Abelian Z-theory: NLSM amplitudes and alpha'-corrections from the open string

In this paper we derive the tree-level S-matrix of the effective theory of Goldstone bosons known as the non-linear sigma model (NLSM) from string theory. This novel connection relies on a recent realization of tree-level open-superstring S-matrix predictions as a double copy of super-Yang-Mills theory with Z-theory --- the collection of putative scalar effective field theories encoding all the alpha'-dependence of the open superstring. Here we identify the color-ordered amplitudes of the NLSM as the low-energy limit of abelian Z-theory. This realization also provides natural higher-derivative corrections to the NLSM amplitudes arising from higher powers of alpha' in the abelian Z-theory amplitudes, and through double copy also to Born-Infeld and Volkov-Akulov theories. The Kleiss-Kuijf and Bern-Carrasco-Johansson relations obeyed by Z-theory amplitudes thereby apply to all alpha'-corrections of the NLSM. As such we naturally obtain a cubic-graph parameterization for the abelian Z-theory predictions whose kinematic numerators obey the duality between color and kinematics to all orders in alpha'.


Introduction
It is well known that string theory provides a powerful and unified framework to study the sea of field theories that arise in the limit when the size of the strings approaches zero; some of the most celebrated examples being the maximally supersymmetric super-Yang-Mills and supergravity theories [1].
Since the gluon and graviton belong to the massless excitations of the string, their scattering amplitudes naturally emerge from low-energy limits of the string-theory S-matrix.
By the same token, one might suspect that scattering amplitudes of field theories absent in the naive 1 string spectrum may be difficult to study within string theory. As we will see, such an expectation is surpassed by a long-hidden double-copy structure, secretly and deftly encoded in open-string theory-a structure which applies universally to a broad set of point-like quantum field theories [3]. Perturbative predictions in double-copy quantum field theories [4,5,6] can be completely fixed by knowing the predictions of two possibly distinct input theories 2 .
In recent work by Broedel, Stieberger, and one of the current authors [3] it was demonstrated that open-superstring amplitudes [8,9] can be understood as a double copy of colorstripped Yang-Mills amplitudes and certain Z-functions which behave like scalar partial amplitudes. These Z-functions are iterated integrals over the boundary of a disk worldsheet and naturally incorporate two notions of ordering. One ordering, Q = (q 1 , q 2 , . . . , q n ), refers to the integrand, and one to the integration domain P = (p 1 , p 2 , . . . , p n ), (see section 2.2) Z P (q 1 , q 2 , . . . , q n ) ≡ α ′ n−3 D(P ) dz 1 dz 2 · · · dz n vol(SL(2, R)) n i<j |z ij | α ′ s ij z q 1 q 2 z q 2 q 3 . . . z q n−1 q n z q n q 1 . (1.1) It was shown in ref. [3] that these doubly-ordered functions obey Kleiss-Kujif (KK) [10] and Bern-Carrasco-Johansson (BCJ) [5] field-theory amplitude relations along its integrand ordering Q, and the string-theory monodromy relations [11,12] along the integration domain P . After dressing the string-monodromy ordering P with the appropriate sum 1 Given the string-theory realizations of principle chiral models through toroidal compactifications along with worldsheet boundary condensates [2], the term "naive string spectrum" refers to string theories in D-dimensional uncompactified Minkowski spacetime. 2 For example, scattering in the N = 5 supergravity theory is completely determined by a double copy consisting of color-stripped amplitudes of N = 4 and N = 1 super Yang-Mills theories. Note that double copy holds at the integrand level [7] for multiloop amplitudes.
over string Chan-Paton factors, we are left with the Q-ordered functions that simply obey field-theory relations. Could these functions be color-ordered scattering amplitudes for some effective field theory ? We conjecture the answer is yes. In this manuscript we provide non-trivial evidence in the abelianized low-energy limit, discovering the D-dimensional color-ordered amplitudes of the non-linear sigma model (NLSM), shown to satisfy the relevant color (flavor)-ordered field-theory scattering amplitude relations in ref. [13].
The KK and BCJ field-theory amplitude relations between ordered scattering amplitudes are incredibly constraining and very special-key to consistency properties of the theories they describe. The fact that the Z-functions (1.1) obey KK and BCJ relations along an ordering suggests the possible interpretation that these functions might themselves be scattering amplitudes for some conjectural and as yet to be identified doublycolored theory, which we will refer to as Z-theory. As such we refer to the Z-functions as Z-amplitudes. This relies on a natural conjecture about factorization properties of these functions supported by the double-copy perspective as well as an open-string worldsheet model which we will describe in more detail in Appendix A.
