The AdS/CFT S-matrix Esperanza L opez sica Te Instituto de F - PowerPoint PPT Presentation

The AdS/CFT S-matrix Esperanza L´ opez ısica Te´ Instituto de F´ orica UAM/CSIC, Madrid R. Hernandez, EL hep-th/0603204 N. Beisert, R.Hernandez, EL hep-th/0609044 The AdS/CFT S-matrix – p.

Outline • Introduction • The S-matrix, integrability and symmetries • Quantum corrections at strong coupling • A crossing symmetric phase • Matching with small coupling • Conclusions The AdS/CFT S-matrix – p.

AdS/CFT Correspondence N = 4 SU ( N ) YM ⇔ Type IIB strings on AdS 5 × S 5 Maldacena 1 /N ⇔ g st λ = R 4 /α ′ 2 λ = g 2 Y M N ⇔ ⇔ gauge th. operators ( ∆ ) string spectrum ( E ) Suppression of string loops ⇔ large N limit • Strong/weak coupling duality: E = E ( 1 ∆ = ∆( λ ) λ small, λ ) λ large √ • String sigma model is very involved The AdS/CFT S-matrix – p.

How to bridge from small to large λ ? • BPS quantities Supergravity approximation • AdS/CFT at large quantum numbers Polyakov Op. with a large R-charge ⇔ Strings on pp-waves Berenstein, Maldacena, Nastase Long operators: Tr( φ 1 φ 2 · · · φ J ) ⇔ Semiclassical strings ◦ Controlled quantum corrections: ∆ − ∆ 0 small ∆ 0 ◦ Operator mixing The AdS/CFT S-matrix – p.

How to bridge from small to large λ ? • Integrability AdS 5 × S 5 classical Integrable structures in perturbative N =4 string is integrable Lipatov; Minahan, Zarembo Bena, Polchinski, Roiban Beisert, Staudacher Hyphothesis: Integrability holds for any λ The AdS/CFT S-matrix – p.

N =4 Yang-Mills and spin chains Dilatation operator: D O = ∆ O At large N, enought with O single trace Equivalent problem: Spin chain dynamics O =Tr( XY Y XY · · · ) X = ↑ Y = ↓ D : spin chain integrable Hamiltonian • D XY, 1 − loop : Heisenberg ferro. spin chain • Long range chain: Interaction range ⇔ Loop order • Dynamical chain: XY Z → ψ 1 ψ 2 The AdS/CFT S-matrix – p.

Integrability and asymptotic Bethe ansatz Integrability ⇔ Factorized scattering p q S(p,q) Central object: 2 → 2 scattering matrix | p � = P J 1 e ilp |↑ · · · ↑↓↑ · · · ↑� l = l Spectrum: periodicity conditions on wavefunctions e ip j J = � k � = j S ( p j , p k ) Asymptotic Bethe ansatz Staudacher • Infinite chain, asymptotic states → S-matrix • Periodicity conditions • Spectrum accurate to order λ J The AdS/CFT S-matrix – p.

The AdS/CFT S-matrix Gauge th. Strings S 5 Tr X J →∞ = Vacuum J →∞ = · · · ↑↑ · · · ↑ · · · ( λ fixed) Excitations φ i , ∂ µ X, ψ k 8b + 8f BMN psu (2 , 2 | 4) Symm. algebra psu (2 | 2) 2 × R Residual symm. psu (2 | 2) 2 × R 3 Enlarged symm. ( introduce momentum | p � ) Beisert S N =4 = S 0 ˆ S su (2 | 2) ˆ S su (2 | 2) = ¯ S 0 ˆ S ′ − → S su (2 | 2) su (2 | 2) ˆ ¯ S su (2 | 2) : uniquely fixed flavour structure S 0 , S 0 : scalar factors The AdS/CFT S-matrix – p.

The dressing phase S 0 ( p, q ; λ ) = e iθ ( p,q ; λ ) where Beisert, Klose ∞ ∞ √ � � � � q r ( p ) q s ( q ) − q r ( q ) q s ( p ) θ = λ c r,s ( λ ) r =2 s>r q r : tower of conserved charges ( q 1 ( p ) = p , q 2 ( p ) ∼ E − J ) • Strong coupling r,s + 1 c r,s ( λ ) = c (0) c (1) √ r,s + · · · λ c (0) 1 Classical strings → r,s = 2 π δ r +1 ,s Arutyunov, Frolov, Staudacher • Small coupling: θ = 0 up to three-loops The AdS/CFT S-matrix – p.

