Non-abelian Z-theory: Berends-Giele recursion for the α ′-expansion of disk integrals

Journal of High Energy Physics, Jan 2017

We present a recursive method to calculate the α ′ -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 α ′7 is made available on the website http://​repo.​or.​cz/​BGap.​git.

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Non-abelian Z-theory: Berends-Giele recursion for the α ′-expansion of disk integrals

Received: November Non-abelian Open Access 0 1 c The Authors. 0 1 0 Am Muhlenberg 1 , 14476 Potsdam , Germany 1 Einstein Drive , Princeton, NJ 08540 , U.S.A 2 Institute for Advanced Study, School of Natural Sciences 3 STAG Research Centre and Mathematical Sciences, University of Southampton We present a recursive method to calculate the 0-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 e ective 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 07 is made available on the website http://repo.or.cz/BGap.git. cMax-Planck-Institut fur Gravitationsphysik; Albert-Einstein-Institut - Berends-Giele recursion for the 0-expansion of disk integrals 1 Introduction Z-theory and double copies Review and preliminaries String disk integrals Open superstring disk amplitudes Symmetries of disk integrals in the integrand Symmetries of disk integrals in the domain 0-expansion of disk amplitudes Basis-expansion of disk integrals Berends-Giele recursion for the eld-theory limit Example application of the Berends-Giele recursion Berends-Giele recursion for disk integrals Symmetries of the full Berends-Giele double currents The 02-correction to Berends-Giele currents of disk integrals Free parameters versus Z-theory equation of motion Manifesting the shu e symmetries of BG currents Simplifying the 02-correction to BG currents The perturbiner description of 0-corrections Perturbiners at higher order in 0 All-order prediction for the BG recursion Local disk integrals in the Z-theory equation of motion Multiple polylogarithms and their regularization Towards the simpset basis 4.3 Integration orders for the simpset elements Polylogarithms and MZVs Polylogarithms and the Koba-Nielsen factor Regularization of endpoint divergences Dependence on the integration order Description of the algorithm Back to the chain basis Description of the algorithm Iterated integrals and integration order Summary and overview example 6 Conclusions and outlook Further directions A Symmetries of Berends-Giele double currents A.1 Shu e symmetry A.2 Generalized Jacobi symmetry A.3 Berends-Giele double current and nested commutators B Ansatz for the Berends-Giele recursion at higher order in 0 C Regular parts of ve-point integrals D Multiple polylogarithm techniques D.1 Polylogarithms and MZVs Methods for shu e regularization D.3 z-removal identities Simple z-removal identities D.3.2 General z-removal identities Five-point pole subtraction E.2 Six and seven-point pole subtraction E.3 The general strategy E Alternative description of regularized disk integrals F Integration orders for the seven-point integrals Introduction It is well known that string theory reduces to supersymmetric eld 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 nite size, set by the length scale p 0. One approach to study such 0-corrections to eld theory is through the calculation of string scattering amplitudes, see e.g. [1, 2]. Within this framework, higher-derivative corrections are encoded in the 0-expansion of certain integrals de ned 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 0corrections to super-Yang-Mills (SYM) eld theory arise from iterated integrals over the disk boundary. These integrals can be characterized by two words P and Q formed from Q = (q1; q2; : : : ; qn) in Z(P jq1; q2; : : : ; qn) D(P ) vol(SL(2; R)) zq1q2 zq2q3 : : : zqn 1qn zqnq1 This paper concerns the calculation of the 0-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 eld and string theories. As the technical novelty of this paper, we set up a Berends-Giele (BG) recursion [3] that allows to compute the 0-expansion of the integrals Z(P jQ) and generalizes a recent BG recursion [4] for their eld-theory limit to all orders of 0. As a result of this setup, nite number of terms in the BG recursion at the wth order in 0 is known, the expansion of disk integrals at any multiplicity is obtained up to the same order 0 . 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 0-corrections at large multiplicities is unprecedented in modern all-multiplicity approaches [5, 6] to the 0-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 e ective1 theory of bi-colored scalar elds 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 de ned coe cients is equivalent to a Berends-Giele recursion2 that computes the 0-expansion of the disk integrals (1.1) as if they were tree amplitudes of an e ective eld theory, Z(A; njB; n) = sA AjB : 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 0-expansion of disk integrals from the equation of motion of , its form to all orders in 0 is proposed to be (z12z23 : : : zl 1;l)(zp;p 1zp 1;p 2 : : : zl+2;l+1) + perm(2; 3; : : : ; p 1) : 1The word \e ective" deserves particular emphasis since the high-energy properties of Z-theory (and its quantum corrections) are left for future investigations. 2For a recent derivation of Berends-Giele recursions for tree amplitudes from a perturbiner solution of the eld-theory equations of motion, see [4, 9]. An older account can be found in [8, 10]. 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 [4, 12] which obey the duality between color and kinematics [13]. Z-theory and double copies The relevance of the disk integrals (1.1) is much broader than what the higher-derivative completion of eld theory might lead one to suspect. They have triggered deep insights into the anatomy of numerous eld theories through the fact that closed-string tree-level integrals (encoding 0-corrections to supergravity theories) boil down to squares of disk integrals through the KLT relations [14]. In a eld-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 identi ed in the tree-level amplitudes of additional eld theories [16]. For instance, classical Born-Infeld theory [17] which governs the low-energy e ective action of open superstrings [18] turned out to be a double copy of gauge theories and an e ective theory of pions known as the non-linear sigma model (NLSM) [19{23], see [24{27] for its tree-level amplitudes. As a string-theory incarnation of the Born-Infeld double copy, tree-level amplitudes of the NLSM have been identi ed as the low-energy limit of the disk integrals in the scattering of abelian gauge bosons [7]. This unexpected emergence of pion amplitudes exempli es that disk integrals also capture the interactions of particles that cannot be found in the naive string spectrum.3 Moreover, the entire tree-level S-matrix of massless open-superstring states can be presented as a double copy of SYM with 0-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- edged 0-dependence describes e ective higherderivative deformations of these two scalar eld theories [7]. As a double-copy component to complete SYM to the massless open-superstring S-matrix, the collection of e ective interactions encompassed by Z-theory deserve further investigations. In this work, we identify the equation of motion (1.4) of the full non-abelian Z-theory, where the integration domain of the underlying disk integrals endows the putative scalars 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 e ective 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 3See [28] for a string-theory realization of the NLSM through toroidal compacti cations in presence of worldsheet boundary condensates. for BCJ-satisfying 0-corrections to the NLSM. The present article extends this endeavor such as to e ciently compute the doubly-partial amplitudes of e ective bi-colored theories with BCJ duality in one of the gauge