Patterns of high energy massive string scatterings in the Regge regime

Abstract

We calculate high energy massive string scattering amplitudes of open bosonic string in the Regge regime (RR). We found that the number of high energy amplitudes for each fixed mass level in the RR is much more numerous than that of Gross regime (GR) calculated previously. Moreover, we discover that the leading order amplitudes in the RR can be expressed in terms of the Kummer function of the second kind. In particular, based on a summation algorithm for Stirling number identities developed recently, we discover that the ratios calculated previously among scattering amplitudes in the GR can be extracted from this Kummer function in the RR. We conjecture and give evidences that the existence of these GR ratios in the RR persists to subleading orders in the Regge expansion of all string scattering amplitudes. Finally, we demonstrate the universal power-law behavior for all massive string scattering amplitudes in the RR. * Electronic address: slko.py96g@g2.

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

“…Moreover, these extended linear relations can be used to reduce the number of independent HSS amplitudes from ∞ down to 1 [14][15][16][18][19][20]. Similar calculation can be performed in the RSS limit [27], and one obtains extended recurrence relations in the RSS limit. These extended Regge recurrence relations again can be used to reduce the number of independent RSS amplitudes from ∞ down to 1 [25,26].…”

“…The s − t channel RSS amplitude of three tachyons and one higher spin state in eq. (2.9) was calculated to be [27] R (n,2m,q)…”

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.…”

“…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. Moreover, an interesting link between Regge SSA and hard SSA was found [24,27], and for each mass level the ratios among hard SSA can be extracted from Regge SSA. This result enables us to argue that the known SL(5; C) dynamical symmetry of the Appell function F 1 [28] is crucial to probe high energy spacetime symmetry of string theory.…”

“…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.…”