Scattering amplitudes

Scattering Amplitudes ! in Three Dimensions Sangmin Lee Seoul National University 27 June 2014
 Strings 2014, Princeton Scattering Amplitudes Recent review [Elvang,Huang 13] SUGRA QCD String SYM Integrability Twistor Talks by Basso, Cachazo, Dolan, Mafra, Sen, Staudacher, Stieberger, Trnka Scope of this talk ABJM theory / tree amplitudes / planar sector Two amplitude-generating integrals Grassmannian A2k (L) = Z dk⇥2k C d(Cmi Cni ) ’km=1 d2|3 (Cmi Li ) vol[GL(k)] M1 (C ) M2 (C ) · · · Mk (C ) Twistor string A2k (L) = Z d2⇥2k s J D ’km=1 d2|3 (Cmi [s ]Li ) vol[GL(2)] (12)(23) · · · (2k, 1) “in four dimensions” = “for N=4 Super-Yang-Mills” This talk is based on SL [1007.4772] ! Dongmin Gang, Yu-tin Huang, Eunkyung Koh, SL, Arthur Lipstein [1012.5032]
 Yu-tin Huang, SL [1207.4851] ! Yu-tin Huang, Henrik Johansson, SL [1307.2222] ! Joonho Kim, SL [1402.1119] special thanks to N Arkani-Hamed Collaborators Henrik Johansson Yu-tin Huang Arthur Lipstein Dongmin Gang Eunkyung Koh Joonho Kim ABJM theory ABJM d = 3, N = 
 superconformal 
 Chern-Simons-matter OSp(6|4) Sp(4, R ) ⇥ SO(6) R ' SO(2, 3) ⇥ SU (4) R Color-ordered amplitudes of ABJM Gauge fields mediate interactions but have no d.o.f Number of external legs must be even ( n = 2k ) Kinematics, on-shell [Bargheer,Loebbert,Meneghelli 10] Null momentum in (1+2)d : pab = pµ (Cgµ )ab = la l b a b h ij i = e ( l ) ( l ) Lorentz invariants : ab i j On-shell super-fields SO(1, 2) ⇥ U (3) ⇢ SO(2, 3) ⇥ SO(6) R (1+3+3+1) = + Cyclic symmetry and Lambda-parity A2k (1, 2, · · · , 2k) = ( 1)k A2k (3, 4, · · · , 2k, 1, 2) A2k (· · · , Li , · · · ) = ( 1)i A2k (· · · , Li , · · · ) L = (la , h I ) BCFW and Grassmannian in 4d Momentum conservation in spinor-helicity 11 l a n n B C i ¯ ¯ ¯  ( pi )a¯ a =  lia˙ la = l1a˙ · · · lna˙ @ A i =1 i =1 lna BCFW deformation is a special case of li ! Si j l j , l¯ i ! l¯ j (S j ) i, [Britto, Cachazo, Feng, Witten 04-05] S GL(n, C ) l¯ (2-plane) P-conservation at each vertex: ! Grassmannian Cmi Gr(k, n) [Arkani-Hamed,Cachazo,Cheung,Kaplan 09] l (2-plane) C (k-plane) BCFW and Grassmannian in 3d Momentum conservation in spinor-helicity l1b n n B C  ( pi )ab =  lia lib = l1a · · · lna @ A i =1 i =1 lnb BCFW deformation is a special case of li ! Ri j l j , R O(n, C ) [Gang,Huang,Koh,SL,Lipstein 10] P-conservation at each vertex: ! orthogonal Grassmannian (a.k.a isotropic Gr)  Cmi Cni = , i [SL 10] Cmi OG (k, 2k) l (2-plane) C (k-plane) Grassmannian Integral in 3d [SL 10] A2k (L) = Z dk⇥2k C d(Cmi Cni ) ’km=1 d2|3 (Cmi Li ) vol[GL(k)] M1 (C ) M2 (C ) · · · Mk (C ) 2k (number of external