Loop Integrands from Ambitwistor Strings

Transcription

1 Loop Integrands from Ambitwistor Strings Yvonne Geyer Institute for Advanced Study QCD meets Gravity UCLA arxiv: , , YG, L. Mason, R. Monteiro, P. Tourkine arxiv: with R. Monteiro work in progress

2 The Double Copy from the Worldsheet Yvonne Geyer Institute for Advanced Study QCD meets Gravity UCLA arxiv: , , YG, L. Mason, R. Monteiro, P. Tourkine arxiv: with R. Monteiro work in progress

3 Quantum fields from ambitwistor strings Worldsheet models of QFT M = (0) + (1) +... = E (g) i = 0 localisation on SE E i = E (g) i = 0

4 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(k i, ɛ i, σ i ) i j i Scattering Equations: For n null momenta k i, define n k iµ P µ (σ) = dσ Ω 0 (Σ, K Σ ). σ σ i i=1 E i = Res σi P 2 k i k j (σ) = k i P(σ i ) = = 0. σ i σ j j i σ 1 σ 2 σ n

5 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 enforce P 2 = 0 on Σ. Möbius invariant dim(m 0,n ) = (n 3) constraints Factorisation: [Dolan-Goddard, YG-Mason-Monteiro-Tourkine,...] j i SE 1 k I 2 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

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

7 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 1 T 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 ) ( ) A C T M = C B A ij = k i k j, B ij = ɛ i ɛ j, C ij = ɛ i k j, σ ij σ ij σ ij A ii = 0, B ii = 0, C ii = C ij. j i

8 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 δx µ = v X µ, δp µ = (vp µ ), δe = v e e v 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 Ψνeik X 2 Q 2 = 0 for d = 10 [Q, V] = 0 k 2 = ɛ k = 0 spectrum: type II sugra

9 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 Ψνeik X 2 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.

10 Loop Integrands from the Riemann Sphere

11 One Loop: Scattering Equations and Integrand SE on the torus: P 2 (z τ) = 0 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 ) Res zi P 2 (z) := 2k i P(z i ) = 0, P 2 (z 0 ) = 0. i [Adamo-Casali-Skinner, Casali-Tourkine] 1 2 z 1 τ 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 invariant: τ τ + 1 1/τ

12 From the Torus to the Riemann Sphere localisation on SE & modular invariance: localisation on q e 2iπτ = 0 τ = i τ Contour argument in the fundamental domain Alternatively: Residue theorem: 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 l n q=0 = δ(k l 2 i P(z i )) I 0. i=2

13 One-loop off-shell scattering equations On the nodal Riemann Sphere: l l n k i P = + σ σ + σ σ σ σ dσ. i Define S = P 2 ( l σ σ + l i=1 σ σ ) 2 dσ 2. σ+ σ One-loop off-shell scattering equations E (nod) i = Res σi S = k i l k i l + σ i σ + σ i σ E (nod) j i = Res σ S = E (nod) + = Res σ+ S = j j k i k j σ i σ j = 0, l k j σ σ j = 0, l k j σ + σ j = 0. Nodal measure: dµ (nod) 1,n = dµ 0,n+2 l 2 =0, where l = l + η, η k i = η l = 0.

14 Integrands Double Copy again One-loop integrand on the nodal Riemann sphere d M (1) d l d n+2 σ = δ ( ) E a (nod) I vol (G) l 2 a=i,± Supersymmetric: I sugra = I 0 Ĩ 0 I sym = I 0 I PT Non-supersymmetric I NS = I (1) NS I(1) NS I YM = I (1) NS IPT I d=4 grav = ( ) I (1) 2 ( NS 2 (σ + ) Pf (M3 ) ) 2. 4 Building blocks Parke-Taylor: I PT = n i=1 tr(t a 1 T a 2...T an ) σ + i σ i+1 i σ i+2 i+1...σ i+n σ + + non-cycl. Susy: I 0 = I (1) + NS I(1) R NS and R: I (1) = NS r Pf (M r NS ), M r NS = Mtree n+2 l 2 =0, ɛ+=ɛ r, ɛ =(ɛ r ) M 3 = M tree Cii =ɛ i P(σ i ) I(1) R = c d Pf (M σ 2 2 ) + σ M 2 = M tree ( ) 1 σ ij 1 σi+ σ j σi σ j+ ij σ i σ + j+ σ i+ σ j C ii =ɛ i P(σ i )

15 The representation of the integrand One-loop integrand on the nodal Riemann sphere d M (1) d l d n+2 σ = δ ( ) E a (nod) I vol (G) l 2 a=i,± Puzzle: Only depends on 1/l 2, remainder l k i, l ɛ i,... Solution: Shifted integrands Repeated partial fractions: Take K a = i I a k i and D a = (l + K a ) 2 1 a D a = i 1 D a b a(d b D a ). a + i i 1 = i i i 1 +

