Revealing the Landscape of Globally Color-Dual Multi-loop Integrands

We report on progress in understanding how to construct color-dual multi-loop amplitudes. First we identify a cubic theory, semi-abelian Yang-Mills, that unifies many of the color-dual theories studied in the literature, and provides a prescriptive approach for constructing $D$-dimensional color-dual numerators through one-loop directly from Feynman rules. By a simple weight counting argument, this approach does not further generalize to two-loops. As a first step in understanding the two-loop challenge, we use a $D$-dimensional color-dual bootstrap to successfully construct globally color-dual local two-loop four-point nonlinear sigma model (NLSM) numerators. The double-copy of these NLSM numerators with themselves, pure Yang-Mills, and $\mathcal{N}=4$ super-Yang-Mills correctly reproduce the known unitarity constructed integrands of special Galileons, Born-Infeld theory, and Dirac-Born-Infeld-Volkov-Akulov theory, respectively. Applying our bootstrap to two-loop four-point pure Yang-Mills, we exhaustively search the space of local numerators and find that it fails to satisfy global color-kinematics duality, completing a search previously initiated in the literature. We pinpoint the failure to the bowtie unitarity cut, and discuss a path forward towards non-local construction of color-dual integrands at generic loop order.


Introduction
Since the turn of the century, our understanding of the S-matrix and its concealed structures has expanded dramatically.At the heart of this progress is the mantra that physical observables are simpler when studied on-shell [1][2][3].However, while inserting on-shell states certainly offers a path towards taming the factorial growth of Feynman diagrams, it also obscures the off-shell simplicity at the heart of many quantum field theories.A shining example of this hidden structure is the duality between color and kinematics [4][5][6], which states that the kinematic numerators of gauge theory amplitudes can be rearranged to obey the same algebraic identities as the constituent color factors.When this duality is realized globally off-shell, it dramatically decreases the combinatorial complexity posed by integrand construction, and offers a path towards efficient assembly of quantum gravity integrands directly from simpler gauge theory building blocks via the double copy [4,5].
Despite the tremendous success in leveraging this duality to compute gauge and gravity observables to high orders in perturbation theory [7][8][9], color-kinematics remains a conjecture at loop level.Indeed, there are many examples in the literature where identifying color-dual representations beyond one-loop has posed a formidable challenge [10][11][12][13].In this work, we use the nonlinear sigma model (NLSM) and Yang-Mills, two theories proven to permit colordual representations at tree-level [14,15], as case studies to advance our understanding of the kinematic algebra at the multi-loop level.
We begin with an overview of color-kinematics duality in section 2, and define the notion of "globally" color-dual integrands in section 2.3.We then construct a manifestly color-dual theory in section 3, which generates D-dimensional color-dual n-point numerators at both tree-level and one-loop.When plugging in appropriate on-shell states, we find that these numerators underpin color-dual representations of self-dual Yang-Mills, NLSM and Chern-Simons theory through one-loop.The construction of this theory, which we dub semi-abelian Yang-Mills, relies on isolating the cubic sector of pure Yang-Mills amplitudes from the fourpoint contacts that are needed to fully realize non-abelian gauge symmetry.Despite the potency of this theory through one-loop, the construction runs into an obstruction at twoloop, which we describe in section 3.3.
Faced with this obstruction, we study the multi-loop sector of both NLSM and Yang-Mills in section 4 using an ansatz-based color-dual bootstrap.In contrast to much of the available literature on color-dual integrand construction at two-loop [10,11,16], our results are completely agnostic to the spacetime dimension; all the polarizations and momenta appearing in our construction will remain formally D-dimensional.This makes our methods and results particularly well suited for algorithmically extracting rational terms from the D-dependence of dimensionally regulated loop momenta.The ansatz approach to constructing numerators generally results in an explosion of terms, but in the case of scalar theories there is an added difficulty because the linear equations become dense.We were able overcome this barrier by employing a custom solver, FiniteFieldSolve, which will soon be made public.
With this solver, we are able to compute a two-loop integrand for NLSM that globally manifests the duality off-shell.This represents a new benchmark for color-dual representations in non-supersymmetric gauge theories.However, in sharp contrast, we find that Yang-Mills does not permit a globally color-dual representation, even when considering the most general polynomial ansatz of Lorentz covariant kinematics. 1 Concretely, the failure point can be pinpointed to a conflict between the "bowtie" cut and the following Jacobi triple: We comment on the implications of this finding in section 5 and discuss alternative approaches where one must relax field theoretic locality in order to accommodate the construction of globally color-dual integrands at generic loop order.

Background
To set the stage, in this section we provide an overview of color-kinematics duality -both the on-shell and off-shell constructions.We then detail a color-dual bootstrap approach for constructing integrands in theories for which the fully off-shell kinematic algebra has not yet been identified, like Yang-Mills and NLSM.