groups and explicitly known eld equations (1.4). This paper is organized as follows: following a review of disk integrals and the BerendsGiele description of their eld-theory limit in section 2, the Berends-Giele recursion for their 0-corrections and the resulting eld 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 elds 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 les complement the discussions in the main text. The BG recursion that generates all terms up to the 07-order in the 0-expansion of disk integrals at arbitrary multiplicity as well as the auxiliary computer programs used in their derivations can be downloaded from [29]. Review and preliminaries In this section, we review the de nitions 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 eld-theory limit in order to set the stage for the generalization to 0-corrections. String disk integrals q1q2 : : : qn of length n as We de ne a cyclic chain C(Q) of worldsheet propagators zij1 with zij zj on words ki + kj + : : : + kp ; Then, the iterated disk integrals on the real line that appear in the computation of opensuperstring tree-level amplitudes are completely speci ed by two words P and Q, p1p2 : : : pn encodes the domain of the iterated integrals, 1 < zp1 < zp2 < : : : < zpn < 1g : Mandelstam variables sij:::p involving legs i; j; : : : ; p are de ned via region momenta kij:::p, and the more standard open-string conventions for the normalization of 0 (which would cause proliferation of factors of two) can be recovered by globally setting erywhere 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 Mobius transformation z ! aczz++db (with ad bc = 1). compensating Jacobian: D(12:::n) vol(SL(2; R)) dzn = z1;n 1z1;nzn 1;n dz2 dz3 : : : dzn 2 : 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 jQ) as doubly partial amplitudes [7]. 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 jQ) and Z(QjP ). This can already be seen from the di erent symmetries w.r.t. variable P at xed Q on the one hand and variable Q at xed P on the other hand. Symmetries of disk integrals in the integrand The manifest cyclic symmetry and re ection (anti-)symmetry of the integrand C(Q) in (2.1) directly propagates to the disk integrals Z(P jq2q3 : : : qnq1) = Z(P jq1q2 : : : qn) ; Z(P jQ~) = ( 1)jQjZ(P jQ) ; is a shorthand for its reversal. Moreover, the disk integrals satisfy [5] the Kleiss-Kuijf relations [30], or equivalently [31, 32], the vanishing of pure shu es in n 1 legs, Z(P jA; 1; B; n) = ( 1)jAjZ(P j1; A~ B; n) ; B; n) = 0 8 A; B 6= ; : The shu e operation in (2.7) and (2.8) is de ned recursively via [33] ; A = A ; = A; Finally, integration by parts yields the same BCJ relations among permutations of Z(P jQ) in Q as known from [13] for color-stripped SYM tree amplitudes [5] 0 = X kq1 kq2q3:::qj Z(P jq2q3 : : : qj q1qj+1 : : : qn) : As a consequence of the form of the integration region D(P ) in (2.3), disk integrals obey Z(p2p3 : : : pnp1jQ) = Z(p1p2 : : : pnjQ) ; Z(P~jQ) = ( 1)jPjZ(P jQ) ; which tie in with the simplest symmetries (2.6) of the integrand Q. However, the KleissKuijf symmetry (2.7) and BCJ relations (2.10) of the integrand do not hold for the integration domain P in presence of 0-corrections. This can be seen from the real and imaginary part of the monodromy relations [34, 35] (see [36] for a recent generalization to loop level) 0 = 0kp1 kp2p3:::pj Z(p2p3 : : : pj p1pj+1 : : : pnjQ) : Nevertheless, (2.12) is su cient to expand any Z(P jQ) in an (n 3)!-element basis Open superstring disk amplitudes The n-point tree-level amplitude Aopen 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, 37]: Aopen(1; P; n 1; n) = FP QASYM(1; Q; n 1; n) While all the polarization dependence on the right hand side has been expressed through the BCJ basis [13] of SYM trees ASYM, the entire reference to 0 stems from dz2 dz3 : : : dzn 2 derivation [11, 37] of (2.13) and (2.14) has been performed in the manifestly supersymmetric pure spinor formalism [38], where the SYM amplitudes ASYM in (2.13) have been identied from their Berends-Giele representation in pure spinor superspace [39]. Hence, (2.13) applies to the entire ten-dimensional gauge multiplet in the external states.4 4A bosonic-component check of the formula (2.13) at multiplicity n 7 within the RNS formalism has been performed in [40]. After undoing the SL(2; R)- xing in (2.5), the integrals FP Q can be identi ed as a linear combination of disk integrals (2.2) [5], FP Q = S[QjR]1Z(P j1; R; n; n 1) ; where P; Q and R are understood to be permutations of 2; 3; : : : ; n 2. The symmetric (n 3)! (n 3)! matrix S[QjR]1 encodes the eld-theory KLT relations [41, 42] (see also [43] for the 0-corrections to S[QjR]1) and admits the following recursive representation [7], S[A; jjB; j; C]i = (kiB kj )S[AjB; C]i; in terms of multiparticle momenta (2.4). Hence, the n-point open-superstring amplitude (2.13) with any domain P can be obtained from the KLT formula, F22 = exp (s12 + s23) = 1 02 2s12s23 + 03 3s12s23(s12 + s23) Aopen(P ) = Z(P j1; R; n; n 1)S[RjQ]1ASYM(1; Q; n 1; n) A~SYM(1; R; n; n 1) Z(P j1; R; n; n 1) [5]. The KLT form of (2.17) reveals the double-copy structure of the open-superstring tree-level S-matrix which in turn motivated the proposal of [7] to interpret disk integrals as doubly partial amplitudes. The speci cation of disk integrals by two cycles P; Q identi es the underlying particles to be bi-colored scalars, and we collectively refer to their e ective interactions that give rise to tree amplitudes Z(P jQ) Note that disk amplitudes of the bosonic string are conjectured in [44] to also admit the form (2.13) or (2.17), with B(1; Q; n 1; n; 0) that also satisfy the KK- and BCJ relations. 0-dependent kinematic factors ASYM(1; Q; n 1; n) ! 0-expansion of disk amplitudes The 0-expansion of disk amplitudes (2.13), i.e. their Taylor expansion in the dimensionless Mandelstam invariants 0sij , involves multiple zeta values (MZVs), 0<k1<k2<:::<kr understood to be additive in products of MZVs). While the four-point instance of (2.14), grals at multiplicity n 5 generally involve MZVs of higher depth r 2, see [45] for a recent closed-form solution at ve points. It has been discussed in the literature of both physics [37, 46, 47] and mathematics [48{50] that the disk integrals (2.2) at any multiplicity exhibit uniform transcendentality: their 0w-order is exclusively accompanied by products of MZVs with total weight w. The basis of functions FP Q in (2.15) is particularly convenient to directly determine the 0-expansion of the open-string amplitudes (2.13) [6] and to describe their pattern of MZVs5 [47, 55]. At multiplicities ve, six and seven, explicit results for the leading orders in the 0-expansion of FP Q are available for download on [56]. Basis-expansion of disk integrals In setting up the Berends-Giele recursion for the fundamental objects Z(AjB) of this work, it is instrumental to e ciently extract their 0-expansion from the basis functions FP 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, Z(1; P; n 1; njR) = where m(AjB) denote the doubly partial amplitudes of biadjoint 3-theory which arise in the eld-theory limit of disk integrals [57] m(AjB) = lim Z(AjB) : Note the striking resemblance of the formulas (2.20) and (2.13), which further point out the similar roles played by the amplitudes Aopen(P ) and Z(P jQ) of string and Z-theory. Berends-Giele recursion for the eld-theory limit The task we want to accomplish in this paper concerns the computation of the 0-expansion of the disk integrals (2.2) in a recursive and e cient manner. In the eld-theory limit 0 ! 0, all-multiplicity techniques have been developed in [37], and a relation to the inverse KLT matrix (2.16) has been found in [5]. The equivalent description of the 0 ! 