Hadronic phenomenology from gauge/string duality

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1 Hadronic phenomenology from gauge/string duality Modern Trends in Field Theory, Joã o Pessoa 09/2006 Nelson R. F. Braga, IF, UFRJ String Theory Strong Interactions Experimental observation ( 1960) of an apparently infinite tower of resonances with mass and angular momenta related by (Regge trajectories): J m 2 α with α 1(GeV ) 2 (Regge slope). 1

2 Scattering amplitudes p 2 p 3 p 1 p 4 p 2 t p 1 p 3 p 4 p 2 + s p 1 p 3 p Mandelstan variables: s = (p 1 + p 2 ) 2, t = (p 2 + p 3 ) 2, u = (p 1 + p 3 ) 2 Veneziano amplitude, postulated in 1968 to reproduce the properties of strong interactions observed at that time (duality s t,....) A(s, t) = Γ( α(s)) Γ( α(t)) Γ( α(s) α(t)) where Γ = Euler gamma function and α(s) = α(0) + α s 2

3 Relativistic bosonic string τ σ Fig. 2: String world-sheet S P olyakov = 1 4πα dτ dσ gg ab a X µ b X µ Excitations show up in representations of Lorentz group with J m 2 α Amplitudes reproduce Veneziano result 3

4 Old fact: many obstacles in relating string theory to hadronic physics New fact: some of them are been removed with the idea of: GAUGE/STRING DUALITIES AdS/CFT correspondence J. Maldacena, Adv. Theor. Math. Phys. 98. Equivalence between string theory in AdS 5 S 5 space and Superconformal gauge theories SU(N) with large N on the corresponding four dimensional boundary (extended supersymmetry N = 4). Important remark by Witten: AdS/CFT is a realization of the Holographic principle: The degrees of freedom of a quantum system with gravity can be represented on the boundary. 4

5 Anti-de Sitter space-time Space of constant negative curvature. Can be seen as a hyperboloid inside a higher dimensional flat space. Poincare coordinates (useful for AdS/CFT) ( 0 z ) ds 2 = R2 z 2 (dz2 + (d x) 2 dt 2 ). The boundary, where the gauge theory lives is at z = 0. (The isometry group of AdS n+1 is isomorphic to the conformal group of n dimensional flat space. (H. Boschi-Filho, N.B., Class. Q. Grav. 2004)). 5

6 Problem: In the AdS/CFT correspondence the gauge theory is conformal (has no energy scale) and supersymmetric. QCD is not conformal. So we need gauge/string dualities where the gauge theory is more similar to QCD. First attempt: Witten 98 Conformal invariance broken by temperature AdS Schwarzschild black hole dual to a non-supersymmetric Yang Mills theory. Glueball masses: Csaki, Ooguri, Oz and Terning, JHEP 99, Ooguri, Robins, Tannenhauser PLB 98,.... Search for EXACT gauge string dualities: Klebanov and A. A. Tseytlin, 2000; Klebanov, Strassler, 2000 Maldacena, Nunes, 2001,... The gauge theory is still supersymmetric (N = 1) the geometries are complicated and yet gauge theory still far from QCD. 6

7 Approximate QCD duals inspired in AdS/CFT: Important result (Polchinski and Strassler PRL 2002) OLD PROBLEM ( 30 years) NEW SOLUTION ( using AdS/CFT): High energy limits of Veneziano amplitude: (1) Regge limit: s, with fixed t : A s α(t) In agreement with experimental results (actually this was the available input at that time!) (2) High energy scattering at fixed angles: s with s/t fixed. Veneziano amplitude: A V en. exp { α sf(θ) } (Soft scattering) Experimental results: A exp. s (4 )/2 (Hard scattering) Reproduced by QCD (Matveev,Muradian,Tavkhelidze;Brodsky,Farrar 1973) 7

8 Idea: reproduce the expected QCD scaling for glueball scattering from string theory using the DUALITY: QCD glueballs AdS dilatons Energy scale: AdS slice as approximately dual to a non conformal theory ds 2 = R2 (z) 2(dz2 + (d x) 2 dt 2 ). with 0 z z max 1/Λ where Λ is the mass of the lightest glueball. 8

