Supersymmetry across the light and heavy-light hadronic spectrum

Transcription

1 Supersymmetry across the light and heavy-light hadronic spectrum Hans Günter Dosch Institut für Theoretische Physik der Universität Heidelberg Based on: G. F. de Teramond, H. G. D., S. J. Brodsky Rev. D 91, (2015) H. G. D., G. F. de Teramond,S. J. Brodsky Phys. Rev. D 91, (2015) H. G. D., G. F. de Teramond, S. J. Brodsky Phys. Rev. D 92, (2015) S. J. Brodsky, G. F. de Teramond, H. G. D., C. Lorcé Phys. Lett. B759,171 (2016) H. G. D., G. F. de Teramond, S. J. Brodsky Phys. Rev. D 95, (2017) **** H.G.D., Erasmo Ferreia Phys. Rev. D,92 (2015)

2 1) Introduction ( Stan s talk) LFHQCD AdS/CFT 2) Dynamical constraints by superconformal algebra Potential fixed, susy of wave functions Filling up the super multiplets: Tetraquarks? 3) Strong breaking of conformal symmetry (Heavy-light hadrons) Constraints from AdS/CFT Constraints from HQET **** A) AdS/CFT and Pomeron trajectories.

3 1 Semiclassical approximations even in QFT essential QED: Dirac and Schrödinger practical advantage, but also structural classical gravitational theory with AdS-metric in 5 dim. Maldacena conjecture superconformal Gauge QFT in 4 dim. with N c Semiclassical Light front Hamiltonian based on this correspondence and on the superconformal algebra Prediction of essential structural features of hadronic spectra. Agreement of predicted spectra with experiment of the order of ±10%. Also applicable for calculation of form and transition factors, but some additional asumptions necessary, R.S. Sufian et al. Phys. Rev. D (2017)

4 The kinematical frame is based on the light front (LF) quantization at fixed LF time x + = x 0 + x 3 Dirac, Rev. Mod. Phys. 21, 392 (1949), Brodsky et al.... Variables: x i, longitudinal momentum fraction the boost invarariant transverse separation ζ = x 1 x 2 b. x 1 P b 7 x 2 P angle ϕ around 3 axis LF angular momentum L Since we have only one variable in AdS/CFT (the 5th dimension) we have eventually to cluster the constituents. e.g. for baryons: x 2eff = x 2 + x 3, b eff = x 2b 2 +x 3 b 3 x 2 +x 3

5 Conformal limit (massless quarks) LF Hamiltonian in the two constituent or constituent +cluster sector does not contain x i explicitely: L LF angular momentum H = d2 dζ 2 + 4L2 1 4ζ 2 + U(ζ); (1) The LF potential U(ζ) in principle determined by QCD Hϕ(ζ) = M 2 ϕ(ζ) M: Hadron mass

6 Dynamical frame AdS 5 /CFT correspondence. AdS 5 : Gravitational Theory in 5-dimensional metric: (ds) 2 = R2 ( z 2 (dx 0 ) 2 (dx 1 ) 2 (dx 2 ) 2 (dx 3 ) 2 (dz) 2) }{{} Minkowski z: Holographic variable AdS action Generating functional propagators Field equations of AdS bound state wave equations with zero width, like in QCD with N c Structure of the field equations from AdS 5 action = structure of LF Hamilton if z ζ S. J. Brodsky and G. F. de Téramond Phys. Lett. B 582, 211 (2004) unperturbed AdS: U(ζ) = 0 No wonder: AdS 5 has maximal symmetry conformal QFT, i.e. no scale

7 Modify AdS action by factor e φ(ζ) (Dilaton) grind through Euler-Lagrange obtain for meson wave function Φ µ1 µj = ϵ J µ 1 µ J ϕ J,L (ζ) ) ( d2 dζ 2 + 4L2 1 4ζ 2 + U(ζ) ϕ J (ζ) = M 2 ϕ J (ζ) (2) U = U[φ] = U(ζ) = 1 2 φ (ζ) φ (z) 2 + 2J 3 2ζ φ (ζ) L is related to the AdS mass by (mr) 2 = (2 J) 2 + L 2. Baryons: Dilaton factor e φ(ζ) does not help! additional Yukawa term ūv (ζ)u in the AdS action, 5-dimensional Fermion action wave function split in positive and negative chirality component: u µ1...µ J 1/2 (p) = 1+γ 5 2 u (p) ψ B+ (ζ) + 1 γ 5 2 u (p) ψ B (ζ). U ± = V 2 ± V + 1+2L V ζ ( d2 dζ 2 + 4(L + 1)2 1 4ζ 2 ) + U (ζ) ψ B (ζ) = M 2 ψ B (ζ) ( d2 dζ 2 + 4L2 1 ) 4ζ 2 + U +(ζ) ψ B+ (ζ) = M 2 ψ B+ (ζ)

