Elliptic multiple zeta values and one-loop superstring amplitudes

We investigate iterated integrals on an elliptic curve, which are a natural genus-one generalization of multiple polylogarithms. These iterated integrals coincide with the multiple elliptic polylogarithms introduced by Brown and Levin when constrained to the real line. At unit argument they reduce to an elliptic analogue of multiple zeta values, whose network of relations we start to explore. A simple and natural application of this framework are one-loop scattering amplitudes in open superstring theory. In particular, elliptic multiple zeta values are a suitable language to express their low energy limit. Similar to the techniques available at tree-level, our formalism allows to completely automatize the calculation.


Introduction
In recent years, we have witnessed numerous fruitful interactions between number theory and particle physics. A particularly rich domain of intersection are iterated integrals, which prominently appear in scattering amplitudes in field theories and string theories. For a large class of Feynman and worldsheet integrals, multiple polylogarithms were recognized as a suitable language to cast results into a manageable form, see e.g. refs. [1][2][3][4]. In a variety of cases, the polylogarithms' Hopf algebra structure [5][6][7][8] paved the way towards efficient manipulations and the recognition of the simplicity hidden in the resulting scattering amplitudes.
However, a growing list of iterated integrals from various field and string theories implies that multiple polylogarithms do not mark the end of the rope in terms of transcendental functions appearing in scattering amplitudes. For example, multiple polylogarithms fail to capture the two-loop sunset integral with non-zero masses [9][10][11], an eight-loop graph in φ 4 theory [12,13] as well as the ten-point two-loop N 3 MHV amplitude in N = 4 super-Yang-Mills (sYM) theory [14]. The sunset integral and its generalization have recently been expressed in terms of elliptic diand trilogarithms [10,11,15], whose connection to the language suggested below remains to be worked out. Considering in addition their appearance in one-loop open-string amplitudes, the situation calls for a systematic study and classification of the entire family of elliptic iterated integrals 1 .
In the present article, we propose a framework for elliptic iterated integrals (or eIIs for short) and the associated periods, elliptic multiple zeta values (eMZVs). The framework aims at expressing scattering amplitudes in a variety of theories, and we here apply the techniques to one-loop amplitudes in open string theory as a first example. The language employed in the present article is primarily inspired by refs. [16,17], while refs. [18][19][20][21][22] contain further information on the mathematical background.
As opposed to multiple polylogarithms, which can be defined using just one type of differential form, elliptic iterated integrals require an infinite tower thereof [16]. These differential forms are based on a certain non-holomorphic extension of a classical Eisenstein-Kronecker series [23,16], and we show how they can be used to naturally characterize and label elliptic iterated integrals as well as eMZVs. We investigate their relations, which results in constructive algorithms to perform amplitude computations.
In the same way as multiple zeta values (MZVs) arise from multiple polylogarithms at unit argument, the evaluation of iterated integrals along a certain path of an elliptic curve leads to structurally interesting periods, the eMZVs [17] mentioned above. These are certain analogues of the standard multiple zeta values, which are related to elliptic associators [24] in the same way as multiple zeta values are related to the Drinfeld associator [25][26][27]. However, the precise connection is beyond the scope of the current article. Given their ubiquitous appearance in the subsequent string amplitude computation, we will investigate eMZVs and discuss some of their properties as well as their Q-linear relations.
The description of string scattering amplitudes via punctured Riemann surfaces at various genera directly leads to iterated integrals at the corresponding loop order. In particular, the disk integrals in open-string 2 tree-level amplitudes closely resemble multiple polylogarithms. Initially addressed via hypergeometric functions in refs. [34,35], the α -expansion of disk amplitudes finally proved to be a rich laboratory for MZVs. Their pattern of appearance has been understood in terms of mathematical structures such as motivic MZVs [7,29] and the Drinfeld associator [36][37][38]. Explicit expressions with any number of open-string states can be determined using polylogarithm manipulations [3] or a matrix representation of the associator [38]. A variety of examples are available for download at the website [39].
The calculation of one-loop open-string amplitudes involves worldsheet integrals of cylinder and Möbius-strip topology [40]. In the current article, we focus on iterated integrals over a single cylinder boundary and leave the other topologies for later. Recognizing the cylinder as a genus-one surface with boundaries, it is not surprising that the α -expansion of one-loop openstring amplitudes is a natural, simple and representative framework for the application of eIIs and eMZVs. We will explicitly perform calculations at four and five points for low orders in α in order to demonstrate their usefulness. Higher multiplicities and orders in α are argued to yield eMZVs and Eisenstein series on general grounds. In summary, one-loop string amplitudes turn out to be an ideal testing ground for the study of eMZVs, in particular because they appear in a more digestible context as compared to their instances in field theory.
This article is organized as follows: In section 2, we start by reviewing multiple polylogarithms and show, how their structure suggests a generalization to genus one. The appropriate differential forms and elliptic functions are discussed and put into a larger mathematical context in section 3. Section 4 is devoted to the application of eIIs and eMZVs to the four-point one-loop amplitude of the open string, while section 5 contains a discussion of its multi-particle generalization.

