On the Scattering Amplitude of Super-Chern-Simons Theory

Transcription

1 On the Scattering Amplitude of Super-Chern-Simons Theory Sangmin Lee University of Seoul [SL, , Phys.Rev.Lett.105,151603(2010)] + work in collaboration with Yoon Pyo Hong, Eunkyung Koh (KIAS), Dongmin Gang, Hee-Cheol Kim (SNU), Arthur Lipstein (Caltech), Yu-tin Huang (UCLA) 16 Oct. 2010, Autumn Symposium in String/M Theory, KIAS

2 Introduction

3 Scattering Amplitudes in Gauge Theories Brue-Force calculation does not work! NLO corrections to three-jet production at LHC ~ 10 4 pages! Final answer ~ 1 page! Individual diagrams are NOT gauge invariant, often more complicated than the final sum.

4 On-Shell Methods: Old & New [Bern-Dixon-Kosower ] (Cachazo) Analytic continuation to complex momenta Color decomposition Supersymmetry identities Perturbative string theory Spinor helicity method Berends-Giele recursion relation 1960 s 1970 s 1980 s Twistor string theory BCFW recursion relation 2000 s MHV vertices AdS/CFT (dual super-conformal symmetry)

5 Main Result of this talk [ArkaniHamed-Cachazo-Cheung-Kaplan 0907] Matrix contour integral for planar d = 4, N = 4 SYM L n,k (W) = d k n C vol [GL(k)] δ 4k 4k (C W) M 1 M 2 M n 1 M n dependence on (super-)momenta proven to produce all tree diagrams. (loops in progress.) Matrix contour integral for planar d = 3, N = 6 SCS [SL 1007] L 2k (Λ) = d k 2k C vol [GL(k)] δ k(k+1)/2 (C C T ) δ 2k 3k (C Λ) M 1 M 2 M k 1 M k dependence on (super-)momenta

6 Three ingredients: 4d vs 3d [Penrose 67][Witten 03] [Britto-Cachzo-Feng 04] [BCF+Witten 05] Twistor BCFW recursion relation Dual super-conformal symmetry [Alday-Maldacena 07] [Berkovits-Maldacena 08] [Beisert-Ricci-Tseytlin-Wolf 08] [Drummond-Henn-Korchemsky-Sokatchev 08]

7 S-matrix of d = 3, N = 6 SCS [Bargheer-Loebbert-Meneghelli 1003][SL 1007]

8 N = 6 super-chern-simons [Aharony-Bergman-Jefferis-Maldacena][Benna-Klebanov-Klose-Smedback] [Hosomichi-Lee 3 -Park][Bagger-Lambert][Schnabl-Tachikawa] L = k 4π L CS + L kin + L Yukawa + L potential. [U(N) U(N)] gauge SU(4) R Z α : (N, N;4), Ψ α : (N, N; 4). L CS = µνρ Tr A µ ν A ρ A µa ν A ρ à µ ν à ρ 2 3 õÃνÃρ L kin = Tr D µ Z α D µ Z α + i Ψ α γ µ D µ Ψ α, L Yukawa = i 2 Tr Z α Z α Ψ β Ψ β Z α Z α Ψ β Ψ β + 2Z α Z β Ψ α Ψ β 2 Z α Z β Ψ α Ψ β + αβγδ Z α Ψ β Z γ Ψ δ αβγδ Z α Ψ β Z γ Ψ δ, L potential = 1 12 Tr 6Z α Z α Z β Z γ Z γ Z γ 4Z α Z β Z γ Z α Z β Z γ Z α Z α Z β Z β Z γ Z γ Z α Z β Z β Z γ Z γ Z a.

