All rational one-loop Einstein-Yang-Mills amplitudes at four points

All four-point mixed gluon-graviton amplitudes in pure Einstein-Yang-Mills theory with at most one state of negative helicity are computed at one-loop order and maximal powers of the gauge coupling using D-dimensional generalized unitarity. The resulting purely rational expressions take very compact forms. We comment on the color-kinematics duality picture and a relation to collinear limits of pure gluon amplitudes.


Introduction and conclusions
It is a classic result in the field of scattering amplitudes that supersymmetric Ward identities force gluon and graviton tree-level amplitudes to vanish if all particles carry the same helicities or at most one state of opposite helicity [1], A n (±, +, +, . . . , +) = M n (±, +, +, . . . , +) = 0 . (1.1) While this result holds at tree level in any quantum field theory, in the presence of supersymmetry the vanishing persists to all loops. In non-supersymmetric field theories, in particular in the "pure" Yang-Mills and gravity theories, the above amplitudes are very interesting as they receive their leading contributions at one loop and are remarkably simple -resembling tree-level expressions, although with more subtle factorization properties [2]. Their unitarity cuts vanish in four dimensions since the helicity configuration of any two-particle cut of the one-loop expressions in (1.1) implies that there is at least one vanishing tree-level piece. Hence, these one-loop amplitudes are finite rational functions of the momentum invariants.
In the case of pure Yang-Mills theory they were efficiently constructed through their analytic properties and even the all-multiplicity expression has been established in the all-plus case [3], resulting in a remarkably compact formula using spinor helicity variables 12 . These one-loop amplitudes are also generated by the self-dual Yang-Mills theory and represent their only non-vanishing amplitudes [5]. The single-minus gluon amplitudes at one loop are also known for all multiplicities and have been constructed using Berends-Giele type [6], as well as BCFW-type recursion relations [2]. Their form is considerably more involved.
All-plus and single-minus helicity amplitudes have also been constructed in pure gravity. A conjecture for the all-plus graviton amplitude at any multiplicity exists [7] and agrees with explicit constructions at n ≤ 7 points. Again, this infinite series of graviton amplitudes is identical to one-loop self-dual gravity. For the single-minus amplitudes, an explicit, yet not very compact expression has been recently derived [8] using a spin-off of the BCFW method known as augmented recursion [9], following earlier work in [10]. As is often the case, the analytic structure, in particular consistency of soft and collinear limits, helped to constrain the ansatz.
In this work we focus on explicit S-matrix elements for mixed graviton and gluon scattering in Einstein gravity minimally coupled to Yang-Mills theory, or EYM for short. In the 1990s EYM amplitudes in four dimensions for the maximally-helicity violating (MHV) case, i.e. two negative-helicity states, were given at tree level in [11]. Only rather recently modern approaches 1 N p is the color weighted number of bosonic minus fermionic states circling in the loop. 2 See [4] for comprehensive reviews.
to scattering amplitudes based on the scattering equation formalism of CHY [12], or the colorkinematic duality relations [13], were applied to the realm of EYM amplitudes, leading to a number of explicit results. Double-copy constructions for gluon-graviton scattering in supergravity theories were given in [14]. However, the most efficient way of establishing EYM amplitudes is by expanding them in a basis of pure gluon amplitudes multiplied by kinematic numerators to be determined (also featuring in color-kinematic duality): This form was initially presented by a string-based construction for one graviton and n-gluon scattering in [15], the field theory proof followed shortly thereafter [16,17] and was further lifted to the sector of three gravitons in [16] employing the CHY formalism. A color-kinematic duality based construction extended this to amplitudes involving up to five gravitons [18]. The complete recursive solution for the numerators n(1, {β}, n) has recently been constructed in the single-trace sector in [19] and for multi-traces in [20]. This, together with the existing result for all tree-level color-ordered gluon amplitudes [21], constitutes the complete solution for the EYM S-matrix at tree level.
