All-order α′-expansion of one-loop open-string integrals

Carlos R. Mafra and Oliver Schlotterer a STAG Research Centre and Mathematical Sciences, University of Southampton, Highfield, Southampton SO17 1BJ, UK and b Department of Physics and Astronomy, Uppsala University, 75108 Uppsala, Sweden. We present a new method to evaluate the α′-expansion of genus-one integrals over open-string punctures and unravel the structure of the elliptic multiple zeta values in its coefficients. This is done by obtaining a simple differential equation of Knizhnik–Zamolodchikov–Bernard-type satisfied by generating functions of such integrals, and solving it via Picard iteration. The initial condition involves the generating functions at the cusp τ → i∞ and can be reduced to genus-zero integrals.


INTRODUCTION
Elliptic analogues of polylogarithms [1,2] and multiple zeta values [3] have become a driving force in higher-order computations of scattering amplitudes in quantum field theories and string theories. The study of their differential equations and their connections with modular forms turned into a vibrant research area at the interface of particle phenomenology, string theory and number theory. In the same way as a variety of Feynman integrals has been recently expressed in terms of elliptic polylogarithms and iterated integrals of modular forms [4,5], the low-energy expansion of one-loop open-string amplitudes introduces elliptic multiple zeta values (eMZVs) [6][7][8].
So far, the appearance of eMZVs in one-loop openstring amplitudes arose from direct integration over the punctures on a genus-one worldsheet of cylinder or Möbius-strip topology. Although there is no conceptual bottleneck in extending the techniques of [6][7][8] to arbitrary multiplicities and orders in the inverse string tension α , in this letter we will present a new method to evaluate these genus-one integrals which is related to elliptic associators [9] and Tsunogai's derivations dual to Eisenstein series [10]. The results are given by eMZVs in their minimal form [3,11] and reveal elegant structures in the α -expansions. More details will be given in a longer companion paper [12].

OPEN-STRING INTEGRALS AT GENUS ONE
One-loop string amplitudes are described by correlation functions of vertex operators in a conformal field theory over a genus-one Riemann surface, the torus. The location of the vertex operator associated with the j th external string state is parameterized by the coordinates z j = u j τ +v j with u j , v j ∈ (0, 1), where τ is the modulus with Im τ > 0, see figure 1, and we define z ij ≡ z i − z j .
Möbius-strip integrals can be reconstructed by specializing planar A-cycle integrals to Re τ = 1 2 , and the cancellation of tadpole divergences from one-loop opensuperstring amplitudes can be analyzed as in [15].
The A-cycle integrand (6) at n points involves n−1 factors of the Kronecker-Eisenstein series (4) at different arguments. The second entry Z τ η ( * |A) specifies permutations A = a 1 a 2 . . . a n ∈ S n of these arguments, and Ω(. . .) at different z aj , η aj are related by the Fay identity Repeated use of (7) and imposing η 1 = − n j=2 η j only leaves (n−1)! independent permutations of the integrand in (6), and we will use a basis of Z τ η ( * |1, B) with permutations B ∈ S n−1 acting on 2, 3, . . . , n.
B. The differential equation: As will be derived in [12], the τ -derivatives of (6) can be written as where the (n−1)! × (n−1)! matrix D τ η is a differential operator w.r.t. η j . Its detailed form will be exemplified in the next section and follows from the properties (5) of the Green function, the vanishing of boundary terms dv j ∂ vj (. . .) and the mixed heat equation (u, v ∈ R) Most importantly, the form of D τ η (B|C) does not depend on the planar or non-planar integration cycle A, and its entries are linear in the dimensionless Mandelstam invariants s ij and therefore in α .
Hence, the α -expansion of the genus-one integrals Z τ η follows from the solution of (8) via Picard iteration, As an initial value, the degeneration Z i∞ η at the cusp τ →i∞ will be expressed in terms of disk integrals with two additional punctures from the pinching of the A-cycle in figure 1.
The decomposition of eMZVs into iterated Eisenstein integrals automatically incorporates all their relations over the rational numbers [11]. Moreover, the derivation of (14) does not rely on any relation among the Mandelstam invariants. The n-point results of this work are valid for 1 2 n(n−1) independent s ij , and one can still impose momentum conservation when applying the αexpansion of Z τ η to string amplitudes.

EXAMPLES FOR DIFFERENTIAL OPERATORS
In this section, we present (n<4)-point examples of the matrix-valued differential operators D τ η in (8), and the four-point case is relegated to the appendix. Allmultiplicity expressions as well as detailed derivations of the differential equations can be found in [12].
A. Two points allow for a single planar and nonplanar A-cycle integral (6) each, Their τ -derivatives resulting from (5), (9) and integration by parts w.r.t. v 2 take the universal form 2πi∂ τ Z τ η2 ( * |1, 2) = s 12 so one can read off the scalar differential operator in (8) and the resulting representation of the derivations, Note that various combinations of iterated Eisenstein integrals drop out from the two-point instance of (14) since commutators [r η2 ( k1 ), r η2 ( k2 )] with k 1 , k 2 ≥ 4 vanish.

