Simple encoding of higher derivative gauge and gravity counterterms

Invoking increasingly higher dimension operators to encode novel UV physics in effective gauge and gravity theories traditionally means working with increasingly more finicky and difficult expressions. We demonstrate that local higher derivative supersymmetric-compatible operators at four-points can be absorbed into simpler higher-derivative corrections to scalar theories, which generate the predictions of Yang-Mills and Gravity operators by suitable replacements of color-weights with color-dual kinematic weights as per Bern-Carrasco-Johansson double-copy. We exploit that Jacobi-satisfying representations can be composed out of other Jacobi-satisfying representations, and show that at four-points only a small number of building blocks are required to generate the predictions of higher-derivative operators. We find that this construction saturates the higher-derivative operators contributing to the four-point supersymmetric open and closed-string tree amplitudes, presenting a novel representation of the four-point supersymmetric open string making this structure manifest, as well as identifying the only four additional gauge-invariant building blocks required to saturate the four-point bosonic open string.

Gravitational quantum scattering amplitudes-the invariant quantum evolution of what distance means in space and time, consistent in the classical limit with Einstein's General Relativity (GR)-are much simpler than expected. This simplicity can be traced to the fact that their perturbative dynamics are completely encoded through a double-copy structure [1][2][3] in the predictions of much simpler gluonic or gauge theories. In turn, these gauge theory predictions are strongly constrained by a similar structure relating kinematics and color-weight, entirely hidden in any standard ways of writing their actions.
While Yang-Mills (YM) theory is famously renormalizable in four dimensions, it ceases to be in higherdimensions, requiring a completion in the UV. It is currently an open question as to whether any fourdimensional (pointlike) quantum field theory of gravity is perturbatively finite. The most promising case, maximally supersymmetric supergravity, is a subject of much current research exploiting double-copy[2, [4][5][6][7][8][9][10][11][12][13][14]. Independent of perturbative finiteness, it is very possible that new physics in the UV could necessitate higher-order corrections, whose predictions using traditional methods are often exhaustive to produce. Our main result is that the predictions due to higher-derivative local gauge and gravity operators -encapsulating novel UV physics -can also be incredibly simple because of this very same doublecopy structure.
Recent work has shown that at tree-level both the supersymmetric and bosonic open string amplitudes admit field theory double-copy descriptions [15][16][17][18][19][20], pulling the higher-derivative corrections to a putative effective scalar bi-colored theory, encapsulating all order α ′ corrections, called Z-theory. Inspired by the existence of Z-theory amplitudes, as a proof of concept, here we consider a bootstrap approach, asking simply what predictions are consistent with unitarity, double-copy structure, gauge invariance, and locality.
We find that all tree-level string corrections to supersymmetric YM and GR at four-points follow from simple field theory considerations. These corrections can be obtained through a simple composition rule that combines color-dual numerators into more complex numerators with the same algebraic properties, promoting colorweights to carry the higher-derivative corrections. This naturally introduces a new type of numerator, mixing color and kinematic factors to satisfy adjoint-type relations in concert. One might expect many possibilities even at four-points, yet we find only three distinct color building blocks. We see that they generate all four-point single-trace gauge-theory predictions consistent with maximal supersymmetry. Concordant corrections to maximal supergravity are even simpler, requiring only permutation invariant kinematic factors.
These considerations only specify the analytic form of higher-derivative corrections. One may choose to fix their coefficients by assuming the asymptotic uniqueness of the Veneziano amplitude (c.f. Ref. [21][22][23]). Similar ideas apply to the open bosonic string, where just five different gauge invariant building blocks, dressed with the same simple modified color factors, are sufficient to generate the full low energy expansion of YM + (DF ) 2 theory [20]. This discussion complements and explains results noted in Ref. [24], demonstrating explicitly through coefficient matching that the low energy four-field effective actions of super and bosonic strings, governed by Z-theory, are highly constrained by field theory color-kinematics duality.
