Scattering Amplitudes! in Three Dimensions Sangmin Lee Seoul National University 27 June 204 Strings 204, Princeton
Scattering Amplitudes Recent review [Elvang,Huang 3] 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 A 2k (L) = Z d k 2k C vol[gl(k)] d(c mi C ni ) k m= d2 3 (C mi L i ) M (C)M 2 (C) M k (C) Twistor string A 2k (L) = Z d 2 2k s vol[gl(2)] J D k m= d2 3 (C mi [s]l i ) (2)(23) (2k,) in four dimensions = for N=4 Super-Yang-Mills
This talk is based on SL [007.4772]! Dongmin Gang, Yu-tin Huang, Eunkyung Koh, SL, Arthur Lipstein [02.5032] Yu-tin Huang, SL [207.485]! Yu-tin Huang, Henrik Johansson, SL [307.2222]! Joonho Kim, SL [402.9] 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 = 6 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 0] Null momentum in (+2)d : Lorentz invariants : p ab = p µ (Cg µ ) ab = l a l b hiji = e ab (l i ) a (l j ) b On-shell super-fields SO(, 2) U(3) SO(2, 3) SO(6) R (+3+3+) = 4 + 4 Cyclic symmetry and Lambda-parity A 2k (, 2,,2k) =( ) k A 2k (3, 4,,2k,, 2) A 2k (, L i, )=( ) i A 2k (, L i, ) L =(l a, h I )
BCFW and Grassmannian in 4d Momentum conservation in spinor-helicity 0 n Â(p i )āa = i= n  i= l iȧ l i a = l ȧ l nȧ B @ l a. l n a C A BCFW deformation is a special case of [Britto, Cachazo, Feng, Witten 04-05] l i! S i jl j, l i! l j (S ) j i, S 2 GL(n, C) l (2-plane) P-conservation at each vertex:! Grassmannian C mi 2 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 0 n Â(p i ) ab = i= n  i= l ia l ib = l a l na B @ l b. l nb C A BCFW deformation is a special case of l i! R i jl j, R 2 O(n, C) [Gang,Huang,Koh,SL,Lipstein 0] P-conservation at each vertex:! orthogonal Grassmannian (a.k.a. isotropic Gr) [SL 0]  i C mi C ni = 0, C mi 2 OG(k,2k) l (2-plane) C (k-plane)
Grassmannian Integral in 3d [SL 0] A 2k (L) = Z d k 2k C vol[gl(k)] d(c mi C ni ) k m= d2 3 (C mi L i ) M (C)M 2 (C) M k (C) 2k (number of external legs) C = k GL(k) O(2k) N=6 SUSY crucial (i) (i + k ) M i (C) =e m m k C m (i) C m 2 (i+) C mk (i+k )
Grassmannian Integral in 3d A 2k (L) = Z d k 2k C vol[gl(k)] d(c mi C ni ) k m= d2 3 (C mi L i ) M (C)M 2 (C) M k (C) Superconformal symmetry L L! manifest 2 L L! killed by d(c CT ) C T bc + bc T C = I 2k 2k LL! L T (C T bc + bc T C) L = 0 Cyclic symmetry A 2k (, 2,,2k) =( ) k A 2k (3, 4,,2k,, 2) C C T = 0 =) M i M i+ =( ) k M i+k M i++k
6-point amplitude [Bargheer,Loebbert,Meneghelli 0][Gang,Huang,Koh,SL,Lipstein 0] Factorization gauge C = 0 @ 0 0 c 4 c 5 c 6 0 0 c 24 c 25 c 26 A 0 0 c 34 c 35 c 36 6-fermion amplitude 2 6 3 4 5 2 6 f 3 4 5 A 6y = +i i (h3ih46i + ih2 p 23 5i) 3 (p 23 ) 2 (h23ih56i + ih p 23 4i)(h2ih45i + ih3 p 23 6i) (h3ih46i ih2 p 23 5i) 3 (p 23 ) 2 (h23ih56i ih p 23 4i)(h2ih45i ih3 p 23 6i). shows the factorization channel clearly: Check via recursion relation! A 6 (, 2, 3, 4, 5, 6)! A 4 (, 2, 3, f ) (p 23 ) 2 A 4( f, 4, 5, 6)+(regular).
