Updates on ABJM Scattering Amplitudes

Transcription

1 Updates on ABJM Scattering Amplitudes Sangmin Lee (Seoul National University) 17 June 2013 Exact Results in String/M Theory (pre Strings 2013 workshop) Korea Institute for Advanced Study

2 Introduction

3 Scattering Amplitudes in Gauge Theories... beyond Feynman diagrams

4 Scattering amplitudes in Twistor string theory MHV vertices expansion BCFW recursion relation Amplitude/Wilson-loop duality BCJ color/kinematics duality [Witten 03] [Cachazo,Svrcek,Witten 04] [Britto,Cachazo,Feng 04] [Britto,Cachazo,Feng,Witten 05] [Drummond,Henn,Korchemsky,Sokatchev 06-08] [Alday,Maldacena 07][Berkovits,Maldacena 08] [Beisert,Ricci,Tseytlin,Wolf 08] [Bern,Carrasco,Johansson 08] Polylogarithms, symbols, Hopf algebras [Goncharov,Spradlin,Vergu,Volovich 10] [Duhr 12] Grassmannians, affine permutations [Arkani-Hamed,Cachazo,Cheung,Kaplan 09] [Arkani-Hamed,Bourjaily, Cachazo,Gonchaorv,Postnikov,Trnka 12]

5 Scattering Amplitudes of ABJM theory N = 4 SYM N = 6 SCS AdS5 x S5 AdS4 x CP3

6 This talk is based on S.L. [ ] Dongmin Gang, Yu-tin Huang, Eunkyung Koh, S.L., Arthur E. Lipstein [ ] Yu-tin Huang, S.L. [ ] Yu-tin Huang, Henrik Johansson, S.L. [1306.nnnn] Joonho Kim, S.L. and friends, work in progress

7 Kinematics

8 Super-conformal symmetry Lorentz: d = 4, N = 4 SYM SO(1, 3) d = 3, N = 6 SCS SO(1, 2) Conformal: SO(2, 4) ' SU(2, 2) SO(2, 3) ' Sp(4, R) Super- Conformal: PSU(2, 2 4) SU(4) R OSp(4 6) SO(6) R Super-charges: 32 24

9 Spinor helicity and Twistor in 4d [Berends, Kleiss, Troost, Wu, Xu s] [Penrose 67, Witten 03] Massless momentum in bi-spinor notation: p aȧ = l a lȧ Lorentz invariants: hiji e ab (l i ) a (l j ) b, [ij] eȧḃ ( l i )ȧ( l j )ḃ Twistor transformation A(, )! Â(, µ) = Z d 2 e iµ A(, ) Twistors linearize conformal symmetry P aȧ = l a lȧ K aȧ = l a! lȧ lȧ µ a! µ a lȧ Z A (, µ ) M A B = Z A Z B (trace) 2 SU(2, 2)

10 Spinor helicity and Twistor in 3d Massless momentum in bi-spinor notation p ab = l a l b hiji e ab (l i ) a (l j ) b Twistor transformation?? A( )! Â(µ) = Twistor?? Z A (l a, / l a ) Z d 2 e iµ A( ) P ab = l a l b K ab = 2 l a l b L b a = l a l b QM analogy! Conformal generators J AB = Z (A Z B) 2 Sp(4, R) act on amplitudes A n (l 1,...,l n ) as if were wave-functions. A n

11 On-shell super-fields SYM4 SU(N) gauge SU(4) R F = A + + h I c I hi h J f IJ e IJKLh I h J h K c L e IJKLh I h J h K h L A helicity SU(4) R SCS6 [U(N) U(N)] gauge SU(4) R Z : (N, N;4), : (N, N; 4). F = f 4 + h I y I e IJKh I h J f K e IJKh I h J h K y 4, F = ȳ 4 + h I f I e IJKh I h J ȳ K e IJKh I h J h K f 4. ( )=(4+4) h I, h J = d J I SO(6) Clifford algebra, only U(3) manifest.

