Loop Amplitudes from Ambitwistor Strings

Transcription

1 Loop Amplitudes from Ambitwistor Strings Yvonne Geyer QCD meets Gravity Workshop Dec 5-9, 2016 UCLA arxiv: , , YG, L. Mason, R. Monteiro, P. Tourkine

2 Motivation: worldsheet models String Theory α 0 (0) + (1) +... integration over moduli space: g 0 M g,n (... ) map: Σ M

3 Motivation: worldsheet models String Theory α 0 (0) + (1) +... integration over moduli space: g 0 (... ) M g,n map: Σ M Worldsheet models of QFT [Witten, Berkovits, RSV; Hodges, Cachazo-YG, Skinner-Mason] = (0) + (1) +... map: Σ T CP 3 4 D = 4, maximal supersymmetry

4 Motivation: worldsheet models String Theory α 0 (0) + (1) +... integration over moduli space: g 0 (... ) M g,n map: Σ M Worldsheet models of QFT [Witten, Berkovits, RSV; Hodges, Cachazo-YG, Skinner-Mason] = (0) + (1) +... map: Σ T CP 3 4 D = 4, maximal supersymmetry

5 Motivation: worldsheet models String Theory α 0 (0) + (1) +... integration over moduli space: g 0 (... ) M g,n map: Σ M Worldsheet models of QFT [CHY, Skinner-Mason, Adamo-Casali-Skinner, CGMMR] E (g) i = 0 = (0) + (1) +...

6 Motivation: worldsheet models String Theory α 0 (0) + (1) +... integration over moduli space: g 0 (... ) M g,n map: Σ M Worldsheet models of QFT [CHY, Skinner-Mason, Adamo-Casali-Skinner, CGMMR] E (g) i = 0 = (0) + (1) +... integration fully localised on the Scattering Equations {E (g) i } map: Σ A = {phase space of complex null rays} upshot: generic for massless QFTs

7 Scattering Equations [Cachazo-He-Yuan, Mason-Skinner] Construction: for n null momenta k i, define P µ (σ, σ i ) Ω 0 (Σ, K Σ ) n k iµ P µ (σ) = dσ. σ σ i i=1 z 1 z 2 z n

8 Scattering Equations [Cachazo-He-Yuan, Mason-Skinner] Construction: for n null momenta k i, define P µ (σ, σ i ) Ω 0 (Σ, K Σ ) n k iµ P µ (σ) = dσ. σ σ i i=1 z 1 z 2 z n Scattering Equations at tree-level Enforce P 2 = 0 on Σ: E i = Res σi P 2 k i k j (σ) = k i P(σ i ) = = 0. σ i σ j j i

9 Scattering Equations [Cachazo-He-Yuan, Mason-Skinner] Construction: for n null momenta k i, define P µ (σ, σ i ) Ω 0 (Σ, K Σ ) n k iµ P µ (σ) = dσ. σ σ i i=1 z 1 z 2 z n Scattering Equations at tree-level Enforce P 2 = 0 on Σ: E i = Res σi P 2 k i k j (σ) = k i P(σ i ) = = 0. σ i σ j j i σ a i E i = 0 for a = 0, 1, 2 SL(2, C) invariant (n 3) independent equations = dim(m 0,n ) (n 3)! solutions factorisation properties

10 Scattering Equations Universality for massless QFTs Scattering Equations at tree-level: E i = Res σi P 2 k i k j (σ) = k i P(σ i ) = = 0. σ i σ j j i

11 Scattering Equations Universality for massless QFTs Scattering Equations at tree-level: E i = Res σi P 2 k i k j (σ) = k i P(σ i ) = = 0. σ i σ j Factorisation: j i SE 1 k 2 I boundary of M 0,n factorisation channel (unitarity, locality)

12 Scattering Equations Universality for massless QFTs Scattering Equations at tree-level: E i = Res σi P 2 k i k j (σ) = k i P(σ i ) = = 0. σ i σ j Factorisation: j i SE 1 k 2 I boundary of M 0,n factorisation channel (unitarity, locality) With σ i = σ I + εx i for i I, the pole is given by i I x i E (I) i = i,j I x i k i k j x i x j = 1 2 k i k j = ki 2. i,j I

13 Scattering Equations Universality for massless QFTs Scattering Equations at tree-level: E i = Res σi P 2 k i k j (σ) = k i P(σ i ) = = 0. σ i σ j Factorisation: j i SE 1 k 2 I boundary of M 0,n Upshot: amplitudes take the form M = σ j E i (σ j )=0 factorisation channel (unitarity, locality) I(σ, k, ɛ) J(σ, k).

