Scattering Forms and (the Positive Geometry of) Kinematics, Color & the Worldsheet

Transcription

1 Scattering Forms and (the Positive Geometry of) Kinematics, Color & the Worldsheet Song He Institute of Theoretical Physics, CAS with N. Arkani-Hamed, Y. Bai, G. Yan, Dec 2017 QCD Meets Gravity Bhaumik Institute, UCLA Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 1 / 26

2 Motivations Search for theory of S-matrix : no obvious notion of locality or time in the asymptotics look for fundamentally new laws [Nima s talk] Fascinating geometric/combinatorial structures underlying the S-matrix, but all in some auxiliary space M g,n : perturbative strings, twistor strings & scattering eqs. Generalized G + (k, n) : amplituhedron for N = 4 SYM Both geometries have factorizing boundary structures, from which locality and unitarity emerge [Yutin s talk] What questions to ask, directly in the kinematic space, to generate local, unitary dynamics? Avatar of these geometries? Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 2 / 26

3 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 & Positive Geometries QCD Meets Gravity 3 / 26

4 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 & Positive Geometries QCD Meets Gravity 4 / 26

5 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 & Positive Geometries QCD Meets Gravity 5 / 26

6 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 s 23 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. Geometric picture: Feynman-diagram expansion = a special triangulation with spurious pole at infinity [Yuntao s talk] Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 6 / 26

7 Kinematic Associahedron The associahedron polytope encodes combinatorial ( factorization ) structures of planar tree graphs or triangulations of n-gon. Kinematic Associahedron in K n : positive region n with all planar variables s i,i+1,,j > 0, intersect with subspace H n defined by positive constants s i j = c i,j for all non-adjacent pairs 1 i, j < n A n := n H n, dim H n = n 3 s+t=c>0 e.g. A 4 = {s > 0, t > 0} { u = c > 0} Ω (1) 4 H 4 = ( ds s dt t ) u=c>0 = ( 1 s + 1 t ) ds Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 7 / 26

8 s 34 s 12 s 56 s 12 s 15 s 34 s 234 s 23 s 123 s 23 s 345 s 61 s 45 s 45 Pullback of Ω n to H n = Canonical form of A n φ 3 planar amplitude e.g. Ω (2) φ 3 H5 = m( ) d 2 s = ( 5 i=1 ) 1 1 ds 12 ds 34 s i i+1 s i+2 i+3 Also for m(α β). New rep. & hidden properties of φ 3! [Yuntao s talk] Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 8 / 26

9 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 & Positive Geometries QCD Meets Gravity 9 / 26

10 Planar Scattering Forms from Pushforward scattering equations as a diffeomorphism 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. = canonical form of A n is the pushforward of ωn WS by summing over (n 3)! sol. of scattering eqs. (equivalent to CHY) [Yuntao s talk] dµ n PT(α) H(α) = m(α α) d n 3 s or sol. sol. ω WS n (α) = Ω (n 3) (α) φ 3 Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 10 / 26

11 Big Kinematic Space New way of thinking about kin. space: collection of all possible propagators (singularities), rather than dot-products of momenta! K n spanned by abstract variables S I s for non-empty I {1, 2,, n}: S I = S Ī where Ī is the complement of I S I = 0 (or constants) for I = 1, n 1 e.g. K4 is spanned by {S 12 = S 34, S 13 = S 24, S 14 = S 23 }, K5 by the 10 S ij s, and K6 by 15 S ij s and 10 S ijk s In general, dim K n = 2 n 1 n 1 > dim K n (highly redundant). How to go back to K n by imposing relations among S I s? Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 11 / 26

12 The 7-term Identity I 1 I 2 I 1 I 4 I 1 I 3 S I1 I 2 S I2 I 3 S I1 I 3 I 4 I 3 I 3 I 2 I 2 I 4 For every partition into four subsets I 1 I 2 I 3 I 4 = {1, 2,, n}, one imposes the identity consisting of at most 7 terms (S IJ := S I J ) S I1 I 2 + S I2 I 3 + S I1 I 3 = S I1 + S I2 + S I3 + S I4 = all we need to identify S I = s I and recover K n (subspace of K n) Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 12 / 26

13 Scattering Forms in Big Space Recall a cubic (tree) graph g in K n n 3 mutually compatible S I s Define d log form for any ordering α compatible with g: n 3 Π (n 3) (g α) := sign(g α) d log S Ia a=1 with sign(g α) flips for every mutation and every vertex flip. Scattering forms generalize planar ones to all (2n 5)!! cubic graphs: Ω (n 3) [N] = cubic g N(g α g ) Π (n 3) (g α g ) with kinematic numerators, N(g), independent of S I s. Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 13 / 26

14 Projective Scattering Forms Scattering forms invariant under local GL(1) transform S I Λ(S)S I δω[n] = a sum over all triplets of graphs, and each term reads n 4 (N(g s I 1 I 2 I 3 I 4 )+N(g t I 1 I 4 I 2 I 3 )+N(g u I 1 I 3 I 4 I 2 )) d log Λ d log S Ia a=1 I 1 I 2 I 1 I 4 I 1 I 3 S I1 I 2 S I2 I 3 S I1 I 3 I 4 I 3 I 3 I 2 I 2 I 4 g s g t g u Projectivity in K n kinematic Jacobi identity for each triplet: N(g s I 1 I 2 I 3 I 4 )+N(g t I 1 I 4 I 2 I 3 )+N(g u I 1 I 3 I 4 I 2 ) = 0 Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 14 / 26