12 Super-twistor SYM4 Super-twistor: W =(, µ, I ) Super-conformal generators: J A B = W A W B SCS6 Super-QM analogy! Configuration space: R 2 3 L =(l, h) Phase space: R 4 6 Z =(L; / L) =(l, h; / l, / h) Super-conformal generators: J AB = Z [A Z B)

13 Recursion Relation [Gang,Huang,Koh,SL,Lipstein 1012]

14 BCFW recursion relation [Britto,Cachazo,Feng(,Witten) 04(05)] Higher point amplitude from lower ones A n = Â r,s,h A h L (z = z rs) i P 2 r s A h R (z = z rs) Idea: on-shell deformation of momenta l j! l j z l l, l l! l l + zl j (p 2 j = 0, p 2 l = 0, p p n = 0 unaffected) A n = A n (z = 0) = Z dz 2pi A n (z) z (deformed contour, fall-off at infinity, residue theorem)

15 BCFW recursion relation Generalization to d > 4 [Arkani-Hamed,Kaplan 08] p j! p j + zq, p l! p l zq p j q = 0, p l q = 0, q 2 = 0 p j =(1, 1, 0, 0; 0,..., 0), p l =(1, 1, 0, 0; 0,..., 0), q =(0, 0, 1, i; 0,..., 0) Naive generalization to 3d fails l j! l j zl l, l l! l l + zl j (??)

16 Recursion relation in 3d [Gang,Huang,Koh,SL,Lipstein 1012] Momentum conservation in 4d vs 3d (4d) (3d)  i  i l i l i = 0 l i l i = 0 GL(n, C) : l i! M i j l j, O(n, C) : l i! R i j l j l i! (M T ) i j l j Use O(2,C) instead of GL(2,C) in 3d! lj l l! cos q sin q sin q cos q lj l l cos q = z + 1/z 2, sin q = z 1/z 2i A n = A n (z = 1) = Z dz 2pi A n (z) z 1

17 Grassmannian Integral Formula [Arkani-Hamed,Cachazo,Cheung,Kaplan 0907][SL 1007]

18 Grassmannian Integral formula in 4d [Arkani-Hamed,Cachzo,Cheung,Kaplan 09] A tree n,k (W) = Z d k n C vol [GL(k)] d 4k 4k (C W) M 1 M 2 M n 1 M n n (C W) m = C mi W i C = M i k GL(k) L GL(n) R (i) (i + k 1) n: total number of external legs GL(k) gauge symmetry k: number of (-) helicity gluons

19 Grassmannian Integral formula in 4d A tree n,k (W) = Z d k n C vol [GL(k)] d 4k 4k (C W) M 1 M 2 M n 1 M n only dependence on (super-)momenta Cyclic + Superconformal symmetry manifest Contour integral over Grassmannian manifold = moduli space of k-planes in C n. = U(n)/[U(k) U(n k)] Proven to produce ALL tree amplitudes. Useful in building up loop integrands.

20 Grassmannian Integral formula in 3d [SL 1007] Z A tree 2k (L) = d k 2k C vol [GL(k)] d k(k+1)/2 (C C T ) d 2k 3k (C L) M 1 M 2 M k 1 M k 2k C = k GL(k) O(2k) (i) (i + k 1) (C L) m = C mi L i 2k: total number of external legs (C C T ) mn = C mi C ni M i = e m 1 m k C m1 (i) C m 2 (i+1) C mk (i+k 1)

21 Grassmannian Integral formula in 3d Z A tree 2k (L) = d k 2k C vol [GL(k)] d k(k+1)/2 (C C T ) d 2k 3k (C L) M 1 M 2 M k 1 M k Contour integral over orthogonal-grassmannian manifold = moduli space of null k-planes in C 2k. = O(2k)/U(k) Contours for ALL tree amplitudes under construction.

