Perturbative formulae for gravitational scattering

Transcription

1 Perturbative formulae for gravitational scattering Lionel Mason The Mathematical Institute, Oxford IHP, Paris, 24/9/2015 With David Skinner. arxiv: , Yvonne Geyer & Arthur Lipstein , , + Monteiro, Tourkine , [Cf. also Cachazo, He, Yuan arxiv: , , , ]

2 Ambitwistors and gravity Ambitwistors were developed in s to generalize Penrose s nonlinear graviton construction to non-self-dual fields. Penrose s scattering formulae [1972]. Yang-Mills Witten and Isenberg, Yasskin & Green [1978]. Kinematics for gravity LeBrun [1983]. Field equations for conformal (and Einstein) gravity Baston, M, [1987] LeBrun [1991]. Over last two years, things have sped up: New gravity perturbative scattering formulae in all dimensions [Cachazo, He & Yuan , ] Arise from ambitwistor strings [M. & Skinner ] Related to I, null geodesic scattering and the BMS group [Geyer, Lipstein & M ].

3 Null infinity and conformal scattering theory For Minkowski space in spherical polars (t, r, ζ, ζ), ζ = e iφ tan(θ/2), let u = t r, R = 1 r and rescale Minkowski metric to find R 2 g = 2du dr + dζd ζ (1 + ζ 2 ) 2 + R2 du 2. Then I + := {R = 0} = S 2 R. Similarly I obtained from v = t + r. i 0 ւ u ր տ v ց i i + (ζ, ζ) Evolution of characteristic data from I I + gives S-matrix via conformal scattering theory [see JPN later]. I + I

4 Gravitational phase space [Ashtekar 1981] On an asymptotically simple (CKN) curved space-time we have R 2 ds 2 = dudr + dζd ζ) (1 + ζ 2 ) 2 + R σdζ2 + c.c. (1 + ζ 2 ) 2 + O(R2 ). Asymptotic gravitational data = σ(u, ζ, ζ) shear. Let H + be phase space of such σ. H + has symplectic structure Ω(σ, σ ) = ( σ u σ du d 2 ζ + c.c.) I (1 + ζ 2 ) 2. Symmetries: BMS group= Lorentz super-translations. 1 Lorentz= Mobius transformations in ζ, 2 (Super-)translations = {δu = f (ζ, ζ)}. Supertranslation Hamiltonian H f [σ] = f u σ 2 du d 2 ζ I (1 + ζ 2 ) 2. So finite energy u σ L 2 I.

5 Christodoulou-Klainerman-Niccolo space-times and gravitational scattering A CKN space (M, g) is a small perturbation of Minkowski space-time. Takes small initial data g 0 on Σ 0 at t = 0 to null infinity: u ր ւ i + I + g 0 σ ± H ± scattering map S : H H + Must solve Goursat problem for small σ ± (not yet done in CKN). i 0 տ v ց i Σ 0 I

6 The classical S-matrix Classical S-matrix is generating function for scattering: given σ H solve Goursat problem for g on M. Then S[σ ] = (renormalized) Einstein action S EG [g] of g on M Scattering σ + = S (σ ) is determined by Ω(σ +, σ ) = d dɛ S[σ + ɛσ ]. ɛ=0 Note: uses identification of I + with I infrared issues. Usually evaluate S-matrix perturbatively For data σ use linearised g in = n j=1 ɛ jg j I For g j take Fourier modes g jµν = ξ jµ ξ jν e ik j x : momentum k j, kj 2 = 0. polarization data satisfies k ξ = 0, ξ ξ + αk. Then n-particle scattering given by M n := M(k 1, ξ 1,..., k n, ξ n ) = Coeff of i ɛ i in S EG [g]

7 , - $ & ' % ( ) * + ( ' Feynman Diagrams Feynman diagrams Amplitudes When forces are arerealized weak theasbasic sums tool offor Feynman describing integrals. interaction of elementary particles is Feynman diagrams. Feynman diagrams are more than pictures. They represent algebraic formulas for the propagation and interaction of particles.!#" % ) & * % )! ' $ " & (. + " / Trees classical, loops quantum. 10

8 Examples of Clever Ideas Need for new ideas Consider the five-gluon tree-level amplitude of QCD. Enters in calculation of multi-jet production at hadron colliders. Described by following Feynman diagrams: If you follow the textbooks you discover a disgusting mess.