I. STRIATING BY f abc STRUCTURES.
We will briefly review adjoint-type color-kinematic representations at four-points. We refer the interested reader to Ref. [3] for a detailed treatment. Yang-Mills amplitudes can be expressed in terms of cubic (trivalent) where s, t, u are four-point momentum invariants following an all outgoing convention as s = s 12 = (k 1 + k 2 ) 2 , t = t 23 = (k 2 + k 3 ) 2 , and u = −s − t = s 13 = k 1 + k 3 ) 2 . The color-weights c g are simple combinations of adjoint color-generators and the kinematic weights n g are Lorentz products between external momenta and polarization vectors. We emphasize that both the color weights and the kinematic weights satisfy Jacobi identities and antisymmetry around vertex flips: As such this is called a color-dual representation of Yang-Mills, specifically manifesting an adjoint-type doublecopy structure. We will parameterize such adjoint type graph weights c g and n g in terms of the the three Mandelstam invariants as follows, The pattern to recognize is j(a|bc|d) = j(s ab , s bc , s cd ). Gauge invariance is maintained by the fact that the color-weights, c g , satisfy anti-symmetry and Jacobi identities. As per double-copy construction, we can replace the color weights with kinematic weights that also satisfy Jacobi identities and anti-symmetry to generate gravity amplitudes invariant under linearized diffeomorphism: Details of state identification for a variety of (super)gravity theories can be found in Ref. [3], but the important point to realize is that the kinematic weights n g and n g need not come from the same theory, and indeed the double-copy construction promotes any global supersymmetry of the kinematic weights into a local supersymmetry of the gravitational amplitude. It will simplify our discussion to introduce the notion of gauge-invariant ordered amplitudes. Let us cast the color-weights in Equation (1) to a minimal basis using the relations in Equation (2), say by eliminating c t and collecting in terms of c s and c u . This results in collections of gauge invariant kinematic terms, called ordered or partial amplitudes, with distinct color-basis prefactors: = c s A YM (s, t) + c u A YM (u, t) .
As the A YM (s ab , s bc ) = A YM (a, b, c, d) = n(a|bc|d)/s ab + n(d|ab|c)/s da appear in the full amplitude with coefficients that are independent color basis elements, they must themselves be individually gauge invariant. Expressing these ordered amplitudes in a basis of kinematic weights n g , say by eliminating n u via Equation (2), demonstrates that the distinct color orders are intimately related. Indeed one immediately identifies the permutation invariant quantity: with the identification of the permutation invariant product stA YM (s, t) = (stu)A YM (s, t)/u a simple consequence. This is the lowest multiplicity manifestation of the (n − 3)! (or the so called BCJ) ordered-amplitude relations [1]. We will first be concerned with how we can express higher-derivative corrections to Yang-Mills by only modifying the color-weights in a manner consistent with this adjoint-type structure. Let us now introduce the notion of a Jacobi identity satisfying composition. If we have functional maps j(a, b, c) and k(a, b, c) that satisfy Jacobi identities (X s = X t + X u ) and antisymmetry X(a, b, c) = −X(a, c, b), then we can define a new antisymmetric and Jacobi-satisfying representation n s as a composition of j and k by At four points, it is natural to ask if there exists a scalar color-dual function that is only linear in the Mandelstam invariants. Indeed one does, which we will refer to as the simple scalar numerator: This corresponds to a scalar charged in the adjoint mediated by a massless vector, e.g. with interaction term f abc A µ (∂ µ φ)φ. What happens when we compose the simple scalar with itself? We find the Jacobi-satisfying kinematic numerator associated with the NLSM, j nl s = s(u − t) = sj ss s ∝ J (j ss , j ss ) .
Any further compositions between j ss and j nl only differ from these numerators by appropriate powers of permutation invariant combinations of the Mandelstam invariants, (s 2 + t 2 + u 2 ) and (stu). It is perhaps not surprising that a gauge-invariant color-dual kinematic numerator representation for Yang-Mills can be written [25] as: The most straightforward modification of the colorweights that preserves anti-symmetry and Jacobi involves simple products of permutation invariant scalar combinations: where we introduce a dimensional parameter α ′ to track mass-dimension. This results in an ordered s-t channel scattering contribution proportional to: As all such modifications result in manifestly permutation invariant scalings of the bi-adjoint ordered amplitude, all field theory relations are automatically preserved. One might be surprised by the appearance of the simple scalar numerator appearing in the expression above, but recall that stA(s, t) must be permutation invariant for 4-point ordered amplitudes that satisfy the (n − 3)! BCJ identities. A natural way of generating permutation invariants given adjoint-type structures like c g is to take a sum over products with other adjoint-type structures. It is straightforward to see that yielding the perhaps more familiar expression for stA bi−adj (s, t). This trivial modification is not the only consistent modification to adjoint color weights. We are free to consider terms that compose scalar kinematic weights and adjoint color-weights to result in consistent modifications. Let us first explore composition with the simple scalar numerators, j ss , Such weights result in an ordered (s, t) channel scattering contribution proportional to: These are completely valid adjoint-type partial amplitudes-satisfying both KK and BCJ relationsthat are quite distinct from those given in Equation (14). Consider now compositions between adjoint colorweights and NLSM numerators, J (c, j nl ). It is clear they do not represent additional operators, being completely redundant with the amplitudes given byĉ X,Y +1 g in Equation (14) for on-shell four-point amplitudes.
