The Radiative Double Copy for Einstein-Yang-Mills Theory

Recently, a double-copy formalism was used to calculate gravitational radiation from classical Yang-Mills radiation solutions. This work shows that Yang-Mills theory coupled to a biadjoint scalar field admits a radiative double copy that agrees with solutions in Einstein-Yang-Mills theory at the lowest finite order. Within this context, the trace-reversed metric $\bar{h}^{\mu\nu}$ is a natural double copy of the gauge boson $A^{\mu a}$. This work provides additional evidence that solutions in gauge and gravity theories are related, even though their respective Lagrangians and nonlinear equations of motion appear to be different.


I. INTRODUCTION
The Lagrangians and equations of motion for gauge and gravity theories appear to be rather different. Nevertheless, there are intriguing double-copy connections between their solutions. This includes the Kawai-Lewellen-Tye (KLT) tree-level relations between gauge and gravity amplitudes in string theory [1] and the Bern-Carrasco-Johansson (BCJ) doublecopy relations between diagrams in quantum field theory [2]. The BCJ double-copy relations are based on color-kinematics duality, which gives particularly simple constructions of gravity amplitudes starting from gauge-theory amplitudes.
With the recent experimental detection of gravitational waves by LIGO [30], precision calculational tools for gravitational wave emission are essential. Exploiting color-kinematics duality to relate radiation solutions between Yang-Mills and general relativity is attractive because general relativity is difficult to solve and the double copy has been shown to work for a wide variety of gravity theories [31][32][33] The connection between radiation solutions of gauge theory and gravity has been described recently [34][35][36][37][38][39][40]. The first example of using the radiative double copy to find nonlinear terms in general relativity utilized perturbative Yang-Mills solutions [41]. Similarly a biadjoint scalar field can be used to find Yang-Mills radiation [42].
This work builds off the radiative double copy for general relativity found by Goldberger et al. [41] to find gravitational radiation in Einstein-Yang-Mills theory. By comparing the differential equations of the sources and fields in gauge theory and gravity, radiative diagrams are used to represent specific algebraic terms. Solutions in gravity can be found from Yang-Mills theory, and the diagrams with three-point vertices can be computed by stitching lowerorder solutions together. At leading order, the trace-reversed metric [43],h µν , is a natural double copy of the Yang-Mills potential A µa [44]. Motivation for a perturbative double copy can be seen at the Lagrangian level, as the linearized gravity Lagrangian is quite similar to the QED Lagrangian, a linearized version of the Yang-Mills Lagrangian. Similarly, these two theories both have an analogous linearized wave equation. Remarkably, radiation solutions of nonlinear gauge and gravity theories are related, at least when iterated perturbatively. A double copy of Yang-Mills-adjoint-scalar theory is also briefly mentioned, which can recover radiation solutions in Einstein-Maxwell theory.
While this paper focuses on classical solutions that could be calculated with more traditional methods [45][46][47][48][49][50][51][52][53][54], the hope is that the radiative double copy could help with difficult calculations that may be more cumbersome to do in general relativity alone. As more experimental data for gravitational radiation is collected, new methods for calculating complicated radiation processes are encouraged. Section (II) calculates radiation in Yang-Mills-biadjoint-scalar theory. Section (III) calculates radiation in Einstein-Yang-Mills theory and the double copy is confirmed by direct calculation. Section (IV) states our concluding remarks. Appendix (A) calculates details of the gravitational contribution to the energy momentum pseudotensor and Appendix (B) gives radiative Feynman rules for simple diagrams with three-point vertices.

