Recurrence relations of higher spin BPST vertex operators for open strings

Abstract

We calculate higher-spin Brower-Polchinski-Strassler-Tan (BPST) vertex operators for an open bosonic string and express these operators in terms of a Kummer function of the second kind. We derive an infinite number of recurrence relations among BPST vertex operators of different string states. These recurrence relations among BPST vertex operators lead to the recurrence relations among Regge string scattering amplitudes discovered recently.Note that the convention for s and t adopted here is different from the original BPST paper in Ref. [2].We first review the calculation of tachyon BPST vertex operator [2]. The s À t channel of an open-string fourtachyon amplitude can be written as *

The Appell function F1 and Regge string scattering amplitudes

“…(31) where a finite sum of Kummer functions was obtained, here we get only one single Appell function in Eq. (20). This simplification will greatly simplify the calculation of recurrence relations among RSSA to be discussed in the next section.…”

“…(20) is a polynomial in s, the result in Eq. (20) is valid only for the leading order in s in the Regge limit. Note that in contrast to the previous calculation [19] in Eq.…”

“…We now proceed to show that the recurrence relations of the Appell function F 1 in the Regge limit in Eq. (20) can be systematically solved so that all RSSA can be expressed in terms of one amplitude. As the first step, we note that in [19] the RSSA was expressed in terms of finite sum of Kummer functions.…”

“…Alternatively, it is interesting to note that the result calculated in Eq. (20) can be directly obtained from an integral representation of F 1 due to Emile Picard (1881) [22] …”

“…Recurrence relations of Kummer functions can then be used to derive some recurrence relations among RSSA [19]. Recurrence relations of higher spin generalization of the BPST vertex operators [15] can also be constructed [20].…”