8 For phenomenological reasons one chooses ad hoc: A. Karch, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. D 74, (2006) S. J. Brodsky and G. F. de Teramond, Phys. Rev. Lett. 96, (2006) I. Kirsch, JHEP 0609, 052 (2006) Z. Abidin and C. E. Carlson, Phys. Rev. D 77, (2008) S.Brodsky, G.de Teramond, H.G.D. and J.Erlich,Phys. Rept. 584, 1 (2015) e φ(ζ) = e λ M z 2 V (ζ) = λ B z mesons : U(ζ) = λ 2 M ζ2 + 2λ M (J 1) baryons : U ± (ζ) = λ 2 B ζ2 + 2Lλ F + λ F ( 1 2 ± 1 2 ) These potentials lead to linear Regge trajectories and a massless pion. For the Delta trajectory one has to choose formally half integer twist (that is to add 1 2 to the angular momentum L) to get agreement with the data.

9 π λm = 0.59 GeV M 2 [GeV 2 ] L ρ λm = 0.54 GeV L N λb = 0.50 GeV M 2 [GeV 2 ] L L 10%, λm λ B S.Brodsky, G.de Teramond, H.G.Dosch and J.Erlich, Phys. Rept. 584, 1 (2015) λb = 0.52 GeV Why e φ(ζ) = e λm z2 V (ζ) = λ B z? Why λ M λ B?

10 2 Dynamical constraints Maldacena: QFT is superymmetric and conformal We search: semiclassical (Quantum mechanical) bound state equations Look for supersymmetric quantum mechanics constructed from superconformal algebra! Supersymmetric QM is very simple E. Witten Nucl. Phys. B 188, 513 (1981). Graded algebra consisting of two fermionic operators (supercharges), Q and Q and a bosonic operator (the Hamiltonian) H with the anti-commutation relations {Q, Q } = H, {Q, Q} = 0, {Q, Q } = 0 and the commutation relations [Q, H] = 0, [Q, H] = 0

11 matrix realization in a Hilbert space with two components: ( ) ( ) 0 q 0 0 Q =, Q = 0 0 q 0 q = d dx + f x + V (x), q = d dx + f x + V (x) where f is a dimensionless constant and V arbitrary V = 0 U = 0, but : For V = 0 conformal symmetry holds, extend to Superconformal Algebra: Supersymmetric algebra of Q, Q, H enlarged by additional supercharges (fermionic operators)s, S In matrix notation S = ( 0 x 0 0 ) (, S 0 0 = x 0 ) {S, S } = K K is the generator of the special conformal tranformation.

12 Following de Alfaro, Fubini and Furlan and Fubini and Rabinovici construct a new Hamiltonian G in the superconformal algebra: with H = {Q, Q } G = {R λ, R λ } R λ = Q + λs, R λ = Q + λs Note that in this generalization of the Hamiltonian a scale λ appears naturally, since S and Q have different physical dimension from Q.

13 The new Hamiltonian is diagonal with G 11 = d2 dx 2 + 4(f )2 1 4x 2 + λ 2 x 2 + 2λ (f 1 2 }{{} ) U, G 22 = d2 dx 2 + 4(f 1 2 )2 4x 2 + λ 2 x 2 + 2λ (f }{{} ) U +. By construction the eigenvalues are identical and we notice two miracles: For f = L and x = ζ : U ± = the potential introduced ad hoc for baryons in LFHQCD for phenomenological reasons. Here a consequence of the algebra. G 11 Hamitonian of a meson with L = J = f G 22 that of the baryon component ψ B+ with L = f 1 2. Fermion and Boson wave functions are transformed into each other by a supercharge, namely R λ, as it should be in supersymmetry! Meson with L M = f is superpartner of Baryon with L B = f 1 2, Mesons with L M = 0 (π e.g.) have no superpartner (L B = 1 excluded) λ B = λ M = λ.

14 Spin effects QM approach: no internal quark spin effects, in contrast to AdS for mesons. G 11 from QM G 11 from AdS G 11 G 11 + λ s Susy: s: total quark spin of the meson. G G + λ s I π: s = 0 susy partner N; ρ: s = 1 susy partner s for baryons: lowest possible internal spin of cluster. Mesons M 2 M = 4λ(n + L M) + 2λ s, Baryons M 2 B = 4λ(n + L B + 1) + 2λ s.

