Nonabelian Ztheory: BerendsGiele recursion for the α ′expansion of disk integrals
Received: November
Nonabelian
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 0expansion of disk integrals arising in treelevel 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 Ztheory, we pinpoint the equation of motion of Ztheory from the BerendsGiele 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.
cMaxPlanckInstitut fur Gravitationsphysik; AlbertEinsteinInstitut

BerendsGiele recursion for the
0expansion of disk integrals
1 Introduction
Ztheory 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
0expansion of disk amplitudes
Basisexpansion of disk integrals
BerendsGiele recursion for the eldtheory limit
Example application of the BerendsGiele recursion
BerendsGiele recursion for disk integrals
Symmetries of the full BerendsGiele double currents
The 02correction to BerendsGiele currents of disk integrals
Free parameters versus Ztheory equation of motion
Manifesting the shu e symmetries of BG currents
Simplifying the 02correction to BG currents
The perturbiner description of 0corrections
Perturbiners at higher order in 0
Allorder prediction for the BG recursion
Local disk integrals in the Ztheory 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 KobaNielsen 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 BerendsGiele double currents
A.1 Shu e symmetry
A.2 Generalized Jacobi symmetry
A.3 BerendsGiele double current and nested commutators
B Ansatz for the BerendsGiele recursion at higher order in 0
C Regular parts of vepoint integrals
D Multiple polylogarithm techniques
D.1 Polylogarithms and MZVs
Methods for shu e regularization
D.3 zremoval identities
Simple zremoval identities
D.3.2 General zremoval identities
Fivepoint pole subtraction
E.2 Six and sevenpoint pole subtraction
E.3 The general strategy
E Alternative description of regularized disk integrals
F Integration orders for the sevenpoint 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 0corrections to eld theory is through the calculation of string
scattering amplitudes, see e.g. [1, 2]. Within this framework, higherderivative corrections
are encoded in the 0expansion of certain integrals de ned on the Riemann surface that
encodes the string interactions.
In this work, we will mostly study treelevel scattering of open strings, where the
Riemann surface has the topology of a disk. As will be reviewed in section 2, the
0corrections to superYangMills (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 0expansion 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 doublecopy structures among eld and string theories.
As the technical novelty of this paper, we set up a BerendsGiele (BG) recursion [3]
that allows to compute the 0expansion of the integrals Z(P jQ) and generalizes a recent
BG recursion [4] for their eldtheory 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 0corrections
at large multiplicities is unprecedented in modern allmultiplicity approaches [5, 6] to the
0expansion of disk integrals.
The conceptual novelty of this article is related to the interpretation of string disk
integrals (1.1) as treelevel amplitudes in an e ective1 theory of bicolored scalar elds
dubbed as Ztheory [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 BerendsGiele recursion2 that computes the
0expansion 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 Ztheory
amplitudes [7] by pinpointing its underlying equation of motion. After this fundamental
conceptual shift to extract the 0expansion 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 highenergy properties of Ztheory (and its
quantum corrections) are left for future investigations.
2For a recent derivation of BerendsGiele recursions for tree amplitudes from a perturbiner solution of
the eldtheory 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 allorder proposal (1.4) has played a fundamental role in the allmultiplicity
derivation of local treelevel numerators [4, 12] which obey the duality between color and
kinematics [13].
Ztheory and double copies
The relevance of the disk integrals (1.1) is much broader than what the higherderivative
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 closedstring treelevel
integrals (encoding 0corrections to supergravity theories) boil down to squares of disk
integrals through the KLT relations [14]. In a eldtheory context, this doublecopy connection
between open and closed strings became a crucial hint in understanding quantumgravity
interactions as a square of suitablyarranged gaugetheory building blocks [13, 15].