Iterated integrals on an elliptic curve
After recalling the definition of multiple polylogarithms as well as several conventions, we will introduce elliptic iterated integrals (eIIs) as their genus-one analogues. While we will limit ourselves to basic definitions and calculational tools in the current section, a thorough introduction to the mathematical background of elliptic functions will be provided in section 3.

Removing the argument z from the label
Starting from an arbitrary iterated integral, the corresponding polylogarithm can not always be determined straightforwardly: whenever the argument appears in the label a, an integration using eq. (2.1) is impossible. Solving this problem requires a rewriting of the multiple polylogarithm G ({0, a 1 , a 2 , . . . , a n , z}; z) (2.9) in terms of polylogarithms whose labels are free of the argument. G(a 1 , . . . , a i−1 , z, a i+1 , . . . , a n ; z) = G(a i−1 , a 1 , . . . , a i−1 ,ẑ, a i+1 , . . . , a n ; z) . . , a n ; t) (2.10c) G(a 1 , . . . , a i−1 , t,â i+1 , . . . , a n ; t) (2.10d) . . , a n ; t) (2.10e) allows to recursively remove the argument z from generic labels of a multiple polylogarithm, because the expressions on the right-hand side either have shorter labels or are free of z. A hat denotes the omission of the respective label, and it is assumed that at least one a j = 0. The availability of a recursive formula like eq. (2.10) is intrinsic to the moduli space of Riemann spheres with marked points [43]. An explicit discussion including algorithms is ref. [44].
As an identity similar to eq. (2.10) will be crucial in deriving relations for eIIs in subsection 2.2 below, let us briefly comment on the application and generalization of eq. (2.10): If the argument z appears multiple times in the label a, the first four terms on the right hand side (terms (2.10a) to (2.10c)) have to be evaluated for each occurrence of z. The reduction will lead to expressions where the labels of the polylogarithms on the right hand side are independent of z or shorter, which is ensured by cancellations between neighboring terms. If a n = z, the term (2.10d) has to be dropped and the term (2.10b) needs to be altered to −G(0, a 1 , . . . , a i−1 ,ẑ; z).
Multiple polylogarithms with a 1 = z require special attention as well. However, in order to keep the exposition simple, we will assume that those polylogarithms have already been taken care of by applying the shuffle regularization rule eq. (2.6).
The following examples (with a j = z) are typical relations derived from the above identity: Proving eq. (2.10) is straightforward. It relies on writing the polylogarithm on the left hand side as the integral over its total derivative and using partial fraction as well as relations (A.1) to (A.3) in appendix A. Finally, let us note that eq. (2.10) preserves shuffle regularization. The complete proof of eq. (2.10) as well as numerous examples are contained in section 5 of ref. [3]. A collection of identities between MZVs can be found in the multiple zeta datamine [45].

Iterated integrals on an elliptic curve
In this subsection we are going to take a first look at eIIs. In the following exposition, we will omit several mathematical details, which will be discussed in section 3 below. As eIIs will turn out to be a generalization of the multiple polylogarithms discussed above, we will follow the structure of the previous subsection closely. In eq. (2.1), the differential dt is weighted by which yields iterated integrals on the genus-zero curve C \ {a 1 , ..., a n }. Here, we propose a generalization to eIIs represented by a torus. An infinite number of weighting functions f (n) of weights n = 0, 1, 2, . . . is necessary, whose appearance will be justified and whose precise definition will be provided in section 3. They lead to eIIs in the same way as does eq. (2.12) at genus zero. Accordingly, the functions f (n) (z, τ ) are doubly periodic with respect to the two cycles of the torus, with modular parameter τ in the upper half plane Below, we are going to suppress the τ -dependence and will simply write f (n) (z). As will be explained in subsection 3.3, the functions f (n) are known for all non-negative integer weights n. In particular they are non-holomorphic and expressible in terms of the odd Jacobi function θ 1 (z, τ ), e.g.
Im z Im τ (2.14) where ∂ and denote a derivative in the first argument of θ 1 . Their parity alternates depending on the weight n: The functions f (n) are defined for arbitrary complex arguments z. Restricting to real arguments z, however, will not only simplify eqs. (2.14) and (2.15) but in addition lead to the system of iterated integrals appropriate for the one-loop open-string calculations in sections 4 and 5 below. Hence, in the remainder of the current section, any argument and label of the eIIs to be defined is assumed to be real. We will comment on the additional ingredients required for generic complex arguments z and relate them to multiple elliptic polylogarithms in subsection 3.1.
Employing the functions f (n) , eIIs are defined in analogy to eq. (2.1) via Γ ( n 1 n 2 ... nr a 1 a 2 ... ar ; z) ≡ z 0 dt f (n 1 ) (t − a 1 ) Γ ( n 2 ... nr a 2 ... ar ; t) , (2.17) where the recursion starts with Γ(; z) ≡ 1. Following the terminology used for f (n) above, the eII in eq. (2.17) is said to have weight r i=1 n i , and the number r of integrations will be referred to as its length.

