Gluon scattering amplitudes/Wilson loops duality in gauge theories - PowerPoint PPT Presentation

Gluon scattering amplitudes/Wilson loops duality in gauge theories Gregory Korchemsky Université Paris XI, LPT, Orsay Based on work in collaboration with James Drummond, Johannes Henn, and Emery Sokatchev (LAPTH, Annecy) arXiv[hep-th]: 0707.0243, 0709.2368, 0712.1223, 0712.4138, 0803.1466 Strong Coupling: from Lattice to AdS/CFT - p. 1/21

Outline ✔ On-shell gluon scattering amplitudes ✔ Iterative structure at weak/strong coupling in N = 4 SYM ✔ Dual conformal invariance – hidden symmetry of planar amplitudes ✔ Scattering amplitude/Wilson loop duality in N = 4 SYM ... x n − 1 p 3 p 2 x n x 3 p 1 . ⇐ ⇒ . . . . . x 1 x 2 p n − 1 p n „ I « � 0 | S | 1 − 2 − 3 + . . . n + � � 0 | tr P exp i dx · A ( x ) | 0 � C ✔ Scattering amplitude/Wilson loop duality in QCD Strong Coupling: from Lattice to AdS/CFT - p. 2/21

On-shell gluon scattering amplitudes in N = 4 SYM ✔ N = 4 SYM – (super)conformal gauge theory with the SU ( N c ) gauge group Inherits all symmetries of the classical Lagrangian ... but are there some ‘hidden’ symmetries? ✔ Gluon scattering amplitudes in N = 4 SYM ✗ Quantum numbers of on-shell gluons | i � = | p i , h i , a i � : . . . i ) 2 = 0 ), helicity ( h = ± 1) , color ( a ) momentum ( ( p µ ✗ On-shell matrix elements of S − matrix A n = S 2 ✗ Suffer from IR divergences �→ require IR regularization 1 n ✗ Close cousin to QCD gluon amplitudes ✔ Color-ordered planar partial amplitudes T a 1 T a 2 . . . T a n ˜ A h 1 ,h 2 ,...,h n ˆ A n = tr ( p 1 , p 2 , . . . , p n ) + [Bose symmetry] n ✔ Recent activity is inspired by two findings ✗ The amplitude A 4 reveals interesting iterative structure at weak coupling [Bern,Dixon,Kosower,Smirnov] ✗ The same structure emerges at strong coupling via AdS/CFT [Alday,Maldacena] Where does this structure come from? Dual conformal symmetry! [Drummond,Henn,GK,Smirnov,Sokatchev] Strong Coupling: from Lattice to AdS/CFT - p. 3/21

Four-gluon amplitude in N = 4 SYM at weak coupling 2 3 a = g 2 YM N c A 4 / A (tree) + O ( a 2 ) , = 1+ a [Green,Schwarz,Brink’82] 4 8 π 2 4 1 All-loop planar amplitude can be split into a IR divergent and a finite part A 4 ( s, t ) = Div ( s, t, ǫ IR ) Fin ( s/t ) ✔ IR divergences appear to all loops as poles in ǫ IR (in dim.reg. with D = 4 − 2 ǫ IR ) ✔ IR divergences exponentiate (in any gauge theory!) [Mueller],[Sen],[Collins],[Sterman],[GK]’78-86 ∞ Γ ( l ) ( ! h i) ( lǫ IR ) 2 + G ( l ) − 1 cusp ( − s ) l ǫ IR + ( − t ) l ǫ IR X a l Div ( s, t, ǫ IR ) = exp 2 lǫ IR l =1 ✔ IR divergences are in the one-to-one correspondence with UV divergences of Wilson loops [Ivanov,GK,Radyushkin’86] l a l Γ ( l ) Γ cusp ( a ) = P cusp = cusp anomalous dimension of Wilson loops l a l G ( l ) G ( a ) = P cusp = collinear anomalous dimension ✔ What about finite part of the amplitude Fin( s/t ) ? Does it have a simple structure? Fin QCD ( s/t ) = [4 pages long mess] , Fin N =4 ( s/t ) = BDS conjecture Strong Coupling: from Lattice to AdS/CFT - p. 4/21