9 Color Decomposition Color ordered amplitude Full amplitude (permutation symmetry) Color-ordered amplitude (cyclic symmetry only) M a 1,a 2,...,a n n (p 1, p 2,...,p n )= σ S n /Z n Tr(T σ(1) T σ(n) )A n (σ(1),, σ(n)) Color ordered Feynman rules (Less diagrams, simpler Feynman rules) Color ordered amplitudes of SCS6 Number of external legs always even (n = 2k) Cyclic symmetry: shift by two sites A 2k (1, 2,,2k) =( 1) k 1 A 2k (3, 4,,2k, 1, 2)

10 Kinematics SO(1, 2) =SL(2, R) =SU(1, 1) Null momentum in (2+1)d : p αβ p µ (Cγ µ ) αβ = λ α λ β Lorentz invariants : real ij = αβ (λ i ) α (λ j ) β real (incoming) pure imaginary (outgoing) On-shell super-field Φ = φ 4 + η I ψ I IJKη I η J φ K IJKη I η J η K ψ 4, Φ = ψ 4 + η I φ I IJKη I η J ψ K IJKη I η J η K φ 4. ( )=(4+4) U(3) subset of SO(6) R-symmetry manifest ζ I = η I, {ηi, ζ I } = δ I J (I, J = 1, 2, 3) (Clifford algebra) Cyclic symmetry Φ(1)Φ(2) Φ(2k 1)Φ(2k) A 2k = A 2k (1, 2,,2k) =( 1) k 1 A 2k (3, 4,,2k, 1, 2)

11 Super-conformal symmetry SO(1,2) x U(3) subgroup manifest Λ =(λ α, η I ) Λ Λ U(1, 1 3) OSp(4 8) Other super-conformal generators ΛΛ or Λ Λ P αβ = λ α λ β, K αβ = λ α λ β, RIJ = η I η J, R IJ = η I η J, etc. Full covariance obscured?

12 Super-conformal Symmetry Metaplectic representation of SO(2, 3) =Sp(4, R) [z A, z B ]=iω AB, (z A ) = z A = T AB = iz (A z B) agrees with P and K once we identify Quantum mechanics analogy z A =(λ α, µ α ) with µ α = i the phase space parametrized by (λ α, µ α ) λ α SO(2, 3) =Sp(4, R) conformal symmetry acts linearly on Amplitudes A n (λ 1,, λ n ) behave like quantum mechanical wave functions

13 Super-conformal Symmetry Quantum mechanics analogy for fermions ζ I = η I, {ηi, ζ I } = δ I J (I, J = 1, 2, 3) (Clifford algebra) SUSY quantum mechanics analogy with Λ =(λ α, η I ). Super-amplitudes are wave-functions: A(Λ). (expand in fermions to recover component amplitudes) Super-conformal symmetry OSp(4 6) represented by bi-linears of Z (Λ, / Λ). (ΛΛ) Λ Λ 2 Λ Λ

14 Twistors (4d vs 3d)

15 Twistors in 4d [Witten 03] Half-way Fourier transform e ip x e ixα α λ α λ α e i µα λ α d 2 λ δ( µ α x α α λ α ) Twistor equation ( incidence relation ) µ = x 1 λ, µ = x 2 λ = (x 1 x 2 ) λ = 0 = (x 1 x 2 ) 2 = 0. Twistors linearize super-conformal symmetry P α α = λ α λ α λ α µ α K α α = µ α λ α λ α λ α PSU(2, 2 4) SU(2, 2)

16 Twistors in 3d Twistor equation Recall µ α = i. λ α Twistor equation as a wave equation (µ α x αβ λ β ) exp i2 x γδλ γ λ δ Half-twistor vs Full-twistor Super-amplitudes z A =(λ α, µ α ) with µ α = i = 0. λ α reminiscent of a 0 = 0 Λ =(λ α, η I ) A(Λ) are wave-functions in half-twistors. Super-conformal symmetry is represented by full-twistors. Z (Λ, / Λ)

17 BCFW-like Recursion Relation

18 BCFW recursion relation [Britto-Cachazo-Feng(-Witten) 04(05)] Higher point amplitude from lower ones A n = 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 λ j λ j z λ l, λ l λ l + zλ j (p 2 j = 0, p 2 l = 0, p p n = 0 unaffected) A n = A n (z = 0) = dz 2πi A n (z) z (deformed contour, fall-off at infinity, residue theorem)

19 BCFW recursion relation Generalization to d > 4 [ArkaniHamed-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 λ j λ j zλ l, λ l λ l + zλ j (??)