22 Grassmannian Integral formula in 3d Z A tree 2k (L) = d k 2k C vol [GL(k)] d k(k+1)/2 (C C T ) d 2k 3k (C L) M 1 M 2 M k 1 M k 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 C C T = 0 =) M i M i+1 =( 1) k 1 M i+k M i+1+k

23 KK & BCJ Relations [Huang,Johansson,SL 1306]

24 Color decomposition Trace-based color decomposition Full amplitude (permutation symm) Color-ordered amplitude (cyclic symmetry only) A tree n (p i, h i, a i )= Â s2s n /Z n Tr(T a s(1) T a s(n) )A n (s(1 h 1),, s(n h n )) Graph vs trace (n-1)! f abc = Tr(T a [T b, T c ]) = Tr(T a T b T c ) Tr(T a T c T b ) f abc f cde = Tr([T a, T b ][T d, T e ]) =(abde) (bade) (abed)+(baed)

25 Color decomposition Jacobi identity [T a, T b ]= f abc T c [[T a 1, T a 2 ], T a 3 ]+[[T a 2, T a 3 ], T a 1]+[[T a 3, T a 1], T a 2 ]=0 f a 1a 2 b f ba 3a 4 + f a 2a 3 b f ba 1a 4 + f a 3a 1 b f ba 2a 4 = f a 1a 2 b + + =

26 Kleiss-Kuijf identities [Kleiss,Kuijf 86] Counting the number of independent graphs after imposing Jacobi identities = n (n 2)!

27 Color/Kinematics duality [Bern,Carrasco,Johansson 08] 4-point amplitude s =(p 1 + p 2 ) 2, t =(p 1 + p 4 ) 2, u =(p 1 + p 3 ) A 4 = s + t u = s s = t t = u u = n sc s s + n tc t t + n uc u u Jacobi identity: c s + c t + c u = 0 Kinematic Jacobi identity! n s + n t + n u = 0

28 Color/Kinematics duality [Bern,Carrasco,Johansson 08] A i = (n 2)! Â j=1 Q ij n j rank(q ij )=(n 3)! Fundamental BCJ

29 Graph combinatorics : Yang-Mills Number of external legs All possible graphs Cyclic traces After applying KK identities After applying BCJ relations n (2n 5)!! (n 1)! (n 2)! (n 3)!

30 Graph combinatorics : ABJM [Huang,Johansson,SL 13] Chern-Simons-matter theory in (1+2) dimensions Quartic vertex with bi-fundamental maps Jacobi-like identity

31 Graph combinatorics : ABJM Legs All graphs Traces n = 2k (3k 3)!k! (2k 1)!2 k 1 k!(k 1)! 2 KK ?? BCJ ??

32 Gravity = (gauge)^2 [Bern,Carrasco,Johansson 08] color kinematics A n (gauge) =Â i c i n i D i (sum over all trivalent graphs) product of scalar propagators =) A n (gravity) =Â i n i ñ i D i (sum over trivalent graphs only!)

33 Gravity = (gauge)^2 in 3d [Huang,Johansson 12] Same gravity from... either Yang-Mills or Chern-Simons!!! color kinematics A n (gauge) =Â i c i n i D i (sum over all trivalent graphs) product of scalar propagators =) A n (gravity) =Â i n i ñ i D i (sum over trivalent graphs only!)

34 Twistor-string-like Integral Formula [Huang,SL 1207]

35 RSV-W formula [Witten 03][Roiban,Spradlin,Volovich 04] A n,k = 1 vol(gl(2)) Z d 2 s 1 d 2 s 2 d 2 s n (12)(23) (n1) d 4k 4k (C[s] W) d 4k 4k (C mi [s]w i ) Z d 4k 4k z n d 4 4 (Z i Cm[s V i ]z m ) i=1 Connected prescription for twistor string theory Integral over moduli space of degree (k - 1) curves in CP 3 4

36 Contours in 4d Parameter counting k(n k) (2n 4) =(k 2)(n k 2) k n - k n - k - 2 k - 2 dim[gr(k, n)] d(c W)/d(P) d n,k = dim[gr(k 2, n 4)] 6- point NMHV amplitude

37 Recursive construction NMHV amplitudes ( k = 3, all n ) d n,k = n 5 A n,3 = Z ~ f =0 d n 5 t h n f 6 f 7 f n 1 f n 1 M 1 M 2 M n 1 M n f j =(j 2, j 1, j)(j, 1, 2)(2, 3, j 2) Soft limit Parity flip (k $ n k) l n! 0, l n fixed 6 n - k A n n 1, 1 n 1, n n,1 A n k

38 Deformation of the integrand h n f 6 f 7 f n 1 f n f j =(j 2, j 1, j)(j, 1, 2)(2, 3, j 2) rewrite = H n F 6 F 7 F n 1 F n F j =(j 2, j 1, j)(j, 1, 2)(2, 3, j 2)(1, 3, j 1) deform = H n F 6 (t) F 7 (t) F n 1 (t) F n (t) F j (t) =(j 2, j 1, j)(j, 1, 2)(2, 3, j 2)(1, 3, j 1) t(1, 2, 3)(3, j 2, j 1)(j 2, j 1, j)(2, j 2, j) Poles move, residues vary, but the integral remains unchanged!

