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Lattice QCD and String Theory Julius Kuti University of California, San Diego Lattice 2005 July 25, 2005 Trinity College, Dublin Confining Force Quark What is this confining fuzz? AntiQuark String in QCD ? String theorists interested in QCD string problem Quenched, but relevant in confinement/string connection large N ADS/CFT On-lattice QCD string excitation spectrum Casimir energy of ground state Goldstone modes Æ collective string variables Effective string theory? Some early developments: Polyakov .. Luscher,Symanzik,Weisz P. Hasenfratz et al. .. Luscher Michael et al., Ford Polchinski and Strominger Gliozzi et al. .. Munster et al. Juge, JK, Morningstar Teper 1977 1980 1980 1981 1990 1991 1996 1997 1997 1998 strings from field theory Wilson loops and QCD string QCD picture of gluon excitation on all scales universal Casimir energy first on-lattice string excitations effective non-critical string (fixes Nambu-Goto) high precision Z(2) free energy two-loop Z(2) interface first comprehensive QCD string spectrum large N This talk: review of recent work on Casimir energy, the excitation spectrum of the Dirichlet string, and the closed string with unit winding (string-soliton) Will not discuss: large N and AdS/CFT connection string breaking Hagedorn temperature K-strings `t Hooft flux quantization Neuberger, Teper, Brower Bali OUTLINE 1. String formation in field theory real time string formation from simulations Z(2) gauge field theory and LW effective string action benchmark checklist 2. Poincare and conformal effective string theory Nambu-Goto string and its problems unless D = 26 Polyakov string and its problems unless D = 26 Poincare invariant and conformal effective string theory any D 3. Ground state Casimir energy D=3 SU(3), SU(2), Z(2), Z(3) D=4 SU(3) 4. Dirichlet string spectrum D=3 Z(2), SU(2) D=4 SU(3) 5. Closed string spectrum (torelon with unit winding) D=4 SU(3) and D=3 Z(2) 6. Conclusions R = 0.675 fm R=6.75 fm D=3 Z(2) lattice gauge theory R=0.675 fm R=6.75 fm quantum string simulation in real time Z(2) lattice gauge theory D=3 Y X φ(x, y, t) is plotted Wilson Surface of 3d Z(2) Gauge Model Similar picture expected in QCD 1 2 β = − ln(tanh β ) Z(2) gauge Ising duality confining phase mass gap 0 gapless surface smooth β R =0.407 ro ug h deconfined β c =0.2216 roughening transition Kosterlitz-Thouless universality class role of skrew dislocations in Wilson surface Loop Expansion Soliton Quantization (string) Effective Schrodinger equation based on fluctuation matrix of string soliton critical region continuum limit (QCD) effective φ4 field theory M = −∇2 + U" (φsoliton ) effective potential long flux limit: spectrum expected to factorize translational zero mode Æ Goldstone spectrum zero energy bound state x y φ(y) ⋅ exp(iqx) approximate here, but exact for torelon π q = n, n = 1, 2,3... L Px = +-1 and Pz = +-1 two symmetry quantum numbers mean field energy eigenvalues and wave functions are determined numerically exact ones from simulations shape and end effects distort the transfer to collective geometric variables ‘classical’ torelon ground state classical solutions suggest to introduce collective coordinates x(x,t) which describe the undulating motion in y-direction: classical soliton φ(x, y, t) = φs (y − ξ(x, t)) + η(x, y − ξ(x, t), t) y x collective variables fluctuation field (quantized) quantum field Path integral can be written in terms of massless x field and massive h field interaction Lagrangian, FP determinants worked out ‘classical’ ground state, fixed end sources flipped at symmetry axis y x ∂ a ξ∂ a ξ∂ y η∂ y φs typical derivative interaction term a labels (x,t) world sheet coordinate indices To get effective string theory, we have to integrate out the massive field h Most important step in deriving correction terms in effective action of Goldstone modes in Z(2) D=3 gauge model: massive scalar η ~ ∂ µ ξ∂ µ ξ Goldstone ξ Goldstone ξ π n << M R n << R ⋅ M / π when q = ∂ a ξ∂ a ξ∂ y η∂ y φs ξ is the D-2 dimensional displacement Boundary operators set to zero in open-closed string duality vector (collective string variables) S LW = σRT + µT + S 0 + S1 + S 2 + S 3 + … T R ⎧ ⎫ 1 ⎪1 ⎪ S0 = d τ ∫ d σ ⎨ ∂ a ξ∂ a ξ ⎬ ∫ ⎪2 ⎪ 2πα ' 0 ⎪ ⎪ ⎩ ⎭ 0 T 1 ∂ µ ξ∂ µ ξ q + M2 2 V ( R ) = σR + µ − π b (D − 2)(1+ ) R 24R π b ∆E = (1+ ) 1 S1 = b ∫ d τ {(∂1ξ∂1ξ )σ=0 + (∂1ξ∂1ξ )σ=R } R R 4 0 T R T R ⎧ ⎫ ⎧1 ⎫ 1 1 ⎪1 ⎪ ⎪ S2 = c2 ∫ d τ ∫ d σ ⎨ (∂ a ξ∂ a ξ )(∂ b ξ∂ b ξ )⎬ S3 = c3 ∫ d τ ∫ d σ ⎪ ⎨ (∂ a ξ∂ b ξ )(∂ a ξ∂ b ξ )⎬ ⎪2 ⎪ ⎪ ⎪ 4 0 4 0 ⎪ ⎪ ⎩ ⎭ ⎪ ⎪ ⎩2 ⎭ 0 0 higher dimensional ops O(1/R3) c term is not independent in D=3 3 Benchmark checklist: - Massless Goldstone modes? - Local derivative expansion for their interactions? from fine structure in the spectrum - Massive excitations? - Breathing modes in effective Lagrangian? - String properties ? Bosonic, NG, rigid, …? OUTLINE 1. String formation in field theory real time string formation from simulations Z(2) gauge field theory and LW effective string action benchmark checklist 2. Poincare and conformal effective string theory Nambu-Goto string and its problems unless D = 26 Polyakov string and its problems unless D = 26 Poincare invariant and conformal effective string theory any D 3. Ground state Casimir energy D=3 SU(3), SU(2), Z(2), Z(3) D=4 SU(3) 4. Dirichlet string spectrum D=3 Z(2), SU(2) D=4 SU(3) 5. Closed string spectrum (torelon with unit winding) D=4 SU(3) and D=3 Z(2) 6. Conclusions Poincare and conformal effective string action One-dimensional string sweeps out two-dimensional world-sheet described by X µ (τ, σ) (t,s) parameters mapped into the string coordinates: (X 0 ( τ, σ), X1 ( τ, σ),…, X d (τ, σ)) D = d + 1 Consistent relativistic quantum theory requires parametrization invariance: action should only depend on embedding in spacetime characterized by the induced metric h ab = ∂ a X µ ∂ b X µ a,b run over (t,s) 1. simplest choice: Nambu-Goto string action SNG = ∫ dτdσLNG , LNG = − M 1 (− det h ab )1/ 2 2πα ' M designates world-sheet symmetries: D-dimensional Poincare group Two-dimensional coordinate (Diff) invariance string tension problems: Light cone quantization which has the correct D-2 oscillators spoils Lorentz invariance outside the critical dimension D=26 Covariant (Virasoro) quantization leads to D-1 oscillators unless D=26 Z(2) vortex, Nielsen-Olesen vortex, and QCD string require Poincare invariance and D-2 oscillators (longitudinal oscillator is not protected as a Goldstone mode of symmetry breaking) 2. Polyakov introduced world-sheet metric γ ab (τ, σ) into the action: SP [X, γ ] = − 1 dτdσ(−γ )1/ 2 γ ab∂ a X µ∂ b X µ 4πα ' M∫ classically SNG and SP are equivalent γ = det γ ab SP has three symmetries: D-dimensional Poincare invariance two-dimensional coordinate (Diff) invariance two-dimensional Weyl invariance new Weyl equivalent metrics correspond to the same embedding in space-time X 'µ (τ, σ) = X µ (τ, σ) γ 'ab ( τ, σ) = exp(2ω( τ, σ)) ⋅ γ ab (τ, σ) ω arbitrary problem : Polyakov quantization contains additional Liouville mode (scalar field) and leads again to D-1 oscillators 3. Polchinski and Strominger: we need a new effective string description For strings emerging from field theory we would like to require D-2 oscillators, Poincare invariance, and dim invariance Goldstone theorem does not protect longitudinal mode from acquiring a mass (breathing mode of Z(2) vortex is an example) PS: to resolve the paradox we could start from path integral of field theory with collective coordinate quantization including measure terms Convert path integral to covarian form with D unconstrained Xm fields S0 = 1 dτ+ dτ−∂ + X µ ∂ − X µ 2πα ' ∫ + determinants The measure derives from the physical motion of the underlying gauge fields and therefore should be built from physical objects, like the induced metric h ab = ∂ a X µ ∂ b X µ same determinant