arXiv:2003.13501v2 [hep-th] 27 Jul 2020 nat-eSte pc 2,2] rgunsatrn on scattering gluon or 21]; functions [20, correlation space Sitter boundary anti-de 4-point in some or (e.g., wave in Compton 14]); plane QED tree-level [13, electromagnetic an at 4-points: in scattering results at trident are best strong backgrounds the in strong analogues) Indeed, its (or S-matrix what backgrounds. the and about backgrounds known trivial is on scattering for formulae back- back- [19]). via (cf., classical operationalised theory exact is perturbation an QFT ground as which treated around are ground settings, fields these In strong correlators [18]. the theory correspondence AdS/CFT field the conformal via the coupled [15]; strongly collisions and ion heavy of of examples computation vicinity QCD few the 14]; [13, a in lasers high-intensity processes Just in processes considered. 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H cteigo losadgaiosi hrlsrn field strong chiral in gravitons and gluons of scattering MHV n setai lwrl nain[6 17]; [16, inflation roll slow in -spectra 2 1 h ahmtclIsiue nvriyo xod X 6GG, OX2 Oxford, of University Institute, Mathematical The colo ahmtc,Uiest fEibrh H F,Un 3FD, EH9 Edinburgh, of University Mathematics, of School i Adamo, Tim n strong − xetdat expected 2 1 Dtd uy2,2020) 28, July (Dated: inlMason, Lionel back- , n pit rmsaetm etrainter;this theory; perturbation space-time from -points h rtalmlilct xrsin in are expressions formulae all-multiplicity in resulting first backgrounds The the gravitational dimensions. and space-time gauge af- four particularly of the a class in on scattering question simple MHV this tree-level strong answer for in firmative we scattering Here, for formulae backgrounds? all-multiplicity find to described longer no functions. rational is sin- by amplitudes the simply that of mean structure background the gularity in functional freedom instance, of For apply- strong degrees techniques. to on innovative Yet obstructions these apparent ing [24–26]). are (e.g., unitar- there worldsheet locality, invariance backgrounds, a of gauge as considerations and from (e.g., ity or theory 23]) alterna- [2, field from Feyn-theory derived of traditional are formulations the they tive with instead, do expansion: to man nothing have grounds theory. perturbation in orders written lowest be very difficult the often is beyond can them with backgrounds Feyn- computations performing While strong explicitly, on [22]. rules background man wave plane Yang-Mills a u ihsgicn oe features. background, novel trivial significant a with result in reminis- but formulae or structurally all-multiplicity are of trivially cent and vanish [27]) which approximations from those (besides ground esnegagbawt etegvnb covariantly a by [28–31]. form given symmetry to centre null vacuum constant commute with the generators algebra properties: whose Heisenberg to symmetry a symmetries solution 5 special a has with or it is motion gluons wave of helicity plane as equations positive A is of backgrounds gravitons. superposition such self-dual. coherent of are a strengths interpretation field physical whose The waves plane of cases waves 2 ihti nmn,i sntrlt s:i tpossible it is ask: to natural is it mind, in this With back- trivial on formulae all-multiplicity However, h togbcgonsw osdrare consider we backgrounds strong The nfu-iesos twl ecnein owr with work to convenient be will it four-dimensions, In cgonsadtercrepnigtwistor corresponding their and ackgrounds n tlSharma Atul and c hcsadlmtt h well-known the to limit and checks ncy luae ncrancia togfields. strong chiral certain in alculated SPs nfu-iesos hs r special are These four-dimensions. in (SDPWs) tntis ohfrua aejs one just have formulae Both tails. iton teigo losadgaiosi the in gravitons and gluons of attering I H HRLBACKGROUNDS CHIRAL THE II. ngueter n eea relativity, general and theory gauge in s h ag hoybackground-coupled theory gauge The h akrudbcmstrivial. becomes background the k-alrfrua u h gravi- the but formula, rke-Taylor ntdKingdom United tdKingdom ited 2 ∗ s any efda plane self-dual togback- strong 2