This state of affairs sets the stage for the investigation of the present paper. Here we compute the remaining rational amplitudes of the EYM theory at the leading one-loop level at multiplicity four. These are the three all-plus helicity amplitudes involving one, two or three gravitons, as well as the six single-minus amplitudes involving one, two or three gravitons. An elegant way to determine such amplitudes consists in employing two-particle unitarity cuts in D = 4 − 2 dimensions [22] (see also [23] for the first uses of D-dimensional generalized unitarity). The main idea is that a rational term in four dimensions, R, will in D dimensions acquire a discontinuity, but to a higher order in the dimensional regularization parameter . Schematically, (1.4) Technically, the calculation is greatly simplified by using the general supersymmetric Ward identity of (1.1) at the one-loop order, which implies that the contribution of an arbitrary state in the loop is proportional to that from a scalar circulating in the loop, It is important to realize that "any state in loop" refers to a "pure" contribution of a definite quantum field excitation (e.g. graviton or gluon) propagating in the loop. This relation may therefore be straightforwardly applied to the EYM situation of a gluon circulating inside the loop of a mixed gluon-graviton amplitude, see Figure 1 for a four-point example: A one-loop single-graviton three-gluon amplitude will have one-loop contributions of order κg 3 and κ 3 g. A generic one-loop m-graviton and n-gluon amplitude will have g-leading contributions of order g n κ m representing only gluons in the loop, whereas the g-subleading contributions g n−2k κ m+2k reflect contributions where 2k gluon propagators are turned into graviton propagators. Note that there is no single-gluon l-graviton vertex. For the contributions to the amplitude maximizing the powers of the gauge coupling constant, i.e. the contributions to A n+m (1, 2, . . . , n; h 1 , . . . , h m ) at order g n κ m , we only have gluons running in the loop, and the relation (1.5) applies with N p = 1, i.e. this contribution may be computed upon replacing the gluon inside the loop by a scalar. The cuts are performed in D dimensions, where a generic loop momentum L satisfies L 2 = 0 = l 2 (−2 ) −l 2 (4) = 0, where l (−2 ) and l (4) represent the (−2 )-and four-dimensional part of L. Because the external kinematics is four-dimensional, at one loop there is just one l (−2 ) . Setting l 2 (−2 ) := µ 2 , one then has l 2 (4) = µ 2 , i.e. all internal D-dimensional scalar can effectively be treated as four-dimensional massive scalar with uniform mass µ 2 , over which one integrates at the end [22].
The "non-pure" contributions of order g n−2k κ m+2k , however, have a mixture of gluons and gravitons running inside the loop. Here the situation is less clear, as (1.5) does not hold. A simple dimensional analysis also reveals that the mixed graviton-gluon contributions in the loop are not represented by (1.5).
Hence in this work we only aim at finding the maximal g contributions to the one-loop rational amplitudes in EYM theory. Here we find intriguingly simple results, to wit 3 It would be interesting to also construct the missing "non-pure" pieces at higher orders in κ as well, even though they will be numerically subleading at energies well below the Planck mass. This should be possible using the double-copy techniques initiated in [18].
The rest of our paper is organized as follows. In the next section we collect all relevant tree-level amplitudes involving gluons, gravitons and massive scalars entering the cuts needed to compute the rational amplitudes we are interested in. Sections 3-5 are devoted to the calculation of all one-loop amplitudes with one graviton and three gluons. A particularly interesting case is that of Section 3, where we find that the all-plus amplitude 1 + 2 + 3 + 4 ++ , although nonvanishing in D dimensions, actually vanishes in the four-dimensional limit. Sections 6-8 discuss the derivation of the amplitudes with two gravitons and two gluons, while Sections 9-10 contain the (vanishing) amplitudes with three gravitons and one gluon. Finally in Section 11 we rederive the curiously vanishing single-graviton all-plus amplitude from a double-copy construction. Two appendices complete the paper. In Appendix A we list the D-dimensional expressions of the relevant integrals and the appropriate limits contributing to the amplitudes of interest, while in Appendix B we derive all the four-point tree-level amplitudes with two massive scalars and gluons/gravitons using recursion relations.

