Scattering on plane waves and the double copy L.J.Mason The Mathematical Institute, Oxford lmason@maths.ox.ac.uk Jurekfest September 19/9/2019 Joint work with Adamo, Casali & Nekovar 2017-8, arxiv:1706.08925, 1708.09249, 1810.05115. Further work to come with Tim Adamo & Atul Sharma.
Calculate amplitudes on plane-wave backgrounds. Conventional motivation: ◮ Construct interacting perturbative QFT on asymptotically simple curved backgrounds. ◮ Plane waves are universal as Penrose limits. ◮ Computable: Plane waves satisfy Huygens & have separable Hamilton-Jacobi and linear fields. ◮ Test Gravity = Double copy of Yang-Mills. We needed data to check 3 point curved background ambitwistor string YM & gravity formulae. [Adamo, Casali, M & Nekovar 1708.09249] [Aim: extend amplitude & ambitwistor-strings to curved space.]
Yang-Mills amplitudes & colour-kinematic duality Scatter n particles, momentum k µ , polarization ǫ µ A µ ( x ) = ǫ µ e ik · x t a , k 2 = 0 , t a ∈ Lie G . k · ǫ = 0 , ◮ Suppose that YM amplitude A ( k i , ǫ i , t i ) , i = 1 , . . . , n arises from trivalent Feynman diagrams N Γ ( k i , ǫ i ) C Γ ( t i ) � A = , Γ ∈ { trivalent diagrams, n legs } . D Γ Γ ◮ N Γ = kinematic factors : polynomials in k i , linear in each ǫ i . i ∈ e k i ) 2 = denominators . ◮ D Γ = � propagators e ∈ Γ ( � ◮ C Γ ( t i ) = colour factor = contract structure contants at each vertex together along propagators and with t i at i th leg. Definition The N Γ are said to BCJ numerators if N Γ satisfy identities when C Γ does via Jacobi identities: C ˜ Γ = C Γ + C Γ ′ ⇒ N ˜ Γ = N Γ + N Γ ′ . Possible at tree-level and up to 4-loops, but not canonical.
Gravity as double copy of Yang-Mills Zvi Bern, J J Carrasco, H Johansson, 2008 Scatter n gravity plane waves h µν = ǫ ( µ ǫ ν ) e ik · x Given BCJ numerators N Γ , the gravity tree-amplitude/loop integrand can be obtained as a double copy of YM amplitude N Γ ( k i , ǫ i ) N Γ ( k i , ǫ i ) � M ( k i , ǫ i , ǫ i ) = D Γ Γ ◮ KLT tree relation gravity amplitudes = ( YM ) 2 from strings. ◮ Proved up to 4-loops. ◮ There are extensions to many theories. ◮ Genuine tool for constructing gravity amplitudes. ◮ No nonperturbative or space-time explanation.
The three point amplitude At three points, there is just one trivalent diagram A = ( ǫ 1 · ǫ 2 ǫ 3 · ( k 1 − k 2 )+ � ) f abc t a 1 t b 2 t c 3 = N ☎ ( ǫ i , k i ) C ☎ ( t i ) For gravity M = N ☎ ( ǫ i , k i ) N ☎ ( ǫ i , k i ) , but very nontrivial: graviton 3-vertex is much more complicated. Can we extend to curved backgrounds? ◮ How do we define momentum eigenstates? ◮ What are momenta and polarization vectors? ◮ How can we relate Yang-Mills and Gravity?
Sandwich plane waves The Brinkman form in d -dimensions of the metric is ds 2 = dudv − Hdu 2 − dx a dx a , a = 1 , .., d − 2 . a := R ab l a l b = 0 for vacuum. with H = H ( u ) ab x a x b , H a out in u v Figure: The sandwich plane wave with x a -directions suppressed, H ab ( u ) � = 0 only in the shaded region with flat in- and out-regions. ◮ H ab = curvature, supported for u ∈ [ 0 , 1 ] (shaded). ◮ These coordinates are global, but: ◮ Space-time not globally hyperbolic! (Penrose).