2-spinor notation, trading tensor indices for a pair of with the plane wave conditions, one finds the metric (in SL(2, C) Weyl spinor indices of opposite helicity (cf., Einstein-Rosen coordinates): [32]). The (complexified) Minkowski line element in light- ds2 =2 dx+ dx− − dz d˜z + f(x−)d˜z2 . (7) front coordinates is We have abused notation to denote the functional free- 2 + −
ds =2 dx dx − dz d˜z , (1) dom by f(x−); it should be clear from the context whether f is Cartan valued (gauge theory) or a pure func- where the transverse coordinate z ˜ is not
required to be tion (gravity). The SDPW metric (7) solves the vacuum the complex conjugate of z. These coordinates are neatly Einstein equations for any f, with Weyl curvature packaged into a 2 × 2 matrix: − Ψ˜ ˙ ˙ = −f¨(x )˜ια˙ ˜ι ˙ ˜ιγ˙ ˜ι ˙ , Ψαβγδ =0 . (8) x+ z˜ α˙ βγ˙ δ β δ xαα˙ = , (2) z x− The sandwich condition for a SDPW space-time is that   f¨ be compactly supported in x−; this ensures existence where (α, α˙ ) are SL(2, C) Weyl spinor indices of negative of a well-defined tree-level graviton S-matrix [31, 34]. and positive chirality, respectively. The line element (1) 2 αα˙ in 2-spinor notation is ds = dx dxαα˙ , with spinor in- dices raised and lowered using the two-dimensional Levi- III. SD PLANE WAVE KINEMATICS Civita symbols ǫαβ, ǫα˙ β˙ , etc. Let Fab be the field strength of a gauge potential. This Strong plane wave backgrounds admit momentum admits a spinor decomposition eigenstate representations for free field perturbations [35– 37] and the Feynman rules for gluons and gravitons ˜ Fααβ˙ β˙ = ǫαβ Fα˙ β˙ + ǫα˙ β˙ Fαβ , (3) can be explicitly constructed from these in any dimen- sion [22, 31]. ˜ where Fα˙ β˙ and Fαβ are the self-dual (SD) and anti-self- a. Yang-Mills A gluon on a Yang-Mills SDPW can dual (ASD) components of the field strength, respec- be reduced to a scalar field of charge e with respect to the tively. They are symmetric in their spinor indices and Cartan-valued background. This is given by eiφK where valued in the adjoint representation of the gauge group. φK also solves the Hamilton-Jacobi equation. Thus The self-duality condition is simply that Fαβ = 0; non- iφK iφK ˜ (∂a − i e Aa)e = i Ka e trivial solutions (i.e., Fα˙ β˙ 6= 0) in Lorentzian signature are only possible for complex gauge fields, which is why defines a null vector field Ka on the background. In 4- we allowz ˜ 6=z ¯. dimensions, the spinor-helicity formalism (cf., [38]) can Combining the self-duality and plane wave conditions be used to further refine the on-shell kinematics of gluon forces the gauge field to be valued in the Cartan of the and graviton perturbations in plane waves [39]. For gauge group [30, 31]. The gauge potential and field SDPW backgrounds, these on-shell kinematics are them- strength are selves chiral. The on-shell 4-momentum of a gluon mov- ing through a SDPW gauge field background admits the − ˜ ˙ − A = −f(x )d˜z , Fα˙ β˙ = f(x )˜ια˙ ˜ιβ˙ , Fαβ =0 , (4) 2-spinor decomposition: − − ˙ − − ef(x ) where f(x ) is Cartan-valued and f ≡ ∂−f. The covari- Kαα˙ (x )= λα Λ˜ α˙ (x ) , Λ˜ α˙ = λ˜α˙ + ˜ια˙ , (9) antly constant null symmetry associated with the back- hι λi ground is generated by the vector α αβ ˜ α˙ ˜ α˙ β˙ ˜ where habi = a bα = ǫ aβbα, [˜a b]=˜a bα˙ = ǫ a˜β˙ bα˙ . 