Removing the argument z from the label
As for the multiple polylogarithms, no arguments z are allowed in the labels {a 1 . . . a r } in order to perform the integration using eq. (2.17). Therefore we need to find relations, which trade eIIs with one or multiple occurrences of the argument z in the label for eIIs where z appears in the argument exclusively. The key idea for finding those relations is to write the eII as the integral of its total derivative This resembles the strategy at genus zero which led to the identity eq. (2.10) between multiple polylogarithms. In the subsequent, we address additional features and subtleties intrinsic to the elliptic case. The feasibility of this approach in the elliptic scenario is discussed in ref. [16], see in particular theorem 26 therein.
Boundary terms. The boundary term at z = 0 usually drops out from eq. (2.24) due to the vanishing volume of the integration domain. However, the special situation when all n j = 1 leads to the appearance of standard MZVs. As will be elaborated on in section 3, the function f (1) is the only source of singularities in the integration variables, as can be seen from its leading behavior f (1) (z) = z −1 + O(z). Hence, the regime z → 0 reproduces multiple polylogarithms as defined in eq. (2.1): follows straightforwardly from eq. (2.17). Slightly more work using ∂ ∂a f (n) (t−a) = − ∂ ∂t f (n) (t−a) as well as eq. (2.27) is required for derivatives with respect to labels a q . Starting with the special cases q = 1 and q = r one finds Deriving with respect to a label a q with q = 1, r yields Total derivatives. Summing the above partial derivatives with respect to the argument z and the labels a q , total derivatives from eq. (2.24) can be expressed in a very efficient way. For a single instance of a q = z, the special cases q = 1 and q = r give rise to For q = 1, r, the integrand of eq. (2.24) takes the form Further examples with repeated appearances of t 0 are displayed in appendix B.1.

Fay-identities.
Having applied the above derivative identities, one is usually left with expressions containing integrals of the form where the integration variable appears in the argument of more than one function f (n) . In the corresponding situation for multiple polylogarithms, with weights of the form eq. (2.12), one would have used partial fraction identities in order to avoid the repeated appearance of the integration variable t. Analogous relations for the more general class of weighting functions f (n) are provided by Fay-identities, which will be put in a larger mathematical context in section 3 below. They relate products f (n 1 ) f (n 2 ) at arguments x, t and x − t and thereby allow to systematically remove repeated appearances of some integration variable. A simple example of a Fay identity relates products of functions f (1) to a sum of functions f (2) The general relation, which is valid for complex arguments x, t as well, while a q = z at a generic position q = 1, r can be addressed via Situations with multiple successive appearance of a j = z are discussed in appendix B.
Examples. At length one, the reflection identity eq. (2.19) implies that which covers all identities at this length. At length two, cases with n 1 = 0 or n 2 = 0 are similarly determined by eq. (2.19), so the simplest non-trivial application of eq. (2.24) is Γ ( 1 1 z 0 ; z). The differential can be derived via eq. (2.28) and simplified using the Fay identity eq. (2.36) as well as eq. (2.41), see eq. (2.20) for the notation on the right hand side. In combination with the boundary term which of course agrees with the general formula eq. (2.38). The same reasoning can be applied recursively to obtain for example as well as In subsection 4.3 these relations turn out to be crucial to express the low energy expansion of one-loop string amplitudes in terms of eMZVs. The most general relation at length two following from eq. (2.38) reads and determines Γ ( n 1 n 2 0 z ; z) through the shuffle identity and eq. (2.41). Analogous relations at length three can be found in appendix B.3.

Relations among elliptic multiple zeta values
Apart from their application to string amplitudes, the above manipulations of eIIs are instrumental to derive relations among eMZVs beyond the obvious reflection and shuffle properties. By definition eq. (2.21), eIIs with all labels a j = 0 yield eMZVs in the limit z → 1 of their argument. At the level of labels a j = z, the limit z → 1 is equivalent to a j → 0 since the f (n) are periodic under z → z + 1, hence lim z→1 Γ ( n 1 n 2 ... nr a 1 a 2 ... ar ; z) = ω(n r , . . . , n 2 , n 1 ) , a j ∈ {0, z} , n 1 , n r = 1 . (2.50) Note that endpoint divergences caused by the simple pole in f (1) might introduce additional MZV constants similar to eq. (2.26), that is why the cases n 1 , n r = 1 are excluded explicitly. At length two, for example, eq. (2.49) implies the following eMZV identity provided that the limit z → 1 is non-singular: At low weights n i , the coefficients in eq. (2.51) are particularly simple such as Similar procedures can be carried out at higher length. Combining e.g. eq. (B.7) and a suitable generalization thereof to length four leads to which will be applied in subsection 4.3.

The elliptic functions f (n)
In this section, we provide the definition and mathematical framework for the functions f (n) , thereby supplementing our heuristic approach in section 2. Before doing so, let us start with some mathematical motivation, in which we explain in particular why we need -in distinction to multiple polylogarithms -an infinite number of them.