15 Projectivity in K n as geometric origin of color-kinematics duality! not true with 7-term identity, so what happens in K n? Exist rep. in K n with N s satisfy Jacobi identity require projective covariance instead: Ω[N] Λ d/2 Ω[N] as p a Λ(p) p a, for N(p, ) with mass dimension d is the simplest projective form: either 0 or 2 of the triplets compatible, for the latter e.g. N(g s 1234) = N(g t 1423) = ±1 Ω (n 3) φ 3 Claim: Any projective scattering form is a linear comb. of Ω (n 3) (α) φ 3 Ω (n 3) [N] = π S n 2 N(g π π) Ω (n 3) φ 3 (1, π(2),, π(n 1), n) where the (n 2)! N(g π ) s form a basis for all kin. numerators [Bern et al] Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 15 / 26

16 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 α) := n 2 v=1 f avbvcv = C(g s )+C(g t )+C(g u ) = 0, triplet n 3 Claim : W (g α) := sign(g α) ds Ia = W (g s )+W (g t )+W (g u ) = 0 a=1 Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 16 / 26

17 A basic fact about K n, as it directly follows from 7-term identity: n 4 (ds I1 I 2 + ds I2 I 3 + ds I1 I 3 ) ds Ia = 0 Fundamental link between color & kinematics (forms on K n ) = Duality between color-dressed amps and scattering forms: a=1 M n [N] Ω (n 3) [N] M n [N] = N(g α g )C(g α g ) cubic g I g Ω (n 3) [N] = cubic g N(g α g )W (g α g ) I g Scattering forms are color-dressed amps without color factors! 1 s I 1 s I Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 17 / 26

18 Color is Kinematics II More is true for U(N)/SU(N): trace decomp. as pullback of forms Recall projecting C(g) to 2 n 2 compatible traces with ±1 coeff.: C(g α) = β ( 1) flip(α,β) Tr(β(1),..., β(n)) = M n [N] = β S n/z n Tr(β(1),..., β(n))m n [N; β] partial amp. M n [N; β] = β planar g N(g β) I g Completely parallel: pullback of W (g) to subspaces H(β) defined by {s β(i) β(j) = const.} for all non-adjacent 1 i, j < n { ( 1) flips(α,β) dv [β] β compatible with g W (g α) H[β] = 0 otherwise 1 s I Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 18 / 26

19 Partial amps = pullback of scattering forms to subspace H(β): Ω (n 3) [N] H[β] = 1 dv [β] = M n [N; β]dv [β] β-pl. g N(g β) I g The same subspace that defines A n (β): same geometry but non-d log forms (A n related to Dynkin diagram of SU(N), coincidence???) Analog of [Del Duca, Dixon, Maltoni]: decompose into (n 2)! KK basis s I π(2) π(3) π(n 1) 1 n Ω (n 3) [N] = W (g π π) M n [N; 1, π(2),..., π(n 1), n] π S n 2 Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 19 / 26

20 How to see projectivity in these rep.? A form is projectively covariant = partial amps from DDM/trace decomp. satisfy BCJ relations! Summary: starting from a generic scattering forms, we impose Projective : SU(N) : both(bcj) : π S n 2 N(g π π) Ω φ 3(1, π(2),, π(n 1), n) π S n 2 W (g π π) M n [1, π(2),..., π(n 1), n] π S n 3 F (π)m n [1, π(2),, π(n 2), n 1, n] where F (π) for π S n 3 are projective forms defined by F (π) = ρ S n 3 m 1 (π ρ) Ω φ 3(1, ρ(2),, ρ(n 2), n 1, n) Theories without color: naturally assign 0-form, i.e. amplitude itself. Double-copy a la BCJ/KLT as a map from Ω L,R to the 0-form. Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 20 / 26

21 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 & Positive Geometries QCD Meets Gravity 21 / 26

22 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 & Positive Geometries QCD Meets Gravity 22 / 26

23 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 & Positive Geometries QCD Meets Gravity 23 / 26

24 Beyond Associahedra: General d log Forms How to find all projective, d log scattering forms (N(g) {±1, 0})? What are the associated polytopes & worldsheet forms/functions? Υ Γ (2n 5)!! : Ω (n 3) Υ = n 3 sign(g) d log s Ia g Υ a=1 for connected Υ with 0 or 2 g s for each triplet (& consistent signs). Full classification unknown, and Υ not always convex polytopes! A nice subset called Cayley polytopes: graph associahedra of (n 1)-pt labelled trees (e.g. A n is from linear tree) [Gao, He, Zhang] (1). Υ = positive region (all s I > 0) subspace H Υ (2). its canonical form = Ω Υ HΥ (3). Ω Υ as pushforward from worldsheet forms LS Υ Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 24 / 26

25 Permutohedra & More P n = n H n, n = {s 1,π(2),,π(a) > } H n = {s ij negative const for 2 i < j n 1} Ω (n 3) P n = sgn(π) π S n n d log s 1π(2) π(a) a=2 LS Pn := dµ n π S n 2 PT n (π) Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 25 / 26

26 Outlook Factorization Polytopes : relations to cluster associahedra, gen. permutohedra [Postnikov] & MHV leading singularities [Cachazo] Non-d log Forms : massive φ 3 φ 4 (sing. at infinity) NLSM? Loops : geometrize color for loops; higher genus [Yvonne s talk] Four Dimensions : amplituhedron in momentum space; forms combining helicity amps & pushforward from twistor string Towards a unified geometric picture for amplitudes & beyond Thank you for your attention! Song He (ITP-CAS) Scattering Forms & Positive Geometries QCD Meets Gravity 26 / 26