A. Equations of Motion and Initial Conditions
In this section, the non-Abelian radiation field for Yang-Mills-biadjoint-scalar field theory is computed to first order in the weak-field approximation. To start, the Lagrangian associated with the Yang-Mills-biadjoint-scalar theory is where f abc and fãbc refer to structure constants of different groups, the biadjoint scalar Φã a has an index associated with each gauge group, and y = −igg/2 relates the conventions of Ref. [27] with the conventions of Refs. [26,42]. In principle, there could be an O(Φ 4 ) term in the Lagrangian, but the coupling constant would have different dimensions than y and is not needed for the double copy. The non-Abelian field strength is given by (2.2) and the the mostly minus metric will be used, such that η µν = diag(1, −1, −1, −1). The covariant derivative is given by 3) The equations of motion for the Yang-Mills field is where J µa (x) is a non-Abelian vector current acting as a source for the Yang-Mills field and is covariantly conserved, such that D µ J µa = 0. The equation of motion for the biadjoint scalar field is For N colliding charged particles, the worldline of particle α is x µ α (τ ) = b µ α +v µ α τ for τ → −∞. These initial conditions specify an impact parameter b µ αβ = b µ α − b µ β and a constant initial velocity v µ α which satisfies v 2 α = 1. For arbitrary times near and after the collision, where z µ α (τ ) is the deflection due to the Yang-Mills and biadjoint scalar fields. The vector source for N colliding charged particles is is the velocity, and c a α (τ ) is the associated adjoint color charge [55]. The biadjoint source Jã a (x) for N particles is where it is assumed that the color charges cã α (τ ) and c a α (τ ) are in two different gauge groups. The Lorenz gauge is taken by setting ∂ µ A µa = 0. In order to simplify these equations, the explicit dependence on the covariant derivatives is removed and gauge dependent sourceŝ J µa andJã a are defined such that where ≡ ∂ ν ∂ ν . With these definitions, the pseudovector source iŝ where the pseudovector is locally conserved, ∂ µĴ µa = 0. The pseudoscalar source is given bŷ Similar to the worldline x µ α (τ ), the color charges are dynamical and are given initial conditions c a α (τ ) = c a α and cã α (τ ) = cã α for τ → −∞. For times near and after the collision, c a α (τ ) = c a α +c a α (τ ), cã α (τ ) = cã α +cã α (τ ), (2.12) wherec a α (τ ) andcã α (τ ) are the corrections due to the Yang-Mills and biadjoint scalar fields. The time evolution of the momentum is dp µ α (τ ) dτ = gc a α (τ )F µνa (x α (τ ))v αν (τ ) − y∂ µ Φã a (x α (τ ))c a α (τ )cã α (τ ), (2.13) and the time evolution of the charges is These summarize all of the equations needed to iteratively solve for radiation in Yang-Millsbiadjoint-scalar theory within a weak-field approximation.

B. Solutions of the Radiation Fields
For weak fields, the lowest-order sources can be found from the initial conditions. The pseudocurrents in momentum space arê which can be utilized to find the Yang-Mills and biadjoint scalar fields to lowest order from Eq. (2.9), giving The lowest-order fields can be used to find the deflections of the sources, given by Plugging in the derivatives of the lowest-order fields gives Note that writing the color charge contraction as c α · c β would be ambiguous with our notation, as c a α c a β and cã α cã β are distinct. The first correction of the color charges to second order in g is given by Once again, plugging in the lowest-order fields gives (2.20) These deflections can be utilized to find the sources to next order, which give the lowest-order radiation fields.
Taking the Fourier transform of Eq. (2.7) and integrating over the delta function gives Expanding these results perturbatively in g and y gives where explicit τ dependence has been suppressed and only terms to second order in g or y are kept. Integrating Eqs. (2.18) and (2.20) allows for the second-order current to be found, which has Yang-Mills and biadjoint scalar contributions given by where an extra integral over l α was added with a momentum conserving delta function such that k = l α + l β and The nonlinear field contributions to the pseudovector are represented by j µa , which gives the following second order contributions While J µa and j µa were computed algebraically, they also can be represented diagrammatically. Fig. (1) depicts the diagrams associated withĴ µa to second order in g and y. The six diagrams are defined as where diagrams (1a), (1b), (1d), and (1e) give J µa (k) and diagrams (1c) and (1f) give j µa (k), both to second order in g and y. The source J µa (k) was split into two types of diagrams, as (1a) represents radiation that was emitted after the particle was deflected, while (1b) represents radiation that was emitted before the particle was deflected. As such, it is anticipated that (1b) and (1e) should be proportional to the undeflected quantities v µ α c a α , while (1a) and (1d) are in terms of corrections such as dz µ α dτ c a α and v µ αc a α . Diagrams (1c) and (1f) are computed in Appendix (B) from the three-point vertex with three vectors and the three-point vertex with two scalars and one vector, respectively. The six diagrams sum to giveĴ µa and satisfy the Ward identity k µĴ µa (k) = 0.
which is the pure Yang-Mills contribution found by Ref. [41]. Summing up the three diagrams The radiative field must be gauge invariant and the above expression satisfies the Ward identity k µĴ µa (k)| O(gy 2 ) = 0, as the identity must be satisfied order by order. Adding the above contributions to Eq. (2.27) gives the total source,Ĵ µa . To find the radiation field A µa rad from the sourceĴ µa [41,56], where k µ = ω(1, x/r).