In the time-honored tradition of periodically reflecting upon the venerable query, "What is string theory?", we find ourselves struck by the ubiquity of double-copy constructions -not only in unifying open-and closed-string predictions a la Kawai, Lewellen and Tye (KLT) [4], but also in constraining the effective-field-theory (EFT) modifications to super-Yang-Mills resulting in open-superstring tree-level predictions. We suspect a strategic path forward may arise from the more modest question driving this current manuscript, "What is Z-theory?".
A critical first clue was given in [3], where it was demonstrated that the α ′ → 0 limit of doubly-ordered Z-functions lands on the inverse of the field-theory KLT matrix. This same limit was later recognized to correspond to the (doubly-partial) tree-level amplitudes of a scalar bi-adjoint theory [14]. Therefore, even though the string spectrum does not include bi-adjoint scalars, string tree-level amplitudes, through Z-amplitudes, contain all their tree-level predictions.
To be clear, we are not uplifting a known field theory's predictions to string-theory amplitudes using some procedure. Rather these conjectural Z-amplitudes are double-copy factors in open-superstring scattering amplitudes which contain all-orders in α ′ and obey field-theory scattering amplitude relations. They are entirely well-defined functions sitting within the open-string predictions. The scalar amplitudes of the conjectural Z-theory retain the fingerprints of string-theory relevance through their second Chan-Paton dressed monodromy-ordering. If our conjecture holds, the (low-energy) α ′ -expansion of these objects should be identifiable as scattering amplitudes in some identifiable field theories, with each successive order in α ′ representing higher-derivative corrections.
Indeed, our primary result in this manuscript is the identification of such a theory: the tree-level S-matrix of Goldstone bosons described by the D-dimensional NLSM can be obtained from open-superstring amplitudes. Denoting the Lie-algebra valued Goldstoneboson field by ϕ, the NLSM Lagrangian in the Cayley parametrization is given by with Lorentz indices µ = 0, 1, . . . , D−1. This theory's predictions emerge when the Chan- In contrast to the doubly-partial amplitudes of bi-adjoint scalars, the NLSM amplitudes arise from nonzero orders (α ′ ) n−2 singled out by the leading low-energy contribution to As will be explained below, the sum over string integrals in (1.3) appears in the n-point amplitude of massless open-string states obtained in [8,9] upon specialization to abelian Chan-Paton degress of freedom. Notice that abelianization refers only to the monodromy 3 In order to avoid cluttering of factors of two, we have rescaled α ′ such that the standard open-string conventions are recovered by setting α ′ → 2α ′ in the equations of this work. 4 For the fixed field-theory color-ordered Z-theory amplitudes one dresses all monodromy orderings with relevant Chan-Paton factors -a sum over (n−1)! orders in the naive trace basis.
In the abelian case we take the Chan-Paton factors trivial, effectively symmetrizing over the integration order of the Z-functions, yielding color-ordered abelian Z-theory amplitudes.
ordering, leaving the field-theory ordering to be dressed with field-theory adjoint weights to represent color/flavor in the full dressed abelian Z/NLSM amplitudes.
The surviving low-energy limit of these amplitudes are color-ordered NLSM amplitudes. In addition, the subleading terms in the α ′ -expansion of (1.3) will be interpreted as stringy higher-derivative corrections to the NLSM amplitudes. This summarizes our primary result for which we report on explicit calculations through nine points. We additionally provide a general approach to generating associated local color-kinematic satisfying numerators from these amplitudes. Along the way we provide a nice recursive form of the KLT matrix S[·|·] 1 , inspired by ref. [15], and propose a strikingly simple form for all-multiplicity n = 2k NLSM color-kinematic satisfying master numerators This manuscript is organized as follows: After a review of disk integrals in section 2, we state and prove our main result (1.3) in section 3. Examples and the systematics of higherderivative corrections to the NLSM are elaborated in section 4, and the construction of explicit BCJ numerators for the α ′ -corrected NLSM can be found in section 5.

