On p-adic string amplitudes in the limit p approaches to one

Abstract

In this article we discuss the limit p approaches to one of tree-level p-adic open string amplitudes and its connections with the topological zeta functions. There is empirical evidence that p-adic strings are related to the ordinary strings in the p → 1 limit. Previously, we established that p-adic Koba-Nielsen string amplitudes are finite sums of multivariate Igusa's local zeta functions, consequently, they are convergent integrals that admit meromorphic continuations as rational functions. The meromorphic continuation of local zeta functions has been used for several authors to regularize parametric Feynman amplitudes in field and string theories. Denef and Loeser established that the limit p → 1 of a Igusa's local zeta function gives rise to an object called topological zeta function. By using Denef-Loeser's theory of topological zeta functions, we show that limit p → 1 of tree-level p-adic string amplitudes give rise to certain amplitudes, that we have named Denef-Loeser string amplitudes. Gerasimov and Shatashvili showed that in limit p → 1 the well-known non-local effective Lagrangian (reproducing the tree-level p-adic string amplitudes) gives rise to a simple Lagrangian with a logarithmic potential. We show that the Feynman amplitudes of this last Lagrangian are precisely the amplitudes introduced here. Finally, the amplitudes for four and five points are computed explicitly.

Regularization of p-adic string amplitudes, and multivariate local zeta functions

“…Gerasimov and Shatashvili showed that in limit p → 1 the well-known non-local effective Lagrangian (reproducing the tree-level p-adic string amplitudes) gives rise to a simple Lagrangian with a logarithmic potential [21]. In [7], we show that the Feynman amplitudes of this last Lagrangian are precisely the Denef-Loser amplitudes.…”

p-adic Mellin amplitudes

“…With the pre-amplitude in hand, we are ready to carry out the two contour integrals in (4.46) to obtain the full Mellin amplitude. One way to do this is to close both contours to the right, and sum over the residues in the c A plane, which occur at 54) and then sum over the residues in the c B plane, occurring at…”

An Introduction to the Theory of Local Zeta Functions from Scratch

“…Actually, the previous equations just give account of the change of variables rule as x → x ′ , in the integral (9), that is…”

“…Then they used these functions to regularize the Koba-Nielsen amplitudes. In [9], the authors discussed the limit p approaches to one of tree-level p-adic open string amplitudes and its connections with the topological zeta functions. There is empirical evidence that p-adic strings are related to the ordinary strings in the limit p → 1.…”

“…Denef and Loeser established that the limit p → 1 of a Igusa's local zeta function gives rise to an object called topological zeta function. By using Denef-Loeser's theory of topological zeta functions, it is showed in [9] that limit p → 1 of tree-level padic string amplitudes give rise to certain amplitudes, that we have named Denef-Loeser string amplitudes.…”