Beta functions of local fields of characteristic zero. Applications to string amplitudes

For local fields $\mathbb K$ of characteristic zero, along with the beta function $\mathbf{B}_{\mathbb K}$ we introduce a new sequence $\mathbf{B}_{\mathbb K}^{(n)}$, $n=1,2,\dots$, of beta functions of $n$ complex arguments expressed in terms of a product of gamma functions $\boldsymbol{\Gamma}_{\mathbb K}$ for arbitrary characters (ramified or not). We consider applications to the 4-particle tree string and superstring amplitudes. It turns out that the tachyon string amplitudes can be expressed in terms of the well-known beta function $\mathbf{B}_{\mathbb K}=\mathbf{B}_{\mathbb K}^{(2)}$. The massless superstring amplitudes can be expressed in terms of the new beta function $\mathbf{B}'_{\mathbb K}=\mathbf{B}_{\mathbb K}^{(3)}$ for non-trivial characters. We establish that the amplitudes of all known strings and superstrings admit adelic formulae.
We give a new proof of the formula relating the 4-particle tree amplitudes for closed strings (generalized Virasoro amplitudes) to the product of two amplitudes for open strings (classical Veneziano amplitudes).