Double-copy construction
Due to the seminal work [4] of KLT, it has long been recognized that the tree-level predictions of open strings entirely encode 5 the tree-level predictions of closed strings. The amplitudes admit a representation in the form of the sum over products of color-stripped gauge (open-string) amplitudes. This made a particular impact in its low-energy limit in the study of field-theory gravitational scattering amplitudes in the 90's where various closed-form representations were identified [18,19]. 5 Given the all-multiplicity tree-level relations between gravity and gauge theory, one might rightfully ask if classical general-relativity solutions are encoded in classical gauge-theory solutions.
Answering this question is an active area of investigation see e.g. refs. [16] for explicit solution relationships, and refs. [17] for considerations of classical symmetries and duality-groups, as well as references therein.
Admittedly, many properties of the KLT double-copy construction were mysterious (including the emergence of the necessary permutation symmetry of the gravity amplitudes), and the original formulation of sums over shuffle-ordered products of permutations could be dizzying, especially at higher multiplicities. Nevertheless, this approach, as it would factor over the state-sum of unitarity cuts, proved critical for gathering information about the spectacular ultraviolet behavior of the maximally supersymmetric supergravity theory through four-loops [20]. In the process of such explorations, the field of scattering amplitudes acquired a new set of insights relevant to the double-copy story when Bern, Johansson, and one of the current authors (BCJ) observed [5] that there was a very direct path to gravity-theory predictions, where the double-copy construction could be made manifest graph by graph. This has particular value at the multiloop integrand level [7], where integrand labeling ambiguities can create obstacles for realizing generalized-gaugeinvariant double-copy relationships, outside of certain kinematic limits like unitarity cuts.
The BCJ double-copy approach relies critically on the realization [5] that gauge-theory predictions (and their supersymmetric partners) admit a color-kinematic duality satisfying representation (color-dual representations are ones where graph by graph color-weights and kinematic weights obey the same generic algebraic properties). The existence of such color-dual representations resulted in the discovery of new relations between color-ordered amplitudes known now as the BCJ relations. With such a dual representation, color-factors could be consistently replaced 6 by kinematic weights, recycling a small set of kinematic predictions to describe a wide variety of theories [7]. Additionally, color-dual kinematic weights could be solved for in terms of color-ordered amplitudes, thus allowing for the generation of generalized KLT relations.
While a Lagrangian understanding of the organizing principle is only available in the four-dimensional self-dual case [22], many theories, including the NLSM [23], in a variety of spacetime dimensions, admit the duality between color and kinematics, and associated double-copy construction [24]. This new perspective on field-theory predictions has proven critical in developing aspects of our understanding of non-planar scattering amplitudes over the last decade, both formally as well as through practical reach in computation. Jacobi relations drastically constrain the independent information relevant to a given scattering calculation. For instance, the closely related double-copy constructions of multiloop gravity 6 Established to some finite multiplicity at tree-level in [5], but later proven via BCFW and the KLT relations in [21]. amplitudes [7] has allowed many explicit calculations that can probe the possible onset of ultraviolet divergences in supergravity theories [25].
String theory continues to provide key insights probing the color-kinematics duality and its associated representation of the double-copy construction: from the powerful proof [11,12] of the (n−3)!-basis of Yang-Mills tree amplitudes as the low-energy limit of the related string monodromy relations 7 , to the elegant construction of explicit local tree-level numerators [27,28], to the construction of string-inspired BCJ numerators [29] at loop level.
The fact that the BCJ-duality also applies to the NLSM [23], can now be appreciated either as a consequence of the BCJ relations satisfied [3] by Z-theory as in (1.3), or as a requirement for the NLSM to be able to participate in Z-theory's construction of the open superstring. Following the recent result of Du and Fu [15] who present an elegant closed-form construction of local color-kinematics satisfying numerators in the NLSM, we will discuss its applicability to all orders in α ′ .
It is worth mentioning a related [30] approach to constructing double-copy representations known as the Cachazo-He-Yuan (CHY) formalism [6,31,14], which generalizes the four-dimensional connected prescription of Roiban, Spradlin, Volovich, and Witten [32,33] to general dimensions. Similar to string theory, scattering amplitudes in the CHY framework are derived from punctured Riemann surfaces 8 . Exploiting a CHY description of Yang-Mills theory and the NLSM model, ref. [38] offered the first double-copy realization of self-dual Born-Infeld [39] scattering amplitudes.
The idea that there can exist a duality between electric and magnetic field densities is as old as gauge theory. Satisfied by sourceless Maxwell electrodynamics, this natural duality has inspired analyses and generalizations that have been key to understanding aspects of supersymmetry, symmetry breaking, and string theory, starting with perhaps most famously the Born-Infeld non-linear generalization of electromagnetics [40]. The emergence of duality invariance in the form of Born-Infeld scattering due to a double-copy interplay between YM and the low-energy limit of abelian Z-theory is remarkable. In concordance with the structure of open-string amplitudes given as a double copy between Yang-Mills constituents and Z-theory disk-integrals [3], the double-copy representation of Born-Infeld 7 See [26] for a recent higher-loop generalization. 8 The CHY integrands for gluon and graviton scattering have direct antecedents in the heterotic string and the type-II superstring, respectively [34,35]. Also see ref. [36] for a careful discussion of subtle differences between CHY (tree-level) integrands for Einstein-Yang-Mills [37] and correlation functions of the heterotic string.
amplitudes as its surviving abelian low-energy limit serves as a key check that our observation (1.3) holds to all multiplicities, beyond the explicit verification at n ≤ 9 points we report on here.