1 The constant null vector k = λ λ˜ is the gluon’s mo- nαα˙ = ια ˜ια˙ , ια = =˜ια˙ . (5) αα˙ α α˙ 0 mentum before it enters the SDPW background. As the   gluon traverses the background, the momentum spinor It is also useful to define the constant spinors oα, o˜α˙ form- ˜ α α˙ α α˙ λα˙ is ‘dressed’: it picks up non-trivial dependence on ing spinor dyads with ι and ˜ι : ι oα =1=˜ι o˜α˙ . lightfront time x− through the background wave profile. It is easyto seethat (4) solves the Yang-Mills equations Chirality of the SDPW background ensures that only the for any Cartan-valued f, but we restrict our attention to positive chirality momentum spinor is dressed: the nega- ‘sandwich’ self-dual plane waves for which f˙ is compactly − tive chirality spinor λα is un-changed by the background. supported in x ; such profiles admit a well-defined gluon This chirality is also present at the level of posi- S-matrix [31, 33]. tive/negative helicity gluon polarizations, which can be In the gravitational context, self-duality is a condition written in lightcone-transverse gauge as: on the Weyl curvature tensor, which admits a spinor de- ι Λ˜ λ ˜ι composition: E(+) = α α˙ , E(−) = α α˙ . (10) αα˙ hι λi αα˙ ˜ ˜ [˜ι λ] Cααβ˙ βγ˙ γδ˙ δ˙ = ǫαβ ǫγδ Ψα˙ β˙γ˙ δ˙ + ǫα˙ β˙ ǫγ˙ δ˙ Ψαβγδ , (6) These obey n ·E(±) =0= K ·E(±), and only the pos- ˜ where Ψα˙ β˙γ˙ δ˙ and Ψαβγδ are the SD and ASD parts of the itive helicity polarization is dressed by the background, − Weyl tensor. Self-duality gives Ψαβγδ = 0 and, combined through Λ˜ α˙ (x ). 3

b. Gravity For a graviton propagating on a SDPW have overall charge conservation i ei = 0 with respect space-time, the kinematics and polarizations are also chi- to the Cartan-valued background. ral. The on-shell graviton 4-momentum is Naively, there should be one residualP lightfront inte- gral for each interaction vertex contributing to a generic ˜ − − − ho λi [˜ι λ] f(x ) Feynman diagram from the background field Lagrangian. Kαα˙ (x )= λα Λ˜ α˙ (x ) , Λ˜ α˙ = λ˜α˙ − ˜ια˙ , hι λi Since both Yang-Mills and gravity behave like cubic the- (11) ories (cf., [41]), this leads to an expected n − 2 residual where kαα˙ = λαλ˜α˙ is the graviton momentum before lightfront integrals for n-point tree amplitudes. However, entering the background. Positive and negative helicity for SDPW MHV amplitudes, a remarkable simplification graviton polarizations (in transverse, traceless lightcone expresses them in terms of a single x−-integral that re- gauge) are places the last component of momentum conservation.

(+) ια ιβ ˜ ˜ ˙ E ˙ = 2 Λα˙ Λβ˙ − i f ˜ια˙ ˜ιβ˙ , ααβ˙ β hι λi A. Gluon amplitude   (12) λα λβ ˜ια˙ ˜ι ˙ E(−) = β . ααβ˙ β˙ [˜ι λ˜]2 We take our particles’ colours to have well-defined charges ei with respect to the Cartan background. The Once again, only the positive helicity configuration is partial amplitude with colour-ordering {1, 2,...,n} for dressed by the background but there is an additional tail MHV gluon scattering on a SDPW gauge background is ˙ term with coefficient f in (12). then given by:

4 +∞ hi ji − IV. MHV AMPLITUDES dx− ei Fn(x ) , (15) h12i h23i···hn − 1 ni hn 1i Z−∞ It is straightforward to prove that all tree-level ampli- where the overall momentum (and charge) conserving tudes in SDPWs with one or zero negative helicity gluons delta functions (13) have been omitted, gluons i and j or gravitons vanish; this is a consequence of the integra- are negative helicity and the remainder are positive he- bility of the self-dual sector in gauge theory and gravity licity. The exponential phase appearing in the residual [40]. Intuitively, this is not surprising: such amplitudes lightfront integration is a n-point generalization of an ob- vanish on a trivial background, and one can imagine the ject appearing at lower points in strong field QED, known SDPW constructed from a coherent superposition of fur- as the Volkov exponent [35, 42]. This is defined by a com- ther positive helicity gluons/gravitons. Hence, the first posite momentum K built from any set of n − 1 distinct non-trivial tree amplitudes in SDPW backgrounds should dressed gluon momenta; for instance, appear for the MHV helicity configuration: two negative helicity gluons/gravitons and an arbitrary number of pos- n−1 Kαα˙ − αα˙ − itive helicity gluons/gravitons. (x ) := Ki (x ) . (16) i=1 Some basic features of tree-amplitudes in strong plane X waves can be deduced from general properties of pertur- The Volkov exponent