Non-abelian $Z$-theory: Berends-Giele recursion for the $\alpha'$-expansion of disk integrals

We present a recursive method to calculate the $\alpha'$-expansion of disk integrals arising in tree-level scattering of open strings which resembles the approach of Berends and Giele to gluon amplitudes. Following an earlier interpretation of disk integrals as doubly partial amplitudes of an effective theory of scalars dubbed as $Z$-theory, we pinpoint the equation of motion of $Z$-theory from the Berends-Giele recursion for its tree amplitudes. A computer implementation of this method including explicit results for the recursion up to order $\alpha'^7$ is made available on the website http://repo.or.cz/BGap.git


Introduction
It is well known that string theory reduces to supersymmetric field theories involving nonabelian gauge bosons and gravitons when the size of the strings approaches zero. Hence, one might obtain a glimpse into the inner workings of the full string theory by studying the corrections that are induced by strings of finite size, set by the length scale √ α ′ . One approach to study such α ′ -corrections to field theory is through the calculation of string scattering amplitudes, see e.g. [1,2]. Within this framework, higher-derivative corrections are encoded in the α ′ -expansion of certain integrals defined on the Riemann surface that encodes the string interactions.
In this work, we will mostly study tree-level scattering of open strings, where the Riemann surface has the topology of a disk. As will be reviewed in section 2, the α ′corrections to super-Yang-Mills (SYM) field theory arise from iterated integrals over the disk boundary. These integrals can be characterized by two words P and Q formed from the n external legs which refer to the integration domain P = (p 1 , p 2 , . . . , p n ) and integrand Q = (q 1 , q 2 , . . . , q n ) in 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) This paper concerns the calculation of the α ′ -expansion of these disk integrals in a recursive manner for any given domain P and integrand Q. This technical accomplishment is accompanied by conceptual advances concerning the interpretation of disk integrals (1.1) in the light of double-copy structures among field and string theories.
As the technical novelty of this paper, we set up a Berends-Giele (BG) recursion [3] that allows to compute the α ′ -expansion of the integrals Z(P |Q) and generalizes a recent BG recursion [4] for their field-theory limit to all orders of α ′ . As a result of this setup, once a finite number of terms in the BG recursion at the w th order in α ′ is known, the expansion of disk integrals at any multiplicity is obtained up to the same order α ′ w . The recursion is driven by simple deconcatenation operations acting on the words P and Q, which are trivially automated on a computer. The resulting ease to probe α ′ -corrections at large multiplicities is unprecedented in modern all-multiplicity approaches [5,6] to the α ′ -expansion of disk integrals.
The conceptual novelty of this article is related to the interpretation of string disk integrals (1.1) as tree-level amplitudes in an effective 1 theory of bi-colored scalar fields Φ dubbed as Z-theory [7]. These scalars will be seen to satisfy an equation of motion of schematic structure, The above equation of motion is at the heart of the recursive method proposed in this paper; solving it using a perturbiner [8] expansion in terms of recursively defined coefficients φ A|B is equivalent to a Berends-Giele recursion 2 that computes the α ′ -expansion of the disk integrals (1.1) as if they were tree amplitudes of an effective field theory, Therefore this paper gives a precise meaning to the perspective on disk integrals as Z-theory amplitudes [7] by pinpointing its underlying equation of motion. After this fundamental conceptual shift to extract the α ′ -expansion of disk integrals from the equation of motion of Φ, its form to all orders in α ′ is proposed to be [Φ 12...l , Φ p,p−1...l+1 ] (z 12 z 23 . . . z l−1,l )(z p,p−1 z p−1,p−2 . . . z l+2,l+1 ) + perm(2, 3, . . . , p−1) .
The detailed description of the above result will be explained in section 4, but here we note its remarkable structural similarity with a certain representation of the superstring disk amplitude for massless external states [11]. The (n−2)!-term representation which led to the all-order proposal (1.4) has played a fundamental role in the all-multiplicity derivation of local tree-level numerators [12,4] which obey the duality between color and kinematics [13]. 1 The word "effective" deserves particular emphasis since the high-energy properties of Ztheory (and its quantum corrections) are left for future investigations. 2 For a recent derivation of Berends-Giele recursions for tree amplitudes from a perturbiner solution of the field-theory equations of motion, see [9,4]. An older account can be found in [8,10].

Z-theory and double copies
The relevance of the disk integrals (1.1) is much broader than what the higher-derivative completion of field theory might lead one to suspect. They have triggered deep insights into the anatomy of numerous field theories through the fact that closed-string tree-level integrals (encoding α ′ -corrections to supergravity theories) boil down to squares of disk integrals through the KLT relations [14]. In a field-theory context, this double-copy connection between open and closed strings became a crucial hint in understanding quantum-gravity interactions as a square of suitably-arranged gauge-theory building blocks [13,15].
Double-copy structures have recently been identified in the tree-level amplitudes of additional field theories [16]. For instance, classical Born-Infeld theory [17] which governs the low-energy effective action of open superstrings [18] turned out to be a double copy of gauge theories and an effective theory of pions known as the non-linear sigma model (NLSM) [19], see [20] for its tree-level amplitudes. As a string-theory incarnation of the Born-Infeld double copy, tree-level amplitudes of the NLSM have been identified as the low-energy limit of the disk integrals in the scattering of abelian gauge bosons [7]. This unexpected emergence of pion amplitudes exemplifies that disk integrals also capture the interactions of particles that cannot be found in the string spectrum.
Moreover, the entire tree-level S-matrix of massless open-superstring states can be presented as a double copy of SYM with α ′ -dependent disk integrals [5]. Their Z-theory interpretation in [7] was driven by the quest to identify the second double-copy ingredient of the open superstring besides SYM. In view of the biadjoint-scalar and NLSM interactions in the low-energy limit of Z-theory, its full-fledged α ′ -dependence describes effective higherderivative deformations of these two scalar field theories [7]. As a double-copy component to complete SYM to the massless open-superstring S-matrix, the collection of effective interactions encompassed by Z-theory deserve further investigations.
In this work, we identify the equation of motion (1.4) of the full non-abelian Ztheory, where the integration domain of the underlying disk integrals endows the putative scalars Φ with a second color degree of freedom. By the results of [5], disk integrals in their interpretation as Z-theory amplitudes obey the duality between color and kinematics due to Bern, Carrasco and Johansson (BCJ) [13] in one of their color orderings. Hence, the effective theories gathered in Z-theory are of particular interest to advance our understanding of the BCJ duality. The abelian limit of Z-theory arises from disk integrals without any notion of color ordering in the integration domain and has been studied in [7] as a factory for BCJ-satisfying α ′ -corrections to the NLSM. The present article extends this endeavor such as to efficiently compute the doubly-partial amplitudes of effective bi-colored theories with BCJ duality in one of the gauge groups and explicitly known field equations (1.4).

Outline
This paper is organized as follows: Following a review of disk integrals and the Berends-Giele description of their field-theory limit in section 2, the Berends-Giele recursion for their α ′ -corrections and the resulting field equations of non-abelian Z-theory are presented in section 3. The mathematical tools to control the equations of motion to all orders in the fields and derivatives by means of suitably regularized polylogarithms are elaborated in section 4. In section 5, the Berends-Giele recursion is extended to closed-string integrals over surfaces with the topology of a sphere before we conclude in section 6. Numerous appendices and ancillary files complement the discussions in the main text.
The BG recursion that generates all terms up to the α ′ 7 -order in the α ′ -expansion of disk integrals at arbitrary multiplicity as well as the auxiliary computer programs used in their derivations can be downloaded from [21].