Color-dual Amplitudes
Color-kinematics duality is a statement about the off-shell graphical simplicity that manifests redundancy in the on-shell observables of certain field theories.Ignoring factors of i and the coupling constant, the starting point to realize the duality is to express generic n-point L-loop amplitudes in a cubic graph representation: Here the spacetime dimension is D.Even though the underlying theory may have quartic (or higher multiplicity) vertices, like YM and NLSM, it is possible to multiply and divide by propagators so that the sum runs over the set of purely cubic graphs, Γ. Overcounting is removed by the internal symmetry factor S Γ , which tracks the number of graph symmetries with external legs held fixed.The color factor C Γ is the string of structure constants f abc associated with the graph Γ,2 the denominator d Γ is the product of propagators associated with the graph Γ, and the numerator N Γ is all of the remaining kinematic factors including dot products of momenta and polarization vectors.In this form the color factors C Γ are not independent and are related to one another via a set of Jacobi identities that look schematically like where i, j, and k are three graphs.Example Jacobi relations are presented in detail in section 2.3.The duality between color and kinematics is the statement that that there exists a way to rearrange factors between the various kinematic numerators N Γ such that the numerators obey the same Jacobi identities as the color factors If color-kinematics duality can be achieved at the off-shell level, then the kinematic numerators should satisfy an algebra just like the color factors.In theories where the kinematic algebra is known, it is often the algebra of volume preserving diffeomorphisms [24][25][26][27][28].When the kinematic algebra is not known explicitly, the kinematic numerators are typically constructed from an ansatz.As an example, the result of such a calculation for four-point NLSM scattering at tree level produces the numerators ) where the coupling has been normalized away.These numerators sum to zero by explicit calculation.Note that the s-channel numerator contains an explicit factor of s that cancels with the propagator and the same goes for the t-and u-channels.The net result is that the amplitude is a local function, just as one would expect for the pion four-point tree amplitude.Once color-dual numerators N Γ have been found, they can be used to replace the color factors (C Γ → N Γ ) in a different amplitude to produce the corresponding amplitude in a new theory, This is known as the double-copy construction [4][5][6].Importantly, the ÑΓ do not need to respect color-kinematics duality.The prototypical example of the double-copy is that YM double copied with YM results in gravity where the gauge invariance of each separate gluon produces the diffeomorphism invariance of the graviton.The double copy can be proven at tree level using the Kawai-Lewellen-Tye (KLT) relations but there is significant evidence that the double copy persists to all loop orders [7-9, 29, 30].In addition to pointing at undiscovered structure hidden in the Lagrangian for gravity, the double copy is immensely useful at a utilitarian level since it is much more efficient at producing M L n than traditional methods like Feynman rules.
Many of the one-loop results presented in section 3 are informed by the tree-level doublecopy so we will review the tree case here.The color structure of tree-level amplitudes is particularly simple.A fully color dressed tree amplitude A n can be decomposed in terms of the trace basis where T a are generators of the gauge group and, because of the cyclicity of the trace, one of the legs is held fixed so that the sum runs over the (n − 1)! permutations of the remaining external legs.The coefficients of the color factors are the color-ordered partial amplitudes A n [...].Any tree-level color factor can be converted into a linear combination of Del Duca-Dixon-Maltoni (DDM) half-ladder color factors associated with the graph by repeated application of the Jacobi identity [31].The fully color-dressed tree amplitude can then be re-expressed as with legs 1 and n fixed.Since the sum only contains (n − 2)! terms, there must be additional relations -the Kleiss-Kuijf (KK) relations [32] -amongst the partial amplitudes appearing in the right-hand side of eq.(2.8).The KK relations are generic to any theory with purely adjoint particles.Color-kinematics duality further implies that the KK basis is overcomplete and can be reduced to a basis of (n − 3)! amplitudes via the fundamental Bern-Carrasco-Johansson (BCJ) identities [4,14] n−1 i=2 In terms of constructing explicit color-dual solutions at tree level, it is enough to specify the numerator of the half-ladder because any diagram can be reduced to a half-ladder through successive applications of the Jacobi identity.At one-loop, every diagram can be related to one topology as well, in this case the n-gon, which can be understood as a forward limit of the half-ladder.Two-loop calculations play an important role in our understanding of colorkinematics duality because this is the first order where multiple basis graphs are required, at least for a generic theory.We refer the reader to Ref. [6] for more background on the topic.

Color-dual Lagrangians
While the BCJ relations are an on-shell statement about the color-dual nature of scattering amplitudes, one might aspire to trivialize the duality by encoding it in an off-shell Lagrangian of the theory.The double-copy construction of eq.(2.7) would suggest a manifestly cubic Lagrangian of the form where V abc and W µνρ are cubic Feynman rules mixing field operators O a µ indexed by quantum numbers a and µ.The Feynman rules for this theory are simply: We have introduced P ab V and P µν W projection operators to encode non-local kinetic structure that could in principle participate in the construction, as is the case for DF 2 theory [17].Generally, these quadratic dressings are simply delta functions, P ab V = δ ab , or flat space metrics, P µν W = η µν .As written, the theory will be color dual if the off-shell cubic vertex is antisymmetric and satisfies the Jacobi identity.Specifically, antisymmetry : Jacobi identity : where, using the Feynman rules of eq.(2.14), the bracketed expression is the half-ladder numerator of the following diagram, and likewise for the V abc dressing.Off-shell descriptions of color-dual theories are rare, and only a few examples are available in the literature [24][25][26][27][33][34][35].Typically, when one goes about constructing color-dual theories directly from a set of cubic interactions, higher point Jacobi and antisymmetry constraints require introducing additional operators [36,37] or propagating fields [34].Indeed, as we will show in section 3, Yang-Mills requires introducing a cubic two-form interaction in order to manifest the kinematic algebra just at four-point.Absent information about the kinematic algebra, one can make progress in the construction of loop-level color-dual amplitudes by employing a color-dual bootstrap.

Color-dual Bootstrap
When an off-shell color-dual Lagrangian is not known, the loop-level amplitude integrand must be constructed from an ansatz.The four conditions we impose on an integrand ansatz are summarized here and elaborated on below. 3After imposing the four constraints below, any remaining coefficients in the ansatz encode "generalized gauge freedom", meaning that the choice of coefficients does not affect the physical integrand or its double copy.
1. Off-shell Locality: Diagram numerators are polynomials in momenta and polarizations, e.g., the diagrams only have poles given by the propagators of the diagrams.

Color-kinematics duality:
The numerators obey the same Jacobi identities as the color factors.This implicitly requires that the integrand is expressed in terms of purely cubic diagrams.
3. Graph symmetries: A diagram's numerator is only a function of the diagram's topology and labeling.Furthermore, the diagram numerators are invariant under the automorphisms of the diagrams including signs that compensate for color-factor sign changes.
4. On-shell Unitarity: The cuts of the ansatz must reproduce the physical unitarity cuts of the theory when internal momenta are taken on-shell.
In order to clarify general statements, the following subsections include examples from a four-point two-loop integrand for some generic color-dual theory like YM.This is the simplest case that demonstrates the full machinery of the color-dual bootstrap.All tree and one-loop processes are generated by single basis diagrams (the half ladder and n-gon respectively) so they lack examples of three-term "boundary" Jacobi relations to be described shortly.Choosing the four-point two-loop integrand also has the advantage that this process appears prominently in this paper.