is then given by bation theory for these backgrounds. One is the absence − − of total 4-momentum conservation: the free function (or x K2 n x − ds (s) Cartan-valued function) defining the background breaks Fn(x ) := = dsKi−(s) , (17) − 2 hι|K(s)|˜ι] Poincar´einvariance in the x -direction. Momentum con- i=1 Z X Z servation in the x−-direction is replaced by lightfront in- the equality following by 3-momentum conservation (14) tegrals that cannot be performed explicitly due to the 2 free function in the background. In the x+, z andz ˜- and the on-shell condition Kn = 0. directions, momentum is still conserved, so there will be The ratio of angle brackets appearing in (15) matches a factor of the Parke-Taylor formula on a trivial background [1], since undotted momentum spinors are not dressed in a n 3 SDPW background. However, it is surprising that there δ+,⊥ ki , (13) is only a single residual lightfront integral, regardless of i=1 ! X the number of positive helicity gluons in the amplitude. in all amplitudes. These surviving components of mo- In a trivial background, a non-local field redefinition en- mentum conservation can be expressed as: ables the Yang-Mills Lagrangian to be recast such that all MHV amplitudes arise from a single interaction ver- n n tex [43–45]. Remarkably, this argument also holds for K (x−)= K (x−) ι ˜ι , (14) iαα˙ i− α α˙ the background Lagrangian on a SDPW, explaining the i=1 i=1 X X simplicity of the residual lightfront integral [46]. where K− is the component of the dressed momentum We have checked this formula against traditional Feyn- in the x−-direction. In the gauge theory setting we also man diagram computations in background perturbation 4 theory at 3- and 4-points, and it is also easy to show that t × t block T are diagonal: the formula has the correct trivial background limit. In [˜ι j] h1 ιi h2 ιi this case, the Volkov exponent and residual lightfront D = diag − , − hι ji h1 ji h2 ji integral reduce to momentum conservation in the x -  j∈∪mam direction. [˜ι j] h1 ji h2 ji Beyond these somewhat piecemeal checks, (15) can ac- T = diag − . tually be derived from the space-time generating func-  hι ji h1 ιi h2 ιi  j∈ /am tional for MHV amplitudes. This generating functional X m=1,...,t   can be lifted to twistor space, where integrability of the For the off-diagonal blocks, h is (n − |a|− 2) × |a| and c SD sector is manifest, enabling an all-multiplicity expan- is t × (n − |a|− 2): sion [40]. This derivation extends to SDPW backgrounds and will be presented in [46]. [Λ˜ Λ˜ ] [˜ιi] h = i j (x−) , c = , ij hi ji mi hιii B. Graviton amplitude where i ∈ a¯ and j ∈ ∪mam. Finally, τ is |a|× t and t is t × |a|: On a trivial background, the tree-level graviton MHV amplitude depends on momentum spinors of both chi- [˜ι j] a [˜ι j] a hι ji if j ∈ m hι ji if j∈ / m ralities [47], and has a compact structure as a once- τjm = , tmj = , ( 0 otherwise ( 0 otherwise reduced (n − 2) × (n − 2) determinant discovered by

Hodges [48]. The graviton MHV amplitude in a SDPW for j ∈ ∪mam and m = 1,...,t. These blocks emerge space-time requires more than a simple dressing – as in through an application of the matrix tree theorem to a (15) for Yang-Mills – and is significantly more compli- space-time generating functional of graviton MHV am- cated. Nevertheless, this added complication has a sim- plitudes, which is also the reason for the vanishing of two ple physical interpretation and the final result is remark- of the off-diagonal blocks in (19). ably simple considering the apparent complexity of per- With these ingredients, the n-point tree-level graviton turbative gravity arising from the Einstein-Hilbert La- MHV amplitude on a SDPW background is: grangian. We find a sum over contributions from once- n−3 reduced (n + t − 2) × (n + t − 2) determinants where t is ⌊ ⌋ +∞ 2 5 the number of interactions with tails in the gravitational − i a a h12i [˜ιi] dx Hi[ 1 ··· t] 2 waves arising from