Recurrence relations of Kummer functions and Regge string scattering amplitudes

Abstract

We discover an infinite number of recurrence relations among Regge string scattering amplitudes [10, 23] of different string states at arbitrary mass levels in the open bosonic string theory. As a result, all Regge string scattering amplitudes can be algebraically solved up to multiplicative factors. Instead of decoupling zero-norm states in the fixed angle regime, the calculation is based on recurrence relations and addition theorem of Kummer functions of the second kind. These recurrence relations among Regge string scattering amplitudes are dual to linear relations or symmetries among high-energy fixed angle string scattering amplitudes discovered previously.

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

“…Similar calculation can be performed in the RSS limit [27], and one obtains extended recurrence relations in the RSS limit. These extended Regge recurrence relations again can be used to reduce the number of independent RSS amplitudes from ∞ down to 1 [25,26].…”

“…In section II, we will first review the extended linear relations in the HSS limit discussed in [8]. We will also work out the corresponding extended recurrence relations [25,26] of the Regge string scattering (RSS) amplitudes [27]. Similar to the extended linear relations in the HSS limit discussed above, the extended recurrence relations can be used to reduce the number of independent RSS amplitudes from ∞ down to 1.…”

“…(2.8), it was shown [25,26] that there existed infinite recurrence relation among RSS amplitudes. For example, the recurrence relation [26] ((N ; q 1 , r 1 ) etc.…”

“…(2.23) can be combined to form one example of the extended recurrence relations in the RSS limit. The possible connection of field theory BCJ relations [3] and Regge string recurrence relations [25] was first suggested in [25]. These relations relate string scattering amplitudes of string states with different spins and different channels in the RSS limit.…”

“…These relations relate string scattering amplitudes of string states with different spins and different channels in the RSS limit. Similar to the HSS limit, it can be shown [25,26] that the complete extended recurrence relations in the RSS limit can be used to reduce the infinite number of independent RSS amplitudes from ∞ down to 1. We have seen in this section that the HSS and the RSS amplitudes calculated previously are consistent with the string BCJ relation in eq.…”