α'-expansion of open superstring amplitudes

On this website we are providing the explicit form of the α'-expansion of superstring disk amplitudes involving N massless states from the gauge multiplet. According to [1106.2645] and [1106.2646], the (N−3)! independent color-ordered open string tree amplitudes A_open are obtained from an (N − 3)! × (N − 3)!-matrix F acting on the (N−3)!-component vector of independent Yang-Mills subamplitudes

A_{YM}\; =\; A_{YM}(1, \sigma(2), \ldots \sigma(N-2), N-1, N)\; ,\hskip2cm \sigma\in S_{N-3}
as:
A_{\rm open}\; =\; F \; \cdot \; A_{YM}\; .

The inverse string tension α' enters F through the dimensionless Mandelstam invariants

s_{ij}:=\alpha' (k_i+k_j)^2,\; \; \; s_i:=s_{i,i+1}, \; \; \; {\rm and} \\[5mm] t_i:=\alpha' (k_i+k_{i+1}+k_{i+2})^2\equiv s_{i,i+1,i+2}, \; \; \; {\rm where} \; \; \; i \simeq i+N \; \; \; {\rm and}\; \; \; k_i^2=0,

i.e. its Taylor expansion in s_ij encodes the low-energy behaviour of the string amplitude. The pattern of multiple zeta values (MZVs)

\zeta_{n_1,\ldots,n_r} \; =\; \sum_{0 < k_1 < \ldots < k_r} \; \; \prod_{l=1}^r k_l^{-n_l} \; , \; n_l \in {\bf N}^+ \; , \; n_r \geq 2
in the α'-expansion of F was analyzed in [1205.1516]. It turned out that the matrix-valued coefficients of single zeta values (in the MZV-basis given in [0907.2557])

M_{2k+1}:=\left. F\;\right|_{\zeta_{2k+1}} \; \; \; {\rm and}\; \; \; P_{2k}:=\left. F\;\right|_{(\zeta_2)^k}

are sufficient to reconstruct the α'-dependence along with any other MZV product, e.g.:

\begin{align*} F = &\; 1 +\zeta_2\; P_2+\zeta_3\; M_3+\zeta_2^2\; P_4+\zeta_2\zeta_3\; P_2M_3 + \zeta_5\; M_5 + \zeta_2^3\; P_6 + \frac{1}{2}\; \zeta_3^2\; M_3 M_3 \notag \\[3mm] + &\; \zeta_7\; M_7+\zeta_2\zeta_5\; P_2M_5 + \zeta_2^2\zeta_3\; P_4 M_3 \notag \\[3mm] + &\; \zeta_2^4\; P_8 + \zeta_3\zeta_5\; M_5 M_3 + \frac{1}{2}\; \zeta_2 \zeta_3^2\; P_2 M_3 M_3 + \frac{1}{5}\; \zeta_{3,5}\; [M_5,M_3] + \ldots\end{align*}

A suitable Hopf-algebra inspired MZV representation paving the way for an extrapolation to all weights is explained in [1205.1516], together with the techniques to also reduce closed-string tree amplitudes to matrices M_w and vectors A_YM. The polynomial structure of the matrices M_w and P_w was systematically addressed in [1304.7267] and [1304.7304].

The purpose of this website is to make the explicit results of these references publicly available:

N=5	matrices at orders 2 to 9 matrices at order 10 to 14 matrices at order 15 to 16 matrices at order 17 to 20 matrix at order 21 matrix at order 22	(57 KB) (561 KB) (750 KB) (4.4 MB) (2 MB) (5 MB)

N=6	matrices at orders 2 to 6 matrix at order 7 matrix at order 8 matrix at order 9	(3.3 MB) (6.1 MB) (5.0 MB) (3.1 MB)

N=7	matrices at orders 2 to 5 matrix at order 6 matrix at order 7	(11.1 MB) (1.3 MB) (4.9 MB)

At higher weights, we keep the data size under control by suppressing redundant information: starting at weight w=9 for N=5 and weight w=6 for N=7, the links lead to the ﬁrst row of the matrices M_w and P_w associated with the canonical color ordering 1, 2, 3, . . . , N . As spelled out in [1304.7267], the remaining entries can be obtained by suitable relabelling.

According to [1304.7304], the complete α'-expansion of the N-point tree amplitude can be obtained from the Drinfeld associator Φ(e₀, e₁) recursively. Herein, e₀ and e₁ are (N − 2)! × (N − 2)!-matrices linear in s_ij acting on worldsheet functions of the (N − 1)-point amplitude. For multiplicities 5 through 9, the matrices can be downloaded here:

	N=5	matrices e₀ and e₁

	N=6	matrices e₀ and e₁

	N=7	matrices e₀ and e₁

	N=8	matrices e₀ and e₁ (Mathematica SparseArray)

	N=9	matrices e₀ and e₁ (Mathematica SparseArray)