A. Equations of Motion and Initial Conditions
The action for the Einstein-Yang-Mills-dilaton theory in consideration is where φ is the dilaton field and dτ = g µν dx µ dx ν . By varying the action above, the energymomentum pseudotensor contributions from the Yang-Mills field and the dilaton are given by According to Dirac, |g|T µν is the density and flux of energy and momentum for matter [57] such that in the presence of gravity, N partices contribute A weak-field approximation is taken by introducing h µν as where h ≡ h ρ ρ and the radiation can be calculated perturbatively in powers of κ. Textbook presentations of gravitational waves often focus on linearized gravity [43], which introduces the trace-reversed metric,h and find that h µν = − κ 2 T µν . If an effective energy-momentum pseudotensorT µν was found to contain contributions from matter, nonlinear gravitational field contributions, and the other fields, then the following equation of motion can be solved iteratively within the context of the weak-field approximation Due to the harmonic gauge condition, the pseudotensor satisfies ∂ µT µν = 0. The field contributions to the pseudotensor t µν ≡T µν −T µν will be found by expanding Eq. (3.2). The pseudotensor slightly differs from the common pseudotensor used by Landau and Lifshitz [43,47,58] and is closer to ones used previously by Einstein and Dirac, givinĝ wheret µν is conveniently defined to absorb (1 − |g|)T µν . In this section, the algebraic method of perturbing Einstein's field equations and iteratively solving for the radiation field is presented, leaving some technical details of the calculation oft µν to Appendix (A). Since the three-point graviton vertex is derived from the Lagrangian of the full theory, diagrams can encode how to find higher order field contributions from linearized field solutions. In Appendix (B), radiative Feynman rules are provided for the diagrams ontributing to t µν .
The Christoffel symbol Γ ρ µν and the Ricci tensor R µν are given by After expanding the metric perturbatively in κ and applying the gauge condition This gives the Ricci scalar R, µν , and all higher order terms in Eq. (3.2) are subtracted to the other side of the equation to be absorbed into the definition of t µν . Splitting these terms between t µν | ∆h , t µν | ∆φ , and t µν | ∆A gives where it is important to raise the indices on R µν − 1 2 g µν R with g µν to get all of the necessary terms.
Similar to the previous section, the position of the particle is given by x µ α (τ ), which has deflections z µ α (τ ) which must be calculated from the field solutions. The matter is also assumed to have a color charge cã α (τ ), but their corrections do not source the lowest-order gravitational radiation field. The Christoffel symbol can be used to find the force on each particle, giving The equation of motion utilized for the dilaton is (3.14) While this equation differs slightly from Ref. [41], both of our total pseudotensors agree and are the physical object that satisfies the gauge-invariant Ward identity. The force due to the gauge field is (3.15) B. Solutions of the Radiation Fields   Fig. (2) shows nine diagrams which contribute to gravitational radiation for Einstein-Yang-Mills theory. Algebraically, the first three diagrams for the pure gravity contributions while the diagrams with internal dilatons algebraically represent Focusing on the energy-momentum tensor |g|T µν , Solving for the appropriate field equations gives h µν , φ, and A µa to lowest order, Plugging the lowest-order field solutions into Eqs. (3.13), (3.14), and (3.15) gives The corrections to the position are useful for finding |g|T µν (k), The lowest order term proportional to v µ α v ν α may be dropped, as it was used find the solution to h µν in Eq. (3.20). Focusing on the corrections due to gravity, Additionally, the dilaton contributes Finally, the gauge boson contributes Appendix (A) works out the algebraic details of computing t µν | ∆h andt µν | ∆h . In summary, gravity contributeŝ Similarly, Eq. (3.12) the dilaton contributeŝ When calculated algebraically from Eq. (3.12), the gauge boson contributeŝ In Appendix (B), the three-point boson vertices of Einstein-Yang-Mills theory are used to find the same results for t µν via radiative Feynman rules.
Summing all contributions to order κ 2 gives the contributions from pure dilaton gravity, which agree with the results found in Ref. [41]. |g|T µν | ∆A andt µν | ∆A give the additional contributions in Einstein-Yang-Mills theory, Adding this result to Eq. (3.29) gives the total source for gravitational radiation for Einstein-Yang-Mills theory. Next, we show that this result agrees precisely with what is found with the radiative double-copy method.