Doublecopy structures have recently been identi ed in the treelevel amplitudes of
additional eld theories [16]. For instance, classical BornInfeld theory [17] which governs
the lowenergy 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 nonlinear sigma model
(NLSM) [19{23], see [24{27] for its treelevel amplitudes. As a stringtheory incarnation
of the BornInfeld double copy, treelevel amplitudes of the NLSM have been identi ed as
the lowenergy 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 treelevel Smatrix of massless opensuperstring states can be
presented as a double copy of SYM with
0dependent disk integrals [5]. Their Ztheory
interpretation in [7] was driven by the quest to identify the second doublecopy ingredient of
the open superstring besides SYM. In view of the biadjointscalar and NLSM interactions
in the lowenergy limit of Ztheory, its full edged 0dependence describes e ective
higherderivative deformations of these two scalar eld theories [7]. As a doublecopy component
to complete SYM to the massless opensuperstring Smatrix, the collection of e ective
interactions encompassed by Ztheory deserve further investigations.
In this work, we identify the equation of motion (1.4) of the full nonabelian Ztheory,
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 Ztheory 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 Ztheory are of particular interest to advance our understanding
of the BCJ duality. The abelian limit of Ztheory 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 stringtheory realization of the NLSM through toroidal compacti cations in presence of
worldsheet boundary condensates.
for BCJsatisfying 0corrections to the NLSM. The present article extends this endeavor
such as to e ciently compute the doublypartial amplitudes of e ective bicolored 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
eldtheory limit in section 2, the BerendsGiele recursion for
their 0corrections and the resulting eld equations of nonabelian Ztheory 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 BerendsGiele recursion is extended to closedstring 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 07order in the 0expansion 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 openstring states. We
also review the recent BerendsGiele approach to their eldtheory limit in order to set the
stage for the generalization to 0corrections.
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 treelevel 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 openstring 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 KleissKuijf
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 colorstripped 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 0corrections. 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 npoint treelevel 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 BerendsGiele representation in pure spinor superspace [39]. Hence, (2.13)
applies to the entire tendimensional gauge multiplet in the external states.4
4A bosoniccomponent 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 eldtheory KLT relations [41, 42] (see also [43]
for the 0corrections 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 npoint opensuperstring
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 doublecopy structure of
the opensuperstring treelevel Smatrix 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 bicolored 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.
0dependent kinematic factors ASYM(1; Q; n 1; n) !
0expansion of disk amplitudes
The 0expansion 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 fourpoint instance of (2.14),
grals at multiplicity n
5 generally involve MZVs of higher depth r
2, see [45] for a
recent closedform 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 0worder 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 0expansion of the openstring 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 0expansion of FP Q are available for download on [56].
Basisexpansion of disk integrals
In setting up the BerendsGiele recursion for the fundamental objects Z(AjB) of this work,
it is instrumental to e ciently extract their 0expansion 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 3theory which arise in
the eldtheory 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 Ztheory.
BerendsGiele recursion for the
eldtheory limit
The task we want to accomplish in this paper concerns the computation of the 0expansion
of the disk integrals (2.2) in a recursive and e cient manner. In the eldtheory limit
0 ! 0, allmultiplicity 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 BerendsGiele
description [4] via biadjoint 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 Ztheory particles
whose interactions give rise to the disk integrals Z(P jQ) as their
doubly partial amplitudes. The nonlinear eld equations in the lowenergy limit
(a0b) = facdf~bgh c g
5After pioneering work in [51{54], the 0expansion of disk integrals at multiplicity n
5 has later
been systematically addressed via allmultiplicity 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 0corrections
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 BerendsGiele 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 treelevel 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 eldtheory 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 threeparticle level.
As shown in [4], the
eldtheory limits of the disk integrals (2.2) and thereby the
doubly partial amplitudes (2.21) are given by the BerendsGiele double currents (A0j)B,
S 1[P jQ]1 =
6See [8, 58, 59] for perturbiner solutions to selfdual sectors of fourdimensional gauge and gravity theories
(see also [10]) and [9] for perturbiners in tendimensional 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 0corrections of string disk integrals.