14 Scattering Equations Universality for massless QFTs Scattering Equations at tree-level: E i = Res σi P 2 k i k j (σ) = k i P(σ i ) = = 0. σ i σ j Factorisation: j i SE 1 k 2 I boundary of M 0,n factorisation channel (unitarity, locality) Upshot: amplitudes take the form M = dµ δ ( ) E i I(σ, k, ɛ). M 0,n i

15 The target space Ambitwistor space - the phase space of complex null geodesics Ambitwistor space A = space of (complex) null rays in M C symplectic quotient of cotangent bundle of spacetime A := { (X µ, P µ, Ψ µ r ) T M P 2 = Ψ r P = 0 } / {D 0, D r } with Hamiltonian vector fields D 0 = P, D r = Ψ r + P Ψr A is a symplectic holomorphic manifold, with symplectic potential Θ = P dx + 1 Ψ r dψ r 2 r C d 1 X S d 2

16 The Ambitwistor String [Mason-Skinner, Berkovits] Construct a theory describing maps Σ A: S = 1 2π P X r Ψ r Ψ r e 2 P2 χ r P Ψ r, where P µ Ω 0 (Σ, K Σ ), Ψ µ i ΠΩ 0 (Σ, K 1/2 Σ ).

17 The Ambitwistor String [Mason-Skinner, Berkovits] Construct a theory describing maps Σ A: S = 1 2π P X r Ψ r Ψ r e 2 P2 χ r P Ψ r, where P µ Ω 0 (Σ, K Σ ), Ψ µ i ΠΩ 0 (Σ, K 1/2 Σ ). Geometrically: action from symplectic potential Θ gauge fields e and χ r impose the constraints reducing to A gauge freedom: δx µ = αp µ, δp µ = 0, δe = α.

18 The Ambitwistor String [Mason-Skinner, Berkovits] Construct a theory describing maps Σ A: S = 1 2π P X r Ψ r Ψ r e 2 P2 χ r P Ψ r, where P µ Ω 0 (Σ, K Σ ), Ψ µ i ΠΩ 0 (Σ, K 1/2 Σ ). Geometrically: action from symplectic potential Θ gauge fields e and χ r impose the constraints reducing to A gauge freedom: δx µ = αp µ, δp µ = 0, δe = α. BRST quantisation: Q = ct + cp 2 + γ r P Ψ r Vertex operators: V = c cδ 2 (γ) ɛ µ ɛ ν Ψ µ 1 Ψν 2 eik X [Q, V] = 0 k 2 = ɛ k = 0

19 Geometry of the Scattering Equations action: S = 1 2π P X r Ψ r Ψ r e 2 P2 χ r P Ψ r, vertex operators: V = c cδ 2 (γ) ɛ µ ɛ ν Ψ µ 1 Ψν 2 eik X

20 Geometry of the Scattering Equations action: S = 1 2π P X r Ψ r Ψ r e 2 P2 χ r P Ψ r, vertex operators: V = c cδ 2 (γ) ɛ µ ɛ ν Ψ µ 1 Ψν 2 eik X Integrate out X in presence of vertex operators: P µ = 2πi k iµ δ(σ σ i )dσ, so P µ (σ) = n i=1 k iµ σ σ i dσ.

21 Geometry of the Scattering Equations action: S = 1 2π P X r Ψ r Ψ r e 2 P2 χ r P Ψ r, vertex operators: V = c cδ 2 (γ) ɛ µ ɛ ν Ψ µ 1 Ψν 2 eik X Integrate out X in presence of vertex operators: P µ = 2πi k iµ δ(σ σ i )dσ, so P µ (σ) = n i=1 k iµ σ σ i dσ. Moduli of gauge field e forces P 2 = 0; Res σi P 2 (σ) = k i P(σ i ) = 0. scattering equations map to A

22 CHY formulae [Cachazo-He-Yuan] The correlation functions yield the CHY formulae: Tree-level S-matrix of massless theories: dσ M n,0 = M0,n n k i k j δ vol G σ i σ j I n i j i

23 CHY formulae [Cachazo-He-Yuan] The correlation functions yield the CHY formulae: Tree-level S-matrix of massless theories: dσ M n,0 = M0,n n k i k j δ vol G σ i σ j I n i j i Integration over M 0,n fixed by imposing the SE E i I n from fermion correlators: I n = Pf (M) Pf ( M) for M = M(k i, ɛ i, σ i )

