Hidden Conformal Symmetry in Tree-Level Graviton Scattering

Florian Loebbert

Humboldt University Berlin

based on arXiv: 1802.05999 with Matin Mojaza and Jan Plefka

Symmetries of S-matrix and Infrared Physics Higgs Centre for Theoretical Physics, Edinburgh July 2018 Graviton Scattering

Recent years show explosion of insights and interest in graviton scattering, e.g. I BCJ Double Copy I CHY and ambitwistor string I BMS symmetries I Gravitational wave physics . . .

What is the symmetry of the quantum gravity S-matrix?

Yang–Mills vs Gravity: I The actions of Yang–Mills theory and Einstein gravity look rather diﬀerent. I Perturbative quantization through gluons and gravitons shows striking similarities in amplitudes.

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 1 / 21 Conformal Symmetry of Gluon Amplitudes 4d Yang–Mills action invariant under conformal transformations: Z 4 µν SYM = d x tr F Fµν

Color-stripped gluon tree amplitudes: p momenta Σ X a1 a2 an polarizations An (ki, i, ai) = Tr T T ...T An(ki, i) a colors noncyc. perm Momenta/polarizations (k, ) traded for spinors/helicities (λ, λ¯, h): ¯ An(ki, i) → An(λi, λi, hi). Spinors λα, λ¯α˙ solve on-shell constraints: αα˙ µ αα˙ α ¯α˙ µ k = k (σµ) = λ λ ⇒ k kµ = 0

a ¯ Witten Conformal Ward identities: J An(λi, λi, hi) = 0 [ 2003 ]

( αβ˙ a ¯β˙ α α 1 α γ a P = λ λ , L β = λ ∂β − 2 δβ λ ∂γ , J ∈ 1 γ 1 ¯γ˙ ¯ ¯ D = 2 ∂γ λ + 2 λ ∂γ˙ , Kαβ˙ = ∂α∂β˙ .

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 2 / 21 Conformal Symmetry in Graviton Scattering?

Hints on hidden symmetries in graviton scattering: I 4d YM trees are conformal and

2 Kawai, ’86 Bern, Carrasco, ... YM ' Gravity [Lewellen, Tye ][ Johansson ’08 ][ ]

Imprint of conformal symmetry on graviton amplitudes? I In curved space at tree level:

Metsaev Maldacena Adamo Einstein gravity ' conformal gravity [ 2011 ][ 2011 ][Mason 2013]

I Graviton amplitudes as conformal correlators on the celestial sphere Cachazo ... [Strominger 2014][ ]

What is the role of conformal symmetry for graviton scattering?

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 3 / 21 Einstein Gravity Conformal symmetry of graviton amplitudes?: Z 1 d √ SEH = 2 d x −g R, gµν = ηµν + κdhµν . κd

Naive answer is no: 2 I coupling constant κd has mass dimension 2 − d. I forbids naive conformal symmetry for d 6= 2. I so if there is a conformal symmetry, it must be hidden.

This talk:

Di Vecchia, Marotta I Motivation: Get inspiration from string soft limits of[ Mojaza ’15-’17 ]. I Method: Experiments with conformal generators and ﬁeld theory graviton amplitudes. I Conclusion: Tree-level graviton amplitudes have conformal symmetry in d dimensions.

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 4 / 21 .

Soft Dilatons and Graviton Scattering Inspiration from Strings

In the α0 → 0 limit, scattering of closed strings yields graviton ﬁeld theory amplitude:

0 α−→→0

Eventually we will be only interested in the ﬁeld theory limit.

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 5 / 21 String Theory Soft Dilatons

Di Vecchia, Marotta Special case of string theory soft limit of[ Mojaza ’15-’17 ]: [see Paolo’s talk] Amplitude for n hard gravitons or dilatons and a soft dilaton φ with small momentum q: inv. string tension 0 h α κd M (k , .., k , q) = √ δ(P )S + q · ∂ δ(P ) S 0 n+φ 1 n d−2 δ P δ i α0 2 + (SW + SV )δ(P ) Mfn + O(q ),

α0→0 I Mn = δ(P )Mfn is the ﬁeld theory amplitude. I Conformal generators enter: µ Sδ = 2 − D∆=0 + qµK∆=0,Sδ0 = 2 − D∆=0.

I as well as certain nonlocal operators (gauge transf. Wi = ki · ∂i ): n n σ X q · i X qρq ρµ ρ SW = − 1 + q · ∂ki Wi,SV = Si Si,µσ + d i ∂i,σ . ki · q 2ki · q i=1 i=1 Understand this formula in detail?