C. The Radiative Double Copy
In order to use the double copy to find gravitational radiation in Einstein-Yang-Mills theory, the same replacement rules used for general relativity [41] may be used with the radiation found in Yang-Mills-biadjoint-scalar theory. The replacement rules are where the momenta q 1 + q 2 + q 3 = 0. Similar to the Ward identity k µĴ µa = 0, we can shiftT µν by terms proportional to either k µ or k ν , such that k µT µν = k νT µν = 0, which shifts the gauge-dependent pseudotensor into the harmonic gauge. Since Ref. [41] showed that the radiative double copy could recoverT µν | O(κ 2 ) and Ref. [40] showed how to use Shifting l µ β → (l β − l α ) µ /2 gives the gauge invariantT µν , Symmetrizing this result gives the appropriate final expression forT µν , where the gauge condition allows for v µ α k ν = 1 2 η µν k·v α . This result agrees precisely with what was found in Eq. (3.31), demonstrating that the radiative double copy holds for Einstein-Yang-Mills theory to leading order.

D. Einstein-Maxwell Theory
Since it is more physically relevant to scatter massive point particles with electric charge rather than particles with weak-isospin or color, an Abelian U(1) gauge symmetry is also worth studying. The action for fields in Einstein-Maxwell theory is When comparing with Einstein-Yang-Mills theory, the Maxwell field A µ can be recovered from a single component of the Yang-Mills field A µã . In order to find results in Einstein-Maxwell theory from Einstein-Yang-Mills theory, care must be taken with the coupling constants. For example, the Maxwell current density for point particles is given by where q α = −1 for electrons, such that eq α represents the electric charge of particle α.
In order to recover Einstein-Maxwell theory from Einstein-Yang-Mills, one must substitutẽ g → e and cã α → q α , given our conventions forg and the normalization of the Lagrangian given in Eq. (3.1). Applying these substitutions to Eq. (3.31) would give gravitational radiation in Einstein-Maxwell theory. At higher orders, fãbc would be sent to zero as well.
In terms of the radiative double copy, an adjoint scalar field Φ a could also be seen as a single component of the biadjoint scalar feld Φã a . Results for Yang-Mills-adjoint-scalar theory can easily be found from Eq. (2.28) by properly sending cã α → q α and reinterpreting y as the coupling constant of the adjoint scalar theory. It is straightforward to see that the double copy of Yang-Mills-adjoint-scalar theory gives solutions in Einstein-Maxwell theory with the replacement rules shown in Eq. (3.31) and y → e.

IV. CONCLUSIONS
In previous work, the double copy has been applied to gravitational radiation in general relativity with a dilaton, which suggested that schematic radiative diagrams may be useful for depicting sources of radiation [41]. Similarly, it was shown that the same replacement rules can be used to find Yang-Mills radiation from biadjoint-scalar radiation [42] and that ghosts can be used to remove the dilaton [40].
In this work, the gravitational radiation produced by colliding color charges was found within the context of Einstein-Yang-Mills theory. Our primary result demonstrates that the double copy can be used to find radiation in Einstein-Yang-Mills theory from Yang-Millsbiadjoint-scalar theory. These calculations provided insight on how a radiative diagrammatic scheme closer to Feynman diagrams used for scattering amplitudes may be possible. Furthermore, radiation in Einstein-Maxwell theory can be found via similar methods. This work suggests that it may be possible to develop systematic rules for constructing radiative diagrams that can be used to calculate radiation to higher orders, at least for initial conditions associated with N particle scattering. It appears that rules for worldline propagators would be needed, in addition to the typical rules used for scattering amplitudes.
In future work, it would be interesting to investigate if the radiative double copy holds for higher orders, as the precise replacement rules are not yet known. Additional efforts to perform the integrals are also needed. The gravitational interactions between the quantized spin of Dirac particles would be an interesting theoretical challenge, while considering the scattering of macroscopic mass distributions with classical angular momentum would be more applicable for experiments such as LIGO. Studying the formation of bound states due to higher order effects would also be important.
The author would like to thank Zvi Bern for constant guidance, as well as Walter Goldberger, Donal O'Connell, Alexander Ridgway, Jedidiah Thompson, and Julio Parra-Martinez for various discussions.

