THE. Secret Life OF SCATTERING AMPLITUDES. based on work with Lionel Mason and with Mat Bullimore. David Skinner - Perimeter Institute

Transcription

1 THE Secret Life OF SCATTERING AMPLITUDES based on work with Lionel Mason and with Mat Bullimore David Skinner - Perimeter Institute Kavli IPMU - 21 st May 2012

2 { Scattering amplitudes are quantum mechanical overlap between states with prescribed asymptotic behaviour β {! Point of contact between theorists and experimentalists Non-linear regime! Encode the dynamics of quantum systems! Live at boundary of α space-time - holographic

3 Perturbatively, scattering amplitudes are usually described in terms of Feynman diagrams In Yang-Mills & gravity - theories we care most about - Feynman diagrams rapidly become very complicated + + +! Complexity directly attributable to gauge redundancy! Necessary consequence of bulk space-time description of massless spin 1 fields

4 Quite remarkably, the amplitude itself was sometimes found to remain very simple + + [Parke,Taylor] + + M MHV n = ij 4 δ 4 ( p i ) n p 2 =0 p α α := p µ σ µ α α = λ α λ α ij := ɛ αβ λ (i) α λ (j) β Standard techniques obscure the true nature of scattering amplitudes

5 Twistors & MHV Diagrams

6 Twistor space is CP 3 CP 3, described by co-ords R 3,1 Z A rz A X Y Z x y x x CP 1 in twistor space Point in space-time Two lines intersect Separation is null X AB = Z [A 1 ZB] 2 X X = 0 X r X

7 One reason twistors provide a good way to describe scattering amplitudes is because they trivialize the external massless field equations } } } } [Penrose] Analytic sol n of wave eq n for massless = Arbitrary holomor phc function of twistors, free field, helicity h homogeneity 2h 2! Depends on three variables - field equations accounted for automatically Φ(x) = λdλ φ(z) X Z A X =(µ α, λ α ) X =(x β α λ β, λ α ) Φ = 2 x µ x µ λdλ φ(z) X = λdλ λ α 2 λ α µ α φ(z) µ α vanishes since antisymmetric! Globally, φ H 1 (CP 3 X, O(2)) on-shell X

8 } Self-dual sol ns } [Penrose; Ward] } Holomorphic vector } of Yang-Mills eq ns bun dls on twistor space! Basis of ADHM construction of instantons; purely algebraic! Related to many integrable systems by choices of symmetry reduction! Similar construction for s.d. gravity - HK / QK manifolds Just as holomorphic functions arise as field equations of S = D 3 Z φ φ, so too holomorphic bundles arise as field equations of holomorphic Chern-Simons theory (A A + 23 ) A3 S = Ω Tr [Witten]! Ω := ɛ is top hol. form on CP 3 4 ABCD Z A dz B dz C dz D d 4 χ A(Z, χ) =a(z)+χ a γ a (Z)+ 1 2 χa χ b φ ab (Z)+ ɛ abcd 3! N =4 multiplet in twistor space [Ferber] χ a χ b χ c γ d (Z)+ ɛ abcd 4! χ a χ b χ c χ d g(z)

9 N =4 S = 1 2 SYM is described on twistor space by the action Ω Tr (A A + 23 A3 ) +g 2 d 4 8 x log det ( + A ) X [Witten; Nair; Boels,Mason,DS]! In axial gauge, Feynman diagrams are MHV diagrams + + +! Reduces to standard form if A is harmonic on each! Similar construction for gravity [Mason, DS] X

10 The Amplitude / Wilson Loop Duality

11 MHV Amplitudes are Null Polygonal Wilson Loops [Alday,Maldacena] p 2 p 2 x 2 x 3 p 1 x i x i+1 = p i p 3 p 1 x 1 p 3 p 4 p 4 x 4 momentum closed massless null conserved polygon particles edges! Amplitude given by area of minimal surface in AdS5! For n =4, 5 agrees with expectation from BDS ansatz