Off-shell Locality
The kinematic numerators must be D-dimensional, Lorentz invariant, local (polynomial) functions of the graph kinematics with the correct power counting and external states.Locality is physically motivated since Feynman rules produce polynomial numerators.Indeed, the most natural way to guarantee locality and avoid spurious poles is to require the off-shell numerators to be polynomial functions of kinematics.However, insisting on a local numerator is partly a matter of convenience as the ansatz for a rational function would quickly grow out of control without a guiding principle for what terms to include.While these assumptions are well motivated from a physical and complexity standpoint, the literature has featured numerators that violate locality, modify naive power counting through the inclusion of higher spin modes, and break manifest Lorentz invariance [10,11,27,33,34,[38][39][40].These developments suggest that relaxing locality is a viable alternative when the condition proves too stringent for an ansatz.However, simply requiring on-shell locality may suffice.This alternative will be discussed in section 5.1.
Color-kinematics duality Color-kinematics duality is a statement about color factors, kinematic numerators, and the accompanying Jacobi relations.Adjoint-type color factors are associated with cubic graphs, so satisfying color-kinematics duality implicitly requires expressing the integrand in terms of such graphs.As mentioned earlier, this is possible to do even for a theory with quartic (or higher multiplicity) vertices by multiplying and dividing by propagators.In the case of the four-point two-loop example promised above, there are The 14 non-pathological four-point two-loop cubic diagrams.
cubic graph topologies (shown in fig. 1) ignoring tadpoles and bubble on external leg (BEL) graphs.
With the integrand expressed in terms of cubic graphs, it is possible to discuss the main ingredient in color-kinematics duality -the Jacobi relations.The color factors in an integrand obey a set of Jacobi identities.Color-kinematics duality states that there exists a way of writing the kinematic numerators of the integrand so that they obey the same set of Jacobi identities.The Jacobi identities can be divided into two categories: defining relations and boundary relations.Defining Jacobi relations can be used to express any graph in terms of a basis.For example, the double box, crossed box, and penta-triangle are related to each other as follows where the shared propagator is shown in red.Using this relation the crossed box can be written (or defined ) in terms of the other two.In this way, all of the 14 cubic four-point two-loop topologies can be written in terms of a basis of the double box and penta-triangle.
(Any two of the double box, crossed box, and penta-triangle would suffice for a basis but the planar basis is chosen for simplicity in this work.)The numerator of each of the two basis graphs receives its own ansatz and the numerator of any other graph can be expressed in terms of these two.After using the defining Jacobi relations to relate every graph to the basis, any remaining Jacobi relations will be referred to as boundary relations.In other words, boundary relations (indirectly) relate basis elements to themselves.For example, the crossed box numerator is related to itself via the following Jacobi relation where the shared propagator is shown in red as before.Each of the crossed box numerators can be written in terms of the double box and penta-triangle so this Jacobi relation places constraints on the basis numerator ansätze.While defining Jacobi relations simplify how many numerators require ansätze, boundary relations place restrictions on those ansätze.In principle there could be Jacobi relations involving tadpoles and BEL graphs, which would be subtle to handle.One option is to force the numerators of these graphs to zero.However, since these graphs will not contribute to physical unitarity cuts, we take a different approach.These graphs are allowed to have non-zero numerators but any Jacobi relations involving them are simply not imposed or solved.See Refs.[41,42] and appendix C for discussions of BELs in the context of color-kinematics duality.What we have just described is sometimes referred to as "global" color-kinematics duality because the Jacobi identities are solved by a single (global) integrand with off-shell internal lines.In order to perform the double copy it is enough to simply satisfy the Jacobi identities on the generalized unitarity cuts, meaning that each graph can be assigned its own ansatz [13].If a global solution cannot be found, this is one way to enlarge the ansatz, where another option is to incorporate non-local terms in the "numerators".A globally color-dual integrand is desirable because of its simplicity and because a color-dual Lagrangian would naively produce precisely this object.
Graph symmetries Every numerator is required to respect all of the symmetries (or automorphisms) of its associated graph.Graph symmetries are an avatar of Bose symmetry.For a pure ansatz construction it is very reasonable to impose graph symmetries, but colorkinematics duality at tree-level is often achieved at the expense of manifest Bose symmetry [27,33,43,44].In an ansatz construction graph symmetries are imposed for every graph, not just the basis graphs.For example, the symmetries of the crossed box are spanned by, In general, internal edge labels must be tracked as well.Furthermore, care must be taken with the signs in a symmetry relation.To compensate for the antisymmetric color factors, f abc , graph vertices are all totally antisymmetric and hence have signs built into their orientations.When used in conjunction with the Jacobi relations, e.g., eq.(2.18), the symmetry constraints of non-basis graphs still impose conditions on the basis ansatz.In fact, since the symmetry properties of a graph are directly linked with transformations of its color dressing, the resulting constraints can be thought of as a subclass of the boundary Jacobi relations.
On-shell Unitarity The final constraints come from ensuring that the ansatz correctly reproduces the generalized unitarity cuts of the desired theory, thus guaranteeing that the numerators encode the correct physical, theory-dependent information.Generalized unitarity has been reviewed in many places [6,13,[40][41][42][45][46][47][48] and was recently optimized for effective field theories like NLSM [49].Generalized unitarity equates products of on-shell tree amplitudes, encoded via a graph γ, to sums over compatible 4 diagram numerators evaluated on the support of the on-shell conditions and weighted by their uncut propagators where V (γ) are the vertices of γ and E(γ) its edges.The sum over states hides much of the complexity of cut construction: for scalars the process is trivial while spinning states greatly complicate matters (see Refs. [42,50] for recent discussion and detailed examples of evaluating state sums).As an example of the two types of expressions appearing in eq.(2.21), the iterated two-particle cut can be represented as a product of trees via or as a collection of diagram numerators via When drawing cut diagrams, we will use gray blobs as in the left-hand sides of eqs.(2.22) and (2.23) and all drawn legs are assumed to be on-shell.When drawing numerator contributions to cuts, we will not use blobs on vertices, and will use dashed lines bisecting edges to denote cut lines and use colored edges to highlight the uncut propagators.From now on we will leave the cut δ(ℓ 2 ) factors implicit.
In color-charged theories, both sides of the cut can be further decomposed according to the color algebra.For theories charged under the adjoint of SU (N ) via f abc -dressed amplitudes, it is convenient to project onto a specific element of the Del Duca-Dixon-Maltoni (DDM) color basis [31].This is done by only inserting color-ordered amplitudes, a la the right hand side of eq.(2.11) in the product of trees, and restricting the set of compatible diagrams to those whose color factors reduce to the appropriate DDM element when summing over only the uncut color contractions.In this situation, the color becomes an overall factor on the entire cut equality and thus can be ignored.Cuts that are organized in this manner are known as "color-ordered" cuts, in the same sense as color-ordered tree amplitudes.Since the DDM basis (and dual Kleiss-Kuijf amplitude basis [32]) have (n − 2)! elements, one generally needs to evaluate (n 1 − 2)!(n 2 − 2)!... separate color-orderings of the same cut, where n 1 , n 2 ... are the multiplicities of the amplitudes making up the cut.As an example, consider the "bowtie" cut topology -one of the factorization channels of the iterated unitarity cut -that has two topologically distinct color-ordered expansions which serve as a complete basis for the full cut.
Even though unitarity methods aim to generate loop-level data from well-defined on-shell information, the procedure can encounter subtleties.Beginning at one loop, there are classes of cuts that are difficult to properly define due to hidden non-physical singularities.The quintessential example of such a "pathological" cut is the 3-1 two-particle cut at one loop which hides a bubble-on-external-leg (BEL) singularity in which one of the uncut propagators is set equal to an external on-shell propagator.Discussions about how to resolve these types of singularities can be found in Refs.[41,42,48,[51][52][53][54] and appendix C. In addition to the BEL cuts, there is a related class of cuts in which an uncut propagator is set to zero by an internal cut condition, for instance, contains a contribution from . (2.26) Fully resolving both of these problems is beyond the scope of the current work, so we will simply not impose any constraints that require handling these types of cuts.

