Scattering amplitudes at strong coupling

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 W= Wednesday, 25 June, 14 X states Cbot ( ) ⇥ e E( )⌧ +ip( ) +im( ) ⇥ Ctop ( ) Refinement : the pentagon way [BB,Sever,Vieira’13] c a v Valid at any coupling = c a v Wednesday, 25 June, 14 " X Y i e i P (0| Ei ⌧i +ipi i +imi i # ⇥ )P ( | )P ( | )P ( |0) Refinement : the pentagon way [BB,Sever,Vieira’13] c a v Valid at any coupling = c a v " X Y i e Ei ⌧i +ipi i +imi i P (0| ⇥ )P ( | )P ( | )P ( |0) To compute amplitudes we need The spectrum of flux-tube states All the pentagon transitions Wednesday, 25 June, 14 i # Beyond the area paradigm Simplest case : hexagon (n = 6) WL classical Wn=6 = f6 288 e 144 p 2⇡ An=6 (1 + O(1/ )) p minimal area Figure 1: The polygon is specified at the AdS boundary by the positions o in k by k = x − These positions are related to an ordered sequence of momenta i i i dimensional a minimal surface streches in the AdS bulk and ends on the pol at the boundary AdS5 quantum Pre-factor p The classical sigma model and Hitchin equ [Alday,Gaiotto,Maldacena’09] [Alday,Maldacena,Sever,Vieira’10] The classical AdS5 sigma model is integrable This can be shown by exhib rameter family of flat connections For our problem, it will be convenient one parameter family in a special way which will simplify its asymptotic b worldsheet In fact, to make this choice we will make use of the Virasoro theory This has been explained in detail in previous papers [22, 23, 24, 25, of repeating the whole discussion, we will present a slightly more abstract version here p 1.04 2⌧ f6 = + O(e ) and Hitchin equations ( + ⌧ )1/72 2.1 General integrable theories [BB,Sever,Vieira’14] Wednesday, 25 June, 14 Let us assume that we have a coset space G/H Let us assume that the has a Z2 symmetry that ensures integrability In other words, imagine that has the decomposition G = H + K so that H is left invariant under the a generator while elements in K are sent to minus themselves We then write currents J = g −1 dg This is a flat current dJ + J ∧ J = We can decompos The flux-tube eigenstates = N particles state (Adjoint) field insertions along a light-ray : create/annihilate state on the flux tube Spectral data p = p(u1 ) + · · · + p(uN ) rapidity E(u) = twist + g p(u) = 2u + g can be found using integrability Wednesday, 25 June, 14 Pentagon/OPE series for hexagon Whex = = XZ du P a (0|u) e E(u)⌧ +ip(u) +im a P a (¯ u|0) i.e in collinear limit Lightest states dominate at large What are they? Wednesday, 25 June, 14 mass Decoupling limit E(p = 0) gluons fermions Scalar mass is exponentially small at strong coupling scalars coupling For ⌧ ! 1/4 m= (5/4) 1/8 e p (1 + O(1/ Wednesday, 25 June, 14 )) ⌧ 1 all heavy flux tube excitations decouple Low energy effective theory : (relativistic) O(6) sigma model Le↵ = p p 4⇡ @X · @X with [Alday,Maldacena’07] X2 = X i=1 Xi2 = The pentagon/twist operator SNG = S 2⇡↵0 q AdS det @↵ xµ @ x⌫ gµ⌫ S5 det @↵ y µ @ y ⌫ gµ⌫ AdS5 square Wednesday, 25 June, 14 pentagon hexagon Hexagon as a correlator of twist operators 3 = corrections from heavy modes irrelevant in collinear limit 6 W6 = h0| D (⌧, ) D (0, 0) |0i + O(e Probes the physics of the O(6) sigma model : WO(6) (z) Large distance z Short distance z⌧1 Wednesday, 25 June, 14 2⌧ p z=m WO(6) = + O(e WO(6) = ? p 2z ) ) + ⌧2 Numerical analysis log W + 1ê36 log z + 1ê24 log a -0.0110 -0.0115 -0.0120 -0.0125 -0.0130 log C = 10 0.01 12 14 a running coupling ↵ = log (1/z) + Wednesday, 25 June, 14 Short distance analysis For z ⌧ (i.e ⌧ ⌧ ⌧ e WO(6) = p /4 C z 1/36 log (1/z) 1/24 ) 1/4 m' (5/4) p z=m + 1/8 An=6 = O(e Wn=6 = f6 Pre-factor Wednesday, 25 June, 14 288 p e 144 2⌧ ) p 2⇡ An=6 (1 + O(1/ )) p 1.04 f6 = + O(e ( + ⌧ )1/72 p 2⌧ ) e + ⌧2 controlled by the gluons p p Infrared/non-perturbative regime O(6) ↵0 expansion m z model equivalently 1/⌧ ⌧ e p /4 Deep (infrared) collinear limit Completely non perturbative Wednesday, 25 June, 14 Cross over O(6) model ↵0 expansion m z⌧1 1/⌧ equivalently 1⌧⌧ ⌧e UV regime of O(6) model : perturbative collinear limit Wednesday, 25 June, 14 p /4 Cross over O(6) model ↵0 expansion m here we could match O(6) analysis with string perturbative expansion Wednesday, 25 June, 14 1/⌧ Full stringy pre-factor O(6) model ↵0 expansion m 1/⌧ full thing : include all heavy modes gluons, fermions, 1.04 f6 = + O(e ( + ⌧ )1/72 Wednesday, 25 June, 14 p 2⌧ ) + O(e 2⌧ Next Strings maybe :) ) Conclusions At strong coupling SA develop a non-perturbative regime in the near collinear limit The string ↵ expansion breaks down for extremely plarge values of ⌧ ⇠ log u2 ⇠ e /4 That’s because flux tube mass gap m becomes extremely small 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 = quadrants glued together 4 = D 2 ⇡ excess angle = P( Wednesday, 25 June, 14 edge | twist operator edge ) =h | D| i Monodromy One can go around the pentagon with mirror rotations ✓5 = ⇡ ✓ + 5i ✓4 = ⇡ ✓ + 4i This is one more than for a square E ! ip ! Wednesday, 25 June, 14 E ! ip ! E ✓ Hexagon as a correlator of twist operators 3 = 6 distance = p + ⌧2 W6 = h0| D (⌧, ) D (0, 0) |0i computed in O(6) sigma model Wednesday, 25 June, 14 Hexagon beyond 2pt approximation W6 = + Z d✓1 d✓2 |P (0|✓ , ✓ )| e 2 (2⇡) m⌧ (cosh ✓1 +cosh ✓2 )+im (sinh ✓1 +sinh ✓2 ) + multi-particle states ✓1 ✓2 ✓3 ✓4 Multi-particle transitions = ⇡1 + ⇡2 + ⇡3 Understood! integrand = Y i