Scattering Forms from Geometries at Infinity

Transcription

1 Scattering Forms from Geometries at Infinity Song He Institute of Theoretical Physics, CAS with N. Arkani-Hamed, Y. Bai, G. Yan, work in progress April Focus Week on Quantum Gravity and Holography IPMU, University of Tokyo Song He (ITP-CAS) Scattering Forms from Geometries April / 25

2 Motivations Scattering Amplitudes: crucial for particle physics, and only known observable of quantum gravity in asymptotically flat spacetime Natural holographic question: is there a theory at infinity that computes S-matrix without local evolution? much harder than in AdS of AdS = ordinary space with time & locality = local QFT No such luxuries for asymptotics of flat spacetime: no time/locality! Mystery: what principles a theory of S-matrix should be based on? This is why S-matrix program failed! New strategy in its revival: look for fundamentally new laws ( new mathematical structures) S-matrix as the answer to entirely different kinds of questions discover unitarity and causality, as derived consequences! Song He (ITP-CAS) Scattering Forms from Geometries April / 25

3 Geometric structures Fascinating geometric structures underlying scattering amplitudes (particles, strings etc.) in some auxiliary space, encouraging this p.o.v. M g,n : perturbative string theory, amplitudes = correlators in worldsheet CFT twistor string theory& scattering equations, similar worldsheet picture without stringy excitations Generalized G + (k, n) : the amplituhedron for N = 4 SYM Both geometries have factorizing boundary structures: locality and unitarity naturally emerge (without referring to the bulk) What questions to ask, directly in the kinematic space, to generate local, unitary dynamics? Avatar of these geometries? Song He (ITP-CAS) Scattering Forms from Geometries April / 25

4 Amplitudes as Forms Scattering amps as differential forms on kinematic space a new picture for amplituhedron [Arkani-Hamed, Hugh, Trnka] & much more! Forms on momentum twistor space bosonize superamplitude in N = 4 SYM: replacing η i by dz i = Ω n (4k) for N k MHV tree (tree) Ampliuehedron = positive region 4k-dim subspaces subspace = canonical form of positive geometry [Arkani-Hamed, Bai, Lam] Ω (4k) n Same for momentum-space forms combining helicity amps ( h 1) This talk: identical structure for wide variety of theories in any dim: Bi-adjoint φ 3 from kinematic and worldsheet associahedra Geometrize color & its duality to kinematics, YM/NLSM etc. Song He (ITP-CAS) Scattering Forms from Geometries April / 25

5 Kinematic Space The kinematic space, K n, for n massless momenta p i (D n 1) is spanned by Mandelstam variables s i j s subject to j i s i j = 0, thus dim K n = ( ) n 2 n = n(n 3) 2 ; for any subset I, s I = i<j I s i j Planar variables s i,i+1,,j for an ordering (12 n) are dual to n(n 3)/2 diagonals of a n-gon with edges p 1, p 2,, p n A planar cubic tree graph consists of n 3 compatible planar variables as poles, and it is dual to a full triangulation of the n-gon Claim: all the n(n 3) 2 planar variables form a basis of K n e.g. {s 12 = s, s 23 = t} for K 4, {s 12, s 23, s 34, s 45, s 51 } for K 5, {s 12,, s 61, s 123, s 234, s 345 } for K 6 Song He (ITP-CAS) Scattering Forms from Geometries April / 25

6 The Associahedron The associahedron polytope encodes combinatorial factorization : each co-dim d face represent a triangulation with d diagonals or planar tree with d propagators (vertices planar cubic trees) Universal factorization structures of any massless tree amps (in particular φ 3 ), but how to realize it directly in kinematic space? Song He (ITP-CAS) Scattering Forms from Geometries April / 25

7 Kinematic Associahedron Positive region n : all planar variables s i,i+1,,j 0 (top-dimension) Subspace H n : s i j = c i,j as positive constants, for all non-adjacent pairs 1 i, j < n; we have (n 2)(n 3) 2 conditions = dim H n = n 3. Kinematic Associahedron is their intersection! A n := n H n Proof : s i j = s i,,j s i,,j 1 s i+1,,j +s i+1,,j 1 = c i,j < 0 = no boundaries for crossing diagonals are allowed, e.g. s 12 = s 23 = 0 forbidden ( c 1,3 = s 13 0 leads to contradiction) Equivalently, one can show A n factorizes to A L A R on every face! Song He (ITP-CAS) Scattering Forms from Geometries April / 25