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 6 / 21 The Operator SW Contribution to the soft expansion:

α0 κ h α0 i 2 M (k , .., k , q) = √ d ··· + S δ(P )M + O(q ), n+φ 1 n d−2 W en

n X q · i 2 SW = − (1 + q · ∂ki ) Wi, +O(q ) Wi = ki · ∂i . ki · q i=1 q∂ e ki

I If one assumes that the amplitude is manifestly gauge invariant, the Di Vecchia, Marotta contribution from SW could be dropped[ Mojaza ’16 ] But: manifestly gauge invariant form not known for n > 4

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 7 / 21 The Operator SW

Contribution to the soft expansion:

α0 κ h α0 i 2 M (k , .., k , q) = √ d ··· + S δ(P )M + O(q ), n+φ 1 n d−2 W en

n X q · i SW = − (1 + q · ∂ki ) Wi,Wi = ki · ∂i . ki · q i=1 q∂ e ki +O(q2)

I If one assumes that the amplitude is manifestly gauge invariant, the contribution from SW could be dropped But: manifestly gauge invariant form not known for n > 4 I Alternative argument: Write n X q · i α0 2 2 − (ki + q) · ∂i Mn (k1, . . . , ki + q, . . . , kn) + O(q ) = 0 + O(q ). ki · q i=1 Note: requires reordering expansion in q!

Thus we assume that SW can be dropped.

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 7 / 21 The Operator SV Contribution to the soft expansion:

α0 κ h α0 i 2 M (k , .., k , q) = √ d ··· + S δ(P )M + O(q ), n+φ 1 n d−2 V en

n σ X qρq ρµ ρ SV = (Si Si,µσ + d i ∂i,σ) . 2ki · q i=1 Use quadratic dependence of amplitude on polarizations: n σ X qρq ρµ ρ α β α0 Si Si,µσ + di ∂iσ δ(P ) i i Men,i,αβ 2ki · q i=1 n α β αβ 2 X h q q (i · i) η (q · i) i α0 = δ(P ) + Men,i,αβ . ki · q ki · q i=1

q q I Interpretation of two terms: +

µ ν µν since for the graviton i · i = 0, while for the dilaton i i ∼ η . Here: Take n hard particles to be gravitons → ﬁrst term vanishes. Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 8 / 21 The Operator SV Contribution to the soft expansion:

α0 κ h α0 i 2 M (k , .., k , q) = √ d ··· + S δ(P )M + O(q ), n+φ 1 n d−2 V en

q q I Interpretation of two terms: +

Amplitude for n gravitons and a soft dilaton φ with small momentum q:

0 h i 0 α κd α M (k , . . . , k , q) = √ δ(P )S + q · ∂ δ(P ) S 0 M n+φ 1 n d−2 δ P δ fn n 2 X (q · i) 0 + √κd δ(P )M α (k , . . . , φ(k ), . . . , k ) + O(q2) , d−2 k · q n 1 i n i=1 i Low engergy string action: √ 1 Z √ 1 1 8 2 d µν − d−2 φ Slow-energy = 2 d x −g R − g ∂µφ∂ν φ − e ∂[µBνρ] . 2κd 2 12

I Dilaton and B-field couple only quadratically to the graviton. I No virtual B-fields in graviton/dilaton amplitudes at tree level. ⇒ In the field theory limit α0 → 0, tree-level amplitudes with n gravitons and an odd number of dilatons vanish!

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 9 / 21 What Remains?

Amplitude for n gravitons and a soft dilaton φ with small momentum q:

0 h i 0 α κd α M (k , . . . , k , q)0√ = δ(P )S + q · ∂ δ(P ) S 0 M n+φ 1 n d−2 δ P δ fn n 2 X (q · i) 0 + √κd δ(P )M α (k , . . . , φ(k ), . . . , k ) + O(q2) , d−2 k · q n 1 i n i=1 i Low engergy string action: √ 1 Z √ 1 1 8 2 d µν − d−2 φ Slow-energy = 2 d x −g R − g ∂µφ∂ν φ − e ∂[µBνρ] . 2κd 2 12

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 9 / 21 Conformal Symmetry?

Thus we are left with

h i α0→0 2 0 = δ(P )Sδ + q · ∂P δ(P ) Sδ0 Mfn + O(q ).

with conformal generators entering as:

µ Sδ = 2 − D∆=0 + qµK∆=0,Sδ0 = 2 − D∆=0.

α0→0 If Mfn were the ﬁeld theory n-graviton amplitude, this would imply conformal invariance! But. . .

