SLIDE 1
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.
SLIDE 2 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.]
SLIDE 3 Yang-Mills amplitudes & colour-kinematic duality
Scatter n particles, momentum kµ, polarization ǫµ Aµ(x) = ǫµeik·xta , k2 = 0 , k · ǫ = 0 , ta ∈ Lie G.
◮ Suppose that YM amplitude A(ki, ǫi, ti), i = 1, . . . , n arises
from trivalent Feynman diagrams A =
NΓ(ki, ǫi)CΓ(ti) DΓ , Γ ∈ { trivalent diagrams, n legs} .
◮ NΓ = kinematic factors: polynomials in ki, linear in each ǫi. ◮ DΓ = propagators e∈Γ( i∈e ki)2 = denominators. ◮ CΓ(ti) =colour factor = contract structure contants at each
vertex together along propagators and with ti at ith leg.
Definition
The NΓ are said to BCJ numerators if NΓ satisfy identities when CΓ does via Jacobi identities: C˜
Γ = CΓ + CΓ′ ⇒ N˜ Γ = NΓ + NΓ′.
Possible at tree-level and up to 4-loops, but not canonical.
SLIDE 4 Gravity as double copy of Yang-Mills
Zvi Bern, J J Carrasco, H Johansson, 2008
Scatter n gravity plane waves hµν = ǫ(µǫν)eik·x Given BCJ numerators NΓ, the gravity tree-amplitude/loop integrand can be obtained as a double copy of YM amplitude M(ki, ǫi, ǫi) =
NΓ(ki, ǫi)NΓ(ki, ǫi) DΓ
◮ 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.
SLIDE 5
The three point amplitude
At three points, there is just one trivalent diagram A = (ǫ1 · ǫ2ǫ3 · (k1 − k2)+ )fabcta
1tb 2 tc 3 = N☎(ǫi, ki)C☎(ti)
For gravity M = N☎(ǫi, ki)N☎(ǫi, ki) , 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?
SLIDE 6 Sandwich plane waves
The Brinkman form in d-dimensions of the metric is ds2 = dudv − Hdu2 − dxadxa , a = 1, .., d − 2 . with H = H(u)abxaxb, Ha
a := Rablalb = 0 for vacuum.
in u v
Figure: The sandwich plane wave with xa-directions suppressed, Hab(u) = 0 only in the shaded region with flat in- and out-regions.
◮ Hab = curvature, supported for u ∈ [0, 1] (shaded). ◮ These coordinates are global, but: ◮ Space-time not globally hyperbolic! (Penrose).
SLIDE 7
Plane wave symmetries: 2d − 3-Heisenberg group
ds2 = dudv − Hab(u)xaxbdu2 − dxadxa
◮ Heisenberg group is transitive on u = const., centre ∂v. ◮ 2d − 4 killing vectors take form ea∂xa − ˙
eaxa∂v s.t. ¨ ea = Habeb , · = d du
◮ Choose d − 2-dimensional abelian subgroup
Di = ea
i ∂xa − ˙
ea
i xa∂v,
i = 1 . . . d − 2 , commuting ⇔ ˙ ea
[iej]a = 0. ◮ Let ei a be inverse matrix, ei aeib = δab.
SLIDE 8 Momentum eigenstates on plane waves: I. Gravity
◮ Choose d − 1 commuting symmetries (∂v, Di) ❀
Separable Hamilton-Jacobi soln, momenta (k+, ki) φk = k+(v +
1 2σabxaxb) + kiei axa + kikjF ij(u) 2k+
, where F ij(u) = u ei
aejadu′ and σab = ˙
ei
aebi ‘shear’. ◮ Then
Φk = eiφk
, |e| = det(ea
i )
solves ✷Φk = 0.
◮ Such a field has a ‘curved’ momentum
KµdX µ := dφk = k+dv + (σabxb + kiei
a)dxa + (. . .)du
Memory: As u → −∞, set ea
i = δa i so Kµ = (k+, ka, kaka/2k+) const.
As u → +∞, ea
i (u) = ba i + uca i , b, c = const., and σab = 0;
so wave fronts φk = const. become curved.
SLIDE 9 Higher spins
◮ We have d − 2 covariantly constant spin raising operators
Ra = duδab∂xb + dxa∂v , ∇µRa = 0 .
◮ Gives linear gauge field on background
A = ǫaRa k+ Φk = εµdX µΦk , with curved polarization εµ, K µεµ = 0, εµdX µ = ǫadxa + ǫa kiei
a
k+ + σabxb
◮ Linear gravity on background
hµνdX µdX ν = ǫaRa(ǫbRbΦk) k2
+
=
k+ ǫaǫbσabdu2
Note potential obstruction to double copy.
SLIDE 10
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|−1/2δ(φk) is clean cut solution to wave equation. ◮ Analogous spin-1 solution is
a = |e|−1/2ǫaRa(φkΘ(φk)) so F = da = δ(φk)|e|− 1
2 ǫaRaφk∧dφk+Θ(φk)|e|− 1 2 ǫaσ0 abdxa∧du
i.e., there is backscattering with a tail.
◮ Similar spin-2 solution has longer tail.
SLIDE 11 Momentum eigenstates on plane waves: II. Yang-Mills
Use same coordinates on flat space-time with gauge potential A = ˙ Aa(u)xadu , F = ˙ Aa(u)dxa ∧ du . Again, take sandwich wave with Supp( ˙ Aa) ⊂ u ∈ [0, 1]. Momentum eigenstate charge e, ✷eAΦk = 0: Φk = e
i
2k+
momentum Kµ(u) =
˙ f(u) 2k+
K · K = 0 ❀ f(u) = u
−∞
(ka + eAa)(ka + eAa)du′ . Memory: Choose Aa = 0 for u < 0, then for u > 1, Aa(u) = const. = 0. Kµ(u) =
2k+
u < 0 ,
˙ f(1) 2k+
u > 1 .