Motivation
The importance of multiple polylogarithms as defined in eq. (2.1) becomes evident, when considering homotopy-invariant iterated integrals on the multiply punctured complex plane C \ {a 1 , ..., a n }: the value of any such integral evaluated on a path γ depends on the homotopy class of the path only and is a C-linear combination of multiple polylogarithms. Instead of the multiply punctured plane, let us now consider the complex elliptic curve E τ = C/(Z + Zτ ) with its origin removed (we write this as E × τ ), where Im (τ ) > 0. One possible definition of multiple elliptic polylogarithms is via iterated integrals on E × τ . Writing the canonical coordinate on E × τ as z = s + rτ with s, r ∈ R, such that r ≡ Im (z) Im (τ ) , two natural differential forms on E × τ read dz and ν ≡ 2πi dr . (3.1) These differential forms, however, are not sufficient to describe all iterated integrals on E × τ . Even worse, iterated integrals employing the differential forms dz and ν only will not be homotopyinvariant in general, i.e. they will depend on the choice of a path in a given homotopy class.
Both problems are overcome simultaneously by supplementing eq. (3.1) by an infinite tower of differentials f (n) (z)dz constructed through a generating function [16] 6 where f (0) (z) ≡ 1. In particular, it has been proven in ref. [16] that every iterated integral in ν and dz can be uniquely lifted to a homotopy-invariant iterated integral over ν and f (n) (z)dz. Conversely, every homotopy-invariant iterated integral on E × τ arises in this way. The form of the generating function and its coefficients f (n) in eq. (3.2) can be fixed by constructing a doubly-periodic connection J satisfying the integrability condition This requirement singles out a unique completion of J = νX 0 + dz X 1 + . . . to a formal power series in non-commuting variables X 0 and X 1 given by [16] It follows from eq. (3.3) that every word in X 0 , X 1 in the formal power series is a homotopy-invariant iterated integral on E × τ , and one can prove that in fact every such iterated integral arises in this way. Therefore, every homotopy invariant iterated integral on E × τ can be written as a special linear combination of iterated integrals of the differential forms f (n) (z)dz and ν. The differential form ν eq. (3.1), however, vanishes on the real integration path γ(t) ∈ R. Hence, the setup in subsection 2.2 based on real variables leads to elliptic multiple zeta values defined in ref. [17] without referring to the differential form ν.
Although homotopy invariance is generically lost for the iterated integral over the forms f (n 1 ) (z 1 )dz 1 . . . f (nr) (z r )dz r on the punctured elliptic curve E × τ , its value at the real path [0, 1] as in eq. (2.21) can in fact be written as a Z-linear combination of coefficients of words in eq. (3.5), again evaluated on the path [0, 1]. In particular, this shows that the eMZVs associated with the path [0, 1] [17] are periods of the fundamental group of E × τ . Hence, the eIIs defined by eq. (2.17) coincide with the elliptic polylogarithms defined in ref. [16] when restricted to the real line. They can be lifted to honest homotopy-invariant iterated integrals on the punctured elliptic curve by means of the differential form ν defined in eq. (3.1).

Elliptic functions and generating series
In this section, we define the functions f (n) through a generating series, closely following ref. [16]. In the sequel, z and α are complex coordinates on E × τ . Simultaneously, α will be used as a formal expansion variable below. The modular parameter often appears in the combination where Im (τ ) > 0 translates into |q| < 1, relevant for convergence issues.

Some elliptic functions
A general reference on elliptic functions is [23]. Let θ 1 denote the odd Jacobi function 7 defined by subject to the following periodicity properties For j ≥ 1 we also define the Eisenstein function E j (z, τ ) and the Eisenstein series e j (τ ) by 8 which are related to the function θ 1 (z, τ ) via (3.10)

Definition of f (n)
We define the functions f (n) entering the eIIs eq. (2.17) through the following Taylor series in α, They are real analytic on the punctured elliptic curve E × τ . As above, we will omit the argument τ and write f (n) (z) or often simply f (n) . Their explicit form is conveniently captured by the following functions 9 E n Note that all En are meromorphic except for E1 (due to the term Im (z)), and that E2(z) = −℘(z) is the Weierstrass function. Higher functions En at n ≥ 3 are related to derivatives of the Weierstrass function, e.g.
These functions result in a simple representation of the generating series and allow for a combinatorial interpretation of f (n) (z, τ ) in terms of the cycle index of the symmetric group S n (see appendix D). Comparison with eq. (3.20) yields the following expressions for the lowest functions f (n) The functions E j can be expressed in terms of ln θ 1 via eq. (3.10), which leads to the representations for f (1) and f (2) provided in eqs. (2.14) and (2.15). As shown in appendix D, the general expression for f (n) following from eq. (3.22) reads and an equivalent recursive representation is given by

Properties of f (n)
The functions f (n) inherit their double periodicity, the form of their antiholomorphic derivative as well as their behavior under modular transformations from the generating series in eqns. (3.16), (3.17) and (3.18): Likewise, the Fay identity eq. (3. 19) implies for f This identity has been used repeatedly to derive relations among eIIs in section 2 (cf. eq. (2.37) above). Given the singular factor θ 1 (0,τ ) θ 1 (z,τ ) = 1 z + O(z) in the Eisenstein-Kronecker series eq. (3.11), one can check that the residue at the simple pole of Ω at the origin is independent on α. Hence, only f (1) has a simple pole at any z = k + τ l for k, l ∈ Z whereas all other weighting functions f (n =1) are regular on the entire elliptic curve: lim z→0 zf (n) (z) = δ n,1 . (3.30) It is this property of the functions f (n) , which is responsible for the z → 0 behavior stated in eq. (2.26).