Z-theory amplitudes
Tree-level scattering amplitudes of open-string states are determined by iterated integrals on the boundary of a disk worldsheet. Massless n-point amplitudes of the open superstring [8] and conjecturally those of the open bosonic string [41] possess cyclic integrands of the following form: The universal and permutation-symmetric Koba-Nielsen factor n i<j |z ij | α ′ s ij built from differences z ij ≡ z i − z j is accompanied by a cyclic product of propagators z −1 q i q i+1 indicated by the labels (q 1 , q 2 , . . . , q n ) on the left hand side. The additional subscript P ≡ p 1 p 2 . . . p n encodes the ordering for the iterated integrals, and thereby the accompanying Chan-Paton trace over gauge-group generators t p 1 t p 2 . . . t p n .
The inverse volume of the conformal Killing group of the disk instructs one to drop any three variables of integration z i , z j , z k and to compensate with a Jacobian z ij z ik z jk , e.g.
The unintegrated variables can then be fixed to any real values such as (z 1 , z n−1 , z n ) = (0, 1, ∞). Finally, the Mandelstam variables are defined in terms of lightlike momenta k i : Their appearance in the open-superstring amplitudes leads us to view the integrals (2.1) as defining the tree-level S-matrix of Z-theory, the collection of putative scalar effective field theories that incorporate all the α ′ -expansion on a disk worldsheet.
Inspired by progress in string-theory scattering organization, it was realized in the 1990's that gauge-theory amplitude calculations simplify tremendously by considering ordered gauge invariants depending only upon kinematics -called color-ordered or colorstripped partial amplitudes. The full color-dressed S-matrix elements could be obtained by summing over a product of these color-ordered amplitudes with appropriate color-weights, either somewhat redundantly in a trace basis, or more efficiently in the Del Duca-Dixon-Maltoni basis of [42]. The advantages in considering stripped or ordered partial amplitudes are enormous -they grow exponentially rather than factorially in local diagram contributions. Doubly-ordered amplitudes like the (2.1) represent a further generalization -stripping out another ordering when one can be defined. One can recover from (2.1) the more familiar field-theory color-ordered amplitudes by summing over the stringy P -orders taking products with Chan-Paton traces.

The field-theory limit
In the field-theory limit α ′ → 0, the disk integrals (2.1) yield kinematic poles that correspond to the propagators of cubic diagrams [45,46]. As a convenient tool to describe the pole structure, we recall the theory of a bi-adjoint scalar φ ≡ φ a|b t a ⊗t b with a cubic interaction Doubly-partial amplitudes m[P |Q] are defined to track the traces of gauge-group generators t a andt b in the tree amplitudes of the above scalar theory [14], Following the all-multiplicity techniques of [9], the field-theory limits of disk integrals have been written in terms of doubly-partial amplitudes as [14], identifying the bi-adjoint scalar theory (2.12) as the low-energy limit 9 of Z-theory, see [28] for an efficient Berends-Giele implementation of (2.14).

Abelian limit
Recall that the color-dressed n-point tree amplitude of the open superstring is given by where the Chan-Paton-stripped amplitudes determined in [8] were later identified in [3] to The symmetric matrix S[ρ|τ ] 1 in (2.16) encodes the field-theory limit (α ′ → 0) of KLT relations [4] to all multiplicities [19] and admits the following recursive definition 10 , where A, B and C are arbitrary multiparticle labels such that |A| = |B|+|C| and the multi- defines the abelian disk integrals or the partial amplitudes of abelian Z-theory whose α ′expansion will be discussed below. 10 The field-theory KLT matrix was originally defined in non-symmetric form in [19], later rewritten in [47,44] with the symmetric form used in [3]. Inspired by equation (3.8) of [15] we arrived at the novel recursive definition (2.17), which generalizes to all orders in α ′ in an obvious manner [48].