20 BCFW-like relation in 3d [Gang-Koh-SL, work in progress] Momentum conservation in 4d vs 3d (4d) i λ i λ i = 0 GL(n) : λ i M i j λ j, λ i (M T ) i j λ j (3d) i λ i λ i = 0 O(n) : λ i R i j λ j BCFW deformation is a particular GL(2) element. Use O(2) instead of GL(2) in 3d! λj λ l cos θ sin θ sin θ cos θ λj λ l cos θ = z + 1/z 2, sin θ = z 1/z 2i

21 Matrix Contour Integral for Planar d = 3, N = 6 SCS [SL 1007]

22 Generating function L 2k (Λ) = d k 2k C vol [GL(k)] δ k(k+1)/2 (C C T ) δ 2k 3k (C Λ) M 1 M 2 M k 1 M k 2k C = k GL(k) O(2k) (i) (i + k 1) (C Λ) m = C mi Λ i (C C T ) mn = C mi C ni M i = m 1 m k C m1 (i) C m 2 (i+1) C mk (i+k 1)

23 GL(k) Gauge Symmetry L 2k (Λ) = d k 2k C vol [GL(k)] δ k(k+1)/2 (C C T ) δ 2k 3k (C Λ) M 1 M 2 M k 1 M k C SC (S GL(k)) d k 2k C = (det S) 2k d k(k+1)/2 (C C T ) = (det S) (k+1) d 2k 3k (C Λ) = (det S) 2+3 (N = 6 crucial!) (M 1 M 2 M k 1 M k ) 1 = (det S) k

24 Cyclic Symmetry L 2k (Λ) = d k 2k C vol [GL(k)] δ k(k+1)/2 (C C T ) δ 2k 3k (C Λ) M 1 M 2 M k 1 M k Cyclic symmetry is not obvious because only (k) out of (2k) minors appear in the denominator. Cyclic symmetry follows from C C T = 0 = M i M i+1 =( 1) k M i+k M i+1+k in accord with the expectation from field theory.

25 O(2k) Global almost Symmetry L 2k (Λ) = d k 2k C vol [GL(k)] δ k(k+1)/2 (C C T ) δ 2k 3k (C Λ) M 1 M 2 M k 1 M k Momentum conservation i λ α i λβ i = 0 Denominator breaks O(2k) into cyclic symmetry. is O(2k) invariant. R 2k C C T = 0 defines (k) holomorphic vectors in. There should exist (k) anti-holomorphic vectors such that C C T = 0, C C T = I k k = C T C + C T C = I 2k 2k. (completeness relation)

26 Superconformal Invariance L 2k (Λ) = d k 2k C vol [GL(k)] δ k(k+1)/2 (C C T ) δ 2k 3k (C Λ) M 1 M 2 M k 1 M k Linearly realized U(1, 1 3) manifest. Λ Λ Two derivative 2 Λ Λ C C T 0 by δ(c C T ) Multiplication (ΛΛ) Λ T Λ = Λ T (C T C + C T C) Λ 0 by δ(c Λ) (completeness relation)

27 4-point amplitude {i = 1, 2, 3, 4} {r = 1, 3} { s = 2, 4} 4-scalar component amplitude: A 4φ = δ(3) (λ r λ r + λ s λ s ) Partial Fourier transform: A 4φ (λ r, µ s )= A 4φ (λ r, λ s )e iµ sλ s d 4 λ s (change of variable) = d 4 c r s δ (3) (δ rp + c r s c p s ) c 14 c 34 e ic r sλ r µ s. λ ᾱ s = c r s λ α r Inverse Fourier transform back to A 4φ plus fermions give L 4 (Λ) = d 2 4 C vol [GL(2)] δ 3 (C C T ) δ 4 6 (C Λ) M 1 M 2 with C = c12 1 c 32 0 c 14 0 c 34 1

28 6-point amplitude [Bargheer-Loebbert-Meneghelli 1003] (6-scalar) color-stripping reduces number of diagrams (6-fermion)

29 6-point amplitude [Bargheer-Loebbert-Meneghelli 1003] 6-scalar 6-fermion Super-conformal symmetry determines all other component amplitudes.

