Scattering Amplitudes at Strong Coupling Beyond the Area Paradigm - PowerPoint PPT Presentation

Scattering Amplitudes at Strong Coupling Beyond the Area Paradigm Benjamin Basso ENS Paris Strings 14 Princeton based on work with Amit Sever and Pedro Vieira Wednesday, 25 June, 14

Wilson loops at finite coupling in N=4 SYM [Alday,Gaiotto,Maldacena,Sever,Vieira’10]     1+1d background : flux tube sourced by two parallel null lines bottom&top cap excite the flux tube out of its ground state Sum over all flux-tube eigenstates X C bot ( ψ ) × e − E ( ψ ) τ + ip ( ψ ) σ + im ( ψ ) φ × C top ( ψ ) W = states ψ Wednesday, 25 June, 14

Refinement : the pentagon way [BB,Sever,Vieira’13] vac Valid at any coupling ψ 3 "Y # X e − E i τ i + ip i σ i + im i φ i × = ψ 2 i ψ i P (0 | ψ 1 ) P ( ψ 1 | ψ 2 ) P ( ψ 2 | ψ 3 ) P ( ψ 3 | 0) ψ 1 vac Wednesday, 25 June, 14

Refinement : the pentagon way [BB,Sever,Vieira’13] vac Valid at any coupling ψ 3 "Y # X e − E i τ i + ip i σ i + im i φ i × = ψ 2 i ψ i P (0 | ψ 1 ) P ( ψ 1 | ψ 2 ) P ( ψ 2 | ψ 3 ) P ( ψ 3 | 0) ψ 1 To compute amplitudes we need vac The spectrum of flux-tube states All the pentagon transitions Wednesday, 25 June, 14

Beyond the area paradigm Simplest case : hexagon (n = 6) WL classical minimal area √ √ λ λ 7 288 e √ 2 π A n =6 (1 + O (1 / W n =6 = f 6 λ − λ )) in 144 − AdS 5 quantum [Alday,Gaiotto,Maldacena’09] [Alday,Maldacena,Sever,Vieira’10] Pre-factor 1 . 04 √ 2 τ ) f 6 = ( σ 2 + τ 2 ) 1 / 72 + O ( e − [BB,Sever,Vieira’14] Wednesday, 25 June, 14

The flux-tube eigenstates ψ = N particles state (Adjoint) field insertions along a light-ray : create/annihilate state on the flux tube Spectral data p = p ( u 1 ) + · · · + p ( u N ) rapidity E ( u ) = twist + g 2 . . . p ( u ) = 2 u + g 2 ... can be found using integrability Wednesday, 25 June, 14

Pentagon/OPE series for hexagon Z X d u P a (0 | u ) e − E ( u ) τ + ip ( u ) σ + im φ P a (¯ = u | 0) W hex = a i.e. in collinear limit Lightest states dominate at large What are they? Wednesday, 25 June, 14

Decoupling limit E ( p = 0) mass gluons fermions Scalar mass is exponentially small at strong coupling scalars coupling 2 1 / 4 → p √ λ Γ (5 / 4) λ 1 / 8 e − 4 (1 + O (1 / m = λ )) ⌧ 1 For all heavy flux tube excitations decouple τ � 1 Low energy effective theory : [Alday,Maldacena’07] (relativistic) O(6) sigma model 6 √ λ X 2 = X X 2 4 π ∂ X · ∂ X L e ff = with i = 1 i =1 Wednesday, 25 June, 14

The pentagon/twist operator 1 q � � � � ∂ α y µ ∂ β y ν g S 5 ∂ α x µ ∂ β x ν g AdS S NG = − − det − det µ ν µ ν 2 πα 0 S 5 AdS 5 square pentagon hexagon Wednesday, 25 June, 14

Hexagon as a correlator of twist operators 3 4 3 4 3 5 5 2 corrections from heavy modes 2 = irrelevant in collinear limit 6 1 6 1 6 √ 2 τ ) W 6 = h 0 | φ D ( τ , σ ) φ D (0 , 0) | 0 i + O ( e − Probes the physics of the p σ 2 + τ 2 W O (6) ( z ) z = m O(6) sigma model : Large distance W O (6) = 1 + O ( e − 2 z ) z � 1 Short distance W O (6) = ? z ⌧ 1 Wednesday, 25 June, 14

OPE as form factor expansion See [Cardy,Castro-Alvaredo,Doyon’07] for similar considerations for computing Insert complete basis of states entanglement entropy in integrable QFT 1 − z P cosh θ i X W O (6) = N ! h 0 | φ D | θ 1 , . . . , θ N i h θ 1 , . . . , θ N | φ D | 0 i e i N Pentagon transition = form factor of twist operator P (0 | θ 1 , . . . , θ N ) = h θ 1 , . . . , θ N | φ D | 0 i We found all these transitions so we can plot Normalization which enforces that h 0 | φ D | 0 i = 1 W O (6) → 1 z → ∞ Wednesday, 25 June, 14

Numerical analysis log W 0.25 n max = 2 0.20 n max = 4 n max = 6 0.15 n max = 8 0.10 log H 1 ê z L 6 8 10 12 14 Plot of the truncated OPE/form factor series representation for log W O (6) Wednesday, 25 June, 14

Short distance analysis Short distance OPE (valid for ) z ⌧ 1 φ D ( τ , σ ) φ D (0 , 0) ∼ log (1 /z ) B φ 7 (0 , 0) z A 3-point function Critical exponent A A = 2 ∆ D − ∆ 7 = 2 ∆ 5 / 4 − ∆ 3 / 2 with the scaling dimension of the twist operator φ k ∆ k c = central charge 12( k − 1 ∆ k = c k ) [Knizhnik’87] 2 π ( k − 1) = excess angle for φ k [Lunin,Mathur’00] [Calabrese,Cardy’04] Wednesday, 25 June, 14