α ′ -expansion
The α ′ -expansion of the disk integrals (2.1) gives rise to multiple zeta values (MZVs), 20) which are characterized by their weight w = n 1 + n 2 + . . . + n r and depth r. More precisely, the order (α ′ ) w of the disk integrals in (2.1) is accompanied by products of MZVs with total weight w (where the weight is understood to be additive in products of MZVs); a property known as uniform transcendentality. This has been discussed in the literature of both mathematics [49] and physics [9,50,51] and can for instance be proven by the recursive construction 11 of disk integrals using the Drinfeld associator [53].
The combination of all integration orders to obtain abelian disk integrals projects out a variety of MZVs from the α ′ -expansion of (2.19). As elaborated in section 4.3, these cancellations include the field-theory limit (2.14) and the coefficients of odd Riemann zeta values ζ 2k+1 without accompanying factors of ζ 2n . Moreover, abelian disk integrals of odd multiplicity vanish at all orders in α ′ by the reflection property (2.10), It turns out that the leading low-energy contribution to abelian disk integrals of even multiplicity n arises from the order α ′n−2 and stems solely from the even Riemann zeta values such as (B 2k are the Bernoulli numbers) The impact of these selection principles on the α ′ -expansion of abelian Z-theory in connection with NLSM amplitudes will be explored in section 4. In light of the ubiquitous appearance of MZVs in both abelian and non-abelian Z-theory, one might be tempted to derive the capital letter in the theory's name from "zeta". 11 At multiplicities five, six and seven, explicit results for the leading orders are available for download on [52], along with the building blocks for eight and nine points.

NLSM amplitudes from string theory
Although the superstring spectrum does not include any bi-adjoint scalar, the doublypartial amplitudes (2.13) emerge naturally from the low-energy limit of the Z-theory amplitudes (2.1) contributing to the open string. Perhaps more familiarly, the color-stripped amplitudes of the bi-adjoint scalar appear as the α ′ → 0 limit of the Chan-Paton dressed doubly-partial Z-theory amplitudes. In this work, we show that the NLSM tree-level amplitudes can be obtained from the abelian disk integrals (2.19).
To see this, note that the Born-Infeld action emerges as the leading low-energy contribution to abelian amplitudes in supersymmetric string theory [54]. Therefore, the expression for M abelian (α ′ ) on the right-hand side of (2.18) must reduce to the Born-Infeld amplitude whose KLT-like double-copy structure has recently been identified by Cachazo, He and Yuan [38], Comparing (2.18) with (3.1) and assuming linear independence of the YM partial amplitudes in the BCJ basis leads to the conclusion that the abelian Z-amplitudes (2.19) reduce to color-ordered NLSM tree amplitudes at low energies, This limit is non-singular due to the cancellation of low-energy orders below (α ′ ) n−2 in abelian n-point Z-amplitudes, see section 4.3 for further details. The emergence of NLSM amplitudes in (3.2) has been explicitly verified up to n = 9, using the expansion method of [53] to probe the α ′6 -order at the highest non-trivial multiplicity n = 8. As an immediate consistency condition for the validity of (3.2), note that the KK and BCJ relations satisfied by the NLSM amplitudes [23] correspond to the following identities of the abelian integrals, (k q 1 · k q 2 q 3 ...q j )Z × (q 2 , q 3 , . . . , q j , q 1 , q j+1 , . . . , q n ) , which are a consequence of (2.7) and (2.8).
Besides reproducing NLSM amplitudes, higher α ′ -orders of abelian disk integrals (2.19) yield natural higher-mass dimension extensions of the NLSM which will all satisfy KK and BCJ relations (3.3). More precisely, the symmetry properties (3.3) hold separately at each order in α ′ , and in fact for the coefficients of any MZV which is conjecturally linearly independent over Q. Hence, abelian disk integrals can be viewed as a factory for effective theories with any number of derivatives, and each such theory obeys the duality between color and kinematics. The discussion of these α ′ -corrections to the NLSM will be the main focus of section 4.
It is straightforward to see using the form of the Veneziano amplitude together with the identities that the four-point abelian integral (4.1) can be written as The abelian integral (4.4) not only reproduces the standard four-point NLSM amplitude A NLSM (1, 2, 3, 4) = −π 2 s 13 at its lowest α ′ order (note the swap of legs 3 ↔ 4), but also implies an infinite series of higher-derivative corrections [55],

Six points
The α ′ -expansion of six-point disk integrals (2.1) was pioneered in [56,57] and later on aligned into systematic all-multiplicity methods using polylogarithms [3] or the Drinfeld associator [53] (see also [58]). When summing over the 5! integration domains to obtain an abelian six-point disk integral (2.19), the leading α ′ -orders associated with s

