 + x 3 = 0 P = dx d x ψ + γ + (i ) + m i + ψ + + interactions P + = dx d x ψ + γ + i + ψ + P = dx d x ψ + γ + i ψ + LF invariant Hamiltonian P = P µ P µ = P P + P P ψ(p ) = M ψ(p ) where ψ(p ) is expanded in multi-particle Fock states n : ψ = n ψ n n Page 4

5 Effective QCD LF Bound-state Equation [GdT and S. J. Brodsky, PRL 0, 0860 (009)] Factor out the longitudinal X(x) and orbital kinematical dependence from LFWF ψ ψ(x, ζ, ϕ) = e ilϕ X(x) φ(ζ) πζ Ultra relativistic limit m q 0 longitudinal modes X(x) decouple and LF Hamiltonian equation P µ P µ ψ = M ψ is a LF wave equation for φ ( d dζ 4L 4ζ Invariant transverse variable in impact space ) + U(ζ) φ(ζ) = M φ(ζ) ζ = x( x)b conjugate to invariant mass M = k /x( x) Critical value L = 0 corresponds to lowest possible stable solution: ground state of the LF Hamiltonian Relativistic and frame-independent LF Schrödinger equation: U is instantaneous in LF time and comprises all interactions, including those with higher Fock states. Page 5

6 () Conformal quantum mechanics and light-front dynamics [S. J. Brodsky, GdT and H.G. Dosch, PLB 79, 3 (04)] Incorporate in -dim effective QFT the conformal symmetry of 4-dim QCD Lagrangian in the limit of massless quarks: Conformal QM [V. de Alfaro, S. Fubini and G. Furlan, Nuovo Cim. A 34, 569 (976)] Conformal Hamiltonian: (p + g x ) H = g dimensionless: Casimir operator of the representation Schrödinger picture: p = i x H = QM evolution ( d dx + g x H ψ(t) = i d dt ψ(t) ) H is one of the generators of the conformal group Conf(R ). The two additional generators are: Dilatation: D = 4 (px + xp) Special conformal transformations: K = x Page 6

7 H, D and K close the conformal algebra [H, D] = ih, [H, K] = id, [K, D] = ik daff construct a new generator G as a superposition of the 3 generators of Conf(R ) and introduce new time variable τ G = uh + vd + wk dτ = Find usual quantum mechanical evolution for time τ dt u + vt + wt G ψ(τ) = i d dτ ψ(τ) H ψ(t) = i d dt ψ(t) G = ( u d dx + g ) x + i4 ( v x d dx + d ) dx x + wx. Operator G is compact for 4uw v > 0, but action remains conformal invariant! Emergence of scale: Since the generators of Conf(R ) SO(, ) have different dimensions a scale appears in the new Hamiltonian G, which according to daff may play a fundamental role Page 7

8 Connection to light-front dynamics Compare the daff Hamiltonian G G = ( u d dx + g ) x + i4 ( v x d dx + d ) dx x + wx. with the LF Hamiltonian H LF H LF = d dζ 4L 4ζ + U(ζ) and identify daff variable x with LF invariant variable ζ Choose u =, v = 0 Casimir operator from LF kinematical constraints: g = L 4 w = λ fixes the LF potential to harmonic oscillator in the LF plane λ ζ U λ ζ Page 8

9 (3) Embedding integer spin wave equations in AdS space [GdT, H.G. Dosch and S. J. Brodsky, PRD 87, (03)] Integer spin-j in AdS conveniently described by tensor field Φ N N J with effective action S eff = d d x dz g e ϕ(z) 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 D M is the covariant derivative which includes affine connection and dilaton ϕ(z) effectively breaks maximal symmetry of AdS d+ ds = R z ( dxµ dx µ dz ) Effective mass µ eff (z) is determined by precise mapping to light-front physics Non-trivial geometry of pure AdS encodes the kinematics and additional deformations of AdS encode the dynamics, including confinement Page 9

10 Physical hadron has plane-wave and polarization indices along 3+ physical coordinates and a profile wavefunction Φ(z) along holographic variable z Φ P (x, z) µ µ J = e ip x Φ(z) 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 Variation of the action gives AdS wave equation for spin-j field Φ(z) ν ν J = Φ J (z)ɛ ν ν J (P ) [ zd J e ϕ(z) z ( e ϕ ) (z) z d J z + ( ) ] µr Φ J = M Φ J z with (µ R) = (µ eff (z)r) Jz ϕ (z) + J(d J + ) and the kinematical constraints to eliminate the lower spin states J, J, η µν P µ ɛ νν ν J = 0, η µν ɛ µνν3 ν J = 0 Kinematical constrains in the LF imply that µ must be a constant [See also: T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega, Phys. Rev. D 85, (0)] Page 0