previous interactions. a hi 1i hi 2i [˜ι 2] t=0 −∞Z i∈¯ The formula involves a summation over partitions of X a1X,...,at the set of positive helicity external gravitons. For the t a a MHV configuration, take gravitons 1 and 2 to be negative × it−| | ei Fn f (| m|−1) . (20) helicity and gravitons {3,...,n} to be positive helicity. m=1 a Y Let m form=1,...,t be t disjoint subsets of {3,...,n} i a a a a The notation |Hi| denotes the determinant of H with that obey | m| ≥ 2 and | 1| + ··· + | t| := | | ≤ n − 3, (d) d − n−3 row and column i removed; f := ∂−f for f(x ) the so that t is a non-negative integer with t ≤ ⌊ 2 ⌋. We denote the complement by: function defining the SDPW background; and Fn is the gravitational Volkov exponent t − x ab a¯ := {3,...,n}\ am . (18) g (s) K (s) K (s) F (x−) := ds a b , (21) m=1 n 2 hι|K(s)|˜ι] [ Z The MHV amplitude involves a sum over t, and further where g is the SDPW metric (7) and K is defined by over all possible partitions a ,..., a for each t. ab a 1 t any set of n − 1 graviton momenta, as in (16). Given t and a ,..., a , define an (n+t−2)×(n+t−2) 1 t Again, there is only a single residual lightfront inte- matrix with block decomposition: gral at arbitrary multiplicity, but unlike the gluon am- H h 0 plitude, the integrand includes substantially more than (d) H[a1,..., at]= 0 D τ . (19) the Volkov exponent: H and f introduce additional  c t T  dependence on x−. Terms in (20) with t ≥ 1 are tail   contributions to the amplitude, arising due to the fail- The (n − |a|− 2) × (n − |a|− 2) block H is given by ure of Huygens’ principle for perturbative gravity on a plane wave space-time [49]. These tail terms represent ˜ ˜ [Λi Λj ] − − h1 ji h2 ji contributions from background gravitons to the scatter- Hij = (x ) , Hii = − Hij (x ) , hi ji h1 ii h2 ii ing, through insertions of (derivatives of) the background j6=i X wave profile f. with indices i, j ∈ a¯; this is a dressed analogue of the The formula (20) has the correct little group weights matrices appearing in [48]. The |a| × |a| block D and in each external graviton, is permutation symmetric in 5 the positive and negative helicity gravitons (the latter background means that standard locality and unitarity follows from the underlying MHV generating functional, arguments cannot be used to constrain the structure of but we have checked it explicitly through 6-points), and tree amplitudes. Recently, it was shown in QED that has been checked against Feynman diagram calculations gauge invariance could provide analytic constraints on in background perturbation theory at 3- and 4-points. amplitudes in strong plane waves [53]; we hope that fur- In the flat background limit only the t = 0 term survives ther developments in this vein could enable robust in- i H12i H and |Hi| → | 12i|, where is now the Hodges matrix on dependent checks (or alternative derivations) of our for- a flat background [48]. The residual lightfront integral mulae. The fact that both formulae contain only a single of the Volkov exponent completes (13) to 4-momentum lightfront integral at arbitrary multiplicity is remarkable, conservation, and the remaining kinematic contribution but explained by their origin in a twistorial expansion of to the amplitude is: local space-time generating functionals [40, 44–46, 52]. There are several directions for future research sug- n n h12i5 [˜ιi] |H123| h12i6 hi 1i [˜ιi] gested by this work. The first is to extend away from H12i = 123 the MHV sector to other helicity configurations. This 12i hi 1i hi 2i2 [˜ι 2] h13i2 h23i2 h12i [˜ι 2] i=3 i=3 topic is under active investigation, and we will present X X 123 6 n 123 6 conjectural NkMHV expressions at arbitrary multiplicity |H | h12i h1|ki|˜ι] |H | h12i = − 123 = 123 , (22) in [46]. Another is to extend to Einstein-Yang-Mills am- h13i2 h23i2 h1|k |˜ι] h13i2 h23i2 i=3 2 X plitudes, extending the formulae of [54, 55] to incorporate self-dual backgrounds of both types. More