legs) GL(k) ⇥ O(2k) k C= N=6 SUSY crucial (i ) (i + k 1) Mi (C ) = em1 ···mk Cm1 (i) Cm2 (i+1) · · · Cmk (i+k 1) KK and BCJ in 4d Kleiss-Kuijf identities A n ( pi , hi , ) =  [Kleiss-Kuijf 89] Tr( T s2Sn /Z n f abc = Tr( T a T b T c ) a s (1) ···T Tr( T b T a T c ) , 1) ! ! ( n Bern-Carrasco-Johansson identities An (gauge) =  I cI nI DI )An (s(1h1 ), · · · , s(nhn )) f abe f e cd + f bce f e ad + f cae f e bd = #(independent amplitudes) : (n ns cs nt ct nu cu A4 = + + s t u as(n) [Bern,Carrasco,Johansson 08] cs + ct + cu = ns + nt + nu = =) Mn (gravity) =  #(independent amplitudes) : (n 2) ! I 2) ! ! ( n n I n˜ I DI ! 3) ! color (Jacobi) kinematics BCJ doubling KK in 3d KK identity follows from bi-fundamental algebra ¯ ¯ a¯ Mb M ¯ c¯ Md f a¯ bcd = Tr( M ¯ f a¯ bcg f g de¯ f ¯ c¯ Mb M ¯ a¯ Md ) M ¯ ¯ f a¯ dcg f g be¯ f = f e¯ f a¯ g f g bcd [Bagger,Lambert 08] “structure constant” ¯ f e¯ f cg f g b a¯ d “fundamental identity” Independent color basis counted; no closed-form formula known external legs 10 12 cyclic/reflection 72 1440 43200 KK identity 57 1144 * n = 2k 1)!/2 k!(k * KK from twistor string [Huang,SL 12][Huang,Johansson,SL 13] A2k (L) = Z d2⇥2k s J D ’km=1 d2|3 (Cmi [s ]Li ) vol[GL(2)] (12)(23) · · · (2k, 1) Under (non-cyclic) permutations, the only non-trivial part is D2k (1, 2, , 2k ) = (12)(34) · · · (2k, 1) All KK identities can be constructed 
 by recursive use of elementary identities: 1 + =0 (ij) ( ji ) (ij) + ( jk) + (kl ) + (li ) + (cyclic) = =0 (ij)( jk)(kl ) (ij)( jk)(kl )(li ) BCJ in 3d Yang-Mills : BCJ relation in any dimension (on-shell) BLG : BCJ relation in 3d only [Bargheer,He,McLoughlin 12] 4- and 6-point 
 BLG doubling for YM and CSm agree! 8-point of higher
 BCJ for BLG, but NOT ABJM Bonus relations and more [Huang,Johansson 12] SUSY truncation On-shell diagrams and positive Grassmannian [Arkani-Hamed,Bourjaily, Cachazo,Goncharov,Postnikov,Trnka 12] All tree amplitudes & ! loop integrands from ! 3-point amplitudes ! via “BCFW bridging” Permutation, ! positive Grassmannian On-shell diagrams in 3d [ABCGPT 12][Huang, Wen1309][Kim, SL 1402] Building blocks d3 ( P ) d6 ( Q ) h14ih34i 4-point on-shell amplitude Z unique integral preserving superconformal symmetry d2 ld3 h Pairing of external particles up to Yang-Baxter equivalence Positive orthogonal Grassmannian [Huang, Wen1309][Kim, SL 1402] Complex OG(k) = moduli space of null k-planes in C2k = O(2k)/U (k) Pure spinor Real OG(k) h = diag( , +, · · · , , +) , C · h · CT = , Positive OG(k) All ordered minors are positive Cmi R OG tableaux [Joonho Kim, SL 1402] 6 5 4 Fix Yang-Baxter ambiguity