q-expansions of f (n)
The Eisenstein-Kronecker series eq. (3.11) is known to have the following power series expansion in q = e 2πiτ [23,16] αF (z, α, τ Disentangling the powers of α yields the holomorphic parts g (n) of the functions f (n) , e.g.
where cot(πz) = 1 πz + O(z) captures the simple pole of f (1) . More generally, we find The antiholomorphic piece in f (n) consisting of factors Im (z) Im (τ ) can be immediately restored via Even though the functions f (n) in the definition eq. (2.17) of eIIs are evaluated at real arguments in the subsequent, we will keep track of the admixtures of Im (z) in eq. (3.37) for further applications beyond this work. For example, another system of eIIs and eMZVs can be defined for the path from 0 to τ instead of the real interval [0, 1] whose properties are crucially affected Re (z) by the factors of Im (z) and the resulting modular properties.

The one-loop four-point amplitude in open string theory
Iterated integrals defined on an elliptic curve in subsection 2.2 appear naturally in superstring theory. Calculating one-loop scattering amplitudes among open string states amounts to evaluating iterated integrals weighted by the functions f (n) defined in section 3. Accordingly, the expansion of one-loop superstring amplitudes in the inverse string tension α involves eMZVs.
The α -expansion of tree-level amplitudes in open string theory is well known to involve standard MZVs, see e.g. ref. [34]. The pattern of their appearance is much simpler as compared to the MZVs and polylogarithms in loop amplitudes of field theory and can be understood in terms of motivic MZVs [29] as well as the Drinfeld associator [38]. Hence, it is not surprising that one-loop string amplitudes furnish a perfect laboratory to study patterns and properties of eMZVs.
Iterated integrals in one-loop open string amplitudes occur on the boundaries of a twodimensional worldsheet of either cylinder or Möbius-strip topology [40]. They describe conformally inequivalent configurations of inserting open string states on the respective boundaries. As a first field of application for eMZVs, we will entirely focus on cylindrical worldsheets in this work with all integrations confined to one boundary 10 . As shown in figure 1, this situation can be described by a torus with purely imaginary modular parameter τ = it with t ∈ R. The cylinder boundaries are then parametrized by Re (z j ) ∈ [0, 1] with Im (z j ) = 0 and Im (z j ) = t 2 , respectively. The configuration of interest with one boundary empty is captured by real insertion points z j ∈ R.

The four-point amplitude
For massless open-string excitations in ten dimensions -gluons and gluinos -supersymmetry requires at least four external states for a non-vanishing one-loop amplitude, so the simplest 10 The interplay between open string worldsheets of different topologies is crucial for the cancellations of infinities [51] and anomalies [52,53] which occur for gauge group SO (32). case to be studied below is the four-point function [54,55], The entire polarization dependence is captured by the four-point tree amplitude of sYM field theory, see [56] for its tensor structure. The worldsheet integral I 4pt (1, 2, 3, 4) depends on the external momenta k i through dimensionless Mandelstam invariants where momentum conservation and the mass-shell condition k 2 i = 0 leave two independent s ij , The dependence on worldsheet positions z j ∈ [0, 1] enters through the genus-one Green function which is related to the singular function f Note that the dependence of I 4pt (1, 2, 3, 4) on s ij and q ≡ e −2πt is suppressed for ease of notation. The non-holomorphic piece in f (1) (z) ≡ ∂ ln θ 1 (z, τ ) + 2πi Im z Im τ drops out for the present cylinder parametrization where all vertices are inserted on the boundary with real coordinates z j . Accordingly, the differential form ν ∼ d Im (z) in eq. (3.1) required for homotopy invariance does not contribute to the cylinder integrals under consideration. However, the admixtures of Im z Im τ in f (n) are crucial for modular invariance of closed-string amplitudes and cylinder diagrams with open string states on both boundaries.
Translation invariance on genus-one surfaces can be used to fix z 1 = 0. In addition, the N -point integration measure which appears for N = 4 in eq. (4.2), is invariant under cyclic shifts z i → z i+1 mod N and, up to a sign (−1) N , under reflection z i → z N +1−i . Some features of the one-loop N -point amplitudes are discussed in section 5. Their integrand then involves factors of f (w i ) (z j − z k ) with overall weight i w i = N − 4. As another generalization of the one-loop amplitude eq. (4.1) in ten spacetime dimensions, one could consider supersymmetry-preserving compactifications on a torus. For each circular dimension of radius R, the associated momentum components are quantized and contribute a correction factor of ∞ n=−∞ e −n 2 πtR 2 /α to the t-integrand [57]. Since this does not affect the z j integrations within I 4pt (1,2,3,4) and the resulting eMZVs, the subsequent results on the α -expansion are universal for any torus compactification to spacetime dimensions D ≤ 10.