In all of the above, for local four-field operators, we have a restriction on the power of the permutations symmetric (stu) term: we must require X ≥ 1 to avoid any cubic propagators in the resulting amplitude. It turns out that with the two distinct building blocks, c X,Y andĉ X,Y,ss , we can build any higher derivative fourpoint amplitude A HD that will involve c s , c t , and c u and that satisfies the (n − 2)! and (n − 3)! field theory relations. Namely stA HD must be permutation invariant in all channels. Such a permutation invariant function of the adjoint color-weights c g and Mandelstam invariants, can always be written in terms of a crossing-symmetric adjoint-type polynomial function in Mandelstam invariants j(a, b, c) as follows: Any such j s = j(s, t, u) can be written as a superposition of simple-scalar and NLSM numerators, schematically, using the following general decomposition, a fact easily verified by recalling the definitions of j ss (s, t, u) = (u − t) and j nl (s, t, u) = s(u − t). What is particularly notable is that their coefficients in Equation (18) are each permutation invariant under all S 3 (s, t, u) by virtue of the adjoint-type properties of j(a, b, c). One might be concerned about potential poles, but it is straightforward to see that both must be local expressions. The simplest argument is to realize b = c is always a zero of the polynomial j(a, b, c) by virtue of antisymmetry, and thus (b − c) must be a factor of j(a, b, c).
taking care of all divisors except for (s − t). But s = t is manifestly a zero of each numerator in these expressions, and thus the remaining divisor (s − t) must be a factor of both. In summary we see that our two building blocks can reproduce every scalar polynomial adjoint-type numerator involving c s , c t , and c u . We have not yet exhausted all potential local operators. Namely, we have not yet considered the possibility that the color-weight information may not be in the adjoint, and could itself be permutation invariant, as per the symmetric symbol: This could be dressed with color-dual scalar weights and permutation symmetric kinematics to generate the predictions of additional distinct operators. Due to redundancy between building blocks, we need only consider adding to our repertoire of globally consistent building blocks at four-points the contributions of scalar weights of the non-linear sigma model, where, due to the propagator canceling prefactor in every j nl g , we are now free to include the cases where X ≥ 0. These building blocks result in (s, t) ordered scattering amplitudes proportional to: again manifestly satisfying the usual field-theory relations by construction. Putatively distinct weights proportional to the simple scalar numerator, can be seen to be redundant, building equivalent amplitudes to those generated fromĉ in Equation (22).
With only three building blocks:ĉ we have exhausted all single-trace higherderivative modifications of color-weight, and so we find that the generic form of such single-trace higherderivative corrections to Yang-Mills to be encapsulated by: where X, Y are integers, and the a i are free parameters encoding distinct operator Wilson-coefficients. All local higher-derivative SUSY-compatible gauge corrections to the four point tree-level amplitude, consistent with adjoint representations, will be given by suchĉ simply as: In Tab. I we provide corresponding higher derivative scalar and gauge operators associated with the variouŝ A through mass dimension four. The supersymmetric open string is a known UV completion to (super) Yang-Mills. It is a fair question to ask whether our simple color-modified building blocks for higher-derivative amplitudes are sufficient to capture the open superstring low-energy expansion [18,26,27]. We will see that the answer is yes, but will delay this discussion until after we have introduced the permutation invariant striation in the next section.
