Exponential Fall-Off Behavior of Regge Scatterings in Compactified Open String Theory

Progress of Theoretical Physics, Nov 2012

We calculate massive string scattering amplitudes of compactified open string in the Regge regime. We extract the complete infinite ratios among high-energy amplitudes of different string states in the fixed angle regime from these Regge string scattering amplitudes. The complete ratios calculated by this indirect method include and extend the subset of ratios calculated previously [J. C. Lee and Y. Yang, Nucl. Phys. B 784 (2007), 22; J. C. Lee, T. Takimi and Y. Yang, Nucl. Phys. B 804 (2008), 250] by the more difficult direct fixed angle calculation. In this calculation of compactified open string scattering, we discover a realization of arbitrary real values L in the identity Eq. (4·18), rather than integer value only in all previous high-energy string scattering amplitude calculations. The identity in Eq. (4·18) was explicitly proved recently in [J. C. Lee, C. H. Yan and Y. Yang, SIGMA 8 (2012), 045, arXiv:1012.5225] to link fixed angle and Regge string scattering amplitudes. In addition, we discover a kinematic regime with stringy highly winding modes, which shows the unusual exponential fall-off behavior in the Regge string scattering. This is complimentary with a kinematic regime discovered previously [J. C. Lee, T. Takimi and Y. Yang, Nucl. Phys. B 804 (2008), 250] which shows the unusual power-law behavior in the high-energy fixed angle compactified string scatterings.

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Exponential Fall-Off Behavior of Regge Scatterings in Compactified Open String Theory

Song He 0 2 ) Jen-Chi Lee 0 1 Yi Yang 0 1 Subject Index: 0 Kavli Institute for Theoretical Physics China , CAS, Beijing 100190, China 1 Department of Electrophysics, National Chiao-Tung University and Physics Division, National Center for Theoretical Sciences , Hsinchu, Taiwan , R.O.C 2 Institute of High Energy Physics, Chinese Academy of Sciences , Beijing 100039, China We calculate massive string scattering amplitudes of compactified open string in the Regge regime. We extract the complete infinite ratios among high-energy amplitudes of different string states in the fixed angle regime from these Regge string scattering amplitudes. The complete ratios calculated by this indirect method include and extend the subset of ratios calculated previously [J. C. Lee and Y. Yang, Nucl. Phys. B 784 (2007), 22; J. C. Lee, T. Takimi and Y. Yang, Nucl. Phys. B 804 (2008), 250] by the more difficult direct fixed angle calculation. In this calculation of compactified open string scattering, we discover a realization of arbitrary real values L in the identity Eq. (418), rather than integer value only in all previous high-energy string scattering amplitude calculations. The identity in Eq. (418) was explicitly proved recently in [J. C. Lee, C. H. Yan and Y. Yang, SIGMA 8 (2012), 045, arXiv:1012.5225] to link fixed angle and Regge string scattering amplitudes. In addition, we discover a kinematic regime with stringy highly winding modes, which shows the unusual exponential fall-off behavior in the Regge string scattering. This is complimentary with a kinematic regime discovered previously [J. C. Lee, T. Takimi and Y. Yang, Nucl. Phys. B 804 (2008), 250] which shows the unusual power-law behavior in the high-energy fixed angle compactified string scatterings. - Recently, following an old suggestion of Mende,18) two of the present authors19) calculated high-energy fixed angle massive scattering amplitudes of closed bosonic string with some coordinates compactified on the torus. The calculation was extended to the compactified open string scatterings.20) An infinite number of linear relations among high-energy scattering amplitudes of different string states were obtained in the fixed angle or Gross kinematic regime (GR). The UV behavior in the GR shows the usual soft exponential fall-off behavior. These results are reminiscent of the existence of an infinite number of massive ZNS in the compactified closed21) and open22) string spectrums constructed previously. In addition, it was discovered that, for some kinematic regime with super-highly winding modes at fixed angle, the so-called Mende kinematic regime (MR), these infinite linear relations break down and, simultaneously, the string amplitudes enhance to hard power-law behavior at high energies instead of the usual soft exponential fall-off behavior. In this paper, we calculate high-energy small angle or Regge string scattering amplitudes23)30) of open bosonic string with one coordinate compactified on the torus. The results can be generalized to more compactified coordinates. It is shown that there is no linear relations among Regge scattering amplitudes as expected. However, as in the case of noncompactified Regge string scattering amplitude calculation,31)33) we can deduce the infinite GR ratios in the fixed angle from these compactified Regge string scattering amplitudes. We stress that the GR ratios calculated in the present paper by this indirect method from the Regge calculation are for the most general high-energy vertex rather than only a subset of GR ratios obtained directly from the fixed angle calculation.19), 20) In this calculation, we have used a set of master identities Eq. (4.18) to extract the GR ratios from Regge scattering amplitudes. Mathematically, the complete proof of these identities for arbitrary real values L was recently worked out in 36) by using an identity of signless Stirling number of the first kind in combinatorial theory. The proof of the identity for L = 0, 1, was previously given in 31)33) based on a set of identities of signed Stirling number of the first kind.35) It is interesting to see that, physically, the identities for arbitrary real values L can only be realized in high-energy compactified string scatterings considered in this paper. All other high-energy string scatterings calculated previously31)33) correspond to integer values of L only. A recent work on string D-particle scatterings34) also gave integer values L. More importantly, we discover an exponential fall-off behavior of high-energy compactified open string scatterings in a kinematic regime with highly winding modes at small angle. The existence of this regime was conjectured in 20). However, no Regge scatterings were calculated there and thus the results for the small angle scatterings extracted from the fixed angle calculation were not completed and fully reliable.31), 32) The discovery of the soft exponential fall-off behavior in this kinematic regime with (...truncated)


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Song He, Jen-Chi Lee, Yi Yang. Exponential Fall-Off Behavior of Regge Scatterings in Compactified Open String Theory, Progress of Theoretical Physics, 2012, pp. 887-901, 128/5, DOI: 10.1143/PTP.128.887