Universality in string interactions

In this letter, we provide evidence for universality in the low-energy expansion of tree-level string interactions. More precisely, in the alpha'-expansion of tree-level scattering amplitudes, we conjecture that the leading transcendental coefficient at each order in alpha' is universal for all perturbative string theories. We have checked this universality up to seven points and trace its origin to the ability to restructure the disk integrals of open bosonic string into those of the superstring. The accompanying kinematic functions have the same low-energy limit and do not introduce any transcendental numbers in their alpha'-corrections. Universality in the closed-string sector then follows from the KLT-relations.


INTRODUCTION
One of the formidable challenges for a theory of quantum gravity is the construction of a gravitational Smatrix which respects unitarity at high energies. Perturbative string theories provide candidate solutions, as its four-point graviton S-matrix is exponentially suppressed in the high-energy limit for fixed-angle scattering [1,2]. In fact, assuming tree-level causality [3] and unitarity [4] imposes stringent constraints, under which string theories provides the only known analytic solutions so far.
Different string theories are understood to be equivalent through a web of strong-weak dualities which relate different orders in the perturbative expansion [5]. At tree level, however, the low-energy description in the form of an effective action with expansion in curvature tensors and covariant derivatives is largely unconstrained by string dualities. More precisely, the coefficients of these higher-dimensional operators are expected to be distinct for different string theories. Thus, if some of these coefficients turn out to be universal, it is then conceivable that such a phenomenon reflects a deeper principle in the theory of quantum gravity beyond the known dualities.
At low energies, closed-string theories yield an effective action that augments the Einstein-Hilbert term S EH with higher-dimensional operators. At tree level, type-II superstring theories exhibit the following expansion in the inverse string tension (or cut-off scale) α ′ , with Einstein-frame conventions for the dilaton couplings e −nφ . The ellipsis · · · represents loop-corrections and higher-order terms in α ′ , while D n R m schematically represent contractions of covariant derivatives and Riemann tensors. The tensor structure of each operator as well as its coefficient furnished by multiple zeta values (MZVs) can be derived by expanding string-theory graviton amplitudes in α ′ . MZVs can be conjecturally categorized according to their transcendental weight n 1 +n 2 +. . .+n r and constitute a fruitful domain of common interest between high-energy physics and number theory. In fact, for type-II theories, the transcendental weight for each coefficient matches the order of α ′ . This property will be referred to as uniform transcendentality, and it also exists for open strings in the type-I theory. The type-I effective action is now an expansion in non-abelian field-strength operators tr(D n F m ). In this light, uniform transcendentality for closed strings is inherited from open strings through the Kawai, Lewellen and Tye (KLT) relations [6].
In this letter, we conjecture that the leading transcendental coefficient at each order in the α ′ -expansion of tree-level amplitudes is universal among all perturbative open-and closed-string theories. We have explicitly verified this up to the seven-point level, and the conjectural all-multiplicity extension is discussed in a companion paper [7]. This remarkable property can be best understood by inspecting the world-sheet correlator of the open-string amplitudes.
It was shown in [8] that the n-point tree amplitude of the open superstring can be cast into an (n−3)! basis of disk integrals, each augmented by Yang-Mills tree amplitudes of different color-orderings. These basis integrals exhibit uniform transcendentality upon α ′ -expansion, see e.g. [9] for a proof. We claim that bosonic open-string amplitudes can be cast upon the very same integral basis where -in contrast to the superstring -the accompanying functions of the kinematic data depend on α ′ . Apart from the Yang-Mills trees recovered in their low-energy limit α ′ → 0, the α ′ -corrections of the kinematic functions exclusively involve rational numbers upon Taylorexpansion, i.e. they do not carry any transcendental weight. Hence, the resulting α ′ -expansion of the bosonic string amplitude will have the same leading transcendental pieces as found for the superstring.
The same property can be extended to closed strings by utilizing the KLT-relations [6], which assemble closedstring tree amplitudes from products of two openstring trees. The accompanying sin-functions with α ′dependent arguments do not alter the uniform transcendentality of the type-II theory. Different double-copies of open bosonic strings and superstrings give rise to three different closed-string theories -bosonic, heterotic and type-II superstrings. Their tree amplitudes are governed by a universal basis of (n−3)! × (n−3)! integrals of uniform transcendentality inherited from the open-string constituents. Only the kinematic coefficients differ between the theories, where the additional α ′corrections specific to open bosonic strings do not introduce any transcendental weight and thereby do not affect the leading-transcendental piece. This completes the argument for universality in closed-string interactions, namely for the O(α ′n ) order of the effective action, the weight-n coefficient is universal for all perturbative closed-string theories.