0 limit in terms of doubly partial amplitudes (2.21) [57] has inspired a recent Berends-Giele description [4] via bi-adjoint scalars (0) t~b. The latter take values in the tensor product of two gauge groups with generators ta and t~b as well as structure constants f acd and f~bgh, respectively. The superscript of the biadjoint scalar (0) indicates that this is the 0 ! 0 limit of the Z-theory particles whose interactions give rise to the disk integrals Z(P jQ) as their doubly partial amplitudes. The non-linear eld equations in the low-energy limit (a0b) = facdf~bgh c g 5After pioneering work in [51{54], the 0-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 [55]). (0) = a1;a2;a3;b1;b2;b3 with d'Alembertian @2 will later be completed such as to incorporate the 0-corrections in Z(P jQ). One can solve (2.22) through a perturbiner [8] expansion6, t~b1 t~b2 tb3 + ~B t eld equations (2.22) [4], sA (A0j)B = BA11BA22==BA and referred to as Berends-Giele double currents deconcatenate B in the same manner. The initial conditions for the recursion in (2.25), The notation P A1A2=A and which resums tree-level subdiagrams and is compactly written as a sum over all words A; B with length jAj; jBj 1 in the last line. We are using the collective notation m(A; njB; n) = sA AjB Given the cyclic symmetry (2.6) of Z(P jQ) in the word Q, one can always choose the last letter of the integrand Q (B; n) to coincide with the last letter of the integration region (A; n) as has been done in (2.28). The recursive de nition of (A0j)B in (2.25) gives rise to an e cient algorithm to obtain the eld-theory limit of disk integrals Z(A; njB; 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], guarantee that (A0j)B vanishes unless A is a permutation of B and yield expressions such as 12j12 = 12j21 = 123j123 = 123j312 = at the two- and three-particle level. As shown in [4], the eld-theory limits of the disk integrals (2.2) and thereby the doubly partial amplitudes (2.21) are given by the Berends-Giele double currents (A0j)B, S 1[P jQ]1 = 6See [8, 58, 59] for perturbiner solutions to self-dual sectors of four-dimensional gauge and gravity theories (see also [10]) and [9] for perturbiners in ten-dimensional SYM.,FTlimit i(0j) = i;j ; Terms such as 1j5 352j132 following from the deconcatenation (2.25) have been dropped from the last equality because the condition (2.26) implies that (105) = 0. In addition, Recursing the above steps until no factor of (A0j)B remains yields, m(13524j32451) = 135j513 = in agreement with the expression for the doubly partial amplitude m(13524j32451) that follows from the methods of [57]. In the next section this method will be extended to compute the 0-corrections of string disk integrals. Berends-Giele recursion for disk integrals In this section, we develop a Berends-Giele recursion7 for the full- edged disk integrals Z(P jQ) de ned in (2.2). The idea is to construct 0-dependent Berends-Giele double AjB such that the integrals Z(P jQ) including 0-corrections are obtained in the same manner as their eld-theory limit in (2.28), And similarly, the 0-corrected BG double currents coe cients of a perturbiner expansion analogous to (2.23), AjB in (3.1) will be given by the Z(A; njB; n) = sA AjB : The computation of the eld-theory limit of the ve-point disk integral m(13524j32451) = lim 2 Z D(13524)vol(SL(2; R)) i<j z32z24z45z51z13 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, that solves non-linear equations of motions which can be viewed as an augmentation of (2.22) by 0-corrections. The eld equation obeyed by the perturbiner (3.2) will be interpreted as the equation of motion of Z-theory, the collection of e ective theories involving 7For a review of the Berends-Giele recursion for gluon amplitudes [3] which is adapted to the current discussion, see section 2 of [4]. bi-colored scalars encoding all the 0-corrections relevant to the open superstring [7]. In addition, the BG double currents above are subject to the initial and vanishing condition ijj = i;j ; AjB = 0 ; unless A is a permutation of B. Given their role in equation (3.1), the words A and B on the BG double current AjB 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 AjB satisfy AjB = ( 1)jAj 1 A~jB = ( 1)jBj 1 upon reversal of either the integration domain A or the integrand B. Similarly, the KleissKuijf relations (2.8) of the disk integrals follow from the shu e symmetry8 of AjB within the integrand B, AjP Q = 0 8 P; Q 6= ; : AjB does not exhibit shu e symmetries in the integration domain A: the 0correction in the monodromy relations [34, 35], more speci cally in the real part of (2.12), yields non-zero expressions9 O(( 0 )2) for P QjB. As a consequence, the perturbiner (3.2) is Lie-algebra valued w.r.t. the t~b generators [31] but not w.r.t. the ta generators. That is why the Z-theory scalar 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 0-corrections of the Berends-Giele double currents, see appendices A and B for further details. 02-correction to Berends-Giele currents of disk integrals Assuming that the 02-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 has dimension of k2, and the 02-terms contain a factor of k4 compared to the leading contribution from 2 in (2.25). Therefore, an ansatz for sA AjB at this order must be based on a linear combination of BA11BA22BA33==BA known as \alternal moulds", see e.g. [60]. plications 2fno+r1a Berends-Giele approach to closed-string integrals. 9Since the monodromy relations only di er from the KK relations by rational multiples of 2n or ( 2)n, the sub-sector of Z(A; njB; n) without any factors of 2 still satis es shu e symmetries, e.g. BA11::::::BA44==BA into non-empty words. By the initial condition (3.3), AjB vanishes unless A is a permutation of B, so there is no need to consider momentum dependence of the form (kAi kBj ) appendix A for the implementation of the shu e symmetry. Then, matching the outcomes of (3.1) with the known 02-order of various integrals at four and rameters, leaving a total of four free parameters. The 02-order of (n ve points xes six pa6)-point integrals does not provide any further input: as we have checked with all the known (n data [56], 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 nite amount of low-multiplicity data | the coe cients of k2 3- and 4-terms (3.6) at the 02-order | determines the relevant order of disk integrals at any multiplicity. BA11BA22==BA Free parameters versus Z-theory equation of motion It is not surprising that the ansatz based on (3.6) is not completely xed (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 the schematic form 2 + O( 3). Self-contractions (kAi kAi ) signal the appearance + : : : on the right hand side, where along with 02 2 2 can be replaced by the entire right hand side. The result in turn leads to another appearance of at higher orders in 0 and the elds. In order to obstruct an in nite iteration of the eld equations, we x three additional parameters The last free parameter re ects the freedom to perform eld rede nitions. Terms of the form 02 2 ( 3) on the right hand side of can be absorbed via 3, i.e. the right-hand side of 0 will no longer contain the term question. This leftover freedom can be xed by requiring the absence of the dot product (kA1 kA3 ) among the leftmost and the rightmost slot-momentum10 in the deconcatenation by a total d'Alembertian are systematically avoided while preserving the manifest parity property (3.4) in A. At the end of the above process, one nds the unique recursion that generates the 0 terms in the low-energy expansion of disk integrals at any multiplicity via (3.1): sA AjB = BA11::::::BA33==BA 10In general, in a p-fold deconcatenation PA=A1:::Ap PB=B1:::Bp , the dot product (kA1 kAp ) among the leftmost and the rightmost momentum will not be included into an ansatz for sA AjB at given order in 0. This freezes the freedom to perform eld rede nitions while preserving the manifest parity property (3.4) BA11::::::BA44==BA For example, applying the above recursion to the disk integral Z(13524j32451) whose