BerendsGiele recursion for disk integrals
In this section, we develop a BerendsGiele recursion7 for the full edged disk integrals
Z(P jQ) de ned in (2.2). The idea is to construct 0dependent BerendsGiele double
AjB such that the integrals Z(P jQ) including 0corrections are obtained in the
same manner as their eldtheory limit in (2.28),
And similarly, the 0corrected 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 eldtheory limit of the vepoint disk integral
m(13524j32451) = lim
2 Z
D(13524)vol(SL(2; R)) i<j
z32z24z45z51z13
using the BerendsGiele 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 nonlinear equations of motions which can be viewed as an augmentation
of (2.22) by 0corrections. The eld equation obeyed by the perturbiner (3.2) will be
interpreted as the equation of motion of Ztheory, the collection of e ective theories involving
7For a review of the BerendsGiele recursion for gluon amplitudes [3] which is adapted to the current
discussion, see section 2 of [4].
bicolored scalars encoding all the 0corrections 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 BerendsGiele 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 nonzero expressions9 O(( 0 )2) for P QjB. As a consequence, the perturbiner (3.2)
is Liealgebra valued w.r.t. the t~b generators [31] but not w.r.t. the ta generators. That is
why the Ztheory scalar
is referred to as bicolored rather than biadjoint.
The symmetries (3.4) and (3.5) will play a fundamental role in the construction of
ansaetze for the
0corrections of the BerendsGiele double currents, see appendices A
and B for further details.
02correction to BerendsGiele currents of disk integrals
Assuming that the 02corrections of the integrals (2.2) can be described by BerendsGiele
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 02terms
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 BerendsGiele approach to closedstring integrals.
9Since the monodromy relations only di er from the KK relations by rational multiples of 2n or
( 2)n, the subsector of Z(A; njB; n) without any factors of 2 still satis es shu e symmetries, e.g.
BA11::::::BA44==BA
into nonempty 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 02order of various integrals at four and
rameters, leaving a total of four free parameters. The 02order 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 BerendsGiele setup kicks in: a
nite amount
of lowmultiplicity data  the coe cients of k2 3 and
4terms (3.6) at the 02order 
determines the relevant order of disk integrals at any multiplicity.
BA11BA22==BA
Free parameters versus Ztheory 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 BerendsGiele
recursion method as the perturbiner solution (2.23) to the Ztheory equation of motion with
the schematic form
2 + O( 3). Selfcontractions (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 righthand 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 slotmomentum10 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 lowenergy expansion of disk integrals at any multiplicity via (3.1):
sA AjB =
BA11::::::BA33==BA
10In general, in a pfold 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
vepoint data entered in the
derivation of (3.7), this recursion allows the computation of 02 terms of disk integrals at
arbitrary multiplicity. The elevenpoint 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 Ztheory 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
02correction 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 righthand
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 0corrections
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 Aslots in TABi11;;BA2i;2::;::;:B:;Apip , while the ordering
of the Bslots 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 BerendsGiele 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 Ztheory 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 Ztheory equation of motion provides the
essential clue for proposing the BerendsGiele 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 bicolored
= P
A. The coe cients
are still Liealgebra 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 BerendsGiele recursion that reproduces
the 02corrections 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 Ztheory
equation of motion up to the 03order:
= [ 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 0corrections to KK relations from (2.12), the Ztheory scalar
is not Liealgebra
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 fourpoint 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 fourpoint
KobaNielsen factor via
jzij j 0sij =
which removes the kinematic poles from the full disk integrals and yields their nonsingular
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 sixpoint integrals.
5 in (3.24) can be traced back to
Allorder prediction for the BG recursion
From the observations in the previous subsection, we propose a closed form for the
contributions to the Ztheory 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 fourpoint 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 allmultiplicity mapping
jP j + jQj = n
preserves all the symmetry properties of its constituents. Finally, the KobaNielsen 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 Ztheory 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 threepoint massless disk amplitude under the mapping (3.29); hV1V2V3i
Extrapolating the above pattern, a natural candidate for the higherorder contributions
; 5; : : : to the Ztheory equation of motion emerges from the integrand of the (n
2)!term representation of the npoint 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 Ztheory 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 KobaNielsen 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
gaugegroup generators ta and t~b which govern the color structure of .