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 10 / 21 Amplitude vs Integral . . . a priori Mf is not the amplitude: α0→0 Mfn becomes the ﬁeld theory amplitude if multiplied by δ(P ). To be more precise: The n-point bosonic string amplitude of massless closed states carrying momenta ki and polarizations i, ¯i is given by α0 α0 Mn = δ(P )Mfn , with the integral

n n−2 Z Q d2z 0 8π κd i Y 0 Mα = i=1 |z − z |α kikj n 0 i j e α 2π dVabc i

2 2 2 d zad zbd zc and dVabc = 2 2 2 , a, b, c ∈ {1, . . . , n} . |za−zb| |zb−zc| |zc−zd| So what does this mean for ﬁeld theory amplitudes?

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 11 / 21 .

Implications for Field Theory Amplitudes Implications for Field Theory Amplitude

Consider soft-dilaton result order by order: α0→0 0 =δ(P )(2 − D∆=0)Mfn µ α0→0 2 + δ(P )qµK∆=0 + q · ∂P δ(P )(2 − D∆=0) Mfn + O(q ) . Leading order: 0 O(q ) : 0 = D d−2 δ(P )Mn, with Di,∆ = ki · ∂ki + ∆ ∆= n amplitude Next-to-leading order: Commute Kµ through δ(P ): n 1 µ d−2 X α0→0 O(q ) : 0 =qµK d−2 δ(P )Mn − n δ(P ) q · ∂ki Mfn , ∆= n amplitude i=1 with the special conformal generator: Kµ = 1 kµ∂2 − (k · ∂ )∂µ − ∆∂µ − iSµν ∂ ,Sµν = iµ∂ν − ν ∂µ i,∆ 2 i ki i ki ki ki i ki,ν j j j j j ⇒ Additional term drops out for d = 2 and ∆ = 0. How about d 6= 2? Try!

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 12 / 21 Field (Theory) Work: Act on Tree Amplitudes

Special conformal generator: µ 1 µ 2 µ µ µν K∆ = 2 k ∂k − (k · ∂k)∂k − ∆∂k − iS ∂k,ν Constraints: I Momenta and polarization vectors obey

ki · ki = 0, ki · i = 0 . I and amplitudes are gauge invariant:

WiAn = 0 , Wi = ki · ∂i . Special conformal generator does not commute with constraints: µ 2 µ µ [K∆, ki ] = (d − 2 − 2∆)ki + 2i Wi, µ µ [K∆, ki · i] = i [(d − 1 − ∆) + i · ∂i ]. [Kµ , W ] = −∂µ W + (1 − ∆)∂µ − iSµν ∂ , ∆ i ki i i i i,ν ∂δ(P ) [Kµ , δ(P )] = (d − D )ηµν + Jµν ∆ ∂P ν ∆ µ ⇒ K∆ takes us oﬀ the kinematical constraint surface.

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 13 / 21 Back to Yang–Mills Amplitudes

Bourjaily Mafra Act on d-dimensional tree amplitudes in {ki, i} by[ ][Schlotterer]. Poincaré symmetry (modulo gauge transformations) dictates: δ-stripped amplitude n n µ X µ X µ K∆ An = i Fi + ki Gi, i=1 i=1 We explicitly resolve on-shell constraints and momentum conservation:

n n n µ X µ 2 X X ka = − ki , ka = ki · kj = 0 , a · ka = −a · ki = 0 , i6=a i6=j6=a i6=a

d−2 and set the scaling dimension to ∆ = 2 in four dimensions:

n µ ∆=1 X µ K∆ An = i Fi6= 0. i=1

⇒ Even Yang–Mills amplitudes in 4d are not invariant in (k, )-space.

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 14 / 21 Yang–Mills Amplitudes in (k, )-Space

Note: Cyclic and reversal symmetry obeyed by YM amplitudes only modulo on-shell constraints:

An(1, 2, . . . , n) = An(2, . . . , n, 1), n An(1, 2, . . . , n) = (−1) An(n, . . . , 2, 1).

Make symmetries manifest:

1 X n Cn[An] := 2n [An(1, 2, . . . , n) + (−1) An(n, . . . , 2, 1)] . Cyc(1,2,...,n)

On shell we have Cn[An] = An but now (checked for n ≤ 6):

µ K∆Cn[An] = 0, for ∆ = 1.

Observation: Manifest symmetries restore conformal properties!