11 Light-front mapping [GdT and S. J. Brodsky, PRL 0, 0860 (009)] Upon substitution Φ J (z) z (d )/ J e ϕ(z)/ φ J (z) and z ζ in AdS WE [ ( ) zd J e ϕ(z) ( ) ] µr e ϕ(z) z z d J z + Φ J (z) = M Φ J (z) z we find LFWE (d = 4) ( d dζ 4L 4ζ ) + U(ζ) φ J (ζ) = M φ J (ζ) y ct z -0 88A3 with U(ζ) = ϕ (ζ) + 4 ϕ (ζ) + J 3 ϕ (ζ) z and (µr) = ( J) + L Unmodified AdS equations correspond to the kinetic energy terms for the partons Effective confining potential U(ζ) corresponds to the IR modification of AdS space AdS Breitenlohner-Freedman bound (µr) 4 equivalent to LF QM stability condition L 0 Page

12 Meson spectrum Dilaton profile in the dual gravity model determined from one-dim QFTh (daff) ϕ(z) = λz, λ = w Effective potential: U = λ ζ + λ(j ) LFWE ( d dζ ) 4L 4ζ + λ ζ + λ(j ) 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 M n,j,l = 4λ λ < 0 incompatible with LF constituent interpretation ( n + J + L ) Page

13 Three relevant points... A linear potential V eff in the instant form implies a quadratic potential U eff in the front form at large distances Regge trajectories U eff = V eff + p + m q V eff + V eff p + m q [A. P. Trawiński, S. D. Glazek, S. J. Brodsky, GdT, H. G. Dosch, PRD 90, (04)] Results are easily extended to light quarks [S. J. Brodsky, GdT, H. G. Dosch and J. Erlich, Phys. Rept. 584, (05) M m q,m q = ( 0 dx m q e λ x + m q x ) ( m q x ( 0 dx m q e λ x + m q x + m q x For n partons invariant LF variable ζ is [S. J. Brodsky and GdT, PRL 96, 060 (006)] ζ = x n x j b j x where x j and x are longitudinal momentum fractions of quark j in the cluster and of the active quark j= ) ) Page 3

14 M (GeV ) (a) π(800) (b) n= n= n=0 n= n= n=0 K (80) π(880) K K (400) (770) π (670) π(300) b (35) K (70) 0 π(40) K(494) A8 0 L 3 0 L 3 6 (a) n=3 n= n= n=0 (b) n= n= n=0 n=3 n= n= n=0 M (GeV ) 4 0 ρ(700) ω(650) ρ(450) ω(40) ρ(770) ω(78) f (300) f (950) a (30) f (70) ρ 3 (690) ω 3 (670) a 4 (040) f 4 (050) K*(680) K*(40) K*(89) K* (430) K* 3 (780) K* 4 (045) M (GeV ) 5 3 φ(70) φ(680) φ(00) φ 3 (850) A9 0 4 L 0 4 L A5 0 4 L Orbital and radial excitations for λ = 0.59 GeV (pseudoscalar) and 0.54 GeV (vector mesons) Page 4

15 (4) Superconformal quantum mechanics and light-front dynamics [GdT, H.G. Dosch and S. J. Brodsky, PRD 9, (05)] SUSY QM contains two fermionic generators Q and Q, and a bosonic generator, the Hamiltonian H [E. Witten, NPB 88, 53 (98)] Closure under the graded algebra sl(/): {Q, Q } = H {Q, Q} = {Q, Q } = 0 [Q, H] = [Q, H] = 0 Note: Since [Q, H] = 0 the states E and Q E have identical eigenvalues E A simple realization is Q = χ (ip + W ), Q = χ ( ip + W ) where χ and χ are spinor operators with anticommutation relation {χ, χ } = In a Pauli-spin matrix representation: χ = (σ + iσ ), χ = (σ iσ ) [χ, χ ] = σ 3 Page 5

16 Following Fubini and Rabinovici consider a -dim QFT invariant under conformal and supersymmetric transformations [S. Fubini and E. Rabinovici, NPB 45, 7 (984)] Conformal superpotential (f is dimensionless ) W (x) = f x Thus -dim QFT representation of the operators Q = χ ( d dx + f x ), Q = χ ( d dx + f x ) Conformal Hamiltonian H = {Q, Q } in matrix form H = d + f(f ) 0 dx x 0 d + f(f+) dx x Conformal graded-lie algebra has in addition to Hamiltonian H and supercharges Q and Q, a new operator S related to generator of conformal transformations K {S, S } S = χ x, S = χ x Page 6