eldtheory limit was computed in (2.32) leads to the following result up to 0 : Z(13524j32451) = It is important to emphasize that, while only four- and ve-point data entered in the derivation of (3.7), this recursion allows the computation of 02 terms of disk integrals at arbitrary multiplicity. The eleven-point example Z(134582679baj123456789ab) = s19absabs345s67 s34 s45 laptop with the program available in [29]. Manifesting the shu e symmetries of BG currents The length of the recursion in (3.7) at the 02 2 order calls for a more e cient representation. In this subsection, we identify the sums of products of AijBj which satisfy the shu e symmetries (3.5) in the Bj -slots. This allows to rewrite the recursion (3.7) in a compact form which inspires the generalization to higher orders and clari es the commutator structure in the Z-theory equation of motion upon rewriting the results in the language of perturbiners (3.2). are annihilated by a linear map is de ned by (Bi) Bi and [31] In order to do this, recall from the theory of free Lie algebras that all shu e products acting on words (B1; B2; : : : ; Bn) of n letters Bi which (B1; B2; : : : ; Bn) (B1; B2; : : : ; Bn 1); Bn (B2; B3; : : : ; Bn); B1 : (B2; B1) and (B1; B2; B3) = (B1; B2; B3) (B2; B1; B3) (B2; B3; B1) + (B3; B2; B1) imply the vanishing of (B1 B2) and ((B1; B2) B3). Therefore, after de ning it is straightforward to check that the following linear combinations A1;A2;:::;An = T B1;B2;:::;Bn 1 A1;A2;:::;An 1 AjB satisfy the shu e symmetries on the Bj -slots [31],11 The rst few examples of (3.13) read as follows, A1;A2;A3;A4 A1;A2;:::;An (B1;B2;:::;Bi) (Bi+1;:::;Bn) = 0 ; i = 1; 2; : : : ; n 11The parenthesis around the B labels signi es that the shu e product treats the (multiparticle) labels sA AjB = A1;A2;:::;An A=A1A2 B=B1B2 A=A1:::A4 B=B1:::B4 and their shu e symmetries (3.14) are easy to verify, starting with T B1;B2 = T B1;B2;B3 + T B1;B3;B2 + T B3;B1;B2 = 0 : Moreover, the -map in (3.10) exhausts all tensors of the type (3.12) subject to shu e symmetry in the Bj -slots it acts on [31, 62]. Hence, a BG recursion which manifests the shu e symmetry in the Bj -slots is necessarily expressible in terms of T A1;A2;:::;An B1;B2;:::;Bn in (3.13). Rather surprisingly, it turns out that the de nition (3.13) not only manifests the shu e symmetries on the Bj -slots but also implies generalized Jacobi identities with respect to the Aj -slots. In other words, the above T B1;B2;:::;Bn satisfy the same symmetries as the A1;A2;:::;An nested commutator [[: : : [[A1; A2]; A3] : : :]; An], see appendix A.2 for a proof. Simplifying the 02-correction to BG currents As discussed in the previous subsection, the BG double current can always be written in terms of T B1;B2;:::;Bn from the de nition (3.13). For example, the expression (3.7) becomes A=A1:::A3 B=B1:::B3 A1;A2;A4;A3 A1;A2;A3;A4 A1;A3;A4;A2 A1;A3;A2;A4 From a practical perspective, it could be a daunting task to convert a huge expression in side of (3.17). Fortunately, since both the BG double current and TAB11;;AB22;;::::::;;ABnn satisfy generalized Jacobi identities in the Aj -slots, an e cient algorithm due to Dynkin, Specht AijBj such as (3.7) into linear combinations of TAB11;;AB22;;::::::;;ABnn on the right-hand and Wever [63] can be used to accomplish this at higher orders in 0. See the appendix A.3 for more details. The perturbiner description of 0-corrections The recursion (3.17) for the coe cients a more compact form by de ning the shorthand [[: : : [[ i1 ; i2 ]; i3 ]; : : : ; ip 1 ]; ip ] AjB of the perturbiner (3.2) can be rewritten in ekA1:::Ap x TABi11;;BA2i;2::;::;:B:;Apip tA1A2:::Ap t~B1B2:::Bp ; (3.18) 12These are the same symmetries in P f i1i2af ai3b : : : f xipy as well as the local multiparticle super elds VP [64] in pure spinor superspace. which exploits the generalized Jacobi symmetry of the Aj -slots in TABi11;;BA2i;2::;::;:B:;Apip . That is, the numeric indices i1; i2; : : : ; ip of the various formal perturbiners i in the commutator match the ordering of the labels within the A-slots in TABi11;;BA2i;2::;::;:B:;Apip , while the ordering of the B-slots is always the same. Finally, the color degrees of freedom enter in a global multiplication order; tA1A2:::Ap The above de nition implies that the Berends-Giele recursion (3.17) condenses to, = [ 1; 2 with the following shorthand for the derivatives: The convention for the derivatives @j is to only act on the position of j , e.g. the perturbiner expansion of @12[[ 3 A=A1A2A3 B=B1B2B3 (kA1 kA2 )TAB31;;AB22;;AB13 . In view of the increasing number of -factors at higher order in 0, we will further lighten the notation and translate the commutators into multiparticle labels P [[: : : [[ i1 ; i2 ]; i3 ]; : : : ; ip 1 ]; ip ] ; which exhibit generalized Jacobi symmetries by construction.12 Hence, any subset of the nested commutators of (3.19) can be separately expressed in terms of ; 4]] = [ 12; 34] = 1243. In this language, the Z-theory equation of motion (3.19) becomes = [ 1; 2 @12[ 1; 32] [ 12; 43] + [ 13; 42] + O( 03) : 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 0. As a reformulation of (3.19) which does not rely on the notion of perturbiners, one can peel o the ta generators13 from the bi-colored = P A. The coe cients are still Lie-algebra valued with respect to the t~b, and this is where the nested commutators act in the following rewriting of (3.19): A1;A2;A3;A4 tA1A2A3 02 2 @23[[ A1 ; A2 ]; A3 ] 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 ta and t~b generators. Perturbiners at higher order in The procedure of subsection 3.2 to determine the Berends-Giele recursion that reproduces the 02-corrections to the disk integrals was also applied to x the recursion at the orders 04 (see appendix B for more details). Luckily, the analogous ansaetze at orders 0w 5 could be bypassed since the general pattern of the eld equations became apparent from the leading orders 0 . To see this, it is instructive to spell out the Z-theory equation of motion up to the 03-order: = [ 1; 2] + 0 3@12(@12 + @23) [ 1; 32] 0 3 @21 + 2@31 + 2@32 + 2@42 + @43 [ 12; 43] 0 3 2@21 + @31 + 3@32 + @42 + 2@43 [ 13; 42] + 2[ 134; 52] + 3[ 14; 523] 13In view of the 0-corrections to KK relations from (2.12), the Z-theory scalar is not Lie-algebra valued in the gauge group of the ta but instead exhibits an expansion in the universal enveloping algebra spanned by tA = ta1 ta2 : : : tajAj . After identifying sij $ @ij , the coe cients of [ 12; 3] and [ 1; 32] in (3.24) are identical to the rst regular terms in the expansion of the four-point disk integrals considered in [5]: 0 z12 m;n=0 0 z32 m;n=0 ( 0s12 ln jz12j)m ( 0s23 ln jz23j)n ( 0s12 ln jz12j)m ( 0s23 ln jz23j)n = 0 2s23 0 2s12 + 02 3s12(s12+s23)+O( 03) regularization prescription denoted by \reg" and explained in section 4. The in nite sums in the above integrands arise from the Taylor expansion of a SL(2; R)- xed four-point Koba-Nielsen factor via jzij j 0sij = 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 0 with the expression for the BG current obtained from an ansatz con rms the pattern, and we will later on see that the terms of order regularized ve- and six-point integrals. 5 in (3.24) can be traced back to All-order prediction for the BG recursion From the observations in the previous subsection, we propose a closed form for the contributions to the Z-theory equations of motion for , to all orders in 0: = [ 1; 2 The integrand in the second line bears a strong structural similarity to the correlation function in the four-point open string amplitude [11, 65] Aopen(1; 2; 3; 4) = with hVP VQVni denoting certain kinematic factors in pure spinor superspace. The precise correspondence between (3.27) and (3.28) maps multiparticle vertex operators VP [64] to perturbiner commutators P de ned in (3.21). Moreover, since VP is fermionic and satis es generalized Jacobi symmetries [64], the all-multiplicity mapping jP j + jQj = n preserves all the symmetry properties of its constituents. Finally, the Koba-Nielsen factor 0sij with sij ! @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 z2 ! 