Amplitudes Conference. Edinburgh Loop-level BCJ and KLT relations. Oliver Schlotterer (AEI Potsdam)

Transcription

1 Amplitudes Conference Edinburgh 2017 Loop-level BCJ and KLT relations Oliver Schlotterer (AEI Potsdam) based on with S. He and with S. He, Y. Zhang

2 Different formulations of double copy 1 color ordered amplitudes manifest gauge-/diffeo invariance cubic diagrams Γ g-loop manifest locality KLT relations 86 M tree grav = A tree YM Ãtree YM establish kin. Jacobis BCJ 08: YM-kinematics n i Mgrav tree = n i ñ i edge α i kα 2 i i Γ tree

3 Different formulations of double copy 2 color ordered amplitudes manifest gauge-/diffeo invariance cubic diagrams Γ g-loop manifest locality KLT relations 86 M tree grav = A tree YM Ãtree YM establish kin. Jacobis BCJ 08: YM-kinematics n i Mgrav tree = n i ñ i edge α i kα 2 i i Γ tree BCJ 10: natural g-loop version M grav (g) = d g d l n i (l) ñ i (l) i Γ g α i kα 2 i (l) need Jacobi-satisfying n i (l) or [see Radu s talk & ] Universal to any number d of spacetime dimensions & #(SUSY)!

4 Different formulations of double copy 3 color ordered amplitudes manifest gauge-/diffeo invariance cubic diagrams Γ g-loop manifest locality KLT relations 86 M tree grav = A tree YM Ãtree YM this talk [He, OS ] M grav (1) d d l = l 2 a(1) YM (l) ã(1) YM (l) any representation of a (1) YM but?? higher-loop KLT?? establish kin. Jacobis rearrange propagators?? BCJ 08: YM-kinematics n i Mgrav tree = n i ñ i edge α i kα 2 i i Γ tree BCJ 10: natural g-loop version M grav (g) = d g d l n i (l) ñ i (l) i Γ g α i kα 2 i (l) need Jacobi-satisfying n i (l) or [see Radu s talk & ] Universal to any number d of spacetime dimensions & #(SUSY)!

5 KLT relations in gravity BCJ relations in YM 4 Consistency condition of KLT formula M tree grav = A tree YM Ãtree YM : BCJ amplitude relations among color-ordered A tree YM (ρ(1, 2,..., n)) = (n 3)! linearly independent choices of ρ S n [Piotr Tourkine s talk; Bern, Carrasco, Johansson 2008]

6 KLT relations in gravity BCJ relations in YM 5 Consistency condition of KLT formula M tree grav = A tree YM Ãtree YM : BCJ amplitude relations among color-ordered A tree YM (ρ(1, 2,..., n)) = (n 3)! linearly independent choices of ρ S n [Piotr Tourkine s talk; Bern, Carrasco, Johansson 2008] Loop-level KLT M grav (1) = d d l l 2 a(1) YM (l) ã(1) YM (l) requires loop-lv BCJ relations... [Boels, Isermann; Du, Luo; Primo, Torres-Bobadilla; Piotr s talk; Vanhove, Tourkine; Hohenegger, Stieberger]

7 KLT relations in gravity BCJ relations in YM 6 Consistency condition of KLT formula M tree grav = A tree YM Ãtree YM : BCJ amplitude relations among color-ordered A tree YM (ρ(1, 2,..., n)) = (n 3)! linearly independent choices of ρ S n [Piotr Tourkine s talk; Bern, Carrasco, Johansson 2008] Loop-level KLT M grav (1) = d d l l 2 a(1) YM (l) ã(1) YM (l) requires loop-lv BCJ relations... [Boels, Isermann; Du, Luo; Primo, Torres-Bobadilla; Piotr s talk; Vanhove, Tourkine; Hohenegger, Stieberger]... among partial integrands a (1) YM (ρ(...)) [He, OS ] = at most (n 1)! linearly independent choices of ρ (will see extra degeneracy that depends on supersymmetry)

