In string theory, high-energy scattering amplitudes describe the interactions of strings at extreme energy scales, such as the Planck scale. Unlike point-particle theories that exhibit power-law behavior, string amplitudes are characterized by a universal, soft exponential fall-off at high energies and fixed angles.
Background The Gross conjecture regarding high energy symmetry of string theory was based on the saddle-point calculation of hard string scattering amplitudes (SSA) of both the closed and open string theories. The conjecture claimed that there existed infinite linear relations among hard SSA of different string states. Moreover, these infinite linear relations were so powerful that they can be used to solve all the hard SSA and express them in terms of one amplitude. Some monographs had made speculations about this hidden stringy symmetry without getting any conclusive results. However, the saddle-point calculation of the hard SSA which was claimed to be valid for all string states and all string loop orders was pointed out to be inconsistent for the cases of the excited string states in a series of works done by the method of decoupling of zero-norm states (ZNS). It was then further shown that even at closed string-tree level, there was no reliable saddle-point in the hard SSA calculation. Three evidences have been given to demonstrate the inconsistency of the saddle-point. So instead of using the saddle-point method, they used the KLT formula to obtain the correct hard closed SSA, which differs from result of Gross and Mende by an oscillation prefactor. This prefactor consistently implied the existence of infinitely many zeros and poles in the hard SSA. Soon later a similar conclusion was made based on the group theoretical calculation of SSA. They found out that up to the string one-loop level the saddle-point calculation was valid only for the hard four tachyon SSA, but was incorrect for other hard SSA of excited string states. For this reason, the authors admitted that they can not consistently find out any linear relations as suggested in Gross conjecture.
Calculations For the case of open bosonic string at the mass level M 2 = 4 {\displaystyle M^{2}=4} , as an example, the hard open SSA of Gross and Manes were miscalculated to be
T T T T ∝ T [ L T ] , T L L T = T ( L T ) = 0 , {\displaystyle T_{TTT}\propto T_{[LT]},T_{LLT}=T_{(LT)}=0,}
which were inconsistent with the Ward identities or the decoupling of zero-norm states (ZNS) in the hard scattering limit to be discussed below. The importance of two types of ZNS was stressed in the massive background field calculation of string symmetries. It was shown that in the weak field approximation (but valid for all energies) an inter-particle symmetry transformation
δ C ( μ ν λ ) = 1 2 ∂ ( μ ∂ ν θ λ ) 2 − 2 η ( μ ν θ λ ) 2 , δ C [ μ ν ] = 9 ∂ [ μ θ ν ] 2 {\displaystyle \delta C_{(\mu \nu \lambda )}={\frac {1}{2}}\partial _{(\mu }\partial _{\nu }\theta _{\lambda )}^{2}-2\eta _{(\mu \nu }\theta _{\lambda )}^{2},\delta C_{[\mu \nu ]}=9\partial _{\lbrack \mu }\theta _{\nu ]}^{2}}
for two propagating states C ( μ ν λ ) {\displaystyle C_{(\mu \nu \lambda )}} and C [ μ ν ] {\displaystyle C_{[\mu \nu ]}} at mass level M 2 = 4 {\displaystyle M^{2}=4} of open bosonic string can be generated by the D 2 {\displaystyle D_{2}} vector ZNS with polarization θ μ 2 {\displaystyle \theta _{\mu }^{2}}
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