as Polyakov, but built from induced metric: 26 − D SL = dτ+ dτ−∂ + φ∂ −φ eiSL Polyakov determinant in conformal gauge 48π ∫ S'L = S PS = 2 2 26 − D + − ∂ + X ⋅ ∂ − X∂ + X ⋅ ∂ − X d τ d τ 48π ∫ (∂ + X ⋅ ∂ − X) 2 substituting induced conformal gauge metric h +− for ⎤ ∂ +2 X ⋅ ∂ − X∂ + X ⋅ ∂ −2 X 1 + − ⎡ 1 µ d d X X τ τ ∂ ∂ + β + O(R −3 ) ⎥ ⎢ + − µ 2 ∫ 4π (∂ + X ⋅ ∂ − X) ⎣ α '/ 2 ⎦ will fix b=0, c2, and c3 in Luscher-Weisz effective action, there will be higher order corrections with new unknown parameters β = βc ≡ D − 26 12 conform invariant Poincare invariant D-2 oscillators anomaly free PS String-soliton with winding number w along the compact dimension R is the length of the compact direction Energy spectrum π 4π 2 n 2 1 2 E N 2 = σ 2 R 2 ⋅ w 2 − σ ⋅ (D − 2) + 4πσ ⋅ (N + N) + + p + O( ) J. Drummond T 3 R2 R3 F. Maresca N − N = n ⋅ w, winding number w = 1 for torelon JK 2π ⋅ n n=0,1,2,… p= R p is the momentum along compact dimension by left and right movers N+N sum of right and left movers Poincare invariant conformal field theory D-2 oscillator states and anomaly free Dirichlet string open string attached to D0 branes in modern language π 2πN E N = σR ⋅ 1 − (D − 2) + 2 12σR σR 2 highly degenerate same as fixed end NG energy spectrum N=0 ground state (Casimir energy) Arvis eφ Caselle OUTLINE 1. String formation in field theory real time string formation from simulations Z(2) gauge field theory and LW effective string action benchmark checklist 2. Poincare and conformal effective string theory Nambu-Goto string and its problems unless D = 26 Polyakov string and its problems unless D = 26 Poincare invariant and conformal effective string theory any D 3. Ground state Casimir energy D=3 SU(3), SU(2), Z(2), Z(3) D=4 SU(3) 4. Dirichlet string spectrum D=3 Z(2), SU(2) D=4 SU(3) 5. Closed string spectrum (torelon with unit winding) D=4 SU(3) and D=3 Z(2) 6. Conclusions .. Luscher-Weisz Casimir Energy Ceff(r) SU(3) V(r) = σ r + const - π(D-2)/24r .. D b=0 Short distance QCD running F(r) = V’(r) CLW (r ) = asymptotic Casimir energy -> string formation D b=0.08 fm 1 3 r F '(r ) 2 −π(D-2)/24 asymptotic r Æ infinity Evidence for string formation in QCD? Quark loops ? Changing dimension D shows significant difference It is D = 2+1 which is more tantalizing NG LW invoked a boundary operator for D = 3+1 Later LW showed that b=0 is set from matching te Polyakov loop correlator spectrum to open-closed string duality full PS universal subleading should not need b NG ? full PS full PS universal subleading universal subleading SU(3) Luscher Weisz SU(2) Caselle Pepe Rago classical + quantum Z(2) D=3 Juge et al. also Majumdar Caselle et al. global fit zero-point oscillators only Conclusions: 1. Casimir energy does not fully match expected universal behavior 2. End distortions matter, sensitive to D, boundary operators? 3. limited global fit to torelon Casimir energy by Meyer and Teper consistency with universal subleading correction is reported 4. It is difficult to discover the string from Casimir energy Casimir energy of baryon string Takahashi, Suganuma D=4 SU(3) Phys. Rev. D70 (2004) 074506 Y shape is favored, first excitation is reported β = 5.8 and 6.0 163 32 lattice de Forcrand and Jahn Nuclear Physics A 755 (2005) 475 D=3 Z(3) gauge model in dual potts spin representation analytic Casimir term by de Forcrand et al. β = 0.60 483 lattice Y shape configuration is favored by Casimir energy Holland, JK β = 0.59 1003 lattice β = 0.56 1003 lattice D=3 Z(3) gauge model in dual potts spin representation work in progress: space-time picture, baryon string excitations wetting at transition point β = 0.554 1003 lattice β = 0.5512 1003 lattice OUTLINE 1. String formation in field theory real time string formation from simulations Z(2) gauge field theory and LW effective string action benchmark checklist 2. Poincare and conformal effective string theory Nambu-Goto string and its problems unless D = 26 Polyakov