All order-systematics
In order to discuss the all-multiplicity systematics of the α ′ -expansion of abelian disk integrals, we recall the patterns of MZVs in open-string amplitudes identified in [51]. A particularly convenient basis for that purpose is furnished by the (n−3)!×(n−3)! integrals 12 appearing in the n-point amplitude (2.16) of the superstring [8,3] Both P w and M w denote (n−3)!×(n−3)! matrices whose entries are degree-w polynomials in α ′ s ij with rational coefficients. The explicit form of these entries can be determined from polylogarithm manipulations [3] or the Drinfeld associator [53], and examples at multiplicity n ≤ 7 are available for download from [52]. As a first non-trivial statement of (4.9), for instance, the coefficient of ζ 2 ζ 3 is given by the matrix product P 2 M 3 combining the constituents at the ζ 2 -and ζ 3 -orders of F .
in the first line of (4.9). Therefore the entire second line of (4.9) -in fact any product of matrices M 2k+1 -preserves the BCJ and KK relations 13 for (M 2k 1 +1 . . . M 2k n +1 )A YM .

Selection rule for the first order in ζ 2
Similarly, the α ′ -deformed BCJ relations of P 2k A YM encoded in the monodromy relations directly carry over to the matrix products P 2k M 2ℓ 1 +1 . . . M 2ℓ m +1 A YM with ℓ j , m ∈ N. This has a direct implication for abelian disk integrals: Whenever the coefficient of ζ k 2 vanishes by the monodromy relations to that order, the same vanishing statement applies to all products of ζ k 2 with the entire second line ∼ ζ M of (4.9), The "KK-like" relations among the ζ 2 -orders of open superstring amplitudes [57,62,63] for instance are known to annihilate permutation sums at multiplicities n ≥ 5 and therefore 13 This argument firstly appeared in the discussion of BCJ relations among amplitudes from higher-mass dimension operators [59], and a similar statement in the context of the heterotic string can be found in [60]. 14 The choice of MZVs at a given weight to represent the α ′ -expansion of disk integrals is ambiguous, and we will follow the conventions of [61,51] to take {ζ 8 , ζ 3 ζ 5 , ζ 2 ζ 2 3 , ζ 3,5 } as the conjectural Q-basis of weight-eight MZVs. Different choices lead to redefinitions of the matrices P w , M w , e.g.

Selection rule for higher orders in ζ 2
Although the symmetry patterns associated with the ζ 4 , ζ 6 , . . .-orders of open superstring amplitudes have not yet been studied, there is an indirect argument to extend the selection rule (4.13) to higher orders: Since the low-energy limit of the abelian amplitude (2.18) is known to stem from the Born-Infeld action [54], the n-point amplitude cannot have contributions of orders below α ′n−2 . In particular, this implies and leads to an infinity of additional vanishing statements by (4.12), with ζ M again referring to any MZV in the second line of (4.9).
Examples of the above selection rules on the low-energy regime of abelian disk integrals are summarized in the subsequent table: 15 Earlier versions of this manuscript described the rather challenging procedure originally used to extract these α ′ -expansions.

Simplifications in the odd zeta sector
It turns out that the laborious procedure to determine the α ′ -expansion of Z × (. . .) can be bypassed for all the M w matrices. Once we have determined the contributions of the type ζ 2k P 2k from the first line of (4.9), then the coefficient of ζ 2k+1 or any other MZV in the second line of (4.9) can be inferred by matrix multiplication Note, however, that the multiplication order of M w matrices is reversed in (4.17) as compared to (4.9). As before, matrix multiplication with any sequence of M 2k+1 propagates the BCJ and KK relations of Z even × to the full integral Z × .
One might wonder if the structure in (4.17) can be refined and if the appearance of any ζ 2k in Z × can be captured by combining a BCJ basis of A NLSM (1, σ(2, . . . , n−2), n−1, n) with polynomials in Mandelstam variables. When insisting on local coefficients for the basis of NLSM amplitudes, this scenario can be ruled out from a simple six-point example: In an ansatz of the form with the left-hand side given by (4.7), the entries of the 6 × 6 matrix M 2 cannot be chosen as degree-two polynomials in α ′ s ij . Hence, there is no local degree-two counterpart M 2 of the M 2k+1 matrices at six-points which preserves the BCJ and KK relations.

