New relations for graviton-matter amplitudes

We present new relations for scattering amplitudes of color ordered gluons, massive quarks and scalars minimally coupled to gravity. Tree-level amplitudes of arbitrary matter and gluon multiplicities involving one graviton are reduced to partial amplitudes in QCD or scalar QCD. The obtained relations are a direct generalization of the recently found Einstein-Yang-Mills relations. The proof of the new relation employs a simple diagrammatic argument trading the graviton-matter couplings to an `upgrade' of a gluon coupling with a color-kinematic replacement rule enforced. The use of the Melia-Johansson-Ochirov color basis is a key element of the reduction. We comment on the generalization to multiple gravitons in the single color trace case.


INTRODUCTION
At the Lagrangian level Einstein's theory of gravity and Yang-Mills (YM) gauge theories look quite different. Nonetheless, in a perturbative quantization on a flat space-time background intimate relations between their S-matrices exist, allowing one to express pure graviton scattering amplitudes through pure gluon scattering data. The first such connection are the Kawai-Lewellen-Tye relations [1] derived from the string theory. Later Bern, Carrasco and Johansson (BCJ) [2][3][4] introduced a double-copy construction for graviton amplitudes from gluons. Here Lie-algebra like relations for the kinematic building blocks of gluon amplitudes were identified. The double copy technique may be used to generate looplevel integrands of gravitational theories from the simpler gauge-theory ones, this being the state-of-the-art method for higher loop computations in (super)-gravity. For the phenomenologically most relevant case of gravity minimally coupled to generic non-abelian gauge and matter fields of spins 1, 1 /2 and 0, our knowledge is less complete. Early results for maximally-helicity violating (MHV) tree amplitudes [5][6][7] were rather recently dramatically extended to the tree-level sector of Einstein-Yang-Mills (EYM) amplitudes. Here the full modern arsenal of amplitude techniques was employed, starting from the field theory limit of string amplitudes [8], field theoretical considerations [9,10] using the Cachazo-He-Yuan (CHY) formalism [11][12][13], as well as double copy methods [14]. In fact, the complete reduction of the EYM tree-level S-matrix to the YM one was accomplished in [15][16][17]. This in combination with the existing result for all tree-level color-ordered gluon amplitudes [18][19][20], constitutes the complete solution for the EYM S-matrix at tree level. First results for pure EYM amplitudes at oneloop level at multiplicity four were recently reported in [21].
In this letter we report on an extension of these results to generic massive non-abelian matter fields, i.e. QCD and scalar QCD with N f flavors minimally coupled to Einstein's gravity. Again, we are able to present compact formulae to express single graviton-quark-gluon and graviton-scalar amplitudes in linear combinations of nongravitational amplitudes. We comment on generalizations to higher graviton multiplicity in the discussion.

EINSTEIN-YANG MILLS
Gluon amplitudes may be color decomposed in various bases. A particular useful and minimal one, as it generalizes to QCD, is the Del Duca-Dixon-Maltoni (DDM) basis [22] which organizes the n-gluon scattering amplitude in a basis of (n − 2)! partial amplitudes with the DDM factors wheref abc = i f abc are the structure constants of the gauge group (our conventions are summarized in the appendix). Clearly, also for an EYM amplitude involving a single graviton and at leading order in gravitational coupling κ (implying a color single trace structure) an identical color decomposition in the DDM basis may be applied where the graviton leg p is of course not participating in the color ordering. In [8] an intriguingly simple representation of this partial EYM-amplitude was derived from a string theory consideration Here the graviton polarization is written as ε µν p = ε µ p ε ν p . Moreover X i = i j=2 k j denotes the region momentum. In a sense one half of the graviton has turned into a gluon evenly distributed amongst the n gluons. Indeed the relation (3) immediately follows from the consistency of soft-limits. Both gravitons [23] and gluons [24] (resp. photons) obey universal factorization properties in the soft limit p → 0 Our convention is that all external momenta are incoming. If one starts with (3) as an ansatz with an a priori undetermined X p one quickly arrives at the consistency conditions upon taking p soft. These relations are solved for the region momenta X i = i j=2 k j . In addition, the second consistency requirement of gauge invariance of (3), ε µ p → p µ , immediately yields the famous BCJ relation [2] an essential ingredient of the double-copy construction [3,4]. So indeed (3) is entirely constrained by soft limits and gauge invariance.