9 Result of a brute force calculation: k 1 k 4 ε 2 k 1 ε 1 ε 3 ε 4 ε 5 23

10 The scattering equations Take n null momenta k i R d, i = 1,..., n, ki 2 = 0, i k i = 0, define P : CP 1 C d σ 2 P(σ) := n i=1 k i σ σ i, σ, σ i CP 1 σ 1 σ n. Solve for σ i CP 1 with the n scattering equations [Fairlie 197?] Res σi (P 2) = k i P(σ i ) = n j=1 k i k j σ i σ j = 0. P 2 = 0 σ. For Mobius invariance P C d K, K = Ω 1,0 CP 1 There are (n 3)! solutions. Arise in large α strings [Gross-Mende 1988] & twistor-strings [Roiban, Spradlin, Volovich, Witten 2004].

11 Amplitude formulae for massless theories. Proposition (Cachazo, He, Yuan 2013,2014) Tree-level massless amplitudes in d-dims are integrals/sums M n = δ d ( i k i ) (CP 1 ) n I l I r i δ(k i P(σ i )) Vol (SL(2, C) C 3 ) where I l/r = I l/r (ξ l/r i, k i, σ i ) depend on the theory. polarizations ξi l for spin 1, ξi l ξi r for spin-2 (k i ξ i = 0... ). ( ) A C Introduce skew 2n 2n matrices M = C t, B A ij = k i k j σ i σ j,, B ij = ξ i ξ j σ i σ j, and A ii = B ii = 0, C ii = ξ i P(σ i ). For gravity, I l = Pf (M l ), I r = Pf (M r ). For Yang-Mills, I l = Pf (M), I r = i C ij = k i ξ j σ i σ j, 1 σ i σ i 1. for i j

12 Chiral bosonic strings at α = 0 Bosonic ambitwistor string action: Σ Riemann surface, coordinate σ C Complexify space-time (M, g), coords X C d, g hol. (X, P) : Σ T M, P K, holomorphic 1-forms on Σ. S B = P µ X µ e P 2 /2. Σ Underlying geometry: e enforces P 2 = 0, P 2 generates gauge freedom: δ(x, P, e) = (αp, 0, 2 α). So target is A = T M P 2 =0 /{gauge}. This is Ambitwistor space, space of complexified light rays.

13 The geometry of the space of light rays Ambitwistor space A is space of complexified light rays. Light rays primary, events determined by lightcones X A of light rays incident with x. Space-time M = space of such X A. Space-time Twistor Space x x X X Z Space-time geometry is encoded in complex structure of A. Theorem (LeBrun 1983 following Penrose 1976) Complex structure of A determines (M, [g]). Correspondence stable under deformations of PA that preserve θ = P µ dx µ.

14 Amplitudes from ambitwistor strings Quantize bosonic ambitwistor string: (X, P) : Σ T M, S B = P µ ( + ẽ )X µ e P 2 /2. Σ Gauge fix ẽ = e = 0, ghosts & BRST Q Introduce vertex operators V i field perturbations. Amplitudes are computed as correlators of vertex ops M n = V 1... V n For gravity add type II worldsheet susy S Ψ1 + S Ψ2 where S Ψ = Ψ µ Ψ µ + χp Ψ. Σ

15 From deformations of A to the scattering equations Gravitons vertex operators V i = def m of action δs = Σ δθ. θ determines complex structure on PA via θ dθ d 2. So: Deformations of complex structure [δθ] H (PA, L). 1 Proposition For perturbation δg µν = e ik x ξ µ ξ ν of flat space-time δθ = δ(k P)e ik X (ɛ P) 2 Ambitwistor repn δ(k P) scattering equs. Proposition CHY formulae for massless tree amplitudes e.g. YM & gravity arise from appropriate choices of worldsheet matter.

16 Evaluation of amplitude Take e ik i X(σ i ) factors into action to give S = 1 P 2π X + 2π ik X(σ i ). Σ i Gives field equations X = 0 and, P = 2π i ikδ 2 (σ σ i ). Solutions X(σ) = X = const., P(σ) = i σ σ i dσ. Thus path-integral reduces to M n = δ d ( i k i ) (CP 1 ) n 3 i (ɛ i P(σ i )) 2 δ(k i P) Vol G We see P(σ) appearing and scattering equations. Unfortunately: amplitudes for S M R + R3. k i

17 Evaluation of amplitude Take e ik i X(σ i ) factors into action to give S = 1 P 2π X + 2π ik X(σ i ). Σ i Gives field equations X = 0 and, P = 2π i ikδ 2 (σ σ i ). Solutions X(σ) = X = const., P(σ) = i σ σ i dσ. Thus path-integral reduces to M n = δ d ( i k i ) (CP 1 ) n 3 i (ɛ i P(σ i )) 2 δ(k i P) Vol G We see P(σ) appearing and scattering equations. Unfortunately: amplitudes for S M R + R3. k i