Relevant tree-level amplitudes
In this section we collect all the tree-level amplitudes entering our calculation. The basic building blocks are the three-point amplitudes involving a gluon or graviton and two massive scalars. The color-ordered gluon-scalar-scalar amplitudes are [24] where p 2 2 = p 2 3 = µ 2 , and µ is the mass of the scalar particles. In these formulae, λ q and λ q are reference spinors, and the amplitudes themselves are independent of their choice. The amplitudes involving a graviton are similarly given by the square of the previous amplitudes 4 We will also need four-point amplitudes involving two gluons/gravitons and two scalars. The amplitudes involving gluons have been derived in [24] using BCFW recursion relations [26] applied to massive scalars, and the relevant amplitudes with gravitons can be obtained similarly (see Appendix B for details). We quote here the expression of the relevant Yang-Mills amplitudes with two gluons and two scalars: while for the amplitudes involving a graviton, a gluon and two scalars we have: 5 (2.7) We have also double-checked these amplitudes through a direct Feynman diagrammatic calculation. The two-graviton/two-scalar amplitudes in turn read (2.9) Note that (2.3), (2.5) and (2.8) manifestly vanish if the scalars are massless.
For later convenience we shall split up (2.5)-(2.9) into a sum of two partial amplitudes which treat the single graviton effectively as if it were color ordered, in the sense that  Figure 2: The s-and t-channel cuts of the all-plus single-graviton amplitude. Cyclic permutations of the labels (1, 2, 3) should also be added. 14) and similarly for the other amplitudes. In the unitarity-based construction of the one-loop amplitudes to be discussed, we then symmetrize explicitly in the graviton leg(s) attached.

11)
3 The 1 + 2 + 3 + 4 ++ amplitude We begin our investigation with the four-point same-helicity amplitude with one graviton and three gluons. We will derive the integrand of this amplitude, as well as its four-dimensional limit. We anticipate the interesting outcome of this computation, namely that this amplitude is zero in the four-dimensional limit -a result that we will also confirm from the double-copy perspective in Section 11. 6 To organize the computation efficiently, we employ the effective "color"-ordered graviton partial amplitudes introduced in the previous section. The diagrams to be considered are shown in Figure 2. As all gluons carry the same helicity, we need only to evaluate the first diagram in Figure 2; the final result will then be obtained by adding the terms obtained by cycling (1,2,3) in the partial result.
For the configuration (1234) of Figure 2 there are two two-particle cuts, in the s 12 = s and s 23 = t channels. We start with the t-channel cut which is given by the product of the two partial amplitudes: where the explicit expressions of the tree-level amplitudes entering the cut are given in (2.3) and (2.5), and the factor of two arises from summing of the possible assignment (φ,φ andφ, φ) for the internal scalar particles.
For the s-channel cut of the (1234)-configuration, one similarly arrives at an integrand The strategy to find the integrand is now to rewrite the t-channel expression in such a way as to reproduce the s-channel expression modulo terms that vanish on the s-cut. For this we first introduce a uniform parametrization of the (1234) box diagram in terms of a single loop momentum l: Using these the, s-and t-channel cuts take the compact forms where s l4 = 4|l|4] = 2 (l · p 4 ), which in turn may be written as We also note the identity 3|l|4] Inserting this into the s-cut expression (3.6) and dropping the D 0 term gives us an integrand which may be lifted off the cuts (with the usual replacement (2π)δ(D) → i/D for the cut propagators): 7 (3.10) 7 The factor of −1 in the following expression arises from reinstating two (uncut) propagators.
The partial one-loop amplitude is thus given by a linear box integral and a scalar triangle.
The final step is to now reduce the linear box integral. Here we use the Mathematica package FeynCalc [27], which efficiently implements the Passarino-Veltman reduction algorithm [28]. Doing this we arrive at the final result 8 where by "perms" we indicate the two permutations (2314) and (3124) of (1234), which interchange the Mandelstam invariants as (s, t, u) → (t, u, s) and (s, t, u) → (u, s, t), respectively. However, we need not do this explicitly as taking the four-dimensional limit using the relations in (A.7) we get a vanishing result: It would be desirable to understand the deeper reason for this curious vanishing.
We also quote an alternative expression of the D-dimensional amplitude, which is given by: (3.13) where the two permutations are the same as in (3.11). The vanishing of (3.13) is of course obtained again upon using the formulae of Appendix A. We also comment that this integrand is manifestly odd under the exchange of any two same-helicity gluons. In color space this means that this amplitude is proportional to f a 1 a 2 a 3 , with no d a 1 a 2 a 3 contribution. We will see that the same property is shared by all amplitudes involving three gluons computed in this paper -they only come with an f a 1 a 2 a 3 color factor.