17 Find enlarged algebra (Superconformal algebra of Haag, Lopuszanski and Sohnius (974)) where the operators {Q, Q } = H, {S, S } = K {Q, S } = f + σ id {Q, S} = f + σ 3 4 id H = ( d dx + f ) σ 3 f x D = i ( d 4 dx x + x d ) dx K = x satisfy the conformal algebra [H, D] = ih, [H, K] = id, [K, D] = ik Page 7

18 Following F&R define a supercharge R, a linear combination of the generators Q and S R = u Q + w S and consider the new generator G= {R, R } which also closes under the graded algebra sl(/) {R, R } = G {R, R} = {R, R } = 0 [R, H] = [R, H] = 0 {Q, Q } = H {Q, Q} = {Q, Q } = 0 [Q, H] = [Q, H] = 0 New QM evolution operator G = uh + wk + uw (f + σ3 ) is compact for uw > 0: Emergence of a scale since Q and S have different units Light-front extension of superconformal results follows from x ζ, f ν +, σ 3 γ 5, G H LF Obtain: H LF = d dζ + ( ) ν + ζ ν + ζ γ 5 + λ ζ + λ(ν + ) + λγ 5 where coefficients u and w are fixed to u = and w = λ Page 8

19 Nucleon Spectrum In block-matrix form H LF = Eigenfunctions d 4ν + λ ζ + λ(ν + ) 0 dζ 4ζ 0 d 4(ν+) dζ 4ζ + λ ζ + λν ψ + (ζ) ζ +ν e λζ / L ν n(λζ ) ψ (ζ) ζ 3 +ν e λζ / L ν+ n (λζ ) Eigenvalues M GeV n 3 n n n 0 M = 4λ(n + ν + ) 4 N 0 Lowest possible state n = 0 and ν = 0 Orbital excitations ν = 0,, = L L is the relative LF angular momentum between the active quark and spectator cluster 3 0 N 900 N 70 N 70 N 680 N 440 Λ 0.49 GeV N L Page 9

20 (6) Superconformal baryon-meson symmetry [H.G. Dosch, GdT, and S. J. Brodsky, PRD 9, (05)] Previous application: positive and negative chirality components of baryons related by supercharge R R ψ + = ψ with identical eigenvalue M since [R, G] = [R, G] = 0 Conventionally supersymmetry relates fermions and bosons R Baryon = Meson or R Meson = Baryon If φ M is a meson state with eigenvalue M, G φ M = M φ M, then there exists also a baryonic state R φ M = φ B with the same eigenvalue M : G φ B = G R φ M = R G φ M = M φ B For a zero eigenvalue M we can have the trivial solution φ(m = 0) B = 0 Special role played by the pion as a unique state of zero energy Page 0

21 Baryon as superpartner of the meson trajectory φ = φ Meson φ Baryon Compare superconformal meson-baryon equations with LFWE for nucleon (leading twist) and pion: ( ( d dx + λ x + λ f + λ + 4(f ) 4x ) φ Baryon = M φ Baryon ( d dζ + λ B ζ + λ B (L B + ) + 4L B ) 4ζ ψ + L B = M ψ + L B d dx + λ x + λ f λ + 4(f + ) 4x ( d dζ + λ M ζ + λ M (L M ) + 4L M ) 4ζ φ LM ) φ Meson = M φ Meson = M φ LM Find: λ = λ M = λ B, f = L B + = L M L M = L B + Page

22 6 6 M (GeV ) 4 b π N N N N 9 N + M (GeV ) 4 a,f ρ 3,ω 3-3 Δ,Δ - a 4,f Δ,Δ,Δ,Δ + Δ A π 0 N + 4 L M = L B A3 ρ,ω 3 Δ L M = L B + Superconformal meson-nucleon partners: solid line corresponds to λ = 0.53 GeV Page

23 Supersymmetry across the light and heavy-light hadronic spectrum [H.G. Dosch, GdT, and S. J. Brodsky, Phys. Rev. D 9, (05)] Introduction of quark masses breaks conformal symmetry without violating supersymmetry 8 D,Λ c,σ c D*,Σ* M (GeV ) M (GeV ) A D D s D s, c, [I] c [I] 0 L M = L B + Σ c Σ c (50) D (40) D* (460) Λ c [I] c [I] D s (536) c (a) D s (460) (c) 0 D* (007) D* s D* s, * c [I] L M = L B + Supersymmetric relations between mesons and baryons with charm c(645) D* s (573) [I] (b) (d) Page 3