12 The duality between scattering amplitudes and Wilson Loops was also found to hold at weak coupling [Drummond,Henn,Korchemsky,Sokatchev;Brandhuber,Heslop,Travaglini; Bern,Dixon,Kosower,Roiban,Spradlin,Vergu,Volovich] p 2 W = 1 N Tr P exp ( g A ) x 2 x 3 p 1 = 1 + g 2 + O(g 4 ) x 1 p 4 x 4 p 3 = M(0) MHV M (0) MHV + M(1) MHV M (0) MHV +! Underlying reason for duality obscure! Not clear how to extend to arbitrary helicity amplitudes

13 Space-time vertices Twistor lines Null edges of polygon Twistor vertices x 1 Z 1 Z 2 x n x 2 Z n X 1 The twistor data is unconstrained: given arbitrary, the twistor lines intersect by construction, so the corresponding space-time vertices are inevitably null separated. Z i

14 The duality extends to all helicities if one constructs a supersymmetric extension of the Wilson Loop ( + A ) X U(σ 1, σ 2 ) = 0 U(σ 1, σ 1 ) = id [Mason,DS] N =4 twistor superfield Formally, we write ( ) U(σ, σ 0 ) = P exp ω A σ 0 σ where ω is unique meromorphic 1-form with simple poles at {σ 0, σ} U(σ 1, σ 2 )U(σ 2, σ 3 ) = U(σ 1, σ 3 ) U(σ 2, σ 1 ) = U(σ 1, σ 2 ) 1 U(σ 1, σ 2 ) g 1 (σ 1 )U(σ 1, σ 2 )g(σ 2 ) concatenation inverse gauge transform

15 [Bullimore, DS] BCFW Recursion for null polygonal superloops Deform external momenta p i p i (r) subject to the constraints pi (r) = 0 p 2 i (r) = 0 p n (r) p 1 (r) Z n (r) Z n Z 1 In twistor space there Z n1 are no constraints - we just vary the Z i freely

16 As the curve varies, the Wilson Loop obeys δw[c] = C ω d Zī δ Z j Tr [ Fī j (Z) P exp ( C )] ω A! Behaviour of correlator is controlled by loop equations [Migdal,Makeenko; Polyakov] Wilson Loops in real Chern-Simons compute knot invariants such as the HOMFLYPT polynomial [Witten] Naively, varying the curve doesn t change this topological quantity Loop equations give derivation of the skein relations - i.e. recursion relations for the knot polynomial [Cotta-Ramusino,Guadagnini,Martellini,Mintchev]

17 In pure holomorphic Chern-Simons theory, the loop equations give [Bullimore,DS] δw[c(r)] = = = ω C ω C integrate by parts = Tr F (0,2) (Z) P exp Tr δs hcs δa(z) P exp ( ( ω A ω A ) ) hcs ω ω δ 3 4 (Z, Z ) W[C 1 ] W[C 2 ] hcs C C only get contribution if hcs C(r) self-intersects as we deform ω ω δ 3 4 (Z, Z ) W[C 1 ] W[C 2 ] C C in planar limit

18 Z n r Z 1 Z n1

19 Z n Z i1 r Z(r i ) Z i Z 1 Z n1 Z i Intersecting twistor lines Null separation Factorization channel p i1 x i p i p 1 (r) x 1 (r) p n (r)

20 For this deformation, the loop equations give W[C n ] = W[C n1 ] + n1 i=3 This is tree-level BCFW recursion. [n1, n, 1,i1,i] W[C i ] W[C i]! The amplitudes are natural holomorphic analogues of knot invariants, with BCFW recursion as a skein relation!! Repeating the derivation for the full twistor action (including MHV vertices) leads to all-loop generalization n1 W[C n ] = W[C n1 ] + [n1, n, 1,i1,i] W[C i ] W[C i] + i=3 DA DB [n1, n, 1, A, B] W[C AB n+2] [Arkani-Hamed,Bourjaily,Cachazo, Caron-Huot,Trnka; Bullimore, DS] Resulting expressions contain spurious, non-local poles