15 M 2 6 [GeV 2 ] π2 L B N M 2 6 [GeV 2 ] a3,f3 a4,f ,3 2 +,5 2 +,7 2 + L B N π b1 N 1/2+ N 1 2,3 2 N 3 2 +, L M ρ,ω a2,f2 1 2,3 2 N 5 2 3/ L M π and N, s B = 1 2 trajectories ρ, ω and, N, s B = 3 2 trajectories H. G. D., G. F. de Teramond, S. J. Brodsky Phys. Rev. D 92, (2015) Achievements of imposing superconformal algebra: LF potential theoretically determined No half integer twist necessary λ M = λ B from theory.

16 Completing the supersymmetric multiplet. Supersymmetric 4plet not yet complete. Partner of ψ B is missing. R λ ϕ M, L B + 1 ψ B+, L B R λ ψ B, L B + 1 ϕ T, L B ( ) ϕm (L Susy quadruplet: B + 1) ψ B (L B + 1) ψ B+ (L B ) ϕ T (L B ) Candidates for complete supermultiplets with fitting quantum numbers are: Meson I(J P ) Baryon I(J P ) Tetraquark I(J P ) b 1 (1235) 1(1 + ) N +, (940) 1 2 ( ) f0 (980) 0(0 + ) a 2 (1320) 1(2 + ) +, (1230) 3 2 ( ) a1 (1260)1(1 + )

17 Beyond the chiral limit Assume: small quark masses do not affect superconformal dymamics, i.e. LF potentials unchanged. Only effect on invariant mass: k 2 i i x i i k 2 i +m2 i x i affects H and the wave function. To first approximation: Mesons M 2 M = 4λ(n + L M) + 2λ s + M 2 2 (m 1, m 2 ), Baryons M 2 B = 4λ(n + L B + 1) + 2λ s + M 2 3 (m 1, m 2, m 3 ). M 2 [m 1,, m n ] = λ2 F df dλ with F [λ] = 1 0 λ 1 dx 1 dx n e ( n i From π and K mass m q = GeV, m s = GeV. m 2 i x i ) δ( n i x i 1).

18 M 2 6 [GeV 2 ] K K K 2 Λ 3 2 +,5 2 + Λ 1 2,3 2, Σ 1 2,3 2 Λ, Σ Λ 7 2 L B Λ L M M 2 6 [GeV 2 ] K K 2+ Σ 3/2+ K 3 K4+ Σ Σ L B L M K and Σ, Λ, s B = 1 2 K and Σ, s B = 3 2 M 2 6 [GeV 2 ] L B M 2 6 [GeV 2 ] L B 4 4 φ η Ξ η φ f 2 Ξ 3/2+ Ξ L M L M η and Ξ, s B = 1 2 ϕ and Ξ, s B = 3 2 Ω (1672) theo: 1760 MeV (+5%) H. G. D., G. F. de Teramond, S. J. Brodsky Phys. Rev. D 92, (2015)

19 The QCD scale parameter λ determined from the different light meson and baryon trajectories. S. J. Brodsky, G. F. de Teramond, H. G. D., C. Lorcé Phys. Lett. B759,171 (2016)

20 3 Strong breaking of conformal symmetry by heavy quark masses In Heavy-Light hadrons, where one constituent is heavy and the rest is light, we have still ultrarelativistic kinematics, but conformal symmetry is strongly broken.? Breakdown of the approach? No, supersymmetry still can survive and can, together with AdS/CFT breaking, give dynamical constraints. Back to Susy algebra: Q = ( 0 q 0 0 ), Q = ( 0 0 q 0 q = d dx + f x + V (x), q = d dx + f x + V (x), with arbitrary V (x) From H = {Q, Q } we can express the mesonic LF potential in terms of V (x) )

21 For a meson with J M = L M = L + 1 we obtain U susy (ζ) = V 2 + 2L + 1 V V. (3) ζ With a dilaton term e φ(ζ) we obtain from AdS/CFT for that meson: with the ansatz: U dil (ζ) = 1 4 (φ ) 2 + 2L M 3 φ + 1 2ζ 2 φ (4) φ = 2λζ α(ζ) V = λζ β(ζ) U dil = λ 2 ζ 2 α 2 + 2Lα + λζα U susy = λ 2 ζ 2 β 2 + 2Lβ λζβ Equating U dil (ζ) = U susy (ζ) and with σ = α(ζ) + β(ζ) we obtain