Plane wave symmetries: 2 d − 3-Heisenberg group ds 2 = dudv − H ab ( u ) x a x b du 2 − dx a dx a ◮ Heisenberg group is transitive on u = const . , centre ∂ v . ◮ 2 d − 4 killing vectors take form e a ∂ x a − ˙ e a x a ∂ v s.t. · = d e a = H ab e b , ¨ du ◮ Choose d − 2-dimensional abelian subgroup D i = e a e a i ∂ x a − ˙ i x a ∂ v , i = 1 . . . d − 2 , e a commuting ⇔ ˙ [ i e j ] a = 0. ◮ Let e i a be inverse matrix, e i a e ib = δ ab .
Momentum eigenstates on plane waves: I. Gravity ◮ Choose d − 1 commuting symmetries ( ∂ v , D i ) ❀ Separable Hamilton-Jacobi soln, momenta ( k + , k i ) a x a + k i k j F ij ( u ) 1 2 σ ab x a x b ) + k i e i φ k = k + ( v + , 2 k + � u e i a e ja du ′ and σ ab = ˙ where F ij ( u ) = e i a e bi ‘shear’. ◮ Then Φ k = e i φ k | e | = det( e a , i ) solves ✷ Φ k = 0 . � | e | ◮ Such a field has a ‘curved’ momentum K µ dX µ := d φ k = k + dv + ( σ ab x b + k i e i a ) dx a + ( . . . ) du Memory: As u → −∞ , set e a i = δ a i so K µ = ( k + , k a , k a k a / 2 k + ) const. As u → + ∞ , e a i ( u ) = b a i + uc a i , b , c = const . , and σ ab � = 0; so wave fronts φ k = const . become curved.
Higher spins ◮ We have d − 2 covariantly constant spin raising operators R a = du δ ab ∂ x b + dx a ∂ v , ∇ µ R a = 0 . ◮ Gives linear gauge field on background A = ǫ a R a Φ k = ε µ dX µ Φ k , k + with curved polarization ε µ , K µ ε µ = 0, � k i e i � ε µ dX µ = ǫ a dx a + ǫ a a + σ ab x b du k + ◮ Linear gravity on background h µν dX µ dX ν = ǫ a R a ( ǫ b R b Φ k ) � ( ε · dX ) 2 − i � ǫ a ǫ b σ ab du 2 = Φ k k 2 k + + Note potential obstruction to double copy.
Tails and Huygens Theorem (Friedlander 1970s) The only space-times that admit clean cut solutions to the wave equation are conformal to plane waves (or flat space). ◮ Φ = | e | − 1 / 2 δ ( φ k ) is clean cut solution to wave equation. ◮ Analogous spin-1 solution is a = | e | − 1 / 2 ǫ a R a ( φ k Θ( φ k )) so F = da = δ ( φ k ) | e | − 1 2 ǫ a R a φ k ∧ d φ k +Θ( φ k ) | e | − 1 2 ǫ a σ 0 ab dx a ∧ du i.e., there is backscattering with a tail. ◮ Similar spin-2 solution has longer tail.
Momentum eigenstates on plane waves: II. Yang-Mills Use same coordinates on flat space-time with gauge potential A a ( u ) dx a ∧ du . A = ˙ F = ˙ A a ( u ) x a du , Again, take sandwich wave with Supp ( ˙ A a ) ⊂ u ∈ [ 0 , 1 ] . Momentum eigenstate charge e , ✷ eA Φ k = 0: � � k + v +( k a + eA a ) x a + f ( u ) i Φ k = e 2 k + , ˙ � � f ( u ) momentum K µ ( u ) = k + , k a + eA a ( u ) , with 2 k + � u ( k a + eA a )( k a + eA a ) du ′ . K · K = 0 f ( u ) = ❀ −∞ Memory: Choose A a = 0 for u < 0, then for u > 1, A a ( u ) = const . � = 0. � � k + , k a , k a k a , u < 0 , 2 k + K µ ( u ) = ˙ � � f ( 1 ) k + , k a + eA a ( 1 ) , , u > 1 . 2 k +
Linear YM fields on the background ◮ ˙ A a ( u ) x a du valued in Cartan subalgebra h of gauge Lie alg.. ◮ Charged linear YM field a µ satisfies D µ D [ µ a ν ] + a µ ∂ [ µ eA ν ] = 0 , D µ = ∂ µ + eA µ . ◮ Colour encoded in charge e = eigenvalue of h × coupling. ◮ Solution a = ˜ ǫ a R a Φ k = ˜ ε µ dX µ Φ k , transverse polarization � dx a + 1 � ε µ ( u ) dX µ = ˜ ( k a + eA a ( u )) du ˜ ǫ a , ǫ a = const .. k + Convention: YM background polarization vectors are tilded.