One-loop cubic construction
Here we will describe why constructing the full Yang-Mills kinematic algebra (outside the self-dual sector) is generically hard, and why the same reason makes NLSM beyond one-loop hard as well.The major obstacle stems from the vector state sum mixing with higher-point contacts.Starting with the Yang-Mills Lagrangian, we can express the YM three-point interaction in terms of the following vertex in Lorenz gauge: 3 This kinematic vertex is antisymmetric in 1 ↔ 2 exchange, and from it we can construct the full Lorenz gauge Yang-Mills vertex by summing over cyclic permutations, Contracting this with the vector state sum, we can see that the cubic Yang-Mills vertex does not satisfy the four-point Jacobi identity, where ε (ij) = ε i • ε j .One way to absorb the remainder is with the four-point contact of eq.(3.1), which preserves non-abelian gauge invariance off-shell.However, this additional term can also be absorbed into the definition of the four-point kinematic numerator in a way that preserves the cubic graph construction as follows, Note that this new definition of the s-channel numerator contains two factors of ε (ij) that contract spacetime indices across the factorization channel, suggesting the inclusion of a spin-2 mode.As such, we use V YM+B to denote the inclusion of an additional two-form, as was studied in Ref. [34] for constructing the NMHV Yang-Mills Lagrangian.After the introduction of the two-form, the numerators now satisfy the Jacobi identity where the new cubic Lagrangian takes the form, Of course, higher multiplicity would likely require further redefinition of the three-point vertex to satisfy Jacobi identities on all internal edges.While introducing successively higher spin states could in principle work at tree level, we will see that it is not consistent with what we find at general loop order.We will comment on this in section 5.2.For now, we will study how to construct one-loop integrands consistent with color-kinematics by isolating the cubic sector of the state sum above.
As can be seen above, the Jacobi identity of eq.(3.4) fails only in terms with two factors of polarization dot products, (εε) 2 .If we restricted ourselves to just considering factors with a single polarization dot product, the Jacobi identity of Yang-Mills would be satisfied off-shell and in arbitrary spacetime dimensions This restriction to the (εε) 1 cubic sector of Yang-Mills is closely related to the MHV decomposition of color-dual numerators [2,[55][56][57].In the MHV sector it is possible to choose the reference vectors so that all dot products of polarization vectors vanish, (εε) → 0, except those involving one of the positive helicity polarizations, (ε + 1 ε − i ) ̸ = 0. We describe this in detail in appendix B. By focusing on (εε) structure rather than specific 4D helicity states, we will be able to construct dimension agnostic color dual integrands at one-loop.
Thus for our approach at one-loop, we aim to build color-dual integrands directly from D-dimensional kinematic factors that are O((εε) 1 ) at tree-level, and O((εε) 0 ) at one-loop.The D-dimensional organizational principle underlying this construction can be understood in terms of a vector amplitude decomposition introduced by one of the authors [58], where we have introduced the shorthand notation,  12), (13), ( 14), ( 23), ( 24), (34)}, and S (1234) = 1, ∆ (1234) = s 13 s 12 . (3.10) We direct the reader to Ref. [58] for further details.The expansion of eq.(3.9) comes with the added advantage of making the transmutation relations of [59] absolutely manifest.By a simple mass-dimension argument [55], we know the tree-level cubic sector of Yang-Mills must be at O((εε) 1 ), while at one-loop, the cubic sector corresponds to O((εε) 0 ) in polarization dot products, When decomposed in this way, the polarization stripped building blocks must obey a set of Ward-identities between different "helicity", or (εε) n , sectors in order for the full amplitude to be gauge invariant, ∆ .
With this in hand, we can construct a Lagrangian description of the of the cubic sector for Yang-Mills and will demonstrate that the resulting amplitudes with manifestly color-dual Feynman rules are equivalent to both SDYM and NLSM through one-loop.

