Residues of bosonic string scattering amplitudes and the Lauricella functions

Abstract

We calculate explicitly residues of all n-point Koba-Nielsen (KN) amplitudes by using on-shell recursion relation of string scattering amplitudes (SSA). In addition, we show that the residues of all SSA including the KN amplitudes can be expressed in terms of the Lauricella functions. This result demonstrates the exact SL(K + 3, C) symmetry of the tree-level open bosonic string theory.Moreover, we derive an iteration relation among the residues of a given SSA. This iteration relation is related to the SL(K + 3, C) symmetry and can presumably be used to soften the well-known hard SSA.

Stringy scaling of n-point hard string scattering amplitudes

“…More recently, it was demonstrated [22,23] that by using the string on-shell recursion relation of SSA [24][25][26][27], one can show that all n-point SSA can be expressed in terms of the Lauricella functions. This result extends the previous exact SL(K + 3, C) symmetry of the 4-point LSSA of three tachyons and one arbitrary string states in Eq.…”

“…( 1) to the whole tree-level open bosonic string theory. Motivated by this remarkable result [22,23], one is tempted to believe that for the higher point (n ≥ 5) LSSA, there exist hard scattering kinematics regimes for which the infinite linear relations persist and the ratios can be solved accordingly.…”

The $SL(K+3,\mathbb{C})$ symmetry of string scatterings from D-branes

“…More recently, by using the string theory extension [14], [15,16] of the field theory BCFW on-shell recursion relations [17,18] , one can show that [19] the residues of all n-point SSA including the Koba-Nielsen (KN) amplitudes can be expressed in terms of the Lauricella functions with nonpositive integer β J . As a result, the above SL(K + 3, C) symmetry group of the 4-point LSSA was extended to the n-point LSSA with arbitrary n [20].…”

“…( 2) can be generalized to LSSA of four arbitrary string states, and to those of n arbitrary string states (see Eq. ( 11)) below [19,20].…”

“…For the general 4-point LSSA, it is straightforward to calculate the operation of these generators on the basis functions and show the SL(K + 3, C) symmetry [24]. For the cases of higher point (n ≥ 5) LSSA, one encounters the operation on the sum of products of the Lauricella functions [19] Residue of n-point LSSA ∼ coefficient (single tensor 4-point LSSA).…”