Gauge/String Duality and Quark Anti-Quark Potential Nelson R. F. Braga, Universidade Federal do Rio de Janeiro Summary Historical facts relating String theory to Strong interactions AdS/CFT, gauge string dualities Approximate dualities: AdS/QCD, glueball masses AdS/QCD, Wilson Loops and quark anti-quark potential Collaborations with Henrique Boschi-Filho, Hector L. Carrion, Carlos Alfonso Bayona and Cristine N. Ferreira Presented at Physics Beyond the Standard Model, Rio 12/06 FIRST RIO-SACLAY MEETING 1
String Theory Strong Interactions Hadronic spectrum 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) (this holds for the lowest energy states corresponding to a given spin J). 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 4 +... 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 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 4
Old fact: many obstacles in relating string theory to hadronic physics (α in string theory is much lower that in hadronic Regge trajectories, so a hadron can not be associated directly to a fundamental string.) New fact: some of them can be removed with the idea of: GAUGE/STRING DUALITIES AdS/CFT correspondence J. Maldacena, Adv. Theor. Math. Phys. 98. Equivalence between string theory in some space and a kind of gauge theory four dimensional flat space. (Very short version) ( 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. ) (Not so short (although not yet complete) version) 5
Very simplified idea about AdS/CFT At low energies only massless excitations show up in string theory. Effective low energy field theory. S = Some action that involves the fields φ, g µν, A µ Horowitz, Strominger, NPB 1991, studied solutions (field configurations) with with horizons in the form of an extended object: black p-branes. In particular: the geometry associated with N coincident D3 branes in 10 dimensions is. ds 2 ( R 4 = 4πgN ) = (1 + R4 r 4 ) 1/2 ( dt 2 + d x 2 ) + (1 + R4 r 4 )1/2 (dr 2 + r 2 dω 2 5 ) 6
Asymptotic form of this space 1) r >> R (Minkowski space) (ds 2 ) = dt 2 + d x 2 + dr 2 + r 2 dω 2 5 2) r << R (AdS 5 S 5 space ) (AdS = Anti-de Sitter space), ds 2 = R2 z 2 (dz2 + (d x) 2 dt 2 ) + R 2 dω 2 5. (changing r to z = R 2 /r) 7
On the other side considering a theory of closed strings that include open strings ending on the D-branes (Type II) and taking the limit when the distance between branes goes to zero (before taking the low energy limit): SU(N) gauge theory (Yang Mills) supersymmetric N = 4 on the branes (3+1 dimensional space). Outside the branes: 10 dimensional Minkowski space. The extended supersymmetry comes from the reduction to four dimensions of the effective ten dimensional string theory. Maldacena : Two different descriptions of the same physical system AdS/CFT correspondence 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. More strong motivation to Quantum Graviy 8
Figure 1: D-branes with open strings 9
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)). 10
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. 11
Approximate QCD duals inspired in AdS/CFT ( AdS/QCD ) 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) 12
Idea: reproduce the expected QCD scaling for (hypothetical) 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. 13
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) 14
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 ), with 0 z z max. Ignoring S 5 directions (zero momentum). Free dilaton (scalar field) with and Dirichlet boundary conditions at z = z max. General form of solution (J = Bessel function) : Φ(z, x, t) = p=1... z 2 J 2 (u p z) {Plane waves in flat directions x, t} N ormalization. u p = χ 2, p z max (momentum associated with z); J 2 (χ 2, p ) = 0 15
On the boundary (z = 0): Scalar glueballs ( states J P C = 0 ++ and their excitations 0 ++, 0 ++,... 0 ++ i,... with masses µ i. 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. (size of the slice) 16
Our estimates compared with SU(3) Lattice e AdS-Schwarzshild (in GeV): 4d Glueball lattice, N = 3 AdS-BH AdS slice 0 ++ 1.61 ± 0.15 1.61 (input) 1.61 (input) 0 ++ 2.8 2.38 2.64 0 ++ - 3.11 3.64 0 ++ - 3.82 4.64 0 ++ - 4.52 5.63 0 ++ - 5.21 6.62 Lattice results: Morningstar and Peardon, PRD 1997; Teper, hep-lat 9711011 AdS-BH supergravity: Csaki, Ooguri, Oz and Terning, JHEP 1999. 17
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 0 ++ 4.329 ± 0.041 4.065 ± 0.055 4.07 (input) 4.07 (input) 0 ++ 6.52 ± 0.09 6.18 ± 0.13 7.02 7.00 0 ++ 8.23 ± 0.17 7.99 ± 0.22 9.92 9.88 0 ++ - - 12.80 12.74 0 ++ - - 15.67 15.60 0 ++ - - 18.54 18.45 18