8 Einstein Yang Mills (EYM): coupling gauge theory to gravity numerous tree-level amplitude relations with the flavour of A tree EYM ( gauge ) gravity = A tree YM ( gauge only! ) (polarizations for extra gravitons) [Stieberger, Taylor; Nandan, Plefka, OS, Wen; de la Cruz, Kniss, Weinzierl; OS; Du, Teng, Wu; Du, Feng, Fu, Huang; Chiodaroli, Günaydin, Johansson, Roiban; Feng, Teng] 7 Consequence of double-copy formulation EYM = YM (YM scalar) [Chiodaroli, Günaydin, Johansson, Roiban , ]

9 Einstein Yang Mills (EYM): coupling gauge theory to gravity numerous tree-level amplitude relations with the flavour of A tree EYM ( gauge ) gravity = A tree YM ( gauge only! ) (polarizations for extra gravitons) [Stieberger, Taylor; Nandan, Plefka, OS, Wen; de la Cruz, Kniss, Weinzierl; OS; Du, Teng, Wu; Du, Feng, Fu, Huang; Chiodaroli, Günaydin, Johansson, Roiban; Feng, Teng] 8 Consequence of double-copy formulation EYM = YM (YM scalar) [Chiodaroli, Günaydin, Johansson, Roiban , ] This talk: one-loop amplitude relations for gauge multiplets in the loop A (1) EYM = d d l l 2 [ a (1) YM (l) (extra polarizations)], also see recent all-loop relations for 1 external graviton [Chiodaroli, Günaydin, Johansson, Roiban ]

10 Derivations: CHY formalism / ambitwistor strings 9 integrands share double-copy structure with superstrings & new perspectives on the field-theory limit α 0 [Cachazo, He, Yuan , , ] [Mason, Skinner ; Adamo, Casali, Skinner 1-loop: torus worldsheet nodal Riemann sphere. 2 n 1 τ i. 2 n 1 n = [Lionel Mason s talk; Geyer, Mason, Monteiro, Tourkine & ]

11 Derivations: CHY formalism / ambitwistor strings 10 integrands share double-copy structure with superstrings & new perspectives on the field-theory limit α 0 [Cachazo, He, Yuan , , ] [Mason, Skinner ; Adamo, Casali, Skinner 1-loop: torus worldsheet nodal Riemann sphere. 2 n 1 τ i. 2 n 1 n = [Lionel Mason s talk; Geyer, Mason, Monteiro, Tourkine & ] natural framework to define partial integrand a (1) YM (...) one-loop BCJ relations from scattering equations one-loop KLT relations from double-copy structure of the integrand [He, OS, Zhang ]

12 Outline 11 I. Partial integrands and one-loop BCJ relations n n = +l l + cyclic(1, 2,..., n) 3 II. One-loop KLT formula and EYM relations ρ,τ S n 1 a(+, ρ(1, 2,..., n 1), n, ) S[ρ τ] l ã(+, τ(1, 2,..., n 1),, n) III. CHY derivation of one-loop amplitude relations a(τ(1, 2,..., n, +, )) = dµ tree n n 1

13 I. 1 New representations of Feynman integrals Ambiguity: which edge carries the loop momentum? 2 4 particularly confusing for l-dependent numerator n i (l) 1 < 5 Resolution: Canonicalize l-dependence via partial fraction (PF) = democratic sum over positions of l n 1 > = l 2 PF shift l 3 = = = n 1 i=0 d d l l 2 (l+k 1 ) 2 (l+k 12 ) 2... (l+k 12...n 1 ) 2 d d n 1 l (l+k 12...i ) 2 d d l l 2 d d l l 2 n 1 i=0 i 1 j=0 j=0 j i 1 (l+k 12...j ) 2 (l+k 12...i ) 2 1 s j+1,j+2,...,i, l n n 1 j=i+1 +l l 1 s i+1,i+2,...,j,+l 3 l or rather l ± k j?? linear in l + cyclic(1, 2,..., n) [Lionel s talk & GMMT ; Q-cuts: Emil s talk & BBBCDF ] 12