22 φ(ζ) = dζ ( λ 2 ζ 2 σ ) (ζ) λζ σ(ζ) λ 2 ζ 2 σ(ζ) + 2(L M 1)λ The dilaton must be free of kinematical quantities and hence be independent L M. σ (ζ) = 0, σ(ζ) = A with some constant A. This yields φ(ζ) = 1 2 λaζ2 + B, V (ζ) = 1 2 λaζ, U(ζ) = Aλζ Susy + AdS: The LF potential even for strongly broken conformal invariance is harmonic. The constant A, however, is arbitrary, the strength is not determined. The resulting hadronic spectrum is : Mesons: M 2 = 4λ Q (n + L) + µ 2 2 Baryons: M 2 = 4λ Q (n + L + 1) + µ 2 3

23 Heavy-Light hadrons with a c-quark H. G. D., G. F. de Teramond, S. J. Brodsky Phys. Rev. D 95, (2017)

24 Heavy-Light hadrons with a b-quark H. G. D., G. F. de Teramond, S. J. Brodsky Phys. Rev. D 95, (2017)

25 π K ρ K D D s D D s B B s B B s Figure 1: The QCD scale parameter λ Q determined from different meson-baryon trajectories.

26 Constraints from Heavy Quark Symmetry If M Q, it decouples and HQS holds decay constant f M of the Heavy-Light pseudoscalar meson MM f M C. With a mass dependence of the wave functions modified according to the transition k i 2 i k i 2 +m2 i x i i we can include the mass dependence in the wave function x i ψ (m) n,l = 1 e 1 ni=1 m i 2 2λ x i ψ (0) N n,l m where ψ(0) chiral w.f. n,l 1 λ 1 f M = dx e m2 /2λ(1 x) Q x(1 x) 1 0 dx e m2 Q /λ(1 x) π 0 Integration by steepest descent : f M = 6 ( ( 1 + erf 1 λ π e 2)) 3/2 mq 2 HQS C M 1/2 M Heavy quark limit: m Q M M λ Q = const M M

27 Thus HQS demands that for heavy quark masses the scale parameter λ increases. Even quantitatively it relates λ b with λ c.

28 Conclusion Maldacena conjecture: Classical Gravitational Theory in AdS 5 Our approach: Classical Gravitational Theory in AdS 5 Superconformal gauge QFT,N c Supersymmetric QM in superconformal Algebra Describes essential features of light hadron spectra: Linear trajectories in orbital and radial excitations, massless pion. Accuracy 10% Meson and baryon (and tetraquark) wave functions are connected by supersymmetry. Supersymmetry survives even if conformal symmetry is broken by heavy quark masses. AdS + Susy harmonic potential λζ 2 also for Heavy-Light hadrons, but scale parameter λ increases with heavy quark mass; in accordance with Heavy Quark Symmetry.

29 AdS/CFT and Diffraction from a Regge point of view σ γ p tot (W, Q 2 ) = 4π2 αf Q 2 2 W 2δ δ(q 2 ) > 0 and increases with Q 2. pqcd: understood; Regge theory: not so clear. Classical (soft pomeron): α P (0) 1 σ W 2(αP(0) 1) const. pqcd: BFKL pomeron: α BFKL 1.4 to high and does not explain δ(q 2 ). Donnachie and Landshoff: Two pomerons soft: α s (0) = 1.07 hard: α h (0) = 1.42 slopes: α s(0) = 0.25 GeV 2, and α h (0) 0.1 GeV 2 Donnachie, Ferreira,HGD: Hard pomeron couples to inner part of hadronic structure (r < 0.2 fm) and soft pomeron to outer parts. Explains a host of data.

30 j New look at pomeron by Brower, Polchinski, Strassler and Tan Pomeron in AdS/CFT. Extremely simplified model: Slope depends on holographic variable r= 1/ z a P Erasmo Ferreira, H.G.D. : Phenomenological extension of that model. intercept depends on ζ small z (U) large z (r) t

31 Procedure: Experiment: F 2 (x, Q 2 ) x λ(q2 ) photon wave function: ζ eff = ζ eff (Q 2 ) α P (0)(ζ 2 eff ). Applied to diffractive VM production. Predicts energy dependence of production cross section without free parameter. γ p ρ p 10 3 ρ production cross section (nb) solid 5479 W 0.18 Q 2 = 0 solid W 0.49 Q 2 = 6 GeV 2 Egloff (79) EMC (88) H1 (96) Zeus (98) Zeus (07) J/Ψ photoproduction cross section (nb) 10 2 γ p J/Ψ p Q 2 = 0 Fixed target (82,84) Zeus (02) H1 (13) LHCb (14) Alice LHC (14) Υ photoproduction cross section (nb) 3.62 W γ p Υ p Q 2 = W 1.04 ZEUS(1998) H1(2000) ZEUS(2009) W (GeV) W (GeV) W (GeV) solid line: final result. dotted line: without shrinkage H.G.D., Erasmo Ferreira, Phys.Rev.D95(2015)