16 The representation of the integrand One-loop integrand on the nodal Riemann sphere d M (1) d l d n+2 σ = δ ( ) E a (nod) I vol (G) l 2 a=i,± Puzzle: Only depends on 1/l 2, remainder l k i, l ɛ i,... Solution: Shifted integrands Repeated partial fractions: Take K a = i I a k i and D a = (l + K a ) 2 1 a D a = 1 D a b a(d b D a ). a Generalisation: Q-cuts [Baadsgaard et al] N ( l, l 2) I qdr = I lin = 1 N ( l K a, 2l K a + K 2 ) a Γ a Γ D a l 2, Γ a Γ b a(d b D a ) l l Ka 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

17 BCJ numerators from the Worldsheet Integrands without solving the Scattering Equations

18 1-loop BCJ numerators from ambitwistor strings The main idea with R. Monteiro, see also [CHY, Fu-Du-Huang-Feng, He-Schlotterer-Zhang] Expand YM and gravity integrands into DDM half-ladder basis I YM n = C(+, ρ, ) I YM (+, ρ, ) ρ(1) ρ(2) ρ(n) ρ S n I grav n = N(+, ρ, ) I YM (+, ρ, ) + ρ S n Then coefficients N(+, ρ, ) satisfy Jacobis! Ambitwistor string integrands: Pf ( M r ) NS = N(+, ρ, ) PT(+, ρ, ) r ρ S n mod E (nod) a Strategy: expand into simpler Pfaffians, whose expansion into PT is known [Fu-Du-Huang-Feng]

19 Pfaffian expansion Pfaffian as sum over permutations: I (1) Pf ( M r ) NS NS = ( 1) sgn(ρ) W IM J...M K. σ r ρ S n+2 + I σ J...σ K tr(i) := tr(f i1...f ini ) for n I > 1 σ i1 i M I = σ I = 2 σ i2 i 3...σ ini i 1 for n I > 1 C ii for n I = 1 σ I = 1 for n I = 1 W I = ɛ r F i1...f ini (ɛ r ) tr ( ) F i1...f ini for n I > 0 = r D 2 for n I = 0 Decompose the sum: I (1) = NS ( 1) sgn(ρ) W I σ I ρ S I + I ( 1) sgn( ρ) M J...M K σ ρ S J...σ K Ī = ( 1) sgn(ρ) Pf ( ) M W Ī I I ρ S I + σ I = W I Ỹ Ī PT(+, ρ, ) ρ S n I } {{ } N(+,ρ, )

20 The algorithm Master numerators N (+ a 1 a 2 a 3 a 4 ): 1 Fix reference ordering RO = ( n ) 2 Dependence of N on RO and CO: split orderings SO Decompose {1,..., n} = I Ī, Ī = R r=1 α(r) s.t. 1 α (r) respects CO 2 last elements α n (r) r respect RO 3 last element α n (r) r smallest in RO Then SO = (+ I α (1)...α (R) ). 3 Calculate N RO (CO) = ( 1) n I W I Y ( α (r)) 1 W I = tr(i) = tr(f i1... F ini ), W I= = d 2 2 Y ( { α (r)) ɛ = a Z a α (r) = {a} ɛ anr F a(nr 1)... F a1 Z a1 α (r) = {a 1,..., a nr }. 3 Z a = i k i i<a in CO and SO I SO r

21 Example: N RO (+1243 ) Ī Split(Ī) SO tr(i) numerator factor Y ( α (r)) {1, 2, 4, 3} {{1}, {2}, {3}, {4}} (+1234 ) (d 2) ɛ 1 l ɛ 2 (l + k 1 ) ɛ 4 (l k 3 ) ɛ 3 (l k 4 ) {{1}, {2}, {4, 3}} (+1243 ) (d 2) ɛ 1 l ɛ 2 (l + k 1 ) ɛ 3 F 4 (l k 3 ) {1, 2} {{1}, {2}} (+4312 ) tr(43) ɛ 1 l ɛ 2 (l + k 1 ) {4, 3} {{3}, {4}} (+1234 ) tr(12) ɛ 4 (l k 3 ) ɛ 3 (l k 4 ) {{4, 3}} (+1243 ) tr(12) ɛ 3 F 4 (l k 3 ).. {1} {{1}} (+2431 ) tr(243) ɛ 1 l.. {} (+1243 ) tr(1243) 1 Remarks: Master numerators: N RO (CO) = N CO (CO). ( ɛ ɛ) All-plus: N RO (CO) = N CO (CO) Amplitude independent of RO

22 Integrands from BCJ numerators N i = N RO (CO) satisfy Jacobi relations: + = 0 Integrands for YM and NS-NS gravity, with linear propagators D i : I YM = Γ i N i C i I NS-NS N i N i = D i D Γ i i Pure gravity: I grav = Γ i N i N i D i (d 2) 2 (d 2) 2 2, d 4 Checks: YM amplitude, known all-plus numerators, NS-NS gravity amplitude, gauge invariance

23 Outlook: Beyond one loop M = E i (σ j ) = 0

24 Outlook: Beyond one loop 1 RNS-Correlator at g = 2, sum over spin structures 2 Riemann surface Σ g residue theorems contract g a-cycles nodal RS 3 NS-sector: I (2) NS? = r,s Pf ( ) M (2) NS If so, then simply extract BCJ numerators by analogous procedure! Nodal operators, see Kai s talk. Relation to BCJ numerators for standard representation of the integrand?

25 Thank you!