0 and z2 ! 1 as discussed in section 4. It is easy to see that the correspondence (3.29) correctly \predicts" the rst term in the the three-point massless disk amplitude under the mapping (3.29); hV1V2V3i Extrapolating the above pattern, a natural candidate for the higher-order contributions ; 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], Aopen(1; 2; : : : ; n) = ( dz2 dz3 : : : dzn V12:::lVn 1;n 2;:::;l+1Vn (z12z23 : : : zl 1;l)(zn 1;n 2zn 2;n 3 : : : zl+2;l+1) +perm(2; 3; : : : ; n 2) ; which appeared in an intermediate step towards the minimal (n 3)!-term expression (2.13). This expression leads us to propose the following Z-theory equation of motion to all orders (z12z23 : : : zl 1;l)(zp;p 1zp 1;p 2 : : : zl+2;l+1) + perm(2; 3; : : : ; p 1) : Apart from the correspondence (3.29) which settles the perturbiner commutators suggested by (3.30), we introduce a formal operator R eom that maps the accompanying disk integrals to local expressions. The precise rules for the map R eom to be explained in the next section include a Taylor expansion (3.26) of the Koba-Nielsen factor as seen in (3.27). Also, R eom incorporates a regularization along with particular parameterization of the ubiquitous domain 0 1 for the p 2 integration variables z2; z3; : : : ; zp 1 which is left implicit in (3.31) for ease of notation. The shorthands i1i2:::ik in (3.31) explained in section 3.3 compactly track the relative multiplication order of the gauge-group generators ta and t~b which govern the color structure of . For example, the equation of motion up to 4-order following from (3.31) reads = [ 1; 2 2 Z eom 4 ] [ 12; 43] + [ 1; 432] +(2 $ 3) +O( 5) ; in this case requires G(z2; z2; z2) = G(0; z2) + i G(0; z2) + i G(z2; 0; z2) = G(0; z2) + i G(0; z2) + 2 G(z2; z2) = G(0; z2) + i ; using G(0; z2; z2) = 2 by (D.10). It is interesting to observe that the last line of (4.35) becomes 12 G(0; z2)2+G(0; z2; z2)+ 12 2, where the term 2 can be traced back to an interplay between two subtle factors of i from very distinct sources: one from the general z-removal identity (4.31) and the other from the = 1 shu e regularization (4.10).23 In addition to the above shu e regularizations, the following z-removal identities based G(z2; z2; 1) = G(0; z2)2 + G(1; z2)2 G(0; z2)G(1; z2) + i G(z2; 0; 1) = 2 2 + i G(0; z2) G(0; 0; z2) + G(0; 1; z2) G(0; z2; 1) = i G(0; z2) + G(0; 0; z2) G(0; 1; z2) G(z2; 1) = G(1; z2) G(0; z2) + i : In combination with the shu e algebra (4.2), the identities in (4.37) yield the following Finally, summing the above results yields the regularized value of the integral (4.29), Using the prescription (3.31), this implies that the Z-theory equation of motion contains 0 4@12[ 1; 5243], in agreement with the Berends-Giele recursion at order 0 previously obtained from an ansatz. 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 23Fortunately, 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 x all these subtleties. This ultimately led us to our nal regularization prescription that has ever since passed many tests at much higher order in 0. of disk integrals (2.2) via two cycles P and Q, any sphere integral in tree-level amplitudes of the type II superstring24 boils down to vol(SL(2; C)) 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 zij ! zij . amplitudes as the corresponding disk integrals [61], While the eld-theory limit of the sphere integrals (5.1) yields the same doubly partial m(A; njB; n) = lim W (A; njB; n) ; only a subset of the 0-corrections in Z(P jQ) can be found in the closed string (5.1). These selection rules obscured by the KLT relations [14] have been identi ed to all orders in [47] and realize the single-valued projection \sv" [81, 82] of the MZVs in the disk integrals [61, 83] W (P jQ) = sv Z(P jQ) : The single-valued map projects Riemann zeta values to their representatives of odd weights, 2 in a manner explained in [81, 82]. As an immediate consequence of (5.3), the Berends-Giele representation W (A; njB; n) = sA sv of closed-string integrals can be derived from the same currents AjB which govern the disk integrals via (3.1). Hence, any tentative \single-valued Z-theory" de ned 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. Note that reality of the sphere integrals W (P jQ) along with the phase-space constraint the following on-shell properties = sv = O(sA) : Hence, one can perform eld rede nitions such as to render the associated perturbiner sv[ ] Lie-algebra valued in both gauge groups. 24The 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 treelevel interactions in these theories whenever their order of 0 ties in with the weight of the accompanying MZV [44, 47]. Conclusions and outlook We have proposed a recursive method to calculate the 0-expansion of disk integrals present in the massless n-point tree-level amplitudes of the open superstring [11, 37]. As a backbone of this method, the disk integrals themselves are interpreted as the tree amplitudes in an e ective eld theory of bi-colored scalars , dubbed as Z-theory in previous work [7]. Its equation of motion (3.31) furnishes the central result of this work and compactly encodes the Berends-Giele recursions that elegantly compute the 0-expansions of the disk integrals at arbitrary multiplicity. More precisely, the Z-theory equation of motion (3.31) is satis ed by the perturbiner series of the Berends-Giele currents, and its structure is shared by an (n 2)!-term representation of the n-point open-string tree amplitude derived in [11]. As a practical result of this work, the BG recursion relations for disk integrals Z(P jQ) with any given words P and Q of arbitrary multiplicity is made publicly available up to order 07 in a FORM [79, 80] program called BGap. In order to ease replication, the auxiliary computer programs used in the derivation of the BG recursion via regularized polylogarithms are also available to download on the website [29]. As a conceptual bene t of this computational achievement, the Berends-Giele description of disk integrals sheds new light on the double-copy structure of the open-string treelevel S-matrix [5]. As manifested by (2.17), disk amplitudes exhibit a KLT-like factorization into SYM amplitudes and disk integrals Z(P jQ). Following the interpretation of Z(P jQ) as Z-theory amplitudes [7], the perturbiner description of the Berends-Giele recursion for disk integrals pinpoints the eld equation (3.31) of Z-theory. Hence, our results give a more precise de nition of Z-theory, the second double-copy component of open superstrings. 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 0-corrections to the NLSM [7] as well as mixed Z-theory amplitudes involving both bi-colored scalars and NLSM pions in future work [84]. 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 [85{ 87] and torus integrals involving modular graph functions [88{96]. Is there an e cient BCFW description of Z-theory amplitudes? Given that BCFW on-shell recursions [97] can in principle be applied string amplitudes [98{100], 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 eld equation 2 of bi-adjoint scalars (obtained from the eld-theory limit 0 ! 