No particle creation or leakage As u → −∞ take linear fields to become flat space-time momentum eigenstates, i.e., e a i = δ a i , and A a = 0; ◮ ± frequency determined by sign of k + , doesnt change with u so no particle creation. ◮ Inner products are u -independent on both backgrounds: � Φ k | Φ k ′ � = 2 k + δ ( k + − l + ) δ d − 2 ( k i − l i ) . Similarly for spin-1 � a 1 | a 2 � = 2 ǫ 1 · ǫ 2 k + δ ( k + − l + ) δ d − 2 ( k i − l i ) and spin-2 � h 1 | h 2 � = 2 ( ǫ 1 · ǫ 2 ) 2 k + δ ( k + − l + ) δ d − 2 ( k i − l i ) . ◮ Failure of global hyperbolicity does not lead to leakage. [Failure in space of null geodesics: those parallel to ∂ v so co-dimension too high.]
Three particle gravity amplitude ◮ Cubic part of action give 3 vertex M 3 = κ � d d X ( h µν 1 ∂ µ h 2 ρσ ∂ ν h ρσ 3 − 2 h ρν 1 ∂ µ h 2 ρσ ∂ ν h µσ 3 )+ perms 4 ◮ Inserting our states yields � 3 � s � � � F ij k ri k rj κ du � � 2 δ d − 1 k r exp � det e a 2 k r 0 i r = 1 r = 1 [( ε 1 · ε 2 ( K 1 − K 2 ) · ε 3 + � ) 2 − ik 1 + k 2 + k 3 + σ ab C a C b ] where ǫ 3 a C a := ε 1 · ε 2 + � k 3 + ◮ First term = (YM 3-pt amplitude) 2 on gravity background. ◮ However: tail term σ ab C a C b seems to obstruct double copy.
Three-point YM amplitude M a [ µ a ν ] D µ a ν d 4 X gives 3 point vertex ◮ Cubic part of action � � 3 � � f r ( u ) � A 3 = du exp i [˜ ε 1 · ˜ ε 2 ˜ ε 3 · ( K 1 − K 2 )+ � ] C ☎ ( t i ) . 2 k r 0 r = 1 ◮ Bracketed term is in flat quantities � �� k 1 + � � �� � k 1 + ǫ a ǫ 1 · ˜ ˜ ǫ 2 ˜ k 2 a − k 1 a + A a e 1 − e 2 + � =: F + C . 3 k 2 + k 2 + ◮ Second term gives background ‘tail’ dependence on A a .
Double copy replacement principle YM to GR uses replacement rules: 1. Flip charges e r → − e r so A 3 = F + C → ˜ A = F − C and |A 3 | 2 := A 3 ˜ A 3 = F 2 − C 2 with F = F ( k r , ˜ ǫ ) and C = C ( k r , ˜ ǫ r , A ) . ǫ ra ) by ( k i e i a , ǫ a ) ❀ F ( k ri e i a , ǫ r ) , C ( k ri e i 2. Replace ( k ra , ˜ a , ǫ r , A ) . 3. Replace � ik r 0 σ ab r = s , e r e s A a A b → i ( k r 0 + k s 0 ) σ ab r � = s . Yields double copy of YM integrand for GR incorporating tails.
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