D = 26

. Insert

m

integrated closed-string tachyon vertex operators

e^{i p_{i} \cdot X}

on the sphere. The zero mode of

X

gives

δ^{(26)} (\sum_{i} p_{i})

, while Gaussian contractions give the Koba-Nielsen factor

\prod_{j < l} ∣ z_{j} - z_{l} ∣^{α^{'} p_{j} \cdot p_{l}}

. Dividing by the conformal Killing group

S L (2, C) / Z_{2}

and using the sphere power of the string coupling gives the displayed amplitude with

g_{s}^{m - 2}

b

Solution

Write the position-dependent factor as

e^{f}

, with

f = α^{'} \sum_{j < l} p_{j} \cdot p_{l} lo g ∣ z_{j} - z_{l} ∣.

In the fixed-angle hard-scattering limit the integral is governed by stationary points. Differentiation gives the scattering equations

j \neq = i \sum \frac{p _{i} \cdot p _{j}}{z _{i} - z _{j}} = 0, j \neq = i \sum \frac{p _{i} \cdot p _{j}}{z ˉ _{i} - z ˉ _{j}} = 0 .

Only

m - 3

complex equations are independent after quotienting by

S L (2, C)

Fix

z_{1} = 0

z_{2} = 1

, and

z_{4} = \infty

, and write

z_{3} = z

\frac{p _{3} \cdot p _{1}}{z} + \frac{p _{3} \cdot p _{2}}{z - 1} = 0,

so in the hard limit

z = \frac{p _{3} \cdot p _{1}}{p _{3} \cdot p _{1} + p _{3} \cdot p _{2}} = \frac{t}{t + u} = - \frac{t}{s} + O (m_{tachyon}^{2} / s) .

Evaluating the Koba-Nielsen factor at this saddle and using

s + t + u = O (m_{tachyon}^{2})

A^{(4, 0)} \sim g_{s}^{2} δ^{(26)} (i \sum p_{i}) exp [- \frac{α ^{'}}{2} (s lo g s + t lo g t + u lo g u + \dots)] .

The omitted terms grow more slowly than the displayed fixed-angle

s lo g s

terms.

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