Four-gluon amplitude in N = 4 SYM at weak coupling II ✔ Bern-Dixon-Smirnov (BDS) conjecture: » Γ cusp ( a ) – ˆ 1 2 ln 2 ( s/t ) + 4 ζ 2 ln 2 ( s/t ) + const all loops + O ( a 2 ) ˜ ⇒ exp Fin ( s/t ) = 1 + a = 4 ✗ Compared to QCD, (i) the complicated functions of s/t are replaced by the elementary function ln 2 ( s/t ) ; (ii) no higher powers of logs appear in ln ( Fin ( s/t )) at higher loops; (iii) the coefficient of ln 2 ( s/t ) is determined by the cusp anomalous dimension Γ cusp ( a ) just like the coefficient of the double IR pole. ✗ The conjecture has been verified up to three loops [Anastasiou,Bern,Dixon,Kosower’03],[Bern,Dixon,Smirnov’05] ✗ A similar conjecture exists for n -gluon MHV amplitudes [Bern,Dixon,Smirnov’05] ✗ It has been confirmed for n = 5 at two loops [Cachazo,Spradlin,Volovich’04], [Bern,Czakon,Kosower,Roiban,Smirnov’06] ✔ Surprising features of the finite part of the MHV amplitudes in planar N = 4 SYM: ☞ Why should finite corrections exponentiate? ☞ Why should they be related to the cusp anomaly of Wilson loop? Strong Coupling: from Lattice to AdS/CFT - p. 5/21

Dual conformal symmetry Examine one-loop ‘scalar box’ diagram ✔ Change variables to go to a dual ‘coordinate space’ picture (not a Fourier transform!) p 1 = x 1 − x 2 ≡ x 12 , p 2 = x 23 , p 3 = x 34 , p 4 = x 41 , k = x 15 x 3 p 2 p 3 d 4 k ( p 1 + p 2 ) 2 ( p 2 + p 3 ) 2 d 4 x 5 x 2 13 x 2 Z Z 24 = k 2 ( k − p 1 ) 2 ( k − p 1 − p 2 ) 2 ( k + p 4 ) 2 = x 2 x 5 x 4 x 2 15 x 2 25 x 2 35 x 2 45 Check conformal invariance by inversion x µ i → x µ i /x 2 i p 1 p 4 [Broadhurst],[Drummond,Henn,Smirnov,Sokatchev] x 1 ✔ The integral is invariant under conformal SO (2 , 4) transformations in the dual space! ✔ The symmetry is not related to conformal SO (2 , 4) symmetry of N = 4 SYM ✔ All scalar integrals contributing to A 4 up to four loops possess the dual conformal invariance! ✔ If the dual conformal symmetry survives to all loops, it allows us to determine four- and five-gluon planar scattering amplitudes to all loops! [Drummond,Henn,GK,Sokatchev],[Alday,Maldacena] ✔ Dual conformality is slightly broken by the infrared regulator ✔ For planar integrals only! Strong Coupling: from Lattice to AdS/CFT - p. 6/21

Four-gluon amplitude from AdS/CFT Alday-Maldacena proposal: ✔ On-shell scattering amplitude is described by a classical string world-sheet in AdS 5 ✗ On-shell gluon momenta p µ 1 , . . . , p µ n define sequence of light-like segments on the boundary xn ✗ The closed contour has n cusps with the dual coordinates x µ i (the same as at weak coupling!) p 2 p 1 x 3 x µ i,i +1 ≡ x µ i − x µ i +1 := p µ x 1 i The dual conformal symmetry also exists at strong coupling! x 2 ✔ Is in agreement with the Bern-Dixon-Smirnov (BDS) ansatz for n = 4 amplitudes ✔ Admits generalization to arbitrary n − gluon amplitudes but it is difficult to construct explicit solutions for ‘minimal surface’ in AdS ✔ Agreement with the BDS ansatz is also observed for n = 5 gluon amplitudes [Komargodski] but disagreement is found for n → ∞ �→ the BDS ansatz needs to be modified [Alday,Maldacena] The same questions to answer as at weak coupling: ☞ Why should finite corrections exponentiate? ☞ Why should they be related to the cusp anomaly of Wilson loop? Strong Coupling: from Lattice to AdS/CFT - p. 7/21