difficult is to with the various equalities following from basic proper- relax the self-duality assumption on the background, con- H ties of and 4-momentum conservation. This leaves pre- sidering general sandwich plane wave backgrounds. This cisely Hodges’ formula for the gravitational MHV ampli- is challenging, since the classical integrability and twistor tude on a flat background [48], in a representation agree- theory underlying the derivation of the formulae here are ing with previous expressions [50, 51]. lost, but it may be possible to use non-chiral ambitwistor The formula (20) can be derived directly from a space- string theory [23] – which encodes the full non-linearity of time generating functional for all MHV amplitudes in a generic strong background [56, 57] – to make progress. general relativity [40], expanded using twistor theory to Colour-kinematics duality and double copy between manifest the integrability of the SD sector as in [52]. gauge theory and gravity on strong backgrounds can be Details of this (which includes a novel derivation of the explored with these formulae. It should now be possible Hodges formula in a flat background) will appear in [46]. to test the proposals of [22, 31] at arbitrary multiplicity, as well as investigate the kinematic algebra associated to the SD sector [58] in the presence of SD background V. DISCUSSION fields. Finally, one might imagine that (15) and (20) are purely ‘academic,’ as the SDPW background is complex; The formulae (15) and (20) constitute the first (non- plane wave backgrounds relevant for phenomenological trivial) all-multiplicity expressions for scattering ampli- strong field QED, QCD or gravity are real. However, it tudes on any strong background. Their derivation brings has been shown that certain back-reaction effects (e.g., together a rich set of geometric techniques which will be beam depletion [59]) in strong field QFT can be com- described in [46], but the results presented here emulate puted through perturbation theory around fixed complex momentum space expressions built from familiar ingredi- background fields [60]. Further, the intrinsic chirality of ents. The distinction from tree-level MHV scattering on introducing spin via the Newman-Janis trick [61] indi- a trivial background lies in the breaking of 4-momentum cates potential applications for these methods in gravi- conservation by the background (captured by the gauge tational wave physics. and gravitational Volkov exponents) and (in the gravi- tational case) explicit tail terms from scattering off the Acknowledgments: We thank Lance Dixon, Maciej SDPW background. Dunajski, Anton Ilderton and Alexander MacLeod for Several features of our formulae are surprising. Firstly, interesting discussions and comments. TA is supported it is not apparent how they arise from tools familiar to by a Royal Society University Research Fellowship. AS the mainstream scattering amplitudes community. The is supported by a Mathematical Institute Studentship, presence of the free function f(x−) describing the SDPW Oxford.

[1] S. J. Parke and T. R. Taylor, An Amplitude for n Gluon 189–258, [hep-th/0312171]. Scattering, Phys. Rev. Lett. 56 (1986) 2459. [3] R. Roiban, M. Spradlin, and A. Volovich, On the tree [2] E. Witten, Perturbative gauge theory as a string theory level S matrix of Yang-Mills theory, Phys. Rev. D70 in twistor space, Commun. Math. Phys. 252 (2004) (2004) 026009, [hep-th/0403190]. 6