24 38 B,Λ b,σ b B*,Σ b * M (GeV ) M (GeV ) A B s B s, b [I] 0 L M = L B + Σ b B (57) Λ b B B* (a) (b) B s (5830) b [I] B* s, * b [I] Σ b * B* (5747) b(5945) B* s (5840) [I] B* s (c) (d) 0 L M = L B + Supersymmetric relations between mesons and baryons with beauty Page 4

25 Emerging SUSY from color dynamics [S. J. Brodsky, GdT, H. G. Dosch, C. Lorcé] Work in progress Superconformal spin-dependent Hamiltonian to describe mesons and baryons (chiral limit) G = {R λ, R λ} + λ I s R Q + λs LFWE for mesons ( d dζ ) 4L M 4ζ + λ ζ + λ(l M + s ) φ Meson = M φ Meson LFWE for nucleons ( d dζ ) 4L B 4ζ + λ ζ + λ(l B + s + ) φ Baryon = M φ Baryon with L M = L B + Spin of the spectator cluster s is the spin of the corresponding meson! Page 5

26 Preliminary Page 6

27 How good is semiclassical approximation based on superconformal QM and LF clustering properties? Preliminary Best fit for the hadronic scale λ from the different sectors including radial and orbital excitations Page 7

28 (6) Light-front holographic cluster decomposition and form factors [S. J. Brodsky, GdT, H. G. Dosch, C. Lorcé] Work in progress LF Holographic FF F τ=n (Q ) expressed as the N product of poles for twist τ = N S. J. Brodsky and GdT, PRD 77, (008) F τ= (Q ) = F τ=3 (Q ) = F τ=n (Q ) = ( ) + Q M ρ ( )( ) + Q + Q M ρ M ρ ( )( ) + Q + Q M ρ M ρ ( + Q M ρ N ) Spectral formula M ρ n 4κ (n + /) Cluster decomposition in terms of twist τ = FFs! F τ=n (Q ( ) = F τ= Q ) ( ) ( ) F τ= 3 Q F τ= N 3 Q Page 8

29 Example: Dirac proton FF F p in terms of the pion form factor F π : F p ( (Q ) = F π Q ) ( ) F π 3 Q (equivalent to τ = 3 FF) Q 4 F p (Q ) (GeV 4 ) A Q (GeV ) But... we know that higher Fock components are required. Example time-like pion FF: π = ψ qq/π qq τ= + ψ qqqq qqqq τ=4 + F π (q ) = ( γ)f τ= (q ) + γf τ=4 (q ) P qqqq =.5% S. J. Brodsky, GdT, H. G. Dosch and J. Erlich, PR 584, (05) log Fπ(s) A6-0 s (GeV ) Page 9

30 Transition form factors for the radial transition n = 0 n = : F n=0 τ= (Q ) = ( + Q M ρ Q Mρ )( + Q M ρ ) F n=0 τ=3 (Q ) = F n=0 τ=n (Q ) = 3 Q M ρ ( )( )( ) + Q + Q + Q M ρ M ρ M ρ N N Q M ρ ( )( ) + Q + Q M ρ M ρ ( + Q M ρ N ) where Fτ=N n=0 (Q ) is expressed as the N product of poles LF cluster decomposition: Express the transition form factor as the product of the pion transition form factor times the N product of pion elastic form factors evaluated at different scales N Fτ=N n=0 (Q ) = N F τ= n=0 ( Q ) ( ) ( ) F τ= 3 Q F τ= (N + ) 3 Q. () Page 30

31 Example: Dirac transition form factor of the proton to a Roper state F p N N : F p N N (Q ) = 3 F π π (Q )F π (equivalent to τ = 3 TFF ) ( ) 5 Q F pn N Q [GdT and S. J. Brodsky, AIP Conf. Proc. 43, 68 (0)] Old JLab data Q GeV Holographic QCD computation including A p / and Sp / : T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega, PRD 87, 0607 (03) Data confirmed by recent JLab data. Possible solution to describe small (Q < GeV ) data: Include τ = 5 higher Fock component qqqqq in addition to τ = 3 valence qqq F τ=3 Q 4, F τ=5 Q 8 Page 3

32 Thanks! For a review: S. J. Brodsky, GdT, H. G. Dosch and J. Erlich, Phys. Rept. 584, (05) Page 3