Review and preliminaries
In this section, we review the definitions and symmetries of the disk integrals under investigations as well as their appearances in tree amplitudes of massless open-string states.
We also review the recent Berends-Giele approach to their field-theory limit in order to set the stage for the generalization to α ′ -corrections.

String disk integrals
We define a cyclic chain C(Q) of worldsheet propagators z −1 ij with z ij ≡ z i − z j on words Q ≡ q 1 q 2 . . . q n of length n as Then, the iterated disk integrals on the real line that appear in the computation of opensuperstring tree-level amplitudes are completely specified by two words P and Q, where P ≡ p 1 p 2 . . . p n encodes the domain of the iterated integrals, Mandelstam variables s ij...p involving legs i, j, . . . , p are defined via region momenta k ij...p , 4) and the more standard open-string conventions for the normalization of α ′ (which would cause proliferation of factors of two) can be recovered by globally setting α ′ → 2α ′ everywhere in this work. In the sequel, we refer to the word P as the integration region or domain and to Q as the integrand of (2.2), where P is understood to be a permutation of Q. The inverse volume vol(SL(2, R)) of the conformal Killing group of the disk instructs to mod out by the redundancy of Möbius transformation z → az+b cz+d (with ad − bc = 1). This amounts to fixing three positions such as (z 1 , z n−1 , z n ) = (0, 1, ∞) and to inserting a compensating Jacobian: Given that the words P and Q in the disk integrals (2.2) encode the integration region D(P ) in (2.3) and the integrand C(Q) in (2.1), respectively, there is in general no relation between Z(P |Q) and Z(Q|P ). This can already be seen from the different symmetries w.r.t. variable P at fixed Q on the one hand and variable Q at fixed P on the other hand.
Finally, integration by parts yields the same BCJ relations among permutations of Z(P |Q) in Q as known from [13] for color-stripped SYM tree amplitudes [5] Note that neither (2.7) nor (2.10) depends on the domain P , and they allow to expand The symmetries (2.6), (2.7) and (2.10) known from SYM interactions crucially support the interpretation of Z(P |Q) as doubly partial amplitudes [7].

Symmetries of disk integrals in the domain
As a consequence of the form of the integration region D(P ) in (2.3), disk integrals obey a cyclicity and parity property in the domain P = p 1 p 2 . . . p n , Z(p 2 p 3 . . . p n p 1 |Q) = Z(p 1 p 2 . . . p n |Q) , Z(P |Q) = (−1) |P| Z(P |Q) , (2.11) which tie in with the simplest symmetries (2.6) of the integrand Q. However, the Kleiss-Kuijf symmetry (2.7) and BCJ relations (2.10) of the integrand do not hold for the integration domain P in presence of α ′ -corrections. This can be seen from the real and imaginary part of the monodromy relations [26,27] (see [28] for a recent generalization to loop level)

Open superstring disk amplitudes
The n-point tree-level amplitude A open of the open superstring takes a particularly simple form once the contributing disk integrals are cast into an (n−3)! basis via partial fraction (2.8) and integration by parts (2.10) [11,29]: While all the polarization dependence on the right hand side has been expressed through the BCJ basis [13] of SYM trees A SYM , the entire reference to α ′ stems from the integrals where P = p 2 p 3 . . . p n−2 and Q = q 2 q 3 . . . q n−2 are permutations of 23 . . . n−2. The original derivation [11,29] of (2.13) and (2.14) has been performed in the manifestly supersymmetric pure spinor formalism [30], where the SYM amplitudes A SYM in (2.13) have been identified from their Berends-Giele representation in pure spinor superspace [31]. Hence, (2.13) applies to the entire ten-dimensional gauge multiplet in the external states 3 .

Z-theory
After undoing the SL(2, R)-fixing in (2.5), the integrals F P Q can be identified as a linear combination of disk integrals (2.2) [5], where P, Q and R are understood to be permutations of 2, 3, . . . , n−2. The symmetric (n−3)! × (n−3)! matrix S[Q|R] 1 encodes the field-theory KLT relations [33,34] (see also [35] for the α ′ -corrections to S[Q|R] 1 ) and admits the following recursive representation [7], upon replacing the right-moving SYM trees viaÃ SYM (1, R, n, n−1) → Z(P |1, R, n, n−1) [5]. The KLT form of (2.17) reveals the double-copy structure of the open-superstring treelevel S-matrix which in turn motivated the proposal of [7] to interpret disk integrals as doubly partial amplitudes. The specification of disk integrals by two cycles P, Q identifies the underlying particles to be bi-colored scalars, and we collectively refer to their effective interactions that give rise to tree amplitudes Z(P |Q) as Z-theory.

α ′ -expansion of disk amplitudes
The α ′ -expansion of disk amplitudes ( The MZV in (2.18) is said to have depth r and weight w = n 1 + n 2 + . . . + n r (which is understood to be additive in products of MZVs). While the four-point instance of (2.14), boils down to a single entry with Riemann zeta values ζ n of depth r = 1 only, disk integrals at multiplicity n ≥ 5 generally involve MZVs of higher depth r ≥ 2, see [37] for a recent closed-form solution at five points. It has been discussed in the literature of both physics [29,38,39] and mathematics [40,41] that the disk integrals (2.2) at any multiplicity exhibit uniform transcendentality: Their α ′w -order is exclusively accompanied by products of MZVs with total weight w.  [39,43]. At multiplicities five, six and seven, explicit results for the leading orders in the α ′ -expansion of F P Q are available for download on [44].

Basis-expansion of disk integrals
In setting up the Berends-Giele recursion for the fundamental objects Z(A|B) of this work, it is instrumental to efficiently extract their α ′ -expansion from the basis functions F P Q . However, solving the mediating BCJ and monodromy relations can be very cumbersome, and the explicit basis expansions spelled out in [5] only address an (n−2)! subset of integrands B. These shortcomings are surpassed by the following formula, where m(A|B) denote the doubly partial amplitudes of biadjoint φ 3 -theory which arise in the field-theory limit of disk integrals [45] m(A|B) = lim

Berends-Giele recursion for the field-theory limit
The task we want to accomplish in this paper concerns the computation of the α ′ -expansion of the disk integrals (2.2) in a recursive and efficient manner. In the field-theory limit α ′ → 0, all-multiplicity techniques have been developed in [29], and a relation to the inverse KLT matrix (2.16) has been found in [5]. The equivalent description of the α ′ → 0 limit in terms of doubly partial amplitudes (2.21) [45] has inspired a recent Berends-Giele a|b t a ⊗t b . The latter take values in the tensor product of two gauge groups with generators t a andt b as well as structure constants f acd andf bgh , respectively. 4 After pioneering work in [42], the α ′ -expansion of disk integrals at multiplicity n ≥ 5 has later been systematically addressed via all-multiplicity techniques based on polylogarithms [5] and the Drinfeld associator [6] (see also [43]).
The superscript of the biadjoint scalar Φ (0) indicates that this is the α ′ → 0 limit of the Z-theory particles Φ whose interactions give rise to the disk integrals Z(P |Q) as their doubly partial amplitudes. The non-linear field equations in the low-energy limit with d'Alembertian ≡ ∂ 2 will later be completed such as to incorporate the α ′ -corrections in Z(P |Q). One can solve (2.22) through a perturbiner [8] which resums tree-level subdiagrams and is compactly written as a sum over all words A, B with length |A|, |B| ≥ 1 in the last line. We are using the collective notation and referred to as Berends-Giele double currents φ (0) A|B . The notation A 1 A 2 =A and B 1 B 2 =B instructs to sum over deconcatenations A = a 1 a 2 . . . a |A| into non-empty words A 1 = a 1 a 2 . . . a j and A 2 = a j+1 . . . a |A| with j = 1, 2, . . . , |A|−1 and to independently deconcatenate B in the same manner. The initial conditions for the recursion in (2.25),
guarantee that φ (0) A|B vanishes unless A is a permutation of B and yield expressions such as  (2.27) at the two-and three-particle level.
As shown in [4], the field-theory limits of the disk integrals (2.2) and thereby the doubly partial amplitudes (2.21) are given by the Berends-Giele double currents φ A|B in (2.25) gives rise to an efficient algorithm to obtain the field-theory limit of disk integrals Z(A, n|B, n) directly from the two words A, B encoding the integrand and integration domain, respectively. Furthermore, the BG double currents allow the inverse of the KLT matrix (2.16) to be obtained without any matrix algebra [4], (2.29)