24 CHY formulae [Cachazo-He-Yuan] The correlation functions yield the CHY formulae: Tree-level S-matrix of massless theories: dσ M n,0 = M0,n n k i k j δ vol G σ i σ j I n i j i Integration over M 0,n fixed by imposing the SE E i I n from fermion correlators: I n = Pf (M) Pf ( M) for M = M(k i, ɛ i, σ i ) progress on evaluation without solving the SE [Cachazo-Gomez, Baadsgaard et al, Sogard-Zhang,...] family of massless models

25 Comment: Colour-Kinematic Duality Gravity YM 2 [Bern-Carrasco-Johansson] Biadjoint scalar: colour c i colour c j Gauge theory: colour c i kinematics n i Gravity: kinematics n i kinematics n i Gauge theory amplitude: n i c i A = d i With c i satisfying the Jacobi identity Γ i + = 0 f dae f ebc f abe f ecd + f ace f edb = 0

26 Comment: Colour-Kinematic Duality Gravity YM 2 [Bern-Carrasco-Johansson] Biadjoint scalar: colour c i colour c j Gauge theory: colour c i kinematics n i Gravity: kinematics n i kinematics n i Gauge theory amplitude: n i c i A = d i With c i satisfying the Jacobi identity Γ i + = 0 f dae f ebc f abe f ecd + f ace f edb = 0 Find n i satisfying Jacobi, then M = Γ i n i n i d i

27 CHY formulae and the Double Copy [Cachazo-He-Yuan] Tree-level S-matrix of massless theories: dσ M n,0 = M0,n n k i k j δ vol G σ i σ j I n i j i Gravity: I n = Pf (M) Pf ( M) Yang-Mills theory: I n = C n Pf (M) Bi-adjoint scalar: with building blocks I n = C n C n Parke-Taylor factor: C n (1,..., n) = tr(ta 1T a 2...T an ) σ σ n 1 n σ n 1 + non-cyclic Reduced Pfaffian: Pf (M) = ( 1)i+j σ ij Pf(M ij ij ), M = M(k i, ɛ i, σ i )

28 Loop Integrands from the Riemann Sphere

29 Upshot Loops Worldsheet models of QFT M = (0) + (1) +... = E (g) i = 0

30 Upshot Loops Worldsheet models of QFT M = (0) + (1) +... = E (g) i = 0 localisation on SE E i = E (g) i = 0

31 Scattering Equations on the Torus Genus 1 moduli space of torus Σ τ : τ τ z z + 1 z + τ τ τ + 1 1/τ Solve P = 2πi i k i δ(z z i )dz: ( θ 1 P µ = 2πi l µ + k (z z ) i) iµ dz, θ 1 (z z i ) i z 1

32 Scattering Equations on the Torus Genus 1 moduli space of torus Σ τ : τ τ z z + 1 z + τ τ τ + 1 1/τ Solve P = 2πi i k i δ(z z i )dz: ( θ 1 P µ = 2πi l µ + k (z z ) i) iµ dz, θ 1 (z z i ) i z 1 Scattering Equations on the torus: P 2 (z τ) = 0 Res zi P 2 (z) := 2k i P(z i ) = 0, P 2 (z 0 ) = 0.

33 The 1-loop Integrand [Adamo-Casali-Skinner, Casali-Tourkine] One-loop integrand of type II supergravity n M (1) SG = d 10 l dτ δ(p 2 (z 0 )) δ(k i P(z i )) M 1,n i=2 } {{ } Scattering Equations Z (1) (z i )Z (2) (z i ) spin struct. } {{ } I q, fermion correlator

34 The 1-loop Integrand [Adamo-Casali-Skinner, Casali-Tourkine] One-loop integrand of type II supergravity n M (1) SG = d 10 l dτ δ(p 2 (z 0 )) δ(k i P(z i )) M 1,n i=2 } {{ } Scattering Equations Z (1) (z i )Z (2) (z i ) spin struct. } {{ } I q, fermion correlator modular invariance: τ τ + 1 1/τ Scattering Equations: P 2 (z τ) = 0 checks: factorisation correct tensor structure at n = 4: t 8 t 8 R 4 Why rational function?