The α -expansion
In this section, we investigate the α -expansion of the t-integrand in eq. (4.1), which encodes the low-energy effective action for the gluon supermultiplet. Expanding in α amounts to Taylor expanding the exponential in eq. (4.2) in all the Mandelstam invariants s ij defined in eq. (4.3) as well as the corresponding worldsheet Green function P ij given by eq. (4.6).
In addition to the power-series expansion in α discussed in the subsequent, the integration region of large t in the amplitude eq. (4.1) gives rise to logarithmic, non-analytic momentum dependence. The associated threshold singularities in s ij are for instance crucial to make contact with the Feynman box integral in the sYM amplitude arising in the point-particle limit [57]. Mimicking the low energy-analysis of closed string one-loop amplitudes [58][59][60][61], we separate the analytic from the non-analytic parts of the amplitude and do not keep track of the non-analytic threshold singularities.
The simplest monomials in P ij inequivalent under cyclic shifts and reflections of the vertex positions z j integrate to At second and third order in α one finds  3 12 , (4.14) these relations have been used to eliminate c 2 5 , c 2 6 as well as c 3 5 , c 3 7 , c 3 12 , c 3 11 from eq. (4.12). Note that the α -expansion of closed string one-loop amplitudes has been analyzed along similar lines in refs. [58][59][60][61]. Since each closed-string insertion point z j is integrated over the entire torus E τ , integrals involving propagators with a free endpoint vanish and therefore much fewer closed-string counterparts of the coefficients c j i arise.

Elliptic multiple zeta values
In this section we convert the constituents of the α -expansion, c j i defined by eqns. (4.9), (4.10) and (4.11), to eMZVs. This will provide a characterization of the particular linear combinations of c j i which appear in eq. (4.12) along with various powers of s 12 and s 23 . Even though the leading term c 0 in eq. (4.9) can be straightforwardly evaluated to yield 1 6 , it is instructive to express it as an eMZV by inserting 1 = f (0) (z i ) along with each dz i : Below, we will repeatedly apply the definitions eq. (2.17) and eq. (2.21) of eIIs and eMZVs, respectively, in order to express the other integrals c j i in the same fashion.

Second order in P ij : integrals c 2 i
At quadratic order in s ij , the rewriting P 1j = (1) (w) dw allows to straightforwardly address any quadratic monomial in P 12 , P 13  where ω(1) vanishes by the reflection identity eq. (2.23).

Integration techniques for P 23 , P 24 , P 34
Green functions P ij where both indices describe a leg to be integrated (legs 2, 3, 4) are more difficult to integrate. Their integral representation eq. (4.6) inevitably gives rise to iterated integrals Γ ( n 1 ... nr a 1 ... ar ; z) with the argument appearing in the labels, that is a i = z. Integration over z 3 and z 4 then requires the techniques of subsection 2.2.2, in particular the recursion formulae eq. (2.38) to eq. (2.40).
The simple corollary Γ ( 1 0 z 0 ; z) = − Γ ( 0 1 0 0 ; z) of the reflection identity eq. (2.19) is sufficient to integrate P 23 and to reproduce eq. (4.16) from a different cyclic representative. The quadratic case c 2 5 = 1234 P 12 P 34 , on the other hand, requires more effort. One obtains The expression for ω(n) can be inferred from order q 0 in the expansions eqs. (3.35) and (3.36).

Third order in P ij : integrals c 3 i
Starting from the third order in Mandelstam variables, relations such as eq. (4.14) are no longer sufficient to reduce the complete list of c 3 i in eq. (4.11) to elementary integrals over monomials in P 12 , P 13 and P 14 . Instead, the inevitable factors of P 23 , P 24 and P 34 require the procedure described in eq. (4.23) together with the recursive identities eq. (2.38) to (2.40) in order to rearrange the labels of the eIIs. This allows to reduce integrals over arbitrary monomials in P ij with 1 ≤ i < j ≤ 4 to eMZVs. The integrals c 3 i , which are cubic in P ij , give rise to where the occurrences of ζ 2 can be traced back to eq. (2.44).

Assembling the results
Momentum conservation only admits particular linear combinations of c j i in the four-point amplitude eq. (4.12). It turns out that for all cases considered divergent eMZVs with the singular integrand f (1) in the first or last position drop out. Up to third order in s ij , we have and the pattern at higher orders is under investigation [62]. The above expressions for β 5 and β 2,3 are obtained using various eMZV relations using the methods of subsection 2.2.3.

On the q-expansion of eMZVs and the string amplitude
The evaluation of eMZVs as initiated in eq. (4.24) will be pursued systematically in [62][63][64]. In this section, we give a glimpse of non-trivial q-dependence in simple cases and provide consistency checks for the constant piece of the low energy expansion eq. (4.26) of the four-point amplitude.

The simplest q-expansions
To determine the q-expansions of the simplest eMZVs, we start from the expansions of f (1) and f (2) spelled out in eq. (3.37), which in turn is based on eqs. (3.32) and (3.33). Using the integrals in appendix C, we arrive at as well as A systematic method is under investigation and will appear in [64]. Note that the q-dependence of all the examples above can be expressed in terms of the function ELi n,m introduced in section 8 of ref. [11] at arguments x = y = 1.