8 e.g. A 4 = {s > 0, t > 0} { u = const > 0} s+t=c>0 A 5 = {s 12,, s 51 > 0} {s 13, s 14, s 24 = const < 0} s 56 s 34 s 12 s 15 s 34 s 234 s 123 s 12 s 23 s 23 s 345 s 61 s 45 s 45 Song He (ITP-CAS) Scattering Forms from Geometries April / 25

9 Planar Scattering Forms The planar scattering form for ordering (12 n) is a sum of rank-(n 3) d log forms for Cat n 2 planar cubic graphs with sign(g) = ±1: Ω (n 3) n := planar g n 3 sign(g) d log s ia,ia+1,,j a Projectivity: invariant under local GL(1) transf. s i,,j Λ(s)s i,,j = Sign-flip rule: sign(g) = sign(g ) for any two planar graphs g, g related by a mutation, i.e. exchange of channel in a 4pt subgraph a=1 2 g 1 3 g s 1,2,3 s 2,3,4, Song He (ITP-CAS) Scattering Forms from Geometries April / 25

10 Projectivity is equivalent to requiring that the form only depends on ratios of variables, e.g. Ω (1) 4 = ds s dt t = d log s t and Ω (2) 5 = ds 12 s 12 ds 34 s 34 + ds 23 s 23 ds 45 s ds 51 s 51 ds 23 s 23 = d log s 12 s 23 d log s 12 s 45 + d log s 12 s 51 d log s 34 Ω (2) 6 = s g=1 ± (d log s)3 = ± d log ratio s It follows immediately that Ω (n 3) is cyclically invariant up to a sign i i+1: Ω (n 3) n ( 1) n 3 Ω (n 3) n, and it factorizes correctly e.g. s 1,,m 0 : Ω n Ω m+1 d log s 1,,m Ω n m+1 Projectivity is a remarkable property of Ω (n 3) n, not true for each diagram or any proper subset of planar Feynman diagrams. Song He (ITP-CAS) Scattering Forms from Geometries April / 25

11 Canonical Form of A n Unique form of any positive geometry= volume of the dual: Ω(A) has d log singularities on all boundaries A with Res = Ω( A) For simple polytopes: v ± d log F for faces F = 0 adjacent to v Canonical form of A n = Pullback of Ω n to H n planar φ 3 amplitude! e.g. Ω(A 4 ) = Ω (1) 4 H 4 = ( ds Ω(A 5 ) = Ω (2) 5 H 5 = t ) u=c>0 = ( 1 s + 1 t ( ) ) ds 1 s 12 s s 51 s 23 ds 12 ds 34 s dt Ω(A n ) = sgn(g) d log s i,,j (s, c) = d n 3 s m(12 n 12 n) Similarly for m(α β): volume of degenerate A n (faces at infinity) Song He (ITP-CAS) Scattering Forms from Geometries April / 25

12 Triangulations & New Rep. of φ 3 Amps Geometric picture: Feynman-diagram expansion = triangulation of the dual into Cat n 2 simplices by introducing the point at infinity Triangulate the dual or itself in other ways new rep. of φ 3 amps! Ω(A 5 ) = d 2 1 s ( s 12 s s 51 s 23 ) ( ) = d 2 s s12 +s 51 s 12 s 34 s 51 + s 12+s 51 s 12 s 51 s 23 + s 12 s 45 +s 23 s 12 s 23 s 45 = sum of 3 triangles of A 5 itself Similar to local or BCFW triangulations of the amplituhedron: manifest new symmetries of φ 3 obscured by Feynman diagrams! Song He (ITP-CAS) Scattering Forms from Geometries April / 25

13 Wordsheet Associahedron A well-known associahedron: minimal blow-up of the open-string worldsheet M + 0,n := {σ 1 < σ 2 < < σ n }/SL(2, R) [Deligne, Mumford] This is non-trivial in σ s but becomes manifest e.g. using cross ratios The canonical form of M + 0,n is the Parke-Taylor form [see also Mizera] ω WS n := 1 vol [SL(2)] n a=1 dσ a σ a σ a+1 := PT(1, 2,, n) dµ n Beautifully planar scattering form of M + 0,n in cross-ratio space. How to connect old and new: M + 0,n A n and their canonical forms? Song He (ITP-CAS) Scattering Forms from Geometries April / 25