Semi-abelian Yang-Mills theory
As we argued above, if we include factors of (εε) n≥1 at tree-level, then we need to keep the Yang-Mills four-point contact for color-kinematics duality to be restored on-shell.However, the contrapositive is also true -if we omit the four-point Yang-Mills vertex, then we only need terms that contribute to the manifestly color-dual cubic sector of the theory.We call this manifestly color-dual theory semi-abelian Yang-Mills, where and A µ is in Lorenz gauge.We can construct this Lagrangian from eq. (3.1) by keeping the right field strength covariant under U (N ), and making the left field strength gauge covariant under U (1) N 2 .Note that since U (1) covariance is identical to U (1) invariance, the amplitudes of this semi-abelian theory vanish under ε(k) → k, where ε is the polarization of an external Āµ vector.
Here we can think of the abelian vector as a background field that sources on-shell A µ currents.Indeed, as we describe in appendix A, semi-abelian Yang-Mills theory is at the heart of Y Z-theory [33], J-theory [26,27], self-dual Yang-Mills [24], and Chern-Simons theory [25].Specifically, semi-abelian YM is simply a clever reinterpretation of the Lagrangian obtained by integrating an auxiliary field into the J-theory equations of motion.The Feynman rule for the cubic vertex is where the incoming arrow is the abelianized gauge field, and the outgoing arrows the on-shell non-abelian vectors.The propagator is simply We can use the Ward identity of eq.(3.13) to check that the amplitudes are indeed gauge invariant when the abelian polarizations are taken to be longitudinal, ε → k.Furthermore, a simple calculation shows that the four-point correlation function of this theory satisfies the Jacobi identity of eq. ( 2.16) off-shell.This ensures that color-kinematics duality holds to all multiplicity and loop order.We also note that semi-YM does not have any ghosts from nonabelian gauge symmetry that could spoil color-kinematics at loop level.Due to the Feynman rules of this theory, the amplitudes are non-vanishing only at tree-level and one-loop.We demonstrate the implications of this property in section 3.3.
As noted above, we can select out the manifestly cubic sector of semi-abelian Yang-Mills from the full theory of eq.(3.1) by selecting only (ε + ε + ) in light-cone gauge (i.e., SDYM) and one-minus at tree-level or by plugging in the on-shell states of J-theory [26,27] and Y Ztheory [33].Indeed, semi-YM is just the following sum over building blocks in the expansion of Yang-Mills given in Ref. [58], and similarly so at one-loop, where [ • • • ] a are terms with appropriate powers of (εk) and (kk).As we describe in detail in appendix A, to recover NLSM at one-loop we need to extract the D-dependent part that corresponds to an internal Ȳ Y -loop from extra-dimensional scalars Why must we take the derivative with respect to D? After all, J-theory and Y Z-theory both have well defined propagators for internal JJ and ZZ propagators.However, as we discuss in appendix A, all the J and Z states must be on-shell in order to produce NLSM amplitudes.Thus, the unitarity cuts of J theory at one-loop will not produce NLSM amplitudes.However, the internal Y Y -loop is a valid forward limit for producing NLSM amplitudes, since as constructed Y Z-theory matches to NLSM for off-shell Y -particles.We now provide explicit expressions for these one-loop amplitudes in the next section.

One-loop color-dual integrands
Our first application of this theory for color-dual construction at loop level is for self-dual Yang-Mills (SDYM).As we have done throughout the text, we will set all coupling constants to unity.At tree-level, the off-shell cubic vertices can be written in terms of light cone coordinates [24] X(p, k) = p u k w − p w k u .
A derivation of this Feynman rule and the definition of light cone coordinates can be found in appendix A. At loop-level, this construction needs to be analytically continued to general dimension in order to apply dimensional regularization at one-loop.This can be acheived by using the cubic semi-YM vertices of the previous section, Plugging in on-shell all-plus helicity states in light-cone gauge will yield precisely the 4D SDYM vertex, up to an unphysical phase where ) in light cone gauge.In the second equality, momentum conservation has been applied to the redefined the spinor bracket, ⟨12⟩ → X(k 1 , k 2 ), whose definition is given in appendix B. Of course, the form of eq.(3.23) has the advantage of permitting a D-dimensional construction of the one-loop integrand.As an example, the four-point box numerator is where and the bracket ⟨ • • • ⟩ indicates that we have applied the gauge fixed state projector, ε µ (+ℓ i ) ε ν (−ℓ i ) = η µν , on all internal polarizations.In general, the n-gon diagram is Recall that all other numerators can be obtained from the n-gon through Jacobi.Due to the state-sum of internal loop factors, ε (+ℓ) ε (−ℓ) ∼ D, the integrand above depends explicitly on the spacetime dimension, D. By taking a derivative 5 , we can recover the integrand needed for the all-plus one-loop amplitudes with an internal scalar This integrand numerator with internal scalar loop is precisely what one would obtain from the Feynman rules of Y Z-theory, absent the ZZ internal vector loop.For more background, we refer the reader to appendix A. When plugging on the on-shell states of Y Z-theory, we thus obtain the following expression for the NLSM one-loop n-gon numerator where we have defined the antisymmetric kinematic variable, We have verified through 10-point one-loop that this n-gon expression is a valid color-dual representation for NLSM.Thus, composing the n-gon numerators of eq.(3.26) and eq.(3.28) would yield D-dimensional integrands that project down to the 4D all-plus Born-Infeld oneloop amplitudes studied in Ref. [60].
Before proceeding, we note that the above definition does give rise to "pathological" bubble-on-external-leg (BEL) diagrams, discussed previously in section 2.3.However, one can show that these diagrams integrate to zero for spacetime dimension, D > 2, and thus can be disregarded as unphysical.For the interested reader, in appendix C we provide a detailed overview of this dimensional regularization of the relevant BEL diagram.

Two-loop obstruction
At two-loop, introducing terms that conspire with four-point contacts is unavoidable.At oneloop, we were able to avoid internal contractions of Āµ A µ by selecting appropriate external states.However, at two-loop when choosing all external A µ states, the amplitude vanishes in semi-YM theory that is, the theory is one-loop exact.In order to produce non-vanishing interactions, we would need to reintroduce D-dimensional vertices from the full Yang-Mills Lagrangian of eq. ( 3.1) that we dropped in our construction semi-abelian YM.In terms of cubic graphs, the additional interaction must necessarily have the opposite number of Āµ and unbarred A µ fields to that of eq.(3.17).Reintroducing these oppositely oriented vertices allows for new internal contractions of Āµ A µ , where we used white dots to indicate interaction vertices of weight6 W[ Ā ĀA] = −1, rather than the isolated semi-YM vertex which always carries weight W[ ĀAA] = +1.This immediately runs into the difficulty of introducing the four-point contact needed for color-kinematics to be satisfied on all internal edges.Therefore, by a simple weight counting argument, one can see that including this wrong sign interactions is unavoidable at two-loop and higher.Thus, to construct two-loop numerators prescriptively, as we have done at one-loop, would require knowledge of the full kinematic algebra for Yang-Mills off-shell.Since this is presently unavailable, we will tackle the two-loop integrand using an ansatz approach.