Constructing this amplitude is a slightly harder task, hence as an introduction we will first rederive the four-point gluon amplitude with a single negative-helicity gluon of [22] and then apply a similar procedure to the more complicated EYM case. The form of the four-gluon integrand is also of use for a double-copy based construction of the EYM amplitudes.
Warmup. As for the case of the all-plus amplitude derived in the previous section, we work with two-particle cuts. Because only gluons are involved, color ordering leaves us with only two channels to consider, see Figure 3. For the s-channel we have Figure 3: The s-and t-channel cuts of the whereas the t-channel cut reads The strategy to find the integrand is now to rewrite the t-channel expression in such a way to reproduce the s-channel one modulo terms that vanish on the s-cut. For this, we will make use of the following identity to rewrite the numerator in (4.2): where s l 1 1 = 1|l 1 |1] = 2 l 1 · p 1 , which in turn may be written as This last expression holds on the t-channel cut. Inserting the expression (4.3) for 1|l 1 |4] into the t-channel cut amplitude A 4 | t of (4.2) then yields an expression which may straightforwardly be lifted off the cut. Thus we get an integrand 9 where we have chosen the loop momentum parametrization as l = l 1 , and Again, the minus sign in front of the following expression arises from two cut propagators.
Note that there is an ambiguity in treating the last term in (4.5). By the logic laid out above we could have also replaced s l 1 1 by D 0 as the resulting expression would agree with (4.2) and (4.1) on the respective cuts. However, only the choice quoted above does reproduce the result in the literature. 10 The final step is to now reduce the tensor integrals appearing in (4.5), which we do again using the Mathematica package FeynCalc [27]. Doing this we find This result agrees with the result in the literature [22]. 11 Single graviton amplitude. After this warmup let us now consider the EYM amplitude for a single graviton and three gluons with one negative-helicity state. Again we shall construct the integrand from two-particle cuts. Now, due to the presence of the graviton 4 ++ which we here include with the effectively colored ordered tree-amplitudes A of (2.10), we will have to consider three distinct type of two-particle cut diagrams. These follow from the particle configurations (1234), (1243) and (1423) pushing the graviton leg 4 ++ through the color-ordered gluons. The full amplitude is then divided into three parts, which we now construct in turn from two-particle cuts.
Diagram (1234). Here we encounter an s-channel and a t-channel cut. For the s-channel of the (1234)-configuration we find where for the diagram (1234) we use the following loop momentum assignments: 10 It would be valuable to understand this seeming ambiguity better. Such an ambiguity does not appear in the procedure of merging cuts employed in later sections, which we have used to confirm all calculations of this paper. In the latter procedure, vanishing integrals are omitted, which may obscure a double-copy interpretation of the results. 11 Had we taken D 2 0 instead of D 0 s l1 in the last term of (4.5) we would on top find a term proportional to [u/(st)] I 3 [µ 4 ] in the above, in disagreement with [22].
Note that we have set l 1 = −l. The t-channel cut of the (1234)-configuration on the other side takes the form We now lift the two expressions (4.9) and (4.11) off the cuts by the same strategy that was applied previously. We rewrite the two l-dependent spinorial expressions in Inserting this into the s-cut amplitude (4.9), and rewriting the Mandelstam invariants s li = 2(l·p i ) as This expression may be straightforwardly reduced to scalar integrals using e.g. FeynCalc. As a matter of fact, one quickly sees that the second term in the above vanishes upon integration.
