  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.