Short distance analysis Short distance OPE (valid for ) z ⌧ 1 φ D ( τ , σ ) φ D (0 , 0) ∼ log (1 /z ) B φ 7 (0 , 0) z A 3-point function Critical exponent A A = 1 since in our case c = 5 36 Critical exponent from one-loop anomalous dimensions B B = − 3 2 A = − 1 24 Wednesday, 25 June, 14

Short distance analysis For z ⌧ 1 include subleading RG logs C W O (6) = z 1 / 36 log (1 /z ) 1 / 24 + . . . Constant is fixed in the IR by C W O (6) → 1 when z → ∞ and is thus non perturbative Wednesday, 25 June, 14

Numerical analysis log W + 1 ê 36 log z + 1 ê 24 log a - 0.0110 - 0.0115 - 0.0120 - 0.0125 - 0.0130 a 6 8 10 12 14 running coupling log C = − 0 . 01 α = log (1 /z ) + . . . Wednesday, 25 June, 14

Short distance analysis λ / 4 ) √ (i.e. 1 ⌧ τ ⌧ e For √ 2 1 / 4 z ⌧ 1 λ Γ (5 / 4) λ 1 / 8 e − m ' 4 C W O (6) = z 1 / 36 log (1 /z ) 1 / 24 + . . . p σ 2 + τ 2 z = m controlled by the gluons √ 2 τ ) A n =6 = O ( e − √ √ 7 λ λ 288 e √ 2 π A n =6 (1 + O (1 / W n =6 = f 6 λ − λ )) 144 − Pre-factor 1 . 04 √ 2 τ ) f 6 = ( σ 2 + τ 2 ) 1 / 72 + O ( e − Wednesday, 25 June, 14

Infrared/non-perturbative regime O (6) σ model α 0 expansion 1 / τ 0 1 m √ equivalently λ / 4 z � 1 τ � e Deep (infrared) collinear limit Completely non perturbative Wednesday, 25 June, 14

Cross over O (6) σ model α 0 expansion 1 / τ 0 1 m √ λ / 4 equivalently z ⌧ 1 1 ⌧ τ ⌧ e UV regime of O(6) model : perturbative collinear limit Wednesday, 25 June, 14

Cross over O (6) σ model α 0 expansion 1 / τ 0 1 m here we could match O(6) analysis with string perturbative expansion Wednesday, 25 June, 14

Full stringy pre-factor O (6) σ model α 0 expansion 1 / τ 0 1 m full thing : include all heavy modes gluons , fermions , ... Next Strings maybe .... :) 1 . 04 √ 2 τ ) + O ( e − 2 τ ) f 6 = ( σ 2 + τ 2 ) 1 / 72 + O ( e − Wednesday, 25 June, 14

Conclusions At strong coupling SA develop a non-perturbative regime in the near collinear limit The string expansion breaks down for extremely large α 0 √ values of λ / 4 τ ∼ − log u 2 ∼ e That’s because flux tube mass gap becomes extremely small m One should think in terms of correlators of twist operators This fixes the collinear limit of SA at strong coupling Wednesday, 25 June, 14

Outlook Higher multiplicity (heptagon, ....)? Next-to-MHV amplitudes? Full one-loop pre-factor? One-loop Thermodynamical-Bubble-Ansatz equations? ... and many other questions... Wednesday, 25 June, 14

THANK YOU! Wednesday, 25 June, 14

BACK UP Wednesday, 25 June, 14

Pentagon as twist operator Asympotically a pentagon = 5 quadrants glued together 4 5 4 5 4 = 1 3 φ D 3 1 2 2 excess angle = π twist operator 2 P ( ψ edge 2 | ψ 0 edge 5 ) = h ψ 0 | φ D | ψ i Wednesday, 25 June, 14

Monodromy One can go around the pentagon with 5 mirror rotations θ 5 γ θ 4 γ θ θ + 5 i π θ + 4 i π = = 2 2 This is one more than for a square γ E − → ip − → − E − → − ip − → E Wednesday, 25 June, 14

Hexagon as a correlator of twist operators 3 4 3 4 3 5 5 2 2 = 6 1 6 1 6 p σ 2 + τ 2 distance = W 6 = h 0 | φ D ( τ , σ ) φ D (0 , 0) | 0 i computed in O(6) sigma model Wednesday, 25 June, 14

Hexagon beyond 2pt approximation Z d θ 1 d θ 2 W 6 = 1 + 1 (2 π ) 2 | P (0 | θ 1 , θ 2 ) | 2 e − m τ (cosh θ 1 +cosh θ 2 )+ im σ (sinh θ 1 +sinh θ 2 ) + . . . 2 multi-particle states θ 1 θ 2 θ 3 θ 4 Multi-particle + + = transitions π 2 π 3 π 1 Understood! w 1 1 Y integrand = P ( θ i | θ j ) P ( θ j | θ i ) × rational θ w 2 i<j w 3 Wednesday, 25 June, 14

Higher multiplicity Higher-point amplitudes correspond to higher-points correlators W n = h 0 | φ D ( τ n − 4 , σ n − 4 ) . . . φ D ( τ 1 , σ 1 ) | 0 i Overall short-distance scaling is controlled by OPE ∼ m − ( n − 4) ∆ ( 5 4 )+ ∆ ( n 4 ) φ ϕ φ D . . . φ D | {z } n − 4 ϕ = 2 π × n − 4 with final excess angle 4 This leads to the addition √ √ √ λ ( n − 4)( n − 5) λ 2 π A n + + o ( λ ) W n = e − 48 n Wednesday, 25 June, 14