Two-loop four-point bootstrap
As we have just seen, it is possible to coerce tree numerators into one-loop numerators at any multiplicity for pions and related theories, but that these methods cannot be reapplied to generate higher-loop numerators.For pions in particular, the two-loop no-go statement is only for one particular representation of the theory, so it does not completely preclude the existence of a two-loop color-dual integrand.We thus turn to the color-dual bootstrap method described in section 2.3 to construct a color-dual representation of two-loop four-point NLSM.Because of the similarity of the problem setup, we will also use the opportunity to revisit the construction of a color-dual representation of pure YM, extending the search space beyond what was covered in Ref. [13] to include the most general local ansatz.
Both NLSM and YM share the same cubic graph basis, and thus their defining Jacobi relations lead to the same graph basis.Ignoring tadpole and BEL graphs, there are 14 cubic four-point two-loop graphs (see fig. 1) which are related to each other via 21 Jacobi relations.
As mentioned in section 2.3, color-kinematics duality ensures that the numerator of every graph can be expressed in terms of a basis of the double box and penta-triangle, Obviously, the actual numerator dressings and physical properties of the two theories differ, so we discuss them separately below.

Two-loop NLSM
For a scalar theory like NLSM the numerator only depends on dot products of momenta.The basis of momentum invariants for both of the graphs can be taken to be where k ij means k i • k j , k 5 = ℓ 1 , and k 6 = ℓ 2 .Every vertex in NLSM scales as k 2 so the numerator of each diagram scales as k 12 .At this mass dimension there are 8,008 independent dot products of momenta.With two basis graphs there is a total of 16,016 ansatz parameters. 7 Imposing the boundary Jacobi relations eliminates all but 4,473 of the original free parameters.Further imposing graph symmetries reduces the number of free parameters to 1,243.
The final essential constraint is unitarity.The two physical cuts that need to be enforced, shown in fig.2, only involve four-point vertices.Notably this means that even at two loops, unitarity is still only probing the on-shell four-point NLSM amplitude.Unitarity cuts besides those shown in fig. 2 are either pathological or vanish because they involve a three-point point on-shell amplitude.In order to manifest the Z 2 symmetry of NLSM at the integrand level, we also require that all non-pathological cuts containing a three-point vertex (such as the maximal cut) vanish.Imposing all relevant cuts leaves 765 free parameters representing the generalized gauge freedom in the answer.
At this point color-kinematics duality has been satisfied, so the double copy will proceed without issue.However, the plethora of generalized gauge freedom parameters leaves the 7 Mathematica is not well suited for solving large sets of equations with sixteen thousand parameters, so a different method is desirable.One possibility is to use a sparse solver, as often employed for integral reduction.However, these solvers typically assume that the matrix has a very low density.While the equations in YM naturally separate into different "helicity" sectors (or ei • ej and ei • kj sectors in D dimensions) which limits the cross-talk between equations and thus the effective density, there is no such separation of sectors in a scalar theory.Thus the NLSM constraint equations were solved using a custom solver, working name FiniteFieldSolve, designed to exactly solve large linear systems of arbitrary density.FiniteFieldSolve will be released publicly shortly.

Figure 2:
The two physical cut topologies shared by NLSM, sGal, BI, and DBIVA option for enforcing additional aesthetic constraints.All remaining generalized gauge parameters could be set to zero, but a simpler and more insightful result can be obtained through physical arguments.N = 4 SYM provides several hints for further conditions to impose on the ansatz.For example, for maximally supersymmetric gauge theory it is possible to enforce the no triangle hypothesis, manifest loop power counting, and, for four-point and up to at least six loops, strip off a factor of stA tree from the integrand [7,41,47,[61][62][63][64][65][66].For the NLSM integrand it is desirable to make color-kinematics duality as manifest as possible.One hope would be to factor out some piece of the one-loop numerator since this at least manifests antisymmetry for the vertices involving external legs.However, a more fruitful direction is to match onto the one known theory with pion power counting that manifests color-kinematics duality to all loop orders, namely, Zakharov-Mikhailov (ZM) theory [28,67].ZM theory is governed by the Lagrangian where more details can be found in appendix A. The color-stripped Feynman rule for the vertex is where off-shell color-kinematics duality to all orders in perturbation theory simply follows from the Schouten identity in 2D.Since the theory is purely cubic and manifests off-shell color-kinematics duality, it is trivial to read off the color-dual numerator for any graph.From the presence of the Levi-Civita tensor ε µν , the theory clearly resides in two spacetime dimensions where scattering is notoriously plagued by infrared regulation issues.Every on-shell particle is either a right mover, with momentum proportional to k µ R ≡ (1, 1), or a left mover, with momentum proportional to k µ L ≡ (1, −1).On-shell ZM amplitudes naturally divide into sectors corresponding to the configuration of left and right movers, where the scattering in many sectors is rather subtle.Only color-ordered amplitudes can be defined unambiguously from the naive Feynman rules.At four-point, only the alternating sector is free of subtleties and the amplitude for this process vanishes.In equations, A[LRLR] = 0 where R corresponds to a right mover and L corresponds to a left mover.In every other configuration of right and left movers, such as A[RRRR], one of the internal propagators is accidentally on-shell since k 2 R = k 2 L = 0.It is tempting to try to match the pion four-point two-loop integrand to ZM on the cuts, but given that the cuts only probe the on-shell four-point amplitude, which is either subtle or vanishes for ZM, it is better to compare the off-shell numerators directly.The numerators of the two basis graphs are where k 12 = k 1 + k 2 and, again, ⟨ab⟩ ≡ p µ a ε µν p ν b is the color-stripped ZM vertex.As local functions, the numerators never suffer from any of the subtleties of on-shell 2D kinematics.We introduce a proportionality constant z between the numerators of the two theories, N NLSM | 2D = z N ZM , because we are only interested in how the kinematical structure of N ZM can be used to eliminate gauge freedom in N NSLM . 8Mechanically, the pion numerator is matched to ZM by first taking every possible assignment of right and left movers for the external particles and then restricting the pion numerator to 2D.The loop momenta are restricted to 2D but left off-shell.After performing the 2D matching, there are only 365 parameters of generalized gauge freedom.
The loop momentum structure of ZM theory provides one final hint for simplifying the pion numerators.A generic term in the ZM numerator (of either basis graph) looks schematically like ℓ m 1 ℓ n 2 k 12−(m+n) where 5 ≤ m + n ≤ 8, ( whereas a generic set of local, cubic Feynman rules could have produced terms with up to m + n = 12 powers of loop momenta.When the pion numerators are forced to have the loop power counting structure in eq.(4.6) of ZM theory, the number of generalized gauge freedom parameters reduces to 58.The pion numerators appearing in the ancillary files make exactly this choice.

