Abstract
In this paper, we studied symmetries of string scattering amplitudes in the high energy limits of both the fixed angle or Gross regime (GR) and the fixed momentum transfer or Regge regime (RR). We calculated high energy string scattering amplitudes (SSA) at arbitrary mass levels for both regimes. We discovered the infinite linear relations among fixed angle string amplitudes and the ifinite recurrence relations among Regge string amplitudes. The linear relations we obtained in the GR corrected the saddle point calculations by Gross, Gross and Mende. In addition, for the high energy closed string scatterings, our results differ from theirs by an oscillating prefactor which was crucial to recover the KLT relation valid for all energies. We showed that all the high energy string amplitudes can be solved using the linear or recurrence relations, so that all the string amplitudes can be expressed in terms of a single string amplitude. We further found that, at each mass level, the ratios among the fixed angle amplitudes can be extracted from the Regge string scattering amplitudes. Finally, we reviewed the recent developments on the discovery of infinite number of recurrence relations valid for all energies among Lauricella SSA. The symmetries or relations among SSA at various limits obtained previously can be exactly reproduced. It leads us to argue that the known S L ( K + 3 , C ) dynamical symmetry of the Lauricella function may be crucial to probe spacetime symmetry of string theory.
Highlights
Quantum Field Theory (QFT) is a powerful theory in modern physics
In addition to the hard limit, i.e., the Gross regime (GR), which we have reviewed in the previous sections, another important regime is the high energy limit with a small scattered angle, i.e., the Regge regime (RR)
The recurrence relations of the Appell function F1 in the Regge limit can be systematically solved, so that one can express all Regge string scattering (RSS) amplitudes in terms of only one amplitude [25,26]. All these results are associated with symmetries of string scattering amplitudes in hard limit discussed in Section 3 [12,13,14,16,17,47,53]
Summary
Quantum Field Theory (QFT) is a powerful theory in modern physics. The microcosmic physics is successfully described by using standard model of particle physics based on QFT. The first one was that at high energies, fixed angle regime or Gross regime (GR) of the theory, there existed an infinite number of linear relations among the string scattering amplitudes of different string states at each order in string perturbation theory. String scattering amplitudes and symmetries or relations among string scattering amplitudes among different string states at various scattering limits calculated previously can be rederived These include the stringy linear relations conjectured by Gross and proved by Taiwan group [12,13,14,15,16,17]. For a more detailed review of high energy string scattering amplitudes and symmetries of string theory, see the recent long review paper [28]
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