18 Take Worldsheet matter for Einstein gravity S = S B + S l + S r where S l, S r are some worldsheet matter CFTs. Total vertex operators given by v l v r δ(k P) e ik X with v l, v r worldsheet currents from S l, S r resp.. Amplitudes become M n = δ d ( i k i ) (CP 1 ) n I l I r i δ(k i P) Vol Gauge where I l, I r are worldsheet correlators of v l s, v r s resp.. For I l/r = Pf (M l/r ) take S Ψ = Σ Ψ µ Ψ µ + χp Ψ and v = δ(k P)(ξ P + ξ Ψk Ψ) 2 e ik X. i.e., S l, S r are worldsheet supersymmetries for gravity. Q 2 = 0 critical dimension d = 10 of standard string theory.

19 Ambitwistor strings with combinations of matter Different choices of worldsheet matter give different theories. CGMMRS S l S r S Ψ S Ψ1,Ψ 2 S ( m) ρ,ψ S (Ñ) CS,Ψ S (Ñ) CS S Ψ E S Ψ1,Ψ 2 BI Galileon S (m) ρ,ψ EM DBI EMS U(1) m U(1) m U(1) m S (N) CS,Ψ EYM ext. DBI EYMS SU(N) U(1) m S (N) CS YM Nonlinear σ EYMS SU(N) U(1) m EYMS SU(N) SU(Ñ) gen. YMS SU(N) SU(Ñ) Biadjoint Scalar SU(N) SU(Ñ) Table: Theories arising from the different choices of matter models.

20 Models from different geometric realizations of A Can start with other formulations of null superparticles Pure spinor version (Berkovits) S = P X + p α θ α d = 4, Twistor-strings of Witten, Berkovits & Skinner A = {(Z, W ) T T Z W = 0}/{Z Z W W } S = W Z + a Z W Σ In 4d have full ambitwistor representation [Geyer, Lipstein, M ] S = Z W W Z + az W Σ Not twistor string: (Z, W ) K 1/2 gives simpler 4d formulae with no moduli. Nonchiral, working with no supersymmetry. Also can adapt to geometry of null infinity and conformal scattering theory.

21 Relation to null infinity, BMS and soft gravitons Take space-time asymptotically simple: Geyer,Lipstein & M (cf. Adamo, Casali & Skinner , ). Real light rays intersect I + and I, A R = T I + = T I. Flat space-time: identification is identity and global. Curved space-time: identification only near real light rays: A = T I + C T I C glued over A R. Infinitesimally glued by Hamiltonian h for light ray scattering Vertex op = δθ = h.

22 Adapting A to I and soft vertex operators Penrose null geodesic scattering formulae Parametrize I by (u, P µ ) (λu, λp µ ) with P 2 = 0; P µ is a projective parametrization of the sphere on I. Correspondingly parametrize T I by (u, P µ, w, q µ ) symplectic potential θ = wdu + q µ dp µ Constraint ru P q = 0. Metric perturbation δg µν = ξ µ ξ ν e ik x gives deformation of gluing generated by Hamiltonian Soft limits: h = w(ξ P)2 e ik q k P as k 0, h w(ξ P) 2 /k P, a supertranslation generator. subleading terms give superrotations.

23 BMS and soft gravitons is worldsheet Ward identity for supertranslation invariance. BMS ± group acts on I ± hence on T I ± with Hamiltonians h. worldsheet generators have same form as Vertex Ops: h = h. Σ Soft gravitons: k 0 then h generates supertranslation Σ δu = (ξ P) 2 /k P. Subleading term generates superrotation. Diagonal subgroup BMS + BMS are symmetries. versions of (subleading) soft graviton thm as Ward identity. Theorem (Following Strominger/Weinberg) Weinberg soft theorem: as k n 0 n 1 M(1,..., n) M(1,..., n 1) i=1 (ξ n k i ) 2 k n k i,

24 Summary Ambitwistor strings provide a reformulation of general relativity that is extremely efficient in the perturbative regime. Worldsheet quantization of ambitwistor string yields gravitational scattering from that of null geodesics. Extends to supergravity and many other theories. Incorporates colour/kinematics Yang-Mills/gravity ideas. Relates Weinberg s soft limits of scattering to supertranslation symmetry via worldsheet cft. Formalism correctly computes 1-loop amplitudes. [Adamo, Casali, Skinner, Geyer, M., Monteiro, Tourkine] Many open questions Proper analytic understanding of conformal scattering, infrared divergences, soft theorems. Scattering on nontrivial backgrounds [Adamo, Casali, Skinner]. Why is this equivalent to Einstein gravity!

25 Thank You!