EINSTEIN-QCD
We now generalize (3) to QCD minimally coupled to gravity (EQCD). For this we need to first discuss the issue of color ordering in the presence of quarks and antiquarks in the fundamental representation T a ij of the gauge group. A very useful color basis for this was provided by Melia [25] and refined by Johansson-Ochirov [26], which we term the MJO-basis. An n-particle QCD amplitude consists of k quark-anti-quark pairs and n − 2k gluons. Without loss of generality we take the flavors of all k quark lines to be distinct. The primitive amplitudes in the MJO-basis are given by Quarks and anti-quarks are marked with under-scores or over-scores, respectively. In the partial QCD amplitude A(1,2, σ) the permutation of the last (n − 2) arguments must form a Dyck word. The most intuitive defintion is that a Dyck word corresponds to well formed bracket expressions with quarks preceded by opening and antiquarks followed by closing brackets. In the MJO-basis a QCD amplitude may then be decomposed as where χ(n, k) = (n−2)! k! is the dimension of the basis [26]. Using the bracket notation the color factors are given by Here a level of 'nestedness' l has been introduced, it is the number of anti-quarks minus the number of quarks to the left of the position in the Dyck word, and reflected in the tensor product structure. The important object Ξ a l takes the form The Ξ a l form a representation of the gauge group Lie algebra [Ξ a l , Ξ b l ] =f abc Ξ c l . The explicit C(1,2, σ)'s are sums of products of k strings of generators (T a1 . . . T ar ) ij contracted over adjoint indices Clearly then, an Einstein-QCD amplitude at leading order in κ (single color trace) involving a single-graviton enjoys a color decomposition in the MJO basis The central result of this letter is, that this partial EQCD-amplitude takes the form in complete analogy to (3). A gauge transformation on leg p yields the BCJ relations (6) in QCD [26] which were proven in [27]. The relation (11) is also consistent with the soft limits as the soft behavior of gravitons and gluons is universal, i.e. (4) also holds for {1, . . . , n} being (anti)-quarks or gluons. These consistencies are strong evidence for the correctness of (11).

DIAGRAMMATIC PROOF
In order to prove (11) consider the Feynman diagrammatic representation of the color dressed A tree n,k;1 of (10). a = gf abc h P ↵ µaµb (pb pa) µc + P ↵ µaµc (pa pc) µb + P ↵ µbµc (pc pb) µa We construct on-shell Feynman with the two gluon one graviton ve

Appendix A: Feynman Rules for Einstein-Yang-Mills-Scalar Theor
The graviton is understood to carry momentum P and Lorentz indices ↵ . All ingoing. The momenta of the fermions and scalars follow their particle's arrows. Starting from a pure QCD color dressed n-point amplitude A tree n,k the EQCD-amplitude may be obtained by attaching the graviton leg with data {ε µ p ε ν p , p ρ } to all propagators and vertices of the lower point QCD amplitude. The relevant vertices are depicted in FIG. 1, their mathematical expressions [28] are collected in the appendix. Putting the graviton leg on-shell results in a subtle simplification. Let us look at the pure gluon-graviton vertices first. Here one observes a decomposition as which would be attached on the corresponding leg. Interestingly, the last term above represents an effective EYM vertex intimately related to the pure YM three-gluon vertex through a color-kinematical replacement rule which we demonstrate in the appendix. In this relation one replaces the color factor f abc by a kinematic expression assigning a momentum to every leg in color space. The convention is that all momenta are inflowing. In this argument the Ward identity has been used for the gluon legs a and b as they attach to invariant subamplitudes. Similarly the graviton-three-gluon vertex decomposes as , where again the dots on the top of the gluon lines denote inverse gluon propagators. Just as before the effective EYM vertex follows from the four gluon vertex upon color-kinematic replacement for generic indices m, n appearing in the color structure of the four gluon vertex. The final graviton dressing to be considered is the graviton-four-gluon vertex with an onshell graviton leg. Here we observe a total decomposition into inverse propagator dressed legs only -(a, b, d, c) cyclic , and no occurrence of a higher order effective gravitonfour-gluon vertex.
Turning to the quark-graviton interactions we observe a similar pattern. The graviton-quark-anti-quark vertex, the graviton leg being on-shell, is already the effective vertex, now with a T p replacement rule whereas the graviton-gluon-quark-anti-quark vertex gives rise to inverse propagator legs only .
Putting all these insights together we see that in the sum over all possible attachments of the on-shell graviton leg to the vertices and propagators of the lower point QCD amplitude, all the inverse propagator (dot on top the gluon leg) terms cancel out and the single-graviton EQCD amplitude may be evaluated by only considering the effective vertices of eqns. 12, (13), (14), which are all generated by the color-kinematic replacement rule from the sum over all attachments of a gluon to the same QCD amplitude. This insight entails immediately the relation for the colored amplitude

COLOR-KINEMATIC REPLACEMENT
What remains to be understood is how our colorkinematic replacement rule acts on the MJO-color basis factors C(1,2, σ). Here we find the key relation C(1,2, σ, p, σ )