14 I. 2 Partial integrands n 13 Each term in partial-fractionized n-gon +l l has a notion of cyclic ordering in n+2 legs (1, 2,..., n, l, +l). Decompose single-trace subamplitudes into n partial integrands A (1) d d l n (1, 2,..., n) = l 2 a(1, 2,..., i,, +, i+1,..., n) i=1 }{{} all propagators linear in l only those diagrams compatible with cycle (1, 2,..., i,, +, i+1,..., n) 2 3 A (1) cyc. orbit of max(1, 2, 3, 4) = n box l 1 4 n = a max (1, 2, 3, 4,, +) = box s 1,l s 12,l s 123,l +l l

15 I. 2 Partial integrands n 14 Each term in partial-fractionized n-gon +l l has a notion of cyclic ordering in n+2 legs (1, 2,..., n, l, +l). Decompose single-trace subamplitudes into n partial integrands A (1) d d l n (1, 2,..., n) = l 2 a(1, 2,..., i,, +, i+1,..., n) i=1 }{{} all propagators linear in l only those diagrams compatible with e.g. max. SUSY: cycle (1, 2,..., i,, +, i+1,..., n) 2 3 A (1) cyc. orbit of max(1, 2, 3, 4) = n box l 1 4 n = a max (1, 2, 3, 4,, +) = box s 1,l s 12,l s 123,l +l l

16 Decompose single-trace subamplitudes into n partial integrands A (1) d d l n (1, 2,..., n) = l 2 a(1, 2,..., i,, +, i+1,..., n) i=1 15 Properties of partial integrand a(1, 2,..., i,, +, i+1,..., n) all propagators made linear in l via partial fraction only those diagrams compatible with cycle (1, 2,..., i,, +, i+1,..., n) decomposition is universal # spacetime dim s d and supercharges can be viewed as forward limit of color-ordered (n+2)-point tree each partial integrand is separately gauge invariant [Cachazo, He, Yuan ]

17 I. 3 Five-point maximal SUSY 16 By the no-triangle property, one pentagon & five boxes per single-trace 3 { 3 4 } A (1) max(1, 2, 3, 4, 5) = cyc(1, 2, 3, 4, 5) l 5 2 Partial integrands only have 4 out of 5 boxes (no massive corner with k 51 ): a max (1, 2, 3, 4, 5,, +) = + & + & & Individual numerators are gauge dependent, e.g. they change under linearized gauge trf. e i k i. & cf. 5pt BCJ numerators in pure-spinor superspace [Mafra, OS ]

18 Explicit five-point numerators for external gluons via generalized t 8 -tensor 17 t 8 (A, B, C, D) = tr(f A f B f C f D ) 1 4 tr(f Af B ) tr(f C f D ) + cyc(b, C, D) with linearized field-strength f mn 1 = k m 1 en 1 kn 1 em 1 & two-particle current e m 12 = em 2 (k 2 e 1 ) km 1 (e 1 e 2 ) (1 2) f mn 12 = k m 12 en 12 s 12 e m 1 en 2 (m n) Gauge trf. t 8 (12, 3, 4, 5) e1 k 1 = s 12 t 8 (2, 3, 4, 5) cancels between diag s t 8 (21,3,4,5) s 12 s 12,l s 123,l s 1234,l l m [ e m 1 t 8 (2,3,4,5) + (1 2,3,4,5) ] s 1,l s 12,l s 123,l s 1234,l [ t8 (12,3,4,5) + (12 13,14,...,45) ] s 1,l s 12,l s 123,l s 1234,l