From gluon amplitudes to Wilson loops Common properties of gluon scattering amplitudes at both weak and strong coupling: (1) IR divergences of A 4 are in one-to-one correspondence with UV div. of cusped Wilson loops (2) The gluons scattering amplitudes possess a hidden dual conformal symmetry ☞ Is it possible to identify the object in N = 4 SYM for which both properties are manifest ? Yes! The expectation value of light-like Wilson loop in N = 4 SYM [Drummond-Henn-GK-Sokatchev] x 4 x 1 „ « 1 I dx µ A µ ( x ) W ( C 4 ) = � 0 | Tr P exp ig | 0 � , C 4 = N c C 4 x 2 x 3 ✔ Gauge invariant functional of the integration contour C 4 in Minkowski space-time ✔ The contour is made out of 4 light-like segments C 4 = ℓ 1 ∪ ℓ 2 ∪ ℓ 3 ∪ ℓ 4 joining the cusp points x µ i x µ i − x µ i +1 = p µ i = on-shell gluon momenta ✔ The contour C 4 has four light-like cusps �→ W ( C 4 ) has UV divergencies ✔ Conformal symmetry of N = 4 SYM �→ conformal invariance of W ( C 4 ) in dual coordinates x µ Strong Coupling: from Lattice to AdS/CFT - p. 8/21

Gluon scattering amplitudes/Wilson loop duality I The one-loop expression for the light-like Wilson loop (with x 2 jk = ( x j − x k ) 2 ) [Drummond,GK,Sokatchev] x 1 x 1 x 1 x 2 x 2 x 2 ln W ( C 4 ) = x 3 x 4 x 3 x 4 x 3 x 4 „ x 2 = g 2  « ff 1 + 1 13 µ 2 ´ ǫ UV + 24 µ 2 ´ ǫ UV ˜ − x 2 − x 2 2 ln 2 13 + O ( g 4 ) ˆ` ` 4 π 2 C F − + const x 2 ǫ UV 2 24 The one-loop expression for the gluon scattering amplitude ln A 4 ( s, t ) = g 2  1 + 1 2 ln 2 “ s ff ´ ǫ IR + ´ ǫ IR i h` ” − s/µ 2 − t/µ 2 + O ( g 4 ) ` − 4 π 2 C F + const IR IR ǫ IR 2 t x µ i,i +1 ≡ x µ i − x µ i +1 := p µ ✔ Identity the light-like segments with the on-shell gluon momenta i : 13 µ 2 := s/µ 2 24 µ 2 := t/µ 2 x 2 x 2 x 2 13 /x 2 IR , IR , 24 := s/t ☞ UV divergencies of the light-like Wilson loop match IR divergences of the gluon amplitude ☞ the finite ∼ ln 2 ( s/t ) corrections coincide to one loop! Strong Coupling: from Lattice to AdS/CFT - p. 9/21

Gluon scattering amplitudes/Wilson loop duality II Drummond-(Henn)-GK-Sokatchev proposal: gluon amplitudes are dual to light-like Wilson loops ln A 4 = ln W ( C 4 ) + O (1 /N 2 c , ǫ IR ) . √ ✔ At strong coupling, the relation holds to leading order in 1 / λ [Alday,Maldacena] ✔ At weak coupling, the relation was verified to two loops [Drummond,Henn,GK,Sokatchev] x 4 2 3 x 1 6 7 6 7 x 2 x 3 6 7 6 7 6 7 6 7 = 1 4 Γ cusp ( g ) ln 2 ( s/t ) + Div 6 7 ln A 4 = ln W ( C 4 ) = 6 7 6 7 6 7 6 7 6 7 6 7 6 7 4 5 ✔ Generalization to n ≥ 5 gluon MHV amplitudes ln A (MHV) = ln W ( C n ) + O (1 /N 2 C n = light-like n − (poly)gon c ) , n ✗ At weak coupling, matches the BDS ansatz to one loop [Brandhuber,Heslop,Travaglini] ✗ The duality relation for n = 5 (pentagon) was verified to two loops [Drummond,Henn,GK,Sokatchev] Strong Coupling: from Lattice to AdS/CFT - p. 10/21