0) has been recently found [101] in an attempt to understand the non-perturbative regime of the double-copy construction. In addition, is it possible to obtain eld equations or e ective actions for massless open- or closed-superstring states along similar lines of (3.31)? In order to approach the 0-corrections to the SYM action, the resemblance of such an equation of motion with the Berends-Giele description of super elds in pure spinor superspace [9, 64] is intriguing. This parallel might for instance be useful in generating the 0-corrections to the on-shell constraint fr ; r g m rm = 0 of ten-dimensional SYM [102{107]. Related to this, it would be desirable to express the Z-theory equation of motion and tentative corollaries for superstring e ective actions in terms of the Drinfeld associator. Given that disk integrals in a basis (2.14) of FP Q have been recursively computed from the associator [6], we expect that suitable representations of its arguments allow to cast the 0-expansion of the Berends-Giele recursion into a similarly elegant form. One could even envision to generate the tree-level e ective action of the open superstring from the SYM action by acting with appropriate operator-valued arguments of the associator. Finally, a rigorous mathematical justi cation for the various prescriptions used in \converting" the open string amplitude (3.30) to the Z-theory equation of motion was not the subject of this paper but clearly deserves further investigation. In particular, it seems mysterious to us at this point why the Z-theory setup selects the regularization scheme for G(0; z); G(z; z), the integration orders, and the change of basis presented in section 4. Acknowledgments We are grateful to Johannes Brodel, John Joseph Carrasco and Ellis Yuan for combinations of valuable discussions and fruitful collaboration on related topics. We are indebted to Erik Panzer for indispensable email exchange and numerous enlightening discussions, in particular for guidance on the subtleties of polylog regularization and di erent orders of integration. Also, we would like to thank Johannes Brodel and Erik Panzer for helpful comments on an initial draft. The authors would like to thank IAS at Princeton where this work was initiated as well as Nordita and in particular Paolo Di Vecchia and Henrik Johansson for providing stimulating atmosphere, support and hospitality through the \Aspects of Amplitudes" program. CRM thanks the Albert-Einstein-Institut in Potsdam for hospitality during the nal stages of this work. CRM is supported by a University Research Fellowship from the Royal Society, and gratefully acknowledges support from NSF grant number PHY 1314311 and the Paul Dirac Fund during the initial phase of this work. Symmetries of Berends-Giele double currents In this appendix we discuss the symmetries obeyed by the Berends-Giele double currents. e symmetry In order to make sure that our ansaetze for BG currents (3.1) for disk integrals satisfy the shu e-symmetry R S = 0; 8 R; S 6= ; : It would be interesting to rigorously derive the symmetry in (A.3) from the properties (A.2) of the deconcatenations in (A.1), possibly along the lines of the appendix of [9]. Generalized Jacobi symmetry The de nition of TAB11;;AB22;;::::::;;ABnn in (3.13) implies the shu e symmetries (3.14) in the Bj slots at xed ordering of the Aj -slots. This raises the question about the dual symmetry properties when the Aj -slots are permuted at a xed ordering of the Bj -slots. For this purpose it is convenient to use the left-to-right Dynkin bracket mapping ` de ned by `(A1) = A1 and [31, 33], `(A1; A2; : : : ; An) = `(A1; A2; : : : ; An 1); An An; `(A1; A2; : : : ; An 1) such as `(A1; A2) = (A1; A2) (A2; A1) and `(A1; A2; A3) = (A1; A2; A3) (A2; A1; A3) (A3; A1; A2) + (A3; A2; A1). One can show that (A.4) projects to the symmetries of nested commutators with Lemma 1. The object TAB11;;AB22;;::::::;;ABnn de ned by (3.13) satis es the generalized Jacobi symmetries in the Aj -slots, i.e. the symmetries of nested commutators appendix of [9]. That is, in a deconcatenation (into non-empty words Xi) of the form P = X1X2=P X1X2X3=P X1X2X3X4=P HX1;X2;X3;X4 + if HX1;X2;:::;Xn satis es shu e symmetries within each individual slot and collectively on all the slots (treating each Xi as a single letter) HX1;X2;:::;A B;:::;Xn = 0 ; H(X1;X2;:::;Xj) (Xj+1;:::;Xn) = 0 ; j = 1; 2; : : : ; n then P in (A.1) is expected to satisfy the shu e symmetry for words of arbitrary length, TAB11;;AB22;;::::::;;ABnn [: : : [[A1; A2]; A3]; : : : ; An] such as TAB11;;AB22 = TAB21;;AB12 and TAB11;;AB22;;AB33 + TAB21;;AB32;;AB13 + TAB31;;AB12;;AB23 = 0. Proof. According to (A.5) it su ces to show that which in turn follows from T`B( A1;1B; A2;2::;::;:B:;Ann) = nTAB11;;AB22;;::::::;;ABnn ; TAB11;;AB22;;::::::;;ABnn = T`d(oAm1;A2;:::;An) niently verify (A.8) by induction: T Bin1t;B2;:::;Bn = T`d(oAm1;A2;:::;An 1);An B1;B2;:::;Bn 1 In the rst line, we apply the recursive de nition (A.4) of the Dynkin bracket operator, followed by the de nition (3.12) of the tensor product T:d::om T:i:n:t in the second line. In passing to the third line, we have used the inductive assumption, i.e. (A.8) at n ! n the resulting expression can be identi ed with the recursive de nition (3.13) of TA1;A2;:::;An B1;B2;:::;Bn nishes the proof. shu e symmetry of the Bj slots and the generalized Jacobi symmetry of the Aj slots, Berends-Giele double current and nested commutators As discussed above, the BG double current satis es generalized Jacobi symmetries within the Aj slots. This means that its expansion in terms of products of AijBj can be written as linear combinations of TAB11;;::::::;;ABnn as, according to Lemma 1, they encode the symmetries of nested commutators. For example, the following terms of order 02 that multiply the factor (kA1 kA2 ) in (3.7) are equal to TAB21;;AB32;;AB13 . This is easy to verify but hard to obtain when the expressions are large. Fortunately, one can use an e cient algorithm due to Dynkin, Specht and Wever (for a pedagogical account, see [63]) to nd the linear combinations of TAB11;;::::::;;ABnn that capture the products of AijBj . The solution exploits the fact that the Dynkin bracket ` gives rise to a Lie idempotent; n n1 `(A1; : : : ; An). Therefore, rewriting each word of length n within a Lie polynomial as n1 `(P ) leads to the answer, e.g., ab ba = 12 `(ab) 12 `(ba) = `(ab). In order to apply this algorithm to products of AijBj , rst rewrite its products such that the Bj labels are always in the same order B1B2B3. For example, (A.11) becomes, L1L2L3 + L2L3L1 where in the second line we used the shorthand notation with non-commutative variables L:::. Applying the idempotent operator n one obtains `(L1; L3; L2) `(L1; L2; L3) + `(L2; L3; L1) `(L3; L2; L1) where we used the property `(a1; a2; i) = `(i; `(a1; a2)) [33]. This algorithm has been used to cast the 0-expansion of the BG double current in terms of the de nition (3.13). Ansatz for the Berends-Giele recursion at higher order in As explicitly tested up to and including order 04, one arrives at a unique recursion for the Berends-Giele double current AjB that reproduces, via (3.1), the disk integrals at various w 2-orders by imposing the following constraints on an ansatz of the form in (3.6): 1. adjusting the powers of momenta and elds to the mass dimensions of the 0w-order 3. absence of dot products (kAi kBj ), (kBi kBj ) and kA2i A=A1A2:::Ap 5. matching the order- 0w recursion with known n-point disk integrals for all n 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 elds and derivatives, p A1jBi1 A2jBi2 : : : Aw+2 pjBiw+2 p p = 0; 1; : : : ; w e.g. the ansatz of the form (3.6) for the 02 2-order generalizes to three types of terms with schematic form k (order 03) $ (kAp kAq )(kAr kAs ) Y Regular parts of ve-point integrals The contributions to of order 4 in the elds are governed by the 0-expansion of regu ve-point integrals, see (3.33). In the regularization scheme explained in section 4, the relevant leading orders are given by = 2 0 3(@24 + @34) + O( 02) 2 + 0 3(@12 + 2@13 + 2@23 + 2@24 + @34) + O( 02) = 2 + 0 3( 2@12 = 2 0 3(@12 + @13) + O( 02) ; while the terms at higher orders in 0 can be found in the ancillary les. Note that the integrals over (z12z23) 1 and (z43z32) 1 have been assembled from the simpset basis (4.19). Multiple polylogarithm techniques Polylogarithms and MZVs Polylogarithms G(a1; a2; : : : ; an; 1) at unit