19 18 I. 4 Amplitude relations of partial integrands From intuition partial integrands forward limit of (n+2)-point trees a(1, 2,..., n,, +) = A tree (1, 2,..., n, φ l, φ l) states φ,φ can export tree-level amplitude relations. KK-relations (n 2)!- basis {A tree (1, ρ(2,..., n 1), n), ρ S n 2 } = non-planar a(..., +,..., ) determined by a(...,, +) from BCJ-relations 0 = n 1 j=2 (k 1 k 23...j )A tree (2, 3,..., j, 1, j+1,..., n) at most (n 1)! independent a(...), with SUSY-dependent degeneracy

20 I. 4 Amplitude relations of partial integrands 19 From intuition partial integrands forward limit of (n+2)-point trees a(1, 2,..., n,, +) = states φ,φ can export tree-level amplitude relations. A tree (1, 2,..., n, φ l, φ l) KK-relations (n 2)!- basis {A tree (1, ρ(2,..., n 1), n), ρ S n 2 } = non-planar a(..., +,..., ) determined by a(...,, +) from BCJ-relations 0 = n 1 j=2 (k 1 k 23...j )A tree (2, 3,..., j, 1, j+1,..., n) 0 = n 1 (l k 12...i ) a(1, 2,..., i, +, i+1,..., n, ) i=1 at most (n 1)! independent a(...), with SUSY-dependent degeneracy

21 II. 1 KLT amplitude relations 20 1st double-copy formula: tree-level KLT relations [Kawai, Lewellen, Tye 1986] M tree grav = kernel S[ρ τ] 1 s n 3 ij ρ,τ S n 3 A tree (1, ρ(2, 3,..., n 2), n, n 1) S[ρ τ] 1 Ã tree (1, τ(2, 3,..., n 2), n 1, n) recursively determined by S[j j] i = k i k j and S[A, j B, j, C] i = k j (k i +k B ) S[A B, C }{{} ] i a 1, a 2,..., a x b 1, b 2,..., b y, c 1,..., c z [Bern, Dixon, Perelstein, Rozowsky 1998] [Bjerrum-Bohr, Damgaard, Feng, Sondergaard, Vanhove 2010] M tree grav is secretly permutation invariant by BCJ relations among A tree (...).

22 21 Identify states Atree (..., φ l,..., φ l) a(...,,..., +) in both gauge-theory copies of (n+2)-point tree-level KLT formula Mgrav 1-loop d d l = l 2 a(+, ρ, n, ) S[ρ τ] l ã(+, τ,, n) ρ,τ S n 1 with e.g. ρ = ρ(1, 2..., n 1), same for τ and tree-level kernel S[ρ τ] l.

23 22 Identify states Atree (..., φ l,..., φ l) a(...,,..., +) in both gauge-theory copies of (n+2)-point tree-level KLT formula Mgrav 1-loop d d l = l 2 a(+, ρ, n, ) S[ρ τ] l ã(+, τ,, n) ρ,τ S n 1 with e.g. ρ = ρ(1, 2..., n 1), same for τ and tree-level kernel S[ρ τ] l. e.g. 4pt supergravity maximal SUSY Mmax 1-loop = t8 (1, 2, 3, 4) 2 d d { } l 1 l 2 + perm(1, 2, 3, 4) s 1,l s 12,l s 123,l is reproduced from KLT formula d d l l 2 a(+, ρ(1, 2, 3), 4, ) S[ρ τ] l ã(+, τ(1, 2, 3),, 4) ρ,τ S 3 with a(+, 1, 2, 3, 4, ) = t 8(1,2,3,4) s 1,l s 12,l s 123,l & 4-term expression for a(+, 1, 2, 3,, 4).