Example application of the Berends-Giele recursion
The computation of the field-theory limit of the five-point disk integral using the Berends-Giele formula (2.28) proceeds as follows. First, one exploits the cyclic symmetry of the integrand to rotate its labels until the last leg matches the last label of the integration region. After applying (2.28) one obtains, Terms such as φ 352|132 following from the deconcatenation (2.25) have been dropped from the last equality because the condition (2.26) implies that φ    32) in agreement with the expression for the doubly partial amplitude m(13524|32451) that follows from the methods of [45]. In the next section this method will be extended to compute the α ′ -corrections of string disk integrals.

Berends-Giele recursion for disk integrals
In this section, we develop a Berends-Giele recursion 6 for the full-fledged disk integrals Z(P |Q) defined in (2.2). The idea is to construct α ′ -dependent Berends-Giele double currents φ A|B such that the integrals Z(P |Q) including α ′ -corrections are obtained in the same manner as their field-theory limit in (2.28), And similarly, the α ′ -corrected BG double currents φ A|B in (3.1) will be given by the coefficients of a perturbiner expansion analogous to (2.23),  [7]. In addition, the BG double currents above are subject to the initial and vanishing condition Given their role in equation (3.1), the words A and B on the BG double current φ A|B will be referred to as the integration domain A and the integrand B, respectively.

Symmetries of the full Berends-Giele double currents
In the representation (3.1) of the disk integrals, their parity symmetries (2.7) and (2.11) can be manifested if the double currents φ A|B satisfy (3.5) 6 For a review of the Berends-Giele recursion for gluon amplitudes [3] which is adapted to the current discussion, see section 2 of [4]. 7 In the mathematics literature, objects T B satisfying the symmetry T P ¡Q = 0 for any P, Q = ∅ are known as "alternal moulds", see e.g. [47].
Note that φ A|B does not exhibit shuffle symmetries in the integration domain A: The α ′correction in the monodromy relations [26,27], more specifically in the real part of (2.12), yields non-zero expressions 8 O((α ′ π) 2 ) for φ P ¡Q|B . As a consequence, the perturbiner (3.2) is Lie-algebra valued w.r.t. thet b generators [23] but not w.r.t. the t a generators. That is why the Z-theory scalar Φ is referred to as bi-colored rather than biadjoint.
The symmetries (3.4) and (3.5) will play a fundamental role in the construction of ansaetze for the α ′ -corrections of the Berends-Giele double currents, see appendices A and B for further details.

The α ′ 2 -correction to Berends-Giele currents of disk integrals
Assuming that the α ′ 2 -corrections of the integrals (2.2) can be described by Berends-Giele double currents as in (3.1), dimensional analysis admits two types of terms at this order.
They have the schematic form k 2 φ 3 and φ 4 since φ has dimension of k 2 , and the α ′ 2 -terms contain a factor of k 4 compared to the leading contribution from φ 2 in (2.25). Therefore, an ansatz for s A φ A|B at this order must be based on a linear combination of into non-empty words. By the initial condition (3.3), φ A|B vanishes unless A is a permutation of B, so there is no need to consider momentum dependence of the form ( The most general linear combination of the terms (3.6) contains 36 + 24 = 60 parameters. Imposing the symmetries (3.4) and (3.5) reduces them to 6 + 4 = 10 parameters, see appendix A for the implementation of the shuffle symmetry. Then, matching the outcomes of (3.1) with the known α ′ 2 -order of various integrals at four and five points fixes six parameters, leaving a total of four free parameters. The α ′ 2 -order of (n ≥ 6)-point integrals does not provide any further input: As we have checked with all the known (n ≤ 9)-point data [44], they are automatically reproduced for any choice of the four free parameters. This is where the predictive power of the Berends-Giele setup kicks in: A finite amount of low-multiplicity data -the coefficients of k 2 φ 3 -and φ 4 -terms (3.6) at the α ′ 2 -orderdetermines the relevant order of disk integrals at any multiplicity. 8 Since the monodromy relations only differ from the KK relations by rational multiples of π 2n or (ζ 2 ) n , the sub-sector of Z(A, n|B, n) without any factors of ζ 2 still satisfies shuffle symmetries, e.g. φ P ¡Q|B ζ 2n+1 = 0, also see [48] for analogous statements for the heterotic string and section 5 for implications for a Berends-Giele approach to closed-string integrals.

Free parameters versus Z-theory equation of motion
It is not surprising that the ansatz based on (3.6) is not completely fixed (yet) by matching the data. The reason for this can be seen from the interpretation of the Berends-Giele recursion method as the perturbiner solution (2.23) to the Z-theory equation of motion with in turn leads to another appearance of Φ at higher orders in α ′ and the fields. In order to obstruct an infinite iteration of the field equations, we fix three additional parameters by demanding absence of (k A i · k A i ) with i = 1, 2, 3 and thereby leave one free.
The last free parameter reflects the freedom to perform field redefinitions. Terms of i.e. the right-hand side of Φ ′ will no longer contain the term α ′ 2 ζ 2 (Φ 3 ) in question. This leftover freedom can be fixed by requiring the absence of the dot product (k A 1 · k A 3 ) among the leftmost and the rightmost slot-momentum 9 in the deconcatenation At the end of the above process, one finds the unique recursion that generates the α ′ 2 terms in the low-energy expansion of disk integrals at any multiplicity via (3.1): In general, in a p-fold deconcatenation among the leftmost and the rightmost momentum will not be included into an ansatz for s A φ A|B at given order in α ′ . This freezes the freedom to perform field redefinitions while preserving the manifest parity property (3.4) in A.
For example, applying the above recursion to the disk integral Z(13524|32451) whose fieldtheory limit was computed in (2.32) leads to the following result up to α ′ 2 : It is important to emphasize that, while only four-and five-point data entered in the derivation of (3.7), this recursion allows the computation of α ′ 2 terms of disk integrals at arbitrary multiplicity. The eleven-point example with the shorthands a = 10 and b = 11 was computed within four seconds on a regular laptop with the program available in [21].

