Ambitwistor strings and the scattering equations at one loop Lionel - PowerPoint PPT Presentation

Ambitwistor strings and the scattering equations at one loop Lionel Mason The Mathematical Institute, Oxford lmason@maths.ox.ac.uk IHES 15 October 2015 With David Skinner. arxiv:1311.2564, and collaborations with Tim Adamo, Eduardo Casali, Yvonne Geyer, Arthur Lipstein, Ricardo Monteiro, Kai Roehrig, & Piotr Tourkine, 1312.3828, 1404.6219, 1405.5122, 1406.1462, 1506.08771, 1507.00321. [Cf. also Cachazo, He, Yuan arxiv:1306.2962, 1306.6575, 1307.2199, 1309.0885, 1412.3479]

Ambitwistors Ambitwistor spaces: spaces of complex null geodesics. • Extends Penrose/Ward’s gravity/Yang-Mills twistor constructions to non-self-dual fields. • Yang-Mills Witten and Isenberg, et. al. 1978, 1985 . • Conformal and Einstein gravity LeBrun [1983,1991] Baston & M. [1987] . Ambitwistor Strings : • Tree S-Matrices in all dimensions for gravity, YM etc. [CHY] • From strings in ambitwistor space [M. & Skinner 1311.2564] • New models for Einstein-YM, DBI, BI, NLS, etc. [Casali, Geyer, M., Monteiro, Roehrig 1506.08771] . • Loop integrands from the Riemann sphere [Geyer, M., Monteiro, Tourkine, 1507.00321]. Provide string theories at α ′ = 0 for field theory amplitudes.

Amplitudes from Feynman diagrams Amplitudes are realized as sums of Feynman integrals. 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. Trees ↔ classical, loops ↔ quantum.

Need for new ideas Result of a brute force calculation: k 1 · k 4 ε 2 · k 1 ε 1 · ε 3 ε 4 · ε 5

The scattering equations Take n null momenta k i ∈ R d , i = 1 , . . . , n , k 2 i = 0, � i k i = 0, • define P : CP 1 → C d σ 2 σ 1 n k i σ n � σ, σ i ∈ CP 1 P ( σ ) := , . σ − σ i i = 1 • Solve for σ i ∈ CP 1 with the n scattering equations [Fairlie 197?] n k i · k j � P 2 � � = k i · P ( σ i ) = = 0 . Res σ i σ i − σ j j = 1 ⇒ 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] .

Amplitude formulae for massless theories. Proposition (Cachazo, He, Yuan 2013,2014) Tree-level massless amplitudes in d-dims are integrals/sums �� � � i ¯ I l I r � δ ( k i · P ( σ i )) M n = δ d k i Vol SL ( 2 , C ) × C 3 ( CP 1 ) n i where I l / r = I l / r ( ǫ l / r , k i , σ i ) depend on the theory. i • polarizations ǫ l i for spin 1, ǫ l i ⊗ ǫ r i for spin-2 ( k i · ǫ i = 0 . . . ). � A � C • Introduce skew 2 n × 2 n matrices M = , − C t B A ij = k i · k j ǫ i · ǫ j C ij = k i · ǫ j , , B ij = , , for i � = j σ i − σ j σ i − σ j σ i − σ j and A ii = B ii = 0, C ii = ǫ i · P ( σ i ) . • For YM, I l = Pf ′ ( M ) , I r = � 1 σ i − σ i − 1 . i • For GR I l = Pf ′ ( M l ) , I r = Pf ′ ( M r ) .

More CHY formulae: “compactify” Gravity BI compactify compactify “compactify” squeeze EM DBI generalize single trace “compactify” EYM YM NLSM compactify φ 4 squeeze YMS corollary generalize gen. YMS Figure: Theories studied by CHY and operations relating them.

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 . � ∂ X µ − e P 2 / 2 . P µ ¯ S B = Σ Underlying geometry: • e enforces P 2 = 0, • P 2 generates gauge freedom: δ ( X , P , e ) = ( α P , 0 , 2 ¯ ∂α ) . So target is A = T ∗ M | P 2 = 0 / { gauge } . This is Ambitwistor space , space of complexified light rays.