24 II. 2 Cubic-graph equivalent of one-loop KLT 23 Rearranged Feynman integrals more cubic diagrams, e.g propagators (l + k 12...j ) 2 only one numerator partial fraction shifts in l l l l. l l l n (l) n (l). n (l)

25 II. 2 Cubic-graph equivalent of one-loop KLT 24 Rearranged Feynman integrals more cubic diagrams, e.g propagators (l + k 12...j ) 2 only one numerator partial fraction shifts in l l l l. l l l n (l) n (l). n (l) One-loop KLT separate squaring of the 5 numerators on the right Mgrav 1-loop d d l = l 2 a(+, ρ, n, ) S[ρ τ] l ã(+, τ,, n) ρ,τ S n 1 d d l n = i (l) ñ i (l) kin. Jacobi rel s l 2 edge α i s αi (l) linear in l i Γ n+2 tree all-multiplicity method for BCJ numerators [He, OS, Zhang ]

26 II. 3 EYM relations at one loop 25 E.g. n external gluons, one ext. graviton {e p, p} & internal gauge states: notion of partial integrand carries over to EYM: A (1) d d EYM (1, 2,..., n p) = l n l 2 a EYM (1, 2,..., i,, +, i+1,..., n p) i=1 }{{} all propagators linear in l forward limit of A tree EYM (... p)

27 II. 3 EYM relations at one loop 26 E.g. n external gluons, one ext. graviton {e p, p} & internal gauge states: notion of partial integrand carries over to EYM: A (1) d d EYM (1, 2,..., n p) = l n l 2 a EYM (1, 2,..., i,, +, i+1,..., n p) i=1 }{{} all propagators linear in l forward limit of A tree EYM (... p) Recycle tree-level relations such as n 1 EYM (1, 2,..., n p) = (e p k 12...j ) A tree (1, 2,..., j, p, j+1,..., n) A tree j=1 [Stieberger, Taylor ] analogous relations for multiple ext. gravitons and color traces. [Nandan, Plefka, OS, Wen; OS; Du, Feng, Fu, Huang; Chiodaroli, Günaydin, Johansson, Roiban; Feng, Teng]

28 One-loop amplitude relation among partial integrands via forward limit 27 n 1 EYM (1, 2,..., n p) = (e p k 12...j ) A tree (1, 2,..., j, p, j+1,..., n) j=1 A tree a EYM (+, 1, 2,..., n, p) = (e p l) a(+, 1, 2,..., n,, p) + n 1 (e p k 12...j ) a(+, 1, 2,..., j, p, j+1,..., n, ) j=1 Gauge invariance under e p p follows from BCJ relations among a(...). Example for recombination to Feynman integral: n = max. SUSY: A (1),max EYM (1, 2, 3; p) = t 8(1, 2, 3, p) d d l (e p l) l 2 (l+k 123 ) 2 (l+k 23 ) 2 + cyc(1, 2, 3) (l+k 3 ) 2

29 One-loop amplitude relation among partial integrands via forward limit 28 n 1 EYM (1, 2,..., n p) = (e p k 12...j ) A tree (1, 2,..., j, p, j+1,..., n) j=1 A tree a EYM (+, 1, 2,..., n, p) = (e p l) a(+, 1, 2,..., n,, p) + n 1 (e p k 12...j ) a(+, 1, 2,..., j, p, j+1,..., n, ) j=1 Gauge invariance under e p p follows from BCJ relations among a(...). Example for recombination to Feynman integral: n = max. SUSY: A (1),max EYM (1, 2, 3; p) = t 8(1, 2, 3, p) d d l (e p l) l 2 (l+k 123 ) 2 (l+k 23 ) 2 + cyc(1, 2, 3) (l+k 3 ) 2

30 III. CHY derivation of one-loop amplitude relations 29 Ambitwistor 1-loop prescription can be localized on nodal Riemann spheres. 2 n 1 τ i. 2 n 1 n = [Lionel Mason s talk; Geyer, Mason, Monteiro, Tourkine & ] Integrands factorize into halves L & R for color or kinematics M (1) d d L R = l n ( l ki n ) k i k j l 2 dσ i δ + Î σ i σ L (l) ÎR(l) ij i=2 j=1 j i

31 III. CHY derivation of one-loop amplitude relations 30 Ambitwistor 1-loop prescription can be localized on nodal Riemann spheres. 2 n 1 τ i. 2 n 1 n = [Lionel Mason s talk; Geyer, Mason, Monteiro, Tourkine & ] Integrands factorize into halves L & R for color or kinematics M (1) d d L R = l n ( l ki n ) k i k j l 2 dσ i δ + Î σ i σ L (l) ÎR(l) ij i=2 Recognize as SL 2 -fixed expression (σ +, σ, σ 1 ) = (0,, 1) with k ± = ±l M (1) d d L R = l l 2 lim dµ tree n+2 I L(l) I R (l) k ± ±l with tree-level measure dµ tree n+2 j=1 j i of the CHY formalism.