SLIDE 12 Linear YM fields on the background
◮ ˙
Aa(u)xadu valued in Cartan subalgebra h of gauge Lie alg..
◮ Charged linear YM field aµ satisfies
DµD[µaν] + aµ∂[µeAν] = 0 , Dµ = ∂µ + eAµ.
◮ Colour encoded in charge e =eigenvalue of h× coupling. ◮ Solution a = ˜
ǫaRaΦk = ˜ εµdX µΦk, transverse polarization ˜ εµ(u)dX µ = ˜ ǫa
k+ (ka + eAa(u))du
ǫa = const.. Convention: YM background polarization vectors are tilded.
SLIDE 13
No particle creation or leakage
As u → −∞ take linear fields to become flat space-time momentum eigenstates, i.e., ea
i = δa i , and Aa = 0; ◮ ± frequency determined by sign of k+, doesnt change with
u so no particle creation.
◮ Inner products are u-independent on both backgrounds:
Φk|Φk′ = 2k+δ(k+ − l+)δd−2(ki − li) . Similarly for spin-1 a1|a2 = 2ǫ1 · ǫ2k+δ(k+ − l+)δd−2(ki − li) and spin-2 h1|h2 = 2(ǫ1 · ǫ2)2k+δ(k+ − l+)δd−2(ki − li) .
◮ Failure of global hyperbolicity does not lead to leakage.
[Failure in space of null geodesics: those parallel to ∂v so co-dimension too high.]
SLIDE 14 Three particle gravity amplitude
◮ Cubic part of action give 3 vertex
M3 = κ 4
1 ∂µh2ρσ∂νhρσ 3 −2hρν 1 ∂µh2ρσ∂νhµσ 3 )+perms ◮ Inserting our states yields
κ 2δd−1 3
kr du
i
exp s
F ijkrikrj 2kr0
- [(ε1 · ε2 (K1 − K2) · ε3+ )2 − ik1+k2+k3+σabCaCb]
where Ca := ε1 · ε2 ǫ3a k3+ +
◮ First term = (YM 3-pt amplitude)2 on gravity background. ◮ However: tail term σabCaCb seems to obstruct double copy.
SLIDE 15 Three-point YM amplitude
◮ Cubic part of action
- M a[µaν]Dµaνd4X gives 3 point vertex
A3 =
3
fr(u) 2kr0
ε1 · ˜ ε2 ˜ ε3 · (K1 − K2)+ ] C☎(ti) .
◮ Bracketed term is in flat quantities
ǫ1 · ˜ ǫ2 ˜ ǫa
3
k1+ k2+ k2a − k1a
k1+ k2+
◮ Second term gives background ‘tail’ dependence on Aa.
SLIDE 16 Double copy replacement principle
YM to GR uses replacement rules:
- 1. Flip charges er → −er so A3 = F + C → ˜
A = F − C and |A3|2 := A3 ˜ A3 = F 2 − C2 with F = F(kr, ˜ ǫ) and C = C(kr, ˜ ǫr, A).
ǫra) by (kiei
a, ǫa) ❀ F(kriei a, ǫr), C(kriei a, ǫr, A).
eresAaAb →
r = s, i(kr0 + ks0)σab r = s . Yields double copy of YM integrand for GR incorporating tails.
SLIDE 17 Four point amplitudes: YM
1810.05115 w/Adamo, Casali, Nekovar
◮ Construct scalar Feynman propagator as
GF(X, X ′) =
k2 + iǫ exp i(˜ φk(X) − ˜ φk(X ′)) where ˜ φk is now solution to massive Hamilton-Jacobi ˜ φk = k+v + (ka + eAa)xa + 1 2k+ u ds[k2 + (k + eA(s))2] with k2 = k+k− − kaka = 0.
◮ For spin-1 solve (✷eA + k2)aµ + 2ieF ν µaν = 0 with
GF
µν =
k2 + iǫPµν(u, u′, k+)ei(φk(X)−φk(X ′)) Pµν = 1 −δac α∆Aa 1 −α∆Ac
α2 2 ∆A2
, ∆A = A(u)−A(u′), α = ie k+ Gravity similar, but much more complicated.
SLIDE 18
BCJ at four points
YM diagrams are 3 exchanges in s, t and u channels and contact 4-vertex.
◮ BCJ form requires opening up 4-vertex into 3-exchange
diagrams using Cs → sCs/s etc..
◮ Then 3-channels, s, t and u in flat space give
A4 = NsCs s + NtCt t + NuCu u with s = (k1 + k2)2, t = (k2 + k3)2, u = (k1 + k3)2.
◮ Jacobi-identity is Cs − Ct + Cu = 0; for BCJ property need
Ns − Nt + Nu = 0 , but this is naively obstructed on a plane wave!
◮ With an additional mapping on scalar propagators, some
nontrivial colour kinematics survives.
SLIDE 19 Conclusions and further developments
◮ Have explicit GR and YM Feynman rules on plane waves. ◮ Double copy is local in momentum space so nonlocal on
space-time.
◮ Nontrivial double copy at 3pts, work in progress at 4pts,
first steps towards nonlinear double copy in space-time.
◮ These formulae verify plane wave background ambitwistor
string computation in arxiv:1708.09249.
◮ Ambitwistor strings manifest double copy. Full nonlinear
version will give optimal nonlinear formulation?
◮ On Cosmological backgrounds?
[Work in progress w/Tim Adamo and Atul Sharma].
SLIDE 20
Thank you! Happy 60th birthday Jurek!