35 From the Torus to the Riemann Sphere Contour argument localisation on scattering equations contour integral argument in fundamental domain modular invariance: sides and unit circle cancel localisation on q e 2iπτ = 0 τ = i τ Contour argument in the fundamental domain

36 From the Torus to the Riemann Sphere More explicitly: residue theorem One-loop integrand of type II supergravity n M (1) SG = d d l dτ δ(p 2 (z 0 )) δ(k i P(z i )) I q M 1,n i=2 σ + σ Residue theorem: elliptic curve nodal Riemann spere at q = 0.

37 From the Torus to the Riemann Sphere More explicitly: residue theorem One-loop integrand of type II supergravity n M (1) SG = d d l dτ δ(p 2 (z 0 )) δ(k i P(z i )) I q M 1,n i=2 σ + σ Residue theorem: elliptic curve nodal Riemann spere at q = 0. M (1) SG = 1 d 10 l dq ( ) n 2πi q 1 δ(k P 2 i P(z i )) I q (z 0 ) i=2 = d 10 1 n l dq P 2 (z 0 ) δ(q) δ(k i P(z i )) I 0 i=2 d 10 l n q=0 = δ(k l 2 i P(z i )) I 0. i=2

38 One-loop off-shell scattering equations On the nodal Riemann Sphere: l l n k i P = + σ σ l + σ σ l σ σ i dσ. Define S = P 2 ( i=1 l σ σ l l + σ σ l ) 2 dσ 2. σ + σ One-loop off-shell scattering equations Res σi S = k i P(σ i ) = k i l k i l σ i σ l + σ i σ l k i k j + = 0, σ i σ j j j i l k j Res σl S = = 0, σ l σ j l k j Res σl + S = = 0. σ l + σ j j

39 The integrand One-loop integrand on the nodal Riemann sphere d M (1) d l d n+2 σ ( = δ l 2 Resσi S ) I vol (G) i,l } ± {{ } off-shell scattering equations type II sugra: I = I q 0, limit of g = 1 correlator manifestly rational upshot: widely applicable for supersymmetric and non-supersymmetric theories

40 The integrand One-loop integrand on the nodal Riemann sphere d M (1) d l d n+2 σ ( = δ l 2 Resσi S ) I vol (G) i,l } ± {{ } off-shell scattering equations Puzzle: Only depends on 1/l 2, remainder l k i, l ɛ i,... Solution: Shifted integrands partial fractions: 1 i D i = i shift: D i l 2 formalised: Q-cuts 1 D i j i(d j D i ) [Baadsgaard et al]

41 The integrand One-loop integrand on the nodal Riemann sphere d M (1) d l d n+2 σ ( = δ l 2 Resσi S ) I vol (G) i,l } ± {{ } off-shell scattering equations Puzzle: Only depends on 1/l 2, remainder l k i, l ɛ i,... Solution: Shifted integrands partial fractions: 1 i D i = i shift: D i l 2 formalised: Q-cuts 1 D i j i(d j D i ) [Baadsgaard et al] Example: 1 l 2 (l + K) 2 = 1 l 2 (2l K + K 2 ) + 1 (l + K) 2 ( 2l K K 2 ) 1 ( ) 1 l 2 2l K + K l K + K 2

42 Integrands Double Copy again One-loop integrand on the nodal Riemann sphere d M (1) d l d n+2 σ = δ(res l 2 σi S) I vol (G) i,l ± Supersymmetric: I sugra = 1 I (σ l + l ) 4 0 Ĩ 0 I sym = 1 I (σ l + l ) 4 0 I PT Building blocks Parke-Taylor: I PT = n i=1 σ l + l tr(t a 1T a 2...T an ) σ l + i σ i+1 i σ i+2 i+1...σ i+n l + non-cycl. Pfaffian: I 0 = r Pf (M r NS ) c d σ 2 l + l Pf (M 2 )

43 Integrands Double Copy again One-loop integrand on the nodal Riemann sphere d M (1) d l d n+2 σ = δ(res l 2 σi S) I vol (G) i,l ± Supersymmetric: I sugra = 1 I (σ l + l ) 4 0 Ĩ 0 I sym = 1 I (σ l + l ) 4 0 I PT Non-supersymmetric I YM = ( r Pf (M r )) IPT NS I grav = ( r Pf (M r NS ))2 α (Pf (M 3 ) q 0) 2. Building blocks Parke-Taylor: I PT = n i=1 σ l + l tr(t a 1T a 2...T an ) σ l + i σ i+1 i σ i+2 i+1...σ i+n l + non-cycl. Pfaffian: I 0 = r Pf (M r NS ) c d σ 2 l + l Pf (M 2 ) α = 1 2 (d 2)(d 3) + 1: d.o.f. of B-field and dilaton