32 1-loop amplitudes of ambitwistor string tree-level CHY measure dµ tree M (1) d d L R = l l 2 lim dµ tree n+2 I L(l) I R (l) k ± ±l 31 n+2 tree-level KLT formula encoded in factorization of integrand I L (l) I R (l) and properties of dµ tree n+2 (yields doubly partial amplitudes of φ3 thy.) [Lionel Mason s talk; Cachazo, He, Yuan , , ] for supersymmetric integrands I L,R (l) I SUSY (l) ( 4 supercharges), result of dµ tree n+2 yields no forward-limit divergences as k ± ±l. proof of tree-level KLT carries over to one loop! [He, OS, Zhang ]

33 1-loop amplitudes of ambitwistor string tree-level CHY measure dµ tree M (1) d d L R = l l 2 lim dµ tree n+2 I L(l) I R (l) k ± ±l 32 n+2 tree-level KLT formula encoded in factorization of integrand I L (l) I R (l) and properties of dµ tree n+2 (yields doubly partial amplitudes of φ3 thy.) [Lionel Mason s talk; Cachazo, He, Yuan , , ] for supersymmetric integrands I L,R (l) I SUSY (l) ( 4 supercharges), result of dµ tree n+2 yields no forward-limit divergences as k ± ±l. proof of tree-level KLT carries over to one loop! [He, OS, Zhang ] explicit BCJ master numerators for n-gons: expand I L,R (l) in terms of n! Parke Taylor factors {PT(, ρ(1, 2,..., n), +), ρ S n }

34 Conclusions & Outlook 33 partial-fraction representation of Feynman integrals identifies partial integrands a(...) as gauge-invariant one loop a(...) = forward limit of trees natural one-loop uplift of tree-level BCJ, KLT and EYM amplitude relations CHY formalism concise derivation of one-loop amplitude relations

35 Conclusions & Outlook 34 partial-fraction representation of Feynman integrals identifies partial integrands a(...) as gauge-invariant one loop a(...) = forward limit of trees natural one-loop uplift of tree-level BCJ, KLT and EYM amplitude relations CHY formalism concise derivation of one-loop amplitude relations How to systematically undo partial-fractioning of Feynman integrals? Desirable to develop integration techniques for propagators linear in l. [Arkani-Hamed, Yuan: in progress]

36 35 need suitable higher-loop partial integrands a (g) for g-loop KLT rel s l 1 +l 2 l 1 < > 1 l2 4 A (2) (1, 2, 3, 4)? g = M grav (g) d d l j = > j=1 l 2 j ρ,τ... x z 1 x z 4 a (2) (1, 2, 3, 4, z, z, x, x) a (g) (...) S[......] ã (g) (...) starting points in [Geyer, Mason, Monteiro, Tourkine 2016; He, OS, Zhang 2017]

37 36 need suitable higher-loop partial integrands a (g) for g-loop KLT rel s l 1 +l 2 l 1 < > 1 l2 4 A (2) (1, 2, 3, 4)? g = M grav (g) d d l j = > j=1 l 2 j ρ,τ... x z 1 x z 4 a (2) (1, 2, 3, 4, z, z, x, x) a (g) (...) S[......] ã (g) (...) starting points in [Geyer, Mason, Monteiro, Tourkine 2016; He, OS, Zhang 2017] loop-level KLT in string theory? connections with monodromy rel s? [Piotr s talk; Vanhove, Tourkine ; Hohenegger, Stieberger ] Thank you for your attention!