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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.
Highlights
Of the external momenta in such a way that the corresponding integrals converge and the corresponding amplitudes are well defined
We established that p-adic Koba-Nielsen string amplitudes are finite sums of multivariate Igusa’s local zeta functions, they are convergent integrals that admit meromorphic continuations as rational functions
In [21], the limit p → 1 of the effective action was studied, it was showed that this limit gives rise to a boundary string field theory (BSFT), which was previously proposed by Witten in the context of background independent string theory [23, 24]
Summary
We show how to extract the four and five-point amplitudes of the GerasimovShatashvili Lagrangian (2.5) found in [21]. In order to do that, we first require to study the interacting theory. The generating functional of the correlation function for the free theory is given by. Where GF (x − x′) is the Green-Feynman function of time-ordered product of two fields of the theory, N is a normalization constant, [det(∆ − 1)]−1/2 is a suitable regularization of the divergent determinant bosonic operator, see e.g. In the standard formalism of QFT [34], the N -point correlation functions are proportional to. Where the φ’s are N local operators (observables) in N different points x1, x2, . XN of the Minkowski spacetime, Z[J] is the generating functional constructed using interacting Lagrangian (2.11). We assert that connected tree-level scattering amplitudes of this theory match exactly with the corresponding amplitudes of the effective action (2.1) in the limit when p tends to one
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