9 The 4-dimensional glueballs amplitude shows up as an effective result of a ten dimensional process Scaling A(p) Λ p 4 QCD like scaling! ( = total scaling dimension of in and out states) Solution to the apparent incompatibility between string theory and the scaling of hadronic scattering amplitudes at high energies with fixed angles. (We also found this scaling from gauge string duality using a mapping between AdS states and boundary states based on the idea of holography: H. Boschi-Filho e N. B., PLB 2003) 9

10 What else can we get from such phenomenological holographic duals to QCD inspired in the AdS/CFT correspondence? ( AdS/QCD ) Simple estimate for glueball mass ratios N.B. and Boschi-Filho JHEP 2003 AdS slice ds 2 = R2 z 2 (dz2 + (d x) 2 dt 2 ) + R 2 dω 2 5, with 0 z z max. Free dilaton with vanishing momentum in the S 5 boundary conditions at z = z max Φ(z, x, t) = d 3 k z 2 J 2 (u p z) p=1 (2π) 3 z max w p ( k) J 3 (u p z max ) {a p ( k) e iwp( k)t+i k x + h.c.}. direction and Dirichlet u p = momentum (discrete) associated with z u p = χ 2, p z max ; J 2 (χ 2, p ) = 0 ; w p ( k) = u 2 p + k 2 10

11 On the boundary (z = 0): Scalar glueballs ( states J P C = 0 ++ and their excitations 0 ++, 0 ++, i,... with masses µ i such that the lightest one is related to the size of the slice z max = C µ 1 ; C = constant So u i = χ 2, i C µ 1 The gauge/string correspondence suggests that the glueball masses are proportional to the dilaton discrete modes: u i µ i = constant So the ratios of glueball masses are related to zeros of the Bessel functions µ i µ 1 = χ 2, i χ 2, 1 Important: THIS RATIOS ARE INDEPENDENT OF z max. 11

12 Our estimates compared with SU(3) Lattice e AdS-Schwarzshild (in GeV): 4d Glueball lattice, N = 3 AdS-BH AdS slice ± (input) 1.61 (input) Lattice results: Morningstar and Peardon, PRD 1997; Teper, hep-lat AdS-BH supergravity: Csaki, Ooguri, Oz and Terning, JHEP

13 Glueball masses in QCD 3 : dilatons is AdS 4 Bessel function J 3/2 µ p µ 1 = χ 3/2, p χ 3/2, 1. Our results from AdS slice compared with lattice QCD and AdS-Schwarzschild black hole supergravity (AdS-BH) 3d Glueball lattice, N = 3 lattice, N AdS-BH AdS slice ± ± (input) 4.07 (input) ± ± ± ±

14 The previous results are just for scalar glueballs. How can we calculate masses for states with higher angular momenta? In standard AdS/CFT one would take spin two glueball to be dual to the graviton,.... Spin zero and two glueballs would be degenerate. Spectrum of light baryons and mesons: Brodsky, Teramond PRL QCD states with different angular momenta considered to be dual to bulk states with different effective masses. This way one can find the Regge trajectories associated with a given hadron by considering just one bulk field. 14

15 8 (a) N (2600) 6 4 N (1700) N (1675) N (1650) N (2250) N (2190) N (2220) (GeV 2 ) N (939) (b) N (1535) N (1520) N (1720) N (1680) (2420) 6 4 (1950) (1920) (1910) (1905) (1232) (1930) 2 (1700) (1620) S=3/2 S=1/ A7 0 2 L 4 6 Figure 1: Prediction for the light baryon spectrum for Λ QCD = 0.22GeV. Guy F. de Téramond 15 and Stanley J. Brodsky, Phys. Rev. Lett 94 (2005)

16 We used a similar approach to estimate masses of glueball states with different spins, find their Regge trajectories and compare with Pomerons, H. Boschi- Filho, N. B. and H. L. Carrion, Phys.Rev. D73 (2006) Motivation: Regge trajectories for Pomerons (Landshoff hep-ph/ ) J M 2 (GeV ) (for baryons and mesons J (0.9)M 2 ). These trajectories do not come from spectroscopy data but from cross sections of hadron-hadron scattering ( pp, pp). These cross sections are phenomenologically represented assuming intermediate colorless states with vacuum quantum numbers: the Pomerons. 16