at the expense of manifest cyclic symmetry 1 6 5 4 OG tableaux [Joonho Kim, SL 1402] 6 5 1 Invertible 6 4 OG tableaux Level = number of crossing Top-cell : unique #(bottom cell) = Ck Canonically positive BCFW coordinates Explicit coordinate charts for POG 0 (C4 , C5 , C6 ) = @0 0 s1 s2 = @ c1 s2 c2 10 1 0A @0 0 c1 s1 10 c2 s1 A @ s2 c1 s1 c2 c3 + c1 s3 c1 c2 c3 s1 s3 s2 c3 10 0A @0 1 c1 c3 + s1 c2 s3 s1 c3 c1 c2 s3 A s2 s3 si = sinh ti , ci = cosh ti , ti R + s2 c2 0 c3 s3 s3 A c3 BCFW coordinates and recursion Integration measure Z dt a ’ sinh ta = a Z ’ a dz a = za Z ’ d log za a ✓ ta z a = ◆ Integration contour problem solved A2k =  2m L 2m R m L + m R = k +1 Alternative coordinates and loop BCFW [Huang,Wen,Xie 1402] Combinatorics and topology of POG Generating function k (k 1)/2 Tk (q) =  l =0 l Tk,l q = k (1 q)k  ( 1) j= k j ✓ ◆ 2k q j( j k+j 1)/2 [Riordan 75] T2 (q) = + q T3 (q) = + 6q + 3q2 + q3 Euler characteristic ck = Tk ( 1) = l ( ) Tk,l =  Combinatorics - Eulerian poset : proven! Topology - Ball : conjectured l [Lam 1406] Open problems - trees Dual superconformal (Yangian) symmetry Momentum twistor ? Fermionic T-duality in AdS4 ? Amplituhedron ? [Huang,Lipstein 10] [Hodges 09] Amplitude/Wilson-loop duality [Drummond,Ferro 10] [SL 10] [Bargheer,Loebbert,Meneghelli 10] [Bianchi,Griguolo,Leoni,Penati,Seminara 14] [Bianchi,Giribet,Leoni,Penati 13] [Adam,Dekel,Oz 10][Bakhmatov 11] [Bakhmatov,OColgain,Yavartanoo 11] [Arkani-Hamed,Trnka 13] Scattering equation ? [Cachzo,He,Yuan 13] Symplectic (a.k.a Lagrangian) Grassmannian in 5d/6d ? Open problems - loops Thanks to feedback from Yu-tin Huang IR divergence 4-point 3-loop and 6-point 2-loop [Caron-Huot Huang 12] [Chen,Huang 11][Bianchi,Leoni,Mauri,Penati,Santambrogio 11][Bianchi,Leoni 14] Unitarity 1-loop = 0, 2-loop 6= Modified unitarity equation? Analyticity [Agarwal,Beisert,McLoughlin 08][Chen,Huang 11] [Bianchi,Leoni,Mauri,Penati,Santambrogio 11] [Jain,Mandlik,Minwalla,Takimi,Wadia,Yokoyama 14] [Bargheer,Beisert,Loebbert,McLoughlin 12] 6-point, 1-loop contains sgn(q2 Non-perturbative resolution? q1 ) [Caron-Huot Huang 12] .. .Scattering Amplitudes Recent review [Elvang,Huang 13] SUGRA QCD String SYM Integrability Twistor Talks... e bd = #(independent amplitudes) : (n ns cs nt ct nu cu A4 = + + s t u as(n) [Bern,Carrasco,Johansson 08] cs + ct + cu = ns + nt + nu = =) Mn (gravity) =  #(independent amplitudes) : (n 2) !... [Arkani-Hamed,Bourjaily, Cachazo,Goncharov,Postnikov,Trnka 12] All tree amplitudes & ! loop integrands from ! 3-point amplitudes ! via “BCFW bridging” Permutation, ! positive Grassmannian On-shell