We have only, thus far, modified color-weights. As global supersymmetry is satisfied by the unmodified Yang-Mills kinematic weights, this exhausts a discussion consistent with the global supersymmetry inherent in YM amplitudes at tree-level. Composition between the above scalar weights and kinematic Yang-Mills weights always satisfies Jacobi, but it is easy to see that the only composition that maintains gauge invariance is redundant with the trivial modification of color-weights with permutation invariant prefactors that we have already considered in Equation (13).
What about non-supersymmetric operators that can be applied to gauge theory? The same discussion carries through, essentially unchanged, by replacing n Y M with other gauge invariant 4-point adjoint color-dual graph weights not trivially related to n Y M , such as the n F 3 numerator weights identified in Ref. [27], or more broadly with the type of n (DF ) 2 weights responsible for the ordered amplitudes defined in the context of the bosonic open string as per Ref. [16,20]. We will return to this discussion in a later section discussing the open bosonic string where we make it clear that only four additional building blocks are required.
Next let us consider counterterms to gravity consistent with an adjoint double-copy representation. From a color-kinematic perspective, it is natural to consider replacing the color weights with Yang-Mills kinematic weights. Let us first treat the familiar replacements of the f abc based color weights c g → n YM g , so that and similarly forn (X,Y,ss) s . In the case ofn (X,Y ) s we encounter no obstacle in the resulting ordered amplitudes; indeed, quite simply one finds A bi−adj (s, t) → A YM (s, t) in Equation (14). Gauge-invariance, however, immediately excludes the amplitudes generated fromn (X,Y,ss) s , for essentially the same reason that we could only include trivial (permutation invariant) higher-derivative modifications to n YM . We are left only with the question as to what permutation invariant quantity to replace d abcd with inĉ (X,Y,d,nl) s to generate gravity amplitudes without introducing unphysical poles. There are two distinct candidates: stA YM (s, t) and A YM (s, t)/u. It turns out that both choices are redundant with the amplitudes generated byn (X,Y ) s , leaving us with only the trivial building blockn (X,Y ) s for adjoint higher-derivative operators consistent with local supersymmetry for N > 4 in fourdimensions. As such, we have a simple argument that the only such higher derivative local operators available to gravity at 4-point give predictions simply proportional to the 4-point graviton amplitude: As these modifications amount to simple factors of permutation invariant kinematics, all of these higherdimensional corrections are consistent with local supersymmetric Ward identities. For operators restricted to on-shell local supersymmetry consistent with N ≤ 4, one can also consider similar arguments where at least one copy has the adjoint color-weights replaced with nonsupersymmetric gauge-invariant adjoint-type graph ones. Such an example is F 3 , whose double-copy to gravity was considered in [27,28]. These are of particular interest because of the possibility of removing anomalies in associated supergravity theories [29][30][31].

II.
STRIATING BY d abcd STRUCTURES.
In addition to admitting an adjoint-type double-copy structure, these corrections admit an alternative decomposition into permutation invariant quantities. This is not the first opportunity to see that a single amplitude may admit multiple distinct double copy descriptions, depending on which algebra color-kinematics duality makes manifest. Indeed, the dimensional reduction of fourdimensional supergravity theories to three-dimensions admits both the adjoint-type double-copy construction of three-dimensional super-Yang-Mills amplitudes, as well as the three-algebra type double-copy construction of BLG amplitudes [32][33][34]. As we see here, the ability to striate along different algebras may be quite general.
Consider the full four-point amplitudes for Yang-Mills and Gravity expressed in terms of Jacobi-satisfying weights in Eqs. (1) and (6). Solving for n i in terms of ordered YM amplitudes and for c i in terms of ordered biadjoint scalar amplitudes leads to the manifestly permutation invariant representations of the four-point Yang-Mills and gravity amplitudes as: All elements (stA bi−adj ), (stA YM ), and (stu) are manifestly permutation invariant, and indeed are recognizable as proportional to full four-point amplitudes of the known theories NLSM, Born-Infeld, and Special Galileon, respectively: −A Spec.Gal = stu = stÃ NLSM (s, t) .
