Derivation of Freund-Witten adelic formula for four-point Veneziano amplitudes
\textit{P. G. O. Freund} and \textit{E. Witten} [Phys. Lett. B 199, 191-194 (1987)] and, independently, \textit{I. V. Volovich} [Lett. Math. Phys. 16, 61-67 (1988; Zbl 0668.43006)] proposed an adelic formula for the four-point crossing-symmetric Veneziano amplitudes (for open strings): \[ B_\infty(- a(s),-a(t)) \prod^\infty_{p= 2} B_p(-a(s), -a(t))= 1,\tag{1} \] where \(B_\infty\) is the Gel'fand-Graev \(B\) function, \(B_p\) is the corresponding \(p\)-adic \(B\) function, \(\alpha(s)= 1+ s/2\), and \(s\), \(t\), \(u\) are Mandelstam variables, \(s+ t+ u= -8\). However, the infinite product in (1) diverges for all complex \(s\) and \(t\), and therefore it is not clear how (1) should be understood. Here, using the adelic group (Tate's formula), the author proposes a regularization for the divergent infinite product of \(p\)-adic \(\Gamma\) functions: \[ \Gamma_p(\alpha)= {1- p^{\alpha- 1}\over 1- p^{-\alpha}},\quad p= 2,3,5,\dots, \] and the adelic formula \[ \text{reg}\prod^\infty_{p= 2} \Gamma_p(\alpha)= {\zeta(\alpha)\over \zeta(1- \alpha)}, \] where \(\zeta(\alpha)\) is the Riemann \(\zeta\) function, is proved.
scientific article; zbMATH DE number 589577
scientific article