Manifesting the shuffle symmetries of BG currents
The length of the recursion in (3.7) at the α ′2 ζ 2 order calls for a more efficient representation. In this subsection, we identify the sums of products of φ A i |B j which satisfy the shuffle symmetries (3.5) in the B j -slots. This allows to rewrite the recursion (3.7) in a compact form which inspires the generalization to higher orders and clarifies the commutator structure in the Z-theory equation of motion upon rewriting the results in the language of perturbiners (3.2).
In order to do this, recall from the theory of free Lie algebras that all shuffle products are annihilated by a linear map ρ acting on words ( For example, it is easy to see that ρ(B 1 , B 2 ) = (B 1 , B 2 ) − (B 2 , B 1 ) and imply the vanishing of ρ(B 1 ¡B 2 ) and ρ((B 1 , B 2 )¡B 3 ). Therefore, after defining it is straightforward to check that the following linear combinations 14) The first few examples of (3.13) read as follows, and their shuffle symmetries (3.14) are easy to verify, starting with Moreover, the ρ-map in (3.10) exhausts all tensors of the type (3.12) subject to shuffle symmetry in the B j -slots it acts on [23,49]. Hence, a BG recursion which manifests the shuffle symmetry in the B j -slots is necessarily expressible in terms of T B 1 ,B 2 ,...,B n A 1 ,A 2 ,...,A n in (3.13). Rather surprisingly, it turns out that the definition (3.13) not only manifests the shuffle symmetries on the B j -slots but also implies generalized Jacobi identities with respect to the A j -slots. In other words, the above T B 1 ,B 2 ,...,B n A 1 ,A 2 ,...,A n satisfy the same symmetries as the

Simplifying the α ′2 -correction to BG currents
As discussed in the previous subsection, the BG double current can always be written in ..,A n from the definition (3.13). For example, the expression (3.7) becomes From a practical perspective, it could be a daunting task to convert a huge expression in ..,A n on the right-hand side of (3.17). Fortunately, since both the BG double current and T B 1 ,B 2 ,...,B n A 1 ,A 2 ,...,A n satisfy generalized Jacobi identities in the A j -slots, an efficient algorithm due to Dynkin, Specht and Wever [50] can be used to accomplish this at higher orders in α ′ . See the appendix A.3 for more details.

The perturbiner description of α ′ -corrections
The recursion (3.17) for the coefficients φ A|B of the perturbiner (3.2) can be rewritten in a more compact form by defining the shorthand which exploits the generalized Jacobi symmetry of the A j -slots in T That is, the numeric indices i 1 , i 2 , . . . , i p of the various formal perturbiners Φ i in the commutator match the ordering of the labels within the A-slots in T ..,A i p , while the ordering of the B-slots is always the same. Finally, the color degrees of freedom enter in a global The above definition implies that the Berends-Giele recursion (3.17) condenses to, with the following shorthand for the derivatives: The convention for the derivatives ∂ j is to only act on the position of Φ j , e.g. the perturbiner In view of the increasing number of Φ-factors at higher order in α ′ , we will further lighten the notation and translate the commutators into multiparticle labels which exhibit generalized Jacobi symmetries by construction 11 . Hence, any subset of the nested commutators of (3.19) can be separately expressed in terms of Φ P ; e.g.
In this language, the Z-theory equation of As will be explained below, this form of the Z-theory equation of motion provides the essential clue for proposing the Berends-Giele recursion to arbitrary orders of α ′ .
As a reformulation of (3.19) which does not rely on the notion of perturbiners, one can peel off the t a generators 12 from the bi-colored fields Φ = A t A Φ A . The coefficients Φ A are still Lie-algebra valued with respect to thet b , and this is where the nested commutators act in the following rewriting of (3.19): Upon comparison with (3.22), the notation in (3.21) can be understood as a compact way to track the relative multiplication orders of the t a andt b generators. 11 These are the same symmetries in P = i 1 i 2 . . . i p obeyed by contracted structure constants . . . f xi p y as well as the local multiparticle superfields V P [51] in pure spinor superspace. 12 In view of the α ′ -corrections to KK relations from (2.12), the Z-theory scalar Φ is not Liealgebra valued in the gauge group of the t a but instead exhibits an expansion in the universal enveloping algebra spanned by t A = t a 1 t a 2 . . . t a |A| .

Perturbiners at higher order in α ′
The procedure of subsection 3.2 to determine the Berends-Giele recursion that reproduces the α ′ 2 -corrections to the disk integrals was also applied to fix the recursion at the orders α ′ 3 and α ′ 4 (see appendix B for more details). Luckily, the analogous ansaetze at orders α ′w≥5 could be bypassed since the general pattern of the field equations became apparent from the leading orders α ′w≤4 . To see this it is instructive to spell out the Z-theory equation of motion up to the α ′ 3 -order: to the first regular terms in the expansion of the four-point disk integrals considered in [5]: The endpoint divergences of these integrals as z 2 → z 1 = 0 and z 2 → z 3 = 1 require a regularization prescription denoted by "reg" and explained in section 4. The infinite sums in the above integrands arise from the Taylor expansion of a SL(2, R)-fixed four-point Koba-Nielsen factor via which removes the kinematic poles from the full disk integrals and yields their non-singular counterparts [5] upon regularization. Comparing the expansion of (3.25) at the next order in α ′ with the expression for the BG current obtained from an ansatz confirms the pattern, and we will later on see that the terms of order Φ 4 and Φ 5 in (3.24) can be traced back to regularized five-and six-point integrals.

All-order prediction for the BG recursion
From the observations in the previous subsection, we propose a closed form for the Φ 3 contributions to the Z-theory equations of motion for Φ, to all orders in α ′ : The integrand in the second line bears a strong structural similarity to the correlation function in the four-point open string amplitude [11,52] A open (1, 2, 3, 4) = −α ′ with V P V Q V n denoting certain kinematic factors in pure spinor superspace. The precise correspondence between (3.27) and (3.28) maps multiparticle vertex operators V P [51] to perturbiner commutators Φ P defined in (3.21). Moreover, since V P is fermionic and satisfies generalized Jacobi symmetries [51], the all-multiplicity mapping preserves all the symmetry properties of its constituents. Finally, the Koba-Nielsen factor 3 i<j |z ij | α ′ s ij with s ij → ∂ ij has been Taylor expanded according to (3.26) in converting (3.28) to (3.27). This projects out the kinematic poles of the integrals to ensure locality of the Z-theory equation of motion, but requires a regularization of the endpoint divergences at z 2 → 0 and z 2 → 1 as discussed in section 4.
It is easy to see that the correspondence (3.29) correctly "predicts" the first term in the right hand side of (3.27) from the well-known [30] expression A open (1, 2, 3) = V 1 V 2 V 3 of the three-point massless disk amplitude under the mapping (3.29); Extrapolating the above pattern, a natural candidate for the higher-order contributions Φ 4 , Φ 5 , . . . to the Z-theory equation of motion emerges from the integrand of the (n−2)!-term representation of the n-point disk amplitude [11], This expression leads us to propose the following Z-theory equation of motion to all orders in the fields and their derivatives (with SL(2, R)-fixing z 1 = 0 and z p = 1): For example, the equation of motion up to Φ 4 -order following from (3.31) reads The above prescription was derived by trial and error through comparison with known data for Z(P |Q) at low number of points, and its consequences extrapolated to arbitrary multiplicity. It remains an open question to find its rigorous mathematical justification.

Multiple polylogarithms and their regularization
In this section, we review selected aspects of the polylogarithm-based setup of [5] to extract local terms (also called regular terms) from the disk integrals 14 in (3.30). The requirement that the eom map must reproduce the correct Z-theory equation of motion induces systematic departures from [5] which will be highlighted in the subsequent discussion.

Polylogarithms and MZVs
We recall that multiple polylogarithms G(A; z) with A = a 1 , a 2 , . . . , a n and a j , z ∈ C are defined by 15 G(a 1 , a 2 , . . . , a n ; z) ≡ z 0 dt t − a 1 G(a 2 , . . . , a n ; t), G(∅; z) ≡ 1, ∀z = 0 , 14 There is a vast body of literature related to iterated integrals on moduli spaces of genus-zero curves with n ordered marked points, see e.g. [41,53,54,55] and references therein. Moreover, their symbolic computation have been recently implemented in computer programs [56,57]. 15 Our conventions for polylogarithms agree with the work [58] of Goncharov as well as for instance reference [59]. See e.g. [60] for other aspects of multiple polylogarithms.