39 Geometry of the deformed contour F j (1) =0 for all j = C = A2 1 A 2 2 A 2 1 n A 1 B 1 A 2 B 2 A n B n A B1 2 B2 2 Bn 2 Veronese map CP 1! CP k (A, B) (A k 1, A k 2 B,, AB k 2, B k 1 ) Veronese map Gr(2, n)! Gr(k, n)

40 Veronese Integral Formula A n,k = 1 vol(gl(2)) Z d 2 s 1 d 2 s 2 d 2 s n (12)(23) (n1) d 4k 4k (C[s] W)

41 Contours in 3d Parameter counting k(k 1) 2 (2k 3) = (k 2)(k 3) 2 k k - 2 dim[og(k,2k)] d(c l)/d(p) d 2k = dim[og(k 2, 2k 4)] Soft limit l 1 + il 2 e! 0 (p 1 + p 2! 0) A 2k 2k 2, 1 2k 2, e e,1 A 2k 2

42 Twistor-string-like integral formula for ABJM [Huang,SL 12] A 2k = 1 vol(gl(2)) Z D 2k [s] =det 2k d 2 s 1 d 2 s 2 d 2 s 2k (12)(23) (2k,1) 1 A 2k 1 j i B j 2k 1 i d j=1 D 2k [s] 2k  i=1 d 2k 3k (C V [s] L) J[s] A 2k 1 j i B j i! J[s] = (2i 1, 2j 1) 1appleiapplejapplek (2k-3)-dim surface in Gr(2,n) Verified by explicit computation up to 8-point amplitude. Consistency check with KK identities cyclic symmetry Twistor string theory interpretation? [Cachazo,He,Yuan 13]

43 KK identities vs Integral formula [Huang,SL 12][Huang,Johansson,SL 13] A 2k = 1 vol(gl(2)) Z d 2 s 1 d 2 s 2 d 2 s 2k (12)(23) (2k,1) D 2k [s] d 2k 3k (C V [s] L) J[s] Under (non-cyclic) permutations, the only non-trivial part is Special KK identities 1 (12)(34) (2k,1)

44 KK identities vs Integral fromula [Huang,SL 12][Huang,Johansson,SL 13] General KK identities via graphical representation

45 KK identities vs Integral fromula [Huang,SL 12] [Huang,Johansson,SL 13]

46 On-shell graphs, Affine permutations, Positive Grassmannian, etc. [Arkani-Hamed,Bourjaily, Cachazo,Gonchaorv,Postnikov,Trnka 12] [Joonho Kim, SL, others, work in progress]

47 On-shell graphs [Arkani-Hamed,Bourjaily,Cachazo,Gonchaorv,Postnikov,Trnka 12]

48 On-shell graphs, permutations [Arkani-Hamed,Bourjaily, Cachazo,Gonchaorv,Postnikov,Trnka 12]

49 Yang-Baxter in 4d and 3d [Arkani-Hamed,Bourjaily, Cachazo,Gonchaorv,Postnikov,Trnka 12]

50 On-shell graphs in 3d [Joonho Kim, SL, others, work in progress] d 3 (P)d 6 (Q) h14ih34i Z d 2 ld 3 h Yang-Baxter via 3d Euler angles! A 6 cf) BCJ duality for dimensional reduction to 2d

51 Summary

52 Summary BCFW-like recursion relation Grassmannian integral formula - formal proof of Yangian invariance KK/BCJ relations Twistor-string-like integral formula Amplitude/Wilson-loop duality? On-shell graph, permutations, cluster algebra,...

53 Thank you