An alternative representation for A (1234) is obtained if one rewrites the t-cut expression (4.11) in terms of the s-cut one plus D 0 terms, arriving at 12 34 which upon Passarino-Veltman reduction indeed matches A (1234) of (4.15). The result after reduction reads: (4.17) Diagram (1243). For the (1243)-contribution we have a u-channel and a s-channel cut, which read and where we have introduced the loop parametrization l := −l 3 along with The s-cut expression may now be lifted off the cut by using the identities (4.22) Again we have an expression in terms of box and triangle tensor integrals amenable to standard integral reduction techniques. An alternative and more compact expression may derived if one rewrites the u-cut in terms of the s-cut followed by a shift in the integration variable l → l + p 3 . One then finds where now Passarino-Veltman reducing (4.22) or (4.23), one arrives at Diagram (1423). The remaining (1423)-contribution carries a u-channel and a t-channel cut. These read and where we identified the loop momentum as l := −l 2 and used the inverse propagators suitable for diagram (1423), However, by inspection we see that A (1423) may be obtained from the (1234)-configuration by simply swapping 2 ↔ 3 (or s ↔ u). Hence we conclude that Taking the four-dimensional limit yields the compact final expression We now consider the rational one-loop amplitude with a single negative-helicity graviton and three positive-helicity gluons A (1) (1 + , 2 + , 3 + ; 4 −− ). For amplitudes containing progressively more negative helicities, the procedure described in previous sections to construct the integrand becomes tedious. Hence, from now on, rather than constructing the integrand, we will use the standard approach of [29,30] where we directly merge all two-particle cuts into a single function. The case at hand is particularly simple given the very symmetric helicity configuration chosen.
Using the tree-level amplitudes in Section 2, we find that the s-cut of the amplitude is given by s-cut: This amplitude also has t-and u-cuts which are obtained by simply cycling the labels (312) → (123) and (312) → (231), respectively. As in the previous sections, we use FeynCalc [27] to perform efficiently all relevant Passarino-Veltman reductions of the three-tensor box in (5.1) (and its permutations). We work first in the s-cut, and focus on the tensor box with particle ordering (1234). We lift the integral off the cut, and perform a Passarino-Veltman reduction. This will generate scalar boxes with particle ordering (1234) (and powers of the (−2 )-momentum µ in the numerator), whose coefficient(s) we will then confirm from the t-cut. It will also generate one-mass triangles and bubbles in the s-channel (again with powers of µ in the numerator), which we keep, as well as spurious one-mass triangles and bubbles with a t-channel discontinuity, which we drop. We then repeat the same operation for the two other box topologies with particle orderings (1243) and (1324). Merging all contributions thus obtained, we arrive at our final result: A (1) (1 + , 2 + , 3 + ; 4 −− ) = 2i 2 [12][34] 12 34 ( 42 [23] 34 ) 3 f (s, t, u) + perms ,

(5.3)
As in the case of the 4 ++ 1 + 2 + 3 + amplitude computed in Section 3, by "perms" we denote the two permutations 2314 and 3124 of 1234, with the the Mandelstam invariants interchanged as (s → t, t → u, u → s) and (s → u, t → s, u → t). Performing the four-dimensional limit using the results of Appendix A, we find: Adding the permutations, we arrive at a very compact final result: Note that the kinematic function in (5.5) is an odd function under any exchange of two gluons, and hence the complete amplitude is even under such an exchange (including a minus sign from the colour factor f abc ), as it should.
6 The 1 + 2 + 3 ++ 4 ++ amplitude In this section we move on to amplitudes which contain two gravitons and two gluons. The simplest case to consider occurs when all particles have the same helicity -a particularly symmetric configuration.
We briefly describe the outline of the derivation, similarly with previous calculations. As usual there are three cut diagrams to consider, in the s-, t-and u-channels. These cuts will give rise to tensor boxes with particle ordering (1234), (1243) and (1324). These are given by: Note that on the right-hand side of the the s-cut in (6.1) we have to include the sum of two color-ordered amplitudes, A 1 + , 2 + , −l 2,φ , −l 1,φ and A 2 + , 1 + , −l 2,φ , −l 1,φ . Indeed, since the left-hand side of the cut is an amplitude with a colorless (two-graviton) external state, both terms contribute to the same color ordering. This will be a recurrent feature of all cuts where one side of the cut is colorless. Moreover, there will be an additional contribution from the cut obtained by swapping φ withφ, which will double up the result of the previous cuts, as usual.