Appendix A: Derivation of Gravitational Radiation from Pseudotensor
In this section, the steps for deriving the gravitational radiation coming from nonlinear gravitational interactions are provided. In Section (III), Einstein's field equations to first order for weak gravitational fields was found to be where the energy-momentum pseudotensorT µν = T µν + t µν = |g|T µν +t µν contains the nonlinear corrections to the linearized field equations, such that the purely gravitational component of the pseudotensor t µν is given by Eq. (3.12) In order to solve for this, the lowest-order solution of the gravitational field is used which gives rise to a source for the nonlinear gravitational interaction via t µν . Each term in t µν is second order in h µν , so one is related to particle α and another to particle β, giving a double sum. The summation and integrals on all terms will have the following form where I µν is the integrand containing many terms. For the integrand, focusing on the 2)) portion of the solution to h µν and manually plug these pieces into Eq. (A2) gives tensor would give the same radiation amplitude. Making such changes gives By considering that α and β are symmetric, all particle labels may be switched for any term, which allows further simplification to give the final result To more easily compare with the diagrammatic method,t µν is found by adding the lowest- Adding this to t µν givest µν ≡ κ 2 2 N α=1 β =α m α m β lα,l β µ α,β (k)Î µν , such that As shown in the next appendix, this result agrees precisely with a diagram involving the three-point graviton vertex.
Appendix B: Some Radiative Feynman Rules

Yang-Mills and Biadjoint-Scalar Theory
A Feynman diagram approach can be used to find the results for diagrams (1c) and (1f), shown in Fig. (1). Expanding the kinetic term of the Lagrangian, the O(A 3 ) term corresponding to the three-point vector boson interaction is This term in the Lagrangian gives the textbook non-Abelian three-point vector boson vertex, given by where A a µ is associated with the momentum k, A b ν is associated with p, and A c ρ is associated with q.
The three-point vertex for two biadjoint scalars and one adjoint vector field can be used to efficiently calculate a piece radiation, which comes from the kinetic term of the biadjoint scalar. Focusing on the terms in the Lagrangian to O(Φ 2 A), Taking the appropriate functional derivatives and properly symmetrizing gives the threepoint vertex for two scalars and one vector, Γã a,νb,cc (k, p, q) = f abc δãc (k ν − q ν ) .
The three-point vertices above can be used to find diagrams (1c) and (1f), giving where a symmetry factor of 1/2 has been added.
The solutions needed for these diagrams were found in Eq. (2.16), giving Plugging in these solutions gives Due to the antisymmetry of f abc c b α c c β , switching α ↔ β for a term multiplied by this factor introduces a minus sign, allowing further simplification, Note how this result agrees with the algebraic method found in Eq. (2.25).
To find the radiative field contribution from this three-point vertex, two instances of the lowest-order field solution will be stitched together with this vertex to find a higher order contribution. The lowest-order field in momentum space is given by The three-point vertex allows for a purely gravitational source to be found, which corresponds to a component of the pseudotensor, t µν . This component of the source that generates radiation is given by t σλ (k) = 1 2 lα,l β V µρ,ντ,σλ (−l α , −l β , k)h µρ (l α )h ντ (l β )δ d (k − l α − l β ).
Since the lowest-order solutions for l α and l β are symmetric, the symmetrization is only needed for indices σ and λ in V µρ,ντ,σλ . Focusing on the integrand and breaking down the two lowest-order solutions into four terms gives (B14) Using Mathematica to perform the index contractions gives where this result gives the integrand of the diagram (2c), as shown in Eq. (3.26).