String/gauge theory duality and QCD M. Kruczenski Purdue University ASU 009
Summary Introduction String theory Gauge/string theory duality. AdS/CFT correspondence. Mesons in AdS/CFT Chiral symmetry breaking Spin chains as a way to see strings in field theory
Introduction String theory ) Quantum field theory: Relativistic theory of point particles. Strong, weak and electromagnetic interactions are described in this way. In particular by gauge theories. Quantum gravity? ) String theory: Relativistic theory of extended objects: Strings Why?
Original motivation: Phenomenological model for hadrons (proton, neutron, pions, rho, etc.) mesons q,,... String picture Quark model q q q Fund. strings ( Susy, 10d, Q.G. ) QCD Large N-limit Strong coupling Effective strings
Regge trajectories M (GeV) 7 6 5 4 3 1 a 3 a 4 0 0 1 3 4 5 6 7 J Simple model of rotating strings gives 5 a 6 6 5 4 3 1 0 1 3 4 5 6 E J improvement m Strings were thought as fundamental, however m
Strings as a fundamental theory Only one fundamental object, strings (open and closed). Different modes of oscillation of the string give rise to the different observed particles. Interactions are the splitting and rejoining of the strings. Tachyons Taking care by supersymmetry Quantum mechanically consistent only in 10 dim. Unified models? Including gravity Too many vacua.
What about hadrons? Instead: bound states of quarks. mesons: qq baryons: qqq Interactions: SU(3); q = ; A = quarks gluons Coupling constant small at large energies (100 GeV) but large at small energies. No expansion parameter. Confinement V=k r (color) electric flux=string? V=- k/r
Idea ( t Hooft) Take large-n limit, q= ; A = N N x N N, g YM N = λ fixed ( t Hooft coupling) 1/N: perturbative parameter. Planar diagrams dominate (sphere) Next: 1/N corrections (torus) + 1/N 4 (-handles) + Looks like a string theory Can be a way to derive a string descriptions of mesons
AdS/CFT correspondence (Maldacena) Gives a precise example of the relation between strings and a gauge theory. (Curiously non-confining). Gauge theory String theory N = 4 SYM SU(N) on R 4 A, i, a Operators w/ conf. dim. IIB on AdS 5 xs 5 radius R String states w/ 1 4 s YM s YM g = g ; R / l = ( g N ) / E = R YM N, λ = g N fixed large small string th. field th.
D-branes (Polchinski) Region of space where strings world-sheets can end. Open strings. Low energy: SU(N) gauge th. Emission of graviton D-branes have mass If N, # of D-branes is large mass large deforms space e.g. D3- brane: [] [6] Suggests an interesting rep. of the large-n limit
AdS/CFT N = 4 SYM II B on AdS 5 xs 5 S 5 : X 1 +X + X 6 = R AdS 5 : Y 1 +Y + -Y 5 -Y 6 =-R How about hadrons and QCD? deform Strings? Escher AdS/CFT Known examples QCD
Other points in AdS/CFT ) Allows gauge theory computations at strong coupling. ) Describes confining gauge theories. Confinement transition in interpreted geometrically. ) Finite temperature quark-gluon plasma is described by a black hole. (e.g. compute viscosity). ) Quantum gravity in AdS is unitary. Usual QM rules apply to gravity!
Mesons (Non-confining case) (w/ D. Mateos, R. Myers, D. Winters) We need quarks (following Karch and Katz) 3+1 bdy z=0 q q D-brane q q bound state=string z ds = = dt + d x r + d z z So, in AdS/CFT, a meson is a string rotating in 5 dim.!
Meson spectrum (N = 4 is conformal Coulomb force) m q Coulomb E = m α / J q Regge (numerical result) J / ( g N) / 1 4 YM The cases J=0, ½, 1 are special, very light, namely tightly bound. (E b ~ m q )
For J=0,1/,1 we can compute the exact spectrum (in t Hooft limit and at strong coupling) scalars (M/M 0 ) = (n+m+1) (n+m+), m 0 1 scalar (M/M 0 ) = (n+m+1) (n+m+), m 1 1 scalar (M/M 0 ) = (n+m+) (n+m+3), m 1 1 scalar (M/M 0 ) = (n+m) (n+m+1), m 1 1 vector (M/M 0 ) = (n+m+1) (n+m+), m 0 1 fermion (M/M 0 ) = (n+m+1) (n+m+), m 0 1 fermion (M/M 0 ) = (n+m+) (n+m+3), m 0 n 0 ; there is a mass gap of order M 0 for m q 0 M L mq = = < < mq for gym N > > 1 R g N 0 YM
Confining case (w/ D. Mateos, R. Myers, D. Winters) Add quarks to Witten s confining bkg. Spectrum is numerical We see U(1) A chiral symmetry geometrically For m q =0 there is a Goldstone boson. (M =0) M m For m q 0 GMOR- relation φ = f q φ ψψ Rot. String (w/ Vaman, Pando-Zayas, Sonnenschein) reproduces improved model : m m
Witten s confining background 3+1 Extra dimension τ. Breaks susy. Two bkgs. (conf. & non conf.) Background metric (not AdS any more): τ U, S 4 D4 branes: 013456789 + D6 branes (quarks)
Chiral symmetry breaking U(1) A iα iα ψ e ψ, ψ e ψ L L R mq = 0 spontaneous mq 0 explicit Here χ-symmetry is geometric, a rotation: R 8 9 D-brane 013456789 D6 ------- ----- m q
Brane embeddings Quark mass When m q =0 there is a massless mode, D-brane rotation
Quark condensate (as a function of quark mass) Creating a condensate costs energy when the quark mass is non-zero
Gell-Mann Oakes Renner relation When m q is non-zero the pion acquires a mass given by: Here we can compute all quantities and it is satisfied. M π is computed using pert. theory. f π is the normalization of φ in the eff. action. was already computed.