Quantum corrections at strong λ � 1 √ J = J � √ √ E st = λ ǫ cl ( J ) + δE ( J ) + O , λ λ δE : sum over fluctuations around the classical solution δE = 1 � ( ω B,n − ω F,n ) 2 Bethe equations: e ip j J = � k � = j S ( p j , p k ) − → spectrum • E cl : thermodynamic limit of BE ( J, #excitations → ∞ ) • Two souces of contribution to δE ◦ Finite size corrections: 1 J → c (1) ◦ Quantum correction to the S-matrix: 1 λ ← √ r,s The AdS/CFT S-matrix – p. 1

Circular strings 2-spin rigid circular strings on R t × S 3 or AdS 3 × S 1 Ji Frolov, Tseytlin The frequency sum can be divide in two pieces at large J 1 2 ( ω B,n − ω F,n ) → e 1 ( n ) , e 2 ( n/ J ) Beisert, Tseytlin; Schäfer-Nameki • Fluctuations with finite mode number n δE 1 = � e 1 ( n ) : finite size correction, O ( 1 J ) ( p = n J → 0 as J → ∞ , n fixed) √ • Fluctuations with finite z = n J = λ p : quantum correction, O ( 1 � δE 2 = J dz e 2 ( z ) λ ) √ The AdS/CFT S-matrix – p. 1

Quantum corrections to the dressing phase r,s = ( − 1) r + s − 1 2( r − 1)( s − 1) c (1) ( r + s − 2)( s − r ) π Hernandez, EL Tests: • 2-spin circular strings On S 3 : checked up to 1 / J 101 !! δE 2 = − m 6 3 J 5 + m 8 3 J 7 − 49 m 10 120 J 9 + 2 m 12 5 J 11 − . . . On AdS 3 × S 1 : up to 1 / J 15 δE 2 = ( m − k ) 3 m 3 1 − 3 k 2 − 8 km + 75 k 4 − 455 k 3 m +679 k 2 m 2 − 153 km 3 +29 m 4 » − · 3 J 5 2 J 2 40 J 4 • 3-spin circular strings Freyhult, Kristjansen • Universality Gromov, Viera The AdS/CFT S-matrix – p. 1

Crossing symmetry ¯ φ = In relativistic integrable QFT φ • S-matrix determined by symmetries up to a global phase • The global phase can be fixed by crossing symmetry AdS/CFT dispersion relation does not have relativistic inv. √ λ � 1 + 16 g 2 sin 2 ( 1 E ( p ) = ± 2 p ) , g = 4 π Beisert But still admits particle/hole interpretation Hyphothesis: Crossing symmetry holds for AdS 5 × S 5 strings Janik The AdS/CFT S-matrix – p. 1

Implementation of crossing Janik x + x + + 1 x + − x − − 1 x − = i x − = e ip , g − → torus • Period ω 1 : p → p + 2 π • Crossing symmetry ( x ± → 1 /x ± ): half period ω 2 � 2 � x − x − 1 − 1 /x − 1 x − 1 − x + ≡ h 2 2 2 2 ( S 0 ) 12 ( S 0 ) ¯ 12 = 12 x + x + 1 − x + 1 − 1 /x + 1 x − 2 2 2 1 → ¯ Double crossing: 1 → ¯ ¯ 1=1 12 /h 12 ) 2 � = ( S 0 ) 12 : non-trivial monodromy ( S 0 ) ¯ 12 = ( h ¯ ¯ Define: θ = θ odd + θ even , S 0 = e iθ = log h 12 θ odd 12 + θ odd θ even + θ even , = log h 12 h ¯ ¯ ¯ 12 12 12 12 h ¯ 12 The AdS/CFT S-matrix – p. 1