For example, the equation of motion up to
4order 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 zremoval
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 zremoval 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 Ztheory equation of motion contains
0 4@12[ 1; 5243], in agreement with the BerendsGiele recursion at order 0
previously obtained from an ansatz.
Closedstring integrals
Our results have a natural counterpart for closedstring scattering, where treelevel
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 treelevel 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 eldtheory 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 0corrections 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 singlevalued projection \sv" [81, 82] of the MZVs in the
disk integrals [61, 83]
W (P jQ) = sv Z(P jQ) :
The singlevalued 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 BerendsGiele representation
W (A; njB; n) = sA sv
of closedstring integrals can be derived from the same currents AjB which govern the disk
integrals via (3.1). Hence, any tentative \singlevalued Ztheory" de ned by reproducing
the closedstring integrals (5.1) as its doubly partial amplitudes is necessarily contained in
the nonabelian Ztheory of this paper.
Note that reality of the sphere integrals W (P jQ) along with the phasespace constraint
the following onshell properties
= sv
= O(sA) :
Hence, one can perform
eld rede nitions such as to render the associated perturbiner
sv[ ] Liealgebra valued in both gauge groups.
24The same kind of organization in terms of (5.1) is expected to be possible in treelevel 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 0expansion of disk integrals present
in the massless npoint treelevel 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 bicolored scalars , dubbed as Ztheory in previous work [7]. Its
equation of motion (3.31) furnishes the central result of this work and compactly encodes
the BerendsGiele recursions that elegantly compute the 0expansions of the disk integrals
at arbitrary multiplicity. More precisely, the Ztheory equation of motion (3.31) is satis ed
by the perturbiner series of the BerendsGiele currents, and its structure is shared by an
(n 2)!term representation of the npoint openstring 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 BerendsGiele
description of disk integrals sheds new light on the doublecopy structure of the openstring
treelevel Smatrix [5]. As manifested by (2.17), disk amplitudes exhibit a KLTlike factorization
into SYM amplitudes and disk integrals Z(P jQ). Following the interpretation of Z(P jQ)
as Ztheory amplitudes [7], the perturbiner description of the BerendsGiele recursion for
disk integrals pinpoints the eld equation (3.31) of Ztheory. Hence, our results give a more
precise de nition of Ztheory, the second doublecopy 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 nonlinear equation of motion (3.31) of Ztheory 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 0corrections to the NLSM [7] as well as mixed Ztheory
amplitudes involving both bicolored scalars and NLSM pions in future work [84].
Do worldsheet integrals over highergenus surfaces admit a similar interpretation as
Ztheory amplitudes? It might be rewarding to approach the lowenergy expansion of
superstring loop amplitudes at higher multiplicity with BerendsGiele methods. At the
oneloop 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 Ztheory amplitudes? Given that BCFW
onshell recursions [97] can in principle be applied string amplitudes [98{100], it would be
interesting to relate the BerendsGiele recursion for Ztheory amplitudes to BCFW methods.
Furthermore, what are the nonperturbative solutions to the full Ztheory equation of
motion (3.31)? A nonperturbative solution to the eld equation
2 of biadjoint
scalars (obtained from the eldtheory limit 0 ! 0) has been recently found [101] in an
attempt to understand the nonperturbative regime of the doublecopy construction.
In addition, is it possible to obtain
eld equations or e ective actions for massless
open or closedsuperstring states along similar lines of (3.31)? In order to approach the
0corrections to the SYM action, the resemblance of such an equation of motion with
the BerendsGiele description of super elds in pure spinor superspace [9, 64] is intriguing.
This parallel might for instance be useful in generating the 0corrections to the onshell
constraint fr ; r g
m rm = 0 of tendimensional SYM [102{107].