string and its problems unless D = 26 Poincare invariant and conformal effective string theory any D 3. Ground state Casimir energy D=3 SU(3), SU(2), Z(2), Z(3) D=4 SU(3) 4. Dirichlet string spectrum D=3 Z(2), SU(2) D=4 SU(3) 5. Closed string spectrum (torelon with unit winding) D=4 SU(3) and D=3 Z(2) 6. Conclusions PS string PS string D=3 SU(2) and Z(2) exhibit similar behavior symbol:circles end distortion PS universal subleading PS Summary of main results on the spectrum of the fixed end Z(2) string first NG term full PS end distortions in field theory first subleading x= σR dimensionless scale variable EN / σ = x 1− D − 22 π + 2π2N 12 x x NG Expand energy gaps for large x R( EN − E0 ) π =N+ a ( N ) b( N ) + 4 +… x2 x First correction to asymptotic spectrum appears to be universal Higher corrections code new physics like string rigidity, etc. Similar expansion for string-soliton with unit winding Data for R < 4 fermi prefers field theory description which incorporates end effects naturally Majumdar ? full PS Juge et al. Juge et al. universal subleading Juge et al. Juge et al. Surprise: data point approach the PS prediction from above Juge et al. universal subleading term suggested the approach from below data follows the full PS curve more closely than expected Juge et al. SU(3) D=4 Opposite fixed color source (antiquark) Three exact quantum numbers characterize gluon excitations: R Λ+ − Λ Æ angular momentum projected along quark-antiquark axis +− +− Fixed color source (quark) S states (Λ =0) Σg P states (Λ =1) Π D states (Λ =2) ∆ . . . +− CP Angular momentum with chirality Chirality, or reflection symmetry for Λ = 0 g (gerade) CP even u (ungerade) CP odd +− Juge, JK, Morningstar SU(3) Dirichlet spectrum with fine structure PRL 90 (2003) 161601 Gluon excitations are projected out with generalized Wilson loop operators on time sclices the spatial straight line is replaced by linear combinations of twisted paths effective mass plot analysis from large correlation matrix Multipole states adiabatically evolve into string states with expected level ordering in large R limit Σu+ , Π u' , ∆ u Σu− Short distance multipole expansion: LW C~1 string? ΠU Bali an Pineda H Cb = H gauge + Q⋅Q ={ g 2Q ⋅ Q − g 3 (Q × Q)a ∫ d 3 rAa (r,t)J(r,t) 4π | r 1 − r 2 | 4 singlet 3 1 + octet 6 − OUTLINE 1. String formation in field theory real time string formation from simulations Z(2) gauge field theory and LW effective string action benchmark checklist 2. Poincare and conformal effective string theory Nambu-Goto string and its problems unless D = 26 Polyakov string and its problems unless D = 26 Poincare invariant and conformal effective string theory any D 3. Ground state Casimir energy D=3 SU(3), SU(2), Z(2), Z(3) D=4 SU(3) 4. Dirichlet string spectrum D=3 Z(2), SU(2) D=4 SU(3) 5. Closed string spectrum (torelon with unit winding) D=4 SU(3) and D=3 Z(2) 6. Conclusions universal subleading Juge et al. full PS no end effects universal subleading Juge et al. full PS no end effects full PS F. Maresca and M. Peardon Trinity, 2004 Typical effective mass plots show good plateaus universal subleading 15 basic torelon operators translated and fuzzed in large correlation matrices Perfect string degeneracies E12 = E02 + 4πσ + p32 p3 = E12 = E02 + 8πσ 2π L Expected string behavior more lattice systematics is needed required technology is in place E = E + 8πσ + p 2 1 2 0 4π p3 = L 2 3 E12 = E02 + 12πσ + p32 p3 = 2π L Conclusions 1. Poincare invariant conformal effective string provides the right framework to analyze the lattice QCD simulations 2. Fine structure in spectrum tests the string properties 3. End effects in Casimir energy and Dirichlet spectrum remain problematic for string theory interpretation 4. Precocious approach to Poincare string spectrum more accurate data is needed to test higher terms 5. It remains a challenge for String Theory to explain the Poincare invariant and conformal effective QCD string