From Grassmannian to twistor string - in 4d Grassmannian : contour integral deformation of integrand partial localization [Arkani-Hamed,Bourjailly,Cachazo,Trnka 09] [Bourjailly,Trnka,Volovich,Wen 0] Veronese map: Gr(2, n)! Gr(k, n) 0 a k a2 k a k n a k 2 b a2 k 2 b 2 a k n 2 s! C[s] = B @... b k b2 k bn k Twistor string : integration over degree (k ) curves in CP 3 4. s = a a n /GL(2, C) 2 Gr(2, n) ' b b n [Witten 03][Roiban,Spradlin,Volovich 04] n points on CP / SL(2, C) b n C A
Twistor string integral for ABJM amplitudes [Huang, SL 2] A 2k (L) = Z d 2 2k s vol[gl(2)] J D k m= d2 3 (C mi [s]l i ) (2)(23) (2k,) s = D = a a n b b n 2k j= d 2k  i= a 2k j i b j i, (ij)=a i b j a j b i, C mi [s] =a k m i b m i!, J = det applei,japple2k a 2k j i b j i applei<japplek (2i, 2j ) dim[gr(2, 2k)] dim(d) =2k 3 cyclic symmetry
8-point amplitude Deform the denominator while preserving the residue M M 2 M 3 M 4! (463)(472) M M 3 [M 2 M 4 (463)(472) M M 3 (467)(475)] Deformed denominator and part of the null constraint d(c mi C ni ) localizes the integral to the image of the Veronese map. General prescription for the contour? Twistor string realization?
Twistor string realization Rational map for 4d SYM and SUGRA dressing by wave function dimensional reduction [Cachazo,He,Yuan 3] Open string theory on CP 2,2 4, truncated to ABJM states [Engelund,Roiban 4]
KK and BCJ in 4d Kleiss-Kuijf identities [Kleiss-Kuijf 89] A n (p i, h i, a i )= Â Tr(T a s() T a s(n) )A n (s( h ),, s(n h n )) s2s n /Z n f abc = Tr(T a T b T c ) Tr(T b T a T c ), f abe f e cd + f bce f e ad + f cae f e bd = 0 #(independent amplitudes) : (n )!! (n 2)! Bern-Carrasco-Johansson identities [Bern,Carrasco,Johansson 08] A 4 = n sc s s + n tc t t A n (gauge) =Â I + n uc u u c s + c t + c u = 0 n s + n t + n u = 0 c I n I D I =) M n (gravity) =Â I n I ñ I D I! color (Jacobi) kinematics BCJ doubling #(independent amplitudes) : (n 2)!! (n 3)!
KK in 3d KK identity follows from bi-fundamental algebra [Bagger,Lambert 08] f āb cd = Tr( MāM b M c M d M c M b MāM d ) structure constant f āb cg f g dēf f ād cg f g bēf = f ēfāg f g b cd f ēf cg f g bād fundamental identity Independent color basis counted; no closed-form formula known external legs 4 6 8 0 2 n = 2k cyclic/reflection 6 72 440 43200 k!(k )!/2 KK identity 5 57 44 * *
KK from twistor string [Huang,SL 2][Huang,Johansson,SL 3] A 2k (L) = Z d 2 2k s vol[gl(2)] J D k m= d2 3 (C mi [s]l i ) (2)(23) (2k,) Under (non-cyclic) permutations, the only non-trivial part is D 2k (, 2,..., 2k) = (2)(34) (2k,) All KK identities can be constructed by recursive use of elementary identities: (ij) + (ji) = 0 (ij)(jk)(kl) + (cyclic) = (ij)+(jk)+(kl)+(li) (ij)(jk)(kl)(li) = 0
BCJ in 3d Yang-Mills : BCJ relation in any dimension (on-shell) BLG : BCJ relation in 3d only 4- and 6-point BLG doubling for YM and CSm agree! [Bargheer,He,McLoughlin 2] [Huang,Johansson 2] 8-point of higher BCJ for BLG, but NOT ABJM SUSY truncation Bonus relations and more