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 P A that preserve θ = P µ d X µ .

Amplitudes from ambitwistor strings Quantize bosonic ambitwistor string: • ( X , P ) : Σ → T ∗ M , � e ∂ ) X µ − e P 2 / 2 . P µ (¯ ∂ + ˜ S B = Σ • Gauge fix ˜ e = 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 Ψ 1 + S Ψ 2 where � ∂ Ψ µ + χ P · Ψ . Ψ µ ¯ S Ψ = Σ

From deformations of A to the scattering equations � Gravitons ↔ vertex operators V i = def’m of action δ S = Σ δθ . • θ determines complex structure on P A via θ ∧ d θ d − 2 . So: • Deformations of complex structure ↔ [ δθ ] ∈ H 1 ∂ ( P A , L ) . ¯ Proposition For perturbation δ g µν = e ik · x ǫ µ ǫ ν of flat space-time δθ = ¯ δ ( k · P ) e ik · X ( ǫ · P ) 2 Proof: Penrose transform. 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.

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

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

Worldsheet matter • Decorate null geodesics with spin vectors, vectors for internal degrees of freedom & other holmorphic CFTs. • Take 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 �� � � ′ ¯ I l I r � δ ( k i · P ) M n = δ d i k i Vol Gauge ( CP 1 ) n i where I l , I r are worldsheet correlators of v l s, v r s resp.. • In good situations, Q -invariance and discrete symmetries (GSO) rule out unwanted vertex operators.

Worldsheet matter models • Worldsheet SUSY: Let Ψ µ ∈ K 1 / 2 , spin 1/2 fermions on Σ , � g µν Ψ µ ¯ ∂ Ψ ν − χ P µ Ψ µ S Ψ = Replace v = ǫ · P by v = ǫ · P + ǫ · Ψ k · Ψ (or u = δ ( γ ) ǫ · Ψ ). Worldsheet correlator I l / r = � u 1 u 2 v 3 . . . v n � = Pf ′ ( M ) . • Free fermions and current algebras: Free ‘real’ Fermions ρ a ∈ C m ⊗ K 1 / 2 � δ ab ρ a ¯ ∂ρ b , S ρ = a = 1 , . . . m , Σ With Lie alg structure const f abc , set v = t a f abc ρ b ρ c . Correlators ❀ ‘Parke-Taylor’ + unwanted multi-trace terms tr ( t 1 . . . t n ) � v 1 . . . v n � = + . . . σ 12 σ 23 . . . σ n 1 where σ ij = σ i − σ j .

Comb system [Casali-Skinner] ρ a , ρ a ∈ g ⊗ K 1 / 2 , bosons q a , y a ∈ g ⊗ K 1 / 2 Use fermions ˜ � [˜ � � ρ, ρ ] ∂ρ a + q a ¯ ∂ y a + χ tr ρ ρ a ¯ S CS = ˜ + [ q , y ] . 2 Σ • Gauge fix χ = 0 ❀ ghosts ( β, γ ) ❀ two fixed vertex operators to end chain of structure contants ‘comb’. • Vertex ops: u = δ ( γ ) t · ρ , u = δ ( γ ) t · ˜ ˜ ρ , (fixed) ˜ v = t · [ ρ, ρ ] , v = t · ([˜ ρ, ρ ] + [ q , y ]) . • To be nontrivial, correlator must have just one untilded VO v n � = C ( 1 , . . . , n ) := tr ( t 1 [ t 2 , [ t 3 , . . . [ t n − 1 , t n ] . . . ]) � u 1 ˜ u 2 ˜ v 3 . . . ˜ . σ 12 . . . σ n 1

The 2013 CHY formulae & ambitwistor models Above lead essentially to original models & formulae: • ( S l , S r ) = ( S ˜ Ψ , S Ψ ) ❀ type II gravity, • ( S l , S r ) = ( S CS , S Ψ ) ❀ heterotic with YM, • ( S l , S r ) = ( S CS , S CS ) ❀ bi-adjoint scalar. The latter two come with unphysical gravity. S CS improves on current algebras in avoiding multi-trace terms and all models critical in 10d.