Double-copy verification
With a cubic color-dual pion representation in hand, we can perform double copies with many theories to extract colorless gravity-like amplitudes.In particular, we produce numerators for special Galileons (sGal), Born-Infeld (BI) theory, and Dirac-Born-Infeld-Volkov-Akulov (DBIVA) by double-copying against pions, pure YM, and N = 4 super-Yang-Mills (sYM) respectively [15,69,70].Such double-copy constructions are important nontrivial checks on the underlying single-copy theories as the Jacobi relations in the single copy conspire to produce the double-copy theory's version of linear diffeomorphism invariance: enhanced 8 Another reason for introducing the parameter z is because the ZM and NLSM are believed to differ at loop level even though they are dual classically in 2D [68].Once all of the constraints from this section are imposed, z is fixed to 216/565 where the four-point pion tree amplitude is normalized to −k1 • k2 and the ZM vertex is normalized to k µ 1 εµν k ν 2 .
shift symmetry for special Galileons and gauge invariance of the BI photon and DBIVA supermultiplet [71].All three of these double-copy theories were recently studied extensively by one of the authors and Carrasco from the perspective of direct unitarity cut construction [49].One of the shared features of these three theories is that they all have the same physical cut topologies in 4D: the two diagrams shown in fig.2, which are only composed of four-point amplitudes.
To construct the double-copy theories, we source the cubic pure YM numerators from Ref. [13] (which additionally satisfies a relaxed form of color-kinematics duality), and the cubic N = 4 sYM representation from the well-known n 2box = n cross-box = s 2 t A tree [72].On the other hand, we use the methods of Ref. [49] to directly compute the needed basis cuts in all three theories without relying on a color-dual pion representation.We find exact agreement in each of the three theories for both physical cuts.Since Ref. [49] has already exhaustively explored the properties of four-point loop amplitudes in these theories, we direct interested readers there for more information.

Two-loop Yang-Mills revisited
Given the close ties between pion and gluon scattering for trees and at one loop, the existence of the two-loop pion numerator prompts us to investigate the most general local numerator for YM, without any of the loop power counting assumptions of Ref. [13].We follow the same general procedure as in section 4.1, again identifying the double-box and penta-triangle as the Jacobi basis graphs and building the most general ansatz for each of their numerators compatible with the assumptions in section 2.3.Because we are now considering pure Yang-Mills, each monomial in the local ansatz must consist of five Lorentz scalar dot products instead of the six for NLSM, and each monomial must be linear in each of the four external gluon polarizations.Thus, the numerator ansatz for both diagrams will be built from terms of the form where the k i take the same definition as described near eq.(4.2).Without any power counting restrictions imposed, both the double-box and the penta-triangle have an ansatz with 10,010 terms each.Imposing diagram automorphisms and maximal-cut gauge invariance on the two basis diagrams, we reduce the number of terms to 2,235 for the double-box, and 4,133 for the penta-triangle.The minimal set of spanning physical cuts is shown in fig.3, but we choose to work within the framework of the method of maximal cuts [73] in order to identify the simplest cut that is in tension with the kinematic Jacobi relations.Maximal cuts and symmetries are then imposed on the 14 non-pathological cubic diagrams shown above in fig. 1. Doing so, we are left with 596 total parameters in the ansatz.
Proceeding to the next-to-maximal cuts, we continue to avoid pathological diagrams including the newly-appearing type discussed in eq.(2.26).After discarding these types The 8 non-pathological 1-particle-irreducible next-to-maximal-cut diagrams used in our cut constraints on the Yang-Mills integrand.
of cuts as well as those involving cuts of external Mandelstams, there are 8 one-particleirreducible cut topologies (see fig. 4).Seven of these next-to-maximal cuts are consistent with the kinematic Jacobi relations and symmetries.Critically, the "bowtie" cut, , cannot be satisfied by the ansatz once Jacobi relations and symmetries are applied.
In fact, we can make the failure extremely precise.First, start with the diagrams shown in the Jacobi relation from eq. (2.18), which includes the double-box, crossed-box, and pentatriangle.Then without defining the crossed-box in terms of the other two diagrams, write down the most generic parity-even local ansatz involving four polarizations and six momenta (any combination of external or loop) for each of the three diagrams.Each ansatz will have 10,010 terms initially.Next impose the symmetry constraints on each of the three diagrams, leaving 2,761 free parameters on the double-box, 2,576 free parameters on the crossed-box, and 5,040 free parameters on the penta-triangle.Finally impose the "bowtie" next-to-maximal cut, which in the non-planar t-u color channel only receives contributions from the double-box After imposing this cut, we find that the original Jacobi relation, eq.(2.18), can no longer be satisfied.In other words, the constraints imposed by the single Jacobi relation and the three sets of diagram symmetries conspire in a way that is inconsistent with the "bowtie" cut.The contrapositive to this surprising result can be summarized in the following diagrammatic statement, which prioritizes the physical unitarity cut: Thus we have identified the minimal failure state for a globally-color-dual two-loop Yang-Mills representation: off-shell locality, symmetry, and the kinematic Jacobi relations for just three diagrams are incompatible with the "bowtie" cut.