14 Scattering Equations as a Diffeomorphism scattering equations as a map from M 0,n to A n With pullback to H n ( s i,j = c i,j ), scattering equations ( s a b b a σ a σ b = 0) provide a one-to-one map from M + 0,n to A n : s a,a+1 = σ a,a+1 1<i+1 a j<n c i,j σ i,j for a = 1,..., n 3 (σ n ) and similarly for s a,a+1,,b : positive iff {σ} M + 0,n (on H n)! One (out of (n 3)!) positive solution iff positive kinematics {s} n. Song He (ITP-CAS) Scattering Forms from Geometries April / 25

15 Pushforward from Worldsheet Theorem: diffeomorphism A B = pushforward Ω(A) Ω(B) y = f(x) as diffeom. from A to B : Ω(B) y = = canonical form of A n is the pushforward of ωn WS over (n 3)! sol. of scattering eqs. (equivalent to CHY) sol. x=f 1 (y) Ω(A) x by summing dµ n PT(α) H(α) = m(α α) d n 3 s or ωn WS (α) = Ω (n 3) (α) φ 3 sol. sol. General : dµ n I n := Ω n [I] Ω n [I] Hα = d n 3 s PT(α) I n CHY Song He (ITP-CAS) Scattering Forms from Geometries April / 25

16 General Scattering Forms General Scattering Forms: sum over all cubic graphs with numerators Ω[N] = g N(g) n 3 I=1 d log s I, e.g. N s d log s + N t d log t + N u d log u N g are kinematic numerators that can depend on other data, for all (2n 5)!! cubic tree graphs, e.g. 15 for n = 5 and 105 for n = 6. Ω φ 3: N g = 0 for non-planar graphs and N g = ±1 for planar ones. Natural Qs: what constraints can we put on these forms and what physics information do they contain? How can we relate them to scattering amplitudes of some theories? Projectivity is the key! Song He (ITP-CAS) Scattering Forms from Geometries April / 25

17 Projectivity and BCJ duality Projectivity: require Ω[N] to be well-defined in projectivized space (treating s I s independently), i.e. covariant under s I Λ(s)s I = kinematic numerators can be chosen to satisfy Jacobi identities N(g S ) + N(g T ) + N(g U ) = 0, e.g. N s + N t + N u = 0 I 1 I 2 I 1 I 4 I 1 I 3 S = s I1 I 2 T = s I2 I 3 U = s I1 I 3 I 4 I 3 I 3 I 2 I 2 I 4 g S g T g U A geometric origin of BCJ duality [BCJ 08], but how to get amps? Song He (ITP-CAS) Scattering Forms from Geometries April / 25

18 Color is Kinematics I f a 1a 2 b f ba 3c f ca 4a 5 ds 12 ds 45 Duality between color factors and differential forms on K n for cubic graphs: C(g) and W (g) satisfy the same algebra! Recall C(g) := n 2 v=1 f avbvcv = C(g S )+C(g T )+C(g U ) = 0, triplet n 3 Claim : W (g) := ± ds I = W (g S )+W (g T )+W (g U ) = 0 I=1 Song He (ITP-CAS) Scattering Forms from Geometries April / 25

19 This is a basic fact of K n directly follows from mom-conservation: s I1 I 2 +s I2 I 3 +s I1 I 3 = s I1 +s I2 +s I3 = (ds I1 I 2 +ds I2 I 3 +ds I1 I 3 ) = 0 Fundamental link between color & kinematics (forms on K n ) = Duality between color-dressed amps and scattering forms: M n [N] Ω (n 3) [N] M n [N] = N(g)C(g) cubic g I g Ω (n 3) [N] = 1 s I cubic g N(g)W (g) I g Scattering forms are color-dressed amps without color factors! 1 s I Song He (ITP-CAS) Scattering Forms from Geometries April / 25

20 Color is Kinematics II More is true for U(N)/SU(N): partial amps as pullback of forms trace decomp. M n [N] = = partial amp. M n [N; β] = β S n/z n Tr(β(1),..., β(n))m n [N; β] β planar g N(g β) I g Completely parallel: Partial amplitude = pullback of scattering form to subspace H(β) = {s β(i) β(j) = const.} for non-adjacent 1 i < j < n Ω (n 3) [N] H[β] = β-pl. g N(g β) I g 1 s I 1 dv [β] = M n [N; β]dv [β] s I Song He (ITP-CAS) Scattering Forms from Geometries April / 25