Polylogarithms and the Koba-Nielsen factor
Using the special cases of the multiple polylogarithms (4.1), can be written as [5] p i<j ; z j ) .

(4.5)
Therefore, the leading orders of the regularized four-point integrals in (3.25) can be traced back to with a ∈ {0, 1}, using (4.1) to perform the z 2 -integral as well as (4.3) to convert the results to MZVs. Divergent cases as exemplified in (D.1) are addressed by the regularization scheme which is denoted by "reg" in (4.6) and will be the subject of the next subsection.

Regularization of endpoint divergences
It follows from their definition (4.1) that multiple polylogarithms diverge at the endpoints of the integration domains whenever a 1 = z or a n = 0, and therefore they need to be regu- formally tends to G(0; z) in the ǫ → 0 limit, and its regularized value ln |z| can be obtained from the right hand side by manually discarding (the source of divergences) ln |ǫ|. Together with a similar reasoning for divergences from the upper integration limit, we specify the following regularized values for divergent integrals at weight one 16 : Further subtleties arise in situations where the endpoint divergence as t → z is approached from above. In this case, one defines where the occurrence of imaginary parts is an artifact of the decomposition of the integration domains in later sections. The choice of sign along with iπ in (4.9) is a convention, and the cancellation of imaginary parts in the Z-theory equation of motion serves as a consistency check of our integration setup.
One can combine (4.8) with (4.9) such as to define the regularized value of G(z; z) via where δ = 0 and δ = 1 if G(z; z) is obtained after integration over t such that t < z and t > z, respectively.
Since the regularization scheme in this work is defined to preserve the shuffle algebra In contrast to the regularizations (4.8) and (4.10) of this work which are selected by the Z-theory equation of motion, the regularization scheme of [5] preserves the scaling property of polylogarithms and implies a vanishing regularized value for G(z; z).  16 We are indebted to Erik Panzer for suggesting this regularization to us.
It turns out that between the two orders of integration displayed in (4.12), the Z-theory equation of motion (3.24) (obtained from an ansatz for the equivalent BG recursion) is reproduced at the α ′ 3 ζ 3 order only if the regularized integral of ln |z 23 |/(z 12 z 13 ) vanishes.
Therefore z 2 must be integrated prior to z 3 in presence of (z 12 z 13 ) −1 in a five-point integrand. By worldsheet parity z j → z 5−j , the integral over (z 24 z 34 ) −1 must then follow the converse order where z 3 is integrated first. The conclusion here is that different integrands require different orders of integration. Adapting the integration order to each integrand will be part of the map eom to be elaborated below.
Similarly, we identified the appropriate integration orders for the 4! six-point integrals in (3.31) at p = 5 by matching with the α ′ 3 and α ′ 4 order of the Berends-Giele recursion obtained from an ansatz. Moreover, an alternative method to determine the desired outcome of regularized integrals to arbitrary orders in α ′ is presented in appendix E which closely follows the handling of poles in [5]. An all-multiplicity algorithm to determine the integration orders which are observed to reproduce the Z-theory equation of motion will be described in section 4.3. As a preparation for this, however, a systematic change of integral bases via repeated use of partial-fraction identities will be introduced in the next section. 17 In order to evaluate the second integral of (4.12) through the definition (4.1) of polylogarithms, the integration limits are rearranged according to

Towards the simpset basis
Our investigations showed that the (n−2)! integrals in the open-superstring amplitude (3.30) need to be rewritten in a very particular basis to define the eom prescription in the Z-theory equation of motion (3.31). In this section, we will introduce a basis where the eom prescription can be associated with appropriate integration orders for the regularized integrals such as to settle the ambiguity seen in (4.12). In order to explain this change of basis 18 it will be convenient to introduce the following chain of worldsheet propagators in which two consecutive z ij factors in the denominator always share a label, with a formal extension Z A ≡ 1 to words of length |A| = 1. One can check that z ij = −z ji and partialfraction identities (4.11) imply the shuffle symmetry [62] Z A¡B = 0 , ∀A, B = ∅ .

Description of the algorithm
At generic multiplicity, the elements of the simpset basis are obtained from the chain basis Z 1P Z (n−1)Q by recursively stripping off factors of Z ij = z −1 ij . At each step, the shuffle symmetry (4.14) is applied to Z 1P and Z (n−1)Q to factor out Z ij , where i and j are the labels in 1P which are maximally apart (i.e. at highest value of |i − j|). This procedure is repeated for the coefficient Z R in the decomposition Z 1P = Z ij Z R , leading to a recursive algorithm.
In a factor of Z 1243 relevant at six points, the labels 1 and 4 constitute the pair which is maximally apart with a separation of |1 − 4| = 3. Therefore, to arrive at the elements in 18 The "basis" of dimension (n−2)! refers to the minimum elements under partial-fraction identities; integration by parts further reduce their number to (n−3)! [11,29]. The reduction of products of z −1 ij via partial fractions to a (n−2)!-dimensional basis is also described in appendix A of [61].
the simpset basis, one needs to rewrite Z 1243 in such a way as to contain the factor z −1 14 . In this case it is easy to show using partial-fraction identities that in which the factor Z 14 = z −1 14 has been stripped off from the chain Z 1243 . The two integrals on the right-hand side of (4.15) belong to the simpset basis since Z 12 Z 34 cannot be written as a single chain factor Z R and the maximally separated labels 2, 4 in Z 24 Z 34 = −Z 243 are already factored out.
The following recursive algorithm implements the change of basis required by the eom map. For each factor of Z R one identifies the pair of labels i and j that are maximally separated and recursively applies the following corollaries of (4.14) and (4.13), 16) which eventually stops at Z ija = Z ij Z ja where the factor Z ij is singled out.
In order to illustrate the algorithm (4.16), consider the seven-point integral characterized by the factor Z 63425 with five labels. Since the labels 2 and 6 are maximally separated, the second identity in (4.16) rewrites it as Z 63425 = Z 6342 Z 25 . The first factor now contains only four labels and iterating the application of the identities in (4.16) yields, In order to arrive at the second line, the factor Z 234 was manipulated w.r.t the maximallyseparated labels 2 and 4 (with similar considerations for the other factor Z 632 ). Therefore, is the transformation from the chain to the simpset basis.
The first non-trivial application of the above algorithm leads the five-point simpset The complete set of denominators in the six-point simpset basis can be found in (4.26) (upon adjoining their parity images under z j → z 6−j ), while the appendix F contains an overview of the seven-point simpset basis.

Back to the chain basis
For completeness, it is straightforward to exploit the shuffle symmetry (4.14) to obtain a recursive algorithm to expand the simpset basis elements back in the chain basis 19 . To motivate the algorithm below, consider the following example: To rewrite Z 12 Z 13 Z 14 in the chain basis note that Z 12 Z 13 = −Z 213 . Next, to make a chain out of Z 213 Z 14 one uses the identity Z A1B = (−1) |A| Z 1(Ã¡B) in the first factor to allow it to be prefixed by completes the basis change to Z 12 Z 13 Z 14 = Z 1234 + perm (2,3,4).
Hence, the general algorithm to expand the simpset basis elements in the chain basis is based on the recursive application of the following two identities, The second identity follows from (4.14) and implies that the basis dimension of Hamilton paths Z P is (|P | − 1)!.

