Ambitwistor Strings beyond Tree-level Worldsheet Models of QFTs

Transcription

1 Ambitwistor Strings beyond Tree-level Worldsheet Models of QFTs Yvonne Geyer Strings 2017 Tel Aviv, Israel arxiv: , , YG, L. Mason, R. Monteiro, P. Tourkine arxiv:170x.xxxx with R. Monteiro

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 Ambitwistor Strings: two-dimensional chiral CFTs auxiliary target space: complexified phase-space of null geodesics

8 Starting Point: Tree-level S-matrix CHY formulae [Cachazo-He-Yuan] dσ M n,0 = M0,n n k i k j δ vol G σ i σ j I n i j i

9 Starting Point: Tree-level S-matrix CHY formulae [Cachazo-He-Yuan] 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 localised onto the scattering equations E i Integrand I n (ɛ, k, σ) K 2 i encodes kinematic data

10 Starting Point: Tree-level S-matrix CHY formulae [Cachazo-He-Yuan] 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 localised onto the scattering equations E i Integrand I n (ɛ, k, σ) K 2 i encodes kinematic data progress on evaluation without solving the SE [Cachazo-Gomez, Baadsgaard et al, Sogard-Zhang,...] family of massless models: Gravity: I n = Pf (M) Pf ( M) Yang-Mills theory: I n = C n Pf (M) Bi-adjoint scalar: I n = C n C n

11 Scattering Equations [Fairlie-Roberts, Gross-Mende][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

12 Scattering Equations [Fairlie-Roberts, Gross-Mende][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

13 Scattering Equations [Fairlie-Roberts, Gross-Mende][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 [Dolan-Goddard, YG-Mason-Monteiro-Tourkine,...]

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 j i

15 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

16 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 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

17 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 Upshot: amplitudes take the form M = σ j E i (σ j )=0 I(σ i, k i, ɛ i ) J(σ i, k i ).

18 The Ambitwistor String [Mason-Skinner, Berkovits] Moduli integral: CHY correlator of CFT on Σ. 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 r ΠΩ 0 (Σ, K 1/2 Σ ).

19 The Ambitwistor String [Mason-Skinner, Berkovits] Moduli integral: CHY correlator of CFT on Σ. 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 r ΠΩ 0 (Σ, K 1/2 Σ ). Geometrically: gauge fields e and χ r impose the constraints P 2 = P Ψ r = 0 target space: Ambitwistor space A gauge freedom: δx µ = αp µ, δp µ = 0, δe = α...

20 The Ambitwistor String [Mason-Skinner, Berkovits] Moduli integral: CHY correlator of CFT on Σ. 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 r ΠΩ 0 (Σ, K 1/2 Σ ). Geometrically: gauge fields e and χ r impose the constraints P 2 = P Ψ r = 0 target space: Ambitwistor space 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 2 = 0 for d = 10 [Q, V] = 0 k 2 = ɛ k = 0 spectrum: type II sugra

21 Localisation and 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

22 Localisation and 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.

23 Localisation and 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; scattering equations map to A Correlator = CHY Res σi P 2 (σ) = k i P(σ i ) = 0.

24 Loop Integrands from the Riemann Sphere

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

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

27 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

28 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.

29 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 )) Z (1) (z i )Z (2) (z i ) i=2 spin struct. } {{ }} {{ } Scattering Equations I q, fermion correlator

30 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 )) Z (1) (z i )Z (2) (z i ) i=2 spin struct. } {{ }} {{ } Scattering Equations 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?

31 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

32 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 i=2 σ + σ Residue theorem: elliptic curve nodal Riemann spere at q = 0.

33 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 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

34 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

35 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

36 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 [Baadsgaard et al] 1 D i j i(d j D i )

37 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 [Baadsgaard et al] 1 D i j i(d j D i ) 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

38 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 )

39 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

40 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

41 BCJ numerators with R. Monteiro Manifest Colour-Kinematics duality: Tree-level: [Fu-Du-Huang-Feng] Loop-level: susy [He-Schlotterer-Zhang] non-susy? Algorithm: 1 Pick an ordered gauge ( n ) 2 For a disjoint split into ρ 1, ρ 2, find all ordered splittings of ρ 2 3 Calculate n = ( 1) n ρ W (+ρ1 ) ρ 1 OS(ρ 2 ) subsets W with W (+ρ1 ) = tr(f ρ1...f ρnρ ) W = ɛ pnp F pnp 1... F p1 Z p1 for OS {p 1...p np } Proof: Expansion of Pfaffian into PT factors

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

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

44 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

45 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 type II sugra Similar complexity as tree amplitudes with n + 4 particles!

46 Conclusion and Outlook Summary: massless QFTs from chiral worldsheet theories moduli integrals localised on scattering equations loop integrands on nodal Riemann spheres M = E i (σ j ) = 0

47 Conclusion and Outlook Summary: massless QFTs from chiral worldsheet theories moduli integrals localised on scattering equations loop integrands on nodal Riemann spheres M = E i (σ j ) = 0 Future directions: Short-term: Medium-term: explore one and two loops model on nodal Riemann sphere

48 Conclusion and Outlook Summary: massless QFTs from chiral worldsheet theories moduli integrals localised on scattering equations loop integrands on nodal Riemann spheres M = E i (σ j ) = 0 Future directions: Short-term: Medium-term: Long-term: explore one and two loops model on nodal Riemann sphere connection with string theory