argument with labels ai 2 f0; 1g can be congences. Divergent iterated integrals G(1; : : : ; 1) and G(: : : ; 0; 1) in this work will be shu ethree, the appearance of 2 and 3 in (3.25) can be traced back to G(1; 0; 1) = + 2 G(1; 0; 0; 1) = G(1; 1; 0; 1) = + 3 G(0; 1; 1) = G(0; 1; 0; 1) = + 2 3; G(1; 0; 1; 1) = G(0; 0; 1; 1) = G(0; 1; 1; 1) = + 3 : The analogous higher-weight relations follow from (4.3), while several identities among MZVs can be found in [108] (obtained using harmonic polylogarithms [109]). Methods for shu e regularization By the shu e 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. G(A; an 1; 0; z) = G(A; an 1; z)G(0; z) G(A 0; an 1; z) ; G(z; a2; A; z) = G(z; z)G(a2; A; z) G(a2; z A; z) ; an 1 6= 0 a2 6= z : ln jzj captures the entire endpoint divergence from the lower integration limit. The same kind of shu e operations includG(0; z) [69]. Analogous statements based on a regularization prescription for G(z; z) can be made for upper-endpoint divergences in integrals like G(z; z; : : : ; z; ak; : : : ; an; z) with ak 6= z. z-removal identities The de nition (4.1) of polylogarithms applies to situations where the integration variable This appendix is devoted to integration techniques for polylogarithms with more general arguments, i.e. with multiple appearances of the integration variable z as G(: : : ; z; : : : ; z) G(a1; : : : ; ai 1; z; ai+1; : : : ; an; z) = G(a1; : : : ; ai 1; t; ai+1; : : : ; an; t) + c(a1; : : : ; ai 1; z^; ai+1; : : : ; an) ; in (D.4) can be evaluated through the di erential equations (^aj means that aj is omitted) G(a2; : : : ; an; z) G(: : : ; a^n 1; an; z) + G(: : : ; a^i 1; : : : ; z) + G(: : : ; a^i+1; : : : ; z) G(: : : ; a^i; : : : ; z) ; i 6= 1; n G(: : : ; an 1; z) : Simple z-removal identities Let us rst address the simpler subset of z-removal identities, where the integration variable is present on both sides of the semicolon, i.e. cases of the schematic form G(: : : ; z; : : : ; z). Inserting the di erential equations (D.5) into (D.4) recursively eliminates the variable z from the labels [5], +G(ai 1; a1; : : : ; ai 1; z^; ai+1; : : : ; an; z) G(ai+1; a1; : : : ; ai 1; z^; ai+1; : : : ; an; z) G(a1; : : : ; a^i 1; t; ai+1; : : : ; an; t) G(a1; : : : ; ai 1; t; a^i+1; : : : ; an; t) G(a2; : : : ; ai 1; t; ai+1; : : : ; an; t); i 6= 1; n ; G(a1; : : : ; an 2; t; t)+ G(a2; : : : ; an 1; t; t) : Similar recursions for repeated appearance of z among the labels as in G(: : : ; z; z; : : : ; z) can be derived from (D.5) and (D.4) in exactly the same manner. The integration constants c(: : : ; z^; : : :) in (D.4) are generically zero unless the labels are exclusively formed from letters aj 2 f0; z^g, in which case they yield MZVs (4.3): c(a1; a2; : : : ; an) = <0 : 9 aj 2= f0; z^g :G( az^1 ; az^2 ; : : : ; az^n ; 1) : aj 2 f0; z^g The simplest nonzero applications of (D.8) at weight two and three are ; c(z^; 0) = + 2; c(0; 0; z^) = c(z^; 0; 0) = ; c(0; z^; 0) = 2 3 and follow from (D.1). For example, the above steps lead to the z-removal identities25 G(a1; z; z) = G(a1; a1; z) G(0; a1; z) G(a1; a2; z; z) = G(a2; 0; a1; z) G(a2; a1; a1; z) + G(a1; a2; a2; z) G(a1; z; a2; z) = G(a1; a1; a2; z) G(a2; 0; a1; z) + G(a2; a1; a1; z) G(a1; 0; a2; z) + G(a2; a1; a2; z) G(0; a1; a2; z) a2;0G(a1; z) 2 + a1;0G(a2; z) 2 G(a2; a1; a2; z) a1;0G(a2; z) 2 + 2 a1;0 a2;0 3 G(a; z; z; z) = G(0; 0; a; z) G(0; a; a; z) G(a; 0; a; z) + G(a; a; a; z) + a;0 3 : (D.10) Note that analogous z-removal identities for G(z; a1; z), G(z; a1; a2; z) and other divergent cases follow from the shu e relation (4.2), see (4.10) for the regularized values of G(z; z) that di er from the choice in [5]. General z-removal identities As exempli ed by (4.12), some of the regularized integrals require di erent orders of integration over the variables z2; z3; : : : ; zn 2. In these situations it can happen that polylogarithms such as G(0; z4; z3) need to be converted to G(: : : ; z4) with no additional instance of z4 in the ellipsis in order to integrate over z4 rst. This requires a generalization of the techniques in the previous subsection. As before, the starting point for a recursion is the di erential equation (D.4) for derivatives in the labels of polylogarithms. The recursion is supplemented by the initial condition G(z1; z2) = G(z2; z1) + G(0; z2) i sign(z2; z1) ; sign(zi; zj ) For example, the rst identity in (D.10) generalizes to G(a1; z1; z2) = G(a1; 0; z2) G(a1; z2; z1) G(0; z2)G(a1; z1) + G(a1; 0; z1) terms of (D.10) in the analogous identities. w = 1 w = 2 w = 3 w = 4 w = 5 w = 6 w = 2 w = 3 w = 4 w = 5 w = 6 w = 3 w = 4 w = 5 w = 6 w = 4 w = 5 w = 6 w = 5 w = 6 w = 6 ` = 0 ` = 1 ` = 2 ` = 3 ` = 4 ` = 5 ` = 0 ` = 1 ` = 2 ` = 3 ` = 4 ` = 0 ` = 1 ` = 2 ` = 3 ` = 0 ` = 1 ` = 2 ` = 0 ` = 1 ` = 0 schematic form of the Berends-Giele double current, the required weight w of z-removal identities (G(a1; : : : ; aw; z)) and the order 0` of the Koba-Nielsen expansion (4.5). +G(a1; z2) G(a1; z1) +i sign(z2; z1)(G(a1; z1) G(a1; z2)) : Note that the polylogarithms on the right hand side are suitable for integration over z1 since there are no instances of z1 among their labels. The use of z-removal identities represents 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 06 6 5 in the Z-theory equation of motion (3.31) arise from integrating the third subleading order Koba-Nielsen factor (4.5) | the o set is due to the factor ( 0)(n 3) in (3.30) | and Alternative description of regularized disk integrals In this appendix, we present a method to determine the 0-expansions for regularized disk integrals selected by the Z-theory equation of motion from the (n 3)! (n 3)! basis FP Q de ned in (2.14). This approach has been very useful to constrain the required regularization scheme via explicit data at high orders of 0, 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 0-expansions of disk integrals at multiplicities and orders where no prior knowledge of FP Q is available. Closely following the lines of [5], the basic idea is to divide disk integrals26 Z(IjP ) into a singular and a regular part with respect to region variables si;i+1:::j in (2.4). The singular parts associated with the propagators of the eld-theory limits can be subtracted with residues given by lower-multiplicity data, and the leftover local expression is identi ed with the regularized integrals in (3.31). However, there are ambiguities in the subtraction scheme by shifting the numerator N ! N + O(s) in the subtracted singular expression examples suggest that changes in the regularization scheme or the integration order can be compensated by the choice of subtraction scheme when reproducing the associated local expressions from regularized integrals over Taylor-expanded Koba-Nielsen factors. In the setup of [5], the regularization scheme for divergent integrals was xed and designed to preserve the shu e algebra and scaling relations of polylogarithms such that 0 instead of (4.10). Moreover, the integration orders were globally chosen as 23 : : : n 2 (i.e. integrating over z2 rst and over zn 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 xed (or \minimal") subtraction scheme for the poles of Z(IjP ). The resulting regular parts | to be denoted by J:r:e:g(: : :) 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, re ecting the di erent choices of regularization scheme and integration orders between this work and [5]. We will regard SL(2; R)- xed combinations of disk integrals Z(P jQ) in the notation Ju1v1;u2v2;:::;un 3vn 3 (k1; k2; : : : ; kn 1) a single pole channel in the eld-theory limit. as functions of n 1 massless momenta kj which determine the sij on the right hand side through their independent dot products. The product k1 kn 1 can be eliminated by This re ects the choice of ansatz in appendix B, where (kA1 kAp ) referring to the outermost slots A1; Ap in a deconcatenation P A=A1A2:::Ap is excluded. In the