  We study closed string one-loop amplitudes in string theory, in particular
the average mass shift for states at given mass and Neveu-Schwarz charges. Our
analysis is based only on well-defined string amplitudes and the exploitation
of symmetries and unitarity properties of the torus amplitudes. We obtain the
result Delta M^2 = - g_s^2 M^(2+(3-D)/2) in D spacetime dimensions for the
average closed string mas-shift (Delta M^2 = - g_s^2 (M^2-Q^2)^(1+(3-D)/4) for
states with non-zero Neveu-Schwarz charges Q). An interesting picture of
one-loop corrections for the string in non-supersymmetric configurations comes
out: the dominant interactions responsible for these corrections are of
long-range type (namely gravitational) and it appears that perturbations theory
is generally reliable on the spectrum of massive string states.