EXAMPLES FOR INITIAL VALUES
This section is dedicated to the degeneration of A-cycle integrals (6) at the cusp τ → i∞ which enters the αexpansion (14) as an initial value.
A. Generalities: The behaviour of A-cycle integrals at the cusp is most conveniently studied in the variables where the planar Green function and Kronecker-Eisenstein series degenerate to (σ ji ≡ σ j −σ i ) Their non-planar analogues take an even simpler form, Since string-theory applications of (14) involve the coefficients w.r.t. η j , we will need the expansions As will be detailed in [12], the σ j -integration in n-point Z i∞ η lines up with explicitly known combinations of N = (n+2)-point disk integrals [18] Z tree (a 1 , a 2 , . . . , a N |1, 2, . . . , N . (27) The two extra punctures n+1 → + and n+2 → − are associated with Mandelstam invariants The α -expansion of (27) and therefore Z i∞ η involves multiple zeta values (MZVs) which can be systematically generated from the all-multiplicity methods of [19,20].

B. Two points:
Planar initial values at two points descend from four-point tree-level integrals, Z i∞ η2 (1, 2|1, 2) = π cot(πη 2 ) 2i sin The factor of 2i sin( πs12 2 ) and similar trigonometric functions below stem from contour deformations detailed in [12]. The gamma functions with standard α -expansion and exemplify that integrals over k factors of G ij in (23) may have up to k kinematic poles. Non-planar three-point initial values in turn boil down to four-point disk integrals with α -expansions in (30),

CONCLUSIONS AND FURTHER DIRECTIONS
In this letter we presented a method to expand a generating series of genus-one integrals (6) relevant to oneloop open-string amplitudes. At each order in the inverse string tension α , our main result (14) pinpoints the accompanying eMZVs in their minimal and canonical representation via iterated Eisenstein integrals. Genus-zero integrals relevant to open-string tree amplitudes obey Knizhnik-Zamolodchikov equations with a characteristic linear factor of α on their right-hand side [19]. This structure is analogous to the ε-form of differential equations among Feynman integrals with dimensional-regularization parameter ε [5,21], suggesting a correspondence between α and ε. By the linearity of the differential operators D τ η in s ij = −2α k i · k j , the Knizhnik-Zamolodchikov-Bernard-type equation (8) also becomes linear in α . So our results generalize this intriguing correspondence to genus one and provide the string-theory analogue of the ε-form for differential equations of elliptic Feynman integrals [5].
The generating functions Z τ η are expected to comprise any moduli-space integral in massless one-loop amplitudes of open bosonic strings and superstrings upon expansion in η j . Accordingly, they are proposed to generalize the universal disk-integrals (27) that appear in the double-copy representation of string tree-level amplitudes [18,22]. Hence, the study of the genus-one integrals Z τ η is an essential step towards universal double-copy structures in one-loop amplitudes of different string theories that generalize those of the superstring [23].
The generating functions Z τ η can be adapted to a closed-string context, encoding the integrals over torus punctures in one-loop amplitudes of type-II, heterotic and closed bosonic string theories. Closed-string analogues of Z τ η will be shown [24] to obey similar differential equations and to shed new light on the properties of modular graph forms [25] including their relation with open-string amplitudes [26].
Moreover, the method of this work to infer modulispace integrals from differential equations should be applicable at higher loops. In the same way as disk integrals were used as the initial value for our one-loop results, higher-genus integrals in string amplitudes are expected to obey differential equations w.r.t. complex-structure moduli such that their separating and non-separating degenerations set the initial conditions. It would be interesting to explore a differential-equation approach of this type to the higher-genus modular graph functions of [27].
In summary, our new approach to one-loop openstring amplitudes via differential equations connects with state-of-the-art techniques in particle phenomenology and provides explicit matrix representations of profound number-theoretic structures. As will be elaborated in [12], our results manifest important formal properties of string amplitudes such as uniform transcendentality, coaction formulae and the dropout of twisted eMZVs from non-planar open-string amplitudes.
Acknowledgements: We are grateful to Johannes Broedel, Jan Gerken, Axel Kleinschmidt, Nils Matthes and Federico Zerbini for inspiring discussions and collaboration on related topics. Moreover, Claude Duhr, Hermann Nicolai, Albert Schwarz and in particular Sebastian Mizera are thanked for valuable discussions, and we are grateful to Sebastian Mizera for helpful comments on a draft. We would like to thank the organizers of the programme "Modular forms, periods and scattering amplitudes" at the ETH Institute for Theoretical Studies in Zürich for providing a stimulating atmosphere and financial support. CRM is supported by a University Research Fellowship from the Royal Society. OS is grateful to the organizers of the workshop "Automorphic Structures in String Theory" at the Simons Center in Stony Brook and those of the workshop "String Theory from a Worldsheet Perspective" at the GGI in Florence for setting up inspiring meetings. OS is supported by the European Research Council under ERC-STG-804286 UNISCAMP.

APPENDIX: FOUR-POINT EXAMPLES
This appendix provides further details on the expansion (14) of four-point A-cycle integrals (6).
Similar to (34), I tree denote combinations of six-point disk integrals (27) which no longer depend on η j , see section 5.5 of [12] for further details. Non-planar four-point initial values reduce to four-and five-point disk integrals, e.g.  (14), we have checked (36), (39) and (40) to reproduce the α -expansions of [6,7] to the orders of α 2 and α 3 in the planar and non-planar sectors, respectively.