The constant piece of eMZVs and the α -derivative
The t-integration in the four-point amplitude eq. (4.1) is divergent unless the choice of gauge group SO (32) leads to cancellations between the cylinder and the Möbius-strip diagram [51]. The divergence is interpreted as a zero-momentum dilaton propagating to the vacuum and therefore proportional to the derivative of the tree level amplitude with respect to α [54]. The latter is given by In the representation of the one-loop amplitude given in eq. (4.26), the divergence originates in the constant part of the eMZVs' power series expansion in q = e 2πiτ = e −2πt . A systematic method to extract the constant term of eMZVs will be described in ref. [62]. The resulting divergence in the above result is given by which is consistent with the α -derivative of the tree amplitude [54] upon comparison with eq. (4.33), (1,2,3,4) . (4.35)

Multi-particle one-loop string amplitudes and f (n)
This section is devoted to one-loop amplitudes involving five and more open string states. We firstly provide the five-point extension of the four-point α -expansion in eq. (4.26). It is secondly demonstrated that the elliptic functions f (n) defined in section 3 naturally enter the calculation of one-loop amplitudes with any number of external legs.

The five-point open string amplitude
In the same way as the four-point open string amplitude in eq. (4.1) allows to factor out the polarization dependence via A tree YM (1,2,3,4), one can express the five-point string amplitude in a basis of color-ordered trees of YM theory [65]. BCJ relations [66] single out two independent subamplitudes A tree YM (1, ρ(2, 3), 4, 5) with permutation ρ ∈ S 2 , and for convenience, we consider the same color orderings in the one-loop string theory counterparts: The 2 × 2 matrix I 5pt (σ|ρ) is the generalization of the four-point scalar integral I 4pt (1,2,3,4). It can be assembled from the kinematic factors which were simplified in ref. [65] using the pure spinor formalism [67], in agreement with the four-point open string tree eq. (4.32). The pattern of eMZVs at higher orders in α as well as the properties of the novel matrices L i will be further studied in ref. [62].

Functions f (n) from the RNS formalism
In this subsection we will show that the elliptic functions f (n) for any n are naturally generated in the one-loop amplitude computation using the RNS formalism [68][69][70]. Their emergence in the parity-even and parity-odd sectors turns out to follow two separate mechanisms.

Parity-even RNS amplitudes
In the parity-even sector of the RNS computation, the functions f (n) arise from the summation over the even spin structures of the fermions on a genus-one worldsheet. We also take this opportunity to use the method of refs. [71,72] to write down explicit results for the N -point spin sum for N > 7. V p (x 1 , . . . , x N ). In the subsequent we use the variables x i ≡ z i − z i+1 for i = 1, . . . , N with the condition z N +1 = z 1 such that N i=1 x i = 0. Using the shorthand Ω i ≡ αΩ(x i , α) it follows from eq. (3.30) that the α p -component of Ω 1 · · · Ω N is an elliptic function with at most p simultaneous single poles in the variables x i . This suggests the following definition

Definition of
Interestingly, the anti-holomorphic recursion eq. (3.27) implies that in the notation of subsection 3.3.3 to establish manifest holomorphicity.
Spin sums in one-loop amplitudes. In the computation of parity-even one-loop amplitudes in the RNS formalism the bosonic worldsheet fields can be straightforwardly integrated out to yield products of f (1) , possibly after integration by parts. Worldsheet fermions, on the other hand, give rise to the following spin sums, where N i=1 x i = 0, S ν is the Szegö kernel and ν denotes the even spin structure with associated Jacobi theta functions θ 2 , θ 3 , θ 4 [73][74][75]49], A method to evaluate such sums was presented in ref. [71] and its explicit results at N ≤ 7 can be written in terms of f (1) (z), the Weierstrass function ℘(z) and its derivatives ∂ k ℘(z), ℘(x j ) . (5.12) One can show that the above results are naturally described by the elliptic functions V p (x 1 , . . . , x N ), Note that the equivalence of this expression for G 6 with the results of [76] can be verified through the Fay identity eq. (2.36). Although the results for N ≥ 8 were not written down explicitly in ref. [71], they also take a natural form when expressed in terms of elliptic functions V p (x 1 , . . . , x N ), The factors of the Eisenstein series e j eq. (3.9) can be systematically computed as well. Following ref. [72], we define Q 0 (℘) = 1, Q 1 (℘) = ℘ and Q k+1 (℘) = ℘ (2k) . For example, where the Weierstrass equation (℘ ) 2 = 4℘ 3 − g 2 ℘ − g 3 has been used to rewrite the 2k th derivative of ℘ as a polynomial in ℘. In the above equation, g 2 = −4(s 1 s 2 + s 2 s 3 + s 3 s 1 ) = 60e 4 , g 3 = 4s 1 s 2 s 3 = 140e 6 are the elliptic invariants and s i are the branch points of the genus-one elliptic curve y 2 = 4(z − s 1 )(z − s 2 )(z − s 3 ) satisfying s 1 + s 2 + s 3 = 0. Defining straightforward calculation leads to 11 F 4 = 3e 4 , F 6 = 10e 6 , F 8 = 42e 8 , F 10 = 168e 10 , F 12 = 627e 12 + 9e 3 4 , which precisely captures the factors of e j in eq. (5.14). We have explicitly checked up to N = 12 11 The Eisenstein series e8, e10 and e12 can be written in terms of e4 and e6 as follows that the spin sums can be uniformly written as,

