The Appell function F1 and Regge string scattering amplitudes

Abstract

We show that each 26D open bosonic Regge string scattering amplitude (RSSA) can be expressed in terms of one single Appell function F 1 in the Regge limit. This result enables us to derive infinite number of recurrence relations among RSSA at arbitrary mass levels, which are conjectured to be related to the known SL(5, C ) dynamical symmetry of F 1 . In addition, we show that these recurrence relations in the Regge limit can be systematically solved so that all RSSA can be expressed in terms of one amplitude. All these results are dual to high energy symmetries of fixed angle string scattering amplitudes discovered previously [4][5][6][7][8].

The string BCJ relations revisited and extended recurrence relations of nonrelativistic string scattering amplitudes

“…Finally, in constrast to the Regge string spacetime symmetry which was shown to be related to SL(5, C) of the Appell function F 1 [26], here we found that the low energy stringy symmetry is related to SL(4, C) [35] of the Gauss hypergeometry functions 2 F 1 .…”

“…For this general case, the s − t channel of the scattering amplitude in the Regge limit was calculated to be [26] R |pn,qm,r l…”

“…These relations relate string scattering amplitudes of string states with different spins and different channels in the RSS limit. Similar to the HSS limit, it can be shown [25,26] that the complete extended recurrence relations in the RSS limit can be used to reduce the infinite number of independent RSS amplitudes from ∞ down to 1. We have seen in this section that the HSS and the RSS amplitudes calculated previously are consistent with the string BCJ relation in eq.…”

The Lauricella functions and exact string scattering amplitudes

“…Recently, it was found that the Regge SSA of three tachyons and one arbitrary string states can be expressed in terms of a sum of Kummer functions U [24][25][26], which soon later were shown to be the first Appell function F 1 [26]. Regge stringy symmetries or recurrence relations [25,26] were then constructed and used to reduce the number of independent Regge SSA from ∞ down to 1.…”

“…As a result, SSA and symmetries or relations among SSA of different string states at various limits calculated previously can be rederived. These include the linear relations first conjectured by Gross [1-5] and later corrected and proved in [6][7][8][9][10][11][12] in the hard scattering limit, the recurrence relations in the Regge scattering limit with associated SL(5, C) symmetry [24][25][26] and the extended recurrence relations in the nonrelativistic scattering limit with associated SL(4, C) symmetry [29] discovered recently. Finally, as an application, we calculate a new recurrence relation of SSA which is valid for all energies.…”