Color-kinematic satisfying numerators
As we will review, the fact that color-stripped NLSM amplitudes satisfy the BCJ relations ensures [68,48] that they admit a color-kinematic satisfying representation at tree-level by virtue of the existence of the KLT decomposition. The fact that this holds to all multiplicity Fig. 1 The Jacobi identity f a 1 a 2 b f ba 3 a 4 + cyc(a 1 , a 2 , a 3 ) implies the vanishing of the color factors C i , C j and C k associated to triplets of cubic graphs. In the above diagrams, the legs a 1 , . . . , a 4 may represent arbitrary cubic tree-level subdiagrams. The duality between color and kinematics states that their corresponding kinematic numerators N i built from polynomials of Mandelstam invariant for the cases of interest can be chosen such that N i + N j + N k = 0 whenever C i + C j + C k = 0 [5].
suggests that the integrands of these theories, effective though they are, should also admit color-kinematic satisfying numerators. This intriguing possibility motivates exploring what various closed forms for color-kinematic satisfying tree-level numerators can be found.
The BCJ relations (3.3) among abelian disk integrals hold separately at each order in α ′ , more precisely for the coefficients of all the MZVs which are conjecturally linearly independent over Q. Following the original derivation of BCJ relations for YM amplitudes from the duality between color and kinematics [5], one should expect each MZV coefficient of Z × (. . .) to admit a cubic-graph organization 16 , where the s ij -dependent numerators satisfy kinematic Jacobi relations. The latter apply to any triplet of cubic diagrams whose color factors under a generic gauge group (obtained from dressing each vertex by a structure constant f abc of a non-abelian gauge group) sum to zero by the Jacobi identity f a 1 a 2 b f ba 3 a 4 + cyc(a 1 , a 2 , a 3 ), see fig. 1.
While the original outline for finding tree-level Jacobi-satisfying numerators relied on manually inverting the propagator matrix and exploiting the residual gauge freedom to establish locality, it was not long before the community realized that the KLT matrix, or momentum kernel, does indeed represent an inversion of the propagator matrix relevant to finding Jacobi-satisfying numerators [68,48]. The prescription is to define the masters as the half-ladder diagrams with external legs k 1 , and k n as fixed farthest rungs, allowing all permutations of legs {2, . . . , n − 1}, as in fig. 2. All such master numerators for all 16 In spite of the Feynman vertices of valence ≥ 4 in the NLSM Lagrangian (1.2), one can always achieve a cubic-graph organization of its scattering amplitudes by introducing targeted propagators 1 = (k i +k j ) 2 (k i +k j ) 2 whose channels i, j are lined up with the accompanying color factors. This kind of bookkeeping has been originally applied to the quartic vertex of Yang-Mills theories [5] and can be straightforwardly extended to deformations of the NLSM by vertices of arbitrary valence and order in derivatives.
All numerators follow via Jacobi from these master numerators. These numerators are manifestly non-local (although of course all physical observables have the appropriate poles).
All poles belonging to the vanishing masters have been absorbed by the non-vanishing masters. As pointed out in [5], as long as the color-stripped amplitudes obey kinematic Jacobi relations on residues, one can find a generalized gauge transformation (cf. ref. [69]) consistent with Jacobi pushing these poles into the appropriate master numerators.
One should expect that if such a local representation is always possible for a theory then there should be a closed form for local masters 17 . Indeed, the authors of [15] present such a closed-form construction for the NLSM, making the key-insight that the symmetric (n − 2)! × (n − 2)! form of the momentum kernel has the necessary freedom to allow for locality, while recognizing the need for an off-shell regulation. The naive on-shell attempt fails as BCJ relations imply that Giele [65] description of NLSM amplitudes [70], one can extend NLSM amplitudes to an 17 Locality of numerators for Jacobi-descendant graphs when expressed in terms of local numerators from masters follows from the color-stripped amplitudes satisfying Jacobi on all poles [5]. off-shell momentum k 2 n = 0 by using an overcomplete set of Mandelstam variables s ij with 1 ≤ i < j ≤ n−1. Accordingly, the sum over ρ in ( which leads to the following expressions at four points, One can verify from (5.4) that the four-point amplitude (1.4) is correctly reproduced.
As pointed out in [71] one can always symmetrize Jacobi-satisfying numerators to arrive at a crossing symmetric function for the generically dressed half-ladder topology in a manner that preserves linear relations (like Jacobi). One can note that fully crossingsymmetric local numerators of [15] were For example, the α ′ -corrections to the abelian four-point disk integrals (4.5) generalize the master numerator (5.4) to (recalling that σ 2 ≡ α ′2 2 (s 2 12 +s 2 13 +s 2 23 ) and σ 3 ≡ −α ′ 3 s 12 s 23 s 13 ) The analogous six-point corrections to (5.5) at the order of ζ 6 and ζ 4 ζ 3 are attached as ancillary files to the arXiv submission of this work; they yield (4.7) and (B.1) according to (5.10).