We have introducedÃ NLSM to emphasize fixing the more standard normalization for chiral-pion numerators relative to Equation (12):ñ NLSM g = 1 3 j nl g . One obvious feature of permutation invariant striations is that full amplitudes for theories serve as natural building blocks. It is, for example, clear that the only permutation invariant higher-derivative modification to A (GR) in Equation (29) that does not affect gauge invariance is to simply include products of permutationinvariant scalar functions. The permutation invariant modifications to the gauge theory are generated by promoting stA bi−adj to sums over combinations of the building blocks stÂ (X,Y ) , stÂ (X,Y ) ss , and stÂ (X,Y ) d,nl introduced earlier, and can all be interpreted as the full amplitudes of various higher-derivative corrections. We note in passing that this permutation invariant color-dual discussion admits the following whimsical departures from the typical: "GR∼YM 2 " slogan at four-points: III. STRING AMPLITUDES AT FOUR POINTS.
We are now prepared to find our building blocks, resummed over all orders in α ′ , in the tree-level fourpoint open supsersymmetric string amplitude. This can be interpreted as answering a field theory question of how atoms of prediction composed into higher-derivative scalar corrections can be made consistent with a UV completion involving massive spin intermediaries of a particular form. We start by recognizing the open superstring amplitude as the field theory double copy between Chan-Paton dressed Z-theory [15,17] At four-points, this amplitude can be represented in a permutation invariant color-dual form as: where all supersymmetric Ward identities are satisfied by virtue of operations on the Yang-Mills factor stA YM (s, t), and A Z (s, t) is the field-theoretic (s-t) partial amplitude of Chan-Paton dressed Z-theory encoding all orders of α ′ corrections. We can build [stA Z (s, t)] starting from the bi-ordered doubly-stripped partial Z-amplitude Z 1234 (s, u), where the subscript ordering refers the Chan-Paton trace ordering, and the parenthetical ordering obeys field-theory relations, We form the field-theory permutation invariant for this Chan-Paton ordering by simply taking the product: s u Z 1234 (s, u) = s t Z 1234 (s, t). By exploiting monodromy relations to permute the subscript orderings, we can generate the Chan-Paton dressed expression, required in Equation (33), stA Z (s, t), stA Z (s, t) = σ∈S3(2,3,4) 3,4) Tr [1σ] sin(πα ′ s 1,σ(3) ) sin(πα ′ s 1,3 ) (stZ 1234 (s, t)) , (35) where we use Tr [ρ] to denote Tr[T a ρ(1) T a ρ(2) T a ρ(3) T a ρ(4) ]. The above is invariant under exchange of any channels, and by expressing the Chan-Paton trace factors in terms of c s , c t , c u , and d abcd (and explicitly symmetrizing as appropriate), we find the following simple color-dual permutation invariant form for the full Chan-Paton dressed open superstring, where the manifestly permutation symmetric Γ {s,t,u} corresponds to a series of higher mass-dimension combinations of Mandelstam invariants with coefficients responsible for familiar ζ contributions to the low-energy expansion, Z adj contains all expressions involving Chan-Paton trace combinations c s , c t , or c u , and Z sym contains all terms proportional to d abcd . These are given as follows: Note that even Z adj is manifestly permutation invariant, as the z g satisfy anti-symmetry and Jacobi-identities in concordance with c g . The z g take a particularly simple form, with and the rest following from relabeling: z t = z s | s↔t and z u = z s | s↔u . In this form, all the coefficients for c (X,Y,d,nl) s may be easily identified already from the lowenergy expansion of Z sym . The remaining two building blocks only require a little teasing out from Z adj , which may be achieved by using Equation (18) to rewrite the z g . This allows us to separate z s into terms proportional to j ss s = (u − t) , and terms proportional to j nl s = s(u − t), (41) where S p denotes sin(πα ′ p), and the Z bi−adj and Z ss higher derivative corrections are given as follows: The j ss terms can be seen to correspond within Z adj to corrections of the form stÂ X,Y (s, t) (c.f. Equation (14)), and j nl terms are likewise associated with stÂ (X,Y ) ss (s, t) (c.f. Equation (16)). Both Z bi−adj and Z ss are manifestly permutation symmetric and local in all orders of an α ′ → 0 expansion, meaning they are completely spanned at any mass-dimension, MD, by a basis in (stu) X and (s 2 +t 2 +u 2 ) Y such that 3X +2Y = MD as per our building blocks. We have therefore exposed within the 4-point open superstring the three unique Jacobi-identity satisfying modifications to the color-weights of Yang-Mills. This can now be written in terms of stA Z (s, t) = stA bi-adj (s, t)f bi (s, t, u)+ stA ss (s, t)f ss (s, t, u) + d a1a2a3a4 f d (s, t, u) , (44) where the higher-derivative expressions through mass dimension six are given by: We use σ 2 and σ 3 to denote (s 2 + t 2 + u 2 ) and (stu) respectively. The individual a X,Y coefficients that fix Equation (25) to the low-energy expansion of the open superstring up through mass-dimension thirteen are given in supplementary Tab. II, and through mass-dimension sixteen in a machine readable auxiliary Mathematica file.