[4] F. Cachazo and D. Skinner, Gravity from Rational scattering equations, JHEP 07 (2014) 048, Curves in Twistor Space, Phys. Rev. Lett. 110 (2013), [arXiv:1311.2564]. no. 16 161301, [arXiv:1207.0741]. [24] R. Britto, F. Cachazo, B. Feng, and E. Witten, Direct [5] F. Cachazo, S. He, and E. Y. Yuan, Scattering of proof of tree-level recursion relation in Yang-Mills Massless Particles in Arbitrary Dimensions, Phys. Rev. theory, Phys. Rev. Lett. 94 (2005) 181602, Lett. 113 (2014), no. 17 171601, [arXiv:1307.2199]. [hep-th/0501052]. [6] F. Cachazo, S. He, and E. Y. Yuan, Scattering [25] N. Arkani-Hamed, L. Rodina, and J. Trnka, Locality Equations and Matrices: From Einstein To Yang-Mills, and Unitarity of Scattering Amplitudes from DBI and NLSM, JHEP 07 (2015) 149, Singularities and Gauge Invariance, Phys. Rev. Lett. [arXiv:1412.3479]. 120 (2018), no. 23 231602, [arXiv:1612.02797]. [7] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, [26] L. Rodina, Scattering Amplitudes from Soft Theorems S. Caron-Huot, and J. Trnka, The All-Loop Integrand and Infrared Behavior, Phys. Rev. Lett. 122 (2019), For Scattering Amplitudes in Planar N=4 SYM, JHEP no. 7 071601, [arXiv:1807.09738]. 01 (2011) 041, [arXiv:1008.2958]. [27] A. Di Piazza, K. Z. Hatsagortsyan, and C. H. Keitel, [8] T. Adamo, E. Casali, and D. Skinner, Ambitwistor Quantum radiation reaction effects in multiphoton strings and the scattering equations at one loop, JHEP Compton scattering, Phys. Rev. Lett. 105 (2010) 04 (2014) 104, [arXiv:1312.3828]. 220403, [arXiv:1007.4914]. [9] Y. Geyer, L. Mason, R. Monteiro, and P. Tourkine, [28] O. R. Baldwin and G. B. Jeffery, The relativity theory One-loop amplitudes on the Riemann sphere, JHEP 03 of plane waves, Proc. Roy. Soc. Lond. A111 (1926) 95. (2016) 114, [arXiv:1511.06315]. [29] J. Ehlers and W. Kundt, Exact solutions of the [10] J. L. Bourjaily, E. Herrmann, and J. Trnka, Prescriptive gravitational field equations, in Gravitation, An Unitarity, JHEP 06 (2017) 059, [arXiv:1704.05460]. Introduction to Current Research (L. Witten, ed.), [11] Y. Geyer and R. Monteiro, Two-Loop Scattering p. 49. Wiley: New York, 1962. Amplitudes from Ambitwistor Strings: from Genus Two [30] A. Trautman, A class of null solutions to the to the Nodal Riemann Sphere, JHEP 11 (2018) 008, Yang-Mills equations, J. Phys. A13 (1980) L1–L4. [arXiv:1805.05344]. [31] T. Adamo, E. Casali, L. Mason, and S. Nekovar, [12] J. L. Bourjaily, E. Herrmann, C. Langer, A. J. McLeod, Scattering on plane waves and the double copy, Class. and J. Trnka, All-Multiplicity Non-Planar MHV Quant. Grav. 35 (2018), no. 1 015004, Amplitudes in sYM at Two Loops, Phys. Rev. Lett. 124 [arXiv:1706.08925]. (2020), no. 11 111603, [arXiv:1911.09106]. [32] R. Penrose and W. Rindler, Spinors and Space-Time, [13] A. Di Piazza, C. Muller, K. Z. Hatsagortsyan, and vol. 1 of Cambridge Monographs on Mathematical C. H. Keitel, Extremely high-intensity laser interactions Physics. Cambridge Univ. Press, Cambridge, UK, 1984. with fundamental quantum systems, Rev. Mod. Phys. 84 [33] J. S. Schwinger, On gauge invariance and vacuum (2012) 1177, [arXiv:1111.3886]. polarization, Phys. Rev. 82 (1951) 664–679. [,116(1951)]. [14] B. King and T. Heinzl, Measuring Vacuum Polarisation [34] G. W. Gibbons, Quantized Fields Propagating in Plane with High Power Lasers, High Power Las. Sci. Eng. 4 Wave Space-Times, Commun. Math. Phys. 45 (1975) (2016) e5, [arXiv:1510.08456]. 191–202. [15] F. Gelis, E. Iancu, J. Jalilian-Marian, and [35] D. M. Wolkow, Uber¨ eine Klasse von Losungen der R. Venugopalan, The Color Glass Condensate, Ann. Diracschen Gleichung, Z. Phys. 94 (1935) 250–260. Rev. Nucl. Part. Sci. 60 (2010) 463–489, [36] F. G. Friedlander, The Wave Equation on a Curved [arXiv:1002.0333]. Space-Time. Cambridge University Press, 1975. [16] V. F. Mukhanov, H. A. Feldman, and R. H. [37] R. S. Ward, Progressing waves in flat space-time and in Brandenberger, Theory of cosmological perturbations, plane wave space-times, Class. Quant. Grav. 4 (1987) Phys. Rept. 215 (1992) 203–333. 