Critical properties of one-dimensional (1D) correlated electron system
s with a mobile impurity are investigated. By applying the finite-size
scaling method to a Bethe-ansatz solvable model, we derive the confor
mal dimensions related to the orthogonality catastrophe. We then apply
the results to the Fermi-edge singularity in quantum wires, and clari
fy how the critical exponent for X-ray absorption depends on the mass
of the core-hole created. A generalization to SU(nu) electron systems
is outlined based on the g-on description of 1D electron systems.