We now turn to the open bosonic string amplitude at four-point tree-level. It was shown in Refs. [16,20] that this amplitude also obeys a field theoretic adjoint-type double-copy description with Z amplitudes as follows: (open bosonic string) = (Z-theory) ⊗ YM + (DF ) 2 , where (DF ) 2 is a massive higher derivative YM theory, compatible with the usual BCJ relations but in violation of supersymmetric Ward identities. It is straightforward to identify that only four new higher mass-dimension gauge-invariant adjoint-type vector building blocks are required to build this amplitude upon dressing with the permutation invariant objects σ 2 and σ 3 , compactly encoded in the following denominator: Machine readable expressions for the four new gaugeinvariant building blocks of these amplitudes are included in an auxiliary Mathematica file.

IV. DISCUSSION
We have shown that at four-points there are simple building blocks, manifest in two algebraic striations of four-point scattering amplitudes, that encode higher derivative corrections to effective gauge and gravity theories. We have demonstrated that these building blocks can be exposed to all orders in α ′ in the open supersymmetric and bosonic string amplitudes. Preliminary exploration confirms [35] that the pattern of identifying color-dual building blocks that admit composition continues at higher-multiplicity, a topic that merits detailed study. Gaining all-multiplicity control would mean that, through unitarity methods, one could build relatively easy to construct higher loop-order scalar integrands that trivially recycle, through double copy, known gauge and gravity integrands to their higher-derivative corrections. It is noteworthy that, at four-points, compatibility with adjoint double-copy structure involving Yang-Mills building blocks ensures compatibility with supersymmetry.
It is worth remarking on a striking fact that Eqs. (21) and (22) make manifest. Consider the SUSY-compatible F 4 amplitude: It was observed [27] that the kinematic factor accompanying individual trace terms, s t A YM (s, t), does not satisfy the (n − 3)! field theory relations associated with adjoint color-kinematic structure. It is possible to misconstrue this result to show that F 4 is incompatible with color-kinematics duality in some broad sense, a question we can address. We should emphasize two points clear now from a perspective informed by many examples [32,[36][37][38] of non-adjoint color-kinematics duality satisfying representations. First, even as written, there is a manifest completely-symmetric color-kinematics duality at work for F 4 : both the color term, d abcd , and the kinematic (Born-Infeld) term, stA YM (s, t), are invariant under all permutations. This seemingly trivial duality even has teeth: there is an associated double-copy construction. Replacing the d abcd term with the permutation invariant kinematic weight stA YM (s, t) generates the gravitational R 4 amplitude consistent with maximal local supersymmetry: with the relationship to the four-graviton scattering amplitude clear from comparison to Equation (29).
Second, we learn from Eqs. (21) and (22) that both A F 4 and A R 4 also manifest a non-trivial adjoint colordual double-copy structure at four-points: The key to realizing this adjoint-type color-dual representation is to allow both color and scalar kinematics to conspire to satisfy the adjoint algebraic relations within the same adjoint-type color-dual weight-a lesson driven home top-down by abelian Z-theory [17], and constructively presented here. Not all effective particles are massless, and not all such particles are single-trace in the adjoint (c.f. QCD, Einstein-Yang-Mills, and the standard model more generally), yet many admit color-dual representations [36][37][38][39][40]. It will be fascinating to see if such simple constructive building blocks are available for higher-derivative corrections to their predictions. Even in the adjoint, we have only focused here on structures involving gaugekinematics in at least one copy. Generalizations of these building blocks should be relevant to exploring higher derivative corrections to more phenomenological effective field theories [41][42][43][44].