A crossing symmetric phase n θ ( n ) θ ( n ) � − � g 1 c ( n ) rs ( q r 1 q s 2 − q r 2 q s 1 ) θ 12 = 12 , 12 = r<s Γ[ 1 Γ[ 1 1) r + s − 1)( r − 1)( s − 1) B n 2 ( s + r + n − 3)] 2 ( s − r + n − 1)] rs = (( − c ( n ) 2 cos( 1 Γ[ 1 Γ[ 1 2 πn )Γ[ n − 1]Γ[ n +1] 2 ( s + r − n +1)] 2 ( s − r − n +3)] Beisert, Hernandez, EL • Even crossing θ even = � g 1 − 2 n θ (2 n ) � � c (0) rs = 2 δ r +1 ,s • Odd crossing c (1) rs = ( − 1) r + s − 1 2( r − 1)( s − 1) θ odd = θ (1) ( r + s − 2)( s − r ) π Odd Bernoulli numbers: B 1 = − 1 2 , B n> 1 = 0 The AdS/CFT S-matrix – p. 1

Problems at small coupling ( c ( n ) • c rs ( g ) has a finite expansion in 1 /g rs =0 , n ≥ s − r +3) • At small coupling q r → g r − 1 q r Regular extrapolation to small g θ even = O ( g 2 ) , θ odd = O ( g 3 ) On the gauge theory side • Trivial phase up to 3-loops: θ = O ( g 6 ) • Analytical in λ ∼ g 2 Worsens the 3-loop discrepancy of θ (0) = O ( g 4 ) The AdS/CFT S-matrix – p. 1

A homogeneuos solution Crossing determines the dressing phase up to θ hom + θ hom = 0 ¯ 12 12 Using c ( n ) 2 πn ) = − 2Γ[ n + 1] ζ ( n ) B n rs ∼ cos( 1 ( − 2 π ) n Γ[ 1 Γ[ 1 1) r + s )( r − 1)( s − 1) ζ ( n ) 2 ( s + r + n − 3)] 2 ( s − r + n − 1)] rs = (1 − ( − c ( n ) ( − 2 π ) n Γ[ n − 1] Γ[ 1 Γ[ 1 2 ( s + r − n +1)] 2 ( s − r − n +3)] θ hom = � n> 1 g − 2 n θ (2 n +1) c hom rs ( g ) does not have a finite expansion − → → adding θ hom will alter the small coupling behaviour − The AdS/CFT S-matrix – p. 1

Connecting to small coupling n ≥ 0 c ( n ) rs g 1 − n c rs ( g ) = � 2 c rs ) − Analytical prolongation at small g ( c rs → g r + s Beisert, Eden, − n ≥ 1 c ( n ) g r + s + n − 1 c rs ( g ) = − � Staudacher rs − c ( n ) ∼ cos( 1 2 πn ) ζ (1+ n ) rs • c ( − n ) = 0 for n> 0 odd: expansion in g 2 rs c ( − 2) • First contribution at O ( g 6 ) : = 4 ζ (3) 23 Matches 4-loop gauge th. calculations! ◦ 4-gluon amplitude → Tr XD S X Bern, Dixon, Kosover, Smirnov ◦ Dilatation op. in SU(2) sector Beisert, Roiban The AdS/CFT S-matrix – p. 1

Analytical structure of the phase Elementary ex. with finite p at strong coupling � 1 + 16 g 2 sin 2 ( 1 2 p ) ∼ g , ∆ ϕ ∼ p E = Classical string solutions: giant magnons Hofman, Maldacena • S cl 0 has branch cuts at p 1 = ± p 2 : condensate of double poles Beisert, Hernandez, Lopez • 2d: on-shell 2-particle exchange → double poles • Magnons can form stable boundstates Dorey S 0 has double poles at 2-magnon boundstate exchange Dorey, Hofman, Maldacena The AdS/CFT S-matrix – p. 1

Twist-two operators: Tr XD S X ∆ = S + f ( g ) log S + O ( S 0 ) , S → ∞ � 73 f ( g ) = 8 g 2 − 8 3 π 2 g 4 + 88 630 π 6 + 4 ζ (3) 2 � g 8 + · · · 45 π 4 g 6 − 16 At strong coupling: folded string rotating in AdS 5 f ( g ) = 4 g − 3 log 2 + O ( 1 g ) π Integral equation for any g : f = 16 g 2 σ (0) Beisert, Eden, Staudacher � ∞ t � � dt ′ K (2 gt, 2 gt ′ ) σ ( t ′ ) K (2 gt, 0) − 4 g 2 σ ( t )= e − 1 0 Numerical and analytical check: smooth interpolation Bena, Benvenuti, Klebanov, Scardicchio Kotikov, Lipatov The AdS/CFT S-matrix – p. 2