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.
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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)