Parity-odd RNS amplitudes
The parity-odd sector of the RNS computation entirely stems from the unique odd spin structure at genus one where the worldsheet spinors obey anti-periodic boundary conditions along both torus cycles and acquire a zero mode. The worldsheet integrand is governed by zero-mode saturation and, probably as a common feature with the Green-Schwarz or pure spinor formalism, OPE contractions of the worldsheet fields which generate N − 4 factors of f For six points, the direct evaluation of the OPEs gives rise to a quadratic factor f for various combinations of labels capturing the behavior of the singularities as the vertices collide. However, we know from the Fay identity eq. (2.36) that these quadratic combinations are not linearly independent and therefore one is naturally led to higher-weight f (n) 's when considering a minimal basis of integrals to evaluate. The simplest example where a higherweight f (n) is generated this way is f 23 + f (2) 31 which can be viewed as generalizing the genus-zero partial fraction identity eq. (2.35). The non-vanishing of the right-hand side provides an important distinction between one-loop and tree-level string amplitudes and it is ultimately related to the gauge anomaly cancellation mechanism in the superstring [52,53]. It will be shown in ref. [62] that the parity-odd part of the six-point amplitude as firstly computed in ref. [77] can be entirely written in terms of f (2) , i.e. that any appearance of f (1) can be removed via eq. (2.36).
More generally, the N −4 powers of f (1) in the N -point amplitude allow, via the Fay identity, the generation of f (p) with up to p = N − 4. In this way the need for a general integration method for the type of iterated integrals on an elliptic curve considered in this paper is justified.

Discussion and further directions
In this article, we have proposed an organization scheme for elliptic iterated integrals and elliptic multiple zeta values (eMZVs), where the key definitions are provided in eqs. (2.17) and (2.21). The infinite family of elliptic functions f (n) appearing in the integrands of section 2 are put into a mathematical context and are related to multiple elliptic polylogarithms in section 3. As a first natural and simple application of this framework, we have identified eMZVs in the α -expansion of one-loop scattering amplitudes in open string theory. The leading orders in the low-energy behavior of the four-and five-point amplitudes in terms of eMZVs are presented in eqs. (4.26) and (5.4). Divergent eMZVs turn out to cancel from our results.
Having demonstrated the potential of the formalism for an initial example, there are numerous open questions to be pursued in the near future. Most obviously, the eMZV content of the low energy expansion of cylinder amplitudes needs to be understood for higher orders in α , which can be done conveniently using the new techniques. Furthermore, the contributions from the cylinder configuration with open string insertions on both boundaries as well as from the Möbius-strip topology shall be determined in terms of the iterated integrals introduced in subsection 2.2. The q-expansion of eMZVs exemplified in section 4.4.1 offers a promising approach to systematically perform the t-integration in eq. (4.1) after summing all topologies for the gauge group SO(32) [51].
On the mathematical side, the network of relations between eMZV explored in subsection 2.2.3 will be further investigated in refs. [62][63][64]. A suitable coaction along the lines of refs. [5][6][7][8]16] might lead to a natural basis choice for eMZVs and might allow to further identify patterns in the one-loop string amplitudes. In the same way as the Drinfeld associator was instrumental in understanding the pattern of MZVs [29] in open string tree-level amplitudes [37] and finally allowed to completely determine their α -expansion in ref. [38], the elliptic associators discussed in ref. [24] might encode the structure of the α -expansion at one-loop. Furthermore, in refs. [78,79] so-called multiple modular values are discussed whose possible relation to the eMZVs studied here needs to be explored.
In multi-particle one-loop open string amplitudes, the pure spinor formalism, in particular the ingredients of ref. [80] are expected to yield a compact description of the kinematic factors associated to the functions f (n) . While the precise superspace kinematic factors along with various powers of f (1) have been derived in ref. [65], the kinematic companions of f (2) in the six-point amplitude will be described in ref. [62] and cases with f (n≥3) are currently under investigation.
Finally, it would be desirable to find a similar scheme for organizing the α -expansion of closed string one-loop amplitudes. In particular, the worldsheet integrals investigated in refs. [59][60][61] might allow for a description in terms of eMZVs and their counterpart defined with respect to the other cycle of the torus. The peculiar linear combinations of torus integrals appearing in the α -expansion of closed-string amplitudes call for an explanation along the lines of the above finding that divergent eMZVs drop out from the open-string expansions.
if λ = 1 a 1 2 a 2 . . . n an is a partition of n (denoted by λ n), the number of permutations with cycle structure λ is [81] n! n i=1 i a i a i ! . (D.1) Note that the coefficients of the monomials E a 1 1 . . . E an n in f (n) given by eq. (3.24) are reproduced by the formula (D.1) with the corresponding cycle structure. This observation can be made more precise with the definition of the cycle index of the symmetric group S n [81], Furthermore, one can also show that [82], Note, in particular, that (D.5) yields an alternative proof of (3.27), Symmetric polynomials. The cycle index of the symmetric group S n also provides a recipe for expressing the complete symmetric function h j in terms of the power sum function p j , i.e., h n = Z(S n ; p 1 , p 2 , . . . , p n ) [81]. Therefore the functional form of h n matches that of f (n) and one can exploit the well-known relation nh n = n i=1 p i h n−i from the theory of symmetric functions to obtain the corresponding recursion formula eq. (3.25) for f (n) .