Using the tree-level amplitudes given in Section 2, we work out the expressions of these cuts, which give rise to three tensor boxes with the different particle orderings (1234), (1243) and (1324). Inspecting all cuts we can reconstruct the amplitude. We find the following results: s-cut:

(6.4)
Note that our cut integrand contains tensor boxes with cut momenta l 1 and l 2 as well as the same contribution but with l 1 and l 2 flipped. At the level of the integral, this will be taken into account by doubling up the contribution of a single copy. The next step consists in combining all cuts, which we will do for each box topology separately. Doing so, we arrive at the following result for the topology (1234): which is obtained from combining the relevant terms in the s-cut given in (6.2) and the t-cut of (6.3). The topology (1243)  The last topology to consider is (1324), which is obtained from combining the relevant terms from the s-and u-cuts, given in (6.2) and (6.4). Doing so we get: (6.7) Finally we take the four-dimensional limit: Combining all terms we arrive at a remarkably simple final result: We note that (6.10) is symmetric under the exchange of the two gluons. This is consistent with the colour factor δ ab of this amplitude -indeed, the complete, color-dressed result should be symmetric under a swapping of the two gluons.
We also quote the compact expression of the D-dimensional result: Here we follow the same strategy as in the previous section, and derive the complete amplitude from merging two-particle cuts. As we will see, this procedure will now give rise to three tensor boxes with different particle orderings as before, with numerators that are up to quartic order in the loop momenta. These will then be Passarino-Veltman reduced as usual.
We now compute the three possible two-particle cuts of the amplitude. We also include the usual factor of two from swapping φ andφ in the loop. The s-cut is given by s-cut: . Again, the appearance of two terms on the right-hand side of the cut, with two different gluon orderings, is due to the fact that the amplitude on the left-hand side of the cut contains a colorless external state. The next cut to look at is: t-cut: obtained from A(4 ++ , 1 − , l 1,φ , l 2,φ )A(2 + , 3 ++ , −l 2,φ , −l 1,φ ). Finally, u-cut: from A(3 ++ , 1 − , l 1,φ , l 2,φ )A(2 + , 4 ++ , −l 2,φ , −l 1,φ ). We also define a convenient spinor prefactor which has the correct spinor weights for the given amplitude: We are now ready to merge the different cuts. From the topology (1234) we get: The box topology (1243) is simply obtained from the topology (1234) in (7.5) by swapping 3 ↔ 4, or (s, t, u) → (s, u, t). Note that J is invariant under this swap, hence the result for the (1243) topology is immediately found to be: Note that in (7.5) and (7.6) the I 2 [µ 2 ] functions only appear in the u-and t-channel.
The last topology is (1324), for which we obtain The expression (7.7) is symmetric in u ↔ t.
Finally we take the four-dimensional limit of (7.5), (7.6) and (7.7) using (A.7), thus getting respectively. Thus, we arrive at the final result for the four-dimensional limit of the amplitude (using the expression of J in (7.4)): The D-dimensional answer is easily obtained by adding (7.5), (7.6) and (7.7).
We proceed similarly to the previous sections and study all two-particle cuts of this amplitude. As in earlier examples, we find three box topologies with tensor numerators. In this case, an appropriate spinor prefactor which has the correct spinor weights for the given amplitude is We construct the two-particle cuts of this amplitude using the tree-level expressions in Section 2.

(8.2)
As in the cases studied in Sections 6 and 7, the s-cut integrand includes the sum of two colorordered tree amplitudes on the right-hand side of the cut, which contribute to the same colorordered amplitude, given that the external state on the left-hand side of the cut is colorless. Using the expressions of the relevant tree-level amplitudes and including a factor of two from the two possible assignments from the internal scalar fields, we obtain the following expressions for the cuts: s-cut: t-cut: As usual, we now merge the cuts focusing separately on the three different box integrals. Merging the s-and t-cut for the topology (1234) we get: The topology (1243) can be obtained by swapping 3 ↔ 4 in (8.6), or (s, t, u) → (s, u, t). Noting that J is invariant under this swap we get: 11s 3 + 59s 2 u + 64su 2 + 22u 3 6s 3 ut 3 .

(8.7)
Next, we merge the u-and t-cuts for the topology (1324): As expected, the expression (8.8) is symmetric in u ↔ t.