How can we calculate masses for states with higher angular momenta? The previous results are just for scalar glueballs. QCD states with different angular spins can be taken as dual to bulk states with different effective masses. We used this approach to estimate Regge trajectories for glueball states. and compare with Pomerons, H. Boschi-Filho, N.B. and H.L. Carrion, PRD 2006. Regge trajectories for Pomerons (Landshoff hep-ph/0108156) J α 0 + α M 2 1.08 + 0.25 M 2 (GeV ) (for baryons and mesons J 0.5 + (0.9)M 2 ). Using Neumann boundary conditions and states with spins J = 2, 4,..., 10 we found a compatible result: α = ( 0.26 ± 0.02 )GeV 2 ; α 0 = 0.80 ± 0.40 19
Wilson Loops and the energy of a q q configuration 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 20
Confining Theory q q L Potential Energy of a pair q q L Total Energy when L 21
String dual to a Wilson Loops in the AdS/CFT case S.J.Rey and J. T. Yee; J.M.Maldacena 1998. 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 22
The energy is proportional to the string world sheet area. 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 0 r 0 r L m q q one finds: E = C 1 L Coulomb like Potential nonconfining (C = 4π2 R 2 Γ(1/4) 4 ). 23
What kind of geometry would lead to a confining gauge dual? Criterium for confinement Y. Kinar, E. Schreiber and J. Sonnenschein, NPB 2000. For a space like ds 2 = g tt (r)dt 2 + g x x (r)dx 2 + g rr(r)dr 2 +... The quantity: 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 24
Phenomenological approach to quark anti-quark potential (at zero temperature) using AdS space with cut-offs H. Boschi-Filho, N.B. and C. N. Ferreira, PRD 2006. 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. 25
b c a x +L crit /2 r L crit /2 r = r 2 r = r 1 Figure 2: Geodesics in an AdS slice. Curve a : geodesic with L < L crit, curve b: L = L crit and c: L > L crit. 26
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 = 0.182 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 27
Figure 3: Energy as a function of string end-points separation L in GeV 1 for AdS slices with quarks at r 1 = nr. For n the energy behaves as the Cornell potential. 28
Free energy of a heavy quark anti-quark pair at finite temperature from gauge string duality H.Boschi-Filho, N.R.F.Braga and C.N.Ferreira, hep-th/0607038. 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,.... 29
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; 1998. 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 position of the cut off. 30
High temperature: r T R Same solution as the case without brane. For large L the world sheet with minimum area has strigh lines and the energy is constant. No confinement!!!!!! 31
r 0 r L r = R r = r T Schematic representation of a geodesic with one minimum r 0. These geodesics appears for small quark anti-quark separation L. 32
r L r = R r = r T Geodesic for high temperature and large quark anti-quark separation L: it reaches the horizon but not the cut off. 33
r L r = r T r = R Geodesic for low temperature and large quark anti-quark separation L: it reaches the cut off but not the horizon. 34
Low temperature: r T < R The string can reach the cut off but not the horizon For large L leading behavior is linear: E L 2πα Confinement! So, if T T C = R T /π R 2, for large L, E σ(t ) L with 1 r4 T R 4. (1) 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 show a decrease in the string tension as it happens lattice calculations: 35
Energy as a function of quark anti-quark distance for different temperatures 36
2 F 1 (r,t) [GeV] 1.5 1 0.5 0-0.5 r [fm] -1 0 0.5 1 1.5 2 2.5 3 P. Petreczky, Eur.Phys.J.C43:51-57,2005. 0.87T 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 37
Conclusions Our model reproduces the transition from a confined to a deconfined phase. Problem: the corrections to the string tension for low temperatures should be like: σ(t ) σ(0) T 2 In our model the corrections are T 4 Different spaces, depending on the temperature: Witten Adv.Theor.Math.Phys. 1998 (based on S. W. Hawking and D. N. Page, Commun. Math. Phys. 1983). High temperatures: AdS Schwarzshild space Low temperatures: Thermal AdS space 38
What happens when there is a cut off in the space? C. P. Herzog, A holographic prediction of the deconfinement temperature, arxiv:hep-th/0608151. Thermal AdS is stable for low temperatures and AdS Schwarzshild for high temperatures. Problem: (it seems that) thermal AdS space does not lead to any corrections in the string tension! We are presently trying to find the corrections to the string tension at low temperatures... 39