Integration orders for the simpset elements
In the simpset basis of integrals attained through the algorithm (4.16), we can now complete the definition of the eom map in (3.31). For each simpset element, the algorithm to be described in this section identifies at least one integration order for which the regularized 20 integrals involving the Koba-Nielsen factor (4.5) are observed to yield the correct Z-theory equation of motion.
It should be emphasized once more that the order of integration must not be confused with the integration domain in (3.31) which is always fixed to be 0 ≤ z 2 ≤ z 3 ≤ . . . ≤ z p−1 ≤ 1. Instead, "order of integration" refers to the decision whether an iterated integral over z 2 , z 3 subject to 0 ≤ z 2 ≤ z 3 ≤ 1 is represented as In the first case, the integration over z 2 is performed first, and we will write 23, whereas the opposite integration order will be referred to through the shorthand 32, with obvious generalization to higher multiplicity.

Description of the algorithm
Let us introduce a formal operator "ord" that takes as input a product of z ij from the denominators in the simpset basis and outputs a combination of words encoding the admissible integration orders. For example, ord(z 12 z 13 ) = 23 for the integrand in (4.12) means that eom requires the integral over z 2 to be performed first, followed by z 3 .
In order to describe a recursive algorithm to determine the order of integration, we associate a graph to each element in the simpset basis where each factor of z ij contributes an edge between vertices i and j. Then, ord(. . .) for a given element of the simpset basis can be obtained by repeated application of two steps: 1. If the graph of z a 1 a 2 . . . z a n a n+1 = ( , apply ord(. . .) to each of its connected subgraphs representing . . z c q c q+1 and shuffle the resulting words, ord(z a 1 a 2 . . . z a n a n+1 ) = ord(z The shuffle between the ordered sequences ijk . . . generated by the individual ord(. . .) operators indicates that the associated integrations commute, e.g., 23¡4 means that any integration order among 234, 243, 423 is allowed.
2. If the element z a 1 a 2 . . . z a n a n+1 z ij is represented by a connected graph where z ij is the factor with maximal separation |i − j|, and j corresponds to the integration variable that has not yet been pulled out of ord(. . .), then ord(z a 1 a 2 . . . z a n a n+1 z ij ) = ord(z a 1 a 2 . . . z a n a n+1 )j . The above ordering means that any permutation of 2345 such that 4 and 5 appear before 3 (e.g. 5243) defines a viable integration order, while the position of 2 is arbitrary.

Examples
Let us list the outcomes of the above algorithm for a few elements. At four points, the two-dimensional basis has a unique order: ord(z 12 ) = 2 , ord(z 23 ) = 2 .

Iterated integrals and integration order
The above algorithm generates the allowed integration orders for all the (n−2)! elements in the simpset basis, with p = n−1 in the Z-theory equation of motion (3.31). Since the integration domain is always 0 ≤ z 2 ≤ z 3 . . . ≤ z p−1 ≤ 1, one can show that the resulting . . a k translate into the following iterated integrals 27) with lower limits b j ≡ max{x ∈ {0, z a j+1 , z a j+2 , . . . , z a k } | x ≤ z a j } as well as upper limits

Summary and overview example
As discussed in the previous subsections, the eom  with Koba-Nielsen insertions (4.5) The above steps will be illustrated through a simple yet representative example The calculations are long and tedious to perform by hand, but they are straightforward to automate in a computer 21 .
The chain basis element Z 5243 under discussion also belongs to the simpset basis with ord(z 52 z 24 z 43 ) = 342. Hence, the eom map instructs to evaluate the regularized integral where the reference to the shuffle regularization scheme (4.8) and (4.10) via "reg" will be left implicit in the remainder of this section. The integration limits in (4.29) associated to 21 We are releasing our code that performs this task via [21]. The evaluation of the 8! = 40.320 integrals in the 10-point simpset basis to their leading order ∼ ζ 7 , ζ 2 ζ 5 , ζ 2 2 ζ 3 takes about two hours on a laptop. The program is written in FORM [63], and improvements to the code are certainly possible and highly welcomed.
In addition to the above shuffle regularizations, the following z-removal identities based on G(0; 1) = G(0, 0; 1) = 0 are needed to perform the final integration over z 2 : In combination with the shuffle algebra (4.2), the identities in (4.37) yield the following results for the remaining integral over z 2 (setting z 5 = 1): Finally, summing the above results yields the regularized value of the integral (4.29), previously obtained from an ansatz. 22 Fortunately, the independent proposal for the regularized value for the integral (4.29) inspired by the methods of [5] and described in the appendix E allowed us to fix all these subtleties. This ultimately led us to our final regularization prescription that has ever since passed many tests at much higher order in α ′ .

Closed-string integrals
Our results have a natural counterpart for closed-string scattering, where tree-level amplitudes involve integrals over worldsheets of sphere topology. Similar to the characterization of disk integrals (2.2) via two cycles P and Q, any sphere integral in tree-level amplitudes of the type II superstring 23 [39] boils down to The inverse volume of the conformal Killing group SL(2, C) of the sphere generalizes (2.5) in an obvious manner, and C(Q) denotes the complex conjugate of the chain (2.1) of worldsheet propagators with z ij → z ij .
While the field-theory limit of the sphere integrals (5.1) yields the same doubly partial amplitudes as the corresponding disk integrals [48], only a subset of the α ′ -corrections in Z(P |Q) can be found in the closed string (5.1). These selection rules obscured by the KLT relations [14] have been identified to all orders in [39] and realize the single-valued projection "sv" [64] of the MZVs in the disk integrals [65,48] W (P |Q) = sv Z(P |Q) . (5. 3) The single-valued map projects Riemann zeta values to their representatives of odd weights, sv(ζ 2n ) = 0 and sv(ζ 2n+1 ) = 2ζ 2n+1 , and acts on MZVs (2.18) of depth r ≥ 2 in a manner explained in [64]. As an immediate consequence of (5.3), the Berends-Giele recursion for closed-string integrals can be derived from the same currents φ A|B which governs the disk integrals via (3.1). Hence, any tentative "single-valued Z-theory" defined by reproducing the closed-string integrals (5.1) as its doubly partial amplitudes is necessarily contained in the non-abelian Z-theory of this paper. 23 The same kind of organization in terms of (5.1) is expected to be possible in tree-level amplitudes of the heterotic string and the bosonic string. This would imply the universality of gravitational tree-level interactions in these theories whenever their order of α ′ ties in with the weight of the accompanying MZV [36].
Note that reality of the sphere integrals W (P |Q) along with the phase-space constraint s A = 0 for n on-shell particles with P = (A, n) implies that single-valued currents obey the following on-shell properties Hence, one can perform field redefinitions such as to render the associated perturbiner sv[Φ] Lie-algebra valued in both gauge groups.

Conclusions and outlook
We have proposed a recursive method to calculate the α ′ -expansion of disk integrals present in the massless n-point tree-level amplitudes of the open superstring [11,29]. As a backbone of this method, the disk integrals themselves are interpreted as the tree amplitudes in an effective field theory of bi-colored scalars Φ, dubbed as Z-theory in previous work [7].