OPEN-STRING AMPLITUDES
A. The open superstring: The tree-level amplitude for n gluon-multiplet states in open superstring theory can be conveniently written as [8] where A S and A YM indicate color-ordered amplitudes of the superstring and super Yang-Mills field theory, respectively. Moreover, ρ, σ with j ρ ≡ ρ(j) denote the (n−3)! distinct permutations with legs 1, n−1, n held fixed, and with z ij ≡ z i − z j . We fix the SL(2) symmetry of the disk by setting (z 1 , z n−1 , z n ) = (0, 1, ∞), and we use dimensionless Mandelstam invariants When viewed as an (n−3)!×(n−3)! matrix, the row-and column indices ρ and σ of F ρ σ label different integration domains and integrands, respectively, where σ acts on the subscripts within the curly bracket in (4). Note that the field-theory limit is recovered as F ρ σ (α ′ ) = δ ρ σ + O(α ′2 ), and the (n−3)!-vector in (3) furnishes a basis of string subamplitudes under monodromy relations [10,11].
The α ′ -expansion of the integrals in (4) yields MZVs (2) whose transcendental weight matches the degree of the accompanying polynomials in s ij . Since A YM do not depend on α ′ , uniform transcendentality of the integrals propagates to the disk amplitude (3). Initially addressed via hypergeometric functions [12], the α ′ -corrections of F ρ σ (α ′ ) at any multiplicity can be recursively generated from the Drinfeld associator [9].
Once undoing the above choice of SL(2) frame, the functions (4) can be identified as a superposition of (n−3)! "single-cycle" disk integrals, where σ and ρ now act on all external legs in the integrand and the integration domain, respectively, and the measure is given by The integral reductions performed in [8] rely on partialfraction manipulations and integrations by parts (IBP) among Z ρ (1 σ , . . ., n σ ). At fixed ρ, these integral relations for different choices of σ can be identified with the KKand BCJ-relations [13] of A YM (. . .) [14]. However, as already exploited in a superstring context [8,15], IBP additionally allows to address closed subcycles of z ij in the integrand such as double poles z −2 ij . Extending these techniques to gluon amplitudes of the bosonic string yields our main result to be reported in the following.

B. The bosonic open string:
The tree-amplitude prescription for n-gluon scattering in the bosonic string introduces significantly more rational functions of z ij of suitable SL(2) weight than captured by the single cycles in (6). Still, repeated use of IBP is expected to reduce all of them to the single-cycle form and thereby to the same integral basis as seen in (3) and (4), e.g.
The denominator on the right-hand side signals tachyon exchange specific to the bosonic string and can be expanded as a geometric series (1 − s ij ) −1 = ∞ k=0 s k ij . In a superstring context, the OPE among supersymmetric vertex operators guarantees that tachyon poles as in (8) are suppressed by numerators 1 − s ij , see e.g. [8,15]. Extending the integral reduction along the lines of (8) to arbitrary multiplicity leads us to conjecture the following structure for the n-gluon tree in bosonic string theory: ×B(1, 2 σ , . . . , (n−2) σ , n−1, n; α ′ ) .
At generic multiplicity n, the B k (. . .)'s have homogeneity degree 4−n+2k in momenta. The simplest instances of the subleading terms occur at the three-point level and signal the F 3 interaction specific to the bosonic string, The higher-point case requires integral reductions as in (8), and the resulting geometric series yield non-zero B k (. . .) for any value of k. In the case of n=4, we find ×s 13 with gauge invariant constituents f ij ≡ (e i · e j )(k i · k j ) − (k i · e j )(k j · e i ) and Note that both s ij and g i carry a power of α ′ when extracting the B k (1, 2, 3, 4)'s from the second line of (12).
It is crucial to note that no MZVs or transcendental weight accompany the α ′ -dependence from B(. . . ; α ′ ). Given the uniform transcendentality of the F ρ σ (α ′ ) and the absence of negative powers of α ′ in the kinematic factor (10), the transcendental weight cannot exceed the accompanying order in α ′ within the bosonic-string amplitude. At fixed order in α ′ , the leading-transcendental part of the open bosonic string follows from picking up B(. . . ; α ′ ) → A YM (. . .) in (10) and therefore agrees with the superstring amplitude. This leads to the conclusion that the leading-transcendental pieces of the tree-level α ′expansion and the resulting tr(D m F n ) interactions are universal in open-string theories.

C. BCJ-symmetries of the kinematic factors:
Although the kinematic factors B k (. . .) in (10) differ from A YM (. . .) in tensor structure and mass dimension, we will now argue that they obey the same KK-and BCJ-relations [13]. The universal monodromy relations [10,11] among bosonic-string subamplitudes have to hold separately at each order in α ′ and along with each transcendentality. Hence, inserting (9)  for any value of k. The idea of imposing monodromy relations order by order has been exploited in [16] to derive BCJ-relations for subamplitudes of the F 3 operators as well as the supersymmetrized D 2 F 4 + F 5 . Moreover, a general argument for the entire gauge sector of the heterotic string has been given in [17]. By the same reason-ing, (13) can be extended to an infinity of α ′ -corrections (M j1 M j2 . . . M jp ) σ τ B k (1, 2 τ , . . . , (n−2) τ , n−1, n) , labelled by j i ∈ 2N + 1. The (n−3)! × (n−3)! matrix M j is the coefficient of ζ j when casting the α ′ -expansion of F ρ σ in (4) into a conjectural basis of MZVs w.r.t. rational numbers Q [18]. The entries of M j are degree-j polynomials in s pq , see [19] for examples at multiplicity n ≤ 7. Note that the symmetry properties (13) of B k (. . .) and their deformations B j1...jp k (. . .) in (14) are inevitable to verify permutation invariance of the world-sheet integrand for the bosonic-string amplitude along with each transcendentality and order in α ′ .
D. Supporting evidence: To confirm the central conjecture (9) implying our universality results, one must prove that the complete bosonic-string integrand including multi-cycle generalizations of (6) can be reduced to the single-cycle case. While a systematic all-multiplicity analysis is relegated to future work [7], the following IBP identities provide substantial support.
At five points, after partial-fractional manipulations, we need following two identities in addition to reduce all the integrals to a single-cycle basis (6): The resulting form of B(1, 2, 3, 4, 5; α ′ ) is rather lengthy, and an auxiliary mathematica notebook containing the full expression is attached to the arXiv submission.
We have checked that indeed all the above six-point integrals can be reduced to single-cycle integrals via IBP, e.g.