17 Pomerons may be related to glueballs. Recent lattice results are consistent with this interpretation (slope and intercept). H. B. Meyer and M. J. Teper, PLB

18 Massive scalars in the AdS slice with mass µ dual to gauge theory glueball states with spin l related by: (µr) 2 = l(l + 4) We found non linear trajectories, but taking a linear approximation J = α 0 + α M 2. For the Dirichlet case, taking the states J = 2, 4,..., 10 we find a linear fit with α = ( 0.36 ± 0.02 ) GeV 2 ; α 0 = 0.32 ± For Neumann boundary conditions our results are compatible with the Pomeron trajectory. For states with spins J = 2, 4,..., 10 we find α = ( 0.26 ± 0.02 )GeV 2 ; α 0 = 0.80 ± 0.40 Remember that for the pomeron α 0.25 GeV 2 and α

19 Dirichlet lightest 1 st excited 2 nd excited glueballs state state state Table 1: Mass ratios with Dirichlet Boundary conditions Neumann lightest 1 st excited 2 nd excited glueballs state state state Table 2: Mass ratios with Neumann boundary conditions 19

20 10 8 J M 2 (GeV 2 ) Figure 2: Neumann Boundary conditions 20

21 10 8 J M 2 (Gev 2 ) Figure 3: Dirichlet Boundary condition 21

22 Wilson Loops 0 T r exp{ig λ i A i µ (y)dyµ } 0 exp{ T (E(L) 2m) } Behaviour of flux associated with gauge field. Nonconfining Theory (e.g. QED) + + L Potential Energy of q q 1/L Total Energy 2m when L 22

23 Confining Theory q q L Potential Energy of a pair q q L Total Energy when L 23

24 Wilson Loops in the AdS/CFT case S.J.Rey and J. T. Yee; J.M.Maldacena Heavy quark anti-quark pair (stationary configuration) at r = r 1 ( ) on the axis x i x separated by a coordinate variation L. (Now: r = R2 z ) The string connecting the quarks corresponds to the geodesic, reaching a minimum at r = r 0. This value is determined in terms of L from the equations of motion. x q 0 r 0 r L q 24

25 The energy is proportional to the string length, along the geodesic. The metric is singular in r (z = 0 where the gauge theory lives), so E. Removing the (infinite) self-energy m of each quark: m q x q r 0 r L 0 m q q One finds: Coulomb like Potential E = 4π2 R 2 Γ(1/4) 4 L nonconfining. 25

26 Criterium for confinement Y. Kinar, E. Schreiber and J. Sonnenschein, NPB For a space The quantity ds 2 = g tt (r)dt 2 + g x x (r)dx 2 + g rr(r)dr g tt (r) g x x (r) should have a minimum at some r = r 0 where it is non vanishing: g tt (r 0 ) g x x (r 0 ) 0 confinement 26

27 Phenomenological approach to quark anti-quark potential (at zero temperature) using AdS spaces with cut-offs H. Boschi-Filho, N.B. and C. N. Ferreira, PRD ds 2 = ( r2 R 2)( dt2 + d x 2 ) + ( R2 r 2 )dr2 with r 2 r r 1 Static string with endpoints located at r 1. In particular, placing the quarks at r 1 the string energy for L L crit is Coulombian: E C/L while for large distances it has a linear leading behaviour: E DL. 27

28 b c a x +L crit /2 r L crit /2 r = r 2 r = r 1 Figure 4: Schematic representation of geodesics in the Randall Sundrum space. Curve a corresponds to a geodesic with L < L crit, curve b to L = L crit and c to L > L crit. 28

29 Choosing the infrared brane at r 2 = R and identifying the string energy with the Phenomenological Cornell potential for a quark anti-quark pair: E Cornell (L) = 4 a 3 L where a = 0.39 and σ = Gev 2 + σl + const.. we find a = 3C 2 1 R2 /2πα with C 1 = 2π 3/2 /[Γ(1/4)] 2. σ = 1 2πα In this case the effectice AdS radius is R = 1.4 GeV 1 29