Appendix B: On-shell and e↵ective Feynman rules
We construct on-shell Feynman rules by contracting the external graviton with its polatrization tensor. We start with the two gluon one graviton vertex: where we defined M µ⌫ (p a , P ) = (" P · p a ) ⌘ µaµb + " µa P P µb " µb P P µa W µ⌫ (p, q) = " µa P q µb " µb P p µa . (B2) We write this as Note that in any tree level computation of a scattering amplitude the term W µaµb (p a , p b ) will vanish due to on-shell Ward identities. It is now straightforward to obtain where k 2σ := k 2 + k σ1 + . . . k σ b with b being the length of σ. To prove this relation we closely follow a proof of a color factor symmetry found in [29]. In fact, this color factor symmetry may be understood as a direct corollary of our relation (11) under a gauge transformation ε µ p → p µ . We first prove (17) for vanishing σ. Here with (8) and using the bracket notation we have We now assume that (17) is true for C ...|pc|... . Zooming in on the position of the insertion we may write as the two gluons are next to each other they appear at the same level of nestedness. Consider now the commutator of C ...|cp|... and C ...|pc|...
By assumption we have C σ|pc|... | Ra p = (ε p · k 2 σ ) C σ|c|... with σ = {σ 1 . . . σ b−1 } and the induction step is performed. We note that the relation (21) points towards a kinematical algebra structure The case where the leg c = σ b prior to p is a quark or antiquark works analogously and is proven in the appendix. This then completes the proof of the claimed relation (11) relating single graviton EQCD amplitudes to QCD ones.

SCALAR MATTER
Our central result eqn. (11) is in fact universal. It also applies to a theory of gravitationally minimally interacting massive, color charged scalars, Einstein-Scalar-QCD.
Clearly, the color structure of a Scalar-QCD amplitude will be captured by the MJO basis as well. The relevant single graviton vertices are collected in FIG. 2. One again shows that the emission of a single graviton from a multi-scalar-gluon or even multi-scalar-gluon-quark process reduces to coupling the graviton with effective vertices similar to the QCD case (see appendix). I.e. the gluon 'upgrading' relation (16) holds analogously and together with the MJO basis reduction (17) this proves (11) also for scalar matter.
An interesting question concerns the general matter theories with Yukawa and φ 4 (or φ 3 ) couplings (in 4d).
Here the issue of a minimal color basis is to be settled. However, considering single graviton emission processes our diagrammatic argument is straightforwardly extended to these couplings as well: There are no onshell graviton couplings to the Yukawa and φ n vertices. Hence the gluon upgrade relation (16) holds here as well.

DISCUSSION
In a sense the key statement of this letter is that for every BCJ-relation there is an associated single graviton amplitude: The BCJ-relation is nothing but the gauge invariance of this graviton. What about multiple gravitons? In [15] the single graviton EYM relation was used as a seed for an all graviton multiplicity reduction of the single trace EYM amplitudes to pure YM ones. The only ingredients were gauge invariance and BCJ relations along with a structural assumption on the form of the higher graviton amplitudes. If one accepts this assumption also for our general matter case the multi-graviton, single trace formulae of [15] directly generalize. The situation for multi color traces is less clear to us. Finally, we believe that (22) points towards a kinematical algebra structure consisting of a "mixed" color-kinematic representation which should be understood in more detail. All our results are dimension independent. We also checked our single graviton relation against known amplitudes involving massive quarks [30] and massive scalars [21] in the literature. Moreover, it is possible to extend our techniques to loop level ampitudes with one external on-shell graviton and no internal gravitons (leading order in κ). It is still possible to use the effective vertex notation except for the case where one inserts a graviton directly into a gluon loop. There it is not possible to use Ward identities and one additional term appears. Still, the new term will be of the same analytic form as the insertion of a gluon to the same gluon loop up to a factor of two. It would be nice to obtain a decomposition similar to (11) for higher loop level integrands using the techniques provided here.

APPENDIX Feynman Rules for Einstein-Yang-Mills-Scalar Theory coupled to Quarks
The theory under consideration is defined by the Lagrangian with the usual definitions where R µν is the usual Ricci tensor, κ is the gravitational coupling constant and D µ ij = ∂ µ δ ij − i g A µ a (T a ) ij is the covariant derivative with g being the YM coupling. Furthermore, we take the textbook normalization of the SU(N) generators We consider the above theory in an expansion around the Minkowski vacuum and identify h µν as the graviton. We choose the Feynman resp. de Donder gauge fixings The form of the ghost Lagrangian is not needed since we exclusively work at tree level. The interested reader is referred to [28]. The graviton is understood to carry momentum P , polarization ε αβ = ε α P ε β P and Lorentz indices αβ. All momenta are ingoing. Gluons are denoted as wiggly lines, gravitons as double wiggly lines, fermions as solid arrows and scalars as dashed arrows.
where we defined The interaction vertices with at most one graviton read [28] On-shell and effective Feynman rules We construct on-shell Feynman rules by contracting one external graviton or gluon with its polarization tensor and using the conditions ε P · P = ε 2 P = 0. Graphically we denote the on-shell leg with a triangle. We start with the two gluon one graviton vertex: where we defined Moreover we used where the first term in the bracket is p 2 = 0. We write this as And identify Note that in any tree level computation of a scattering amplitude the term W µaµ b (p a , p b ) will vanish due to on-shell Ward identities. It is now straightforward to obtain