Further directions
To conclude, we would like to mention an incomplete selection of the numerous open questions raised by the results of this work.
The non-linear equation of motion (3.31) of Z-theory gives rise to wonder about a Lagrangian origin. Moreover, the form of (3.31) is suitable for (partial) specialization to abelian generators in gauge group of the integration domain. Hence, we will explore the implications of our results for the α ′ -corrections to the NLSM [7] as well as mixed Z-theory amplitudes involving both bi-colored scalars and NLSM pions in future work [66].
Do worldsheet integrals over higher-genus surfaces admit a similar interpretation as Z-theory amplitudes? It might be rewarding to approach the low-energy expansion of superstring loop amplitudes at higher multiplicity with Berends-Giele methods. At the one-loop order, this concerns annulus integrals involving elliptic multiple zeta values [67] and torus integrals involving modular graph functions [68].
Is there an efficient BCFW description of Z-theory amplitudes? Given that BCFW on-shell recursions [69] can in principle be applied string amplitudes [70], it would be interesting to relate the Berends-Giele recursion for Z-theory amplitudes to BCFW methods.
Furthermore, what are the non-perturbative solutions to the full Z-theory equation of motion (3.31)? A non-perturbative solution to the field equation Φ = Φ 2 of bi-adjoint scalars (obtained from the field-theory limit α ′ → 0) has been recently found [71] in an attempt to understand the non-perturbative regime of the double-copy construction.
In addition, is it possible to obtain field equations or effective actions for massless open-or closed-superstring states along similar lines of (3.31)? In order to approach the α ′ -corrections to the SYM action, the resemblance of such an equation of motion with the Berends-Giele description of superfields in pure spinor superspace [51,9] is intriguing.
This parallel might for instance be useful in generating the α ′ -corrections to the on-shell constraint {∇ α , ∇ β } − γ m αβ ∇ m = 0 of ten-dimensional SYM [72]. Related to this, it would be desirable to express the Z-theory equation of motion and tentative corollaries for superstring effective actions in terms of the Drinfeld associator.
Given that disk integrals in a basis (2.14) of F P Q have been recursively computed from the associator [6], we expect that suitable representations of its arguments allow to cast

Appendix A. Symmetries of Berends-Giele double currents
In this appendix we discuss the symmetries obeyed by the Berends-Giele double currents.

A.1 Shuffle symmetry
In order to make sure that our ansaetze for BG currents (3.1) for disk integrals satisfy the shuffle-symmetry φ A|P ¡Q = 0, we will need the generalization of the result proven in the appendix of [9]. That is, in a deconcatenation (into non-empty words X i ) of the form if H X 1 ,X 2 ,...,X n satisfies shuffle symmetries on each individual slot and collectively on all the slots (treating each X i as a single letter) H X 1 ,X 2 ,...,A¡B,...,X n = 0 , H (X 1 ,X 2 ,...,X j )¡(X j+1 ,...,X n ) = 0 , j = 1, 2, . . . , n − 1 ,

A.2 Generalized Jacobi symmetry
The definition of T B 1 ,B 2 ,...,B n A 1 ,A 2 ,...,A n in (3.13) implies the shuffle symmetries (3.14) in the B jslots at fixed ordering of the A j -slots. This raises the question about the dual symmetry properties when the A j -slots are permuted at a fixed ordering of the B j -slots. For this purpose it is convenient to use the left-to-right Dynkin bracket mapping ℓ defined by ℓ(A 1 ) = A 1 and [23,25],   Note that ρ 2 (A 1 , . . . , A n ) = nρ(A 1 , . . . , A n ) [23] and (A.5) imply a duality between the shuffle symmetry of the B j slots and the generalized Jacobi symmetry of the A j slots, A. 3 Berends-Giele double current and nested commutators As discussed above, the BG double current satisfies generalized Jacobi symmetries within the A j slots. This means that its expansion in terms of products of φ A i |B j can be written as linear combinations of T B 1 ,...,B n A 1 ,...,A n as, according to Lemma 1, they encode the symmetries of nested commutators. For example, the following terms of order α ′ 2 that multiply the . This is easy to verify but hard to obtain when the expressions are large. Fortunately, one can use an efficient algorithm due to Dynkin, Specht and Wever (for a pedagogical account, see [50]) to find the linear combinations of T B 1 ,...,B n A 1 ,...,A n that capture the products of φ A j |B j . The solution exploits the fact that the Dynkin bracket ℓ gives rise to a Lie idempotent; θ n ≡ 1 n ℓ(A 1 , . . . , A n ). Therefore, by rewriting each word of length n within a Lie polynomial as 1 n ℓ(P ) leads to the answer, e.g., ab − ba = 1 2 ℓ(ab) − 1 2 ℓ(ba) = ℓ(ab).
In order to apply this algorithm to products of φ A i |B j , first rewrite its products such that the B j labels are always in the same order B 1 B 2 B 3 . For example, (A.11) becomes, where in the second line we used the shorthand notation with non-commutative variables L ... . Applying the idempotent operator θ n one obtains where we used the property ℓ(a 1 , a 2 , i) = −ℓ(i, ℓ(a 1 , a 2 )) [25]. This algorithm has been used to cast the α ′ expansion of the BG double current in terms of the definition (3.13).
Appendix B. Ansatz for the Berends-Giele recursion at higher order in α ′ As explicitly tested up to and including order α ′ 4 , one arrives at a unique recursion for the Berends-Giele double current φ A|B that reproduces, via (3.1), the disk integrals at various α ′ w≥2 -orders by imposing the following constraints on an ansatz of the form in (3.6): 1. adjusting the powers of momenta and fields to the mass dimensions of the α ′ w -order 2. reflection symmetry in both slots A and B as well as shuffle symmetry in the B slot 3. absence of dot products (k A i · k B j ), (k B i · k B j ) and k 2 A i 4. absence of dot products (k A 1 · k A p ) referring to the outermost slots in A=A 1 A 2 ...A p 5. matching the order-α ′ w recursion with known n-point disk integrals for all n ≤ w + 3 By dimensional analysis and triviality of the three-point integral, the BG recursion of the disk integrals at a given order is captured by the following number of fields and derivatives, e.g. the ansatz of the form (3.6) for the α ′2 ζ 2 -order generalizes to three types of terms with schematic form k 4 φ 3 , k 2 φ 4 , φ 5 along with α ′3 ζ 3 , The analogous higher-weight relations follow from (4.3), while several identities among MZVs can be found in [73] (obtained using harmonic polylogarithms [74]).

D.2 Methods for shuffle regularization
By the shuffle algebra (4.2), the regularized values (4.8) and (4.10) for weight-one cases G(0; z) and G(z; z) propagate to divergent multiple polylogarithms at higher weight, e.g.

D.3 z-removal identities
The definition (4.1) of polylogarithms applies to situations where the integration variable z only appears on the right of the semicolon in G(a 1 , a 2 , . . . , a n ; z), i.e. to labels a j = z.
Note that analogous z-removal identities for G(z, a 1 ; z), G(z, a 1 , a 2 ; z) and other divergent cases follow from the shuffle relation (4.2), see (4.10) for the regularized values of G(z; z) that differ from the choice in [5].

D.3.2 General z-removal identities
As exemplified by (4.12), some of the regularized integrals require different orders of integration over the variables z 2 , z 3 , . . . , z n−2 . In these situations it can happen that polylogarithms such as G(0, z 4 ; z 3 ) need to be converted to G(. . . ; z 4 ) with no additional instance of z 4 in the ellipsis in order to integrate over z 4 first. This requires a generalization of the techniques in the previous subsection. As before, the starting point for a recursion is the differential equation (D.4) for derivatives in the labels of polylogarithms. The recursion is supplemented by the initial condition For example, the first identity in (D.10) generalizes to G(a 1 , z 1 ; z 2 ) = G(a 1 , 0; z 2 ) − G(a 1 , z 2 ; z 1 ) − G(0; z 2 )G(a 1 ; z 1 ) + G(a 1 , 0; z 1 ) + G(a 1 ; z 2 ) G(a 1 ; z 1 ) − G(0; z 1 ) − 2δ a 1 ,0 ζ 2 + iπ sign(z 2 , z 1 )(G(a 1 ; z 1 ) − G(a 1 ; z 2 )) .