30 Figure 5: Energy in GeV as a function of string end-points separation L in GeV 1 for AdS slices with r 1 = nr and r 2 = R. For n the energy behaves as the Cornell potential. 30

31 Heavy quark anti-quark potential at finite temperature from gauge string duality H.Boschi-Filho, N.R.F.Braga and C.N.Ferreira, hep-th/ Thermal effects in gauge/string duality: AdS Schwarzshild black hole dual to finite temperature N = 4 gauge theory ds 2 = ( r2 R 2)( f(r) dt2 + d x 2 ) + ( R2 r 2 ) 1 f(r) dr2 + R 2 d 2 Ω 5, where f(r) = 1 r 4 T /r4, r T = π R 2 T. At T = 0 this is AdS space. This space was sucessifully used to find the viscosity of Quark Gluon Plasma: G. Policastro, D. T. Son and A. O. Starinets PRL 2001; P. Kovtun, D. T. Son and A. O. Starinets, PRL 2005,

32 Static strings in this space were discussed in detail by S. J. Rey, S. Theisen and J. T. Yee; A. Brandhuber, N. Itzhaki, J. Sonnenschein and S. Yankielowicz; This space is not confining. Following our phenomenological approach to reproduce confinement we introduced an infrared cut off brane at r = R. The energy of a static string with endpoints at r separated by coordinate distance x = L depends on the relation between horizon radius (temperature) and the brane position. 32

33 High temperature: r T R Same solution as the case without brane. For small quark anti-quark separations world sheets with minimum area corresponds to U-shaped curves and energy increases with L. E = 1 πα { 1 y 4 r4 T r 4 ( 0 y4 1 1) r 0dy r 0 } (subtracting quark masses, chosen as m q = (1/2πα ) 0 dr ) Note that L is related to r 0 by L(r 0 ) = 2 R2 r 0 1 r4 T r dy (y 4 1) (y 4 r4 T r 4 0 ) For L > L = 0.75R 2 /r T (corresp. to r 0 = r T /0.65 ) world sheet with minimum area are Box-shaped. The energy is constant E = r T π α (The constant depends on the definition of quark masses) 33

34 r 0 r L r = R r = r T Schematic representation of a U-shaped geodesic with minimum r 0 far from the brane and from the horizon. This kind of geodesic appears for small quark anti-quark separation. 34

35 r L r = R r = r T Schematic representation of a Box-shaped geodesic which reaches the horizon but not the brane. This is a typical situation at high temperature (r T > R) and large quark anti-quark separation L. 35

36 r L r = r T r = R Schematic representation of a degenerated U-shaped geodesic with minima along the brane. 36

37 Low temperature: r T < R The string can reach the brane but not the horizon For quark distances L < L(r 0 ) r0 =R U-shaped geodesics with energy as in the high T case. For L > L(r 0 ) r0 =R U-shaped curve with degenerated minima along the brane. E = 1 πα 1 y 4 r4 T R ( 4 y4 1 1) Rdy R (L L(R)) 1 r4 T R 4. For large L leading asymptotic behavior is linear: E L 2πα So there is confinement. Remark: for 0.85R < r T < R there are actually three kinds of geodesics, depending on L but the assymptotic behaviour for large L is the same r4 T R 4. (1)

38 Energy as a function of quark anti-quark distance for different temperatures 38

39 So if T T C = R T /π R 2, for large L, E σ(t ) L with σ(t ) = 1 1 (πrt ) 4 2πα Taking the AdS radius R to have the value considered in the zero temperature case we find T C 230MeV These results are compatible with lattice calculations: P. Petreczky, Eur.Phys.J.C43:51-57,

40 2 F 1 (r,t) [GeV] r [fm] T c 0.91T c 0.94T c 0.98T c 1.05T c 1.50T c 3.00T c Color singlet free energy in quenched QCD, the solid line is the zero temperature, from: Heavy quark potentials and quarkonia binding P. Petreczky, Eur.Phys.J.C43:51-57,2005 F 1 [GeV] T [MeV] r [fm] Color singlet free energy in three flavor QCD, the solid line is the zero temperature, from: Heavy quark potentials and quarkonia binding P. Petreczky, Eur.Phys.J.C43:51-57,