Conclusions and outlook
In this paper, we interpret the disk integrals in open-string tree-level amplitudes as the S-matrix of a collection of putative scalar effective theories we refer to as Z-theory. Our key result (1.3) establishes that the low-energy limit of abelian Z-theory amplitudes yields the n-point amplitudes of the NLSM at order α ′ n−2 , while the next orders define new higher-derivative corrections which admit color-dual representations. Using this setup, we obtain α ′ -corrections to the local BCJ-satisfying numerators recently identified by Du and Fu [15], and indeed, a novel all-multiplicity expression (1.6) for the master numerators of the NLSM.
Given the high orders of α ′ involved in extracting the NLSM amplitudes and corrections from the Z-amplitudes, the straightforward organization of the string-theory calculations presented in this paper is not optimized to probe high multiplicities. Here, we chose to instead emphasize the relationship between open-string predictions, abelian Z-amplitudes, and explicit α ′ -corrections. In future work [64,67], efficient calculations will be addressed by the corollaries of monodromy relations presented in [72] and a Berends-Giele recursion for the α ′ -expansion of non-abelian disk integrals using an extension of the method described in [28].
Towards identifying patterns within Z-theory, we find ourselves encouraged to investigate the relevant higher-derivative corrections to the Lagrangian description of the NLSM which reproduces the higher α ′ -corrections of the amplitudes discussed in this work. Preliminary considerations suggest that these are not the only higher-derivative corrections consistent with color-kinematics. Accordingly, additional guiding principles may need to be invoked to arrive at the selection rules and the patterns of MZVs realized by worldsheets of disk topology. It has not escaped our notice that apprehending such guiding principles could indeed prove a fruitful line of inquiry.
In addition to providing higher derivative corrections to the NLSM we have through double copy, en passant, generated predictions for a set of higher-derivative corrections to Born-Infeld, and its supersymmetric partners including Volkov-Akulov from the fermionic sector [73,74]. It would be interesting to contrast with higher-derivative corrected Born-Infeld-type theories existing in the literature, cf. the set of self-dual theories constructed in ref. [75].

A.1. Via double copy
Here we sketch an argument why one might expect all participants of a double-copy scattering amplitude to support factorization on its massless poles based on quite general double-copy properties. Up to the orders in α ′ explored through eight points this is obviously the case between multiplicities for the color-ordered abelian Z-amplitudes presented here. The interested reader can verify these properties continue to hold for higher point amplitudes and higher derivative α ′ corrections, including non-trivial Chan-Paton factors, using the doubly-ordered Z-amplitudes presented in [64].
Let us use X = Y ⊗Z to represent the massless-adjoint BCJ double-copy construction of scattering amplitudes where Y and Z may encode states but such encoding is absolutely independent. The massless states of X are taken to be the outer product of the states of Y and Z. We take as given that X factorizes on massless channels. This means that the residues of X on any of its massless channels are given as the sum over X states of products of lower-point X amplitudes evaluated with on-shell momenta. We take that Y represents the kinematic information from color-stripped amplitudes which also factorizes on massless channels. This means that Y ⊗φ 3 represents color-dressed scattering amplitudes in a Y theory. For the case of abelian Z-functions presented here, X would represent the abelianized open string, Y would represent supersymmetric Yang-Mills, and Z would represent the α ′ dependent kinematic information associated with abelian Z-functions.
Since X factorizes, and Y factorizes, we can expect the numerator weights of Z to factorize on massless channels up to only terms that vanish due to algebraic properties satisfied by Y . Since we expect Y and Z states to be independent, we expect any such putative zeroes to represent generalized gauge freedom -terms that vanish simply by virtue of a double copy with any theory that satisfies Jacobi and anti-symmetry. As such colordressed abelian Z-amplitudes: Z ⊗φ 3 should factorize since the biadjoint scalar amplitudes obey Jacobi and anti-symmetry in each color-weight. Color-stripping these color-dressed Z-amplitudes to color-ordered Z-amplitudes should in turn be independent of any such zeros, as the zeros can not care about what gauge group was being stripped. It should follow that any such color-ordered Z-amplitudes will factorize. This argument neglects the interesting, but irrelevant to the order-by-order finite α ′ exploration of the putative massless effective field-theory, consideration of the factorization on the infinite tower of higher-spin massive modes, which is entirely contained within the Z-factor of the open superstring.
Restricting the unitarity analysis to the terms of uniform transcendentality, we conclude that Z-amplitudes factorize correctly on their massless poles.