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

Abstract

We review and extend high energy four point string BCJ relations in both the fixed angle and Regge regimes. We then give an explicit proof of four point string BCJ relations for all energy. This calculation provides an alternative proof of the one based on monodromy of integration in string amplitude calculation. In addition, we calculate both s − t and t − u channel nonrelativistic low energy string scattering amplitudes of three tachyons and one higher spin string state at arbitrary mass levels. We discover that the mass and spin dependent nonrelativistic string BCJ relations can be expressed in terms of Gauss hypergeometry functions. As an application, for each fixed mass level N, we derive extended recurrence relations among nonrelativistic low energy string scattering amplitudes of string states with different spins and different channels.

The Lauricella functions and exact string scattering amplitudes

“…which was proved by monodromy of integration of string amplitudes [34,35] and explicitly proved recently in [29]. We can now do a change of variable…”

“…More recently, the extended recurrence relations [29] among nonrelativistic low energy SSA of a class of string states with different spins and different channels were constructed by using the recurrence relations of the Gauss hypergeometric functions with associated SL (4, C) symmetry [30]. These extended recurrence relations generalize and extend the field theory BCJ [31] relations to higher mass and higher spin string states.…”

“…As a result, all these SSA and symmetries or relations among SSA of different string states at various limits calculated previously can be rederived. These will be presented in sections 3, 4 and 5 which include the recurrence relations in the Regge scattering limit [25,26] with associated SL(5; C) symmetry, the linear relations conjectured by Gross [3,4] and corrected and proved in [6][7][8][9][10][11][12] in the hard scattering limit and the extended recurrence relations in the nonrelativistic scattering limit [29] with associated SL(4; C) symmetry discovered very recently. However, since not all Lauricella functions F (K) D with arbitrary independent arguments can be used to represent SSA, it remained to be studied how the basis states of each SL(K +3, C) group representation for a given K relates to SSA [33].…”

“…we can rederive the string BCJ relation [29,34,35] 25) which gives another form of the (s, t) channel amplitude…”

“…(2.20) can be rewritten as [1][2][3][4][5] and proved in [6][7][8][9][10][11][12] in the hard scattering limit, the recurrence relations in the Regge scattering limit [24][25][26] and the extended recurrence relations in the nonrelativistic scattering limit [29] discovered recently. In this section, we first calculate the Regge scattering limit.…”