We can compute meson spectrum at strong coupling. In the confining case results are similar to QCD, including qualitative features. (Also Sakai-Sugimoto) How close are we to QCD? 5-dim Ideal sit. E In practice E QCD confinement M KK 4-dim QCD quarks gluons mesons glueballs 5-dim M KK 4-dim dim. red.
Can we derive the string picture from the field theory? Study known case: N = 4 SYM Take two scalars X = 1+ i ; Y= 3 + i 4 O = Tr(XX Y..Y X), J 1 X s, J Y s, J 1 +J large Compute 1-loop conformal dimension of O, or equiv. compute energy of a bound state of J 1 particles of type X and J of type Y (but on a three sphere) R 4 S 3 xr E
Large number of ops. (or states). All permutations of Xs and Ys mix so we have to diag. a huge matrix. Nice idea (Minahan-Zarembo). Relate to a phys. system Tr( X X Y X X Y ) operator conf. of spin chain mixing matrix op. on spin chain J j+ 1 j= 1 λ 1 r r H = S S 4 j 4π Ferromagnetic Heisenberg model!
Ground state (s) Tr( X X X X X X ) Tr( Y Y Y Y Y Y ) First excited states k π n i k l = e......, k = ; ( J = J1 + J ) l J λ ε( k) = ( + cos k) 1 k 0 J λn J More generic (low energy) states: Spin waves (BMN)
Other states, e.g. with J 1 =J Spin waves of long wave-length have low energy and are described by an effective action in terms of two angles, : direction in which the spin points. 1 Seff. = J dσ dτ cosθ τ φ λ dσ dτ σθ + θ σ φ 3π J [ ( ) sin ( ) ] Taking J large with /J fixed: classical solutions
According to AdS/CFT there is a string description particle: X(t) string: X(,t) We need S 3 : X 1 +X +X 3 +X 4 = R J 1 J CM: J 1 Rot: J Action: S[ (,t), (,t) ], which, for large J is: (agrees w/ f.t.) 1 λ Seff. = J dσ dτ cos θ τ φ ( σ ) + sin ( σ ) 3π J [ θ θ φ ]
Suggests that (, ) = (, ) namely that S is the position of the string Examples point-like
Strings as bound states Fields create particles: X x, Y y Q.M. : ψ = cos(θ /) exp(i φ / ) x + sin(θ /) exp(- i φ /) y We consider a state with a large number of particles i=1 J each in a state v i = ψ (θ i, φ i ). (Coherent state) Can be thought as created by O = Tr (v 1 v v 3 v n )
x θ = 0 Ordered set of points looks like a string y θ = π/ S Internal space Strings are useful to describe states of a large number of particles (in the large N limit)
Rotation in AdS 5? (Gubser, Klebanov, Polyakov) 1 3 4 Y + Y + Y + Y Y Y = R 5 6 sinh ; [ 3] ρ Ω cosh ρ ; t ds = cosh ρ dt + dρ + sinh ρ dω [ ] 3 = t λ E S + S S π ln, ( ) ( Φ S Φ ) O = Tr, x = z + t + +
Verification using Wilson loops (MK, Makeenko) The anomalous dimensions of twist two operators can also be computed by using the cusp anomaly of light-like Wilson loops (Korchemsky and Marchesini). In AdS/CFT Wilson loops can be computed using surfaces of minimal area in AdS 5 (Maldacena, Rey, Yee) z The result agrees with the rotating string calculation.
Generalization to higher twist operators (MK) 1 1 1 0 1 1 0 1 1 1 E S n + S S λ π ln, ( ) O ( S/ n S / n S / n S/ n Φ Φ ΦK Φ ) = Tr + + + + Also Belitsky, Gorsky, Korchemsky
Strings rotating on AdS 5, in the field theory side are described by operators with large spin. Operators with large spin in the SL() sector Spin chain representation s i non-negative integers. Spin S=s 1 + +s L Conformal dimension E=L+S+anomalous dim.
Belitsky, Korchemsky, Pasechnik described in detail the L=3 case using Bethe Ansatz. It can be generalized to all loops (Beisert, Eden, Staudacher E = S + (n/) f(λ) ln S Large spin means large quantum numbers so one can use a semiclassical approach (coherent states).
Conclusions AdS/CFT provides a unique possibility of analytically understanding the low energy limit of non-abelian gauge theories (confinement). Two results: Computed the masses of quark / anti-quark bound states at strong coupling. Showed a way in which strings directly emerge from the gauge theory.