Conclusions and Outlook
In summary, we have found that there is a tension between color-kinematics duality and an off-shell local construction of the two-loop four-point pure Yang-Mills integrand in Ddimensions.At tree level, we found that the duality can be manifested by breaking manifest Bose symmetry at the Lagrangian level, either by explicitly picking different gluon states (as in SDYM) or by picking states that only restore Bose symmetry in the final amplitude (as in NLSM).Both of these theories can be encapsulated in the semi-abelian YM theory presented in eq.(3.14) which obscures the Bose symmetry of the external states due to the presence of the two gauge fields A µ and Āµ .
In addition to the tree-level construction, we found that kinetic mixing between the two gauge fields permits globally color-dual one-loop integrands.After plugging in suitable on-shell states, the integrands produced by semi-abelian YM correspond to those of SDYM in eq.(3.26) and NLSM in eq.(3.28), along with Chern-Simons theory, which we discuss in appendix A.5. Due to the kinetic mixing that stems from the broken Bose symmetry, eq.(3.30) shows why it is impossible to construct a two-loop integrand for semi-abelian YM theory.In section 4, we overcame this obstruction by successfully constructing two-loop NLSM numerators with an ansatz-based bootstrap.However, we found that the same approach fails for two-loop pure Yang-Mills, and pin-pointed the failure to a particular unitarity cut captured by eq.(4.9).Thus, obtaining a globally color-dual integrand beyond one-loop then requires sacrificing more than just Bose symmetry, and in this paper we have argued that offshell locality9 must be abandoned for color-dual constructions of multi-loop Yang-Mills.To expand the available function space, we propose that one should consider including rational functions of kinematics, rather than just polynomials.
We concede building loop-level numerators from rational functions of the kinematics is rather unnatural from the perspective of point-like quantum field theories.After all, operators that produce rational functions are typically non-local in their construction.The archetypal example of non-local quantum operators are string vertex operators with α ′ corrections.These operators promote local tree-level amplitudes to stringy extended objects by integrating over disc integrals of the open string worldsheet The resulting Veneziano factor of Gamma functions produces rational functions of kinematics, while preserving color-kinematics duality [74][75][76][77].While introducing some type of worldsheet formulation of color-dual numerators might seem unjustified given the results of this work, our findings certainly suggest that we must do something to relax the constraint of off-shell locality, and realize locality only in the limit of on-shell kinematics.This could be achieved by either modifying the kinetic term or potentially something new and more exotic.Below we provide some simple examples at tree-level of what one might consider for implementing such a construction.

Non-local construction of scattering amplitudes
As an exemplar of an off-shell non-local structure, consider the simple four-point example of a color-dual representation of Yang-Mills theory.We can define a functional numerator, N YM (12|34) , as follows, where N s = N (12|34) , N t = N (14|23) and N u = N (13|42) .By construction, this functionally symmetric numerator is antisymmetric and obeys the Jacobi identity However, it must also factorize to kinematics that are consistent with the local Feynman rules of eq.(3.1).Evaluating the residue on the s = s 12 → 0 pole we find with t = s 23 and u = s 13 .Plugging these into a cubic graph representation of Yang-Mills, the tree level version of eq. ( 2.1), we find where we have applied the color structure Jacobi identity, c s + c t + c u = 0.In light of our findings in eq.(4.9), building color-dual numerators in this way where locality is only realized on-shell might be a more natural approach.While we can absolutely apply a generalized gauge transformation [4] to restore field theoretic locality to the cubic numerators, this could merely be an aesthetic choice that is only permissible at tree-level.Moreover, by abandoning traditional notions of off-shell locality, we incidentally have gained enough functional freedom to massage the color-dual numerators into a form that is manifestly gauge invariant for all particles.As an organizational principle, pulling out overall gauge-invariant factors comes with the added advantage of possibly simplifying a loop-level construction of color-dual numerators for Yang-Mills.

Future Directions
With an eye towards generalizing the non-local construction above to future multi-loop studies, we note that the four-point half-ladder of Yang-Mills secretly makes use of the color-dual structure of NLSM.At four-point, we can construct permutation invariants from the single BCJ basis amplitude of both NLSM and pure YM theory as follows stA YM (s,t) = t 8 F 4 stA NLSM (s,t) = stu . (5.8) Using this, we can redefine the non-local numerators of eq. ( 5.3) so that the vector structure of YM is captured in a permutation invariant prefactor and the kinematic Jacobi identity is entirely due to the NLSM numerators stu . (5.9) where four-point pion numerator, N NLSM s , is given in eq.(2.4).One advantage of this construction is that it reduces some of the D-dimensional complexity of vector theories that arises due to the mixing between external polarization and internal loop momenta.But maybe more importantly, it puts all the heavy lifting of functional Jacobi relations on the scalar kinematic numerator, N NLSM (12|34) .Thus, rather than building an ansatz from the irreducible scalar products of eq.(4.7), one might instead consider the following construction where O i are the on-shell gauge invariant tensor basis elements of [77,78], and R (i) m,L are rational functions of irreducible scalar products of eq.(4.2).It would be interesting if a similar construction as our tree-level example could be uplifted to two-loop using the globally color-dual NLSM integrand that we have computed in this work.We see this as a natural future direction worth investigating that is now made possible by our findings.appropriate numerator.Weight counting tells us that the n-gon master numerator must have n on-shell Z-particles.Unitarity requires that there are three distinct contributions to the n-gon: one from an off-shell Y -loop particle and then two more from different orientations of a ZZ-loop.Thus, Y Z theory gives us the following one-loop n-gon numerator: where ⟨ • • • ⟩ indicates an internal contraction over the Y Y and ZZ loops and T and F were given in eq.(A.19) and eq.(A.23), respectively.In terms of the external momenta k i and the loop momenta ℓ i , where ℓ i flows into k i and out of k i−1 , the NLSM numerator is which is the same as eq.(3.28) after using 2(k i •ℓ i ) = ℓ 2 i −ℓ 2 i−1 .Similarly, there is a pure vector contribution that is identical to the semi-abelian YM n-gon numerator that we constructed in eq.(3.26) after projecting external states along longitudinal modes ε → k, The dimension-dependent factor essentially counts that number of internal vector states.While this n-gon is manifestly color-dual, it does not produce the right cuts for NLSM.However, the scalar contribution, that comes dressed with an overall factor D does manifest the duality globally, and satisfies all the desired pion cuts due to the factorization of eq.(A.34) and eq.(A.35).In order for the Feynman rules for YZ theory compute one-loop color-dual numerators consistent with NLSM cuts, one would have to add additional states to cancel off the spurious poles, while preserving color-kinematics duality.We leave this as a direction of future work.While the ZZ vector loop spoils color-kinematics off-shell, the Ȳ Y loop alone gives us the desired expression for the n-gon.However, as we noted in the text, the n-gon numerator has the strange property that it produces non-vanishing bubble-on-external-leg (BEL) graphs.However, the BEL diagram integrates to zero after integral reduction.As an example, the four-point BEL diagram can be reconstructed from the n-gon, where we have introduced a mass regulator that will be proportional to the on-shell momentum inside the BEL, µ 2 ≡ k 2 1 .Thus, in sufficiently large dimension, D > 2, this integral suppresses the µ −2 divergence appearing in the denominator of the BEL diagram.
and S 2|k σ is the set of k pairs of external legs appearing in the color-ordered label list, σ.For example, at four-point, S 2|1 σ = {(

Figure 3 :
Figure 3: The three spanning physical unitarity cuts of pure Yang-Mills at two loops