775–778. [17] J. M. Maldacena, Non-Gaussian features of primordial [38] L. J. Dixon, A brief introduction to modern amplitude fluctuations in single field inflationary models, JHEP 05 methods, in Proceedings, 2012 European School of (2003) 013, [astro-ph/0210603]. High-Energy Physics (ESHEP 2012): La Pommeraye, [18] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, Anjou, France, June 06-19, 2012, pp. 31–67, 2014. and Y. Oz, Large N field theories, string theory and arXiv:1310.5353. gravity, Phys. Rept. 323 (2000) 183–386, [39] T. Adamo and A. Ilderton, Gluon helicity flip in a [hep-th/9905111]. plane wave background, JHEP 06 (2019) 015, [19] L. F. Abbott, Introduction to the Background Field [arXiv:1903.01491]. Method, Acta Phys. Polon. B13 (1982) 33. [40] L. J. Mason and D. Skinner, Gravity, Twistors and the [20] E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis, MHV Formalism, Commun. Math. Phys. 294 (2010) and L. Rastelli, Graviton exchange and complete four 827–862, [arXiv:0808.3907]. point functions in the AdS / CFT correspondence, Nucl. [41] Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson, Phys. B562 (1999) 353–394, [hep-th/9903196]. and R. Roiban, The Duality Between Color and [21] S. Raju, Four Point Functions of the Stress Tensor and Kinematics and its Applications, arXiv:1909.01358. Conserved Currents in AdS4/CFT3, Phys. Rev. D85 [42] D. Seipt, Volkov States and Non-linear Compton (2012) 126008, [arXiv:1201.6452]. Scattering in Short and Intense Laser Pulses, in [22] T. Adamo, E. Casali, L. Mason, and S. Nekovar, Plane Proceedings, Quantum Field Theory at the Limits: from wave backgrounds and colour-kinematics duality, JHEP Strong Fields to Heavy Quarks (HQ 2016): Dubna, 02 (2019) 198, [arXiv:1810.05115]. Russia, July 18-30, 2016, pp. 24–43, 2017. [23] L. Mason and D. Skinner, Ambitwistor strings and the arXiv:1701.03692. 7

[43] F. Cachazo, P. Svrcek, and E. Witten, MHV vertices of amplitudes on backgrounds from gauge invariance and tree amplitudes in gauge theory, JHEP 09 (2004) and the infra-red, arXiv:2001.10553. 006, [hep-th/0403047]. [54] T. Adamo, E. Casali, K. A. Roehrig, and D. Skinner, [44] L. J. Mason, Twistor actions for non-self-dual fields: A On tree amplitudes of supersymmetric Derivation of twistor-string theory, JHEP 10 (2005) Einstein-Yang-Mills theory, JHEP 12 (2015) 177, 009, [hep-th/0507269]. [arXiv:1507.02207]. [45] P. Mansfield, The Lagrangian origin of MHV rules, [55] K. A. Roehrig, Chiral splitting and N = 4 JHEP 03 (2006) 037, [hep-th/0511264]. Einstein-Yang-Mills tree amplitudes in 4d, JHEP 08 [46] T. Adamo, L. Mason, and A. Sharma, To appear, . (2017) 033, [arXiv:1705.09315]. [47] F. A. Berends, W. T. Giele, and H. Kuijf, On relations [56] T. Adamo, E. Casali, and D. Skinner, A Worldsheet between multi-gluon and multi-graviton scattering, Phys. Theory for Supergravity, JHEP 02 (2015) 116, Lett. B211 (1988) 91–94. [arXiv:1409.5656]. [48] A. Hodges, A simple formula for gravitational MHV [57] T. Adamo, E. Casali, and S. Nekovar, Yang-Mills theory amplitudes, arXiv:1204.1930. from the worldsheet, Phys. Rev. D98 (2018), no. 8 [49] A. I. Harte, Tails of plane wave spacetimes: Wave-wave 086022, [arXiv:1807.09171]. scattering in general relativity, Phys. Rev. D88 (2013), [58] R. Monteiro and D. O’Connell, The Kinematic Algebra no. 8 084059, [arXiv:1309.5020]. From the Self-Dual Sector, JHEP 07 (2011) 007, [50] Z. Bern, L. J. Dixon, M. Perelstein, and J. S. Rozowsky, [arXiv:1105.2565]. Multileg one loop gravity amplitudes from gauge theory, [59] D. Seipt, T. Heinzl, M. Marklund, and S. S. Bulanov, Nucl. Phys. B546 (1999) 423–479, [hep-th/9811140]. Depletion of Intense Fields, Phys. Rev. Lett. 118 [51] D. Nguyen, M. Spradlin, A. Volovich, and C. Wen, The (2017), no. 15 154803, [arXiv:1605.00633]. Tree Formula for MHV Graviton Amplitudes, JHEP 07 [60] A. Ilderton and D. Seipt, Backreaction on background (2010) 045, [arXiv:0907.2276]. fields: A coherent state approach, Phys. Rev. D97 [52] T. Adamo and L. Mason, Conformal and Einstein (2018), no. 1 016007, [arXiv:1709.10085]. gravity from twistor actions, Class. Quant. Grav. 31 [61] E. T. Newman and A. I. Janis, Note on the Kerr (2014), no. 4 045014, [arXiv:1307.5043]. spinning particle metric, J. Math. Phys. 6 (1965) [53] A. Ilderton and A. J. MacLeod, The analytic structure 915–917.