30 6-point amplitude from the generating function [to appear] Cyclic gauge 1 6 C = c 21 1 c 23 0 c 25 0 c 41 0 c 43 1 c 45 0 c 61 0 c 63 0 c fermion amplitude A 6ψ = +i i [(p 135 ) 2 ] 3 ( i2 p 135 5)( i4 p 135 1)( i6 p 135 3) [(p 135 ) 2 ] 3 (1346 i2 p 135 5)(3562 i4 p 135 1)(5124 i6 p 135 3) agrees with the Feynman diagram computation!

31 6-point amplitude from the generating function [to appear] Factorization gauge C = c 14 c 15 c c 24 c 25 c c 34 c 35 c 36 6-fermion amplitude f A 6ψ = +i i ( i2 p 123 5) 3 (p 123 ) 2 ( i1 p 123 4)( i3 p 123 6) (1346 i2 p 123 5) 3 (p 123 ) 2 (2356 i1 p 123 4)(1245 i3 p 123 6). shows the factorization limit clearly: A 6 (1, 2, 3, 4, 5, 6) A 4 (1, 2, 3, f ) 1 (p 123 ) 2 A 4( f, 4, 5, 6)+(regular).

32 Toward all tree amplitudes L 2k (Λ) = d k 2k C vol [GL(k)] δ k(k+1)/2 (C C T ) δ 2k 3k (C Λ) M 1 M 2 M k 1 M k Net number of integration variables k 2k k 2 k(k + 1)/2 2k + 3 =(k 2)(k 3)/2 C GL(k) δ(c C T ) δ(c λ) δ(λλ) Up to 6-point: delta functions 8-point: one integral variable, ordinary contour integral.

33 8-point amplitude + more [Gang-Koh-SL, work in progress] Contour integral boils down to a sum of residues over 4 pairs of poles. Cyclic symmetry and lambda-parity picks out the unique contour. BCFW for 8 >> confirmed. Choice of contour for ALL TREE amplitudes yet to be done. cf) [ArkaniHamed-Bourjaily-Cachazo-Trnka 0912]

34 Dual Super-conformal & Yangian Symmetry

35 4d: Dual Super-conformal / Yangian Dual super-conformal symmetry y i+1 y i = p i y n+1 y 1 y n p 1 p 2 y 2 First noticed in perturbation theory [Drummond-Henn-Korchemsky-Sokatchev 07] y 3 Physical explanation from AdS/CFT, T-duality & Wilson loops. [Alday-Maldacena 07; Berkovits-Maldacena 08, Beisert-Ricci-Tseytlin-Wolf 08] Original + dual super-conformal >> Yangian [Drummond-Henn-Plefka 09]... mutually non-local and non-linear (spin-chain analogy)

36 Dual super-conformal in 3d?? Fermionic T-duality in AdS4 not enough. [Adam-Dekel-Oz 09, Grassi-Sorokin-Wulff 09] y n+1 y 1 p 1 y n p 2 y 3 y 2 Yangian can be studied without direct reference to dual super-conformal [Drummond-Henn-Plefka 09] Explicit check for 4-point and 6-point amplitudes shows Yangian invariance. [Bargheer-Loebbert-Meneghelli 1003, Huang-Lipstein 1004]

37 Integrability via Yangian Symmetry L 2k (Λ) = d k 2k C vol [GL(k)] δ k(k+1)/2 (C C T ) δ 2k 3k (C Λ) M 1 M 2 M k 1 M k Level one Yangian generators [Drummond-Henn-Plefka 09] J A B = ( 1) C Ji A C Jj C B Jj A C Ji C B i<j, J A i B = Z A i Z Bi, Z A i =(Λ i, / Λ i ). Rewrite the Yangian generators [Drummond-Ferro 1001] J A B = i<j Zi A Z Bj Zi CZ Cj izi A Z Bi (i j). O(2k) generator on Λ δ(c Λ) O(2k) generator on C (integration by parts) Non-local quartic terms cancel against local quadratic terms Full Yangian Invariance Non-trivial action on the denominator

38 Dual super-conformal vs Yangian Original + dual super-conformal >> Yangian Is the converse true in 3d? Likely... [work in progress with Hong, Kim / Huang, Lipstein]

39 Discussion

40 4d vs 3d Twistor BCFW recursion relation Dual super-conformal symmetry

41 Thank you