On-shell diagrams and positive Grassmannian [Arkani-Hamed,Bourjaily, Cachazo,Goncharov,Postnikov,Trnka 2] All tree amplitudes &! loop integrands from! 3-point amplitudes! via BCFW bridging Permutation,! positive Grassmannian
On-shell diagrams in 3d [ABCGPT 2][Huang, Wen309][Kim, SL 402] Building blocks d 3 (P)d 6 (Q) h4ih34i 4-point on-shell amplitude Z d 2 ld 3 h unique integral preserving superconformal symmetry Pairing of external particles up to Yang-Baxter equivalence
Positive orthogonal Grassmannian [Huang, Wen309][Kim, SL 402] Complex OG(k) = moduli space of null k-planes in C 2k. = O(2k)/U(k) Pure spinor Real OG(k) h = diag(, +,,, +), C h C T = 0, C mi 2 R Positive OG(k) All ordered minors are positive
OG tableaux [Joonho Kim, SL 402] 2 5 6 6 3 4 5 4 2 3 Fix Yang-Baxter ambiguity at the expense of manifest cyclic symmetry 2 5 6 6 3 4 5 4 2 3
OG tableaux [Joonho Kim, SL 402] 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 Invertible
OG tableaux Level = number of crossing Top-cell : unique #(bottom cell) = C k
Canonically positive BCFW coordinates Explicit coordinate charts for POG (C 4, C 5, C 6 ) = = 0 @ 0 0 0 0 0A @ 0 0 0 0 c s A @ c 0 2 s 2 0 s 2 c 2 0A @ 0 0 0 c 3 s 3 A 0 0 0 s c 0 0 0 s 3 c 3 0 @ s s 2 s c 2 c 3 + c s 3 c c 3 + s c 2 s 3 c s 2 c c 2 c 3 s s 3 s c 3 c c 2 s 3 A. c 2 s 2 c 3 s 2 s 3 s i = sinh t i, c i = cosh t i, t i 2 R +
BCFW coordinates and recursion Integration measure Z a dt a sinh t a = Z a dz a z a = Z a d log z a z a = tanh t a 2 Integration contour problem solved A 2k = Â m L +m R =k+ 2m L 2m R Alternative coordinates and loop BCFW [Huang,Wen,Xie 402]
Combinatorics and topology of POG Generating function T k (q) = k(k )/2  l=0 T 2 (q) =2 + q T k,l q l = ( q) k k  ( ) j 2k k + j j= k j(j )/2 q [Riordan 75] T 3 (q) =5 + 6q + 3q 2 + q 3 4 3 Euler characteristic 2 2 c k = T k ( ) =Â( ) l T k,l = l 5 Combinatorics - Eulerian poset : proven! [Lam 406] Topology - Ball : conjectured
Open problems - trees Dual superconformal (Yangian) symmetry [Drummond,Ferro 0] [SL 0] [Bargheer,Loebbert,Meneghelli 0] Momentum twistor? [Hodges 09] [Huang,Lipstein 0] Amplitude/Wilson-loop duality Fermionic T-duality in AdS4? [Bianchi,Griguolo,Leoni,Penati,Seminara 4] [Bianchi,Giribet,Leoni,Penati 3] [Adam,Dekel,Oz 0][Bakhmatov ] [Bakhmatov,OColgain,Yavartanoo ] Amplituhedron? [Arkani-Hamed,Trnka 3] Scattering equation? [Cachzo,He,Yuan 3] 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 2] [Chen,Huang ][Bianchi,Leoni,Mauri,Penati,Santambrogio ][Bianchi,Leoni 4] Unitarity -loop = 0, 2-loop = 0 Modified unitarity equation? [Agarwal,Beisert,McLoughlin 08][Chen,Huang ] [Bianchi,Leoni,Mauri,Penati,Santambrogio ] [Jain,Mandlik,Minwalla,Takimi,Wadia,Yokoyama 4] Analyticity [Bargheer,Beisert,Loebbert,McLoughlin 2] 6-point, -loop contains sgn(q 2 q ) Non-perturbative resolution? [Caron-Huot Huang 2]