Scaling behavior of regularized bosonic strings

TL;DR

This paper studies the scaling behavior of regularized bosonic strings using a proper-time UV regularization, deriving effective actions, and identifying two distinct scaling limits that connect to previous regularization methods.

Contribution

It introduces a proper-time UV regularization for the Nambu-Goto string and analyzes its scaling limits, linking to existing regularization approaches and canonical quantization.

Findings

Identifies two scaling limits for the regularized string as the cutoff goes to infinity.

One scaling limit matches hypercubic and dynamical triangulation regularizations.

The other scaling limit reproduces canonical quantization results.

Abstract

We implement a proper-time UV regularisation of the Nambu-Goto string, introducing an independent metric tensor and the corresponding Lagrange multiplier, and treating them in the mean-field approximation justified for long strings and/or when the dimensions of space-time is large. We compute the regularised determinant of the 2d Laplacian for the closed string winding around a compact dimension, obtaining in this way the effective action, whose minimisation determines the energy of the string ground state in the mean-field approximation. We discuss the existence of two scaling limits when the cutoff is taken to infinity. One scaling limit reproduces the results obtained by the hypercubic regularisation of the Nambu-Goto string as well as by the use of the dynamical triangulation regularisation of the Polyakov string. The other scaling limit reproduces the results obtained by canonical quantisation of the Nambu-Goto string.