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 15 / 21 (Pathological) Three-Point Example

The three-point delta-stripped YM amplitude takes the form

A3 = (1 · k2)e23 − (2 · k1)e31 − (3 · k2)e12 , eij = i · j,

with explicit implementation of momentum conservation using k3 = −k1 − k2 and (3 · k1) = −(2 · k2). One ﬁnds µ µ µ µ K∆=1A3 = e123 − e231 + e312 6= 0, Cyclic and reversal symmetrized form of amplitude

1 1 1 C3[A3] = 2 1 · (k2 − k3) e23 + 2 2 · (k3 − k1) e31 + 2 3 · (k1 − k2) e12 , is annihilated: µ K∆ C3[A3] = 0. Works also for n = 4, 5, 6 for ∆ = 1 (for n = 3 ∆ is actually arbitrary).

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 16 / 21 Gravity Tree-Amplitudes

Scattering data from Yang–Mills via KLT:

M3(1, 2, 3) = iA3(1, 2, 3)A3(1, 2, 3) ,

M4(1, 2, 3, 4) = −is12A4(1, 2, 3, 4)A4(1, 2, 4, 3) ,

M5(1, 2, 3, 4, 5) = is12s34A5(1, 2, 3, 4, 5)A5(2, 1, 4, 3, 5)

+ is13s24A5(1, 3, 2, 4, 5)A5(3, 1, 4, 2, 5) , M6(1, 2, 3, 4, 5, 6) = −is12s45A6(1, 2, 3, 4, 5, 6) s35A6(2, 1, 5, 3, 4, 6) + (s34 + s35)A6(2, 1, 5, 4, 3, 6) + P(2, 3, 4) .

Similar situation as for Yang–Mills amplitudes: δ-stripped amplitude canonical dimension in 2d n n n µ X µ X µ ∆=0 X µ K∆Mn = i Fi + ki Gi = i Fi6= 0. i=1 i=1 i=1

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 17 / 21 Gravity Tree-Amplitudes

Note: Full permutation symmetry obeyed by graviton amplitudes only modulo on-shell constraints:

Mn(1, 2, . . . , n) = Mn(P(1, 2, . . . , n)).

Make symmetries manifest:

1 X P [M (1, . . . , n)] = M (1, . . . , n). n n n! n P(1,...,n)

We have on shell that Pn[Mn] = Mn but now (checked for n ≤ 6):

µ K∆Pn[Mn] = 0, for any value of ∆.

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 18 / 21 Conformal Symmetry of Graviton Amplitudes

Special conformal generator: µ 1 µ 2 µ µν µ K∆ = 2 k ∂k − (k · ∂k)∂k − iS ∂k,ν − ∆∂k . Independence of ∆ means that n X ∂µ P [M ] = 0. ki n n i=1 Consistent with soft-dilaton statement: n µ d−2 X α0→0 0 = qµK d−2 δ(P )Mn − n δ(P ) q · ∂ki Mfn , ∆= n i=1

Hence, the manifestly symmetric amplitude Pn[Mn] furnishes a rep- resentation such that

µ µ K d−2 δ(P )Pn[Mn] = 0, D d−2 δ(P )Pn[Mn] = 0. ∆= n ∆= n

Together with translations Pµ and Lorentz rotations Lµν this yields full conformal invariance.

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 19 / 21 Summary

(d) Invariance of full amplitude An = δ (P )An under dilatations D∆ and (d) vanishing commutator of K∆ with δ (P ) requires d − 4 d − 2 YM : ∆ = + 1, Gravity : ∆ = . n n µ Pn µ Pn µ Poincaré invariance implies K∆An = i=1i Fi + i=1ki Gi . I YM: Gi = 0 for ∆ = 1. I Gravity: Gi = 0 for ∆ = 0. µ Non-zero Fi related to incompatibility of K∆ with constraints. Make the physical symmetries, i.e. cyclic/reversal or full permutation symmetry, manifest such that (checked for n ≤ 6) YM: in d = 4 and for ∆ = 1 we ﬁnd F = 0 ) I i µ K An = 0 I Gravity: for any value of d and ∆ we ﬁnd Fi = 0 ∆

Conclusion: Graviton scattering shows conformal symmetry at tree level.

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 20 / 21 Puzzles for the Future

I Meaning of the d-dimensional conformal symmetry? I Derivation from the Einstein–Hilbert action? Twistor methods? I Relation to CHY, double copy, celestial sphere? I Similar symmetry for d-dimensional Yang–Mills theory? I Use the new symmetry?

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 21 / 21 Puzzles for the Future

Thank you!

Florian Loebbert: Hidden Conformal Symmetry in Tree-Level Graviton Scattering 21 / 21