Related to this, it would be desirable to express the Ztheory 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
0expansion of the BerendsGiele recursion into a similarly elegant form. One could even
envision to generate the treelevel e ective action of the open superstring from the SYM
action by acting with appropriate operatorvalued 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 Ztheory 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 Ztheory 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 AlbertEinsteinInstitut 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 BerendsGiele double currents
In this appendix we discuss the symmetries obeyed by the BerendsGiele double currents.
e symmetry
In order to make sure that our ansaetze for BG currents (3.1) for disk integrals satisfy the
shu esymmetry
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 lefttoright 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 nonempty 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,
BerendsGiele 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 noncommutative 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 0expansion of the BG double current in terms of the de nition (3.13).
Ansatz for the BerendsGiele recursion at higher order in
As explicitly tested up to and including order 04, one arrives at a unique recursion for the
BerendsGiele double current
AjB that reproduces, via (3.1), the disk integrals at various
w 2orders 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 0worder
3. absence of dot products (kAi kBj ), (kBi kBj ) and kA2i
A=A1A2:::Ap
5. matching the order 0w recursion with known npoint disk integrals for all n
By dimensional analysis and triviality of the threepoint 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 2order generalizes to three types of terms with
schematic form k
(order 03) $ (kAp kAq )(kAr kAs ) Y
Regular parts of
vepoint integrals
The contributions to
of order 4 in the elds are governed by the 0expansion of
regu
vepoint 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 higherweight 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 weightone 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 upperendpoint divergences in integrals like G(z; z; : : : ; z; ak; : : : ; an; z)
with ak 6= z.
zremoval 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 zremoval identities
Let us rst address the simpler subset of zremoval 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 zremoval 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 zremoval 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 zremoval 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 BerendsGiele double current, the required weight w of zremoval identities
(G(a1; : : : ; aw; z)) and the order 0` of the KobaNielsen 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 zremoval 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 BerendsGiele
recursion is given in table 1. For example, terms at the order of 06 6 5 in the Ztheory
equation of motion (3.31) arise from integrating the third subleading order
KobaNielsen 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 0expansions for regularized
disk integrals selected by the Ztheory 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 BerendsGiele 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 Ztheory equation of motion: the ultimate goal and achievement of this
work is to compute 0expansions 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 eldtheory limits can be subtracted
with residues given by lowermultiplicity 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 Taylorexpanded KobaNielsen 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 scalingpreserving regularization. In these adjustments of the
subtraction scheme, certain regular admixtures were incorporated by systematically shifting
the arguments of the lowerpoint 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 Ztheory 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 eldtheory 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 fourpoint case, the eldtheory 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 0expansion 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.
Fivepoint pole subtraction
The regular parts Jirjeg(: : :) in (E.2) are by themselves functions of three lightlike momenta
under spq ! kp kq and can later on be promoted to massive momenta kP provided that
At ve points, generic eldtheory limits of Z(P jQ) yield two simultaneous propagators,
and by factorization on fourpoint integrals, the residue on single poles in sij still involves all
orders in 0. As elaborated in [5], the 0dependence of the singular pieces can be removed
using the regular fourpoint 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 0dependence of the local fourpoint expressions Jirjeg(: : :) is
accessible from F22, their vepoint 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 0expansion of fF2323; F2332g as pioneered in [51{54] are available
from the allmultiplicity methods based on polylogarithms [5] and the Drinfeld
associator [6]. Moreover, recent advances based on their hypergeometricfunction
representation [45, 110] render even higher orders in 0 accessible, also see [45] for a closedform
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 highermultiplicity.
Six and sevenpoint pole subtraction
The above vepoint examples shed light on various aspects of the regularization scheme
selected by the Ztheory equation of motion including the integration orderings and the
zremoval 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 allweight information on the regularization scheme for vepoint
integrals in (3.31) by demanding the 0expansion 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 07order 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 0expansion 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 eldtheory 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 zij1factors 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 sevenpoint integrals
In this appendix, we explicitly list the results of section 4.3 on the integration orders for
regularized sevenpoint 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 sevenpoint R eomintegrals are su cient to determine the
order of the Ztheory equation of motion (3.31) to any order in 0 and the 04order of
disk integrals at any multiplicity.
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