44 Integrands Double Copy again One-loop integrand on the nodal Riemann sphere d M (1) d l d n+2 σ = δ(res l 2 σi S) I vol (G) i,l ± Supersymmetric: I sugra = 1 I (σ l + l ) 4 0 Ĩ 0 I sym = 1 I (σ l + l ) 4 0 I PT Non-supersymmetric I YM = ( r Pf (M r )) IPT NS I grav = ( r Pf (M r NS ))2 α (Pf (M 3 ) q 0) 2. Building blocks Parke-Taylor: I PT = n i=1 σ l + l tr(t a 1T a 2...T an ) σ l + i σ i+1 i σ i+2 i+1...σ i+n l + non-cycl. Pfaffian: I 0 = r Pf (M r NS ) c d σ 2 l + l Pf (M 2 ) α = 1 2 (d 2)(d 3) + 1: d.o.f. of B-field and dilaton Checks: numerical, factorisation

45 The proof in a nutshell Recall: Poles (and residues) of M from boundary of moduli space M 0,n+2g SE s I

46 The proof in a nutshell Recall: Poles (and residues) of M from boundary of moduli space M 0,n+2g SE s I At one loop, there are two cases of interest: Separating: Q-cut:

47 The proof in a nutshell Recall: Poles (and residues) of M from boundary of moduli space M 0,n+2g SE s I At one loop, there are two cases of interest: Separating: Q-cut: Pole structure: ki 2 s I = 2l k I + ki 2 separating non-separating

48 The proof in a nutshell Q-cuts [Baadsgaard, Bjerrum-Bohr, Bourjaily, Caron-Huot, Damgaard, Feng] Starting from the Feynman diagram expansion, M (1) FD = N(l) D 1 (l)...d m (l), l l + η with η l = η k i = 0, η 2 = z. D i (l) D i (l) + z Cauchy Residue theorem appropriate shifts of l, remove forward limit singularities

49 The proof in a nutshell Q-cuts [Baadsgaard, Bjerrum-Bohr, Bourjaily, Caron-Huot, Damgaard, Feng] Starting from the Feynman diagram expansion, M (1) FD = N(l) D 1 (l)...d m (l), l l + η with η l = η k i = 0, η 2 = z. D i (l) D i (l) + z Cauchy Residue theorem appropriate shifts of l, remove forward limit singularities Q-cut expansion M (1) FD = I l I I = l+k I I M (0) I M (0) Ī l 2 (2l K I + K 2 I )

50 Question: Beyond one loop? M = E i (σ j ) = 0

51 Heuristics: Higher genus Riemann surface Σ g residue theorems contract g a-cycles nodal RS fixes g moduli remaining 2g 3 2g new marked points mod SL(2, C) 1-form P µ P = ω r (g) = (σ r + σ r ) dσ (σ σ r +)(σ σ r +) g r=1 l r ω (g) r + i k i dσ, σ σ i : basis of g global holomorphic 1-forms

52 The Scattering Equations The multiloop off-shell scattering equations are Res σa S = 0, A = 1,..., n + 2g g S(σ) := P 2 lr 2 ω 2 r + a rs (lr 2 + ls 2 )ω r ω s. r=1 r<s

53 The Scattering Equations The multiloop off-shell scattering equations are Res σa S = 0, A = 1,..., n + 2g g S(σ) := P 2 lr 2 ω 2 r + a rs (lr 2 + ls 2 )ω r ω s. r=1 r<s At two-loops: Factorisation: a 12 = 1 Two-loop integrand M (2) = 1 l 2 1 l2 2 M 0,n+4 d n+4 σ Vol G n+4 A=1 δ ( Res A S(σ A ) ) I, integrands known for n = 4 sym and sugra Similar complexity as tree amplitudes with n + 4 particles!

54 Conclusion and Outlook Summary: Worldsheet models describing scattering in massless QFTs moduli integrals localised on scattering equations framework to derive loop integrands on nodal Riemann spheres M = E i (σ j ) = 0 This formalism implies that n-point g-loop scattering amplitudes have the same complexity as n + 2g-point tree-level amplitudes! Outlook: sharpen ambitwistor string derivation at higher loop order model on nodal Riemann sphere integrals vs integrands?

55 Thank you!