four-point case, the eld-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 J2r1eg(k1; k2; k3) = J21(k1; k2; k3) J3r2eg(k1; k2; k3) = J32(k1; k2; k3) are analytic in sij and coincide with the regularized integrals (3.25) [5] in any regularization scheme of our awareness. Their 0-expansion is straightforwardly determined by F22 in (2.19) (also see [110] for a neat representation in terms of G(0; : : : ; 0; 1; : : : ; 1; 1)), J21(k1; k2; k3) = J32(k1; k2; k3) = no reference to kP2 is expected. Five-point pole subtraction The regular parts Jirjeg(: : :) in (E.2) are by themselves functions of three light-like momenta under spq ! kp kq and can later on be promoted to massive momenta kP provided that At ve points, generic eld-theory limits of Z(P jQ) yield two simultaneous propagators, and by factorization on four-point integrals, the residue on single poles in sij still involves all orders in 0. As elaborated in [5], the 0-dependence of the singular pieces can be removed using the regular four-point expressions in (E.2) with composite momenta kij J2r1eg;43(k1; k2; k3; k4) = J21;43(k1; k2; k3; k4) J3r1eg;42(k1; k2; k3; k4) = J31;42(k1; k2; k3; k4) J2r1eg;31(k1; k2; k3; k4) = J21;31(k1; k2; k3; k4) J3r2eg;31(k1; k2; k3; k4) = J32;31(k1; k2; k3; k4) Following the dot products of momenta, arguments k12; k3; k4 in the above Jirjeg instruct to replace any s12 and s23 in their expansion from (E.2) and (E.3) by s13 + s23 and s34, respectively [5]. Note that the counterpart of J2r1eg(k1; k23; k4) in [5] required a di erent replacement s12 ! s123 instead of the prescription s12 ! s12 + s13 in (E.4). This kind of of the [5] with G(z; z) In the same way as the 0-dependence of the local four-point expressions Jirjeg(: : :) is accessible from F22, their ve-point counterparts Jirje;gpq(: : :) can be expanded as soon as the right hand side of (E.4) is expressed in terms of the basis functions fF2323; F2332g, J21;43(k1; k2; k3; k4) = J31;42(k1; k2; k3; k4) = J21;31(k1; k2; k3; k4) = J32;31(k1; k2; k3; k4) = Explicit results on the 0-expansion of fF2323; F2332g as pioneered in [51{54] 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 [45, 110] render even higher orders in 0 accessible, also see [45] for a closed-form solution. Once we adjoin the parity images Jprqeg;rs(k1; k2; k3; k4) = 0 2 Z eom 4 J3r2eg(k1; k2; k3)J3r2eg(k123; k4; k5) s23s123s45 J2r1eg;43(k1; k23; k4; k5) Again, the arguments sij ! ki kj of Jprqeg;rs can be promoted to massive momenta ki ! kP as we will now see in the pole subtractions at higher-multiplicity. Six and seven-point pole subtraction The above ve-point examples shed light on various aspects of the regularization scheme selected by the Z-theory equation of motion including the integration orderings and the z-removal identities in appendix D.3. However, the appearance of i in (4.10) cannot be seen from integrals below multiplicity six, so the J:r:e:g(: : :) at (n 6)-points have been an instrumental window to infer these particularly subtle ingredients of the regularization scheme. In this section, we present one example each at multiplicity six and seven: J3r1eg;32;54(k1; k2; : : : ; k5) = J31;32;54(k1; k2; : : : ; k5) J3r1eg;32(k1; k2; k3; k45) one can extract valuable all-weight information on the regularization scheme for ve-point integrals in (3.31) by demanding the 0-expansion of (E.4) and (E.6) to match with Note that also the counterparts of J2r1eg(k1; k23; k45) and J2r1eg;43(k1; k23; k4; k5) seen in [5] exhibit additional contributions s23 in their arguments. In the J:r:e:g(: : :) under discussion, At seven points, the local integral used in [5] to generate the expansion of FP Q up to and including the 07-order stored on the website [56] matches with J2r1eg;31;41;65 = +J21;31;41;65 : s12s123s1234s56 s12s123s1234 s123s1234s56 s12s1234s56 s12s123s56 J2r1eg;31(k1; k2; k3; k4) J2r1eg(k1; k2; k3)J2r1eg(k123; k4; k56) J2r1eg(k1; k2; k3)J3r2eg(k1234; k5; k6) J2r1eg;43(k123; k4; k5; k6) J2r1eg;31(k12; k3; k4; k56) J2r1eg(k12; k3; k4)J3r2eg(k1234; k5; k6) J2r1eg;31;41(k1; k2; k3; k4; k56) J2r1eg;31;54(k12; k3; k4; k5; k6) J2r1eg(k1; k2; k3)J2r1eg;43(k123; k4; k5; k6) J2r1eg;31(k1; k2; k3; k4)J3r2eg(k1234; k5; k6) J31;32;54 = J21;31;41;65 = s23s45s123 The general strategy s1234s56 s12s123 The 0-expansion of the right hand sides of (E.8) and (E.9) is available from the following decompositions into basis functions FP Q: The choice of labels and momenta for the J:r:e:g(kA1 ; kA2 ; : : : ; kAm 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 limit. Each factor of zij1 in the integrand is associated with one of the n 3 propagators of the eld-theory diagram, and the pole subtraction exhausts all 2n 3 possibilities to relax a subset of these propagators. The residue of diagrams with less than n 3 propagators is a J:r:e:g(: : :) labeled by the zij1-factors associated with the relaxed propagators, i.e. each relaxed propagator increases the multiplicity of the associated J:r:e:g(: : :) by one. The massive momenta in its arguments can be read o 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:r:e:g(: : :) at sij $ ki kj are compatible with the integrals (3.31) in the regularization scheme and integration orders of this work, Jure1gv1;u2v2;:::;up 2vp 2 (k1; k2; : : : ; kp) = ( 0)p 2 A variety of alternative regularization schemes and integration orders including those of [5] are expected to correspond to a modi ed choice of arguments for J:r:e:g(kA1; kA2; : : : ; kAn 1), where selected dot products kAp kAq are shifted by (half of) kA2i. Integration orders for the seven-point integrals In this appendix, we explicitly list the results of section 4.3 on the integration orders for regularized seven-point integrals in the simpset basis (see section 4.2). The rst topology z15z12z13z14 ! 2345; z15z13z23z14 ! 2345; z15z12z13z45 ! (23 4)5; z15z13z23z45 ! (23 4)5; z15z12z35z45 ! (43 2)5; z15z12z34z35 ! (43 2)5; z15z25z35z45 ! 4325; z15z25z34z35 ! 4325; z15z12z34z14 ! (2 3)45; z15z13z24z14 ! (2 3)45; z15z14z24z35 ! (24 3)5; z15z12z14z35 ! (24 3)5; z15z13z24z25 ! (42 3)5; z15z13z25z45 ! (42 3)5; z15z25z23z45 ! (3 4)25; z15z25z24z35 ! (3 4)25; z15z23z24z14 ! 3245; z15z24z34z14 ! 3245; z15z13z14z25 ! (34 2)5; z15z14z34z25 ! (34 2)5; z15z14z25z35 ! (32 4)5; z15z14z23z25 ! (32 2)5; z15z25z24z34 ! 3425; z15z25z24z23 ! 3425: z12z13z14z56 ! 234 5; z13z23z14z56 ! 234 5; z12z13z15z46 ! 235 4; z13z23z15z46 ! 235 4; z12z14z15z36 ! 245 3; z14z24z15z36 ! 245 3; z13z14z15z26 ! 345 2; z14z34z15z26 ! 345 2; z12z34z14z56 ! ((2 3)4) 5; z13z24z14z56 ! ((2 3)4) 5; z12z35z15z46 ! ((2 3)5) 4; z13z25z15z46 ! ((2 3)5) 4; z12z45z15z36 ! ((2 4)5) 3; z14z25z15z36 ! ((2 4)5) 3; z13z45z15z26 ! ((3 4)5) 2; z14z35z15z26 ! ((3 4)5) 2; z23z24z14z56 ! 324 5; z24z34z14z56 ! 324 5; z23z25z15z46 ! 325 4; z25z35z15z46 ! 325 4; z24z25z15z36 ! 425 3; z25z45z15z36 ! 425 3; z34z35z15z26 ! 435 2; z35z45z15z26 ! 435 2: (F.2) following integration orders, z12z13z46z56 ! 23 54; z13z23z45z46 ! 23 54; z12z14z35z36 ! 24 53; z15z45z26z36 ! 45 32; z13z23z46z56 ! 23 54; z12z14z36z56 ! 24 53; z14z24z35z36 ! 24 53; z13z14z25z26 ! 34 52; z15z25z36z46 ! 25 43; z13z15z26z46 ! 35 42; z15z35z24z26 ! 35 42; z14z15z23z26 ! 45 32; z12z13z45z46 ! 23 54; z14z24z36z56 ! 24 53; z13z14z26z56 ! 34 52; z14z34z25z26 ! 34 52; z12z15z34z36 ! 25 43; z15z35z26z46 ! 35 42; z14z15z26z36 ! 45 32; z15z45z23z26 ! 45 32; and the remaining topologies of the simpset basis at seven points follow from (F.1) and (F.2) via parity zj ! z7 j . The seven-point R eom-integrals are su cient to determine the order of the Z-theory equation of motion (3.31) to any order in 0 and the 04-order of disk integrals at any multiplicity. Open Access. 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Carlos R. Mafra, Oliver Schlotterer. Non-abelian Z-theory: Berends-Giele recursion for the α ′-expansion of disk integrals, Journal of High Energy Physics, 2017, 31, DOI: 10.1007/JHEP01(2017)031