(D.13)
Note that the polylogarithms on the right hand side are suitable for integration over z 1 since there are no instances of z 1 among their labels.
The use of z-removal identities represent the most expensive step in the computation of regularized integrals as they tend to increase the number of terms considerably. An overview of the weights of the identities required at a given order of the Berends-Giele recursion is given in Table 1. For example, terms at the order of α ′ 6 ζ 6 Φ 5 in the Z-theory equation of motion (3.31) arise from integrating the third subleading order ∼ α ′ 3 of the Koba-Nielsen factor (4.5) -the offset is due to the factor (−α ′ ) (n−3) in (3.30) -and require z-removal identities for G(P ; z) at weight |P | = 5.
n-pts MZVs BG current z-removal Koba-Nielsen Table 1. Summary of the contributions from regularized n-point integrals, the order of MZVs, the schematic form of the Berends-Giele double current, the required weight w of z-removal identities (G(a 1 , . . . , a w ; z)) and the order α ′ℓ of the Koba-Nielsen expansion (4.5).

Appendix E. Alternative description of regularized disk integrals
In this appendix, we present a method to determine the α ′ -expansions for regularized disk integrals selected by the Z-theory equation of motion from the (n−3)! × (n−3)! basis F P Q defined in (2.14). This approach has been very useful to constrain the required regularization scheme via explicit data at high orders of α ′ , without the need to obtain the Berends-Giele recursion from an ansatz at these orders. However, we only understand this method as an intermediate tool to determine the appropriate regularization scheme selected by the Z-theory equation of motion: The ultimate goal and achievement of this work is to compute α ′ -expansions of disk integrals at multiplicities and orders where no prior knowledge of F P Q is available.
Closely following the lines of [5], the basic idea is to divide disk integrals 25 Z(I|P ) into a singular and a regular part with respect to region variables s i,i+1...j in (2.4). The singular parts associated with the propagators of the field-theory limits can be subtracted with residues given by lower-multiplicity data, and the leftover local expression is identified with the regularized integrals in ( In the setup of [5], the regularization scheme for divergent integrals was fixed and designed to preserve the shuffle algebra and scaling relations of polylogarithms such that G(z; z) ≡ 0 instead of (4.10). Moreover, the integration orders were globally chosen as 23 . . . n−2 (i.e. integrating over z 2 first and over z n−2 in the last step). In all examples under consideration in [5], it was possible to choose a scheme for pole subtraction such that the resulting regular parts could be reproduced by integration in the canonical order 23 . . . n−2 within the given scaling-preserving regularization. In these adjustments of the subtraction scheme, certain regular admixtures were incorporated by systematically shifting the arguments of the lower-point integrals in the above numerators N .
Here, by contrast, we work with a fixed (or "minimal") subtraction scheme for the poles of Z(I|P ). The resulting regular parts -to be denoted by J reg ... (. . .) in the sequel -turn out to exactly reproduce the desired Z-theory equation of motion upon insertion into (3.31). As will become clear from the following examples, this subtraction scheme is canonical in the sense that the aforementioned regular admixtures of [5] are completely avoided, reflecting the different choices of regularization scheme and integration order between this work and [5].
We will regard SL(2, R)-fixed combinations of disk integrals Z(P |Q) in the notation J u 1 v 1 ,u 2 v 2 ,...,u n−3 v n−3 (k 1 , k 2 , . . . , k n−1 ) ≡ α ′ n−3 0≤z 2 ≤z 3 ≤...≤z n−2 ≤1 dz 2 dz 3 . . . dz n−2 n−1 i<j |z ij | α ′ s ij z u 1 ,v 1 z u 2 ,v 2 . . . z u n−3 ,v n−3 (E.1) 25 For the sake of simplicity, the discussion of [5] and the current appendix is restricted to linear combinations of disk integrals Z(I|P ) with the canonical domain I = 12 . . . n, where the choices of P only leave a single pole channel in the field-theory limit. as functions of n−1 massless momenta k j which determine the s ij on the right hand side through their independent dot products. The product k 1 · k n−1 can be eliminated by momentum conservation and is absent in (E.1) by the SL(2, R)-fixing z 1 = 0 and z n−1 = 1.
This reflects the choice of ansatz in appendix B, where (k A 1 ·k A p ) referring to the outermost slots A 1 , A p in a deconcatenation A=A 1 A 2 ...A p is excluded.
In the four-point case, the field-theory limit of (E.1), which follows from the rules in section 4 of [5] or from (2.28), already exhausts the singular part. Hence, the expressions are analytic in s ij and coincide with the regularized integrals (3.25) [5] in any regularization scheme of our awareness. Their α ′ -expansion is straightforwardly determined by F 2 2 in (2.19) (also see [75] for a neat representation in terms of G(0, . . . , 0, 1, . . . , 1; 1)), The regular parts J reg ij (. . .) in (E.2) are by themselves functions of three light-like momenta under s pq → k p · k q and can later on be promoted to massive momenta k P provided that no reference to k 2 P is expected.

E.1 Five-point pole subtraction
replacement s 12 → s 123 instead of the prescription s 12 → s 12 + s 13 in (E.4). This kind of dependence on k 2 23 = 2s 23 was inevitable to accommodate with the regularization scheme of the [5] with G(z; z) ≡ 0.
In the same way as the α ′ -dependence of the local four-point expressions J reg ij (. . .) is accessible from F 2 2 , their five-point counterparts J reg ij,pq (. . .) can be expanded as soon as the right hand side of (E.4) is expressed in terms of the basis functions {F 23 23 , F 23 32 }, Explicit results on the α ′ -expansion of {F 23 23 , F 23 32 } as pioneered in [42] are available from the all-multiplicity methods based on polylogarithms [5] and the Drinfeld associator [6].
Moreover, recent advances based on their hypergeometric-function representation [75,37] render even higher orders in α ′ accessible, also see [37] for a closed-form solution. Once we adjoin the parity images Again, the arguments s ij → k i · k j of J reg pq,rs can be promoted to massive momenta k i → k P as we will now see in the pole subtractions at higher-multiplicity.

E.3 The general strategy
The choice of labels and momenta for the J reg ... (k A 1 , k A 2 , . . . , k A m−1 ) in the above pole subtractions follows from an algorithm explained in section 4.3 of [5]. This algorithm applies to integrals J ... (. . .) of the form (E.1) with a single cubic diagram in their field-theory limit.
Each factor of z −1 ij in the integrand is associated with one of the n−3 propagators of the field-theory diagram, and the pole subtraction exhausts all 2 n−3 possibilities to relax a subset of these propagators. The residue of diagrams with less than n−3 propagators is a J reg ... (. . .) labeled by the z −1 ij -factors associated with the relaxed propagators, i.e. each relaxed propagator increases the multiplicity of the associated J reg ... (. . .) by one. The massive momenta in its arguments can be read off from the structure of the leftover propagators in the diagram. The reader is referred to [5] for further details, examples and diagrammatic illustrations.
From these rules, it is straightforward to extract the local parts of integrals at arbitrary multiplicity. We have checked up to and including the 5! integrals at seven points that these J reg ... (. . .) at s ij ↔ k i · k j are compatible with the integrals (3.31) in the regularization scheme and integration orders of this work, (E.11) A variety of alternative regularization schemes and integration orders including those of [5] are expected to correspond to a modified choice of arguments for J reg ... (k A 1 , k A 2 , . . . , k A n−1 ), where selected dot products k A p · k A q are shifted by (half of) k 2 A i .