Finally we take the four-dimensional limit of (8.6), (8.7) and (8.8). These are given by respectively. Thus, we arrive at the final result for the four-dimensional limit of the amplitude, using the expression for J in (8.1), Similarly to the previous section, we can easily show that the amplitude 1 + 2 ++ 3 ++ 4 −− vanishes upon integration. Consider for instance its s-channel cut. This is given by s-cut: 11 The 1 + 2 + 3 + 4 ++ amplitude from the double copy The color-kinematic duality or double copy [13] was extended in the works [14,18] also to the domain of mixed graviton-gluon amplitudes in the Einstein-Yang-Mills theory. In particular [18] exposed explicitly how to construct an Einstein-Yang-Mills amplitude through a double copy from Yang-Mills and Yang-Mills + φ 3 theory: The latter Yang-Mills-Scalar theory contains biadjoint scalars φ Aâ next to the gluons Aâ µ and is defined through the Lagrangian As a one-loop application of (11.1), we wish to derive the vanishing of the 1 + 2 + 3 + 4 ++ amplitude, which we observed with a direct computation in Section 3. Thus we need to construct integrands for the two amplitudes A (1) (1 + , 2 + , 3 + , 4 + ) and where color ordering is performed in both cases with respect to the hatted gauge group index. The first one, the all-plus helicity four-gluon amplitude, is well-known and takes the form As this is a pure box-integral, in the construction of the one-loop YM + φ 3 amplitude integrand we only need to construct the box-contribution to the A (1) (1 A φ , 2 B φ , 3 C φ , 4 + ) amplitude as well: (1,2,3) . (11.4) Here we have simply inserted the scalar-scalar-on-shell-gluon vertex of (2.1) in the south-east corner with a reference spinor λ q . The numerator emerging from this integrand respects colorkinematics duality as it is built entirely from three-valent graphs. Employing the double-copy prescription [18] of (11.1) we are therefore led to the following representation of the all-plus single-gluon EYM-amplitude [12][34] 12 34 + cycl(1,2,3) .
(11.5) Passarino-Veltman reducing the integral one arrives at the D-dimensional expression Going to four dimensions simplifies this result considerably, and one arrives at Finally, we comment on the question whether the amplitude relations of Stieberger and Taylor [15,31] relating pairs of collinear gluons to gravitons extend to the one-loop level for the one-loop rational amplitudes we have considered in this paper.
We will test this for the simplest case of the all-plus amplitude with one graviton. For such a relation to be true, the vanishing four-dimensional result must follow from the specific collinear limit proposed by Stieberger and Taylor on the five-point all-plus rational amplitude in pure Yang-Mills. In analogy to the tree-level relation, in four dimensions we expect to have: YM (1 + , 5 + , 2 + , 4 + , 3 + ) + cycl(1, 2, 3) , (11.8) where the equality would hold in the collinear limit {p 4 → xP, p 5 → (1 − x)P } on the righthand side of (11.8), and G(x) is an undetermined function of the momentum splitting fraction x which is expected to be independent of the helicities of the particles. Note that G(x) has been determined for tree amplitudes in [31]. We have also added cyclic permutations of the three gluons to secure cyclic symmetry in these particles. Using the well-known expression for the all-plus five-point rational amplitude in Yang-Mills [32], we see that the right-hand side of (11.8) contains the factor Performing the above-mentioned collinear limit on (11.9), followed by a cyclic permutation of the three gluon legs in order to reflect the anticipated color structure, and relabelling P → p 4 (with p 4 being the momentum of the graviton leg), we arrive at lim p 4 p 5 This is curiously proportional to the x-independent part of the right-hand side of (11.10), which was obtained from the Stieberger-Taylor collinear limit. Given the vanishing of our final result in four dimensions, also the triangle contribution in (3.11) can be written in a similar way: In conclusion, even though the amplitude (3.11) vanishes in four dimensions, we find the similarities between (11.12) (or (11.13)) and (11.10) intriguing, and worth further investigation.
for the bubble and one-mass triangle, while for the zero-mass box function one has [33] The in complete agreement with results of [22,7] (after taking into account the opposite sign in the definition of triangle functions compared to those papers).

B Tree-level amplitudes via recursion relations
In this appendix we derive the relevant tree amplitudes involving gravitons, gluons and massive scalars which enter the one-loop calculations in EYM performed in earlier sections.
Soft limits of the A(4 ++ , 1 + , 2 φ , 3φ) amplitude It is an interesting check to confirm that the amplitude obtained in this way has the correct soft limits. To this end we consider the case with gluon 1 + becoming soft. We then expect the amplitude to factorize as A(4 ++ , 1 + , 2 φ , 3φ) − −− → where the soft function is which is identical to the result for A(4 ++ , 1 + , 2 φ , 3φ).