21 Gravity Amplitudes and Double Copy How about theories without color, such as gravity amplitude? A 0-form or equivalently top form, Ω top = M n d n(n 3)/2 s Define dual forms for every scattering form: e.g. the dual for φ 3 Ω φ 3(1, 2,, n) := 1 i<j 1<n 1 ds i,j, = pullback to partial amp, e.g. Mn YM (α)d n(n 3)/2 s = Ω YM Ω φ 3(α) Natural language for BCJ double-copy : top form for e.g. gravity is literally the (wedge) product of a Ω YM and its dual Ω YM : Ω top GR = Ω(n 3) YM Ω (n 2)(n 3)/2 YM = d n(n 3) s g I g N(g)Ñ(g) s I Song He (ITP-CAS) Scattering Forms from Geometries April / 25

22 Scattering Forms for Gluons and Pions Permutation invariant forms encoding full amps of gluon/pion scattering (e.g. can be directed constructed from Feynman rules ) Remarkably rigid, fundamental objects for YM & NLSM (lowest dim): (2n 5)!! Ω (n 3) YM/NLSM = g n 3 N YM/NLSM (g) a=1 d log S Ia N NLSM ({s}) of degree-(n 2) in s i,j ; N YM ({ɛ, p}) from contractions of momenta & polarizations with no more than (n 2) (ɛ i p j ) (rep. dependent and can be chosen to satisfy Jacobi). For n = 4: Ω (1) NLSM = sdt tds = tdu udt = uds sdu Ω (1) YM = T 8(ɛ,p) stu Ω (1) NLSM = N YM(g 1234 )d log s t + N YM(g 1324 )d log u t Song He (ITP-CAS) Scattering Forms from Geometries April / 25

23 Uniqueness of YM and NLSM Forms Gauge invariance: Ω YM invariant under every shift ɛ µ i ɛµ i + αpµ i Adler zero: Ω NLSM vanishes under every soft limit p µ i 0 Key: forms are projective = unique Ω YM and Ω NLSM! Stronger than the amp uniqueness theorem [Arkani-Hamed, Rodina, Trnka]: (n 1)! parameters for amp vs. unique form up to an overall const. Proof: (1). Ω (n 3) ansatz = π S n 2 W (g π )A n (π) with partial amps A n (π) sum over π-planar graphs (2). A n (π) = α π M n (π) by amp uniqueness, and α π = α since A n (π) s must satisfy BCJ relations by projectivity! Direct proof/deeper reason for uniqueness of forms? Extended positive geometry for gluons ( geometrize polarizations) & pions? Song He (ITP-CAS) Scattering Forms from Geometries April / 25

24 YM and NLSM Forms from the Worldsheet Despite lack of geometric meaning, non-d log projective forms can also be obtained as pushforward of non-d log worldsheet forms Ω[N] = π N(g π)ω φ 3(π) = Ω[N] = sol. ω[n], ω[n] dµ n π N(g π) PT n (π), means = up to scattering eqs Ω YM/NLSM as pushforward of canonical, rigid worldsheet objects: Ω (n 3) YM = sol. dµ n Pf Ψ n ({ɛ, p, σ}) Ω (n 3) NLSM = dµ n det A n ({s, σ}) sol. = at this order, Pf Ψ n (or det A n ) is the unique gauge inv. (or Adler zero) worldsheet function, on support of scattering eqs! Question: why Pf Ψ n also determines complete superstring amps? Song He (ITP-CAS) Scattering Forms from Geometries April / 25

25 Outlook Factorization Polytopes : relations to cluster associahedra, gen. permutohedra [Postnikov] & MHV leading singularities [Cachazo] Loops : geometrize color for loops; ambitwistor strings Four Dimensions : amplituhedron in momentum space; forms combining helicity amps & pushforward from twistor string Beyond amplitudes: Witten diagrams, cosmological polytopes etc. Towards a unified geometric picture for amplitudes & more Thank you for your attention! Song He (ITP-CAS) Scattering Forms from Geometries April / 25