21 Superloop in sd N =4 Tree superamplitude x i X j X i x j Superloop in full N =4 Planar superamplitude X X j x i x X i x j

22 Grassmannians

23 twistor space momentum twistor space conformal symmetry dual conformal symmetry momentum space Dual descriptions are each representations of a more invariant underlying picture + ks

24 Grassmannian contour integral D k(nk) C Γ (12 k)(23 k+1)... (n1 k1) [Arkani-Hamed,Cachazo et al.; Mason,DS] k r=1 δ 4 4 (C ri Z i ) takes identical form on both sides; neither is preferred! Describes scattering without relying on a space-time interpretation! Different amplitude decompositions = residue theorems b 2 1 Contains all-loop information a 1 b 1 1 b 1 a 2 1 a 2 b 2 n1 a n

25 Original Superconformal Symmetry Dual + Superconformal = Symmetry Infinite Dimensional Yangian [Beisert,Ricci,Tseytlin,Wolf; Berkovits,Maldacena; Drummond,Henn,Plefka]! Reflection of integrability of planar N =4SYM The Yangian is represented on (either) twistor space by J (0) = i Z i Z i J (1) = i>j Z i (Z j Z i ) Z j (i j) plus infinitely many higher generators! Grassmannian formula is uniquely fixed by Yangian [Drummond,Ferro]

26 The amplitudes are defined by a polytope [Hodges] Spurious pole as vertex = =! Different triangulations give different representations

27 At loop level, l-loop amplitudes are given in terms of polylogarithms of transcendentality 2l! Polylogarithms represent relative homology classes on Grassmannians [Goncharov; Hain,MacPherson] Li 1 (z) = z 0 dt 1 t z Li 2 (z) = z CP 1 CP 2 0 dt 1 t ds s 0 1 Suggests loops proper may be obtained from the Grassmannian integral on contours with boundary

28 Towards Strong Coupling

29 At strong coupling the MHV amplitude is given by a minimal surface in AdS5 [Alday,Gaiotto,Maldacena,Sever,Vieira]!"#$%&"'&( "'%)*+',-! Integrable system related to both harmonic maps and wall-crossing in N =2 gauge theories [Gaiotto,Moore,Neitzke]! " & "! "%$! "#$ The strongly-coupled superamplitude should be given by IIB action for a minimal surface on supercoset! Much simpler than including vertex operators & perturbing around bosonic background work in progress...

30 Z CP 3 4 defines a totally null super-ray Z i+1 (x i+1, θ i+1 ) Z i1 Z i (x i, θ i ) x i+1 µ α =ix α α λ α x i χ a = θ αa λ α The corresponding space-time superloop is ( ) 1 Tr P exp i A A = A N α α dx α α + Γ αa dθ αa where superconnection obeys integrability conditions λ α λ β [ bos α α, ferm ] βb =0 λ α λ β { ferm αa, ferm } βb =0

31 Integrability along super null rays is worldline κ-symmetry! Agrees with restriction of Type IIB worldsheet κ-symmetry to boundary [Ooguri,Rahmfeld,Robins,Tannenhauser] Sigma model into coset described by graded Lax connection. Pohlmeyer reduce to account for Virasoro constraints of string [Grigoriev,Tseytlin; Mikhailov,Schafer-Nameki] [Witten] [Gaiotto,Moore,Neitzke] A! Expect a flat PSL(4 4; C) connection on CP 1, wildly ramified at a single point! Different supertwistors associated to each Stokes sector work in progress...

32 Conclusions

33 Polytopes & Knot Invariants Topological Strings Twistor Theory Grassmannian Polylogarithms AdS/CFT Scattering Amplitudes Yangians & Integrability Vast increase in technical power New structures at the heart of QFT

34 Scattering amplitudes have many remarkable properties that are completely invisible from the perspective of Feynman diagrams They are also among precious few observables that can still exist in a diffeomorphism invariant quantum theory Reformulating amplitudes in ways that do not rely on space-time is important preparation for the case that there is no space-time Hopefully, the great technical progress is a good sign that we re on the right track...